ELECIRDMC STRUCTURE or
mama INTERMEDEATES: .
CARBONYL mama AND LITHIUM
(mamas; THE ZERO-FIELD spumue
PARAMETERS or METHYLENE

Dissertation for the Degrae of Ph. D.
MECHieAN STATE UNWERSITY
RICHARD CHARLES UEDTKE
1975

LIBP RY

 

  

ichard Charles Liedtke
,a: J
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has been accepted towards fulfillment
of the requirements for

Ph.D.

\ degree in

Chemical Physics

   
 

Major professor

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ABSTRACT

.527 ELECTRONIC STRUCTURE OF REACTIVE INTERMEDIATES:
as! CARBONYL CARBENES AND LITHIUM CARBENES;

l
L

THE ZERO-FIELD SPLITTING PARAMETERS OF METHYLENE

By
Richard Charles Liedtke

Minimal basis Q initio SCF-CI calculations on the lowest
states of the carbonyl carbene, :CHCOH, and its fluorine-
substituted forms, :CHCOF and :CFCOH, yielded the state order-
ing E(3A")<E(1A')<E(1A") for :CHCOH and :CHCOF, with the dif-
ferential effects of correlation energy expected to make the
1A' state the ground state for :CFCOH. The state energy
separations in :CHCOH were calculated to be 1.09 eV for
1A' + 3A" and 1.39 eV for 1A" 4- 3A". A geometry search
for the :CHCOH states predicted equilibrium H-C-(COH) angles
of m125°(3A"), m103°(1A') , and m133°(1A") , very similar to
the :CH2 values. The effects of fluorine substitution on
the formyl group (:CHCOF) was found to have a minimal effect
on the electronic structure of the low-lying states, while
direct fluorine substitution (:CFCOH) resulted in a dramatic

stabilization of the 1A' state relative to the A" states.

Significant polarization of the carbonyl group was found to

1

a“Olupélny either substitution. The A' state Of =CHC0H

8flowed no spontaneous tendency to rearrange to ketene.
In addition, using a double-zeta quality basis plus a set
of oPtimized Li p functions, restricted open-shell SCF calcu-

lations were done on the triplet ground states of the

Richard Charles Liedtke

lithium-substituted methylenes, :CHLi and :CLi and the first

2:
excited triplet state (311) of :CHLi. The low-lying singlet
states were constructed via a single-excitation CI within the
set of virtual 1r orbitals from the ground state SCF. All
thezani states examined were found to have linear equilibrium
geometries, a characteristic expected to apply more strongly
mathe states of :CLiZ. Open-shell SCF calculations on the
stable fragments of both molecules indicated that all states
except the 12+ state of :CLi2 were at least nominally stable
to dissociation.

A refinement of a previous calculation of the spin-spin
mnfimibution to the zero-field splitting parameters of 331
“Ethylene was also done. The gaussian lobe function basis
was retained, but the CI wave function was enlarged to the
"best" 100 structures and improved via the iterative natural
orbital technique. The ZFS parameters obtained (D=O.722 cm-l;
E=0.053 cm-l) were in close agreement with the previous values
and,*with the inclusion of the most recent values for the spin-

orbit contribution, in agreement with the range of experimental

values.

ELECTRONIC STRUCTURE OF REACTIVE INTERMEDIATES:
CARBONYL CARBENES AND LITHIUM CARBENES;

THE ZERO-FIELD SPLITTING PARAMETERS OF METHYLENE

BY

Richard Charles Liedtke

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics
Program in Chemical Physics

1975

To My Parents

ii

ACKNOWLEDGMENT

The author wishes to express his sincere appreciation

to Dr. J. F. Harrison for his interest and guidance during

the course of this research.

iii

TABLE OF CONTENTS

Chapter Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES. . . . . . . . . . . . . . . . . . . ix
INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1
CHAPTER I - CARBONYL CARBENES. . . . . . . . . . . . 4
A. Preliminaries. . . . . . . . . . . . . . . . 4
B. :CHCOH . . . . . . . . . . . . . . . . . . . 7
C. :CHCOF . . . . . . . . . . . . . . . . . . . 22
D. :CFCOH . . . . . . . . . . . . . . . . . . . 34
E. Discussion . . . . . . . . . . . . . . . . . 44
CHAPTER II - LITHIUM CARBENES. . . . . . . . . . . . 48

CHAPTER III - THE ZERO-FIELD SPLITTING PARAMETERS

OF 331 METHYLENE . . . . . . . . . . . 65

A. The Electron Spin-spin Interaction
and the ZFS Parameters . . . . . . . . . . . 65

B. The Spin-spin Contribution to D and-
E for a CI Wave Function . . . . . . . . . . 68

C. The ZFS Parameters of 3

Bl Methylene. . . . . 71
APPENDIX I . . . . . . . . . . . . . . . . . . . . . 76
A. The Closed Shell SCF (CSSCF) . . . . . . . . 76
B. The Basis Set. . . . . . . . . . . . . . . . 79
C. The CI . . . . . . . . . . . . . . . . . . . 80
D. Natural Orbitals . . . . . . . . . . . . . . 101
APPENDIX II. . . . . . . . . . . . . . . . . . . . . 123

A. Basis Set. . . . . . . . . . . . . . . . . . 123

iv

 

Chapter

B. The Restricted Open Shell SCF (ROSSCF)
C. The Single Excitation CI (SECI). . . .
APPENDIX III . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY O C O O O O O C O O O O O O O O O

Page

123
127
135

141

Table

10

11

12

LIST OF TABLES

Page
Bond Lengths Fixed in Carbonyl Carbene
Calculations . . . . . . . . . . . . . . . . 6
:CHCOH SCF—MO's at 3A" cis Equilibrium
Geometry . . . . . . . . . . . . . . . . . . 8

:CHCOH Equilibrium Geometries and CI
Energies for the Three Lowest States,
3A", lA', and 1A". . . . . . . . . . . . . . 11
:CHCOH Largest Contributing Occupancies

in the Three Lowest States . . . . . . . . . 15
:CHCOH (cis) Active NO's and Occupation

Numbers for Three Lowest States. . . . . . . l8
:CHCOH (cis) Gross Mulliken Populations

and Net Charges. . . . . . . . . . . . . . . 21
:CHCOF Active MO's and Orbital Energies. . . 24
:CHCOF Principal Contributing CI

Structures for the Lowest States. . . . . . . 28
:CHCOF Active NO Occupation Numbers. . . . . 30
:CHCOF Gross Mulliken Populations

and Net Charges. . . . . . . . . . . . . . . 33
:CFCOH Active MO's and Orbital Energies

at the 3

A" cis Geometry. . . . . . . . . . . 35
:CFCOH Principal Contributing CI Struc-

tures for the Three Lowest States. . . . . . 41

vi

Table

13

14

15

16

17

18

19

20
21
22
23
24
25
26

27

:CFCOH Active NO Occupation
Numbers . . . . . . . . . . . . . .
:CFCOH Gross Mulliken Populations
and Net Charges . . . . . . . . . .

Occupied MO's at Ground State

Equilibrium Geometries: a) LiCH (32‘

3—
and b) Lizc ( Zg) . . . . . . . . .
Ground and Excited State Energies
and Equilibrium Bond Lengths for

LiCH and Lizc . . . . . . . . . . .

Gross Mulliken Populations and Net

Charges for LiCH and Li C . . . . .

2
Carbonyl Carbene Basis Set: STO—3G
Coefficients and Scaled Exponents .
:CHCOH Determinantal Basis for All

States. . . . . . . . . . . . . . .

:CHCOF 3A" CI Structures. . . . . .
:CHCOF 1A' CI Structures. . . . . .
:CHCOF 1A" CI Structures. . . . . .
:CFCOH 3A" CI Structures. . . . . .
:CFCOH 1A' CI Structures. . . . . .
:CFCOH 1A" CI Structures. . . . . .
Basis Set for LiCH and Lizc

Calculations. . . . . . . . . . . .

1

LiCH A SECI Structures . . . . . .

vii

)

Page

43

.102
.108
.115

.124

.129

Table

28
29
30
31
32

LiCH
LiCH
LiCH
Li C

Li C

SECI

SECI

SECI

SECI

SECI

Structures
Structures
Structures
Structures

Structures

viii

Page

130
131
132
133

134

Figure

10

LIST OF FIGURES

Coordinate System for Carbonyl

Carbenes . . . . . . . . . . .

0-0 Energies for :CHCOH (cis) States

:CHCOH Excitation Energies at 3

cis Geometry . . . . . . . . . .

:CHCOH (cis) Electron Density Contour

Maps for the Active a' NO's and the

Total Density. . . . . . . . . .

:CHCOF (cis) CI Energies for the

3 1 1

States A", A', and A" . . . .

:CHCOF (cis) Excitation Energies at

the 3A" Geometry . . . . . . .

:CHCOF (cis) Electron Density Contour

Maps for the Active a' NO's and the

Total Density. . . . . . . . . .

:CFCOH (cis) CI Energies for 3

1A', and 1

A.' C O I O O I O O O O

:CFCOH (cis) Excitation Energies at

3

the A" Geometry . . . . . . . .

:CFCOH (cis) Excitation Energies at

the 1A' Geometry . . . . . . . .

ix

A"

A",

page

13

14

19

25

27

31

37

38

39

Figure Page

11 :CFCOH Electron Density Contour

Maps for Active NO's and Total Density

at the 3A" cis Geometry . . . . . . . . . . 42
12 Effect of Fluorine Substitution on

the Energy Levels of Carbonyl

Carbenes. . . . . . . . . . . . . . . . . . 46
13 LiCH (32-) Electron Density Contours

for Non-Core MO's and the Total

Density . . . . . . . . . . . . . . . . . . 51
14 Li2C (32;) Electron Density Contours

for Non-core MO's and the Total

Density . . . . . . . . . . . . . . . . . . 52

15 Excitation Energies at the Ground

State Geometry for LiCH and LiZC. . . . . . 55
16 LiCH State Energies gs eLi-C-H’ . . . . . . 57
17 LiCH Orbital Energies gs eLi_C_H. . . . . . 59
18 Correlation of LiCH Levels and Dissocia-

tion Fragments. . . . . . . . . . . . . . . 62
19 Correlation of Li2C Levels and

Dissociation Fragments. . . . . . . . . . . 63
20 Relation of D and E to the Zero-Field

Level Splittings of a Triplet State . . . . 68

, 3
21 .CH2 ( B1) Energy 23 GHCH . . . . . . . . . 74
22 :CH2 (3B1) Spin-spin ZFS Parameters

Kg 8 75

HCH O O O O O O O O O O O O O O O O O O

INTRODUCTION

Methylene, :CHZ, is the parent species of a family of
compounds called carbenes, which have the distinguishing
property that they contain a formally divalent carbon.
The term methylene appeared in chemistry as early as the
1830's, and the halocarbene, :CC12, was proposed by Geutherl
in 1862 as an intermediate in the hydrolysis of chloroform.
However, it was only about 1910 through the work of Staud-
inger that firm ground was laid in understanding the mechan-
isms of carbene chemistry. It was Staudinger and Kupfer2
who obtained the first compelling evidence for the existence
of free methylene. Except for the suggestions of Mulliken3
in 1932, the theoretical investigation of carbene structure
was virtually non-existent until the 1950's. After 1950,
a prolific literature of divalent carbon has been seen to
develop in both the experimental and theoretical aspects
of carbene structure.

It was in 1950 that the first substantial spectrosc0pic

4'5 In 1959,

evidence of the carbene :CF2 was obtained.
Herzberg and Shoosmith6 succeeded in recording the first
absorption spectrum of methylene produced in the flash
photolysis of diazomethane at low pressure. Herzberg

7'8 made a detailed analysis of the high resolution

and Johns
absorption spectra of both the singlet and triplet states,

concluding that the ground state of :CH2 was a linear triplet

(3 =103°)

1A

2;), and the first excited state a highly bent (eHCH

10
Consistent theoretical structure calculations of increas-

9,10

ing accuracy, supported by the first successful observa-

tion of an esr spectrum of matrix-isolated :CH2 by two
11,12

groups in 1970, prompted a reinterpretation of the
absorption spectra by Herzberg in 1971.13 He found that
under the assumption of a bound 3A2 state, heterogeneously

predissociated by a 3

B2 state, an interpretation consistent
with a bent 3B1 ground state could be made. Furthermore,
under the new interpretation, the spectroscopic data pre-

dicted a 3

Bl angle of 136° and a C-H bond length of 1.078 A,
in striking agreement with the theoretical values. This
event was important in lending credibility to the predic-
tive power of quantum chemical calculations.

It has been in the last two decades that quantum chem-
ical calculation has developed into the powerful tool it
is today. This fact is a consequence of the phenomenal
development of high-speed reliable computing machines --
a development which has accelerated through the 1960's.
Calculations on methylene have progressed from the extended

14 to the most recent and

15

Huckel results of Hoffman gt 3;.
accurate ab initio study by Staemmler.
It was in the light of the history of carbenes and the
continuing interest in carbenes as reactive intermediates
in organic chemistry that the work presented in this thesis

was begun. The basic motivation was to extend the theoretical

understanding of carbene structure by examining the effects
of substitution on carbonyl carbenes, the simplest example

of which was taken to be :CHCOH. Of primary interest was

the electronic character and the equilibrium geometry of

the ground state and the lowest excited states, as well

as the ordering and approximate excitation energies of those
states. In addition, calculations were done on the lithium-
substituted methylenes, :CHLi and :CLiz, to examine the
effect of a strong electron donor as a substituent on methyl-
ene. Also included is a refined calculation of the Zero-

Field Splitting parameters of 3B methylene, first done by

16

1
Harrison. The importance of these parameters is connected
with the esr work cited above, the value of which is un-
questionable.

Good reviews of the structure and chemistry of carbenes
may be found in the literature; a few of the most compre—

hensive are cited in the bibliography.l7'18'19

CHAPTER I

CARBONYL CARBENES

A. Preliminaries

Ab initio SCF-CI studies of three carbonyl carbenes were
carried out to examine the effects on the carbene electronic
structure arising from fluorine substitution. The three
carbenes involved were :CHCOH, :CHCOF, and :CFCOH, written
more generally as :CRCOR'. The coordinate system and

relevant geometrical parameters are illustrated in Figure

l.

 

V [x-axis into paper]

Figure 1. Coordinate System for Carbonyl Carbenes.

The molecules were taken to be planar for the states of
interest, and only four of the many degrees of freedom were
allowed to vary. As shown in Figure l, the four parameters
included all the angles and one bond length. Two of the
angles, 8F and TF, define the formyl group geometry; the
third angle, 8, is what was called the "carbene angle".

Except for the carbon-carbon bond length, RCC' all bond

lengths were fixed at predetermined values. Table 1 gives
the bond length values used for each molecule.

For 8 # 180°, it is useful to distinguish between two
cases. Since it was desired to restrict 8 to be positive
and 1 180°, the convention was adopted that for the sub-
stituent R and the oxygen on the same side of the carbon-
carbon bond, the term "cis" would apply; the alternate
case was termed "trans".

Appendix I contains a description of the basis func-
tions used for this series of molecules as well as the
closed shell self-consistent field (CSSCF) theory used to
obtain the molecular orbitals (MO's). Also given is a
description of the configuration interaction method (CI)
for extending the SCF results. Briefly, the basis used
was a minimal set of STO-3G functions described by Pople
gt 31.,20 constructed in a primitive basis of nuclear-
centered cartesian gaussian functions (NCGTF). Appendix
I also defines the approximate natural orbitals (NO's)
discussed in this chapter and used to construct the atomic
populations associated with the total CI electron density.

In this chapter, as well as the rest of the thesis,
all results were obtained in atomic units (au), but some

results are given in other units where noted.

Table 1. Bond Lengths Fixed in Carbonyl Carbene Calcula-

 

 

tions.a
Molecule :CHCOH :CHCOF :CFCOH
b
RCO 2.340 2.340 2.340
C

RCF (formyl) ----- 2.532 -----

RC3 (formyl) 2.060b ----- 2 060

RCH 2.0787 2.0787 -----
d

RCF ---------- 2.457

 

 

3All values are in atomic units (bohrs).

bShalhoub and Harrison.21

CCsizmadia.22

d 23

Powell and Lide.

B. :CHCOH

This molecule is the simplest carbonyl carbene (excluding
:CCO), and, like its precursor :CHZ, was found to have a
bent triplet ground state. The MO's used to characterize
the molecule were constructed via a CSSCF calculation.
These MO's, their orbital energies at the ground state
equilibrium geometry and approximate character are listed

in Table 2. Following Hoffmann,l4

the MO of a' symmetry
localized as a "lone pair" orbital on the carbene carbon

is denoted "a", while the a" MO localized roughly as an
atomic p function on the carbene carbon is called "p".
Three other MO's were given descriptive symbols reflecting
their atomic associations: the la" (NCO) appears mainly

as a CO bonding function in the formyl group; the 9a' (00)
is strongly localized "p function" on oxygen perpendicular
to the CO bond; and the 3a" MO (n*) is a virtual MO (i.e.,
unoccupied in the SCF determinant) displaying primarily
anti-bonding character between the formyl carbon and oxygen.
These five MO's are the only MO's whose occupation was
allowed to change in the CI wave function, and therefore
they are called the "active" MO's. The eight lowest-energy
MO's were considered to represent the molecular core. The
occupation of these "core" orbitals was "frozen" in the

CI at two each. The electronic structure of the lowest
excited states is usually determined by excitations from

the valence electrons and not the "core". An unfortunate

 

 

 

 

Table 2. :CHCOH SCF MO's at 3A" cis Equilibrium Geometry

MO Character -e(au)
la' 9 20.24
2a' 11.15
3a' 11.02
4a' 1.30

CORE
5a' 0.90
6a' 0.73
7a' i 0.58
8a' 0.51
1a ECO 0.47
9a' 00 0.32
2a" p 0.25
10a' 0 -0.12
3a" Tr* -0031
lla' -0.58
Unused virtual MO's
12a' -0.70

 

 

discovery near the end of the work on this molecule was that
the SCF determinant used was not the lowest root. The lowest-
energy single determinant has a doubly-occupied 0 MO and an
empty p MO. As a result, there was some question of the
description of the 0 MO. Some consideration of the orthogon-
ality constraints on the a system above the nine occupied

a' MO's, the apparent character of the 0 MO itself, and the
power of the CI to adjust for such effects led to some
confidence that the net result of the less desirable MO's

was small. With regard to this discussion it is convenient
to adopt another notational convention used by Hoffmann.
Since it is usual in carbenes that the CSSCF determinant has
either the 0 or p MO as the highest occupied MO and the other
as the lowest unoccupied MO, Hoffmann calls the carbene a

"02 carbene" if the highest occupied MO in the CSSCF is

2 carbene" if the p MO holds that position.

2

the 0 MO and a "p

In those terms :CHCOH could be called a o carbene.
To obtain the wave functions representative of the

molecular electronic states, a CI calculation was done in

a basis of 100 determinants constructed by permuting the

six active electrons (3a and 33) among the ten active spin-

orbitals (5 spatial MO's). A list of the determinants used

can be found in Appendix I. Since only three a" MO's (one

for each first-row atom) could be constructed in the minimal

basis, all three were included in the active set to insure

as much flexibility as the CI could afford.

For each of the three lowest states of :CHCOH (3A",

10
1A', 1A"), equilibrium geometries were found by a limited.
optimization of the four geometrical parameters with respect
to the total energy. Each state was found to be bent
(8 f 180°), and the cis and trans equilibria were nearly
degenerate in energy, with the equilibrium values of the
geometrical parameters essentially identical. These results
are displayed in Table 3. In contrast to a 1972 study of

24

this molecule by Bodor and Dewar, the first excited state,

lA', did not display any spontaneous tendency to rearrange
to ketene. Their work, using the semi-empirical MINDO/Z
technique and a one-determinant closed shell representation

of the 1

A' state, predicted a barrierless migration of the
formyl hydrogen to form singlet ground state ketene. Since

ab initio SCF-CI results for :CH have shown that the first

2
singlet (1A1 for :CH2) is a strong mixture of both the
02 and p2 closed shell determinants, results of calculations

ignoring this fact are at least suspect.
The basic results of the :CHCOH, as reported here, are

3

that :CHCOH has a A" strongly bent ground state (8 2 125°);

the first excited state is an even more strongly bent (B =
103°) 1A'; and the second excited state, 1A", is still
significantly bent (B 2 133°). In addition, the effect
of the formyl group is not significant in distinguishing
the cis and trans conformations, the difference in energy
between the two forms being 5 2 kcal/mole for any state.
The results thus far are very close to the structural

properties of the parent methylene.

11

 

 

 

 

 

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o~.HmH oh.mHH Hmm.~ om.mOH mmmmm.mvan mama» .ma
om.m~H oH.omH mum.m ob.moa vnmmm.meal mwo .«H
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me me louroeeuum a “stem oueum
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12

The predicted 0-0 transition energies are shown in
Figure 2. Since the trans levels are nearly degenerate
with their cis counterparts, they are not shown in the
3

figure. The vertical excitation energies at the A" cis

equilibrium geometry are displayed in Figure 3, which also

3A"

includes the third and fourth excited states at the
cis geometry.

The electronic structure of each state was analyzed by exam-
ining the constituent contributions to the CI wave function.
The determinants are the obvious constituents of the
wave function, but more fundamental are those combinations
of determinants which are spin eigenfunctions and have the
same symmetries as the state of interest. Such entities
are called "structures". Each structure is composed of
determinants defining the same distribution of electrons
among the spatial MO's, i.e., defining a definite orbital
occupancy. Table 4 lists the largest contributing "occu-
pancies” and their weights in the CI wave function. In
the table, only the active orbital occupations are given,
the "core" being understood to be completely filled in
each determinant. From the table, it can be seen that
the A" states are dominated by the occupancy involving
two unpaired electrons, one in the 0 MO and one in the p
'MO. The next most important occupations in these two
cases involve single and double excitations with respect
to the dominant term, i.e., one or two electrons in the

dominant occupancy may be considered to have been moved

40“

35'

30"

20‘

15‘

AE (keel/mole)

 

O"

13

 

 

 

 

 

3 II
A

Figure 2. 0-0 Energies for :CHCOH (cis) States.

14

70‘

60‘ r 1A

 

 

 

 

 

30q 1.49 eV

20*

AE (keel/mole)

10‘

 

 

 

o 3A”

 

Figure 3. :CHCOH Excitation Energies at 3A" cis Geometry.

15

 

 

 

 

 

Table 4. :CHCOH Largest Contributing Occupancies in the
Three Lowest States.
Structure Occupation
State “CO 0 p 0 fl* Coefficient

2 2 l 1 0.936
3A" cis 2 1 l 2 -0.182
1 2 l l 1 0.176
1 2 2 1 -0.150
2 2 2 0.890
2 2 2 -0.272
1A' cis 1 2 1 2 -0.218
2 2 2 -0.176
2 l 2 l -0.l75
2 2 1 1 0.908
2 2 l 1 0.249
1A" cis l 2 1 1 1 (a) 0.211
1 2 l l l (b) 0.163
2 l l 2 0.148

 

 

16

to higher energy MO's of the same symmetry. The excitations
serve two roles in the CI wave function, they correct the

MO description, which may be inappropriate for the state

of interest, and they allow for the correlation of electrons
of opposite spins - an effect not achieved completely by

the SCF wave function. The single excitations usually

only contribute via the first role, while the doubles partici—
pate in both. For the two A" states of :CHCOH under considera-
tion, the MO's are derived from a closed shell SCF wave-
function, and the large contributors outside of the dominant
occupancy are principally involved with refining the MO's

to describe open shell states.

1

Turning to the A' state, a different situation becomes

apparent; namely, that the second largest contributing oc-
cupancy is not describable as a double excitation refining

the 0 MO. The MO excited to is of a different symmetry type.

2

Hence, the second occupancy listed, the p determinant,

contributes a completely different aspect of the electronic

1

distribution in the A' state than does the dominant 02

2

determinant. The c and p2 determinants represent different

character in terms of preferred geometries; the p2 has a

2

linear (B 2 180°) equilibrium, while the a is strongly

bent (8 3 100°). The conclusion is that just one of these

determinants could not even qualitatively represent the

1

essential electronic structure of the A' state of :CHCOH.

1

In reference to the A" state entries in Table 4, the two

rows labelled a) and b) indicate that for four unpaired

17

electrons there are two completely different ways to couple

1A" structure.

the spins so as to form a

Another way to analyze a CI wavefunction is to construct
its NO's, i.e., the eigenfunctions of the one-particle density
matrix. They provide the simplicity of a set of one-particle
functions, usually more localized than the MO's, but with
non-integral occupations. Along with their eigenvalues,
called occupation numbers, the NO's yield the same electron
density distribution as the CI wave function, and therefore
are convenient for use in Mulliken atomic population analyses.
For the 3A", 1A', and 1A" states of :CHCOH, Table 5
lists the NO's corresponding to the MO active set along
with their occupation numbers. Electron density contour
maps of the a' NO's and the total CI density of the 3A"
cis state are shown in Figure 4. The contour values range
from 0.005 electrons/(bohr)3 to 0.075 electrons/(bohr)3
in steps of 0.01 electrons/(bohr)3. These contour values
are used consistently throughout the thesis and seem to be
appropriate to electron densities in the valence region.

For the A" states, the o and p NO's are strongly localized
on the carbene carbon, and their occupation numbers (both
essentially one) indicates the unpaired spin is therefore

1A' state,

also associated with the carbene carbon. For the
it is apparent that although the 0 NO is very important,
the p NO contains a substantial contribution. The sig-

nificance of the p NO is important in the chemical properties

of carbenes, since the electrophilicity of singlet carbenes

18

 

 

 

Table 5. :CHCOH (cis) Active NO's and Occupation Numbers
for Three Lowest States
State
NO 3A" 1A' 1A"
NCO 1.870 1.859 1.881
00 1.994 1.989 1.991
p 1.002 0.229 1.003
c 1.004 1.829 1.008
n* 0.129 0.093 0.117

 

 

19

 

TOTAL

 

Figure 4. :CHCOH (cis) Electron Density Contour Maps for
the Active a' NO's and the Total Density.

20

is tied directly to the amount of a" density on the carbene
carbon, i.e., less density indicates greater electrophilicty.
Hence, this property is not dealt with preperly at the SCF
level of description. Also of note, although not shown, is

1A' state is delocalized over

the fact that the p N0 of the
the whole a" system.

As mentioned above, the NO's provide a very convenient
means for assigning electron populations to atomic centers.

25 and

This was done, according to the method of Mulliken,
the resulting gross atomic populations are given for each
state in Table 6, divided into a' and a" contributions.

The total gross atomic populations are just the sum of the

a' and a" parts for each atom. Also given is the net charge
on each atomic center. Again, in the results given so far,
no significant differences were seen for the trans configura-
tions. In the table, the carbene carbon and its associated
hydrogen are listed first and subscripted with a "c".

The population analysis reveals small net charges on any
atomic center (indicative of covalent bonding), a carbene
carbon slightly negative in the A" states and positive

in the A' state, and a consistently negative oxygen (not
surprising since oxygen is the most electronegative element
present). Here again the significance of the a" density

1A' state

(0.20 electron) on the carbene carbon in the
indicates a less electrophilic character than the 02 SCF
would predict.

The equilibrium geometries and energy ordering of the

21

 

 

 

 

 

 

 

vo.+ Incl mm. vo.+ III: om. vo.+ III: mm. m
oo.l mo.H hm.m mo.u wo.a no.5 mo.| oa.a mm.m o
No.1 mo.H oo.m Ho.u mm. oa.m vo.+ mm. Ho.m O
o
no.1 mm. mH.m mo.+ om. Nb.m «0.1 mm. mo.m U
o
mo.+ Ina: em. no.1 III: mo.H mo.+ Illa mm. m
:m .m 8” .m 3” .m
o s c e s .s o c : Housoo
EH .5 2m
.momumsu #02 was msofiumasmom :oxfiaasz mmouo Amflov moomuu .m wanna

22

low-lying states, as well as the essential electronic struc-
ture of :CHCOH was seen to be very close to those of the
precursor, :CHz. It should be remarked that differences
in the correlation energies of the three states investigated

1A' state

would have the probable result of lowering the
energy relative to the A" states. This relative lowering,
expected to be mainly due to the pair correlation of the

1A' state, was anticipated to

doubly-occupied 0 MO in the
be several tenths of an eV, certainly less than 1 eV, and
therefore not enough to change the prediction of a 3A"
ground state. The effect of the formyl group was not

disruptive of the methylene-like electronic structure.

C. :CHCOF

Fluorine was substituted for hydrogen in the formyl
group of :CHCOH. In the series of molecules treated in
this chapter, :CHCOF may be called the indirectly—substi—
tuted carbene, the substitution site not being on the
carbene carbon itself. The C—F bond length was taken as
that listed in Table l. A preliminary geometry search

3A" cis equilibrium geometry of :CHCOH indicated

3

near the
negligible differences in geometry for the :CHCOF A"
equilibrium. With this datum in mind and the intent of
determining the electronic structure of these compounds

rather than their detailed geometries, it was decided to

adopt the equilibrium geometries of the states of :CHCOH

23

in the calculations on :CHCOF and :CFCOH (discussed in the
next section).

The basis set increase associated with the fluorine re-
sulted in one a" and three a' functions added to the MO set.
Of these new MO's, the fluorine non-bonding a' MO and the new
a" (essentially an atomic p function) were added to the active
MO's in the CI. Table 7 lists the active MO's for :CHCOF,
their symbolic description, and their orbital energies at

the 3

A” cis assumed equilibrium geometry. The :CHCOF active
MO's, although given similar symbols, differed in some sig-
nificant aspects from the corresponding MO's of :CHCOH. The
"CO was more like a F-O anti-bonding function; the p MO became
more delocalized onto the formyl carbon; and the N* became
essentially a C-C antibonding MO. The CO and o MO's retained
their localized character. In short the fluorine substitu-
tion generally effected a re-characterization of the a" or-
bitals in the CSSCF description.

The CI wave function was constructed using the set of
seven active MO's {pF 0F ECO 00 o p n*} to form structures of

the required symmetries. With the o doubly-occupied, all

F
single and double excitations off the SCF determinant (02)

were formed using six active MO's. Then two singlet and two
triplet structures involving excitations from the CF were
added. This procedure resulted in a total of 214 structures.
A more detailed breakdown is given in Appendix I.

3 1 1

The CI energies for the A", A', and A" states are

compared for the cis geometry in Figure 5. Again the energy

24

Table 7. :CHCOF Active MO's and Orbital Energies.

 

 

 

MO Character -e(au)
la" pF 0.597
lla' OF 0.480
12a' 00 0.421
2a" H 0.413
13a' 0 0.289
3a" p -0.131

4a" H -0.357

 

 

25

 

 

 

 

4O ( ) IA" -247.l2062 au
. 41.5

30-

(266) ‘A’ —247.14440 au

13
O
E 20-
b
C
u
35
L1J 10-
<

O-J 3A" -247.18682 au

Figure 5. :CHCOF (cis) CI Energies for the States 3A":
1A', and 1A".

26

differences between the cis and trans geometries for each state

3A" state and less than

are less than 2 kcal/mole for the
1/2 kcal/mole for the singlets. Because of the problem of
selection of the SCF determinant for :CHCOH mentioned earlier,

1

the lowering of the A' state relative to the A" states was

difficult to interpret in terms of the fluorine substitution
effects. The vertical excitation energies at the 3A" cis
geometry are illustrated in Figure 6, which indicates the
appearance of the first excited triplet (3A') at an excita-
tion energy of about 3.23 eV. In terms of relative energies,
the indirectly-substituted fluorine did not cause any sig-
nificant departures from the :CHCOH results.

An examination of the electronic structure was begun
by an analysis of the important structures in the CI wave
function for each state. Table 8 lists the principal struc-
tures by specifying their active MO occupation and includes
their weight in the wave function. It is immediately
apparent that the A" states contain a strong second struc-
ture. The effect of the second structure in both cases is
to correct the a" MO's, which are presumably not close to
optimal as a one-particle basis to describe the a" density

2 1A' state has

of the A" states. The a structure in the
increased in weight relative to :CHCOH, but, again, that
may easily be due to the :CHCOH SCF problem, so no conclu-
sions are drawn.

The information in the CI wave function was again ex-

tracted through the construction and examination of its

27

 

(87.8)

80-

 

] 1*
(73.2)?) A
2.69 eV

60-

 

40- (43-3) 1.61 eV
(325) 11—

 

AE (keel/mole)

 

 

 

 

<:)J fizxra

 

Figure 6. :CHCOF (cis) Excitation Energies at the 3A"

Geometry.

28

Table 8. :CHCOF Principal Contributing CI Structuresfkn:the
Lowest States.

 

Structure Occupation

 

 

 

State pF 0F 00 “CO 0 p fl* Coef.
2 2 2 2 1 1 .874
3A" cis 2 2 2 2 1 1 -.365
2 2 2 1 2 1 .132
2 2 2 l 1 2 -.127
2 2 2 1 1 2 -.114
2 2 2 2 2 .930
2 2 2 2 2 -.199
1A' cis 2 2 2 2 1 1 -.l47
2 2 2 2 2 -.132
2 2 2 2 l 1 .110
2 2 2 2 1 1 .859
2 2 2 l 1 2 -.347
1A" cis 2 2 2 2 1 l .165
2 2 2 1 l 1 1 .163
l 2 2 2 1 2 -.l45

 

 

29

NO's. The active NO's and their occupation numbers for each
state are listed in Table 9. The A" states are described
essentially by doubly-occupied NO's up to the o and p NO's,
which each host one electron. The 0 NO is slightly de-
localized onto the oxygen, but is roughly unchanged from its
MO counterpart, as can be seen in Figure 7. The significant
changes induced by the CI were to localize the a" unpaired
electron in the A" states on the carbene carbon, the p NO
for :CHCOF appearing almost identical to the p NO for
:CHCOH. Thus, the distribution of unpaired spins is essen-
tially restricted to the carbene carbon and isolated from
the influence of the substituted fluorine. The effect on

the 1 1

A' state is small. The 0 NO for the A' state is
mainly restricted to the carbene carbon and the p NO is
very similar to the :CHCOH p NO.

The apparent effects of indirect substitution seemed
very small, and an examination of the charge distribution
was made using a Mulliken population analysis of the NO's for
each state. The results are shown in Table 10. Comparing
the populations of :CHCOF with those of :CHCOH, in the A"
states there is seen to be a general shift of a' density
from the hydrogen and both carbons to the more electronega-
tive fluorine and oxygen, accompanied by a small return
of charge from fluorine in the a" system. The electro-
negativity of the oxygen seems to have been enhanced in

the presence of the fluorine. The result for the formyl

carbon, attached to both oxygen and fluorine, is the most

30

 

 

 

Table 9. :CHCOF Active NO Occupation Numbers
State
NO 3A" 1A' 1A"
OF 2.000 2.000 2.000
pF 1.999 1.999 1.999
00 1.994 1.995 1.990
ECO 1.874 1.859 1.878
o 1.005 1.870 1.009
p 1.002 0.083 1.005
r
H 0.126 0.193 0.119

 

 

33

 

 

 

 

 

 

 

ma.1 mm.a H~.e mH.1 oo.a mm.e ma.1 ~o.a H~.e m
4H.1 mo.H eo.e mH.1 so.H mo.e oa.1 HH.H mo.e o
mm.+ oo.a «5.4 mm.+ mm. we.e em.+ Ho.H me.e u
no.1 mo. oo.m Ho.+ as. Ne.m ao.1 mo.H em.e oo
ao.+ 1--- mo. mo.+ 1111 so. ao.+ 1111 no. or

m ems .ms 0 so: .mc m soc .mc noucoo

-. g.” o «H .- <m
.momumso uoz use meoeumaseoo corneas: umouo moors" .oH manta

34

profound, causing it to be about a quarter of an electron
deficient. For the 1A' state, the gross result is similar,
with the interesting additional result that the a" popula-
tion on the carbene carbon appears to have decreased slightly,
suggesting an electrophilicity equal to or greater than

that of the 1

A' state of :CHCOH. An obvious phenomenon,
common to all three states, is the strong polarization of
the CO group in the direction C+O-. It would seem that the
CO group "insulates" the carbene carbon vicinity, harboring
the c and p electrons, from the strong effects of the

fluorine, and thus ameliorating greatly any changes in

the basic electronic structure of the low-lying states.

D. :CFCOH

In this case, the fluorine was substituted directly
on the carbene carbon. The C—F bond length assumed is given
in Table 1. As done for :CHCOF, the geometries used were
those found for :CHCOH.

Although the same number of MOfs resulted from the SCF
calculation as in :CHCOF, the presence of the fluorine
adjacent to the carbene carbon required a greater flexibility
of the UP MO in CI excitations. The set of active MO's,
listed in Table 11, was increased to nine MO's, including

the two highest a' virtual MO's.

35

Table 11. :CFCOH Active MO's and Orbital Energies at the
3A” cis Geometry

 

 

 

MO Character -e(au)
la" (pF) Fluorine p orbital .571
lla' (0F) Fluorine non-bonding .570
2a" (NCO) C formyl-O bonding .461
12a' (00) Oxygen non-bonding .410
13a' (0) Carbene "lone pair" .264
3a" (p) mCarbene p orbital -.l49
4a" (U*) C-C anti-bonding -.365
14a' (0') a' virtual -.559

15a' (0") a' virtual -.689

 

 

36

The structures possible using the active MO's were too num-
erous to handle within the limits of technology and expense,
so that the arrangements of the 5d and 58 electrons among
the 18 available spin-orbitals were restricted as follows.
All possible excitations were formed within the first seven
MO's, deleting all excitations beyond singles and doubles

2 determinant. Then several selected double

off the SCF o
excitations into the o' and o" MO's were added. The result
was a total of 319 structures of all four possible spin

and spatial symmetries. Appendix I gives the distribution
over symmetry type and lists all the structures.

The O-O transition energies and absolute state energies
for the three lowest states at their cis geometries are
illustrated in Figure 8. As before, the cis and trans
configurations of each state differed by an insignificant
amount -- less than 1 kcal/mole in this molecule. The

1A' state relative

dramatically increased stability of the
to the A" states is immediately apparent and reflects some
significant change in the basic electronic structure in
this directly-substituted molecule relative to the two
previous molecules. In fact, the differential effects of
the inclusion of correlation energy would almost certainly

lA' ground state. For this reason, the vertical

3

predict a
excitation energies at both the A" cis and 1A' cis geom-
etries have been included (Figures 9 and 10, respectively).

The principal structures (on the basis of coefficient)

37

 

 

 

 

1 ll
. —247.l2772 au
(35.2) A
30-1
A 4
.22
O
E 20-
b
O
U
x
\_/
L1] 10-
<1
Vac
-247.17744 au
(4.0)
O‘ 3A" —247.18383 au
Figure 8. :CFCOH (cis) CI Energies for the States 3A", lA',
and A".

38

 

 

 

 

 

 

 

 

 

(57.2) /[\

33 2'48 eV
3

40" 1 N
\ A
'6' (37.0) 1.150 eV
U
.1
Lu 20" 1 I
<‘ «24> .I‘A

<:)J :aBCO

Figure 9. :CFCOH (cis) Excitation Energies at the 3A"

Geometry.

39

 

 

 

 

 

2‘3 .
C) 'VACI
E 40‘ (42°) 1.84eV
'3
u .
x
“J 20-
G
3 n
(:)J (4J2) (£SEI- 1‘3:

 

 

 

Figure 10. :CFCOH (cis) Excitation Energies at the 1A'

Geometry.

40

in the CI wave functions of each state are given in Table
12. Due to the delocalized nature of the a" MO's, the A"
states require a large orbital correction structure, ap-

pearing in each case as a single excitation within the a"

system. The 1A' state is more strongly dominated by the
02 SCF determinant, the 0 MO having become significantly
2

stabilized by the adjacent fluorine; and the p determinant
has faded in significance as a component in the qualitative

electronic structure.

1 3

Partly accounting for the decreased A" + A" energy
is the delocalization of both the o and p NO's onto the
fluorine. This argument can be made in terms of the NO's
because, if the CI were redone in the NO basis, the op
structure would be much more dominant than it was for the

1A" + 3A" energy for a op occupancy is

MO's. Since the
just 2ch (twice the exchange integral for the o and p
orbitals), which diminishes in magnitude as the average
distance between 0 and p electrons, the delocalization of
the o and p NO's is a significant factor in decreasing

1 3

the A" + A" energy. Figure 11 shows electron density

contours for the a' NO's and the total electron density

at the 3

A" cis geometry. The active NO's and their occupa-
tion numbers are listed in Table 13. The relative unim-
portance of the 0' and c" NO's in the A" states is apparent
from their near-zero occupation numbers, reflecting a very

small mixing of the 0' and o" MO's in the lower a' MO's.

*
It is interesting to notice that the n NO (mostly C-O

41

Table 12. :CFCOH Principal Contributing CI Structures for the
Three Lowest States.

 

 

Structure Occupation

 

 

 

State pF OF ECO co 0 p n* Coef.
2 2 2 2 1 1 .864
2 2 2 2 l l —.375
3A" cis 2 2 2 1 2 1 .161
2 2 1 2 1 2 -.152
2 2 2 1 1 2 -.129
2 2 2 l 2 1 .119
2 2 2 2 2 .950
2 2 2 2 l 1 .169
1A' cis 2 2 2 2 2 -.161
2 2 2 2 2 -.103
2 2 2 2 2 -.098
2 2 2 2 1 1 .868
2 2 1 2 1 2 .344
2 2 1 2 l 1 l .173
1A" cis 2 2 2 2 1 1 -.164
2 2 2 1 2 1 .113
2 2 l 2 l 2 -.107

 

 

42

 

 

3A" cis

:CFCOH Electron Density Contour
NO's and Total Density at the

11.

Figure

43

 

 

 

Table 13. :CFCOH Active NO Occupation Numbers
State
NO 3A" ‘lA' 1A"
OF 1.999 2.000 1.998
pF 1.998 1.989 1.996
00 1.986 1.996 1.984
ECO 1.874 1.863 1.890
0 1.012 1.959 1.017
p .999 .044 1.008
*
n .132 .149 .108
0' .000 --- .000
o" .000 --- .000

 

 

44

1

anti-bonding) has a larger occupation in the A' state than

does the p (localized on the carbene carbon) -- a situation

1A' state and associated with the

also found in the :CHCOF
increased ionic character of the C—0 bond in the fluorine-
substituted species.

The Mulliken population analysis, summarized in Table
14, shows the effects on the charge density due to direct
fluorine substitution. As in :CHCOF, the fluorine draws
substantial charge through the a' system and returns a
smaller amount to the a" system, which doubles the carbene

1A' state relative to :CHCOF and

1

carbon a" density in the
implies a much less electrophilic A' than either :CHCOF
or :CHCOH. There is still a significant C-O polarization,
although not as severe as in :CHCOF. The effect on the
electronic structure is undoubtedly the stabilization of
the 0 orbital by the fluorine, leading to the transition

energies shifts and a singlet, instead of a triplet, ground

state.

E. Discussion

 

The effects of fluorine substitution on the relative
energies of the low-lying states are summarized in Figure
12. It should be repeated that the expected result of
including the correlation energy is to lower the 1A' state
relative to the A" states for each molecule, leading to

a 1A' ground state for :CFCOH. The results of direct

45

 

 

 

 

 

 

 

so.+ 1--- no. mo.+ 1111 mm. oo.+ 1111 am. e
oH.1 mo.H oo.e mH.1 No.a ma.» oH.- mo.H mo.e o
oo.+ mo.H Hm.e mo.+ om. om.e eo.+ Ho.H no.4 o
ma.+ em. me.e HH.+ em. mm.m NH.+ No.H ee.e oo
oo.1 ee.H oH.s ae.1 ee.a mm.e oa.1 oo.H o~.e a
a. m pm on” an n.“ -m
0 C C 0 C G v C C Hmfich
=¢H naifi 34m
.momumnu uoz one msofiumaomom coxwaaoz mmouo moumuu .va manna

46

 

 

 

 

 

 

/ \
40. 1 / 41.5 \
A ' / \\
37.8 \
‘ ¥
35.2
1 l
A 30. A
m 29.6 \\\
g . 26.6 \\
_\_ ‘1‘
U 20' 1
g 1
Lu 1‘
<1 \
10‘ \
\
1
1
1 \.
4.0
3 I!
0‘ A

 

 

 

 

:CHCOH __ ICHCOF “ :c FCOH

Effect of Fluorine Substitution on the Energy
Levels of Carbonyl Carbenes.

Figure 12.

47

fluorine substitution are very close to those found in

26 The effects of indirect substitution

studies on methylene.
are significant in demonstrating the insulating properties

of the carbonyl group and the resulting minimal effect on

the electronic structure of the low—lying states. A parallel
study on carbonyl nitrenes (isoelectronic with the carbenes)

21 revealed a similar effect. In

by Shalhoub and Harrison
their study, CH3 and OCH3 were also used as substituents,
with the result that the A" states were insensitive to such
substitutions. In their study the analogous p and o orbitals
on nitrogen were localized on that center. It is this
localization of the critical one particle densities, common
to both the carbenes and nitrenes which makes the carbonyl
group so effective in minimizing substituent effects on
the gross electronic structure and perhaps the chemical
reactivity.

Two interesting studies should be pursued to extend
this work. One is the investigation of the barrier to for-
mation of ketene, discussed earlier. Experimentally, there
is no evidence for the presence of the reactive singlet
carbonyl carbene, but ketene is found to be present. In
this regard, a theoretical investigation of the rearrange-

ment barrier in the 1

A' state would be helpful. A second

rearrangement reaction which could be followed theoretically
is the formation of the heterocycle oxirene. The stability
and electronic characterization of that species as an inter—

mediate is also of interest in reactions involving the Wolff

rearrangement.

CHAPTER II

LITHIUM CARBENES

In this chapter, the results of an ab initio SCF study
of lithium-substituted methylene are presented. Lithium
has as its first unoccupied atomic orbital a low-lying p-
type function. Hence, the two molecules studied, LiCH
and Lizc, were expected to be characterized by low-lying
excited states. Also, in contrast to the mainly electro-
negative substituents of Chapter 1, lithium is a good
electron donor and it was anticipated that a substantial
amount of c electron density would shift from lithium to
carbon, and some amount would be returned to lithium via
the n system.

The level of calculation for the triplet ground states
of both molecules and the first excited triplet of LiCH
was that of a restricted open-shell SCF (ROSSCF). The
basis set of contracted nuclear-centered cartesian gaussian
functions was of double zeta quality and included one 4-
component lithium p function, whose exponents and coef-
ficients were optimized by William527 in an SCF calculation

2P state. The other excited state wave-

of the lithium
functions were constructed via a single excitation CI (SECI)
calculation using the ground state MO's. Further details

of the basis set, CI wave functions and calculational methods

can be found in Appendix II.

48

49

All the states examined for these molecules were linear
and at least nominally bound. The ground states were found

to be of 3

2- symmetry (32- for LiCH, 32; for Li2C). Table
15 contains a list of the ground state MO's and their or-
bital energies for both molecules at their equilibrium
geometries. Figures 13 and 14 display electron density
contour maps of the MO's above the atomic core orbitals and
the total molecular density for LiCH and LiZC, respectively.
In LiCH, the two MO's which primarily determine the bonding
characteristics are the 30 (covalent C-H bonding) and the
highly ionic 40 (C-Li bonding). For Li2C, which has no
covalent bonds, two ionic C-Li bonding orbitals, similar
to the 40 in LiCH, can be formed as linear combinations of
the Zen and 30g MO's. These characteristics are apparent
in the contour maps.

From the ground state electronic configurations of
LiCH and LiZC, each having an occupancy describable as (0
core)2(wx)l(wz)1, with the two single electrons triplet

1A and a non-degenerate 12+

coupled, a doubly degenerate
state can be constructed by alternate couplings of the

n electrons. Virtual orbital representations of these
states were constructed and improved via SECI calculations
within the U orbitals. Table 16 gives the ground and ex-
cited state energies and equilibrium bond lengths for both
molecules. The fact that no 0 excitations were allowed

in the SECI wave functions undoubtedly contributed to the

nearly identical geometries of the three lowest states

Table 15. Occupied MO's at Ground State Equilibrium
’ Geometries: a) LiCH (32‘) and b) LiZC (32;).

 

 

 

 

a)
MO Character -e(au)
10 Carbon ls 11.123
1 20 Lithium ls 2.461
30 C-H bonding .581
4c C-Li bonding (ionic) .305
EX Carbon px .042
Hz Carbon pz .042
b)
109 Carbon ls 11.093
209 Li-Li bonding o 2.409
lou Li-Li anti-bonding o 2.409
Bog Carbon 25 .395
Zou Carbon py (lonlc) .164
"Xg Carbon px .006
U2 Carbon pz .006

 

 

 

51

.muwmson mac m
can m.oz 0&001soz mom musoucou aoamsoo couuooam flaw venom“
I M .

 

 

.me mesons

 

51

one o

 

 

.mueusoo Hence on»

OS 0H001coz mom whoousoo mpwmcoo souuooam A1wmv moan

.ma enemas

 

52

.auflmsoo

Hmuoe on» use m.oz ouoo1coz you musousoo Swanson sonuooam Amwmv UNHQ

.ee mesons

 

53

 

 

 

Table 16. Ground and Excited State Energies and Equilibrium
Bond Lengths for LiCH and LiZC.
Molecule State Energy(au) RC-Li(b°hrs) RC_H(bohrs)
LiCH 3 -45.697l3 4.40 2.08
1 + -45.69438 3.58 2.06
1 -45.73662 3.59 2.06
3 ‘ -45.78154 3.59 2.07
L12C
1 ; -52.53350 3.73 ----
1 g -52.56700 3.76 ----
3 ; -52.60479 3.78 ----

 

 

54

of each molecule. As a consequence, the 0-0 transition
energies are the same as the vertical excitation energies.
An energy level diagram illustrating both sets of energies
is given in Figure 15 for both molecules. It is seen that
the excitation energies, E(12;)-E(1Ag) and E(1Ag)-E(3Z;),
in Lizc are smaller than the corresponding separations

in LiCH. This result is due to the delocalization of the
occupied n MO in LiZC compared to LiCH. Outside of small
corrections due to the single excitations, the two dif-
ferences are largely determined by the exchange integral

connecting the two occupied n MO‘s:
AE < 2KTr n .
x

For LiCH, 2K = 1.27 eV, and for Li C, 2K = 1.08
anz 2 Trx'"z

eV. Hence, the extent to which n density can become more

diffuse on carbon or delocalize to the lithium has a rather

significant effect on the predicted relative energies of

the lowest excited states. The expected effect of the

inclusion of correlation energy would be to preferentially

1A state relative to the 2 states by some fraction

lower the
of an eV, which would not be sufficient to alter the pre-
dicted state ordering, but it would reduce the singlet-
triplet splitting. This correlation energy difference is
principally the pair correlation energy associated with

the U MO.

LiCH was bent from its linear equilibrium geometry

55

 

 

 

 

 

 

 

 

 

 

 

547 1‘ 12+

A . 2 37 0V 1
(l) E
_ 44.7 g
CE) 40. I4 0V
2.

« 1
g) 28.4 F A 'A
\/ 123eV 23] g
LU 20. OEV
<1

J 3 — , 3 —

O 2 29

LiCH LIZC

Figure 15. Excitation Energies at the Ground State Geometry
for LiCH and LiZC.

56

holding the bond lengths fixed. The resulting energy
dependence gs angle is shown in Figure 16. The preference
for a linear configuration is difficult to rationalize
because several small effects are involved, and their
magnitudes are not reliably estimated at present. Consid-
eration of the stabilization of the in—plane U MO through
acquisition of s-character, stabilization of the bonding

3c and 4c MO's due to loss of s-character, the effects of
incomplete orbital following, and bond-bond repulsion

must all be taken into account with a good estimate of
their relative importance. Without attempting such a

task here, it does seem possible to give an argument as to
why these ionic lithium-substituted methylenes should prefer
linear geometries as compared to methylene itself, which is
predicted by theoretical calculations to be bent in the
states analogous to those considered here. The differences
are mainly due to the ionic nature of the bonding in the
lithium carbenes. The ionic bond is manifested as an
electron pair closely associated with the carbon and
hybridized in a roughly sp manner. Upon bending, the
nuclear-nuclear repulsion between Li and H is more effec-
tive with the Li positively-charged. Also, if the ionic

MO is less faithful in following the Li nucleus than a
covalent MO would be (although plausible, this effect needs
verification), the result would be less s-character ac-
quired by the in-plane w MO. All of these differences

imply an increased destabilization on bending compared

57

 

 

.80 - . - , - j
180° 160° 140° 120°

®LiCH

Figure 16. LlCH State Energies y_s_ 0Li-C-H'

58

to methylene. For LiZC, with two ionic bonds, the effect
would be expected to increase. A Walsh diagram of the
LiCH MO's above the 2a' MO is included in Figure 17.

In examining the charge distribution of the lithium
carbenes, a Mulliken population analysis was made of the
ground state wavefunction. The results are summarized in
Table 17, which lists the net atomic populations broken down
into the o orbital contribution, the U orbital contribution,
and the net atomic charge. It is seen in LiCH that the Li
loses about 0.9 electrons through the 0 system and regains
only about 0.1 electron via the w MO's. Since the carbon
extracts about 0.1 electron from H, it is left with an ex-
cess charge of ~0.9 electron. In going to the Li2C system,
the excess charge on C has increased to 1.2 electrons, which
is the result of drawing over 0.7 electrons from each Li
through the o MO's and losing about 0.12 electrons to each
Li via the n MO's. The overall result is less 0 charge
shifted to C and a greater n delocalization for Li2C as
compared to LiCH. However, this n delocalization is not
apparent in the H MO contour plots because in neither case
does the Li contribution reach the density of the lowest
contour, 0.005 e1ectrons/(bohr)3.

The 3H state of LiCH was examined at the ROSSCF level
because its electronic configuration was essentially dif-
ferent than the three lowest states and because as the
first allowed transition from the ground state, a better

transition energy was desired. The results showed its C-Li

 

59

 

 

 

 

 

 

1
I
r
229-1
4a’
A -.30-1
3 40'
8 it
“I
-.5
8 4% 3a'
30
159‘
' 1 ' 1 ' i
180° 160° 140° 120°

Figure 17.

®LiCH

LlCH Orbital Energies vs eLi-C-H'

60

Table 17. Gross Mulliken Populations and Net Charges for
LiCH and Li C.

 

 

 

 

2
LiCH 32'
Atom n n n 5
0 fix U2
C 4.995 0.950 0.950 -0.895
Li 2.110 0.050 0.050 +0.789
a 0.894 ---------- +0.106
1122
C 5.453 0.878 0.878 -1.208

Li 2.274 0.061 0.061 +0.604

 

 

61

bond length to be dramatically longer than the 32- (4.40
bohr compared to 3.59 bohr). It was also found to be bound
by only 5 kcal/mole with respect to Li(ZS)+CH(2H), into
which it smoothly dissociates. Hence, it was expected that
a vertical transition from the ground state would result
in dissociation. The potential energy was found to be a
very broad and shallow function of the C-Li bond length.

32-

The vertical transition energy to the 3H from the
equilibrium is estimated to be about 60 kcal/mole (2.6 eV),
which would lie in the energy continuum of the 3H surface.

The bonding in the 3

H state was found to be essentially
different from the other states examined in that there

was little charge displacement, i.e., the Li was almost
neutral. The long and shallow Li-(C-H) stretching potential
and the substantial sp hybridization of the Li "28" electron
indicates a bonding based on a dipole—induced dipole attrac-
tion.

Heats of formation of the two molecules were estimated
by determining the ROSSCF ground state equilibrium energies
of the most stable dissociation fragments and calculating
the dissociation energies. The calculated AE's differ from
the corresponding AH's by an amount roughly 1 to 2 kcal/
mole,which was less than the resolution of these calcula-
tions. FigUres 18 and 19 correlate the levels of LiCH
and LiZC, respectively, to their lowest energy stable frag-

3

ments. It is seen that Z- LiCH is much stabler to dissoci-

ation than Li2C, and all the states examined, barring the

100

 

 

 

60-
A

(D 1
O

> 40-
O
U

x 1
V
LlJ

< 20-

1

0.1

Figure 19.

63

L12C

‘25 (45)
‘\\ (37)
\\2L1(2s)+C(°P)

 

 

 

//F
/////7L12(129)+C(3P)
Ag (24)// 0” (34)

 

Correlation of Lizc Levels and Dissociation
Fragments.

64
Li2C 12;, appear at least nominally stable.

Although lithium carbenes have not been observed ex-
perimentally, their generation does not a priori seem un-
attainable. As with several other carbenes produced in
the laboratory, a promising technique would be the photolysis
(or pyrolysis) of a diazo precursor, such as NZCLi2 in this

case. The synthesis of that precursor may not be possible,

but that question would require an experimental answer.

CHAPTER III
THE ZERO-FIELD SPLITTING PARAMETERS
3

OF Bl METHYLENE

A. The Electron Spin-spin Interaction and the ZFS Parameters

Each electron has a magnetic dipole moment, 5) associated
with its intrinsic spin, 5; the relation connecting the two
is F'= -gB§. In any system containing two or more electrons,
each pair of electronic magnetic dipoles interact through
their associated magnetic fields. This interaction is
called the electron spin-spin interaction, and the corres-

ponding quantum mechanical energy operator is given by

 

 

all all A A A A A A
e e“ E -'s' 3(s ~35 )(‘5 -'f )
g _ 1 282 2 2' K A _ K K1 1 K1
as A3 15
K 1 K1 rKl

where g is the electronic g-factor, and B is the Bohr mag-
neton (B = eM/Zmec). rKl is the distance between electrons
K and A. The range of the interaction is quite short,
dropping off as l/rgx.

In practice, an empirical "spin Hamiltonian" is used
which includes both the spin-spin and spin-orbit interac-

tions. This empirical Hamiltonian has the form

where S is the total electronic spin of the molecular system,

65

66

and D and E are constants into which all the information
concerning the spatial distribution of the spin density has
been compressed. It has been demonstrated that 988 can be

28,29

reduced to this form, in which case D and E are given

(for a two-electron wave function ¢(l,2)) by:

£2 -322
D = % 9282<¢| 135 12|¢>
r12
§2 -fi2
E = % 9282<¢l 135 12|¢>
r12

It is apparent that if the x and y directions are indis-
tinguishable in the molecule (axial symmetry about 2), then
B s 0. Also, if there is complete Spatial isotropy in the
electronic distribution, D E 0. To the extent that D and
E differ from zero, their magnitudes are a measure of the
diffuseness of the net unpaired electronic spin density.
Other properties of the spin Hamiltonian can be gleaned
from an examination of the general matrix element

2 A2
x

A . . A 1 A2 A2
' ' -
<SM8| HSIS Ms>' Writing S S as 7(Sl + S_), it is seen

that
A _ 1 -
H |00> - 0(0 - — - 0) + E(0) = o,
s 3
and
A 1 1 1 l 3 _
HSI-Z- 1'3) < D";- ' '3‘ ' '4") + E(0) : 0.

67

Thus there is no correction to the molecular energy levels

for molecules having 8 < 1. Also,
A a '
HSISMS> ISMS>,

indicating that mixing of spin states occurs only within a
spin multiplet and does not mix states of different S. The
energy shift of each level within a spin multiplet is given
by D(M: - % S(S+1)) and is a result of the first term in
38. The second term causes a mixing of the multiplet com—

ponents which differ by 12 in their MS value:

 

A _ E — _
<SM8| HSISMst2> .. 2\I(S+MS+2) (S+Mst2+l).

In general, then, it is apparent that the spin-spin inter-
action causes a splitting of the Zeeman multiplets for S 3 1
in molecules possessing spatial anisotropy -- even in the
absence of externally applied fields. This last observa-
tion justifies the name "Zero-field Parameters" (ZFS) for
D and E.

For molecules containing only "low Z" nuclei, the spin-
spin interaction is the dominant contributor to the ZFS
parameters. However, the spin-orbit interaction will also

4, where

contribute, and its strength increases roughly as 2
"2" refers to the charge of the nuclei in the molecule
considered. Since both of these effects are small in common

molecules of interest in, say, organic chemistry, (i.e., in

68

the order of cm.1 energies), they are treated effectively
in terms of perturbation theory. In those terms, the spin-
spin interaction contributes in first order while the spin-
orbit effects don't appear until second order corrections
are accounted for.

A useful diagram of the perturbed multiplet levels can
be constructed which relates the level splittings to the

measured quantities D and E:

 

 

 

 

I X
, T b [
/.muuwfmun. 2E xa and xb are
E: 9 I 1 linear combinations
/
n xo,x1.x_l 45/ xa 0f x+1 and x_1]
e \
r \\ D
9 \
\
y \
\
\
\ X0

 

 

Figure 20. Relation of D and E to the Zero-Field Level
Splittings of a Triplet State.

B. The Spin-Spin Contribution to D and E for a CI Wave

Function

In demonstrating the essential equivalence of ass and

A

HS, McLachlan29 develops a function, 5(FI,F2), which is
identical to the "coupling anisotropy"function of McWeeny.3o

As McLachlan derives it,

69

._ ._ _ 1 A A _; .2 _.__
3(r1,r2) — §T§§3TT<SSGI Z {BSzuszv su sv}6(ru r1)
u#v

x5(E§-Eé)|53a>,

where ISSa> is the state of total spin 8 in its "standard
state”, i.e., with M8 = S, and a denotes its non-spin
quantum numbers. Since the Ms =0 component was preferred

in the techniques used in this thesis, S(Fi,?2) was used
with a normalization factor appropriate to M8 = 0 wave func-
tions, i.e., 1/S(2S-l) was replaced by -l/S(S+1). Corres-
pondingly, a wave function describable as IS 0 a> replaced
|SSa>. Of course, this choice of M8 = 0 would not be appli-
cable in the case of a wave function having non-integral
spin.

S(§i,?2) can be used to construct D and E as follows:

 

 

2 2
-32
_ 3 2 2 r12 12 1— .— -. —
D - -4- 9 BH( r5 )S(r1,r2)dr1dr2
12
y2 _x2
_ 3 2 2 12 12 —. — 1— —
E - 71- g 81]} 5 )S(r1,r2)dr1dr2,

r12

where S(fi,?2) can be written explicitly as:

_ -1 *M =0
S(r1,r2) - Wf'fws 8 (Tl'T2"'°'Tn)
N
X{ Z'[3s

u¥v

A

dT dT d1

1 2 3...dTN, (III.B.l)

70

where N is the number of electrons in the system, and the
integration is done over the space and spin coordinates,
Ti, of all electrons. The wave functions employed in this

work were CI wave functions and have the general form

=Zc
K
where DK is a Slater determinant of one-particle functions.

For such wave functions, 3(f1,Fé) reduces to

_ ._ t _ _ * __ _

s(rl,r2) - {[22 aijk£¢i(rl)¢j(r1)¢k(r2)¢2(r2), (111.3.2)
ijzm

where the ai ijkz are determined by the CK

for w and by factors arising from the spin integrations over

in the expression

the ¢'s in each term of S. Hence, D takes the form (and

E, in a similar way)

2 2 2-3222
D=z<3 B {lilaijkgffi (rlmj (r1)<:-1—— r5 2)¢]:(r2 )¢£('r.2)d? drz.
ijkz 12
The author wrote a program to construct the set of aijkz from

a CI wave function; and using an existing routine for calcu-
lating the spatial integrals in a gaussian lobe function
(GLF) basis, D and E were obtained for the ground 381 state
of :CHZ. For further details on the construction of the

aijkl' see Appendix III.

71

C. The ZFS Parameters of 331 Methylene

The apparent conflict which developed between theory and
experiment concerning the electronic structure of methylene
has been gradually finding resolution. The initial inter-

pretation of the electronic spectra of methylene by Herz-

7 suggested a linear 32; ground state, while the rapidly-

improving theoretical calculations indicated a bent triplet

berg

ground state (331; 6 N140°). This situation remained stale-

11,12

mated until the esr spectrum was observed in 1970,

supporting the common belief in a triplet ground state. The
ZFS parameters were the critical measurements in determin-
ing the question of the bond angle. The experimental values

of D and E for methylene trapped in a matrix at low tempera-

1

ture were found to be 0.69 cm” and 0.003 cm-1, respectively.

A previous semi-quantitative estimate had been made by

31

Higudhi. The low experimental value of E, suggesting a

nearly linear triplet state, must be corrected for motional

effects. The result is an experimental value of 0.074 cm-1

1

in a xenon matrix and 0.021 cm- in SF6. An ab initio

calculation using an eight structure CI was carried out by

32

Harrison, who found the spin-spin contribution to D and

E at eHCH = 132.5° (the determined equilibrium bond angle)
to be 0.71 cm-1 and 0.05 cm_1, respectively. When the spin-
orbit contribution (expected to be much less significant)

is added to Harrison's value for D, the theoretical predic-

tion of 0.776 cm.1 (using D = 0.066 cm.1 from the latest

SO

72

33 9

calculation by Hameka and Hall at = 135°) agrees well

HCH
with the average experimental value of 0.76 1 .02 cm—l.
The results obtained in a xenon lattice are not included
in the average because of the abnormally large spin-orbit
contribution induced by the xenon host.

The calculations described in this chapter were under-
taken to verify the essential correctness of Harrison‘s
results. The double zeta set of GLF was retained but the
CI procedure was extended to include the 100 most energet-
ically important structures (based on their first order
contribution to the 331 energY) selected from the set of
single and double excitations with respect to the dominant

331 structure represented by the orbital occupation

(la1)2(2a1)2(lb2)2(3a1)1(1b1)1.

0f the 14 MO's resulting from the closed shell restricted
SCF calculation, the 1a1 was held doubly occupied and the
7a1 and 8a1 were maintained empty in the construction of the
excitations. Using the NO's obtained from the CI wave func-
tion as a basis; the selection of the "best" 100 structures
was repeated, and a CI calculation done. This procedure,
first introduced by Bender and Davidson34 is called the
interative natural orbital technique and has the property
that it produces a CI energy lowering for one or more itera—
tions but, at some point, begins to diverge, i.e., the CI

energy increases with iteration. In the case described

73

here, one iteration gave the lowest CI energy; the second
iteration energy was somewhat higher (see Figure 21).

With a C-H bond length fixed at 1.05 A, the minimum CI
energy was interpolated to correspond to a eHCH value of
132.5°. The calculation of the ZFS parameters at this bond
angle, using the first iteration CI wave function (CI-N01),

gave D = 0.722 cm.1 and E = 0.053 cm-1

, supporting the earlier
eight structure CI results. Figure 22 exhibits the angular
dependence of D and E as calculated from the CI wave function
for each iteration, and it also includes the eight structure
results (C10). The results of this extended calculation are
also consistent with the range of experimental results for
D, using the same spin-orbit contributions as above.
Subsequent to the results described here, Langhoff and
Davidson35 have calculated the spin-spin values of D and E
based on a 575 configuration iterative NO CI wave function at
1 l

and E8 = 0.050 cm” .

o _ -
135 . They report DSS — 0.781 cm 8

Although this puts the theoretical value of D somewhat higher
than the experimental results, it does help confirm the
essential stability of the theoretical method.

With the theoretically predicted spin-spin contributions
fairly well defined, the appearance of a sound value of the
spin-orbit part is of most importance in making a firm com-

parison with the experimental results.

-38.956T

 

74

 

.9584
\'-/' Cl-MO
’5
(0
" .9604
)p-
0
CE
UJ
E .962“ CI-NOZ
NI. /
/ Cl-NO1
.964-
13?‘ 155' 1:10"
('9ch

. 3
Figure 21. .CH2 ( Bl) Energy XE eHCH’

D (cm")

E (cm“)

Figure 22.

.74:1

.72."—“/
+

.06 ‘

.05"I

.7.J

.76

j

 

T

EXPERIMENTAL

.1

 

0 Clo
A Cl-MO
v CI'NO’]

o Cl-NO2

 

 

2

.041
I I 1
130° 135° 140°
9mm
:CH (3B1) Spin-spin ZFS Parameters XE eHCH'

APPENDICES

APPENDIX I

A. The Closed Shell SCF (CSSCF)

 

The quantum mechanical description of molecules is commonly
done in terms of the solutions of the time-independent

Schrodinger Equation with the purely electrostatic hamil-

 

tonian
N M N N N
"_ 1 2 a 1 I 1
H - -'2 Z 61 - Z r . + 2 Z 2 r1
i=1 a=l i=1 a1 i=1 j=l 3
M M
+12 2' Zazb
2 r b '
a=l =1 a
where atomic units are employed (e = 1, m = 1, h = l).

e
N is the number of electrons in the system, M the number of

nuclei, Z is the multiple of the unit charge carried by

a
each nucleus, and the r's are interparticle distances. The
nuclei are assumed to be fixed at some pre-chosen positions
so that their coordinates become parameters in the problem.
With the rab fixed, the last term, the nuclear repulsion
energy, is simply a constant and doesn't affect the wave
function solution by being neglected until the end. Of

the first three terms, the first two are sums of one-elec-
tron operators (depending on the coordinates of only one
electron). Because of the way the third term couples the

coordinates of pairs of electrons, there is no exact mathe-

matical way to write the part of P involving electron

76

77

coordinates as a sum over one-electron operators. Hence,
there is no finite procedure which can yield an exact solu-

tion, V(§i,§é,...§fi), for the Schrédinger Equation

In)

W = EV. (E constant).

Approximate solutions are obtained by the Hartree-Fock method
which replaces the exact electron-electron potential with

an "effective" electron repulsion potential, in which each
electron feels not the force of other electrons as point
charges but only their "smeared out" charge densities.

This modification requires a knowledge of the solutions
before constructing the potentials used in 3 and transforms
E into a sum of effective one-electron operators, which
permits separation of the Schrdedinger Equation into a
coupled set of one-particle equations. The dependence of
the hamiltonian on the solution makes the problem a pseudo-

eigenvalue problem, expressed by:

This equation (or set of equations) must be iteratively solved,
beginning with an initial estimate of T. When, or if, the
procedure converges, the solution, W, is called the self-
consistent field (SCF) wave function. Multiple nuclei impose
difficulties which restricted the numerical calculations

to atomic systems. Molecular systems became generally

78

treatable when Roothaan36 demonstrated the validity of a
reformulation of the problem. In the alternate form, the
one-particle solutions are written as expansions in a com-
plete set of analytic functions satisfying the general
boundary conditions of the problem; the expansion coef-
ficients are determined by a variational minimization of
the total energy, E. The variational energy is equal to

the Hartree—Fock energy, E and the variationally-deter-

HF'
mined coefficients describe the one-particle Hartree-Fock
solutions. The numerical solution of a set of coupled
integro-differential equations is thus transformed into a
set of algebraic equations, solved by methods of linear
algebra. Although the new formulation allows solution of
molecular systems and yields solutions in terms of analytic
functions (instead of numerical tables), in practice it can
only be carried out approximately; any complete set of
expansion functions is necessarily infinite for these
problems, and the choice of a truncated set is a very ser-
ious qualification on the results obtained.

The SCF wave functions obtained for the carbonyl car-
benes were constructed using the Hartree-Fock-Roothaan
(HFR) method discussed above for the case of a restricted,
closed—shell wave function, i.e., each a spin electron
was associated with a 8 spin electron having the same
spatial distribution, and there were always an even number

of electrons involved. The one particle SCF spatial functions

are
911

316

P3

79

are called molecular orbitals (MO's); if the MO is multi-
plied by one of the S2 eigenfunctions (called a and B),

the function is called a spin-orbital (SO).

B. The Basis Set

The basis functions employed in the carbonyl carbene
calculations were the STD-3G functions of Pople E£.21-20
These functions are linear combinations of nuclear-centered
cartesian gaussian functions,least-squares fitted to indi-
vidual unscaled Stater-type functions (STO's). The scaling
factors, Ci' used for each atomic shell were those suggested
by Pople as the "best molecular" scale factors. The form

of each such simulated STO is

2

3 2
= -c-aikr
Xi(r) Ni 2 dik e 1

k=1

where Ni is a normalization constant, dik are the least-
squares determined coefficients of each gaussian in the
linear combination, aik are the least-squares determined
exponents, Ci is the scale factor appropriate to the basis
function Xi' and r is the radial distance of the electron
from the nucleus on which xi is centered. As Xi is written
above, it has the angular isotropy of an s-type function.
The spherical harmonics, YE, having 2 > 0 (in real form)
are obtained by incorporating appropriate products of

cartesian coordinates into Xi' e.g., a pX would contain the

80

additional factor x:

3
2
(x,y,z) = Ni x Z dik e Clalkr
k=1

2

X

px

Table 18 contains the coefficients, dik' and scaled ex-
ponents, aik = Ciaik' used for the basis functions on each
type of nucleus included in the series of carbenes involved.
The basis set for each molecule was minimal, which is to

say that the number of basis functions included for each

atom involved in the molecules was just enough to account

for the occupied (n,£) subshells. Thus, H has only one basis
function, an s-type function, while C through F have five --
two 8 functions and a set of three p—type functions (px, py,
and pz).

C. The CI

One method used to extend the wave function beyond the
Hartree-Fock level (HF), is to construct a trial function as
a linear combination of single determinants, formed by re-
placing some of the 80's in the HF determinant by some of
the "virtual orbitals" resulting from the SCF procedure.

The inclusion of these "excitations" allow the electrons
a flexibility not obtainable in the HF wave function and
this consequently allows the recovery of the Coulomb correla-

tion energy missing from the HF total energy, E Also,

HF'
this extended wave function is sometimes necessary to represent

81

Table 18. Carbonyl Carbene Basis Set: STO-3G Coefficients

and Scaled Exponents

 

 

 

Basis Function 0' d(s) d(p)
Carbon ls: 3.53053 .444635
13.04509 .535328
71.616818 .154329
Carbon 28,2p: .22229 .700115 .391957
.68348 .399513 .607684
2.94125 -.0999672 .155916
Oxygen ls: 6.44364 Same as
23.80886 Carbon ls
130.70929
Oxygen 28,2p: .38039 Same as Same as
1.16959 Carbon 25 Carbon 2p
5.03315
Fluorine ls: 8.216857 Same as
30.360801 Carbon ls
166.67909
Fluorine 28,2p: .488589 Same as Same as
1.502279 Carbon 25 Carbon 2p
6.464805
Hydrogen ls: .15814 Same as
.58431 Carbon ls

3.20783

 

 

82

molecular systems which are not even qualitatively described
by a single determinantal wave function. This extended wave
function is variationally determined by minimizing the expec-

tation value of H,

E = <¢CI|HI¢CI>/<¢CI|¢CI>:

with respect to the coefficients of each determinant. The
basis for the CI wave function is then the set of all deter-

minants derivable from the HF determinant by replacement of

“at...

w»?

SO's occupied in that determinant by virtual SO's. However,
it is also possible to alter this basis (as was done for the
CI wave functions of :CHCOF and :CFCOH) by chosing linear
combinations of the determinants. This is done to facilitate
the interpretation of the wave function. In the calculations
in this thesis, the new basis elements of the CI are linear
combinations of determinants having definite symmetry under
the elements of the molecular point group and the spin
operators Sz and Sz; such linear combinations of determinants
are called "structures"

It is important to note that usually the number of de-
terminants in the basis is finite (if the SCF basis is finite),
but usually it is too large to be fully employed, so a subset
of the basis is used. That subset vd11.be termed the CI
basis here.

For :CHCOH, a determinantal basis was used. Since, given

the SCF MO's, a determinant is completely specified by a list

83

of the occupied spin-orbitals, the description of the CI
basis in the following tables consists in associating a

"." (unoccupied) or a 1 (occupied) with each SO. The "ac-
tive" MO's are those with variable occupation in the basis
used. Hence, there is also a "core", consisting, in this
work,of MO's always doubly occupied in the basis determinants.
In the tables, each MO is associated with two adjacent posi-
tions, the first for the a spin-orbital, the second for the
B spin-orbital. The core is de—emphasized by leaving no
blanks between its MO's; the active MO's are separated by
blanks. The coefficient given in front of each determinant
is its weight in the associated structure. Since for :CHCOH
structures were not used, each determinant has a coefficient
of 1.0.

*
The active MO's for :CHCOH were the set {w p o n }.

co “0
There were six electrons (3a, 38) distributed among these
10 80's. The total number of arrangements was

s 5 51 2_ 2
(3H3)=(§'1'7T) -10

100.

These 100 determinants are given in Table 19.

A structure basis was chosen for :CHCOF. The set of ac-
tive MO's was taken as {pF OF 00 "CO 0 p n*} and hosted 10
electrons (5a and 58). First, leaving the CF doubly occupied,
all determinants were formed having 4a and 4B electrons distribu-
ted among the remaining six MO's. There are (g)(2) = 225

such determinants. When structures are formed from these

."
Li‘s-2 ‘1

Table 19.
1 10
2 10
3 1o
4 lo
5 1o
6 1o
7 10
3 10
9 10
10 10
11 10
12 10
13 1o
14 10
15 10
16 10
17 1o
18 10
19 10
2O 1.
21 1o
22 1o
23 10
24 1o
25 1o
26 1o
27 10
28 1o
29 10
3O 10
3! 1o
32 1o
33 1o
34 1o
35 10
36 1o
37 1o
38 10
39 10
40 10

:CHCOH Determinantal Basis for all States.

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111111
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

84

CO

1!
11
11
1o
1!
11
I.
11
10
1o
11
1!
11
lo
11
11
lo
11
1.
lo
11
11
11
lo
11
1!
1.
11
1.
lo
o!
01
0!
o!
.1
o!

11
11
1.
1!
1!
1o
11
lo
11
lo
1!
11
1.
11
1!
lo
1!
lo
11
lo
01
o!
o!
01
o!
01
1!
1!
10
1!
1!
1.
11
10
11
lo

11
lo
11
1!
lo
11
11
lo
I.
11
01
01
01
01
01
o!
11
lo
1!
1!
1.
11
11
lo
1.
11
1!
1.
11
11
1.
1!
11
lo
10
11

Q

Conny-.3... O Ono-o...

”0.0.0.0000

”p”
flpfl

hint-obo-oo—O—
O unfit-n. O O

gnu-ppm...
O O 0 up...

—~”ufl~h
gnu—u... on...“

an“...
00.

any".
Hpfl

OOOOOOOOOOOOOOOOOOOOOOO
ppppnuOOOQppp—wfiOOOOo-nu

L. I
__‘|_- _~

85

Table 19 - Continued

 

"CO 00 p o n*
4! 1. 1111111111111!!! 1! 1! .1 .. 1.
42 1. 1111111111111!!! 1! 11 .. .1 1.
43 1. 1111111111111!!! 11 1. .1 .1 l.
44 1. 1111111111111!!! 1. 11 .1 .1 1.
45 1. 1111111111111!!! 11 1! .. .. 11
46 1. 1111111111111!!! 1! 1. .1 .. 11
47 1. 1111111111111!!! 1. 1! .1 .. 1!
48 1. 1111111111111!!! 11 1. .. .1 11
49 1. 1111111111111!!! 1. 1! .. .1 1!
50 1. 1111111111111!!! 1. 1. .1 .1 11
5! 1. 1111111111111!!! 1! .1 1! .. 1.
52 1. 1111111111111!!! 11 .1 1. .1 1.
53 1. 1111111111111!!! 1! .. 1! .1 1.
S4 1. 1111111111111!!! 1. .1 1! .1 1.
55 1. 1111111111111!!! 11 .1 1. .. 1!
56 1. 1111111111111!!! 1! .. 1! .. 11
57 1. 1111111111111!!! 1. .1 1! .. 1!
58 1. 1111111111111!!! 11 .. 1. .1 1!
59 1o 1111111111111!!! 1. .1 1. .1 11
6O 1. 1111111111111!!! 1. .. 11 .1 11
6! 1. 1111111111111!!! .1 11 11 .. 1.
62 1. 1111111111111!!! .1 1! 1. .1 1.
63 1. 1111111111111!!! .1 1. 11 .1 1.
64 1. 1111111111111111 .. 1! 11 .1 1.
65 1. 1111111111111!!! .1 1! 1. .. 1!
66 l. 1111111111111!!! .1 1. 1! .. 1!
67 1. 1111111111111!!! .. 1! 1! .. 1!
68 1. 1111111111111!!! .1 1. 1. .1 11
69 1. 1111111111111!!! .. 1! 1. .1 1!
7O 1. 1111111111111!!! .. 1. 1! .1 1!
7! 1. 1111111111111!!! 1! .1 .1 1. 1.
72 1. 1111111111111!!! 1! .1 .. 1! 1.
73 1. 1111111111111!!! 1! .. .1 11 1.
74 1. 1111111111111!!! 1. .1 .1 1! 1.
75 1. 1111111111111!!! 1! .1 .. 1. 1!
76 10 1111111111111!!! 11 .. .1 1. 1!
77 1. 1111111111111!!! 1. .1 .1 1. 1!
78 l. 1111111111111!!! 11 .. .. l! 1!
79 1. 1111111111111!!! 1. .1 .. 11 11
80 1. 1111111111111!!! 1. .. .1 1! 11

Table 19 - Continued

1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.
1.

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!

86

.1
.1
01
.1
01
.1
.1
01
.1
.1
01
.1

11
11
1.
11
11
1.
11
1.
11
1.
.1
.1
.1
.1
.1
.1

.1
.1
.1
.1
.1
.1
11
1.
11
11
1.
1!
11
1.
1.
11

1.
11
11
11
1.
1.
l.
11
11
11
1.
11
1!

1.
1.
1.
11

11

1.
1.
1.
1.
11
1!
11
11
11
11
1.
1.
1.
1.
11
11
1!
1!
11
11

- "-z YI-

 

87

determinants, the following distribution of symmetries is

 

 

 

 

obtained:
Singlet Triplet Quintet Total
A' 57 49 7 113
A" 48 56 8 112
Total 105 105 15 225

 

 

 

 

 

 

 

The quintet spin structures were not included in the CI

wave function. In addition, two singlet and two triplet
structures involving single excitations from the OF were
included, bringing the total number of structures in the
CI basis to 214. In the tables listing these structures,

two "core" MO's appear above the o and pF active MO's.

1

F
Tables 20, 21 and 22 list the structures of 3A",
1

A', and
A" symmetry, respectively.
The active orbital set for :CFCOH included nine MO's:

*

{o o p n o' 0"}. In the CSSCF determinant, the

F P]? 1'CO °o
first five were doubly occupied. The structure basis was
constructed by, first, forming all excitations within the
first seven active Mo‘s excluding any quadruple excitations
with respect to the SCF determinant and all structures of
quintet spin. In addition, a few selected double excitations

involving the o' and 0" virtual MO's were added. The

following breakdown of the structures obtained (the quintets

 

Table 20.
1 1.
1 1.
2 1o
2 1o
3 1-
3 1.
4 1-
4 1.
5 10
5 1.
6 1.
6 10
7 1.
7 1-
8 1.
8 1o
9 1o
9 1o
10 1.
10 1.
11 1o
11 1o
12 1.
12 1.
13 1.
13 1.
14 1.
14 1.
15 10
15 1-
16 10
16 1o

:CHCOF 3A"

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

88

pF

1!
11

l!
11

11
1!

11
11

11
11

1!
11

11
11

O.

01
1.

01
1.

01
1.

.1
1.

0F
11
11

11
11

11
11

11
11

11
11

11
11

11
11

11

on...
an..-

11
11

1!
11

1!
11

1!
11

11
11

11
11

11
11

CI Structure.

1111
1111

111!
11!!

1111
1111

11!!
1111

1111
1111

1111
1111

1111
1111

1111
1111

111!
1111

1111
1111

1111
11!!

1111
111!

1111
11!!

1111
1111

1!!!
11!!

111!
111!

00 “CO C p

11
1!

1.
01

.1
1.

1.
.1

.1
1.

11
1!

1!
1!

11
1!

1.
.1

1.
.1

1.
.1

1.
.1

11
11

11
11

.1
1.

.1
1.

1.
.1

11
1!

11
11

11
11

.1
1.

.1
1.

11
11

11
!1

1!
11

1.
.1

11
11

1.
.1

1.
01

1!
1!

1.
.1

1.
.1

1.
.1

11

1.
.1

.1
1.

11
1!

11
11

11
1!

11
11

1!
1!

1!
11

11
11

11
1!

11
1!

11
11

 

Table 20 - Continued

17
17

18
18

19
19

20
2O

2!
21

22
23
23

24
24

25
25

26
26

28
28

29
30
3O

3!
31

32
32

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1.
1.

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111!!!

11111111!
1111111!!!

11!!!!

1111111111111!!!
1111111111111!!!

1111111111!

11!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

111111111111!
111111111111!

0....

11
11
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

89

pF
.1
1.

.1
1.

1!
11

on... an...
up
on...

1111

1!!!
11!!

1111
1111

1111
111!

1111
1111

00 1'co 0

1.
.1

1!

on...
no...

1.
.1

11
1!

1.
.1

.1
1.

11
11

1!

11
1!

.-.-.
”n

-
”fl

1!
11

11
11

.1
1.

11
11

1.
.1

P

11

on..-
one...

1.
.1

1!
11

1!
1!

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

11
1!

 

Table 20 - Continued

33
33

34
34

35
35

36
36

37
37

38
38

39
39

a.
a.

41
41

42
42

43
43

44
44

45
45

46
46

47
47

a.
a.

1.
1.

1.
-1.

1.

-1.

1.

-l.

1.

-l.

1.

-l.

1.

-‘.

1.

-1.

1.
-!.

1.

-1.

1.
-1.

1.
-1.

1.
-l.

1.

-l.

1.

-1.

1.

-1.

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
111111111111111!
11111111111111!
11111111111111!

”h

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

90

11
11

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1!
1!

11
11

11
11

11

1!
11

1!
1!

1.
.1

1.
.1

1.
.1

13}? OF

1.
.1

11
11

11
11

11
11

1!
11

11
11

11
11

11
11

11
11

11
11

11
11

11
11

11
1!

11
1!

11
11

11
11

111!
11!!

o-g-o
p...

0011'

1!
11

.1
1.

.1
1.

1.
.1

1!
1!

!1
11

1!
11

1!
11

11
11

11
11
1.
.1

1.
.1

1.
.1

1!
1!

11
11

11
11

CO
11
11

.1
1.

1.
.1

.1
1.

.1
1.

.1
1.

1.
.1

1.
.1

1.
.1

1.
.1

.1
1.

.1
1.

1.
.1

11
1!

11
1!

1!
11

O

1!
11

11
11

11
1!

1!
1!

.1
1.

1.
.1

.1
1.

.1
1.

.1
1.

1.

.1

11
1!

11
11

11
1!

.1
1.

.1
1.

1.
.1

917*

.1
1.

1.
.1

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

1.

.1

.1
1.

11
11

1!
11

11
11

11
1!

11
11

1!
11

1.
.1

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

 

Table 20 - Continued

49
49

50
50

51
51

52
52

S3
53

54
54

55
55

56
56

57
57

1.

-!.

1.

-1.

1.

-1.

1.

1.
-‘.

1.
-1.

1.

-1.

1.

-1.

1.

-1.

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1!!!!111!!!!!!!!
!!!!!!!!!!!!!!!!

91

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

11
11

1!
11

1!
1!

11
1!

11
11

11
11

1111
1111

111!
111!

1111
111!

1!!!
111!

1111
111!

1111
1!!!

1!!!
111!

1111
1111

1111
1111

0‘0 7&0 °

.1
1.

.1
1.

1.
.1

1!
1!

.1
1.

.1
1.

1.
.1

1!
1!

1!
1!

1!
11

.1
1.

.1
1.

1.
.1

.1
1.

1.
.1

.1
1.

1!
1!

11
11

.1
1.

1.
.1

.1
1.

11
11

1!
11

11
1!

.1
1.

1.
.1

.1
1.

“i

1.
.1

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

1.
.1

.1
1.

.1
1.

 

Table 21.
1 1.
2 1.
3 1.
4 1.
5 1.
6 1.
7 1.
8 1.
9 1.
10 1.

1! 1.
12 1.
13 1.
14 1.
15 1.
16 1.
16 -1.
17 1.
17 -1.
18 1.
18 -1.
19 1.
19 -1.

2O 1.

20 -1.

21 1.

21 -1.

:CHCOF 1

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
11!

1111111111111

1111111111111
1111111111111
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

92

1!

11

1!

!1

1!
1!
1!
1!
11

11

1.
.1

1.
.1

.1
1.

.1
1.

1.
.1

1.
.1

11
11
11
11
11
1!
11
11
11
11
1!
11

1!

A' CI Structures.

1111
1111
1111
1111
1111
1111
111!
1111
1111
1!!!
1!!!
1111
1111
11!!
1111

1111
111!

“o "co

0'

11 11 1!

11 11

11 .

11 1

11 11

11 .
.. 1
11 1
11 .

11 1

11 .
.. 1

!! .
1! !

O. l

!

1

1
1

1
1

1
1

1

1!
11
1!
11
1!
1!

11

11
1!
11
11
1!
1!
11
1!
1!

1!

 

a,

“Lie”:

93

Table 21 - Continued

0 11 o p n*

22 1. 1111111111111!!! 1. 11 1111 11 1! .. 11 .1
22 -1. 1111111111111!!! .1 11 1111 11 11 .. 11 1.

23 1. 1111111111111!!! 1. 11 1111 11
23 -1. 1111111111111!!! .1 11 1111 1!

.. .111
..1.11

3......
fins...

24 1. 1111111111111!!! 1. 11 1111 11 .1 .. 11 11
24 '1. 1111111111111!!! .1 11 111! 11 1. .. 1! 11

 

25 1. 1111111111111!!! 1. 11 1111 11 .1 11 .. 11
25 '1. 1111111111111!!! .1 11 111! 1! 1. 11 .. 11
26 1. 1111111111111!!! 1. 11 111! 11 11 1! .. .1
26 -1. 1111111111111!!! .1 11 11!! 11 1! 11 .. 1.
27 1. 1111111111111!!! 1. 11 111! 11 1! 1! .1 ..
27 -1. 1111111111111!!! .1 11 1!!! 11 1! 1! 1. ..
28 1. 1111111111111111 .. 11 1111 11 1. 11 .1 1!
28 ~1. 1111111111111!!! .. 11 1!!! 1! .1 11 1. 1!

29 1. 1111111111111111 .. 11 1111 1. 1! .1 11 11
29 -1. 1111111111111111 .. 11 1111 .1 11 1. 11 11

3O 1. 1111111111111111 .. 11 1111 11 11 11 1. .1
30 -1. 1111111111111111 .. 11 1111 1! 11 11 .1 1.

31 1. 1111111111111!!! .. 11 1111 1! 1. 11 11 .1
31 -1. 1111111111111111 .. 1! 1111 11 .1 1! 11 1.

32 1. 1111111111111!!! 1! 1! 1111 .. 1. 11 1! .1
32 -1. 1111111111111!!! 11 1! 1111 .. .1 11 11 1.

33 1. 1111111111111!!! 11 11 1111 11 1. .. 11 .1
33 -1. 1111111111111!!! 11 11 1111 11 .1 .. 1! 1.
3A 1. 1111111111111!!! 1! 11 1111 .. 11 1! 1. .1
34 -1. 1111111111111!!! 11 11 1111 .. 11 1! .1 1.

3S 1. 1111111111111!!! 1! 11 1111
35 -1. 1111111111111!!! 11 11 1111

36 1.

1111111111111!!! 1! 11 1111 11 1. 11 .. .1
36 -1. 1111111111111!!! 1! 11 111! 11 .1 11 .. 1.

37 1. 1111111111111!!! 11 11 1111 11 .. 11 1. .1
31 -1. 1111111111111!!! 1! 11 1111 11 .. 11 .1 1.

Table 21 - Continued

38
38

39
39

40
4O

41
4!

42
42

43
43

44
44

45
45
45
45
45
45

46
46
46
46
46
46

47
47
47
47
47
47

48
48
48
48
48
48

1.

-l.

1.

1.

1.

-1.

1.

1.

-l.

1.

-1.

2.
-1.
-1.
-1.
-1.

2.

2.
-1,
-1.
-1.
-1.

2.

2.
-1.
-1.
-1.
-1.

2.
2.

-l.
-I.
-1.
-1.

2.

1111111111111!!!
1111111111111!!!

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1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

94

11
11

11
1!

11
11

9F OF

1!
11

11
11

11
1!

11
11

1!
1!

11
11

1111
11!!

1111
1!!!

1111
11!!

1111
111!

1111
1111

1111
11!!

1111
1!!!

11!!
1111
1111
111!
1111
1111

1111
111!
1111
111!
111!
1111

1111
1111
11!!
1111
1111
111!

1111
1111
1111
1!!!
1111
11!!

00 “CO or p

11
11

.1
1.

1.
.1

1!
11
11
1!

1!

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
.1
.1
1.
1.
.1

.1
1.

11
1!

11
1!

.1
1.

1.
.1

11
11

1.
.1
.1
1.
1.
.1

”mono—u..—
hon—uh..-

.1
.1
1.
1.
.1

11
11
1!
11
1!
11

11
11

1.
.1

.1
1.

1.
.1

1.
.1

11
1!

1!
11

.1
1.

11
1!

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

1!
11
11

11
1!

1!
11
11
11
1!
11

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

5.1.:-

 

m _— - ”‘17..

fiqu

Table 21 - Continued

-1.
-1.
-l.
-1.

2.
2.

-1.
-l.
-1.
-1.

2.

2.
-1.
-1.
-1,
-1.

2.

1.
-1.
c}.

1.

1.
-1.
-1.

1.

1.
-1.
-1.

1.

1.
-1.
o‘.

1.

1.
a!.
-1.

1.

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1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
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1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
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1111111111111!!!
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95

11
1!
11
11
1!
11

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
1.
.1
.1

11
1!
1!

11
1!
11
11

1.
1.
.1
.1

11
11
11
1!

00 "CO 0

1.
1.
.1
.1
.1

1.
.1
.1
1.
1.
.1

1.
.1
.1
1.
1.
.1

11
11

11

1.
1.
.1
.1

1.
1.
.1
.1

.1
.1
1.
1.

1.
1.
.1
.1

1.
.1
.1
1.
1.
.1

11
1!
11
11
11
1!

.1
1.
.1
1.
.1
1.

.1
.1
1.

pong-.—
”flu”

.1
.1
1.
1.

1!
11
1!
11

.1
.1
1.
1.

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

.1
.1
1.
.1
1.
1.

11
11
11
11

.1
.1
1.
1.

1.
.1
1.
.1

1.
.1
1.
.1

1.
.1
1.
.1

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

11
11
1!
1!
1!
11

1.
.1
1.
.1

1.
.1
1.
.1

11
1!
1!
11

11
11
11
11

.1
1.
.1
1.

 

Table 21 - Continued

57
S7
57
57

58
58

58

1.
-l.
-1.

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1111111111111!!!

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96

1.
1.
.1
.1

1.
1.
.1
.1

1!!!
1111
1!!!
1!!!

1111
111!
11!!
1111

Q
...-..-0

0...... O

.1
1.
1.

uptown
.-..-..-..-.o

.1
1.
.1

O

1.
.1
1.
.1

.1
1.
.1
1.

91*

.1
1.
.1
1.

1!
11
11
11

11
11

1!

11
1!
11
1!

Table 22.
1 1.
1 '1.
2 1.
2 ‘1.
3 1.
3 ‘1.
4 1'.
l1 '1.
5 1.
5 '1.
6 1.
6 ’1.
7 1.
7 ‘1.
8 1.
8 '1.
9 1.
9 '1.
10 1.
10 -1.
11 1.
11 ‘1.
12 1.
12 ‘1.
13 1.
13 '1.
111 1.
14 '10
15 1°
15 ‘1.
16 1°

16

-1.

l

97

:CHCOF A" CI Structures.

1111111111111!!!
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91? OF

1!
11

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

1.
.1

11
11
1'1
11

1!
1!

11
1!

11
11

1!
1!
1!
1!
11
11

1!
11

11
1!

11
11

11
1!

11
11

1!
11

11
1!

11
11

1111
111!

11!!
111!

1111
1!!!

no...
—~
on...

111!
111!

111!
1111

11!!
1!!!

1111
1111

-
”0‘
fl”

1!!!
11!!

111!
1!!!

1111
111!

1111
1!!!

11!!
11!!

111!
1!!!

1!!!
111!

C’0 1Tco °

1!
11

.1
1.

.1
1.

1!
11

11
11

1.
.1

1.
.1

11
1!

1.
.1

1.
.1

1.
.1

1!
1!

1!
11

11
1!

.1
1.

1!
11

o!
1.

.1
1.

.1
1.

1!
11

.1
1.

.1
1.

1.
.1

1.
.1

11
11

1!
1!

P

1.
.1

1!
1!

1!
11

1!
1!

1!
1!

1!
11

1!
1!

1!
11

1!
11

11
1!

1!
11

.1
1.

.1
1.

.1
1.

.1
1.

 

Table 22 - Continued.

17
17

18
18

19
19

20
20

21
21

22
22

23
23

24
24

25
25

26
26

28

28

29

30
3O

31
31

32
32

1.
-l.

1.
-1.

1.
-1.

1.
-1.

1.

-‘.

1.
-1.

1.

-1.

1.

-l.

1.
-1.

1.

1.

-1.

1.

-1.

1.

-‘.

1.

-1.

1.

-l.

1.

-l.

1! 1 1 1! 1
11 1 1 1! 1

an...

111 11 1 11
111 11 1 11
1111111111111!!!
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1111111111111!!!

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1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!

1111111111111!!!
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98

11
11

11
11

11
11

.1
1.

11
1!

11
1!

11
11

1!
1!

11
1!

11
11

11
11

11
11

11
1!

1!
11

11
1!

11
11

1!
11

II..-
on...

11
11

11
11

11
11

11
11

11
11

1!
11

11
11

11
11

H“
”H
h..-
D‘I‘

1111
111!

1111
11!!

.1
1.

1.
.1

1.
.1

1!
1!

1.
.1

.1
1.

.1
1.

.1
1.

11
11

1!
11

1.
.1

1.
.1

11
11

1.
.1

1.
.1

1.
.1

11
11

.1
1.

.1
1.

11
1!

1.
.1

11
11

1!
11

11
1!

11
11

11
1!

1.
.1

1.
.1

1.
.1

1.
.1

71*
.1
1.

.1
1.

.1
1.

.1
1.

11
1!

11
1!

1!
11

11
11

11
1!

1!
11

11
11

 

I fi‘fi'd

Table 22 - Continued

33
33

34
34
34
34
34
34

35
35
35
35
35
35

36
36
36
36
36
36

37
37
37
37
37
37

38
38
38
38
3'
38

39
39
39
39
39
39

40
4O
4O
4O
4O
4O

1.
-1.

2.
-1,
-1,
-1,
-1,

2.

2.
-1.
-1,
-1.
-1.

2.

2.
-1.
-1,
-1,
-1,

2.

2.
-1.
-1,
-1.
-1,

2.

2.
-1.
’1.
-1.
-1.

2.

2.
-g,
-1.
-1.
-1.

2.

2.
-1,
-1,
-1.
-1,

2.

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1111111111111!!!
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1111111111111!!!
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1111111111111!!!

1111111111111!!!
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1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!

1111111111111!!!
1111111111111!!!
1111111111111!!!
1111111111111!!!
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1111111111111!!!

1111111111111!!!
1111111111111!!!
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1111111111111!!!

1111111111111!!!
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99

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1!
11
11
11
11
1!

11
1!
1!
11
11
11

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

11
11
!!
11
11
1!

11
11
1!
11
1!
1!

1!

11
11
1!
1!

1!
1!
1!
11
11
11

11
1!
1!
11
1!
1!

11
11
1!
1!
1!
11

1!
11
11
11
1!
1!

0.."
“fl
-
OI.“

1111
1111
1111
1!!!
1111
111!

1!!!
1111
111!
1111
1111
1111

111!
1111
1111
1111
111!
1111

111!
1!!!
1111
1111
11!!
111!

1111
11!!
1111
111!
1111
1111

1111
1111
1111
1111
111!
1111

1111
1111
1111
1111
1111
1111

1.
1.
1.
.1
.1
.1

11
1!
1!
11
11
11

1.
.1
.1
1.
1.
.1

11
1!

.1
1.
.1
1.
.1
1.

1.
.1
.1
1.
1.
.1

1.
.1
.1
1.
1.
.1

p“ O
H”

11
1!
1!
11
11
11

1.
.1
.1
1.
1.
.1

11
1!
11

1!
1!

1.
.1
.1
1.
1.
.1

11
11
11
1!
11
1!

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

1.

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

11
1!
11
11
11
1!

11
1!
11
1!
1!
1!

.1
.1
1.
.1
1.
1.

“Mk

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

11
11
11
11
11
11

 

,1

Table 22 - Continued

4!
4!
4!
41
41
41

42
42
42
42

43
43
43
43

44
44
44
44

45
45
45
45

46
46
46
46

47
47
47
47

48
48
48
48

49
49
49
49

2.
-1.
-1.
-1.
-1.

2.

1.
-1.
-1.

1.

1.
-1.
-l.

1.

1.
-1.
-l.

1.

1.
-l.
-!.

1.

1.
-1.
-1.

1.
1.

-!.
-!.

1.

1.
-1.
-1.

1.

1.
-1.
-1.

1.

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100

pF OF

1.
1.
1.
.1
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

11
11
1!
1!

11
1!
1!
11

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1!
11
1!
11
11
1!

11
11
1!
1!

11
1!
11
1!

1!
11
!1
11

11
1!
11
11

11
1!
11
1!

11
11
1!
11

11
11
11
11

1!
1!
11
11

11!!
111!
1111
1111
111!
1!!!

1!!!
1111
1!!!
1!!!

111!
1111
1111
1111

11!!
1!!!
1111
1!!!

1111
1111
1!!!
1111

1111
1111
1111
1!!!

11!!
1111
1111
1111

1111
1!!!
11!!
1111

1111
1111
111!
111!

0'0 “CO 0'

1.
.1
.1
l.
1.
.1

.1
.1
1.
1.

1!
1!
11
1!

1.
1.
.1
.1

1!
11
1!
11

.1
.1
1.
1.

11
1!
11
1!

11
11
11
1!

.1
.1
1.
1.

.1
1.
.1
1.
.1
1.

guano-.-
up”...

o-o-pfi-
an”...

.1
.1
1.
1.

1.
1.
.1
.1

1.
.1
1.
.1

.1
.1
1.
1.

.1
.1
1.
1.

1.
.1
1.
.1

11
11
11
11
11
11

11
11
11
11

.1
.1
1.
1.

11
1!
11
1!

.1
.1
1.
1.

11
11
11
1!

1.
.1
1.
.1

1.
.1
1.
.1

11
11
1!
11

P

.1
.1
1.
.1
1.
1.

1.
.1
1.
.1

1.
.1
1.
.1

1.
.1
1.
.1

1.
.1
1.
.1

1!
1!
11
1!

1!
11
1!
11

.1
1.
.1
1.

.1
1.
.1
1.

17*

1!
11
11
1!
1!
11

.1
1.
.1
1.

.1
1.
.1
1.

.1
1.
.1
1.

.1
1.
.1
1.

.1
1.
.1
1.

.1
1.
.1
1.

11
11
11
11

11
11
1!
1!

 

101

listed were not included in the CI):

 

 

 

 

Singlet Triplet Quintet Total
A' 76 81 16 173
A" 74 88 14 176
Total 150 169 30 349

 

 

 

 

 

 

 

3 1 1

The structures for the symmetries A", A', and A" are

displayed in Tables 23, 24, and 25, respectively.

D. Natural Orbitals

The natural orbitals (NO's) used in the population
analyses and displayed in the electron density contour maps
were obtained as the eigenfunctions of the one-particle den-
sity matrix for the CI wave function for each state. This
procedure was carried out using a program written by the
author of this thesis. It is technically inaccurate to
call the NO's obtained "the NO's for the molecule" since
that description is reserved for the eigenfunctions of the
exact one-particle density matrix. However, the NO's con-
structed may be called "approximate NO's" or perhaps the
NO's "for this approximate wave function". The occupation
numbers are the eigenvalues of the one-particle density

matrix corresponding to each NO. In a limited CI, the

102

pF OF Trco 0o

1!
11

11
11

11
11

11
1!

11
11

11
11

11
1!

11
1!

11
11

1!
1!

11
11

11
11

1!
11

11
1!

11

11
11

11
11

11
1!

1!
11

11
11

11
1!

11
1!

11
11

11
1!

1.
.1

1!
11

1.

11
1!

1!
1!

1!
11

.1
1.

1.
.1

.1
1.

.1
1.

.1
1.

1.
.1

11
11

1.
.1

11

Table 23. :CFCOH 3A" CI Structures.
1 1. 1111111111111111111!
1 1. 11111111111111111111
2 1. 11111111111111111111
2 l. 11111111111111111111
3 1. 11111111111111111111
3 1. 11111111111111111111
4 l. 11111111111111111111
4 1. 11111111111111111111
S 1. 11111111111111111111
S -1. 11111111111111111111
6 1. 11111111111111111111
6 1. 11111111111111111111
7 1. 11111111111111111111
7 1. 11111111111111111111
8 1. 11111111111111111111
8 1. 11111111111111111111
9 1. 11111111111111111111
9 1. 11111111111111111111
10 1. 1111111111111111111!
10 1. 11111111111111111111
1! 1. 11111111111111111111
1! -1. 11111111111111111111
12 1. 11111111111111111111
12 ~1. 1111111111111111111!
13 l. 11111111111111111111
13 -l. 11111111111111111111
la 1. 11111111111111111111
14 -1. 11111111111111111111
15 1. 11111111111111111111
15 -1. 11111111111111111111

11

.1

11

11
11

.1
1.

11
11

11
11

11
1!

.1
1.

11
11

11
11

11
11

1.
.1

11
11

11
11

1.
.1

1.
.1

O

.1
1.

.1
1.

1.
.1

.1

pap.
n.—

.1
1.

.1
1.

1.
.1

1.
.1

1.
.1

p n* o' 0"

1.
.l

1.
.1

1!
11

1.
.1

11
11

1.
.1

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

1.
.1

1.
.1

1!

.1
1.

.1
1.

.1
1.

103
Table 23 - Continued

pF OFTTCO 00 o p 1r* 0'
16 l. 111111111111111111111.111.1111.1.1.
16 -1. 11111111111111111111.111.111111. 1. .

17 1. 11111111111111111111 1. 1. 11 1! 11.1.1.
17 -1. 1111111111111111111!.1.1111!111. 1. .

18 1. 11111111111111111111111. 1111.11. .1 .
18 -1. 1111111111111111111!11.111111. .11. .

19 1. 111111111111111111111.11111. .111.1.
19 -1. 11111111111111111111.11111.11.111. .

20 1. 11111111111111111111111.1.11.111.1.
20 -1. 1111111111111111111111.1.!!!1.11!. .
2! 1. 111111111111111111111 1. .1 .
21-1. 11111111111111111111!

22 1. 1111111111111111111!111.11.!!! 1. .1 .
22 -1. 1111111111111111111!11.1111. 11.11. .

23 1. 111111111111111111111.1111.1!. 11.1.
23 '1. 11111111111111111111.111111. .1111. .

24 1. 1!!!!111111111111111111.1. .11111.1.
24 -1. 1111111111111111111111.1.11.1111!. .

25 1. 111111111111111111111111.11.

111. .1 .
25 -1. 1111111111111111111111111. .111

.11. O

1. .1.

26 1. 111111111111111111111. 11.11!!!
11011. .

26 '1. 11111111111111111111.1111.11

27 1. 111111111111111111111! 1. .1111. 11.1.
27 '1. 1111111111111111111111.11. 11.1111. .

11.1.

28 1. 11111111111111111111111. .11.11
11111. .

28 -1. 1111111111111111111111.11. .1

29 1. 111111111111111111111. 1. .1111111.1.
29 *1. 11111111111111111111.!.11. 1111111. .

3O 1. 1111111111111111111111.111111. 11.. .
30 1. 11111111111111111111 11 1. 1! 11.! 11.. .

104

Table 23 - Continued

31
31

32
32

33
33

34
34

35
35

36
36

37

37'

38
38

39
39

4O
4O

41
41

42
42

43
43

44
44

45
45

1.
1.

1.

.x.

-1.

1.
-1.

1.
-1.

1.
9‘.

1.
-1.

1.
1.

1.
1.

1.
1.

1.

-1.

1.
-1.

1.
-1.

1.

-l.

1.

-1.

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

91? OF "co 00 O

11
11

11
11

1.
.1

11
11

11
11

1.
.1

.1
1.

.1
1.

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1.

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1.

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1.

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1.

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1.

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1.

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1.

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1.

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1.

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1.

11
11

11
11

1.
.1

11
11

1.
.1

11
11

1.
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11
11

11
11

11
11

11
11

11
11

1.
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1.
.1

1.
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11
11

1.
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11
11

1.
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11
11

11
11

11
11

1.
.1

11
11

1.
.1

11
11

11
11

11
11

11
11

11
11

11
11

1.
.1

11
11

1.
.1

11
11

1.
.1

11
11

11
11

11
11

11
11

11
11

1.
.1

1.
.1

1.
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11
11

poo-o
“on.

1.
.1

11
11

11
11

1.
.1

1.
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11
11

no...
pay-n

.1
1.

pfl*

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1.

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1.

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1.

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1.

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1.

.1
1.

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1.

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1.

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1.

.1
1.

11
11

105

Table 23 - Continued

46
46

47
47

48
48

49
49

SO
50

51
51

52
52

53
S3

S4
54

SS
55

56
SC

57
57

58
58

S9
S9

60
60

1.

-1.

1.
-1.

1.
-1.

1.
1.

1.
-1.

1.

-1.

1.
-1.

1.
1.

1.
1.

1.
-1.

1.

o!.

1.
-1.

1.
1.

1.
1.

1.
1.

1111111111
1111111111

1111111111
1111111111

cunt-o

111111111
111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

1111111111111111111
1111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

P

11
11

1.
.1

11
11

11
11

11
11
11
11
1.
.1

11
11

11
11

11
11

11
11

1.
.1

.1
1.

.1
1.

.1
1.

1.
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11
11

1.

on...
0......

1.
.1

1.
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1.
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.1
1.

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1.

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1.

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1.

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1.

1.
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1.
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1.
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1.

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1.

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1.

11
11

1.
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1.
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1.
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F OF Trco C’0

11
11

.1
1.

.1
1.

1.
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11
11

1.
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11
11

1.
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.1
1.

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11
11

11
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1.

.1
1.

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1.

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1.

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1.

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1.

11
11

1.
.1

106

Table 23 — Continued

* I
O'FTTCOOO a pn 0 o
11111.1. .111.. o.
11 11 .1 .1 1. 11 .. ..

61 1. 11111111111111111111 .
61 *1. 11111111111111111111 1

62 1. 11111111111111111111 .1 1. 1. 11 11 .1 11 .. ..
62 '1. 11111111111111111111 1. .1 .1 11 11 1. 11 .. ..

63 1. 11111111111111111111 11 11 11 .1 .. 11 1. .. ..
63 1. 11111111111111111111 11 11 11 1. .. 11 .1 .. ..

64 1. 11111111111111111111 11 11 11 .. .1 11 1. .. ..
64 1. 11111111111111111111 11 11 11 .. 1. 11 .1 .. ..

65 1. 11111111111111111111 11 11 11 .1 .. 1. 11 .. ..
65 1. 11111111111111111111 11 11 11 1. .. .1 11 .. ..

66 1. 11111111111111111111 11 11 11 .. .1 1. 11 .. ..
66 1. 11111111111111111111 11 11 11 .. 1. .1 11 .. ..

67 1. 11111111111111111111 11 .. 11 11 .1 11 1. .. ..
67 1. 11111111111111111111 11 .. 11 11 1. 11 .1 .. ..

68 1. 11111111111111111111 11 .. 11 11 .1 1. 11 .. ..
68 1. 11111111111111111111 11 .. 11 11 1. .1 11 .. ..

69 1. 11111111111111111111 .1 11 11 11 .. 11 1. .. ..
69 1. 11111111111111111111 1. 11 11 11 .. 11 .1 .. ..

7O 1. 11111111111111111111 .. 11 11 11 .1 11 1. .. ..
7O 1. 11111111111111111111 .. 11 11 11 1. 11 .1 .. ..

71 1.

11111111111111111111 .1 11 11 1 .. 1. 11 .. ..
71 1. 11111111111111111111 1. 11 11 1 .. .1 11

72 1. 11111111111111111111 .. 11 11 11 1. .1 11 .. ..

73 1. 111111111111111111111111.. .111111. .. ..
73 1. 11111111111111111111 11 11 .. 1. 11 11 .1 .. ..

74 1. 11111111111111111111 11 11 .. .1 11 1. 11 .. ..
74 1. 11111111111111111111 11 11 .. 1. 11 .1 11 .. ..

75 1. 11111111111111111111 11 .. 11 .1 11 11 1. .. ..
75 1. 11111111111111111111 11 .. 11 1. 11 11 .1 .. ..

107

Table 23 - Continued

91? 0F Trco 0‘0

76 1. 11111111111111111111 11
76 1. 11111111111111111111 11

77 1. 11111111111111111111 .1
77 1. 11111111111111111111 1.

78 1. 11111111111111111111 ..
78 1. 11111111111111111111 ..

79 1. 11111111111111111111 .1
79 1. 11111111111111111111 1.

80 1. 11111111111111111111 ..
80 1. 11111111111111111111 ..

81 1. 11111111111111111111 .1
81 1. 11111111111111111111 1.

82 1. 11111111111111111111 .1
82 1. 11111111111111111111 1.

83 1. 11111111111111111111 .1
83 1. 11111111111111111111 1.

84 1. 11111111111111111111 .1
84 1. 11111111111111111111 1.

85 1. 11111111111111111111 11
85 1. 11111111111111111111 11

86 1. 11111111111111111111 11
86 1. 11111111111111111111 11

87 1. 1 11 11111 111 1
87 1. 1 11 11111 111 1

~—
an...

1 1 11 111 1
1 1 11 111 1

88 1. 11111111111111111111 11
88 1. 11111111111111111111 11

11
11

11
11

11
11

11
11

11
11

11
11

11
11

11
11

.1
1.

11
11

11
11

11
11

11

pun.
poo-o

p”
u..-

1.
.1

1.
.1

11
11

1.
.1

1.
.1

1.
.1

11
11

11
11

1.
.1

11
11

1.
.1

11
11

Table 24.
1 1.
2 1.
3 1.
4 1.
5 1.
6 1.
7 1.
8 1.
9 1.
1O 1.
11 1.
12 1.
12 ’1.
13 1.
13 -1.
14 1.
14 -1.
15 1.
15 '1.
16 1.
16 ‘1.
17 1.
17 '1.
18 1.
18 -1.
19 1.
19 ‘1.

108

:CFCOH lA'

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

111111111111111111
111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

1

1

1

1

11

11

1

1

1

1

11

11

CI Structures.

pF 1Tco 0o

11

11

11

11

11

11

11

11

1.

1.
.1

11
11

11

11

11

11

11

11

11

11

11

1.
.1

11
11

1.
.1

11
11

11
11

11
11

11
11

1.
.1

11

11

11

11

11

11

o
11

11

11

11

11

11

11

11

1

1

1

11

11

.1

1

1

1.

1

1.

11

11

11

11

11

.1
1.

.1
1.

.1
1.

.1
1.

109

Table 24 - Continued

20 1. 11111111111111111111 11
20 -1. 11111111111111111111 11

21 1. 11111111111111111111 1.
21 -1. 11111111111111111111 .1

22 1. 11111111111111111111 11
22 -1. 11111111111111111111 11

23 1. 11111111111111111111 11
23 -1. 11111111111111111111 11

24 1. 11111111111111111111 11
24 -1. 11111111111111111111 11

25 1. 11111111111111111111 11

25 -1. 11111111111111111111 11
26 1. 11111111111111111111 11
26 -1. 11111111111111111111 11

27 1. 11111111111111111111 11
27 -1. 11111111111111111111 11

28 1. 11111111111111111111 11
28 -1. 11111111111111111111 11

29 1. 11111111111111111111 11
29 -1. 11111111111111111111 11

3O 1. 11111111111111111111 ..
30 -1. 11111111111111111111 ..

31 1. 11111111111111111111 ..
31 -1. 11111111111111111111 ..

32 1. 11111111111111111111 ..
32 -1. 11111111111111111111 ..

33 1. 11111111111111111111 11
33 -1. 11111111111111111111 11

34 1. 11111111111111111111 1.
34 ’1. 11111111111111111111 .1

11
11

11
11

1.
.1

1.
.1

11
11

1.
.1

11
11

11
11

1.
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11
11

11
11

11
11

1.
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11
11

1.
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1.
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11
11

11
11

11
11

11
11

11
11

11
11

11
11

out...
w...

11
11

1.
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1.

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1.

11
11

11
11

1.
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11
11

O! O. O.

l. O. O.

.1 .. ..

l. O. O.

.1.. O.

l. O. .0

.1 .. o.

1. .. ..

.1.. O.

‘0 O. O.

.1.. .0

l. O. O.

.1.. .0

l. O. O.

.1.. O.

l. O. O.

.1.. O.

1. .. ..

.1.. O.

1. .. o.

11.. ..
11.. ..

pan.

hot-
I
O
I
O

110

Table 24 - Continued

35
35

36
36

37
37

38
38

39
39

4O
4O

41
41

42
42

43
43

44
44

45
45
45
45
45
45

46
46
46
46
46
46

47
47
47
47
47
47

1.

-1.

1.

-1.

1.

-1.

1.

-1.

1.
-1,

1.
-1.

1.

-1.

1.
-1.

1.
-1.

1.
-1.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

OI‘UI‘Q
hob-'11

1.
.1

11
11

p...—
gnu-o

11
11

11
11

11
11

1.
1.
1.
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.1
.1

1.
1.
1.
.1
.1
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1.
1.
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9? "co

11
11

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11
11

11
11

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1.
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1.
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1.

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11
11

11
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1.
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11
11
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....-..-..-..-n.-.
punch—“p

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1.

Ody-#0....—
wflflnpp

111

Table 24 - Continued

48
48
48
48
48
48

49
49
49
49
49
49

50
50
50
50
50
50

51
51
51
51
51
51

52
52
52
52
52
52

53
53
53
53
53
53

54
54
54
54
54
54

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1.
‘1.
-1,
-1.

2.

2.
-1,
-1,
-1,
-1,

2.

2.
’1.
-1,
-1.
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

1.
1.
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11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

1.
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11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

upon-nun...
”flu-out...“

11
11
11
11
11
11

F p]? "co 00

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1.
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1.
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11
11
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.1
.1
1.
1.
.1

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

.1
1.
.1
1.
.1

Haunt-Inpu-
hogan—op..-

.1
1.
.1
1.
.1
1.

P

.1
.1
1.
.1
1.
1.

.1
1.
.1
1.
.1
1.

112

Table 24 - Continued

55
55
SS
55
55
55

56
S6
56
56
56
56

57
S7
57
57
57
57

58
58
58
58
58
58

S9
59
59
59
S9
S9

60
60
60
6O
60
60

61
61
61
61

2.
-1,
-1,
-1,
-1,

2.

2.
-l,
-l,
'1.
-l,

2.

2.
-1,
‘1.
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-l,
-l,
-1,
-l.

2.

1.
1.
-l,
-l,

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

tight-thin
penning-no.0...

1.
1.
1.
.1
.1
.1

1.
1.
.1
.1

PF 1Tco 00

1.
.1
.1
1.
1.
.1

11
11
11
11
11
11

1.
.1
.1
1.
1.
.1

1.
.1
.1
1.
1.
.1

1.
1.
1.
.1
.1
.1

1.
.1
.1
1.
1.
.1

11
11
11
11

<5 P

.1 .1
1. .1
.1 1.
1. .1
.1 1.
1. 1.

1 .1
1 1.
1 .1
1 1.
11 .1
11 1.

.1 .1
1. .1
.‘ 1.
1. .1
.1 1.
1. 1.

.1 11
1. 11
.1 11
1. 11
.1 11
1. 11

1. .1
.1 1.
.1 .1
1. 1.
1. .1
.1 1.

11 .1
11 1.
11 .1
11 1.
11 .1
11 1.

1. 11
.1 11
1. 11
.1 11

71*

11
11
11
11
11
11

.1
.1
1.
.1
1.
1.

11
11
11
11
11
11

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.

hog-Op...
..~.-

.1
1.
.l
1.

113

Table 24 - Continued

62
62
62
62

63
63
63
63

64
64
64
64

65
65
65
65

66
66
66
66

67
67
67
67

68
68
68
68

69
69
69
69

70
7O
7O
7O

1.
1.
col.

-1,

1.
1.
-1.

-1.

1.
1.
-1,
-1,

1.
1.
-1,

-1.

1.
1.
-1.

-1,

1.
1.
-1.

-1.

1.
1.
-1.

-1,

1.
1.
-1,

-1,

1.
1.
-1.

-l,

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

0'

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

11
11
11
11

11

.1
.1
1.
1.

.1
1.

.1
.1
1.
1.

11
11
11
11

11
11
11
11

11
11
11
11

11
11
11
11

11
11
11
11

11
11
11
11

“pg...—

1“ pF‘Tcoo’o 0 p

Q.

114

Table 24 - Continued

71
71
71
71

72
72
72
72

73
73
73
73

74
74
74
74

75
75
75
75

76
76
76
76

1.
1.
-1.

-l,

1.
1.
-1,
-l,

1.
1.
-1.

-1.

1.
1.
-1,

-1.

1.
1.
-1,

-1,

1.
1.
-l,

-1.

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11
11
11
11

11
11
11
11

11
11
11
11

11
11
11
11

11
11
11
11

1.
1.
.1
.1

PF "co 0‘0

1.
1.
.1
.1

1.
1.
.1
.1

11
11
11
11

11
11
11
11

11
11
11
11

11
11
11
11

.1

1.

1.
1.
.1
.1

1.
1.
.1
.1

11
11
11
11

11
11
11
11

.1
.1
1.
1.

11
11
11
11

.1
.1
1.
1.

.1
.1
1.
1.

1.
1.
.1
.1

.1
.1
1.
1.

<5 P

1.
.1
1.
.1

11
11
11
11

1.
.1
1.
.1

1.
.1
1.
.1

.1
1.
.1
1.

1.
.1
1.
.1

.1
1.
.1

--
you...”

1.
.1
1.
.1

1.
.1
1.
.1

Table 25.
1 1.
1 ’1.
2 1.
2 '1.
3 1.
3 ‘1.
4 1.
4 '1.
5 1.
5 '1.
6 1.
6 ’1.
7 1.
7 -1.
8 1.
8 '1.
9 1.
9 '1.

10 1.

10 '1.

11 1.

11 ‘1.

12 1.

12 ’1.

13 1.

13 '1.

14 1.

14 -1.

15 1.

15 '1.

115

1

:CFCOH A" CI Structures.

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

OFPFTTCO
1111

11

11
11

1.
.1

11
11

11
11

1.
.1

11
11

11
11

11
11

1.
.1

11
11

11
11

11
11

1.
.1

1.
.1

11

11
11

11
11

11
11

11
11

11
11

11
11

1.
.1

11
11

11
11

11
11

1.
.1

11
11

11
11

11
11

11
11
11
11

11
11

11
11

11
11

11
11

1.
.1

11
11

11
11

11
11

1.
.1

11
11

11
11

11
11

.1
1.

O

11
11

1.
.1

11
11

11
11

1.
.1

11
11

11
11

1.
.1

11
11

.1
1.

O

1.
.1

11
11

11
11

.1
1.

.1
1.

11
11

11
11

1.
.1

11
11

P

.1
1.

.1
1.

.1
1.

11
11

11
11

11
11

11
11

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

as

116

Table 25 - Continued

16
16

17
17

18
18

19
19

2O
20

21
21

22
22

23
23

24
24

25
25

26
26

28
28

30
30

1.

-l,

1.

1.

-1.

1.

1.

-1.

1.

-1.

1.

-1.

1.
-1.

1.

1.

-1.

1.

-1,

1.

o‘.

1.

-1.

1.

-1.

1.
-1,

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11
11

1.
.1

1.
.1

11
11

11
11

1.
.1

11
11

11
11

1.
.1

a...
no...

11

9F
11
11

11
11

11
11

.1
1.

11
11

11
11

11
11

1.
.1

11
11

11’

3......
cup—-

11
11

11
11

11
11

11
11

11
11

1.
.1

11
11

11
11

11
11

1.
.1

11
11

1.
.1

11
11

11
11

1.
.1

11
11

1.
.1

11
11

.1
1.

.1
1.

G

1.
.1

1.
.1

11
11

11
11

1.
.1

11
11

.1
1.

.1
1.

P

11
11

11
11

11
11

11
11

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

11
11

11
11

11
11

11
11

11
11

117

Table 25 - Continued

31
31

32
32

33
33

34
34

35
35

36
36

37
37

38
38

39
39

40
40

41
41

42
42

43
43

44
44

45
45

1.

-1.

1.

-1.

1.
-1,

1.
-1.

1.
-1,

1.

-l,

1.

-1.

1.

-1,

1.

-1.

1.

-l,

1.
-1,

1.

1.
-1,

1.
-1,

1.

-1,

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

111111

11111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111

1.
.1

11
11

1.
.1

1.
.1

11
11

11
11

1.
.1

11
11

11
11

11
11

11
11

11

11

11

11
11

11
11

11
11

11
11

11
11

11
11

c...-
an...

OI..—
an...

1.
.1

11
11

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

.1
1.

o...—
pun.

gun...
”on

118

Table 25 - Continued

46
46

47
47
47
47
47
47

48
48
48
48
48
48

49
49
49
49
49
49

50
50
50
50
50
50

51
51
51
51
51
51

52
52
52
52
52
52

1.

-1,

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

11
11

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

1.
1.
1.
.1
.1
.1

OF 91‘ N00

11
11

11
11
11
11
11
11

1.
.1
.1
1.
1.
.1

11
11
11
11
11
11

1.
.1
.1
1.
1.
.1

1.
.1
.1
1.
1.
.1

1.
.1
.1
1.
1.
.1

11
11

1.
.1
.1
1.
1.
.1

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

11
11
11
11
11
11

11
11
11
11
11
11

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

1.
.1
.1
1.
1.
.1

.1
1.
.1
1.
.1
1.

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

gnawing-j...
pong-noun..."

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

.1
.1
1.
.1
1.
1.

.1
.1
1.
.1
1.
1.

 

119

Table 25 - Continued

53
53
53
53
53
53

54
54
S4
54
54
54

55
55
55
55
55
55

S6
56
56
56
56
56

57
57
57
S7
57
57

58
58
58
58
58
58

S9
59
59
59
59
59

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

2.
-1,
-l,
-1,
-1,

2.

2.
-1,
-1,
-1,
-1,

2.

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

1.
.1
.1
1.
1.
.1

1.
.1
.1
1.
1.
.1

1T

8

...-~
.-..-..-o. 0 O

1.
1.
1.
.1
.1
.1

11
11
11
11
11
11

1.
.1
.1
1.
1.

Hflpno—t‘
O‘I‘flflo—Ofl

11
11
11
11
11
11

.1
1.
.1
1.
.1
1.

U

11
11
11
11
11
11

1.
.1
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10
1.
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1.
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1.
1.
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11
11
11
11
11
11

1.
.1
.1
1.
1.
.1

11
11
11
11
11
11

11
11
11
11
11
11

O

1.
.1
.1
1.
1.
.1

pug-Huck.
pomp-non...-

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

.1
1.
.1
1.
.1
1.

11
11
11
11
11
11

hint-unti—
Human...—

-O~OO
000-...‘0-3

-—.-.

‘gvanz

TableZS
60 2.
60 '1.
60 ‘1.
60 '1.
60 ‘1.
60 2.
61 1.
61 1.
61 '1.
61 '1.
62 1.
62 1.
62 '1.
62 '1.
63 1.
63 1.
63 ‘1.
63 ‘1.
64 1.
64 1.
6‘1 '1.
64 ‘1.
65 1.
65 1.
65 '1.
65 '1.
66 1.
66 1.
66 ’1.
66 ’1.
67 1.
67 1.
67 "1.
67 “1.
68 1.
68 1.
68 '1.
68 ‘1.

120

- Continued

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

OF

1.
1.
1.
.1
.1
.1

Only-oh...
o—pQ-sg-n

--
~fl-

11
11
11
11

11
11
11
11

11
11
11
11

PFTT

1.
.1
.1
1.
1.
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
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1.
1.
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1.
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11
11
11
11

11
11
11
11

CO
.1

1.
1.

1.
1.
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.1

1.
1.
.1
.1

1.
.1
1.
.1

11
11
11
11

0.....-—
gnu-own...

11
11
11
11

.1
1.
.1
1.

.1
1.
.1
1.

 

A
h
-. .-
I
Q
t
V
.1
h‘ ‘
.
.1 ,, .

121

Table 25 - Continued

69
69
69
69

7O
7O
7O
70

71
71
71
71

72
72
72
72

73
73
73
73

74
74
74
74

1.
1.
-1,

'1.

1.
1.
-1,

-1,

1.
1.
-l,
-1,

1.
1.
-l,

-1,

1.
1.
-l,

-1.

1.
1.
-1,

-l,

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

11111111111111111111
11111111111111111111
11111111111111111111
11111111111111111111

OF

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

1.
1.
.1
.1

gun—npfl

uwflfl
0.9-ah.”

11
11
11
11

.1
.1
1.
1.

11
11
11
11

11
11
11
11

11
11
11
11

1.
.1
1.
.1

flung-ou—
boo—pot—

1.

poo—pp.
flint—OI.

1.
.1
1.
.1

.1
1.
.1
1.

1.
.1
1.
.1

gnu-on...-
outfit—.—

.1
1.
.1
1.

n*<fl

.1
1.
.1
1.

.1
1.
.1
1.

11
11
11
11

.1
1.
.1
1.

.1
1.
.1
1.

11
11
11
11

122

NO's and their eigenvalues contain a large amount of concise
information about the CI wave function and are of value as

a diagnostic device as well as a source of physical informa-

tion about the electronic structure.

APPENDIX I I

A. Basis Set

 

The basis chosen for the lithium carbenes was of roughly

double zeta quality and consisted of a set of contracted

nuclear-centered cartesian gaussian-type functions. Each '37“
function in the basis has the form
ni
X1: 2 dim 9‘aim’r“
m=1 ‘VJ

 

where gnaim;r) is a primitive normalized cartesian gaussian
having an exponent aim' and where dim is the contraction co-
efficient of that primitive in the basis function xi. The
exponents and contraction coefficients defining each basis
function in the set used are given in Table 26. Except for
the 4-component p function on Li, taken from Williams,27
the prfinitives were the revised set of Huzinaga36 contracted
according to the method of Dunning.37 Specifically, for
carbon a (985p) set was contracted to a [4s/2p] set; for

lithium, an (85) set of primitives was contracted to [33],

and a (43) set for hydrogen to [23].

B. The Restricted Open Shell SCF (ROSSCF)

 

The SCF results for the lithium carbenes were obtained
using the ROSSCF routine (PA41) of the POLYATOM package of
programs.

123

Table 26 .

124

Basis Set for LiCH and Li C Calculations.

 

 

 

 

 

2
Atom s-functions p-functions
a. d. a. d.
1 1 l 1
1354.159 .00084731232 1.5343 .032763
203.30116 .0065139116 .27499 .139005
46.323493 .032726414 .073618 .500402
13.133489 .11848646 .024026 .508552
Li 4.2477542 .29643380
1.4872743 .44723288
.54097490 1.0
.047841890 1.0
4240.3098 .0012152226 18.099144 .014760512
637.77827 .0092731586 3.9769145 .091649350
146.74534 .045279235 1.1450768 .30392714
42.531428 .15492334 .36188831 .50711806
14.184804 .35808349
C 2.0072531 .14932812 .11460548 1.0
5.1756943 1.0
.49677422 1.0
.15348718 1.0
19.2406 .032828
2.8992 .231208
H .6534 .817238
.1776 1.0

 

 

 

125

For a closed shell wave function (all electrons spin-
paired) the set of one-particle equations are mixed by a
matrix of LaGrangian multipliers. A unitary transformation
on the occupied functions can be found which does not alter
the physical meaning of the total wave function but effects
a diagonalization of the multiplier matrix. The result is
a set of pseudo-eigenvalue equations for the individual
MO's. In the open shell problem, attempts to apply the same
technique met with difficulties in eliminating the off-
diagonal multipliers connecting the closed and open shell
functions, which cannot be mixed in a unitary transformation
without altering the total wave function. No exact method of
constructing open shell Hartree-Fock-Roothaan wave functions
existed until Roothaan38 found a formulation of the problem
requiring no approximations. It is Roothaan's method which
is the basis for the program employed here.

An Open shell wave function based on the independent
particle approximation usually cannot be written as a
single Slater determinant, and in the ROSSCF method, an
expression representing the energy expectation value of the
multi-determinantal wave function is variationally minimized
with respect to the coefficients, Cij' which relate the MO's
to the fixed set of basis functions. This energy expression,

in the Roothaan formulation, has the form

E = 2 X Hk + 1: (2Jk2-Kkz) + f[2 Z Hm
k k1 m

+ £2: (2a Jmn-men) + 2 Z (2
mn km

ka-Kkm)]’

126

where the indices k and 2 run over the set of closed MO's,
m and n run only over the open shell MO's, and f, a, and b
are parameters which are chosen to make the expression
identical to the energy expectation value of the multi-
determinantal wave function of interest. It should be
noted that a set of f, a, and b cannot be found for all

multi-determinantal functions.

 

As an example, consider the LiCH 32- ground state func- E
tion. In terms of Slater determinants, a

¢(32-) =‘JL {I1016202O303c4OZCNXFZ

/2

+ I1016202o3o3o4046?xnzl}

where I...| represents a Slater determinant of one—particle
spin functions, the bar over a spatial function indicates

a 8 spin function, no bar means a spin. Here the spatial
MO's {10,20,3c,4o}are the set of closed orbitals, and the
degenerate pair {flx,flz} are the partially filled open set
(only 2 of the 4 possible spin-orbitals are occupied). The
expectation value of the electronic energy for this function,
written in terms of one-electron integrals, Hi, and coulomb,

Jij’ and exchange Kij' integrals is:

127

e1 _

1'11
I
N
-hfl
:13
H.
+
54
:3
L1
1...
I
N

+
hfl
:F

+

L1
:

Q

X z
m
+ Z (J -K. +J -K )
i lrflx lrflx Iflz lrflz

irj = {10120130140}

m = {lefl21-

Roothaan's energy expression will be identical if f = 1/2,

a = 1, and b = 2.

C. The Single Excitation CI (SECI)

 

To obtain variational wave functions for the excited

states, which again are open-shell functions, without the

ROSSCF calculation to optimize the Mo‘s, a multideterminantal
trial function was explicitly constructed, and the expecta-

tion value of the electronic Hamiltonian over this function

was minimized variationally with respect to the weighting
coefficients of the determinants. In actuality, the trial
function was constructed of structures, which are linear
combinations of determinants and have a definite spatial

and spin symmetry:

128

where the Dj are determinants and the aij are fixed in rela-

tive size for a particular choice of spatial and spin sym-
metry. The two-determinant function 1(3X-) given above is

an example of a structure, with a1 = 1, a2 = 1.

. . +
The structures for the LiCH linear states 1A and 12

1 1

are listed in Tables 27 and 28. The A', A" structures

for bent LiCH appear in Tables 29 and 30. Tables 31 and

32 list the Li C structures of 1A and 12+ symmetry. The

2 9 9
format of each determinant and structure is as given in

Appendix I.

Table 27.

H...

(0101010

00000.1

129

LiCH 1A SECI Structures

Cine x1

1. 11111111................ 11

'1. 11111111................ ..

1. 111111110............... 1.
‘1. 11111111................ .1
-1. 11111111................ ..

l. 11111111000000.000000000 O.

1. 11111111................ 1.
-1. 11111111................ .1

-1. 111111“OOOOOOOOOOOOOOOO O.

1. 11111111................ ..

O. 11..

.. 1. .1
.. .11.

.1.. .0
1. O. .0
O. l. O.

.1
1.

130

Table 28. LiCH 12+ SECI Structures.

a core

1 1. 11111111................
1 1. 11111111................

2 1. 11111111................
2 ’1. 11111111................
2 1. 11111111................
2 '1. 11111111................

3 1. 11111111................
3 ‘1. 11111111................
3 1. 11111111................
3 '1. 11111111....o...........

1.
.1

1.
.1

1.
.1

Thhk329.

U‘U" kh 0U

00

10
10

11
11

12
12

13
13

14
14

l.

l.

1.
-l.

1.
-1.

l.
-1.

1.

-1.

l.

-1.

l.
-1.

1.
-1.

l.

‘1.

1.
-1.

l.
-l.

l.

-1.

l.
-1.

Lflfiil

CONE
11111111

11111111
11111111
11111111
11111111
11111111

11111111
11111111

11111111
11111111

111111
1

11
11111 11

11111111
11111111

11111111
11111111

11111111
11111111

11111111
11111111

11111111
11111111

A'EECI

l.
01

1.
01

l.
01

l.
01

l.
.1

l.
01

l.
.1

I.
.1

I.
01

1.
.1

Stnxnmmes.

131

5‘10 ailal'za 13 514 315a" ‘3'";

11

l.
.1

l.
01

01
1.

a3

Table 30 .

12
12

13
13

l.
-1.

l.
-1.

l.
-1.

1.
-1.

1.
-1.

l.
-1.

1.
-1.

1.
-1.

1.
-1.

1.
-1.

l.
-1.

l.
-1.

l.
-‘.

LiCH 1

(DRE
111111
111111

on...

1
1

11
11

on...

11111
11111
11111111
11111111

11111111
11111111

11111111
11111111
11111111
11111111
11111111
11111111

1111111
1

1
1111111

11111111
11111111

1111
1111

11111111
11111111

11111111
11111111

A" SKI

l.
01

1.
01

l.
01

SW5 .

132

I I I I I I I 1'
a9 a10 an 5‘12 ‘3113 8‘14 315 a1

0.

.1
1.

01
I.

.1
1.

01
l.

01
1.

.1
1.

01
l.

01
l.

.1
1.

.1
1.

.1
1.

a2 a3
.1..
l. ..
.. .l
.. l.

Table 31 .

Ono-n

00001» [0101010

bbbb

1.
-1.

1.
-1.
-1,

1.

1.
-1.
-1.

1.

1.
-1.
-1.

1.

1

133

Li C Ag SEBI Structures.

2

CORE

1111111111
1111111111

1111111111
1111111111
1111111111
1111111111

1111111111
1111111111
1111111111
1111111111

1111111111
1111111111
1111111111
1111111111

11

l.
01

l.
.1

1.
01

lxglxu

.1
1.

21:9 qu 12g

11

1.
.1

1.
.l

1.
.1

Table 32 .

no...

0000110 10101010

bbbb

1.
1.

1.
-1.
1.

-1.

1.
-1.
1.

1.
-1.
l.

-1.

. l
L12C

CORE
1111111111
1111111111
1111111111
1111111111

1111111111
1111111111

1111111111
1111111111
1111111111
1111111111

1111111111
1111111111
1111111111
1111111111

134

1:; SECI Structures

11

1.
.1

1.
.1

1.
.1

APPENDIX III

To construct the aijkl in Equation III.B.2, given
s(?1,?2) in the form of Equation III.B.1, it is sufficient
for illustration to evaluate one term in the sum over the

pairs of determinants:

N N

<DK|Z {asuzsvz-a‘u-EV) 5 (Eu-El) 5 (Ev-f2) |13L >.

ufv

First, however, it is necessary to make some preliminary
explanatory remarks essential to clarity.

The determinants are assumed to be of the form

DK = A(¢l(1)¢2(2)¢3(3)...¢N(N)).

where A is the N-particle anti-symmetrizer, and the ¢i
are spin-orbitals forming an orthonormal set. Although each
pair of spin-orbitals, ¢i and ¢j' are distinct for i¢j,
their space parts, xi and Xj' may be identical.

Also it is convenient to rewrite the spin part of the
two particle operator as follows:

A A
A ‘ — A
3s 3 -s -s =25 s —
uz vz u v uz vz 2

A 1A A A A
—

s 5 +5 5
( u+ v- u- v+

).

The contribution due to a pair of determinants over a
two-particle operator is given by the well-known Slater-

‘Condon rules, by which three cases can be distinguished.

135

136

Defining

0(u,v)E[Zs s u+sv-

1A A A —- .- — .-
uz v2. 5(8 +8u-Sv+)]5(ru.r1)5(rv-r2)'

the three cases are as follows:

a) DK 5 DL:
2<DKIZ l6( ,V)IDK>
U>V

= 22 X<¢k1u1¢21v1IO<u.v1(1-Puv)l¢k(u)¢,(v)>;
k>£
in which R and 2 run over the sets of occupied spin-orbitals
in each determinant, and fiuv is the transposition operator

for the particles u and v. It is through this case that

every determinant has the potential to contribute.

b) In place of ¢a in DK' ¢é appears in D (l dissonant

L
orbital, 1DO):

2<DKIX {6(UIV)IDL>
u>v

= 2 Z <¢k<u1¢a(v)l6(u.v)(i—fiuv)l¢k(u)¢;(v)>.
kfa

c) ¢
(ZDO):

and ¢b in D are replaced by ¢é and ¢£ in D

a K L

2<DKI Z 6(u,v)|DL>
u>v

= 2<¢a<u>¢b(v>|8(u.v)(i-Euv)|¢5(u)¢g(v>>.

137

Any other situation yields a result of zero.
Since each case reduces to a sum of integrals of or-

bital products over 6(u,v)(l-§u ), it is useful to evaluate

V

a general integral of that type. For the integral

<¢a‘“)¢b“” |6<u.v) (i-fiuv) I¢p<u1¢q(v>>.

1“
two possibilities arise; the four spin-orbitals involved
have the same spin or both a and 8 spin are involved. Con—
sidering the first possibility (like spins): J
J

<¢a(u)¢b(v)l(2§uz§vz- 1f§u+§v-+§u-§v+"a(?fi‘f1’
x6 (Ev-E2) (i-EW) |¢p(u) ¢q(v)>
=<¢a(u) ¢b(v) |[2 1%1- 111(0’15(?u"f1)5 ('fv'i‘z’ (i-fiuv) |¢p1u1¢q1v1>
= %{xa (E1) Xb (Y2)xp ('51) xq 52) -xa (3'51) xb (f2) xq 51’ xp “‘2’ } °

For mixed spins, i.e., one a and one B in each orbital pro-

duct, the result is:

d. B A A 1" A A A _ _
<¢a(u%3¢b(v)a|[zsuzsvz-‘§(Su+Sv-+su-sv+)]6(ru-rl)

x5 (Ev-3E2) (’i-fiuv) | ¢p1u1g¢q (v)§>

= <¢a(u)g¢b(V)§|5 (Eu-E1161'Ev-‘r'21{2(- ‘1'"1‘i’uv)l¢p‘“’g¢q(">5>

138
2(1 Puv"¢p(u)a¢q(V>e>}

= - %Xa<?1)Xb(?2)xp(?i)xq(ffi)‘ %[-xa(E1)xb(E2)xq(?1)xp(F2)]

-1%xa1F11xb(?21(i-fi121xp1Fl1xq<E2).

Using these expressions, it is now possible to write

out the results for each pair of determinants.

Case a):

u>v

= Z Z xka(El)x£a(E2)[1-P121Xka(?l)xia(E2)
k >1
(1 C1

+ Z Z XRBEINILBGZ)[1'P123XkBGI’X1352’
kB>£B

ka>£B

- Z Z XkB (37-1))(20' (E2[1-P12]xk8 (171))(10' (E2)

kB>2a
With respect to the index running over the spin-orbitals in
the orbital product of DK’ Rd is the subset labelling the

, and 2 .

a spin-orbitals and similarly for 2a, kB 8

Case b):

2<DK|Z ZO(u,v)|DL>
u>v

139

(-1)PKL*(‘“8"" Z xa('f1)xka(?2) (1-13121x511r'11xka1E21

ka#a

-(-11PKL+““8‘*’ z xa<E11kaGE21 (1-1’5121x51‘511xk85‘2).
kB#a

PKL is the number of transpositions required to bring the

dissonant orbitals, ¢a and ¢;, to the same position in the

k and DL' Ms is the

S2 eigenvalue of the dissonant orbitals (% for a; - % for

orbital products of each determinant, D

8).

Case c):

2 DK|Z [C(u.v)|DL

u>v
= ~~<~11qKL (E) <")(’1‘-f> 1'1‘f1'1?)
' Xa 1 Xb r2 12 xa 1 Xb 2
URL is the number of transpositions required to bring the

dissonant orbitals into corresponding positions in their
respective determinants; such that their spin functions
match in the case of mixed spins. The (+) applies if all
the spins are the same; the (-) if both spin functions are
involved.

The above completes the details of how each integral over
each pair of determinants contributes to the function
S(?1,EZ). It is significant to note that although the sum-
mations involve all the spin-orbitals in each determinant,

in practice there is usually a "core" of closed shells,

140

spatial Mo‘s doubly occupied in every determinant. Due to

39 the core, as a contributor to the

a theorem by McConnell,
spatially symmetric part of the two-particle density matrix,
can be ignored as a cause of ZFS. This will also be true
for any closed shell common to both determinants when
evaluating the contributions from the sum over determinant

pairs. Depending on the wave function employed, these

considerations can effect significant savings in computation.

BIBLIOGRAPHY

10.

11.

12.

13.

14.

15.
16.
17.

BIBLIOGRAPHY

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Standinger, H., and Kupfer, 0., Ber., 25, 501 (1912).
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Venkateswarlu, P., Phys. Rev., 22, 79, and 676 (1950).

 

Andrews, F. B., and Barrow, R. F., Nature (London),
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Herberg, G., and Shoosmith, J., Nature (London), 183,
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Herzberg, 6., Proc. Roy. Soc. (Ser. A), 262, 291 (1961).

 

Herzberg, G., and Johns, J. W. C., Proc. Roy. Soc.
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Bender, C. F., Schaefer, H. F. III, O'Neil, S. V., J.
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Bernheim, R. A., Bernard, H. W., Wang, P. S., Wood, L. S.,
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Herzberg, G., and Johns, J. W. C., J. Chem. Phys. 55,
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Hoffman, R., Zeiss, G. D., and VanDine, G. W., J. Am.
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Staemmler, V., Theor. Chim. Acta., 31, 49 (1973).

 

 

Harrison, J. P., Int. J. Quant. Chem., 5, 285 (1971).

 

Hine, J., Divalent Carbon, The Ronald Press Co., New
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A1

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142

Kirmse, W., Carbene Chemistry (2nd Ed.), Academic Press
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Pople, J. A., Hehre, W. J., Stewart, R. F., J. Chem.
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Shalhoub, G., and Harrison, J. P., J. Am. Chem. Soc., _1,
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Csizmadia,I. G., J. Chem. Phys., 33, 1849 (1966).

 

Powell, F. X., Lide, D. R., Jr., J. Chem. Phys., 32,
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Bodor, N., and Dewar, M. J. S., J. Am. Chem. Soc., 35,
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Mulliken, R. S., J. Chem. Phys., 33, 1833 (1955).

 

Harrison, J. P., J. gm. Chem. Soc., JJ, 4112 (1971).

 

Williams, J. E., Chem. Phys. Lett., 3g, 507 (1974).

 

 

Griffith, J. S., Mol. Phys., J, 79 (1960).

 

McLachlan, A. D., Mol. Phys., 6, 441 (1963).

McWeeny, R., J; Chem. Phys., 23, 399 (1961).

 

Higuchi, J., J. Chem. Phys., _3_8, 1237 (1963); 313, 1339
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Harrison, J. P., J. Chem. Phys., JJ, 5413 (1971).

 

Hameka, H. F., and Hall, W. R., J. Chem. Phys., JJ,
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