
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00003920/00001
Material Information
 Title:
 Optical and electron spin resonance spectroscopy on matrixisolated silicon and manganese species
 Creator:
 Ferrante, Robert Francis, 1951
 Publication Date:
 1977
 Language:
 English
 Physical Description:
 xiii, 237 leaves : ill. ; 28 cm.
Subjects
 Subjects / Keywords:
 Atoms ( jstor )
Electron paramagnetic resonance ( jstor ) Electrons ( jstor ) Magnetic fields ( jstor ) Magnetism ( jstor ) Matrices ( jstor ) Molecular spectra ( jstor ) Molecules ( jstor ) Orbitals ( jstor ) Symmetry ( jstor ) Electron paramagnetic resonance spectroscopy ( lcsh ) Manganese ( lcsh ) Silicon ( lcsh ) City of Springfield ( local )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 ThesisUniversity of Florida.
 Bibliography:
 Includes bibliographical references (leaves 232236).
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Robert Francis Ferrante.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 026333748 ( ALEPH )
AAX4725 ( NOTIS ) 04080446 ( OCLC )

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Full Text 
OPTICAL AND ELECTRON SPIN RESONANCE SPECTROSCOPY
OF MATRIXISOLATED SILICON AND MANGANESE SPECIES
By
ROBERT FRANCIS FERRANTE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
To my parents
Digitized by the Internet Archive
in 2010 with funding from
University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation
http://www.archive.org/details/opticalelectrons00ferr
ACKNOWLEDGEMENTS
The author extends his deep appreciation to Professor
William Weltner, Jr. whose encouragement, professional
guidance, patience, and support made this work possible.
Thanks are also due to all the members of Professor
Weltner's research group, particularly Dr. W. R. M.
Graham and Dr. R. R. Lembke, for their collaboration,
assistance, and advice during the research.
The author greatly appreciates the expert craftman
ship displayed in fabrication of experimental apparatus
by A. P. Grant, C. D. Eastman, and D. J. Burch of the
Machine Shop and R. Strasburger and R. Strohschein of the
Glass Shop, as well as the maintenance of electronic
equipment by R. J. Dugan, J. W. Miller, and W. Y. Axson.
The author would also like to acknowledge the support
of the Air Force Office of Scientific Research (AFOSR) and
the National Science Foundation (NSF) during this work.
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF TABLES vi
LIST OF FIGURES viii
ABSTRACT xi
CHAPTERS
I INTRODUCTION 1
The MatrixIsolation Technique 1
References Chapter I 9
II EXPERIMENTAL 11
Introduction 11
Experimental 11
Apparatus 11
General Technique 26
References Chapter II 29
III ESR THEORY 30
Introduction 30
Atoms and the Resonance Condition 30
The Hyperfine Splitting Effect 33
2E Molecules 43
The Spin Hamiltonian 43
The g Tensor 47
The A Tensor 52
Randomly Oriented Molecules 53
Molecular Parameters and the
Observed Spectrum 61
3E Molecules 7
The Spin Hamiltonian 76
4Z Molecules 94
The Spin Hamiltonian 95
E Molecules 105
The Spin Hamiltonian 106
References Chapter III 113
IV SILICON SPECIES 117
Introduction 117
Experimental 118
ESR Spectra 119
SiN2 119
SiCO 125
Si2 133
Optical Spectra 133
Si and Si2 133
SiN2 137
SiCO 144
Si(CO)2 150
Discussion 152
References Chapter IV 171
V MANGANESE SPECIES 175
Introduction 175
Experimental 177
ESR Spectra 179
Mn Atoms 179
Mn+ 179
MnO 184
MnO2 190
MnO3 196
Mn04 205
Discussion 207
Mn Atoms and Mn 207
MnO 212
MnO2 215
MnO3 217
MnO4 220
References Chapter V 232
BIOGRAPHICAL SKETCH 237
CHAPTERS
Page
LIST OF TABLES
TABLE PAGE
I ESR data of SiN2 and SiCO in their 3
ground states invarious matrices at 40K 131
II Si2 absorption bands in argon matrices at 40K 135
14
III Ultraviolet absorption spectrum of Si N2
in an argon matrix at 40K 140
IV Vibrational frequencies and calculated
force constants (mdyn/A) for SiNN and
SiCO molecules in their ground 3E states 143
12
V Absorption spectrum of Si CO in an argon
matrix at 4K 147
VI Comparisgn of stretching force constants
(mdyn/A) for relevant molecules XYZ 154
VII Total density matrix elements for SiCO,
SiN2, and the free ligands CO and N2 157
VIII Spin densities in SiCO and SiNN 164
IX Comparison of vibrational frequencies and
electronic transitions of CXY and SiXY
molecules 167
X Field positions (in gauss) of observed fine
and hyperfine structure lines of Mn+:Ar
at 40K. A = 275 G; v = 9390 MHz 185
XI Magnetic parameters, observed and calculated
line positions for the I+1/2)+I/2)
perpendicular transition of MnO ( E) in Ar 191
XII Magnetic parameters, observed and calculated
line positions for the I+1/2) >I1/2)
perpendicular transition of MnO2 (4E) in Ar 195
XIII Spin Hamiltonian matrix for the states M, m)
for Mn03 (2A) including interaction with
the 55Mn (I = 5/2) nucleus 199
TABLE PAGE
2
XIV Magnetic parameters of MnO (2A ) in Ne;
observed transitions in Ne an Ar 203
XV Magnetic parameters, observed and calculated
line positions for MnO4 (2T1) in Ne 208
XVI Summary of magnetic parameters and derived
quantities for manganese and some
manganese oxides 210
vii
LIST OF FIGURES
FIGURE PAGE
1 Basic design features of the liquid
helium dewar used for ESR studies 13
2 Variabletemperature modification of
liquid helium dewar used for ESR studies 14
3 Basic design features of variabletemper
ature liquid helium dewar used for
optical studies 17
4 Basic design features of cryotip assembly
used for optical studies 20
5 Zeeman energy levels of an electron
interacting with a spin 1/2 nucleus 36
7
6 Zeeman energy levels of a SS ion with
I = 5/2 37
7 Absorption and first derivative lineshapes
of randomly oriented molecules with
axial symmetry 58
8 Energies of the triplet state in a magnetic
field for a molecule with axial symmetry;
field parallel to molecular axis 82
9 Energies of the triplet state in a magnetic
field for a molecule with axial symmetry;
field perpendicular to molecular axis 84
3
10 Resonant fields of a E molecule as a func
tion of the zero feild splitting 87
11 Theoretical absorption and first derivative
curves for a randomly oriented triplet
state molecule with axial symmetry 90
12 Theoretical absorption and first derivative
curves for a randomly oriented triplet
state molecule with orthorombic symmetry 92
4
13 Energy levels for a Z molecule in a magne
tic field; field parallel to molecular
axis 102
viii
14 Energy levels for a 4 molecule in a mag
netic field; field perpendicular
to molecular axis 103
15 Resonant fields of a 4E molecule as a
function of the zero field splitting 104
16 Energy levels for a 6E molecule in a
magnetic field for 0 = 00, 300, 600,
and 90 109
17 Resonant fields of a 6 molecule as a
function of the zero field splitting 111
18 ESR spectra of SiN2 molecules in argon
matrices at 40K 120
19 ESR spectra of SiN2 molecules in various
matrices at 40K 124
20 ESR spectra of SiCO molecules in argon
matrices at 40K 126
21 Effect of temperature upon the ESR spectrum
of Sil3CO in an argon matrix 128
22 ESR spectra of SiCO molecules in Ar and
CO matrices 130
23 Ultraviolet absorption spectrum of SiN2
molecules in an agron matrix at 40K 138
24 Infrared bands of SiN in argon and
nitrogen matrices a? 4K 142
25 Absorption spectrum of SiCO molecules in
an argon matrix at 40K 145
26 Infrared spectra at 40K of an argon matrix
containing vaporized silicon atoms and
13CO/Ar = 12CO/Ar = 1/375 148
27 Infrared spectra at 40K of an argon matrix
containing vaporized silicon atoms and
Cl60/Ar = C180/Ar = 1/375 149
28 ESR spectrum of Mn in argon at 40K 180
29 Zeeman levels and observed transitions for
Mn+ in argon 185
FIGURE
PAGE
FIGURE PAGE
30 ESR spectrum of MnO in argon at 40K 187
31 ESR spectrum of MnO2 in argon at 4K 193
32 ESR spectrum of MnO3 in neon at 40K 197
33 ESR spectrum of MnO3 in argon at 40K 202
34 M = 3/2 component of MnO in argon at
various temperatures, and in neon 204
35 ESR spectrum of MnO4 in neon at 4K 206
36 Molecular orbital correlation diagram
for MnO3 218
37 Molecular orbital correlation diagram
for MnO4 223
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
OPTICAL AND ELECTRON SPIN RESONANCE SPECTROSCOPY
OF MATRIXISOLATED SILICON AND MANGANESE SPECIES
By
Robert Francis Ferrante
December, 1977
Chairman: William Weltner, Jr.
Major Department: Chemistry
The 3 molecules carbonyl silene, SiCO, and diazasilene,
SiNN have been prepared by vaporization and reaction of
silicon atoms with N2 or CO and trapped in various matrices
at 40K. The electron spin resonance (ESR) spectra indicate
that some or all sites in some matrices induced slight
bending in the molecules, and that the species undergo
torsional motion in the solids. Isotopic substitution of
13C, 15N, 180, and 29Si was employed to obtain hyperfine
coupling data in the ESR and shifts in the optical spectra.
1
In solid neon, assuming gl =gi=ge, D = 2.28 and 2.33 cm1
for SiN2 and SiCO, respectively. Hyperfine splitting in
14 15 29
argon yield A = 17 ( N), 21 (15N) and 95 (29Si) MHz for
SiN2 and 84 (29Si) and <14 (13C) MHz for SiCO. These
confirm calculated results, in the complete neglect of
differential overlap (CNDO) approximation, that the electron
spins in both molecules are largely in the pn orbitals of Si.
Optical transitions (with vibrational progressions) were
0 1
observed beginning at 3680A (SiN stretch, 470 cm ) and
0 1 o
3108A (NN stretch, 1670 cm ) for SiN and 4156A (SiC
stretch, 750 cm1; C0 stretch, 1857 cm1) for SiCO in Ar.
Infrared (IR) spectra in Ar indicate vNN = 1733 and
NN
1 n 1
V = 485 cm for SiN and v = 1899 cm for SiCO.
SiN 2 C0
Calculated stretching force constants are k = 2.0,
SiN
kN = 11.8, kCO = 15.6, and ksiC = 5.3 mdyn/A, the
1
latter assuming vSiC = 800 cm The CNDO calculations
suggest r bonding of Si to the ligand, which is stronger
in SiCO than SiN2, and some ligand + dSi backdonation,
also stronger in SiCO. An attempt was made to correlate
these vibrational and electronic data with those for CCO
and CNN. Annealing an argon matrix containing SiCO to
350K led to the observation in the IR of 1 Si(CO)2, a
n! 1
silicon counterpart of carbon suboxide, with v = 1899.6 cm1
A corresponding treatment of a SiN2 matrix did not produce
N2SiN2, nor was N2SiCO observed when both ligands were present.
The molecules MnO, MnO2, MnO3, and MnO4 have also been
prepared, by the vaporization and reaction of manganese atoms
with 02, N20, or 03, and isolated in various inert gas matrices
at 40K. ESR has been used to determine magnetic parameters
which are interpreted in terms of molecular geometry and
electronic structure. MnO is confirmed to have a 2 22 6 +
ground state with gl = 1.990(7), assuming gl = ge, and a
zero field splitting in accord with the gas phase value
IDI = 1.32 cm Hyperfine splitting due to the Mn (I = 5/2)
nucleus are JAI I = 176(8) and IA1! = 440(11) MHz. MnO2 is
S 4 2
a linear E molecule with probable configuration U6 ,
xii
D = 1.13 cm1 (assuming g11 = g = 2.0023), IAI I = 353(11),
IAi = 731(11) MHz. Mn03 exhibits very large hyperfine
splitting IAII = 1772(3) and JAI = 1532(3) MHz indicative
2
of an sdz2 hybrid 2A ground state of D3h symmetry. The
spectrum of MnO4 is consistent with a C 3 molecule
2
distorted from a T1 electronic state in tetrahedral symmetry
by a static JahnTeller effect. The g and A tensors are
slightly anisotropic: gll = 2.0108(8), gl = 2.0097(8);
JAIII = 252(3), JAII = 196(3) MHz. The electron hole is
almost entirely in an oxygen rbonded orbital with one
oxygen atom displaced along its MnO bond axis. Warming
to 350K did not induce thermal reorientation.
xiii
CHAPTER I
INTRODUCTION
The MatrixIsolation Technique
Molecular spectroscopy, the primary tool for the investi
gation of intimate details of molecular geometry and electronic
structure, has been routinely applied to a large assortment
of stable chemical species in the solid, liquid, and gaseous
phases. The advent of highspeed electronic instrumentation
has extended the range of spectroscopic techniques to allow
the study of unstable or shortlived molecules and fragments.
However, some molecules are still very difficult or even
impossible to observe because of their short lifetimes,
reactivity and/or method of preparation. Among these are
molecules that exist only under high temperature conditions,
such as stellar atmospheres or in arcs, and fragments whose
reactivity precludes production of sufficient quantities for
normal analyses. Even when such species are observed, anal
ysis of their spectra is often greatly complicated by their
production in a multitude of electronic, vibrational, and
rotational states, as can occur in laboratory methods of
generation of such radicals via arcs, flash photolysis, etc.
Utilization of the matrixisolation technique can overcome
many of these difficulties.
The technique of matrixisolation spectroscopy was pro
posed independently by Norman and Porter (1) and by Whittle,
Dows,, and Pimentel (2) in 1954. Basically, the high
temperature species, reactive molecules, or radical frag
ments are prepared and trapped as isolated entities in inert,
transparent solids, or matrices, at cryogenic temperatures.
They do not undergo translational motion, but are immobilized,
thus preventing further reaction and preserving the specimen
for conventional spectroscopic analysis. The common technique
of treating solid samples by diluting with KBr and forming a
compressed disk of finely dispersed solid in a KBr matrix
can be considered a crude form of matrixisolation. It has
been demonstrated that the information gained by this tech
nique is gaslike within a few percent.
The matrix material can be any gas which will not react
with the trapped species and which can be readily and rigidly
solidified. Many different substances have been used for this
purpose, including CH4, CO, N2, CS2, SF6, 02, as well as
aliphatic and aromatic organic, However, the solid rare
gases Ne, Ar, Kr, and Xe are usually employed because they
are relatively inert chemically, transparent to radiation
over a wide wavelength region, and offer a wide range of
melting points and atomic sizes. The choice of matrix gas
is determined to a large extent by the effect the solid matrix
will have upon the trapped molecule. Neon, because it is the
least polarizable, is expected to perturb the molecule least
and generally makes the best matrix. Unfortunately, its
trapping efficiency is not as great as the other rare gas
solids, and it is difficult or impossible to achieve isolation
of some species in solid Ne. Argon is better in this
respect, and is generally used as the matrix medium. The
heavier rare gases are found to perturb the trapped molecules
to a greater extent than Ne or Ar, and are therefore, less
desirable as matrix materials.
Temperatures sufficiently low to condense the matrix
gas can be attained with either physical refrigerants (nitro
gen, hydrogen, or helium in the liquid state) or mechanical
closedcycle refrigerators utilizing JouleThompson expansion
of high pressure hydrogen or helium gas. Readily available
and inexpensive, liquid N2 (boiling point 77.40K) is useful
for some matrix materials, but is limited to relatively
stable guest species. Liquid H2 (boiling point 20.4K)
is often used, but it entails a fire hazard in addition to
the normal dangers of handling cryogenic fluids. Thus liquid
He is the most suitable of the physical refrigerants and the
only one useful for condensation of solid Ne, which melts at
240K and permits solid state diffusion at about half that
temperature. Temperatures of liquids H2 and He can be
reduced to 150K and 1.20K, respectively, by pumping on the
liquid. Closedcycle refrigerators which can attain temper
atures near the boiling point of He are commercially avail
able. Their advantages include convenience, elimination of
the need to replenish cryogenic fluids while experiments
are in progress, and low cost of operation after the initial
investment.
The substrate on which the matrix is condensed is
usually chosen to be transparent in the spectroscopic region
of interest. Some suitable materials are CsI, KBr, NaCI, and
Au (for reflectance systems) in the infrared region, quartz,
sapphire, and CaF2 for the visible and ultraviolet regions,
LiF for the vacuum ultraviolet and sapphire or other non
conducting material for microwave spectroscopy, including
electron spin resonance. The polished crystal plates are
mounted on a cold block which makes good thermal contact
with the liquid refrigerant reservoir or the expansion cham
ber of the JouleThompson refrigerator. In variabletempera
ture dewars, the cold block is isolated from the reservoir,
but cooled by introducing a controlled leak of refrigerant
into the mounting block.
Several methods can be employed for production of the
guest species which are trapped on such substrates. One
common procedure is vaporization of a nonvolatile material
from a hightemperature Knudsen cell in a vacuum furnace.
These cells can be constructed of carbon, or the refractory
metals, Mo, Ta, or W. To prevent degradation of the cell
material from contact with the hot vapor, the crucible can
be lined with C, A1203, or BN. The cells are heated by
resistance or induction methods, and temperature up to 29000K
can be achieved in this manner. The vapor effusing through
a small orifice is collimated into a crude molecular beam
and deposited simultaneously with the matrix gas in such
proportions that M/R (the ratio of the number of moles of
matrix material to the number of moles of the trapped species)
is 500:1 or greater. In many cases the compositions of
vapors so obtained have been characterized mass spectrometri
cally, greatly facilitating the analysis of the resulting
spectra. The matrix gas can also be doped with a reactant
gas by standard manometric techniques to produce other
unstable reaction products. Another high temperature source
is thermolysis of a gaseous compound by passing it through
a hot W or Ir tube, the resulting products being cocondensed
with the matrix gas as above. An additional common method
for generating unstable species is to subject a volatile
parent compound to highenergy radiation, as produced by
microwave or electric discharges, ultraviolet lamps, laser
sources, or gamma rays, or by electron or ion bombardment.
This exposure can be performed during or after sample deposi
tion, and the resulting changes observed spectroscopically.
Combination of the hightemperature and photolytic procedures
can produce other unstable species for study.
A large array of spectroscopic techniques can be applied
to matrix samples, including infrared (IR) and Raman,
visible (VIS) and ultraviolet (UV) absorption and emission,
electron spin resonance (ESR) and Mossbauer spectroscopy.
Matrix isolation spectroscopy has several advantages over
gas phase work. First of all, there is the ability to
observe normally unstable or highly reactive species at
leisure, using conventional or only slightly modified spectrom
eters. Another prime advantage, particularly important for
hightemperature species, is that the molecules are always
trapped in their ground electronic and vibrational states.
With most work done below 250K, there are no "hot" bands, as
l
the thermal energy is only 17 cm Because states other
than the ground state are thermally inaccessible, sensitivity
is increased over that of hightemperature gasphase work,
and analysis is aided since the originating level for spectro
scopic transitions is always the ground state. With long
deposition times (up to 48 hrs with automatic flow control),
sufficient concentrations of molecules can be accumulated
in order to observe species of low abundance or spectral
features with low absorption coefficients. In addition,
controlled diffusion experiments can be conducted in order
to follow the formation of new species, polymers, or clusters.
Finally, it is possible to observe preferential orienta
tion of some species in matrices; the equivalent effect is
observed in single crystals, but for most species considered
in matrix work, these would be impossible to prepare.
Of course, there are some disadvantages to the technique,
primarily the frequency shift from gasphase values, caused
by perturbing effects of the matrix. Frequencies in neon
matrices are generally higher than those in argon, and the
data in these solids often bracket the gasphase value. Neon
does give the closest agreement, with vibrational frequencies
i
shifted 10 cm or less. Electronic transitions show a
similar trend, the Ne values differing from gasphase by
up to 200 cm. For trapped molecules, all transitions
up to 200 cm .For trapped molecules, all transitions
exhibit shifts of the same order of magnitude, and usually
the same direction; trapped atoms show no such regularity,
and often include absorptions that have no apparent corre
spondence to gasphase transitions. Magnetic parameters are
also influenced by matrix effects, usually exhibiting, trends
related to the atomic number of the matrix gas. Theoretically,
hostguest interactions causing these perturbations are not
well characterized, although some effort has been extended
in this direction. Samples of such attempts can be found in
(312).
Another common observation in matrix work is that the
shapes and widths of bands vary widely, depending upon the
extent of interaction between the absorbing species and
the matrix. Usually Ne shows the narrowest lines, up to
I
10 cm full width at half maximum (FWHM) in the IR, up to
0
20 A FWHM for electronic absorptions. The broadening effect
usually increases with the atomic number of the matrix gas,
but lines in Ne are occasionally rather broad, also. Line
shapes are also somewhat matrixdependent. However, for a
given matrix, the perturbations are useful in identification
of progressions of vibrational lines in a particular excited
electronic state, since such bands always show the same
shape. The departure of the lineshapes from the usual
Lorentzian form are generally attributed to occupation of
multiple sites of similar energy in the rare gas lattice,
and/or the simultaneous excitation of lattice modes (phonons).
Site splitting of a few wavenumbers, Angstroms, or gauss
are common, but can often be partially or completely
eliminated by the irreversible process of annealing the
matrix. This is done by allowing the matrix to warm up,
and then rapidly quenching it to the original low temperature.
A final disadvantage is the loss of information about
rotational levels of trapped species. Some molecules, such
as H20, HC1, NH2, and NH3 do rotate in matrices, but rota
tional sturcture is usually lost in broad vibrational
envelopes.
This introduction is not designed to discuss in depth
the various aspects of the matrix isolation technique, but
to illustrate the method and possibilities of its applica
tion, as well as enumerate some advantages and disadvantages.
More extensive details can be found in several recent reviews
(6, 7, 11, 1322) and references contained therein.
References Chapter I
1. I. Norman and G. Porter, Nature, 174, 508 (1954).
2. E. Whittle, D.A. Dows, and G.C. Pimentel, J. Chem.
Phys., 22, 1943 (1954).
3. M. McCarty, Jr. and G.W. Robinson, Mol. Phys., 2,
415 (1959).
4. M.J. Linevsky, J. Chem. Phys., 34, 587 (1961).
5. G.C. Pimentel and S.W. Charles, Pure Appl. Chem.,
7, 111 (1963).
6. B. Meyer, "Low Temperature Spectroscopy," Elsevier,
New York, 1971.
7. A.J. Barnes, "Vibrational Spectroscopy of Trapped
Species" (H.E. Hallam, ed.), Wiley, New York, 1973,
p. 133.
8. R.E. Miller and J.C. Decius, J. Chem. Phys., 59,
4871 (1973).
9. A. Nitzan, S. Mukamel, and J. Jortner, J. Chem. Phys.,
60, 3929 (1974).
10. G.R. Smith and W. Weltner, Jr., J. Chem. Phys.,
62, 4592 (1975).
11. S. Cradock and A.J. Hinchcliffe, "Matrix Isolation,
A Technique for the Study of Reactive Inorganic
Species," Cambridge University Press, Cambridge,
1975.
12. B. Dellinger and M. Kasha, Chem. Phys. Lett., 38,
9 (1976).
13. A.M. Bass and H.P. Broida, "Formation and Trapping
of Free Radicals," Academic, New York, 1960.
14. W. Weltner, Jr., Science, 155, 155 (1967).
15. J.W. Hastie, R.H. Hauge, and J.L. Margrave, "Spectro
scopy in Inorganic Chemistry," Vol. 1 (C.N.R. Rao
and J.R. Ferraro, eds.), Academic, New York, 1970,
p. 57.
16. W. Weltner, Jr., "Advances in High Temperature
Chemistry," Vol. 2 (L. Eyring, ed.), Academic,
New York, 1970, p. 85.
17. D. Milligan and M.E. Jacox, "MTP International Review
of Science, Physical Chemistry, Series I," Vol. 3
(D.A. Ramsay, ed.), Butterworth, London, 1972, p. 1.
18. A.J. Barnes, Rev. Anal. Chem., 1, 193 (1972).
19. A.J. Downes and S.C. Peake, Mol. Spectrosc., 1,
523 (1973).
20. L. Andrews, Vib. Spectra Struct., 4, 1 (1975).
21. B.M. Chadwick, Mol. Spectrosc., 3, 281 (1975).
22. G.C. Pimentel, New Synth. Methods, 3, 21 (1975).
CHAPTER II
EXPERIMENTAL
Introduction
The general experimental procedure including cryogenic,
hightemperature, spectroscopic, and photolytic apparatus is
discussed in this section. Specific details peculiar to
individual molecules investigated will be presented with
the discussion of those species.
Experimental
Apparatus
In this research, three separate cryogenic systems
were employed, two for optical and IR studies and a third
for ESR experiments. An ESR and an optical dewar utilizing
liquid He as physical refrigerant were adapted from the design
of Jen, Foner, Cochran, and Bowers (1), and modified for
variabletemperature operation as described by Weltner and
McLeod (2). Both systems are comprised of an outer liquid
N2 dewar, which serves as a heat shield, surrounding the inner
liquid He dewar. The sample substrate is attached to a
copper block suspended from the bottom of the inner dewar
by a small tube which permits passage of a controlled leak
of refrigerant into the block. The inner dewar is positioned
such that the trapping surface is directly in the path of
the sample inlets.
Pertinent details of the ESR dewar, described by Easley
and Weltner (3), and Graham, et al.(4), are indicated in
Figures 1 and 2. The stainless steel inner dewar, capacity
approximately 2.1 liters, is jacketed at the lower end by
a copper shroud which is part of the outer (liquid N2) reser
voir surrounding the upper portion. This jacket extends down
to encase the microwave cavity and maintain it near 770K.
A slot approximately 3.5 cm long and 0.6 cm wide allows the
matrix gas and furnace vapor to reach the sample substrate.
At 900 to either side of this slot, two rectangular openings
of approximately 2.4 cm x 0.6 cm are provided. These points
correspond to the location of interchangeable windows, sealed
by Viton "O" rings to the outer vacuum chamber, which permit
visual examination of the matrix and serve as ports for
photolysis or spectroscopic observations.
The trapping surface is single crystal sapphire, obtained
from Insaco, Inc. It is a flat rod, 3.3 cm long, 3.1 mm
wide, and 1.0 mm thick, securely enbedded by Wood's metal
solder into the copper block, as indicated by Figure 2.
A Chromel vs Au0.02 at. % Fe thermocouple is also attached
to the copper block so that the temperature can be monitored.
Although the temperature of the sample substrate itself is
never determined, single crystal sapphire has high thermal
conductivity at 40K so that it rapidly equilibrates to the
temperature of the mounting.
Also shown in Figure 2 is the construction of the
variabletemperature modification. The copper lower can is
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connected to the main liquid He reservoir by a thin stainless
steel tube; a vent for the lower can is also provided to
exhaust the gaseous He as it evaporates. Outside the vacuum
vessel, the He outlet is equipped with a valve to control the
flowrate of liquid He into the lower can. The main liquid He
reservoir is pressurized to about 2.5 psi to supply an
uninterrupted flow through the mounting block. To vary the
temperature, the outlet valve is closed, and as the He
evaporates, it forces the liquid refrigerant out of the
lower chamber, allowing it to warm. The change in tempera
ture is monitored with either a Leeds and Northrup model
8687 potentiometer or a Newport model 2600 digital thermometer.
After sample deposition is completed, the entire
inner dewar assembly is lowered approximately 3.8 cm with
respect to the fixed outer can and vacuum chamber, utilizing
a vacuumtight bellows arrangement mounted at the top and
not indicated in the Figures. In this manner, the rod is
positioned in the center of the copper Xband (=9.3GHz)
microwave cavity; this location corresponds to the maximum
intensity of the circulating magnetic field of the microwave
radiation injected into the cavity. The front end of the
cavity is slotted and aligned with another interchangeable
window in the outer vacuum chamber located just below the
sample inlets. In this way, the sample can be photolyzed
and ESR spectra recorded simultaneously. The back end of
the cavity is fitted with a standard copper waveguide coupling
mounted just outside a mica window, which serves to maintain
high vacuum conditions in the cavity.
With the sapphire rod in position, the dewar is separated
from the furnace assembly by means of a gate valve and
disconnected from it. The entire dewar assembly is then
rolled on fixed tracks to the proper position between the
poles of the ESR magnet. When so aligned, the alternating
magnetic field of the microwave radiation is oriented
perpendicular to the static field of the external magnet.
However, the inner (liquid He) dewar can be rotated 3600
on bearings to permit detection of resonance signals with
the flat surface of the rod oriented at any angle with respect
to the external field.
Following the same basic design, the dewar used for
optical studies is diagrammed in Figure 3. The stainless
steeliriner (liquid He) dewar, of capacity 8.6 1, is surrounded by
a'liquid N2 container and copper sheath. Four circular open
ings in the sheath, each of 3.5 cm diameter, are located at
the level of the sample substrate mounted on the lower cham
ber of the inner dewar. At one of these openings, the fur
nace and matrix gas inlets are attached on the outer vacuum
chamber. This opening can be sealed with a gate valve when
deposition of vapor from the furnace is not desired. At
90 to either side of the sample inlet, the openings form
part of the optical path of the spectrometric instruments.
Interchangeable windows (approximately 4 cm diameter) are
mounted with Viton "O" rings to the outer vacuum chamber at
these points. The window materials are chosen to match the
spectral region of interest; CaF2 is used for the visible
VALVES
Figure 3. Basic design features of variable temperature
liquid helium dewar used for optical studies.
0
and UV regions (20007000 A), and CsI is used for visible
0
and IR studies (3500 A50p). All optical crystals were ob
tained from Harshaw Chemical Co. The fourth port is located
at 1500 to the sample inlets, and interchangeable windows
can also be mounted on the outer chamber at this point.
These are usually either quartz or LiF, and serve to admit
photolyzing radiation to the sample in the ultraviolet or
vacuum ultraviolet regions, respectively. Such sample
photolysis cannot be accomplished simultaneously with deposi
tion. As in the case of the ESR dewar, the entire inner
assembly can be rotated on bearings through 3600, to align
the sample window with any of the above.
The sample substrate is a polished optical crystal,
2.2 x 1.1 cm, chosen to match those on the outer vacuum
casing and the spectral region of interest. It is mounted
in the lower chamber of the variabletemperature inner dewar
with all four sides in contact with the cold copper block.
At all points where the window is in contact with the copper
heat sink, a thin gasket of indium metal is inserted. This
material has good thermal conductivity and is sufficiently
plastic that it conforms to all contours of both surfaces
when the window mounting frame is firmly screwed into the
copper block from above the substrate. A chromel vs Au0.02
at % Fe thermocouple is mounted to the copper immediately
adjacent to the window. Temperatures at this point are
measured with either the potentiometer or a Cryogenic Tech
nology Inc. digital thermometer /controller. Variable
temperature operation is achieved in the same manner as
described above for the ESR dewar.
The third piece of cryogenic apparatus employed in this
research utilizes an Air Products model DE202 Displex
cryotip. This is a twostage, closedcycle He refrigerator
which makes use of the JouleThompson effect as compressed
gas at 300 psi is expanded, with a pressure drop of over
200 psi. The vacuum housing for the cryotip is very similar
in design to the liquid He optical dewar, except that the
fourth window discussed above is located at 1800 to the
sample inlets; thus photolysis can be conducted simultaneously
with sample deposition.
Internally, there are a few modifications. The cryotip
unit (shown in Figure 4) is constructed of stainless steel,
with the exception of the final expansion chamber. There
is no liquid N2 outer dewar, but its function as a heat
shield is taken over by a nickelplated copper shroud attached
to the first expansion stage, maintained at 40600K. This
extends down to and surrounds the sample substrate holder,
with two openings cut at 1800 apart. The entire unit is
rotatable through 1800, to align the sample window with any
two opposite ports in the external vacuum housing.
The second expansion stage is terminated with a copper
cold tip. The copper sample window holder is firmly screwed
into this tip, and good contact is assured with an indium
gasket. The circular sample windows (2.6 cm diameter) are
of the same materials as employed in the other optical dewar,
ELECTRICAL
He GAS
<C
THERMOCOUPLE
and
HEATING WIRES
EXPANDER
Ist STAGE
2nd STAGE
COPPER COLD TIP
TARGET WINDOW
RADIATION SHIELD
MATRIX GAS INLET
GATE VALVES
Figure 4.
ROTATABLE
JOINT
FURNACE
ASSEMBLY
VACUUM
PUMPS
Basic design of cryotip assembly used for
optical studies.
and are secured with indium gaskets and a copper retaining
ring. The chromel vs Au 0.02 at.% Fe thermocouple is mounted
on the sample holder. The temperature is varied between 100K
and ambient by an electrical heating wire wrapped around the
second stage expander cold tip. The temperature is measured
and automatically maintained at any preset value with the
CTI thermometer/controller.
The vacuum chambers of all three of the above systems
are pumped by 2 inch oil diffusion pumps, with liquid N2 cold
traps, backed by mechanical forepumps. Dewars employing
liquid He refrigerant attain an ultimate vacuum of approx
8
imately 5 x 10 torr. All pressures are measured with
BayertAlpert type ion gauge tubes and Veeco RG31X control
circuits.
Vaporization of nonvolatile materials was accomplished
in vacuum furnaces of identical design attached to each of
the cryogenic systems. These are watercooled brass cylinders
20.3 cm long and 15.2 cm in diameter. Furnaces associated
with the liquid He dewars are pumped by an oil diffusion
pump of minimum two inch diameter intake, backed with a
mechanical forepump and equipped with liquid N2 cold traps,
6
attaining pressures below 1 x 10 torr. The furnace mounted
on the Displex cryotip utilized the same pumping system as
the cryotip assembly.
A schematic of the furnace apparatus is illustrated in
Figure 1. Interchangeable, demountable flanges of various
design are inserted into the furnace body. For resistance
heating of samples, these flanges are equipped with water
cooled copper electrodes. Tantalum cell holders are securely
bolted with Ta screws to the ends of the electrodes, and
inserted in the holders are Ta cylinders 2.5 cm long, 6.4 mm
O.D., and of varying wall thicknesses, filled with the solid
to be vaporized. The open ends of the cell were sealed
with tightfitting Ta plugs.. The effusion orifice of 1.6 mm
diameter is directed towards the target window. These cells
could be aligned at any angle to the horizontal; the vertical
position is necessary for samples which are melted to produce
sufficient vapor pressure for deposition. Not shown in the
figure is a watercooled copper heat shield placed 1 2 cm
in front of the cell and fitted with a central 2 cm hole to
allow passage of some of the sample vapor to the target.
The cell was heated by passing up to 500 amps at up to 6 volts
through the electrodes. Cell temperatures were measured with
a Leeds and Northrup vanishing filament optical pyrometer
through a flangemounted Pyrex window which was shuttered
to prevent deposition of a film on the window.
Induction heating of samples was performed in the same
furnace body, but equipped with a flange holding 10 turns
of 6.5 mm hollow copper coil, the axis of the helix aligned
towards the target window. A Ta Knudsen cell, 2.2 cm long,
1.4 cm O.D., 3 mm wall, is supported, coaxially with the coil,
on three W rods (1.5 mm diameter and 8 cm long) attached to
a screw mechanism mounted on the furnace flange in place of
the electrodes. In this way the cell position could be
adjusted in the coils to provide maximum coupling. Tempera
ture measurements were achieved as above, except that the
cell was provided with a blackbody hole, which eliminates
the necessity of emissivity corrections. The watercooled
RF coils were attached to a Lepel 5 kW high frequency induc
tion heater. With both methods of heating, the distance from
cell orifice to the trapping surface was approximately 12 cm.
Matrix gases or gas mixtures were usually admitted to
the dewars through the copper inlet shown in the figures.
This was connected to a copper manifold equipped with
fittings to connect to Pyrex sample bulbs. The manifolds
were pumped by a 2 inch oil diffusion pump, equipped with
liquid N2 trap, and backed by a mechanical forepump, which
5
gave pressures less than 1 x 10 torr. Flow rates were adjusted
with an Edwards needle valve, and pressure changes monitored
with a Heise Bourdon tube manometer. Gas mixtures were
produced in a similar vacuum system by standard manometric
techniques. The rare gases, neon and argon, were Airco
ultrapure grade, and used without further purification except
for passage through a liquid N2 cold trap immediately prior
to deposition.
Electron spin resonance measurements were made with an
Xband Varian V4500 spectrometer system employing super
heterodyne detection. A12 inch electromagnet useful from
0 13 kG provided the static magnetic field, which was
modulated at low (200 Hz) frequency. The output of the
instrument was recorded on a Moseley model 2D2 XY recorder.
When signals were weak, a Nicolet model 1072 signal average,
equipped with SW71A sweep and SD72A analogtodigital
converter plugin units was used to improve the signalto
noise ratio. The magnetic field was measured with either
an Alpha Scientific model AL67 or a Walker Magnemetrics
model G502 NMR gaussmeter in conjunction with a Beckman
6121 counter. The microwave cavity frequency was determined
with a HewlettPackard highQ wavemeter.
0
Absorption spectra were recorded from 7000 2000 A
using a Jarrell Ash 0.5 meter Ebert mount scanning mono
chromator. Gratings ruled with 1200 lines/mm and blazed at
0
5000 and 3000 A gave a reciprocal linear dispersion of
0
16 A/mm in first order. Detectors used were the RCA 7200,
0
for the range 3700 2000 A, and either the RCA 1P21 or
0
931A, for the range 7000 3500 A, each operated at 1000 VDC.
The photomultiplier output was processed by a Jarrell Ash
82110 electronic recording system and displayed on a Bristol
model 570 strip chart recorder. Continuum light sources
were a General Electric tungsten ribbonfilament lamp for
the visible and a Sylvania DE 350 deuterium lamp for the UV.
Radiation from these sources was passed through the matrix
and focused onto the spectrometer slit with quartz optics.
The spectra were calibrated with emission lines from a Pen
Ray low pressure Hg arc lamp. A PerkinElmer 621 spectro
photometer (purged with dry N2 gas) was used in the IR region
from 4000 300 cm with an accuracy of 0.5 cm1
Photolyzing radiation in two spectral regions was avail
able from either a high pressure Hg arc lamp or a flowing
H2He electrodeless discharge lamp. The Hg lamp consists of
a watercooled General Electric type AH6 Hg capillary
lamp operated at 1000 W, the output of which was focused
onto the sample with quartz optics. The radiation from this
lamp consists of the characteristic Hg lines and a strong
base continuum. When this lamp was in use, the dewars were
equipped with quartz optical windows for transmitting the
radiation to the sample.
The flowing H2He electrodeless discharge lamp is
constructed after the design of David and Braun (5). It
consists of a quartz tube, 15 cm long, fused 4 cm from the
end with a larger diameter quartz tube to form an annular
space 6 cm in length. The annulus has provision for inlet
of the 10% H2 in He gas mixture (Air Products), and the
central tube is connected to a mechanical pump, which can
2
evacuate the entire system to about 3 x 102 torr. This
effectively seals the quartz body against an LiF optical
window by means of a brass fitting equipped with "0" rings.
The LiF window is mounted in the dewar photolysis ports.
The gas flow is adjusted to give a pressure of about 1 torr
with a gas regulator. An 85W Raytheon PGM 10 microwave genera
tor, operating at 2450 MHz, was used with a tunable cavity
to excite a discharge in the flowing gas. The emission was
0
characterized by the intense Lyman a line at 1216 A. Color
centers which developed in the LiF due to the high energy
radiation could be removed by annealing the windows at
AIr Or1 4:M Ci.i^ I,^,,,
General Technique
Preparation of the matrix samples was achieved in the
following manner. The dewars, furnace assemblies, and gas
manifolds were readied at least one day before an experiment
was run, and allowed to pump out overnight. If a good vacuum
was maintained, the liquid N2 cold traps associated with the
diffusion pumps were filled; this brought the furnace and
gas manifold assemblies near their ultimate vacuums. The
sample cells were then slowly heated and the samples allowed
to outgas at low temperatures (about 2000C below deposition
temperatures) while the dewars were prepared.
While preparation of the Displex cryotip involved only
checking the He and cooling water pressures, and switching
on the device, preparing the liquid He dewars was somewhat
more involved. First the outer, and then the inner dewar
was filled with liquid N2. The inner dewar was constantly
purged with dry N2 gas when not in use to prevent formation
of ice in the narrow channels of the variabletemperature
chamber. Filling this container with liquid N2 served to
precool it, and minimize the quantity of liquid He wasted
for this purpose. After the lower chamber, on which the
sample substrate was mounted, had reached liquid N2 tempera
tures, that refrigerant was pumped out by pressurizing the
chamber with N2 and He gas. When the liquid N2 had been re
moved and recovered, the dewar was flushed with He gas and
allowed to warm up 10200K. This assured complete removal
of the N2, which could solidify as liquid He was added. It
was found to be very important that a positive pressure of
dry gas was applied to both dewar openings, especially when
it was cold. When the dewar had warmed slightly, liquid
He was transferred by a vacuuminsulated tube into the dewar,
which was sealed with a pressure cap when transfer was
completed. These preliminary activities took approximately
one hour to perform. A charge of liquid He lasted approxi
mately 4 8 hours, depending on the rate of flow through
the lower chamber, which was set with a needle valve. The
Displex cryotip also took about one hour to reach operating
temperature, but it could maintain that temperature indefi
nitely.
With the deposition surface at a sufficiently low
temperature, the gas manifold was sealed from its pumps
and filled with the matrix gas or gas mixture. To prevent
formation of a solid residue from vaporization on the
surface, the gaseous sample alone was deposited on each side
of the substrate for approximately five minutes. The rate
of gas deposition through the entire run was controlled with
an Edwards needle valve to obtain a flow of about 0.3 mmole/
min. During this time the nonvolatile sample was heated
to its deposition temperature; formation of the metal film
on heat shield and furnace viewing port indicated that
sufficient material was being vaporized. With the initial
deposit on the sample substrate, the gate valve separating
furnace and dewar was opened and furnace vapors were co
condensed with the matrix gas. During sample deposition,
typical pressures observed in the furnace and dewar were
4 5
1 x 10 and 2 x 10 torr, respectively. Deposition times
varied from 1/2 to 2 hrs, depending on the species being
formed; the sample substrate was rotated 1800 periodically,
to form an even coating on the surface. When deposition
was completed, the dewar and furnace were isolated, the
gas flow stopped and hightemperature cell allowed to
cool. The matrix samples thus prepared were then observed
spectroscopically.
29
References Chapter II
1. C.K. Jen, S.N. Foner, E.L. Cochran, and V.A. Bowers,
Phys. Rev., 112, 1169 (1958).
2. W. Weltner, Jr. and D. McLeod, Jr., J. Chem. Phys.,
45, 3096 (1966).
3. W.C. Easley and W. Weltner, Jr., J. Chem. Phys., 52,
197 (1970).
4. W.R.M. Graham, K.I. Dismuke, and W. Weltner, Jr.,
Astrophys. J., 204, 301 (1976).
5. D. David and W. Braun, Appl. Opt., 7, 2071 (1968).
CHAPTER III
ESR THEORY
Introduction
The interactions of paramagnetic atoms and molecules
with magnetic fields, which gives rise to the electron spin
resonance phenomenon, is discussed in this chapter. Details
of the theory applicable to atoms (or ions) and doublet,
triplet, quartet, and sextet state molecules encountered
in this research will be presented in separate sections.
More extensive treatments of the basic theory presented
here can be found in a number of excellent references (113),
Atoms and the Resonance Condition
The paramagnetic substances with which we are concerned
are those which possess permanent magnetic moments of atomic
or nuclear magnitude. In the absence of an external field
such dipoles are randomly oriented, but application of a
field results in a redistribution over the various orienta
tions in such a way that the substance acquires a net magnet
ic moment. Such permanent magnetic dipoles occur only when
the atom or nucleus possesses a resultant angular momentum,
and the two are related by
p = yG [i]
where p is the magnetic dipole moment vector, G is the angular
momentum (an integral or halfintegral multiple of h/2n = "i
where h is Planck's constant), and y is the magnetogyric
ratio. The motion of these vectors in a magnetic field H'
consists of uniform precession about H at the Larmor precession
frequency
w =yH. [2]
The component of G or p along H remains fixed in magnitude,
so that the energy of the dipole in the field (the Zeeman
energy)
W = VIH [3]
is a constant of the motion.
The magnetogyric ratio which relates the magnetic moment
to the angular momentum according to Eq. [1] is given by
Y = g(e/2mc) [4]
where e and m are the electronic charge and mass, respectively,
and c is the speed of light. The factor g = gL is unity for
orbital angular momentum and g = gS = 2.0023 for spin angu
lar momentum. Including this factor with Eq. [1] and defining
the Bohr magneton as 8 = ei/2mc, we have (along the field
direction)
UL= gLmL [5a]
[5b]
S = gsmS
Because the angle of the vector p with respect to the applied
field H is space quantized, only 2G + 1 orientations are
allowed. These allowed projections along the magnetic field
are given by mG where mG is the magnetic quantum number
taking the values
mG = G, Gl, ..., G. [6]
This accounts for the appearance in Eqs. [5] of the factors
mL for orbital angular momentum and mS for spin angular
momentum.
2
If only spin angular momentum arises, as in a S1/2 atom,
the 2S + 1 energy levels separate in a magnetic field, each
of energy
E = mSH (7)
EmS geBmSH, [7]
with equal spacing ge H. However, the angular momentum does
not generally enter as pure spin, so that the g factor is
an experimental quantity and mS an "effective" spin quantum
number, since some orbital angular momentum is usually mixed
into the wavefunction. In orbitally degenerate states
described by the strong (RussellSaunders) coupling scheme,
J = L + S, L + S 1, ...,IL Sland
E = gi m H
where
S+ S(S + 1) + J(J + 1) L(L + 1)
J 2J J + 1)
is the Lande splitting factor. This reduces to the free
electron value for L = 0.
Taking the simplest case of a free spin, mj = ms = 1/2,
and there are only two levels. Transitions between these
levels can be induced by application of magnetic dipole
radiation obtained from a second magnetic field, at right
angles to the fixed field, having the correct frequency to
cause the spin to flip. Thus the resonance condition is
hv = geSHo [10]
where H is the static external field and v is the fre
0
quency of the oscillating magnetic field associated with
the microwave radiation; this frequency is about 9.3 GHz
2 2
for the Xband. Thus for a S or P atom, the ESR spectrum
will consist of one line corresponding to the particular g
value of the atom.
The Hyperfine Splitting Effect
If only one line were observed in the general case, the
ESR technique could offer only a limited amount of informa
tion, the g value. However, there are other interactions
to consider which increase the number of spectral lines and
the information that can be obtained. One of the most impor
tantisthe nuclear hyperfine interaction.
Usually at least one isotope of an element contains a
nucleus having a nonzero magnetic moment. The magnetic
moment of the odd electron can interact with this nuclear
moment and split the single ESR line into hyperfine struc
ture. This effect can be pictured as follows. The magnetic
field "felt" by the electron is the sum of the applied
external field and any local fields. One such local field
will be that caused by the moment of the magnetic nucleus;
this is, in turn, governed by the nuclear spin state. It
is then clear that, in the case of nuclear spin I = 1/2,
for example, the local field in which the electron finds
itself will be one of two contributed by the nucleus, since
there are 21 + 1 nuclear levels. Hence, there will be two
values of the external field which satisfy the resonance
condition, that is,
r (H' ) = (H' AM) [11]
where A/2 is the value of the local magnetic field, A
being the hyperfine coupling constant, and H' is the
resonant field for A = 0. One example of this phenomenon is
1 2
the H atom. This is a pure spin system, S1/2, with I = 1/2;
its ESR spectrum consists of two lines separated by A = 508G,
centered around g = ge = 2.0023, as shown in Figure 5.
A more detailed look at the paramagnetic species with
nonzero nuclear spin in a mangetic field indicates that
there are several interactions at work. One is the inter
action of the external field with the electrons, which has
already been considered. An analogous term results from
the precession of the nuclear magnetic moment in the exter
nal field. The nuclear magnetic moment pI is related to the
nuclear g factor gI by the relation
g = [12]
N
where 8N = eli/2M is the nuclear magneton and M is the proton
mass. The third term describes the interaction between the
electrons and nuclei. Thus the Hamiltonian can be written
H = gHJ + hAI.J g B HI [13]
where the underscore indicates that the quantities are
operators. Except in very strong fields, the interaction
of the nuclear moment with the external magnetic field (the
nuclear Zeeman term), which is represented by the last term
in Eq. [13], is small, and will be neglected. Also omitted
from this Hamiltonian are even smaller effects, such as the
nuclear electric quadrupole interaction.
Reference to Figures 5 and 6 will indicate the behavior
of the levels as a function of external field strength.
The two limiting cases of very weak and very strong field
are of particular interest.
The Zeeman effect in weak fields is characterized by
an external field splitting which is small compared to the
natural hyperfine splitting; that is, hAI.J > gBHJ in Eq. [13].
In this case, the orbital electrons and the nuclear magnet
N
I
0)
o ro
O N
I II 4
0)0
0 ,
0 0
z Z
tl
_n) I i
N2 "
z a
Wj ,. 4J
S'\\/4
0 u
w z
U LJ \0
Z l< E
0)
(1)
Cl
*il
(H
b
I)
0 7
L C\OJ C0J c\j J C\J
r+
I
C\J
+
.0
L)
Q
C)
0
0
C\
O
0
0
O
O
I
remain strongly coupled. A total angular momentum F = I + J
exists, which orients itself in the external field. F takes
the values I + J, I + J 1, ..., II JI. The component
of F along the field direction, mF, has 2F + 1 allowed
values, the integers between F and +F. In the case presented
in Figure 6, with both pN and A positive, these components
are arranged, in order of decreasing energy, mF = F, F 1,
..., F. Each individual hyperfine level splits up into
2F + 1 equidistant levels in the weak field, giving (2J + 1)*
(21 + 1) Zeeman levels altogether. Note that in both Figures
5 and 6, the levels are not all degenerate even at zero
field. This effect, produced by the hAIJ term, is called
the zero field splitting.
In the PaschenBack or strong field region, the splitting
by the external field is large compared to the natural hyper
fine splitting. The strong interaction with the external
field decouples I and J, which now process independently
around H. F is no longer a good quantum number, but there
exist m and mi, the components of J and I along the field
direction. In this case, each Zeeman level of the multiple
characterized by a fixed m is split into as many Zeeman
hyperfine lines as there are possible values of mi, that
is, (21 + 1). Since there are still (2J + 1) levels for
a given J, there are, exactly as in the weak field, (2J + 1)*
(21 + 1) total energy states. In contrast to the weak field
situation, the levels here form a completely symmetric pattern
around the energy center of gravity of the hyperfine multiple.
This pattern manifests itself in Figure 6. Also recorded in
that Figure are the m values of the different hyperfine
groups. Values of m are, in order of decreasing energy,
5/2, 3/2, 1/2, 1/2, 3/2, 5/2 for mj = 0, 1, 2, 3;
this order is inverted for the remaining mj groups.
The situation in intermediate fields is somewhat more
complicated. The transition between the two limiting cases
takes place in such a way that the magnetic quantum number
m is preserved. In weak field, m = mF; in strong field,
m = mI + m In this region, the Zeeman splitting is of
the order of the zero field hyperfine splitting.
For a 2S1/2 state, as in Figure 5, the general solution
for the energy levels over all fields is given by the
BreitRabi equation (14). In terms of the quantum numbers
F and m = mF, it is
W(F, AW I Hm AW 4m X2 21/2 14
W(Fm) = 2(2 + 1) I + ( + 21 + 1[14a
hA
where AW = (21 = 1) [14b]
andX ( J/J + II /I) H0
and X = [14c]
The plus sign in Eq. [14a] applies for F = I + 1/2 and
m = +(I + 1/2), ..., (I 1/2) and the minus sign for
F = I 1/2 when m = (I 1/2), ..., (I 1/2). The zero
field hyperfine splitting is AW. The limiting cases of weak
2 2
and strong fields correspond to X <<1 and X >>1, respectively.
For the general case of intermediate fields, the energy
values of the Zeeman levels can be derived from the following
key equation given by Goudsmit and Bacher (15):
Xm + 1 1 (I + m + 1)(J mj + 1)]
m + 1, m 1 I
I J
[A
Xm 1 m (I m + 1)(J + m + 1)] = 0,
M 1, m + 1 2 I J
I J
where A is the hyperfine coupling constant, gj and g'I = I
are the electronic and reduced nuclear gfactors, respectively,
and the XXA are coefficients in the expansion of the wave
function; the other symbols have their usual meaning. Here,
AWH is the energy of the level with respect to the center of
gravity of the hyperfine multiple. This relation yields
one system of homogeneous equations in XX for each value
of m = mI + m The resulting secular equations are solved
for the energies of the Zeeman levels at any field. Such
a calculation was performed to produce Figure 6.
With a multitude of levels available, it is necessary
to explain the observed ESR spectra in terms of the selection
rules. Since transition between Zeeman levels involve changes
in magnetic moments, we must consider magnetic dipole transi
tions and the selection rules pertaining to them. In the
pure spin system with I = 0, the single line observed
corresponds to the m = 1/2  m = 1/2 transition. In
s s
general, the criterion is that Amj = 1, corresponding to
a change in spin angular momentum of T. Since a photon has
an intrinsic angular momentum equal to I, only one spin
(nuclear or electronic) can flip on absorption of the photon,
in order to conserve angular momentum. With the fields
and frequencies ordinarily used in ESR work, the transition
usually observed is limited to the selection rules Amj = 1,
Am = 0; the opposite is true in NMR work. It is, however,
possible to observe the Amj = 0, Am = 1 NMR transition
with ESR apparatus, if the zero field splitting (propor
tional to the hyperfine coupling constant A) is large
enough relative to the microwave frequency. If this does
not occur, only the ESR lines will be observed, resulting
in a multiple of 21 + 1 hyperfine lines for each fine
structure (Amj = 1) transition. Thus the H atom spectrum
(Figure 5) will consist of two lines, while that of Mn
( S3, I = 5/2) will contain 36 individual lines, if all
are resolved.
These interactions of the electron with a nucleus are
related to fundamental atomic parameters, which can be
deduced from the observed spectrum. They are most simply
categorized as isotropic and anisotropic interactions.
The anisotropic interaction has its roots in the classi
cal dipolar coupling between two magnetic moments. This
interaction is given by
42
e'N *3( e'r) ( N' r)
E 35[16]
3 5
r r
where r is the raduis vector from the moment e to N and
e N
r is the distance between them. The quantum mechanical
version is obtained by substitution of the operators, gBS
and gNN I, for the moments e and N', respectively, yielding
I(LS) 3( r) (Sr)
Hdip = rg N 3 5 [17]
r r
For a hydrogenic atom with nonzero orbital angular momen
tum (that is, p, d, ... electrons), this yields
L(L + 1) 1
aJ e N J(J + I) ; [18'
a more exact relativistic treatment also adds a multiplica
tive factor [F(F + 1) I(I + 1) J( J + 1)]. For s elec
trons, a similar dipolar term yields
2
a = gg (3cos 8 1) [19
a = geBgiSN 3 91
r
where 0 is the angle between the magnetic field direction
and a line joining the two dipoles. However, the electron
is not localized and the angular term must be averaged over
the electron probability distribution function. For an
s orbital, all angles are equally probable due to the
spherical symmetry, and the average of cos2 over all 0
causes the function to vanish. Thus the classical dipolar
term cannot be responsible for the hyperfine structure of
the 2S1/2 hydrogen atom.
The actual interaction in the selectron case is
described by the Fermi contact term (16). This isotropic
interaction represents the energy of the nuclear moment in
the magnetic field produced at the nucleus by electric cur
rents associated with the spinning electron. Since only
s orbitals have finite electron density at the nucleus,
this interaction only occurs with s electrons. This yields
the isotropic hyperfine coupling constant
a = 7T 9g Io l2 [20)
as 3 e~ (0) [20]
where the last term represents the electron density at the
nucleus. This term has no classical analog. The a value
s
is proportional to the magnetic field at the nucleus, which
can be on the order of 105 G. Thus unpaired s electrons
can give very large hyperfine splitting.
2
E Molecules
The Spin Hamiltonian
The discussion to this point has involved only those
terms in the Hamiltonian of the free atom or ion which are
44
directly affected by the magnetic field. However, it will
be useful to begin the discussion of paramagnetic molecules
by consideration of the full Hamiltonian, which can be
written, in general,
H = + H LS+ S + SH + IH [21]
E LS SI SH IH
Each of the terms can be described as follows. The term
H expresses the total kinetic energy of the electrons,
:E
the coulombic attraction between the electrons and nuclei,
and the repulsions between the electrons:
2 2
P ZAe E e
H =  + [22]
E i 2m A,i r. i>j r [22
1 i]
where pi is the momentum of the ith electron, r. is the
i 1
distance from electron i to nucleus A of atomic number ZA,
and r.i is the distance between electrons i and j. The
BornOppenheimer approximation (17) has already been invoked
to separate out the nuclear motions and nuclearnuclear
repulsions. This term yields the unperturbed electronic
levels before spin and orbital angular momentum are con
sidered. Eigenvalues of this term are on the order of 105 cm1
The energy due to the spinorbit coupling interaction
is usually expressed in the form
H = XLS [23]
LS
where L and S are the orbital and spin angular momentum
operators and A is the molecular spinorbit coupling
constant. The magnitude of this interaction is of the order
2 3 1
of 10 10 cm .
The hyperfine interaction arising from the electronic
angular momentum and magnetic moment interacting with any
nuclear magnetic moment present in the molecule may be
expressed as
SI I= gI Nge + 3
3 5
r r
S I 8T6 (r)S*
+ [24]
r 3
This term corresponds to the sum of the isotropic and
anisotropic hyperfine interactions discussed for atoms
above. The Dirac 6function indicates the isotropic
part which has a nonzero value only at the nucleus. The
2 1
hyperfine interaction has a magnitude of about 10 cm
The electronic Zeeman term, SH' is primarily responsible
for paramagnetism. It accounts for the interaction of the
spin and orbital angular moment of the electrons with
the external magnetic field, according to
H = B(L + g S)*H [25]
These energies are on the order of 1 cm
These energies are on the order of cm1
These energies are on the order of 1 cm
The interaction of the nuclear moments with the
external field, that is, the nuclear Zeeman effect, is
given by
H =? Y Y H), [26]
3 1
and is usually too small (10 cm ) to be significant.
It is evident that this total Hamiltonian (which still
excludes higherorder terms and field effects which could
be observed in crystals) is very difficult to use in
calculations. However, experimental spin resonance data
obtained from the study of the lowestlying levels can be
described by use of a spin Hamiltonian in a fairly simple
way which does not require detailed knowledge of all
the interactions. These levels are generally separated
i
by a few cm (by the magnetic field), and all other
electronic states lie considerably higher. The behavior
of this group of levels in the spin system can be described
by such a spin Hamiltonian, and the splitting, which may
usually be calculated by first and second order perturba
tion theory, are precisely the same as if one ignored the
orbital angular momentum and replaced its effect by an
anisotropic coupling between the spin and the external
magnetic field. This was first shown by Abragam and Pryce (18).
Just as the g factor becomes anisotropic and not necessarily
equal to g = 2.0023, the S cannot represent a true spin
but is actually an "effective" spin. This is defined, by
convention, to be a value such that the observed number
of levels equals (2S + 1), just as in a real spin multiple.
Thus we can relate all the magnetic properties of a system
to this effective spin by the spin Hamiltonian, since it
combines all of the terms in the general Hamiltonian which
are sensitive to spin. Nuclear spins can be treated in
the same manner, and the spin Hamiltonian corresponding to
Eq. [21] can be written (neglecting the nuclear Zeeman
term)
H pin= 8HS + IAS [27]
spin 0
where the double underscore indicates a tensor quantity.
The g Tensor
As alluded to previously, the anisotropy of the gtensor
arises from the orbital angular momentum of the electron
through spinorbit coupling. Even in the case of E states,
which have zero orbital angular momentum, the interaction
of a presumably pure spin ground state with certain
excited states can admix a small amount of orbital angular
momentum into the ground state, and change the values of
the components of q. This interaction is usually inversely
proportional to the energy separation between the states.
The spinorbit interaction can be described as (5)
H = AL*S = A[L S + L S + LS ]
LS  xx yy zz
[28]
This term is added to the Zeeman term in the Hamiltonian,
thus
H = H. (L + g S) + AL*S .[29]
For an orbitally nondegenerate ground state represented
by IG, MS>, the first order energy is given by the diagonal
matrix element
(1) = <,MSIg HzSz G,Ms> + ISIHz + ASz Ms , [30]
where the first term is the spinonly electronic Zeeman
effect. Because the ground state is orbitally nondegenerate,
<ILzI > = 0, and the second term vanishes. The second
order correction to each element in the Hamiltonian matrix
is given by
H ZI2
!M M 1 [31]
s S n (0) (0)
n G
where the prime designates summationover all states except
the ground state. Since = 0, the matrix elements of
g98H.S will vanish. The operator matrix can then be
expanded to
HM M []
S1 S (0) (0)
n W W [32]
n G
and the quantity
Axz
Ayz
Azz
_'_
(0) (0)
n G
n G
Axx
Axy
Axz
factored out where the ijth element of the
tensor is given by
=
ij W"" H'0
n (0) (0)
w 
n G
= A [33]
second rank
[34]
where i and j are any of the cartesian coordinates. This
simplification yields
HMs = M (MS2i H.A.H + 2AH*A.S + A 2 SAMs' > .
sg
[35]
The first operator represents a constant contribution to
the paramagnetism and need not be considered further.
The other terms represent operators which act only on
spin variables. When combined with the Zeeman term of
Eq. [30], the result is the spin Hamiltonian
2
H = B. (g 1 + 2AA)*S + X2S*A*S
spin e
= BHg'S + S*D.S ,
[36]
where
S= g 1 + 2AA [37]
and
D = X2A [38]
with 1 being the unit tensor.
The SD*S term is operative only in systems with S 1 1
and will be considered later. The other term in Eq. [36]
is the spin Hamiltonian in the absence of hyperfine inter
action. It is evident that the anistropy of the g tensor
arises from the spinorbit interaction due to the orbital
angular momentum of the electron. This may be expanded
to show the gtensor as
BSgqH = B[S S S I g g g g H [39]
xyz xxxyxz x [39
gyx yy yz y
zxgzygzz z
where S S and S are the components of the effective spin
along the axes. Strictly, g is a 3x3 matrix and is referred
to as a symmetrical tensor of the second order (the symmetry
implies that the unpaired electrons are in a field of
central symmetry). The double subscripts on the gtensor
elements may be interpreted as follows: gxy is the contri
bution to g along the xaxis when the magnetic field is
applied along the yaxis. These axes are not necessarily
the principle directions of the gtensor, but a suitable
rotation of the axes will diagonalize it; then the diagonal
components are the principle directions of the gtensor
with respect to the molecule. It is noteworthy that, if
the molecule has axes of symmetry, they must coincide with
the principle axes of g; if it has symmetry planes, they
are perpendicular to the principle g axes.
Three cases of interest, with regard to molecular
symmetry can be outlined. If the system is truely a spin
only system, g will be isotropic and the diagonal elements
equal to g If it is isotropic but contaminated with
orbital momentum, the principle components will be equal
but unequal to g e In the former case
spin xyz ex
H0 g H [40
= ge[Hx S + H S + H S ].
S xx yy zz
For a system containing an nfold axis of symmetry (n > 3),
two axes are equivalent. The unique axis is usually designated
z and the value of g for H II z is called g ". For H I z,
the value is gi. Thus
Hspin = (gHS + g HYS + g HS ).
spin Ixx I yy zz
[41]
Finally, for systems where there are no equivalent axes
(orthorhombic symmetry), gxx yy g 9zz and
H spin (g H S + g H S + g H S ). [42]
spin xx xx yy yy zz zz
The A Tensor
It has already been noted that the hyperfine inter
action is composed of isotropic and anisotropic parts.
While the isotropic coupling is that which is observed
in liquids, the anisotropy due to dipoledipole interac
tions can be observed in fixed systems, such as molecules
in a rigid matrix. If Eq. [24] is expanded (and the L*I
term is dropped since this is a Z state), the interaction
can be seen to assume the form of a tensor
2 2
H r / rAr2 3x \ /3xy\ 3xz \
dip xz \Sf Kr5 [x
dip = ( NSxS Sz r x r 5 r
3 r2 3y2 /3yz [43]
r r r [43]
2/3xy\ 3y r23z2
3" \ (r z z
r r r
= hS*T*I
When the isotropic part is added, the spin Hamiltonian
with hyperfine becomes
H spin= BSgH + hS.A*I
spin
[44]
where
A = A 1 + T [45]
with A0 being the isotropic hyperfine coupling constant
and 1 the unit tensor. Thus the element
A.. = A. + T.. [461
13 iso 13
and, as with the gtensor, certain cases can be selected due
to the molecular symmetry. An isotropic system has A =
xx
A = A = A. Systems with axial symmetry are
yy zz iso
characterized by A = A = A = A. + T and Az
xx yy iso xx zz
All = Ais + Tz
iso zz
Randomly Oriented Molecules
Having discussed some basic theory of ESR in mole
cUles, one must consider how these effects manifest them
selves in the spectrum. Anisotropy will appear in the
spectrum of a rigidlyheld molecule, but there is a
difference between the effects observed in crystals and
in matrices. In a single crystal, with a paramagnetic
ion or defect site, for example, the sample can be aligned
to the external field and spectra recorded at various angles
of the molecular axis to the field. In the matrix, the
samples ordinarily have a random orientation with respect
to the field, and the observed absorption will have contri
butions from molecules at all angles. This was first
considered by Bleaney (19, 20), and later by others (2126).
Solving the spin Hamiltonian,Eq. [42],in the ortho
rombic case and assuming the g tensor to be diagonal, the
energy of the levels will be given by
2 2 2 2 .2 2 2 21/2
E = HH(gl sin Ocos2 + g2 sin Osinc + g cos2)/2
= BgHSHH [47]
where SH is the component of the spin vector S along H,
gH is the g value in the direction of H, 6 is the angle
between the molecular z axis and the field direction, and
{ is the angle from the x axis to the projection of the
field vector in the xy plane. For axial symmetry g
2 2 2 2 1/2
(g1 sin + g11 cos ) and the energy of the levels
is given by
2 2 2 2
E = BSHH(gl sin 6 + g 1 cos 6). [48]
Thus the splitting between energy levels, and therefore
transitions between them, are angularly dependent.
Consider first the case of axial symmetry. As a
measure of orientation, it is convenient to use the solid
angle subtended by a bounded area A on the surface of a
sphere of radius r. The solid angle is the ratio of the
surface area A to the total area of the sphere, that is,
2
S= A/4wr If all orientations of the molecular axis
are equally probable, the number of axes in a unit solid
angle is equal for all regions of the sphere. If the
sphere is in a magnetic field, the orientation of the axes
will be measured by their angle 6 relative to the field.
Taking a circular element of area for which the field
axis is the z direction, the area of the element is
27(r sin6)rd6, and the solid angle dQ it subtends is
2
2nr sind6
dQ 2  = 1/2 sin6d6, [49]
47r2
and if there are N0 molecules, the fraction in an angular
increment d8 is
N0
dN =  sinEd6. [50]
Assuming the transition probability is independent of
orientation, which is approximately the case, the absorption
intensity as a function of angle is proportional to the
number of molecules lying between 6 and 8 + dO.
Since g is a function of 6 for a fixed frequency v,
the resonant magnetic field is
hv 2 2 2. 21/2
H = (g2 cos 6 + g sin 2 [51]
and from this
s2 (g0H0/H) g
sin 6 = 0 2 [52]
g 2
gi 11
where go = (gl + 2gi)/3 and HO = hv/g0B.
Therefore
2 2
g0 H0O
sinede = H
3
H
2 ( 2 0 [o 2 2H 1/2
S) H. [53]
The intensity of absorption in a range of magnetic field
dH is proportional to
dN IdN Ie
HdH ILdH
[54]
where the two factors on the right are obtained from Eqs. [50]
and [53], respectively. From the above equations
H = hv/gll = g0Ho/gl at 6 = 00
H = hv/gB = g0H0/g9 at 0 = 900.
[55a]
[55b]
At these two extremes, the absorption intensity varies from
IdN N 091.1
2g H0(g11 2g12
dN = m
dH
at 0 = 0
[56]
at 6 = 90. [57]
Tf plotted against magnetic field, this absorption (after
considering natural linewidth) takes the appearance of
Figure 7a, for gll>gj. Since the first derivative is usually
observed in ESR work, the spectrum would appear as in
Figure 7b. If the g tensor is not too anisotropic, gll and
gi can be readily determined as indicated. In general,
the perpendicular component can easily be distinguished
in such a "powder pattern;" the weaker parallel peak is
often more difficult to detect.
If there is also a hyperfine interaction in the randomly
oriented molecules, the pattern of Figure 7b will be
split into (21 + 1) patterns, if one nucleus of spin I is
involved. Such a spectrum for I = 1/2 is shown in Figure 7c,
where, because gl1 % g, and Ai < Al the lines for m =+1/2
and 1/2 point in opposite directions. If instead, gl were
shifted upfield, relative to gll and A1 % All the spec
trum would appear as two patterns similar to Figure 7b,
and separated by the hyperfine splitting.
The spin Hamiltonian for an axially symmetric molecule
is similar to that for an atom, but now incorporating
parallel and perpendicular components of g and A:
H sp = B[g H S + g (HS + H S )]
spin II zz xx yy
[58]
+ Ai S I + A (SxI + S I )
II zz xx yy
0
a
0 .0
*H H rQ
0 0
wc)
>1> >
Or 
S*H 
(0 M
O  
0 4J
t i rnv
M .SH 
0 0
4) 4 U U
I Q
>U 0a)
C4 tP
0 U C
rOO
> W r1
0) >i
O O (*
4' >i U>
() r4 (d
rl (t (L) H
0 4 4 Q)
0 O0)
SH H
t r
rX
C4
 07
This omits the small nuclear Zeeman term and assumes the
symmetry axis is the z axis. Equation [51] can be rewritten
to include the nuclear hyperfine effect (8) to first
order as
hv K
H 8 g mI [59]
where
2 2 2 2 2
g = g cos 6 + g sin 2 [60]
and
2 2 2 2 2 2 2 2
Kg = All gl cos + Ai g1 sin [61]
The intensity of absorption IdN/dHI can again be derived
and is found to be
22 2
dN 0 2cos6 (g0 _g )goH
'dH' 2 2
9 2g [62]
+ m g11 Al 2 g2 A12 K(g g ) 1
+I  n9*I U^L
+ ml g
2K g
Here sinOd6 cannot be solved explicitly so that dN/dH cannot
be written as a function of only magnetic parameters.
Equations [59] and [62] must be solved for a series of
values 8 to obtain resonant fields and intensities as a
function of orientation. However,
H = (g0H0/gl ) (mIAll /Bgll ) at 0 = 00 [63a]
H = (g0H0/g ) = (miA /Bg) at 6 = 900 [63b]
and again IdN/dHI  Thus a superposition of the
typical powder pattern results, with the relative phase
of the lines determined by the magnitudes of the magnetic
parameters, as discussed above. In Figure 7c, the lines
do not overlap and analysis is simple, but this is not
always the case. In general, the best approach is to
solve the given equations by computer for a trial set of
g and A values, and match the calculated spectrum to the
observed.
A similar treatment can be applied to molecules of
orthorombic symmetry and instead of the two turning points
at g11 and qg, there will be three corresponding to gl'
92, and g3. Such a spectrum is considered, including
hyperfine interaction with a spin 1/2 nucleus, by Atkins
and Symons (11) and Wertz and Bolton (5).
As mentioned in Chapter I, molecules trapped in
rare gas matrices do not always assume random orientations.
During condensation of a beam of reactive molecules in
solid neon or argon matrices, some preferential orientations
of the molecules relative to the flat sapphire rod may occur,
and in some cases the alignment can be extreme. This non
random orientation can easily be detected by turning the
matrix in the magnetic field; a change in the ESR spectrum
indicates some degree of preferential orientation. The
degree of orientation appears to depend upon the size and
shape of the molecule, the properties of the matrix, and
other factors which are not completely understood (27).
Two examples are the molecules Cu(N03)2 (28) and BO (29).
The latter case shows very strong orientation such that
with the magnetic field perpendicular to the rod surface,
the parallel lines were strong and the perpendicular,
weak. With the rod parallel to the field, the perpendicu
lar lines became very strong and the parallel components
disappeared entirely. This indicated that the BO molecules
were trapped with their molecular axes normal to the plane
of the condensing surface. This orientational behavior is
analagous to that usually observed in singlecrystal work.
Molecular Parameters and the Observed Spectrum
Having discussed the nature of the interactions appearing
in the spin Hamiltonian and the form of the observed spectrum,
it is time to consider the relationships between the spectral
features and the paramagnetic species themselves. This will
begin with the exact solution to the spin Hamiltonian in
axial symmetry, and presentation of the secondorder solutions
which are usually adequate, and conclude with the molecular
information revealed through g and A components.
A thorough discussion of the spin Hamiltonian
Hspin g 11H S + gi (H S + HIS )
+ A I S + A (I S + IS ) [64]
jII zz xx yy
has been given by several authors (8, 30, 31). Considering
the Zeeman term first, a transformation of axes is per
formed to generate a new coordinate system x', y', and z',
with z' parallel to H. If the direction of H is taken
as the polar axis and 6 is the angle between z and H, then
y can be arbitrarily chosen to be perpendicular to H and
hence y = y'. Thus only x and z need to be transformed.
With H = Hsin9, H = HcosO, and H = 0
x z Y
H = 8[gl cos8Sz + glsinS x]H. [65]
Choosing the direction cosines z = g cos9/g and x =
gisin6/g, with g defined by Eq. [60], then
H = gg[ S + S ]H [66]
and the Zeeman term becomes
and the Zeeman term becomes
H = gS''H
z
[67]
where
S = S + S
z zz xx
S = . S + 2 S
X XZ ZX
S = S
y y
For the hyperfine terms
hf = A S I + A (SII + S I ),
hf  zz xx yy
and I is rotated by
I = nI nI
Z zz xx
I = nI + .nI
X XZ ZX
I = I '
y y
where the n. are the direction cosines for the nuclear
1
coordinate system relative to the electronic coordinates.
By inverting Eq. [68] to obtain the S. analogs of Eq. [70],
1
and substituting into the Hamiltonian of Eq. [69], the
Hamiltonian of Eq.[64] is transformed into
[68a]
[68b]
[68c]
[69]
[70a]
[70b]
[70c]
S= ghS + KI 'S + KI I 'S '
z z z K x x
2 2
+ A A sinecoseI S + A I'S [71]
K z x iy Y
with the definitions = A1 g1 cose/Kg, = A g sin9/Kg,
2 2 2 2 2 2 2 2
and K g = All g1 cos + A g1 sin 8. Dropping the primes
and using the ladder operators S = S + iS and S = S iS
x y x y,
this can be rewritten in the final form
2 2 + 
A A2 g S+S S
Hspin signSz + KSz +kO 
S = gHSz + KS Iz [ + I 2 cosesin 2 z
+ I AI + ) (s+I+ + SI )
4K 4
+ ( A+ A (S+I + SI) [72]
4K 4
This Hamiltonian matrix operates on the spin kets IMs,MI),
and can be solved for the energies at any angle. An
example of the use of this exact solution will be presented
later.
The exact solution is difficult to solve at all angles
except e = 0, but elimination of some of the offdiagonal
elements (those not immediately adjacent to the diagonal)
results in some simplification and is usually adequate. The
solution is then correct to second order, and can be used
when gH>>A11 and A as is often the case. The general
secondorder solution is given by Rollmann and Chan (32)
and by Bleaney (20) as
2 2 2
A All + K 2
AE(M,m) = g H + Km + [ 1 [I(I + 1) m ]
8G K
A 2
+ 4( A)(2M l)m [73]
where K is A and A1 at 6 = 0 and 900, respectively, and
G = gH/2. Also, M is the electron spin quantum number of
the lower level in the transition, and m is the nuclear spin
quantum number. Note that the first two terms on the right
result from the diagonal matrix elements and yield equi
distant hyperfine lines; this is the first order solution
given in Eq. [59]. The last two terms cause increasing
spacing of the hyperfine lines at higher field which is
referred to as a "secondorder effect." This solution can
routinely be applied because the hyperfine energy is
usually not comparable to the Zeeman energy.
The g tensor has appeared in the derivation of the
spin Hamiltonian, and it is seen, from the above equations,
how the values of the principle axis components can be
determined from the ESR spectrum. Now we shall consider
its relationship to a molecular wavefunction in the linear
combination of atomic orbitals (LCAOMO) approximation.
The usual form applied, Eq. [37], is a result of the
secondorder perturbation treatment, yielding the first
order corrections to the g components
<0 ILn) )
g g 6. 2E [74]
ij e 13 E
n n
where the primed summation is over all excited states n
which can couple to the ground state 0 and E is the energy
of that state above the ground state.. The Kronecker delta is 6...
13
Because the correction is caused by the spinorbit inter
action, only certain states can couple with the ground state.
Specifically, these are the states such that (33)
Fn e L 00 + A lg [75]
1
that is, the direct product of the irreducible representations
of the ground and excited state with the representation of
the angular momentum operator (which transforms as rotations)
must include the totally symmetric representation of the
symmetry group. A specific example, which will be encountered
later, is that a Z state can only mix with a H state. Usually
there will only be one such state of energy low enough
to make the term significant. The spinorbit coupling
constant C can be assumed positive or negative, depending
on whether the excited state involves excitation of an
electron or a "hole," respectively. Both this constant
and the orbital angular momentum operator L can be written
as sums of atomic values:
67
ZL. = k [76a]
k1 k k
i = k [76b]
where k indicates a particular atom in the molecule. Actually,
,k decreases rapidly for large rk (ar3) so that k is
essentially zero except near atom k, where it may be assumed
to have a fixed atomic value Xk. Thus Eq. [74], for the
perpendicular component of a E state diatomic molecule,
becomes
g9 = Z nikk Ix ( lkI) > [77]
n k,k'
Here, E and H are the LCAO wavefunctions
@ = Eaix(i) [78a]
1
j = Zb X(j) [78b]
j
where x(i) and X(j) are A.O.'s in the ground and excited
states, respectively. Then the second matrix element in
Eq. [77] will reduce, for a diatomic, to sums of terms
involving integrals of the type
(X(i)kl kxIx(J)k) (atom k only) [79a]
x
x(i)kIk'. Ix(j)k ) (both atoms) [79b]
(X(i)k l,' X()k> (both atoms) [79c]
k'
(X(i)k' X(J)k,) (both atoms) [79d]
The first matrix element in Eq.[77] can be simplified if
Ak is assumed constant near k and zero elsewhere. Then
it becomes similar to the integrals of Eq. [79a].
The integrals in Eqs. [79c, d] require that the origin
of the operator kx be moved from atom k to atom k'. This
introduces a linear momentum term according to
x = k +i RP [80]
k ky
where R is the interatomic distance. Fortunately, the
elements involving P are usually zero or small, so that
the term can be neglected and the integrals in Eqs. [79]
are then all of similar form.
Eq.[79a, b] involve the application of the angular
momentum operator to the atomic functions involved. The
nonzero elements have been tabulated (34) and are given
below:
(PxLy lP> )
x y zy z z xy
(d IL Id 2 2)
(dxz Lxdyz) = (dz2 LxIdyz) = i [81c
(dxyl dyz) = (dyz Lz dzx) = i [81d]
(d xL xd y) = (dx2 2 L Id = (dx 2 21L d z)= i
[81e]
(ilL qi') = (i' Lq i> .[81f]
Overlap integrals which appear because the A.O's are
centered on different atoms have also been tabulated (35, 36).
Detailed discussions of this approach can be found in
Stone (37) and Atkins and Jamieson (38).
Secondorder corrections to the gfactor have been
determined by Tippins (39), utilizing thirdorder perturba
tion theory. This degree of the theory must be used to
calculate corrections to gI and the result, analogous
to Eq. [74], is
( <(nl L 10 2
gil = g 1/2  EE [82]
n 0 
Thus it can be seen that gll is always very close to or
less than g since the correction factor is small and
squared. On the other hand, gl can be greater or less
than g and the difference can be quite significant.
Thus, if wavefunctions are available, or can be
constructed, the values of Ag = g ge can be calculated
and compared to those determined experimentally. Examples
of this approach can be found in references (3, 5, 11, 40),
and in the discussion of the MnO molecule to follow.
Alternately, one can approximate the energy separation of
the lowest interacting level from the ground state.
The hyperfine coupling constant has been shown to
consist of isotropic and anisotropic parts, and its tensor
nature has been discussed. The spin Hamiltonian, with
hyperfine interaction, can be written as
H s = HII.g.S + aLI + bI*S + cI S [83]
spin zz
where
a = gegNn 3~ [84a]
Sc 2 s 1 84b
SN g N \ 2r
c = 3gg KN N 3 [84c]
2r
where the angular brackets are, as usual, quantum mechanical
averages. This definition has been given by Frosch and
Foley (41). Neglecting the small L.I term, we can compare
Eq.[83] with Eq.[58] and identify the observed splitting
as
A = b + c [85a]
Ai =b [85b]
The isotropic part can be written in terms of the parallel
and perpendicular components as
A + 2A
A. II c 8rr 2
Aiso 3 =b + 3 8 NBN [86]
The anisotropic or dipolar component is given by
A A 2
Adp = c 3cos 01 [87
dip 3 3 gePN N\ 2r3
Thus the observed spectrum is related to the fundamental
quantities I(0) 2 and (3cos281/2r3) for interaction with
that nucleus. If the L*I term is included, the values become
All = b + c Agl a [88a]
AI = b + Ag a
[88b]
where the Agi = gi ge have already been discussed. These
small corrections can be approximated from the observed
Agi value and the value (1/r3) for the particular nucleus.
The dipolar coupling constant can be considered
further. The dipolar part of the Hamiltonian (Eq.[171) can
be written as
g N (3 cos 1) SI [89a]
3
r
and the energies of the levels IMs,MI) are given by
3cos 81
E = geN MI r3 1). [89b]
For an electron in an orbital centered on the nucleus in
question, the anisotropic hyperfine coupling follows
Eq. [89], but has an additional term to represent the average
direction of the electron spin vector in the orbital.
The hyperfine splitting is the separation between adjacent
levels IMSMI) and IMs,MI1) and equals
A 3cos20 (3cos2l) [901
dip =e 9NcoN 3 3s) [90]
where a is the angle between r and the principal axis of
the orbital, and 0 is the angle between the latter and the
direction of the nuclear magnetic moment vector. The
value of (3cos2a1) can be evaluated for the atomic orbital
functions; the values are 4/5 for any p orbital and 4/7, 2/7,
and 4/7 for the dz2, dxz,yz, and dx2y2 xy orbitals,
respectively. These quantities are used to give the principle
value of Adip for an orbital, that is A Atomic values of
the quantities A. and Ap have been evaluated for many
iso dip
atoms, and a useful table can be found in Ayscough (1) or
Goodman and Raynor (42).
Equation [90] assumes that the nuclear moment vector
PI is aligned with the external field, that is, the applied
field is much stronger than the field at the nucleus due
to the electron. This strongfield approximation is actually
valid only when (a) the applied field is large, (b) the
anisotropic coupling is small (and hence the field at the
nucleus small), and (c) the isotropic coupling is large
(since the field caused by the electron reinforces the
applied field). It is found that the field at the nucleus
due to the electron is much larger than the field at the
electron due to the nucleus. If the strong field approxi
mation is not valid, the dipolar interaction will vary as
2 1/2
(3cos6+1)/2. However, the numerical values of the aniso
tropic coupling at the turning points will be the same, and
only the signs will differ. In a crystal, the difference
will be discernable, but for the powder patterns obtained in
matrices, the strongfield analysis is sufficient. A more
thorough discussion can be found in references (5, 42).
The calculated atomic A. and A dip values are often
iso dip
used with experimentally determined molecular values to
derive coefficients or spin densities on a particular atom
in a molecule. Using an LCAOMO wavefunction described as
= ZaiXi, the values of As. and Adip at a particular
nucleus x can be written
dip e= g NN (iI (3cos2 1/2rx)1> [91a]
x 87r 9 Nlg2 [91b]
iso 3g e N Nl (0)x2 [91b
Since these integrals are expected to be small except near
atom x
Ax = ai(P d )2A (atom) [92a]
dip i x x dip
A = a.(s )2 A (atom) [92b]
1so I x iso
where the ai are the coefficients. Although this implies
that atomic properties remain unchanged in the molecule, which
is unlikely, it is quite useful in comparing trends to model
wavefunctions. Examples of its utility will be given later.
A more quantitative approach to calculating A tensor components,
based on the intermediate neglect of differential overlap
(INDO) molecular orbital approximation, has been developed
(4346).
3 Molecules
A theorem due to Kramers states that, for all systems
with an odd number of electrons, at least a twofold
degeneracy will exist which can only be removed by applica
tion of a magnetic field. This would apply to cases with
S = 1/2, 3/2, 5/2, ...; the first has been considered in
detail.
In the triplet case, S = 1, and two noninteracting
electrons can be described by four configurations:
a(l)a(2), c(1)8(2), 8(1)a(2), and 8(1)B(2). In a molecule
of finite size, interactions will occur and the configurations
can be combined into states which are symmetric or anti
symmetric to electron interchange. These states are
a(1)a(2) [93a]
(1/T ) [a(l)B(2) + 8(l)a(2)] (1/1) [a(l)8(2) B(1)a(2)]
[93b]
8(1) P(2) [93c]
The multiplicity of the symmetric states (on the left) is
(2S + 1) = 3; this is a triplet state. Because of the Pauli
principle, this state may exist only if the two electrons
occupy different spatial orbitals.
For systems of two or more unpaired electrons, the
degeneracy of these spin states may be lifted even in the
absence of a magnetic field; this is termed the zerofield
splitting (ZFS). If the number of unpaired electrons is
even (S 1 i, 2, ...), the degeneracy may be completely
lifted in zero field. Additional terms in the Zeeman
Hamiltonian H = BH.g.S are required to account for this.
The Spin Hamiltonian
It was shown in the derivation of the spin Hamiltonian
(Eq.[36]) that the anisotropic part of the spinorbit
coupling produces a term S*D*S which is operative only in
systems with S > 1. However, at small distances, two
unpaired electrons will experience a strong dipoledipole
interaction, such as has been considered in the anisotropic
hyperfine interaction:
Sg2 S *S 3(S *r)(S *r)
HSS 3 2 5 [* 94]
r r
In this equation, F is the vector connecting the electrons,
and the Si are the spin operators of the individual electrons i.
If the scalar products are expanded, and the Hamiltonian
expressed in terms of a total spin operator S = S + S2,
1 2
2 2 2 2
and considering that r =x + y + z the matrix form of
77
the Hamiltonian can be written
2 2 2 
S=/2) [ s s s r 3x 3xy 3xy
H =(1/2)g 8 S [ S ] 55 S
SS xyz 5 5 5 x
r r r
2 2
3xy r 3y 3yz S
5 5 5 y
r r r
2 2
3xz 3yz r 3z2
5 5 5 z
r r r .
=S DS [95]
This term, representing the dipolar interaction of the
electron spins, should be compared to the last term of
Eq. [36]; the latter evolved from Eq. [31] by treating the
spinorbit coupling interaction as a perturbation on the
Zeeman energy and assuming that the space and spin parts of
the electronic wavefunctions were separable. It can be
seen that the two terms are identical in form, except for
a numerical constant. These spinorbit and spinspin
contributions to D cannot be distinguished experimentally.
Whatever the origin of the interaction, the D tensor can
be diagonalized and the fine structure term becomes
2 2 2
H=D S + D S D S [96]
xxx yyy z zz
where D + D + D = 0, that is, D is a traceless tensor.
xx yy zz
This can be written in terms of the total spin as
2 2 2
SD.S = D[S 2 (1/3)S(S + 1)] + E(S S 2
z _x y
+ C/3)(D + D + D ) S(S + 1), [97]
xx yy zz
where D = D (D + D )/2 and E = (D D )/2.
zz xx yy xx yy
The last term is a constant, proportional to the trace of D
which is zero for pure spinspin interaction, and does not
appear in the spin Hamiltonian. The terms involving D
and E account for the removal of the degeneracy of the three
triplet wavefunctions in the absence of an external magnetic
field.
While the effects of spinspin and spinorbit inter
actions are not separable, some qualitative statements can
be made about them. It is found, in general, that organic
triplets show little influence from spinorbit coupling.
The g values of such molecules are usually very close to
ge, and the zero field splitting can be ascribed almost
completely to spinspin dipolar interactions. Then one
would expect the value of D to be approximately inversely
proportional to the molecular volume; this trend can be
seen in the methylene derivatives C6 H C C6H5 and
1 1
NC C CN, where D is 0.4 cm (47) and 1.0 cm (48),
respectively. Introduction of a heavy atom increases the
spinorbit coupling (since XaZ4 ), thus D for CN2 is 1.16 cm
i
(49) but for SiN2, discussed below, it is 2.28 cm1. If the
molecule involves transition metals, which usually have
large spinorbit coupling constants, the zero field splitting
is due mainly to the spinorbit interaction.
The solution to the general Hamiltonian for triplet
states of randomly oriented molecules is given by Wasserman,
Snyder, and Yager (50). Here, the special case of linear
molecules will be briefly discussed.
Neglecting hyperfine and other spinorbit interactions,
the spin Hamiltonian for a linear triplet will be
spin = g9) Hz + g(HS + HS ) + D(Sz2 2/3) [98]
where z is the molecular axis. If y is chosen arbitrarily
to be perpendicular to the magnetic field, then H = 0 and
2
Hspin = g BH zS + g0(HS ) + D(S 2 2/3). [99]
Choosing as a basis the orthonormal spin wavefunctions
S+ 1) =I'aal) [100a]
1 0 (') =(l//l2B2 +1"2) [100b]
I 1) = 1 ,2) [100c]
and considering the effect of the spin operators on the func
tions as
S a =(1/2) S x =(1/2),a S a =(1/2)iB S 8 =(1/2)ia
x x y y
2 2
S a =(1/2)a S z =(1/2) S a =(1/4)a S 2B =(1/4)
z z Z Z
[101a]
[101b]
then the Hamiltonian matrix will be
I + 1)
D/3 + G
Gx /2
x
S0 )
G
Gx/2
2/3 D
Gx/ 2
I 1)
[102]
D/3 Gz
z
where G
z
= g 8Hz and Gx = gjBHx.
The eigenvalues for
Hi z (Hz = H, H = 0) are
W+1 = D/3 + gl BH
W0 =(2/3)D
W 1 = D/3 g 11 H.
1+ 1)
I 0o
I 1)
[103a]
[103b]
[103c]
At zero field, the + 1) and 1) states are degenerate,
and the appropriate wavefunctions are
T =(1/2 )(I + 1> I 1)) [104a]
T =(1/V2)(I + 1) + 1>) [104b]
Tz = 0) [104c]
In these wavefunctions the spins are quantized along the
X y, and z axes, respectively: S IT ) = S T y = S IT ) = 0.
S' x x yy z z
The eigenvalues for Hilz are plotted against H in Figure 8,
and all yield straight lines.
For H z, H = H and H = 0, and the roots of the secular
Sx z
determinant are
W1 = D/3 [105a]
W2 = (1/2) D/3 + (D2 + 4g 2H2 ) /2] [105b]
W3 = (1/2)D/3 (D2 +4g 2H2) 1/2 [105c]
with the eigenvectors
x = 1/i) + 1) ) [106a]
I
HI
O o
0 
,I
r4
0II
I~ LLJ J
4O "0
4O
rU
0 0
 I4 0
C~~j ,I
OaO
4 ( 
roD Q )4
4) G)
 4" AJI I II  o 4
Cr
 'N
I
X~ N
I co
f
1
py = cosa jt + 1) + ) + sinal0) [106b]
1 F 1
z = sinea I + 1) + I ) +cosaI0), [106c]
where tan 2a = 2gl3H/D. As H approaches zero, where a = 0,
these functions reduce to the functions T, Ty, and T
x z
given in Eq.[104]. The eigenvalues for H z are plotted versus
H in Figure 9, and only at high fields, where ar/4,
do the lines become straight. In the intermediate region,
y and iz are mixed and lead to a curvature of the energy
with H.
It is evident that the energy levels, and hence the
fields at which transitions between them occur, are very
dependent upon the orientation of the axis of the molecule
with respect to the fixed magnetic field. This will,
in general, cause the spectra of randomly oriented molecules
to be broad and difficult to observe.
The relative transition probabilities are given by
2 2 2 2
hij k = g cos i yl( IS j)I [107]
where i, j, and k are any of the molecular axes (z is axial)
and y is the angle between the oscillating magnetic field
of the microwave radiation (perpendicular to the fixed
field) and k. Then transitions are allowed between levels
characterized by the following wavefunctions:
84
I Ll
0
0
t
Vb
N 0
1 I
3 WU
K
N
3^
0
I:
NQU
4
S r( N o N
+ f + + I I I
>~KN
lI
(13
X
4,
0
4
0
o'
'0
E
r41
*H H
0
4J;
r4
U
H
(0
34 04
m
)rl *
r X
4
0
>1
H 4
*oh4
a) ^
cl nj
i0 r
(1) T T z
(2) Ty T z
y z 2
(3) y < X xy2
(4) z  X xYI'
where the symbols on the right are the usual designations
given to the observed ESR lines. These transitions are
indicated in Figures 8 and 9, and all correspond to AM = 1
transitions. Also indicated in Figure 9 is a dashed line
representing the forbidden AM = 2 transition. This transi
tion (0y c 'z) is allowed for Hosc H, but is has a finite
transition probability (9) when H is not parallel to any
of the x, y, or z axes, even if Hos IH, as in the apparatus
employed here. The AM = 2 notation is acutally a misnomer,
since the spin functions I + 1) and 1), corresponding
to the infinite field Ms values, mix significantly at
finite fields. Thus M is not a good quantum number and the
transition could actually be described as AM = 0, since
the eigenfunctions each contain contributions from spin
functions of the same M. Thus if D is not too large, the
transition will be observable.
i. Employing the exact solution to the spin Hamiltonian
matrix for triplet molecules, which may be bent (E ; 0), the
resonant fields of the transitions are (50)
86
H = T [(hVD) E ] 11
zl g 8
1 2 2 1/2
H [(hv+D) E21/
z2 g [I
1
H 
xl gjy
1
Hy2 gi
y2 = 9
H = 
AM=2
[(hvD+E) (hv+2E) 1/2
[(hv+DE) (hv2E)] 1/2
[(hvDE) (hv2E) I1/2
[(hv+D+E)(hv+2E)] 1/2
2 2 2
1 (hv) D +3E2 1/2
gB 4 3
In Figure 10, the resonant fields of these transitions
for linear molecules (E = 0) are plotted as a function of
D, for a fixed microwave frequency of 9.1 GHz, where the
energy equals 0.3 cm1. The z and xy lines are so marked.
As can be seen from the Eqs.[108], the effect of a nonzero
E term in the spin Hamiltonian is to split each xy line into
[108a]
[108b]
[108c]
[180d]
[108e]
[108f]
[108g]
0.9
0.8
0.7
0.6
0.2
0.
Figure 10.
I 2 3 4 5 6 7 8
Hr (Kilogauss)
3
Resonant fields of a 3 molecule as a function
of the zero field splitting.

Full Text 
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INGEST IEID ESPMIQN1B_7VWCJX INGEST_TIME 20120309T03:56:30Z PACKAGE AA00003920_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
OPTICAL AND ELECTRON SPIN RESONANCE SPECTROSCOPY
OF MATRIXISOLATED SILICON AND MANGANESE SPECIES
By
ROBERT FRANCIS FERRANTE
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1977
To my parents
Digitized by the Internet Archive
in 2010 with funding from
University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation
http://www.archive.org/details/opticalelectronsOOferr
ACKNOWLEDGEMENTS
The author extends his deep appreciation to Professor
William Weltner, Jr. whose encouragement, professional
guidance, patience, and support made this work possible.
Thanks are also due to all the members of Professor
Weltner's research group, particularly Dr. W. R. M.
Graham and Dr. R. R. Lembke, for their collaboration,
assistance, and advice during the research.
The author greatly appreciates the expert craftman
ship displayed in fabrication of experimental apparatus
by A. P. Grant, C. D. Eastman, and D. J. Burch of the
Machine Shop and R. Strasburger and R. Strohschein of the
Glass Shop, as well as the maintenance of electronic
equipment by R. J. Dugan, J. W. Miller, and W. Y. Axson.
The author would also like to acknowledge the support
of the Air Force Office of Scientific Research (AFOSR) and
the National Science Foundation (NSF) during this work.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS iii
LIST OF TABLES VÃ
LIST OF FIGURES viii
ABSTRACT xi
CHAPTERS
IINTRODUCTION 1
The MatrixIsolation Technique 1
References  Chapter I 9
IIEXPERIMENTAL 11
Introduction 11
Experimental 11
Apparatus 11
General Technique 26
References  Chapter II 29
III ESR THEORY 30
Introduction 30
Atoms and the Resonance Condition 30
The Hyperfine Splitting Effect 33
Molecules 43
The Spin Hamiltonian 43
The g Tensor 47
The A Tensor 52
Randomly Oriented Molecules 53
Molecular Parameters and the
Observed Spectrum 61
Molecules 75
The Spin Hamiltonian 76
Molecules 94
The Spin Hamiltonian 95
IV
CHAPTERS
IV
V
Molecules
Page
105
The Spin Hamiltonian
106
References  Chapter
III
113
CON SPECIES
I'
117
l
Introduction
117
Experimental
118
ESR Spectra
119
SiN2
119
SiCO
125
S i 2
133
Optical Spectra
133
Si and Si2
133
SiN2
137
SiCO
144
Si(CO)2
150
Discussion
152
References  Chapter
IV
171
ANESE SPECIES
175
Introduction
175
Experimental
177
ESR Spectra
179
Mn Atoms
179
Mn+
179
MnO
184
Mn02
190
MnO 3
196
Mn04
205
Discussion
207
Mn Atoms and Mn
207
MnO
212
Mn02
215
MnO 3
217
MnO 4
220
References  Chapter
V
232
SKETCH
2 37
v
LIST OF TABLES
TABLE
PAGE
I
3
ESR data of SÃN2 and SiCO in their E
ground states in various matrices at 4Â°K
131
II
SÃÂ£ absorption bands in argon matrices at 4Â°K
135
III
14
Ultraviolet absorption spectrum of Si ^
in an argon matrix at 4Â°K
140
IV
Vibrational frequencies and calculated
force constants (mdyn/A) for SiNN and
SiCO molecules in their ground states
143
V
12
Absorption spectrum of Si CO in an argon
matrix at 4Â°K
147
VI
Comparison of stretching force constants
(mdyn/A) for relevant molecules XYZ
154
VII
Total density matrix elements for SiCO,
SÃN2, and the free ligands CO and ^
157
VIII
Spin densities in SiCO and SiNN
164
IX
Comparison of vibrational frequencies and
electronic transitions of CXY and SiXY
molecules
167
X
Field positions (in gauss) of observed fine
and hyperfine structure lines of Mn+:Ar
at 4Â°K. A = 275 G; v = 9390 MHz
185
XI
Magnetic parameters, observed and calculated
line positions for the  +1/2) 1 ~Â¿/2)
perpendicular transition of MnO ( E) in Ar
191
XII
Magnetic parameters, observed and calculated
line positions for the +1/2) 11/2)
perpendicular transition of MnC>2 (^E) in Ar
195
XIII
Spin Hamiltonian matrix for the statesM, m)
for MnO3 (^a^), including interaction with
the 55^ (i = 5/2) nucleus
199
vi
TABLE
PAGE
XIV
2
Magnetic parameters of MnO~ ( A
observed transitions in Ne an
â– j) in Ne;
d Ar
203
XV
Magnetic parameters, observed and calculated
line positions for Mn04 (^Tj) in Ne
208
XVI
Summary of magnetic parameters
quantities for manganese and
manganese oxides
and derived
some
210
vil
LIST OF FIGURES
FIGURE PAGE
1 Basic design features of the liquid
helium dewar used for ESR studies 13
2 Variabletemperature modification of
liquid helium dewar used for ESR studies 14
3 Basic design features of variabletemperÂ¬
ature liquid helium dewar used for
optical studies 17
4 Basic design features of cryotip assembly
used for optical studies 20
5 Zeeman energy levels of an electron
interacting with a spin 1/2 nucleus 36
6 Zeeman energy levels of a ion with
1=5/2 J 37
7 Absorption and first derivative lineshapes
of randomly oriented molecules with
axial symmetry 58
8 Energies of the triplet state in a magnetic
field for a molecule with axial symmetry;
field parallel to molecular axis 82
9 Energies of the triplet state in a magnetic
field for a molecule with axial symmetry;
field perpendicular to molecular axis 84
3
10 Resonant fields of a Z molecule as a funcÂ¬
tion of the zero feild splitting 87
11 Theoretical absorption and first derivative
curves for a randomly oriented triplet
state molecule with axial symmetry 90
12 Theoretical absorption and first derivative
curves for a randomly oriented triplet
state molecule with orthorombic symmetry 92
4
13 Energy levels for a Â£ molecule m a magneÂ¬
tic field; field parallel to molecular
axis 102
viii
FIGURE
PAGE
14
4
Energy levels for a E molecule in a magÂ¬
netic field; field perpendicular
to molecular axis
103
15
4
Resonant fields of a E molecule as a
function of the zero field splitting
104
16
Â£
Energy levels for a Z molecule in a
magnetic field for 0 = 0Â°, 30Â°, 60Â°,
and 90Â°
109
17
Resonant fields of a molecule as a
function of the zero field splitting
111
18
ESR spectra of SiN^ molecules in argon
matrices at 4Â°K
120
19
ESR spectra of Sil^ molecules in various
matrices at 4 Â°K
124
20
ESR spectra of SiCO molecules in argon
matrices at 4Â°K
126
21
Effect of temperature upon the ESR spectrum
of Sil3C0 in an argon matrix
128
22
ESR spectra of SiCO molecules in Ar and
CO matrices
130
23
Ultraviolet absorption spectrum of SiN^
molecules in an agron matrix at 4Â°K
138
24
Infrared bands of SiN^, in argon and
nitrogen matrices at 4Â°K
142
25
Absorption spectrum of SiCO molecules in
an argon matrix at 4Â°K
145
26
Infrared spectra at 4Â°K of an argon matrix
containing vaporized silicon atoms and
13C0/Ar = 12cO/Ar = 1/375
148
27
Infrared spectra at 4Â°K of an argon matrix
containing vaporized silicon atoms and
Cl6o/Ar = Cl8o/Ar = 1/375
149
28
ESR spectrum of Mn+ in argon at 4Â°K
180
29
Zeeman levels and observed transitions for
Mn+ in argon
185
IX
FIGURE
PAGE
30 ESR spectrum of MnO in argon at 4Â°K 187
31 ESR spectrum of Mn02 in argon at 4Â°K 193
32 ESR spectrum of MnO^ in neon at 4Â°K 197
33 ESR spectrum of MnO^ in argon at 4Â°K 202
34 Mj = 3/2 component of MnO^ in argon at
various temperatures, and in neon 204
35 ESR spectrum of MnO^ in neon at 4Â°K 206
36 Molecular orbital correlation diagram
for MnO^ 218
37 Molecular orbital correlation diagram
for MnO^ 223
x
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of
the Requirements for the Degree of Doctor of Philosophy
OPTICAL AND ELECTRON SPIN RESONANCE SPECTROSCOPY
OF MATRIXISOLATED SILICON AND MANGANESE SPECIES
By
Robert Francis Ferrante
December, 1977
Chairman: William Weltner, Jr.
Major Department: Chemistry
3
The Z molecules carbonyl silÂ»ne, SiCO, and diazasilene
SiNN have been prepared by vaporization and reaction of
silicon atoms with N2 or CO and trapped in various matrices
at 4Â°K. The electron spin resonance (ESR) spectra indicate
that some or all sites in some matrices induced slight
bending in the molecules, and that the species undergo
torsional motion in the solids. Isotopic substitution of
13 15 18 29
C, N, 0, and Si was employed to obtain hyperfine
coupling data in the ESR and shifts in the optical spectra.
In solid neon, assuming g 11 =gjj=ge, D = 2.2 8 and 2.33 cm ^
for SiN2 and SiCO, respectively. Hyperfine splittings in
argon yield A= 17 (^4N), 21 (^N) and 95 (2^Si) MHz for
SiN2 and 84 (2^Si) and <14 (12C) MHz for SiCO. These
confirm calculated results, in the complete neglect of
differential overlap (CNDO) approximation, that the electron
spins in both molecules are largely in the pr orbitals of Si
Optical transitions (with vibrational progressions) were
Â° 1
observed beginning at 3680A (SiN stretch, 470 cm ) and
xi
Â° _ 1 o
3108A (NN stretch, 1670 cm ) for SiN^ and 4156A (SiC
stretch, 750 cm ^; C0 stretch, 1857 cm for SiCO in Ar.
Infrared (IR) spectra in Ar indicate v" .T = 1733 and
NN
v . = 485 cm 1 for SiNâ€ž and v" _ = 1899 cm ^ for SiCO.
SiN 2 C0
Calculated stretching force constants are = 2.0,
SxN
O
kN_N = 11.8, kc_Q = 15.6, and ks^_c = 5.3 mdyn/A, the
" 1
latter assuming  800 cm . The CNDO calculations
suggest tt bonding of Si to the ligand, which is stronger
in SiCO than SiN_, and some tt. . , *â– dâ€ž. backdonation,
2 ligand Si
also stronger in SiCO. An attempt was made to correlate
these vibrational and electronic data with those for CCO
and CNN. Annealing an argon matrix containing SiCO to
35Â°K led to the observation in the IR of ^E Si(CO)2, a
silicon counterpart of carbon suboxide, with v^' ^ = 1899.6 cm .
A corresponding treatment of a SiN^ matrix did not produce
N^SiN^, nor was N^SiCO observed when both ligands were present.
The molecules MnO, Mn02, MnO^, and MnO. have also been
prepared, by the vaporization and reaction of manganese atoms
with 02, N20, or 0^, and isolated in various inert gas matrices
at 4Â°K. ESR has been used to determine magnetic parameters
which are interpreted in terms of molecular geometry and
2 2 6 +
electronic structure. MnO is confirmed to have a air 6 , E
ground state with g^ = 1.990 (7), assuming g j = ge, and a
zero field splitting in accord with the gas phase value
D = 1.32 cm . Hyperfine splittings due to the Mn (I = 5/2)
nucleus are A11  = 176(3) and AjJ = 440 (11) MHz. Mn02 is
4  2
a linear E molecule with probable configuration erÃ³ ,
Xll
D = 1.13 cm 1 (assuming gj = = 2.0023), Aj  = 353(11),
 AjJ = 731(11) MHz. MnO^ exhibits very large hyperfine
splittings  A   = 1772(3) and  AjJ = 1532(3) MHz indicative
2
of an sdz2 hybrid A^ ground state of symmetry. The
spectrum of MnO^ is consistent with a molecule
2
distorted from a electronic state in tetrahedral symmetry
by a static JahnTeller effect. The g and A tensors are
slightly anisotropic: g11 = 2.0108(8), g^ = 2.0097(8);
 A   = 252 (3),  AjJ = 196(3) MHz. The electron hole is
almost entirely in an oxygen rbonded orbital with one
oxygen atom displaced along its Mn0 bond axis. Warming
to 35Â°K did not induce thermal reorientation.
xiii
CHAPTER I
INTRODUCTION
The MatrixIsolation Technique
Molecular spectroscopy, the primary tool for the investiÂ¬
gation of intimate details of molecular geometry and electronic
structure, has been routinely applied to a large assortment
of stable chemical species in the solid, liquid, and gaseous
phases. The advent of highspeed electronic instrumentation
has extended the range of spectroscopic techniques to allow
the study of unstable or shortlived molecules and fragments.
However, some molecules are still very difficult or even
impossible to observe because of their short lifetimes,
reactivity and/or method of preparation. Among these are
molecules that exist only under high temperature conditions,
such as stellar atmospheres or in arcs, and fragments whose
reactivity precludes production of sufficient quantities for
normal analyses. Even when such species are observed, analÂ¬
ysis of their spectra is often greatly complicated by their
production in a multitude of electronic, vibrational, and
rotational states, as can occur in laboratory methods of
generation of such radicals via arcs, flash photolysis, etc.
Utilization of the matrixisolation technique can overcome
many of these difficulties.
The technique of matrixisolation spectroscopy was proÂ¬
posed independently by Norman and Porter (1) and by Whittle,
1
2
Dows,, and Pimentel (2) in 1954. Basically, the high
temperature species, reactive molecules, or radical fragÂ¬
ments are prepared and trapped as isolated entities in inert,
transparent solids, or matrices, at cryogenic temperatures.
They do not undergo translational motion, but are immobilized,
thus preventing further reaction and preserving the specimen
for conventional spectroscopic analysis. The common technique
of treating solid samples by diluting with KBr and forming a
compressed disk of finely dispersed solid in a KBr matrix
can be considered a crude form of matrixisolation. It has
been demonstrated that the information gained by this techÂ¬
nique is gaslike within a few percent.
The matrix material can be any gas which will not react
with the trapped species and which can be readily and rigidly
solidified. Many different substances have been used for this
purpose, including CH^, CO, N2, CS2, SF^., 0^, as well as
aliphatic and aromatic organics. However, the solid rare
gases Ne, Ar, Kr, and Xe are usually employed because they
are relatively inert chemically, transparent to radiation
over a wide wavelength region, and offer a wide range of
melting points and atomic sizes. The choice of matrix gas
is determined to a large extent by the effect the solid matrix
will have upon the trapped molecule. Neon, because it is the
least polarizable, is expected to perturb the molecule least
and generally makes the best matrix. Unfortunately, its
trapping efficiency is not as great as the other rare gas
solids, and it is difficult or impossible to achieve isolation
3
of some species in solid Ne. Argon is better in this
respect, and is generally used as the matrix medium. The
heavier rare gases are found to perturb the trapped molecules
to a greater extent than Ne or Ar, and ar^ therefore, less
desirable as matrix materials.
Temperatures sufficiently low to condense the matrix
gas can be attained with either physical refrigerants (nitroÂ¬
gen, hydrogen, or helium in the liquid state) or mechanical
closedcycle refrigerators utilizing JouleThompson expansion
of high pressure hydrogen or helium gas. Readily available
and inexpensive, liquid (boiling point 77.4Â°K) is useful
for some matrix materials, but is limited to relatively
stable guest species. Liquid (boiling point 20.4Â°K)
is often used, but it entails a fire hazard in addition to
the normal dangers of handling cryogenic fluids. Thus liquid
He is the most suitable of the physical refrigerants and the
only one useful for condensation of solid Ne, which melts at
24Â°K and permits solid state diffusion at about half that
temperature. Temperatures of liquids and He can be
reduced to 15Â°K and 1.2Â°K, respectively, by pumping on the
liquid. Closedcycle refrigerators which can attain temperÂ¬
atures near the boiling point of He are commericially availÂ¬
able. Their advantages include convenience, elimination of
the need to replenish cryogenic fluids while experiments
are in progress, and low cost of operation after the initial
investment.
4
The substrate on which the matrix is condensed is
usually chosen to be transparent in the spectroscopic region
of interest. Some suitable materials are Csl, KBr, NaCl, and
Au (for reflectance systems) in the infrared region, quartz,
sapphire, and CaF^ for the visible and ultraviolet regions,
LiF for the vacuum ultraviolet and sapphire or other nonÂ¬
conducting material for microwave spectroscopy, including
electron spin resonance. The polished crystal plates are
mounted on a cold block which makes good thermal contact
with the liquid refrigerant reservoir or the expansion chamÂ¬
ber of the JouleThompson refrigerator. In variabletemperaÂ¬
ture dewars, the cold block is isolated from the reservoir,
but cooled by introducing a controlled leak of refrigerant
into the mounting block.
Several methods can be employed for production of the
guest species which are trapped on such substrates. One
common procedure is vaporization of a nonvolatile material
from a hightemperature Knudsen cell in a vacuum furnace.
These cells can be constructed of carbon, or the refractory
metals, Mo, Ta, or W. To prevent degradation of the cell
material from contact with the hot vapor, the crucible can
be lined with C, A^O^, or BN. The cells are heated by
resistance or induction methods, and temperature up to 2900Â°K
can be achieved in this manner. The vapor effusing through
a small orifice is collimated into a crude molecular beam
and deposited simultaneously with the matrix gas in such
proportions that M/R (the ratio of the number of moles of
5
matrix material to the number of moles of the trapped species)
is 500:1 or greater. In many cases the compositions of
vapors so obtained have been characterized mass spectrometri
cally, greatly facilitating the analysis of the resulting
spectra. The matrix gas can also be doped with a reactant
gas by standard manometric techniques to produce other
unstable reaction products. Another high temperature source
is thermolysis of a gaseous compound by passing it through
a hot W or Ir tube, the resulting products being cocondensed
with the matrix gas as above. An additional common method
for generating unstable species is to subject a volatile
parent compound to highenergy radiation, as produced by
microwave or electric discharges, ultraviolet lamps, laser
sources, or gamma rays, or by electron or ion bombardment.
This exposure can be performed during or after sample deposiÂ¬
tion, and the resulting changes observed spectroscopically.
Combination of the hightemperature and photolytic procedures
can produce other unstable species for study.
A large array of spectroscopic techniques can be applied
to matrix samples, including infrared (IR) and Raman,
visible (VIS) and ultraviolet (UV) absorption and emission,
electron spin resonance (ESR) and tlossbauer spectroscopy.
Matrix isolation spectroscopy has several advantages over
gas phase work. First of all, there is the ability to
observe normally unstable or highly reactive species at
leisure, using conventional or only slightly modified spectromÂ¬
eters. Another prime advantage, particularly important for
6
hightemperature species, is that the molecules are always
trapped in their ground electronic and vibrational states.
With most work done below 25Â°K, there are no "hot" bands, as
the thermal energy is only 17 cm ^. Because states other
than the ground state are thermally inaccessible, sensitivity
is increased over that of hightemperature gasphase work,
and analysis is aided since the originating level for spectro
scopic transitions is always the ground state. With long
deposition times (up to 48 hrs with automatic flow control),
sufficient concentrations of molecules can be accumulated
in order to observe species of low abundance or spectral
features with low absorption coefficients. In addition,
controlled diffusion experiments can be conducted in order
to follow the formation of new species, polymers, or clusters
Finally, it is possible to observe preferential orientaÂ¬
tion of some species in matrices; the equivalent effect is
observed in single crystals, but for most species considered
in matrix work, these would be impossible to prepare.
Of course, there are some disadvantages to the technique
primarily the frequency shift from gasphase values, caused
by perturbing effects of the matrix. Frequencies in neon
matrices are generally higher than those in argon, and the
data in these solids often bracket the gasphase value. Neon
does give the closest agreement, with vibrational frequencies
shifted 10 cm ^ or less. Electronic transitions show a
similar trend, the Ne values differing from gasphase by
up to 200 cm For trapped molecules, all transitions
7
exhibit shifts of the same order of magnitude, and usually
the same direction; trapped atoms show no such regularity,
and often include absorptions that have no apparent correÂ¬
spondence to gasphase transitions. Magnetic parameters are
also influenced by matrix effects, usually exhibiting trends
related to the atomic number of the matrix gas. Theoretically,
hostguest interactions causing these perturbations are not
well characterized, although some effort has been extended
in this direction. Samples of such attempts can be found in
(312) .
Another common observation in matrix work is that the
shapes and widths of bands vary widely, depending upon the
extent of interaction between the absorbing species and
the matrix. Usually Ne shows the narrowest lines, up to
10 cm ^ full width at half maximum (FWHM) in the IR, up to
O
20 A FWHM for electronic absorptions. The broadening effect
usually increases with the atomic number of the matrix gas,
but lines in Ne are occasionally rather broad, also. Line
shapes are also somewhat matrixdependent. However, for a
given matrix, the perturbations are useful in identification
of progressions of vibrational lines in a particular excited
electronic state, since such bands always show the same
shape. The departure of the lineshapes from the usual
Lorentzian form are generally attributed to occupation of
multiple sites of similar energy in the rare gas lattice,
and/or the simultaneous excitation of lattice modes (phonons).
Site splittings of a few wavenumbers, Angstroms, or gauss
8
are common, but can often be partially or completely
eliminated by the irreversible process of annealing the
matrix. This is done by allowing the matrix to warm up,
and then rapidly quenching it to the original low temperature.
A final disadvantage is the loss of information about
rotational levels of trapped species. Some molecules, such
as Ã^O, HC1, NH^, and NH^ do rotate in matrices, but rotaÂ¬
tional sturcture is usually lost in broad vibrational
envelopes.
This introduction is not designed to discuss in depth
the various aspects of the matrix isolation technique, but
to illustrate the method and possibilities of its applicaÂ¬
tion, as well as enumerate some advantages and disadvantages.
More extensive details can be found in several recent reviews
(6, 7, 11, 1322) and references contained therein.
9
References  Chapter I
1. I. Norman and G. Porter, Nature, 174, 508 (1954).
2. E. Whittle, D.A. Dows, and G.C. Pimentel, J. Chem.
Phys., 22, 1943 (1954).
3. M. McCarty, Jr. and G.W. Robinson, Mol. Phys., 2^,
415 (1959).
4. M.J. Linevsky, J. Chem. Phys., ^4, 587 (1961).
5. G.C. Pimentel and S.W. Charles, Pure Appl. Chem.,
7, 111 (1963).
6. B. Meyer, "Low Temperature Spectroscopy," Elsevier,
New York, 1971.
7. A.J. Barnes, "Vibrational Spectroscopy of Trapped
Species" (H.E. Hallam, ed.), Wiley, New York, 1973,
p. 133.
8. R.E. Miller and J.C. Decius, J. Chem. Phys., 59,
4871 (1973) .
9. A. Nitzan, S. Mukamel, and J. Jortner, J. Chem. Phys.,
6_0, 3929 (1974) .
10. G.R. Smith and W. Weltner, Jr., J. Chem. Phys.,
6_2, 4592 (1975) .
11. S. Cradock and A.J. Hinchcliffe, "Matrix Isolation,
A Technique for the Study of Reactive Inorganic
Species," Cambridge University Press, Cambridge,
1975.
12. B. Dellinger and M. Kasha, Chem. Phys. Lett., 38,
9 (1976) .
13. A.M. Bass and H.P. Broida, "Formation and Trapping
of Free Radicals," Academic, New York, 1960.
14. W. Weltner, Jr., Science, 155, 155 (1967).
10
15. J.W. Hastie, R.H. Hauge, and J.L. Margrave, "SpectroÂ¬
scopy in Inorganic Chemistry," Vol. 1 (C.N.R. Rao
and J.R. Ferraro, eds.), Academic, New York, 1970,
p. 57.
16. W. Weltner, Jr., "Advances in High Temperature 1'
Chemistry," Vol. 2 (L. Eyring, ed.), Academic,
New York, 1970, p. 85.
17. D. Milligan and M.E. Jacox, "MTP International Review
of Science, Physical Chemistry, Series I," Vol. 3
(D.A. Ramsay, ed.), Butterworth, London, 1972, p. 1.
18. A.J. Barnes, Rev. Anal. Chem. , .1, 193 (1972).
19. A.J. Downes and S.C. Peake, Mol. Spectrosc., 1_,
523 (1973) .
20. L. Andrews, Vib. Spectra Struct., A_, 1 (1975).
21. B.M. Chadwick, Mol. Spectrosc., _3, 281 (1975).
22. G.C. Pimentel, New Synth. Methods, 3, 21 (1975).
CHAPTER II
EXPERIMENTAL
Introduction
The general experimental procedure including cryogenic,
hightemperature, spectroscopic, and photolytic apparatus is
discussed in this section. Specific details peculiar to
individual molecules investigated will be presented with
the discussion of those species.
Experimental
Apparatus
In this research, three separate cryogenic systems
were employed, two for optical and IR studies and a third
for ESR experiments. An ESR and an optical dewar utilizing
liquid He as physical refrigerant were adapted from the design
of Jen, Foner, Cochran, and Bowers (1), and modified for
variabletemperature operation as described by Weltner and
McLeod (2). Both systems are comprised of an outer liquid
N2 dewar, which serves as a heat shield, surrounding the inner
liquid He dewar. The sample substrate is attached to a
copper block suspended from the bottom of the inner dewar
by a small tube which permits passage of a controlled leak
of refrigerant into the block. The inner dewar is positioned
such that the trapping surface is directly in the path of
the sample inlets.
11
12
Pertinent details of the ESR dewar, described by Easley
and Weltner (3), and Graham, et al.(4), are indicated in
Figures 1 and 2. The stainless steel inner dewar, capacity
approximately 2.1 liters, is jacketed at the lower end by
a copper shroud which is part of the outer (liquid reserÂ¬
voir surrounding the upper portion. This jacket extends down
to encase the microwave cavity and maintain it near 77Â°K.
A slot approximately 3.5 cm long and 0.6 cm wide allows the
matrix gas and furnace vapor to reach the sample substrate.
At 90Â° to either side of this slot, two rectangular openings
of approximately 2.4 cm x 0.6 cm are provided. These points
correspond to the location of interchangeable windows, sealed
by Viton "0" rings to the outer vacuum chamber, which permit
visual examination of the matrix and serve as ports for
photolysis or spectroscopic observations.
The trapping surface is single crystal sapphire, obtained
from Insaco, Inc. It is a flat rod, 3.3 cm long, 3.1 mm
wide, and 1.0 mm thick, securely enbedded by Wood's metal
solder into the copper block, as indicated by Figure 2.
A Chromel vs Au0.02 at. % Fe thermocouple is also attached
to the copper block so that the temperature can be monitored.
Although the temperature of the sample substrate itself is
never determined, single crystal sapphire has high thermal
conductivity at 4Â°K so that it rapidly equilibrates to the
temperature of the mounting.
Also shown in Figure 2 is the construction of the
variabletemperature modification. The copper lower can is
Basic design features of the liquid helium dewar used for ESR studies.
Figure 1.
T.C. Connection
p r
Figure 2. Variable temperature modification of liquid helium dewar used for ESR
studies.
15
connected to the main liquid He reservoir by a thin stainless
steel tube; a vent for the lower can is also provided to
exhaust the gaseous He as it evaporates. Outside the vacuum
vessel, the He outlet is equipped with a valve to control the
flow rate of liquid He into the lower can. The main liquid He
reservoir is pressurized to about 2.5 psi to supply an
uninterrupted flow through the mounting block. To vary the
temperature, the outlet valve is closed, and as the He
evaporates, it forces the liquid refrigerant out of the
lower chamber, allowing it to warm. The change in temperaÂ¬
ture is monitored with either a Leeds and Northrup model
8687 potentiometer or a Newport model 2600 digital thermometer
After sample deposition is completed, the entire
inner dewar assembly is lowered approximately 3.8 cm with
respect to the fixed outer can and vacuum chamber, utilizing
a vacuumtight bellows arrangement mounted at the top and
not indicated in the Figures. In this manner, the rod is
positioned in the center of the copper Xband (=9.3GHz)
microwave cavity; this location corresponds to the maximum
intensity of the circulating magnetic field of the microwave
radiation injected into the cavity. The front end of the
cavity is slotted and aligned with another interchangeable
window in the outer vacuum chamber located just below the
sample inlets. In this way, the sample can be photolyzed
and ESR spectra recorded simultaneously. The back end of
the cavity is fitted with a standard copper waveguide coupling
mounted just outside a mica window, which serves to maintain
high vacuum conditions in the cavity.
16
With the sapphire rod in position, the dewar is separated
from the furnace assembly by means of a gate valve and
disconnected from it. The entire dewar assembly is then
rolled on fixed tracks to the proper position between the
poles of the ESR magnet. When so aligned, the alternating
magnetic field of the microwave radiation is oriented
perpendicular to the static field of the external magnet.
However, the inner (liquid He) dewar can be rotated 360Â°
on bearings to permit detection of resonance signals with
the flat surface of the rod oriented at any angle with respect
to the external field.
Following the same basic design, the dewar used for
optical studies is diagrammed in Figure 3. The stainless
steel inner (liquid He) dewar, of capacity 8.6 1, is surrounded by
a liquid N2 container and copper sheath. Four circular openÂ¬
ings in the sheath, each of 3.5 cm diameter, are located at
the level of the sample substrate mounted on the lower chamÂ¬
ber of the inner dewar. At one of these openings, the furÂ¬
nace and matrix gas inlets are attached on the outer vacuum
chamber. This opening can be sealed with a gate valve when
deposition of vapor from the furnace is not desired. At
90Â° to either side of the sample inlet, the openings form
part of the optical path of the spectrometric instruments.
Interchangeable windows (approximately 4 cm diameter) are
mounted with Viton "O" rings to the outer vacuum chamber at
these points. The window materials are chosen to match the
spectral region of interest; CaF2 is used for the visible
17
Figure
VALVE
WINDOW
SECTION aa
3. Basic design features of variable temperature
liquid helium dewar used for optical studies.
18
and UV regions (20007000 A), and Csl is used for visible
O
and IR studies (3500 A50y). All optical crystals were obÂ¬
tained from Harshaw Chemical Co. The fourth port is located
at 150Â° to the sample inlets, and interchangeable windows
can also be mounted on the outer chamber at this point.
These are usually either quartz or LiF, and serve to admit
photolyzing radiation to the sample in the ultraviolet or
vacuum ultraviolet regions, respectively. Such sample
photolysis cannot be accomplished simultaneously with deposiÂ¬
tion. As in the case of the ESR dewar, the entire inner
assembly can be rotated on bearings through 360Â°, to align
the sample window with any of the above.
The sample substrate is a polished optical crystal,
2.2 x 1.1 cm, chosen to match those on the outer vacuum
casing and the spectral region of interest. It is mounted
in the lower chamber of the variabletemperature inner dewar
with all four sides in contact with the cold copper block.
At all points where the window is in contact with the copper
heat sink, a thin gasket of indium metal is inserted. This
material has good thermal conductivity and is sufficiently
plastic that it conforms to all contours of both surfaces
when the window mounting frame is firmly screwed into the
copper block from above the substrate. A chromel vs Au0.02
at. % Fe thermocouple is mounted to the copper immediately
adjacent to the window. Temperatures at this point are
measured with either the potentiometer or a Cryogenic TechÂ¬
nology Inc. digital thermometer /controller. Variable
19
temperature operation is achieved in the same manner as
described above for the ESR dewar.
The third piece of cryogenic apparatus employed in this
research utilizes an Air Products model DE202 Displex
cryotip. This is a twostage, closedcycle He refrigerator
which makes use of the JouleThompson effect as compressed
gas at 300 psi is expanded, with a pressure drop of over
200 psi. The vacuum housing for the cryotip is very similar
in design to the liquid He optical dewar, except that the
fourth window discussed above is located at 180Â° to the
sample inlets; thus photolysis can be conducted simultaneously
with sample deposition.
Internally, there are a few modifications. The cryotip
unit (shown in Figure 4) is constructed of stainless steel,
with the exception of the final expansion chamber. There
is no liquid outer dewar, but its function as a heat
shield is taken over by a nickelplated copper shroud attached
to the first expansion stage, maintained at 4060Â°K. This
extends down to and surrounds the sample substrate holder,
with two openings cut at 180Â° apart. The entire unit is
rotatable through 180Â°, to align the sample window with any
two opposite ports in the external vacuum housing.
The second expansion stage is terminated with a copper
cold tip. The copper sample window holder is firmly screwed
into this tip, and good contact is assured with an indium
gasket. The circular sample windows (2.6 cm diameter) are
of the same materials as employed in the other optical dewar,
20
Basic design of cryotip assembly used for
optical studies.
Figure 4.
21
and are secured with indium gaskets and a copper retaining
ring. The chromel vs Au 0.02 at.% Fe thermocouple is mounted
on the sample holder. The temperature is varied between 10Â°K
and ambient by an electrical heating wire wrapped around the
second stage expander cold tip. The temperature is measured
and automatically maintained at any preset value with the
CTI thermometer/controller.
The vacuum chambers of all three of the above systems
are pumped by 2 inch oil diffusion pumps, with liquid cold
traps, backed by mechanical forepumps. Dewars employing
liquid He refrigerant attain an ultimate vacuum of approx
_ g
imately 5 x 10 torr. All pressures are measured with
BayertAlpert type ion gauge tubes and Veeco RG31X control
circuits.
Vaporization of nonvolatile materials was accomplished
in vacuum furnaces of identical design attached to each of
the cryogenic systems. These are watercooled brass cylinders
20.3 cm long and 15.2 cm in diameter. Furnaces associated
with the liquid He dewars are pumped by an oil diffusion
pump of minimum two inch diameter intake, backed with a
mechanical forepump and equipped with liquid N2 cold traps,
_ g
attaining pressures below 1 x 10 torr. The furnace mounted
on the Displex cryotip utilized the same pumping system as
the cryotip assembly.
A schematic of the furnace apparatus is illustrated in
Figure 1. Interchangeable, demountable flanges of various
design are inserted into the furnace body. For resistance
22
heating of samples, these flanges are equipped with water
cooled copper electrodes. Tantalum cell holders are securely
bolted with Ta screws to the ends of the electrodes, and
inserted in the holders are Ta cylinders 2.5 cm long, 6.4 mm
O.D., and of varying wall thicknesses, filled with the solid
to be vaporized. The open ends of the cell were sealed
with tightfitting Ta plugs.. The effusion orifice of 1.6 mm
diameter is directed towards the target window. These cells
could be aligned at any angle to the horizontal; the vertical
position is necessary for samples which are melted to produce
sufficient vapor pressure for deposition. Not shown in the
figure is a watercooled copper heat shield placed 1  2 cm
in front of the cell and fitted with a central 2 cm hole to
allow passage of some of the sample vapor to the target.
The cell was heated by passing up to 500 amps at up to 6 volts
through the electrodes. Cell temperatures were measured with
a Leeds and Northrup vanishing filament optical pyrometer
through a flangemounted Pyrex window which was shuttered
to prevent deposition of a film on the window.
Induction heating of samples was performed in the same
furnace body, but equipped with a flange holding 10 turns
of 6.5 mm hollow copper coil, the axis of the helix aligned
towards the target window. A Ta Knudsen cell, 2.2 cm long,
1.4 cm O.D., 3 mm wall, is supported, coaxially with the coil,
on three W rods (1.5 mm diameter and 8 cm long) attached to
a screw mechanism mounted on the furnace flange in place of
the electrodes. In this way the cell position could be
23
adjusted in the coils to provide maximum coupling. TemperaÂ¬
ture measurements were achieved as above, except that the
cell was provided with a blackbody hole, which eliminates
the necessity of emissivity corrections. The watercooled
RF coils were attached to a Lepel 5 kW high frequency inducÂ¬
tion heater. With both methods of heating, the distance from
cell orifice to the trapping surface was approximately 12 cm.
Matrix gases or gas mixtures were usually admitted to
the dewars through the copper inlet shown in the figures.
This was connected to a copper manifold equipped with
fittings to connect to Pyrex sample bulbs. The manifolds
were pumped by a 2 inch oil diffusion pump, equipped with
liquid N2 trap, and backed by a mechanical forepump, which
5
gave pressures less than 1 x 10 torr. Flow rates were adjusted
with an Edwards needle valve, and pressure changes monitored
with a Heise Bourdon tube manometer. Gas mixtures were
produced in a similar vacuum system by standard manometric
techniques. The rare gases, neon and argon, were Aireo
ultrapure grade, and used without further purification except
for passage through a liquid N2 cold trap immediately prior
to deposition.
Electron spin resonance measurements were made with an
Xband Varian V4500 spectrometer system employing superÂ¬
heterodyne detection. A 12 inch electromagnet useful from
0  13 kG provided the static magnetic field, which was
modulated at low (200 Hz) frequency. The output of the
instrument was recorded on a Moseley model 2D2 XY recorder.
24
When signals were weak, a Nicolet model 1072 signal averager,
equipped with SW71A sweep and SD72A analogtodigital
converter plugin units was used to improve the signalto
noise ratio. The magnetic field was measured with either
an Alpha Scientific model AL67 or a Walker Magnemetrics
model G502 NMR gaussmeter in conjunction with a Beckman
6121 counter. The microwave cavity frequency was determined
with a HewlettPackard highQ wavemeter.
O
Absorption spectra were recorded from 7000  2000 A
using a Jarrell Ash 0.5 meter Ebert mount scanning monoÂ¬
chromator. Gratings ruled with 1200 lines/mm and blazed at
O
5000 and 3000 A gave a reciprocal linear dispersion of
O
16 A/mm in first order. Detectors used were the RCA 7200,
O
for the range 3700  2000 A, and either the RCA 1P21 or
O
931A, for the range 7000  3500 A, each operated at 1000 VDC.
The photomultiplier output was processed by a Jarrell Ash
82110 electronic recording system and displayed on a Bristol
model 570 strip chart recorder. Continuum light sources
were a General Electric tungsten ribbonfilament lamp for
the visible and a Sylvania DE 350 deuterium lamp for the UV.
Radiation from these sources was passed through the matrix
and focused onto the spectrometer slit with quartz optics.
The spectra were calibrated with emission lines from a Pen
Ray low pressure Hg arc lamp. A PerkinElmer 621 spectroÂ¬
photometer (purged with dry ^ gas) was used in the IR region
 300 cm â– *â– , with an accuracy of Â± 0.5 cm .
from 4000
25
Photolyzing radiation in two spectral regions was availÂ¬
able from either a high pressure Hg arc lamp or a flowing
H2~He electrodeless discharge lamp. The Hg lamp consists of
a watercooled General Electric type AH6 Hg capillary
lamp operated at 1000 W, the output of which was focused
onto the sample with quartz optics. The radiation from this
lamp consists of the characteristic Hg lines and a strong
base continuum. When this lamp was in use, the dewars were
equipped with quartz optical windows for transmitting the
radiation to the sample.
The flowing H2~He electrodeless discharge lamp is
constructed after the design of David and Braun (5). It
consists of a quartz tube, 15 cm long, fused 4 cm from the
end with a larger diameter quartz tube to form an annular
space 6 cm in length. The annulus has provision for inlet
of the 10% H2 in He gas mixture (Air Products), and the
central tube is connected to a mechanical pump, which can
2
evacuate the entire system to about 3 x 10 torr. This
effectively seals the quartz body against an LiF optical
window by means of a brass fitting equipped with "O" rings.
The LiF window is mounted in the dewar photolysis ports.
The gas flow is adjusted to give a pressure of about 1 torr
with a gas regulator. An 85W Raytheon PGM 10 microwave generaÂ¬
tor, operating at 2450 MHz, was used with a tunable cavity
to excite a discharge in the flowing gas. The emission was
O
characterized by the intense Lyman a line at 1216 A. Color
centers which developed in the LiF due to the high energy
radiation could be removed by annealing the windows at
A O O Â£/NV* 4T.T/*N
26
General Technique
Preparation of the matrix samples was achieved in the
following manner. The dewars, furnace assemblies, and gas
manifolds were readied at least one day before an experiment
was run, and allowed to pump out overnight. If a good vacuum
was maintained, the liquid N2 cold traps associated with the
diffusion pumps were filled; this brought the furnace and
gas manifold assemblies near their ultimate vacuums. The
sample cells were then slowly heated and the samples allowed
to outgas at low temperatures (about 200Â°C below deposition
temperatures) while the dewars were prepared.
While preparation of the Displex cryotip involved only
checking the He and cooling water pressures, and switching
on the device, preparing the liquid He dewars was somewhat
more involved. First the outer, and then the inner dewar
was filled with liquid N2. The inner dewar was constantly
purged with dry N2 gas when not in use to prevent formation
of ice in the narrow channels of the variabletemperature
chamber. Filling this container with liquid N2 served to
precool it, and minimize the quantity of liquid He wasted
for this purpose. After the lower chamber, on which the
sample substrate was mounted, had reached liquid N2 temperaÂ¬
tures, that refrigerant was pumped out by pressurizing the
chamber with N2 and He gas. When the liquid N2 had been reÂ¬
moved and recovered, the dewar was flushed with He gas and
allowed to warm up 1020Â°K. This assured complete removal
of the N2, which could solidify as liquid He was added. It
27
was found to be very important that a positive pressure of
dry gas was applied to both dewar openings, especially when
it was cold. When the dewar had warmed slightly, liquid
He was transferred by a vacuuminsulated tube into the dewar,
which was sealed with a pressure cap when transfer was
completed. These preliminary activities took approximately
one hour to perform. A charge of liquid He lasted approxiÂ¬
mately 48 hours, depending on the rate of flow through
the lower chamber, which was set with a needle valve. The
Displex cryotip also took about one hour to reach operating
temperature, but it could maintain that temperature indefiÂ¬
nitely .
With the deposition surface at a sufficiently low
temperature, the gas manifold was sealed from its pumps
and filled with the matrix gas or gas mixture. To prevent
formation of a solid residue from vaporization on the
surface, the gaseous sample alone was deposited on each side
of the substrate for approximately five minutes. The rate
of gas deposition through the entire run was controlled with
an Edwards needle valve to obtain a flow of about 0.3 mmole/
min. During this time the nonvolatile sample was heated
to its deposition temperature; formation of the metal film
on heat shield and furnace viewing port indicated that
sufficient material was being vaporized. With the initial
deposit on the sample substrate, the gate valve separating
furnace and dewar was opened and furnace vapors were coÂ¬
condensed with the matrix gas. During sample deposition,
28
typical pressures observed in the furnace and dewar were
4 5
1 x 10 and 2 x 10 torr, respectively. Deposition times
varied from 1/2 to 2 hrs, depending on the species being
formed; the sample substrate was rotated 180Â° periodically,
to form an even coating on the surface. When deposition
was completed, the dewar and furnace were isolated, the
gas flow stopped and hightemperature cell allowed to
cool. The matrix samples thus prepared were then observed
spectroscopically.
29
References  Chapter II
1.C.K. Jen, S.N. Foner, E.L. Cochran,
Phys. Rev., 112, 1169 (1958).
2. W. Weltner, Jr. and D. McLeod, Jr.,
45, 3096 (1966) .
3. W.C. Easley and W. Weltner, Jr., J.
197 (1970).
4. W.R.M. Graham, K.I. Dismuke, and W.
Astrophys. J., 204, 301 (1976).
and V.A. Bowers,
J. Chem. Phys.,
Chem. Phys., 52,
Weltner, Jr.,
5.
D. David and W. Braun, Appl. Opt., 7, 2071 (1968).
CHAPTER III
ESR THEORY
Introduction
The interactions of paramagnetic atoms and molecules
with magnetic fields, which gives rise to the electron spin
resonance phenomenon, is discussed in this chapter. Details
of the theory applicable to atoms (or ions) and doublet,
triplet, quartet, and sextet state molecules encountered
in this research will be presented in separate sections.
More extensive treatments of the basic theory presented
here can be found in a number of excellent references (113).
Atoms and the Resonance Condition
The paramagnetic substances with which we are concerned
are those which possess permanent magnetic moments of atomic
or nuclear magnitude. In the absence of an external field
such dipoles are randomly oriented, but application of a
field results in a redistribution over the various orientaÂ¬
tions in such a way that the substance acquires a net magnetÂ¬
ic moment. Such permanent magnetic dipoles occur only when
the atom or nucleus possesses a resultant angular momentum,
and the two are related by
y = yG [i]
where y is the magnetic dipole moment vector, G is the angular
momentum (an integral or halfintegral multiple of h/2ir = "R
30
31
where h is Planck's constant), and y is the magnetogyric
ratio. The motion of these vectors in a magnetic field H
consists of uniform precession about H at the Larmor precession
frequency
[2]
io =yH.
The component of G or y along H remains fixed in magnitude,
so that the energy of the dipole in the field (the Zeeman
energy)
[3]
W = yll
is a constant of the motion.
The magnetogyric ratio which relates the magnetic moment
to the angular momentum according to Eq. [1] is given by
y = g(e/2mc)
[4]
where e and m are the electronic charge and mass, respectively,
and c is the speed of light. The factor g = g is unity for
orbital angular momentum and g = g^ = 2.0023 for spin anguÂ¬
lar momentum. Including this factor with Eq. [1] and defining
the Bohr magneton as 3 = e'R/2mc, we have (along the field
direction)
[5a]
ys = _gspras
[5b]
32
Because the angle of the vector y with respect to the applied
field H is space quantized, only 2G + 1 orientations are
allowed. These allowed projections along the magnetic field
are given by mJÃ±, where mâ€ž is the magnetic quantum number
taking the values
m^, = G, Gâ€”1, G. [6]
This accounts for the appearance in Eqs. [5] of the factors
m for orbital angular momentum and mc for spin angular
JLi ÃD
momentum.
2
If only spin angular momentum arises, as in a ^1/2 atom,
the 2S + 1 energy levels separate in a magnetic field, each
of energy
Em= = 9e6msH, !7i
with equal spacing g^BH. However, the angular momentum does
not generally enter as pure spin, so that the g factor is
an experimental quantity and m an "effective" spin quantum
number, since some orbital angular momentum is usually mixed
into the wavefunction. In orbitally degenerate states
described by the strong (RussellSaunders) coupling scheme,
J = L + S, L+Sl, ..., IL  SI and
EJ = gjSmjH
[8]
33
where
S(S + 1) + J(J + 1)  L(L + 1)
2JTJ + 1)
[9]
is the LandÃ© splitting factor. This reduces to the free
electron value for L = 0.
Taking the simplest case of a free spin, irtj = mg = Â±1/2
and there are only two levels. Transitions between these
levels can be induced by application of magnetic dipole
radiation obtained from a second magnetic field, at right
angles to the fixed field, having the correct frequency to
cause the spin to flip. Thus the resonance condition is
[10]
where H is the static external field and v is the freÂ¬
o
quency of the oscillating magnetic field associated with
the microwave radiation; this frequency is about 9.3 GHz
2 2
for the Xband. Thus for a S or P atom, the ESR spectrum
will consist of one line corresponding to the particular g
value of the atom.
The Hyperfine Splitting Effect
If only one line were observed in the general case, the
ESR technique could offer only a limited amount of informaÂ¬
tion, the g value. However, there are other interactions
to consider which increase the number of spectral lines and
the information that can be obtained. One of the most impor
tant is the nuclear hyperfine interaction.
34
Usually at least one isotope of an element contains a
nucleus having a nonzero magnetic moment. The magnetic
moment of the odd electron can interact with this nuclear
moment and split the single ESR line into hyperfine strucÂ¬
ture. This effect can be pictured as follows. The magnetic
field "felt" by the electron is the sum of the applied
external field and any local fields. One such local field
will be that caused by the moment of the magnetic nucleus;
this is, in turn, governed by the nuclear spin state. It
is then clear that, in the case of nuclear spin I = 1/2,
for example, the local field in which the electron finds
itself will be one of two contributed by the nucleus, since
there are 21+1 nuclear levels. Hence, there will be two
values of the external field which satisfy the resonance
condition, that is,
Hr = (H' 1 f5 = (H' " AMI) UU
where A/2 is the value of the local magnetic field, A
being the hyperfine coupling constant, and H' is the
resonant field for A = 0. One example of this phenomenon is
1 2
the H atom. This is a pure spin system, ^1/2' 1 = 1/2
its ESR spectrum consists of two lines separated by A = 508G,
centered around g = g& = 2.0023, as shown in Figure 5.
A more detailed look at the paramagnetic species with
nonzero nuclear spin in a mangetic field indicates that
there are several interactions at work. One is the interÂ¬
action of the external field with the electrons, which has
35
already been considered. An analogous term results from
the precession of the nuclear magnetic moment in the exterÂ¬
nal field. The nuclear magnetic moment y is related to the
nuclear g factor g by the relation
â€” ^1
gi1 = [12]
where 8N = eh/2M is the nuclear magneton and M is the proton
mass. The third term describes the interaction between the
electrons and nuclei. Thus the Hamiltonian can be written
H = gBH*J + hAl*J  g 3 H*I [13]
where the underscore indicates that the quantities are
operators. Except in very strong fields, the interaction
of the nuclear moment with the external magnetic field (the
nuclear Zeeman term), which is represented by the last term
in Eq. [13], is small, and will be neglected. Also omitted
from this Hamiltonian are even smaller effects, such as the
nuclear electric quadrupole interaction.
Reference to Figures 5 and 6 will indicate the behavior
of the levels as a function of external field strength.
The two limiting cases of very weak and very strong field
are of particular interest.
The Zeeman effect in weak fields is characterized by
an external field splitting which is small compared to the
natural hyperfine splitting; that is, hAI*J > g(3HJ in Eq. [13].
In this case, the orbital electrons and the nuclear magnet
ZEEMAN ENERGY LEVELS OF
AN ELECTRON INTERACTING
WITH A SPIN 1/2 NUCLEUS
Zeeman energy levels of an electron interacting with a spin 1/2 nucleus,
U>
Figure 5.
Figure 6.
5/2.
38
remain strongly coupled. A total angular momentum F = I + J
exists, which orients itself in the external field. F takes
the values I+J, I+Jl, ..., lj. The component
of F along the field direction, m^, has 2F + 1 allowed
values, the integers between F and +F. In the case presented
in Figure 6, with both and A positive, these components
are arranged, in order of decreasing energy, m = F, F  1,
r
..., F. Each individual hyperfine level splits up into
2F + 1 equidistant levels in the weak field, giving (2J + 1) *
(21 + 1) Zeeman levels altogether. Note that in both Figures
5 and 6, the levels are not all degenerate even at zero
field. This effect, produced by the hAI_*J term, is called
the zero field splitting.
In the PaschenBack or strong field region, the splitting
by the external field is large compared to the natural hyperÂ¬
fine splitting. The strong interaction with the external
field decouples I and J, which now precess independently
around H. F is no longer a good quantum number, but there
exist mT and m , the components of J and I along the field
direction. In this case, each Zeeman level of the multiplet
characterized by a fixed m is split into as many Zeeman
hyperfine lines as there are possible values of m^, that
is, (21 + 1). Since there are still (2J + 1) levels for
a given J, there are, exactly as in the weak field, (2J + 1)â€¢
(21 + 1) total energy states. In contrast to the weak field
situation, the levels here form a completely symmetric pattern
around the energy center of gravity of the hyperfine multiplet.
39
This pattern manifests itself in Figure 6. Also recorded in
that Figure are the m values of the different hyperfine
groups. Values of are, in order of decreasing energy,
5/2, 3/2, 1/2, 1/2, 3/2, 5/2 for mT = 0, 1, 2, 3;
u
this order is inverted for the remaining m groups.
The situation in intermediate fields is somewhat more
complicated. The transition between the two limiting cases
takes place in such a way that the magnetic quantum number
m is preserved. In weak field, m = m ; in strong field,
r
m = m^ + rrij. In this region, the Zeeman splitting is of
the order of the zero field hyperfine splitting.
2
For a ^1/2 state' as in Figure 5, the general solution
for the energy levels over all fields is given by the
BreitRabi equation (14). In terms of the quantum numbers
F and m = m_., it is
r
.> _ AW pi .. , AW ,. 4m â€ž2. 1/2 .... .
W(F,m) 2(21 + 1) I H0m  2 ^1+2I+1X+X) [l4al
A
where AW = y(2l = 1)
[14b]
and X =
(Uj/J + Pj/I) Hq
AW
[14c]
The plus sign in Eq. [14a] applies for F = I + 1/2 and
m = +(I + 1/2), ..., (I  1/2) and the minus sign for
F = I  1/2 when m = (I  1/2), ..., (I  1/2). The zero
field hyperfine splitting is AW. The limiting cases of weak
2 2
and strong fields correspond to X <<1 and x"â€™ >>1, respectively.
40
For the general case of intermediate fields, the energy
values of the Zeeman levels can be derived from the following
key equation given by Goudsmit and Bacher (15):
'*mT + 1, m  1 I J 11 + ml + 1)(J  mJ + 1)1
L u
+*m m 1
L J
â– XmT  1, mT + 1 1 J 11  mI + 11 (J + mJ + 1)1 = Â°<
where A is the hyperfine coupling constant, gT and g' = 5=
are the electronic and reduced nuclear gfactors, respectively,
and the x^ are coefficients in the expansion of the wave
function; the other symbols have their usual meaning. Here,
AW is the energy of the level with respect to the center of
gravity of the hyperfine multiplet. This relation yields
one system of homogeneous equations in for each value
of m = m + m . The resulting secular equations are solved
for the energies of the Zeeman levels at any field. Such
a calculation was performed to produce Figure 6.
With a multitude of levels available, it is necessary
to explain the observed ESR spectra in terms of the selection
rules. Since transition between Zeeman levels involve changes
in magnetic moments, we must consider magnetic dipole transiÂ¬
tions and the selection rules pertaining to them. In the
pure spin system with 1=0, the single line observed
41
corresponds to the m = 1/2 m = 1/2 transition. In
s s
general, the criterion is that Am^ = Â±1, corresponding to
a change in spin angular momentum of Â±Ti. Since a photon has
an intrinsic angular momentum equal to TÃ, only one spin
(nuclear or electronic) can flip on absorption of the photon
in order to conserve angular momentum. With the fields
and frequencies ordinarily used in ESR work, the transition
usually observed is limited to the selection rules Am = Â±1,
Arrij = 0; the opposite is true in NMR work. It is, however,
possible to observe the Am = 0, Am = Â±1 NMR transition
J i.
with ESR apparatus, if the zero field splitting (proporÂ¬
tional to the hyperfine coupling constant A) is large
enough relative to the microwave frequency. If this does
not occur, only the ESR lines will be observed, resulting
in a multiplet of 21+1 hyperfine lines for each fine
structure (Am = Â±1) transition. Thus the H atom spectrum
u
(Figure 5) will consist of two lines, while that of Mn+
7
( S^, I = 5/2) will contain 36 individual lines, if all
are resolved.
These interactions of the electron with a nucleus are
related to fundamental atomic parameters, which can be
deduced from the observed spectrum. They are most simply
categorized as isotropic and anisotropic interactions.
The anisotropic interaction has its roots in the classi
cal dipolar coupling between two magnetic moments. This
interaction is given by
42
E
3 (y â€¢ r) (y â€¢ r)
e N
5 â€œ
r
[16]
where r is the raduis vector from the moment y to y__, and
â– e N
r is the distance between them. The quantum mechanical
version is obtained by substitution of the operators, gBS
and g^BN^, for the moments yg and yN, respectively, yielding
H,.
dip
"gBgNBN
I*(LS)
3
r
3(Iâ€¢r)(Sr)
5
[17]
For a hydrogenic atom with nonzero orbital angular momenÂ¬
tum (that is, p, d, ... electrons), this yields
aj = ge6giBN
L(L + 1) /_1_\
J(J + 1) \r3/ '
[18]
a more exact relativistic treatment also adds a multiplicaÂ¬
tive factor [F(F + 1)  1(1 +1)  J( J + 1)]. For s elecÂ¬
trons, a similar dipolar term yields
a
(3cos^0  1)
3
r
[19]
where 0 is the angle between the magnetic field direction
and a line joining the two dipoles. However, the electron
43
is not localized and the angular terra must be averaged over
the electron probability distribution function. For an
s orbital, all angles are equally probable due to the
2
spherical symmetry, and the average of cos 0 over all 0
causes the function to vanish. Thus the classical dipolar
term cannot be responsible for the hyperfine structure of
2
the ^1/2 hydrogen atom.
The actual interaction in the selectron case is
described by the Fermi contact term (16). This isotropic
interaction represents the energy of the nuclear moment in
the magnetic field produced at the nucleus by electric curÂ¬
rents associated with the spinning electron. Since only
s orbitals have finite electron density at the nucleus,
this interaction only occurs with s electrons. This yields
the isotropic hyperfine coupling constant
as = 3 ge3gi3N^(0) I2 t20]
where the last term represents the electron density at the
nucleus. This term has no classical analog. The a value
^ s
is proportional to the magnetic field at the nucleus, which
5
can be on the order of 10 G. Thus unpaired s electrons
can give very large hyperfine splittings.
2
Z Molecules
The Spin Hamiltonian
The discussion to this point has involved only those
terms in the Hamiltonian of the free atom or ion which are
44
directly affected by the magnetic field. However, it will
be useful to begin the discussion of paramagnetic molecules
by consideration of the full Hamiltonian, which can be
written, in general,
H = H + H + H + H_â€ž + H
â€” â€”E â€”LS â€”SI â€”SH â€”IH
[21]
Each of the terms can be described as follows. The term
H expresses the total kinetic energy of the electrons,
â€”
the coulombic attraction between the electrons and nuclei,
and the repulsions between the electrons:
H Â£Pi
ÃœE i 2m
E
A,i
V
r.
i
+
i>l r. .
iD
[22
where p. is the momentum of the ifc^ electron, r. is the
i
distance from electron i to nucleus A of atomic number Z,,
A
and r. . is the distance between electrons i and j. The
BornOppenheimer approximation (17) has already been invoked
to separate out the nuclear motions and nuclearnuclear
repulsions. This term yields the unperturbed electronic
levels before spin and orbital angular momentum are con
5 1
sidered. Eigenvalues of this term are on the order of 10 cm
The energy due to the spinorbit coupling interaction
is usually expressed in the form
â€”LS =
[23]
45
where L and S are the orbital and spin angular momentum
operators and A is the molecular spinorbit coupling
constant. The magnitude of this interaction is of the order
of 102  103 cm"1.
The hyperfine interaction arising from the electronic
angular momentum and magnetic moment interacting with any
nuclear magnetic moment present in the molecule may be
expressed as
â€”SI = gAge8
L1
~T
r
+
3(S*r)(rI
5
r
SI
r
3tt6 (r) S* I
3
[24]
This term corresponds to the sum of the isotropic and
anisotropic hyperfine interactions discussed for atoms
above. The Dirac 6function indicates the isotropic
part which has a nonzero value only at the nucleus. The
2 1
hyperfine interaction has a magnitude of about 10 cm
The electronic Zeeman term, H , is primarily responsible
â€”bH
for paramagnetism. It accounts for the interaction of the
spin and orbital angular momenta of the electrons with
the external magnetic field, according to
â€”SH = 6(E + Sell'Â»
These energies are on the order of 1 cm
[25]
46
The interaction of the nuclear moments with the
external field, that is, the nuclear Zeeman effect, is
given by
3 1
and is usually too small (10 cm ) to be significant.
It is evident that this total Hamiltonian (which still
excludes higherorder terms and field effects which could
be observed in crystals) is very difficult to use in
calculations. However, experimental spin resonance data
obtained from the study of the lowestlying levels can be
described by use of a spin Hamiltonian in a fairly simple
way which does not require detailed knowledge of all
the interactions. These levels are generally separated
by a few cm ^ (by the magnetic field), and all other
electronic states lie considerably higher. The behavior
of this group of levels in the spin system can be described
by such a spin Hamiltonian, and the splittings, which may
usually be calculated by first and second order perturbaÂ¬
tion theory, are precisely the same as if one ignored the
orbital angular momentum and replaced its effect by an
anisotropic coupling between the spin and the external
magnetic field. This was first shown by Abragam and Pryce (18).
Just as the g factor becomes anisotropic and not necessarily
equal to g= 2.0023, the S cannot represent a true spin
but is actually an "effective" spin. This is defined, by
convention, to be a value such that the observed number
47
of levels equals (2S + 1), just as in a real spin multiplet.
Thus we can relate all the magnetic properties of a system
to this effective spin by the spin Hamiltonian, since it
combines all of the terms in the general Hamiltonian which
are sensitive to spin. Nuclear spins can be treated in
the same manner, and the spin Hamiltonian corresponding to
Eq. [21] can be written (neglecting the nuclear Zeeman
term)
H . = 3H gS + IAS [27]
where the double underscore indicates a tensor quantity.
The g Tensor
As alluded to previously, the anisotropy of the gtensor
arises from the orbital angular momentum of the electron
through spinorbit coupling. Even in the case of E states,
which have zero orbital angular momentum, the interaction
of a presumably pure spin ground state with certain
excited states can admix a small amount of orbital angular
momentum into the ground state, and change the values of
the components of g. This interaction is usually inversely
proportional to the energy separation between the states.
The spinorbit interaction can be described as (5)
Ht_ = AL*S = A [L S + L S + LS]
â€”LS â€”xâ€”x â€”yâ€”y â€”zâ€”z *
[28]
48
This terra is added to the Zeeman term in the Hamiltonian,
thus
H = 3HÂ» (L + geS) + AL*S . [29]
For an orbitally nondegenerate ground state represented
by  G, the first order energy is given by the diagonal
matrix element
WG1} = ^'MS I I GMS> + I 6Hz + ASzlMsXGliizG>, [30]
where the first term is the spinonly electronic Zeeman
effect. Because the ground state is orbitally nondegenerate,
<Â£  Lz  (Â£> = 0, and the second term vanishes. The second
order correction to each element in the Hamiltonian matrix
is given by
y
I
n

[31]
W(0)  Nri0)
n G
where the prime designates summation over all states except
the ground state. Since <(31 n^> = 0, the matrix elements of
ge8HÂ«S will vanish. The operator matrix can then be
expanded to
[] []
(0)
n
n
[32]
49
and the quantity
E
'
w<Â°>  w!0)
n G
th
tensor is given by
Axx
Axy
Axz
Axy
Ayy
Ayz
Axz
Ayz
Azz
slement
of the
second
L I nXnl
Â£ilG>
= A [33]
[34]
" w(0)  w<Â°>
n G
where i and j are any of the cartesian coordinates. This
simplification yields
M M ' = (MSB2HA*H + 2A3HAS + A2SASms' ) . [35]
S' S
The first operator represents a constant contribution to
the paramagnetism and need not be considered further.
The other terms represent operators which act only on
spin variables. When combined with the Zeeman term of
Eq. [30], the result is the spin Hamiltonian
H . = 8H(g 1 + 2AA) *S + A2SA*S
â€”spin e = â€” â€” â€” â€”
= 6H*g*S + SyDS
[36]
50
where
2 = gcl + 2 AA
[37]
and
D =
[38]
with 1 being the unit tensor.
The S*D*S term is operative only in systems with S > 1
and will be considered later. The other term in Eq. [36]
is the spin Hamiltonian in the absence of hyperfine interÂ¬
action. It is evident that the anistropy of the g tensor
arises from the spinorbit interaction due to the orbital
angular momentum of the electron. This may be expanded
to show the gtensor as
3S*g_*H
6[S S S ]
xyz
2 2 2
â€™h
xx xy xz
X
2 2 2
H
yx yy yz
y
2 2 2
H
[_ zx zy^zz J
z _
[39]
where S , S , and S are the components of the effective spin
x y z
along the axes. Strictly, g is a 3x3 matrix and is referred
to as a symmetrical tensor of the second order (the symmetry
implies that the unpaired electrons are in a field of
central symmetry). The double subscripts on the gtensor
elements may be interpreted as follows: g is the contriÂ¬
bution to g along the xaxis when the magnetic field is
51
applied along the yaxis. These axes are not necessarily
the principle directions of the gtensor, but a suitable
rotation of the axes will diagonalize it; then the diagonal
components are the principle directions of the gtensor
with respect to the molecule. It is noteworthy that, if
the molecule has axes of symmetry, they must coincide with
the principle axes of g; if it has symmetry planes, they
are perpendicular to the principle g axes.
Three cases of interest, with regard to molecular
symmetry can be outlined. If the system is truely a spin
only system, g will be isotropic and the diagonal elements
equal to ge If it is isotropic but contaminated with
orbital momentum, the principle components will be equal
but unequal to g . In the former case
H
â€”spin
8[S S S ]
â€” xâ€”y â€” z
3
o
o
'H
"e
X
o
Q
o
H
e
y
0 0 g
H
^e
z
[40]
= g 8 [H S +
ye xx
H S + H S
yy zz
For a system containing an nfold axis of
two axes are equivalent. The unique axis
z and the value of g for H j z is called
the value is g^. Thus
H . = 8 (g, H S + g i H S + g >. H S
spin JJ_ xx ^ J_ yy ^  zz
symmetry (n > 3),
is usually designated
g  . For H J_ z,
). [41]
52
Finally, for systems where there are no equivalent axes
(orthorhombic symmetry), g ^ g ^ g and
H . = Big H S +
â€”spin ^xx xx
g H S + g H S ). [42]
yy yy zz z~z
The A Tensor
It has already been noted that the hyperfine interÂ¬
action is composed of isotropic and anisotropic parts.
While the isotropic coupling is that which is observed
in liquids, the anisotropy due to dipoledipole interacÂ¬
tions can be observed in fixed systems, such as molecules
in a rigid matrix. If Eq. [24] is expanded (and the L* 1^
term is dropped since this is a E state), the interaction
can be seen to assume the form of a tensor
H . = (gBgâ€žAJ [S s s ]
dip ^ N xyz
" 2 _ 2
m 
\r
o>
â€” â€”
I
â€”x

2 2
(a:>
I
y
(E
23z2\
r5 /J
i
â€”z
= hS*T*I
When the isotropic part is added, the spin Hamiltonian
with hyperfine becomes
H . = 6S*g*H + hS*AI
â€”spin â€” â€” â€” â€” = â€”
[43]
[44]
53
where
A = Aq1 + T
[45]
with Aq being the isotropic hyperfine coupling constant
and 1 the unit tensor. Thus the element
A.. = A. + T..
lj iso lj
[46]
and, as with the gtensor, certain cases can be selected due
to the molecular symmetry. An isotropic system has A
xx
A = A = A. . Systems with axial symmetry are
yy zz iso 2 1
A i = A. + T and A
ISO XX ZZ
characterized by A = A
2 xx yy
Am â€” A. + T
iso zz
Randomly Oriented Molecules
Having discussed some basic theory of ESR in mole
cÃºles, one must consider how these effects manifest themÂ¬
selves in the spectrum. Anisotropy will appear in the
spectrum of a rigidlyheld molecule, but there is a
difference between the effects observed in crystals and
in matrices. In a single crystal, with a paramagnetic
ion or defect site, for example, the sample can be aligned
to the external field and spectra recorded at various angles
of the molecular axis to the field. In the matrix, the
samples ordinarily have a random orientation with respect
to the field, and the observed absorption will have contriÂ¬
butions from molecules at all angles. This was first
considered by Bleaney (19, 20), and later by others (2126).
54
Solving the spin Hamiltonian, Eq. [42],in the ortho
rombic case and assuming the g tensor to be diagonal, the
energy of the levels will be given by
E
= 6SHH(g;L
2 2 2
sin 9cos
2 2 2
sxn 0sin
2 2a.1/2
g^ cos 0)
= 6gHSHH
[47]
where S is the component of the spin vector S along H,
gH is the g value in the direction of H, 0 is the angle
between the molecular z axis and the field direction, and
is the angle from the x axis to the projection of the
field vector in the xy plane. For axial symmetry gtI =
2 2 2 2 1/2
(g^ sin 0 t g 11 cos 0) , and the energy of the levels
is given by
E = BSHH(g^2sin20
2 2..
cos 0 )
[48]
Thus the splittings between energy levels, and therefore
transitions between them, are angularly dependent.
Consider first the case of axial symmetry. As a
measure of orientation, it is convenient to use the solid
angle subtended by a bounded area A on the surface of a
sphere of radius r. The solid angle is the ratio of the
surface area A to the total area of the sphere, that is,
2
ft = A/4nr . If all orientations of the molecular axis
are equally probable, the number of axes in a unit solid
55
angle is equal for all regions of the sphere. If the
sphere is in a magnetic field, the orientation of the axes
will be measured by their angle 0 relative to the field.
Taking a circular element of area for which the field
axis is the z direction, the area of the element is
27r(r sin0)rd0, and the solid angle dft it subtends is
=1/2 sin0d0
[49]
and if there are NQ molecules, the fraction in an angular
increment d0 is
[50]
Assuming the transition probability is independent of
orientation, which is approximately the case, the absorption
intensity as a function of angle is proportional to the
number of molecules lying between 0 and 0 + d0.
Since g is a function of 0 for a fixed frequency v,
the resonant magnetic field is
H
h
and from this
2
2
sin 0 =
g
2
[52]
56
where gQ = (
Therefore
sin0d6

[Vol
2 2
H
 gx j
.
[53]
The intensity of absorption in a range of magnetic field
dH is proportional to
dN i = i dN i ,_d0
dH1 'd0'â€˜dH
[54]
where the two factors on the right are obtained from Eqs, [50]
and [53], respectively. From the above equations
H = hv/g i. B = g0H0/gi at 0 = 0Â° [55a]
H = hv/g^B = gQH0/g at 0 = 90Â°.
[55b]
At these two extremes, the absorption intensity varies from
dN
dH
Nog
2goHo(gÂ¡ 2gi
at 0 = O'
[56]
to
dN
dH
at 0 = 90Â°. [57]
57
Tf plotted against magnetic field, this absorption (after
considering natural linewidth) takes the appearance of
Figure 7a, for g j j >gâ€¢ Since the first derivative is usually
observed in ESR work, the spectrum would appear as in
Figure 7b. If the g tensor is not too anisotropic, g j and
g^ can be readily determined as indicated. In general,
the perpendicular component can easily be distinguished
in such a "powder pattern;" the weaker parallel peak is
often more difficult to detect.
If there is also a hyperfine interaction in the randomly
oriented molecules, the pattern of Figure 7b will be
split into (21 + 1) patterns, if one nucleus of spin I is
involved. Such a spectrum for I = 1/2 is shown in Figure 7c,
where, because g 11 % g^ and Aj^ < A , the lines for = +1/2
and 1/2 point in opposite directions. If instead, g^ were
shifted upfield, relative to g 11 , and Aj^ % A11 , the specÂ¬
trum would appear as two patterns similar to Figure 7b,
and separated by the hyperfine splitting.
The spin Hamiltonian for an axially symmetric molecule
is similar to that for an atom, but now incorporating
parallel and perpendicular components of g; and A:
H . = B tg II H S + g i (H S + H S ) ]
â€”spin II zâ€” z xâ€”x yâ€”y
[58]
SI + A i (S I + S I ) .
zz l'xx yy
+ A
I
5,1
U1
oo
Figure 7. (a) Absorption and (b) first derivation lineshapes of randomly oriented
molecules with axial symmetry and gi <5 11 ; (c) first derivative lineshape
of randomly oriented, axially symmetric molecules (g_Â¡_=
hyperfine interaction with a spin 1/2 nucleus (A. < An ).''
59
This omits the small nuclear Zeeman term and assumes the
symmetry axis is the z axis. Equation [51] can be rewritten
to include the nuclear hyperfine effect (8) to first
order as
hv
g6 " g6 ml
[59]
where
= g
2 2 Q
cos 0
2 â€¢ 2fl
sm 0
[60]
and
2 2 2 2 2n 2 2 . 2n
Kg = A  gj cos 0 + Aj^ g^ sin 0.
[61]
The intensity of absorption [dN/dH can again be derived
and is found to be
2cos 0
2
g
* goHo
2g
[62]
+
2 2
gl A1
Here sin0d0 cannot be solved explicitly so that dN/dH cannot
be written as a function of only magnetic parameters.
60
Equations [59] and [62] must be solved for a series of
values 0 to obtain resonant fields and intensities as a
function of orientation. However,
H = (g0H0/g * â€œ (miA //^g ^ at 0 = 0Â° [63a]
H = (9qHq/gj^) = (mjAj^/Bg^) at 0 = 90Â° [63b]
and again  dN/dH  *â– Thus a superposition of the
typical powder pattern results, with the relative phase
of the lines determined by the magnitudes of the magnetic
parameters, as discussed above. In Figure 7c, the lines
do not overlap and analysis is simple, but this is not
always the case. In general, the best approach is to
solve the given equations by computer for a trial set of
g and A values, and match the calculated spectrum to the
observed.
A similar treatment can be applied to molecules of
orthorombic symmetry and instead of the two turning points
at g11 and gj, there will be three corresponding to g^,
g^, and g^. Such a spectrum is considered, including
hyperfine interaction with a spin 1/2 nucleus, by Atkins
and Symons (11) and Wertz and Bolton (5).
As mentioned in Chapter I, molecules trapped in
rare gas matrices do not always assume random orientations.
During condensation of a beam of reactive molecules in
solid neon or argon matrices, some preferential orientations
61
of the molecules relative to the flat sapphire rod may occur,
and in some cases the alignment can be extreme. This non
random orientation can easily be detected by turning the
matrix in the magnetic field; a change in the ESR spectrum
indicates some degree of preferential orientation. The
degree of orientation appears to depend upon the size and
shape of the molecule, the properties of the matrix, and
other factors which are not completely understood (27).
Two examples are the molecules CufNO^^ (28) and BO (29) .
The latter case shows very strong orientation such that
with the magnetic field perpendicular to the rod surface,
the parallel lines were strong and the perpendicular,
weak. With the rod parallel to the field, the perpendicuÂ¬
lar lines became very strong and the parallel components
disappeared entirely. This indicated that the BO molecules
were trapped with their molecular axes normal to the plane
of the condensing surface. This orientational behavior is
analagous to that usually observed in singlecrystal work.
Molecular Parameters and the Observed Spectrum
Having discussed the nature of the interactions appearing
in the spin Hamiltonian and the form of the observed spectrum,
it is time to consider the relationships between the spectral
features and the paramagnetic species themselves. This will
begin with the exact solution to the spin Hamiltonian in
axial symmetry, and presentation of the secondorder solutions
which are usually adequate, and conclude with the molecular
information revealed through g and A components.
62
A thorough discussion of the spin Hamiltonian
H â€¢ = g ti BH S + g i B (H S + H S )
â€”spin J  zâ€”z xâ€”x yâ€”y
+ A
+ A,(I S + I S ) [64]
J_ â€”xâ€”x â€”yâ€”y
has been given by several authors (8, 30, 31). Considering
the Zeeman term first, a transformation of axes is perÂ¬
formed to generate a new coordinate system x', y', and z',
with z' parallel to H. If the direction of H is taken
as the polar axis and 0 is the angle between z and H, then
y can be arbitrarily chosen to be perpendicular to H and
hence y = y'. Thus only x and z need to be transformed.
With H = HsinB, H = Hcos0, and H =0
z
y
X
[65]
Choosing the direction cosines Â£ = gn cos0/g and Â£ =
z
x
gisin0/g, with g defined by Eq. [60], then
H = gBUS + Â£ S ] H
â€” ^ zâ€”z xâ€”x
[66]
and the Zeeman term becomes
H = gBS'H
â€” â€”z
[67]
63
where
S ' = Ãœ S + i S
â€”Z Zâ€”Z Xâ€”X
[68a]
s ' = a s + i s
â€”x xâ€”z zâ€”x
[68b]
S ' = S
y y
[68c]
For the hyperfine terms
H, = A,. S I + A i (S I +SI),
â€”hf  â€”zâ€”z â€”xâ€”x â€”yâ€”y
[69]
and I is rotated by
Â£ni '
zâ€”z
Â£ni'
xâ€”x
[70a]
I =
ani ' + Â£ni '
x xâ€”z
zâ€”x
[70b]
I = I '
y y
[70c]
where the Ã.1? are the direction cosines for the nuclear
i
coordinate system relative to the electronic coordinates.
By inverting Eq.[68] to obtain the analogs of Eq. [70],
and substituting into the Hamiltonian of Eq. [69], the
Hamiltonian of Eq.[64] is transformed into
64
H = g6hSz ' + KI 'S ' +
A,, A
K
1\
I 'S '
â€”x x
/VA,,2!
gll gj]
K
g
sin6cos0I 'S ' + Ail 'S ' [71]
â€”z â€”x y â€”y
with the definitions Â¿z = A g j cos0/Kg, = Aj^gj^sin0/Kg,
22 222 222
and Kg = A 9 cos 6 + A_[_ g__ S'*'n Dropping the primes
and using the ladder operators S+ = S + iS and S = S iS
â€” â€”x â€”y â€” x â€”y i
this can be rewritten in the final form
H
spin
gpHS + KS I +
^ â€”z â€”zâ€”z
Al2'All2.gllgi
K
(Â§!+Â§I, i'
cos0sin0 ' 2 â€”z
+
Aâ€žA_i_+
4 K
(S+I+ + S I)
A"a1 + ^
4K
(S+I + S"l+)
[72]
This Hamiltonian matrix operates on the spin kets M ,Mj),
and can be solved for the energies at any angle. An
example of the use of this exact solution will be presented
later.
The exact solution is difficult to solve at all angles
except 0=0Â°, but elimination of some of the offdiagonal
elements (those not immediately adjacent to the diagonal)
results in some simplification and is usually adequate. The
solution is then correct to second order, and can be used
when g8H>>Ai and Ai, as is often the case. The general
65
secondorder solution is given by Rollmann and Chan (32)
and by Bleaney (20) as
AE (M,m)
gÂ£H + Km +
+ K
K
2,
[1(1 + 1)
2
m
]
+
4G
(2M  1)m
[73]
where K is A11 and Aj^ at 0 = 0Â° and 90Â°, respectively, and
G = g3H/2. Also, M is the electron spin quantum number of
the lower level in the transition, and m is the nuclear spin
quantum number. Note that the first two terms on the right
result from the diagonal matrix elements and yield equiÂ¬
distant hyperfine lines; this is the first order solution
given in Eq. [59]. The last two terms cause increasing
spacing of the hyperfine lines at higher field which is
referred to as a "secondorder effect." This solution can
routinely be applied because the hyperfine energy is
usually not comparable to the Zeeman energy.
The g tensor has appeared in the derivation of the
spin Hamiltonian, and it is seen, from the above equations,
how the values of the principle axis components can be
determined from the ESR spectrum. Now we shall consider
its relationship to a molecular wavefunction in the linear
combination of atomic orbitals (LCAOMO) approximation.
The usual form applied, Eq. [37], is a result of the
secondorder perturbation treatment, yielding the first
[74]
order corrections to the g components
^(olkilnXnkL. 0>
gij = ge6ij " 2
n
n
where the primed summation is over all excited states n
which can couple to the ground state 0 and is the energy
of that state above the ground state. The Kronecker delta is 6^ j .
Because the correction is caused by the spinorbit interÂ¬
action, only certain states can couple with the ground state.
Specifically, these are the states such that (33)
Fn 0 rLiÂ° r0 " Alg' [75]
that is, the direct product of the irreducible representations
of the ground and excited state with the representation of
the angular momentum operator (which transforms as rotations)
must include the totally symmetric representation of the
symmetry group. A specific example, which will be encountered
later, is that a E state can only mix with a II state. Usually
there will only be one such state of energy low enough
to make the term significant. The spinorbit coupling
constant Â£ can be assumed positive or negative, depending
on whether the excited state involves excitation of an
electron or a "hole," respectively. Both this constant
and the orbital angular momentum operator L can be written
as sums of atomic values:
67
EÂ£.
, k k
k
[76a]
L.
i
[76b]
where k indicates a particular atom in the molecule. Actually,
3
decreases rapidly for large r^ (ar ) so that is
essentially zero except near atom k, where it may be assumed
to have a fixed atomic value A, . Thus Eq. [74], for the
perpendicular component of a E state diatomic molecule,
becomes
9i = ge   L <Â£*jjn> . [77]
n k ,k'
Here, E and II are the LCAO wavefunctions
= Â£a^x (i) [78a]
i
^ = Eb . x(j) [78b]
j J
where x(i) aad X(j) are A.O.'s in the ground and excited
states, respectively. Then the second matrix element in
Eq. [77] will reduce, for a diatomic, to sums of terms
involving integrals of the type
68
kl\x
lx(j)k)
(atom k only)
[79a]
X
1 x(j)k. )
(x(i)kl\.
(both atoms)
[79b]
1 X(j)k>
(both atoms)
[79c]
1 x(j)k,>
(both atoms)
[79d]
The first matrix element in Eq.[77] can be simplified if
Ak is assumed constant near k and zero elsewhere. Then
it becomes similar to the integrals of Eq. [79a].
The integrals in Eqs. [79c, d] require that the origin
of the operator be moved from atom k to atom k' . This
introduces a linear momentum term according to
Zk = Zk' +** 1rI\
[80]
where R is the interatomic distance. Fortunately, the
elements involving are usually zero or small, so that
the term can be neglected and the integrals in Eqs. [79]
are then all of similar form.
Eq. [79a, b] involve the application of the angular
momentum operator to the atomic functions involved. The
nonzero elements have been tabulated (34) and are given
below:
69
(plp) = (p l p ) = (plp) = i
' x1 y1 z' N y z' x' xirz' x' y'
[81a]
(d L Id 2 2) 0.
' xy1 z1 xy ' = 2i
[81b]
(d  L  d ) = (d 2 I L I d ) =
x'XZ XlyZ/ ' Z , X1 yz'
[81c]
(d IL Id ) = (d  L Id ) = i
\ Xy1 y1 yz' \ yz1 z1 zx'
[ 81 d ]
(d IL Id > = (d 2 2 IL Id ) = (d 2 2 IL Id )= i
' zx1 x1 xy' ' xy 1 x1 yz7 \ xy 1 y1 xz'
y 1 x1 yz
Y 1 Y
[ 81e ]
(i ILqIi > =
[ 81 f ]
Overlap integrals which appear because the A.O.'s are
centered on different atoms have also been tabulated (35, 36).
Detailed discussions of this approach can be found in
Stone (37) and Atkins and Jamieson (38).
Secondorder corrections to the gfactor have been
determined by Tippins (39), utilizing thirdorder perturbaÂ¬
tion theory. This degree of the theory must be used to
calculate corrections to g11 , and the result, analogous
to Eg. [74] , is
= 9e  ^
Y (nÂ£L+10)
2
n
E E^
n 0
g
[82]
70
Thus it can be seen that g11 is always very close to or
less than g , since the correction factor is small and
squared. On the other hand, can be greater or less
than ge, and the difference can be quite significant.
Thus, if wavefunctions are available, or can be
constructed, the values of Ag = g  gg can be calculated
and compared to those determined experimentally. Examples
of this approach can be found in references (3, 5, 11, 40),
and in the discussion of the MnO molecule to follow.
Alternately, one can approximate the energy separation of
the lowest interacting level from the ground state.
The hyperfine coupling constant has been shown to
consist of isotropic and anisotropic parts, and its tensor
nature has been discussed. The spin Hamiltonian, with
hyperfine interaction, can be written as
H . = (?H *g *S + aL*I + bI*S + cl S [83]
â€”spin ^ â€” â€” â€” â€” â€” â€”zâ€”z
where
[84a]
[84b]
[84c]
71
where the angular brackets are, as usual, quantum mechanical
averages. This definition has been given by Frosch and
Foley (41) . Neglecting the small L*I^ term, we can compare
Eq.[83] with Eq.[58] and identify the observed splittings
as
An = b + c
[85a]
[85b]
The isotropic part can be written in terms of the parallel
and perpendicular components as
A ,
+ 2A
A.
ISO
1 =
b +1
8tt
3
ge3gN6N
iMO)
[86]
The anisotropic or dipolar component is given by
A, â€¢
dip
c o ,c /3cos20l\
3 = geSgNEN ^3 )
[87]
Thus the observed spectrum is related to the fundamental
2 2 3
quantities  ip (0)  and (3cos 0l/2r ) for interaction with
that nucleus. If the L*I term is included, the values become
b + c  Ag11 a
[88a]
b +
a
[88b]
72
where the Ag^ =  ge have already been discussed. These
small corrections can be approximated from the observed
Ag^ value and the value (l/r ) for the particular nucleus.
The dipolar coupling constant can be considered
further. The dipolar part of the Hamiltonian (Eq.[17]) can
be written as
9e6gNeN(l_cos_ezl) .J
r
[89a]
and the energies of the levels m ,M ) are given by
O L
E = Wn MlMs,
'3cos 01
[89b]
For an electron in an orbital centered on the nucleus in
question, the anisotropic hyperfine coupling follows
Eq. [89], but has an additional term to represent the average
direction of the electron spin vector in the orbital.
The hyperfine splitting is the separation between adjacent
levels Jms,Mj.) and M ,, and equals
Adip ^e^N^N
'3cos 01
2r~
^>(3cos2al )
[90]
where a is the angle between r and the principal axis of
the orbital, and 0 is the angle between the latter and the
direction of the nuclear magnetic moment vector. The
73
2
value of (3cos ctl) can be evaluated for the atomic orbital
functions; the values are 4/5 for any p orbital and 4/7, 2/7,
and 4/7 for the d ?, d , and do ? orbitals,
zz xz,yz' x^yz, xy
respectively. These quantities are used to give the principle
value of A,. for an orbital, that is A . Atomic values of
dip zz
the quantities A. and A,. have been evaluated for many
^ iso dip J
atoms, and a useful table can be found in Ayscough (1) or
Goodman and Raynor (42).
Equation [90] assumes that the nuclear moment vector
jjj is aligned with the external field, that is, the applied
field is much stronger than the field at the nucleus due
to the electron. This strongfield approximation is actually
valid only when (a) the applied field is large, (b) the
anisotropic coupling is small (and hence the field at the
nucleus small), and (c) the isotropic coupling is large
(since the field caused by the electron reinforces the
applied field). It is found that the field at the nucleus
due to the electron is much larger than the field at the
electron due to the nucleus. If the strong field approxiÂ¬
mation is not valid, the dipolar interaction will vary as
2 1/2
(3cos 0+1) . However, the numerical values of the anisoÂ¬
tropic coupling at the turning points will be the same, and
only the signs will differ. In a crystal, the difference
will be discernable, but for the powder patterns obtained in
matrices, the strongfield analysis is sufficient. A more
thorough discussion can be found in references (5, 42).
74
The calculated atomic A. and A,. values are often
iso dip
used with experimentally determined molecular values to
derive coefficients or spin densities on a particular atom
in a molecule. Using an LCAOMO wavefunction described as
ii> = Ea. y . , the values of A. and A.. at a particular
r iAi iso dip ^
nucleus x can be written
= ge3gN3N ('H <3cos2 V2r^) ^> [91a]
iso
8 TT
3
i(i(0)
X
[91b]
Since these integrals are expected to be small except near
atom x
a.(p ,d )2 A*
i ^x' x dip
(atom)
[92a]
â€žx . . 2,x , , .
A. = a.(s ) A. (atom)
iso i x iso
[92b]
where the a^ are the coefficients. Although this implies
that atomic properties remain unchanged in the molecule, which
is unlikely, it is quite useful in comparing trends to model
wavefunctions. Examples of its utility will be given later.
A more quantitative approach to calculating A tensor components,
75
based on the intermediate neglect of differential overlap
(INDO) molecular orbital approximation, has been developed
(4346).
3
E Molecules
A theorem due to Kramers states that, for all systems
with an odd number of electrons, at least a twofold
degeneracy will exist which can only be removed by applicaÂ¬
tion of a magnetic field. This would apply to cases with
S = 1/2, 3/2, 5/2, . ..; the first has been considered in
detail.
In the triplet case, S = 1, and two noninteracting
electrons can be described by four configurations:
a(l)a(2), a(l)3(2), B(l)a(2), and 3(1) 3(2) . In a molecule
of finite size, interactions will occur and the configurations
can be combined into states which are symmetric or antiÂ¬
symmetric to electron interchange. These states are
ot(l) a(2) [93a]
(1A/2) [
[93b]
3(1)3(2) [93c]
The multiplicity of the symmetric states (on the left) is
(2S + 1) = 3; this is a triplet state. Because of the Pauli
76
principle, this state may exist only if the two electrons
occupy different spatial orbitals.
For systems of two or more unpaired electrons, the
degeneracy of these spin states may be lifted even in the
absence of a magnetic field; this is termed the zerofield
splitting (ZFS). If the number of unpaired electrons is
even (S ~ 1, 2, ...), the degeneracy may be completely
lifted in zero field. Additional terms in the Zeeman
Hamiltonian H = 3H*g*S are required to account for this.
The Spin Hamiltonian
It was shown in the derivation of the spin Hamiltonian
(Eq , [36]) that the anisotropic part of the spinorbit
coupling produces a term S*D*S which is operative only in
systems with S 1. However, at small distances, two
unpaired electrons will experience a strong dipoledipole
interaction, such as has been considered in the anisotropic
hyperfine interaction:
3(S1r)(S2â€¢r)
[94]
5
r
In this equation, r is the vector connecting the electrons,
and the are the spin operators of the individual electrons i.
If the scalar products are expanded, and the Hamiltonian
expressed in terms of a total spin operator S = + S^,
. I
77
the Hamiltonian can be written
â€”ss =
r 2 2 ~i
r  3x 3xy 3xy
C
5 5 5
o
â€”x
r r r
3xy r2  3y2 3yz
c
5 5 5
â€”y
r r r
2 2
3xz 3yz r  3z
C
5 5 5
r r r J
o
â€”z
= S D S
[95]
This term, representing the dipolar interaction of the
electron spins, should be compared to the last term of
Eq. [36]; the latter evolved from Eq. [31] by treating the
spinorbit coupling interaction as a perturbation on the
Zeeman energy and assuming that the space and spin parts of
the electronic wavefunctions were separable. It can be
seen that the two terms are identical in form, except for
a numerical constant. These spinorbit and spinspin
contributions to D cannot be distinguished experimentally.
Whatever the origin of the interaction, the D tensor can
be diagonalized and the fine structure term becomes
= D S
xxâ€”x
+ D S
yyy
D S
zzâ€”z
H
2
2
2
[96]
78
where D + D + D =0, that is, D is a traceless tensor,
xx yy zz =
This can be written in terms of the total spin as
S*D*S = D [S 2  (1/3 )S (S + 1)] + E (S 2  S 2)
â€” = â€” z â€”x â€”y
+ CL/3)(D + D + D ) S(S + 1), [97]
XX yy zz' '
where D=D (D +D )/2 and E = (D  D )/2.
zz xx yy xx yy
The last term is a constant, proportional to the trace of D
which is zero for pure spinspin interaction, and does not
appear in the spin Hamiltonian. The terms involving D
and E account for the removal of the degeneracy of the three
triplet wavefunctions in the absence of an external magnetic
field.
While the effects of spinspin and spinorbit interÂ¬
actions are not separable, some qualitative statements can
be made about them. It is found, in general, that organic
triplets show little influence from spinorbit coupling.
The g values of such molecules are usually very close to
g , and the zero field splitting can be ascribed almost
completely to spinspin dipolar interactions. Then one
would expect the value of D to be approximately inversely
proportional to the molecular volume; this trend can be
seen in the methylene derivatives C,H,_  C  and
6 5 6 5
NC  C  CN, where D is 0.4 cm ^ (47) and 1.0 cm ^ (48),
respectively. Introduction of a heavy atom increases the
79
4
spinorbit coupling (since AaZ ), thus D for CN^ is 1.16 cm
(49) but for SiN^, discussed below, it is 2.28 era ^. If the
molecule involves transition metals, which usually have
large spinorbit coupling constants, the zero field splitting
is due mainly to the spinorbit interaction.
The solution to the general Hamiltonian for triplet
states of randomly oriented molecules is given by Wasserman,
Snyder, and Yager (50). Here, the special case of linear
molecules will be briefly discussed.
Neglecting hyperfine and other spinorbit interactions,
the spin Hamiltonian for a linear triplet will be
where z is the molecular axis. If y is chosen arbitrarily
[99]
Choosing as a basis the orthonormal spin wavefunctions
I + 1> = I Â«!<Â»â– ,_)
[100a]
 0 > =(l/V3a1B2 +B1a2)
[100b]
I  i> = I e1Â»2>
[100c]
80
and considering the effect of the spin operators on the funcÂ¬
tions as
S a =(1/2)8 S 6 = (1/2),a S a =(l/2)iB
â€”x â€”x â€”y
Sza =(1/2) a SzB =(1/2)8 Sz2ct =(l/4)a
then the Hamiltonian matrix will be
SyB =(1/2) ici
Sz28 =(1/4)8
[101a]
[101b]
 + l)  0 >   l)
+1>
D/3 + G
z
G /\Ã±
X
0
0>
G /v/2
X
2/3 D
G /\fl
X
[102]
1>
0
G /v/2
X
D/3  G
z
where G = giBH_ and G = giBH .
z ^ 11 z x _L x
H  z (Hz = H, Hx = 0) are
W+1 = D/3 + g 11 8H
WQ =(2/3) D
W;L = D/3  g  BH.
The eigenvalues for
[103a]
[103b]
[103c]
31
At zero field, the  + l) and   l) states are degenerate,
and the appropriate wavefunctions are
T =(1/^2 )(  + 1>   1>) [104a]
Ty = (l/yfl )( + l) +  1>) [104b]
Tz =  0 ) . [104c]
In these wavefunctions the spins are quantized along the
x, y, and z axes, respectively: S It ) = S I'T ) = S It ) = 0.
The eigenvalues for Hz are plotted against H in Figure 8,
and all yield straight lines.
For HiZ, H = H and H =0, and the roots of the secular
J_ x z
determinant are
W = D/3
W2 = (1/2)
D/3 + (D2 + 4g2B2H2)1/2
W3 = (1/2)
D/3  (D2 +4g82H2)1/2
with the eigenvectors
*x = 0A/2)
+ l)    1>
[105a]
[105b]
[105c]
[106a]
H Hz , D POSITIVE
E = 0
gn/SH/D
Energies of the triplet state in a magnetic field for a molecule with axial
symmetry; field parallel to molecular axis.
Figure 8.
83
1
cosa
1
_SÃna ~\Ã2
 + l) +   l)
I + l) + I 
sinao) [ 106b]
+ cosao), [106c]
where tan 2a = 2gi6H/D. As H approaches zero, where a = 0,
these functions reduce to the functions T , T , and T
x y z
given in Eq. [104]. The eigenvalues for Hj^z are plotted versus
H in Figure 9, and only at high fields, where a+TT/4,
do the lines become straight. In the intermediate region,
ip and ip are mixed and lead to a curvature of the energy
with H.
It is evident that the energy levels, and hence the
fields at which transitions between them occur, are very
dependent upon the orientation of the axis of the molecule
with respect to the fixed magnetic field. This will,
in general, cause the spectra of randomly oriented molecules
to be broad and difficult to observe.
The relative transition probabilities are given by
lyijlk = gi2cos2Yl<^iSk^j)2 [107]
where i, j, and k are any of the molecular axes (z is axial)
and y is the angle between the oscillating magnetic field
of the microwave radiation (perpendicular to the fixed
field) and k. Then transitions are allowed between levels
characterized by the following wavefunctions:
w
D
I Ty >
Tx >
Tz>
Figure 9
0 12 3 4
Hlz, D POSITIVE
E = 0
gx/3H/D
Energies of the triplet state in a magnetic field for a molecule with axial
symmetry; field perpendicular to molecular axis.
CO
85
(1) T T
x z
z
1
(3)
(4)
Xy2
xyXf
where the symbols on the right are the usual designations
given to the observed ESR lines. These transitions are
indicated in Figures 8 and 9, and all correspond to AM = Â±1
transitions. Also indicated in Figure 9 is a dashed line
representing the forbidden AM = 2 transition. This transiÂ¬
tion (ip Â«> ipz) is allowed for Hosc I H, but is has a finite
transition probability (9) when H is not parallel to any
of the x, y, or z axes, even if Hqsc J_H, as in the apparatus
employed here. The AM = 2 notation is acutally a misnomer,
since the spin functions  + l) and   l) , corresponding
to the infinite field Mg values, mix significantly at
finite fields. Thus M is not a good quantum number and the
transition could actually be described as AM = 0, since
the eigenfunctions each contain contributions from spin
functions of the same M. Thus if D is not too large, the
transition will be observable.
i. â€¢ Employing the exact solution to the spin Hamiltonian
matrix for triplet molecules, which may be bent (E ^ 0) , the
resonant fields of the transitions are (50)
86
1 22 1/2
H , = [(hvD)Â¿  EZ]X/ [108a]
zi g II B
H o = ~â€”~p [(hv+D)2  E2]1/2 [108b]
z2 g i 3
H
xl g^B
â€” [ (hvD+E) (hv+2E) ]1//2
[108c]
H
x2 g^B
1 [(hv+DE)(hv2E)]1/2
[180d]
II , = [(hvDE) (hv2E) ]1/2
y1 g i p
i'
[108e'
H = ^ [ (hv+D+E) (hv+2E) ] 1^2
y2 g^B
[10 8f]
_ 1 f (hv)2 D2 + 3E2 ,1/2
AM=2 gB 1 4 3 J
[10 8g]
In Figure 10, the resonant fields of these transitions
for linear molecules (E = 0) are plotted as a function of
D, for a fixed microwave frequency of 9.1 GHz, where the
energy equals 0.3 cm . The z and xy lines are so marked.
As can be seen from the Eqs.[108], the effect of a nonzero
E term in the spin Hamiltonian is to split each xy line into
D (cm'
87
Figure 10. Resonant fields of a molecule as a function
of the zero field splitting.
88
separate x and y components, separated from the xy line
by Â±3E/2. Thus, when E=0,H,=H,=H ,, and
1 xl yl xyl
similarly for H 2 â€¢ As D increases from zero and approaches
the fixed hv, the z^ line approaches H = 0, and then
appears at increasing fields. The xy^ and AM = 2 lines
similarly approach H = 0 at D = hv, and then can no longer
be observed. This behavior is understandable by reference
to Figures 8 and 9. For the case of Hz, as D increases
and the energy separating the  + l),   l) states and the
0) state gets larger, the Z2 line will continue to move
out to higher fields. However, the z^ line occurs at H = 0
when D = hv, after which the transition will occur between
the o) and   l) states, instead of the o) and  + l)
states. As D increases still further, this line appears at
higher fields, and both z lines involve transitions between
the same states,  0)â€”â€”  l) . In the case of H_[_z, the
xy^ and AM = 2 lines similarly approach H = 0, but then
disappear as the zerofield splitting (D) increases beyond
hv. Then the xy^ line is the only transition possible. If
the molecule is nonlinear, the same effect occurs for the
corresponding x and y lines. In the case of D>hv, if
E < 0.009 cm ^, D can be approximated by
g^V
2h2Cv
2 2
H + H
x2 y 2
 2
hv
D
[109]
89
The lineshapes for randomly oriented, axially symmetric
molecules have been considered previously, and Wasserman,
Snyder and Yager (50) have treated the problem of randomly
oriented triplets, neglecting any angular variation of the
transition probabilities and line broadening. As with
the doublet species, the unnormalized absorption intensity
I(H) follows the relation
1(H) a sin0 [110]
(dHr/d0)
where Hr = Hq Â± [(D'/2)sin20  D'cos20], with D' =
and Hg = hv/g^B, in spherical polar coordinates. Thus
I(H)a[(D1/2) + (H  H0)]â€œ1/2 [111a]
I (H) =0  H  Hq I > D' . [Hlb]
The upper sign refers to the region about Hq from D' to
D'/2, and the lower to the region D'/2 to +D'. The total
absorption is the sum of the two terms. As indicated in
Figure 11, there is a step in the curve at Â±Dâ€˜. This correÂ¬
sponds to absorption by molecules with Hz. The absorption
rises without limit at Â±D'/2 due to these triplets where
H lies in the x, y plane. The total absorption is the sum
of the two "powder patterns" indicated in that Figure,
and the first derivative is infinite or effectively very
90
Figure 11. (a) Theoretical absorption and (b) first deriÂ¬
vative curves for a randomly oriented triplet
state molecule with axial symmetry.
91
large at the four field positions mentioned. This feature
permits the detection of randomly oriented triplets; the
spectrum would appear as in Figure lib.
As discussed previously, the effect of a nonzero E
term (nonlinear molecules) is to split the xy lines. Thus
the peaks at Â±D'/2 are separated such that a step appears
at Â± (D1 + 3E')/2 due to the molecules in which y  H, and the
absorption rises without limit at Â±(D'  3E')/2 due to
those molecules with xH. The steps occur along the y
and z axes as any changes in the orientation of the molecule
moves the resonance in the same direction. Figure 12 shows
the absorption pattern, which appears as the superposition
of two powder patterns of nonsymmetric (g^ Â¿ g2 Â¿ g^)
molecules, and the expected spectrum.
Figures 11 and 12 contain only the region of magnetic
field corresponding to the AM = Â± 1 transitions. Due to
the small anisotropy of the AM = 2 transitions, that is,
a small value of dH/d0, these transitions exhibit relatively
large amplitudes.
As previously noted, the hyperfine interaction is
generally small compared to the fine structure (D) and
electronic Zeeman terms. Thus first order perturbation is
sufficient to account for the hyperfine splitting. If
the usual hyperfine term An S I + Ai(S I + S I ) is added
II â€”zâ€”z J_ â€”xâ€”x â€”yâ€”y
to the spin Hamiltonian given by Eq. [98], and utilizing the
same wavefunctions as above, one can obtain the contribution
of the hyperfine interaction.
92
b.
Figure 12. (a) Theoretical absorption and (b) first deriÂ¬
vative curves for a randomly oriented triplet
state molecule with orthorombic symmetry.
93
For H  z, the nonvanishing matrix elements are
<+lszl+1) = +1> <â€œ 11 S z  1 > = 1, and (mlzm) = m
so that the hyperfine contributions to the energy levels
of Eqs. [103] are
W+i = A  m
[112a]
o
II
o
[112b]
W_ ^ = Aj m.
[112c]
Note that at zero field the levels +l) and l) are degenerÂ¬
ate and the hyperfine splitting vanishes; this was not the
case in doublet molecules. A AM = Â±1 transition of hv
then appears at  W_^ = hv such that
Hr = [hv + D  A11 m]/g j 6.
[113]
For hyperfine interaction with a spin I =
1/2 nucleus, the
hyperfine splitting is then
AHr  A11 /g11 6.
[114]
The AM = 2 lines, if observed, would exhibit the same splitting
For Hj^z, using the eigenfunctions of Eq. [106] and
the nonzero matrix elements of the hyperfine Hamiltonian
94
(m'I m) = â€” (m'l+ + I m), where m' = m Â± 1, the new
eigenvalues are
Wx' = V [115a]
, Aisin 2a
W2 = W2Â± J= [115b]
, A i sin 2a
W = W + â€”Â± [115c]
2
where W^, W2, and were given in Eq.[105] and â€˜sin 2a =
[1  (D/hv) ] . This arises from consideration of the
Hamiltonian matrix, Eq.[102], which must be expanded to a
6 x 6 to act on the wavefunctions ^,m). In the limit of
small D relative to hv, AHr approaches A^/giB, analogous to
Eq. [114]. However, when D and hv are comparable,
(hv + D/2)
(hv [hv + D])1/2.
[116]
4
Z Molecules
Although there are exceptions, molecules of highspin
(S > 1) systems generally fall into the realm of transition
metal compounds. As indicated above, the presence of the
metal atom, with its large spinorbit coupling constant,
usually implies a large D, or zero field splitting value.
95
To this dominant LS^ interaction is added a sizable spin
spin interaction, particularly if there are few ligands
to participate in bonding to the metal atom, since this
confines a large number of unpaired electrons to a small
molecular volume. The resulting large value of D may preclude
observation of many ESR lines of such molecules.
The Spin Hamiltonian
3 4
As in the E case, the E molecule will exhibit fine
structure in the spectrum, and two Kramers doublets are
expected, with values of Â±3/2 and Â±1/2. Neglecting
hyperfine structure, the spin Hamiltonian for an axially
4
symmetric E molecule is
H
sp:
completely analogous to Eq.[99] for the triplet case.
Using the nonvanishing matrix elements
(m Is m ) = m
' s 1 â€”z 1 s7
s
[118a]
[118b]
= [S (S + 1)
M (M Â±1)]1/2
s s
[118c]
96
and S = (S+ + S )/2, the spin matrix can be calculated:
X
1+3/2) 1+1/2) II/2) I 3/2)
where G = g11 gH and G = g1gH .
z 11 Z X lx
For H   z , H = H, H =0, the four eigenvalues are
Z X
WÂ±3/2 = DÂ± I g  gH [120a]
WÂ±i/2 = DÂ± \ q  gH, [120b]
and at zero field the Â±3/2 degenerate level is separated
by 2D from the Â±1/2 level. As for triplet molecules, the
energy levels vary linearly with magnetic field for H
parallel to the molecular axis.
For HiZ, H = H, H =0, and the offdiagonal terms
x z
are nonzero, making the eigenvalues more difficult to find.
If the eigenvalue matrix is expanded, the quartic equation
may be written as
97
e4  1/2(1 + 15x2) e2 + 3x2e +(1/16)(1 + 6x2 + 81x4) = 0 [121]
where e = W/2D and x = g^BH/ (2v/3D) . A more general form of
the expression, applicable to any angle, is given by Singer
(51). If H/D or x is small, e may be expanded as e = a + bx
2
+ cx + ..., and the eigenvalues for H^z are
WÂ±3/2 = D 2 + â€¢â€¢â€¢ [122a]
wÂ±l/2 = D Â± g^8H  ^(g^8H)2 + ..., [122b]
with the levels indexed by the lowfield quantum numbers.
As another approach, if D >gBH*S, all matrix elements
of the type (Â± 3/2  Hspin  Â± l/2> = <Â± 1/2  Hspin  Â± 3/2) vanish.
Thus, the matrix in Eq. [119] reduces to
+3/2> 11/2) l/2> 3/2>
+3/2>
0
0
+l/2>
11/2)
0
G
x
0
[123]
0
G
x
0
13/2)
0
0
0
98
and the eigenvalues become
W+ 3/2 = Â° Â± (3/2>g11 3Hcos6 [124a]
22 221/2
WÂ±l/2 = _D 1 (1/2)3H (g  cos 0 + 4g^ sin 0)i/z [124b]
since H = Hcos0 and H = Hsin0; as usual, 0 is the angle
between the molecular axis and the applied field. This
introduces the "apparent" or "effective" gvalue, which
is often used to designate where a transition occurs. This
is defined by assuming the resonance is occurring within
a doublet, that is between Ms = Â±1/2 levels with g = gg = 2.00.
The resonance field is then given by = hv/gr , with g^
the effective g value. Thus for a large zero field
splitting, the g values of the observable transitions
 3/2) I 3/2) and 11/2) **â– I I/2) become
Ms = Â±3/2 g ] = 3g   6.0, gj = 0.0 [125a]
Ms = Â±1/2 gj = g  2.0, gj = 2g^4.0 [125b]
where the prime indicates the effective g value. Even if the
D value were still low enough to allow population of the Â±3/2
levels and the induction of transitions between them by the
microwave radiation, the absorption pattern corresponding
to the g values for this transition would be extremely
broad, and the derivative signal probably undetectable.
99
Thus, only the +1/2)+>11/2) transition is observed. In
fact, the assumption that H/D is small implies that the
transition (+3/2) 13/2) is forbidden.
Kasai (52) and Brom et al. (53) have considered this
case in the VO and NbO molecules, respectively. They
found that the spin Hamiltonian
H
â€”spin
BHS + g,B(H S +HS)
zâ€”z xâ€”x yâ€”y
+ A,
I S
â€”zâ€”z
+ A i (I S + I S ) + D [S "
â€”xâ€”x â€”yâ€”y â€” z
 (1/ 3) S (S + 1) ]
[126]
could be rewritten as an effective spin Hamiltonian for the
 +l/2)Â«â€”â–º  â€”1/2) transition in this way:
H . = g BH S + 2g ,'B (H S + H S ) + A,, I S
â€”spin ^  zâ€”z xâ€”x yâ€”y  â€”zâ€”z
+ 2 A i (I S + I S )
1 xx yy
[127]
Note that S is now taken to be 1/2, and the D term vanishes.
This is an example of the fictitious or effective spin
mentioned earlier in the derivation of the spin Hamiltonian.
In the same manner as described previously in obtaining
Eq . [72], this effective spin Hamiltonian can be rearranged
to be diagonal in the Zeeman terms:
100
H . = gBHS + AI S +
â€”spin ^ â€”z â€”zâ€”z
(4A^  A  ) /2g 11 g
A
sin0cos0I S
â€”zâ€”z
+ (1/2) A
1
A 11 + A'
A
(I+S + I S + )
+ (1/2) A
1
 A
A
(iV + I S")
[12 8]
2 22 22 2 222
where g = g11 cos 0 + 4g^ sin 0 and A = (A11 g j /g ) â€¢
2 2 2 2 2
cos 0 + (16A^ g^ /g ) sin 0. This equation can be solved
analytically at 0 = 0 and by a continued fraction method
at 0 = 90Â°. When such an exact solution to the spin
Hamiltonian is required, an iterative procedure is used
to match observed and predicted spectral lines, and arrive
at accurate values of the g and A components. This was
necessary in the two cases cited above, where the hyperfine
splittings were too large to be amenable to the perturbation
treatment of Eq. [73].
Although this effective spin treatment ignores the fine
structure term, it is still possible to obtain the D value
from the effective g value by perturbation theory. KirkÂ¬
patrick, MÃ¼ller, and Rubins (54) considered ions of halfÂ¬
integral spin (S > 3/2) in an axial field, and developed the
expression for the effective g value:
e r 2,.2 2 2, . 2 Q.1/2
g = [g  + (rn gj  g  ) sin 0]
n2(g.3H)2
1 =*=5 F(0)
4(2D)^
, [129]
101
where
F(0) = sin20
â– 2 2
(m g. + 2gn ) sin 0  2g
2 2
(m g^  g  ) sin 0 + g^
[130]
which simplifies if g11 = g^ (as it usually is) to
F(0) = sin20
(m2 + 2)sin20  2
(m2  1)sin20 + 1
[131]
In these equations, H is the observed resonant field and m
and n are defined as m = [S(S + 1) + l/4]"*â€˜//2,
n = [S(S + 1)  3/4]1/2.
As in the triplet case, the energy levels and observed
transitions between them are strongly dependent on 0 and D.
Figures 13 and 14 indicate the levels as a function of field
for the parallel and perpendicular orientations, respectively,
with g 11 = g^ = 2.00 and v = 9.1 GHz; the D values and
AM = Â± 1 transitions are labelled in each case. In Figure 14,
two transitions are indicated between the same two levels at
690 and 1840 G. The reason for this behavior can be clearly
4
seen m Figure 15, which is the Â£ counterpart of Figure 10.
Here, the xy^ line is shown as an arc which reaches a maximum
at about 1000 G, and for some D values, between 0.15 and
0.18 cm \ approximately, there are two transitions between
the 3/2 and +1/2 levels (lowfield quantum numbers). As
in Figure 10 the lines are designated z^ and xy^, with i
102
4
Energy levels for a E molecule in a magnetic
field; field parallel to molecular axis.
Figure 13.
H (Kilogauss)
Energy levels for a molecule in a magnetic
field; field perpendicular to molecular axis.
Figure 14.
104
H (Kilogauss)
Figure 15.
Resonant fields of a molecule as a function
of the zero field splitting.
increasing from low field to high at low D. This designaÂ¬
tion is convenient, but somewhat ambiguous, since the xy and
z lines of a given i do not refer to the same transition.
This is readily apparent by reference to Figure 11; there,
for example, z^ is the parallel component of the powder
pattern absorption whose perpendicular component is xy^â€¢
It is a general effect that, to first order in the fine
structure, the resonant fields are given by (7, 8)
H = Hq  (Ms  1/2)[3cos20  1] [132]
where is the resonant field at D = 0, Mg is the spin
quantum number of the upper level in the transition, and
it is assumed that g jj = g^ = g^. This approximation is
valid at very low D ( 0.01 cm ^), and the result is that
the lowest field z line is associated with the highest
field xy line. Unfortunately, as D increases to about
0.1 cm \ the generalization is no longer valid. Hence all
designations of z or xy lines included in this work will
refer to the convention used in the Figures.
A final note about this diagram is to point out that,
just as predicted by Eq. [125b], if D>>hv only a perpendicuÂ¬
lar line near g' = 4 (xy^) and a parallel line at g = gg
is expected to be observed.
E Molecules
+ 3 +2
Due to the Fe and Mn 10ns present in many coordinaÂ¬
tion complexes, the ESR of spin 5/2 systems has been greatly
106
studied In many cases, the high spin ion is subjected to
a field of octahedral or tetrahedral symmetry, with the
result that the five unpaired electrons occupy t2 and e
orbitals in either symmetry, yielding an orbitally nonÂ¬
degenerate ground state. The lowest excited states will
not mix strongly through spinorbit coupling, hence D should
be rather small and g close to g . However, the crystal
field operators (5, 7, 34) can remove the spin degeneracy,
since they can couple states with Mg values differing by
Â±4. This results in an additional quartic term (3, 5, 7,
31, 34, 55, 56) in the spin Hamiltonian, of the form
4 4 4
a [S + S + S ], where the value of a and the exact form
â€”x â€”y â€”z
of the operator depends on the symmetry. However, for
g
axial symmetry and large D, as in the E molecule considered
here, this term is not significant, and the treatment is
just an extension of that previously described.
The SpinHamiltonian
g
Taking the Â£ analog to Eqs. [99] and [117], the spin
Hamiltonian for axial symmetry can be written, including
all angles
H . = g 11 3HS cos0 + gi3H S sin0 + D(S
â€”spin ^ [ â€”z xâ€”x â€”z
[133]
Utilizing the nonzero matrix elements given in Eqs. [118]
the energy eigenmatrix is
107
+5/2>
+3/2>
1+1/2)
11/2)
3/2>
15/2)
l+!>
g +^rD
2 z 3
Ã6
2 x
0
0
0
0
i+!>
V5
4g
2 x
3 2
4g 4d
2 z 3
0
0
0
2 x
[134]
i4>
0
2 x
yG â€œfD
2 z 3
K
0
0
I !\
0
0
g
2 x
kfD
^G
0
' 2'
2 x
1
0
0
0
ÃG
2 x
fGzfD
is
2 x
1 2'
lÂ§>
0
0
0
0
%
2 x
g +^D
2 z 3
where
G = g II
z y 
BHcosG and
G = giBHsinO. For
x
6= 0 (
case), all
offdiagonal elements are zero and the eigenvalues are
[135a]
[135b]
[135c]
At zero field, three levels appear, separated by 2D and
4D. Application of a magnetic field splits these into
three Kramers' doublets, which diverge linearly with field
EÂ±5/2 = T DÂ± I 91 1311
EÂ±3/2 = " 3 DÂ± 2 g
eÂ±1/2 = 3 DÂ± 2 g  e H.
108
and with slopes proportional to M . This is the same effect
4
as seen in the Z case.
For 0 = 90Â°, there is a mixing of states and no simple
solution of the secular equation is possible. The matrix
could be expanded to a sixthdegree polynomial, and presumably
the continued fraction method would be applicable. However,
a direct solution by computer is more convenient, and
involves no assumptions. Such a calculation has been done
by Dowsing and Gibson (57), Aasa (58), and Sweeney et al. (59),
for S = 5/2 systems with axial symmetry. Sweeney, Coucouvanis,
and Coffman (59) have also considered D<
4
approach, which incorporates the quartic spin operators S^
to account for crystal field effects; this shall not be
discussed further.
The eigenvalues of Eq. [134], calculated by computer
diagonalization of the secular determinant at many fields
and four angles, are shown in Figure 16. The effect of
mixing of spin states as the angle of the molecular axis
with respect to the field is increased to 90Â° is quite
striking. The lines do not cross except at 0 = 0Â° (51).
Transitions of 9.4 GHz are indicated, and it is evident
that the resonant field for some transitions is strongly
angle dependent. It must be kept in mind that the observed
transitions in polycrystalline samples occur at the
turning points of the powder pattern absorption, that is,
at 0 = 0Â° and 90Â°, and the total absorption pattern is
spread between these points. Thus, for two transitions of the
109
Energy levels
field for 0 =
for
0Â° ,
30Â° ,
molecule
60Â°, and
in a magnetic
90Â° .
Figure 16
f
110
same transition probability and state populations, that which
has a narrower range of resonant fields is more readily
detected. Note also that the  + 3/2)â– **â–  â€” 3/2) and +5/2)
 â€” 5/2) transitions indicated are strictly forbidden at 0 = 0Â°
g
Figure 17 is the D vs. diagram for E molecules with
g11 = g^ = ge and v = 9.1 GHz. It was prepared by solution
of the Hamiltonian matrix at many fields and D values for
0=0Â° and 90Â°. The characteristics of such diagrams have
already been discussed. Note that for D>>hv, the xy^
line at g Ã 6 and the line at g = 2 will be the most
readily observed. That these both correspond to the +1/2
II/2) transition can be seen in Figure 16 and predicted in
4
a manner analogous to the treatment of the E case in the
limit of high D. Thus if the offdiagonal elements are
neglected, the eigenvalues of Eg. [134] for = Â±1/2 are
E
Â±1/2
 DÂ± 1.0H(g ^cos^0 + 9gj^sin^0)
[136]
yielding the effective values g^ = 6, g j  =2.
Dowsing and Gibson (57), and Aasa (58), have treated
the spin 5/2 system in nonaxial symmetry, and presented
D vs. diagrams for such cases. Aasa (58) has also
considered the relative transition probabilities, ., which
can be calculated from
P.. = 
[137]
Ill
/T
Resonant fields of a E molecule as a function
of the zero field splitting.
Figure 17.
112
where ib.
ri
levels.
cular to
P . .
il
and ib. are the wavefunctions of the ith and jt51
1
With HoscJ_H' and averaging over a plane perpendi
H, this becomes
= K^l^k.)!2 +  <^iS~^j) I2 [138]
113
References  Chapter III
1. P.B. Ayscough, "Electron Spin Resonance in Chemistry,"
Methuen, London, 1967.
2. C.P. Poole, Jr., "Electron Spin Resonance," Wiley
Interscience, New York, 1967.
3. A. Carrington and A.D. McLachlan, "Introduction to
Magnetic Resonance," Harper and Row, New York, 1967.
4. C.P. Slichter, "Principles of Magnetic Resonance,"
Harper and Row, New York, 1963.
5. J.C. Wertz and J.R. Bolton, "Electron Spin Resonance:
Elementary Theory and Practical Applications,"
McGrawHill, New York, 1972.
6. H.M. Assenheim, "Introduction to Electron Spin Resonance,"
Plenum, New York, 1966.
7. A. Abragam and B. Bleaney, "Electron Paramagnetic Resonance
of Transition Ions," Oxford University Press, London, 1970.
8. W. Low, "Paramagnetic Resonance in Solids," Supplement 2,
Academic, New York, 1960.
9. S.P. McGlynn, T. Azumi, and M. Kinoshita, "Molecular
Spectroscopy of the Triplet State," Prentice Hall,
Englewood Cliffs, N.J., 1969.
10. "The Triplet State," Proceedings of the International
Symposium Held at the American University of Beirut,
Lebanon, February 1419, 1967. Cambridge University
Press, Cambridge, 1967.
11. P.W. Atkins and M.C.R. Symons, "The Structure of Inorganic
Radicals," Elsevier, New York, 1967.
12. G.B. Pake, "Paramagnetic Resonance," Benjamin, New York,
1962.
13. H. Kopferman, "Nuclear Moments," Academic, New York, 1958.
14. G. Breit and I. Rabi, Phys. Rev., 38^, 2002 (1931).
15. S. Goudsmit and R. Bacher, Phys. Rev., 34, 1499 (1929).
114
16. E. Fermi, Z. Physik, 60.' 320 (1930).
17. M. Born and J.R. Oppenheimer, Ann. Phys., 84, 457 (1927).
18. A. Abragam and M.H.L. Pryce, Proc. Roy. Soc., 205A,
135 (1951) .
19. B. Bleaney, Proc. Phys. Soc. (London), A63, 407 (1950).
20. B. Bleaney, Phil. Mag., 42^, 441 (1951).
21. R.H. Sands, Phys. Rev., 9J9, 1222 (1955).
22. J.W. Searl, R.C. Smith, and S.J. Wyard, Proc. Phys.
Soc., 74, 491 (1959) .
23. B. Bleaney, Proc. Phys. Soc., A75, 621 (1960).
24. F.K. Kneubuhl, J. Chem. Phys., 33, 1074 (1960).
25. J.A. Ibers and J.D. Swalen, Phys. Rev., 127, 1914 (1962).
26. M.M. Malley, J. Mol. Spectr., 17, 210 (1965).
27. W. Weltner, Jr., "Advances in High Temperature Chemistry,"
Vol. 2 (L. Eyring, Ed.), Academic, New York, 1970, p. 85.
28. P.H. Kasai, E.B. Whipple, and W. Weltner, Jr., J. Chem.
Phys. , 4_4 , 2581 (1966) .
29. L.B. Knight, Jr., W.C. Easley and W. Weltner, Jr., J.
Chem. Phys., 54, 1610 (1971).
30. "Hyperfine Interactions" (A.J. Freeman and R.B. Frankel,
eds.), Academic, New York, 1967.
31. K.D. Bowers and J. Owen, Reports Progr. Phys., 18,
304 (1955) .
32. L.D. Rollmann and S.I. Chan, J. Chem. Phys., 50,
3416 (1969).
33. F.A. Cotton, "Chemical Applications of Group Theory,"
WileyInterscience, New York, 1971.
34. C.J. Ballhausen, "Introduction to Ligand Field Theory,"
McGrawHill, New York, 1962.
35. R.S. Mulliken, C.A. Rieke, D. Orloff, and H. Orloff,
J. Chem. Phys., 12, 1248 (1949).
36. R.S. Mulliken, J. Chem. Phys., 1^/ 900 (1951).
115
37. A.J. Stone, Proc. Roy. Soc. (London) , A271, 424 (1963).
38. P.W. Atkins and A.M. Jamieson, Mol. Phys., 14_, 425 (1967).
39. H.H. Tippins, Phys. Rev., 160, 343 (1967).
40. W.C. Easley and W. Weltner, Jr., J. Chem. Phys., 52,
197 (1970).
41. R.A. Frosch and H.M. Foley, Phys. Rev., 8j8, 1337 (1952).
42. B.A. Goodman and J.B. Raynor, "Advances in Inorganic
Chemistry," Vol. 13 (H.J. Emeleus and A.G. Sharpe, eds.),
Academic, New York, 1970, p. 135.
43. D.L. Beveridge and J.W. Mclver, Jr., J. Chem. Phys.,
54, 4681 (1971) .
44. J.A. Pople, D.L. Beveridge, and P.A. Dobosh, J. Amer.
Chem. Soc., 9_0, 4201 (1968).
45. J.A. Pople and D.L. Beveridge, "Approximate Molecular
Orbital Theory," McGrawHill, New York, 1970.
46. J.D. Memory, "Quantum Theory of Magnetic Resonance
Parameters," McGrawHill, New York, 1968.
47. E. Wasserman, A.M. Trozzolo, W.A. Yager, and R.W.
Murray, J. Chem. Phys., 4_0, 2408 (1964).
48. E. Wasserman, L. Barash, and W.A. Yager, J. Amer. Chem.
Soc., Â£7, 2075 (1965).
49. G.R. Smith and W. Weltner, Jr., J. Chem. Phys., 6 2,
4592 (1975).
50. E. Wasserman, L.C. Snyder, and W.A. Yager, J. Chem.
Phys., 41, 1763 (1964] .
51. L.S. Singer, J. Chem. Phys., 23, 379 (1955).
52. P.H. Kasai, J. Chem. Phys., 4_9, 4979 (1968).
53. J.M. Brom, Jr., C.H. Durham, Jr., and W. Weltner, Jr.,
J. Chem. Phys., 61_, 970 (1974).
54. E.S. Kirkpatrick, K.A. Muller, and R.S. Rubins, Phys.
Rev., 135, A86 (1964) .
B. Bleaney and R.S. Trenam, Proc. Roy. Soc. A, 223, 1
(1954) .
55.
116
56. B. Bleaney and K.W.H. Stevens, Reports. Progr. Phys.,
16, 108 (1953).
57. R.D. Dowsing and J.F. Gibson, J. Chem. Phys., 50,
294 (1969).
58. R. Aasa, J. Chem. Phys., 52, 3919 (1970).
59. W.V. Sweeney, D. Coucouvanis, and R.E. Coffman, J.
Chem. Phys., 59, 369 (1973).
CHAPTER IV
SILICON SPECIES
Introduction
Carbonylsilene, SiCO, diazasilene, SiNN, and dicarbonyl
silene, Si (CO)2> are the silicon analogues of the carbon
molecules CCO, CNN, and C(CO)2, which have received considÂ¬
erable study. Consideration of these species is important
in terms of their molecular electronic structure and the
nature of the bonding involved. Carbonylcarbene (CCO) and
diazacarbene (CNN) are isoelectronic triplet molecules with
distinctly different bonding (1  4) and zero field splitting
(5, 6). Carbon suboxide (C^02) is a stable molecule which
has been discussed at length by Herzberg (7). It has been
noted (8) that there is no more striking example of an
enormous discontinuity in general properties than between
the first and second row elements; little of the chemistry
of silicon can be inferred from that of carbon. Thus a
study of these silicon species, and comparison with the
known properties of their carbon counterparts, is of great
interest. In addition, because of the relatively large
cosmic abundance of silicon, and the postulated or observed
presence of small siliconcontaining molecules in stars
or interstellar space, the problem merits consideration. For
example, SiC2, a related species which comprises a small
proportion of the vapor over hot silicon carbide (9), has
117
118
been observed (10  13) in class C stars. Circumstellar
SiC particles have been tentatively identified (14) , and it
could exist in interstellar dust clouds (15) where SiO and
SiS have been observed, along with CO, N^H+, and many other
inorganic and organic species (16). The silicon diatomics
SiH, SiF, and SiN have also been reported in class M
stars (17). Thus the possibility exists that some of the
species studied here either are present in such environments,
or play some role in the formation of the molecular species
observed there.
The object of this research, then, was to investigate
the electronic structure and spectra of the molecules SiCO,
SiN^, and Si(CO)2, for the purpose of identification and
comparison with their carbon analogues.
Experimental
The molecules SiCO and Si(CO)2 were prepared by the
reaction of Si atoms with pure CO, or mixtures of CO in Ar
or Ne with (Ar/CO) = (Ne/CO) = 100200. Similarly, Si^
was prepared by reaction of Si atoms with pure N2 and mixÂ¬
tures of N2 in Ar or Ne. A beam of Si atoms was produced
by vaporization of silicon powder (99.9999% pure , Spex
Industries, Inc.) from a Ta cell at temperatures that varied
from 1500 to 1900Â°C. Both resistance heating and induction
heating were used. The Ta crucibles have been described above;
it was found that for resistanceheated samples, a thick
walled (1 mm) cell was necessary to withstand destructive
SiTa alloying. Reagent and matrix gases employed were all
119
Aireo ultrapure grade, and these were used without further
purification except for passage through a liquid nitrogen
trap prior to deposition.
Isotopic substitution was employed to obtain hyperfine
coupling data in the ESR and shifts in the optical spectra.
2 9
In two experiments 95% enriched Si powder (Oak Ridge
National Laboratory) was mixed with an equal part of natural
abundance Si and vaporized from a resistivelyheated cell.
13 16
Isotopically enriched samples of CO (Merck, Sharpe
12 18
and Dohme of Canada, Ltd., 92.6 atom % enriched), C 0
15
(Miles Laboratory, 96.88 atom % enriched), and N2 (Merck,
Sharpe and Dohme of Canada, Ltd., 95.0 atom % enriched) were
used in addition to 12C'L^0 and "*"4N2 as reagents.
ESR Spectra
SiN2
When silicon vapor is trapped in an argon matrix
14
containing (M/A = 100), the strong line shown in
Figure 18 (top) appeared at 9503 G in the ESR. A small
triplet hyperfine splitting of 6.0 Â± 1 G due to interaction
14
with one N (I = 1) nucleus is discernible. A broad weak
feature also appears about 50 G upfield from the strong line,
and this did not disappear after several annealing cycles
between 25 and 4Â°K.
As discussed above and indicated in Figures 7 and 11,
the derivative shape of the strong line is characteristic of
the molecules oriented perpendicular to the static magnetic
field. In light of the known ground state of CN2, and with
120
ESR spectra of SiN2 molecules in argon matrices
at 4Â°K.
Figure 18.
121
reference to Figure 10, this line is assigned to the xy2
3
perpendicular line of a linear E SiNN molecule. The weak
band at higher field could be attributed either to a few
linear molecules in alternate sites in the matrix, or
to some slightly bent molecules in the matrix. The shape of
the band, and the fact that such weak lines were observed
in most, but not all, matrices, and always at higher
field to the strong line, suggests that the latter explanaÂ¬
tion is correct. Thus the matrix may induce the molecule
to be slightly bent, particularly in molecules where the
bending force constant is small, as may be the case for
SiN21 If the molecule is bent, that is, the E term in the
spin Hamiltonian is nonzero, the xy2 line will split into
separate x2 and y2 lines. The effect can be seen by a
comparison of Figures 11 and 12; the predicted lineshapes
and relative positions of the x and y lines are also indicated
there for Dhv, the symmetry about Hg is lost,
along with one transition, but the lineshapes remain the
same (18). This is in agreement with the observed spectrum,
if one considers that the xy^ (or x^ and y^) line will not
be seen for a triplet with a high D value, and the z lines
will not appear in the same scan because they occur at much
higher fields (see Figure 10). Then the weaker broad line
observed is attributed to a y2 perpendicular line of the
small population of molecules which are slightly bent in
their sites.
122
For the spin Hamiltonian
H . = g
â€”spin
3H S +
z z
g i B(H S + H S ) + D(S  2/3)
^ 1 xâ€”x yâ€”y â€” z
T
+ E (S
 S )
y
+ I.[Am i1s
1  zz
+ A I (I1 S + I1s ) ] ,
â€”xâ€”x â€”yâ€”y
[139]
where the sum is over all nuclei i with magnetic moments,
and utilizing Eqs. [108d, f], with the assumption that g^ = gg,
one finds D = 2.206 cm ^ for the linear molecule, and
D = 2.214, E = 0.00087 cm ^ for the bent molecules. The
assumption g^ = gg must be made because not enough lines
were observed to evaluate all the parameters. The line
positions for all triplet molecules discussed in this work
were measured according to the criterion of Wasserman et al.
(18). Here, the line position is obtained by choosing a
point on the side of the more intense peak in the derivative
signal which is closest to H^. This point is selected such
that the distance of the point above the peak is equal to
the magnitude of the less intense peak. Hence the line
positions reported here are slightly to the lowfield side
of the position of the minimum in the observed derivative
signal. The hyperfine splittings for these silicon species
are all rather small, and first order theory is sufficient
to account for them. Thus, with the assumption that
g i' = ge, hyperfine data was obtained from
AH = A./g.3
xyÂ¿ J/
[140]
123
where AH is the separation between the xy_ lines of
xy 2 2
different hyperfine components; this relation stems from
Eq.[59], and may be used with less than 1% error (6).
Substitution of "^N2 (95% enriched) yiÃ©lded the spectrum
in Figure 18 (middle), where the (1 = 1/2) doublet
splitting is found to be 7.4 Â± 1 G. Again, only the hyper
15
fine interaction with one N nucleus is detected, and
the resolution is sufficient to prove that the nitrogen
atoms are not symmetrically disposed about the Si, since
that conformation would produce three lines in a 1:2:1
intensity pattern. According to Eqs.[16] and [17], the
splitting is proportional to the nuclear magnetic moment,
which is related by Eq.[l] to the magnetogyric ratio of the
nucleus. Thus when the splitting is multiplied by the
magnetogyric ratio factor Y^/Y^ = 0.713, the result of
14
5.3 G for the t N splitting is obtained, which is in
14
satisfactory agreement with the observed value for Si .
2 8 29
Finally, when an equal mixture of Si and Si(I = 1/2)
15
was trapped with N2 in argon, the spectrum shown at the
bottom of Figure 18 was obtained. This exhibits both the
2 8 15
Si N2 spectrum of Figure 18 (middle), and a doublet of
29
doublets due to the Si hyperfine splittings of 34 Â± 1 G.
When trapped in neon, ^Â®Si^N2 exhibited a much broader
line at 9632 G with no evidence of a weak y2 line at higher
field, as shown in Figure 19. The halfwidth of the neon
line was 56 G, as compared to 12 G in argon. (The halfline
width here is measured between the field at which the derivative
124
SÂ¡N2
Figure 19. ESR spectra of Si^ molecules in various
matrices at 4Â°K.
125
ESR signal crosses the zero line and the field at the
negative peak.) On one occasion a narrow line (about 19 G
wide) at 9624 G was observed on the low field side of the
broad neon line; its peak was about 50 G lower in field than
the peak of the broad line. (A similar shape was seen for
l
SiCO'in neon.) On other broad neon lines, there was only
the indication of a shoulder on the low field side.
Â¡In a pure matrix, it appears that most of the moleÂ¬
cules are bent, since a strong line of characteristic shape
appears at 9806 G as shown in Figure 19. The entire spec
l
trum is shifted upfield by about 200 G relative to that
measured in argon. In nitrogen, then, the D value for
SiN^ increases to 2.343 cm \ and E = 0.00152 cm ^.
Using Eqs.[108a, b] and the observed zero field splitting
(D, E) values, and assuming g11 = gg, resonant fields for the
z^ and z^ lines were predicted. A search of these regions
yeilded no signals in all matrices, even after signal
averaging. This is not unusual for linear triplets since
these lines are very weak and broadened by motional effects
(19). A summary of the D and E values derived from the
spectra of SiN2 in various matrices is given in Table I.
SiCO
. . 28 29 12 13
Similar experiments with Si, Si, CO, and CO
in argon yielded the ESR spectra shown in Figure 20. For
2 8 12
Si CO, an x2 line with halfwidth 5G was obtained at
9488 G with a y2 line at 9627 G, as measured at the points
indicated in Figure 20 (top). A broad very weak line
126
Figure 20. ESR spectra of SiCO molecules in argon matrices
at 4 Â°K.
127
appears between the two at about 9584 G. Substitution
of CO causes only line broadening as in Figure 20 (middle),
indicating that the 13C (I = 1/2) hyperfine splitting is
less than approximately 5 G. The linewidths are about twice
12 ...
as large here as with CO, and the line positions are
shifted about 3 G to higher field. Utilizing the same
equations as described above, and again assuming g^ = gg,
one obtains D = 2.236 cm ^ for Si^CO and 2.237 cm ^ for
13 1
Si CO, with E = 0.00215 cm for both species. An intenÂ¬
sive search for z lines at the predicted magnetic field
values in the range of the magnet yielded no other resonances.
28 29
A 50:50 mixture of Si and Si yielded the spectrum
in Figure 20 (bottom), where the hyperfine splitting is
29
clearly resolved for both the x^ and y2 lines. The A( Si)
value for the x2 lines is 30 Â± 1 G, and that for the y2 line
is the same within the larger experimental error for the
weaker line.
Several annealings to 20  25Â°K did not change the
general pattern of the lines of SiCO near 9500 G when observed
at 4Â°K. When the spectrum was measured at 25Â°K, the spectrum
changed from that at 4Â°K in Figure 21A to that in Figure 21B.
The entire spectrum became much weaker, the high field lines
apparently disappearing, and the x2 line shifting
7 G to lower field, as would be expected for some motional
averaging. Quenching to 4Â°K gave the spectrum in A. MeasureÂ¬
ment of the spectrum at intermediate temperatures showed only
a gradual diminution of the entire pattern and slight
Figure 21.
29SÂ¡ l3C0/Ar
13
Effect of temperature upon the ESR spectrum of Si CO in an argon matrix.
128
129
shifts to lower fields with increasing temperature until
at 25Â°K the spectrum in Figure 21B was obtained.
As with SiN^, trapping SiCO in neon at 4Â°K gave a
much broader spectrum of the same shape as that shown in
Figure 19. It appeared to also consist of a narrow line
superimposed on the low field side of a broad line at about
9700 G; the linewidth here was about 70 G. The upfield
shift of the spectrum from that in Ar results from an
increase in D by about 4%, to 2.333 cm ^ As with Si^ in
i
Ne, there was no evidence of a separate y^ line upfield
from the main signal.
In pure CO, the spectrum obtained is shown in Figure
22. It is much broader than in Ar, but the linewidth is
still only half as large as that observed for the neon signal.
Here, the signal moves still farther upfield than the neon
line, indicating a D value of 2.402 cm ; a separate y^ line
from nonlinear molecules is again absent. However, a
new feature at low field appeared; this was not observed
in any other spectra. Neither this nor the shoulder on the
high field side of the main line was removed by annealing.
As with Si^/ even the much greater intensity of the
signals which resulted from the use of pure CO as the
matrix reagent did not permit observation of z lines within
the range of the magnet.
A summary of the D and E values, line positions, and
widths, and observed hyperfine data for SiCO trapped in
various matrices is also included in Table I.
130
SiCO
Figure 22.
ESR spectra of SiCO molecules in Ar and CO
matrices.
131
"3
Table I. ESR data of SiN^ and SiCO in their E ground states
in various matrices at 4Â°K
X2 or X2 line
Molecule Matrix xy2 line (G)a y2 lines (G) halfwidth (G)
Si14N2
Ar
9503
(^9559)
12
Ne
9632
56
N2
9706
9806
15
Si15N2
Ar
9509
Ne
9632
28'29Si15N2
Ar
Si12CO
Ar
9488
(^9584), 9627
5
CO
9863
38
Si13CO
Ar
9491
(^9588), 9630
10
Ne
9720
70
28'29SiCO
Ar
9489
9627
aLine position determined by the criterion of reference 18.
^Halfwidth is here measured between the field at which the
derivative signal crosses the zero line and the field at
the negative peak,
c
Calculated assuming g^ = g0 =
2.0023.
132
Table I,
v (GHz)
extended
D (cm â– *')C
E (cm â– *") C
(G)
9.3900
2.206
(0.00087)
6.0 Â± 1 (14N)
9.3890
2.276
9.3880
2.343
0.00152
9.3910
2.209
7.4 Â± 1 (15N)
9.3890
2.276
9.3895
7 Q
34 Â±1 ( ^Si)
9.3890
2.236
0.00215
9.3865
2. 402
9.3900
2.237
0.00215
13
< 5 (1JC)
9.3910
2.333
9.3880
2.236
0.00215
30 Â±1 (29Si)
133
â€”2
3
Although Si , ground state Â£ , is certainly present
z. g
in most of the matrices as indicated by their ultraviolet
spectra (see below), it was not observed in the ESR. With
a maximum attainable field of 14 kG for the magnet, and
utilizing the equations for resonant field positions employed
above, the D value for Si2 must have a lower limit of about
4 cm ^ to put its xy signal outside the range of magnetic
fields available here.
Optical Spectra
Si and Si^
The mass spectrometric work of Honig (20) has established
that atomic silicon is the predominant vapor species over
hot silicon, and that Si2, Si^, and Si^ are present in about
1 to 2%. For these cluster species, the relative proportion
in the vapor decreased as the number of atoms in the cluster
increased. Although Si and Si2 often appeared strongly in
the optical matrix spectra, no bands attributable to other
pure silicon molecules were detected, even after warming to
allow some diffusion to occur.
Â° 1
Two strong, broad bands at 2320 A (43 090 cm ) and
O _ I O
2190 A (45 648 cm ), and a weaker one at about 2270 A
(44 039 cm ^) in argon matrices appear to be due to Si atoms.
In the gas phase, one strong line, originating from the
Â° 13 0
ground state, is observed at 2514.32 A (39 760 cm , P^ +â€¢
2PQ) , and a weaker line at 2207.97 A (45 276 cm \ <2Pq)
also appears (21  23). In general, atomic transitions are
134
blueshifted and sometimes split in matrices (23, 24).
Assignments are also complicated by the fact that intensities
can vary from the gas phase, with intensity perturbations
not necessarily the same for all bands. However, these
are generally within a factor of 5 to 10 of those for the
gaseous atoms, and it seems reasonable that the two strong
matrix bands correspond to the only observed absorptions for
the gaseous ground state atoms. No definite assignment can
be made, but the above correlation would indicate matrix
shifts V.  v of + 3330 and + 370 cm ^. Such shifts
Ar gas
are in the right direction for atoms isolated in Ar matrices,
and their magnitudes are within the range of shifts usually
observed; these are on the order of 1000 cm but shifts
from 80 to 4100 cm ^ have been reported (25). They do,
however, run counter to a general tendency for Ar matrix shifts
to increase with increasing energy (23). These lines
assigned to Si atoms were present in all matrices observed
in the ultraviolet region, and the bands were never
completely removed by warming to allow diffusion.
The SÃ2 spectrum is well known (26, 27) and, unlike
Si atoms, has been observed in matrices previously (28, 29).
3  3 
On occasion, extensive progressions of the Si^ H E^ *X E^
(4021  3562 A) and K3eâ€œ Â« X3E â€œ (3257  3011 A) transitions
u. y
were observed here in solid argon. In Table II, they are
compared with those reported by Milligan and Jacox (28),
which were prepared by photolysis of various silanes in
argon matrices. The differences probably lie within the
135
Table II. Si^ absorption bands in argon matrices at 4Â°K
This work
/ li a
v (cm )
Av
Milligan and Jacox
b
v (cm
Av
H + X
24863
25174
25408
25680
25920
26194
26421
26686
26918
27153
27384
27621
27831
28066
311
234
273
239
274
227
265
232
236
231
236
211
235
24937
25150
25425
25686
25953
26205
26448
26673
26910
27166
213
275
261
267
252
243
225
237
256
K Â« X
30694
449
435
30647
31143
31095
31578
31537
31950
32414
398
32812
33202
390
33213
33581
33981
34378
448
442
413
368
400
397
a 1
Estimated error = Â± 15 cm
^From data on photolyzed SiH^ in an argon matrix in Table II
of reference 28. Estimated error = Â± 45 cm
136
uncertainity in the measurements of band peaks, since the
lines were rather broad and the transitions occurred in a
region where the source lamp output and detector response
were falling rapidly. Use of a different detector and
source for the higher members of the progression (>27000 cm 4)
resulted in fair agreement with the expression (30)
AGv'+1/2 = we " 2{vâ€™+ 2) ueXe [141]
where AG^, + is the spacing and the frequency coe and
anharmonicity w x are given by references (26, 27).
Weltner and McLeod (29) report a much more regular progression
for the H Â« X transition of Si2 in solid Ne. The observed
K *X system is quite regular and in good agreement with
the above expression employing the data of Verma and
Warsop (27).
3 3
The D IT Â«x Â£ system was not observed in these
g g
experiments, possibly because of the appearance of a strong,
unidentified progression, with a vibrational spacing of about
870 cm \ in the same region of the spectrum, from 2790 to
O
2580 A. This latter system was produced in varying intenÂ¬
sity when Si was evaporated from a Ta cell and trapped in
Ar. Many attempts were made to identify this system. Its
relative intensity did not vary consistently with the observed
systems of Si2, suggesting that it was not a pure silicon
species. The vibrational frequency is probably too high
to be identified with a SiTa intermetallic compound which
137
might be suggested by the destruction of the cell material.
Another possibility was reaction of Si with impurities in
the Ta, the most abundant of which are oxygen and carbon.
Vaporization of SiC or mixtures of Si and C did not intenÂ¬
sify the observed signal. Formation of a siliconoxygen
compound is then a good possibility, considering that.the
1 1 . .
SiO IRx E transition was often observed between 2100 and
O
2300 A in the matrix spectra. In addition, work on vaporiÂ¬
zation of SiC>2 (31, 32) indicated the presence of an (SiO) 2
dimer with many vibrations near 800 cm 1 in the IR. If
the reaction producing the species is oxygenlimited, as
in the case of a small 0 impurity, and rich in Si and Si2,
the carrier of this band may be the SiSi0 molecule.
SiN2
A progression of bands separated by about 450 cm ^ was
O
observed beginning at about 3680 A when Si vapor was deposited
in an Ar/N2 = 200 matrix, as indicated in Figure 23. The
first three observed members of the progression are not
shown in that Figure, but the large anharmonicity in the
higher members is clearly evident. Also attributed to
O
SiN2 are two large single bands at 3109 and 2955 A, separated
by 1670 cm ^. This spacing is about what might be expected
for the NN stretching frequency in the excited state, but
it is not definite that these two bands belong to the
same electronic transition, since a third band at about that
spacing could not be found. However, the shapes of these
bands are similar, which, as described previously, is a
Ultraviolet absorption spectrum of Si^ molecules in an argon matrix at 4Â°K.
Arrow indicates increase in slit width.
Figure 23.
138
139
useful indicator of bands in the same transition in matrix
spectra. Also, the rapidly decreasing intensity of the
O
line at 2955 A suggests that the FranckCondon factors
are changing rapidly.
Unfortunately, the substitution of did not
establish the assignments. The vibrational frequency of
about 450 cm ^ (presumably the SiN stretching frequency)
in the first system did not shift very much when the
heavier isotope was substituted, and the NN stretching freÂ¬
quency does not appear in that system. An average of two
runs did indicate a slight decrease in vibrational spacing,
but the difference of less than 20 cm ^ is within the errors
of measurement of the broad bands, and so cannot be considÂ¬
ered definitive. In the second system, averaging the '*â€™5N2
values indicates a decrease in vibrational frequency which
is outside the experimental error. However, the concenÂ¬
trations in these two experiments were quite different,
which appeared to result in further broadening and line
shifts, so they may not be directly comparable.
Despite this difficulty, the infrared evidence (discussed
below) indicates reasonable correlation with the UV absorptions,
and thus these bands are tentatively assigned to two electronic
transitions of SiN2, as shown in Table III. The (0, 0, 0)
O
band of the A^X system at 3680 A cannot be definitely
specified, because the first few observed members of the
progression are quite weak, and could lie at longer waveÂ¬
lengths .
140
Table III. Ultraviolet absorption spectrum of Si^w in
an argon matrix at 4Â°K
1 a
v'
A' (A)b
v1 (cm ^
AGv + b <
0
^ 3680
A + X System
^ 27170
1
3622
27600
% 430
2
3562
28070
470
3
3503
28540
470
4
3448
28993
453
5
3398
29424
431
6
3352
29825
401
7
3313
30178
353
0
3108.4
B X System
32162
1672
1
2954.7
33834
aThe v' values assigned to the A Â« X system could be low
since the (0,0) band may have been too weak to be observed.
bAverages of measurements on two spectra. Estimated errors
are Â± 3 A and Â± 20 cm1 in the A + X system and Â± 1 A and
Â± 10 cm1 in the B < X system
141
The SiN2 spectrum was also observed, along with that
of SiO, Si^O (?), and Si2, when N20 (Aireo, purified) was
used as the reactive gas rather than N2.
When Si vapor was codeposited with a mixture of 1%
14
N2 in Ar, the infrared spectrum appeared to consist of
a strong, sharp band near 1730 cm ^ and a weak absorption
at 485 cm ^. At higher resolution, the lines were observed
15
to split, as indicated in Figure 24. The N2/Ar spectrum
14
is similar, but shifted from the N2 spectrum to lower
frequencies. When Si was deposited into a pure nitrogen
matrix, the splittings became more pronounced, and the
bands shifted in position, as shown in Figure 24. It is
difficult to anneal the nitrogen matrix in the liquid He
dewar, but the Ar matrix was annealed, resulting only in a
sharpening of lines. The splittings in both matrices are
then attributed to matrix effects, the guest species being
trapped in energetically inequivalent sites.
In Ar, then, the vibrations are assigned as (SiN),
485 cm and v. (NN) , 1731 cm \ for the Si^N2 species.
15
These data, along with the N2 data were used in a normal
coordinate analysis computer program, incorporating the
Wilson FG matrix method (33). The calculations were
repeated to produce the best fit to the observed data.
The results of these calculations are indicated, along with
the experimental data, in Table IV. The stretching force
O
constant values k^ = 11.83 and k^^ = 2.02 mdyn/A produce
an excellent fit when the interaction force constant is
zero. The bending frequency was not observed but, because
142
Ar/N2 = 200
l4N2 Matrix
v in cm
Infrared bands of SiN2 in argon and nitrogen
matrices at 4Â°K.
Figure 24.
143
Table IV. Vibrational frequencies and calculated force
constants (mdyn/Ã) for SiNN and SiCO molecules
in their ground states
SiNN
kSiN  2021'
kNN
= 11.825
SiCOa
kSiC = 5â€™3'
kco
15.6, ksicCO = 2,4
Frequency (cm â– *")
V3
Molecule
Obsd
Caled
Obsd
Caled
281414
1731
1732.6
485
484.1
281515
1676
1674.2
475
475.9
281216
1899.3
1899.3
(800)a
800.0
281316
1855.3
1855.4
793.7
281218
1856.4
1856.5
785.3
19 i r
aSiC stretching frequency in Si C 0 was assumed to be
800 cm1 since it was not observed but inferred from the
value in the excited state (see text).
it is of a different symmetry species, a value for it was
not required to perform the calculation.
The vibrational frequencies in the excited states are
in accord with decreasing bonding in the NN and SiN
bonds in the B<X and AÂ«X electronic transitions, respectively.
In neither transition is there an indication of a progression
in a bending frequency. This was thoroughly searched for
in the IR, especially after considering the ESR analysis.
The line must either be very weak, so as not to appear
even in the pure nitrogen matrix, or so low that is is
outside the effective operating range of the infrared
spectrometer.
SiCO
The absorption spectrum of this molecule is shown in
O
Figure 25. It begins at 4160 A and exhibits a progression
with spacing of about 750 cm \ A second progression ap
O
pears to begin at 3860 A, indicating that another vibraÂ¬
tional frequency, 1857 cm \ is also excited in the upper
electronic state. The members of the progression in the
lower frequency, assigned to an SiC stretch in the upper
state, are quite intense, and this progression begins to
repeat, starting at the second member in the C0 vibrational
progression of larger frequency. This pattern, a progression
of progressions, is just what is expected for the excitaÂ¬
tion of two vibrations in the excited electronic state.
The absence of a third progression in a bending frequency
indicates a linearlinear transition. The welldefined
ABSORPTION
3700
4100
I
O
X (A)
3900
Figure 25. Absorption spectrum of SiCO molecules in an argon matrix at 4Â°K.
145
146
shoulder at the high frequency side of each band could not
be removed by annealing, and is attributed to matrix
effectspossibly interactions with lattice phonons (34).
No other electronic transitions of SiCO were observed
O
between 8000 and 2000 A. The observed bands and their
assignments are given in Table V.
13 18
Isotopic substitution with CO and C O was, as in
the case of SiN2, not successful in confirming these
assignments. The same problem of broad lines and decreasing
efficiency of the spectroscopic equipment in this waveÂ¬
length region prevented observation of the expected small
shifts to lower vibrational frequencies.
In the IR spectra of argon matrices containing 1% CO,
two new bands appeared near 1900 cm 1, a strong, sharp line
at 1899.3 cm ^ and a similar line at 1928 cm \ which had
only about one quarter the intensity of the strong line.
13 18
Isotopic substitution with CO and C O produced the
same pattern, shifted to lower frequency, as shown in
Figures 26A and 27A. The weaker high frequency line in each
pattern is assigned to Si(CO)2, as described below. Thus,
only the C0 stretching frequencies at 1899.3, 1855.3, and
1856.4 cm ^ for the isotopic molecules Si^C^O, si^C^O,
12 18
and Si C 0, respectively, could definitely be assigned.
Considerable effort was expended to observe a SiC stretchÂ¬
ing frequency below 1000 cm \ but no band was detected in
neon or argon matrices which could be assigned to SiCO.
A matrix was also prepared by vaporization of Si into pure
147
Table V.
12
Absorption spectrum of Si
at 4Â°K
CO in an
argon matrix
(vi'v2'v3}
Aâ€™ (A)a
v1 (cm ^)a
Av^ (cm
b) Av (cm 1)
(0,0,0)
4155.7
24056
O
CO
CM
rH
(24218)
750
rH
O
O
4030.2
24806
(4007.0)
(24949)
737
1857
(0,0,2)
3913.9
25543
(3892.7)
(25682)
726
(0,0,3)
3805.7
26269
(1,0,0)
3858.0
25913
751
(1,0,1)
3749.3
26664
724
(1,0,2)
3650.2
27388
01
Maximum estimated error is Â± 1 A and Â± 10 cm
bBands in parentheses are the shoulders on the high frequency
sides of the strongest bands (see Fig. 25).
148
Figure 26. Infrared spectra at 4Â°K of an argon matrix conÂ¬
taining vaporized silicon atoms and l^co/Ar =
12co/Ar = 1/375: (a) initial spectrum;
(b) spectrum after warming to 15Â°K and cooling
to 4Â°K; (c) spectrum after warming to 35Â°K and
cooling to 4Â°K.
149
4 0 K.
150
CO, and a large IR band at 1909 cm ^ indicated that it conÂ¬
tained a high concentration of SiCO. A very weak low
frequency absorption did appear at 803 cm ^ in this case,
but it cannot be definitely assigned to SiCO, because of
the presence of a significant concentration of the dicarÂ¬
bonyl species in this medium at 1933 cm 1 (see below), and
other bands at higher frequencies due to more complex
molecules. Reasoning from the electronic spectra of SiCO,
the SiC frequency in the ground state is expected to lie
above and near 750 cm ^. The observed absorption in the
CO matrix is in this region, thus the value of vl^ == 800 cm ^
is assumed for computational purposes as the SiC stretch in
SiCO. The results of the normal coordinate analysis, along
with the calculated and observed frequencies of the different
isotopically substituted SiCO molecules, are given in
Table IV. The large interaction force constant probably
reflects to some extent the inaccuracy of the assumed
II
value of v^.
Si(CO)2
Formation of the Si(CO)2 molecule was suggested by
the appearance of a weak band at higher frequency than SiCO
for each of the isotopic variations, and the effects of
annealing the pure CO matrix containing silicon vapor. In
the latter case, annealing the matrix to 40Â°K caused the
1909 cm ^ SiCO band to disappear completely, with a concomiÂ¬
tant large increase in the intensity of the 1933 cm 1 band.
The relative separation of these lines can be seen to
151
agree closely with separations between the identified SiCO
bands and their weaker counterparts at higher frequencies.
That these features indeed represent formation of the
Si(CO)2 molecule can be seen most clearly in Figures 26
and 27.
To produce the spectrum in Figure 26A, a mixture of
12 13
equal parts of CO and CO in argon was reacted with Si,
12 13
yielding the two strong bands of Si CO and Si CO at
1899.3 and 1855.3 cm ^, respectively. Annealing to succesÂ¬
sively higher temperatures then gave the spectra in Figures
26B and 26C. It can be seen that the SiCO bands decrease
in intensity, and are replaced by the bands at 1928.0,
1899.6, and 1886.0 cm ^, with the middle one stronger than
the other two. These bands are attributed to the formation
of OCSiCO, the silicon counterpart of carbon suboxide.
13
The middle band is assigned to the mixed molecule, (O C) 
12
Si( CO), to be in accord with the expected 1:2:1 relative
intensity ratio predicted by the statistical probability
of reacting with the two ligands. Again, no bands at lower
frequencies could be definitely assigned to the molecules,
but weak higherfrequency lines did intensify on annealing,
as shown in the Figures. These must be attributed to more
complex species which increase in concentration as diffusion
occurs.
Figure 27 shows the IR spectra of a similar experiment
16 18
in which equal parts of C O and C O in argon were reacted
with silicon vapor. The three C0 stretching frequencies
152
here occur at 1928.0, 1897.5, and 1882.5 cm ^, and are
assigned to (C160)Si(C160), (C160)Si(C180), and (C180)Si(C180),
respectively. Because only one frequency was observed
in each case, the data for these molecules was not amenable
to the normal coordinate analysis treatment.
It should be noted, in connection with the Si(CO)2
species, that no analogous molecule was formed when N2
was used as the ligand. Similar results did not appear
even in pure nitrogen matrices. Further, attempts were made
to produce the N2SiCO molecule by reacting Si vapor with a
mixture of both ligands in Ar, but these were unsuccessful.
Thus no evidence was found for the formation of either
of these species, in analogy with the instability of the
corresponding N2CCO and C(N2)2 molecules.
Discussion
Since for SiCO, the SiC stretching frequency was not
detected in the IR, it was assumed to lie near 800 cm
because of the observed value of 750 cm ^ in the exctied
state. On that basis the stretching force constants in
Table IV were calculated. Both frequencies were detected
for SiN2, so there was no ambiguity there, and an adequate
fit to the data for two isotopically substituted molecules
could be made assuming the interaction force constant was
zero.
Then, although the SiC force constant in the ground
state of SiCO can only be inferred from the spectral data
153
obtained here, both the SiC and C0 bonds appear to be
quite strong, and very close to the carbon analogue
CCO, as shown in Table VI. Another illustration of the bond
strengths in these molecules can be obtained by considering
the force constants of the relevant diatomics. The diatomic
force constants, using the spectroscopic data of Rosen (35)
1/2
and the harmonic oscillator approximation v = l/27r(k/y) ,
where k is the force constant and y the reduced mass, are
O
19.0 and 12.2 mdyn/A for CO and , respectively, and
O
7.3 and 9.2 mdyn/A for SiN and SiO, respectively. The same
O
analysis yields a value k = 7.4 mdyn/A for SiC, based on
the estimate of 1226 cm ^ given by Weltner and McLeod (29).
Although the SiC molecule has not been observed, its bond
strength is expected to be in the same range as SiN and
SiO, so this estimate is reasonable. Thus it appears from
Table VI that the electronic redistribution accompanying
formation of a second bond on the central atom reduces the
bond strengths in CCO, CiCO)^ (36) and
SiCO very little, when compared to the diatomic components.
This also appears to be true with C^ (37, 38) and SÃC2 (29),
although the reduction in bond strength here is greater
(see Table VI).
For Si^, on the other hand, the changes are quite large.
The same procedure utilized above yields the diatomic force
O
constants 16.3 and 22.9 mdyn/A for CN and N2, respectively.
For CNN, it can be considered that the CN bond strength
increases at the expense of the NN bond, compared to the
diatomics, but both are still quite strong. The bond
154
Table VI. Comparison of stretching force constants (mdyn/A)
for relevant molecules XYZ
Molecule
k
xy
kyz
CNNa
19.5
14.7
SiNN
2.0
11.8
ccob
6.0
14.9
SiCOC
5.3
15.6
C(CO)2d
11.8
15.4
Sicce
7.4
8.0
C f
C3
10.3
interaction force constant = 3.9, reference 2.
interaction force constant = 1.4, references 1,3.
C\j. assumed to be 800 cirT^ (see Table IV and text) .
aReference 36.
0
Reference 29.
interaction force constant = 0.54, reference 37. (See
reference 38 for a summary of data).
155
strengths in the silicon species deviate markedly from the
diatomics, and, at least in the XY bond, a great deal from
the carbon analogue as shown in Table VI.
An interesting approach to aid in understanding the
differences between the SiCO and SiN^ molecules is to
perform molecular orbital calculations. This has been done,
at the complete neglect of differential overlap (CNDO)
level of approximation. It must be noted that the CNDO
approximation is too extreme to give a proper account of
the spin polarization contribution to unpaired spin density.
This is because it neglects onecenter atomic exchange
integrals which quantitatively introduce the effect of
Ilund's rule, according to which electrons in different
atomic orbitals on the same atom will have a lower repulsion
energy if their spins are parallel. Thus the intermediate
neglect of differential overlap (INDO) method is the lowest
level of approximation which can account for unpaired spin
densities. Unfortunately, at its present stage of developÂ¬
ment, the INDO method is unable to accommodate heavy atoms
with d orbitals, such as silicon. However, both the CNDO
and INDO methods have been shown to be capable of accommodat
ing total electronic charge distributions in a satisfactory
manner, and this is the type of information of interest here
A thorough discussion of these methods, their utility, and
their limitations is given by Pople and Beveridge (39).
The diagonal elements of the charge and bond order
(total density) matrix P for SiCO, SiN_, and the free
1 y v 2
156
ligands CO and N2 are listed in Table VII. These values
stem from a calculation at the minimumenergy geometry,
which is linear for both SiCO and SiN2, with internuclear
o o o
distances râ€ž. _ = 1.8 A, r_ _ = 1.21 A, râ€ž. .. = 2.0 A, and
SiC C0 SiN
O
r , â€ž = 1.14 A. The internuclear distances for the calcula
tions of the free ligand have been chosen to be the same
as in the silicon molecules; these differ from the actual
values, taken from the tables of Herzberg (40), which
O O
are rN_N = 1.094 A, and ^q_q = 1.128 A.
Using these values, in a manner analogous to that of
Olsen and Burnelle in their treatment of CCO (41), one can
compare total a and ptt charge densities of the CO and N2
moieties in the silicon species with those of the free
ligands. Here, the a charge density is taken as the sum
of the s + pz atomic densities on C or N , and the pr density
is the sum of the atomic p or p orbitals on C and 0, and
on and N^. The subscripts a and 6/ of course, only apply
to the bound species, a being closest to the Si atom. Then
the effect of odonation to form the SiX bond, X being
C or N , is given by the difference between the a density
in the free and bound ligands. For CO, the decrease in
o density from the free ligand is 2.80200  2.22646 =
0.57554, and for N2, it is 3.00000  2.75137 = 0.24863. Then
the effect of adonation is 2.3 times greater in CO than
in N2. Simultaneously, there is an increase in ptt density
in forming a bond with the Si atom, and this is given by
2.26964  2.00000 = 0.26964 for CO, and 2.03837  2.00000 =
157
Table VII. Total density matrix elements for SiCO, SiNâ€ž,
and the free ligands CO and ^
SiCO3
COb
SiN2C
N d
2
Si
s
1.75724
1.88128
Px
0.70481
0.95706
Py
0.70481
0.95706
Pz
0.83659
0.38817
d 2
z
0.02488
0.01738
d
xz
0.02555
0.00457
d
y z
0.02555
0.00457
do o
XZ_y^
0.0
0.0
d
0.0
0.0
xy
C or
s
1.26100
1.72637
1.51864
1.70792
N
a
Px
0.80180
0.55399
1.04205
1.00000
py
0.80180
0.55399
1.04205
1.00000
pz
0.96546
1.07563
1.23272
1.29208
0 or
s
1.73307
1.73454
1.70833
1.70792
N3
Px
1.46784
1.44601
0.99632
1.00000
p
1.46784
1.44601
0.99632
1.00000
y
1.29208
Pz
1.42177
1.46346
1.25348
CNDO values at energy minimum where rsÂ¿_c = l8
rC0 = 1,21 Aâ€™
} Â°
CNDO values where rc_Q = 1.21 A.
â€ž O
'CNDO values at energy minimum where rgi_N = 2.0 A,
rNN = 114 S
l 0
CNDO values where ^N_N = 1.14 A.
158
0.03837 in N^. The density as a result of nbonding with
the Si is then seven times greater with CO than ^â€¢ "ConsiderÂ¬
ation of the p or p orbitals on Si indicate that this
*y
excess electron density comes mostly from the p orbitals,
as the decreases in density there, from 1.0 in the free atom,
are commensurate with the increases in pu density on the
ligands. Then it appears that the silicon is actually
engaging in ttbonding contrary to the statement (8) that
the Group IVA elements, except carbon, "...do not form pu
multiple bonds under any circumstances." Similar bonding
has been found in Si and Si C molecules (29).
x x y
It can be seen also, by reference to Table VII, that the
participation of d orbitals is quite small in both species,
but the du density in d and d is six times greater for
SiCO than Si^. This brings up another unusual feature
of the bonding in these molecules, if they are compared to
transition metal carbonyl and dinitrogen complexes. The
situation is formally analogous to the du backbonding in
those systems, where the filled metal d orbitals push
electron density into the lowest ligand antibonding (it*)
orbitals (8, 4248). In silicon, however, the d orbitals
are initially vacant, and it appears that the bonding
ir orbitals in the ligands transfer electron density in the
opposite direction, that is, into the d orbitals. Caulton
et al. (42), in their comparison of CO and Nj as ligands in
transition metal complexes, did consider a udonation effect
from the Itt (bonding) ligand orbitals, but concluded that
159
this effect is counterbalanced by backdonation from the
filled metal d orbitals; the result being that the Itt
occupation was unchanged from that of the free ligand. With
the empty d orbitals of Si, this d^TT donation effect is
absent, and the ir>d effect is small, but significant, and
more pronounced in SiCO than SiN2> It should be noted that
both of these facts, that is, greater p orbital and d
orbital participation in the bonding of Si with CO than
N2, are in accord with the observation that Si(CO)2 does
form, but its counterpart Si(N2)2 was definitely not
observed.
From the ESR data it appears that SiN2 and SiCO molecules
are bent in some sites in some matrices. Judging from the
relative intensities of the x2 and y2 lines, SiN2 in pure
N2 matrix and SiCO in argon were cases where almost all
molecules appeared to be nonlinear. The CNDO calculations
on the two molecules confirmed that small departures from
linearity can be expected to be unstable with respect to
the linear conformation. A secondary minimum in the CNDO
energies did appear for both molecules when the SiXY
angle was between 60 and 90Â°, the exact minimum depending
on the bond lengths used. However, the quality of the
fit to the observed IR absorptions produced by the normal
coordinate treatment assuming linearity, the absence of a
progression in a bending frequency in the optical spectra
and, at least for SiN2, the observed hyperfine splitting
pattern in accord with nonequivalent N atoms, as well as
160
the known geometries of CCO and CNN, all indicate that the
species observed here are linear. Then the inference is
that the molecular bending force constant is quite low,
and the constraints in the matrix sites induce bending.
Wasserman et al. had similarly observed the spectrum of a
bent triplet molecule in an ESR study of C^N^ in a hexa
fluorobenzene matrix (5), and they interpreted it in the
same way. The low bending frequency of that molecule (36)
makes it a reasonable assumption, and it also seems
reasonable here. The bending frequency in SiC2 is 147 cm ^
(49), and it is probably even lower in SiCO. If this is
the case, it is quite clear why the bending vibration was
not observed with the IR instrument employed.
It is also clear that there are motional effects
upon the ESR spectra in the various matrices. In neon, which
is the least polarizable and usually the least perturbing
matrix, the ESR signals are unusually broad, and presumably
this stems from such effects. An extreme case of this effect
was observed by Ron (50) in matrixisolated C>2, where the
linewidth of the xy2 absorption varied from 25 G to
10,000 G, depending on the matrix material and, in some
instances, on the concentration. The assumption there was
that the molecular oxygen was showing some torsional
oscillation in the matrix around the equilibrium position,
with the potential well given by V = Vq (1cos20),
0 being the angle between the molecular axis and the
equilibrium direction. Such a treatment has also been applied
161
+o oxygen in clathrate compounds (51) , to CH2 in solid Xe
(52), and to CCO and CNN (6) in various rare gas matrices.
From Ron (50), the effective zero field splitting parameter
D' is given by
D' = D [1 + 3(av  l)/4] [142]
where D is the gas phase value. Since the resonant field
is related to D' by Eqs. [108d, f], there will be a downfield
shift of the resonance corresponding to the mean angular
deviation of the molecular axis from the equilibrium position,
which is perpendicular to the magnetic field. Thus the
observed resonance will be broadened. Increasing the
temperature will increase the average angular deviation and
give the same effect, and this was observed for SiCO (above).
2 1/2
This model (51) also gives (l  cos20)av = {Ã± /IVq) 7 / where
I is the moment of inertia of the trapped species (51). Thus
if D (the gas phase value) were known, the barrier height
could be calculated, but unfortunately it is not known for
these molecules. This relation also indicates that there
should be an isotope effect, with D', and therefore the
resonant field, increasing on substitution of a heavier
isotope. This is observed to occur in Ar matrices (see
Table I) for both SiCO and SiN2 with ^CO and ^5N2
substitutions, respectively. The isotopic shifts of the D
value are considered more thoroughly by Smith and Weltner
(6) .
162
Further consideration of Table I indicates that there
are other matrix effects on D, since the same molecule
in different matrices exhibits different zero field splitting
values. Kon (50) has found that, for O^, the position
and linewidth of the resonance signal still varies in
matrices which have very similar potential barriers. Smith
and Weltner (6) have treated the theory of such matrix
effects on the zero field splitting values of CCO and
CNN, and found two effects which contribute to an increase
in D, for positive D: (1) exclusion forces leading to inÂ¬
creased spinspin interaction by effectively compressing
the molecular volume, and (2) direct spinorbit effects due
to the electrons of the molecule entering the field of
the electrons and nucleus of the perturber (matrix). For
those moleules, these effects were small and dominated
by motional averaging and spinorbit mixing of singlet and
triplet states enhanced by the matrix, both of which led
to a decrease in D from the gas phase values. Motional
averaging appears to be present here, as discussed above.
Again, the gas phase values for SiCO and SiN2 are unknown,
but D does decrease, at least for SiN2 in going from Ne
to Ar matrices, in agreement with the theory that D decreases
as the spinorbit coupling constant of the matrix gas atoms
increase, therefore increasing singlettriplet mixing.
This effect is not apparent for SiCO, and no statements can
be made about the CO and N2 matrices, which have not been
adequately treated theoretically. The situation is probably
163
even more complex because of the matrixinduced bending
in these molecules. At any rate, the data here seems to
parallel that for CCO and CNN sufficiently to state that
the D values are obtained most reliably from the Ne data,
which probably provides a lower limit, with the gas phase
values perhaps 0.1 cm 1 higher.
From Table I, then, the neon values are 2.28 and
2.33 cm for SiN^ and SiCO, respectively, as compared to
1.16 and 0.74 cm ^ for CNN and CCO (6). The larger D
values for the silicon molecules are understandable
because of the larger spinorbit coupling constant of Si
(75 cm compared to C (26 cm . Consideration of the spin
densities points out some difficulties with this simple
conclusion.
Experimental and calculated spin densities are given
in Table VIII. The observed hyperfine splitting values were
converted to approximate spin densities by dividing by
the atomic data, as discussed in connection with Eq. [92a];
29 13
the atomic values are 40.7 G for Si, 38.2 G for C, and
14
19.8 G for N (53). The calculated values are the diagonal
elements of the CNDO spin density matrix for the pertinent
atomic orbitals, and must be considered approximate in light
of the discussion of the limitations of the calculation given
above. They are, however, in essential agreement with the
observed values for both SiCO and SiN2 Almost all unpaired
spins in SiCO are on Si and 0. Experimentally, hyperfine
splittings due to only one N nucleus were observed in
Table VIII. Spin densities in SiCO and SiNN
SiCO
Caled*5
SiN
nq
a 8
Obsda
Obsdc
Caled'
Si
pTT
0.74
0.657
Si
piT
0.84
0.939
dTT
0.015
diT
0.000
C
P 7T
<0.13
0.030
N
a
PIT
0.061
0
pir
0.298
Ne
piT
0.30
0.122
aHfs from Table I divided by 40.7 G for Si and 38.2 G for
(from reference 53). Here all observed spin density
on Si has been assumed pir.
b Â°
CNDO values at energy minimum where râ€ž. = 1.8 A, r_ n
1.21 A. Si L C U
CHfs from Table I divided by 40.7 G for Si and 19.8 G for
(from reference 53). All observed spin density on
nitrogen has here been placed on to conform to the CNDO
results.
r\ n
aCNDO values at energy minimum where r. = 2.0 A, rN_N
1.14 A.
165
the ESR spectra of SiN9, and the CNDO calculations indicate
that this is , as is also the case in CNN (6). However,
the calculated spin density on N^ is considerably smaller
than the experimental value. The CNDO calculations also
indicate that the unpaired it electrons have very little d
character in both molecules, and so all of the observed
spin density on Si has been assumed to be pn.
The relatively large spin density in the Si pTT orbitals
in these molecules amplifies the spinorbit contribution to
D mentioned above (54). It would appear, then,that the D
value of SiN2 should be larger than that of SiCO, based on
the spin density distributions given in Table VIII. Since
the different contributions to the zero field splitting
cannot be experimentally distinguished, as mentioned in
discussion of the D tensor above, one can only conjecture that
the spinspin contribution may be smaller in SiN2 than in
SiCO, even though the spinorbit contribution is greater.
The reasoning is that the larger spinspin contribution in
SiCO arises because of the relatively large spin density on
the small 0 atom, whereas the larger spinorbit contribuÂ¬
tion arises in SiN2 because of the greater spin density on
the heavier Si atom.
In analogy with C20 (4) and CN2, the ground state con
2 4
figuration of SiCO and SiN2 can be written as ... (4a) (2ir)
2 2 3 
(5a) (37T) , E , where small contributions from d orbitals
on Si may now enter into the molecular orbitals. The excited
3
ni state observed by Devillers and Ramsay (4) for C20 arises
166
2 4
for the silicon molecules from the configuration ... (4a) (2^)
1 3 3 1
(5a) (3tt) , II. , IT, and there were indications of vibronic
3 
mixing with an excited E state which can now arise from
the configuration . . . ( 4a) ^ (2 tt ) ^ (5a) ^ (3 tt ) ^ , ^Â£_, â€˜'"E.
The molecular term symbols here are all derived by the
standard methods outlined by Herzberg (38). The optical
data observed for all four molecules, including additional
information on CN^ (55) and the molecules SiC^ (49) and
(56) are summarized in Table IX.
For SiN2, one would tend to associate the transition
1 3 1
at 27 200 cm with a IL upper state and that at 32 200 cm
3 1
with the Â£ excited state. This would correlate the 470 cm
progression in the A+X system with a 3TrÂ«5a electron excitation,
weakening the SiN bond, and the 1670 cm ^ progression of the
B*~X system with a 3ttÂ«2tt excitation, weakening the NN bond.
If accurate g values were available, the first assignment
might be borne out, since interaction with a JL state is
expected to make g^ > g^, as discussed in connection with
Eq. [74].
For SiCO, the assignment is more difficult to make,
since the transition energy is more than twice that of the
3 3
Ih'X Â£ system of CCO, and therefore, might not involve
similar orbitals. However, it has been suggested that
substitution of Si for C results in an increase in a bonding
at the expense of tt bonding (29). This occurs as follows.
In general, a molecule with n odd has completely
filled shells, with a bonding it orbital lying highest, so
Table IX. Comparison of vibrational frequencies and electronic transitions of
CXY and SiXY molecules
Electr. Vibr.
State Mode
SiN2
CNNa
SiCO
CCOb
SiC2C
c d
3
V1
1733
2847
1899
1978
1742
2040
Ground v"
393
379
147
63
V.
485
1241
(800)
1074
852
1230
T
27200s
23800g
24100
11700
20100
24700
oo
x:
(cm1)
32200
V1
167 0f
1857
2046
1464
Excited v'
(340)g
608
145
308
v 3
470S
(1325)g
750
^1270
499
1086
aFrom references 2
and 3.
^From reference 4.
3
Ground state
Z, excited
state JH.
, Renner
parameter
Â£
= 0.172
0
From reference 49.
Ground state
bZ, excited
I1
state II
, Renner
parameter
Â£
= +0.023
^From reference 56.
Ground state
bZ, excited
state II
, Renner
parameter
Â£
= +0.537
0 3
Upper state possibly IT..
f 3 1
Upper state possibly Z.
9From references 2 and 55. Ground state JZ , excited state assumed to be II., Renner
parameter e probably near 0.17. (Note that for NCN s = 0.168, essentially the
same as CCO). vl, is derived here from the observed difference of 596 cm1 in
reference.55, assuming e = 0.17.
167
168
that its ground state is E^ (57). When a carbon atom is
added to make an even numbered molecule, two valence electrons
. . 3
enter a a orbital and two enter a tt orbital, giving a Â£
ground state of less stability. This has been confirmed
to some extent; C2 is a ground ^E state, but with a very
low
lying triplet excited state (58), is a E molecule
3
(37, 59, 60), and has a E ground state (61). For the
Si^ series the opposite is true; the lower members contain
another a orbital which lies lower than the bonding tt orbital,
3 
so that SÃ2 (26, 27) and Si^ (29) have ground states.
The higher members with n even add two a electrons and
3 
two tt electrons to a E molecule to give filled shells
9
and a ^E^ ground state. Thus there is a relative increase
in a bonding or decrease in it bonding when Si replaces C.
The variation in stability of the Si^ series appears in the
mass spectrometric results of Honig (20).
Then for an increase in the cttt energy, the transition
energy would rise, and the changes in vibrational frequencies
in the two molecules during excitation might be expected
to occur as in Table IX, that is, those of CCO all rising
and those of SiCO all falling. Thus the differences can be
rationalized. This weakening of the it bonding upon substituÂ¬
tion of Si for C is also expected in SiNN, but presumably it
is less drastic. This seems reasonable, based on the CNDO
charge densities discussed above, since SiCO had much more
involvement of silicon piT orbitals than did Si^
169
3 3
One difficulty with both of the IIÂ«X Â£ assignments is
the lack of discernible multiplet triplets, split by perhaps
50  70 cm \ in the ultraviolet spectra. These would arise
3
because the spinorbit interaction splits the upper ( II) states
the electronic energies of the multiplet components are
given by (38) = Tq + AAÂ£, where A is the orbital angular
momentum of the state, Â£ is the component of the spin S
along the internuclear axis, and A is the molecular spin
orbit coupling constant, which is probably near the atomic
value for Si. The inability to observe this splitting must
be rationalized as due to either unresolved structure hidden
under the broad bands, or to matrix effects leading to
distortion and/or shifts in the multiplet components. The
latter is prevalent in atomic spectra in matrices, as
discussed above.
It is noteworthy that none of the upperstate progresÂ¬
sions exhibit irregularities, indicating that if any of
those states are indeed II states, the Renner effect (62)
is small. This is a vibronic interaction which corresponds
to the splitting of the degenerate electronic state when
the bending vibration is excited. The apparent absence of
this effect is in accord with the very large drop in the
Renner parameter e, which is a measure of the electronic
vibrational interaction, from e = 0.537 for (56) to
e = 0.023 for SiC2 (49) .
For SiC2 and , which contain two less valence elecÂ¬
trons than the other molecules in Table IX, the observed
170
transition is but it still involves excitation of
a 5a (or 4a) electron into an empty 3tt (or 2tt) orbital,
where the orbitals in parentheses correspond to the carbon
molecule. This silicon molecule is consistent with the
other silicon species presented in the table, since
excitation into a 3tt orbital leads to a decrease in the
observed vibrational frequencies in each molecule. The
carbon molecules, on the other hand, appear to exhibit
more irregularities.
171
References  Chapter IV
1.
M.E. Jacox, D.E. Milligan, N.G. Moll, and W.E
Thompson, J. Chem. Phys., 43, 3734 (1965).
â€¢
2 .
D.E. Milligan and M.E. Jacox, J. Chem. Phys.,
2850 (1966) .
44,
3.
R.L. DeKock and W. Weltner, Jr., J. Amer. Chem. Soc.,
93, 7106 (1971) .
4 .
C. Devillers and D.A. Ramsay, Can. J. Phys.,
2839 (1971).
49,
5.
E. Wasserman, L. Barash, and W.A. Yager, J. Amer.
Chem. Soc., 87, 2075 (1965).
6.
G.R. Smith and W. Weltner, Jr., J. Chem. Phys
4592 (1975).
. , 6_2
r
7.
G. Herzberg, "Molecular Spectra and Molecular
II. Infrared and Raman Spectra of Polyatomic
Van Nostrand, New York, 1945.
Structure,
Molecules,
8 .
F.A. Cotton and G. Wilkinson, "Advanced Inorganic
Chemistry, A Comprehensive Text," Interscience,
New York, 1972.
9.
J. Drowart, G. de Maria, and M.G. Inghram, J.
Phys. , 29., 1015 (1958) .
Chem
â€¢
10.
P.W. Merrill, Publ. Astron. Soc. Pacific, 38,
175
(1926)
11.
R.F. Sanford, Publ. Astron. Soc. Pacific, 38,
177
(1926)
12.
C.D. Shane, Lick Obs. Bull., 13, 123 (1928).
13.
A. McKellar, J. Roy. Astron. Soc. Canada, 41,
147
(1947)
14 .
R. Treffers and M. Cohen, Astrophys. J., 188,
545
(1974)
15.
B.L. Lutz and J.A. Ryan, Astrophys. J., 194,
753 (1974).
16.
E. Herbst and W. Klemperer, Physics Today, 2_9
, 32
(1976)
17.
W. Weltner, Jr., Science, 155, 155 (1967).
172
18. E. Wasserman, L.C. Snyder, and W.A. Yager, J. Chem.
Phys., 41, 1763 (1964) .
19. R.A. Bernheim, R.J. Kempf, J.V. Gramas, and P.S. Skell,
J. Chem. Phys., 43_, 196 (1965).
20. R.E. Honig, J. Chem. Phys., 2_2, 1610 (1954).
21. C.H. Corliss and W.R. Bozman, Natl. Bur. Stand. (Ãœ.S.)
Monogr., No. 53^ (196 2).
22. C.E. Moore, "Atomic Energy Levels," Natl. Bur. Stand.
(U.S.), Circ. 467, Vol. 1, U.S. Government Printing
Offioe, Washington, 1949.
23. D.H.W. Carstens, W. Brashear, D.R. Eslinger, and
D.M. Gruen, Appl. Spec., 2J5 , 184 (1972).
24. B. Meyer, "Low Temperature Spectroscopy," Elsevier,
New York, 1971.
25. D.M. Mann and H.P. Broida, J. Chem. Phys., 5_5, 84
(1971) .
26. A. E. Douglas, Can. J. Phys., 3J3, 801 (1955).
27. R.D. Verma and P.A. Warsop, Can. J. Phys., 41,
152 (1963).
28. D.E. Milligan and M.E. Jacox, J. Chem. Phys., 52,
2594 (1970).
29. W. Weltner, Jr. and D. McLeod, Jr., J. Chem. Phys.,
41 , 235 (1964) .
30. J.I. Steinfeld, "Molecules and Radiation," Harper and
Row, New York, 1974.
31. R.F. Porter, W.A. Chupka, and M.G. Inghram, J. Chem.
Phys., 23, 216 (1955).
32. J.S. Anderson and J.S. Ogden, J. Chem. Phys., 51,
4189 (1969).
33. E.B. Wilson, J.C. Decius, and P.C. Cross, "Molecular
Vibrations," McGrawHill, New York, 1955.
34. J.M. Brom, Jr., W.D. Hewett, Jr., and W. Weltner, Jr.,
J. Chem. Phys., 6_2, 3122 (1975).
35. B. Rosen, "International Tables of Selected Constants 17,
Spectroscopic Data Relative to Diatomic Molecules,"
Pergamon, Oxford, England, 1970.
173
36. W.H. Smith and G.E. Leroi, J. Chem. Phys. , Â£Â£
1767 (1966).
37. W. Weltner, Jr. and D. McLeod, Jr., J. Chem. Phys.,
40, 1305 (1964) .
38. G. Herzberg, "Molecular Spectra and Molecular Structure,
III. Electronic Spectra and Electronic Structure of
Polyatomic Molecules," Van Nostrand, New York, 1966.
39. J.A. Pople and D.L. Beveridge, "Approximate Molecular
Orbital Theory," McGrawHill, New York, 1970.
40. G. Herzberg, "Molecular Spectra and Molecular Structure,
I. Spectra of Diatomic Molecules," Van Nostrand,
New York, 1950.
41. J.F. Olsen and L. Burnelle, Tetrahedron, 2_5 , 5451 (1969).
42. K.G. Caulton, R.L. DeKock, and R.F. Fenske, J. Amer.
Chem. Soc., Â£Â£, 515 (1970).
43. A.D. Allen, R.O. Harris, B.R. Loescher, J.R. Stevens,
and R.N. Whiteley, Chem. Rev., 73, 11 (1973).
44. K.F. Purcell, Inorg. Chim Acta, Â£, 540 (1969).
45. J.K. Burdett, J. Chem. Soc., Faraday Tras. II, 70,
1599 (1974).
46. G.R. Dobson, I.W. Stolz, and R.K. Sheline, "Advances
in Inorganic Chemistry and Radiochemistry," Vol. 8
(H.J. Emeleus and A.G. Sharpe, eds.), Academic,
New York, 1966, p. 1.
47. L.M. Haines and M.H.B. Stiddard, "Advances in Inorganic
Chemistry and Radiochemistry," Vol. 12 (H.J. Emeleus and
A.G. Sharpe, eds.), Academic, New York, 1969, p. 53.
48. R.D. Johnston, "Advances in Inorganic Chemistry and
Radiochemistry," Vol. 13 (H.J. Emeleus and A.G. Sharpe,
eds.), Academic, New York, 1970, p. 471.
49. R.D. Verma and S. Nagara j , Can. J. Phys., 5_2 , 1938 (1974).
50. H. Kon, J. Amer. Chem. Soc., _95, 1045 (19 73).
51. H. Meyer, M.C. M. O'Biien, and J.H. Van Vleck, Proc.
Roy. Soc. (London) , A243, 414 (1958) .
52. E. Wasserman, W.A. Yager, and V.J. Kuck, Chem. Phys.
Lett., 7, 409 (1970).
174
53. C.M. Hurd and P. Coodin. J. Phys. Chem. Solids, 28,
523 (1967).
54. H.F. Hameka, "The Triplet State," Proceedings of the
International Symposium Held at the American University
of Beirut, Lebanon, February 1419, 1967. Cambridge
University Press, Cambridge, 1967, p. 1.
55. W. Weltner, Jr. and D. McLeod, Jr., J. Chem. Phys.,
45, 3096 (1966).
56. L. Gausset, G. Herzberg, A. Lagerqvist, and B. Rosen,
Astrophys. J., 142, 45 (1965).
57. K.S. Pitzer and E. Clementi, J. Amer. Chem. Soc.,
81, 4477 (1959).
58. E. A. Ballik and D.A. Ramsay, Astrophys. J., 137,
84 (1963).
59. L. Gausset, G. Herzberg, A. Lagerqvist, and B. Rosen,
Disc. Faraday Soc., ^5, 113 (1963).
60. W. Weltner, Jr., P.N. Walsh, and C.L. AngelÃ, J. Chem.
Phys. , 4_0, 1299 (1964 ) .
61. W.R.M. Graham, K.I. Dismuke, and W. Weltner, Jr.,
Astrophys. J., 204, 301 (1976).
62.R. Renner, Z. Physik, 92, 172 (1934).
CHAPTER V
MANGANESE SPECIES
Introduction
The series of manganese oxide molecules presented here,
MnO where x is 1 to 4, have all recieved some consideration
X
in the past, in connection with widely different experiments
and observations. The terminal species in this series,
MnO and MnO^ (or its anionic counterparts), have received
the most attention. In both cases, the interest arises
through attempts to explain the bonding and spectra of these
molecules.
The detailed structure of molecular bonds involving
transition elements is still imperfectly known, even in
the simplest species such as diatomic compounds. This is
because of the multitude of states which can arise in such
compounds, making theoretical calculations quite difficult.
HartreeFock (HF) calculations on ScO(l), Ti0(2), V0(3), and
Fe0(4) have illustrated the problem of obtaining accurate
relative locations of electronic states, particularly where
the large number of valence electrons gives rise to a large
number of lowlying configurations.
The MnO molecule has also generated theoretical
interest, with studies ranging in sophistication from semi
empirical (5  7) to a full scale selfconsistent field
175
176
calculation with configuration interaction (SCFCI) included
(8). These have been undertaken in attempts to account
for the green system in the MnO electronic spectrum, origiÂ¬
nally observed by Das Sharma (9) and later by others (10  12).
Pinchemel and Schamps (12) have rotationally analyzed the
10 band of this system, and identified it as part of the
A^E+^X^Z+ transition. This identification of the ground
state is supported by the matrixisolation study of the
same MnO optical spectrum by Thompson et al. (11). The SCFCI
calculation (8) supports the ground state as ^Z+, the high
spin arising essentially from the five 3d electrons on the
manganese atom.
The MnO^ and MnO^ molecules have not been observed
preÃ¡ously, and have apparently not been studied theoretically.
Cations of these species have been proposed to occur on the
basis of chemical reasoning, but have not been characterized
3+ +
spectroscopically. Thus MnC^ and MnO^ have been suggested
as intermediates resulting from neutron irradiation of
KMnO^ crystals or solutions (13  15); the latter ion has
also been discussed as the manganese species responsible for
the green color of strongly acidic KMnO^ solutions (16, 17).
Royer (16) has presented a molecular orbital (MO) scheme
for MnO^+ which is consistent with the optical spectra of
these solutions.
After the first observations of the MnO^ spectrum by
Teltow (18, 19), and the first explanation of the spectrum
using MO theory by Wolfsberg and Helmholz (20), an extensive
177
theoretical literature developed on the properties of the
xâ€”
tetroxyanion series MnO^ , where x = 1, 2, 3 (2131).
Most of these studies viere initiated to explain the spectra
of MnO^ , or to gain some insight into estimation of paraÂ¬
meters for use in semiempirical calculations of other
transition metal complexes. The observed members of this
series seem to be very similar in electronic structure to
MnO^ . The neutral MnO^ molecule has never been reported
and it has received theoretical consideration only recently
(31). It appears that the neutral species has been observed
in this research, although its assignment is somewhat
tenuous, and that its properties can be interpreted by the
same MO structure as found for MnO^ .
This research, then is an ESR investigation of the
four manganese oxides MnO, MnO^, MnO^, and MnO^, as well
as the Mn+ ion, which has been previously observed by Kasai
(32, 33), in inert gas matrices at 4Â°K. The observed magnetic
parameters will be interpreted in terms of molecular
geometry and electronic structure.
Experimental
The species of interest were prepared by the condensation
of Mn atoms with samples of Ar or Ne containing 0.5 to 2.0%
02, N?0, or O2 reagent. A beam of Mn atoms was produced by
vaporization of 99.9% pure manganese powder (Spex Industries,
Inc.) from a Ta cell of the type previously described, but
with 0.5 mm wall thickness, which was resistively heated to
about 1075Â°C. The beam was cocondensed with the reagent/inert
178
gas mixture on a flat sapphire rod as described above. In
most cases, depositions were continued for 1 hour with gas
flow maintained at a rate of ^0.3 mmole/min.
The Ne and Ar matrix gases used were Aireo ultrapure
grade. These were mixed with the reactive gases O2 (Aireo,
ultrapure) or ^0 (Aireo, purified) by standard manometric
techniques. For one experiment, a mixture of Ar with
17 18
C>2 = O2 ~ was prepared similarly by dilution of an
isotopically enriched sample from Miles Laboratory (52.4
17 18
at.% 0, 47.2 at. % 0) . Such samples were used without
further purification except for passage through a liquid
nitrogen cooled trap prior to deposition.
Samples of 0^ were produced by Tesla coil discharge
of C>2 gas in a Pyrex bulb at a pressure near 75 torr for
about 1.5 hours. The resulting ozone was condensed to a
liquid in a Pyrex finger cooled with liquid N2, outgassed
at 77Â°K, further discharged and recondensed as described by
Spiker and Andrews (34). The final product was then diluted
with Ar or Ne. No infraredactive contaminants were deÂ¬
tected in these mixtures. Ozone samples were always used
within one day of preparation. Due to reaction with the
copper gas manifold described previously, O^/rare gas mixÂ¬
tures were admitted to the system via a Pyrex delivery tube,
the flow rate controlled with a Teflon needle value. These
samples were not cooled to 77Â°K prior to deposition.
Sample photolysis was accomplished with either the
General Electric AH6 high pressure mercury arc lamp, or
179
the microwavepowered flowing H2~He discharge lamp; both have
been described above, and the latter also by Milligan and
Jacox (35). In some experiments, both lamps were used.
Irradiation was conducted during or after sample deposition,
or both, for periods of 45 minutes to 4 hours, through windows
of quartz, CaF^, or LiF; the first was used only with the
Hg lamp. A Corning 754 glass filter was used with the Hg
lamp in some instances in order to limit the amount of
infrared radiation reaching the sample.
ESR Spectra
Mn Atoms
Immediately upon deposition, all matrices studied here
exhibited the strong sextet of lines due to the presence of
manganese atoms in their S5/2 cJroun<^ state (see Figure 28) .
55
In 100% natural abundance, Mn (I = 5/2) create the sixline
hyperfine pattern centered at g = 2 with A = 27.9 G, as
reported by Kasai (32, 33). As with that work, these
signals were never completely removed by photolysis or
reaction. However, in reactions with 0^ in solid Ne,
they were strongly diminished, to the point of no longer
being the most intense features in the spectrum.
Mn+
Photolysis of argon matrices containing Mn and 02,
N20, or 0^ all produced spectra attributable to singly
ionized Mn, similar to that shown in Figure 28. These
features were never observed in neon matrices. Five hyperfine
groups, each containing three to six fine structure lines,
VIn+: Ar
Mn
2200 2600 3000 3400 3800
Figure 28. ESR spectrum of Mn+ in argon at 4Â°K.
1B0
181
could be resolved. The variable spacings between fine
structure lines of different hyperfine groups is attributed
to an incomplete PaschenBack effect, as indicated by Kasai
(32, 33), and previously described. Utilization of any
of the above gases as electron acceptors produced spectra of
*
Mn of varying intensity, depending to some extent on the
type and concentration of the gas, photolysis duration, and
excitation energy. These facts are all in accord with the
solid state electrontransfer mechanism proposed by Kasai
for such matrices (33).
This mechanism can be described as follows. The donor
species D with ionization potential Epp is trapped in
the Ar matrix along with the acceptor A, of electron affinity
E . The concentrations are such that each species is
isolated from the others in its ground state, but neighboring
molecules in their excited states have potential curves
which overlap substantially. The host matrix serves
to keep the species rigidly fixed at their respective positions.
Then it is possible to transfer an electron from D to A
with excitation energy AE considerably lower than the
ionization potential of D. The energy required is
E = EIp(D)  Eea(A)  e2/eR [143]
where e is the dielectric constant of Ar and R the distance
between the resulting ions. The last term is then the
Coulombic potential energy between the anioncation pair,
182
and can become as large as 2eV for an average 78 A separation
between the ions. Calculations (36) show that the resulting
ions have electronic distributions very close to those of
the free ions. This is substantiated by the fact that the
spectra usually showed a more isotropic appearance than in
Figure 28, which was chosen for clarity of the observed
lines. The anisotropy is presumably due to the presence of
too high a concentration to provide complete isolation of
the resulting ions. Signals expected from the 0^ or 02 ions
were never observed, probably because they would appear
under the very strong Mn*"* sextet (32, 33).
Calculations were undertaken to determine the origin
of the fine and hyperfine structure lines exhibited by
Mn+. The energies of the levels are given by Eq. [15], as
described previously. There are then twelve sets of equations,
corresponding to m = m + m = Â±11/2, Â±9/2, ...,Â±1/2.
To illustrate the use of this equation, several of the sets
are given. For m = +11/2 (m^. = 5/2, m = 3) , it is
5/2,3
(AWh  A (5/2 ) ( 3)  3Gj +(5/2) G]
= 0
[144]
where G = g RH, G = gjgH, and the other symbols have been
u u XX
previously defined. For m = +9/2(m^ = 3/2, m^ = 3, and
mj. = 5/2, irij = 2), they are
 X
5/2, 2
(Â§) ( +  + !) (3  3 + i;
X 3/2, 3
+
(AWh  A(r)(3)  3Gj + ^z)
= 0
[145a]
183
X t:
5/2, 2
(AWr  A()(2)  2Gj + iGj)
3/2, 3
() (   + i) (3 + 2 + 1)
= 0
[145b]
For m = 7/2 (m = 1/2, in = 3; m = 3/2, m = 2, and m = 5/2,
L U _L u L
m = 1), they are
U
 X
3/2, 2
() ( + i + i) (3  3 + 1)
1/2, 3
(AWr A(j) (3)  3Gj +
= 0 [146a]
X
5/2, 1
() ( +  + D (3  2 + 1)
+ X 3/2, 2
(awh 
A(j) (2)
2Gj +
 X1/2, 3
() (   + 1) (3 + 2 + 1)
= 0 [146b]
5/2, 1
<%  a  gj + K
3/2, 2
() (   + 1) (3 + 1 + 1)
= 0. [146c]
The other sets of equations follow similarly. The matrices
(which are not symmetric) for each m value were expanded and
the resulting polynomials solved at different values of
H and A by computer iteration. Repetition of these calculaÂ¬
tions produced a fit to the observed spectrum, with energy
differences calculated to within 1% of the experimental micro
wave quantum energy. Assignments for the observed transitions
obeying the selection rules Am = Â± 1, An = 0 are listed
J i.
in Table X, and indicated in Figure 29. The m^ = 1/2 lines
were obscrued by the strong manganese atom signals. Of the
36 possible transitions, 23 were obscured and fit in this
manner, with the calculated hyperfine coupling constant
A = 275 G.
MnO
The ESR spectrum of MnO trapped in its 6X ground state
in solid Ar is shown in Figure 30. The spectrum was observed
in a variety of Ar matrix experiments, but was never
detected in solid Ne. Vacuum ultraviolet irradiation of 02/Ar
mixtures produced weak MnO signals, partially obscured by
a much stronger system attributed to MnO^, and in one experiÂ¬
ment, very weak MnO signals were observed in the unphotolyzed
sample. Optical spectra showing the A^E^X^E bands of MnO
have been observed in such matrices by Thompson et al. (11),
working in this laboratory.
Mercury lamp photolysis of Ar matrices containing
N^O and Mn atoms yielded far more intense signals for MnO
and allowed observation of all members of the sextet hyperÂ¬
fine pattern by reducing the probability of forming the
dioxide species. Magnetic parameters deduced from these
experiments are presented in Table XI and further described
below.
ENERGY (GHz)
Figure 29.
185
Table X. Field positions (in gauss) of observed fine and hyperfine structure lines
of Mn+:Ar at 4Â°K. A = 275 G; v = 9390 MHz
m.
m,
5
2
3
2
1
2
1
2
3
2
5
2
2
3
2413a
4154
1
2
2525
2767
3052
3702
4080
0
1
2605
2843
3109
3690
4023
1 ~
0
2668
2898
3149
3676
3981
2
1
2715
2939
3175
3945
3
2
2974
3193
Estimated uncertainty = Â± 5 G for all field positions.
186
H (Gauss)
Figure 30. ESR spectrum of MnO in argon at 4Â°K.
187
188
Reaction of Mn atoms with 0^ led directly to observation
of MnO without photolysis, along with MnO^ and signals pecuÂ¬
liar to Mn0^ reactions. The MnO spectrum was weak but
quite distinct immediately upon deposit, and became much
more intense after mercury lamp photolysis. In one experiÂ¬
ment, the variabletemperature feature of the dewar was
utilized to maintain the rod near 20Â°K during the deposiÂ¬
tion. This treatment produced quite strong signals of MnO
which did not increase in intensity on subsequent photolysis.
The absence of the signal in neon, which does not quench as
efficiently as argon, and the intensity behavior when
deposited at a high temperature, probably indicate that
the reaction occurs quite readily on the surface of the matrix,
before quenching to 4Â°K in Ar; the reaction could not be
halted rapidly enough to isolate MnO in the "softer" neon
matrix.
Omitting the weak nuclear electric quadrupole and nuclear
Zeeman interactions, the spin Hamiltonian for this molecule
can be written
H . = g
â€”spin
<3H S
zâ€”z
+ g 6 (H S
T
xâ€”x
H S )
yy
D (S â€˜
â€”z
 S(S + 1) )
+A
S I
â€”z;â€”z
+ A i (S I + S I ) ,
J_ â€”xâ€”x â€”yâ€”y
[147]
189
which, for the ^E molecule, is just the electronic spin
Hamiltonian of Eq.[133] plus the hyperfine Hamiltonian in
axial symmetry. The zero field splitting parameter D
can be obtained from the spinspin interaction constant
derived by rotational analysis of optical spectra. A
value of A" = 0.66 cm ^ has been determined by Pinchemel and
r â€” 1
Schamps (12) for the X E state of MnO, yielding D = 1.32 cm
As described previously, for D>>hv, as in the case here
(hv = 0.3 cm "*â– ) only the 1+^)Â«~> ^) transitions will be
observed. The solutions for the allowed transitions, to
second order in the hyperfine energy, are given by Eq. [73];
the perturbation treatment is valid for A
Thus the ESR spectrum should consist of an xy (perpendicÂ¬
ular) line near g = 6 and a z (parallel) line near g = 2,
each split into six components by interaction with the
magnetic nucleus. The possible highfield xy line, which
is predicted by Figure 16 to occur near 8500 G, represents
an apparent transition between the levels with "high field"
quantum numbers = Â±3/2. Because there is significant
mixing of levels, it is allowed, and with a relative transiÂ¬
tion probability near that of the g = 6 line (37). It can
be seen from Figure 16, with represents the exact solutions
to the spin Hamiltonian for D = 1.32 cm \ that the absorption
envelope for this transition, for randomly oriented molecules,
is spread out over a range of 7300 G, from 1200 G for the z
line, to 8500 G for the xy line. In contrast, the absorption
pattern for the +1/2)1/2) transition is spread over only
190
2000 G. As discussed previously, it is only the turning
points that are seen in the first derivative spectrum of
randomly oriented molecules; the turning point (at 0 = 90Â°)
for the highfield line is, because the absorption covers
such a broad range of fields, not as sharp as for the
g = 6 line, and the signal was not detected. The parallel
component (z^) of the +1/2) Â«>11/2) absorption pattern is
superimposed on the intense Mn^ lines and cannot be resolved.
Thus only the xy^ line at g = 6 was observed. A comparison
of the observed field positions with those calculated
from Eq. [73] is given in Table XI, and the fit is within
experimental error.
0
Since D and the effective g^ values are known, the
treatment of Kirkpatrick et al. (38) can be applied, and
Eqs.[129] and [131] yield the value g^ = 1.990 from the
0
experimental g^ = 5.952. The negative value of Ag^ =
g^  ge indicates significant interaction with excited II
states. This was mentioned above, and will be discussed in
further detail below for the case of MnO.
Isotropic and dipolar hyperfine coupling constants,
derived from the experimental data through Eqs. [86] and
[87] are also given in Table XI, and will be discussed
more fully below.
Mn02
The strongest features in the spectrum resulting from
the reaction of Mn atoms with 02 in solid argon are attrib
4
uted to MnC>2 molecules in a ground X state. The sixline
191
Table XT.
Magnetic parameters, observed and
line positions for the 1+1/2)
pendicular transition of MnO (^Â£)
calculated
I/2) per
in Ar
A (G) = 157 (4)a
A. (MHz)
= 353 (
1
iso
A( (G) = 63(3)
AdiP
= 87 (
g = 1.990 (7)b
D (cm b)
= 1.32
v (MHz)  9380(1)
Mi
H , , (G)d
obsd
Hcalcd
5/2
718
717
3/2
848
849
1/2
993
993
1/2
1151
1151
3/2
1318
1320
5/2
1506
1504
aAll components of A tensor are for interactions with the
S^Mn nucleus; A., and A assumed positive,
b II 1
Assuming g. = g .
c
Reference 12.
uEstimated uncertainty = Â± 2 G.
192
multiplet centered at g = 3.99 is shown in Figure 31?
this Figure also shows, at the high field side, several very
prominent lines of Mn+ and, at low fields, members of the
weak MnO progression. Mercury or hydrogen lamp photolysis
for periods of one hour or more appeared to be equally
efficient in production of this species. This molecule
was never observed in Ne matrices, but could be readily
formed in solid Xe. In this case, the spectrum exhibited
extreme broadening, probably due to both motional effects
and interactions with various magnetic Xe nuclei; the
signal here disappeared completely on warming to only
25 Â°K.
While MnC^ was never detected when N^O was used as
the oxygen donor, it did form directly on deposit when 0^
was used for that purpose. In that experiment, the weak
MnC>2 lines were partially obscured by the strong MnO
pattern; both signals intensified somewhat on photolysis.
Analogous to the VO (39) and NbO (40) molecules,
4
the strong perpendicular line at g  4 suggests a Â£
molecule with D > hv; reference to Figure 15 also suggests
that this is the case. Utilizing Eqs.[129] and [131] for
S = 3/2, and assuming g^ = g j  = gg, the expression
3 ig^pn; 2 
J1 â€œ â€™
' 3 (g1BH)
16 â€ž2
D
[148]
193
Mn02 in Ar at 4Â°K
600 1000 1400 1800 2200 2600
H (Gauss)
ESR spectrum of MnC>2 in argon at 4Â°K.
Figure 31.
194
0
is obtained. Then for the observed gj^ = 3.99, the zero
field splitting parameter D = 1.13 cm ^, considerably
larger than the microwave quantum of about 0.3 cm ^.
4
As discussed above in the section on E molecules, and
in analogy to the ^E case, for D > hv only the perpendicular
transitions between the M = Â±1/2 levels are observable;
u
the expected parallel (z2) line near g = 2 is again
obscured by the stronger MnÂ° signals. Other possible
perpendicular lines would occur outside the range of the
magnet, or be undetectable because of the broad absorption
patterns previously described. Since A < ggH, the second
order solution to the resonant fields is again valid. Then
the application of Eqs. [73], [86], and [87] yield the
magnetic parameters and calculated fields listed in
Table XII.
It is noteworthy that introduction of ^70 (i = 5/2)
into the molecule had no detectable effect. The three
possible isotopic products of a reaction of Mn atoms with
17 18
equal parts of C>2 and 02 are expected to produce a total
of eleven lines centered on each member of the Mn hyperfine
55
multiplet (assuming the Mn splitting is greater than the
170 splitting). While the splittings may be beyond
resolution, one expects at least a broadening effect if
the unpaired spins interact with the oxygen nuclei; none,
however, was detected.
195
Table XII. Magnetic parameters, observed and calculated
line positions for the 1+1/2)  â€”1/2 ) perÂ¬
pendicular transition of MnC^i^E) in Ar
A (G) = 261 (4)a
A. (MHz) =
603(22)
1
ISO
A (G) = 126(4)
A,. (MHz) =
dip
126 (22)
g  2.0023b
D (cm â– *â– ) =
1.13C
v (MHz) = 9390(1)
M
I
H , ,
obsci
H . .
caled
(G)
5/2
993
996
3/2
1203
1207
1/2
1440
1442
1/2
1698
1703
3/2
1990
1988
5/2
2298
2298
aAll components of A tensor are for interaction with the
55Mn nucleus; A and A assumed positive,
b II x
Assumed value.
c
With gâ€ž = g = g . See text.
d ^ e
aEstimated uncertainty = Â± 2 G
196
Mn03
Among the systems under consideration here, the only
one to lead to products isolated in solid Ne is that of
Mn and 0^. The resulting ESR spectrum is displayed in Figure
32. It is interesting to note that a large central sextet
pattern due to Mn atoms is not present in this spectrum,
and such a strong multiplet was never observed in Ne matrices.
Again, this is probably due to the inefficiency of quenching
in neon. The most important feature of this spectrum is
the series of six strong perpendicular lines spanning the
range of magnetic field from 1700 to 4700 gauss, separated
by nearly 550 G, and exhibiting a very large secondorder
effect. Weaker parallel lines are detectable near each
member of the previous series, both groups centered near
g = 2. These lines can best be interpreted as arising from
an axially symmetric molecule of spin S = 1/2; the presence
of one Mn atom accounts for the sixline hyperfine patterns.
The positions of g j and g^, as well as the relative
magnitudes of A j j and A^, account for the change of phase
in the lines observed at the third member of the series
from the lowfield side. These lines are unique to matrices
containing 0^, and appeared directly upon deposit in both
Ne and Ar. Photolysis with the high pressure mercury lamp
had little effect on the neontrapped species, other than
sharpening of the lines. Similar photolysis of Ar matrices
intensified these lines, which are attributed to MnO^.
197
Figure 32. ESR spectrum of MnC>3 in neon at 4Â°K.
198
Here the second order solution of Eq.[73] is not valid
because of the larger hyperfine splittings, and an exact
treatment was necessary. For axial symmetry, the spin
Hamiltonian is given by Eq. [72], and the nonvanishing eleÂ¬
ments of the Hamiltonian matrix are (41), operating on
the twelve spin state M,m)
m
[149a]
(MÂ±l,mSÂ±I_' M,m) = m [S (S+l) M (MÂ±l) ] 1//2 [149b]
(Mil, mÂ±lIS1!1M,m) = [S(S+l)M(M+l)]l^2â–
[I(1+1)m(mÂ±l)
1/2
[149c]
(Mil, m+l  SÂ±I+  M, m) = [S (S + l)M(Mil) ]1//2 â€¢
[I(1+1)m(m+l)]
1/2
[149d]
where M is the electron spin quantum number and m is the
nuclear spin quantum number. For 0 = CP and 0 = 90Â°,
the sin@cos0 term in Eq.[72] vanishes, and the resulting
matrix, which is symmetric about the diagonal, is given
in Table XIII. Here the symbols G = g^8H/2, where i is
J_ for 0 = 90Â° and  for 0 = 0Â° and X+ = A Aj/ (4K) iAj/4,
= Aj^ or A11 for 0 = 90Â° or 0Â°, respectively,
where K
199
Table XIII. Spin Hamiltonian matrix for the states M,m) for
MnO,(2A.), including interaction with the 55jy[n
(I = 5/2) nucleus
1 5
2'2
I! 1
12'2
I 1
2'2
>
>
>
1 5N
1 5\
I1 3N
1 3N
1 XN
2'2/
2'2/
1 2'2 /
2'2/
2'2/
G + f K
0
0
/5X_
0
G 
5
4 K
/5X+
0
0
G + k 0 0 0
G  k /8X+ 0
1
2'2/
1 IN
2â€™2/
I IN
2'2/
I IN
2'2 /
I IN
2'2 /
1
2â€™2/
I ^
2'2 /
1 lN>
2'2 /
200
Table XIII, extended
1 !\
1
1 3\
1 3\
1 5\
T'2/
2'2 /
2'2 /
2'2/
2'2 /
0
0
/8X
0
0
0
0
0
0
3X_
0
0
G +
0
0
0
0
0
0
0
/8X+
0
0
0
0
0
/8X_
0
0
0
G + k
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
/5X_
0
0
G + k
MU1
201
are used for clarity. Notice that rows and columns with
the same values of M + m are adjacent. The secular deterÂ¬
minant was solved at the magnetic fields of the observed
transitions by computer iteration. Varying the magnetic
parameters A11 , A^, g 11 , and g^, the calculations were
repeated until acceptable accuracy in the energy differences
was obtained. Field positions for the MnO^ lines in Ne and
the derived magnetic parameters are included in Table XIV.
Also listed in Table XIV are the line positions for
observed transitions of MnO^ in solid Ar. The spectrum
is presented in Figure 33, and, even in this broad scan,
it is evident that many of the lines are doubled in this
spectrum, compared to Figure 32. This is attributed to the
trapping of MnO^ in two different sites in the Ar matrix.
The site structure could not be annealed away by warming
to temperatures as high as 28Â°K. However, it appears that
one site is selectively populated at higher temperatures,
as illustrated in Figure 34. This shows the m = 3/2
component of MnO^ in both Ar and Ne matrices, and the effect
of temperature on the site splitting in solid Ar. Trace
(a) shows the original spectrum, run at 4Â°K. As the sample
is warmed to 23Â°K (trace (b)), the lower field components
of both the parallel and perpendicular lines are disproporÂ¬
tionately reduced in intensity. This spectrum was run
after the rod temperature had been maintained near 20Â°K
for approximately 10 minutes. When the sample is returned
to 4Â°K, all lines are intensified again, but the population
202
Figure 33. ESR spectrum of MnC>3 in argon at 4Â°K.
203
2
Table XIV. Magnetic parameters of MnO^ ( A^) in Ne; observed
transitions in Ne and Ar
A (G) = 633 (1) a
A.
iso
(MHz) = 1613(6
AÂ± (G) = 547(1)
A . â€¢
dip
(MHz) = 81(3
g( = 2.0036(8)
g = 2.0084
r l
V (MHz)
= 9383(1)
kt b
Ne
. c
Ar
H (G) (G)
Hn ,G
) HÂ± (G)
5/2
1640 1828
1671
1861
1698
1896
3/2
2082 2150
2108
2181
2134
2216
1/2
2617 2580
2662
2603
2678
2628
1/2
3254 3132
3266
3146
3286
3159
3/2
3994 3804
3994
3804
4028
5/2
4826 4586
4790
4550
4808
4571
aAll components of the A tensor are for interaction with
the 55f4n nucleus; A., and A assumed positive.
L II J
Estimated uncertainty = Â± 1 G
CEstimated uncertainty = Â± 2 G
204
Figure 34.
Hi =
(b)
trum
at 23Â°k, ai
in neon at
Mn03 in
returned
argon at (a
to 4Â°K; (d)
) 4 Â°K
spec
205
of the higherfield site is increased relative to that of
the other. Trace (d) shows the spectrum in Ne for comparison.
Note that for this final curve, the spectrometer gain is less
than onethird that of the other scans.
MnO.
  4
In addition to the MnO^ absorption another sixline
multiplet centered at g  2 appeared as a result of Mn
atom reactions with 0^ in Ar and Ne. The pattern is similar
to that observed for MnO^, but the magnitudes of the A
components are substantially reduced. Figure 35 shows
the spectrum of this species trapped in solid Ne. The
lines appear similar in Ar, but the innermost hyperfine
components are obscured by strong Mn atom signals.
These features appeared immediately on deposit in
both Ne and Ar matrices; ultraviolet photolysis in both
cases appeared to cause an intensity decrease, although
it was much more dramatic in Ne. Because such lines were
never observed in the other oxygendonor systems, and the
intensities appeared to behave inversely to those of MnO^
on photolysis, it seems reasonable that the molecule is a
species of the form MnO , where x>3. The apparent axial
symmetry rules out the possibility of a nonsymmetric
MnO^ conformation, and the sixline pattern is evidence
that only one Mn nucleus interacts with the unpaired
electron. Further discussion of the nature of this molecule
will appear below.
Figure 35.
VI N +0,
E
H (Gauss)
ESR spectrum of Mn04 in neon at 4Â°K. * indicates Mn atom lines; + Â± and
% indicate perpendicular and parallel components of one MnC>3 hyperfine line.
206
207
Treating the species as MnO^ with a doublet ground state
and axial symmetry, we can apply the secondorder solution
to the spin Hamiltonain, Eq. [73], to derive magnetic
parameters and predict resonance fields. The results of
these calculations are included in Table XV. The fit
is quite good except for the m^ = 1/2 component, where
distortion caused by the Mn atom line makes it difficult
to determine the field position. The unusual shape of the
perpendicular m^ = 5/2 line, which is even more distorted
in Ar matrices, is caused by overlap with an unidentified line.
Discussion
There is extensive literature on the ESR investigations
of manganese ions, either as substitutional impurities
in various lattices or as pure salts grown as single crystals
(4246). In general, these studies are largely concerned
+ 2
with the Mn ion in various environments and are not
meaningfully comparable to this investigation of molecules.
However, the anions of MnO^, studied in crystals (22, 23,
29, 47), are relevant to the proposed detection of neutral
MnO^ discussed below. The Mn+ ion with its large hyperfine
splitting, observed earlier by Kasai (32, 33), is also
an allied species.
Mn Atoms and Mn+
The spectrum of Mn atoms trapped in matrices (Figure 28)
5 2 6
is in accord with the 3d 4s ( S^.^) ground state. The
S state has zero orbital angular momentum and, as discussed
g
above in connection with Z molecules, a highspin system
208
Table XV.
Magnetic parameters, observed and calculated
line positions for MnO^ (^T^) in Ne
A
II
(G) = 90 (1)a
Aiso
(MHz)
= 216(3)
A
1
(G) = 70(1)
Adip
(MHz)
= 20(3)
g = 2.0108
(8)
gi
= 2.0097(8)
V (MHz)
= 9383(1)
MI
IlÂ°bsd (G)b
caled
II
(G) IlÂ°bSd
1
(G) b
Hcalcd (G)
5/2
3108
3108
3160
3158
3/2
3194
3195
3223
3224
1/2
3284
3283
3294
3293
1/2
3382
3372
3371
3363
3/2
3463
3463
3435
3435
5/2
3555
3556
3509
3509
aAll components of A tensor are for interaction with the
55Mn nucleus; A. and A assumed positive.
Id II
Estimated uncertainty = Â± 1 G.
209
(S = 5/2) will only be split by small amounts through
higherorder spinspin interactions caused by the crystalline
field (48). Then in the absence of a strong crystalline
field, the five finestructure lines will all occur at the
same magnetic field; that is, the D value becomes zero and
the lines coalesce. The only interaction which can split
the lines now is the nuclear hyperfine effect. Thus for
Mn atoms, the sextet pattern results from the hyperfine
55
interaction with the Mn (I = 5/2) nucleus. The very small
value of the isotropic hyperfine coupling constant A^so
(see Table XVI) is consistent with the limited unpaired s
electron density at the nucleus predicted by the ground
state electron configuration. The fact that it is nonzero
has been accounted for by the configuration interaction or
spin polarization mechanisms (48), which are equivalent. The
former involves contamination of the ground state wavefunc
tion with a slight admixture of excited configurations
containing unpaired s electrons. The polarization effect
can be described as follows (48). In this spin S = 5/2
system, the d electrons are all parallel; all the spins can
be considered "up." Then the electrostatic repulsion between
an s electron in an inner shell, with spin up, and these
3d electrons will not be the same as that for an s electron
with spin "down." This is a consequence of the exclusion
principle, which prevents two electrons with parallel spins
from occupying the same position in space. Calculations
show that the electronic densities behave as if electrons
Table XVI. Summary of magnetic parameters and derived quantities for manganese and
some manganese oxides
MnÂ°
Mn+
MnO
MnO 2
Mn03
MnO^
Point Group
R3
C
ooy
Dooh
Â°3h
C3v  9
Ground State
S5/2
7 c
h3
6^ +
4Z"
2 7\
A1
(2tx)
A (G)a'b
63(3)
126 (4)
633(1)
90(1)
A" (G)a'b
157(4)
261 (4)
547(1)
70(1)
A. (G)a
27.9C
275(5)
126(4)
215(8)
576(2)
77 (1)
1SO (MHz)
78. lc
770(14)
353(11)
603(22)
1613(6)
216(3)
A . (G)a
31(2)
45(8)
29(1)
7 (1)
Q1P (MHz)
87(6)
126(22)
81 (6)
20(3)
gf
5.952 (21)
3.990(2)
2.0084(8)
2.0097 (8)
gÂ±
1.990(7)
2.0023d
2.0084(8)
2.0097(8)
Agx
+0.012(7)
0.0
+0.0061(8)
+0.0074(8)
g.i
2.0023d
2.0023d
2.0036(8)
2.0108 (8)
Agâ€ž â€¢,
0.0
0.0
+0.0013(8)
+0.0085(8)
I D 1 cm
1.32e
1.13f
T(0)2 (a.u.)
0.0708
0.698 (13)
0.320(10)
0.547 (20)
1.463(6)
0.196(3)
/ fiâ€” 1 \
( 3 )(a.u.)
1.32(9)
1.92 (33)
1.23(9)
0.304(46)
3. 5 5
All hyperfine tensor components are for interaction with the Mn nucleus.
^Assumed positive.
Q
References 32 and 33.
aAssumed value.
0
Reference 12.
^Calculated assuming gj = gÂ± = See text.
^Axial symmetry indicated by spectrum. See text.
210
211
with parallel spins attract each other, thus polarizing
the inner shell, and introducing some net unpaired spin
density.
Starting with the Mn atom, oneelectron ionization to
5 7
the 3d 4s ( S^) configuration is expected to drastically
increase unpaired s electron density, and, hence, raise
the value of A. . This is confirmed in the observations of
iso
Mn+. However, because the Zeeman splitting is of the same
order as the hyperfine splitting, the energy levels can
no longer be treated in the strong field limit, as was done
for Mn atoms. The fine structure lines do not collapse
together, and both fine and hyperfine lines are observed
in the spectrum. It is interesting to note that the Mn+
signals were the same in all Ar matrices with the presumably
different electron acceptors. The good fit assuming that
the ion is spherically symmetric (ZFS = 0) and unperturbed
by a neighboring anion supports the picture of wellseparated
ion pairs. It also indicates that the crystal field
parameters appearing in the spin Hamiltonian of S state
ions (41, 48), which are responsible for ZFS j 0, are
negligible here.
Failure to observe the spectrum of Mn+ in neon matrices
with any electron acceptor, even with sufficiently energetic
photolysis, remains to be explained. Brus and Bondybey (49)
have improved the electrontransfer model described above
by considering the solvation energies of the resulting cations
and electrons in the matrix; they found that solvation
212
energies in neon are small, relative to the other inert
gas matrices, so a smaller donoracceptor distance is necesÂ¬
sary to make electron transfer feasible. However, the ionizaÂ¬
tion energy of Mn atoms is only 7.4 eV (50), so in these
matrices, with the electron acceptors present, there seems
to be no energetic reason for not forming ion pairs. The
probable reason is simply poor isolation, since even the Mn
atom signals are severely diminished in neon matrices.
MnO
Pinchemel and Schamps (8) have recently shown that the
electronic structure of the MnO molecule is very closely
related to that of the Mn atom. In fact, their population
analysis indicates that the 9a, 4tt, and 16 molecular orbitals
are essentially the axially symmetric components of the 3d
(Mn) atomic shell. They find that under such conditions,
5
Hund's rule, which applies to the 3d atomic configuration,
also applies to the whole group of molecular states arising
from the (9a^ 4iTm 16n) configurations (1 + m + n = 5) . Thus
the ground state is the state among those configurations
2
which maximizes the number of parallel spins, ... 8a 9a
2 2 6 +
47T 16 ( Â£ ) . If we consider the axially symmetric distorÂ¬
tion caused by bonding to an oxygen atom, and yet keep the
unpaired electrons in nonbonding orbitals arising from the
d levels of Mn, the observed magnetic parameters in Table
XVI are plausible. The distortion leads to a zero field
splitting, which produces three Kramers' doublets, and, due
to the magnitude of D, transitions are only observed between
the Â±1/2 levels.
213
The experimental values of the isotropic (A^sq) and
anisotropic (A^ ) hyperfine coupling constants, assuming
A  and are positive in sign, are given for MnO in
Table XVI. These are related to the wavefunction of the
molecule by Eqs. [91a, b] and the derived values of
((3cos20l)/r2) and iJj(0) 2 are also listed in that Table.
Using a very simple model for MnO, similar to that
for VO (39), the contributions to A. and A.. can be
' iso dip
2 2
calculated. The configuration is an 5 , where a is a
sd 2 hybrid, and the other orbitals are essentially Mn
z
atomic d orbitals. The procedure is just a summation of the
terms in Eqs. [92a, b], with the atomic contributions broken
down into the different types of orbitals, and the coefficients
chosen to produce the configuration described above. Thus
Adip = (l/2)(l/5)a^p + (2/5>ajâ€œp+ (2/5) a"p [150a]
Aiso * U/SHi/SU^o [150b]
The atomic values a,. and a. are obtained from the
dip iso
5 2
3d 4s state of the free manganese atom (45) and must
be considered as reasonable approximations. Then a^so =
1093 G, and a,. = +56, +28, and 56 G for d ?, d ,
dip zÂ¿ xz, yz
and d^2_y2 Xy' respectively (45). The reasons for the
different contributions from different orbitals have
been discussed previously. Using these relations, then,
A,. = 6 G and A. = 109 G, in fair agreement with
dip iso ^
214
observations (see Table XVI). The fit is somewhat improved
if the electron densities are taken from the published
groundstate wavefunction of Pinchemel and Schamps (8),
when the sums of the coefficients squared are normalized
12 2
to give the 9o 4tt 15 configuration. In this case,
A,. = 11 G, and A. = 114 G. In either case, the results
dip iso
lend support to the assumption that both A11 and Aj^ are
positive quantities.
The perturbation treatment, Eq.[73], and the exact
solution of the spin Hamiltonian, Eq.[133], produce an
excellent fit to the observed spectrum with the values
g 11 = 2.0023, g^ = 1.990 , and D = 1.32 cm 1. The deviation
of gj^ from gg is caused by spinorbit coupling to excited
6n states, and is given to second order by Eq. [74]. The
nonzero matrix elements involved were given in Eqs.[81b f].
Utilizing the model for electron distribution described
above, and the calculations of Pinchemel and Schamps (8),
g
the pertinent wavefunctions for the X ground state and lowest
6n excited states can be described by
*0
= Â¿L (4s)
Â«10
+ Lr (90)
"VIO
E = 0
[151a]
*1
1_
t/To (4s)
+ p= (9o)
VI0
+ =â€”(4 tt )
V5
+ p(10o)
V5
E = 24,263 cm
[151b]
215
5 5
(16) + i (5tt4
V5
E = 40,807 cm
1
[151c]
With the spinorbit coupling constant for monovalent manganese
is in excellent agreement with the observed g^ = 1.990 Â± 0.007.
The bonding in MnO, which is relevant to the following
discussion of MnC^, essentially involves the 4s (Mn) plus
2p(0) orbitals and may be depicted as (8) "... roughly
Mn+ (3d54s) + O" (2p5)
MnO
2
The ESR spectrum of MnO^ is characteristic of a linear
molecule in that the strong perpendicular absorptions centered
at g  4 give no indication of splitting into separate x
and y components. It is also apparent that the unpaired spin
on the 0 atoms is negligible, since there was no perceptible
hyperfine or broadening effect produced by isotopic sub
17
stitution of 0.
The geometry of MnO^ can be considered by reference to
MnF2, this molecule has been shown to be linear by Buchler,
et al. (51), and recent work in this laboratory (52) has
confirmed that geometry. It is a highspin molecule with
a Â£ ground state (51  53), and Hayes (53) suggests that
the five unpaired electrons are in antibonding orbitals
tending to increase the FMnF angle. This was arrived
at by construction of a Walsh diagram (54) for AB^ molecules,
216
modified to include d orbitals. One would then expect
that the removal of two electrons to form MnO^ would still
leave the oxide molecule in a linear conformation, as observed
here, since the theory implies that more than 18 valence
electrons favors the linear structure (55) .
The isotropic hyperfine splitting in MnO^ (see Table
XVI) is about twice that in MnO, which must be attributed
to increased 4sa character. Then one expects its ground
2
state to have an effective configuration of either tÃ o or
2 4 .
6 a, leading to the observed Â£ ground state. The anisotropic
parameter is also increased over that of MnO, and its
magnitude suggests that the most probable configuration is
2
6 a. For this configuration of MnO,,, again assuming the a
orbital is an sd 2 hybrid,
Adip = (l/2)(l/3)a^p +
dÃ“
(2/3)adip = 28G
[152a]
4s
A. * (1/2)(1/3)a. = 182G,
iso iso
[152b]
where the a,. values were given above in the discussion of
MnO. These calculated parameters are, again, in accord with
the experimental values when A11 and Aj^ are assumed to be
2
positive. The air configuration yields A^ = 4 7G, and
cannot account for the observed magnitude and sign of A^
in this model.
217
As indicated above, the value of D was found by
assuming g 11 = g^ = gQ, yielding d = 1.13 cm ^. While
this results from an approximation, it appears certain that
D = 1 cm \ since this minimum value is obtained by perÂ¬
forming the same calculation and varying g within reasonable
limits. One would expect a smaller value of D in MnC^
relative to MnO because of the fewer spins, and perhaps due
to some increased delocalization in the triatomic species.
As described above, however, this argument pertains to the
spinspin contribution to D, which should be of lesser
importance than the spinorbit effect in these molecules
with heavy atoms; in addition, all spin appears to be quite
localized at the Mn atom in both species.
MnO ^
With a value of A. = 575.6 G, Mn0o appears to exhibit
iso 3 ^
the largest hyperfine coupling of manganese in any molecule
or crystal previously reported. In order to account for
this large value and the observed axial symmetry, the
unpaired spin is best described as occupying an sd.^2 hybrid
orbital, resulting in a molecule of planar symmetry and
2
A^ ground state. This sd hybridization indicates that the
e" (7T*) and a (a^*) energy levels given by Royer (16)
are inverted for MnO^ relative to MnO^+. Figure 36 indicates
that molecular orbital correlation diagram, including this
modification. The diagram is purely qualitative, and based
on a consideration of the MnO^ (20, 21) and SO^ (56) molecules,
which are electronically and geometrically similar, respecÂ¬
tively. As described by Royer (16), the levels up to a^(nb)
218
Figure 36 .
Molecular orbital correlation diagram for MnO
3*
219
are relatively certain, and fit the observed spectrum of
MnO^+. The relative energies of the first three excited
levels, which are essentially perturbed dorbitals on the
central Mn, could be shifted from the ordering normally
produced by a trigonal field (e", a', e' in increasing
energy) if the e" atomic orbitals interact strongly; then
the lowest two of these levels would invert. This appears
to be the case, with the manganese e" (d^z ) orbitals
engaging in strong covalent bonds with the 0 orbitals.
The geometry and the sdz2 hybridization are character
istics shared with the isoelectronic species TiF^ recently
investigated in this laboratory (57). Assuming an equal
mixture of s and dz2 orbitals,
A,. * (l/2)a do = 23 G [153a]
dip dip
A. * (l/2)a.4s = 546 G. [153b]
ISO ISO
These compare quite favorably with the observed values
listed in Table XVI.
Atkins and Symons (58) considered the gshift for
such 25 valence electron radicals, and predicted g 11 g^_>ge
However, as they pointed out, the parallel component of
Ag, which arises from coupling with an A^ level centered
on the ligands, may not be very large because of the small
overlap with the ground state, which involves a (Mn) sdz2
220
orbital. Thus it is reasonable that, within experimental
error, gj =gg.
However, the deviation Agj^= 00061 from the free elecÂ¬
tron value is significant, and is given by Eq.[74]. As
explained above, for Ag^ to be a positive quantity, the
terms in the summation giving the largest contributions must
involve lowlying states formed by excitation from a bonding
orbital to the singlyoccupied a^ level; these states must
be of E symmetry. In this respect, the situation is unlike
that encountered in TiF^. For that molecule, Agj^=0.12 37,
and the excited E state is calculated to lie =2000 cm ^
above the ground state (57). If the MO scheme of Figure 35
corresponds to TiF^, the E state would be that arising from
the ...(e')^(a21)^(a^')^(e")^ configuration, and for MnO^,
3 2 2
it would be the ... (e') (a2 ') (a^') configuration. It has
been noted (59) that a positive value of Agj^ is indicative
of d orbital involvement in covalent bonding; this is in
accord with the above conclusions. The TiF^ molecule is
quite ionic (57) and might not be expected to demonstrate
this effect.
MnO.
4
The spectrum shown in Figure 35 has been attributed to
the molecule MnO^ mainly on the basis of the nature of the
matrixisolated species present and the effects of photolysis
on the spectrum. As indicated above, the sixline pattern
demonstrates that only one Mn nucleus interacts with the
unpaired electron. The parallel and perpendicular lines
221
are characteristic of a molecule with axial symmetry, but
indicate nothing about the nature of the ligands. The only
reasonable possibilities are oxygen atoms and oxygen and
ozone molecules. Because the molecule is not formed
when oxygen is used as a reagent, a possible 0^ ligand can
be eliminated. A manganese atom interacting with an
molecule could be expected to form a ozonide, Mn+0^ ,
similar to the alkali metal ozonides studied by Spiker and
Andrews (34, 60). The ESR signals of Mn+ would then be
a prominent feature, but these were observed only in a few
cases of photolyzed Ar matrices, when other signals peculiar
to Mn0^ reactions also appeared. However, considering the
ease with which the covalent species MnO^ is formed during
deposition, it is not unreasonable that MnO^ should also
occur. Further evidence is that both species are formed
more readily in solid neon than in argon; presumably because
the former matrix quenches less effectively and allows
more surface reactions to occur during deposit. Finally,
the fact that UV photolysis causes an intensity decrease in
the MnO^ signals, while enhancing those of MnO^ suggests
that the latter is formed from MnO^ by elimination of one
oxygen ligand.
While MnO^ is expected to have tetrahedral symmetry,
a C^v structure is postulated in order to explain the
axial symmetry apparent in the spectrum. If the distortion
is not too great, the species would be an analogue of the
Mn0^x series, where x = 1, 2, 3, which has received considerÂ¬
able study.
222
In tetrahedral symmetry, MnO^ is expected to have either
2 2
a A1 (31) or (29) ground state. However, in order
to exhibit the JahnTeller distortion (61) which is
proposed here, it must be orbitally degenerate, so that a
2
state is indicated. Subramanian and Rogers (29) eariler
considered MnO^ as a possible species formed on yirradiation
of KMnO^. Using the MO scheme of Ballhausen and Liehr (21),
which is presented in Figure 37, they considered it as having
2
a ground state, and because that implied certain propÂ¬
erties for its magnetic parameters, they rejected it as
the source for their ESR spectrum, which they assigned to
2 2
MnO^ . For a state one expects a "hole" delocalized
on the oxygen atom it orbitals, positive isotropic g shifts,
and only small isotropic hyperfine coupling with the
central manganese atom. Similar effects are observed in the
2
isoelectronic species VO^ (61) and CrO^ (62, 63). With
slight modifications arising from a proposed JahnTeller
effect, these expected parameters are observed in the ESR
spectrum assigned here to MnO^. In that spectrum (Figure 35
and Table XVI), the g shifts are positive and (Mn)
is only 76.6 G, but both the g and A tensors are slightly
anisotropic. Although the absolute error in measuring each
gtensor component is probably Â± 0.001, the difference
g 11 gj^ = Â± 0.0011 is considered accurate to about Â± 0.0006.
The spectrum indicates that the molecule is axial with
at least a threefold axis of symmetry.
22 3
3a,
Mn04
Figure 37.
Molecular orbital correlation diagram for Mn04.
224
It is possible that the radical being observed here
may be tetrahedral MnO^ distorted by the surrounding matrix
cage. It has already been shown that some linear molecules
with low bending force constants can be induced to bend
in some matrices. This is unlikely for this species because
the spectra observed in Ar and Ne matrices are very similar,
and if the matrix were the cause, one might expect different
degrees of distortion in the different lattice sites.
Secondly, a study of the argon spectrum taken at various
temperatures between 4Â°K and 35Â°K showed no change in the
shapes or positions of the lines. The highest temperature
here is a very adequate annealing temperature in solid argon,
O
at which thermal expansion should cause about a 0.02 A inÂ¬
crease in the nearestneighbor spacing over that at 4Â°K (64).
A higher temperature leads to rapid diffusion and diminuÂ¬
tion in the intensity of the spectrum.
Then, accepting the tentative suggestion that the MnO.
radical is being observed, one can rationalize the ESR
spectrum if the tetrahedral symmetry is distorted to C^v
by the JahnTeller effect. Simply stated, the JahnTeller
theorem states that any nonlinear molecular system in an
orbitally degenerate electronic state will be unstable and
will undergo some kind of distortion that will lower its symmetry
and split the degenerate state (65). The only exceptions
in this class of systems are in molecules for which no
geometric distortion can remove the degeneracy; thus the
Kramers spin degeneracy is stable. The theorem only predicts
225
that, for degenerate states, a distortion must occur; it
gives no indication of the geometric nature or magnitude of
such distortions. Like the Renner effect described above,
the JahnTeller effect has its origins in vibronic interÂ¬
actions, that is, the breakdown of the BornOppenheimer
approximation, which separates the vibrational and electronic
parts of the wavefunction.
Although it is not appropriate to MnO^, a simple case
of the JahnTeller interaction can be described for a moleÂ¬
cule with symmetry in a doublydegenerate E' electronic
state (66). The distortion which removes electronic degenÂ¬
eracy must itself be degenerate (67), and here we consider
a vibrational mode of e' symmetry, with normal coordinates
Qa and . Here and are the two electronic eigenfuncÂ¬
tions of the degenerate electronic state, and the Hamiltonian
Hc is dependent on the normal coordinates as parameters.
Using the complex normal coordinates and eigenfunctions
+
[154a]
!+> = â™¦a + i*b
[154b]
[154c]
the electronic energy of the system is determined by
H
++
E
H
+
 +
0 [155]
H
H
E
226
where the matrix elements are H~ = (Â±  H^  Â±) and H_+= (Â±   +).
If the Hamiltonian is expanded in a power series of the normal
coordinate of interest, and neglecting terms of second
order and higher, then
He = Hq + H+Q~ + hâ€q+ +
[156]
and using the matrix elements above,
HÂ±Â±=<Â±HqÂ±) + qâ€œ<Â±H+Â±> + Q+(Â± hâ€œÂ±> [157a]
HÂ±+= <Â± I Hq  +>+Q_<Â± I Il  +>+Q+ (Â±  H~ I +) . [157b]
In the expansion, the terms of different superscripts appear
together because the Hamiltonian must be totally symmetric
under all operations, and, with the rotation, for example,
the effect is multiplication of H+, +), (, and Q+ by
the factor eand for h , ), (+[, and Q , multiplicaÂ¬
tion by e 2iri/3^ The integrals in Eqs. [157a, b] will
be nonzero if the integrands are invariant under the operations
of the point group. Thus, for H++ and H , the linear terms
in Q vanish, while one of the linear terms, Q+ in H+ and
Q in H +, does not vanish in Eq. [157b] . Then, using Eq. [155],
with H++ = Wq = H and H4~ = CQ~, we obtain
WqE CQ+
CQ~ WQE
= 0
[158]
22 7
Then the energy E = W_ Â± C ~^Q+Q , or
E = WQ Â± Cr [159]
2 2 1/2
where r = (Q + Q, ) is the magnitude of the normal coordi
3 D
nate,and C is a measure of the coupling of vibrational and
electronic energies. The net effect, then, is that the
degenerate electronic potential, which could be represented
as two superimposed parabolic surfaces with their minima at
a value of zero for the normal coordinate, have separated,
such that the potential minima are not in the symmetrical
position, but at some value r along the normal coordinate.
In this approximation (neglecting quadratic and higher
terms in Q), the potential well will be a circular trough
of constant depth around the origin. If the higherorder
terms are included, the trough develops into three minima
separated by saddle points; the minima correspond, essentially,
to stretching or compressing one of the three bonds in the
molecule, This is the deformation from the symmetrical
conformation predicted by the JahnTeller theorem. The
JahnTeller effect has been described more fully in several
papers and texts (48, 6672).
This has been a rather lengthy discussion, for a situaÂ¬
tion that does not apply to MnO^, but the physical descripÂ¬
tion pertaining to this case is more complicated, since it
is inseparably sixdimensional (70). The reason is as
follows. Although the JahnTeller theorem states nothing
228
about the type of deformation, it is possible by symmetry to
determine what the nature of the distortion would be for
vibronic involvement of different vibrational modes. The
2
condition is that the symmetric direct product [r] of
the degenerate electronic state must contain a representation
also found in the normal vibrational modes of the molecule
(68). Ballhausen (68), among others, has listed the possible
vibrational modes which will break the degeneracy in a T
state of a tetrahedron, and these are either e or t2 vibrations.
Interaction with the e mode will produce a tetragonal disÂ¬
tortion, but to obtain the trigonal distortion, which retains
axial symmetry as observed here, the t^ modes must be involved
(67, 68). Because there are two of these modes in tetrahedral
species, and each is triply degenerate, the movements in
space are sixdimensional, as mentioned above.
Another facet of the JahnTeller effect is that it can
be described as either a "static" or a "dynamic" effect
(67, 69), and the same species could exhibit both types,
depending on the conditions of the experiment. The static
effect is such that the mean distortion is finite when
averaged over a certain period of time. The dynamic effect
involves a thermally activated reorientation of the system
which shows a static effect at very low temperatures. Thus
the characteristic time scale of the experiment and the
temperature will affect the category into which the effect
falls. Then the observed temperature behavior in solid argon,
which was discussed above, would indicate that the vibronic
229
coupling is strong enough to be described as the static
effect. If only one t9 mode is involved, the result of the
JahnTeller coupling (in analogy with the molecule
described above) will be to have four potential minima, one
corresponding to elongation or compression of each of the
bonds. Increasing the temperature would then be expected
to excite the molecule over the potential barrier of the
saddle point between the different wells. Experimentally,
raising the temperature to 35Â°K only amounts to a thermal
energy of about 24 cm . Now the JahnTeller energy, which
is the difference in energy of the distorted and symmetric
2 51
configurations, can reach from 10 to 10 cm for an orbitally
degenerate level of a transition metal ion (73, 74). The
general view (73, 75) that JahnTeller energies very much
less than the zeropoint vibrational energy (typically 100 cm ^)
can have no observable effects has been borne out by calculaÂ¬
tion (71). The height of the saddle is quite variable, since
it depends on anharmonic effects, but is typically about
200 cm ^ at least for doublydegenerate ions in octahedral
complexes (76). Thus it is very reasonable that the thermal
energy at 35Â°K is insufficient to overcame the barrier.
Although there has been some work done on the interÂ¬
actions of T states with t2 vibrations in tetrahedral
symmetry (77  81), the theory has not been completely
solved, due to the complexity of the problem. In addition,
there is a paucity of experimentally observed examples of
this type of coupling. Indeed, there are only two examples
230
which Sturge (67) believes can definitely be attributed to
a trigonal distortion of a tetrahedral molecule. These
2 +
are V in CaF^ (82) and substitutional N donors in diamond
(83  85). The latter case seems quite similar to the
MnO. results.
4
In the tetrahedral molecule, the t^ MO is a linear
conbination of tt orbitals on the four oxygen atoms, and is
occupied by five electrons (see Figure 37). The distortion
can be visualized as an elongation of one Mn0 bond with the
"hole" then being essentially shared among the tt orbitals
of the other three equivalent oxygen atoms. Or the hole could
be on the odd 0 which would be synonymous with an ionic model
Mn0^+0 . The latter is appealing because of previous evidence
for MnO^+ (15, 16). This also seemed to fit the N in
diamond case, where the electron was localized along one
CN bond (83  85). In either case, there is a fourfold
degeneracy due to the four equivalent trigonallydistorted
conformations of the molecule with each 0 axial. It appears
to be a static JahnTeller case, at least within the temperÂ¬
ature range of these measurements. Reorientation in the
diamond system was only observed above 570Â°K (85).
Since the electron hole is essentially localized on
the light oxygen atoms in MnO^, one expects only small spin
orbit effects. In general, large spinorbit coupling in
systems of halfintegral spin reduces the configurational
(JahnTeller) instability caused by the orbital degeneracy
(66, 68). Jahn (73) has shown that in such a case of large
2 31
spinorbit coupling, it is the antisymmetric (rather than the
symmetric) direct product of the species of the spinorbit
function which must contain the same species as a non
totally symmetric normal mode, in order to make the Jahn
Teller instability possible. Utilizing the symmetry tables
of Herzberg (66), the spinorbital species, given by the
direct product of the spin 1/2 species and the orbital
species, is = E\/2' G3/2* spinorbit
coupling were important, the molecule would be in one of
these resulting states. Which species it will be cannot
be easily determined, but it can be said that, at least
in the ^2/2 state' JahnTeller effect is still possible,
through coupling with the same vibrations considered
above. Thus the assumption of small spinorbit effects is
not critical.
In review, the axial symmetry apparent in the spectrum
suggests that MnO^ undergoes a trigonal JahnTeller disÂ¬
tortion which reduces the symmetry from to C^. The
Mn hyperfine splitting is small enough to be accounted for
by the type of spin polarization described above; orbitals
on Mn are not directly involved in the t^ MO. The g shifts
are slightly positive, as expected for excitation of a
hole. The slight anisotropy in the g and A tensors is
probably more in accord with the nonionic model of disÂ¬
torted MnO^, since one might expect larger anisotropies
in Mn0^+0 . All magnetic parameters, then, are in accord
with such a model.
232
References  Chapter V
1. K. D. Carlson, E. Ludena, and C. Moser, J. Chem. Phys.,
43, 2408 (1965) .
2. K. D. Carlson and R. K. Nesbet, J. Chem. Phys., 41,
1051 (1964).
3. K. D. Carlson, and C. Moser, J. Chem. Phys., 4_4 , 3259
, (1966).
4. P. S. Bagus and H. J. T. P. Preston, J. Chem. Phys.,
59, 2986 (1973).
5. R. A. Berg and O. Sinanoglu, J. Chem. Phys., 32^, 1082
(1960).
6. C. K. Jorgensen, Mol. Phys., 1_, 417 (1964).
7. C. J. Cheetham and R. F. Barrow, "Advances in High
Temperature Chemistry," Vol. 1 (L. Eyring, ed.),
Academic, New York, 1967, p.7.
8. B. Pinchemel and J. Schamps, Chem. Phys., 1^, 481 (1976).
9. J. M. DasSharma, Z. Physik., 157, 98 (1959).
10. M. B. Garrett, P. ShuTi Lee, and J. G. Kay, J. Chem.
Phys., 45, 2698 (1966).
11. K. R. Thompson, W. C. Easley, and L. B. Knight, Jr.,
J. Phys. Chem., 7^7, 49 (1973).
12. B. Pinchemel and J. Schamps, Can J. Phys., 5_3, 431
(1975).
13. W. F. Libby, J. Amer. Chem. Soc., 6Â£, 1930 (1940).
14. D. J. Apers and G. Harbottle, Radiochim. Acta, 1_,
188 (1963).
.15. W. C. Lecington and C. W. Owens, Radiochim. Acta, 1_,
212 (1967).
16. D. J. Royer, J. Inorg. Nucl. Chem., 17, 159 (1961).
2 33
17. U. A. Spitzer, T. W. Toone, and R. Stewart, Can. J.
Chem. , 5_4 , 440 (1976) .
18. J. Teltow, Z. Phys. Chem. B, 4J), 397 (1938).
19. J. Teltow, Z. Phys. Chem. B, 4_3, 198 (1939).
20. M. Wolfsberg and L. Helmholz, J. Chem. Phys., 20,
837 (1952).
21. C. J. Ballhausen and A. D. Liehr, J. Mol. Spec., 2^
342 (1958).
22. A. Carrington, D. J. E. Ingram, K. A. K. Lott, D. S.
Schonland, and M. C. R. Symons, Proc. Roy. Soc. A,
254, 101 (1959).
23.D. S. Schonland, Proc. Roy. Soc. A, 254, 111 (1959).
24. A. Carrington and M. C. R. Symons, Chem. Revs., 63,
443 (1963).
25. H. D. Gray, Coord. Chem. Rev., 1^, 2 (1966).
26. L. Oleari, G. DeMichelis, and L. DiSipio, Mol. Phys.,
10, 111 (1966).
27. G. DeMichelis, L. Oleari, L. DiSipio, and E. Tondello
Coord. Chem. Rev., 2, 53 (1967).
28. J. P. Dahl and H. Johansen, Theoret. Chim. Acta, 11,
13 (1968).
29. S. Subramanian and M. T. Rogers, J. Chem. Phys., 57,
2192 (1972).
30. J. H. Murrell and I. G. Vincent, J. Chem. Soc.,
Faraday Trans. II, 7JlÂ» 890 (1975).
31. H. Hsu, C. Peterson, and R. M. Pitzer, J. Chem. Phys.
64, 791 (1976).
32.
P.
H.
Kasai,
Phys.
Rev.
Lett., 2JL, 67
(1968) .
33.
P.
H.
Kasai,
Acct.
Chem.
Res., 4, 329
(1971) .
34.
R.
C.
Spiker
, Jr.
and L
. Andrews, J.
Chem. P
1851 (1973).
35. D. E. Milligan and M. E. Jacox, J. Chem. Phys., 47,
5146 (1967).
36. A. C. Wahl, Sci. Amer., 222, 54 (1970).
234
37.
R.
Aasa, J. Chem. Phys., 52,
3919 (1970) .
38.
E. S.
Phys.
Kirkpatrick, K. A. MÃ¼ller, and R. S.
Rev., 135, A86 (1964) .
Rubins,
39.
P.
H.
Kasai, J. Chem. Phys.,
49, 4979 (1968).
40.
J.
Jr
M.
â€¢ /
Brom, Jr., C. H. Durham
J. Chem. Phys., 61, 970
, Jr., and W.
(1974) .
Weltner,
41.
J.
C.
Wertz and J. R. Bolton,
"Electron Spin Resonance
Elementary Theory and Practical Applications,"
McGrawHill, New York, 1972.
42.B. Bleaney and D. J. E. Ingram, Proc. Roy. Soc. A,
205, 336 (1951).
43. L. M. Matarrese and C. Kikuchi, J. Phys. Chem. Solids,
1, 117 (1956).
44. V. M. Vinokurov, M. M. Zaripov, and V. G. Stepanov,
Fiz. Tverd. Tela, (5 , 1125 (1964); see Soviet Phys.
Solid State (Eng. Translation) Â£, 866 (1964 ).
45. B. A. Goodman and J. B. Raynor, "Advances in Inorganic
Chemistry and Radiochemistry," Vol. 13 (H. J. Emeleus
and A. G. Sharpe, eds.), Academic, New York, 1970,
p . 135.
46. H. J. Stoller, G. Rist, and H. H. GÃ¼nthard, J. Chem.
Phys., 57, 4651 (1972).
47. K. J. White, U.S. Army Ballistic Research Laboratories
Report No. 1482, 1970.
48. A. Abragam and B. Bleaney, "Electronic Paramagnetic
Resonance of Transition Ions," Oxford University Press
London, 1970.
49. L. E. Brus and V. E. Bondybey, J. Chem. Phys., 63,
3123 (1975).
50. C. E. Moore, "Atomic Energy Levels," Natl. Bur. Stand.
(U.S.), Circ. 467, Vol. II, U.S. Government Printing
Office, Washington (1952).
51. A. BÃ¼chler, J. L. Stauffer, and W. Klemperer, J. Chem.
Phys., Â£0, 3471 (1964).
52. T. C. DeVore, R. J. VanZee, and W. Weltner, Jr.,
Submitted to J. Chem. Phys., 1977.
53. E. F. Hayes, J. Phys. Chem. Â£0, 3740 (1966).
235
54. A. D. Walsh, J. Chem. Soc., 2266 (1953).
55. W. Weltner, Jr., Submitted to Ber. Bunsenges. Phys.
Chem., 1977.
56. A. D. Walsh, J. Chem. Soc., 2301 (1953) .
57. T. C. DeVore and W. Weltner, Jr., J. Amer. Chem.
Soc., 99, 4700 (1977).
58.P. W. Atkins and M. C. R. Symons, "The Structure of
Inorganic Radicals," Elsevier, Amsterdam, 1967.
59.
J. Fiddone
(London),
and
73,
K.
116
W. H. Stevens
(1959).
, Proc
. Phys.
Soc.
60.
L. Andrews
(1973).
and
R.
C. Spiker, J.
Chem.
Phys.,
59, 1863
61.
H. A. Jahn
(1937) .
and
E.
Teller, Proc.
Roy.
Soc. A,
161, 220
62.
P. R. Edwards,
S.
Subramanian,
and M.
C. R.
Symons,
Chem. Comm., 799 (1968).
63. D. K. Lister and M. C. R. Symons, J. Chem. Soc. A,
782 (1970).
64. G. L. Pollack, Rev. Mod. Phys., 3_6, 748 (1964).
65. F. A. Cotton and G. Wilkinson, "Advanced Inorganic
Chemistry, A Comprehensive Text," Interscience, New
York, 1972, p. 590.
66. G. Herzberg, "Molecular Spectra and Molecular StrucÂ¬
ture, III, Electronic Spectra and Electronic Structure
of Polyatomic Molecules," Van Nostrand, New York, 1966.
67. M. D. Sturge, "Solid State Physics," Vol. 20 (F. Seitz,
D. Turnbull, and H. Ehrenreich, eds.), Academic, New
York, 1967, p. 91.
68. C. J. Ballhausen, "Introduction to Ligand Field Theory,"
McGrawHill, New York, 1962.
69. R. Englman, "The JahnTeller Effect on Molecules and
Crystals," WileyInterscience, New York, 1972.
70. A. D. Liehr, J. Phys. Chem., 6J7, 389 (1963).
71. F. S. Ham, Phys. Rev., A138, 1727 (1965).
72.
F. S. Ham, "Electron Paramagnetic Resonance," (S.
Geschwind, ed.), Plenum, New York, 1972, p.l.
236
73. H. A. Jahn, Proc. Roy. Soc., A164, 117 (1938).
74. V. I. Avvakumov, Opt. I Spectroskopiya, 1_3, 588
(1962); see Opt. Spectry. (USSR), (Eng.Translation), 13,
332 (1962).
75. J. H. Van Vleck, J. Chem. Phys., 1_, 72 (1939).
76. U. Opick and M. H. L. Pryce, Proc. Roy. Soc., A238,
425 (1957).
77. M. Caner and R. Englman, J. Chem. Phys., 4_4, 4054
(1966).
78. J. Fridel, M. Lannoo, and G. Leman, Phys. Rev., 164,
1056 (1967).
79. M. C. M. O'Brien, Phys. Rev., 187, 407 (1969).
80. C. A. Bates and P. E. Chandler, J. Phys. C.: Solid
St. Phys., 4, 2713 (1971).
81. L. S. Dang, R. Romestein, Y. M. d'Aubigne, and A.
Fukuda, Phys. Rev. Lett., 3_8 , 1539 (1977).
82. U. T. HÃ³chli, Bull. Am. Phys. Soc., 203 (1966).
83. W. V. Smith, P. R. Sorokin, I. L. Gelles, and G. J.
Lasher, Phys. Rev., 115, 1546 (1959).
84. J. H. N. Loubser and L. DuPreez, Brit. J. Appl. Phys.,
16, 457 (1965).
85. L. A. Shul'man, I. M. Zaritskii, and G. A. Podzyarei,
Fiz. Tverd. Tela, 8, 2307 (1966): see Soviet Phys.
Solid State (Eng. Translation), Q, 1842 (1967).
BIOGRAPHICAL SKETCH
Robert Francis Ferrante was born on November 26, 1951,
in Philadelphia, Pennsylvania, and is the son of Mr. and
Mrs. Joseph C. Ferrante, of Springfield, Pennsylvania. He
graduated from Springfield Senior High School, Springfield,
Pennsylvania, in June, 1969, and entered Villanova
University in September, 1969. In June, 1973, he was
graduated from Villanova University with a Bachelor of
Science in Chemistry. Since September, 1973, he has
pursued a course of study leading to a Doctor of Philosophy
degree in Chemistry at the University of Florida,
Gainesville, Florida.
2 37
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
yy ^ V / \
William Weltner, Jr., Chairman'
Professor of Chemistry
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
ClL V. CGv
JohrJ R. Eyler Â¡J
As^/stant Professor of Chemistry
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Stephen T. Gqttesman
Associate Professor of Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Earle E. /Muschlitz, Jr.
Professor of Chemistry
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
R. Carl Stoufer Â¿y
Associate Professor of Chemistry
This dissertation was submitted to the Graduate Faculty
of the Department of Chemistry in the College of Arts and
Sciences and to the Graduate Council, and was accepted as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
December, 1977
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 1262 08553 9814
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