 Title Page 
 Dedication 
 Acknowledgement 
 Table of Contents 
 List of Tables 
 Abstract 
 Introduction 
 Electron binding energies 
 Excitation energies 
 Appendix: Correlation selfene... 
 Bibliography 
 Biographical sketch 

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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 Dissertations, Academic  Chemistry  UF 

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bibliography ( marcgt ) nonfiction ( marcgt ) 
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Statement of Responsibility: 
by Joseph Vincent Ortiz. 

Thesis: 
Thesis (Ph. D.)University of Florida, 1981. 

Bibliography: 
Bibliography: leaves 7679. 

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Typescript. 

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Vita. 
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UF00099238 

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VID00001 

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University of Florida 

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alephbibnum  000295452 oclc  07895204 notis  ABS1798 

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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
Page 34
Page 35
Page 36
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
Page 56
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Page 63
Page 64
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Appendix: Correlation selfenergies
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
11 C12 and C12 CGTO Basis.................................. 14
12 Total Energies for C12 and C12 below 918.0 a.u.......... 16
13 HartreeFock Total Energy Lowering for C12 and C12 ....... 17
14 Vertical Electron Affinities for C12... ....... ....... .. 18
15 Adiabatic Electron Affinity of C12 ...... .............. 19
16 Adiabatic Electron Affinity of C12...................... 20
17 Adiabatic Electron Affinity of C1 ........................ 21
18 CN and CN CGTO Basis ................................... 23
19 Total HartreeFock Energies for CN and CN ............... 25
110 HartreeFock Total Energy Lowering for CN and CN ........ 26
111 SCF Equilibrium Nuclear Geometries ....................... 27
112 Vertical Ionization Energy of CN......................... 28
113 Hydrogen CGTO Basis (NH2, PH2).......................... 30
114 CGTO Basis for NH2 .................................... 31
115 Vertical Ionization Energies for NH2 ..................... 33
116 Adiabatic Ionization Energies for NH2 and PH2 ........... 35
117 CGTO Basis for PH ...................................... 37
118 Vertical Ionization Energies for PH2 ..................... 39
119 26 CGTO Basis for H20 .. .................................. 41
120 Vertical Ionization Energies of H20....................... 42
121 Neon Gaussian Basis Set .................................. 43
122 Ionization Energies for Ne............................... 44
Table Page
123 Vertical Ionization Energies of NH2 and CN Calculated
with the Transition Operator Method and SelfEnergy
Corrections.............................................. 45
21 LiH Basis............................................... 60
22 Convergence Behavior for AGP Wavefunction Optimization for
LiH 1Z+ at R = 3.015 bohr................................ 62
23 AGP Geminal Coefficients and Principal Natural Orbital
Occupation Numbers for LiH................................. 64
24 LiH Total Energies and Correlation Percentages for 1 +
Ground State.......................................... 67
25 Total Energy Differences Along the Ground State 1 +
Potential Curves with Respect to the Total Energy at
R = 3.015 bohr............................................ 68
26 AGPRPA 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 selfenergy diagrams enables the
construction of electron propagators that include relaxation contribu
tions to all orders. Correlation selfenergies are obtained in second
order, third order, and in the diagonal 2ph TammDancoff 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 BornOppenheimer 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. HartreeFock 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 HartreeFock 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 quasiparticle concepts of elec
tron binding energies and the introduction of energydependent correla
tion potentials.
Excitation energies, or energy differences between electronic states
with the same number of electrons, are also amenable to propagator inter
pretations. HartreeFock 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 spinorbital
basis has the elements
f n*(i)f (j) gn*(i)g (j)
Gij(E) = EE (N+1)+E (N)+ia + E+E (N1)Eo(N)ia
The total energy of the N electron ground state is EO(N), and E (N1) is
the total energy of the nth 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) = (12)
where the field operators satisfy the relations
4
[ai,a] = [a t,a ] = [ai,aj ]+ 6 (13)
In this notation, the manyelectron Hamiltonian is
H = hiji a+ aii aa ak (14)
1 j i j k 1
1P
where =
1 2
NUCLEI Z
h = (15)
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 (16)
in the spin orbital basis that diagonalizes the Fock matrix,
Fj = hij + I (17)
k I
where
= (18)
so that
Fij = Eiij
(19)
Since the model Hamiltonian contains only oneelectron terms, the elec
trons do not interact. The model ground state is
N t
N> = I ai vac> (110)
i=l1
where
N ovac> = n.ivac> = I ai ailvac> = 0.vac> (111)
op 1
The ground state total energy is
= \ i = a cini (112)
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(N1) = n (113)
which are negative ionization energies, and
EPOLE = En(N+1) EO(N) = En (114)
which are negative electron affinities. The overlap amplitudes are
f (i) = = din
(115)
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) = <> (116)
the identities
1 E (N+1)EO(N)
E[E En(N+1) + EO(N)] 1 + EE (N+1)+E(N)
(117)
(En(N+1) EO(N)) = ,
and the corresponding relations for the N1 electron case imply that
E<>E = <[ai,a] > + <<[ai,H];a>>E (118)
This equation can be iterated to give
<>E = E1<[ai,a ]+> + E2<[[ai,H],a t]> (119)
+ E3<[[ai,H],H],at]> + ...
The definition of superoperators (Goscinski and Lukman, 1970), 1 and R
IX = X HX = [X,H] (120)
which act on the set of field operators (Redmon et al., 1975)
X = {a,aa aka a akaam'...}) (121)
X =~ia jka
7
and the definition of a scalar product (Goscinski and Lukman, 1970)
(XY) = <[Y,X ]+> (122)
convert the expansion in equation 119 to the form
<>E = (aj (E1H)1ai) (123)
The inverse operator which occurs in the propagator matrix,
G(E) = (aI(EIH)la) (124)
can be circumvented by the inner projection technique (L6wdin, 1965):
G(E) = (alh)(hl(EIA)h)l (hl a) (125)
where the operator manifold is complete, i.e.,
h = {a,a aa,a a aaa,...} (126)
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} (127)
such that
(alf) = 0 (aaI) = (ff) = 1 (128)
implies that
G(E) = (al(EIP)a) (aIf)(fI(EIH)f)1(fIHa) (129)
For the model uncorrelated system where
IN> = HartreeFock> (130)
and where the model Hamiltonian is
H Y lEa'ai (131)
the exact inverse propagator is
Go'(E) = (aj(EIR0)a) (132)
since the second term in the righthand side of equation 129 vanishes.
Koopmans's theorem appears once again since
[GO1(E)]ij = (aii(ETAo)aj) = (Ei)5ij (133)
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> = IHartreeFock> + Icorrelation> (134)
then the propagator becomes
G(E) = El (aHa)HF (afHa)CORR (135)
(a f)(fl(Ei)f)l(f(Ha) ,
where (Iaa)HF is (aIHa) taken only with respect to the HartreeFock
contribution to the ground state, and where the remainder of (aiHa) is
(alHa)CORR. In other words,
G1(E) = GOI(E) I' '(E) .
(136)
The superoperator matrix element
(ajlAai) = hji + i (137)
kl
depends only on the oneelectron reduced density matrix and is indepen
dent of E. Taken together, the energy dependent and energy independent
corrections to the uncorrelated propagator,
+' + '(E) = (E) (138)
are the selfenergy and appear in the Dyson equation,
G (E) = GO'(E) (E) (139)
The selfenergy 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 HartreeFock 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 = ( ) + (140)
The components of the selfenergy also derive from diagrammatic
perturbation theory (Cederbaum and Domcke, 1977). In this approach, the
terms of the selfenergy 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
HartreeFock 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 ringladder 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 selfenergy, 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 selfenergy.
Yet another method of analysis of the selfenergy is the identifica
tion of socalled relaxation and correlation terms by perturbation theory
(Born et al., 1978). Relaxation terms are terms that occur in the
difference between HartreeFock 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 HartreeFock 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 selfenergy
which are not relaxation terms are correlation terms. Since all self
energy contributions treat correlation effects with respect to the
uncorrelated HartreeFock propagator, the use of the term correlation
selfenergy 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 HartreeFock 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 HartreeFock orbitals of the
same symmetry in the overlap amplitudes are obliterated when the diagonal,
or quasiparticle, approximation is made. Previous studies of the non
diagonal propagator have also indicated that these mixings are small
(Schirmer et al., 1978). The quasiparticle 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 selfenergy
included in the ASCF energy difference, the renormalized quasiparticle
propagators under consideration will have the form (Kurtz and Ohrn, 1978)
12
G.l .(E) = E (ASCF)i CORRii(E) (141)
The quasiparticle assumption permits direct iteration of equation 141
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)
Gii(ci) = ci (ASCF)i CORRii(i) (142)
will be evaluated and reported. Typically, when the quasiparticle
approximation is valid, dG1(E)/dE is small. Consequently, the results
of 141 and 142 closely agree (within 0.1 election volts). Diagrammatic
theory suggests the second and third order correlation selfenergies.
(See the Appendix for correlation selfenergy formulas.) Another approx
imation, which derives from the inner projection formalism with the f3
operator product manifold and a HartreeFock ground state, ignores off
diagonal terms in the matrix (Born and Ohrn, 1978, 1979 and 1980)
(f3(ER)f3)1 (143)
This approximation is the socalled diagonal 2ph TammDancoff approxima
tion (2ph 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 ringladder diagrams. (For a discussion
of the nondiagonal 2ph TDA, see Mishra and Ohrn, 1980.) All of the
following applications of the ASCF plus correlation selfenergy 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 11); 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 selfenergy.
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
CoIn
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
HI I 1I f I T
CO
CO 
tfl Ila t L L
15
The correlation selfenergy exhibits the aggregate effect of these
apparent instabilities.
As the data in Tables 12 and 13 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 selfenergy are also listed
in Table 14. 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 2ph TDA correlation selfenergies 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 15. Adiabatic electron affinities using
various combinations of potential energy curves and vertical electron
affinities are in Tables 15, 16 and 17. The restricted HartreeFock
potential curve is inadequate because it separates into ions instead of
co IcOL C n iLin coC 0 0 co co c
cnot 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
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rl rr 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
cOfcL Lnoc co CO m Cl M0 O
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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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In
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mc v
Ci
CT 0*
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In C CD
*r0 CM
CO 0O
0
0
CO
CI
o
0
(,
o
C\j
I 0
00
Q) =
i 0
O
I 
0
4
C
0
I
CO 0
m LO
cni
C
0
I in
cc
4 
O c
0 I
+ . 0
4 C
CO CO
C\M CM
O O
Q) Q)
cC 0 c00
CO C C! 0 CO9 .01
m On Cl) n Crl0nfLfL
0
4)
C
m
U
c i
CMm
= 0
+ II
0ti'.
0 0
In  ' CO
l O
u!03
ln 0
In i 1
yi i+ c
*i d/ i
C
0
0 0
r1 0
m ii 3
f0 CQ = u
CQ Lj
C
C 0
0 0
40 IA
+ i
U 0
o c
=3
U. 4r
0/1 CD
C
n I
4 0
4 I
*r* i C .
3
0
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m* a
*T nc (
(>fO fl3
fOCQ
CQ  L
LO ln M m m (n LO co m
0000 oo o o oo o
~I 555 5555 5
C) CD
I I
>  o 0 0 m c o m cD m
0000 00 0 O 1 00
> I I I IO V I I I I I
LL
I
CL
u
hi
0
3
o
mo in M iO ro in i Ln
cOOM
Sm 0
S> CM Co DCD m
U CD C)
z
0.
CM
Ci
I . .
n *
<0 Co1 o 0 i u
o* Ci 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 lcj C\oj C cm 
m o .n o m r
LO  N 0C InN N:I
1 U 0NJi 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 toc LoL
O0js  NOf Oil LO 
cr o r coj 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 rmcOcc Sor0 DO CI LOO
0 CO3.4 OC'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
fC
0 m
0
U U
c I
4
0 *
4I V
01
s
C 0
0.
U U
c lc
CI 
,nO I
( C!
CE C
] 0)
fO 
T(J 0 
C
N
Ci
0
4
vl/
I~ *0
aC
U
4'
ai s
S r
5i
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)
40
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
4C
+ 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 17
Adiabatic Electron Affinity of C12 (eV)
Basis Verticala V(C12) EA(adiab)
A Eq. (A1) 3.08 1.59
Eq. (A2) 2.62 1.13
B Eq. (Al) 3.60 2.11
Eq. (A2) 3.19 1.70
E Eq. (Al) 4.00 2.51
Eq. (A2) 3.74 2.25b
D(C12) = 1.49 eV (see Douglas et al., 1963)
a Eq. (Al) means that V(C12) is calculated as I + ZC(2). Eq (A2)
means that V(C12) is calculated as I + E2ph 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 111.) The total
energies listed in Tables 19 and 110 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 18.) 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 HartreeFock 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 112). 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
4I
5
di
cc
di
0
1
CM Co CO
0 co o cc
Ln 00 c
UI 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)
drd dOdiJ
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
II 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 2ph 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 113.
The second basis consists of an 8s, 7p, 2d nitrogen basis (Dunning, 1970)
and a 7s, Ip hydrogen basis (Huzinaga, 1965). (See Table 114.) 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
IC
CO CON
CM
i^ . .N
Table 114
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 selfenergy calculation.
(See Table 115.)
The results of the calculations indicate that the diagonal 2ph 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 116.) 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
dagonal 2ph TDA correlation selfenergy 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 117.) 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 113. The nuclear geometries of the neutral
states and of the anionic state are optimized with this basis. The third
order selfenergy is certainly no better than the diagonal 2ph TDA
selfenergy. The agreement with experiment for the vertical electron
Table 115
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 115 Extended
(ii(ASCF)
li(ASCF) +EC(2)ii(E))
1.10
1.09
(Ii(ASCF)
+EC(2ph 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 43.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 Ci
4' C50 cii<
o 4 5 5r
a a ac 'c
a ac ci4
ca rci
Table 117
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 113)
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 118). The
adiabatic splitting of the PH2 states is satisfactory in the diagonal
2ph TDA, although the third order approximation is not so far off this
time as it was with NH2. (See Table 116.)
The computational experience gained in these studies indicates that
ASCF + diagonal 2ph TDA selfenergy 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 spinorbital i, a choice of spin
orbital occupation numbers (Purvis and Ohrn, 1976),
0 j unoccupied
n. = j = i (144)
1 j occupied
defines a grandcanonical oneelectron reduced density matrix (Abdulnur
et al., 1972). This density matrix gives the form of a transition
operator matrix with the elements
Tij = hij + (145)
where the subscripts are spinorbital indices. This matrix is set up and
diagonalized iteratively in much the same manner as the Fock matrix in
UHF spinorbital optimizations. At convergence, the spinorbitals
satisfy
I= DIAGONAL "
(146)
I C)
w
S~
C)N
I C)
+
C) W
N
C)C
I C)
C
4I
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
4I
0
a)
2
s
a)
0~
a)
1z
4'
a)
cl
a)
s
2
OJ
CJ
a)
The ith spinorbital 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
spinorbital and suggests a reference state for a propagator calculation.
Specifically, the transition operator method (TOM) spinorbitals and
energies are the basis of the selfenergy. The ground state density is
the grandcanonical density formed from the occupation numbers of
equation 144. It is possible, then, to evaluate second order and
diagonal 2ph TDA corrections to the TOM spinorbital energy in the
quasiparticle approximation, but without excluding terms in the self
energy that correspond to relaxation when HartreeFock orbitals are
used. Since the relaxation is itself renormalized into the orbital,
there is no need to restrict the summations in the selfenergy.
1.11 Applications of the transition operator reference
TOM spinorbital energies and spinorbital energies plus selfenergy
corrections are in Table 120, along with results in the same basis for
H20 (Dunning, 1970; Huzinaga, 1965). (See Table 119.) Among the other
approximations are the ASCF plus correlation selfenergy results. The
TOM plus diagonal 2ph TDA selfenergy results are superior to the other
approximations.
For Ne, the TOM plus selfenergy calculations employ a large Gaussian
basis set (Table 121). While the 2s and 2p ionizations are satisfacto
rily predicted, the Is prediction is much poorer. The diagonal 2ph TDA
result is much worse than the second order result (see Table 122); this
effect could be due to the necessity for precisely describing certain
Type
Table 119
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 120
Vertical Ionization Energies of H20 (eV)
26 CGTO Basis Set
TOM
TOM + 2nd order
TOM + diag. 2ph TDA
Orbital energy
ASCF
ASCF + 2nd order
ASCF + diag. 2ph TDA
Second ordera
Diag. 2ph 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 121
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 122
Ionization Energies for Ne (eV)
Is
TOM
TOM + 2nd order
TOM + diag. 2ph 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 123
Vertical Ionization Energies of NH2 and CN
Calculated with the Transition Operator Method
and SelfEnergy 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 2ph TDA
3.86
2.51
0.30
CN : Basis D from Table 18.
NH : First Basis from Table 115.
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 123.)
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 selfenergy improvements.
1.12 Interpretation
Quasiparticle 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 HartreeFock canonical orbitals.
There are many poles that correspond to each orbital, but the poles have
different weights. Thus, certain socalled 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 Nrepresentability 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 Nrepresentability in propagator
theory can be formulated in terms of socalled 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 HartreeFock ground
state and the operator manifold consisting of simple field operators has
electron addition operators
N+1> = a pN> (21)
which generate N+1 electron states and electron annihilation operators
IN1> = ah N> (22)
which generate N1 electron states. The adjoints of these operators
annihilate the ground state, i.e.,
ap N> = ah N> = 0 (23)
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
G1 (E) = ESij (hij + ) = (El F) (24)
(see equations 15 for an explanation of the symbols), then this matrix
is easily related to the Fock matrix which is constructed in a Hartree
Fock selfconsistent field calculation (see McWeeny and Sutcliffe, 1969).
When the eigenvectors of this propagator are reinserted into the
propagator, an identical selfconsistent calculation ensues. The
HartreeFock SCF procedure is a selfconsistent electron propagator
calculation. This selfconsistent 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 oneelectron density matrix, but it is not
necessarily Nrepresentable.
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 spinorbitals, is given by the expression
<0ai aaln> <0aktalin> (O>
ijkl(E) = EEn+Eo+i+ E+E EOia
Note that the poles
E = (En EO) (26)
and the residues
(27)
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 BetheSalpeter 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) (28)
where
h = f2 = {akal,kl} (29)
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 HartreeFock ground state produces the timedependent
HartreeFock 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(2i1,2i) (210)
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 HartreeFock 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 selfconsistent 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 AGPRPA.
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 twoelectron 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(2i1,2i) (211)
OS i=
is formed from the geminal, g(1,2), which is expressed in terms of a
spinorbital basis as follows,
M M
g(1,2) = C I C i [i(1)p' (2) V' (1)> (2)] (212)
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)] (213)
i=1 1 1 1
where unprimed 0 stands for a spinorbital with alpha spin and primed t
stands for a spinorbital with beta spin. The HartreeFock 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 oddelectron
systems, the generalized AGP
N+K N/2
AC Xi(i) 1 g(2i1,2i) (214)
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(N1) 'iN/2(N)} (215)
1 1 2 2 N/2 N/2
and includes certain double, quadruple, hextuple and higher evenfold
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)) (216)
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)) (217)
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)) (218)
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 oneelectron density matrix of the
AGP
Nij = = Niiij (219)
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 (220)
S1
2
By introducing the symmetric function of order N/2 of the squares of the
coefficients, gi, the occupation numbers of the oneelectron density
matrix of the geminal,
(221)
Cl = 1 ...>2> gi l2 gi212 .gi N2,
11 2 N/2 1 2 N/2
the elements
expressed as
of the oneelectron density matrix of the AGP can be
fN
1 DC NN
Nii = g A
_Z
The elements
in a similar
(222)
of the twoelectron density matrix that are not zero follow
way according to the formulas
1 N
where i and j are the spinorbitals, and
1 
where i = i with opposite spin. The total energy is
where i = i with opposite spin. The total energy is
(223)
(224)
55
E = (2h +)N + I (42) (225)
+ 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 225
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 oneelectron,
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 paircorrelation theories (Meyer, 1977 and references therein)
and manybody perturbation theory (MBPT), in which pair correlation
energies provide corrections to the HartreeFock total energy (Bartlett
and Shavitt, 1977; Bartlett and Purvis, 1978). These pair correlation
terms are produced by matrix elements of double excitations with the
closedshell HartreeFock configuration. In MBPT, the contributions
from the first and second summations in equation 225 that do not come
from the reference configuration vanish according to the linkedcluster
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 manybody approaches.
2.5 Coefficient optimization
Because of the simplifications in the twoelectron 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 217), 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
RayleighRitz variational quotient,
E = (226)
be written as
E = E' (227)
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 (228)
subject to the constraint that the geminal be normalized,
g*jg = 1 (229)
J
then the necessary condition
[E (Z g*jgj1)] = 0 (230)
1 *i j
is obtained, where X is a Lagrange multiplier. By noticing that both the
normalization of equation 227 and the unnormalized total energy satisfy
the Euler formula for symmetric functions,
E' = 2 gi i (231)
1 BC
C N2 j g (232)
J gj
the necessary condition for stationarity of the total energy can be
rewritten
S 1 E'l E BC2 (233
1 E' (233)
Bg*. ag*i [cI]12 Sg*.
= E2C g (234)
SI Hg (235)
'2 M
where Hij = H*.j. Now with the introduction of the constraint on the
normalization of the wavefunction, equation 230 implies that
1 Hig = Ag (236)
Equation 236 is aJ standard eigenvalue problem that is iteratively
Equation 236 is a standard eigenvalue problem that is iteratively
solved. At selfconsistency, the value of A is zero since
7 H. g = E = 0 (237)
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 (238)
(Dalgaard and Jorgensen, 1978), where
A = i I kl(a kai altak) (239)
The resulting total energy after such a transformation on 0> is
E(X) = <0OeiA HeiA 0> = <0HIO0> + i<0[[A,H]I0> (240)
+ {<0IA,[H,A]]0>
+ <01[[A,H],A]0>} + .. .
A stationary point occurs when 6E(X) = 0, or
6E(X) = i + i{ + (241)
+ } + ....
Wrs = c<0[ar as as ar,H]lO> (242)
and let
Ars,r's, = { (243)
+ .
With the neglect of terms higher than second order, equation 241 becomes
A = A1W (244)
The solutions, X, are increment vectors and are converted to unitary
orbital transformation matrices according to
X = eA = Ucosh(d)Ut XUsinh(d)d1Ut (245)
where
2 = UdUt (246)
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 21). The total energies from
these calculations and the total energy differences along the ground
state potential energy curve are compared to results from HartreeFock,
second order MBPT, an MCSCF wavefunction that contains the same config
urations as AGP, but not the constraints on the coefficients, and to the
60
Table 21
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
HartreeFock 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 HartreeFock 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 22). 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
HartreeFock 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 22. The use of a damping factor of 0.5 in the coefficient
Table 22
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 1010 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 (247)
rs rs
are also employed as convergence criteria. When these tolerances are
both set equal to 1012, 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 23). 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 HartreeFock 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 AA '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
11
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 24). 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 25). 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 HartreeFock. 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
41 0I 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 4I 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 (248)
ij gjjai giaI ,' (249)
where j = j with opposite spin, then the polarization propagator will be
block diagonal,
P(E) = (250)
PE = 0 <>E
for a selfconsistent 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 250 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 251). In Table 26, they are compared to full CI excita
tion energies. Near the equilibrium internuclear distance, the agreement
between the AGPRPA 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)} + (252)
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 (253)
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 rI 04 :I. 0
"4 IZI rD 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 (254)
where
a1 = Li,ls (255)
a2 = Li,2s + H,ls (256)
3 = tLi,2s H,s (257)
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 SELFENERGIES
The following terms should be excluded from the selfenergy in the
calculation of electron binding energies when the ASCF plus correlation
selfenergy formulas are used. These exclusions apply only to the
diagonal elements of the selfenergies. The indices refer to spin
orbitals. Indices a,b,c,... refer to occupied spinorbitals, and indices
p,q,r,... refer to unoccupied spinorbitals.
The second order correlation selfenergy is
(A1)
C,22(E) = 1 + 1
ii ai b p E+i p ab 2 pii q ai a 'p q
The diagonal 2ph TDA correlation selfenergy is
(A2)
C,DIAG 2ph TDA abllip>
Sai bi E+EpC b+4
1
2 a pi q/E+ea p q ++
74
The third order correlation selfenergy is
(A3)
C,3 1 (16pr~pi)
S(E) = ( 2 2E+eac sC E1E+E_ E
ii a p q r s arsE+aEpq
1 (16ac6ia)
a b c d p E+pcdJ E+pa
((16ab6ia)(16pr6ip)+(l6qi)]
a b q p E+aap E+bSqrJ
((16acia)(l6,pqip)+(16bi))
+ c E+ pEabJ E+EqEbEc
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 qE
1
4a b p q r E+ep Eaeb a +b _q r
S(16q )
a b p q r b p E+EbpEq a+cb Fp Fr
(16ri)
abp q r E+b prJ :a+cbpq
(16ci)
(E+c Ec a C p
a b c p q E+q a c] [a+Ebp qJ
(16bi)
abcp q E+pa b (a+cpq
75
S1
a b p q r oa Eb pqpqJ (Ea bq 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 cbEp 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 PlainviewOld 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

