• TABLE OF CONTENTS
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 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 Abstract
 Introduction
 Electron binding energies
 Excitation energies
 Appendix: Correlation self-ene...
 Bibliography
 Biographical sketch














Title: Propagator calculations on molecular ionization and excitation processes
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Title: Propagator calculations on molecular ionization and excitation processes
Physical Description: ix, 80 leaves : ; 28 cm.
Language: English
Creator: Ortiz, Joseph Vincent, 1956-
Copyright Date: 1981
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Subject: Electrons   ( lcsh )
Excited state chemistry   ( lcsh )
Chemistry thesis Ph. D
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Statement of Responsibility: by Joseph Vincent Ortiz.
Thesis: Thesis (Ph. D.)--University of Florida, 1981.
Bibliography: Bibliography: leaves 76-79.
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General Note: Vita.
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Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
    Abstract
        Page viii
        Page ix
    Introduction
        Page 1
        Page 2
    Electron binding energies
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
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        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
    Excitation energies
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
        Page 55
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        Page 58
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        Page 60
        Page 61
        Page 62
        Page 63
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        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Appendix: Correlation self-energies
        Page 73
        Page 74
        Page 75
    Bibliography
        Page 76
        Page 77
        Page 78
        Page 79
    Biographical sketch
        Page 80
        Page 81
        Page 82
        Page 83
Full Text











PROPAGATOR CALCULATIONS ON MOLECULAR
IONIZATION AND EXCITATION PROCESSES










BY

JOSEPH VINCENT ORTIZ


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

UNIVERSITY OF FLORIDA

1981


































To Karen

Come my Way, my Truth, my Life















ACKNOWLEDGEMENTS


I acknowledge the counsel of Professor Yngve Ohrn, but I am more

grateful for the example he has set. Professor P. 0. Lbwdin enabled my

attendance at the 1978 Quantum Chemistry Summer School in Uppsala, Sweden.

Conversations with Professor Brian Weiner have expedited my research.

Doctor Gregory Born and Doctor Henry Kurtz have rendered technical

assistance. Laura Wagner typed the dissertation. The National Science

Foundation has provided grant support for part of this research.
















TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS................................... iii

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

ABSTRACT............................................ viii

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

ELECTRON BINDING ENERGIES ............................ 3
1.1 The electron propagator......................... 3
1.2 An uncorrelated propagator..................... 4
1.3 Toward correlated electron propagators......... 6
1.4 Approximations................................. 9
1.5 Approximate propagator expressions............. 11
1.6 Vertical and adiabatic electron affinities of
C12 ......... ............. .................. 13
1.7 The vertical electron affinity of CN........... 22
1.8 The ionization energies of NH2 ................ 29
1.9 The ionization energies of PH2 ................ 32
1.10 The transition operator method................. 38
1.11 Applications of the transition operator
reference...................................... 40
1.12 Interpretation................................ 46


CHAPTER TWO EXCITATION ENERGIES....................................
2.1 Consistent ground states and the random phase
approximation.................................
2.2 The antisymmetrized geminal power wavefunction.
2.3 Density matrices and the total energy expres-
sion .................... ......................
2.4 Interpretations of the total energy expression.
2.5 Coefficient optimization......................
2.6 Orbital optimization...........................


CHAPTER ONE










Page

2.7 Results for lithium hydride..................... 59
2.8 The random phase approximation of the polariza-
tion propagator..................... .........6 66
2.9 Polarization propagator calculations on lithium
hydride......................................... 70

APPENDIX............................................ 73

BIBLIOGRAPHY....................................... 76

BIOGRAPHICAL INFORMATION......................... .... 80















LIST OF TABLES


Table Page

1-1 C12 and C12 CGTO Basis.................................. 14

1-2 Total Energies for C12 and C12- below -918.0 a.u.......... 16

1-3 Hartree-Fock Total Energy Lowering for C12 and C12 ....... 17

1-4 Vertical Electron Affinities for C12... ....... ....... .. 18

1-5 Adiabatic Electron Affinity of C12 ...... .............. 19

1-6 Adiabatic Electron Affinity of C12...................... 20

1-7 Adiabatic Electron Affinity of C1 ........................ 21

1-8 CN and CN CGTO Basis ................................... 23

1-9 Total Hartree-Fock Energies for CN and CN ............... 25

1-10 Hartree-Fock Total Energy Lowering for CN and CN ........ 26

1-11 SCF Equilibrium Nuclear Geometries ....................... 27

1-12 Vertical Ionization Energy of CN-......................... 28

1-13 Hydrogen CGTO Basis (NH2-, PH2-).......................... 30

1-14 CGTO Basis for NH2 .................................... 31

1-15 Vertical Ionization Energies for NH2 ..................... 33

1-16 Adiabatic Ionization Energies for NH2 and PH2 ........... 35

1-17 CGTO Basis for PH ...................................... 37

1-18 Vertical Ionization Energies for PH2 ..................... 39

1-19 26 CGTO Basis for H20 .. .................................. 41

1-20 Vertical Ionization Energies of H20....................... 42

1-21 Neon Gaussian Basis Set .................................. 43

1-22 Ionization Energies for Ne............................... 44










Table Page

1-23 Vertical Ionization Energies of NH2 and CN Calculated
with the Transition Operator Method and Self-Energy
Corrections.............................................. 45

2-1 LiH Basis............................................... 60

2-2 Convergence Behavior for AGP Wavefunction Optimization for
LiH 1Z+ at R = 3.015 bohr................................ 62

2-3 AGP Geminal Coefficients and Principal Natural Orbital
Occupation Numbers for LiH................................. 64

2-4 LiH Total Energies and Correlation Percentages for 1 +
Ground State.......................................... 67

2-5 Total Energy Differences Along the Ground State 1 +
Potential Curves with Respect to the Total Energy at
R = 3.015 bohr............................................ 68

2-6 AGP-RPA Excitation Energies............................... 71















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


PROPAGATOR CALCULATIONS ON MOLECULAR
IONIZATION AND EXCITATION PROCESSES


By

Joseph Vincent Ortiz

June 1981


Chairman: N. Yngve Ohrn
Major Department: Chemistry


Renormalizations, which sum classes of diagrams to all orders in

perturbation theory, are applied to the electron propagator. The sepa-

ration of relaxation and correlation self-energy diagrams enables the

construction of electron propagators that include relaxation contribu-

tions to all orders. Correlation self-energies are obtained in second

order, third order, and in the diagonal 2p-h Tamm-Dancoff approximation,

which is also a renormalization. These propagators are applied to the

electron binding energies of CN-, NH2, PH2 and C12. An alternative

propagator renormalization through the transition operator method

reference state is applied to the electron binding energies of Ne, H20,

NH2 and CN.

The antisymmetrized geminal power wavefunction, which is the form

of the consistent ground state of the random phase approximation of the

polarization propagator, is variationally optimized for the LiH ground










state potential curve. Random phase approximation polarization

propagator calculations yield excitation energies.

Physical concepts suggested by these theories are discussed.















INTRODUCTION


Modern chemical investigation inevitably revolves about the axis of

molecular electronic structure theory. The quantum mechanical investiga-

tion of the structure and properties of isolated molecules lays a founda-

tion for studies of complex chemical systems. Energy differences between

states are properties of continuing importance in molecular electronic

structure theory, for they provide insight into macroscopic chemical

phenomena. Propagator theory is a theoretical technique that concen-

trates on determining energy differences and transition probabilities.

The concepts that are introduced with propagator theory therefore explain

the energy differences themselves without necessarily treating the total

energies or wavefunctions of the states of interest.

Molecular energy differences are associated with nuclear and elec-

tronic rearrangements. The Born-Oppenheimer approximation permits a

separation of electronic and nuclear motion that facilitates quantum

mechanical calculations based on the time independent Schr6dinger

equation. With the nuclei fixed at given positions, the electronic

states of the molecule are studied. A series of nuclear configurations

defines an adiabatic potential energy surface, where the total electronic

energy plus the internuclear repulsion energy is plotted against the

nuclear positions. Ground state potential energy surfaces have typically

been calculated using the variational method; propagator theory has made

few contributions to this field. However, for the calculation of energy

differences between potential energy surfaces, propagator theory is

competitive with current variational techniques. In the chapters that







2

follow, propagator theory will be applied to energy differences between

electronic states. The calculations will be made in a limited basis of

square integrable functions and there will be no explicit treatment of

the continuum.

Electron binding energies are energy differences between N electron

and N1 electron states, i.e., ionization energies and electron affini-

ties. Hartree-Fock theory provides predictions of electron binding

energies based on Koopmans's theorem (Koopmans, 1933) which states that

orbital energies are interpretable as ionization energies or electron

affinities. Unfortunately, the invariance of the Hartree-Fock wavefunc-

tion under unitary transformations of the occupied (or the unoccupied)

orbitals among themselves places in doubt the validity of the orbital

energy interpretation. Propagator theory justifies and delimits the

simple orbital model and suggests ways to improve it. These ways include

the extension of the orbital concept to quasi-particle concepts of elec-

tron binding energies and the introduction of energy-dependent correla-

tion potentials.

Excitation energies, or energy differences between electronic states

with the same number of electrons, are also amenable to propagator inter-

pretations. Hartree-Fock theory also gives a model for excitation

energies that can be extended with perturbative propagator corrections,

but propagator theory also suggests models that abandon the orbital

picture as a point of reference. Correlated model Hamiltonians provide

physical pictures for the ground and excited states and explain the

nature of the interactions present in each state.

These applications illustrate how propagators provide direct deter-

mination of energy differences and introduce concepts that are, so far,

inaccessible to purely variational theories.














CHAPTER ONE

ELECTRON BINDING ENERGIES


1.1 The electron propagator

Electron propagators efficiently determine and explain electron

binding energies. The poles of the electron propagator are energy

differences between the N electron ground state and the N1 electron

states, i.e., electron binding energies. The residues of the propagator

are overlap amplitudes between the N electron and N1 electron states

and describe the electronic structure of the ground state. The electron

propagator matrix (Linderberg and Ohrn, 1973), G, in a spin-orbital

basis has the elements


f n*(i)f (j) gn*(i)g (j)
Gij(E) = E-E (N+1)+E (N)+ia + E+E (N-1)-Eo(N)-ia


The total energy of the N electron ground state is EO(N), and E (N1) is

the total energy of the n-th N1 electron state. The poles of the

propagator are real energy differences when the positive parameter o

approaches zero. The continuum can be treated with the electron propaga-

tor in conjunction with other techniques (Mishra, 1981). The overlap

amplitudes (Purvis and Ohrn, 1975) between the ground and N1 electron

states are


f (i) = gn(i) = (1-2)


where the field operators satisfy the relations







4

[ai,a] = [a t,a ] = [ai,aj ]+ 6 (1-3)


In this notation, the many-electron Hamiltonian is


H = hiji a+ aii aa ak (1-4)
1 j i j k 1


1-P
where =
1 2

NUCLEI Z
h = (1-5)


This Hamiltonian operates in Fock space and is independent of the number

of electrons in the states on which it operates.

1.2 An uncorrelated propagator

Consider the model Hamiltonian


HO = iai a (1-6)


in the spin orbital basis that diagonalizes the Fock matrix,

Fj = hij + I (1-7)
k I

where


= (1-8)


so that


Fij = Eiij


(1-9)









Since the model Hamiltonian contains only one-electron terms, the elec-

trons do not interact. The model ground state is

N t
N> = I ai vac> (1-10)
i=l1


where


N ovac> = n.ivac> = I ai ailvac> = 0.vac> (1-11)
op 1


The ground state total energy is


= \ i = a cini (1-12)
1 1


There are N orbitals where ni = 1; the rest of the orbitals have ni = 0.

The N1 electron states differ from the ground state in the choice of

occupation numbers. The poles of the model electron propagator are


EPOLE = EO(N) En(N-1) = n (1-13)


which are negative ionization energies, and


EPOLE = En(N+1) EO(N) = En (1-14)


which are negative electron affinities. The overlap amplitudes are


f (i) = = din
(1-15)

gn(i) = = in


The purpose of this exercise is to recover Koopmans's theorem

(Koopmans, 1933) and the attendant orbital concept of electronic






6

structure. The remaining discussion considers how the electron propagator
can be extended to describe correlated electron dynamics.

1.3 Toward correlated electron propagators
Given the notation for the propagator


Gij(E) = <> (1-16)


the identities

1 E (N+1)-EO(N)
E[E En(N+1) + EO(N)]- 1 + E-E (N+1)+E(N)

(1-17)
(En(N+1) EO(N)) = ,


and the corresponding relations for the N-1 electron case imply that


E<>E = <[ai,a] > + <<[ai,H];a>>E (1-18)


This equation can be iterated to give


<>E = E-1<[ai,a ]+> + E-2<[[ai,H],a t]> (1-19)

+ E-3<[[ai,H],H],at]> + ...


The definition of superoperators (Goscinski and Lukman, 1970), 1 and R

IX = X HX = [X,H] (1-20)

which act on the set of field operators (Redmon et al., 1975)


X = {a,aa aka a akaam'...}) (1-21)
X =~ia jka






7

and the definition of a scalar product (Goscinski and Lukman, 1970)


(X|Y) = <[Y,X ]+> (1-22)


convert the expansion in equation 1-19 to the form


<>E = (aj (E1-H)-1ai) (1-23)


The inverse operator which occurs in the propagator matrix,


G(E) = (aI(EI-H)-la) (1-24)

can be circumvented by the inner projection technique (L6wdin, 1965):


G(E) = (alh)(hl(EI-A)h)-l (hl a) (1-25)

where the operator manifold is complete, i.e.,


h = {a,a aa,a a aaa,...} (1-26)

and where the superoperator matrix elements are evaluated with respect to
the exact ground state. A partitioning (Ldwdin, 1965 and 1971) of h,

h = {aJf} (1-27)

such that

(alf) = 0 (aaI) = (f|f) = 1 (1-28)

implies that


G-(E) = (al(EI-P)a) (aIf)(fI(EI-H)f)-1(fIHa) (1-29)

For the model uncorrelated system where








IN> = |Hartree-Fock> (1-30)

and where the model Hamiltonian is


H Y lEa'ai (1-31)


the exact inverse propagator is


Go-'(E) = (aj(EI-R0)a) (1-32)


since the second term in the right-hand side of equation 1-29 vanishes.
Koopmans's theorem appears once again since


[GO-1(E)]ij = (aii(ET-Ao)aj) = (E-i)5ij (1-33)

For the full Hamiltonian, approximations of the exact ground state and
choices of a subspace of f are necessary. If the ground state is written
as

IN> = IHartree-Fock> + Icorrelation> (1-34)

then the propagator becomes


G-(E) = El (aHa)HF (afHa)CORR (1-35)

(a f)(fl(Ei-)f)-l(f(Ha) ,

where (Ia|a)HF is (aIHa) taken only with respect to the Hartree-Fock
contribution to the ground state, and where the remainder of (aiHa) is
(alHa)CORR. In other words,


G-1(E) = GO-I(E) I' '(E) .


(1-36)









The superoperator matrix element

(ajlAai) = hji + i (1-37)
kl

depends only on the one-electron reduced density matrix and is indepen-

dent of E. Taken together, the energy dependent and energy independent

corrections to the uncorrelated propagator,

+' + '(E) = (E) (1-38)


are the self-energy and appear in the Dyson equation,


G- (E) = GO-'(E) (E) (1-39)

The self-energy contains all correlation effects.

1.4 Approximations

Choices for the ground state and for a subset of f define approxi-

mate propagators. A specific physical problem or computational feasibil-

ity may motivate these choices. Redmon et al. (1975) and Nehrkorn et

al. (1976) thoroughly analyzed the corrections to Hartree-Fock from

single and double excitation contributions to the second order Rayleigh-

Schridinger perturbation theory wavefunction. Several authors have

studied choices for the operator manifold (Pickup and Goscinski, 1973;

Purvis and Ohrn, 1975; Redmon et al., 1975; Nehrkorn et al., 1976;

Herman et al., 1980a and 1980b; Simons, 1977 and 1978; McCurdy et al.,

1977). An example is f3:


f3 fklm} = {(ak alam) + alkm am m)Nklm '

Nklm = ( ) + (1-40)









The components of the self-energy also derive from diagrammatic

perturbation theory (Cederbaum and Domcke, 1977). In this approach, the

terms of the self-energy are classified by their order of electron

interaction. Born has discussed the connections between the order by

order approach and the inner projection formalism (Born, 1979). Operator

product manifolds, such as f3 or f,5 sum classes of terms in the diagram-

matic expansion to all orders in electron interaction. The infinite

summations, or renormalizations, are useful when the convergence of the

diagrammatic series is in doubt. For example, the choice of the

Hartree-Fock ground state and the f3 operator product manifold sums all

terms in second order and also includes classes of diagrams called rings,

ladders, and mixed ring-ladder diagrams to all orders. If the ground

state includes double excitations from the first order RSPT wavefunction

and single excitations from the second order RSPT wavefunction, then the

renormalization sums all diagrams through third order, in addition to

other diagrams in higher orders. The two complementary modes of analysis

of the self-energy, from diagrammatic theory or from the approximation

of the inner projection formula, enable verification of approximate self-

energies. They are alternative terminologies which suggest improvements

of the self-energy.

Yet another method of analysis of the self-energy is the identifica-

tion of so-called relaxation and correlation terms by perturbation theory

(Born et al., 1978). Relaxation terms are terms that occur in the

difference between Hartree-Fock total energies for the N electron and

N1 electron states. (The ASCF method is discussed in Bagus, 1965.) The

perturbation is the difference between the Fock operators of each state.

In general, the total energies are unrestricted Hartree-Fock total

energies. The relaxation terms appear in the diagonal elements of the










propagator matrix when E = ci, where E. is the orbital energy pertaining

to the N1 electron state of interest. Perturbation theory connects the

SCF total energy of the N1 electron states with the orbitals and orbital

energies of the N electron SCF calculation. The terms in the self-energy

which are not relaxation terms are correlation terms. Since all self-

energy contributions treat correlation effects with respect to the

uncorrelated Hartree-Fock propagator, the use of the term correlation

self-energy will refer only to contributions that do not originate from

relaxation effects.

A further approximation proceeds from investigations (Schirmer et

al., 1978) which demonstrate the unimportance of nondiagonal elements of

the propagator matrix when the matrix is constructed in the basis of

canonical Hartree-Fock orbitals. (This observation holds for outer

valence and core binding energies of molecules near their equilibrium

geometries.) Neglecting the nondiagonal elements sacrifices little

numerical accuracy. Mixings of canonical Hartree-Fock orbitals of the

same symmetry in the overlap amplitudes are obliterated when the diagonal,

or quasi-particle, approximation is made. Previous studies of the non-

diagonal propagator have also indicated that these mixings are small

(Schirmer et al., 1978). The quasi-particle approximation causes many

poles to be assigned to each canonical orbital. In the uncorrelated

propagator, only one pole is assigned to each orbital.


1.5 Approximate propagator expressions

With the orbital energy and the relaxation terms in the self-energy

included in the ASCF energy difference, the renormalized quasi-particle

propagators under consideration will have the form (Kurtz and Ohrn, 1978)







12

G-.l .(E) = E (ASCF)i CORRii(E) (1-41)


The quasi-particle assumption permits direct iteration of equation 1-41

with respect to E, i.e., there is no need to diagonalize with every new

value of E. However, since relaxation terms are strictly defined only

when E = Ei, the expression (Kurtz and Ohrn, 1978)


G-ii(ci) = ci (ASCF)i CORRii(i) (1-42)


will be evaluated and reported. Typically, when the quasi-particle

approximation is valid, dG-1(E)/dE is small. Consequently, the results

of 1-41 and 1-42 closely agree (within 0.1 election volts). Diagrammatic

theory suggests the second and third order correlation self-energies.

(See the Appendix for correlation self-energy formulas.) Another approx-

imation, which derives from the inner projection formalism with the f3

operator product manifold and a Hartree-Fock ground state, ignores off

diagonal terms in the matrix (Born and Ohrn, 1978, 1979 and 1980)


(f3(E-R)f3)-1 (1-43)


This approximation is the so-called diagonal 2p-h Tamm-Dancoff approxima-

tion (2p-h TDA) (Cederbaum and Domcke, 1977; Schirmer and Cederbaum, 1978)

and is no more difficult to compute than second order. It renormalizes

certain ring, ladder and mixed ring-ladder diagrams. (For a discussion

of the nondiagonal 2p-h TDA, see Mishra and Ohrn, 1980.) All of the

following applications of the ASCF plus correlation self-energy propaga-

tors are discussed in Ortiz and Ohrn, 1980 and 1981.







13

1.6 Vertical and adiabatic electron affinities of C12

A vertical electron binding energy is an energy difference between N

electron and N1 electron states at the same nuclear geometry. An

adiabatic electron binding energy is an energy difference between the

same states evaluated at their equilibrium nuclear geometries. For C12,

the change in the equilibrium internuclear distance upon the addition of

an electron is about 1.3 bohr. To calculate the adiabatic electron

affinity of Cl2 requires the difference between the energies at the

internuclear distances of the neutral molecule and of the anion. Either

C12 or C12 curves, combined with the appropriate vertical electron

affinity, give the adiabatic electron affinity.

Basis sets for all the systems consist of contracted Gaussian type

orbitals (CGTOs). Dunning and Hay's (1977) chlorine basis set is the

simplest choice (Basis A in Table 1-1); d functions and diffuse s and p

functions are subsequent augmentations. As the basis set is enlarged,

numerical convergence of the results occurs. Furthermore, similarities

in numerical convergence between theoretical approximations establish

criteria for choosing basis sets in subsequent applications. For example,

numerical convergence with respect to basis sets in the ASCF approxima-

tion might be indicative of convergence in the correlation self-energy.

For propagator calculations, where one basis set describes more than

one state, the use of different basis sets for each state in the ASCF

calculations is undesirable. When convergence of SCF total energies for

each state is attained with basis set augmentations, the occupied

orbitals and their energies have been adequately characterized. Virtual

orbitals, however, may change markedly as the basis set is improved.

Diffuse basis functions have a pronounced effect on the virtual orbitals.



















LI-


u


LL
Co


r--









ci



u

r-

0






co















- O
SCD


W C
1 I

0 0
/1)







(4 Ia
Ou (-


0 0 0


Ll-


Co


C
Ci Ci rN- r- CY '-I
* o- ( O 0 Ln CO 0O

*r- nM (L LO L I
4- Co oa M n

0 00000







CM LDi

0 0 ) LOD l LO
Q r N> CJ 0 CO

LU 0 1. t
ci-


o -- Nr-

cO LO c-
0 CO LI)

C o -












In m C
Co-In


0 0 0 0 0 0


mc o r- co
co o cCM co
Co 10 CD LCo
(n I M o
1.0 (' 0 0


L L CO

o C CO









0
10 I) LD)
CO Lo o ro

i Ln --
CM


0 0 0 0 0 0 0 0

H-I I- 1--I f-- I-- T---


CO
CO -


tfl Ila t L L







15

The correlation self-energy exhibits the aggregate effect of these

apparent instabilities.

As the data in Tables 1-2 and 1-3 demonstrate, the augmentations may

affect the SCF total energy of one state more than the other. One p and

two diffuse s functions are added to basis A to form basis B. These

additions cause a much greater energy lowering for C12 than for Cl2.

Further improvements of this kind lead to basis C, but the ASCF electron

affinities change little. The addition of d functions to the original

basis (Basis D and Basis E) has an important effect on the ASCF electron

affinity at the equilibrium internuclear distance of Cl2, but has a less

important effect at the C12- equilibrium distance. Results for the

second order relaxation contributions to the self-energy are also listed

in Table 1-4. These terms change similarly to ASCF with respect to

basis set improvements. (See also the restricted SCF study of Gilbert

and Wahl, 1971.)

Second order and diagonal 2p-h TDA correlation self-energies are

constructed with the orbitals from the C12 SCF calculation. The improve-

ment of the correlation description decreases the vertical electron

affinity, but improved basis sets increase it.

For Cl2, as with many molecules, experimentalists have measured

only the adiabatic electron affinity (Dispert and Lacmann, 1977). Energy

differences, calculated or empirical (Douglas et al., 1963), along the

C12 or Cl2 potential curves are necessary. Energy differences in the

SCF approximation for the C12 curve, as well as second order RSPT calcu-

lations for Cl2,are in Table 1-5. Adiabatic electron affinities using

various combinations of potential energy curves and vertical electron

affinities are in Tables 1-5, 1-6 and 1-7. The restricted Hartree-Fock

potential curve is inadequate because it separates into ions instead of





















co IcOL C n -iLin- coC 0 0 co co c
cn-ot C P co)0- O co CM co N NC

N 0C OlO- O C O 0 Cl oo
cli m co o o o m ) o 30 clr r- mri Lo























0 1 0 O -cMO oCM O
rl- r-r- O C 0- O CO! D CO C) 0 t r C rJ m C CD0























~co -ilOl o 1 C 01 CO 01 CO 01 01 0101
cO-fcL Lnoc co CO m Cl M0- O


010'nCM -l n CMi Ln CJ 0 CM CO CMo 00Cmm


I I I I I II I I I I I I I I I


CM CM CM CM




ca 0: 0 a a

CO iO i c0) 0
CLO 0 C9o9Ln
m lr Ln n coLO iLO


O I
3= Q
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0CIA
10 "








mc v
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0

0

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CI


o
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i- 0

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0
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C



0

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CO 0

m LO













cni
C




0

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cc
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O c


0 I

+ .-- 0
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CO CO


C\M CM

O O

Q) Q)
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CO C C! 0 CO9 .0-1

m On Cl) n Crl0-nfLfL


0

4-)
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m
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CMm


= 0

+ II


0-ti'.
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l O






u!03
ln 0
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C
0






0 0






r-1 0


m ii 3
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C



C 0
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40 IA

+ i-

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o c


=3
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3

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m*- a
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LO ln M m m (n LO co m


|0000 oo o o oo o
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I I


> - o 0 0 m c o m cD m

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LL
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3


o


mo in M iO ro i-n i Ln


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S> CM Co DCD m
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n *
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o* C-i O Lc 0






3 00O Lo
CM-



S. .---
+ fLOCM SCf

LL- S0 0 c

-I


















-o /-








2] N-..- C')
( -3 oCMCO ri
/ ii D .tf-.r

i 0 0 0 -


't m DM lc-j C\oj C cm --

m o .-n o m r-























LO - N 0C InN N:I-
1 U 0NJ-i C'] 0C C c'-
Co -r0 0Cin oo co Io






C -c'] -imo cj i ofll cOi3-






) -It r- c mo cn c c-
cLOCr- ']- C\ co cN O o
















COn CoO t-oc LoL-
O0j-s -- NO-f Oi-l LO -






cr o r- co-j o-. oCo
00 -- Oi- 0 LO












N nln c O tr LnO

0 0 - O .-4 0 OT -


S co Ln Co c 'L, LD m) mm -1 Ir Gc'
CM CSOC'-4 -CJcCbO CM]3- COCO rQL)
- 1 rmcOc-c Sor0- DO C-I LOO
0 CO3.--4 O-C'C.i OC' O,4 OC\JN
C I
5]J -


CoCo- co coo coo

S,--)LLn n cO LO C LO


cn Ln Lo








WI


C- I LUI


X I
U'





-0
O














o
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0 m
0












U U
c I



















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4I V
01



s-

C 0
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U U





c lc





CI -





,nO -I

( C!

CE C

] 0)





fO -
T(J 0 -


C
N

C-i

0
4-

vl/




I~ *0





aC
U




4-'
ai s

S r







5-i
CO
SLL





4-
^.
>1


















u


)c 0 M

C-



L.J




-o



-o

























I -
L CM o
U c'I
LU





I *-







4NJ
F -
-o
4-





















UU U
- 4n

a,


o -Q
a C M


















In c
4- C


a,
cc


NO CM CO M


M CM c M
OS LN N~ L) 0 II










S a)
4--0

O co



C,





4-




o
4 0
C\J r- C\J r- -






u
O 0

















o
a aa-
LO CM C 4-



F^ -

I U














CM C U
Co











-, a) u
w LU -




o .0 0
(NJ n3


aa,
Ul 4-




















C c- j
-M0 U,- S.-




CO h CT - r 4
*- W u
















W E 4- LIS
o* rug u
LU LL L

u- u
I ar a,













i CM a- a, 3a




















Tn 0
lo C 0 0C -
SC C C ,





II 0








c u W















4-











MM
11..
4-,




















2
CM On In QI In






41 0 -



c- 4-
4-C
+ 2

olo






- L -

LU IdU












i- I n CO C f











cO


IJ =ci In 0 U I U











InO IC .0
r- 4- m- COw I
C_ ll J
m- <^i <^ occ o







21

Table 1-7

Adiabatic Electron Affinity of C12 (eV)


Basis Verticala V(C12) EA(adiab)

A Eq. (A-1) 3.08 1.59

Eq. (A-2) 2.62 1.13


B Eq. (A-l) 3.60 2.11

Eq. (A-2) 3.19 1.70


E Eq. (A-l) 4.00 2.51

Eq. (A-2) 3.74 2.25b


D(C12) = 1.49 eV (see Douglas et al., 1963)


a Eq. (A-l) means that V(C12-) is calculated as -I + ZC(2). Eq (A-2)
means that V(C12-) is calculated as -I + E2p-h TDA

b EXPERIMENT EA(adiab) = 2.4 0.2 eV (Dispert and Lacmann, 1977).







22

into neutral chlorine atoms. The second order RSPT results contain some

correlation. The experimental energy difference or the second order

RSPT energy difference along the C12 curve, combined with the vertical

electron affinity at 5.0 bohr, give an adiabatic electron affinity that

is within experimental error. A smaller basis set combined with a

simpler treatment of correlation could have produced a prediction that

was also within experimental tolerance through a fortuitous cancellation

of errors.


1.7 The electron affinity of CN

The vertical electron affinity of CN is negligibly different from

the adiabatic electron affinity since there is little change in the bond

length upon addition of an electron. (See Table 1-11.) The total

energies listed in Tables 1-9 and 1-10 show the small change in the SCF

total energy from one equilibrium distance to the other. Therefore,

vertical electron affinities are computed at only one point per basis

set. (See Table 1-8.) As the total energies indicate, the d functions

produce the greatest improvements. The addition of diffuse functions

is not as important since basis A has already been optimized for anionic

molecules (Pacansky and Liu, 1977). Griffing and Simons also studied

the electron affinity of CN using the equation of motion technique

(Griffing and Simons, 1976). They chose Hartree-Fock plus first order

RSPT corrections for the ground state and the f3 operator product mani-

fold. They also varied the exponents of the s and p Gaussian basis

functions to minimize the 50 orbital energy (see the critique of Liu,

1977) and included no d functions whatsoever.

The two smallest basis sets permit third order correlation self-

energy calculations (see Table 1-12). For the smallest basis set, the


















0 (


+-,
C CO Ci C
U o o
S(Z1 c c-
a 000

u o cc i





(2









LO CD
D0
0
H-





10 LO U -


0 0 0 0 0


C,
C-


c ci-
CM CMJ
C\0 1

0 0


CD
CO 0

(J

c z
I-
-o
c
fI-
z
0


9 C C


01















M c


- -


a. a Q


a 0


0 0 0 0 0


a a. Q. 0 L


o co
000
a a a
a a aD




















cc co

CD m m m m
CD U C I n mC


[ cc ccl Cci cci c0
LC I I1 1 1
cc


co


-0 CDj

I c

3










LU
I C
u U- 0c


L)





a,
4-




c

I- C -

H0 C

0 ^
U- Q


5-
-4-I
5-
di
cc


di

0
1-


CM Co CO
0 co o cc
Ln 00 c
U-I -I If CO
CO CO) CM 0c
CM CM CM C
CM CM CM CM

CM CM CM CM
C ci cc ci


- cc
cM -

CM CM


.- .



fu L 0


mU co C l


u L.3 05L-)
d-rd dO-di-J
TOa 0- TO
CQm aC g


0





di


0






i,- I-.
CM




+
4-'--.







V* r
r-- 1 1


C.








4-

U










'"5
CJ



U-
o












mc
4--

In -


















0 0









CO
i co Ln0



U 0 0 0





000











4-*I





CC
0 3



m o 00



D D 0 0


C)
I









S-
0 0










.I 0 0o 0









1:0 C lI -
oj o o
C3 u .
0, 0
I-I d













1( C








0)u

ff
+-' /^











27












cn L~ c

o C :J


c- C-

CD



Q-- r-- N
0




-I








< < m 0. c^


a 0 CM. )- 0n 0. co
U -
0










-o









(/i

OS: LJ 3:


0)
c- + +
'3 C-)
4- -
0l


c- C cO


0
01
I g I I CM
U, Z CO CM ~ CM
C) C) CO
0 j CO


CUNi


C co
" C) C'











28







SI I *i


4- -=
u







*R
CV) r- M



<0



C) ai
I * * C Cn






> 0 3 0 w-


CM C' ci C (
L .







-0. 1
V -'


+





V) A







I 0
- &- Q-










LO0
S-- 4- 01 1









ci O r
0 CMC
.. 0- L -














CD O m LO 4- 10 4 I

_C- a)** a)
i. + M Ci Ca 4-'









- S-


SC*




-4- C) C



SLI
o -^ 01

t~~1- CM C CM -
















i 0, 0 00 0 r.
I (A m - *Ci C0














*- 0) +


0 00 S CM
I 01
















tn c3: cQ cQ LJ -C L
"3-







29

third order result is in closest agreement with experiment, but the

addition of the d functions casts doubt on the balance of this approxima-

tion. If the trends of the basis set improvement continue, the third

order approximation will be inferior to the diagonal 2p-h TDA result as

the basis set is improved.


1.8 Ionization energies of NH2

The lowest ionization energy of NH2 is the electron affinity of NH2.

The second ionization energy of NH2 yields an excited final state of NH2

and, in combination with the lowest ionization energy, an excitation

energy. For the first ionization, there is little nuclear rearrangement

and, as with CN, the vertical electron binding energy at one nuclear

geometry is sufficient. For the second ionization, however, there is a

large change in the bond angle and a small change in the bond distances.

Once again, energy differences along the N electron or N1 electron

surface are necessary. Other investigators (Bender and Schaefer, 1971;

Heaton and Cowdry, 1975) have calculated energy differences using SCF

and CI wavefunctions. These calculations indicate that SCF calculations

are reliable for the surfaces of interest. (Heaton and Cowdry calculated

ASCF electron binding energies of NH2 with a basis set optimized only for

uncharged molecules.) The first basis consists of the nitrogen 5s, 4p,

Id basis from the CN study plus the 3s, Ip hydrogen basis of Table 1-13.

The second basis consists of an 8s, 7p, 2d nitrogen basis (Dunning, 1970)

and a 7s, Ip hydrogen basis (Huzinaga, 1965). (See Table 1-14.) The

clear superiority of the second basis over the first does not manifest

itself by large changes in the electron binding energies. The differences

for all theoretical approximations between the basis sets are small and

indicate that the first basis is adequate. [Anderson and Simons (1976)











30





o O
r *N "O
SNN







o C\ C
y- C CM
















I m 00 m-
































u Il CC
CC
u







D C- CO C CM
-oJ
L CCO M-







010 C: 7
I-C










CO CO-N










CM
i^ . .N









Table 1-14

CGTO Basis for NH2


NITROGENa


Type Exponent Coefficient Type Exponent Coefficient

s 13520. 0.000760 p 35.91 0.040319
1999. 0.006276 8.48 0.243602
440. 0.032847 2.706 0.805968
120.9 0.132396
38.47 0.393261 p 0.9921 1.0
13.46 0.546339
p 0.3727 1.0
s 13.46 0.252036
4.993 0.779385 p 0.1346 1.0

s 1.569 1.0 p 0.059 1.0

s 0.58 1.0 p 0.033 1.0

s 0.1923 1.0 p 0.21 1.0

s 0.088 1.0

s 0.048 1.0 d 1.5 1.0

s 0.30 1.0 d 0.95 1.0


HYDROGENb

s 68.16 1.0 s 0.082217 1.0

s 10.2465 1.0 s 0.03580 1.0

s 2.34648 1.0

s 0.673320 1.0 p 1.0 1.0

s 0.22460 1.0



a Dunning, 1970.

b Huzinaga, 1965.









found that d functions are important in describing the ground state

potential energy surface, but their effect on equation of motion calcula-

tions for the electron affinity of NH2 is small.] The first basis is

small enough to permit a third order correlation self-energy calculation.

(See Table 1-15.)

The results of the calculations indicate that the diagonal 2p-h TDA

is the most reliable predictor. Although the third order results are not

markedly inferior for the vertical ionization energy of NH2 they give

less accurate values for the adiabatic splitting of the NH2 states.

(See Table 1-16.) The adiabatic electron affinity obtained with the

anionic surface is slightly better than that estimated with the neutral

surface. The satisfactory approximation of the electron affinity of NH2

and the energy splitting of the NH2 states indicate that the ASCF plus

d-agonal 2p-h TDA correlation self-energy is a balanced treatment of

electron affinities.


1.9 Ionization energies of PH2

The states of PH2 and PH2 are similar to those of NH2 and NH2.

These calculations show if the approach taken for NH2 with regard to

the basis set choice and to the treatment of correlation is valid for a

molecule with a third period atom. The phosphorus basis set (McLean and

Chandler, 1980) is approximately as good as the first basis set for NH2

(See Table 1-17.) The phosphorus basis set has been optimized for the

anionic atom as was the first nitrogen basis of the NH2- study. The

hydrogen basis is in Table 1-13. The nuclear geometries of the neutral

states and of the anionic state are optimized with this basis. The third

order self-energy is certainly no better than the diagonal 2p-h TDA

self-energy. The agreement with experiment for the vertical electron










Table 1-15

Vertical Ionization Energies for


Bond Distance
R (bohr)


1.92


1.92c


Bond Angle Symmetry
0 (deg.) i


103.9


102.6c


a Compare difference with the observed splitting of 2A1 and B1 (1.98 to

b Compare with experiment (0.779 + 0.037 eV); Celotta et al. (1974).

c Heaton and Cowdry (1975).


NH2 (eV)


Basis


N: 5s,4p,ld

H: 3s,lp




N: 8s,7p,2d

H: 7s,1p


1.28

3.63


1.32

3.87









Table 1-15 Extended


-(-ii(ASCF)
li(ASCF) +EC(2)ii(E))


-1.10


-1.09


-(-Ii(ASCF)
+EC(2p-h TDA)(E))


0.61b

2.50


0.67a,b


2.4 eV); Dressier and Ramsay (1959).
Anderson and Simons (1976) obtain 0.42 eV.


-(-Ii(ASCF)
+EC(3)ii(E))


0.52

3.56



















LL.
L LU

< -- CM o3 ao

10 CM CM.


[ ]
+m O C\


U.



c> I
S 1/1









+


a) a) cU


CM..




- 4- I- -


I LU




L C


0

4-
'a
N

C






.0 n
u 4 C


4 -S -Q 3 03


S/-
41~


4- 4- 4-
LO LO m
CO 0 C0











C c(\j n

























LO m












-0 aQ U
('1 10 1


*"-I CM C
oj I
c 0 M


V)


a)
4-
a)

(D c
o- OJ




S-

(1) cu
LC

4-
0 '










0
>) .C
a)
CD
M

a) a(U
4-

o a
0 0


.0 aIC
0

c


















L) -D
0 a)
o +-'
-0

+a) C








a) 4 a

40 4-
o o


El E-























>~ ,
27 C C)

= ) aI
m -c












4-'W 4-3.C
0 +C-' 04-

4- -













ai m
0 0=
C:, CL C)


- SI aC S-)
m> r
Scm i- fT


-o a) S.-
+J (U4r .


- uC- -
- ajc o c




aO)L
) U0 -



+j s.- S...
CL U CU U

L) aS4- S4r-
> 0 > C
a)
r OW aa4-'

-= 4-'a) 4-'oC
4-' '4-


*T n 1(-S

a r) (U

fT3 W.. 03Q4-'
C4-' C
0 *C *>- O


LU f0 ( a)
a3 -Q 0 0-











36









o0

U





u















0 Lo' 'A
O. CM





c%-i






SD 00
+i 0






+C 3 4 <
Qi > I 5


























< 31
C c0 In




















4-

~ CO J-
O m



































4- 4s -
+ -i LL *-









-- <3 0 u

a c ci
3 4-' r-
3- '.0
o 4-> n ci
3 cc -cr

ci mC c '4-
0 CM 4 4' 0

*0 E E

















S 65- (1 C* CaS-
o -- +- 0.


















4-- C S X

i ij 4-'
i CM i-
T3 -




r- S- >


4-' 4- ci **


o C-i

4-' C50 cii<



o 4- 5- -5-r-

a a ac 'c-










a ac ci4-


-ca rci










Table 1-17

CGTO Basis for PH2


PHOSPHORUSa


Type Exponent

s 77492.4
11605.8
2645.96
754.98
248.76
91.157


s 91.157
36.226
15.211


Coefficient

0.00781
0.006068
0.031160
0.123431
0.378209
0.563262


0.160255
0.627647
0.263849


Type Exponent

p 384.84
90.552
29.134
10.886


Coefficient

0.009206
0.069874
0.292470
0.728103


p 4.3526 0.628349
1.7771 0.428044


0.6970 1.0


p 0.2535 1.0


4.7942 1.0


1.8079 1.0


0.3568 1.0


0.1148 1.0


0.0685 1.0


d 0.43c 1.0


HYDROGENb

(see Table 1-13)


a McLean and Chandler, 1

b Huzinaga, 1965.

c Dunning and Hay, 1977.









affinity of PH2 is not as good as with NH2 (see Table 1-18). The

adiabatic splitting of the PH2 states is satisfactory in the diagonal

2p-h TDA, although the third order approximation is not so far off this

time as it was with NH2. (See Table 1-16.)

The computational experience gained in these studies indicates that

ASCF + diagonal 2p-h TDA self-energy is a useful tool for studying

electron affinities. It is sufficiently simple to allow application to

larger molecules.

1.10 The transition operator method

For considering the electron binding energies corresponding to the

attachment or removal of an electron in spin-orbital i, a choice of spin-

orbital occupation numbers (Purvis and Ohrn, 1976),


0 j unoccupied
n. = j = i (1-44)
1 j occupied

defines a grand-canonical one-electron reduced density matrix (Abdulnur

et al., 1972). This density matrix gives the form of a transition

operator matrix with the elements

Tij = hij + (1-45)


where the subscripts are spin-orbital indices. This matrix is set up and

diagonalized iteratively in much the same manner as the Fock matrix in

UHF spin-orbital optimizations. At convergence, the spin-orbitals

satisfy


I= DIAGONAL "


(1-46)























I C)






w








S-~






C)N

I C)
+




C) W



N

C)C

I C)


C
4-I
E -o

E F- -0


a)

0*
C T-

0 ^
o ~
C
OCD


aO I
L)

c, -*



f0
O o


U0


C
LO
01









-J









m a
r
ar
a-
*^ a






0 -

.a

-'

NJ



















N..
a0
C
C
C



+1


N.


CM
4-I
0








a)
2
s








a)
0~








a)
1-z

4-'-





a)
cl








a)
s-

2
OJ





CJ
a)









The i-th spin-orbital energy agrees with ASCF through third order except

for some correction terms (Goscinski et al., 1973; Born et al., 1978).

These terms have been evaluated in a number of systems and are small.

(They are less than 0.01 electron volts for all the basis sets of CN"

and C12.) This method incorporates many relaxation effects into a single

spin-orbital and suggests a reference state for a propagator calculation.

Specifically, the transition operator method (TOM) spin-orbitals and

energies are the basis of the self-energy. The ground state density is

the grand-canonical density formed from the occupation numbers of

equation 1-44. It is possible, then, to evaluate second order and

diagonal 2p-h TDA corrections to the TOM spin-orbital energy in the

quasi-particle approximation, but without excluding terms in the self-

energy that correspond to relaxation when Hartree-Fock orbitals are

used. Since the relaxation is itself renormalized into the orbital,

there is no need to restrict the summations in the self-energy.


1.11 Applications of the transition operator reference

TOM spin-orbital energies and spin-orbital energies plus self-energy

corrections are in Table 1-20, along with results in the same basis for

H20 (Dunning, 1970; Huzinaga, 1965). (See Table 1-19.) Among the other

approximations are the ASCF plus correlation self-energy results. The

TOM plus diagonal 2p-h TDA self-energy results are superior to the other

approximations.

For Ne, the TOM plus self-energy calculations employ a large Gaussian

basis set (Table 1-21). While the 2s and 2p ionizations are satisfacto-

rily predicted, the Is prediction is much poorer. The diagonal 2p-h TDA

result is much worse than the second order result (see Table 1-22); this

effect could be due to the necessity for precisely describing certain















Type


Table 1-19

26 CGTO Basis for H20


Exponent

17.37
2.6273
0.58994


0.16029


1.0


7816.5
1175.8
273.19
81.17
27.184
3.4136


9.5322


0.9398


0.2846


35.183
7.904
2.3051
0.7171


0.2137


1.0


0.002031
0.015436
0.073771
0.2476
0.61183
0.2412


0.01958
0.12419
0.39473
0.6273


Oxygen s and p basis from Dunning, 1970.

Hydrogen s basis from Huzinaga (1965).


Coefficient

0.032828
0.23121
0.81724









Table 1-20

Vertical Ionization Energies of H20 (eV)
26 CGTO Basis Set


TOM

TOM + 2nd order

TOM + diag. 2p-h TDA

Orbital energy

ASCF

ASCF + 2nd order

ASCF + diag. 2p-h TDA

Second ordera

Diag. 2p-h TDAa

Second order

Third order

Experiment



a Diagonalization of full


lal 2al

540.35 33.97

540.51 33.36

33.56

559.39 36.62

540.49 33.86

540.70 32.78

540.62 31.28

33.4

33.7

32.93

35.10

540.2 32.2


propagator matrix (Born and Ohrn, 1979).


b Diagonalization of full propagator matrix (Cederbaum et al., 1973).

c Photoelectron spectroscopy (Siegbahn et al., 1969).


1b2

17.41

18.56

18.94

19.35

17.93

18.99

19.20

18.0

18.5

17.70

19.22

18.6


3al

13.03

14.13

14.61

15.66

13.15

14.20

14.44

13.4

14.1

13.18

15.18

14.7


Ibl

10.75

11.84

12.38

13.67

10.89

11.93

12.19

11.1

11.8

10.92

13.03

12.6










Table 1-21

Neon Gaussian Basis Seta


Type Exponent

s 279341.011322
43638.324426
9155.291348
2399.53
744.787
260.221
100.115
41.7663
18.6344
8.76275
4.21282
1.7713
0.723060
0.285620


p 354.616931
82.0552
25.6929
9.44944
3.82817
0.70396
0.30883
0.13229


d 2.5
2.0
0.8



a The basis employs no contractions.

See van Duijneveldt (1971) for details
of basis set optimization.









Table 1-22

Ionization Energies for Ne (eV)


Is


TOM

TOM + 2nd order

TOM + diag. 2p-h TDA

Second order

Third order

Valence approximation

Experiment


868.14

868.98

873.1







870.2


49.09

48.37

48.57


48.26


a Photoelectron spectroscopy (Carlson et al., 1971).

b Cederbaum and Domcke (1977).


19.54

20.91

21.58

19.86

22.40

21.56

21.56







45

Table 1-23

Vertical Ionization Energies of NH2 and CN
Calculated with the Transition Operator Method
and Self-Energy Corrections (eV)


System Orbital TOM TOM + 2nd order


50 2.80

3al 0.78

Ibl -1.37


3.57

1.93

-0.24


TOM + diag 2p-h TDA

3.86

2.51

0.30


CN : Basis D from Table 1-8.

NH : First Basis from Table 1-15.


CN

NH2

NH2







46

integrals involving core orbitals which arise in the denominator shift.

Such precision may be lacking when a Gaussian basis set is employed (see

Cederbaum and Domcke, 1977).

In the case of the electron affinity of CN, the results using the

largest basis set from the previous group of calculations are superior

to all other results. (See Table 1-23.)

Finally, less satisfactory results are obtained for the NH2- ioniza-

tions. Since ASCF calculations are worse than Koopmans's theorem predic-

tions in this case, the relaxation renormalization may hinder the

convergence of self-energy improvements.


1.12 Interpretation

Quasi-particle approximations are conservative methods for extending

the orbital energy concept to explain electron binding energies without

invoking the full apparatus of electron propagator theory. The overlap

amplitudes remain proportional to simple Hartree-Fock canonical orbitals.

There are many poles that correspond to each orbital, but the poles have

different weights. Thus, certain so-called shakeup peaks in a photoelec-

tron spectrum may be assigned to a simple orbital. The success of these

methods implies that it may not be necessary to completely abandon the

simple orbital picture to explain binding energies, particularly in the

core and outer valence regions.















CHAPTER TWO

EXCITATION ENERGIES


2.1 Consistent ground states and the random phase approximation

The previous chapter's study of the uncorrelated electron propagator

and its extension to correlated electron dynamics is an example of how

approximate propagators are constructed to give improved energy differ-

ences and transition probabilities. Avoiding the construction of wave-

functions for states other than the ground state is the salient differ-

ence between the propagator approach to correlation and variational

theory. This characteristic is at once the strength and the enigma of

propagator theory (Linderberg and Ohrn, 1977). Since improved ground

state density matrices from the propagator are useful for the calculation

of properties (Jorgensen, 1975; Purvis and Ohrn, 1975), it is important

to know if there exists a corresponding wavefunction. This is the so-

called N-representability problem (Davidson, 1976 and references therein)

which also occurs in other theories that seek to circumvent the explicit

use of wavefunctions, such as density matrix and density functional

theories (Ohrn, 1976). The question of N-representability in propagator

theory can be formulated in terms of so-called consistency requirements

or in terms of sum rules, which, when satisfied, imply the existence of

wavefunctions for the ground and excited states generated by the propaga-

tor (Linderberg and Ohrn, 1968).

To elucidate the points made above, the model electron propagator

of the previous chapter will serve as an example of a consistent







48

propagator. The model electron propagator with the Hartree-Fock ground

state and the operator manifold consisting of simple field operators has

electron addition operators


|N+1> = a pN> (2-1)


which generate N+1 electron states and electron annihilation operators


IN-1> = ah N> (2-2)


which generate N-1 electron states. The adjoints of these operators

annihilate the ground state, i.e.,


ap N> = ah N> = 0 (2-3)


Thus, the model electron propagator provides an explicit wavefunction for

the ground state and the N1 electron states. If the model electron

propagator is constructed in a basis so that its inverse matrix elements

are


G-1 (E) = ESij (hij + ) = (El F) (2-4)



(see equations 1-5 for an explanation of the symbols), then this matrix

is easily related to the Fock matrix which is constructed in a Hartree-

Fock self-consistent field calculation (see McWeeny and Sutcliffe, 1969).

When the eigenvectors of this propagator are reinserted into the

propagator, an identical self-consistent calculation ensues. The

Hartree-Fock SCF procedure is a self-consistent electron propagator

calculation. This self-consistent procedure is justified because it is

known that the consistent ground state for the simple field operator









manifold is a determinantal wavefunction. Consistent ground states are

states where the adjoints of the excitation operators (or ionization

operators in the case of the electron propagator) destroy the ground

state. If the operator manifold is extended beyond simple field opera-

tors, then the consistency requirements are no longer fulfilled. The

Dyson equation will yield correlated binding energies and overlap

amplitudes, but there will be no explicit wavefunctions for the N1

electron states and no consistent ground state. The propagator can yield

an improved ground state one-electron density matrix, but it is not

necessarily N-representable.

A correlated consistent ground state is associated with the particle-

hole or polarization propagator. This propagator produces excitation

energies and overlap amplitudes between N electron states as the electron

propagator produces ionization energies, electron affinities and overlap

amplitudes between the ground state and N1i electron states (Linderberg

and Ohrn, 1973). The polarization propagator, expressed as a matrix in

a basis of spin-orbitals, is given by the expression


<0|ai aaln> <0aktalin> (O>
ijkl(E) = E-En+Eo+i+ E+E -EO-ia


Note that the poles

E = (En EO) (2-6)


and the residues

(2-7)


have an interpretation for excitation processes that is completely







50

analogous to the interpretation of the poles and residues of the electron

propagator for ionization processes. As the electron propagator was

developed to give the Dyson equation from the inner projection technique,

the polarization propagator can yield the Bethe-Salpeter equation with the

aid of inner projection theory. The simplest operator manifold that is

inspired by the inner projection technique uses the f2 operator product

manifold:


P(E) = (a alh)(hl (EI -H)h)-1(hl a) (2-8)

where


h = f2 = {akal,kl} (2-9)


Improvements can be realized by including higher operator products, such

as f4 or f6. Now the question of the ground state choice arises. The

choice of the Hartree-Fock ground state produces the time-dependent

Hartree-Fock approximation (Jorgensen and Linderberg, 1970). This ground

state is not consistent and leads to inquiries concerning the form of the

ground state that is consistent with f2 (Linderberg and Ohm, 1977; Ohm

and Linderberg, 1979; Goscinski and Weiner, 1980; Weiner and Goscinski,

1980a and 1980b). The form of the consistent ground state is the anti-

symmetrized geminal power (AGP),

N/2
Ground state> = OAS NI g(2i-1,2i) (2-10)
i=I

where N is the number of electrons and 0AS = N- (-1)P, which suggests

that a variationally optimized AGP might be an interesting ground state

for a polarization propagator calculation with the f2 operator product







51

manifold. All polarization propagator calculations that employ the f2

operator manifold are called random phase approximation calculations.

This name is also applied when a Hartree-Fock reference state is used.

The name extended RPA is applied to calculations where there are improve-

ments in the ground state or in the operator manifold (Shibuya and McKoy,

1970). Finally, the term self-consistent RPA has been applied to calcu-

lations where improved ground state densities from the polarization

propagator have been calculated iteratively (Jorgensen and Oddershedde,

1972). In the present calculations there is no assurance that the

variationally determined AGP will be the ground state that fulfills all

the consistency requirements. Therefore, the polarization propagator

calculations that follow are called AGP-RPA.

The AGP has previously been studied for reasons that are unrelated

to propagator theory (Kutzelnigg, 1964 and 1965; Bratoz and Durand, 1965;

Bessis et al., 1967 and 1969; Coleman, 1965; Fukutome, 1977; Fukutome et

al., 1977). The sparseness of its two-electron density matrix and the

incorporation of correlation into a single geminal have made AGP attrac-

tive to many investigators. The following discussion treats the mathe-

matical properties of the AGP and the interpretation of electronic

structure that it provides. In addition, the AGP is used as a reference

state in a polarization propagator calculation.


2.2 The antisymmetrized geminal power wavefunction

The AGP

N/2
0AS g(2i-1,2i) (2-11)
OS i=


is formed from the geminal, g(1,2), which is expressed in terms of a









spin-orbital basis as follows,


M M
g(1,2) = C I C i [i(1)p' (2) V' (1)> (2)] (2-12)
i=1 j=1

where M is the dimension of the orbital basis. This geminal can also be

written in natural form


M
g(1,2) = I gi[, I(1)1' (2) i(1)ti(2)] (2-13)
i=1 1 1 1

where unprimed 0 stands for a spin-orbital with alpha spin and primed t

stands for a spin-orbital with beta spin. The Hartree-Fock wavefunction

is included as a special case of the AGP by restricting M to be the

number of electrons divided by two. For the case of odd-electron

systems, the generalized AGP


N+K N/2
AC Xi(i) 1 g(2i-1,2i) (2-14)
Si=N+1 i=1


where the system has N + K electrons, is the wavefunction that fulfills

the consistency requirements of the RPA. Returning to the simple AGP,

a superposition of configurations is generated when M is greater than

half the number of electrons. This expansion of configurations is





det{C (1)0' 1 (2)i2 (3) 2(4)... N/2(N-1) 'iN/2(N)} (2-15)
1 1 2 2 N/2 N/2

and includes certain double, quadruple, hextuple and higher even-fold

excitations from any determinant which is chosen as the reference







53

configuration. In this study, the geminal will be optimized in its

natural form and the further restrictions to the singlet, restricted case

M
g(1,2) = Y gi i(l1) i(2) 1(a(1)B(2)-B(1)a(2)) (2-16)
i=1 1 2


will be made, where {Pi) stand for spatial orbitals.

2.3 Density matrices and the total energy expression

The singlet, restricted geminal,

M M
g(1,2) = Ciji(1)( (2)--l(a(1)(2)-(1)a(2)) (2-17)
i=1 j=1 J/

may be transformed to natural form,

M
g(1,2) = l g ii(1)>i(2)--(a(1)8(2)-8(1)a(2)) (2-18)
i=1 1 1 1 2


where the natural orbitals of the geminal are {pi} and the natural
2
orbital occupation numbers of the geminal are Igi2 The superposition

of configurations derived from the AGP contains no configuration that is

singly excited with respect to any other determinant when the geminal is

in natural form. The elements of the one-electron density matrix of the

AGP


Nij = = Niiij (2-19)


are zero unless the indices are equal. Therefore, the natural orbitals

of the AGP are the natural orbitals of the geminal. The diagonal element

N.i is the sum of the squared absolute values of the coefficients of the

configurations in which the orbital i appears:









Nii : I lil2 1 ...
11 2> >I -1
ti i 2


Igi 2 gi 12...IgiN 12 (2-20)
S-1
2


By introducing the symmetric function of order N/2 of the squares of the
coefficients, gi, the occupation numbers of the one-electron density

matrix of the geminal,


(2-21)


Cl = 1 ...>2> gi l2 gi212 .gi N2,
11 2 N/2 1 2 N/2


the elements

expressed as


of the one-electron density matrix of the AGP can be


fN
1 DC NN
Nii = -g- A
_Z


The elements

in a similar


(2-22)


of the two-electron density matrix that are not zero follow

way according to the formulas


1 N



where i and j are the spin-orbitals, and



1 -



where i = i with opposite spin. The total energy is
where i = i with opposite spin. The total energy is


(2-23)


(2-24)







55

E = (2h +)N + I (4-2) (2-25)


+ 2Re{y Y }
i>j J 1 ij


(Linderberg, 1980), with i and j now spatial orbital indices.


2.4 Interpretations of the total energy expression

The total energy expression can be interpreted with the terminology

of configuration interaction. The first two summations in equation 2-25

arise from Hamiltonian matrix elements between identical configurations.

Each configuration's total energy is weighted by the absolute value

squared of its coefficient in the CI expansion generated by the AGP.

Thus, the integrals that occur in the first two terms are one-electron,

Coulomb and exchange integrals. In the third summation, the integrals

cannot be classified as Coulomb or exchange integrals; they arise from

matrix elements in the CI expansion between configurations that differ

by two spin orbitals.

Another framework in which the total energy may be analyzed is in

terms of pair-correlation theories (Meyer, 1977 and references therein)

and many-body perturbation theory (MBPT), in which pair correlation

energies provide corrections to the Hartree-Fock total energy (Bartlett

and Shavitt, 1977; Bartlett and Purvis, 1978). These pair correlation

terms are produced by matrix elements of double excitations with the

closed-shell Hartree-Fock configuration. In MBPT, the contributions

from the first and second summations in equation 2-25 that do not come

from the reference configuration vanish according to the linked-cluster

theorem (Brueckner, 1955a and 1955b; Goldstone, 1957). The contributions

from the third term derive from the same type of CI matrix elements as

the pair energies of the various many-body approaches.









2.5 Coefficient optimization

Because of the simplifications in the two-electron density matrix

that obtain by considering the natural form of the geminal, both the

coefficients, gi, of the geminal and the natural orbitals of the geminal

must be optimized. If the general form of the geminal had been employed

(see equation 2-17), it would have been sufficient to only vary the

coefficients, cij.

The coefficient variation procedure is obtained by imposing a

necessary condition for the stationarity of the total energy. Let the

Rayleigh-Ritz variational quotient,


E = (2-26)



be written as


E = E' (2-27)




where E' is the unnormalized AGP total energy, and C(N/2) is the normal-

ization, or the symmetric function of order N/2 of the squares of the

absolute values of the coefficients of the geminal. If the first varia-

tion of the energy with respect to the coefficients is zero,


3- E = 0 (2-28)


subject to the constraint that the geminal be normalized,


g*jg = 1 (2-29)
J


then the necessary condition









[E (Z g*jgj-1)] = 0 (2-30)
1 *i j

is obtained, where X is a Lagrange multiplier. By noticing that both the

normalization of equation 2-27 and the unnormalized total energy satisfy

the Euler formula for symmetric functions,


E' = 2 gi i- (2-31)


1 BC
C N2 j g (2-32)
J gj


the necessary condition for stationarity of the total energy can be

rewritten


S 1 E'l E BC2 (2-33


1 E' (2-33)
Bg*. ag*i [cI]12 Sg*.



= E2C g (2-34)



SI Hg (2-35)

'2 M

where Hij = H*.j. Now with the introduction of the constraint on the
normalization of the wavefunction, equation 2-30 implies that


1 Hig = Ag (2-36)
Equation 2-36 is aJ standard eigenvalue problem that is iteratively


Equation 2-36 is a standard eigenvalue problem that is iteratively









solved. At self-consistency, the value of A is zero since

7 H. g = -E = 0 (2-37)
N[N ij 13 9*i


provided that gi is itself not zero.

2.6 Orbital optimization
A unitary transformation of the orbitals in the AGP may be written as

ar = e ar e- (2-38)

(Dalgaard and Jorgensen, 1978), where


A = i I kl(a kai altak) (2-39)


The resulting total energy after such a transformation on |0> is


E(X) = <0Oe-iA HeiA 0> = <0HIO0> + -i<0[[A,H]I0> (2-40)
+ {<0IA,[H,A]]|0>
+ <01[[A,H],A]|0>} + .. .

A stationary point occurs when 6E(X) = 0, or

6E(X) = -i + i{ + (2-41)

+ } + ....









Wrs = c<0[ar as as ar,H]lO> (2-42)


and let


Ars,r's, = { (2-43)


+ .


With the neglect of terms higher than second order, equation 2-41 becomes


A = -A-1W (2-44)


The solutions, X, are increment vectors and are converted to unitary

orbital transformation matrices according to


X = e-A = -Ucosh(d)Ut XUsinh(d)d-1Ut (2-45)


where


2 = UdUt (2-46)


The integrals are transformed to the new basis before the next orbital

or coefficient iteration.

2.7 Results for lithium hydride

Calculations for AGP wavefunction optimizations are attempted with

a small Gaussian basis set (see Table 2-1). The total energies from

these calculations and the total energy differences along the ground

state potential energy curve are compared to results from Hartree-Fock,

second order MBPT, an MC-SCF wavefunction that contains the same config-

urations as AGP, but not the constraints on the coefficients, and to the







60

Table 2-1

LiH Basis

13 CGTO Basisa


Type Exponent Coefficient

Li,s 642.41895 0.00214
96.79849 0.01621
22.09109 0.07732
6.20107 0.24579
1.93512 0.47019
0.63674 0.34547

Li,s 2.19146 0.03509
0.59613 0.19123

Li,s 0.07455 1.00000

Li,s 0.02079 0.39951
0.00676 0.70012

Li,s 0.08948 1.00000

Li,p 2.19146 0.00894
0.59613 0.14101
0.07455 0.94535

Li,p 0.08948 0.15559
0.02079 0.60768
0.00676 0.39196

H,s 18.73110 0.00349
2.82539 0.83473
0.64012 0.81376

H,s 0.16128 1.00000


a Kurtz and Ohm, 1978.







61

full CI result. The percentage of the correlation energy is also

monitored.

The usual starting point for beginning the optimizations is the

Hartree-Fock wavefunction in the canonical orbital basis. The choice of

coefficients is zero or one according to whether the orbital in question

is occupied or not in the Hartree-Fock wavefunction. (The coefficients

are assumed to be real.) Coefficient optimizations are begun; usually

five coefficient iterations are sufficient to begin the orbital optimiza-

tion. Only one orbital optimization is made before the next round of

coefficient optimizations is started (see Table 2-2). The motivation

for favoring the coefficient optimizations stems from the comparatively

smaller effect of the orbital optimizations on the total energy and from

the necessity of performing an integral transformation after each orbital

optimization. For the regions of the potential energy curve where the

Hartree-Fock determinant dominates the full CI expansion, this procedure

is satisfactory and converges in about 20 cycles, where each cycle is

composed of five coefficient iterations and one orbital iteration. The

convergence behavior for other regions of the curve grows worse as the
2 2
doubly excited determinant 02 -+ 032 becomes more important in the CI

expansion of the wavefunction. The AGP thus begun also has a tendency

to converge to an excited state. For this region of the potential curve

(beyond R = 4.0 bohr), the natural orbitals of the full CI result were

used, and the AGP calculation converged as quickly as the previous

results. The full CI natural orbitals were also used to start AGP

calculations in the interior of the potential curve (within R = 4.0 bohr)

as a check on the original results. The AGP iterations converged to the

original result. A typical example of the convergence behavior is given

in Table 2-2. The use of a damping factor of 0.5 in the coefficient









Table 2-2

Convergence Behavior for AGP Wavefunction
Optimization for LiH 1Z+ at R = 3.015 bohr


Cycea


Energy (a.u.)


-7.96216

-7.97133

-7.97416

-7.97512

-7.97514

-7.97515

-7.97515

-7.97515


53658

53664


a One cycle is five coefficient
by one orbital iteration.


iterations followed









optimizations has been found to assure convergence, although it is not

necessary to impose a damping factor in all cases. The convergence

behavior listed in the table is probably not the most rapid possible with

this method. The convergence criterion for the total energy of 10-10 is

generally the first criterion to be satisfied. The sum of the squares

of the orbital transformation matrix elements and the sum of the squares

of the elements of the orbital gradient matrix,


Y I XrsI 2 and I I|<0O[artas,H]|0> 2 (2-47)
rs rs


are also employed as convergence criteria. When these tolerances are

both set equal to 10-12, the former criterion is invariably the last to

be satisfied, often requiring five iterations after the satisfaction of

the total energy requirement.

Inspection of the coefficients of the geminal at each internuclear

distance reveals a trend (see Table 2-3). Near the equilibrium inter-

nuclear distance, two of the coefficients dominate the geminal. This

dominance is revealed by the table in which natural occupation numbers

for the AGP are listed. The other terms in the geminal are quite small

and have the opposite sign as the dominant coefficients. As the bond

length is increased, a third coefficient becomes important. This coef-

ficient has a negative sign and approximates the lower of the two other

important coefficients. The absolute values of the second and third

largest coefficients decrease as the distance between the nuclei

increases. The largest coefficient attains the maximum value of one.

The AGP coefficients are forcing the AGP to approximate a two configura-

tion wavefunction where both electrons of the highest occupied sigma

orbital in the Hartree-Fock wavefunction are promoted to the lowest










64






Cc C) cl cc cc c o CD CD c c:. c c c Co
C X X X X X X X X X X X X X
Nc oC N- LO Z m 00 m m m -` n
CD O cO Cc o J N 0 c D co C D CC0 OC
CI O O O O O O O O DO DO O O



co cC r- m N- N o Nc o No


x x x x x x x x X x x x- X
CD m- o O o HO o N -A HO 0 cc 0, n- n M
o- I c c o c\i cc HO H Cc LO Ho CD Cj Co C-




0 r- HO m 0 0 0 0 0 00 H 0 Cn
It H N H NO N N- L-


o x x x x x x x X X X X x x
C -O Nl CD r CD iD CD HO ) C> CD CD C:, c -
S C I I I o r o

QC O O HO N.O O OJ O HO N. N- O N- HO
co c\
0 I

Scc Ln rI I I I Io r i
S c0 0 0 0 0 0 0 0 0 0 0 0 0 0

O X X X X X X X X X X X X XX


m I- N c j0 0 M 00 C: n M 1- 0L
m m
o I I I II III










3c
S- - - - -



C -1 cc c- cc 00 N- m- N- l HO HO HO HO



N- co cc N- i m- o n CD D C D c N cc N- c-
3 CC i HO cj HO N- N- LO O N- O AC O --i
11o c cci r HO N- N- ro 0c L-) CL 1- Ho N- Ho
( m
1- HI I I I I I I I I I I I









a< c - - - - - -` -


a X X x X X X X < X X X X X X
M cC\j O A c c N ro C rN- O CO O M





MC iD D 1O N N- mt HO :T N- cc cc
II No N- H N- O co HO O





0H c f- r- N- H O cc cc cc -O




" HO mc rH A-A 'j- HO N- N- N- NO NJ-
C. I I I I I I I I I I
H I- cI i I I O i- I' co m I I
-= 1c 1c c c cc c cc c cc cc cc c c

CN- N- N- N- HO CO N N cO O HO CO C





















od- C C
CD o o CD

o V V V







1-1

10 I I I
C C C







S0 0- -



0 I o












CM C C
C V V 0
o 0









- CD
C m .o 1
LnC I I 0


CO v V V









n CD
ro 0
J 0 v


CLn C c l a- al
C V V V V V V v


r- o
m -i
C V

C:)






C C
C I

0 V


0 o 03 0 o
0 I I 0 I

C 0 0











0 C) 0 i-
0 V V 0 V


C0 0



0 C C


C 0 0 I 0 I 0 I
InO Co C C C CO COO CO
U- C C 0 C 0 -
o C D o ,C -
0 0 0 0 0


- m '

C C:) I
C (n C


C CO
o o


m C Cm


CD CC


O CO


CO I o o to C D C 0 : o C C

T O iV V O O O O V0 Ii 0
CI aI 0 0 0 0 C0 CI
C cN C C C C co C C C= CL C>
m C7 - -I CO C) CO C) CD CD









C) C i i C CoMo C C c CD
SI I I 0 0 0 0 I I


o CMc m o 0 C) CM m


C

0

A r-~

0
co Co
C: On
J m



z
0




0 C
L- C
C)


- M m d.ln LO








2 2 2 2
unoccupied sigma orbital, i.e., o1 o2 -- a1 3 This type of excitation

is included in the AGP wavefunction. The next most important configura-

tion in the AGP is the configuration where all four electrons are
2 2
assigned to the sigma orbitals that were just mentioned: a2 23 The

core orbitals on lithium are not occupied in this configuration. In the

limit of infinite separation of nuclei, such a configuration should have

a coefficient of zero in the CI wavefunction since it corresponds to a

Li+ atom and a H- atom. The AGP reduces the importance of this unphysi-

cal configuration by making the second and third largest coefficients

very small. When the ratio of the largest coefficient to the next

largest coefficient approaches infinity, the AGP wavefunction approaches

a generalized AGP wavefunction (Weiner and Goscinski, 1981).

The AGP at the equilibrium internuclear distance provides slightly

less than half of the correlation energy (see Table 2-4). This result

is similar to the results obtained by other investigators (Jensen, 1980;

Bessis et al., 1967 and 1969; Linderberg, 1980; Kurtz et al., 1981).

At the internuclear distance of 20.0 bohr, however, the percentage of the

correlation energy rises to 92. What is of greatest importance is the

treatment of total energy differences along the potential curve (see

Table 2-5). The results from AGP give a curve that is similar to the

full CI curve. The AGP wavefunction predicts dissociation into

uncharged atoms, unlike restricted Hartree-Fock. This dissociation

property makes the AGP a useful ground state for the investigation of

potential energy surfaces.


2.8 The random phase approximation of the polarization propagator

The AGP is now used as a ground state for the determination of exci-

tation energies with the random phase approximation of the polarization











67



-o in u r 1- 0
L CO e N Z O
t In a NC CM Cn a CM 6^ a
I C Co o Co m o NO N O NCCM
C cO CM C C C Co M

-I I I I i I I








QJ
41I

-1 in co o co *- r- Co
S Lu- M c C m Ln co cr Cm
u C- 0 1 C- 0 0


o d C M p N. -. O. OC M
D) 03 co me a m a 4 e o q Z









S C

V)

Sm r^ o 0L) L
M0 O Ln Ln co cO
4-1 0-I CO h 1 D C O C C 0 C
S* c c C 0 Z C



*0
D 00 C m
C)



- C




I-. -n C CM cn co Z 1z



o co *
13 1 r- ir- I- r-- r1-









CUm
S- -S
Ci U

i C CM C Co C- IC) Co


n0 4-I rn- CM 1- r c- CZ. C D-
l- s.- I I I I I I












CM
Q;


















0 0 C) c'd z- CO CM


I 0 0 0 0 0 0

& 0 0 0 0 0 0











S L m co n r- C
a CO LO L a-










m 0S CD














a 0
CS-
C











00
+ II
c r C- o o

S a) i C -D C )















4- 0
C3
nn s




























c a) 0 C m3
Ci S
m- 0
C)









C) C! 9 9 9 9 9
4-











n W in CM ii CM c
C- I io U COc in h. C'

W 0 O O I- .-



o I-









in


cMr c d c- in cc c o
CJ









propagator. If the operator manifold is transformed so that


ij = g*ja'a g*iaj ai (2-48)



ij gjjai giaI ,' (2-49)


where j = j with opposite spin, then the polarization propagator will be

block diagonal,



P(E) = (2-50)
PE = 0 <>E


for a self-consistent RPA polarization propagator (Weiner and Goscinski,

1980a and 1980b). The excitation energies are obtained from


<

While the AGP that has been calculated for use in this calculation has

been variationally optimized, there is no guarantee that it is the AGP

that is the consistent ground state of the RPA. The previously cited

proofs only state that the consistent ground state is of the AGP form.

Nonetheless, if the upper block of equation 2-50 is constructed and

diagonalized, the eigenvectors form excitation operators and their

adjoints destroy the AGP ground state. To this extent, the variationally

determined AGP is a consistent ground state. Of course, the other blocks

of the propagator have been neglected and are, in general, not equal to

zero.









2.9 Polarization propagator calculations on lithium hydride

The lowest excitation energies and the corresponding excitation

operators are calculated according to the previously discussed procedure

(see equation 2-51). In Table 2-6, they are compared to full CI excita-

tion energies. Near the equilibrium internuclear distance, the agreement

between the AGP-RPA calculation and full CI is good. As the internuclear

distance is increased, the results grow worse. This perhaps is due to

the need for elements of f4 in the operator manifold since more than one

configuration is important in the ground state. Specifically, near the

equilibrium internuclear distance, the single determinant, molecular

orbital description of electronic structure is qualitatively correct.

Excitations are approximately single replacements from a reference deter-

minant. As the internuclear distance grows, configuration interaction

in the ground state becomes important. The two configuration wavefunc-

tion that is appropriate for such regions of the potential energy curve,
22 22
1 2 01 23 can be approximately written as a valence bond wavefunc-
tion,

det{ Li,1s(1)a(1)kLi ,s(2)8(2)Li,2s.(3)a(3)PH, s(4)B(4)} + (2-52)

det{.Lils()t1a ()lLils(2)g(2)OH, s(3)a(3).Li,2s(4)8(4)}


and the lowest excitation is easily characterized as


[Li,2stH,ls + H,1sLi,2s] H,ls H,1s (2-53)

which is a single excitation from the valence bond wavefunction. In the

delocalized molecular orbital treatment, the same excitation must be

written as











71






tlO m om
'- ) h 1 Lo Lo


0 0 0 0 0 0 0
C:) CO C=;O C O DO CO










co CO )Cj -z0 zt
o 0 C) :I. Ln

Ar ai O Lf 0 O O O
C D fl CD r- r-. 04 r CO
rr) O O CO CO CO CO













Ln co C D



C) CD CD C) CD C) CO
I c




4- 0 CO CD O CO CD


0









o CD r-I 04 :I. 0










"4 IZI r-D m o D


U- a) Cl C) CO CO CO CO
4 CM CM o 0 o Co CO
M ~ 0 0
<:dd d






O M 'm C D 'd -
CM ^t r L n i- o
r- rD CM O Oi L r> iO L
r- i1 .- 0 0 0 0







72

2 2 2 2 22 22 2 2
1 2 1 a 3 -i a1 2 + 1 3 1 2 23 31 23 2 (2-54)


where


a1 = Li,ls (2-55)


a2 = Li,2s + H,ls (2-56)


3 = tLi,2s H,s (2-57)

This change in the nature of the electron distribution suggests that

perhaps a transformation of the basis of excitation operators might be

useful in cases where more than one configuration is dominant in the CI

wavefunction.














APPENDIX

CORRELATION SELF-ENERGIES

The following terms should be excluded from the self-energy in the

calculation of electron binding energies when the ASCF plus correlation

self-energy formulas are used. These exclusions apply only to the

diagonal elements of the self-energies. The indices refer to spin-

orbitals. Indices a,b,c,... refer to occupied spin-orbitals, and indices

p,q,r,... refer to unoccupied spin-orbitals.
The second order correlation self-energy is

(A-1)

C,22(E) = 1 + 1
ii ai b p E+i -p a-b 2 pii q ai -a 'p- q

The diagonal 2p-h TDA correlation self-energy is

(A-2)
C,DIAG 2p-h TDA abllip>

Sai bi E+Ep-C -b+4--

1
2 a pi q/E+ea p q- -++






74
The third order correlation self-energy is

(A-3)
C,3 1 (1-6pr~pi)
S(E) = ( 2 2E+eac sC -E1E+E-_ -E
ii a p q r s a-r-sE+a-Ep-q

1 (1-6ac6ia)
a b c d p E+p-c-dJ E+p-a-

((1-6ab6ia)(1-6pr6ip)+(l-6qi)]
a b q p E+a-ap- E+b-Sq-rJ

((1-6acia)(l-6,pqip)+(1-6bi))
+ c E+ p-Ea-bJ E+Eq-Eb-Ec


4abcp q cE+Ec- q a+b p q

1 y pq>
4 a b p q E+ea-_p qEb +c- pE q

+ 1
Sab p q r E+e --ebJa+ :b -q-E

1


4a b p q r E+ep -Ea-eb a +b _q r

S(1-6q )
a b p q r b p E+Eb-pEq a+cb Fp Fr

(1-6ri)
abp q r E+b- p-rJ :a+cb-p-q

(1-6ci)
(E+c -Ec- a C p
a b c p q E+q -a c] [a+Eb-p -qJ

(1-6bi)
abcp q E+p-a b (a+c-p-q






75

S1
a b p q r oa Eb pqp-qJ (Ea b-q- rJ

1
2 L L L L +-C -{- ( +e -E- F
Sa b c p q a +b p q b+Ec p q

1 r
2 abp q r a +b q r a

1 I I pq
2 a b c p q b c p q a p

+
2a b p q r cb-Ep q a- r

1 1 r I I
2 a b c p q a +b -p- q -C Ep















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BIOGRAPHICAL INFORMATION


Joseph Vincent Ortiz was born on April 26, 1956 in Long Island,

New York. In 1973 he graduated from Plainview-Old Bethpage High School

and was a New York State Regents' Scholar. He received the Bachelor of

Science with High Honors from the University of Florida in 1976. In

1979 he married the former Karen Fagin.










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. Yngv Ohrn, Chairman
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.






Thomas L. Bailey
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. Micha
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.


I I


William Weltner, Jr.
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.






Joh/R. Sabin
Professor of Physics and 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 1981
Dean for Graduate Studies and Research




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