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Alternative decouplings of the electron propagator

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Title:
Alternative decouplings of the electron propagator
Creator:
Born, Gregory Jay, 1951- ( Dissertant )
Ohrn, N. Yngve ( Thesis advisor )
Broyles, Arthur A. ( Reviewer )
Micha, David A. ( Reviewer )
Person, Willis B. ( Reviewer )
Sabin, John R. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1979
Language:
English
Physical Description:
x, 164 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Alkynes ( jstor )
Approximation ( jstor )
Diagrams ( jstor )
Electrons ( jstor )
Ionization ( jstor )
Molecules ( jstor )
Orbitals ( jstor )
Photoelectrons ( jstor )
Photoionization ( jstor )
Water tables ( jstor )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Optoelectronic devices ( lcsh )
Photoionization ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 157-163.
General Note:
Typescript.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
023054398 ( AlephBibNum )
05697751 ( OCLC )
AAK4427 ( NOTIS )

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ALTERNATIVE DECOUPLINGS OF THE
ELECTRON PROPAGATOR













By

GREGORY J. BORN













A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA















ACKNOWLEDGMENTS

After twenty-one years of formal education which is culminating

with this dissertation, it is impossible to individually thank every

teacher who assisted me in this pursuit. The largest debt of gratitude

however, is owed to my dissertation supervisor, Professor Yngve Ohrn,

whose constructive criticism and constant encouragement has guided the

direction of this research and my graduate education. His generous

financial assistance over the years has also been greatly appreciated.

I would like to thank Professor Per-Olov L6wdin for his stimulating

series of lectures in quantum theory as well as the national and inter-

national contacts he has made available to me and the other members of

the Quantum Theory Project through the invitation of visiting scientists

and the organization of the Sanibel Symposia. My attendance at the

Summer School in Quantum Theory held in Uppsala, Sweden,and Dalseter,

Norway (August 1976),was made possible by monetary awards secured by

Prof. Lbwdin for which I am also greatly appreciative.

Next I would like to thank Professor Jack Sabin, who supervised me

during some preliminary investigations, and the other members of my

supervisory committee for contributing their time and for occasional

letters of recommendation.

Without the friendships and intellectual stimulation of other

members of the Quantum Theory Project whom I have known, my graduate

education would not have been as enjoyable or as rewarding. To several

people I am indebted for the use of various computer subroutines which

I gratefully acknowledge.









It is with regret that I must posthumously acknowledge my indebted-

ness to Professor Boris Muslin. Prof. Muslin supervised my undergraduate

research in quantum theory at Southern Illinois University and was

largely responsible for guiding me into this field.

I would finally like to thank my parents for their constant encour-

agement and financial assistance.

Special thanks are owed to Miss Brenda Foye for her painstaking

efforts in typing this manuscript.

I also wish to take this opportunity to acknowledge the Northeast

Regional Data Center of the State University System of Florida for the

use of their facilities to obtain the numerical results presented here

and the American Institute of Physics for permission to reproduce

Figure 1 and Appendix 1 from the paper by G. Born, H. A. Kurtz, and

Y. Ohrn in the Journal of Chemical Physics, Vol. 68, p. 74 (1978).















TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . ........ . .... .. ii

LIST OF TABLES . . . . . . . . . . ... . vi

LIST OF FIGURES . . . . . . .. . .. . . viii

ABSTRACT ............ . . . . .... .ix

INTRODUCTION . . . . . . . . .. .. ... 1

CHAPTER 1: OPERATOR PRODUCT DECOUPLINGS .. . . . . 10

1.1 Definition, Spectral Representation, and Equation
of Motion of the Electron Propagator . . 10
1.2 The Superoperator Notation and Inner Projection
Technique . . ... .. ...... . 15
1.3 The Hartree-Fock Propagator . . . .... .18
1.4 Operator Product Decoupling . . . .... .21
1.5 Method of Solution . . . . . . 23
1.6 Analysis and Limitations of the Operator
Product Decoupling . ... ....... .27

CHAPTER 2: MOMENT CONSERVING DECOUPLINGS .. . . . .. 31

2.1 Pade' Approximants and the Extended Series
of Stieltjes .... ............ 31
2.2 Moment Conserving Decoupling . . ... . . 36
2.3 Method of Solution . . . . . . . 40
2.4 Computational Considerations and Applications . 43
2.5 Evaluation of the Moment Conserving Decoupling 50

CHAPTER 3: DIAGRAM CONSERVING DECOUPLINGS . . . . 54

3.1 The Diagrammatic Expansion Method . ... . 54
3.2 Perturbation Theory . . . . . . ... 58
3.3 Equivalence of the Superoperator Formalism and
the Diagrammatic Expansion Method ... . .. 62
3.4 Diagram Conserving Decoupling . . . ... .68
3.5 Approximations and Applications . . . .. 72
3.6 Evaluation of the. Diagram Conserving Decoupling 82









TABLE OF CONTENTS (Continued)


Page


CHAPTER 4: RENORMALIZED DECOUPLINGS .. . ......

4.1 Renormalization Theory .. . .......
4.2 Derivation of the 2p-h TDA and Diagonal 2p-h
Equations . . . .
4.3 Diagrammatic Analysis .....
4.4 Computational Applications and Evaluation of
Diagonal 2p-h TDA Self-Energy ....

CHAPTER 5: PHOTOIONIZATION INTENSITIES ....

5.1 Introduction . . . . . . . .
5.2 Derivation of Computational Formulae for the
Total Photoionization Cross-Section .
5.3 Discussion of Approximations .......
5.4 Computational Applications .... ....

CONCLUSIONS AND EXTENSIONS .. . . .........

APPENDIX 1 . . . . . . . . . . . .

APPENDIX 2 . . . . . . . . . . . .

APPENDIX 3 . . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . .

BIOGRAPHICAL SKETCH ........

















Table


1. Contracted Gaussian Basis for Nitrogen . . . . .

2. Principal Ionization Energies for the Nitrogen
Molecule Resulting from the [1,0] and [2,1]
Propagator Approximants . . . . .

3. Contracted Gaussian Basis for Water . . . . .

4. Principal Ionization Energies for Water Resulting
from the [1,0] and [2,1] Propagator Approximants . .

5. Rules for Constructing Self-Energy Diagrams . . .

6. Principal Ionization Energies of Water Computed
with the 14 CTGO Basis . . . . . . . .

7. Basis Set Effects on the Ionization Energies of
Water Computed with a Second-Order Self-Energy
Approximation . . . . . . . .


Contracted Gaussian Basis for Formaldehyde . .

Principal Ionization Energies for Formaldehyde .


LIST OF TABLES


Page


46



48

49


51

67


74



76


. . 78

. . 79


10. Comparison of Principal Ionization Energies for
Water Obtained with the Second-Order and the [1,1]
Self-Energies Using the 14 and 26 CGTO Basis Sets .

11. Water Results Obtained with the Diagonal 2p-h TDA
and Diagonal 2p-h TDA Plus Constant Third-Order
Self-Energies . . . . .

12. Formaldehyde Results Obtained with the Diagonal
2p-h TDA Self-Energy . . . . . . . . .

13. Relative Photoionization Intensities for Water
Excited by Mg Ka (1253.6 eV) . . . . . . .

14. Relative Photoionization Intensities for Water
Excited by He (II) (40.81 eV) ......









LIST OF TABLES (Continued)

Table Page

15. Valence Ionization Energies for Acetylene
(24 CGTO's) . . . . . . . . . . 126

16. Valence Ionization Energies for Acetylene
(42 CGTO's) . . . . . . . . . .. .. 128

17. Relative Photoionization Intensities for Acetylene
Excited by Mg Ka (1253.6 eV) . . . . . .129

18. Relative Photoionization Intensities for Acetylene
Excited by He (II) (40.81 eV) . . . ... .. . 130















LIST OF FIGURES


Figure Page

1. Relaxation and Correlation Errors for Each of the
Principal Ionizations in the Water Molecule . . .. 6

2. A Sketch of the Energy Dependence of the Function
Wk(E) . . . . . . . . . . . . 26

3. A Sketch of W2a in the Energy Region of the 2al
Ionization for Water...... . . . . . 85

4. Fourth-Order Self-Energy Diagrams Arising From the
2p-h TDA . . .... ........ . 100

5. A Plot of the Theoretical ESCA Spectrum for the
Valence Ionizations of Water . . . . . .... .124

6. A Plot of the Theoretical ESCA Spectrum for the
Valence lonizations of Acetylene . . ... . . . 133

7. A Plot of the Photoionization Cross-Sections Versus
Photon Energy for the Valence Orbitals of Acetylene . 135

8. Orbital Plots for the 2au, 3og, and 1ru Feynman-Dyson
Amplitudes of Acetylene . . . . . . . . 138

9. A Density Difference Plot Between the 3og Feynman-Dyson
Amplitude and the 3og Hartree-Fock Orbital of Acetylene 140

10. A Density Difference Plot Between the I1u Feynman-Dyson
Amplitude and the 1ru Hartree-Fock Orbital of Acetylene 142

11. A Density Differenct Plot Between the 2ou Feynman-Dyson
Amplitude and the 20u Hartree-Fock Orbital of Acetylene 144















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


ALTERNATIVE DECOUPLINGS OF THE
ELECTRON PROPAGATOR

By

Gregory J. Born

June 1979

Chairman: N. Yngve Ohrn
Major Department: Chemistry

Several alternative decouplings of the electron propagator are

investigated in this dissertation in an attempt to derive more accurate

and more tractable computational schemes for extracting the physical

information contained in the electron propagator. When the electron

propagator is defined as a single-time Green's function, the decoupling

approximation and the choice of reference state average are shown to be

independent approximations, and the use of uncorrelated, Hartree-Fock

reference states is advocated. The derivation of each decoupling

approximation utilizes the superoperator formalism and emphasizes

elementary algebraic manipulations. In Chapter 1, operator product

decouplings are reviewed and critically discussed. In Chapter 2, the

moment conserving decouplings which consist of Pade' approximants to

the propagator moment expansion are investigated. In Chapter 3, a

partitioning of the superoperator Hamiltonian is invoked, and a pertur-

bation expansion of the superoperator resolvent is developed. This









development leads straightforwardly to the derivation of the Dyson

equation and permits an identification of wave and reaction superoper-

ators. Two types of diagram conserving decouplings are then examined,

and equivalences with the diagrammatic expansion method are demonstrated.

Finally in Chapter 4, renormalized decouplings are considered, and the

two particle-one hole, Tamm-Dancoff approximation is specifically derived

and investigated. In each of the first four chapters, the decoupling

approximations are evaluated on the basis of computational applications

in which the propagator poles are compared to experimentally determined

ionization energies for several molecules. In order to avoid a possible

bias with this evaluation criterion, the quality of the Feynman-Dyson

amplitudes is examined in Chapter 5 via the calculation of relative

photoionization intensities. The four decoupling approximations are

finally summarized as various approximations to the wave and reaction

superoperators, and several extensions of these investigations are

proposed.















INTRODUCTION

Since subatomic particles are beyond the limit of human sensory

perception, our knowledge of atomic structure is based on the interpre-

tation of measurements with auxiliary probes. The accumulation and

interpretation of data from these types of measurements have led to the

conception and axiomatization of quantum theory. Using the calculus of

this theory, quantum mechanics, there is evidence to believe that it is

possible, at least in principle, to calculate the statistical result of

any experimental measurement. Unfortunately, the mathematical complexity

of this calculus precludes exact solutions for all but a few, relatively

trivial applications; consequently, the predictive value of the theory

is limited. Owing to this limitation, one aspect of current theoretical

research involves the formulation and evaluation of accurate mathematical

approximations which are relevent to the interpretation of specific

experiments.

With the development of photoelectron spectroscopy (Turner et al.,

1970, Siegbahn et al., 1969), photoionization has become an extremely

useful probe of atomic and molecular structure and has stimulated much

theoretical interest (Cederbaum and Domcke, 1977, and references therein).

In the photoionization experiment, light is shone on an atomic or molec-

ular sample and the kinetic energy of the ionized electrons or photoelec-

trons which are ejected is then analyzed. From energy conservation, the

binding energies of the photoelectrons may be deduced.









An ab initio, theoretical interpretation of photoelectron spectra

requires the calculation of ionization energies. These calculations

reflect fundamental assumptions about the complex nature of the many-

electron interactions that occur in atoms and molecules and may be per-

formed at various levels of sophistication. One conceptually simple

scheme is the Hartree-Fock self-consistent field (HF-SCF) method (see

e.g. Pilar, 1968). In this method, an orbital energy is calculated for

each electron in an N-electron system by assuming that that electron

interacts only with an average electron density formed by the remaining

N-1 electrons. By thus averaging out the instantaneous electron-electron

interactions, the original N-electron problem is reduced to N one-electron

problems. The negative of the orbital energies obtained in this calcula-

tion can then be related to ionization energies via Koopmans' theorem

(Koopmans, 1933).

Ionization energies obtained at the Hartree-Fock level of approxi-

mation are rarely accurate and occasionally predict even the wrong se-

quence of ionization. In order to obtain more accurate ionization ener-

gies, the electron-electron interactions must be treated more realisti-

cally. In the Hartree-Fock approximation, the N-1 electrons of the ion

state are assumed to be "frozen" at the same energies they had in the

ground state. Conceptually, each electron screens to some extent the

electrostatic attraction between the positively charged nuclei and all

the other electrons in the system. As can be easily rationalized, this

screening is most effective for deep-lying or core electrons which have

a high probability density near the nucleus than for more diffuse,

valence electrons. Nevertheless, if one electron is ionized, all the

others should experience a stronger nuclear attraction and will contract









producing a lower total energy. This rearrangement defines the relaxa-

tion energy, and it may be easily incorporated in the ionization energy

calculation. Performing separate Hartree-Fock calculations for both the

ground and ion states and obtaining a total energy by adding orbital

energies with corrections for the overcounting of interelectronic repul-

sions, an improved ionization energy can be obtained by subtracting total

energies. This level of approximation is known as the AE(SCF) method

(Bagus, 1965) and generally yields reliable core electron ionization

energies.

The remaining discrepency between the AE(SCF) ionization energies

and the ionization energies obtained from the exact solution of a non-

relativistic, many-electron formulation can be defined as the correla-

tion energy. This correction arises from the tendency of any pair of

electrons in an atom or molecule to correlate their motion so as to min-

imize the electron-electron repulsion. Electron correlation can be con-

ceptualized as various virtual scattering events between bound electrons

in both the N- and (N-l)-electron systems. The simplest of these virtual

processes is a particle-hole excitation in which one bound electron

absorbs a virtual photon emitted by another electron and is excited from

its original Hartree-Fock orbital to a more diffuse orbital of higher

energy. A hole or vacancy is simultaneously created and propagates in

the system. At some later time, the excited electron may decay re-

emitting the virtual photon which may then be reabsorbed by the first

electron. Adopting the convention that holes propagate backwards in

time while electrons propagate forward in time, a particle-hole excita-

tion in the (N-l)-electron ion state can be diagrammatically represented

as
















where the dotted lines represent the virtual photon exchanges. Although

diagrams of this type represent quasi-classical, virtual processes which

are not experimentally observable, they do provide a conceptual model

and,as we will later see in Chapter 3, categorize specific algebraic

expressions that will be derived to calculate correlation energy correc-

tions. Finally, it should be noted that since the electron motions will

be correlated to different extents in the N-electron ground state and

(N-1)-electron ion states, it is not possible to predict, a prior, the

effect of this correction on the ionization energies (see Fig. 1).

One many-electron formulation which provides a systematic procedure

for incorporating both relaxation and correlation corrections into the

calculation of ionization energies and which is the basis of this invest-

igation, is the non-relativistic, single-particle Green's function or

electron propagator (Linderberg and Ohrn, 1973). This method has the

advantage of yielding ionization energies directly unlike other many-

electron formulations which necessitate the total energy calculation for

both ground and ion states and which yield the ionization energy as the

difference. In the latter methods, significant loss of accuracy is

inherent in the subtraction of two, nearly equal total energies to obtain

a much smaller, ionization energy. Care must also be taken to avoid

disparate levels of approximating electron correlation in the calculation

of each different state.





























4-


0 00
4- 0-
0V 0

U -



0 0 +
S- E 5- 0-- Q


0 (4 -
0 0 0 (14U


0-.0 0 4

S- -E




0-C05

0O O--

-1-- -. E 01
-0 O I



SO C~-'


0 00*r- <7 .





+.. aj in aj|i

.0 "'S




*4- E -' U
(U 0. 0

0- 0 r 0



i- 0 1- o .0Q






L-


'u-










I I I I I I I I


_Q
L I_
Li-

c-Q ) i
{-Q
(NJ


0

In


UHl
ur


I l I i


(AG) d0 d3


U--I


>-


Ri


I I


I I I I I'









The electron propagator can be written as a function of the space

and spin coordinates of one electron and of a complex energy variable.

For each bound electron, this function describes mathematically all of

the complicated many-electron interactions between that electron and the

remaining N-1 electrons. This function also contains information about

the interaction of an additional electron with the N bound electrons if

this electron were to be added to the system in any of several possible

orbitals. In a discrete basis representation of the electron propagator,

this information manifests itself as simple poles or singularities at

those values along the real energy axis which correspond to electron ion-

ization energies or electron affinities, that is, electron detachment or

attachment, respectively. The residues at these poles yield the single-

particle reduced density matrix from which the N-electron ground state

average of any one-electron operator may be calculated or which may be

related to transition probabilities for electron detachment or attachment

provided some description of the removed or added electron is included.

The electron propagator can be calculated in several ways. The

procedure adopted here is derived from the electron propagator equation

of motion, but one aspect of this investigation will demonstrate the

formal equivalence between this method and the diagrammatic expansion

technique. The equation of motion relates the electron propagator to

the more complicated two-particle propagator. This two-particle propa-

gator also satisfies an equation of motion which relates it to the three-

particle propagator, and so on. This hierarchy of equations finally

terminates with the N-particle propagator, but in order to make any

practical calculation, this hierarchy must be approximated at some lower

level, generally by expressing an M-particle propagator in terms of an









(M-1)-particle propagator. This approximation is called decoupling the

equations of motion and is not unique. The accuracy of calculated ioni-

zation energies and the computational effort in obtaining them depend

critically on the decoupling. This investigation proposes and evaluates

several alternative methods.

In Chapter 1, the electron propagator is formally defined, and its

equation of motion is derived. After an introduction of the superoperator

formalism, the electron propagator is approximated by an inner projection,

and the decoupling problem is studied in terms of the selection of an

inner projection manifold. The remainder of this chapter discusses the

computational procedure for solving the propagator equations and presents

a critical evaluation of the operator product decoupling.

Chapter 2 describes some general aspects of the Pade' approximant

method and its application to the calculation of the electron propagator.

Owing to the conservation of various moment matrices in the propagator

equation of motion these decouplings are known as moment conserving

decouplings (Goscinski and Lukman, 1970). Numerical results obtained

from the [1,0] and [2,1] Pade' approximants are presented and discussed.

A partitioning of the superoperator Hamiltonian and a perturbation

expansion of the superoperator resolvent in the operator space is devel-

oped in Chapter 3. A superoperator Dyson equation is derived and wave

and reaction superoperators are identified in analogy with ordinary

resolvent operator techniques. Truncations of the wave superoperator

operating on simple annihilation and creation operators are shown to

yield inner projection manifolds that result in Pade' approximants to

the self-energy. These Pade' approximants conserve various orders of

the perturbation expansion for the self-energy and are therefore









categorized as diagram conserving decouplings. Two approximations based

on this decoupling scheme are discussed and evaluated.

Chapter 4 is devoted to renormalized decouplings. These decouplings

sum certain types of self-energy diagrams to all orders. The two-particle,

one-hole Tamm-Dancoff approximation (2p-h TDA),which sums all ring and

ladder diagrams, is explicitly derived and discussed in terms of the

superoperator formalism. The diagonal 2p-h TDA previously proposed by

other authors (Cederbaum, 1974, Purvis and Ohrn, 1974) is re-examined

and is shown to neglect certain diagonal contributions. Both approxima-

tions are analyzed diagrammatically, and numerical results are presented

and discussed.

The evaluation of each decoupling approximation in the first four

chapters is ultimately based on a comparison of propagator poles to ex-

perimental ionization energies. Chapter 5, on the other hand, attempts

to corroborate this evaluation criterion by an examination of the quality

of the Feynman-Dyson amplitudes. This is indirectly accomplished via

the calculation of relative photoionization intensities and their com-

parison with experimental data. The requisiteequations for the photo-

ionization cross-section are derived in terms of the Feynman-Dyson ampli-

tudes, and the most critical approximations are discussed. Finally,

numerical results are presented and are also discussed.














CHAPTER 1
OPERATOR PRODUCT DECOUPLINGS

1.1 Definition, Spectral Representation, and Equation of Motion of the
Electron Propagator

The electron propagator is most commonly defined as a double-time

Green's function which involves an exact N-electron ground state average

of a time-ordered product of electron field operators, p(x,t) and

i(x',t') (Linderberg and Ohrn, 1973). These field operators are gener-

ally expressed in the Heisenberg representation,


p(x,t) = exp(iHt) p(x,O)exp(-iHt) (1.1)

and are functions of the combined space-spin and time coordinates of the

electrons. The operator, H, in the exponentials is the N-electron Hamil-

tonian. The field operators i(x,t) and i(x',t') have the property of

annihilating and creating, respectively, an electron at the space-spin
N
and time coordinates, x,t (x',t'). Letting I|Y> denote an exact eigen-

state of the N-electron Hamiltonian, y y -> and +1> denoting exact

eigenstates of the (N-I)- and (N+1)-electron, ion Hamiltonians, these

properties are expressed as


-(x,t)|T> z cj exp{-i(E0 E01)t} 1> (1.2)
J
and

(x' ) = c exp{- (E+1 E )t'}Y +> (1.3)
J









If the N-electron Hamiltonian is time independent, it is easy to show

that the double-time Green's function depends only on the time difference,

t-t' (Fetter and Walecka, 1971). In all practical calculations, however,

this definition proves to be a severe restriction since the exact N-

electron ground state is rarely known and an approximate ground state

average is usually employed. With an inexact ground state average, the

electron propagator will depend on both t and t'. To avoid this restric-

tion, it is possible to define the electron propagator as a single-time

Green's function by choosing t' equal to zero (Simons, 1976),


< = -ie(t)

+i9(-t) (1.4)

This definition insures the dependence on only the time difference even

when the ground state average is inexact. The brackets, < . >, in

Eq.(1.4) represent an average which may be either a pure-state average or

an ensemble average,


< . > = Z PK < I .. K> (1.5)
K

To elucidate the analytic properties of the electron propagator, it

is convenient to derive the spectral or Lehman representation (Linderberg

and Ohrn, 1973). This representation is obtained by first expanding the

eigenstates, IK>, in terms of the exact N-electron eigenstates, |Py>,


S = c | '> (1.6)


Using resolutions of the identity in terms of the exact (N-I)- and (N+1)-

electron eigenstates, Eq.(1.4) can be written:









< = E E P Ci cKj {-ie(t)
K i,j,k

x < N+1 t(x' O) > + ie(-t)< N (x' ) -1 ( ,-t) iT >} .(1.7)

Explicitly introducing the time dependence of the field operators, Eq.(1.7)
becomes

<<*i(x',O); ((x,t)>> = z z P cKi c
K i,j,k
x {-ie(t)<|txO)YN+1TN+1 t(x',O)|>exp[-i(EN+-E )t]

+ ie(-t)<'w |lt(x',0)|Yk N >< N-l (x,0) Y.>exp[i(E -Ej- )t]} (1.8)

Since the calculation of ionization energies (electron affinities) from
the electron propagator will be treated as a transition between station-
ary states of the N-electron ground state and N-I (N+1)-electron ion
states, it is convenient to Fourier transform Eq.(1.8) into an energy
representation,

<>E = <<@(x',0)(x,t)>>exp(iEt)dt (1.9)
-co

which yields the spectral representation:


fik(x)fkj(x')
<>E = lim E Z P c c
KE +L K i,j,k K E-(Ek -E) + in



+ gik(x)gkj(x')
NE-(E-E-1) (1.10)
E-(Ej-Ek ) in








where fik(x) = < l (x,O)JN+1> (1.11)


and gik(x') = < It (x',0)jY-1> (1.12)

are referred to as the Feynman-Dyson amplitudes. From Eq.(1.1O) it is

observed that the electron propagator has simple poles along the real

energy axis corresponding to the difference between exact eigenvalues of

the N-electron Hamiltonian and the (N+1)-electron Hamiltonians. The

poles of this function, therefore, have a physical interpretation as

ionization energies and electron affinities.

Since atomic and molecular computations are most conveniently per-

formed in a Hilbert space, we introduce a complete, orthonormal set of

one-electron spin orbitals, {ui(x)}. In this basis, the electron field

operators are represented by the expansion,


ip(x,t) = z ai(t)ui(x) (1.13)
i
which, with the expansion of the adjoint, defines the spin orbital anni-

hilation and creation operators, ai(t) and a-(O). At equal times, these

operators satisfy the usual anti-commutation relations,


[ai'a ] = ij (1.14)

[ai, + [aijaj]+ = 0
[a a ] = [aq,a ] = 0 (1.15)

In this discrete representation, the causal electron propagator can be

written


<> = u (x')<>u (x) (1.16)

where
where








<> = -iO(t) +iie(-t) (1.17)

Although a computational scheme for obtaining ionization energies
and electron affinities could now be established from the spectral reso-

lution, it is more convenient to develop a scheme based on the equation
of motion (Linderberg and Ohrn, 1973),

i <
> = 6(t)
at 1 3 3 1

+ <
>, (1.18)
i '-
which follows directly from Eq.(1.17). The quantity <
is a two-particle propagator, and the N-electron Hamiltonian appearing

in the commutator has the following Hilbert space representation:

H = hrsaras + 4 aa as as (1.19)
r,s r,r',s,s
where

hrs = ur(1)[-V2(1 Z Z- ~ ]us(1)dr1 (1.20)
a rla

and

= u(1)Ur,(2)r12(1-Pl2)u(1)us(2)dT1dT2 (1.21)

With a notation similar to that used in Eq.(1.9), the energy trans-

forms of the various quantities in Eq.(1.18) can be defined, e.g.


<
>E = <> exp(iEt)dt. (1.22)


Substituting the inverse transforms:

i <>E exp(-iEt)dE, (1.23)
It ''' 2fff E









6(t)
= <[ai .,a ]>exp(-iEt)dE, (1.24)


and

= <- a (0); [a (t),H]_>>Eexp(-iEt)dE,

-m (1.25)
into Eq.(1.18), we obtain


2rJ {E<
>E <[a.,a'] > <>E} exp(-iEt)dE = 0.(1.26)

From the general properties of Fourier transforms, it can be shown

(Morse and Feshbach, 1953) that Eq.(1.26) implies

E<
>E = <[ai,a']+> + <>E (1.27)

which represents the energy transform of the equation of motion. The

iteration of this equation yields N coupled equations relating the single-

particle (electron) propagator to each of the higher-particle propagators.

Successive substitution of these more complicated propagators back into

Eq.(1.27) yields

<>E = E -<[ai'aj]> + E2<[aiH]_,aj]+>

+ E-3<[[[a ,H]_,H]_,at] > + . (1.28)


1.2 The Superoperator Notation and Inner Projection Technique

The use of superoperators has antecedents in the work of Zwanzig

(1961) and Banwell and Primas (1963) in statistical physics and was intro-

duced into atomic and molecular propagator theory by Goscinski and Lukman

(1970). As a notational simplification, the definition of a superoperator








Hamiltonian and identity, H and I, provides a convenient representation

of the nested commutators appearing in Eq.(1.28). More formally, this

notation provides a connection with the time-independent resolvent

methods introduced into many-body theory by Hugenholtz (1957).

The superoperator Hamiltonian and identity are defined to operate

on the spin orbital annihilation and creation operators through the

relations

Ha. = [aiH]_ (1.29)

and

Iai = ai (1.30)

Powers of the superoperator Hamiltonian are defined by successive appli-

cation of this superoperator, i.e.


H ai. = H[a.,H] = [[ai,H]_,H]_, (1.31)

and will always yield linear combinations of odd (Fermion-like) products

of the simple field operators, a. and a'.. This set of all Fermion-like
1 J
operator products, {Xi}, forms a linear space and supports a scalar

product defined by

(XjlXi) = Tr{p[Xi,Xj]+} (1.32)

where p is a normalized, but otherwise arbitrary, density operator corre-

sponding to the N-electron ground state average of the electron propagator.

Using the preceding definitions and notation, Eq.(1.28) can be re-

written as

<
+ ) + (133)
S+ 3(ajHai.) + .. .. (1.33)








Collecting all annihilation operators in a row matrix and all creation

operators in a column and formally summing the expansion in Eq. (1.33),

the matrix equation of motion for the electron propagator becomes:

G(E) = (aj(EI-H)-a). (1.34)

The superoperator resolvent in Eq. (1.34) can now be represented in

closed form by a matrix inverse using the inner projection technique

(Liwdin, 1965, Pickup and Goscinski, 1973). Introducing a projection

operator,

0 = I f)(fI -1(f (1.35)

where 0 = 0 and 0 = 0, the inner projection of a positive definite,

self-adjoint operator, A, is given by

A' = A1 0 A2 ; A > 0 (1.36)

Making the substitution

If) = A2- h) (1.37)
the inner projection of Eq. (1.36) becomes (Bazley, 1960)

A' = Ih)(h Ahl)-1(hj (1.38)

and satisfies the operator inequalities (Lwdin, 1965)

0 < A' < A (1.39)

Since the superoperator resolvent in Eq. (1.34) is an indefinite opera-

tor, it is not valid to discuss an inner projection of the type in Eq.

(1.36). Equation (1.38) however, which does not contain A2, is still an

acceptable definition of the inner projection provided A is nonsingular.

Using this definition for an indefinite operator, the equality in Eq.

(1.39) will still hold when h is complete, but the bounding properties

will now be lost with incomplete manifolds. Using the Bazley inner








projection, the electron propagator has the following form

G(E) = (alh)(hl(EI-H)h)-1(hla) (1.40)

in which the decoupling problem has now been transformed into the

problem of choosing an appropriate inner projection manifold.



1.3 The Hartree-Fock Propagator

Before proceeding to formulate more sophisticated decoupling schemes,

it is convenient at this point to recapitulate the approximations under-

lying all propagator calculations and to demonstrate the algebraic manip-

ulations which are involved by examining one simple decoupling in some

detail. Implicitly assuming the clamped nuclei and non-relativistic

approximations, there are basically three additional approximations in-

volved in any scheme for computing the electron propagator. The first

is the truncation of the complete (infinite) set of spin orbitals,

{ui(x)}, to some finite subset. This approximation is also characteris-

tic of the more conventional wavefunction formulations and has received

considerable attention. Standard basis sets of various sizes and quali-

ties are available in the literature (Roetti and Clementi, 1974, Huzinaga,

1965, Dunning, 1970, Dunning and Hay, 1977). The second approximation

is the specification of the N-electron ground state average or equiva-

lently, a density operator (Eq. (1.32)), in terms of which the electron

propagator is defined. The final approximation is the specification of

the inner projection manifold or the actual decoupling of the equations

of motion.

The simplest approximation to the inner projection manifold, h, in

Eq. (1.40) is just the set, {ai}, of simple field operators. With this

choice, Eq. (1.40) simplifies to








G(E) (a (EI-H)a)-1 (1.41)

since

(ala) = 1 (1.42)

which can be verified by evaluating a specific matrix element:

.1-
(a. a ) = Tr{p[a ,a ] } = 6. Tr{p} = 6i (1.43)

One particularly convenient density operator, which corresponds to

an independent particle, ensemble average, is the grand canonical density

operator (Abdulnur et al., 1972, Linderberg and Ohrn, 1973),

p = H [1 + (2 1) akak]. (1.44)
k

This density operator yields the following results for various operator

averages:


Tr{pQa } = 6rs (1.45)


Tr{pa rasa s = [rsr's' r rs sr', nr> r (1.46)

and reduces to a pure state average when occupation numbers, , of

zero or one are chosen.

Considering the i,j-th matrix element of the electron propagator,

G(E)i = [E ij (ajlHa.)]-1, (1.47)

the remaining operator scalar product, (ajlilai), can be evaluated by

first operating with the superoperator Hamiltonian (Eq. 1.29)

Ha. = [ai,H] = Z hisas + a ,a ,a s (1.48)
s r',s,s'

then anti-commuting with a. (Eq. 1.32)
J








(a lHa.) = h.j + Tr{pa1,as, (1.49)
Sr',s'

Using the grand canonical density operator to evaluate the trace

(Eq. 1.45), we obtain


(aj.Ha.) = h.i + Z . (1.50)


The particular basis of simple field operators,


ai = xikak (1.51)

which satisfies the equation


Hai = ciai (1.52)

diagonalizes the matrix, (alHa), and must be obtained self-consistently.

This is an equivalent statement of the conventional Hartree-Fock procedure

since the transformation in Eq. (1.51) also defines the canonical Hartree-

Fock spin orbitals


u Z x (1.53)
Ui XikUk

The eigenvalues, e., are the Hartree-Fock orbital energies.

Substituting these results into Eq. (1.47), we obtain


G(E)ij = (E-ci)-1 ij (1.54)

for this simple decoupling scheme. The poles of this function occurring

at E = ei correspond to the Koopmans' theorem (Koopmans, 1933) approxi-

mation to the ionization energies. Because of the analogy between this

decoupling and the conventional Hartree-Fock procedure,Eq. (1.54) is

referred to as the Hartree-Fock propagator and will constitute the start-

ing point for more exact approximations.


~_









1.4 Operator Product Decoupling

As was pointed out in the introduction, the Koopmans' theorem approx-

imation to ionization energies is frequently unreliable and necessitates

the incorporation of many-electron relaxation and correlation corrections.

The simple decoupling scheme which yielded the Hartree-Fock propagator in

the preceding section can be extended to incorporate relaxation and corre-

lation by extending the inner projection operator manifold, h, in Eq.

(1.40). One extension of this manifold, proposed by Pickup and Goscinski

(1973), is the union of all operator subspaces containing different

Fermion-like products of simple field operators:


S {ak}U{aalam}U{ajaalaman}U . (1.55)


In terms of the equations of motion, Eq. (1.28), this type of decoupling

is equivalent to expressing a higher-particle Green's function in terms

of lower ones and was originally discussed in the context of atomic and

molecular theory by Linderberg and Ohrn (1967).

The formulation of explicit electron propagator approximations with

this extended operator product manifold is simplified by the use of the

partitioning technique (Lbwdin, 1962). Having already derived an ex-

pression for the Hartree-Fock propagator with a manifold consisting of

simple field operators, it is convenient to make the partition


h = aUF (1.56)


where a is the subspace of simple field operators and f represents the

orthogonal complement containing all higher, Fermion-like operator

products. This partitioning is imposed through the relations








(aja) = (f|f) = 1 (1.57)
(alf) = (fla) = 0 (1.58)

and leads to a blocked matrix equation for the electron propagator:


(a[(EI-H)a) -(a|Hf)
G(E) = (1 0) -(f( (1.59)
-(f|Ha) (f|(EI-H)f)

Solving for the upper left block of the inverse matrix, an equation for
G- (E) is easily obtained

G- (E) = (al(EI-H)a) (ajHf)(f (EI-H)f)- (flHa) (1.60)

The first term on the right hand side of Eq. (1.60) is the inverse

of the Hartree-Fock propagator and the second term represents the relax-

ation and correlation corrections to the Koopmans' theorem ionization
energies. Based on the resemblance of Eq. (1.60) with the Dyson equation

derived in quantum electrodynamics (Dyson, 1949), the Hartree-Fock propa-
gator can be identified as a zeroth order, i.e. uncorrelated, approxima-

tion to G(E) while the remaining term is identified as the self-energy,

G- (E) = 0 (E) Z(E) (1.61)


A number of approximations to G(E), based on different choices of
the operator manifold in Eq. (1.55), have been reported in the literature.
Pickup and Goscinski (1973) chose their manifold to consist of single-
and triple-operator products and replaced the superoperator Hamiltonian
in the self-energy by the Fock superoperator defined by


FX = [X,F] ; F = Z ekakak
k


(1.62)








This approximation was applied to the calculation of ionization energies

for helium and nitrogen by Purvis and Ohm (1974) and was later extended

to include the full superoperator Hamiltonian (Purvis and Ohrn, 1975a).

Redmon et al. (1975) have derived an approximation which includes

single-, triple-, and quintuple-operator products in h and have computed

the ionization energies of neon. Finally, several approximations have

been reported using an inner projection manifold of single and triple

products in conjunction with correlated reference states. (Purvis and

Ohrn, 1975b, Jdrgensen and Simons, 1975).


1.5 Method of Solution

The solution of Eq. (1.60) consists of finding the poles and Feynman-

Dyson amplitudes of the electron propagator and writing a spectral repre-

sentation similar to that of the exact propagator in Eq. (1.10). The

procedure for obtaining the spectral representation from the Dyson equa-

tion has been discussed by Layzer (1963) and Csanak et al. (1971) and

begins with a solution to the energy dependent eigenvalue problem:


L(E)t(E) = i(E)W(E), (1.63)

where

L(E) = (ajHa) + E(E) (1.64)

{(E)ji(E*) = 1 (1.65)

and W(E) is a diagonal eigenvalue matrix. Expanding in terms of the

eigenfunctions, ((E), G(E) assumes the form


G(E) = a(E)[EI-W(E)]-1 i(E) ,


(1.66)








and the poles are those values of E satisfying the equation,


Ek = Wk(Ek) (1.67)

The energy dependence of the eigenvalues, Wk(E), is sketched in Fig. 2

which shows that the poles occur at the intersections of these curves

with a line of unit slope passing through the origin. When the inner

projection manifold is energy independent, the slopes of the Wk(E) curves

arealways negative since


dE (E) = -(aJHf)(fl(EI-H)f)-2(fHa) (1.68)
dE '

and the number of propagator poles between any pair of self-energy poles

is equal to the number of basis functions in that symmetry (Purvis and

Ohrn, 1974).

From the spectral representation, it was noted that the exact propa-

gator has only simple poles, and it is easily shown that the residues at

the poles are precisely the Feynman-Dyson amplitudes. Assuming that the

approximate propagator in Eq. (1.60) also has only simple poles, the

residues can be obtained from elementary residue calculus as

lim (E-Ek)Gij(E) = rkik(Ek) k(Ek) (1.69)
E+E
E-E k

where

-1 (E)(1.70)
k = [ W- dE Wk( E=EE

According to the Mittag-Leffler theorem (Mittag-Leffler, 1880), the elec-

tron propagator can now be written as































Figure 2. A sketch of the energy dependence of the function
Wk(E) between self-energy poles (indicated by
vertical dashed lines). Propagator poles occur
at the intersections of these curves with a line
of unit slope.











T ik(E )W (E )
G (E) = kk (k J k (1.71)
k (E-Ek


which has the form of the spectral representation.



1.6 Analysis and Limitations of the Operator Product Decoupling

In order to analyze approximate electron propagator expressions as

obtained in Eq. (1.60), it is necessary to compare the propagator poles

with the exact energy differences between the corresponding N- and N-I

(N+1)-electron states. This analysis can be performed, in principle,

in one of two ways. Since a full configuration interaction (CI) calcu-

lation will yield the exact total energy to any finite dimensional, model

problem, the total energies of the N- and N-1 (N+l)-electron state could

be calculated and then subtracted to yield the ionization energy (electron

affinity). A full CI, however, is not practical except for systems well

described by small basis sets (<10) because the number of configurations

in the CI expansion, given by Weyl's formula (Shavitt, 1977), increases

roughly as N2 (2M1e/N)N where N is the number of electrons, M is the size

of the spin orbital basis, and e is the constant 2.718. Furthermore, the

CI solution, which is expressed as a determinental expansion, is not

readily amenable to detailed analysis. On the other hand, a perturba-

tion expansion of the N- and N-I (N+1)-electron total energies also

yields the exact solution (in a non-zero region of convergence), but in

addition, allows an order by order analysis of the total energy contribu-

tions in terms of orbital energies and two-electron integrals.

Using Rayleigh-Schrbdinger perturbation theory (RSPT) to represent

the N- and (N-1)-electron states through second order, Pickup and








Goscinski (1973) derived a second-order electron propagator expression

for the energy difference. Since a Hartree-Fock self-consistent field

solution was assumed as the unperturbed problem, the orbital basis sets

of the N- and (N-1)-electron states are different. Before the total en-

ergy expressions may be subtracted, therefore, the orbitals of the (N-I)-

electron problem must be expanded in terms of the N-electron orbitals.

This procedure has been extended by Born et al. (1978) through third

order, and the resulting third-order self-energy is listed in Appendix 1.

Each third-order term in Appendix 1 is characterized by a diagram and

may be alternatively obtained using diagrammatic techniques, but for

pedagogic reasons this discussion will be deferred until Chapter 3.

Although the results of Purvis and Ohrn (1974, 1975a) and Redmon

et al. (1975) represent significant improvements to the Koopmans' theorem

and AE(SCF) approximations for the ionization energies they computed, the

operator product decoupling was demonstrated to have certain computation-

al and formal limitations or ambiguities. Computationally, the most

severe limitation is the dimension of the inverse matrix in the matrix

product of Eq. (1.42). The dimension of this matrix increases rapidly

with the size of the spin orbital basis and prohibits an exact inversion

for all but the smallest basis sets. In contrast to the CI matrix where

only the lowest few eigenvalues of each symmetry are usually computed,

the inversion of this matrix requires all the eigenvalues and eigen-

vectors. As an illustration of the size problem, when f consists of only

triple products, the dimension is roughly proportional to NM(M-I) where

N is the number of electrons and M is the size of the spin orbital basis.

This limitation necessitates the approximation of the inverse matrix in

diagonal or near-diagonal form (Purvis and Ohrn, 1974).








The formal limitations of the operator product manifolds involve

certain ambiguities when extending the decoupling approximation and the

difficulty in performing an order analysis. Although Manne (1977) has

proven that the set of all Fermion-like operator products forms a com-

plete set, the extension to higher operator products is not consistent

with an order by order extension of the perturbation analysis. In fact,

Redmon et al. (1975) have suggested that perhaps some quintuple products

should be preferentially included before all triple products. This, they

felt, was particularly important in describing shake-up processes, i.e.

ionization plus a simultaneous excitation of the (N-1)-electron ion.

This observation has recently been confirmed by Herman et al. (1978) in

calculations employing the closely related equation of motion (EOM)

method (Rowe, 1968, Simons and Smith, 1973).

As mentioned in Section 1.3, correlation corrections may be included

in either the density operator or the inner projection manifold. This

dichotomy leads to another ambiguity: should larger operator products be

chosen to extend the decoupling or a more highly correlated reference

state? It has been shown by J0rgensen and Simons (1975) that in order

to obtain a decoupling approximation correct through third order, an

inner projection manifold consisting of single and triple products must

be chosen as well as a reference state which includes all single and

double excitations. Unfortunately, this combination of approximations

makes the order analysis unnecessarily complicated, as we will later

show in Chapter 3.

Finally, the Hermiticity problem should be mentioned. With a den-

sity operator that commutes with the Hamiltonian, the Hermiticity of the

superoperator Hamiltonian can be expressed as









(Xi HXj) = (HXilXj) = (Xj HX )* (1.72)


With density operators that do not commute with the Hamiltonian, however,

Eq. (1.72) is generally not satisfied and leads to both computational

and formal complications. This problem has been studied by Nehrkorn

et al. (1976) who observed computationally that the non-Hermitian terms

which arise when the density operator is correlated to first and second

order in RSPT are cancelled when the reference state was improved to

second and third order respectively. A general proof was given by

Linderberg which states that the Hermiticity error is of order n+1 when

the reference state is correlated through order n.

In the following chapters, alternative decoupling procedures will

be proposed and investigated with the intention of remedying the various

limitations inherent in the operator product decoupling as discussed

here, yet which retain a quantitative description of ionization processes.



*See Ref. (14) in Nehrkorn et al. (1976).














CHAPTER 2
MOMENT CONSERVING DECOUPLINGS

2.1 Pade' Approximants and the Extended Series of Stieltjes

The evaluation of special functions assumes a central role in

applied mathematics. A large number of these functions, from the simple

trigonometric and exponential functions to the more complex, hypergeomet-

ric functions and Green's functions, have power series expansions. Their

evaluation,therefore, consists of summing the corresponding series expan-

sion. When the series is slowly convergent or when only a limited number

of expansion coefficients are known (as e.g. through perturbation theory),

it may not be practical, or even possible, to evaluate the series term by

term until a desired accuracy has been achieved. In these cases, optimal

approximations based on a limited number of expansion coefficients are

sought. This general problem was first studied by Tchebychev (1874) and

Stieltjes (1884) for the series which bear their namesand is referred to

as the problem of moments. (For more recent reviews of this problem see

e.g. Wall, 1948, Shohat and Tamarkin, 1963, or Vorobyev, 1965). A general

solution of this problem was given by Pade' (1892) and is known as the

Pade' approximant method (Baker, 1975).

Given a function, f(z), (z complex) which admits the formal, but

not necessarily convergent, power series expansion


k
f(z) = ak (2.1)
k=O








the [N,M] Pade' approximant is defined as a rational fraction of the

form P(z)/Q(z) where P(z) is a polynomial of degree M and Q(z) is a

polynomial of degree N. The coefficients of these polynomials are

uniquely determined by equating like powers of z in the equation


f(z)Q(z)-P(z)=0 (through order zN+M) (2.2)


with the auxiliary condition Q(0)=1. The expansion of P(z)/Q(z), there-

fore, coincides with Eq. (2.1) through the (N+M)-th power of z and pro-

vides an approximation to the remaining terms.

The term by term convergence of Eq. (2.1) is replaced by the con-

vergence of sequences of approximants (such as [N,N], N=1, 2, 3, )

in the Pade' approximant method, and although general convergence

theorems are difficult to prove for arbitrary series, there exist several

extensive special cases for which convergence has been proven. For these

series, the Pade' approximant can often be shown to extend the natural

region of convergence (Baker, 1975) and may be viewed as a method of

approximate analytic continuation. A sequence of Pade' approximants,

therefore, may converge rapidly when the original series expansion con-

verges slowly or not at all.

Two series which have been extensively studied in the problem of

moments and for which sequences of Pade' approximants have been proven

to converge are the series of Stieltjes (Stieltjes, 1894) and the ex-

tended series of Stieltjes (Hamburger, 1920, 1921a, 1921b, also known as

the Hamburger moment problem). A series is of the Stieltjes type if and

only if the coefficients, ak in Eq. (2.1), can be identified as moments

of a distribution









ak = xkd(x) (2.3)
0

where ip(x) is a bounded, non-decreasing function with infinitely many

points of increase in the interval [o,-). The extended series of

Stieltjes is defined similarly for the extended interval (-m,m).

The extended series of Stieltjes has particular significance owing
to its intimate relationship with resolvents of Hermitian operators.
For any operator, A, we can define the operator function

R(zA) = (1-zA)-1 (2.4)

which is triviallyrelated to the resolvent of A. When A is Hermitian,

the spectral theorem (Riesz and Sz.-Nagy, 1955) insures a unique integral

representation of R(zA) having the form
dE(X) dE(25
R(zA) (2.5)


The operator E(X) is called an orthogonal resolution of the identity,

and when A has only a discrete spectrum, it can be written

E(A) = E 6(A-ak) O k> k

where an and n are the eigenvalues and eigenfunctions of A. For any
vector f in the domain of An for all n, the function

T dEf (A)
= R (z) = -zk- (2.7)


represents either an extended series of Stieltjes or a rational fraction








depending on whether Ef(A) has an infinite or finite number of points of

increase (Masson, 1970).

In view of possible applications of the Fade' approximant method to

the superoperator resolvent, we state two theorems regarding the extended

series of Stieltjes and discuss some properties of two particular se-

quences of Pade' approximants to these series.

Theorem 1: (Wall, 1948, theorem 86.1) A necessary and sufficient
condition for f(z) to be an extended series of Stieltjes is

0 al . a

det a2 +1 >0 ;n= 1, 2 . (2.8)



an an+1 a 2n

Theorem 2: (Masson, 1970, theorem 4) If f(z) is an extended series
of Stieltjes and the associated moment problem is determinant*,
then, for fixed j=0,+1,+2, . +m, the sequence [N,N+2j+1]
of Pade' approximants converges to f(z) for Im (z)/ 0. The
convergence is uniform, i.e.

lim JI[N,N+2j+l]-f(z) l = 0 (2.9)
N-

with respect to z in any compact region in the upper or lower
half-z plane.

In addition to being uniformly convergent, sequences of [N,N] and

[N,N-1] Pade' approximants to extended series of Stieltjes have two

other features which make them particularly attractive for computational

applications. First, these approximants are closely related to



*The moment problem is said to be determinant if there is a unique,
bounded, non-decreasing function p(x) satisfying the moment conditions
in Eq. (2.3) and the supplementary conditions i(-m)=0 and

(x) = lim '{p(x+e) + i(x-e)}
e-O








variational methods (Nuttall, 1970, 1973). When the operator R(zA) (de-

fined in Eq. (2.5)) is positive definite, the [N,N] and [N,N-1] approxi-

mants provide the following bounds to Rf(z) (Goscinski and Brandas, 1971):

[N,N] > Rf(z) > [N,N-1] (2.10)

For resolvent operators such as the superoperator resolvent which are

indefinite, bounding properties are more difficult to establish. Vorobyev

(1965) has shown, however, that the inverse poles of the [N,N-1] approx-

imant to Rf(z) are equivalent to the eigenvalues obtained from the usual

Rayleigh-Ritz variational problem

extr < > (2.11)

where c = c0f + clAf + .. . c AN- f, and the coefficients {ci} are

variationally determined. In this sense, the poles of [N,N-1] to Rf(z)

are variationally optimum, but they have no definite bounding properties.

The second attractive feature of the [N,N] and [N,N-1] approximants

is the ease with which they may be computed. Rather than solving Eq.

(2.2) to obtain the coefficients of the polynomials P(z) and Q(z), these

approximants may be expressed directly in terms of the series coefficients

{ak} using matrix formulae derived by Nuttall (1967) and Goscinski and

Brrndas (1971). For the [N,N-1] approximant, we have

[N,N-1] = a[Ao-zA1 V1 (2.12)

where, in general, ai is a column vector with the elements ai, ai+ ,

ai+N-, and A. is an N x N square matrix with the columns ai,

ai+, a i+N-I. Similarly for the [N,NI approximant we can write


[N,N] = a0 + za'[Al-zA2] -1( .


(2.13)








2.2 Moment Conserving Decoupling

Expanding the superoperator resolvent in Eq. (1.34) and multiplying

both sides of the equation by E, the electron propagator can be expressed

as the moment expansion


EG(E) = E k(a Hka) (2.14)
k=0


Before the Pade' approximant method may be applied to this equation, how-

ever, the conventional definition of the Pade' approximant must be gener-

alized to matrix Pade' approximants (Baker, 1975). This generalization

is achieved by replacing the moment coefficients by the corresponding

moment matrices and noting that these matrices do not commute when per-

forming subsequent algebraic manipulations. Using Eq. (2.12) to repre-

sent the [N,N-1] approximant to EG(E), we obtain


EG(E) = m(M-E-1 )-1Y (2.15)


or multiplying each side of this equation by E-1, Eq. (2.15) becomes


G(E) = mV(EM -M1 m0 (2.16)


where m is now a column matrix with block elements

-1


1 = (; c = (ajH1a) (2.17)

^f--1



If c has the dimensions 11 x M, M is an NM x NM square matrix with

columns mi, mi+1 m i+N-1







There is a close relationship between Eq. (2.16) and the inner pro-

jection of the superoperator resolvent

G(E) = (alh)(hI (EI-H)h)-l (hl a) (2.18)

Goscinski and Lukman (1970) have shown that if the inner projection man-

ifold is chosen to consist of

h = {ak}U{Hak}U . U{HN-ak (2.19)

the inner projection and the [N,N-1] Pade' approximant are equal. Since,

in general, the [N,M] Pade' approximant conserves the first N+M+1 moments

in the moment expansion, this choice of inner projection manifolds for

the superoperator resolvent is called a moment conserving decoupling of

the equation of motion.

An examination of the [N,N-1] approximant to the electron propagator

shows that its poles are given by the eigenvalues of

1c = Mcd (2.20)

The matrix M corresponds to a metric matrix and by virtue of the opera-

tor scalar product, is always positive definite


S= (hj) = Tr{p[hh+h h]}>0 (2.21)

The determinants of the metric matrices corresponding to various trunca-

tions of the moment conserving inner projection manifold, i.e.

det (hI0h) > 0 h = {a} (2.22)

det (hllhl) > 0 h = {a}U{Ha} (2.23)









provide the necessary and sufficient conditions of Theorem 1 to prove

that the electron propagator is an extended series of Stieltjes. Con-

sequently, the sequence of [N,N-1] Pade' approximants to the electron

propagator should be uniformly convergent and should have variationally

optimum properties.

The spectral representation of the electron propagator (Eq. (1.10))

consists of two summations, one which has poles in the lower half of the

complex E-plane corresponding to ionization energies and one which has

poles in the upper half plane corresponding to electron affinities.

Based on the physical argument that the removal of an electron from a

stable atomic or molecular system always requires energy, we might sus-

pect a separation of the superoperator resolvent which yields a nega-

tive definite operator for these processes. If this were possible, the

poles of the [N,N-1] approximants would then be upper bounds to the exact

ionization energies obtainable with a given basis. This separation

has not been explicitly demonstrated but an overwhelming amount of

numerical data seemsto substantiate this conjecture. In particular, the

[1,0] approximant, which is easily verified to be the Hartree-Fock propa-

gator, generally yields poles larger in absolute value than experimental

ionization energies. One possible exception to this rule may be the

near Hartree-Fock limit calculation of Cade et al. (1966) on diatomic

nitrogen. In this calculation, the magnitude of the ITu orbital energy

is slightly (0.2 eV) smaller than the experimental Inu ionization energy.

If on the other hand, the X ion state was fortuitously better described

than the ground state with their extended basis, this result may still be

an upper bound to the exact ionization energy in that basis.








Relaxation and correlation corrections are incorporated in any

[N,N-1] Pade' approximant beyond the 11,0] or Hartree-Fock approximant.

In particular, we have studied the [2,1] approximant in some detail.

This approximant corresponds to the truncation


h = {a}U{Ha} (2.24)


of the inner projection manifold and conserves the first four moment

matrices. The operators {filfi=Hai}, which were evaluated in Eq. (1.48),

consist of a sum over all triple products of simple field operators with

each operator product in the sum weighted by an antisymmetrized, two-

electron integral. These linear combinations provide a significant re-

duction in the subspace of triple operators thus overcoming one major

limitation of the operator product decoupling.

Another type of moment conserving decoupling of the electron propa-

gator equations of motion has been analyzed by Babu and Ratner (1972).

This decoupling is achieved by truncating the moment expansion after the

m-th moment and replacing the m-th moment matrix with

c = cG(E) .(2.25)
-m -me-

Solving the truncated moment expansion for G(E) yields

m-1
(Em+1-c )G(E) = X Em-kc (2.26)
k=O

or

1 m-1 -k
G(E) = (E+11-c )- Em-k (2.27)
k=0

These rational approximants formally conserve m moments but are not of

the Pade' type (as Babu and Ratner incorrectly identify them) since the








auxiliary equation, Q(0)=1, is not satisfied, i.e.


Q(0) = -cm 1 (2.28)
-m


The auxiliary equation guarantees the uniqueness of the Pade' approx-

imants: only one [N,N-1] Pade' approximant will conserve exactly m

moments. The nonuniqueness of Babu and Ratner's decoupling scheme is

easily demonstrated by replacing Eq. (2.25) with


m m
S-k -k
E E c = E ck G(E) 0 < n < m (2.29)
k=n k=n

Solving for G(E),


m n-1
[El E Ekc G(E) = Z E -c (2.30)
k=n 1=0


m n-l
G(E) = [Em1 Z Em-k ]-1 Em-l (2.31)
k=n 1=0


we obtain m rational approximants (n=l, m) which formally conserve

m moments. Because these approximants are not uniquely defined, we will

only consider Pade' approximants in this chapter.



2.3 Method of Solution

The first step in obtaining the spectral representation of the elec-

tron propagator with the moment conserving decoupling is the evaluation

of the necessary moment matrices. The first four moment matrices which

are necessary to construct the [2,1] approximant have been evaluated by

Redmon (1975) using the grand canonical density operator (Eq. (1.44)).








An independent check of these derivations, however, revealed an error

in the matrix elements of (alH3a) (Redmon, 1975, Eq. (11.30)). The

correct result has subsequently been verified by Redmon (1977) and

appears in Appendix 2.

Once the moment matrices have been evaluated, the matrices M and
-c
MI are constructed and the corresponding eigenvalue problem, Eq. (2.20),

must be solved. In general, the dimension of the eigenvalue problem in-

creases linearly with the size of the inner projection manifold, i.e. the

[N,N-11 approximant presents an eigenvalue problem of dimension NM where

M is the size of the spin orbital basis. For the [2,1] approximant,

therefore, the dimension of this problem is only twice the size of the

spin orbital basis. This means that for even rather large basis sets,

standard matrix eigenvalues techniques may be employed to solve this

problem in nonpartitioned form. As a consequence, all the poles and

the spectral density of the electron propagator are obtained from a

single matrix diagonalization thus avoiding the energy-dependent pole

search.

Denoting the eigenvectors by c, where

t -1
cc = (2.32)

and the eigenvalues by the diagonal matrix d, the spectral representation

of the electron propagator can be derived,

G(E)= m -1' c EM -M1 -l cc 1 1 (2.33)


= m c[E1-d]-1c-1M1oo (2.34)

Defining the matrix


(2.35)








which is rectangular with the dimensions M x NM, Eq. (2.34) becomes

G(E) = x(E1-d)-x (2.36)

This equation conserves the first 2N moment matrices of the moment expan-

sion which implies, in particular,

xx = (ala) = 1 (2.37)

from the conservation of the first moment.

The complete solution of the electron propagator which is conven-

iently obtained with this decoupling can be used to determine a self-

consistent,single-particle reduced density matrix (1-matrix). The i,

j-th element of the 1-matrix can be computed from the contour integral

(Linderberg and Ohm, 1973)

= (2n.)1 c G(E)ijdE (2.38)


The contour, c, runs from -m to m along the real axis and encloses only

poles of G(E)ij that lie below the chemical potential (u) when finally

closed in the upper half of the complex E-plane. The integral is then

evaluated using the Cauchy residue theorem (Morse and Feshbach, 1953)

= E lim (E-dk)G(E)ij (2.39)
k

S i ikxjk (2.40)

wing to the orthonormality of the spectral density elements (Eq. (2.37))
Owing to the orthonormality of the spectral density elements (Eq. (2.37))


x x. = i xik. 2,
ik jk = .ij


(2.41)








it follows that the 1-matrix is diagonal with occupation numbers deter-

mined by


= C rxik 2 (2.42)
k
Using pure state occupation numbers of zero and one in the grand canoni-

cal density operator for the initial computation of G(E), then occupation

numbers determined from Eq. (2.42) on subsequent computations, a self-

consistent set of occupation numbers can be sought.



2.4 Computational Considerations and Applications

The most time consuming step in the construction of the [2,1] Pade'

approximant to the electron propagator is the construction of the moment

matrices. The fourth moment matrix (given in Appendix 2) is particularly

difficult since it involves five unrestricted orbital summations plus

another two symmetry restricted, orbital summations of two-electron inte-

grals. Using direct summation techniques, the time needed to construct

this matrix is roughly proportional to N7. This is a formidable computa-

tional problem, but one that must be accepted in favor of the more manage-

able matrix dimensions.

Fortunately, the N7 problem is not as intimidating as it might seem

on first appearance. The "brute-force" summation of two-electron inte-

grals in the moment matrices resums certain partial sums which may appear

in more than one term or matrix element. These redundant summations can

be avoided with considerable savings in computer time by computing the

partial sums once and reusing them. Two specific partial sums we have

employed are

[ijlkl] = Z (2.43)
ss,s








{ij kkl} = E (2.44)
s,s'

Since these partial sums contain a double summation which is performed

only once, the original N7 problem is effectively reduced to N5. The

construction of the moment matrices is now comparable in difficulty to

the transformation of the two-electron integrals from the primitive basis

to the computational (usually Hartree-Fock) basis which is also roughly

proportional to N5

When the number of two-electron integrals is too large to be held

in core, their random access from peripheral storage becomes relatively

time consuming. The partial sums are much more efficiently constructed

from ordered lists of two-electron integrals which can be read into

primary (core) storage when needed. For the partial sums defined above,

the two-electron integrals must be sorted into ordered lists of the type

and where indicates all orbital indices which yield a

non-zero integral for the corresponding i,j-th distribution.

The integral sorts are performed using the Yoshimine sorting tech-

nique (Yoshimine, 1973). Briefly summarized, this technique involves a

partition of available core into a number of buffers. Each buffer holds

integrals corresponding to a specific i,j distribution, e.g. .

(When the number of distributions is large, several may be held in each

buffer.) Reading through the two-electron integral list, integrals are

then sorted into the appropriate buffers. As each buffer fills, it is

written to direct access, peripheral storage and assigned a record number.

All record numbers corresponding to integrals from the same buffer are

saved in a "chaining" array for that buffer. After the entire integral

list has been processed and all buffers have been dumped, it is then








possible to chain back through the direct access records, copying inte-

grals of the same distribution back into core. These integrals may then

be further sorted within distributions, e.g. k<1 for each i,j, and

finally saved sequentially on a peripheral storage device.

Diatomic nitrogen was the first molecule to be studied with the

[2,1] Pade' approximant. Owing to its abundance in the atmosphere, nitro-

gen has great chemical interest and has been extensively investigated

both experimentally and theoretically. It is an ideal test case for cal-

culating ionization energies from a correlated, many-electron formalism

such as propagator theory since both the Hartree-Fock and AE(SCF) approx-

imations incorrectly predict the order of the 3o and ITu ionizations
g u
(Cade et al., 1966). Only when correlation corrections are included is

the correct ordering obtained (Cederbaum and Domcke, 1977 and references

therein).

A double zeta, contracted basis of Gaussian type orbitals (GTO's)

was employed in this calculation. This basis consisted of Huzinaga's

9s,5p set of primitive orbitals (Huzinaga, 1965) which was contracted to

4s,2p (Dunning, 1970). This basis has been optimized by Dunning on the

nitrogen atom and is listed in Table 1. The corresponding one- and two-

electron integrals were calculated at the experimental internuclear

separation, R=2.068 a.u. (Herzberg, 1955), using the MOLECULE integral

program (Almlbf, 1974).

The Hartree-Fock calculation and the two-electron integral trans-

formation were performed with the program GRNFNC (Purvis, 1973). The

Hartree-Fock total energy with this basis was E(HF)= -108.8782 a.u.

which is about 3 eV higher than the result of Cade et al. (1966).

There is also a discrepancy in the Hartree-Fock orbital energies. While


















Table 1. Contracted Gaussian Basis for Nitrogen.

Nitrogen s orbitals


Contraction
Coefficients


0.002001
0.015310
0.074293
0.253364
0.600576
0.245111
1.000000
1.000000
1.000000


Nitrogen p orbitals


Contraction
Exponents Coefficients


26.7860
5.9564
1.7074
0.5314
0.1654


0.018257
0.116407
0.390111
0.637221
1.000000


Exponents

5909.4400
887.4510
204.7490
59.8376
19.9981
2.6860
7.1927
0.7000
0.2133








the calculation of Cade et al. (incorrectly) predicted the Ir orbital

energy to be 0.53 eV below the 3g this calculation predicts the Inu

energy to be 0.05 eV higher. The correct ordering of the 3o and In

ionizations with this basis is merely fortuitous, since based on a total

energy criterion, the basis of Cade et al. is more accurate.

The next step of the calculation involved the integral sorts, par-

tial summations, and the construction of the moment matrices. The poles

and spectral density were finally computed as outlined in the previous

section and are presented along with the [1,0] results in Table 2. The

ionization energies of both approximants seem to be upper bounds to the

experimental results of Siegbahn et al. (1969), but without exception,

the results of the [2,1] approximant are worse than the [1,0] approximant.

In an attempt to incorporate some ground state correlation into the grand

canonical density operator, new occupation numbers were computed from the

spectral density and the [2,1] approximant was recalculated. This cal-

culation, however, yielded no significant improvements in the ionization

energies.

In order to ascertain whether the poor results from the [2,1] approx-

imant for nitrogen are representative of other calculations or just the

consequence of a pathological test case, the water molecule was chosen

for a second application. Similarly to the calculation for nitrogen, a

double zeta contracted basis of GTO's was also employed in this calcula-

tion. Huzinaga's 9s,5p primitive basis for oxygen and 4s primitive basis

for hydrogen were contracted with Dunning's coefficients to 4s,2p and

2s, respectively. The orbital exponents for the hydrogen atoms were

scaled by 1.14 to more realistically represent the effective nuclear

charge in the molecule,and the final basis appears in Table 3. Again,




















Table 2. Principal Ionization Energies for the Nitrogen
Molecule Resulting from the [1,0] and [2,1]
Propagator Approximants.


Orbital

1iC
g
2(r
g
3o
9


lu
20
U


[1,0]

427.7

41.6

17.0

17.0

427.6

21.0


[2,11

472.4

46.5

30.4

23.1

478.8

30.9


Exp.a

409.9

37.3

15.5

16.8

409.9

18.6


E(HF) = -108.8782 H.


aSiegbahn et al. (1969).













Table 3. Contracted Gaussian Basis for Water.


Hydrogen s sets

Contraction
Exponents Coefficients

13.3615 0.032828
2.0133 0.231208
0.4538 0.817238
0.1233 1.000000


Oxygen s sets

Contraction
Exponents Coefficients


7816.5400
1175.8200
273.1880
81.1696
27.1836
3.4136
9.5322
0.9398
0.2846


0.002031
0.015436
0.073771
0.247606
0.611832
0.241205
1.000000
1.00000
1.000000


Oxygen p sets

Contraction
Exponents Coefficients


35.1832
7.9040
2.3051
0.7171
0.2137


0.019580
0.124189
0.394727
0.627375
1.000000








the integrals were computed with the MOLECULE program at the equilibrium

internuclear geometry, R(OH)= 1.809 a.u., 4HOH = 104.50 (Benedict et al.,

1956). A total energy of E(HF)= -76.0082 a.u. was computed with the

Hartree-Fock portion of GRNFNC and was followed by the two-electron inte-

gral transformation. Finally, the integral sorts and partial sums were

performed, the moment matrices constructed, and the poles and spectral

density obtained for the [2,1] approximant. The results for both the

[1,0] and [2,1] approximants are presented in Table 4 and appear to be

upper bounds to the experimental ionization energies. Once more, the

[2,1] results are consistently worse than the [1,0] results. A few

iterations on the occupation numbers yielded no significant improvements.



2.5 Evaluation of the Moment Conserving Decoupling

Formally, the moment conserving decoupling is an attractive decou-

pling procedure. Being closely related to the Fade' approximant method,

this decoupling allows the application of numerous results from the clas-

sical moment problem to propagator theory. In particular, it was proven

that the sequence of [N,N-1] approximants converge uniformly to the

exact electron propagator in a given basis, and it was shown that these

approximants represent a variationally optimum choice of the inner pro-

jection manifold. Why then are the results of the [2,1] approximant so

much worse than the results of the [1,0] approximant? To answer this

question, it is necessary to analyze the three approximations identified

in Section 1.3, namely: basis quality, density operator, and decoupling

procedure.

First of all, since computational economy and not high accuracy was

the criterion for the test calculations on nitrogen and water, polarization



















Table 4. Principal Ionization Energies for Water Resulting
from the [1,0] and [2,1] Propagator Approximants.

Orbital [1,0] [2,1] Exp.a

la1 559.4 619.2 540.2

2a1 37.0 44.7 32.2

3a1 15.4 29.7 14.7

lb1 13.8 32.6 12.6

Ib2 19.5 29.6 18.6

E(HF) = -76.0082 H.

aSiegbahn et al. (1969).








functions were intentionally excluded from the basis sets. Polar-

ization functions are diffuse, virtual orbitals which can be very impor-

tant in describing electron relaxation and correlation (Purvis and Ohrn,

1974, Cederbaum and Domcke, 1977). It is reasonable to expect that the

addition of polarization functions will improve both the [1,0] and [2,1]

approximants to varying degrees; however, with the same basis and with

the same density operator, the larger inner projection manifold (if

judiciously chosen) should yield a more accurate decoupling. Since this

was not the situation in these test calculations, any improvements in

the basis sets did not seem worthwhile.

Second, it is possible that significant ground state correlation

may have been neglected with our choice of the grand canonical density

operator. With the spin orbital annihilation and creation operators

expanded in the Hartree-Fock basis and using pure state occupation

numbers of zero or one, this density operator yields the uncorrelated,

Hartree-Fock ground state average. Rather than explicitly correlating

the density operator (as e.g. through perturbation theory), an attempt

was made to estimate the effect of correlation through the self-consis-

tent determination of the occupation numbers as described in Section 2.3.

This procedure was not pursued to true self-consistency since each iter-

ation required a complete recalculation of the [2,1] approximant. It

was obvious, however, after the first few iterations that no significant

improvements had been obtained.

Based on the preceding implications, the third approximation--the

inner projection manifold truncation--seems to be primarily responsible

for the poor numerical results. Owing to the complicated operator sums

in this manifold, an order analysis (as discussed in Section 1.6) is not








readily possible. Consequently, it is extremely difficult to identify

the problem with this decoupling procedure. It can only be concluded

that the number of moments conserved is not a useful criterion for decou-

pling. This conclusion is consistent with the uniform convergence of

the [N,N-1] sequence since uniform convergence is not necessarily mono-

tonic, but it suggests that more accurate decouplings require the incor-

poration of more information about the moment expansion than just the

moment matrices. The additional information needed is indeed available

and,in the next chapter, we will demonstrate how it may be extracted

using perturbation theory.



*Babu and Ratner (1972) reported the same conclusion which was based
on an application of their rational approximants to the Hubbard model.














CHAPTER 3
DIAGRAM CONSERVING DECOUPLINGS

3.1 The Diagrammatic Expansion Method

The superoperator formalism which is employed in the previous two

chapters is by no means the only formalism available to formulate decou-

pling approximations for the electron propagator. Two commonly used,

alternative methods are the functional differentiation method (see e.g.

Csanak et al., 1971) and the diagrammatic expansion method (see e.g.

Mattuck, 1967, Fetter and Walecka, 1971, or Cederbaum and Domcke, 1977).

Of these latter two methods, the diagrammatic expansion method has proven

to be particularly effective. This method avoids some of the algebraic

tedium involved in deriving propagator decoupling approximations by

establishing certain rules for constructing and manipulating diagrams

which represent the underlying algebraic structure.

The diagrammatic expansion of the electron propagator is usually

derived using time-dependent perturbation theory. The N-electron Hamil-

tonian is partitioned into an unperturbed part plus a time-dependent

perturbation


H = H0 + exp(-eltl)V (3.1)


where e is a small positive quantity. The unperturbed Hamiltonian, HO,

is chosen to yield an exactly solvable, eigenvalue problem


Ho 0 > = Eo o> (3.2)







and the time dependence of the unperturbed eigenstates is given by

Dg0(t)> = exp(-iHot) 0> (3.3)

In order to simplify the remaining problem of finding the fully
perturbed eigenstates I'(t)>, it is convenient to introduce the "inter-
action representation" (Fetter and Walecka, 1971) by the transformation

|l' (t)> = exp(iHot)lI (t)> (3.4)


In this representation, the Schrddinger equation has the form

i J- |I (t)> = exp(-cltl)V(t)[I i(t)> (3.5)

where

V(t) = exp(iHot)Vexp(-iH0t) (3.6)

The time dependence of the interaction eigenstates can be expressed as

|I(t)> = Ue(t,t o )TI(to)> (3.7)

where U (t,t0) is the time-evolution operator. Substituting Eq. (3.7)
into Eq. (3.5), the evolution operator is found to satisfy the differen-
tial equation


i U (t,t ) = exp(-Eltl)V(t)U (t,to) (3.8)

with the initial condition

UE(tot0) = 1 (3.9)

It is more convenient to solve for U (t,t0) by first transforming Eq.
(3.8) into an integral equation







t
U( t,t ) = 1 i f dt1 exp(-cl|t )V(tl)U (tl,t ) (3.10)
to

This integral equation has the form of the Volterra equation of the
second kind (Ldwdin, 1967) and is solved iteratively
t
UE(t,tO) = 1 i f dt1 exp(- ltl)V(t1)
to
t tI
+ (-i)2 dt1 f dt2 exp{-E(It +It2 )}V(tl)V(t2)U (t2't ) (3.11)
t t

1 n-1

0 0 0
-in d [ exp{-c [t I+ItF+ . .



x V(t )V(t2) ... V(tn) t>t1t2> . > tn (3.12)


Eq. (3.12) can be generalized slightly by modifying the limits of inte-
gration and introducing the time ordering operator, T,

t t t
U (tto)= E (-i)n dt dt . dt
t0 t0 t0

x exp{-e(ItiI+It21+ . ItJn )} T[V(t1)V(t2) . V(tn)] (3.13)


The time ordering operator rearranges the product of perturbation opera-
tors such that the left-most term is the latest in chronological order.
The perturbed eigenstates 'iY(tO)> can now be expressed in terms of
the unperturbed eigenstates by noting that as t0 --, I J(tO)>I 0'>, and
as tO increases from -m to zero, the perturbation is "adiabatically
switched on"







I1'I(0)> = U (o,- )l 0o> (3.14)

According to a theorem of Gell-Mann and Low (1951), if

U (o,- >)l| > I| I(0)>
lim (3.15)
S0 <0 IUE(o,'-m)I > ITI(0)> (3.15)

exists, then it is an eigenstate of HI

H|i (0)> E 'rI(0)>
(3.16)


These results can now be used to determine the electron propagator.
In Chapter 1, the propagator was defined as the ground state average of
a time-ordered product of field operators in the Heisenberg representa-
tion


iG (t) =-
Using Eq. (3.15) and the fact that IyH>=II(0)>, this average can be ex-
pressed in the interaction representation as

< o|UC(=,t)T[ai (t)aI(o)]u (t,- ) O>0
iG.i (t) < !U ->= (3.18)

Using the expansion of the evolution operator (Eq. (3.13)) and taking
the limit e-O, it can be shown (Fetter and Walecka, 1971) that


iG.i (t) = (-i)n 1 dt . dt
13 n= n n


0 (
x m) i 0> (3.19)








The final step in the diagrammatic expansion method is to expand

the numerators of each term in Eq. (3.19) using Wick's theorem (Wick,

1950) and to represent them diagrammatically (e.g. Fetter and Walecka,

1971). The denominator of Eq. (3.19) must also be expanded and dia-

grammed, and when this is done, all disconnected diagrams

arising from the expansion of the numerator will cancel (Abrikosov

et al., 1965).

Formally, the diagrammatic expansion method and the superoperator

formalism appear strikingly dissimilar. The diagrammatic method is

formulated in the causal representation while the superoperator formal-

ism utilizes the energy representation. The diagrammatic method employs

a pictorial representation of the algebraic structure while the super-

operator formalism emphasizes the algebraic structure directly. Yet

the primary goal of each formalism is the same: an accurate prediction

of ionization energies and electron affinities. Therefore, the two

formalisms are inherently equivalent. It is our desire in this chapter

to explicitly demonstrate the equivalence between these two formalisms

and to re-examine the superoperator decoupling approximations in terms

of a diagrammatic analysis.



3.2 Perturbation Theory

The unifying feature of the diagrammatic expansion method and the

superoperator formalism is perturbation theory (Born and Ohrn, 1978).

Since the commutator product is distributive with respect to addition,

we can define a partitioning of the superoperator Hamiltonian into an

unperturbed part plus a perturbation,

SH = + V (3.20)








One convenient partitioning, which will be shown to readily yield the

Hartree-Fock propagator as the unperturbed electron propagator, is the

M0ller-Plesset partitioning (MOller and Plesset, 1934). With this par-

titioning, H0 has the form


H a a (3.21)
r r,r'

and the perturbation is expressed as

V = E [ra a',a a-6r ,aa ] +
r,r',s,s' r r r s r r r,r'
(3.22)
Of course, when the commutator product is formed for the superoperators,

the constant term in these definitions will cancel.

Other partitionings of the Hamiltonian may also be assumed and may

lead to superior convergence properties (Claverie et al., 1967). One

alternative partitioning which has been employed in the perturbation

calculation of correlation corrections to the total energy is the Epstein-

Nesbet partitioning (Epstein, 1926, Nesbet, 1955a, 1955b). In propagator

applications, the work of Kurtz and Ohm (1978) may be roughly interpreted

in terms of a partitioning where the unperturbed Hamiltonian incorporates

all relaxation contributions to the ionization energy. It is difficult

to define this unperturbed Hamiltonian explicitly, but it formally satis-

fies the eigenvalue equation


H0ak = AEk(SCF)ak (3.23)

in contrast to


Hoak = Ckak (3.24)

for the M11er-Plesset partitioning. The method of Kurtz and Ohrn yields







excellent ionization energies and electron affinities with a simple

second-order self-energy, however it has not been formally analyzed in

detail.

Corresponding to the partitioning of the superoperator Hamiltonian,

we can introduce a partitioning of the operator space defined by the

projection superoperators 0 and P,

0 = E ak)(akl = |a)(aj (3.25)
k

P I 0 (3.26)

These superoperators operate on elements of the operator space through

the relations

OXi = |lak)(akIXi) (3.27)
k

PX. = Xi OXi (3.28)

and are idempotent (02 = 2 = P), self-adjoint ( -t = 6, P = P), and

mutually exclusive (OP=PO=0). The superoperator 0 projects from an ar-

bitrary operator product that part which lies in the model subspace,

i.e. that part which is spanned by the eigenelements of HO. The super-

operator P projects onto the orthogonal complement of the model subspace,

i.e. that part which we have no a priori knowledge about.

To obtain a perturbation expansion of the superoperator resolvent,

we consider its outer projection (Liwdin, 1965) onto the model subspace,
I ^
G(E) = O(EI-H) -1 (3.29)

= 0(EI-HQ-V)-1 (3.30)
= 0(6-]0- ) 1 (3.30)


By iterating the identity







(A-B)-1 = A 1 + A-1B(A-B) (3.31)

the inverse in Eq. (3.30) can be expanded as

G(E) = (EI-H 0)-O + (Ei-H) 0)1V(EI-HO) 1O

+ (EI-Ho -0oV(EI-Ho1(EI-HO)-o1 + . (3.32)

where the property

[HO,0]- = 0 (3.33)

has been used. Now since 0 plus P form a resolution of the identity,
each resolvent of HO occurring between perturbation superoperators, V,
can be rewritten as a sum of its projections on the model subspace and
the orthogonal complement,

(EI-HO)-1 = (EI-H0)-10 + (EI-HO)-1P (3.34)

= GO(E) + TO(E) (3.35)

With this notation, Eq. (3.32) becomes

G(E) = GO(E) + GO(E)VGO(E) + Go(E)V[Go(E) + TO(E)JVG0(E)

+ GO(E)V[GO(E) + TO(E)]V[GO(E) + T0(E)]VGO(E) + . .(3.36)

and can be resumed to yield

G(E) = Go(E) + G0(E)[V + VTO(E)V + VTO(E)VTo(E)V + .. ]G(E) (3.37)

Defining the reduced resolvent of the full superoperator Hamiltonian as
T(E) = P[a0 + P(EI-H)P]1P (cWO) (3.38)
= TO(E) + TO(E)VT(E) (3.39)







Eq. (3.37) can be written in closed form


G(E) = GO(E) + GO(E)[V + VT(E)V]G(E) (3.40)

Alternatively, we can define wave and reaction superoperators through

the equations (cf. Ldwdin, 1962, or Brandow, 1967)

W(E) = I + T(E)V (3.41)

t(E) = VW(E) (3.42)

The reduced resolvent, wave, and reaction superoperators introduced

in this section are functions of the superoperators I, HO, and V and as

a consequence, operate in a more complicated way. To apply a superoper-

ator function to an operator in the operator space, it must first be ex-

panded in terms of the superoperators I, HO, and V which are then suc-

cessively applied to the operator. For example,


W(E)Xi = [I + T(E)V]Xi (3.43)

= [I + TO(E)V + TO(E)VTO(E)V + ... ]Xi (3.44)

where

TO(E)VXi = [E-1 + E-2 H + E H + . PVXi (3.45)

etc.


3.3 Equivalence of the Superoperator Formalism and the Diagrammatic
Expansion Method

Eqs. (3.37) and (3.40) represent the superoperator form of the Dyson

equation (Dyson, 1949), and the reaction superoperator (Eq. (3.42)) can

be identified as the self-energy. To demonstrate that Eq. (3.37)








corresponds term by term with the diagrammatic propagator expansion, we

must first form the operator average of G(E) to obtain the matrix Dyson

equation, next evaluate all necessary operator averages, and finally

diagram the resulting algebraic formulae. Owing to the complicated

operator averages that must be evaluated in third and higher orders of

the perturbation superoperator, the equivalence between these two for-

malisms has only been explicitly demonstrated through third order and

is assumed in all higher orders.

The matrix Dyson equation is obtained by forming the operator aver-

age of G(E) with respect to the basis elements of our model subspace


G(E) = (a|G(E)a) (3.46)

= G2(E) + (E)(E)(E(E)E (3.47)

where


Z(E) = (ajVa) + (aJVTo(E)Va)


+ (aIVTO(E)VTO(E)Va) + . . (3.48)

Since HO was chosen to be the Fock superoperator, the appropriate den-

sity operator to employ in the evaluation of the operator averages is

the Hartree-Fock density operator. Realizing that the grand canonical

density operator (Eq. (1.44)) reduces to the Hartree-Fock density oper-

ator when pure state occupation numbers of zero or one are chosen, we

shall employ this density operator.

Beginning with the evaluation of matrix elements for the unperturbed

propagator, Go(E), the Hartree-Fock propagator is easily obtained (cf.

Section 1.3)








Go(E)ij = (aj (EI-H )-lai) (3.49)

= E-1(ajai) + E-2 IH ) EE-3(ajIq2ai) + . (3.50)

-1 2 -3 2
= E ij + E iij + E3 c ij + . (3.51)


GO(E)ij = (E- i)-1 ij (3.52)

The evaluation of each term in the self-energy expansion requires the

initial evaluation of Va.,

Vai = 14 7 [ai,arar,a s a s
r,r',s,s r r
+
[ai,aras ] (3.53)
r,s,s'

= a a ,a a (3.54)
r,s,s' r s s S s

With this result, the first-order term (aj|Vai) is obtained without much

additional effort

(1)(E)i = (aj Va i) (3.55)

= 2 Tr{p[a ar,asaj]+}


E Tr{p[as,a ] } (3.56)
,S '

= 6 rs, (3.57)
r,s s

Y(1)(E)j = 0 (3.58)

When the effective, single-particle potential used in the unperturbed

problem is the Hartree-Fock self-consistent field potential, all single-

particle corrections vanish (Bartlett and Silver, 1975a).







The evaluation of the second- and higher-order self-energy matrices

requires the evaluation of TO(E)Vai and VT(E)Vai. The first of these

quantities can be expanded as

T0(E)Vai = (EI-H )-PVai (3.59)


= (EI-HO)'Va z (EI-HO) lak)(akl Vai) (3.60)
k
using Eq. (3.26). It follows from the previous result for (a.jVai) that

the second term in Eq. (3.60) vanishes. The first term can now be evalu-

ated by expanding the resolvent of H0 and realizing that any operator

product is an eigenelement to HO, i.e.


a aas,as (r- s -Es)a asas (3.61)

Consequently, we obtain

To(E)Vai = (E+cr-cs- s ,' a asas
r,s,s s

E (E- s)-l < is' lss'>as (3.62)
s,s'
with the help of Eq. (3.45). The remaining application of V and the
I-
average value evaluation is straightforward and yields

(2)(E)ij = (aj VTo(E)Vai) (3.63)


ir[+'-J (3.64)
r,s,s (+r r s r

for the matrix elements of the second-order self-energy.

The Hartree-Fock average is now obtained by choosing occupation
numbers of zero and one. An examination of the occupation number factor

in Eq. (3.64) reveals that with this restriction, it will be non-vanishing







only when the summation index r runs over occupied spin orbitals and s

and s' run over unoccupied spin orbitals or when r runs over unoccupied

spin orbitals and s and s' run over occupied spin orbitals. Denoting

a, b, c, . as summation indices over occupied spin orbitals; p, q,

r, .. for unoccupied spin orbitals; and i, j, k, . for unspecified

spin orbitals, Eq. (3.64) can now be written as two terms which involve

restricted spin orbital summations


E(2)(E) = E
a,p,q (E+a- p q

+ < <~ ab> (3.65)
pab (E+ -C-
pa,b p a b

The conversion of Eq. (3.65) into diagrams is a straightforward pro-

cedure for which we shall use the rules and diagram convention of Brandow

(1967) and Bartlett and Silver (1975b). This convention represents the

synthesis of the antisymmetrized vertices of the Hugenholtz (1957) or

Abrikosov (1965) diagrams with the extended interaction lines of the

Goldstone (1957) diagrams, and the rules for constructing these diagrams

are given in Table 5. The application of these rules to the terms in

Eq. (3.65) yields the following diagrams:

,, (3.66)
2ap,q (E+c a-CP-Cq)
a,p,q VL "a 'p ` q'

1. E j
p,a,b E p-a-c b) (3.67)

These diagrams are precisely the same as those obtained in the second-

order diagrammatic expansion after a Fourier transformation into the

energy representation (Cederbaum and Domcke, 1977).








Table 5. Rules for Constructing Self-Energy Diagrams.

1. Each antisymmetrized two-electron integral factor in the
numerator is represented by an interaction line with a vertex
(dot) at both ends. The number of interaction lines denotes
the order of the term.

2. Using the Dirac bra-ket notation, both indices in the bra are
represented by lines leaving a vertex while those of the ket
are represented by lines entering a vertex. There must be only
one outgoing and one incoming line per vertex, therefore, assign
the index of electron coordinate one to the left vertex and the
index of electron coordinate two to the right vertex of each
interaction line.

3. Summation indices running over hole states (occupied orbitals)
are directed downward, indices running over particle states
(unoccupied orbitals) are directed upward, and external indices
(not summed) are drawn horizontally.

To Check Diagrams:

4. The energy denominator of the diagrammed expression should be
obtained by first connecting the external lines and assigning
a factor of E to this directed segment. Second, imagine hori-
zontal lines drawn between each pair of interaction lines.
Each horizontal line corresponds to a multiplicative, denominator
factor obtained by summing the orbital energies of each hole
(downgoing) line that intersects it minus the sum of orbital
energies for particle (upgoing) lines that intersect it. Treat
the factor E of the connected external lines as an orbital energy.

5. Numerical factors should be obtained by assigning a factor of !
for each pair of equivalent internal lines. Equivalent internal
lines are two lines which begin on the same interaction line,
end on the same interaction line, and go in the same direction.

6. The overall sign factor should be obtained by assigning a factor
of minus one to each internal hole line segment and a minus one
to each closed loop.








The evaluation of the third-order self-energy matrix is similar to

the second-order matrix but much more tedious and the result is presented

in Appendix 3. As was done for the second-order expression, the occupa-

tion numbers must again be restricted to zero and one to obtain the

Hartree-Fock average. When this restriction is made, the unrestricted

spin orbital summations in Appendix 3 will reduce to summations involving

occupied, unoccupied, and unspecified spin orbitals. Using the algebraic

identity

A A 1 1
(E-a)(E-bT Ta- T - TET-b] (3.68)

it is possible to combine terms in such a way that expressions involving

only occupied and unoccupied spin orbital summations are obtained. These

expressions are presented in Appendix 1. The corresponding diagrams in

Appendix 1 again are precisely those occurring in the third-order, dia-

grammatic self-energy expansion.



3.4 Diagram Conserving Decoupling

The wave and reaction superoperators identified with the help of

perturbation theory in Section 3.2 have special importance in the develop-

ment of decoupling approximations for the electron propagator. As we

have already seen, the reaction superoperator generates the diagrammatic

self-energy expansion. A truncation of this expansion offers one viable

decoupling scheme. The wave superoperator, on the other hand, has the

property of generating eigenelements to the full superoperator llamilto-

nian from the eigenelements of the unperturbed superoperator Hamiltonian


(EI-H)W(E)a = 0


(3.69)







This property is easily proven by first using Eq. (3.41) to expand W(E)

and then premultiplying both sides of Eq. (3.69) by P

(EI-H)W(E)a = (EI-H)a + (EI-H)T(E)Va (3.70)


P(EI-H)W(E)a = P(EI-H)a + P(EI-H)T(E)Va (3.71)

Using the identity

P(EI-H)T(E) = P (3.72)

and the property Pa = 0, Eq. (3.71) simplifies to


P(EI-H)W(E)a =- PVa + PVa = 0 (3.73)

which implies the validity of Eq. (3.69)

It is of interest at this point to show a connection between the

superoperator formalism and the Equations of Motion (EOM) method for

determining ionization energies (Simons and Smith, 1973). In this

method, one seeks solutions of the equation

[H,Q]_ = ojQ (3.74)

which is precisely Eq. (3.69). Here the operator Q is interpreted as a

correlated ionization operator that generates, in principle, the exact

(N-1)-electron ion states from the exact N-electron reference state.

One approach to solving Eq. (3.74) involves the application of Rayleigh-

Schrbdinger perturbation theory (Dalgaard and Simons, 1977). By parti-

tioning the Hamiltonian operator, expanding both the ionization operator,

Q, and the ionization energy, (o, in terms of a perturbation parameter,

and collecting terms of the same order, a set of perturbation theory

equations are obtained. The solution of these equations yields an expan-

sion for Q which is analogous to the superoperator equation








h = W(E)a (3.75)

The only difference is that E is replaced by wO which is a consequence
of using Rayleigh-Schridinger rather than Brillioun-Wigner perturbation
theory.

Returning now to the inner projection of the superoperator resolvent,

G(E) = (aih)(hl(EI-H)h)-1(hJa) (3.76)

we may view Eq. (3.75) as an alternative prescription for choosing the

inner projection operator manifold. Recalling from Section 1.6 that since
the density operator describing the unperturbed (model) problem does not

commute with the full Hamiltonian, the operator scalar product will not
in general exhibit Hermitian symmetry. Consequently, we define


(h]=(al t(E) (3.77)

and note that

(aJW (E) / (W(E)aI (3.78)

Approximate electron propagator decouplings can now be obtained by
truncating the expansion of the wave superoperator,

W(E) = I + To(E)V + To(E)VT0(E)V + . (3.79)

Truncation of this expansion, with only the superoperator identity, triv-
ially yields the Hartree-Fock propagator, therefore we next consider

W(E) = I + TO(E)V (3.80)

Noting that the subspaces {ak} and {fklf = TO(E)Vak} are mutually
orthogonal, Eq. (3.76) can be readily solved for G- (E)







G (E) = G (E) (E) ,(3.81)

where

Z(E) = (ajVT0(E)Va)(aIVT0(E)(EI-H)TO(E)Va) (aIVTO(E)Va) (3.82)

Making the following identifications from Section 3.3:

(aVT 0(E)Va) = (2)(E) (3.83)


(aIVT0(E)(EI-H0)T0(E)Va) = (2)(E) (3.84)

and

(aJVT0(E)VT0(E)Va) = Z(3 (E) (3.85)

Eq. (3.82) can be rewritten

z(E) = Z(2)(E)[E(2)(E) Z(3)(E)]-Z(2)(E) (3.86)

Expanding the inverse of Eq. (3.86), we easily see that this self-

energy approximant coincides with the diagrammatic expansion through

third order but additionally yields contributions to all higher orders.

If the exact self-energy is rewritten as a moment expansion in terms of

a perturbation parameter, A,

X-h (E) = xk(alV(T0(E)V)ka) (3.87)
k=O

we see that Eq. (3.86) represents the [1,1] Pade' approximant to this

expansion. Owing to the close connection between Pade' approximants and
the inner projection technique as demonstrated in Chapter 2, this result

is not surprising. These Pade' approximants to the self-energy, however,
will have entirely different convergence properties than those studied in

Chapter 2.








3.5 Approximations and Applications

Computational applications of the [1,1] Pade' approximant to the

self-energy require the evaluation of the second- and third-order self-

energy matrices.The second-order matrix is relatively easy to evaluate.

The third-order matrix, on the other hand, is exceedingly more difficult

and can presently be only approximately calculated without excessive

computational effort. An examination of the formulae in Appendix 3 re-

veals that unlike the fourth moment matrix in the moment conserving

decoupling, the third-order self-energy matrix is energy dependent.

This additional complication makes the partial summation technique used

in the moment conserving decoupling ineffectual since the third-order

self-energy matrix will generally need to be resumed with different

values of E hundreds of times in the search for poles of the propagator.

The first approximation that we will examine is the complete neglect

of the third-order self-energy matrix. With this approximation, the

[1,1] approximant in Eq. (3.86) reduces to a second-order truncation of

the diagrammatic self-energy expansion,

Z(E) = Z(2)(E) (3.88)

This second-order self-energy approximation is interesting not only be-

cause it contains the most important relaxation and correlation correc-

tions to Koopmans' theorem (in a perturbation theoretical sense), but

also because it exhibits the same analytic form as the exact self-energy

(Hedin and Lundqvist, 1969, Cederbaum and Domcke, 1977). Furthermore,

since several second-order, ionization energy calculations have been

reported in the literature, this approximation will afford both a con-

venient check of new computer code and the computational experience

necessary to implement more refined approximations.








The first computational application of this decoupling approximation

was to the water molecule using the same basis and internuclear geometry

as described in Section 2.4. The results of this calculation are pre-

sented in Table 6 along with the Koopmans' theorem, AE(SCF), and experi-

mental values for the ionization energies. Two ionization energies have

been tabulated for the 2al ionization with their corresponding pole

strength (rk of Eq. (1.70)) in parentheses. The occurence of two, rela-

tively strong propagator poles for this ionization represents a breakdown

in the quasi-particle description of inner valence ionizations (Cederbaum,

1977) and makes assignments of principal and shake-up ionizations ambig-

uous. In general, the second-order ionization energies are quite en-

couraging and represent significant improvements to each of the Koopmans'

values. Furthermore, these results are comparable in accuracy to the

AE(SCF) results but possess the convenience of being obtained in a single

calculation whereas the AE(SCF) results required six separate Hartree-

Fock calculations.

The relatively poor agreement of the 3al and Ib1 ionization energies

with the experimental values in Table 6 seems attributable to basis in-

completeness. Despite the lack of polarization functions, this suspicion

is supported by the facts that the 3a orbital is the highest occupied

orbital in that symmetry and that this basis contains only two contracted

Gaussian orbitals of b1 symmetry. In order to study the basis dependence

of the second-order self-energy approximation, two additional calculations



*The ESCA spectrum of the water molecule (Siegbahn et al., 1969)
substantiates this phenomenon since the 2a peak is quite broad and asym-
metric. Experimentally, it appears that the lower energy ionization
should have a larger pole strength (in contrast with the results of Table
5) since the peak is skewed to higher binding energies.

















Table 6. Principal Ionization Energies of Water Computed
with the 14 CGTO Basis.


Orbital

la1

2a1


3a1

IbI

lb2


aGoscinski

Siegbahn


Koopmans

559.4

37.0


15.4

13.8

19.5


AE(SCF)a

540.8

34.6


13.0

11.0

17.8


(2)
z(E)

539.4

34.0 (.61)
32.6 (.28)

12.9

10.8

18.1


b

540.2

32.2


14.7

12.6

18.6


et al. (1975).

et al. (1969).








were performed with larger basis sets. The first of these calculations

employed a 26 contracted orbital basis which augmented the original 14

orbitals (Table 3) with a set of p-orbitals on the hydrogen atoms and a

set of d-orbitals on oxygen--all with unit exponents. The Hartree-Fock

total energy obtained with this basis was E(HF)= -76.0459 H. The second

calculation employed a 38 contracted orbital basis which included all of

the orbitals in the 26 orbital basis plus an additional set of diffuse

p-orbitals on the hydrogen atoms (a = 0.25) and a set of diffuse d-orbit-

als on oxygen (a = 0.40). This basis yielded a Hartree-Fock total energy

of E(HF)= -76.0507 H.

The most significant propagator poles calculated in the valence

region (0-40 eV) with each of the three water basis sets are presented in

Table 7 along with the second-order results of Cederbaum (1973a). The

inclusion of polarization functions not only improves the 3a1 and lb1

ionization energies, it also reverses the relative pole strengths of the

two dominant 2a, poles bringing the theoretical results into better agree-

ment with experimental observations (see footnote on page 73).

Cederbaum's second-order results were obtained with a basis comparable

in size and quality to the 26 orbital basis in Table 7. He deletes

several virtual orbitals from this basis before computing the ionization

energies, however. This approximation may account for the small discrep-

ancies between his results and those reported here.

The formaldehyde molecule was chosen for a second application of

the second-order self-energy approximation. Ionization energies were

calculated using two basis sets. The first consisted of Huzinaga's 9s,

5p primitive basis sets for oxygen and carbon (Huzinaga, 1965) contracted

to 4s and 2p functions with Dunning's contraction coefficients (Dunning,



















Table 7. Basis Set Effects on the Ionization Energies of Water
Computed with a Second-Order Self-Energy Approximation.


Symmetry

al


14 CGTO's

36.5 (.005)
34.0 (.607)
32.6 (.279)
12.9 (.913)


26 CGTO's

37.1 (.003)
33.4 (.288)
32.1 (.592)
13.4 (.908)


38 CGTO's


33.2 (.231)
31.9 (.628)
13.5 (.903)


Ced.a



32.9
13.2


b 34.9 (.005) 35.1 (.005)
10.8 (.909) 11.1 (.904)

b2 40.6 (.003) 40.8 (.004)
18.1 (.931) 18.0 (.922)


E(HF) -76.0082


aCederbaum (1973a).


-76.0459


11.2 (.900) 10.9


18.0 (.919) 17.7


-76.0507


-76.0419








1970). The orbital exponents of Huzinaga's 4s primitive basis for hydro-

gen were scaled by a factor of 1.2, and the resultant orbitals were con-

tracted to 2s functions as recommended by Dunning. The complete basis

appears in Table 8. The second basis augmented the first by the addition

of one set of p-orbitals on the hydrogen atoms and one set of d-orbitals

on both the oxygen and carbon atoms. Unit exponents were chosen for the

p-orbitals on hydrogen while exponents of 0.8 were chosen for the d-orbit-

als. One- and two-electron integrals were computed with the MOLECULE

program (AlmlVf, 1974) at the experimental equilibrium geometry: R(CO)=

2.2825 a.u., R(CH)= 2.1090 a.u., and 4(HCH)= 116.520 (Oka, 1960, Takagi

and Oka, 1963), and the Hartree-Fock calculations and two-electron inte-

gral transformations were performed with GRNFNC (Purvis, 1973). The

Hartree-Fock total energy for the smaller, 24 orbital basis (no polari-

zation) was E(HF)= -113.8257 H., and for the larger, 42 orbital basis

(with polarization) E(HF)= -113.8901 H. The Hartree-Fock orbital energies

and second-order self-energy results for both basis sets are presented

in Table 9 for the principal ionizations along with the second-order

results of Cederbaum et al. (1975) and the experimental values.

The results in Table 9 typify two general features of ionization

energy calculations. The first is that Koopmans' theorem yields values

which are usually higher than experimental ionization energies. Second,

the inclusion of second-order relaxation and correlation corrections

generally overcorrects the Koopmans' estimate and yields values which

are usually lower than experiment. For several ionizations in Table 9,

the second-order deviations from experiment are as large as the Koopmans'

values only opposite in sign.- Although it is possible that the larger,

polarized basis used in the second calculation may still lack adequate














Table 8. Contracted Gaussian Basis for Formaldehyde.


Carbon s sets


Contraction
Exponents Coefficients

4232.6100 0.002029
634.8820 0.015535
146.0970 0.075411
42.4974 0.257121
14.1892 0.596555
1.9666 0.242517
5.1477 1.000000
0.4962 1.000000
0.1533 1.000000


Carbon p sets

Contraction
Exponents Coefficients


18.1557
3.9864
1.1429
0.3594
0.1146


0.018534
0.115442
0.386206
0.640089
1.000000


Oxygen s sets

Contraction
Exponents Coefficients

7816.5400 0.002031
1175.8200 0.015436
273.1880 0.073771
81.1696 0.247606
27.1836 0.611832
3.4136 0.241205
9.5322 1.000000
0.9398 1.000000
0.2846 1.000000


Oxygen p sets

Contraction
Exponents Coefficients


35.1832
7.9040
2.3051
0.7171
0.2137


0.019580
0.124189
0.394727
0.627375
1.000000


Hydrogen s sets

Contraction
Exponents Coefficients


19.2406
2.8992
0.6535
0.1776


0.032828
0.231208
0.817238
1.000000













Table 9. Principal Ionization Energies for Formaldehyde.


24 42

(2) (2)
Orbital Koopmans E(E) Koopmans Z(E) Ced. Exp.

la1 560.12 538.93 559.81 538.62 539.43b

2a1 309.09 297.27 308.87 296.90 294.21b

3a1 38.94 33.66 38.18 32.56 34.2c

4a1 23.39 20.97 23.30 21.03 21.15c

5a1 17.29 13.98 17.38 14.38 14.42 16.2d

Ib1 14.56 13.83 14.45 13.72 13.50 14.5d

lb2 19.47 17.16 19.08 17.07 16.63 17.0d

2b2 12.06 9.04 11.93 9.30 9.25 10.9d

E(HF) -113.8257 -113.8901 -113.9012


aSecond order results of Cederbaum et al. (1975).

bJolly and Schaaf (1976).

CHood et al. (1976).

dEstimated center of gravity (Cederbaum and Domcke, 1977) from spectrum
of Turner et al. (1970).







polarization functions, the rather large discrepancies between the
second-order results and experiment more probably indicate that third-
(and higher) order self-energy corrections are now sizable. The general
conclusion that a second-order self-energy approximation is inadequate
for an accurate calculation of ionization energies has been previously
concluded by Cederbaum (1973b) and necessitates a re-examination of the

approximation made in Eq. (3.88).
Rather than completely neglecting the third-order self-energy matrix,
let us now consider an approximation that includes at least part of these
contributions. Which third-order self-energy diagrams should be included?
There are two well-established results that are relevant to this ques-
tion: Studies of the electron gas model have shown that in the limit of
high electron density, the so-called ring diagrams dominate the self-
energy expansion (Pines, 1961), while in the limit of low electron den-
sity, the so-called ladder diagrams dominate (Galitskii, 1958). In order
to determine whether atomic and molecular self-energies can be approxi-
mated by specific third-order diagrams (e.g. rings or ladders), we need
to evaluate all third-order diagrams for some representative systems.
Cederbaum (1975) has done this for several simple systems and has found
that both ring and ladder diagrams dominate the third-order self-energy.
This result implies that atoms and molecules lie somewhere between the
high and low density extremes. It is therefore essential to include
both ring and ladder diagrams in any third-order self-energy approxima-
tion. These diagrams are

rings ladders



1% -D (3.89)








and correspond to the algebraic expressions labeled A, B, C, and D in

Appendix 1.

We include six additional diagrams in our third-order self-energy

approximation because of the computational efficiency with which they

are evaluated. These diagrams are the energy independent diagrams





V-U V,-v >[ V~i(3.90)
corresponding to expressions M-R in Appendix 1. For these six diagrams,

it is feasible to employ the partial summation technique since they must

be evaluated only once.

Approximating the full third-order self-energy matrix by only ring,

ladder, and constant energy diagrams, let us now consider the solution

of the Dyson equation with the [1,1] Pade' approximant to the self-energy

expansion. Owing to the fact that the inner projection manifold from

which the [1,1] approximant was derived is energy dependent (Eq. (3.80)),

the simple analytic form of the self-energy eigenvalues, illustrated in

Fig. 2, is lost. Furthermore, the self-energy poles are now given by


det (Z(2)(E) E(3)(E)) = 0 (3.91)


rather than by an eigenvalue problem and are consequently more difficult

to obtain. For these reasons, the pole search described in Chapter 1

and used with the second-order self-energy approximation is no longer

an efficient or reliable procedure. An alternative method of solution

used in the following applications was to use the Hartree-Fock orbital

energy as an initial guess to the propagator pole and to iterate Eq.

(1.67) to convergence. When convergent, this procedure invariably yields








a principal propagator pole and its corresponding pole strength. Al-

though the [1,1] self-energy approximant does not guarantee a positive

pole strength, this was never a problem in any of the calculations re-

ported here.

The principal ionization energies for the water molecule were cal-

culated using both the 14 and 26 CGTO basis sets in order to evaluate

the [1,1] Pade' approximant to the self-energy expansion, and the results

appear in Table 10. The most significant feature of these results is that

each ionization energy has been shifted from its second order value to

higher energy and is now in better agreement with the experimental value.

It is further noticed that the valence ionization energies are still

smaller than the experimental values while the la1 (core) ionization

energy is now larger than experiment. Apparently, the diagrams included

in the third-order self-energy matrix overestimate the actual relaxation

and correlation effects for this ionization.

Convergence difficulty was experienced for the 2al ionization

energy using the 14 orbital basis. A schematic plot of W2a (E) is pre-

sented in Fig. 3 and reveals that there are no propagator poles in this

energy region. This anomaly is no doubt a consequence of some quirk in

the basis since the 26 orbital basis yields a very accurate 2a1 ioniza-

tion energy.



3.6 Evaluation of the Diagram Conserving Decoupling

The algebraic structure of the superoperator formalism has been

successfully exploited in this chapter to yield several new insights into

the decoupling problem. The application of perturbation theory has

demonstrated that the electron propagator equation of motion can be




















Table 10. Comparison of Principal Ionization Energies for Water
Obtained with the Second-Order and the [1,1] Self-Energies
Using the 14 and 26 CGTO Basis Sets.


Orbital


539.4


2a 34.0 (.607)
32.6 (.279)


[1,1]

541.6

no convergence
(see text)

13.1

11.1

18.4


(2)
T(E)


[1,1]


Exp.a


539.2 540.9 540.2


32.2


13.6

11.3


18.0 no results


aSiegbahn et al. (1969).































Figure 3. A sketch of W2al in the energy region of the
2al ionization obtained with the [1,1] self-
energy approximant using the 14 CGTO basis.


































-1.215


- 1.175








resumed to yield the equivalent of the diagrammatic expansion. This

resummation also allows the identification of wave and reaction super-

operators which have special importance in decoupling approximations.

We have shown that when the inner projection manifold of the superoper-

ator resolvent is chosen to consist of the first-order truncation of the

wave superoperator, a [1,1] Pade' approximant to the self-energy expan-

sion is obtained. This approximant is correct through third order and

contains a geometric approximation to all higher orders. In general,

the Nth-order truncation of the wave superoperator will yield an [N,N]

Pade' approximant which is correct through the (2N+1)st order in the

self-energy expansion. One final insight afforded by this decoupling

is the realization that electron correlation can be described exclusively

in the operator space. We argued in Chapter 1 that when the propagator

was defined as a single-time Green's function, the density operator was

arbitrary. We have now demonstrated in this chapter that any desired

order in the self-energy expansion may be obtained using as a specific

choice,the uncorrelated, Hartree-Fock density operator.

Computational applications of the diagram conserving decoupling

have been encouraging. These applications have confirmed previous con-

clusions (Cederbaum, 1973b) that a second-order self-energy is generally

inadequate for obtaining accurate ionization energies. It is important

if not essential that third-order ring and ladder diagrams be included

in any self-energy approximation although the errors arising from basis

incompleteness may be of equal magnitude and hence cannot be ignored.

The inclusion of the third-order ring, ladder, and constant energy dia-

grams in the [1,1] self-energy approximant has succeeded in improving the

second-order results but even these results are not consistently better

than the Koopmans' theorem values.








One important feature of the [1,1] self-energy approximant is that

even though it is constructed from only the second- and third-order self-

energy matrices, it contains a geometric approximation of all higher

orders in the self-energy expansion. Certainly, some fourth- or higher-

order terms may be just as important as third-order terms; therefore,

this approximation is highly desirable. The fourth- and higher-order

terms arising from the [1,1] self-energy approximant, however, are not

readily analyzed diagrammatically. In fact, being a purely algebraic

approximation, the [1,1] approximant may not yield any valid fourth- or

higher-order diagrams. Given the fact that ring and ladder diagrams

dominate the third-order self-energy matrix, one can argue that they may

also dominate the higher orders of the self-energy expansion. An appro-

priate modification of this decoupling scheme might then allow the sum-

mation of these specific diagrams in all orders. Approximations of this

type are referred to as renormalized decouplings and are examined within

the superoperator formalism in the next chapter.














CHAPTER 4
RENORMALIZED DECOUPLINGS

4.1 Renormalization Theory

In Chapter 3, we tacitly assumed that the application of perturba-

tion theory to the calculation of ionization energies and electron

affinities was valid and that the resulting self-energy expansion was

convergent. Historically however, it was discovered that in both the

nuclear many-body problem and the electron gas model, the simple self-

energy expansions are divergent. In order to remove these divergencies,

it is necessary to sum certain appropriate classes of diagrams to all

orders. This method of partial summations is known as renormalization

theory (see e.g. Kumar, 1962 or Mattuck, 1967) and may be viewed as an

analytic continuation of the perturbation expansion. Although a variety

of renormalization procedures exist, such as propagator renormalizations,

interaction renormalizations, and vertex renormalizations, the distinc-

tions mainly depend on the types of diagrams included in the partial

summation and are not particularly important for our consideration.

One renormalization that we are already familiar with is the [1,1]

self-energy approximant derived in the preceding chapter. In fact, any

rational self-energy approximant may be regarded as a renormalization

since its geometric expansion will approximate all orders oF the pertur-

bation expansion. One problem encountered with the [1,1] approximant

and that occurs in general for rational approximants derived via purely

algebraic considerations is that their geometric expansions may contain








no readily identifiable diagrams (at least beyond the lowest orders).

Since specific diagrams often dominate the self-energy expansion (such

as ring and ladder diagrams for atoms and molecules) it is valuable to

investigate whether the superoperator formalism can be adapted to yield

renormalized self-energy expressions that sum specific diagrams. The

solution as we shall see is rather simple and involves a restriction in

the types of operator products allowed to span the orthogonal complement

of the model subspace. As a specific example, the two particle-one hole

Tamm-Dancoff approximation (2p-h TDA), (Schuck et al., 1973, Schirmer

and Cederbaum, 1978), is derived from an effective interaction which is

logically obtained by a projection of the perturbation superoperator onto

the subspace spanned by 2p-h type operators (Born and Ohrn, 1979).

Finally, the diagonal approximation to the full 2p-h TDA self-energy

previously derived and applied to the calculation of ionization energies

is shown to neglect terms which, in fact, are diagonal and are necessary

to prevent an overcounting of all diagrams containing diagonal ladder

parts.



4.2 Derivation of the 2p-h TDA and Diagonal 2p-h TDA Equations

Recalling some of the results of the previous chapter, we had ob-

tained the matrix Dyson equation


G(E) = (E) + G (E) (E)G(E) (4.1)

where the self-energy matrix, Z(E), had the following expansion


L(E) = (aV+VT (E)V+VTo(E)VTO(E)V+ . a) .


(4.2)








Introducing the reduced resolvent of the full superoperator Hamiltonian,

T(E), which is just a projection of the superoperator resolvent on the

orthogonal complement


T(E) = P[ac+(EI-H P-PVP -1 (4.3)


the self-energy expansion was written in closed form


Z(E) = (aJVa) + (aIVT(E)Va) (4.4)


It was further shown that when the grand canonical density operator is

used to evaluate the operator averages, the first-order term vanishes.

When P is the exact projector of the orthogonal complement,


P = I 0 (4.5)

the term PVP in Eq. (4.3) is responsible for generating the operator

products that span this subspace. The expansion of this term from the

inverse and its repeated application in each order of the perturbation

expansion yields larger and larger operator products which are only

limited by the number of electrons in the reference state. If instead

of allowing all possible operator products, we restrict them to some

simple types which occur in each order, it may be possible to identify

and sum specific diagrams in all orders of the perturbation expansion.

The restriction of the operator products in the orthogonal comple-

ment is achieved by approximating the orthogonal projector as


P = If)(fj (4.6)


where the manifold {f} contains the desired operator products. The pro-

jector P now has the effect of projecting from the perturbation expansion




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ALTERNATIVE DECOUPLINGS OF THE
ELECTRON PROPAGATOR
By
GREGORY J. BORN
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1979

ACKNOWLEDGMENTS
After twenty-one years of formal education which is culminating
with this dissertation, it is impossible to individually thank every
teacher who assisted me in this pursuit. The largest debt of gratitude
however, is owed to my dissertation supervisor, Professor Yngve Ohm,
whose constructive criticism and constant encouragement has guided the
direction of this research and my graduate education. His generous
financial assistance over the years has also been greatly appreciated.
I would like to thank Professor Per-Olov Lowdin for his stimulating
series of lectures in quantum theory as well as the national and inter¬
national contacts he has made available to me and the other members of
the Quantum Theory Project through the invitation of visiting scientists
and the organization of the Sanibel Symposia. My attendance at the
Summer School in Quantum Theory held in Uppsala, Sweden,and Dalseter,
Norway (August 1976), was made possible by monetary awards secured by
Prof. Lowdin for which I am also greatly appreciative.
Next I would like to thank Professor Jack Sabin, who supervised me
during some preliminary investigations, and the other members of my
supervisory committee for contributing their time and for occasional
letters of recommendation.
Without the friendships and intellectual stimulation of other
members of the Quantum Theory Project whom I have known, my graduate
education would not have been as enjoyable or as rewarding. To several
people I am indebted for the use of various computer subroutines which
I gratefully acknowledge.

It is with regret that I must posthumously acknowledge my indebted¬
ness to Professor Boris Muslin. Prof. Muslin supervised my undergraduate
research in quantum theory at Southern Illinois University and was
largely responsible for guiding me into this field.
I would finally like to thank my parents for their constant encour¬
agement and financial assistance.
Special thanks are owed to Miss Brenda Foye for her painstaking
efforts in typing this manuscript.
I also wish to take this opportunity to acknowledge the Northeast
Regional Data Center of the State University System of Florida for the
use of their facilities to obtain the numerical results presented here
and the American Institute of Physics for permission to reproduce
Figure 1 and Appendix 1 from the paper by G. Born, H. A. Kurtz, and
Y. Ohrn in the Journal of Chemical Physics, Vol. 68, p. 74 (1978).

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS n
LIST OF TABLES vi
LIST OF FIGURES viii
ABSTRACT ix
INTRODUCTION 1
CHAPTER 1: OPERATOR PRODUCT DECOUPLINGS 10
1.1 Definition, Spectral Representation, and Equation
of Motion of the Electron Propagator 10
1.2 The Superoperator Notation and Inner Projection
Technique 15
1.3 The Hartree-Fock Propagator 18
1.4 Operator Product Decoupling 21
1.5 Method of Solution 23
1.6 Analysis and Limitations of the Operator
Product Decoupling 27
CHAPTER 2: MOMENT CONSERVING DECOUPLINGS 31
2.1 Pade' Approximants and the Extended Series
of Stieltjes 31
2.2 Moment Conserving Decoupling 36
2.3 Method of Solution 40
2.4 Computational Considerations and Applications . . 43
2.5 Evaluation of the Moment Conserving Decoupling . 50
CHAPTER 3: DIAGRAM CONSERVING DECOUPLINGS 54
3.1 The Diagrammatic Expansion Method 54
3.2 Perturbation Theory 58
3.3 Equivalence of the Superoperator Formalism and
the Diagrammatic Expansion Method 62
3.4 Diagram Conserving Decoupling 68
3.5 Approximations and Applications 72
3.6 Evaluation of the. Diagram Conserving Decoupling . 82
iv

TABLE OF CONTENTS (Continued)
Page
CHAPTER 4: RENORMALIZED DECOUPLINGS 88
4.1 Renormalization Theory 88
4.2 Derivation of the 2p-h TDA and Diagonal 2p-h TDA
Equations 89
4.3 Diagrammatic Analysis 95
4.4 Computational Applications and Evaluation of the
Diagonal 2p-h TDA Self-Energy 102
CHAPTER 5: PHOTOIONIZATION INTENSITIES 108
5.1 Introduction 108
5.2 Derivation of Computational Formulae for the
Total Photoionization Cross-Section 109
5.3 Discussion of Approximations 116
5.4 Computational Applications 119
CONCLUSIONS AND EXTENSIONS 146
APPENDIX 1 152
APPENDIX 2 154
APPENDIX 3 155
BIBLIOGRAPHY 157
BIOGRAPHICAL SKETCH 164
v

LIST OF TABLES
Table Page
1. Contracted Gaussian Basis for Nitrogen 46
2. Principal Ionization Energies for the Nitrogen
Molecule Resulting from the [1)0] and [2,1]
Propagator Approximants 48
3. Contracted Gaussian Basis for Water 49
4. Principal Ionization Energies for Water Resulting
from the [1,0] and [2,1] Propagator Approximants .... 51
5. Rules for Constructing Self-Energy Diagrams 67
6. Principal Ionization Energies of Water Computed
with the 14 CTGO Basis 74
7. Basis Set Effects on the Ionization Energies of
Water Computed with a Second-Order Self-Energy
Approximation 76
8. Contracted Gaussian Basis for Formaldehyde 78
9. Principal Ionization Energies for Formaldehyde 79
10. Comparison of Principal Ionization Energies for
Water Obtained with the Second-Order and the [1,1]
Self-Energies Using the 14 and 26 CGTO Basis Sets ... 83
11. Water Results Obtained with the Diagonal 2p-h TDA
and Diagonal 2p-h TDA Plus Constant Third-Order
Self-Energies 104
12. Formaldehyde Results Obtained with the Diagonal
2p-h TDA Self-Energy 106
13. Relative Photoionization Intensities for Water
Excited by Mg 1^ (1253.6 eV) 120
14. Relative Photoionization Intensities for Water
Excited by lie (II) (40.81 eV) 121
vi

LIST OF TABLES (Continued)
Table Page
15. Valence Ionization Energies for Acetylene
(24 CGTO's) 126
16. Valence Ionization Energies for Acetylene
(42 CGTO's) 128
17. Relative Photoionization Intensities for Acetylene
Excited by Mg Ka (1253.6 eV) 129
18. Relative Photoionization Intensities for Acetylene
Excited by He (II) (40.81 eV) 130

LIST OF FIGURES
Figure Page
1. Relaxation and Correlation Errors for Each of the
Principal Ionizations in the Water Molecule 6
2. A Sketch of the Energy Dependence of the Function
Wk(E) 26
3. A Sketch of W2a, in the Energy Region of the 2ai
Ionization for jWater 85
4. Fourth-Order Self-Energy Diagrams Arising From the
2p-h TDA 100
5. A Plot of the Theoretical ESCA Spectrum for the
Valence Ionizations of Water 124
6. A Plot of the Theoretical ESCA Spectrum for the
Valence Ionizations of Acetylene 133
7. A Plot of the Photoionization Cross-Sections Versus
Photon Energy for the Valence Orbitals of Acetylene . . . 135
8. Orbital Plots for the 2ou, 3ag, and lrru Feynman-Dyson
Amplitudes of Acetylene 138
9. A Density Difference Plot Between the 3ag Feynman-Dyson
Amplitude and the 3aq Hartree-Fock Orbital of Acetylene . 140
10. A Density Difference Plot Between the 1ttu Feynman-Dyson
Amplitude and the lng Hartree-Fock Orbital of Acetylene . 142
11. A Density Differenct Plot Between the 2ou Feynman-Dyson
Amplitude and the 2cru Hartree-Fock Orbital of Acetylene . 144
vi i i

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ALTERNATIVE DECOUPLINGS OF THE
ELECTRON PROPAGATOR
By
Gregory J. Born
June 1979
Chairman: N. Yngve Ohrn
Major Department: Chemistry
Several alternative decouplings of the electron propagator are
investigated in this dissertation in an attempt to derive more accurate
and more tractable computational schemes for extracting the physical
information contained in the electron propagator. When the electron
propagator is defined as a single-time Green's function, the decoupling
approximation and the choice of reference state average are shown to be
independent approximations, and the use of uncorrelated, Hartree-Fock
reference states is advocated. The derivation of each decoupling
approximation utilizes the superoperator formalism and emphasizes,
elementary algebraic manipulations. In Chapter 1, operator product
decouplings are reviewed and critically discussed. In Chapter 2, the
moment conserving decouplings which consist of Fade' approximants to
the propagator moment expansion are investigated. In Chapter 3, a
partitioning of the superoperator Hamiltonian is invoked, and a pertur¬
bation expansion of the superoperator resolvent is developed. This
IX

development leads straightforwardly to the derivation of the Dyson
equation and permits an identification of wave and reaction superoper¬
ators. Two types of diagram conserving decouplings are then examined,
and equivalences with the diagrammatic expansion method are demonstrated.
Finally in Chapter 4, renormalized decouplings are considered, and the
two particle-one hole, Tamm-Dancoff approximation is specifically derived
and investigated. In each of the first four chapters, the decoupling
approximations are evaluated on the basis of computational applications
in which the propagator poles are compared to experimentally determined
ionization energies for several molecules. In order to avoid a possible
bias with this evaluation criterion, the quality of the Feynman-Dyson
amplitudes is examined in Chapter 5 via the calculation of relative
photoionization intensities. The four decoupling approximations are
finally summarized as various approximations to the wave and reaction
superoperators, and several extensions of these investigations are
proposed.
x

INTRODUCTION
Since subatomic particles are beyond the limit of human sensory
perception, our knowledge of atomic structure is based on the interpre¬
tation of measurements with auxiliary probes. The accumulation and
interpretation of data from these types of measurements have led to the
conception and axiomatization of quantum theory. Using the calculus of
this theory, quantum mechanics, there is evidence to believe that it is
possible, at least in principle, to calculate the statistical result of
any experimental measurement. Unfortunately, the mathematical complexity
of this calculus precludes exact solutions for all but a few, relatively
trivial applications; consequently, the predictive value of the theory
is limited. Owing to this limitation, one aspect of current theoretical
research involves the formulation and evaluation of accurate mathematical
approximations which are relevent to the interpretation of specific
experiments.
With the development of photoelectron spectroscopy (Turner et al.,
1970, Siegbahn et a]_., 1969), photoionization has become an extremely
useful probe of atomic and molecular structure and has stimulated much
theoretical interest (Cederbaum and Domcke, 1977, and references therein).
In the photoionization experiment, light is shone on an atomic or molec¬
ular sample and the kinetic energy of the ionized electrons or photoelec¬
trons which are ejected is then analyzed. From energy conservation, the
binding energies of the photoelectrons may be deduced.
1

2
An aja initio, theoretical interpretation of photoelectron spectra
requires the calculation of ionization energies. These calculations
reflect fundamental assumptions about the complex nature of the many-
electron interactions that occur in atoms and molecules and may be per¬
formed at various levels of sophistication. One conceptually simple
scheme is the Hartree-Fock self-consistent field (HF-SCF) method (see
e.g. Pilar, 1968). In this method, an orbital energy is calculated for
each electron in an N-electron system by assuming that that electron
interacts only with an average electron density formed by the remaining
N-l electrons. By thus averaging out the instantaneous electron-electron
interactions, the original N-electron problem is reduced to N one-electron
problems. The negative of the orbital energies obtained in this calcula¬
tion can then be related to ionization energies via Koopmans' theorem
(Koopmans, 1933).
Ionization energies obtained at the llartree-Fock level of approxi¬
mation are rarely accurate and occasionally predict even the wrong se¬
quence of ionization. In order to obtain more accurate ionization ener¬
gies, the electron-electron interactions must be treated more realisti¬
cally. In the Hartree-Fock approximation, the N-l electrons of the ion
state are assumed to be "frozen" at the same energies they had in the
ground state. Conceptually, each electron screens to some extent the
electrostatic attraction between the positively charged nuclei and all
the other electrons in the system. As can be easily rationalized, this
screening is most effective for deep-lying or core electrons which have
a high probability density near the nucleus than for more diffuse,
valence electrons. Nevertheless, if one electron is ionized, all the
others should experience a stronger nuclear attraction and will contract

3
producing a lower total energy. This rearrangement defines the relaxa¬
tion energy, and it may be easily incorporated in the ionization energy
calculation. Performing separate Hartree-Fock calculations for both the
ground and ion states and obtaining a total energy by adding orbital
energies with corrections for the overcounting of interelectronic repul¬
sions, an improved ionization energy can be obtained by subtracting total
energies. This level of approximation is known as the AE(SCF) method
(Bagus, 1965) and generally yields reliable core electron ionization
energies.
The remaining discrepency between the AE(SCF) ionization energies
and the ionization energies obtained from the exact solution of a non-
relativistic, many-electron formulation can be defined as the correla¬
tion energy. This correction arises from the tendency of any pair of
electrons in an atom or molecule to correlate their motion so as to min¬
imize the electron-electron repulsion. Electron correlation can be con¬
ceptualized as various virtual scattering events between bound electrons
in both the N- and (N-l)-electron systems. The simplest of these virtual
processes is a particle-hole excitation in which one bound electron
absorbs a virtual photon emitted by another electron and is excited from
its original Hartree-Fock orbital to a more diffuse orbital of higher
energy. A hole or vacancy is simultaneously created and propagates in
the system. At some later time, the excited electron may decay re¬
emitting the virtual photon which may then be reabsorbed by the first
electron. Adopting the convention that holes propagate backwards in
time while electrons propagate forward in time, a particle-hole excita¬
tion in the (N-l)-electron ion state can be diagrammatically represented
as

4
ID
where the dotted lines represent the virtual photon exchanges. Although
diagrams of this type represent quasi-classical, virtual processes which
are not experimentally observable, they do provide a conceptual model
and,as we will later see in Chapter 3, categorize specific algebraic
expressions that will be derived to calculate correlation energy correc¬
tions. Finally, it should be noted that since the electron motions will
be correlated to different extents in the N-electron ground state and
(N-l)-electron ion states, it is not possible to predict, a priori, the
effect of this correction on the ionization energies (see Fig. 1).
One many-electron formulation which provides a systematic procedure
for incorporating both relaxation and correlation corrections into the
calculation of ionization energies and which is the basis of this invest¬
igation, is the non-relativistic, single-particle Green's function or
electron propagator (Linderberg and Ohrn, 1973). This method has the
advantage of yielding ionization energies directly unlike other many-
electron formulations which necessitate the total energy calculation for
both ground and ion states and which yield the ionization energy as the
difference. In the latter methods, significant loss of accuracy is
inherent in the subtraction of two, nearly equal total energies to obtain
a much smaller, ionization energy. Care must also he taken to avoid
disparate levels of approximating electron correlation in the calculation
of each different state.

Figure 1. The relaxation error (r) and the correlation error (c) for
each of the principal ionizations in the water molecule.
The numbers are estimated from the experimental results of
Siegbahn et_ aj_. (1969) and from the AE(SCF) results reported
by Goscinski et al_. (1975).


7
The electron propagator can be written as a function of the space
and spin coordinates of one electron and of a complex energy variable.
For each bound electron, this function describes mathematically all of
the complicated many-electron interactions between that electron and the
remaining N-l electrons. This function also contains information about
the interaction of an additional electron with the N bound electrons if
this electron were to be added to the system in any of several possible
orbitals. In a discrete basis representation of the electron propagator,
this information manifests itself as simple poles or singularities at
those values along the real energy axis which correspond to electron ion¬
ization energies or electron affinities, that is, electron detachment or
attachment, respectively. The residues at these poles yield the single¬
particle reduced density matrix from which the N-electron ground state
average of any one-electron operator may be calculated or which may be
related to transition probabilities for electron detachment or attachment
provided some description of the removed or added electron is included.
The electron propagator can be calculated in several ways. The
procedure adopted here is derived from the electron propagator equation
of motion, but one aspect of this investigation will demonstrate the
formal equivalence between this method and the diagrammatic expansion
technique. The equation of motion relates the electron propagator to
the more complicated two-particle propagator. This two-particle propa¬
gator also satisfies an equation of motion which relates it to the three-
particle propagator, and so on. This hierarchy of equations finally
terminates with the N-particle propagator, but in order to make any
practical calculation, this hierarchy must be approximated at some lower
level, generally by expressing an M-particle propagator in terms of an

(M-l)-particle propagator. This approximation is called decoupling the
equations of motion and is not unique. The accuracy of calculated ioni¬
zation energies and the computational effort in obtaining them depend
critically on the decoupling. This investigation proposes and evaluates
several alternative methods.
In Chapter 1, the electron propagator is formally defined, and its
equation of motion is derived. After an introduction of the superoperator
formalism, the electron propagator is approximated by an inner projection,
and the decoupling problem is studied in terms of the selection of an
inner projection manifold. The remainder of this chapter discusses the
computational procedure for solving the propagator equations and presents
a critical evaluation of the operator product decoupling.
Chapter 2 describes some general aspects of the Pade1 approximant
method and its application to the calculation of the electron propagator.
Owing to the conservation of various moment matrices in the propagator
equation of motion these decouplings are known as moment conserving
decouplings (Goscinski and Lukman, 1970). Numerical results obtained
from the [1,0] and [2,1] Pade' approximants are presented and discussed.
A partitioning of the superoperator Hamiltonian and a perturbation
expansion of the superoperator resolvent in the operator space is devel¬
oped in Chapter 3. A superoperator Dyson equation is derived and wave
and reaction superoperators are identified in analogy with ordinary
resolvent operator techniques. Truncations of the wave superoperator
operating on simple annihilation and creation operators are shown to
yield inner projection manifolds that result in Pade' approximants to
the self-energy. These Pade' approximants conserve various orders of
the perturbation expansion for the self-energy and are therefore

9
categorized as diagram conserving decouplings. Two approximations based
on this decoupling scheme are discussed and evaluated.
Chapter 4 is devoted to renormalized decouplings. These decouplings
sum certain types of self-energy diagrams to all orders. The two-particle,
one-hole Tamm-Dancoff approximation (2p-h TDA),which sums all ring and
ladder diagrams, is explicitly derived and discussed in terms of the
superoperator formalism. The diagonal 2p-h TDA previously proposed by
other authors (Cederbaum, 1974, Purvis and Ohrn, 1974) is re-examined
and is shown to neglect certain diagonal contributions. Both approxima¬
tions are analyzed diagrammatically, and numerical results are presented
and discussed.
The evaluation of each decoupling approximation in the first four
chapters is ultimately based on a comparison of propagator poles to ex¬
perimental ionization energies. Chapter 5, on the other hand, attempts
to corroborate this evaluation criterion by an examination of the quality
of the Feynman-Dyson amplitudes. This is indirectly accomplished via
the calculation of relative photoionization intensities and their com¬
parison with experimental data. The requisite equations for the photo¬
ionization cross-section are derived in terms of the Feynman-Dyson ampli¬
tudes, and the most critical approximations are discussed. Finally,
numerical results are presented and are also discussed.

CHAPTER 1
OPERATOR PRODUCT DECOUPLINGS
1.1 Definition, Spectral Representation, and Equation of Motion of the
Electron Propagator
The electron propagator is most commonly defined as a double-time
Green's function which involves an exact N-electron ground state average
of a time-ordered product of electron field operators, ip(x,t) and
^(x',t') (Linderberg and Ohrn, 1973). These field operators are gener¬
ally expressed in the Heisenberg representation,
if(x,t) = exp(iHt) p(x,0)exp(-iHt) (1.1)
and are functions of the combined space-spin and time coordinates of the
electrons. The operator, H, in the exponentials is the N-electron Hamil¬
tonian. The field operators ip(x,t) and ^(x',t') have the property of
annihilating and creating, respectively, an electron at the space-spin
and time coordinates, x,t (x1,t'). Letting denote an exact eigen¬
state of the N-electron Hamiltonian, and denoting exact
eigenstates of the (N-l)- and (N+l)-electron, ion Hamiltonians, these
properties are expressed as
if(x,t)|^J> = l cj exp{-i(E- - Ej~1)t}|'f'Jj~1>
(1.2)
and
(1.3)
10

11
If the N-electron Hamiltonian is time independent, it is easy to show
that the double-time Green's function depends only on the time difference,
t-t' (Fetter and Walecka, 1971). In all practical calculations, however,
this definition proves to be a severe restriction since the exact N-
electron ground state is rarely known and an approximate ground state
average is usually employed. With an inexact ground state average, the
electron propagator will depend on both t and t'. To avoid this restric¬
tion, it is possible to define the electron propagator as a single-time
Green's function by choosing t‘ equal to zero (Simons, 1976),
«^(x',0); > = -i0(t)
+i9(-t) . (1.4)
This definition insures the dependence on only the time difference even
when the ground state average is inexact. The brackets, <...>, in
Eq.(1.4) represent an average which may be either a pure-state average or
an ensemble average,
< . . . > = Z . (1.5)
K
To elucidate the analytic properties of the electron propagator, it
is convenient to derive the spectral or Lehman representation (Linderberg
and Ohrn, 1973). This representation is obtained by first expanding the
eigenstates, |k>, in terms of the exact N-electron eigenstates, jT1^,
Using resolutions of the identity in terms of the exact (N-1)- and (N+l)-
electron eigenstates, Eq.(1.4) can be written:

12
«Il>+(x\0); > = z r, pk CK;i c . {-ie(t)<^|^(x,t)|H^+1>
k i.j.k
x <^+1]^+(x,,0)|'Fj> + i0(-t)<^'1|i{)(x)-t)|^>} .(1.7)
Explicitly introducing the time dependence of the field operators, Eq. (1.7)
becomes
«i|)+(x’,0); ifj(x,t)» = I T. PK c . c .
k i,j,k 'J
x {-i0(t)<'f^+1|^+(x' ,0) |Tj>exp[-i (E^+1-E^)t]
+ i6 (-1)14;+ Cx1 ,C) | 11 (x,0)I^>exp[ i (E^-E^1)t]} . (1.8)
Since the calculation of ionization energies (electron affinities) from
the electron propagator will be treated as a transition between station¬
ary states of the N-electron ground state and N-l (N+l)-electron ion
states, it is convenient to Fourier transform Eq.(1.8) into an energy
representation,
co
«^(x1); í)(x)»E = j «>p^(x‘ ,0)'f(x,t)»exp(i Et)dt (1.9)
which yields the spectral representation:
, * r fjk(x)fk-(x')
< E n-o+ k i i k K K1 KJ *â–  r ,rN+l rN, ^ .
no k i ,j ,k E- (E. -E,) + in
k i â– 
9ik(x)gk.j(x‘)
c /pN rN-l, .
E"(Ej”Ek } - 1,1
}
(1.10)

13
where
fik(x) = <¥}|*(x,0|Y¡¡+1>
(1.11)
and
gik(x‘) = (1.12)
are referred to as the Feynman-Dyson amplitudes. From Eq.(l.lO) it is
observed that the electron propagator has simple poles along the real
energy axis corresponding to the difference between exact eigenvalues of
the N-electron Hamiltonian and the (N+l)-electron Hamiltonians. The
poles of this function, therefore, have a physical interpretation as
ionization energies and electron affinities.
Since atomic and molecular computations are most conveniently per¬
formed in a Hilbert space, we introduce a complete, orthonormal set of
one-electron spin orbitals, {u.¡(x)}. In this basis, the electron field
operators are represented by the expansion,
T(x, t) = T. a1- (t)ui(x)
(1.13)
which, with the expansion of the adjoint, defines the spin orbital anni¬
hilation and creation operators, (t) and aJ(0). At equal times, these
operators satisfy the usual anti-commutation relations,
(1.14)
(1.15)
[ai,aj]+ = [ai5aj]+ = 0
In this discrete representation, the causal electron propagator can be
written
<«fr+(xl,0); ip(x,t)» = T. u*(x' )«aj(0); a1-(t)»u1-(x) (1.16)
i .j
where

14
«aj(O); ai(t)>> = -ie(t) + ie(-t) . (1.17)
Although a computational scheme for obtaining ionization energies
and electron affinities could now be established from the spectral reso¬
lution, it is more convenient to develop a scheme based on the equation
of motion (Linderberg and Ohrn, 1973),
i -ft" «aj(0)i = 6(t)
+ «al(0); [ai(t),H]_>>, (1.18)
which follows directly from Eq. (1.17). The quantity <>
is a two-particle propagator, and the N-electron Hamiltonian appearing
in the commutator has the following Hilbert space representation:
H= E hrsaras + 2 a a ' ,a , a (1.19)
r,s 5 r s r,r' ,s,s' r r s s
where
hrs = f u*(l)[-'iV2(l) -s ^]us(l)dt1 (1.20)
1 a la
and
= || u*(l)u*,(2)rj2(l-P12)us(l)us,(2)dT1dT2 (1.21)
With a notation similar to that used in Eq. (1.9), the energy trans¬
forms of the various quantities in Eq. (1.18) can be defined, e.g.
CO
«at; ai»E = j <> exp( i Et)dt . (1-22)
-oo
Substituting the inverse transforms:
co
i <> = i | E«aj; ai»£ exp(-iEt)dE, (1.23)

15
00
ó(t) =
aj]+>exp(-iEt)d E, (1.24)
and
«3j (0); [a. (t) ,H]_>> = i
«at(O); [ai(t),H]_»Eexp(-iEt)dE,
(1.25)
into Eq. (1.18), we obtain
CO
2tt
—oo
From the general properties of Fourier transforms, it can be shown
(Morse and Feshbach, 1953) that Eq.{1.26) implies
(1.27)
which represents the energy transform of the equation of motion. The
iteration of this equation yields N coupled equations relating the single¬
particle (electron) propagator to each of the higher-particle propagators.
Successive substitution of these more complicated propagators back into
Eq.(1.27) yields
(1.28)
1.2 The Superoperator Notation and Inner Projection Technique
The use of superoperators has antecedents in the work of Zwanzig
(1961) and Banwell and Primas (1963) in statistical physics and was intro¬
duced into atomic and molecular propagator theory by Goscinski and Lukman
(1970). As a notational simplification, the definition of a superoperator

16
Hamiltonian and identity, H and I, provides a convenient representation
of the nested commutators appearing in Eq. (1.28). More formally, this
notation provides a connection with the time-independent resolvent
methods introduced into many-body theory by Hugenholtz (1957).
The superoperator Hamiltonian and identity are defined to operate
on the spin orbital annihilation and creation operators through the
relations
Hai = [a.,H]_ (1.29)
and
la-j = a,- • (1-30)
Powers of the superoperator Hamiltonian are defined by successive appli¬
cation of this superoperator, i.e.
H2ai = H[ai ,H)_ = [ [a]. (1.31)
and will always yield linear combinations of odd (Fermion-like) products
.1.
of the simple field operators, a^ and aj. This set of all Fermion-like
operator products, {X.}, forms a linear space and supports a scalar
product defined by
(Xj|Xi) = Tr{p[X.,xt]+} (1.32)
where p is a normalized, but otherwise arbitrary, density operator corre¬
sponding to the N-electron ground state average of the electron propagator.
Using the preceding definitions and notation, Eq.(1.28) can be re¬
written as
«aj > V>E = G(E)ij = E-1(ajlai> + E~2(aj | Ha ■)
+ E'^ajlilV) + . .
(1.33)

17
Collecting all annihilation operators in a row matrix and all creation
operators in a column and formally summing the expansion in Eq. (1.33),
the matrix equation of motion for the electron propagator becomes:
G(E) = (a|(EÍ-H)_1a). (1.34)
The superoperator resolvent in Eq. (1.34) can now be represented in
closed form by a matrix inverse using the inner projection technique
(Lowdin, 1965, Pickup and Goscinski, 1973). Introducing a projection
operator,
6 = |f)(f|f)_1(f| , (1.35)
A2 -f
where 0=0 and 0=0, the inner projection of a positive definite,
self-adjoint operator, A, is given by
A' = fl!i 0 A*3 ; A > 0 . (1.36)
Making the substitution
ID = A'!l|h) , (1.37)
the inner projection of Eq. (1.36) becomes (Bazley, 1960)
A' = |h)(h|A"1h)"1(h| (1.38)
and satisfies the operator inequalities(Ldwdin, 1965)
0 < A' < A . (1.39)
Since the superoperator resolvent in Eq. (1.34) is an indefinite opera¬
tor, it is not valid to discuss an inner projection of the type in Eq.
(1.36). Equation (1.38) however, which does not contain A1, is still an
acceptable definition of the inner projection provided A is nonsingular.
Using this definition for an indefinite operator, the equality in Eq.
(1.39) will still hold when ji is complete, but the bounding properties
will now be lost with incomplete manifolds. Using the Bazley inner

18
projection, the electron propagator has the following form
G(E) - (a|h)(h|(EÍ-H)h)-1(h|a) (1.40)
in which the decoupling problem has now been transformed into the
problem of choosing an appropriate inner projection manifold.
1.3 The Hartree-Fock Propagator
Before proceeding to formulate more sophisticated decoupling schemes,
it is convenient at this point to recapitulate the approximations under¬
lying all propagator calculations and to demonstrate the algebraic manip¬
ulations which are involved by examining one simple decoupling in some
detail. Implicitly assuming the clamped nuclei and non-relativistic
approximations, there are basically three additional approximations in¬
volved in any scheme for computing the electron propagator. The first
is the truncation of the complete (infinite) set of spin orbitals,
lu-(x)}, to some finite subset. This approximation is also characteris¬
tic of the more conventional wavefunction formulations and has received
considerable attention. Standard basis sets of various sizes and quali¬
ties are available in the literature (Roetti and Clementi, 1974, Huzinaga,
1965, Dunning, 1970, Dunning and Hay, 1977). The second approximation
is the specification of the N-electron ground state average or equiva¬
lently, a density operator (Eq. (1.32)), in terms of which the electron
propagator is defined. The final approximation is the specification of
the inner projection manifold or the actual decoupling of the equations
of motion.
The simplest approximation to the inner projection manifold, h^, in
Eq. (1.40) is just the set, {a^}, of simple field operators. With this
choice, Eq. (1.40) simplifies to

19
G(E) = (a|(EÍ-H)a)_1 (1.41)
si nee
(aja.) = I (1.42)
which can be verified by evaluating a specific matrix element:
(aj |a.) = Tr{p[a . ,aj] + } = Ó^.Tríp} = 6^. . (1.43)
One particularly convenient density operator, which corresponds to
an independent particle, ensemble average, is the grand canonical density
operator (Abdulnur ert aj_., 1972, Linderberg and Ohrn, 1973),
[1 - + (2 - 1) akak].
(1.44)
This density operator yields the following results for various operator
averages:
TripaJa$)
6
rs r
(1.45)
Trfpa'a^Ja ,a } = [6 6 , ,-6 ,6 ,] (1.46)
H r r s s 1 rs r s rs sr 1 r r ' '
and reduces to a pure state average when occupation numbers, , of
zero or one are chosen.
Considering the i,j-th matrix element of the electron propagator,
G(E)ij = [E6-j - (ajlHa.)]"1, (1.47)
the remaining operator scalar product, (a . |lla^), can be evaluated by
first operating with the superoperator Hamiltonian (Eq. 1.29)
Ha . = fa.,H] = 51 h, a +h l al, a , ac ,
1 1 ‘ s 1S s r' ,s,s1 r s s
then anti-commuting with aj (Eq. 1.32)
(1.48)

20
(a.|Ha -) = h,. + l Tr{pa',a ,}. (1.49)
J i r'si r s
Using the grand canonical density operator to evaluate the trace
(Eq. 1.45), we obtain
(a . ¡ Ha.) = h., + 2 .
J 1 1 J ^ I
The particular basis of simple field operators,
which satisfies the equation
(1.50)
(1.51)
Hai = c.a. (1.52)
diagonalizes the matrix, (ajllaj, and must be obtained self-consistently.
This is an equivalent statement of the conventional Hartree-Fock procedure
since the transformation in Eq. (1.51) also defines the canonical Hartree-
Fock spin orbitals
ui = l xikV (1.53)
The eigenvalues, c^, are the Hartree-Fock orbital energies.
Substituting these results into Eq. (1.47), we obtain
G(E) • j = (E-ci)'16ij (1.54)
for this simple decoupling scheme. The poles of this function occurring
at E = e.¡ correspond to the Koopmans' theorem (Koopinans, 1933) approxi¬
mation to the ionization energies. Because of the analogy between this
decoupling and the conventional Hartree-Fock procedure, Eq. (1.54) is
referred to as the Hartree-Fock propagator and will constitute the start¬
ing point for more exact approximations.

21
1.4 Operator Product Decoupling
As was pointed out in the introduction, the Koopmans' theorem approx
imation to ionization energies is frequently unreliable and necessitates
the incorporation of many-electron relaxation and correlation corrections
The simple decoupling scheme which yielded the Hartree-Fock propagator in
the preceding section can be extended to incorporate relaxation and corre
lation by extending the inner projection operator manifold, h^, in Eq.
(1.40). One extension of this manifold, proposed by Pickup and Goscinski
(1973), is the union of all operator subspaces containing different
Fermion-like products of simple field operators:
h = {ak}U{aJa1ara}U{aj!ak+a1anian}U .... (1.55)
In terms of the equations of motion, Eq. (1.28), this type of decoupling
is equivalent to expressing a higher-particle Green's function in terms
of lower ones and was originally discussed in the context of atomic and
molecular theory by Linderberg and Ohrn (1967).
The formulation of explicit electron propagator approximations with
this extended operator product manifold is simplified by the use of the
partitioning technique (Lowdin, 1962). Having already derived an ex¬
pression for the Hartree-Fock propagator with a manifold consisting of
simple field operators, it is convenient to make the partition
h = aUr (1.56)
where a is the subspace of simple field operators and f represents the
orthogonal complement containing all higher, Fermion-like operator
products. This partitioning is imposed through the relations

22
(ill) = (III) = I , (1.57)
(ill) = (Hi) = 0 (1.58)
and leads to a blocked matrix equation for the electron propagator:
G(E) = (1 0)
(a|(EI-H)a) -(ajHf)
-(II Ha)
(f|(EI-H)f)
(1.59)
Solving for the upper left block of the inverse matrix, an equation for
^(E) is easily obtained
G."1 (E) = (a|(EI-H)a) - (a| Hf) (f | (EI-H)f )_1(f | Ha) . (1.60)
The first term on the right hand side of Eq. (1.60) is the inverse
of the Hartree-Fock propagator and the second term represents the relax¬
ation and correlation corrections to the Koopmans1 theorem ionization
energies. Based on the resemblance of Eq. (1.60) with the Dyson equation
derived in quantum electrodynamics (Dyson, 1949), the llartree-Fock propa¬
gator can be identified as a zeroth order, i.e. uncorrelated, approxima¬
tion to G(E) while the remaining term is identified as the self-energy,
G."1 (E) = G¿!(E) - E(E) . (1.61)
A number of approximations to G(E), based on different choices of
the operator manifold in Eq. (1.55), have been reported in the literature.
Pickup and Goscinski (1973) chose their manifold to consist of single-
and triple-operator products and replaced the superoperator Hamiltonian
in the self-energy by the Fock superoperator defined by
’ F = ^ ekakak '
FX = [X, F]
(1.62)

23
This approximation was applied to the calculation of ionization energies
for helium and nitrogen by Purvis and Ohrn (1974) and was later extended
to include the full superoperator Hamiltonian (Purvis and Ohrn, 1975a).
Redmon et aj_. (1975) have derived an approximation which includes
single-, triple-, and quintuple-operator products in and have computed
the ionization energies of neon. Finally, several approximations have
been reported using an inner projection manifold of single and triple
products in conjunction with correlated reference states. (Purvis and
Ohrn, 1975b, Jorgensen and Simons, 1975).
1.5 Method of Solution
The solution of Eq. (1.60) consists of finding the poles and Feynman-
Dyson amplitudes of the electron propagator and writing a spectral repre¬
sentation similar to that of the exact propagator in Eq. (1.10). The
procedure for obtaining the spectral representation from the Dyson equa¬
tion has been discussed by Layzer (1963) and Csanak ej; al_. (1971) and
begins with a solution to the energy dependent eigenvalue problem:
L(E)«t(E) = ¿(E)W(E), (1.63)
where
L(E) = (a|Ha) + 1(E) , (1.64)
i(E)£'(E*) = I , (1.65)
and W(E) is a diagonal eigenvalue matrix. Expanding in terms of the
eigenfunctions, £(E), GjE) assumes the form
G(E) = i(E)[EI-W(E)]'V’(E) ,
(1.66)

24
and the poles are those values of E satisfying the equation,
Ek = Wk(Ek) ' (1-67)
The energy dependence of the eigenvalues, (E), is sketched in Fig. 2
which shows that the poles occur at the intersections of these curves
with a line of unit slope passing through the origin. When the inner
projection manifold is energy independent, the slopes of the W^(E) curves
arealways negative since
gp 2(E) = -(i|Hf)(f|(EI-H)f)'2(f|Ha) , (1.68)
and the number of propagator poles between any pair of self-energy poles
is equal to the number of basis functions in that symmetry (Purvis and
Ohrn, 1974).
From the spectral representation, it was noted that the exact propa¬
gator has only simple poles, and it is easily shown that the residues at
the poles are precisely the Feynman-Dyson amplitudes. Assuming that the
approximate propagator in Eq. (1.60) also has only simple poles, the
residues can be obtained from elementary residue calculus as
!“ (E-Ek)G¡j(E| ’ Vi'dEk)4(V
L*Lk
where
rk = ti - ^ Wk(E)]E=El '
(1.69)
(1.70)
According to the Mittag-Leffler theorem (Mittag-Leffler, 1880), the elec¬
tron propagator can now be written as

Figure 2. A sketch of the energy dependence of the function
W^(E) between self-energy poles (indicated by
vertical dashed lines). Propagator poles occur
at the intersections of these curves with a line
of unit slope.

26

27
* k'N'k^k^ik^k^
G- -(E) = Z â–  -Jk â– â– 
1J
which has the form of the spectral representation.
(1.71)
1.6 Analysis and Limitations of the Operator Product Decoupling
In order to analyze approximate electron propagator expressions as
obtained in Eq. (1.60), it is necessary to compare the propagator poles
with the exact energy differences between the corresponding N- and N-l
(N+l)-electron states. This analysis can be performed, in principle,
in one of two ways. Since a full configuration interaction (Cl) calcu¬
lation will yield the exact total energy to any finite dimensional, model
problem, the total energies of the N- and N-l (N+l)-electron state could
be calculated and then subtracted to yield the ionization energy (electron
affinity). A full Cl, however, is not practical except for systems well
described by small basis sets (<10) because the number of configurations
in the Cl expansion, given by Weyl's formula (Shavitt, 1977), increases
roughly as N ^(2Me/N)^ where N is the number of electrons, M is the size
of the spin orbital basis, and e is the constant 2.718. Furthermore, the
Cl solution, which is expressed as a determinental expansion, is not
readily amenable to detailed analysis. On the other hand, a perturba¬
tion expansion of the N- and N-l (N+l)-electron total energies also
yields the exact solution (in a non-zero region of convergence), but in
addition, allows an order by order analysis of the total energy contribu¬
tions in terms of orbital energies and two-electron integrals.
Using Rayleigh-Schrbdinger perturbation theory (RSPT) to represent
the N- and (N-l)-electron states through second order, Pickup and

28
Goscinski (1973) derived a second-order electron propagator expression
for the energy difference. Since a Hartree-Fock self-consistent field
solution was assumed as the unperturbed problem, the orbital basis sets
of the N- and (N-l)-electron states are different. Before the total en¬
ergy expressions may be subtracted, therefore, the orbitals of the (H-1)-
electron problem must be expanded in terms of the N-electron orbitals.
This procedure has been extended by Born et_ al_. (1978) through third
order, and the resulting third-order self-energy is listed in Appendix 1.
Each third-order term in Appendix 1 is characterized by a diagram and
may be alternatively obtained using diagrammatic techniques, but for
pedagogic reasons this discussion will be deferred until Chapter 3.
Although the results of Purvis and Ohrn (1974, 1975a) and Redmon
ert al_. (1975) represent significant improvements to the Koopmans1 theorem
and AE(SCF) approximations for the ionization energies they computed, the
operator product decoupling was demonstrated to have certain computation¬
al and formal limitations or ambiguities. Computationally, the most
severe limitation is the dimension of the inverse matrix in the matrix
product of Eq. (1.42). The dimension of this matrix increases rapidly
with the size of the spin orbital basis and prohibits an exact inversion
for all but the smallest basis sets. In contrast to the Cl matrix where
only the lowest few eigenvalues of each symmetry are usually computed,
the inversion of this matrix requires all the eigenvalues and eigen¬
vectors. As an illustration of the size problem, when £ consists of only
triple products, the dimension is roughly proportional to NM(M-N) where
N is the number of electrons and M is the size of the spin orbital basis.
This limitation necessitates the approximation of the inverse matrix in
diagonal or near-diagonal form (Purvis and Ohrn, 1974).

29
The formal limitations of the operator product manifolds involve
certain ambiguities when extending the decoupling approximation and the
difficulty in performing an order analysis. Although Manne (1977) has
proven that the set of all Fermion-like operator products forms a com¬
plete set, the extension to higher operator products is not consistent
with an order by order extension of the perturbation analysis. In fact,
Redmon et al_. (1975) have suggested that perhaps some quintuple products
should be preferentially included before all triple products. This, they
felt, was particularly important in describing shake-up processes, i.e.
ionization plus a simultaneous excitation of the (N-l)-electron ion.
This observation has recently been confirmed by Herman e_t aj_. (1978) in
calculations employing the closely related equation of motion (EOM)
method (Rowe, 1968, Simons and Smith, 1973).
As mentioned in Section 1.3, correlation corrections may be included
in either the density operator or the inner projection manifold. This
dichotomy leads to another ambiguity: should larger operator products be
chosen to extend the decoupling or a more highly correlated reference
state? It has been shown by Jorgensen and Simons (1975) that in order
to obtain a decoupling approximation correct through third order, an
inner projection manifold consisting of single and triple products must
be chosen as well as a reference state which includes all single and
double excitations. Unfortunately, this combination of approximations
makes the order analysis unnecessarily complicated, as we will later
show in Chapter 3.
Finally, the Hermiticity problem should be mentioned. With a den¬
sity operator that commutes with the Hamiltonian, the Hermiticity of the
superoperator Hamiltonian can be expressed as

30
(Xi|HXj) = (HX-IXj) = (Xj |HX1)* . (1.72)
With density operators that do not commute with the Hamiltonian, however,
Eq. (1.72) is generally not satisfied and leads to both computational
and formal complications. This problem has been studied by Nehrkorn
et al_. (1976) who observed computationally that the non-Hermitian terms
which arise when the density operator is correlated to first and second
order in RSPT are cancelled when the reference state was improved to
second and third order respectively. A general proof was given by
•k
Linderberg which states that the Hermiticity error is of order n+1 when
the reference state is correlated through order n.
In the following chapters, alternative decoupling procedures will
be proposed and investigated with the intention of remedying the various
limitations inherent in the operator product decoupling as discussed
here, yet which retain a quantitative description of ionization processes.
*See Ref. (14) in Nehrkorn et al_. (1976).

CHAPTER 2
MOMENT CONSERVING DECOUPLINGS
2.1 Pade' Approximants and the Extended Series of Stieltjes
The evaluation of special functions assumes a central role in
applied mathematics. A large number of these functions, from the simple
trigonometric and exponential functions to the more complex, hypergeomet¬
ric functions and Green's functions, have power series expansions. Their
evaluation,therefore, consists of summing the corresponding series expan¬
sion. When the series is slowly convergent or when only a limited number
of expansion coefficients are known (as e.g. through perturbation theory),
it may not be practical, or even possible, to evaluate the series term by
term until a desired accuracy has been achieved. In these cases, optimal
approximations based on a limited number of expansion coefficients are
sought. This general problem was first studied by Tchebychev (1874) and
Stieltjes (1884) for the series which bear their namesand is referred to
as the problem of moments. (For more recent reviews of this problem see
e.g. Wall, 1948, Shohat and Tamarkin, 1963, or Vorobyev, 1965). A general
solution of this problem was given by Pade1 (1892) and is known as the
Pade1 approximant method (Baker, 1975).
Given a function, f(z), (z complex) which admits the formal, but
not necessarily convergent, power series expansion
f(z) = ? a,zk , (2.1)
k=0 K
31

32
the [N,M] Pade' approximant is defined as a rational fraction of the
form P(z)/Q(z) where P(z) is a polynomial of degree M and Q(z) is a
polynomial of degree N. The coefficients of these polynomials are
uniquely determined by equating like powers of z in the equation
f(z)Q(z)-P(z)=0 (through order z^+^) (2.2)
with the auxiliary condition Q(0)=1. The expansion of P(z)/Q(z), there¬
fore, coincides with Eq. (2.1) through the (N+M)-th power of z and pro¬
vides an approximation to the remaining terms.
The term by term convergence of Eq. (2.1) is replaced by the con¬
vergence of sequences of approximants (such as [N,N], N=l, 2, 3, ... )
in the Pade' approximant method, and although general convergence
theorems are difficult to prove for arbitrary series, there exist several
extensive special cases for which convergence has been proven. For these
series, the Pade' approximant can often be shown to extend the natural
region of convergence (Baker, 1975) and may be viewed as a method of
approximate analytic continuation. A sequence of Pade1 approximants,
therefore, may converge rapidly when the original series expansion con¬
verges slowly or not at all.
Two series which have been extensively studied in the problem of
moments and for which sequences of Pade1 approximants have been proven
to converge are the series of Stieltjes (Stieltjes, 1894) and the ex¬
tended series of Stieltjes (Hamburger, 1920, 1921a, 1921b, also known as
the Hamburger moment problem). A series is of the Stieltjes type if and
only if the coefficients, a^ in Eq. (2.1), can be identified as moments
of a distribution

33
ak = | xk#(x) (2.3)
o
where iji(x) is a bounded, non-decreasing function with infinitely many
points of increase in the interval [o,°°). The extended series of
Stieltjes is defined similarly for the extended interval (-°°,°°).
The extended series of Stieltjes has particular significance owing
to its intimate relationship with resolvents of Hermitian operators.
For any operator, A, we can define the operator function
R(zA) = (1-zA)'1 (2.4)
which is trivially related to the resolvent of A. When A is Hermitian,
the spectral theorem (Riesz and Sz.-Nagy, 1955) insures a unique integral
representation of R(zA) having the form
co
R(zA) = f . (2.5)
-CO
The operator E(A) is called an orthogonal resolution of the identity,
and when A has only a discrete spectrum, it can be written
E(A) = l 0(A-ak)|k>«t,k| , (2.6)
where aR and are the eigenvalues and eigenfunctions of A. For any
vector f in the domain of An for all n, the function
r dE (>,)
= Rf(z) = (2.7)
represents either an extended series of Stieltjes or a rational fraction

34
depending on whether Ef(A) has an infinite or finite number of points of
increase (Masson, 1970).
In view of possible applications of the Pade' approximant method to
the superoperator resolvent, we state two theorems regarding the extended
series of Stieltjes and discuss some properties of two particular se¬
quences of Pade1 approximants to these series.
Theorem 1: (Wall, 1948, theorem 86.1) A necessary and sufficient
condition for f(z) to be an extended series of Stieltjes is
det
ra0 al
al a2
. a
. a
" \
n+l
>0 ; n=0, 1, 2, .
(2.8)
.a a . i . . . a~
1 n n+l 2n
Theorem 2: (Masson, 1970, theorem 4) If f(z) is an extended series
of Stieltjes and the associated moment problem is determinant*,
then, for fixed j=0,+\L,f_2, . . . +m, the sequence [N,N+2j+l]
of Pade’ approximants converges to f(z) for Im {z)f 0. The
convergence is uniform, i.e.
lim ||[N,N+2j+l]-f(z)|| = 0 , (2.9)
Itw
with respect to z in any compact region in the upper or lower
half-z plane.
In addition to being uniformly convergent, sequences of [N,N] and
[fl.N-l] Pade1 approximants to extended series of Stieltjes have two
other features which make them particularly attractive for computational
applications. First, these approximants are closely related to
*The moment problem is said to be determinant if there is a unique,
bounded, non-decreasing function fi(x) satisfying the moment conditions
in Eq. (2.3) and the supplementary conditions i|i(-°°)=0 and
ip(x) = lim !,{iHx+e) + if(x-e)} .
e->0

35
variational methods (Nuttall, 1970, 1973). When the operator R(zA) (de¬
fined in Eq. (2.5)) is positive definite, the [N,N] and [N,N-1] approxi-
mants provide the following bounds to R^-(z) (Goscinski and Brandas, 1971):
[N,N] > Rf(z) > [N.N-l] . (2.10)
For resolvent operators such as the superoperator resolvent which are
indefinite, bounding properties are more difficult to establish. Vorobyev
(1965) has shown, however, that the inverse poles of the [N,N-1] approx-
imant to R^(z) are equivalent to the eigenvalues obtained from the usual
Rayleigh-Ritz variational problem
extr
«M A |
(2.11)
N-l
where tp = Cgf + c^Af + . . . c^A f, and the coefficients {c-} are
variationally determined. In this sense, the poles of [N,N-1] to R^(z)
are variationally optimum, but they have no definite bounding properties.
The second attractive feature of the [N,N] and [N,N-1] approximants
is the ease with which they may be computed. Rather than solving Eq.
(2.2) to obtain the coefficients of the polynomials P(z) and Q(z), these
approximants may be expressed directly in terms of the series coefficients
{a.} using matrix formulae derived by Nuttall (1967) and Goscinski and
Brandas (1971). For the [N,N-1] approximant, we have
[N.N-l] = aJtAg-zAj]'1^ , (2.12)
where, in general, is a column vector with the elements a^, a^+^,
. . . a.,M ,, and A. is an N x N square matrix with the columns a.,
i+N-1 —i —l
a. + 1, . . . cq.+N_j. Similarly for the [N.N] approximant we can write
[N,N] = aQ + zaJlA^-zA^]-1^ .
(2.13)

36
2.2 Moment Conserving Decoupling
Expanding the superoperator resolvent in Eq. (1.34) and multiplying
both sides of the equation by E, the electron propagator can be expressed
as the moment expansion
EG(E) = E E"k(a|Hka) . (2.14)
k=0
Before the Pade' approximant method may be applied to this equation, how¬
ever, the conventional definition of the Pade1 approximant must be gener¬
alized to matrix Pade1 approximants (Baker, 1975). This generalization
is achieved by replacing the moment coefficients by the corresponding
moment matrices and noting that these matrices do not commute when per¬
forming subsequent algebraic manipulations. Using Eq. (2.12) to repre¬
sent the [H,N-1] approximant to EG(E), we obtain
EG(E) = ¡¡¿(Mq-E'1^)'1^ , (2.15)
or multiplying each side of this equation by E , Eq. (2.15) becomes
G(E) = ¡¿(EMq-Mj)'1^ (2.16)
where m^ is now a column matrix with block elements
; £i = (ajH'a) • (2-17)
If £. has the dimensions II x M, M. is an NM x NM square matrix with
columns m., m.+1> . . .
â– ^0
^tl-1

There is a close relationship between Eq. (2.16) and the inner pro¬
jection of the superoperator resolvent
G(E) = (a|h)(h|(EÍ-H)h)'1(h|a) .
(2.18)
Goscinski and Lukman (1970) have shown that if the inner projection man¬
ifold is chosen to consist of
h = {ak}U{Hak)U . . . U(HN_1ak} ,
(2.19)
the inner projection and the [N,N-l] Pade' approximant are equal. Since,
in general, the [N,M] Pade' approximant conserves the first N+M+l moments
in the moment expansion, this choice of inner projection manifolds for
the superoperator resolvent is called a moment conserving decoupling of
the equation of motion.
An examination of the [N,N-1] approximant to the electron propagator
shows that its poles are given by the eigenvalues of
(2.20)
The matrix corresponds to a metric matrix and by virtue of the opera¬
tor scalar product, is always positive definite
1^ = (hjh) = Tr{p[hh++h+h]}>0 •
(2.21)
The determinants of the metric matrices corresponding to various trunca¬
tions of the moment conserving inner projection manifold, i.e.

38
provide the necessary and sufficient conditions of Theorem 1 to prove
that the electron propagator is an extended series of Stieltjes. Con¬
sequently, the sequence of [N,N-l] Pade1 approximants to the electron
propagator should be uniformly convergent and should have variationally
optimum properties.
The spectral representation of the electron propagator (Eq. (1.10))
consists of two summations, one which has poles in the lower half of the
complex E-plane corresponding to ionization energies and one which has
poles in the upper half plane corresponding to electron affinities.
Based on the physical argument that the removal of an electron from a
stable atomic or molecular system always requires energy, we might sus¬
pect a separation of the superoperator resolvent which yields a nega¬
tive definite operator for these processes. If this were possible, the
poles of the [M,N-1] approximants would then be upper bounds to the exact
ionization energies obtainable with a given basis. This separation
has not been explicitly demonstrated but an overwhelming amount of
numerical data seemsto substantiate this conjecture. In particular, the
[1,0] approximant, which is easily verified to be the Hartree-Fock propa¬
gator, generally yields poles larger in absolute value than experimental
ionization energies. One possible exception to this rule may be the
near Hartree-Fock limit calculation of Cade et_ ajj. (1966) on diatomic
nitrogen. In this calculation, the magnitude of the lrru orbital energy
is slightly (O.B eV) smaller than the experimental 1ttu ionization energy.
2
If on the other hand, the X ion state was fortuitously better described
than the ground state with their extended basis, this result may still be
an upper bound to the exact ionization energy in that basis.

39
Relaxation and correlation corrections are incorporated in any
[N,N-1] Pade1 approxirnant beyond the [1,0] or Hartree-Fock approximant.
In particular, we have studied the [2,1] approximant in some detail.
This approximant corresponds to the truncation
h = {a}U{Ha}
(2.24)
of the inner projection manifold and conserves the first four moment
matrices. The operators {f^jf-=Ha .}, which were evaluated in Eq. (1.48),
consist of a sum over all triple products of simple field operators with
each operator product in the sum weighted by an antisymmetrized, two-
electron integral. These linear combinations provide a significant re¬
duction in the subspace of triple operators thus overcoming one major
limitation of the operator product decoupling.
Another type of moment conserving decoupling of the electron propa¬
gator equations of motion has been analyzed by Babu and Ratner (1972).
This decoupling is achieved by truncating the moment expansion after the
m-th moment and replacing the m-th moment matrix with
(2.25)
Solving the truncated moment expansion for G(E) yields
(E'
â– m+1
(2.26)
or
, m-1 ,
„,r\ /r-m+1, ,-l rni-k
G(E) = (E 1-c ) lie.
1,1 k=0 k
(2.27)
These rational approximants formally conserve m moments but are not of
the Pade' type (as Babu and Ratner incorrectly identify them) since the

40
auxiliary equation, Q (0 ) = 1, is not satisfied, i.e.
0.(0) = -c^ t 1 . (2.28)
The auxiliary equation guarantees the uniqueness of the Pade1 approx-
iniants: only one [N,N-1] Pade' approximant will conserve exactly m
moments. The nonuniqueness of Babu and Ratner's decoupling scheme is
easily demonstrated by replacing Eq. (2.25) with
m k m l.
I E C| = I E V G(E) 0 < n < m .
k=n k=n
Solving for G(E),
m , n-1 ,
[El- E E Kc ] G(E) = E E" c,
k=n K 1=0 1
m i , n — 1 ,
G(E) = [EmI- E E KcJ-1 E Em'£, , (2.31)
k=n 1=0 1
we obtain m rational approximants (n=l, . . . m) which formally conserve
m moments. Because these approximants are not uniquely defined, we will
only consider Pade' approximants in this chapter.
2.3 Method of Solution
The first step in obtaining the spectral representation of the elec¬
tron propagator with the moment conserving decoupling is the evaluation
of the necessary moment matrices. The first four moment matrices which
are necessary to construct the [2,1] approximant have been evaluated by
Redmon (1975) using the grand canonical density operator (Eq. (1.44)).
(2.29)
(2.30)

41
An independent check of these derivations, however, revealed an error
in the matrix elements of (aj H a_) (Redmon, 1975, Eq. (11.30)). The
correct result has subsequently been verified by Redmon (1977) and
appears in Appendix 2.
Once the moment matrices have been evaluated, the matrices M. and
Mj are constructed and the corresponding eigenvalue problem, Eq. (2.20),
must be solved. In general, the dimension of the eigenvalue problem in¬
creases linearly with the size of the inner projection manifold, i.e. the
tN,N-1] approximant presents an eigenvalue problem of dimension NM where
M is the size of the spin orbital basis. For the [2,1] approximant,
therefore, the dimension of this problem is only twice the size of the
spin orbital basis. This means that for even rather large basis sets,
standard matrix eigenvalues techniques may be employed to solve this
problem in nonpartitioned form. As a consequence, all the poles and
the spectral density of the electron propagator are obtained from a
single matrix diagonalization thus avoiding the energy-dependent pole
search.
Denoting the eigenvectors by c, where
(2.32)
and the eigenvalues by the diagonal matrix ci, the spectral representation
of the electron propagator can be derived,
(2.33)
(2.34)
Defining the matrix
(2.35)

42
which is rectangular with the dimensions H x NM, Eq. (2.34) becomes
G(E) = x(El-d)-1x+ . (2.36)
This equation conserves the first 2fl moment matrices of the moment expan¬
sion which implies, in particular,
xx+ = (aja) = 1 (2.37)
from the conservation of the first moment.
The complete solution of the electron propagator which is conven¬
iently obtained with this decoupling can be used to determine a self-
consistent, single-particle reduced density matrix (1-matrix). The i,
j-th element of the 1-matrix can be computed from the contour integral
(Linderberg and Ohrn, 1973)
= (27i.)_1 | G (E) 1 j d E . (2.38)
The contour, c, runs from -•» to °° along the real axis
poles of G(E)... that lie below the chemical potential
closed in the upper half of the complex E-plane. The
evaluated using the Cauchy residue theorem (Morse and
and encloses only
(y) when finally
integral is then
Feshbach, 1953)

k h x., x .,
key lk Jk
Hu (E-dk)G(E). .
Mk '
(2.39)
(2.40)
Owing to the orthonormality of the spectral density elements (Eq. (2.37))
Xikxjk = 5ij'xikl
(2.41)

43
it follows that the 1-matrix is diagonal with occupation numbers deter¬
mined by
k Using pure state occupation numbers of zero and one in the grand canoni¬
cal density operator for the initial computation of G(E), then occupation
numbers determined from Eq. (2.42) on subsequent computations, a self-
consistent set of occupation numbers can be sought.
2.4 Computational Considerations and Applications
The most time consuming step in the construction of the [2,1] Pade1
approximant to the electron propagator is the construction of the moment
matrices. The fourth moment matrix (given in Appendix 2) is particularly
difficult since it involves five unrestricted orbital summations plus
another two symmetry restricted, orbital summations of two-electron inte¬
grals. Using direct summation techniques, the time needed to construct
this matrix is roughly proportional to N^. This is a formidable computa¬
tional problem, but one that must be accepted in favor of the more manage¬
able matrix dimensions.
Fortunately, the problem is not as intimidating as it might seem
on first appearance. The "brute-force" summation of two-electron inte¬
grals in the moment matrices resums certain partial sums which may appear
in more than one term or matrix element. These redundant summations can
be avoided with considerable savings in computer time by computing the
partial sums once and reusing them. Two specific partial sums we have
employed are
[ij|kl] = I < s s1||k1>
s, s'
(2.43)

44
{i j | kl} = l . (2.44)
s, s1
Since these partial sums contain a double summation which is performed
7 c
only once, the original N problem is effectively reduced to N . The
construction of the moment matrices is now comparable in difficulty to
the transformation of the two-electron integrals from the primitive basis
to the computational (usually Hartree-Fock) basis which is also roughly
5
proportional to N .
When the number of two-electron integrals is too large to be held
in core, their random access from peripheral storage becomes relatively
time consuming. The partial sums are much more efficiently constructed
from ordered lists of two-electron integrals which can be read into
primary (core) storage when needed. For the partial sums defined above,
the two-electron integrals must be sorted into ordered lists of the type
and where * indicates all orbital indices which yield a
non-zero integral for the corresponding i,j-th distribution.
The integral sorts are performed using the Yoshimine sorting tech¬
nique (Yoshimine, 1973). Briefly summarized, this technique involves a
partition of available core into a number of buffers. Each buffer holds
integrals corresponding to a specific i,j distribution, e.g. .
(When the number of distributions is large, several may be held in each
buffer.) Reading through the two-electron integral list, integrals are
then sorted into the appropriate buffers. As each buffer fills, it is
written to direct access, peripheral storage and assigned a record number.
All record numbers corresponding to integrals from the same buffer are
saved in a "chaining" array for that buffer. After the entire integral
list has been processed and all buffers have been dumped, it is then

45
possible to chain back through the direct access records, copying inte¬
grals of the same distribution back into core. These integrals may then
be further sorted within distributions, e.g. k finally saved sequentially on a peripheral storage device.
Diatomic nitrogen was the first molecule to be studied with the
[2,1] Fade' approximant. Owing to its abundance in the atmosphere, nitro¬
gen has great chemical interest and has been extensively investigated
both experimentally and theoretically. It is an ideal test case for cal¬
culating ionization energies from a correlated, many-electron formalism
such as propagator theory since both the Hartree-Fock and AE(SCF) approx¬
imations incorrectly predict the order of the 3o^ and lir^ ionizations
(Cade et_ aj_., 1966). Only when correlation corrections are included is
the correct ordering obtained (Cederbaum and Domcke, 1977 and references
therein).
A double zeta, contracted basis of Gaussian type orbitals (GTO's)
was employed in this calculation. This basis consisted of Huzinaga's
9s,5p set of primitive orbitals (Huzinaga, 1965) which was contracted to
4s,2p (Dunning, 1970). This basis has been optimized by Dunning on the
nitrogen atom and is listed in Table 1. The corresponding one- and two-
electron integrals were calculated at the experimental internuclear
separation, R=2.068 a.u. (Herzberg, 1955), using the MOLECULE integral
program (Almlof, 1974 ).
The Hartree-Fock calculation and the two-electron integral trans¬
formation were performed with the program GRNFNC (Purvis, 1973). The
Hartree-Fock total energy with this basis was E(HF)= -108.8782 a.u.
which is about 3 eV higher than the result of Cade et al. (1966).
There is also a discrepancy in the Hartree-Fock orbital energies. While

46
Table 1. Contracted Gaussian Basis for Nitrogen.
Nitrogen s orbitals
Contraction
Exponents
Coefficients
5909.4400
0.002001
887.4510
0.015310
204.7490
0.074293
59.8376
0.253364
19.9981
0.600576
2.6860
0.245111
7.1927
1.000000
0.7000
1.000000
0.2133
1.000000
Nitrogen
p orbitals
Contraction
Exponents
Coefficients
26.7860
0.018257
5.9564
0.116407
1.7074
0.390111
0.5314
0.637221
0.1654
1.000000

47
the calculation of Cade £t aj_. (incorrectly) predicted the In orbital
energy to be 0.53 eV below the 30^, this calculation predicts the ln^
energy to be 0.05 eV higher. The correct ordering of the 3o^ and Itt^
ionizations with this basis is merely fortuitous, since based on a total
energy criterion, the basis of Cade ejt aJL is more accurate.
The next step of the calculation involved the integral sorts, par¬
tial summations, and the construction of the moment matrices. The poles
and spectral density were finally computed as outlined in the previous
section and are presented along with the [1,0] results in Table 2. The
ionization energies of both approximants seem to be upper bounds to the
experimental results of Siegbahn £t al_. (1969), but without exception,
the results of the [2,1] approximant are worse than the [1,0] approximant.
In an attempt to incorporate some ground state correlation into the grand
canonical density operator, new occupation numbers were computed from the
spectral density and the [2,1] approximant was recalculated. This cal¬
culation, however, yielded no significant improvements in the ionization
energies.
In order to ascertain whether the poor results from the [2,1] approx¬
imant for nitrogen are representative of other calculations or just the
consequence of a pathological test case, the water molecule was chosen
for a second application. Similarly to the calculation for nitrogen, a
double zeta contracted basis of GTO's was also employed in this calcula¬
tion. Huzinaga's 9s,5p primitive basis for oxygen and 4s primitive basis
for hydrogen were contracted with Dunning's coefficients to 4s,2p and
2s, respectively. The orbital exponents for the hydrogen atoms were
scaled by 1.14 to more realistically represent the effective nuclear
charge in the molecule,and the final basis appears in Table 3. Again,

48
Table 2. Principal Ionization Energies for the Nitrogen
Molecule Resulting from the [1,0] and [2,1]
Propagator Approximants.
Orbital
[l.oi
[2,11
Exp.3
%
427.7
472.4
409.9
2o
g
41.6
46.5
37.3
3og
17.0
30.4
15.5
liru
17.0
23.1
16.8
lau
427.6
478.8
409.9
2 a
u
21.0
30.9
18.6
E(HF) = -108.8782 H.
aSiegbahn et al. (1969).

Table 3.
Contracted Gaussian Basis for Water.
Hydrogen s sets
Exponents
Contraction
Coefficients
13.3615
2.0133
0.4538
0.1233
0.032828
0.231208
0.817238
1.000000
Oxygen s sets
Exponents
Contraction
Coefficients
7816.5400
1175.8200
273.1880
81.1696
27.1836
3.4136
9.5322
0.9398
0.2846
0.002031
0.015436
0.073771
0.247606
0.611832
0.241205
1.000000
1.000000
1.000000
Oxygen p sets
Exponents
Contraction
Coefficients
35.1832
7.9040
2.3051
0.7171
0.2137
0.019580
0.124189
0.394727
0.627375
1.000000

50
the integrals were computed with the MOLECULE program at the equilibrium
internuclear geometry, R(OH) = 1.809 a.u., jHOH = 104.5° (Benedict et al.,
1956). A total energy of E(HF)= -76.0082 a.u. was computed with the
llartree-Fock portion of GRNFNC and was followed by the two-electron inte¬
gral transformation. Finally, the integral sorts and partial sums were
performed, the moment matrices constructed, and the poles and spectral
density obtained for the [2,1] approximant. The results for both the
[1,0] and [2,1] approximants are presented in Table 4 and appear to be
upper bounds to the experimental ionization energies. Once more, the
[2,1] results are consistently worse than the [1,0] results. A few
iterations on the occupation numbers yielded no significant improvements.
2.5 Eval_uation of the Moment Conserving Decoupling
Formally, the moment conserving decoupling is an attractive decou¬
pling procedure. Being closely related to the Fade1 approximant method,
this decoupling allows the application of numerous results from the clas¬
sical moment problem to propagator theory. In particular, it was proven
that the sequence of [N,N-1] approximants converge uniformly to the
exact electron propagator in a given basis, and it was shown that these
approximants represent a variationally optimum choice of the inner pro¬
jection manifold. Why then are the results of the [2,1] approximant so
much worse than the results of the [1,0] approximant? To answer this
question, it is necessary to analyze the three approximations identified
in Section 1.3, namely: basis quality, density operator, and decoupling
procedure.
First of all, since computational economy and not high accuracy was
the criterion for the test calculations on nitrogen and water, polarization

51
Table 4. Principal Ionization Energies for Water Resulting
from the [1,0] and [2,1] Propagator Approximants.
Orbital
[1,01
[2,1]
Exp.a
lai
559.4
619.2
540.2
2al
37.0
44.7
32.2
3a ^
15.4
29.7
14.7
lbl
13.8
32.6
12.6
lb2
19.5
29.6
18.6
E(HF) = -76.0082 H.
aSiegbahn et al_. (1969).

52
functions were intentionally excluded from the basis sets. Polar¬
ization functions are diffuse, virtual orbitals which can be very impor¬
tant in describing electron relaxation and correlation (Purvis and Ohrn,
1974, Cederbaum and Domcke, 1977). It is reasonable to expect that the
addition of polarization functions will improve both the [1,0] and [2,1]
approximants to varying degrees; however, with the same basis and with
the same density operator, the larger inner projection manifold (if
judiciously chosen) should yield a more accurate decoupling. Since this
was not the situation in these test calculations, any improvements in
the basis sets did not seem worthwhile.
Second, it is possible that significant ground state correlation
may have been neglected with our choice of the grand canonical density
operator. With the spin orbital annihilation and creation operators
expanded in the Hartree-Fock basis and using pure state occupation
numbers of zero or one, this density operator yields the uncorrelated,
Hartree-Fock ground state average. Rather than explicitly correlating
the density operator (as e.g. through perturbation theory), an attempt
was made to estimate the effect of correlation through the self-consis¬
tent determination of the occupation numbers as described in Section 2.3.
This procedure was not pursued to true self-consistency since each iter¬
ation required a complete recalculation of the [2,1] approximant. It
was obvious, however, after the first few iterations that no significant
improvements had been obtained.
Based on the preceding implications, the third approximation--the
inner projection manifold truncation--seems to be primarily responsible
for the poor numerical results. Owing to the complicated operator sums
in this manifold, an order analysis (as discussed in Section 1.6) is not

53
readily possible. Consequently, it is extremely difficult to identify
the problem with this decoupling procedure. It can only be concluded
that the number of moments conserved is not a useful criterion for decou-
pling. This conclusion is consistent with the uniform convergence of
the [N.N-l] sequence since uniform convergence is not necessarily mono-
tonic, but it suggests that more accurate decouplings require the incor¬
poration of more information about the moment expansion than just the
moment matrices. The additional information needed is indeed available
and,in the next chapter, we will demonstrate how it may be extracted
using perturbation theory.
*Babu and Ratner (1972) reported the same conclusion which was based
on an application of their rational approximants to the Hubbard model.

CHAPTER 3
DIAGRAM CONSERVING DECOUPLINGS
3.1 The Diagrammatic Expansion Method
The superoperator formalism which is employed in the previous two
chapters is by no means the only formalism available to formulate decou¬
pling approximations for the electron propagator. Two commonly used,
alternative methods are the functional differentiation method (see e.g.
Csanak et aj_., 1971) and the diagrammatic expansion method (see e.g.
Mattuck, 1967, Fetter and Walecka, 1971, or Cederbaum and Domcke, 1977).
Of these latter two methods, the diagrammatic expansion method has proven
to be particularly effective. This method avoids some of the algebraic
tedium involved in deriving propagator decoupling approximations by
establishing certain rules for constructing and manipulating diagrams
which represent the underlying algebraic structure.
The diagrammatic expansion of the electron propagator is usually
derived using time-dependent perturbation theory. The U-electron Hamil¬
tonian is partitioned into an unperturbed part plus a time-dependent
perturbation
H = Hq + exp(-c111 )V
(3.1)
where e is a small positive quantity. The unperturbed Hamiltonian, Hq,
is chosen to yield an exactly solvable, eigenvalue problem
(3.2)
54

55
and the time dependence of the unperturbed eigenstates is given by
14>0(t)> = exp(-iHQt) |$Q> . (3.3)
In order to simplify the remaining problem of finding the fully
perturbed eigenstates it is convenient to introduce the "inter¬
action representation" (Fetter and Walecka, 1971) by the transformation
|Yj(t)> = exp(iHQt)|Y(t)> . (3.4)
In this representation, the Schrodinger equation has the form
i §f |'fj(t)> = exp(-e 11 j) V (t) jfj (t)> (3.5)
where
V(t) = exp(iHQt)Vexp(-iHgt) . (3.6)
The time dependence of the interaction eigenstates can be expressed as
¡Yj(t)> = Ue(t,t0)|¥j(t0)> (3.7)
where U (t,tg) is the time-evolution operator. Substituting Eq. (3.7)
into Eq. (3.5), the evolution operator is found to satisfy the differen¬
tial equation
i |f UE(t,t0) = exp(-e|t|)V(t)U£(t,t0) (3.8)
with the initial condition
uE(t0,t0) = 1 • (3.9)
It is more convenient to solve for U (t,tg) by first transforming Eq.
(3.8) into an integral equation

56
t
UE(t,t0) = 1 - i | dt1 exp(-c|t|)V(t1)Ue(t1,t0) . (3.10)
This integral equation has the form of the Volterra equation of the
second kind (Lowdin, 1967) and is solved iteratively
t
U£(t,t0) = 1 - i j dt1 exp(-e|t[)V(t1)
t0
t t^
+ (-i)2 j dtj j dt? exp{-e(|t1|+|t2|)}V(t1)V(t2)Ue(t2>t0) (3.11)
t0 t0
t t j tn_^
= L (-i)n j dtj J dt2 . . . | dtn exp{-e(|t1|+|t2|+ . . . |tj)}
n_^ t t t
u0 0 0
x V(tj)V(t2) . . . V(tn) t>tj>t2> . . . > tn . (3.12)
Eq. (3.12) can be generalized slightly by modifying the limits of inte¬
gration and introducing the time ordering operator, T,
t t t
Ue(t’V7nH)n n! 1 ritl \ dt2 • ■ ' 1 dtn
^ to t0
x exp{-r.(|t1| + |t2[+ . . . | tn |)} T[V(t1)V(tz) . . . V(tn) ] . (3.13)
The time ordering operator rearranges the product of perturbation opera¬
tors such that the left-most term is the latest in chronological order.
The perturbed eigenstates l'*'j(tg)> can now be expressed in terms of
the unperturbed eigenstates by noting that as tg-»-+», |'fj(tg)>-,-|i’g>, and
as tg increases from -°° to zero, the perturbation is "adiabatically
switched on

57
|'1,I(0)> = Ue(o,-»)|$0> . (3.14)
According to a theorem of Gell-Mann and Low (1951), if
U£ (o, -“>) ] 4’q> _ | 'f j (0) >
llo <*0lue(o»-“)l®0> E <^0ITI(°)> (3'15)
exists, then it is an eigenstate of II
H11 j (0) > E | T j (0) >
«I.gl’fjíOlT = <4.0|¥j(0)> • (3-16)
These results can now be used to determine the electron propagator.
In Chapter 1, the propagator was defined as the ground state average of
a time-ordered product of field operators in the Heisenberg representa¬
tion
iGij(t)

(3.17)
Using Eq. (3.15) and the fact that |'l'^>=|¥j(0)>, this average can be ex¬
pressed in the interaction representation as
<4>0|Ue(«,t)T[ai(t)a!(0)]Ue(t,-“)|
lGij(t) |
(3.18)
Using the expansion of the evolution operator (Eg. (3.13)) and taking
the limit e-f-0, it can be shown (Fetter and Walecka, 1971) that
oo , ,
1Gij(t) = nf0(‘Í)n n! J dtl ' ' • J dtn
— OO — oo
<0|T[V(t1) . . . V(tn)a.(t)a](0)]|*0>
<'í’g|U(
X
(3.19)

58
The final step in the diagrammatic expansion method is to expand
the numerators of each term in Eq. (3.19) using Wick's theorem (Wick,
1950) and to represent them diagrammatically (e.g. Fetter and Walecka,
1971). The denominator of Eq. (3.19) must also be expanded and dia¬
grammed, and when this is done, all disconnected diagrams
arising from the expansion of the numerator will cancel (Abrikosov
et al., 1965).
Formally, the diagrammatic expansion method and the superoperator
formalism appear strikingly dissimilar. The diagrammatic method is
formulated in the causal representation while the superoperator formal¬
ism utilizes the energy representation. The diagrammatic method employs
a pictorial representation of the algebraic structure while the super¬
operator formalism emphasizes the algebraic structure directly. Yet
the primary goal of each formalism is the same: an accurate prediction
of ionization energies and electron affinities. Therefore, the two
formalisms are inherently equivalent. It is our desire in this chapter
to explicitly demonstrate the equivalence between these two formalisms
and to re-examine the superoperator decoupling approximations in terms
of a diagrammatic analysis.
3.2 Perturbation Theory
The unifying feature of the diagrammatic expansion method and the
superoperator formalism is perturbation theory (Born and Ohrn, 1978).
Since the commutator product is distributive with respect to addition,
we can define a partitioning of the superoperator Hamiltonian into an
unperturbed part plus a perturbation,
(3.20)

59
One convenient partitioning, which will be shown to readily yield the
Hartree-Fock propagator as the unperturbed electron propagator, is the
Mdller-Plesset partitioning (Muiller and Plesset, 1934). With this par¬
titioning, H0 has the form
H = E £ a'a - h I
0 „ r r r , 11 r r1
i r j r
(3.21)
and the perturbation is expressed as
V= Z [l;a V ,a ,a -6 , ,a„a ] + >s E
r,r',s,s' r r s s r s r r s r>r.
(3.22)
Of course, when the commutator product is formed for the superoperators,
the constant term in these definitions will cancel.
Other partitionings of the Hamiltonian may also be assumed and may
lead to superior convergence properties (Claverie et_ aj_., 1967). One
alternative partitioning which has been employed in the perturbation
calculation of correlation corrections to the total energy is the Epstein-
Nesbet partitioning (Epstein, 1926, Nesbet, 1955a, 1955b). In propagator
applications, the work of Kurtz and Ohrn (1978) may be roughly interpreted
in terms of a partitioning where the unperturbed Hamiltonian incorporates
all relaxation contributions to the ionization energy. It is difficult
to define this unperturbed Hamiltonian explicitly, but it formally satis¬
fies the eigenvalue equation
H¿ak = AEk(SCF)ak (3.23)
in contrast to
Vk = ckak (3‘24'
for the Mdller-Plesset partitioning. The method of Kurtz and Ohrn yields

60
excellent ionization energies and electron affinities with a simple
second-order self-energy, however it has not been formally analyzed in
detail.
Corresponding to the partitioning of the superoperator Hamiltonian,
we can introduce a partitioning of the operator space defined by the
projection superoperators 0 and P,
O = Iz |ak)(ak| = |a)(a| (3.25)
P = I - 6 (3.26)
These superoperators operate on elements of the operator space through
the relations
0Xi = X |ak)(a|<|Xi.) (3.27)
PX. = X- - 0Xi (3.28)
and are idempotent (0^ = O, P^ = P), self-adjoint (0‘ = 0, P1 = P), and
mutually exclusive (0P=P0=0). The superoperator 0 projects from an ar¬
bitrary operator product that part which lies in the model subspace,
i.e. that part which is spanned by the eigenelements of H^. The super¬
operator P projects onto the orthogonal complement of the model subspace,
i.e. that part which we have no a priori knowledge about.
To obtain a perturbation expansion of the superoperator resolvent,
we consider its outer projection (Lowdin, 1965) onto the model subspace,
G(E) = 6(EI-H)-16
(3.29)
= 6(ei-h0-v)-1o
(3.30)
By iterating the identity

61
(A-B)"1 = A'1 + A_1B(A-B)_1
the inverse in Eq. (3.30) can be expanded as
(3.31)
G(E) = (EI-H0)_1Ó + (EI-H0)_10V(EI-H0)_10
+ (EÍ-H0)“10V(EÍ-H0)_1V(EÍ-H0)~16 + . . .
where the property
N-l 3
(3.32)
(H0,0]_ = 0
(3.33)
has been used. Now since 0 plus P form a resolution of the identity,
each resolvent of Hg occurring between perturbation superoperators, V,
can be rewritten as a sum of its projections on the model subspace and
the orthogonal complement,
(EI-Hg)'1 = (EI-H0)_16 + (EI-M0)_1P
Gq(E) + T0(E)
(3.34)
(3.35)
With this notation, Eq. (3.32) becomes
G(E) = G0(E) + G0(E)VG0(E) + GQ(E)V[GQ(E) + TQ(E)]VGQ(E)
+ Gq(E)V[G0(E) + T0(E)]V[G0(E) + TQ(E)]VGQ(E) + . . .
and can be resummed to yield
(3.36)
G(E) = G0(E) + Gq(E)[V + VT0(E)V + VTQ(E)VTQ(E)V + . . . ]G(E) (3.37)
Defining the reduced resolvent of the full superoperator Hamiltonian as
T(E) = P[aO + P(EI-H)P]-1P
= Tq(E) + Tq(E)VT(E) ,
(tfO)
(3.38)
(3.39)

62
Eq. (3.37) can be written in closed form
G(E) = Gq(E) + G0(E)[V + VT(E)V]G(E) . (3.40)
Alternatively, we can define wave and reaction superoperators through
the equations (cf. Lowdin, 1962, or Brandow, 1967)
W(E) = Í + T(E)V (3.41)
t(E) = VW(E) . (3.42)
The reduced resolvent, wave, and reaction superoperators introduced
in this section are functions of the superoperators I, Hq, and V and as
a consequence, operate in a more complicated way. To apply a superoper¬
ator function to an operator in the operator space, it must first be ex¬
panded in terms of the superoperators I, Hq, and V which are then suc¬
cessively applied to the operator. For example,
W(E)X. = [Í + T(E)V]X. (3.43)
= [I + Tq(E)V + T0(E)VTQ(E)V + . . . ]X. (3.44)
where
T0(E)VXi = [E~11 + E~2 Hq + E'3 H2 + . . . JPVX^ , (3.45)
etc.
3.3 Equivalence of the Superoperator Formalism and the Diagrammatic
Expansion Method
Eqs. (3.37) and (3.40) represent the superoperator form of the Dyson
equation (Dyson, 1949), and the reaction superoperator (Eq. (3.42)) can
be identified as the self-energy. To demonstrate that Eq. (3.37)

63
corresponds term by term with the diagrammatic propagator expansion, we
must first form the operator average of G(E) to obtain the matrix Dyson
equation, next evaluate all necessary operator averages, and finally
diagram the resulting algebraic formulae. Owing to the complicated
operator averages that must be evaluated in third and higher orders of
the perturbation superoperator, the equivalence between these two for¬
malisms has only been explicitly demonstrated through third order and
is assumed in all higher orders.
The matrix Dyson equation is obtained by forming the operator aver¬
age of G(E) with respect to the basis elements of our model subspace
G(E) = (a|G(E)a) (3.46)
= Gq(E) + Gq(E)S(E)G(E) , (3.47)
where
KE) = (a|Va) + (a|VTQ(E)Va)
+ (a|VT0(E)VT0(E)Va) + . . . . (3.48)
Since Hq was chosen to be the Fock superoperator, the appropriate den¬
sity operator to employ in the evaluation of the operator averages is
the Hartree-Fock density operator. Realizing that the grand canonical
density operator (Eq. (1.44)) reduces to the Hartree-Fock density oper¬
ator when pure state occupation numbers of zero or one are chosen, we
shall employ this density operator.
Beginning with the evaluation of matrix elements for the unperturbed
propagator, G^(E), the Hartree-Fock propagator is easily obtained (cf.
Section 1.3)

64
VE>ij ' ‘*jl«EI-Ho> »1>
■ r’lajla,) t E_Z , r\ * rt,.u * r\‘
VE>ij ■ (£-.,)%
(3.49)
(3.50)
(3.51)
(3.52)
The evaluation of each term in the self-energy expansion requires the
initial evaluation of Va..,
Vai = U T. ^rr'I |ssl>[ai,arar,as,as]_
r, r j s, s
5 [a . .aras]_
r,s,s'
l a a ,a - l a
i rs s i s s
y^jS^S s, s
(3.53)
(3.54)
With this result, the first-order term (aj | Va ^) is obtained without much
additional effort
£Í1J(E)ij = (“jlV^i) (3.55)
= !á 7 Tr{p[a„a ,a ,a .] }
r,s,s' J
-l Tr{p[a ,at] } (3.56)
s, s1 J
= >: 6 , - T. (3.57)
r,s s *
i(1)(E)ij = 0 . (3.58)
When the effective, single-particle potential used in the unperturbed
problem is the Hartree-Fock self-consistent field potential, all single¬
particle corrections vanish (Bartlett and Silver, 1975a).

65
The evaluation of the second- and higher-order self-energy matrices
requires the evaluation of T0(E)Vai and VT0(E)Va.. The first of these
quantities can be expanded as
T 0(E) V a. = (EI-H0)"1PVai
(3.59)
= (EI-H0)'1Vai - z (EÍ-H0)'1|ak)(ak|Va.)
(3.60)
using Eq. (3.26). It follows from the previous result for (a^ |Va^) that
the second term in Eq. (3.60) vanishes. The first term can now be evalu¬
ated by expanding the resolvent of Hq and realizing that any operator
product is an eigenelement to Mq, i.e.
Vras'as = (WEs)aras'as '
(3.61)
Consequently, we obtain
T0(E>Vai = - z i(E+Vcs-Es,)'1 ajas,as
r, s s s
- Z (E-e )-1a
s, s1
(3.62)
with the help of Eq. (3.45). The remaining application of V and the
i-
average value evaluation is straightforward and yields
Z(2)(E)i:j = (aj|\/T0(E)Va.)
(3.63)
= 7. >+1 > r,s ,s ■ (E+Er-cs-es,) r s s' r s’
(3.64)
for the matrix elements of the second-order self-energy.
The Hartree-Fock average is now obtained by choosing occupation
numbers of zero and one. An examination of the occupation number factor
in Eq. (3.64) reveals that with this restriction, it will be non-vanishing

66
only when the summation index r runs over occupied spin orbitals and s
and s' run over unoccupied spin orbitals or when r runs over unoccupied
spin orbitals and s and s' run over occupied spin orbitals. Denoting
a, b, c, . . . as summation indices over occupied spin orbitals; p, q,
r, . . . for unoccupied spin orbitals; and i, j, k, . . . for unspecified
spin orbitals, Eq. (3.64) can now be written as two terms which involve
restricted spin orbital summations
y(2)m = ,, r
¿ U ij 2 L (E+e -e ~e )
a.p.q v a p V
+ 1. v
P.a.b
(3.65)
The conversion of Eq. (3.65) into diagrams is a straightforward pro¬
cedure for which we shall use the rules and diagram convention of Brandow
(1967) and Bartlett and Silver (1975b). This convention represents the
synthesis of the antisymmetrized vertices of the Uugenholtz (1957) or
Abrikosov (1965) diagrams with the extended interaction lines of the
Goldstone (1957) diagrams, and the rules for constructing these diagrams
are given in Table 5. The application of these rules to the terms in
Eq. (3.65) yields the following diagrams:
j. l
a,p,q ^VVV
1.0
y
(E+e -E -£,)
p a b
p, a , b
(3.66)
(3.67)
These diagrams are precisely the same as those obtained in the second-
order diagrammatic expansion after a Fourier transformation into the
energy representation (Cederbaum and Domcke, 1977).

67
Table 5. Rules for Constructing Self-Energy Diagrams.
1. Each antisymmetrized two-electron integral factor in the
numerator is represented by an interaction line with a vertex
(dot) at both ends. The number of interaction lines denotes
the order of the term.
2. Using the Dirac bra-ket notation, both indices in the bra are
represented by lines leaving a vertex while those of the ket
are represented by lines entering a vertex. There must be only
one outgoing and one incoming line per vertex, therefore, assign
the index of electron coordinate one to the left vertex and the
index of electron coordinate two to the right vertex of each
interaction line.
3. Summation indices running over hole states (occupied orbitals)
are directed downward, indices running over particle states
(unoccupied orbitals) are directed upward, and external indices
(not summed) are drawn horizontally.
To Check Diagrams:
4. The energy denominator of the diagrammed expression should be
obtained by first connecting the external lines and assigning
a factor of E to this directed segment. Second, imagine hori¬
zontal lines drawn between each pair of interaction lines.
Each horizontal line corresponds to a multiplicative, denominator
factor obtained by summing the orbital energies of each hole
(downgoing) line that intersects it minus the sum of orbital
energies for particle (upgoing) lines that intersect it. Treat
the factor E of the connected external lines as an orbital energy.
5. Numerical factors should be obtained by assigning a factor of %
for each pair of equivalent internal lines. Equivalent internal
lines are two lines which begin on the same interaction line,
end on the same interaction line, and go in the same direction.
6. The overall sign factor should be obtained by assigning a factor
of minus one to each internal hole line segment and a minus one
to each closed loop.

68
The evaluation of the third-order self-energy matrix is similar to
the second-order matrix but much more tedious and the result is presented
in Appendix 3. As was done for the second-order expression, the occupa¬
tion numbers must again be restricted to zero and one to obtain the
Hartree-Fock average. When this restriction is made, the unrestricted
spin orbital summations in Appendix 3 will reduce to summations involving
occupied, unoccupied, and unspecified spin orbitals. Using the algebraic
identity
a a r i i
(E-a) (E-bT ' Ta^bJ [ " TTO
it is possible to combine terms in such a way that expressions involving
only occupied and unoccupied spin orbital summations are obtained. These
expressions are presented in Appendix 1. The corresponding diagrams in
Appendix 1 again are precisely those occurring in the third-order, dia¬
grammatic self-energy expansion.
3.4 Diagram Conserving Decoupling
The wave and reaction superoperators identified with the help of
perturbation theory in Section 3.2 have special importance in the develop¬
ment of decoupling approximations for the electron propagator. As we
have already seen, the reaction superoperator generates the diagrammatic
self-energy expansion. A truncation of this expansion offers one viable
decoupling scheme. The wave superoperator, on the other hand, has the
property of generating eigenelements to the full superoperator Hamilto¬
nian from the eigenelements of the unperturbed superoperator Hamiltonian
(EI-H)W(E)a = 0 .
(3.68)
(3.69)

69
This property is easily proven by first using Eq. (3.41) to expand W(E)
and then premultiplying both sides of Eq. (3.69) by P
P(EI-H)W(E)a = P(EI-H)a + P(EI-H)T(E)Va . (3.71)
Using the identity
P(EI-H)T(E) = P (3.72)
and the property Pa_ = 0, Eq. (3.71) simplifies to
P(EI-H)W(E)a = - PVa + PVa = 0 (3.73)
which implies the validity of Eq. (3.69)
It is of interest at this point to show a connection between the
superoperator formalism and the Equations of Motion (EOM) method for
determining ionization energies (Simons and Smith, 1973). In this
method, one seeks solutions of the equation
[H,Q]_ = ojQ (3.74)
which is precisely Eq. (3.69). Here the operator Q is interpreted as a
correlated ionization operator that generates, in principle, the exact
(N-l)-electron ion states from the exact N-electron reference state.
One approach to solving Eq. (3.74) involves the application of Rayleigh-
Schrodinger perturbation theory (Dalgaard and Simons, 1977). By parti¬
tioning the Hamiltonian operator, expanding both the ionization operator,
Q, and the ionization energy, io, in terms of a perturbation parameter,
and collecting terms of the same order, a set of perturbation theory
equations are obtained. The solution of these equations yields an expan¬
sion for Q which is analogous to the superoperator equation

70
h = W(E)a . (3.75)
The only difference is that E is replaced by which is a consequence
of using Rayleigh-Schrodinger rather than Brillioun-Wigner perturbation
theory.
Returning now to the inner projection of the superoperator resolvent,
G(E) = (a|h)(h|(EI-H)h)-1{h|a) (3.76)
we may view Eq. (3.75) as an alternative prescription for choosing the
inner projection operator manifold. Recalling from Section 1.6 that since
the density operator describing the unperturbed (model) problem does not
commute with the full Hamiltonian, the operator scalar product will not
in general exhibit Hermitian symmetry. Consequently, we define
(h|=(a|W+(E) (3.77)
and note that
(a_|W+(E) f (W(E)aJ . (3.78)
Approximate electron propagator decouplings can now be obtained by
truncating the expansion of the wave superoperator,
W(E) = I + Tq(E)V + TQ(E)VT0(E)V + . . . . (3.79)
Truncation of this expansion, with only the superoperator identity, triv¬
ially yields the Hartree-Fock propagator, therefore we next consider
W(E) = Í + T0(E)V .
doting that the subspaces (a^J and (fjf^ = T^(E)Va^} are mutually
orthogonal, Eq. (3.76) can be readily solved for G~*(E)
(3.80)

/I
G."1 (E) = G^(E) - £(E) , (3.81)
where
7(E) = (a|VT0(E)Va)(a.|VT0(E)(EÍ-H)T0(E)Va)'1(a|VT0(E)Va) (3.82)
Making the following identifications from Section 3.3:
(aJVT0(E)Va) = E(2)(E) , (3.83)
(a)VT0(E)(EÍ-H0)T0(E)Va) = £(2)(E) , (3.84)
and
(a|VT0(E)VT0(E)Va) = £(3)(E) , (3.85)
Eq. (3.82) can be rewritten
£(E) = £^2^(E)[^2^(E) - I^3)(E)]'V2^(E) . (3.86)
Expanding the inverse of Eq. (3.86), we easily see that this self¬
energy approximant coincides with the diagrammatic expansion through
third order but additionally yields contributions to all higher orders.
If the exact self-energy is rewritten as a moment expansion in terms of
a perturbation parameter, A,
X_1E(E) = z Ak(a|V(T (E)V)ka) , (3.87)
k=0 U
we see that Eq. (3.86) represents the [1,1] Pade1 approximant to this
expansion. Owing to the close connection between Pade' approximants and
the inner projection technique as demonstrated in Chapter 2, this result
is not surprising. These Pade' approximants to the self-energy, however,
will have entirely different convergence properties than those studied in
Chapter 2.

72
3.5 Approximations and App1ications
Computational applications of the [1,1] Pade' approximarit to the
self-energy require the evaluation of the second- and third-order self¬
energy matrices. The second-order matrix is relatively easy to evaluate.
The third-order matrix, on the other hand, is exceedingly more difficult
and can presently be only approximately calculated without excessive
computational effort. An examination of the formulae in Appendix 3 re¬
veals that unlike the fourth moment matrix in the moment conserving
decoupling, the third-order self-energy matrix is energy dependent.
This additional complication makes the partial summation technique used
in the moment conserving decoupling ineffectual since the third-order
self-energy matrix will generally need to be resummed with different
values of E hundreds of times in the search for poles of the propagator.
The first approximation that we will examine is the complete neglect
of the third-order self-energy matrix. With this approximation, the
[1,1] approxiniant in Eq. (3.86) reduces to a second-order truncation of
the diagrammatic self-energy expansion,
Z.(E) = E^(E) . (3.88)
This second-order self-energy approximation is interesting not only be¬
cause it contains the most important relaxation and correlation correc¬
tions to Koopmans' theorem (in a perturbation theoretical sense), but
also because it exhibits the same analytic form as the exact self-energy
(Hedin and Lundqvist, 1969, Cederbaum and Domcke, 1977). Furthermore,
since several second-order, ionization energy calculations have been
reported in the literature, this approximation will afford both a con¬
venient check of new computer code and the computational experience
necessary to implement more refined approximations.

73
The first computational application of this decoupling approximation
was to the water molecule using the same basis and internuclear geometry
as described in Section 2.4. The results of this calculation are pre¬
sented in Table 6 along with the Koopmans' theorem, AE(SCF), and experi¬
mental values for the ionization energies. Two ionization energies have
been tabulated for the 2a^ ionization with their corresponding pole
strength (r^ of Eq. (1.70)) in parentheses. The occurence of two, rela¬
tively strong propagator poles for this ionization represents a breakdown
in the quasi-particle description of inner valence ionizations (Cederbaum,
1977) and makes assignments of principal and shake-up ionizations ambig-
â– k
uous. In general, the second-order ionization energies are quite en¬
couraging and represent significant improvements to each of the Koopmans'
values. Furthermore, these results are comparable in accuracy to the
AE(SCF) results but possess the convenience of being obtained in a single
calculation whereas the AE(SCF) results required six separate Hartree-
Fock calculations.
The relatively poor agreement of the 3a^ and lb^ ionization energies
with the experimental values in Table 6 seems attributable to basis in¬
completeness. Despite the lack of polarization functions, this suspicion
is supported by the facts that the 3a^ orbital is the highest occupied
orbital in that symmetry and that this basis contains only two contracted
Gaussian orbitals of symmetry. In order to study the basis dependence
of the second-order self-energy approximation, two additional calculations
*The ESCA spectrum of the water molecule (Siegbahn et al., 1969)
substantiates this phenomenon since the 2a, peak is quite broad and asym¬
metric. Experimentally, it appears that the lower energy ionization
should have a larger pole strength (in contrast with the results of Table
5) since the peak is skewed to higher binding energies.

Table 6. Principal Ionization Energies of Water Computed
with the 14 CGTO Basis.
Orbital
Koopmans
AE(SCF)a
(2)
2(E)
Exp.b
lal
559.4
540.8
539.4
540.2
2a i
37.0
34.6
34.0 (.61)
32.6 (.28)
32.2
3d 2
15.4
13.0
12.9
14.7
ibi
13.8
11.0
10.8
12.6
lb2
19.5
17.8
18.1
18.6
aGoscinski ert al_. (1975).
bSiegbahn et al_. (1969).

75
were performed with larger basis sets. The first of these calculations
employed a 26 contracted orbital basis which augmented the original 14
orbitals (Table 3) with a set of p-orbitals on the hydrogen atoms and a
set of d-orbitals on oxygen--all with unit exponents. The Hartree-Fock
total energy obtained with this basis was E(HF)= -76.0459 H. The second
calculation employed a 38 contracted orbital basis which included all of
the orbitals in the 26 orbital basis plus an additional set of diffuse
p-orbitals on the hydrogen atoms (a = 0.25) and a set of diffuse d-orbit¬
als on oxygen (a = 0.40). This basis yielded a Hartree-Fock total energy
of E(HF)= -76.0507 H.
The most significant propagator poles calculated in the valence
region (0~40 eV) with each of the three water basis sets are presented in
Table 7 along with the second-order results of Cederbaum (1973a). The
inclusion of polarization functions not only improves the 3a^ and lbj
ionization energies, it also reverses the relative pole strengths of the
two dominant 2a^ poles bringing the theoretical results into better agree¬
ment with experimental observations (see footnote on page 73).
Cederbaum's second-order results were obtained with a basis comparable
in size and quality to the 26 orbital basis in Table 7. He deletes
several virtual orbitals from this basis before computing the ionization
energies, however. This approximation may account for the small discrep¬
ancies between his results and those reported here.
The formaldehyde molecule was chosen for a second application of
the second-order self-energy approximation. Ionization energies were
calculated using two basis sets. The first consisted of Huzinaga's 9s,
5p primitive basis sets for oxygen and carbon (Huzinaga, 1965) contracted
to 4s and 2p functions with Dunning's contraction coefficients (Dunning,

76
Table 7. Basis Set Effects on the Ionization Energies of Water
Computed with a Second-Order Self-Energy Approximation.
Symmetry
14 CGTO's
26 CGTO's
38 CGTO's Ced.
al
36.5
(.005)
37.1
(.003)
34.0
(.607)
33.4
(.288)
33.2
(.231)
32.6
(.279)
32.1
(.592)
31.9
(.628)
32.9
12.9
(.913)
13.4
(.908)
13.5
(.903)
13.2
bl
34.9
(.005)
35.1
(.005)
10.8
(.909)
11.1
(.904)
11.2
(.900)
10.9
40.6
(.003)
40.8
(.004)
18.1
(.931)
18.0
(.922)
18.0
(.919)
17.7
E(HF)
-76.0082
-76.0459
-76.0507
-76.0419
aCederbaum (1973a).

77
1970). The orbital exponents of Huzinaga's 4s primitive basis for hydro¬
gen were scaled by a factor of 1.2, and the resultant orbitals were con¬
tracted to 2s functions as recommended by Dunning. The complete basis
appears in Table 8. The second basis augmented the first by the addition
of one set of p-orbitals on the hydrogen atoms and one set of d-orbitals
on both the oxygen and carbon atoms. Unit exponents were chosen for the
p-orbitals on hydrogen while exponents of 0.8 were chosen for the d-orbit-
als. One- and two-electron integrals were computed with the MOLECULE
program (Almlof, 1974) at the experimental equilibrium geometry: R(C0)=
2.2825 a.u., R(CH)= 2.1090 a.u.,and -)(HCH)= 116.52° (Oka, 1960, Takagi
and Oka, 1963), and the Hartree-Fock calculations and two-electron inte¬
gral transformations were performed with GRNFNC (Purvis, 1973). The
Hartree-Fock total energy for the smaller, 24 orbital basis (no polari¬
zation) was E(HF)= -113.8257 H., and for the larger, 42 orbital basis
(with polarization) E(HF)= -113.8901 H. The Hartree-Fock orbital energies
and second-order self-energy results for both basis sets are presented
in Table 9 for the principal ionizations along with the second-order
results of Cederhaum et al_. (1975) and the experimental values.
The results in Table 9 typify two general features of ionization
energy calculations. The first is that Koopmans1 theorem yields values
which are usually higher than experimental ionization energies. Second,
the inclusion of second-order relaxation and correlation corrections
generally overcorrects the Koopmans' estimate and yields values which
are usually lower than experiment. For several ionizations in Table 9,
the second-order deviations from experiment are as large as the Koopmans'
values only opposite in sign. Although it is possible that the larger,
polarized basis used in the second calculation may still lack adequate

Table 8. Contracted Gaussian Basis for Formaldehyde.
Carbon
s sets
Oxygen
s sets
Exponents
Contraction
Coefficients
Exponents
Contraction
Coefficients
4232.6100
0.002029
7816.5400
0.002031
634.8820
0.015535
1175.8200
0.015436
146.0970
0.075411
273.1880
0.073771
42.4974
0.257121
81.1696
0.247606
14.1892
0.596555
27.1836
0.611832
1.9666
0.242517
3.4136
0.241205
5.1477
1.000000
9.5322
1.000000
0.4962
1.000000
0.9398
1.000000
0.1533
1.000000
0.2846
1.000000
Carbon p sets
Oxygen p sets
Exponents
Contraction
Coefficients
Exponents
Contraction
Coefficients
18.1557
0.018534
35.1832
0.019580
3.9864
0.115442
7.9040
0.124189
1.1429
0.386206
2.3051
0.394727
0.3594
0.640089
0.7171
0.627375
0.1146
1.000000
0.2137
1.000000
Hydrogen
s sets
Exponents
Contraction
Coefficients
19.2406
0.032828
2.8992
0.231208
0.6535
0.817238
0.1776
1.000000

79
Table 9.
Principal
Ionization
Energies for
Formaldehyde.
24
42
(2)
(2)
Orbital
Koopmans
3(E)
Koopmans
2(E)
Ced.a
Exp.
lal
560.12
538.93
559.81
538.62
-
539.43b
2al
309.09
297.27
308.87
296.90
-
294.21b
3a |
38.94
33.66
38.18
32.56
-
34.2C
4a i
23.39
20.97
23.30
21.03
-
21.15c
5a ^
17.29
13.98
17.38
14.38
14.42
16.2d
lbl
14.56
13.83
14.45
13.72
13.50
14.5d
lb2
19.47
17.16
19.08
17.07
16.63
17.0d
2b2
12.06
9.04
11.93
9.30
9.25
10.9d
E(HF)
-113.8257
-113.8901
-113.9012
aSecond order results of Cederbaum et al_. (1975).
bJolly and Schaaf (1976).
cHood et a]_. (1976).
^Estimated center of gravity (Cederbaum and Domcke, 1977) from spectrum
of Turner et al. (1970) .

80
polarization functions, the rather large discrepancies between the
second-order results and experiment more probably indicate that third-
(and higher) order self-energy corrections are now sizable. The general
conclusion that a second-order self-energy approximation is inadequate
for an accurate calculation of ionization energies has been previously
concluded by Cederbaum (1973b) and necessitates a re-examination of the
approximation made in Eq. (3.88).
Rather than completely neglecting the third-order self-energy matrix,
let us now consider an approximation that includes at least part of these
contributions. Which third-order self-energy diagrams should be included?
There are two well-established results that are relevant to this ques¬
tion: Studies of the electron gas model have shown that in the limit of
high electron density, the so-called ring diagrams dominate the self¬
energy expansion (Pines, 1961), while in the limit of low electron den¬
sity, the so-called ladder diagrams dominate (Galitskii, 1958). In order
to determine whether atomic and molecular self-energies can be approxi¬
mated by specific third-order diagrams (e.g. rings or ladders), we need
to evaluate all third-order diagrams for some representative systems.
Cederbaum (1975) has done this for several simple systems and has found
that both ring an_d ladder diagrams dominate the third-order self-energy.
This result implies that atoms and molecules lie somewhere between the
high and low density extremes. It is therefore essential to include
both ring and ladder diagrams in any third-order self-energy approxima¬
tion. These diagrams are
rings ladders
(3.89)

81
and correspond to the algebraic expressions labeled A, B, C, and D in
Appendix 1.
We include six additional diagrams in our third-order self-energy
approximation because of the computational efficiency with which they
are evaluated. These diagrams are the energy independent diagrams
corresponding to expressions M-R in Appendix 1. For these six diagrams,
it is feasible to employ the partial summation technique since they must
be evaluated only once.
Approximating the full third-order self-energy matrix by only ring,
ladder, and constant energy diagrams, let us now consider the solution
of the Dyson equation with the [1,1] Pade1 approximant to the self-energy
expansion. Owing to the fact that the inner projection manifold from
which the [1,1] approximant was derived is energy dependent (Eq. (3.80)),
the simple analytic form of the self-energy eigenvalues, illustrated in
Fig. 2, is lost. Furthermore, the self-energy poles are now given by
det (2{2^(E) - 2^(E)) = 0 (3.91)
rather than by an eigenvalue problem and are consequently more difficult
to obtain. For these reasons, the pole search described in Chapter 1
and used with the second-order self-energy approximation is no longer
an efficient or reliable procedure. An alternative method of solution
used in the following applications was to use the Hartree-Fock orbital
energy as an initial guess to the propagator pole and to iterate Eq.
(1.67) to convergence. When convergent, this procedure invariably yields

82
a principal propagator pole and its corresponding pole strength. Al¬
though the [1,1] self-energy approximant does not quarantee a positive
pole strength, this was never a problem in any of the calculations re¬
ported here.
The principal ionization energies for the water molecule were cal¬
culated using both the 14 and 26 CGTO basis sets in order to evaluate
the [1,1] Pade' approximant to the self-energy expansion, and the results
appear in Table 10. The most significant feature of these results is that
each ionization energy has been shifted from its second order value to
higher energy and is now in better agreement with the experimental value.
It is further noticed that the valence ionization energies are still
smaller than the experimental values while the la^ (core) ionization
energy is now larger than experiment. Apparently, the diagrams included
in the third-order self-energy matrix overestimate the actual relaxation
and correlation effects for this ionization.
Convergence difficulty was experienced for the 2a^ ionization
energy using the 14 orbital basis. A schematic plot of W9 (E) is pre-
2a i
sented in Fig. 3 and reveals that there are no propagator poles in this
energy region. This anomaly is no doubt a consequence of some quirk in
the basis since the 26 orbital basis yields a very accurate 2a^ ioniza¬
tion energy.
3.6 Evaluation of the Diagram Conserving Decoup 1ing
The algebraic structure of the superoperator formalism has been
successfully exploited in this chapter to yield several new insights into
the decoupling problem. The application of perturbation theory has
demonstrated that the electron propagator equation of motion can be

83
Table 10. Comparison of Principal Ionization Energies for Water
Obtained with the Second-Order and the [1,1] Self-Energies
Using the 14 and 26 CGTO Basis Sets.
14
26
Orbital
(2)
£(E)
[1,1]
(2)
S(E)
[1,1]
Exp.a
lal
539.4
541.6
539.2
540.9
540.2
2al
34.0 (.607)
32.6 (.279)
no convergence
(see text)
32.1
32.2
32.2
3a j
12.9
13.4
13.4
13.6
14.7
Ibi
10.8
11.1
11.1
11.3
12.6
lb2
18.1
18.4
18.0
no results
18.6
aSiegbahn et al. (1969).

Figure 3. A sketch of W£ai in the energy region of the
2aj ionization ^obtained with the [1,1] self-
energy approximant using the 14 CGTO basis.

85
-1.215
- 1.175

86
resummed to yield the equivalent of the diagrammatic expansion. This
resummation also allows the identification of wave and reaction super¬
operators which have special importance in decoupling approximations.
We have shown that when the inner projection manifold of the superoper¬
ator resolvent is chosen to consist of the first-order truncation of the
wave superoperator, a [1,1] Pade' approximant to the self-energy expan¬
sion is obtained. This approximant is correct through third order and
contains a geometric approximation to all higher orders. In general,
the Nth-order truncation of the wave superoperator will yield an [N,N]
Pade1 approximant which is correct through the (2N+l)st order in the
self-energy expansion. One final insight afforded by this decoupling
is the realization that electron correlation can be described exclusively
in the operator space. We argued in Chapter 1 that when the propagator
was defined as a single-time Green's function, the density operator was
arbitrary. We have now demonstrated in this chapter that any desired
order in the self-energy expansion may be obtained using as a specific
choice,the uncorrelated, Hartree-Fock density operator.
Computational applications of the diagram conserving decoupling
have been encouraging. These applications have confirmed previous con¬
clusions (Cederbaum, 1973b) that a second-order self-energy is generally
inadequate for obtaining accurate ionization energies. It is important
if not essential that third-order ring and ladder diagrams be included
in any self-energy approximation although the errors arising from basis
incompleteness may be of equal magnitude and hence cannot be ignored.
The inclusion of the third-order ring, ladder, and constant energy dia¬
grams in the [1,1] self-energy approximant has succeeded in improving the
second-order results but even these results are not consistently better
than the Koopmans1 theorem values.

8;
One important feature of the [1,1] self-energy approximant is that
even though it is constructed from only the second- and third-order self¬
energy matrices, it contains a geometric approximation of all higher
orders in the self-energy expansion. Certainly, some fourth- or higher-
order terms may be just as important as third-order terms; therefore,
this approximation is highly desirable. The fourth- and higher-order
terms arising from the [1,1] self-energy approximant, however, are not
readily analyzed diagrammatically. In fact, being a purely algebraic
approximation, the [1,1] approximant may not yield any valid fourth- or
higher-order diagrams. Given the fact that ring and ladder diagrams
dominate the third-order self-energy matrix, one can argue that they may
also dominate the higher orders of the self-energy expansion. An appro¬
priate modification of this decoupling scheme might then allow the sum¬
mation of these specific diagrams in all orders. Approximations of this
type are referred to as renormalized decouplings and are examined within
the superoperator formalism in the next chapter.

CHAPTER 4
RENORMALIZED DECOUPLINGS
4.1 Renormalization Theory
In Chapter 3, we tacitly assumed that the application of perturba¬
tion theory to the calculation of ionization energies and electron
affinities was valid and that the resulting self-energy expansion was
convergent. Historically however, it was discovered that in both the
nuclear many-body problem and the electron gas model, the simple self¬
energy expansions are divergent. In order to remove these divergencies,
it is necessary to sum certain appropriate classes of diagrams to all
orders. This method of partial summations is known as renormalization
theory (see e.g. Kumar, 1962 or Mattuck, 1967) and may be viewed as an
analytic continuation of the perturbation expansion. Although a variety
of renormalization procedures exist, such as propagator renormalizations,
interaction renormalizations, and vertex renormalizations, the distinc¬
tions mainly depend on the types of diagrams included in the partial
summation and are not particularly important for our consideration.
One renormalization that we are already familiar with is the [1,1]
self-energy approximant derived in the preceding chapter. In fact, any
rational self-energy approximant may be regarded as a renormalization
since its geometric expansion will approximate all orders of the pertur¬
bation expansion. One problem encountered with the [1,1] approximant
and that occurs in general for rational approximants derived via purely
algebraic considerations is that their geometric expansions may contain
88

89
no readily identifiable diagrams (at least beyond the lowest orders).
Since specific diagrams often dominate the self-energy expansion (such
as ring and ladder diagrams for atoms and molecules) it is valuable to
investigate whether the superoperator formalism can be adapted to yield
renormalized self-energy expressions that sum specific diagrams. The
solution as we shall see is rather simple and involves a restriction in
the types of operator products allowed to span the orthogonal complement
of the model subspace. As a specific example, the two particle-one hole
Tamm-Dancoff approximation (2p-h TDA), (Schuck et aj_., 1973, Schirmer
and Cederbaum, 1978), is derived from an effective interaction which is
logically obtained by a projection of the perturbation superoperator onto
the subspace spanned by 2p-h type operators (Born and Ohrn, 1979).
Finally, the diagonal approximation to the full 2p-h TDA self-energy
previously derived and applied to the calculation of ionization energies
is shown to neglect terms which, in fact, are diagonal and are necessary
to prevent an overcounting of all diagrams containing diagonal ladder
parts.
4.2 Derivation of the 2p-h TDA and Diagonal 2p-h TDA Equations
Recalling some of the results of the previous chapter, we had ob¬
tained the matrix Dyson equation
G(E) = Gq(E) + EgfEWEME) , (4.1)
where the self-energy matrix, T_(E), had the following expansion
1(E) = (a|V+VT0(E)V+VT0(E)VT0(E)V+ . . . |a)
(4.2)

90
Introducing the reduced resolvent of the full superoperator Hamiltonian,
T(E), which is just a projection of the superoperator resolvent on the
orthogonal complement
T(E) = P[aO+(EÍ-H0)P-PVP]_1P , (4.3)
the self-energy expansion was written in closed form
2(E) = (aJVa) + (aJVT(E)Va) . (4.4)
It was further shown that when the grand canonical density operator is
used to evaluate the operator averages, the first-order term vanishes.
When P is the exact projector of the orthogonal complement,
P = Í - 6 , (4.5)
the term PVP in Eq. (4.3) is responsible for generating the operator
products that span this subspace. The expansion of this term from the
inverse and its repeated application in each order of the perturbation
expansion yields larger and larger operator products which are only
limited by the number of electrons in the reference state. If instead
of allowing all possible operator products, we restrict them to some
simple types which occur in each order, it may be possible to identify
and sum specific diagrams in all orders of the perturbation expansion.
The restriction of the operator products in the orthogonal comple¬
ment is achieved by approximating the orthogonal projector as
P = |f)(f| (4.6)
where the manifold {f} contains the desired operator products. The pro¬
jector P now has the effect of projecting from the perturbation expansion

91
only those operator products which lie in the subspace spanned by {f}.
The approximation to P in Eq. (4.6) must of course preserve the proper¬
ties of the exact projector and should be idempotent, self-adjoint, and
orthogonal to 0.
Our previous experience with the operator product decouplings
suggests that the set of triple operator products {a^a^a^} be chosen
as a first approximation to P. There is a stronger motivation for using
this operator product, however. If the third-order ring and ladder dia¬
grams in Eq. (3.70) are examined, it can be seen that between any two
interaction lines there occurs only two particle lines (upgoing) and one
hole line (downgoing) or vice versa. This implies that the intermediate
or virtual states that are represented by these diagrams consist of only
2p-h or 2h-p excitations of the reference state. Both of these excita¬
tions are described with the triple-operator products.
The set of triple products {a^a^a^} is not orthogonal to the simple
operators of the model subspace, hence these two subspaces must be or-
thogonalized. Using the Gram-Schmidt orthogonalization procedure (see
e.g. Pilar, 1963), we define
f i, = N. ,2 [a, a, a -E(a la. a .a )a ]
klm klm1 k 1 m ' n1 k 1 nr nJ
-l- +
1. j [a, a,a +6, a,-6, ,a ]
klm1 k 1 m km k 1 kl km
(4.7)
(4.8)
where
N. , = --+
klm k k 1 k ni 1 m
(4.9)
The projector
P = E |
k,l ,m
fklm^fkliJ
l (4.10)

92
is now idempotent, self-adjoint, and orthogonal to 0. The projection of
the perturbation superoperator on the 2p-h subspace, Pl/P, which occurs in
Eq. (4.3) can now be regarded as an effective interaction. The expansion
of Eq. (4.3) with P defined as in Eq. (4.10) should yield all diagrams
containing 2p-h and 2h-p excitations of the reference state.
The necessary operator averages needed to evaluate PVP are:
(4-ID
and
(ak'al'Vl5akalaJ = Nk'l'm'{6kk'(1-)
--)-<5^, (-)
+<5m'1(-)+6nil,(-)
+6kl+5krn}
+Nklm{6m1k’+51,k,} (4.12)
Substituting these expressions and performing some cancelation yields:
E
c',1
1
N'!f N
, k 1 in
' 2
k' 1
'm
.6kk
i(l- > -
m
11 'm' ^klm
1 '<
m
E
â– ' >1
1
tcf m
, klm
k'l
'm
, 1m||km'>6^ ^
â– ( - m>)lfk'l
'm'^kltJ
1 '<
m‘
?:
;M
1 ,m'
N''5 N
, klm
k'l
'm
, 11 i Ik1'>6
1 1 mm
,(
- l>)lfk'l
'm' ^kklnJ
1 '<
m*
E
M'
3m'
N,-'5 W
klm
1,
'2
k'l
'nr
, 'mllkl'.dim
â– ( - fk111
'm' ^klrJ
1 '<
ni'
E
M '
sm*
n:’5 N
klm
2
k'l
1 m'
1 ! |km'>6ml
-( - l>)lfk‘T
m' ^kkliJ
1 < m 1 1 < m1
(4.13)

93
Additional simplification can be achieved at this point by anti-commuting
the operators a-j, and am, in I f^, -j ,m■ ) °f the last two terms in Eq. (4.13)
with the appropriate change of sign,
If
k' 11 m
If
k' m11
(4.14)
After interchanging dummy indices 1'+->■ m’, the fourth and fifth terms be¬
come equal to the second and third terms, respectively, and since the
diagonal terms l'= m1 vanish, the summations with 11< m' and 1'> m1 can
be combined as unrestricted summations over 1' and m'. Since the first
term in Eq. (4.13) is obviously symmetric in 1' and m1, the restriction
l'< m' in that term may be removed by multiplying the sum by a factor
of The remaining restriction, 1 ing another factor of ^ since
PVP = E E NkímNk,l'm,í,5,5kk,(1"")
K j I jlil K ^ I )III
1 < m
-6,,, (-)-6 , (-)} I f, ,, , ,)(f. , I
11 11 k m ' 11 mm 1 k 1 ' 1 k 1 m k1m1
(4.15)
is symmetric in these indices as can be verified by interchanging l+-+m
and relabeling dummy indices 1 '<-+ m1. Expanding the ket-bra superoperator
(4.16)
'S ' f I f V 11 >m1) (f
k,l,m k1,11,m1 K 1 m
klnr
out of the inverse, evaluating the remaininq operator averages, and
resumming the expansion yields the 2p-h TDA self-energy
2p-h TDA
>:(E)iJ -
k,1,m k1,11 ,m
v, ,NklrnNk-Tm*<1kl llm>
x {(f j (EI-H0)f)-(f |Vf)}-j|[)(k,rm, (4.17)
where

94
--)--)} (4.18)
Although different in appearance, this self-energy expression is formally
the same as that obtained by Purvis and Ohrn (1975a) using the operator
product decoupling and by Cederbaum (1975) and Schirmer and Cederbauin
(1978) using the diagrammatic method. The present derivation clearly
illuminates the parallelism between the two formalisms.
Owing to the large dimension of the {f, , } operator subspace and
the associated difficulty in diagonalizing (f|Vf), computational appli¬
cations of the 2p-h TDA have usually involved additional approximations.
One approximation which has facilitated computational applications is
known alternatively as the shifted Born collision (SBC) approximation
(Purvis and Ohrn, 1974, 1975a) or the diagonal 2p-h TDA (Cederbaum,
1974, 1975 and Cederbaum and Domcke, 1977). This purportedly "diagonal"
approximation restricts the spin-orbital summation indices in Eq. (4.13)
to k’ = k, l' = l, and m'=m thereby neglecting the last two summations
and yielding the following self-energy expression
(4.19)
where
A=(l--)-(-)-(-). (4.20)
mi Km k i
By neglecting the last two summations in Eq. (4.13), however, this approx¬
imation actually neglects some diagonal contributions to (fk■t *m■IVfkim)•
As we have explicitly demonstrated in the derivation of Eq. (4.17) and

95
(4.18), the 2p-h TDA self-energy sums are symmetric in both 1,m and
1',m'; consequently, the last two summations in Eq. (4.13) contain pre¬
cisely the same contributions as the second and third summations, re¬
spectively. If the diagonal approximation (k'=k, l'=l, m'=m) is made in
Eqs. (4.17) and (4.18), this symmetry is properly accounted for and the
resulting self-energy expression is
(4.21)
where
A=!2^ml | |ml>(l--)-(-)-(-). (4.22)
ni i k m k i
Eq. (4.22) differs from Eq. (4.20) by the factor of % in the first term.
The inclusion of this factor in the diagonal 2p-h TDA is necessary to
prevent an overcounting of diagrams with diagonal ladder parts (see next
section) and typically shifts ionization energies 0.3-0.4 eV higher in
energy (Born and Ohrn, 1979).
4.3 Diagrammatic Analysis
In order to determine precisely which self-energy diagrams are in¬
cluded in the 2p-h TDA self-energy, it is necessary to expand the effec¬
tive interaction matrix, (£|V_f), from the inverse in Eq. (4.17) and dia¬
gram the resulting algebraic expressions in each order. The expansion
of Eq. (4.17) yields the following terms in lowest orders
2p-h TDA
7(E),. = 8 >: >: N
1J U 1 m L- ' 1 ' m 1
k, 1, m k1,1 ', m'

96
+ Z
K,X,P
%Y?ltVv:?IVrv
+
• ]■ <1 'm1 | | jk1 > .
(4.23)
As was done in Section 3.3, the terms in Eq. (4.23) can be simplified by
first restricting the summations over all spin orbitals to summations
over occupied or unoccupied spin orbitals such that the occupation
1. U
number factor NklmNk'rm' is nonvanishing. Doing this in the first term
of Eq. (4.23) and then summing over the delta functions yields
is 7
a,p,q

(E+e -e -E )
a p q'
(4.24)
+H 7
p,a,b

(E+e -e -e, )
p a b;
(4.25)
which are the same second-order self-energy diagrams as obtained in Eqs.
(3.66) and (3.67).
Restricting and spin orbital summations in the second term of Eq.
(4.23) yields the following expressions
h z
a,p,q
7
b,r,s

(fbrslífaPq)

(4.26)
{!VpV
lEVfSy
+!S Z
a,p,q
7
r,b,c

^rbcl^apq)

(4.27)
+H Z
p,a,b
7
d,q,r

(fcqr!ífPab)

^Vv^b7
(E+e -c -e )
c q ry
(4.28)
+‘s 7
p,a,b
7
q,c,d

Vd^W

(4.29)
(E+VVebJ
TF+VVcdT
Now substituting Eq. (4.18) for the effective interaction matrices, we
find that the delta functions in Eq. (4.18) further restrict the spin

97
orbital summations in such a way that only expressions (4.26) and (4.29)
are nonvanishing. After some simplification, the nonvanishing contribu¬
tions are found to be
h 2
a
-h 2
a ,b
!S 2
a ,b
!i 2
a,b,c,d
+*2 2
a,b,c
+!2 2
a,b,c

P»q,r,s
(E+WV
he+vvs)
2
P.q>r
|arxpr[ | jb>
TeTVVV
,^E+Eb-Ver)
2
P>q>r
larxrql |jb>
v a p q;
E+c.-e -c )
b r q'
2
P
1 dcxcd | |,ip>
(E+ep-ec-£d)
2
p>q

2
p,q
< i p 1 |abxqa|
1 pcxcb | | jq>
(E+e -e,-e )
q b c
(4.30)
(4.31)
(4.32)
(4.33)
(4.34)
(4.35)
Expressions (4.30) and (4.33) are the only ones which represent valid
third-order diagrams as written, however, by interchanging dummy indices
p +-* q in Eq. (4.32) and a « b in Eq. (4.35), expressions (4.31) and
(4.32) can be combined to yield
2
a, b
7
P,q,r

(E+Vrp'ViE+Cb:V£rT
(4.36)
and expressions (4.34) and (4.35) combined to yield

2
a,b,c
2
p>q
(4.37)

98
which now correspond to the two third-order ring diagrams as indicated.
These results verify that one of our original objectives, which was to
include all third-order ring and ladder diagrams, has been achieved.
The diagrammatic analysis of the third term in Eq. (4.23) proceeds
in the same way as that of the first two terms, but since the effective
interaction matrix appears twice, it involves considerably more algebra.
For this reason, we simply display the resulting diagrams in Fig. 4.
It is significant to realize that the fourth-order diagrams in Fig. 4
include not only ring and ladder diagrams but also mixed diagrams which
consist of both ring and ladder parts. In the third-order analysis, the
first term in Eq. (4.18) was responsible for yielding the ladder diagrams
while the second and third terms yielded the ring diagrams. If we there¬
fore denote the first term as a ladder part and the second and third
terms as ring parts, the mixed diagrams in fourth order are found to
arise from the product of a ladder part and a ring part. Inducing the
results of the fourth-order analysis to higher orders, we conclude that
our second objective, which was to sum all ring and ladder diagrams in
all orders of the self-energy expansion, has been exceeded: not only
are all ring and ladder diagrams included in the 2p-h TDA self-energy,
but also the mixed diagrams which exhibit both ring and ladder parts.
The diagonal 2p-h TDA self-energy may also be analyzed diagrammat-
ically. This analysis is even simpler than for the full 2p-h TDA since
the denominator shifts are now scalars rather than matrices. A compar¬
ison of diagrams obtained with the denominator shift in Eq. (4.30)
versus that in Eq. (4.22) will reveal the significance of the factor of
’s. Considering the approximation in Eqs. (4.21) and (4.22) first, we
obtain the following third-order expressions

Figure 4
Fourth-order self-energy diagrams arising from
the 2p-h TDA.

100

k £
a,p,q
(4.38)

(E+£a-£p-eq)2
i, j,
4a,b,p (EWeb)2

u 9
a,p,q (E+e -e -e )
a p q
j. < ip I |abxpb| |pbxab| |jp>
a,b,p (E+ep-Ea-eb)2
(4.39)
(4.40)
(4.41)
The only difference between these diagrams and the third-order diagrams
of the full, 2p-h TDA is that the incoming lines on the middle interac¬
tion line have the same labels as the outgoing lines. Now analyzing the
approximation in Eqs. (4.19) and (4.20), we obtain the following third-
order expressions

(4.42)
' 2 L
a,p,q
(E+ea’ep~Eq)2
— '2 1
a ,b,p

(E+Ep'£a-£b)2
(4.43)
T,
a,p,q

(4.44)
(E+£a-Ep-Eq)2
+ T,

, 7
(4.45)

102
Expressions (4.44) and (4.45) are identical to (4.40) and (4.41) respec¬
tively; however, expressions (4.42) and (4.43) both differ by a factor
of from (4.38) and (4.39). This discrepancy is a direct consequence
of the missing factor in the denominator shift, Eq. (4.20), and leads
to an overcounting of these third-order diagonal ladder diagrams since,
by rule 5 in Table 5, there should be a factor of % for each pair of
equivalent lines. Similarly, it is rather easy to show that this approx¬
imation overcounts all higher order diagrams containing this diagonal
ladder part.
4.4 Computational Applications and Evaluation of the Diagonal 2p-h TDA
Self-Energy
The main attraction of the diagonal 2p-h TDA self-energy for the
calculation of ionization energies and electron affinities is its pseudo
second-order structure. Computational experience with the diagram con¬
serving decouplings has taught us that the second-order self-energy
approximant is both easily constructed and evaluated. The diagonal 2p-h
TDA requires only the additional evaluation of Eq. (4.22) which merely
shifts the second-order self-energy poles. As a consequence the diagonal
2p-h TDA self-energy mimics the exact self-energy by possessing only
simple poles. Another consequence of the pseudo second-order structure
is that the energy dependence will have the simple analytic form illus¬
trated in Fig. 2. This property, which was absent in the [1,1] self-
energy approximant, simplifies the pole search for the electron propaga¬
tor. In spite of the simplicity of its pseudo second-order structure,
however, the diagonal 2p-h TDA self-energy incorporates diagonal ring,
ladder, and mixed diagrams in all orders of the self-energy expansion

103
as was shown in the previous section. This property encourages specula¬
tion that significantly more relaxation and correlation will be accounted
for in this decoupling than with the second-order decoupling, but the
accuracy of the corresponding ionization energies or electron affinities
can only be evaluated via actual computational applications. As was
done with the diagram conserving decouplings, ionization energies for
the water and formaldehyde molecules were computed using the diagonal
2p-h TDA self-energy.
For the water molecule, calculations were performed with both the
14 and 26 contracted Gaussian orbital basis sets described in Chapters
2 and 3. The principal ionization energies computed with the diagonal
2p-h TDA and the diagonal 2p-h TDA plus third-order constant energy
(2) (2) (3)
diagrams (denoted 2(E)SHIFT and 2(E)SHjpT + £(»)) are presented in Table
11. Comparing these results with those in Table 9 for the second-order
and [1,1] self-energy approximants reveals that each ionization has been
shifted to higher energy. This shift has led to a significant improve¬
ment in the valence ionization energies (3ap, lb^, and lb,,) which are
now within approximately 0.5 eV of the experimental results. For the
inner valence (2a^) and core (la^) ionizations, however, this energy
shift leads to worse agreement. In addition, the diagonal 2p-h TDA
results for the la^ ionization exhibit an enormous basis dependence.
The addition of polarization functions in the 26 orbital basis has
yielded nearly 13 eV in additional relaxation. The most probable explana¬
tion for this basis dependence is that the 2p-h TDA self-energy poles
are determined by Hartree-Fock orbital energies and two-electron inte¬
grals rather than by orbital energies alone as with the second-order
self-energy. The orbital energies are rather insentitive to basis
changes, whereas the two-electron integrals are not.

Table 11. Water Results Obtained with the Diagonal 2p-h TDA and Diagonal 2p-h TDA plus Constant
Third-Order Self-Energies.
14 CGTO's 26 CGTO's
Koopmans
(2)
z(e)shift
(2)
2(E)
(3)
SHIFT +
Koopmans
(2)
z(e)shift
(2:
E(E]
1 (3)
'shift +
Exp.a
lal
559.45
554.91
556.04
559.40
542.78
543.51
540.2
2al
37.04
34.62
35.00
36.52
33.69
33.93
32.2
3a j
15.43
13.61
13.95
15.66
14,12
14.31
14.7
lbl
13.78
11.64
11.97
13.67
11.83
12.00
12.6
lb2
19.52
18.51
18.80
19.34
18.46
18.61
18.6
aSiegbahn al_. (1969).

105
Calculations for the formaldehyde molecule were performed with the
24 and 42 orbital basis sets described in Chapter 3. Ionization energies
were computed using the diagonal 2p-h TDA and are presented in Table 12.
The third-order constant energy diagrams were not evaluated for this
molecule. Similar to the water results, the diagonal 2p-h TDA results
for formaldehyde are also consistently higher in energy than the second-
order results (Table 9). This shift considerably improves the valence
ionization energies; however, the average deviation from the experimental
results remains approximately 0.8 eV. The core ionizations within the
diagonal 2p-h TDA suffer a small deterioration in accuracy but do not
exhibit the extreme basis dependence which was observed in the water
calculations. Part of the discrepancies between the diagonal 2p-h TDA
and the experimental results can certainly be eliminated by further basis
saturation; however, in the next chapter, we will propose that even ioni¬
zation energies of 1.0 eV accuracy are usually sufficient to unambiguously
interpret photoelectron spectra if combined with a calculation of rela¬
tive photoionization intensities.
Cederbaum and co-workers have recently developed computer programs
which implement the full, nondiagonal 2p-h TDA to the self-energy and
have reported several molecular applications (Cederbaum ert aj_., 1977 ,
Schirmer ert aj_., 1977, Cederbaum £t aj_., 1978, and Schirmer et al.,
1978). In these calculations, they claim only a 1.0 eV accuracy and rely
heavily on vibrational analyses to assist with the interpretation of
photoelectron spectra. The off-diagonal matrix elements seem to have
little importance in the valence region because the propagator poles are
relatively well-separated. In the inner valence and core regions, how¬
ever, where principal ionization poles and shake-up poles overlap and

106
Table 12. Formaldehyde Results Obtained with the Diagonal 2p-h
TDA Self-Energy.
24CGT0's 42 CGTO's
Koopmans
(2)
Z^SHIFT
Koopmans
(2)
^^SHIFT
Exp.
lal
560.12
542.86
559.81
542.09
539.43'
2a i
309.09
299.82
308.87
299.31
294.21'
3a j
38.94
35.60
38.18
34.31
34.2b
4a i
23.39
21.75
23.30
21.79
21.15*
5a j
17.29
14.99
17.38
15.33
16.2C
lbl
14.56
14.11
14.45
14.03
14.5C
lb2
19.47
17.91
19.08
17.77
^4
O
n
2b2
12.06
9.92
11.93
10.11
10.9C
aJo 11y and Schaaf (1976).
bHood et al_. (1976).
cEstimated center of gravity (Cederbaum and Domcke, 1977) from
spectrum of Turner et al_. (1970).
d14.38(8) VIP (Turner etal., 1970).

107
interact, level shifts and intensity changes are observed. In these
regions, even the nondiagonal 2p-h TDA is not fully satisfactory since
ion-state relaxation and hole-hole interactions, neither of which are
described by this self-energy approximation, may also be important
(Wendin, 1979).

CHAPTER 5
PHOTOIONIZATION INTENSITIES
5.1 Introduction
The evaluation of each decoupling approximation in the preceding
chapters was based on the comparison of propagator poles to experimental
ionization energies. This criterion represents a particular bias since
it does not reflect the quality of the Feynman-Dyson amplitudes (defined
in Section 1.1). The Feynman-Dyson amplitudes determine the spectral
density function (Linderberg and Ohrn, 1973)
A(x,x 1 ;E) =
f r fk(x)f (x
1)6(E-Ek)
f gk(x)g (x')6(E-Ek)
E > p
E <
(5.1)
which contains a plethora of useful information. This is evidenced by
the relation of the spectral density to the single-particle, reduced
density matrix (Linderberg and Ohrn, 1973)
y(x,x') = [ A(x,x';E)dE .
JC
(5.2)
It is important, therefore, to choose a decoupling approximation which
not only yields accurate ionization energies but also an accurate spectral
density. The quality of the spectral density is somewhat more difficult
to evaluate since there are no experimental data with which it can be
directly compared. One evaluation procedure, however, might involve the
calculation of averages of specific one-electron operators from the
reduced density matrix. The averages could then be compared with
108

109
experimental results. Another procedure which is more closely related
to our goal of interpreting photoelectron spectra is the calculation of
photoionization intensities or cross-sections. A theoretical prediction
of relative photoionization intensities can simplify orbital assignments
when the ionization energies are not accurately given. A theoretically
predicted variation in relative intensities with photon energy is par¬
ticularly useful if photoelectron spectra are available with different
ionization sources (Katrib et aj_., 1973). In the following sections,
we derive computational expressions which relate the Feynman-Dyson ampli¬
tudes to the total photoionization cross-section, discuss the major
approximations assumed, and then present several applications.
5.2 Derivation of Computational Formulae for the Total Photoionization
Cross-Section
The differential cross-section for photoionization derived from
first-order, time-dependent perturbation theory using a semi-classical
model for the interaction of radiation and matter is (Bethe and Salpeter,
1957, Kaplan and Markin, 1968, Smith, 1971)
da 4/|L|
dfi*
c IA01 w

(5.3)
In this equation, is the momentum of the ejected photoelectron and
it = S it. is the vector potential. For a closed-shell system, the initial
k K
state |N> can be represented by an antisymmetrized N-electron wavefunction
[N> - 0q(Xj,X2, - - • x^j) (5.4)
and the final state |N,s> is represented by

110
;N,s> = (N/2)2 0AS [v(kf,r)a(c)í's0(x1,x2, . . . x^)
- v^.i^eteH^íx^x^ . . . xN_j)] . (5.5)
Here $ and $ are the two, doublet spin components of the (N-l)-
SCt S p
electron ion, v(k^,r) denotes the wavefunction of the photoelectron, and
is an antisymmetrizer
,-1 r
1-1
°AS N PkNJ '
(5.6)
The form of Eq. (5.5) guarantees that the singlet spin symmetry of the
system is preserved.
Evaluating the matrix element in Eq. (5.3) yields (Purvis and Ohrn,
1975a)
= SZ J ?s gs(r)dr
+ /? J v (íf,r)ps(r)dr (5.7)
where
N'^CrMi) = | (Xj, . . . ’tN_1)«’0(x1, . . . xN1 ;r,x(c))
x dXj . . . dxN_j (5.8)
and
N'1;ps(r)x(c) = (N-l) | * (Xj, . . . xN_1)^1- ^ ®0(xr • • • xN_j;
r,x(c))dXj . . . dxN_1 . (5.9)
The first term in Eq. (5.7) relates the Feynman-Dysori amplitude g (r) to
the photoionization cross-section and the second term arises from the
nonorthogonality of v(lt^,r) and i>g. When v(k^,r) is strongly orthogonal
to $q, this term vanishes. Even when v(£f,r) is not strongly orthogonal
to $q, the first term in Eq. (5.7) will dominate if the kinetic energy

Ill
of the photoelectron is much greater than its binding energy (Rabalais
et al., 1974), therefore we shall neglect this term:
8^|kf)
c|Aq|2ü)
v (kf,r)í$- gs(r)dr 2
(5.10)
Further simplification can be obtained by neglecting retardation of
the photoelectron momentum by the photon momentum (also called the dipole
approximation in analogy with photon induced transitions in the discrete
spectrum, Bethe and Salpeter, 1957, Steinfeld, 1974). The vector poten¬
tial t has the following plane wave decomposition in terms of the inci¬
dent photon momentum f and polarization n
ís = \tQ\ exp (i^ ■ f$)n . (5.11)
If the wavelength of the incident photon is large compared with molecular
dimensions, the exponential may be approximated by the first term in its
Taylor series expansion (unity)
With this approximation, the differential photoionization cross-section
becomes
8tt2 I ítf |
CO)
|n • ?|2 .
where
(5.13)
? - | v (k>f,r)Vgs(r)dr .
(5.14)

112
Owing to the random orientation of molecules in a gaseous sample,
the experimentally observed photoionization intensities in the molecular
reference frame represent an average over all incident photon directions.
Furthermore, if the incident photon beam is unpolarized, we must also
average over photon polarizations. Making the appropriate averages in
Eq. (5.13) (Smith, 1971), we obtain
^r~J ¿ I- {l"i• V + • (5-15>
Since the two polarization directions, and n^, and the incident photon
direction, ky|kj, form a right-handed system of axes, we can write
nr = iv V + V V + V V / iV
and Eq. (5.15) becomes
(5.16)
V
dfi-
8tt^ I ] V
C(i)
8"2|*fl
CM
8-n-2 | ( ¿I
If!
iV .
,2 ±
8tt
£ ■ £\¿ / |1< ridn
0) 1 1 to1 w
{1 - cos e }dtt
OJ 0)
(5.17)
(5.18)
(5.19)
In order to evaluate |?|2, some form for the photoelectron wave-
function v(k^.,r) must now be chosen. In principle, the photoelectron
wavefunction could be obtained by the solution of the Bethe-Sal peter
equation for the polarization propagator where the superoperator resol¬
vent has been modified to include the time-dependent interaction of the
radiation and matter fields (see e.g. Csanak et al., 1971). A solution

113
of this type would require the use of continuum functions as well as
discrete functions in the molecular basis and is not yet feasible owing
to several formal and practical difficulties. Alternatively, we seek a
simple but accurate, analytic representation of the photoelectron wave-
function. For photoionization of atoms or molecules with high (tetra¬
hedral or octahedral) symmetry, the electronic potential of the ion is
nearly spherically symmetric, and v(k^,r) may be asymptotically repre¬
sented by a plane wave plus incoming Coulomb waves (see e.g. Smith, 1971).
For molecules with lower symmetry, distortions of the electronic poten¬
tial enormously complicate the nature of the incoming waves, in this
derivation, the incoming waves are neglected, and the photoelectron is
simply represented by the plane wave part
v(j (5.20)
The applicability and implications of this approximation will be dis¬
cussed in the following section. With this choice, Eq. (5.14) can be
integrated by parts and yields
(5.21)
(5.22)
In our computational scheme, the Feynman-Dyson amplitudes in Eq.
(5.22)are represented by a linear combination of Hartree-Fock orbitals.
The Hartree-Fock orbitals can be decomposed into contracted Cartesian
Gaussian functions on each atomic center, and each contracted Gaussian
function can be further decomposed into a sum of primitive Gaussian
functions. Ultimately therefore, the Feynman-Dyson amplitudes can be

114
represented as some linear combination of primitive Gaussian functions
on each atomic center
z
a, k
o I
ak
^ak
(? - ta) .
(5.23)
Here a represents the sum over atomic centers and k the sum over primitive
Gaussian functions on each center. Transforming the photoelectron posi¬
tion vector to the coordinate frames of each atomic center by the sub¬
stitution r' = r-ft , Eq. (5.22) can be re-expressed in terms of the prim¬
itive computational basis
-i£f(2TT)'3/2 exp(-iicf • ft )4 | exp(-ilcf ■ ?' )4>ak(1"' )dr' .
(5.24)
The integral over r' in Eq. (5.24) represents a Fourier transform of
each primitive Gaussian function, and formulae for its evaluation are
derived by Kaijser and Smith (1977).
The final step in this derivation is to integrate the differential
photoionization cross-section (Eq. (5.19)) over all photoelectron direc¬
tions in the solid angle di2f
Since £ involves a sum over atomic centers, Eq. (5.25) will contain both
one- and two-center contributions (Schweig and Thiel, 1974)
as =
3ao
£
a,k
£
6,1
'ak
C61 Qak,Bl
(5.26)
The one-center terms (a=B) have the form

115
where O^fi^) represents the Fourier transform
*ak(fy = (2tt)_3/2 J exp(-iiak(r')dr'
and the two-center terms (a^S) are given by
Qak,81 = I *ak(ltf)*Blil?f)exPf-il'f ' ^aB)dnf
(5.27)
(5.28)
(5.29)
where = Sa - Owing to the orthogonality of the spherical har¬
monics which describe the angular dependence of the Fourier transforms,
the one-center terms are easily evaluated. The two-center terms are
complicated slightly by the exponential factor, but these too can be
evaluated analytically. For this purpose, it is convenient to define
the integral
1(1, 12m2^
(0f,<¡>f)y1^(0f,iíf)exp(-i£f • ft„R)dilf, (5.30)
where yq m and ^mg are real spherical harmonics (Harris, 1973) repre-
the angular
Using the expansion
senting the angular dependence of the transforms and respectively.
exp(±itf • ¡taS)
m 1
4tt £ £ (±i)
1=0 m=-l
JVkfRaf
)klm(ef,f)ylm(0,lf), (5.31)
Eq. (5.30) becomes
m 1 , m.m„-m
1(11m1 ,l2m2) = 4ir^ ^M-i) JV kfRap^Cl ^ 2 1 ^In/0’^’
(5.32)
In these equations, (6^,<¡i^) represents the photoelectron direction in
the atomic reference frame, (0,<|>) represents the direction of R^gin the
molecular frame, ji(k) is a spherical Bessel function of 1-th order,
mjn^-m
and Cq ^ q g q is a Clebsh-Gordan coefficient defined by (Harris, 1973)

116
rmln,2m
L1 1 1
12
{ ^Ijm^f ,4>f2m2 (ef’'í’fÍyimief^dSif
(5.33)
When the distance vector 1$ „ coincides with the molecular z-axis, the
«6
expansion in Eq. (5.31) simplifies to
exp(±i£f • iía6) = (4rr)'s r, (±i)1(21+l)*sj1(kfRaB)yio(0f,«f) (5.34)
and Eq. (5.30) becomes
IW
i j rn i m o 0
inim1,l?m?) = (4Tr)’s6_ 2 (-i)1 (21+1) % V,
1 1 ¿ ¿ mlm2 1 = 111g| 11121
(5.35)
For arbitrary directions of ft . which do not coincide with the molecular
ap
z-axis, a new coordinate frame can be defined in which S . does coincide
Cip
with this axis, and the spherical harmonics can be transformed to this
coordinate frame using the familiar rotation matrices (Schweig and Thiel,
1974). In this way, Eq. (5.35) can be used to evaluate all two-ccnter
integrals arising from any molecular geometry.
5.3 Discussion of Approximations
Relatively little work has been devoted to the theoretical calcula¬
tion of photoionization cross-sections for molecules compared with that
for atoms (see e.g. Marr, 1967, Steward, 1967, Kelly, 1976). The major
impediments until recently have been a lack of sufficiently accurate
molecular wavefunctions and an absence of accurate analytic representa¬
tions for the photoelectron wavefunction (Kaplan and Markin, 1967,
Schweig and Thiel, 1974). With the development of efficient, molecular
integral and Hartree-Fock programs, Hartree-Fock wavefunctions are now
readily available for a large number of molecules. This availability
in turn has stimulated several theoretical calculations of molecular

117
photoionization cross-sections using the frozen orbital approximation
(Rabalais et aK, 1974, Dewar et aj_., 1975, Allison and Cavell, 1978,
and Cavell and Allison, 1978).
In the frozen orbital approximation, the N-electron reference state
is assumed to be the Hartree-Fock ground state, and the ion states are
constructed by removing the orbital corresponding to the ionized elec¬
tron. This approximation neglects both correlation corrections in the
ground and ion states and ion state relaxation (see Introduction).
Following Purvis and Ohrn (1975a), we have replaced the frozen orbital
approximation by a many-electron treatment which incorporates both relax¬
ation and correlation corrections. This treatment derives from the use
of the Feynman-Dyson amplitudes obtained from the electron propagator to
compute the photoionization cross-section.
The form of the photoelectron wavefunction is still a major problem
in the calculation of molecular photoionization cross-sections and repre¬
sents the most critical step in our derivation. The plane wave approxi¬
mation was chosen for its simplicity, however, it has several serious
limitations. The most serious limitation is its failure to correctly
predict experimentally observed angular distributions (Bethe and Salpeter,
1957, Schweig and Thiel, 1974). This deficiency is not readily apparent
when the differential cross-section is averaged over all photoelectron
directions, and in our computational applications, we compute only spher¬
ically averaged, total cross-sections. Lohr (1972) has shown that by
retaining the second term in Eq. (5.7), qualitatively correct angular
distributions may be obtained. The retention of this term is equivalent
to orthogonalizing the plane wave to every one-electron function in the
N-electron ground state and is known as the orthogonalized plane wave
(OPW) approximation.

118
A second limitation of the plane wave approximation is the implicit
neglect of electrostatic interactions between the photoelectron and the
molecular ion. For ionization processes near threshold where the photo¬
electron leaves with low kinetic energy, these interactions are espe¬
cially important, and the wavefunction of the photoelectron exhibits a
decrease in wavelength as r 5. The OPW approximation again partially
corrects this deficiency (Lohr, 1972) but exhibits a rather abrupt change
in wavelength which is more characteristic of a short-range potential
rather than the long-range Coulomb potential. Other representations of
low energy photoelectrons, e.g. multicentered Coulomb wave expansions,
have also been proposed (Tuckwell, 1970). As the kinetic energy of the
photoelectron tends to higher energies, the effect of electrostatic
interactions on the total cross-section becomes negligible, and the plane
wave approximation becomes sufficiently accurate (Bethe and Salpeter,
1957).
Unfortunately, as the plane wave approximation becomes more suitable
at high photoelectron energies, the dipole approximation (Eq. (5.12))
deteriorates and must be re-examined. Bethe and Salpeter (1957) have
shown that when retardation effects are included in the calculation of
differential photoionization cross-sections, the lowest order correction
is proportional to (|v|/c), where v is the photoelectron velocity. This
correction, however, only effects the angular distribution and vanishes
when the differential cross-section is averaged over all incident photon
directions. Higher-order corrections for retardation are proportional
to (|v|/c) as are the relativistic corrections, but these are usually
negligible even when the photon wavelength is comparable to the molecular
dimensions.

119
5.4 Computational Applications
In this section, the relative photoionization intensities for the
water and acetylene molecules are computed using the Feynman-Dyson ampli¬
tudes obtained in the second-order, diagram conserving and the diagonal
2p-h TDA, renormalized decoupling approximations. Comparative calcula¬
tions corresponding to a Mg photon source (tim = 1253.6 eV), for which
the plane wave approximation is expected to be good, and a He (II) photon
source (tioo = 40.81 eV), for which the plane wave approximation is expected
to be less accurate, are presented. These results are further compared
with intensities obtained using the frozen orbital approximation in order
to assess the magnitude of many-electron correlation and relaxation cor¬
rections. For acetylene, the dependence of intensity on photon energy
for the valence orbitals is plotted, and orbital and density difference
plots are presented and discussed.
The relative photoionization intensities for water corresponding to
a Mg photon source and a He (II) photon source are presented in Tables
13 and 14, respectively. These results were obtained using the 26 con¬
tracted Gaussian orbital basis at the experimentally determined equilib¬
rium geometry as described in detail in Section 3.5 and are scaled rela¬
tive to the 3aj intensity. As expected, the relative intensities com¬
puted for the Mg source in Table 13 compare reasonably well with those
obtained experimentally (Rabalais et al_., 1974). The orthogonal i zed
plane wave results of Rabalais et al_. (1975) are presented for compari¬
son and also correctly order the relative intensities. It is interesting
to note the larger discrepancy between the lb^ and lbg intensities in
our calculations compared to Rabalais et al_. than in the 2a^ and 3a^
intensities. Since for X-ray photons the OPW corrections are not expected

Table 13. Relative Photoionization Intensities for Water
Excited by Mg
Ka (1253.6 eV).
(2)
(2) (3)
Orbital
F0a
2(E)
SHIFT +
0PWD
Exp.*3
2a i
4.69
1.32
4.15
4.50
3.84
2.57
3a i
1.00
1.00
1.00
1.00
1.00
Ibl
0.62
0.52
0.55
0.18
0.36
lb2
0.40
0.35
0.37
0.09
0.31
aFrozen orbital approximation.
^Rabalais et al. (1974).

121
Table 14. Relative Photoionization Intensities for Water
Excited by He(II) (40.81 eV).
Orbital
F0a
(2)
2(E)
(2) (3)
Z^SHIFT +
0PWb
Exp.C
2a i
0.55
0.28
0.66
0.84
0.26
not observed
3a j
1.00
1.00
1.00
1.00
1.00
lbl
1.16
1.15
1.15
0.86
0.96
lb2
0.83
0.84
0.83
1.10
0.80
aFrozen orbital
approximation.
bRabalais
et a]_
. (1975)
cRabalais
et al
. (1974)

122
to be large, this discrepency is most likely a basis set effect. Our
basis contained d-type polarization functions on oxygen as well as p-type
polarization functions on the hydrogen atoms, whereas Rabalais et al.
have used only a minimal basis.
The relative intensities presented in Table 14 do not correctly
match the experimentally observed intensities. This is not particularly
surprising owing to the inadequacy of the plane wave approximation at
low photon energies. It is more surprising that the OPW results also
yield an incorrect ordering of the intensities. The failure of the OPW
approximation indicates the necessity for a more accurate photoelectron
wavefunction.
Many-electron correlation and relaxation corrections appear negli¬
gible in the intensities of both Tables 13 and 14. The deviations be¬
tween the frozen orbital approximation (FO), the second-order self-energy
(2)
approximation (2(E)), and the diagonal 2p-h TDA with third-order constant
(2) (3)
energy diagrams (2(E)j^jpj + 2(°°))> are small and may be attributed to
differences in the photoelectron momenta. Different ionization energies
obtained with different decoupling approximations were used to compute
the photoelectron momenta from the conservation of total energy
|lcf| = [2(m - I.P.JP (in atomic units) , (5.36)
consequently, deviations in the I.P.'s result in deviations in |K,|.
The relative intensities computed with the diagonal 2p-h TDA plus
third-order constant energy diagrams for the Mg K photon source are
plotted in Fig. 5, In order to compare the theoretical spectrum to the
experimental ESCA (Electron Spectroscopy for Chemical Analysis) spectrum
(Siegbahn et al_., 1969) which is sketched as an insert, the ionization

Figure 5. A plot of the theoretical ESCA spectrum for the valence
ionizations of water. The experimental spectrum of
Siegbahn eT (1969) is sketched in the insert.

[T'l; >j(i
il'i,: ‘NVi-i1
a1. Mi'ii
m

125
lines were given a Lorentzian width estimated from the experimental
spectrum. The major features of the experimental spectrum are well-
reproduced with the exception of the weak peak at approximately 23 eV.
This peak has been attributed to the ionization of 2a^ electrons by the
Mg K(.£3 4 satellite in the photon source (Siegbahn et ¡TL, 1969) and
therefore should not appear in the theoretical spectrum since it was
calculated for a purely monochromatic source.
A recent experimental study of relative photoionization intensities
of acetylene with various photon sources (Cavell and Allison, 1978)
motivated our examination of this molecule. As preparatory steps to the
calculation of relative intensities, Hartree-Fock and electron propagator
calculations were performed at the experimental equilibrium geometry,
R(C-C) = 2.279 a.u. and R(C-H) = 2.005 a.u. (Buenker et a_l_. , 1967), with
two different basis sets. The first basis consisted of Huzinaga's 9s,5p
primitive basis for carbon and 4s basis (unsealed) for hydrogen (Huzinaga,
1965) contracted with Dunning's coefficients (Dunning, 1970) to 4s,2p
on carbon and 2s on hydrogen (see Table 8). The complete molecular basis
consisted of 24 contracted Gaussian orbitals and yielded a Hartree-Fock
total energy of E(HF) = -76.7948 H. The second basis augmented the first
by the addition of a set of d functions on each carbon atom (oj = 0.60)
and a set of p functions on each hydrogen atom (oip = 0.75). The addition
of these diffuse, polarization functions brought the size of the basis
to 42 contracted Gaussian orbitals and yielded a Hartree-Fock total
energy of E(HF) = -76.8267 H.
Valence ionization energies were computed with the second-order
self-energy approximation and the diagonal 2p-h TDA for each basis. The
results for the 24 orbital basis are presented in Table 15 and the

126
Table 15.
Valence Ionization
(24 CGTO's).
Energies for Acetylene
Orbital
(2)
Koopmans Z(E)
(2)
^^SHIFT
Ced.a
r 1
Exp.
2ag
-
38.29
-
28.9
27.6
Shake-up
2ag
28.42
25.47
26.48
23.9
23.5
3og
18.55
16.39
17.08
16.4
16.8
11.32
11.20
11.29
10.8
11.4
2a
u
20.81
18.21
19.12
18.0
18.7
aCederbaum
et al.
(1978).
^Cavell and
Al 1ison
(1978).

127
results for the 42 orbital basis appear in Table 16. In each table the
full, nondiagonal 2p-h TDA results of Cederbaum et aj_. (1978) and the
experimental results of Cavell and Allison (1978) are included for com¬
parison. In contrast to previous calculations reported here, the ioni¬
zation energies obtained for acetylene are larger (with the exception of
the lr ionization) than the experimental values. A breakdown in the
quasi-particle picture for the 2a^ ionization which is evidenced by the
shake-up pole may account for the particularly poor results for these
ionizations. The off-diagonal 2p-h TDA contributions considerably im¬
prove these two ionization enerqies (Cederbaum et al_., 1978), however,
the shake-up energy still disagrees with the experimental value by more
than an electron volt.
Relative photoionization intensities for acetylene are presented in
Table 17 for a Mg photon source and in Table 18 for a He (II) photon
source. In both tables the intensities were scaled relative to the In
u
intensity. As for water, the intensities corresponding to the Mg
source are in reasonable agreement with experimental intensities and show
only a slight diminution when correlation and relaxation effects are
included. The OPW result of Cavell and Allison (1978) for the 2a
9
ionization in Table 17 seems unexplicably high, and the 3a ^ ionization
is not observed experimentally (Cavell and Allison, 1978).
The relative intensities for acetylene computed with the lie (II)
source exhibit better agreement with the experimental results than did
the water results. Here, only the relative intensities of the two
weakest ionizations, the 2a^ and 2au, were reversed. The OPW results
predict the correct ordering but attribute a much weaker intensity to
the 2
128
Table lfi. valence Ionization Energies for Acetylene
(42 CGTO's).
Orbital
Koopmans
(2)
1(E)
2 o
-
38.20
Shake-up
2ag
28.07
24.56
3o
18.46
16.56
3
lTru
11.16
11.13
2a
u
20.90
no results
(2)
(E)
1,1 ;SHIFT
Ced.a
Exp.b
34.45
28.9
27.6
25.73
23.9
23.5
17.24
16.4
16.8
11.24
10.8
11.4
19.47
18.0
18.7
aCederbaum et. aj_. (1978).
bCavell and Allison (1978).

129
Table 17. Relative Photoionization Intensities for Acetylene
Excited by Mg Kq (1253.6 eV).
Orbital
F0a
(2)
E(E)
(2)
z<^e^shift
0PWb
r b
Exp.
2ag
18.90
17.75
17.79
26.86
13.3
3ag
0.51
0.48
0.48
0.65
not observed
1.00
1.00
1.00
1.00
1.00
2a
u
7.74
-
7.50
11.18
9.5
aFrozen orbital approximation.
bCavell and Allison (1978).

130
Table 18. Relative Photoionization Intensities for Acetylene
Excited by He(II) (40.81 eV).
Orbita 1
F0a
(2)
2(E)
(2)
r^SHIFT
0PWb
c b
Exp.
2a
9
0.45
0.36
0.72
0.09
0.26
3ag
0.45
0.44
0.87
0.77
0.68
llTu
1.00
1.00
1.00
1.00
1.00
2au
0.29
-
0.50
0.41
0.29
aFrozen orbital approximation.
bCavell and Allison (1978).

131
The relative intensities in Table 17 computed with the diagonal
2p-h TDA and Mg source have been plotted in Fig. 6. As in Fig. 5,
the lines have been given Lorentzian widths and the experimental spec¬
trum appears as an insert. The most striking difference between these
two spectra is the absence of the intense shake-up in the theoretical
spectrum, hot only was this peak predicted to lie 7 eV higher than the
experimental peak in energy, it also yielded a relative intensity several
orders of magnitude smaller than the 2a^ intensity (at 23.4 eV). The
weak experimental peak at about 14.5 eV arises from the ionization of
2 absent in the theoretical spectrum. The 3a^ peak which should occur at
16.8 eV is not observed experimentally but can be identified as a shoulder
on the 2ou peak in the theoretical spectrum.
Figure 7 shows the dependence of the photoionization cross-section
on photon energy from 0-20(1 eV. Over this energy range, ionization from
the 1tiu orbital is predicted to be most probable, and this is verified
by the experimental data of Cavell and Allison (1978) obtained at 21.2 eV
(He ( I )), 40.8 eV (He (II)), and 151.4 eV (Zr M.,). Another interesting
feature of this figure is the slight minimum exhibited by the 2a^ curve
around 125 eV. Minima of this type in the photoionization cross-section
were first predicted by Cooper (1962) and are referred to as Cooper
minima. Cooper minima have been observed in the cross-sections for a
number of atoms by means of X-ray and ultraviolet absorption spectroscopy
(see e.g. Codling, 1976); however, at present there are no photoelectron
spectroscopic data of this type owing to the limitation of photon sources
which are available. The application of synchrotron radiation to photo¬
electron spectroscopy should soon offer a means of studying the energy

Figure 6. A plot of the theoretical ESCA spectrum for the valence
ionizations of acetylene. The experimental spectrum of
Cavell and Allison (1978) is sketched in the insert.

133

Figure 7.
A plot of the
photon energy
in the region
photoionization cross-sections versus
for the valence orbitals of acetylene
0-200 eV.


136
dependence of the orbital cross-sections in a continuous energy interval
from 0 to about 200 eV (Codling, 1973).
Orbital plots are presented for the 2o^, 30^, and Itt^ Feynman-Dyson
amplitudes in Fig. 8. Since the dominant component in each of these
amplitudes is the corresponding Hartree-Fock orbital (i.e. 2g , 3c?g, and
1ttu, respectively) the correlation and relaxation corrections are not
readily observable. In order to examine these many-electron contributions
more readily, density difference plots between the amplitudes of Fig. 8
and their corresponding Hartree-Fock orbital were made and are presented
in Figs. 9-11. Since the Feynman-Dyson amplitudes are not normalized,
it is necessary to normalize them before computing the density difference.
In all of these plots, the Hartree-Fock density is negative which means
any positive distortions imply density enhancement in the propagator am¬
plitude while negative distortions imply density diminution. Figure 9
shows a density diminution in both the C-C and C-H bonding regions with
a density enhancement in the anti-bonding region. Figure 10 shows an
enhancement in the C-C pi bonding, and Fig. 11 shows an enhancement in
the C-H bonding with a slight diminution of anti-bonding character.
It is apparent from the numerical results presented in this section
that the calculated relative photoionization intensities, at least within
the plane wave approximation, are not very sensitive to improvements in
the Feynman-Dyson amplitudes. It is likely that more accurate photoelec¬
tron wavefunctions will improve the sensitivity of these quantities but
not at the orthogonalized plane wave level. Although the OPW approxima¬
tion is more .justifiable formally, the results of Rabalais et_ aj_. ( 1974)
and Cavell and Allison (1978) do not exhibit any marked improvements to
the plane wave results in the two molecules studied here. Qualitative

Figure 8. Orbital plots for the 2au, 3c>g, and liru Feynman-Dyson
amplitudes of acetylene. Approximate scales on the
plots are ±0.72 a.u. for 2üu, ±0.28 a.u. for 3aa, and
±0.29 a.u. for liru.

138

Figure 9. A density difference plot between the 3oq Feynman-Dyson
amplitude and the 3og Hartree-Fock orbital of acetylene.
Each contour represents a density increment of about
1.5 x 10-4 atomic units.

140
m
i.i
\ - *
t'k
Ikik
H&a
’x'V'&iif
â– :U*'
/§|v
c'iAlr:.' ' ’
VMv ip^si |P
't» "if

Figure 10. A density difference plot between the liru Feynman-Dyson
amplitude and the 1ttu Hartree-Fock orbital of acetylene.
Each contour represents a density increment of about
3.6 x 10~3 atomic units.

1
zn

Figure 11. A density difference plot between the 2au Feynman-Dyson
amplitude and the 2au Hartree-Fock orbital of acetylene.
Each contour represents a density increment of about
4.4 x 10~4 atomic units.

HO D H

145
changes in the propagator amplitudes may be observed via density
difference plots; however, more quantitative comparisons must await
other computational applications.

CONCLUSIONS AND EXTENSIONS
The primary objective of this dissertation has been the investiga¬
tion of alternative decouplings which may allow more accurate calcula¬
tions of the electron propagator with less computational expenditure
than the operator product decouplings. Contrary to previous contentions
that decouplings of the propagator equations of motion and choice of
reference state averages are interrelated (Oddershede and Jorgensen,
1977), we have demonstrated that they are in fact independent approxima¬
tions, hence justifying the systematic improvement of one or the other.
We have chosen to examine the decoupling approximation while maintaining
a Hartree-Fock reference state since these reference states have a simple
theoretical representation and are efficiently generated by a number of
available computer programs. Since Hartree-Fock reference states ignore
electron correlation (by definition), the correlation and relaxation
corrections to the electron propagator must be exclusively incorporated
through the decoupling approximation.
The superoperator formalism which was originally introduced as a
notational convenience has a rich algebraic structure that has been
largely unexploited. As we demonstrate in Chapter 3, the superoperator
Hamiltonian may be partitioned and a perturbation theory may be developed
in the operator space. The perturbation expansion of the superoperator
resolvent readily yields the Dyson equation and allows an identification
of wave and reaction superoperators. The definition of these auxiliary
superoperators elucidates the parallelism between the superoperator
146

147
formalism and the diagrammatic expansion method and permits a unified
conceptual framework.
When the superoperator resolvent is approximated by an inner pro¬
jection, various choices of the inner projection manifold which corre¬
spond to different decouplings may be viewed as different approximations
to the wave superoperator in the following equation (Born and Ohrn, 1977)
[h) = W(E) |aj .
In particular, the operator product decoupling corresponds to the choice
W = I + {a^a^}
W = í + {a^} + ia+aja^}
* * * )
the moment conserving decoupling corresponds to
W = í + H
W = í + H + H2
... 5
and the diagram conserving decoupling of the Pade' type corresponds to
W(E) = 1 + T0(E)V
W(E) = I + T0(E)VT0(E)V
Alternatively, the superoperator resolvent may be approximated by
an outer projection in which case decoupling approximations correspond
to various approximations to the reaction (or self-energy) superoperator.
Two decouplings of this type which have been presented are the simple

148
diagram conserving decoupling which corresponds to
t(E) = V + VT(E) V
wi th
T(E) = T0(E)
T(E) = T0(E) + T0(E)VT0(E)
and the renormalized decoupling which corresponds to
t(E) = V + VT(E)V
with
T(E) = Tq(E) + T0(E)MT(E)
and where M is an effective potential obtained by projecting the pertur¬
bation superoperator onto a subspace of the full operator space. When
this subspace is chosen to consist of simple operator products, this
decoupling is formally equivalent to the operator product decoupling,
however, other approximations may be envisioned.
Each of the above-mentioned decoupling approximations has been
studied in detail computationally and has been evaluated on the basis
of a comparison of the propagator poles with experimental ionization
energies. The moment conserving decouplings discussed in Chapter 2 were
found to yield unacceptably poor numerical results. It is plausible
that since the moment matrices involve averages of various powers of
the full superoperator Hamiltonian, the moment expansion may not be con¬
vergent anywhere in the complex plane. If this is the case, analytic
continuation cannot be performed, and the Pade1 approximant method will
not improve convergence. In any event, it may be generally concluded

149
that the number of conserved moments is no indication of the accuracy
of this decoupling.
A more successful approach is the diagram conserving decouplings
presented in Chapter 3. A simple second-order self-energy truncation
was found to overcorrect the Hartree-Fock orbital energy estimate of
the ionization energy and yielded results that were generally too small.
The inclusion of third-order diagrams, particularly rings and ladders,is
necessary to obtain reasonable agreement with experiment. Energy shifts
resulting from the addition of diffuse, polarization functions to the
computational basis were observed to be nearly the same magnitude as the
shifts obtained from third-order ring and ladder diagrams, hence care
must be taken to insure basis saturation.
The renormalized 2p-h TDA decoupling was found to be the most satis¬
factory decoupling studied. Although only a diagonal approximation was
implemented, principal ionization energies accurate to within an electron
volt were generally obtained. Shake-up energies were less accurately
described owing to the neglect of ion state relaxation and hole-hole
correlation in the 2p-h TDA.
The numerical results of Chapter 5 indicate that the calculation of
relative photoionization intensities using a plane wave approximation
for the photoelectron is not a sensitive reflection of the quality of
the Feynman-Dyson amplitudes. The severity of the plane wave approxima¬
tion most likely obliterates the many-electron relaxation and correlation
corrections. More accurate photoelectron representations such as orthog-
onalized plane waves or Coulomb waves should afford a more accurate eval¬
uation of the Feynman-Dyson amplitudes, or alternatively, one-electron
properties may be computed using the single-particle, reduced density
matrix obtained from the spectral density.

150
There are several other possible extensions of this research that
should be mentioned. Most obvious is an application of these decoupling
ideas to the polarization propagator. Currently, decouplings of this
propagator are based on operator products. The odd or Fermion-like
operator products in the electron propagator theory must be replaced by
even or Boson-like operator products to account for the particle-conserv¬
ing nature of the polarization propagator theory, but with this minor
modification, the application of the diagram conserving and renormalized
decouplings also seems feasible.
More accurate decouplings are still needed for the electron propa¬
gator in order to better describe shake-up energies. With this goal in
mind, Cederbaum and co-workers have recently implemented the full, non¬
diagonal 2p-h TDA by exploiting spin and orbital symmetry to reduce the
dimension of the 2p-h operator subspace. Preliminary results indicate
that this approximation provides some improvement but is still inadequate.
The need for higher operator products, particularly quintuple products,
has been demonstrated by Bagus and Viinikka (1977) in Cl studies and may
necessitate the implementation of a 3p-2h renormalization.
Another attractive avenue for extension is a combination of the
2p-h TDA renormalization with the Hamiltonian partitioning of Kurtz and
Ohrn. Some work along this line is in progress by Ortiz and Ohrn (1979)
and preliminary results are very encouraging. One criticism of this
approximation is that if one interprets it as an alternative partitioning
of the superoperator Hamiltonian in which the eigenvalues of the unper¬
turbed Hamiltonian correspond to AE(SCF) energies (as in Eq. (3.23)),
then the denominators of the self-energy which are obtained from
(EI-Hq)"* (some operator product)

151
should also involve AE(SCF) energies. For example, the second-order
self-enerqy denominator would be
[E + AEjJSCF) - AE] (SCF) - AE^SCF)]'1 .
Since it is probably not feasible to perform AE(SCF) calculations for
all doublet (N-l) and (N+l)-electron states, simple approximations to
the AE(SCF) energies such as
ak>
or
might be considered.
Finally, the problem of open shell reference states has not been
considered here. The retention of a restricted Hartree-Fock reference
state in this regard will be a tremendous simplification, and the various
spin and orbital symmetry couplings between the reference state and
operator manifold may be treated with the theory of tensor operators.

2SI
(M11j>I\ A)
3 - 4 ? ~ * * + "’)(** “ * i ~ “ * * '/) r—T *-y A A '~7
(hi | |.(») wn \i> — if) ryryAAry ^
~ > - q 3 t "a)(q3 - *^3 ~ J > + ;./) r—7 ry A A ^7 b
<(v"l \‘1‘) A AAA A [
d > 3 ? t ** >) ( > - *A+.7) ry A A A A? b
<«i| |j<*<7iT".'> A.xAAA i
S qifA)( 3 ~ - J3 ~* //) f-7 r~7 A A A b
aaaaa ¡
‘ * 3 - > + .7H* * “ * + 7) A A ry A A
{’If 11{ > - B3~tA+.7)(°3-a?-ti3-Ay) A A y7 A A
ATT^/Tk--^111 >jA kAAA A
A'iii'-iX/'-'i i'i«k‘i"''nA) kAAAA [
(°5-<5-'>+3)(‘3-i5-"5+,V)
_("7 I 1 ar.i >"<-s:.rI j ¿./)(ZÜ?"|'| u I) '
XI0N3ddV

Oí)
(O)
un
(O)
(N)
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APPENDIX 2
■ 45íj
+ ;; (e. + e, + e + e , - e )
r,s,s' J
x [,í + !5-]
+ X <1111 | jr>
r,s,s1,1,1 '
x [h - !j + J5 - *5
+ - - h
r s 1 1 1 s s s
+ !s]
+ X <1 's | | jl>
r,s ,s 1,1,11
x [2 + - -
1 s rls rl rl
+ +2 -
rl s r 1 1 ' Is r
- - 2 -
+ ]
+ X <1111 |rs'>
r,s,s1,1,11
x [h - -
r s 1 1 r s 1
+ 'j + + ,?
+ 'í - l
1 1 s s s1 1 1
154

APPENDIX 3
(a|VT0(E)VT0(E)Va)
ij
r,s,s 1,1,11
+ X
r,s,s ‘,1,1
+ X
r,s,s' ,1 ,1 1
+ X
r,s,s' ,1,1
.... <.Ír.l.l.ss ' ><1 T | | jr> u >_l.
( E + e^, - es - es ,) (E + er - - e-| ,) nr 2 nr nl
+ !á - J2 + - %]
<11'|Ijr> „
(E + er - Eg - es , ME - Ej + Et + ej ! - es - e$ . ) N "l^V*
- 3j - > + !2]
<11 ' | | rs ' > p > (E “ ej er - es^(E ‘ £j zr es 1 ' el ~ eW r s
- ^n-ixn-i ,> - + %
+ 3g]
<11 ’ 11 rs' >
(E - Ej + Er'- e$)(E Ej + e-j + E-j I - es - e$ , ) ['s
- ¡5 + h + xn^ , >
- ]
s s 1
+ X
r,s ,s ' ,1,11
<1 s ,j
m.—LL^ H-ft r -
(E Er' S - V )(£ E1 ' Es - E1 1) 1 s r 1
+ + -
r 1 s r 1 1 1 s r
- - + >1
155

156
+ l
r,s,s' ,1,1'
- cryxr^ , > + cr^xr^ , > + ci-i^xn-j , xn-j > -
- ]
Te+^
<1 ' s | l,jl>
£s - ^
ej + er + £l'-cs'
J [
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BIOGRAPHICAL SKETCH
Gregory J. Born was born on July 1, 1951,in Joliet, Illinois. He
was raised in Coal City, Illinois,where he attended public schools until
his graduation from Coal City High School in May 1969. From September
1969 through June 1973, he attended Southern Illinois University at
Carbondale as an Illinois State Scholar and majored in chemistry with
math and physics minors. Throughout his senior year, he participated in
an undergraduate research program working under the supervision of the
late Professor Boris Muslin. After completing his Bachelor of Science
degree, Mr. Born enrolled in graduate school at the University of Florida.
From September 1973 to June 1976, he was a teaching assistant in the
Department of Chemistry. In February 1974, he joined the Quantum Theory
Project and has been a research assistant working under the supervision
of Professor Yngve Ohrn from June 1976 to the present. During the summer
of 1976, Mr. Born attended the Summer School in Quantum Theory held in
Uppsala, Sweden,and Dalseter, Norway. He is the co-author of four
research publications and has presented several contributed talks at
various conferences.
164

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
N. Yngve Oh
ProfessorV
Chairman
Chemistry and Physics
I certify that 1 have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Arthur A. Broyles
Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
David A. Mifha"
Professor of Chemistry dnd Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
/a£>. [
Willis B. Person
Professor of Chemistry
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
nd Chemistry

This dissertation was submitted to the Graduate Faculty of the
Department of Chemistry in the College of Liberal Arts and Sciences
and to the Graduate Council, and was accepted as partial fulfillment
of the requirements for the degree of Doctor of Philosophy.
June 1979
Dean, Graduate School

I




PAGE 1

ALTERNATIVE DECOUPLINGS OF THE ELECTRON PROPAGATOR By GREGORY J. BORN A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IIVERSITY OF FLORIDA 1979

PAGE 2

ACKNOWLEDGMENTS After twenty-one years of formal education which is culminating with this dissertation, it is impossible to individually thank every teacher who assisted me in this pursuit. The largest debt of gratitude however, is owed to my dissertation supervisor, Professor Yngve Ohrn, whose constructive criticism and constant encouragement has guided the direction of this research and my graduate education. His generous financial assistance over the years has also been greatly appreciated. I would like to thank Professor Per-Olov Lowdin for his stimulating series of lectures in quantum theory as well as the national and international contacts he has made available to me and the other members of the Quantum Theory Project through the invitation of visiting scientists and the organization of the Sanibel Symposia. My attendance at the Summer School in Quantum Theory held in Uppsala, Sweden, and Dalseter, Norway (August 1976), was made possible by monetary awards secured by Prof. Lowdin for which I am also greatly appreciative. Next I would like to thank Professor Jack Sabin, who supervised me during some preliminary investigations, and the other members of my supervisory committee for contributing their time and for occasional letters of recommendation. Without the friendships and intellectual stimulation of other members of the Quantum Theory Project whom I have known, my graduate education would not have been as enjoyable or as rewarding. To several people I am indebted for the use of various computer subroutines which I gratefully acknowledge.

PAGE 3

It is with regret that I must posthumously acknowledge my indebtedness to Professor Boris Muslin. Prof. Muslin supervised my undergraduate research in quantum theory at Southern Illinois University and was largely responsible for guiding me into this field. I would finally like to thank my parents for their constant encouragement and financial assistance. Special thanks are owed to Miss Brenda Foye for her painstaking efforts in typing this manuscript. I also wish to take this opportunity to acknowledge the Northeast Regional Data Center of the State University System of Florida for the use of their facilities to obtain the numerical results presented here and the American Institute of Physics for permission to reproduce Figure 1 and Appendix 1 from the paper by G. Born, H. A. Kurtz, and Y. Ohrn in the Journal of Chemical Physics, Vol. 68, p. 74 (1978).

PAGE 4

TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT INTRODUCTION CHAPTER 1: OPERATOR PRODUCT DECOUPLINGS 1.1 Definition, Spectral Representation, and Equation of Motion of the Electron Propagator 1.2 The Superoperator Notation and Inner Projection Technique 1.3 The Hartree-Fock Propagator 1.4 Operator Product Decoupling 1.5 Method of Solution 1.6 Analysis and Limitations of the Operator Product Decoupling CHAPTER 2: MOMENT CONSERVING DECOUPLINGS 2.1 Pade' Approximants and the Extended Series of Stieltjes 2.2 Moment Conserving Decoupling 2.3 Method of Solution 2.4 Computational Considerations and Applications . . 2.5 Evaluation of the Moment Conserving Decoupling CHAPTER 3: DIAGRAM CONSERVING DECOUPLINGS 3.1 The Diagrammatic Expansion Method 3.2 Perturbation Theory 3.3 Equivalence of the Superoperator Formalism and the Diagrammatic Expansion Method 3.4 Diagram Conserving Decoupling 3.5 Approximations and Applications 3.6 Evaluation of the. Diagram Conserving Decoupling . 10 15 18 21 23 27 31 31 36 40 43 50 54 54 58 62 68 72 82

PAGE 5

TABLE OF CONTENTS (Continued) CHAPTER 4: RENORMALIZED DECOUPLINGS Page 4.1 Renormalization Theory 88 4.2 Derivation of the 2p-h TDA and Diagonal 2p-h TDA Equations 89 4.3 Diagrammatic Analysis 95 4.4 Computational Applications and Evaluation of the Diagonal 2p-h TDA Self-Energy 102 CHAPTER 5: PH0T0I0NIZATI0N INTENSITIES 108 5.1 Introduction 108 5.2 Derivation of Computational Formulae for the Total Photoionization Cross-Section 109 5.3 Discussion of Approximations 116 5.4 Computational Applications 119 CONCLUSIONS AND EXTENSIONS 146 APPENDIX 1 152 APPENDIX 2 154 APPENDIX 3 155 BIBLIOGRAPHY 157 BIOGRAPHICAL SKETCH 164

PAGE 6

LIST OF TABLES Table 1. Contracted Gaussian Basis for Nitrogen 2. Principal Ionization Energies for the Nitrogen Molecule Resulting from the [1,0] and [2,1] Propagator Approximants 3. Contracted Gaussian Basis for Water 4. Principal Ionization Energies for Water Resulting from the [1,0] and [2,1] Propagator Approximants . 5. Rules for Constructing Self-Energy Diagrams . . . 6. Principal Ionization Energies of Water Computed with the 14 CTGO Basis 7. Basis Set Effects on the Ionization Energies of Water Computed with a Second-Order Self-Energy Approximation 8. Contracted Gaussian Basis for Formaldehyde . . . . 9. Principal Ionization Energies for Formaldehyde . . 10. Comparison of Principal Ionization Energies for Water Obtained with the Second-Order and the [1,1] Self-Energies Using the 14 and 26 CGTO Basis Sets 11. Water Results Obtained with the Diagonal 2p-h TDA and Diagonal 2p-h TDA Plus Constant Third-Order Self-Energies 12. Formaldehyde Results Obtained with the Diagonal 2p-h TDA Self-Energy 13. Relative Photoionization Intensities for Water Excited by Mg ^ (1253.6 eV) 14. Relative Photoionization Intensities for Water Excited by He (II) (40.81 eV) Page 46 48 49 51 67 74 76 78 79 83 104 106 120 121 VI

PAGE 7

LIST OF TABLES (Continued) Table Page 15. Valence Ionization Energies for Acetylene (24 CGTO's) 126 16. Valence Ionization Energies for Acetylene (42 CGTO's) 128 17. Relative Photoionization Intensities for Acetylene Excited by Mg K a (1253.6 eV) 129 18. Relative Photoionization Intensities for Acetylene Excited by He (II) (40.81 eV) 130

PAGE 8

LIST OF FIGURES Figure , Page 1. Relaxation and Correlation Errors for Each of the Principal Ionizations in the Water Molecule 6 2. A Sketch of the Energy Dependence of the Function W k (E) 26 3. A Sketch of W 2ai in the Energy Region of the Za\ Ionization for Water 85 4. Fourth-Order Self-Energy Diagrams Arising From the 2p-h TDA 100 5. A Plot of the Theoretical ESCA Spectrum for the Valence Ionizations of Water 124 6. A Plot of the Theoretical ESCA Spectrum for the Valence Ionizations of Acetylene 133 7. A Plot of the Photoionization Cross-Sections Versus Photon Energy for the Valence Orbitals of Acetylene ... 135 8. Orbital Plots for the 2a u , 3a g , and 1tt u Feynman-Dyson Amplitudes of Acetylene 138 9. A Density Difference Plot Between the 3o g Feynman-Dyson Amplitude and the 3a g Hartree-Fock Orbital of Acetylene . 140 10. A Density Difference Plot Between the 1tt u Feynman-Dyson Amplitude and the Tmj Hartree-Fock Orbital of Acetylene . 142 11. A Density Differenct Plot Between the 2a u Feynman-Dyson Amplitude and the 2o u Hartree-Fock Orbital of Acetylene . 144

PAGE 9

Abstract of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ALTERNATIVE DECOUPLINGS OF THE ELECTRON PROPAGATOR By Gregory J. Born June 1979 Chairman: N. Yngve Ohrn Major Department: Chemistry Several alternative decouplings of the electron propagator are investigated in this dissertation in an attempt to derive more accurate and more tractable computational schemes for extracting the physical information contained in the electron propagator. When the electron propagator is defined as a single-time Green's function, the decoupling approximation and the choice of reference state average are shown to be independent approximations, and the use of uncorrelated, Hartree-Fock reference states is advocated. The derivation of each decoupling approximation utilizes the superoperator formalism and emphasizes elementary algebraic manipulations. In Chapter 1, operator product decouplings are reviewed and critically discussed. In Chapter 2, the moment conserving decouplings which consist of Pade' approximants to the propagator moment expansion are investigated. In Chapter 3, a partitioning of the superoperator Hamiltonian is invoked, and a perturbation expansion of the superoperator resolvent is developed. This IX

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development leads straightforwardly to the derivation of the Dyson equation and permits an identification of wave and reaction superoperators. Two types of diagram conserving decouplings are then examined, and equivalences with the diagrammatic expansion method are demonstrated. Finally in Chapter 4, renormalized decouplings are considered, and the two particle-one hole, Tamm-Dancoff approximation is specifically derived and investigated. In each of the first four chapters, the decoupling approximations are evaluated on the basis of computational applications in which the propagator poles are compared to experimentally determined ionization energies for several molecules. In order to avoid a possible bias with this evaluation criterion, the quality of the Feynman-Dyson amplitudes is examined in Chapter 5 via the calculation of relative photoionization intensities. The four decoupling approximations are finally summarized as various approximations to the wave and reaction superoperators, and several extensions of these investigations are proposed.

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INTRODUCTION Since subatomic particles are beyond the limit of human sensory perception, our knowledge of atomic structure is based on the interpretation of measurements with auxiliary probes. The accumulation and interpretation of data from these types of measurements have led to the conception and axiomatization of quantum theory. Using the calculus of this theory, quantum mechanics, there is evidence to believe that it is possible, at least in principle, to calculate the statistical result of any experimental measurement. Unfortunately, the mathematical complexity of this calculus precludes exact solutions for all but a few, relatively trivial applications; consequently, the predictive value of the theory is limited. Owing to this limitation, one aspect of current theoretical research involves the formulation and evaluation of accurate mathematical approximations which are relevent to the interpretation of specific experiments . With the development of photoelectron spectroscopy (Turner et al., 1970, Siegbahn et al_. , 1969), photoionization has become an extremely useful probe of atomic and molecular structure and has stimulated much theoretical interest (Cederbaum and Domcke, 1977, and references therein). In the photoionization experiment, light is shone on an atomic or molecular sample and the kinetic energy of the ionized electrons or photoelectrons which are ejected is then analyzed. From energy conservation, the binding energies of the photoelectrons may be deduced.

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An ab initio, theoretical interpretation of photoelectron spectra requires the calculation of ionization energies. These calculations reflect fundamental assumptions about the complex nature of the manyelectron interactions that occur in atoms and molecules and may be performed at various levels of sophistication. One conceptually simple scheme is the Hartree-Fock self-consistent field (HF-SCF) method (see e.g. Pilar, 1968). In this method, an orbital energy is calculated for each electron in an N-electron system by assuming that that electron interacts only with an average electron density formed by the remaining N-l electrons. By thus averaging out the instantaneous electron-electron interactions, the original N-electron problem is reduced to N one-electron problems. The negative of the orbital energies obtained in this calculation can then be related to ionization energies via Koopmans' theorem (Koopmans, 1933). Ionization energies obtained at the Hartree-Fock level of approximation are rarely accurate and occasionally predict even the wrong sequence of ionization. In order to obtain more accurate ionization energies, the electron-electron interactions must be treated more realistically. In the Hartree-Fock approximation, the N-l electrons of the ion state are assumed to be "frozen" at the same energies they had in the ground state. Conceptually, each electron screens to some extent the electrostatic attraction between the positively charged nuclei and all the other electrons in the system. As can be easily rationalized, this screening is most effective for deep-lying or core electrons which have a high probability density near the nucleus than for more diffuse, valence electrons. Nevertheless, if one electron is ionized, all the others should experience a stronger nuclear attraction and will contract

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producing a lower total energy. This rearrangement defines the relaxation energy, and it may be easily incorporated in the ionization energy calculation. Performing separate Hartree-Fock calculations for both the ground and ion states and obtaining a total energy by adding orbital energies with corrections for the overcounting of interelectronic repulsions, an improved ionization energy can be obtained by subtracting total energies. This level of approximation is known as the AE(SCF) method (Bagus, 1965) and generally yields reliable core electron ionization energies. The remaining discrepency between the AE(SCF) ionization energies and the ionization energies obtained from the exact solution of a nonrelativistic, many-electron formulation can be defined as the correlation energy. This correction arises from the tendency of any pair of electrons in an atom or molecule to correlate their motion so as to minimize the electron-electron repulsion. Electron correlation can be conceptualized as various virtual scattering events between bound electrons in both the Nand (N-l)-electron systems. The simplest of these virtual processes is a particle-hole excitation in which one bound electron absorbs a virtual photon emitted by another electron and is excited from its original Hartree-Fock orbital to a more diffuse orbital of higher energy. A hole or vacancy is simultaneously created and propagates in the system. At some later time, the excited electron may decay reemitting the virtual photon which may then be reabsorbed by the first electron. Adopting the convention that holes propagate backwards in time while electrons propagate forward in time, a particle-hole excitation in the (N-l)-electron ion state can be diagrammatical ly represented as

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where the dotted lines represent the virtual photon exchanges. Although diagrams of this type represent quasi-classical, virtual processes which are not experimentally observable, they do provide a conceptual model and, as we will later see in Chapter 3, categorize specific algebraic expressions that will be derived to calculate correlation energy corrections. Finally, it should be noted that since the electron motions will be correlated to different extents in the N-electron ground state and (N-l)-electron ion states, it is not possible to predict, a priori , the effect of this correction on the ionization energies (see Fig. 1). One many-electron formulation which provides a systematic procedure for incorporating both relaxation and correlation corrections into the calculation of ionization energies and which is the basis of this investigation, is the non-relativistic, single-particle Green's function or electron propagator (Linderberg and Ohrn, 1973). This method has the advantage of yielding ionization energies directly unlike other manyelectron formulations which necessitate the total energy calculation for both ground and ion states and which yield the ionization energy as the difference. In the latter methods, significant loss of accuracy is inherent in the subtraction of two, nearly equal total energies to obtain a much smaller, ionization energy. Care must also be taken to avoid disparate levels of approximating electron correlation in the calculation of each different state.

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PAGE 16

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The electron propagator can be written as a function of the space and spin coordinates of one electron and of a complex energy variable. For each bound electron, this function describes mathematically all of the complicated many-electron interactions between that electron and the remaining N-l electrons. This function also contains information about the interaction of an additional electron with the N bound electrons if this electron were to be added to the system in any of several possible orbitals. In a discrete basis representation of the electron propagator, this information manifests itself as simple poles or singularities at those values along the real energy axis which correspond to electron ionization energies or electron affinities, that is, electron detachment or attachment, respectively. The residues at these poles yield the singleparticle reduced density matrix from which the N-electron ground state average of any one-electron operator may be calculated or which may be related to transition probabilities for electron detachment or attachment provided some description of the removed or added electron is included. The electron propagator can be calculated in several ways. The procedure adopted here is derived from the electron propagator equation of motion, but one aspect of this investigation will demonstrate the formal equivalence between this method and the diagrammatic expansion technique. The equation of motion relates the electron propagator to the more complicated two-particle propagator. This two-particle propagator also satisfies an equation of motion which relates it to the threeparticle propagator, and so on. This hierarchy of equations finally terminates with the N-particle propagator, but in order to make any practical calculation, this hierarchy must be approximated at some lower level, generally by expressing an M-particle propagator in terms of an

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(M-l)-particle propagator. This approximation is called decoupling the equations of motion and is not unique. The accuracy of calculated ionization energies and the computational effort in obtaining them depend critically on the decoupling. This investigation proposes and evaluates several alternative methods. In Chapter 1, the electron propagator is formally defined, and its equation of motion is derived. After an introduction of the superoperator formalism, the electron propagator is approximated by an inner projection, and the decoupling problem is studied in terms of the selection of an inner projection manifold. The remainder of this chapter discusses the computational procedure for solving the propagator equations and presents a critical evaluation of the operator product decoupling. Chapter 2 describes some general aspects of the Pade' approximant method and its application to the calculation of the electron propagator. Owing to the conservation of various moment matrices in the propagator equation of motion these decouplings are known as moment conserving decouplings (Goscinski and Lukman, 1970). Numerical results obtained from the [1,0] and [2,1] Pade' approximants are presented and discussed. A partitioning of the superoperator Hamiltonian and a perturbation expansion of the superoperator resolvent in the operator space is developed in Chapter 3. A superoperator Dyson equation is derived and wave and reaction superoperators are identified in analogy with ordinary resolvent operator techniques. Truncations of the wave superoperator operating on simple annihilation and creation operators are shown to yield inner projection manifolds that result in Pade' approximants to the self-energy. These Pade' approximants conserve various orders of the perturbation expansion for the self-energy and are therefore

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categorized as diagram conserving decouplings. Two approximations based on this decoupling scheme are discussed and evaluated. Chapter 4 is devoted to renormalized decouplings. These decouplings sum certain types of self-energy diagrams to all orders. The two-particle, one-hole Tamm-Dancoff approximation (2p-h TDA), which sums all ring and ladder diagrams, is explicitly derived and discussed in terms of the superoperator formalism. The diagonal 2p-h TDA previously proposed by other authors (Cederbaum, 1974, Purvis and Ohrn, 1974) is re-examined and is shown to neglect certain diagonal contributions. Both approximations are analyzed diagrammatical ly, and numerical results are presented and discussed. The evaluation of each decoupling approximation in the first four chapters is ultimately based on a comparison of propagator poles to experimental ionization energies. Chapter 5, on the other hand, attempts to corroborate this evaluation criterion by an examination of the quality of the Feynman-Dyson amplitudes. This is indirectly accomplished via the calculation of relative photoionization intensities and their comparison with experimental data. The requisite equations for the photoionization cross-section are derived in terms of the Feynman-Dyson amplitudes, and the most critical approximations are discussed. Finally, numerical results are presented and are also discussed.

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CHAPTER 1 OPERATOR PRODUCT DECOUPLINGS 1 • 1 Definition, Spectral Representation, and Equation of Motion of the Electron Propagator The electron propagator is most commonly defined as a double-time Green's function which involves an exact N-electron ground state average of a time-ordered product of electron field operators, i/j ( x , t ) and ^ + (x',t') (Linderberg and Ohrn, 1973). These field operators are generally expressed in the Heisenberg representation, >Mx,t) = exp(iHt) ijj(x,0)exp(-iHt) (1.1) and are functions of the combined space-spin and time coordinates of the electrons. The operator, H, in the exponentials is the N-electron Hamiltonian. The field operators 4>(x,t) and i|V(x',t') have the property of annihilating and creating, respectively, an electron at the space-spin and time coordinates, x,t (x',t'). Letting k-> denote an exact eigenstate of the N-electron Hamiltonian, |rf _1 > and \^ +l > denotinq exact eigenstates of the (N-l)and (N+l)-electron, ion Hamiltonians, these properties are expressed as ^(x,t)|^> = l c. expI-KE^ E^ _1 )t> |v F ^ | 1 > (i.2) and * + (x\t')|^> = £ Cj exp.{-1(Ej +1 E5 J )t'}|^ +1 > . (1.3) 10

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11 If the N-electron Hamiltonian is time independent, it is easy to show that the double-time Green's function depends only on the time difference, t-t' (Fetter and Walecka, 1971). In all practical calculations, however, this definition proves to be a severe restriction since the exact Nelectron ground state is rarely known and an approximate ground state average is usually employed. With an inexact ground state average, the electron propagator will depend on both t and t'. To avoid this restriction, it is possible to define the electron propagator as a single-time Green's function by choosing t' equal to zero (Simons, 1976), <<^ + (x' ,0) ; iHx,t)» = -ie(t) +i9(-t)+(x\0) *(x,-t)> • (1.4) This definition insures the dependence on only the time difference even when the ground state average is inexact. The brackets, <...>, in Eq.(1.4) represent an average which may be either a pure-state average or an ensemble average, <>= I P K . (1.5) K To elucidate the analytic properties of the electron propagator, it is convenient to derive the spectral or Lehman representation (Linderberg and Ohrn, 1973). This representation is obtained by first expanding the eigenstates, |k>, in terms of the exact N-electron eigenstates, |H'^>, l<> = * c Ki |^> . (1.6) i Using resolutions of the identity in terms of the exact (N-l)and (N+l)electron eigenstates, Eq .(1.4) can be written:

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12 <«J> + (x',0); i|;(x,t)» = Z Z P K c Ki c K { i 6 ( t ) <^^ J |^ ( x , t ) \^ +1 > k i,j,k x <^ + V(x',0)|^ N > + ie(-t)<^| ! { ) + (x 1 ,0)|^1 ><^1 |^(x,-t)|^>} .(1.7) Explicitly introducing the time dependence of the field operators, Eq .(1.7 becomes «f i "(x',n); i|)(x,t)» = Z Z P c . c . k i,j,k ] J x {-ie(t)<^V(x,0)|^ +1 ><^ +1 |^(x , ,0)|^ N >exp[-i(E^ 1 -E5 , )t] + ie(-t)<^|^ t (x , ,C)|l'^ 1 >exp[i(E5 , -E^ 1 )t]} . (1.8) Since the calculation of ionization energies (electron affinities) from the electron propagator will be treated as a transition between stationary states of the N-electron ground state and N-l (N+l)-electron ion states, it is convenient to Fourier transform Eq.(1.8) into an energy representation, «^ + (x'); 4j(x)» e = f «^ + (x',0)^(x,t)»exp(iEt)dt (1.9) -00 which yields the spectral representation: * r f-il^( x )fn(x' ) «^ + (x'); iKx)» c = lim Z Z P c.c.t ^ "k u i + gjk( x )9kj( x ') | E-tEj-EJJ" 1 ) in ; ^10 )

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13 where f.^x) = <^|tHx,0)|^ +1 > (1.11) and g. k (x') = <^|^t( x j0) | • N-1> (1-12) are referred to as the Feynman-Dyson amplitudes. From Eq.(l.lO) it is observed that the electron propagator has simple poles along the real energy axis corresponding to the difference between exact eigenvalues of the N-electron Hamiltonian and the (N+l)-electron Hamil tonians. The poles of this function, therefore, have a physical interpretation as ionization energies and electron affinities. Since atomic and molecular computations are most conveniently performed in a Hilbert space, we introduce a complete, orthonormal set of one-electron spin orbitals, {u n -(x)}. In this basis, the electron field operators are represented by the expansion, (x,t) = I a^tju^x) (1.13) i which, with the expansion of the adjoint, defines the spin orbital annihilation and creation operators, a^t) and aj(0). At equal times, these operators satisfy the usual anti-commutation relations, [a i ,a+] + = &.. (i.i4) [a-,aj] + = [a i ,a j ] + = (1.15) In this discrete representation, the causal electron propagator can be written «ff-(x',0); ip(x,t)» = T. u*(x')«at(0); a.(t)»u i (x) (1.16) i,j J J where

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14 «at(0); a i (t)» = ie(t)
+ ie(-t) . (1.17) Although a computational scheme for obtaining ionization energies and electron affinities could now be established from the spectral resolution, it is more convenient to develop a scheme based on the equation of motion (Linderberg and Ohrn, 1973), i ft «at(0); a.(t)» = 6(t) + «at(0); [a i (t),H]_», (1.18) which follows directly from Eq. (1.17). The quantity «a ; .(0); [a.,H] » is a two-particle propagator, and the N-electron Hamiltonian appearing in the commutator has the following Hilbert space representation: H= S h ala + \ I aV.a, a (1.19) r,s rs r s r,r',s,s' r r s s where h rs = f u*(l)[->,V 2 (l)-Z ^-]u s (l)dT 1 (1.20) ' a la and = ^ u*(l)u* 1 (2)r^(l-P 12 )u s (l)u s ,(2)dT 1 dT 2 (1.21) With a notation similar to that used in Eq. (1.9), the energy transforms of the various quantities in Eq. (1.18) can be defined, e.g. «at; a i » E = «at(0); a.(t)» exp(iEt)dt . (1.22; -00 Substituting the inverse transforms: i ft «aj(0);a.(t)» = ^[ E«at; ai » E exp(-iEt)dE, (1.23)

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15 oo 6(t)
= ~ j <[a i ,at] + >exp(-iEt)dE, (1.24) and «at(0); [a.(t),H]_» = ij «at(0); [a. (t),H]_» E exp(-iEt)dE, (1.25) into Eq. (1.18) , we obtain 00 -M t t t 2tt J {E«a.;a.» E <[a.,a '.] > «a ;[a,,H]_» F } exp(-iEt)dE = 0.(1.26) _oo « 'J J ' From the general properties of Fourier transforms, it can be shown (Morse and Feshbach, 1953) that Eq.(1.26) implies E«at,a.» E = <[a.,at] + > + «at; [a 1 ,H]_» £ . (1.27) which represents the energy transform of the equation of motion. The iteration of this equation yields N coupled equations relating the singleparticle (electron) propagator to each of the higher-particle propagators. Successive substitution of these more complicated propagators back into Eq.(1.27) yields «at ;ai » E = E" 1 <[a i ,at] + > + E" 2 <[ [a. ,H]_,aT] + > + E" 3 <[[[a.,H]_,H]_,a^'] + > + . . . . (1.28) 1.2 The Superoperator Notation and Inner Projection Technique The use of superoperators has antecedents in the work of Zwanzig (1961) and Banwell and Primas (1963) in statistical physics and was introduced into atomic and molecular propagator theory by Goscinski and Lukman (1970). As a notational simplification, the definition of a superoperator

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16 Hamiltonian and identity, H and I, provides a convenient representation of the nested commutators appearing in Eq. (1.28). More formally, this notation provides a connection with the time-independent resolvent methods introduced into many-body theory by Hugenholtz (1957). The superoperator Hamiltonian and identity are defined to operate on the spin orbital annihilation and creation operators through the relations and Ha. = [a i ,H]_ (1.29; la. = a. . (1.30; Powers of the superoperator Hamiltonian are defined by successive application of this superoperator, i.e. H 2 a.j = H[a.,H]_ = [ [a. ,H]_,H]_, (1.31) and will always yield linear combinations of odd (Fermion-1 ike) products of the simple field operators, a. and a'. This set of all Fermion-like operator products, {X.}, forms a linear space and supports a scalar product defined by (Xj|X.) = Tr{p[X.,x]] + } (1.32) where p is a normalized, but otherwise arbitrary, density operator corresponding to the N-electron ground state average of the electron propagator. Using the preceding definitions and notation, Eq .(1 .28) can be rewritten as «at; a i » E = G(E) ij = E" 1 (a j .|a i ) + E" 2 (a j .|Ha j ) + E" 3 (a.|H 2 a 1 ) + . . . . (1.33)

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17 Collecting all annihilation operators in a row matrix and all creation operators in a column and formally summing the expansion in Eq. (1.33), the matrix equation of motion for the electron propagator becomes: G(E) = (a|(Ei-H) _1 a). (1.34) The superoperator resolvent in Eq. (1.34) can now be represented in closed form by a matrix inverse using the inner projection technique (Lowdin, 1965, Pickup and Goscinski, 1973). Introducing a projection operator, = IWIirVl 3 (1.35) where 0=0 and 0=0, the inner projection of a positive definite, self-adjoint operator, A, is given by A' = K 2 6 f\ h ; A > . (1.36) Making the substitution ID = A" % |h) , (1.37) the inner projection of Eq. (1.36) becomes (Bazley, 1960) A' = |jl) (ill A" 1 J h)~ 1 (h L | (1.38) and satisfies the operator inequalities (Lowdin, 1965) < A' < A . (1.39) Since the superoperator resolvent in Eq. (1.34) is an indefinite operator, it is not valid to discuss an inner projection of the type in Eq . (1.36). Equation (1.38) however, which does not contain A 2 , is still an acceptable definition of the inner projection provided A is nonsingular. Using this definition for an indefinite operator, the equality in Eq. (1.39) will still hold when h^ is complete, but the bounding properties will now be lost with incomplete manifolds. Using the Bazley inner

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18 projection, the electron propagator has the following form G(E) U|]l)(h.| (EI-H)h)~ 1 (_h|a.) (1.40) in which the decoupling problem has now been transformed into the problem of choosing an appropriate inner projection manifold. 1.3 The Hartree-Fock Propagator Before proceeding to formulate more sophisticated decoupling schemes, it is convenient at this point to recapitulate the approximations underlying all propagator calculations and to demonstrate the algebraic manipulations which are involved by examining one simple decoupling in some detail. Implicitly assuming the clamped nuclei and non-relativistic approximations, there are basically three additional approximations involved in any scheme for computing the electron propagator. The first is the truncation of the complete (infinite) set of spin orbitals, {u.j (x)}, to some finite subset. This approximation is also characteristic of the more conventional wavefunction formulations and has received considerable attention. Standard basis sets of various sizes and qualities are available in the literature (Roetti and Clementi, 1974, Huzinaga, 1965, Dunning, 1970, Dunning and Hay, 1977). The second approximation is the specification of the N-electron ground state average or equivalents, a density operator (Eq. (1.32)), in terms of which the electron propagator is defined. The final approximation is the specification of the inner projection manifold or the actual decoupling of the equations of motion. The simplest approximation to the inner projection manifold, h, in Eq. (1.40) is just the set, {a^.}, of simple field operators. With this choice, Eq. (1.40) simplifies to

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19 G(E) = (ll(EI-H)^)" 1 (1.41) since (a|a) = I (1.42) which can be verified by evaluating a specific matrix element: (a^la.) = Tr{p[a.,at] + } = 6. .Tr{p} = <5.. . (1.43) One particularly convenient density operator, which corresponds to an independent particle, ensemble average, is the grand canonical density operator (Abdulnur et al_. , 1972, Linderberg and Ohrn, 1973), p = n [1 + (2 1) a[a k ]. (1.44) This density operator yields the following results for various operator averages: Tr{pa r a s } = 6 rs (1.45) Tr{pa a, a ,a } = [6 8 , ,-6 ,6 ,] (1.46) r r s s l rs r s rs sr J r r \->--^ u / and reduces to a pure state average when occupation numbers, , of zero or one are chosen. Considering the i,j-th matrix element of the electron propagator, G(E). . = [E6. . (a.lHa.)]" 1 , (1.47) the remaining operator scalar product, (a . 1 1 la • ) , can be evaluated by first operating with the superoperator Hamiltonian (Eq. 1.29) Ha i = [a r H]_ = l h is a s + h I a^,a ,a , (1.48) s r ' , s , s ' " then anti -commuting with a. (Eq. 1.32)

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20 (a^ | Ha i ) = h.. + I _ Tr{paJ l a s ,}. (1.49) Using the grand canonical density operator to evaluate the trace (Eq. 1.45), we obtain (a j -|Ha i ) = h^ + £ . (1.50) The particular basis of simple field operators, a i = I x ikV (1.51) which satisfies the equation Ha i = e i a i (1.52) diagonalizes the matrix, (ajHa), and must be obtained self-consistently. This is an equivalent statement of the conventional Hartree-Fock procedure since the transformation in Eq. (1.51) also defines the canonical HartreeFock spin orbitals U i = I X ikV (I-") k The eigenvalues, c, are the Hartree-Fock orbital energies. Substituting these results into Eq. (1.47), we obtain G(E).. = (E-e^-^j (1.54) for this simple decoupling scheme. The poles of this function occurring at E = e. correspond to the Koopmans' theorem (Koopmans, 1933) approximation to the ionization energies. Because of the analogy between this decoupling and the conventional Hartree-Fock procedure, Eq. (1.54) is referred to as the Hartree-Fock propagator and will constitute the starting point for more exact approximations.

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21 1.4 Operator Product Decoupling As was pointed out in the introduction, the Koopmans' theorem approximation to ionization energies is frequently unreliable and necessitates the incorporation of many-electron relaxation and correlation corrections. The simple decoupling scheme which yielded the Hartree-Fock propagator in the preceding section can be extended to incorporate relaxation and correlation by extending the inner projection operator manifold, h_, in Eq. (1.40). One extension of this manifold, proposed by Pickup and Goscinski (1973), is the union of all operator subspaces containing different Fermion-like products of simple field operators: h = {V U{ 4Vm }U{ 4 a k a l a m a n }U • ^55 ) In terms of the equations of motion, Eq. (1.28), this type of decoupling is equivalent to expressing a higher-particle Green's function in terms of lower ones and was originally discussed in the context of atomic and molecular theory by Linderberg and Ohrn (1967). The formulation of explicit electron propagator approximations with this extended operator product manifold is simplified by the use of the partitioning technique (LSwdin, 1962). Having already derived an expression for the Hartree-Fock propagator with a manifold consisting of simple field operators, it is convenient to make the partition h = aUf (1.56) where a is the subspace of simple field operators and f represents the orthogonal complement containing all higher, Fermion-like operator products. This partitioning is imposed through the relations

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22 (a | a) = (l\f) = 1 , (1.57) (ill) = (Hi) = (1.58) and leads to a blocked matrix equation for the electron propagator: AaJ(EI-H)a) -(a|Hf) G(E) = (10) A ~ 7 ) [ " | . (1.59) \^-(f|Ha) (f|(EI-H)f; Solving for the upper left block of the inverse matrix, an equation for G_ (E) is easily obtained G _1 (E) = (aJ(EI-H)a) (a | Hf ) (f | (EI-H)f ) _1 (f | Ha) . (1.60) The first term on the right hand side of Eq. (1.60) is the inverse of the Hartree-Fock propagator and the second term represents the relaxation and correlation corrections to the Koopmans' theorem ionization energies. Based on the resemblance of Eq. (1.60) with the Dyson equation derived in quantum electrodynamics (Dyson, 1949), the Hartree-Fock propagator can be identified as a zeroth order, i.e. uncorrected, approximation to G(E) while the remaining term is identified as the self-energy, G _1 (E) = G^(E) i(E) . (1.61) A number of approximations to G(E), based on different choices of the operator manifold in Eq. (1.55), have been reported in the literature. Pickup and Goscinski (1973) chose their manifold to consist of singleand triple-operator products and replaced the superoperator llamiltonian in the self-energy by the Fock superoperator defined by FX = [X,F]_ ; F = Z e^ . (1.62)

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23 This approximation was applied to the calculation of ionization energies for helium and nitrogen by Purvis and Ohrn (1974) and was later extended to include the full superoperator Hamiltonian (Purvis and Ohrn, 1975a). Redmon et al_. (1975) have derived an approximation which includes single-, triple-, and quintuple-operator products in h and have computed the ionization energies of neon. Finally, several approximations have been reported using an inner projection manifold of single and triple products in conjunction with correlated reference states. (Purvis and Ohrn, 1975b, Jtfrgensen and Simons, 1975). 1.5 Method of Solution The solution of Eq. (1.60) consists of finding the poles and FeynmanDyson amplitudes of the electron propagator and writing a spectral representation similar to that of the exact propagator in Eq. (1.10). The procedure for obtaining the spectral representation from the Dyson equation has been discussed by Layzer (1963) and Csanak et a]_. (1971) and begins with a solution to the energy dependent eigenvalue problem: L(E)i(E) = i(E)W(E), (1.63) where L(E) = (a [ Ha) + E(E) , (1.64) £(E)£ + (E*) = 1 , (1.65) and W(E) is a diagonal eigenvalue matrix. Expanding in terms of the eigenfunctions, £(E), G(E) assumes the form G(E) = i(E)[Ei-W(E)]"V'(E) , (1.66)

PAGE 34

24 and the poles are those values of E satisfying the equation, E k = W • (1-67) The energy dependence of the eigenvalues, W.(E), is sketched in Fig. 2 which shows that the poles occur at the intersections of these curves with a line of unit slope passing through the origin. When the inner projection manifold is energy independent, the slopes of the W,(E) curves are always negative since 3E 2(E) = -(a|Hf)(f|(EI-H)f)2 (f|Ha) , (1.68) and the number of propagator poles between any pair of self-energy poles is equal to the number of basis functions in that symmetry (Purvis and Ohrn, 1974). From the spectral representation, it was noted that the exact propagator has only simple poles, and it is easily shown that the residues at the poles are precisely the Feynman-Dyson amplitudes. Assuming that the approximate propagator in Eq. (1.60) also has only simple poles, the residues can be obtained from elementary residue calculus as lirn (E-E k )G ij( E) = ^(E^^) (1.69) where F k 1= t 1 -dTVE^E, • (1-70) k According to the Mi ttag-Leffler theorem (Mi ttag-Leffler, 1880), the electron propagator can now be written as

PAGE 35

Figure 2. A sketch of the energy dependence of the function WijE) between self-energy poles (indicated by vertical dashed lines). Propagator poles occur at the intersections of these curves with a line of unit slope.

PAGE 36

26

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27 r * (e H* (e ) G u (E) = I (E-il k ) J (1 71 : which has the form of the spectral representation. 1.6 Analysis and Limitations of the Operator Product Decoupling In order to analyze approximate electron propagator expressions as obtained in Eq. (1.60), it is necessary to compare the propagator poles with the exact energy differences between the corresponding Mand N-l (N+l)-electron states. This analysis can be performed, in principle, in one of two ways. Since a full configuration interaction (CI) calculation will yield the exact total energy to any finite dimensional, model problem, the total energies of the Nand N-l (N+l)-electron state could be calculated and then subtracted to yield the ionization energy (electron affinity). A full CI, however, is not practical except for systems well described by small basis sets (<10) because the number of configurations in the CI expansion, given by Weyl's formula (Shavitt, 1977), increases roughly as N" (2Me/N) where M is the number of electrons, M is the size of the spin orbital basis, and e is the constant 2.718. Furthermore, the CI solution, which is expressed as a determinental expansion, is not readily amenable to detailed analysis. On the other hand, a perturbation expansion of the Nand N-l (N+l)-electron total energies also yields the exact solution (in a non-zero region of convergence), but in addition, allows an order by order analysis of the total energy contributions in terms of orbital energies and two-electron integrals. Using Rayleigh-Schrodinger perturbation theory (RSPT) to represent the Nand (N-l)-electron states through second order, Pickup and

PAGE 38

28 Goscinski (1973) derived a second-order electron propagator expression for the energy difference. Since a Hartree-Fock self-consistent field solution was assumed as the unperturbed problem, the orbital basis sets of the Nand (N-l)-electron states are different. Before the total energy expressions may be subtracted, therefore, the orbitals of the (N-l)electron problem must be expanded in terms of the N-electron orbitals. This procedure has been extended by Born et aj_. (1978) through third order, and the resulting third-order self-energy is listed in Appendix 1. Each third-order term in Appendix 1 is characterized by a diagram and may be alternatively obtained using diagrammatic techniques, but for pedagogic reasons this discussion will be deferred until Chapter 3. Although the results of Purvis and Ohrn (1974, 1975a) and Redmon et a]_. (1975) represent significant improvements to the Koopmans' theorem and AE(SCF) approximations for the ionization energies they computed, the operator product decoupling was demonstrated to have certain computational and formal limitations or ambiguities. Computationally, the most severe limitation is the dimension of the inverse matrix in the matrix product of Eq. (1.42). The dimension of this matrix increases rapidly with the size of the spin orbital basis and prohibits an exact inversion for all but the smallest basis sets. In contrast to the CI matrix where only the lowest few eigenvalues of each symmetry are usually computed, the inversion of this matrix requires all the eigenvalues and eigenvectors. As an illustration of the size problem, when £ consists of only triple products, the dimension is roughly proportional to NM(M-N) where N is the number of electrons and M is the size of the spin orbital basis. This limitation necessitates the approximation of the inverse matrix in diagonal or near-diagonal form (Purvis and Ohrn, 1974).

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29 The formal limitations of the operator product manifolds involve certain ambiguities when extending the decoupling approximation and the difficulty in performing an order analysis. Although Manne (1977) has proven that the set of all Fermion-like operator products forms a complete set, the extension to higher operator products is not consistent with an order by order extension of the perturbation analysis. In fact, Redmon et al_. (1975) have suggested that perhaps some quintuple products should be preferentially included before all triple products. This, they felt, was particularly important in describing shake-up processes, i.e. ionization plus a simultaneous excitation of the (N-l)-electron ion. This observation has recently been confirmed by Herman et aJL (1978) in calculations employing the closely related equation of motion (EOM) method (Rowe, 1968, Simons and Smith, 1973). As mentioned in Section 1.3, correlation corrections may be included in either the density operator or the inner projection manifold. This dichotomy leads to another ambiguity: should larger operator products be chosen to extend the decoupling or a more highly correlated reference state? It has been shown by J0rgensen and Simons (1975) that in order to obtain a decoupling approximation correct through third order, an inner projection manifold consisting of single and triple products must be chosen as well as a reference state which includes all single and double excitations. Unfortunately, this combination of approximations makes the order analysis unnecessarily complicated, as we will later show in Chapter 3. Finally, the Hermiticity problem should be mentioned. With a density operator that commutes with the Hamil tonian, the Hermiticity of the superoperator Hamil tonian can be expressed as

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30 (X i |HXj) = (HX-IXj) = (Xj |HX 1 ) . (1.72) With density operators that do not commute with the Hamiltonian, however, Eq. (1.72) is generally not satisfied and leads to both computational and formal complications. This problem has been studied by Nehrkorn et a]_. (1976) who observed computationally that the non-Hermitian terms which arise when the density operator is correlated to first and second order in RSPT are cancelled when the reference state was improved to second and third order respectively. A general proof was given by * Linderberg which states that the Hermiticity error is of order n+1 when the reference state is correlated through order n. In the following chapters, alternative decoupling procedures will be proposed and investigated with the intention of remedying the various limitations inherent in the operator product decoupling as discussed here, yet which retain a quantitative description of ionization processes, 'See Ref. (14) in Nehrkorn et al. (1976).

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CHAPTER 2 MOMENT CONSERVING DECOUPLINGS 2.1 Pade' Approximants and the Extended Series of Stieltjes The evaluation of special functions assumes a central role in applied mathematics. A large number of these functions, from the simple trigonometric and exponential functions to the more complex, hypergeometric functions and Green's functions, have power series expansions. Their evaluation, therefore, consists of summing the corresponding series expansion. When the series is slowly convergent or when only a limited number of expansion coefficients are known (as e.g. through perturbation theory), it may not be practical, or even possible, to evaluate the series term by term until a desired accuracy has been achieved. In these cases, optimal approximations based on a limited number of expansion coefficients are sought. This general problem was first studied by Tchebychev (1874) and Stieltjes (1884) for the series which bear their namesand is referred to as the problem of moments. (For more recent reviews of this problem see e.g. Wall, 1948, Shohat and Tamarkin, 1963, or Vorobyev, 1965). A general solution of this problem was given by Pade' (1892) and is known as the Pade' approximant method (Baker, 1975). Given a function, f(z), (z complex) which admits the formal, but not necessarily convergent, power series expansion Hz) = ? a z k , (2.1! k=0 K 31

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32 the [N,M] Pade' approximant is defined as a rational fraction of the form P(z)/Q(z) where P(z) is a polynomial of degree M and Q(z) is a polynomial of degree N. The coefficients of these polynomials are uniquely determined by equating like powers of z in the equation f(z)Q(z)-P(z)=0 (through order z N+M ) (2.2) with the auxiliary condition Q(0)=1. The expansion of P(z)/Q(z), therefore, coincides with Eq. (2.1) through the (N+M)-th power of z and provides an approximation to the remaining terms. The term by term convergence of Eq. (2.1) is replaced by the convergence of sequences of approximants (such as [N,N], N=l, 2, 3, ... ) in the Pade' approximant method, and although general convergence theorems are difficult to prove for arbitrary series, there exist several extensive special cases for which convergence has been proven. For these series, the Pade' approximant can often be shown to extend the natural region of convergence (Baker, 1975) and may be viewed as a method of approximate analytic continuation. A sequence of Pade' approximants, therefore, may converge rapidly when the original series expansion converges slowly or not at all. Two series which have been extensively studied in the problem of moments and for which sequences of Pade' approximants have been proven to converge are the series of Stieltjes (Stieltjes, 1894) and the extended series of Stieltjes (Hamburger, 1920, 1921a, 1921b, also known as the Hamburger moment problem). A series is of the Stieltjes type if and only if the coefficients, a, in Eq. (2.1), can be identified as moments of a distribution

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33 a k = I x #(x) !2.3: where ijj(x) is a bounded, non-decreasing function with infinitely many points of increase in the interval [o,»). The extended series of Stieltjes is defined similarly for the extended interval (-~,»). The extended series of Stieltjes has particular significance owing to its intimate relationship with resolvents of Hermitian operators. For any operator, A, we can define the operator function R(zA) = (i-za; 1 !2.4) which is trivially related to the resolvent of A. When A is Hermitian, the spectral theorem (Riesz and Sz.-Nagy, 1955) insures a unique integral representation of R(zA) having the form R(zA) dE(A) 1-zA [2.5] The operator E(X) is called an orthogonal resolution of the identity, and when A has only a discrete spectrum, it can be written E(A) = E e(A-a k )| k >«J> k | (2.6) where a n and are the eigenvalues and eigenfunctions of A. For any vector f in the domain of A n for all n, the function = R (z) dE f (x; ~T^zT !2.7) represents either an extended series of Stieltjes or a rational fraction

PAGE 44

34 depending on whether E f (A) has an infinite or finite number of points of increase (Masson, 1970). In view of possible applications of the Pade' approximant method to the superoperator resolvent, we state two theorems regarding the extended series of Stieltjes and discuss some properties of two particular sequences of Pade' approximants to these series. Theorem 1 : (Wall, 1948, theorem 86.1) A necessary and sufficient condition for f(z) to be an extended series of Stieltjes is det a Q a x a l a 2 . a . a n+1 a a . i . . . a n n+1 2 -/ >0 ; n=0, 1, 2, . !2.8] Theorem 2 : (Masson, 1970, theorem 4) If f(z) is an extended series of Stieltjes and the associated moment problem is determinant*, then, for fixed j=0,+l,+2, . . . +m, the sequence [N,N+2j+l] of Pade' approximants converges to f(z) for Im {z)f 0. The convergence is uniform, i.e. lim ||[N,N+2j+l]-f(z)|| = :2.9! with respect to z in any compact region in the upper or lower half-z plane. In addition to being uniformly convergent, sequences of [N,N] and [N,N-1] Pade' approximants to extended series of Stieltjes have two other features which make them particularly attractive for computational applications. First, these approximants are closely related to *The moment problem is said to be determinant if there is a unique, bounded, non-decreasing function ip ( x ) satisfying the moment conditions in Eq. (2.3) and the supplementary conditions i|j(-°°)=0 and ij»(x) = lim k{\p(x+e) + ip(x-e)} . e+0

PAGE 45

35 variational methods (Nuttall, 1970, 1973). When the operator R(zA) (defined in Eq. (2.5)) is positive definite, the [N,N] and [N.N-1] approximants provide the following bounds to R f (z) (Goscinski and Brandas, 1971): [N,N] > R f (z) > [N.N-1] . (2.10) For resolvent operators such as the superoperator resolvent which are indefinite, bounding properties are more difficult to establish. Vorobyev (1965) has shown, however, that the inverse poles of the [N,N-1] approximant to R^(z) are equivalent to the eigenvalues obtained from the usual Rayleigh-Ritz variational problem where ip = c Q f + c,Af + . . . c N _,A " f, and the coefficients {c} are variationally determined. In this sense, the poles of [N,N-1] to R f (z) are variationally optimum, but they have no definite bounding properties. The second attractive feature of the [N,N] and [N,N-1] approximants is the ease with which they may be computed. Rather than solving Eq. (2.2) to obtain the coefficients of the polynomials P(z) and Q(z), these approximants may be expressed directly in terms of the series coefficients {a, } using matrix formulae derived by Nuttall (1967) and Goscinski and Brandas (1971). For the [N,N-1] approximant, we have [N.N-1] = a^-z/y" 1 ^ , (2.12) where, in general, a. is a column vector with the elements a., a.,,, — i i i+l . . . a. +N ,, and A. is an N x N square matrix with the columns a_. , a_. + 1 , . . . a+N _.. Similarly for the [N,N] approximant we can write [N,N] = a Q + zajt^-z^]" 1 ^ . (2.13)

PAGE 46

36 2.2 Moment Conserving Decoupling Expanding the superoperator resolvent in Eq. (1.34) and multiplying both sides of the equation by E, the electron propagator can be expressed as the moment expansion EG(E) = Z E" k (a|H k a k=0 (2.14) Before the Pade' approximant method may be applied to this equation, however, the conventional definition of the Pade' approximant must be generalized to matrix Pade' approximants (Baker, 1975). This generalization is achieved by replacing the moment coefficients by the corresponding moment matrices and noting that these matrices do not commute when performing subsequent algebraic manipulations. Using Eq. (2.12) to represent the [N,N-1] approximant to EG(E), we obtain EG(E) = ^(Mq-E" 1 ^)" 1 ^ (2.15) or multiplying each side of this equation by E , Eq. (2.15) becomes G(E) = ^(EMq-Mj)" 1 ^ (2.16! where m~ is now a column matrix with block elements % M ^N-l ; £. = (al^a) If c. has the dimensions M x M, M. is an columns m. , m. , , , . . . m. ... , . -l -l+l -i+N-1 (2.17 x NM square matrix with

PAGE 47

37 There is a close relationship between Eq. (2.16) and the inner projection of the superoperator resolvent G(E) = (a|h)(h|(Ei-H)h) _1 (h|a) . (2.18) Goscinski and Lukman (1970) have shown that if the inner projection manifold is chosen to consist of h = {a k }U{Ha k }U . . . U{H N_1 a k } , (2.19) the inner projection and the [N.N-1] Pade' approximant are equal. Since, in general, the [N,M] Pade' approximant conserves the first N+M+l moments in the moment expansion, this choice of inner projection manifolds for the superoperator resolvent is called a moment conserving decoupling of the equation of motion. An examination of the [N,N-1] approximant to the electron propagator shows that its poles are given by the eigenvalues of M.c = M Q cd . (2.20) The matrix NL corresponds to a metric matrix and by virtue of the operator scalar product, is always positive definite Mq = (h|h) = Tr{p[hh + +h + h]}>0 . (2.21) The determinants of the metric matrices corresponding to various truncations of the moment conserving inner projection manifold, i.e. det (h^) > h^ = {a} (2.22) det (hjlhj) > hj = {a_}U{Ha_} , (2.23)

PAGE 48

38 provide the necessary and sufficient conditions of Theorem 1 to prove that the electron propagator is an extended series of Stieltjes. Consequently, the sequence of [N,N-1] Pade' approximants to the electron propagator should be uniformly convergent and should have variational ly optimum properties. The spectral representation of the electron propagator (Eq. (1.10)) consists of two summations, one which has poles in the lower half of the complex E-plane corresponding to ionization energies and one which has poles in the upper half plane corresponding to electron affinities. Based on the physical argument that the removal of an electron from a stable atomic or molecular system always requires energy, we might suspect a separation of the superoperator resolvent which yields a negative definite operator for these processes. If this were possible, the poles of the [N , N-l ] approximants would then be upper bounds to the exact ionization energies obtainable with a given basis. This separation has not been explicitly demonstrated but an overwhelming amount of numerical data seemsto substantiate this conjecture. In particular, the [1,0] approximant, which is easily verified to be the Hartree-Fock propagator, generally yields poles larger in absolute value than experimental ionization energies. One possible exception to this rule may be the near Hartree-Fock limit calculation of Cade et ah (1966) on diatomic nitrogen. In this calculation, the magnitude of the Itt orbital energy is slightly (0.2 eV) smaller than the experimental Itt ionization energy. 2 If on the other hand, the X tt ion state was fortuitously better described u J than the ground state with their extended basis, this result may still be an upper bound to the exact ionization energy i_n that basis .

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39 Relaxation and correlation corrections are incorporated in any [N,N-1] Pade' approximant beyond the [1,0] or Hartree-Fock approximant. In particular, we have studied the [2,1] approximant in some detail. This approximant corresponds to the truncation h = {a_}U{Ha_} (2.24) of the inner projection manifold and conserves the first four moment matrices. The operators {f . If^Ha.}, which were evaluated in Eq. (1.48), consist of a sum over all triple products of simple field operators with each operator product in the sum weighted by an antisymmetrized, twoelectron integral. These linear combinations provide a significant reduction in the subspace of triple operators thus overcoming one major limitation of the operator product decoupling. Another type of moment conserving decoupling of the electron propagator equations of motion has been analyzed by Babu and Ratner (1972). This decoupling is achieved by truncating the moment expansion after the m-th moment and replacing the m-th moment matrix with c = c G(E) . (2 25) Solving the truncated moment expansion for G(E) yields (E m+1 I-c )G(E) =V E m " k c, (2.26) ~™ k=0 -* G(E) = (E'^I-c)1 'V E m k c, (2.27) '" k=0 ~* These rational approximants formally conserve m moments but are not of the Pade 1 type (as Babu and Ratner incorrectly identify them) since the

PAGE 50

40 auxiliary equation, Q(0) = 1, is not satisfied, i.e. Q(0) = -c^ f 1 . (2.28) The auxiliary equation guarantees the uniqueness of the Pade 1 approximants: only one [N,N-1] Pade' approximant will conserve exactly m moments. The nonuniqueness of Babu and Ratner's decoupling scheme is easily demonstrated by replacing Eq. (2.25) with m -k m k Z E V = I E K c, G(E) < n < m . (2.29) k=n "^ k=n "* Solving for G(E) , m n-1 [El E E K c, ] G(E) = E E'c, (2.30] k=n "* 1=0 -l G(E) = [E m II E m \]1 V E m -\, , (2.31) k=n "* 1=0 ' we obtain m rational approximants (n-1, . . . m) which formally conserve m moments. Because these approximants are not uniquely defined, we will only consider Pade' approximants in this chapter. 2 . 3 Method of Solution The first step in obtaining the spectral representation of the electron propagator with the moment conserving decoupling is the evaluation of the necessary moment matrices. The first four moment matrices which are necessary to construct the [2,1] approximant have been evaluated by Redmon (1975) using the grand canonical density operator (Eq. (1.44)).

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An independent check of these derivations, however, revealed an error in the matrix elements of (ajH a) (Redmon, 1975, Eq. (11.30)). The correct result has subsequently been verified by Redmon (1977) and appears in Appendix 2. Once the moment matrices have been evaluated, the matrices NL and M, are constructed and the corresponding eigenvalue problem, Eq. (2.20), must be solved. In general, the dimension of the eigenvalue problem increases linearly with the size of the inner projection manifold, i.e. the [N,N-1] approximant presents an eigenvalue problem of dimension NM where M is the size of the spin orbital basis. For the [2,1] approximant, therefore, the dimension of this problem is only twice the size of the spin orbital basis. This means that for even rather large basis sets, standard matrix eigenvalues techniques may be employed to solve this problem in nonpartitioned form. As a consequence, all the poles and the spectral density of the electron propagator are obtained from a single matrix diagonalization thus avoiding the energy-dependent pole search. Denoting the eigenvectors by c, where cc + = Ml 1 , (2.32) and the eigenvalues by the diagonal matrix d_, the spectral representation of the electron propagator can be derived, G(E) = mj cc' 1 [EM -M 1 ]" 1 M cc" 1 ^" 1 ^ (2.33) = mj cIEl-dfV 1 ^ 1 !^ . (2.34) Defining the matrix 2L = Eq £ ' (2.35)

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42 which is rectangular with the dimensions M x NM, Eq. (2.34) becomes 6(E) = x(EI-d) _1 x + . (2.36) This equation conserves the first 2N moment matrices of the moment expansion which implies, in particular, xx = (aja) = I (2.37) from the conservation of the first moment. The complete solution of the electron propagator which is conveniently obtained with this decoupling can be used to determine a selfconsistent, single-particle reduced density matrix (1-matrix). The i, j-th element of the 1-matrix can be computed from the contour integral (Linderberg and Ohrn, 1973) = (2tt.) _1 | G(E) ij dE . (2.38) The contour, c, runs from -« to °° along the real axis and encloses only poles of G(E).. that lie below the chemical potential (u) when finally closed in the upper half of the complex E-plane. The integral is then evaluated using the Cauchy residue theorem (Morse and Feshbach, 1953) = Z lim (E-d k )G(E).. (2.39) J k
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43 it follows that the 1-matrix is diagonal with occupation numbers determined by £ x. 2 k (2.43) s,s'

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44 {ij|kl}= E . (2.44) S , S ' Since these partial sums contain a double summation which is performed only once, the original N problem is effectively reduced to N 5 . The construction of the moment matrices is now comparable in difficulty to the transformation of the two-electron integrals from the primitive basis to the computational (usually Hartree-Fock) basis which is also roughly 5 proportional to N . When the number of two-electron integrals is too large to be held in core, their random access from peripheral storage becomes relatively time consuming. The partial sums are much more efficiently constructed from ordered lists of two-electron integrals which can be read into primary (core) storage when needed. For the partial sums defined above, the two-electron integrals must be sorted into ordered lists of the type and where * indicates all orbital indices which yield a non-zero integral for the corresponding i,j-th distribution. The integral sorts are performed using the Yoshimine sorting technique (Yoshimine, 1973). Briefly summarized, this technique involves a partition of available core into a number of buffers. Each buffer holds integrals corresponding to a specific i,j distribution, e.g. . (When the number of distributions is large, several may be held in each buffer.) Reading through the two-electron integral list, integrals are then sorted into the appropriate buffers. As each buffer fills, it is written to direct access, peripheral storage and assigned a record number. All record numbers corresponding to integrals from the same buffer are saved in a "chaining" array for that buffer. After the entire integral list has been processed and all buffers have been dumped, it is then

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45 possible to chain back through the direct access records, copying integrals of the same distribution back into core. These integrals may then be further sorted within distributions, e.g. k
PAGE 56

46 Table 1. Contracted Gaussian Basis for Nitrogen. Nitrogen s orbital s Contraction Exponents Coefficients 5909.4400 0.002001 887.4510 0.015310 204.7490 0.074293 59.8376 0.253364 19.9981 0.600576 2.6860 0.245111 7.1927 1.000000 0.7000 1.000000 0.2133 1.000000 Nitrogen p orbital s Contraction Exponents Coefficients 26.7860 0.018257 5.9564 0.116407 1.7074 0.390111 0.5314 0.637221 0.1654 1.000000

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47 the calculation of Cade et aj_. (incorrectly) predicted the In orbital energy to be 0.53 eV below the 3a , this calculation predicts the Itt 9 u energy to be 0.05 eV higher. The correct ordering of the 3a and Itt g u ionizations with this basis is merely fortuitous, since based on a total energy criterion, the basis of Cade et al . is more accurate. The next step of the calculation involved the integral sorts, partial summations, and the construction of the moment matrices. The poles and spectral density were finally computed as outlined in the previous section and are presented along with the [1,0] results in Table 2. The ionization energies of both approximants seem to be upper bounds to the experimental results of Siegbahn et aj_. (1969), but without exception, the results of the [2,1] approximant are worse than the [1,0] approximant. In an attempt to incorporate some ground state correlation into the grand canonical density operator, new occupation numbers were computed from the spectral density and the [2,1] approximant was recalculated. This calculation, however, yielded no significant improvements in the ionization energies. In order to ascertain whether the poor results from the [2,1] approximant for nitrogen are representative of other calculations or just the consequence of a pathological test case, the water molecule was chosen for a second application. Similarly to the calculation for nitrogen, a double zeta contracted basis of GTO's was also employed in this calculation. Huzinaga's 9s, 5p primitive basis for oxygen and 4s primitive basis for hydrogen were contracted with Dunning's coefficients to 4s, 2p and 2s, respectively. The orbital exponents for the hydrogen atoms were scaled by 1.14 to more realistically represent the effective nuclear charge in the molecule, and the final basis appears in Table 3. Again,

PAGE 58

Table 2. Principal Ionization Energies for the Nitrogen Molecule Resulting from the [1,0] and [2,1] Propagator Approximants. 48 Orbital la. 2cj 3a. Itt lo 2a [1,0]

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49 Table 3. Contracted Gaussian Basis for Water. Hydrogen s sets Contraction Exponents Coefficients 13.3615 0.032828 2.0133 0.231208 0.4538 0.817238 0.1233 1.000000 Oxygen s sets

PAGE 60

50 the integrals were computed with the MOLECULE program at the equilibrium internuclear geometry, R(0H)= 1.809 a.u., $H0H = 104.5° (Benedict et al_. , 1956). A total energy of E(HF)= -76.0082 a.u. was computed with the Hartree-Fock portion of GRNFNC and was followed by the two-electron integral transformation. Finally, the integral sorts and partial sums were performed, the moment matrices constructed, and the poles and spectral density obtained for the [2,1] approximant. The results for both the [1,0] and [2,1] approximants are presented in Table 4 and appear to be upper bounds to the experimental ionization energies. Once more, the [2,1] results are consistently worse than the [1,0] results. A few iterations on the occupation numbers yielded no significant improvements. 2.5 Evaluation of the Moment Conserving Decoupling Formally, the moment conserving decoupling is an attractive decoupling procedure. Being closely related to the Pade 1 approximant method, this decoupling allows the application of numerous results from the classical moment problem to propagator theory. In particular, it was proven that the sequence of [N,N-1] approximants converge uniformly to the exact electron propagator in a given basis, and it was shown that these approximants represent a variational ly optimum choice of the inner projection manifold. Why then are the results of the [2,1] approximant so much worse than the results of the [1,0] approximant? To answer this question, it is necessary to analyze the three approximations identified in Section 1.3, namely: basis quality, density operator, and decoupling procedure. First of all, since computational economy and not high accuracy was the criterion for the test calculations on nitrogen and water, polarization

PAGE 61

51 Table 4. Principal Ionization Energies for Water Resulting from the [1,0] and [2,1] Propagator Approximants. Orbital la. [1,01 2a. 3a. lb, lb, [2,1] E(HF Exp.' 559.4

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52 functions were intentionally excluded from the basis sets. Polarization functions are diffuse, virtual orbitals which can be very important in describing electron relaxation and correlation (Purvis and Ohrn, 1974, Cederbaum and Domcke, 1977). It is reasonable to expect that the addition of polarization functions will improve both the [1,0] and [2,1] approximants to varying degrees; however, with the same basis and with the same density operator, the larger inner projection manifold (if judiciously chosen) should yield a more accurate decoupling. Since this was not the situation in these test calculations, any improvements in the basis sets did not seem worthwhile. Second, it is possible that significant ground state correlation may have been neglected with our choice of the grand canonical density operator. With the spin orbital annihilation and creation operators expanded in the Hartree-Fock basis and using pure state occupation numbers of zero or one, this density operator yields the uncorrected, Hartree-Fock ground state average. Rather than explicitly correlating the density operator (as e.g. through perturbation theory), an attempt was made to estimate the effect of correlation through the self-consistent determination of the occupation numbers as described in Section 2.3. This procedure was not pursued to true self-consistency since each iteration required a complete recalculation of the [2,1] approximant. It was obvious, however, after the first few iterations that no significant improvements had been obtained. Based on the preceding implications, the third approximation—the inner projection manifold truncation—seems to be primarily responsible for the poor numerical results. Owing to the complicated operator sums in this manifold, an order analysis (as discussed in Section 1.6) is not

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53 readily possible. Consequently, it is extremely difficult to identify the problem with this decoupling procedure. It can only be concluded that the number of moments conserved is not a useful criterion for decoupling. This conclusion is consistent with the uniform convergence of the [N,N-1] sequence since uniform convergence is not necessarily monotonic, but it suggests that more accurate decouplings require the incorporation of more information about the moment expansion than just the moment matrices. The additional information needed is indeed available and, in the next chapter, we will demonstrate how it may be extracted using perturbation theory. *Babu and Ratner (1972) reported the same conclusion which was based on an application of their rational approximants to the Hubbard model.

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CHAPTER 3 DIAGRAM CONSERVING DECOUPLINGS 3. 1 The Diagrammatic Expansion Method The superoperator formalism which is employed in the previous two chapters is by no means the only formalism available to formulate decoupling approximations for the electron propagator. Two commonly used, alternative methods are the functional differentiation method (see e.g. Csanak et al_. , 1971) and the diagrammatic expansion method (see e.g. Mattuck, 1967, Fetter and Walecka, 1971, or Cederbaum and Domcke, 1977). Of these latter two methods, the diagrammatic expansion method has proven to be particularly effective. This method avoids some of the algebraic tedium involved in deriving propagator decoupling approximations by establishing certain rules for constructing and manipulating diagrams which represent the underlying algebraic structure. The diagrammatic expansion of the electron propagator is usually derived using time-dependent perturbation theory. The N-electron Hamiltonian is partitioned into an unperturbed part plus a time-dependent perturbation H = H Q + exp(-e|t|)V , (3.1) where e is a small positive quantity. The unperturbed Hamiltonian, H„, is chosen to yield an exactly solvable, eigenvalue problem w = W ' ( 3 2 : 54

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55 and the time dependence of the unperturbed eigenstates is given by |* (t)> = exp(-iH t)| . (3.3) In order to simplify the remaining problem of finding the fully perturbed eigenstates |•(t)>, it is convenient to introduce the "interaction representation" (Fetter and Walecka, 1971) by the transformation |Yj(t)> = exp(iH t)|Y(t)> . (3.4) In this representation, the Schrbdinger equation has the form i ft IV t}> = exp(-e|t|)V(t)|Yj(t)> (3.5) where V(t) = exp(iH t)Vexp(-iH Q t) . (3.6) The time dependence of the interaction eigenstates can be expressed as |•j(t)> = U £ (t,t )|f I (t )> (3.7) where U £ (t,t ) is the time-evolution operator. Substituting Eq. (3.7) into Eq. (3.5), the evolution operator is found to satisfy the differential equation i ft U e (t,t ) = exp(-e|t|)V(t)U e (t,t ) (3.8) with the initial condition U E (t Q ,t ) 1 . (3.9) It is more convenient to solve for U (t,t„) by first transforming Eq. (3.8) into an integral equation

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56 U e (t,t ) =1-1 dtj exp(-e|t|)V(t 1 )U e (t 1 ,t O 1 ' (3.10; This integral equation has the form of the Volterra equation of the second kind (Lowdin, 1967) and is solved iteratively t U e (t,t ) = 1 i J dtj exp(-e|t|)V(t 1 ) ^2 + (""•) j dt x J dt 2 exp{E (|t 1 | + |t 2 |)}V(t 1 )V(t 2 )U e (t 2 ,t ) (3. IT t t 00 ( ( 1 1 (-i) n dt. dt 9 n=0 > v > l l l 'n-1 dt exp{-e( 1 1, |+|tp|+ t I)} n i / x v(t 1 )v(t 2 : V(t ) v n 7 tHj^tp^ . > t (3.12) Eq. (3.12) can be generalized slightly by modifying the limits of integration and introducing the time ordering operator, T, U (t,t n )= Z (-i) n -. £ ° n=0 n! dt l J dt 2 ' • J dt n t t x exp{-e(|t 1 | + |t 2 |+ . . . |tj)} T[V(t 1 )V(t 2 ) . . . V(t n )] . (3.13! The time ordering operator rearranges the product of perturbation operators such that the left-most term is the latest in chronological order. The perturbed eigenstates I ^ j ( t Q )> can now be expressed in terms of the unperturbed eigenstates by noting that as t Q -*-±°°, |'Mt n )>^|$ n >, and as t Q increases from -°° to zero, the perturbation is "adiabatically switched on"

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57 h = u e (o,-°°)|$ > . According to a theorem of Gell-Mann and Low (1951), if lim K I (0)> <* n |U (o 5 -«»)|$ > E «f n |f T (0) U e (o,-)|* > e+0 *0' e 1 'I exists, then it is an eigenstate of H Hh'j(O): E|Vj(0)> '^ I^ I (0)> ~ <$ |fj(0): (3.14) (3.15) :3.16! These results can now be used to determine the electron propagator. In Chapter 1, the propagator was defined as the ground state average of a time-ordered product of field operators in the Heisenberg representation iG. .(t) ij (3.17; Using Eq. (3.15) and the fact that |T H >= \V, (0)>, this average can be expressed in the interaction representation as iG. .(t) * n |U J-,t)T[a,(t)at(0)]U Jt,-»)|* n > '0 1 £ r-LL-J .,--<$ n U (00 , 00 ) $ n > 1 e ' [3. 18) Using the expansion of the evolution operator (Eq. (3.13)) and taking the limit e-K), it can be shown (Fetter and Walecka, 1971) that iG..(t)= M-i)^, jd tl . . .{dt n <$ |T[V(t 1 ) . . . V(t n )a 1 (t)a](0)]|$ : (3.19;

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58 The final step in the diagrammatic expansion method is to expand the numerators of each term in Eq. (3.19) using Wick's theorem (Wick, 1950) and to represent them diagrammatical ly (e.g. Fetter and Walecka, 1971). The denominator of Eq. (3.19) must also be expanded and diagrammed, and when this is done, all disconnected diagrams arising from the expansion of the numerator will cancel (Abrikosov et _§_]_. , 1965). Formally, the diagrammatic expansion method and the superoperator formalism appear strikingly dissimilar. The diagrammatic method is formulated in the causal representation while the superoperator formalism utilizes the energy representation. The diagrammatic method employs a pictorial representation of the algebraic structure while the superoperator formalism emphasizes the algebraic structure directly. Yet the primary goal of each formalism is the same: an accurate prediction of ionization energies and electron affinities. Therefore, the two formalisms are inherently equivalent. It is our desire in this chapter to explicitly demonstrate the equivalence between these two formalisms and to re-examine the superoperator decoupling approximations in terms of a diagrammatic analysis. 3.2 Perturbation Theory The unifying feature of the diagrammatic expansion method and the superoperator formalism is perturbation theory (Born and Ohrn, 1978). Since the commutator product is distributive with respect to addition, we can define a partitioning of the superoperator Hamiltonian into an unperturbed part plus a perturbation, H = H Q + V . (3.20)

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59 One convenient partitioning, which will be shown to readily yield the Hartree-Fock propagator as the unperturbed electron propagator, is the Mtfller-Plesset partitioning (Miller and Plesset, 1934). With this partitioning, hL has the form H = Z £ r a r a r " ^ E (3.21) r r,r' and the perturbation is expressed as v= E [ka f al,a ,a -5 , ,a'la] + h I r,r',s,s' r r s s r s r r s r ^, (3.22) Of course, when the commutator product is formed for the superoperators, the constant term in these definitions will cancel. Other parti tionings of the Hamiltonian may also be assumed and may lead to superior convergence properties (Claverie et al . , 1967). One alternative partitioning which has been employed in the perturbation calculation of correlation corrections to the total energy is the EpsteinNesbet partitioning (Epstein, 1926, Mesbet, 1955a, 1955b). In propagator applications, the work of Kurtz and Ohrn (1978) may be roughly interpreted in terms of a partitioning where the unperturbed Hamiltonian incorporates all relaxation contributions to the ionization energy. It is difficult to define this unperturbed Hamiltonian explicitly, but it formally satisfies the eigenvalue equation H^a k = AE k (SCF)a k (3.23) in contrast to Vk = e k a k < 3 24 ) for the M011er-Plesset partitioning. The method of Kurtz and Ohrn yields

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60 excellent ionization energies and electron affinities with a simple second-order self-energy, however it has not been formally analyzed in detail . Corresponding to the partitioning of the superoperator Hamiltonian, we can introduce a partitioning of the operator space defined by the projection superoperators and P, 6 = X |a k )(a k | = |a)(a| (3.25) P = I (3.26) These superoperators operate on elements of the operator space through the relations OX. = E |a k )(a k |X.) (3.27) PX. = X. OX. (3.28) and are idempotent (6 2 = 6, P 2 = P), self-adjoint (0 + = 6, P f = P), and mutually exclusive (0P=P0=0). The superoperator projects from an arbitrary operator product that part which lies in the model subspace, i.e. that part which is spanned by the eigenelements of f-L. The superoperator P projects onto the orthogonal complement of the model subspace, i.e. that part which we have no a priori knowledge about. To obtain a perturbation expansion of the superoperator resolvent, we consider its outer projection (Lowdin, 1965) onto the model subspace, G(E) = 6(EI-H)~ 1 6 (3.29) = 6(Ei-H -V) -1 6 (3.30) By iterating the identity

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61 (A-B)" 1 = A" 1 + A _1 B(A-B) _1 , (3.31) the inverse in Eq. (3.30) can be expanded as G(E) = (Ei-H ) -1 + (EI-H )" 1 0V(EI-H )" 1 + (EI-H )" 1 6v(Ei-H )" 1 V(Ei-H )~ 1 6 + . . . (3.32) where the property [H ,6]_ = (3.33) has been used. Now since plus P form a resolution of the identity, each resolvent of EL occurring between perturbation superoperators, V, can be rewritten as a sum of its projections on the model subspace and the orthogonal complement, (EI-Hq)" 1 = (EI-H ) _1 + (EI-H ) -1 P (3.34) = G Q (E) + T Q (E) . (3.35) With this notation, Eq. (3.32) becomes G(E) = G Q (E) + G Q (E)VG (E) + G Q (E)V[G Q (E) + T Q (E) ]VG Q (E) + G (E)V[G Q (E) + T (E)]V[G (E) + T Q (E)]VG (E) + . . . (3.36) and can be resummed to yield G(E) = G Q (E) + G Q (E)[V + VT (E)V + VT (E)VT Q (E)V + . . . ]G(E) (3.37) Defining the reduced resolvent of the full superoperator Hamiltonian as T(E) = P[a0 + PCEI-H)?]"1 ? (a^O) (3.38) = T Q (E) + T Q (E)VT(E) , (3.39)

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62 Eq. (3.37) can be written in closed form G(E) * G Q (E) + G Q (E)[V + VT(E)V]G(E) . (3.40) Alternatively, we can define wave and reaction superoperators through the equations (cf. Lowdin, 1962, or Brandow, 1967) W(E) = I + T(E)V (3.41) t(E) = VW(E) . (3.42) The reduced resolvent, wave, and reaction superoperators introduced in this section are functions of the superoperators I, hL, and V and as a consequence, operate in a more complicated way. To apply a superoperator function to an operator in the operator space, it must first be expanded in terms of the superoperators I, FL, and V which are then successively applied to the operator. For example, W(E)X. = [I + T(E)V]X. (3.43) = [I + T n (E)V + T n (E)VT n (E)V + . . . ]X. (3.44) where T Q (E)VX i = [E _1 I + E" 2 H Q + E" 3 H 2 + . . . ]PVX. , (3.45; etc. 3 . 3 Eq uivalence of the Superoperator Formalism and the Diagrammatic Expansion Method Eqs. (3.37) and (3.40) represent the superoperator form of the Dyson equation (Dyson, 1949), and the reaction superoperator (Eq. (3.42)) can be identified as the self-energy. To demonstrate that Eq. (3.37)

PAGE 73

63 corresponds term by term with the diagrammatic propagator expansion, we must first form the operator average of G(E) to obtain the matrix Dyson equation, next evaluate all necessary operator averages, and finally diagram the resulting algebraic formulae. Owing to the complicated operator averages that must be evaluated in third and higher orders of the perturbation superoperator, the equivalence between these two formalisms has only been explicitly demonstrated through third order and is assumed in all higher orders. The matrix Dyson equation is obtained by forming the operator average of G(E) with respect to the basis elements of our model subspace G(E) = (a|G(E)a) (3.46) = G^E) + G^E^EJGJE) , (3.47) where E(E) = (aJVa) + (a|VT Q (E)Va) + (a|VT Q (E)VT (E)Va) + . . . . (3.48) Since H Q was chosen to be the Fock superoperator, the appropriate density operator to employ in the evaluation of the operator averages is the Hartree-Fock density operator. Realizing that the grand canonical density operator (Eq. (1.44)) reduces to the Hartree-Fock density operator when pure state occupation numbers of zero or one are chosen, we shall employ this density operator. Beginning with the evaluation of matrix elements for the unperturbed propagator, G^(E), the Hartree-Fock propagator is easily obtained (cf. Section 1.3)

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64 G (E) ij = ( aj |(EI-H )-V) = E~ l (a J |a 1 ) + E~ 2 ( aj |H a n . ) + E^a^H^.) + f\ + E-%6^ + rt? fi + . . . G Q (E) ir (E-e^Vj (3.49) (3.50) (3.51) (3.52) The evaluation of each term in the self-energy expansion requires the initial evaluation of Va . , Va. = k Z [a . ,a a ' ,a ,a ] 1 r.r'.s.s' L i r r s s J £ [a.,a'a _] r,s,s' ^ r s E a a ,a c I a r,s,s' r s s s ,s' s s (3.53) (3.54) With this result, the first-order term (a.|Va.) is obtained without much additional effort .(1 E . . = a. Va. ij J 1 i (3.55) h l Tr{p[a a c .a_,aT] } r,s,s' r s s j + • I Tr{p[a ,a.] } s,s' J (3.56) = 7 5 r,s S rs i: 7
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65 The evaluation of the secondand higher-order self-energy matrices requires the evaluation of T Q (E)Va i and VT Q (E)Va.. The first of these quantities can be expanded as T Q (E)Va i = (EI-H )" 1 PVa i (3.59) = (EI-H )" 1 Va i Z (EI-H )" 1 |a k )(a k |Va i ) (3.60) using Eq. (3.26). It follows from the previous result for (a.|Va.) that the second term in Eq. (3.60) vanishes. The first term can now be evaluated by expanding the resolvent of hL and realizing that any operator product is an eigenelement to l-L, i.e. Vr a s' a s = (e rE s'e s )a r a s' a s ' < 3 61 ) Consequently, we obtain T (E)Va. = h T, ( (E+E r -e s -e s ,)" 1 aja sl a s E (E-e )" 1 a (3.62) s,s' b b with the help of Eq. (3.45). The remaining application of V and the average value evaluation is straightforward and yields £ (2) (E) i:J = ( aj |VT (E)Va.) (3.63) _ v r , ,, . ,_ ... = T, — t^l — lu — p-< n >+i 5< n ,>-] (3.64) i h+c-e-c, r s s r s ' r,s,s' v r s s' ; for the matrix elements of the second-order self-energy. The Hartree-Fock average is now obtained by choosing occupation numbers of zero and one. An examination of the occupation number factor in Eq. (3.64) reveals that with this restriction, it will be non-vanishing

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66 only when the summation index r runs over occupied spin orbitals and s and s' run over unoccupied spin orbitals or when r runs over unoccupied spin orbitals and s and s' run over occupied spin orbitals. Denoting a, b, c, . . . as summation indices over occupied spin orbitals; p, q, r, . . . for unoccupied spin orbitals; and i, j, k, . . . for unspecified spin orbitals, Eq. (3.64) can now be written as two terms which involve restricted spin orbital summations Z (2)/ E) = ,, „ ij 2 a,p,q ^VVV +h z _ (3 _ 65) p,a,b u e p~ a~V The conversion of Eq. (3.65) into diagrams is a straightforward procedure for which we shall use the rules and diagram convention of Brandow (1967) and Bartlett and Silver (1975b). This convention represents the synthesis of the anti symmetrized vertices of the Hugenholtz (1957) or Abrikosov (1965) diagrams with the extended interaction lines of the Goldstone (1957) diagrams, and the rules for constructing these diagrams are given in Table 5. The application of these rules to the terms in Eq. (3.65) yields the following diagrams: y | ^ { a.p.q {E+ W £ q ) i v Va,b t^VVV < ^ (3.07; These diagrams are precisely the same as those obtained in the secondorder diagrammatic expansion after a Fourier transformation into the energy representation (Cederbaum and Domcke, 1977).

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67 Table 5. Rules for Constructing Self-Energy Diagrams. 1. Each anti symmetrized two-electron integral factor in the numerator is represented by an interaction line with a vertex (dot) at both ends. The number of interaction lines denotes the order of the term. 2. Using the Dirac bra-ket notation, both indices in the bra are represented by lines leaving a vertex while those of the ket are represented by lines entering a vertex. There must be only one outgoing and one incoming line per vertex, therefore, assign the index of electron coordinate one to the left vertex and the index of electron coordinate two to the right vertex of each interaction line. 3. Summation indices running over hole states (occupied orbitals) are directed downward, indices running over particle states (unoccupied orbitals) are directed upward, and external indices (not summed) are drawn horizontally. To Check Diagrams: 4. The energy denominator of the diagrammed expression should be obtained by first connecting the external lines and assigning a factor of E to this directed segment. Second, imagine horizontal lines drawn between each pair of interaction lines. Each horizontal line corresponds to a multiplicative, denominator factor obtained by summing the orbital energies of each hole (downgoing) line that intersects it minus the sum of orbital energies for particle (upgoing) lines that intersect it. Treat the factor E of the connected external lines as an orbital energy. 5. Numerical factors should be obtained by assigning a factor of h for each pair of equivalent internal lines. Equivalent internal lines are two lines which begin on the same interaction line, end on the same interaction line, and go in the same direction. 6. The overall sign factor should be obtained by assigning a factor of minus one to each internal hole line segment and a minus one to each closed loop.

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68 The evaluation of the third-order self-energy matrix is similar to the second-order matrix but much more tedious and the result is presented in Appendix 3. As was done for the second-order expression, the occupation numbers must again be restricted to zero and one to obtain the Hartree-Fock average. When this restriction is made, the unrestricted spin orbital summations in Appendix 3 will reduce to summations involving occupied, unoccupied, and unspecified spin orbitals. Using the algebraic identity a _ a r i i i (E-a)(E-b) " Ta^bT [ TI^T ' T^bT J (3.68; it is possible to combine terms in such a way that expressions involving only occupied and unoccupied spin orbital summations are obtained. These expressions are presented in Appendix 1. The corresponding diagrams in Appendix 1 again are precisely those occurring in the third-order, diagrammatic self-energy expansion. 3.4 Diagram Conserving Decoupling The wave and reaction superoperators identified with the help of perturbation theory in Section 3.2 have special importance in the development of decoupling approximations for the electron propagator. As we have already seen, the reaction superoperator generates the diagrammatic self-energy expansion. A truncation of this expansion offers one viable decoupling scheme. The wave superoperator, on the other hand, has the property of generating eigenelements to the full superoperator llamiltonian from the eigenelements of the unperturbed superoperator Hamiltonian (EI-H)W(E)a = . (3.69)

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69 This property is easily proven by first using Eq. (3.41) to expand W(E) and then premul tiplying both sides of Eq. (3.69) by P (EI-H)W(E)a = (EI-H)a + (EI-H)T(E)Va_ (3.70) P(EI-H)W(E)a = P(EI-H)a + P(EI-H)T(E)Va . (3.71) Using the identity P(EI-H)T(E) = P (3.72) and the property Pa_ = 0, Eq. (3.71) simplifies to P(EI-H)W(E)a =PVa_ + PVa = (3.73) which implies the validity of Eq. (3.69) It is of interest at this point to show a connection between the superoperator formalism and the Equations of Motion (EOM) method for determining ionization energies (Simons and Smith, 1973). In this method, one seeks solutions of the equation [H,Q]_ = wQ (3.74) which is precisely Eq. (3.69). Here the operator Q is interpreted as a correlated ionization operator that generates, in principle, the exact (N-l)-electron ion states from the exact N-electron reference state. One approach to solving Eq. (3.74) involves the application of RayleighSchrodinger perturbation theory (Dalgaard and Simons, 1977). By partitioning the Hamiltonian operator, expanding both the ionization operator, Q, and the ionization energy, to, in terms of a perturbation parameter, and collecting terms of the same order, a set of perturbation theory equations are obtained. The solution of these equations yields an expansion for Q which is analogous to the superoperator equation

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70 h = W(E)a (3.75) The only difference is that E is replaced by u) Q which is a consequence of using Rayleigh-Schrb'dinger rather than Brillioun-Wigner perturbation theory. Returning now to the inner projection of the superoperator resolvent, G(E) = (alhKhKEI-lOhr^hJa) (3.76) we may view Eq. (3.75) as an alternative prescription for choosing the inner projection operator manifold. Recalling from Section 1.6 that since the density operator describing the unperturbed (model) problem does not commute with the full Hamiltonian, the operator scalar product will not in general exhibit Hermitian symmetry. Consequently, we define (h|=(a|W f (E) (3.77) and note that (a|W f (E) t (W(E)aJ . (3.78) Approximate electron propagator decouplings can now be obtained by truncating the expansion of the wave superoperator, W(E) = I + T Q (E)V + T Q (E)VT (E)V + . . . . (3.79) Truncation of this expansion, with only the superoperator identity, trivially yields the Hartree-Fock propagator, therefore we next consider W(E) = I + T Q (E)V . (3.80) Noting that the subspaces {a. } and (fJfV = T Q (E)Va, } are mutually orthogonal, Eq. (3.76) can be readily solved for G (E)

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71 G _1 (E) = G^(E) 1(E) , (3.81) where E(E) = (aj\/T (E)Va)(a|VT (E)(EI-H)T (E)Va) _1 (a[VT (E)\/a) (3.82) Making the following identifications from Section 3.3: (a_|VT (E)Va) = E Uj (E) , (3.83; .(2! (a|VT Q (E)(EI-H )T (E)Va) = Z^ ; (E) , (3.84) and (a|VT (E)VT Q (E)Va) = Z (3) (E) , (3.85) Eq. (3.82) can be rewritten 1(E) = E (2) (E)[E (2) (E) E (3, (E)f V^E) . (3.86) Expanding the inverse of Eq. (3.86), we easily see that this selfenergy approximant coincides with the diagrammatic expansion through third order but additionally yields contributions to all higher orders. If the exact self-energy is rewritten as a moment expansion in terms of a perturbation parameter, A, X~h(E) = E A k (a|V(T n (E)V) k a) , (3.87) k=0 u we see that Eq. (3.86) represents the [1,1] Pade' approximant to this expansion. Owing to the close connection between Pade' approximants and the inner projection technique as demonstrated in Chapter 2, this result is not surprising. These Pade' approximants to the self-energy, however, will have entirely different convergence properties than those studied in Chapter 2.

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12 3.5 Approximations and Applications Computational applications of the [1,1] Pade' approximant to the self-energy require the evaluation of the secondand third-order selfenergy matrices. The second-order matrix is relatively easy to evaluate. The third-order matrix, on the other hand, is exceedingly more difficult and can presently be only approximately calculated without excessive computational effort. An examination of the formulae in Appendix 3 reveals that unlike the fourth moment matrix in the moment conserving decoupling, the third-order self-energy matrix is energy dependent. This additional complication makes the partial summation technique used in the moment conserving decoupling ineffectual since the third-order self-energy matrix will generally need to be resummed with different values of E hundreds of times in the search for poles of the propagator. The first approximation that we will examine is the complete neglect of the third-order self-energy matrix. With this approximation, the [1,1] approximant in Eq. (3.86) reduces to a second-order truncation of the diagrammatic self-energy expansion, Z(E) = E (2) (E) . (3.88) This second-order self-energy approximation is interesting not only because it contains the most important relaxation and correlation corrections to Koopmans' theorem (in a perturbation theoretical sense), but also because it exhibits the same analytic form as the exact self-energy (Hedin and Lundqvist, 1969, Cederbaum and Domcke, 1977). Furthermore, since several second-order, ionization energy calculations have been reported in the literature, this approximation will afford both a convenient check of new computer code and the computational experience necessary to implement more refined approximations.

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73 The first computational application of this decoupling approximation was to the water molecule using the same basis and internuclear geometry as described in Section 2.4. The results of this calculation are presented in Table 6 along with the Koopmans' theorem, AE(SCF), and experimental values for the ionization energies. Two ionization energies have been tabulated for the 2a, ionization with their corresponding pole strength (r. of Eq. (1.70)) in parentheses. The occurence of two, relatively strong propagator poles for this ionization represents a breakdown in the quasi-particle description of inner valence ionizations (Cederbaum, 1977) and makes assignments of principal and shake-up ionizations ambig* uous. In general, the second-order ionization energies are quite encouraging and represent significant improvements to each of the Koopmans' values. Furthermore, these results are comparable in accuracy to the AE(SCF) results but possess the convenience of being obtained in a single calculation whereas the AE(SCF) results required six separate HartreeFock calculations. The relatively poor agreement of the 3a, and lb, ionization energies with the experimental values in Table 6 seems attributable to basis incompleteness. Despite the lack of polarization functions, this suspicion is supported by the facts that the 3a, orbital is the highest occupied orbital in that symmetry and that this basis contains only two contracted Gaussian orbitals of b, symmetry. In order to study the basis dependence of the second-order self-energy approximation, two additional calculations *The ESCA spectrum of the water molecule (Siegbahn et aj_. , 1969) substantiates this phenomenon since the 2a, peak is quite broad and asymmetric. Experimentally, it appears that trie lower energy ionization should have a larger pole strength (in contrast with the results of Table 5) since the peak is skewed to higher binding energies.

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74 Table 6. Principal Ionization Energies of Water Computed with the 14 CGTO Basis. Orbital

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75 were performed with larger basis sets. The first of these calculations employed a 26 contracted orbital basis which augmented the original 14 orbitals (Table 3) with a set of p-orbitals on the hydrogen atoms and a set of d-orbitals on oxygen--all with unit exponents. The Hartree-Fock total energy obtained with this basis was E(HF)= -76.0459 H. The second calculation employed a 38 contracted orbital basis which included all of the orbitals in the 26 orbital basis plus an additional set of diffuse p-orbitals on the hydrogen atoms (a = 0.25) and a set of diffuse d-orbitals on oxygen (a = 0.40). This basis yielded a Hartree-Fock total energy of E(HF)= -76.0507 H. The most significant propagator poles calculated in the valence region (0~40 eV) with each of the three water basis sets are presented in Table 7 along with the second-order results of Cederbaum (1973a). The inclusion of polarization functions not only improves the 3a, and lb, ionization energies, it also reverses the relative pole strengths of the two dominant 2a^ poles bringing the theoretical results into better agreement with experimental observations (see footnote on page 73). Cederbaum' s second-order results were obtained with a basis comparable in size and quality to the 26 orbital basis in Table 7. He deletes several virtual orbitals from this basis before computing the ionization energies, however. This approximation may account for the small discrepancies between his results and those reported here. The formaldehyde molecule was chosen for a second application of the second-order self-energy approximation. Ionization energies were calculated using two basis sets. The first consisted of Huzinaga's 9s, 5p primitive basis sets for oxygen and carbon (Huzinaga, 1965) contracted to 4s and 2p functions with Dunning's contraction coefficients (Dunning,

PAGE 86

76 Table 7. Basis Set Effects on the Ionization Energies of Water Computed with a Second-Order Self-Energy Approximation, Symmetry

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77 1970). The orbital exponents of Huzinaga's 4s primitive basis for hydrogen were scaled by a factor of 1.2, and the resultant orbitals were contracted to 2s functions as recommended by Dunning. The complete basis appears in Table 8. The second basis augmented the first by the addition of one set of p-orbitals on the hydrogen atoms and one set of d-orbitals on both the oxygen and carbon atoms. Unit exponents were chosen for the p-orbitals on hydrogen while exponents of 0.8 were chosen for the d-orbitals. Oneand two-electron integrals were computed with the MOLECULE program (Almlof, 1974) at the experimental equilibrium geometry: R(C0)= 2.2825 a.u., R(CH)= 2.1090 a.u., and }(HCH)= 116.52° (Oka, 1960, Takagi and Oka, 1963), and the Hartree-Fock calculations and two-electron integral transformations were performed with GRNFNC (Purvis, 1973). The Hartree-Fock total energy for the smaller, 24 orbital basis (no polarization) was E(HF)= -113.8257 H. , and for the larger, 42 orbital basis (with polarization) E(HF)= -113.8901 H. The Hartree-Fock orbital energies and second-order self-energy results for both basis sets are presented in Table 9 for the principal ionizations along with the second-order results of Cederbaum et al_. (1975) and the experimental values. The results in Table 9 typify two general features of ionization energy calculations. The first is that Koopmans' theorem yields values which are usually higher than experimental ionization energies. Second, the inclusion of second-order relaxation and correlation corrections generally overcorrects the Koopmans 1 estimate and yields values which are usually lower than experiment. For several ionizations in Table 9, the second-order deviations from experiment are as large as the Koopmans' values only opposite in sign.Although it is possible that the larger, polarized basis used in the second calculation may still lack adequate

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78 Table 8.

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79 Table 9. Principal Ionization Energies for Formaldehyde. 24 42 Orbital

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polarization functions, the rather large discrepancies between the second-order results and experiment more probably indicate that third(and higher) order self-energy corrections are now sizable. The general conclusion that a second-order self-energy approximation is inadequate for an accurate calculation of ionization energies has been previously concluded by Cederbaum (1973b) and necessitates a re-examination of the approximation made in Eq. (3.88). Rather than completely neglecting the third-order self-energy matrix, let us now consider an approximation that includes at least part of these contributions. Which third-order self-energy diagrams should be included? There are two well-established results that are relevant to this question: Studies of the electron gas model have shown that in the limit of high electron density, the so-called ring diagrams dominate the selfenergy expansion (Pines, 1961), while in the limit of low electron density, the so-called ladder diagrams dominate (Galitskii, 1958). In order to determine whether atomic and molecular self-energies can be approximated by specific third-order diagrams (e.g. rings or ladders), we need to evaluate all third-order diagrams for some representative systems. Cederbaum (1975) has done this for several simple systems and has found that both ring ajid ladder diagrams dominate the third-order self-energy. This result implies that atoms and molecules lie somewhere between the high and low density extremes. It is therefore essential to include both ring and ladder diagrams in any third-order self-energy approximation. These diagrams are rings ladders "L 71 ..3 1-3 3.89

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81 and correspond to the algebraic expressions labeled A, B, C, and D in Appendix 1. We include six additional diagrams in our third-order self-energy approximation because of the computational efficiency with which they are evaluated. These diagrams are the energy independent diagrams * m l:ih ,,,, corresponding to expressions M-R in Appendix 1. For these six diagrams, it is feasible to employ the partial summation technique since they must be evaluated only once. Approximating the full third-order self-energy matrix by only ring, ladder, and constant energy diagrams, let us now consider the solution of the Dyson equation with the [1,1] Pade' approximant to the self-energy expansion. Owing to the fact that the inner projection manifold from which the [1,1] approximant was derived is energy dependent (Eq. (3.80)), the simple analytic form of the self-energy eigenvalues, illustrated in Fig. 2, is lost. Furthermore, the self-energy poles are now given by det (E (2) (E) E (3) (E)) = (3.91) rather than by an eigenvalue problem and are consequently more difficult to obtain. For these reasons, the pole search described in Chapter 1 and used with the second-order self-energy approximation is no longer an efficient or reliable procedure. An alternative method of solution used in the following applications was to use the Hartree-Fock orbital energy as an initial guess to the propagator pole and to iterate Eq. (1.67) to convergence. When convergent, this procedure invariably yields

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82 a principal propagator pole and its corresponding pole strength. Although the [1,1] self-energy approximant does not quarantee a positive pole strength, this was never a problem in any of the calculations reported here. The principal ionization energies for the water molecule were calculated using both the 14 and 26 CGTO basis sets in order to evaluate the [1,1] Pade' approximant to the self-energy expansion, and the results appear in Table 10. The most significant feature of these results is that each ionization energy has been shifted from its second order value to higher energy and is now in better agreement with the experimental value. It is further noticed that the valence ionization energies are still smaller than the experimental values while the la 1 (core) ionization energy is now larger than experiment. Apparently, the diagrams included in the third-order self-energy matrix overestimate the actual relaxation and correlation effects for this ionization. Convergence difficulty was experienced for the 2a., ionization energy using the 14 orbital basis. A schematic plot of VL (E) is presented in Fig. 3 and reveals that there are no propagator poles in this energy region. This anomaly is no doubt a consequence of some quirk in the basis since the 26 orbital basis yields a very accurate 2a, ionization energy. 3 . 6 Evaluation of the Diagram Conserving Decoupling The algebraic structure of the superoperator formalism has been successfully exploited in this chapter to yield several new insights into the decoupling problem. The application of perturbation theory has demonstrated that the electron propagator equation of motion can be

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83 Table 10. Comparison of Principal Ionization Energies for Water Obtained with the Second-Order and the [1,1] Self-Energies Using the 14 and 26 CGTO Basis Sets. 14 26 Orbital la. 2a. 3a. lb, lb. (2) E(E)

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Figure 3. A sketch of W2 a , in the energy region of the 2a j ionization obtained with the [1,1] selfenergy approximant using the 14 CGTO basis.

PAGE 95

85 W 2a (E) -1.215 -1.175

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resummed to yield the equivalent of the diagrammatic expansion. This resummation also allows the identification of wave and reaction superoperators which have special importance in decoupling approximations. We have shown that when the inner projection manifold of the superoperator resolvent is chosen to consist of the first-order truncation of the wave superoperator, a [1,1] Pade' approximant to the self-energy expansion is obtained. This approximant is correct through third order and contains a geometric approximation to all higher orders. In general, the Nth-order truncation of the wave superoperator will yield an [N,N] Pade' approximant which is correct through the (2N+l)st order in the self-energy expansion. One final insight afforded by this decoupling is the realization that electron correlation can be described exclusively in the operator space. We argued in Chapter 1 that when the propagator was defined as a single-time Green's function, the density operator was arbitrary. We have now demonstrated in this chapter that any desired order in the self-energy expansion may be obtained using as a specific choice, the uncorrelated, Hartree-Fock density operator. Computational applications of the diagram conserving decoupling have been encouraging. These applications have confirmed previous conclusions (Cederbaum, 1973b) that a second-order self-energy is generally inadequate for obtaining accurate ionization energies. It is important if not essential that third-order ring and ladder diagrams be included in any self-energy approximation although the errors arising from basis incompleteness may be of equal magnitude and hence cannot be ignored. The inclusion of the third-order ring, ladder, and constant energy diagrams in the [1,1] self-energy approximant has succeeded in improving the second-order results but even these results are not consistently better than the Koopmans' theorem values.

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87 One important feature of the [1,1] self-energy approximant is that even though it is constructed from only the secondand third-order selfenergy matrices, it contains a geometric approximation of all higher orders in the self-energy expansion. Certainly, some fourthor higherorder terms may be just as important as third-order terms; therefore, this approximation is highly desirable. The fourthand higher-order terms arising from the [1,1] self-energy approximant, however, are not readily analyzed diagrammatical ly. In fact, being a purely algebraic approximation, the [1,1] approximant may not yield any valid fourthor higher-order diagrams. Given the fact that ring and ladder diagrams dominate the third-order self-energy matrix, one can argue that they may also dominate the higher orders of the self-energy expansion. An appropriate modification of this decoupling scheme might then allow the summation of these specific diagrams in all orders. Approximations of this type are referred to as renormalized decouplings and are examined within the superoperator formalism in the next chapter.

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CHAPTER 4 RENORMALIZED DECOUPLINGS 4. 1 Renormal ization Theory In Chapter 3, we tacitly assumed that the application of perturbation theory to the calculation of ionization energies and electron affinities was valid and that the resulting self-energy expansion was convergent. Historically however, it was discovered that in both the nuclear many-body problem and the electron gas model, the simple selfenergy expansions are divergent. In order to remove these divergencies, it is necessary to sum certain appropriate classes of diagrams to all orders. This method of partial summations is known as renormal ization theory (see e.g. Kumar, 1962 or Mattuck, 1967) and may be viewed as an analytic continuation of the perturbation expansion. Although a variety of renormal ization procedures exist, such as propagator renormal izations, interaction renormal izations, and vertex renormal izations , the distinctions mainly depend on the types of diagrams included in the partial summation and are not particularly important for our consideration. One renormal ization that we are already familiar with is the [1,1] self-energy approximant derived in the preceding chapter. In fact, any rational self-energy approximant may be regarded as a renormal ization since its geometric expansion will approximate all orders of the perturbation expansion. One problem encountered with the [1,1] approximant and that occurs in general for rational approximants derived via purely algebraic considerations is that their geometric expansions may contain

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no readily identifiable diagrams (at least beyond the lowest orders). Since specific diagrams often dominate the self-energy expansion (such as ring and ladder diagrams for atoms and molecules) it is valuable to investigate whether the superoperator formalism can be adapted to yield renormalized self-energy expressions that sum specific diagrams. The solution as we shall see is rather simple and involves a restriction in the types of operator products allowed to span the orthogonal complement of the model subspace. As a specific example, the two particle-one hole Tamm-Dancoff approximation (2p-h TDA), (Schuck et al_. , 1973, Schirmer and Cederbaum, 1978), is derived from an effective interaction which is logically obtained by a projection of the perturbation superoperator onto the subspace spanned by 2p-h type operators (Born and Ohrn, 1979). Finally, the diagonal approximation to the full 2p-h TDA self-energy previously derived and applied to the calculation of ionization energies is shown to neglect terms which, in fact, are diagonal and are necessary to prevent an overcounting of all diagrams containing diagonal ladder parts. 4.2 Derivation of the 2p-h TDA and Diagonal 2p-h TDA Equations Recalling some of the results of the previous chapter, we had obtained the matrix Dyson equation G(E) = G^E) + Gq(E)Z(E)G(E) , (4.1) where the self-energy matrix, EJE), had the following expansion Z(E) = (a|V+VT (E)V+VT (E)VT (E)V+ . . . |a) . (4.2)

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90 Introducing the reduced resolvent of the full superoperator Hamil tonian, T(E), which is just a projection of the superoperator resolvent on the orthogonal complement T(E) = P[a6+(EI-H )P-PVP]" 1 P , (4.3) the self-energy expansion was written in closed form 2(E) = (a|Va) + (aJvT(E)Va) . (4.4) It was further shown that when the grand canonical density operator is used to evaluate the operator averages, the first-order term vanishes. When P is the exact projector of the orthogonal complement, P = I , (4.5) the term PVP in Eq. (4.3) is responsible for generating the operator products that span this subspace. The expansion of this term from the inverse and its repeated application in each order of the perturbation expansion yields larger and larger operator products which are only limited by the number of electrons in the reference state. If instead of allowing all possible operator products, we restrict them to some simple types which occur in each order, it may be possible to identify and sum specific diagrams in all orders of the perturbation expansion. The restriction of the operator products in the orthogonal complement is achieved by approximating the orthogonal projector as P = ll)(ll (4.6) where the manifold (f) contains the desired operator products. The projector P now has the effect of projecting from the perturbation expansion

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91 only those operator products which lie in the subspace spanned by {f}. The approximation to P in Eq. (4.6) must of course preserve the properties of the exact projector and should be idempotent, self-adjoint, and orthogonal to 0. Our previous experience with the operator product decouplings suggests that the set of triple operator products {a.^a } be chosen k 1 m as a first approximation to P. There is a stronger motivation for using this operator product, however. If the third-order ring and ladder diagrams in Eq. (3.70) are examined, it can be seen that between any two interaction lines there occurs only two particle lines (upgoing) and one hole line (downgoing) or vice versa. This implies that the intermediate or virtual states that are represented by these diagrams consist of only 2p-h or 2h-p excitations of the reference state. Both of these excitations are described with the triple-operator products. The set of triple products {a.a,a } is not orthogonal to the simple operators of the model subspace, hence these two subspaces must be orthogonal ized. Using the Gram-Schmidt orthogonal ization procedure (see e.g. Pilar, 1968), we define _!. .1. J. f n m = N i,i 2 [a,a,a -S(a la, a, a )a ] (4.7) klm klnr k 1 m n 1 k 1 nr n v ' = H.J [a, a n a +6. a,-6. -,a 1 (4.8) klm 1 k 1 m km k 1 kl k m v ' where N Um = --+ . (4.9) klm k k 1 k m 1 m v ' The projector P= k j 5 Jf klm )(f kl J , Km (4.10)

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92 is now idempotent, self-adjoint, and orthogonal to 0. The projection of the perturbation superoperator on the 2p-h subspace, PVP, which occurs in Eq. (4.3) can now be regarded as an effective interaction. The expansion of Eq. (4.3) with P defined as in Eq. (4.10) should yield all diagrams containing 2p-h and 2h-p excitations of the reference state. The necessary operator averages needed to evaluate PVP are: (a' 1 a 1I a ml |5a ] ) = (a, IVa+.a^.)* = N^ v (4.11) and (a k' a l * a m' ' Va k a l a m ) = N k'l'm' { « kk .(l-) -6, -, , (-)-6 ,(-) 11 11 k m ' ' mm v k 1 ; +6 , 1 (-)+6 -, , (-) 11 m 1 k m ' ' ml k 1 ' +6 kl +6, } +N klm {6 ni , k ,+6 ] . k .} (4.12) Substituting these expressions and performing some cancelation yields: p ^\j imk ,j, jm / ( kl N k'r m '<™'iln.'r>6 kk ,(i-<„ ni >-< ni >)|f k ,,, m ,)(f k1m | 1 < m 1 '< m ^j, m k^^, m ' N » mn ^' m,6 ' 1 ' (' N^N?.-,. .6 ,(-)|f, ,, , ,)(f., I k 1 m k' 1' m' k'l'm' M mm' v k 1 " k 1 m klm 1 1 < m 1 ' < m ' + T. I N.-'-fN; 2 ,,, ,6 n , (-) If. ,, , , ) (f. , I klmk'l'm' m 1 < m 1 ' < m ' + z l N M m N b-T m 6 11 (-)|f, ,,, ,)(f., I klmk'l'm' 1 < m l'< m' (4.13)

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93 Additional simplification can be achieved at this point by anti -commuting the operators a ] , and a m , in I ^^ . -j . m . ) of the last two terms in Eq. (4.13! with the appropriate change of sign, IVl'm') = " l f kVl' } • < 4 14 ) After interchanging dummy indices 1 '<->m', the fourth and fifth terms become equal to the second and third terms, respectively, and since the diagonal terms 1'= m' vanish, the summations with l'< m 1 and l'> m' can be combined as unrestricted summations over 1' and m'. Since the first term in Eq. (4.13) is obviously symmetric in 1' and m', the restriction l'< m' in that term may be removed by multiplying the sum by a factor of kThe remaining restriction, 1 6 kk' (1 ) k,l,m k',r,m' Klm K ' m kk m ' 1 < m -6 11 ,(-)-6 ,(-)}|f, M , ,)(f., I 11 k m ' ' mm v k 1 ' ' k' lm v klm 1 (4.15) is symmetric in these indices as can be verified by interchanging Wm and relabeling dummy indices l'-w-m'. Expanding the ket-bra superoperator hi E l f k'Tm' )(f klml (4 16 ^ k,l,m kM',m' K ' m klm out of the inverse, evaluating the remaining operator averages, and resumming the expansion yields the 2p-h TDA self-energy 2p-h TDA , , 2(E). = h Y, l N, s , N, 3 ,, , , 1J k,l,m k',l',m' klm kVm x {(fKEi-H^fJ-dlVDJ-^^^^^Tm'llJk^ (4.17) where

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94 (f k'Tm'K EI H o)W = ( E+e ke le J 6 kk' 6 ir 6 mm f^H'm'l 5 W BN i;W'rm'^ ral ll m,1,>fi kk'< 1 ) -6 11 ,(-)-6 mm , (- ) } (4.18) Although different in appearance, this self-energy expression is formally the same as that obtained by Purvis and Ohrn (1975a) using the operator product decoupling and by Cederbaum (1975) and Schirmer and Cederbaum (1978) using the diagrammatic method. The present derivation clearly illuminates the parallelism between the two formalisms. Owing to the large dimension of the {f, , } operator subspace and the associated difficulty in diagonal izing (f|Vf), computational applications of the 2p-h TDA have usually involved additional approximations. One approximation which has facilitated computational applications is known alternatively as the shifted Born collision (SBC) approximation (Purvis and Ohrn, 1974, 1975a) or the diagonal 2p-h TDA (Cederbaum, 1974, 1975 and Cederbaum and Domcke, 1977). This purportedly "diagonal" approximation restricts the spin-orbital summation indices in Eq . (4.13) to k' = k, l' = l, and m'=m thereby neglecting the last two summations and yielding the following self-energy expression 2p-h TDA .. | ,, , ,,., E(E).. -H I N., <;k||1m> (4 } 1J k,i, m k1m ( £+ v £ rvA where A=( l--)-(-)- (-) . (4.20) in i K ill K 1 By neglecting the last two summations in Eq. (4.13), however, this approximation actually neglects some diagonal contributions to (f. ,-,, , |Vf, , ). 3 3 v k 1 m ' klm ; As we have explicitly demonstrated in the derivation of Eq. (4.17) and

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95 (4.18), the 2p-h TDA self-energy sums are symmetric in both 1 ,m and l',m'; consequently, the last two summations in Eq. (4.13) contain precisely the same contributions as the second and third summations, respectively. If the diagonal approximation (k'=k, l'=l, m'=m) is made in Eqs. (4.17) and (4.18), this symmetry is properly accounted for and the resulting self-energy expression is 2p-h TDA ., , ,, , I,, Z(E) = h T N <1m |jk> (4 n) {) " \j>^I^WJ^ (4 ' 21) where A= ? 2( l--)-(-)-(-) . (4.22) II \ m i ) II v y m / II v y ] / \ / Eq. (4.22) differs from Eq. (4.20) by the factor of h in the first term. The inclusion of this factor in the diagonal 2p-h TDA is necessary to prevent an overcounting of diagrams with diagonal ladder parts (see next section) and typically shifts ionization energies 0.3-0.4 eV higher in energy (Born and Ohrn, 1979). 4.3 Diagrammatic Analysis In order to determine precisely which self-energy diagrams are included in the 2p-h TDA self-energy, it is necessary to expand the effective interaction matrix, (jfJVfj, from the inverse in Eq. (4.17) and diagram the resulting algebraic expressions in each order. The expansion of Eq. (4.17) yields the following terms in lowest orders 2p-h TDA , E(E). = h >: >: hL N, ? n , , ij it i i i i i km km ' ' J k , 1 , m k , 1 , m r 6i i ifin .5 . (f, n i i i Vf i i ) i kk II ' mm v k 1 m ' klnr (E+e, -Ei -e ) (E+c, -e-,-e ) (E+e, ,-e-, , -e v k 1 nr v k 1 nr v k 1 m"

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96 ( Vivi v W ( W v W k,A,m T " E+E ke r e m ,(E+ V E X-S ,(E+e k'E Te m' T + . . . ] <1V| |jk'> . (4.23) As was done in Section 3.3, the terms in Eq. (4.23) can be simplified by first restricting the summations over all spin orbitals to summations over occupied or unoccupied spin orbitals such that the occupation \. i, number factor N 2 , N, 2 ,,, , is nonvanishing. Doing this in the first term k im k I m a ° of Eq. (4.23) and then summing over the delta functions yields h y k ^ (4 ?/j a,p,q 'a p q ; * * h try + n <''» p,a,b p a b' which are the same second-order self-energy diagrams as obtained in Eqs. (3.66) and (3.67). Restricting and spin orbital summations in the second term of Eq. (4.23) yields the following expressions V y y (f I v-F ) (a oa\ 'a.p.q b.r.s ^aV? (f b,sl Vf a W > (B,^) (4 26) +v y y (f iy f n ,. ?? , \p., r.b.c <^a ! V e q» '^W ^vW (4 ' 27) +' r r <1 "Pl l ab> (f I v-f 1 <5rJJj' c> 2 p,a,b d.q.r KV? W (E+^-e,.) (4.28) +h T T , < ipll ab> . (f 1 v-f ) .. , , n i' k „ i ^ [E+c -e -e. ) ^qcd |VT pab j TE+e -c -eJ [ ' yj p,a,b q,c,d p a b' M p v q c d' Now substituting Eq. (4.18) for the effective interaction matrices, we find that the delta functions in Eq. (4.18) further restrict the spin

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97 orbital summations in such a way that only expressions (4.26) and (4. 29' are nonvanishing. After some simplification, the nonvanishing contributions are found to be h i „ a . (^l-^fel^) r\ ' 4 3 ° a p,q,r,s a p q' x a r s I v , . a,b p,q,r v a p q !K b p r E <1|^ (432) a,b p,q,r [t G a e p e q M b r V r 1T ( , a.b.c.d P (E+£ p' e a£ b )(E+ VV e d T +1 ( } a,b,c p,q (! Wb }W qVc y a,b,c p,q U e p G a V U e q G b V Expressions (4.30) and (4.33) are the only ones which represent valid third-order diagrams as written, however, by interchanging dummy indices p +* q in Eq. (4.32) and a ^+ b in Eq. (4.35), expressions (4.31) and (4.32) can be combined to yield ' 'I I 7, a ,b p,q ,r < ia| |pqxbq l larx P rHJb> L U and expressions (4.34) and (4.35) combined to yield ,,_„„„, . " "0.. a,b,c p,q T (E^-f-fll^r" 1 ^) f A < 4 37 > ,b,c p,q (t E p e a E b ,(E+e q S V _A .17

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98 which now correspond to the two third-order ring diagrams as indicated. These results verify that one of our original objectives, which was to include all third-order ring and ladder diagrams, has been achieved. The diagrammatic analysis of the third term in Eq. (4.23) proceeds in the same way as that of the first two terms, but since the effective interaction matrix appears twice, it involves considerably more algebra. For this reason, we simply display the resulting diagrams in Fig. 4. It is significant to realize that the fourth-order diagrams in Fig. 4 include not only ring and ladder diagrams but also mixed diagrams which consist of both ring and ladder parts. In the third-order analysis, the first term in Eq. (4.18) was responsible for yielding the ladder diagrams while the second and third terms yielded the ring diagrams. If we therefore denote the first term as a ladder part and the second and third terms as ring parts, the mixed diagrams in fourth order dre found to arise from the product of a ladder part and a ring part. Inducing the results of the fourth-order analysis to higher orders, we conclude that our second objective, which was to sum all ring and ladder diagrams in all orders of the self-energy expansion, has been exceeded: not only are all ring and ladder diagrams included in the 2p-h TDA self-energy, but also the mixed diagrams which exhibit both ring and ladder parts. The diagonal 2p-h TDA self-energy may also be analyzed diagrammatically. This analysis is even simpler than for the full 2p-h TDA since the denominator shifts are now scalars rather than matrices. A comparison of diagrams obtained with the denominator shift in Eq. (4.30) versus that in Eq. (4.22) will reveal the significance of the factor of 'i. Considering the approximation in Eqs. (4.21) and (4.22) first, we obtain the following third-order expressions

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Figure 4. Fourth-order self-energy diagrams arising from the 2p-h TDA.

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100 'G TG ....3 1_.0 0... TIL ..:.:o '0.. TO. <*-r ..J>

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101 ! 4 i 4 a,P,q ( E+ W £ q )2 .j, v L a,p,q (E+e -e -e ) 2 a p q , -^.3 q (4.38) (4.39) (4.4o; + E a.b.p ( E+VV e b ) 2 (4.41) The only difference between these diagrams and the third-order diagrams of the full, 2p-h TDA is that the incoming lines on the middle interaction line have the same labels as the outgoing lines. Now analyzing the approximation in Eqs. (4.19) and (4.20), we obtain the following thirdorder expressions jj E 2 a, P ,q (E+VVV 2 (4.42) __,, y 2 a,b,p ( E ^p-^-%) 2 14.43) t. na I I pax-ap | |apxpq| | ja> 2 a,p,q (E+e -e -e ) K ' v a p q ; (4.44) + v a.b.p ( E+V c a £b ) 2 (4.45;

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102 Expressions (4.44) and (4.45) are identical to (4.40) and (4.41) respectively; however, expressions (4.42) and (4.43) both differ by a factor of h from (4.38) and (4.39). This discrepancy is a direct consequence of the missing factor in the denominator shift, Eq. (4.20), and leads to an overcounting of these third-order diagonal ladder diagrams since, by rule 5 in Table 5, there should be a factor of H for each pair of equivalent lines. Similarly, it is rather easy to show that this approximation overcounts all higher order diagrams containing this diagonal ladder part. 4.4 Computational Applications and Evaluation of the Diagonal 2p-h TDA Self-Energy ~ " ' The main attraction of the diagonal 2p-h TDA self-energy for the calculation of ionization energies and electron affinities is its pseudo second-order structure. Computational experience with the diagram conserving decouplings has taught us that the second-order self-energy approximant is both easily constructed and evaluated. The diagonal 2p-h TDA requires only the additional evaluation of Eq. (4.22) which merely shifts the second-order self-energy poles. As a consequence the diagonal 2p-h TDA self-energy mimics the exact self-energy by possessing only simple poles. Another consequence of the pseudo second-order structure is that the energy dependence will have the simple analytic form illustrated in Fig. 2. This property, which was absent in the [1,1] selfenergy approximant, simplifies the pole search for the electron propagator. In spite of the simplicity of its pseudo second-order structure, however, the diagonal 2p-h TDA self-energy incorporates diagonal ring, ladder, and mixed diagrams in all orders of the self-energy expansion

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103 as was shown in the previous section. This property encourages speculation that significantly more relaxation and correlation will be accounted for in this decoupling than with the second-order decoupling, but the accuracy of the corresponding ionization energies or electron affinities can only be evaluated via actual computational applications. As was done with the diagram conserving decouplings, ionization energies for the water and formaldehyde molecules were computed using the diagonal 2p-h TDA self-energy. For the water molecule, calculations were performed with both the 14 and 26 contracted Gaussian orbital basis sets described in Chapters 2 and 3. The principal ionization energies computed with the diagonal 2p-h TDA and the diagonal 2p-h TDA plus third-order constant energy (2) (2) (3) diagrams (denoted e(E) shjft and Z(E) SHIFT + Z [<*>)) are presented in Table 11. Comparing these results with those in Table 9 for the second-order and [1,1] self-energy approximants reveals that each ionization has been shifted to higher energy. This shift has led to a significant improvement in the valence ionization energies (3a,, lb., and lb ? ) which are now within approximately 0.5 eV of the experimental results. For the inner valence (2a,) and core (la.) ionizations, however, this energy shift leads to worse agreement. In addition, the diagonal 2p-h TDA results for the la, ionization exhibit an enormous basis dependence. The addition of polarization functions in the 26 orbital basis has yielded nearly 13 eV in additional relaxation. The most probable explanation for this basis dependence is that the 2p-h TDA self-energy poles are determined by Hartree-Fock orbital energies and two-electron integrals rather than by orbital energies alone as with the second-order self-energy. The orbital energies are rather insentitive to basis changes, whereas the two-electron integrals are not.

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104 T"!

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105 Calculations for the formaldehyde molecule were performed with the 24 and 42 orbital basis sets described in Chapter 3. Ionization energies were computed using the diagonal 2p-h TDA and are presented in Table 12. The third-order constant energy diagrams were not evaluated for this molecule. Similar to the water results, the diagonal 2p-h TDA results for formaldehyde are also consistently higher in energy than the secondorder results (Table 9). This shift considerably improves the valence ionization energies; however, the average deviation from the experimental results remains approximately 0.8 eV. The core ionizations within the diagonal 2p-h TDA suffer a small deterioration in accuracy but do not exhibit the extreme basis dependence which was observed in the water calculations. Part of the discrepancies between the diagonal 2p-h TDA and the experimental results can certainly be eliminated by further basis saturation; however, in the next chapter, we will propose that even ionization energies of 1.0 eV accuracy are usually sufficient to unambiguously interpret photoelectron spectra if combined with a calculation of relative photoionization intensities. Cederbaum and co-workers have recently developed computer programs which implement the full, nondiagonal 2p-h TDA to the self-energy and have reported several molecular applications (Cederbaum et al_. , 1977, Schirmer et aj_. , 1977, Cederbaum ejt aj_. , 1978, and Schirmer et al . , 1978). In these calculations, they claim only a 1.0 eV accuracy and rely heavily on vibrational analyses to assist with the interpretation of photoelectron spectra. The off-diagonal matrix elements seem to have little importance in the valence region because the propagator poles are relatively well-separated. In the inner valence and core regions, however, where principal ionization poles and shake-up poles overlap and

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106 Table 12. Formaldehyde Results Obtained with the Diagonal 2p-h TDA Self-Energy. 24CGT0's 42 CGTO's (2) (2) Koopmans i(EL,, T[ --r Koopmans E(E),-utct Exp. la l

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107 interact, level shifts and intensity changes are observed. In these regions, even the nondiagonal 2p-h TDA is not fully satisfactory since ion-state relaxation and hole-hole interactions, neither of which are described by this self-energy approximation, may also be important (Wendin, 1979).

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CHAPTER 5 PHOTOIONIZATION INTENSITIES 5.1 Introduction The evaluation of each decoupling approximation in the preceding chapters was based on the comparison of propagator poles to experimental ionization energies. This criterion represents a particular bias since it does not reflect the quality of the Feynman-Dyson amplitudes (defined in Section 1.1). The Feynman-Dyson amplitudes determine the spectral density function (Linderberg and Ohrn, 1973) £ f k (x)f*(x')S(E-E k ) E > u k A(x,x ' ;E) = 'I 9 k (x)g"(x')6(E-E k ) E < u (5.1) which contains a plethora of useful information. This is evidenced by the relation of the spectral density to the single-particle, reduced density matrix (Linderberg and Ohrn, 1973) Y(x.x') = A(x,x';E)dE . (5.2) It is important, therefore, to choose a decoupling approximation which not only yields accurate ionization energies but also an accurate spectral density. The quality of the spectral density is somewhat more difficult to evaluate since there are no experimental data with which it can be directly compared. One evaluation procedure, however, might involve the calculation of averages of specific one-electron operators from the reduced density matrix. The averages could then be compared with 108

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109 experimental results. Another procedure which is more closely related to our goal of interpreting photoelectron spectra is the calculation of photoionization intensities or cross-sections. A theoretical prediction of relative photoionization intensities can simplify orbital assignments when the ionization energies are not accurately given. A theoretically predicted variation in relative intensities with photon energy is particularly useful if photoelectron spectra are available with different ionization sources (Katrib et aj_. , 1973). In the following sections, we derive computational expressions which relate the Feynman-Dyson amplitudes to the total photoionization cross-section, discuss the major approximations assumed, and then present several applications. 5.2 Derivation of Computational Formulae for the Total Photoionization Cross-Section The differential cross-section for photoionization derived from first-order, time-dependent perturbation theory using a semi-classical model for the interaction of radiation and matter is (Bethe and Salpeter, 1957, Kaplan and Markin, 1968, Smith, 1971) da 4tt Ik J s_ _ ' f ' f c|Ap.| co k K K (5.3) In this equation, k f is the momentum of the ejected photoelectron and A = I A, is the vector potential. For a closed-shell system, the initial k K state |N> can be represented by an anti symmetrized N-electron wavefunction |N> = $ (x 1 ,x 2 , . . . x N ) (5.4) and the final state |N,s> is represented by

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110 |N,s> = (N/2)' 5 AS [v(t f ,r)a(c)* se (x 1 ,x 2 , . . . x^j) v(t f) f)3(?)$ sa (x r x 2 , . . . x^)]. (5.5) Here $ and $ are the two, doublet spin components of the (N-l)electron ion, v(lL,r) denotes the wavefunction of the photoelectron, and Or,,is an antisymmetrizer °AS = N ~ 1 ^ \l t P kN ] • ( 5 6 ) The form of Eq. (5.5) guarantees that the singlet spin symmetry of the system is preserved. Evaluating the matrix element in Eq. (5.3) yields (Purvis and Ohrn, 1975a) = /I | v*(t ff r)^ s ^ g s (r)d^ + /? J V*(t f ,f)p s (f)dr (5.7) where N' % g s (r) x (c) = | 0* (x r . . . x N _ 1 )$ (x r . . . x N _ i; r, x (c)) x dXj . . . dx N _ } (5.8) and N _is P s (r) x (c) = (N-l) | $* (x p . . . Y 1 )^-V 1 $ (x 1 , . . . x N _ r r,x(c))dXj . . . dx N _ 1 . (5.9) The first term in Eq. (5.7) relates the Feynman-Dyson amplitude g (r) to the photoionization cross-section and the second term arises from the nonorthogonal ity of v(k f .,r) and $«. When v(£ f ,~r) is strongly orthogonal to $ , this term vanishes. Even when v(k^,f) is not strongly orthogonal to $ , the first term in Eq. (5.7) will dominate if the kinetic energy

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Ill of the photoelectron is much greater than its binding energy (Rabalais et aj_. , 1974), therefore we shall neglect this term: da s 3ir 2 |t c|A r v*(t f ,r)X s -^ s g s (r)df 5.10! Further simplification can be obtained by neglecting retardation of the photoelectron momentum by the photon momentum (also called the dipole approximation in analogy with photon induced transitions in the discrete spectrum, Bethe and Salpeter, 1957, Steinfeld, 1974). The vector potential A s has the following plane wave decomposition in terms of the incident photon momentum k and polarization n /t s = \t Q \ exp (il< u • r s )n . (5. 11) If the wavelength of the incident photon is large compared with molecular dimensions, the exponential may be approximated by the first term in its Taylor series expansion (unity) A |A |n 5.12 With this approximation, the differential photoionization cross-section becomes da s d£L where ? = Btt 2 I ]<, n v (k f ,r)Vg s (r)dr . (5.13) (5.14)

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112 Owing to the random orientation of molecules in a gaseous sample, the experimentally observed photoionization intensities in the molecular reference frame represent an average over all incident photon directions. Furthermore, if the incident photon beam is unpolarized, we must also average over photon polarizations. Making the appropriate averages in Eq. (5.13) (Smith, 1971), we obtain do [ 8tt 2 ||< |^ , Since the two polarization directions, n, and rL, and the incident photon direction, ky[kj, form a right-handed system of axes, we can write i^i 2 = iv?i 2 + iv^i 2 + iV^i 2 / ^f < 5 16 ) and Eq. (5.15) becomes [-nr-J mj^ 2 -\Kf \ 2 ntjh^ (5.i7) da s_ (1 cos 2 9 }dfi (5.18) WO) v ' da 8tt 2 | ft f d^ = -EST 1 " HT • (5.19) In order to evaluate |f*| , some form for the photoelectron wavefunction v("k f ,r) must now be chosen. In principle, the photoelectron wavefunction could be obtained by the solution of the Bethe-Salpeter equation for the polarization propagator where the superoperator resolvent has been modified to include the time-dependent interaction of the radiation and matter fields (see e.g. Csanak et aJL , 1971). A solution

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113 of this type would require the use of continuum functions as well as discrete functions in the molecular basis and is not yet feasible owing to several formal and practical difficulties. Alternatively, we seek a simple but accurate, analytic representation of the photoelectron wavefunction. For photoionization of atoms or molecules with high (tetrahedral or octahedral) symmetry, the electronic potential of the ion is nearly spherically symmetric, and v(t f ,f) may be asymptotically represented by a plane wave plus incoming Coulomb waves (see e.g. Smith, 1971). For molecules with lower symmetry, distortions of the electronic potential enormously complicate the nature of the incoming waves. In this derivation, the incoming waves are neglected, and the photoelectron is simply represented by the plane wave part v(l< f ,r) = (2tt)" 3/2 exp{it f r) . (5.20) The applicability and implications of this approximation will be discussed in the following section. With this choice, Eq. (5.14) can be integrated by parts and yields ? = (2^)" 3/2 | g s (r)v^exp(-i£ f • r)dr (5.21) = -i£ f (2^)" 3/2 I exp(-Ht f • r)g s (r)dr . (5.22) In our computational scheme, the Feynman-Dyson amplitudes in Eq. (5.22) are represented by a linear combination of Hartree-Fock orbitals. The Hartree-Fock orbitals can be decomposed into contracted Cartesian Gaussian functions on each atomic center, and each contracted Gaussian function can be further decomposed into a sum of primitive Gaussian functions. Ultimately therefore, the Feynman-Dyson amplitudes can be

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114 represented as some linear combination of primitive Gaussian functions on each atomic center g s (r) = I c^ ^^ (r ft J . (5.23! a,k 'ak ak • ,
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115 >s Q \ t = $ u(M $ i(k*)dn# (5.27) ak,al J ak f al f f w,t '' where $ . (k^) represents the Fourier transform * ak (t f ) » (2tt) _3/2 J exp(-it f • r' )* ak (r' )dr ' (5.28) and the two-center terms (a^3) are given by Q ak,61 = j Eq. (5.30) becomes m 1 -, m..m„-m 1(1 m l ? m ) = 4tt l l (-i ) 'j (k f R JC, V , y lm (e,rf>). (5.32) 1 1=0 m=-l '' '12 In these equations, {$*,§„) represents the photoelectron direction in the atomic reference frame, ( B , cb ) represents the direction of R „in the r aB molecular frame, j -. ( k^R ) is a spherical Bessel function of 1-th order, m]m2-m and C] i i is a Clebsh-Gordan coefficient defined by (Harris, 1973)

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116 m,m„m \i z i " J \ m ^r^ y i^r^ y iJ^f)^ f (5.33) When the distance vector ft coincides with the molecular z-axis, the expansion in Eq. (5.31) simplifies to exp(±lt f -ft a0 ) = (4-rr)F, (±1 ) ] (21+1)^ (y^g)^^ ,* f ) (5.34) 1 and Eq. (5.30) becomes 1 1 -, + l^j i, 1 ^ i i m 1 m 9 1(1 l m l' , 2" , 2»-<*'>\m 2 ,.„*_ , i'"" (21+1)SC l}l 2 l V k fV (5.35) For arbitrary directions of ft which do not coincide with the molecular z-axis, a new coordinate frame can be defined in which ft „ does coincide with this axis, and the spherical harmonics can be transformed to this coordinate frame using the familiar rotation matrices (Schweig and Thiel , 1974). In this way, Eq. (5.35) can be used to evaluate all two-center integrals arising from any molecular geometry. 5.3 Discussion of Approximations Relatively little work has been devoted to the theoretical calculation of photoionization cross-sections for molecules compared with that for atoms (see e.g. Marr, 1967, Steward, 1967, Kelly, 1976). The major impediments until recently have been a lack of sufficiently accurate molecular wavefunctions and an absence of accurate analytic representations for the photoelectron wavefunction (Kaplan and Markin, 1967, Schweig and Thiel, 1974). With the development of efficient, molecular integral and Hartree-Fock programs, Hartree-Fock wavefunctions are now readily available for a large number of molecules. This availability in turn has stimulated several theoretical calculations of molecular

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117 photoionization cross-sections using the frozen orbital approximation (Rabalais et aj_. , 1974, Dewar et al_. , 1975, Allison and Cavell, 1978, and Cavell and Allison, 1978). In the frozen orbital approximation, the N-electron reference state is assumed to be the Hartree-Fock qround state, and the ion states are constructed by removing the orbital corresponding to the ionized electron. This approximation neglects both correlation corrections in the ground and ion states and ion state relaxation (see Introduction). Following Purvis and Ohrn (1975a), we have replaced the frozen orbital approximation by a many-electron treatment which incorporates both relaxation and correlation corrections. This treatment derives from the use of the Feynman-Dyson amplitudes obtained from the electron propagator to compute the photoionization cross-section. The form of the photoelectron wavefunction is still a major problem in the calculation of molecular photoionization cross-sections and represents the most critical step in our derivation. The plane wave approximation was chosen for its simplicity, however, it has several serious limitations. The most serious limitation is its failure to correctly predict experimentally observed angular distributions (Bethe and Salpeter, 1957, Schweig and Thiel, 1974). This deficiency is not readily apparent when the differential cross-section is averaged over all photoelectron directions, and in our computational applications, we compute only spherically averaged, total cross-sections. Lohr (1972) has shown that by retaining the second term in Eq. (5.7), qualitatively correct angular distributions may be obtained. The retention of this term is equivalent to orthogonalizing the plane wave to every one-electron function in the N-electron ground state and is known as the orthogonal ized plane wave (OPW) approximation.

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118 A second limitation of the plane wave approximation is the implicit neglect of electrostatic interactions between the photoelectron and the molecular ion. For ionization processes near threshold where the photoelectron leaves with low kinetic energy, these interactions are especially important, and the wavefunction of the photoelectron exhibits a decrease in wavelength as r -> fj. The OPW approximation again partially corrects this deficiency (Lohr, 1972) but exhibits a rather abrupt change in wavelength which is more characteristic of a short-range potential rather than the long-range Coulomb potential. Other representations of low energy photoelectrons, e.g. multicentered Coulomb wave expansions, have also been proposed (Tuckwell, 1970). As the kinetic energy of the photoelectron tends to higher energies, the effect of electrostatic interactions on the total cross-section becomes negligible, and the plane wave approximation becomes sufficiently accurate (Bethe and Salpeter, 1957). Unfortunately, as the plane wave approximation becomes more suitable at high photoelectron energies, the dipole approximation (Eq. (5.12)) deteriorates and must be re-examined. Bethe and Salpeter (1957) have shown that when retardation effects are included in the calculation of differential photoionization cross-sections, the lowest order correction is proportional to (|v|/c), where v is the photoelectron velocity. This correction, however, only effects the angular distribution and vanishes when the differential cross-section is averaged over all incident photon directions. Higher-order corrections for retardation are proportional to (|v|/c) as are the relativistic corrections, but these are usually negligible even when the photon wavelength is comparable to the molecular dimensions.

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119 5 . 4 Computational Applications In this section, the relative photoionization intensities for the water and acetylene molecules are computed using the Feynman-Dyson amplitudes obtained in the second-order, diagram conserving and the diagonal 2p-h TDA, renormalized decoupling approximations. Comparative calculations corresponding to a Mg K photon source (tioj = 1253.6 eV), for which the plane wave approximation is expected to be good, and a He (II) photon source (nw = 40.81 eV), for which the plane wave approximation is expected to be less accurate, are presented. These results are further compared with intensities obtained using the frozen orbital approximation in order to assess the magnitude of many-electron correlation and relaxation corrections. For acetylene, the dependence of intensity on photon energy for the valence orbitals is plotted, and orbital and density difference plots are presented and discussed. The relative photoionization intensities for water corresponding to a Mg « a photon source and a He (II) photon source are presented in Tables 13 and 14, respectively. These results were obtained using the 26 contracted Gaussian orbital basis at the experimentally determined equilibrium geometry as described in detail in Section 3.5 and are scaled relative to the 3aj intensity. As expected, the relative intensities computed for the Mg K source in Table 13 compare reasonably well with those obtained experimentally (Rabalais et al_. , 1974). The orthogonal ized plane wave results of Rabalais et al_. (1975) are presented for comparison and also correctly order the relative intensities. It is interesting to note the larger discrepency between the lb, and lb ? intensities in our calculations compared to Rabalais et a]_. than in the 2a, and 3a-, intensities. Since for X-ray photons the OPW corrections are not expected

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120 Table 13. Relative Photoionization Intensities for Water Excited by Mg K (1253.6 eV). Orbital

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121 Table 14. Relative Photoionization Intensities for Water Excited by He(II) (40.81 eV). a (2) (2) ( 3 ) h Orbital F0 d l(E) z(E) SHIFT + E(») 0PW D Exp. 2a l

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122 to be large, this discrepency is most likely a basis set effect. Our basis contained d-type polarization functions on oxygen as well as p-type polarization functions on the hydrogen atoms, whereas Rabalais et al_. have used only a minimal basis. The relative intensities presented in Table 14 do not correctly match the experimentally observed intensities. This is not particularly surprising owing to the inadequacy of the plane wave approximation at low photon energies. It is more surprising that the OPW results also yield an incorrect ordering of the intensities. The failure of the OPW approximation indicates the necessity for a more accurate photoelectron wavef unction. Many-electron correlation and relaxation corrections appear negligible in the intensities of both Tables 13 and 14. The deviations between the frozen orbital approximation (FO), the second-order self-energy (2) approximation (z(E)), and the diagonal 2p-h TDA with third-order constant (2) (3) energy diagrams (x(E) SHIFT + E(°°)), are small and may be attributed to differences in the photoelectron momenta. Different ionization energies obtained with different decoupling approximations were used to compute the photoelectron momenta from the conservation of total energy |k f | = [2(w I.P.)] ?S (in atomic units) , (5.36) consequently, deviations in the I.P.'s result in deviations in |k\-|. The relative intensities computed with the diagonal 2p-h TDA plus third-order constant energy diagrams for the Mg K photon source are plotted in Fig. 5. In order to compare the theoretical spectrum to the experimental ESCA (Electron Spectroscopy for Chemical Analysis) spectrum (Siegbahn e_t al_. , 1969) which is sketched as an insert, the ionization

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Figure 5. A plot of the theoretical ESCA spectrum for the valence ionizations of water. The experimental spectrum of Siegbahn et a]_. (1969) is sketched in the insert.

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Kii.uft.Lii'i^ii!) mm m>mm \m mm 124 JO Ti 20 \i .;--h :-'!'")i. %> "^ f'rcioi 0.'/H! ; /.'h 0LUQ.

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125 lines were given a Lorentzian width estimated from the experimental spectrum. The major features of the experimental spectrum are wellreproduced with the exception of the weak peak at approximately 23 eV. This peak has been attributed to the ionization of 2a, electrons by the Mg Kqj satellite in the photon source (Siegbahn et al_. , 1969) and therefore should not appear in the theoretical spectrum since it was calculated for a purely monochromatic source. A recent experimental study of relative photoionization intensities of acetylene with various photon sources (Cavell and Allison, 1978) motivated our examination of this molecule. As preparatory steps to the calculation of relative intensities, Hartree-Fock and electron propagator calculations were performed at the experimental equilibrium geometry, R(C-C) = 2.279 a.u. and R(C-H) = 2.005 a.u. (Buenker et al_. , 1967), with two different basis sets. The first basis consisted of Huzinaga's 9s, 5p primitive basis for carbon and 4s basis (unsealed) for hydrogen (Huzinaga, 1965) contracted with Dunning's coefficients (Dunning, 1970) to 4s, 2p on carbon and 2s on hydrogen (see Table 8). The complete molecular basis consisted of 24 contracted Gaussian orbitals and yielded a Hartree-Fock total energy of E(HF) = -76.7948 H. The second basis augmented the first by the addition of a set of d functions on each carbon atom (ex , = 0.60) and a set of p functions on each hydrogen atom (a = 0.75). The addition of these diffuse, polarization functions brought the size of the basis to 42 contracted Gaussian orbitals and yielded a Hartree-Fock total energy of E(HF) = -76.8267 H. Valence ionization energies were computed with the second-order self-energy approximation and the diagonal 2p-h TDA for each basis. The results for the 24 orbital basis are presented in Table 15 and the

PAGE 136

126 Table 15. Valence Ionization Energies for Acetylene (24 CGTO's). Orbital

PAGE 137

127 results for the 42 orbital basis appear in Table 16. In each table the full, nondiagonal 2p-h TDA results of Cederbaum et aj_. (1978) and the experimental results of Cavell and Allison (1978) are included for comparison. In contrast to previous calculations reported here, the ionization energies obtained for acetylene are larger (with the exception of the 1it u ionization) than the experimental values. A breakdown in the quasi-particle picture for the 2c ionization which is evidenced by the shake-up pole may account for the particularly poor results for these ionizations. The off-diagonal 2p-h TDA contributions considerably improve these two ionization energies (Cederbaum et aj_. , 1978), however, the shake-up energy still disagrees with the experimental value by more than an electron volt. Relative photoionization intensities for acetylene are presented in Table 17 for a Mg K photon source and in Table 18 for a He (II) photon source. In both tables the intensities were scaled relative to the Itt u intensity. As for water, the intensities corresponding to the Mg K source are in reasonable agreement with experimental intensities and show only a slight diminution when correlation and relaxation effects are included. The OPW result of Cavell and Allison (1978) for the 2a 9 ionization in Table 17 seems unexplicably high, and the 3a ionization is not observed experimentally (Cavell and Allison, 1978). The relative intensities for acetylene computed with the He (II) source exhibit better agreement with the experimental results than did the water results. Here, only the relative intensities of the two weakest ionizations, the 2a and 2a , were reversed. The OPW results predict the correct ordering but attribute a much weaker intensity to the 2a ionization than is observed experimentally.

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128 Table 16. Valence Ionization Energies for Acetylene (42 CGTO's). Orbital

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129 Table 17. Relative Photoionization Intensities for Acetylene Excited by Mg K (1253.6 eV). Orbital

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130 Table 18. Relative Photoionization Intensities for Acetylene Excited by He(II) (40.81 eV). Orbital

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131 The relative intensities in Table 17 computed with the diagonal 2p-h TDA and Mg K source have been plotted in Fig. 6. As in Fig. 5, the lines have been given Lorentzian widths and the experimental spectrum appears as an insert. The most striking difference between these two spectra is the absence of the intense shake-up in the theoretical spectrum. Not only was this peak predicted to lie 7 eV higher than the experimental peak in energy, it also yielded a relative intensity several orders of magnitude smaller than the 2a intensity (at 23.4 eV). The weak experimental peak at about 14.5 eV arises from the ionization of 2a electrons by Mg K^ . radiation (Cavell and Allison, 1978) and is absent in the theoretical spectrum. The 3a peak which should occur at 16.8 eV is not observed experimentally but can be identified as a shoulder on the 2a peak in the theoretical spectrum. Figure 7 shows the dependence of the photoionization cross-section on photon energy from 0-200 eV. Over this energy range, ionization from the Itt orbital is predicted to be most probable, and this is verified by the experimental data of Cavell and Allison (1978) obtained at 21.2 eV (He ( I )), 40.8 eV (He (II)), and 151.4 eV (Zr M ). Another interesting feature of this figure is the slight minimum exhibited by the 2a curve around 125 eV. Minima of this type in the photoionization cross-section were first predicted by Cooper (1962) and are referred to as Cooper minima. Cooper minima have been observed in the cross-sections for a number of atoms by means of X-ray and ultraviolet absorption spectroscopy (see e.g. Codling, 1976); however, at present there are no photoelectron spectroscopic data of this type owing to the limitation of photon sources which are available. The application of synchrotron radiation to photoelectron spectroscopy should soon offer a means of studying the energy

PAGE 142

Figure 6. A plot of the theoretical ESCA spectrum for the valence ionizations of acetylene. The experimental spectrum of Cavell and Allison (1978) is sketched in the insert.

PAGE 143

133 Kiiu^^jWii mm mmmm m\ nmnmrn . on ij . G:"i :]/.U 0.40 i].^n a. in a . ou 30 25 ?[\ IS 10 mmm mm* vm •fr.UH, r ? •. i f jiji-.ton IV'.-:!oj 0. T /S':-/7J l'IV:r.t

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Figure 7. A plot of the photoionization cross-sections versus photon energy for the valence orbitals of acetylene in the region 0-200 eV.

PAGE 145

135 KITKK GULII^LE S3 t IS, u **> n E3 M Pi ea 1 10 Li D.4 0.8 1.2 mim mwjm mm isi INENi Hclnlosh Plot. Sy5t.an te
PAGE 146

136 dependence of the orbital cross-sections in a continuous energy interval from to about 200 eV (Codling, 1973). Orbital plots are presented for the 2a , 3a , and Itt Feynman-Dyson u g u J amplitudes in Fig. 8. Since the dominant component in each of these amplitudes is the corresponding Hartree-Fock orbital (i.e. 2o , 3o , and Itt , respectively) the correlation and relaxation corrections are not readily observable. In order to examine these many-electron contributions more readily, density difference plots between the amplitudes of Fig. 8 and their corresponding Hartree-Fock orbital were made and are presented in Figs. 9-11. Since the Feynman-Dyson amplitudes are not normalized, it is necessary to normalize them before computing the density difference. In all of these plots, the Hartree-Fock density is negative which means any positive distortions imply density enhancement in the propagator amplitude while negative distortions imply density diminution. Figure 9 shows a density diminution in both the C-C and C-H bonding regions with a density enhancement in the anti-bonding region. Figure 10 shows an enhancement in the C-C pi bonding, and Fig. 11 shows an enhancement in the C-H bonding with a slight diminution of anti-bonding character. It is apparent from the numerical results presented in this section that the calculated relative photoionization intensities, at least within the plane wave approximation, are not very sensitive to improvements in the Feynman-Dyson amplitudes. It is likely that more accurate photoelectron wavefunctions will improve the sensitivity of these quantities but not at the orthogonalized plane wave level. Although the OPW approximation is more justifiable formally, the results of Rabalais et aj_. (1974) and Cavell and Allison (1978) do not exhibit any marked improvements to the plane wave results in the two molecules studied here. Qualitative

PAGE 147

Figure 8. Orbital plots for the 2c? u , 3a g , and 1tt u Feynman-Dyson amplitudes of acetylene. Approximate scales on the plots are ±0.72 a.u. for 2o u , ±0.28 a.u. for 3a a , and ±0.29 a.u. for 1tt u . y

PAGE 148

138 Mills ^-"£22^ =%. 2o 1 ^ t , f^ ^ .r^^''J^-iJ. l | f.. , /? w 5 SJPfi I AwA •?=!il tea in i m. I Iff liuilf™ \\n 'lilnvii'i'iiii'ii'iililiii IIIIIMIillllllllilllll' liliii.ilili'i'.lliiri lil In, V,-:'. J3'M I ! 'i!':'-'' i',i|i' ! i; Vi';i 5>^ShW»2*!i lTT

PAGE 149

Figure 9. A density difference plot between the 3a q Feynman-Dyson amplitude and the 3og Hartree-Fock orbital of acetylene. Each contour represents a density increment of about 1.5 x 10-4 atomic units.

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140 4 A'YiJ i I ft iiipBiii Wi, . . .'rl'-n . ST" ' / / / W :;;:'' !ffe,: f. it'll /.:y>zi \ " : mm, mm 1 ; » -a JUu '.',.1,1 \\ 1 .'I I \\ ..'-"V / i ilifMp)!J ''.v.'',';a Ik \lSf1piIf|| : :S'/ V wot \! Win "•

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Figure 10. A density difference plot between the 1tt u Feynman-Dyson amplitude and the 1tt u Hartree-Fock orbital of acetylene. Each contour represents a density increment of about 3.6 x 10"3 atomic units.

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142 ® W, Hill i//,*;^ .iv,i. mBi miwm Wi)\W ^' !n if iH'.'l •vi 1 ,^: :yu$\\]:li / , .....,hI I I : ! 111 4R •II Ml A ./I i I''.'.' H \\i ", i V 1 ,' ii'i'Vf. wh .' I u .-. r ^ .>,';h\V',i '.'>'. 'i' '"' '.'" I in',.. ". . ", l ,V,.,..'<'.;\ -rprp-' \7 ( ( I--" ;''; l!."'' •'''.' ','',',' 'i'i'Vi;,'.'. ' ;v , v L .' ' ! Hi;. '"-:r::.[].v:rTi>.

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Figure 11. A density difference plot between the 2a u Feynman-Dyson amplitude and the 2a u Hartree-Fock orbital of acetylene. Each contour represents a density increment of about 4.4 x 10 _ 4 atomic units.

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144 *K « I \ v \ \ mk\ :i \ \ \ ll'v: .'".\'!tii:::. , :vv.-. 5 "^"'-r^Wil.'Hj,,,. ui&ti

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145 changes in the propagator amplitudes may be observed via density difference plots; however, more quantitative comparisons must await other computational applications.

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CONCLUSIONS AND EXTENSIONS The primary objective of this dissertation has been the investigation of alternative decouplings which may allow more accurate calculations of the electron propagator with less computational expenditure than the operator product decouplings. Contrary to previous contentions that decouplings of the propagator equations of motion and choice of reference state averages are interrelated (Oddershede and J0rgensen, 1977), we have demonstrated that they are in fact independent approximations, hence justifying the systematic improvement of one or the other. We have chosen to examine the decoupling approximation while maintaining a Hartree-Fock reference state since these reference states have a simple theoretical representation and are efficiently generated by a number of available computer programs. Since Hartree-Fock reference states ignore electron correlation (by definition), the correlation and relaxation corrections to the electron propagator must be exclusively incorporated through the decoupling approximation. The superoperator formalism which was originally introduced as a notational convenience has a rich algebraic structure that has been largely unexploited. As we demonstrate in Chapter 3, the superoperator Hamiltonian may be partitioned and a perturbation theory may be developed in the operator space. The perturbation expansion of the superoperator resolvent readily yields the Dyson equation and allows an identification of wave and reaction superoperators . The definition of these auxiliary superoperators elucidates the parallelism between the superoperator 146

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147 formalism and the diagrammatic expansion method and permits a unified conceptual framework. When the superoperator resolvent is approximated by an inner projection, various choices of the inner projection manifold which correspond to different decouplings may be viewed as different approximations to the wave superoperator in the following equation (Born and Ohrn, 1977] |h) = W(E) |a) . In particular, the operator product decoupling corresponds to the choice W = I + {a^} ~ a + -j+ W = I + {a, a,} + {a, 1 a, a a } k 1 u ran the moment conserving decoupling corresponds to W = I + H W = I + H + H 2 * . . , and the diagram conserving decoupling of the Pade' type corresponds to W(E) = I + T Q (E)V W(E) = I + T (E)VT Q (E)V Alternatively, the superoperator resolvent may be approximated by an outer projection in which case decoupling approximations correspond to various approximations to the reaction (or self-energy) superoperator. Two decouplings of this type which have been presented are the simple

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148 diagram conserving decoupling which corresponds to t(E) = V + VT(E)V with T(E) = T Q (E) T(E) = T Q (E) + T (E)VT Q (E) * 5 and the renormalized decoupling which corresponds to t(E) = V + VT(E)V with T(E) = T Q (E) + T Q (E)MT(E) and where M is an effective potential obtained by projecting the perturbation superoperator onto a subspace of the full operator space. When this subspace is chosen to consist of simple operator products, this decoupling is formally equivalent to the operator product decoupling, however, other approximations may be envisioned. Each of the above-mentioned decoupling approximations has been studied in detail computationally and has been evaluated on the basis of a comparison of the propagator poles with experimental ionization energies. The moment conserving decouplings discussed in Chapter 2 were found to yield unacceptably poor numerical results. It is plausible that since the moment matrices involve averages of various powers of the full superoperator Hamiltonian, the moment expansion may not be convergent anywhere in the complex plane. If this is the case, analytic continuation cannot be performed, and the Pade' approximant method will not improve convergence. In any event, it may be generally concluded

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149 that the number of conserved moments is no indication of the accuracy of this decoupling. A more successful approach is the diagram conserving decouplings presented in Chapter 3. A simple second-order self-energy truncation was found to overcorrect the Hartree-Fock orbital energy estimate of the ionization energy and yielded results that were generally too small. The inclusion of third-order diagrams, particularly rings and ladders, is necessary to obtain reasonable agreement with experiment. Energy shifts resulting from the addition of diffuse, polarization functions to the computational basis were observed to be nearly the same magnitude as the shifts obtained from third-order ring and ladder diagrams, hence care must be taken to insure basis saturation. The renormalized 2p-h TDA decoupling was found to be the most satisfactory decoupling studied. Although only a diagonal approximation was implemented, principal ionization energies accurate to within an electron volt were generally obtained. Shake-up energies were less accurately described owing to the neglect of ion state relaxation and hole-hole correlation in the 2p-h TDA. The numerical results of Chapter 5 indicate that the calculation of relative photoionization intensities using a plane wave approximation for the photoelectron is not a sensitive reflection of the quality of the Feynman-Dyson amplitudes. The severity of the plane wave approximation most likely obliterates the many-electron relaxation and correlation corrections. More accurate photoelectron representations such as orthogonal ized plane waves or Coulomb waves should afford a more accurate evaluation of the Feynman-Dyson amplitudes, or alternatively, one-electron properties may be computed using the single-particle, reduced density matrix obtained from the spectral density.

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150 There are several other possible extensions of this research that should be mentioned. Most obvious is an application of these decoupling ideas to the polarization propagator. Currently, decouplings of this propagator are based on operator products. The odd or Fermion-1 ike operator products in the electron propagator theory must be replaced by even or Boson-like operator products to account for the particle-conserving nature of the polarization propagator theory, but with this minor modification, the application of the diagram conserving and renormalized decouplings also seems feasible. More accurate decouplings are still needed for the electron propagator in order to better describe shake-up energies. With this goal in mind, Cederbaum and co-workers have recently implemented the full, nondiagonal 2p-h TDA by exploiting spin and orbital symmetry to reduce the dimension of the 2p-h operator subspace. Preliminary results indicate that this approximation provides some improvement but is still inadequate. The need for higher operator products, particularly quintuple products, has been demonstrated by Bagus and Viinikka (1977) in CI studies and may necessitate the implementation of a 3p-2h renormal ization. Another attractive avenue for extension is a combination of the 2p-h TDA renormalization with the Hamiltonian partitioning of Kurtz and Ohrn. Some work along this line is in progress by Ortiz and Ohrn (1979) and preliminary results are yery encouraging. One criticism of this approximation is that if one interprets it as an alternative partitioning of the superoperator Hamiltonian in which the eigenvalues of the unperturbed Hamiltonian correspond to AE(SCF) energies (as in Eq. (3.23)), then the denominators of the self-energy which are obtained from ( EI-HJL ) (some operator product)

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151 should also involve AE(SCF) energies. For example, the second-order self-energy denominator would be [E + AE k (SCF) AE^SCF) AE^SCF)]" 1 Since it is probably not feasible to perform AE(SCF) calculations for all doublet (N-l) and (N+l)-electron states, simple approximations to the AE(SCF) energies such as AE k (SCF) s E|< E a,p e e a p or pa> + might be considered. Finally, the problem of open shell reference states has not been considered here. The retention of a restricted Hartree-Fock reference state in this regard will be a tremendous simplification, and the various spin and orbital symmetry couplings between the reference state and operator manifold may be treated with the theory of tensor operators.

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APPENDIX 1 s,W = j EEEEE .^'-!M^l | '^''M : i ; (a, V (/• + t „t r£ ,)(£ +£ „-S-%) 1 6 c d p yy 4 /_> t-^ V V y y V \J»h){„h\ \ //r )(qj-l\ip)_ 1 yyyyy W j| ^own i^x^i i;/>> 152 vrrvv /rl l, "'^" / ' 1 '/"'X/"/ 1 L' 7 ') yyyyy ^//»l ly-i->
><,„,l I hi) yyyyy <»/' I "'•><„/, I !/„,)( /,r I I /ft) _A ((') 1 („.||„/,)(„ftll/, f/ )(y,^|/ r )__ -U-[ (!') 1 y y y y y { '" 1 IM/L^i" h <±^ ] Liz>_ *(,-[) jfj (ID (K)

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153 1 ryyrr (il>\\j a){bc\ \l>q)(aqWbc) iftcpi? v 6 C p ?1 Iff 6)( ff6l lc ?) 2 vvvvv («.+«. -s -«.)(<.-«»> (L) (M) yyy^yy (R)

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APPENDIX 2 (a|H 3 a) ,h + E (e. + e. + e„ + £ . e ) r.s.s' 1 J s S r x [^ + %-] L r s s r s + E <11 ' | | jr> r,s,s',l,T r r I 1 1 r s + 1 s + ; 2] + E r,s,s',l,T x [2 + - Is r 1 s r 1 r 1 + + 2 r 1 ' s rll 1 s r -2 1 s r 1 s 1 1 s s + ] + E r,s,s' ,1 ,1 ' x t 3 z 1 r s 1 1 r s 1 1 I r 1 I s s s + % ] 1 I s s s 1 154

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APPENDIX 3 (a|VT (E)VT (E)Va) i:J I llld Iss'xss' I |11 '><1T 1 |jr> " r.s.s'.l.l' ( E + e r £ se s ,)(E + e re r e r [^ *2 1 r r 1 + I r + % h + J j] I 1 r s r s 1 1 1 s J <11 ' I I jr> Vn > ! $ + 5g] s s 1 1 ' s s s 1 r,s,s ,1 ,1 v j r s' v j r s 1 i — v [Jg 'r VM i + , p] 'r v,, i E TT ,

+ % + % ] s s 1 T __ — ~ ttft "*" v [- r.s.sM.T (E + E re se s l)(E+e l " E s " E l } ] s r ] + + r 1 s r 1 1 1 s r + ] 155

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156 „ <1 's I I jl> r,s,s',l,V [t + e r e s e s ,Mh e j e r e T S ' e l J ' s + + I s 1

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BIBLIOGRAPHY Abdulnur, S. F., Linderberq, J., Ohrn, Y., and Thulstrup, P. W. (1972) Phys. Rev. 6, A889. Abrikosov, A., Gorkov, L., and Dzyaloshinskii , J. (1965) Quantum Field Theoretical Methods in Statistical Physics , Pergamon Press, Oxford. Allison, D. A. and Cavell, R. G. (1978) J. Chem. Phys. 68, 593. Almlof, J. (1974) Univ. of Stockholm Institute of Physics Report 74-29. Babu, S. V. and Ratner, M. A. (1972) J. Chem. Phys. 57, 3156. Bagus, P. S. (1965) Phys. Rev. 139, 619. Bagus, P. S. and Viinikka, E.-K. (1977) Phys. Rev. A 15, 1486. Baker, G. A. (1970) in The Pade' Approximant in Theoretical Physics , p. 1, G. A. Baker and J. L. Gammel , Eds., Academic Press, New York. Baker, G. A. (1975) Essentials of Pade' Approximants , Academic Press, New York. Banwell, C. N. and Primas, H. (1963) Mol. Phys. 15, 225. Bartlett, R. J. and Silver, D. M. (1975a) J. Chem. Phys 62, 3258. Bartlett, R. J. and Silver, D. M. (1975b) Intern. J. Quantum Chem. S9, 183. Bazley, N. W. (1960) Phys. Rev. 120, 144. Benedict, W. S., Gailar, N., and Plyler, E. K. (1956) J. Chem. Phys. 24, 1139. Bethe, H. A. and Salpeter, E. E. (1957) Quantum Mechanics of Oneand TwoElectron Atoms , Academic Press, New York. Born, G. and Ohrn, Y. (1978) Intern. J. Quantum Chem. S12, 143. Born, G. and Ohrn, Y. (1979) Chem. Phys. Lett. 61,, 307 Born, G., Kurtz, H. A., and Ohrn, Y. (1978) J. Chem. Phys. 68, 74. Brandow, B. H. (1967) Rev. Mod. Phys. 39, 771. 157

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160 Kaijser, P. and Smith, Jr., V. H. (1977) in Advances in Quantum Chemistry , Vol. 10, p. 37, P.-0. LSwdin, Ed., Academic Press, New York. Kaplan, I. G. and Markin, A. P. (1968) Opt. Spectrosc. 24, 475. Katrib, A., Debies, T. P., Colton, R. J., Lee, T. H., and Rabalais, J. W. (1973) Chem. Phys. Lett. 22, 196. Kelly, H. P. (1976) in Photoionization and Other Probes of Many-Electron Interactions , p. 83, F. J. Wuilleumier, Ed., Plenum Press, New York. Koopmans, T. A. (1933) Physica 1_, ( 104. Kumar, K. (1962) Perturbation Theory and the Nuclear Many Body Problem , North-Holland, Amsterdam. Kurtz, H. A. and Ohrn, Y. (1978) J. Chem. Phys. 69, 1162. Layzer, A. J. (1963) Phys. Rev. 129, 897. Linderberg, J. and Ohrn, Y. (1967) Chem. Phys. Lett. U 295. Linderberg, J. and Ohrn, Y. (1973) Propagators in Quantum Chemistry , Academic Press, London. Lohr, Jr., L. L. (1972) in Electron Spectroscopy , p. 245, D. A. Shirley, Ed., North-Holland, Amsterdam. Lowdin, P.-0. (1962) J. Math. Phys. 3, 969. LSwdin, P.-0. (1965) Phys. Rev. i39, A357. Lowdin, P.-0. (1967) in Advances in Quantum Chemistry , Vol. 3, p. 324, P.-0. Lowdin, Ed., Academic Press, New York. Manne, R. (1977) Chem. Phys. Lett. 45, 470. Marr, G. V. (1967) Photoionization Processes in Gases , Academic Press, London. Masson, D. (1970) in The Pade' Approximant in Theoretical Physics , p. 197, G. A. Baker and J. L. Gammel , Eds., Academic Press, New York. Mattuck, R. D. (1967) A Guide to Feynman Diagrams in the Many-Body Problem , McGraw-Hill, New York. Mittag-Leffler, M. G. (1880) Acta Soc. Scient. Fennicae K), 273. M011er, C. and Plesset, M. S. (1934) Phys. Rev. 46, 618. Morse, P. M. and Feshbach, H. (1953) Methods of Theoretical Physics , McGraw-Hill, New York. Nehrkorn, C, Purvis, G. D. and Ohrn, Y. (1976) J. Chem. Phys. 64, 1752.

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162 Rowe, D. J. (1968) Rev. Mod. Phys . 40, 153. Schirmer, J. and Cederbaum, L. S. (1978) J. Phys. B: Atom. Molec. Phys. 11' 1889 Schirmer, J., Cederbaum, L. S., Domcke, W., and von Niessen, W. (1977) Chem. Phys. 26, 149. Schirmer, J., Domcke, W., Cederbaum, L. S., and von Niessen, W. (1978) J. Phys. B: Atom. Molec. Phys. U, 1901. Schuck, P., Villars, F., and Ring, P. (1973) Nucl. Phys. A208, 302. Schweig, A. and Thiel, W. (1974) J. Chem. Phys. 60, 951. Shavitt, I. (1977) in Methods of Electronic Structure Theory , p. 189, H. F. Schaefer III, Ed., Plenum Press, New York. Shohat, J. A. and Tamarkin, J. D. (1963) The Problem of Moments , American Mathematical Society, Providence, Rhode Island. Siegbahn, K., Nordling, C, Johansson, G., Hedman, J., Hed£n, P. F., Hamrin, K. , Gelius, U., Bergmark, T., Werme, L. 0., Manne, R., and Baer, Y. (1969) ESCA Applied to Free Molecules , North-Holland, Amsterdam. Simons, J. (1976) J. Chem. Phys. 64, 4541. Simons, J. and Smith, W. D. (1973) J. Chem. Phys. 58, 4899. Smith, K. (1971) The Calculation of Atomic Collision Processes , WileyInterscience, New York. Steinfeld, J. I. (1974) Molecules and Radiation: An Introduction to Modern Molecular Spectroscopy , Harper and Row, New York. Stewart, A. L. (1967) in Adv. At. Mol . Phys., Vol. 3, p. 1, D. R. Bates and I. Estermann, Eds., Academic Press, New York. Stieltjes, T. J. (1894) Ann. Fac. Sci . Toulouse 8, J, 1. Takagi, K. and Oka, T. (1963) J. Phys. Soc. Japan 18, 1174. Tchebychev, P. (1874) J. Math. Phys. Appl . 19, 157. Tuckwell, H. C. (1970) J. Phys. B: Atom. Molec. Phys. 3, 293. Turner, D. W. , Baker, C, Baker, A. D. , and Brundle, C. R. (1970) Molecular Photoelectron Spectroscopy , Wiley, New York. Vorobyev, Yu. V. (1965) Method of Moments in Applied Mathematics , Gordon and Breach, New York.

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163 Wall, H. S. (1948) A nalytic Theory of Continued Fractions , Chelsea Publishing Co., Bronx, N.Y. Wendin, G. (1979) preprint of article to be published in Structure and Bonding , Springer Verlag, Berlin. Wick, G. C. (1950) Phys. Rev. 80, 268. Yoshimine, M. (1973) J. Comp. Phys. 11, 449. Zwanzig, R. (1961) in Lectures in Theoretical Physics , W. E. Brittain, B. W. Downs, and J. Downs, Eds., Vol. 3, p. 106, Interscience, New York.

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BIOGRAPHICAL SKETCH Gregory J. Born was born on July 1, 1951, in Joliet, Illinois. He was raised in Coal City, 111 inois, where he attended public schools until his graduation from Coal City High School in May 1969. From September 1969 through June 1973, he attended Southern Illinois University at Carbondale as an Illinois State Scholar and majored in chemistry with math and physics minors. Throughout his senior year, he participated in an undergraduate research program working under the supervision of the late Professor Boris Muslin. After completing his Bachelor of Science degree, Mr. Born enrolled in graduate school at the University of Florida. From September 1973 to June 1976, he was a teaching assistant in the Department of Chemistry. In February 1974, he joined the Quantum Theory Project and has been a research assistant working under the supervision of Professor Yngve Ohrn from June 1976 to the present. During the summer of 1976, Mr. Born attended the Summer School in Quantum Theory held in Uppsala, Sweden, and Dalseter, Norway. He is the co-author of four research publications and has presented several contributed talks at various conferences. 164

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. N. Yngve Ohnh , Chairman Professor or Chemistry and Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. / /,-) Arthur A. Broyles Professor of Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Q^u^^sS-^x David A. Miifha Professor of Chemistry and Physics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. /fj}jjL* %>. \^jU^n^_ Willis B. Person Professor of Chemistry I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jpjjfi *R. Sabin ^fessor of Physics and Chemistry

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This dissertation was submitted to the Graduate Faculty of the Department of Chemistry in the College of Liberal Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. June 1979 Dean, Graduate School

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IIMIUBB4ITV OF FLORIDA Ilf 3 1262 08553 149B