EEEETEEEETTEE EEEEETE STEETES OF GROUND
ENE EXCITED STATES OF TOEEETEMIC MOLECULES

TEESES F812 TEE BESREE 8F Fix: 3.
MECHIEAN STATE UNEVERSITY

MELVYN HACKMEYER
1970

{Ht-.555

This is to certify that the
thesis entitled

Configuration Interaction Studies of Ground

and Excited States of Polyatomic Molecules

presented by

Melvyn Ha ckmeyer

has been accepted towards fulfillment
of the requirements for

Doctor of Philosophy degree in Chemistry

  

Major professor

 

   

 

 

 

 

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ABSTRACT

CONFIGURATION INTERACTION STUDIES OF GROUND AND EXCITED STATES OF
POLYATOMIC MOLECULES

by Melvyn Hackmeyer

An ab initio configuration interaction (CI) procedure is
fbrmulated, and it is applied to the study of the electronic spectra of

formaldehyde (HZCO), glyoxal (H o ), and pyrazine (HhCuNz). The

202 2
general features of this procedure are the generation of a set of
configurations, {oi}, the accurate evaluation of the Hamiltonian matrix
of elements (mi 'H3l¢3>, where
H = 2[-%vi - 220(ria)'l] +,2 2 (ri.)'l,
i a i J J
i<j
and the diagonalization of the Hamiltonian matrix to obtain electronic
wavefunctions and energies. Previous theoretical investigations of the
electronic spectra of formaldehyde, glyoxal, and pyrazine have been
carried out, but these investigations used semiempirical methods employing
only a small number of configurations and nonrigorous evaluation of the
Hamiltonian matrix elements.
Since the number of configurations which can be generated from
a given basis set generally is too large to be fully utilized, a number
of approximations are made in the present work which limit the number of
configurations to approximately four hundred for each state. The
configurations are fermed by promoting electrons from ground state SCF
occupied orbitals to either ground state virtual orbitals or to a
transformed set of virtual orbitals. A set of low-lying doubly occupied

orbitals, called the fixed-core, is chosen which is common to each of the

Melvyn Hackmeyer

configurations.

wavefunctions and energies are generated in a multistep process.
The initial wavefunctions consist of linear combinations of those
configurations which are expected to be important in describing the low-
lying electronic states. The descriptions are successively improved by the
selective addition of configurations which are chosen based on their
ability to lower the energies of the statasconsidered. This energy

lowering is measured by the value of the expression

Euiuxneiwz ' ,
l<+1§i>lHlT§f>>~ <E§*’IHIE§1>>I

H
II

th electronic state at the ith level

'where tfii) is the wavefunction of the k
of refinement, méi) is a configuration which is obtained by a single or
double promotion from a configuration of the set used to construct téi),
and 153. All configurations are accepted for which In is greater than or
equal to a predetermined threshold, 6, for some value of k, where k is
usually less than five. The final set of configurations is augmented so
that the resulting wavernctions are eigenfunctions of 32.

For large matfices, an approximate method of diagonalization,
called "successive diagonalization", is developed in which a number of
smaller matrices are successively diagonalized.

.Much experimental information exists about the electronic spectra
of’fbrmaldehyde, glyoxal, and pyrazine. HOwever, a definitive determin-
ation of the nature of the orbital promotions involved in most of the
states is lacking; also, nOne of the triplet states except the lowest are
observed experimentally.

The calculated transition energies of the lowest energy singlet

and_triplet states of formaldehyde and pyrazine are within 0.5 eV of the

Melvyn Hackmeyer
experimental band origins, while those of glyoxal are slightly above the
upper limits of the experimental ranges. The calculations predict that
these lowest excited states are intravalence n4u* states, in agreement
with previous interpretations. However, the relative positions of the
different n*u* states in the spectra and the nature of some of the
orbital.promotions disagree with some previous interpretations.

Except for the lB2u(fl*11*) state of pyrazine, no low-lying
singlet "Hu* states are found. Three possible explanations are offered
fbr the lack of these states in the calculated spectra. The first
involves a reinterpretation of these states as Rydberg states, the
second involves a supposed deficiency in the basis sets, the third
involves displacement of the minimum in the potential curves of the
excited singlet n+d* states.

A number of higher singlet, triplet, and quintet states and
double promotion states are predicted in the calculations, but no

experimental information exists about these states.

CONFIGURATION INTERACTION STUDIES OF GROUND

AND EXCITED STATES OF POLYATOMIC MOLECULES

BY

Melvyn Hackmeyer

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1970

GET/e73

This thesis is dedicated to my parents
Betty Hackmeyer
and
the late

David Hackmeyer

ii

ACKNOWLEDGEMENTS

I acknowledge the help and encouragement of Professor J. L.
Whitten who laid the foundations of this work, and whose ideas and insight
helped me throughout its course. I also thank him for his extreme
patience during the course of my study.

I also express my appreciation of the Chemistry Department of
.Michigan State University for allowing me to take a leave of absence in
order to fellow Professor Whitten to the State University of New York at
Stony Brook.

I am.also grateful to the Chemistry Department of the State
university of New Ybrk at Stony Brook and its chairman Professor F. T.
Bonner for allowing me to pursue my study in the department during my
leave of absence from Michigan State.

I thank the computing centers of Michigan State and Stony Brook
fer the use of their facilities and for helpful suggestions.

Finally, I thank my colleagues: Mr. J. D. Petke, for his
‘permission to use his orbital hybridization programs, integral programs,
and several helpful suggestions; Dr. A. W. Douglas, for the use of a
number of programs which he wrote or modified and for many helpful
suggestions; and Mr. D. L. Wilhite, for several useful conversations

during the course of this work.

iii

TABLE OF CONTENTS

INTRODUCTION

I. THEORETICAL BACKGROUND

A. The Configuration Interaction Method
. Exact SchrBdinger theory
. Approximate Schradinger theory
. The Hartree-Fock-Roothaan Self Consistent Field (SCF)
Method
. Configuration interaction (CI) method
. A simple theoretical description of singlet and
triplet states
6. Comparison of the CI and SCF methods
B. Previous CI Calculations
C. The Theory of the Electronic Spectra of Polyatomic
Molecules

U147 WNW

II. FORMULATION OF THE CI METHOD

A. Formation of the Set of Configurations
1. General description of configurations and their
Hamiltonian matrix elements
2. General discussion of the limitations of the number
of configurations
3. Generation of configurations
h. Discussion of the restrictions
B. A Discussion of Eigenfunctions of S
C. Obtaining the CI Wavefunctions
l. Perturbation theory
2. Successive Matrix Diagonalization

III..APTLICATIONS AND RESULTS

.A. Intrductory Remarks
B. Formaldehyde
1. Experimental spectrum of formaldehyde
2. Formaldehyde calculations and discussion
a. Description of basis set
b. SCF calculations
c. CI calculations
3. Comparison of calculated and experimental results

iv

\J'I

O\\N\n

10
16

21

32
32
32

33
35
39
kt
52
52
5h

Page

C. Glyoxal 92
1. Experimental spectrum of glyoxal 92
2. Glyoxal calculations 97
3. Comparison of calculated results with experiment 10?
D. Pyrazine 113
1. Experimental spectrum of pyrazine 113
2. Pyrazine calculations 118

3. Discussion of pyrazine results and comparison with
experiment 132
IV. CONCLUSIONS lhO
Bibliography lh9
Appendix A 151+
Appendix B 156
Appendix C 160
Appendix D 166

Table 1.

Table 2.

Table 3.

Table h.

Table 5.

Table 6.

Table 7.

Table 8.

Table 9.

Table 10.

Table 11.

Table 12.

LIST OF TABLES

Comparison of successive diagonalization procedure
with more accurate matrix diagonalization for the
lowest singlet B2u state of pyrazine 57

Experimental electronic absorption spectrum of
formaldehyde 6l

Formaldehyde basis orbital parameters 6%

A comparison of orbital energies from three different
SCF calculations on the singlet A1 ground state and
triplet A2 excited state of H200 7 66

A comparison of n and n* molecular orbitals of H200
from ground and excited state SCF treatments 67

.Molecular orbital and total energies from ground state
calculations on HQCO for three basis sets: (I) split

group, (II) split group plus C 3pn, O 3pn, and O 3pr,

and (III) split group plus C 3s and O 38 69

{Additional basis functions of formaldehyde 72

Self consistent field molecular orbital energies for

the ground state of formaldehyde using a basis set of
thirty six functions, including five one component d
functions on carbon and on oxygen, and a one component

pa and pn function on each hydrogen 73

A comparison of energies from constrained SCF
calculations on the triplet A2 state of H2C0 with
single promotion and full SCF energies 75

Energies of HQCO ground and excited states from CI
treatments based on ground state SCF molecular orbitals
and a split group basis set 80

Energies of H200 ground and excited states from.CI
treatments based on ground state SCF molecular orbitals.

TA split group plus 3prtorbitals on C and O and a 3pb2
orbital on 0 basis set is used for the Al and A2 state,
and a split group plus 33 orbitals on C and 0 basis set

is used for the B1 and B2 states. 81

Optimum 2b1 and 3bl (fl) molecular orbitals from single
excitation n-m* and THW" calculations 83

vi

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

13.

lha.

lub.

15.

16.

17.

190

20a.

20b.

21b.

Page

Energies of H2CO ground and excited states from CI
treatments based on molecular orbitals determined by
single excitation CI calculations 85

Major contributions to the ground and Al and A2
excited state CI wavefunctions for HQCO 86

Major contributions to the Bl and 32 excited state CI
wavefunctions for H200 87

A comparison of calculated transition energies and
other properties of H2CO with experimental results 88

The experimental electronic absorption spectrum of
glyoxal 96

Ground state self consistent field molecular orbital
energies and total energies of cis and trans glyoxal,
expressed in atomic units 98

Total electronic energies of glyoxal ground and
excited states based on ten variably occupied ground
state SCF molecular orbitals 100

Total electronic energies of glyoxal ground and
excited states based on sixteen variably occupied
ground state SCF molecular orbitals 101

Total electronic energies of glyoxal ground and

excited singlet states based on twelve variably

occupied molecular orbitals, ten of which are from

the ground state SCF, and the other two are constructed
from 3p basis functions on the oxygen atoms 102

Total electronic energies of glyoxal ground and

excited triplet states based on twelve variably
occupied.molecular orbitals, ten of which are from

the ground state SCF, and the other two are constructed
from.3p basis functions on the oxygens 103

CI wavefunctions of the Ag electronic states of trans
glyoxal obtained in the calculations corresponding to
Table 20 10h

CI wavefunctions of the BE and.Au states of trans

glyoxal obtained in the calculations corresponding to
Table 20 105

vii

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

Table

21c.

22.

23.

2h.

25.

27a.

27b.

28a.

28b.

29a.

29b.

306.

30b.

CI wavefunctions for the Eu electronic states of trans
glyoxal obtained in the calculations shown in Table 20

Additional SCF results on glyoxal using 3s and 3p
orbitals

A comparison of the calculated and experimental
transition energies of glyoxal

The experimental electronic absorption spectrum of
pyrazine

Pyrazine basis orbital parameters

Ground state SCF molecular orbital energies of
pyrazine for each basis set

Comparison of single configuration transition energies
of pyrazine

Comparison of n* orbitals from ground and excited state

SCF calculations

Energies of pyrazine ground and excited singlet states
based on ground state SCF molecular orbitals and a
fixed-group basis set

Energies of pyrazine ground and excited triplet states
based on ground state SCF molecular orbitals and a
fixed-group basis set

Energies of pyrazine ground and excited singlet states
based on molecular orbitals constructed from the split
pfl'basis set

Energies of pyrazine excited triplet states based on
molecular orbitals constructed from the split pn'basis
set

CI wavefunctions of the A , B3u,.Au, and B2 states of
pyrazine from the calcula%ions performed usTng the
extended basis set

CI wavefunctions of the B
of pyrazine from.the calc
using the extended basis set

tions thch were performed

viii

Page

106

108

110

ill;

119

120

121

122

125

126

127

128

129

130

Table 31.

Table 32.

Table 33.

Results of calculations on the lowest lAg, lB311, lBgu,
and 3B3u states of pyrazine, using a threshold of
0.0000228, and no approximation in diagonalizing
matrices

Mulliken.population analysis of the n- (Sblu) and.n+
(6ag) orbitals of pyrazine

Comparison of calculated and experimental spectrum of
pyrazine

ix

131

135

137

Figure 1.

Figure 2a.

Figure 2b.

Figure 2c.

Figure 3.

Figure ha.

lFigure hb.

Figure hc.

Figure 5a.

Figure 5b.

LIST OF FIGURES

Page
Schematic representation of the molecular orbitals of
formaldehyde. ll
Franck-Condon transitions. 23
Nuclear distribution functions and most probable
transitions. 26
Illustration showing how a steep potential curve of
the upper electronic state can cause more than one
band to have maximum intensity. 29
Nuclear coordinates (in atomic units), nuclear
repulsion energy, and symmetry orbitals of
formaldehyde. 60
Nuclear coordinates (in atomic units) amd nuclear
repulsion energies of trans and cis glyoxal. 93
Symmetry orbitals of trans glyoxal (Cgh) expressed as
linear combinations of functions on the carbon and
oxygen atoms. 98
Symmetry orbitals of cis glyoxal expressed as linear .-
combinations of s and p functions on the carbon and
oxygen atoms. 95
Nuclear coordinates (in atomic units) and nuclear
repulsion energy of pyrazine. 115
Symmetry orbitals of pyrazine expressed as linear
combinations of equivalent functions on the carbon
and the nitrogen atoms. 116

Appendix A.

Appendix B.

LIST OF APPENDICES
Page

The determination of the approximate energy
contribution of a configuration to the total energy 15h

Proof that all calculated energies of ground and
excited states are upper bounds to the true energy 156

Appendix c. A discussion of m" states and the frequent

Appendix D.

disagreement of calculated singlet-singlet transition
energies with experiment 160

Gaussian lobe functions 166

xi

1

INTRODUCTION

The work described in this thesis concerns the application of
quantum mechanics to the description of electronic spectra of polyatomic
molecules, the ultimate goal being the prediction of spectra solely from
theoretical arguments. According to the quantum.mechanical theory, an
exact description of an electronic state and of its observable properties
can be obtained from the wavefunction, ln’ for that state, where in is a

solution of the Schrgdinger equation,

Hln = Enln’
for the system with Hamiltonian H.

For studies of the electronic structure of a many electron
system for a fixed nuclear geometry, the Hamiltonian can be simplified to
the fbllowing form which includes only electron kinetic energy and electro-
static potentials. In atomic units,

H=E<-=:-v12-E;2) +§§(rfi.

i<J ‘
where i and 3 refer to electrons, a to nuclei with charge 20, and rkl is
the distance between particles k and L.

Although a direct solution of the Schrfidinger equation is out
of the question for a many electron system, approximate solutions can be
developed using the energy variation principle. Wavefunctions, in, of the
electronic states can be expanded in terms of a set of N-electron

functions (configurations), {TL}’

ln(1:29---9N) = g c£n¢L(l’2""’N)

2
In principle, by using this method with a complete set of functions, {‘91,},
the exact wavefunction can be obtained. In the present work, the set of
functions used is not complete; instead, a finite number is chosen in a
complicated manner to obtain the best possible description for the basis
set employed.

The only property which is directly optimized is the total
energy; however for large expansions, it is hoped that the properties of
the wavefunction will also approach.those of the true wavefunction. In
practice, the energy convergence is very slow, and extensive calculations
yield diminishing returns. Thus, a great deal of effort has been made to
develop expansion procedures which are based on an efficient and generally
small set of functions.

In view of the precise experimental spectral measurements now
being performed on excited states of polyatomic molecules, it becomes in-
creasingly important to develop accurate abhipitig procedures to assess
the validity of qualitative concepts about excited electronic states
which are often used to interpret the spectral data.

The present work can be characterized as an initial step toward
the development of a tractable ab initig configuration interaction method
which is capable of yielding accurate descriptions of excited states of
polyatomic molecules, The method is applied to spectral studies of several
molecules; formaldehyde (HQCO), glyoxal (H20202), and pyrazine (HAChN2)’
A variety of electronic transitions have been observed for these systems
and there is an extensive amount of information available on certain of
the excited states. However, there are also several states of these

molecules which are not yet well understood.

3

Many configuration interaction (CI) formulations have been used
in the past and these generally can be divided into two groups; EE.$E$EE2
and semiempirical. The distinguishing feature of ab initio calculations
is the precise evaluation of all numerical quantities. The various ab
‘igitig methods are distinguished from each other by their choice of basis
sets and configurations. Each ab initio formulation determines the
optimum energy wavefunctions from its chosen set of configurations. In
the case of large scale 52 initig_CI calculations, the agreement between
calculated and experimental results is determined almost entirely by the
adequacy of the basis set.

The semiempirical methods generally use non rigorous mathe-
matical approximations coupled with experimentally measured properties
to calculate necessary numerical quantities. These techniques were
developed befbre advances in computer technology made large scale_§b
.EEEEEQ CI calculations feasible; however, sudh methods often are still
useful for calculations which involve large molecules where rigorous
.methods are quite cumbersome. For example, in terms of the number of
electrons, the largest molecules on which large scale ab initio CI
spectra calculations have been reported are benzene (reference 1) and
pyrazine (this work). The basis functidns which are used in most semi-
empirical methods are atomic valence orbital basis functions, while most
‘gb’igitig calculations use more flexible basis sets. Also, only a very
limited set of configurations is employed in semiempirical formulations.

Certain of the excited states studied in this thesis have been
calculated previously using semiempirical methods. However, the reli-

ability of semiempirical spectral assignments must be gauged in part by

h

comparison with.more accurate calculations, and not only by comparison
with experiment. There are two reasons fOr this assertion. First, if
accurate calculations are performed which parallel the semiempirical
calculations in every way except that the semiempirical approximations
are replaced by accurate evaluations, the spectra results may be
different. For example, compare the semiempirical (References 2 and 3),
and 52 igitig (Reference 1) calculations of the spectrum of benzene.
The experimental spectrum is shown in Reference 1. Therefore, although
agreement with experiment may be achieved, the approximations themselves
may be invalid. Second, since many of the semiempirical calculations rely
on experimental spectra to evaluate numerical quantities, the calculated
results may be biased toward agreement with experiment where such agree—
ment is not possible in principle given the basic structure of the
formulation. In either case, the agreement of a semiempirically calculated
energy of.a transition from the ground to an excited state with the
position of an excited spectral state is not conclusive evidence that the
calculated state is the same as the spectral state.

It is therefore important to reexamine the theory of the spectra
of polyatomic molecules in terms of a more quantitative method of
calculation. Such a method is formulated in this work and is applied to

the several molecules listed previously.

I. THEORETICAL BACKGROUND

A. The Configuration Interaction Method

 

l. Exact Schrodinger theory

 

According to the Schrodinger development of the quantum.mechanical
theory, an exact description of all of the electronic properties of the
nth electronic state of a molecule can be achieved if both the exact
electronic wavefunction, in, of that state, and the quantum mechanical
operators {Ti}, corresponding to the properties are known. The average

value of a property is expressible as the integral,
(in [F1 an) =

*

'Itn (x1, x2,....,xn) F(xl, x2,....,xn) tn(xl, x2,...,xn) dTl dvz....dTn,
'where x1 represents the spatial and spin coordinates of electron i, and
dvi is the volume element associated with electron i.

The exact wavefunction, t can be found by solving the Schrodinger

n)

equation,

min = En‘in

where En is the total energy and H is the quantum.mechanical Hamiltonian,

H=He +Hn+Hen+Hmag+Hrel
where He is the kinetic energy and electrostatic repulsion operator of
the electrons, HD is the corresponding operator for the nuclei, Hen is the
electrostatic attraction operator of the electrons and the nuclei, Hmag is
the energy operator for magnetic interactions, and erl is a relativistic
correction. It is evident that the Schrodinger equation is of such
complexity fOr a many electron system that an exact solution is not

possible.

6
2. Approximate Schrodinger theory
For studies of the electronic structure of a molecule including
electronic spectra, it is possible to neglect all terms of the
IHamiltonian except the terms He and Hen without significantly affecting
the accuracy of the description}IF Thus, the Schrddinger equation

becomes

(He + Hean - Envn

1 2 Z“ =
_[z(- -Vi firfiie) + :irl 15 Entn

where ~%V&2 is the kinetic energy operator of electron i and_ 1 is the
electron repulsion operator of electrons i and J. at:

The physical picture corresponding to this Hamiltonian is that of
electrons moving in the framework of fixed nuclei, and interacting
electrostatically with each other and with each of the fixed nuclei. Even
with this great simplification, the Schrodinger equation still cannot be
solved exactly. .

If the sum of the two-electron operators, § §'Y_;’ is replaced by
a sum of one-electron operators, E V(x1), the wavefunctions can be
expressed as orthonormal antisymmetrized products of molecular spin
orbitals,

on = det[a1(n)(l)a2(n)(2)...anfin)(N)]
‘Where a spin orbital, a1, is the product of a spatial orbital, bk, and a
spin function, a or B. The antisymmetry property which.must be satisfied
by the exact wavefunction is imposed on the approximate wavefunctions,

‘where here, the exact wavefunction is understood to mean the exact

solution of the fixed nuclei electrostatic Schrodinger equation.

7

The spin orbitals can be expressed as linear combinations of a mathemati-
cally convenient set of basis functions. Such antisymmetrized product

functions are called configurations, and they are approximate wave-

 

functions for the molecular electronic states.

The specific linear combinations of the basis functions in terms
of which the spin orbitals are expressed can be determined by the varia-
tion method, in which the energy expectation value, <¢n IH l¢h>’ for wn
normalized, is minimized.

The physical interpretation of a single determinant wavefunction
is that of a set of electrons occupying the antisymmetrized spatial

(n)

(n)
distributions {aiai, i = 1....N}. The energy of such a wavefunction is,

-1
gain) I-svi‘ - goes) as» +§<§t<n<i>ai<1>t§l a3<a>a3<a>>

.2.

-(9-1(1)aj (1) [1‘13

lai(J)&J(J))]

where

(“1(1) lr<1> Iaj<i>) efai*<1>r<i)aj<1>ati

and

(§1(1)a,<1) 1111.3) lemmas»:-
ja1*(1>a;(a)r<1,a)a,<1)a,<s>avidr,

3. The Hartree-Fock-Roothaan Self Consistent Field (SCF)
.MEEEQQ
The set of molecular orbitals, a1, 1 = l...N, in the single
determinant wavefunction which minimizes the energy satisfies the eigen—
value problem

Fla1 a liai

8

where the operator F1, called the Hartree—Fock Hamiltonian of electron i,

can be expressed as,

a 2 N l-Pi
F1 51 +§fl(aj(d)l~;jlaj(d))s

where P13 interchanges the electronic coordinates xi and x, of the

i' which is called

the orbital energy, conceptually is the energy of an electron that

molecular orbitals on which it operates.5 The quantity c

occupies the orbital a1.

There is no loss of generality of the total electronic wave-
function, and at the same time there is quite a simplification of the
problem, if the ai's are constrained to be orthonormal symmetry functions.

The exact solution of these equations yields the Hartree-Fock
molecular orbitals, the Hartree-Fock wavefunction, quF and the Hartree-
Foch energy, (CPH'F [H WHIP).

For most problems, however, the Hartree—Fock-Roothaan equations
cannot be solved exactly, making it necessary to expand the molecular
orbitals in terms of a mathematically convenient and physically reason-
able set of basis functions. The self-consistent-field (SCF) method
is the one which is most frequently used to determine the molecular
orbitals. It is an iterative procedure in which a trial set of
molecular orbitals is used to construct the Hartree-Fock Hamiltonian,
the eqmtions are then solved yielding a new set of molecular orbitals
and a new Hartree-Fock Hamiltonian, and (the process is, repeated until the
input set of molecular orbitals is the same as the set which results (i.e.,
until self-consistency is achieved). A number of unoccupied (virtual)

orbitals also results from this procedure.

9

For small basis sets, the SCF results depend on the basis set
which is used. However, for large basis sets the SCF results approach
the Hartree-Fock limit.

’4. Configuration interaction (CI) method
If the complete set of solutions, {on}, of acne-electron

Schrodinger equation with proper boundary conditions,
{ [(-l-v 2)-2‘f." II + zv(x )} ==
§ 2 1 yr; 1 1 (9n Encpn
are known, then the exact solution of the correct Hamiltonian equation

2 Z l
f§[(-%Vi ) ‘- 53%;] +51: gain - Enin,
i<.j

can be expressed as linear combinations of the functions, {on}. Such

a procedure where the wavefunctions are expressed as linear combinations
of configurations, is called a configpration interaction (CI) procedure.
(For a discussion of the CI method see References 6 and 7, and references
contained therein.) ‘

The CI formulation which is presented here uses -a finite number
of configurations, and the energies and wavefunctions are found by
diagonalizing the matrix of elements ((91 In Inn). All of these matrix
elements are calculated accurately, and for this reason the CI
formulation is called an ab _i_r_l_i_t_i_q CI formulation. The features that
distinguish this particular formulation from other £13 39252 CI
formulations are as follows: the means by which configurations are con-
structed (i.e., the choice of basis set and molecular orbitals), the
selection and limitation of configurations, and the approximate method
by which large matrices are diagonalized. A detailed discussion of

these topics is presented in the second chapter of this thesis.

lo
5. A simple theoretical description of singlet and triplet
223.22

Several_types of molecular orbitals are important in the molecules
which are studied in this work. In approximate order of increasing orbital
energy these orbitals can be classified as inner shell orbitals, c orbitals,
‘7 orbitals, n (non-bonding) orbitals, 3s and 3p Rydberg orbitals, TT*
orbitals, and 0* orbitals. The different types of orbitals are illustra-
ted in Figure l. The first four types are doubly occupied in the ground
state, and the other three are involved only in the excited states.

The single configuration descriptions of most of the low lying
excited states are formed by promotiflg an electron from a a, 17, or n
orbital to a Rydberg, 17*, or 0* orbital. Although both the ground and
excited state configurations contain the same types of orbitals (except
for the change brought about by the single promotion), they do not
necessarily contain orbitals of the same precise shape. The nomenclature
for designating electronic states, which will. be used throughout this
thesis, is based on the single promotion description such as n-OBs, n-'3p,
rm", n-to*, "411*, etc.

At a very simple level of description, a level which indeed gives
rise to the notion that triplet states arising from a given orbital pro- ‘
motion are lower in energy than the corresponding singlet states, the
singlet state and the I.- 8.‘ +l,O,-l triplet states can be expressed in
terms of four configurations, each containing the same set of spatial
orbitals. Consider a state which involves the promotion of an electron
from orbital: i to orbital a. The states which arise from such a pro-

motion are characterized as i-«a states. Starting from a closed shell

Hi

Atomic orbitals
so and sc are the 2s orbitals on oxygen and carbon respectively,
(pg, °, pg) and (pfi, pg}, pg) are the 2p orbitals on oxygen and carbon
respe ively, » and 31 and s2 are the ls orbitals on hydrogen atoms 1 and 2
respectively.

Molecular orbitals

I
I

 

“CO ~ 1:180 4- 1:259 - 11392 + khpg
0:10 ~ 1:230 - klsc + mg + k3pg
OCH ~mis° + maple, + m3pg + mh(sl + 82)
0*... s°+ p°+ c- (3 +3)
cs “'1 ‘2 3* 1“31°24 mu 1 2
1'! ~ nlp; + nng, where n2>nl
*
o
n N ”21’: ' n21px
n ~ 31p§ + 32p?" where 32>31

The 38 and 3p Rydberg orbitals are linear combinations of 3s and 3p atomic
orbitals on the carbon and oxygen atoms .

Figure 1. Schematic representation of the molecular orbitals of
formaldehyde.

12
ground state configuration, a promotion such as i-‘a leads to four electron

configurations, described as

1a4gp,1a-Ow, 13 43.513433,
where a and B are electronic spin functions which correspond to :118 values
of «Ii-,and -% respectively.

Four linear combinations of the above configurations can be formed,
each of which will be an eigenfunction of the operator 82 as is required
for a Hamiltonian containing no spin operators. The correct normalized

linear canbinations are,

1. 1043.3 3.1, m8 =.]_

2. lEia-baa-iB-Daaj s.=l,ms=0
fl

3. 18*a0! Bul,'m8=l

1}. lEia-Oacd-ip-Oafl] 3:0,msuo'
[2'

If the Hamiltonian that is used is

2
H-2[(- -zza ]+)32..L
1 yi) a 13—11. 1‘3 r139

in atomic units, then the energy expression for triplet states (states 1,
2, and 3) is

sad - ago: In lk) + (1 In I 1) + (a In la) +E §[2(kklu)— (xx. [111)]
'+ £12051: I11) - (ki In) + 2(kk laa) - (ka ‘ka)]+(ii laa)

"' (15 ‘13)»

where (wx lyz) is the integral (w(l)x(l.) lrat— Iy(2)z (2)), the index k
1‘12

spans the doubly occupied spatial molecular orbitals , and (x [h ly) is

13

the integral (x(l) I-é‘z - gig.“ I y(l)). The energy of the singlet
(state it) is

35:0 -2E(kIhIk) + (iIh Ii) + (thIa) +EIi[2(kkI£L)-(RLIKL)]

+§[2(1¢ Iii) - (k1 Iki) + 2(kk Iaa) - (ka I ka)] + (ii Iaa) + (ia Iis)

The ground state energy is

Emma .2§ (kIhIk) +§§E2(kklu) - (quL)],

where k and t span the entire set of occmied spatial orbitals.
111118, at this level of approximation, the energy of transition
firm the ground state to the excited 114a state (i.e., the singlet state

corresponding to the promotion i-oo) is given by the expression
Esp, - EM - {(a IhIa) +§[2(n Iaa) - (s. Ika)]} -
{(1 In I1) +§[2(rklii) - (ki In): 4. (ii Iii)} + (iiIaa) + (iaIia)
= {3, - [2(11 Isa) - (ia Iis)]} - {C1} + (11 I aa) + (ia I is)

= on - s1 - (ii I aa) 4» 2(ia I ia)

where the index k spans the doubly occupied orbitals of the 11-h. state,
and a, is the energy of nolecular orbital 1.. Similarly, the difference
in energy between the 31". states and the ground state is

3,.1 - aground . c, - :1 - (ii Ia),
and the singlet-triplet splitting is 2 (ia Iia).

Since (ia Iia) is always positive, the triplet state is always
lower in energy than the singlet state at this level of approximation.
For the singlet and triplet states of the molecules which were studied

lh
in this work, ‘a - 'i is of the same order of magnitude as (ii Iaa) and
thus the difference in orbital energy is not a good approximation to the
difference in energy between configurations.

6. Comparison of the CI and SCF methods

 

There are several reasons for prefering the CI over the SCF
method in the treatment of excited states. The most obvious reason,
although not the most important in practice, is the fact that if both
treatments are carried out to their full limits of accuracy, the CI treat-
ment can, in.principle, yield exact results, while the SCF results will
always be approximate. The SCF procedure meets with practical difficulties
when applied to the calculation of‘molecular electronic spectra since the
procedure more accurately determines the energy of an open shell con-
figuration, (e.g., a singly excited configuration) than the energy of a
closed shell configuration. This discrepancy results from the inability
of the single configuration SCF procedure to account properly fer electron
correlation effects since the SCF solution corresponds to the substitution
of an average one-electron electron repulsion operator for the proper
two-electron operator in the Hamiltonian. In.most, but not all, cases the
correlation error associated with.two electrons which occupy the same
spatial orbital (i.e., a closed-shell) is greater than for two electrons
in different spatial orbitals (open shell). Therefore, the ground state,
which often has one more closed shell than the single—promotion excited
states:may‘be less well described than those excited states at the SCF
I level. If this effect continues to exist at the Hartree-Fock limit then
calculated transition energies will be below those of experiment.

Another failing of the single configuration SCF procedure is its

15
inability to describe adequately states which involve strong contributions
from more than one configuration. A similar inadequacy appears in the
description of open-shell singlet states involving singly occupied spatial
orbitals of the same symmetry.

The CI procedure affords a simple way to obtain simultaneously a
number of orthonormal electronic wavefunctions and energies of states of
a given symmetry and multiplicity while an SCF procedure requires separate
calculations for each of these states, and indeed may require several
iterations for each state.- In addition the requirement of orthogonality
of the different states of the same symmetry imposes serious difficulties.
on the SCF procedure.

Assuming that the SCF treatment can be performed rigorously on a
given state, then the results provide nearly as good a description of one—
electron properties as does the CI procedure. This conclusion is based
on the fact that configurations which are single promotions from the SCF
configuration of an electronic state mix only very slightly with the SCF
configuration. Such a conclusion can be illustrated as follows. Consider
two configurations mi and mé, where ml is a SCF configuration of an
electronic state, and $2 is the same as ml except that orbital bi has been
replaced by orbital ba. Then the CI wavefunction, t = clTl + c2m2, can
be expressed as a single configuration mg, in which the molecular orbitals
are the same as those of ml except that orbital b1 is replaced by chi + c2ba.
However, since mi is determined by the SCF procedure, no alteration of its
molecular orbitals can result in further lowering of the energy; there-
fore, 02 will be equal to zero. Thus, a CI wavefunction can be expressed

, ab ab
approximately as clml +-§g Cid $13 + higher exc1tation configurations,
' a

16
where (9:: is obtained from cpl by promoting electrons from orbitals i and
J to orbitals a and b. If the contributions of higher order configurations
to the wavefunction are neglected, and if a one—electron property
corresponding to an operator I‘ is considered, then the average value of
this property, (1 IF I y) is °12<‘P1 I I‘ Iol>+2cl§jejf§<cpl Ir lei?)

ab

ab cd ab cd

+2 )3 613 cu, (T13 ‘1"ka ).

13 kt

ab cd

ab

The integral (cpl I F In”) is identically equal to zero, and since the
contribution of the SCF configuration to the CI expansion will be con-
siderably greater than the contributions of the other configurations,
c1))c:§. The SCF description of the property, (cpl I I" I cpl), thus will be
only slightly altered by the CI procedure. However, if more than one
configuration is important in describing the state, this argument is no
longer valid.

B. Previous CI Calculations

 

A number of ab M CI calculations have been performed on atoms,
diatomic molecules, and polyatomic molecules,3"9’:l‘-I":1'2"20 mainly for
ground electronic states. For an extensive list of ab initio calculations
on molecular electronic states, some of which use CI, see References 21
and 22. Weissllperformed large scale (from nineteen to fifty five
configurations) CI calculations on the lowest singlet and triplet S
states of the helium atom, the lowest doublet S states of lithium and of
the corresponding isoelectronic series up to and including atoms with a
nuclear charge of eight, and the lowest ls state of beryllium and the

corresponding isoelectronic series up to and including a nuclear

17
charge of eight. He used a basis set which included 8, p, d, f, and g
functions and obtained energies essentially equal to the exact eigen-
values of the electrostatic fixed-nucleus Hamiltonian. These are
extremely important results which show that the CI method works in
practice. Certainly, reaching correct energy values for polyatomic
molecules is a much more difficult task, but the effectiveness of the
CI method on small atoms gives encouragement that with some modification
the CI method could be used for polyatomic molecules.'

Among the CI methods which have been reported, there are
various methods of choosing configurations. Bender and Davidson12
performed calculations on several of the electronic states of hydrogen
fluoride. They chose the configurations on the basis of the sum of the
orbital energies of the molecular orbitals of the configurations; those
fifty configurations which had the lowest sum were used in the CI
procedure. Promotions from inner shell is orbitals were not considered.

Shaeffer and Harris,13 in their study of sixty two electronic .
states of the oxygen molecule, generated all of the configurations con-
sistent with their basis set except those arising from promotions from
the inner shell orbitals.

Buenker, Whitten, and Petke,:L in their study of the electronic
states of benzene, used large CI expansions in which they restricted the
number of configurations by considering in stages only configurations
which differed from the ground state Configuration by single, double,
and higher promotions.

Idpano and ShavittIUSin work published during the course of the
present study used a criterion which enabled them to determine the

approximate contribution that a triply or quadruply excited configuration

18
makes to the lowering of the ground state energy by examining its co-

efficients, E in the CI expansion. The energy contribution, AEi, of a

1
configuration, mi, was assumed to be

2
AEi I ci 1 (HOG—H11)

where

Hoo

where 90 is a linear combination of the ground and singly and doubly

(to [H {to} and H11 = ($1 [H Fri)

excited configurations which describes the ground state.

Sinanogluaahas shown that, although configurations which result
from double promotions from the Hartree—Fock determinant may contribute
significantly to the description of an electronic state, the effect of
those resulting from.triple promotions is negligible. He has shown that
the coefficient, °i§j?a,b,c,d, of the configuration resulting from the
quadruple promotion 124a,b, and 324c,d is Approximately equal to the

product of the coefficients, 01?a,b and c d, of the configurations re-

2 fie.
sulting from the double promotions 1 4a,b, and 324c,d. This approxi-
mation can be used to construct wavefunctions which include quadruply
excited configurations, when only doubly excited configurations are
taken into consideration explicitly. The size of the coefficient of a
quadruply excited configuration can also be used as a criterion for
accepting or rejecting it as an expansion function.

Another procedure has been developed called the multiple con-
figuration SCF (MC-SCF) procedure which combines both the SCF and CI
‘principles. In this approach the molecular orbitals of a multi-
configuration wavefunction are varied until the corresponding energy is

minimized. Such a procedure enables the results of long CI expansions

to be achieved with a smaller number of configurations. Indeed, if

19
only a limited number of configurations is important in the description

of an electronic state, the MC-SCF method can be very profitably applied.
If many configurations are important, however, the MC-SCF procedure
tends to become intractible and solutions of comparable accuracy can be
more easily obtained by CI methods of the type employed in the present
work. Fer a given basis set, the limit of accuracy of the MC-SCF method
and the one formulated here is the same. For a description of the details
of the MC-SCvaethod, and for references to calculations, see References
8, 9, and 10.

Sabelli and Hinze9 reproduced some of W'eiss'll calculations using
the MC-SCF technique. They were able to achieve ninety seven percent
of the correlation energy (i.e., the energy difference between the eigen-
value of the electrostatic fixed-nucleus Hamiltonian and the Hartree-Fock
energy limit) of the lowest ls state of the two-electron atoms and ions
‘with only nine configurations, and with only ten configurations they were
able to account for eighty six percent of the correlation energy of the
four-electron atoms and ions. This shows a significant improvement of the
conciseness of the wavefunctions compared to those obtained by the ordinary
CI method.

Another method which can be used to improve the conciseness of
a wavefunction is that which uses natural orbitals to construct configu-
rations.29 These orbitals result from diagonalization of the first order
density matrix and if configurations are constructed from such orbitals,
a relatively small number of configurations is required to describe an
electronic state. It is generally necessary to know the CI wave—
function before these natural orbitals can be determined. However,

the natural orbitals are definite mathematical entities (as distinguished

2O
from‘virtual orbitals which are dependent on basis set) and therefore it
is useful to express the wavefunction in terms of them.

In all of the CI calculations which have been mentioned so far,
molecular electronic integrals are calculated accurately, and for this
reason, they are called ab initio CI calculations. However, most of the
calculations reported in the literature do not involve accurate evaluation
of integrals and these have been called non-empirical if for example only
numerical approximations such as the Mulliken approximation* or the zero
differential overlap approximation+ are used, or semiempirical if
experimental quantities such as ionization potentials and electron
affinities are also relied upon to calculate integrals. For typical
examples of non-empirical and semi-empirical calculations see

References 3, 23-27.

 

*Mulliken approximation:
(ab lcd) = sawscd Mac I cc) + (aa 1 dd) +_ (bb Ice) + (bb ldd)]
where
Sab = (a lb)
and
a,b,c and d are atomic orbital basis functions
+Zero differential overlap approximation:
(ab lcd) e (aa lcc) 5ab5cd

Where
a,b,c, and d are atomic orbital basis functions and 5ab is the

Kronecker 5

_. if =b
55.1: “ {t if 3e)

21
C. The Theory_of;the Electronic Spectra of Polyatomic Mo1ecules

 

Each electronic state contains a number of vibrational levels,
which in turn contain rotational sublevels. An electronic transition con-
sists of a number of transitions from the vibrational levels of a lower
electronic state to those of an upper electronic state. Such transitions
- are called vibronic (vibrational-electronic) transitions, and each
vibrational level of an electronic state is called a vibronic state. Each
vibronic transition appears as a band (a number of individual spectral
lines with complex spacing) in the spectrum. The individual lines of a
band arise from the rotational transitions which occur between the two
vibronic states.

The series of bands which give rise to a single electronic
transition may extend over a considerable range; however, the theoretical
calculations of the type presented here designate only a single quantity
as the transition energy. An association between this single calculated
value and one of the bands can be developed. It will be shown that in
most cases the corresponding band is the one of maximum intensity, and
that such a band is not necessarily the 0-0 band.*

At temperatures such that kT is small compared to the vibrational
spacing almost all of the molecule are in the zeroth vibrational level of
the ground state, and vibronic transitions involve transitions from this
level to the various vibronic states of the excited electronic states.

According to the Franck-Condon Principle, an electronic transition occurs

—hL

*fl.vibronic transition will be called a v-v' transition, where v and v’
are the vibrational quantum numbers of the lower and upper vibronic

states respectively.

22
in a.much shorter time than the period of a nuclear vibration, and there—
fbre, electronic transitions occur with nearly fixed nuclei. Such transi-

tions are called Franck-Condon transitions (see Figure 2a). The specific

 

transitions which are determined in the calculations presented in this
work are those in which the nuclei are in the ground state equilibrium
geometry before and after the transition.

The above concept of a Franck-Condon transition is somewhat quali-
tative, and a more exact description of an electronic transition requires
a more quantitative argument of the type presented in Reference 30, p. 60
as outlined below.

If an electric dipole allowed electronic transition occurs from
the zeroth (v=0) vibrational level of the ground electronic state (G)
to the kfih (v’=k) vibrational level of an excited electronic state (E),
the intensity of the corresponding band is proportional to the square
of the transition moment (t;EO,Ek '2). If each vibronic wavefunction

is separable into an electronic part and a vibrational part,

VGO(<1,Q) = ¢G(<1.Q) XGO(Q)
and

ka ((1,0) = vE(q.Q) he)
where q represents the set of electronic coordinates, and Q,represents
the set of nuclear coordinates, the transition moment can be expressed

8.8

<» * *
MGO,Ek = swede (we 3,) vE(q,q)aq1xEk<Q)dQ

-x- 4 PX-
.9
where r; and r: are the position vectors of the ith electron and the nth

nucleus respectively, and 20 is the charge of the nth nucleus.

23

.Energy

 

 

 

 

 

V

Internuclear distance of diatomic molecule

Figure 2a. Franck-Condon transitions.

21+

The second term.vanishes because fer the same value of Q the
electronic wavefunctions are orthogonal. Therefore, if the electronic

transition moment is defined as

" -9
MG,E(Q) = JvG*<q.Q) (g r1) vE(q.Q>dq

then

4 «x -9
”60,31; = IXGO (Q) MGE(Q) XEk(Q)dQ

.a
If it is assumed that the variation of MG E(Q) with respect to
4 ’ 3
Q, is slight, then MG,E(Q) can be replaced. by some average value MGO,E,

then
lMGO,Ek ' " ”tom ' 3 90,31;

where S is the overla inte ral between X and . Therefore the
GO,Ek P g GO XEk ’
intensity of an electronic transitionuis approximately proportional to
the square of the overlap of the vibrational wavefunctions.
The intensity of a band or of a system of bands is usually re—

ported in terms of a dimensionless quantity called the oscillator strength*

r = gooogzin' J‘ evd‘fi = 1;.3 x 10’9 Ievd'fi,
Ne -

A where V equals v/c, where c’ is the speed of light, a is the
mass of an electron,N is:-Avogadro‘s number,e is the charge of an electron,
v is the w’avenumber (cm’J) and at; is the molar extinction coefficient in
units of liters mole.-l cm.-l. This quantity can have values from zero
(for forbidden transitions) to one (for very strong transitions). The

oscillator strength can be measured accurately for an electronic transi-

tion, only if the band or system of bands is isolated from all other band

 

*Reference 30, p. 9

25
systems; otherwise, it must be determined approximately.
From the theory of electnanic transitions, the expression for
the oscillator strength of an electronic transition from a state m to a
state n is

.. ’4 2
fmn =-§ Emn ’Mhn I

_. 4
where Emn’ in atomic units, is the average transition energy and an’

in atomic units, is the average transition moment.

The vibrational wavefunction ofra vibronic state determines the
probability of finding the nuclei in each of the positions which they
pass through during a vibration. If, for a certain position of the
nuclei, the vibrational wavefunction is large, then the nuclei are likely
to be in that position, if it is small, then they are unlikely to be in
that Position. In the zeroth vibrational level of an electronic state,
the position of maximum.probability is the equilibrium position of the
vibration, while for higher vibrational levels, the two classical turning
points are the positions of highest probability.

Two extreme cases of electronic potential curves can be imagined.
In one case, the curves of the two electronic states are exactly the same
and the sets of vibrational wavefunctions are exactly the same. Because

of the orthogonality of the set of vibrational wavefunctions, the zeroth
i'vibrational wavefunction of the ground state will have a nonvanishing
overlap integral only with the zeroth vibrational level of the upper
electronic state, and only the 0-0 band will appear. In the other extreme
case, the electronic potential curves will be so different that the
electronic transition will lead to dissociation of the molecule.

In the intermediate cases, the size of the overlap integral of

 
 
  
 

*
XE! OXE lo

Most probable transition
‘X’
XEOXEO

Most pr bable transition
Energy

*
XGnXGn

 

*
XGlXGl

 

 

Internuclear distance of diatomic molecule

Figure 2b. Nuclear distribution functions and most probable transitions.

27
the vibrational wavefunctions depends on the extent to which both wave-
functions have large value in the same regions of space. Such a situation
is likely to occur if the nuclear geometries of the two states,

E
Q and Q , for which X “and X have their maxima, are the same (i.e.,
X m GO Ek

ma ax
Q? = Q3 ). This will be the case if the ground state equilibrium
max max

nuclear configuration corresponds either to the excited state equilibrium
nuclear configuration for the case of the 0-0 band having maximum intensity,
or to the classical turning point of a vibration corresponding to a O-v'
band having.maximum intensity, where v'>O.

In qualitative terms, the band of maximum intensity can be
thought of as arising from.a vertical transition, originating while the
nuclei are in their ground state equilibrium geometry, and ending in that
excited Vibronic state in which the vibrational wavefunction is a maximum
when the nuclei are in their ground state equilibrium geometry. Such a
vertical transition will terminate in the zeroth vibrational level of the
excited electronic state if the equilibrium nuclear geometries are similar
in both electronic states or in a higher vibrational level in which a
classical vibrational turning point occurs when the nuclei are in the
positions corresponding to equilibrium in the ground state. A band which
is not of maximum intensity can be thought of as arising from a transition
which occurs from the zeroth vibrational level of the ground electronic
state when the nuclei are remOVed from their equilibrium geometry, and
which terminates in that state in which the vibrational wavefunction is
a.maximum When the nuclei are in these same positions. This concept is
illustrated in Figure 2b. In terms of this viewPoint, a calculated

Franck-Condon transition corresponds to a band of maximum intensity.

28

In some cases, a number of bands which extend over a consider—
able region of the spectrum may have nearly equal intensities. In terms
of the simplified viewpoint of electronic transitions which was presented
above, such a phenomenon can be viewed as arising because the turning points
of a number of vibrations of the upper electronic state occur when the
nuclei are approximately in the ground state equilibrium geometry. This
is illustrated in Figure 2c. It becomes difficult to associate the
calculated electronic transition with a specific band in this case.

The intensity of a system of bands is an important clue to the
type of orbital promotion giving rise to that band?os3l’3hln general, in
order of increasing intensity, the most important low—lying intravalence
transitions of unsaturated organic molecules are those involving the lnanf
(falO'h-lO'z), ln40*, ‘lcdo* and 'ln4n* (f=O.l—l.0) states. The
theoretical explanation of this ordering is that the size of the electronic

transition moments of the transitions to these states generally occur in

this order. These transition moments can be expressed as before

‘4 .‘ .+
MG,E ”I *G*(q,Q) (in) ¢E(q,Q) dq

In terms of the simple molecular orbital theory usually used by
spectrosc0pists, in which electronic states are described as single con-
figurations (or by a pair of configurations in the case of open-shell
singlets), and where excited states are derived directly from the ground
state by means of single orbital promotions, the transition moment

integral reduces to

-+ * 1*
MG,E ~f bi (k) rK ba(k) d'rk

“where the excited state, in this case, involves the promotion 14a. If this

29

 

Energy

 

 

 

m\
,

Internuclear distance of diatomic molecule

Figure 2c. Illustration showing how a steep potential curve of the upper

electronic state can cause more than one band to have maximum
intensity.

30
integral is to be large, orbitals bi and he must have appreciable values
in the same region of space. In the case of.n4n* transitions, the non-
bonding orbitals and the n? orbitals usually do not have density in the
same regions of‘space, while in the case of n*o* transitions, both
orbitals are in the same plane, and although the n orbital is more
localized than the 0* orbital, there are regions, nevertheless, where
the two orbitals do have density, The n and n* or O and 0* orbitals may
have considerable density in the same regions of’space.

There are ways of experimentally distinguishing between n4fl*
and "*fl# transitions; the solvent effect, the effect of conjugative
substitution, and the effect of acid solution on the transition energies.

Hydrogen containing solvents tend to form.hydrogen bonds with
the nonebonding electrons, which have the effect of lowering the non-
bonding orbital energy, while not affecting the n* orbital energy,
resulting in an increase in the n+n* transition energy. In contradistinc-
tion to the effect of hydrogen bonding solvents on the nanf transition
energies, sudh solvents have little effect on the n4n* transition energies.

The effect of solvents of high dielectric constant is to raise
the transition energy of the n4n*transition and to lower that of the
flwu*‘transition. This is thought to be due to the greater dipole moment
of the:fi*fl* state over that of the ground state, and the corresponding
lower dipole moment of the n4n* state compared with that of the ground
state.

The effect of conjugative electron donating substituents is to
raise the n4nf transition energy, while lowering the flnn* transition energy.
In the case of a single conjugated system.it has been argued that such

substituents have the effect of raising the energy of the W* orbital more

31
than that of the fi orbital, while not greatly affecting the non—bonding
orb ital energy. *
Rydberg transitions fall into series, where the wavenumbers of

the transitions can be related by the formula

' '53- = :13 - R(n+q)-2, n=l,2...,
where :2’13 the series limit, R is the Rydberg constant, and q is the
Rydberg correction factor. The lowest (n=l) member of such a series is
usually quite intense.
The Rydberg transitions can also be identified by the reduction
of their intensity or their diappearance in dense media (high pressures
or solutions). This occurs because a Rydberg orbital is quite diffuse,

and thus is significantly influenced by intramolecular forces.

*See Reference 30, p. h32, and Reference 31+, p. 176

32
II. FORMULATION OF THE CI METHOD

A. Formation of the Set of Configurations

 

1. General description of configurations and their

 

Hamiltonian matrix elements

 

The N-electron wavefunction for a given electronic state is

formmlated as a linear combination of configurationa,on,

*k = Ecnkwn
where the on are antisymmetrized products (determinantal functions) of

molecular spin orbitals, a2,

n n n n n n
on = mn(ala2...aN) = det[a1(l)a2(2)....aN(N)]
n _ n
ai — bk(a or B)
The spatial part of each molecular spin orbital bfi is expanded in terms
of the fundamental set of basis functions {gm} which in the present work

is the set of Gaussian group functions

bi = ass
(Rumiderations here are restricted to an orthonormal final set of molecular
orbitals. This restriction greatly simplifies the calculations without
imposing a constraint on the total wavefunction.

wavefunctions are determined by a variational minimization of
the energy expectation value, Ek = (#k ’H’ltk), with respect to available
Parameters subject to the orthogonality constraint (tk 'Vn) = 6kn’ where

H is the non relativistic Hamiltonian

Matrix elements between configurations reduce to the following formulas:

Case 1. ¢k=tn
N
(tk IH I vn> = Hkn =§=l(ilh li)+~213§[(n law-(1:1 11.1)]

i<3

Case 2. *k equals *n except for one spin orbital, where *k contains a

spin orbital m.*n contains n.

H1m = (mlh In) +§=l[(iilmn) - (mlinn
i¢m,n

Case 3. 'k equals tn except for two spin orbitals, where *k contains

spin orbitals ml and m2, In contains I11 and n2

Hm = (minl lmznz) - (min, lmgnl)

Case R. *k differs from tn by more than two spin orbitals

2. General discussion of the limitations of the number of
configurations
In order to study intravalence transitions, a basis set must
tachosen with_at least as many basis functions as the number of inner
8hell and'valence orbitals of the atoms in the molecule under consideration.
Functions analogous to higher principlecquantum number atomic orbitals

also become necessary if Rydberg states are to be studied.

3h

A given set of basis functions {gk, k = l...R} for an N-electron
system.can be transformed into a set of orthonormal symmetry orbitals,
which in turn give rise to a set of ER orthonormal spin orbitals. From
this set of spin orbitals, a set of @2 mS = O configurations can be
constructed. ‘Molecular electronic wasefunctions can then be constructed
as linear combinations of these configurations. For a finite basis,
coefficients of configurations can be determined by energy minimization
and if all possible configurations of molecular orbitals are constructed
the result will be referred to as a complete CI calculation for the fixed
set of basis functions. The result will be the lowest energy wave-
functions achievable with the given basis set.

In order to approach the true energy, the basis set can be
varied and extended and the above calculation repeated until no further
loweringeffect occurs. In the limit of a complete basis set, which
cannot be attained in practice, however, the exact wavefunction can be
obtained.

6 Usually, in practice, even for a fixed basis set, the total
possible number of configurations is too large to handle. For example,
in formaldehyde there are twelve valence and inner shell atomic orbitals,
and sixteen electrons. If only twelve atomic-like basis functions were
used, there would be a total of (182)2's21t5 ,025 configurations. For thirteen
basis functions, this number would increase to 1,656,369. Therefore,
for a reasonably sized basis set, the number of configurations quickly
becomes too large to handle completely.

There are several ways by which the number of configurations

35

can be reduced. The symmetry of the molecule allows the partitioning of
the set of configurations into a number of smaller sets. These smaller
sets will contain members of one irreducible representation of the point
group of the molecule. This occurs because the integral (*k 'H l¢n>
vanishes if *k and In are basis functions of different irreducible
representations. Such a partitioning operation is performed in the
present work. Another limitation which can be imposed is that of con-
sidering only single, double, and n—tuple excitations from the ground
state configuration, where n is maintained sufficiently small. In
contrast to this approach, a rather more sophisticated scheme is adopted
in the present work in order to select those configurations which are
most useful for minimizing the energy of a specific state. A different
set of configurations produced by the method outlined in the next
section is used for each symmetry and multiplicity.

3. Generation of configurations

The main objective of the present work is the determination
of wavefunctions for excited electronic states; thus, a procedure was
developed for generating configurations which is referenced to the
particular excited state of interest, as opposed to a scheme which is
organized around certain types of excitations from the ground configurations,
single, double, triple, etc. All such excitation procedures are closely
related if a sufficient number of multiple excitations is allowed, but
since configurations which are important contributors to the excited
state are accessible by single and double excitations from.the major
excited state configurations, it seems advantageous to generate the con-
figurations directly in this way rather than performing all excitations

(at least through triple excitations for the low-lying states of interest)

36
from the ground state.

The procedure used to generate configurations for states of
given spatial symmetry and multiplicity (ground or excited states) can be
outlined as follows:

(1) A set of M parent configurations, {tél)}, is chosen,
which consists of the configurations expected to be the most important
contributors to the state of interest. This set is subject to refine-
ment subsequently‘based on purely analytical criteria. If several states
of the same spatial symmetry and multiplicity are of interest, these must
be treated simultaneously to satisfy orthogonality requirements.

(2) A second set of configurations, {t£2)}, is generated
by performing single and double excitations from.each parent, 1&1),

subject to a threshold criterion

J(!k(2)LHL*1Kl)) 12 ) 5 for at least one
(“Jain H157“) - of” [H NJ”) > 3 = 1,2....,M.
where 6 is chosen to be small subject to tractability considerations,
typically 10"3 - 10’5.

(3) The combined set of configurations, {¢(l)} U {#(2)} is
augmented at this point to include all configurations rejected by the
interaction-criterion which are necessary to obtain precise eigen-
fUnctions of the spin angular momentum operator, 82, e.g., if the mS=O
configurations are considered, all ms=0 permutations of a given open-

Shell spatial configuration must be included. .Assuming the {t£2)} has

been so modified, the total wavefunction is expanded.

2 (i)
= 2 >2 V
V i=1 k cki k

37
and the Hamiltonian matrix is diagonalized to obtain energies EJ and
wavefunctions #3.

(M) The M‘ lowest-energy CI wavefunctions are defined as the
parent set {§§1)} and step 2 is repeated using the entire CI wavefunctions
for '31). . The new configurations {$19)} plus those contained in 113. (l)
are used, after augnentation to include missing ms=0 components, as a
basis for the final CI expansion.

This procedure is the result of considerable development
in which a number of different procedures were tested; the final
method appears tobe quite efficient and accurate.

The most time consuming part of the CI procedure is the
calculation of two-electron integrals (bibj ’I‘iJé lbmbn). The number of
these integrals that must be calculated depends on the number of molecular
orbitals that are allowed to have variable occupancy where'hbyvariabde
occupancy is meant occupancy by either zero, one, or two electrons. All
other molecular orbitals are either doubly occupied or never occupied
in all of the configurations). For a molecule with no symmetry, the number

3+ 3M2+2M)/8

of such two-electron integrals over molecular orbitals is (ML: 2M
where M is the number of molecular orbitals which are allowed to have
variable occupancy. 'Jhis number increases approximately as M“ and thus,
for tractability reasons, it is very important to limit the number of
molecular orbitals used.

In the present work, the number of molecular orbitals which are
allowed to have variable occupancy is limited in two ways: by the establish-
ment of a core of orbitals which are doubly occupied in each of the con—

figurations, and by a contraction of the set of virtual orbitals, i.e,

by restricting certainvvirtual orbitals to have zero occupancy. These

V
NIX» fiflhv

limitations are illustrated below:

 

ground state
SCF orbitals

38

 

 

. A
_ ground 6
+ (M N) state
. virtual
orbitals
+ 3
+ 2
+ l fixed—core
excited state
SCF'S \
/’
ground state
occupied
orbitals

b!
N +'(M"N) contracted set of
virtual orbitals
M-N .
. b‘ = 2 c b
o izloooMt’N)
b!
N + 2
bl
N + l

bk +-(N-k) non-core ground
. state occupied

. orbitals
bk+2
bk +vl
bk
. core
orbitals
b2
b1

CI orbitals

39
The core and lower non core occupied orbitals are ground state SCF
orbitals. The higher non core orbitals are obtained by fixed core
SCF calculations on the singly excited states. In these SCF treat—
ments all orbitals are restricted to be the same as in the ground
state except the highest singly occupied orbital. This orbital is
expressed as that linear combination of virtual orbitals which mini-v
.mizes the energy of the state. A number of virtual orbitals from each
of these SCF calculations also results and the higher non core orbitals
are‘chosen from this set.

h. Discussion of the restrictions

In order to discuss the validity of the restrictions
mentioned in the previous section some preliminary remarks must be
made.

It is convenient to separate the lowering in energy whidh
occurs on performing a CI calculation into those amounts which are due
to different types of orbital promotions from the most important con»
figuration (e.g., i'+ a, 12 4 a,b). The significance of single
promotions is that the "full" CI calculation involving only single
promotions from a given orbital is similar to the SCF treatment described
in the previous section. To show this, take the case of a configuration
a .

pi = det(a§ aé.....a§)

where a set of orbitals {ap, p = N + l....N+M} remains available for
excitations Consider a set of configurations which are formed by successive-

i . i i i
13Lexciting an electron from orbital an; to orbitals aN+l’ aN+2""§NsM .

If the total CI wavefunction is expressed as that linear combination of

ho
the above configurations which minimizes the energy of the state, and if
the coefficients c1, 1 = O,l,....M, are associated with these configura-

tions, then the wavefunction t of the state can be expressed as

i i l i i i
* - det(al 8.2....ooooaN_l[coaN+cla~N+l+ooooo+cM 314])

which is Just a single determinant wavefunction. In other words, the

sum of determinants, each of which can be expressed as a single promotion
from a given orbital of a given determinant, is also a single determinant.
Thus single excitation CI treatments give rise to fixed core SCF results.

If ground state molecular orbitals are used, then a single excitation CI

on the ground state would yield coefficients c0 = 1, c1 = O, i = l...M
because air is already optimized in this case. However, for excited states,
significant mixing of singly excited configurations can occur.

.Another type of promotion is the ioiflfl+ ade (12'4 a,b)
double promotion, in which contributions to the pair correlation energy
associated with orbital i can be found. Strictly speaking the orbital i
should be SCF optimized in the state under consideration, for this effect
to be truly distinguishable from the SCF effect. Hewever, it is very
time consuming to use a different set omeolecular orbitals for each state.
Therefbre, if possible, it is advantageous to use one set of near optimum
orbitals for all states and to depend on the configuration interaction
process to account for small changes in orbitals in different states.
Other types of excitations also occur in this work but these will be
considered later.

The core restriction and the virtual orbital contraction
potentially can lead to a higher accuracy of the ground state description

over that of the excited states; however, this point can be examined for

hi

each specific state as shown in applications reported in Chapter 3.

The analysis of the effect on the calculated speCtrum of the
fixing of a core and the contraction of the set of virtual orbitals at the
12 4’a,b promotion stage is much more complex. In order to simplify the
analysis, a simple model of the energy lowering effect of a configuration
will be proposed. This model is based on the perturbation theory-type
approximation described in Appendix A.

If cpl: is the configuraticn which is the major contributor
to a state k, then the energy lowering effect of any configuration o:
is approximately given by

KQPEIHMEZF
1ak-<m§lnlqn§>l-

 

ll

Ii
where Bk is the total energy resulting from optimizing the total wave-
functiOn fk = f cgoik with respect to the coefficients'c': assuming that
(Q: '1?) e O for all i,,j,=2,3....(see Appendix A).

Assuming now that we are concerned only with the effect of
the core restriction, and of the virtual orbital contraction on the
spectrum, the questions are whether the energy of the core is different
in the different states, and whether the virtual orbital contraction
affects one state above another. é.

If Cpl: is obtained by a double excitation from (pl; then the
integral ((9: [H lo?) depends only on the orbitals which are involved in
the excitation (see p.33). If it is assumed that Ek - (of [H Io?) =
En - ((9? 'H lo?) a AE where (p? is obtained from cp: by the same double
excitation that yielded. (9: from cpE then I}: = I2. Based on this obser-
vation” the difference in energy of the different states I: is- due to

the differences in the types of excitations that can be achieved in the

1+2
different states.

If m? is obtained from the ground state configuration by the
promotion 1 4>a and a? by j 4'b, and if m.is an occupied molecular orbital,
then the orbital promotion involving orbital m which appears in state k,
but not in state n is m2 4>b2, while the promotion which appears in state n,
but not in k, is m2 4»a2. Thus the difference in contributions to the
energy of orbital m in the singlet and triplet states arising from the
orbital promotions i 4’a, and j 4‘b from the ground state is equal to

web = [(mb lmb)2 - (ma [ma)2]/AE

If m is an inner Shell orbital, the regions in which it has
significant density are not the same as those of the valence shell orbitals.
Therefore, the two electron integrals involving orbital m.and orbitals a
or b would be expected to be small in this case and the difference AEab
will also be small. If valence shell molecular orbitals are also included
in the core and m is a valence shell orbital, then the numerator of AEab
will not necessarily be small and AEab will be guaranteed small only for
highly energetic excitations such that each denominator is large.

Similar remarks may be made about the effect on spectra
"calculations of virtual orbital contraction at the double promotion stage.
The contraction will affect all states equally, except for the fact that
certain orbital promotions are possible for one state and not for another.
For example, if because of the contraction of the virtual orbital set, a
set {51, i=1...N} of orbitals are discarded, the State k previously dis-
cussed will be missing configurations which are obtainable by promotions
32 4 a‘ba} from the major configuration, while the state n will not be
nussing these configurations. Similarly, state n and not R will be missing

those configurations, which are obtainable by promotions i2 4’a:,a; from

1+3
the major configuration. Thus the difference in energy between excited
states that is due to virtual orbital contraction can be expressed approxi-

mately as

“ab = (1.111.322 _ (aiuahg
[Bk ' ((912%1’33113 JTisz’ag‘fliEn " (figural; [H ' (szflgyaljffl.
Similar statements can be made about double promotions to orbitals, one of
which is a member of the set which is allowed to have variable occupancy in
the CI procedure, and the other is a member of the discarded set. A
If the molecule is small enough that only very high energy
virtual orbitals are discarded, the denominators will be relatively large.
This occurs because the configurations formed by double excitations to
these high virtual orbitals will also be high in energy. Similarly, under
these same conditions the numerator will be small, because the diffuseness
of the. virtual orbitals will cause the two electron integrals to be small.
Under these conditions, the contraction of the virtual set is Justified.
As the discarded virtual orbitals decrease in energy, these arguments
clearly become less valid.
In slnmnary, the fixing of a core, and the contraction of the set
of virtual orbitals affect the resulting spectra calculations, at both
the SCF (single promotion) stage, and the double promotion stage. It is
argued that in certain cases this effect will be small and in other cases

somewhat larger, but still relatively small. The argument is based on the

perturbation theory-type treatment discussed in Appendix A.

My

Numerical results on the effects of these restrictions are
presented in the sections on formaldehyde and glyoxal in Chapter III.

B. A Discussion of Eigenfunctions of 82

In the present work, 1118 = O configurations are utilized in
order to obtain all possible multiplicities corresponding to a given
spatial orbital excitation. The sets of configurations which are used
to construct wavefunctions, in the present study, are complete with
respect to the spin angular momentum operator, $2. The Hamiltonian
which is used commutes with 82, assuring that the eigenfunctions of the
Hamiltonian matrix will also be eigenfunctions of s2.

Corresponding to each configuration (pi in the final set there is a
configuration $1, called the complement of (pi, which is the same as “Pi
except that o and 8 spin functions in the non-core are interchanged.

For a given 2n open shell arrangement, a total of a?) configuraticn-
complement pairs must be included in order to assure that the wave

functions will be eigenfunctions of 82. In order to reduce the dimensions
of the Hamiltonian matrix approximately by a factor of two, a given

matrix of elements, ((pi [H lepi), (cp1 IH [263), (c—pi 'H loj) and (351 [H [253.) is
replaced by two smaller matrices of elements ((cpi + 361) [H I (th +3953» and
«(Pi - $1) '11 ' (cp;l - 253)) respectively. This can safely be done because

all matrix elements of the form ((ijL + $1) I H I (cp.j - 253)) are identically

equal to zero. The proof of this assertion follows.

I =((T1"$1) lHl(<PJ‘$J)>=<‘P1'H"°3>‘<‘P1'H'$J>+@i'Hl'%>'@i WWII)

115
The integrals between configurations reduce to the molecular
-l
spin orbital integrals (ai(l) Ih '33 (1)) and (ai(l)aj(l) I (r12) lam(2)an(2))

which are non zero only if a and a 3 have the same spin functions and a

1 m

and an have the same spin functions. Thus, if all of the or and B spin
functions were interchanged the same molecular spin orbital integrals would
be non zero and the integ‘als between the configurations would be unaffected.
In the studies presented here, only the spin functions of the
non-core orbitals were interchanged. In this case the Anon-core integals
are unaffected by the interchange. Since the core contains only doubly
occupied orbitals, interchanging spin functions in the core merely changes
columns of the determinantal function and therefore at most only changes
the 8191 of the function. If such a sign change occurs for one of the
configurations involved in the integral, it will also occur for the
other configuration because they both contain the same core. Therefore,
the interchange of spin functions in the core has no effect on the
integral, and thus interchanging spin functions in the non core is
equivalent to interchanging all of the spin functions as far as the
integral is concerned. Therefore, since interchanging all of the spin

functions has no effect on the integral,

(0P1'H'CPJ> = @i'H'$j>
@f'HicpJ) = (epilHWJ)
and I =0

Thus, two classes of'wavefunctions are ibrmed, those which are linear
combinations of the functions {4:14-61}, and those which are linear combina-
tions of the functions {$1.51}; It will be shown that depending on the
number of non-core electrons, either the set {cp1 + $1} will give rise

to singlets and quintets, etc., (s=0,2,h,6...), while the set

h6
{qfiefia} will give rise to triplets, septets etc., (s=l,3,5...), or vice

versa. If NC is the number of non-core electrons, then the general rule

is
32*(¢14$1) = 5(S+1)¢(¢it$i), where s=0,2,h,... for Nc/2 even,
i and s:l,3,5,... for Nc/2 odd,and
Sefiq’sz-i) = S(S+l)¢(<pi--<Ei), where s=l,3,5,... for NC/Z even,

and s=0,2,h,... for NC/2 odd,
where “(piss-pi) is a wavefunction which is a linear combination,
2c1(<p1:t$i). Thus the parity of the quantum number s of a given
:avefunction can be easily deduced. Finally, it will be shown how to
distinguish singlets (s=0) from quintets (8:2), and triplets (s=l) from
V septets (s=3) in the final set of wavefunctions.‘

Consider a configuration méz) and its complement, 5:2) having
two open shells.
o§2)=det(bl(1)'6'l(2)...bp(2p-1)'SP(2p)di(1')Ei(2')...d;.(2r'-1)E;(2r')

f£(l")f§(2"))
—§2)=det(bl(l)31(2)- . .bp(2P-l)-5P(2p)3i(l' )di(2’). . .E;(2r'-1)d11(2r')
¥§<r>¥g<2m
where the set of orbitals {bi} are the core orbitals, the set {d5} are

" and f" are the non—core

the non—core doubly occupied orbitals, and fl 2

.sineg occupied orbitals.
2 (2)“(2) _ 2 2 ’2“
s (:91 +cpi )-(s+S_, + sz.- sz)(cp§ Wis 1:
Since the value of the m8 quantum number associated with these

configurations is zero, Si and Sz have zero eigenvalues. Therefore,

l+7
32(cp£2)@§2)) = S+S_.(.Pi(2)+$i(2))

1 -= 2+3 +....
wqere Si: Sli- 2:1:

S operating on a closed shell results in a determinant with two equal

columns, which is identically zero

say f1 = hid, f2 = h2B, fl = hlB, f, = h or

82¢£m16+dettbié$gl<2fi..bp(2p—l) bp(2p)di‘ (1')}‘31 (21)”.(3; (2r'-l)'a;. (21")
r3<1">r2 <2"§J

.. det[b1(l)bl(2)...bfi(2p-l)bp(2p)di'(l'fdi (2')...d; (2r'-1)‘d.} (21")
f;L(1")f;p(2")] +

"" _ _ h 1 "t g '- g_ _.' 3"" 11-" n
det[b1(l)bl(2)...bp(2p l)bp(2p)dl (1 )d1 (2 )...dr (2r 1)dr (2r )fl(l )f2 (2 )J
The first determinant above is (pg-2) , the second is the same

as Egg-macaw: that the d; 4, column has been interchanged with the di' column,

—!

the (1' column with the d2" column. . .and the (1;. column with the 311' column.

2

Interchanging columns of a determinant changes the sign of the determinant.

Since the second determinant is the same as Egg) except for 33...: r'
interchanges of columns, then the second determinant is equal to (-l)r"$j(-2).
If q' is the total number of non-core electrons, then since r'-q'-2,

I
the second determinant is equal to {-1) '3‘ '1 :91. Therefore,

sip?) = «42> + <-1> 9‘54 26?)
and similarly,

825:2): $12) + 01%" -‘LCP§_2)
Thus,

sew?) + $12)) = op§2>+<-1>%‘ “1?p§2)+$§2)+<-1>%1'1cp§2)

u O(O+l)(¢§2)+¢§2)) for even 9;
2

1&8

= 1(1+1)(cp(2)+'q3§2)) for odd ‘1'

2
In other words, (¢(2 )+p§2 )) is a (fififigigfi) if one-half the number of non-

core electrons is (Eggn). Similarly, (m(2)— $(2)) is a (:ifigigt) if one-

half the number of non-core electrons is gggn). Thus, since configurations
with two open shells do not contribute quintets and septets, the above
theorem is proven for the case of two open shells.

For the case of four, six, and higher numbers of open shells, the
linear combinations {¢§n)£$(n)} themselves are no longer eigenfunctions of
Se. However, it can still be proven that {¢(n)fl gn)} are basis functions

(2) —(2)

for the same spin multiplicity functions as are {mi imi }. The proof is
based on two facts. First, since H commutes with 82, the wavefunctions
which result from.the diagonalization of the Hamiltonian matrix are also

eigenfunctions of 82.

Secondly, if <¢§n-2)g§n—2) IH ltp§n>i§§n)) is unequal
to zero, and (fin-65.55111-” is a basis fimction for certain eigenfunctions
of 82, then ¢(n) fién) is a basis function for the same eigenfunctions of

82*?

. Thus, it remains to prove that the integral is not zero for certain
i and J, for any value of n. Then, since we know that for n=2, {¢(2)fl (2)}
are eigenfunctions of 82, the theorem can be proven by induction for the

singlet (8:0) and triplet (s=l) cases.

(n )5 f) IH [w (m)

*Note e, «p14¢3m)) is always zero, as was proven previously.
*This statement is not necessarily true if the symbols n and n-2 are
interchanged. The reason for this is that the eigenfunction of 82 with
highest 8 quantum.number that can be constructed from configurations with

n open shells cannot receive any contributions from.configurations with

n-2 open shells, even if the integral is nonzero.

19

Every configuration.¢§n) having n open shells, can be

-2
obtained from a configuration having n-2 open shells, min )

bya
single promotion. In this case mg“) will be the same as wgn'2l except
that where mg”) has spin orbital ak, mgn‘g) has spin orbital am.

Therefore,

«991-2) 13 My”) = «Bin-2) l H 1369))

N
a (ak 'h lam>+§:l[(ajaJ lakam)-(ajak Iajam)]
J¢k,m
where N is the total number of electrons in the molecule. (mén-2)'le£n)>
then, is not equal to zero except by coincidence. However, the integral

«Pin-'2) 'H ’25:”) may or may not be equal to zero. Therefore,

((win'mdrfifflh '3 l (cpgnwgmw =

(gin-2) 'H “pg-11))..(zpin-2) 'H [$Sn)»<$§n-2) ‘H [cp§n))+((p'£n-2l 'H :fqn»
.. gain-2) [H 1.§n>>+2<¢§n-2> [H 12,39»
which is unequal to zero except by coincidence. Similarly

(mfg-fine) 1H [aim-QM) = 2<<p§n'2) 1H Mn» -2<cp£“'2‘ 1313391))
which is unequal to zero except by coincidence.

Therefore, {¢£n)45£n)} net are basis functions for (€%§§i§¥§)
if one half the number of non-core electrons is (833”). By induction
from the above statements for min) and ¢£n-2) it follows that the same
statement can be made about {orbs-159)} where n=6,8. .. . .

The next question is which set {¢§n)4$£n)} or {$1n);s£n)} n=h
will be basis functions for quintet states. This question can be

answered in the following way:

f!

«I

0"

so

For a four electron four open shell problem, there are six configurations
that can be constructed:
,1 - b1(1)a(l) b2(2)oz(2) b3(3)8(3) 1100800
'51 = b1(1)a(1) b2(2)s(2) b3(3)a(3) madam)
p2 = bl(l)0'(l) b2(2)B(2) b3(3)a(3) by(1+)B(‘+)
$2 .. b1(1)B(l) b2(2)a(2) b3(3)s(3) bu(1+)a(1+)
°P3 e b1(1)a(1) b2(2)B(2) b3(3)B(3) bu(l+)a(1+)
$3 = b1(1)8(1) b2(2)<r(2) b3(3)cr(3) 3008(4)
The quintet state with n.8«==o can be formed by using the lowering operator
twice on the configuration

bl(1)a(1) b2(2)a(2) b3(3)a(3) >bu(’+)a(1+)
which is a- quintet state with mg:2.Upon doing this, the quintet state
with m =0 results
(M) +33?) Hpéu) +59) +(pgb') +$§h)

(Pl

.Consider another four open shell problem, but with some closed
shells also. Say there are q' electrons in the non-core. Since the

lowering operator affects only open shells, the ms=0 quintet becomes

2 2
min) + (a); "2 '36:“) + mg) + (-1)? ’2 5;”) + mg“) + (4)329: ‘2 35%“)
Therefore, {rink $5.11)} n=’+, is a basis function for quintets if one half
the number of non-core electrons is 'even (reign) or odd (~sign). Similar
results hold true for the sets
{(9:11) i $91)} with n=6, 8.
Similar results hold for the septets and higher multiplicity states.
Thus, the functions (<pjL + $1} are basis functions for even

numbered s states when é-N c is even and for odd numbered s states when ~12-Nc

is odd; similarly, (cpl - Ei) are basis functions for odd numbered 3 states

51

fOr éNc even, and even numbered s states for %Né odd. Within each group
the specific eigenvalues of 82 must be identified. This process is very
simple.

Singlet states can be distinguished from states of higher even
numbered values of the quantum number s, because only to singlet states do
closed shell and doubly open-shell configurations contribute. Thus, if a
wavefunction with even 3 quantum.number contains contributions from closed,
land/or doubly open shell configurations, it is a singlet.

Similarly, quintets (s=2) can be distinguished from nonets (s=h)
and higher states, because s=2 states contain contributions from con-
figurations with fOur open shells, but no contributions from closed or doubly
open-shell configurations; also, the absolute values of the coefficients of
each of the six configurations which result from a given four open shell
arrangement are equal in the quintet state.

Similarly, triplet states (s=l) can be distinguished from septet
states (s=3) and states of higher odd quantum number s, because doubly
open-shell configurations contribute only to s=l states and not to the
higher ones. The septets can be distinguished from higher states because
of contributions from configurations with six open shells, but none from
configurations with two or four open shells; also, the absolute values of
the coefficients of each of the twenty configurations which result from a
given six open shell arrangement are equal in the septet state. Thus,
the states of different multiplicity can be readily distinguished.

For example, if the number of non-core electrons is twelve

52

{(8:0)} " singlets
{Ti2) i-¢§2)} - * singlets
triplets
{¢(h) i $(h)} * singlets quintets
i i triplets
{Tim i $§6)} -o singlets quintets

triplets septets

C. Obtaining the CI wavefunctions

l. Perturbation Theory:

The problem of obtaining the lowest energy states for the large
numbers of configurations utilized here was attacked in two ways. The
first way involves second-order perturbation theory. The lowest energy
wavefunctions involving the first M most important configurations were
Obtained by accurate diagonalization of the Hamiltonian matrix, where typically
M .3 100. The effect of the remainder of the configurations was gauged by an
iterative perturbation theory type procedure.

If an NxN'Hamiltonian matrix of elements Hhm:Hmk is replaced by

an approximate Hamiltonian matrix of elements Hhm =2H’k such that
m.

H'k = H k e 1,2...N

1 kn,

H' =H k m=2 ...N
km m§km ’ ’3

the resulting energy eigenvalue (see Appendix A) can be expressed implicitly
as
2
_szl'
k Hkk-E
If this expression for E is used in the secular equation

2:"H-E =0
R ¢k( kl 6kl)

E = H11

53

 

then
Hk 2
l
- + -—————-— .... =
cl(Hll Hll E 'Hkk — E + C2H2l + CNHNl) 0’
implying that
C =:.I_._{u__——-— :: Hkl .—
k _ E i?—————- k — 2,3,...,N
Hkk - Hkk
Thus,
0
(t [H In >
*1 ~' $10 + X l k Tk (norm)
‘ 5: k=2 E - Hkk

In the above expression, ti itself is a linear combination of configurations
determined by the initial diagonalization, and each mk is a configuration (or

a configuration and its-complement) such that

“'10 he = o and a In = a.

The energy expression which corresponds to this wavefunction

 

 

 

18 E = (*1 'H Iii)
«1 My
- N H 2 H ' H
2 l
1 2 ml ml nl

This equation was solved for E by an iterative procedure. The
value of E was first set equal to H11, a new value of E was obtained from
the above equation, and the process was repeated until the value of E converged.
Once a value of E is obtained the wavefunction #1 is determined. If the matrix
elements, (wk lH tom) ktm, are sufficiently small, the value of E obtained by

the above procedure should closely approximate the Eexact obtained by

complete diagonalization; in any case, E 2 Eexact.

5h

2. Successive Matrix Diagonalization

A more accurate way of constructing wavefunctions from a given
set of configurations was developed in this work and is called a successive
diagonalization procedure. It is particularly useful in case more than
one eigenvalue of the Hamiltonian matrix is of interest. This method takes
advantage of the facts that only the lowest energy configuration wave-
functions are of real interest as representations of molecular states, and
that the approximate importance of a configuration-complement pair can be
gauged before the diagonalization calculation. The energy contribution of a
configuration fé2) to a state *3 is measured by evaluating the interaction

expression

ij = [($(2) 1H lv§l> F
((3) IHHE) >- fig” MILE”)

 

where *(l ), J=l,2,...M is a linear combination of a relatively small number
of configurations which represent *3 approximately, e.g., $§1)might be the
CI wavefunction generated by considering the most important set of configura-
tions (See Section II.A. 3LIf more than one state *3 is of interest, the
importance of 9(2 ) is measured by the largest value of ij, 321,2 ...M.

All m.S = O configurations with identical closed and open shell
spatial orbital arrangements must be considered simultaneously, so that
the wavefunctions will be eigenfunctions of 82. For example, a Spatial
arrangement of any number of closed shells, and four open shells gives

rise to six linearly independent configurations.

55

(closed shells) aabflcada

(closed shells)‘ aBbacha

(closed shells) aabachB

(closed shells) aBbflcada

(closed shells) adecha

(closed shells) afibacddB

If any one of these configurations is important, then they all
must be included. Thus, the importance of the group as a whole is
measured by the largest value of the interaction expression for all possible
members of the group.

The groups of configurations are ordered according to their
importance, so that the most important ones can be treated accurately,
while the ones of lesser importance will be treated approximately. The
augmented combined set of configurations {t(l)}U{#(2)} used in part (3)
of the generation procedure (Section II.A.3)are ordered first in the list.
.After this, the groups of configurations are listed in the order of the
sizes of the interaction expression ij with which they are associated.

The "successive diagonalization" of the resulting NxN
Hamiltonian matrix is accomplished in a series of‘p smaller diagonalization
steps Ni x N1

(N2 +-k)

(N... +k>

(up +k>

in which the k lowest energy linear combinations resulting from.a

(Né +'k)

(hm +-k)

”a. “no.”

(N +-k)
p

particular step are passed on to the next step, and

N B;E N
EFl .m

56
Thus, the step mrl.produces the k lowest energy wavefunctions

*1 =3 cmiwm 1 = 1,2,...k

and in the mth step, the wavefunction is considered as a linear combination
of‘th+vk.terms

k Nm
V _ i=1 citi + i=1 dm cPIWI

in whiCh the first k terms are constrained to be the k linear combinations

resulting from the m-l Step, and q=Nl +~N +3..N' . At each step the hi

2 m-l
are chosen to include all necessarymS = 0 spin permutations to insure a
proper eigenfunction of 82.
The above diagonalization procedure allows some adjustment of the
k lowest energy linear combinations at each step, but only to an extent
possible by mixing with constndned linear combinations actually present.
The interest in the present work is only in the one, two, or
three lowest energy wavefunctions for each spatial symmetry and
multiplicity, and for this purpose k:6 was used; likewise, in order to obtain
. matrices for which diagonalization would not be unduly time consuming, the
NR were chosen arbitrarily to be less than or equal to eighty. The energies
which result are all upper bounds to the correct energies (see Appendix B).
A comparison of this method with one involving an accurate matrix
diagonalization shown in Table 1 shows that the two methods give very
similAr results. It is also interesting to note that the lowest energy
solution obtained by the perturbation theory procedure in Section Cl was
also fbund to be accurate in a numerical test case. Such numerical tests,

however, clearly depend on an assumed distribution of’matrix elements and

hence cannot be taken as a definitive proof of the method for a general

57

Table 1. Comparison of successive diagonalization procedure with - '-
'more accurate mat ix diagonalization” for the lowest singlet Bgu
state of pyrazine. The threshold and the number of configuration
pairs appear in parentheses. Only those coefficients of 0.1 or
greater are shown. Plus signs indicate<y spin and minus signs
indicate 5 spin. +

Configuration Coefficients Coefficients
accurate successive
diagonalization diagonalization

(2.28xlO'5, 209)69 (5x10'5, 26579

21422 .6963 .7011;

19-23 .5381 .51+2l+

(17, 21)..(22)2 -.1882 -.1855
(-17, +19)-»(+22, -23) .11u72 .1hh6
(+17, -l9)-'(+22, -23) -.1li27 -.1l+21
(19)2-*(23, 25) -4271; -.1273
(+19. -21)-*(+22, -25) -.1611+ -.1592
(-19, +21)'-o(+22, -25) .1h9i .'.11+56
(17, 21)-o(23)2 -.:L390 -.13l+8
(21)2-.(23, 25) .1392 .1362

Total electronic energy (a.u.) 472.6055 -1+72.6oh7

*For the accurate diagonalization, the subroutine Nesbet (QCPE 93) which
was obtained from the Quantum Chemistry Program Exchange was used.

+Two different sets of configurations were used in these calculations, so
comparison of the results is somewhat difficult. The larger set includes
the smaller set.

In the less accurate calculation, the three lowest energy states were
sought, while in the more accurate calculation, only the lowest energy
state was sought.

58
matrix diagonalization. On the other hand, in the present applications,
the decreasing importance of higher configurations in the ordered list

gives rise to matrices which can be diagonalized sufficiently accurately

by approximate methods.

59
III. .APPLICATIONS AND RESULTS

A. Introductory Remarks

 

The remainder of this thesis is concerned with the application
of the CI fbrmulation developed in Chapter 2 to the study of the excited
states of polyatomic molecules. The molecules which are investigated are
formaldehyde (H200), glyoxal (H2C202), and pyrazine (HuCth).

The results of these calculations will be used to provide a
theoretical interpretation of the excited electronic states of the
molecules investigated and, in addition, to give a numerical indication
of the reliability of the approximations which have been made in the
formulation of the CI method. The results also will be a reflection on
the adequacy of the basis set used to describe the various types of
excited states.

B. Formaldehyde

 

1. ‘Experimental spectrum.of formaldehyde

 

The nuclear coordinates and symmetry orbitals of formaldehyde
‘ in the equilibrium geometry of the ground electronic state are shown in
Figure 3-

The electronic spectrum of formaldehyde has been studied both

exp” 1mm 1] 35-h2,1+L+-h9w 33,51-62

d theoretically by several investigators.
The experimental results are summarized in Table 2.

The lowest observed excited state occurs in the range 3.13-3.hh eV

32’35-'39This state

above the ground state, with a band origin at 3.12 eV.
has been described as a 3A2(n*1#‘) state. The equilibrium geometry of

the molecule in this state is also found to be different from that of

St:

"in

‘\N

no
A»

1:.

s

‘0.

Nuclear coordinates: H , Hg (0, :tltlul, -l.09), c(o, o, o),
oto, , 2.286h)

Nuclear repulsion energy: 31.1138 a.u.

Symmetry orbitals expressed as signs of coefficients of s and_p_functions
centered on the atoms

5; orbital (shown as a linear combination of s functions)
- +
C O

+
+ + H
C O or ,
H
+

+m m+

bE orbital (py.functions on C and O and 3 functions on H1 and H2)

1'. + u- -
H + + H - + H + + H - +
C 0, C O, C O, or C O
H H H H
- - + +
bi orbital.(px functions)
H + +
C 0, or H - +
H C 0

Figure 3. Nuclear coordinates (in atomic units), nuclear repulsion
energy, and symmetry orbitals of formaldehyde.

J

5.4

61

Table 2. Experimental electronic absorption spectrum of formaldehyde.

Energies are expressed in electron volts.

Electronic
transition

1B2*J‘A1
1A14M

5241a
1A1+1A1
1B2AM
tA2*;A1

3A2+1A1

a. Reference 32

Range8

9.63
9.58
9.25
8.87
8.1M
7.97
7.08’7 7.51
3.51 - 5-39
3.13 - 3.hh

Banda'

origin
9.63
9.58
9.25
8.88
8.1a
7.97
7.08
3.1+9
3.12

Oscillator
strength

References

32
32
32
32, an
32
32, an
010'4 32, a, 1+6
2x10"1+ 32, 36, ho-h2

32, 35-39

62

the ground state.* The dipole moment in the lAZ state was found to be
1.56 th compared.to 2.3h D59 in the ground state. The Franck-Condon
transition between the ground state and the 1A2 excited state is forbidden
by the electric dipole mechanism, Peple and Sidman6O and Henderson62
showed that the transition could occur by vibronic coupling, but Callomon
and Inneshl later showed that the intensity of the transition was mainly
due to the magnetic dipole mechanism.

The next lowest state observed occurs in the region 7.08-7.51 ev,

with the band origin at 7.08 ev.32’m“’1‘6

The nature of the electronic
transition which gives rise to this state has not been clearly established.
It has been called both a 1B2 (nflo*)33 and a 1B2 (n+3sal)32 Rydberg state.
The next band occurs at 7.97 eV. It has been classified both as a lAl
(n+n*)6° and a tAl (nfipbz)32 Rydberg state. There are also several
higher Rydberg transitions which have been reported.32

2. Formaldehyde calculations and discussion

_ a. Discussion of basis set

The type of basis set employed in this work and described in
tAccording to Reference #3, page 210, the equilibrium geometries of the

 

ground and A2 excited states are:

(CH)'+1.09, (00) = 1.22, (HCH) a 120°, (1120-0) a 180°

GROUND -
312 - (CH)-o1.08, (co) = 1.3212, (HCH) a 121°, (H2C-O) = ll+2°
1A2 - (CH)-ol.08, (co) = 1.326, (HCH) == 121+0, (1120-0) = 159°

where the arrows indicate assumed quantities, distances are expressed in
Angstromm, and the A2 classification is based on the ground state

symetry, C2v'

63
Table 3 has been described previously in an application to the first-row
atoms and the ethylene molecule where numerical values of atomic orbital
parameters are tabulated.63 Atomic orbitals of approximate Hartree-Fock
quality are constructed using linear combinations of Gaussian lobe
functions? first-row ls, 2s and 2p expansions contain 10, 10, and 5
Gaussian components, respectively. In the present work several different
basis sets are used. The minimal size basis, which is only slightly more
flexible than a basis of near Hartree—Fock ls, 2s and 2p atomic orbitals,
utilizes three fixed s groups (constrained linear combinations of Gaussians)
consisting of 3,3, and h Gaussian components, respectively, and one S-term
group for each p orbital (see Reference 63). The second basis set, called
a split group basis is designed to allow considerable distortion of atomic
orbitals in the presence of the molecular field.and is obtained by treat—
ing the longest-range Gaussian component in the 3 groups and in the p
group as independent basis functions to give a final basis set of h 5
groups, (h,3,2, and 1 component) and 2 p groups (h and 1 component)
(see Reference 63). In all cases, on hydrogens, a five-component
representation of the 1s orbital with scale factor n =.J§'is used.63
In certain of the molecular calculations, additional basis functions are
included to represent atomic 3s and 3p orbitals, thereby permitting the
description of certain low-lying Rydberg-type states. Except for one
calculation on the ground state of formaldehyde, the present basis
sets are deficient, however, in the omission of d orbitals, and p-type
polarization functions on hydrogens.

b. SCF calculations

 

SCF calculations were performed on the ground state using both

the fixed group and the split group basis set. An SCF calculation was
*See Appendix D.

69

Table 3. Formaldehyde basis orbital parameters. Exponents, coefficients,
and lobe displacements for C, O, and H s- and p-type group
functions are shown in the rows.+

Oxygen ‘
p h6.2879 9.0369 2.3606 0.7031
1 1.0000 15.7917 8u.9397 202.7770
0.0h17 0.0501 0.06h1 0.0756
132, p 0.2120, 0.0300
3 1.0000, 1.0000
0.1178, 0.1732
31 290.8205 1u2u.06u3 u6h3.hh85
1.0000 0.13uu 0.0323
8 0.9311 9.70hh
2 1.1526 -0.1538
83 0.2825
1.0000
3 31.3166 12.8607 n.6037 76.2320
h 1.0000 1.8781 1.0838 0.6261
Carbon 3
p 19.0660 5.0720 1.h305 0.h1u2
1 1.0000 9.2113 62.7606 208.058u
0.5910 0.06h7 0.0750 0.08h6
p , p 0.1216, 0.0200
2 3 1.0000, 1.0000
0.0990, 0.2121
81 159.627u 781.6h95 25h8.7256
1.0000 0.13uu 0.032u
s? 0.h735 b.93hh
’ 1.1h12 ‘ -0.1585
. s 0.1h80
3 1.0000
sh 17.1893 7.0591 2.5269 u1.8u27
1.0000 1.8651 1.1090 0.6308
Hydrogen
31 2.2956 0.6517 0.2059 9.9195 6h.7869
1.0000 2.h751 1.8971 0.2283 0.0315

+The function p3 is the principle component of the 3p A0; the same value
of the exponent is calculated for the 85 (38) basis function. Basis sets:
I. Split group (C and 0 p1, p2, 81, 32, 33, sh and H 31); II. Split group
' plus 0p3 (n), 0p3 (b2), and Cp3 (11). III. Split plus 035 and C35.

65
also performed on the 3A2 (n*fi*) state using the split basis set. The
results are shown in Table h.
The results in Table h show that splitting the basis results
in an energy lowering of 0.1 a.u. from -ll3.70h8 a.u. to -ll3.809h a.u.

The Hartree—Fock limit has been estimated56

to be -113.96 a.u., and
therefbre, the splitting has accounted for about forty percent of the
difference between the Hartree Fock energy and the energy of the fixed
group basis. The occupied orbital energies which result from the two
calculations are very similar. The orbital energies of the 211 orbitals
are also quite close. However, the closeness of orbital energies does
"not necessarily imply close agreement of the molecular orbitals theme
selves. An examination of Table 5 which displays the orbital coefficients
‘of the lbl (fl) and the 2bl (n*) orbitals obtained from the three SCF
calculations shows a close similarity in the shapes of the two ground
state lbl orbitals, but not of the two ground state 2bl orbitals. The
Zbl orbital.which was obtained from the split group calculation is much
more diffuse than the one obtained via the fixed group calculation. The
reason fer this is that the splitting of the basis set allows the virtual
fl} orbital to become much more diffuse on optimization in the field of
the ground state occupied orbitals.

Excited state configurations of molecules have often been
rationalized as single promotions from.the ground state SCF configuration.
This approximation can be examined, in the case of the 3A2 state of
formaldehyde by comparing the configuration obtained by the promotion
lb142b1 from the ground configuration with the 3A2 SCF configurations.

It is especially interesting to compare the ground state virtual 2b

1
orbital with the excited state 2bl orbital, since the argument in favor

66

Table h. A comparison of orbital energies from three different SCF
calculations on the 1A1 ground state and 3A2 excited state of
formaldehyde. Energies are in atomic units, la.u. = 27.21 eV.

Molecular Ground state Ground state 3A2 state
orbital fixed group basis split group basis split group basis
symmetry
(02.)
1131 1.88 1.86
5b2 1.11. 1.07
lOal V 0.99 0.99
hbl 0.97 '0.91
981 0.8u 0.88
ab, 0.82 0.77
8a1‘ 0.77 ’ 0.77 '
751 0.72 0.5M 0.57
3b1 0.50 0.h7
3b2 0.78 0.h3 0.hh
6a1 0.65 0.29 0.29
2b1(fi*) 0.1h 0.11 -o.30
2b2(n) -0.h6 -0.hh -0.5u
lbl(fl) -0.55 -0.53 -0.59
5a1 -0.6h -0.6h -0.65
1b2 -0.72 -0.70 -0.66
“‘1 -0.88 -0.86 -o.85
331 -1.hh -1.u2 -1.u3
2a1 -11.h7 -11.35 -11.28
lal -20.59 -20.58 -20.61
Total energy -113.70u8 -113.809h -113.7151

67

Table 5. .A comparison of n and n* molecular orbitals of H2C0 from ground
and excited state SCF treatments. Subscripts s and L denote
short and long range p groups on carbon and oxygen; coefficient
ratios of short and long range groups on the same nucleus are
variable only in the split basis calculations.

C and 0 Ground state SCF

p basis fixed group split s,p Excited state SCF
groups basis basis split s,p basis
08' 0.53 0.56 0.65
0, 0.22 0.26 0.27
lbl (n) .
c8 0.h3 0.u1 0.31
Cl 0.20 0.15 0.09
03 -0.63 -0.uh -0.h3
* 0‘ -0.26 -0.53 -O.36
Zbl (" )
c 0.67 0.hh 0.66

0,, 0.31 0.7M ‘ 0.52

68
of the similarity of these two orbitals is particularly weak. The shape
of neither the fixed group nor the split group ground state 2bl virtual
orbital is particularly close to that or the 312 excited state 2hl
orbital. It is not clear, based on this evidence alone, which virtual
orbital is a better description of the excited state orbital.

Although the shape of a ground state virtual orbital may differ
from that of the corresponding excited state orbital, a single promotion
configuration involving such an orbital may be considerably lower in
energy than other configurations of the same symmetry, and in this
' respect it may be uoed along with other low energy configurations to
form an initial description of an excited state. This diffugeness can be
gauged from the ratio of the coefficients in Table 5 of the long and short
range p H'basis function on each center. The lbl orbital which was
obtained from the SCF calculation on the 3A .state is somewhat more

2
contracted than either of the other lb orbitals, but it is still fairly

1
similar in shape.

The inadequacy of spectral arguments based solely on orbital
energies Obtained from extended basis set SCF calculations can be
appreciated from an examination of Table 6. This table compares the
energies of the ground state molecular orbitals of formaldehyde which
were obtained from calculations using three different basis sets. The
first basis set is the split basis set. The second set is the split
basis set with the addition of one-component 3p n and ‘3pb2 orbitals
on oxygen and a 3p "'orbital on carbon. The third basis set is a split
group basis set with an additional one component 33 orbital on oxygen

and on carbon. The exponents of these 33 and 3p orbitals were determined

69

Table 6. Molecular orbital and total energies from ground state calculations
on HZCO for three basis sets: (I) split group, (II) split group
plus 0 3pn, 0 3pn, and 0 3pb2, and (III) split group plus 0 3s and
0 38. Orbitals allowed to have variable occupancy in CI calculations,
performed subsequently, are indicated by an asterisk. Energies
are in atomic units.

Basis set+ Basis set+ Basis set+
I. II. III.
Ilal 1.88 11sl 1.88 5b2 1.11*
5b2 1.11 6b2 1.11* ubl 0.97*
10al 0.99 6bl l.01* ubz 0.82%
hbl 0.97* 108.1 0.99 3bl 0.50*
l 0.8M 5b2 0.85% 3b2 0.h3*
b2 0.82* 9&1 0.8M 2bl 0.11*
8a 0.77* 8&1 0.77 7al 0.11*
7e} 0.5h* 5bl 0.56* 6sl 0.07*
3bl 0.50* 7al 0.5M 2b2 -0.uh*
3b2 0.u3* hoe 0.h9* lbl -0.53*
6al 0.29* 6al 0.29 59.1 -0.6h*
Zbl 0.11* ubl 0.13* 1b2 —0.70*
2b2 -0.hh* 3hl 0.12* hal -0.86
1bl -0.53* 3oz 0.0 3al -1.h2
5s.1 -0.6h* 2bl 0.0 * 2sl -ll.35
lb2 -0.70* 2b2 -0.hu* lal —20.58
tel -0.86* 1bl -0.53* ET -1l3.809u
3al -l.h2 5al -0.6u
2a.l -ll.35 lb2 -0.70*
Isl -20.58 hal —0.86
ET -113.809h 3&1 -l.h2
A 2&1 -31035
lal ~20.58
ET -113.8097

+BCF calculations were performed for basis sets I and II; for basis set III,
C 38 and 0 38 orbitals were simply Gram-Schmidt orthogonalized to the Mo’s
obtained from basis set I. The nuclear coordinates and the nuclear repulsion
energy are given in Table II.

70
by optimizing the 3(2p " 3p) and 3(2p '* 3s) configuration of the carbon
and oxygen atoms. Numerical.parameters for all orbitals are given in
Table 3.

SCF calculations were performed with both the first and the
second basis set. no SCF calculation was performed using the third basis
set, the 3s molecular orbitals merely being GramrSchmidt orthogonalized
to the SCF orbitals obtained using basis set I.

The results in Table 6 show that the long range Rydberg orbitals
affect the molecular orbital energy diagram merely by adding new virtual
orbitals; all other molecular orbitals remain approximately the same as
those obtained from the calculation using basis set I. The new Rydberg-
type virtual molecular orbitals have rather low orbital energies; 0.0%
(Zbl n) 0.09 (3102) and 0.13 (mol n) a.u. These orbitals contribute
practically nothing to the description of intravalence transitions, they
‘merely add new transitions (i.e., Rydberg transitions).to the spectrum.
Indeed, the lowest energy n*u* configuration that can be formed from the
i SCF orbitals of basis set II is the one involVing the promotion 2b2*3bl,
While the orbital.energy diagram may lead one to believe that it would be
2bg+2bl.

Although the splitting of the basis set lowers the energy it
does not insure a more correct wavefunction overall; indeed, the dipole
moment of formaldehyde which was obtained using the ground state SCF
wavefunction of basis set I is 3.h6 D in rather poor agreement with the
experimental.value of 2.3M D50. The dipole moment obtained using a
fixed group basis is actually better at 2.93 D, although as mentioned

previously the total energy is 0.1 a.u. higher than that of the split

group basis.
The addition of p functions on hydrogen.and d functions on

carbon and oXygen generally improves both the energy and dipole moment.

A SCF calculation was perfbrmed on the ground state of formaldehyde using
a basis set that had a one component p o and p H basis function on each of
the hydrogen atoms, and five one-component d functions on carbon and on
oxygen. The numerical.values of the exponents and lobe separations of
these additional functions are shown in Table 7. The SCF results are
shown in Table 8. The wavefunction which resulted from this calculation
yielded a dipole moment of 2.95 D, a considerable improvement over that
of the split basis set.

Thus, as expected, the addition of basis orbitals with higher
azimuthal quantum.nnmbers gave rise to a considerable improvement of the
formaldehyde wavefunction. Such functions are also expected to enable
more accurate transition moments to be calculated. In the CI calculations
on excited states presented here, such additional p- and d-functions were
not used, and Rydberg orbitals were not fully optimized. Therefore,
calculated transition moments (and oscillator strengths) of many of the
transitions are not clearly reliable. This difficulty could be overcome
in future calculations, however.

The reliability of the approximations which were made in the
formulation of the CI method, namely: the fixing of a core, the truncation
or contraction of the set of virtual orbitals, the threshold criterion
fbr limiting the number of configurations, and the successive diagonaliza-
tion approximation will be examined numerically based on the results of

calculations that are presented in this section. Prior to presenting the

72

Table 7. Additional basis functions of formaldehyde. Exponents and lobe
separations are shown for one-component d functions centered
on carbon and oxygen atoms, and one-component pn and p0 functions
centered on the hydrogen atoms. Lobe separations, si, of the
p functions are related to the exponents, a1, by the formula

Bi = 0.03/(ai)§. The corresponding formula for the d functions
is 31 = (0.03/ai)§.

Basis function Exponent Lobe separation
carbon d 0.3 0.3162
oxygen d 0.5 0.2uu9
hydrogen.pn’ 1.8 0.022h

hydrogen po 1.0 0.0300

73

Table 8. Self consistent field molecular orbital energies for the
ground state of formaldehyde, using a basis set of thirty six
functions, including five one component d functions on carbon
and on oxygen, and a one component pa and pn function on each

hydrogen.
Molecular Molecular
orbital orbital
symmetry energy
3 h.16
7:: h.01
16a1 2.87
8b2 2.82
1531 2.52
7102 2.01:(
6bl 1.89
lhal 1.89
6b2 1.71
138.1 1.66
2a2 1.50
12al l.hg
1.2
“ii”: 1.01
hbl 0.92
13:: 0.32
O. 2
la2 0.73
3&1 0072
ha 0.69
8a1 0.59
3131 0 . 50
79.1 0.h8
3oz 0.h3
68.1 o. 30
2bl 0.12
2132 -0 . 141+
lb]. ‘00 53
Sal -0.6h
1b2 -0.69
h‘fi -0086
3a1 -1.h0
2‘]- ”no 35
la:L ~20. 58

Total energy: -113.8577 a.u.

1'9!

.v..‘

“if

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395';

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bone

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71+
CI results, the results and implications of a simple SCF calculation of the
3A2 state of formaldehyde will be discussed. This calculation was performed
as a test of the fixed core, and virtual truncation approximations.

The ability of a calculation which is performed within the frame-
work of a fixed core and truncated virtual set to reproduce the full CI
result is not practicable to gauge rigorously. The reason for this is that
the number of configurations which results from the full CI calculation is
too large to be manageable. Hewever, a question similar to the one
presented above, namely, the ability of a SCF calculation, which is performed
within the same framework as the CI calculation, (i.e., fixed core and
truncated virtual set), to achieve the full SCF result can be readily
answered.

In pursuit of an answer to this question, for the case of the
3A2 (n+n*) state of formaldehyde, a series of SCF calculations on the 3A2
state were performed. In each of these calculations, a fixed core
consisting of all ofothe doubly occupied molecular orbitals except the
lb

lbl (n) and 2b2 (n) was assumed, and the lb2, lb 2b and 2b

2’ 1’ 2’ l
orbitals were determined by linear transformations of the ground state

1b 1b and 2b orbitals plus specified numbers of higher b1 and b

2’ l 2
virtual orbitals.

2

The energies from this series of calculations are compared in
Table 9 with the m1 3A2 excited state scr limit.

These results exhibit two important features. First, there is a
significant lowering of the energy at each step up to the inclusion of hbl
and hb2 MD's. This result implies that much of the flexibility needed to

reproduce the excited state bl and b2 gO's is trapped in the higher ground

75

Table 9. A comparison of energies from constrained SCF calculations on
the 3A2 state of HQCO with single promotion and full SCF
energies. In the constrained SCF calculations, all a1 orbitals
are the same as produced by the ground state SCF treatment and
variable numbers of the ground state b1 and b2 orbitals are used
as symmetry orbitals.

 

 

Symmetry;orbital basis fig, energy a.u.,
(single promotion, no SCF) -lhh.7722
kbl, kb2 (k: 1, 2) -1uu,78u9
kbl, kbe (k = 1, 2, 3) -11+1+.798h
kbl, 1:132 (k = 1, 2, 3, h) -11m.8189

(flu—1 SCF) -iuuszuo

76
state virtuals and that the energy differences (given in Table 9) which
are due to truncation of the virtual spectrum.w0uld not necessarily be
recovered by CI. Examdnation of the SCF M0 coefficients reveals that the
n MD's are the ones most significantly altered.through mixing with the

higher virtuals; the 2b orbital is not changed appreciably even though

2
this orbital is now singly occupied in the 3A2 state. The second important
feature of the series of calculations is the documentation of the adequacy
(at least energetically) of a large core of unperturbed MD's, resulting

in an energy within 0.006 a.u. of the completely optimized SCF. It is

also interesting to note that the 3A2 tlAl transition energy determined

by a split basis SCF calculation on each state is 2.6 eV and, similarly,

2 wavefunction, the

calculated ;A2 +I;Al transition energy is 3.2 ev; thus, both transition

energies are lower than experimental values corresponding to the band

if the 3A2 SCF MD‘s are used to construct the 1A

origins of the transitions, 3.1 and 3.5 ev, respectively. This result
implies that the SCF technique using the present basis set affbrds a
somewhat better description energetically of both excited states than of
the ground state. Such a situation is reasonable in view of the smaller
correlation energy associated with the particular open shell configuration
of the A2 excited states; this interpretation is borne out by studies,
discussed in the next section, in which the energies of the A. excited

2

states are lowered less by CI (based on 3A SCF MO’s) than is,the energy

2
of the ground state fbr a CI treatment based on ground state SCF Mo’s.

c. CI calculations

 

Concerning the general feasibility of the procedure outlined

above in investigating the excited states of large molecules, it should

77
be mentioned that such constrained SCF calculations are not prohibitively

difficult and can be used to advantage, before starting the CI treatment,
to investigate the adequacy of a proposed core arrl the point at which to
truncate the virtual spectrum. It is also possible, of course, to base the
CI formulation on the molecular orbitals produced by such limited SCF
treatments. Since the major defect of the ground state Mo’s is in the virtual
spectrum, these SCF treatments could be limited to a transformation of the
virtual m's only.

Likewise, it should be noted that a transformation of the virtual
m's could be accomplished by single-excitation CI treatments or fixed-core
SCF treatments which could be performed advantageously before generation
of the full set of M) integrals needed in the fiml CI treatment. For
example, for the 311 states of H200, the set of ground—state virtual
orbitals Zbl, 3bl, and ubl, could be replaced by the transformed set,
Zbi', 3bi‘ , and hbi", determined by a CI calculation involving only the
single excitation configuratiors lbl 4 2bl, lbl. *' 3bl, and lbl -+ l#bl.
That is, if the resulting three CI wavefunctions, denoted by index k,

' orbital

have coefficients cnk for the configuration cp(nbl) (lb:L 4 nbl) the kb:L

is given by
kb' 93 c ( b ) k l 2

lfn=l nkcpnl’ _”3
Similarly, the bé and a1 MD‘s could be determined by separate CI calcula-
tions on configurations resulting from a specified choice of ground-state
orbitals from which single, excitations would be performed. While it is
true that the diffuseness of the lower virtuals would be reduced by such

transformations, and that initial convergence of CI expansions based on

- the new m's presumably would be enhanced, the orbital optimizations out-

78
lined are definitely limited in scope and are not clearly highly
advantageous. .A numerical investigation of the merits of the over—all
procedure is presented subsequently.

Configuration interaction treatments of the low-lying excited
states of H200, many of which have been observed experimentally and
have been qualitatively described as n'4 n*, n + “7, n'4 33, and n + 3p
excitations will.now be considered. The treatments are based in part on

the findings above concerning the 3A state, mainly, the adequacy of a

2
relatively large core of double occupied.MD‘s and the importance of

including essentially all ground state virtual b orbitals in the CI

1
formulation. Since these conclusions have not been established for the
other states of interest, a somewhat larger number of MD's is allowed to
have variable occupancy in the present CI treatments. In addition, in
certain of the calculations, the split basis set is augmented by the
inclusion of additional basis functions which.provide a representation

of 3p and 3s atomic orbitals on carbon and.oxygen. The basis orbitals for
H260 are listed in Table 3. This basis is somewhat minimal fOr the
investigation of spectral excitations and specifically does not include
drtype polarization functions. As mentioned.previously, such functions
are known to be important in the computation of ground state molecular

56 and are likely also important in the evaluation of transition

properties
moments.
Configuration interaction treatments based on ground state SCF
.MD's are presented for the three basis sets discussed previously: a split-
group basis, a split—group basis plus 3pn orbitals on carbon and oxygen
and a 3p nonbonding (b2) orbital on oxygen, and a split-group basis plus 33

orbitals on carbon and oxygen. The SCF results are given in Table 6 in

79
which those molecular orbitals which are allowed to have variable

occupancy in the CI calculations performed are indicated by an asterisk.
Ground. and excited state energies from CI treatments, performed in the
manner outlined in Chapter2 where the CI formulation is discussed in
detail are reported in Tables 10 and 11. The first step in these CI
treatments involves the selection of the principal single excitation
configurations (parents) from.which a relatively small number of
additional configurations is generated.by'perfOrmdng single and double
excitations, subject to the interaction threshold criterion. The re-
sulting lower energy CI wavefunctions (including up to four expansions
per state) are then treated as parents and additional configurations are
generated to form a basis for the final CI calculation. The lowering of
energies whidh occurs in the final CI step, as shown in the tables,
partly is due to the lowering of the interaction threshold and partly to
the inclusion of configurations which interact strongly with configurations
not present in the initial set of parents. In comparing the CI treatment
based on the split group basis set (Table 10) with the CI treatment based
on the split group plus 3p or 3s basis set (Table 11) it should be noted
that the former allows variable occupancy of the hal to 8al Mo‘s, which
is not the case in the latter treatment. This fact is responsible for
the lower total energies for comparable states obtained in the former
treatment, although the differences between ground and excited states are
approximately the same in both CI treatments. .Also given in Table 12 are
3A2 energies obtained from a CI treatment using 3A2 SCF MD's; these
energies are lower than the previous results at the single—excitation

and initial CI levels, but the final CI energy of the 3A2 state is only

0.008 a.u. lower than the result obtained using ground-state SCF Mo‘s.

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81

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82
For purposes of comparison with the above CI treatments,

elementary studies along the lines mentioned previously were conducted
to investigate the transformation of virthal orbitals by single-excitation
CI calculations to enhance the convergence of the CI expansions. Although
the technique could be applied specifically to each state of a given
symmetry and multiplicity, in the present work, an attempt was made to
obtain a single set of virtual orbitals to be used insofar as possible
for all Al and A2 states. In these states, it is the singly occupied
2b1(n) orbital which requires the most attention. The differences in the
optimum 2bl orbital (in the single configuration, virtual only trans-
formation sense) in the various states is investigated by performing single-
excitation CI calculations on the states, 1A2 (n-+n*), 3A2 (n-HT*) ,
1A1 (n-On*), and 3A1 (M*). The mixing of the single-excitation configura—
tions is reported in Table 12. Since all configurations for a given state
are produced by a single excitation from the same ground state orbital, the
CI result is precisely the same as would be obtained by a restricted SCF
calculation in which all occupied orbitals except the Zbl (n) are con-
strained to be the same as produced by the ground-state SCF. The results
shown in Table 1h indicate that the optimum 2b

1
in the 3A2, 1A2,and 3A1 states, but differ considerably from the optimum

orbitals are quite similar

Zbl orbital in the 1A1 excited state, the latter being rather similar to
the ground state virtual. Zbl orbital.

The final CI treatment of the lal states is based on the 2bl to
6b1 m's determined by the interaction of the 1A1 configurations, lbl 4 201,
.3b1, hbl, Sbl, and 6b1, . and on the 3b2 to 6b2 Mb's determined by the

interaction of the tAl conrtgurations, 2b2 4’3b2, hb2, 5b2, 6b2.

83

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8h

For the ¥A2, 3A2, and 3A1 states, the CI treatment is based on the 2b

to 6bl m's determined by, the interaction of the 1A

l

2 configurations

t
Egbl’ 3bl, hbl, Sbl, 6b1, and on the 3b2 to 6b2 M) 3. Since the new

orbitals are only linear transformations of the original Mo's, the CI

2b

treatment of course does not contain in principle any additional

capability over the former CI; thus, the only'possible advantage would be

a reduction in the number of configurations for comparable numerical
accuracy. The results of the CI treatment are reported in Table 13. A
comparison of these results with the results in Table ll shows the

expected considerable improvement in the energy at the single configuration
level, moderate improvement in the energy of most states for generally
fewer configurations at the initial CI step, and essentially no significant
change in the final CI results. Those differences in energy which do V
exist are most significant for those states in.which the 2bl orbital differs
most significantly from the ground-state virtual 2bl MO. For all states,
however, the agreement between the two CI calculations must be considered
quite close, particularly since the use of the same interaction threshold
might tend to favor the CI based on the transformed set of MD's; thus,
these results provide some numerical evidence to support the reliability of
the over-all CI procedure outlined previously.

In Table 1% the main contributions to the calculated ground.and
excited state CI'wavefunctions are reported, and in Table 15, calculated
transition energies are compared with experimental values. In certain
cases, calculated oscillator strengths and dipole moments are also reported;
however, as mentioned.prev10usly, these quantities are not expected to be

well described in view of the d-orbital deficiency of the basis set.

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86

Table lha. Major contributions to the ground and A1 and A2 excited state
CI wavefunctions for H200. Only those configurations with
coefficients greater than 0.09 in magnitude in the final CI
wavefunctions are included in the list; in the case of open
shell configurations, the coefficient listed is for the pair
of configurations 2Ԥ(qdn0; see text. Configurations are
described in terms of orbital promotions from the ground
configuration based on molecular orbitals numbered according
to increasing orbital energy as given in Table 6.

.A] states
configuration state and coefficient
1A1 1A1 1A1 3A1 3A1
ground 0.97
8 4 10 0.93 0.9a
7 4 9 0.61; 0.31
7 4 11 0.62 0.79
72 4 112 0.11 0.15
8 4 19 0.ll+
72 411, 12 0.10 0.10
72 4 9, 12 0.09
7 8 411, 10 0.09
7é 4 9, 1.1 0.13
5 4 10 0.12 0.12
7 4 12 0.25 0.147
7 4 16 0.16
7, 8 410, 12 0.10 0.09
7 4’21 0.11
8 4>lh 0.10
A2 states 1A2 1A2 3A2 3A2
8 4 11 0.76 0.1+0 0.76 0.1a
8 4 9 0.33 0.811 0.33 0.81:
8 4 12 0.1+2 0.11 0AM 0.10
8 4 16 0.11 0.11
7, 8 4 112 0.1M 0.12
7, 8 49, 12 0.09
5 4 11 0.1h 0.12

87

Table lub. Major contributions to the Bl and B2 excited state CI
wavefunctions for HQCO. Only those configurations with
coefficients greater than 0.09 in magnitude in the final CI
WEVefunctions are included in the list; in the case of open
shell configurations, the coefficient listed is for the pair
of configurations 2“2(qn40; see text. Configurations are
described in terms of orbital promotions from.the ground
configuration based on molecular orbitals numbered according
to increasing orbital energy as given in Table 6.

.§1 states
configurations state and coefficient
1131 1131

7 4 10 0.78
6 4 11 0.91;

7 4 9 0.56
6é 7 4 112 0.22

7 410, ll 0.15
6 4 1h 0.15

72 4 9, 11 0.1h
lB state

_2____..

8 4 10 0.85

8 4 9 0.39

7, 8 410, 11 0.12

S 4 10 0.11

72, 8 4 112, 10 0.09

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89
3. Comparison of Calculated and experimental results
Concerning the comparisons with experiment, four of the excited

states calculated.have been observed experimentally and haVe been classified

1 3
as A2, A2, 2 2v
1

The .A2 and 3A2 states are the ones most definitely established and these

and ¥Al'based.on the C point group of the ground state.
are known to have nonplanar equilibrium.geometries“3 qualitatively, the
states have been described as n+fi* excitations, involving a promotion from
the:nonbonding (b2) orbital of oxygen to the CO fl* orbital.» Transitions
to these states from the ;Al ground state are forbidden to first order,
but can occur in the ¥A2 case byra magnetic dipole or vibronic mechanism,

h1,6o,62

and in 3A case by spin-orbit mixing with excited singlet states.

2
In neither case does the available experimental information correspond
directly to the Franck-Condon transition energy calculated; however, for
both states the calculated transition energies (3.38 eV for 3A2+;Al and
3.80 for tAzttAl) are within 0.3 eV of the band origins of the transitions,'
and as such, the agreement with experiment must be considered excellent.

;A2 state is in moderate agreement

The calculated dipole moment of the
‘with experiment which is as satisfactory as can be expected from the
present basis set. The principal contributions to the wavefunction,
reported in Table 1% confirm the description of the 1’3A2 states as
involving n*fi*;promotions.

The next excited state fOr which experimental evidence is
available is the 1B2, where the calculated transition energy of 7.h8 eV
for the transition 132+;Al(ground) is within the reported experimental
range and differs by'only 0.h ev from the reported band origin. As seen
in Table 1%, this state involves mainly a n43s excitation on oxygen, and

thus the agreement with experiment is perhaps surprisingly good in view

90
of the simple description of the 3s atomic orbital. The exceedingly small
calculated oscillator strength is likely unreliable, however.*

The experimentally observed transition at 7.97 eV has been
described variously as a THIT* or a n+3p (Rydberg) transition to a 1A1 state.
{the lowest 1A1 state calculated is at 8.30 eV which would identify quite
satisfactorily with a state with a band origin at 7.97 eV. From the wave-
function reported in Table 15, this state is seen to be principally
1143pr on oxygen; the mixing with the rr-vn" configurations, although
substantial, is insufficient to produce a large value for the lAli-lAl
(ground) oscillator strength which is calculated as 0.03. The next
111 state calculated which is principally W* is quite high in energy
at 11.31 eV with an oscillator strength of 0.1+. Thus, the present
calculations on formaldehyde clearly favor the existence of a n43p, Al,
state at .o 8 all, but do not show a 11%;“ singlet state in this region of
the spectrum. The problem of the lack of a lfl'flT* state in the proper
region of the calculated spectrum is discussed further in Appendix C.

Although, there is no evidence which would suggest that the
above conclusions regarding the 1A1 states are not reliable, several
areas of uncertainty in the present treatment should be noted. The first
is obviously at the basis function level where, although a split-youp

basis plus carbon and oxygen 3p11 orbitals is used, the present basis might

 

* The final value of the J‘thlAl oscillator strength is the result of

many numerical cancellations, and thus is likely strongly dependent on

the precise form of the 3s A0.

91

be incapable of providing a sufficiently accurate representation of
molecular orbitals in the n%n* excited state. .A second area of un-
certainty would be the possibility of a CI deficiency which is peculiar
to the ln*u* state. Conceptually, for example, double excitation from
the 11 to the low-lying 11* orbital, "94182, is quite important
energetically in stabilizing the ground configuration. In the singly.w
excited "-011", 1A

1
if a similar type of correlation effect is important in the excited

state, suCh an excitation is no longer possible; thus,

state, the only orbitals to which excitation could be performed are the
higher p-type virtual components since d functions are not included in
the basis set. Finally, the possibility also exists that the excited
state has an equilibrium geometry which is sufficiently different from that
of the ground state to cause extension of the luau" transition (in Car)
down to the 8 cv region; however, significant intensity in this region
would likely also require a Francerondon excitation considerably lower
than the 11.31 eV'value calculated.

The remaining states for which the basis could be expected to
be reasonably adequate are the 3A1 and the 131 which are calculated at
5.66 and 9.35 ev, respectively. Unlike the lowest *Al excited state, the
lowest 3A1 state_is almost entirely n4fl*, as shown in Table 1h. .Also
shown in the table, is the 3131 composition which is mainly sen" as
opposed to "+38. The remaining states listed in Table 15 are not clearly
.meaningful since they are, to a large measure, only low-lying orthogonal
complements to the lower energy states of the same symmetry and
multiplicity. Likewise, at most, only the lower two or three wavefunctions
of a.particular symmetry'and.mu1tiplicity are listed in the table; for the
present basis sets, a.proliferation of additional states occurs starting

around 8 eV.

92

Cu Glygxal
l. firmwtal Spectrum of glyoxal

The nuclear coordinates and symmetry classification of glyoxal

molecular orbitals in its ground state geometry, C , are shown in

2h
Figure h.

The experimentally observed electronic absorption spectrum
of glyoxal is summarized in Table 16. The lowest transition,65’66
classified as 3A“ (n+n*)*tAg, occurs in the region 2.16 - 2.hl ev,
with a band origin at 2.38 ev; it consists of weak sharp bands. The

next lowest transition,6b"65’67,68

is described as the corresponding
1A“ (n'9n*)*lAg transition, and it occurs in the region 2.30 e 3.18 eV,
with a band origin at 2.72 cv. The band origin is found to be the band
of maximum.intensity. This transition also consists of a series of weak
sharp bands, although they are much stronger than those corresponding
to the lowest transition which is spin forbidden. The rotational
analysis of the singlet-singlet transition67 reveals that the geometry
of the-excited state is very similar to that of the ground state. The
next observed absorption system consists of diffuse bands in the
region 3.87 - 5.39 ev. No detailed analysis of these bands is reported.
There is a weak absorption system from 6.05 - 6.70 ev, a fairly strong
band at 7.08 cv, with a weak satellite at 7.12 ev, and another strong
band at 7.hh ev§9 There is also a very diffuse band at 7.7h ev. .At
higher energies there are two bands, one at 9.15 ev, and the other at
9.36 ev.69
0n the basis of qualitative molecular orbital arguments by
Walsh,69 the band system from 6.05 - 6.70 ev was attributed to a n*o*

transition and the band at 7.hh ev was attributed to either a n+n*

    

l
trans glyoxal

Nuclear coordinates: H1, 32 ($1.79, 0, 120142): C1: 02 (03 0’ Til-39),

01, 02 (41.97, 0, 12.53)

Nuclear repulsion new 103.3003 a.u.

\1/

H1 Hz

cis glyoxal

Molar coordinates: H1, H2 (-l.79, 0, '-'I-'2.h2), C1’ C2 (0, 0, #139),

01. 02 (1.97, 0. 12.53)

Nuclear repulsion enerfl: 105.3531 a.u.

Figure ha. Nuclear coordinates (in atomic units) and nuclear repulsion
energies of trans and cis glyoxal.

914

a8 orbital (shown as a linear combination of 8 functions)

+ +
+ + 0 - -_ 0
+ C C or + C C
O O
bu orbital (s functions)
4- +
- + o + - o
- C C or - C C
O 0
b8 orbital ( pn functions)
+ +
- + 0 + - 0
- C C or - C C
O O
au orbital (p11 functions)
+ +
+ + o - - o
+ C C or + C C
0 0

Figure hb. Symmetry orbitals of trans glyoxal (C2h) expressed as linear
combinations of functions on the carbon and oxygen atoms . The
plus and minus signs denote the signs of the coefficients of
the functions in the symmetry orbitals. The positive lobes of
the pfl’ funtions are above the plane of the paper.

95
a1 orbital (shown as a linear combination of 3 functions)

+ + + +
0 + + 0 or 0 - - 0
C C C C
be orbital (3 functions)
+ - + -
0 + - 0 or 0 - + 0
C C C C
a»; orbital (p1! functions)
+ - + -
0 + - 0 or 0 - + 0
C C C C
bl orbital (p'fl functions)
+ + 3 + +
0 + + 0 or 0 - - 0
C C - C C

Figure hc. Symmetry orbitals of cis glyoxal expressed as linear
combinations of s and p functions on the carbon and oxygen
atoms. The plus and the minus signs are the signs of the
coefficients of the functions in the symmetry orbitals. The
positive lobes of the pfl functions are above the plane of the

paper .

96

Table 16. The experimental electronic absorption spectrum of glyoxal.
Energies are expressed in electron volts.

State Rangea‘ Banda Maximum References
origin
, 9.15-9.36 9.15 69
7.00—8.h3 7.07 69
6.05-6.70 6.07 69
3.87-5a39 3.8h 68
¥Au(n+n*) 2.30-3.18 2.72 2.72 6h,65,67,68
hum-m") 2.16-2.1+l 2.38 ' 65,66,

a. Reference 32

V‘ 1'
.-

l. P v F.” Wlw . U I: .“I5 ”do
9 7. v o b I in s “V. I’mb
:0 J I

53.0

.”1

 

 

97

transition or a transition to a Rydberg state. The bands at 9.15 eV and
9.36 ev were attributed to Rydberg transitions.69

The only states which appear to be well characterized in the
electronic spectrum are the ¥Au (n4n*) and the 3A“ (n4n*) states. The
electronic description of each of the other states (e.g., uen*, and
n*o*) is speculative. In the case of n+n* transitions, it has also been
assumed in previous work52 that the two existing non-bonding molecular
orbitals (6bu, 7ab) are degenerate, and that consequently the two n+fi*

l l
transitions, A.u (7ag42au) and B8 (6bu42au) are degenerate.

2. Glyoxal calculations

 

The same split group basis set used for formaldehyde was also
used for glyoxal; SCF calculations were performed on the ground state for
both the cis and the trans geometries. The SCF results given in Table 17
show that the energy of the trans configuration is lower than the cis by
.0115 a.u.

All CI spectra calculations were performed assuming that the
molecule was in the trans geometry in each state. Configurations were
constructed either directly from the ground state SCF molecular orbitals
or in certain cases by first transforming virtual orbitals by fixed core
excited state SCF treatments of the type described in Section II.A.3.

The CI calculations were performed for three different sets of
:molecular orbitals which.were allowed to have variable occupancy. The
first set consisted of the ground state lau(n), lbg(fl), 6bu(n), 7ag(n),
2au(fl) 3au(n), hau(n), 2bg(fl), 3bg(fl), and hbg(fl) SCF.molecu1ar orbitals.
In the calculations involving this set, only the description of the

lowest energy state of each symmetry and multiplicity was sought. The

98

Table 17. Ground state self consistent field molecular orbital energies
and total energies of cis and trans glyoxal, expressed in
atomic units.

 

 

Trans glyoxal Cis glyoxal
Mblecular orbital Orbital Molecular
orbital energy energy orbital
(0211) (e...)
no“ .9867 1.0183 hag
the .9232 .9299 hbl
3bg .5290 .5333 3a2
3au .h650 .u636 3bl
9s; .h206 .ho2u 9al
Bag .3762 .3807 8b2
8bu .31h9 .275h 8al
Wm .mmv .2n0 We
2bg(n*) .1883 .2017 2s2(n*)
zeu(n*) .0365 .031h 2bl(n#)
7ag(n+) -.hhh6 -.hhhl 7al(n+)
Ibg(n) -.5220 -.5119 6b2(n_)
6bu(n_) -.5229 -.5137 1a2(n)
la,“ 11) - .6014-1 " . 5991 1b]. (11)
6a.8 -.6u61 -.6h91 5b2
5b.. - . 6997 - .6730 6al
Bag -.7312 -.77hl 5&1
hhu -.8u12 -.8057 hbz
has -.9810 -.9926 hsl
3bu -l.hh97 -1.h3u9 3b2
3s.g -1.h710 -1.h733 3sl
2tu -l_'L. 3735 -11. 3751 21.2
2cg -11.37h3 -11.3758 2a1
1bu -20.5995 -20.5988 1sl

Total.molecular energy of trans glyoxal: -226.h893 a.u.
Total.molecular energy of cis glyoxal: -226.h778 a.u.

99

calculated ground and excited state energies are shown in Table 18.
The second set of molecular orbitals included the first set,

and in addition contained the ground state orbitals 5bu, 7b 8b“, 6ag,

u’
Bag, and 9ag. .Again, in the calculations involving this set, only a
description of the lowest state of each symmetry and multiplicity was
sought. The calculated ground and excited state energies are shown in
Table 19.

The third set of'molecular orbitals investigated included the.
first set, and also two additional orbitals which are classified as 3pag
and 3pbu. To construct these latter molecular orbitals, two 3p basis
functions were used, one on each oxygen stony each in the plane of the
nuclei and perpendicular to a C0 axis. The exponent and lobe separation
of the 3p functions were the same as those used for formaldehyde 3p
orbitals. In the calculations using this set of molecular orbitals, a
description of several higher energy states in the spectrum was sought.

The calculated ground and excited state energies and wavefunctions are
shown in Tables 20 and 21.

Several additional calculations were performed on states involving
3p orbitals on oxygen and also 38 orbitals on oxygen while orbital para-
meters were chosen to be the same as used for formaldehyde. It is reason-
able to assume that the nflBs and ndap states in glyoxal would appear
approximately in the same regions of the spectrum as they did in formalde-
hyde. Indeed a lBu state has been reported in the region 7.00 - 8.h3 ev
above the ground state. Such a state can be achieved by'a u4u* promotion,

a promotion 7ag-'(3sl-’382), or by the promotion 7ag+(3p1+3p2) where the latter
orbitals are the in plane 3p orbitals defined above. Fixed-core SCF

calculations, in which all molecular orbitals were identical to those of

100

Table 18. Total electronic energies of glyoxal ground and excited states
based on ten variably occupied ground state SCF molecular
orbitals. Total energies are in atomic units, transition energies
are in electron volts. The interaction threshold,6, is shown at
the top of the appropriate columns, and the total number of
configuration pairs for each calculation is shown in parentheses.
Only the lowest energy state of each symmetry and multiplicity was

sought.
State and Single Initial Final Spectral Type
orbital configuration step CI CI energy of
promotion energy energy energy orbital
(0.002) (5x10-5) promotion
1A8 —329.7896 -329.8h82 -329.8727 ground
ground (10) (53)
1.541 -329.6222 -329.7ou8 -329. mu 3.57 n-m-X-
7ag-vaau (27) (198)
113 329.51% ' -329.6619 -329.6985 'h.7h n-onx-
6bu-ogau (27) (237)
‘ lsu -929. 3961 -329.M+56 -329.h770 10.7 mm
l‘bg-02au (6) (97)
3% -329.61+15 -329.7255 -329.7590 3.10 n+rr*
728422“ (28) (201)
3B - 329 . 5615 -329 . 621m -329. 7115 1+ . 31 moms
6bu~gau . (27) (232)
3A8 - 329.5038 -329.6280 ~329.61+99 6.06 m
lau*2au (1h) (69)
BBu -329. 58th - 329.6820 -329. 7012 h.67 mm
lbg-OZau (12) (68)

no

101

Table 19. Total electronic energies of glyoxal ground and excited states

based on sixteen variably occupied ground state SCF molecular
orbitals. Total energies are in atomic units, transition energies
are in electron volts. The interaction threshold, 6, is shown at
the top of the appropriate columns, and the total number of
configuration pairs for each calculation is shown in parentheses.
Only the lowest energy state of each symmetry and multiplicity

was sought.
State and Single
orbital configuration
promotion energy
ground
1
-329.6222
7.2.
l
B -329.5hh6
ebueéku
lBu -329.3961
lbg*2au
3Au -329.6u15
7ag*2&u
33 -329.56h5
ebuafiiu
3A -329.5038
lauvghu
33a -329.58hh

1b8*2au

Initial
step CI
energy

(.0006)

-329.8639
(25)

-329.7157
(#1)

-329.6616
()1)

-329.h690
(2h)

-329.735h
(#2)

-329.6812
(#2)

-329.6338
(25)

-329.6869

(20)

Final
CI

energy

<5x10‘5)

-329.8786
(flu)

-329.7h8u
(258)

-329.7010
(237)

-329.h959
(205)

-329.7657
(267)

-329.7165
(273)

-329.6 2k
(150?

-329.7oh3
(1%)

Spectral Type

energy

3.5M

n.8h

10.h

3.07

h.h1

6.16

of
orbital
promotion

ground

n-HT*

n4n*

nwn*

n4fl*

3>l‘

3;

Table 203.

State and Single Initial Final Spectral Type
orbital configuration step CI CI energy of
promotion energy energy energy orbital
( . 005) (. 0002) promotion
¥Ag (h: 58) (h, 301)
double -329.h175 -329.5700 -329.6323 6.52 nan-n"?
excitation
double _. -329.3926 -329.5o73 -329.5331 9-20 (n . n2 "
excitation ("£2 "E
7ag+3pag -329 . 3731 -329 . 11,618 -329 . 5097 9 - 85 1143p».g
1A.. (2, 15) (2, at»)
732mm ~329-6226 -329.6868 -329.7397 3.59 n-m*
6bu-2bg -329.3572 -329.1+373 42951125 8.96 n-m*
6mm ~329.51+51 -329.6375 -329.6935 h.85 n+n*
738%,; -329.hh23 -329.5107 -329.5810 7.91 - mm"
In. (2, 8) <2. 186)
7‘s*3Pbu -329.l+127 -329.l+600 -329.5377 9.09 n-’3pbu
lbg‘hu -329.3967 -329.h312 -329.h789 10.7 W

102

Total electronic energies of glyoxal ground and excited singlet
states based on twelve variably occupied molecular orbitals, ten
of which are from the ground state SCF, and the other two are
constructed from 3p basis functions on the oxygen atoms. Total
energies are in atomic units, transition energies are in electron
volts. The interaction threshold,5, is shown at the top of the
appropriate columns, and the number of parents and the total
number of configuration pairs for-each calculation are shown in
parentheses. Several states of each symmetry and multiplicity
were sought.

Table 20b.

103

Total electronic energies of’glyoxal calculated excited triplet
states based on twelve variably occupied molecular orbitals, ten
of which are from the ground state SCF, and the other two are
constructed from 3p basis functions on the oxygens. Total energies
are in atomic units, transition energies are in electron volts.
The interaction threshold, 6, is shown at the top of the
appropriate columns, and the number of parents and the total
number of configuration pairs for each calculation are shown in
parentheses. Several states of each symmetry and multiplicity

were sought. '

State and Single Initial Final Spectral Type
orbital configuration step CI CI energy of
promotion energy energy energy orbital
(.005) (.0002) promotion
3A8 ()4, 55) (3, 397)
iauaeau -329.50h2 -329.599h -329.6h76 6.10 n~n*
double -329.3926 -329.u772 -329.5062 9.91 $111, 3%?»
excitation Hi, 4
lbg42bg -329.3916 -329.h205 -329.5033 10.0 n+w*
7age3pag ~329.379o ~329.367o
31.1 (2, 13) (2, 213)
7aé*2au -329.6u20 -329.6959 -329.7573 3.11 n+n*
6131,3213g -329.3722 -329.uuuu -329.550h 8.73 n*n*
3B“ (3, 21) (3’ 290)
1b3422u -329.5850 -329.65u2 -329.6996 h.68 fi+n*
double . -329.5100 -329.5662 -329.620h 6.8h
excitation
7ag43pbu —329.h167 -329.h630 -329.5u10 9.00 naspnu
338 (2. 1h) (2, 198)
613,929.u -329,5650 ~329.65uh -329.7098 h.h1 n4n*

7ag42bg -329.h5u3 -329.5135 -329.5871 7.7h n4n*

101+

Table 21a. CI wavefunctions of the.A electronic states of trans glyoxal
obtained in the calculati ns corresponding to Table 20. Only
those configurations with coefficients whose absolute values
are greater than 0.07 are listed. Orbitals 12 to 23 are the
6bu: lbg’ 73g: 2311: 213g: 33119 3bg: 1mg, nan, 3Paiz, and 3Pbu
orbitals respectively. The plus and minus signs denote a and B
spins respectively.

Configuration Absolute values of Coefficients
of configurations in states
1 ..
Ag 1Ag 518 3.1%
ground 0.96
(1h)2+(16)2 0.16
(+12, +lh)-9(+16, +17) 0.12
(+12 ~11+ +(-16, +17) 0.10
12)é-0(16 2 0.09
(15)2"(16)2 0.8a
(13)?” 16 2 . 0112
(192-9 17 2 0.15
11+, 152)4(162, 17) 0.12
(+13, -15)-r(+16, -17) 0.09

(-13, +15)-b(+16, 47;
(+13, +ls)-b(+l6, +17
(+12, +13, -15)+(162, +17)
(+12, -13, -15)"(l62, -17)
(+12, -13, +15)-o(162, +17)
(+13, -15)-'(+16, -19)

~13, +1sg-v(+16, 49;

+1 , +15 4( +16, +19

(13 11+)» 162, 17)
(Bf-K17)

15 2”(16. 21)

(15%(16, 18)

12"].6

1%" 2
12 , 1h 4 16 17
1M1 ) ( . )

c>c>c>c>§>c>c>c>c>
888 El: 1:111:11??-

9990
000
moot:

0
mm

9
(12, 1h2)-+(16, 172)
11920

12*18
12421
(+12, -1u)+(+16, -17)

090090090

OOOOI—‘L-l‘i-‘UIQ

«ICIHOKOC

105

Table 21b. CI wavefunctions of the B and Au states of trans glyoxal
obtained in the calculations corresponding to Table 20. Only
the configurations with coefficients that have absolute values
greater than or equal to 0.07 are listed. Orbitals 12 to 23 are
the Ian, 6131,, lb , 7a , 2%, 2bg, 334,, 3b8, hbg, ha“, 3pa , and
3pbu orbitals regpectgvely. The plus and minus signs deno e 01 and
B spins respectively.

Configurations Absolute values of coefficients
of configurations in states

138 132 3B8 333
13416 0.83 0.13 0.83 0.19
15417 0.37 0.68 0.112 0.6M
1h, 15)-o(16)2 0.30 0.51 0.21 0.55
+12, ~15)+(+16, ~17) 0.09
~13, +1h +(~16, +17 0.08 0.12
(11+ , 15)-2(162, 17) 0.08 0.09
15-919 0.08 0.09 0.08 0.08
~12, +15)~-+(+16, ~17) 0.08 0.07
12, 13)-o(16)2 0.29 0.28
+13, +1h)-9(+l6, +17) 0.20 0.19
(1h, 15)-o(17)2 0.19 0.17
(+13, ~1h)-9(-16, +17) 0.1M 0.15
(122, 15)-o(162, 17) 0.09 0.09
(+12, +15)-0(+16, +17) 0.09 0.08
13021 0.08

1Au lAu 3 3%
15-016 0.91 0.91
13417 _ 0.21: 0.59 0.26 0.58
13, 1h)-v(16)2 0.22 0A2 0.17 0.12
~12, +1h, +15)-+(162, +17) 0.08 0.07
12 15)-0(16)2 0.1+9 0.1+7
(+1 , +15)+(+16, +17) 0.27 0.211
+1h, ~15)-o(+16, ~17) 0.20 0.23
13, 11+)»(17)2 0.19 0.18
~12, +13)-o(+16, ~17) 0.10 0.11
122, 13)-o(162, 17) 0.10 0.10
12, 15)-o(17)2 0.09 0.10
(+12, +13)-’(+16, +17) 0.08
~1l+, +15)-»(+16, ~17) 0.07 . 0.11;
15-21 0.07

 

__,.—~_-
._—

 

Table 21:

106

Table 21c. CI wavefunctions for the Bu electronic states of trans glyoxal
obtained in the calculations shown in Table 20. Only the
configurations with coefficients that have absolute values
greater than or equal to 0.07 are listed. Orbitals 12 to 23 are
the lau, 6bu, 18g, 7a , 2a“, 213%, 39.1, 3bg, hbg, hen, 31mg, and
3pbu orbitals respectgvely. The plus and minus signs denote a
and B spins respectively.

Configurations Absolute values of coefficients
. of configurations in states

13., 3Bu 3Bu 3Bu
15423 0.92 ' 0.93
1h-oi6 0.15 0.90
(+13, ~1h)-+(~16, +23) 0.15 0.17
(+13, +1h)~.(+16, +23 0.12 0.1h
(11+ , 15)+(162, 23) 0.12 0.11
122, 15 4(162, 23) 0.10 0.10
1322 0.08 0.08
12-017 0.33
12, 1124(16)? 0.13
12, 1h )+(162, 17) 0.12
(1h)2-»(16, 17) 0.09
114-21 0.09
12.19 0.09
(122 1104(16, 172) 0.08
11:41 0.07
(13, 15)» 16)2 0.93
13. 15) 17)"2 0.21
~13, +1h, +15)+(162, +17) 0.1h
(~13, ~1l+ +15)-o(162 ~17) 0.1h
+13, ~15§~(+16, ~21) 0-07
(~13, +15 +(+16, ~21 0-07

107
the ground state except the highest singlyboccupied orbital, were per-
formed on both the 1Bum-93p) and the lBu(n-'3s) states. The results are
given in Table 22.
3. Cgmparison of calculated results with experiment

It is seen from a comparison of Tables 17 and 18, that the six
non n'molecular orbitals which were added to the basis had very little
effect on the calculated energies of the lowest excited state of a given
symmetry and multiplicity. This was definitely not the case in fOrmalde-»
hyde, where calculations using only n virtual orbitals gave poor results.
This perhaps implies that as the number of centers which contribute
valence shell.pn orbitals to a conjugated n system increases the non n
virtual orbitals become less important in determining spectra in the region
of low transition energy.

The calculated results of Table 20 and the experimental results
are compared in Table 23. The calculated transition energies for the
3AufEAg and ;Auf}kg transitions are higher than those of experiment, but
not by more than one electron volt. The most important configuration
in each of these states is the one involving the excitation 7agfi2au(n4n*).
In an attempt to investigate a possible lowering of the transition
energies of these states, additional p-n basis functions were added to
oxygen. Separate SCF calculations were;performed using the;p-n'basis
functions having exponents 0.03, 0.1, and 0.7, respectively; none of
these calculations showed significant improvement over the one using the
original basis set.

The calculated transition energy of the lBg+tAg transition is
‘within the experimental range of the third electronic transition. The

:main configuration is the one involving the single excitation 6bufl2au.

108// ".
Table 22. Additional SCF results on glyoxal using 35 and 3p orbitals. In
each of the calculations, the energy is minimized by varying the

highest singly occupied orbital. All other orbitals are ground
state SCF orbitals .

State Transition energy

lBu(n"3Pbu) 9.8+

lBu (n'93 sbu) 10 . 32

110

Table 23. A comparison of the calculated and experimental transition
energies of glyoxal. Energies are expressed in electron volts.

Electronic
state

Transition
energy

9.20
9.09
9.00
8.96
8.73
7.91
7.71

' 6.81

6.52
6.10
1.85
1.68
1.11
3.59
3.11

0.00

Experimental
range

7.00-8.13*

3.87-5.39*

20 30'3018
2.16-2.11

Band
origin

7-07

3.8h

2.72

2.38

*The identity of this state has not been clearly established.

111

The present study shows that there is neither a degeneracy
of the non-bonding orbitals (6bu, and 7ag), nor of the two n-m*transitions,
in disagreement with the assumptions of Sidman.52

the calculated lBu(n-O3p)"lA g transitions appears to be below the
calculated lBu(n-’3s)¢-l.lig transition, which is opposite to what would be
expected on the basis of comparison with formaldehyde. The transition
energy of the calculated lBu(n"3p)"lA8 transition is slightly above the
upper limit of the range of the experimentally reported lfiufilAg transition.
Halvever, this transition was reported as a We!” on the basis of
qualitative molecular orbital arguments. The lsu state which is identified
with a m* promotion is calculated in this work at 10.1+ eV above the
ground state. The problem of the lack of a 111+an state in the proper
region of the calculated spectrum is similar to the situation encountered
in the study of formaldehyde and is discussed further in Appendix C.

The n43p and/or the n+3s transitions can be associated with the
bands at 9.15 eV-and 9.36 eV.

A number of forbidden transitions which are not observed
experimentally are reported in Table 20. The lowest non-observed state

is the 3

Be at 1.11 ev in which the main configuration involves the
promotion 6bu-02au. The next lowest calculated state is the 33., state
which occursat l+.68 eV. This state consists mainly of the configuration
involving the promotion lb 842a“. The next state, the 3118 state at 6.10 eV
is a mixture of the configuration involving the promotions 195.1% and
lbu‘02bu. Another 1A8 state occurs at 6.52 eV, and it consists mainly of
the doubly excited configurations involving the promotion (7a8)2-'(2au)2.
There is also a considerable contribution fran the (6bu)2"(28u)2

configuration to this state.

1.12
A 3Bu state occurs at 6.8% eV. This is a double excitation state

. 2
which mainly involves the configuration (6bu, 7a g)-9(2au) . The next state

is the 3138 at 7.7% eV which is mainly a miximre of the configurations 7ag-i2bg,
and (lbs, 7ag)-O(2au)2 the 1388 state at 7.91 eV is described similarly.

The 3Au state at 8.73 eV, and the 1% state at 8.96 all are described by the
three configurations 6hu-t2hg, (lau, 7mg)-Ieuu)2 and (6bu, lbg)-*(2au)2.

The 313u and 13“ states at 9.00 and 9.09 ev, respectively, are those
involving the n43p configuration (i.e., 7eg+3phu). The 5.18
involves the six configurations which are obtained from the promotion

(61.. vague-u. 21g).

at 9.20 eV

113
D- mes
. 1. Experimental ppectrum of jmzine

he pyrazine molecule, chnhnz, has been the subject of several
experimental and theoretical investigations 52’71.'92for a review of the
experimental work see Reference 32, page. 55133 and 660, (and Reference 70,
M9 13859. Table 2h smarizes the experimental spectral data. The
nuclear coordinates and symetry orbitals of pyrazine used in the present
study are shown in Figm'e 5 for the molecule in its ground state
equilibrium geonetry, Dal. '

The lowest experimentally observed transition‘of pyrazine con-
sists of a system of week sharp bands in the region 3.30 - 3.118 “.82-86
The systan is described as arising from a 333“ (n-011"")0:|-ii8 transition. An
observed phosphorescence spectrum also is thought to involve the 3133,;
(ii-m") state.

The next lowest observed transition consists of moderately

82,83 s87'89m3 transition

strong sharp bands in the region 3.76 - h.28 eV.
has been classified as involving the 33311 (M*)*1A 8 transition.
Hochstrasser and Marzzacco,9° and El-Sayved9l have also argued
in favor of the existence of a 3‘31u(fl-fl'r*) state beheen the 333“ and
:33“ states based on indirect evidence. Hochstrasser and Marzzacco are
led to their claim by the appearance of a perharbation of. part of the
observed 33311.}18 spectnm which “they attribute to the influence of a
3131‘1 (Tr-011*) state in the same region of the spectrum. El-Bayed asserts
that the 351“ (W*) state is involved in the mechanism that leads to
phosphorescence following excitation of the molecule to the 1‘83“ state.

the mechanism proposed is as follows:

11%

Table 2h. The experimental electronic absorption spectrum of pyrazine.
Energies are expressed in electron volts.

Electronic
transition

Rangea

(lseu, ialu)+» 7.53-7.90

‘As
lBZua-lAg

llamas
lBeu“;Ag
lB3u‘J'Ag

3
B3uf;lg

6.72-7.56

6.70-6.91

6.30-6.8h

h059-5006

3.76-n.28

3.30-3014‘8

a. Reference 32

Band“

Band

Oscillator

origin maximum strength

7.52

6.8h

6.69

6.30

n.80

3.83

3.32

h.80

3.83

1.0

0.25

0.1h5

0.10

0.01

References

92

02

82, 83,
87-89

82-86

115

 

Nuclear coordinates of DMZ—LIL: 01’ Ca, C3, Ch (0, :12.1272, $13020),
N1, N2 (0, o, £2.65h6),
H1, H2, H3, Hu (0, i3.8hoh, i2.303l)

Nuclear repulsion enerfl: 210.h1+07 a.u. 0

Figure 5a. Nuclear coordinates in atomic units and nuclear repulsion
energy of pyrazine.

a orbital (expressed as a linear

8
combination of 8 functions)

+ e-
+ + + +
or
+ + + +
+ -
b38 orbital (8 functions)

- +

+ -

blu orbital (8 functions)

01‘

b2u orbital (8 functions)

- +

b3u orbital (p11 functions)

01'

an orbital (p17 functions)
- +
+ -

b2 orbital (pn function)

8

01‘

b18 orbital (pn functions)

— +

- ‘ +

Figll‘I'e 5b. Symmetry orbitals of pyrazine expressed as linear combinations
of equivalent functions on the carbon and on the nitrogen
atoms. The plus and minus signs are the signs of the
cOefficients of the functions in the symmetry orbitals. The
positive lobes of the p11 functions are above the plane of

the paper.

118

 

 

intersystem internal phospho-
* crossing 3 * conversion 3 * rescence l
123“ (am > > on (ma ) _> B3u(n—m )____—> Ag<sround>

The above process is Justified by El-Sayed on the basis of the strong
spin orbit coupling which exists between the 1B311 and the 3Blu states.
Parkin and Innes92 have reported five additional band systems
in the vacuum.ultraviolet. A system was also found which consists of
fairly strong diffuse bands in the region h.59 - 5.06 ev.82 These bands

have been attributed to a 11 .. .3" transition.

2. Pyrazine calculations

 

Self-consistent-field and configuration interaction calculations
on.pyrazine were performed for two different choices of basis set.— The
first basis, called a fixed group basis, contained functions which were
capable essentially of representing only atomic Hartree-Fock orbitals for
the inner and valence shells. In the second basis, additional one term
long range pn functions were added to the nitrogen and carbon atoms.
Parameters for both basis sets are given in Table 25.

The results of the SCF calculations are shown in Table 26 for
both choices of basis set. SCF treatments were also performed on the exci-
ted“ states. In these calculations only the n* orbital was optimized,
all other molecular orbitals were constrained to be the same as in the
ground state. The results of these excited state SCF calculations are
shown in Table 27.-

In the CI calculations, a set of'molecular orbitals was chosen
which would be doubly occupied in each of the configurations generated.
These orbitals are shown in Table 25. I. For each basis set, certain
molecular orbitals were chosen which would be allowed to have variable

occupancy in forming configurations. These sets of orbitals contain the

119

Table 25. Pyrazine basis orbital parameters. Exponents, coefficients, and
lobe displacements for N, C, and H s- and p- type group
functions are shown in the rowsf

Nitrogen
pl 27.7513 7.h787 2.1519 0.6170 0.1758
1.0000 9.3502 62.hh8h 205.9778 218.8h91
0.0h8h 0.0535 0.061u 0.0696 0.0823
pg 0.1758
1.0000
0.0823
sl 218.5h99 1070.1762 3h89.5253
1.0000 0.13h5 0.0323
s2 0.2079 0.678% 7.0699
1.0000 1.1527 -0.1569
83 23.53h2 9.66h8 3.h597 57.2879
1.0000 1.86uh 1.07u1 0.6305
Carbon
p1 19.0660 5.0720 1.h305 0.h1h2 0.1216
1.0000 9.2113 62.7606 208.058h 216.6769
0.0591 0.06h7 0.0750 0.08h6 0.0990
p2 0.1216
1.0000
0.0990
s1 159.627h 781.6h95 25u8.7256
1.0000 0.13hh 0.032h
s2 0.1h80 0.h735 h493hh
1.0000 1.1h12 -0.1585
83 17.1893 7.0591 2.5269 1181427
1.0000 1.8651 1.1090 0.6308
Hdenogen
a 2. 2956 0.6517 0.2059 6h.7869 9. 9195
1.0000 1.8971 0.0315 0.2283

2.h751

*The basis orbitals p2 on carbon and nitrogen were used only in the
second basis set.

120

Table 26. Ground state SCF molecular orbital energies of pyrazine for
each basis set? Energies are in atomic units.

Orbital

Total energy

Basis set I*

.0856’
.0788'

-.h399

-.hh68’

-.h8l8
-.5h07
-.6016
-.6338
-.6862
-.7392
-.7h63
-.7885
-.9322
-.957M
-l.1216
-1.2789
-1.3712
-ll.hh18
-11.l+1+18
-11.l+h29
-ll.hh30
-15.68h8
-15.68h8

~262.25h7

 

Orbital energies

Basis set II *

.3806

.2279
-.0309

.5096
-.1865
-.2528
-.h353
-.h379
-.h770
—.5329
-.5921
-.6367
-.6761
-.730u

-.7357.

-.7796
-.9228
-.9h72
-l.lllO
-l.2698
-1.3615
-ll.h206
-11.h206
-11.h2l7
-11.h218
-15.67hh
-15.67hh

262.2681

*For basis set I the lowest energy fifteen orbitals were included in the

3core.

1for basis set II the lowest fifteen and the seventeenth orbital were
included in the core. The orbital energies of orbital 2b3u and higher

(except the 2b

orbital) were determined in the fixed core SCF

calculations in which these orbitals were obtained. The 2b

orbitals were obtained in the 1(5b1u42b

in the 1(5b1u+

) son, the 1au an

orbital in the ground state SCF calculation.

3‘12

and 3b
au orb

a. The calculations using basis set I are taken from.Reference 80.

itals

) SCF, the 2b2g in the (Ib2g42b2g) SCF, and the 2blg

121

Table 27a. Comparison of single configuration transition energies of
pyrazine. Energies are in electron volts.

Transition energies

 

aw

.J

Electronic
state Single Single Single
orbital configurations configurations configurations
promotion (fixed gro (extended (fixed-core
basis set) basis set) SCF's)c
3133u 6ag 4 2b3u use 1+.l+2 11.112
3131u 1blg 4 1au 5.19 5.01 n.96
1b2g 4 2b3UL 5.1+7 5.20 5.20
3132u lbig 4 2b3u 6.15 5.60 5.51
1b2g 4 1au 7.1+3 7.07 7.06
3Au 6ag 4 lau 6.93 6.1+0 6.1+0
3323 '5qu 4 2b311 6.91 6.56 6.56
31313 5qu + lau 9.83 9.27 9.27
1133u 6ag 4 2b3u 5.70 5.22 5.22
1A“ 6ng 4 1n.u 7.06 6.52 6.52
132s 5qu 4 2b311 (8.03 7.62 7.62*
1
lb 4 2b3u 8.80 8.17 80+
B2“ 1b: 4 lau 10.0 9.11% 9.1a
1
B111 lblg 4 9.79 9.15 9.11
lb28 4 2b3u 11.1 10.3 10.1
Big 5qu 4 lau 10.0 9112 9.u2*

xcited configurations were obtained by virtual orbital.promotion from

the ground state configuration from the fixed group basis set SCF.

shown in Table 26.

bExcited configurations were Obtained by single promotions from the
extended basis set ground state configuration to the unoccupied orbitals

cExcited configurations were obtained by fixed core SCF calculations.

*This orbital was subsequently allowed to have variable occupancy in the

CI calculations.

122

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ommo. memm. eHmm. mer. owns. mace. cams. moHe. Home.

cFE?z?E?

 

 

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0666. 6000. 6600. 6000. oooo. 0060. 6060. oooo. 6060. Hz
ammm. eemH. memm. more. mmsm. Home. came. mmmm. mHom. Ho
3.. H5” HH 5 om .5 ma w."
o.o mm aH 4w aw mm mH am aH
eoesoHesH eoeone one so e.aom acne eooHoeoo aneaono as“ ssH
aemm. Form. ‘ ommm. moon. mmwm. comm. H:0m. room. mean. oz
mmem. mmam. manm. boom. mmmm. :Hom. Hmom. Hmmm. mmmm. no
eHom. more. amen. mmnH. coon. emmH. team. mmoH. mmrH. Hz
mmmH. mmeo. rooH. mono. ammo. Homo. meoH. ero. more. Ho

5N SN 5m 5m .5. 5H mm mm
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$3330.70 mom open.» oopaoxo cod canon» 5C nganho : codename. no 83.3930 AFN 0.34.9
*

123
higher occupied ground state orbitals and also orbitals constructed from
the ground state virtual orbitals. For the calculations using the first
basis set, there occurred only one unoccupied n orbital of b3u symmetry,
one of au symmetry and one of b2g symmetry. These orbitals along with
the 7ag were used without modification in the CI calculations.

For the second basis set there are nine n virtual orbitals from
the ground state SCF treatment. Three of these orbitals are of b3u
symmetry, three of b2g symmetry, two of an symmetry and one of blg symmetry.
Fixed core SCF calculations were perfbrmed on excited states in which the
n* orbitals of b3u, an, andb28 symmetry were optimized.

Table 27 shows the energy difference between the single promotion
configurations and the ground state configuration resulting from the
calculations uSing the first basis set,and those resulting from the second
basis set with and without the use of the fixed core SCF modification.

The coefficients of the basis functions in the optimized orbitals are also
shown in this table.

From the set of b3u’ an, and beg orbitals which resulted from
these calculations, a set of orbitals was chosen which would be allowed to
have variable occupancy in each of the CI configurations. This set con-

sisted of the two lowest energy b orbitals from the 1(5b1n32b3u) fixed-

3u
core SCF calculation, the two au orbitals from the 1(5blfiflafi)fixed-core
scr calculation, and the ~-. lowest energy b2g orbital from the 3(lb28%28)
fixed core SCF calculation. The b3". and au orbitals which were chosen are
quite similar to the b3n and au orbitals obtained in the other fixed core
SCF calculations. Examination of the lowest energy configurations of each
symmetry showed that none involved.promotion to the 2b28 orbital. There—

fore, a 2b2g orbital was sought which would.be effective in increasing the

12%
matrix elements between the most important configurations and those
which differed from them.by promotion to 2b2g orbital. This condition
would be satisfied if the 2b28 orbital had appreciable value in the
same regions of space as the ground state occupied orbitals from.which
excitations are performed. Since the occupied orbitals are generally
confined to relatively small regions of space, the 2b2g orbital that
was chosen was the one whidh was expected to have appreciable value
in these same regions of space, i.e., the one which had the largest ratio
of the coefficients of the short range pn functions centered on the
carbon and nitrogen atoms to the long range pn functions centered on the
same atoms. The fixed—core SCF calculation.which produced this orbital was
the triplet (lb2g42b2g) SCF calculation.

CI calculations were then performed in which the following
molecular orbitals were allowed to have variable occupancy in forming
configurations: the three occupied fl and.the two non-bonding orbitals
from the ground state scr treatment, the blg virtual orbital from the
ground state SCF treatment, plus two b3u orbitals, two an, and ab2g
orbital selected in the manner described above. Energies of the various
excited states obtained from the CI treatments are Shown in Tables 28 and
29 fer the first and second basis sets respectively; The wavefunctions
obtained using the expanded basis set are shown in Table 30 for the
low-lying states. Several additional calculations were performed in
‘which the lowest state of each of the symmetries ;Ag’ 1B3“, 3B3u, and
lhgu were investigated by accurate diagonalization of large matrices
instead of using the stepwise diagonalization procedure discussed in

section II.C.2. Calculated energies are shown in Table 31.

125

Table 28a. Energies of pyrazine ground and excited singlet states based
on ground state SCF molecular orbitals and a fixed-group basis
set. The orbitals allowed to have variable occupancy in forming
excited configurations are described in the text. The
interaction threshold, 6, and the number of parents and the
total number of configuration pairs (in parentheses) are shown.
Total energies are in atomic units, transition energies are in
electron.volts.

Electronic Single Initial Final CI Transi- Desc-
state and configuration step CI energies tion ription
orbital energies energies 6:0.00005 energies
promotion 5 =0 . OOl
1A8 (l, 17) (l, 77)
ground -h72.6953 -h72.7772 -h72.7897
133“ (1, 27) (1, 170)
6a8 *’2b3u -h72.h856 —h72.596o -u72.621h h.58 n+n*
1B2u (2. 30) (2, 132)
lblg‘+ 2b3u -h72.3720 -h72.5761 -h72.5878 5.h9 in+fi*
Ib28‘+ lau -h72.3259 -h72.3783 -h72.h011 10.57 quintet
tau (1, 29) (1, 166)
6ag *’1au -h72.h359 -h72.563h -h72.585o 5.57 n4n*
1328 (1, 3o) (1, lh3)
5b1u‘* 2b3u -h72.h000 -h72.53h3 -h72.5699 5.98 n+n*
1313 (l, 32) (1, 169)
5‘01,1 -' 1a“ -l+72.327h -h72.1+8h8 41.72.5180 7.39 n-m'"
lslu (2, 32) (2, 17h)
llLblg 4»1au -h72.3356 -u72.u172 -h72.h256 9.91 nan*
'l-sz +2193u —h72.2876 -h72.3h27 -h72.3636 11.59 n+fi*

126

Table 28b. Energies of pyrazino calculated excited triplet states based

on ground state SCF molecular orbitals and a fixed-group basis
set. The orbitals allowed to have variable occupancy in forming

excited configurations are described in the text. The

interaction threshold, 6, and the number of parents and the

total number of configuration pairs (in parentheses) are shown.
Total energies are in atomic units, transition energies are in
electron volts.

Electronic
state and
orbital
promotion

333u
6ag 4 2b3u
3Blu
1'blg"' lau
l'b28 4’2b3u
3B28
S'blu 4 2b3u
3A..
6ag 4’1au
3B2u
11:18 4 21:3u
l'bgg 4'lau

3318

5bluau“

Single

configuration

energies

-h72.5l7h

—u72.50h7
-h72.h9h1

-h72.hhl2

-h72.hh06

-h72.h69h

-h72.h222

-h72.3338

Initial
step CI
energies
6:0.001

(1, 25)
-h72.62h9
(2, 3o)
-h72.628h
-h72.5721
(1, 27)
-h72.56h7
(l, 28)
-h72.5636
(2, 27)
-h72.562h
-h72.hh18
(l, 32)
-h72.h8h3

Final CI
energies
5:0.00005
(l, 167)
-h72.6h85
(2, 131)
-h72.6361
-h72.5827
(l, lh3)
-h72.5975
(l, 167)
-h72.5877
(2, 156)
-h72.5823
-h72.h532
(1, 172)
-h72.52ho

Transi-

tion

energies

3.8%

n.18
5.63

5.23
5.50

5.65
9.16

7.23

Desc-
ription

n4n*

n4fi*

33

n*n

l2?

Table 29a. Energies of pyrazine ground and excited singlet states based
on molecular orbitals constructed from.the split prrbasis set.
The molecular orbitals which are allowed to have variable
occupancy in forming excited configurations are described in
the text. The interaction threshold, 6, the number of parents,
and the total number of configuration pairs are shown in
parentheses. The wavefunctions corresponding to these states
are shown in Table 30. Total electronic energies are in atomic
units, and transition energies are in electron volts.

Electronic Single Initial Final CI Transition
state configuration step CI energies energies
energies energies (0.0001)

(0.003)
1A3 . (3, M4') (39 255)
—h72.7088 -h72.7800 -h72.7986
-h72.2982 -h72.h778 _u72.5290 7.33
-u72.2h6h -h72.h28h* -u72.u57u* 9.28
133u (2, ks) (2, 216)
-h72.5168 -h72.6l61 -h72.6h33 h.22
-h72.2398 ~h72.h256* -u72.u5u1* 9.37
1Au (1, 3o) (1, 170)
-h72.u692 -h72.5726 -h72.6067 5.22
1328 (h, 58) (3, 2&3)
~u72.h288 -h72.56l2 —h72.5907 5.65
-h72.3l58 -u72.u707* -h72.h986* 8.16
-h72.27hh -h72.h620 -h72.h872 8.h7
-h72.2717 -u72.366h
lBZu (3, 57) (3, 265)
-h72.h082 _u72.57u2 ~u72.60h0* 5.29
-h72.36l6 -h72.hh5h* -h72.h650 9.07
-h72.3286 -h72.h096 -h72.h327 9.95
13
1 (2 26) (1 95)
u .-h72.372h -h§2.hh73 -h72.h639 9.10
1B1. (2, hi) (1, 177)
-h72.3625 -h72.5111 -h72.5396 7.0M
-h72.2807 -h72.3957*

*Quintet state

128

Table 29b. Energies of pyrazine excited triplet states based on
molecular orbitals constructed from the split pn basis set.
The molecular orbitals which are allowed to have variable
occupancy in forming excited configurations are described in
the text. The interaction threshold, 6, the number of parents,
and the total number of configuration pairs are shown in
parentheses. The wavefunctions corresponding to these states
are shown in Table 30. Total electronic energies are in atomic
units, and transition energies are in electron volts.

Electronic Single Initial Final CI Transition
state configuration step CI energies energies
energies energies (0.0001)

(0.003)

3s3u (2, A2) (2, 241)
-u72.5h63 -h72.6362 -h72.6678 3.56
-u72.2398 ~h72.h001 —u72.h299 10.0h
3Blu (3, 33) (3, 25h)
-u72.52u5 -h72.6030 —h72.6h75 h.ll
~h72.5175 -h72.5670 -h72.5995 5.hl
-h72.hl79 -h72.h83h -u72.51uu 7.73
3B2u (3. M6) (3, 3M9)
-u72.5028 -h72.5552 -h72.6006 5.39
-h72.hh89 -h72.u6l3 —h72.h90h 8.38
—h72.3286 -h72.h227 —h72.hh58 9.60
3Au (2, kl) (1, 165)
-u72.h735 -h72.5816 —u72.6098 5.1a
-h72.2329 ~h72.3539
3
B2 (h 5”) (3 2&2)
g -h72.h675 -h$2.5833 -h72.6150 n.99
-h72.3l58 ~h72.h55h -h72.h929 8.31
-h72.292l ~h72.hh39 -h72.h75h 8.79
-h72.27hh -h72.3807
3513 (2, hi) (1, 183)
-h72,3679 -h72.51h2 -h72.5h53 6.89
-h72.2872 -h72.377h
3Ag (2, 22) (1, 9h)
-h72.333h -h72.h9l7 -h72.5l76 7.6M
-h72.2628 -h72.305h
3B38 (2, 29) (1, 91)
-h72.3070 -h72.h903 -u72.5l22 7.79
-h72.2866 -h72.3223

129

Table 30a. CI wavefunctions of the A , B3u, Au, and B2g states of Pyrazine
from the calculations per ormed using the extended basis set.
Only those configurations with coefficients of absolute value
greater than or equal to 0.2 are shown. The molecular orbitals are
numbered as in Table 26. The plus and minus signs denote a and B
spins respectively.

Configurations Absolute values of coefficients
of configurations in states

1118 lag 5A8 3Ag
ground 0.9M
(20)2-+(22)2 0.81
(18)2->(22)2 0.1+7
+19, —21)-+(—22, +23) 0.52 0.116
-19, +21)-+22, +23) 0.52
+19, +21)—(+22, +23) 0.52
17421 0.66
19425 0.18

LI33a 3B3u
20422 0.90 0.91
(18, 19)-+(22)2 0.27

1A1). 3Au
20423 0.90 0.90
18422 0.81 0.8h
(l9, 20)-~(22)2 0.141 0.37 0.26 0.15
(+20, -21)-'(-22, +23) 0.52 0.39 0.1a 0.51)
(+20, -21) +22, -23) 0.52 0.30 0.h9
(+20, +21) +22, +23 0.52 OMB
2025 0.611 0.31 0.61

130

Table 30b. CI wavefunctions of the B2u’ Blu: Blg, and 83g states of pyrazine
from the calculations which were performed using the extended
basis set. Only those configurations with coefficients of absolute
value greater than or equal to 0.2 are shown. The molecular
orbitals are numbered the same as in Table 26. The plus and minus
signs denote a and B spins respectively.

Configurations Absolute values of coefficients
of configurations in states
lB2u 5B2u 3B2u 3B2u
2l+22 0.70 0.76 0.56
19423 0.5M 0.50 0.79
(+18, -20)+(-22, +23) 0.53 0.20
(+18, -20)+(+22, -23) 0.53
+18, +20)+(+22, +23) 0.53
113111 313In 3Blu 313111
21423 0.87 0.69 0.61
19+22 0.h2 0.62 0.68
17425 0.20
(+17, -21)+(-22, +23) 0.20 '
(18, 20)-+(22)2 0.93
1312 3312
18423 0.82 0.80
(+19, -20)+(+22, -23) 0.22 0.23
(+19, +20)-+(+22, +23) 0.22 0.21
(20, 21)+(22)2 0.21 0.23
3B32
17423 0.59
21+25 0.53
(19 21)-+(22)2 0.27
(17324422, 23) 0.26
(19)24(22, 23) 0.21;
(19, 21)*(23) 0.22

131

Table 31. Results of calculations on the lowest lA 1B3u9 lB , and
3B3u states of'pyrazine, using a threshold of3 0 0008328, and
no approximation in diagonalizing matrices.*

Total electronic energy Transition energy

State (a.u.) (eV)
1A h 2

g - 7 .79962
333,, -h72.67098 3.50
1B3u —l+72.6l+705 1+.15
l

132u -h72.6055h 5.28

*The subroutine NESBET (QCPE 93) which was obtained from the Quantum

Chemistry Program Exchange was used.

132
3. Discussion of pyrazine results and comparison

with experiment

 

A comparison of the SCF results shown in Table 28 indicates that
the total energy of the ground state is lowered only by 0.013 a.u. as a
result of the addition of rong range pn basis functions to the basis set;
however, the use of additional pn functions introduces greater flexibility
into the description of excited states.

An important feature of these SCF results is the separation of
the n- orbital (Sblu) and the n+ orbital (6ag), the n_ orbital having a
much lower orbital energy. Some previous workers72 have argued qualita-
tively with some spectral evidence that these orbitals were either degener-
'ate, or that the n+ orbital is lower than the n_ orbital. Indeed, at a
qualitative level as argued below, each of these assertions is reasonable.

The claim of degeneracy of the two orbitals is reasonable if,
as was assumed, the n+.and the n; orbitals are localized on the nitrogen
atoms. In this case, the n+ and the n. orbitals can be expressed as the
sum and the difference of two functions, one localized on one of the

nitrogen atoms, the other localized on the other nitrogen atom:

n+,= N+(nl+'n2)

D
l

_ - N_(nl- n2)

 

133
where N+ and N_ are normalization constants. If I11 and n2 orbitals are

localized on the rather widely separated N atoms then (nl InZX: O and

N+ =:Ng.=,2 .

NIH

The orbital energies then become

2N+?{<nl IF 'n1> + (“1 'F ln2>}

(n+.lF tn+>

II

8+

(n_ [F 'n_) 2N_2{(nl 'F 'n1) 4 (nl IF {n2)}

:9
II

where F is the Hartree-Fock Hamiltonian.
(nl IF 'n2) is approximately equivalent to the energy of an electron in
a charge distribution which is the product of the functions nl and n2. If
nl and n2 are highly localized, then each function tends to vanish in the
region where the other function has significant density. Therefore, under
these conditions,

e+~ c_

The assertion that cr<¢_ is also reasonable, since the orbital n.
would have a node, whereas n+.would not. In such a case n+.would be like a
bonding orbital and n_ like an antibonding orbital, in which case it is
reasonable to expect that c+.would be less than c_. More quantitatively, if
the interaction were stabilizing, (n1 IF ’n2)<0, then it would follow for a
solution of the secular equation that che_; the splitting being determined
by the magnitude of the interaction.

One.of the serious weaknesses in each of these arguments is the
assumption that n+ and n_ can be expressed as n1 +-n2 and nl-n2,
respectively, where 111 and n2 are highly localized. Indeed, an examination
of the n+_and the n_ orbitals which-were determined from the SCF

calculations of this work, shows that this assumption is not correct. The

6&8 non-bonding orbital is considerably more delocalized than the 5blu

131+
orbital as shown by the population analysis in Table 32.

An examination of Table 29 shows that most of the 2b3u(n*) orbitals
which were obtained by fixed core SCF calculations on various states are
similar. Those 2b3u orbitals which are exceptions are the ones which were
obtained from SCF calculations on the-1B 1B and 3B states. 0f the

1d 2u 2u
eight lau orbitals which were fOrmed, the ones which are most strikingly
different from the others are those that were determined by the lu and

3B SCF calculations. The lau orbital which was determined by the 1B

In lu
SCF calculation' is the most diffuse and is very similar to the ground state
virtual orbital. A number of different 2b2g orbitals result from the
fixed core SCF calculations; however,there is no low lying excited con—
figuration which involves this orbital.

The importance of examining these fixed-core SCF calculations is
twofold. First, for the first time in the molecular studies presented here,
the full set of n orbitals is not variably occupied in the excited con-
figurations. It is, therefore, important to examine the {ability of a
ltmited set of W orbitals to reproduce the fixed-core SCF results of all
of the states. An examination of Table 29 shows that energy-wise the

discrepancy can be no more than 0.l3 eV at the fixed—core SCF stage. The

maximum discrepancy occurs in the case of the 1B2u (lbg42b3u) n9n* state,

where the 2b3u orbital which is chosen for the CI calculation was quite
different from the one which was optimized in the lB2u (lblgw2b3u) fixed-
core SCF calculation. Much of this 0.13 ev energy difference can be
retrieved, however, through mixing of the lblg48b3u configuration at the
CI stage.

In the fixed-core SCF calculations involving promotions to each

135
.Table 32..Mulliken population analysis of the n_(5blu) and n+(6ag) orbitals

of pyrazine.
Orbital Carbon Nitrogen Hydrogen
n- O. 029 0.932 0 . 001+

n+ 0.108 0.71M 0.035

136
of the n* orbitals, all of the 0* orbitals resulting from the n*“*
calculations and some of those resulting from “40* calculations appeared
to have quite similar shapes. However, a number of 0* orbitals resulting
from n4fl* calculations appeared to be quite different. In the study of
formaldehyde, the "* orbital resulting from.the fl4fl* calculation was
considerably different from the other "* orbitals, and at the same time
the calculated 1n*n* state was not located in the expected region of the
spectrum, while the other calculated states involving the “*orbital were
quite reasonably located.

The n+n* states in which the n* orbitals were different than the
other n* orbitals were also too high in energy in the calculated spectrum.
Indeed, in pyrazine, no lBlu (n+n*) states are calculated in the regions
6.30-6.8h eV or 7.53-7.90 eV (see Table 26). Neither were there lB2n (n+n*)
states calculated in the regions 6.72-7.56 ev and 7.5h-7.90 eV (see
Table 2h). The problem of the lack of calculated n*n* states in the expected
regions of the spectrum is discussed further in Appendix C.

A comparison of the CI calculations using the two different
basis sets shows that qualitatively they yield the same results. All of
the total energies were lower, of course, in the case of the calculations
that were performed using the larger basis set. .The more important feature
of the comparison is that the transition energies were all lowered by
the addition of the long range pn functions. The CI lowering of the
excited states relative to the ground state, however, is in general greater
for the smaller basis set. Thus, the lowering of the transition energies
occurs at the SCF stage where the primary effect of the long range pn functions

is an improvement of the description of the fl* orbitals.

The calculated and experimental transition energies are compared

in Table 33. The calculated transition energies of the 3B3urlAg and the

137

Table 33. Comparison of calculated and experimental spectrum of pyrazine.
Energies are in electron volts.

Electronic Transition Experimental Band Oscillator
States energies range origin strength
calculated calc. expt'l.

5A8 9.28
113111 9.10 6.30-6.8M 6.30 0.115
5132u 9.07
31328 8.79
11328 8A7
3132u 8.38
51328 8.16
3131,, 7.73
3A8 7.61;
1A3 7.33
lBlg 7.0M
3Blg 6.89
1B2g 5.65
3131u 5.1+l
3132,, 5. 39 *
132“ 5.29 11.59-5.06 b..80 0.01 0.10
1A,, 5.22 -
3A11 5.11:.
3B2g ”-99
1133,, #22 3.76-M28 3.83 0.01 0.01
33111 hull
333,1 3. 6 3.30-3A8 3.32

A . . . . .

8

*Two configurations which have opposite transition moment vectors are
important in describing this state. -

138

1B u+$Ag transitions agree well with experiment. However, the lB3u and

3

3B n4n* states involve the orbital promotions 6ag42b u and not, as has

3u 3
been suggested by others, the promotion Sblur2b2g (see Reference 32, pp 553).
The other n4n* states, the 3B2g (n+n*) and the 182% (n4n*) involve the
promotion 5b1fi*2b3u, and occur more than one eV above the lower n*n* state.
The transition lBegt'lAg is not allowed by the electric dipole mechanism.
However, an interpretation of the spectrum by El—Sayed and Robinson72 led

them to believe that they had located the lngtlA transition in the same

8
region of the spectrum as the lB3u+lAg transition. They also presented
qualitative theoretical arguments of the type described earlier in this
section in support of the degeneracy of these transitions. It now appears
that the original interpretation of the spectrum is not Justified?)4

It is interesting to note that the calculated transition energies
to the 3Au (n+n*) and tau (n*n*) states are both lower than that of the
1B28"J7Ag transition. The Au states involve the promotion 6ag4lau. Thus,
the three l(n-"T*) states mentioned above, in order of increasing energy, are
the 1B311 (6ag*2b3u), the 1Au (6ag4lau) and the 1B2g (5blu42b3u). The
energies of these three states are in the region h.22 - 5.65 eV above that
of the ground state.

The location of the 3Blu state shown in Table 33 gives support to
the intersystem.crossing and internal conversion mechanism proposed by
El-Sayed7h to explain the pyrazine phosphorescence. He showed clearly that
the 1B u state was mixed with the3Blu state by spin orbit interaction.

3
What he could nbt substantiate was the assertion that the energy of

the 3Blu state was between that of the lB3u state and that of 3

This assertion is crucial to the validity of the mechanism, and the

B311 state 0

139

present calculation clearly supports it.

The calculated transition energy of the 1B2u (n%n*)+;Ag
transition is only slightly above the experimental range. This is the
only low-lying l(us-n") state which has been found in any of the three
molecules investigated in this thesis. The lBQu state is described as a
mixture of two configurations; one involves the orbital promotion

lblgazb , and the other the promotion lb264lau. The 3B2u (n4n*) state

3u
involving these same configurations is higher in energy than the singlet.
This might seem unusual in the sense that at the single configuration

stage triplets are always lower in energy than singlets and that the
singlet-triplet splitting is especially great in the case of n4u* states.
Hewever, the CI results show the B2u states to be a mixture of two

important single configurations and thus a single configuration argument
about the singlet-triplet splitting is not valid.

The four highest observed transitions shown in Table 26 are not
found in the calculated results. The lowest calculated allowed transition
above that of the ¥B2uftAg is the lBlu (n4n*)+;Ag transition which has a
transition energy 2.5 ev above that of experiment. In view of the fOrmalde-
hyde results, there is a possibility that there may be a low-lying Rydberg
state of B1‘“ symmetry. Such a state could involve the orbital promotion ‘
6ag-Iblu or 5blu4ag.

The examination of Tables 29 and 3l which give the energies of the
lowest tAg, l13311, 3B3u and 132“ states shows that the method of successive
diagonalization is quite accurate for the energy. The wavefunctions
resulting from this method were also found to be quite close to those
obtained by more precise matrix diagonalization. In Table l wavefunctions

of the lB2u state obtained by the two procedures are compared.

11+0
IV . CONCLUSIONS

Since the configuration interaction (CI) method is ultimately
capable of yielding accurate energies and.wavefunctions for molecular
states, all of the uncertainties which are inherent in the present treat-
ments are due to the approximations which were discussed in Section II.
These approximations are:

l. The choice of a finite basis set.

2. The fixed-core approximation.

3. The limitation of the number of virtual orbitals.

h. The limitation of the number of configurations by means

of an energy criterion.

5. The use of an approximate method for diagonalizing large

matrices.

Each of these approximations has the effect of increasing the
difference between the calculated and the exact total energies. There
are no apparent cancellations of errors energetically resulting from
these approximations, as there are in some semiempirical.methods. For the
calculations which are presented in this thesis, the approximations 195
are listed in an order of decreasing likelihood of effect on the spectral
results.

Approximations 2 and 3 were examined for the 3A2 state of
formaldehyde, and.the results in Table h Show that most of the flexibility
of the basis set which is important for excited state SCF calculations is
contained in a few of the occupied and virtual ground state SCF orbitals.
The importance of including all of the fl'molecular orbitals in the CI

calculation is especially clear from these results.

11+].

A test of the approximations 2 and 3 at the CI level is found by
comparing calculations which involved different sized sets of core and
discarded virtual orbitals (see Tables 10 and 11). In the fbrmaldehyde
calculations, one set of variably occupied orbitals included orbitals of
a1 symmetry, another did.not. Although the total energies were lower with
the larger set, the spectral.predictions of the calculations based on each
of these sets of orbitals was the same, indicating the lack of importance
of the al orbitals in the CI calculations. Similarly, in one set of
glyoxal calculations, a core of twelve orbitals and a set of ten.variably
occupied orbitals was used, while in another set of calculations a core of
ten orbitals and a set of sixteen variably occupied orbitals was used. The
two calculations yielded essentially the same spectra (see Tables 18 and
19). Thus, these results on formaldehyde and glyoxal show the feasibility
of performing calculations with a fixed core and a reduced set of'virtual
orbitals.

Approximation h could not be explicitly tested, since a complete
CI calculation was not possible with the basis sets used in this work.
However, the energy threshold gives an indication of the maximum ability
of each of the discarded configurations to reduce the energy. Since the
energy thresholds used are quite small, generally on the order of 10'h a.u.,
’ the total contribution of discarded configurations is not likely to
constitute a serious source of error in the calculated spectra.

Concerning approximation 5, numerical tests were made of the
successive diagonalization procedure in the pyrazine calculations, where
the Hamiltonian matrices of a number of symmetries and multiplicities were

diagonalized accurately and by the successive diagonalization procedure.

112
The resulting eigenvalues and eigenvectors were found to be in close
agreement (see Tables 1, 29, and 31).

The successive diagonalization procedure has the useful property
of successively displaying the energy lowering resulting from groups of
configurations which are listed in the order of their assumed ability to
lower the energy; thus, it gives an indication of the convergence value of
the energy which would result if all possible configurations were included
in the CI calculation. ‘

In summary, it appears that approximations 2-5 are not likely to
produce significant errors in the calculated spectral results for those
v cases studied in the present work. Approximation l, which concerns the
choice of basis set, ranains to be considered and this can be accomplished
by a careful examination of the results of the spectral treatments of
formaldehyde, glyoxal and pyrazine. The conclusions should have some
bearing in general on the adequacy of certain descriptions of low lying
excited electronic states of polyatomic molecules.

Because of the relatively small size of the formaldehyde molecule,
and the formldehyde basis set, the severity of approximations 2-5 were
still further relaxed. The accuracy of the calculated spectrum, there-
fore, should be reflective of the reliability of approximation 1 concerning
the capabilities of the basis set. Comparison of the calculated spectrum
of formaldehyde with that of experiment (Table .15) shows that the

1

positions of four of the calculated states, 3A A 1132, and 1A1 agree

2’ 2’
with those of experiment. In addition, a number of forbidden transitions
that are not observed experimentally are predicted by the calculation;

also, a number of experimentally observed transitions are not accounted for

1h3
in the calculations. The description of the calculated A2 states as
n-m* agrees with previous ideas about this transition; however, the

description of the lowest 1B and 1A1 states as n43s and n43pb2,

2
respectively, disagrees with previous descriptions of these states as
n-v* and “411* transitions, (see Section III.A'.1. and Section III.A.3.).

The failure to find a n-m*, 1A1, state in the region around 8 ev
is indeed a serious difficulty of the present treatment. In Appendix C,
arguments are presented to explain the possible absence of a 1W* state
in the calculated spectrum "of formldehyde. It was suggested that,
because of the greater contribution of ionic terms in the lrr-ofl* state
compared with that of the ground state, (1 functions likely would be
important in lowering the 1A1 (fl-m*) *- lAl (ground) transition energy.

Another possible explanation of the apparent absence of a
calculated singlet 11-m* state in formldehyde at 8 eV is that the
association of the F‘ranck—Condon calculated transition with the maximm
intensity band in the spectrum is not Justified. In Section LC. the
qualitative theory on which such associations are based was presented,
along with the more quantitative theory which showed that the association
of a calculated Franck—Condon transition with a spectral band is not -
unambiguous.

The singlet rm" state of formaldehyde should have an equilibrium
geometry in which the co bond distance is greater than in the ground state.
This is expected because the replacement of a bonding n orbital by an
antibonding 11* orbital weakens the co bond. Increasing the (:0 bond length
significantly lowers the energy of the singlet n-m* state, while it raises

the energy of the ground state only slightly, thus lowering the

11m
1A1 (TI-fir") *- 1A1 (ground) transition energy.

It is reasonable to conclude that a combination of the effect of
d function additions to the basis set and the variation in geometry of the
excited state might produce a significant Mering of the energy of the
1A1 (fl'fiT*) ‘- 1A1 (ground) transition of formaldehyde.

If the absence of a singlet u—m* state in the calculated spectrum
of formaldehyde is in part due to a deficiency in the basis set, such a
deficiency will also be present in the basis sets of glyoxal and pyrazine
since these basis sets are similar to those of formaldehyde. This is
especially true of glyoxal, where the same split group basis functions are
used.

In pyrazine, the calculated singlet and triplet B3n (n-M*) 0- lAg(ground)
transitions are in very close agreement with experiment, while in glyoxal
the calculated and emerimental singlet and triplet Au (n+n*) «- 1A8 (ground)
transition energies are only in fair agreement. However, in each molecule,
the singlet and triplet n-m* states are the lowest energy excited states,
and in this respect there is agreement with previously held notions.

There is disagreement between the present work and previous
theoretical investigations on the issue of the degeneracy of the n-m"
transitions i of glyoxal and pyrazine. The present calculation indicates
that there is a separation of 1.26 eV in the two singlet n-I'n* transitions
of glyoxal and a separation of 1.18 eV in the two singlet w* transitions
of pyrazine. The nature of the orbital promotions involved in the n-m*
states of pyrazine were found to be different from those which were found
in previous investigations (see Section III.C.3.).

The only TH"* transition energy calculated in the present work which

11+5
was feund to be in agreement with experiment was that involving the lBZu
state of pyrazine. A calculated lBu (n43pbu) Rydberg transition of
glyoxal at 9.09 eV also possibly corresponds to an experimental spectral
state at 9.15 or 9.36 eV. Nam of the experimentally observed states of
formaldehyde, glyoxal, and pyrazine do not correspond to any of the
calculated states (see Tables 2, 15, 16, 23, 2% and 33). In some cases
the experimentally observed states are thought to correspond to certain '
Mberg states which could not be described by the present basis sets; in
other cases, however, the reason for the disagreement is not clear. Mary
of the transitions which were determined in the calculations are either
symetry or spin forbidden, and therefore, the reliability of their
calculated locations cannot be gauged accurately from the experimental
spectrum. It is interesting to note that some of the lowlying forbidden
transitions which were determined by the calculations correspond to
two-electron excitations. The low lying double excitation states that

l

were found in glyoxal were the Ag state at 6.52 eV and the 3Bu state at

9.00 ev, and in.pyrazine the lAg state at 7.33 ev and the 3328 state at
8.79 eV. These states would not appear in the experimental spectrum since
the transition moment connecting these states to the ground state would
be near zero even if such transitions were multiplicity allowed. a

Also found in the calculated spectra were a number of quintet
states below 10 ev, a 5A8 in the glyoxal spectrun.at 9.20 ev, and a"
5323’ a 532“, and a 5A8 in the pyrazine spectrum at 8.16, 9.07, and 9.28 ev
respectively. .

Comparimn of pyrazine CI results based on the fixed group basis
set, withthose CI results based on the split pn basis set (Tables 27, 28,

and 29) shows that the effect of splitting the p11 basis functions is mainly

116

to improve the SCF description of the electronic states, rather than to
affect the calculation at the CI stage. Although all of the transition
energies calculated with the extended basis set are less than those
calculated with the fixed group basis set at the CI stage, both sets of
calculations yield quite similar spectral results, thus indicating the
feasibility of performing CI calculations with fixed group basis sets.

Concerning the question of the necessity of using large final
sets of configurations in constructing CI wavefunctions for excited
electronic states, the results presented in this thesis shdv that for
those states in which there is agreement bet-men calculation and
experiment, the transition energies improve at each stage of the three
step process for generating configurations. The time required to generate
CI expansions containing several hundred configurations is negligible
compared with the time required to generate the most basic information,
integrals over basis functions and integrals over molecular orbitals .
The deterrent to large scale ab initio CI application thus remains at
the integral stage.

Finally, several remarks should be made concerning the
comparison of the present CI method and results for fbrmaldehyde ,
glyoxal, and pyrazine with corresponding semiempirical methods and
results. Only a small mmber of configurations is generally used in
semiempirical calculations, and the Hamiltonian matrix elements between
pairs of these configurations are evaluated by means of non rigorous
approximations. The major area of disagreement between the results of
semiempirical calculations of the spectra of formaldehyde, glyoxal, and
pyrazine and those of the present study is the location of singlet TM?"

1 2
states5 ’5 ’76. The semiempirical calculations generally predict low-lying

1M7
singlet “ha!" states, while the present study does not.

It was argued in the introduction of this thesis that if the
predictions of semiempirical calculations concerning the nature of excited
states are to be regarded as reliable, the results should not be changed
significantly if the approximate Hamiltonian matrix elements are replaced
by accurate natrix elements. In the present calculations, all Hamiltonian
matrix elements are evaluated accurately; however, a larger number of I
configurations and a more flexible basis set were used than employed in
the semiempirical studies. Thus, the direct comparison 'of the present
results with those of semiempirical calculations is somewhat difficult;
however, an indirect comparison will be presented below.

The results of the present calculations on pyrazine show that
the splitting of the p1! functions and increasing the size of the set of
configurations lowers the transition energies, thus closing the gap
beWeen calculated and experimental transition energies (see Tables 28
and 29). The results obtained using this more flexible basis set and
larger numbers of configurations therefore should be more reliable than
those that would be obtained merely by correctly evaluating the specific
Hamiltonian mtrixelements produced by a smaller basis set. In
semiempirical calculations, such small basis sets, usually consisting only
of valence shell atomic orbitals, are used, and therefore it must be
concluded that the agreement between semianpirically calculated spectra
and experimental spectra in the case of singlet thr* transitions is
better than can be expected from the basic framework of the theory. It
follows that the associations between semiupirically. calculated excited
states and experimental excited states are not clearly reliable . in the

specific cases mentioned above. Indeed, the present work suggests that

. 1&8
many 1H1* singlet excited states previously thought to involve simple

excitations to valence shell molecular orbitals must instead be described
in terms of molecular orbitals involving a significant distortion of valence
shell basis orbitals. On the other hand, n-On* transitions and certain
mrdberg transitions can be described fairly simply at the level of

sophistication employed in this work.

17.F

19.

20.

119

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91+. M. A. El-Sayed, Private communication.

15h
.Appendix A. The determination of the approximate energy contribution
of a configuration to the total energy

Consider a set of configurations, (mi, i=1,2,...,N}, and a linear

combination of configurations, 20, where H00 = (‘0 (H '00),Hii = ($1 'H (pi),
i=1,2,...,N. Let the function *ch tom: lcicpi be such that (g ‘H I 1)
V i

is minimized. Then the coefficients, ci, result from the diagonalization of

the matrix H.
Consider an approximation H’ to the matrix H, such that HéjyeH

d I = h _ = 2.. P ‘ -
an H13, éiJHii ,ten l H E II 0 -——=-) j(1)JPJ[(HOJO E6030)

(H 13 -E§lj )...(HnJE W)] Owhere the sum is over all of the permutations
1 l n
P3 of the indices 3, pan is the parity of the 3t hpermutation (pj equals one

fOr odd permutations and two for even permutations), and I is the unit NXN

matrix.

. N - N
lH'-E'I l= 0: 120 (Hii-E') + §=1('1)1H03H30u; (HM-E05)

N
Dividing the above equation by 2 (Hii-E')¢0 gives
i=0

I; (4)1 Renae

 

— O
3:1 (HOG-E') (HJJ-E')
N [H l2
or E' = H60 4-2: ___lfil___. ,
' J=l E' - H

33
Energy contributions from a configurationcpJ can be approximated
by '330 l2 E' is not known, but E'S H00, and therefbre,

E. - H33

155

2 . Thus, the maximum contribution to the

'3' " ”3:1 I THOO'HJJ I

energy that a configuration can make within the framework of this

 

 

 

approximation is 'HJO ' 2 .
H - H,
00 JJ

The above type of argument is commonly used to derive by the

variational principle energy expressions of the same form as obtained

by perturbation theory.93 .

156

Appendix B. Proof that all calculated energies of ground and excited

states are gpper bounds to the true energies

Theorem: If {#k} and {Ek} are, respectively, the complete set of exact
orthonormal wavefunctions and the set of exact energies of a quantum
mechanical system where El S E2 S...S Ek S...,then if cpl and cp2 are
approximations to $1 and t2 such that (Cpl 'cpl) = ((132 [(92) = l, (cpl [’92) = O,
and (cpl IH Icp2>*= 0, then the approximate )energies satisfy (cpl 'H ' cpl)-E120
and (cp2 IH "92) - E22 0. In other words, (cpl IH lcpi) and {we '3 H32) are

upper bounds to the true energies El and E2, respectively.

Proof:
A proof will first be given that (Cpl [H ltpl)-E 2 0.
Q

q) = 2': Ckl'k since the set {1k} is complete.
k=l

*-
<<p1 In 1ch = (E cuvk IH l§ 031%) =§§ Gui-11“]; land)

=zzc 0 E6 =zlc [2E
13.311131ng k kl k

(cpl lH lcpl) - E1 = § [031! 2(EJ-El)20 since

._ 2 ..
EJ 2E13-l,2...and§'(331l — 1.

The proof that (cp2 In 1'92) - E 20 will now be given. Let cpl denote the

2
West energy approximate wavefunction,

 

* .These properties hold true if the functions cpl and (92 are obtained by

diagonalization of the same Hamiltonian matrix.

157

Q Q
- -- 2
= C t +2" 17 where )3" I = l
‘91 1.1 1 k=2 0x11: k=l Ckl'
and

(“’1 lchpl) S («pg 1111432)

Casel: Cn=0
. . - =35 ..
«91,11,91) E2 kélcklIZEkEZ

w 2
= 2: Ian] (ER-E2) 2 0 since
k=2

no
23 'Ckl'2=l inthis case.
ké

Therefore, (cpl [H 'cpl) - E2 2 O and since (m2 '3 [$2)2(cpl lH 'cpl>, then
(o2 [H [‘92) - E2 20

use 2. C11 * O

i =_:_L__ cPl '__1_ 3 Ckfik
C11 Cu k=2

a Q ‘
and ‘pz :21 Ck2'k = ‘1 q’1 1:2 duh:

cp2 = dcp:L +X2

(¢21H1¢2> = Idlz (cpllHlopl> +d*((pl'HlX2>

+d<X2'chpl) + (leHlxz)

((Pl '11 'CP2> = d<¢1lH"Pl) + ((91 III [1(2) = O by assumption

158
therefore, (cpl [H |X2~> = -d<cp:L 1H |<P1>

«pg In l<p1> = d*<cpl lH Ml) + <x2 IH Isl) = 0
therefore (X2 lH {cpl} = -d*<<Pl [H I cpl)

which also follows from the Hermitian property of H.

therefore (cp2 lH'qJ2> = Id 12<cpl 1H Mal) - Id le<cpl lHl=P1>
- ldl2<q>lthol> + (X21H lx2>
=- Id |2<<Pl lchPl> + (X21313?)
now (<92 [‘92) =1 =ld12 +d*<<pllX2>+d (X21r91>+<X2lX2>

but since (cpl I ape) = d+(cpl 'X2) = O, (cpl 1X2) = —d

and. (<92 "91) = d* +<x21q>l> =0, (X2'cp1)= —d*
the . <c21<92>=r= ldlz-ldlz-ldlz+<x2!x2>
=1=-ldl2 + (x21X2>

therefore <ch l H [top-E2 = <cp2 [H l $2)-E2(<Pg Mug)
=<x2 lHlx2>-‘E2<x21x2>+sz 1.112- M 12<cpl lH 1:91)
=(1E=2dk2*k l H l inedkztk)-E2 <12;de I gawk)
+1612 (file-(cpl llel»

or tawny-sf? laje lean-EndsPeg-«plinth»

=2

159
if ($1 lH l¢l>SE2 then ($2 1H |¢2>—E220
if (ml in Imlfi>E2 then since
($2 lH lv2> 2 (ml lH Isl)
then (”2 [H ‘w2)-Eé> o
Q.E.D.

Similarly, it can be proven.that

(mi IH Imi>-Eizo for

i = 3,h,....if the corresponding assumptions are made,

(oi lHlopJ.) =0 and (cpi 1ch =0, J<i

A different approach to the proof of this theorem can be

found in reference 6 pp. 265-266 and references contained therin.

160
Appendix C. A discussion of "+11" states and the frequent disagreement of
calculated singlet-singlet transition energies with experiment

Assuming that 1(TI‘"¥T*) states do exist in the regions of‘the spectrum
to which they have previously been assigned experimentally, it is important
to determine the reason for the inability of the present calculation to
account for them. The 3(rr-m*) states, on the other hand, seem to be quite
reasonably located. An examination of the wavefunctions of the singlet and
triplet states at a simple level of approximation provides considerable in-
sight into potential difficulties of valence shell CI treatments.

If all doubly occupied orbitals except the n orbital being excited
from are neglected (they contribute a factor which is common to both the
singlet and triplet states) then two electron mBuO anefunctions for singlet
and triplet states can be constructed from the orthonormal singly occupied

11 and 11* molecular orbitals as follows
film?) = [11(1) n*(2> +n*<1) 11(2)] Ea<1>e<2> -. 8(l)a(2)]2'1
#(3Wn*) = ["(l) 11*(2) - "*(l) 11(2)] [a(l)B(2)i+ 3(1)a(2)]2-l

At the linear combination of atomic orbitals, (LCAO), level the n and 17*

orbitals can be expressed as linear combinations of pn basis functions, one
from each atom,

"=Eckpk
11*:de
333

Therefore, the spatial wavefunction can be expanded

«la-m > = [(123 ckpk<1>)<§d,pj<2>>+<§ d,p,<1>)<1§ckpk<2)>1

161

= z
E ckdkpk(1)pk(2) +12: J ckdjpk(1)pj(2)

k¢J

2 d 1 2 + E 2 d l (2
+1: kckpk( )pk( ) k3 Jckp3( )pk )

ktj
= + Z 2 d +d l 2
2 E ckdkpk(l)pk(2) k3 [Chg kcjipg )pJ( )
ktj
ionic terms covalent terms
H3

n*=[2 12d 2-Zd 12c 2]
.m) (k <=kpk(>)<:j Jpjm (J jrdmxk k1ok< >>

ckdkpk(l)Pk(2) +‘E i cdePk(1)Pj(2)
k¢3

- z dkckpk(l)pk(2) - z 2 dJcka(l)pk(2)
k k 3
k#,j

= E g [cde-dkcjlpk(l)pj(2)
kt;
only covalent terms
Thus, the above analysis shows that the singlet wavefunction possess
ionic contributions (two electrons in the same p-orbital on a nucleus) while
the triplet wavefunctions do not. This, of course, is a consequence of the
Pauli exclusion principle at the simple LCAO level of description.
Concerning the formaldehyde molecule, if the ground and singlet and
triplet fl4n* excited states are analyzed in the manner described above, then
the ionic character of the ground state is intermediate between that of the
singlet and that of the triplet n*n* states. If pc andp.o are the pn'basis

functions on the carbon and oxygen atoms respectively, and if the n and n*

orbitals are represented by

162
11 = s,pc + p0
4(-
11 = PC " 13130:
then the two electron spatial wavefunctions of the ground and singlet and

triplet n%n* states can be expressed as

# [(apcmo) (apcmon

ground

+a +a
achPc PcPo PchtPoPo

#1“; [(apaino) (Pcrfipo) +(pc-bpo) (ape-Inc”

a - a -b
pcpc bpcpniivopc popo

+ - .-
apcpcflgcpo baPOPC bPOpO

2a;pcjpc+(l-ab)pcpo+-(l-ba)popc

2bpopo

13m? [(apcmo) (pa-bpo)- (pa-bpo) (anemofl

ll

apcpcmabpfi'pOl+p0p(3mbPOPOUa'PCBP(Z'IQGPO
baPOpcqlprPO

+

- +ab + l+b re ectivel .
(1 )PcPO( amenc 8p y

In these expressions and in the following, the orderpmpn refers to mel)
pn(2), -In the-case that a =‘b = l which is approximately true for formalde-
hyde at the LCAO level, the ground state has equal ionic and covalent
character, and the lnan* state has only ionic character. The 3n+fi* state
always has only covalent character at this level of approximation of the
wavefunction. It should also be noted that the covalent character of the

ground state increases at the CI stage where the ground state configuration

163

is strongly mixed with the fle-rrr*2 configuration.

In the case of pyrazine, the energy of the J132,1)‘(TT-m'rflp) state was
calculated fairly accurately, while that of the l'Blu(TT"TT*) was not. There
are two configurations which are important in each of these states. The
singlet and triplet wavefunctions which result from each of these con-
figurations will be analyzed in terms of contributions from ionic and.
covalent terms.

If the pn basis functions are numbered
2.

6

then the unnormalized orbitals ‘which are involved in the singlet promotions
mich give rise to the four important configurations of these states can
be expressed as

1‘02g (111) = P1 + apz + p3 - pg - ans ~ 136

lb;g ("2) = p1 - p3 -_ pa + P6
2be1 (n3) =pl +ap2 +p3 +ph" +335 +p6

inn ("1.) =91 - p3 +pg - p6

Then
*113 ("Z-mil.) = 2 [pip]. "' P1133 "' P3Pl + P3P3 " Puph + 131,196
lu
+ Pepi; - p6p6]
'13 ("l-m3) = 2E1’11) 1 + “Pipe + I’13’3 + apepl + sap-£2
lu '

+ ap2p3 + P3P), + 3133132 + P3P3 "’ Phph

" aP’4P5 " aPLFP6 "' aPBPh - aapsps

- apspé - D6131, - ap6p5 - p6p6]

16h

i1 ("2""3) = 2131191 + 8431p2 + aplps + 291136 - ap3p2
Ben
- 2p3p3 — 2P3Pl+ - ap3p5 - a-php2 - 291333 - ZPhpu
" “Pups " 2P6Pl + 396132 + 8{36135 + 2P6P6
+ a“19.2131 ' a‘P2P3 " 3132131; + ELI’21’6 ' aPuPy,
+ apspl - ap5p3 - apSph + ap5p6 + p61):-
EB ("14%) ‘ 291p). " 215.96 + aPZPl ' ap2P3 + ”2% " 8LPepe
2“ - 2p3p3 + 2p3ph + 2php3 - 213th - apspl + ap5p3 -

- ap5p1+ + ap5p6 - 296101 + 296% + 8131132

+ 5133135 + 3p)-92 " aPHPS ' aP6132 4” aP6P5

Two points can be made about the comparison of these wavefunctions.

First, the 1B2u'wavefunctions do not have ionic terms on the nitrogen atoms,

In
together contain a much larger number of covalent terms (forty) than do the

while one of the 1B wavefunction does. Second, the lB2u states taken

In
covalent terms is 5/8 fbr the 1B

113 states (sixteen). The ratio of the number of ionic terms to the number of
in states and 1/5 for the lB2u states.

Thus, the inability of the CI calculation to determine the correct.
energy of a 1fl*n* wavefunction seems to correlate with the amount of ionic

character of the wavefunction; the greater the ionic character, the poorer

the calculated energy.

In order to examine the effect of additions to this basis set on the

ground, ln+n* and 3

n#n* states, consider an additional basis function'wc on
carbon and an additional basis function wb on oxygen. If *ground’ *3 *,

and V3 * are the single determinant descriptions of the ground, ln4n*, and
n4n

165

’3n¢n* states described earlier in this Appendix, then double promotion
configurations of the from.wcwc, wowo, wcwo, and wowc can be formed which
will interact with the configurations pcpc, Popo’ pcpo, and popc' The

interaction matrix elements of H are the two electron integrals;

(Piwmklgi I ijn)

where
i = c,o
3:0:0
m = c,o
and n = c,o

This integral is likely to have its largest value if pi’PJ’ Wm
and wn are all on the same center and thus the wavefunction with the
greatest ionic character is likely to be affected most by such additions.
If‘wc and “t have significant density in the some regions of space as pc
Iand p0, they would.be expected to be especially effective in lowering the
energies of the ionic wavefunctions. Since certain short range d functions
can have significant density in the sane regions of space as pn functions
and since the most highly ionic wavefunction of the three states mentioned

1n-m* state, the (1 functions might have an important

above is that of the
preferential effect on this state.

Therefore, in.principle, the addition of d functions can be expected
to have an effect in lowering the transition energy of the l(n+n*) state in
formaldehyde. Similar remarks can also be made about the preferential
contribution of d functions to highly ionic states of other molecules
leading to an effect on calculated spectra which might be appreciable.
However, it should be emphasized that the lowering of the energy of a

given state by the present mechanism is restricted to be less than the

correlation energy associated with the pair of electron undergoing excitation.

166

Appendix D. Gaussian lobe functions

 

The basis functions are linear combinations of Gaussian lobe
functions, {hm}' Gaussian s-, p-, and d—type lobe functions have the

ferns:

via») (fiestas-roe]
was) N;1(%b5"{exp[-b(r-r1)2] - expE-b(r-r2)211
v30») = male??? {expt-b(r-r1)2] - exp[-b(r-r2)2] +

exPE-b(r-r3)2] - eXPE-b(r-rg)2]}.

I!

where points 1, 2, 3, and h are shown in the Figure for the pZ and dxy

functions.

 

 

a

 

 

Figure. Origins and lobe displacement vectors of Gaussian lobe p2 and

dxy functions.

.
5
- . '
~.' ,.
‘yi-l
. r
b-
o

 

   

"'William"[illflfflifll‘iiiflfg’iifl'ES