, . V & Rafi .3 . . . . .4 , . V , sit
a Vm. . 1. . V. , . : V, V as

, , V . t 2 figfifihfi‘
.. V . . V. . +2 «5..» g VV
. . 2&4: LET» has, 2

”w? . ,. . : . an?
.

:35 L 3 .. .
. V ., , a? fiwufihmnfiufix . . .V :3
. ...4 .. , .... , ... .V
a V . .eafimfix a . a , 3k ”9%. .sz V. Vamqazdwié . V
v . . Vu,Wow-§fiytflrfigm..wm hr . . . ..,. V V «$.15 3%.! . . . .5.ka 1. 3
Mn: 1... . . . . fiaivmwfinm.¥m. um V .. VV . . : ,V . V ., 2 V .
11 dot 1...: , .. . V V . , 9 u 1t.¢v:..%t V V
(L017. .95”... V . ‘ . . . . , . . in» r .
. v . , ‘ (3.3. 1...

. .. » afi$Ttiai
. ... %T#;.L+
. 45...».

0
1. ,
:5;

Stir}... 211:.
1.31.71.13.97?!
r i t v3.

v!-

~z
E.
;
x

..7 .

‘3
I
4

Bfifificnygiix
ii. :wknflzu
Sin:

«£2.05,

N .1,
(
' 1:33;: .
: 7-,“;
+2135:

, , . P V . V
V . . i If ..L.,~.r..ll\ (\r v ,
x:%i xy.1.2¢3£hi.§ iii: - V V, .V
Nut. 2.; 1M... ..x.u§h..ifiwu.un,flrxl_Luna $3.5:
:2, .lhrimwg . 2.3% a .21“ , ‘ L3..VJ:..m.£u3m..{SVs
1t . . l . 11.3,)». . .?1k .
g. Ti, V} {a . t. .>fi3!fir:rf.§2 .
y . , V A
;....v.1:.« . . .
.. .. r l . .. . .
6.... Wenfmwfi V . . : . . . . , . . . ,
. , . V . . : gflu‘
., . v V V: V a 5
. . .. . , .. . 1 £011??wa
V V .YIVIL..L:Y.I . 111
Ll is. EYES

7.715.! ,
Lil-{4L} It}; V. 2- t¥?.lx\.l

LlLLéti .7.
.1!§}v&wetvtw 1A . ..
,§lt.t._{.¥r11§ . A1311“.
5 it LL‘ Fr! . 1..

4‘1; , .Nrfl
11% .

11.91.11 £41
Etéiléxg
. . V . I 53.41%,71 . ‘11:}!
:tarfinrtaz. it. A} :«hfiaflit. .EAti uh.
A . . LA xx): - Li , V
.fifianfiutfilzfinwfirt . i. pig?
,1 1:51.! 39%;: 1 Kt? .
. . I LIAfiWNLLCK .flnlsé. V
. . . .53... .TIIEL (Luxg‘
8: V l 71.12 J .1le
Ai; ?E‘..1H.L§1 .
in? Lil .CL‘A Quit! . .
£25.14. (EV €5.31
irlelt. Wu LL..$.~:1,€.£.$tjgfl+msflvfifixiéu
L... .732: ,_ E‘L..x.e.1 ....I.t 1.9?!
Whflr.$WWYT?£F.BL figidflwwi. nfifinhgihfitttx
\ . “HwTCotLELkA-aufinrflfl 1... An. ‘71:!)
‘ x.‘ . . #1 “11:71.?
‘ .1. A2 LII-03.1?
{Vacanmi thfiuflkif:&
£J1!xwfi§v§~k.fi.1'girlifl. . 974. .u)

in .IP. 3:311.

 

 

Eagan. . . .z . . _

A I?! E 31.4.) If)? .v( .3. ,
.‘éu fi>.vp..90r§{!.).)¥~>: (3‘.

.. LL.1.3..:DI§ a

 

 

   
 

W llHI“illllHlllHIWllllIlllHlllHHIHIHHHIHIHHI
3 0063 4967

1293 1

 

This is to certify that the

thesis entitled
ELECTRON SPIN RESONANCE STUDIES OF COPOLYMER—
SUPPORTED CHLORO—CYCLOPENTADIENYL—NIOBIUM(IV)
COMPOUNDS, AN EFFICIENT POWDER PATTERN
SIMULATION TECHNIQUE, DERIVATION OF EQUATIONS,
AND THEIR USE IN A MOLECULAR ORBITAL STUDY
presented by

Neal Harold Kilmer

has been accepted towards fulfillment
of the requirements for

Ph .D. deg”? in Chemistry

Major professor

Date w 9

0—7639

LIBRAlx r

Michigan SLAB:
' . University

OVERDUE FINES ARE 25¢ PER DAY
PER ITEM

Return to book drop to remove
this checkout from your record.

 

 

 

 

 

 

ELECTRON SPIN RESONANCE STUDIES OF COPOLYMER-SUPPORTED
CHLORO-CYCLOPENTADIENYL-NIOBIUM(IV) COMPOUNDS,
AN EFFICIENT POWDER PATTERN SIMULATION TECHNIQUE,
DERIVATION OF EQUATIONS, AND THEIR USE
IN A MOLECULAR ORBITAL STUDY

By

Neal Harold Kilmer

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1978

 

 

 

ELECTRON SPIN RESONANCE STUDIES OF COPOLYMER-SUPPORTED
CHLORO-CYCLOPENTADIENYL-NIOBIUM(IV) COMPOUNDS,
AN EFFICIENT POWDER PATTERN SIMULATION TECHNIQUE,
DERIVATION OF EQUATIONS, AND THEIR USE
IN A MOLECULAR ORBITAL STUDY

By

Neal Harold Kilmer

AN ABSTRACT OF A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Chemistry

1978

 

/\l\.l.: [‘1 .

 

,r t ‘V “v -r'- U"
m f 81""? i {C}

ABSTRACT
ELECTRON SPIN RESONANCE STUDIES OF COPOLYMER-SUPPORTED
CHLORO-CYCLOPENTADIENYL-NIOBIUM(IV) COMPOUNDS,
AN EFFICIENT POWDER PATTERN SIMULATION TECHNIQUE,
DERIVATION OF EQUATIONS, AND THEIR USE
IN A MOLECULAR ORBITAL STUDY
By

Neal Harold Kilmer

Experimental electron spin resonance (ESR) spectra of
trichloro(n5-cyclopentadienyl)niobium attached to styrene-
divinylbenzene copolymer (20% cross-linked) (I) at ~126°
and dichlorobis(ns-cyclopentadienyl)niobium attached to
styrene-divinylbenzene copolymer (20% cross-linked) (II)
at -l26° have been obtained.

Two computing packages, ESRSIM-FAST and ESRMAP, GATHER,
and PEAKS have been written to enable simulation of ESR
powder pattern spectra of systems having total electron spin
of 1/2 (at any given paramagnetic site) and hyperfine split-
ting by one nucleus and to enable calculation of magnetic
resonance field values of simulated features that can be com-
pared readily with corresponding features in experimental
spectra. A special interpolation procedure for alloting
simulated ESR absorption is used in program GATHER, which
enables "gathering" magnetic resonance field values calcu~
lated by program ESRMAP and production of a simulated ESR

spectrum. Execution of either computing package produces a

 

 

Neal Harold Kilmer

simulated spectrum having an excellent signal-to-noise ratio
at a low computing cost. These computing packages have been
used to help determine values of spin Hamiltonian parameters

of I and II. The ESR spectrum of I at -126° can be described

approximately by the spin Hamiltonian

A A A

it = ‘ -
Bengxbex + gyyHySy + zszSz]

0“

+ TXXIXSX + TnyySy + TZZIZSZ
= i 0 = O t O = O
with gxx 1.981 0 002, gyy l 990 0 002, gZZ l 908
e 0.002, Txx = -0.0105 2 0.0001 cm", Tyy = ~0.0093 :
0.0001 cm", and T22 = -0.021u : 0.0001 cm”. (The x and y
patterns might be interchanged.) The ESR spectrum of II at

-126° can be described approximately by that spin Hamiltonian

with gxx = 1.976 t 0.002, gyy = 1.952 t 0.002, Txx = -0.0106

2 0.0001 cm", and Tyy = -0.0159 2 0.0001 cm’l. Experimental
data for II do not seem adequate for a confident determina-
tion of gzz and Tzz (for which values of 2.017 and -0.00528
cm"), respectively, have been used in producing simulated
spectra).

Equations relating spin Hamiltonian parameters to ex-
pressions containing molecular orbital coefficients have
been derived. These equations are exact if the molecular

orbital containing the unpaired electron consists entirely

of d orbitals of one atom.

 

 

Neal Harold Kilmer

The unpaired electron of I is assumed to be in a molecu-
lar orbital composed chiefly of niobium hdxy or/and "dxz-yz
atomic orbitals, and other orbitals are assumed to be mixed
in by spin-orbit perturbation. These assumptions, derived
equations, and the spin Hamiltonian parameters determined
for I allow the Kramers doublet containing the unpaired

electron in the ground state of I to be described approxi-

mately by
w *‘(a w + a v )8 + (b v + b v )a
Q l dxy 2 dx2_y2 l dxz 2 dyz
and
xy x - xz yz

or by the same expressions with xy and xz-y2 as well as xz
and yz interchanged. If the absolute values of a1, a2, bl,
and b2 are 0.986, 0.00, 0.133, and 0.019, respectively, and
are used with values of 0.0135 cm" and 1.0a for the hyper-
fine parameters P and K, respectively, then the calculated
value of each spin Hamiltonian parameter differs from the
experimental value by less than twice the previously-
estimated uncertainty. The s orbital character (which could
include both 8 orbital contribution to the orbitals contain-
ing the unpaired electron and spin polarization of electrons

in filled orbitals that consist at least partly of Nb 3 or-

bitals) in I has been calculated to be 6.2%.

 

 

 

 

 

To Mom

 

 

11

 

 

ACKNOWLEDGMENTS

I would like to thank Dr. Carl H. Brubaker, Jr., my
research preceptor, for his patience and guidance and his
assistance in proofreading and revising the manuscript.

I also would like to thank Dr. James F. Harrison for
helpful discussions and for making suggestions as the second
critical reader of the manuscript.

I am grateful to Dr. Chak-po Lau for providing samples
of compounds vital to this research project.

I also am grateful to Michigan State University for use
of its computer facilities and services.

I appreciate partial financial support from the Depart-_
ment of Chemistry at Michigan State University.

I would like to thank Dr. Thomas V. Atkinson and Timothy

 

G. Kelly for their help in printing Appendices A through E.
I also would like to thank Dr. Donald L. Ward for in-
structions on producing Figure 18.

I appreciate the conscientious Job of typing done by

Mrs. Nancy Bennis.

I would like to give special thanks to my mother, Mrs.

Luella Sharp Kilmer, for her love, support, and encourage-

ment .

iii

TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . vii
LIST OF SELECTED SYMBOLS . . . . . . . . . . . . . ix
CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . l

The General Hamiltonian . . . . . . . . 2
The ESR Effect . . . . . . . . . . . . 6
Powder Pattern ESR Spectra . . . . . . . . . 9
ESR Powder Pattern Lineshapes . . . . . . . . 12
Extraction of Spin Hamiltonian Parameter rs

from ESR Spectra . . . . . . . . . . . . 15
Computer Simulation of ESR Spectra . . . . . 18
Molecular Orbital Methods . . . . . . 25
Molecular Orbital Studies of szMLZ and

Similar Compounds . . . . . . . . . . . . 30

CHAPTER 2. EXPERIMENTAL ESR SPECTRA . . . . . . . 35
CHAPTER 3. COMPUTER METHODS . . . . . . . . . . . AA

Calculating Experimental Magnetic Field

Values . . . . . . . . . . . . . . . . . . AA
Initial Estimation of Spin Hamiltonian

Parameters for Copolymer—Attached

CprCl3 . . . . . . . . . . . . . . . . . . Us
Computer Simulation of ESR Powder Pattern

Spectra . . . . . . . . . . . . . A7
Early Versions of ESRSIM . . . . . . . . . A7
ESRSIMeFAST and PEAKS . . . . . . . . . . . . 50
_Programs ESRMAP and GATHER . . . . .'. . . . 51
Using a Distribution of Values for One

Parameter . . . . . . . . . . . . . . . . . 56
Numerical Solution of Equations . . . . . . . 56
Program M. 0. DATA SORT . . . . . . . . . . . 57

CHAPTER A. ESR SPECTRA SIMULATION RESULTS . . . . 59

Experimental vs. Simulated ESR Spectra --

Copolymer-Attached CprCl3 . . . . 79
Experimental vs. Simulated ESR Spectra -- .
Copolymer-Attached Cp2Nb012 . . . . . . . . 80-

iv

 

 

 

Page
CHAPTER 5. MOLECULAR ORBITAL TREATMENT
I. DERIVATION OF EQUATIONS . . . . . . . 81
CHAPTER 6. MOLECULAR ORBITAL TREATMENT
II. RESULTS FOR COPOLYMER—ATTACHED CprCl 102

3 . .
Z Axis, Zeroth-Order Ground State, and
Identification of Z Pattern of Copolymer-
Attached CprCl3 . . . . . . . . . . . . . 102
Determination of Molecular Orbital
Coefficients of Copolymer-Attached
CprCl3 . . . . . . 107
Calculation of s Orbital Character . . . . . 116

CHAPTER 7. SUMMARY AND CONCLUSIONS . . . . . . . 117
APPENDIX A. LISTING OF PROGRAM MAGFLD . . . . . 120
APPENDIX B. LISTING OF ESRSIM-FAST . . . . . . . 121

APPENDIX C. LISTING OF PROGRAMS ESRMAP, GATHER,
AND PEAKS C O O O O O O O C O O C O I O I C O Q lu3

APPENDIX D. LISTING OF SUBROUTINES EQN AND

KILMER O O O C O O O O O O O O O O O O O I I O 160
APPENDIX E. LISTING OF PROGRAM M. 0. DATA
SORT C . C O O O . . 0 . I Q C O O O O C . 165
APPENDIX F. COMPARISON OF ESRMAP, GATHER, AND
PEAKS WITH SIMlAA . . . . . . . . . . . . . . 167
Discussion of Efficiency in Computer
Programs . . . . . . . . . . . . . . . . . 182
Conclusions . . . . . . . . . . . . . . . . 183

APPENDIX G. PLOTTING SIMULATED DATA . . . . . . 185

BIBLIOGRAPHY I . Q C O O O O O Q I O O . Q C O I 187

 

LIST OF TABLES

Table Page

1.

Angular Factors a for Real Atomic d Orbitals
(from Morton and Preston23) . . . . . . . . . . 30

Spin Hamiltonian Parameters Used in Simulating
ESR SpeCtra O O O O O O O 0 O O O O O O O O O O 61

Intermediate Solutions for Molecular Orbital
Coefficients, K, and P of Copolymer-Attached
CprCl3 o o o o o o o o o o o o o o o o o o o o 113

Spin Hamiltonian Parameters of Copolymer-
AttaChed CprClB o o o o o o o o o o o o o o o 115

Final Solutions for Molecular Orbital
Coefficients, K, and P of Copolymer-Attached
CprC13 O O O O O O 0 O O O O O O O O 0 O O O O 115

Spin Hamiltonian Parameters Used in Calculating
Simulated ESR Spectra . . . . . . . . . . . . . 169

vi

 

LIST OF FIGURES

Figure
1. Definition of the Angles 9, ¢E’ and ¢ . . . . .
2. A Typical ESR Powder Pattern (Simulated) . . .
3. Experimental ESR Spectrum of Copolymer-Attached
CprCl3 0 O O O O O O O O O O 0 O O O O O O O O
A. Experimental ESR Spectrum of Copolymer-Attached
CprCl3 O O O O O 0 O 0 0 O O O O O 0 O O O O O
5. Experimental ESR Spectrum of Copolymer-Attached
Cp2Nb012 O 0 O O O O O 0 O O O O O O O 0 O 0 0
6. Example of Area Apportioning Performed by
Program GATHER . . . . . . . . . . . . . . .
7. Simulated ESR Spectrum of Copolymer-Attached
CprCl3 -- ESRSIM-FAST Data 0 o o o c o o o o o
8. Simulated ESR Spectrum of Copolymer-Attached
CprCl3 -"" ESRSIM‘FAST Data 0 o o o o o o o o
9. Simulated Unbroadened ESR Spectrum of
COPOlymer-AttaChed CprCl3 o o o o c o c o o o
10. Simulated ESR Spectrum of Copolymer-Attached
'11. Simulated ESR Spectrum of Cepolymer-Attached
CprCl3 -- ESRMAP and GATHER Data . . . . . . .
12. Simulated ESR Spectrum of Copolymer-Attached
szNbCl2 "" ESRSIM‘FAST Data 0 o o o o o o o o
13. Simulated Unbroadened ESR Spectrum of
Copolymer-Attached Cp2NbC12 . . . . . . . . . .
1A. Simulated ESR Spectrum of C0polymer-Attached
Cp2NbC12 -- ESRMAP and GATHER Data . . . . . .
15. Experimental and Simulated ESR Spectra of

Copolymer-Attached CprCl3 . . . . . . . . . .

vii

Page

10
1A

39
no
' Al
54
6A
65
66
67
68
59
70
71

73

 

viii

Figure . Page
16. Experimental and Simulated ESR Spectra of
Copolymer-Attached CprCl3 . . . . . . . . . . 75
17. Experimental and Simulated ESR Spectra of
Copolymer-Attached Cp2NbCl2 . . . . . . . . . 76
18. Copolymer-Attached CprCl3 or the Hypothetical
MeCprCl3 Molecule . . . . . . . . . . . . . . 10A
19. Attempt To Simulate ESR Spectrum of Copolymer-
20. Simulated I = 7/2 ESR Spectrum -- SIMlAA
Data 0 O I O O O O O I O O O O O O O O O O O O 173
21. Simulated I = 7/2 ESR Spectrum -- ESRMAP and
GATHER Data 0 O O 0 O O O 0 O O O O 0 O O O O 175
22. Simulated ESR Spectrum of Copolymer-Attached
CprCl3 -- SIMlAA Data . . . . . . . . . . . . 176
23. Simulated ESR Spectrum of Copolymer-Attached
CprCl3 -- SIMlL’A Data 0 o o o o o o o o o o o 177
2A. Simulated ESR Spectrum of Copolymer-Attached
25. Simulated ESR Spectrum of Copolymer-Attached

CprCl3 -- SIMluA Data 0 o o o o c o o o o o o 180

 

LIST OF SELECTED SYMBOLS

A.

Ax, Ay, Az (as used in
Appendix F)

A1, A2, A3

a, axy, ax2_y2, axz, ayz,
azz’ 31’ a2

AH

A12: A13’ A23

b, bxy, bx2-y2, bxz, byz,
bzz’ b1, b2

C

C, ny, ex2_y2, sz, Cyz,
2
OZ

CD
2
d, d1, d2, d3, (1“, (15

e (in Equations 1—39 and
1-Ao)

e (elsewhere without
exponent)

Isotropic hyperfine interaction
or A (e,¢ ), the effective hyper-
fine coupIing (angularly depen-
dent)

Hyperfine interaction tensor

Principal values of hyperfine
interaction tensor in Gauss

Principal values of hyperfine
interaction tensor (These equal
the corresponding values of the
T tensor multiplied by hc.)

Molecular orbital coefficients

Vector potential for magnetic
field

Defined by Equation l-3A

Molecular orbital coefficients

Speed of light in vacuum

Molecular orbital coefficients

Cyclopentadienyl
Fine structure interaction tensor

Molecular orbital coefficients

Molecular orbital coefficient

Charge of electron

ix

 

 

 

 

*i—7

e (with exponent)

.E

exy’iexz-yz: exz: eyz: ez2

exp[...]-
F(H')

g

5.
EN
gxxs Syy: gzz: 812 g2: g3

(gzz)N

$12: 813: 823

H

H or H'

III-1

:9

X

Base of natural logarithms
(2.71828...)

Electric field in which elec-
tron moves

Molecular orbital coefficients

Same as e[°"J
Line-shape function

Isotropic g-factor or g(6,¢ ,
effective g-factor (angularIy
dependent) for electron

Electron spin gyromagnetic tensor
Nuclear g factor

Principal values of electronic 5
tensor

A principal value of nuclear g
tensor

Defined by Equation 1-35

Independent variable correspond-
ing to local magnetic field
strength

Observed magnetic field strength
(i.e., lgl)

Constant external applied magnet-
ic field ,

Hamiltonian or spin Hamiltonian
operator

Planck constant
h/(2w)

Components of magnetic field or
magnetic field strengths at which
given resonance conditions are
fulfilled exactly

Smallest of Hx’ H , and H2

y
Middle value of Hx, Hy, and HZ

____J

 

 

 

 

 

 

 

 

H

3

12’ H13, H

H 23

H) H>H

H)

i (when not used as a sub-
script)

k, k' (when not used as a
subscript)

K

K(k) or K(k')

1

i

1,, iy, 12
i4-

1

xi
Largest of Hx’ Hy, and H2

Magnetic field strengths at
which extra singularities are
observed

Hx + iHy

Hx - iHy

The magnetic field strength at
which a given resonance condition
is fulfilled exactly

Nuclear spin

Nuclear spin operator

Nuclear spin operator for nucleus
1 .

Components of nuclear spin opera-
tor

Nuclear spin "raising operator"
Nuclear spin "lowering operator"
A square root of -1

Functions of H, H1, H2, and H3
defined by Equations l-23 and
1-26

Degrees Kelvin

Complete elliptic integral of the
first kind

orbital angular momentum quantum
number

Angular momentum operator for one
electron

Components of angular momentum
operator for electron

Angular momentum "raising opera-
tor" for electron

Angular momentum "lowering opera-
tor" for electron

 

 

 

 

 

 

 

M or m
s 3

Me
P!
pmdm

P0, P1, P

A

S

IO

Q1

.. 3 _ 3
> < >
(1" 01" I‘ av

xii

Peak-to-peak Gaussian line width
Mass of electron

Nuclear magnetic quantum number
Orbital magnetic quantum number
of an electron in an atomic
orbital (~1 é,m1 ; 1)

Electron magnetic quantum number
of one or more electrons having
total spin S (-S é Ms ; 3)
Methyl

AnisotrOpic hyperfine parameter

Defined by Equation 1—12

Metal ion d-orbital population in
the molecular orbital Y

Molecular orbital parameters

Linear momentum operator for
electron

Nuclear quadrupole moment tensor

Nuclear quadrupole coupling param-
eter of nucleus of transition
metal atom

Distance between the unpaired
electron and the nucleus that de-
fines the natural center

Average (mean) of the reciprocal
of the cube of the distance be-
tween the unpaired electron and
‘the nucleus of the central metal
atom

Distance between nucleus 1 and
electron k

Distance vector between nucleus
1 and electron k

Total electron spin

Electron spin-momentum operator

 

 

 

 

 

i
I
l
l
J.

 

 

xiii

3 Components of electron spin-
momentum operator

Electron spin-momentum "raising
Operator“

§ Electron spin-momentum "lowering
operator"

S (H) Kneubfihl function

T Tensor describing the hyperfine
interaction that involves the
nucleus of the central metal atom

T Principal values of hyperfine

interaction tensor in cm‘

Txx’ yy’ Tzz

x, y, and z (subscripts) These relate to the x, y, and z
axes, which are defined as a set
of axes that have a mutual origin
at the nucleus of the central
metal atom and that diagonalize
the g_and hyperfine tensors.

 

Spherical harmonic associated with
l quantum numbers 1 and ml

 

1m

Electron spin eigenvector for i
electron spin of +1/2 7

Angular factors of anisotrOpic
hyperfine interaction

Molecular orbital coefficients

Electron spin eigenvector for I
electron spin of -l/2

 

Bohr magneton

Gaussian broadening parameter
Nuclear magneton

Magnetogyric ratio of nucleus ex-
cept where defined otherwise in
text (as in Equation 1-36)

Proton gyromagnetic ratio

Dirac delta function

   

 

 

 

xiv

e, d Spherical polar coordinate angles
for the unpaired electron

e, ¢E Euler angles of external magnetic
field H relative to principal
axes of g and T_(or A) tensors

K _ Parameter such that -KP represents
the isotropic part of the hyper-
fine interaction energy

v Microwave frequency

Vp Proton resonance frequency

v0 Constant applied microwave fre-
quency

¢l, ¢2, ¢3, PA: ¢5 One-electron orbitals

Yz(0) Square of the wave function P at

the nucleus

 

T* (in Equation l-AO) Higher energy wave function of
Kramers doublet

 

W* (elsewhere in Introduc- An antibonding orbital

tion) 5
¥* (elsewhere) Complex conjugate of W ;
y Nuclear spin eigenfunction for a }

mI nucleus with nuclear magnetic é
quantum number mI and nuclear 3
spin I j

The wave functions of the ground 4
state Kramers doublet that con-
tains the unpaired electron. YR
has higher energy than WQ in a
magnetic field.

 

This represents the definite
integral of T1*V2 over the entire
range of all applicable coordi—
nates.

Degrees Celsius (Centigrade)

 

 

 

 

 

CHAPTER 1

INTRODUCTION

Electron spin resonance (ESR) has sometimes been used
as a probe of molecular orbitals that contain unpaired elec-
trons.1 In general, experimental ESR results can be de-
scribed by a spin Hamiltonian involving only electron-spin
and nuclear-spin Operators multiplied by constants whose
values may be obtained from ESR spectra. Since the spin
Hamiltonian could be obtained from the wave function, one
sometimes can solve for the molecular orbital coefficients
using the experimental data.

In this work, a form is assumed for the wave functions
describing the pair of molecular orbitals that contain the
unpaired electron in a compound whose dl transition metal
atom causes hyperfine splitting to be observed in the ESR
spectrum. Equations are derived by substituting the terms
(of these wave functions) that involve d orbitals of the
transition metal atom into expressions involving appropriate
parts of the Hamiltonian given by McGarveyl and performing
the algebra involved. Substituting the experimental spin
Hamiltonian constants for trichloro(n5-cyclopentadieny1)-

niobium attached to 20% cross-linked styrene-divinylbenzene

copolymer beads into resulting equations and making some

 

 

 

 

 

 

 

 

2
further assumptions lead to some conclusions about the

molecular orbital coefficients.

The General Hamiltonian

 

The general Hamiltonian can be written1

" h A a A A
- Z V + V + i + +

+~JCLI +-XQ + terms that can be neglected (1-1).

The absolute values of the energies associated with

hz 2
- 2 Vk +

2me k

V are much larger than any of those associated

with the following terms, which are given in their general
order of magnitude.

V is the electrical potential energy, which includes
attractions between electrons and nuclei and repulsions be-

tween electrons.l

A

:KLS is the part of the Hamiltonian that gives rise to

the spin—orbit interaction

A eh A e A
JCLS = ”ii—{E X [p + <—-) 5-H] .5 (1’2),

2me C
where E is the electric field in which the electron moves,
6 is the linear momentum Operator for the electron, s is the
spin-momentum operator in units of h, and AH is the vector
potential for any magnetic field present. The term involv-
ing AH can make a small contribution to the experimental g

tensor.l

 

 

 

 

 

3

A

HE describes the Zeeman interaction. For a compound”
that has a natural center about which the angular momenta of
the electrons can be calculated,le can be written

:Kz = Be (L + 2.0023s)o§ - BN a}? gNi Ii (1-3)

when a term that is not significant to ESR is discarded.1

fl 1k (l-A)

and

CD)
ll

i Ek (1-5).
The constant Be is the Bohr magneton, ik is the angular—
momentum operator for electron k, and ék is the electron
spin—momentum Operator in units of h for electron k. H is
the external applied magnetic field, gN1 is the nuclear g
factor for nucleus 1, and 8N is the nuclear magneton. I1 is
the nuclear spin operator for nucleus 1. The last term of
Equation 1—3 is relatively small and is often disregarded.
When that is done and a compound having only one unpaired
electron is considered, the remaining part of Equation 1-3

can be written1

x2 = Be(1z + 2.0023sz)Hz
+(1/2)Be[(i+ + 2.0023§+)H_

+ (i_ + 2.0023§-)H+] (1-6),

 

 

 

 

 

 

L;
where
1+ = 1X t 11y (1’7),
5: = sx i 15y (1-8),
and
H: = Hx i iHy . (1-9).

The symbols 1x, 1y, and 12 represent the components of the
angular-momentum operator for the unpaired electron. The
symbols sx, Qy, and SZ represent the components of the spin-
momentum operator for the unpaired electron. Hx, Hy, and H2
are the components of the magnetic field.

figs is the interaction between electron magnetic dipoles.
This can be omitted from the Hamiltonian of a compound that

has only one unpaired electron.

({SI’ the Hamiltonian for the electron spin-nuclear spin

interaction, is1

A 2 A A
.HSI = ~2.0023 BeBN 12k gni [rik(3k°li)

,

'3(§k°£ik)(ii°£ik)]rIR

817 A A
+(-3—-> 2.0023 BeBN 12k gNi5<rik)Ii’Sk (1&0).

The scalar Pik is the distance between electron k and nucleus
i. The vector Eik is the distance vector between nucleus 1
and electron k. The function 6(rik) is the Dirac delta func—
tion, which gives the value of the wave function at rik = 0

when integrated with the wave function. The first term in

 

 

 

 

5
Equation 1-10 is the ordinary dipole-dipole interaction for
two dipoles that are not too close to each other and is ap-
propriate for p, d, and f electrons, which are not found at
the nucleus. The second term (the contact term) applies when
the electron is close to the nucleus. The second term is
therefore appropriate for s electrons, which have nonzero
probabilities of being at the nucleus. However, spin polar-
ization and/or admixture of excited states in which the un—
paired electron is in s orbitals often cause this term to be
significant even when the unpaired electron is not predomi-
nantly in an s orbital.‘ If there is only one unpaired elec-
tron, and if the hyperfine interaction involving the unpaired
electron occurs with only one nucleus, theanSI can be re-

written1 as

RSI = P'[(%1) 6(r)’s\z + (3 cos20-1)r'3§2

..i¢"

+ sinecoser-3(e s + ei¢s_)]Iz

mlw

+

+ 9:1:(9371) 6(r)§+ +%—(1-3 cosze)r-3§+

; 2 21¢ -3“ i
+ A sin 0e r + 2

s_ sinecoseei¢r-3§z]I_

+ P'[(C%1) 5(r)§_ + %(1-3 cosze)r-3§_

3 2 -2i¢ -3A ; ~1¢ -3“ A
+ A sin 6e r s+ + 2 sinecosee r SZJI+

(1-11),

where

P' = 2.0023gNBeBN (1-12).

 

 

 

 

 

6
3£LI represents the nuclear spin-orbit interaction, which

is

“L1 = 23.2% 2 8N1 rik

3 (iik-ii) (1-13).

JTQ is the nuclear quadrupole interaction.

The terms that can be neglected in Equation 1-1 include

the nuclear spin-nuclear spin dipole interactions and the

chemical-shift terms.l

 

The ESR Effect

The linear momentum and electrical terms in Equation 1—1
lead to energy level separations much greater than energies
involved in electron spin resonance. Spin-orbit coupling
can lead to further splitting of energy levels. Energies of
spin-orbit coupling interactions are often in the order of

2 or 103 cm"l,l’2 which are also much greater than energies

10
observed in ESR. Small distortions (Jahn-Teller distortions)
remove degeneracies in compounds that otherwise would have
degenerate states, except that each energy level of a com-
pound having an odd number of electrons is left doubly de-
generate. Such doubly degenerate levels are called Kramers
doublets. The application of an external magnetic field to
such a compound would remove this degeneracy via the Zeeman
interaction. A magnetic field of 3000 G (which is comparable
to field strengths used in much ESR work) would cause a sepa-

ration Of approximately 0.3 cm'“1 between the levels of the

ground state Kramers doublet. Under the conditions of the

 

 

 

 

 

 

 

7
ESR experiment, a quantum having approximately that energy
(which is in the microwave region) can induce a transition
between these levels. At equilibrium, the lower energy
level is more populated than the higher energy level. If
the probability that a single molecule in the lower energy
level undergoes a transition equals the corresponding prob-
ability for a single molecule in the higher energy level,
then a net absorption of microwave energy is expected. This
absorption is measured in the ESR experiment.

When a typical ESR spectrum is obtained, the magnetic
field strength is varied and the microwave frequency remains
constant (or practically so). Therefore, the energy differ-
ence between the two levels of the ground state Kramers
doublet varies during an ESR scan. When this difference be-
comes equal to the energy of a quantum of the microwaves be-
ing used, absorption is observed. The magnetic field at which
this absorption occurs is referred to as the resonance field.

If a molecule has more than one unpaired electron, elec-
tron dipole (Hss) interactions, which are often of the same
order of magnitude as the electronic Zeeman interaction,
would greatly affect the ESR spectrum.~ Since only dl sys-
tems are considered in depth in this dissertation, details
oszss effects on spectra are omitted.

The hyperfine interaction, if present, further splits
each level of the ground-state Kramers doublet. If the hyper-
fine splitting is due to a nucleus that has a nuclear spin

of I, then 21 + 1 lines corresponding to allowed transitions

 

 

 

 

 

 

8

can be observed. These transitions involve a change of :1
in the total electron magnetic quantum number (Ams = :1)
and no change in the nuclear magnetic quantum number

(AmI = 0).

Although the nuclear Zeeman term (describing interac-
tion of the nuclear moment with the applied field) and the
interaction of the nucleus with a field produced by the in-
duced magnetization of the electronic charge distribution by
the external field are small and often neglected, they can
combine with a highly anisotropic hyperfine interaction to
permit observation of "forbidden" Ams = 11, AmI = t1 hyper-
fine lines for some orientations in strong external magnetic
fields.3 In such a case, the very large hyperfine anisotro-
py is believed to cause the axis of spin quantization of the
nucleus to differ appreciably from the direction of the ap-
plied field. Then terms containing ix (a component of the
nuclear spin Operator) can become comparable to those con-

A A

taining Iz' Expressing Ix as

’1‘, = §<i+ + i ) (1.11:).

where I+ is the nuclear spin "raising Operator" and I_ is
the nuclear spin "lowering operator", shows how such terms
can give rise to "forbidden" transitions.

Another mechanism, which has been regarded as usually
being responsible for "forbidden" transitions in ESR spectra,
involves interaction of a nuclear quadrupole moment with an

electric field gradient.3’u

 

 

 

 

 

Powder Pattern ESR Spectra

 

For a given microwave frequency, paramagnetic center,
and ESR transition, the resonance field (the strength of the
magnetic field at which this transition occurs) depends upon
the orientation of the magnetic field with respect to the
coordinate axes at this paramagnetic center.5 This orienta-
tion can be described by the Euler angles 0 and 0B, which
are illustrated in the top part of Figure l. The spherical
polar coordinate angles 0 and 0 are defined at the bottom
part of Figure 1. If 6 is less than or equal to 180°, 6
has the same value in both cases. However, 0 and 0E are not
the same.

Solutions obtained by matrix diagonalization (for the
Zeeman term) and second-order perturbation treatment (for
other terms) of the spin Hamiltonian have been given5 for
the resonance condition for the mS-l ++ mS allowed ESR tran-
sition. One of these solutions, applicable to systems that
have only one unpaired electron and exhibit only hyperfine

and electronic Zeeman interactions in ESR spectra, is5

 

 

 

 

 

 

10

I3:
All-ha

 

 

 

 

psi

lw
b-N

 

 

 

‘_
"‘
--
dd

FI-------- --------

 

Figure 1. Definition of the Angles 0, 0E, and 0

 

 

 

11

m 2sinze
hv = gBeH + AmI + _1______ .
2gBeH

2 2 2 2 2
g1 g2 (A1 ”A2 ) 2 2
sin 0E cos OE

gaZgZ A2
E 2 2 2 2
+ g 8 (A -B )
a. 3 3 00326 +

gt A2

 

 

[1(1 + 1) - mlz] A12A22 + A3282
AgBeH B2 A2

2 2 2 2 2- 2 2
+ g1 g2 33 A3 (A2 A1 ) coszesin2(2¢E)
“gaugz BZAZ
(1-15)
(This equation has been corrected. One of the exponents had

been omitted in the published review.)

The equation defining g is

 

g =‘//(glzsin2¢E + g22cosz¢E)sin26 + g32cose (l-l6).

A is defined by

 

 

 

=.1 2 2 2 2 2 2 _
A gW/[ga B sin 0 + g3 A3 cos 9 (1 17).

The equation for ga2 is

 

2 = 2 2 2 2 _
ga gl sin 0E + g2 cos 0E (l 18).

B is defined by

 

= 1_ 2 2 2 2 2 2 _
B ga'V/gl Al sin 0E + g2 A2 cos 0E (l 19).

     

 

 

IIIIIIIIIIIIIllll---:::———————_______,

12
The quantities g1, g2, and g3 are the principal values of
the electronic g tensor. A1, A2, and A3 are the principal
values of the hyperfine interaction (A) tensor. (See Equa-
tion 1-31 and the text immediately preceding that equation.)
The Planck constant and the Bohr magneton are denoted by h
and Be, respectively. The frequency Of the applied oscil-
lating magnetic field (i.e., microwave frequency) is v. H is

the resonance field (i.e., the magnetic field strength at

 

which the resonance condition is satisfied exactly). The
nuclear magnetic quantum number is ml. The Euler angles of
the external magnetic field relative to the principal axes
of the g and hyperfine (A) tensors are e and ¢E' (See the
top part of Figure l.) The nuclear Spin is represented by
the letter I. Equation 1-15 applies only if the principal
axes of the g tensor are identical to the corresponding ones

of the hyperfine (A) tensor.

ESR Powder Pattern Lineshapes

 

Now consider a magnetically dilute sample whose para-

 

 

magnetic centers are at many random orientations with reSpect

to the direction of the externally applied magnetic field.

 

Such a sample may be a glassy sample, a polycrystalline sam-
ple whose crystals are powder size, or a polymer onto which
paramagnetic centers have been attached. The type of ESR
spectrum exhibited by this kind of sample is called a "powder
pattern"5 (a designation consistent with the second-mentioned

example of such a sample). An unbroadened, broadened, and

   

 

 

13

first derivative powder pattern (simulated) is presented in
Figure 2. In this figure, the horizontal coordinate repre-
sents the external magnetic field strength and has an iden-
tical scale for all three parts of the figure. In the upper
two plots, the vertical coordinate represents ESR absorption,
but the first derivative of ESR absorption with respect to
the external magnetic field strength is displayed vertically
in the lower plot. (No attempt was made to keep the size of
the vertical unit identical for all three plots.)

Lineshapes exhibited in ESR powder pattern spectra have

been discussed and illustrated in the literature.1’5"'9 A pow-

der pattern results from the summation of ESR spectra exhib-
ited by paramagnetic centers at many different orientations.
Solving Equation 1-15 for H (by multiplying the whole equa-
tion by H and solving for the appropriate root in the qua-
dratic equation that results) gives an expression for the
resonance field as a function of the nuclear magnetic quan-
tum number mI and the orientation angles 6 and ¢E' Also in
that expression are a number of constants, including Spin
Hamiltonian constants and the microwave frequency v. If the
transition probability is independent of orientation, and if
there were no processes that cause line broadening, then sum-
ming over all possible values of 0 and 0E -- or integrating
over the domain of 0 and 0E -- would reveal a powder pattern
ESR spectrum with no broadening for each value of ml. The
top illustration Of Figure 2 would be typical. However, line

broadening processes do exist. If a convolution with a

 

 

 

 

1A

 

 

 

 

 

 

 

 

 

A
Typical ESR Powder Pattern (Simulated)

Figure 2.

 

 

I»,

 

 

 

15
Gaussian broadening function is performed on the simulated
spectrum, the appearance becomes similar to the illustration
in the middle of Figure 2. Experimental ESR spectra are
usually displayed as the first derivative; therefore, in
Figure 2, the bottom plot, which is the first derivative of
the middle section of this figure, would be compared with
experimental spectra.

If hyperfine splitting is exhibited by one nucleus with
nuclear spin I, then the ESR powder pattern spectrum is a
sum of 21 + 1 spectra (one for each of the allowed values of
mI) similar to the spectrum pictured in Figure 2. At and
above the temperature of liquid nitrogen, the thermal energy
is so much greater than the hyperfine interaction energies
that energy levels corresponding to all allowed values of
mI are approximately equally pOpulated. Thus, nearly equal
numbers of paramagnetic centers would have each value of mI.
This implies that each of the superimposed Spectra would have
a weight nearly identical to each of the others in the sum-

mation over mI.

Extraction of Spin Hamiltonian

Parameters from ESR Spectra

A necessary step in progressing from an ESR spectrum to
conclusions about the paramagnetic species is extracting spin
Hamiltonian parameters from the spectrum. If the spectral
features are all well resolved, this step can be easy. In

case all of the features are not resolved, then those that

 

 

 

16
are can be used in making initial guesses. Thus, after an
ESR spectrum is obtained, the first step in interpreting it
is to determine the resonance fields of features that are at
least reasonably well-isolated.

Consider how powder pattern features arise in ESR spec-
tra.l’5‘9 The case involving completely anisotropic g and
hyperfine tensors, whose principal axes are collinear, is
used to illustrate how a powder pattern arises for one value
of mI. A situation that is very common and relatively easy
to understand is one in which the following three conditions
are obeyed:

(1) The maximum resonance field occurs when one of the prin-
cipal axes of the g and hyperfine tensors is aligned with the
field, (2) the minimum resonance field occurs when one of A
the other such principal axes is so aligned, and (3) there
are no turnabouts in the dependence of the resonance field

on orientation except when one of the principal axes is a-
ligned with the field. For clarity of illustration, this
example has a linewidth much smaller than the separations be-
tween the three distinct features of the spectrum. Thus,
when ESR absorption is plotted against the external magnetic
field strength, there is a sudden onset of absorption at the
minimum resonance field, which (as indicated by condition 2)
corresponds to one pair of principal components of the g and
hyperfine tensors. This absorption is exhibited as a shoul-
der, which appears as a peak in the derivative spectrum.

(See the features connected by the left dashed line in

1;! '.
:1,

 

 

 

 

17
Figure 2.) Obviously, the relative maximum at this peak
occurs at the resonance field that corresponds to the pre-
viously-mentioned pair of principal components. At the other
end of this powder pattern, there is a sudden end to the
absorption at the maximum resonance field, which (as indi-
cated by condition 1) corresponds to another pair of prin-
cipal components of the g and hyperfine tensors. This is
also exhibited as a shoulder, but the derivative of this
shoulder appears as a valley. (See the features connected
by the right dashed line in Figure 2.) The resonance field
corresponding to this pair of principal g and hyperfine com-
ponents is the magnetic field at which the relative minimum
of this valley is exhibited. Now Consider the resonance
field that corresponds to the remaining pair of principal
components of the g and hyperfine tensors. This is not only
the resonance field along the appropriate principal axis but
is also the resonance field for many other orientations.
An extremely large number of orientations exhibit resonance
at or very close to this field strength, giving rise to a
divergence in the unbroadened spectrum. (See the features
connected by the middle dashed line in Figure 2.) Broaden-
ing and taking the first derivative of this feature produces
a lineshape that has a peak, baseline crossing, and valley.
The magnetic field at which the derivative spectrum crosses
the derivative equals zero axis (baseline) is the desired
resonance field for this feature. It is stressed that this

analysis applies only if the resonances are sufficiently

 

 

 

 

 

 

18
isolated to avoid distortion of their lineshapes. In case
of overlapping features, it may be more reliable to compare
(simulated vs. experimental) the magnetic field strength
at one-half the vertical peak—to-valley distance (instead
of at the baseline crossing) for this type of feature.

Sometimes the dependence of the resonance field on
orientation exhibits turnabouts at orientations that do not
align any of the principal axes of the g and hyperfine ten-'
sors with the magnetic field. ESR powder pattern spectra
may then contain singularities in addition to those observed
when the field lies along the principal axes directions.9’10

These "extra" singularities defy cursory interpretation.
Computer Simulation Of ESR Spectra

It is common to obtain computer-simulated ESR spectra.
A simulated spectrum allows one to Judge how well the as-
sumed values of the parameters reproduce the experimental

spectrum. If the agreement is poor, then some or all of the

parameters can be adjusted and another simulation can be per
_formed. If the spin Hamiltonian describes the actual situa-
tion and the line—broadening phenomena are accurately repre-
sented, then repeated simulations and intelligent adjustment
of parameters eventually should lead to a simulated Spectrum

that matches the experimental spectrum at least rather well.

Some computer programs that simulate powder pattern ESR spec-

tra have been described in the literature.7’9’ll-l3

 

 

 

—’—

19
Some other computer programs, which simulate ESR pow-
der pattern spectra and which are available from the Quantum

1” SIMlA,15 and

Chemistry Program Exchange,are FIBRE,
SIMlAA.15 Of these three programs, an appropriately modified
version of SIMlAA seems to be best suited for simulating com-
pletely anisotropic ESR powder pattern spectra of niobium
compounds. SIMlAA has been found to be unsuitable for simu-
lating ESR spectra for cases in which the y component of the
diagonal hyperfine tensor is negative and not equal to the x
component. However, the reason for that unsuitability has

16 Details and a comparison

with the trio of programs ESRMAP, GATHER, and PEAKSl6 are

been identified and corrected.

given in Appendix F.

One way to simulate a powder pattern ESR spectrum is
to calculate single-crystal spectra for a large number of
orientations and to superimpose them. However, if this
method is used alone, without some type of interpolation, a
very large number of orientations and consequently an exces-
sively long computation time often is required to obtain a

spectrum with a good signal-to-noise ratio. A recent arti-

 

cle9 describes a method (applicable for total electron spin
S up to 5/2) that interpolates between calculated orienta-
tion angle and Ha values Of each transition by using inter-
polation functions of e for axial symmetry or interpolation
functions of 0 and 0 for symmetry that is less than axial.

3H
This method uses partial derivatives, including (-§§3) and

( 21:0!) , where Ha is the magnetic field strength at which a

 

 

 

 

20

given resonance condition is fulfilled exactly, in perform-
ing relatively economical simulations of powder pattern ESR
spectra. "Turning points", including points at which
(2%) =10 at 4: = 0° or 90° and at which (gig) = 0 at
e = 90°, in the angular dependence of the resgnance positions
determine the positions of singularities in the powder pat-
tern spectrum. This well—known relationship is illustrated
nicely by figures in that article.9

A method that enables spin Hamiltonian parameters to be
extracted from certain (11 systems that have at least C2v sym—
metry has been described.10 If, in such a (11 system that
contains one transition metal atom, (l) the principal axes
of the T (hyperfine) and g tensors coincide and (2) the
quadrupolar and Zeeman interactions of the transition metal
nucleus are both small, then the spin Hamiltonian can be
written in the diagonal form

H§+ HS +g HS]
eéxxx PM .2

+T s

xxIx x + TnyySy zz z Z

If; - %~I(I + 1)] - (gzz)N8NHz (1-20),

I'_l

K = §
S

£38m

where x, y, and z are the principal axes of the g and T ten-
sors. In this method, only the ‘Ams = 11, AmI = 0’ transi-
tions are considered in order to account for powder pattern
ESR spectra. If it is further assumed that the transition

probability is independent of orientation and that the broad-

ening function has a Gaussian form, then the lineshape func-

tion F(H'), which defines the ESR absorption spectrum as a

21
function of the Observed magnetic field H’, can be writ-
tenlo’l7 as

+I +0°

_i s (H)
F<H’> = (2.) 2 1E: ‘jr _E;___

mI=-I -°° 8G

-(11’-H)2
exp 28G2 dH (1-21).

SmI(H) is the Kneubfihl function.6 The exponential divided
by 8G is a broadening function, in which 8G is an adjustable
parameter related to linewidth. If H1 < H2 < H3 and there
are no "extra" singularities, then Sm (H) can be calculated
according to Equations 1—22 and 1-25: 10

In the field range H1 ; H < H2,

Sm (H) = 2H1H2H3 K(k) 2
I "H2[(H32—H2)(H22—H12)]l/ (1-22>.

where

k _ [ (H32—H22)(H2—H12):]1/2

(H32—H2)(H22~H12) (1-23),

and K(k) is the complete elliptic integral of the first kind

defined by

"/2 -1 2
K(k) = v//p (l—kzsinzx) / dx (1-2a).
o

In the field range H2 < H i H3,

 

22

 

Sm (H) = 2H1H2H3K(k’) ‘_____
sz[(H32-H22)(H2-H12)]l/2 (1‘25):

where

k, = [ (H32-H2)(H22-H12)]1/2

(H32-H22)(H2-H12) (1-26)

and

K(k’) =o/:’fl/2 [1-(k’)2sir12x]'l/2 dx (1-27).
In Equations 1-22 and 1-25, H1, H2, and H3 are Hx, Hy, and

H2 arranged so that H1 < H2 < H3. Hx, Hy, and Hz are the
resonant field strengths when the magnetic field lies along
the x, y, and z axes, respectively. During a simulation,

these are calculated by

 

 

 

hv hCTxme
Hx = ______ _ ________
gxxBe gxxBe
2 2 2 _ 2
_ he [Tyy + Tzz ][I(I + l) mI J
A gxxBev
hc(Q’)2[2I(I + 1) - 2mI2 — llmI
2 gxxBeTxx (1—28),
hv th m
H = — .__lQL;L
y B

gyyBe gyy e
2 2 2 _ 2 '
_ hc [Txx + TZZ J[I(I + 1) ml]
u gyyBeV
hc(Q’)2[21(I + 1) — 2mIz -1]mI

2 gyyBeTyy (1—29),

23

H = hv _ thzzmI

gzzBe gzzBe

 

2 2 ' 2 2
_ hc [Txx + Tyy ][I(I + 1) - mI J

 

a @2238“ (1-30).

The principal components of the g tensor are gxx: gyy, and
gzz. Txx, Tyy, and T22 are the principal components of the
T (hyperfine) tensor in units of cm-l. Q’ is the nuclear
quadrupole coupling parameter for the transition metal atom.
The velocity of light is denoted by c.

The hyperfine components T11 (which are in units of cm-l)

could be converted to the hyperfine components A1 (which

could be divided5 by giBe to obtain values in units of Gauss)

by applying
A1 = hCTii (1-31).

in which i is either one of the integers 1, 2, and 3 or one
of the letters x, y, and z.

In some cases, ESR powder pattern spectra exhibit singu-
larities (shoulders, whose first derivatives appear as peaks
or/and valleys, or/and divergences) in addition to those ob-
served when the field lies along the principal axes direc—
tions.10 Conditions resulting in the occurrence of such
"extra" singularities and equations for calculating the mag-

netic field strengths at which those singularities occur have

been given5 for the spin Hamiltonian

 

 

2A

312 z 85.5.5 1 55.5.; (1-32).

Although the spin Hamiltonian was solved exactly for the
electronic Zeeman term, the hyperfine term was treated to
only first order in perturbation theory. Therefore, the
resulting equations are probably not adequate for an accu-
rate quantitative calculation of magnetic field strengths
at "extra" singularities when patterns are as wide as those
for a number of niobium compounds. In the case of complete
asymmetry, extra singularities occur at approximately the
magnetic fields H13 if A13 is between A1 and A3 (1 = 1,2

and j = 2,3). H13 and A11 are defined by

H.. = hvo - AiJmI

 

 

lJ
gijBe giJBe (1‘33):
where
2 .
1,. = {3:9 . [(19) + 21% - my] “2
J I 2
“m1 . “m1 2(gi2 ‘ g3 ) (1-3A),
1/2
gij = [£712 + (gjz - €12) COSZXJ (1-35),
and
COSZK = T//[ l + 352(A32 - A132) ]
2
312(A132 ’ A1 ) (1-36).

The constant applied frequency is v0. Equation 1-36 has been

corrected by adding a pair of parentheses to the numerator of

the fraction in the expression in the denominator. Further

 

 

 

25

conditions are given5 to predict whether such an "extra"

singularity would be a divergence or a shoulder.

The preceding discussion has assumed that the principal
axes of all the relevant spin Hamiltonian tensors coincide.

However, if the symmetry of a compound is very low, then

the principal axis systems may be non-collinear. An energy

formula obtained as a second-order perturbation solution has

been given18 for the spin Hamiltonian

H = Be§-g-A + §.D-§ + f-A-s + foo-1 (1-37)

when the principal axes of the different tensors in the spin

Hamiltonian possess arbitrary directions. An equation for

the energy difference involved in a AMS = 1 (MS + MS + l),

AmI = 0 transition has been obtained19 via a second-order

perturbation solution to the general spin Hamiltonian

(1-38)

(A)

H = Begog-s + s-D-s + I°A°

when the axes of the g, Q, and A tensors may be non-collinear.
Effects of the relative orientation of the g and hyperfine

tensors7’19 and of the g and Q tensorslg on powder pattern

spectra have been studied and illustrated.
Molecular Orbital Methods

Once spin Hamiltonian constants are determined experi-
mentally, they can be used to infer some information about
the molecular orbital in which the unpaired electron resides.

A method is discussed in a review by McGarvey. An expression

 

—i—

26

containing unknown coefficients is assumed to represent the

molecular orbital wave function. Matrix elements of the ap—

propriate parts of the exact Hamiltonian are calculated using

this wave function. Expressions containing the unknown co-

efficients are obtained. These expressions are equated to

the corresponding spin Hamiltonian matrix elements, which

are obtained by using spin—only wave functions. The experi-

mentally-determined values of the spin Hamiltonian constants
can then be substituted into the resulting equations. These
equations could then be solved for the molecular orbital

coefficients if there are not too many unknowns compared

with the number of equations.

An expression representing the ground state wave func-
tion or a reasonable approximation of it is necessary in this

treatment. Some approaches are given in a review by Kuska

and Rogers. These approaches include (1) a perturbation

approach, (2) solving for the ground state, (3) interpreta-

tion Of ligand hyperfine splittings, and (A) interpretation
of quadrupole coupling. In the perturbation approach, an

approximate ground state is assumed. A correction term,

which considers the effect of mixing some of the higher

states into the ground state by spin-orbit coupling, is added

to the approximate ground state. The second approach can

give one a better idea of the ground state, but less electron

delocalization information, than the first approach.

McGarveyl describes use of a perturbation approach in

detail. Symmetry is considered in writing molecular orbitals

 

27
for the ground state and nearby higher states that could be
mixed into the ground state by spin-orbit coupling. Then
perturbation theory is used to include the effect Of mixing
those higher states into the ground state. Since the energy
difference between each of those higher levels and the ground
state helps to determine the amount of the higher level wave
function that is mixed in, these energy differences are
They are typically determined from optical spectra.

needed.

However, if the physical state of the compound is such that

reliable optical spectra cannot be obtained, then another

approach to representing the ground state wave function may

be more appropriate.
Swalen, Johnson, and Gladney21 have written the ground

state Kramers doublet wave functions for a compound having

DAh symmetry as

v = a¢la + b¢3a + ic¢2a - id¢u8 - e¢5B (1-39)
and
w* = i(a¢lB + b¢3B — ic¢28 - id¢ua + e¢5a) (1—u0).
where
21 3 Va (A) (1-u1),
Z2
’b
T2 = wd (B1) (1.42),
xy
¢ 9 A (A) (1 A3)
3_d22 - ’
x -y
2 (1-AA),

 

 

 

28

’b
= w (B2) <1-45).

Coefficients to be determined are a, b, c, d, and e. Co-

valency is "buried" in 01, $2, ¢3, 0“, and ¢S, as they in-

clude ligand orbitals. The normalization condition, experi-

mental values of gxx’ gyy, and gzz, plus an assumption re-
= 0.9d)

lating coefficients (such as assuming e - d or e

were used in solving for the five coefficients in Equations
1-39 and l-AO. The determined values were substituted into

equations involving hyperfine constants, which were then

used in solving for K and P. Fluorine transferred hyperfine

constants were used in determining fractional orbital occu-

pation of the fluorine atoms.

Equations that relate spin Hamiltonian parameters to

expressions containing parameters Po’ P1’ and PS and coeffi-

d d d”, and d5 have been reported.22 The

l, 2’ 3’
parameters P0, P1, and PS relate to the contact and dipole-

The

cients d

dipole interactions of the electron with the nucleus.

coefficients d1, d2, d3, d“, and d5 are coefficients of

spherical harmonics for the azimuthal quantum number 1 = 2

in a two—component spinor.

For a number of nuclei, the atomic parameter W2(0) and
the isotropic hyperfine interaction A for unit spin density
in the valence s orbital have been calculated.23 Tabulatedzu

values of the normalized radial wave function vs. a reduced

distance parameter have been used23 in determining T2(0) for

 

 

 

 

 

 

29
5 wave functions. Then A was obtained23 by computing
(8ngBea/3)W2(O) times a relativistic correction factor.
Apparently, g was assumed to be equal to the electronic
free-spin value. The symbol 3 represents the magnetogyric
ratio of the nucleus. For_93Nb, WSSZ(O) was calculated23
to be 4.736 a.u.'3, and A for the 55 orbital was calculated23
to be 6590 MHz.

Comparison of A with an isotropic hyperfine interaction
in a compound may be used23 to estimate the percent of s
orbital character in the molecular orbital containing the
unpaired electron.

The atomic parameter <r'3> was calculated23 for the
valence p, d, or f wave function of each of many elements.
Numerical integration of a function defined as the square of
the appropriate normalized radial wave function divided by
the cube of a reduced distance parameter was used to obtain
a result from which <r’3> was easily found. The anisotropic
hyperfine parameter P was obtained23 by computing gBeX<r-3>
for one wave function of each of many elements. Again, the
electronic free-spin value apparently was used for g. By
this method, the value of P for the Ad atomic sub—shell of
93Nb was calculated23 to be A57.3 MHz.

To infer which atomic orbital or orbitals could be the
main atomic orbital or orbitals contributing to the molecular
orbital containing the unpaired electron, the anisotropic
portion of the hyperfine interaction can be compared with the

, and a 2' If the anisotropic

angular factors axx’ ayy z

 

30
portion can be attributed to at most one atomic orbital per
atom, then the contribution of this atomic orbital to that
molecular orbital can be estimated quantitatively by com-
paring the hyperfine interaction with axxP, any, and
azzP.23 The angular factors “xx: ayy, and azz have been
given23 for real atomic p, d, and f orbitals. The angular
factors for real atomic d orbitals are given in Table 1.

Table l. Angular Factors a for Real Atomic d Orbitals (from
Morton and Preston23).

 

 

Principal Values

 

 

Orbital axx “yy “22
dxy 2/7 2/7 -4/7
dx2-y2 2/7 ' ‘ 2/7 -u/7
dxz 2/7 -u/7 2/7
dyz -u/7 2/7 - 2/7
dzz -2/7 -2/7 “/7

 

 

Table 1 clearly shows that it is possible to have an axial

hyperfine tensor in which the unique axis is NOT the z axis.

Molecular Orbital Studies of Cp2ML2

and Similar Compounds

Stewart and Porte17 have done molecular orbital studies
on (fl-C5H5)2VL2 (where L = Cl, SCN, CON, and ON) and

Or-C5H5)2NbL2 (where L = Cl, SCN, 0-C5H5, and ON).

 

 

a- M"WPU

31
(05H5 = Cp = cyclopentadienyl.) These studies are based
largely on ESR spectra of these compounds in chloroform:
ethanol (9:1) glass at 77 K. These spectra were fitted to
the first derivative of Equation 1-21 with respect to H’.

The spin Hamiltonian used was

K = BeEgXXHXSX + gyyHysy + gzszSz]

AA AA

+ Txx x x + TnyySy + Tzz Z z (1-46).

Some of the bands in the visible-ultraviolet spectra were
used to determine d + d transition energies. These compounds
were all assumed to have local 02v symmetry. The z axis was
assumed to be the C2 axis. The x axis apparently was de-
fined to be in the Cl-M-Cl (M = V or Nb) plane. The y axis
apparently was taken to be in the plane defined by the metal
atom and the centers of the two cyclopentadienyl rings.
'Standard Hfickel calculations for (n-05H5)2v012 indicated
that the unpaired electron lies in an antibonding orbital
W*(A1) if spin-orbit coupling, Zeeman, and hyperfine inter-
actions are all ignored. When only the important contribu-

tions to this orbital were considered, its form was simpli-

fied to
- * +
W*(Al) - a1 (aWd 2 2 dezz)
x-
*(I) (1‘u7)3
+ a1 TCle[A1]

in which

 

 

 

32
(3*, 01*(1), a, and b are molecular orbital coefficients.

The wave functions Vd and Yd are for metal ion orbit-
x2 z2

- 2
als. The bonding and aniibonding orbitals of Bl and 82
symmetry at energy levels within 12 ev of that of W*(Al)
were listed with that orbital in this order of increasing
energy: Y(Bz), W(Bl), V*(A1), W*(B2), and W*(Bl).

Equating perturbation matrix elements of the Zeeman
and hyperfine interactions to the corresponding matrix ele-

ments of the spin Hamiltonian1 (Equation 1416)17 and making

the approximations

(l-U8)

I
[a

Pmd(32*) + Pmd(B2)
and

(l'u9)’

I
F:

Pmd(Bl*) + Pmd(Bl)

where Pmd(w) is the metal ion d-orbital population in the

17
molecular orbital w, lead to:

= 2.002 — {2(al*)2(a + b/3)2€v}

 

 

 

 

gxx
Pmd(B2*) - [1 ' Pmd(B2*)J :}
{:AE[T*(B2)+W*(A1)] AE[w*(Al)+w(82)] (1-50),
2 _ 2
gyy = 2.002 - {2(a1*) (a b/3) Ev}
Pmd(Bl*) - [1 ' Pmd(Bl*)] },
AE[Y*(Bl)+W*(Al)] AE[W*(A1)+W(BI)J (1-51),

 

 

 

 

 

33
2.002 (1-52),

P‘{-K +
xx

/ .
u7§ab(c1*)2 + (gxx - 2.002) +

I?

*9
ll

2 _ 2 2 _
(a b )(al*)

«WV

1 (3a + b/3)
[20002 "
la (a - b/3) ‘ gyy

 

) -

'N[H
SDIU'

(2.002 - gzz)}’ (1-53).

= _ 2, 2_2 *2
Tyy P{K+7(a b)(al) +

*3 2
u __ a * + - 2.002 +
7 ab( 3 ) (gyy )

 

1 (3a - b/3) l b

** 2.002 - + --r 2.002 -

l“ (a + b/3) ( gxx) 7 a ( gzz)jr( u
1-5 )9

Tzz = P{-K - 3% (a2 - b2)(al*)2

1 (3a + b/3)
_‘ 2.002 " 3 "'
14 (a - b/3) ( gyy)

_1 (3a - b/3)
in (a + b/3) (2.002 - gxx) + (gzz - 2.002l}~ (1_55).

 

The constant 5v is the spin—orbit coupling constant for the
metal ion in the appropriate valence state. P was indicated
to be 2.002gNBeBN < Yd 2*Ir'3lvd 2>. In the niobium series,
P was 0.01086 cm'l, an: Ev estimzted at +490 cm'l. The in-

fluence of ligand spin-orbit coupling on magnetic properties

was neglected.

 

 

 

 

 

3h
Experimental spin Hamiltonian parameters for

(n-CSH5)2NbC12 in chloroformzethanol (9:1) glass at 77K

were17
gxx = 1.980,
gyy = 1.9A0,
gzz = 2.000 a 0.001,
Txx = -0.01066 2 0.00002 cm'l,
Tyy = -0.01598 2 0.00002 cm'l,
and
T22 = -0.00528 2 0.00002 cm‘l.
Absolute values of molecular orbital coefficients also were
presented.17 However, later work by Petersen and Dahl25

suggests that the x and z axes defined by Stewart and Portel7

do not coincide respectively with the x and z axes of the g
and 1 (hyperfine) tensors. Instead, a single-crystal ESR
study of (ns—CSHNCH3)2V012 (CSHQCH3 = methylcyclopentadienyl)
doped in (ns—CSHHCH3)2T1012 reveals that the direction of the
x component of the g and T tensors bisects the Cl-V—Cl bond
angle, and the 2 component is normal to the plane which bi-
sects the Cl-V#Cl bond angle.25 Molecular orbital coeffi—
cients, including apparently-revised molecular orbital coef-
ficients, including apparently—revised molecular orbital co-
efficients for (w-CSH5)2NbCl2, are also given by Petersen
and Dahl.25 ’
Some other interesting molecular orbital studies on

-2
similar compounds or fragments have also been reported.26 8

 

 

 

 

 

 

CHAPTER 2

EXPERIMENTAL ESR SPECTRA

Samples of trichloro(ns-cyclopentadienyl)niobium at-
tached to 20% cross—linked styrene-divinylbenzene copolymer
beads (hereafter referred to as copolymer-attached CprClB)
and dichlorobis(ns—cyclopentadienyl)niobium, also attached
to 20% cross-linked styrene—divinylbenzene copolymer beads,
(hereafter referred to as copolymer-attached Cp2NbC12) were
obtained from Chak-po Lau. Syntheses of these compounds
have been described by Lau.29

Portions of these compounds were transferred into Pyrex
tubes in an argon-filled glove box. Those tubes were tempo-
rarily sealed with modeling clay and/or rubber septum stop-
pers in that glove box. After being removed from that glove
box, those tubes were permanently sealed by a torch. The
Pyrex tubes were subsequently used as ESR sample tubes.

(The ESR absorption due to Pyrex occurs at magnetic field
values very different from those at which these samples ab-
sorb, as Pyrex glass exhibits apparent g-values of 6.2 and
1:230)

ESR spectra to be used for quantitative interpretation
were carefully obtained by using a Varian EPB Spectrometer

and varying the magnetic field strength. A Hewlett-Packard
35

 

 

 

 

 

36
frequency converter and electronic counter were used to
determine microwave frequency accurately. A proton marker
was used to determine magnetic field values accurately.

ESR spectra were obtained in sections. A typical scan
covered only 200 - 250 G -- sometimes less -- of the total
magnetic field range. This procedure was adopted to enhance
accuracy by having a smaller change in magnetic field per
horizontal unit of chart paper, by using the proton marker
relatively close to the spectral features, by insuring that
the scan rate-time constant errors were small, and by having
smaller variations in instrument parameters between uses of
the proton marker. However, keeping the sample at a low
temperature in the ESR cavity eventually led to deteriora-
tion of constancy of microwave frequency (probably due to
ice formation). Thus, neither of the spectra obtained for
quantitative interpretation was obtained entirely in one
session. The microwave frequency during a given session was
often slightly different from microwave frequencies during
other sessions. Easily-perceived variations in microwave
frequency have been accepted instead of having larger magni-
tudes of other errors, which could have been more difficult
to determine.

Microwave frequency was fairly constant while the mag-
netic field strength was varied. For quantitatively inter-
preted spectra of copolymer~attached CprCl3, the microwave
frequency was between 9.098 and 9.111 gigahertz (GHz). For

the spectral portions at which the very best signal-to-noise

 

 

 

 

37
ratios were enjoyed, the microwave frequency was usually
9.099 or 9.100 GHz. However, for much of the rest of the
spectrum, the microwave frequency was 9.108 — 9.110 GHz.
Such variation in microwave frequency could lead to errors
of approximately .002 in g tensor values and approximately
.OOOOH cm"1 in hyperfine tensor values.

The sample temperature was maintained at -l26° while
spectra to be interpreted quantitatively were obtained.
This temperature was used for both copolymer-attached CprCl3
and copolymer-attached szNbClZ. This temperature was deter-
mined by calibrating with a thermocouple under conditions
similar to those in effect when ESR spectra were being ob-
tained for quantitative interpretation. When those quantita-
tive ESR spectra were actually being obtained, no thermo-
couple was used; instead, a temperature dial setting and
nitrogen gas flow rate corresponding to those used in cali-
bration were consistently employed.

To provide one check on the accuracy of magnetic field
strength values and microwave frequency values, an "experi-
mental g value" was determined for a standard sample of pitch
in KCl. This "experimental g value" was 2.00303. Comparing
this with the known g value of 2.0028 indicates an error of
0.01% in microwave frequency to magnetic field strength ratio
or/and in the procedure used to determine the magnetic field
strength of the ESR absorption from a spectrum on chart pa-

per. This magnitude of error, which amounts to about 0.3 or

0.4 G, is acceptable.

 

 

38

Magnetic field strength values were determined at
selected points on piecewise experimental spectra. These
selected points included peaks, valleys, and points whose
vertical coordinates were halfway between those of peaks
and the next valleys. Many of these magnetic field strengths
were later compared with corresponding values from simula-
tions. (Details are given in Chapter A.)

The sections of spectra obtained for accurate interpreta-
tion do not provide as aesthetically excellent pictures as
desired for illustrations. In fact, the ESR spectra pre-
sented in Figures 3, 4, and 5 were obtained with a varian
E-A EPR Spectrometer, which is a different instrument from
the one with which very accurate work was done. The scan
rates were much faster, and other conditions were slightly
different, when the illustrated spectra were obtained. Nei-
ther a proton marker nor a frequency counter was used when
these illustrated spectra were produced. The instrumental
"field set" and "scan range" values were used with chart pa-
per divisions to estimate magnetic field strengths. However,
the field strength values so obtained are not reliable for
“000 G scans done with that instrument. Figures 3 and N are
parts of scans of that width. Pitch (g = 2.0028) was used
as a standard to estimate microwave frequencies for Figures
A and 5. In Figures 3, A, and 5, the vertical coordinate is
the first derivative of ESR absorption intensity with respect

to magnetic field strength, and the horizontal coordinate is

the magnetic field strength.

 

 

 

39

«Se

 

zao umcocup<cgmszpoaou so Ezcuumam mmm Paucwswamaxm

 

 

 

 

 

 

 

.m mgsmwu

 

m
a -
Fonz u vmgueua< Lass—once mo Esauomam mmm qucmepgmaxm .e seamed

 

 

 

 

 

 

141

 

NPunzNau umcumau<ngmezpoaou mo Eaguumam «mu qucme_amaxm

 

 

 

 

.m menus;

 

 

 

 

~120
net1<
accu1
late<
tion
son-
feat
quen
ting
with
a di
tua]
scal
stre
lus

obt

fir

the
For
n01
9.1
GB:

ag:

 

 

 

#2

An ESR spectrum of copolymer—attached CprCl3 at

—l20 2 no is illustrated in Figures 3 and A. If the mag—
netic field strengths read from chart paper were perfectly
accurate, then the microwave frequency would have been calcu-
lated to be 9.351 t 0.006 GHz for Figure A. Pitch calibra-
tion was not used for the spectrum in Figure 3, but compari-
son of the horizontal chart positions of selected spectral
features and comparison of the approximate microwave fre—

quencies indicated by the instrument microwave frequency set-

 

ting show that the microwave frequencies are identical to
within the stated uncertainty. However, there appears to be
a discrepancy of approximately 6% between the stated and ac—

tual Gauss per horizontal division on chart paper when the

 

scans illustrated in Figures 3 and N were obtained. It is
stressed that Figures 3, H, and 5 are presented only as il-
lustrations and that the accurately interpreted data were
obtained from piecewise spectra that were produced with the
first-mentioned instrument set-up.

For quantitative spectra of copolymer—attached szNb012,
the microwave frequency was between 9.1081 and 9.1092 GHz.
For the spectral portion at which the very best signal-to-
noise ratios were exhibited, the microwave frequency was

9.1091 GHz. At this frequency, this uncertainty of .0011

 

 

GHz could contribute uncertainties of about .0003 to the di-
agonal elements of the g tensor and uncertainties of about

.000005 cm'”1 to the diagonal elements of the hyperfine tensor.

 

 

 

A
-126 1
streng
then 1
this :
streng

with11

 

 

“3

An ESR spectrum of copolymer-attached Cp2NbC12 at
-126 2 4° is shown in Figure 5. If the magnetic field
strengths read from chart paper were perfectly accurate,
then the microwave frequency was 9.197 t 0.010 GHz. For
this scan (which was only a 2000 G scan), the magnetic field

strengths read from chart paper do seem to be reliable to

within 10 G.

 

__,-___..— ~_.—

 

 

 

bee:
ete:
in
of
in i

(198:

als:

Appq
Vide
' Dur]
been
tim

1181

Get!

SDE<

 

 

 

CHAPTER 3

COMPUTER METHODS

A number of computer programs and subroutines have

been written16

to aid in extracting spin Hamiltonian param-
eters from ESR spectra and in interpreting those parameters
in terms of molecular orbital coefficients. Selected ones

of these programs and subroutines, which have been written

in Fortran IV Extended language (CDC dialect), are briefly
described in this chapter and listed in Appendices A through
E. KINFITA and the framework of subroutine EQN, which are
also mentioned in this chapter, have been described previous-
ly.31 Subroutine EQN is one of the subroutines included in
Appendix D. The framework of subroutine EQN has been pro-
vided31 for use with KINFITM, a version of KINFIT, a general
-purpose curvefitting program. An early version of KINFIT has
been described in the literature, and an example of asubrou-
tine EQN appropriate for that version of KINFIT has been
listed.32

Calculating Experimental Magnetic Field Values

As mentioned previously, a proton marker was used to

determine accurate magnetic field values when piecewise ESR

SPectra were obtained for quantitative interpretation. A

AA

 

 

 

 

 

sin
to

mag

in

01‘

PI"
th
t1
ma
An

I18

pr

 

45
simple calculation, described by Equation 3-1, has been used

to convert the proton resonance frequency v in MHz to the

P
magnetic field strength H' in Gauss.

H' = KP VP (3-1).

in which KP is 2“
.0267527

 

or 1 Gauss/MHz.
.00425759

A short computer program, MAGFLD, uses two observed
proton marker frequencies and the abscissa values at which
they are observed on ESR chart paper to determine the quané
titative relationship between those abscissa values and the
magnetic field strengths if a linear relationship is assumed.

Any other desired abscissa values are then converted to mag-

netic field strengths. (See Appendix A for a listing of this

program.)

Initial Estimation ofg§pin Hamiltonian Parameters

for Copolymer-Attached CprCl3

It was desired to have good estimates of spin Hamiltonian
parameters before computer simulations (other than those done
to test programs) of ESR spectra were attempted. For copoly-

mer-attached CpZNbClZ, the previously reported17 values for

CpngCl2 (not attached to a polymer) sufficed as initial es-
timates.

 

 

 

 

 

 

an :
suf:
min.-
tro'
Suc
ter
to

98C

of

pa
par
thf
the
co:
be:
t1.
0t
st
be
ma

un

 

 

#6

From an ESR spectrum of copolymer-attached CprCl3,
an attempt was made to select resonance features that were

sufficiently isolated to permit reasonably accurate deter-
mination of resonance field values as described in the in-
troduction for a first derivative powder pattern spectrum.

Such features that seemed to belong to the three main pat-

terns were classified in three groups. An attempt was made

to guess the nuclear magnetic quantum number associated with

each of these features.
A pair of subroutines, ESRKF2, one of which is a version

of EQN, was used with KINFITA to estimate spin Hamiltonian

parameters from data obtained as described in the preceding

paragraph. For each data point used as input to KINFITA and

this pair of subroutines, four numbers are required. One is
the magnetic resonance field strength, and another is a
coded number that indicates the nuclear magnetic quantum num-
ber and which one of the axes (x, y, or z) is in the direc-

tion of the magnetic field. In the logic of KINFITA, the

other two numbers represent variances. The magnetic field

strength is a continuous variable. However, the coded num-
bers used to specify nuclear magnetic quantum numbers and

magnetic field directions are not continuous; in fact, the

units digit (after rounding off) is used to determine whether

Hx’ Hy’ or Hz is calculated. Therefore, it was desired that

KINFITA not change those coded numbers enough that they would

be interpreted incorrectly by any subroutines. Thus, a very

small number was used to represent the variance of each of

 

 

 

 

 

thos

cert

par

pro

bet

’47
those coded numbers, and nearly all of the experimental un-

certainty was attributed to the magnetic field strength.
Computer Simulation of ESR Powder Pattern Spectra

Once reasonably good estimates of spin Hamiltonian
parameters exist, they can be used as input to a computer
program that simulates ESR spectra. Quantitative comparison
of the experimental and simulated spectra then can lead to

better estimates of the spin Hamiltonian parameters.
Early Versions of ESRSIM

A computer program, ESRSIM, has been written16 to simu~
late an ESR powder pattern spectrum of a magnetically dilute
compound that has only one unpaired electron and hyperfine
splitting by at most one nucleus. That nucleus must have a
nuclear spin I of 9/2 or less. It is assumed that the g
tensor and T (hyperfine) tensor axes are collinear. Txx’
Tyy, and Tzz may all be unequal, as may gxx, gyy, and gzz.
For a given nuclear magnetic quantum number mI, Hx, Hy, and
Hz must all be unequal. There are more than one version of
ESRSIM, but each version is based largely on a reportedlo
method.

As the experimental spectra are first derivative spectra,
the first derivative of the lineshape function F(H') with re-
spect to the observed magnetic field strength H' has been ob-
tained from Equation 1-21, which is a generalized form of an

equation reported by Stewart and Porte.lo

 

 

 

The re

gig

Here,
Kneul
quan‘
in E
calc

in c

the
mag]
Th1:
as
ete
mag
lat
tel
we]
ma;
of
la

SU

AS

The resulting equation is

dF(H') 1 +1 u/P+m 3mI(H) H-H'
dH' =fi: —m_[2].

mI--I

—(H'—H)2
exp 2362 dH (3-2L

Here, BG is a Gaussian broadening parameter. SmI is the
Kneubuhl function6 corresponding to the nuclear magnetic
quantum number equal to mI. This function has been described
in Equations 1-22 through 1-27. Resonant field strengths
calculated according to Equations 1-28 through 1-30 are used
in computing SmI(H).

A simulated ESR spectrum produced by ESRSIM shows only
the singularities that correspond to those observed when the
magnetic field lies along the principal axes directions.

This is a drawback in the ESRSIM series of computer programs,
as certain inter-relationships of the spin Hamiltonian param-
eters can lead to other ("extra") singularities.lo In ESRSIM,
magnetic field values for "extra" singularities are calcu-
lated (if any are found to exist according to published5 cri-
teria) according to Equations 1-33 through 1-36. Printout
warns of these extra singularities and tells the approximate
magnetic field strengths at which these would occur, but none

of the extra singularities are incorporated into the simu-

 

lated spectrum. It is thought that higher-order effects in

such wide patterns as many niobium compounds exhibit make

 

 

 

 

 

Equations 1'
(Extra sing
and GATHER,
Numeri
the method
this method
computer ti
ter time; 1
sired comp‘
If th
1-21, then
can be ap;
propriate
3-2 can tr

-Z‘

n

This appr
ESRSIM.

QUired to
the numbe
level. 1
quired c<

trum wi t1

 

1&9
Equations 1-33 through 1-36 inadequate for quantitative use.
(Extra singularities pose no problem when programs ESRMAP
and GATHER, which are described later, are used.)

Numerical integration was originally performed by using
the method of approximating area with trapezoids. Although
this method worked, it required a rather large amount of
computer time. Using fewer trapezoids conserved some compu-
ter time; however, significant error occurred before the de-
sired computer time reduction was achieved.

If the exponential dominates the function in Equation
l-2l, then, for a sufficiently narrow interval, Equation 3-2
can be approximated by substituting a constant (with an ap-
propriate value) for Sm (H)/BG. The integral in Equation
3-2 can then be approximated by

- H'-H 2
— Z (constant) exp _(____.'_'_)_
n 2 -
n 28G

This approximation was incorporated into a newer version of

ESRSIM. Significantly fewer terms in this summation were re-
quired to achieve a desired accuracy level, as compared with

the number of trapezoids required to achieve the same accuracy
level. This newer version, called ESRSIM-ACCURATE, still re-

quired considerable computer time to simulate an entire spec-

trum with high accuracy.

 

 

 

 

 

of E5
of ti
into
all :
firs
Plate
Exec
exec
trur
whe1
is

the

lar

DPC
03.]

Si!

ge
at

an

ESRSIM-FAST and PEAKS

A number of changes were made to create another version
of-ESRSIM. This version, ESRSIM-FAST, enables calculation
of the unbroadened simulated ESR spectrum and its collection
into channels one Gauss wide. After this has been done for
(all allowed values of mI,.a vector convolution33 with the
first derivative of a Gaussian function is performed to simu-
'late broadening while taking an approximate first derivative.
Execution of ESRSIM—FAST costs only a fraction as much as
execution of ESRSIM—ACCURATE when an entire simulated spec-
trum is produced in each case. Loss of accuracy suffered
when using ESRSIM-FAST is very apparent when numerical output
is compared with that produced by ESRSIM—ACCURATE. However,
the Calcomp plots produced by both programs appear very simi-
lar. Therefore, the error in a simulated spectrum produced
by ESRSIM—FAST is considered to be tolerable. This program
produces two unformatted data files, TAPE21 and TAPE22, which
can be cataloged for later use (such as for replotting the

simulated spectrum or an expanded portion of it).

Program PEAKS enables use of the TAPE21 and TAPE22 files
generated by ESRSIM-FAST to determine magnetic field strengths
at which relative maxima, relative minima, and points that
are 25%, 50%, and 75% of the vertical distance between con-
secutive relative extrema occur in the simulated spectrum.
This type of quantitative information is readily compared

with corresponding data from a quantitative experimental

50

 

 

spe
bee

APP

eqx
anc
obe
us:
th:
pr.
ad'
b0

SD

st
eq
de
la

81

 

 

51
spectrum. The companion programs ESRSIM-FAST and PEAKS have
been made into overlays. A listing of them is included in

Appendix B.
Prqgrams ESRMAP and GATHER

If the nuclear quadrupole coupling parameter is set
equal to zero, if Txx’ Tyy, and T22 all have the same sign,
and if the restrictions necessary for using ESRSIM-FAST are
obeyed, then a pair of programs, ESRMAP and GATHER, can be
used to simulate an ESR powder pattern spectrum. Although
the systems whose spectra can be simulated by this pair of
programs are more restricted, this pair of programs has the
advantage of automatically including all singularities --
both normal and extra -- in the plots of simulated ESR
spectra.

ESRMAP calculates the value of H (the resonance field
strength) obtained as the appropriate root of a quadratic
equation derived from Equation 1-15 (which incorporates
definitions in Equations l-l6 through l-l9)5 for each simu-
lated orientation (0° 0 ; 90°, 0° éch ; 90°) for each de-

--
_-

sired value of mI (the nuclear magnetic quantum number). A

map of calculated resonance field strengths for allowed
(AmI = 0) ESR transitions is sometimes given as printed out-

put for each desired m Execution of this program produces

I.
unformatted data files TAPEAl and TAPEA2 to be read by pro—

gram GATHER.

 

i______

 

 

 

 

simu
simu
one
take
A C:
two
fil.
pro

etc

pro
int
Cor
and
one
st}
re:
th
te
ax

on

 

 

52

Program GATHER enables reading TAPEH2, "gathering"
simulated data from TAPEfll, and efficient production of a
simulated ESR powder pattern spectrum in two versions ~-
one of the unbroadened spectrum before the derivative is
taken and one of the broadened first derivative spectrum.

A Calcomp plot and printed output are given for each of these
two versions. Also produced is another unformatted data
file, TAPE21, which can be cataloged for later use (such as
providing data for further plotting or having the extrema,
etc., determined by an apprOpriate version of PEAKS).

An important part of program GATHER is the interpolation
procedure, which transforms a limited number of data points
into a simulated spectrum with a good signal-to-noise ratio.
Consider the surface of a Sphere sliced by lines of longitude
and lines of latitude. All intersections of these lines in
one octant represent points whose H (magnetic resonance field
strength) values have been calculated in program ESRMAP and
read by program GATHER for a given ml. The axis of this hypo-
thetical sphere represents the z axis of the g and hyperfine
tensors, which are assumed to have the same axes. The x and y
axes of these tensors pass through the equator. A given point
on the surface of this sphere represents an orientation of the
magnetic field with respect to the tensor axes. Now consider
a polyhedron inscribed in this sphere. This hypothetical
polyhedron has a corner at every intersection of lines on the
surface of the sphere. Every corner of the portion of this

polyhedron that is inscribed in one octant represents a

 

 

 

 

simu]
poly}
else1
into
thet
cula

be p

not
pro:
is z
app¢
thr.

cha
the
res
tic

ing

thz
otl
ne
th
ii
at

11

 

 

53
simulated data point. The surface of this one-eighth of a
polyhedron consists of triangles at the pole and trapezoids
elsewhere. Imagine all of the trapezoids being bisected
into triangles. At this point, there are a number of hypo-
thetical triangles, each corner of which represents a cal—
culated H value, and the special interpolation process can
be performed.

It is assumed that the ESR transition probability does
not vary with orientation. Also, the error involved in ap-
proximating the surface of a sphere with that of a polyhedron
is neglected. For each hypothetical triangle, the area is
apportioned to the appropriate channel or channels. If all
three corners represent the same resonance field strength
(H), the entire area of the triangle is apportioned to one
channel. In Figure 6, a hypothetical example is given for
the case in which the three corners represent three different
resonance field strengths. In this case, linear interpola-
tion is used to select a point, which is on the line connect-
ing the corners representing the two extreme values of H
(2995.7 G and 3006.4 G in Figure 6) and which is located such
that the H value at this point is identical to that at the
other corner of the triangle. A line is then defined to con-
nect that other corner and the point thus defined. Within
the accuracy of the approximations made, this resonance
field strength (which is 3003.2 G in Figure 6) is the same
at every point on this line. Parallel to this latter-defined

line, other lines are defined such that they represent

 

 

 

 

m1,

 

 

54

3006.4 G

3005.5 G

n
00

3003.2 G 3002 5

3001.5 G
3000.5 G
2999. 5 G
2998.5 G
2997. 5 G

2996.5 G

 

2995.7 G

Figure 6. Example of Area Apportioning Performed by Program
GATHER .

 

 

 

 

resc
In 1
Gene
fie]
Two
are
ang]
of c
defi
of 1
fig
por1
are:
ing
the
the
hour
abs<
300!

at;

are;
The
the
by :

at

 

 

 

55

resonance field values at the boundaries of the channels.
In program GATHER, these channels are considered to be one
Gauss wide and to represent integral values of the resonance
field. Thus, the boundaries are at half-integral values.
Two series of triangles, which are similar within each series,
are thus defined. The ratio of areas of any two similar tri-
angles is equal to the ratio of the squares of the lengths
of corresponding sides. The areas of strips such as those
defined in Figure 4 are calculated by calculating the area
of the largest triangle of each set of similar triangles,
figuring the areas of the other triangles by using the pro-
portional relationship Just mentioned, and subtracting the
area of every other triangle of that set in turn, progress-
ing from the next-to-largest triangle to the smallest, from
the area that remains after the preceding calculatidnl In»
the example illustrated in Figure 6, the area of the strip
bounded by 3006.u G and 3005.5 0 is apportioned.to the ESR
absorption at 3006 G. The area of the strip bounded by
3005.5 G and 3004.5 G is apportioned to the ESR absorption
at 3005 G, and so on.

The unbroadened spectrum produced by a summation of

33 with a Gaussian broadening vector.

areas is convoluted
The first derivative of the ESR absorption with respect to
the magnetic field strength at (H+0.5) Gauss is approximated

by subtracting the simulated ESR absorption at H from that

at (H+l.0) Gauss.

 

 

 

 

of

ESE

to
Cpl
ra:
ti
et

an

 

 

56
A listing of ESRMAP, GATHER, and an appropriate version
of PEAKS (which is slightly different from the one used with

ESRSIM-FAST) is given in Appendix C.
Using a Distribution of Values for One Parameter

The assumption of constant line broadening did not seem
to describe the true situation for copolymer-attached
Cp2NbCl2. It is sometimes possible for a compound with
randomly-oriented paramagnetic centers to exhibit a distribu-
tion of values for at least one spin Hamiltonian param-
eter.ll-l3 Modifications were made so that programs ESRMAP
and GATHER could simulate an ESR spectrum when one of the
spin Hamiltonian parameters has a distribution of values.
However, when such a distribution was used, the simulated
ESR spectrum still did not mimic the line width behavior of
copolymer-attached Cp2Nb012 very well. (The version of
ESRMAP and GATHER chosen for inclusion in Appendix C allows

only one input value for any given parameter.)
Numerical Solution of Equations

Derived equations relating molecular orbital coeffi-
cients to spin Hamiltonian parameters are given in Chapter
5. These equations were solved numerically for molecular
orbital coefficients, K, and P by using a pair of subroutines
with KINFIT4, Which is a version of KINFIT. An early version
of KINFIT is described in the literature,32 where an example

of subroutine EQN apprOpriate for that version of KINFIT is

 

 

 

 

 

 

 

list

exan

eig}

orb:

frc
weI

KII

of

ve:

th

 

 

57
listed.32 KINFIT4 has been described,31 and an appropriate
example of subroutine EQN has been listed.31

Of the ten unknowns that were not set equal to zero,
eight were determined numerically by KINFIT4. Two molecular
orbital coefficients were expressed as an algebraic solution
in terms of other unknowns and were calculated during each
iteration from expressions containing those other unknowns
in a subroutine used with KINFIT4.

Different initial estimates led to different solutions
from KINFIT4. However, exact or nearly exact solutions that
were obtained when Equation 5-39 was omitted from subroutine
KILMER could be classified in eight classes. (See Chapter 6.)

The pair of subroutines (one of which is a revised form
of EQN) used with KINFIT4 to solve for molecular orbital co-
efficients, K, and P is listed in Appendix D. A much-used
version that does not include Equation 5-39 is not listed in

this dissertation.

Program M. 0. DATA SORT

 

Since KINFIT4 did not include the values of the two
algebraically solved unknowns in the output, these were calcu~
lated in another program. Also, the sum of squares of molecu-
lar orbital coefficients were calculated for each of the two
Kramers doublet orbitals. Sets of numbers which had been ob-
tained as solutions from KINFIT4 were used as input data to
this program, which is referred to as M. 0. DATA SORT. Each

set of numbers was classified according to which (if any) of

 

 

 

 

the

bee

 

abl

 

 

58
the eight classes of solutions it fit. This program has
been writtenlé for a very specific problem and is not suit-

able for general use. A listing of M. 0. DATA SORT is given.

in Appendix E.

 

 

 

CHAPTER 4

ESR SPECTRA SIMULATION RESULTS

ESR spectra of copolymer-attached Cpr01 and copolymer-

3
attached szNbCl2 have been fitted to the spin Hamiltonian
given by Equation 1-20. The assumptions stated in this para-
graph have been made for simulating spectra according to the
spin Hamiltonians given by Equations l-20 and 1-46. Only one
type of paramagnetic center is assumed to be present. Only
AMS = 21, AmI = 0 transitions are considered. (Actually, the
excess of absorption due to AMS = +1, AmI = 0 transitions
over emission due to AMS = -l, AmI = 0 transitions; that is,
net allowed ESR absorption for a S.= l/2 system; has been con-
sidered.) The transition probability is assumed to be inde-
pendent of orientation. (This is a good approximation when
the g tensor is nearly isotropic.) Constant Gaussian broad-
ening is assumed. Hyperfine splitting by only the 93Nb nucle-
us has been considered. The principal axes of the hyperfine
tensor are assumed to be identical with the corresponding 5
tensor axes.

Whenever applicable in this work, both the axis of elec-
tron spin quantization and the axis of 93Nb nuclear spin
quantization are assumed to be identical with the z axis, and

thus with each other.
59

 

 

 

 

60
The spin Hamiltonian parameters given in Table 2 for
copolymer-attached CprCl3 and copolymer-attached Cp2Nb012

5.10 corresponding to the spin

have been used with solutions
Hamiltonian given by Equation 1-46 to simulate ESR spectra
of copolymer-attached CprCl3 and copolymer-attached
szNbClZ. (Actually, ESRSIMrFAST uses a solution applicable
to the spin Hamiltonian given by Equation 1-20, except that
the nuclear Zeeman term is disregarded. Specifying that Q'
be zero makes the solution appropriate for the spin Hamil-
tonian given by Equation l-46.) Equations 1-28 through 1-30,

10 have been in-

which have been given by Stewart and Porte,
corporated into the ESRSIM-FAST computing package. Equation
1-15, which is a solution given for hv by Taylor, Baugher,
and Kriz,5 has been regarded as a quadratic equation in H
and has been solved16 for H. The realistic root, which is
the expression with a plus sign preceding the square root
symbol, has been incorporated into computer program ESRMAP.
(Equations 1-16 through 1-19, which define variables used in
Equation 1-15, are also incorporated into ESRMAP.)

Careful comparison of the magnetic field values at 28
selected features (selected incidences of relative maxima,
relative minima, and points occurring at half the vertical
distance between consecutive extrema) on sections of experi-
mental spectra vs. a spectrum simulated by ESRSIM—FAST has
shown that the spin Hamiltonian parameters given for copoly-
mer-attached Cpr013 in Table 2, except that .000183 cm“1

has been used as the value of 0', lead to a good fit

 

 

 

 

 

 

Table 2. Spin Hamiltonian Parameters Used in Simulating
ESR Spectra.
Copolymer- Copolymer- Cp NbC12 in
attached attached ch oroform:
CprCl szNbClg ethanol (9:1)
Compound at -12 ° at -126 glass at ~196°
gxx 1.9897 1.976 1.980
gyy 1.9806 1.952 1.940
gzz 1.9082 2.017 2.000
Txx(in cmfl) . -0.009278 -o.01059 -o.01066
Tyy(in cm‘l) -0.01ou77 -0.01586 -0.01598
Tzz(in cm'l) —o.021350 -0.00528 -0.00528
Reference This work This work 17

 

(simulated vs. experimental).

When some correction has been

applied to compensate somewhat for variation of experimental

microwave frequency, the root-mean-square simulated vs. ex-

perimental difference has been found to be 3.1 G.

Using zero

as the value of Q' has produced a simulated spectrum having

.only slight quantitative differences from the corresponding

simulated spectrum having 0' = .000183 cm-l.

These differ-

ences do not seem significant when compared with experimen-

tal vs. simulated spectral differences.

Therefore, the value

used for 0' seems insignificant, and the term involving 0'

has been dropped from the spin Hamiltonian.

Thus, the spin

Hamiltonian given by Equation 1-46 eventually was used for

describing ESR of copolymer-attached CprCl

 

3.

However, better

 

 

me:
or!
vai
in
Ch:

de
se
ex
mi
A1
11

et

 

62

results from relating spin Hamiltonian constants of copoly-
mer-attached CprCl3 to expressions containing molecular
orbital coefficients have been obtained when the gxx and gyy
values in Table 2 are interchanged and the Txx and Tyy values
in Table 2 also are interchanged for this compound. (See
Chapter 6.)

The spin Hamiltonian of Equation 1-46 has been used for
describing ESR of copolymer-attached CpZNbClZ. There do not
seem to be enough resolved features of the 2 pattern in the
experimental spectrum of copolymer-attached Cp2Nb012 to per-
mit a confident and precise determination of gzz and Tzz'
Also, the Gaussian broadening does not seem even approximate-
ly constant in this experimental spectrum. Therefore, less
effort has been expended in improving the precision of the
fit for this compound (compared with the effort expended in
improving the precision of the fit for copolymer-attached
CprCl3) after a rather good semiquantitative fit was ob-
tained.

The spin Hamiltonian constants used in simulating the
spectra presented in Figures 7, 8, 9, 10, ll, 15, and 16 as
simulated ESR spectra of copolymer-attached CprCl3 are

listed in a column of Table 2. The spin Hamiltonian param-

 

eters used in simulating the spectra presented in Figures l2,
l3, l4, and 17 as simulated ESR spectra of copolymer-attached
CpZNbCl2 are listed in another column of Table 2. For com-
parison, the spin Hamiltonian parameters reported by Stewart

and Portel7 for niobocene dichloride (Cp2Nb012, not attached

 

 

 

to

als

Cpl

1fi

a l

11:

da‘

pry

 

V8.1

11

F1

h:

 

 

 

63
to any polymer) in chloroform:ethan01 (9:1) glass at -l96°
also are listed in that table.

Figures 7 and 8 show a simulated copolymer-attached
CprCl3 ESR spectrum as plotted from data produced by a mod-
ified version of ESRSIM-FAST. (For simulation of these data,
a version of ESRSIM-FAST has been changed from the version
listed in Appendix B to a version that directs simulation of
data without generation of any plots. All simulated spectra
presented in this dissertation have been plotted by executing
various versions of program ESRPLOT, as stated in Appendix G.)

Figure 9 depicts a simulated ESR spectrum of copolymer-
attached CprCl3 with neither application of Gaussian broad-
ening nor calculation of the first derivative. After data
used in producing Figure 9 are subjected to convolution with
a Gaussian broadening vector and calculation of an approxi-
mate first derivative with respect to magnetic field strength,
then the simulated spectrum illustrated in Figures 10 and 11
results. Simulated data used for Figures 9, 10, and 11 have
been generated by computer programs ESRMAP and GATHER.

Figures 12, 13, and 14 illustrate simulated ESR spectra
of copolymer-attached Cp2NbC12. The simulated spectrum shown
in Figure 12 has been replotted from data generated by ESRSIM-
FAST. The simulated spectra depicted in Figures 13 and 14
have been replotted by using data generated by ESRMAP and
GATHER. Figures 12 and 14 illustrate simulated first deriva-

tive spectra with constant Gaussian broadening.

 

 

 

 

 

64

 

 

 

puma Bm<mISHmmmm II m

mwnca 2H Ipczmmpw ogmHm QHHmzomz
soon .omne. .ommer omen. .omom. .omnu

— b h -

Hopzao pogomuu<upoEmH0doo no Esnuoon mmm noumassum .s shaman

09

L p p b L P b . .

b -

 

 

 

NOIldHOSQU 893 JO BAIiUAIHEU 1981;! 8

 

 

 

65

some em<mtszmmm nu maoozoo oononoo<uao2sfionoo no essoooom mmm ooeaneHm

wmaco 2H reazmmpw oanuo oHemzomz
omme. come. soon comm c.2m sass as

b L b

.m onswfim

 

- n b -

 

 

 

 

 

 

 

 

 

NOIldHOSQU 883 :JO 3AIlUAIHEU 1981;! 8

 

 

 

66

HQDZQQ ponomuu<t9oszaoaoo ho saucepan mmm.pcc0pmopnc: pmumfisefim

wmsmo 2H Ebzumfiw ninth. qumzcmz

cont Dome comm comm oofim comm
r n p b b . — — — . n —

b n p p n b n n

.m opswfim

oomw

 

b

Noilaaoseb 393

 

67

been masses can iazmmm u- maoozoo obnosneeuaoEaHoooo no subpoena mmm ooosassam

moccc 2H Ipczmmhw comcu ccbmzocz

coco come coco comm coco comm
—||D D r D b P D D D D D D D D — D D D D - D D D D — D D D

 

 

 

cch

 

.oa opswfim

N01188OSQU 883 JO 3A118A1830 13813

 

68

mama mmma<c vow m<2mmm II maonzco omnowuu<lhoshaoaoo ho Esauoocm mmm cwumHSEHm

wwocc 2c I
8:. . 8.9.. . £8.5czmfimmm coco doowm Hmzccwflwsu

.HH madman

—| D D D D D D OOMN

 

 

 

 

 

 

 

t:

 

 

 

 

NOIid80888 883 30 3A118Al830 18813

 

69

some amesqummmm In «Hoo2mno cocosoe<usc5aflocoo co eassooom mmm obnoassfim

mmocc zH IHcZUMFm oomcm chmzccz

cmue omcm 035m ommm ommm cmam

omow och oomN

b

.ma ouswfim

 

 

 

 

 

N011880888 sea 10 3A118AI830 19311

 

 

70

 

macachc concepu<rncsmaoooo no saucepan mmm cccocmopncc cmumassam .mH madman
mwomo zo zeozmmem ommou eooemzomz
cage omcm coho comm comm om__m coma ombu oomN
F F! D D D D . D D D . D

 

 

 

 

 

 

NOIld80888 883

 

 

 

damn 53320 new mgmmm II NHOQZNQO Umnowvfialhmfihaocoo .Ho Eshuomam mmm cmumHSEHm £2” @55me

mmoco 2H Ihczmmhm couch. copmzccz

come comm ombm ommm cmmm omum omcm

P; b s _ .

cmbm comm

D D D D

 

D

71

 

 

 

N011880888 as; 10 3A118AI830 19311

 

     

 

 

 

Figure 15.

72

Experimental and Simulated ESR Spectra of
Copolymer-Attached CprCl3

 

 

 

 

 

 

 

 

 

73

ooom
p .

mmoco 2H IPoZMMHm ccmom copmzccz

. . .cc_m¢. . . .oc_c¢ cmmm cmcm cc_mm

on

_om

 

-—--“--——---—- ‘
\~

N011880888 883 30 3AI18AI83O 18813

 

74

Figure 16. Experimental and Simulated ESR Spectra of
Copolymer-Attached CprCl3

 

 

 

75

 

mmocc 2H Ipozmmp

m ocmcm cohmzocz

comm

D 1-

D

comm

 

 

 

_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
.
_
_
_
_

 

 

  

_~-_—----—~—_-

 

 

 

~‘----_
—-——_—n-—————————

 

 

     

N011380888 883 30 3A11UA183O 18813

 

 

 

 

 

 

«Poozmco occusup<tioexroooo mo sspumom «mm ompmpzewm can —eucmswimaxo .up mszovm

mmocc 2H :Hozmmpm ccmom coewzcmz

comm comm comm cowm comm ochm
_ _ _ c e, . to «t

D D L P

ocu¢ cmmm

 

 

 

 

 

 

 

 

 

on

 

65
cu

N011380888 883 30 3AI1HAI83U 18813

 

 

 

 

 

77

The simulated spectrum shown in Figure 13 represents an ESR
absorption spectrum with no Gaussian broadening (no signifi-
cant variation in local environments). If data depicted in
Figure 13 are subjected to constant Gaussian broadening and
an approximate first derivative with respect to the magnetic
field strength is calculated, then one resulting simulated
spectrum is shown in Figure 14.

A comparison of Figures 9 and 13 can illustrate the ef-
fect of the number of 6 and ¢E orientations on simulated un-
broadened ESR spectra. (In each of these figures, the most
intense absorption peak has been cut off and the vertical
scale has been expanded to show details of the other features
more clearly.) Data used in producing Figure 9 were simu-
lated when 61 6 orientations times 61 oE orientations were
used for each value of ml, the nuclear magnetic quantum num-
ber of the 93Nb nucleus. However, data used in producing
Figure 13 were simulated when only 19 6 orientations times

19 0E orientations were used for every value of m Some

I’
noise can be seen in Figure 13. However, convolution with a
Gaussian broadening vector smooths out this noise. The
broadened derivative spectrum, which may be compared with
the experimental spectrum, has an excellent signal-to-noise
ratio, even when only 19 0 orientations times 19 0E orienta-
tions are used for each mI. (See Figure 14.)

Figures 15 and 16 illustrate experimental (solid lines)
and simulated (dashed lines) ESR spectra of copolymer-

attached CprCl The experimental spectra in these figures

3.

 

 

 

 

 

78
also are shown in Figures 3 and 4, respectively. The sim-
ulated spectra depicted in Figures 15 and 16 have been re-
plotted from data generated by executing ESRMAP and GATHER.
These simulated spectra also have been plotted in Figures
10 and 11, respectively.

Figure 17 illustrates experimental (solid line) and
simulated (dashed line) ESR Spectra of copolymer-attached
CpZNbClz. This experimental spectrum also is shown in Fig-
ure 5. For the simulated spectrum pictured in Figure 17, a
simulated baseline exhibiting a linear drift has been added
to data generated by using ESRMAP and GATHER. This simulated
spectrum has been plotted without the baseline drift in
Figure 14.

As mentioned in Chapter 2, 4000-0 scans performed by
the varian E-4 EPR Spectrometer have strayed from the indi-
cated Gauss per inch scale. Therefore, for presenting both
experimental and simulated spectra in Figure 15 and in Fig-
ure 16, two peaks, each of which is located at a magnetic
field strength in the experimental spectrum very close to
the location of the corresponding peak in the simulated spec-
trum, have been used to match the experimental and simulated
horizontal scales. (See Chapter 2 for related comments and

for other information regarding experimental conditions.)

 

 

 

 

 

 

Experimental vs. Simulated ESR

 

Spectra -- Copolymer-Attached CprCl3

The experimental ESR spectrum of copolymer-attached
CprCl3 at -l26° exhibits some extra features that do not
appear in the simulated spectrum. These features appear
relatively minor. The two such extra features closest to
the most prominent feature have been interpreted as arising
from "forbidden" transitions, i.e., transitions involving
changes in both the electron magnetic quantum number and the
93Nb nuclear magnetic quantum number. No explanation is of-
fered for the other "extra" features.

There also appear to be some systematic differences in
the magnetic field strengths at which the most prominent
features occur. These differences are thought to arise

from a 93

Nb nuclear quadrupole effect, which is not ade-
quately described in Equation 1-20. If these differences
actually are due to a 93Nb nuclear quadrupole effect, then
failure to represent this effect in the spin Hamiltonian
(Equation 1-46) leads to estimated errors of approximately
.00005 cm"1 in the principal values of the hyperfine tensor
but to no significant error in the principal values of the
g tensor.

Overall estimated uncertainties in spin Hamiltonian
parameters of copolymer-attached CprCl3 are approximately

.002 for gxx, g , and gzz and approximately .0001 cm'1 for

yy

T , and Tzz'

XX’ Tyy

79

 

 

 

Experimental vs. Simulated ESR

 

Spectra -- 00polymer-Attached Cp2NbC12

 

The experimental ESR spectrum of copolymer-attached
CpZNbClZ at -126° does not seem to exhibit constant broaden-
ing. However, a constant Gaussian broadening function was
used in producing the simulated spectrum.

There appear to be some systematic differences in the
magnetic field strengths (around 2900 - 3200 G in Figure 17)
at which some features occur. These differences are similar

to those for copolymer-attached CprCl and again are thought

3
to arise from a 93Nb nuclear quadrupole effect. If these
differences for cepolymer-attached Cp2NbC12 actually are due
to a 93Nb nuclear quadrupole effect, then failure to include
this effect in ESR simulation leads to estimated errors of

approximately .00005 cm"1

in the principal values of the
hyperfine tensor but to no significant error in the principal
values of the g tensor.

Overall, the uncertainties in spin Hamiltonian parameters
of copolymer-attached Cp2NbCl2 are estimated to be approxi-

l for

mately .002 for gxx and gyy and approximately .0001 cm-
Txx and Tyy. The experimental spectrum does not seem to ex-
hibit enough resolved features of the 2 pattern to permit a
confident determination of gzz and Tzz. Thus, the values
listed in Table 2 for gzz and T22 of this compound are not

considered reliable.

80

 

 

CHAPTER 5

MOLECULAR ORBITAL TREATMENT
I. DERIVATION OF EQUATIONS

This chapter applies only to compounds whose paramag-
netic centers each have exactly one unpaired electron and
one natural center (such as exactly one transition metal
atom) about which the angular momentum of that electron can
be calculated. 7

The portion of the spin Hamiltonian describing the

Zeeman interaction for one electron is gfiegs, which can be

1

written as

.EBeflfi g gxxBeHxSx + gyyBeHySy + gzzBeHzSz (5‘1)

when the principal axes of the g tensor are defined such
that the off-diagonal elements are all zero.
Solving Equations 1-8 and 1-9 for 8x, 8y, Hx’ and Hy

yields these equalities:

ex = g— (a, + s_) (5-2),

«3y = i- <-§+ + §_> <5-3).
1

Hx = '5 (11+ + H_) (5’14):

81

 

 

82

<-H+ + H_> (5-5).

{It
II
N IP-

Substituting Equations 5-2 through 5-5 into Equation 5-1

produces
Eflefls = gzzBeHzSz
J: A A A A
+ L1 [(S+ + S_)gxx + (-34. + S_)gnyBeH+
+-l—[(§ +8) +(§-§)g 18H
u + - gxx + - yy e - (5-6).
Consider the normalized spin wave functions dWm and
I
BYm . Tm is the nuclear Spin eigenfunction for a nucleus
I I

with nuclear magnetic quantum number mI and nuclear spin 1,
a is the electron spin eigenvector for an electron spin of
+l/2, and B is the electron spin eigenvector for an electron
spin of -l/2. Equation 5-6, which considers only the elec-

tronic Zeeman interaction, has been used to obtain

1

<BWmIIEBe§SIBVmI> = ‘ E gzzBeHz (5-7),
<dWmIIEBeE§IaYmI> = + E gzzBeHz (5-8),
<0L‘l’mI [589.118 I B‘YmI)

- —; [(gxx ~ gyy)H+ + (gxx + gyy)H-] (5‘9)’

and

 

83

<BWmII58eE§IGYmI> a

B

‘77 “gm + gym, + (gxx " gyym-J (5-10).
These are spin Hamiltonian matrix elements. Eventually,
gxx’ gyy, and gZZ are to be replaced with experimental values.

TQ and TR represent the wave functions of the ground

state Kramers doublet that contains the unpaired electron:
WQ has the lower energy in a magnetic field. Expressions
for WQ and TR can be substituted into <WQ|EZIWQ>c
<‘1‘RIEZI‘P >, WRIEZI‘PQ , and <rQ|iEz|rR>, where 1132 is given1
by Equation 1-6, and the results can be equated to the right!
hand sides of Equations 5-7 through 5~10, respectively. This
new set of equations can be solved algebraically for gxx:
gyy, and gzz. (Sometimes two different expressions may be

obtained for gzz -- one by using TQ and one by using YR.)

Since the values of spin Hamiltonian parameters gxx,
gyy, and gzz do not depend on any of the components of the
external magnetic field, no terms involving any components
of the external magnetic field should appear in the final
solutions for these spin Hamiltonian constants. If such a
term does appear, the offending term could be removed by ex-
pressing it as a magnetic field factor multiplied by an ex-
pression that does not contain any elements of the magnetic
field, and then setting the latter expression equal to zero.
Another equation would then exist to aid in solving for un-

known molecular orbital coefficients.

 

 

84
The portion of the spin Hamiltonian describing the hyper-

fine interaction between one electron and one nucleus can be

written1 as

A A

ESI = TXXSXIX + Tyy ny + TZZ ZIZ (5-11)

if the T tensor axes are chosen such that all of the off-

diagonal elements of T are zero.

If 1, and i_ are defined by

PD
N
FD
+
[A
F!

+ x y (5-12)
and

I- = Ix - in (5‘13):
then

I = i (I + I )

x 2 + - (5-14)
and

A =i -I\ A

Iy 2 ( 1+ + I-) (5'15).

Substituting Equations 5-2, 5-3, 5-14, and 5-15 into Equa—

tion 5-ll produces

AA AA

TZZ ZIZ

la
m
#1
H

+ %»[TXX(§+ + s-) + T (’s‘+ - §-)JT-

(§ + g ) + T (‘§+ + §—)Ji+ (5-16).

 

85
Equation 5-16 has been used to obtain algebraic expressions

AA

for matrix elements of TsI. These expressions are given in

Equations 5-17 through 5-20. For one electron and one nucle-

us with nuclear Spin 1,

l

(WmIBIESIIWmIB> = I E mITzz (5—17),
<9 angIIT d> = + l'm T
m ._ mI 2 I 22 (5-18),
<Wm +1913§i|wm 3) =
I I
1 1/2
- + - + 1 T - T
4 [I(I l) mI(mI )1 I xx yy] (5_19)’
and
<2 angIlT 8> =
mI mI+l
l [I(I + l) - m (m + 1)]l/2[T + T ]
4 I I xx yy . (5-20).

Solving Equations 5-17 through 5—20 for the diagonal elements
of T, replacing TsI with (RSI +;HLI), substituting VQ for B,
and substituting YR for a lead to Equations 5-21 through 5—24:

 

=-§— A A
Tzz mI (WmeII(KSI +1HLI)IYmeI> (5-21),
T = + 3— <r r |<fi +13 )lr v >
zz mI R mI SI LI R mI (5—22),
2 c A
T = [<r r (K +'K ) r r >
XX [I(I+l)-mI(mI+l)11/2 R mI+1| SI LI I Q mI

+ <memI|(wSI + niI)IWQWmI+1>3 (5-23).

 

 

 

86
and
T = 2 1/2 (<1?er WESI + JELIH‘PQrm +1>
yy [I(I+l)-mI(mI+l)] I I
' ‘wn‘yml + 1Im31 * mLIHrQrmIfl (5-24).
H61 and “II are given1 by Equations l-ll and 1-13, respective-

ly. These can be combined to give
A = —_ 6 “
+ “LI P'[ 3 (r)sz +

(3 cosze - Dr’s;z +

3 sin9 cosB r'3(e-i¢§+ + e1¢§_)
2 ~3“ A
+ —————— r l I +
2.0023 2] Z

P'[o£1 6(P)S+ + l (1 _ 3 cosze)r—3S+
3 4

2 -3A .3A
id)r s_ + ; sine cose ei¢r sz

2
0
sin e 2

+

Jolt»

P'l: 5% 6(r)§_ + % (l - 3 cosze)r'3§

+ i sinze e"21¢r'3s + i-sine cose e“i¢r'3§
u + 2 Z
l -3“ A
+ _—
2.0023 r 1"] I“ (5'25)°

P' is defined by Equation 1-12.

 

 

87
The wave functions WQ and WR are defined to be
+ a

W
Q xy dxy xz-y2 dxz—yz xz dxz

*E
N
A
m
'6

+ ayzwd + azzwd 2 + asTS)B
yz z

+ (b v + b ,_ ,v + b r
d x y dxz 2 xz dxz

xy xy -y

+ byzwd + 1322de2 + bSWS)a

yz

+ terms having the form (constant) .

(ligand wave function) . (electron spin eigenvector)

(5-26)
and
r = (c r + c 2_ 2m + c w
R xy dxy x y de-yz xz dxz
+ cyzvd + czz‘yd 2 + csws)a
yz z
+ (e T + e 2 2? + e T
xydxy x y dx2_y2 xz dxz
+ eyzwd + ezz‘i’d 2 + eSTS)8
yz z
+ terms having the form (constant) ~ (ligand wave
function) . (electron spin eigenvector) (5-27).
The wave functions Yd , id 2 2, 7d , Wd , and Yd 2 are
xy x -y xz yz z

wave functions for the indicated d orbitals of the central
metal atom. The symbol TS represents a wave function for an
s orbital of the central metal atom or a combination of wave

functions for s orbitals of the central metal atom.

 

88

By definition, the x, y, and z axes are the principal
axes of the diagonal g tensor. It is assumed that the
principal axes of the diagonal T (hyperfine) tensor are
identical to the corresponding principal axes of the diagonal
g tensor. Also, it is assumed that the z axis is the axis
of quantization of electron spin, electronic orbital angular
momentum, and nuclear spin. The following set of equations
apply only when both of these assumptions are true.

In addition to those assumptions and the restrictions
given in the first paragraph of this chapter, some approxima-
tions have been made in deriving the following set of equa-
tions. The terms involving VS or ligand orbitals have been
neglected except for the contribution of terms containing
W8 to the isotropic part of the hyperfine interactions.

Also, all of the isotrOpic part of the hyperfine interactions

is collected into terms involving KP, where P is defined by

__ -3
P - 2.0023 gNBeBN <r >av (5~28).
Contributions to this isotropic interaction are expected
from spin polarization in addition to contributions from
terms containing TS. Therefore, no attempt has been made to

retain a bs, cs, and e in the derived equations. All

8’ S

overlap integrals involving VS or ligand orbitals have been

neglected.

In the absence of any experimentally-observed ligand
hyperfine interactions, the neglect of terms involving ligand

orbitals appears to be a good approximation when considering

 

89

hyperfine interactions. In a given direction from the
nucleus of the central metal atom, the anisotropic part of
the hyperfine interactions involving that nucleus is inverse-
ly proportional to the cube of the distance of the unpaired
electron from that nucleus. Also, the isotropic part of the
hyperfine interactions is not significant except at the
nucleus. However, neglecting terms involving ligand orbit-
als and 93 may introduce significant error when Zeeman inter-
actions are considered. For cases in which these approxima-
tions introduce too much error, the right hand sides of the
following set of equations (with omission of the terms con-
taining K if a more detailed interpretation of the isotrOpic
part of the hyperfine interactions is performed) can be in-
cluded as parts of a more general treatment.

Inspection of Equation 5-25 reveals that
<TJI(ESI +J€LI)|Wk> (where J = Q or R and k = Q or R) would
contain a term independent of r (isotropic or contact term)

plus terms having P (as defined by Equation 5-28) as a com-
mon factor. Other than the <r">av factor in P. r would not
appear in such integrals after integration is performed.
Inspection of Equation 1-6 reveals that r would not appear
in <rJ|fiz|wk> (where J = Q or n and k = Q or R). Thus, in-
tegration over r can be considered complete when the ~KP
isotropic (contact) term and the factor <r'3>av (which is in-
cluded in P) are introduced into the hyperfine expressions

and the appropriate spherical harmonics or linear combina-

tions of spherical harmonics are substituted for d orbital

 

90
wave functions. Therefore, it has been sufficient to con-
sider only the integrations over 9 and 0 in detail. To
simplify evaluation of these integrals, expressions for YQ
and WR (equivalent to those given in Equations 5-26 and 5-27
when only terms that include central atom d orbital wave
functions are included) have been written in terms of spher-
ical harmonics with complex and real coefficients. When

ligand orbitals and s orbitals are omitted, these expressions'

are

i, 1
rQ [_ (ax2_y2 + axyi)Y2_2
.I2
1

+—

./2 M
+ —l (-a + i)Y
/2 xz ayz 21

l
+ 72- (ax2_y2 "' axy1)Y22] B

1
+[ 72- (bX2-'y2 + bxy1)Y2_2

l
+ 35'(bxz + byzi>Y2-l + bzzY2O
+ —l (-b + b i)Y
y2 xz yz 21
+ —i (b 2 2 - b i)Y c
/2 x -y xy 22 (5-29)

and

91

+
_ 2 Oxyi)Y2-2

(c + c zi)Y2_

1
--— +
/2 XZ y czzY

l 20

l
—— - +
)2 ( cxz cyz1)Y2l

1
+ ./2 (cx2_y2 ' °xy1)Y22] °‘

1
+ [ 75 (ex2_y2 + exy1)Y2_2

l
/2 + eyzi)Y2_l + ezzY2O

1
7E (-exz + eyzi)Y2l

1
+175 (ex2_y2 - exy1)Y22:]B (5'30)-

By definition, 1 is a square root of -1. All of the twenty
coefficients represented by the letters a, b, c, and e with
subscripts on the right hand sides of Equations 5-29 and
5-30 are real numbers. In these two equations, Ylml are
spherical harmonics associated with quantum numbers 1 (or-

bital angular momentum quantum number) and ml (magnetic quan-

tum number). These spherical harmonics are1

1 15 1/2 _2
Y2-2 = 74‘ (‘2?) Sinze 6 im (5-31),
1/2
1 l
Y2-1 = '9: (315;) sine cos6 e-iq> (5-32),

(E?) (3 ”S 9 " 1) (5-33).

92

- 1 15 1/2 i¢
Y - _ .__
21 2 (2n) sine cose e (5-34),
and
1/2
=.i .l3 2 21¢
Y22 h 211) Sin 6 e (5‘35).

The approximations given by Equations 5-29 and 5-30
were used for TQ and WR in deriving all of the equations
presented between this paragraph and the end of this chapter.
Also, -KP has been added to every expression shown as being
equal to a hyperfine tensor component.

Equating <TQIEZIYQ> to the right hand side of Equation

5-7 leads to

09
ll

2.00232[(a ,' 2)2 + (a )2
22 x —y xy

2 2 2
+ (axz) —+ (ayz) + (322)

2
- (bx,_y,)2 - (bxy)2 - (bXz

— (byz)2 - (022)21 (5-36).

Equating <vR|K2|wR> to the right hand side of Equation

5-8 leads to

93

2.00232[(cx2__yz)2 + (cxy)2

+ (cxz>2 + (cyz)2 + (cz.)=

2)2 - (e )2 "' (e )2

- (e 2 xy xz

x “y

2
‘ (eyz)2 - (922) J

Equating the right hand side of Equation 5-9 to

(5-37).

<rRLKz|rQ> and the right hand side of Equation 5-10 to

<TQLHZIYR>, performing some algebra, and making the require-

ment given by Equation 5-40 produces

09
ll

2.00232Eaxz_ 2c 2 2 +

a me
xx y x -y xy xy

3 c + a c + a 20 2
xz xz yz yz z z

+ b e + b
x2_ 2 2

2 e + b e
y x -y xy xy xz xz

+
byzeyz bzzezz]

and

(5-38)

94

g = 2[a e -
yy xz-y2 xz axzexz-y2 + axyeyz

a e + b c
YZ xy Xz-y2 xz bxzcxz-y2

+b ]

C C
xy yz yz xy

4.

[2-73][axzezz - azzexz

+ b c - b
xz z z xz

+ 2.002 2
3 [ax2_y2c 2

X ‘y

4.

a c e + a 0
xy xy xy xy xz xz

I bxzexz + ayzcyz I byzeyz
+ a c - b e ] .
22 22 z2 22 (5-39).

A precursor to Equation 5-39 included an expression
multiplied by Hx/Hy’ Since gyy must not depend on Hx/Hy and

Hx cannot be set to zero arbitrarily, that expression is re-

quired to equal zero. This requirement is stated as
axzexy I axyexz + axz-yzeyz I ayzexZ-y2

+ bxzcxy I bxycxz I bxz-yzcyz I byzcxz-y2

1/2 _ _
+ (3) (azzeyz ayzezz + bzzcyz byzczz)

(5-40).

ll
0

95
Substituting Equation 5-29 into Equation 5-21 and adding
the isotropic term (-KP) to the right hand side of the re-

sulting equation leads to

T = -KP +
zz

4
{P}{- 7 “3x232” + (axym

4 2 2
+ ?'[(bxz_y2) + (bxy) 1

+ % [(axz)2 + (ayz)2]

2 2
- - [0)sz + (tn) 1

7
+%(a22)2 --f‘,—(bzz)2
-_6.a b2 2"_6.a b
7 xz x -y 7 yz xy

2-J3
-<g%£%>(azszz) I(I;II>(axzb22)

- 6 - 6
7 axz-yszz 7 axybyz} (5-41).

Substituting Equation 5-30 into Equation 5-22 and adding
the -KP term to the right hand side of the resulting equation

yields

96

 

2
+ 7 [(cxz)2 + (cyz)2]
+[—-‘27 [(e )2 + (e )2]

7 d xz yz

1'1 I 2 2
+ [ éfi] IC 2 e + C e

7 L x -y2 xz xz xz-y2

 

C e + C e ]
xy yz yz xy

2 1/2 t
+ [T?.] [3] [cxzez2 I czzex;]}- (5-42).

Substituting Equations 5-29 and 5-30 into Equation 5-23
produces a complex expression on the right hand side. Since
Txx is real, it may be regarded as Txx + Oi. Equating Txx
to the real portion of that complex expression plus the con-

tact term (-KP) leads to

97

T = -KP +

xx
2]? i [a 2 C + a C - a C
7 x -y2 xz-y2 xy xy 22 22
+ b e + b e - b e
x2.y2 x2_y2 xy xy 22 22]
l 1 2
+ ? axzcxz ‘ ;’ayzcyz + 7'bxzexz - 7'byzeyz
(3)1/2
+ T— ["azzcxz-y2 ' axz-yzczz ‘ bzzexz--y2 " bx2_y2ezz]
+’3

I; [-axzex2_y2 - ayzexy - ax2_yzexz - axyeyz

J

+ +
bxzcxz-yz + byzcxy bxz-yzcxz bxycyz

1 2 1
+ [3] / [TE-h bxzczz + bzzcxz - axzezz - azzeXZJ}
Equating the imaginary part of that complex expression

to zero (01) leads to

(3)1/2
o = 2P1{1——;—-— [azzc + a c 2 ' bx e 2 ‘ bz2e 3

xy xy 2 y 2 xy
+ -§ [-a c - a c + b e + b e J
11; xz yz yz xz xz yz yz xz
2

+ §f35§§§ [axyexz ' axZ-yzeyz + ayzexz-y2 “ axzexy

- + - b c
+ bxycxz xz-yzcyz byzcxz-y2 xz xy]
+ ___§___ [3]1/2[a e 2 - b 2c
2.00232 yz z z yz

+ byzczz -' azzeyZJ} (S-Lll-l).

98
Substituting Equations 5-29 and 5-30 into Equation 5-24
also leads to a complex expression on the right hand side.
Tyy is equated to the real portion of that complex expression

plus the contact term (-KP). In this way, T is found to be

yy
'I'yy = -KP +
2Py{— [axz cx 2-y2 axycxy azzczz
- - +
bx2_y2ex2_y2 bxyexy bzze 21
2 1 2 l
- -’ + — + —-b e - — b e
7 axzcxz 7 ayzcyz 7 xz xz 7 yz yz
+ “;——— [azzcxz _y2 + axz -y2cz 2 - bx2_yzezz- b H26 2_ yz]
+ ___§___ [ + a e - a e - a e
2.00232 axz-yzexz xy yz xz xz-y2 yz xy

+ bxz-yzcxz + bxycyz - bxzcxz-yz - byzcxy]

1/2 2 _ j}
+ [3] [2.00232:H:axzez2 - azzexz + bxzcz2 bzzcxz
Equating the imaginary part of that complex expression

to zero (01) produces

99

(3)1/2
0 - 2P1 —-;7--- [-azzcxy - axyczz

J

- b e 2 - bzze

Xy Z xy

+ 3

a c + a c + b +
14 xz yz yz xz xzeyz byzexz

a e a e + a e
yz xz-y2 xz xy xz-y2 yz

a e
xy XZ

+ b c + b
xz-y2 yz xzcxy byzcxz-y2

]

c
xy xz

_1_ 1/2 _
+ l“ [3] [byzczz azzeyz

+ b22°yz ‘ ayzezzl} (54:6).

Equations 5-38 and 5-“6 have been used to solve alge-
‘braically for cxy and cxz-yz in terms of other unknowns and

Sxx. To facilitate performing algebra and expressing the re-

sults, F1’ F3, and D13 have been defined by

g
F - __§£_—— - [a c + a c

1 ‘ 2.00232 xz xz yz yz

+ a c + b e + b e
22 z2 xz-yz xz-y2 xy xy

+ + + b e
bxzexz byzeyz z2 22] (5-h7),

 

100

F3 = 2(axyczz + bxyezz + bzzexy)

+ (3)1/2Ca e + a e - a 2 2e
xy xz xz xy x -y yz

 

 

- - +
ayzexz-yz xycxz bxz-yzcyz
' axzcyz ‘ ayzcxz ' bxzeyz ' byzexz)
+ - -
(ayzezz + azzeyz byzc22 bzzcyz) (5-h8),
and
D13 — Zaxz’yzazz (3) axybyz (3) ax2_y2 xz
(5-N9).
Then cxy and cx2_y2 are given by
1/2
D
13 (5-50)
and
1/2
cx2_y2 = F3axy + 2Flaz2 ' (3) Flbxz
D1
3 (5-51).

None of the assumptions made in deriving Equations 5-36
through 5-51 have specified which of the d orbitals are the
main ones represented in the Kramers doublet containing the
unpaired electron in the ground state. However, these equa-

tions contain 22 unknowns to be determined. Comparing this

101
number of unknowns with the number of equations indicates a
necessity of making more assumptions either to decrease the
number of unknowns or to increase the number of relation-
ships involving them. Some of the unknowns can be eliminated
by specifying one or two d orbitals involved in the zeroth—
order approximation of the ground state and allowing only
terms involving these orbitals and terms mixed in by spin-
orbit coupling in the anisotropic parts of TQ and TR. Two
d orbitals have been selected to define a zero-order approx-
imation of the Kramers doublet of molecular orbitals contain~
ing the unpaired electron in copolymer-attached CprCl3 in
the ground state. The choice is supported in Chapter 6, in

which assumptions are stated and the chosen d orbitals are

indicated.

 

 

CHAPTER 6

MOLECULAR ORBITAL TREATMENT
II. RESULTS FOR
COPOLYMER-ATTACHED CprCl3
Z Axis, Zeroth-Order Ground StateL
and Identification of Z Pattern

of Copolymer-Attached CprCl3

 

A difficulty typically encountered in ESR powder pat-
tern studies is a shortage of experimentally-obtained values
compared to the number of unknowns. In such studies, rea-
sonable assumptions and/or approximations are often made.
Such assumptions and approximations have been made in this
.work. Much of the unknown area in this work involves rela-
tionships among sets of axes and the molecular geometry.
Even the molecular geometry of copolymer-attached CprCl3
(which is not crystalline) is unknown. A reasonable set of
approximations and assumptions has led to a logically con-
sistent (although not certain) identification of the z axis,

g , and Tzz' These identifications, which are described in

22
the following paragraphs, permit some quantitative conclu-
sions to be drawn about some molecular orbital coefficients
of copolymer~attached CprCl3.

102

 

 

 

Th:
axes.
when Q'
fine th
been as
Furthel
and th«
sumed
invoke

1
dienyi

magnef

. simil

copol
MeCpl
tends
hyp0‘
the

Den-
att
axi
Pix

Dal

103

The x, y, and z axes have been defined as the g tensor

 

axes. However, the mathematical equivalence of x, y, and 2
when Q' equals zero implies that the g tensor does not de-
fine the axes labels. The set of hyperfine tensor axes has
been assumed to coincide with the set of g tensor axes.
Furthermore, both the axis of quantization of electron spin
and the axis of quantization of nuclear spin have been as-
sumed to coincide with the z axis. This assumption is not
invoked to determine which axis is the z axis.

The hypothetical compound trichloro(ns-methylcyclopenta-
dienyl)niobium (MeCprCl3) has been defined because its para-
magnetic center is expected to experience a local geometry

similar to that in the vicinity of a paramagnetic center of

 

copolymer-attached CprCl and because the hypothetical

3
MeCprCl3 molecule is sufficiently simple to permit an ex-
tended Huckel molecular orbital (EHMO) calculation. The

hypothetical MeCprCl -molecule is pictured in Figure 18 if

3
the * in that figure represents a hydrogen atom. If that *
represents attachment to styrene-divinylbenzene copolymer,
then Figure 18 represents an assumed structure of a
-CHZCprCl3 unit in copolymer-attached CprCl3.

In the hypothetical MeCprCl molecule, the five cyclo-

3
pentadienyl ring C atoms and the four H atoms and one O atom
attached to that ring were defined to be coplanar. The z
axis was defined to be perpendicular to the plane of this
ring. The three Cl atoms were defined to lie in a plane

parallel to the plane of the ring (and therefore perpendicular

 

10h

 

 

 

 

 

Figure 18. Cepolymer-Attached CprCl3 or the Hypothetical
MeCprCl3 Molecule

 

 

to the
which t
pass t1
nucleu:
by the
on thi
the an
orbit:
state
would
hdxy
of a
are \
orbi1
is si

ment

Drir
attz
biuJ
are
sor
If
con
tiw
bu

t1

105
to the z axis) but on the other side of the Nb nucleus, at
which the origin was defined. The 2 axis was defined to
pass through the center of the five-carbon ring, the Nb
nucleus, and the center of the equilateral triangle formed
by the three Cl atoms. An EHMO calculation was performed
on this hypothetical MeCprCl3 molecule in order to estimate
the main atomic orbital or orbitals involved in the molecular
orbital occupied by the unpaired electron in the ground
state. According to this calculation, the unpaired electron
would occupy a molecular orbital consisting chiefly of a Nb
hd atomic orbital or a molecular orbital consisting chiefly

xy
of a Nb hdx2_y2 atomic orbital. These two molecular orbitals
are very close to each other in energy. Another molecular
orbital, which consists mainly of a Nb “dzz atomic orbital,
is slightly higher in energy than the two previously-
mentioned molecular orbitals.

ESR simulations by ESRSIM-FAST have shown that the three
principal components of the hyperfine tensor of copolymer-
attached CprCl3 all have the same sign. As a typical nio-
bium compound has hyperfine tensor principal components that
are negative, the principal components of the hyperfine ten-

sor of copolymer-attached CprCl have been assumed negative.

3
If this assumption is true, then the almost unique principal
component of the anisotropic hyperfine interaction is nega-
tive, and the other two principal components are positive

but have smaller absolute values. Comparison of this descrip-

tion with Table l eliminates the possibility of having a dz;

 

 

 

orbit:
that
in th

copo]
pertI
of t
circ
won]
ula:
men‘
sul
ato
sex
so
to
pe
te

106
orbital as the major contributor to the molecular orbital
that contains the unpaired electron. However, at this point
in the logic, four possibilities remain..

To a first approximation, the spin Hamiltonian of
copolymer-attached CprCl3 may be considered as slightly
perturbed from axial. If the methylcyclopentadienyl part
of the hypothetical MeCprCl3 molecule were replaced with a
circle defined by the five ring C atoms, then this molecule
would have C3v symmetry. When made in a study of the molec—
ular orbital containing the unpaired electron, this replace-
ment is a reasonable approximation, largely because EHMO re-
sults have indicated this orbital to consist mainly of a Nb
atomic orbital. By analogy (which is supported by the ab-
sence of ligand hyperfine splitting in the experimental ESR
spectrum), this approximation of C3" symmetry is considered
to be good enough in actual c0polymer-attached CprCl3 to
permit identification of the z axis of the spin Hamiltonian
tensors with the approximate local 03v axis. This axis is
analogous to the z axis of the hypothetical MeCprCl3 mole-
cule. Invoking EHMO results for the hypothetical MeCprCl3

molecule, one can infer that the unpaired electron of actual

copolymer-attached CprCl3 resides in a molecular orbital

composed mainly of a niobium Adxy orbital or a molecular
orbital composed mainly of a niobium fldx2_y2 orbital. Table

1 shows23 that the anisotropic hyperfine angular factors axx’

“yy’ and a are 2/7, 2/7, and ~4/7, respectively, for each

zz
of these two cases. Therefore, the almost unique pattern in

 

 

the ESR
the z p
the spi
disser‘
T
labeli
ity ha
orbits
been c
have
CprC
inter
The e
iden‘
mole

88811

to

att
an<
KI
ha

107
the ESR spectrum of this actual compound is identified as
the 2 pattern. (This identification is consistent with
the spin Hamiltonian parameters reported elsewhere in this
dissertation.)

There is not enough information to permit confident
labeling of the other two (x and y) patterns. This ambigu-
ity has been carried over into the determination of molecular
orbital coefficients: Molecular orbital coefficients have

been determined for the case in which gx T x’ and T

x’ gyy’ x yy
have the values given in Table 2 for copolymer-attached
CprCl3 and also for the case in which x and y have been
interchanged to obtain the values presented in Table A.

The experimental and EHMO data are not adequate to permit
identification of the x and y axes with respect to the

molecular geometry, except that the x, y, and z axes are

assumed to be mutually perpendicular.

Determination of Molecular Orbital
Coefficients of Copolymer-Attached CprC13

 

Subroutines EQN and KILMER have been used with KINFITM
to solve for molecular orbital coefficients of copolymer—

attached CprCl Equations 5-36. 5-37. 5-39 through 5-“5,

3.
and 5-47 through 5-51 have been incorporated into subroutine
KILMER (except that the nonzero factor 2P1 in Equation S-uh
.has been omitted in subroutine EQN).

The zeroth-order approximation of the Kramers doublet

of molecular orbitals containing the unpaired electron in

 

 

 

 

 

the Er°\
Nb dx2_:
electrO'
orbital
higher-
is assx
these 1
orbitai
cyz, c
subrou
cxy ar
other:
known
turne
mates
set c
set <
fer ‘

0ft

ofa

wit!
0m1‘
mai

ret

f0l

108

the ground state is assumed to consist of only Nb dx and/or

y

Nb dxz 2 atomic orbital wave functions multiplied by the

-y
electron spin eigenvector B for the lower-energy molecular

orbital or by the electron spin eigenvector a for the
higher-energy molecular orbital. Only spin-orbit interaction
is assumed important in mixing other wave functions into
these two molecular orbitals. Therefore, the molecular

orbital coefficients axz’ ayz, azz, bxy’ bx2-y2, bzzs 0x2,

c 2, e 2, and e 2 have been equated to zero in

xy’ exz—y z

subroutine KILMER. Two other molecular orbital coefficients,

yz’ cz

cxy and cxz_y2, are expressed algebraically in terms of
others in that subroutine. Thus, KINFITA has only eight un-
known parameters to adjust. Nevertheless, the solutions re-
turned by KINFITA have been sensitive to the initial esti-
mates of unknown parameters. A different solution for this
set of unknowns has been returned for almost every different
set of initial estimates. ("Solution" as used here may re-
fer to the set of values returned as a solution by KINFITM
or to the last or best set of estimates preceding occurrence
of a fatal execution error.) I

A series of solutions was obtained by using KINFITA
with subroutine EQN and a version of subroutine KILMER that
omits equation 5-39. Many of the solutions satisfy the re-
maining equations almost exactly. Only these solutions were
retained.

Although many different sets of solutions have been

found, some consistencies have been observed in them. These

soluti<
divisil
parame
attach
these

two gr
value

having
of the
of 802
(the .

tive!

<W

Also,

(\

Howe
EQua

a or

hate

109
solutions have been grouped into eight groups. The main
division has been made between two sets of spin Hamiltonian
parameters -- the set listed in Table 2 for copolymer-
attached CprCl3 and the set presented in Table A. Each of
these two groups of solutions has been divided further into
two groups: a group of solutions each having an absolute
value of axy greater than 0.9 and a group of solutions each
having an absolute value of axZ-yz greater than 0.9. Each
of these four groups has been divided further into two groups
of solutions according to whether the main "a" coefficient
(the one whose absolute value is greater than 0.9) is posi-

tive or negative.

Since
<wQ|vQ> = 1 (6-1).
(axy)2 + (ax2_y2)2 + (bxz)2 + (byz)2 é 1 (6—2).

Also, since
<vR|vR> = 1 (6-3).
(cxy)2 + (cx2_y2)2 + (exz)2 + (eyz)2 é 1 (6-4).

However, many of the preliminary solutions did not obey
Equations 6-2 and 6-“. Thus, these two equations furnished
a criterion for eliminating unsatisfactory solutions.

Even after Equations 6-2 and 6-A had been used to elimi-

nate unsatisfactory solutions, eight classes of solutions

 

remained
table wt
intermec'
solutior
counter]
1R. Th1
the foli
If

and

YZ

If

as is

 

110

remained. These solutions, which were regarded as accep-
table when Equation 5-39 was omitted, are designated as
intermediate solutions. For each of these intermediate
solutions, each molecular orbital coefficient in TQ has a
counterpart with approximately the same absolute value in
YR. Throughout all eight cases of intermediate solutions,
the following relationships hold:

If

>
Iaxy' laxz-yZI

(6-5).
as is true in four cases, then
bezl > lbyzl (6-6),
axy ” cxy (6-7).
axz_yz¢v -cx2_y2 (6-8),
bxz as exz (6-9),
and
byzfit --eyz (6-10).
If
lax,_y,l > Iaxyl (6-11),

as is true in the other four cases, then

 

 

 

 

 

 

yz

xyl'

32
X

b
xz
and

byz

ll
molecu:
contail

by exp:

”Q

and

01‘ as

 

 

 

 

 

 

-y -y
b kfi-e
xz xz
and
byz “9 eyz

 

(6-1"):

(6-15).

(6-16).

The intermediate solutions for coefficients of the two

molecular orbitals of the ground-state Kramers doublet that

contains the unpaired electron can be pooled and summarized

by expressing those molecular orbitals (yQ and VB) as

TQ fi=(ale + asz 2 )B
x x -

yZ

+ (blvd + wad )a
xz yz

and

WR ’3 (alwd ‘ 32‘” )0‘

xy dxz- 2

y

+ (blvdxz - b2vdyz)8

O]? as

 

(6-17)

(6-18)

 

 

lo

and

The
solu
eter
Tabl
Tab]
01‘:
and
Tabi
of
in
whe
The
ete

Tat

soi

Sm

 

 

 

112

VQ a5(al‘i’d + ang )8
xz-y2 xy
+(bvI +b‘l’ )u _
1 dyz 2 dxz (6 19)
and
YR Q‘(a1Td - ade )a
xz-y2 xy
+(b‘i' ~b‘i’ )8 ..
1 dyZ 2 dxz (6 20).

The second column of Table 3 summarizes the intermediate
solutions obtained when the set of spin Hamiltonian param-
eters listed for c0polymer-attached CprCl3 at ~126° in
Table 2 (gxx = 1.9897, etc.) are used. The third column of
Table 3 summarizes the intermediate solutions when the set
of spin Hamiltonian parameters obtained by interchanging x
and y (gxx = 1.9806, etc.) is used. The fourth column of
Table 3 summarizes the intermediate solutions for both sets
of spin Hamiltonian parameters. The uncertainties indicated
in Table 3 reflect only variations in intermediate solutions
when exact sets of spin Hamiltonian parameters are used.
The experimental uncertainties in the spin Hamiltonian param-
eters contribute more uncertainty, which is not reflected in
Table 3.

Molecular orbital coefficients from each intermediate
solution were used in Equation 5-39 to obtain a calculated
g . Every calculated gyy obtained by this procedure was

yy

smaller than the experimental gyy. However, molecular

 

 

 

 

 

orbital
gyy V8.11;
mental 1
one of 1
tonian j
that is
set 113

At
routine
ing Eq1
routin:
used w:

molecu

Table

 

 

113

 

orbital coefficients from seven solutions yielded calculated

gyy values that were each within .002 (the estimated experi-

mental uncertainty in gyy) of the experimental gyy. Every

one of those seven solutions is for the set of spin Hamil-
tonian parameters presented in the second column of Table A,
that is, the set obtained by interchanging x and y in the

set listed in Table 2 for copolymer-attached CprCl3.

After intermediate solutions had been obtained, sub-

routine KILMER was completed by adding statements represent-

ing Equation 5-39. Subroutine EQN and the completed sub-

routine KILMER, which are listed in Appendix D, have been

used with KINFITD in attempting to solve numerically for

molecular orbital coefficients, K, and P. Selected

Table 3. Intermediate Solutions for Molecular Orbital
Coefficients, K, and P of Copolymer-Attached

CprCl

 

 

 

3
gxx 1.9897 1.9806 Both
Iall 0.9867 0.9856 0.9862
10.0003 10.0002 20.0008
|a2| <0.057 <0.074 <0.074
lb I 0.107 0.104 0.146
1 20.007 :0.010 10.012
Ibzl 0.02 a 0.01 0.016 2 0.011 0.02 2 0.01
K 0.96 1 0.03 1.0a : 0.04 1.00 2 0.07
P (in cm-l) 0.0139 0.0135 0.0137
10.000A 20.0005 20.0007

 

 

 

 

intern
mates
tainet
nearl
cantl
culat
solut
three
para]
is,
set
Tab]
give
Cle
CpN
Tab
spi
col
in
£01
to:
we
in
Ta
Te

114
intermediate solutions were used to provide initial esti-
mates of parameters. However, none of the solutions ob-
tained when Equation 5-39 was included were exact or even
nearly exact. Three of those solutions exhibited signifi-
cantly smaller weighted sums-of—squares of differences (cal-
culated minus experimental values) than did any of the other
solutions obtained when Equation 5-39 was included. These
three solutions all apply to the set of spin Hamiltonian
parameters presented in the second column of Table A, that
is, the set obtained when x and y are interchanged in the
set of parameters given for copolymer-attached CprC13 in
Table 2. Therefore, the set of spin Hamiltonian parameters
given in Table A is favored slightly for copolymer—attached
CprCl3, but the set given in Table 2 for copolymer-attached
CprCl3 has not been eliminated. The uncertainties given in
Table A are repeated from Chapter A. Appropriately rounded
spin Hamiltonian parameters are presented in the third (last)
column of Table u. The three best solutions are summarized
in Table 5. These three solutions represent three of the
four cases of solutions obtained when the set of spin Hamil-
tonian parameters listed in the second column of Table A
were used. Therefore, all four cases have been retained
implicitly in the summarization presented in Table 5. As in
Table 3, the uncertainties indicated in the second column of
Table 5 represent only variations in solutions when an exact
set of spin Hamiltonian parameters is used. The experimental

uncertainties in the spin Hamiltonian parameters

 

 

 

 

Table
xx
git
gzz
Txx
Tyy
Tzz

Tabl

ar

No

ta

115

Table A. Spin Hamiltonian Parameters of Copolymer-
Attached CprCl3

 

 

gxx 1.9806 I: 0.002 1.981 a 0.002
gyy 1.9897 2 0.002 1.990 a 0.002
gzz 1.9082 2 0.002 1.908 t 0.002
Txx -0.010u77 : 0.0001 cm“1 -0.0105 2 0.0001 cm”
Tyy -0.009278 2 0.0001 cm" -0.0093 2 0.0001 cm"
Tzz -0.021350 1 0.0001 cm" -0.0214 1 0.0001 cm"

 

Table 5. Final Solutions for Molecular Orbital Coefficients,
K, and P of Copolymer-Attached CprC13

 

 

 

Iall 0.985832 2 0.000016 0.986
Iazl 0.000050 2 0.000004 0.00
lbll 0.13285 2 0.00013 0.133
|b2| 0.01905 2 0.00010 0.019
K 1.0u1 : 0.015 1.04
P (in cm") 0.013u7 : 0.00013 0.0135

 

are expected to contribute relatively large uncertainties.
No quantitative attempt has been made to translate uncer-
tainties in spin Hamiltonian parameters into uncertainties
in molecular orbital coefficients. However, the third col~
umn of Table 5 shows a set of rounded values, the last digit
of each of which is regarded as uncertain. The solutions

given in Table 5 lead to calculated spin Hamiltonian

 

 

 

para
expe
exp:
eve]
and
Tab
of

co;
mai
Ta]
of
eq
sm

0;

116

parameter values that do not differ from the corresponding
experimental values by more than two times the estimated
experimental uncertainties. This agreement is valid for
every spin Hamiltonian parameter, for each of the four cases,
and for both the solution given in the second column of
Table 5 and the rounded numbers given in the third column
of Table 5.

These results indicate that the unpaired electron of
copolymer-attached Cpr01 is in a molecular orbital that is

3
mainly a Nb dxy or dxz‘yz orbital. The value of P given in
Table 5 has been compared to that for the 4d atomic sub-shell
of 93Nb, which has been calculated23 to be “57.3 MHz, which
equals .01525 cm". This comparison indicates that <r'3> is
smaller for the unpaired electron in copolymer-attached

CprCl3 than for a 4d electron in a 93Nb atom.

Calculation of s Orbital Character

 

The isotropic hyperfine interaction A for unit Spin
density in the 53 orbital of the 93Nb atom has been calcu-
lated23 to be 6590 MHz, which equals 0.2198 cm'l. Dividing
this number into -0.0137 cm'l, which is (Txx+Tyy+TZZ)/3 for
copolymer-attached CprC13, yields -0.062, which implies
6.2% s orbital character. It is not asserted that 6.2% of
the Kramers doublet orbitals that contain the unpaired elec-
tron is a Nb s orbital, as spin polarization of electrons in

filled orbitals that consist at least partly of Nb 3 orbitals

could account for some of this s orbital character.

 

 

 

at -l
obtai
have
giver
been
list.
cept
x an
toni
mine
cm
22
sim
the

rep

wri
001

qu

mo

CHAPTER 7

SUMMARY AND CONCLUSIONS

Experimental ESR spectra of copolymer-attached CprCl3
at -126° and copolymer-attached Cp2Nb012 at -126° have been
obtained carefully. ESR spectra of both of these compounds
have been described approximately by the spin Hamiltonian
given as Equation 1-46. Spin Hamiltonian parameters have
been determined for copolymer-attached CprCl3 and are
listed in the last (third) column of Table 4. Another ac-
ceptable set of parameters may be obtained by interchanging
x and y. For cepolymer-attached Cp2NbClZ, the spin Hamil-

tonian parameters T x’ and T have been deter-

mined to be 1.976 0.002, 1.952 i 0.002, -0.0106 2 0.0001

H

1

cm“ , and -0.0159 1 0.0001 cm", respectively. Although

gzz and T22 input values listed in Table 2 have been used in
simulating the ESR spectrum of copolymer-attached Cp2Nb012,
the experimental data do not seem adequate for confidently
reporting those values as the actual gzZ and Tzz'

Some original computer programs and routines have been
written. One computing package (ESRMAP, GATHER, and PEAKS)
contains a special interpolation procedure and enables high-

quality simulation of powder pattern ESR spectra at low or

moderate computing costs.

117

 

 

 

 

 

E
pressi
been C
molece
unpaii
atom.
orbit
deriv

relia

Zeene

assw
atta
cal

hype
of t
SUB!
mole
Ndx
Dai

and

ter
bee
wa~
ti
to

118

Equations relating spin Hamiltonian parameters to ex-
pressions containing molecular orbital coefficients have
been derived. These equations are exact if the pair of
molecular orbitals in the Kramers doublet that contains the
unpaired electron consists entirely of d orbitals of one
atom. If that pair of molecular orbitals also has some s
orbital character from that same atom, then the equations
derived by considering hyperfine tensor elements remain
reliable, but the equations derived by considering the
Zeeman interaction become inaccurate.

The derived equations have been used with some further
assumptions in a molecular orbital treatment for copolymer-

attached CprCl EHMO calculation results for a hypotheti-

30
cal model compound (MeCprCl3) and consideration of the

hyperfine anisotropy of 00polymer-attached CprCl in light

3
of the expected sign of the hyperfine tensor elements have

suggested that the unpaired electron resides in a pair of

molecular orbitals consisting chiefly of niobium udx or/and

y

4d 2 atomic orbitals. A zero-order approximation to this

2

x -y

pair of molecular orbitals has been assumed by using Va 8
xy

and Wd 2 28 as basis functions for one molecular orbital and

X -y

W a and T a for the other molecular orbital. Only

dxy dx2_y2

terms that could be mixed in by spin-orbit perturbation have

been added to this zero-order approximation. The resulting
wave functions are special cases of TQ and TR, for which equa—
tions have been derived. Experimental values of spin Hamil-

tonian parameters have been used with those equations to

 

 

 

 

 

enable
attacl
first
valid
preci
molec

valie

inte
with
thro
unpa
con:
trol
cha:
iza
tro
mol
are

is

 

119
enable description of these wave functions for copolymer—
attached CprCl3 by Equations 6-17 through 6-20 and the
first and last columns of Table 5. This description is
valid if all of the assumptions, including the lack of ap-
preciable s, p, and ligand orbital contributions to the
molecular orbitals that contain the unpaired electron, are
valid.

EHMO calculation results and consideration of hyperfine
interaction anisotropy lead to identification of the z axis
with an axis that is approximately a local 03v axis passing
through the Nb nucleus in cOpolymer-attached CprCl3. The
unpaired electron seems to be in a molecular orbital that
consists chiefly of a Nb dxy or dx2_y2 orbital. The iso-
tropic hyperfine interaction seems to reflect 6.2% s orbital
character, at least some of which could be due to spin polar-
ization of electrons in filled orbitals. The unpaired elec-
tron is thought to be a largely nonbonding electron in a
molecular orbital whose highest electron density is in some
areas in or near a plane which includes the Nb atom and which

is normal to the z axis indicated in Figure 18.

 

 

 

 

cccccccccccccccc

 

OOQOOOOOOGOOOOOG

APPENDIX A
LISTING OF PROGRAM MAGFLD

PROGRAM MAGFLD (INPUT.OUTPUT)
X REFERS TO THE SCALE MARKED ON THE EPR CHART PAPER.

FIRST DATA CARD. IN 4F8. 4. I2 FORMAT. SHOULD HAVE (IN THIS ORDER).
... kUE AT WHICH PROTON MARKERI
... X VALUE AT WHICH PROTON MARKER 2 MARKS:
... FREQUENCY OF PROTON RESONANCE I.
... FREQUENCY OF PROTON RESONANCE 2. AND
... NUMBER OF X VALUES (NOT COUNTING PROTON MARKER POINTS) THAT

TO BE TRANSLATED INTO MAGNETIC FIELD VALUES.

A NEGATIVE VALUE FOR THIS ”NUMBER 0? X VALUES' IS USED TO INDICATE

END OF DATA.

NEXT DATA CARD(S). IN IOF8.4 FORMAT. SHOULD HAVE X VALUES THAT ARE
TO BE TRANSLATED INTO MAGNETIC FIELD VALUES (IN GAUSS).

REAL INTERC
REAL INTERZ
DIMENSION X(IIO). )F(110)
DIMENSION FF(
CONST= 2000. 1* 03. 1415926536 / 26.7527
CONST?‘ 1.0 / 4. 25759E-0 3
2, XPMI. XPMZ. PMI. PM2, NOPTS
FORMAT (4F8.4.12)
IF (NOPTS. LT. 0) GO TO 9
D0 4 J = 1.10
K = (J-I) * IO
READ 3, (X(K+L).L=I.10)
3 FORMAT (IOFB 4)
IF (NOPTS.LE.10*J) GO TO 5
CONTINUE
FPMI = CONST * PMI
FPM2 = CONST * PM2
SLOPE = (FPMZ-FPMI) / (XPMZ-XPMI)
INTERC = - (SLOPE*XPMI) + FPMI

N I-

club

FFPMI = CONST2 * PMI

FFPMZ = CONST2 * PM2

SLOPE2 = (FFPM2-FFPMI) / (XPMZ-XPMI)
INTER2 = - (SLOPE2*XPMI) + FFPMI
NUMBER = l

WRITE 6. NUMBER.XPMI.PMI,FPMI.FFPMI
NUMBER = 2

WRITE 6, NUMBER, XPMZ. PM2 FPMZ; FFPM2

FORMAT (* PROTON MARKER* ,I2 X= *,F7. 4. IOX. *FREQU
UENCY =*, F8. 4. 10X. *MAGNETIC FIELD IN GAUSS =*, F8. 2.* OR *.F8. 2/)
D0 8 .l = I.NOP TS

F(J) = (SLOPE*X(J)) + INTERC

FF(J) = (SLOPE2*X(J)) + INTER2

WRITE 7, X(J).F(J).FF(J)

7 FORMAT (20X.*AT X =*.F7.4.*, MAGNETIC FIELD IN GAUSS =*.F8.2.* 0
OR *,F8.2/)

8 CONTINUE

0‘

I
9 CONTINUE
END

120

 

 

ccc

-
cccccc

000

000000

000

10
20

3O

APPENDIXIB

LISTING OF ESRSIM - FAST

OVERLAY (ESRSIM,0.0)
PROGRAM ESRSIM (INPUT‘65. OUTPUT5513. TAPE2I=5I3. TAPE22=65.
+ TAPE60=INPUTZ TAPE61=OUTPUT)

SEE COMMENTS AT BEGINNING OF PROGRAM UNITS ESRSIMF AND PEAKS.

COMMON /SPEEDY7 BETA,BIGK.DFCALC.ELLIPT(202).
2 HP,HX,HY.HZ.HI.H2,H3.IGO.JB.LOOP,NPUSED,NUQUAN.
3 PI,QNNUCL,SMALLK,SPINNU,V.VAL.YCALC

CALL OVERLAY (6HESRSIM.1.0)

BEGIN DUMMY STATEMENTS.

THE FOLLOWING STATEMENTS, THROUGH STATEMENT NUMBER 30, ARE
INCLUDED SO THAT CERTAIN ROUTINES WILL BE IN THE MAIN OVERLAY.

IF (ELLIPTCG?).NE.-5.6789E+94) GO TO 30
WRITE (61.10) LOOP. RX. HY. NPUSED

FORMAT (* DISREGARD*.IIO.FIO.6.IPE18.5.A15)
READ (60,20) NUQUAN. SPINNU. YCALC

FORMAT (I2,F10.6.E10.3)

READ (21) IGO. HI

WRITE (22) JB. H2

REWIND 21

CALL SYSTEM

CONTINUE

END DUMMY'STATEMENTS.

CALL OVERLAY (6HESRSIM52.0)
CALL OVERLAY (6HESRSIMJ3.0)
END

121

 

 

122

OVERLAY (Esns1n.1,o)
PROGRAH'ELLINT
common xspmnnvr BETA.BIGK,DFCALC.ELL1PT(202).
2 HP.HX.HY.HZ.H1.HZ.33.Ico,J3.Loor.nrusnn,nuonAn
3 PI,QNNUCL.SMALLK.SPINNU.V.VAL.YCALC ’
DIMENSION ELLIP(92)

 

 

ANTLOGIX) = EXP ( X * ALOG( 10.0) )
ELLIP(I) = 0.196120
ELLIP(2) = 0.196153
ELLIP(3) = 0.1962'2
ELLIP(4) = 0.196418
ELLIP(5) = 0.196649
ELLIP(6) = 0.196947
ELLIP(7) = 0.197312
ELLIP(B) = 0.197743
ELLIP(9) = 0.198241
ELLIP(IO) = 0.198806
ELLIP(II) = 0.199438
ELLIP(12) = 0.200137
ELLIP(13) = 0.200904
ELLIP(14) = 0.201740
ELLIP(15) = 0.202643
ELLIP(16) = 0.203615
ELLIP(17) = 0.204657
ELLIP(18) = 0.205768
ELLIP(I9) = 0.206948
ELLIP(20) = 0.208200
ELLIP(21) = 0.209522
ELLIP(22) = 0.210916
ELLIP(23) = 0.212382
ELLIP(24) = 0.213921
ELLIP(25) = 0.215533
ELLIP(26) = 0.217219
ELLIP(27) = 0.218981
ELLIP(28) = 0.220818
ELLIP(29) = 0.222782
ELLIP(30) = 0.224723
ELLIP(31) = 0.226793
ELLIP(32) = 0.223943
ELLIP(33) = 0.231173
ELLIP(34) = 0.233485
ELLIP(35) = 0.235880
ELLIP(36) = 0.238359
ELLIP(37) = 0.240923
ELLIP(38) = 0.243575
ELLIP(39) = 0.246315
ELLIP(40) = 0.249146
ELLIP(41) = 0.252068
ELLIP(42) = 0.255085
ELLIP(43) = 0.258197
ELLIP(44) = 0.261406
ELLIP(45) = 0.264716
ELLIP(46) = 0.268127
ELLIP(47) = 0.271644
ELLIP(48) = 0.275267
ELLIP(49) = 0.279001
ELLIP(50) = 0.282848
ELLIP(51) = 0.286811
ELLIP(52) = 0.290895
ELLIP(53) = 0.295101
ELLIP(54) = 0.299435
ELLIP(55) = 0.303901
ELLIP(56) = 0-308504
ELLIP(57) = 0.313247
ELLIP(58) = 0.318138
ELLIP(59) = 0.323182
ELLIP(60) = 0.328384
ELLIP(61) = 0.333753
ELLIP(62) = 0.339295
ELLIP(63) = 0.345020
ELLIP(64) = 0.350936

 

 

00000000000

301

302

303

304

ELLIP(65)
ELLIP(66)
ELLIP(67)
ELLIP(68)
ELLIP(69)
ELLIP(70)
ELLIP(71)
ELLIP(72)
ELLIP(73)
ELLIP(74)
ELLIP(75)
ELLIP(76)
ELLIP(77)
ELLIP(78)
ELLIP(79)
ELLIP(BO)
ELLIP(BI)
ELLIP(82)
ELLIP(83)
ELLIP(84)
ELLIP(85)
ELLIP(86)
ELLIP(87)
ELLIP(BS)
ELLIP(89)
ELLIP(90)
DO 301 JA
ELLIPT(JA)
CONTINUE

JB =
ELLIPT(JB)
CONTINUE

II II II II II II II 11 II II II II ll 11 II II II II II II II II II II II II II

0.357053
0.363384
0.369940
0.376736
0.383787
0.391112
0.398730
0.406665
0.414943
0.423596
0.432660
0.442176
0.452196
0.462782
0.474008
0.485967

'0.498777

0.512591
0.527613
0.544120
0.562514
0.583396
0.607751
0.637355
0.676027
0.735192
1,66

 

123

ANTLOG (ELLIP(JA))
DO 302 JA = 67.81

JA + JA - 66

ANTLOG (ELLIP(JA))
D0 303 JA = 82,86

 

 

JB = (5*JA) - 309

ELLIPT(JB) = ANTLOG (ELLIP(JA))
CONTINUE

DO 304 JA = 87.90

JB = (10*JA) ~ 739

ELLIPT(JB) = ANTLOG (ELLIP(JA))
CONTINUE

ELLIP(JA) CORRESPONDS TO JAPI DEGREES.

ELLIPT(JB) CORRESPONDS T0.....

65 + 1 (JB-66)/2 ) DEGREES FOB .
80 + < (JB-9é)/5 ) DEGREES F0R.6g7%EEg§fiFEfi96'
35 + 1 (JB—1211/10 ) DEGREES FOR 122 LE 33 '12"
39 DEGREES AND 2*(JB—l6l) MINUTES FDR 162'LE'161'
89 DEGREES AND JB-141 MINUTES FOR 182.LE.3EELgPé%§‘IBI'

ELLIPT(67) = 2.3261

ELLIPT<69) = 2.3622

ELLIPT(71) = 2.4001

ELLIPT(78) = 2.4401

ELLIPT(75) = 2.4325

ELLIPT(77) = 2.5273

ELLIPT(79) = 2.5749

ELLIPT(81) = 2.6256

ELLIPT(83) = 2.6796

ELLIPT(85) = 2.7375

ELLIPT(87) = 2.7993

ELLIPT(89) = 218669

ELLIPT<91> = 2.9397

ELLIPT(93) = 3.0192

ELLIPT(95) = 3.1064

ELLIPT(97) = 3.1729

ELLIPT(98) = 3.1920

ELLIPT(99) = 3.2132

ELLIPT(100) = 3.2340

 

 

ELLIPT(102)
ELLIPT(103)
ELLIPT(104)
ELLIPT(105)
ELLIPT(107)
ELLIPT(108)
ELLIPT(109)
ELLIPT(110)
ELLIPT(112)
ELLIPT(113)
ELLIPT(114)
ELLIPT(115)
ELLIPT(117)
ELLIPT(118)
ELLIPT(119)
ELLIPT(120)
ELLIPT(123)
ELLIPT(125)
ELLIPT(127)
ELLIPT(129)
ELLIPT(133)
ELLIPT(135)
ELLIPT(137)
ELLIPT(139)
ELLIPT(143)
ELLIPT(145)
ELLIPT(147)
ELLIPT(149)
ELLIPT(153)
ELLIPT(155)
ELLIPT<157)
ELLIPT(159)
ELLIPT(164)
ELLIPT(167)
ELLIPT(170)
ELLIPT(173)
ELLIPT(176)
ELLIPT(179)
ELLIPT<183)
ELLIPT(189)
ELLIPT(195)
ELLIPT(122)
ELLIPT(124)
ELLIPT€126)
ELLIPT(128)
ELLIPT(130)
ELLIPT<132)
ELLIPT(134)
ELLIPT(136)
ELLIPT(138)
ELLIPT(140)
ELLIPTC142)
ELLIPT(144)
ELLIPT(146)
ELLIPT(148)
ELLIPT(150)
ELLIPT<152)
ELLIPT(154)
ELLIPT(156)
ELLIPT(158)
ELLIPT(160)
ELLIPT(162)
ELLIPT(163)
ELLIPT(165)
ELLIPT(166)
ELLIPT(168)
ELLIPT(169)
ELLIPT(171)
ELLIPT(172)
ELLIPT(174)
ELLIPT(175)

IIIIIIIIDIIIIIIIIIIIIIIIIIIIIIIIIIIIIIllIlllllI'll"111111111111IlllllllllIIIIIOIIIIIIIIIIIIIIIIIIIIllllllIIIIIIIIIIIIIIIIIIIIIINIIIIIIIIIIII

3.2771
3.2995
3.3223
3.3458
3.3946
3.4199
3.4460
3.4728
3.5288
3.5581
3.5884
3.6196
3.6852
3.7198
3.7557
3.7930
3.8721
3.9142
3.9583
4.0044
4.1037
4.1574
4.2142
4.2744
4.4073
4.4811
4.5609
4.6477
4.8478
4.9654
5.0988
5.2527
5.5402
5.6579
5.7914
5.9455
6.1278
6.3509
6.6385
7.0440
7.7371
3.852
3.893
3.936
3.981
4.028
4.078
4.130
4.185
4.244
4.306
4.372
4.444
4.520
4.603
4.694
4.794
4.905
5.030
5.173
5.340
5.469
5.504
5.578
5.617
5.700
5.745
5.840
5.891
6.003
6.063

124

 

 

 

 

ELLIPT(177)
ELLIPT(178)
ELLIPT(180)
ELLIPT(181)
ELLIPT(182)
ELLIPT(184)
ELLIPT(185)
ELLIPT(186)
ELLIPT(187)
ELLIPTC188)
ELLIPT(190)
ELLIPT(191)
ELLIPT1192)
ELLIPT(193)
ELLIPT(194)
ELLIPT<196)
ELLIPT(197)
ELLIPT(198)
ELLIPT(199)
ELLIPT(200)
END

6.197
6.271
6.438
6.533
6.584
6.696
6.756
6.821
6.890
6.964
7.131
7.226
7.332
7.449
7.583
7.919
8.143
8.430
8.836
9.529

125

 

 

 

 

 

1126

OVERLAY (ESRSII'I.2.0)
PROGRAM ESRSIMF

**************3*******8***¥***¥1¥*¥******8***¥*******************8*8**¥

THIS PROGRAM PRODUCES A SIMULATED FIRST-DERIVATI '
TO THE MAGNETIC FIELD STRENGTH. HP) ELECTRON :31ng20ng“

SPECTRUM OI" A MAGNETICALLY DIL
GLASS SAMPLE. UTE POLYCRYSTAILINE OR FROZEN

ELECTRON SPIN (S. NOT REPRESENTED IN PROGRAM) = 1/2
HYPERFINE SPLI'ITING OCCURS PROM ONE NUCLEUS, moss NUCLEAR
( I.REPRESENTED IN THIS PROGRAM AS SPINNU) 13 9/2 OR LESSSPm
THE PARAMAGNETIC SPECIES HAS AT LEAST 02v SYMMETRY ‘
MUCH OF THIS PROGRAM IS BASED ON INFORMATION GIVEN'ON PAGE
AN ARTICLE BY C. P. STEWART AND A. L. PORTE IN J CHEM ‘663 01'
DALTON TRANS.. 1972. PAGES 1661-1666. ' ’ soc.

THE VALUES OF THE PARAMETERS U( 1) THROUGH (
ON THE FIRST DATA CARD IN 3F10.6 FORMATKB) m mmm To BE

G TENSOR VALUES. . . . .

UH) = mm
mm = GYY

U(3) = Gzz

T (HYPERFINE) TENSOR VALUES (IN WAVENUHBERS '
0(4) = TXX ‘ "E" R15"‘31""~0(Hflll- CID.
11(5) = TYY ‘

0(6) = Tzz

U( 7) = NUCLEAR GUADRUPOLE COUPLING PARAMETER

GAUSSIAN BROADENING PARAMETER

U( 8)

THE VALUES OP CONSTU). SPINNU. HHEGIN. HERD, 3
MORE ARE EXPECTED TO BE IN THAT ORDER ON 73683363022?“ A1")
THE FORMAT OP NHICH IS TO BE E10.3.F4.1.4F6.0.12. A CAR”)

CONST(1) = V = MICROWAVE FREQUENCY (IN HER'IZ)
SPINNU = NUCLEAR SPIN
IIBEGIN = BEGINNING MAGNETIC FIELD STRENGTH (IN GAUSS) FOR PRINTED

OUTPUT
BEND = ENDING MAGNETIC FIELD STRE .

OUTPUT NGTH. ( IN GAUSS) FOR PRINTED
BEGPLT = BEGINNING MAGNETIC FIELD STRENGTH ( 1N
ENDPLT = ENDING MAGNETIC FIELD STRENGTH ( IN 3:333; Egg $1113;

MORE = O INDICATES THAT THIS Is THE LAST SIMULATION
MORE = 2 INDICATES THAT THERE IS AT LEAST ONE MORE éIMULATION

THIS PROGRAM WAS WRITTEN BY NEAL KILMER AT MICHIGAN STATE UNIV

000000000000000000000000000000000000000000000

00000

**********************************************************Ԥ******

9 q - ‘******
COMMON /HSAVE/ H123IIO 3) IHGTcz) IHLT 2

JUMP = -IOO ' ° ’ ( )’JU"P’LA1LBILBE.LAH

LA = 2

CALL CCPLOT

END

 

 

C *** CAUTION ***

127

SUBROUTINE EPR (U,XX.CONST)

COMMON /SPEEDY7 BETfi.BIGK.DFCALC.ELLIPTT202)o
2 NP,NX,HY,BZ.Hl,H2.N3.1G0,JB.LOOP.NPUSED.NUQUAN.
3 PI.QNNUCL,SMALLK.SPlNNU.V.VAL.YCALC
COMMON /CONVOL/ MCA(3COO),YCB(3000).YCA(4025),B(1025)
COMMON /HSAVE/ H123(10.3).IHGT(2).IHLT(2),JUMP.LA.LB.LBR.LAB
DIMENSION U(20).XX(4).CONST(16)

IF (IGO.GT.0) GO TO 2
BETA = 0(8)
IF (BETA.GT.68.26)

8 FORMAT (xsx,* ----- NARNING .....
IF BETA Is GREATER THAN 68,

UT OF BOUNDS.*/)

WRITE (61.8)

SOME SUBSCRIPTS OF 8(10 MAY BE 00

LB = (15.0$BETA) + .001

LBT = (LB+.001)/2.0

IF ((Lsz).EQ.(LBT*2)) LB = LB + 1
LEE = LB/2

IIP=LBII+1

no 42376 J = 1.LB

H = J

K=LB+I-J

B(K) =

42376 CONTINUE

up;

IF (JUMP.GT.-1) GO TO 2
JUMP = O
2 CONTINUE
HP = XX(1)
V = CONST(1)
IF (IGO.GT.0) GO TO 6
TERMN = SPINNU + 1.0
MAXNQ = (2.0*SPINNU) + 1.01
4 CONTINUE
IF (JUMP.LT.1) GO TO 10

DO 9 J = 1.LAB
9 YCA(J) = 0.0
O CONTINUE

DO 1 NUQUAN = 1.MAXNQ
QNNUCL = NUQUAN - TERMN

IF (JUMP.GT.1) GO TO 3
CALL RDCALC (U)
3 CONTINUE
CALL ESR
IF (LA.LT.O) GO TO 12
1 CONTINUE
IF (JUMP.GT.I) GO TO 6
JUMP = 2
HMIN = H123(1.1)
IIMAX = I1123( 1.3)
DO 5 J = 2,MAXNQ

IF (HB23(J.1).LT.HMIN) HMIN =

IF (H!23(J.3).GT.HMAX) HMAx =
5 CONTINUE ,

IHGTII) = RMIN + 0.499999

IHLTII) = NMAx + 0.500001

LA = IHLT(1) - 1HOT(l) + 1

IF (LA.GT.3OOO) WRITE (61.7)

7 FOIU'IAT (/5X,$ ————— WARNING .....

WILL BE OUT OF BOUNDS.*/)
IHGT(2) = IHGT(1) - LBH
IHLT(2) = IIILT( 1?) + LBII
LAB =‘LA + LB - 1
GO TO 4

6 CONTINUE

JSUB = (INT(HP+0.5)) -

IF (JSUB.LT.1) GO TO 11
IF (JSUB.GT.LAB) GO TO 11
DFCALC = YCA(JSUB)

GO TO 12

DFCALC = 0

CONTINUE

RETURN

END

NH

Hl23(J.1)
H123(J,3)

REDIMENSION B AND YCA.

((H'HP)/(BETA*BETA)) * EXP((HP-H)*(H-HP)/(2.*BETR*BETA))

SOME SUBSCRIPTS FOR.YCB AND MCA W

IHGT(2) + 1

 

128

SUBROUTINE RDCALC (U)

REAL MI

COMMON /SPEEDY/ BETA. BIGK. DFCALC. ELLIPT(202) .
2 HP,HX.HY.HZ.II1,II2.II3.IGO,JB,LOOP.NPUSED,NUO.UAN.
3 PI,QNNUCL,SMALLK,SPINNU.V.VAL.YCALC

COMMON /TENSOR/ GXX. GYY. GZZ, TXX. TYY.‘I'ZZ. B

DIMENSION U(20)

M1 = QNNUCL

GXX = U( l)

GYY = U(2)

GZZ = 0(3)

TIOI = U(4)

TYY = U(5)

TZZ = U(6)

0.? = U(7)

B = 9.2732E-21

C = 2.9979245623+10
H = 6.6256E-27

(SP INNUIH SP INNU-l- 1) ) - ( MI*MI)

(H*V/( GZZ*B) ) - (BXC*TZZ*M1/( GZZ*B) )

HZ - (H*C*C*( ( 'I‘XXSETXX) +( TYYa‘R'I'YY) ) *QN/(4. 0*GZZ*B*V))
(H*V/( GXX‘a‘iB) ) - ( 11*C*TXX*MI/(GXX*B) )

EX - (II*C*C*( (TYY*TYY) +( 'I'ZZWI'ZZ) )*0.N/( 4. 0*GXX*B*V) )
RX - (H*C*O.P*QP*( (2 . 0*ON)- 1)*MI/( 2 . 0*GXX*B*TXX))
(H*V/( GYY*B) ) - (H*C*TYY*MI/( GYY*B) )

HY - (H*C*C*( (TXX’RTXX) +(1'ZZ*TZZ) )*0.N/( 4 . OXGYYIKBIKV) )
HY - (H*C*QP*QP*( (2 . 0*QN) ‘- 1) *MI/( 2 . 0*GYY*B*TYY))
VAL = H * C

RETURN

END

ifififiifififig

 

 

129

SUBROUTINE ESR
REAL J12.LI3
COMMON /SPEEDY/ BETA. B IGK. DFCALC . ELLIPT( 202) .
2 IIP.HX.HY,HZ.H1.H2.H3. IGO.JB, LOOP.NPUSED,NUQUAN.
3 PI,QNNUCL,SMALLK,SPINNU.V.VAL.YCALC
COMMON /CONVOI/ MCA( 3000) . YCB( 3000) .YCA( 4025) .B( 1025)
COMMON /HSAVE/ H123(10,3). IHGT(2), IHLT(2) .JUMP.LA.LB.LBH.LAB
COMMON /TENSOR/ GXX. GYY. GZZ. TXX. TYY. 'IZZ, BET
DIMENSION A(3).G(3),LI3(2).AIJ(2,3.2)
SDM(DEG,DMIN) = SIN ( ( DEG + DMIN/60.0 ) 13 CD )
IF (JUMP.GT.1) GO TO 20
CD = .017453292519943
IF (HX.GT.HY) GOTO 11

H1 - HX
HZ = BY
A( 1) = TXX
A(2) = TYY
G(1) = GXX
G(2) = GYY
GOTO 12

11 H1 = HY
112 = HX
A(l) = TYY
A(2) = TXX
G(1) = GYY
G(2) = GXX

12 IF (HZ.GE.82) GOTO 18
H3 = 112
A(3) = A(2)
0(3) = G(2)
IF (HZ.GE.H1) GO'I‘O 17
112 = HI
H1 = 82
A(2) = A( 1)
A(l) = TZZ
G(2) = G(1)
G(1) = GZZ
COTO I9

17 H2 = HZ
A(2) = 122
G(2) = 022
GOTO 19

18 H3 = HZ
A(3) = TZZ
0(3) = G22

19 CONTINUE

C
C************* ********¥******u*******M**************************8*m*

000000000

000

X'I'RA NGULARITIES ACCORDING TO CRITERIA GIVEN BY P. C.
TEFIEON? NINE"? BAUGHER. AND H. M. KRIZ ON PAGE 223 OF CHEMICAL
REVIEWS, 1975, VOLUME 75. ARTICLE ON PAGES 203-240. AND CALCULATING
APPROXIMATE MAGNETIC FIELD STRENGTHS FOR SUCH RESONANCES (IF ANY ARE
FOUND) , ETC. , ACCORDING TO INFORMATION ALSO GIVEN ON THAT PAGE. . . . .

CHANGING UNITS TO CORRESPOND WITH THOSE USED BY TAYLOR. BAUGHER. AND

@1200...
DO 201 J = 1,3
A(J) = A(J) >1: VAL

201 CONTINUE

= 6.6256E—27
THIS USE OF H AS PLANCK'S CONSTANT WILL NOT INTERFERE WITH ITS

COMPLETELY DIFFERENT USE AS ARG. IN CALLS TO VAL12 AND VAL23.

DO 203 I = 1.2
B0 202 J : 2&3 '10 202
IF (I.EO..J
CL).GT.1.0E-03) GO TO 225
FROM (HEABIII'SZ(%MSJT. 225 IS TO AVOID A ZERO DENOMINATOR, WHICH WOULD
OCCUR WHEN NUCLEAR MAGNETIC QUANTUM NUMBER IS ZER‘SLNOT DEF!
AIJ(I,J,1) IS SET EQUAL TO ZERO. AND AIJ(I,J,2) I NED

 

130

g ***** Nogfig¥¥3N(SI:CE IT COULD EQUAL +-INFINITY).
:33: THIS
AIJ(I,J,1) = 0.0 CASE HAS NOT BEEN TESTED. #3333
LMAX.= 1
GO TO 226
225 CONTINUE
LMAX = 2
R = H * V / (4.0*QNNUCL)
S = (G(I)*G(I)*A(I)*A(I)) - (G(J)*G(J)*A(J)*A(J))
IF (G(I)*G(I).EQ.G(J)*G(J)) GO TO 202
S = S / (2.0*((G(I)*G(I))-(G(J)*G(J))))
SQ = (RXR) + S
IF (SQ.LT.0.0) GO TO 202
SQ = SQRT(SQ)
AIJ(I,J,1)
AIJ(I.J.2)
226 CONTINUE
DO 204 L = 1,LMAX
IF (AIJ(I,J,L)-A(I)) 205,204,206
205 IF (AIJ(I,J,L)-A(J)) 204,204,207
206 IF (AIJ(I,J,L)-A(J)) 207,204,204
207 CONTINUE
DO 230 IBOTH = 1.2
HIJ = 1.0E+99

R - SQ
R.+ SQ

C IF HIJ IS NOT OTHERWISE DEFINED THIS ROUND. PRINTOUT WILL
C CONTAIN ASTERISKS INSTEAD OF A VALUE FOR HIJ.
IF (IBOTH.EQ.2) GO TO 231
WRITE 232
232R§$§£NT (1X.* ----- RESULT OBTAINED FROM USING EQUATION GIVEN INIR
O...’*)
T = (G(J)*G(J) * A(J)*A(J)) - (AIJ(I.J,L)*AIJ(I,J{L))
GO TO 234
231 WRITE 233 -
233 FORMAT (1X,*---- RESULT OBTAINED WHEN PARENTHESES ARE ADDED TO

EEQUATION GIVEN IN REVIEW.....*)
T = (G(J)*G(J))*((A(J)*A(J)) - (AIJ(I.J.L)*AIJ(I,J,L)))

234 CONTINUE
T = T / ((G(I)*G(I))*((AIJ(I,J,L)*AIJ(I,J,L))-(A(I)*A(I))))

COSZPH = 1.0 / (1.0+T)
GIJ = (G(I)*G(I)) + ((G(J)*G(J))-(G(I)*G(I)))*COS2PH
IF (GIJ.LT.0.0) GO TO 210

GIJ = SQRT(GIJ)
HIJ (H*V/(GIJ*BET)) - (AIJ(I.J.L)*QNNUCL/(GIJ*BET))

IF (ABS(QNNUCL).LT.1.0E-03) GO TO 210

IF (J.EQ.3) GO TO 211

R1 = (H$V*((G(3)*G(3))-(GIJ*GIJ))) / QNNUCL

R2 = (G(3)*G(3)*A(3)*A(3)) - (GIJ*GIJ*AIJ(1,2,L)*AIJ(1,2,L))
R2 = R2 / AIJ(1,2,L)

R3 = 2.0*AIJ(1.2.L)*((G(3)*G(3))—(GIJ*GIJ))

J12 = R1 + 82 - R3
IF (J12) 213.210.212
(H*V*((G(1)*G(1))-(G(2)*C(2)))) / QNNUCL

211 R1 =
R2 = ((G(1)*G(1)*A(1)*A(1))-(G(2)*G(2)*A(2)*A(2))) / AIJ(I,3.L)
R3 = 2.0*AIJ(I,3,L)*((G(1)*G(1))-(G(2)*G(2)))

LI3(I)=R1+R...-R3
IF (I.EQ.2) GO TO 214
IF (LI3(I)) 213,210,212
214 IF (LI3(2)) 212,210,213
215 FORMAT (3X}*AN EXTRA SINGULARITY (DIVERGENCE) *,
F *OCCURS AT H*.2Il,*(*,ll,*) = APPROXIMATELY*,F9.3,* GAUSS FOR.N
NUCLEAR MAGNETIC QUANTUM NUMBER OF*,F4.1)
216 FORMAT (3X,*AN EXTRA SINGULARITY (SHOULDER) *, .
F . *OCCURS AT H*,211,*(*,Il,*) = APPROXIMATELY*,F9.3,* GAUSS FOR N
NUCLEAR MAGNETIC'QUANTUM NUMBER OF*.F4.1)

217 FORMAT (3x,*‘.IIN EXTRA SINGULARIT‘Y 21:,
F *OCCURS AT H*.211,*(*,11,*) = APPROXIMATELY*,F9.3,* GAVSS FOR N

NUCLEAR MAGNETIC QUANTUM NUMBER OF*.F4.1)
220 FORMAT (16H *** WARNING *** . * THE SIMULATED DATA DO NOT INCLUDE
E THIS EXTRA SINGULARITY. ALSO. THE PLOT DOES NOT INCLUDE THIS EXT

TRA SINGULARITY.*/)
212 WRITE 215,1,J,L,HIJ,QNNUCL

 

131

GO TO 218

213 WRITE 216,I.J,L,HIJ.QNNUCL
GO TO 218

210 WRITE 217,1,J,L,HIJ,QNNUCL

218 WRITE 220

230 CONTINUE

204 CONTINUE

202 CONTINUE

203 CONTINUE

C
g*******3***************************$***$$¥**3**$********$**********m3

 

H1
H2
H3

H123(NUQUAN,1)
H123(NUQUAN.2)
H123(NUQUAN,3)
GO TO 9999
20 CONTINUE
DO 31 J = 1
‘YCB(J) = .
MCA(J) =
31 CONTINUE
DO 4 JB
DEG = JB
DMIN = 0.0
SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)
CALL VAL12 (H)
4 CONTINUE
DO 5 JB = 67,96
DEG = 65.0 + ( (JB-66.0)/2.0 )
DMIN = 0.0
SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)
CALL VAL12 (H)
5 CONTINUE
DO 6 JB = 97,121
DEG = 80.0 + ( (JB-96.0)/5.0 )
DMIN = 0.0
SMALLK = SDM(DEG.DMIN)
BIGK = ELLIPT(JB)
CALL VAL12 (H)
6 CONTINUE
DO 7 JB = 122.161
DEG = 85.0 + ( (JB-I2I.0)*0.1 )
DMIN = 0.0
SMALLK.= SDM(DEG.DMIN)
BIGK = ELLIPTCJB)
CALL VAL12 (H)
7'CONTINUE
DO 8 JB = 162,181
DEG = 89.0
DMIN = 2.0 * (JR-161.0)
SMALLK = SDM(DEG,DMIN)
BICK = ELLIPT(JB)
CALL VAL12 (H)
8 CONTINUE
DO 9 JB = 182.200
DEG = 89.0
DMIN = JB - 141
SMALLK = SDM(DEG.DMIN)
BICK = ELLIPT(JB)
CALL VAL12 (H)
9 CONTINUE
DO 21 JB = 1.66
DEG = JB - 1
DMIN = 0.0
SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)
CALL VAL23 (H)
21 CONTINUE
DO 22 JB = 67.96

.LA
0

00

1,66
1

22

23

24

25

26

44

45

 

132

DEG = 65.0 + ( (JBP66.0)/2.0 )
DMIN = 0.0

SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)

CALL VAL23 (H)
CONTINUE

DO 23 JB = 97.121

DEG = 80.0 + ( (JBP96.0)/5.0 )
DMIN = 0.0

SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)

CALL VAL23 (H)
CONTINUE

DO 24 JB = 122,161

DEG = 85.0 + ( (JB-121.0)*0.1 )
DMIN = 0.0

SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)

CALL VAL23 (H)
CONTINUE

DO 25 JB = 162.181

DEG = 89.0

DMIN = 2.0 * (JR-161.0)
SMALLK = SDM(DEG.DMIN)
BIGK = ELLIPT<JB)

CALL VAL23 (H)
CONTINUE

DO 26 JB = 182.200

DEG = 89.0

DMIN = JB - 141

SMALLK = SDM(DEG,DMIN)
BIGK = ELLIPT(JB)

CALL VAL23 (H)
CONTINUE

STPI = SQRTC2.0$PI)

DO 45 J = 1.LA

IF (MCA(J).GT.0) GO TO 44
GO TO 45

YCB(J) = YCB(J) / MCA(J)
YCB(J) = YCB(J) / STPI
YCA(J) = YCA(J) + YCB(J)
CONTINUE

MAXNQ = (2.0*SPINNU) + 1.01
IF (NUQUAN.LT.MAXNQ) GO TO 1
DO 14 J = 1.LA

YCB(J) = YCA(J)

CONTINUE

DO 15 J = 1.LAB

YCA(J) = 0.0

CONTINUE

CALL VCONVO (LA,LB)
CONTINUE

ICO = IGO + 2

RETURN

END

 

2

9999

3

 

133

SUBROUTINE VAL12 (H)

REAL K

COMMON /SPEEDY7 BETA.BIGK,DFCALC,ELLIPT(202).
HP,HX.HY,HZ.H1,H2.H3,IGO.JB,LOOP.NPUSED,NUQUAN,
PI.QNNUCL,SMALLK.SPINNU,V,VAL.YCALC

COMMON /CONVOL/ MCA(3000).YCB(3000),YCA(4025),B(1025)

COMMON /VALREM7 HOLD,VALOLD,KSUB

COMMON /HSAVE/ H123(10,3),IHGT(2).IHLT(2).JUMP.LA,LB.LBH.LAB

HI = H123(NUQUAN,1)

H2 = H123(NUQUAN,2)
= HI23(NUQUAN.3)

= SMALLK

T = (KXK) * ((H2*H2)-(H1*H1)) / ((H38H3)*(H2*H2))
= SQRT ( ( (H1*R1) + (T3H3*H3) ) / ( 1.0 + T ) )

SMI = 2.0*H1*H2*H3 / (PI*H*H)

VAL 3 SMI*BIGK7(SQRT(((H3$H3)-(H*H))*((H2*H2)-(H1*H1)))#BETA)

JSUB = INT(H+0.5) - IHGT(1) + 1

YCB(JSUB) = YCB(JSUB) + VAL

MCA(JSUB) = MCA(JSUD) + 1

IF (JB.EQ.1) GO TO 9

IF ((JSUB-KSUB)*(JSUB-KSUB).LT.2) GO TO 9

J1 KSUB + 1

J2 JSUB ‘ 1

DO I J = J1.J2

RC = J + IHGT(1) - 1

MCA(J) = 1

SLOPE = (VAL-VALOLD)/(H-HOLD)

YCB(J) = (SLOPE*(HC-HOLD)) + VALOLD

CONTINUE

CONTINUE

KSUB = JSUB

HOLD = H

VALOLD = VAL

RETURN

END

 

 

 

 

 

\DH

9999

 

1311

SUBROUTINE VAL23 (H)
REAL K
COMMON /SPEEDY7 BETA.BIGK.DFCALC,ELLIPT(202).

2 HP.HX,HY,HZ.HI,H2,H3,IGO.JB,LOOP,NPUSED,NUQUAN.
3

PI,QNNUCL,SMALLK,SPINNU,V,VAL,YCALC
COMMON /CONVOL/ MCA(3000). YCB(3000). YCA(4025). 8(1025)
COMMON /VALREM/ HOLD, VALOLD. KSUB
COMMON /HSAVE/ 8123(10, 3), IHGT(2), IHLT(2), JUMP, LA, LB, LBH, LAB
HI = H123(NUQUAN, 1)

H2 = Hl23(NUQUAN,2)

H3 = HI23(NUQUAN.3)

K SMALLK

T KRKR((H3*H3)-(H2*H2)) / ((H2*H2)-(H1*H1))
H SQRT ( ((T*H1*H1)+(H3*H3)) / (191.0) )
SMI 2.0*H1*H2*H3 / (PI*H*H)

VAL SMI*BIGK7(SQRT(((H3*H3)-(HZ*H2))*((HxH)-(H1*H1)))*BETA)
JSUB = INT(H+0.5) - IHGT(I) + 1

YCB(JSUB) = YCB(JSUB) + VAL

MCA(JSUB) = MCA(JSUB) + 1

IF (JB.EQ.1) GO TO 9

IF ((JSUB-KSUB)*(JSUB-KSUB).LT.2) GO TO 9
J1 = JSUB + 1

J2 = KSUB - 1

DO 1 J = J1.J2

NC = J + IHGT(1) - I

MCA(J) = 1

SLOPE = (VAL-VALOLD)/(H-HOLD)

YCB(J) = (SLOPE*(HC-HOLD)) + VALOLD
CONTINUE

CONTINUE

KSUB = JSUB

HOLD = H

VALOLD = VAL

RETURN

END

 

.135

SUBROUTINE CCPLOT
COMMON /SPEEDY7 BETA,BIGK,DFCALC,ELLIPT(202),
2 HP,HX.HY,HZ,H1,H2.H3,IGO.JB,LOOP.NPUSED,NUQUAN,
3 PI.QNNUCL,SMALLK.SPINNU,V,VAL,YCALC
COMMON /HSAVE/ H123(10,3),IHGT(2),1HLT(2),JUMP,LA,LB,LBH.LAB
COMMON /CONVOL/ MCA(3000),YCB(3000),YCA(4025).B(1025)
DIMENSION U(20),XX(4)
DIMENSION CONST(16)
DIMENSION Y(10)
DIMENSION XPLOT(4025).YPLOT(4025),IBUF(1025)
EQUIVALENCE (XPLOT,MCA),(YPLOT.YCA),(IBUF,B)
P1 = 3.14159 26535 898
120 MORE = 0

REWIND 21
IF (JUMP.GT.0) JUMP = 0
CALL YOURS(U.CONST.SPINNU.HBEGIN.HEND.BEGPLT}ENDPLT.MDRE)

225 FORMAT (1H1.28X.*THESE VALUES HAVE BEEN USED IN CALCULATING THE FO
OLLOWING SIMULATED DATA.....*/)
226 FORMAT (56X.*G TENSOR VALUES.....*)
227 FORMAT (58X,*GXX =*,F10.6)
22B FORMAT (58X,*GYY =*,F10.6)
229 FORMAT (58X,*GZZ =*,F10.6/)
230 FORMAT (26X.*T (HYPERFINE) TENSOR VALUES (IN VAVENUMBERS. I.E.. RE
ECIPROCAL CENTIMBTERS).....*)
231 FORMAT (58X.*TXX =*,F10.6)
232 FORMAT (58X.*TYY =*.F10.6)
233 FORMAT (58X,*TZZ =*,F10.6/)
234 FORMAT (41X.*NUCLEAR OUADRUPOLE COUPLING PARAMETER. =*.F10.6/)
235 FORMAT (44X,*GAUSSIAN BROADENING PARAMETER =*.F12.6/)
236 FORMAT (43X.*MICROWAVE FREQUENCY (IN HERTZ) =*,1PE13.5/)
237 FORMAT (56X,*NUCLEAR.SPIN =*,F5.l//////)
WRITE 225
WRITE 226
WRITE 227.0(1)
WRITE 223.0(2)
WRITE 229.0(3)
WRITE 230
WRITE 231.0(4)
WRITE 232.0(5)
wRITE 233.0(6)
[WRITE 234.0(7)
WRITE 235.U(8)
WRITE 236,CONST(1)
WRITE 237.SPINNU

100 = -1 - (4*SPINNU)
HINCR = l

XXNK = 250.0 * HINCR
XXNN = 5.0 * HINCR
XXKK = HINCR

XXCON = HBEGIN - XXNK - XXNN
101 FORMAT (1H1.16X,*H*,15X,*AT H*,13X,*AT H +*,F7.3,8X,*AT H +*,F7.3,

A 8X.*AT H +*,F7.3,8X,*AT H +*,F7.3/)
102 FORMAT (F21.3,1PE21.7,4E2I.7) .

JSUB = 0
DO 109 NK = 1,100
DO 108 NN = 1.50

XX(1) = (XXNKRNK) + (XXNN*NN) + XXCON
IF (XX(1).GT.HEND) GO TO 9999
XXS = XX(1)
DO 107 KK = 1,5
XX(1) = XXS + (XXKIGMKK-U)
CALL EPR (U.XX.CONST)
IF (LA.LT.0) GO TO 240
IF (NN+KK.GT.2) GO TO 220
D0 106 IN = 1,4
Y(IN) = IN * HINCR
106 CONTINUE
WRITE (61,101) (Y(IN).IN=1.4)
220 CONTINUE
Y(KK) = DFCALC
IF (XX(1).LT.BEGPLT) GO TO 107

 

107
108

109
9999

110

240

250

A
A

136

IF (XX(1).GT.ENDPLT) GO TO 107
= JSUB + 1

JSUB

WRITE (21)

JSUB

WRITE (21) XX(1),DFCALC
CONTINUE

WRITE (61.102)

CONTINUE
CONTINUE
CONTINUE
= JSUB + 1
WRITE (21) JEOT
REWIND 21

CONST(16) =
REWIND 22

WRITE (22)

JSUB,CONST(16).U(1).U(2),U(3),U(4),U(5),U(6),U(7),U(8),SPINNU
REWIND 22
CONTINUE

JEOT

2

READ

(21)

XXS,(Y(IN).IN=1.5)

CONST(I) * 1.0E-09

NK

READ (21) XPLOT(NK).YPLOT(NK)

IF (NK.LT.JSUB)

REWIND 21
CALL PLOTS (IBUF.1025.2)

CALL PLOT (0.0.1.7.-3)

CALL SCALE (XPLOT.8.5.JSUB.1)

CALL SCALE (YPLOT.6.0.JSUB.1)

CALL AXIS (0.0.0.0,34HFIRST DERIVATIVE OF ESR.ABSORPTION,34.6.0.

GO TO 110

90.0,YPLOT(JSUB+1).YPLOT(JSUB+2))

CALL AXIS (0.0.0.0,23HOBSERVED FIELD IN GAUSS.-23,8.5,

0.0.XPLOT(JSUB+1).XPLOT(JSUB+2))

CALL LINE (XPLOT.YPLOT,JSUB.1.0.11)

CALL

CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL
CALL

SYMBOL

Z=.0.0.35)

NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
NUMBER
SYMBOL
SYMBOL
NUMBER
SYMBOL
NUMBER

(0.00,7.14,0.140,35HM1CROWAVE FREQUENCY IN GIGAHERTZ

(5.18.7.14,0.I40,CONST(16).0.0,4)
(0.00.6.81,0.140,5HG =,0.0,5)
(0.14,6.76,0.098.2HXX.0.0,2)
(0.98,6.81.0.140.U(1).0.0.4)
(3.00,6.81,0.140,5HG =,0.0,5)
(3.14,6.76,0.098.2HYY,0.0,2)
(3.98,6.81,0.140,U(2).0.0.4)
(6.00,6.81.0.140,5HG .0,5)
(6.14,6.76,0.098,2HZZ,
(6.98,6.81,0.140,U(3),
(0.00,6.48.05140,4HT
(0.14,6.43.0.098,2HXX.
(0.70,6.48,0.140,U(4),
(3.00,6.48,0.140,4HT
(3.14,6.43,0.098,2HYY,
(3.70,6.48,0.140,U(5),
(6.00,6.48,0.140,4HT
(6.14,6.43,0.098,2HZZ.
(6.70,6.48,0.140,U(6),
(0.00,6.15,0.140,4HI
(0.84,6.15.0.140.SPINN
(3.00,6.15.0.140,4HQ
(3.14.6.15,0.136,16,0.0
(3.70,6.15,0.140,U(7),0
(6.00,6.15,0.140,6HBETA
(6.98,6.15,0.I40,U(8),0.

0 || 0 I 00060000000006“

0 0 C 0

(Iqllocaucpollecpnapo

O O O 0 O O

PLOT (11.0,-1.7,999)

CONTINUE
IF (MORE.GE.1) GO TO 120

WRITE 250
FORMAT (1H1)
RETURN

END

[9:—

137

SUBROUTINE YOURS(U.CONST.SPINNU,RBEGIN,BEND.BEGPET.ENDPLT}MORE)
DIMENSION U(20),CONST(16)

READ E,(U(J),J=1.8)

READ 2,CONST(1).SPINNU.HBEGIN,HEND,BECPLT,ENDPET.MORE
FORMAT (8F10.6)

FORMAT (E10.3,F4.1,4F6.0.I2)

RETURN

END

DO

 

138

SUBROUTINE VCONVO (LA.LB)
COMMON /CONVOL/ MCA(3000).A(3000).C(4025).B(1025)
LDA = 3000
LDB = 1025
LDAB = 4025
IF (LA.GT.LDA) GO TO 7
IF (LB.GT.LDB) GO TO 7
LAB = LA + LB - 1
IF (LAB.GT;LDAB) GO TO 7
DO 2 J=1,LAB
DO 1 K=1,J
IF (KLGT.LA) GO TO 1
JMKPI = J - KI+ 1
IF (JMKP1.GT.LB) GO TO 1
C(J) = C(J) + (A(K)*B(JMKP1))
1 CONTINUE
2 CONTINUE
GO TO 8
7 WRITE 9
WRITE 10
9 FORMAT (1H1.5X.*SUBSCRIPT WOULD BE OUT OF BOUNDS IN SUBROUTINE VCO
ONVO.*///)
10 FORMAT (6X5*EXECUTION OF SUBROUTINE VCONVO IS ABORTED.*///)
LA = -12
LB = -12
NEGATIVB VALUE RETURNED FOR.EITRER.LA OR LB INDICATES ABORTION.
CALLING PROGRAMISROULD TEST LA OR LB AFTER RETURN.
8 CONTINUE
RETURN
END

 

 

C50

 

139

OVERLAY ( ESRS III. 3 . 0)
PROGRAM PEAKS

*******************************8¥**************************8***********

0:50:50

THIS PROGRAM USES DATA FROM THE TAPE21 AND TAPE22 FILES GENERATED
BY CERTAIN ESRSIM PROGRAMS.

APPROXIMATE MAGNETIC FIELD VALUES AT WHICH PEAKS. VALLEYS. AND
25 PERCENT, 50 PERCENT. AND 75 PERCENT OF EACH VERTICAL PEAK-
TO-PEAK DISTANCE OCCUR ARE GIVEN IN OUTPUT.

NO DATA CARDS ARE REQUIRED.

INGREDIENT BETWEEN CONSECUTIVE MAGNETIC FIELD VALUES ON TAPE21 MUST
BE ONE GAUSS.

THIS PROGRAM WAS WRITTEN BY NEAL KILMER AT MICHIGAN STATE UNIV.

********************************************$**************************

0 Of) (50(30CDOCDOC50CD

C30

COMMON /EXTR/ YEXTRM

DYNSTO IS USED TO PROVIDE A REFERENCE POINT FOR DYNAMIC STORAGE.
COMMON DYNSTO

DIMENS ION MINMAX( 500) . XTREM( 500)
DIMENSION CONST( 16) . U( 20) . XPLOT( 4025) .YPLOT( 4025)

DIMENSION PCT( 3)

SETFL IS CALLED TO SET FIELD LENGTH DOWN.
CALL SETFL (DYNSTO)

REWINI) 22
READ (22)
JSUB,CONST( 16).U(1).U(2),U(3),U(4),U(5).U(6).U(7).U(8).SPINNU

J = 0

REWIND 21

CONTINUE

READ (21) NK

READ (21) X.Y

2

J = J + 1
XPLOT(J) = X
YPLOT(J) = Y
IF (NK.LT.JSUB) GO TO 110
K = O

JXMAX = J
LASTP = 0
KPTO = -1

X1 = XPLOT(1)
X2 = XPLOT(2)
X3 = XPLOT(3)
Y1 = YPLOT(1)
Y2 = YPLOT(2)
Y3 = YPLOT(3)
JX = 3

CONT NUE
IF (ABS(X3-X2-I.0) .GT. 1 .E-07) GO TO 9998

IF (ABS(Y2) .GT.I.0E-12) GO TO 10
IF (ABS(Y1).LE.1.0E-l2) GO TO 20
IF (ABS(Y3).LE.I.OE-12) GO TO 20
CONTINUE

IF ((Y3-Y2)*(Y2-Y1).GT.0.0) GO TO 20
K = K + 1

IF ( (Y3-Y2) - (Y2-Yl) ) 11.12.13
MINMAXUO = 7HMAXIMUM

XTREM(K) = EXTREM(X1.Y1,Y2.Y3)
XLMAX " XTREM(K)

JLMAX JX

YLMAX - YEXTRM

IF (LASTP.EQ.-l) KPTOP=1

XPI = XLMIN

 

11:0

XP2 = XLMAX
YPI = YLMIN
YP2 = YLMAX
LASTP = 1
JP = JLMIN
GO TO 14

12 MINMAX(K) = 4HFLAT
XTREM(K) = 1.0E+99
GO TO 15
13 MINMAX(K) = 7HMINIMUM
XTREM(K) = EXTREM(X1.Y1.Y2.Y3)
XLMIN = XTREMfK)
JLMIN = JX
YLMIN = YEXTRM
IF (LASTP.EQ.1) KPTOP=1

XPI = XLMAX

XP2 = XLMIN

YPl = YLMAX

YP2 = YLMIN

JP = JLMAX

LASTP = -1

14 CONTINUE

IF (KPTOP.LE.0) GO TO 15
KPTOP = -l

XTREM(K+3) = XTREM(K)
MINMAX(K+3) = M]NMAX(K)

KBASE = K ‘ 1

MINMAXIK) = 6H25 PCT
MINMAX(K+1) = 6H50 PCT
MINMAX(K%2) = 6H75 PCT

PCT(I) = YPl + (0.25*(YP2-YP1))

PCT(2) = Y?! + (O.50*(YP2-YP1))
PCT(3) = YPI + (0.75*(YP2-YP1))
JY = I

16 CONTINUE

XP2 = XPLOT(JP)
YP2 = YPLOT(JP)
DISTI = YPI - PCT(JY)
DIST2 = YPZ - PCT(JY)
IF (DIST1*DIST2) 17.17.19
17 DENOM = ABSCDISTI) + ABS(DIST2
K = KBASE + JY .
DISTI = ABS(DIST1)
DIST2 = ABS(DIST2)
XTREM(K) = (DIST1*XP2) + (DIST2*XP1)
XTREM(K) = XTREMIK) / DENOM
GO TO 21
19 JP = JP + 1
XPI = XP2
YPI = YP2
GO TO 16
21 IF (JY.GE.3) GO TO 22
JY = JY + 1
GO TO 16
22 K = K + 1
I5 CONTINUE
20JX=JX+1
IF (JX.GT.JXMAX) GO TO 30

X1 - X2
X2 = XS
Y1 = Y2
Y2 = Y3
X3 = XPLOT(JX)
Y3 = YPLOT(JX)

GO TO 1
30 KMAX = K .
WRITE 33 E OF FEA
33 FORMAT (31X.*MAGNETIC FIELD STRENGTH IN GAUSS*.5X;*TYP TUBE
E$/)
31 FORMAT (51X.F12.2.A15)
DO 32 K = 1.KMAX

 

O 000

32

9998
9997
9999

 

1111

WRITE 31.XTREM(K) .MINMAX(D

CONTINUE

GO TO 9999

WRITE 9997

FORMAT (/10X,¥DATA ARE NOT SUITABLE FOR THIS PROGRAH.*/)
CONTINUE

REWIND 21

REWIND 22

RESETFL IS CALLED T0 RESET FIELD LENGTH FOR A RELIABLE RETURN '10
THE MAIN OVERLAY.
CALL RESETFL

END

 

1142

FUNCTION EXTREM(X1,Y1,Y2.Y‘3)

THIS FUNCTION RETURNS THE VALUE OF X AT WHICH THE PARABOLA DEFINED
BY THE (X.Y) POINTS (XI.Y1). (X1+1.Y2). AND (XI+2.Y3) HAS AN
EXTREMUM (MINIMUM OR MAXIMUM) .

THE Y-VALUE AT THAT EXTREMUM IS MADE AVAILABLE VIA LABELED COMMON.

COMMON /EXTR/ YEXTRM

0000

A = (Y3-(2.*Y2)+Y1) / 2.

B = Y2 - Y1 - (2.*A*x1) - A

C = Y! - (A*X1*Xl) - (Bxxn

EXTREM = - B / (2.2m)

x = EXTREM

YEXI‘RM = (A*X*X) + (Basic + c
RETURN

END

 

 

 

 

C50

OCDOCTOCDDCDOCDOC30630f30C30C5OCDOC30€30650630C50f30€3OCDOCUOCUOCUOffiOCUOCDOCDOC’OCIOCfi

APPEND IX C

LISTING OF PROGRAMS ESRMAP. GATHER. AND PEAKS

PROGRAM ESRMAP (INPUT=65.0UTPUT=513.TAPE41.TAPE42=5I3)

******************************************************************3****

THIS PROGRAM PRODUCES A MAP OF SIMULATED RESONANCE FIELD VALUES
EXPECTED IN ELECTRON SPIN RESONANCE OF A RANDOMLY-ORIENTED
SAMPLE.

ELECTRON SPIN (S. NOT REPRESENTED IN PROGRAM) = 1/2.

HYPERFINE SPLITTING OCCURS FROM ONE NUCLEUS. WHOSE NUCLEAR SPIN
(LREPRESENTED IN THIS PROGRAM AS SPINNU) IS 9/2 OR LESS.
THE PRINCIPAL AXES OF THE G TENSOR ARE ASst TO COINCIDE WITH

THE CORRESPONDING ONES OF THE HYPERFINE TENSOR.

ONLY THE ALLOWED TRANSITIONS ARE CONSIDERED.

THE VALUES OF THE PARAMETERS U( 1) THROUGH U(B) ARE EXPECTED TO BE
ON THE FIRST DATA CARD IN 8F10.6 FORMAT.....

G TENSOR VALUES. . . . .

U( 1) = GXX
U(2) = GYY
U(3) = GZZ

T (HYPERFINE) TENSOR VALUES (IN WAVENUMBERS. I.E.. RECIPROCAL CM).
TXX

U(4)
U(5) TYY
TZZ

U(6)
TXX. TYY. AND 'IZZ MUST ALL HAVE THE SAME SIGN.

U(7) = NUCLEAR QUADRUPOLE COUPLING PARAMETER
SINCE THIS PROGRAM DOES NOT CONSIDER NUCLEAR QUADRUPOLE
COUPLING. U(7) IS SET EQUAL TO ZERO.

U( 8) = GAUSSIAN BROADEN ING PARAMETER

U( 8) IS NOT USED IN THIS PROGRAM FOR GENERATING PRINTED OUTPUT.

THE VALUES OF CONST(1).SPINNU.FIRSTM.AND LASTM ARE EXPECTED TO
BE IN THAT ORDER ON THE SECOND DATA CARD. THE FORMAT OF WHICH
IS TO BE E10.3,3F4.1.

CONST( l) = V = MICROWAVE FREQUENCY (IN HER'I‘Z)

SPINNU = NUCLEAR SPIN

FIRSTM = FIRST NUCLEAR MAGNETIC QUANTUM NUMBER TO BE USED IN
PROGRAM

LASTM ..... NUCLEAR MAGNETIC QUANTUM NUMBER WILL BE INCREMENTED-
BY ONE AND SIMULATED DATA WILL BE PRODUCED FOR EACH NUCLEAR
MAGNETIC QUANTUM NUMBER UNTIL NUCLEAR MAGNETIC QUANTUM NUMBER
EXCEEDS LASTM.

THE THIRD DATA CARD IS TO CONTAIN THE VALUES OF MTH AND MFI IN
213 FORMAT. IF EITHER IS GREATER THAN 19. MAP IS NOT PRINTED.

THE FIELD LENGTH REQUIREMENT FOR THIS PROGRAM IS INCREASED BY
MTH * MFI.

MTH = NUMBER OF DIFFERENT VALUES FOR ORIENTATION ANGLE THETA
(DEFAULT = 19)

MFI = NUMBER OF DIFFERENT VALUES FOR ORIENTATION ANGLE PHI
(DEFAULT = 19)

TAPE41, WHICH WOULD INCLUDE SIMULATED MAGNETIC FIELD VALUES. CAN
BE CATALOGED FOR POSSIBLE LATER USE.

143

 

 

 

1141!

C THIS PROGRAM CALCULATES MAGNETIC FIELD (H) VALUES BY USING A
C SOLUTION (THE ONE THAT INVOLVES ADDING A POSITIVE SQUARE ROOT)
C TO THE QUADRATTC EQUATION RESULTING FROM A CORRECTED VERSION OF
C EQUATION (Arll) ON PAGE 233 IN AN ARTICLE BY P.C. TAYLOR, J.F.
C BAUGHER. AND H.M. KRIZ. CHEMICAL REVIEWS. I975, 75(2).203-240.
C
C THIS PROGRAM WAS WRITTEN BY NEAL KILMER AT MICHIGAN STATE UNIV.
C
cm********************************************************************
C

REAL LASTM

COMMON /MAP/ CONST(2).EXTREMI3.12),U(IO).SPINNU.FIRSTM5LASTM

COMMON DYNSTO(1)

REWIND 41

READ l.(U(J).J=l.8)

I FORMAT (BFIO.6)

U(7) = 0.0
C THIS PROGRAM DOES NOT CONSIDER.NUCLEAR.QUADRDPOLE COUPLING
C PARAMETER.

READ 2.CONST(1).SPINNU.FIRSTM.LASTM
2 FORMAT (E10.3.3F4.I)
READ 9. MTH. MFI
9 FORMAT (213)
IF (MTH.LT.2) MTH = 19
IF (MFI.LT.2) MFI = 19
IDYNS = MTH * MFI + I
CALL SETFL (DYNSTO(IDYNS))

IONE = I
CALL MAP (DYNSTO(IONE).MTH.MFI)
END

 

 

1145

SUBROUTINE MAP (H.MWH.MFI)

REAL LASTM. M

COMMON /MAP/ CONST( 2) .EXTREM(3. 12) .U( 10) .SPINNU.FIRSTM. LASTM
DIMENSION HCMTH,MFI)

V = CONST(I)

LASTM = LASTM 'I" .001

GXX = U(l)
GYY = U(2)
GZZ = U(3)
TXX '-' U(4)
TYY’= U(5)
122 = U(6)

C = 2.997924562E+IO

PI = 3.14159 26535 898
BETA = 9.2732E—21
HPLANC = 6.625613-27
THNOS = 2 * (MTH-l)

'FINOS 2 * (MFI-I)
THINCR = P1 / 'I'HNOS
FIINCR = PI / FINOS
THINCD = 180. / TIINOS
FIINCD = 180. / FINOS

CONST(Z) = CONST(I) * 1.0E-O9
WRITE (41)
4 MTH,MFI.CONST(2),U(l).U(2).U(3),U(4),U(5).U(6).U(?).SPINNU _
225 FORMAT’CIHI.28X.*TRESE VALUES HAVE BEEN USED IN CALCULATING THE F0
OLLOWING SIMULATED DATA.....*/)
226 FORMAT (56X.*G TENSOR VALUES.....*)
227 FORMAT (58X.*GXX =*,FIO.6)
228 FORMAT (58X.*GYY =*.FIO.6)
229 FORMAT (58X.*GZZ =*.FlO.6/)
230 FORMAT (26X.*T (HYPERFINE) TENSOR VALUES (IN WAVENUMBERS. I.E.. RE
ECIPROCAL CENTIMETERS).....*)
231 FORMAT (58X.*TXX =*,F10.6)
232 FORMAT (58X.*TYY =*,F10.6)
233 FORMAT (58X,£TZZ =*,FlO.6/)
234 FORMAT (41X.*NUCLEAR,QUADRUPOLE COUPLING PARAMETER =*.FIO.6/)
236 FORMAT (43X.*MICROWAVE FREQUENCY (IN HERTZ) =*.IPEI3.5/)
237 FORMAT (56X.*NUCLEAR.SPIN =*.F5.1////z/)
WRITE 225
WRITE 226
WRITE 227.U(1)
WRITE 228.U(2)
WRITE 229.0(3)
WRITE 230
WRITE 231.0(4)
WRITE 232.U(5)
WRITE 233,U(6)
WRITE 234,U(7)
WRITE 236.CONST(I)
WRITE 237.8PINNU
AI HPLANC * C * TXX
HPLANC * C * TYY
RPLANC * c *‘REZ
GXX
GYY
GZZ
= A1 $ Al
A2 * A2
A3 * A3
01*01
02$02
— 03*03
M = FIRSTM
JEXTRA = 0
JEXTRM = O
REWIND 42
7 CONTINUE
TH = -THINCR
IF (MTH.GT.19) GO TO 8
IF (MFI.GT.I9) COTOB

O
p
II II II II II II

53
O
y...
I II II II II

 

1H6

WRITE 6.M
6 FORMAT (1H1923X.*RESONANCE FIELD VALUES (IN GAUSS) FOR.NUCLEAR.MAG

6NETIC QUANTUM

NUMBER = *.F4.1//)

8 CONTINUE

11

:9- Cl

15

WRITE
JEXTRM =
EXTREM(1,JEXTRM) =
ITH=1.MTH

DO 4

(41) M
JEXTRM + 1
M

TH = TH + THINCR

SINTH =
COSTH =
SINTH2
COSTH2
-FIINCR

FI =

SIN(TH)
COS(TH)
= S INTIIzm2
= COSTHxe

DO 3 IFI=I.MFI

FI =
SINFI
COSFI

SINFI2
COSFI2
SIN2F2

0A2 =
GA =
G:
00:

PI + FIINCR

= SIN(FI)
= COS(FI)

= SINFI*$2

= COSFI**2

= (2.*SINFI*COSFI)**2
(GIG1*SINF12) + (G2G2*COSFI2)

SQRT(GA2)
SQRT((GA2*SINTH2)+(G3G3*COSTH2))
G*G

B = (SQRT((GIG1*A1A1 *SINFI2)+(GZG2*A2A2 *COSF12)))/GA

BB =

B * B

AANUME = ((GA2*BB*SINTH2)+(GSG3*A3A3*COSTH2)I
AA = AANUI‘IE / (GJRG)

A = (SQRT(AANUME)) / G

AAA = G * BETA

BBB = (A*M) - (HPLANC*V)

TERM] = M3M¥SINTH2 / (2.*G*BETA)

PART2 = ((A1A1-A2A2)*SINFI2*COSFI2) / (A*A)

PART2 = PART2 * (A1AI-A2A2)

PART2 = PART2 * GIGI * G202 / (GA2*GG)

PARTS = GA2 * G363 * ((A3A3-BB)**2) *COSTH2 / (GG*GG*AA)
TERMI = TERMI * (PART2+PART3)

TERM2 = G1G1 * G2G2 * GGG3 / (4.*GA2*GA2*GG)

TERM2 = TERM2*A3A3*((A2A2-A1A1)**2) / (BB*AA)

TERM2 = (TERM2*COSTH2*SIN2F2)+(AlAI*A2A2/BB)

TERM2 = TERM2 + (A3A3*BB/AA)

TERMZ = TERM2 * ((SPINNU*(SPINNU+1.))-(M*M))/(4.*G*BETA)
CCC = TERMI + TERMZ .
H(ITH.IFI) = (-BBB+SQRT((BBB*BBB)~(4.*AAA*CCC)))/(2.*AAA)

WRITE (41) ITH.IFI.H(ITH.IFI)

IF (ITH+IFI.GT.2) GO TO ll

EXTREM(2,JEXTRM) = H(1.1)

EXTREM(3.JEXTRM) = H( I. I)

GO TO 12

CONTINUE

IF (H(ITH,IFI).LT.EXTREM(2.JEXTRM)) EXTREMI2.JEXTRM) = HIITH’IFI;

IF (H(ITH.IFI).GT.EXTREM(3.JEXTRM)) EXTREM(3.JEXTRM)

CONTINUE
CONTINUE
IF (MTH.GT.19) GO TO 4
IF (MFI.GT.19) GO TO 4

WRITE 5.(II( ITH. IFI) . IFI=1.MFI)

FORMAT (//6X519F6.0)

CONTINUE

INDVAR = 2

DO 19 ITH = I.MTH

D0 18 IFI = 2,MPI

IF (IFI.GT.2) GO TO 15 _
PD = H( ITH,2) - H(ITH.1) I
GO TO 18

CONTINUE

PDO = PD

PD = H(ITH,IFI) - H(ITH.IFI-1)

IF (PDO*PD.GT.0.0)
IF (PDO.EQ.0.0)

GO TO 18
GO TO 18

 

1‘47

JEXTRA = JEXTRA.+ 1

IMM = -1

IF (PDO.GT.0.0) IMM = I

THDEG = (ITH-1) * THINCD

FIDEG = (IFI-I) * FIINCD

WRITE (42) M. H( ITH, IFI) ,THDEG.FIDEG, INDVAR, IM

18 CONTINUE

19 CONTINUE
INDVAR = 1
DO 29 IFI = 1.MPI
DO 28 ITH = 2.MTH

IF (ITH.GT.2) GO TO 25
PD = H(2,IFI) - H(1,IFI)
GO TO 28
25 CONTINUE
PDO = PD
PD = H(ITH.IF1) - H(ITH-1.IFI)
IF (PDO*PD.GT.0.0) GO TO 28
IF (PDO.EQ.0.0) GO TO 28
JEXTRA = JEXTRA + 1
IMM = -1
IF (PDO.GT.0.0) IMM = 1
THDEG = (ITH-I) * THINCD
FIDEG = (IFI-I) * FIINCD
WRITE (42) M.H(ITH.IFI).THDEG.FIDEG.INDVAR.IMM
28 CONTINUE
29 CONTINUE
IF (M.LE.LASTM) GO TO 7
M = -1500.0
WRITE (41) M
WRITE 32
32 FORMAT (/////)
WRITE (41) JEXTRM
DO 39 J = I.JEXTRM
WRITE (41) EXTREM(1.J).EXTREM(2.J).EXTREM(3.J)
WRITE 33. (EXTREM(JJ.JI.JJ=1.3)
33 FORMAT (2X,*FOR NUCLEAR MAGNETIC QUANTUM.NUMBER.OF*,F5.I,*, MAGNET
TIC FIELD RESONANCE VALUES RANGE FROIfl.F10.3,* TO¥.F10.3/)
'IF (J.GT.1) GO'TO 35
HMIN = EXTREM(2.J)
HMAX = EXTREML3,J)
GO TO 39
35 CONTINUE
IF (EXTREM(2.J).LT.HMIN) HMIN = EXTREM(2.J)
IF (EXTREM(3.J).GT.HMAX) HMAX = EXTREM(3.J)
39 CONTINUE
WRITE 225
WRITE 226
WRITE 227.U(1)
WRITE 228.U(2)
WRITE 229,U(3)
WRITE 230
WRITE 231,U(4)
WRITE 232.U(5)
WRITE 233.U(6)
WRITE 234.U(7)
WRITE 236.CONST(1)
WRITE 237,SPINNU
WRITE (41) JEXTRA ‘
IF (JEXTRA.EQ.O) GO TO 99
REWIND 42
DO 97 J =1.JEXTRA
READ (42) M,HEXTRA.THDEG.FIDEG.INDVAR,IMM
THDE = THDEG‘~(THINCD*(2-INDVAR))
FIDE = FIDEG - (FIINCD*(INDVAR91))
WRITE (41) HEXTRA,THDE.THDEG.FIDE.FIDEG.IMM
WRITE 91.M.HEXTRA
91 FORMAT (/8X.*A SINGULARITY IS EXPECTED FOR.NUCLEAR.MAGNETTC QUANTU
UM NUMBER 0F*.F5.I.* AT A MAGNETIC FIELD 0F*.FIO.3.* GAUSS.!)
WRITE 92.THDE,THDEG

 

 

 

1148

92 FORMAT (17X;*ZERO PARTIAL DERIVATIVE OCCURS AT*.F6.2.

2 * DEGREES .LE. THETA .LE. *.F6.2,* DEGREES.*)
IF (IMM.LT.O) WRITE 94
IF (IMM.GT.O) WRITE 95
94 FORMAT (1H+,41X,*(MINIMMM)*)
95 FORMAT (1H+,41X.*(MAXTMUM)8)
WRITE 93,FIDE.FIDEG
93 FORMAT (17X.*ZERO PARTIAL DERIVATIVE OCCURS AT*.F6.2.
2 * DEGREES .LE. PHI .LE. *.F6.2,* DEGREES.*)
IF (IMM.LT.O) WRITE 94
IF (IMM.GT.O) WRITE 95
97 CONTINUE
99 CONTINUE
REWIND 42
WRITE (42) HMIN.HMAX
WRITE (42) U(B)
WRITE 34.HMIN.HMAX
34 FORMAT (//17X.*OVERALL. MAGNETIC FIELD RESONANCE VALUES RANGErFROM
M*,F14.7,* TO*,F14.7,* GAUSS.*//)
RETURN
END

 

 

 

 

1149

PROGRAM GATHER ( TAPE4 1 . TAPFA2=65 . OUTPUT-=5 13 . TAPE6 1 =OUTPUT. TAPE2 1 )
C
C***********************************************$$¥$8**************$*3*3

THIS PROGRAM PRODUCES A SIMULATED F IRST-DERIVATIVE (WITH RESPECT
TO THE MAGNETIC FIELD STRENGTH. HP) ELECTRON SPIN RESONANCE
SPECTRUM OF A MAGNETICALLY DILUTE POLYCRYSTALLINE OR FROZEN
GLASS SAMPLE.

ELECTRON SPIN (S. NOT REPRESENTED IN PROGRAM) ‘-' 1/2.

HYPERPINE SPLITTING OCCURS FROM ONE NUCLEUS. WHOSE NUCLEAR SPIN
(I.REPRESENTED IN THIS PROGRAM AS SPINNU) IS 9/2 OR LESS.

THE VALUES OF THE PARAMETERS U( 1) THROUGH U( 8) ARE EXPECTED TO BE
ON TAPE41 AND TAPE42 AS WRITTEN BY PROGRAM ESRMAP.

G TENSOR VALUES. . . . .

U( l) = GXX

U(2) = GYY

U(3) = GZZ

T (HYPERFINE) TENSOR VALUES ( IN WAVENUMBERS. I.E.. RECIPROCAL CID.
U(4) = TXX

U(5) = TYY

U(6) = TZZ

U( 7) = NUCLEAR QUADRUPOLE COUPLING PARAMETER

U( 8) = GAUSSIAN BROADENING PARAMETER

CONST( 1) = V = MICROWAVE FREQUENCY (IN HERTZ)
SPINNU = NUCLEAR SPIN

TAPE21. WHICH WOULD CONTAIN DATA FROM THESE SIMULATIONS. CAN BE
CATALOGED FOR POSSIBLE LATER USE.

THIS PROGRAM WAS WRITTEN BY NEAL KILMER AT MICHIGAN STATE UNIV.

**********************************************************************3

Of)O(50(50CDOCDOCDOC50C10C30650630C30C30610

ocaocaoca

COMMON /ALL/ BEGPLT.BETA.CONST(2).ENDPLT.HMAX.HMIN.IHGT.SPINNU.
+U(10)

COMMON DYNSTO(I)

REWIND 42

READ (42) HMIN.HMAX

READ (42) U(8)

IHGT = INT(HMIN+.4999)

IHLT = INT(HMAX}.5001)

LA = IHLT - IHGT + 1

REWIND 41

READ (41)
41 MTH.MFI.CONST(2).U(1).U(2).U(3).U(4),U(5).U(6).U(7).SPINNU

1 I

12 11 + LA

13 I2 + MTH

IDYNS = 13 + MTH * MFI

CALL SETFL (DYNSTO(IDYNS))

CALL CHANNL(DYNSTO(I1).DYNSTO(12).DYNSTO(I3).LA.MTH.MFI.LB)

LYCA = LA + LB - 1

I3 = I2 + LB

IDYNS = 13 + LYCA

CALL SETFL (DYNSTO(IDYNS))

CALL OUTPT (DYNSTO(II).DYNSTO(I2).DYNSTO(I3).LA.LB.LYCA.JSUB)

LPLOT = JSUB + 2

I2 = II + LPLOT ‘

13 = I2 + LPLOT

IBSUB = 1025

IDYNS = 13 + IBSUB

CALL SETFL (DYNSTO(IDYNS))

ENHL CCPLOT(DYNSTO(11).DYNSTO(I2).DYNSTO(I3).LPLOT.IBSUB)

II II II

 

 

00 00000

150

‘figggogTINE CHANNL (H.AREA.HH.LA.MTH.MFI.LB)
EOMEON IALL/ BEGPLT.BETA.CONSTKZ).ENDPLT.HMAX.HMIN.IHGT.SPINNU.
+

DIMENSION H(LA). AREA(MTH). HH(MTH.MFI)

PI = 3.14159 26535 898

THNOS = 2 * (MTHrl)

FINOS = 2 * (MFI-1)

THINCR = PI / THNOS

FIINCR = PI / FINOS

V = CONST(2) * 1.0E+09

CONST(I) = V

SUM = 0.0
DO 2 J = 1,MTH
RJ = J - 1

TH = THINCR * RJ

BASE = FIINCR * SIN(TH)

AREA(J) = BASE * THINCR./ 2.0

SUM = SUM + AREA(J)
2 CONTINUE

SUM = 2.*SUM - AREA(MTH)

SLICES = MFI - 1

DO 3 J = I.MTH

AREA(J) = AREA(J) / (SUM*((2.0*SPINNU)+1.0)*SLICES)
3 CONTINUE

D0 7 J = 1.LA

H(J) = 0.0
7 CONTINUE

IN THIS PROGRAM. H(J) REPRESENTS INTENSITY OF ESR.ABSORPTION AT

MAGNETIC FIELD OF IHGT + J - 1 GAUSS.
JMAX = MTH * MFI

6 CONTINUE
READ (41) M
IF (M.LT.-IOO0.0) GO TO 49

WHITE 4.M
4 FORMAT (10X.*DATA WERE READ FOR NUCLEAR MAGNETIC QUANTUM NUMBER 8
R$.F5.I)
DO 5 J = I.JMAX
READ (41) K.L.HYAL
HH(K,L) = HVAL
5 CONTINUE
DO 39 ITH = 2.MTH
D0 35 IFI = 2.MPI
ITH AND IFI SUBSCRIPTS DISTINGUISH POINTS DEFINED ON ONE-EIGHTH-
SPHERE BY THETA AND PHI. THOSE POINTS DEFINE (ITH-2)*(IFI-l)
TRAPEZOIDS AND (IFI-l) TRIANGLES. WHICH ARE TREATED AS
TRAPEZOIDS WITH ONE BASE EQUAL TO ZERO. INSCRIBED IN THE
SURFACE OF THIS ONE-EIGHTH-SPHERE.
30=3I END 2’REPRESENT TWO TRIANGLES FORMED BY DIAGONAL BISECTING
TRAPEZOID.
IF (J.GT.1) GO TO 8
HY’= HH(ITH-I.IFI)

GO TO 9

8 CONTINUE
HY = HH(ITH.IFI-l)

9 CONTINUE
HX.= HH(ITH-I,IFI-I)
HZ = HH(ITH,IFI)
IF (HX.GT.HY) GOTO 11
H1 = HX
H2 = HY
GOTO 12

11 III = HY
H2 = RX

12 IF (HZ.GE.H2) GOTO 18
H3 = H2
IF (HZ.GE.H1) GOTO 17
H2 = H1
HI = HZ

GOTO I9

 

GOO

GOO

00000

on

17

18
19

2O

21

22

26

27

30
31
35
39

49

151

H2 = HZ

GOTO 19

H3 = HZ

CONTINUE

IAREA = ITH + J - 2

IF (H3-H1.NE.0.0) GO TO 20

IF (H3-Hl) EQUALS ZERO. THEN THIS ENTIRE TRIANGLE REPRESENTS ONE
MAGNETIC FIELD. AND THIS ENTIRE AREA I: ALLOTTED TO THE
APPROPRIATE H(J).

JH = (INT(H3+O.5)) - IHGT + I

H(JH) = H(JH) + AREA( IAREA)

GO TO 31

CONTINUE

EACH TRIANGLE IS FURTHER DIVIDED BY'A LINE FROM'PHE H2 VERTEX‘TO
THE POINT ON THE OPPOSITE SIDE AT WHICH A LINEARLY-VARYING
MAGNETIC FIELD WOULD EQUAL H2.

COMFAC = AREA(IAREA) / (H3-H1)

AREAA COMFAC * (HZ-H1)

AREAB COMFAC * (H3-H2)

IHA = INT(H2+O.5)

HA = IHA

JHA = IHA ‘ IHGT‘+ 1

BETWEEN THIS COMMENT AND STATEMENT 25. THE AREA (REPRESENTING SOME
CONTRIBUTION TO ESR ABSORPTION INTENSITY) OF A CERTAIN TRIANGLE
IS ALLOTTED TO MAGNETIC FIELD VALUE(S). THIS TRIANGLE IS ONE
OF THOSE FORMED ACCORDING TO THE COMMENT IMMEDIATELY FOLLOWING
STATEMENT 20.

IF (H2-H1.EQ.0.0) GO TO 25

AREAR = AREAA

UNIT = AREAA / ((H2-Hl)*(H2-H1))

HR = HA - 0.5

JHB = JHA

CONTINUE

IF (HB.LE.H1) GO TO 22

H(JHB) = H(JHB) + AREAR.- (UNIT*(HB-H1)*(HB-HI))

AREAR = UNIT * (RB-H1) * (BB-HI)

JHB = JHB - 1

HR = HB - 1.0

GO TO 21

CONTINUE

H(JHB) = AREAR + H(JHB)

CONTINUE
BETWEEN THIS COMMENT AND STATEMENT 30. THE AREA OF ANOTHER

TRIANGLE IS ALLOTTED T0 MAGNETIC FIELD VALUE(S).
IF (H3-B2.E0.0.0) GO TO 30
AREAR = AREAB
UNIT = AREAB / ((B3-H2)*(H3-H2))
BB = HA + 0.5
JHB = JHA
CONTINUEE BS) GO TO 27
I (BB.G . ~
HFJHB) = H(JHB) + AREAR.- (UNIT*(B3-HB)*(H3—HB))
AREAR.= UNIT * (Ha-BB) * (HS-BB)
JHB = JHB + 1
HB = HB + 1.0
GO TO 26
CONTINUE
H(JHB) = AREAR + H(JHB)
CONTINUE
CONTINUE
CONTINUE
CONTINUE
GO TO 6
CONTINUE
BETA = U(8)
LB = (15.0*BETA) + .001
LBT = (LB+.OOl)/2.0
IF ((LBxZ).EO.(LBT*2)) LB = LB + 1
RETURN
END

 

 

 

152

SUBROUTINE OUTPT (H. B. YEA. LA. LB. LYCA. JSUB)
+3??g?fl /ALL/ BEGPLT.BETA.CONST(2). ENDPLT. HMAX. HMIN. IHGT. SPINNU.
DIMENSION H(LA). H(LB). YCA(LYCA)
DIMENSION XX(4). YTIO)

LBH= LB/2

HP = BH

DEFINE 1GAUSSIAN CONVOLUTION VECTOR.
B(LBH+

D0 2201J = I. LBH

F:

K = LB + 1 -

H(K) = EX§§CHP-P)*(F-HP) / (2. *BETA*BETA))

22

9
O
O
2
E
2
fi

= 250.0 * HINCR
XXNN = 5.0 * HINCR
INCR

XXKK =

225 FORMAT (1H1.28X,*THESE VALUES HAVE BEEN USED IN CALCULATING THE F0

OLLOWING SIMULATED DATA.....*/)
226 FORMAT (56X.*G TENSOR VALUES.....*)
227 FORMAT (58X.*GXX =*.F10.6)
228 FORMAT (58X.*GYY =*.F10.6)
229 FORMAT (58X.*CZZ =*,F10.6/)
230 FORMAT (26X.*T (HYPERFINE) TENSOR VALUES (IN WAVENUMBERS. I.E.. RE

ECIPROCAL CENTIMETERS).....*)
231 FORMAT (58X.*TXX =*.F10.6)
232 FORMAT (58X.*TYY =*.F10.6)
233 FORMAT (58X.*TZZ =*,F10.6/)
234 FORMAT (41X,*NUCLEAR QUADRUPOLE COUPLING PARAMETER =*.F10.6/)
235 FORMAT (44X.*GAUSSIAN BROADENING PARAMETER =*.F12.6/)
236 FORMAT (43X.*MICROWAVE FREQUENCY (IN HERIZ) =*.IPE13.5/)
237 FORMAT (56X,*NUCLEAR SPIN =*.F5.l//////)
101 FORMAT (1H1.16X.*H*.15X.*AT H*.18X.*AT H +*.F7.3.8X.*AT H +*.F7.3.

8X.*AT H +*,F7.3.8X.*AT H +*.F7.3/)

102 FORMAT (F21.3.1PE21.7.4E21.7)

DO 250 JTWICE = 1.2

WRITE 225

E 226
WRITE 227,U(I)
WRITE 228.U(2)
WRITE 229.U(3)
WRITE 230
WRITE 231.U(4)
WRITE 232.U(5)
WRITE 233. U(6)
WRITE 234, U(7 )
IF (JTWICE. GT. 1) GO TO 241
ZERO = 0. 0
WRITE 235, ZERO
HBEGIN = AINT ((HMIN-11.0)/10.0)
HBEGIN = 10.0 * HBEGIN
HEND ' AINT((HMAX+21.0)/10.0)

WRITEN 235.U(8)
HBEGIN — (7. 0*BETA) - 5. O

HBEGN =
HBEGIN = AINT(HBEGIN/10. 0)
HBEGI = 10.0 * HBEGIN
BEGPLT = HBEGIN
ND = HEND + (7.0*BETA) + 5.0
HEND = AINT (BEND/10.0)
HEND = 10.0 * HEND
ENDPLT = HEND

309

242

106

370
371

330
331

311
312

107

108
109
9999

340

3153

XXCON = HBEGIN - XXNK‘- XXNN
LAB = LA + LB - 1

D0 309 J = 1.LAB

YCA(J) = 0.0

CALL VCONVO (H.B.YCA.LA.LB.LAB)
LAB = LAB “ 1

CONTINUE

WRITE 236.CONST(1)

WRITE 237.SPINNU

JSUB = 0

D0 109 NK = 1.100
DO 106 IN = 1.4
Y(IN) = IN 3 HINCR
CONTINUE

WRITE (61.101) (Y(IN).IN=I.4)

DO 108 NN = 1.50

X)“ 1) = (XXNIGKNIO + (XXNNIKNN) + XXCON
IF (JTWICE.GT.1) XX(1) = XX(1) + 1.0
IF (XX(1).GT.HEND) GO TO 9999

IF (JTWICE.GT.1) XX(1) = XX(1) - 1.0
XXS = XX(1)

D0 107 XX = 1.5

XX(1) = XXS + (XXKK*(KK-1))

HP = YX(1)

IF (JTWICE.GT.I) GO TO 370

JH = (INT(HP+0.5)) - IHGT + 1

GO TO 371

JH = (INT(HP+0.5)) - IHGT + LBH + 1
CONTINUE

IF (JH.LT.1) GO TO 311

IF (JH.GT.LAB) GO TO 311

IF (JTWICE.GT.1) GO TO 330

DFCALC = H(JH)

GO TO 331

DFCALC = YCA(JH+1) - YCA(JH)

'X'X(1) = XX(1) + 0.5

2

CONTINUE

GO TO 312

DFCALC = 0.0

CONTINUE

Y(KK) = DFCALC

IF (XX(1).LT.BEGPLT) GO TO 107
IF (XX(1).GT.ENDPLT) GO TO 107
JSUB = JSUB + I

WRITE (21) JSUB

WRITE (21) XX(1).DFCALC
CONTINUE

IF (JTWICE.GT.1) XXS = XXS + 0.5
WRITE (61.102) XXS.(Y(IN).IN=1.5)
CONTINUE

CONTINUE

CONTINUE

JEOT = -JSUB

WHITE (21) JEOT

IF (JTWICE.GT.1) GO TO 340
UBSAVE = U(8)

U(8) = 0.0

CONTINUE

WRITE (21) ’U(2),U(3),U(4).U(5),U(6).U(7).U(8).SPINNU

JSUB.CONST(2).U(1)
IF (JTWICE.GT.1) GO TO 341
U(8) = UBSAVE

341 CONTINUE
250 CONTINUE

RETURN
END

00 T
335 READ (21)
2

GO
350 CONTINUE

1514

SUBROUTINE CCPLOT (XPLOT.YPLOT.IBUF.LPLOT.IBSUB)
COMMON /ALL/ BEGPLT.BETA.CONST(2).ENDPLT.HMAX.HMIN.1HGT.SPINNU.

+U( 10

gégEgSION XPLOT(LPLOT). YPLOT(LPLOT), IBUF(IBSUB)
D0 260 JTHICE = 1.2

CONTINUE

READ (21) NK
IF (NK.LE.0) co T0 335 ‘
READ (21) XPLOT(NK).YPLOT(NK)

o no ‘

JSUB.CONST(2).U(1).U(2).U(3).U(4).U(5).U(6).U(7).U(8).SPINNU

RANGE = ENDPLT - BEGPLT

SCALU = 1.0

IF (RANGE.GT.1000.0) SCALU = 10.0

XSTART = AINT(BEGPLT/SCALU)

XPLOT(JSUB+1) = XSTART * SCALU

RANGE = RANGE + (9.5*SCALU)

HINCR = AINT(RANGE/(8.5*SCALU))

XPLOT(JSUB+2) = SCALU * HINCR

CALL PLOTS (IBUF. 1025) 5)

CALL PLOT (0. 0.I.7.-

CALL SCALE (YPLOT. 6. 0. JSUB. 1)

IF (JTWICE.EQ.1) GO TO 350

CALL SYMBOL (— 0. 80. 0. 00. 0. 098. 6IH(SIMULATED DATA WERE PRODUCED BY
YESRMAP AND GATHER--METHOD2 .). 90.0, 61 )

CALL AXIS (O. 0 0. 0, 34HFIRST DERIVATIVE 0F ESR ABSORPTION. 34. 6. 0.

90.0. YPLOT(JSUB+1). YPLOT(JSUB+2))
351

CALL AXIS (0.0.0.0,14HESR ABSORPTION.14.6.0.
90.0.YPLOT(JSUR+1).YPLOT(JSUB+2))

351 CONTINUE

36
36

26

0
1

0

CALL AXIS (0.0.0.0.23HOBSERVED FIELD IN GAUSS.-23.8.5.
A 0. 0. XPLOT(JSUB+I). XPLOT(JSUB+2))

CALL LINE (XPLOT.YPLOT.JSUB.1.0.

CALL0 SYMBOL (0.00.7.14.0.140.35HMICROWAVE FREQUENCY IN GIGAHERIZ
Z=. .35)

CALL0 NUMBER (5. 18. 7. 14, 0. I40. CONST(2). 0. 0. 4)
CALL SYMBOL (0. 00. 6. 81. 0. I40. 5H0 =,0.0 5)
CALL SYMBOL (0. I4. 6. 76. 0. 098. 2HXX. 0. 0. 2)
CALL NUMBER (0.98.6. 81.0. 140. U(I).0. 0.4)
CALL SYMBOL (3. 00. 6. 81. 0. 140. SEC = 00. 5)
CALL SYN“OL (3. 14. 6. 76. 0. 098. 2HYY. 0. 0, 02)
CALL NUMBER (3. 98. 6 81. 0. I40. U(2). 0. 0. 4)
CALL SYMBOL (6 00. 6. 81. 0. 140.536 .0. 0, 5)
CALL SYMBOL (6 14. 6. 76, 0. 098.2HZZ. O. 0. 2)
CALL NUMBER (6. 98, 6. 81. 0. I40. U(3), 0. 0, 4)
CALL SYMBOL (0. 00, 6. 48. O. 140. 4HT ,0. O, 4)
CALL SYMBOL (O. 14, 6. 43. 0. 098, 2NXX. O. 0. 2)
CALL NUMBER (0."0. 6. 48, 0. 140. U(4). 0. 0. 6)
CALL SYMBOL (3. 00. 6. 48. 0. 140. 4HT = .0. 0. 4)
CALL SYMBOL (3. 14. 6. 43. 0. 098.2HYY. 0.0.2)
CALL NUMBER (3. 70. 6. 48. 0. 140. 0(5), 0.0.6)
CALL SYMBOL (6. OO. 6 48, 0. 140,4HT =,0.0.4
CALL SYMBOL (6 14. 6. 43. 0. 098.2NZZ.0.0.2)
CALL NUMBER (6.70. 6. 48. O. 140. U(6). 0. 0.6)
CALL SYMBOL (0.00.6.15.0.140.4HI =,0.0,4)
CALL NUMBER (0.84.6.15.0.140.SPINNU.0.0.1)
CALL SYMBOL (3. 00. 6. l5. 0. 140. 4HQ =.0.0.4)
CALL SYMBOL (3.24,6.15.0.136.16.0 0.-1)
CALLN NUMBER (3.70. 6 15.08 140. U(7). 0.0.6)

IF (JTWICE.EQ.1) CO T0360

CALL SYMBOL (6. 00. 6. 15.0. 140. 6HBETA= ,0. 0. 6)
CALL NUMBER (6. 98. 6. 15.0. 140. U(8), 0. 0. 3)

GO

CALEOSYMBOL (6.00. 6. 15.0 140.13NDELTA H=1.0,0.0.13)
CONTINUE

CALL PLOT (11.0,-l.7.999)

CONTINUE

RETURN

END

NH

155

SUBROUTINE VCON'VO (A.B.C.LA.LB.LAB)
DIMENSION A(LA). H(LB). C(LAB)
DO 2 J=1.LAB

DO I K=I.J

IF (K.CT.LA) GO TO I

JMKPI = J - K + I

II" (JI‘IKPI.GT.LB) GO TO 1
C(J) = C(J) + (A(IO*B(JMKP1))
CONTINUE

CONTINUE

RETURN

END

 

156

PROGRAM PEAKS (TAPE21.0UTPUT:513)

 

***********************************************8***********888*******¥*

THIEAPROERAM USES DATA FROM‘THE TAPE21 FILE GENERATED BN'PROGRAM

ONLY DATA FROM THE SECOND SET OF DATA IN TAPEZI ARE USED IN
CALCULATIONS.

THUS. ANOTHER VERSION OF PEAKS SHOULD BE USED IF TAPEZI WAS
PRODUCED BY A VERSION OF GATHER.THAT WRITES ONLY'ONE SET OF
DATA ONTO TAPE21.

APPROXIMATE MAGNETIC FIELD VALUES AT WHICH PEAKS. VALLEYS. AND
25 PERCENT. 50 PERCENT. AND 75 PERCENT OF EACH VERTICAL PEAK?
TO-PEAK DISTANCE OCCUR ARE GIVEN IN OUTPUT.

NO DATA CARDS ARE REQUIRED.

INCREMENT BETWEEN CONSECUTIVE MAGNETIC FIELD VALUES 0N TAPEZI MUST
BE ONE GAUSS.

THIS PROGRAM.WAS WRITTEN BY'NEAL KILMER AT MICHIGAN STATE UNIV.

CDCCDOC)OC50C§OC§DCDOC§O

********************$***************************$$***$*****************

0(50630C30C50

COMMON /EXTR/ YEXTRM
DIMENSION MINMAX(500).XTREM(500)
DIMENSION CONST(16).U(20).XPLOTC4025).YPLOT(4025)

DIMENSION PCT(3)

'REWIND 21
DO 336 JTWICE = 1.2
J = 0

110 CONTINUE
READ (21) NK
IF (NK.LT.0) GO TO 335
READ (21) X.Y
J = J + l
XPLOT(J) = X
YPLOT»: J) = Y
GO TO 110
335 READ (21)
2 JSUB.CONST(16),U(1).U(2),U(3).U(4).U(5),U(6).U(7).U(8).SPINNU
336 CONTINUE
K = 0
JXMAX
LASTP
KPTOP
X1
X2
X3
Y1
Y2

J

0

-l
XPLOT( 1)
XPLOT(2)
XPLOT(3)
YPLOT( l)
YPLOT(2)
Y3 YPLOT(3)
JX 3

1 CONTINUE '
IF (ABS(X3-X2-1.0).LT.1.0E-07) GO TO 340

IF (ABS(Y1).GT.I.0E-12) GO TO 9998

IF (ABS(Y2).GT.I.0E-12) GO TO 9998

IF (ABS(Y3).GTZI.0E-l2) GO TO 9998

340 CONTINUE

IF (ABS(Y2).GT.I.0E-12) GO TO 10

IF (ABS(Y1).LE.1.0E-12) GO TO 20

IF (ABS(Y3).LE.I.0E-12) GO TO 20
10 CONTINUE

IF ((Y3-Y2)*(Y2-Yl).GT.0.0) GO TO 20

K = K + 1

IF ( (Y3-Y2) - (Y2-Y1) ) 11.12.13
11 MINMAX(K) = 7HMAXIMUM

XTREM(K) = EXTREM(X1.Y1.Y2.Y3)

XLMAX = XTREM(K)

12

13

14

16

l7

19

21

22
15
20

30

157

JLMAX = JX

YLMAX = YEXTRM

IF (LASTP.EQ.-1) KPTOP=1
XPI XLMIN

XP2 XLMAX

YPI YLMIN

YP2 YLMAX

LASTP = 1

JP = JLMIN

GO TO 14

MINMAX(K) = 4HFLAT

XTREM(K) = 1.0E+99

GO TO 15

MINMAX(K) = 7HMINIMUM
XTREM(K) = EXTREM(X1.Y1.Y2.Y3)
XLMIN ' XTREM(K)

JLMIN JX

YLMIN YEXTRM .

IF (LASTP.EQ.1) KPTOP=1
XPI XLMAX

XP2 XLMIN

YPI YLMAX

YP2 YLMIN

JP = JLMAX

LASTP = -1

CONTINUE

IF (KPTOP.LE.0) GO TO 15
KPTOP = -1

XTREM(K+3) = XTREMUO
MINMAX(K+3) = M]NMAX(K)
KBASE = K - 1

MINMAX(K) = 6H25 PCT
MINMAX(K+1) = 6H50 PCT
MINMAX(K}2) = 6fl75 PCT

PCT(l) = YPl + (0.25*(YP2-YP1))
PCT(2) = YP1 + (0.50*(YP2-YP1))
PCT(3) = YPI + (0.75*(YP2-YP1))
JY = I

CONTINUE

XP2 = XPLOT(JP)

YP2 = YPLOT(JP)

DISTl = YPI v PCT(JY)

DIST2 = YP2 - PCT(JY)

IF (DIST1*DIST2) 17.17.19
DENOM = ABS(DIST1) + ABS(DIST2)
K'= KBASE + JY

DISTl = ABS(DIST1)

DIST2 = ABS(DIST2)

XTREM(K) = (DIST1*XP2) + (DIST2*XP1)
XTREM(K) = XTREM(K) / DENOM

GO TO 21

JP = JP + 1

XPI = XP2

YPI = YP2

GO TO 16

IF (JY.GE.3) GO TO 22

JY = JY + 1

GO TO 16

K = K + 1

'CONTINUE

IF (Jx.cT.Jm15m. 00 '10 30
X1 ’

x2
Y1
Y2
X3 XPLOT(JX)
Y3 YPLOT(JX)
(:0 T0 1
KMAX = K
WRITE 33

Y3

II II II II II II
4
l0

 

33

158

Egogmn (31X.*MAGNETIC FIELD STRENGTH IN GAUSS*.5X.*TYPE 0F FEATURE

/
FORMAT (51X.F12.2.A15)
I) 1

O 32 K = .KMAX
WRITE 31.XTREM(K) .MINMAX(K)
CONTINUE
GO TO 9999
WRITE 9

997
FORMAT (/10X.*DATA ARE NOT SUITABLE FOR THIS PROGRAMJh’)
CONTINUE
END

-:"(--.

0000

159

FUNCTION EXTREM(X1.Y1.Y2.Y3)
THIS FUNCTION RETURNS THE VALUE OF‘X AT WHICH THE PARABOLA DEFINED
BY THE (X.Y) POINTS (X1.Y1). (X1+1.Y2). AND (X1+2.Y3) HAS AN

EXTREMUM (MINIMUM OR MAXIMUPD .
THE Y‘VALUE AT THAT EXTREMUM IS MADE AVAILABLE VIA LABELED COMMON.
COMMON /EXTR/ YEXTRM
A = (Y3-(2.*Y2)+Y1) / 2.
B = Y2 - Y1 - (2.*A*X1) - A
C = Y] - (A*X1*X1) - (B*X1)

EXTREM = - B / (2.*A)

X = EXTREM

YEXTRM = (A*X*X) + (B3100 + C
RETURN

 

APPENDIXID

LISTING OF SUBROUTINES EQN AND KILMER

SUBROUTINE EQN

C
C**********************************************$************************

C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C

C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C
C

***************************************

THIS PAIR OF SUBROUTINES IS INTENDED FOR USE WITH KINFITfi.

PURPOSE:.....

TO DETERMINE VALUES OF MOLECULAR.ORBITAL COEFFICIENTS FOR THE.
PAIR OF MOLECULAR ORBITALS THAT CONTAIN THE UNPAIRED ELECTRON.

RESTRICTIONS .....
(l) COMPOUND MUST CONTAIN EXACTLY ONE UNPAIRED ELECTRON.

(2) VALUES OF SPIN-HAMILTONIAN PARAMETERS MUST BE AVAILABLE
AND IIKHJWUED IN INPUT AS INDICATED HEREV.....

GXX AS CONST(I)
GYY AS CONST(2)
GZZ AS CONST(3)
TXX AS CONST(4)
TYY AS CONST(5)
TZZ AS CONST(6)

ASSUMPTIONS .....

(1) EACH OF THE TWO MOLECULAR ORBITALS CONTAINING THE UNPAIRED
ELECTRON CAN BE DESCRIBED BY AN ISOTROPIC TERM PLUS 10

OTHER TERMS, EACH OF WHICH HAS THE FORM
COEFFICIENT * D‘ORBITAL WAVEFUNCTION * ELECTRON SPIN
WAVEFUNCTION.

(2) THE LOWERrENERGY MOLECULAR.ORBITAL IS PRIMARILY A DXY’AND
DX2Y2 CENTRAL ATOM ORBITAL WITH ELECTRON SPIN = -1/2.

(3) THE HIGHERFENERGY MOLECULAR ORBITAL IS PRIMARILY A DXY AND
DX2Y2 CENTRAL ATOM ORBITAL WITH ELECTRON SPIN = +1/2.

(4) ONLY SPIN-ORBIT INTERACTION IS IMPORTANT IN MIXING OTHER

WAVEFUNCTIONS INTO THESE TWO MOLECULAR ORBITALS.
THUS. HALF OF THE M.O. COEFFICIENTS CAN BE SET EQUAL TO

ZERO.

SPIN-ORBIT INTERACTION IS CONSIDERED TO ONLY FIRST ORDER.

(5)
THUS. A22 AND C22 ARE SET EQUAL TO ZERO.

SPECIAL INPUT INSTRUCTIONS .....
INCLUDE A DATA CARD WITH VALUES FOR X(1.1) AND VAR(1.1) .

(THE X(I.1) AND VAR(1.1) VALUES ARE MEANINGLESS OR.ESSENTIALLY
MEANINGLESS. BUT KINFIT4 EXPECTS THEM.)

THE VAR(1,1) VALUE SHOULD BE A POSITIVE NUMBER.

SEE KINFIT INSTRUCTIONS FOR FORMAT AND FOR OTHER INPUT REQUIRED.

----- INEAL KJLMEKI
********************************

160

 

161

COMMON KOUNT.ITAPE.JTAPE.IWT,LAP.XINCR.NOPT.NOVAR.NOUNK.X.U.ITMAX.
.TEST,I.AV.RESID.IAR.EPS.ITYP.XX.RXTYP.DX1I.EOP.F0.FU.P.ZL.TO.E

2IGVAL,XST,T.DT.L,M.JJJ,Y.DY.VECT.NCST,CONST,NDAT.JDAT.MOPT.LOPT.
3YYY, CONS TS

COMMON/FREDT/I MTH

COMMON/POINT/KOPT. JOPT. XXX

COMMON /SQRT3/ SQRB

DIMENSION X(4.300).U(20),WTX(4.300).XX(4).FOP(300).F0(800).FU(300)
1 P(20. 21), VECT(20. 21). ZL(300). T0(20). EIGVAL(20). XST(300).Y(10).
2DY(10), CONSTS(50. 16) NCST(50). RXTYP(50). DXlI(50)
3 MOPT(50) LOPT(50). YYY(50) CONST(16). XXX<I5

GO TO (2,3,4.5.1.7.8.9.10.11.12) ITYP

CONTINUE

p.

Mm"!
DO
00
22
2:
22
g C

P!

CONTINUE
CALL KILMER
IF(IMETH.NE.‘1) GO TO 35

RETURN
CONTINUE
RETURN
CONTINUE
RETURN

3

$0001
0
O
2
'-I
b-fl
E

CONTINUE
IF(IMETH.NE.-1) GO TO 20
RETURN

GI

20 CONTINUE
RN

0
O
O
2
"3
v—I
2
a

_-
n
O
2
"I
t—
2
E

l

N
o
o
2
3
2
g

l

0000 0000 COD

0000

0000

162

SUBROUTINE KILMER

REAL

COMMON FILL1(1209).U(20).FILL2(1204).RESID.F1113(2562).CONST(16).
2 FILL4(952)

COMMON /POINT/ KOPT,FILL5(16)

COMMON /SORT3/ SQR3

EQUIVALENCE (GXX,CONST(1)), (GYY.CONST(2)). (GZZ,CONST(3))
EQUIVALENCE (TXX,CONST(4)). (TYY.CONST(5)). (TZZ.CONST(6))
EQUIVALENCE (AXY.U(1)). (AX2Y2.U(2)), (BXZ,U(3)). (BYZ,U(4))
figgIVALENCE (EXZ,U(5)), (EYZ.U(6)), (KAP.U(7)). (P.U(8))

AYZ
A22

IHIII
$96

:92

9

ms:
E‘
5

E22 = .
DATA FREE /2.00232/
CXY AND CX2Y2 ARE NEEDED. SO CALCULATE THEIR VALUES.

D = 2.*AX2Y2*AZZ - SQR3*(AXY*BYZ+AX2Y2*BXZ)
IF (D.NE.0.0) GO TO 20

TO AVOID TRYING TO DIVIDE BY ZERO. SET D EQUAL TO A VERY SMALL
NUMBER AND PROCEED.

D = 1.0E-5O
20 CONTINUE
NOW THAT D HAS BEEN CALCULATED OR APPROXIMATED. CALCULATE CXY AND
0X2 .
F1 = (GXX/FREE) - (AXZ*CXZ+AYZ*CYZ+AZ2*CZ2+BX2Y2*EX2Y2+BXY*EXY
+BXZ*EXZ+BYZ*EYZ+BZ2*EZZ)
T2 = AXY*CZZ + BXY*E22 + BZZ*EXY
T3 = AXY*EXZ + AXZ$EXY ‘ AX2Y2*EYZ - AYZ$EX2Y2 - BXY$CXZ
2 + BX2Y2*CYZ - AXZ*CYZ - AYZ*CXZ - BXZXEYZ - BYZ*EXZ
T1 = AYZ*EZ2 + AZ2*EYZ - BYZXCZZ - BZZ*CYZ
F3 = 2.*T2 + SQRS$T3 + T1
CXY = (-F3*AX2Y2-SQR3$F1*BYZ) / D -
CX2Y2 = (F3*AXY+2.*Fl*AZ2-SQR3*F1*BXZ) / D
THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST
EXACTLY SATISFIED.
T4? = BX2Y2**2 + BXY**2 - AX2Y2$$2 - AXY**2 + A22**2 - BZZ**2
T27 = AXZ**2 + AYZ**2 - BXZ**2 - BYZ**2
T6? = - AXZ$BX2Y2 - AYZ$BXY - AX2Y2$BXZ - AXY*BYZ
a = - 2*BXZ — AXZ$B22
$32: = (A§.*T47 + 2.*127 + 6.*T67 + 2.*SQR3*T237 ) / 7.
CALC = P * ( CALC - KAP )
RESID = CALC — TZZ
RESID = RESID * 3.3E+O4
TOTAL = RESID**2
5O CONTINUE

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST
EXACTLY SATISFIED.

C = FREE * ( AX2Y2**2 + AXY**2 + AXZ**2 + AYZ**2 + A22**2
CAL - BX2Y2**2 - BXY**2 - BXZ**2 —BYZ**2 - BZZ**2 )

RESID = CALC - GZZ
RESID = RESID * 500.
= TOTAL + RESID**2

 

anon anon

0000

GOOD

noon

7O
80

100

110

163

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH TO
EXACTLY SATISFIED. IS BE ALMOST

RESID = AXZ*EXY - AXY*EXZ + AX2Y2*EYZ - AYZ*EX2Y2 + BXZ*CXY
~ BXY*CXZ + BX2Y2*CYZ - BYZ¥CX2Y2 + SQR3 * ( AZZ*EYZ
- AYZ*E22 + BZ2*CYZ - BYZ*CZZ )
RESID = RESID * 500.
= TOTAL + RESID**2
CONTINUE
CONTINUE

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST
EXACTLY SATISFIED.

CALC = FREE * ( CX2Y2**2 + CXY**2 + CXZX$2 + CYZ**2 + C22**2
- EX2Y2$$2 — EXY**2 - EXZ**2 - EYZ**2 - EZZ**2 )

RESID = CALC - GZZ

RESID = RESID * 500.

TOTAL = TOTAL + RESID**2

CONTINUE

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST

EXACTLY SATISFIED.

T17 = AX2Y2*CX2Y2 + AXY$CXY - A22*CZZ + BX2Y2$EX2Y2 + BXY*EXY
- BZZ*EZZ + AXZ*CXZ + BXZXEXZ

T27 = - AYZ*CYZ - BYZ*EYZ

T37 = - A22*CX2Y2 - AX2Y2$CZ2 - BZ2*EX2Y2 - BX2Y2$EZ2

T314 = - AXZ*EX2Y2 - AYZ*EXY - AX2Y2$EXZ - AXY$EYZ + BXZ*CX2Y2
2 + BYZ*CXY + BMY2>KCXZ + BXY>I=CYZ

T3114 = BXZ*C22 + BZZ$CXZ - AXZ*EZ2 - A22*EXZ

CALC = T17 + 2.*T27 + SQR8*T37 ) / 7.

CALC = CALC + ( 3.*T314 + SQR3*T3114 ) / l4.

CALC = P * ( 2. * CALC - KAP )

RESID = CALC - TXX

RESID = RESID * 3.3E+O4

TOTAL = TOTAL + RESID**2

CONTINUE

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST
EXACTLY SATISFIED.

T17 = AX2Y2*CX2Y2 + AXY*CXY - AZ2*CZ2 - BX2Y2*EX2Y2 - BXY*EXY
2 + BZ2*EZ2 + AYZ$CYZ - BYZ*EYZ .

T27 = - AXZ*CXZ + BXZ*EXZ

T37 = AZZ*CX2Y2 + AX2Y2*CZZ - BX2Y2*E22 - BZZ*EX2Y2

T22 = AX2Y2$EXZ + AXY*EYZ - AXZ*EX2Y2 - AYZ*EXY + BX2Y2*CXZ
2 + DXY*CYZ - BXZ*CX2Y2 - BYZ*CXY

T322 = AXZ*EZ2 - AZ2*EXZ + BXZ*CZ2 - BZZ*CXZ

CALC = ( T17 + 2.*T27 + SQR3*T37 ) / 7.

CALC = CALC + ( T22 + SQR3>I=T322 ) a: 2. / FREE

CALC = P * ( 2. * CALC - KAP )

= CALC - TYY
RESID = RESID X 3.8E+O4
TOTAL = TOTAL + RESID$$2

THE FOLLOWING CORRESPONDS TO AN
EXACTLY SATISFIED.

T37 = AZZ*CXY + AXY$CZ2 - BXY*EZ2 - BZZEYEYEXZ

T314 = - AXZ*CYZ - AYZ*CXZ + BXZ*EYZ + *

T22 = AXY$EXZ - AX2Y2*EYZ + AYZ$EX2Y2 - AXZ*EXY + BXY*CXZ
~ BX2Y2*CYZ + BYZ*CX2Y2 - BXZ*CXY

EXPRESSION WHICH IS TO BE ALMOST

2
T392 ' AYZ$EZ2 - BZ2*CYZ + BYZ*CZ2 - A22*EYZ

SQRB$T37/7. + 3.*T314/14. + (T22+SQR3*T322)*2./FREE

RESID = RESID * 330.
= TOTAL + RESID**2

0000

0000

1611

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST
EXACTLY SATISFIED.

T2 = AX2Y2$EXZ - AXZ$EX2Y2 + AXY*EY2 - AY2*EXY + BX2Y2$CX2
2

- BXZ*CX2Y2 + BXY*CY2 - BYZ$CXY
T23 = AXZ*E22 - AZ2*EXZ + BX2*CZZ - BZ2*CXZ

TFREE = AX2Y2*CX2Y2 - BX2Y2*EX2Y2 + AXY*CXY BXY* EXY + WCXZ
2 - BXZ*EXZ + AYZ*CY2 - BYZ*EYZ + A22*C22 - BZ2* E22

CALC = 2. *T2 + 2. *SOR3*T23 + FREE*TFM E

RESID = CALC - GYY

RESID = RESID * 500.

TOTAL = TOTAL + RESID**2

CONTINUE

140

THE FOLLOWING CORRESPONDS TO AN EXPRESSION WHICH IS TO BE ALMOST
EXACTLY SATISFIED.

T4? = - CX2Y2**2 - CXY**2 + EX2Y2**2 + EXY**2 + CZ2**2 - E22**2
T27 = CX2**2 + CYZ**2 - EXZ**2 - EYZ**

T67 = CX2Y2*EX2 + CXZ*EX2Y2 + CXY*EYZ + CYZ*EXY

T32 = CXZ*E22 + CZ2*EX2

CALC = ( 4.*T47 + 2.*T27 + 6.*T67 + 2.*SOR3*T327 ) / 7.
CALC = P * ( CALC - KAP )

RESI = CALC - TZZ

RESID = RESID * 3.3E+04

TOTAL = TOTAL + RESID**2

RESID = SORT (TOTAL)

RETURN

END

CO

I
l

110

01

APPENDIX E
LISTING OF PROGRAM M. 0. DATA SORT

PROGRAM MO (INPUT=513.0UTPUT.TAPE1=257.TAPE2=257.TAPE3=257.
+ TAPE4=257.TAPE5=257.TAPE6=257.TAPE7=257.TAPE8=257.TAPE65=INPUT)

REAL KAP

DIMENSION NPTS(8)

DATA FREE / 2.00232 /
FORMAT ( / 10X. 5E20.7 )

O FORMAT ( 5E13 7 )

5

1

[Or-

a

3

2
2

4

0

01°

9

FORMAT (3E13.7.26X.F7.4.A8)
DATA AXZ / O. /
DATA AYZ / O. 1
DATA A22 / O. /
DATA B / 0. /
DATA BX2Y2 / O. /
DATA BZ2 / O. /
DATA C22 / O. /
DATA E / O. /
DATA EX2Y2 / O. /
DATA EZ / O. /
DO 1 J = 1.8

REWIND J

WRITE (J.1101) J
FORMAT (I3)

NPTS(J) = O
CONTINUE

CONTINUE

WRITE 3

FORMAT (1H1)
READ IO, AXY, . BXZ. BYZ. EXZ -
IF (EOF(65).NE.0.0) GO TO 900

H5 EYZ, KAP. P. GXX. JOBNO

FORMAT ( / 43X. 10H ---------- ,5X.A8. 6X.10H -------- ////// )
WHITE 5, AXY. AX2Y2, sz. AYZ, A22

WRITE 5. BXY, BX2Y2. sz. BYZ, BZ2

WHITE 5. EXY, Ex2Y2. Exz. EYZ, 1522

WHITE 5. 022. KAP. P. GXX

CALCULATE CX2 AND CYZ.

CX2 = BXZ * EX2Y2 / AX2Y2

CYZ = BYZ * EXY / AXY

CXY AND CX2Y2 ARE NEEDED, SO CALCULATE THEIR VALUES.

D = 2.*AX2Y2*AZ2 - SOR3*(AXY*BYZ+AX2Y2*BXZ)

NOW THAT D HAS BEEN CALCULATED 0R APPROXIMATED. CALCULATE CXY AND
CX2 .

F1 :2(GXX/FREE) - (AX2*CXZ+AYZ*CYZ+A22*C22+BX2Y2*EX2Y2+BXY*EXY
2 +BXZ$EX2+BYZ$EYZ+B22$E22)
AXY*CZ2 + BXY*E22 + B22*EXY

T2:

T3 = AXY*EX2 + AX2*EXY - AX2Y2*EY2 - AYZ*EX2Y2 - BXY*CX2
2 + BX2Y2*CY2 ~ AXZ*CYZ - AY2*CX2 — BXZ$EYZ - BY2*EX2
T1 = AYZ$EZ2 +‘AZ2*EYZ ‘ BY2*CZ2 - 822*CY2

F3 = 2.*T2 + SOR3*T3 + T1

CXY = (—F32‘:AXZY2-SQR3*F1*BYZ) / D
CX2Y2 = (F3*AXY+2.*F1*AZZ-SQBS*F1*B)E) / D

WRITE 20. CXY. CX2Y2

FORMAT (//////5ox.*cxy =*,El5.7 /// 50X.*CX2Y2 =*,E15.7 xx)
WRITE 25, cxz, cyz

FORMAT (50x.*cxz =*.E15.7 /// 50x,*cyz =*,E15.7 // )

PK = KAP a: P

WHITE 40 PK

FORMAT ('//// 48x, IOEKAP * P =. E15.7 // )

SSO = AXY**2 + AX2Y2**2 + mm + AYZ**2 + A22**2

165

166

 

O. + BXY**2'+ BX2Y2**2 + BXZ**2 + BYZ**2 + B22**2
SSR.= CXY**2 + CX2Y2**2 + CXZ**2 + CYZX*2 + CZ2**2
R, + EXY**2 + EX2Y2**2 + EXZ**2 + EXZ**2 + E22**2
WRITE 50. SSO. SSR
5O FORMAT ( ////// 35X,*SUM.OF SQUARES OF A AND B COEFFICIENTS =*,
5 015.7 /// 35X.*SUM.OF SQUARES OF C AND E COEFFICIENTS =3.
5 G15.7 /// )

DETERMINE WHICH CASE THESE M.O. COEFFICIENTS BELONG TO.

000

IF (GXX.EQ.1.9897) GO TO 110
IF (GXX;EO.1.9806)‘ GO TO 120
WRITE 180 '
130 FORMAT (*1 --~ GXX‘VALUE DOES NOT FIT ANY OF THE DEFINED CASES.
+ ---* / )
GO TO 999
110 ICASE = 1
GO TO 140
120 ICASE = 5
140 CONTINUE
IF (ABS(AXY).GT.O.9) GO TO 160
IF (ABS(AX2Y2).CTSO.9) GO TO 150
WRITE 170
I70 FORMAT (*1 --- UNDEFINED CASE ---* / )
GO TO 999
150 ICASE = ICASE + 2
AMAJOR = AX2Y2
GO TO 180
160 AMAJOR.= AXY
180 CONTINUE
IF (AMAJOR.LT.O.) GO TO 190
ICASE = ICASE + I
190 CONTINUE
WRITE 200. ICASE
200 FORMAT ( / 48X3*CLASSIFICATION... CASE =*.IZ)
NP'I‘S( ICASE) = NP'IS( ICASE) + 1
NP = NPTS(ICASE)
WRITE (ICASE.1102) NP. AXY. AX272. BXZ. BYZZ CXY. szYz. EXZ,
+ EYZ, SSQ, SSR. '
1102 FORMAT (13.10F12.9)
GO TO 2
900 CONTINUE
DO 910 J = 1.8
REWIND J
910 CONTINUE
WRITE 980
980 FORMAT (*1 --- NORMAL END -—-*)
999 CONTINUE
END

 

 

 

APPENDIX F

COMPARISON OF ESRMAP, GATHER,
AND PEAKS WITH SIMIHA

A computer program, SIMllIA,15 can be used to simulate
first or second derivative powder pattern ESR spectra for
S = 1/2 systems that can be described by the spin Hamil-
tonian

A

J{ = Be(gzssz + gxstx + gySyHy)

+ AA AA + AA
AzszIz + Axstx AysyIy

I 2 r
+ Q [i2 - I(I + 1)/3] (F-l).

Up to five nuclei can be simulated axially, but only the
first three of these would be calculated to second order with
quadrupole terms included. Only two nuclei can be simulated
non-axially, and these terms are to second order and include
quadrupole terms. As many as seven different non-equivalent
paramagnetic sites can be added together in user—specified
ratios. Separate line widths may be specified for parallel
and perpendicular directions. The line shape may be either
Lorentzian or Gaussian. Transition probabilities are consid-
ered in calculations. The resolution (in Gauss) may be speci-

fied by the user.

167

 

 

 

168

On the other hand, the pair of programs ESRMAP and
GATHERl6 is more specific. This pair of programs considers
hyperfine splitting for only one nucleus, considers only
one site, and does not include quadrupole terms. Further-
more, only one user-specified line width is used, and the
line shape must be Gaussian. The transition probability is
assumed independent of orientation. The resolution is set
at one Gauss.

In both SIMIQA and the pair of programs ESRMAP and
GATHER, apparently the three principal components of the
hyperfine tensor all must have the same sign. However,
ESRSIM-FASTl6 may be used for any combination of hyperfine
signs, although extra singularities are not plotted by
ESRSIM-FAST.

Program PEAKS,l6 which provides digital information
about locations of features that can be compared readily
with those in experimental spectra, has no analogue in pro-
gram SIMlNA.

SIMlHA was modified slightlyl6 without changing any of
the computational logic. Some maximum allowed subscript
values were changed in a DIMENSION statement and in labeled
COMMON statements, some plot-related statements were altered
and some were inserted, and statements were inserted to store
data for replotting by another program. Spin Hamiltonian
parameters of copolymer-attached CprCl3 were used in the in—
put. These values, which are equivalent to values reported

earlier in this dissertation, are given in the "19 (input

 

 

169

 

npoazocfia cmflmmsmu xwmatou

 

 

IVmwmm N 3H I

 

 

. o.ma o.mH . . .1::::::::
0 ma m.m \m m.» 0 ma 0 ma Awe” .
Na Ne m S N: Na we a m
. Hmm.a Hmm.m mA=.m a~:.m . . H
Hmm m o o o o Hmm m Hmm m Assoc»
ommamo. u mmaoo. 5 use
sssoao. a AHHo. M.nswvsse
mpmaoo. . spec. A.-sowxxs
ss.amms ms.mmm- mm.ss so.am~- ms.mmm- .- “mesa
Hm.msa- Hm.mflau mH.ANH mm.mm . Hm.MHH- Ase.“
mm.mm - mm.am - Hw.me mm.mm - ww.mm . go we
mmom.H mmom.a Nwom.a ooo.m ooo.~ mmom.~ mmom.a wnm
momm.d momm.H oomm.a HNQ.H Hum.H mowm.H momm.d m%w
Ammm.a Ammm.a Amwm.a omm.a mmm.~ Ammm.a Ammm.a xxw
Aoom:

mm zHHm=pom nmmsam>

use mozam>v usacav
mm am mm Hm om ma ma magmas

 

 

appomam mam ompmassam wcfipmasoflmo ca owns

 

mumposwuwm amazouaaemm swam

.m magma

 

 

 

 

170

values)" column of Table 6. Resulting simulated data were
replotted by another program, and the result is shown in
Figure 19. Comparison with the experimental spectrum pre-
sented in Figure A reveals that this simulation is very poor.
The mismatch could not be attributed entirely to a poor
signal-to-noise ratio.

The reason (other than the presence of noise) for this

mismatch has been identified.16

If Ay (the y component of
the diagonal hyperfine tensor, which is expressed in Gauss)

is zero or negative, SIMINA resets A to equal Ax' There is

y
no printout that informs the user of this action, except that
the "NONAXIAL HFS" message is absent. In fact, in this last-
mentioned simulation attempt, both the printed output and

the plot gave the user-specified value of Ay instead of the
value that actually was used in calculations. Therefore,

the original version of SIMllIA15 is unsuitable for simulation
of spectra of compounds having Ay that is less than zero and
also not equal to Ax'

A simple modification has been made16 in SIMlflA to make
this program suitable for simulating ESR spectra when Ay is
less than zero and also not equal to Ax. The revised program
also remains suitable for all cases for which the original
program15 is suitable. The first "125" in the card image
that is labeled "SIM 360" in the original program has been
changed16 to "126". The resulting statement has become

" IF (0(1)) 126,125,126"

 

 

span 5.3sz II m

Honzmo cmnomuueupmssaoaoo do suppomam mmm mumflssam oe unamuua .mH shaman

mmzao 2H :Hozmxpw ohmHL qumzacz
. omme. . ommm. .‘ommm. . omsm. . omsw. on

n n

09.5%

 

171

 

 

 

NUIldHOSQU 883 JD BAIIUAIUBU 19813 3

 

 

 

172
and causes the statement that resets Ay (which is C(I) in the
program) to be skipped unless Ay equals zero.

Spin Hamiltonian parameters of vanadocene dichloride
(Cp2VClz),l7 except that the signs of the hyperfine compo-
nents were all changed so that these components were positive,
were used in input to a version of SIMINA that was more nearly
like the original15 version. This compound was chosen largely
because none of the array sizes in SIMluA had to be changed,
thus permitting a test of a version of SIMINA that was more
nearly like the original version than would be possible if the
spectrum of a niobium compound were simulated. (The original
version of SIMlMA allows a maximum nuclear spin multiplicity
of 8 for the first nucleus. 51V has a nuclear spin of 7/2
and thus a nuclear spin multiplicity of 8, but 93Nb has a
‘nuclear spin of 9/2 and thus a nuclear spin multiplicity of
10.) A simulated ESR spectrum was produced in a computer run,
during which SIMllIA required 273.412 CP (central processor)
seconds to execute. Simulated data from this run were replot—
ted by another program. The resulting plot is shown in Figure
20. Equations 1-28, 1-29, and 1-30 suggest that this simulated

Spectrum should be similar to the experimental and simulated

l7
Cp2V012 ESR spectra reported by Stewart and Porte, even

though the hyperfine tensor components of Cp2VCI2 are all

negative and those used in simulating data plotted in Figure

20 are all positive. Indeed, these spectra are similar, al-

though Figure 20 displays some noise near the ends of the

simulated spectrum. Spin Hamiltonian parameters (which are

 

173

Name .3:sz II Ezhuommm mmm NE. I H UOOMHSEHW .om onswdm

mwsmm zH IFozmmHm ahmum uHhmzomz
one. . mmmn comm seem oemm comm seem comm comm cosh oeom

nph-bnub-—-—-—b-——-bh— pn—nbpp—npnh—

On

- p n n — n h - —

 

 

 

 

 

NOIldHOSBU 893 JO BAIIUAIHBU 18813 3

 

_—_—7

17“
listed in Table 6) equivalent to those used in generating
data plotted in Figure 20 were used as input to ESRMAP.
Requiring a total of 32.736 CP seconds to execute, the pair
of programs ESRMAP and GATHER produced simulated data that
were stored. (Program PEAKS, which has no analogue in SIMlHA,
required an additional 2.841 CP seconds of execution time.)
These stored data were replotted by another program to pro-
duce the simulated spectrum shown in Figure 21. Figures 20
and 21 appear very similar, except that Figure 21 has an ex-
cellent signal-to-noise ratio throughout the whole spectrum.
Therefore, SIMlNA produces a reasonable simulated spectrum in
the case involving all positive hyperfine tensor components.
Also in this case, the pair of programs ESRMAP and GATHER
has produced a simulated spectrum with a much better minimum
signal-to-noise ratio in less than 1/8 of the CP time required
by SIMlUA.

A revised16 version of program SID/1114A15 was used to simu-
late an ESR spectrum of copolymer—attached CprCl3. This ver-
sion included the previously—mentioned revision of an IF
statement and modifications16 that did not change the compu-
tational logic. For this simulation, the resolution was set
at one Gauss (which is.the resolution used by GATHER), and

the maximum number of 6 values (72) was used. This expensive

simulation, which required 1505.531 CP seconds to execute,

produced very good simulated data. These data were replotted

by another computer program. Resulting plots are shown in

Figures 22 and 23. It has thus been established that this

 

175

mama mmm8¢w 0cm m<2mwm II Sappooam mmm N\N u H umumHSEHm .Hm mhswfim

wmsmo 2H rpozumpw oquu louzomz
ommmL,. mem. . wmmm. . mmmm comm comm oovm oomm comm oofim

. b n - — n h - p — n p - u —

Doom On

b n — — n p n — - n n - — n h - - — I b n -

 

 

 

 

 

NOIldHOSBU 883 JO BAIlUAIHBU 18813 3

 

176

Mama <3H2Hm II

.ooom
p .

P n

m

omm¢

 

 

 

Hon2do emcompp<upmssaoaoo so suppomcm mmm emumfissmm

.mm musmfim

ooow
1:
I
HO
9
l
flu
34
MO
I
A
Hu
1
I
An
.11.
0
I...
fl:
nb
ud
Hu
8
OD
nu
ma
Id
Ill—
I
nu
MW

wmsma 2H 5.02.”.me 04m: 3:225
.33. . . .oommk . _ L38 8.8

LP lb

 

 

 

 

 

 

 

177

 

coma <cHsz l- maopzao monomoo<uam5ofioooo oo Eosouoam mam ooomficsmm
wmzca 2H Ipazmmpm oomouo coHHmzacz
omoe. .oomv. . comm. comm. . .oocw. on

 

 

 

 

 

 

.mw wpswfim

NOILAHOSQU 883 JO 3A118AI830 19815 8

 

 

178
revised version of SIMlNA works properly when given an ideal
set of input parameters.

It is obvious that SIMlNA is more flexible and allows
simulation of more complex spectra than the pair of programs
ESRMAP and GATHER. However, a comparison has been made in
the area where these two computing packages overlap. As has
already been shown for one set of Spin Hamiltonian constants
in Figures 20 and 21 and associated text, ESRMAP and GATHER
produced a spectrum that was better (in terms of having an
excellent signal-to-noise ratio throughout the entire spec-
trum) than the one produced by SIMlAA, although execution of
SIMlAA required much more computer time.

After SIMluA was amended to enable it to simulate com-

pletely anisotropic spectra when A is negative, another such

y
comparison was made. Input values were chosen such that com—
puting charges for executing SIMlAA would be somewhat com—
parable to computing charges for executing the entire ESRMAP,
GATHER, and PEAKS package when copolymer—attached CprCl3
spectra were simulated. The spin Hamiltonian parameters used
are listed in Table 6. The resolution in SIMluA was chosen
to be 5 G (compared with a fixed resolution of 1 G in GATHER).
It was decided to use 34 6 values and 15 ¢ or ¢E values when
using SIMluA and to use 31 6 values and 31 ¢E values when
using ESRMAP and GATHER. When these numbers were used, execu—
tion of ESRMAP and GATHER required a total of 105.893 0? sec—
onds and produced the simulated spectrum shown (as replotted

by another program) in Figure 24. (Execution of PEAKS, which

 

 

179

some omno<o com mozmmm u: mfloczoo omcomppqnpmsofloooo do Esspomom mmm omommcsmm

ombv

 

wwama 2H Ipwzmmpw QJMHmo cQHHMZOEZ

.oomv

PL!

comm

Down.

P

.OOPN.

0n

 

 

 

 

 

 

NOILJHUSQU 883 JO HAIlUAIHBU 13813 3

 

 

 

 

N969 ¢-HEHW II MHUQZQD UOSOMHDJWIHOEZHOQOO .HO Eflhuomflm mwm UOpGHSEHm omN whfiwfim

wmaco zH :Hozmmhw momHmo qumzomz
cams. comm .oomm. comm .oocm. on

. L . P . .

 

 

180

 

 

 

 

NOIldHUSQU 893 JO HAIlUAIUBU 19813 8

 

 

 

 

181
has no analogue in SIMlAA, took an additional 6.505 C? see-
onds.) Execution of SIMlAA required 143.A36 CP seconds and
produced the simulated spectrum shown (as replotted by another
program) in Figure 25. The maximum field length used by each
of the programs ESRMAP, GATHER, and PEAKS in this comparison
was much smaller than the field length required by SIMlAA.
This smaller field length requirement led to further computing
economy beyond that reflected in comparison of CM time used.
Although a portion of the spectrum simulated by SIMlAA ap-
peared to be of good quality, another part of this spectrum
appeared to be virtually inundated with noise. On the other
hand, the spectrum produced by ESRMAP and GATHER appeared to
be very good throughout. Furthermore, these data generated
by ESRMAP and GATHER appeared to be of quality comparable to
that of corresponding data (depicted in Figures 22 and 23)
that had been simulated in a 1505.531-CP-second-long execu-
tion of SIMlAA.

Thus, two sets of spin Hamiltonian parameters have been
used in comparing the efficiency (in simulating a spectrum
having a large signal-to-noise ratio throughout the entire
spectrum without requiring excessive computer time) of SIMIAA
with that of the pair of programs ESRMAP and GATHER, and the
ESRMAP and GATHER combination has been found to be much more

efficient for each of these sets.

 

 

 

Discussion of Efficiency in Computer Programs

 

In subroutine CHANNL of program GATHER, the simulated
ESR absorption is apportioned by a special interpolation
procedure within the innermost loop of the nested nuclear
magnetic quantum number, 6, and ¢E loops. (There is, in es-
sence, a nuclear magnetic quantum number loop, although it
is not defined by a DO statement.) Thus, it is sufficient to
execute this three-deep nest of loops in subroutine CHANNL
only once during calculation of an entire simulated spectrum.
On the other hand, program SIMlAA executes a similar nest of
loops for every point of the simulated spectrum. This dif-
ference is believed to be a major cause of the much greater
efficiency of the ESRMAP and GATHER combination relative to
SIMlAA. The special interpolation procedure (which appor-
tions simulated ESR absorption within the innermost loop of
the nested nuclear magnetic quantum number, 6, and ¢E loops
in subroutine CHANNL of program GATHER and which is described
in Chapter 3 and listed in Appendix C of this dissertation)
is claimed to be a significant advance in generating high
quality simulated powder pattern spectra at low or moderate
computing costs.

Another example of time-saving strategy employed in pro-
gram GATHER and its subroutines is the complete calculation
of an unbroadened simulated spectrum before vector convolu-
tion33’3u with a Gaussian broadening vector is performed.

The exponential function EXP is used in generating the

182

_;

 

 

183
Gaussian broadening vector. This strategy contrasts with the
use of EXP within the innermost loop of a nest of loops in
program SIMlAA.

The time-savings described could not be employed com-
pletely in SIMlAA if that program is to retain all of its
flexibility. However, if the parallel and perpendicular
linewidths were equal, then introduction of the special in-
terpolation procedure and use of a delayed vector convolution
broadening calculation would be possible in the resulting

program.

922%

Although the combination of programs ESRMAP and GATHER
and the program SIMlHA both produce simulated ESR powder pat-
tern spectra of systems having a total electron spin (at any
single paramagnetic site) of 1/2, each of these computing
packages has a different area of superiority. SIMlNA is much
more flexible and can be used to simulate spectra of more
complex systems. This program would be desirable for simula-
tions that are beyond the capability of ESRMAP and GATHER.
For systems that are within the more narrow restrictions re-
quired by the ESRMAP and GATHER combination, the ESRMAP and
GATHER computing package performs much more efficient simula-
tions than SIMlAA does. Thus, each of these two computing
packages has an area in which it is preferred.

If SIMlAA is to be used to simulate a spectrum of any

system having a completely anisotropic (non-axial) hyperfine

 

 

 

 

18A
tensor with Ay less than zero, then the previously-mentioned
change in card image "SIM 360" must be made.
Program PEAKS has no analogue in SIMlAA. However, PEAKS
may be adapted for use with a version of SIMluA that produces

a data file.

 

 

 

 

 

 

 

APPENDIX G

 

PLOTTING SIMULATED DATA

 

All of the simulated spectra shown in this dissertation
have been plotted by executing some versions of program

ESRPLOT. This program has been written16

to use data from
one or two unformatted data files (produced by an ESR simula-
tion program) to produce a plot in a format chosen by the
user. (Although ESRSIM-FAST, GATHER, and SIMlflA all enable
plotting of simulated ESR spectra, a user may exercise more
control of plot format when stored simulated data are plotted
by ESRPLOT.) Except for Figure 17, the only changes that
using ESRPLOT has made in any simulated data have been to
truncate ordinate values above a user-specified limit to

equal that limit and ordinate values below another user-

specified limit to equal this latter-mentioned limit. This

 

procedure has the effect of simulating a recorder pen's being
"pegged" when a portion of the plot would otherwise lie be-
yond the specified vertical limits. These cutoffs can be
used with vertical scale expansion to allow simulated spec-
tral features that are less prominent to be seen clearly on
the plot while the entire plot remains within desired bound-
aries. For example, compare Figure 7, in which none of the

data are truncated, with Figure 8, which is a plot of the

185

 

 

 

186
same data, except for truncation as has been described, on a
different scale.
For Figure 17, a simulated linear (negative) baseline
drift has been added to the ordinate (first derivative of

ESR absorption) of the simulated (dashed line) spectrum.

 

 

BIBLIOGRAPHY

l. B. R. McGarvey. Transition Metal Chem., Ser. Advan.
1966, 3, 89-201. '

2. A. Abragam and B. Bleaney. "Electron Paramagnetic
Resonance of Transition Ions". Clarendon Press, Oxford,
1970. Pages 399-“15~

3. 3.6H. Choh and G. Seidel. Phys. Rev. 1967, _63, 412-
l .

 

4. L. D. Rollmann and S. 1. Chan. J. Chem. Phys. 1962,
fig, 3u16-3u31.

 

5. P. C. Taylor, J. F. Baugher, and H. M. Kriz. Chem. Rev.
1215, 15, 203-240. "““"“‘

6. F. K. Kneubuhl. J. Chem. Phys. 1960, 33, 107u-1078.

 

7. R. Lefebvre and J. Maruani. J. Chem. Phys. 1965, 52,

8. P. H. Rieger in "Physical Methods of Chemistry, Pt. 3A:
Optical, Spectroscopic, and Radioactivity Methods
(Techniques of Chemistry, Vol. 1) (Incorporating
"Technique of Organic Chemistry, Vol. 1: Physical
Methods of Organic Chemistry. 4th ed.")." A. Weissberger
and E. w. Rossiter, Editors. Wiley-Interscience, New
York, N. Y., 1972. Pages N99-59 .

9. G. van Veen. J. Magn. Reson. 1978, 39, 91—109.

10. C. P. Stewart and A. L. Porte. J. Chem. Soc. Dalton
Trans. 1972, 1661-1666.

11. D. L. Griscom, P. C. Taylor, D. A. Ware, and P. J. Bray.

12. P. C. Taylor and P. J. Bray. "Lineshape Program Manual".
Physics Department, Brown University, Providence, R. I.,
1968 (unpublished). Revised 1969. Addenda 197H.

13. P. C. Taylor and P. J. Bray. J. Magn. Reson. 1979, g,
305-331.

 

 

 

1A.

15.

l6.

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

188

T. Porumb and E. F. Slade. Quantum Chemistry Program
Exchange program number 295.

G. Lozos, B. Hoffman, and C. Franz. Quantum Chemistry
Program Exchange program number 265. (Both programs
SIMlA and SIMINA are included as number 265.)

N. H. Kilmer, this work.

C. P. Stewart and A. L. Porte. J. Chem. Soc. Dalton
Trans. 1973, 722-729.

A. Rockenbauer and P. Simon. J. Magn. Reson. 1973,
11, 217-218.

R. M. Golding and W. C. Tennant. Mol. Phys. 1973, 25,
1163-1171.

H. A. Kuska and M. T. Rogers. Spectrosc. Inorg. Chem.
1971. a, 175-196.

J. D. Swalen, B. Johnson, and H. M. Gladney. J. Chem.
Phys. 1970, 52, A078-A086.

. Istomin and M. Ya. Shcherbakova. Zh. Strukt. Khim.
, 11, 808-813. Translated in J. Struct. Chem.
(USSR), 1976, $1, 695—699.

J. R. Morton and K. F. Preston. J. Magn. Reson. 1978,
i9, 577-582-

F. Herman and S. Skillman. "Atomic Structure Calcula—
tions". Prentice-Hall, Englewood Cliffs, N. J., 1963.

J. L. Petersen and L. F. Dahl. J. Am. Chem. Soc. 1975,
91, 6422—6A33.

J. L. Petersen and L. F. Dahl. J. Am. Chem. Soc. 1975,
21, 6A16-6422.

J. L. Petersen, D. L. Lichtenberger, R. F. Fenske, and
L. F. Dahl. J: Am. Chem. figgo 1975, 21, 6u33-6uhl.

J. W. Lauher and R. Hoffman. J. Am. Chem. Soc. 1976,
gfi, 1729—17u2. ““‘"‘*"‘—’——— -—-

C. P. Lau. Ph. D. Dissertation. Michigan State Univer-
sity, East Lansing, MI, 1977.

R. H. sands. PhZSo REV. 1955’ 22, 1222—1226.

 

 

 

31.

32.

33.

3A.

189

J. L. Dye, V. A. Nicely, F. Tehan, R. Cochran, and J.
Leckey. "KINFITD -- A General Non-Linear Curvefitting

'Program". Department of Chemistry, Michigan State

University, East Lansing, MI, 1977 (unpublished).

J. L. Dye and V. A. Nicely. J. Chem. Educ. 1971, 48,
AA3-AA8. _—-—___———__— ’——_' _—

Subroutine VCONVO, which directs a vector convolution,
has been patterned mainly after a description of an IMSL
(International Mathematical and Statistical Library)
routine having the same name.

The idea of using one convolution over the entire spec-
trum (or integral) is NOT original in this work.

Taylor and Bray12 have described a computer program in
which the average over orientations is calculated before
the average over local environments is calculated, that
is, summation over 6 and o is performed before the
broadening calculation is Epplied.