THESIJ

This is to certify that the

thesis entitled

Theoretical and Experimental studies of ESR Spin Hamiltonian
parameters of Transition metal oxohalo complexes.

presented by

K. K. Sunil

has been accepted towards fulfillment
of the requirements for

Ph . D 0 degree in Chemi Stry

 

 

Date October 31,1980

 
 

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charge from circulation records

 

 

THEORETICAL AND EXPERIMENTAL STUDIES OF ESR
SPIN HAMILTONIAN PARAMETERS OF TRANSITION

METAL OXOHALO COMPLEXES

By

K. K. Sunil

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemistry

1980

ABSTRACT

THEORETICAL AND EXPERIMENTAL STUDIES OF ESR
SPIN HAMILTONIAN PARAMETERS OF TRANSITION
METAL OXOHALO COMPLEXES

By
K. K. Sunil

A detailed study of the electronic structure of a
series of d1 transition metal oxohalo complexes [MOXnJm‘,
where M = V, Nb, Cr, Mo, W and X = F, Cl, Br (n = A,5),
has been carried out by the self-consistent field multiple-
scattering Xa (SCF-MS-Xa) method. The results of the study
provide values of the d-d transition energies, and also
give some understanding of the similarities and dif-
ferences in bonding characteristics, of the penta- and
hexacoordinated complexes. The g and hyperfine inter-
action (A) tensor components were computed using the
SCF-MS—Xa wavefunctions and values of spin-orbit coupling
constants and <r-3> values computed for the appropriate
valence configuration of the atoms in the molecule using

atomic Xa wavefunctions. Comparison with the g and A

K. K. Sunil

tensor components of [CrOClujl- computed using extended
Hfickel wavefunctions shows the importance of using good
quality wavefunctions in estimating spin-Hamiltonian

parameters as well as the need to estimate the required

spin orbit coupling constants and <r'3> values in a non-
empirical manner. The various factors which determine
the magnitudes and signs of the ESR spin-Hamiltonian
parameters of transition metal oxohalo complexes are also
discussed.

In this thesis, the results of ESR studies of three
pentacoordinated dl transition metal oxohalo complexes
are also discussed. The single-crystal ESR spectra of
[VOFMJZI and [MoOFujl- were studied in single crystals of
(NHu)28bF5 and [MoOClujl- in single crystals of (NHu)2SbClS.
The spin-Hamiltonian parameters are compared with those of

the corresponding hexacoordinated species.

TO MY PARENTS

ACKNOWLEDGMENTS

I wish to express my sincere appreciation to Professors
M. T. Rogers and J. P. Harrison for their inspiring guid-
ance, encouragement and for the freedom allowed during the
course of this work.

It is a pleasure to thank Professor D. A. Case of
the University of California at Davis for supplying a copy
of his X Multiple-scattering program and for many helpful
advice on its use.

Of the many friends and colleagues who have helped
me I want to thank Dr. T. H. Pierce III for numerous
valuable discussions.

Thanks are extended to Michigan State University for
financial support as a graduate-assistant throughout the
course of this research and also to the Department of Energy
for support as a research assistant. Finally, I would like
to thank Peri-Anne Warstler for excellent typing and Beverly

Adams for the fine graphics work.

iii

TABLE OF CONTENTS

Chapter Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . V11
LIST OF FIGURES... . . . . . . . . . . . . . . . . Xi
CHAPTER I - INTRODUCTION . . . . . . . . . . . . . 1
CHAPTER II - AN INTRODUCTION TO THE
THEORY OF ESR SPECTRA . . . . . . . A
A. Spin Hamiltonians. . . . . . . . . . . 7
l. JCLS: Spin-Orbit Interaction . 7
2. JCZ: Zeeman Interaction. 9
3. JCSI: Electron Spin-Nuclear Spin
Interaction. 9
A. JCLI: Nuclear Spin-Orbit Inter-
action . O C O O O O O O O O O 0 O O 0 lo
5- JCSS: Electron Dipole Interaction. . . 11
6. }CQ: Nuclear Quadrupole Interaction. . 11
B. Experimental Methods for Obtaining
the Spin-Hamiltonian Parameters. . . . . . 1“
Chapter It- References . . . . . . . . . . . . . . 17
CHAPTER III - THEORY OF g AND HYPERFINE
INTERACTION TENSORS. . . . . . . . . 19
A. Theory of the g Tensor . . . . . . . . . . 19
B. Theory of the Hyperfine Interaction (A). . 2A
C. Evaluation of g and A Tensor
Components . . . . . . . . . . . . . . . . 27

iv

Chapter Page

Chapter III - References . . . . . . . . . . . . . 30

CHAPTER IV - AN INTRODUCTION TO THE SELF—
CONSISTENT FIELD MULTIPLE

SCATTERING Xa THEORY. . . . . . . . . 31

A. Hartree-Fock Equations and the

Xa Approximation . . . . . . . . . . . . . 32
B. Determination of o in the Xa

Method . . . . . . . . . . . . . . . . . . “2
C. Interpretation of Xa Eigenvalues and

Slater Transition States . . . . . . . . . “A
D. Multiple Scattering SCF Method . . . . . . A9

I. Muffin-tin Approximation . . . . . . . 50

2. Secular Equations. . . . . . . . . . . 53
3. The Self-consistent Potential
Field. . . . . . . . . . . . . . . . . 6A
A. Overlapping-Sphere Model . . . . . . . 68
5. Evaluation of One-electron
Properties . . . . . . . . . . . . . . 70
Chapter IV - References. . . . . . . . . . . . . . 75
CHAPTER V - ELECTRONIC STRUCTURE AND ESR
PARAMETERS OF [CrOClujl' . . . . . . . 77
A. Introduction . . . . . . . . . . . . . . . 77
B. Methods. . . . . . . . . . . . . . . . . . 78
C. Technical Details. . . . . . . . . . . . . 83
D. Electronic Structure of [CrOClujl' . . . . 87

E. Theory of g and Hyperfine Inter—
action Tensors . . . . . . . . . . . . 92

"IJ

Evaluation of g and A Tensor
Components . . . . . . . . . . . . . . . . 101

Chapter

G. Conclusions. . . . . . . .

Chapter V — References

CHAPTER VI - AN SCF—MS- Xq STUDY OF d1

TRANSITION METAL OXOHALO

COMPLEXES . . . .

A. Introduction . . . . . .
B. Methods.

(1) The SCF-MS-Xa Method.
(ii) Computational Details

C. Electronic Structure .

D. Electronic Excitation Energies

E. Evaluation of g and Hyperfine

Interaction Tensor Components.

F. Conclusion . . . . . . . . .
Chapter VI - References. . . . .

CHAPTER VII - ESR STUDY OF [voruj2
and [MoOClujl-

A. Experimental

B. Results. . . . .

l. Tetrafluoro Complexes of 0x0—
vanadium(IV) and Cxomolybdenum(V).

2. Tetrachlorooxomolybdate(V) Ion

C. Discussion .
Chapter VII - References

APPENDIX A

vi

,[MoOFujl-

Page

11A

115

118
118
120
120
122
126

139

1AA
157
159

162
16A
166

166
187
196
2OU
207

Table

Chapter V

10

LIST OF TABLES

Page
Geometry, sphere radii and
a values . . . . . . . . . . . . . . . . 82
Extended Hfickel parameters and
basis functions. . . . . . . . . . . . . 85
One-electron eigenfunctions
and eigenvalues. . . . . . . . . . . . . 86
Chromium-oxygen bonding in some
oxochromium compounds. . . . . . . . . . 88
Charge distribution in some oxo-
chromium compounds . . . . . . . . . . . 89
Electronic transition energies
from EHT and SCF-MS-Xo calculations. . . 9O
Chromium spin orbit coupling
constants (3d) and <r'3>3d values. . . . 103
ESR parameters of [CrOClujl- . . . . . . 107
Estimated chromium spin-orbit
coupling constants (A3d) and
<r-3>3d. . . . . . . . . . . . . . . . . 108
Extended Hfickel results for g and A
tensor components. . . . . . . . . . . . 112

vii

Table Page

 

Chapter VI
1 Geometrical data for [MOanm'
complexes. . . . . . . . . . . . . . . . 121
2 Sphere radii (R), a values and

virial ratios for [MOXnJm- com-

plexes . . . . . . . . . . . . . . . . . 123
3 Molecular orbitals of [VOFHJ2' . . . . . 127
A Molecular orbitals of [VOCluj2'. . . . . 128
5 Molecular orbitals of [VOF5]3_ . - . . . 129
6 Molecular orbitals of [voc1513‘. . . . . 130
7 Charge distribution in [Moxnjm'

complexes. . . . . . . . . . . . . . . . 13A
8 Electron distribution in the

metal orbitals of [Moxnjm‘

complexes. . . . . . . . . . . . . . . . 135
9 Electronic transition (d-d)

energies (cm-l) in [MOXnJm'

complexes. . . . . . . . . . . . . . . . 138
10 Computed values of Afid,

<r-3>IGd and Pfid. . . . . . . . . . . . . 1A9
11 Computed values of gll and

51 for [Moxnjm' complexes. . . . . . . . 151
12 Computed values of All and AL for

[Moanm‘ . . . . . . . . . . . . . . . . 155

viii

 

 

Table
Chapter VII
1 Single-crystal ESR parameters
of [VOFu12-. . . . . .
2 Single-crystal ESR parameters
of [MoOFMJl- .
3 Single crystal ESR parameters
of [MoOClu11-. .
A Molecular orbital coefficients
for molybdenyl complexes
5 Molecular orbital coefficients
for vanadyl complexes.
6 Spin Hamiltonian parameters for
vanadyl complexes. .
7 Spin Hamiltonian parameters for
molybdenyl complexes .
Appendix A
1 Molecular orbitals of [CrOFu]1-.
2 Molecular orbital of [Cr0C1532‘.
3 Molecular orbitals of [CrOF5J2'.
A Molecular orbitals of [MoOFu]1'.
5 Molecular orbitals of [MoOClujl—
6 Molecular orbitals of [MoOBrujl-
7 Molecular orbitals of [MoOFSJ2'.
8 Molecular orbitals of [MoOClSJZ-

ix

Page

17“

183

19A

198

199

201

202

207
208
209
210
211
212
213
21A

Table Page

9 Molecular orbitals of [NbOFujl-. . . . . 215

10 Molecular orbitals of [WOFujl- . . . . . 216
11 Molecular Orbitals of [WOC1u]1'. . . . . 217

LIST OF FIGURES

 

 

Figure Page
Chapter IV
1 Partition of space for a tri-
atomic molecule. . . . . . . . . . . . . 51
Chapter V
1 Coordinate system for [CrCClujl‘ . . . . 96
2 Plot of spin orbit coupling

constant (X83) versus charge

 

on chromium (QCr)° . . . . . . . . . . . 109
3 Plot of spin orbit coupling
constant (133) versus <r73>gg. . . . . . 110
Chapter VI
1 Percentage d contribution to M—0

0 and n bonds in MOn+ and [MOijn'

where M = V, Cr, Mo, X = F, C1 and

m = A and 5. . . . . . . . . . . . . . . 132
2 Plot of electronic transition

(d—d) energies for [MOXSJHI where

M = V, Cr, Mo and X = F, C1. . . . . . . 1A0

xi

Figure Page

3 Plot of electronic transition

energies for [MOXujn- where

M = V, Cr, Mo, W and X =

F, C1. . . . . . . . . . . . . . . . . . 1A1
A Plot of electronic transition

energies for [MOXQJn- where

M = Mo, W and X = F, Cl, Br. . . . . . . 1A2

Chapter VII

1 Crystal structure of (NHu)28bF5

with internuclear distances given

in Angstrom units. . . . . . . . . . . . 165
2 Coordinate system for the

analysis of g and A tensors of

pentacoordinated transition

metal oxohalo complexes. . . . . . . . . 167
3 Variation of g with the magnetic

field in the ab plane for [VOFMJ2'

in (NHu)ZSbF5. . . . . . . . . . . . . . 169
A Variation of A with the magnetic

field in the ab plane for [VOFu]2-

in (NH4)2SbF5. . . . . . . . . . . . . . 170
5 Variation of g with the magnetic

field in the ho plane for [VOFUJ2—

in (NHu)2SbF 171

5.

xii

Figure Page

6 Variation of A with the magnetic
field in the ho plane for [voruj2‘
in (NHu)2SbF5. . . . . . . . . . . . . . 172
7 ESR spectrum of polycrystalline
sample of [V0F432‘ in (NHu)28bF5
at room temperature. . . . . . . . . . . 176
8 Simulated ESR spectrum of poly—
crystalline sample of [VOFujz‘ in
(NHu)2SbF5 . . . . . . . . . . . . . . . 177
9 Variation of g with magnetic field
in the be plane for [MoOFujl- in
(NHA)ZSbFS . . . . . . . . . . . . . . . 178
10 Variation of g with magnetic field

in the ab plane for [Moorujl‘ in

(NHu)2SbF5 . . . . . . . . . . . . . . . 179
11 ESR spectrum of [MoOFu]l- in

(NHu)2SbF5 for 0 = 0° and

o = 0° . . . . . . . . . . . . . . . . . 181
12 ESR spectrum of [MoOFujl- in

(NHu)2SbF5 for e = 90° and

d = 0° or 90°. . . . . . . . . . . . . . 182
13 ESR spectrum of polycrystalline

sample of [MoOFu]l- in (NHA)2SbF

5
at 77°K. . . . . . . . . . . . . . . . . 185

xiii

Figure Page

1A Crystal structure of (NHu)ZSbC15

with internuclear distance given

in Angstrom. . . . . . . . . . . . . . . 188
16 Variation of g with magnetic

field in the bc* plane for [MooClull'

in (NHu)ZSbC15 . . . . . . . . . . . . . 189
17 Variation of g with magnetic field

in the ab plane for [MoOClu]l- in

(NHu)ZSbC15. . . . . . . . . . . . . . . 190
18 Variation of A with magnetic field

in the ab plane for [MoOClujl- in

(NHu)28bC15. . . . . . . . . . . . . . . 191
19 Variation of A with magnetic field

in the bc* plane for [MoOClujl' in

(NHu)28bC15. . . . . . . . . . . . . . . 192
20 Variation of A with magnetic field

in the ac* plane for [MoOClujl-

in (NHu)2SbCl5 . . . . . . . . . . . . . 193

xiv

CHAPTER I

INTRODUCTION

The nature of bonding in molecules has been an active
area of study since the early days of chemistry and will
continue to be so, since more and more powerful experi-
mental and theoretical techniques are being developed.

The ultimate goal of all spectroscopic studies is to
understand the basic forces that hold atoms together in
the form of molecules, ions and radicals and to explain
the observed trends in physical and chemical properties.
The main aim of the various theoretical models of molecu-
lar electronic structure theory is essentially the same.

Among the numerous spectroscopic techniques available
to study the basic features of bonding in transition
metal complexes, electron spin resonance spectroscopy
is considered to be one of the most powerful. The analysis
of electron spin resonance spectra provides valuable in—
formation concerning molecular symmetry, spin distribu-
tion and the nature of the ground and low lying excited
states. Hence the electron spin resonance parameters
which depend on the details of molecular electronic struc-

ture are interpreted in terms of the molecular orbitals

of the system.

A detailed study of the electronic structure of a
series of dl transition metal oxohalo complexes [MOXnJm',
where M = V, Nb, Cr, Mo, W and X = F, Cl, Br (n = A,5),
has been carried out by the self consistent field mul-
tiple scattering Xa (SCF-MS-Xa) method. The electronic
structure studies were carried out in the SCF-MS-Xa model
primarily because it allows an approximately quantita-
tive description of the electronic structure of systems
with a large number of electrons. The results of the
study provide values of the d-d transition energies, and
also give some understanding of the similarities and dif-
ferences in bonding characteristics, of the penta- and
hexa coordinated complexes. The g and hyperfine inter-
action (A) tensor components were computed using the
SCF-MS-Xa wavefunctions and values of spin-orbit coupling

3

constants and <r' > values computed for the apprOpriate
valence configuration of the atoms in the molecule using
atomic Xa wavefunctions. Comparison with the g and A
tensor components of [CrOClujl- computed using extended
Hackel wavefunctions shows the importance of using good
quality wavefunctions in estimating spin-Hamiltonian
parameters as well as the need to estimate the required

3

Spin orbit coupling constants and <r- > values in a non-
empirical manner. The various factors which determine

the matnitudes and signs of the ESR spin Hamiltonian

parameters of transition metal oxohalo complexes are also
discussed.

In this thesis, ESR spectra of three penta coordinated
dl transition metal oxohalo complexes are discussed. The
single-crystal ESR spectra of [VCFuj2- and [MoOFu]1- in
single-crystals of (NHu)28bC15. The spin-Hamiltonian

parameters are compared with those of the corresponding

hexacoordinated species.

CHAPTER II

AN INTRODUCTION TO THE THEORY OF

ESR SPECTRA

An electron has a spin angular momentum of one-half
which, in the absence of a magnetic field gives rise to
a doubly-degenerate spin energy level. The degeneracy
of the spin states is removed by a magnetic field and

the energy separation AE, of the two states is then given

by

AB = hv = gBB, (l)

where h is Planck's constant, v is the frequency of the
electromagnetic radiation required to induce a transi-
tion between these two energy states, g = 2.0023 is a
constant for a free electron and B is the magnitude of
the applied magnetic field. An unpaired electron in a
molecule cu° ion which moves in the force field of nuclei
possess orbital angular momentum in addition to the spin
angular momentum. The interaction between the orbital

and spin angular momenta of the electron, which is

referred to as spin-orbit interaction, makes a

contribution to the g value, thus making 'g' a charac-
teristic property of the molecule or ion containing the
unpaired electronl

The spin angular momentum of an electron interacts
with the nuclear magnetic moment giving rise to hyper-
fine structure in the ESR spectra of molecules or ions
containing nuclei with nonzero nuclear spin. An electron
interacting with a nucleus of spin angular momentum I
gives rise to (2I+1) lines in the ESR spectrum. The
hyperfine coupling constant can be measured from the
spacing between the lines. The measured values of g and
of the hyperfine coupling constants provide valuable in-
formation concerning the molecular symmetry, symmetry of
the orbital containing the unpaired electron, the spin
distribution and the nature of bonding.

The analysis and interpretation of ESR spectra are
customarily done in terms of a spin HamiltonianZ, since
the spectral transitions arise from induced changes in
the spin state of the system. The spin Hamiltonian
arises from the replacement of the complete Hamiltonian1
by an effective Hamiltonian which includes the applied
magnetic field, the spin operators and a set of param-
eters which characterize the ESR spectra. The expressions
for the magnetic energy levels of the system can be
worked out in terms of these ESR parameters. The analysis

of an ESR spectrum thus reduces to the specification

of the appropriate values of the parameters which appear
in the spin Hamiltonian for the system and the inter-
pretation of the spectrum is concerned with understanding
the magnitudes and signs of the parameters in terms of
the molecular electronic structure of the system.

There are many surveys of the transition metal ESR
literature3-6. The use of molecular orbital theory to
interpret ESR results has been reviewed by McGarveyl,

7

, and Kuska and Rogersa. There are two reviews
9,10

Konig

with emphasis on first row transi-
11

by Kuska and Rogers
tion metal complexes. A review by Goodman and Raynor

gives comprehensive coverage of the d1 to d9 ions for the

12

entire transition metal series. Recently Kohin has

reviewed ESR studies of vanadyl ion in crystalline solids.
In addition to these the current literature is reviewed
in the annual reports of the Chemical Societyl3, in the
series "Spectroscopic properties of Inorganic and Organo-

1A

metallic Compounds" and in the Journal "Magnetic Reson-

ance Reviews"15.
There is also a large number of books on ESR. The

books by Carrington and McLachIan16 and by Slichterl7

give complete introduction to magnetic resonance while

those by AthertonlB, Ortonlg, Pake2O and Wertz and Bolton21

deal with only ESR. A comprehensive account of the ESR

of transition metal ions is given in the book by Abragam

and Bleaney22 and a detailed mathematical account of the

theory of transition metal ions is available in the book
by Griffith23. Books dealing with experimental tech-

niques include those by Poole214 and by Alger25.

A. Spin Hamiltonians

For a molecule with fixed nuclei (Born-Oppenheimer
approximation), the general Hamiltonianl which takes into

account all the magnetic and electric interactions that

arise in ESR spectroscopy can be written as
2 2
R: ‘2'm 2 Vi + V +JLCLS ”CZ +}CSI +chLI +J'CSS ”Co
(2)
where the first term is the kinetic energy operator for

the electrons and V is the electrostatic potential which

includes both attractive potentials between electrons and

nuclei and repulsive potentials between electrons. The

remaining terms in the Hamiltonian are discussed below:

1. JQLS: Spin-Orbit Interaction23

The spin-orbit interaction is a relativistic effect
arising from interaction between the spin magnetic moment
of the electron and the magnetic field produced by the
motion of the nucleus. In relativistic quantum theory

this interaction is represented as

J-C = °°2 {§X[++9KJ}-S (3)
LS T2 9 C ’

2m C
where E is the electric field in which the electron moves,
3 is the linear momentum operator for the electron, S
is the spin-angular momentum operator in units of‘h,
and A is the vector potential for any magnetic field
present. In most cases the term involving A is neglected,

owing to its small magnitude. Since the electric field

is spherically symmetric for a single ion or atom,

 

ELS=(22)E(r)E°§=ECr)E-S, (u)

where E is the orbital angular momentum of the electron
in units of h- For the case of an atom in a molecule,

where there is more than one center of the electric field,

it is customary to assume that the spin-orbit interaction

can be written as

, (5)
LS1K i

where r1K is the distance of electron i from nucleus K,
.p
11K is the orbital angular momentum Operator for electron

i centered at nucleus K and Si is the spin angular momentum

operator for the electron 1. Equation (5) is based on
the assumption that the main contribution tO-HIS comes
from the region close to the nucleus, since €(r) varies
as <r-3>, and that near the nucleus the electric field

can be regarded as approximately spherical.

2. }£z: Zeeman Interaction

The interaction between the magnetic field and the
spin and orbital angular momentum is called the Zeeman
interaction. This interaction is represented by the

following term in the Hamiltonian

UH

JCZ = seBeE- + Beg-f - gNBNE-I , (5)

where ge is the free electron g value, Be is the electronic

Bohr magneton, B is the magnetic field, gN the nuclear g

factor and 5N the nuclear Bohr magneton.

3. }£ Electron Spin-Nuclear Spin Interaction

SI‘
This interaction which is called the hyperfine inter-

action can be written as

_ + + + 2 -5
}CSI ‘ gegNBeBN 12K [3(Si°riK)(TK°riK) ‘ riK(§i°TK)]riK

,

+ '83.!!- gegNBeBN iZK 6(riK)IK.Si 3 (7)

.3

10

where gN is the nuclear g value, BN the nuclear Bohr
magneton and giK the vector connecting electron i with
nucleus K. The summation index i represents the summa-
tion over all electrons in the system and for K the sum-
mation is over all the nuclei with nonzero nuclear spin.
In Equation (7) 6(riK) is the Dirac delta function which,
when integrated with the wavefunction, gives the value of
the wavefunction at r1K = 0. The two terms in Equation
(7) are the two limiting forms of the same interaction.
The first term represents the dipole-dipole interaction
for two dipoles that are not too close to each other.

It is the proper form ofLHgl for electrons in p,d, and

f orbitals which have nodes at the nucleus. The second
term, which is referred to as the contact term in the
hyperfine interaction, represents the interaction between
the nucleus and an electron which has a finite probability

density at the nucleus.

A. JQLI: Nuclear Spin—Orbit Interaction

 

_ -3 I +
JCLI ‘ gegNBeBN Z P1K(£iK°IK) (8)
i,K
This term is important only in that it gives a second order
contribution to the hyperfine interaction by allowing the
nuclear spin and electron spin to couple indirectly through

the orbital angular momentum.

11

5. JQSQ: Electron Dipole Interaction

U

This interaction gives rise to the zero field or the

spin-spin splitting, in the ESR spectra and arises from

a dipole-dipole type interaction between the electrons.

_ 2 2 2 .+ + .+ .+ -5
}CSS - seBe JZKErJK(§j SK) - 3<sj rJK)(§K ij)1rJK (9)
where rJK is the distance between two electrons. This

interaction arises only for systems with more than one

unpaired electron.

6. JgQ: Nuclear Quadrupole Interaction

If the nucleus has a quadrupole moment Q, there is
an electrostatic interaction between the electrons and
the nuclear quadrupole. This interaction is represented

by the following term in the Hamiltonian

.. 2 2 ‘* * 2 ‘5
JCQ - 1£K[e QK/2IK(2IK-l)JEPiKIK(IK+1) ‘ 3(riK°IK) JriK :
(10)

where QK is the electric quadrupole moment of nucleus K.
Some small terms have not been included in the total

Hamiltonian (Equation (2» since these terms were not found

necessary to account for the observed ESR spectra. The

terms so neglected are the nuclear Spin-spin coupling

12

term and the nuclear chemical shift term. For systems
with only one unpaired electron, including those which
are the subject ofthis thesis, the electron-dipole inter-
actions do not make any contribution and the nuclear Zee-
man interaction and the nuclear quadrupole interaction have
been found to be generally negligible. Hence these inter-
actions will not be considered further.

The complete Hamiltonian for a system with one un-
paired electron, where the nuclear Zeeman and nuclear
quadrupole interactions make negligible contributions,

can be written as

R zxo +J'CLS ”ct ”'CSI +J'CLI ’ (11)
where
R--h2zv2+v (12)
o - EH 1 i
' + + + +
JCZ = geBeB 0 S + BeB - L (13)

and the remaining terms in Equation (11) are defined in

Equations (5), (7) and (8). If u§ is the ground state

electronic wavefunction, i.e.,

mm = mm , w

13

then the wavefunction corrected to first order for the

spin-orbit interaction gives rise to a pair of Kramer's

 

 

doublets26 as shown below
0 O
<w IJC [w a>
+> = lea> = lw§a> + z lw§> K 0 L30 N
K36N EN- EK
o 0
<1» |}C Iw B>
N
-> = IA 8> = Iw°B> + 2 Iw°> K LS , (15)
N N K O 0

These first-order spin-orbit corrected wavefunctions

|+> and |->, are not eigenfunctions of the spin angular

momentum operator S.

Now we define a fictitious spin

angular momentum operator g, the components of which are

defined to act on the states |+> and I-> in exactly the

same way as the true spin operators act on the spin func-

tions Ia> and IB>; that is

SZ |+> =
§z I“) =

and so on. Then the

in terms of only the

A l
|+>, §XI+> = -2- |->

NIH

(16)
-%l->, §|+>=%|->

total Hamiltonian can be rewritten

spin (fictitious) Operators and the

magnetic field as follows

1A

.§+

:0»

° fiK ° IK , (17)

09>)

J—C=BB° 2
K
where 2 and A are the g and hyperfine interaction tensors,
respectively. The Hamiltonian of Equation (17) is the ESR
spin Hamiltonian for a system containing only one un-
paired electron. The spin Hamiltonian is an artificial

but useful concept which has become the crossroad for the
path followed by the experimentalists and the theoretic-
ians. Experimentally, the spin Hamiltonian and the param-
eters which define it, are determined from the ESR spectra,

whereas, theoretically, the parameters are computed from

the wavefunctionl.

B. Experimental Methods for Obtaining the Spin-Hamiltonian

Parameters

Electron spin resonance measurements are usually made
on magnetically dilute samples. Measurements made with
pure paramagnetic samples are less informative because of
exchange broadening of the lines. Magnetic dilution is
usually done in solution by preparing a dilute solution of
the material in a suitable solvent. In the solid state it

is accomplished by doping the paramagnetic sample to an

extent of about 1% in a diamagnetic solid.
For the transition metal oxohalo complexes studied

here the spin-Hamiltonian is

15

A

}c= gIIBeBZSZ + giseaaxsX + BySy)

M A A M
+ A||IZSZ + Al(IxSX + IySy)

M M M M M
= A a A = A = A
I l _|_ X

3|! 3 gzz’ gi,= gxx = gyy’ A 22 x yy’

‘where only the interaction between the unpaired electron

and the metal atom is included and Si’ Ii’ 1 = x,y,z are

the components of the electron and nuclear spin operators,
respectively.

The principal components (gll, g', AT', AT) of both the
g and the hyperfine interaction (A) tensors can be obtained

from single crystal studies. In Schonland's method27

of determining the principal components of g and A tensors,
ESR spectra should be obtained for rotations of the
crystal about three orthogonal axes. The need to obtain
spectra for rotations about three orthogonal axes is only
a limitation of Schonland's method. The method of Waller
28

and Rogers for the general case of rotations about any
three axes can be used to determine the principal components
of both the g and A tensors. Even though it is difficult,
the principal components of g and A tensors can be ob-

tained from powder and frozen solution samples with fairly

good accuracy29. The spectra of powder and frozen solution

16

samples can be simulated to facilitate the analysis and

to obtain accurate values for the spin Hamiltonian
parameters. The ESR spectra of low viscosity solution
samples provide only the average values of g and A tensors
because of the rapid molecular motions. These average
values of g and A are related to their principal components

by the equations

09
ll
LIOIH
>4
N
+
09
‘<
‘<:
+
0‘?
N
N
v

Since the dipolar part of the hyperfine interaction tensor
is traceless, the measured hyperfine coupling constant
from solution spectra is equal to the Fermi contact coupling

constant.

REFERENC ES

10.

11.

12.
13.

REFERENCES

B. R. McGarvey, "Transition Metal Chemistry", Vol. 3,
Ed., R. L. Carlin, Marcel Dekker, New York, 1966.

K. w. H. Stevens, "Magnetism", Ed. G. J. Rado and H.
Suhl, Vol. 1, Academic Press, Inc., New York, 1968.

B. Bleaney and K. M. H. Stevens, Rep. Progr. Phys.
lg. 108, 1953.

K. D. Bowers and J. Owen, Rep. Progr. Phys. 18, 30A,
1955.

J. Owen and J. H. M. Thornley, Rep. Progr. Phys. 29,
675, 1966.

A. Carrington and H. Longuet-Higgins, Quart. Rev. IA,
A27, 1960.

E. Konig, "Physical Methods in Advanced Inorganic
Chemistry", Ed., H. A. 0. Hill and P. Day, Interscience,
New York, 1968.

H. A. Kuska and M. T. Rogers, "Spectroscopy in In-
organic Chemistry", Ed., J. R. Ferraro and C. N. R.
Rao, Academic Press, New York, 1971.

H. A. Kuska and M. T. Rogers, "Radical Ions", Ed.,
E. T. Kaiser and L. Kevan, Interscience Publishers,
New York, 1968.

H. A. Kuska and M. T. Rogers, "Coordination Chemistry",
Ed., A. E. Martell, Reinhold, New York, 1971.

B. A. Goodman and J. B. Raynor, "Advances in Inorganic
Chemistry and Radiochemistry", Ed., H. J. Emellus

and A. G. Sharpe, Academic Press, New York, 1970.

R. P. Kohin, Mag. Res. Rev., 5, 75, 1979.

Chem. Soc. Ann. Repts., London (1950- ).

Spectrosc. PrOp. Inorg. Organomet. Compounds, Chemical
Society, London (1967- ).

15.

16.

17.

18.

19.

20.

21.

22.

23.

2A.

25.

26.

27.
28.

29.

18

Mag. Res. Rev.,(1972- ).

A. Carrington and A. McLachlan, "Introduction to Mag-
netic ResonanceJ'Harper and Row, New York, 1967.

C. Slichter, "Principles of Magnetic Resonance",
Harper, New York, 1963. 2nd Edition, Springer
Verlag, 1978.

N. M. Atherton, "Electron Spin Resonance: Theory and
Applications", Halsted Press, New York, 1973.

J. W. Orton, "Electron Paramagnetic Resonance",
Iliffe Books, London, 1968.

G. E. Pake, "Paramagnetic Resonance in Solids",
Academic Press, New York, 1960.

J. E. Wertz and J. R. Bolton, "Electron Spin Resonance:
Elementary Theory and Practical Applications", McGraw
Hill, New York, 1972.

A. Abragam and B. Bleaney, "Electron Paramagnetic
Resonance of Transition Ions", Oxford, 1970.

J. S. Griffith, "The Theory of Transition Metal Ions",
Cambridge, 1961.

C. Poole, Jr., "Electron Spin Resonance", Interscience,
New York, 1967.

R. Alger, "Electron Paramagnetic Resonance: Tech-
niques and Applications", Wiley, New York, 1968.

M. Tinklam, "Group Theory and Quantum Mechanics",
McGraw-Hill, 196A.

D. S. Schonland, Proc. Phys. Soc. (London) 13, 788, 1959.

W. G. Waller and M. T. Rogers, J. Mag. Reson. 2, 92,
1973.

P. W. Atkins and M. C. R. Symons, "The Structure of
Inorganic Radicals", Elsevier, New York, 1967.

CHAPTER III

THEORY OF g AND HYPERFINE INTERACTION TENSORS

The theory of g values and hyperfine interactions in
the electron spin resonance spectra of transition metal
complexes is based on the perturbation theory treatment of
Abragam and Prycel for the crystal field model, later
modified to include covalency effects. The general
subject of the interpretation of spin Hamiltonian param-

2-A

eters has been discussed in detail by McWeeny and the

theory of g values and hyperfine interaction has been the
subject of detailed reviews5‘7. So in this chapter only

outlines of the theories are given to Show the dependence

of g and hyperfine interaction tensor components on the

molecular electronic structure of the system.

A. Theory of thepg Tensor

The electronic Zeeman interaction term in the spin

Hamiltonian which accounts for the observed g value is

fc=s§~§o§, (1)

19

20

where g is the g tensor, which can be isotropic or aniso-
tropic depending upon the system. If the g value calcu-
1ated from the g tensor (average of the principal com-
ponents of the g tensor) differsfimmithe free electron
value of 2.0023, then S cannot possibly represent the true
spins. The experimental g value deviates considerably from
the free electron value. Hence the spin operator S in
Equation (1) should be replaced by a fictitious spin Opera-
tor ? as shown in Chapter I.

The spin and orbital angular momenta interact with the
magnetic field as given in the complete Hamiltonian (Chap-
ter I). Hence the actual total electronic Zeeman inter-

action is given by
}C = Be(L + geS) . B (2)

Since the electronic Zeeman interaction of Equations (1)

and (2) represents the same interaction, we have

(3)

II
m
(D
U?)
o
09»
(M)
u

BeCL + geS) ° B

where S is the so called fictitious spin angular momentum
operator.

For a molecule with a single unpaired electron (spin
a) in an orbitally nondegenerate ground state the Zeeman

interaction energy is given by

21

[:11
ll

<1A’oo‘le + geSZ|w0a>BeB

BeB<wOlLZIw > + geBeB<a|SZ|a>

O

BeB<Lz> + 2 geBeB.S <Lz> = 0

where $0 is the molecular orbital containing the unpaired

electron. A similar calculation for the state mos gives

seBeB; <LZ> = 0

NIH

E8 = BeB<Lz> —

E - E8 = geBeB = hv

From this one concludes that the g value should always

be equal to the free electron g value which is contradic-
tory to the experimental observation.

The shift in the g value from ge==2.0023 is attributed
to the fact that the electron possesses orbital angular
momentum in addition to the spin angular momentum. The
odd electron can acquire orbital angular momentum via the
spin-orbit interaction. The ground state wavefunctions
|¢09> and I¢08>, corrected for the spin—orbit interactions
through first order in the perturbation, takes the follow-

ing form:

22

 

 

 

 

<w |§(r)fi It > <w g(r)£ w >
|+>=IWOG> _ E 2 nE _ Ez o W a> _ % n; E+l o B>
n¢0 n o n n#0 n ' o n
(1“)
1 <wnlatr)£zlwo> 1 <wnla(r)fi-lwo>
-> = w B>+ — z B> - - Z w o>
I O 2nR‘O En ' Eo n 2n7¥0 En ' Eo n

where the summation index n refers to all the excited single
particle states,l%1arethe energies of the excited single
particle states labelled n, and 5(r) the spin-orbit coup-
ling operator. The two states obtained by correcting for
the spin-orbit interaction are no longer eigenstates of

the true spin Operator S. Now we define the fictitious

spin Operator S as an Operator, the components of which

act on the states |+> and I-> in exactly the same way as

the components of the true spin operator S act on the spin

functions |a> and |B>, i.e.,

A

 

In a magnetic field directed along the z-axis the spin

Hamiltonian for the Zeeman interaction takes the form

23

A A A A A

}C = BeB<gxz§x + gyz§y + gzz§z)

and Equation (3) can be written as follows:

Be(Lz+geSz)B = BeB(gXZ§X+gyz§y+gzz§z) (5)

Using the expression for |+> of Equation (A), evaluating
the matrix element
YZ~Y

<+[BeB(I‘z+geSz)|+> = <+IBeB<gxz§x+g S +gzz§z) +>

and solving for gzz we get

<wolt(r)£zlwn><wnlizlwo>

n#0 En — E0

 

gzz = ge ' 2

It can be shown8 easily that the general expression for the

elements of the g tensor is given by

2 2 <woIE(5)/iiI‘A’nx‘pnlf’jH’o
n¢0

 

$11 = $9 [1.3 = x,y,z]

E - E
o
n (6)
From the above general expression for the components

of the g tensor it follows that they are determined by

2A

the ground state electronic wave function, the excited
state wavefunctions of appropriate symmetry, the energy
differences between the ground and the various excited
states and the spin-orbit coupling constants of the various
atoms. The evaluation of g—tensor components will be dis-

cussed later.

B. Theory of the Hyperfine Interaction (A)

The hyperfine interaction tensor (A) may be written as
a sum of the anisotropic part arising from the dipolar
interaction between the unpaired electron and the nuclear
magnetic moments plus an isotropic part attributed to a
Fermi contact interaction. Since the anisotropic part of
the hyperfine interaction tensor is traceless, the experi-
mentally measured hyperfine tensor can be factored into the
anisotropic and isotropic parts.

For a transition metal complex with the unpaired elec-
tron in the d orbital one would expect the isotropic hyper—
fine coupling constant to be zero since the square of the
d orbital wavefunction is zero at the nucleus. But experi-
mentally it is seen that isotropic hyperfine interaction
does exist in transition metal complexes. This can be

accounted for by invoking either configuration interaction

or core polarizationg.

The hyperfine interaction term in the spin Hamiltonian

is

25

1m >
.
:x>>>
.
H)
u

J-C = (7)

where A is the hyperfine interaction tensor, S the fic-

titious spin Operator and I the nuclear spin operator.

In this discussion we consider only the anisotropic part

of the A tensor. In the complete Hamiltonian (Chapter II),
the interaction of the spin and the orbital angular momenta
of the unpaired electron with the magnetically active

(I # 0) nucleus is given by

RPert =}CIL +J'CDD (8)
L -E
_ i i
Rm P'i .3

1 =
P gegnBeBn ’

where.R&L and;HbD represent the nuclear spin orbit and the
electron-nuclear dipolar interactions, respectively. In
the expression forLHbD, Fij is a linear combination of
normalized real spherical harmonics. Since Equations (7)

and (8) represent the same interaction, we have

26

 

A.’A\.A = A A = '
§ A I E §1AiJIj P 3[

Using the expression for |+> of Equation (A), evaluating

the matrix element

i i r
i i A 11 A
+ E I S 1|+> (10)

 

<+I i 81A1313|+> = P'<+| if
and solving for Azz’ we get

£
<wol£(r)Lzlwn><wn ;§lwo>

 

F
_ zz
Azz - P'[<WO|‘:3-WO> + A {2

 

n#0 EO - En
PBZ
<woIt(r)Lalwn><wnl—3plwo>
I"
}1
+ Z is ’
a,8 aBz EO - En

where EaBz is the Levi-civita symbol. A rather lengthy

computation of matrix elements similar to the one in Equa-
tion (10) can be used to evaluate the various elements of
the anisotropic part of the A tensor. The general ex-
pression for the anisotropic hyperfine interaction tensor

(AGB) is given by

27

 

 

A i
Foe <w0|€<r>Lalwn><wnl;%lwo>

A” 2 Ptt<w°lfilw°> + ;:!o{2 E E
n -

O n
,. Fae

<w0|€(r)LY|wn><wnl-leo>

+ 2 166a 0 r }1 (11)
y,5 E - E

O n

From the general expression for the anisotropic hyperfine
coupling constant of Equation (11), it follows that the
factors which determine the magnitude of hyperfine inter-
action (dipolar) are the ground and excited state wave-
functions, the energy difference between the ground and
the various excited states of appropriate symmetry, the
spin-orbit coupling constants and <r'3> values of the
atomic orbital containing the unpaired electron. The
expression of Equation (11) had been derived earlier by

10 using a different method.

Keijzers pp 1.

C. Evaluation of g and A Tensor Components

The evaluation of the principal components of g and
A tensors using the general expressions of Equations (6)
and (11) requires the estimation of various types of matrix
elements. Some approximations are made in the evaluations

of certain types of matrix elements and are given below

(i) <wol€(r)ia|wn>

28

In the one-particle approximation being used the spin-

orbit coupling operator E(r) is given by

 

 

where Z' is the effective nuclear charge, with this ap-
proximation for the spin-orbit operator only one center
integrals are retained in evaluating the above matrix
element, since the contribution to <r'3> are small and may
be neglected in the overlap region.

F

L
K B K K 6 K
(11) <<wnl;3|wo> and <wn|::%|mo>

Only the matrix elementscfi‘the following type are retained

A

L F
B K K 6a K
|¢o> and «phi—r3 l¢0>

K
<¢nl_3
r

3
where K refers to the nucleus for which the hyperfine inter-
action is computed,since r is the distance between the
nuclei of interest and the unpaired electron.

The expressions relating theg- and A—tensor components
to the electronic excitation energies, atomic spin-orbit
-3

coupling constants, <r > values and the coefficients of

atomic orbitals in the molecular orbitals for systems with

29

CAV symmetryaxregiven in Chapter V and the relative im-
portance of the various parameters for transition metal

oxohalo complexes are discussed there.

REFERENCES

10.

REFERENCES

A. Abragam and M. H. L. Pryce, Proc. Roy. Soc (London),

R. McWeeny, "Spins in Chemistry", Academic Press,
New York, 1970.

R. McWeeny and B. T. Sutcliffe, "Methods of Molecular
Quantum Mechanics," Academic Press, New York, 1969.

R. McWeeny, "Orbital Theories of Molecules and Solids",
Ed. N. H. March, Oxford, 197A.

B. R. McGarvey, "Transition Metal ChemistryJ'Vol. 3,
Ed. R. L. Carlin.

H. A. Kuska and M. T. Rogers, "Spectroscopy in Inorganic
Chemistry", Ed. J. R. Ferraro and C. N. R. Rao, Academic
Press, New York, 1971.

D. E. O'Reilly and J. H. Anderson, "Physics and Chem-
istry of the Organic Solid State", Ed. D. Fox, M. M.
Laber and A. Weissberger, John Wiley and Sons, New
York, 1965.

A. Carrington and A. McLachlan, "Introduction to
Magnetic Resonancef'Harper and Row, New York, 1967.

A. Abragam and B. Bleaney, "Electron Paramagnetic
Resonance of Transition Ions", Oxford, 1970.

C. P. Keifjers and E. deBoer, J. Chem. Phys. 51, 1277,
1972.

30

CHAPTER IV

AN INTRODUCTION TO THE SELF-CONSISTENT FIELD

MULTIPLE-SCATTERING Xa THEORY

The electronic structure and properties of any mole—
cule, in any of its stationary states, can in principle
be determined by solving the SchrOdinger (time-independent)
equation. The exact solution of the Schrbdinger equation
has only been possible for atoms and molecules with one
electron because of the mathematical and computational
complexities involved in its solution. So approximate
solutions of the SchrOdinger equation are generated to
obtain information of chemical value from theory.

There are many different procedures based on the varia-
tion principle available for solving for approximate solu-
tions of the SchrOdinger equation. These schemes for
developing approximate solutions can be classified into
two categories, the ab initio and the semi-empirical
methods. The ab initio method has the well known Hartree-
Fock self-consistent field theory as its basis while the
semi-empirical methods attempts to mimic the ab initio
method. The X-a multiple scattering self-consistent field

method fits well into the traditional gap between the

31

32

ab initio and the semi-empirical approaches.

The multiple-scattering Xc self-consistent field
method is a computationally convenient method for develop-
ing an approximately quantitative description of the elec-
tronic structure of many-electron systems. The two main
features of the MS—Xa-SCF method are the Xc approximation
for the exchange contribution to the total potential and
the multiple scattering method of solving the modified
one-electron equations. Although the two approximations
are often used in conjunction, they are logically distinct
and each may be used without the other.

A complete discussion of the historical development
of the MS-Xa-SCF method can be found in Slater's auto-
biographyl. The Xa approximation and its application,
with emphasis on atoms and solids are given in detail in
Volume IV'of "Quantum Theory of Molecules and Solids"
by Slater2. The complete derivation of the system of equa-
tions for the MS—Xa-SCF method has been given in the re-
views by Johnson3 and by Weinberger pt 11.“. In addition
to these, there are a few reviews dealing with the outline

of the method and application to moleculess'7.

A. Hartree-Fock Equations and the Xa Approximation2

 

For a system of N electrons moving in the potential

field due to nuclei, the SchrOdinger equation is

33
}Cw(1,2...N) = Ew(l,2,...N) (l)
where;H2 is the Hamiltonian operator,

f(i) + 2 g(i,J). (2)
1 1>j

R =

IIMZ

1

Here the first term is the one-electron operator and the

second the two-electron operator

2 M Zu
f(i) = -V - Z -- (3)
i u=1 rui
2
(iaj) = _ ’ ' (Li)
S r13

where the first term in Equation (3) is the kinetic energy
operator and the second term represents the electron nuclear
attraction. The summation over u in Equation (3) takes
into account the electron nuclear interaction involving all
the nuclei in the system. The two-electron operator
g(i,j) of Equation (A) accounts for the electron—electron
repulsion. The above Hamiltonian is in Rydberg units.

The N electron wavefunction m(1,2,...N) is written as
a single determinantal function in terms of N one-electron
functions Xi’ each of which is a product of a spatial func-

tion and a spin function,

3A

w<1,2,...N> = AX1(1)X2(2)....XN(N) ,

where

and A is the antisymmetrizer. The orthonormal set of one-
electron functions {xi}1N=lare determined using the varia-

tion principle so that
E = <w(l,2,...N)|}C|¢(1,2,...N)> (6)

is a minimum. On varying the set of function {Xi}§=l
independently with the orthogonality constraint (Equation

5) we get the Hartree-Fock equation for the spin orbital

Xi:
{f + JElIXJ(2)g(l’2)(l-P12)XJ(2)dT(2)}xi(l) = 51X1(l)

(7)

where P12 is a permutation operator.
The second term in the Hartree-Fock equation (Equa-

tion (7)) for Xi can be rewritten as

35

N
z fdr(2)x3(2)s(1,2)(1-P12)x,(2)xi<l)

31

II
n M2

1 Xi(1)fdt(2)X3(2)ECl,2)XJ(2)

ItMZ

xj(l)fdr(2)x§(2)s(l,2)xi(2) (8)

J 1

The spin integration in Equation (8) results in a factor
of +1 multiplying the spatial integral for the first term
on the R.H.S. of Equation (8), while the second term van—
ishes unless Xi and Xj have the same spin component.
Hence the potential energy contribution (excluding the
nuclear attraction) in the Hartree-Fock equation can be

written as

ll _
z xi(.l)fx3‘(2)s (1, 2)xj (2)dt(2)

+ xix 0)fx*(2)s(1.2)x (2)dt(2)
J i J 3

ll
2 <1) x*(2)s(l,2)x C2)dt(2) (9)
‘j A J i

The terms in Equation (9) can be written in the following

forms (Equations (9)-(12)):

36

H
2 >&(l)fx?(2)s(l,2)x-(2)dT(2)
J J J
H
= Xi(l)fo (2)2(la2)dV(2) (10)
ll H

where p (2) = 2 de(2)x3(2)xJ(2) is the density of elec-
trons With the same spin as Xi at the terminus of a position

vector F2. The integration in the expression for p (2)

is the spin integration.

§1x1(1)fx3(2)s(1,2)x3(2)dT(2) = x1(l)foi(2)s(1,2)dV(2)

(11)

where ol(2) = {Ad£(2)x3(2)xj(2) is the density of electrons

with spins opposite to that hosted by Xi'

H
- z xj(l)fx§(2)s(l,2)xi(2)dr(2l =
J

 

ll x1Cl) * _
- f xj(l) £2237-fo(2)s(l,2)xi(2)dr(2)
(x (l)x*(2)x (2))
= - xi(1)f2|l J J i g(1,2)dT(2)
J Xi(l)
(12)

We now define the exchange charge density pix(l,2) as

37

X (11

and using Equations (10)-(12) can write the Hartree—Fock

equation determining x1 as

{f + In (2)s(l,2)dV(2) + fpl(2)s(l,2)dV(2)
_ foex(l,2)g(l,2)dv(2)}xi(l) = cixi(1). (13)
Since
I I l
fp (2)dv(2) = Nll and fp (2)dv(2) = Ni,

where NII + N1 = N, the total number of electrons. The
electron hosted by orbital Xi thus seems to interact with
itself via Coulombs law, which is physically unrealistic.
This self interaction is cancelled by a part of the exchange

charge density. The number of electrons in the exchange

charge density is

II x (1)
ex = j
fp (1,2)dv(2) i Xi(l) fX3(2)Xi(2)dT(2)
II x (l)

or exactly one electron, and as particle 2 tends to 1

38

' H x (1)
Lim ex 1

II II
a 2 fx3(l)xj(l)d€(l) = o (1)

L1.

the exchange charge density is equal to the density of
electrons with the same spin as electron 1.

Hence, in the immediate vicinity of electron 1 the
exchange charge density is equal to the density of elec-
trons with the same spin as electron 1. The exchange charge
is dense around 1 = 2 and goes to zero rapidly as the

electrons are separated.

Inthiszregion, therefore, x1 is determined by

(?(1) + f Epfil de21)Xi(1) = eixi(l)

for 1 close to 2.

Thus the following picture then emerges within the Hartree-
Fock model: "Each electron moves in the field of the nucleus,
the electrons of the opposite spin, and those of parallel
spin outside an 'exchange hole' or 'Fermi hole', which
follows around wherever it goes".8

The exchange charge density has a different form
for each wavefunction. The total exchange charge, however,

equals one electronic charge in each case, and its value

39

when point 1 equals point 2 is in each case equal, so that
the net size of the Fermi hole must be approximately the
same for each wavefunction, even though it may differ in
shape and other details. Thus, a great error may not be
made if the actual Fermi holes, which are different for
each xi, are replaced by an average value taken to be the
same for all xi's. This forms the basis of Xa self-con-
sistent field calculations.

In the Xa approximation the different exchange charge
density for each orbital of the Hartree-Fock model is
replaced by a weighted mean of the exchange charge den-
sities, the weighting factor being the probability that an
electron found at the position symbolized by 1 should be
found in the ith spin orbital. The density of charge of
spin-up electron at position 1, arising from the ith spin
orbital, is niX§(1)Xi(1), and the total density of spin-
up electron at this position is £1nKX§(l)XK(l)‘ Here it is
assumed that x1 is a spin orbital corresponding to spin-
up electron. Hence the probability that an electron at
position 1 with spin-up should be in the ith spin orbital
is

nix§(l)xi(l)

 

it-
2 n X (l)x-(l)
K, K K K

The desired weighted mean of the exchange potential is then

A0

2 2:ninjfx§(l)x§(2)g(l,2)xj(l)xi(2)dV(2)

i+ +
[VXHFi+(l)]av = ' J

*-
é; nKxK(l)xK(l)

where n1, nJ and nK are the occupation numbers of the
respective orbitals. A similar expression can be written
for the spin-down case. The above expression for the ex-
change potential can be evaluated exactly for a free
electron gas for which the spin orbitals are plane waves.

If one carries through the calculationg, the result is

[VXHFMQHav = -6[§%p+(l)11/3 = VXS+(1)
h p+(l) = 2 *(l) 1_.
w ere 1+ nixi XiC )

In the Xa approximation the exchange potential is re-
written as

VXa+(1) = aV = -6ot§%o+<l>11/3 ,

xs+
where a is an adjustable parameter. In Slater's approxi-
mation10 for the exchange potential, the exchange potential
of the Hartree-Fock equation is replaced by its average,
and this in turn is replaced by its value for a free elec-
tron gas. On the other hand if one replaces the exchange

potential in the total energy expression by its statistical

A1

equivalent as Gaspar, Kohn and Shaml2 did, and then varies

the spin orbital Xi in this statistical expression for

total energy,one obtains a one-electron equation for the

spin orbital with sz+(l) having only two-third of the
value obtained by Slater's method. If in the statistical

expression for total energy, the exchange potential term
is replaced by %a times the Gaspar-Kohn-Sham value, the

following expression for the total energy is obtained
= 91'
<EXa> i nifxi(1)fxi(l)dv(l)

A fp(1)0(2)s(l,2)dV(l)dV(2)

+

gHAP/3mm>1A/3+Ip+<1>i“/3}dv<i>
(1A)

If in this expression for the total energy the spin or-
bitals are varied to minimize energy, the following one

electron equation is obtained

2
[‘71 + Vc(l) + VXa+(l)JXit(l) = EinXi+(l) (15)
where
' -_2_?;
Vc(l) - i + 3 njfx§(2)xj(2)s(l,2)dV(2)

VXa+(l) = -6a£g%o+<l)11/3

A2

The Equations (1A) and (15) are the total energy expres-
sion and the equation for the one electron orbitals in

the Xa approximation, respectively.

B. Determination of a in the Xa Method

In the total energy expression of the Xc model,

<Exa> i nifx;(1)fxi(l)dv(l)
+ % f0(1)p(2)g(1,2)dv(1)dv(2)
- §a(%—)1/3rip+<l)1“/3 + £p+<l>1“/3}dv<l>

The Xc exchange term appears with a negative sign and is
linear in a. So a cannot be determined by minimizing the
energy. There are three sets of a values available for
the various atoms. One of the criteria used for determin-

ing a is to choose a so that <E > is exactly equal to the

Xa
configuration averaged Hartree-Fock total energyl2.
The a values so determined show a smooth variation with
atomic number; a decreases with atomic number, being around
0.78 for the two electron atom, decreasing to a range of
0.72-0.70 for the 3d transition series and remaining
almost constant thereafter at 0.69. The a's determined

in this manner are the most widely used in molecular

calculations. The second procedure is to use Xd atomic

A3

orbitals and Hartree-Fock operators to compute the virial

ratio (—V/T) and choose a to get the best number for the

13.

virial ratio In the third procedure for determining

a, a linear variation of Fermi-hole density is assumed

1A The a values

with the appropriate boundary values
depend on the number of electrons with each spin type

as shown in the following equations

 

2 i- +%
(1+ = 5%.(1‘77'41/1— “i (16)
+ 3
a = (n+a+ + nyo,)/(n+ + ny), (17)

where n+ is the number of electrons with spin up (a spin),
n+ the number of electrons with spin down (8 spin) and

9+ is the a value for spin up electrons. The expression
for c+ is similar to that of Equation (16). These so-
called theoretical a values reproduce the atomic number
dependence of a values determined by the first procedure
in all details. The a values determined by this procedure

also fall in the range 1.0 to 0.6666.

AA

C. Interpretation of Xc Eigenvalues and Slater Transition

States15

 

The eigenvalue aiHF of the Hartree-Fock method is a
finite difference of energies <EHF> for the two states
for which the occupation number ni of the ith spin orbital
differstnrunity. On the other hand, the eigenvalue

6 of the KC method is given by

1Xd

_ 3
EiXa _ 3n

 

<E >, (18)

i Xa

i.e., a partial derivative instead of a finite difference.

The total energy in the Xa method can be written as

E = £ni<ili> + A}: In
1 ij

2 nj<i||j> + ca{<pi/3(l)>+<o$/3(1)>},

Xa i

(19)

where 11K is the occupancy of the spin orbital XK’ whiCh

is a solution of the following one-electron equation

it + §fx3(2)xj(2)g(l, ,2>dv<2> + §Cao+/3}XK+(1)= —erK,(l>,

(20)

Where f1 and g(l,2) are the one- and two-electron Operators

defined earlier and

M5

<1Ii> = fx:(l)fxi(l)dv(1)
<1||J> = ffx§(l)xi(l)g(l,2)x§(2)xj(2)dv(1)dv(2)
<o(n)> = fo(n)dVCn) 0(n) = i nKx§(n)xKCn)
c ='- % (E%)l/3.

From Equation (20) the Xa eigenvalue can be written as
e = <KIK> + z n <KIIJ> + flca<x (l) pl/3(l)>- (21)
K J j 3 K+ XK+ +

The unrelaxed ionization potential IEr for the Kth or-
bital can be calculated by setting nK = l for the neutral
state and nK = O for the ion in Equation (20) and taking
the difference, keeping the xi's the same as in the

neutral case. Thus

Ifir = E+ - E0 = —<K[K> - z n,<K||J> - %<KIIK> +
JfiK J
oa<p§§3(1) - ofi/3<1)>, (22)
where

pK+ = p+(l) - nKX;(l)XK(l)

U6

The last term on the R.H.S. of Equation (22) can be re-

written as

 

x§(1)xK(l))U/3

ca<ofi/3(1)E(1 -
p+(l)

- 1]) o

Binomial expansion and neglect of all higher powers of

X*(1)XK(l)/0+(l), which are expected to be small, enables
K

us to write this term as

- % Ca<x§<1>xK(1)o}/3<1)> .

Thus,
Ifir = -<KIK>—J;KnJ<KIIJ>- §<KIIK>- %Ca<x§(1>xx(1)o1/3<1)>
(23)
and from Equations (21) and (23), we get
Ifir = - 6K + % <KIIK> . (2“)

In the Hartree-Fock theory, the unrelaxed ionization po-

tential (KOOpmans' approximation) is given by

ur _ _ HF = + _ o

where E;p(k) is the Hartree-Fock energy of the system with

U7

h orbital removed and E0 that of the

the electron in the Kt
neutral system.
Equations (2“) and (25) show that the Xa eigenvalue
does not have the same physical significance as in the
Hartree-Fock theory. In the Xa method, the eigenvalue
differs from the ionization potential by the self—inter-
action term %<KIIK>. It is easy to trace the origin of
this discrepancy. In the Hartree-Fock theory, the self-
interaction term is exactly cancelled by part of the ex-
change term; but in the Xa method, because of the exchange
approximation, no such cancellation occurs and this un-
physical term appears in the expression for IK'
In the Slater transition-state method of calculating
ionization potential the eigenvalue 8K of the orbital of
interest is calculated for a state midway between the
neutral and ionized states, namely when half an electron
is removed from the Kth orbital. The unrelaxed eigen-

value 6K for the transition state nK = % can be written

as

€K(nK=%)=<KIK>+J:KnJ<jIIK>+%<KIK>+§ca<x§(1)xx(1)pi/3(l)>
(26)

after expanding the exchange term and neglecting all highe
powers of x§(l)xK(l)/20+(l). On comparing with Equations
(23) and (2“) we get

However, in actual transition-state calculations, the
SCF eigenvalue equations are solved for the occupancy
nK = %, which therefore involves some relaxation (complete
relaxation is not included since the whole electron is
not removed). These eigenvalues, which we denote by
tr are then identified with the negative of the relaxed

€K ,
ionization potential, liel

, i.e.,

Irel = _ tr
K

There is empirical Justification for this assumption from

the reported16 agreement between efir values and

AE(=(E+(K) - E0) where the E's are the total energies)
values. Then the relaxation contribution to the ioniza-
tion potential is given by

rel _ ur _ rel = _ 1 tr
K - IK IK 8K + §<KIIK> + 8K

AE

It has been shown that the main effect of the transi-
tion-state procedure is to correct for the self energy of
the electron in the Xa approximation and that the transi-
tion state eigenvalues do not include any correlation ef-
fects15 as had been suggested earlierlY. Recently it has
again been shown that the Xa approximation does not include

18

any correlation effects

“9

D. Multiple Scattering SCF Method3’l4

The multiple scattering SCF method is based on the
geometrical partitioning of a molecule or a cluster
into three fundamental types of regions, namely, atomic,
interatomic and extramolecular. The one-electron Schro-
dinger equation is numerically integrated within each
region in the partial-wave representation for spherically-
averaged potential in the atomic and extramolecular region
and volume-averaged potential in the interatomic region.
The wavefunctions and their first derivatives are Joined
continuously throughout the various regions of the cluster
via multiple-scattering theory. This procedure leads to
a rapidly convergent set of equations which are numerically
solved for the molecular orbital energies and wavefunc-
tions. This entire numerical procedure is repeated, using
the wavefunctions obtained at each iteration to generate
a charge density and new potential, until self-consistency
is attained.

The three fundamental types of regions into which a

molecule is partitioned are:

(i) Atomic: The region within nonoverlapping spheres
centered on the constituent atoms (spheres can be made to
overlap too).

(ii) Interatomic: The region between the ’inner'

atomic spheres and an 'outer' sphere surrounding the

SO

entire molecule.
(iii) Extramolecular: The region exterior to the

outer sphere.

The one-electron Schrodinger equation (in Rydberg

units)
[-v2 + V(r)]w(r) = Ewcr)

is solved in each of the different regions for the

appropriate local potential V(r)
V(r) = Vc(r) + VXa(r).

The local potential includes the Xa approximation to the
exchange potential in addition to the Coulomb potential.

Z + anfx3(2)xj(2)g(l,2)dv(2) g(l,2) = __2_
J

N

Vc(1) = .—

"3

1 r'12

an<1) = -6a[E%o(l)Jl/3.

(l) Muffin-tin Approximation:

In the muffin-tin approximation the local potential
field of V(r) of a molecule or a cluster is replaced by

a set of individual, non-overlapping spherically sym-

metric potentials Vi(ri) around each atomic site Ri’

51

 

.mHSOoHoE anOpwfipu w you woman no soapfiupmm

 

.H mpswfim

I

52

where Hi refers the atomic site to the origin of the
cluster (Figure l). A spherically-averaged potential is
used in the region outside the outer sphere while a volume-

averaged or a constant potential is used in the inter-
atomic region.

Thus, within the 'muffin—tin' approximation, the po-
tential field is replaced by a model potential of the

V(r) = VOCIr-ROI); Ir-Rol=rO > bO

V 3 otherwise.

The potential energy at an arbitrary point 3 of the

molecule can be expanded as a superposition

v6) = z vJui-Zfi I)
J J
of free-atom or free ion SCF—Xa potentials centered at
DOSitiOHS R3. The potential energy is represented inside
each atomic region I by expanding the superposition in

a series of spherical harmonics.

VI(;) = E VL(F)YL(?) L = (£,m)

53

The muffin—tin approximation consists of using only the
first term L = (0,0) (spherically symmetric) in the
spherical harmonic expansion of the superposed potential.
In the muffin-tin approximation the potential in a par-
ticular atomic region includes not only the contribution
of the atom located there, but also the spherically-
averaged contribution of all the other atomic potentials

to that region.

2. Secular Equations

Consider a molecule with the geometry as shown in
Figure l. The outer sphere will be denoted with an index
i = O, and atomic spheres with i > O.

The wavefunction w(r) in regions of spherically-sym-
metric potential (interior to the ith (i > O) sphere and

exterior to the outer sphere) can be written as
wtr) = z CiRi(r 'E)Y (r ). L = (2 m) (27)-
L L L i’ L i ’ ’

O 1 r1 < b for all i > O

i

b < r < m for i = 0,
where the quantities Ci are coefficients to be determined,
YL(ri) are real spherical harmonics and R%(ri;E) solutions

of the radial Schrodinger equation in region 1, corresponding

54

to the trial eigenvalue E and angular momentum 2:

1. d 2 d 2(1+1) i i _
{— jazz—r1 5-11- + -——2—'— + V (r1) - E}R (ri,E) - 0
r1 1 1 r1

(28)

For all regular scattering potentials Vi(ri) i.e.,

11m
[rivi(ri)1 = o
P1+O

Two independent solutions of Equation (28) exist which

i 2-1

behave at the origin 1 as r1 and r; , respectively.

Two boundary conditions are necessary to make a solu-
tion of Equation (28) unique. The first is a condition
of regularity for RiCri,E), and is sufficient to identify
one of the two solutions mentioned up to a multiplicative

constant, i.e.,

i
2

2

R i; i > O

(ri,E) + const. r
I’+O
i
which characterizes the asymptotic behavior of R%(ri,E).
In the intersphere region (i.e., region of constant
potential) the Schrodinger equation takes the following
form

2

{-v + K2}w(r) = 0, K2 = (Viz). (29)

55

The solution of Equation (29) around any scatterer

i > O can be divided into an incident wave winc(r) and

a scattered wave wéc(r),
i i
WU") = Winch”) + Wsch”); i > O (30)

i
E BLi£CKri)YL(r1), E < V, i > O

i
winc(r) =
1 _
EBszCKri)YL(ri)’ E > v; i > o (31)
i (1) —
E ALK£ (Kri)YL(ri), E < V, i > 0
i
Wsc(r) = (32)
i . .
EALu£(Kri)YL(ri)’ E > v, i > o
32(X) - spherical Bessel function
u£(X) - spherical Neumann function
12(X) = i£J£(X) - Modified spherical Bessel function

K§1)(X) = -i-£h§l)(ix) - Modified Henkel function of

the first kind.

For any scatterer i > O the incident wave winc(r) is
regular when expanded about the center of the scatterer,
while the corresponding scattered wave wSC(P) is irregular

at the center of the scatterer when analytically con-

tinued.

56

The second boundary condition is to match continu-
ously the wavefunction and its first derivative interior
to a sphere i > 0, at the sphere boundary bi with the cor-
responding quantities derived from the field [Equation (30)]
around this sphere. In order to satisfy this condition,
it is necessary and sufficient to equate the amplitudes
and the logarithmic derivatives taken at the sphere boun-

dary r1 = bi’

 

i i 1
ie AL = t£(E)BL
[i,cxb,>,R§Cbi,Eu _
- (15(Kb icb 15)]3 E < V
ti(E) = (3“)
i
_ [32(Kbi),R£Cbi,E)] E > v

 

[u,<Kbi>,Ri<bi,E)J

where the square bracket in en) symbolizes the following

Wronskian form
[f(x),gcx)] = f(x) 3; (x) - g(x) g-gg- (x) ;

t:(E) is frequently called the 'scattering factor'.

We can express the amplitudes A: of the scattered

waves wéc(r) in terms of the coefficients Ci of Equation

57

(27) by matching the wavefunctions at the sphere boundaries.

For example, for E <‘V we obtain

 

CiR:(bi,E) = Bii£(Kbi) + A%K(1)(Kbi)
_ i -1 (1) i
_ {t£(E) i£(Kbi) + K (Kbi)}AL
[1 (Kb ),Ri(b ,E)]
Ai = 2 1 2 1 Ci (35)

[12(Kb1),K§1)(Kbi)J

(l) _ 2+1 2
[i£(Kbi),K2 (Kbi)] — {-1) /Kbi
[i,CKbi),u,(Kbi)J = 1/Kbi
_ 2+1 1 ,
AL = Kb CE (36)
i , -
[i,(Kbi).R,<bi,E)J. E > V
For the region exterior to the outer-sphere region
_ o o
¢(r) - E CLRlCrO,E)YL(rO) (37)

and the regularity of the solution is required for

large r

58

ie lim 0

ro+mR2 (ro,E) = 0.

Hence, we have a different situation from that in the

i > 0 case for solution of Equation (29) in the inter-

sphere region
wCr) = winccr) + wECCr) , (38)

namely, the incident wave winCCr) is irregular at the

origin of the outer sphere and the scattered wave wgc(r)

regular;
0 (l) . —

E BLK£ (KrO)YL(rO), E < V

wincm = (39)
E Bin, (KrO)YL(rO); E > v
z AEi£CKrO)YL(rO); E < v
L

wgc(r) = (MO)

0. . —
ALJ£(KrO)YL(rO), E > v

F'M

By analogy to the i > 0 case, we can express the ampli-

tudes A: of the scattered wave in terms of BB or CE,

59

o _ o o

(l) o
[Ki (Kbo),R£(bO,E)].

[i,(xbo),R3(bo,E>J

 

E < V

0 -
t£(E) - (ui)

[uzcxbo),RZCbo,E)J

' o
[J.,(Kbo).R,(bO,E)J

 

(—1)£+1£Ri<bo,E>,K§1)(Kbo)]; E <‘V

o _ 2 0
AL - KbOCL (U2)

[Ri(bo,E),u£(KbOJ; E >

<|

Method of constructing a unique wavefunction w(r) for the

whole cluster.

1. The incident wave corresponding to any site i > 0
equals the superposition of the scattered waves from all
other scatterers, including the one from the outer sphere:

ZAit:(E)-li£(Kri)YL(ri) = g(l‘513)gfiAJ'K§})(KrJ)YL'<r )

L J

+ 6% AE'12'(Kro)YL'Cro)3 E <‘V (“3)

i i —1. _ J

+ fit AE.J£.(KrO)YL,(rO); E > V . (Mu)

60

2. The incident wave for i = 0 has to match the sum

of the scattered waves from all scattering regions i > 0.

z A°t°(E)’lK§l)<KrO)YL(rO)

L L 2
J (l) -
= Z A 1K! (Kr )Y .(I‘ )3 E < V (’45)
J,L. L 2 J L J
EAEti(E)-lu£(Kro)YL(ro) = Jii'AiyU£.(Krj)YL,(rJ); E > v

(H6)

In order to treat only the center i in Equations (#3) and

(nu) we expand all scattered waves, j > O, J # i, as inci-
dent waves at the site i by means of the following scat-
tered-wave expansion theorems, and according to the follow-

ing coordinate relations3,

Ké1)(K|r2-rl|)YL(r2—rl) = un2a(—1)£+“'

L 2 ILH(L;L') X

L"

K§%)<Krl>YLn<rl>i,.(Kr2>YL.<r2) ri>P2

(l) _ 2441' .
Kfi (KIP2-rll)YL(P2-rl) - ufl§a(-l) ELIL"(L,L') X

i£n(Kr1)YLn(rl)Ké%)(Kr2)YL,(P2) P1<P2

61

+ Y
i£CK|r2-rlI)YL(r2-rl) = 1:an, (-l)2 2 L;.~"IL..<L;L')

i£n(Krl)YLn(rl)i£v(Kr2)YLI(r2)

u (Klr -r |)Y (r -r ) = u z 2""21"‘"I (L'L')
2 21 L21 “L, L" L",

“g"(Krl)YL"(rl)J£(Kr2)YL'(r2) P1>P2

£'—£ _£n
Ll£(Klr2-rll)YL(r2-rl) = ufl'Zi- 21. I (L;L')

L! L" L"

j2."(Krl)YL"(rl) HQgCKI'2)YL,(I'2) I‘l<I‘2

_. 5L'-IL -2"
J£(Klr2-rl|)YL(r2-rl) - unzi Xi

Lv Ln IL"(L3L')

J£"(Krl)YLn(rl)jlv(Kr2)YLg(r2)

2w

1T
where IL,,(L;L') = fo dofo sinedeYL"(e,¢)YL(e,o)YL.(e,¢)

ll
‘1
+
:13

l
:11
II

r:J = r - RJ

The constants IL"(L;L') are called 'Gaunt' numbers and can

62

be expressed in terms of Clebsch-Gordan coefficients.

We can similarly express the scattered wave for

i = 0 in Equations (#3) and (an) as an incident wave at the
center i. We can rewrite Equations (#3) and (an) in terms

of the incident waves for a chosen site i, for E <‘V as

i i -1 - 3 JJ
EALt£(E) 12(Kr1)YL(ri) - flepALGLL'i£(Kri)YL(ri)

o 10 _
+-IlaflALALL.i£(Kri)YL(ri); E < v (H7)
’

EAEL,ti(E)-1Ké1)(KrO)YL(rO) = 33L i,AiAE%'K9El)(Kro)YL(ro);

E <‘V (H8)
where

(-1)“"'L2"1L..(L;L')KfileRijmum”); an?

Gii'=(l-613 )LflT

£-£ £"

v -
i Z i

+ + _
L" ILn(L§L')U£n(K|RiJ|)YL"(RiJ), E>V

v
(-l)£+2’ ngyIL"(L;L'>12,"(K|§10|)YL"(§10); E<V

io
ALL' ' u“

_ I _ n _
12’ 2 LGi 2’ IL"(L;L')JQ,"(Kl§iO|)YL"(§iO); E>V

63

and similar equations for E >‘V can be written. Equating
likewise coefficients in Equations (H7) and (U8) we ob-
tain the following set of equations for E < V, as well

as for E > V

i i -l _ J G33 0 10
A E) - AL + A
Lt£( 3,2L1L' GLL! L2! L1 ALL:
AEti(E)'l = fIL'Aj.AEi. . (“9)
,

which can be writen in a more compact form as

J -1GJJ J _ 10 _
LZ' §{t£(E) 51351-413, - GLLL'}A ALL'AL' ' 0
z {EAL 03 Ai.} - to H(E) $.5LL, = o . (50)

L' J

This set of linear equations (50) determines the ampli-
i

tudes of AL and Ag of the scattered waves and hence also
the amplitudes of the incident waves and the coefficients
CE in Equation (27). The homogeneous system of equations
(Equation (50)) has a non-trivial solution if, and only
if,the determinant vanishes. Since all the terms of Equa-
tion (50) are energy dependent, it is necessary to evaluate
explicitly this determinant as a function of the energy E
and to find the zero locations of this function. Thus,

molecular orbital energies correspond to those parameters

6M

E for which non-trivial solutions of Equation (50) exists.
The determinant for a given parameter E, and conse-

quently the location of zero, depends on the number of L

terms to be included in the summation in Equation (50).

It turns out that this summation is very rapidly convergent

and the set 2 = 0,1,2 is most likely to be sufficient for

many applications to polyatomic molecules including those

involving transition metals.

3. The Self-consistent Potential Field19

For each root 6K of the secular determinant (Equation

(50)) we get the expansion coefficients for the solution

of one-electron Schrodinger equation in the three dif-

ferent types of regions. The corresponding charge

density pi(ri,eK) in scattering region 1 being

2C

_ 11+ + i**+ * .
i,eK) - L LR£ (ri,€K)YL(ri)CL R2 (ri,eK)YL,(ri), 1:0,

(51)

where the spherically-averaged charge density 01(ri,eK)
is given by
1<

0'

ri,eK) = pri(ri,eK)r2, i > O. (52)

The expansion coefficients Ci are still unnormalized, as

are the charge densities in Equations (51) and (52).

65

The wavefunction corresponding to the energy 6K be denoted

by w(r,eK) and must be normalized as follows

mg? = f‘P*(P,€K)‘P(r,€K)dV

b

11 co
= Z l" o
i>0£> o ( 1:8K)dri + i) o (ro,gK)drO + f w*(r’€K)w(r’€K)dvin

O n (53)

where N(eK) is the normalization constant for this or-
bital. The last integral defines the fraction of charge
in the interstitial region.

The normalized total charge density in a scattering

region 1 Z 0 is given by a sum over all occupied states

1 _ i
0 (r1) - i nK/N(eK)c (ri’eK) , (5“)
with nK being the occupation number of the orbital K.
The total charge within a sphere i > 0 is defined as
b

Q =f
i o

ioi(ri)dri <55)

and exterior to the outer sphere as

Q0 = fb o°(ro)dro (56)
O

66

Using these quantities, we can distribute uniformly the

remainder Qin of the total molecular charge

Q = Z (Z - Q ) - Q (57)
in i>0 i i 0

throughout the volume of the interatomic region, 0

in’
_ Ln 3 3
Q
in
o = —- (59)
c Qin

In Equation (57), Z1 is the atomic number for a site
i > O.

From the quantities 01(r1),p c and Qi’ we can derive
a new Coulomb potential ViCri) as in Equation (61) for a
scattering region i > O by solving Poisson's equation
interior sphere i > 0 (first three terms of Equation (61))
and adding the various Coulomb contributions arising from
a hypothetical system of 'effective' point-charge 51,

61 = Q1 + %g cab; (60)

located at sites i > O, in a uniformly negative background

90 together with the corresponding arrangement for the

outer sphere.

67

Z I‘i b
vi<r > = 2{ - —l + l; x oicrz> or' +; ioi(r')d£nr'
C i 1 r1 0 1 i i i
I'i
5 co
2 2w 2 j 2 o
+ 2np b - p IR I - Z -2wo b + f o (r')dlnr'
(61)

Similarly, the Coulomb potential exterior to the outer

sphere is given as

1”

Q

0 = _ .2. 2L. 0 O ' t m 0 v v
Vc(ro) 2{ o + r0 4) o (P0)drO + £1 0 (ro)d2nrO .(62)
O O

In order to obtain the Coulomb part Vc of the constant
potential V'in the interatomic region, we have to average
the various Coulomb contributions from our system of ef-
fective point charges and the outer sphere charge,

respectively, over the volume Qin' Thus, we obtain

- 2 co 0( l 161T2 bi (bi
V =4wo b + 2f 0 r )dinr - -——{—-—-o [—— - Z
2 3 2 2 u 2 8 b3:
+Rb)3+zoj[unb—uno.—-ER —-12 J}.
I il 1 i>0 i o 1 3I iJ 3 j>0 R13
Ji‘i (63)

The exchange part of the potential in any type of region

will be a function of the charge density only:

68

Vicki) = -6ai[83;pi(ri)]l/3; i Z 0
FM = -6E[83—fl-QC]1/3 . (on)

In Equation (64) the constants a1 and E are the exchange
parameters for the scattering regions 1 and the inter-

atomic region, respectively. The new molecular potential

field of 'muffin-tin' form is then given by

i __ i i ,
v (r1) - vccr-i) + vxacri), i _>_ 0

V = Vc + VXa'

u. Overlapping:Sphere Model

The principal source of error in the SCF—Xa-MS method
is the assumption of non-overlapping atomic spheres,
the muffin-tin approximation. This problem can be circum-
vented, to high order of approximation, through the straight-
forward use of overlapping atomic spheres. It has been
shown that the use of overlapping spheres with the muffin-
tin approximation for the potential, can be formally
Justified through the analytic continuation of the multiple
scattered wave expansions.20.
The secular equations of the scattered wave technique

contain two fundamental types of matrix elements, the on-

diagonal atomic "scattering elements" and the off-diagonal

69

"structure elements". The former quantities depend solely
on the partial wave solutions of the radial Schrodinger
equation for each individual spherical potential, regard-
less of its range, and therefore are independent of whether
or not a neighboring spherical potential overlaps. 0n

the other hand, the structure elements depends only on

the molecular geometry, and their mathematical form is
derived on the basis of certain standard multipole-type
expansion theorems in the angular momentum representa-
tions3. Similar expansions theorems are also used in the
solution of Poisson's equation of classical electro-
statics, which is an integral part of the SCF iterative

20 that these theorems are

procedure. It has been shown
valid over regions of space that can be spanned by both
overlapping and non-overlapping spheres. The only restric-
tions are that each atomic sphere does not overlap a neigh-
boring atom beyond its nucleus and that the outer sphere
does not overlap the peripheral atoms beyond their
respective nuclei.

The essential idea behind the overlapping sphere model
is to distribute the charge in the intersphere region among
the various atoms. The nonempirical scheme for choosing
overlapping sphere radii now being used is based on the

virial theorem2l. In this procedure the sphere radius

is assigned the value of the radius of the sphere around

each atom in the initial molecular charge distribution

70

which encloses the atomic number of electrons. These

sphere radii are then varied, with the relative sizes

fixed, to get the best value of the virial ratio at self

consistency.

5. Evaluation of One-electron Properties22

The calculation of expectation values over the MS-Xa-

SCF wavefunctions can in principle be done using the or-

bital representations

M?) = 22m 03”,] czowgm (Sphere a) (65)

a

g,m J£CPa)Y?(;a) (Intersphere) (66)

oCF) = 2 Z A
a ,m

l

where C:,m are the expansion coefficients determined by

the matching conditions at the sphere boundary, the

P%(r) are the appropriately normalized solutions of the
radial Schr5dinger equation and Y?(r) are real spherical
harmonics. The wavefunction for the intersphere region

is a multicenter representation where the sum over a cor-
responds to the sum over all the atoms, Aim are the expan-
sion coefficients and J£(ra) are spherical Bessel functions.

The major problem with a direct numerical integration is the

complicated shape of the intersphere region and the fact

that the charge density can vary widely within it.

The integrals involving one-electron operators are

71

simple to calculate inside each atomic spheres; the
angular integrals over spherical harmonics can be evaluated
analytically and the one-dimensional radial integrals can
be calculated numerically. In order to avoid the dif-
ficulties arising from the integration over the intersphere
region, Case and Karplus22 had proposed a method to divide
the intersphere charge among the various atoms by extend-
ing the range of the radial variables beyond the atomic
sphere radii ba; that is, the intersphere charge is ap-
proximated by a sum of expressions of the form of Equation
(65) with the assumption that the overlap between atomic
charge distributions can be neglected. Procedure for par-
titioning the intersphere charge: The intersphere charge
is partitioned in proportion to the average charge density
at the surface of each atomic sphere multiplied by the part
of that surface bordering the intersphere region. The
average charge density of the molecular orbital ¢(r) at

the surface of the atomic sphere a of radius ba is
2Emmimpgwafi .

Since the wavefunction is continuous across the sphere
boundary, the additional charge in the sphere due to its
expansion by an infinitisimal amount is proportional to

average density. This quantity has to be multiplied by

the surface area of the sphere bordering the intersphere

72

region. For tangent spheres the area is Hub: and the frac-

tional seduction in surface area caused by each overlapping

sphere 3 is
f = {b2 - (R -b )2}/uR b R < (b +b )
a8 8 a8 a a8 a’ as a B
= 0 RGB > (ba+b8)

where Rae is the distance between atoms a and 3. If the
total intersphere charge is divided among the atoms in

this way, the extra charge Aqi for each partial wave can

be obtained from

_ -1 a a 2
Ag?” N qintunb: {1‘ aisa faB}r§{C£,mP£Cba)} :

where the normalization constant N is given by

N

5 gfmuwb: {l-Biafa B}{C£mP £(ba )}2
and qint is the total intersphere charge. The parameter
qint is calculated by muffin-tin procedures, and its ap-
pearance in the expression for qu insures that the
normalization of the molecular orbital is consistent with
this approximation.

The functional form of the radial function beyond the
sphere boundary is determined by extending the radial

function inside an atomic sphere by using the leading

73

term of the multicenter expansion in the intersphere

region, i.e.,

a _
P£(r) - B J2Cr), for r > be
where
J£(r) = JiCKr) for E > Vint
(l)

with hK = |2m(E - vint)|. Here v is the value of the

int
constant intersphere potential. The constant B is chosen
to make the expansion continuous at r = ba; the first
derivative is in general discontinuous. It is possible
to assume a more flexible functional form and match both
the function and its derivative at the sphere boundary.
Many properties appear to be rather insensitive to the
choice of the functional form.

In order to include the amount of extra atomic charge

Aq: the sphere radius is increased to b: for each value

of 2. The function P% as determined above between bu

and b: are used as the standard muffin-tin orbitals and

the functions are assumed to vanish beyond the radius
2
a'
The calculation of one-electron properties with the

b

7A

r“
orbitals defined above is equivalent to the conventional
LCAO calculation with the neglect of differential over-
lap. This procedure has been used by Cook and Karplus23
to calculate the one-electron prOperties of LiH. The
charge-partitioning procedure was found to introduce
errors on the order of those in the Xa wavefunction itself
and to improve in accuracy with parameter variations that

improve the Xa wavefunctions.

REFEREMC ES

l.

10.
ll.

l2.

13.

REFERENCES

J. C. Slater, "Solid State and Molecular Theory, A
Scientific Biography", Wiley-Interscience, 1975.

J. C. Slater, "Quantum Theory of Molecules and Solids",
Vol. IV. McGraw-Hill, 197U.

K. H. Johnson, Adv. Quantum. Chem. 1, 143 (1973).

P. Weinberger and K. Schwarz, International Review of
Science-Physical Chemistry-Series Two — Vol. 1, Ed.

A. D. Buckingham and C. A. coulson, Butterworth, London—
Boston, 1975.

K. H. Johnson, Ann. Rev. Phys. Chem. 26, 39 (1975).

J. W. D. Connolly, "Modern Theoretical Chemistry"
Vol. IV, Ed. G. A. Segal, Plenum, 1976.

K. H. Johnson, J. G. Norman and J. W. D. Connolly,
"Computational Methods for Large Molecules and Localized
States in Solids", Ed. F. Herman, A. D. McLean and R.

K. Nesbet, Plenum, New York, 1973.

M. Tinkham, "Group Theory and Quantum Mechanics",
McGraw-Hill, 196A.

J. C. Slater, "Quantum Theory of Atomic Structure",
Vol. I, McGraw-Hill, New York, 1960.

J. C. Slater, Phys. Rev. 81, 385 (1951).
(a) R. Gaspar, Acta Physica 3, 263 (195A).

(b) W. Kohn and L. J. Sham, Phys. Rev. 140, A1133
(1965).

(a) K. Schwarz, Phys. Rev. BE, 2A66 (1972).

(b) I. Lindgren and K. Schwarz, Phys. Rev. A5, 5A2
(1972).

(a) M. Berrondo and O. Goscinski, Phys. Rev. 189,.
10 (1969).

(b) D. J. McNaughton and V. H. Smith, Int. J. Quantum.
Chem. 35, 775, 1970.

76

1H. M. S. Gopinathan, M. A. Whitehead and R. Bogdanovic,
Phys. Rev. A1”, 1 (1976).

15. M.EL Gopinathan, J. Phys. gig, 521 (1979).
16. K

17. H. F. B. Nelson, Chem. Phys. Letts. 12, 290 (1973).
18. J

19. W. D. Connolly and J. R. Sabin, J. Chem. Phys. 56,
5529 (1972)

20. F. Herman, A. R. Williams and K. H. Johnson, J. Chem.
Phys. 6;, "530 (197")-

. Schwarz, Chem. Phys. l, 100 (1975).

H. Wood, J. Phys. 813, l (1980).

21. J. G. Norman, Jr., Mol. Phys. 31, 1191, (1976).

22. D. A. Case and M. Karplus, Chem. Phys. Lett. 32, 33
(1976).

23. M. Cook and M. Karplus, J. Chem. Phys. 13, 7 (1980).

CHAPTER V

ELECTRONIC STRUCTURE AND ESR PARAMETERS
OF [CrOClqu-

A. Introduction

 

The electronic structures of the transition metal oxo-
halo complexes of the type [MOXnJm‘, where M = V, Nb, Cr,
Mo, W and X = F, Cl, Br, I (n = A or 5), have been exten-
sively studied experimentallyl-10 for more than a decade.
The basis for the interpretation of the experimental data
has been the well-known discussion of the electronic

2+ complexes of Ballhausen and Gray11 based

structure of V0
on extended Hfickel calculations. These have been followed
by similar calculations on the oxopentaquovanadium (IV)
ion12 and on the oxotetrachloro- and oxopentachlorovanadium
(IV) ionsl3’1u. Recently some ab initio molecular orbital
studies have been described for CrO3+ complexes15’16.
Since there are still many ambiguities in the interpreta-
tion of experimental results,we have undertaken a fairly

detailed study of the electronic structure of this class

of compounds in the hope of providing an understanding

of the bonding and consequently a sound basis for the

77

78

interpretation of experimental observations.

There are at least two approaches to the study of the
bonding characteristics of transition metal complexes
when detailed ESR data are available. One may assume an
atomic expansion basis for the molecular orbitals and
determine the linear expansion coefficients (the MO
vector) so as to reproduce the available ESR data,u-8’l7
or conversely one constructs as accurate a function as
possible for the system under consideration and then cal-
culates the ESR parameters from this function18'22. Of
the two, this latter approach is clearly preferred as it
has fewer biases and it is the approach we take in this
study. In particular, we use the results of SCF-MS
Xa23’25 calculations on [CrOCluj'l to interpret experi-
mental ESR data. We compare our SCF-MS-Xa results with
those we have obtained using the EHT26 as well as with

earlier theoretical studies.

B. Methods

 

In this section we describe the SCF-MS-Xd method and
the various modified extended Hfickel methods used in the

present study.

79

(i) The SCF-MS-Xa Method

 

The SCF-MS-Xa method is a technique used to approxi-
mate the solutions to the Hartree—Fock equations for a

many electron system. Its underlying assumptions and pro-
cedures, along with the results for many systems, are avail-
able in the 1iterature23'27. The two characteristic ap-
proximations of the method are the use of the Xa approxi-
mation for the exchange potential and the muffin-tin ap-
proximation for the potential. In this latter approxima-
tion, the most severe of the two, the molecule is parti-
tioned into three fundamental types of regions: atomic,
interatomic and extramolecular (this is the region outside
a sphere which encloses the "entire" molecule). The po-
tential in the atomic and extramolecular region are spheri-
cally averaged while a constant potential is used in the
interatomic region. The one electron equations are solved
numerically in each of the three regions and the resulting
wavefunctions and their first derivatives are joined con-
tinuously throughout the various regions.

The use of the Km approximation for the exchange
potential makes the interpretation of one-electron eigen-
values different from that of the Hartree-Fock eigen-
values. The correspondence between the SCF-MS-Xa eigen-
values and the Hartree—Fock eigenvalues has been estab-
lished28’29. In the SCF-MS-Xd calculation the molecular

orbitals are characterized by the charge associated with

80

each atomic sphere, and with the region outside the outer-
sphere, in terms of the percentage of s,p,d, etc. character
and the charge in the intersphere region. The total charge
associated with each atom in a molecule is taken to be the
sum of the charge inside the atomic sphere around the atom
and a fraction of the intersphere charge. The latter was
obtained by the interSphere charge partitioning scheme of
Case and Karplus31 in which the intersphere charge is
partitioned among the basis functions centered on the
various atoms in proportion to the average charge density
at the surface of each atomic sphere multiplied by the
area of that surface bordering the intersphere region.

Thus the fractional s,p,d, etc. character of the contribu-

tion of each atom to the molecular orbital can be calculated.

(ii) The Extended Hfickel Method

In the extended Hfickel method the molecular orbitals

(W1) are expanded as linear combinations of atomic

orbitals (¢J) and are assumed to be eigenfunctions of

an effective one-electron Hamiltonian, fieff‘ On minimizing
the total energy with respect to the molecular orbital

coefficients,the following secular equation is obtained

w = X C ¢
1 j 13 J
[HiJ - ESij] = o ,

81

where H13 and 813 are elements of the Hamiltonian and over-

lap matrices, respectively,

Hij <¢ilHeffl¢j>

S

13 ‘ <¢1|¢J>

The secular equation is solved to obtain the orbital ener-
gies and the molecular orbitals.

There are many schemes available for approximating
the diagonal and off-diagonal Hamiltonian matrix elements

and we have explored the following three.

1. In the first, the diagonal matrix elements Hii
are set equal to the negative of the valence shell ioniza-
tion energies (VSIE), which are presumed to be functions

of the charges q on the atoms,

Hii = VSIE(q)

2
H11 = -(Aq + Bq + C).

The values of A, B and C which characterize the valence

orbitals and configurations are taken from Reference

30. The off-diagonal elements are approximated by the

Wolfsberg-Helmholz relation32,

82

 

pmpcfiauuzo

5"

SO

namm>.o :mmm.o u p m

30

mmsm.m u s m

 

Hos u mmmmg.o mmas.m n Hum ow.mm u Hangouov ommm.m u Hum om.:oH u Houpouov

on u A:::A.O mnzw.fl u om «mm.m u Aaoupovp ABAA.H u om nozm.m u Aaoucovs

cos n mmmda.o omHm.m u pom moo.H u Aoupovc ommfl.m u Com moam.a u Aoupovp
a Asmv m %LpoEomc Asmv H mpmeomw

Haomm muonam

Haomm mpmsdm

 

.mmzam> a ocw HHUME opmcdm .mpumEooo .H canoe

83

_ l
2

1.1

where K is an empirical constant.

2. In our second method we have used the same

parameterization scheme for diagonal elements while a

weighted HiJ formula wasused for the off-diagonal ele-
33

ments

3. In the third method, a term representing the inter-
action of the electron in an orbital i with the electro-
static field arising from the non-zero net charges on the
other atoms in the molecule is added to the diagonal
elements Hii and the corresponding changes are made in
the way the off-diagonal elements Hij are evaluated for

molecules with non-zero net charge. The details of this

Madelung correction have been discussed by Hay gt al.3u.

C. Technical Details

The calculation has been done for two different

geometries in Cuv symmetry. The atomic arrangements for

both the geometries are given in Table 1. Geometry 1 is

based on the crystallographic data for [AsPhMJECrOCluJ35

while geometry 2 is the one used in the previous ab initio
16

studies . In both cases the chromium atom is above the

basal plane passing through the chlorine atoms.

8A

In the SCF-MS-Xa calculations the atomic a parameters
(Table l) were those of Schwartz36 and a weighted average
of the atomic a values was used in the interatomic and
extramolecular region. The sphere radii were taken to
be 90% of the atomic number radii, in accord with the sug-
gestion of Norman37, and are given in Table 1. Although the
sphere radii were not varied to obtain a correct virial
ratio, fairly good results were obtained for -V/T: 1.99992
for geometry 1 and 1.99981 for geometry 2. The outer
sphere with origin at the center of nuclear charge of the
molecule was chosen to touch the chlorine spheres and
serve as a watson sphere38 with +1 charge in order to
simulate the stabilizing effect of the environment on
the ion.

The partial wave expansions were truncated at a = A
for the outer sphere, A = 2 for chromium and l = l for
oxygen and chlorine. The core energy levels, i;g;,
ls, 2s, 2p on chromium, ls on oxygen and ls, 2s, 2p on
chlorine, were calculated in each iteration using only
the surrounding atomic potentials. The calculations were
carried out self-consistently with the convergence cri-
terion that the maximum relative change in potential be-
tween two consecutive iterations was lower than 10'“.

The extended Hackel calculations were carried out with
the Slater type double-zeta functions for the chromium 3d

orbital39 and single exponent functions for As, Up on

85

.LmuoEmpma wczaoomz 0:» ma >

.HOMH .mom :0 omcfiumo mum 0cm >0 CH ohm > 620 o .m .<

 

 

 

+
000.0 0.H 0.0a 00.0H 00.0H 0.0
000.0 0.H 0.0a 00.00 00.0H 0.0 00 H0
000.0 0.H 0.0a 0.0a 0.0H 0.0 00
000 m 0.H 0.0H 0.0m 0.0a 0.0 mm 0
00.0 000.0 000.0
00.: 000.0 000.0
000.0 0.0 0.0 00.0 000.0 000.0 00
00.0 000.0 000.0
00.0 000.0 000.0
000.H 0.0 0.0 00.0 00H.» 000.0 00
00.0 00.0H 00.0
00.H 0000.0 0H.0 00.HH H0.H
00.0 0000.0 0.0 00.0 000.0 000.0 00 E0
heeeoexm hemaeammmoo 0 0 0 0 Hmeahno soh<
eoaeeesm Hmaeem
.mCOfipoczm mamas 0:0 mhmpoemhmo meowm poccouxm .m @0909

+

86

 

 

00.00 00.0 00.0 00.00 0000.0- 000

00.00 00.0 0000.0- 000

00.00 00.0 0000.0- 00

00.00 00.0 0000.0- 000

00.00 00.0 00.0 00.0 00.00 0000.0- 000

00.0 00.00 00.00 0000.0- 00

00.00 00.00 0000.0- 000

00.00 00.0 00.0 00.0 0000.0- 0.00

00.00 00.00 0000.0- 000

m0.00 om.m 00.0 0m.m 0000.0I 00

00.00 00.0 0000.0- 000

00.00 00.0 0000.0- 00

00.00 0000.0- 000

00.00 00.0 00.0 00.0 0000.0- 00

00.00 0000.0- 000

00.00 00.00 0000.0- 000

00.00 00.00 mo.m0 m00m.o- 00

00.00 00.0 00.00 0000.0- 000

00 00 00 mm 00 00 00 0000 02
02000050 cmwzxo 85080050 0000mm

R .HmuOMchSO OHEOU¢

 

.mm300>:mw0o paw mCOHpocszmeT coppoTHC oco .m @0909

87

chromium, 2s,2p on oxygen and 3s, 3p on chlorine. We had
used the normal Hfickel constant, K = 1.75. The parameters
used in the extended Huckel calculations are given in Table
2.

D. Electronic Structure ofLCrOClujl-

The SCF-MS-Xa calculation of the ground electronic
state of [CrOClull' predicts the ground state to be 282
in agreement with the earlier ab initio studyl6. The un-
paired electron is in the 2b2 orbital which is primarily

a 3dx orbital on chromium (Table 3). The fractional

y
charges given in Table 3 for geometry 1 are based on the

charge partitioning scheme of Case and Karplus3l. The
highest energy occupied orbitals (Ne-laz) are primarily
chlorine 3p in character and are essentially non-bonding.
The orbitals 1b2 and 2bl which follow this non-bonding set
account for the bonding between chromium and the chlorine
atoms. The next two low-lying orbitals contribute to the
bonding between chromium and oxygen and involve two sig-
nificantly strong bonds: a 0(5a1) and a W(3e) bond. There
are substantial differences between the SCF-MS—Xd and the
ab initio descriptions of the Cr-O bonding. In order to
understand the Cr-O bonding changes in detail we have
carried out SCF-MS-Xa calculations on CrO and CrO3+ U0

and have tabulated the fractional atomic character of the

orbitals describing the o and n bonds between chromium

88

 

 

Table A. Chromium-oxygen bonding in some oxochromium com-
pounds.
0(Cr-O) v(Cr-O)
Atomic Character % Atomic Character %
Cr 0 Cr 0
3d22 sz 3dxz,yz 2px,y
CrO 22.84 63.59 15.84 76.80
CrO3+ 30.52 57.96 39.98 58.61
a[CrOClu]1— 26.53 62.29 29.91 6u.u6
b[CrCClujl' 26.96 63.10 26.70 62.82
CECrOClujl- 50.00 35.00 16.00 76.00

 

aGeometry 1.

bGeometry 2.

cAb initio result from Reference 16.

89

Table 5. Charge distribution in some oxochromium compounds.*

 

 

Molecule Chromium Oxygen Chlorine
CrO 0.361 -o.361
Cro3+ 2.379 0.621
aECrOClu]l_ 0.957 -o.156 -o.u50
bECrOClujl- 0.974 -o.229 -o.u36

 

aGeometry 1.

bGeometry 2.

s
The outer sphere charge distributed between Cr and O is in
the ratio of atomic numbers in CrO and CrO3+.

90

Table 6. Electronic transition energies from EHT and
SCF-MS-Xa calculations.

 

 

Geometry E(2b2 + 7e) E(2b2 + Dbl)
and
Method KK KK
SCF-MS-Xd
Geometry l 13.83 19.97
Geometry 2 8.u5 20.30

Extended Hfickel

Geometry 1

Method 1 9.92 30.82
2 9.59 39.32
3 11.96 38.72

Geometry 2

Method 1 “.52 31.27
2 “.69 30.90
3 5.97 39.nu

 

91

and oxygen in Table “, along with those of [CrOClujl'.
From the results of Table “ we conclude that both the o
and w Cr-O bonding in [CrOClu]1_ ‘hs weaker, compared to
that of Cr03+, than would be anticipated.

The charge distributions in CrO, Cr03+ and [CrOClujl-
obtained from the SCF-MS-Xa calculations are tabulated in
Table 5. The extended Hackel charge distribution is
critically dependent on the way the calculation is done.
For example, the charges on chromium for geometry 1 are
+0.36“, +0.206 and +1.222 for methods 1, 2 and 3, respec-
tively.

The relative energies of the empty d-orbitals follow
the sequence d

< d 2 2 < d 2 in SCF-MS-Xa calculations
x Z

1-10

xz,yz _y

in agreement with the order generally used as the basis

of interpretation of electronic spectra of CrO3+ complexes,

but not with the order found in the ab initio studiesl6.

The extended HUckel calculation predicts that the empty

d-orbitals lie in the order d < dz? < dx2 2. The

xz,yz _y
SCF-MS—Xa results for the d—d transition energies calcu-
lated by the transition-state procedure23"25, along with

those obtained from EHT calculations are given in Table
6. According to the SCF-MS-Xa results, the first absorption
maximim at 13.1 KK in the electronic spectrum should be
assigned to the transition (2b2 * 7e),which can be considered

a d-d transition 3dXy + 3d the 2b2 and 7e orbitals

xz,yz’

being predominantly 3d in character (Table 3). This

92

assignment is in agreement with the generally accepted
interpretation of the electronic spectra of this class
of compounds and with the recent ab initio studies. The
same transition energy calculated for geometry 2 predicts
an absorption at 8.“5 KK in fairly good agreement with

the corresponding CI result16

of 9.9 KK,but lower in
energy than the experimental result. Our conclusion from
this observation is that the 3dXy + 3dxz,yz transition
energy is critically dependent on the angle between the
chromium-oxygen and chromium-chlorine bonds and on the
chromium—oxygen distance. The next lowest energy d-d
transition 2b2 + “bl calculated by the SCF-MS-Xa method
predicts an absorption at 19.9 KK,in good agreement with
the experiment (18.1 KK). This assignment of the second
absorption band, though in agreement with the generally

usedl-lo interpretation, doesn't agree with the polarized

single-crystal spectra and CI studies of Garner et al.16.
The extended Hfickel results for the d-d transition ener-
gies continue to show a strong dependence on the method of

calculation (Table 6).

E. Theory of g and Hyperfine Interaction Tensors

 

The theory of g and hyperfine interaction tensors of

transition metal complexes is generally based on the per-

“1

turbation method introduced by Abragam and Pryce for

the crystal field model and later modified to include

93

17.

covalency effects This general topic has been the

subject of many reviews“2 and the perturbation method has
been applied to the specific case of transition metal oxo-
halide complexes of C“v symmetry by DeArmond gt _1.u. In
this section we give an outline of our treatment of g and
hyperfine interaction (A) tensors indicating the dif-
ferences from earlier versions.

The following general expressions for the components
of the g and A tensors have been obtained from the standard

second order perturbation theory treatment of g143 and A22

tensors,

<wO|§(r)£a|wn><¢n|£B|wo>
e n#0 En - EO

 

(l)

A

A L '
2<¢O|g(r)La|wn><wn|-%|wo>
r

 

A = P'[<¢ [Egélw > + 2 {
a8 (3 r3 O n#0 E - E

 

F
" .98
<wolatrmy|¢n><¢nl r3 [00>
+ Z isydd } + K (2)
v.5 EO - En
H - P' z I F—O‘é S ’3)
DD 0,8 d r3 8

9“

where 00 is the orbital containing the unpaired electron,

the summation over n is over all excited states, Ed is the
a-th component of orbital angular momentum and g(r) is

the one-electron spin—orbit coupling operator. In the
expression for AaB’ F08 and F68,are linear combinations

of normalized real spherical harmonics arising from the
electron nuclear dipolar Hamiltonian (fiDD)’as in Equation

1
(3). P is a constant equal to gegnBeBn, where ge is

the free electron a value, the g value for the nucleus,

EN

3 and 8n are the electron and nuclear Bohr magneton,

respectively, and 6760 is the Levi-Civita symbol. The

e

term K in Equation (2) takes into account the Fermi contact
interaction.

The unpaired electron in the system under considera-
tion here is in an orbital of b2 symmetry (C“v)° So the
computation of g and A tensor components requires the
evaluation of matrix elements between orbitals of b2
symmetry 30d orbitals Of bl and 6 symmetry. The molecular

orbitals required to computegy-and A—tensor components

are:
- M _ _
wbl €1¢dx2_y2 €2¢bl(8) + €3¢bl(x,y) E“¢bl(z)
M
0 v 6'0 + 6'0 - 6'0 - 6f0
b1 1 dx2_y2 2 bl(s) 3 bl(x,y) 4 bl<z)
l 2 3
(pb (X,Y) py " pX - py + pX

l 2
¢bl(s) = (S “S +3

Ct’bl(x,y)

= l_ 3 . 1,3 = l 3 , 1,3 _ l 3
(s s ), 0er) (px+px), ¢e(y - (py+py>
1,3 = l 3

3_s“)

l 2 u
= (px+py-pi-py)

¢b1(z) = (pi-p:+pg-p:)
B145:1qu - B2¢b2(x,y)
O‘1‘15I“Ipx - a2¢fipy + a3¢gxz - au¢§yz - 0593px
O'6¢’3py + “7¢§Z3) ‘ “8¢§i§) + a9¢§i§>
“10¢:Zg) + “11'22:> - “12¢gii) + “13¢SZ;>
“lu¢:?:)
M AM M 0
ai¢upx + a'¢upy + a§¢dxz + afi¢3¥z aé¢2px
a6¢gpy * “%¢:?§) ' aé¢eii) ' at¢ii§> ' “io¢eZ§>
2,“ 2,“ 2,u 2,0

all¢e(s) ‘ O'1'L2"’e(x) - “i3¢e(y) + O'iu‘bem :

96

‘0‘)
N

 

6:2 I /

Cr ’7 Us

 

 

 

ClI C|4

/

X

Figure 1. Coordinate system for [CrOClu]1'.

97

where the 0M and 00 are orbitals centered on the metal and
oxygen atoms, respectively, while the remaining 0's are
the symmetry adapted linear combination of chlorine or-
bitals for the coordinate system shown in Figure l. The
¢§,“ in 081 and 082 have the same form as the 0i’3 given
above. The molecular orbitals 061 and 082 are the lowest—
lying degenerate empty orbitalscfi'e symmetry, the orbital
wbl is the next-lowest-energy empty orbital and 0b. is
a doubly-occupied orbital of bl symmetry. 1

On evaluating the required matrix elements retaining
only single center contributions,while evaluating matrix
elements involving the spin orbit coupling operator,we get
the following expressions for the principal components
of the g and A tensors

AII = ge - gzz

2(28 6 A -“B 6 A )
_ l l M 2 3 L r _ .

 

+

28163S(bl(x,y);d 2 2) - 2Bl6us(bl(z);d

)

+

282813(b2(X:Y)3dxy) - 82828(bl(s);b1(x,y))

“B2620 - B2€3S(bl(x,y);bl(x,y))], (“)

A||(bi)

Ai(el) =

98

-2(2818iAM+“82€éAL)
AE(bi + b2)

 

r v + r .
L2816l 28162S(b1(s),dX2-y2)

23le§s<bl(x,y); d 2) - 2BleiS(b1<z>sdx2_y2>

x2-y

282€iS(b2(X,y);dxy) + 826é8(bl(s);bl(x,y))

 

a7BlS(el’3(s); dxz) - 68818(e1’3(x); dx

0'1051

0‘1032

“BZEéQ + BZEéSCbl(X,Y)Sbl(X,Y)J, (5)
2(a 8 A +2a B A )
3 l M 10 2 L O,
AE(b2 + e) [(1381 - GBBlSC2pX’ dxz)

)

Z

s<e2’“<z>;dxz> - 012813(82’u(x)3 dxz)

- 01823(“px;el’3(2))

ngel’3(Z)) + 0582S(2p§:;el’3(2))

S(el’3(z);el’3(z))l, <6)

99

B2 2(2BlelXM-HB2EBAM) 8181 + 3
AE(b2 + bl) 7AE(b2 + e7

 

 

M “
A Z<bl) = ’PC7

{(a3BlAM+201082AL)Bld3 + (d§eliM+2dioeexL)elaé

+

(0“81AM+2GI“B2AL)81 a” + (duBlAM+2diu 1901118l } +K]
(7)

M _ M _ 2 2 2
Axx ‘ Ayy “ ‘PE‘ I81 + AE(b2 + e) {(Bla3xM+2B2a10AL)a3Bl

 

+

(8103AM+282a'lO AL)Bla3}

 

3
- 7AECb2+ ET {(auBlAM +20 luBZAL)Blau +
+ (auBlA M+2a1“B2AL)Bla“} + K] (8)

where AM and AL are the single-electron spin-orbit coup--
ling constants for the metal d and halogen p orbitals,
respectively. The 8 terms which appear in the expres-
sions for gl' and gi are the group overlap integrals. In
the equation for AEZ,only the contributions from the un-
occupied orbital of bl symmetry are included. P

and Q in the above equations are defined as

100

_ ’3 M
P - gegNBeBN <r >3d

D
II

R<nsI§§1 npK> (K = x, y or z),

where R is the distance between the metal and halogen
atoms and ns, np are Slater type atomic orbitals.

The major difference in the expressions for g|l,
$1, ATI and A? between our treatment and that of DeArmond,
23.31. arises from the fact that the molecular orbitals we
have used in deriving the equations take into account all
possible contributions from various atoms. The inclusion
of halogen contributions in orbitals of e symmetry makes
gi dependent on the ligand spin-orbit coupling constant
AL. In deriving these equations we have retained all
the overlap integrals. The contribution from any low
lying occupied b1 orbital to gll can be calculated using
expressions very similar to that for A||(bl)' The expres-
sion for Al(e2) which is not given in the above set of
equations, is very similar to that for Aliel). The same
set of equations can be used to compute the g and A tensor
components for hexacoordinated transition metal oxohalo
complexes by adding a term involving the overlap between
the metal atom and the axial halogen atom to the expres-

sion for gi.

101

F. Evaluation of g_and A Tensor Components

In the SCF-MS-Xa method the population analysis is
done by expanding the nonatomic part of the wavefunction
about the atoms and thus the calculation corresponds to
a zero-overlap model. Hence we have used the following
expressions, obtained by neglecting the terms involving

overlap integrals, to compute g and A tensor components

All = 6e - 622; A1 = 6e - 6xx

2(8161AM-“8283AL)(28161-“B2e3)

 

A||(bl) = (9)

zealeiweaeaneneieeaep

 

A||(bi) = - (10)

2(a3BlAM+2alOBZAL)(6381+201082)

 

Ai(8) = (11)

AEQb2 + e)

2(28 6 A —“B 6 A )8 6
_ “ 2 l l M 2 3 L l l

 

+ + K] (12)

 

7 AE(b2 + e)

(a B A +2a A A )
Al = -P[- 58; + %i 3 1 M 10 2 L 9381 + K] . <13)
AE<b2 '+ G)

 

102

where A||(b1) includes only the contribution from the
empty bl orbital. The contribution from other orbitals
of bl symmetry to All can be included by adding terms
similar to the second term in the expression for All.

To take into account the charge in the outer sphere
region we have distributed the outer sphere charges as-
sociated with the molecular orbitals needed for computing
g and A tensor components among the ligand atoms in the
ratioscd‘their atomic numbers. The additional charge was
further partitioned into s and p portions for each atom
in the ratio of net s and p populations. In the SCF-MS-
Xa model the square of the molecular orbital coefficients
in Equations (9)-(l3) has been identified with the frac-
tional charge associated with the corresponding partial
wave27.

The computation of gll, gi, All and Al requires the

magnitudes for AM, A <r-3>3d and the electronic excita-

L’
tion energies, in addition to the molecular orbital co-
efficients. There is no obviously "correct" value of the
spin orbit coupling constant of the metal or the chlorine
to use in evaluating the ESR parameters. In most analyses
of EPR data the spin-orbit coupling constants are assigned
values based on assumed values of the charges on the atoms,
or on some other empirical procedureuu. This method of
assigning values for AM doesn't make any significant dif-

ference in results if the variation of AM with charge on

103

Table 7. Chromium spin orbit coupling constants

 

 

and <r-3>3d values.
Spin orbit coupling
<r-3>3d (au) constant (A3d) cm"l
Config- Hartree
uration Charge Xa Fock Xd Expt.
3d5“s1 0 2.910 206.80 185
3d5 3.011 2.97“ 255.21 190
3d“ 2 3.530 3.053 301.75 230
3d3 3 “.063 3.961 351.03 275
3d? u “.609 “.“82 “02.85 325
3d1 5 5.170 0.993 “57.58 380

 

10“

the metal is not drastic, as in the case of copper where

l l

ACu varies only from 817 cm- for Cu(3d9“s2) to 828 cm-

for Cu2+(3d9). On the other hand, for metals like chrom-
ium for which the spin-orbit coupling constant varies
from 185 cm-1 for Cr(3d5“sl) to 380 cm'1 for Cr5+(3dl)u5
such assignments are difficult to make since gll and gi
are critically dependent on the choice of the value for
AM. This makes it difficult to interpret the observed
trends in measured ESR parameters.

The value of <r-3>§Id required to calculate A tensor
components is generally taken from the (r—3>Md values

calculated using the atomic Hartree-Fock wavefunctions.

The spin-orbit coupling constant ACr for the various ox-

idation states of chromium were computed using the Xawave-

function and the single particle approximation for
3d

 

ACr, i.e.,
3d 2 e2h2 1 dV _ e2fi2 , —3
)‘Cr '2 2' <‘ a? ' 2 2 Z <I' >3d ’
2m 0 11 3d 2m c 1

where z' is the effective nuclear charge; the values are

given in Table 7. Even though this simple approximation

doesn't correctly take into account the exchange ef-

fectsu6, the relative values and the variations are com-

parable to the experimental numbersus. The A33 and

<r-3>gg required to calculate the ESR parameters were

calculated for the formal valence configuration of Cr

105

in [CrOClujl- of 381.96“ 3p5.786 3du.“53 “$0.33“ up0.505.

By this procedure we obtained A3: = 289.36 cm-

<r-3>gg = 3.390 au for the valence configuration of chrom-
l.96“ 3p5.7863d“.“53 “80.33“ “p

l and

ium 33 0'505 for geometry

1 of [CrOClujl- and A3: = 288.51 cm'1

3.382 au for the valence configuration of chromium
3sl'975 3P5.861 3d“.359 “$0.338 upo.“90

and <r'3>3d =
Cr

for geometry

2 of [CrOClujl'. We had used for the ligand spin orbit

1

coupling constant (AL) a value of 667.05 cm- obtained

from an X0 calculation on the chlorine atom.

We consider first the calculation of g||,which is

2B

2
ground state (Equation (9)). Using the electronic excita-

determined by mixing of 281 excited states into the

tion energy for the transition 2b2 + ”bl (Table 6),and
computed spin—orbit coupling constants,we obtained a value
of 1.9793 for g'l which is not in agreement with the exper-

imental value gll = 2.006. The fact that the experi-.

mental gll is greater than the free-electron g value indi-

cates that the contributions from low_1ying 2B1 states

arising from the promotion of an electron from a doubly

occupied bl orbital to the singly-occupied 2b2 orbital
should be important. We had calculated(Equation (10))

the contribution to gll from two such states, namely the

one arising from the excitation 3bl + 2b2 with the excita-

1

tion energy of 15,350 cm- and the one from 2bl + 2b2

with the excitation energy 28,530 cm-l. On adding the

106

contributions from all the three states we get a value

of 2.0“0 for gl|,which is greater than the free electron

g value and in fairly good agreement with the experimental

value. If gll is computed including only the contribu—

tion from the chromium from all the three states the value

is 1.980. This indicates the importance of including

the ligand contribution in evaluating g tensor components.
The magnitude of gl is determined primarily by the

contribution from the low-lying empty orbital of e sym-

l6

metry . The value obtained for gl using the excitation

energy of 13,830 cm"1 for the transition 2b + 7e and the

2
calculated spin—orbit coupling constants is l.978,in
excellent agreement with the experimental value of 1.979.
In the spin-restricted molecular orbital approximation
in which we had done all the calculations,the Fermi con-

tact interaction parameter K vanishes. So K was determined

using the following equation

A0 = — KP - (ge-g)P ,

where A0 is the isotropic hyperfine coupling constant,

g is the free-electron g value and g the experimental

6
average g value for the complex. The values of gll,

gi, AET and Air calculated using SCF-MS-Xa wavefunctions
and computed spin—orbit coupling constants, along with

the results of ab initio studies16 and available

107

.mme Hmucmsfigmdxm Eopm pmmeHpmm mm; :mm.o n y

n
.mmmmnp:0LMQ CH C0>Hm

0pm mmfiwnmcm coauHmCMLu omumasoamo map wcfiamom pmpmm.mwag mocmpmmmm mo mpazmmp 0:90

.mmmmnpcmhmm :fi 20>Hw 0pm mcofiusnwppcoo ncmwfia 0:» wcfiuomawmc Umcfimpoo muHSmmp 0:8

 

 

n
.UmoSHQCH mampfinpo Ho: 66m .Hnm .Hnm Eopw mCOsznHLucoo 0:90
H.wH mmm.a msm.a moo.m eccefiscdxm
Asmm.fiv Assm.fiv Asoo.mv
msm.fi mom.H mom.H econ chHcH c<
Aosm.fiv Ammm.fiv Aowm.flv axcmzumom
cmm.a mmm.fi Hzo.m m senescco
Aomm.av Aomm.HV Aomm.fiv exumznmom
mm.OH mm.mm mam.H msm.a o:o.m H accesses
chad as __< m an __m
AHIEo :OHxV

occapowpmuCH mcfimpmazm no

n.m.lam:HQOL0u mo mpmpmempma mmm .m manna

108

Table 9. Estimated chromium spin-orbit coupling constants
(A3d) and <r’3>3d.

 

Charge on Spin-orbit coupling <r‘3>3d

Method chromiuma constant (cm-l) (au)

 

Geometry 1

Method 1 0.36“ 177.20 2.877
2 0.206 170.77 2.808
3 1.222 212.13 3.2“9

Geometry 2

Method 1 0.366 177.28 2.878
2 0.212 171.01 2.811
3 1.196 211.07 3.238

 

aFrom extended Hfickel calculations.

109

 

 

3d
>‘Cr ' A QCr 4’ B
500 "- A s40.7l . B! I62.38

400—-
3d
x'Cr

300~‘

200*-

100-—

1 J l J l
0 LC 2.0 3.0 4.0 5-0

QCr" Charge on Chromium

3d

Plot of spin orbit coupling constant (AC r)

.Figure 2.
versus charge on chromium (QCr)’

110

 

 

A2d84A<&'3>§d+E3
Cr C"
500 " A -93.a|
B ' '9267
400 ""
A3d
Cr
300 '—
200 —
I 00 t"
l l l l J
0.0 | .0 2 .0 3.0 4.0 5.0

.3 3d I
(r >Cr ou

Figure 3. Plot of spin orbit coupling constant (A33)
versus <r-3>gg.

111

experimental results, are given in Table 8.

In the extended Hfickel model for calculating g and A

tensor components one has no choice but to guess the

Ag: and <r-3>gg values based on the formal charge on
chromium obtained from the molecular orbital calculation.

The A33 values were therefore obtained from a plot (Figure

2) of A33 for the various oxidation states versus the cor-

responding charges on chromium, assuming a linear rela-

tionship of A3: with the charge on chromium (Table 8).

The <r-3>gg values were obtained from a plot A33 versus

<r-3>gg Hartree-Fock values“7 for the various oxidation

states of chromium (Figure 3) again assuming a linear de-
pendence of Ag: on <r-3>gg. Using this procedure for a
charge of +0.36“ on chromium in [CrOClujl-, we obtain from
Figure 2, A3: = 177.20 cm'l; the value of <r-3>gg cor—
responding to this value of Ag: from Figure 3 is 2.877
au. The spin-orbit coupling constant obtained by this
method turns out to be lower than the value for the neu-

tral chromium (3d5“s1) atom. We had used the set of

3d
ACr
there doesn't seem to be any other systematic way to assign

numbers for given in Table 9 for our calculations as

the values for A33. For the chlorine spin orbit coupling

<2onstant we used a value of 587 cm-l.u8 The results for

izhe g- and A-tensor components (Equations (“-8)), using
TShe extended Hfickel wavefunctions obtained by the three

Ciifferent methods we have described earlier, are given in

112

 

 

Table 10. Extended Hfickel results for g and A tensor
componentsa.
Cr Hyperfine coupling
constantsb (xloucm'l)
Method gll gi Al! Al
Geometry 1
Method 1 2.091 1.971 26.11 10.11
2 2.088 1.977 27.07 9.““
3 2.057 1.985 32.3“ 10.28
Geometry 2
Method 1 2.08“ 1.931 26.88 11.20
2 2.072 1.93“ 27.12 10.51
3 2.055 1.936 32.36 11.52

 

aThe contributions from two occupied orbitals of bl sym-
metry also included.

b

K = 0.52“ was estimated from experimental data.

113

Table 10 .
From the results in Table 8 it is clear that gl cal-
1—

<3ulated for geometry 2 of [CrOClu] is lower than the

(experimental number. This dependence of the gl value on
ggeometry can be explained based on the fact that the

2b2 + 7e transition energy required to calculate gi is
critically dependent on geometry.

The gll values calculated from the extended Hfickel
Inethods l and 2 are greater than the free-electron g
'value even when only the contribution from the low-lying
«empty orbital of b1 symmetry is included. This problem
<3an be taken care of by using A83 values greater than 20“.O
<3m71, which corresponds to a charge of greater than +1.0
can chromium. This type of dependence of computed g-tensor
(zomponents on spin-orbit coupling constant makes the cal-
<3ulation of ESR parameters involving empirically de-
tzermined values for spin-orbit coupling constants un-

Ireliable.

We have shown the importance of including the contribu-
1:ions to g tensor components from occupied orbitals. It
kbecomes practically impossible, without making too many
Eissumptions, to determine the molecular orbital coef-
17icients from the expressions for g and A tensor com-
Fxonents if the contributions from occupied orbitals are
ildcluded. Thus, even in the case of molecules with fairly

1'ligh symmetry,the first approach to study the bonding

ll“

c:haracteristics of transition metal complexes in which

c>ne estimates MO coefficients from ESR data appears to be

1:00 difficult.

(3. Conclusions

We conclude from this study that reasonably good esti-
rnates of the g tensor components can be obtained from
ESCF-MS-Xa wavefunctions and the required spin orbit coup-

Zling constants calculated for the appropriate valence

(zonfiguration of the atom on which the unpaired electron

:18 centered using atomic Xa wavefunctions. Another con-

czlusion that we have reached is that the ligand contribu-
tsion to g tensor shift is important in addition to the
(zontributions to g tensor components from occupied or-
tDitals. We think the difficulties involved in calculating
g; tensor components using extended Hfickel wavefunctions and

enmpirical values for spin orbit coupling constants have

t>een clearly demonstrated.

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3”

2L

3.

31:3.

1.“

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P. J. Hay, J. C. Thibeault and R. Hoffmann, J. Amer.
Chem. Soc. 91, “88“ (1975).

C. D. Garner, L. H. Hill, F. E. Mabbs, D. L. McFadden
and A. T. McPhail, J. C. s. Dalton, 853 (1977).

Schwarz, Phys. Rev. B 9, 2“66 (1971).

. G. Norman, Jr., M01. Phys. 9;, 1191 (1976).

. W. Richardson, W. C. Nieuwpoort, R. R. Powell and

K
J
R. E. Watson, Phys. Rev. B, 1108 (1958).
J
W. F. Edgell, J. Chem. Phys. 99, 1057 (1962).

The calculations for CrO and CrO3+ were done with the
internuclear distance taken to be the same as in [CrOClujl-,
“inter = ”out = 0.72899 and the sphere radii RCr =

2.1119 au, R0 = 1.777“ an and Rout = 3.3799 au. The
chromium and oxygen sphere radii are chosen to be the

average of the chromium and oxygen sphere radii in
[Crorujl‘ and [CrOClu]1‘.

A. Abragam and M. H. L. Pryce, Proc. Roy Soc. London.
A205, 135 (1951); 206, 16“ (1951).

B. R. McGarvey, Transition Metal Chem., R. L. Carlin,

A. Carrington and A. D. McLachlan, "Introduction to
Magnetic Resonance", Harper and Row, New York, 1967.

G. deBrouchere, N. J. Trappeniers and C. A. tenSeldom,
Physica 8“B, 295 (1976).

T. M. Dunn, Trans. Faraday. Soc. 91, 1““1 (1961).

M. Blume and R. E. Watson, Proc. Roy. Soc. London.
A270, 127 (1962); 271, 565 (1963).

B. A. Goodman and J. B. Raynor, Adv. Inorg. Chem.,
Academic Press, New York, 29, 156 (1970).

D. S. McClure, J. Chem. Phys. 21, 905 (19“9).

CHAPTER VI
AN SCF-MS-Xa STUDY 0F (11 TRANSITION METAL

OXOHALO COMPLEXES

1%.. Introduction

The elements at the beginning of the transition metal

Eseeries have a remarkable ability to form oxycations with
Tzlle general formula M0n+. Most of these oxycations, even

TSIIOugh not stable, form a wide variety of complexes, the

inn<>st extensively studied being those of vanadium, niobium,

<311romium, molybdenum and tungsten. The electronic struc-

tZLLres of the halogen complexes of these oxycations have

't>eeen.the subject of detailed experimental studies for the

3-ELst several yearsl-lu, especially since the pioneering

eExrtended Hfickel molecular orbital studies on V02+ complexes

(Dif‘ Ballhausen and Gray.15 Their work has been followed by

16’17 and, more recently, by studies

11,12

51 few similar ones
employing ab initio methods In spite of this ac-
t31svity, there are still many ambiguities in the interpre-
tation of experimental spectroscopic results. We have

t1Jerefore undertaken a fairly detailed study of the elec-

t3I‘onic structure of the transition metal oxohalo complexes

118

119

of the type [moxnjm‘, where M = v, Nb, Cr, Mo, w and X = F,
Cl, Br (n = “ and 5), in the hope of understanding the
basic features of bonding and therefore providing a sound
basis for interpreting the experimental results for this
c: lass of compounds.

The relatively small number of ab initio studies of
the electronic structure of systems as large as the one
under investigation here is primarily because of the
enormous computational effort required for such studies.
We have chosen to carry out our studies in the Xa approxi-

18-20

mation , which was developed from the beginning with

the transition metal systems in mind and has been used
numerous times with considerable successlg'el. The goals
Of this study are four-fold: first, to understand the
Similarities and differences in bonding characteristics

1 oxycation complexes;

Of the penta- and hexacoordinated d
Eseecond, to explain the observed trends in g and metal-hyper-
fine interaction tensor components, and thus to resolve
the existing ambiguities in the interpretation of electron
Spin resonance parameters; third, to use computed d-d
transition energies to perhaps resolve the ambiguities in

the assignment of bands in the electronic spectra; and

fOurth, to test the capability of SCF-MS-Xa method to

reproduce the observed trends accurately.

120

B. Methods

In this section a brief description of the SCF-MS-

Xa method and the computational details are given.

(1) The SCF-MS-Xd Method

 

The SCF-MS-Xa method is a technique used to approxi-
mate the solutions to the Hartree Fock equations for many

electron systems. Its underlying assumptions, procedures
and results for many systems are available in the litera-

turelB—zo.

The two characteristic approximations of the
method are the use of the Xd approximation for the exchange
potential and the muffin-tin approximation for the poten-
tial. In this latter approximation, the most severe of
the two, the molecule is partitioned into three fundamental
types of regions; atomic, interatomic and extramolecular;the
latteris the region outside a sphere which encloses the
"entire" molecule. The potential in the atomic and extra-
molecular region is spherically averaged while a constant
potential is used in the interatomic region. The one-
electron equations are solved numerically in each of these
regions and the resulting wavefunctions and their first
derivatives are Joined continuously throughout the various
regions.

The use of the Xa approximation for the exchange po-

tential makes the interpretation of one-electron eigen-

values different from those of the Hartree—Fock

121

 

 

 

Table l. Geometrical dataa for [MOanm- complexes.
M-O M-X M-X <Q_M-X
_ eq ax eq

[MOanm <8) <8) <8) <°>
[VOF “32' 1.63 1.97 98.5°
[V0C1u321.63 2.“2 98.5
[VCF5 33' 1.63 1.97 2.16 90
[V0C15J3‘ 1.63 2.“2 2.67 90
[CrOFujt 1.519 1.79 10“.5
[CrOCluJT 1.519 2.2“0 10“.5
[CrOFs]: 1.519 1.79 2.01 90
[CrOClSJ21.519 2.2“ 2.389 90
[MoOFuji 1.610 1.88 105.2
[MoOClu]? 1.610 2.333 105.2
[MoOBru1T1.610 2.“7 105.2
[MOOF5]21.610 1.88 2.08 90
[M60C15121.610 2.333 2.600 90
[WOFHJI 1.63 1.90 106
[w0C1uJ:1.63 2.35 106
[WOBrujl 1.63 2.“9 106
[NbOFuJ21.68 1.8“ 99
[NboF5 13 1.68 1.8“ 2.06 90
8‘Some of these values are experimental and others have been

estimated; see text.

122

eigenvalues. The correspondence between the SCF-MS-Xa
eigenvalues and the Hartree-Fock eigenvalues has been

established22’23.

In the SCF-MS-Xa calculation the molec-
ular orbitals are characterized by the charge associated
with each atomic sphere, and the region outside the outer—
sphere, in terms of the percentage of s, p, d, etc. charac-
ter and the charge in the intersphere region. The total
charge associated with each atom in a molecule is taken to
be the sum of the charge inside the atomic sphere around
the atom and a fraction of the intersphere charge. The
latter has been obtained by the scheme of Case and

2“ in which the intersphere charge is partitioned

Karplus
among the basis functions centered on the various atoms

in proportion to the average charge density at the surface
of each atomic sphere, multiplied by the area of that sur-
face bordering the intersphere region. Thus the per-

centage s, p, d, etc. character of the contribution of

each atom to the molecular orbital can be calculated.

(ii) Computational Details

 

We have carried out SCF-MS-Xd calculations with over—
lapping spheres for the transition metal oxohalo complexes
of the type [MOXnJm- in C“v symmetry for geometrical data
given in Table 1. Despite the fact that this class

of compounds has been the subject of detailed studies,

there are very few compounds for which crystallographic

123

 

 

 

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mmmmH.o u Hos
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125
data are available. The atomic arrangements for [VOCluJ2_,25

[CrOClu]l-,lu and [MoOClujl-lu were obtained from the
crystallographic data, while for [NbOFSJ3- the geometry

2-26

of [Nb0F5] was used. Since the available geometrical

data show that the metal atom is above the plane formed
by the halogen atoms in the case of pentacoordinated com—
plexes our calculations on pentacoordinated species em=
ployed a similar distorted square pyramidal geometry. For

all hexacoordinated complexes the calculations were done

with the metal in the plane formed by the equatorial halo-
2-

51

geometrical data for complexes for which no structural

gen atoms, based on the structure of [NbOF .26 The

data are available were estimated based on the structure

of related systems.

The values for the atomic a parameter (Table 2) for
all complexes,with the exception of tungsten complexes,
were taken from Schwartz27. For all atoms in the tungsten
complexes we had used the theoretical a values of Gopina-
than gt_§;,28, since the Schwartz 0 value is not available
for tungsten. In the intersphere and the outersphere
regions weighted averages of the atomic a values were used.
The sphere radii were taken to be 90% of the atomic number
radii, following the suggestion of Norman29, and are given
in Table 2. Although the sphere radii were not varied to
obtain the correct valuescfi'the virial ratios, fairly good

results were obtained for the virialratios (Table 2).

The outer sphere, with origin at the center of nuclear

126

charge of the molecule, was chosen to touch the halogen
sphere and serve as a Watson sphere30 with a positive
charge equal in magnitude to that of the anion, in order to
simulate the stabilizing effect of the environment on the
ion. In order to get a converged solution for [NbOFujz-
and [Nb0F5]3.,the charge on the Watson sphere had to be
increased to +3 and +“, respectively.

The partial wave expansions were truncated at 2 = “
for the outer sphere, 2 = 2 for the metals except tungsten
and 2 = l for oxygen and the halogens. The core energy
levels, for example ls, 2s, 2p on chlorine, were calculated
in each iteration using only the surrounding potentials.

In the case of tungsten complexes the completely filled
tungsten “f orbitals were not treated as core orbitals be-
cause they had an energy higher than the 53 orbitals. All
the calculations were carried out self consistently,with
the convergence criterion that the maximum relative change
in potential between two consecutive iterations was lower

than 10.“.

C. Electronic Structureg

The SCF-MS-Xa calculations of the ground electronic
statescm‘all the penta- and hexacoordinated complexes
predict the ground state to be 2B2 in agreement with the

11,12,15-17

earlier molecular orbital studies and the

available spectroscopic data. In these complexes, where

127

 

 

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131

the metal atoms have a formal dl

configuration, the unpaired
electron is in an orbital of b2 symmetry which is primarily
a metal dxy orbital. The occupied molecular orbitals and
orbital energies for [VOFuj2-, [VOC1“]2-’ [VOFSJ3' and
[VOC1513-, which are representative examples of this class
of molecules, are given in Tables 3, “, 5 and 6, respec-
tively; the fractional charges were obtained by the charge
partitioning scheme of Case and Karpluszu. The molecular
orbitals of the remaining complexes are given in Appendix
1.

The molecular orbitals of the pentacoordinated com-
plexes (Tables 3 and “) can be divided into four groups.
The first set of orbitals are the three lowest-lying
virual orbitals, plus the orbital of b2 symmetry containing
the unpaired electron, each with a substantial metal d
contribution. The occupied orbitals which follow this
set are essentially non-bonding in nature and are halogen
p type orbitals. These are followed by an orbital of bl
symmetry which accounts for the bonding between the metal
and the halogen atoms. The next two lower-energy orbitals
contribute to the metal oxygen bonding a 0 and a pair of
n bonds. The orbitals still lower in energy are the non-
bonding oxygen 25, halogen s type orbitals and the low-
lying metal 8 and p type orbitals.

The hexacoordinated complexes (Tables 5 and 6) have

four more occupied molecular orbitals in addition to those

132

 

 

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Figure 1. Percentage d contribution to M-0 0 and 6 bonds
in M0n+ and [moxmjn‘ where M = v, Cr, Mo,
K = F, Cl and m = “ and 5.

133

described for the pentacoordinated complexes. Of these
three, two of e symmetry and one of al symmetry are higher
in energy than the equatorial-halogen p orbitals. The
orbital of al symmetry corresponds to a rather weak 0 bond
between the metal and the axial halogen atom and involves
the lowest unoccupied metal p type orbital and the halogen
p-type orbital. An interesting feature of this metal-
axial halogen bonding is that it does not involve any
contribution from metal d orbitals.

There are some common features of the bonding in the
penta- and hexacoordinated complexes. The metal-oxygen
bonding involves both 0 and H type bonding and is sig-
nificantly stronger than the metal equatorial halogen bond-
ing and the metal—axial halogen bonding in hexacoordinated
complexes. In order to understand the metal-oxygen bond-
ing in these complexes we have carried out SCF-MS-Xa cal-
culations on the corresponding metal oxycations.31 As
expected, in the oxycations there is multiple bonding
between the metal and oxygen, a o and a pair of 0 bonds,
but there are differences compared to the corresponding
complexes. The variation in the metal d orbital contribu-
tion to the metal-oxygen bonding orbitals in vanadium,
chromium and molybdenum complexes and the corresponding
oxycations are illustrated in Figure l. The metal d

orbital contribution to both the 0 and n bonding orbitals

follows the sequence Cr>Mo>W and in the case of both

13“

 

 

Table 7. Charge distribution in [MOXnJm— complexes.
Complex Metal Oxygen Halogen(Eq) Halogen(Ax)

v02+ 1.81“ +0.186

[VOFu]21.056 -0.“80 -0.6““

[VOCluJZ 0.807 -0.331 -0.619

[VOF 5] 1.059 -0.5“3 -0.688 -0.766
[v0C15J3 0.830 -0.358 -0.673 -0.780
N602+ 2.176 -0.176

[NbOFuj2— 1.522 -0.8“7 -0.669

Cr03+ 2.380 0.620

[CrOFujl 1.319 -0.235 -0.521
[00001,]10.957 -0.156 -0.u50

[Cr0F5J2’ 1.316 -0.317 -0.578 -0.688
[Cr0C15J2 0.99“ -0.182 -0.537 -0.661

M603+ 2.673 0.327

[MoCFth 1.639 -0.“19 -0.556

[MoOClu]: 1.222 -0.198 -0.506
[MoOBru]11.096 -0.813 —0.“78

[MooF5J2' 1.660 —0.“89 -0.616 —O.708
[M00015]21.2“5 —0.2“0 -0.567 -0.735
wo3+ 2.676 0.323

[WCFqu‘ 1.36“ —0.377 -0.“98

[WOClu]? 0.989 -0.153 -0.“59

[wosrqu 0.881 -0.132 -0.““7

 

135

 

 

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137

vanadium and niobium complexes the metal contributions are
lower than those in molybdenum complexes. In the fluoro
complexes the metal-oxygen o bonding orbitals have a
larger contribution from the metal d orbitals than they
have in the corresponding chloro and bromo complexes.

0n the other hand, the extent of metal-oxygen w bonding

is sensitive to the nature of the ligand atoms. The
molecular orbital which represents the n bonding between
metal and oxygen in the fluoro complexes has a significant
contribution from a fluorine p type orbital (3e orbitals
of Table 3 and “). The metal participation in the metal-
oxygen w bonding in the fluoro,chloro and bromo complexes
is lower than in the corresponding metal oxycation. The
metal-halogen bonding, though weaker than the metal-oxygen
bonding, increases in strength in the sequence F<C1<Br, as
would be anticipated based on the electronegativity of the
ligands.

A few general trends in the nature of bonding in this
Class of molecules can be obtained from the charge distribu-
tion (Table 7) and the electron distribution on the metal
atoms (TableED. In the fluoro complexes the metal atoms have
larger positive charge compared to the corresponding chloro
and bromo complexes, reflecting the greater ionic character
of the fluoro complexes. The charge on the metal varies
by a very small amount on going from a pentacoordinated

complex to the corresponding hexacoordinated complex,

138

 

 

Table 9. Electronic transition (d-d) energiesa (cm-

in [MOXnJm- complexes.
Complex E(dxy7dxz,yz) E(dxy+dx2_y2) Ref.b
[vorul2 7350 15“70
[VOClu] 2’ 9220 (11750) 12950 (13500) 32
[VOFSJ3- 6150 (13660) 15580 (18200) 3n
[v001513 8230 (15500) 13060 (16“00) 33
[CrOFujl 11960 23u80
[CrOClu]l_ 13830 (13100) 19980 (18100)0 12
[CrOF512 7350 (8350) 2“800 (22330)c 10,11
[Cr0C15J212320 (12380) 207“0 (18530) 10
[MoOFu1115360 320“0
[MoCClqu17uso (1“300) 252uo (22600)C 1“
[MoOBrqu 17560 (1“810) 2“1“0 (22730) 35
[M00F512 8890 (12700) 3““60 (21600) 35
[M60015J212290 (13700) 26670 (22“00) 35.11
[WOFujll 15800 32590
[W0C1u3117890 26120
[WOBrujl 18000 2“580

 

aThe numbers in parentheses are the experimental d-d

transition energies.

b

CThese absorption bands were not assigned to the d
dx2-y2 transition in the experimental work.

The references given are for the experimental number.

Xy

139

indicating a rather weak metal-axial halogen bonding. The
low occupation numbers of the low-lying unoccupied metal

s and p type orbitals indicate that these metal orbitals
don't have a significant role in the bonding of these

complexes.

D. Electronic Excitation Energies

The three lowest-lying virtual orbitals of both the
penta- and hexacoordinated complexes have a large metal d
character and the relative energies of these metal d or-

bitals follow the sequence d in agreement

xz,yz< My 22’

with the order generally used1 10 15 1 as the basis for

interpretation of the electronic spectra of this class of
compounds. The SCF—MS—Xa results for the d-d transition
energies calculated by the transition-state procedure18'2O
are given in Table 9, along with available experimental
results.

The lowest energy absorption maximum in the electronic
spectra of this class of molecule is assigned according to
the experimental, as well as the molecular orbital studies,
to the excitation of the unpaired electron in the molecular
orbital Of b2 symmetry to the lowest-energy unoccupied or-
bital of e symmetry, which has a large metal d orbital con-

tribution. Our calculated excitation energies for this

transition (Table 9) are in fairly good agreement with

l“0

40.0 _-

35.0—
dxz -y2

8
o
l

L_ d x2.y2

N
,0
O

20.0 ~—

Electronlc Transition (d-d) Energy KK

 

 

XaF
15.0 -—
x-CI

dxz,yz

no.0 —
X= Cl/ dxz.yz

5.0 ~—

00 l 3__ 1 L

[voxs] [Croxslz‘ [MoOX5] 2'

Figure 2. Plot of electronic transition (d-d) energies

for [MOXSJn- where M = V, Cr, Mo and X = F

3

Cl.

350

30£i

gy KK

N
.o 5‘}.
o '0

Electronic (d-d) Transition Ener
ii
c>

0£>

Figure 3.

 

lUl

 

*-= dx2_y2

 

 

I“

 

1 _5: 1 1 1 dxy
[V0X4] [009]" [MoOXJ' [wox4]"'

Plot of electronic transition energies for

[moxujn‘ where M = v, Cr, Mo, w and X - F, Cl.

1&2

 

 

 

 

350-
IW'MP\
f
scat-M M°
X:
I!
>.
9250
a: . "-
IE ‘jggyz
".5
“O
c 20.0—
.2
’5'. WW f % dxzarz
C /
2
*- |5.0— ,
U M-Mo
'E
o
.:
.§ «lo-
u:
210-
1 I 1 dxy
0'0 [MOF " " "
4] [MOCI4] [MOBn4]
Figure H. Plot of electronic transition energies for

[moxujn‘ where M = M0, w and x F, Cl, Br.

1M3

experiment for a large number of molecules. As we had
shownul, the lowest energy excitation energy is dependent
on the angle between the metal-oxygen and metal-halogen
bond and on the metal-oxygen distance. It is therefore not
really surprising to note that for some complexes the com-'
puted dXy + dxz,yz transition energy doesn't agree with the
eXperimentally observed value as well as for other com-
plexes, since the geometries of quite a few of these com-
plexes were guessed based on the available data for similar
complexes. The lowest d-d transition energies follow the
sequence x = F<Cl<Br for [MOXujn-, and [moxu]n“ >
[M0X5](n+1)' for any halogen (Figures 2, 3, and u).

The variations in the second d-d transition dx +

y
dx2-y2 are essentially the same as those for the dxy +
dxz,yz transition energy. Recently, the two lowest-energy

bands in the electronic spectra of [CrOClu]1‘, [CrOF512'
and [MoOClujl' had been assigned differently from the gen-

erally used identification of the first and the second-lowest

energy absorption bands to the dxy + dxz yz and dxy *
’

dx2_y2 transitions based on single-crystal polarized spectra

11-1u

and ab initio studies The first absorption band in

l- .
[CrOClu] was assigned to the usual dxy + dxz,yz tranSi-
tion while the second band was assigned to Cr-O(n) +
Cr-O(o*)l2. According to the SCF-MS—Xa results the transi-
tion Cr-0(nl + Cr-O(o*) corresponds to an energy greater

than 80,000 cm-l, while the computed d-d transition ener-

gies are in good agreement with the two lowest energy

th

41

bands in the electronic spectra of [CrOClujl-. For

[CrOstz' our computed d-d transition energies are in agree-
ment with the electronic spectral assignments of Ziebarth

§£_§l.lo In the case of [MoOClujl', the SCF-MS—Xa value for
the transition energy Mo-0(w) + Mo-O(c*) assigned by Garner

14

t 1. to the band at 22,600 em"l is greater than 70,000

cm'l, while the computed d-d transition energies seem to be

in fair agreement with the two lowest-energy absorption bands

in the electronic spectra (Table 9).

E. Evaluation of giand Hyperfine Interaction Tensor

Components

The principal components of the g and the hyperfine
interaction (A) tensors which characterize the electron
spin resonance spectra of transition metal complexes have
been used widely to study the nature of bonding in these com-
plexes and the changes in bonding within a class of compounds.
The two approaches generally used in utilizing the measured
ESR spectral parameters to investigate the bonding are: (i)
the coefficients of the atomic orbitals in the molecular
orbital containing the unpaired electron and in a few low-
lying virtual orbitals, are determined using the experimental
values of the g and A tensor components;3-5’8’9’36 (ii)
secondly, one computes the principal components of the g and

hyperfine interaction tensors using the molecular orbitals

obtained by one of the (usually semi-empirical) molecular

1u5

orbital methods and compares them with the experimental
values. When the two sets agree the orbitals used in the
computation are assumed to give a good description of the

37-140.

bonding in the compound under investigation Even
for molecules with as high a symmetry as the penta- and
hexacoordinated transition metal oxohalo complexes, the
first approach requires more experimental data than one
has, while the second method is critically dependent on
the molecular orbitals used. Note however, as we have
shown earlierul, that the latter is clearly the most
objective approach and so is the one used in this study.
Both the procedures depend heavily on the choice of values
for parameters such as the spin-orbit coupling constants

3

and <r- > values.
There are some interesting similarities and differences
in the experimental values of the principal components of
the g and A tensors of the penta- and hexacoordinated
transition metal oxohalo complexes of vanadium, chromium,
molybdenum and tungsten. For all vanadyl complexes the
value of glI is less than that of gi, while gll is greater
than gi for the chloro- and bromo-complexes of chromium
and molybdenum. In addition some of the chromium and
molybdenum complexes have a gll greater than the free-
electron g value of 2.0023. It has been suggested by

Kon gt al.2 that for chromium complexes the contribution

from the low-lying occupied orbital of bl symmetry should

1H6

be included in evaluating gll, while Manoharan t 1.“

included the ligand contribution to gll to account

for the above-mentioned difference in the principal compon-
ents of theg;tensor for this class of compounds. We have
investigated the relative merits of both suggestions

using g and A tensor components calculated from the SCF-

MS-Xa wavefunctions.

The following equations whichhave been discussed

 

 

 

 

 

earlieru;,were used in the computation of gll, g1, ATI
M
and A :
i
Al' = ge - gzz3 Ai_= ge ' gxx
2(28 6 A -M3 6 A 1C2B e -“8 e )
Allcbl) = l 1 M 2 3 L l 1 2 3 (1)
ll 1 AE(bi + be)
A (e) = 2("‘3BIAIVI‘QO‘lOBéAL)("‘351+2°‘1052) (3)
2(28 8 A -H8 5 A )
M
AZZLbl) = -P[%Bi + 1 1 M 2 3 L 8121 +
AE(b2 + b1)
(a B A +2a B x )
g 3 1 M 10 2 LG381 + K] (u)

AE<b2 + e)

1H7

 

M M 2 2 11 (“331*M+2“1032*L)
A = A = -P[— -B + —— a e + K],
xx yy 7 l 7 AE(b2 + e) 3 l
(5)
-3 nd

P = gegNBeBN<r >M 3

where 81 and 52 are the coefficients of the metal d and

halogen p -type orbitals in the molecular orbital (b2)

x,y
containing the unpaired electron, El and 53 the correspond-

ing coefficients for the lowelying unoccupied orbital of

b1 symmetry and a3 and alO are the metal d and halogen

xz,yz

p orbital coefficients of the low-lying unoccupied orbital

Z

Of e symmetry; 6i and 8% are the coefficients of the metal

d and halogen p orbitals of the occupied orbital of b1

Xay
symmetry; XM and XL are the one-electron spin-orbit coupling

constants for the metal d and halogen p orbitals, respec-

tively, K takes into account the Fermi contact interaction,

and the apprOpriate excitation energies are denoted by AB.
The contribution from any occupied orbital of b1

symmetry to gll can be calculated using expressions very

similar to that for Al|Cbl). In the equation for Ag

2(01)
only the contribution from the unoccupied orbital of bl
symmetry is included. The contributions from the occupied

orbital of bl symmetry to Azz can be included by adding

terms similar to the second term in the expression for

M
Azszl)‘

In order to take into account the charge in the outer

1A8

sphere region,we have distributed the outer sphere charge
associated with the molecular orbitals needed for computing
g and A tensor components among the ligand atoms in the
ratio of atomic number. The additional charge was further
partitioned into s and p fractions for each atom in the
ratio of the net 8 and p populations. In the SCF-MS-Xd
model the squarescfi'the molecular orbital coefficients in
Equations (1-5) have been identified with the fractional
chargesassociated with the corresponding partial waves.

The computation of gll, gi, Ali and Al requires
knowledge of the magnitude of AM’AL’ <r'3>M and of the
electronic excitation energies,in addition to the molecular

orbital coefficients. The required electronic excitation

energies were calculated by the transition-state pro-

18-20

cedure The magnitude of computed ESR parameters

depends critically on the values assigned for KM, A and

L
<r-3>M. -The spin orbit coupling constants (KM) and values of
<r-3>M for the metal d orbital are sensitive function of
the valence electron configuration of the metalul. This
makes the choice of values for these parameters, from those
available for the various oxidation states and configura-
tions, very difficult in any systematic manner. The spin
orbit coupling constants AM were computed for the formal
valence configurations of the metals (Table 8), using the

atomic Xd wavefunction for the appropriate configuration

and the single-particle approximation for AM, i.e.,

1A9

Table 10. Computed values of l§d, <r 3>15}d and Pgd.

 

 

Complex Agd (cm-1) <r_3>§11d (au) Pgd (x10)4 cm-l)
[voruj2' 220 2.762 129.u3
[V001,]2 209 2.625 123.00
[v0F513’ 221 2.779 129.99
[v001513 210 2.639 123.66
[CrOFqu; 308 3.59M -36.13
[0r001uJT 289 3.382 -3u.00
[CrOFSJ22 309 3.609 -36.28
[CrOCl5J2 292 3.A1A -3A.32
[MoOFuji 895 H.851 -56.2u
[MoOClujf 835 “.532 -52.55
[MoOBru]? 820 u.u53 -51.63
[M00F5]: 902 u.890 —56.7o
[M00015128A2 9.572 -53.00
[WOFujl 325A 8.919
[WOC1u113170 8.69M

[WOBru113158 8.650

 

150

2 2 dV e2112 Z1

_1__ - -3
<r ' 2C2 <r >’

 

>
dr 2m

where z' is the effective nuclear charge; these are given
in Table 10. The ligand spin orbit coupling constants

lip were calculated for the neutral atoms in a similar
manner (Table 10). The <r.3>M values used in computing

the A tensor components were also calculated for the formal
valence configurationscu‘the metalsixlthe complexes (Table
10). Note that for tungsten valence electron configurations
given in Table 8, since we couldn't locate the 5f orbital
which is very high in energy, the lad were computed for two
configurations where the 5f p0pulation was added to the 6s
and 6p orbitals,respectively. The lad computed for these

1 and we used

configurations differ only by about 30 cm-
the higher lad value in evaluating the ESR parameters for
the tungsten complexes.

The values of gll, which are determined by mixing of
281 excited states into the 282 ground state (Equation (1)),
have been calculated using the computed AM values (Table
10) and electronic excitation energies (Table 9). We first
consider the evaluation of gll for the vanadium complexes.
In this case, since the 281 states arising from the promo-
tion of an electron from the occupied molecular orbitals

of bl symmetry to the orbital containing the unpaired

electron had very large electronic excitation energies and

151

 

 

: HH¢.H mom.a new.a mom.a ummmaoezu
esm.a H=H.m newscmoozi
ms n:m.a mzm.H mmm.a emm.H nameaooozi
m: mma.a ama.a mme.a mam.a -amsaoozi
m esm.fl mom.a moo.m m:o.m ummmaooaou
ea. . . . mem.a mea.a Hea.a mea.a -mmmaoeei
: . . . . mom.a mmm.a
m: mem.a omm.a woo.m o:o.m nameaooeoi
: mem.a ewm.fi mmm.a mmm.H -Hflsaoao_
m mwa.a Ammo.av mem.a mam.fi imam.av a:m.a :Mmmaoo>i
: sum.a Amsm.av a:m.a amm.a Ammm.Hv nmm.a umflmmo>i
mm aea.a Aoaa.ao wem.a mzm.a Aomm.av mea.a -mmeaeo>i
ms mem.a iema.av mmm.a mma.a isma.av mmm.a -Nfleeeei
.mmm .me .oamo .me .oamo moxoaoeoo
aw __w

 

.moxoadsoo Ismcxosu pom Hm can __w no monam> moopSQEoo .HH magma

152

.cofipsofigpcoo ocmwfia on» moaosflocfi psozufiz oopmHSOHmo who: mononucopmd :H newness teem

 

 

m mom.H oam.a ome.a mae.a namsamozu

m me.fi Hom.a me.H mme.a tamaaoozi

m mem.a H:~.H mom.a smm.H tamsaozu

: o:m.a sam.fi mom.H 0mm.fi -mmmfiooozg

.mom .me .oamo .oxm .oamo moxoaasoo
Hm __w

IDIIIII vi

.eezcaecoo .HH edema

153

thus make only negligible contribution to gll, only the

contribution from the lowest unoccupied orbital of bl

symmetry was included. For [VOCIHJ2' gll calculated includ-

ing the ligand contribution is l.9u8, which is in very
good agreement with the experimental value of 1.9480. If

the ligand contribution is neglected, however, g'l has
a value of 1.9201 indicating the importance of the ligand

contribution in calculating gll (see Table 11). The gll

valuesfku’thefluoro complexes of chromium (Table 11) and
molybdenum (Table 11) were calculated by including only

contributions from the lowest—lying unoccupied orbital of

bl symmetry, Just as in the case of the vanadium complexes.

For the evaluation of gll for [CrOClSJ , [MOOClu]l-,

[M00C15]2- and [MoOBrujl', it was found necessary to include

the contributions from the occupied orbitals of b1 sym—

metry, as shown earlier for ICrOClujl-fl'in order to account

for the fact that for these complexes gll has a value
greater than gl. For all the chloro complexes of chromium

and molybdenum, as well as for IMoOBrujl', there are two

231 states which arise from the occupied orbitals of 01

symmetry having electronic excitation energies comparable

2

to the B1 state from the lowest-energy unoccupied

orbital of bl symmetry. Thus, for example, for [MoOClHJl-

the computed g'l is 1.9963, including the contributions

2

from all the three Bl states, while the corresponding

2E state

value including only the contribution from 1

154

arising from the lowest unoccupied orbital is 1.9112.

This latter value is smaller than the computed gi value

of 1.9u62. For [MoOClu]1-, though the computed gll value
of 1.9963 is not in very good agreement with the experimental
value of 1.965, the experimental observation that gll

> gi is predicted correctly. If gll for [MoOClu]l- is com-
puted neglecting the ligand contribution but including all
the three 281 states, the value is l.926l which is again
lower than gl. This indicates the importance of includ-
ing the ligand contributions as well as the contributions
from bOth the 231 states arising from the occupied orbitals
of bl symmetry. The values of gll for all the chloro

and bromo complexes of chromium and molybdenum given in
Table 11 were computed including the contributions from

2

all the three B states mentioned above. The magnitude

1
of gl,which is primarily determined by the low-lying un-

occupied orbital of e symmetry12 was computed for all the
complexes (Table 11).

For all the tungsten complexes, only the contributions
to 8" from the 281 state arising from the lowest-lying
unoccupied b1 orbital was included, since the other 2Bl
states were found to be rather high in energy. We note
however,that for the tungsten complexes relativistic ef-
fects, which were not included in the present study, may
be important in determining the relative energies of

various excited states. The computed principal components

155

.mme Hmpcoefipoaxo one now who moocopomop one

o

.uHHsHooozu coo once mH och> c ones

 

 

s mHm.o mm.c= om.mm ss.sm s.:s om.os umHmHoooz_
s sso.o co.Hc mH.ms H=.sz mm.mm mc.mm umHmaoozu
swam.o sm.mm mm.sc HHzomoozu

ms mem.e mm.ms me.sm cm.mm mm.ms es.ms MHsHooozi
ewes.e cm.wm em.mm wH.Hm -HHseooza

m mHm.o om.mH ms.OH cs.mm AH mmooooi
s smm.o mo.mm mm.sH Hs.o: NH maooou
sH mHm.o OH.wH cm.OH He.mm MmsHooooi
: mmm.o m=.Hm oH.mH mm.mm :HHsaoooi
m mas.o om.oOHu om.mcu sm.scu oo.msHu om.scHu :mH mHoot
s m:s.o om.m0Ha mo.scu mm.mmu om.wsHu MH.msHu mH mao>i
mm mes.o oo.mm u om.mon mw.:cu om.mcHn sw.mcHt mHsHoo>u
ms aes.o mH.moH- os.cc- mm.os- oo.mmH- sm.ssH- -mHseo>E
n.mom y .me .me .ono .me .ono moonoEoo

o H. __
HIEoAsOHxV a -soH30Hxv a Husofisonv a

 

.IEmcxozu pom H< 0cm __< mo mozam> condosoo

.NH oHQmB

156

of the g tensor of [WOXu]1" CX = F, Cl, Br) are compared
with the corresponding values for the hexacoordinated
species since there are no experimental results available
for the pentacoordinated complexes. For [WOFujl' and
[WOBrujl- the computed 8" value is less than gi, as ob-
served experimentally for IWOFSJZ‘ and [WOBrSJ2'.8 For
[WOCIMJl- the computed gll is less than gl and for
[WOCISJ2' gll is found experimentally to be greater than
Si-

The Fermi contact interaction parameter K in the ex-
pressions for the principal components of the A tensor,
vanishes in the spinerestricted molecular orbital approxi-
mation in which we had carried out all the calculations.

K was determined using the following equation:

A0 = -KP - (ge-g) P,

where A0 is the isotropic hyperfine coupling constant, ge
is the free-electron g value and g the experimental average
g value for the complex. The values of P, which was de-
fined earlier, and of the experimental isotropic hyperfine
coupling constant used in evaluating K are given in Table
10. The computed values of the principal components of

the A tensors for vanadium, chromium and molybdenum com-
plexes are given in Table 12. For the chloro complexes

of chromium and molybdenum, as well as for [MoOBru]1-,

157

the contributions from all the three 281 states to AII
were included. All the computed All values are smaller
than the experimental values, while the Ai values are
larger than the corresponding experimental values. Even
then the experimental trends are reproduced fairly well
(Table 12).

The computed principal components of the g tensor
reproduce the experimental trends very well, even though
the numerical values for some complexes are not as good as

2' the calculated

for others. For [CrOFSJ2' and [M00F5]
relative values of gll and gl are the reverse of the experi-
mental values. This could be due to the inaccuracies in

the geometrical data used in the calculation, since the
magnitude of gi has been shown”1 to depend on the angle

between the metal-oxygen and metal-halogen bonds.

F. Conclusion

We conclude from this study that there are many
similarities in the electronic structure and properties
of the halide complexes of the d1 oxycations of vanadium,
niobium, chromium, molybdenum and tungsten. The one
major difference between the vanadium oxohalide complexes
and those of chromium and molybdenum is that for both the
fluoro and chloro complexes of vanadium, the 281 excited

states arising from the occupied orbitals of b1 symmetry

are much higher in energy than those of the corresponding

158

chromium and molybdenum complexes. This accounts for the
observation that for all vanadyl complexes gll values are
lower than gi. We have also shown that it is again the
rather high energy of the 281 excited states arising from
occupied bl orbitals of fluoro complexes of chromium and
molybdenum that makes the significant differences in the
ESR spectra of these complexes compared to the correspond-
ing chloro and bromo complexes. The importance of includ-
ing the ligand contributions in computing the g tensor
components has been clearly demonstrated. We agree with the

11‘1” that it is not possible

conclusions of Garner et al.
to assume that the two lowest—energy absorption bands in
the electronic spectra are d-d transitions for all the oxo-
halo complexes of vanadium, chromium and molybdenum. In
spite of the numerous limitations of the SCF-MS—Xa method,
the trends in the principal components of the g and A

tensors of this class of complexes have been calculated to
a surprisingly high degree of accuracy. The SCF-MS-Xa
model thus appears to be a useful theoretical model for

systems as large as those studied here, which are not

readily amenable to ab initio studies.

REFERENCES

10.

ll.

12.

13.

14.

150

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39: 236 (1977).

D. A. Case and M. Karplus, Chem. Phys. Lett. 39, 33
(1976); M. Cook and M. Karplus, J. Chem. PhysT‘Zg,

7 (1980).

L. 0. Atovmyan and Z. G. Aliev, Zh. Strukt. Khim. 11,
782 (1970).

G. Z. Pinsker, Sov. Phys. Crystallogr. 11, 639 (1967).
K. Schwarz, Phys. Rev. B5, 2966 (1971).

M. S. Gopinathan, M. A. Whitehead and R. Bogdanovic,
Phys. Rev. A19, 1 (1976).

J. G. Norman, Jr., Mol. Phys. 31, 1191 (1976).
R. E. Watson, Phys. Rev. B, 1108 (1958).

The calculations for v02+, Nb02+, Cr03+, Moo3+ and w03+
were done with the internuclear distance chosen to be the
same as in the halo complexes, and the atomic d values the
same as in Table 2; a valuesfkm‘the intersphere and
outersphere were the average of the atomic a values.

The following sphere radii were used: RV = 2.3509 a.u.

32.

33.

39.

35.

36.

37.

38.

39.

90.

91.

92.

93.

161

2+ 5 . _
R0 = 1.8791 au, for V0 , RNb = 2.2930 au, R0 — 1.82::
au for Nb02+, RCr = 2.119 au, R0 = 1.7776 au for CrO :
3+ _
— = and R - 2.9108
RMo - 2.3253 au, RO 1.7590 au for M00 W

+
au and R0 = 1.7986 au for W03 .

J. M. Flowers, J. C. Hempel, W. E. Hatfield and H. H.
Dearman, J. Chem. Phys. 58, 1979 (1973).

R. A. D. Wentworth and T. S. Piper, J. Chem. Phys.
91, 3889 (1969).

T. R. Ortolano, J. Selbin and S. P. McGlynn, J. Chem.
Phys. 91, 262 (1969).

E. A. Allen, B. J. Brisdon, D. H. Edwards, G. W. A.
Fowles and R. G. Williams, J. Chem. Soc. 9699 (1963).

A. H. Maki and B. R. McGarvey, J. Chem. Phys. 29,
31 (1958).

C. P. Keijzers, H. J. M. deVries and A. van der Avoird,
Inorg. Chem. 11, 1338 (1972).

C. P. Keijzers and E. deBoer, M01. Phys. 29, 1007
(1975).

P. T. Manoharan and H. B. Gray, Inorg. Chem. 5,
823 (1966).

S. Vijaya and P. T. Manoharan, Farad. Soc. Trans. 857
C1979).

K. K. Sunil, J. F. Harrison and M. T. Rogers, un-
published results.

C. D. Garner, I. H. Hillier, F. E. Mabbs, C. Taylor
and M. F. Guest, J. C. S. Dalton, 2258 (1976).

K. K. Sunil and M. T. Rogers, unpublished results.

CHAPTER VII

EPR STUDIES OF [VOFu32-, [MoOFull'

and [Mo001u11'

The transition metal oxohalo complexes of the type
[moxsjn‘, where M = V, Nb, Cr, Mo, w and x = F, Cl, Br
and I, have been the subject of detailed EPR studies.l-2O
The g, metal hyperfine, and ligand hyperfine tensors have
been used to investigate the nature of bonding in these
compounds. On the other hand very little work has been
done on the corresponding pentacoordinated oxohalo com-
plexe321’22, Since EPR spectr0500py provides a very
sensitive probe for the detection and measurement of the
effects of small changes in bonding, we have undertaken
a fairly detailed study of the single-crystal EPR spectra
of [VOFu]2-, [MoOFujl' and [MoOClHJ1' in the hope of
getting a better understanding of the differences in the
bonding between the penta- and hexacoordinated transition
metal oxohalo complexes. 9

The pentacoordinated transition metal complexes form
a class of compounds of considerable interest with the

structures possessing a diversity of forms between the

two limiting symmetries of trigonal bipyramidal (D3h

162

163

22-29 The

symmetry) and square pyramidal (C9v symmetry).
energy barrier between these two structures is pre-
dicted25 to be small for species with five equivalent
ligands, and examples are observed to occur in both sym-
metry classes.26’27 On the other hand, complexes having
an axial ligand different from the other four tend to

29

form square pyramidal complexes. The pentacoordinated

oxohalo complexes under investigation here fall in the
latter category.

21

5 and

28havebeen found to provide penta-

2+ 3+

ESR studies of V0 and Cr in (NHu)28bC1

Fe3+ in (NHu)2SbF5
coordinated species [VOCluj2-, IMoOFujl’ and [FeF5j2-,
respectively, while for Cr3+ in K2SbF5 the species which
predominates29 is trigonally distorted [CrF6J3'. ESR
spectra of [MoOClujl-have been studied in solution31

as well as in a diluted single crystal of [AsPhHJINbOClu].32
We have carried out the ESR studies of the fluoro complexes
of V02+ and M003+ in ammonium pentafluoroantimonate(III)

and the chloro complex of M003+ in ammonium pentachloro-
antimonate(III). The ligand hyperfine interactions have
been observed for the fluoro complexes at room temperature
and for the chloro complex at low temperature. We have

used the results from our earlier SCF—MS-Xd studies on this

class of compounds32 to interpret the observed g and metal

hyperfine interaction tensors.

169

A. Experimental

 

Ammonium pentafluoroantimonate(III) was made by evap-
orating a solution of 3 moles of NHuF and 1 mole of SbF3
in distilled water. Ammonium pentachloroantimonate(III)

was made by evaporation of a solution containing SbCl3

and NHuCl in the molar ratio 3:9 in dilute hydrochloric
acid.3Q The single crystals of (NHu)28bF5 containing

2+

about 1% by weight of V0 were obtained by slow evapora—

tion of a solution of (NHu)2SbF and NHAF in the molar

5
ratio of 1:1 with about 1-2% by weight of VOSOu-7H20.
The single crystals of (NHMZSDF5 containing [MoOFujl'
were made by dissolving (NHu)ZSbF5 and NHuF in water in
mole proportion of 1:1, adding a solution of ammonium
molybdate in hydrofluoric acid reduced with metalic tin
and allowing the solution to evaporate slowly. The single
crystals of (NHulzstlS containing [MoOClujl- were made by
dissolving SbCl3 and NHuCl in dilute hydrochloric acid
in the molar ratio of 3:9, adding a solution of ammonium
molybdate in hydrochloric acid reduced with mettalic
tin and allowing the solution to evaporate slowly.

EPR spectra were recorded for the single crystals
using a Varian E-9 X-band spectrometer. The powder measure-

ments were made using powdered samples of the single

crystals.

165

    
 

Figure 1. Crystal structure of (NHu)ZSbF5 with inter-
nuclear distances given in Angstrom units.

 

166

B. Results

1. Tetrafluoro Complexes of Oxovanadium(IV) and

Oxomolybdenum(V)

The diamagnetic host lattice used in single-crystal
studies of fluoro complexes has ammonium pentafluoroanti-
monate. This forms orthorhombic crystals with each anti-
mony ion at the center of a distorted octahedron in which
five of the vertices are fluoride ions and the sixth is
the sterically-active lone pair associated with trivalent
antimony.31 The details of the structure are shown in
Figure l, where it may be noted that the axial-fluoride-
Sb-lone pair direction is parallel to the b-axis of the
crystal (Sb-Fax = 1.916 A) with the antimony ion displaced
0.382 A from the center of the rectangle formed by the four
axial fluorines (Sb-F = 2.075 A) towards the lone pair. Two
classes of antimony sites related by a center of inversion,
and magnetically equivalent, are defined in this way
(Figure 1).

The vanadyl (V02+) and molybdenyl (M003+) ions can
replace either Sb3+ or [Sb-F]2+ of the [SbFSJ2' ions in
(NHu)2SbF5 to form either the hexa- or the pentacoordinated
complex, or a mixture of both. The ESR study of the single

2+

crystals of both the systems, V0 in (NHu)2SbF and M003+

5
in (NHu)2SbF5, shows intense resonances associated with
only one site of a magnetic species. The angular varia-

tion of the ESR spectra for each system was studied by

167

 

 

Figure 2. Coordinate system for the analysis of g and A
tensors of pentacoordinated transition metal
oxohalo complexes.

168

recording the spectra at room temperature for every 10°
rotation about the crystal a, b and c axes. For both the
systems, the rotation study about the crystal b-axis shows
spectra which are independent of the rotation angle. This
shows that V02+ and Moo3+ replace either Sb3+ or [Sb-Fax]2+

2' ion and that the metal-oxygen bond is either

of the [SbFs]
along the b-axis or directed on the surface of a cone
making a fixed angle to the b-axis. Since the EPR spectra
are independent of the angle in the ac plane one can

further conclude that both the systems are axially sym-

metric.

(i) Tetrafluoro oxovanadateCIV) Ion - The Zeeman and
metal hyperfine interaction tensors are considered to
originate at the metal nuclei and a coordinate system is
chosen with the z-axis along the metal-oxygen bond and the
x and y axes in the equatorial plane formed by the four
halide ligands (Figure 2). The origin for each halogen
hyperfine interaction tensor is the halogen atom and the
coordinate system has the z axis parallel to the metal-
oxygen bond, the x-axis along the metal-halogen bond and
the y axis chosen to form a right—handed coordinate
system (Figure 2).

The electronic Zeeman and metal hyperfine tensors

obey the relationships

169

.m

mbmmHsmzv cH

 

ummsmo>u Hoe o:ch be the :H cHoHo oHeoswos one neHz w Ho coHocHsd> .m oesmHm
3024
a: I n =1
00. cm. CV. ON. 00. . 00 00 O¢ ON 0 .
H H H H _ . _ .mH .q an.
1 V3
1 mm.—
1 0m.. 6
1 had
1 0%.
1 $0..

 

170

IN

mamo>u LCM mcmHQ mm 03» CH Ufimflh 0Humcme mcp SUHB < M0 COHuwHLm> : .

w..oz<

o .. I a .. I
0m. 00. OV. ON. 00. . Om Om OV ON O

- q .4 q - . . . . _ Ch

1 OO.
1 ON.

1 CV.

(moo) v

1 0w.

1 Om.

 

OON
O.N

171

.mmemmHsmzv cH
mmzmo>. How ocmHQ on one CH oHon cauocwwe on» npfiz w mo coHumHHm> .m ogstm

3024

o..I 2....

0m. 8. CV. ON. OO. . 00 00 0V ON 0
H H H H H H H H H ”0..

 

1 Va.

1 mm.

1 mm.

1 50..

1 mm.

 

172

.m

mommflzzzv CH

umHHso>. coo occHd on one cH cHoHe oHoocwcs one eoHs a co coHooHoc> .c ocsmHa

w..oz<

2.: 2.:
co. cc. o: 8. oo. . om cc 8 8 o
1

5!. H H H H . . H H

L

C)
05

ON

8

C)
y :

(ssnoby v

c>
3.

OO.

Om.

 

OON
O_N

173

g = ($300529+g:sin290082¢+8§sin265in2¢)1/2 (1)
and
M _ 2 2 2 2 2 2 2 2 2 2 2 1 2
SA - (Azgzcos 8+AngSin ecos ¢+Aygysin esin o) / ,

(2)
where the angle 8 and ¢ relate the external magnetic field
vector B0 to the z and x axes, respectively (Figure 2).
The ESR spectrum for 8 = 0° corresponds to gz and AZ
while that for e = 90° and ¢ = 0° corresponds to gx
and Ax and that for 6 = 90° and 0 = 90° corresponds to
g and A . In the case of V02+ in (NH4)2SbF5, it was

y Y

found that 8x = g and AX = A ,since the spectra are

y y

angle independent for rotations about the b axis, as

would be expected for an axially symmetric system. The
angular variation of g and AV for rotations about the

crystal a and c axes are given in Figures 3, 9, 5 and 6,
respectively. In the plot of g versus the rotation angle

for rotations about the crystal a and c axes (Figures 3 and
5) there are two maxima separated by about 30°. The values of
Gll’ G1’ All and Al were determined from the measured
magnetic field values for the various mI transitions by

a least-squares fitting procedure using the following

equations, which are correct to second order:

[I(I+l) - mi]

 

(8=O°)

2
= ,1 '7'
BO B(mI) + A||WII A".

2B(mI)

Table 1. Single-crystal ESR parameters of IVOFu]

179

2-

 

Rotation Axis

 

gII g1 All A1
a 1.9318 1.9718 -199.61 -79.36
b 1.9728 -72.95
0 1.9325 1.9792 -198.91 -73.11

 

175

2 + 2 [I(I+l)-m§] 0
B0 = BORI) + Aim]: 4' (All Al) h—Bonl) (6:90 )

 

B0 = hv/BB
The values of gll, gi, All and Al determined in this way
from angular variation studies in ab and b0 planes are
given in Table l,a10ng with the values of gi and A1
determined from b-axis rotation studies. The spin-Hamil-
tonian parameters determined from different planes agree
within eXperimental error.

We have observed the fluorine hyperfine interaction at
room temperature. When the applied magnetic field is
along the crystal b axis each of the vanadium hyperfine
lines is split into five lines with a separation of 10 G
and intensity ratios of approximately l:9:6:9:l. The
fluorine hyperfine interaction was not observed at any
other orientation of the crystal with respect to the ap-
plied magnetic field at room temperature. We have assign-
ed the observed ESR spectra to the species [VOFHJZ-
based on the observed fluorine hyperfine interaction and
the fact that one would have observed ten fluorine hyper-
fine lines had the species been IV0F533‘.1°

The EPR spectra at low temperatures were too complex
to analyze,as we could not follow the angular variations
of the many different sets of vanadium hyperfine lines

that appear at low temperatures.

176

mmommHHmzv cH

IN

.opsumHoQEop Eoop um
mzmo>. no oaaemm oCHHHmpm>Lo>Hoa mo SappooCm mmm

 

 

 

 

 

 

.H ocsaHs

177

 

 

 

 

 

 

 

 

 

 

 

 

Figure 8. Simulated ESR spectrum of polycrystalline sample
2-

v.

a
H ”cabmmHszzv :H
lam @002. How ocmad on 0:» CH oHon oHuocme cqu m mo coHpmHHm> .m ogszm

 

u = I n = I
00. O@_ OV— ON. 00. - 8 CO 0? ON LO .
1 H H H H H H H H me.

00..

. .m..

178

Na. .0

 

1 mm.

 

179

.meeansmzo eH

lawnmoozu pom ocde no on» :H oHon oHuocmmE npfiz m mo COHpmHHm> .oH ogszm

 

M1524
n = I O = I n .. I
00. OO- OV. ON. 00. H ON 00 OV ON My .
7 H H H H H H H H mm.
o
1 ON.
1 _m..
6
1 NO.

1 mm.

 

180

The analysis of the ESR spectra of the powder sample
(Figure 7) gave spin Hamiltonian parameter values close
to those obtained from the single crystal studies. The
powder spectrumvnussimulated using the spin Hamiltonian
parameter values obtained from single crystal studies

(Figure 8).

(ii) Tetrafluoro oxymolybdate(IV) Ion - The molybdenyl
(MoO3+) ion in single crystals of (NHH)ZSbF5 is a system
very similar to that of vo2+ in (NHu)2SbF5. For the
axially symmetric MoO3+ in (NHu)ZSbF5 system, the coor—
dinate axes of Figure 2were used in the analysis. ESR
spectra were recorded for every 10° rotation in the
crystal be, ac and ab planes. Even though fairly well
resolved spectra were obtained at room temperature, the
molybdenum hyperfine lines were observed only at certain
orientations of the crystal with respect to the applied
magnetic field because of the large fluorine hyperfine
interaction.

The variation of g with angle in the crystal be and
ab planes is given in Figures 9 and 10. In the be plane
g remains a constant from 6 = 80° to 9 = 100°. So gll

and gi were determined by fitting the experimental g2

values to the equation

g2 = d + Bcos2e + ysin26

181

.00 u 6 6C6 00

o soc mmnmmHzsz :H

j

 

1H

 

 

mzmoozg mo sappooam mmm

.HH mtsaHm

 

182

00

e ocm com

a .Hom

m

mnmmszzv :H

1H

. 000 .HO
mzmoozg do EztpomCm mmm

 

 

 

. NH mLSEfiL

 

183

Table 2. Single-crystal ESR parameters of [MoOFu]1-.

 

 

 

Rotation Axis gll gi A?I ii
a 1.89U8 1.9253 95.5
b 1.9256
c 1.89u5 1.925“ 95.5

d

Hyperfine coupling constants are in gauss.

18u

where e is the angle the applied magnetic field makes with
the crystal b axis. Using the a, B and y values deter-
mined by the least-squares procedure,g2 values were com-
puted for all angles and were found to have the minimum

at 9 = O°and the maximum at 6 = 90° corresponding to gll
and gi, respectively. The same procedure was carried

out for data in the ab plane and the minimum and maximum

2 were found to occur again at e = G’and e = 90°.

in g
The gll and gi values so obtained from rotation studies
in the two different planes agree within experimental
error and are given in Table 2. The g value for rotation .
about the crystal b axis was found to be invariant to the
rotation angle and is equal to the Si value determined
from the bc and ab planes (Table 2). For 9 = O°the molyb-
denum hyperfine interaction could be measured in both the
ab and be planes and was found to be 95.5°. This value
was assigned to All. The Al value could not be measured
from the single-crystal studies as very intense fluorine
hyperfine lines mask the relatively weak molybdenum hyper-
fine lines.

For 6 = O°and ¢ = 03 no fluorine hyperfine inter-
action was observed (Figure ll) thus indicating that
AZ (19F) was smaller than the linewidth of the spectrum.
For 6 = 90°and ¢ = U’or 90°the fluorine hyperfine struc-
ture on the molybdenum I = 0 line consists of a nine

line pattern (Figure 12). For the case Ax (19F) # AV

ClgF) # 0 one expects nine fluorine hyperfine lines with

 

 

 

 

 

 

 

 

 

Figure 13. ESR spectrum of polycrystalline sample of
[MoOFu]l- in (NHu)2SbF5 at 77° K.

186

.Eopummcn CH co>fiw
mpmaho
m a mAzmzv no manpoSMQm H
Ammaoscpmpcfi spa: Henm
mocmumao

.u .
.42 ®

am AU
ON.Q: a Q

 

 

 

 

 

  

.:H whamfim

187,

relative intensity ratios l:2:l:2:U:2:l:2:l,which is ap-
proximately what was observed (Figure l2). From these
data Ax (19F) and Ay (19F) were assigned the values

-15 G and 55 G, respectively.

The ESR spectrum of the powder sample (Figure 13) is
not well enough resolved to do a complete analvsis. The
spectrum could not be analyzed by simulation as the simu-
lation program available does not properly take into

account the ligand hyperfine interaction.

2. Tetrachlorooxomolybdate(V) Ion

Ammonium pentachloroantimonate forms monoclinic
crystals with the b axis coinciding with the needle axis.
Each antimony is at the center of an approximately octa-
hedral configuration of ligandsixiwhich five vertices are
occupied by chloride ions and one vertex is occupied by
the lone pair of electrons associated with antimony in
a 3+ oxidation state. Antimony ions, surrounded by four
chlorine ligands (so-Cli = 2.62 K i = l—u), lie in sheets
with the fifth chloride (Sb—015 = 2.36 3) either above
or below the sheet as shown in Figure l“. Antimony
sites are equivalent, with the b axis parallel to the
longer side of the rectangle formed by four chloride
ligands and the Sb-Cl5 axis parallel to an axis “0°

from a in the ac plane. If no distortion occurs upon

substitution of paramagnetic ions into antimony sites,

188

 

 

.maoommHemzv ca
Iamzaooozg mom onwaa $05 on» CH cflofim afipocwms npfiz w mo coaumfipm> .mH opswflm

 

U =I
.x 2.1

00. ow. 03 ON. OO. 00 8 Ce. ON 0 .
H H H H H _ H, H H H .u¢5w_
i 000..

o
1 com.—

o
o

4 0mm.

 

 

 

189

.maopmmfizmzv ca
Iamzaooozu Low ocmfia *on mnp :H oaoflm oaumcmme spa: m mo coflumfipm> .wa opsmflh

n =1 so?
00. OO. O! ON. OO. 00 OO OV ON 0
H H H H H H H H H H 8a..

Ohm;

 

OOQ.

 

190

 

1H

Om.
H

 

.mHoommHemzv :H
mzaooozm pom ocean pm on» CH caofim ofipocwms 39H: a mo coaumflpm>

a maozq
oo. o: 8. 8.2.1 om om or 8

c — q u .4 u _ - A

 

.NH opzmwm

6::

0
8m.-

Ohm;

 

1 00¢.

 

191

.maoommHesz ca
Ifimzaooozg pom ocmad no map :a UHon afluocwme spa: < no coaumfipm> .mH ossmfim

Haze. a __ I
8. om. ov. om. oo. .2: co co 2. cm 0 .
H H H H H . H _ H H .unyu

 

.uAvv

 

nVan

 

4060

(SU‘V

192

e .mfioommHemzv :H
lam Hooozg pom ocmaa *on on» Ca onHm oapmcwwa Sufi: < mo :oHpmHLw> .mH opzmflm

 

3oz< ao ._ I
om. oo. 9.. cm. 8. a; 8 cm 8. ow o .
1 _ _ _ u q q - q u 0 ON

ode

(9H1

0.00

 

 

i 0.00

193

.mHoommHemzv ca
:Hmeaooozu soc wooed *oo one :H oaoac oaoocwhs crHs < do coaooaho> .om magmas

 

wJGZ< 621
cm. oo. o: om. ooze: om 8 9 8 o.
H, H H H H . H H H H Aunyw
1.0.3
1 0.8
40.8

 

(5))‘7

194

Table 3. Single crystal ESR parameters of [MoOClu]l-.

 

gxx = l.9u6l

All = 83.190:
gyy = l.9u7u

A = .7 G

1 37 5
g = 1.9650

ZZ

 

195

the sites should be indistinguishable by ESR, as was

observed experimentally.
ESR spectra were recorded every 10° for rotation of

the single crystal about the a, b and c* axes. The spectra

in all the crystal planes indicated the presence of only

one magnetic site. ESR spectra at all the crystal orienta—
tions with respect to the applied magnetic field consisted

of a central intense line corresponding to the molybdenum

I = 0 nucleus and three relatively weak hyperfine lines

on either side of the central line arising from M095’97

2

(I = 5/2). The measured g valuesimieach planewere least-

squares fitted to the equation

g2 = d + Bcos26 + ysin26

2

and the maximum and minimum g values were determined.

The g2 tensor was diagonalized using Schonland's method
to obtain the principal components of the g tensor (Table

3). The gxx and g values determined by Schonland's

yy
method are 1.9961 and 1.997“, respectively. Since the

differences between gXX and g are small, M003+ in

yy
(NHu)28b015 is to a first approximation, an axially sym-
metric system. The angular variation of g and A in the

crystal bc*, ac* and ab planes is given in Figures 15, 16, l7,

18, 19 and 20. The g value in the ac plane has a maximum

at 40° from the a axis corresponding to gll indicating

196

that the Mo-O axis is oriented along Sb—Cl5 axis. The All
and Al values were determined by fitting the measured g2A2
values from the ac* plane to the equation

g2A2 = a + 800820 + ysin26

and are given in Table 3.

The ligand hyperfine interaction even though observed
at low temperature could not be analyzed, as all the lines
were not well resolved. According to Boorman gt 11.32
the gav values of [MoOClujl-, [M0001u(H20)]l- and [M0001512'
are 1.951, 1.9“? and l.9u0, respectively. Our measured gav
value of 1.9528 indicates that the magnetic species is

[1400011,]1 in the present case.

C. Discussion

The interpretation of the ESR spin Hamiltonian
parameters of transition metal oxohalo complexes of the
type [MOanm‘, where M = v, Nb, Cr, Mo, w and x = F, Cl,
Br, I (n = u or 5) if; generally based on the discussion
of the electronic structure of vanadyl complexes by Ball-
hausen and Gray33, and similar studies on chromyl and
molybdenyl complexes by Gray gt al.3u’35, all based on
extended Hfickel calculations. For deriving expressions

relating the spin-Hamiltonian parameters to the molecular

197

orbitals of the system, it was generally assumed that the
complexes have Cuv symmetry and that the unpaired electrons
are in orbitals of b2 symmetry. The molecular orbitals

necessary for the discussion are then written as
IB2> = B2ldxy> H B2l®b2>

IBl> = 8lldx2-y2> + Bl|¢bl>

|E> = eld - e'|¢ 0

>
xz’dyz ex’ ey ’

where the ligand orbitals ¢ are group orbitals of approp-
riate symmetry. These molecular orbitals are used to
derive expressions for gll, gi, All and A1 using the
standard second-order perturbation theory treatment of

Abragam and Pryce39. For the transition metal oxohalo

6

complexes, DeArmond gt 3;. have derived the required

expressions. The expressions for gll, gi, All and A1

are functions of metal and ligand spin-orbit coupling
constants and <r-3> values,in addition to the molecular
orbital coefficients. It is customary in using these
expressionstxasolve for the molecular orbital coefficients
using experimental values of spin-Hamiltonian parameters
and assumed values for spin-orbit coupling constants and
<r-3> values. The molecular orbital coefficients so

obtained depend critically on the choice of values for

198

 

 

 

.wH mocopomom Eopmm
mm:m.o Hom.o mHsm.o omm.o mmmm.o mmzm.o o
esms.o ems.o ohms.o Ham.o amos.e maem.o Hm
memm.o moa.o :HHa.o omm.e mmom.o aHme.e we
oxlmslmom a.uaxm oXImSImom m.paxm dxlmzlmom oxumzlmom
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200

spin-orbit coupling constants and <r-3> values. In addi-
tion to this it was found necessary to include contribu-
tions from occupied orbitals of bl symmetry to account for
the observed g tensor components of chloro and bromo
complexes of chromium and molybdenum (Chapter VI). It
becomes practically impossible, without making too many
assumptions, to solve for the molecular orbital coeffic-
ients from the expressions forgr-and A-tensor components
if the contributions from occupied orbitals are included.
The derivation of the expressions for spin Hamiltonian
parameters and the methodsused.in the computations, are
discussed in Chapters V and VI. The molecular orbital co-
efficients 82, 81 and a computed from experimentalg- and
A tensor components of vanadyl and molybdenyl complexes
are given along with those obtained from SCF-MS-Xa cal-
culations for comparison in Tables 4 and 5. The molecular
orbital coefficients estimated from experimental data are
larger than the values obtained from the SCF-MS-Xa

method. The coefficient of the metal dX orbital 82

y
in the molecular orbital containing the unpaired electron

is larger for the hexacoordinated vanadium complexes than
for the pentacoordinated vanadium complexes and the
reverse order is observed for molybdenum complexes (Tables
U and 5). Another interesting observation is that 81

is larger for vanadium complexes than for molybdenum

complexes.

201

Table 6. Spin Hamiltonian parameters for vanadyl complexes.

 

 

 

a a
Complex gH gi All AJ- RGf.
[V0(H20)5]2+ 1.9331 1.9813 182.8 72.0 33
[voFuj2' 1.932 1.973 182.0 66.7 This work
[VOClu12- 1.9A78 1.9793 168.8 62.8 21
[v08533‘ 1.937 1.977 178.5 6u.05 15
[V001513‘ 1.9450 1.98u7 173.0 63.8 6
u -1

aHyperfine coupling constants are given in 10' cm

202

 

 

Table 7. Spin Hamiltonian parameters for molybdenyl
complexes.
gH gi All Al REf.
[MoOFujl- 1.895 1.925 85.38 This work
[MoOC1ujl- 1.9650 l.9u68 75.85 3u.u2 This work
[MoOFSJ2- 1.87u 1.911 92.93 u5.l3 16
[Mo001532' 1.9632 l.9uo 7u.7 32.6 16

 

 

 

203

For all vanadium oxohalo complexes gll is less than
gi while gll is greater than gi for the chloro complexes of
molybdenum. There have been two proposals to account for
this observation. Kon and Sharpless were of the opinion
that gll was greater than gl because for chloro complexes
there is more than one 2Bl state which makes contribution
to gll while Manoharan and Rogers proposed that it was
the large chlorine spin-orbit coupling constant that
caused this reversal of the relative magnitudes of gll

and gi. From our SCF—MS-Xd studies we conclude that the
chloro complexes of molybdenum have more than one 2Bl
state that contributes to the $11 value, while for vanadium
complexes only one 281 state arising from a low-lying
virtual orbital of bl symmetry was observed. It was

found necessary to take into account the ligand contribu-
tions to the g-tensor components to explain the observed
trends. A detailed discussion of these factors is given

in Chapter VI.

There is only a very slight difference between the
spin-Hamiltonian parameters of penta— and hexacoordinated
vanadium complexes. The A-tensor components of the tetra-
fluorooxovanadium complex are larger than those of the
pentafluorooxovanadium complex while for the chloro com—
plexes the reverse is observed (Tables 6 and 7). For

molybdenum complexes the g-tensor components are larger

for the penta- than for the hexacoordinated complexes.

 

REFERENCES

10.

11.

12.

13.

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APPENDIX A

207

 

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