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dissertation entitled

THE QUADRUPOLE MOMENTS OF THE FIRST AND SECOND ROW
HOMONUCLEAR DIATOMICS AND THE SPECTROSCOPIC
PROPERTIES OF METAL LITHIDES

presented by

Daniel B. Lawson

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1M chlHC/DateDuopG5—p.“

THE QUADRUPOLE MOMENTS OF THE FIRST AND SECOND ROW
HOMONUCLEAR DIATOMICS

AND
THE SPECTROSCOPIC PROPERTIES OF METAL LITHIDES
By

Daniel B. Lawson

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1997

ABSTRACT

PART I. ON THE DISTANCE DEPENDENCE OF THE QUADRUPOLE
MOMENTS OF THE FIRST AND SECOND ROW HOMONUCLEAR
DIATOMICS

PART II. THE ELECTRONIC STRUCTURE OF AlLi, ScLi, TiLi, VLi, CrLi, CuLi
AND THEIR POSITIVE IONS

By

Daniel B. Lawson

In Part I of this thesis, we estimate the complete basis set limits for the
Hartree-Fock, MP2, CASSCF, and CASSCF + 1 + 2 wavefimctions for the first and
second row homonuclear diatomics and calculate the molecular quadrupole moment as
a fiinction of bond length. Our recommended values for (v = 0, J = 0), compare
favorably to the current experimental values and previous high-level calculations. To
aid in the analysis of the relationship between the molecule’s electronic structure and
quadrupole moment, we introduce the concept of a quadrupole moment density that
permits one to write the molecular quadrupole moment as a sum of the separated
atoms quadrupole and a purely molecular contribution. The quadrupole density
provides a (reference state dependent) means of determining the contribution to (-9
fi'om various regions in the molecule and gives considerable insight into the
relationship between the electron density and the magnitude and sign of G), and it
allows a detailed assessment of the contribution of electron correlation to G).

In Part II, we begin with a study of the ground and low-lying excited states of
the transition-metal lithides ScLi, TiLi, VLi, CrLi, and CuLi and their mono-positive

ions, and report bond energies, bond lengths, and vibrational frequencies, calculated

using MRCI and ACPF techniques. The ground states of ScLi (3A) and TiLi (4CD)
trace their lineage to the ground 52dn configuration of the transition element and are
weakly bound. The ground states of CrLi (62+) and CuLi (12+) trace their lineage to
the ground sdn+1 configurations and are more strongly bound. VLi (5A) traces its
lineage to the excited sd4 configuration and is strongly bound, relative to this
asymptote. The positive ions ScLi+ (2A), TiLi+ (3CD), VLi+ (4A), and CrLi+ (72+)
are more strongly bound than their neutral precursors, while CuLi+ (223+) is more
weakly bound than its neutral precursor. In general, the structure of those transition-
metal lithides that dissociate to the 52dn asymptote are very sensitive to electron

correlation, while those that dissociate to an sdn+1 asymptote are less sensitive. The
bonding in the positive ions is primarily ionic. And, in the last chapter of Part II, the
ground state (12*) and several low lying excited states (including the 12' , 32' , lI'I , and
3H ) of LiAl are characterized by MCSCF, MRCI, and ACPF techniques using a large

correlation consistent basis set.

To my wife, Jeanne

Acknowledgments

I would like to take this opportunity to thank Dr. James F. Harrison for his
unending support and guidance during the work that led to this thesis. I am indeed
grateful to him both as a mentor and fiiend. I would also like to thank my guidance
committee, Dr. Ned Jackson, Dr. Gary Blanchard, and Dr. Katherine Hunt. I would
like to acknowledge the faculty and staff of Michigan State University’s Department of
Chemistry. Special thanks to Paul Reed and Dr. Tom Atkinson for maintaining the
departmental computers along with answering my many questions. I would like to
thank the many fi'iends that I have acquired at Michigan State University, especially
those of the Harrison and Hunt research groups. Without their contributions and
support, I certainly would not have completed my studies.

Finally, I would like to thank my entire family. They were helpful, supportive,

and inspirational throughout my years in graduate school.

TABLE OF CONTENTS

List of Tables

List of Figures

Keys to Symbols and Abbreviations
PART I

CHAPTER 1
INTRODUCTION

CHAPTER 2
ON THE DISTANCE DEPENDENCE OF THE QUADRUPOLE
MOMENTS or H2, N2, 02, AND F2

A. Introduction

B. Discussion

C. MPZ Results

D. Hellmann-Feynman Theorem

E. Quadrupole Moment Functions

F. H2

G. On the Sign of the Quadrupole Moment

H. Correlation Effects on (93,01 and 63.01

1. Conclusion
J. References

CHAPTER 3
THE QUADRUPOLE MOMENT OF P2, 82, AND C];

A. Introduction

B. Computational Technique

C. Discussion

D. On the Distance Dependence of $2 and C];
E. The Quadrupole moment of N2 and P2

ix

xix

25
26
28
50
52

52

57
58

59
62
63
63
64
79

F. Conclusion 95

G. References 100
CHAPTER 4
THE QUADRUPOLE MOMENT OF H2, Liz, AND Naz 102
A. Introduction 103
B. Methods 103
C. Results 104
D. H2, Liz, and Na; 112
E. The Quadrupole Density of Liz and Na; 112
F. Conclusion 1 19
APPENDIX A
INTRODUCTION T 0 THE QUADRUPOLE MOMENT 121
A Multipole Moments 122
B. Experimental Evaluation 125
C. Theoretical Evaluation 126
l. Ab-Initio Methods 126
2. Expectation Values 128
3. Point Charge Method 129
4. Finite Field Method 131
D. References 133
APPENDIX B
TABLES OF BASIS SETS 135
A. Description 136
B. References 141
APPENDIX C
TABLES OF PROPERTIES 189
APPENDIX D
TABLES OF DISTANCE DEPENDENCE OF PROPERTIES 202
PART 11
CHAPTER 5
THE ELECTRONIC STRUCTURE OF ScLi, TiLi, VLi, CrLi, AND CuLi
AND THEIR POSITIVE IONS 232

A. Introduction 233

vii

B. Technical Details and WavefiJnction Construction 234
C. Neutral Lithides 236
D. Positive Lithides 252
E. Previous Work 261
F. Conclusions 264
G. References 266

CHAPTER 6

THE ELECTRONIC STRUCTURE OF AlLi AND ITS POSITIVE ION 267
A. Introduction 268
B. Computational Methods 268
C. Results 271
D. Excited States 274
E. Positive Ion 277
F. AlLi and ScLi 277
G. Conclusions 280
H. References 281

viii

LIST OF TABLES

Table

CHAPTER 2

2-1.

2-3.

2-4.

2-5.

2-6.

2-7.

2-8.

2-9.

Equilibrium value of the first and second derivatives of the molecular
quadrupole moment function for H2, N2, 02, and F2, calculated with

various wavefirnctions.

. Selection of H2 quadrupole moments.

Selection of N2 quadrupole moments.

Comparison of experimental and theoretical quadrupole-moment
derivatives.

Selection of 02 quadrupole moments.
Selection of F2 quadrupole moments.
CASSCF+1+2 results for Ropt and 9.

Comparison of MPZ and CASSCF+1+2 results.

Expectation Value quadrupole moment and Point Charge quadrupole
moment.

CHAPTER 3

3-1.

Equilibrium value of the first and second derivatives of the molecular
quadrupole moment function for P2, 82, and C12, calculated with various

wavefunctions.

Selection of C12 quadrupole moments.

. CASSCF + 1 + 2 results for Ropt and (-9.

Comparison of MP2 and CASSCF + 1 + 2 results.

ix

Page

11

13

18

22
23
25

26

27

66

78
78

100

CHAPTER 4

4-1.

First and second derivatives of O(R) near Rap, using various models and the
V52 basis for Li; and the VQZ basis for N82.

CHAPTER 5

5-1.

5-3.

5-4.

5-5.

5-6.

Energy separation between the lowest terms of the 52d" and sdn+1
configurations. Calculated (MRCI) and experimental separations are
presented with the experimental results connected by a dotted line.

. Dissociation energy, bond length, and vibrational frequency for states of

ScLi, TiLi, VLi, CrLi, and CuLi asymptotic to the transition metal 4sz3d“
and 433dn+1 atomic states.

Estimate of Exchange Energy Loss.
Calculated characteristics of various Transition-Metal Lithide Cations.
Population Analysis of M-Li+.

Comparison of Experimental and Theoretical Results.

W

6-1.

6-2.

6-3 .

6-4.

6-5.

6-6.

Adiabatic ionization potentials as S => S + e'.
Characteristics of low lying AlLi electronic states.
Population analysis of low lying states of AlLi.
Calculated properties of MRCI roots.

Properties of low lying states of the positive ion.

Population analysis of AlLi“.

107

238

239

243

257

260

263

270

272

274

276

279

279

LIST OF FIGURES

Figure Page
CHAPTER 2
1. Model dependence of G) for H2 as a function of basis set. 6

2. Distance dependence of G) for H2 calculated with various models
using the aug-cc-vaz basis. 8

3. Model dependence of G) for N2 as a function of basis set. 12

4. Distance dependence of O for N2 calculated with various models
using the aug-cc-pqu basis. 15

5. Model dependence of G) for 02 as a function of basis set. 17

6. Distance dependence of O for 02 calculated with various models
using the aug-cc-pqu basis. 19

7. Model dependence of O for F2 using various basis sets. 21

8. Distance dependence of G) for F2 calculated with various models
using the aug-cc-pqu basis. 24

9. MRCI potential energy curves for N2, 02, and F 2 using the

aug-cc-pqu basis and a 10 CSF, CASSCF potential energy curve
for H2 using the aug-cc-vaz basis. 29

10. Molecular quadrupole moments of H2, N2, 02, and F2 as a
function of bond length. 30

11. Distance dependence of the molecular quadrupole moments of N2,
02, and F2 partitioned into the 95 and On components. 33

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

Distance dependence of the quadrupole moment of N2 partitioned

into its 6:0, and (-920, components.

Distance dependence of 6,7,0] for 02 partitioned into its (920, and

0:0, components.

Distance dependence of Gmol for F2 partitioned into its (93,0, and

(9,3,0, components.

Nodes of - 1/2 (3 cos2 9-1) (solid lines) and parabolas separating
the Berlin bonding and antibonding regions.

Electron-density difference and the associated quadrupole-density
contours for N2 at 2.1 a0 calculated with the aug-cc-pvtz basis. The
contour values are 2N*10'3(atomic units) with N = 0-6. The red
contours are negative, the blue positive, and the nodes are in black.

811., and 59., for N2 at 2.1 a0 calculated with MRCI using the
aug-cc-pvtz basis. Contours values and colors are same as in
Figure 16.

6n, and 59,. for N2 at 2.1 a0 calculated with MRCI using the
aug-cc-pvtz basis. Contours values and colors are same as in
Figure 16.

Electron-density difference and the associated quadrupole-density
contours for 02 at 2.3 a0 calculated with the aug-cc-pvtz basis.
Contours values and colors are same as in Figure 16.

5n, and 59., for 02 at 2.3 a0 calculated with MRCI using the
aug-cc-pvtz basis. Contours values and colors are same as in
Figure 16.

6n. and 56),. for 02 at 2.3 a0 calculated with MRCI using the
aug-cc-pvtz basis. Contours values and colors are same as in
Figure 16.

Electron-density difference and the associated quadrupole-density

contours for F2 at 2.7 a0 calculated with the aug-cc-pvtz basis.
Contours values and colors are same as in Figure 16.

xii

36

37

38

39

41

42

43

44

45

46

47

23.

24.

25.

26.

27.

28.

5n, and 59,, for F2 at 2.7 a0 calculated with MRCI using the
aug-cc-pvtz basis. Contours values and colors are same as in
Figure 16.

6n, and 69,. for F2 at 2.7 a0 calculated with MRCI using the
aug-cc-pvtz basis. Contours values and colors are same as in
Figure 16.

Electron-density difference, 6n, and the associated quadrupole
density, 86), contours for H2 at 1.4 a0 calculated with a 10 CSF

MCSCF wavefunction and the aug-cc-vaz basis. Contour
values and conventions are as in Figure 16.

Effect of electron correlation on the magnitude and distance
dependence of D, Go and On for N2.

Effect of electron correlation on the magnitude and distance
dependence of O, (-96 and On for 02.

Effect of electron correlation on the magnitude and distance
dependence of (-9, (9° and On for F 2.

HAPTER 3

Model dependence of O for P2 as a fiinction of basis set.

Distance dependence of G) for P2 calculated with various models
using the aug-cc-pqu.

The quadrupole moment of P2 as function of internuclear
separation with various models using the aug-cc-pqu basis.

Model dependence of O for S2 as a function of basis set.

Distance dependence of O for S2 calculated with various models
using the aug-cc-pqu.

The quadrupole moment of 82 as function of internuclear separation
with various models using the aug-cc-pqu basis.

Model dependence of G) for C12 as a firnction of basis set.

xiii

48

49

51

54

55

56

65

67

68

70

71

72

73

8. Distance dependence of O for C12 calculated with various models
using the aug-cc-pqu.

9. The quadrupole moment of C12 as function of internuclear separation
with various models using the aug-cc-pqu basis.

10. CASSCF+1+2 potential energy curves for N2, 02, F2, P2, S2, and
C12 using the aug-cc-pqu basis.

11. Molecular quadrupole moments of N2, 02, F2, P2, S2, and C12
using the CASSCF+1+2 and the aug-cc-pqu basis.

12. Distance dependence of the sigma component of the quadrupole
moment of N2, 02, F2, P2, S2, and C12.

13. Distance dependence of the pi component of the quadrupole
moment Osz, 02, F2, P2, 82, and C12.

14. Distance dependence of the quadrupole moment of P2 partitioned
into its 9°...“ and Ohm. components.

15. Distance dependence of the quadrupole moment of S2 partitioned
into its 9°mor and 0"”! components.

16. Distance dependence of the quadrupole moment of C12 partitioned
into its 9°“. and 6"”; components.

17. Electron-density difference and the associated quadrupole-
density contours for N2, at 2.1 a0 calculated with MRCI wavefunction
and the aug-cc-pqu. Contour values are 0.0, $0.001, $0.002,
$0.004, $0.008, $0.016, $0.032, and $0.064. Blue represent
positive contours and red represents negative.

18. Electron-density difl‘erence and the associated quadrupole-
density contours for 02, at 2.3 a0 calculated with MRCI wavefiinction
and the aug-cc-pqu. Contour values are 0.0, $0.001, $0.002,
$0.004, $0.008, $0.016, $0.032, and $0.064. Blue represent
positive contours and red represents negative.

19. Electron-density difference and the associated quadrupole-density
contours for F2, at 2.6 a0 calculated with MRCI wavefimction and
the aug-cc-pqu. Contour values are 0.0, $0.001, $0.002, $0.004,
$0.008, $0.016, $0.032, and $0.064. Blue represent positive
contours and red represents negative.

xiv

76

77

80

81

83

84

86

87

88

89

9O

91

20. Electron-density difference and the associated quadrupole-density

21.

22.

23.

24.

25.

26.

contours for P2, at 3.6 a0 calculated with MRCI wavefunction and
the aug-cc—pqu. Contour values are 0.0, $0.001, $0.002, $0.004,
$0.008, $0.016, $0.032, and $0.064. Blue represent positive
contours and red represents negative.

Electron-density difference and the associated quadrupole-density
contours for S2, at 3.6 a0 calculated with MRCI wavefunction and
the aug-cc-pqu. Contour values are 0.0, $0.001, $0.002, $0.004,
$0.008, $0.016, $0.032, and $0.064. Blue represent positive
contours and red represents negative.

Electron-density difi‘erence and the associated quadrupole-density
contours for C12, at 3.7 a0 calculated with MRCI wavefirnction
and the aug-cc-pqu. Contour values are 0.0, $0.001, $0.002,
$0.004, $0.008, $0.016, $0.032, and $0.064. Blue represent
positive contours and red represents negative.

CASSCF+1+2 potential energy curves for N2 and P2 using the
aug-cc-pqu basis.

Distance dependence of the molecular quadrupole moments of N2
and P2 using the CASSCF+1+2 and the aug-cc-pqu basis.

Distance dependence of the molecular quadrupole moment of N2
and P2 partitioned into 6)., and O, components.

Distance dependence of the molecular quadrupole moment of
N2 and P2 partitioned into ®°moi and 9",...“ components.

CHAPTER 4

1.

Distance dependence of (-9 for Li2 near Rap. calculated with various
models using the VSZ basis.

Distance dependence of (-9 at large R for Li2 calculated with various
models using the V52 basis.

Distance dependence of O for Na2 near Rap. calculated with various
models using the VQZ basis.

Distance dependence of G) at large R for Na2 calculated with
various models using the VQZ basis.

92

93

94

96

97

98

99

105

108

110

111

. CASSCF+1+2 potential energy curves for H2, Li2, and Na2 using

the aug-cc-vaz, ASZ, and VQZ basis, respectively.

Distance dependence of G) for H2, Li2, and Na2 using the
CASSCF+1+2 wavefunction and the aug-cc-vaz, VSZ, and VQZ
basis, respectively.

Electron-density difference and the associated quadrupole-density
contours for H2, at 1.4 a0 calculated with CASSCF wavefirnction
and the aug-cc-vaz. Contour values are $00004, $0.001, $0.002,
$0.004, $0.008, $0.016. Blue represent positive contours and red
represents negative.

Electron-density difference and the associated quadrupole-density
contours for Li2, at 5.0 a0 calculated with CASSCF wavefirnction
and the VSZ. Contour values are $00004, $0.001, $0.002, $0.004,
$0.008, $0.016. Blue contours represent positive density and red
contours represent negative density.

Electron-density difference and the associated quadrupole-density
contours for Na2, at 6.0 a0 calculated with CASSCF wavefilnction
and the VQZ. Contour values are $00004, $0.001, $0.002,
$0.004, $0.008, $0.016. Blue contours represent positive density
and red contours represent negative density.

APPENDIX A

1.

The potential at an arbitrary point P created by a static molecular
charge distribution.

2. Schematic representation of Point Charge method.

APPENDIX B

1.

Comparison of O2 RKR data to CASSCF+1+2 potential energy
curve using the cc-pvtz basis.

Comparison of the ROHF and UMP2 total energy for 02 using
the cc-pvtz and the aug-cc-pvtz basis.

. Comparison of the ROHF and UMP2 quadrupole moment of 02

using the cc-pvtz and the aug-cc-pvtz basis.

113

114

115

116

117

122

131

138

139

140

CHAPTER 5

1. The energy separation between the lowest terms of the 32d“ and

sdn'*‘1 configurations. Calculated (MRCI) and experimental
separations are presented with the experimental results connected
by a dotted line. 235

2. The orbitals involved in the MCSCF wavefiinctions for the neutral
and mono-positive transition-metal lithides. 237

3. The calculated (MCSCF) potential energy curves for the lowest
quartet and doublet states of TiLi that correlate with the ground-state

products, Ti (3F) + Li (ZS). 239

4. The calculated (MCSCF+1+2) potential energy curves for the lowest
quartet and doublet states of TiLi that correlate with the ground-state

products, Ti (3F) + Li (2S). 240

5. The 4CD ground-state potential curve of TiLi, calculated with the
MCSCF, MCSCF+1+2 and ACPF techniques. 241

6. The 62"" ground-state potential curves of CrLi, calculated with the
MCSCF, MCSCF+1+2, and ACPF techniques. 242

7. The two 5A states of VLi that correlate with the s2d3 and sd4

asymptotes. 247
8. A comparison of the neutral and positive ion states of ScLi, calculated

at the MRCI level. 248
9. A comparison of the neutral and positive ion states of TiLi, calculated

at the MRCI level. 249
10. A comparison of the neutral and positive ion states of VLi, calculated

at the MRCI level. 250
11. A comparison of the neutral and positive ion states of CrLi, calculated

at the MRCI level. 251
12. The MRCI ground-state potential curves for the neutral lithides. 253

13. The experimental energy levels for the low-lying asymptotic products
M+Li, M++Li, M+Li+, and M*+Li+. 254

1 4. The 4A ground-state potential curves of VLi+, calculated with the
MCSCF, MCSCF+1+2, and ACPF techniques. 255

1 5. The MRCI ground-state potential curves for the mono-positive
lithides. 256

1 6. Variation of the calculated (MRCI) De’s of the mono-positive lithides

with equilibrium bond length. 262
CHAPTER 6
1. Low lying excited atom energies. 269

2. Potential energy curves of the low lying electronic states of AlLi
using MCSCF and MRCI. 273

3. Potential energy curves of the low lying electronic states of AlLi
using MRCI. 275

4. Potential energy curves of the low lying electronic states of AlLi+
using MRCI. 278

xviii

Smbols

2
e/ao

2
eao

e

ea0

e/ai

Abbreviations
ACPF
CASSCF
CASSCF+1+2

CBS
CISD
CSF
EEL
IVIPZ
MCSCF
MCI

RHF/UHF
ROHF

Ra
Ra

KEYS TO SYMBOLS AND ABBREVIATIONS

Quadrupole Moment
Sigma component of the molecular quadrupole

Sigma component of the molecular quadrupole moment

referenced to its asymptotic atomic contribution.
Pi component of the molecular quadrupole

Pi component of the molecular quadrupole moment
referenced to its asymptotic atomic contribution.

Field Gradient
Atomic Units of Length

Atomic Units of Energy

Atomic Units of Quadrupole Moment

Atomic Units of Quadrupole Moment Second
Derivative

Atomic Units of Quadrupole Moment First
Derivative

Atomic Units of Field Gradient

Averaged Coupled Pair Functional

Complete Active Space Self Consistent Field
Complete Active Space Self Consistent Field
Singles and Doubles Configuration Interaction
Complete Basis Limit

Single and Double Configuration Interaction
Configuration State Functions

Exchange Energy Loss
Second Order Moller-Plesset Perturbation

Multi-Configuration Self Consistent Field
Multi-Reference Singles and Doubles Configuration

Interaction
Restricted/Unrestricted Hartree-Fock

Restricted Open Shell Hartree-Fock

Optimal Model Separation
Experimental Separation

xix

CHAPTER 1

Introduction

A. Introduction

The purpose of Part I of this thesis is to discuss the calculation and
interpretation of the quadrupole moments of the first and second row homonuclear
diatomics, and their distance dependence. The primary interest is in developing a
qualitative understanding of the relationship between a molecule’s quadrupole moment
and electronic structure, and, in order to do so reliably, we have explored the
sensitivity of the calculated quadrupole moment to basis sets as well as electron
correlation. Using Self-Consistent Field (SCF), Multi-Configurational Self-Consistent
Field (MCSCF/CASSCF), Moller-Plesset Second Order Perturbation (MP2), and
Multireference Configuration Interaction (MRCI/CASSCF+1+2), we calculate
quadrupole moments, as a function of internuclear separation, for a sequence of basis
sets obtained from the Dunning Augmented Correlation Consistent basis sets by
deleting g and higher symmetries.

The goal is to provide accurate calculated quadrupole moments of the first and
second row homonuclear diatomics, an interpretation of the magnitude and sign of the
quadrupole moment, the effects of correlation, and the distance dependence of the
quadrupole moment.

The work in Part I divided into three chapters. Chapter H is a description of
the distance dependence of the quadrupole moment for H2, N2, 02, and F2. The next
Chapter is a discussion of the quadrupole moments and distance dependence of P2, S2,

amd C12, and a comparison with their first row conjugates. In the fourth chapter of
ths work we discuss and compare the quadrupole moment and distance dependence

Osz, Li2, and N32.

Appendix A outlines the quadrupole moment as a molecular property by giving
a derivation of the multipole moments, and a description of how multipole moments
are determined. Appendix B contains the basis sets used in these calculations.
Appendix C contains the dissociation energy, quadrupole moment and field gradient as
firnction of internuclear separation. Appendix D gives complete tables of the
Homonuclear Diatomics for which calculations were done, but may not have been
discussed.

In part H of this thesis, we describe the calculations of the excited states and
positive ions of certain metal lithides. Chapter V describes the ground and low-lying
excited states of the transition-metal lithides ScLi, TiLi, VLi, CrLi, and CuLi and their
mono-positive ions, and report bond energies, bond lengths, and vibrational
fi-equencies. Chapter VI includes the ground and excited states of AlLi and its positive

ions.

CHAPTER 2

On the Distance Dependence and Spatial Distribution of the Molecular
Quadrupole Moments of H2, N2, 02, and F2

A. Introduction

We first discuss the quadrupole moments of the titled molecules at the
calculated equilibrium separations and compare these with experiment and previous
calculations. Then, we examine the quadrupole moment functions (variation of the
quadrupole moment with the internuclear separation), using MCSCF and MRCI
techniques. Finally, we develop an interpretation of the molecular quadrupole moment
as the sum of atomic contributions associated with the free (non-interacting) atoms
and a molecular contribution that depends on the shift in the electron density due to
molecule formation (the deformation density). Using the deformation density, we
define a quadrupole density whose integral is the molecular contribution to the
molecular quadrupole moment. The distance dependence of the quadrupole moment,
the differing contributions to G from various regions of the charge distribution, and
the role of electron correlation in determining O are analyzed in terms of the

quadrupole density.

B. Discussion

Let’s look, first, at the H2 results. Figure 1 shows the graphical representation
of the quadrupole data in Table C-1 of Appendix C. Several characteristics are
evident. First, the convergence to the CBS limit is only monotonic after the aug-cc-
pvdz basis and, accordingly, we use the last three basis sets to extrapolate to the CBS
limit. Within these three basis sets, increasing the flexibility of the basis decreases, as
does increasing the level of correlation. The aug-cc—vaz basis set results are close to

the CBS limit for each model wavefirnction.

0.50- 1 +
H2( 2g )

0.49 - / \

 

 

i.
5
[i
a

 

 

 

8?
m0
3 - CISD limit
ca 0.45 A
0.44 A . ——I— RHF
- —~— MP2
0.43 - —A— CISD
A
0.42 I | T I
2 3 4
aug-cc-vaz

Figure 1. Model dependence of G) for H2 as a firnction of basis set.

Note that the CBS-CISD limit predicts an equilibrium bond length Rom =

1.4010 a0 and a dissociation energy De = 4.740 ev in excellent agreement with the

experimental values1 1.4011 a0 and 4.749 ev. Our CBS-CISD value of O is + 0.455
cog and agrees with the essentially exact theoretical result2 of + 0.457 eag' . The

experimental value for O, + 0.460 $ 0.021 eag, is an indirect value assembled from

the magnetic susceptibility anisotropy and the molecular g value, assuming a

vibrationless H2.2‘4 As such, it agrees well with our CBS-CISD estimate of the
vibrationless G), + 0.455 eag . Buckingham and Cordle5 have estimated the vibrational

(v = O, J = 1) value to be 0.4853 eag. Figure 2 shows the variation of ('9 with R for

the SCF and CISD wavefilnctions, using the aug-cc-pqu basis. These data were fit to
a quadratic function of the form

(R—Rop,)+(@—]o M (3)

90?) = ®(R°P') + (3%) dR2 2

op! p,
and the parameters are reported in Table 2-1. These data are useful in correcting our

calculated values to other bond lengths for comparison with previous calculations.

Additionally, they allow us to estimate the vibrational dependence of 6), using the

formula5:6

Be ago), d9
(9v = Gopt +2)— 3[1+ B: )Rop (ER?)

e

 

+46%) (+4
R

0P‘

e Rap!

and experimental values1 for the spectroscopic parameters one, we, and Be. Using

Equation 3 and our Rap, for the HF and CISD wavefunctions, we corrected 9 to the

 

 

 

0.7 - 1 + I.
2
. H2( g )
6) VS. R-R .5
opt
0.6 -
0.5 -
$0
(0 .
3
e
0.4 -
0.3 — ------ - ------ SCF
l " ------'~-CASSCF
' —A— CISD
0.2 1 l . | ' l ' l
-0.4 -02 0.0 02 0.4
R - Ropt (a0)

Figure 2. Distance dependence of O for H2 calculated with various models using the
aug-cc-vaz basis.

Table 2-1. Equilibrium value of the first and second derivatives of the molecular
quadrupole moment function for H2, N2, 02, and F2, calculated with various

wavefunctions.

 

 

. (99-) (mo) [$2] (8)
Molecule Wavefunctron Ropt(a0) dR Rap! Rap!
H2 SCF 1.3865 0.6064 0.2622
H2 CISD 1.4022 0.5174 0.1028
N2 SCF 2.0133 1.4017 0.5594
N2 MCSCF 2.0825 0.9481 0.0548
N2 MRCI 2.0806 0.9593 0.0892
02 ROSCF 2.1747 1.4797 — 0.0744
02 MCSCF 2.2939 1.3720 — 0.1998
02 MRCI 2.2864 1.4738 — 0.2160
F2 SCF 2.5071 1.1656 — 0.6212
F2 MCSCF 2.7632 1.1399 — 0.9718
F2 MRCI 2.6825 1.2134 — 0.9142

experimental Rexp = 1.4011 a0 at which most other calculations were done. Our HF
result at Rexp is 0.4937 eag , in excellent agreement with the numerical HF result
(0.4934 egg) of Larksonen, Pyykkc‘i, and Sundholrn.7 Using (4) and the data in Table
2-1, we estimate the vibrational dependent CBS-CISD value of O to be

(9(1772 ; v) = +0455 + 0.051(v + i)

10

@(Hz ;v = 0) = 0.481ea3‘ , which is in good agreement with the v = 0 Wolniewicz8
value, 0.484 eag and the v = 0, J = 1 value, 0.4853 eag, of Buckingham and Cordle.5
There are a vast number of calculationsz+3+9'14 of O (H2), and we collect, in Table 2-2,

a representative collection of ab initio values for O (H2), along with the experimental
values and our CBS-HF and CBS-CISD results.

From Figure 2, we see that the slope of 9 (Hz) around Rom is positive; and, as
we will see, this is also true for N2, 02, and F2. Accordingly, to the extent that the HF
model limit for Rem is less than Rexp, the HF model limit for G) is always less than the
HF value calculated at Rexp. Note that electron correlation decreases 6) relative to the

HF value. Precisely how much depends on whether one compares the HF and

correlated value at the experimental bond length or at the optimal bond length (Rom)

corresponding to each model. Since correlation corrections to the HF wavefunction

are responsible for changing the predicted Rom, measuring correlation

effects relative to the experimental bond length obscures this important efi‘ect. For
example, 0 (CISD) at the experimental bond length is 7.8 percent smaller than D (HF)
at this bond length, while G) (CISD) and 9 (HF) differ by 6.0 percent when each is

referred to its model limit Rom.
Our N2 results are summarized in Table C-6 and Figure 3 and compared with
selected calculations12+15‘25 and experiment826+27 in Table 2-3. As with H2,

increasing the quality of the basis set within a model reduces the calculated O, and

adding correlation decreases 9, relative to the SCF values. Our CBS-HF limit is

L'i

Table 2-2. Selection of H2 quadrupole moments.

11

 

If

 

R(a0) @(eag ) Reference Comment
T4 0.493 1 1 Hartree Fock limit
1 - 4 0.4934224 7 Numerical HF
1.3863 0.4845 This (333-1413; Ropt
Work
1.4016 0.4937 This CBS-HF; Exp. Rfixp
Work
1.4 0.457 9 essentially exact wavefunction
1 .40 1 0.437 12 numerical DFT
1.40 0.4438 13 MP4 Sadlej15 basis
1.40 0.4414 13 CISD Sadlej basis
1 .3 895 0.4649 This CBs-MP2 R0
pt
Work
1 .
.4016 0.4712 This CBS-MPZ Exp. Rexp
Work
1.4010 0.4552 This CBS-CISD RO
pt
Work
vibrational average 0.516 This CBS-HF; v = 0, J = 0
Work
Vibrational average 0.481 This CBS-CISD; v = 0, J = 0
Work
vibrational average 0.477 14 essentially exact wf-integrate radial
wavefirnction
experimental 0.460 a 0.021 2 derived from exp. data -
non-vibrating molecule
experimental 0.485 5 vibrational average of magnetic

\

anisotropy and g factor; v = 0, J = 1

12

094-. I N (12 +)
-0.96- 2 g
-0.98-
-1.00-
-1.02-i I mm
-1044
4.06-
-1.08-
4.104
4.12;

-1.14 _ \\ "‘6 .......................... o MRCI 11ml!

-1.16 - \
—:I

'1-13 j i. MP2 limit
-1.20 - i
-1.22 -
-1 .24 -

'1.26 I l

2 3 4
aug-cc-vaz

 

 

G) (eaoz)

 

 

 

 

 

—1
-

Figure 3. Model dependence of O for N2 as a function of basis set.

13

Table 2-3. Selection of N2 quadrupole moments.

 

 

R(aO) @(eag ) Reference Comment
2.07432 — 0.9310 15 numerical HF
2.068 — 0.9400 16,17 numerical HF
2.07432 — 0.9054 18 SCF large basis
2.07430 — 0.9285 19 SCF large basis
2.068 — 0.937 20 SCF
2.0133 — 1.0158 This Work SCF at Ropt
2.07435 — 0.9306 This Work CBS-SCF limit
2.075 — 1.137 12 numerical DFT
2.068 — 1.1426 21 numerical HFS
2.07430 — 1.1289 22 large basis set DFT
2.068 — 1.15 23 numerical HFS
2.07432 — 1.1131 18 SDQ-MPPT(4); 6s4p3d1f
2.07430 - 1.0905 19 MRSD-CI
2.068 — 1.154 20 MRSD-CI
2.068 — 1.16865 24 CCSD-Sadlej’s15 Ss4p2d basis
2.0856 — 1.1755 This Work CBS-CASSCF; Ropt
2.0810 — 1.1270 This Work CBS-CASSCF+1+2; Rom
2.07432 — 1.1334 This Work CBS-CASSCF+1+2; Rexp
vibrational average — 1.118 This Work CBS-CASSCF+1+2;v=0, J=0
vibrational average — 1.1557 25 CCSD, v=0, J=0; Sadlej’s

5s4p2d basis

experimental — 1.09 $ 0.07 26 Optical Birefringence
experimental — 1.05 $ 0.06 27 Optical Birefiingence

 

l4

-l.0158 eag at Ropt of2.0133 a0 and — 0.9306 gag at R = 2.07432 a0, in excellent

agreement with the numerical HF result20 of — 0.9310 eag at this bond length. The

convergence of the MP2, MCSCF, and MRCI results all suggest that the aug-cc-pqu

basis produces a quadrupole moment that differs from the CBS limit by less than 0.2

%, suggesting that the individual CASSCF and CASSCF+1+2 values of — 1.1753 cog

and -- 1.1247 eag are near the model limit. The distance dependence of 9 around
Rep. is shown in Figure 4 for the SCF, MCSCF, and‘MRCI models, using the aug-cc-
pqu basis. Note the similarity between the MCSCF and the MRCI 0 versus R
curves. The SCF curve has significantly larger first and second derivatives, and these

are in good agreement with calculations by Truhlar28 and Maroules and Bishop.29
Using the MRCI derivatives in Table 2-1, we estimate the vibrational dependence of

the CBS-CASSCF+1+2 quadrupole moment as

®(N2;v) = —1.1247+0.0137 (6+1)
2

and so our recommended vibrationally corrected quadrupole moment is O (N 2 ; v = O)

= -1.118 eag. This is in good agreement with the reported experimental quadrupole
moments gathered in Table 2-3. The several experimental estimates of the quadrupole
derivative available in the literature33'36 average 0.95 eao, and these may be compared
with our SCF, MCSCF, and MRCI results of 1.402 eao, 0.948 eao, and 0.959 eao.

The SCF result is clearly much too large, while the correlated values agree with the

average of the experimental values. The data are collected in Table 2-4. Note that the

15

-0.6 -4 .-"x

2
G) (ea0 )

    
 

...... ....... SCF
------- MCSCF
- ,” —‘— MCSCF+1+2

'1.6 ‘2’ r r 1 I I T I I
-0.4 -0.2 0.0 0.2 0.4
R - R (a0 )

 

 

Figure 4. Distance dependence of G) for N2 calculated with various models using the
aug-cc-pqu basis.

16

significant reduction in dO/dR, when a correlated wavefirnction is used, is implicit in

the SCF vs GVB results reported by Cartwright and Dunning.3O

Our 02 results are summarized in Table C-7 and Figure 5. As with H2 and N2,

increasing the quality of the basis set decreases the quadrupole moment. However,

unlike H2 and N2, adding correlation increases the quadrupole moment (makes it less

negative). The opposing effects of basis-set quality and correlation permits a limited
correlation wavefirnction with a small basis set to predict a quadrupole moment
comparable with the CBS-MRCI limit. We have not found a reported Hartree F ock

limit for O (02) with which to compare either our ROHF results

(Rap, = 2.1747 a0 ,6 = —0.4188 egg) or our UHF results

(Rap, = 2.1885a0 ,6) = —0.3427 cog). Our ROHF and UHF results at R = 2.28 ao are

-0.264 eag and —0.218 cog , respectively and are in reasonable agreement with a large

basis SCF calculation, by Bundgen et al.,20 at R = 2.2819 a0 that predicts G) = — 0.249
ea?) . There are two numerical HFS calculations of O, both at R = 2.28 a0. The first,

by Becke,23 is an unrestricted calculation that predicts = — 0.36 ea?) , while the

second, by Laaksonen et (11.,21 is a spin-restricted calculation predicting G) = —0.3885

ea?) .

Our CBS CASSCF and CASSCF+1+2 results are — 0.2885 cc?) and — 0.2530

ea?) , respectively. It is fascinating that ROHF calculations at the experimental bond

length predict a O (— 0.264 mg ), which differs from our best correlated result

G) (eaoz)

l7

 

 

 

 

 

 

 

0.10: ‘ O (32 -)
0.12 4 c 2 g
0.14 - ' —'— ROHF
0.16 - —'— U HF
0.18 - _‘— UMP2
.020 .3 —V— MCSCF
. O . .
0.24 -. \ ‘w llmlt
0.26 \ ‘ L
. \ ‘\ MRCI limit
0.28 - .\
'0-30 j \ MCSCF limit
0.32 4 °\
.0 34 j .\e
' - UHF limit
0.36 -
0.38 -
0.40 - \-
'°'42 . RoI-IF limit
'0-44 r l l I
2 3 4
aug-cc-vaz

Figure 5. Model dependence of (9 for 02 as a function of basis.

18

Table 2-4. Comparison of experimental and theoretical quadrupole-moment

 

 

derivatives.
(50/071) Re (eao) Reference Comment

+ 0.94 33 quadrupole absorption

+ 0.97 34 collision-induced

+ 0.95 ' 35 collision-induced

+ 0.933$0.039 36 quadrupole absorption
0.959 This Work MRCI aug-cc-pqu; Rom
0.948 This Work MCSCF aug-cc—pqu; Rom
1.402 This Work SCF aug-cc-pqu; Ropt

 

(—0.253 ea?) ) by only 4 percent. The reason for the insensitivity of O (02) to correlation

effects is a consequence of the difference in the response of the c and 1t electrons in 02 to

electron correlation and will be discussed after the quadrupole density is introduced. The

distance dependence of O is shown in Figure 6. Unlike in N2, the MCSCF and MRCI

curves are not nearly as parallel as are the MRCI and SCF curves. The first and second

derivatives of these curves are collected in Table 2-1.

19

(+9 (e302)

 

 

 

Figure 6. Distance dependence of G) for O2 calculated with various models using the aug-
cc-pqu basis.

20

Experimental data on 02 is sparse. Buckingham et al.37 report — 0.3 $ 0.1 cog
from induced birefringence measurements, and Cohen and Birnbaum32+37 report K9] =

0.25 cog obtained from the interpretation of pressure-induced far-infrared spectra. These,

and selected theoretical results, are shown in Table 2-5. Using the derivatives in Table 2-

1, we estimate the vibrational correction to our CBS - CASSCF+1+2 result is

@(02 ; v) = —0.2530+ 0.0257 (v + l)
2

resulting in G) (02 , v = 0) = —0.2273 eag , which is in general agreement with the highly
uncertain experimental values. The experimental estimate33 of (dB/$08 for 02, obtained

fiom an analysis of the quadrupole absorption spectrum is + 1.6 eao and is in reasonable
agreement with our ROHF, MCSCF, and MRCI values of 1.8, 1.4, and 1.5 eao,

respectively.

Our F 2 results are collected in Table C-8 and Figure 7. As with H2, N2, and 02,
the quadrupole moment decreases within a model with increasing quality of basis set and,

as in 02, electron correlation increases G. Our CBS-RI-IF limit is + 0.3081 egg at
Ropt = 2.5064 a0 and + 0.501 cog at R= 2.68 (10. This latter value is in good agreement
with the numerical Hartree-Fock result of McCullough, 17 + 0.505 e03 , at R = 2.68 a0.

Our CBS-CASSCF+1+2 value of + 0.7131 eag (Rom = 2.6853 a0) is in good agreement

HM-H.‘ \ A..-

21

 

 

 

 

 

 

1.05 1
"l 2 g
0.95 -
0.90 1
0.85 -
0.80 - A
0 75 ' \F\A—Mgmm" it
o - —\
. \ '\L
0.70 - j . .
A MP2 /MRCI llmlt
N ..
(5° 0.65 —
m .-
‘6 0.60 -
0,551 —l— RHF
0.50.: . —9— MP2
M, 2 + MRCI
0.35 "' \-
‘ \L
0+3°j RHF lirrnt
0.25 I I I I
2 3 4 5
aug-cc-vaz

Figure 7. Model dependence of O for F2 using various basis sets.

22

Table 2-5. Selection of Oz quadrupole moments.

 

 

R(a0) (9(eag ) Reference Comment
2.1747 — 0.4188 This Work ROHF; Ropt
2.1885 — 0.3427 This Work UHF; Ropt
2.28 — 0.2634 This Work ROHF
2.28 — 0.188 20 SCF Sadlcj15 basis (5s3p2d)
2.28 — 0.249 20 SCF
2.2819 — 0.271 38 CI
2.282 — 0.356 12 numerical DFT (unrestricted)
2.28 — 0.3885 21 numerical HFS (or = 0.7)
2.28 — 0.36 23 numerical HFS (restricted)
2.2970 — 0.2885 This Work CBS-CASSCF; Ropt
2.2873 — 0.2530 This Work CBS-CASSCF+1+2; Ropt
vibrational average — 0.240 This Work CBS-CASSCF+1+2
experimental 1025' 32,37 pressure-induced, far-infiared
spectrum
experimental — 0.3 :l: 0.1 31 Optical Birefringence

 

23

with the numerical HF S calculations of Laaksonen et al.,21 0.6911 eag at R = 2.68 ao,

and those ofBecke,23 + 0.69 eag also at R = 2.68 a0. Our CBS limit at R = 2.68 a0 is

+0.7068 cog . The distance dependence of O (F 2) is shown in Figure 8. Correlation

corrections effects more than double 6) (F 2), a much larger effect than in H2, N2, and 02.

As we will see subsequently, the correlation corrections to 0, due to the c and 7:

electrons, are both in the same direction, and, rather than cancel as in 02, they reinforce

one another. Using the data in Table 2-1, we write the vibrational averaged (9 as

@(F2 ; v) = +0.7131+ 0.0276(v + 2)

and, so, 9(F2 ;0) = +0.7269 gag . We collect the values of G from selected calculations

in Table 2-6.

Ta ble 2-6. Selection of F2 quadrupole moments.

 

3(ao) eleai)

 

Reference Comment
2- 6 8 0.505 17 numerical HF
2 - 6 8 0.659 35 SCF
2- S 064 0.3081 This Work CBS-HF; Ropt
2 - 6 8 0.501 This Work CBS-HF; Rexp
2- 68 0.6911 17 numerical HFS (or = 0.7)
2- 68 0.69 19 numerical HFS (or = 0.7)
2- 6853 0.7131 This Work CBS-CASSCF+1+2; Ropt
2-68 0.707 This Work CBS-CASSCF+1+2; Rex‘p
vibrational average 0.727 This Work CBS-CASSCF+1+2; v=0, J=0

\

24

1.2-
1.1—
1.0-

@(eao)

 

 

 

Figure 8. Distance dependence of (~) for F2 calculated with various models using the aug-
cc-pqu basis.

25

There are no experimental measurements of 0 (F2). We have collected the CASSCF+1+2

values of O along with the estimated CBS limit and vibrational corrections in Table 2-7.

Table 2-7. CASSCF+1+2 results for Ropt and O.

 

 

 

 

Molecule aug-cc-pqu estimated complete basis set limit
Ropt(aO) @013! (eag) Rop,(a0) 90m (ea: ) vibration correction
H2 1.4014 + 0.4569 1.4009 + 0.4552 + 0.051 (v + 2)
N2 2.0827 —1.1247 2.0810 —1.1270 + 0.0137 (v + g)
02 2.2907 — 0.2368 2.2873 — 0.2530 + 0.0257 (v + %)
F2 2.6853 + 0.7165 2.6853 + 0.7131 + 0.0276 (v + y,)
C MP2 Results

It’s apparent, from Figures 1, 3, 5, and 7, that MP2 is a significant improvement
over the SCF model with little additional effort. We collect our MPZ results in Table 2-8
and compare them to the corresponding CBS CASSCF+1+2 results. The comparison is

striking and suggests strongly that the corrections due to MP3 and MP4 cancel one

another significantly as seen by Wolinski et 01.6 and Maroulis and Thakkar14.

26

Table 2-8. Comparison of MPZ and CASSCF+1+2 results.

 

Molecule MPZ (estimated CBS) CASSCF+1+2 (estimated CBS)

Roman) 90.448) Ropt("0) 044803)

 

 

H2 1.3895 + 0.4699 1.4009 + 0.4552
N2 2.0970 — 1.1751 2.0810 —1.1270
02 2.3032 — 0.2546 2.2873 — 0.2530
F2 2.6452 0.7178 2.6853 + 0.7131

 

D. Hellmann-Feynman Theorem

When a perturbation on a Hamiltonian is the effect of an applied field of strength
A, the perturbed Hamiltonian can be written as
I? = 19° + 2.0
where O is for example, a dipole moment operator and H 0 is the full many-electron

Hamiltonian for the unperturbed molecule. The Hellmann-Feynman theorem.“°’“’42 The

Feynman is commonly written as,

(d—Eifll;<wll++>

d2

where O is a one-electron operator of a given property that responds directly to the field
and ‘1’ is the exact wavefunction. This equality is also true for certain approximate
methods such as HF and MCSCF where all variables are variationally optimized within a
given space. In the CI and MP wavefiinctions, however, the configurational coefficients
are variationally optimized, but the molecular orbital expansion coefficients are not so the

equality is not assured.42

27

To assess the difference between <‘PlOI‘I’> and (dE(/1)/d/i) 0, calculations have
been performed using both HF and MPZ methods to calculate the quadrupole moment of

various diatomics. <‘1’i0l‘1’> is determined as an expectation value (EV) while

(dE(.i)/d/l) 0 is referred to as the point charge method (PC). Application of these

techniques are described in Appendix A. As shown in Table 2-9 the MP2 wavefirnction
using an aug-cc-pvtz basis set brings the EV quadrupole moment to within 2% of the

point PC quadrupole moment. The MP2 results can be compared to the HF

Table 2-9. Expectation Value Quadrupole moment and Point Charge Quadrupole

 

 

Moment
HF MP2
Epc EEV o/ODIII Epc EEV o/oDIIf
H2074) 0.502 0.500 0.34 0.478 0.476 0.42
N2(‘2;) 0.910 0.920 0.11 -1171 -1.158 -111
0262,) -0.160 -0.163 -1.85 -0.262 0.264 0.76
F262;) 0.530 0.534 0.75 0.769 0.784 1.91

 

results also given in Table 2-9. With the exception of 02, the difference in the HF-EV
quadrupole moments and HF-PC quadrupole moments are smaller. Since the Hellmann-
Feynman equality holds for HF wavefunctions, improving the ‘completeness’ of the basis
sets should lessen the difference between EV quadrupole moments and the PC quadrupole
moments for the other approximate wavefilnctions and the results listed in Table 2-9

indicate this to be correct.

28

E. Quadrupole Moment Functions

We study the distance dependence of O (quadrupole-moment function), using the
aug-cc-pqu basis and the MCSCF and MRCI wavefunctions. The calculated potential
energy curves for the molecules of interest are shown in Figure 9, and the calculated

properties (Re, De, cue) are very close to the CBS CASSCF+1+2 limits. These

wavefimctions provide a reasonably accurate description of the molecule’s electronic
structure over a large range of internuclear distances, and we expect the calculated
quadrupole-moment filnctions to be realistic.

The quadrupole moment functions for the four molecules of interest are shown in

Figure 10, from which we see that H2 is unique——having the only quadrupole-moment
function that is everywhere positive. 0 for H2 and N2, is zero at large R because both
molecules separate to atoms in S states. 9, for F2 and 02, separates to the sum of the
atomic quadrupole moments of the atoms. For F2, the F atoms are in the 2FIFO (2P)
state, loosely corresponding to the configuration,

1522322p32pi2p;
with z labeling the internuclear axis. For 02, the atoms are in the 3 IT M|=1 levels, which,

in a real representation, corresponds to one atom in

1s22522p,2cp)l,pzl

 

Energy (e 2/a0)

 

 

29

 

 

2 _
1 .—
0 .l
- /
-1 ..
-2 - F2(128+)MRCI
-3 —I
4 - 02028) MRCI
-5 ..
‘ H (12 +)CASSC1=
-6 1 2 g
-7 q
.8 .. N2(1£g+) MRCI
.9 _
'10 1 1 l I I I I I l r r I 1
1 2 3 4 5 6 7 8
R (a o )

Figure 9. MRCI potential energy curves for N2, 02, and F2 using the aug-cc-pqu basis
and a 10 CSF, CASSCF potential energy curve for H2 using the aug-cc-vaz basis.

30

 

 

1.5-
F2(12+)MRCI
1.04 g
. 3 -
02( Zg)MRCI

0.5 - \

H2(12‘.g+) CASSCF

 

   

 

 

0.0-i
No 7
to
m -O'SI N 1 MRCI
V 2+
(9 _ ,< g)
-10-
-L5-
-2-0 I l/[I l1 lfi l I l U I IT. I I I U ' EFT]
0123456789101112

R(a0)

Figure 10. Molecular quadrupole moments of H2, N2, 02, and F2 as a function of bond
length.

"L

.1

h H

31

and one in

13223222125323; .
We will first consider N2, 02, and F 2, returning to H2 latter. Let’s first write the electron
density at internuclear separation R as

n(f';R) = n}; (F)+ng(r‘)+5n(r‘, R) (5)
where 712 and n3 are the electron densities of the two non-interacting atoms placed at the
appropriate nuclear positions. Note that 511 is defined by this equation. As a practical

matter, 713 and 723 are obtained from the natural orbitals of the MCSCF or MRCI
wavefimctions at large values of R and translated, intact, to the internuclear separation of
interest.

Using Equation 5, the quadrupole moment defined by Equation 1 can be rewritten

as

(-:-)(A2 ; R) = 26°(A) + j 6(9(F;R)dV (6)
where ®0(A) is the quadrupole moment of the separated atom A in the diatomic A2

1

(90(A) = —§Jn2(7)(322 —r2)dV (7)

and 59 is the quadrupole-moment density
1 — 2 2

(5(9=—-2-&2(r;R)(3z —r ) . (8)

Note that the nuclear contribution to @(A2;R) is now implicit in 6n('r°,R). Note, also,

that ®(A2 ;R) is now written as the sum of a (constant) atomic contribution, 200(A) and

a contribution Om], due to molecule formation

nu.

32

em, (R) = j ao(F;R)dV. (9)

Since we may easily partition 6n into 0 and 7t contributions,

6n = 810 + an,“ (10)
we may also write

0(A2;R)=00(A2;R)+®,,(A2;R), (11)
where

90(42;R)= 29%(A)+ [89.(F.R)dV. (12)
with an analogous expression for 6) ”(A2 ;R). The 0 and 1: components of the
quadrupole-moment curves for N2, 02, and F2 are shown in Figure 11 for a MRCI
wavefirnction in an aug-cc-pqu basis. Note that (90 is always negative and ('9,t is always

positive; and, while the Sign of 0 depends on the relative magnitudes of these
contributions, 9 usually decreases from its asymptotic value with decreasing R. The
relative asymptotic values are easily understood, in terms of an orbital model. For
example, the zz component of the atomic quadrupole moment of (oriented, m = 0) F is

given by
®(F; 2P2): @(2p2)+2®(2py)+29(2px)

where

@(Zpa) = ’% [(2p,)2(322 02).”.

33

4.0 -l
3.5 -

.1

3.0 -

2.5 -
2.0 -l
1.51

1.0 -
0.5 4
,

0.0
-0.5 -
-10 4

d

-1.5-

d

-2.0 -

-2.5 4
-3.0 l
-3.5 J
-4.0 4
-4.5 J

‘5.0 I I T I
1.0 1.5 2.0

 

 

6) (e302)

 

 

 

I I j I

l l I'I'H‘T'l
3.0 3.5 4.0 4.5 5.0 5.5 6.0
Ra

(0)

r
2.5

Figure 11. Distance dependence of the quadrupole moments of N2, 02, and F2 partitioned
into the (9., and 0,. components.

34

By symmetry

@(pr) = 9(2Py)
and @(pr)+@(2py)+®(2pz) = 0.
With 2 as the internuclear line,

6,,(F) = 29(2px)+2@(2py) = -2®(2pz) a —200(F).
It’s easy to show

@(ZPz) = 960’") < 0,
so, in an orbital model where all 2p orbitals are radially equivalent,

6,,(F) = —2®0(F) > 0.
Our asymptotic values are

6),,(17) = +2.517 eag

90(17) = —l.170 e03.
These are not precisely in the symmetry determined ratio because our wavefunction has

D21l symmetry and our correlated wavefunction results in asymptotic p functions that are

not equivalent.

In a similar fashion, the O (3 ITMlzl) quadrupole moment is given by

@(0; 3}1A41=1) = 20(2px)+®(2py)+@(2pz)
6,,(0)=2o(2p,)+o(2p,)

G90(0) = (9(2pz)

35

6,,(0) = 3(—-21—o,(0)) = — 3/2o(,(o).
Our asymptotic values are

(9,,(0) = +2435 egg

00(0) = —1.499 mg

Differing from the simple orbital ratios for reasons described earlier.
If we reference each molecular quadrupole-moment fiinction to its asymptotic

atomic contribution, we obtain Figures 12-14, which are simply (9 m01(R) and its

component (95,0, and (9:50,. Some insight into why @201 is always negative and 07,201 is

always positive may be obtained by examining 590(7, R) and a 6,,(7, R). Recall

6O 2 —%c5’na(r",R)(3z2 -—r2) = —%r2 5na(F,R)(3 cos 26— l) (13)

0'

where the origin is the molecular midpoint and dis measured relative to the internuclear
line as the polar axis. This equation relates electron shifts in the 0 system, upon bond

formation to the molecular contribution to the quadrupole moment. Note that the factor

- -;—(3 cos 20 - I) partitions the molecular space into two regions, labeled N and P and

shown in Figure 15. It is interesting to note that the parabolas delineating the Berlin“5
bonding and antibonding regions are asymptotically tangent to the nodal surfaces
separating the N and P regions. In the conical regions labeled N, to the rear of the nuclei,

the angular factor is negative, and, thus, a positive Snc in this region results in a negative

contribution to @301. In N2, the 0 bond involves a large sp hybridization, moving charge

towards the midpoint of the molecule. Simultaneously, the opposite-

 

36

 

 

 

 

2.0-
.N2(12 +) ,_ ......
1.5- g
1.0- .\
0.5-
a all
as" 00-
3
E 0.5-4
9 .
11
® “1.0%
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J
-2.0-
-2.5- .........
‘ MRCI
'30 I I T I I I I I r j f I I I l I
10 15 20 2.5 3.5 40 45 50

Figure 12. Distance dependence of the quadrupole moment of N2 partitioned into its
0;, and 0;, components.

37

2.0 1

(e802)

6)
mol
I
O
‘01
.1

   

MRCI
l l ' F ' l ' T ' I
1 .0 1 .5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

R(ao)

 

 

 

It
mol

Figure 13. Distance dependence of Omar for O2 partitioned into its (9:0, and 0
components.

38

' F (12+)
2 g

 

(eaoz)

 

mol

0)
S
l

 

 

MRCI
1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

 

 

Figure 14. Distance dependence of Omar for F2 partitioned into its 6):”, and

0:”, components.

39

 

 

 

 

—6 0 5
6 I 1 4 p r I 1 1 l r 1 6
d P region .
0 ‘ N region N region r 0
1 P region '
-6 T U r I I I I I U U r 6
-6 0 6

Molecular Axis (22/R a0)

Figure 15. Quadrupole density nodes represented by solid lines. Berlin bonding and
antibonding regions separated by parabolic curves.

40

phase sp hybrid pooches out to the rear of the nuclei contributing to a positive 8nO and,
therefore, a negative 596. Note that this effect is enhanced by the r2 term in Equation 13,
which weights heavily the farther reaches of 8nd. Along with the 0 bond in N2, we, of

course, have the 1: bond, which results in 5n7t being positive in the region between the

. . . 1 . .
nuclei and above the molecular km; and, srnce 5 O,r = —§r2(‘)‘n,r (3 cos 261 — 1), thls lS
precisely where the angular factor is positive, and, therefore, an accumulation of charge in
the 1: system (positive Sn“) results in a positive value of 56'),t and contributes toward a

positive value of 0:0,. These effects are vividly illustrated in Figure 16-18, which show
Jn, 6:10, and 611,, and the associated (X9, 6(1),, , and 69,, densities for N2. Note that the
increase in charge density around the molecular midpoint contributes little to O mob as it is
multiplied by r2 (small in this region), and much cancellation results fi'om the integration
over 3 cos 26— 1. The situation with 02 and F 2 differ only in degree. sp hybridization
decreases in going from N2 to 02 to F2, and this is reflected in a less negative value of
6:0, in 02 and F2, relative to N2.

The 1t systems in 02 and F 2 are qualitatively different from Nz’s, and this is
reflected in the 07,201 curves shown in Figures 12-14. For 02, 672,, is smaller than in N2
and, thus, (9:0,(02) is less positive. For F2, an” is almost zero and 9:”,(F2) is small.

These characteristics of 02 and F2 are illustrated by the contour plots in Figures 19—21 and

22-24. One striking feature of 0:0, in N2 and 02 is the maximum (Figures 12 and

41

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50

13), which we understand as follows. As the nuclei come together, It electron density

accumulates between them (in the P region of Figure 15) and, as R becomes comparable
to Ropt, some of this density begins to spill over into the N region and thus 9:0, begins to
decrease. The same situation obtains in 02 and F2. In 02, one has less accumulation, a
smaller increase in €97,201, but the same spilling of 6n” into the N region. 611,, for F2 is
rather flat, although one still has a slight maximum in 9%,. The equilibrium bond length

in these molecules is smaller than the internuclear distance where @201 is a maximum, so,

the slopes of both the 1: and 0 component of (9 are both positive around Ropt.

EH2

A similar analysis for H2 shows that, as the two H atoms approach to form a bond,
the density difference (3n,I is positive between the nuclei with a large negative region to
the rear of each nucleus. Because of the 3 cos 2 6—1 factor in the quadrupole density
integrand, the positive region between the nuclei integrate to a small contribution to
@201, while the decreased density in the N region of Figure 13 contributes to a large
positive @301. This characteristic of 6nd is common to s-s bonds such as H2, Li2, Na2,
etc. and opposite to the sp-sp bonds characteristic of N2, 02, and F2. The maximum in
(93,01 obtains when large positive components of 6110 in the N region spill over into the P

region and this happens a little before the equilibrium separation. Clearly, G) will go to

51

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52

zero as R —+ O and n(F;R) approaches the united atom limit. These observations are

illustrated in Figure 25.

G. On the Sign of the Quadrupole Moment

From the preceding discussion, we see that the sign of (9 for N2, 02, and F2
depends on the relative values of @a (negative) and (9,, (positive). In F2, for example,

one has a large positive 7: contribution at 00, due to the separated atoms, which changes

little as the molecule forms because 5n” is small and essentially independent of R. The

asymptotically negative 6 contribution is reduced further by 0 bond formation but not

enough to change the sign of G, which remains positive. The F2 0' contribution is
anemically negative because of the very slight sp hybridization in F2. In 02, the
asymptotic value of (9 is less positive than in F 2, and the increased sp hybridization is able
to overcome a very positive 1: contribution and results in a negative quadrupole moment.
In N2, the large sp hybridization causes the (90 to be dominant and G) is decidedly
negative. The situation in H2 is fundamentally different. The sign is always positive
because at large R the s-s bond results in 6'20 being negative in the N region, and this
situation obtains until the maximum in G) (H2;R), after which (9 decreases toward zero, as

described above.

H. Correlation Effects on 33,01 and 6,730,

Further insight into the relationship between electron correlation and the

quadrupole moment obtains from an analysis of (9c, and 9”. In Figure 27, we plot these

53

quantities for the SCF, MCSCF, and MRCI wavefunction of N2 (aug-cc-pqu basis). The
distance dependence of 6),, (around Ropt) for all three wavefunctions is similar, with the
SCF and MCSCF contributions being remarkably so. The MRCI value of 90 is the
largest of the three and reflects the effect of dynamic correlation in increasing 5’70- in the
region between the nuclei. The correlated distance dependence of 6),, differs markedly
from the SCF value and is significantly smaller. These results suggest that the reasonable
values of G) calculated fiom SCF functions (Table 2-3) obtain because of a cancellation of
errors, 0,, (SCF) being too negative and 0,, (SCF) too positive. Our calculated 6)

(SCF) is —0.9306 eag, while (9 (CASSCF+1+2) is —1 . 1334 egg, both at R = 2.07432 a0.

These differ by 18 percent, and most of the error is in the 1: contribution.

The corresponding data for 02 and F2 are given in Figures 28 and 29. Note that

the scales in these plots are identical and the magnitude of the effect of electron

correlation on Go and 9,, is similar. In 02, like N2, (9,, (SCF) is too negative, while
6,, (SCF) is too positive, resulting in a similar cancellation of errors. At 2.28 a0, we
calculate 9 (oz; ROHF) = — 0.2634 egg and o (0,; CASSCF+1+2) = — 0.2530 egg,

a difference of only 4 percent, as noted earlier. In F2, both (9,, (SCF) and (9,, (SCF) are

too small, but rather than cancel they add and result in a G) (SCF) (0.501 eag at R = 2.68

54

uuuuuuu
.....................
..........
. n.
.......
n' .
.........
a .
.....
'o-_
"u.

    

.‘
-

‘-

—

Q--
-
.- -—--_

.... ----_-

n

2- 1 +
N,< 2,) a

 

 

 

 

Figure 26. Effect of electron correlation on the magnitude and distance dependence of (9,
@o, and G), for N2.

55

 

 

 

 

 

 

2.0 2.5 3.

Figure 27. Efi‘ect of electron correlation on the magnitude and distance dependence of G),
(90, and (9,. for 02.

56

 

 

 

 

 

 

Figure 28. Effect of electron correlation on the magnitude and distance dependence of (9,
@02 311d 93 for F2.

57

a0) compared to the (9 (CASSCF+1+2) value of 0.707 eag (a difference of 29 percent) at

the same bond length.

1. Conclusions

We have studied the quadrupole moments of H2, N2, 02, and F2 and have
estimated the CASSCF+1+2 basis-set limit for the latter three and the CISD limit for H2.

These are in excellent agreement with comparable calculations by others and in good
agreement with the existing experimental data. The rather large values of the quadrupole
moment derivatives, shown in Table 2-1, result in the quadrupole moment being a very
sensitive fiinction of R around Rom. We have written the global quadrupole-moment
function as the sum of an atomic contribution and a molecular contribution (9",01. The
atomic contribution is simply the sum of the quadrupole moments of the constituent
(oriented) free atoms, while the molecular contribution is an integral over a quadrupole-

density fiinction of the form
(X9 = —%(3cos 26—1)r26n

in which the nuclear contributions are implicit in 6n. While not unique, this partitioning of
(9 allows us to separate molecular effects from additive atomic efl‘ects and provides a
deeper understanding of the variation of (9 with R. In particular, the nodal structure of

3 cos 26—1 allows us to partition the space in a diatomic molecule into N regions, to the
rear of the nuclei and complimentary P regions. Because 5(9 is linear in 5n, one may
define quadrupole densities associated with 6nd. and 672,, and examine the contribution of

the o and 1: deformation densities to the molecular quadrupole moment. Increases in 5er

58

in the N region contribute to make 9%,), negative, while increases in 572,, in the P region

make 97,201 positive, and, thus, the molecular contribution to the quadrupole moment is
the sum of two opposite-signed terms. The total molecular quadrupole moment is the sum
of this molecular contribution and the moment due to the sum of the separated atoms.

This perspective is useful in understanding the difference in the quadrupole moments of
related systems, such as N2 and P2, 02 and $2, C6H6 and C6F6, etc.44 Additionally, the
effect of electron correlation on 9 may be partitioned into 0 and 1: contributions via (91,,
and 6:1,.

We note that the parabolas that delineate the Berlin bonding and antibonding
regions are asymptotically tangent to the nodal surfaces separating the N and P regions
(Figure 15). Indeed, the largest contribution to 9:01 and 9:0, come from a density
difference that is largely localized in the Berlin45 antibonding and bonding regions,
respectively.

Additional contour plots and three-dimensional images of on and 59, as well as
detailed numerical values of 9 as a function of R, are available via our web page,

www.cem.msu.edu/~harrison.

59

J. References

l.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

Huber, K. P. and Herzberg, G., “Molecular spectra and molecular structure,”
Vol. 4, Constants of Diatomic Molecules (Van Nostrand-Reinhold, New York,
1979)

McLean, A. D. and Yoshimini, M., J. Chem. Phys. 1966, 45, 3676.

Stogryn, D. E. and Stogryn, A. P., Mol. Phys. 1966, 11, 371.

Ramsy, N. F., Molecular Beams (Oxford University Press, London, 1956).
Buckingham, A. D. and Cordle, J. E., M01. Phys. 1974, 4, 1037.

Amos, R. D., M01. Phys. 1980, 39, 1.

Laaksonen, L., Pyykko, P., and Sundholm, D., Comput. Phys. Rep. 1986, 4, 313.
Wolniewicz, L., J. Chem. Phys. 1966, 45, 515.

Kolos, W. and Wolniewicz, L., J. Chem. Phys. 1965, 43, 2429.

Wolinski, K., Sadlej, A. J., Karlstrom, G., M0]. Phys. 1991, 72, 425.

Kolos, W. and Roothaan, C. C. J ., Rev. Mod Phys. 1960, 32, 219.

Dickson, R. M. and Becke, A. D., J. Phys. Chem. 1996, 100, 16105.
Diercksen, G., Sadlej, A. J., Theoret. Chim. Acta 1983, 63, xxxx.

Kolos, W. and Wolniewicz, L., J. Chem. Phys. 1964, 41, 3674.

Sundholm, D., Pyykko, P., and Laaksonen, L., M01. Phys. 1985, 56, 1411.
Maroulis, G. and Thakkar, A. J., J. Phys. B 1987, 20, L551.

McCullough, E. A., Jr., M01. Phys. 1981, 42, 943.

Maroulis, G. and Thakkar, A. J., J. Chem. Phys. 1988, 88, 7623.

Feller, D., Boyle, C. M., and Davidson, B. R., J. Chem. Phys. 1987, 86, 3424.

Bundgen, R, Green, F., and Thakker, A. J., J. Molecular Structure 1995, 334, 7.

Laaksonen, L., Sundholm, D., and Pyykko, P., Int. J. of Quant. Chem. 1985, xxvii,
601.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

41.

60

Duffy, P. and Chong, D. P., J. Chem. Phys. 1995, 102, 3312.
Becke, A. D., J. Chem. Phys. 1982, 76, 6037.

Piecuch, P., Kondo, A. E., Spirko, V., and Paldus, J ., J. Chem. Phys. 1996, 104,
4699.

Spirko, V., Piecuch, P., Kondo, A., and Paldus, J ., J. Chem. Phys. 1996, 104,
4716.

Buckingham, A. D., Graham, C., and Williams, J. H., Mol. Phys. 1983, 49, 703.
Graham, C., Pierrus, J ., and Raab, R. E., Mol. Phys. 1989, 67, 939.

Truhlar, D. G., Int. .1. Quantum Chem. 1972, VI, 975.

Maroulis, G. and Bishop, D. M., Mol. Phys. 1986, 58, 273.

Cartwright, D. C. and Dunning, T. H., J. Phys. B 1974, 7, 1776.

Buckingham, A. D., Disch, R. L., and Dummur, D. A., J. Am. Chem. Soc. 1968,
90,3104.

Cohen, R. E. and Bimbaum, G., J. Chem. Phys. 1977, 66, 2443.

Camy Plyret, C., F laud, J. M., Delbouille, L., Roland, G., Brault, J. W., and
Testerman, L., J. Physique Lettres 1981, 42, L279.

Reddy, S. P. and Cho, C. W., CanadianJ. Phys. 1965, 43, 2331.
Shapiro, M. M. and Gush, H. P., Canadian J. Phys. 1966, 44, 949.
Reuter, D. and Jennings, D. E., J. Mol. Spec. 1986, 115, 294.
Bimbaum, G. J. and Cohen, E. R., Mol. Phys. 1976, 32, 161.

Rijks, W., van Heeringen, M., and Wormer, P. E. S., J. Chem. Phys. 1989, 90,
6501.

Wahl. A. C., J. Chem. Phys. 1964, 41, 2600.
Hellmann,H., Einfuhrung in die Quantenchemie (Deuticke, Leipzig, 1937)

Feynman,R.P., Phys.Rev. 1939, 41,721

42.

43.

44.

45.

61

Raghavachari,K., and Pople, J.A., Int. J. Quant. Chem, 1981, 20, 1067
Amos, R.D., Adv. Chem. Phys, 1987
Lawson, D. and Harrison, J. F., to be published.

Berlin, T., J. Chem. Phys. 1951, 19, 208.

CHAPTER 3

The Quadrupole Moments of P2, 32, and C12

a-..

A. Introduction

In Chapter 2, the quadrupole moments of H2, N2, 02, and F2 were discussed in
detail. This chapter provides an extension of that discussion to the second row
homonuclear diatomics. With the exception of Clz, very little theoretical or experimental
data could be found for molecules containing heavy atoms. In this Chapter we present
calculated quadrupole moment data for the second row homonuclear diatomics and
compare them with the first row. Using methods developed in Chapter 2, we also will
look at the sign and magnitude of the quadrupole moment as a function of the electronic

structure.

B. Computational Technique

The MCSCF and MRCI calculations were done using the COLUMBUSl system of
codes, while the full CASSCF and CASSCF+1+2 were done using MOLPROZ. The UHF,
ROHF, and Moller-Plesset calculations were done using g943. Tables 11-13 of Appendix
C contain the quadrupole moments of the titled molecules for various wavefiinctions and

basis sets. The MPZ wavefimctions include excitations from the 40g, 4ou , 508, Sou,
21tux, 21ruy, 21th, and 21tgy orbitals of P2, S2, and C12. The MCSCF wavefunctions are
CASSCF functions over the MO’s derived from the valence p orbitals (the 568, 2nux,
21tuy, Zugx, any). The MRCI incorporates all single and double excitations from these
six orbitals, as well as all double excitations from the 40g and 4ou orbitals. The CASSCF

wavefimctions include all valence electrons and the 8 orbitals described above, and the

CASSCF+1+2 wavefunction consists of all single and double excitations from the

63

64

CASSCF reference space. Quadrupole moments were computed in atomic units using the
Buckingham convention described in Appendix A. The Dunning basis sets used in this

study are described in Appendix B.

C. Discussion

The results of P2 are summarized in Table C-11 and Figure 1. Unlike H2, N2, 02,
and F 2 in Chapter 2, the quadrupole moment does not decrease systematically with
increasing basis set. This is an indication that the basis, although fully optimized for the

energy, may not be fiilly optimized for this particular property. Our best SCF value, given

as the value associated with the aug-cc-vaz basis, is 0.7920 eag at 3.4950 a0. This

does not compare well with Glaser, Horan, and Haney4 SCF value of 0.5212 eag at

3.5773 aO , however, this value was calculated with a relatively small basis set. At the

experimental separation of 3.5780 a0 our SCF quadrupole was 1.0629 eag. This value

should be very close to the numerical HF result. Following the SCF calculation, the post-

SCF techniques also do not yield CB S-limit quadrupole moments. Our aug-cc-vaz
CASSCF and CASSCF+1+2 values are 0.3162 ea; at 3.6288 a0 and 0.5241 ea: at

3.6055 a0 , respectively. Using the aug-cc-vaz basis at the experimental geometry, the

CASSCF gives 0.2039 ea: and the CASSCF+1+2 gives 0.4608 eag.

At the SCF level, the aug-cc-pvdz quadrupole moment is 8.7 % smaller than the
aug-cc-vaz while the aug—cc-pvtz quadrupole moment is 0.6% smaller. For the RHF,

MPZ, and CASSCF+1+2, the aug-cc-vaz CASSCF+1+2 value is less than 5.0% higher

 

65

 

3.70 -
‘ (12 +)
- g
3.65 -
\\\:\
NA
‘ \o
\Nv
3.60 4
A0
CU .
V0 —-— RHF
0: —0— MP2
3-55 ‘ —A— CASSCF
.\ —V— CASSCF+1+2
.l
I
3.50 - \IR.
3-45 f I l f
2 3 4 5
aug-cc-vaz

Figure 1. Model dependence of G) for P; as a function of basis set.

§

66

than the aug—cc-pqu quadrupole moment and 3.8% lower than the aug-cc-pvtz
quadrupole moment. Adding correlation to the RHF wave fiinction by MPZ and
CASSCF+1+2 lowers the value of the quadrupole moment by similar amounts. As in the
last chapter MPZ predicts the quadrupole moment comparatively well at little overall
computational cost. The CASSCF wavefunction has the greatest differences in the
quadrupole moments for each basis set. The CASSCF, also, overestimates the correction
to the quadrupole moment, lowering it more than either the MPZ or the CASSCF+1+2.
The distance dependence of (*9 around Rap. is shown in Figure 2 and 3 for the SCF,
CASSCF, and CASSCF+1+2 models, using the aug-cc-pqu basis. The slopes of the
CASSCF and CASSCF+1+2 are parallel to each other, and this is to be found true for 82

and C12 as can be seen in Table 3-1.

Table 3-1. Equilibrium value of the first and second derivatives of the molecular
quadrupole moment fimction for P2, S2, and C12, calculated with various wavefunctions.

 

 

(a [g]

Molecule Wavefunction Ropt(a0) dR Rap! (“0) dR2 Ropt
P2 SCF 3.5007 3.3894 -0.8168
P2 CASSCF 3.6330 2.1529 -1 .6946
P2 CASSCF+1+2 3.6114 2.0972 -1.6068
82 ROSCF 3.5216 1.3023 -1.0148
S; CASSCF 3.6457 1.2266 -1.4504
S2 CASSCF+1+2 3.6101 1.2559 -1.4362
C12 SCF 3.7309 2.2393 -1.0038
C12 CASSCF 3.8574 2.4962 -1.1354
C12 CASSCF+1+2 3.8050 2.3801 -1.1788

 

67

(9 (e302)

 

 

 

_0 5 _ e/ _I_ SCF
' /
. r —4— CASSCF
—A— CASSCF+1 +2
'1-0 ' l ' r ' l ' 1
-0.4 -02 R R 0.0 02 0.4
- opt (a0)

Figure 2. Distance dependence of G for P2 calculated with various models using the aug-
cc-pqu basis.

68

 

2
® (ea0 )

 

.5:
_4 _ '3‘; ............... SCF

_5 j j] --------- CASSCF

. —— CASSCF+1 +2

 

 

I I 1

l'l'l'l'l'1
789101112

1 l l j
1 2 3 4 5 6
R (a0)

Figure 3. The quadrupole moment of P; as a fiinction of internuclear separation with
various models using the aug-cc-pqu basis.

69

The SCF curve has significantly larger first derivative, while the SCF second derivative is
of a smaller magnitude. In comparing P2 with N2, the N2 second derivatives for all
methods were positive, while the P2 second derivatives are negative. This can be better
understood by looking at the quadrupole moment as a firnction of R, @(R). The N2 @(R)
curve is zero at R=oo and everywhere else negative, while the P2 @(R) curve is zero at
R=oo, positive at large R, and becomes negative as R becomes smaller than Row. The
reasons for the difference between N; and P; will be described later. Using the
CASSCF+1+2 derivatives in Table 3-1, we estimate the vibrational dependence of the

aug-cc-vaz CASSCF+1+2 quadrupole moment as

9(1)“) = 0.5241+ 0.0197(v+ %)
and so our recommended vibrationally corrected quadrupole moment is 69(ng = 0):
+0.5340 eag. There is no experimental value with which to compare our results.

The CBS-ROHF quadrupole moment of S2 is 1.3966 ea: at 3.5143 a0 and our

CBS-UHF quadrupole moment is 1.3909 eag at 3.5171 a0. The data at Rap did not
extrapolate well for the quadrupole moments at the experimental separation of 3.5701

a0 . The ROHF quadrupole moment with the aug-cc-pv52 basis is 1.5404 ea: and the
aug-cc-pv52 UHF quadrupole moment is 1.5315 eag. The CBS-CASSCF and CBS-
CASSCF+1+2 results were 1.4612 ea: at 3.6354 a0 and 1.4585 ea: at 3.5978 a0,
respectively. The aug—cc-vaz CASSCF and CASSCF+1+2 where 1.3143 ea: and
1.3918 ea: , respectively, at the experimental separation. The CBS-Unrestricted-MPZ

gave a quadrupole moment of 1.3909 ea: at 3.5171 a0. The results for each method and

70

1.8- ‘ 82 (32%;)

 
  
  

 

 

 

 

1.7 -
—-— UHF
° + ROHF
A 1.6 - —0— MP2
No —A— CASSCF
3 —v— CASSCF+1 +2
. * ~#
1.4 - \IR.
1.3 - \F *9
I 1 l l
2 3 4 5

aug-cc—vaz

Figure 4. Model dependence of ('9 for S; as a fiinction of basis set.

(9 (e302)

 

71

 

2.50 -1
1 S 32 '-
2.25 - 2 ( g )
2.00 4
.
1.75 -
1.50 -
1.25 4
1.00 - "
0.75 - ‘
0.50 -
. ——I— ROHF
0.25 - —~— CASSCF
- —-— CASSCF+1 +2
0.“) I l I l r l T l
-0.4 -02 0.0 02 0.4
R - Ropt (a0)

Figure 5. Distance dependence of G) for S2 calculated with various models.

72

2
® (ea0 )

 

‘ --------- CASSCF
- —— CASSCF+1 +2

I'u'I'uTrfil'l
6 7 8 9 101112
R(ao)

 

 

Figure 6. The quadrupole moment of S2 as a function of internuclear separation with
various models using the aug-cc-pqu basis.

73

 
   
 
 
   
 

 

 

 

3.1 -
1 , (:1 (12 +)
3.0 - 2 g
2.9 .4 o
4 V
2.8 4
‘ ' —-— RHF
A J —0— MP2
foo 2'7 —A— CASSCF
3 ‘ —v— CASSCF+1 +2
N 2.6 u
N
(D .
A
2.5 - \A\
A
2.4J
.. \'
2.3 - I
II \- I
22 l T l l
2 3 4
aug-cc-vaz

Figure 7. Model dependence of 9 for C]; as a function of basis set.

74

basis set are collected in Table C-12 and Figure 4. No other theoretical or experimental
data exists to compare with our S; quadrupole moment.

A plot of the @(R) in the region of Rap. is given with the aug-cc-pqu for various
wavefunctions in Figure 5. From the plot we can see the CASSCF and CASSCF+1+2
quadrupole functions are on top of one another while the ROHF quadrupole fiinction has
a steeper slope. From Table 3-1, we see that the slope @(R) SCF slope is larger than both
the CASSCF and CASSCF+1+2, while the SCF second derivative is a smaller negative
number. The CASSCF+1+2 first derivative is +1.2559 and the CASSCF+1+2 second
derivative is -1.4362. Using the CASSCF+1+2 derivatives, our vibrationally corrected S2
quadrupole moment is

@(Sz;v)=1.4585+0.0102(v+12—)
and so our recommended vibrationally corrected quadrupole moment is @(S2;v = 0)=
+1.4636 eag. The overall @(R) curve is given in Figure 6 and is very similar to the 0;
curve in Chapter 2. From Chapter 2, the ROHF quadrupole moment of 02 was 18 %
larger than from the UHF quadrupole moment. For 52, however, the ROHF and UHF
quadrupole moments are much closer. This is related to the strong dependence of the
quadrupole moment on the internuclear separation. The S2 ROHF internuclear separation
is only 0.003 a0 larger than the UHF bond length, while the 02 ROHF internuclear
separation is 0.015 ao smaller than the UHF separation.

Our C12 results are collected in Table C-13 and Figure 7. As in the previous

SJittems the quadrupole moment decreases within a model with increasing quality of basis.

COlTelation increases the quadrupole moment from the SCF value and the CASSCF+1+2

75

decreases the quadrupole moment from its CASSCF value. Our CBS-RHF limit is 2.2373

eag at 3.7267 a0. The aug-cc-vaz RHF quadrupole moment at the experimental

geometry of 3.7567 a0 is 2.2884 eag. There are no numerical-HF data for C12, however,

our values compare well with other large basis SCF data.“ With the aug-cc-vaz at the
experimental geometry, our CASSCF and CASSCF+1+2 quadrupole moments are 2.3165
ea: and 2.2896 eag, respectively.
The distance dependence of @(C12) near the optimized geometry is shown in

Figure 8 and 9. Correlation correction effects are relatively small on the first and second
derivatives. The SCF, CASSCF, and CASSCF+1+2 first derivatives are positive and SCF,
CASSCF, and CASSCF+1+2 second derivatives are negative. The SCF first and second
derivatives are 2.23 93 and -1 .003 8, respectively, while the CASSCF+1+2 first derivative

is 2.3801 and the second derivative is -1.1788. Using the CASSCF+1+2 first and second

derivatives and equation 4 from Chapter 2, we write the vibrational averaged G) as

@(C12;v) = 2.3559 + 0.0301(v+—21)

and, so or vibrationally corrected CASSCF+1+2 quadrupole moment for C1; is 2.371 eag.
This compares well with Buckingham, Graham, and Williams8 experimental value of
2.405i0. 120 eag. We collect the experimental and select calculated values of (9 in Table

3-2. We have collected the CASSCF+1+2 values of G) along with estimated CBS limit

vibrational corrections in Table 3-3.

76

3.0q
1 +
.C12( 2g)

Q
2 .8 ~ {.7/ / ,/'

q

I I ,
l a
v I '
.4 x
2.6 - x . I
I '1 ‘0
I, f l’
5 I '
I ‘
, fl
.. I ,’
II I ’I
’ I
I I I
I I /
2 4 _. ’ ’ I
. I I ,
I I I
I I /

I I I
I I
22- . . ,
I
I

2
(~) (ea0 )

2 .0 _ :9"; I", fl“

1.3- —-— SCF
‘ ' _._ CASSCF

.4 - —A— CASSCF+1+2
126 . . , .

0.0
R - Ropt (a 0)

 

 

l f I

0.2 0.4

Figure 8. Distance dependence of (9 for C1; calculated with various models using the
aug-cc-pqu basis.

77

 

6) (e302)

 

_ 1 +
C12(Zg)

4- -------------- SCF
' --------- CASSCF
- —— CASSCF+1 +2

I l I '71 l l l I T F ' Ij
6 7 a 9 10 11 12
R (ao)

 

 

Figure 9. The quadrupole moment of C12 as a function of internuclear separation with
various models using the aug-cc-pqu basis.

78

Table 3-2. Selection of C12 quadrupole moments.

 

 

R(a0) @(eag ) Reference Comment

3.7568 2.2696 4 SCF

3.757 2.686 5 SCF

3.7568 2.270 6 SCF

3.7568 2.2884 This Work aug-cc—vaz HER“,

3.7267 2.23 73 This Work CBS-HERO,“

3.757 2.652 5 SCF+1+2

3.7568 2.244 6 SDQ-MP4;vibrational
average

3.7568 2.274 10 SDQ-MP4;vibrational
average

3.7875 2.3559 This Work CBS-CASSCF+1+2; Ropt

3.7568 2.2896 This Work aug-cc-vaZ
CASSCF+1+2; Rexp

experimental 3 6810.38 9 optical birefiingence

experimental 2.405i0. 120 8 optical birefringence

vibrational average 2.3860 This Work CBS-CASSCF+1+2

 

Table 3-3. CASSCF + 1 + 2 results for Ropt and 9.

 

Molecule

aug-cc-pqu

estimated complete basis set limit

 

P2

82
C12

*

@opt (ea: )

+0.5035

+1.467O
+2.380l

Ropt(‘1'0)

3.6055

3.5978
3.7875

@opt(eag) vibration correction
0.5241 +0.0197 (v + x)
1.4585 +0.0102(v + x)
2.3559 +0.0301 (v + x)

79

D. On the Distance Dependence of Sz and C12

We study the distance dependence of 0, using the aug-cc-pqu basis and the SCF,
CASSCF, and CASSCF+1+2 wavefunctions. The calculated potential energy curves for
N2, P2, 02, S2, F2, and C12 are shown in Figure 10, and the calculated properties (R., De,
and to.) are very close the CBS CASSCF+1+2 limits. The calculated spectroscopic
properties, also compare well with the experimental values taken from Herzberg.ll The
wavefunctions provide a reasonably accurate description of the molecule’s electronic
structure over a large range of internuclear distances, and we expect the calculated
quadrupole-moment function to be realistic.

The quadrupole moment fiinctions for the molecules of interest are shown in
Figure 11. The behavior of (9 of the second row molecules at large R is the same as
discussed for N2, 02, and F2 in Chapter 2. As with H2 and N2, (9 for P2 is zero at large R
because the molecule separates to atoms in S states. (9 for S2 and C12 separates to the sum
of the atomic quadrupole moments of the atoms. For S2, the oriented S atoms are in
3PM=1 state. For C12, the oriented C] atoms are in the 2PM=0 state. As shown in Figure 10,
the minimum in the potential energy of the second row homonuclear molecules is at larger
R then the first row molecules. In Figure 11, ('9 begins descending at larger R relative to
the first row molecules. The core electrons of the second row molecules shield the
bonding electrons making the electronic distribution of the second row molecules more
diffuse. The more diffuse the electronic distribution the greater the magnitudes of the

various

80

 

Energy (eV)

 

 

J N CASSCF+1+2

 

'I'T'I'TfF'fi
789101112

1 F l
3 4 5 6
R (a0)

Figure 10. CASSCF+1+2 potential energy curves for N2, 02, F2, P2, S2, and C12 using the
aug-cc-pqu basis.

 

 

81

 

  

CASSCF+1+2

'l'l'l'l'l
89101112

 

 

Figure 11. Molecular quadrupole moments of N2, 02, F2, P2, S2, and C12 using the
CASSCF+1+2 and the aug-cc-pqu basis.

82

quadrupole components. This effect is also enhanced by the r2 term in Equation 13 of
chapter 2.

In describing the distance dependence of (9, we first consider S2 and C12, returning
to P2 latter. As in Chapter 2, we first partition (9 into its 0 and 1: components. For
technical reasons, this is done using the natural orbitals from the MRCI wavefunctions.
By setting the occupations of the 1: or o electrons to zero, we can select which symmetry
components of the electronic distribution we are interested in. These are not precisely in
the symmetry determined ratio because our wavefiinction has D211 symmetry and our
correlated wavefunction results in asymptotic p fiinctions that are not equivalent.

The 0 components of the quadrupole-moment curves for N2, P2, 02, S2, F2, and
C12 are given in Figure 12 for a MRCI wavefiinction in an aug-cc-pqu basis. The 1t
components of the quadrupole-moment curves for the same molecules are given in Figure
13. Note that for all molecules 6)., is always negative and (9,. is always positive. The sign
of (9 depends on the relative magnitudes of these contributions and (9 usually decreases
from its asymptotic value with decreasing R. The relative asymptotic values of S2 and C12
depend on the signally occupied atomic orbitals and is structurally identical to those of O2
and F2, respectively. For S2 we have, in an orbital model

®.(S) = (9(3p.)

(93(8) = -3/2 (90(8)
and for C12

(95(Cl) = @(3pz)

o..(c1) = -2 0.,(01)

83

 

 

(9 (ea02)

 

 

 

Figure 12. Distance dependence of the sigma component of the quadrupole moment of
N2, 02, F2, P2, 82, and C12.

84

 

 

(~) (eaoz)

 

 

 

 

Figure 13. Distance dependence of the pi component of the quadrupole moment of N2,
02, F2, P2, 52, and C12.

85

Our numeric asymptotic values for O2 and S2 are

(9.10) = +2435 ea: e..(S) = +6326 ea;

03(0) = -1.499 ea; (90(3) = -4.261 eag
and the relative asymptotic values for F2 and C12 are

9.07) = +2517 eag e..(C1) = +6575 eag

0.0?) = -1.170 eag (90(0) = —3.473 eag
Again, note relative magnitudes of (9., and (9,. of the second row molecules are much
larger their first row counterparts due the larger spatial distribution of the electronic
charge.

Referencing each molecular quadrupole-moment fimction to is asymptotic atomic
contribution, we obtain Figures 14-16, which are simply 0...,(11) and its component 9°...“
and (9"m1. In the region near R0,”, for S2 there is an increase in (9"...01, which is a result of
spilling of Sn, (see Chapter 2) into the P region. For @“ml in C12 there is a small decrease
due to 5n,I spilling into the N region. These characteristics are illustrated by the contour
plots in Figures 17 - 22. The equilibrium bond length in these molecules is smaller than
the internuclear distance where (9"...01 is maximum, so, the slopes of both the 1: and 6
component of (9 are both positive around Re. The fiinctions for S2 and C12 are very similar
to those of O2 and F2 . The sign of the quadrupole moment is determined by the relative
internuclear separation while the magnitude of the quadrupole moment is dominated by

the (9,. contribution.

86

opt

 

2
(ea0 )
1‘

 

®=C9
mol
L. o
I . 1
‘2
3

 

- MRCI
fir ' 1 . l - I ' u
7 9 10 11 12
(a0)

 

I I

 

' I I f I
1 2 3 4 5 6 8
R

Figure 14. Distance dependence of the quadrupole moment of P2 partitioned into its
(9;, and (9,1,0, components.

87

2
(ea0 )
O
1

mol
I
—l
l

(9

 

 

 

 

Figure 15. Distance dependence of the quadrupole moment of S2 partitioned into its
(9;, and (9:0, components.

88

54 C1 (12 +)
2 g

(6802 )

 

(9
mol
0
J l

 

 

 

 

I
89101112

Figure 16. Distance dependence of the quadrupole moment of C12 partitioned into its
(9;, and (9;, components.

89

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95

E. The Quadrupole Moments of N 2 and P2

The isovalent molecules of N2 and P2 have significantly different quadrupole
moments. In Chapter 2, we developed methods of analyzing the quadrupole moment in
terms of the bonding density. It is our intention to use these techniques to evaluate the
significant differences between the quadrupole moment functions of N2 and P2.

First lets consider the quadrupole moments of the separate atoms. Both the N
atom and the P atom are in the 4S state and therefore the quadrupole moment is zero. The
N; and P2 potential energy surface and the quadrupole moments as a fimction of
internuclear separation is given in Figures 23 and 24. As two nitrogen atoms come
together to form N2, the quadrupole moment immediately becomes negative. With two
approaching P atoms, however, the quadrupole moment first shifts positive. Then as the

two P atoms approach the energy optimized separation (sz < 5.0 a0) the magnitude of
the quadrupole moment begins to decrease. The reason for this can be seen by breaking
the quadrupole moment in o and 1!: symmetry contributions. At infinity the (9,. and (9:,
must sum to zero for both N2 and P2, as shown in Figure 25. Our numeric asymptotic

values of 9., and (9,. for N2 and P2 are

(MN) = +2.041 ea: (9,0)) = +5.592 ea:

(MN) = -2.041 eag (90(1)) = 65% ea:
Referencing each molecular quadrupole-moment fimction to is asymptotic atomic
contribution as in Figure 21, as R approaches Rap, 9,. increases more in P2 then in N2. In
the region near Rom, for P2 there is a greater increase in @"mol, which is a result of spilling

of 8nfl into the P region. The effect in P2 is magnified by the greater spatial distribution

96

 

Energy (eV)

 

CASSCF+1+2

 

 

 

Figure 23. CASSCF+1+2 potential energy curves for N2 and P2 using the aug-cc-pqu
basis.

97

2.0 -
Ropt(N2) R (P)

opt 2

 

 

-0.5-

{4‘ q
o

8‘: -1.0-:

® 4.5-1

-2.04

'2‘“. CASSCF+1+2

 

 

-3.0- 1

35 - 5' N2( 29+)

- . - 5 ............... 1 +
4 P2( 29)

 

 

Figure 24. Distance dependence of the molecular quadrupole moments of N2 and P2
using the CASSCF+1+2 and the aug-cc-pqu basis.

98

P o... ..... N29 =+2.041

m
.3. 2

_ N2 9 =-2.041
4 -l

‘5‘ on =-5.592

 

 

. MRCI

I

 

 

I l r r I I

ll
89101112

Figure 25. Distance dependence of the molecular quadrupole moments of N2 and P2
partitioned into (9., and (9,, components.

99

 

®=®mol (eao)

 

. 2 P R N & P MRCI
. 2 opt 2 2

 

 

 

 

'l'l'l'l'l'l

I
6(789101112

Figure 26. Distance dependence of the quadrupole moment of N2 and P2 partitioned into
their 9;, and 6):”, components.

100

and the r2 term of the operator, thus giving P2 its positive quadrupole moment. Since this
spatial distribution increases moving down the periodic table, the magnitude of the

quadrupole moment should also increase.

F. Conclusion

We have studied the quadrupole moments of P2, 82, and C12 and have estimated
the CASSCF+1+2 basis-set limit for the latter two. The C1; quadrupole moments are in
excellent agreement with calculated values by others and in good agreement with the
existing experimental data. The rather large values of the quadrupole moment derivatives,
shown in Table 3-1, result in the quadrupole moment being a very sensitive function of R

around Re. We have provided the global quadrupole-moment function as the sum of an
atomic contribution and a molecular contribution 0mg]. We have also provided reasons

for the differences in the quadrupole moment between the first and second row molecules.
Additional contour plots and three-dimensional images of Sn and 69, as well as
detailed numerical values of G) as a firnction of K are available via our web page,

www.cem.msu.edu/~harrison.

Table 3-4. Comparison of MP2 and CASSCF + 1 + 2 results.

 

 

 

Molecule MP2 (estimated CBS) CASSCF + 1 + 2 (estimated CBS)
Ropt(a 0) eopt (ea: ) Ropt(aO) 90m (ea: )
P; 3.6171 +0.5317 3.6055 +0.5241
52 3.5801 +1.4812 3.5978 +1.4585

C12 3.7440 +2.3566 3.7875 +2.3559

 

101

G. References

1.

10.

11.

Shepard, R., Shavitt, I., Pitzer, R. M., Comeau, D. C., Pepper, M., Lischka, H.,
Szalay, P. G., Ahlrichs, R., Brown, F. B., and Zhao, J. G., Int. J. Quantum Chem.
Symp, 1988, 22, 149.

MOLPRO is a package of ab initio programs written by H.J.Wemer and
P.J.Knowles, with contributions from J .Almof, R.D.Amos, M.J.O.Deegan,
S.T.Elbert, C.Hampel, W.Meyer, K.Peterson, R.Pitzer, A.J.Stone, P.R.Taylor,
R.Lindh, M.E.Mura, and T.Thorsteinsson

Gaussian 92, Revision A, Frisch, M. J ., Trucks, G. W., Head-Gordon, M., Gill, P.
M. W., Wong, M. W., Foresman, J. B., Johnson, B. G., Schlegel, H. B., Robb, M.
A., Replogle, E. S., Gomperts, R., Andres, J. L., Raghavachari, K., Binkley, J. S.,
Gonzalez, C., Martin, R. L., Fox, D. J., Defrees, D. J., Baker, J ., Stewart, J. J. P.,
and Pople, J. A., Gaussian, Inc., Pittsburgh PA, 1992.

Glasier R., Horan Cl, and Haney P.E., J.Phys.Chem. 1993, 97, 1835

G. Maroulis, J.Mol.Struct.(THEOCI-IEM), 279 (1993) 79

J .F.Williams and R.D.Amos, Chem.Phys.Lett. 70 (1980) 162

G.Maroulis, Chem.Phys.Lett. 199 (1992) 244

AD. Buckingham, C.Graham and J .H. Williams, Mol. Phys, 1983, 49, 703.
R.J.Emrich and W. Steele, Mol.Phys. 40 ( 1980) 469.

E.F.Archibong and A.J.Thakkar, Chem.Phys.Lett. 201 (1993) 485

Huber, K. P. and Herzberg, G., “Molecular spectra and molecular structure,”
Vol. 4, Constants of Diatomic Molecules (Van Nostrand-Reinhold, New York,
1979)

CHAPTER 4

The Quadrupole Moment of H2, Liz, and Na;

A. Introduction

From the preceding discussions of Chapters II and III, we see that the sign of G)
for N2, 02, F2, P2, 52, and Cl; depends on the relative values of (Bar (negative) and 6),,
(positive). The relative magnitudes of (9., and 6),. depends on the extent of hydridization.
The magnitude of (9,. depends on the distance between the nuclei and the number of 1c

orbitals involved in the bonding. The F2 6 contribution, for instance, is less negative then
N2 because of only slight sp hybridization in F2. In N2, the large sp hybridization causes

the 90 to be dominant and (9 is decidedly negative. For systems such as H2, Liz, and
N32, however, the situation is fimdamentally different. As previously discussed in Chapter
H, the sign in H2 is always positive because at large R the s-s bond results man, being
negative in the region where the quadrupole operator (cosze-l) inverts the sign to

positive. As the nuclear separation decreases @(HzR) increases and reaches a maximum

before R reaches Rom, after which (9 decreases toward zero. This Chapter presents our

results for Li; and Na; and compares these results with H; of Chapter 2.

B. Methods

The Dunning cc-vaz basis sets are unavailable for Li and Na, so a (14s7p4d3f) Li
basis contracted to [6sSp4d3f] was derived from vanDuijenveldtl plus J. Williams2 p
fimctions and augmented with an evenly spaced set of d and f functions. This basis will be
refered to as the V52 basis. A (19512p3d2f) contracted to [6sSp3d2f] VQZ basis fi'om
the MolPro96 library3 was employed for Na. An AN 0 basis fiom Widmark, Malmquist,

and Roos4 (WMR) was used for both Li and Na as comparison basis sets. These basis

103

104

sets consisted of (14s9p4d3f) contracted to [655p4d31] for Li and a (17512p5d4t)
contracted to [7s6p4d3f] for Na. Both, the V52 basis for Li; and the VQZ basis for Na;
gave a lower total energy then did the WMR basis, therefore, we use these results and our
‘best’ results.

The CASSCF for Li; consisted of 8 orbitals including 208, 2c“, 308, 3o“, 2mm,
21:”, 21%, and the 27:”. The CASSCF+1+2 wavefirnction include all single and double
excitations out of the CASSCF reference space. The Na; CASSCF wavefunction
consisted of similar 8 orbital space including the 408, 46“, 503, Son, 37:“, Buy“, 37%, and
the 3a,,8 orbitals. Again, the CASSCF+1+2 wavefimction was derived from the CASSCF
reference space and included all single and double excitations. This wavefunction is
deemed the best due to its flexibility. All RHF, CASSCF and CASSCF+1+2 calculations

were done using the MolPro96 package3.

C. Results

The values for calculated properties of U; are collected in Table C-2 of Appendix
C. Our SCF quadrupole moment for U; was 11.1478 eag at the equilibrium geometry of

5.2610 a0 using our largest basis, the VSZ. This compared well with SCF quadrupole of
11.1486 ea: calculated with the WMR basis at ROP. of 5.2609 a0. The SCF quadrupole
moment for both the V52 basis and the WMR basis calculated at the experimental

geometry of 5.0510 a0 was 10.6432 ea: and 10.6411 eag, respectively.

105

 

 

 

11.50 -
11.25 -
11.00 -
(“go 10.75 -
3
® .
10.50 -
10.25 - """ " """ RHF
--------0 CASSCF
—A— CASSCF+1 +2
10-00 I w I ' I T I . I
-0.4 -02 0.0 02 0.4
R-Ropt (ao)

Figure 1. Distance dependence of G) for Li; near Rap. calculated with various models
using the VSZ basis.

106

By comparing the optimized SCF geometry, total energy, and dissociation energy to other

large basis calculations”, we conclude that our values are very close to the SCF limit.
Our CASSCF quadrupole moment was 10.8891 eag at the optimized geometry of 5.0986

a0 using the V5Z basis, while the values determined with the WMR basis was 10.8846
eag at an Rap, of 5.0972 a0. The CASSCF quadrupole moment at the experimental

separation using the V5Z basis and the WMR basis was 10.8016 ea: and 10.8034 ea: ,

respectively. Correlation from configuration interaction of the CASSCF+1+2 changes the

quadrupole by a small amount. The CASSCF+1+2 quadrupole moment using the V5Z

basis was 10.9213 eag at the optimized separation of 5.0971 a0. Using the WMR basis
the optimal separation was the same and the quadrupole moment was 10.9210 eag. The
CASSCF+1+2 quadrupole moment at the experimental geometry was 10.8367 ea: using

the V52 basis and 10.8372 ea; using the WMR basis.

A plot of the @(R) in the region of R0,, is given with the VSZ for the 3
wavefunctions employed is shown in Figure 1. From the plot we can see the CASSCF and
CASSCF+1+2 quadrupole functions are virtually on top of one another while the RHF has
a much larger slope then either the CASSCF or the CASSCF+1+2. The first and second
derivatives of @(R) near Rep. are given in Table 4-1. The first derivative of the SCF ©(R)

was 2.4293 eao and the second derivative was -0. 1976 e. The first derivatives of the

CASSCF and CASSCF+1+2 was 1.7718 ea0 and 1.7974 eao,

107

respectively, while the second derivatives for the CASSCF and CASSCF+1+2 were
-0.7220 e and -0.7154 e. Using the CASSCF+1+2 derivatives, we estimate the vibrational

dependence of the CASSCF+1+2 quadrupole moment as

@(Liz; v) = +109213 + 0.0650(v+ -;-)

and so our vibrationally corrected quadrupole moment is @(Li2;v = 0)= +10.9538 eag

There is no reported experimental quadrupole moment for Li2. Plots of @(R) over larger
distances is given in Figure 2. The RHF does well near the optimized geometry but fails at
larger separations. The CASSCF corrects the SCF curve and follows the CASSCF+1+2

closely.

Table 4-1. First and second derivatives of @(R) near Rap, using various models and the
V52 basis for Li; and the VQZ basis for N32.

 

[9L9] (e)
Molecule Wavefunction Ropt(ao) (213%,, (eao) dR2 Ropt

 

Li2 SCF 5.2609 2.4293 -0. 1976
Li2 CASSCF 5.0986 1.7718 -0.7220
L12 CASSCF+1+2 5.0971 1.7974 -0.7154
N32 SCF 6.0365 2.7023 -0. 1583
Naz CASSCF 6.0118 1.6013 -1.0010
N32 CASSCF+ 1 +2 6.0030 1.6377 -0.9963

 

Our results for Na; are collect in Table C-3. Our calculated RHF quadrupole

moment for Na; was 12.1456 eag at 6.0365 a0 using the VQZ basis. The RHF results
with the WMR basis was 12.1456 ea: at a separation of 6.0414 a0. The RHF results at

the experimental geometry of 5.8182 a0 was 11.5392 eag with WMR basis

108

1

. 1 + , ....................
L12( 2g )

(9 (eaoz)

 

 

 

 

2- --------- CASSCF

' CASSCF+1+2

0 . r I I I I I I I I I I I I

2 4 6 8 10 12 14 16
R(a0)

Figure 2. Distance dependence of 9 at large R for Li2 calculated with various models
using the V5Z basis.

109

and 11.5359 ea: with the VQZ basis. Our WMR basis, CASSCF quadrupole moment
was 11.6614 eag at a nuclear separation of 6.0141 a0 and our VQZ, CASSCF

quadrupole moment was 11.5966 eag at a separation of 6.01 18 a0. At the experimental
geometry, both the WM CASSCF and the VQZ, CASSCF gave quadrupole moments of

11.3183 ea: and 11.2665 eag. The CASSCF+1+2 quadrupole moments for the WMR
and VQZ basis was 11.6905 eag at 6.0055 a0 and 11.6264 eag at 6.0030 a0,
respectively. The CASSCF+1+2 results at the experimental geometry were 11.3565 eag

for the WMR and 11.3054 ea: for the VQZ.

A plot of the @(R) in the region of Rep. is given with the VQZ basis for the various
wavefunctions in Figure 3. The plot shows the CASSCF and CASSCF+1+2 @(R) are

very similar while the RHF curve is above and has a larger slope. The RHF first derivative

is 2.7023 ea0 and the second derivative is 01583 e. The CASSCF first derivative is
1.6013 eao , and as with Li2, the N32 CASSCF+1+2 first derivative has a similar value of
1.6377 eao. The second derivatives for the CASSCF and CASSCF+1+2 @(R) are -
1.0010 e and -0.9963 e, respectively. Using the CASSCF+1+2 first and second

derivatives our vibrationally corrected quadrupole moment for Na; is

@(Na2 ;v) = +11.6264 + 0.0215(v + g)

G) (eaoz)

13.25 -

 

110

   
  
    
    
    

 

1 Na (12 +) .-
13.00 — 2 g .-

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12.75 - .-"

u .3”
12.50 a

.'I

.1 I.'
12.25 -

u- .‘l
12.00 4
11.75 - ,

. 3'"
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—A— CASSCF+1 +2
10.75 I I l I ' l I 1
-0 4 -02 0.0 02 0.4
opt (30)

Figure 3. Distance dependence of (9 for Na; near R0,. calculated with various models
using the V5Z basis.

25-

20-

 

111

1 +
Naz(2g)

.....
..........
.“ ‘

 

 

 

10-
5-
............... RHF
. --------- CASSCF
CASSCF+1+2
0IIIIIIIIIIIII1
2 4 6 8 10 12 14 16

R (a0)

Figure 4. Distance dependence of (II) for Na; at large R calculated with various models
using the VQZ basis.

112

and so our vibrationally corrected quadrupole moment is ®(Na2;v = 0)= +1 1.63 72 eag.

The large magnitude of the quadrupole moment makes the vibrational corrections
relatively small for both Li2 and Na2 . There is no reported experimental quadrupole
moment for Na2 dimer. Plots of @(R) over larger distances is given in Figure 4. The
figure shows how closely the CASSCF and CASSCF+1+2 follow one another, while the

RHF does rather well near the Rap. but fails at larger separations.

D. H2, Liz, and Na2

Figure 5 contains the relative potential energy functions and Figure 6 contains
@(R) for H2, Li2, and Na2. The dissociation energy of H2 is much larger than Li2 or Na2,
while the quadrupole moment is much smaller. A maximum in @(R) occurs on each curve
before the nuclear separation reaches the optimized geometry. This maximum in @(R) is
attributed to a maximum in the decrease of electron density on either side of the nuclei
along the bond axis. As the bond length moves towards zero, the electron density moves
towards spherical symmetry and the associated quadrupole density approaches zero. All
three molecules have similar shaped @(R) and differ only by magnitude. Li2 and Na2 have
a significantly larger quadrupole moment due to their difl‘use charge distribution and is

discussed in the next section.

E. The Quadrupole Density of Li2 and Na2

The left-hand diagrams of Figures 7, 8, and 9 are contour plots of the density

difference 5n(r) of H2, Li2, and Na2. Looking first at H2, we see a density increase,

113

 

 

Energy (eV)

 

 

. H CASSCF+1+2

1 I I l I l fl I I I l r I I l

1
0 2 4 5 8101214161820
Ra

(0)

 

 

Figure 5. CASSCF+1+2 potential energy curves for H2, Li2, and Na2 using the aug-cc-
pv52, ASZ, and VQZ, respectively.

(9 (eaoz)

 

2:
R

114

IrlTl ~~~~~
1214

.0
- ‘0
-.-.—‘.'L -------

18

' I I
10 16
1a,)

Figure 6. Distance dependence of (9 for H2, Li2, and Na2 using the CASSCF+1+2
wavefimcvtion and aug-cc-vaz, V5Z, and VQZ basis, respectively.

115

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118

denoted by blue contours, mostly between the nuclei. Outside of this region is the
depletion of density, denoted by the red contours. This depletion region is much more
diffuse for Li; and N212 then for H2. The minimum in this depletion is also closer to the
nuclei in H2, then in either U; or N32. When we apply the quadrupole operator on 5n(r),
the r2 term has a greater effect on U; and N32 due to their greater diffusive charge then on
H2. The contour diagrams for the quadrupole moment density for H2, Liz, and Na; are
presented in the diagrams on the right of Figures 7, 8, and 9. The large magnitude of the
quadrupole moment of Liz and Na; are well represented by the blue contours within these
figure.

The density difference plots also reveal a complex nodal structure for Liz and Na;
near the core electrons that is not present in the density difference of H2. When 2 H atoms
approach, electron density shifts between the nuclei. This is caused by electron-nuclear
attraction pulling the density towards each nucleus. The nuclei are in-tem shielded from
one another by the density increase. In Li2, a similar situation occurs with the valence
electrons, however, Li2 also has 4 core electrons. The increase in density near the center
of the molecule creates a repulsive force on the core electrons. The core electrons
respond by shifting away from the center to either side of the nuclei as seen in Figure 8.
Again N32 has a similar increase in electron density near the center of the molecule.
Looking at the Figure 9, first at the center of the bond and moving along the z axis
towards one of the nuclei, we see 5n(r) begins positive then becomes negative then again
positive up to the nucleus. In bond formation, density from the core is first repelled by the
density increase of the bonding region. This depletion in density de-shields the nuclei and

attracts electron density closer to the nuclei. Due to the nodal structure of the quadrupole

119

density, these complex features of the bonding have little effect on the quadrupole. The

quadrupole moment is primarily determined by the electron density away from the nuclei.

F. Conclusion

We have studied the quadrupole moments of Liz and Na; and related them to the
quadrupole moment of H2. There are no experimental values for the quadrupole moments

of Li2 or Na2, however, we believe our values to be close to the ‘true’ quadrupole

moments. The CASSCF+1+2 quadrupole moment of U; was +10.9863 ea: while the

CASSCF+1+2 quadrupole moment of N82 was +11.6479 eag. Using techniques
developed in Chapter II, we have described why Liz and Na; have both the sign and
magnitude of the quadrupole moment. The lack of sp hybridization keeps @(R) positive

over all internuclear separations.

120

G. References

1.

2.

Williams, Jr., J. E.; Streitwieser, Jr., A. Chem. Phys. Lett. 1974, 25, 507.
van Duijneveldt, F. B. IBM Research Report No. R1945, 1971.

P.-O.Widmark, P.-A.Malmquist and B0. Widmark, Malmquist, and R005,
Theor. Chim. Acta 77, 291 1990

MOLPRO is a package of ab initio programs written by H.J.Wemer and
P.J.Knowles, with contributions from J .Almof, R.D.Amos, M.J.O.Deegan,
S.T.Elbert, C.Hampel, W.Meyer, K.Peterson, R.Pitzer, A.J.Stone, P.R.Taylor,
R.Lindh, M.E.Mura, and T.Thorsteinsson

D.W.Davies and G.J.R.Jones, Chem. Phys. Lett. 81, 179 1981

R.P.Subramanian, M.A.Lee, K.E.Schmidt, J .W.Moskowitz, J .Chem.Phys. 97,
2600 1992

APPENDIX A

Calculation of the Quadrupole Moment

A. Multipole Moments
The potential at an arbitrary point P created by a static molecular charge

distribution can be written as

_ _— 77(7) 3‘ nuclei Zk
V(R)- [W r+ ; lR—Fkl (1)

where the first term is the potential due to the electronic charge distribution 7]( i" ) and

 

the latter term represents the contribution due to the nuclei Zk. This form of the

electrostatic potential is written in terms of an arbitrary origin 0 where the distance

IR — Fl is the distance between R and the corresponding 1“ , as shown in the Figure.

P

 

 

Figure A-l. The potential at an arbitrary point P created by a static molecular charge
distribution.

The point P is taken as infinitely away fi'om the charge distribution and the nuclei are
considered as fixed point charges. The nuclear contributions can thus be computed
analytically as classical charges and the electrostatic potential can be evaluated

independently of the origin."2

122

123

If we consider the nuclear component first, 9,, can be taken as the angle

between R and Fk , where 7k is the correspond nuclear coordinate, by law of cosines
|R - Fl becomes
— _ 2 2 2

= R2[1— (2 cosep)(%’f) + (Iii—)2] (2)

Taking the square root and reciprocal gives

-%
1 1 r r 2
IR— ‘l 2 RP — (2c056p fi)+(_1%) :l (3)

Using the series expansion

 

113.311,“...

(1+a)" =1+na+n

(4)

gives

2
fi—E—é—l = 7141 + cosfipgfj + %(3 cos2 6,, — 06%) +...] (5)

with this expansion the potential at the point P becomes

V(R) = %i[l + cosflpffij + “3 cos2 6,, —1)(%)2+...]zk (6)

k=1

which can be written as

 

- N 1 N N 3 26 -1
V(R)=lZZk+—2-ZZ,,rkcostSlp+-1—32Zkrk2 cos P I”
Rk=l R k=l R k=1 2

(7)

For a continuous cylindrically symmetric charge distribution as in the electrons about

the nuclei, the corresponding expansion of the potential can be written as

124

 

2 2
HR) = - I-ngd3r+I—r2(2r) cosa) d3r+Ir ggr)[3cosza)— ljd3’+'”](8)

where cosa) = cos(9 - 6p) The negative sign reflects the negative charge of the

electronic distribution. The first terms, %. , is the monopole term, the second term
, A2 , is the dipole, the third term, A3 , is the quadrupole; the next is the octopole,
etc...

Writing sum of both the nuclear and electronic components for a molecule'in

Cartesian form gives

Q=-l n<F>dr+§Z. (9)
p. =21 «molt +§Z.r. (10)

Gal, 2 —%Iq(F)[3rarfl —r26afi]dr+:i=:Zk[3rarfl —r25afl]

 

 

m (11)
=—}/ F 23cosztfil—l 1+ Zr2 3c0526—l 2
21m )4 Z ( )/
oringeneral:
:__(_ 1)m- 2m+l a”. (1)
M05 d
I“) are, .a T
(12)
+""“( "3"" ——Zr 2m+1 5'" (l)
! " a,a,...a,, r

where Q, 11..., Gap, and Map”... are components of the charge, dipole, quadrupole, and in

general'm‘h rank tensors, respectively.

125

B. Experimental Evaluation

Most experimentally determined values of molecular quadrupole moments
were obtained by indirect methods involving the interaction between two or more
molecules. Thus pressure broadening of the pure rotational and inversion spectra of
gases can lead to information about molecular quadrupole moments.‘"5 Other indirect
methods involve the use of the second virial coefficients6, heats of sublimation of
single crystals at 0 °K 7, and dielectric constants!”9 Quadrupole moments determined
by any of these methods depend on assumptions made about the sign of the
quadrupole and the interaction potential. 1°

In 1958, AD. Buckingham presented a direct experimental technique to
determine the molecular quadrupole moment of gaseous non-dipolar molecules.11 The
apparatus consisted of a glass tube enclosing four parallel wires acting as condensers.
The gaseous molecules between the wires experience a torque in the field gradient
created by the wires with charge. The quadrupole moment of the molecules can then

be evaluated from the index of refraction by the relationship

 

n, - ny = 27zNF,;[B’ + 29(a” _ an] (13)

1 SkT

where (a // - a .L) is the anisotropy in the molecular polarizability; B describes the

change in the effective polarizability induced by the field gradient, Fxx’ is the externally

generated fieled gradient, and T is temperature. This technique has given N2 (12;) a

quadrupole moment of -1.05 803 .12 This direct experimental technique, however,

126

has proven to be extremely difficult, especially for diatomic gases that are highly
reactive. 12'”
C Theoretical Evaluation
1. A b-Initio Methods
Throughout this work, we use the Augmented Dunning Correlation Consistent

16‘1“ and refer to them as aug-cc-vaz, where X is a cardinal number (2-5)

basis sets
characterizing the basis. Because of technical limitations in the COLUMBUS
properties program, we deleted angular functions of g or greater. The SCF, Moner-
Plesset and CISD calculations were done using g92/DFT[17], MCSCF and MRCI
calculations used the COLUMBUS” suite of codes and the fun CASSCF and
CASSCF+1+2 were done using MOLPRO”. We use the traceless form of the

quadrupole-moment operator‘, which, for a homonuclear diatomic, reads
" 1 N' 2 2 1 2
(9,, Eek—5262,. —r, )+EZR (14)

where Z is the atomic number of the nuclei in the diatomic and R is the internuclear

separation. We use atomic units throughout, except where explicitly noted. In these

units, one atomic unit of quadrupole moment (eag) contains 1.344911 Buckinghams,

4.4866x10-40 CMZ, and 1.344911 x 10-26 esu cm2.

We estimate the complete basis set(CBS) limit!“ of a property by fitting the
property values to a function of the form
P(X)= P(oo)+Be'CX (15)

where X is the cardinal number of the basis set and P00 is the property calculated with

that basis.

127

Appendix C contains the collected tables of the quadrupole moments and other
properties of 11262;), Li2(‘>:,*), 13262,), C262,“), N2 (12;), 02(328'), F2 (12;), Ne2
(12;), 16.12623), $262,), P2 (12;), $262,), and Cl; (12;) for various wavefimctions
as a fimction of basis set. The MPZ wavefimctions include excitations from the log

and lou orbitals of H2 through F2. For molecules N2 through F2, the MCSCF

wavefunctions are CASSCF fimctions over 6 MO’s derived from the valence p orbitals
(the 36g, 30“, lnux, lnuy, lrtgx, Ingy). The MRCI incorporates all single and
double excitations from these six orbitals, as well as all double excitations from the
20g and Zou orbitals. The CASSCF wavefunctions include the 203 and 26.. orbitals
along with the 6 orbitals described above, and the CASSCF+1+2 wavefunction
consists of all single and double excitations from the CASSCF reference space. For
02, both ROI-IF and UHF results are reported.

For the second row homonuclear diatomics (Na; through C12), The MPZ

wavefirnctions include excitations from the 40g and 4011 orbitals. The CASSCF
wavefiJnctions include all valence electrons and the 8 orbitals (the 463, 4c“, 50g, Sou,
21tux, 27tuy, Zagx, any), and the CASSCF + 1 + 2 wavefirnction consists of all single

and double excitations from the CASSCF reference space. As with 02, both ROHF
and UHF results are reported for 82. Of the wavefunctions used, the best is expected
to be the CASSCF+1+2. This wavefimction utilizes the optimized orbitals from the
CASSCF to provide much of the correlation energy and a large active space able to

accurately describe the valance region of the molecule.

128

2. Expectation Values
The general procedure for calculating physical properties requires each of the
dynamical variables of the system to be represented in the formalism by a linear
operator. The process of evaluating a physical quantity then corresponds to operating

on the wave firnction by the corresponding Operator. Consider an arbitrary dynamical

variable which is represented by the operator 0. Then the ‘expectation value’ of 6,

when the system is in the state 111, is given by

(0) = IT'O‘I’dT (16)

This is the mean value of 0 which will be obtained by making repeated measurements
when the system is known to be in the state w prior to each measurementzo'”.

Permanent multipole moments of molecules are first order properties that can

be given as expectation values of the corresponding one-electron operator.
93%Z(3’ic’ip —6aflri2)q, (17)
resulting in
(9,, = —%jn(r)[3rar, — #5,, ]dr (18)
where ra and I") are a Cartesian coordinates and r2 = (x2 + y2 + 22). For a

cylindrically symmetric system
—%Oa=®n=® =—%O (19)

W

The corresponding quantum mechanical expectation value is

N
Z -}2[3’ia’tp ‘ 525a51w> + GZUC (20)
i=1

 

129

where
om: Z 3 :25 21
:2 %Z A[ralrfll ’31 afl ( )
,1

where 7t is the index of the nucleus. Given a wavefiinction determined by ab initio
techniques, the quadrupole moment can be determined by effectively operating the

wavefunction on the appropriate operator and adding in the nuclear contribution.”

3. Point Charge Method

An alternate approach to the calculation of moments and polarizabilities uses
one or more fixed charges placed in a vicinity of the molecule.23 One or more charges
are placed around the molecule in regions where the molecular wavefunctions are
negligible. This technique allows for the incorporation of uniform and nonuniform
electric field contributions due to gradients and higher order field derivatives. The
static properties are then determined as extrapolations fi'om the diminishing interaction
energies. It is important that the basis set used be adequate to describe any
polarization of the molecule in the presence of this field.24

In this method the field applied should be small enough to let a truncated
expansion accurately represent the moment or energy and strong enough to ensure an
efi‘ect can be calculated with numerical significance. Generally charges are place 30 -
60 a0 from the center of the molecule depending on the magnitude of the charge, and
the resulting uniform field components range from 0.002 to 0.05 eao. If the charges
are placed in special arrangements around the molecule, the contributions from the

field gradients can be eliminated. For instance, with a homonuclear diatomic, two

130

charges could be placed along the molecular axis at distances equal but opposite in
magnitude to the center of the molecule. This would prevent an additional induced

dipole and dipole-quadrupole contribution to the potential.
The Hamiltonian of a molecule If” -in the presence of a field may be expanded

about the field free Hamiltonian [30 in terms of the molecular multipole moment

tensors as
Hp = 190 —i1aFa _%®aflFafl—%SaaflyFafly "” (22)
and the corresponding perturbed energy[25]

E” = E0 'flaFa —%®¢Fafi -%,Q¢7Fafi7
‘ )6 AafirFaFflr " % Cafi.75FaflFn5 "" (23)
— %Bam,FanFfl,+-~

where Fa, F043, are the field, field gradient, etc. at the origin. E0, 1.1., O, Q are the
energy, dipole, quadrupole, and octopole moment tenors, respectively and A, B, and C
are the pure dipole, dipole-quadrupole, and pure quadrupole polarizability tensors of
the field free molecule.26

From Equation 23 it can be seen that the energy of a charge system in an
electrostatic field is the sum of the energies of a point charge in a potential, a dipole in
a uniform field, a quadrupole in a field gradient, and an octopole in the gradient of a
field gradient. In general, there will be a torque acting on a rigid charged body in an
electric field, the torque about any point being proportional to the multipole moments
about that point as origin.

As stated earlier, if the charges are placed in special arrangements around the

molecule, the contributions fiom the field gradients can be eliminated and induced

131

dipoles can be prevented in such a way to conveniently determine the quadrupole
moment for a homonuclear diatomic. By placing two point charges at equi-distant yet
opposite sides of the origin along the z axis, the quadrupole moment can be

determined by the interaction energy as shown in the figure below.

 

 

 

Where IRLl = lRRl, for a symmetric charge distribution about the Z axis

E”=E°-#E-%®F;-%QE;~ (24)
becomes
E’—E°=-}§®F,;—%QF;... (25)
where
‘ = 4 - "‘ 1
F. /R., F... cc 4. (26)

where 6 becomes the intercept of the line

y.(AE)R3 =—®—% , (27)

4. Finite Field Method
The ab initio calculation of moments and polarizabilites can be accomplished
directly from the solution of the Schrodinger equation with an augmented

Hamiltonian. The finite-field approach incorporates the electric field terms directly

132

into the Hamiltonian for a fixed field strength and the wavefunction is solved at that
strength.25 This method has not been employed in this work and is mentioned only
briefly. This method is more thoroughly described by Sadjle26 in and the references

therein.

133

D. References

l.

10.

11.

12.

13.

14.

15.

1 6a.

16b.

A. Shadowitz, “The Electromagnetic Field”, Dover Publications, Inc., New
York, 1988, p.242

R. Becker,”Electromagnetic F ielda and Interactions”, Dover Publications, Inc.,
New York, 1982, p.80

A.D. Buckingham, Trans. Faraday Soc. 1956, 52, 747

Gordy, Smith, and Trambarulo, “ Microwave Spectroscopy”, John Wiley &
Sons, Inc., New York, 1953, Chap. 7

W.V. Smith, J.Chem. Phys. 1956, 25, 510

J .A. Pople, Proc. Roy. Soc. (London) 1954, A221, 508

Jansen, Michels, and Lupton, Physica 1954, 20, 1235

AD. Buckingham and IA. Pople, Trans. Faraday Soc. 1955, 51, 1029
R.W. Zwanzig, J. Chem. Phys. 1956, 25, 211

DE. Stoygryn and AP. Stogryn, Mol. Phys, 1966, 11, 371

AD. Buckingham, J.Chem.Phys., 1959, 30, 1580.

AD. Buckingham, C. Graham, and J .H. Williams, Mol. Phys, 1983, 49, 703

AD. Buckingham, R.L. Disch, and D.A.Dunmur, J. Amer. Chem, 1968, 90,
3104

C. Graham, J. Pierrus, and RE. Raab, Mol. Phys, 1989, 67, 939
RA. Bonham and M. Fink, Phys. Rev. A 1986, 33, 1569

Dunning, T. H., Jr., J. Chem. Phys. 1989, 90, 1007; Kendall, R. A., Dunning,
T. R, Jr., and Harrison, R. J., J. Chem. Phys. 1992, 96, 6796; Peterson, K. A.,
Kendall, R. A., and Dunning, T. H., Jr., J. Chem. Phys. 1993, 99, 9790.

Basis sets were obtained from the Extensible Computational Chemistry
Environment Basis Set Database, Version 1.0, as developed and distributed by
the Molecular Science Computing Facility, Environmental and Molecular
Sciences Laboratory, which is part of the Pacific Northwest Laboratory, P. O.
Box 999, Richland, Washington 99352, USA, and firnded by the U. S.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

134

Department of Energy. The Pacific Northwest Laboratory is a multi-program
laboratory operated by Battelle Memorial Institute for the U. S. Department of
Energy under contract DE-ACO6-76RLO 1830. Contact David Feller, Karen
Schuchardt, or Don Jones for firrther information.

Gaussian 92, Revision A, Frisch, M. J ., Trucks, G. W., Head-Gordon, M.,
Gill, P. M. W., Wong, M. W., Foresman, J. B., Johnson, B. G., Schlegel, H.
B., Robb, M. A., Replogle, E. S., Gomperts, R., Andres, J. L., Raghavachari,
K., Binkley, J. S., Gonzalez, C., Martin, R. L., Fox, D. J., Defrees, D. J.,
Baker, J ., Stewart, J. J. P., and Pople, J. A., Gaussian, Inc., Pittsburgh PA,
1992.

Shepard, R., Shavitt, 1., Pitzer, R. M., Comeau, D. 0, Pepper, M., Lischka, H.,
Szalay, P. G., Ahlrichs, R., Brown, F. B., and Zhao, J. G., Int. J. Quantum
Chem. Symp. 1988, 22, 149.

MOLPRO is a package of ab initio programs written by H.J.Wemer and
P.J.Knowles, with contributions from J .Almof, R.D.Amos, M.J.O.Deegan,
S.T.Elbert, C.Hampel, W.Meyer, K.Peterson, R.Pitzer, A.J.Stone, P.R.Taylor,
R.Lindh, M.E.Mura, and T.Thorsteinsson

P.A.M.Dirac,The Priciples of Quantum Mechanics. Oxford Science
Publications, 4th Ed., 1991

E.Merzbacher, Quantum Mechanics, John Wiley & Sons, 2nd Ed., 1970

A. Szabo and N.S.Ostlund, Modern Quantum Chemistgg, McGraw-Hill
Publishing, lst. Ed. rev., 1982

AD. Buckingham, in Intermolecular Interactions: From Diatomics to
Biopolymers, edited by B. Pullman (Wiley, New York, 1978), p. 1.

GO Maitland, M. Rigby, E.B.Smith, and WA. Wakeharn, Intermolecular
Forces: Thier Origin and Determination (Clarendon, Oxford, 1981).

AD. Buckingham, Adv. Chem. Phys, 1967, 12, 107

(a) A.J. Thakkar, J .Chem. Phys.,1981, 75, 4496; A.Koide, W.J. Meath, and
A.R.Allnatt, J .Phys.Chem., 1982, 86, 1222; (b) D.M. Bishop and B.Lam,
Phys. Rev.A, 1988, 37, 464

APPENDIX B

Basis Sets

A. Description

The Dunning correlation consistent basis sets have been defined as to contain
all of the correlating functions which lower the correlation energy by similar amounts
as well as all correlating firnctions which lower the energy by larger amounts.1 For
example, the Nitrogen (9s4pld) primitive set contracted to [3 s2pld] is the simplest
correlation consistent basis set, since the 3s and 2p spatial firnctions lower the
correlation energy by approximately the same amount as the 1d set; all other functions
lower the correlation energy by substantially smaller amounts. This is referred to as
the correlation consistent polarized valence double-zeta (cc-pvdz) set. Other
correlation consistent basis sets for Boron through Fluorine include a (1055p2dlf) set
contacted to [453 p2d1f] referred to as the correlation consistent polarized valence
triple-zeta (cc-pvtz), a (1236p3 d2fl g) contracted to [554p3 d2fl g] referred to as the
correlation consistent polarized valence quadrupole-zeta (cc-pqu) set, and a
(1438p4d3f2g1h) contracted to [6sSp4d3f2g] as the correlation consistent polarized
valence 5-zeta (cc-vaz). Due to technical limitations, calculations in this work have
only included 5, p, d, and, f basis firnctions unless otherwise stated.

Augmenting these basis sets with spatially diffuse basis firnctions dramatically
improves the description of certain properties, especially those strongly influenced by
the outer electron charge distribution. Dunning provides a series of augmented basis
functions, constructed specifically for the evaluation of electron affinities of atomsz'3
For each correlation consistent basis set, an additional 5, p, d, f, g, h basis function is

available. Since these functions are spatially diffiJse, their contribution to the shape of

the molecular potential energy surface is negligible. They do, however, have a

136

137

significant effect on the charge distribution, and therefore, properties, away from the
nuclei. Figure 1 shows a comparison of the molecular potential energy surface of the
cc-pvtz basis set and a CASSCF+1+2 wavefunction to RKR data“. Figure 2 is a
comparison of the total energy of 02(323') using the augmented and non-augmented
basis sets at the ROHF and MPZ level. As can be seen in the figure, both the
augmented and non-augmented converge rather well to the basis set model limit. The
quadrupole moment, however, is much more sensitive to the basis sets used, as can be
seen in Figure 3. Figure 3 is a plot comparing 02(328') ROHF and MP2 quadrupole
moments computed with augment to non-augmented basis firnctions. By looking at
Figure 3, its immediately apparent to the importance of augmenting the basis functions
when modeling a property such as the quadrupole moment.

This appendix lists the Dunning Correlation Consistent Basis Sets5 used in the
main component of this thesis. Attached at the end of each basis set are the
augmented functions optimized for each particular set. The basis sets are listed in a

format that is compatible with the input file for the Gaussian946 suite of codes.

cm -1

 

138

 

4mm- ....................................
3mm-
- 3 -
30000 . 9‘ O2 ( 2g)
25000- I I --------- CASSCF+1+2
. 9’ —0—-RKR
200004 . é
o 0"
1mm4 5
‘ ' CASSCF+1+2 / Experiment
100W 9: 1" Re= 1.2018 Ang. / 1.208 Aug.
‘3 7 De=5.1197eV / 5.241 eV
_ c c
5000 3, (0 =1616cm'1 /1556 cm-1
1 t e
0- ‘0’
I 1 I ' I ' l ' l T l ' l
2

5
R (a0)

Figure 1. Comparison of 02 RKR data to CASSCF+1+2 potential energy curve using

the cc-pvtz basis.

139

 

 

ROI-IF limit

 

)
L.
5‘3
8
I

-14995 _ ...... .. ..... cc_vaZ
- o... —-— aug-cc-vaz

Energy (ea
'55
'8

 

 

 

 

J.
.8
N
o

r
2 3 4
aug-cc-vaz

Figure 2. Comparison of ROHF and UMP2 total energy for 02 using the cc-pvtz and
aug-cc-pvtz basis sets.

140

 

 

 

 

 

-0.10 _
- a O (32 _)
2 g
0.15 _
-0.20 _
-0.25 - "\E UMP2 hrrut
'- . ................. .
0.30 - . .......
(210° -0.35 _
3 .
6) 0.40 4 l
\E R0 t
.045 - .......... .. ................. .. ,,,,,,,,,,,
d .... .....
—0.5o _ .
.. ...... O ..... CC-pVXZ
-0.55 A —a— auQ_Cc_vaZ
‘ 0'
‘060 I j l T l
2 3 4 5 6
aug'CC-pvxz

Figure 3. Comparison of the ROHF and UMP2 quadrupole moment of 02 using the
cc-pvtz and aug-cc-pvtz basis sets.

141

B. References

1.

2.

Dunning, T. R, Jr., J. Chem. Phys. 1989, 90, 1007

Woon,D.E., and Dunning, T. H., Jr., J. Chem. Phys. 1993, 98, 1358
Woon,D.E., and Dunning, T. H., Jr., J. Chem. Phys, 1993, 99, 3730
Flugge,S., Walger,P., and Weiguny, A., J. Mol. Phys, 1967, 23, 243

Basis sets were obtained from the Extensible Computational Chemistry
Environment Basis Set Database, Version 1.0, as developed and distributed by
the Molecular Science Computing Facility, Environmental and Molecular
Sciences Laboratory, which is part of the Pacific Northwest Laboratory, P. O.
Box 999, Richland, Washington 993 52, USA, and funded by the U. S.
Department of Energy. The Pacific Northwest Laboratory is a multi-program
laboratory operated by Battelle Memorial Institute for the U. S. Department of
Energy under contract DE-AC06-76RLO 1830. Contact David Feller, Karen
Schuchardt, or Don Jones for further information.

Gaussian 94, Revision D.3,M. J. Frisch, G. W. Trucks, H. B. Schlegel, P.
MW. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A.
Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G.
Zakrzewski, J. V. Ortiz, J. B. Foresman, J. Cioslowski, B. B. Stefanov, A.
Nanayakkara, M. Challacombe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W.
Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J.
S. Binkley, D. J. Defrees, J. Baker, J. P. Stewart, M. Head-Gordon, C.
Gonzalez, and J. A. Pople, Gaussian, Inc., Pittsburgh PA, 1995.

Hydrogen
aug-cc-pvdz
H 0.000000
S 4 1 .00
1 3.01 0000
1 .962000
0.444600
0.122000
S 1 1 .00
0.122000
P 1 1 .00
0.727000
8 1 1 .00
0.029740
P 1 1 .00
0.141000

0.019685
0.137977
0.478148
0.501240
1.000000
1.000000
1.000000

1 .000000

142

aug-cc-pvtz
H

S 5 1.00
S 1 1.00
S 1 1.00
P 1 1.00
P 1 1.00
D 1 1.00
S 1 1.00
P 1 1.00
D 1 1.00

0.000000

33.870000

5.095000
1 .1 59000
0.325800
0.102700
0.325800
0.102700
1 .407000
0.388000
1 .057000
0.025260
0.102000

0.247000

0.006068
0.045308
0.202822
0.503903
0.383421
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

Hydrogen
aug—cc-pqu

S

0 U '0

TI

0.000000

82.640000
12.410000

2.824000
0.797700
0.2581 00
0.089890
0.797700
0.258100
0.089890
2.292000
0.838000
0.292000
2.062000
0.662000
1 .397000
0.023630
0.084800
0.1 90000

0.360000

0.002006
0.01 5343
0.075579
0.256875
0.497368
0.296133
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

143

aug—cc-vaz
H 0.000000

S 8 1.00
402.000000
60.240000
13.730000
3.905000
1.283000
0.465500
0.181100
0.072790

S 1 1.00
1.283000

S 1 1.00
0.465500

S 1 1.00
0.181100

S 1 1.00
0.072790

P 1 1.00
4.516000

P 1 1.00
1.712000

P 1 1.00
0.649000

P 1 1.00
0.246000

D 1 1.00
2.950000

D 1 1.00
1.206000

D 1 1.00
0.493000

F 1 1.00
2.506000

F 1 1.00
0.875000

G 1 1.00
2.358000

S 1 1.00
0.020700

P 1 1.00
0.074400

D 1 1.00
0.156000

F 1 1.00
0.274000

G 1 1.00

0.543000

0.000279
0.0021 65
0.01 1201
0.044878
0.1 42299
0.330979
0.436269
0.1 76440
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

Boron
aug-cc-pvdz
B 0.000000
S 9 1.00
4,570.000000
685.900000
156.500000
44.470000
14.480000
5.131000
1.898000
0.332900
0.104300
S 9 1.00
4,570.000000
685.900000
156.500000
44.470000
14.480000
5.131000
1.898000
0.332900
0.104300
S 1 1.00
0.104300
P 4 1.00
6.001000
1.241000
0.336400
0.095380
P 1 1.00
0.095380
D 1 1.00
0.343000
S 1 1.00
0.031050
P 1 1.00
0.023780
D 1 1.00
0.090400

0.000696
0.005353
0.027134
0.101380
0.272055
0.448403
0.290123
0.014322
-0.003486

-0.000139
-0.001097
-0.005444
-0.021916
-0.059751
-0.138732
-0.131482

0.539526

0.580774

1 .000000
0.035481
0.198072
0.505230
0.479499
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

144

aug-cc-pvtz
8 0.000000
S 12 1.00
23,870.000000
3,575.000000
812.800000
229.700000
74.690000
26.81 0000
10.320000
4.1 78000
1 .727000
0.470400
0.189600
0.073940
S 12 1.00
23,870.000000
3,575.000000
812.800000
229.700000
74.690000
26.81 0000
10.320000
4.178000
1 .727000
0.470400
0.189600
0.073940
S 1 1 .00
0.470400
S 1 1 .00
0.189600
S 1 1.00
0.073940
P 6 1 .00
22.260000
5.058000
1 .487000
0.507100
0.1 81200
0.064630
P 1 1 .00
0.507100
P 1 1 .00
0.181200
P 1 1 .00
0.064630
D 1 1 .00
1 .1 10000

0.000088
0.000687
0.003600
0.014949
0.051435
0.143302
0.300935
0.403526
0.225340
0.015407
-0.003955
0.001 124

-0.000018
-0.000139
-0.000725
-0.003063
-0.010581
-0.031365
-0.071012
-0.132103
-0.123072

0.261819

0.586662

0.290494

1 .000000
1 .000000
1 .000000
0.005095
0.033206
0.1 32314
0.331 81 8
0.472063
0.257979
1 .000000
1 .000000
1 .000000

1 .000000

145

1 .00
0.402000
1.00
0.145000
1.00
0.882000
1.00
0.311000
1.00
0.673000
1.00
0.027210
1.00
0.018780
1.00
0.046600
1.00
0.113000
1.00
0.273000

1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1.000000 ’
1 .000000
1 .000000

1 .000000

Boron
aug-cc-pqu
8 0.000000
S 101.00
5,473.000000
820.900000
186.800000
52.830000
17.080000
5.999000
2.208000
0.587900
0.241500
0.086100
S 101.00
5,473.000000
820.900000
186.800000
52.830000
17.080000
5.999000
2.208000
0.587900
0.241500
0.086100
S 1 1.00
0.587900
S 1 1.00
0.086100
P 51.00
12.050000
2.613000
0.747500
0.238500
0.076980
P 1 1.00
0.238500
P 1 1.00
0.076980
D 1 1.00
0.661000
D 1 1.00
0.199000
F 1 1.00
0.490000
S 1 1.00
0.029140
P 1 1.00
0.020960
D 1 1.00
0.060400

0.000555
0.004291
0.021949
0.084441
0.238557
0.435072
0.341955
0.036856
-0.009545
0.002368

-0.0001 12
-0.000868
-0.004484
-0.017683
-0.053639
-0.1 19005
-0.165824

0.120107

0.595981

0.411021

1 .000000
1 .000000
0.0131 1 8
0.079896
0.277275
0.504270
0.353680
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

146

aug-cc-vaz
8 0.000000
S 14 1 .00
68,260.000000
10,230.000000
2,328.000000
660.400000
216.200000
78.600000
30.980000
12.960000
5.659000
2.556000
1 .175000
0.424900
0.171200
0.069130
S 14 1 .00
68,260.000000
10,230.000000
2,328.000000
660.400000
216.200000
78.600000
30.980000
12.960000
5.659000
2.556000
1 .175000
0.424900
0.171200
0.069130
S 1 1 .00
1 .175000
S 1 1.00
0.424900
S 1 1.00
0.171200
S 1 1.00
0.069130
P 8 1.00
66.440000
15.710000
4.936000
1 .770000
0.700800
0.290100
0.121 100
0.049730
P 1 1 .00

0.000024
0.000185
0.000970
0.004056
0.014399
0.043901
0.1 1 3057
0.233825
0.353960
0.301 547
0.087521
0.002819
0.000043
0.000078

-0.000005
-0.000037
-0.0001 96
-0.000824
-0.002923
-0.0091 38
-0.0241 05
-0.054755
-0.096943
-0.1 37485
-0.044565

0.324345

0.570414

0.243444

1 .000000

1 .000000

1 .000000

1 .000000

0.000838
0.006409
0.028081
0.092152
0.224164
0.369915
0.374441
0.139086

F

1 1.00

0.163000

1.000000

147

I G) 0 "1

(0

0.700800
1.00
0.290100
1.00
0.121 100
1.00
0.049730
1.00
2.010000
1.00
0.796000
1.00
0.316000
1.00
0.125000
1.00
1.215000
1.00
0.525000
1.00
0.27000
1.00
1.124000
1.00
0.461000
1.00
0.834000
1.00
0.026100
1.00
0.015700
1.00
0.043100
1.00
0.084300
1.00
0.202000
1.00
0.384000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

Carbon
aug-cc—pvdz
C 0.000000
S 9 1.00
6,665.000000
1 ,000.000000
228.000000
64.710000
21.060000
7.495000
2.797000
0.521500
0.159600
S 9 1.00
6.665.000000
1 ,000.000000
228.000000
64.710000
21.060000
7.495000
2.797000
0.521500
0.159600
S 1 1.00
0.159600
P 4 1.00
9.439000
2.002000
0.545600
0.151700
P 1 1.00
0.151700
D 1 1.00
0.550000
8 1 1.00
0.046900
P 1 1.00
0.040410
D 1 1.00
0.151000

0.000692
0.005329
0.027077
0.101718
0.274740
0.448564
0.285074
0.015204
-0.003191

-0.000146
-0.001 154
-0.005725
-0.023312
-0.063955
-0.149981
-0.127262

0.544529

0.580496

1 .000000
0.038109
0.209480
0.508557
0.468842
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

148

aug-cc-pvtz
C 0.000000

8 101.00
8236000000
1 235000000
280.800000
79.270000
25.590000
8.997000
3.319000
0.905900
0.364300
0.128500

S 101.00
8236000000
1 235000000
280.800000
79.270000
25.590000
8.997000
3.319000
0.905900
0.364300
0.128500

S 1 1.00
0.905900

S 1 1.00
0.128500

P 51.00
18.710000
4.133000
1.200000
0.382700
0.120900

P 1 1.00
0.382700

P 1 1.00
0.120900

D 1 1.00
1.097000

D 1 1.00
0.318000

F 1 1.00
0.761000

S 1 1.00
0.044020

P 1 1.00
0.035690

D 1 1.00
0.100000

0.000531
0.004108
0.021087
0.081853
0.234817
0.434401
0.346129
0.039378
-0.008983
0.002385

-0.0001 13
-0.000878
-0.004540
-0.01 81 33
-0.055760
-0.1 26895
-0.1 70352

0.1 40382

0.598684

0.395389

1 .000000
1 .000000
0.01 4031
0.086866
0.29021 6
0.501 008
0.343406
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

149

F 1 1.00
0.268000 1 .000000

Carbon
aug-cc-pqu

C

S

0.000000
12 1 .00
33,980.000000
5,089.000000
1 .157.000000
326.600000
106.100000
38.1 10000
14.750000
6.035000
2.530000
0.735500
0.290500
0.1 1 1100
12 1 .00

33,980.000000
5,089.000000
1 .157.000000
326.600000
106.100000
38.1 10000
14.750000
6.035000
2.530000
0.735500
0.290500
0.1 11 100

1 1.00
0.735500

1 1.00
0.290500

1 1.00
0.1 1 1100

6 1 .00
34.510000
7.915000
2.368000
0.813200
0.289000
0.100700

1 1.00
0.813200

1 1.00
0.289000

1 1.00
0.100700

1 1.00
1 .848000

1 1.00

0.000091
0.000704
0.003693
0.01 5360
0.052929
0.147043
0.305631
0.399345
0.217051
0.015894
-0.003084
0.000978

-0.000019
-0.000151
-0.000785
-0.003324
-0.011512
-0.034160
-0.077173
-0.141493
-0.118019

0.273806

0.586510

0.285430

1 .000000
1 .000000
1 .000000
0.005378
0.0361 32
0.142493
0.3421 50
0.463864
0.250028
1 .000000
1 .000000
1 .000000

1 .000000

150

aug-cc-vaz

C
S

S

0.000000
14 1 .00
96770000000
14500000000
3,300.000000
935.800000
306.200000
1 1 1 .300000
43.900000
1 8.400000
8.054000
3.637000
1 .656000
0.633300
0.254500
0.1 01900
14 1 .00

96,770.000000
14,500.000000
3,300.000000
935.800000
306.200000
1 1 1.300000
43.900000
18.400000
8.054000
3.637000
1 .656000
0.633300
0.254500
0.101900

1 1 .00
1 .656000

1 1 .00
0.633300

1 1 .00
0.254500

1 1 .00
0.101 900

8 1 .00
101 .800000
24.040000
7.571 000
2.732000
1 .085000
0.449600
0.187600
0.076060

1 1 .00

0.000025
0.000190
0.001000
0.004183
0.014859
0.045301
0.1 16504
0.240249
0.358799
0.293941
0.077665
0.002333
0.000505
0.000030

-0.000005
-0.000041
-0.00021 3
-0.000897
-0.003187
-0.009961
-0.026375
-0.060001
-0.106825
-0.144166
-0.024644

0.349009

0.558737

0.228102

1 .000000

1 .000000

1 .000000

1 .000000

0.000891
0.006976
0.031669
0.104006
0.241633
0.371946
0.354200
0.131568

0.649000
1 .00
0.228000
1 .00
1 .419000
1 .00
0.485000
1 .00
1 .01 1000
1 .00
0.041450
1 .00
0.032180
1 .00
0.076600
1 .00
0.187000
1 .00
0.424000

1 .000000

1 .000000

1.000000

1.000000

1.000000

1.000000

1 .000000

1 .000000

1 .000000

1 .000000

151

TI

0 G)

1 .085000
1 .00
0.449600
1 .00
0.187600
1 .00
0.076060
1 .00
3.134000
1 .00
1 .233000
1 .00
0.485000
1 .00
0.191000
1 .00
2.006000
1 .00
0.838000
1 .00
0.350000
1 .00
1 .753000
1 .00
0.678000
1 .00
1 .259000
1 .00
0.039400
1 .00
0.027200
1 .00
0.070100
1 .00
0.138000
1 .00
0.319000
1 .00
0.586000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

Nitrogen
aug-cc-pvdz
N 0.000000
S 91.00
9,046.000000
1 ,357.000000
309.300000
87.730000
28.560000
10.210000
3.838000
0.746600
0.224800
8 91.00
9,046.000000
1 ,357.000000
309.300000
87.730000
28.560000
10.210000
3.838000
0.746600
0.224800
3 1 1.00
0.224800
P 4 1.00
13.550000
2.917000
0.797300
0.218500
P 1 1.00
0.218500
D 1 1.00
0.817000
S 1 1.00
0.061240
P 1 1.00
0.056110
D 1 1.00
0.230000

0.000700
0.005389
0.027406
0.103207
0.278723
0.448540
0.278238
0.015440
-0.002864

-0.000153
-0.001208
-0.005992
-0.024544
-0.067459
-0.1 58078
-0.121 831

0.549003

0.57881 5

1 .000000
0.039919
0.217169
0.51031 9
0.462214
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

152

aug-cc-pvtz
N 0.000000
8 10 1 .00
11 ,420.000000
1 ,712.000000
389.300000
1 10.000000
35.570000
12.540000
4.644000
1 .293000
0.51 1800
0.1 78700
S 10 1 .00
1 1 ,420.000000
1 ,712.000000
389.300000
1 10.000000
35.570000
12.540000
4.644000
1 .293000
0.51 1800
0.178700
S 1 1 .00
1 .293000
S 1 1 .00
0.1 78700
P 5 1 .00
26.630000
5.948000
1.742000
0.555000
0.172500
P 1 1 .00
0.555000

.0
—A

1 .00

0.172500
1 1 .00
1 .654000

0 U

1 .00
0.469000

.11
A

1.00
1.093000

S 1 1.00
0.057600

P 1 1.00
0.049100

D 1 1.00
0.151000

0.000523
0.004045
0.020775
0.080727
0.233074
0.433501
0.347472
0.041 262
-0.008508
0.002384

—0.0001 15
-0.000895
-0.004624
-0.01 8528
-0.057339
-0.1 32076
-0.1 7251 0

0.1 51 814

0.599944

0.387462

1 .000000
1 .000000
0.01 4670
0.091 764
0.298683
0.498487
0.337023
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

153

F 1 1 .00
0.364000 1 .000000

Nitrogen
aug-cc-pqu
0.000000
S 12 1.00
45,840.000000
6.868.000000
1 ,563.000000
442.400000
144.300000
52.1 80000
20.340000
8.381 000
3.529000
1 .054000
0.41 1800
0.1 55200
S 12 1.00
45,840.000000
6,868.000000
1,563.000000
442.400000
144.300000
52.180000
20.340000
8.381 000
3.529000
1.054000
0.41 1800
0.1 55200
S 1 1.00
1.054000
S 1 1 .00
0.41 1800
S 1 1.00
0.1 55200
P 6 1 .00
49.330000
1 1 .370000
3.435000
1 .1 82000
0.417300
0.142800
P 1 1.00
1 .1 82000
P 1 1.00
0.41 7300
P 1 1.00
0.142800
D 1 1 .00
2.837000
D 1 1.00

0.000092
0.000717
0.003749
0.015532
0.053146
0.146787
0.304663
0.397684
0.217641
0.016963
-0.002745
0.000953

-0.000020
-0.000159
-0.000824
-0.003478
-0.01 1966
-0.035388
-0.080077
-0.146722
-0.1 16360

0.279919

0.585481

0.284028

1 .000000
1 .000000
1 .000000
0.005533
0.037962
0.149028
0.348922
0.458972
0.244923
1 .000000
1 .000000
1 .000000

1 .000000

154

aug-cc-vaz
N 0.000000

S 141.00
129,200.000000
19,350.000000
4,404.000000
1,248.000000
408.000000
148.200000
58.500000
24.590000
10.810000
4.882000
2.195000
0.871500
0.350400
0.139700

S 141.00
129,200.000000
19,350.000000
4,404.000000
1 ,248.000000
408.000000
148.200000
58.500000
24.590000
10.810000
4.882000
2.195000
0.871500
0.350400
0.139700

S 1 1.00
2.195000

S 1 1.00
0.871500

S 1 1.00
0.350400

S 1 1.00
0.139700

P 81.00
147000000
34.760000
11.000000
3.995000
1.587000
0.653300
0.268600
0.106700

P 1 1.00

0.000025
0.0001 97
0.001 032
0.004325
0.01 5380
0.046867
0.1201 16
0.245695
0.361379
0.287283
0.070171
0.001 831
0.000835
-0.000006

-0.000006
-0.000043
-0.000227
-0.000958
-0.003416
-0.01 0667
-0.028279
-0.064020
-0.1 13932
—0. 1 46995
-0.007251

0.3661 83

0.547908

0.216645

1 .000000

1 .000000

1 .000000

1 .000000

0.000892
0.007082
0.032816
0.1 08209
0.248094
0.37451 3
0.348414
0.1 28340

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.968000

0.335000

2.027000

0.685000

1 .427000

0.054640

0.044020

0.111000

0.245000

0.559000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

155

G) (D 'T'

I

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1 .587000

0.653300

0.268600

0.1 06700

4.647000

1.813000

0.707000

0.276000

2.942000

1 .204000

0.493000

2.511000

0.942000

1 .768000

0.051800

0.036900

0.097100

0.1 92000

0.436000

0.788000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

Oxygen
aug-cc-pvdz
0 0.000000
S 91.00
1 1 ,720.000000
1,759.000000
400.800000
113.700000
37.030000
13.270000
5.025000
1.013000
0.302300
S 91.00
1 1 .720.000000
1 ,759.000000
400.800000
113.700000
37.030000
13.270000
5.025000
1.013000
0.302300
S 1 1.00
0.302300
P 4 1.00
17.700000
3.854000
1.046000
0.275300
P 1 1.00
0.275300
D 1 1.00
1.185000
S 1 1.00
0.078960
P 1 1.00
0.068560
D 1 1.00
0.332000

0.000710
0.005470
0.027837
0.104800
0.283062
0.448719
0.270952
0.015458
—0.002585

-0.000160
-0.001263
-0.006267
-0.025716
-0.070924
-0.16541 1
-0.1 16955

0.557368

0.572759

1 .000000
0.043018
0.228913
0.508728
0.460531
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

156

aug-cc-pvtz
0 0.000000

8 10 1.00
1 5,330.000000
2,299.000000
522.400000
147.300000
47.550000
16.760000
6.207000
1.752000
0.688200
0.238400

8 10 1.00
1 5,330.000000
2,299.000000
522.400000
147.300000
47.550000
16.760000
6.207000
1.752000
0.688200
0.238400

S 1 1.00
1.752000

S 1 1.00
0.238400

P 5 1.00
34.460000
7.749000
2.280000
0.715600
0.214000

P 1 1.00
0.715600

P 1 1.00
0.214000

1.00
2.314000

0 U
..l

1 1.00
0.645000

11
.5

1.00
1.428000

8 1 1.00
0.073760

P 1 1.00
0.059740

D 1 1.00
0.214000

0.000508
0.003929
0.020243
0.0791 81
0.230687
0.4331 18
0.350260
0.042728
-0.0081 54
0.002381

-0.0001 15
-0.000895
-0.004636
-0.01 8724
-0.058463
-0. 1 36463
-0.1 75740

0.160934

0.60341 8

0.378765

1 .000000
1 .000000
0.01 5928
0.099740
0.31 0492
0.491 026
0.336337
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

157

F 1 1 .00
0.500000 1.000000

Oxygen
aug-cc-pqu
0 0.000000
S 12 1 .00
61 ,420.000000
9,199.000000
2,091 .000000
590.900000
1 92.300000
69.320000
26.970000
1 1 .100000
4.682000
1 .428000
0.554700
0.206700
S 12 1 .00
61 .420.000000
9,199.000000
2,091.000000
590.900000
192.300000
69.320000
26.970000
1 1 .100000
4.682000
1 .428000
0.554700
0.206700
S 1 1 .00
1 .428000
S 1 1 .00
0.554700
S 1 1 .00
0.206700
P 6 1 .00
63.420000
14.660000
4.459000
1 .531000
0.530200
0.1 75000
P 1 1 .00
1 .531000
P 1 1 .00
0.530200
P 1 1 .00
0.1 75000
D 1 1 .00
3.775000
D 1 1 .00

0.000090
0.000698
0.003664
0.015218
0.052423
0.145921
0.305258
0.398508
0.216980
0.017594
-0.002502
0.000954

-0.000020
-0.0001 59
-0.000829
-0.003508
-0.01 21 56
-0.036261
-0.082992
-0.1 52090
-0.1 1 5331

0.288979

0.586128

0.277624

1 .000000
1 .000000
1 .000000
0.006044
0.041799
0.1 61 143
0.356731
0.448309
0.244940
1 .000000
1 .000000
1 .000000

1 .000000

aug-cc—vaz
O 0.000000
S 14 1 .00
164,200.000000
24,590.000000
5, 592.000000
1.582.000000
51 6.1 00000
1 87.200000
73.930000
31 .220000
1 3.810000
6.256000
2.776000
1 .1 38000
0.460000
0.1 82900
S 14 1 .00
164,200.000000
24,590.000000
5,592.000000
1 ,582.000000
516.1 00000
187.200000
73.930000
31 .220000
1 3.810000
6.256000
2.776000
1 .138000
0.460000
0.182900
S 1 1 .00
2.776000
S 1 1 .00
1 .138000
S 1 1 .00
0.460000
S 1 1 .00
0.182900
P 8 1 .00
195.500000
46.160000
14.580000
5.296000
2.094000
0.8471 00
0.336800
0.128500
P 1 1 .00

0.000026
0.000205
0.001076
0.004522
0.016108
0.049085
0.124857
0.251686
0.362420
0.279051
0.063552
0.001063
0.001 144
-0.000040

-0.000006
-0.000046
-0.000244
-0.001 031
-0.003688
-0.01 1 514
-0.030435
-0.068147
-0.120368
-0.148260

0.009905

0.384286

0.536805

0.202667

1 .000000

1 .000000

1 .000000

1 .000000

0.000918
0.007368
0.034958
0.1 15431
0.256803
0.373938
0.343447
0.129706

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1 .300000

0.444000

2.666000

0.859000

1 .846000

0.069590

0.053480

0.154000

0.324000

0.714000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

159

I C) O 71

(D

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

2.094000

0.847100

0.336800

0.128500

5.879000

2.307000

0.905000

0.355000

4.016000

1 .554000

0.601000

3.350000

1.189000

2.319000

0.065500

0.044600

0.131000

0.237000

0.517000

1 .024000

1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

Fluorine

aug-cc-pvdz
F 0.000000

S 91.00
14,710.000000
2,207.000000
502.800000
142.600000
46.470000
16.700000
6.356000
1.316000
0.389700

S 91.00
14,710.000000
2,207.000000
502.800000
142.600000
46.470000
16.700000
6.356000
1.316000
0.389700

S 1 1.00
0.389700

P 41.00
22.670000
4.977000
1.347000
0.347100

P 1 1.00
0.347100

D 1 1.00
1.640000

S 1 1.00
0.098630

P 1 1.00
0.085020

D 1 1.00
0.464000

0.000721
0.005553
0.028267
0.106444
0.286814
0.448641
0.264761
0.015333
-0.002332

-0.000165
-0.001308
-0.006495
-0.026691
-0.073690
-0.170776
-0.1 12327

0.562814

0.568778

1 .000000
0.044878
0.235718
0.508521
0.4581 20
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

160

aug-cc-pvtz
F 0.000000

8 101.00
19,500.000000
2,923.000000
664.500000
187.500000
60.620000
21.420000
7.950000
2.257000
0.881500
0.304100

8 101.00
19,500.000000
2,923.000000
664.500000
187.500000
60.620000
21.420000
7.950000
2.257000
0.881500
0.304100

S 1 1.00
2.257000

S 1 1.00
0.304100

P 51.00
43.880000
9.926000
2.930000
0.913200
0.267200

P 1 1.00
0.913200

P 1 1.00
0.267200

D 1 1.00
3.107000

D 1 1.00
0.855000

F 1 1.00
1.917000

S 1 1.00
0.091580

P 1 1.00
0.073610

D 1 1.00
0.292000

0.000507
0.003923
0.020200
0.079010
0.230439
0.432872
0.349964
0.043233
-0.007892
0.002384

-0.0001 17
-0.000912
-0.004717
-0.019086
-0.059655
-0.140010
-0.176782

0.171625

0.605043

0.369512

1 .000000
1 .000000
0.01 6665
0.1 04472
0.31 7260
0.487343
0.334604
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

 

 

161

F 1 1 .00
0.724000 1 .000000

Fluorine
aug-cc-pqu
F 0.000000
S 12 1 .00
74,530.000000
11.170.000000
2,543.000000
721 .000000
235.900000
85.600000
33.550000
1 3.930000
5.915000
1 .843000
0.712400
0.263700
S 12 1 .00
74,530.000000
1 1 ,170.000000
2,543.000000
721 .000000
235.900000
85.600000
33.550000
1 3.930000
5.91 5000
1 .843000
0.712400
0.263700
S 1 1 .00
1 .843000
S 1 1 .00
0.712400
S 1 1 .00
0.263700
P 6 1 .00
80.390000
18.630000
5.694000
1 .953000
0.670200
0.216600
P 1 1 .00
1 .953000
P 1 1 .00
0.670200
P 1 1 .00
0.216600
D 1 1 .00
5.014000
D 1 1 .00

0.000095
0.000738
0.003858
0.015926
0.054289
0.149513
0.308252
0.394853
0.21 1031
0.017151
-0.002015
0.000869

-0.000022
-0.0001 72
-0.000891
-0.003748
-0.012862
-0.038061
-0.086239
-0.1 55865
-0.1 1 0914

0.298761

0.585013

0.271 159

1 .000000
1 .000000
1 .000000
0.006347
0.044204
0.168514
0.361 563
0.4421 78
0.243435
1 .000000
1 .000000
1 .000000

1 .000000

162

aug-cc-vaz
F 0.000000
S 14 1.00

21 1 ,400.000000
31 ,660.000000
7,202.000000
2,040.000000
666.400000
242.000000
95.530000
40.230000
17.720000
8.005000
3.538000
1.458000
0.588700
0.232400

S 141.00
21 1 ,400.000000
31 ,660.000000
7,202.000000
2,040.000000
666.400000
242.000000
95.530000
40.230000
17.720000
8.005000
3.538000
1.458000
0.588700
0.232400

S 1 1.00
3.538000

S 1 1.00
1.458000

S 1 1.00
0.588700

S 1 1.00
0.232400

P 81.00
241.900000
57.170000
18.130000
6.624000
2.622000
1.057000
0.417600
0.157400

P 1 1.00

0.000026
0.000201
0.001056
0.004432
0.015766
0.0481 12
0.123232
0.251519
0.364525
0.279766
0.063545
0.001 1 1 1
0.001258
-0.000035

-0.000006
-0.000047
-0.000244
-0.001031
-0.003683
-0.01 1 513
-0.030663
-0.069572
-0.123992
-0.1 50214

0.016460

0.392550

0.532164

0.196524

1 .000000

1 .000000

1 .000000

1 .000000

0.001002
0.008054
0.038048
0.123779
0.266060
0.371796
0.332890
0.128491

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1 .725000

0.586000

3.562000

1 .148000

2.376000

0.085940

0.065680

0.207000

0.460000

0.924000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

163

'TI

6) C)

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

2.622000

1 .057000

0.41 7600

0.157400

7.760000

3.032000

1 .185000

0.463000

5.398000

2.078000

0.800000

4.338000

1.513000

2.995000

0.080600

0.055000

0.1 72000

0.331000

0.663000

1 .326000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

Aluminum
aug-cc-pvdz
Al 0.000000
8 12 1 .00
64,1 50.000000
9,617.000000
2.189.000000
620.500000
202.700000
73.1 50000
28.550000
1 1.770000
3.300000
1 .1 73000
0.175200
0.064730
S 12 1 .00
64,1 50.000000
9,617.000000
2,189.000000
620.500000
202.700000
73.150000
28.550000
1 1.770000
3.300000
1 .1 73000
0.175200
0.064730
S 12 1 .00
64,1 50.000000
9,617.000000
2,189.000000
620.500000
202.700000
73.1 50000
28.550000
1 1.770000
3.300000
1 .173000
0.1 75200
0.064730
S 1 1 .00
0.064730
P 8 1 .00
258.800000
60.890000
1 9.140000
6.881000
2.574000
0.957200

0.000290
0.002251
0.011646
0.046738
0.146299
0.330283
0.415861
0.189253
0.011589
-0.001284
0.000426
-0.000199

-0.000076
-0.000582
-0.003081
-0.01231 1
-0.041978
-0.103371
-0.1 96308
-0.083000

0.541040

0.578796

0.028815
-0.009538

0.000018
0.0001 34
0.000712
0.002843
0.009768
0.0241 85
0.047499
0.020362
-0.1 58788
-0.31 1694
0.620147
0.520943

1 .000000

0.004068
0.030682
0.129149
0.320831
0.453815
0.275066

164

aug-cc-pvtz
AI 0.000000
S 1 5 1 .00
205,500.000000
30,780.000000
7,006.000000
1,985.000000
649.1 00000
235.000000
91 .620000
37.670000
1 5.910000
5.850000
2.542000
1 .057000
0.2931 00
0.145500
0.056500
S 15 1 .00
205,500.000000
30,780.000000
7,006.000000
1 ,985.000000
649.1 00000
235.000000
91 .620000
37.670000
15.910000
5.850000
2.542000
1 .057000
0.293100
0.145500
0.056500
S 1 5 1 .00
205,500.000000
30,780.000000
7,006.000000
1 985000000
649.1 00000
235.000000
91 .620000
37.670000
1 5.910000
5.850000
2.542000
1 .057000
0.2931 00
0.145500
0.056500

0.000068
0.000527
0.002762
0.01 1473
0.039819
0.1 15040
0.260887
0.396386
0.284597
0.044458
-0.004898
0.00261 3
-0.001 089
0.000722
-0.000178

-0.000018
-0.000137
-0.000719
-0.003011
-0.010601
-0.032135
-0.080316
-0.156794
-0. 168376
0.126879
0.561494
0.436613
0.035729
-0.01 1456
0.002202

0.000004
0.000032
0.000166
0.000695
0.002455
0.007446
0.01 8825
0.037277
0.041 950
-0.035438
-0. 1 751 32
-0.276203
0.1 08729
0.652809
0.394587

8 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

0.209900
0.059860

258.800000
60.890000
19.140000

6.881000
2.574000
0.957200
0.209900
0.059860
0.059860
0.189000
0.023100
0.015300

0.053500

0.019081
-0.003128

-0.000748
-0.005458
-0.024537
-0.058214
-0.098376
-0.026006

0.464020

0.648870

1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

165

S
S

p

p

U U 13

TI

1 1.00

1 1.00

9 1.00

9 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

0.293100

0.056500

444.400000
1 05.1 00000
33.470000
12.330000

4.869000
1 .961 000
0.783400
0.1 88800
0.055570

444.400000
1 05.100000
33.470000
1 2.330000

4.869000
1 .961 000
0.783400
0.1 88800
0.055570
0.783400
0.055570
0.1 09000
0.333000
0.244000
0.0221 00
0.014600
0.035600

0.085800

1 .000000

1 .000000

0.001626
0.013069
0.061234
0.187870
0.360452
0.408454
0.188640
0.009765
-0.001151

-0.000286
-0.002423
-0.01 0866
-0.036431
-0.0641 07
-0.097224
0.01 4744
0.503448
0.597984
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

Aluminum
aug-cc-pvtz
Al 0.000000
S 16 1 .00
419,600.000000
62,830.000000
14,290.000000
4,038.000000
1 ,312.000000
470.500000
1 81 .800000
74.460000
31 .900000
1 3.960000
5.1 80000
2.265000
0.966400
0.244700
0.1 18400
0.050210
S 16 1 .00
419,600.000000
62,830.000000
14,290.000000
4,038.000000
1 ,312.000000
470.500000
1 81 .800000
74.460000
31 .900000
13.960000
5.180000
2.265000
0.966400
0.244700
0.1 18400
0.050210
8 16 1 .00
419,600.000000
62,830.000000
14,290.000000
4,038.000000
1 ,312.000000
470.500000
181 .800000
74.460000
31 .900000
1 3.960000
5.180000
2.265000
0.966400

0.000028
0.000216
0.001 138
0.004796
0.017239
0.053807
0.141326
0.289268
0.384825
0.232852
0.029333
-0.003006
0.001667
-0.000606
0.000431
-0.000123

-0.000007
-0.000056
-0.000297
-0.001249
-0.004551
-0.014439
-0.040346
-0.092262
-0.164510
-0.141296
0.195365
0.572475
0.374041
0.023412
-0.009052
0.0021 12

0.000002
0.000013
0.000069
0.000288
0.001 053
0.003339
0.009392
0.021605
0.039587
0.034918
-0.052842
-0.191876
-0.2541 15

166

aug-cc-pqu
Al ' 0.000000
8 20 1 .00
3,269,000.000000
489,400.000000

1 1 1 ,400.000000
31 ,560.000000
1 0,320.000000
3,731 .000000
1 ,456.000000
604.1 00000
263.500000
1 19.800000
56.320000
27.1 90000
1 3.260000
6.052000
2.981 000
1 .476000
0.733400
0.244700
0.1 08800
0.046720

S 20 1 .00
3 269,000.000000
489,400.000000
1 1 1 ,400.000000
31 ,560.000000
1 0,320.000000
3,731 .000000
1 ,456.000000
604.1 00000
263.500000
1 19.800000
56.320000
27.1 90000
1 3.260000
6.052000
2.981 000
1 .476000
0.733400
0.244700
0.1 06800
0.046720

8 20 1 .00
3 269,000.000000
489,400.000000
1 1 1 ,400.000000
31 ,560.000000
1 0,320.000000

0.000002
0.00001 7
0.000088
0.000369
0.001 339
0.004356
0.01 2896
0.034820
0.084353
0.1 75907
0.292091
0.32820
0.1 86927
0.031 043
-0.000509
0.001488
-0.000278
0.0001 19
-0.000066
0.00001 8

-0.000001
-0.000004
-0.000023
-0.000096
-0.000348
-0.001 138
-0.003387
-0.00931 5
-0.023302
-0.052349
-0.099950
-0.1 50560
-0.1 1 9121

0.1 08091

0.41 1 129

0.45721 4

0.1 75938

0.007706
-0.001662

0.000445

0.000000
0.000001
0.000005
0.000022
0.000081

p

1 1.00

1 1.00

1 1.00

11 1.00

0.244700
0.118400
0.050210
0.244700
0.118400

0.050210

891 .300000
21 1 .300000

11 1.00

68.280000
25.700000
1 0.630000

4.602000
2 .01 5000
0.870600
0.297200
0.1 10000
0.039890

891 .300000
21 1 .300000

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

68.280000
25.700000
10.630000

4.602000
2.01 5000
0.870600
0.297200
0.1 10000
0.039890
0.297200
0.1 10000
0.039890
0.080400
0.1 99000
0.494000
0.1 54000
0.401000

0.357000

0.275070
0.604743
0.287629

1 .000000

1 .000000

1 .000000

0.000492
0.0041 58
0.021254
0.076406
0.194277
0.334428
0.375026
0.204041
0.021374
-0.002021
0.00081 7

-0.000089
-0.000746
-0.003870
-0.013935
-0.036686
-0.062780
-0.078960
-0.028859
0.238256
0.551 363
0.354385
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

167

p

p

3,731 .000000
1 ,456.000000
604.1 00000
263.500000

1 19.800000
56.320000
27. 1 90000

1 3.260000

1 1.00

1 1.00

1 1.00

1 1.00

12 1.00

6.052000
2.981000
1 .476000
0.733400
0.244700
0.108800
0.046720

0.733400

0.244700

0.1 08800

0.046720

1 .461 .000000
346.200000
1 12.200000

12 1.00

42.51 0000
1 7.720000

7.852000
3.571 000
1 .637000
0.738200
0.257700
0.097730
0.036900

1 ,461.000000
346.200000
1 12.200000

1 1.00

42.51 0000
1 7.720000

7.852000
3.571000
1 .637000
0.738200
0.257700
0.097730
0.036900

0.738200

0.000263
0.000784
0.002150
0.005420
0.012169
0.023682
0.036094
0.030328
-0.030903
-0.1 1 9126
-0.21 1 145
-0.157944
0.339038
0.613097
0.235168

1 .000000

1 .000000

1 .000000

1 .000000

0.000209
0.001810
0.009734
0.037827
0.1 10898
0.234295
0.345245
0.331430
0.147064
0.012141
-0.000872
0.000438

-0.000037
-0.000329
-0.001 743
-0.006948
-0.020281
-0.044866
-0.064328
-0.075267
0.000616
0.28781 9
0.543030
0. 301 865

1 .000000

1.00

1.00

1.00

1.00

1.00

0.018300

0.012100

0.028200

0.058200

0.153000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

168

G) G)

I

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.257700

0.097730

0.036900

1.317000

0.526000

0.210000

0.084000

0.1 30000

0.258000

0.513000

0.252000

0.543000

0.446000

0.017700

0.011500

0.029400

0.050900

0.106900

0.227000

1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1.000000
1 .000000

1 .000000

Silicon
aug-cc-pvdz
Si 0
S 12 1 .00
78,860.000000
1 1 ,820.000000
2,692.000000
763.400000
249.600000
90.280000
35.290000
1 4.51 0000
4.053000
1 .482000
0.251700
0.092430
S 12 1 .00
78,860.000000
1 1 ,820.000000
2,692.000000
763.400000
249.600000
90.280000
35.290000
14.51 0000
4.053000
1 .482000
0.251 700
0.092430
S 12 1 .00
78860000000
1 1 ,820.000000
2,692.000000
763.400000
249.600000
90.280000
35.290000
14.510000
4.053000
1 .482000
0.251 700
0.092430
S 1 1 .00
0.092430
P 8 1 .00
31 5.900000
74.420000
23.480000
8.488000
3.21 7000
1 .229000

0.000270
0.002097
0.010851
0.043675
0.137653
0.316644
0.418581
0.210212
0.014495
-0.002036
0.000624
-0.000283

-0.000072
-0.000555
-0.002938
-0.01 1769
-0.040291
-0.1 00609
-0.1 96528
-0.1 02382

0.5271 90

0.593251

0.033265
-0.009737

0.00001 9
0.000142
0.000752
0.003023
0.01 0368
0.026256
0.052399
0.029096
-0.1 78003
-0.346874
0.623020
0.537712

1 .000000

0.003927
0.029881
0.127212
0.320943
0.455429
0.268563

169

aug-cc-pvtz
Si 0
S 1 5 1 .00
254,900.000000
38,190.000000
8,690.000000
2,462.000000
804.800000
291 .300000
1 1 3.600000
46.750000
1 9.820000
7.708000
3.340000
1 .402000
0.438700
0.207000
0.079440
S 1 5 1 .00
254,900.000000
38,1 90.000000
8,690.000000
2,462.000000
804.800000
291 .300000
1 13.600000
46.750000
1 9.820000
7.708000
3.340000
1 .402000
0.438700
0.207000
0.079440
S 1 5 1 .00
254,900.000000
38,1 90.000000
8,690.000000
2,462.000000
804.800000
291 .300000
1 1 3.600000
46.750000
1 9.820000
7.708000
3.340000
1 .402000
0.438700
0.207000
0.079440

0.000063
0.000486
0.002545
0.01 0587
0.036879
0.1 07479
0.247936
0.390927
0.302026
0.055924
-0.004024
0.002580
-0.001 038
0.000608
-0.0001 54

-0.00001 7
-0.0001 29
-0.000679
-0.002841
-0.01 0055
-0.030577
-0.077726
-0.1 54236
-0.1 80368
0.079822
0.547441
0.4801 19
0.047485
-0.01 0700
0.0021 99

0.000004
0.000033
0.000174
0.000728
0.002583
0.007864
0.020216
0.040732
0.049936
-0.024940
-0.190350
—0.318350
0.094804
0.681 180
0.395672

8 1.00

0.296400
0.087660

315.900000

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

74.420000
23.480000

8.488000
3.21 7000
1 .229000
0.296400
0.087680
0.087680
0.275000
0.033200
0.025000

0.062300

0.018834
-0.002624

-0.000858
-0.006303
-0.028826
-0.069456
-0.1 19493
-0.019958

0.510268

0.600382

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

170

S
S

p

p

U U '0

'TI

1 1.00

1 1.00

9 1.00

0.438700

0.079440

481 .500000
1 13.900000

9 1.00

36.230000
1 3.340000

5.252000
2.120000
0.8561 00
0.252800
0.078890

481 .500000
1 13.900000

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

36.230000
1 3.340000

5.252000
2.1 20000
0 .8561 00
0.252800
0.078890
0.8561 00
0.078890
0.1 59000
0.481 000
0.336000
0.033000

0.023700

0.055600 '

0.1 25000

1 .000000

1 .000000

0.001920
0.01 5355
0.071399
0.21 3052
0.390354
0.393721
0.1 32565
0.003956
0.000332

-0.000405
-0.003359
-0.01 5286
-0.048922
-0.085501
-0.1 121 37
0.061 827
0.551 91 9
0.523492
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

Silicon
aug-cc-pqu
Si 0
S 16 1 .00
513,000.000000
76,820.000000
17,470.000000
4,935.000000
1 ,602.000000
574.1 00000
221 .500000
90.540000
38.740000
16.950000
6.452000
2.874000
1 .250000
0.359900
0.169900
0.070660
S 16 1 .00
513,000.000000
76,820.000000
1 7,470.000000
4,935.000000
1 ,602.000000
574.1 00000
221 .500000
90.540000
38.740000
16.950000
6.452000
2.874000
1 .250000
0.359900
0.169900
0.070660
S 16 1 .00
51 3,000.000000
76,820.000000
17,470.000000
4,935.000000
1 ,602.000000
574.1 00000
221 .500000
90.540000
38.740000
16.950000
6.452000
2.874000
1 .250000

0.000026
0.000203
0.001067
0.004506
0.016236
0.050891
0.135155
0.281292
0.385336
0.245651
0.034315
-0.003349
0.001876
-0.000693
0.000438
-0.000123

-0.000007
-0.000054
-0.000285
-0.001202
-0.004384
-0.01 3978
-0.039352
-0.091428
-0.165609
-0.1 52505
0.168524
0.569284
0.398056
0.0291 51
-0.008490
0.001996

0.000002
0.000014
0.000073
0.000308
0.001126
0.003584
0.010173
0.023752
0.044348
0.041904
-0.050250
«0.216578
-0.286448

171

aug-cc-vaz
Si 0
S 20 1.00

3,948,000.000000
591 ,100.000000
134,500.000000
38,120.000000
12,460.000000
4,504.000000
1 ,758.000000
729.1 00000
31 8.000000
144.600000
67.970000
32.820000
16.030000
7.396000
3.661000
1 .823000
0.914700
0.339300
0.1 50000
0.064360

8 20 1 .00
3948,000.000000
591 ,1 00.000000
134,500.000000
38,120.000000
12,460.000000
4,504.000000
1 ,756.000000
729.1 00000
31 6.000000
144.600000
67.970000
32.820000
16.030000
7.396000
3.661000
1 .823000
0.914700
0.339300
0.1 50000
0.064380

S 20 1 .00
3948,000.000000
591 ,100.000000
134,500.000000
38,120.000000
12,460.000000

0.000002
0.000016
0.000083
0.000351
0.001277
0.004152
0.012303
0.033310
0.080985
0.170290
0.286879
0.330340
0.196602
0.035454
-0.000535
0.001615
-0.000373
0.000146
-0.000079
0.000019

-0.000001
-0.000004
-0.000022
-0.000094
-0.000340
-0.001 11 1
-0.003309
-0.0091 16
-0.022879
-0.051 712
-0.099909
-0. 1 52747
-0.1 27508

0.094696

0.41 4036

0.467934

0.1 73927

0.008439
-0.000998

0.000362

0.000000
0.000001
0.000006
0.000024
0.000087

P

1 1.00

1 1.00

1 1.00

11 1.00

0.359900
0.169900
0.070660
0.359900
0.169900

0.070660

1 ,122.000000
266.000000

11 1.00

85.920000
32.330000

1 3.370000
5.800000
2.559000
1 .124000
0.398800
0.1 53300
0.057280

1 ,122.000000
266.000000

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

85.920000
32.330000

1 3.370000
5.800000
2.559000
1 .124000
0.398800
0.1 53300
0.057280
0.398800
0.1 53300
0.057280
0.120000
0.302000
0.760000
0.212000
0.541000

0.461000

0.263256
0.634900
0.290832

1 .000000

1 .000000

1 .000000

0.000448
0.003816
0.01981 1
0.072702
0.189839
0.335672
0.379365
0.201 193
0.020852
-0.001703
0.000750

-0.000096
-0.000812
-0.004301
-0.01 5750
-0.042954
-0.075257
-0.097145
-0.022751
0.291 988
0.550670
0.29761 8
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

172

p

p

4,504.000000
1 ,758.000000
729.1 00000
31 8.000000
144.600000
67.970000
32.820000
16.030000
7.396000
3.661000
1 .823000
0.914700
0.339300
0.1 50000
0.064380

1 1 .00
0.914700

1 1 .00
0.339300

1 1 .00
0.150000

1 1 .00
0.064380

12 1 .00
1 ,780.000000
421 .600000
1 36.700000
51 .81 0000
21 .600000
9.563000
4.350000
2.006000
0.920500
0.350000
0.1 381 00
0.053380
12 1 .00

1 ,780.000000
421 .800000
1 36.700000
51 .810000
21 .600000
9.563000
4.350000
2.006000
0.920500
0.350000
0.1 381 00
0.053380

1 1 .00
0.920500

0.000284
0.000850
0.002335
0.005905
0.01 3346
0.026289
0.040743
0.036148
-0.030392
-0.135961
-0.250144
-0.158050
0.369655
0.61 7718
0.222514

1 .000000

1 .000000

1 .000000

1 .000000

0.000201
0.001749
0.009481
0.037231
0.1 10763
0.237933
0.353691
0.328839
0.132373
0.010330
-0.0001 50
0.000266

-0.000043
-0.000377
-0.002022
-0.008128
-0.024227
-0.054383
~0.079905
-0.088896
0.01 8400
0.335096
0.532288
0.254374

1 .000000

1.00

1.00

1.00

1.00

1.00

0.027500

0.020000

0.043500

0.084600

0.212000

1 .000000

1 .000000

1.000000

1 .000000

1 .000000

173

U U U

U

I O 0

(D

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.350000

0.138100

0.053380

0.126000

0.321000

0.817000

2.082000

0.169000

0.341000

0.688000

0.320000

0.705000

0.583000

0.026000

0.019200

0.046800

0.073500

0.151000

0.323000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

Phosphorous
aug-cc-pvdz
P 0.000000
S 12 1 .00
94,840.000000
14,220.000000
3,236.000000
91 7.1 00000
299.500000
1 08.1 00000
42.1 80000
17.280000
4.858000
1 .81 8000
0.337200
0.1 23200
S 12 1 .00
94,840.000000
14,220.000000
3,236.000000
917.1 00000
299.500000
108.1 00000
42.1 80000
17.280000
4.858000
1 .818000
0.337200
0.123200
S 12 1 .00
94,840.000000
14,220.000000
3,236.000000
917.1 00000
299.500000
108.1 00000
42.1 80000
1 7.280000
4.858000
1 .818000
0.337200
0.123200
S 1 1 .00
0.123200
P 8 1 .00
370.500000
87.330000
27.590000
10.000000
3.825000
1 .494000

0.000256
0.001982
0.010276
0.041482
0.131984
0.308662
0.420647
0.222878
0.016404
-0.002543
0.000748
-0.000331

-0.000070
-0.000535
-0.002837
-0.01 1398
-0.039293
-0.099636
-0.197983
-0.1 14860

0.51 8595

0.601 847

0.036861
-0.009708

0.000019
0.000147
0.000778
0.003145
0.010820
0.027996
0.056398
0.035819
-0.193387
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0.624246
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1 .000000

0.003950
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0.327594
0.456992
0.253086

174

aug-cc-pvtz
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312,400.000000
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1 0,650.000000
3,018.000000
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357.400000
1 39.600000
57.630000
24.600000
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357.400000
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1 1.00

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1 1.00

1 1.00

1 1.00

0.392100
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370.500000
87.330000
27.590000
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3.825000
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0.113000

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175

S

S

p

p

1 1.00

1 1.00

9 1.00

9 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

0.615800

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504.900000
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504.900000
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5.457000
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0.422439
1 .000000
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1 .000000
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1 .000000

1 .000000

Phosphorous
aug-cc-pqu

0.000000

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615,200.000000
92,120.000000
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5,920. 000000
1 ,922.000000
688.000000
265.000000
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0.000025
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0.000002
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0.003743
0.010682
0.025266
0.047928
0.047710
-0.046653
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176

aug-cc-vaz

P 0.000000

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698,600.000000
159,000.000000
45,040.000000
14,720.000000
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375.700000
1 70.800000
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38.770000
1 8.930000
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4.358000
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p

0.488800
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1 1 1 .00
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177

5.323.000000
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S 1 1 .00
1 .095000

S 1 1 .00
0.440000

S 1 1 .00
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S 1 1 .00
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P 12 1 .00
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476.300000
154.400000
58.510000
24.400000
10.800000
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2,010.000000
476.300000
154.400000
58.510000
24.400000
10.800000
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1 .043000
0.431300
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1 .043000

0.000301
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178

'0 U U U

I 0 C) '0

(D

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1 .00

1.00

1.00

1.00 .

1.00

1.00

1.00

1.00

0.431300

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1 .054000

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1 .000000

1 .000000

1 .000000

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1 .000000

1 .000000

Sulfur
aug-cc-pvdz
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110,800.000000
16,610.000000
3,781 .000000
1,071 .000000
349.800000
126.300000
49.260000
20.160000
5.720000
2.182000
0.432700
0.157000
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1 10,800.000000
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349.800000
126.300000
49.260000
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1 10,800.000000
16,610.000000
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349.800000
126.300000
49.260000
20.160000
5.720000
2.182000
0.432700
0.157000
3 1 1 .00
0.157000
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399.700000
94.190000
29.750000
10.770000
4.1 19000
1 .625000

0.000248
0.001920
0.009962
0.040298
0.128604
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0.230781
0.017897
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0.513194

0.607120

0.039675
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0.000020
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0.632840
0.556924

1 .000000

0.004475
0.034171
0.144250
0.353928
0.459085
0.206383

179

aug-cc-pvtz
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374,100.000000
56,050.000000
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3,615.000000
1 ,183.000000
428.800000
1 67.800000
69.470000
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374,1 00.000000
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3,61 5.000000
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69.470000
29.840000
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374,1 00.000000
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3,61 5.000000
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428.800000
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69.470000
29.840000
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0.1 32200

0.000054
0.000421
0.002207
0.0091 93
0.0321 12
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0.223630
0.374393
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0.000004
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0.0001 78
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8 1.00

1 1.00

1 1.00

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1 1.00

0.472600
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399.700000
94.1 90000
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10.770000
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1 .000000

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180

S

S

p

P

1 1.00

1 1.00

9 1.00

9 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

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0.776700

0.1 32200

574.400000
135.800000
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6.208000
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574.400000
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6.208000
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0.216000

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1 .000000

Sulfur
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727,800.000000
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31 3.400000
1 27.700000
54.480000
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0.627000
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1 27.700000
54.480000
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727,800.000000
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1 27.700000
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0.000024
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0.001 923

0.000002
0.00001 5
0.000078
0.000327
0.001 197
0.003848
0.01 1054
0.026465
0.050877
0.053003
-0.042552
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181

aug-cc-vaz

S 0.000000

8 20 1 .00
5,481 ,000.000000
820,600.000000
186,700.000000
52,880.000000
17,250.000000
6,226.000000
2,429.000000
1 007000000
439.500000
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45.340000
22.1 50000
1 0.340000
5.1 1 9000
2.553000
1 .282000
0.545000
0.241 100
0.1 03500

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5,481 ,000.000000
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439.500000
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5.1 19000
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0.545000
0.241 100
0.1 03500

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0.000002
0.000015
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0.000224

0.000000
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p

0.627000
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1 1 .00
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1 1 .00
0.287300

1 1 .00
0.1 17200

1 1 1 .00
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1 ,546.000000
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1 1.00

0.090140
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1 1.00

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1 .000000

182

6,226.000000
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1 99.800000
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45.340000
22.1 50000
1 0.340000
5.1 19000
2.553000
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0.545000
0.241 100
0.1 03500

S 1 1 .00
1 .282000

S 1 1 .00
0.545000

S 1 1 .00
0.241 100

S 1 1 .00
0.103500

P 12 1 .00
2,200.000000
521 .400000
169.000000
64.050000
26.720000
1 1.830000
5.378000
2.482000
1 .1 16000
0.484800
0.200600
0.079510
P 12 1 .00

2,200.000000
521 .400000
169.000000
64.050000
26.720000
1 1 .830000
5.378000
2.482000
1 .1 16000
0.484800
0.200600
0.079510

P 1 1 .00
1 .1 16000

0.000313
0.000936
0.002578
0.006541
0.014963
0.029894
0.047695
0.044956
-0.029301

' -0.168916

-0.31 1014
-0.136491
0.423195
0.613888
0.200473

1 .000000

1 .000000

1 .000000

1 .000000

0.000239
0.002077
0.01 1236
0.044069
0.129168
0.269083
0.37861 1
0.296779
0.077966
0.002052
0.001436
~0.000060

-0.000061
-0.000530
-0.002879
-0.01 1440
-0.034276
-0.073581
-0.1 07782
-0.087977
0.108261
0.407082
0.468994
0.1 84877

1 .000000

1.00

1.00

1.00

1.00

1.00

0.042800

0.031700

0.074800

0.140000

0.297000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

183

I C) G) "'1

(I)

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.484800

0.200600

0.079510

0.205000

0.512000

1 .281000

3.203000

0.255000

0.529000

1 .096000

0.463000

1.071000

0.872000

0.042000

0.029400

0.079400

0.118800

0.220000

0.472000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

Chlorine
aug-cc-pvdz
Cl 0.000000
S 12 1 .00
127,900.000000
19,170.000000
4,363.000000
1 236000000
403.600000
145.700000
56.810000
23.230000
6.644000
2.575000
0.537100
0.193800
S 12 1 .00
127,900.000000
19,170.000000
4,363.000000
1 236000000
403.600000
145.700000
56.810000
23.230000
6.644000
2.575000
0.537100
0.193800
S 12 1 .00
127,900.000000
19,170.000000
4,363.000000
1 236000000
403.600000
145.700000
56.810000
23.230000
6.644000
2.575000
0.537100
0.193800
S 1 1 .00
0.193800
P 8 1 .00
417.600000
98.330000
31 .040000
1 1 .190000
4.249000
1 .624000

0.000241
0.001871
0.009708
0.039315
0.125932
0.299341
0.421886
0.237201
0.019153
-0.003348
0.000930
—0.000396

-0.000068
-0.000522
-0.002765
-0.01 1 154
-0.038592
-0.099485
-0.201392
-0.130313

0.509443

0.610725

0.042155
-0.009234

0.000021
0.000158
0.000834
0.003399
0.01 1674
0.030962
0.062953
0.046026
-0.219312
-0.408773
0.638465
0.562362

1 .000000

0.005260
0.039833
0.164655
0.387322
0.457072
0.151636

184

aug-cc-pvtz

Cl 0.000000

8 1 5 1 .00
456,100.000000
68,330.000000
15,550.000000
4,405.000000
1 ,439.000000
520.400000
203.1 00000
83.960000
36.200000
1 5.830000
6.334000
2.694000
0.976800
0.431 300
0.162500

S 1 5 1 .00
456.1 00.000000
68,330.000000
1 5,550.000000
4,405.000000
1 ,439.000000
520.400000
203.1 00000
83.960000
36.200000
1 5.830000
6.334000
2.694000
0.976800
0.431 300
0.162500

8 1 5 1 .00
456,1 00.000000
68,330.000000
1 5,550.000000
4,405.000000
1 ,439.000000
520.400000
203.1 00000
83.960000
36.200000
1 5.830000
6.334000
2.694000
0.976800
0.431 300
0.162500

0.000049
0.000383
0.002009
0.008386
0.029470
0.087833
0.21 1473
0.365364
0.340884
0.1 021 33
0.0031 17
0.001058
-0.000376
0.0001 56
-0.000051

-0.000014
-0.0001 07
-0.000565
-0.002361
-0.008459
-0.025964
-0.068636
-0.141 874
-0.1 99319
-0.01 9566

0.499741

0.563736

0.079033
-0.008351

0.002325

0.000004
0.000032
0.0001 71
0.000714
0.002567
0.007886
0.021 087
0.044226
0.065167
0.006030
-0.206495
-0.405871
0.075956
0.725661
0.394423

8 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

0.532200
0.162000

417.600000
98.330000
31 .040000
11 .190000

4.249000
1 .624000
0.532200
0.162000
0.162000
0.600000
0.060800
0.046600

0.1 96000

0.001816
0.001883

-0.001436
-0.010780
-0.047008
-0.1 1 1030
-0.153275
0.089461
0.579444
0.483272

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

185

S

S

P

p

1 1.00

1 1.00

9 1.00

9 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

1 1.00

0.976800

0.162500

663.300000
1 56.800000
49.980000
1 8.420000

7.240000
2.922000
1 .022000
0.381 800
0.1 301 00

663.300000
1 56.800000
49.980000
1 8.420000

7.240000
2.922000
1 .022000
0.381800
0.130100
1 .022000
0.130100
1 .046000
0.344000
0.706000
0.059100
0.041900
0.135000

0.312000

1 .000000

1 .000000

0.002404
0.019215
0.088510
0.256020
0.436927
0.350334
0.058550
-0.004584
0.002270

-0.000652
-0.0051 94
-0.024694
-0.07281 7
-0.1 34030
-0.094774
0.262289
0.564667
0.341 250
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

Chlorine
aug-cc-pqu
Cl 0.000000
S 1 6 1 .00
834,900.000000
125,000.000000
28,430.000000
8,033.000000
2,608.000000
933.900000
360.000000
147.000000
62.880000
27.600000
1 1 .080000
5.075000
2.278000
0.777500
0.352700
0.1431 00
S 16 1 .00
834,900.000000
125,000.000000
28,430.000000
8,033.000000
2,608.000000
933.900000
360.000000
1 47.000000
62.880000
27.600000
1 1 .060000
5.075000
2.278000
0.777500
0.352700
0.1431 00
S 16 1 .00
834,900.000000
125,000.000000
28,430.000000
8,033.000000
2,608.000000
933.900000
360.000000
147.000000
62.880000
27.600000
1 1 .080000
5.075000
2.278000

0.000023
0.000180
0.000948
0.004001
0.014463
0.045659
0.123248
0.264369
0.382989
0.270934
0.047140
-0.00371 8
0.002192
-0.000850
0.000425
-0.000120

-0.000007
-0.000050
-0.000266
-0.001 125
-0.004105
-0.01 31 99
-0.037534
-0.089723
-0.1 67671
-0.1 74763
0.1 14909
0.56361 8
0.441 606
0.042670
-0.006672
0.001 907

0.000002
0.000015
0.000081
0.000340
0.001246
0.003996
0.01 1475
0.027550
0.053292
0.057125
-0.039520
-0.264343
-0.349291

186

aug-cc-vaz
CI 0.000000
S 20 1.00
6,410,000.000000
959600000000
218300000000

61 ,810.000000
20,140.000000
7,264.000000
2,832 .000000
1 ,175.000000
51 2.600000
233.000000
1 09.500000
52.860000
25.840000
12.1 70000
6.030000
3.012000
1 .51 1000
0.660400
0.292600
0.125400

S 20 1 .00
6,410,000.000000
959,600.000000
218,300.000000
61 ,810.000000
20,140.000000
7,264.000000
2 ,832.000000
1 ,175.000000
51 2.600000
233.000000
1 09.500000
52.860000
25.840000
1 2 .1 70000
6.030000
3.012000
1 .51 1000
0.660400
0.292600
0.125400

8 20 1 .00
6,410,000.000000
959,600.000000
218,300.000000
61 ,810.000000
20,140.000000

0.000002
0.000014
0.000074
0.000314
0.001146
0.003739
0.011095
0.030115
0.073915
0.158258
0.274753
0.334066
0.217589
0.045728
-0.000135
0.001639
-0.000522
0.000181
-0.000100
0.000019

-0.000001
-0.000004
-0.000021
-0.000088
-0.000322
-0.001 053
-0.0031 42
-0.008664
-0.021 935
-0.050258
-0.099541
-0.1 57647
-0. 1 46024
0.069223
0.43041 2
0.490802
0.1 58390
0.008399
0.000768
0.000227

0.000000
0.000001
0.000006
0.000027
0.000097

p

0.777500
0.352700
0.143100

1 1 .00
0.777500

1 1 .00
0.352700

1 1 .00
0.143100

1 1 1 .00
1 ,703.000000
403.600000
130.300000
49.050000
20.260000
8.787000
3.919000
1 .765000
0.720700
0.283900
0.106000
1 1 1 .00

1 ,703.000000
403.600000
130.300000
49.050000
20.260000
8.787000
3.919000
1 .765000
0.720700
0.283900
0.106000

1 1 .00
0.720700

1 1.00

0.283900

1 1.00

0.106000

1 1.00

0.254000

1 1.00

0.628000

1 1.00

1.551000

1 1.00

0.423000

1 1.00

1 .089000

1 1.00

0.827000

0.269671
0.676073
0.284679

1 .000000

1.000000

1 .000000

0.000474
0.004064
0.021336
0.079461
0.208927
0.364945
0.371725
0.146292
0.010791
0.001 170
0.000339

-0.000128
-0.001094
-0.005834
-0.021926
-0.060139
-0.106929
-0.122454

0.038362
0.385256
0.507265
0.22721 8
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000
1 .000000

1 .000000

187

7,264.000000
2,832.000000
1 ,175.000000
51 2.600000
233.000000
1 09.500000
52.860000
25.840000
12.1 70000
6.030000
3.012000
1 .51 1000
0.660400
0.292600
0.125400

S 1 1 .00
1 .51 1000

S 1 1 .00
0.660400

S 1 1 .00
0.292600

S 1 1 .00
0.125400

P 12 1 .00
2.548.000000
603.700000
1 95.600000
74.1 50000
30.940000
1 3.690000
6.229000
2.878000
1 .282000
0.5641 00
0.234800
0.093120
P 12 1 .00

2,548.000000
603.700000
1 95.600000
74.1 50000
30.940000
1 3.690000
6.229000
2.878000
1 .282000
0.5641 00
0.234800
0.093120

P 1 1 .00
1 .282000

0.00031 8
0.000952
0.002624
0.006682
0.01 5360
0.030943
0.050064
0.048978
-0.026081
-0.1 78426
-0.332324
-0.132008
0.440556
0.61 1891
0.1 931 72

1 .000000

1 .000000

1 .000000

1 .000000

0.000236
0.002052
0.01 1 154
0.043962
0.129994
0.272959
0.383690
0.291870
0.070446
0.001288
0.001 826
0.000020

-0.000064
-0.000553
-0.003026
-0.01 2065
-0.036635
-0.079076
-0.1 1742
-0.086094
0.140308
0.422336
0.446363
0.166634

1 .000000

1.00

1.00

1.00

1.00

1.00

0.051900

0.037600

0.095200

0.217000

0.378000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

188

I G) C) 71

(I)

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

1.00

0.564100

0.234800

0.093120

0.250000

0.618000

1 .529000

3.781000

0.320000

0.656000

1 .345000

0.556000

1 .302000

1 .053000

0.047900

0.034800

0.1 00300

0.164000

0.277000

0.607000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

1 .000000

APPENDIX C

Tabulated Properties

In this appendix tables are given of properties for various types of wavefirnctions and
basis sets. Selected experimental values are given for comparison. The properties
computed are: re the internuclear separation in atomic units at the model energy
minimum, E the model total energy minimum, Dc the energy difference between model

minimum and asymptotic limit, On the expectation of quadrupole
t-l-Z 32—r2' "k f dth
momen 2 q,- z,- ,- 1n atomic umts ta en at center 0 mass, an qa e
i

2 2
32. —- r.
expectation value of the field gradient 2 q ,- ——'A 5 [A

but riA

in atomic units taken at

nucleus. In both the quadrupole moment and the field gradient q. denotes the charge
of particle i, and the summation index runs over nuclei and electrons. The last table
contains experimental spectroscopic values used in the determination of vibration

corrections of calculated properties.

190

Table C-l. Calculated and experimental constants for the ground state, X ‘22:, of H2

 

 

Method/basis r. (00) E1 (e/ao) D. (ev) 9.: (eag ) (la (e/ag )
RHF

aug-cc-pvdz 1.4137 -l.128826 3.5066 0.4613 -0.3665
aug-cc-pvtz 1.3880 -1. 133056 3.6207 0.4916 -0.3639
aug-cc-pqu 1.3865 -1. 133508 3.6330 0.4867 -0.3608
aug-cc-vaz 1.3863 -l.l33648 3.6368 0.4852 -0.3591
CBS limit 1.3863 -l.l336 3.6355 0.4845 -0.3589
MP2

aug-cc-pvdz 1.4271 -1. 156216 4.2509 0.4401 -0.3513
aug-cc-pvtz 1.3933 -1. 165023 4.4906 0.4719 -0.3550
aug-cc-pqu 1.3912 -1.166740 4.5373 0.4679 -0.3520
aug-cc-vaz 1.3902 -1. 167191 4.5496 0.4662 -0.3507
CBS limit 1.3893 4.5498 0.4649 -0.3504
CASSCF

aug-cc-pvdz 1.4332 -1. 149819 4.0769 0.4589 -0.3350
aug-cc-pvtz 1.4273 -1. 151743 4.1292 0.4455 -0.3l64
aug-oc-pqu 1.4262 -1. 152121 4.1395 0.4422 -0.3147
aug-cc-vaz 1.4258 -1. 152259 4.1433 0.4408 -0.3l44
CBS limit 1.4256 4.1439 0.4404 -0.3142
CISD .

aug-cc-pvdz 1.4392 -1. 164900 4.4873 0.4275 -0.3383
aug-cc-pvtz 1.4041 -1. 172636 4.6978 0.4616 -0.3447
aug-cc-pqu 1.4022 -1. 173867 4.7313 0.4585 -0.3393
aug-cc-vaz 1.4014 -1. 174175 4.7397 0.4569 -0.3380
CBS limit 1.4010 4.7403 0.4552 -0.3374
CASSCF+1+2

aug-cc-pvdz 1.4117 -1. 168170 4.5762 0.4666 -0.3606
aug-cc-pvtz 1.4043 -1.172910 4.7052 0.4620 -0.3014
aug—cc-pqu 1.4021 -1. 173905 4.7323 0.4587 -0.3392
aug-cc-vaz 1.4014 -1. 174187 4.7400 0.4570 -0.3380
CBS limit 1.4012 4.7405 0.4567 -0.3377
Exptl. 1.401112 4.748712 0.460

:t 0.02110

 

192

Table C-2. Calculated and experimental constants for the ground state, X 12;. of Liz

 

 

Method/basis re (ao) ET (e/ao) D. (av) @u (808 ) Clzz (e/ (13)
RHF

roos 5.2609 -l4.87l8858 0.1758 11.1486 0.005577
352 5.2609 -l4.8719081 0.1759 11.14507 0.005468
CASSCF

roos 5.0972 -l4.9032651 1.0272 10.8846 0.004657
352 5.0986 -l4.9033815 1.0273 10.8891 0.004653
CASSCF+1+2

roos 5.0971 -l4.9037444 1.0396 10.9210 0.004896
852 5.0971 -14.9037781 1.0397 10.9213 0.004653
Exptl. 5.0510 1.046

 

Table C-3. Calculated and experimental constants for the ground state, X 12;, of Na;

 

 

Method/basis r. 020) E.- (e/ao) D. (av) 69.. (eag) qn(e/a3)
RHF

roos 6.0414 -323.715883 -0.01458 12.1456 0.02574
qu 6.0365 -323.7l6628 -0.01973 12.1212 0.02541
CASSCF

roos 6.0141 -323.742751 0.7117 11.6614 0.02054
qu 6.0118 -323.743497 0.7079 11.5966 0.02041
CASSCF+1+2

roos 6.0055 -323.743156 0.7227 11.6905 0.02134
qu 6.0030 -323.743890 0.7175 11.6264 0.02112
Exptl. 5.8182 0.720

 

193

Table C-4 Calculated and experimental constants for the ground state, X 328', of B2

 

 

Method/basis r. (a0) Er (e/ao) D. (ev) (9.. (eag) qu(e/ag)
UHF

aug-cc-pvdz 3.1188 49.088087 0.7331 1.5809 -0.1310
aug-cc-pvtz 3.0942 49.092796 0.7743 1.4246 -0. l 132
aug-cc-pqu 3.0900 49.094648 0.7805 1.4000 -0.1126
aug—cc-vaz 3.0897 49.094949 0.7807 1.3758 -0.1123
CBS limit 3.0867 0.7808 1.2103 —0.1122
MPZ

aug-cc-pvdz 3.0642 49.229733 2.4950 1.0072 0.034542
aug-cc-pvtz 3.0205 49.255513 2.7043 0.8777 -0.007246
aug-cc-pqu 3.0118 49.261555 2.7245 0.8528 0.005252
aug-cc-vaz 3.0103 49.263164 2.7309 0.8476 -0.006043
CBS limit 3.0081 2.7338 0.8462

CASSCF

aug-cc-pvdz 3.0698 49.2144360 2.4825 1.0066 0.03152
aug-cc—pvtz 3.0525 49.2192017 2.2614 0.9064 0.04575
aug-cc-pqu 3.0507 492210594 2.2781 0.8887 0.04870
aug-cc-vaz 3.0503 492213994 2.2792 0.8843 0.04804
CBS limit 3.0503 0.8832 0.04800
CASSCF+1+2

aug-cc-pvdz 3.0753 492815040 2.2868 1.0414 0.01255
aug-cc-pvtz 3.0283 492995716 2.4153 0.9260 0.03609
aug-cc-pqu 3.0214 493030189 2.4347 0.8931 0.04108
aug-cc-vaz 3.0198 493039741 2.4358 0.8858 0.05683
CBS limit 3.0152 0.8802

Exptl.

 

194

Table C-S. Calculated and experimental constants for the ground state, X '22:, of C2

 

 

Method/basis r. (no) 131 (e/ao) D. (ev) 6.. (egg) q..(e/a§)
RHF

aug-cc-pvdz 2.3673 -75.388372 0.6017 2.9023 -0.8553
aug-cc-pvtz 2.3442 -75.401798 0.7653 2.7844 -0.8274
aug-cc-pqu 2.3408 45.405643 0.7891 2.7693 -0.8265
aug-cc-pv52 2.3406 45.406302 0.7894 2.7657 -0.8239
CBS limit 2.3405 0.7896 2.7645 -0.8231
MP2

aug-cc-pvdz 2.4153 45.706322 6.1068 2.2683 -0.6051
aug-oc-pvtz 2.3805 45.760793 6.5844 2.1823 -0.5828
aug-cc-pqu 2.3737 45.772458 6.6526 2.1615 -0.5843
aug-cc-vaz 2.3726 45.775764 6.6730 2.1556 -0.5848
CBS limit 2.3721 6.7104 2.1533 -0.5852
CASSCF

aug-cc-pvdz 2.3877 45.6266524 4.7080 2.3403 -0.4309
aug-oc-pvtz 2.3704 45.6396363 5.9879 2.2628 -0.4051
aug-cc-pqu 2.3688 45.6432811 6.1743 2.2505 -0.4002
aug-cc-pvsz 2.3685 -75.64404ll 6.1765 2.2463 -0.3987
CBS limit 2.3683 2.2425 -0.3980
CASSCF+1+2

aug-oc-pvdz 2.3963 45.7429896 4.7480 2.3700 -0.4815
aug-cc-pvtz 2.3631 45.7839690 6.0386 2.2987 -0.4554
aug-cc-pqu 2.3575 45.7914593 6.2266 2.2757 -0.4489
aug-cc-vaz 2.3562 457936075 6.2287 2.2694 -0.4470
CBS limit 2.3557 2.2684 -0.4464
Exptl. 2.074412 9.906512 409100731

 

195

Table C-6. Calculated and experimental constants for the ground state, X '53:, of N2

 

 

Method/basis r. (am ET (e/ao) D. (ev) 0.. (egg) q..(e/ag)
RHF

aug-cc-pvdz 2.0377 408.961925 4.9576 -0.9490 1.1237
aug-cc-pvtz 2.0163 408.987796 5.2404 4.0090 1.3857
aug-cc-pqu 2.0135 408994616 5.2836 4.0118 1.4060
aug-cc-vaz 2.0133 408995999 5.2870 4.0149 1.4042
CBS limit 2.0133 5.2891 4.0158

MPZ

aug-cc-pvdz 2.1388 409280650 9.4100 4.1262 0.8417
aug-cc-pvtz 2.1053 409364800 10.0293 4.1626 1.0637
aug-cc-pqu 2.0985 409383055 10.1363 4.1710 1.0858
aug-cc-vaz 2.0976 409388586 10.1731 4.1747 1.0854
CBS limit 2.0970 10.1724 4.1751

MCSCF

aug-oc-pvdz 2.1076 409097018 8.6337 4.1683

aug-cc-pvtz 2.0853 409120694 8.8568 4.2260

aug-cc-pqu 2.0825 409127412 8.8973 4.2292

CBS limit 2.0821 8.9063 4.2294

CASSCF

aug-cc-pvdz 2.1063 409111025 9.0131 4.1263 1.0345
aug-cc-pvtz 2.0873 409133920 9.2165 4.1675 1.1462
aug-cc-pqu 2.0860 409139872 9.2364 4.1753 1.1763
aug-cc-vaz 2.0856 409141326 9.2759 4.1790 1.1780
CBS limit 2.0857 4.1795 1.1870
MRCI

aug-oc-pvdz 2.1144 409284004 8.7673 4.1079

aug-cc-pvtz 2.0866 409367229 9.5037 4.1390

aug-cc-pqu 2.0806 409383822 9.5698 4.1464

CBS limit 2.0789 9.5721 4.1487
CASSCF+1+2

aug-cc-pvdz 2.1135 409305121 8.9126 4.0878 0.9881
aug—cc-pvtz . 2.0855 409372107 9.5294 4.1162 1.1095
aug-cc-pqu 2.0827 409386811 9.6725 4.1247 1.1432
aug-cc-vaz 2.0802 409388891 9.7291 4.1322 1.1478
CBS limit 2.0810 4.1345 1.1518
Exptl. 2.074412 9.906512 4.09100731

 

Table C-7. Calculated and experimental constants for the ground state, X 323', of Oz

196

 

 

Method/basis r. (a0) ET (e/ao) D. (ev) (9.. (gag) q..(e/a3)
ROHF

aug-cc-pvdz 2.1840 449625220 1.1784 4.2833 4.4415
aug-cc-pvtz 2.1779 449660506 1.2923 4.3928 -2.4657
aug-cc-pqu 2.1749 449670308 1.3110 -0.4085 4.4927
aug-cc-vaz 2.1747 449672974 1.3186 4.4183 4.5006
CBS limit 2.1739 1.3181 04188 4.5012
UHF

aug-cc—pvdz 2.1979 449646215 1.7497 4.2341 4.2547
aug-cc-pvtz 2.1915 449682470 1.8899 4.3134 4.2472
aug-oc-pqu 2.1885 449692404 1.9124 4.3273 4.2591
aug-cc-vaz 2.1885 449695099 1.9207 -0.3356 4.2578
CBS limit 2.1889 1.9172 4.3427 4.2565
UMP2

aug-cc-pvdz 2.3314 450.011480 5.2852 4.1107 4.3375
aug-cc-pvtz 2.3138 450.128401 5.6078 4.2178 4.3771
aug-cc-pqu 2.3064 450.153761 5.6322 4.2421 4.3934
aug-cc-vaz 2.3049 450.162153 5.6548 4.2537 4.3901
CBS limit 2.3032 5.6472 4.2546 4.3894
MCSCF

aug-cc-pvdz 2.3022 449718078 3.7982 4.1457

aug-cc-pvtz 2.2975 449752814 3.9146 4.2618

aug-cc-pqu 2.2939 449763505 3.9606 4.2858

CBS limit 2.2936 3.9906 4.2920

CASSCF

aug-cc-pvdz 2.3065 449725943 3.9798 4.1669 4.3470
aug-cc-pvtz 2.3009 449759501 4.0959 4.2650 4.3442
aug-oc-pqu 2.2973 449768890 4.1072 4.2822 4.3465
aug—cc-vaz 2.2970 449771592 4.1161 4.2930 4.3532
CBS limit 2.2957 4.1189 4.2921 4.3548
MRCI

aug-cc-pvdz 2.3072 450003808 4.4229 4.1213

aug-cc-pvtz 2.2921 450114527 4.9575 4.2156

aug-cc-pqu 2.2864 450138411 5.0034 4.2474

CBs limit 2.2830 5.0077 4.2636
CASSCF+1+2

aug-cc-pvdz 2.3069 450027600 4.7815 4.1448 4.3594
aug-cc-pvtz 2.2944 450.118085 4.8974 ' 4.2269 4.3543
aug-cc-pqu 2.2907 450.140112 5.0056 4.2368 4.3605
aug-cc-vaz 2.2868 450142995 5.0148 4.2612 4.3639
CBS limit 2.2854 5.0152 4.2588 4.3643
Exptl. 2.281812 5.231812 4.310136

 

Table 08. Calculated and experimental constants for the ground state, X ‘ZJ, of F2

197

 

 

Method/basis r. (a0) ET (e/ao) D. (ev) 0.. (egg) q..(e/a3)
RHF

aug-cc-pvdz 2.5288 498.703251 1.3847 0.5229 6.4144
aug-cc-pvtz 2.5099 -l98.760936 1.1764 0.3558 6.6284
aug-cc-pqu 2.5090 498774915 1.1770 0.3321 6.7109
aug-cc-vaz 2.5071 498779081 1.1786 0.3149 -6.7190
CBS limit 2.5079 498.7801 1.1792 0.3180 6.7198
MP2

aug-cc-pvdz 2.6959 499126917 1.5039 0.9254 -5.7669
aug-cc-pvtz 2.6490 499290907 1.7987 0.7588 -6.1000
aug-cc-pqu 2.6471 499326593 1.7931 0.7374 -6.1953
aug-oc-vaz 2.6452 499338340 1.8004 0.7199 6.2094
CBS limit 2.6460 499.3399 1.7970 0.7236 6.2109
MCSCF

aug-cc-pvdz 2.8257 498777612 0.6388 0.9873

aug-oc-pvtz 2.7643 498832379 0.7677 0.7899

aug-cc-pqu 2.7632 498846561 0.7658 0.7651

CBS limit 2.7632 498.8515 0.7681 0.7616

CASSCF

aug-cc-pvdz 2.8021 -l98.780304 0.7041 0.9585 -6.0449
aug-oc-pvtz 2.7654 498834724 0.7305 0.7778 6.2151
aug—cc-pqu 2.7586 498848763 0.8257 0.7611 6.2788
aug-cc-vaz 2.7560 498852728 0.8261 0.7426 6.2969
CBS limit 2.7560 0.8262 0.7481 6.3098
MRCI

aug-cc-pvdz 2.7520 499119575 1.2603 0.9370

aug-oc-pvtz 2.6838 499277054 1.4875 0.7356

aug-cc-pqu 2.6825 499310003 1.4879 0.7111

CBS limit 2.6827 4993187 1.4906 0.7077

CASSCF+1+2

aug—cc-pvdz 2.7329 499158133 1.2056 0.9032 -5.9777
aug-cc-pvtz 2.6955 499285251 1.4407 0.7352 -6. 1519
aug-cc-pqu 2.6853 499311735 1.4421 0.7165 6.2275
aug-oc-vaz 2.6802 499320173 1.4490 0.6927 6.2496
CBS limit 2.6791 1.4496 0.6985 6.2699
Exptl. 2.668112 1.691612 None Available

 

198

Table C-9. Calculated and experimental constants for the ground state, X 323’, of A12

 

 

Method/basis r. too) 131 (e/ao) D. (ev) 0.. (egg) q..(e/a3)
UHF

aug-cc-pvdz 4.8475 483.762718 0.3919 7.9896 -0.3016
aug-cc-pvtz 4.8180 483.774231 0.4322 7.6578 4.3168
aug-cc-pqu 4.8048 483.776836 0.4351 7.5442 4.2935
aug-cc-vaz 4.8027 483.777656 0.4365 7.5292 -O.3275
CBS limit 4.8023 7.5269

MP2

aug-oc-pvdz 4.7366 483853878 1.1071 7.2889 -0.2365
aug-oc-pvtz 4.6765 483879735 1.2532 7.1173 -0.2381
aug-cc-pqu 4.6599 483884449 1.2616 7.0273 -0.2205
aug-cc-vaz 4.6553 483.777050 1.2625 7.0348 -0.2422
CBS limit 4.5542

CASSCF

aug-cc-pvdz 4.7984 483826309 1.1890 6.6022 -0.2122
aug-cc-pvtz 4.7766 483837438 1.2381 6.3208 ~0.2006
aug-cc-pqu 4.7661 483840329 1.2431 6.2268 -0. 1965
aug-cc-vaz 4.7634 483841074 1.2511 6.2208

CBS limit 4.7625 6.2204

CASSCF+1+2

aug-cc-pvdz 4.7957 -483 888899 1. 1989 6.6022

aug-cc-pvtz 4.7263 483.910444 1.3468 6.3208

aug-cc-pqu 4.7086 483.914596 1.3906 6.1937

aug-cc-vaz 4.7063 483.915684 1.5268 6.1914

CBS limit 4.7060 6.1912

Exptl. 4.660 1.55

 

199

Table C-10. Calculated and experimental constants for the ground state, X 323', of Si;

 

Method/basis r. (ao) Er (e/ao) De (CV)

Ga (eag) qu (e/ag)

 

UHF
aug-cc-pvdz 4.2256 677.764760 1.71 13 4.6444 1.6932
aug-cc-pvtz 4.2016 677.779332 1.7981 4.7239 2.0001
aug-cc-pqu 4.1927 677.783621 1.8183 4.7584 1.8211
aug-cc-pvsz 4.1884 -577.785048 1.8317 4.7514 2.0860
CBS limit 4.1880

MP2

aug-oc-pvdz 4.3122 677897246 2.7258 4.2857 1.4929
aug-oc-pvtz 4.2689 677939991 3.0221 4.1696 1.7909
aug-cc-pqu 4.2541 677947457 3.0519 4.1973 1.6408
aug-cc-vaz 4.2517 677951384 3.1002 4.1834 1.8302
CBS limit 4.2511

CASSCF

aug-oc-pvdz 4.3555 677814916 2.6587 4.3142

aug-cc-pvtz 4.3300 -577.82833O 2.7862 4.3936

aug-cc-pqu 4.3205 677832431 2.8095 4.4238

aug-cc-pvsz 4.3162 677833570 2.8241 4.4151

CBS limit 4.3158

CASSCF+1+2

aug-cc-pvdz 4.3429 677932991 2.6835 4.0188

aug-cc-pvtz 4.2952 677970127 3.0642 4.0175

aug-cc-pqu 4.2788 677976584 3.1821 4.0593

aug-cc-vaz 4.2750 677978619 3.2318 4.0425

CBS limit 4.2741

Exptl. 4.2443 3.21

 

200

Table C-1 1. Calculated and experimental constants for the ground state, X 12;, of P2

 

 

Method/basis r. (ao) Er (e/ao) D. (eV) 92 (e613) 0:. (e/ag )
RHF

aug-cc-pvdz 3.5345 681.470428 1.3826 0.7231 1.6350
aug-oc-pvtz 3.5096 -681.491898 1.6232 0.7870 1.8942
aug-cc-pqu 3.5007 681.498395 1.6788 0.7785 1.8316
aug-cc-vaz 3.4969 681.500786 1.7288 0.7920 2.0363
CBS limit 3.4950

MPz

aug-cc—pvdz 3.6895 -68l.707199 4.1800 0.4203 1.1365
aug-cc-pvtz 3.6421 -681.775016 4.7259 0.5137 1.3700
aug-cc-pqu 3.6256 681.786020 4.8047 0.5080 1.3519
aug-oc-vaz 3.6201 681.790680 4.8396 0.5317 1.5039
CBS limit 3.6171

CASSCF

aug-cc-pvdz 3.6698 -681.564365 3.9380 0.3758 1.4625
aug-cc-pvtz 3.6425 681.583508 4.1160 0.3042 1.5462
aug-cc-pqu 3.6330 681589364 4.1543 0.3001 1.5705
aug-oc-vaz 3.6288 681591273 4.1882 0.3162 1.5712
CBS limit 3.6268

CASSCF+1+2

aug-cc-pvdz 3.6704 681.731599 4.0398 0.4989 1.3901
aug-cc-pvtz 3.6259 -68l.79l347 4.5810 0.5166 1.5007
aug-cc-pqu 3.6114 681.800869 4.6585 0.5035 1.5323
aug-cc-vaz 3.6077 -68l.804154 4.7094 0.5241 1.5319
CBS limit 3.6055

Exptl. 3.5780 5.033

 

201

Table C-12. Calculated and experimental constants for the ground state, X 328’, of S;

 

 

Method/basis r. (00) ET (e/ao) D. (ev) 9.1 (eag ) qn(e/ag )
ROHF

aug-cc-pvdz 3.5654 -795.054156 2.1420 1.4897 -1.7751
aug-cc-pvtz 3.5321 -795.083129 2.4563 1.4286 -1.7635
aug-cc-pqu 3.5216 -795.090308 2.5118 1.3940 -1.7337
aug-cc-vaz 3.5146 -795.093390 2.5791 1.3931 -1.7639
CBS limit 3.5123 1.3930

UHF

aug-cc-pvdz 3.5712 -795.067319 2.5002 1.7471 -0.8210
aug-ec-pvtz 3.5373 -795.098276 2.8685 1.4342 -0.7869
aug-ec-pqu 3.5268 -795. 105711 2.9309 1.4001 -0.7198
aug-cc-vaz 3.5194 -795. 109046 2.9768 1.3911 -0.7730
CBS limit 3.5171 1.3910

UMP2

aug-oc-pvdz 3.6679 -795.323675 3.9239 1.7466 -0.9704
aug-oc-pvtz 3.6109 -795.420906 4.0521 1.5373 -0.9648
aug-cc-pqu 3.5937 495.435561 4.4509 1.4916 4.8856
aug-oc-vaz 3.5831 -795.44196l 4.4968 1.4845 -0.9510
CBS limit 3.5801 1.4831

CASSCF

aug-cc-pvdz 3.6975 -795. 104169 3.1746 1.8238 -1. 1505
aug-cc-pvtz 3.6570 -795. 130461 3.4201 1.4922 -1.0727
aug-ec-pqu 3.6457 -795. 137012 3.4541 1.4660 -1.0454
aug-ec-vaz 3.6372 -795.139513 3.5000 1.4614 -1 .0226
CBS limit 3.6354 3.4927 1.4604 4.0112
CASSCF+1+2

aug-cc-pvdz 3.6780 -795.337615 3.6018 1.7322 -l.1505
aug-cc-pvtz 3.6258 -795.423487 3.9994 1.5075 -l.0249
aug-cc-pqu 3.6101 -795.435364 4.0591 1.4670 -0.9931
aug-cc-vaz 3.6005 -795.440183 4.1289 1.4602 -0.9689
CBS limit 3.5978 4.1191 1.4585 -0.9656
Exptl. 3.5701 4.3693

 

202

Table C-l3. Calculated and experimental constants for the ground state, X 12;, of C12

 

 

Method/basis r. (40) ET (e/ao) D. (ev) 0.. (egg) q..(e/a3)
RHF

aug-cc-pvdz 3.7915 -918.966152 0.7945 2.7868 6.5182
aug-cc-pvtz 3.7496 -919.000062 1.0793 2.2923 -5.8076
aug-CC-pqu 3.7309 -919.007029 1.1005 2.2401 -5.7488
aug-CC-pVSZ 3.7308 -919.009640 1.1441 2.2400 -6.0858
CBS limit 3.7267 2.2400 -6.0866
MP2

aug-cc-pvdz 3.8532 -9l9.259803 2.0128 2.8937 -5. 1629
aug-cc-pvtz 3.7770 -919.387079 2.4579 2.4349 -5.4837
aug-cc-pqu 3.7538 -919.404287 2.4938 2.3833 6.4457
aug-cc-vaz 3.7471 -9l9.410504 2.5252 2.3476 -5.7545
CBS limit 3.7440 2.3412 -5.7520
CASSCF

aug-cc-pvdz 3.9130 -918.993656 1.7089 3.0242 -6.1467
aug-cc-pvtz 3.8652 -919.024862 1.7526 2.5262 -6.0555
aug-cc-pqu 3.8554 -9l9.031537 1.7668 2.4962 -6.0503
aug-CC-pVSZ 3.8439 -919.033687 1.7982 2.4725 -6.0136
CBS limit 3.8434 1.7935 2.3833 -6.0168
CASSCF+1+2

aug-cc-pvdz 3.8846 -9l9.277887 1.8873 2.8560 -5.9721
aug-CC-pvtz 3.8182 -919.388888 2.2294 2.4251 -5.8841
aug-cc-pqu 3.8050 -919.402437 2.2633 2.3801 -5.8801
aug-CC-pVSZ 3.7881 -9l9.407181 2. 3082 2.3468 -5.83 89
CBS limit 3.7875 2.3159 2.3214 -5.8390
Exptl. 3.7566 2.4794 2.405zt0. 120

 

APPENDIX D

Distance Dependence of Select Properties

204

This Appendix contains lists of the total energy(E), quadrupole moment((~))
and the field gradient (qu) in atomic units as functions of internuclear separation
for SCF, CASSCF, and CASSCF+1+2. The CASSCF and CASSCF+1+2
wavefunctions as those described in chapter 2,3, and 4. Molecules with
multiplicities greater than 1 use the Restricted Open Shell Hartree-Fock
wavefilnction. All calculations use the Dunning aug-cc-pqu basis sets which are
described in Appendix B. The quadrupole moment and the field gradient were

computed using the MOLPRO package.

205

 

 

 

 

 

 

 

 

 

 

 

 

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CHAPTER 5

The Electronic Structure of ScLi, TiLi, VLi, CrLi, and CuLi
and Their Positive Ions

A. Introduction

In an earlier study, Harrisonl showed that those states of ScLi that dissociate to
ground state Li and Sc in a 4323d configuration are characterized by a small (<O.27 eV)

dissociation energy, while those states that correlate with Sc in a 453d2 configuration are

strongly bound, relative to this configuration. This is understood as follows: As Sc and
Li approach, the initial encounter is between the Sc 452 pair and the Li 23 orbital. This is
a repulsive interaction, and a bond will not form until Sc has hybridized the 4s, 4p into
two sp hybrids. This takes energy and the resulting bond energy reflects this cost of
hybridization. The 453 d2 configuration, however, can form a bond using the singly
occupied 43 and does not suffer a hybridization2 penalty. It does, however, suffer an

exchange energy loss3 (EEL) (since the 4s and 3d’s are all high-spin coupled) in
uncoupling the 4s electron from the two high-spin 3d’s.

Generalizing these observations to the remaining early transition metal atoms, it is
expected that the cost to hybridize the 432 pair to two sp hybrids will be related to the
separation between the 4523dn and 454p3dn configurations. Since the lowest 4s4p3dn
states are ~ 2 eV above4 the 52dn states for Se, Ti, and V, we expect the hybridization
cost will be approximately constant across this series, and the bond energies relative to
the szdn configuration will be similar and essentially independent of the number of 3d
electrons. In contrast, because of EEL, the bond energy associated with the sd““1
configuration will decrease, as the number of 3d electrons increases in going from Sc to
Cr. Note that the exchange energy loss is a maximum for Cr and is zero for Cu, and we

will study CuLi as an important reference. Of course, superimposed on these

233

234

considerations are orbital effects such as the variation of the size of the 4s orbital in going
from Sc to Cu.

The variation of the experimental4/calculated 52d“ - sdn+1 separation for the
elements considered here is shown in Figure 1. The drop in this separation, as one goes
from Sc to Cr, suggests that either Ti or V lithide might have ground states that correlate
with the excited sdn+1 configurations. As we shall see, the ground state of TiLi is 4(1) and

traces its lineage to the 5de configuration, while the ground state of VLi is 5A and traces

its lineage to the sd4 configuration.

Also presented in this work is a study of the mono-positive ions, MLi+. Ionizing
the lithides dramatically changes the nature of the bonding. The easiest way to see this is
to note that lithium’s ionization potential is smaller than that of the first-row transition
elements, so the lowest energy asymptotic fragments are Li+ and M in its ground state.
The resulting molecular states retain this qualitative description and are bound, due

primarily to the large polarizability of the transition element.

B. Technical Details and Wavefunction Construction

The basis sets used for the transition metal atoms consisted of a 14s9p5d
primitive set of Wachters5 augmented with two additional p fimctions and a diffiise d as
recommended by Hay.6 This set of functions was contracted to 554p3d, using the

Raffenetti7 contraction scheme. Two additional f firnctions were added to correlate the d

electrons, and their exponents were determined variationally at the MCSCF+1+2 level. A

235

 

 

2'0 " Relative Atomic Energies
4 1.26
15 .. _4F (453.12)
4 143-... .
1.0 a 0§§
‘ _5F (453d3)
i oer-...,“
05 ~ 035
,2 . '--_ 60 (4s3d4)
> 0.24..
8, 0.0 d —Zo (4s23d) —3F (4523d2) —4F._g4523d3) — so (4s23d4)
Lu ,
<
-O.5 -
231.00
-1.0 1 : 7s (453d5)
. -1.08
'1 '5 l l l l
Sc Ti V Cr

Figure 1. Energy separation between the lowest terms of the 52dn and sdn+1
configurations. Calculated(MRCI) and experimental separations are presented with the
experimental results connected by a dotted line.

236

second f set of four primitive functions, contracted to 3, were taken from Langhofi‘ and
Bauschlicher8 and were used in the CuLi calculations.

For the Lithium basis, a Williams and Streitwieser9 4p representation of the Li 2p
orbital was added to a 12 s primitive set of van Duijneveldt.10 The Lithium basis set was
contracted to 352p and proves flexible enough to describe both the ground- and the low-
lying excited 2P state of Lithium.

All calculations were performed on Silicon Graphic Workstations, using the
Columbu592ll set of codes. Multiconfigurational Self-Consistent Field wavefimctions
(MCSCF) were constructed, using two orbitals to describe the bonding electron pair and
permitting the d electrons to occupy orbitals so that the angular momentum (electronic
symmetry) was accurately described. For the Configuration Interaction wavefunction, all
single and double excitations were calculated from the MC SCF wavefirnction,
MCSCF+1+2. Unlinked clusters were accounted for by an Averaged Coupled-Pair
Functional12 (ACPF) over the same MCSCF+1+2 CSF space. In what follows, we will
refer to the MCSCF+1+2 as the multireference CI or MRCI. We show, in Figure 2, the
basic form of the MCSCF wavefunction and, in Table 5-1, the number of configuration

state functions (C SF’s) used in the MC SCF and MRCI wavefimctions.

C. Neutral Lithides

We will consider, first, the states that result when a Li atom interacts with the
ground state of the transition elements. This means we will consider the szdn

configurations for Sc, Ti, and V and the sdn”1 configuration of Cr and Cu. To orient the

discussion, Figures 3 and 4 show the potential energy curves for the states of TiLi that

237

Ground State MCSCF WaveFunctions

Neutral Ions

ScLi‘I’ mcscf— —(core) (b b*)2 cr3d5+l

TiLi ‘Pmcscf— —(core) (b b"‘)2 o[3d,rx 3d5+-3d7ry 3d5.]2
VLi rum: (core) (b b*)2 3d..1 3d5+1 3d,...l 3d,,1

CrLi W.,... = (core) (b b*)2 3d,.l 3a..1 3d5+1 3d,...l 3d,,l

CuLi \rlmcscf = (core) (b b*)2 3.1.."- 3a..2 3.15.2 3d...2 3a.,2

Positive Ions

ScLiW mcscf— (core) (b b"‘)2 3d5+l

TiLi+ Wm... = (core) (b b*)2 [3.1.x 3d8+ - 3d,, 3d.» ]2
VLi+ Wm... = (core) (b b*)2 3.15.1 3d,...l 3d...1

CrLi+ Irmcscf= (core) 0' 3d,,1 3d5.1 3.15.1 3d,...l 3.1.,l

CuLi+ rpm... = (core) 0' 3d,2 3.1.} 3d5+2 3an 3d...2

Figure 2. Orbitals involved in the MCSCF wavefimctions for the neutral and
monopositive transition-metal lithides.

238

Table 5-1. Energy separation between the lowest terms of the 52d“ and sd'"l
configurations. Calculated (MRCI) and experimental separations are presented with the
experimental results connected by a dotted line.

 

 

 

Molecule State High-Spin Electron CSF ’5
Configuration

MCSCF MRCI
ScLi 3A od5 4817
ScLi+ 2A d5 4 1619
ScLi+ 4A oold5 1 721
TiLi 4(1) Gdnds 12 17723
TiLi 4A dnxdnydg 6 9770
TiLi+ 3CD dnda 4622
TiLi+ 5A o dflxdnyds 1 1293
VLi 5A o dnxd.,[yd5 7 17641
VLi 5A do dnxdflydgg 7 17641
VLi+ 4A duxdnyd5 6 9770
VLi+ 62+ 0 dfixdnydm d5_ 1 2056
CrLi 6H o dodnyd5+ d5_ 8 27338
CrLi 62+ d5 8 27338
CrLi+ 52+ dflxdflydg+ d5_ 7 16970
CrLi+ 72+ od5 1 2833
CuLi 3A od5 5 98030
CuLi 12+ — 3 24747
CuLi+ 2A d5 4 49166
CuLi“ 22+ 6 4 49166

 

239

. TiLi MC SCF

‘1‘“. "

Energy (m Hartree)

 

 

 

 

Figure 3. Calculated (MCSCF) potential energy curves for the lowest quartet and
doublet states of TiLi that correlate with the ground-state products, Ti(3F) + Li(ZS).

240

41

TiLi MCSCF+1+2

Ti (39mm

 

Energy (mHartree)
r.

 

 

 

Figure 4. Calculated (MCSCF+1+2) potential energy curves for the lowest quartet and
doublet states of TiLi that correlate with the ground-state products, Ti(3 F) + Li(2S).

Energy (m Hartree)
O
l

 

241

TiLi 4(1)

Ti (534p3d2f) + Li (332p)

_..-
—-__..
l.—
..

    
 

............... mCSCf
......... mcscf+1 +2
acpf

 

 

Figure 5. 4(1) ground-state potential curve of TiLi, calculated with the MCSCF,
MCSCF+1+2, and ACPF techniques.

242

    
  

 

 

40 _
30 _ _
.1 '. Cr (554p3d2f) + Li (352p)
20 _
717 10 _
e ‘I
t ' z
‘1“ .
0 q ................................
é ....................
3 d ...........
53 -1o _
c
m -
20 J ............... mcscf
--------- mCScf+1+2
. acpf
'30 ..
I I I I ' I ' I ' I ' I I T t l I I

 

Figure 6. 62+ ground-state potential energy curves of CrLi, calculated with MCSCF,
MCSCF+1+2, and ACPF techniques.

243

 

 

 

 

 

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244

correlate with the 3F ground state of Ti. The MCSCF results in Figure 3 suggest
strongly that this level of theory is incapable of describing bond formation in this
molecule. The MRCI results in Figure 4, however, are much improved and predict that
TiLi is bound in all of the quartet and doublet states that derive from Ti(3F) + Li(ZS).
Note, also, the remarkably small binding energies and the large density of states predicted
for this system. The ACPF12 approximation corrects for unlinked cluster effects in the
MRCI, and we compare, in Figure 5, the results of this technique with the MCSCF and
MRCI prediction for the ground 4(1) state of TiLi. This figure illustrates a recurring
theme when Li interacts with a transition metal atom in the szdn configuration; i. e., most
of the bond energy is correlation energy. Similar results obtain for ScLi (3A) and VLi
(5A).

The ground-state terms of both Cr and Cu are derived from the sdn+1
configuration, and when Li interacts with these terms, we form 62* and 12* states,
respectively. The potential energy curves for CrLi are shown for various levels of
calculations in Figure 6. Note that, in this instance, the MC SCF seems qualitatively
correct, although correlation effects are crucial to the accurate representation of these
interactions. A similar result obtains for CuLi in the 12“ state. Summarized in Table 5-2,
the dissociation energy, bond length, and vibrational frequency for these states and
several others yet to be discussed.

The ground state of ScLi, TiLi, CrLi, and CuLi each correlate with the lowest

term of the transition element. The ground state of VLi is not the 5A (szd3) state

described above but a second 5A state that correlates to the sd4 configuration. These two

245

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246

states are shown in Figure 7. The MRCI binding energy for the 5A (sd4) state is 0.868
eV, relative to the 6D asymptote, while the binding energy of the 5A (szd3) state is 0.264
eV, relative to the 4F asymptote. Since our calculated $2d3-sd4 splitting is 0.342 eV, the
5A (sd4) is the ground state by 0.262 eV. This raises the question of how far above the
ground states of the other transition metal lithides are the states that correlate with the
excited state of the transition metal? To address this, wavefimctions were constructed
for TiLi that trace their lineage to the 5F (sd3) configuration of Ti and for CrLi that traces
their lineage to the 5D (szd4) of Cr. We were unsuccessful in constructing the
corresponding multireference CI functions that correlate with the 4F (sdz) state of Sc or
the 2D (52d9) state of Cu. The results are collected in Table 5-2. Figures 8-11, the
summarized results of bonding to the 52d“ or sdn“l states, for all of the lithides except
CuLi.

From Table 5-2, we see that those states that correlate with the den
configuration of the transition element have bond energies (at the MRCI level) between
0.2 and 0.3 eV; and further, the De’s increase with increasing bond length. Also, the
vibrational frequencies are all ~ 200 cm]. The states that correlate with the sdn+1
configurations have somewhat different characteristics. The bond strengths, relative to
the adiabatic asymptote, are larger than those of the 52dn states, and the vibrational
frequencies are also uniformly larger.

It is expected that, in order for Li to bond to a high-spin state of the sdn+1
configuration, the transition metal must uncouple the 4s spin from the remaining high-

spin 3d electrons. The energy involved in this randomization of the 4s spin, the exchange

247

   
 

15 —
VLi 5A MCSCF+ 1+2
‘ v (5s4p3d2f) + Li (382p)
1o - V(GD) + Li(28)
'l
5 A

V(4F) + LJ'(Z’S)

Energy (mHartree)
I

 

 

7 8 9 10 11 12
R (a 0)

Figure 7. Two 5A states of VLi that correlate with the 52d3 and sd4 asymptotes.

248

 

 

.+ .
8ch So SCLI
0.08 ev
_.__ 4F(spd)/
\ ..-...
I ’ \
/ 4 2
F d \
0.94 ev / (S ) l
/ \
/ \
/ 1.69 ev ‘
4 I \ 1.47«ev

 

 

 

 

 

 

2‘
_ l 20(32d1
1.92 ev l \\ 0.60 ev
,/ 0.30 ev\
/ .L__ ___ aA
/ .
/
l
/ l
’ 1.09 ev
2A __l I __._l

Figure 8. Comparison of the neutral and positive ion states of ScLi, calculated at the
MRCI level.

 

249

 

 

 

 

TiLi+ Ti TiLi.
__ .. 5F sd3 .____
I \\ l
1 I, \‘ 890
1.13 ev / 0.914 ev \ 0. av
/ \
/ \
/ X
/ 3F(32d2) \ 4
/ I i x A
5,, ... / I 0305 SW. 0.324 ev
l __L _ __ \__L__ 4(1)
l /
/ . v
/
0.784 ev /
I l’ 1 00 ev
3 ’ —---

Figure 9. Comparison of the neutral and positive ion states of TiLi, calculated at the
MRCI level.

 

250

VLi+ v

60(sd4)
I"_"-_]—\
I’ 0.342 ev\

"l-

 

 

.0.933 ev
I I
/ / _l__
I, I”
6 + _ / I
E _l— ,
4 0.323 ev,’ 0.914 ev

A __.l__’ _L.._

VLi

fF(82d3) \\
’ ' 0'264 e0\‘\ 0.868 ev

\\ 5
A
\ 0.262 ev
\___l_

5A

Figure 10. Comparison of the neutral and positive ion states of VLi, calculated at the

MRCI level.

 

 

251

 

 

.+ .
CrLI Cr CrLi
5D(82d4)
_ .. I x .. ......
1 I, l . \ 0.20 ev 6H
/
0.88 ev /’ 1.08 ev
/
/ 1.56 ev
52+ ' I
78(sd5)
ll \
/ \\
1.10 ev I, 0-68 eV \\
/

 

 

/ l \x 6 +
7 + L/ 0.90 ev'”'""' _— E
z I ....L-

Figure 11. Comparison of the neutral and positive ion states of CrLi, calculated at the
MRCI level.

252

energy loss,3 is easily estimated as half of the difference between the state in which the 45
is high-spin coupled to the 3d“1 system and in which it is low-spin coupled. For
example, in Sc, the 4F term corresponds to the 45 being high-spin coupled to the high-
spin 3d2 core, while the lowest 2F term corresponds to it being low-spin coupled to this
same high-spin 3d2 core. This energy difference4 is 0.424 eV and, therefore, the EEL is
0.212 eV. Continuing in this way, we construct Table 5-3 in which we tabulate the total
EEL for Sc-Cr as well as the BEL per 3d electron. Note that this latter quantity
decreases slightly in going from Sc-Cr, reflecting the contracting 3d shell. The calculated
De(both the MRCI and ACPF values) was augmented with the experimental EEL to
produce the tabulated “intrinsic” bond energies. These results suggest that the intrinsic

bond strength of Li and a 45 transition element orbital is ~ 1.62 eV (ACPF result) or 1.40

eV (MRCI result). This allows us to estimate the as—yet-calculated De for the 32‘ (sdz)
state of ScLi as 1.4 - 1.2 eV. Summarized in Figure 12 are the ground-state potential

curves for these lithides.

D. Positive Lithides
The asymptotic products for M-Li+ are M + Li+, M+ + Li, and M“ + Li+, where
M* is an excited state of the transition metal. The relative experimental energies of these

asymptotes are shown in Figure 13. The lowest asymptote for all transition elements is
the ground-state M, interacting with Li+, and we will consider this interaction first. Once
again, there are two classes of interaction. With Sc, Ti, and V, the Li+ ion encounters a

52dn configuration, while, for Cr and Cu, it is an sdn+1 configuration.

253

   

 

 

 

20- .
M - L1
106
04
a
g -10— .
(In ‘ “‘ TI
‘1‘ ‘-.‘ Cr
6% _20 _ l“ '3,“ \,"
a; ‘3‘. - -'
21:3 ‘ ‘1. --------------- ScLi 3A (4323d1)
L“ -30— ...... V --------- TiLi4<I> (4Sz3d2)
- ------------- VLi5A (4813d4)
-40 1 CrLi 6):*'(4s1 3d5)
, Cu Li 123“(4Sl3d10)
'50 fil'f'l'l'l'l‘l'l'l
3 4 5 6 7 8 9 10 11 12
R (30)

Figure 12. MRCI ground-state potential curves for the neutral lithides.

254

 

 

 

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6 '1 .................. ,5&3-5'32-(30).I.."‘£6d5183)m.
M + Lit ' 134952“ (20) _52d2(~‘F) :32d3(4F) _s2d4(so) _sd1°(?S)
5 21
4 d
m 1
v 3 _,
“J 1
<
2 2.
Mt + U 1 EWUF) L19 82(19 (20)
1 d ................. 0.?! sd3 (5F) ..... .EOde‘ISD)
. . ................ 124-6‘00)
M + Li 0 .. _ 52d (20) _52d2(3F) _ s2d3(4F) _ sd5 (75) _sd1°(~’S)
q
'1 I I l r 1
SC Ti V Cr Cu

Figure 13. Experimental energy levels for the low-lying asymptotic products M + Li, M
+ Li, M + Li*, and M* + Li“.

255

5‘. VLi+ 4A

0 4 v (5s4p3d2f) + Li (332p)

-10...

Energy (m Hartree)

 

-35- " I, --------------- mcscf
. ......... mcscf+1+2
acpf

 

 

 

Figure 14. 4A ground-state potential curves of VLi”, calculated with the MCSCF,
MCSCF+1+2, and ACPF techniques.

256

30—

20-1

104

O
I

L

-20 2

Energy (mHartree)
<13

 

 

 

 

. ~----«-----~ VLi+ 4A
-40- '''' --------- CrLi+ 72+

. CuLi+ 22+
'50 ' I ' I ' I I I T I ' I I I r I . I

3 4 5 6 7 8 9 10 1 1 12

Figure 15. MRCI ground-state potential energy curves for the monopositive lithides.

257

 

 

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258

In the presence of Li+, the 2D of Sc generates 22+, 2H, and 2A symmetries; the 3F
of Ti generates 32', 3H, 3A, and 3CD, and the 4F of V generates 42‘, 4IT, 4A, and 4(1). We
have considered the 2A of ScLi+, the 3CD of TiLi+, and the 4A of VLi+. The potential
energy curves for each of these states were calculated at several theoretical levels, and
representative curves for VLi+ are shown in Figure 14. It is interesting that, relative to
the more correlated techniques, the MCSCF level of theory overestimates the interaction
energy between L‘i+ and these transition metals in the 32d“ configurations. Since this
interaction energy will depend strongly on the in-situ charge on Li, we speculate that the
MC SCF function does not permit as much charge transfer from M to Li+ as the ACPF
and MRCI calculations. Another viewpoint is that the CI techniques more accurately
order the various mono-positive asymptotes shown in Figure 13. These results also
indicate that, unlike the neutral lithides, these systems can be adequately described
without significant electron correlation. The potential energy curves for these systems
are compared in Figure 15, while the dissociation energies, bond lengths, and vibrational
frequencies calculated at the MRCI level are collected in Table 5-4.

In each of these states, the transition element starts out with a 52dn configuration.
If the interaction was purely ionic, the number of electrons on Li at the equilibrium
separation in MLi+ would be 2, the 152 pair. From the population analysis shown in
Table 5-5, we see that Lithium has gained between 0.39 and 0.35 electrons, attesting to a
significant covalent interaction. Cr and Cu are fimdamentally different, in that their
ground states have a 5d“1 configuration and the bare Li+ interacts with a configuration
without any singlet coupled electron pairs. When interacting with Li+, the 7S state of Cr

generates a 72+ state, while the 2S state of Cu generates a 22*“ state. The potential

259

curves for these states are also shown in Figure 15, and their dissociation energies, bond
length, and vibrational frequencies are collected in Table 5-4. The population analysis
(Table 5-5) suggests that, in addition to the long-range electrostatic interaction, a small
but significant amount of charge is transferred to Li+, resulting in a one-electron, two-
center bond.

The data in Table 5-4 show that the bond energies of the ground-state M-Li+ ions
increase with increasing bond length. The correlation is almost linear, as we can see in
Figure 16. This is remarkable, considering that the bonding in the ScLi+, TiLi+, and
VLi+ is quite different from CrLi+ and CuLi+. To pursue this further, we constructed
wavefunctions for the states derivable from the M“ + Li+ asymptote. For Cr and Cu, this
means Li+ will interact with the 5D and 2D (szdn) asymptote, respectively, so, we studied
the 521+ state of CrLi+ and the 2A state of CuLi+. These results are shown in Table 5-4
and the corresponding population analysis in Table 5-5. Note that the correlation
between the bond length and bond energies is maintained. It is also noteworthy that,

while the bond energies change by a factor of two and the bond lengths by 0.4 A, the
vibrational frequencies of all of the 32d“ states changes by only 7 cm"1 in going fi'om the
lowest (240 cm'l) to the highest (247 cm'l).

States derivable from the M“ + Li+ asymptote for Sc, Ti, and V were also
considered. For Ti and V, this means that Li+ will interact with the 5F (sd3) and 6D (sd4)
states, respectively, so also studied was the 5A state of TiLi+ and the 62* state of VLi+.

These results are collected in Tables 5-4 and 5-5. The 5A (sd3) state of TiLi+ is bound by

 

260

 

 

 

 

83+ 83+ 83+ 83+ 83+ 83+ 83+ 83+ 83+ 83+ 3 o
83 83 83 83 83 83 83 83 3 3 .3 8 2
3 83 83 83 83 3 83 3 3 x33 5
83 83 W 83 83 W 83 83 W 83 83 W 3 3 a. 8 2
.3 83 83 83 83 3 83 3 3 3 x33 2

3 83 83 83 83 3 83 3 3 3 333 5

u . . u . . u . . u . . 3
83 83 m 83 3 83 3 m 83 _ 38 _ m 83 3 38 3 m 3 3 a 8 2
3 83 M 83 83 M 83 .3 M 83 3 3 3 3% 2
33 £3 83 :3 83 83 83 3:3 83 83 D33 3
83 83 83 83 83 83 33 83 83 83 mm 5
83 83 W 83 83 W 83 83 M 83 83 W 83 83 +38 2
83 83 m 83 383 W 83 83 M 83 3.3 w 32 3 33 o8 2
83 83M 83 33M 33 83M 83 33W 383 23 832
E3 83 83 83 83 83 83 83 E3 83 8 2
AS??? @313 Q” Fumv +w> $33 +Nm 98v +wo @333. A233 <n 933 Sn €qu <v ~33 <~

+56 M +56 M +35 M +3... m +58 335

 

3-2 8 885 88383 .3 23.2.

 

261

1.13 eV, relative to its adiabatic asymptote, while the 3CD (szdz) state is bound by 1.00
eV. Since the 52d2--sd3 separation is 0.96 eV, the 5A state is bound, relative to ground-
state Ti and Li+. A similar situation obtains for the 62* state of VLi+. The calculated
szd3-sd4 separation in V is 0.34 eV and the 62+ state is bound by 0.93 eV, placing it well
below the asymptote ground-state products, V (4F) and Li+. Scandium, once again, has
been difficult to characterize. We attempted to calculate the interaction of the 4F (sdz)
state of Se with Li“ and selected the 4A component of the term. Our results are
summarized in Tables 5-4 and 5-5. We see that, instead of the desired 45 do (is
configuration, we have converged to a state with a large 4 p0 component and very little

3do. This is, of course, the result of the 4F (4s4p3 d) term of Sc being only 0.07 eV

(calculated) above the 4F (453d2) term and is the same problem we encountered in trying

to determine the bond energy of 3Z- ScLi.
These energetics are summarized, and compared with the neutral lithides, in

Figures 8-11.

E. Previous Work

Very little is known about the transition—metal lithides. There are two published

experimental studies. Neubert and Zmbovl3 have estimated the dissociation energy of

CuLi to be 1.96 :1: 0.09 eV, using a Knudsen effusion technique combined with analysis of
the vapor composition. Van Zee, Braumann, and Weltner14 have identified the ESR
spectrum of CrLi (62+) in an argon matrix at 4 K. They determined the g-tensor,

hyperfine, and zero-field splitting parameters. There have been three previous

262

- Bond Energy/ Bond Length

¥ ScL'r’

38
.1

(A
O
1

Bond Energy (mHartree)
$3
I

N
co
:4.

 

26 _
- L CuLi” Da = 17.695 "' Re - 69.773

 

 

24 I I I l I l I I I T I I I I I l I I I I
5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4
Bond Length (a0)

Figure 16. Variation of the calculated (MRCI) De's of monopositive lithides with
equilibrium bond length.

263

Table 5-6. Comparison of Experimental and Theoretical Results

 

 

 

 

 

ScLi (3A)
Reference De(eV) Re(A) Comments
Harrisonl 0.27 3.28 NIRCI
Beckmann14 et a], 0.05 3.49 MRCI/pseudo potential
Lawson & Harrison 0.30/O.33 3.31/3.28 MRCI/ACPF
CuLi (12+)

De(eV) Rem.) (De(cm'l) Comments
Beckmannl4 et a]. 1.30 2.65 — —
Bauschlicher15 et al. 122/174 234/2.31 377/392 SDCI/CPF
Lawson and Harrison 1.30/1.62 247/2.43 332/356 NIRCI/ACPF
Neubert and Zmbov12 1.96 :t 0.09 — — Exp

 

theoretical studies. The first, by Harrison1 (3 MRCI study), discussed the structure of

ScLi and was followed by a study by Beckmann15 et a]. in which they examined the

electronic structure of ScLi, CuLi, and PdLi. These authors employed a pseudo potential

for the Sc and Cu, Argon cores and a MRCI for the remaining valence electrons. The

last is by Bauschlicherl6 et al. in which various metal dimers and trimers, including CuLi,

were studied. These authors used a large expansion basis and the SDCI and CPF

techniques. The experimental and theoretical results are summarized in Table 5-6.

264

F. Conclusions

First, consider the neutral lithides. Our results demonstate:
' Bonding is primarily between transition metal 4s and Li 25 electrons.
Significant 4s,4p hybridization in the ground states of ScLi and TiLi. These states
trace their lineage to the lowest state of the 4523dn transition metal configuration.
Virtually no 4s,4p hybridization in the ground states of VLi, CrLi, or CuLi. These
states trace their lineage to the lowest state of the 453dn+1 transition metal
configuration.
Bonding is extremely sensitive to electron correlation. Bond lengths decrease with
electron correlation, while bond energies increase significantly.
Bond strength increases significantly in going from ScLi (0.335 eV) to CuLi (1.62
eV), while bond length decreases from ScLi (3.305 A) to CuLi (2.471 A).

Now, the positive ions:
Bonding is primarily ionic and to first order seems to be Li+ interacting with the
highly polarizable neutral transition metal atoms.
The bond energies decrease as one goes from ScLi+ (1.09 eV) to CuLi+ (0.705 eV).
The bond lengths decrease as one goes from ScLi+ (3.259 A.) to CuLi+ (2.913 A)
Bonding is insensitive to electron correlation. The MCSCF, MRCI, and the ACPF

methods all predict similar Re, De, and (De’s.

The behavior upon ionization varies considerably across the series. For example,

ScLi+ and TiLi+ are more strongly bound than their neutral ground-state precursors

265

VLi+ has a bond strength comparable to its neutral precursor but has a much larger
bond length.
CrLi+ (72,") has a larger dissociation energy and a longer bond length than CrLi (62+)

CuLi+ (2A) is more weakly bound and has a longer bond length than CuLi (12+).

266

G. References

1.

2.

10.

11.

12.

13.

14.

15.

16.

Harrison, J. F. J. Phys. Chem. 1983, 87, 1323.

Brewer, L; Winn, J. S. Faraday Symp. Chem. Soc. 1980, 14, 126.
Alvarado-Swaisgood, A. E.; Harrison, J. F. J. Mol. Struct. 1988, 169, 155.
Moore, C. E. “Atomic Energy Levels,” National Bureau of Standards,
Washington, DC, 1949, Natl. Bur. Stand. (U. S. ), Circ. N0. 467, Vol. 1 & 2.
Wachters, A. J. H. J. Chem. Phys. 1970, 52, 1033.

Hay, P. J. J. Chem. Phys. 1977, 66, 4377.

Raffenetti, R. C. J. Chem. Phys. 1973, 58, 4452.

Langhofi‘, S. R.; Bauschlicher, Jr., C. W. Chem. Phys. Lett. 1986, 124, 241.
Williams, Jr., J. E.; Streitwieser, Jr., A. Chem. Phys. Lett. 1974, 25, 507.

van Duijneveldt, F. B. IBM Research Report No. R1945, 1971.

Shepard, R.; Shavitt, 1.; Pitzer, R. M.; Comeau, D. C.; Pepper, M. T.; Lischka,
H.; Szalay, P. G.; Ahlrichs, R.; Brown, F. B.; and Zhao, J. G. Intemat. J.
Quantum Chem. 1988, S22, 149.

Gdanitz, R. J .; Ahlrichs, R. Chem. Phys. Lett. 1988, 143, 413.

Neubert, A.; Zmbov, K. F. Trans. Faraday Soc. 1974, 70, 2219.

Van Zee, R. J .; Baumann, C. A., Weltner, Jr., W. Chem. Phys. Lett. 1985, 113,
524.

Beckmann, H.; Geanfranco, P. ; Jeung, G. Chem. Phys. Lett. 1985, 116, 423.
Bauschlicher, Jr., C. W.; Langhofl‘, S. R.; Partridge, H.; Walch, S. P. J. Chem.

Phys. 1987,86, 5603.

CHAPTER 6

The Electronic Structure of LiAl and Its Positive Ions

A. Introduction

In a recent experimental study1 of some spectroscopic states of AlLi, the
ground state was determined to be 12+. Two other excited states where also detected,
however, the symmetries of these states where not determined. Other theoretical
work agrees with the 123+ as the ground state, however, the location and symmetry of
the excited states are still indeterminate. This study evaluates the spectroscopic
constants and potential energy curves of the AlLi molecule. A comparison is made

between the electronic ground state of ScLi and AlLi.

B. Computational Methods

A Dunning correlation consistent polarized quadrupole zeta (cc-pqu) basis
was augmented with an additional 282p2d Double Rydberg Dunning Hay2 basis giving
185] 3p5d2f primitive firnctions on the Al atom. Adding the Rydberg Dunning Hay
functions improved the flexibility of the Al basis enough to characterize the low lying
atomic states of interest including the 3524s1 or 2S configuration. This basis set was
contracted to 8s7p5d2f providing a compact basis set with enough contractions to
characterize various atomic as well as molecular excited states.

Lithium basis sets came from the van Duijenveldt4 1158p primitive with the
addition of 5d functions basis fimctions. The basis set was contracted to 635p4d and
gave near Hartree-Fock limit energy for the ground state of the Li atom. The
wavefunctions for various excited states of the Al and Li atoms were evaluated and

their energies are summarized in Figure 1. As shown in the plot, these energies

268

30000 1

25000 —

20000 -r

Energy (crn-1 )
‘8‘
8

10000-

5000-

 

Figure 1. Low lying excited atom energies.

269

Electronic States of Li and Al

152381 (28)

1522p1 (2P0)

152231 (28)

Moore Li 655p4d

—3s13p2 (4P‘)__

352451 (28)

3s23p1 (2P0)

Moore A] 857p5d2f

 

270

compare well with the transition energies in Moore's tables. The ionization potentials
of Al and Li were determined to be 5.976 ev and 5.380 ev which compare well with
the experimental values of 5.984 ev and 5.390 ev. The ionization potentials for each

atom with various wavefirnctions are summarized in Table 6-1.

Table 6-1. Adiabatic ionization potentials as S => S+ + e'

 

MC SCF MRCI ACPF

A] 5.4281 5.9623 5.9755
Li 5.4320 5.3796 5.3804
AlLi 5.1354 6.5034 7.2461

 

Initially, Multireference-Self-Consitent-Field(MCSCF) wavefunctions were
generated fiom the valance space consisting of Al 3s3pz3px3py and the Li 25 space.
Multireference-Configuration-InteractionmeI) wavefunctions were constructed
from the MCSCF reference space. A larger set of MRCI active space calculations
were derived from the same MCSCF reference space, in which, single and double
excitations were also included out of Al 252px2py2pz orbitals and from the Li ls
orbital. Averaged Coupled Pair Functional(ACPF) wavefiinctions used the same
MCSCF reference space both with and without the single and double excitations from
the core electrons as in the MRCI calculations. All calculations were performed in sz

symmetry using the Columbus suite of codes.

 

271

C. Results

Three MCSCF wavefimctions were considered to describe electronic states
that dissociate to the ground state atoms and 2 MCSCF wavefunctions dissociate to
the first excited state of Li. Table 6-2 shows the various spectroscopic properties and
Figure 2 gives the relative potential energy surfaces of each electronic state. The
ground state was determined by MCSCF to be the 'Y with a bond energy of
De=0.959 ev relative to the ground state atoms and a vibrational frequency of me=3 19
cm'1 and a bond length of Rapt=5.379 a0. Mulliken population analysis shows the bond
to be from the atomic Al 3p, orbital and a Li 252pz hybrid orbital as expected. Four
other states are also summarized by Table 6-2 and Figure 2.

The MRCI and ACPF results are also given in Table 6-1 along with results of
the larger active space calculations. MRCI correlation increases the bond energy by
+0055 ev and decreases the. bond length by -0.011 a0. The relative populations of the
5 low lying states are provided in Table 6-3. The Al atom in the ground state acquires
a charge of -0.206 and this state has the smallest charge on A1 of any of the low lying
states. The larger active space of the MRCI and ACPF wavefiinctions increase the
MCSCF bond energy of the ground state by 0.060 ev and 0.112 ev respectively. Both
the MRCI and the ACPF gave the fundamental vibrational frequency as coe=3 03 cm“.
Though changing the total energy by as much as 21.1 Hartrees, these large active
space calculations indicate the insignificance of correlation in bonding to the core

orbitals.

272

Table 6-2. Characteristics of low lying states

 

 

State Method Total Energy De (mH / eV) ch (a0) u(debye)
(De(cm'l)
'2‘“ MCSCF -249.344441 35.229 / 0.9587 5.3787 3.752 319
MRCI -249.404096 40.034/ 1.0894 5.3800 3.083 318
-249.444066 37.450/ 1.0191 5.3522 3.279 308
ACPF -249.431216 38.398/ 1.0449 5.3437 303
-249.452287 39.340/ 1.0753 5.3107 308
expt.‘ 27.729 / 0.7545 318
Boldyrev, Gonzales, and Simons5 37.1 / 1.01
12' MCSCF -249.295088 28.273 / 0.7694 4.6222 4.236 391
MRCI -249.355194 52.347/ 1.4245 4.5537 4.226 419
-249.393786 51.019/ 1.3883 4.4931 4.074 439
ACPF -249.361025 57.129/ 1.5546 4.6093 399
-249.404229 58.432/ 1.5900 4.5583 407
32' MCSCF -249.311093 44.279/ 1.2049 4.5359 4.033 412
MRCI -249.394522 69.896/ 1.9020 4.5041 4.056 428
-249.412987 70.208/ 1.9150 4.4836 4.153 431
ACPF -249.398735 72.418/ 1.9706 4.5087 428
-249.419624 74.918 / 2.0387 4.4854 432
1II MCSCF ---.--- --.---- / --.---- - ---- ---
MRCI -249.398901 6.641 / 0.1807 5.2181 5.007 242
-249.416317 6.839 / 0.1861 5.2056 5.121 241
ACPF -249.402313 8.368 / 0.2277 5.1979 251
-249.421863 9.524 / 0.2592 5.1689 255
31'1 MCSCF -249.343916 9.433 /0.2567 5.0684 5.485 317
MRCI -249.420417 28.158 / 0.7662 4.9769 5.260 341
-249.437743 28.737 / 0.7820 4.9680 5.276 382
ACPF -249.423809 29.864 / 0.8126 4.9443 367
-249.443119 32.614 / 0.8875 4.9390 374

 

Energy (eV)

2.2 -

2.0 ~

1.8 -

1.6 -

1.4 5

12 -

1.0 4

0.8:

.

0.6 n

0.4 -'

02 -

0.0 —

-o.2 -
-o.4 -
-0.6 5
-o.8-3
-1.o-'

 

-1.2

273

Alw-r-LiGPO)

...------:::.- ..........
................
......
,.

1:,-

NHHC‘FP) + L? (’3)

o
......
...
....
I
oooooooooooooooo
...
........
........................
0'.
.-
D
I.
‘.
‘-
o

AlLi

--------------- MCSCF
—— MRCI

 

I 1

'l‘l
1314

l I ' l ' l
a 9 11 12
R (a0)

1
10.

Figure 2. Potential energy curves of the low lying electronic states of AlLi using
MCSCF and MRCI.

 

274

Table 6-3. Population analysis of low lying states of AlLi

 

 

 

112* 1‘2' 132' m I311
Al 35 1.9370 1.8112 1.7467 1.8319 1.8469
A1 3pz 1.1673 0.0823 0.0824 0.4983 0.5997
Al 3py 0.0187 0.7902 0.8259 0.9757 0.8398
Al 3px 0.0187 0.7902 0.8259 0.0339 0.0373
Li 25 0.7082 0.0505 0.0625 0.5348 0.4048
Li 2p, 0.0530 0.0454 0.0601 0.0602 0.0833
Li 2py 0.0148 0.1835 0.1780 0.0102 0.1292
Li 2px 0.0148 0.1835 0.1780 0.0002 0.0000
Q(M) +0.2092 +0.5049 +0.5214 +0.3944 +0.3825

The bond dissociation energy is inconsistent with the experimental value of
Duncan and co-workers (0.755 ev, 6089 cm") and a previous theoretical value of
Rosenkrantz (De=0.793 ev). However, it is consistent with Boldyrev and co-workers
(De=1.01 ev). Duncan regards his value as an estimate since it is based on a fit of four
vibrational hotband energy levels to a Morse potential. Based on initial calculations of
the excited states A1 atom, a large diffuse basis set is required to characterize the

electronic states with any degree of accuracy.

D. Excited States

A variety of other excited states of AlLi have been characterized by optimizing
the higher roots fi'om the MRCI wave. Figure 3 gives the potential energy curves of
all states computed by MRCI. The asymptotes of the potential energy curves are
ordered according to the energies of the corresponding atomic products. The ground

state is labeled I 12+ and higher roots of the same multiplicity and electronic symmetry

 

275

 

 

 

 

3
>5
U)
L—
OJ
C
NJ
0.0_ 132? ___A1(ZP°)+Li(28)
fi
1
-0.5-4 In
NH
\ .
-1.0- 112* .
' A1L1
rI'I'I'I'l'l'l'r'l'l'l
3 4 5 6 7 8 91011121314
R a
(0)

Figure 3. Potential energy curves of the low lying electronic states of AlLi using
MRCI.

are labeled II, 111,... in order of increasing energy. Both of the A states come from the

276

second roots of the singlet and triplet A2 wavefunction.

electronic state.

Table 6-4. Calculated characteristics of MRCI roots

Table 6-4 provides the total energies of the roots corresponding to each

 

 

State MRCI Total Energy TE(eV) De (mH / eV) Req (a0) mc(cm‘1)
112+ ROOT 1 -249.404096 0.0 40.034 / 1.089 5.3800 317
1112.:+ ROOT 2 -249.318327 2.3372 52.602 / 1.431 5.4734 206
11112:+ ROOT 3 -249.308791 2.6620 55.188 / 1.502 4.9678 457
112' ROOT 1 -249.355194 1.5504 52.347/ 1.424 4.5538 419
11A ROOT 2 -249.284764 3.2966 9.985 / 0.272 6.7297 130
132:- ROOT 1 -249.394522 0.5172 69.884/ 1.902 4.5041 428
13A ROOT 2 -249.290997 3.2610 16.203 /0.441 7.5073 355
111-1 ROOT 1 -249.377533 0.7233 7.010 / 0.191 5.2638 238
1111-1 ROOT 2 -249.317899 2.3633 36.965 / 1.006 5.6686 254
111111 ROOT 3 -249.284391 3.4157 3.771 /0.1026 7.4928 73
~ 131-1 ROOT 1 -249.398905 0.1751 28.362 / 0.772 5.0056 337
1131-1 ROOT 2 -249.321009 2.3598 40.915 / 1.113 6.1799 222
111311 ROOT 3 -249.299366 3.9665 20.677 / 0.563 5.8474 254

 

The adiabatic transition energy, optimized separation, and vibrational frequency are
also provided. All of the computed transition energies are within 4 ev of the ground
state. The potential energy curves of the first three electronic state are asymptotic to
the ground state atoms. The next 10 excited states dissociate to a ground state Al

atom and the first excited state of the Li atom. The remaining 3 electronic states

277

dissociate to products of a ground state Li atom and various electronic configurations

of the Al atom.

E. Positive Ion

The MCSCF first adiabatic ionization potential of AlLi was 5.1354 ev. MRCI
and ACPF give an ionization potential of 6.5034 ev and 7.2461 ev respectively. The
ground state of AlLi+ was determined to be the 222+ state. The positive ion was bound
at the MCSCF level by De=1. 1837 ev. MRCI and ACPF both increased the bonding
by less than 6 mev. The positive charge in the system reduces the bonding to an
electrostatic interaction of a positive Li ion and a neutral polarizable Al atom. Thus
correlation offers little change in the bonding energy. The correlation difference
between the MCSCF and MRCI ionization potential is attributed to the increase in
total energy of the neutral system with correlation.

Two other states, 211 and 411, also obtained minima in their potential energy
curves. As shown in Figure 4, the 211 dissociates to a ground state Al atom and a Li+,
while the asymptotic limit of the ‘II is an Al atom with a 333p2 configuration and a

positive Li atom. Populations for all bound states are given in Table 6—6.

F. AlLi AND ScLi
The ground state orbital configuration of the Sc atom is 4523dl while the Al
atom has a similar ground state of 3523p’. Similarities between the bonding of ScLi

and AlLi ground states might be expected, however, the two molecules have

278

3.5 ~ Li+ (IS) + A1 (4P)

 

Li*(15) + A1 (2P)

Dissociation Energy (eV)

.0

O
l
N
:1

j

5'3
0|
1

1

_.s
'0
J

N
*2

AlLi+MRC1
l'l'l'l'l'l'l'l

7 8 91011121314
R(a0)

 

 

.L

bl
U
0|—
0)—

Figure 4. Potential energy curves of the low lying electronic states of AlLi" using
MRCI.

279

Table 6-5. Characteristics of Low Lying States of the Positive Ion

 

 

State Method Total Energy De (mH / eV) ch (a0) c0¢(cm'l)
22* MCSCF —249. 156876 43.499/ 1.1837 5.3824 206
MRCI -249.166473 43.717/ 1.1896 5.8449 663
ACPF -249.166382 43.620/ 1.1869 5.9138 321
Boldyrev, Gonzales, and Simons5 37.131 / 1.0104 5.6786 242
211 MCSCF -249010094 26.567 / 0.7229 5.3799 269
MRCI -249.180183 8.599 / 0.2340 5.4433 208
ACPF -249.199630 9.751 /0.2653 5.3419 228
Boldyrev, Gonzales, and Simons5 5.3309 205
41'1 MCSCF -249.059376 30.198 / 0.8217 5.8064 232
MRCI -249.077067 30.192 / 0.8216 5.7890 235
ACPF -249.077286 30.194 / 0.8216 5.7897 234

 

Table 6-6. Population anaylsis of A1Li+

 

 

 

22+ 22- 4“
Al 35 1.9894 1.2591 0.9983
Al 3pz 0.7323 0.0306 0.8093
A] 3py 0.0003 0.6266 0.9478
Al 3px 0.0003 0.6266 0.0040
Li 25 0.2555 0.0089 0.1467
Li 2pz 0.0002 0.0114 0.0162
Li 2py 0.0000 0.0272 0.0225
Li 2px 0.0000 0.0272 0.0001
Q(M) +0.2555 +0.1097 +0.1785

280

significantly different chemistry. As stated in Chapter 5, there is a repulsive
interaction between the Sc 452 and the Li 25 orbitals and remains repulsive until
the Sc hybridizes. The singly occupied Sc 3d is unable to bond with the Li 25
due to its compact nature, or rather the diffuse nature of the Li 25 orbital
prevents it from passing the Sc 35. The energy required for hybridization of the
Sc 35 leaves an optimal bonding configuration for the ground state of ScLi as a
Sc atom with 4s‘3d2 configuration and the Li 251 atomic configuration. For an
A1 atom the 3s2 orbital is also repulsive to an approaching Li 25, however, the
single occupied 2p orbital extends well beyond the Al 35. There is no energy
1055 required for hybridization and the Li 25 orbital readily forms a bond with the
A1 3p orbital. It is also of interest to point out that the energy need for an Sc
atom to go fi'om a 4523dl state to a 4s'3d2 state is 1.427 ev while the energy
needed for an Al atom to go from 3523pl configuration to a 3513p2 configuration

is 3.605 ev.

G. Conclusion

In this study we computed the spectroscopic properties (re, we, Te, and
De) of 16 electronic states of AlLi using MCSCF followed by MRCI and ACPF.
The ground state of the AlLi was determined to be 12* state with a bond length
of ROP‘=5.311 a0 (2.8103 Angstroms), a vibrational frequency of (0e=308 cm'1
and a dissociation energy of De=1.071 ev relative to the ground state

asymptotes.

281

H. References

1.

2.

L. R. Brock, J .S. Pilgrim, M.A. Duncan, Chem. Phys. Lett. 1994, 230, 93

ME. Rosenkrantz, University of Dayton Interim Report No. PL-TR-3088,
Phillips Laboratory, Edwards Air Force Base, CA(1992)

A.I. Boldyrev, J. Simons and P. Von Schleyer, J .Chem.Phys., 1993, 99,
8793

HP. Cheng, R.N. Barnett and U. Landman, Phys. Rev. B, 1993, 48, 1820
AI. Boldyrev, N. Gonzales, and J. Simons, J. Phys. Chem., 1994, 98, 2098

CE. Moore, Atomic Energy Levels, NSRDS-NBS, Circular No. 35 (US
GPO, Washington, 1971)

van Duijneveldt, F. B., IBM Research Report No. R1945, 1971.

Basis sets were obtained from the Extensible Computational Chemistry
Environment Basis Set Database, Version 1.0, as developed and
distributed by the Molecular Science Computing Facility, Environmental
and Molecular Sciences Laboratory which is part of the Pacific Northwest
Laboratory, P.O. Box 999, Richland, Washington 993 52, US, and fimded
by the US. Department of Energy. The Pacific Northwest Laboratory is a
multi-program laboratory operated by Battelle Memorial Institue for the
U. S. Department of Energy under contract DE-AC06-76RLO 1830.
Contact David Feller, Karen Schuchardt, or Don Jones for further
information.

Shepard, R.; Shavitt, 1.; Pitzer, R. M.; Comeau, D. C.; Pepper, M. T.;
Lischka, H.; Szalay, P. G.; Ahlrichs, R.; Brown, F. B.; and Zhao, J. G.
Intemat. J. Quantum Chem. 1988, S22, 149.

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