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Dedication

To my farnily, who were always there to cornfort and support me. To Doctor

Israel Unger, for believing in me and helping me find my own way in life.

Abstract

Electronic spectra of the cobalt monofluoride molecule have been obsenred for the

£kt t h e . Laser-Ïnduced fluorescence in a pulsed molecular bearn was used to find several

rnoIecuIar bands between 450 and 540 nm. A CoF ground state vibrational fiequency of

662.6 cm-' was determined using the dispersed fluorescence technique. Three molecular

bands, centred at 18780.435 cm-', have been rotationally assigned and the electronic

transition has been determined to be 3<bi - x3aP Well determined constants for these two

States have been obtained by a nonlùiear least squares fit of the data.

High-resolution spectra were obtained for two bands, allowùig analysis of the

hyperfïne splitting caused by the cobalt nuclear spin. The Frosch and Foley hyperfine

parameters were assessed fi-om this data. The a parameter was determined to be 0.0168

cm-' for the upper state and 0.0 1 13 8 cm-' for the lower state. No data for a A i 2 4 band

were collected, so the b and c parameters were inseparable. Upper state values for 6+c

were - 0.0 1 703 cm-' and the lower state result 0.063 5 6 cm".

The hyperfïne tensors of X2X+ Sc0 and T N were computed using the

multi-reference single and double excitation configuration interaction (MRSD-CI) method.

The accuracy of this method was studied as the number of double excitations and

reference configurations in the CI wave function was increased. The computed b,CSSc)

and bF(47~i) parameters were within 92 and 93% of the experimental values, respectively,

while those of cesSc) and c(~'T~) were within 97 to 99%. Thus, the MRSD-CI technique

seems to be a feasible tool for predicting the hyperfine parameters in this class of

diatomics.

Acknowledgments

The author wishes to thank the University of New Brunswick for many years of

financial aid. Thanks are also due to the CheMstry Department for the chance to teach,

l e m and grow. Doctor Michael Sears offered many years of tutelage, and showed the

task of rearing freshmen both hectic and gratifying. Finally, Doctor Man Adam and

Doctor Saba Mattar are warmly thanked for having the patience to deal with a stubbom

student.

Table of Contents

TitIePage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Abbreviations and Acronyms

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 -3 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Spectroscopy Theory . . . . . . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction

2.2 Types of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Consequences of Interaction

2.3 Total Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Kinetic Energy

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Potential Energy

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Rotational and Other Energies

2.3 -4 Rotational Fine Structure ................................... 25

2-3 -5 Ladder Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2-4SpinInteractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.lSpin-Orbit 27

2.4.2 Spin-Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.3 Spin-Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 28

2.4.4 Second-Order Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Hyperfine Interactions .......................................... 29

2-6 Matrix Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6.1 Linear Least-Squares Fitting Procedures . . . . . . . . . . . . . . . . . . . . .. .. 34

2.6.2 Non-Linear Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 .1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 .2 TheoryandFocus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 -3 Configuration Interaction .. 43

3 .3.1 The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 -3 -2 Cornputational Details .. 47

3.3 .2.a Configuration Interaction Limitations and Strengths . . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . . . . . . . . . . 3 -3 .2.b Atornic and Molecular Orbitals 4 8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 .3.2.c Syrnmetry Considerations 49

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 .4 Introduction to MELD 50

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 MELD Components 51

4.4.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.4.2 Other Experirnental Considerations . . . . . . . . . . . . . . . . . . . . . . . . . .. 81

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Spectroscopie Results 83

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction 83

5.2 Generd Method of Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . . 84

5.3 Prelirninary Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Low-Resolution Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.4.1 Ground State Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5 .4.2 Analysis and Assigrunent of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.4.2.a Prelirninary Evaluation Using a Hund's Case(c) Mode1 . . . . . . . . . 98

. . . . . . . . . . . . . . . . . . . 5 A.2. b Evduation Using Hund' s Case(a) Mode1 100

5 .4.2.c Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

. . . . . . . . . . . . . . . . . . . . . . . . . . 5 -5 Rotationally-Resolved Vibrational Andysis 106

5.5.1 Case (a) Rotational Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

. . . . . . . . . . 5.6 High-Resolution Andysis Including Cobalt Hyperfine Structure 113

. . . . . 5 .6.1 Matrix Elements for High-Resolution Andysis, Hund's case (ap) 120

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Andysis of Hypefine Results 123

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Computational Results 127

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction 127

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Computational Details 130

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 ~ h e ondin^ in X2Z'Sc0 and T ~ N 131

. . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Hyperfine Coupling Constants of S c 0 .. 134

... Vlll

. . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Hypef ie Coupling Constants of T a 142

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary and Conclusions 144

7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 CobaltFluonde 146

7.2 Configuration Interaction of Scandium Oxide and Titanium Nitride . . . . . . . . 147

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Future Considerations 147

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 References 150

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 158

List of Tables

4-1 Laser Dyes Used in Experiments . . . . . . . . . . . . . . . . . . . . . . .. .... . . . . . . . . . 77

5-1 Observed Cobalt Fluoride Molecular Bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5-2 Dispersed fluorescence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5-3 Linear Regression for Low-Resolution Scan . . . . . . . . . . . . . . . . . . . . . . . . . .. . 94

5-4 T, and B, for Excited and Ground '@ States . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5-5 ~ a t r i x ~lements for the 3@ State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5-6 Molecular Constants for the Excited and Ground 3@ States . . . . . . . . . . . . . . . . 105

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-7 Unassigned Bands 108

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 vibrationai Constants for the 3@ ~ a n d s 109

. . . . . . . . . . . . . . . . . . . . 5-9 Case (a) Non-linear Ieast-squares fit for the '0 Bands 111

. . . . . . . . . . . . . . . . . . . . . . . . . . . 5- 10 Observed Hyperfine Transition Frequencies 121

. . . . . . . . . . . . . . . . . . . . 5-1 1 Hyperfine and Molecular Constants for the 'a,, Band 122

5-12 Hyperfine and Molecular Constants for the 'a, Band . . . . . . . . . . . . . . . . . . . . 124

6-1 Orbital Occupations and Correlation Coefficients for Sc0 and T N . . . . . . . . . . . 133

. . . . . . . . . . . . . . . . . . . . . . . . 6-2 Magnetic Hyperfine Parameters for S c 0 and T a 143

. . . . . . . . . . . . . . . . . . . . . . A-1 Observed Line Frequencies for the 3@4 X 'a, Band 158

. . . . . . . . . . . . . . . . . . . . . . A-2 ~bserved ~ i n e Frequencies for the 3@3 x 3@3 and 159

. . . . . . . . . . . . . . . . . . . . . . A-3 Obsenred Line Frequencies for the X 'a, Band 160

. . . . . . . . . . . . . . A-4 Line Frequencies For the Combined Low-Resolution Dataset 161

. . . . . . . . . . . . . . A-5 Observed Hyperfine Line Frequencies for the '0, X 304 Band 163

. . . . . . . . . . . . . . A-6 Obsenred Hyperfine Line Frequencies for the 3@3 X 'a3 Band 167

X

List of Figures

2-1 Potential energy curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2-2 Molecule-ked fiame of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2-3 L and S projections on rnolecdar fiame of reference . . . . . . . . . . . . . . . . . . . . . . 18

2-4 Hund's case (a) coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2-5 Hund's case (c) coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2-6 Hund's case (%) coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1

4-1 Pulsed-dye laser apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4-2 Vacuum rack assernbly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4-3 Reaction charnber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4-4 Timing and Data Collection Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Spectral Coverage of Laser Dyes 78

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-6 ReactionZone 80

. . . . . . . . . . . . . . . . . . . . . . . . . . 5-1 Molecular Orbital Diagram for Cobalt Fluoride 90

. . . . . . . . . . . . . . . . . . . . . . 5-2 Molecular Band Spectum with P, Q and R branches 92

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-3 LOW-~esolution ~ c a n of 3 @ , - X '4D4 and 95

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Low-Resolution Scan of 'a3 - X 3 ~ 1 3 Band 96

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-5 LOW-~esolution ~ c a n of 'a2 - x '0, and 97

5-6 First R, Q and P transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5-7 Scheme for AR = - 1 fluorescence experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 212

5-8 Combined High-Resolution Scan of Several Rotational Transitions . . . . . . . . . . . 114

5-9 High-Resolution Scan of R-branch Rotational Transition . . . . . . . . . . . . . . . . . . 116

5-10 Hyperfine Transitions for the R(5) Rotational Transition . . . . . . . . . . . . . . . . . . 117

5-1 1 HyperFine Transitions for the Q(4) Rotational Transition . . . . . . . . . . . . . . . . . 118

5- 12 Q-Branch Hyperfine Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6-1 Fenni contact of 45Ti versus SARC'S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

6-2 Fenni contact of 170 versus SARC's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

6-3 ~ e r m i contact of 4 5 ~ i versus-log T, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

6-4 ~ermi contact of 170 versus-log TE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

List of Abbreviations and Acronyms

CI

CISTAR

CPF

EPR

GTO

HFSD-CI

LCC

LDF-LCAO

LIF

MELD

MOLINT

MOMAVGLT

MRSD-CI

MRSD-CI-NO

Configuration Interaction

Configuration interaction /perturbation theory program

Coupled Pair Functional

Electror Paramagnetic Resonance

Gaussian-Type Orbitals

Hartree-Fock Single and Double excitation Configuration

Interaction

Improved Virtual Orbitais

Linearized Couple Cluster

Local Density Functional Linear Combination of Atornic

Orbitals

Laser Induced Fluorescence

Many Electron Description

CI molecular properties prograrn

S p herically Averaged Momentum Distribution program

segment

Multi-Reference Single and Double excitation ConGguration

Interaction

Multi-Reference Single and Double excitation Configuration

Interaction using Naturai Orbitds

program segment to analyze orbital populations

Restricted Hartree-Fock Self-Consistent Field

ROHF

RTSIM

SARC

SCF

SINT

SORTIN

SPNORB

STO

TMOM

TRNX

Restrkted Open shell Hartree-Fock

Davidson method sparse matrix eigenvalue/eigenvector

PrOgran'l

Spin Adapted Reference Cofigurations

Self-Consistent Field

Cartesian Gaussian integraVpseudopotential prograrn

Sort routine for transformed integrals in the CI prograrn

Effective Spin Orbit operator program segment

Slater-Type Orbital

Transition Moment prograrn segment

Transformation of integrals over molecular orbitals program

1 INTRODUCTION

1.1 Philosop hy

Practitioners of the physicd sciences have a desire to understand the world fiom

the perspective of fùndamental concepts. How is it that things are put together? If we

c m expand our knowledge of the microscopic, perhaps we can better grasp the

macroscopic world around us. The exponential increase in computing powers over the

Iast haIf-century, and irnprovements in electronics, optics, and so forth have enabled

modern investigators to delve ever more deeply into the intemal workhgs of rnoIecules

that make up chernical physics.

The fùndamental mathematical tenets that comprise our understanding of

intermolecular interactions were developed in the early part of this century. From these

early works, Iifetimes of philosophical reflections have honed these ideas, embellished

them with new, exciting concepts, expanded Our picture until it reached the current state

of comprehension. With the introduction of enormous computing powers, these

mathematical principles went beyond the mere 'thought' experiment and occasional

numerical rigour. We now have the ability to carry out a staggering amount of calculation

in fractions of a second. This has allowed investigators to examine highly complex

systems in an dl-inclusive mamer. Any discrepancies between computation and

observation can often be accounted for by noting physical limitations in experimental

techniques or inaccuracies in the mathematical model.

Much of the same history that advanced Our understanding of the mathematics and

physics of rnicroscopic events also benefitted experimental research work. Improvements

in the fields of electronics and optics aiiowed better equipment to be designed for

investigations of the observable, physical effects in small ensembles of atorns and

molecules. More precise measurement ofmicroscopic interactions enables researchers a

better view into the intemal workings of the building blocks of nature.

The study of small molecules has burgeoned in the last few decades.

Investigations take many foms, including expenrnental and theoretical endeavours. One

might expect the results Eom such explorations to be rather simple or at least

straightfonvard, but this is seldom the case as the constituents become 'heavy.' Many of

the small molecules containing atoms of the first rows in the penodic table are quite well

understood. Studies of their behaviour generally involve exacting cornputational

examinations of high-order electronic effects or investigations into the physical,

macroscopic properties they manifest.

#en the next period is approached, however, things irnmediately take on rnuch

more complexity. The study of transition metal atoms and molecules containing these

species is more difficult due to the addition of the M-shell, or 3d electrons. This collection

of up to 10 electrons interacts with nearly al1 the effects exhibited by the rather pedestrian

s- and p-shell electrons, as well as a codksing selection of cross-terms with other

electrons of the 3d shel and the nuclei they surround. Hence, the study of compounds

containing transition metal atoms is fraught with subtle but important effects that need to

be accounted for so that proper analysis can be completed.

1.2 Raiionale

The scope of this thesis was to examine a smali molecule from both expenmental

and theoretical viewpoints. It was hoped, by investigating such a species in this manner,

that the dBerences between physical, experirnental evidence and computational theoretical

results could be critically exarnined. The choice of cobalt fluoride as candidate for such a

study seemed reasonable: no experimentai evidence of this compound was known, and

only a rough approximation of the theoretical underpinnings had been proposed [l]. The

available experimental apparatus would aiIow us to attempt creation of this species using

well-known methodologies[2]. Although the nurnber of electrons was substantial at 36, it

was thought that available computational techniques and resources would be adequate to

the task.

1.3 Methodologies

For the experimental work, laser facilities at the University of New Brunswick

were utilized, in collaboration with Dr. Colan Linton of the Physics Department. The

pulsed-dye and ablation lasers were new equipment set up shortly before this work began,

while the continuous wave laser apparatus was legacy equipment. Cornputer control of

the CW laser was dso fairly dated technology, utilizing an Apple IIe for data collection.

Further details are found in Chapter 4.

Many cobalt fluoride band systems were observed during the course of research.

The theoretical background to spectral analysis of this form is discussed in Chapter 2.

Most of the observed band systems were studied in detail and a compendium of these

results is given in Chapter 5.

An examination of the computational theoretical approach to small molecde

investigations is presented in Chapter 3. Some of the most important spectroscopie

constants of paramagnetic transition metal diatomics are the Frosch-Foley hyperfine

constants a, b and c. Unfortunately, as pointed out in Chapter 6, they are the most

ditncult magnetic properties to compute. Only a handful of a, b, b, and c computations

exist. They have all been done in Our laboratory. The X3Q CoF is particularly difEcult to

cornpute since it cannot be expressed as a single determinant configuration of Cartesian

orbitals. Consequently, single determinant techniques such as local density functional or

Hartree-Fock approximations are not applicable. This is compounded by the fact that the

Co moiety has a Iarge nurnber of d electrons that require extensive electron correlation to

properly describe the electronic States. Finally, the Co atom is heavy enough for

relativistic effects to play a discemabIe contribution to the hyperfine parameters.

Therefore, before attempting to compute the Frosch-Foley pararneters of a complicated

high spin state like X3<D CoF, we set out to establish whether multi-reference

configuration interaction computations are capable of accurately computing doublet state

transition metal-containing molecules where correlation is minimal. The candidates that fit

these cntena are the isoelectronic molecules S c 0 and Tm. Results of this study are

presented in Chapter 6.

A summary of the experimental and theoretical findings contained in this work is

presented in Chapter 7.

2 SPECTROSCOPY THEORY

2.1 Introduction

A variety of modem techniques are now available that allow researchers to

investigate atoms and molecules. Atoms consist of a charged core contauiing positive and

neutral particles, protons and neutrons, respectively, and an outer sheli of negatively

charged electrons, which balance out the positive charge in neutral atoms. The interaction

of these particles with other atoms cause attractive forces to f o m bon& which give rise to

diatomic and polyatomic molecuIes. The electrons within atomic and molecular species

reside in orbitds having specific energy states or levels. By observing transitions of

electrons fkom level to level within these species, important information is obtained about

the electronic makeup of atoms and molecuIes. The dflerent types of molecular

transitions are described below.

2.2 Types of Energy

To properly describe the energy within an atom or molecule requires consideration

of both potential and kinetic energy contributions. Simply put, potential energy is the

ability of a systern to rnove fiom a state of higher energy to one of Iower energy.

Projected on the atomic or molecular level, potential energy may also be considered as the

ability to change an electron'sposilion from one energy state to another. Potential energy

transitions are large, usually a few thousand wavenumbers. More subtle interactions occur

within the molecular h e w o r k and are summarized by the kinetic energy terms.

5

Kinetic energy involves molecular momenta. This motional energy may take on

several foms summarized as translational, vibrational and rotational. Translational energy

describes the actual motion of molecules through space, and is determined, for the most

part, by externai factors. As such, little molecular information cm be obtained f?om the

study of translational energy. The other two forms of energy make up the bulk of

spectroscopic investigations. While polyatomic molecules have a large number of

available energetic modes which make analysis difficult, the simplest molecules, diafomics

exhibit well-behaved patterns of energy transfer. Rotational energy is contained within the

fiamework of spinning molecules about sorne moment of inertia. The rate of rotation can

be determined in a quantitative fashion. In addition to this tumbling motion, molecules

also tend to vibrate or oscillate about the centre of mass. This vibrationai energy can also

be quantized. The quantization and charactekation of these energy modes are discussed

below.

2.2.1 General

In molecules, atomic orbitals combine to give characteristic bonding, nonbonding

and antibonding molecular orbitais. In a diatomic molecule, a pair of orbitals of roughly

equal energy and the same symrnetry wilI combine to form a bonding and an antibonding

orbital pair, with the bonding orbital lower in energy than the component atomic orbitals

(hence, a stabilizing influence in lowering the overall energy). Unrnatched orbitals tend to

remain localized on an atom and are usuaily considered non-bonding with little change in

energy fiom atomic to molecular States. For a molecule to exist, the gain in stabilization

energy (bonding) must exceed the destabilizing effects of the antibonding orbitals, which is

why atoms with partidly fiiied orbitals make the best molecuIes- the electrons wiIl fïll the

bonding orbitals first, leading to a more stable state, while leaving the antibonding orbitals

ernpty. Other effects, such as thermal energy absorption, weak bonding and so forth, may

lead to mefastable or short-lived molecules. While the investigation of transient molecules

is often ditncult and requires specidized experimental apparatus, modem rnethods and

equipment make such studies possible. These rnethods are further elaborated in

Chapter 4.

While the concept of molecular bonciing is well understood, the ordering of

orbitds is a more complex situation. SmalI molecules with few electrons are usuaily well-

behaved (Le., they may be easily described, with orderly filhg of the molecular orbitals).

When larger atoms containing comrnensurately more electrons are investigated, the picture

becomes less clear. The energy level separations are srnaIl so that a descriptive 'picture' of

the bonding is not easily obtained. It is often the case that the ground state of larger

molecules cannot be obtained using such a sirnplistic approach.

2.2.2 Consequences of Interaction

A good mathematical rep~esentation of the system, which includes al1 signdicant

interactions between the different forces within the molecule, is essential to the study of

rnolecular energetics. Practically speaking, one can onIy account exact@ for most

interactions, with the remaining effects treated in an approximate fashion or by use of

perturbation methods. Construction of the mode1 must be done with extreme care, as an

oversight of any significant terms will invaiidate the method. For a diatomic molecule,

spin, rotation, vibration, orbital anguiar momentum, and naclear eEects must a ï i be

considered. It is not enough to include these terms in an isolated manner, as interactions

between these effects will also occur.

The following three sections examine the details of these interactions, and

mathematical methods used for malysis are discussed in section 2.6.

2.3 Total Energy

CIassically, the energy of a system can be descnbed by [3]:

where E,, denotes total energy, and Ev and ET represent potentiaI and kinetic energy,

respectively. Since the energy of a system depends on the momenta of its electrons and

nuclei, as well as the interactions between molecular pzrticles, the problem of describing

this energy can b e extreme.

In the 1920's, quantum mechanics and matrix mechanics were deveIoped to

improve the theoretica1 description of molecules. The two approaches were found to be

equivalent treatments to this problem. Electron behaviour may be described using a wave-

like rnathematical formula. Schrodinger postulated a wavefinction that would describe

such a system. This wavefünction must describe the system with respect to the

coordinates of a11 particles in the system, as well as include the effects of the system

chmges over tirne. Schrodinger's tirne-dependent equation for an n-particle system is

given by[4] :

where Y denotes the state function and h = hnn. The last tenn accounts for the potential

of the system and the kinetic term %m 9 has been replaced by momentum equivalents:

with p, in operator form:

A moIecule or atom in an isolated system c m be considered independent of time:

where the surn is over n particles and a solution to the wavefinction ry is cailed a

stationary state,

Finally, we d e h e the electronic Hamiltonian as:

where the first terrn is the electronic kinetic energy and the second the potentiai energy of

the electrons. The symbol V2 denotes:

And the compact expression is then:

Next, the molecular energy terms of the Harniltonian are examined.

2.3.1 Kinetic Energy

The kinetic energy of a molecule is a function of the momenta of nuclei and

electrons. The Born-Oppenheimer approximation allows us to treat these two types of

particles separately. S ince a proton or neutron is nearly two thousand times heavier than

an electron, and the nuclear centre contains many of these particles, the electrons

essentially see fixed point charges for the nuclei, and the nuclei see the fast electron

motion as a smeared out distribution of electronic charge [SI. By investigating a molecule

having fixed nuclei, the translationai kinetic energy operator for the nuclei is effectively

omitted from the analysis, leaving only potential energy and electronic khetic energy

terms to be examined:

The kinetic energy may be fiirther broken up into rotational and vibrational contributions

as shown below.

2.3.2 Potential Energy

The potential energy is described by a series of terms:

EV = V*, + Vm + V,

where the first term is attractive and represents the nuclear-electron interaction, the

second and third terms are repulsive and describe nuclear-nuclear and electron-electron

interactions. This energy is a function of the coordinates of the nuclei and electrons.

While multinuclear molecules are three-dimensional, we cari treat diatomics as essentially

two-dimensional systems, since out-of-plane potential energy interactions average to zero.

This allows us to examine the interatomic motion as a simple harmonic osciilator problem

govemed by Hooke's law. As the atoms move toward and away f?om each other, the

potentiai rises steeply on either side of the equilibrïum. A distortional term arises fkom the

dissociation behaviour of the atoms in the molecule. See Figure 2-1.

Simple Harrnonic Oscillator

l Anharmonic Distortion

Bond Length (r)

Figure 2-1. Potential energy curve showing harmonic and anharmonic behaviour.

This potentid is given classicaiiy by:

v = %kX2

where k is the force constant (bond energy) and x de :scribes the displ

( 11)

acernent fi-om the

equiiibrium bond distance ( x = r-r,). The oscillation fiequency is given by:

where p is the reduced rnass, and the oscillation fiequency is in Hertz 161. Vibrational

energies are typicdly given in wavenurnbers:

- - k - L - Cm -1 osc 2xc p

It is found that vibrational energies are quantized, so the Schrodinger equation can be used

to obtain [6] :

where eV is the vibrational energy for a given vibrational quantum number v. Successive

vibrational levels will be evenly spaced @y the amount GoSc). The constant tenn %GoSc

results in a zero-point energy (ground vibrational state).

The paraboiic curve arising fi-om the harmonic osciIlator mode1 is generally a poor

description of the real behaviour of a system. As the curve reaches the level of the

13

dissociation energy, it deviates markedly f?om the harmonic osciHator curve. A

distortional correction to this mode1 must be included. A better approximation of the

distortion is obtained if an exponential Morse function is used [7]:

where De is the dissociation energy and a is a molecular constant. When this is

incorporated into the Schr6dinger equation, an anharmonic oscillator function is obtained:

Higher order distortion terms may also be included to improve the 'fit' of the calculated

potential energy curve to observed data. With sufficient experimental information, the

equilibrium frequency and anharmonicity constants may be obtained. Vibrational energy

changes within an electronic transition will result in a senes of vibrational progressions.

Vibrational progressions in the States observed for this thesis are discussed in section 5.5.

Potential energy curves exist for excited state systems. Transitions occur f?om one

state to another, sign@ing a change in electronic orbital occupations. Consequently, a

change in angular momenta, vibrational and rotational energies may occur with this

electronic promotion.

2.3.3 Rotational Energy

Rotational motion involves the change in nuclear coordinates for a fixed bond

length r, so a ngid-rotor approximation [8] cm be used to describe this effect. Since

vibrational energy describes a change in bond length, these two terms are effectively

separated. Use of the Born-Oppenheimer approximation d o w s us to determine the

instantaneous interaction energies for these terms. The translational energy wiii be a

constant within this fiamework. We have already discussed the vibrational behaviour of

molecules and will now examine rotational energy.

The rotational kinetic energy describes the 'end-over-end' motion in a diatomic.

This term represents the momentum of both the electrons and nuclei:

where P is a vector denoting the momentum and m the mass for a nucleus n or an electron

e. Since nuclei are much heavier than electrons, momentum is well accounted for by the

following:

where m denotes nuclear mass and Fdistance fiom the origin (centre of mass) for each of

the n nuclei.

In a diatomic molecule, bonding between atoms creates an 'interatomic' axis. This

gives a frame of reference in which to describe the interactions of the nuclei and electrons

from the two atoms. See Figure 2-2. Unfortunately, the framework for observables

(experimental spectra) is fixed macroscopically within the laboratory, whereas rnolecutar

interactions are fixed microscopically within the molecule. The main properties that

describe electronic structure in a molecule aise from the orbital angular momentum and

15

Figure 2-2. Molecule-fked Eame of reference, showing no net off-axis mornentum.

16

spin of its electrons. Paired electrons (Le., fiIled orbitals) are not used to describe the

molecular state as the opposed spins effectively cancel their interactions. Electrons in

u a e d orbitais will contribute to the description of the overaii electronic state. The

electronic orbital and electron spin components of the molecule interact with the axial

electric and magnetic fields produced within the molecule[9]. Projection of the orbital

angular momentum L onto this internuclear axis is denoted A. Projection of the spin

component S results in the projection quantum number Z See Figure 2-3. The strength of

these interactions with each other and their behaviour under molecular rotation lead to

severai unique situations. Hund [l O] proposed five cases of angular momentum coupling,

labelled (a) to (e), whîch encompass these difYerent scenarios. Cases (a) and (c), whîch

are important to this thesis, will be examined shortly, but first the behaviour of (diatornic)

molecules interacting with extemal radiation must be exarnined.

Rotational energy within a molecule is quantized, with a value dependent on the

moment of inertia. For diatornic molecules, rotation around the internuclear axis carries

no net effect (Il, = O), so only rotation perpendicular to this axis is considered. This

rotationd motion interacts with the spin and orbital angular momenta, as shown below-

The rotationd Harniltonian operator may be stated as:

A

where p is the reduced mass, r the internuclear distance, R the rotational operator and B is

the rotational constant.

Figure 2-3. Angular rnomentum (L,) and spin ( S . ) projections (A and C, respectively) on a molecuIe-fked axis for a diatomic rnolecule. Where both terms are properly quantized, their sum is given by a.

Explicitly, the rotational operator may be *en as:

where Ri is the vector for the ith axis. For a diatomic molecule with no z-axis rotation,

we have:

To explain experimental spectra, a number of molecular interactions must be

included. The strengths of these interactions determine which mathematical mode1 best

describes the system. Hund [IO] ordered the different combinations of interactions into

specific cases. Cases (a) and (c) are most important to this discussion. Hund's case (a)

describes a diatomic rnolecule wherein both the electronic orbital and spin angular

rnomenta are strongly coupled (quantized) to fields dong the internuclear axis. The

3 + rnomentum vectors L and S are projected ont0 the internuclear axis to give A and C,

respectively. These in turn add to give an overd projection momentum vector Q. This

+

couples with the end-over-end molecular rotation vector R to form a resultant y, the total

angular rnomentum (excluding hyperfine interactions). See Figure 2-4.

In case (a) systerns, since A, 2 and J are al1 quantized, they are said to have 'good

n quantum numbers,' Le. their operators cornmute with H. This means that wavehnctions

+ + 3

Figure 2-4. Hund's case (a) type interaction: Rotational vector R couples with L and S +

(projection Q) to form the resultant rotational projection operator J.

for these angular momentum operators forrn an accurate basis hnction description of the

rotational state. In ket notation, rotational energies are given by:

-, 3 3

The rotational vector may be restated in terms of J, L and S as:

which, for diatomics, gives the rotational operator the following form:

Recall, % rotation around the internuclear axis, is neglected. This c m be rewritten in

terms of raising and lowenng operators and projections ont0 the internuclear (2) axis as:

Inserting this into the rotational Harniltonian (eq. 1 9) results in an operator:

where B denotes B(r), the rotational constant for a fixed value of r. The eigenvalues of J2

are found to be:

Sirnilarly, the quantum numbers for L2, S2, J& L: and Sj are given by[13]:

When these are substituted into the rotational Hamiltonian, we obtain[l3]:

where h2 is absorbed into the rotational constant, B. Recaii, these terms are al1 well

quantized in a case (a) basis set. To descnbe a molecule using Hund's case (c), one can

imagine the electronic spin and angular momentum couphg together with a strength of

interaction that is stronger than the separate interactions with the axial fields [12]. In this

+ * -b

case, L and S will form a resultant J, which will in turn have a projection on the

-* -*

intemuclear axis, R . The vector J, will add with R to give the overail total angular

4

momentum J . See Figure 2-5.

Raising and lowering operators, as the name suggests, are terms that connect

States whose projections differ by one. For instance, the operator L+ connects a state

having projection A with one having a value of A+l.

+ + +

Figure 2-5. Hund's case (c) molecule showing L coupling directly with S to give J,. -. +

This couples with the rotational vector R to form J .

Commutation relations give the raising and lowerùig operators the following

definitions[13] :

Unlike L and S operators, which are invariant under rotation, the J operator involves a

rotation in the space-fked framework. This leads to the reversal of roles (and signs) for

the J ladder operators, Le., J+ is a lowering operator and J- the raising operator [14]. This

is due to the transformation from a molecule-hced to space-ked coordinate system. The

coordinates fiom one system are transfonned to the other by a unitary matrix containing

the direction cosines.

The L?+I.,,~ tenn is dficult to assess[l3], so is usuaily incorporated into the term

energy. Using the sirnpler form of equation (26) and rearranging the terms, gives:

The spin and rotation elements have been grouped in the first tenn. This will be

considered the pure rotational operator, with cross terrns treated separately. The second

term is a spin-rotation interaction and connects states whose spins differ by one. At high

rotational energy, this term causes Hund's case (a) molecules to migrate to case (b)

systems. This occurs due to the electronic spin decoupling fiom the intemuclear axis [15].

The third tenn connects states of different A and is called the etectronic Coriolis tenn..

The last term includes both orbital projection terms and spin-orbit interactions.

A closer examination of the mixed-operator terms must be performed before an

accurate picture of the molecular interactions can be formed. Fust, terms in a valid

mathematical fiarnework need to be assessed. A good method is to use Dirac notation

(see Equation (27)). The rnatrk form of the diagonal (pure rotationai) operator is given

by Zare [16] as:

where B, is defined as:

where B, is the rotational constant and D, the centrifugai distortion constant for the

vibrational Ievel v. Off-diagonal rotationd terms also exist, and are discussed in

section 2.3.5.

2.3.4 Rotational Fine Structure

When rotational transitions occur within an electronic transition, rotationalflne

struclzrre results [log]. These rotational transitions obey selection mles that allow only

certain changes in the rotational quantum number J. Transitions will occur for AJ = O and

il. This change in J is always measured from the lower state. For AJ = +1, transitions

are labelled R-branch. When AJ = O, Q-branch lines occur and for AJ = - 1, P-branch lines

are observed. See section 5.4.2 for rotational fine structure experimental results.

2.3.5 Ladder Operators

Using Dirac notation [17], ladder operators for spin take on the form:

Equation (34) shows that the S, ladder operator connects States whose projections C

dif5er by one. The other ladder operators have a similar form:

Note, however, the sign reversai of the J, ladder operator. The rotational Hamiltonian

contains terms off-diagonal in C [16]:

with sirnilar elements for (El,,,),, , diffenng only in the leading B term:

For molecules having unpaired spin, the spin rnay couple with the orbital and

rotationai momenta. This leads to three distinct terms: spin-orbit, spin-rotation and spin-

spin interactions. The Hamiltonian operators for these terms are given below.

2.4 Spin Interactions

2.4.1 Spin-Orbit

For the spin-orbit interaction, the overail expression is found to be [ 161:

The first tem is one of the most important of the fine structure elements. For multiplet

systems, this A term defines the spacing between energy levels which differ only in their Q

values. In a good case (a) molecule, where L and S are well-behaved, this tenn may be

easily isolated. The spin-orbit ladder terms connect States that d s e r in A by *1. It is

usually combined with the Coriolis term (JJJ of the spin-rotation Hamiltonian to

describe an effect calied A-doubling. This effect is important for low-spin molecules

(singlets and doublets). For diatomic molecules having high spin or high angular

momentun% as was the case in this study, the spin-orbit Hamiltonian typically consists of

just the first diagonal term[l6]:

Recall, the eigenvalues for the L, and S, operators have been given previously and lead to

the tenn A- for this operator.

2.4.2 Spin-Rotation

The spin-rotation Hamiltonian &ses 6om the interaction of the electron spin with

the weak magnetic field generated by the rotation of the molecule. The effect is important

when dealing with Hund's case @) molecules [18] but is quite weak for Hund's case (a)

molecules, at least at low I. The term is described by [Il]:

where y, denotes the spin-rotation constant for the vibrational level v. For a case (a)

molecule, the diagonal matrix element for this term becomes [16]:

An off-diagonal spin-rotation term has the form [16]:

(HSR)ml = -%ywl [J(J+ 1) - R(Gt1)]'[S(S + 1) - Z(B1)IK

where the coefficient y,,, wiU dze r from the diagonal coefficient.

2.4.3 Spin-Spin

The spin-spin Hamiltonian wiU only arise for molecules with Sr 1 (at Ieast 2

unpaired electrons to interact). It has the generd form [I 11 :

Where A,, is the spin-spin constant and q is a higher-order parameter. This interaction rnay

be considered a dipole-dipole interaction between two unpaired electron spin magnetic

components (hence the requirement for S z 1). For linear molecules, ody the first term

exists. The fïrst order term requires a little explanation to bring it into line with older

treatments. The parameter Y3 l+, has been used rather than a, as is more cornmon to the

discussion of diatomic spectroscopy [Il]. For case (a), Zare [16] summarizes this ma&

element as:

2 H,, = - 5[3z2 - S(S+l)] 3

2.4.4 Second-Order Interactions

Second-order cross-terms arising fiom centrifuga1 distortion of the molecule are

grouped together in the following form [IL]:

This interaction leads to a large number of off-diagonal matrix elements.

2.5 Kypeflne Interactions

For molecules whose atoms have nuclear spin, additional terms may be required to

properly assess molecular interactions. This is usually not an important effect unless high-

resolution equipment is available or the nuclear spin causes a large energy level splitting.

For the current discussion, these terms are important, as cobalt has nuclear spin I=7/2 and

dues cause distinct splitting within each rotational transition. As with electronic spin,

nuclear spin is quantized and will take on values of +I to -1 so, for cobalt, eight distinct

values are found (I, = *7/2, S / 2 , *3/2, and *1/2). Fluorine, which makes up the other

halfof oür diatomic, also has nuclear spin (I=1/2), but the effects are small (undetectable)

and will not be considered here. Inclusion of the nudear spin interactions into a basis set

forms a Hund7s case (ap) model. See Figure 2-6. The matrix elements needed in this

forrnaiism are discussed in section 5.6.1-

Significant nuclear spin Ieads to two important interactions: nuclear spin angular

momentum with orbital angular momentum, and nuclear spin with electron spin. Nuclear

spin-rotation and nuclear spin-spin components are much smaller than these interactions,

and will not be elaborated.

For an analysis of hyperfine interactions, the landmark work of Frosch and Foley is

recommended [19]. In this paper, they discuss the different Hund's coupling cases with

respect to these interactions. Using Dirac notation for the electronic motion in the

molecular potential field, they derive a suitable model. We will discuss the different

hyperfine components, and emphasize their origin.

Frosch and Foley [19] began with Breit and Doermann7s [20] approximately

correct 2-component Pauli equation derived from the Dirac equation of an electron in the

field of a nucleus. Next, the interaction is expressed in terms of electron orbital angular

momentum, transformed into molecule-fixed cylindrical coordinates and integrated over

the wavefunction for the molecuIe. They obtained elements diagond in A. Off-diagonal

elements also appear, but the effects are typically much larger than the homogeneous

+ Figure 2-6. Hund's case (a& Rotational projection operator J couples with the nuclear

* -*

spin 1 to form the resultant operator F.

interactions, so may be treated separately.

For a case (a) molecule, Frosch and Foley obtained the following hyperfine

operator:

For a case (a) rnolecule, t fis operator may be rewritten:

H, = *ILz/ + b(l,S,/ + I,B,l) + @ + C)I~S,I (47)

The rniddle term is non-zero only for matrix elements off-diagonal in R. For a good case

(a) molecule, this term will be srnail. Neglecting it gives an effective Hamiltonian of:

=&ff = [ah + @ + c)Z]Izl (48)

Recall, for good case (a) coupling, both L and S are well-defined and have projections of

A and Z, respectively. Experirnentdy, unless transitions involving AQ +O are observed, b

and c wiU not be separable. The first term is a nuclear spin-electronic angular orbital

element with:

where gf is the Landé splitting factor for the nucleus, the terms p, and p, denote the Bohr

and nuclear magneton, and riJ is the distance fiom the nucleus to the interacting electron,

averaged over the spatial coordinates of the states Cl 91.

The b and c tenns are spin-spin interactions, between nucleus and electron, The

second type of hyperfine interaction &ses fiom the nuclear spin (I) coupling with the

electronic spin (S). The 14 interaction is magnetic in character, with a hyperfine

interaction occurring between the nuclear electric quadrupole and the electron spin. The

b term has magnitude regardless of the orbital type, as it contains spherical and non-

sphencd components [19]:

The angle between the nucleus-electron vector r, ' and the axis is given by

tadp'/z,'. The 5 terms are electric field components excluding the interacting particles.

This is also called the Fermi contact interaction. The bracketed term contains two parts:

the first gives a contribution only in the region of the nucleus closer than the classical

electron radius, and has spherical symrnetry (an s-type orbital). The second terni arises

fiom the deviation of the electronic wavefunction fiom spherical symmetry within this

approximation (a non-s orbital). The last term in the hyperfine Harniltonian, cSZ4 (see

equation (47)), represents the electron dipole-nuclear dipole magnetic interaction. Frosch

and Foley [19] obtained:

A fiil1 explanation of terms may be found in the reference[l9]. Note that this parameter is

33

quite sirnilar to b. For s-type orbitals, the last term in both b and c go to zero. Using the

argument of sphericai symmetry nea. the nucleus, Frosch and Foley reduced the b and c

terms: no s-type contribution led to the equaiity 3b+c=0, so 3b+c+0 indicates some degree

of s-orbital character. This term is proportional to the expectation vdue of the unpaired

electron orbital at the nucleus, Iq(0)12. Typicdy, spectroscopists refer to the parameter

b, = b + [21]:

2.6 Ma& Treafmen f

From the preceding discussion, we c m see that inclusion of ali pertinent terms to

the system, as well as the range of values each interaction can take on, leads to a very

complex mathematical problem. For this reason, these interactions are often presented in

rnatrix form, so that al1 diagonal and off-diagonal elements may be included in one scheme.

The general procedure for linear least-squares fitting of spectroscopic values will now be

presented, followed by limitations and modiiïcations necessary for non-linear fitting.

2.6.1 Linear Least-Squares Fittîng Procedures

Analysis of rotational spectra entails determination of many molecula. effects,

descrïbed in terms of rotational, spin orbit and electron spin interactions. By using the

Hund's coupling case appropriate for a given molecule, based on the behaviour of

molecular interactions, an accurate or 'best approximation' rotational (ancilor hyperfine)

Hamihonian rnay be constructed. It remains to be shown how this mathematical mode1

can be used to obtain molecular constants describing these interactions fiom the analysis

of the experirnental data.

One approach to this problem is to design a matrix containing ali the significant

interactions. The model Hamiltonian would then have the form [22]:

where C is a vector containing x molecular constants and Mis the skeleton HamiItonian

matrix consisting of the mathematical formulae used to describe the molecular constant

dependencies (on J, S, etc.). The dimensions of M will be (x, number of constants) by (O,

number of observations).

The calculated energy eigenvalues are obtained in matnx form by diagonalization

of the model Harniltonian [22]:

where U is the unitary matrix and (Tt the adjoint ofU (conjugate of the transpose U3.

Proper diagonalization of this equation makes Ut = U-'.

Least-squares fitting is only valid when the model Hamiltonian is linear. While the

equations may be non-linear in many aspects, the model will be linear if the change in

cdculated energies with a change in molecular constants can be described by the matrix M

where y is the caiculated energy for the observable o. For simple models containing only

vibrational and rotational information, higher order terms will be zero (Le., a linear fit will

suffice). Thus, calculated energy eigenvalues are obtained fiom the muitiplication of the

skeleton matrix with the molecular constant vector [22]:

We are concemed with the difference between the calculated values obtained in y

WC) and the observation values Wb3 Minimization of the dinerence between calculated

and observed values Ieads to the best fit for the given molecular constants. The sum of the

squared differences (or residuals) is given by [22]:

for O observations. We have explicitly included the skeleton matrïx and vector of

molecular constants. When is rninirnized [22] :

Differentiating equation (57) and substituthg into (58) gives 1221:

Simplification of this equation yields [22]:

0 P 0 = x ( Mil M, C, - y:bs Min) for all n

i=l l=I

Rearranging equation (60) gives [22]:

0 P O C C Mil Min Cl = x yiobS Min for aZl n i = l l = l i= l

In mat& form, the equations can be rewritten as [22]:

The noma1 (p x p) matrix N is defined as [22]:

N z M t M

Inserting equation (63) into (62) followed by rearrangement yields [22]:

C = N-' M t yobs (64)

This is now in the desired form, with the molecular constant vector on the left and

measured and calculable quantities on the right. This form wili work provided that the

matrix M is Iinearly independent (det IN1 +O, so that N-L exists).

A measure of the 'goodness' of a fit is the standard deviation [22]:

where 9 is the sum of the squared residuals (equation (57)) and O-p w u be the degrees of

fieedom (number of observations less number of molecular constants). For a good fit, a

will be of the same magnitude as the measurement uncertainty. The standard error is

given by [22] :

where oW-' is known as the variance-covariance matnx. For datasets having more than

10 degrees of fieedom, the fitted values will be withui *26Ci of the true values at the 92%

confidence level [22].

The off-diagonal elements of the variance-covariance matrix aW-' determine the

independence of the molecular constants. A correlation rnatrix describing how the error in

one parameter affects another is given by [22]:

Values near *1 show strong correlation between the two given parameters (Le., they are

38

not separable in the current model). For correlation values near 0, little interdependence

of the constants is found and confidence in the obtained values is high.

2.6.2 Non-Linear Considerutions

If the model Karniltonian contains terms of order higher than one (&"yi/6Cnm *O), a

simple Ieast-squares fit is not possible. Instead, the molecular constants must be obtained

through an iterative process of refining the parameterized values to rninimize the residuals.

The model Harniltonian may still be parsed into the components described in equation

(53). By separating the Hadtonian in this fashion, the Hellm--Feynman theorem [23]

can be applied. Starting ftom equation (54), dflerentiation with respect to changes in the

molecular constants gives [22]:

where Di is the derivative for an energy E,. For each cornponent Hamiltonian, Di

expresses the change in calculated energy (eigenvalues) with changes in the parameterized

molecular constants. Taken over all n (molecular constants), a derivatives rnafrix is

required [22] :

Approximate solutions to the derivatives matrix may be calculated using an

iterative rnethod, where the matrix equation (64) will have the form:

AC, = N-l Mt Ay

here, Ay is the vector of residuals between observed and calculated values and AC,, is

detennined as the change in molecular constant values fiom the previous iteration:

the 'new' constants (superscript 1 after 1 iteration, 2 after 2, etc.) are obtained fiom the

constants fiom the previous iteration plus the determined change. Iterations continue until

AC, (or Ay, see equation 70) f d s below a preset threshold, typically taken as the

experirnental precision.

For analysis of experimentally obtained spectra, a FORTRAN 77 program

containing this iterative procedure is needed. A custom-made subroutine (and major

portion of the program) describing the interactions between the ground and excited 3 ~ i

states of cobalt fluoride was constructed and compiled with a pre-existing least squares

routine [24] to furnish an executable, reusable program. The input file for this program

contains initial guess values for the molecular constants as well as switches to fix these

constants at the given values. This construction allows refinement of the more si@cant

constants before implernentation of a full optirnization. The flexibility in the program

structure helped ensure convergence to the global minimum rather than some local

minimum on the energy hypersurface. The input file also contained a listing of transition

energies with their assigned rotational J-values for the upper and lower states. From these

assignments and initial guess values of the molecular constants, a set of irnproved

constants and residuals were generated. Assignrnents of transitions were assessed

(verified) and any measurement or transcription errors easily detected (Le., unrealistic

residuals in the output file). A discussion of the analysis is found in chapter 5.

3 COMPUTATXONAI, APPROACH

3.1 Introduction

The Iast quarter century has seen a remarkable acceleration in the ability of

computers to process a staggering number of instructions, Current methods of

computation typically invofve desktop computers in the forxn of PC's and workstations

networked together or working as 'stand-alone' machines. The underlying theov for

most modem computational methods has deep roots based on old methods. Early

investigations into molecular fine and hyperfine structure suffered fiom the capabilities and

capacities of the computers to process ail the interactions, so that most studies had to

focus on small molecules with few electrons. In an effort to hurdle this limitation, many

sirnplified, approximate 'softer' methods arose, semi-empincal or parametric in nature.

These rnethods allowed study of many molecular phenornena, but lacked the ability to

properly quant* s m d electronic, nuclear or relativistic effects due to the nature of the

approximations used. Modem computers have led to a renaissance in computational

theoretical chemistry, with new limitations arising from the capabilities of programming

code to process huge problems as well as the barriers inherent in the methods of

approaching the problem.

As Davidson noted in his contribution to "The World of Quantum Chemistry"[25]:

"As difficult as these technological problems were, the conceptuai

problems were in some ways more difiïcult to solve since they

required discarding many preconceptions and then, by numerical

experimentation aided by perturbation theory analysis, trying to

discover systematic procedures for selecting basis functions, for

selecting transformation coefficients fiom basis functions to

molecdar orbitals and for selecting configurations for inclusion in

the wavefùnction."

So, while the fnistrating aspects of huge computations being segregated to 'off-hours' has

disappeared, a rethinkùig of the underlying processes has had to occur to take advantage

of the new technology. Ifcare is taken in construction of the theoretical model, very

exacting calculations c m be perforrned on relatively large, electron-rich molecules.

3.2 Theory and Fucus

Chapter 2 described the means to investigate hyperfhe interactions experimentally.

This method involves expensive materials and powerfùl lasers. Data collection is

laborious and analysis somewhat tedious and often difficult. While the argument can be

made that this is the only way to investigate 'real' phenornena, it must be noted that

theoretical or 'virtual' approaches to the problem are also available. Computationd

methods are capable of detennining these minute interactions, but are largely untested for

transition-metal ligand diatornics. It remains to be seen how well computational

techniques c m describe hyperfine interactions.

Metal ligand diatormics contain a large number of valence electrons, fiom both the

3d and 4s shells of the transition metal and the unfïlled valence orbitals of the ligand.

These molecules are therefore poor candidates for rnethods based on a single-determinant

fomalism, such as the local density functional ap proach. Self-consistent field (SCF)

rnethods based on the Hartree-Fock formalism are a good starting point for advanced

calculations, but fail to correlate electron-electron interactions in molecules [26]. Instead,

an electron is treated as a simgle particle in an average electric field composed of the

remaining electrons* This averaging of electronic interactions leads to large errors in the

total energy. Due to the number of interacting electrons in metal ligand diatomics, the

correlation energy is expected to be si@cant [27]. Correlation also improves the

description of rnolecular dissociation (Le., separated-atom limit) [28]. To properly assess

the subtle interactions between valence electrons requires the system be treated in a

rigorous manner using multi-determinant wavefùnctions. There are many methods that

include correlated wavefiinctions. One of the most popular is codiguration interaction

(CI). With this in mind, multi-reference configuration interaction has been chosen as the

means to approach the current problem. An examination of CI theory follows, with the

results obtained in this reseaxch presented in chapter 6.

3.3 Configuration InteractSon

The ground state of an open-sheIl molecule cannot be properly described by a

single electronic configuration (determinant). This is due to interactions between the main

(ground) state configuration and other configurations of sunilar energy, spin, etc. [29].

The 'tme' rest state for a molecule, therefore, is some amalgam of various configurations,

For srnall molecules composed of tightly bound s and p electrons, a singie codiguration

may account for 90% of a ground state system as little interaction with nearby states

occurs [30]. As molecules get heavier, higher angular spin orbitals are required to contain

the electrons. The increased degeneracy of d (and $ etc.) orbitals leads to a higher density

of states having similar enerw This density of states leads to increased interactions

between the single, lowest configuration and these siightly excited states. The interactions

tend to 'relax' the overall configuration, dowing the total energy to f d . The contribution

fiom the lowest state to the overall ground state is reduced with a cornmensurate increase

in contributions fl-om other states. To obtain a good description of the molecular ground

state, therefbre, these additiond contributions must be included. This is done by

constructing a linear combination of Slater determinants, discussed in the following

section.

3.3. i The Bnsics

For configuration interaction computations, a configuration is defined as a

syrnmetry-adapted linear combination of Slater determinants [3 11. The electronic

wavefùnction for a molecular state is composed of any number of codigurations,

described by a linear combination of orbitals and their electron occupations. This

collection of different configurations is usually cont ained in determinant form [3 23 :

Symrnetry-adapted determïnants will contain all the symmetry properties of the molecular

state they describe. The syrnrnetry of a molecular state wili be descnbed only by the

electrons found in unfilled shells, as closed shells are symmetric by nature.

The condensed fonn for the electronic wavefùnction is given by [3 11:

where the q's are an orthonormal set o fn electron configurations. The coefficients ci are

optimized so as to minimize the energy [3 11:

Configuration interaction methods in general may be simplifïed by considering

molecular configurations rather than Slater determinants [3 11, as several configurations

may be constructed fi-om a determinant but only certain of these configurations are

permissible interactions, thereby simpliSing the caiculation. The problem may be solved

using the variation principle. The eigenvalue problern is obtained [3 11 :

where H is the Hamiltonian matrix of interactions between configurations, 4 is the CI

identity matrix, and C the coefficient vector containing the configuration 'weights.' By

solving (79, the contributions fÎ-orn ali states included in the CI wavefbnction cm be

obtained. The interaction matrix II contains elements between the configurations included LI

in the description of the electronic wavefùnction:

Elements of this matrix are zero unless both configurations are of the same symmetry, at

least through second-order effects 1331- The extent of interaction between the reference

or starting state(s) and other states is categorized by the differences in their

configurations. If one can generate the state in question by promoting an electron nom a

ground state orbital to a virtual orbital, this is termed a single excitafion. If promotion of

two electrons fiom ground to virtual orbitals is required, this is a double excifation, and so

on- To maintain molecular symmetry, promotion of electrons must remain symmetric, i.e-,

no change in overall spin or angular rnomentum can occur. This type of promotion leads

to the common o-o', x-x', etc. excitations known as 'allowed transitions.' It is only

upon inclusion of triple or quadruple excitations that elements of dif5ering syrnrnetnes will

contribute to the CI wavefbnction. For fairly simple systems, 95% of the correlation

energy rnay be accounted for by including only single and double excitations [30]. As we

will see in the discussion of computed results, more complex systems require higher

excitations to account for most of the correlation energy.

3.3.2 Compu tafr.0~ ai Details

The introduction to configuration interaction theory should impress upon the

reader the complexiv of thÏs form of computation. Implementation of programmuig

routines is a herculean task, and need not be constructed on-site. Instead, a number of

well-suited prograrns are available to researchers keen to use these methods. One of the

most popular collections of computational programs is due to the efforts of IBM at its

Research Center in Kingston, NY. From this facility, a series of presented research

reports have been published as MOTECC, or M o d e m TEchniques in Computationai

Chernistry [34]. The scope of this publication includes configuration interaction routines

and methods presented by Davidson [3 5,361.

The major portion of the computational work performed for this thesis used the

ensemble of programs known as MELD. This program series is discussed in section 3.4.

3.3.2.a Configurafion Interaction Limitations und Strengths

The configuration interaction method of analysis has the major improvement over

other methods o f computation in correctly accounting for the correlation energy between

electrons. This ability causes the cornputationai time to be exponentially greater for CI

than SCF or pararnetnc methods. Cornparison of the dinerent methods shows N~

dependence for pararnetric (eg, LDF), N4 for HF-SCF and NS for configuration interaction

methods, where N denotes the number of basis fiinctions used to describe the system [37].

The design of configuration interaction routines typically uses symmetry aspects of the

molecules to reduce the number of actual interactions which must be computed 'fiom

scratch.' Due to the complexity of high s y m m e q point groups, CI often uses lowered

symrnetxy for diagonabation of the interaction mat& p 61. Small molecules with high

symmetry are often 'misrepresented' in these lesser point groups. It is ditncult to properly

descnbe high spin-orbit systems in terms of the irreducible representations of the lower

symmetry point groups, as 'mirCingY of high symrnetry groups occurs on reducing the

representative point group. See section 3.3.2.c for an explanation of this problem.

MELD has been designed to perform symmetxy blocking using the D, point group [3 61.

3.3.2. b Atomic and Molecular Orbitals

The general approach to molecular computations is to choose a set of basis

functions representing the atomic orbitals, then combine these to form molecular orbitals.

These molecular orbitals are 'modi£ied7 or 'weighted' in the rninimization of the molecular

energy. The most cornmon function used to represent atom-centred orbitals are Slater-

type orbitals (STO's) after Slater [38] proposed their use. They have the form [39]:

where A is a normalization factor, n the principle quantum number and 6 the orbital

exponent or screening parameter. Equation (77) shows only the radial dependence (r) for

an orbital. The angular dependence is accounted for by multiplying by the sphencal

harmonic Y, (9,cp). STO' s af3ord an accurate picture of atornic orbitds, but suffer fiom

discontinuity at the atornic centre. This leads to d E c u l t assessrnent of these firnctions

during computation [40]. Additionally, two-electron integrals are extremely dficult to

compute using STO's 19, 161.

In 1950, Boys [41] proposed the use of Gaussian hctions as an alternative orbital

description. Gaussian-type orbitals (GTO's) avoid the discontinuity problem associated

with STO7s by varying srnootldy as r - 0.

This fùnction has the form [39]:

The anguIar dependence is usually introduced by an expansion of coefficients, replacing

the normalization coefficient B by (C xP yq 2 ). These factors are termed the Cartesian

Gaussians, and can be set to mimic the different orbitals. For instance, an s-type orbital

will have p, q and s = O, a p, orbital has p = 1 and q = s = O, a 4, orbital has p = q = 1 and

s = 0, etc. Anotber attractive feature of this method is the ability to have non-atomic

centred functions [39]. One complication that arises is the need to use several Gaussians

to accurately mimic the STO. Whereas 4 s-type Slater functions cm describe the 1s to 4s

orbitals in an atom, a dozen or more Gaussian fùnctions are required to give an accurate

description. Widespread use of Gaussian basis functions has become the nom. Huge

compendia of optimized basis fimctions for rnany atorns are available, including

Huzinaga's [42] and Poirier's collections [43].

3.3.2.c Symmetry Considerations

MELD requires a symmetry equivalent reference space or main configuration, that

is, a wavefunction that is totally symrnetric. For systems having few valence electrons,

this poses iittle problem. Difficulties aise when attempting to describe systerns having an

odd number of electrons in the higher spin orbitais. This is due to the degenerate orbitais

of Cm,, symrnetry falling into different non-equivalent irreducibles in the lowered symmetry

used by the program. For example, the C, imeducible representation il consists of atomic

orbitals p, and p, 4, and 4, etc., which are energetically degenerate, painvise. In C,

symmetry, these orbitals are found in separate irreducible representations @, and 4, in B,,

p, and d, in BJ. Thus, a a electron will be found in one of two distinct irreducibles in C,

symrnetry, which greatly complicates the description of relatively simple systems. These

dserent configurations typically do not show degenerate energetics, due to their mapping

in Cartesian space. Thus, configurations which are symmetnc in the C., point group may

not be considered symrnetric in the C, point group. This will cause errors when

attempting to correlate al1 configurations of like symmetry in the CI calculation. This

problem rnay often be overcome by selection of a symrnetric reference state which is close

to the "ground" state. In this fashion, the perceived or desired ground state may be

included in the next step when additional reference states are added to the computation.

3.4 Introduction tu MELD

A complete description of the capabilities of this package and sample input files

can be found in reference 34. As this suite of programs is quite extensive and

accommodates a wide variety of computational approaches, a review of the compIete

package was not considered. Lnstead, the components used for this work wilI be

discussed. A mathematical treatment of the involved concepts is given by the same

author[3 61.

For Our research, we have used the foilowing prograrns of MELD to perform the

computational trials.

3-41 MELD Components

The design of this package is a series of independent prograrns which draw the

necessary computational information £îom shared data files. When used as a complete

computational approach, this package forms a powerfùl tool with extensive ffexibility,

dowing the investigation of fairly large molecular systerns. A typical approach uses the

following components.

3.4.1.a SINT- Cartesian Gaussian integraVpseudopotentia1 program.

Action: Basis sets are read in, as well as several control variables including the

number of symmetry unique basis function sets, the number of contracted basis functions

in the input (contraction scheme), the number of atoms and integral assessment choices

(i.e., cornpute al1 2-electron properties, pseudopotential switch, 2-electron integral cut-O&

symmetry of molecule, etc.)

3.4.l.b RBFSCF- Closed and open shell SCF program.

Acrion: The SCF is computed in this step. Method of assessing integrais,

accuracy, damping factors and nurnber of iterations are cornrnon input variables.

This program segment k d s the SCF wavefunctions for closed shelis as weil as

space and equivalence restricted open shells. Options within this program include a choice

of vimial orbitals and the method of calculation. Coupled Hartree-Fock calculations may

be ped?ormed with a variety of operators and can be constrained to use sphencal hannonic

combinations of Cartesian Gaussians.

Starting with a generalized form for the molecular orbital [ 36 ] :

where "a" is a specific irreducible representation, g the group orbitals (see section

3 -3.1 .d ) and c the coefficients or weights, the Hartree-Fock equation will then have the

forrn [36] :

with E the energy eigenvalues and the Fock matrix F representing [ 36 ] :

h denotes the one-electron Hamiltonian matnx, the J matrix contains correlation

interactions [3 61 :

J," = [iajal kblb]~; b k l

K the matrix of exchange interactions 1361:

K,; = [ i aka~b lb ]~ ,b b k l

and P is a measure of the charge density, obtained as a product of orbital coefficients and

occupation number [3 61 :

The Hartree-Fock equation is solved iteratively using the transformation [44, 451:

G = W ~ F W (85)

where W is the eigenvector matrix from the previous iteration. It must satis@ the

condition [44, 451:

The initial W matrix may be constructed by specwng the orbital weights in the C matrix,

obtained fiom an output file of a previous computation, or detennined by Hückel

calculation. The active subspace for the calculation can also be specified by providing the

Linear combination of basis fùnctions to be used.

The iterations may be prevented fiom osciliating by using a darnping factor which

averages the Fock operators. This can improve computational performance and becomes

smdl near convergence. Extrapolation of the Fock matrix may also be performed on Fock

operators fiom successive iterations. This speeds convergence on well-behaved systems

that are slow to do so otherwise. Another option that can be invoked is to fieeze orbitai

coefficients afler the initial Schmidt orthogonalization [45].

The program has a number of methods available to generate molecular orbitals.

Normal canonid virtual orbitals, irnproved vixtual orbitals (NO'S) [46] or K orbitals can

be produced. The problem with virtud orbitals fiom an SCF calculation is that these are

not eigenvalues for excited state orbitais of the molecule in question but rather represent

the energy leveIs for the negative ions [47]. Improved virtua.1 orbitals are variationaiiy

correct approximations to the SCF orbitals for excited states. Hunt and Goddard [46]

devised these by removing the self-interaction coulomb and exchange operators for the

Hartree-Fock Harniltonian [47]. The LVO's lead to better descriptions of excited states

than the regular virtual orbitals afEord. K orbitds are obtained by diagonalizing hF - K

(h-0.04) within the virtual space [45]. The K orbitals generated are a good starting point

for CI cornputations. In MELD, the £irst calculation is perforrned on the Gaussian

fùnctions used to describe the main configuration. This k s t iteration generates a set of

nalurd orbitals which are then used to pedorm fiirther iterations. The naturai orbitals

have the property of allowing the energy to converge much faster than it would ifjust the

Gaussian functions were used to describe the molecular orbitais. See section 3 -4.1 .g for a

more detaiIed discussion of natural orbitals.

Open-shell spin-restricted computations rnay be performed using either the typical

"one-HamiltonianY' operator or dternatively a "two-Harniltonian" method. The two-

Hamiltonian rnethod utilizes an additional Fock operator for each fiactionaily occupied

orbital. The original operator may also have fiactional occupation numbers for degenerate

systems or less than 2 electrons in doubly occupied orbitals. Further details can be found

in reference 23. The variation equation then takes the form [45]:

where the last tenn is a Lagrange multiplier used to preserve orthogonality. The Fock

matrk for open-shell systems has the above form with an orbital subscript, F:. Each open

shell is diagonalized in the virtual space left over Eom the previously determined orbitals

[45]. For open-shell singlets, a stable method is empIoyed that avoids "variational

collapse" of the wavefirnction by allowing the orbitals to be non-orthogonal- The two-

Harniltonian method is not typically used for electron hole or particle States, and special

steps are taken with the one-Hamiltonian formalkm to accommodate these.

3.4.l.c TRNX- Transformation of integrals over molecular orbitals [45].

Action: Using the output of RHFSCF (Canonical filled orbitals and virtual K-

orbitals, for instance), TRNX transforms these integrals fiom atomic to rnolecular centres

(LCAO-MO) for use by the configuration interaction routine called Iater. Additionally,

orbitals may be assigned as "fiozen core" at this stage, effectively removing them fiom

consideration in the CI routine. This transformation is affected by moving the orbital

effects for the selected fiozen orbitds into a modifïed nudear-nuclear repulsion term:

and a modified nuclear-electron attraction term:

W f N e ) , = (9 , I V ~ e + J c o m - ~ K c o r e l Oj)

The remaining orbitals are then passed to the next prograrn segment.

3.4.1.d SORTIN- Sorts the transformed integrals for the CI prograrn [45].

Actzorz: SORTIN groups the transformed integrds into symmetry blocks of g r o q

orbitals, used to reduce integral computation times within the configuration interaction

program. These integrals are transformed via matrix multiplication, with the resdtant

integrals blocked into those required for diagonal rnatrix elements, those needed for single

excitations, and finaliy the ones required for double excitation terms. They are stored in a

file for use by the CI program step.

3.4.1.e CISTAR- Configuration interaction Perturbation theory program [45].

Action: The main part of MELD, this program performs the configuration

interaction computations. Starting variables include the nurnber of reference states,

overall symmetry, spin multiplicity, perturbation theory switch, etc.

The reference space configurations may be implemented in several ways for this

program. They may simply be read fiom output, or 'seed' configurations may be specified

with dl necessary single and double excitations then generated, or a list of orbitals and

occupation preferences specified with possible excitations fkom these generated. In the

Iast method, a subset of the generated configurations are selected based on rules

goveming excitation spectra. If a "fiozen core" approximation has been chosen, certain

inner or core orbitals are designated as "non-interacting" so that no excitations are

allowed fiom these orbitals. The program also has a physical limit with regards to spin

(56) and number of open shells (18).

The configuration interaction program takes the input configurations and performs

a preliminary computation to generate the (zeroth order) starting wavefùnction. The

program then selects doubly-excited configurations outside the reference space based on

one of two criteria: either fkom the estimated second-order Epstein-Nesbet energy [48]:

or fiom the first-order wavehnction coefficients. Any single-excitations are normally also

kept. Al1 significant configurations collected in this procedure are then used to generate a

CI matrix.

The CI program can be used to find the quasi-degenerate second order estimation

of the energy [49]. Once the reference space is determined, the effective Hamiltonian

takes the form:

3.4.1.f RTSIM- Davidson method sparse matrix eigenvalue/eigenvector prograrn [45].

Action: Once the CISTAR program has formed the preliminary CI matrix, the

RTSIM prograrn takes over. This step starts by determining the lowest few eigenvectors

and eigenvalues by expansion of the true eigenvectors through the equality:

where B is a j by K matrix whose columns are composed of a set of orthonormal vectors:

[bilE, (93)

and cj is related to the energy equation by:

B'HBC~ = Ejcj

Ej values are related to the true eigenvalues 4 of H by:

Ej " l;

The matnx B can be augmented by additional b vectors, which will lower the energy

58

eigenvalues:

so that the energies are monotonically convergent as B is enhanced [45].

The b vectors are chosen by first order perturbation theory. Starting with the

residual (between cdculated and true eigenvalues) ri defined as:

Reordering this gives (when Ei + Ha):

The current vector space is described by:

y = (1 - BB~)x(~)

and the next b, is chosen fkom outside this representation:

RTSIM adds one b vector each iteration. Choice of x, is deterrnined by Iargest c, fiom

the previous iteration, see equation (92).

The program iterates until the c, f d s below a predetemiined tolerance value, T.

The correlation energy is then converged to a relative error of T'. The energy for the

system is then estimated using:

where E, is the zeroth order energy detemiined by CISTAR, E, the RTSIM eigenvalue,

EQ) the second order pemirbation theory result, the energy obtained for

codigurations retained in the CI step and C: the sum of the squares of the coefficients

f?om reference space configurations. The first quotient accounts for the energy f?om

discarded configurations while the second accounts for the contributions from higher

(undetennined) excitations.

One major dra-wback to perturbation selected CI compared to singles and doubles

CI is that it is not size consistent. A size consistent method is one in which the calculated

energy scaies linearly with the number of particles [29], or retains the correct energies

upon separating the molecule into its' component atoms. These problems are addressed

by partitioning the CI matrix:

( h G') where Ho is the reference space matrix, G the selected space matrix and h the connecting

matrix. The eigenvector c m then descnbed by:

with aTa=l. This gives partitioned eigenvalue equations O£

Thus, for h=E,, the matrix b is given by the Hylleraas variation perturbation theory result

[SOI :

The vector a is obtained either fiom luiearized couple ciuster &CC) theory [SI]:

Hoa = Eoa (106)

where:

where 1 is chosen as an eigenvalue of Ho or an expectation value of a T ~ , a .

The LCC method is size-consistent, but inaccurate unless a is the eigenvector of Ho.

Alternatively, if a is the eigenvector for He, it wiil be more accurate for nearly degenerate

systems but loses its size consistency.

3.4.1.g MOLINT- CI molecular properties program [45].

Action: Once the CI matrix routine is finished, one-electron properties such as

dipole and quadmpole moments, nuclear force and field gradients and nuclear delta

functions can be cornputed from their respective operators.

The first step is to form the density ma& using the molecular orbitd basis.

Diagonakation fûrnishes the naturd orbitds for iterative NO calcu~ations- Naturai

orbitals reduce the density matrix to diagonal form [33]:

obtained by diagonalizing the matrix coefficients for the configurations. The coefficients

b, are termed the occupation numbers and indicate the relative importance of the

interactions. Natural orbitals improve the convergence behaviour of configuration

interaction computations since only configurations with large occupation numbers will

have signifïcant contributions to the overall wavefùnction. Next, the total spin density and

the unpaired spin density are used to obtain the electronic spin properties. The field

gradient and anisotropic hyperfine tensors are diagonalized to produce the irreducible

components.

The parameters necessary to describe the hyperfine interactions in diatomics are

computed in this prograrn segment. These are discussed in section 2.5 and are

summarized below. The hyperfke parameters b and c for an atom X are related to the

hyperfine tensor components [52, 531 by:

and

where the fiee electron gas constant g = 2.0023 and P is used rather than p for the Bohr

magneton. The wavefunction represents the CI expansion and the operators are summed

over al1 contributing configurations. Rearranging (1 Il), the hyperfine parameter 'c' may

be assessed.

The isotropic hyperfine (Fermi contact) term is given by:

The MOLINT program furnishes (among other terms) the isotropic and anisotropic

hyperfine tensor components. The Fermi contact term, a rneasure of the coupling between

hyperfine cornponents on the two atoms, can be obtained using the above formula. After

these have been derived, the 'b' pararneter can be computed. The 'a' hypefine pararneter

(see section 2.5 and equation 5 1) can also be computed, although this term is not

fûrnished by the program. In tems of the CI wavefiinction, this parameter is given by [52,

531:

Other properties can be investigated. One is the transition moment (program

segment TMOMJ, with additional program segments devoted to evaluating the transition

matrix elements of the effective spin orbit operator (SPNORB) and the sphericaliy

averaged momentum distribution (MOMAVGLT) fiom the Dyson orbital. Another

program segment, OCCUP, analyzes orbital populations (Mulliken or Lowdin) from

either the SCF or CI results.

3.4.2 Computational Choices

There are practical limits to be considered when constructing wavefùnctions in this

fashion, as an infinite, exact description would take an infinite amount of t h e to compute.

By including al1 single and many double excitations in the reference space, a relatively

complete picture cm be obtained. The results obtained in this research are presented in

Chapter 6.

4 EXPERIMENTAL

4.1 Procedure Overvieiv

Cobalt fluoride is created using laser-ablation techniques and investigated with two

distinct laser probe sources. The general procedure will be explained foiiowed by an

examination of the specific cornponents.

A high-intensity Nd:YAG laser fumishes the ultraviolet radiation (Model HY-400:

Lumonics, Inc.: Kanata (Ottawa) ON, Canada) which is used to vapourize atoms £iom a

cobalt metal rod (Goodfellow: Cambridge Science Park, England) housed in a vacuum

chamber. The resultant plasma is entrained in a mixture of carrier and reactant gases,

typically 1% reactant gas (SF, Eom Matheson of Canada: Whitby ON, Cmada) in helium

(Liquid Air). The vacuum charnber is designed in such a way that the reactant/carrier gas

mixture is expanded through a pulsed-valve (General Valve Corporation: Fairfield, NJ) at

the top of the chamber and pumped away using a BaIzers d ias ion pump at the bottom.

Between the reaction zone and the pulsed valve, a second laser bearn is introduced to

interrogate the reaction plasma. This is where the difference in experimental setup mises:

a pulsed-dye laser (Lumonics HD-500 pumped with a YM-600) is used for low-resolution

studies while a continuous-wave (CW) ring dye laser (Model CR-699 Ring Laser.

Coherent: Palo Alto, CA) pumped with an ion laser (1 100-20 Ion Laser. Coherent) is

used for high-resolution work . The different laser apparatus are discussed below. The

intersection of the probe bearn and plasma is aligned such that fluorescence is detected

through a window on the side of the vacuum chamber. A monochromator is positioned at

65

the window to reduce the scatter and plasma background. This monochromator may be

fitted with narrow slits which increase resolution but aiso reduce noise and signal.

Positioned behind the monochromator is a thermoelectrically-cooled photomultiplier tube

or PMT (Model TE104RF. Products for Research, Inc.: Danvers, MA) used to collect the

radiation. The output is sent to a signal processor Nodel 428 Current Amplifier.

Keithley Instruments, Inc.: Cleveland, OH), amplified and sent to a recorder or cornputer.

Timing of the experiment is accomplished using a Cchannel digital delay/pulse generator

(Stanford Research Systems, Inc. Model DG535). A general schematic of the laser setup

is given in Figure 4-1.

4.2 merimental Apparatus

4.2.1. PulsecC-Dye Laser

The bulk of the experimental work was performed with a pulsed-dye laser

apparatus. This consisted of an ablation laser (Lumonics, Inc. Kanata, ON. Model HY-

400) which delivered 3 r d of ultraviolet ( k 3 5 5 nrn) radiation. A second laser was used to

probe the experiment. This pump laser (Lumonics, Inc. Kanata, ON. Model YM-600)

delivered 60-80 mJ at 355 nm for pumping of UV absorbing dyes or about 80 r d at 532

nrn for pumping green absorbing dyes. The pulsed lasers as well as the pulsed molecular

beam valve are run at a rate of 10 pulses per second (10 Hz). The laser pulses are 10 ns in

duration. This pulse was directed into the dye-laser apparatus (Lumonics, Inc. Model HD-

500) which controls the frequency of the output beam. Typical output power fiom the

pulsed-dye laser apparatus was 1 to 8 ml per 10 ns pulse. This converts to 100-800 kW

for the duration of a pulse1

The pulsed-dye apparatus is controlled by a scan control unit (Lumonics, Inc. KD-

50 SCU for HD-300). The SCU allows input of start and end wavelength (or

wavenumber, typically) as well as the step or increment rate. It can also slew (adjust) the

Iaser to an input fi-equency, which is usefùl for alignrnent and timing procedures. For

rapid scans, the increment is set for 0.1 to 2 cm-'s", but for measurable scans, the rate is

set to a small step value of 0.012 cm-'. At this rate, a scan takes nearly 1% minutes per

wavenumber. A band system has measurable peaks spanning 50 cm*' or more, so care has

to be taken to obtain as much information as possible before the gas pressure in the

"bomb" drops below acceptable values. This apparatus is discussed below.

For calibration purposes, a small fraction of the pulsed-dye laser output is

redirected to reference cells. See section 4.2.5 for details.

4.2.2 Continuous- Wuve Ring-Dye Laser

For high-resolution work, the Coherent mode1 CR699-29 ring dye laser is used.

This continuous-wave laser is capable of outputting up to 1 W of power, depending on the

dye used. Special optics must be employed for different wavelength regions, due to the

nature of the apparatus. With the optics available to us, we were able to span from 520

nm to 680 nrn. Within this range, only 2 systems were of suitable strength to allow us to

Laser

Tunable DY^ Laser

HY400 Nd:YAG Laser

1, or U Reference Cell

' \ Steering &''X Mirrors 1 Pickoff 1 1 1

1 1

Vacuum Ultraviolet Chamber Visible

Figure 4-1 Laser apparatus used in ablation experiments. Arrows represent the course of radiation from the laser table to the vacuum assembly and, b y use of'pickoff mirrors, to the reference cell.

obtain useful results. For Our purposes, the laser dye C-6 (Coumarin 540) is used.

Frequency calibration was obtained with the ccAutoscan" system which has a specined

absolute frequency accuracy of k200 MHz and a precision of 160 MHz.

4.2.3 Vacuum Line

The carrier gas mixture used in the experiment is prepared on-site using a vacuum

rack apparatus attached to the side of the reaction charnber frame. See Figure 4-2. Met

valves are attached to regulated gas cylinders. Due to the limitations on available

regulators, the helium pressure cannot be raised above 120 psi. This is used as a 'iirniting

factor' in determining total gas pressures. The inlet pressure is regulated by adjusting the

inlet valves (Swagelok: Solon, OH) on the rack. The usual method of preparation is to

close off the 'front end' (Le., fiom the regulator to the pulsed-valve head), then introduce

the reactant gas to a pressure of 1 to 1.2 psi. This is monitored with a Matheson test

gauge (30 psi maximum pressure: P/N 63-563 1). Next, 100 to 120 psi of helium carrier

gas is added. The pressure is monitored with a 400 psi gauge (Scott Specialty Gases).

The resultant mixture is contained in a Whitey (Swagelok: Solon, OH) stainiess steel

cylinder with an inherent valve. This cylinder aIlows preparation of about 1 litre of gas,

including the vacuum line volume, which typically lasts 1 to 1 M hours before the pressure

drops below the regulated value and the signal begins to diminish.

Other +@- gases

Pressure gauges

To reaction chamber

Reinforced \ steel Regulator cannister

Figure 4-2 Rack assembly used to prepare gas mixtures. The various gauges allowed preparation of low concentration SF, in He 'bombs' which were contained in the steel cannister before use. The regulator allowed control of gas pressure to the expenment.

The pulsed-valve used to introduce the gas mixture into the reaction chamber is

controlied by electronic components built in-house. Reactant gas sent to the pulsed-valve

is regulated at 40 psi using a USGauge regulator (100 psi mm). See section 4.2.6 for

fiirther details.

4.2.4 Reaction Chamber

The reaction charnber has four round access ports. These aiIow airtight seals to

the vacuum chamber as well as to the flanges holding the pulsed-valve nozzle and inport

and outport windows. See Figure 4-3. The windows, fitted at the ends of tubes, are

designed to allow polarized Light to pass through the window. The tubes are used to aUow

more precise alignment of the probe laser beam into the reaction zone. Similarly, there is

a flat window above the inport tube that ailows the ablation beam to enter. The back

scatter fiom this window must be blocked. There is an 'observation port' on the fkont

face of the chamber which is used to collect the fluorescence signd.

4.2.5 Reference Sources

The calibration signal which arises f?om the directed 'pickoff,' Figure 4-1, is sent

into an iodine gas ce11 or uranium hollow cathode (Catheodeon, England. P/N 3 -UAX U,

15mA max. current). The iodine reference was designed in-house and requires special

handling. It has to be draped in black cloth and the impinging and radiant beams carefully

adjusted. The hollow cathode is much easier to use- it plugs into a baseplate, can be run

in arnbient light and has a flat face (window) and target cathode which are easily

'iew Port 0

Fluoresc

Ablation Beam

Interrogatiot Bearn

... . . y . - . - ., ... .....S........

Output

Exhaust to Vacuum Pump

Figure 4-3 The reaction chamber, viewed fiom the £?ont of the apparatus. The UV radiation enters through the flat window on the right of the chamber, just above the in-port for the scanning bearn. Reactant gas enters fiom above and is exhausted with high-efficiency pumps below. The experimental data is coilected at right angles through an observation port in fiont of the reaction zone.

aligned with the pickoff source. The generated signal is sent to the second boxcar for

processing.

The reason for using two dEerent reference sources is quite simple. While the

iodine ce11 gave a wealth of reference peaks, it became sparse toward 20 000 cm-' (500

nm) and the reference [54, 551 atlas did not list peaks below this wavelength. The

uranium reference is sufficient below this wavelength, but the density of lines f?om the

uranium and argon buffer gas make calibration somewhat dEcult-

4.2.6 Timing Sources

Timing of the experïmentd procedure is critical to efficient collection of data. The

gas must be injected and the signals collected in synchronization with the laser pulses.

This is accomplished using a digital delay generator, which allows timing signals to be sent

at different times on separate channels. See Figure 4-4.

The pulsed lasers used to ablate the metal rod 0 and pump the dye laser are run

at a frequency of 10 Hr. Thus, pulses are 100 ms apart. AU the events are triggered

within a 1 ms envelope. Starting at a tirne T,, the triggered events go as foliows: The

pulsed-valve driver is opened to allow the gas mixture into the reaction chamber.

Approximately 300-500 ps later, the reference signal and experimental signal (PMT)

boxcars are triggered, as well as the ablation laser. Finally, the pulsed-dye laser is

tnggered about 100 ps later. The boxcar integrators, as well as the pulsed-valve driver,

are al1 equipped with adjustable sampling windows (time domain). This aliows for the

necessary fine adjustments to maximize the signal. The timing generalizations corne from

1 Signal Generator

Amplifier To pulsed-dye laser

To UV laser

1 1 Trigger sig$al ln lntegatorl To CW laser

C

To chart recorder

To chart recorder

Reference cell 1 input

Oscilloscope . Pulsed-Valve Driver

I To pulsed valve

High Voltage

Figure 4-4 Electronics used to time the experimental apparatus. The oscilloscope is used for 'fine tuning' signals, boxcars for data collection (experimental and reference) and the pulsed-valve driver to inject reactant gas. Typical timing profiles are listed at lower lefi.

adjusting these parameters for each particular experiment.

4.2.7 Data Collection EIectronics

The low-resolution @ulsed-dye) laser scans are coliected on a chart recorder.

These bands then have to be measured using an ocular and Light table. The magnined

scale is 20 mm in width and allows measurements to 2 decirnai places (hO.Olmm). The

reference peaks are measured and tabulated with their actual frequencies, as noted £tom

the iodine atlas [54, 551 or Corn uranium [56] and argon [5 71 line listings. These peaks

are treated statistically and the linear regression results furnish slope (scan rate) and

intercept vaIues (v,). The cobalt h o r i d e spectra are measured and the relative offsets are

adjusted by the dinerence in pen positions between reference and expenmental signals.

The linear regression values are then implemented to furnish f?equencies for the peaks, and

further analysis of the band systems continues from these generated values.

High-resolution data is coliected using an Apple IIe cornputer comected to an

Autoscan unit (Coherent) which is used to control the stepping of the laser frequency.

These scans are typically 20 GHz in size (0.67 cm-'). Successive scans are overlapped to

fùrnish a 'continuous' scan of the cornplete band systems.

4.3 Laser Dyes

4.3.1 Spectral Coverage

Different laser dyes are used to cover the entire spectral range of the high and low-

resolution lasers. The dynarnic range of the pulsed-dye laser system includes the full

visible spectrum, so a series of dyes are prepared. Each dye has a distinct dynamic range

and wavelength of maximum power, &. The dyes used, their working ranges and

maximum wavelengths are given in Table 4- 1. A chart of these dyes is given in

Figure 4-5.

The continuous-wave laser requires separate optics to probe different ranges. Due

to the unavailability of a large nurnber of these optics, and the general signai strength of

the observed transitions, only the region near the 528 n m band could be probed. For this

band, the laser dye Coumarin 540A (C6) is used. Preparation of this dye includes both

methanol and ethylene glycol solvents. The ethylene glycol is an additive that irnproves

the viscosity of the laser dye. This property is important as the dye 'jet' must spray across

the path of the laser bearn. The improved viscosity allows this to occur with a minimum

of spillage. More importantiy, this ais0 allows the jet to have a flat profle without the

presence of air bubbles.

In the high-resolution experiments, a continuous-wave laser (Co herent . Mode1 CR-

699) delivers approximately 100-200 mW of power near 530 nm using an argon ion purnp

laser power fkmishing 6.6 W. The need for special mirrors for a given wavelength,

coupled with the reduced signal strength of successive bands, does not allow us to obtain

uiformation on any but the strongest bands (at 18912 and 18798 cm-'). Only the strongest

band is completely analyzed. The second strongest band, at about 532 nm, although much

weaker, allowed several Lines to be eventually measured. See Chapter 5 for further

information on low- and high-resolution data.

Table 4-1. Laser Dyes Used to Investigate Cobalt Fluoride"

Stilbene 420 412 444 425

Coumarin 500 483 559 507

Coumarin 540A 516

Rhodarnine 590 552

Kiton Red 620 578 606 584

DCM

LDS 698

LDS 751 714 790 750

a. Pulsed-dye (Nd:YAG) Iaser, methanol solvent. Stilbene 420 to C540A used a 3 55

nrn pump, the rest required 532 nm pumping.[Exciton Laser Dyes Catalog. Exciton,

Inc.: Dayton, OH (1 992) ]

Laser Dyes Used

Wavelength (nm)

Figure 4-5 Laser dyes and their spectral coverage.

78

4-3.2 Spectral Resolution

The apparatus allows cobalt fluoride lïnewidths of about 0.2 cm-' with the pulsed-dye

laser probe. These are less resolved than the laser bandwidth (-0.07 cm-') which indicates

the presence of unresolved hyperfhe structure. This is confirmed by the fa& that some

low-J rotational lines are more than 0.4 cm-' in width. The focusing lens on the outport

of the vacuum charnber has been optimized to give the best resolution. The continuous

wave laser gives an optimal resolution of 180 MHz (0.006 cm-') which is lirnited by

residual DoppIer width in the molecular beam.

4.4 Experimental Details

4.4.1 Procedure

The reaction proceeds as follows: A high-purity (99.9%) cobalt rod 50 mm x 6.5 mm

(Goodfellow) is comected to a motor rnicrometer (Oriel Corporation: Stratford, CT)

within the vacuum chamber. The motor rnicrometer is used to rotate and translate the

metal rod in its' housing. The housing has a pinhole on one face that is used to focus the

ultraviolet radiation on the rod. See Figure 4-6. By rotating and translating the rod, a

fairly even Wear is attained, minimizing pitting and grooving of the rod, which leads to

uneven generation of plasma. From the top of the chamber, where the pulsed valve is

situated, the reactant gas travels 1 0 mm to the rod. This expansion channel is 17 mm long

by 1.5 mm wide overall. The cobalt plasma reacts with SF, to form CoF, among other

products. This reactant mumire then expands into the vacuum chamber. The cooling

plasma stream is probed 5 cm downstream &om the expansion channel exit, where tunable

Reactant gas from pulsed valve

Motor Neoprene micrometer collar

Co rod

1 Monochromator

To signal processor

plasma Probe laser

Figure 4-6 Reaction zone within the vacuum chamber, not to scale. The left side of this diagrarn is the front of the vacuum assembly.

radiation frorn a dye laser (pulsed or continuous wave) is used to induce fluorescence. The

laser-induced fluorescence signal is coiiected at right angles to both the laser beam and

molecular beam by means of a lens used to focus the fluorescence on a 0-25 m

monochromator. The bandpass of the monochromator is about *15 nm but may be

reduced to *2 nm by use of narrow slits. The reduced bandpass is used primarily for

dispersed fluorescence experiments.

The light passing through the monochromator is detected with a cooled

photomuItiplier tube (housing by: Products for Research, Inc. Danvers, MA. Model:

TE104RF). The output signal is generated using a high-voltage power supply (Harshaw

Nuclear Systems. Model: NV-26 A) mnnùig at 1300 to 1600 V. This signal is sent to a

current amplifier (Keithley Instruments, Inc. Cleveland, OH. Model 428)' which is used to

ampli& the signal a millionfold, then sent to a boxcar integrator (built in-house). The

boxcar allows averaging of signal. Time constants of 0.3 s to 5 s are available, which

allow enhancement of the signal and reduce the effects of random 'noise.' The integrated

signal is then sent to a 2-channel chart recorder (Linseis, Inc. Princeton Junction NJ.

Model L6512B). The second charnel of the recorder is used to display a calibration

spectrum.

4.4.2 Other aperimentul considerations

The interconnectivity of band systems is probed by means of dispersed fluorescence.

In this procedure, the probe laser is set to the fiequency of a band origin, thus 'pumping' a

given transition. The monochromator is fiequency scanned for observations. Detected

transitions must occur Çom the excited state to some other low-lying electronic state or

excited vibrational levels of the gound state. Results of this experiment are given in the

next chapter-

5 SPECTROSCOPIC RESULTS

5.1 Introduction

How metals and non-metals interact is a primary consideration in construction,

fabrication, industrial cataiysis processes, and everyday life. An obvious example is the

rusting of metal by air and water, but this type of interaction occurs fi-equently in other

forms as well. The primer in automotive paint, for instance, acts as a bridge or glue

between very dBerent compounds, the metal shelt of the car and the inorganic

composition of the paint. The means by which molecules interact with metals is of great

interest to manufacturers and scientists alike. Interstellar media contain metal-ligand

fragments, and astrophysicists use experimentally obtained data to help describe their

observations of distant events. While the physical, macroscopic properties of such systems

may be known, the microscopie interactions are less than completely understood.

The universe contains a great deal of metal, and our understanding of the interactions

that occur between metal atoms or with other particles is lirnited. Recent years have seen

a large amount of theoreticai research into metal diatomics [58-6 11, and experimental

study of metal-ligand diatomics is not new [62], but Our knowledge of these systems is far

from complete. While 3 d transition metal-oxides have been thoroughly catalogued 1631

and results on other metal-ligand systems have been reviewed[62], the volume of data for

these diatomics is less than adequate.

Metal-halide systems make interesting subjects, as their high electronegativities

suggest fiee halide atoms will readily bond with the electron-rich metal atoms. Given the

83

right experimental conditions, bond strengths in these metal-halides should be sufficient to

alIow inspection of the transient species. Many of these metal-halides have been presented

[l], but the diatomic cobalt fluoride had no experimental (spectroscopic) results reported

when this research began. Since this time, Bernath and Rarn have reported their

experimental findings [64, 651. There is no mention of CoF in Huber and Herzberg's book

[66]. The only other report of this molecule in the fiterature was an X311 ground state

prediction [l]. Whether the paucity of experimental data on CoF is due to availability or

intractability is uncertain.

The experimental apparatus available here allowed investigations of the fundamentai

relationships found in metal-ligand diatomics. For this study, cobalt fluoride was

examined, although many other systems have also been investigated using this apparatus

[2], and the methods employed are used elsewhere for sirnilar investigations [67-701. The

results presented here will revisit the prediction for the cobalt fluoride ground state [l] and

introduce other experimentally obtained values.

5.2 General Method of Investigation

Using the apparatus discussed in Chapter 4, we began Our experiments by assuring the

W ablation beam was strikuig the cobalt rod, the 1% rnix of SF, in helium was correctly

regulated and flowing, the probe beam and the PMT aligned. With the power supply for

the PMT at about 80%, scans were conducted at moderate speeds (0.2 cm-'-s-'). Survey

scans were conducted fiom 400 to 700 nm, and any potential band systems encountered

were investigated further. This approach proved very fiuitfùl, and many spectroscopic

bands were obtained. In this fashion, the fast scans were used to elicit the approximate

positions of the systems. Bands were then scanned at slower speeds, typically 0.012

cm-'-s-' (about 80 s per wavenumber), to obtain accurate (measurable) data. A few scans

were collected at a very slow scan rate of 0.008 cm-'d. This namowed the collection

window (Le., less range could be measured) and gave minimal irnprovement in scan

quality. Carefùl calibration of the reference signal allowed accurate assignment of

wavenumbers for many systems. These bands were collected using the 2-charnel chart

recorder discussed in Chapter 4. Analysis of the individual band systems entailed the use

of a light table and an ocular, a small eyepiece with a 2 cm scale, which allowed

measurements to 0.2 mm resolution (see section 4.2.7). Since most peaks were not

symmetric, the fiequency was obtained by taking the average of left and right

measurements at half-height.

5.3 Preliminary Findings

The normal method of scanning was to set the monochromator to the sarne

wavelength as the middle of the pending scan. In this fashion, it was felt that this would

give best coverage of the scan area. The monochromator was not motorized, and an

attempt to use a stepping motor proved less than satisfactory. Any possible structure f?om

the original scan was then rescanned much more carefilly.

At this point, the method of investigation becomes more ngorous. Survey scans were

again performed, fixing the monochromator at wavelengths of the suspected peaks. Using

slower scan speeds, a more accurate reading of the band positions was obtained. Finally,

using a scan speed of 0.012 cm-'=s-' allowed about 50 cm-' to be coiiected on a 'fidl ttank'

of gas mixture. The original survey gave quite a wealth of potential systens. Table 5- 1

lists the bands observed in the original survey.

5.4 Low-Resolution Spectral Anaiysis

The survey for cobalt fluoride fluorescence spectra ranged fiom 680 nm to 450 nm

and used a number of laser dyes. Molecular features, however, were only observed in the

region below 540 nrn. A total of twenty-one molecular bands were seen. Their

wavelengths and relative intensities are given in Table 5-1. Dispersed fluorescence data

have been taken for 8 of the 21 bands, an example of which is given in Table 5-2. Readily

apparent in this data is a ground state vibrational progression which has an approximate

fiequency of 660 cm-'. The electronic transitions are assumed to originate fiom the

ground state since the molecule is produced in a cold molecular beam. From ail Our data,

we have been able to obtain a value of o = 662.6 * 17.4 cm-' for the CoF ground state

vibrational fiequency. The Iarge error on this value cornes £tom the limited bandwidth of

Our small monochromator.

The first three band systems studied were at 528 nm, 532 nm, and 535 nrn and are

discussed below. Using Hund's case (c), a band by band fit of the scans was performed.

Good rotational branch separation allowed us to assign P, Q and R lines to the bands-

Table 5-1. Observed CoF band positionsa.

Band Position htensity Band Position Intensity

(nm) (nm)

medium

very weak

very strong

medium

medium

strong

very strong

very strong

weak

weak

weak

strong

very weak

strong

very strong

weak

very strong

very strong

weak

very weak

very weak

values below 500 nm.

Table 5-2. Dispersed fluorescence data of the CoF 494 nm band.

Relative Strength 0 (cm-') Aîï (cm-')

494.09 medium 20239

strong

medium strong

weak

very weak

very weak

666.26 very weak

It seemed apparent that these bands had the same P, Q, and R structure and most probably

were due to excitation fiom the ground Gate to a cornmon excited electronic state.

Another reason for concentrating on these bands was that a high-resolution Ar-ion

pumped ring dye laser was available for work in this wavelength region. This laser would

allow us to study the hypef ie structure apparent in the spectrum (I=7/2 for cobalt). High

resolution data have been collected for the 528 nm band and the 532 MI band, A

combination of weak laser power and band positions outside our wavelength coverage

area did not dlow further analysis. A case (c) basis set was used for band-by-band

analysis, modified with case (a) hyperfine elements. A discussion of the high-resolution

results is given in section 5.6 below.

5.4.1 Ground State Assignment

To properly assess the experimental data, an electronic configuration needed to be

determined. The collected data suggested a ground state. Due to the presence of the

unfïlled valence orbitals in this configuration, an inverted series with the 'a, manifold as

ground state was suggested [71]. Figure 5-1 contains a proposed valence orbital diagram

that could account for this molecular ground state. A a-bond forms between singly-

occupied fluorine 2p and cobalt 3d orbitals. The remaining 8 valence orbitals on cobalt

distribute themselves arnongst the 90,4x and 16 molecular orbitals. These orbitals are

largely cobalt in composition, so will remain relatively unchanged energetically and

contribute little to the bonding picture. Bonding within the molecule is largely accounted

Cobalt

valence - - . - . - . - - - - . - - . - . - - - . . . . .

core ... 7+ 2+

CoF Fluorine

Figure 5-1 Proposed molecular orbital diagram for cobalt fluoride. The core orbitals arise fiom the 1 s2 2s2 2p6 3s2 3p6 inner orbitals of cobalt and the 1s2 2s2 orbitals of fluorine. The main bonding molecular orbital is the 80 formed by the interaction of the 3d . cobalt orbital with the 2p, orbital of fluorine. The 90 is non-bonding (mostiy Co in character).

for by the shared o-bond. A donation-back donation scheme may exist between the 3 2

(£tom fluorine) and 4 2 (cobalt) orbitals a£Eording the molecule some degree of

stabilization. The ordering of the molecu1a.r orbitals is uncertain, so the 9a may Lie below

the n and 6 orbitals. The 3s valence orbital fiom f l u o ~ e is most likely too high to interact

with the 90 molecular orbital. The valence x and 6 orbitals allow cobalt fluoride to have a

rich and varied manifold of nearly degenerate low-lying excited states.

To fùrther assert this ground state configuration, we examined a simiIar molecule,

CoH. Theoretical[72] and experimental[73] results on cobalt hydride agree on a 3@,

ground state. The (CAS)SCF/CI results of Freindorf et al [72] describe a large manifold

of low-lyhg excited states for CoH. A sirnilar picture could be constructed for the

isoelectronic cobalt fluoride. Cobalt hydride has the same valence configuration as cobalt

fluoride, with a 0-bond forming between the hydrogen 1s orbital and the 3d, orbital in

cobalt. The remaining valence electrons behave as described for CoH, with a dli163

configuration being most stable. Finaily, other experimental results suggesting the sarne

ground state have been published since our work first appeared [64, 651.

5.4.2 Anaiysis and Assignment of Lines

Well-resolved transitions were first detected fkom survey scans, followed by more

refined scans as necessary, much like 'zooming in' on a band. While highly resolved data

makes for good analysis material, lower resolution scans allow for a better concepmal

grasp of the situation, A good example of this is given in Figure 5-2, which shows the low

resolution survey of the band centred at 18908.97 cm-'. A well-formed P-branch extends

to Iower frequency, the Q-branch is 'piled up' and the R-branchr shows resolved lines as

well as a branch head. Details are given below.

Slow scans for the structure of the band were performed. Sgcan rates for measurable

spectra were typicdy 0.008 cm%-' fiom the pulsed-dye laser assernbly. The spectra were

recorded using a chart recorder tracking 2 cm of paper per minute so that one

wavenumber covered just over 4 cm of paper. This gave acceptable peaks to measure,

with the Q-head 75 mm in height and many P- and R-branch h e s 50 mm or more hi& and

bases generally Iess than 20 mm wide. Three separate measured scans were used to

collate the data for the band centred at 18908.97 cm-'. These scans had good overlap,

which ailowed for acceptable synchronization. The measured R, Q and P-branches are

given in Appendix A-1. Measured reference peaks from an 1, source were compared to

their literature values [54, 551 by performing hea r regression o n the data sets. A typical

regression output is given in Table 5-3.

The rotation* resolved spectrum of the 528 nrn band is shown in Figure 5-3. The

'a, - X30, system is presented in Figure 5-4. Line listings for tIais systern are found in

Appendix A-2. The 53 5 nm band is depicted in Figure 5-5 and the lines tabulated in

Appendix A-3. The intensities of these bands are roughly in the ratio of 10:2: 1

respectively, indicating that the 528 nm band is fkom the lowest Iying R-component of the

ground state manifold. The separations of the first lines and the intensities of the P, Q,

and R branches in these spectra essentially assign the J and R vdues of these transitions

for us. The spectra al1 have a strong Q branch with weaker P amd R branches that are of

roughly the sarne intensity. This is indicative of M = O type tramsitions. See section 5.4.3

Table 5-3. Linear Regression Output for a 529 m Band Scan.

Regression Output:

Constant 18869.9719"

Std Err of Y Est 0.015933

R Squared

No. of Observations

Degrees of Freedom

Std Err of Coef

a. Start of measurernent fiequency (on the chart).

b, Number of reference peaks

c. Correlation coefficient

d, Related to chart measurements

for examples of M + O type transitions. Given that the ground state is proposed to be a

'a state, then the excited state must also be '@. On this basis, these three transitions must

be 'a4 - X304 , 'O3 - x3@, ,and '0, - X3@,. The k s t R branch lines within these

transitions will be the R(4), R(3) and R(2) lines, respectively. The frst Q lines are Q(4),

Q(3) and Q(2), while the frst P lines are P(5), P(4), and P(3). The first transitions for the

- X304 series are shown in Figure 5-6. The ratio of the spacings for the first R and Q

lines in these bands should be 5 : 4 : 3. Measurements taken directly offthe chart paper

gave values of roughly 5 -25 : 4.17 : 3. The assignments corroborate Our assumption that

the strongest band at 528 nm is the 'a4 - X3@, transition, the 532 nm band is the 'O3 -

X3@, transition, and the 535 nm band is the 'a, - X30, transition. This means that the

ground state is an hverted state, 30i. The final proof for these assignments, however, was

obtained by fitting the data to determine the molecular constants.

5.4.2. a Preliminaty Evaluation Using a Hund's Case@) Mode1

Our first efforts to extract enough data to present a case (a) b asis function were

unsatisfactory, so we began by using the simpler Hund's case (c) model. The equation

used to fit the three subbands is given by:

T, + Ben(Jcl) - D e d 2 (J + 1)2 (114)

Estimates of the rotational constants, B and Dy in the ground and excited States of

each subband were obtained fiom combination differences. The transitions of each

subband were then fit with the case (c) formula. Lt was found that while including higher-

order centrifugai distortion terms should irnprove the fit of the data, they were

Figure 5-6 First R, Q and P transitions for the 3 @ c ~ 3 @ 4 senes.

99

unrealistically large and often the wrong sign (about - 1 ~ 1 0 - ~ cm-'). Therefore, they were

not included in any of the fits and actudy were not needed to adequately describe the

data. The band origins and effective B values for the three subbands are given in

Table 5-4. The rms errors of al1 three fits were in the range of 0.049 to 0.056 cm-'. These

errors are of the same order as the bandwidth of the pulsed dye laser. Examination of the

constants in Table 5-4 shows that the ground state value of B, of all three subbands is

0.39 c d to two decimal places but the value for the 'a, - x3@, subband differs slightly

fiom the other two in the third place. This is not too surprising since the signal-to-noise

ratio for this subband was the poorest of the three and we were unable to resolve any Q

transitions for this subband. The upper state values are all of the order of 0.37 cm-' but do

differ slightly from each other. Given Our simple case (c) equation and the fact that there

is unresolved hyperhe structure distorting the iineshapes, not much can be read into these

differences. High-resolution scans of the bands at 18908.97 cm-' and 18780.76 cm-'

allowed a more accurate measure of these and other constants. The results are discussed

below.

5.4.2.6 Evulua fion Using Hund's Case(a) Mo&

When sufficiently good data had been coliected, rotational transitions based on the

proposed states were assigned. Great care had to be taken in the rneasurement methods as

these line values would next be used in a fitting procedure. The rotational Hamiltonian

Table 5-4. T, and effective B values for the excited and ground 'a states of cobalt

f l ~ o r i d e , ~ ~

a- Vaiues in cm-'.

b. Errors in parentheses are 3c.

operator for a Hund's case (a) basis set, in the appropriate R~ [74, 751 formalism is gïven

b y:

H=AL,S,+%h(3 S - s 2 ) +B(J2 -T+S2 -S) -(B -%y)(J+S-+J-S-)

where A is the £irst-order spin-orbit parameter, h is the second-order spin-orbit pararneter

including the dipolar electron spin-spin interaction, B is the rotational constant, D is the

centrifugal distortion correction, y is a spin-rotation pararneter, and AD and h, are

centrifugal distortion corrections to the spin-orbit interactions. The '@ electronic state

matrix element s for this Hamiltonian have been published[76]. This interaction matnx is

reproduced in Table 5-5. The fitting routine used to analyze Our data is examined in the

next section.

5.4 2. c Mir frix Elements

The first step in fitting the data was to perform a nonlinear least squares fit to

combination differences fiom both the ground and excited States. Combination differences

are the energy diEerences that can be obtained by subtracting the energies fiom different

rotational lines that have the same upper or lower state. For example, the R(4) Iine in

Figure 5-6 has an upper state J d u e of 5, lower state J 4 . By subtracting the Q(4)

transition energy, we obtain the upper state spacing between the J=4 and J=5 rotational

levels. Similarly, the difference between Q(4) and P(5) would yield the lower state

spacing between J=4 and J=5. Our experimental low-resolution results do not allow us to

Table 5-5. Matrix Elements for Rotational and Electronic Spin Parts of the Hamiltonian for the 'QI State."

a. A-doubling has been neglected. x=J(J+l).

deterrnine the transition energies from Q-branch lines. Instead, ciifferences fiom di three

lower or upper substates were fit simultaneously. The results are given in Table 5-6. The

constants were well deterrnined with rms errors of 0.10 cm-' and 0.15 cm-' for the ground

and excited state fits, respectively. These errors are weli within the experimental Luie

widths of the transitions. The distortion constant, D, could not be detennined in either the

lower or upper state. In order to determine the value of T, for this transition, the

transitions of the 'O3 - 'a, subband were fit separately. Close examination of the diagonal

matrk elements in Table 5-5 indicates that transitions in this subband do not depend on

the A spin-orbit parameters. Keeping the constants fixed at the values given in Table 5-6,

and using just the diagonal matrix elernents f?om Table 5-5, allowed a value of

T, = 18780.435 * 0.013 cm-' to be determined with an mis error of 0.074 cm-'. The other

two subbands depend upon the spin-orbit parameters of the upper and lower States;

unfortunately, we had no information at fïrst that would allow us to determine these two

constants separately. Since the ground state is inverted, we expect the upper state to be

Iikewise inverted. Therefore, we were only able to deterrnine the difference between the

parameters, AA. The two subbands were fit using diagonal matrix elements to give

estimates for AA. A value of 42.6 cm" was obtained for the 'a4 - 'a4 subband, while

the 'O, - 'a2 subband gave 33.1 cm-'. An attempt to fit al1 three subbands simultaneously

with just AA varying was made. We were unable to get a good fit to the data without

releasing other parameters. It is interesting to note that we were able to fit either the '@, -

'@, and '@, - 'a, subbands or the '0, - '0' and 'a, - 3 ~ , subbands simultaneously with

quite good rms errors of about 0.07 cm-' and 0.17 cm-'.

Table 5-6. Molecular constants for the excited and ground States of cobalt fiuonde.'

a. Values in cm-'. Errors in parentheses are 3a.

These two fits gave values of AA = 42.6 cm-' and AA = 32.2 cm-', respectively. Our best

estimate, therefore, for the daerence between the upper and lower state spin-orbit

pararneters was 37.4 * 5.2 cm?

The next step was to obtain information on a connected (Ai2 +O) state. With

information of this type, severai of the correlation difoculties experienced in the original

assessrnent could be Iifted- A discussion of this experiment is given in section 5.5.1-

5.5 Rotationally-Resolved Vibrationai Analysis

Of the bands detected, two series of bands were determined to be upper state

vibrational progressions. The fïrst series arose f?om Our main band at 18908.97 cm-'.

Upper state vibrational progressions of this system were determined to occur at

19573.38 cm-' (1,0), 20229.76 cm-' (2,O) and 20877.02 cm-' (3,O). In addition to this

system, it was determined that another, similar system was observed. The bands centred

at 19235.47 cm", 19875.76 cm-' and 205 10.34 cm' are proposed as another upper state

vibrational progression (0,0), (1,O) and (2,O). This assignment was made based on the

similarity of band structure, transition strengths (progressing Eom very strong to weak),

spacing and rotational constants. From the assignments, vibrationd constants are

determinable. A simple model, including the first-order anharmonicity constant, for a

given transition is given by[77]:

where vc is the transition frequency (Tcr- Tc") and v,, the fiequency of the 0-0 transition

(Tot- Tou), ru0 and w~ the zero-point f7equency and anharmonicity, respectively.

By fitting our band system to this equation, we can obtak the vibrational constants a,'

and o,'x,'[77] :

The anharmonicity constant o,'ar= oo'x,,' as insuEcient information exists to determine

higher order distortion terms.

The vibraticmil B value can be fit to the expression [78]:

to obtain the equilibrium values for the rotational constant B and anharmonicity constant,

a. T,, effective B and r values and equilibrium vibrational constants for the two series are

presented in Table 5-7 and Table 5-8.

The equilibrium bond length is deterrnined by[78]:

where h is Planck's constant, c the speed of light and ,u the reduced mass of cobaIt

fluoride.

Using the above equations, the equilibrium bond lengths for the excited States

were deterrnined. These are presented in Tables 5-7 and 5-8.

To assess the ground state configuration in this manner would require a ground

Table 5-8, Transition series vibrational constants for the second set of bands.

Equilibrium Constants

a. Errors in parentheses are 3 0 . b. Insuficient data to obtain this constant. c. Error values unavailable for these constants.

state vibrational progression (0-0, 0-1, 0-2, etc.) which was not detected during this work.

Combination differences were compiled and a non-linear least-squares fit was

performed. See the first two columns of Table 5-9 for these results. Heavy correlation

between upper state and lower state parameters forced us to zero or fk many of the

coupled and distortion terms. This gave us approximate values for the preliminary analysis

and a starting point for fùrther calculations. The advanced treatrnent using Hund's case (a)

and a Iarger data set are discussed in the next section.

5.5.1 Case (a) Rotational Analysis

M e r our initial analysis, we were left with bands that did not belong to any of the

transition manifolds under study. The band at 19484 cm-' was not a typical system as it

exhibited a strong P branch, weaker Q and Iittle discemable R branch information. This

shape is indicative of a AR= - Z transition. To determine connections between bands, we

had to approach this problem a little differently. Typicaily, we scanned a certain range

with the laser and set the monochromator at the scans' mid-point. This alIowed us to

detect irnrnediate fluorescence fiom bands directly. When we excite the molecules into the

upper state with the scan laser, they Buoresce down to many lower states. Systems

'comected' or sharing the same upper state to the one in question wili exhibit band

structure.

The band at 19484 cm-' was observed during a scan when the monochromator was

set on the 18780 X3<D3) system. The observed band was therefore assigned to

the '0,- X30, transition. Figure 5-7 contains a diagram of these transitions.

Figure 5-7 Scheme used to determine the ground and excited States contributing to the system at 19484 cm". This system was scanned with the excitation laser. Fluorescence fiom the system at 1878 1 cm" was detected.

Combination differences for upper and lower states were compiied using the 3@ - x3@ manifold and additional information obtained fiom the AR= - lsystem. See

Table 5-9, columns 3 and 4 for the results. Agaîn, correlation problems did not allow the

parameters to be totaily freed. Note that fixed and floated values are quite similar.

Finally, in the rotational analysis, the line positions themselves were used as a

dataset. Ap pendix A-4 contains the Iine positions for these systems. Nine parameters

were allowed to float, and the r.m.s. error was reduced to 0.05. See the last column of

Table 5-9 for these results,

5.6 High-Resolution Analyss Including Cobalt Hyperfine Sîructure

It was detennined that high-resolution scans could be attempted for some of the

detected bands in the 500 to 540 nm range. See section 4.1 and section 4.2.2 for details of

the Coherent continuous-wave ring dye laser used to collect this data. Use of Coumarin

540A limited the possible bands of study to those above 5 16nm, whiIe available optics, the

energy profile of the dye and overall band strengths reduced our 'window' fiirther still.

Only scans of the 3~,-X3<b, band (TO=l 89 12 cm") and 3@3-X30, band (To=l 878 1 cm-')

transitions were collected.

The high-resolution spectra for cobalt fluoride allow a glimpse at the hyperfine

workings within the molecule. Individual rotational lines are seen to be split by the

nuclear spin of the cobalt atom (I='/J into 21+1 or 8 lines. A fine example of this is given

in Figure 5-8 for the R(4) to R(8) transitions in the 3 @ , - ~ 3 @ , band. This figure is a

compilation of many hi&-resolution data sets. Linewidths are about 180 MHz which was

the optimum resolution obtained in the study. To properly describe these phenomena, we

must tum to a Hund's case (ap) basis set which includes hyperfine matrix elements. This

basis set further extends the interactions within the molecule by having the nuclear spin

vector 1 couple with the rotational J to fonn a resultant vector F. A fU-resolution scan

for an R-branch transition is given in Figure 5-9 and shows the hyperfïne splitting for the

R(5) branch in the 3 ~ 4 - ~ 3 @ 4 series. Within the nuclear-spin split rotational transition,

eight distinct lines may be seen. These represent the AF = +l transitions. Additional

transitions are apparent in the thkd through eighth lines showing AF = O transitions.

A diagram of these transitions is given in Figure 5-1 O. For a P-branch transition, the main

series is composed of eight AF = - Z lines. Up to seven satellite lines denote the AF = O

transitions. In Q-branch transitions, the main lines will be AF = AJ = O. Both AF = +l and

AF = - 1 satellites c m be seen, so a total of 22 distinct transitions make up each

Q-branch rotationd line. A diagram of possible Q-branch transitions is shown in

Figure 5-1 1. The Q-head for Our 'a,-X3<b, band is presented in Figure 5-12 and

represents several combined high-resolution scans. Transitions fiom the first six Q lines

are marked.

These figures show that nuclear splitting does occur within the cobalt fluoride

molecule, but the bewildering collection of lines reveals little hard data until rigorous

analysis of the line positions is undertaken. The next section describes the matrix elernents

that must be included in the model, and the following section assesses the results of the

analysis.

- - - -

18912,6677 cm-i

Figure 5-9 High-resolution scan of the R(5) branch in the 30c~30, band system. The full scan represents ,6667 cm-' or 20 Ghz. The F values are listed for each hyperfine transition.

Figure 5-10 Hyperfine transitions for the R(5) transition showing the eight AF = +l (AJ = +l, A I = O) main transitions and 7 AF = O (AJ = +l, AI = - 1) satellite transitions for a total of 15 hyperfine transitions per rotational Iine.

Figure 5-1 1 Hyperfine transitions for the Q(4) rotational iine showing the eight AF=AJ=O main transitions and AF;.AI=&l satellite transitions (7 each) for a total of 22 hyperfine transitions per line.

S. 6.1 M& EIements for High-Remlution Analysis, Hund's case (ab

Recdl from section 2.3.3 and section 2.4 the rotational Hamiltonian, its £kst and

second order correction terms in the case (a) basis set. In addition to these terms, we

must include terms specific to the case (as) model that take into account the magnetic

hyperfine interactions of J, 1 and F. The matrïx elements diagonal in IT are [76, 791:

and

As in equation (49), h = ah + (b+c)C. Elements off-diagonal in Cl, needed to fit muitiple

bands, are also available 1761.

AfIer measurement of the hyperfine lines in individual rotational transitions, a data

set is built containing the positions and indices describing the upper and lower state F

values corresponding to each line. The non-linear least-squares program code used for

analysis was modified then recompiled to include the case (a,J terms noted above. From

this computation, values for the upper and lower state h parameters are obtained.

Table 5-1 0 contains a representative data set showing the hyperfine line positions for the

R(5) branch of the band centred at 18908.97 cm-'. This rotational line has a low-

resolution position of 18912.402 cm". The entire data set contains 524 separate hyperfine

transitions. The results of the fit are presented in Table 5-1 1. It is interesting to note that

sufficient data existed to dIow determination of the distortion term D, which was not

120

Table 5-10. Observed line fiequencies for the '<D, - X3@, CoF electronic system.

Hyperfine transitions of the R(5) rotational Iine."

a. Values in cm",

Table 5-1 1. Hyperfine and rnolecular constants for the excited and ground 'a, states of

cobalt fluoride."

a. Values in cm-=. Errors in parentheses are 3 a.

attaïnable with the low-resolution data set.

The band centred at 18780.76 cm-' was also analyzed with a case (ap) model. The

weaker signal strength did not allow determination of nearly as many hyperfine transitions.

The data set contained 148 points which proved insufficient to determine a lower state

distortional term. The results for this series are presented in Table 5-12. The data set

contained an incomplete set of lines fiom R(14) to R(3)? Q(5) to Q(3) and P(4) to P(14).

5.6.2 Analysis of Hyperfine Results

As noted above, the evaluation of hyperfine data yields information about the

electronic bonding picture within the molecule. The terms needed to describe this bonding

are very sophisticated and require careful consideration of their composition to accurately

assess the results. The hypedne values obtahed fiom the 'a, and 'ou>, band systems were

used to isolate individual constants for both upper and lower state configurations, as

described below. Starting with the formula h = a A +(b +c)Z , we get for the lower

state:

for the 'Qi, (A=3, Z=1, and S 2 4 ) state and for the 'a, (A=3,Z=0, and R=4) state.

Inserting the h" values obtained, we get:

which gives immediately a" = 0.0 168 cm-' and ( b " + c" ) = -0.0 1703 cm-'. Similar

Table 5- 12. Hyperfïne and molecular constants for the excited and ground 3@3 States of

cobalt fluoride,"

a. Values in cm-'. Errors in parentheses are 30 .

equations for the upper state yield af=O.O1 138 cm-' and ( b' + cf ) = 0.06356 cm-'.

To understand what these values represent, we begin with an examination of the a

term. From equation (SO), we see that a is inversely proportional to the interacting

electron's distance from the nucleus. As a gets smaller on excitation, this translates as the

electron moving away fiom the cobalt nucleus.

The b+c terms, as described in chapter 2, equations (50) to (52), represent the 1 4

interactions within the molecute. The composition of c will not change much between

ground and excited state systems since the dependence is sirnilar to a except for an angular

contribution. The Fermi contact parameter b, is defined by b, = b + %c and wiIl be

essentially zero ifthere are no unpaired s electrons. If our '@ state arises fiorn the 16 and

the 4n electrons, this will be true.

Ifthe Fermi contact parameter is zero, then b = -'/3c and b+c = 543~. Since V3c" =

-0.017 cm-', c" = -0.026 cm-: or -780 MHz. The parameter c will scale sirnilarly to a, cf =

-0.0 18 cm-'. Our computations yield (bf+c') = 0.064 cm-' or b' = 0.082 cm' (2460 MHz)

and b,' = b' + 1hc' = 0.082 - 0.006 = 0.076 cm-' or 22800 MHz.

The Fenni contact parameter b, has gone from essentially zero in the Iower state

to a decidedly non-zero value in the upper state. As this is a measure of the s-orbital

character, we can assume this promotion is from a non-s to s-type orbital within the

molecule. This can be rationalized by starting with a da or pa electron and prornoting it

into an so molecular orbital largely of cobalt character. Promotion of a o electron will

leave the 16 and 4n orbitals alone, retaïning the angular momentum necessary to yield a 'a

excited state configuration. The decrease in a would imply that the electron rnoves to an

orbital further £iom the nucleus- Both cm be accomplished by promotion into the 10a or

1 Io molecular orbital as one of these will be largely 4s cobalt in character. Using the

same analogy, promotion into a higher o orbital could also occur. The simple 1-electron

picture begins to break down at this point. We cannot determine the composition of the a

orbital (Co s, p, or d character), or what contribution F makes to the orbital. It is aIso

difficult to deterrnine the actual transition fiom the molecular orbital diagram as we have

no calculations on these energies. However, we can use the CoH results of Freindorf et al

[72] to rationalize this suggestion. As noted by Ram et al [65], the energetics observed in

cobalt fluoride mirnic the energy ordering scheme obtained for cobalt hydride

configurations. While there cannot be an explicit correspondence between the two

disparate species, Rarn et al [65] suggest their 3@i - X3<Di system scales to the 2 '<D state

reported in the cobalt hydride computations[72]. The next cobalt hydnde 'a state would

therefore scale to just under 19000 cm", where our transition occurred. As this

configuration has the same components (~s'o'R'6~) as the state 'claimed' by Rarn et al[65],

it is most likely that Our transition is not the 90 to 100 transition, but rather a o electron

promotion fiom the 90 orbital to a higher orbital, perhaps corresponding to the 5s orbital

on cobalt. Akematively, the transition in question could aise fi-om promotion of an

'imer' o orbital, perhaps the 80, to the valence 100 or 1 la orbitals. This last would

translate as a 'bonding' electron tramferring to an essentially non-bonding orbital, largely

atomic 5s cobalt in character. The changes in bond length are thus accounted for as the

bond gets longer on excitation. The results of Freindorf et al [72] also point out that the

one-electron promotion description is simplistic. Detailed theoretical calculations are

necessary to sort out this problem.

6 COMPUTATIONAL RESULTS

6.1 Introduction

The study of the complex spectroscopy and bonding properties of the diatomic

oxides [63, 801, carbides [81, 821, and nitrides [83, 843 that contain first row transition

metal atorns is presently an active field of experimental researcb In general, the electronic

states of most of these molecules arise fiom more than a single electronic configuration.

Consequently, simple molecular orbital theory will usually fail to quantitatively predict

their spectral properties. Post Hartree-Fock ab initio methods are required to accurately

descnbe these properties and must take into consideration the sensitive balance of their

electron-eIectron exchange and correlation [85, 861. This rnay be achieved by employing

configuration interaction techniques.

The situation is somewhat simpler when the transition metals involved are fiom

the right and left ends of the periodic table, such as Sc, Ti, Cu and Zn. Their diatomics

may usually be considered as arising fiom one predominant electronic configuration [83,

84, 871. Even for such molecules, the metal atoms have quite a few low-fying states [88]

that may combine with either C, N or O to give rise to complex eIectronic spectra [80,

841.

Recent advances in experimental molecular beam techniques and high resolution

gas phase spectroscopy have also resulted in the reliable determination of the hyperfhe

tensor components for a large number of 3d transition metal diatomics [84]. Such

magnetic hyperfine tensor components are extremely sensitive to the quality of the

127

electronic wave function and are one of the most diflicdt properties to cornpute. Both the

core and valence electrons must be ufiozen in the MRSD-CI computations. In addition,

the core and vaience components of the CI wave fùnction must be balanced and welC

described by extended basisfunctions to yield sufficiently good results. The majority of

the accurate hyperfÏne tensor computations have been carried out for smail first row

diatomics and triatomics [89-921. To the best of our knowledge, oniy a very limited

number of magnetic hyperfine tensors have been calculated for diatomic and triatomics

that contain 3d transition metal atoms [52, 53,931- Due to the small nurnber of published

computations on the 3d transition metal diatomics, the use of sophisticated ab inifio

techniques to reproduce the expenmentai values for these systems has not yet been

established.

In the serninal work of Davidson and Feller on molecules contaking the first row

main group elernents, agreement of 80-90% between the experirnental and cornputed

hyperfine tensors was expected [89]. With the recent advances in cornputer hardware and

software this agreement is now within the 90-99% accuracy range [91]. However, for the

more complicated diatomics that contain 3d transition metal atoms such an agreement is

expected to be more difficult to achieve. Previous computations on X)A VN [52], VCH

[53] and Ti0 [94] have indicated that the MRSD-CI method can account for

approximately 88% of the Fermi contact interaction.

This chapter is part of an effort to increase the number of hyperfine coupling

constants of paramagnetic 3d transition metal diatomics computed by the MRSD-CI

technique. This work attempts to determine the minimum requirements imposed on the CI

wave function such that it can predict the experimentai hyperfïne tensors to within 8595%

accuracy. If such agreement can be obtained for a large number of these diatomics, these

computational methods can be used with confidence to predict and interpret the hyperfhe

structure obtained fiom gas phase high resolution laser induced fluorescence (LE) and

electron pararnagnetic resonance @PR) spectra.

The isoelectronic Sc0 and T X molecules are good starting candidates to test the

accuracy of the computed hyperfine tensor components of 3d transition metal diatomics.

They are expected to give the closest agreement between theory and experiment for the

following reasons. S c 0 and T N are the simplest pararnagnetic species of the series with a

single unpaired electron (2Z+ ground states). The scandium and titanium atoms have

relatively mal1 spin-orbit coupling constants compared to other first row transition metal

atorns such as Cr, Mn, Fe, Co, Ni and Cu. Consequently, their relativistic effects are

expected to be smaller. Since Sc and Ti are early transition metals, relatively few

configurations are required to represent their total wave tùnctions [87]. Finally, the

hyperfine coupling constants for both Sc0 and TiN have been measured with a high

degree of accuracy and are available for cornparison [80, 841. The third member of this

isoelectronic family, VC, will not be considered since it has a different ground state (X*A)

and its hyperfine coupling constants have not been accurately measured in the gas phase.

6.2 Cornpurarional details

The basis sets used are sirnilar to the ones used previously for VN [52] and

VCH [53]. The Sc and Ti 14s/9p/Sd primitives of Wachters [95] were contracted to

62 1 1 1 1 1 1 s/3 3 l2p/32d. They were further augmented by the p and d polarization

fùnctions of Bauschiicher et al [96]. The van Duijneveldt 13s/8p oxygen and nitrogen

basis sets [97] were contracted to 621 11 1 Id421 1p and firther augmented by p and d

polarization fiinctions [98].

The MRSD-CI calculations used the MELD senes of prograrns [36]. The

computations were carried out using the experimental gas phase geometries [84, 991 and

Ç, symmetry. The initial wave function was obtained fiom a restricted open sheli

Hartree-Fock (ROHF) self-consistent-field (SCF) computation correspondhg to the

1 9013x4> single determinant. The canonical virtual orbitais were converted to K orbitals

by the method of Feller =d Davidson [100] and were subsequently used as initial

eigenvectors for a preliminary MRSD-CI calculation (TE = 1 .O pEJ. The final MRSD-CI

used to compute the properties were started fiom the natural orbitals (NO) generated in

the preliminary run.

An iterative method was used to select the single and double excited

configurations included in the reference space of MRSD-CI calculations. Initialiy, a

singles and doubles configuration interaction (HFSD-CI) cornputation employing a single

configuration in the reference space was carried out. From the resulting CI wave finction,

the con£ïgurations with the largest expansion coefficients were added to the reference

space, usualIy five at a tirne. A MRSD-CI computation was then performed and the next

five configurations with the largest expansion coefficients were selected and added to the

reference space. This iterative selection procedure was repeated until the desired number

of configurations in the reference space was reached.

The sum of the squares of the coefficients of the reference configurations is an

indicator of the quality of the CI wave fiinction. In the largest MRSD-CI-NO

computations this parameter was 0.94 for both Sc0 and TN.

6.3 Ine bonding in XZr S c 0 and TiN

The Sc0 ground state has been found experimentaliy to be of 2Z' symrnetry [80,

IO 1, 1 021. HFSD-CI, coupled pair fùnctional (CPF) Cl031 and locd density functional

(LDF-LCAO) [IO41 computations have confirmed this state. The value of the "SC Fermi

contact term was determined, fiom rnatrix-isolation experiments, to be 20 10 MHz Cl0 1,

1021. This was corroborated by recent gas phase studies where it was found to be 1947

MHz [80].

Formally S c 0 has a double bond while TïN has a triple bond. Although their

bonding is quditatively different, their overall electronic structure is the same. In order to

form TiN the Ti atom must first be promoted from its 3F (3d24s2) ground state to the 5F

(3 d34s1) excited state. The degenerate 3 &(Ti) and 3 %(Ti) atomic orbitals then combine

with the 2px(N) and 2py(N) atomic orbitals to form two x bonds. A third o bond is formed

by the interaction of the 3dL(Ti) with the 2pz(N) orbitals. In this situation, a triply bonded

moIecu1e is formed with a non-boriding 90 molecular orbital that is mainly 4s(Ti) in

character [83, 84, 104-1063. Alternatively, TiN is envisaged as a VN diatomic that has

lost its highest occupied non-bonding 16I.t electron resulting in a 9a13n4 dominant

electronic configuration. In the case of ScO, the Sc atom is promoted to its first excited

state and has the electronic configuration Sc(3d<rL 3dz1 4s1) [103]. This then transfers its

3 dx electron to the highly electronegative oxygen atom to form a bond that is partially

ionic and of the form Sc'(3da14s1)0-(2pd 2px4) [103]. Therefore, for both TïN and Sc0

the net products of the bonding are x2Z' rn~lecule~ with (90' 37c4) predominant electronic

configurations where the 90' is rnainly 4s in character.

The &(Sc) and 4s(Ti) electrons have a finite probabiiity density at the nuclei and

hence the effective total spin operator, S, interacts strongly with the nuclear spin of the

rnetal. This leads to strong nuclear hyperfine couplings and Hund's case bps molecules.

The sirnilanty in the electronic structure of Sc0 and TïN is also apparent in the

present cornputations. The first few leading conQurations of the MRSD-CI-NO wave

function are listed in Table 6-1. They show that, for both molecules, the composition of

the CI wave fiinctions is very similar. Thus most of their one- and two-electron properties

are expected to be comparable. Table 6-1 also reveals that, as expected, the coeEcient of

the 90 '31 electronic configuration is large (0.92 - 0.94). However, as is shown later, the

values of the hypefine tensors still change as the degree of correlation introduced in the

wave fiinction is increased. Thus a quantitative description of these sensitive one-electron

properties requires a MRSD-CI treatment.

Table 6-1 : Occupancies and Coefficients of the Leading Electronic Configurationsa for the

S c 0 and TiN X2Ç* States.

Electronic Configuration CI Coefficient

1 2 2 1 4 4 0.9394 0.9205

2 b 2 1 1 f 4 3 I 0.0993 O- 1127

3' 2 2 1 4 2 2 0.094 0.1047

4 2 1 2 4 4 0.083 0.1068

5 b 2 2 I 4 2 2 0.0823 0.1022

6 2 2 1 2 4 2 0.051 0.048

7 b 1 2 1 1 4 3 1 0.035 0.034

8 I 1 1 1 1 4 4 0.031 0.037

a. Obtained from MRSD-CI-NO computations using 73 SARC and a selection threshold

b. Doubly degenerate configuration.

c. Explicit x configuration is 2q2 2%'3q1 3s1 4%' 4%'.

6.4 The hyperfine couphg constants of Sc0

In general, for a gas phase diatornic the effective spin Hamihonian is given by:

where L and R are the effective orbital and rotation operators respectively. The nuclear

spin operator of the jth nucleus is denoted by IO'). The aG), bF(j), c(i) and d(j) Frosch-

Foley parameters [19] represent the nuclear spin-orbital, Fermi contact, dipolar and par@

doubling constants. For a 'E' diatomic in its ground state, where L = O and S = E, a(j)

and da) are irrelevant and wiil not be considered fürther.

The Fermi contact term for a nucleus j in a rnolecule that contains M electrons is

given by:

where & and gj are the electronic and nuclear g tensors, and pN are the electronic and

nuclear magnetons and Y is the electronic wave fùnction. The <Y 1 6(rj) 1 Y> term,

containing the Dirac delta fùnction, is the normalized net spin density at the jth nucleus.

An alternate form of Equation (126), written in terms of the one-electron spin operator,

s(k) is:

b F ~ ) = %&~PP,I 3 +) k=l 5 6(ik)szn<)lY) (127)

where S is the total spin multiplicity.

Ideally for the best agreement with experiment, a relativistic full CI computation

with complete basis sets is required. In a simple ROHF treatment the contributions to

b,CSsc) and b,(170) are due to the s character in the 9$ one-electron orbital of the

predominant 9013ir4 electronic configuration. This treatment does not make any

provisions for core polarization effects. Most of the core polarization at the nuclei may be

taken into account by including the single excitations generated f?om the predomuiant

electronic configuration. This usuaily increases the net spin density at the nuclei. The

b,(4S~c) and b,(170) values computed at this level are labeled S-CI in Figure 6-1 and 6-2,

respectively.

The introduction of dynamic correlation should, in p ~ c i p l e , improve the

agreement between the cornputed and experimental hyperfine vdues. The sirnplest way to

include these effects is to add to the CI wave function al1 double excitations with respect

to the ROHF configuration. This decreases the net spin density at the nuclei and the

resulting b, values are smaller in magnitude (and worse) than those obtahed by S-CI

computation. They are illustrated in Figure 6-1 by the points labeled SD-CI. The poor

agreement of the SD-CI results with experiment is due to the insuficient correlation

recovery fiom the valence electrons, at this Ievel.

The situation may be rernedied by perfonning MRSD-CI computations fiorn a

carefùlly seIected set of reference configurations as described in section 6.2.

Computations using this procedure usually y-ield good results and use less than 1% of the

total number of excitations required for a full CI treatrnent 189, 9 11. This method

1790-• u u m a 1 m

a

a

a m

4

a 4

. a

I

3 D ..

1750- - . rn

. a SD-CI

1740-, fi a rn a I I

O 30 60 90 120 150 Number of SARC

Figure 6-1 Behaviour of b,("Sc) as a function o f the number of spin adapted reference configurations (SARC) included i n the variational MRSD-CI-NO treatment. The energy seIection threshold, TE, is 1 .O p h .

30 60 90 120 Number of SARC

Figure 6-2 Behaviour of b,(170) as a fûnction of the number of spin adapted reference configurations (SARC) included in the variational MRSD-CI-NO treatment. The energy selection threshold, TE, is 1.0 FE,.

also rnimics the incorporation of triple and quadruple excitations, with respect to the

ROHF configuration, in the CI wave hnction-

Figure 6-1 shows the variation of the scandium isotropic Fermi contact term with

the number of spin-adapted-reference configurations (SARCs), while Figure 6-2 is the

corresponding graph for "0. From these graphs it is clear that, as the number of SARCs

are increased, the absolute values of b,(4sSc) and b,("O) increase. When more than 20

SARCs are used these values level off and change very little. The anisotropic tensor

components are less sensitive to the number of reference configurations and pose no

additional restrictions on the number of SARCs used. Although 34 SARCs are probably

adequate, we have opted to perfonn computations that employ 73 SARCs and natural

orbitais to investigate the hypefine coupling constants as a fùnction of the increasing

number of double excitations included in the CI wave fùnction. It may also be assumed

that the use of more than 3 4 SARCs is enough to take into consideration the effects of

triple and quadruple excitations on the hyperfine pro perties.

When a multireference approach is adopted, the inclusion of all double excitations

in the CI wave fùnction becomes a prohibitive process. Therefore, only the double

excitations that contribute a certain energy threshold, TE, are included in the CI wave

fùnction. In other words, decreasing TE has the effect of increasing the number of double

excitations in the CI wave fùnction.

The variation of b,("Sc) as a function of -log TE is given in Figure 6-3. It starts

at 1757.5 MHz and gradually increases as the selection threshold, T, is decreased.

IdealIy lower TE values should be used until b,(4SSc) saturates and becomes independent

Figure 6-3 The "Sc Fermi contact interaction as a function of -logTE. Seventy three SARCs generated from natural orbitals were used.

of the number of double excitations. Due to hardware and software iimitations, the

computations were o d y carried out to a TE value of ~ . O X ~ O - ~ E, resulting in a bF(45Sc) of

1790.6 MHz. This represents approxhately a 2.0% increase in the magnitude of the

hyperfine spiitting.

The small variation in b c S c ) due to the decrease in TE does not necessarily

imply that the computed and expenmental bF(4SSc) values are close. The computed values

must be directly compared with experimental results in order to determine their accuracy.

For this reason, the gas phase results listed by Childs and Steirnle are used. They

detemiined experimentally bFCSSc) to be 1946.8 MHz [80]. Consequently, at the highest

level of calculation, b,(45Sc) is within 92% of the experimental value.

Presently there are no experimental values available for bF(170). The plot of

b,(170) versus -logTE, is s h o w in Figure 6-4. This parameter smoothly varies fiom - 17.6

MHz to - 18.8 MHz as TE decreases frorn 6.0~1 Eh to 4.0x10-~ %. It almost levels out

at smail TE vaIues indicating that it has become practically independent of the number of

double excitations included in the CI wave fünction. From Figure 6-4, one can estimate

that at the full CI limit bF(170) = -18.8 MHz. It is large enough to be detectable by

experiment ifthe appropriate "O isotopic substitution is performed.

Accurate values for the anisotropic hyperfine tensor components that are close to

the expenmental ones are easily obtained even at the ROKF level. The CG) Frosch-Foley

parameter is related to the anisotropic tensor component, TA), via the simple relation:

Y).

Figure 6-4 The ''0 Fermi contact interaction as a function of -logTE. Seventy three SARCs generated from natural orbitals were used.

The variation of cCSSc) as TE decreases fiom 6 . 0 ~ 1 0 ~ E, to 4.0x10-' E, is less than 2

MHz. The largest computation yields a c("Sc) of 76.4 MIIZ and only overestimates the

experirnental value by 2.7%. Should relativistic and spin-orbit coupling effects be included

in the caiculation an agreement with experiment closer to 100% would be expected.

In sumrnary, the large magnitude of bF(45Sc) can be attributed to the fact that the

dominant eiectronic configuration is found to be the 9a13d. In this configuration, the

unpaired electron is located in the 9cr orbital, which is mainly Sc 4s in character.

Consequently, the spin density at the Sc nucleus is large and results in a large Fermi

contact term. This large 4s(Sc) is at the expense of the p and d character of both atoms.

It also causes the Zs(0) character to be very smd. This, in turn, leads to a relatively

small c(~'SC), bF(I70) and c("0). For example, the total spin density at the oqgen

nucleus is approximately two orders of magnitude smalier than that at the scandium

nucleus. In addition, the c(170) value was found to be essentiaily zero (-0.1 MHz) because

of the small p and d character on the oxygen centre.

6.5 The Iiyperfine coupiing constants of TiN

The changes in the TiN b, and c hyperfïne parameters as a fùnction of the number

of SGRCs and double excitations Included in the CI wave function are very sirnilar to

those of ScO. Therefore, to be concise, the graphs for TiN, corresponding to those of

Sc0 in Figures 6-1 to 6-4, will not be shown. Instead, the hyperfine parameters for both

molecules are listed in Table 6-2.

Table 6-2. Magnetic Hyperhe Parametersa for the S c 0 and T N x2X' States.

Atom

Experimentai Computed

a- Ail values in MHz.

b- Reference 2.

c- Reference 6.

Experimental Computed

The agreement with experiment and the accuracy of bF(47Tii) and c(~'T~) are better

than their "SC counterparts. The computations c m account for 93% of the b,(47Ti)

experimental value. On the other hand there is a 99.9% agreement between theory and

experiment for the c('~T~) hypelfine parameter.

Table 6-2 indicates that although the MRSD-CI-NO computations can only

account for approxhately 85% of the b,(14N) pararneter the ciifference between the

experirnentai and calculated values is only 2.7 MHz. This is very similar to the accuracy

obtained for b,(14N) in the case of the X2n NO molecule where very large and extensive

calculations were perfonned [9 11.

Finally it is very difficult to estimate the accuracy of c(14N) since both the

experimentai and computed values are very srnall ( l e s than 1 MHz).

6.6 Summaty and conclusions

Although S c 0 and T a , in their xZ2+ ground States, are considered to have a

double and a triple bond respectively, the MRSD-CI-NO computations show that their

predominant electronic configuration is 9013d with an unpaired electron that is mainly

metal in character. Cornparison of the first eight leading reference configurations for the

two molecules shows that they are very similar and explains why they have comparable

electronic properties.

Systematic investigation of their hyperfine tensors indicates that, starting with

natural orbitals, relatively few SARCs are required to obtain good agreement with

experiment. At the highest level of caiculation, the computed b, parameters for Sc and Ti

are estimated to be withh 92-93% of the experimental values. They arise rnainly fiom the

large 4s character in the 90 orbital of the predorninant 9u137t4 electronic co&guration.

The agreement of the c parameters with experiment is even better (97 to 99%). The srnail

c values of the "0 and 1 4 ~ centres are rnainly due to the s m d 2p character in their singly

occupied rnolecular orbitais.

The present study shows that, thus far, MRSD-CI computations seern to be a

viable method to help the experimentaiist predict and interpret the h y p e h e tensor

cornponents of this cfass of diatomics. Inclusion of relativistic and spin-orbit coupling

effects should bring results closer to 100% agreement.

CONCLUSION

7.1 Cobalt fluoride

The experimentd results obtained for cobalt fluoride were quite satiswg, in iight

of the fact that the ground state of this molecule was uncertain. The wealth of band

systems obtained during the course of study allowed an enviable g h p s e into the intemal

workings of this diatomic. SufEicient data were collected during the initial studies to

enable rough determination of many rotationai constants for this system, and fùrther

examination of the available band systems furnished clues to the intercomectivity between

these transitions and improved the assessrnent of spectroscopic constant values. A rather

complete picture of the 'ai ground state for cobalt fluoride was obtained fiom the

rotational spectra collected during ttiis work, and information on several excited States

was assessed. Use of different methods of laser scanning dowed insight into the

connections between band systems and helped determine the correct assignments for the

systems.

The high-resolution work that followed gave a picture of the finer details, the

interaction of cobait's nuclear spin with the previously determined rotation vectors. These

spectra allowed a first look at the extent of hyperfïne interactions within the rnolecule. By

examining changes within a transition, a measure of the hyperfine properties was

obtained. These results gave the first experimental measure of Fermi contact parameters

for cobalt fluoride and clues to the effects involved in visible molecular transitions. The

hyperfine constants indicate that the transition most likely arises fiom promotion of do or

po b e r bonding electrons into valence SC orbitals contained on the cobalt.

7.2 Configuration Interaction of Scandium Oxide and Titunium Nitrr.de

Computational results on Sc0 and T N show that a complete description for

effects as small as the hyperfine interactions berneen two atoms is measured mainly by the

number of configurations included within the dataset. A good result was obtained by

single excitation CI based on a restricted open-shell Hartree-Fock configuration.

Improvements to the hyperfine description were made by including selected double

excitation configurations. By choosing these coniïgurations rather than attempting to

include ail contributors, a huge savings in computational time was obtained. Additionally,

the anaiysis of the results became much more manageable. A measure of the completeness

of this description was determined by surnming the squares of the coefficients. More than

90% of the overall picture was obtained by using 73 SARCys.

7.3 Future Considerations

Cobalt fluonde was one of the first molecules successfiilly studied with the UNB

laser ablation apparatus. Other molecuIes have been studied since this work. Molecules

studied past and present include CoC, YbF, YbC1, TiF, and ZrN. There is still much work

to be done on the cobalt fluoride molecule. Since this work was completed, high-

resolution spectra have been collected and analyzed for the 51 1 nm band. This was found

to be the 1-0 band of the '0, - X3Q4 band system studied in this work. High-resolution

spectra have been collected and anaiyzed for the 5 19 nm bandCl 071. It is another

30, - 3<D, system (the 0-0 band). The 503 nm band has been determined to be the 1-0 band

of the 519 nm system. High-resolution spectra have also been collected for the

484 nm band. It was found to be a 'T, - 'Q4 system that shows fluorine hyperfine spiïtting

in addition to the cobalt hyperfhe structure. Analysis of this more compIex spectrum will

entail adding several more matrix elements to the molecular Harniltonian to describe the

coupling of another nuclear spin angular momentum. Currently, attempts are underway to

collect results in the 469 nrn region which contains the 1-0 band of the 484 nm system.

These new spectroscopie results for cobalt monofluoride have been made possible due to

new laser equipment installed since this thesis research was pefiorrned. It has led to new

studies and will continue to do so because the blue region of the spectrum is now available

for high-resolution spectroscopy at UNB. As mentioned in chapter 4, this spectral region

could not be accessed with the older laser system. Computerization of the data colIection

will improve the tum-around time between experiment and publication by removing the

tedium of labour-intensive measurement of the low-resolution scans. Probe radiation

sources of other wavelengths could be used to hrther uncover the electronic composition

of cobalt fluoride and similar small molecules. Research f?om elsewhere has shown this to

be the case, as infrared spectroscopy of this compound has been reported [64, 651. In

collaboration with researchers at other locations, rnany such molecules could be

investigated. By combining the equipment and knowledge f?om several locales, exciting

methods of investigation and insights into small molecules can be achieved.

From a computational perspective, we can expect furtkier investigations into

similar diatomics. Electron-rich transition metals still present one of the greatest

chalIenges to theoretical chernical physics. The extensive configuration interaction seen

for Sc0 and T N will most likely occur in other transition metal diatomics. A study into

the nature of excited States and magnetic properties for cobalt carbide has been

compIeted[208]. Other chaiienging molecules are being considered for f h r e

investigation.

8 REFERENCES

DeVore, T. C.; Van Zee, R J.; Weltner, Jr. W. Proc.-Electrochem. Soc- 78-1

(Proc. Symp. High Temp. Met. Halide Chem., 1977) (1 W8), p. 187.

Barry, J. A Ph. D. thesis, University of British Columbia (1 987). Cramb, D. T.

Ph. D. thesis, University of British Columbia (1990). Peers, J. R. D. Ph. D.

thesis, University of New Brunswick (1996).

Levine , 1. N. Physical Chemisby, McGraw-Hiu Book Co.:New York (1 W8),

p. 560.

Ibid, p. 545.

Ibid, p.601.

Banwell, C. N. Funhnentals of Molecular Spectroscopy, 3rd edition. McGraw-

Hill Book Co., Ltd.: London (1983), p. 74.

Ibid, p. 77.

Ref. 3, p. 563.

Herzberg, G. Spectra of Diatomzc Moledes. 2nd edition. Van Nostrand

ReinhoId: New York (1950), p. 212.

Hirota, E. High-Resolution Spectroscopy of Transient Molecules, Springer

Series in Chernical Physics 40, Schafer, F.P., editor. Spnnger-Verlag: Berlin

(1985), p. 18.

Ibid, p. 20f.

Kroto, H. W. Molenrlar Rotation Spectra. John Wdey & Sons, Ltd. London

(1975), p. 229.

Merer, A J. Mol. Phys. 23(2), 309-3 15 (1972).

Brown, J. M.; Howard, B. J. Mol. Phys. 31(5), 15 2 7-1525 (1976).

Herzberg, G. Spectra of Diatomic Molecules, 2nd edition. Van Nostrand

Reinhold: New York (1 9SO), p. 23 1.

Zare, R N- Molecular Spectroscopy: Modern Research, Rao, K- N.; Mathews,

C. W., editors- Academic Press: New York (1972), 207-221.

Dirac, P. A M. The Principles of Quantum Mechanics, 4'" edition. Oxford

University Press: London (1958).

Kroto, H. W. Molecular Rotation Specha. John Wdey & Sons: London (1975),

p. 232.

Frosch, R- A-; Foley, K. M. Phys. Rev. 88(6), 1337 (1952).

Breit, G.; Doermann, F. W. Phys. Rev. 36, 1732 (1930).

Adam, A. G.; Azurna, Y.; Barry, J. A.; Merer, A. J.; Sassenberg, U.; Sc-hoder, J.

O.; Cheval, G-; Féménias, J. L. J. Chem- Phys. 100(9), 6240-6262 (1994).

Lefebvre-Brion, H.; Field, R. W. Perturbations in the Spectra of Diatomic

Molecules. Academic Press: Orlando (1986), p. 150.

La Paglia, S . R. Introductory Quantum Chemistry. Harper & Row: New York

(1971). Appendk E.

Merer, A. J., by permission.

Davidson, E. R. The WorId of Quantum C h e m i s ~ , Daudel, R and Puliman, 8-,

editors. Reidel Publishing Co.: Dordrecht (1974). p. 18.

Slater, J. C. Quanlum litleory of Atomic Structure. McGraw-EU Book

Company, Inc. New York (1960), p. 3 1.

Schaefer, Ha F. m. The Electrunic Structure of Atoms and Moledes: A Survey

of Rigorous Quantum Mechanicd Results. Addison-Wesley Publishing

Company: Reading, Massachusetts (1 972), p. 29.

Ref. 26, p. 32.

Ref. 27, p. 19 f.

Ref. 27, p. 37.

Ref. 27, p. 31 f.

Ref. 27, p. 4.

Ref. 27, p. 40f.

MOTECC. Modern Techniques in Cornputarional Chemistry. Clementi, E.,

editor. Escom: Leiden (1 990).

Davidson, E. R. MELD: A Many Electron Description. Input & Output

Documentafion fio m MOTECC-90. IBM Corp. Center for Scientzc and

Engineering Cornputations. Kingston: New York (1 WO), p. 4 17.

Davidson, E. R. MELD: A Mmy Electron Description fiom MOTECC. Modern

Techniques in Compuiaiional Chernistry, Clementi, E., editor. Escom: Leiden

(1990), p- 553.

Andzelrn, J.; Radzio, E.; Sdahub, D. R. J. Comp. Chem. 6(6), 520-532 (1985).

Slater, J. C . Phys. Rev- 36, 57 (2930).

Ref. 27, p- 56 f.

Steinborn, E. O. Methods in ComputationaZ MoZecuZar Physics, NATO AS1

series, no. 1 13. Diercksen, G. H. F. and Wilson, S., editors. ReideI Publishing

Co.: Dordrecht (2983), p. 37.

Boys, S. F. Proc. Roy. Soc. (London) A200, 542 (1950).

Hutinaga, S .; Andzelrn, 1; Klobukowski, M-; Radzio-Andzelrq E. Gmssian

Basis Sets for Molecular Calarla~ions. Elsevier: Amsterdam ( 1 984).

Poirier, R; Kari, R.; Csizmadia, 1. G. Hmdbook of Gaussim Basis Sets,

Elsevier: Amsterdam (1 98 5).

Ref. 36, p. 580.

Davidson, E. R. Me thoh in Computntional Molenrlar Physics, NATO AS1

series, no. 113. Diercksen, G. H. F. and Wilson, S., editors. Reidel Pubiishing

Co.: Dordrecht (1983), p. 97.

Hunt, W. J.; Goddard, W. A. Chem. Phys. Lett. 3, 414 (1969).

Ref. 27, p. 305.

Davidson, E. R. J. Chem. Phys. 48, 3169 (1968).

Nitzsche, L. E-; Davidson, E. R. J. Chem. Phys. 68, 3 1 O3 (1 978).

Cave, R J.; Davidson, E. R. J. Chem. Phys. 88(9), 5770-5778 (1988).

Ref. 36, p- 589.

Mattar, S. M.; Doleman, B. J. Chem. Phys. Lett. 216, 369-374 (1993).

Mattar, S. M.; Kennedy, C . Chem. Phys. Lett- 238,230-235 (1995).

S. Gerstenkorn, P. Luc, Atlas du qecire d'absorption de la molécule de I'iode

entre 14800- 20000 cm-', CNRS: Paris (1978).

S. Gerstenkorn, P. Luc, Rev. Phys. Appl. 14, 791 (1979) .

Palmer, B. A-; Keller, R A.; Engleman, Jr., R An Allm of Uranium Emission

Intensities in a Yellow Cathode Discharge. UNversity of California at Los

Alamos LA-825 1-MS Momd Report UC-34a (1 980).

Wiese, W. L.; Smith, M. W.; Miles, B-M. Atomic Transition Probabilities-

Volume I I NSRDS-NBS22 US. Government Print Office (1969).

Weltner, W. W.; Van Zee, R- J. Ann. Rev. Phys. Chem. 35, 29 1 (1984) .

Morse, M. D. Chem- Rev. 86, 1049 (1986).

Salahub, D. R. Ab Inilio Methodr in Quantum Chemise Vol. II. Lawley,

K. P., editor. John Wsey & Sons, Ltd.: New York (1987). p. 447.

Langhoff, S. R-; Bauschlicher, k, C. W. Ann. Rev. Phys. Chem. 39 (1988) 181.

Suchard, S. N. ; Metzer, J. E. Spectroscopie Data - I Heteronuclear Diatomic

Molecules. EI/Plenum: New York 1975.

Merer, A. J. Annu. Rev. Phys. Chem. 40, 407 (1 989).

Ram, R. S.; Bernath, P. F. J. Mol. Spect. 173, 158 (1995).

Rarn, R. S.; Bernath, P. F.; Davis, S.P. J. Chem. Phys. lO4(18), 6949 (1996).

Huber, K. P.; Herzberg, G. MolecuIar Spectra andMoZecuIar Structure, Vol.

IV. Van Nostrand Reinhld Inc. : New York (1 979).

Simard, B.; Mitchell, S. A.; Humphries, M. R.; Hackett, P. A. J. Mol. Spectrosc.

129, 186 (1988).

Ebben, M.; Meijer, G.; ter Meulen, J. J. Appl. Phys. B. 50, 3 5 (1990) .

Whitham, C. J.; Soep, B.; Visticot, J.-P.; Keller, A J. Chem Phys. 93, 99 1

(1990).

Barnes, M.; Fraser, M. M.; Hajigeorgiou, P. G.; Merer, A. J.; Rosner, S. D.

J. Mol. Spectrosc. 170, 449 (1995) .

Ref 9, p. 216.

Freindorf, M.; Marian, C . M.; Hess, B. A J. Chern. Phys. 99, 1215 (1993).

Ram, R. S.; Bernath, P. F.; Davis, S. P. J. Mol. Spect. 175, 1 (1996).

Van Vleck, J. H. Rev. Mod. Phys. 23,213 (1951) .

Zare, R. N.; SchmeltekopS A L.; Harrop, W. J.; Albritton, .J- Mol. Spectrosc.

46, 37 (1973).

Azurna, Y.; Barry, J. A-; Lyne, M. P. J.; Merer, A J.; Schroder, J. 0. J. Chem,

Phys. 91, 1 (1989).

Ref. 9, p. 153.

Ref. 9, p. 106.

Carrington, A.; Dyer, P. N.; Levy, D. H. J. Chem. Phys. 47, 1756 (1967).

Childs, W. J.; Steirnle, T. C. J. Chem. Phys. 88, 6168 (1988).

[8 11 Barnes, M.; Merer, A J.; Metha, G. F. J. Chem. Phys. 103, 8360 (1995).

[82] Balfour, W. J.; Cao, I.; Prasad, C. V. V.; Qian, C.X.W. J. Chem. Phys. 103,

4046 (1995).

[83] Steimle, T. C.; Shirley, J. E.; Jung, K. Y.; Russon, L. R.; Scurlock, C . T.

J. Mol. Spectrosc., 144, 27 (1990).

Fletcher, D. A.; Scurlock, C. T.; Jung, K- Y-; Steirnle, T. C. J. Chem. Phys. 99,

4288 (1993).

Bauschlicher Jr., C. W.; WaIch, S. P.; Langhoff, S . R Quantum Chernisfry: ï?ze

Challenge of Transition Me tak and Coordination Chemistry. VeiUard, A.,

editor. NATO AS1 Series, Senes C. Reidel: Dordrecht (1986).

Bauschlicher Jr., C. W.; Walch, S. P-; Partndge, EL J. Chem. Phys. 76, 1033

(1982).

Bauschlicher Jr., C. W.; Nelin, C. J.; Bagus, P. S. J. Chem. Phys., 82, 3265

(1985).

Moore, C. E. Atornic Energy Levels as Derivedfi.om the Analyses of Optical

Spectra, Volumes 1 and II. U.S. National Bureau of Standards, U.S. Government

Print Office NSRDS-NBS 35. Reprint of NBS Circdar 467 ( 1952).

FeIler, DD.; Davidson, E. R J. Chem, Phys. 80, 1006 (1984).

Engles, B.; Peyerirnhoff, S. D. Mol. Phys., 67, 583 (1989).

Feller, D.; Glendening, E. D.; McCullough, Jr., E. A.; Miller, R. J.

J. Chem. Phys. 99, 2829 (1993).

Mattar, S. M.; Bourque, C.; Kennedy, C. Chem. Phys. Lett. 227, 539 (1995).

Davidson, E. R-, unpublished results on VO. Personal communication to S.M.M.,

June 1996.

Mattar, S. M.; Hamilton, W. D.; Kingston, C. T. Chem. Phys. Lett 271, 125

(1997).

Wachters, A. J. H. J. Chem. Phys. 52, 1033 (1970).

Bauschlicher Jr., C . W.; Langhoq S. R; Barnes, L. A J. Chem. Phys. 91,2399

(1989).

van Duijneveldt, F. B. IBM Res. J. 16437, 945 (1972).

Lie, G. L.; Clementi, E. J. Chem. Phys, 60, 1275 (1974).

Ref. 9, pp 93-109.

Feller, D.; Davidson, E. R J. Chem. Phys. 84, 3977 (1981).

Kasai, P. H.; Weltner Jr., W. J. Chem. Phys. 43, 2553 (1965).

Weltner Jr., W.; McLeod, D.; Kasai, P. H. J. Chem. Phys. 99,3 172 (1967).

Bauschlicher, Jr., C. W.; Langhoff, S. R. J. Chem. Phys. 85, 5936 (1986).

Mattar, S. M. 5. Phys. Chem. 97, 3171 (1993).

Bauschlicher, Jr., C. W. Chem- Phys. Lett. 100, 515 (1983).

Harrison, J. F. J. Phys. Chem. 100, 3513 (1996)-

Adam, A. G. Private communication.

Mattar, S. M. Manuscript submitted for publication.

Ref. 9, p. 168.

APPENDIX

Tuble A-l Observed Iine frequencies for the 3@4 - CoF electronic systema

a. Values in cm-'.

Table A-2 Observed Iine frequencies for the 'G3 - X3G3 CoF electronic system '

blended

blended

bIended

18780-032

18779.771

1 8779,486

18779.143

18778.805

18778-365

18777-92 1

Values in cm".

Table A-3 O b s e d Iine frequencies for the 'a, - X3rP, CoF electronic system'

- a. Values in cm-'.

b. No Q-branch lines were resolved in this system.

160

Table A-4 Line frequencies for the combinedfiîting of 3@4 - X3@& - Xe3, 3@z - X 3 q and '<P, - X3G4 CoF electronic systemsa

Table A-5 Hyperfine linefiequencies for the '!Pd - X3e4 CoF electronic system'

b. * denotes AF=+l denotes bF=-1

Table A-6 Hyperfne Iine frequencies for the 'a3 - CoF elecironic systema

in cm-'.

b. * denotes AF=+l denotes AF=-1