Spectroscopic investigations of metal clusters and metal carbonyls in rare gas matrices


Material Information

Spectroscopic investigations of metal clusters and metal carbonyls in rare gas matrices
Physical Description:
xiv, 187 leaves : ill. ; 28 cm.
Bach, Stephan Bruno Heinrich, 1959-
Publication Date:


Subjects / Keywords:
Matrix isolation spectroscopy   ( lcsh )
Metal crystals   ( lcsh )
Metal carbonyls   ( lcsh )
Fourier transform infrared spectroscopy   ( lcsh )
Electron paramagnetic resonance   ( lcsh )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1987.
Includes bibliographical references (leaves 178-186).
Statement of Responsibility:
by Stephan Bruno Heinrich Bach.
General Note:
General Note:

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Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 001036908
notis - AFB9323
oclc - 18287365
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Full Text









The author wishes to extend his deepest thanks and

appreciation to Professor William Weltner, Jr., whose

patience, understanding, encouragement, and professional

guidance have made all of this possible. Thanks are also due

to Professor Bruce Ault for the initial opportunity do matrix

work and the encouragement to continue on to do graduate work

in the field. Thanks need also be given to Professor

Weltner's research group, specifically to Dr. Richard Van

Zee, whose help and guidance were invaluable in completing

this work.

The author also wishes to acknowledge the assistance of

the electronics, machine, and glass shops within the

Department of Chemistry. They kept the equipment

functioning, and fabricated new pieces of apparatus when

necessary making it possible to perform the desired

experiments. Thanks are also due to Larry Chamusco for many

enlightening conversations regarding the present work and

also for his help in preparing this work for publication.

Thanks are also due to Ngal Wong for his assistance in

preparing the final version of this publication.

The author also wishes to acknowledge the support of

the National Science Foundation (NSF) for this work as well

as Division of Sponsored Research for support in completing

the work for this project.



ACKNOWLEDGEMENTS ...................................... ii

LIST OF TABLES .......... ......... ..................... vii

LIST OF FIGURES ........................................ ix

ABSTRACT .................................................. xiii


I INTODUCTION .................................. 1

Matrix Isolation ........................ 1
Theory of Cluster Formation ........... 19
ESR Theory .............................. 30
The Hyperfine Splitting Effect ..... 34
Doublet Sigma Molecules ............ 40
The spin Hamiltonian ............ 40
The g tensor ...................... 42
The A tensor ...................... 46
Randomly oriented molecules ...... 49
Molecular parameters and
the observed spectrum .......... 51
Spin densities .................. 56
Quartet Sigma Molecules (S=3/2) .... 58
The spin Hamiltonian ............ 59
Sextet Sigma Molecules ............ 66
The spin Hamiltonian .............. 66
Infrared Spectroscopy ................ 67
Theory ................................ 70
Fourier Transform IR Spectroscopy .. 76

II METAL CARBONYLS ............................... 79

ESR of VCO, Molecules ................. 79
Introduction ..................... 79
Experimental ......................... 80
ESR Spectra ....................... 81
V CO .. .. .. .. .. 8 1
51VCO(A) and 5VCO(a) in argon ... 81
51V13CO(A) and 51V13CO(a) in argon 82
51VCO(A) in neon ................ 82
51VCO(a) in krypton .............. 88
51V(12CO)2 and V(13CO)2 in neon 89
51V(12C0)3 and 51V(13CO)3 in neon 89

Analysis ......
VCO, (A) and
V(CO)2 ......
V(CO)3 ......
Discussion ....
VCO .........
V(CO)2 ......
V(CO)3 ......
Conclusion ....
Infrared Spectrosc
Transition Metal
Spectra .......
Discussion ....
Conclusion ....


a ) .. .

py of First
py of First
Carbonyls .


Introduction ..........................
Experimental ..........................
ESR Spectra ...........................
Analysis and Discussion ...............

IV ESR OF METAL SILICIDES .....................

ESR of AgSi and MnSi ......
Introduction ...........
Experimental ...........
ESR Spectra ............
AgSi .................
MnSi .................
Analysis and Discussion
A gS i ...... .. ..... ....
MnSi .................
ESR of Hydrogen-Containing
Silicon Clusters ........
Introduction ...........
Experimental ...........
ESR Spectra ............
HScSiHn ..............
H2ScSIHn .............
Analysis and Discussion
HScSiHn ..............
H2ScSiHn .............







.. o












... ,




V CONCLUSION.... ............................... 172

REFERENCES ........................................... 178

BIOGRAPHICAL SKETCH .................................. 187


Table Page

II-1. Observed and calculated line positions (in G)
for VCO (X6E) in conformation (A) in argon at
4 K. v = 9.5596 GHz ............................. 83

11-2. Observed and calculated line positions (in G)
for VCO (X6O) in conformation (a) in argon at
4 K. v = 9.5596 GHz ............................. 84

11-3. Observed line positions (in G) for VCO (X6O)
in conformation (A) in neon at 4 K.
v = 9.5560 GHz................................... 90

11-4. Observed and calculated line positions (in G)
for V(CO)2 (X4 g) isolated in neon at 4 K.
v = 9 .5560 GHz ................................. 91

11-5. Calculated and observed line positions and
magnetic parameters of the V(CO)3 molecule
in neon matrix at 4 K. v = 9.55498 GHz........... 92

II-6. Carbonyl Stretching Frequencies for the First
Row Transition Metal Carbonyls................... 112

III-1. Calculated and observed ESR lines of 109Ag7 in
solid neon at 4 K (v = 9.5338 GHz)
See Figure III-3 ................................. 125

III-2. Magnetic parameters and s-electron spin
densities for 109Ag7 cluster in its A2
ground state.(a) ............................. 131

III-3. Comparison of the magnetic parameters of
107Ag7 (this work) with those of Howard,
et al s 107Ag5 cluster ......................... 135

III-4. Spin densities (s-electron) compared for the
2A2 ground states of the Na7, K7, and Ag7....... 136

IV-1. Observed and Calculated Line Positions (in Gauss)
for AgSi in argon at 14 K. (v = 9.380 GHz)...... 140

IV-2. Observed and Calculated Line Positions (in Gauss)
for MnSi in argon at 14 K. (v = 9.380 GHz)...... 141

IV-3. Hyperfine parameters and calculated spin densities
for MnSI in argon at 14 K. (v = 9.380 GHz)...... 147



IV-4. Observed and calculated line positions (in Gauss)
for HScSiHn in argon at 14 K. (v = 9.380 GHz)... 151

IV-5. Hyperfine parameters and calculated spin densities
for HScSiHn in argon at 14 K. (v = 9.380 GHz)... 152

IV-6. Observed and Calculated Line Positions (in
Gauss) for H2ScSiHn, the (A) site, in argon
at 14 K. (v = 9.380 GHz) ........................ 158

IV-7. Hyperfine parameters and calculated spin
densities for H2ScSiHn, site (A), in argon
at 14 K. (v = 9.380 GHz)......................... 159

IV-8. Observed and Calculated Line Positions (in
Gauss) for H2ScSIHn, the (a) site, in argon
at 14 K. (v = 9.380 GHz) ........................ 160

IV-9. Hyperfine parameters and calculated spin
densities for H2ScSiH,, site (a), in argon
at 14 K. (v = 9.380 GHz)......................... 161



Figure Page

I-1. The furnace flange with copper electrodes and
a tantalum cell attached......................... 10

1-2. The EPR cavity and deposition surface within
the vacuum vessel. The apparatus is capable
of cooling the rod to 4 K, because of the
liquid helium transfer device (Heli-Tran)
on the top ........................................ 11

I-3. The ESR cavity and deposition surface within
the vacuum vessel. The apparatus is capable of
cooling to 12 K because of the closed cycle
helium refrigeration device (Displex) on the
t o p . . . . .. . 1 2

1-4. The vacuum vessel, deposition surface, Displex,
and furnace assembly for infrared experiments... 13

I-5. Zeeman energy levels of an electron interacting
with a spin 1/2 nucleus ......................... 52

1-6. (a) Absorption and (b) first derivative line-
shapes of randomly oriented molecules with
axial symmetry and gj (c) first derivative lineshape of randomly
oriented, axially symmetric molecules
g- with a spin 1/2 nucleus (A
1-7. Energy levels for a 4E molecule in a magnetic
field; field perpendicular to molecular axis.... 63

1-8. Energy levels for a 4E molecule in a magnetic
field; field parallel to molecular axis......... 64

1-9. Resonant fields of a 4E molecule as a function
of the zero field splitting...................... 65

1-10. Energy levels for a 6E molecule in a magnetic
field for 9 = 0, 30, 600, and 90 ............ 68

1-11. Resonant fields of a 6E molecule as a function
of the zero field splitting...................... 69


II-1. ESR spectrum of an unannealed matrix at 4 K
containing 51VCO(A), with hfs of about 100 G,
and 51VCO(a), with hfs of about 60 G. For the
conformation (A) two perpendicular lines and
an off principle axis line are shown.
v = 9.5585 GHz.................................. 85

11-2. ESR spectrum of an annealed argon matrix at 4 K
containing only 51VCO in conformation (a). Two
perpendicular lines and off principal axis
line are shown. v = 9.5585 GHz.................. 86

11-3. ESR spectrum of the perpendicular xy1 line of
51V13CO in conformation (a) in an argon matrix
at 4 K. v = 9.5531 GHz ......................... 87

11-4. ESR lines in a neon matrix at 4 K attributed to
51VCO in conformation (A). v = 9.5560 GHz...... 93

11-5. ESR lines in a neon matrix at 4 K attributed to
51V(CO)2. v = 9.5560 GHz ....................... 94

11-6. (Top) ESR spectrum near g=2 in a neon matrix
at 4 K attributed to an axial 51V(CO)3 molecule.
v = 9.5584 GHz.
(Bottom) Simulated spectrum using g, A(51V)
parameters and linewidths given in the text.... 95

11-7. Molecular orbital scheme for the 6z VCO
molecule (modeled after Fig. 5-43 in DeKock
and Gray (84 )) .................................. 101

11-8. Infrared spectrum of CrCO using both 12CO and
13CO in argon. The top trace has a 1:200 CO/Ar
concentration and the bottom has a 1:1:200
12C0/13CO/Ar concentration ..................... 113

11-9. Infrared spectrum of Mn and CO codeposited into
an argon matrix. The bottom trace is after
annealing the matrix to about 30 K and cooling
back down to 14 K............................... 114

II-10. Plot of the CO stretching frequencies in the
first row transition metal monocarbonyl
molecules MCO (circled points are tentative).
Also shown is the variation of the energy of
promotion corresponding to 4s23dn-2 to 4sl3dn-,
where n is the number of valence electrons(91). 117


III-1. The pentagonal bipyramid structure ascribed to
Ag7 in its 2A2 ground state. It has D5h symmetry
with two equivalent atoms along the axis and
five equivalent atoms in the horizontal plane.. 126

III-2. The ESR spectrum of 109Ag in solid neon matrix
at 4 K (v = 9.5338 GHz). The top trace is the
overall spectrum, and the bottom traces are
expansions of the three regions of interest
after annealing. Notice the large intensity
of the silver atom lines at 3000 G before
annealing ...... ................................ 127

III-3. The ESR spectrum of 109Ag7 in a solid neon
matrix at 4 K (v = 9.5338 GHz). The fields
indicated are the positions of the four
hyperfine lines corresponding to
IJ,Mj> = 11,1>, 11,0>, 10,0>, and 11,1>
(see Table III-1). The spacing within each of
the four 6-line patterns is uniformly 8.7 G.
The few extra lines in the background of the
lines centered at 3006 G are due to residual
109Ag atom signals.............................. 128

IV-1. The ESR spectrum of AgSI is an argon matrix at
12 K. The line positions of the impurities (CH3
and SlH3) are noted. v = 9.380 GHz ............. 142

IV-2. The ESR spectrum of MnSI in an argon matrix at
12 K. v = 9.380 GHz ............................ 143

IV-3. The ESR spectrum of Sc codeposited with Sl into
an argon matrix at 12 K. The eight sets of
doublets are shown for HScSiHn. v = 9.380 GHz.. 153

IV-4. The mi= -1/2 and -3/2 transitions for HScSiHn
at 12 K. Two different rod orientations are shown.
The top is with the rod parallel to the field and
the bottom trace is for the rod perpendicular to
the field. v = 9.380 GHz ....................... 154

IV-5. The ESR spectrum in the g=2 region after
annealing a matrix containing both Si and Sc
(v = 9.380 GHz). The doublets due to HScSiHn have
disappeared. The only remaining lines are due to
impurities and H2ScSiHn (noted) ................ 155


IV-6. The ESR spectrum (mi=7/2 and 5/2) for H2ScSiHn
in argon at 12 K after annealing to about 30 K.
The top trace is for the rod perpendicular to
the field and the bottom trace is for the rod
parallel to the field. v = 9.380 GHz ........... 162

IV-7. Same as Figure IV-6 except the mi=3/2 and -3/2
transitions are shown........................... 163

IV-8. Same as Figure IV-6 except the m=l1/2 transition
is shown........................................ 164

IV-9. Same as Figure IV-6 except the mi=-5/2 and -7/2
transitions are shown .......................... 165


Abstract of Dissertation Presented to the
Graduate School of the University of Florida
in Partial Fulfillment of the Requirements for
the Degree of Doctor of Philosophy




December, 1987

Chairman: Professor William Weltner, Jr.
Major Department: Chemistry

Three vanadium carbonyls were formed by codeposition of

vanadium vapor and small amounts of 12CO and 13CO in neon,

argon, and krypton. Two of the species were high spin (S>1)

molecules. For VCO (S=5/2) two conformations of almost equal

stability were trapped in various matrices. The dicarbonyl

was also observed and found to have an S=3/2 ground state and

a zero field splitting parameter DI = 0.30 cm-1. Also

observed only in a neon matrix was V(CO)3. The ground state
2 2 depending on
for this axial molecule is either A or 2A depending on

whether it has planar D3h or pyramidal C3V symmetry.

Other first row transition metal carbonyls were studied

using Fourier transform infrared spectroscopy. When chromium

and CO were codeposited into an argon matrix, a molecule was

formed with CO for which a stretching frequency at 1977 cm-1


was observed. An attempt is made to relate the bonding of

first row transition metal monocarbonyl molecules to the

observed infrared CO stretching frequencies of these


A cluster of seven silver atoms was produced when the

109 isotope of silver was vaporized and deposited into a neon

matrix and analyzed using electron spin resonance

spectroscopy. The product signals were strongest after the

matrix had been annealed. From the observed hyperfine

splitting it was determined that the cluster has a
2 "
pentagonal bipyramidal (D5h symmetry) structure with a A2

ground state. Its properties are shown to be similar to

those found by other workers for the Group IA alkali metal


Pure metal clusters were isolated when silicon was

codeposited with silver and manganese. Silver silicide was

isolated in an argon matrix and found to have a doublet

ground state. Manganese silicide was observed in both argon

and neon matrices and found to have an S=3/2 ground state.

Hyperfine parameters have been determined for both species.

Silicon-containing scandium hydrides have also been observed

in argon matrices upon codeposition of silicon and scandium

vapor. The two species identified contained one and two

hydrogens attached to the scandium. Both molecules were

found to have doublet ground states. Hyperfine parameters

were determined for both species.



Matrix Isolation

Matrix isolated metal clusters and metal carbonyls can

be studied in a variety of ways and are thought to aid in the

understanding of metal catalysts. The methods of studying

these compounds are as diverse as the compounds themselves,

varying from optical methods such as infrared. Raman,

uv-visible, to electron spin resonance and magnetic circular

dichroism spectroscopies which probe the cluster for its

magnetic and electronic properties. Using these varied

techniques, it is possible to determine the electronic

structure of the metal clusters. This information can then

be tied together with theoretical calculations to elucidate

the properties of the metal cluster.

Before the advent of matrix isolation, the study of

metal clusters was carried out either in the gas phase or in

solutions. Matrix isolation was developed in the mid-1950's

by George Pimentel and coworkers (1). The technique was

developed as a means of studying highly unstable reaction

intermediates which would, under standard conditions, be too

short lived to be observed. It has since been applied to a

wide variety of systems which have one thing in common: The

species of interest are too unstable to be studied under

normal laboratory conditions.


Several things are required in order to do matrix

isolation. First, the experiments require a high vacuum

environment. This means a pressure below 1x10-6 torr must be

maintained in order to minimize the amount of atmospheric

impurities that will be trapped within the matrix. This type

of a vacuum is usually achieved using an oil diffusion and a

mechanical pump with a liquid nitrogen cold trap.

Matrix isolation experiments are usually carried out

between 4 and 15 K. This temperature range can be achieved

by one of two methods depending upon the desired minimum

temperature. Using commercially available closed cycle

helium refrigerators is one way to cool the deposition

surface to the desired temperature. The only drawback of

this system is that the minimum temperature the refrigerator

can achieve lies at about 12 K (Recent advances have produced

a closed cycle system which is capable of 4 K, but their cost

is prohibitive). An alternative to this method is to simply

use liquid helium to cool the deposition surface. This can

be done by using either a dewar or a commercially available

transfer device such as the Air Products Hell-tran. These

devices can achieve a low temperature in the neighborhood of

4 K.

The final consideration in setting up this type of

experiment is the substrate on which to deposit the matrix.

Types of material for this surface range from CsI (or other

suitable alkali halide salts), to sapphire, or to polished


metal surfaces. Factors to be considered when choosing the

substrate depend on the type of experiment being done, but

all such solids must have high thermal conductivity. Also,

optical properties must be considered when doing absorption

or emission studies, whereas magnetic susceptibilities are of

concern in ESR and MCD. Obviously, the purity of the solid

substrate is important since even small amounts of some

impurities can cause strong absorptions or magnetic


The term "matrix isolation" comes about because the

molecules of interest are trapped in a matrix of inert

material, usually a noble gas such as neon, argon, krypton,

or nitrogen. The trapping site is usually a substitutional

site or an imperfection in the crystalline structure of the

solid gas, and the trapped species, seeing only inert nearest

neighbors, are isolated and can not react further.

Trapping the metal atom in the matrix is fairly

straightforward once the method of atomizing the metal has

been determined. But, a problem arises in producing metal

dl, tri, and higher-order species. It is unusual to simply

deposit a matrix and get a species other than a monomer or a

dimer. In order to get these higher order species of

interest several techniques can be used.

The most common technique used is to simply anneal the

matrix. Annealing involves depositing the matrix, and then

warming it. The amount that the matrix is warmed depends on


two things, the matrix gas being used and the trapped

species. A matrix can usually be annealed up to a

temperature equal to approximately one third that of its

melting point without solid state diffusion occurring. The

problem that occasionally arises is that once the temperature

of the matrix has begun to rise, it is possible for some

reactions to occur due to diffusion of smaller atoms or

molecules that are trapped. This might induce an exothermic

reaction, causing the matrix to heat more rapidly than

intended, exceeding the capability of the cooling system to

dissipate the heat produced. The pressure then rises and

rapid evaporation of the matrix occurs (2).

Photoaggregation is another method employed to produce

metal clusters after the matrix has been deposited. This

method usually involves photo-excitation of the metal which

causes local warming of the matrix as the metal atom

dissipates the excess energy. This partial warming loosens

the matrix around the metal atom allowing it to diffuse and

possibly interact with other metal atoms in the vicinity.

This method has been used to successfully produce silver

clusters by Ozin and coworkers (3).

Using different matrix gases can also give differing

results as to the size of the metal molecules formed. These

differences arise for several reasons. The most obvious is

that the different solid "gases" will have different sized

substitutional and interstitial spaces. Another consideration

is the rate at which the matrix freezes. This will depend

not only on the capacity of the cooling system to dissipate

the excess energy, but also on the freezing point of the gas.

It is important to remember that the amount of energy that

the cooling system can dissipate depends on the temperature

to which it must cool the deposition surface. The Displex or

Helium dewar can dissipate substantially more energy at a

higher temperature, such as that necessary to freeze krypton

(melting point 140 K) rather than neon, which freezes at

about 20 K. This difference in freezing rates will allow a

varying amount of time for the atoms to move around on the

surface of the matrix which is in a semi-liquid state. The

longer the atoms can move on the surface of the matrix, the

greater the chance for the aggregation and formation of small

metal molecules (2). The kinetics of cluster formation will

be discussed later in greater detail.

In the last 30 years a wide variety of methods has been

applied to the study of molecules and atoms which have been

trapped in matrices. One technique used to obtain data is

resonance Raman spectroscopy. In this case a polished

aluminum surface is used as the deposition surface for the

matrix. The metal is vaporized by electrically heating a

metal ribbon filament and codepositing the vapor with the

matrix gas. The aluminum deposition surface is contained in

a pyrex or quartz bell to facilitate viewing and irradiating

the matrix with an argon laser (4).

In this type of an experiment one can limit the amount

of metal entering the matrix to half of the aluminum surface,

leaving the other half virtually free of metal atoms. It is

then possible to probe various parts of the matrix to

determine the distribution of metal in the matrix (Moskovits

purposely screened part of the metal stream so as to achieve

a concentration gradient within the matrix) (4). From

resonance Raman experiments it is possible to determine the

vibrational frequency (at the equilibrium internuclear

separation (we)), and the first order anharmonicity constant

(oexe). Typical molecules which have been investigated using

this technique are Fe2, NiFe, V2, Ti2, Ni3, Sc2, Sc3, and

Mn2. Another common way of determining the presence of metal

in the matrix is the color of the matrix. Most matrices

containing metal atoms or molecules will have a

characteristic color (4).

Magnetic circular dichroism (MCD) spectroscopy is

another technique which has been used to study matrix

isolated metal clusters. MCD is the differential absorbance

of left and right circularly polarized light by a sample

subjected to a magnetic field parallel to the direction of

propagation of the incident radiation. A one inch diameter

CaF2 deposition window is used, and the magnet (.55T) is

rolled up around the vacuum shroud surrounding the deposition

surface. Before an MCD spectrum is taken, a double beam

absorption spectrum is usually taken, the reference beam


being routed around the vacuum shroud through the use of

mirrors, in order to maximize the signal to noise ratio of

the MCD spectrum. The optimum absorbance value has been

found to be 0.87, and deposition times are controlled

accordingly (5). The information gained from this type of

experiment is very useful in assigning the electronic ground

state of the species under study. The MCD technique also has

the advantage of being able to assign spin-forbidden

electronic transitions. Properties of excited electronic

states have also been investigated utilizing MCD (5).

Optical absorption spectroscopy has also been done on

matrix isolated samples; for example, PtO and Pt2 have been

studied in argon and krypton. Atomic platinum lines were

also observed. A KBr cold surface was used as a deposition

surface. A hollow cathode arrangement was used to vaporize

platinum wire, which was being used as the anode. This was

then put into a stainless steel vacuum vessel equipped with

an optical pathway. Deposition times were varied from a few

minutes for Pt up to two hours to make PtO and Pt2. The

absorption spectrum was then taken (6).

The present work has utilized two types of analysis,

namely electron spin resonance spectroscopy and Fourier

transform infrared spectroscopy. When doing ESR, two types

of deposition surfaces are usually used, either a copper or a

single-crystal sapphire rod. Both are magnetically inert,

and they, like other deposition surfaces, have good thermal


transport properties. In order to do ESR the sample has to

be placed into a homogeneous magnetic field. This is usually

accomplished by mounting the vacuum shroud surrounding the

deposition surface on rails. The matrix can then be

deposited outside of the confines of the magnet's pole

faces (7,8).

In order to perform a typical matrix isolation

experiment using electron spin resonance to analyze the

matrix, several pieces of specialized equipment are

necessary. Measurements on the matrix take place between the

pole faces of an electromagnet. This inherently restricts

the size of the vacuum chamber and deposition surface. The

set-up used is typically in two parts. One half contains the

metal deposition set-up or "furnace" (Figure I-1). The

second part contains the deposition surface and the ESR

cavity (Figures I-2 and 1-3). Figure I-2 shows the system

configured with the Hell-Tran liquid helium transfer device

from Air Products, and Figure I-3 has the set-up configured

with the Air Products Displex closed cycle helium

refrigeration system. The two halves are separated by a set

of gate valves so that they can be disconnected from each

other without compromising the high vacuum conditions

maintained in each. Once separated the rod is lowered into

the ESR cavity with the aid of pneumatic pistons. After the

rod is in the cavity, the half containing the ESR cavity and

the rod is rolled into the magnet so that the ESR cavity and

rod are located between the pole faces of the magnet.

Infrared work can also be done in a fashion similar to

that used for ESR. The primary difference is that the

deposition surface is usually CsI or quartz because of their

optical properties (No significant absorptions between 4000

and 200 wavenumbers). For this type of work the vacuum

shroud containing the deposition window usually sits in the

sample compartment of the infrared spectrometer aligned so

that the sample beam passes through the deposition window.

The apparatus for doing infrared experiments has some

similarities to that used for the ESR experiments. There is

a furnace and a dewar, and gate valves separating the two

(Figure 1-4). But a much smaller vacuum shroud can be used

because only a deposition window is contained inside of it.

It is important that the infrared beam passes into the vacuum

vessel, through the deposition surface (in most cases), and

back out again so that the beam can reach the detector, which

means that the matrix, as well as the windows through which

the infrared beam must pass, needs to be able to transmit

radiation in the infrared region. The deposition window

usually remains in the infrared instrument for the entire

experiment, which then enables one to follow the deposition

of the matrix.

Several options are available to vaporize the metal

sample. The usual method is resistive heating. The metal is

Figure I-1. The furnace flange with copper electrodes
and a tantalum cell attached.

Figure I-2. The EPR cavity and deposition surface
within the vacuum vessel. The apparatus is capable of
cooling the rod to 4 K, because of the liquid helium
transfer device (Heli-Tran) on the top.


Figure 1-3. The ESR cavity and deposition surface
within the vacuum vessel. The apparatus is capable of
cooling to 12 K because of the closed cycle helium
refrigeration device (Displex) on the top.












Figure 1-4. The vacuum vessel, deposition surface,
Displex, and furnace assembly for infrared experiments.

placed into a cell made of a high melting metal with good

electrical properties. (Mixed metal species may sometimes

arise in high temperature work because a significant portion

of the cell may also vaporize with the sample.) Heating in

this fashion, it is possible to achieve temperatures in

excess of 2000 C. An alternative method is to put the

sample cell into an inductive heating coil; in this manner,

comparable temperatures can usually be attained. The

temperature of the furnace is estimated by using an optical

pyrometer; more accurate measurements require an estimate of

the emissivity of the hot surface.

The determination of what has been trapped can

sometimes be simple or, at other times, rather complex. In

the case of Sc2 it was rather straightforward. The ESR

spectrum was measured by Knight and co-workers (9). Since

the Sc nucleus has a spin of 7/2 (1=7/2), the hyperfine

structure observed identified the trapped species. A

resonance Raman experiment determined the vibrational

frequency of the ground state molecule to be 238.9 cm-1 (10).

From this information it was then possible to determine

whether a chemical bond exists between the trapped species.

In the case of discandium, a single bond and not van der

Waals forces binds the two atoms (11).

A more controversial diatomic molecule, dichromium, is

not quite as straightforward; it has a singlet ground state

and is therefore not observable using ESR. Theoretical

studies of Cr2 indicate a variety of bonding configurations.

A resonance Raman study has examined both di and tri chromium

(10). For the dichromium species it was first necessary to

decide which of the spectral features belonged to dichromium

and which belonged to trichromium. This was done in two

ways. The change in relative intensities of the bands was

observed as the concentration was varied (the assumption

being that a more concentrated matrix would favor a larger

cluster), and a high resolution scan of one of the observed

lines was fit with the calculated isotopic fine-structure

spectrum, assuming the carrier of the line to be Cr2. The

vibrational frequency could then be determined from the

spectra, and from this a force constant indicating the

strength of bonding. The results from the experiment

indicate that multiple bonding does exist (k=2.80 mdyne/A).

(Dicopper with a single bond has a k=1.3 mdyne/A.) The

extent of the multiple bonding can not be determined from

these results (11).

Divanadium has also been investigated using the

resonance Raman technique to yield an equilibrium vibrational

constant of 537.5 cm-1 (4), but mass spectrometric data were

needed to complete the picture. It was determined that the

dissociation energy for divanadium is about 1.85 eV (11).

From spectroscopy done in a two-photon-lonization mass

selective experiment, on a supersonically expanded metal
beam, a value of 1.76 A for the equilibrium distance was

measured (12). The short bond distance coupled with a high

vibrational frequency shows that the molecule is strongly

bonded by 3d electrons (11).

Higher order metal clusters such as Mn5 have also been

trapped, and ESR spectra measured in matrices. In this case

several equally spaced (300 G) lines were observed in the

spectrum. From the number of these fine-structure lines it

was determined that the molecule has 25 unpaired electrons

(Hyperfine splitting were not resolved.). On this basis it

is possible to postulate the cluster size. Smaller clusters

are improbable on the basis of S=25/2. It is also important

to remember that the larger clusters are unlikely to form in

the matrix initially. The structure of Mn5 is thought to be

pentagonal with single bonds between each of the manganese

atoms and with each of the atoms having five unpaired

electrons. The ESR spectrum indicates that all of the

manganese atoms in the molecule are equivalent; a pentagonal

structure fulfills this requirement (13).

A similar problem arises in the case of doing a high

concentration scandium experiment. In this case it is

thought that a molecule with 13 Sc atoms is made. The ESR

spectrum contains over 60 lines in the g=2 region of the

spectrum, which usually indicates one unpaired electron in

the molecule. Because the intensity of the lines drops off

at the fringes of the hyperfine structure, it is difficult to

ascertain exactly how many lines exist. With the observed

lines there are at least nine Sc atoms in the molecule. The

Sc13 molecule seems likely because of theoretical

calculations done on transition metal clusters with 13 atoms.

A single Self-Consistent Field Xa-Scattered wave calculation

has been done for Sc13 giving a single unpaired electron, in

agreement with the ESR results (14).

Matrix isolation is more of a technique than a method

of analysis. It can be used in conjunction with various

analytical tools which then can be used to determine the

composition and structure of the trapped compound. It is

important to use some forethought in choosing the method of

analyzing the trapped molecule because the method chosen will

determine what information can be obtained from the

experiment. Data from various methods will tend to

complement each other. For example, if the molecule of

interest does not contain an unpaired electron, then it would

not be worthwhile to do ESR since this method requires the

presence of at least one unpaired electron in order to

produce a spectrum. Analyzing the molecule for its

vibrational structure by using resonance Raman or infrared

spectroscopy will only give the molecule's vibrational modes.

From these modes it may or may not be possible to determine

the molecule's structure, depending on the complexity of the

molecule and the degeneracy of the modes. Another problem,

when dealing with clusters, is that it is sometimes difficult

to determine the size of the cluster from the observed


spectra. Also, once the molecule is trapped, its trapping

environment may not be uniform throughout the matrix. This

will cause splitting in the observed lines of the spectrum

due to different trapping sites. The trapping site may also

cause a lowering of the observed point group of the molecule,

which will cause lines to split because they are no longer

degenerate. Matrix isolation is a useful tool which aids in

determining the structure of unstable species, but it is best

used in conjunction with other techniques if accurate

structures are to be determined.

As can be seen from what other workers have done,

matrix isolation can be used to trap very reactive and also

very interesting species. We set out to use this technique

to further elucidate the properties of transition metals and

transition metal carbonyls. Following this line of interest

has lead us to study various first row transition-metal

carbonyls using both ESR and FTIR. We continued our work

with transition metals by investigating the group IB metals

and attempted to produce larger clusters. We finally turned

our attention to the first row transition-metal silicides.

Our hope in these endeavors was to produce various metal

containing species in order to determine structure and to

obtain possible enlightenment as to the reaction processes

occurring to form them.

Producing these metal species as well as analyzing the

resultant spectra tends to be a rather complicated process.

A review of the kinetics of cluster formation is very helpful

in pointing out and understanding some of the processes

involved in producing these exotic species. The analysis of

the ESR spectra can sometimes be rather simple when one is

dealing with only a few lines. But when the trapped species

produces many lines, the analysis rapidly becomes complicated

and a review of relevant theory becomes mandatory. A brief

review of infrared spectroscopy will also be presented.

Theory of Cluster Formation

In recent years the area of metal cluster chemistry

has become rather active. The main reason behind this is the

hope that the metal cluster will be useful in the

investigation of the chemistry that occurs at metal surfaces.

This interest has lead to two major thrusts, one involving

the reproducible production of these clusters and the other

concerning itself with the mechanisms involved in the

evolution of the clusters. Both of these areas are now being

actively pursued by various workers (15-19).

Experimentally, the production of these clusters and

their identification have proven extremely challenging.

Three primary methods of vaporizing the metal exist, laser

vaporization, resistively heating of a cell containing the

metal of interest, or heating a wire made of the appropriate

metal. Of these the most successful has been the use of

lasers. A major problem in determining the kinetics involved


in cluster formation is the reproducibility of the

distribution of cluster sizes from experiment to experiment.

Several groups have had some success at this and even have

begun to react these metal clusters with various types of

reactants (20,21).

In developing a general mean-field kinetic model of

cluster formation one must look first at the aggregation

process in the thermal vaporization source. Second, a method

needs to be found to calculate probabilities for cluster

formation taking into account atom-atom collisions to form

dimers as well as collisions between clusters. This would

have to include not only aggregation but also cluster

fragmentation from collisions, structural stabilities of

certain clusters, how energy is dissipated upon collision,

and possible transition states of the clusters. It should

also be able to explain the cluster distribution found in

mass spectra of these systems.

The metal clusters are produced by laser vaporization

in a supersonic nozzle source and then allowed to enter a

fast-flow reactor, before being mass analyzed. The source of

the metal of interest is a rod about 0.63 cm in diameter.

The rod is placed in the high pressure side of a pulsed

supersonic nozzle, operating with a ten atmosphere back

pressure. The frequency doubled output of a Q-switched

Nd:YAG laser (30 to 40 mJ/pulse, 6 ns pulse duration) is

focused to a spot approximately 0.1 cm in diameter on the


target rod, and fired at the time of maximum density in the

helium carrier gas pulse. The target rod is continually

rotated and translated, thus preventing the formation of deep

pits, which would otherwise result in erratic fluctuations in

the sizes of the metal clusters. The helium-metal vapor

mixture then flows at near sonic velocity through a

cluster-formation and thermalizatlon channel, 0.2 cm in

diameter and 1.8 cm in length, before expanding into a 1 cm

diameter, 10 cm long reaction tube. Effectively, all cluster

formation in such a nozzle source is accomplished in the

thermalization channel since expansion into the 1 cm diameter

reaction tube produces a 25 fold decrease in density of both

the metal vapor and the helium buffer gas (20).

The reaction tube has four needles which can be used to

inject various reactants into the flowing mixture of carrier

gas and metal. Following the reaction tube, the reaction gas

mixture is allowed to expand freely into a large vacuum

chamber. A molecular beam is extracted from the resulting

supersonic free jet by a conical skimmer and collimated by

passage through a second skimmer. The resulting well-

collimated, collisionless beam is passed, without

obstruction, through the ionization region of a time-of-

flight mass spectrometer (TOFMS). Detection of the metal

clusters and their reaction products is accomplished by

direct one photon ionization in the extraction region of the

TOFMS (20).


With the advent of this type of a device, it is now

possible to produce metal clusters under relatively

controlled conditions with a fairly reproducible distribution

of cluster sizes. The reproducibility of cluster size

distribution between experiments has made it possible to

compare the results to kinetic studies dealing with the

formation of metal clusters. The kinetic analysis of the

clusters has been able to explain why some cluster sizes are

favored, to suggest the relative importance of kinetic and

thermodynamic effects, and to shed some light on the possible

influence of ionization of the clusters.

The kinetic theory applicable is that of aggregation

and nucleation. The mean-field rate equations governing the

aggregation of particles developed by Smoluchowski (22) are

i-1 x i = N
(I-1) Xi j~1Kjii Xj X i_ 1 Xj X I = 1,...,N

In equation I-1, Xi denotes the concentration of clusters of

size i. The aggregation kernel, Kij, determines the

time-dependent aggregation probability. The first term on

the right hand side of the equation describes the increase in

concentration of clusters size I due to the fusion of two

clusters size j and i-j. The second term describes the

reduction of clusters size i due to the formation of larger

clusters. The equation must be generalized in order that the

neutral-positive, neutral-negative, and positive-negative

cluster fusions are included because the vaporization process

will produce some ions. But, since the electrons and the

ions are attracted by long range Coulomb forces, the

recombination processes are very fast, leading to a

population of neutral atoms that is much larger than the

population of ions. Therefore, the probability for

positive-negative cluster fusion is much smaller than that of

neutral-positive and neutral-negative cluster fusion and can

be neglected (15).

The terms due to the formation of positive clusters are

.0 0+ 0 +
(1-2) Xi = Xi Kij Xi Xj


.+ 1i-1 0+ 0 + N 0+ 0 +
(1-3) Xi j 1 Kji X Xi-j Kji X Xi i = 1,.....N

where Xi is defined by the right hand side of equation I-1
.0 .+
and Xi and Xi denote the concentration of the neutral and

positively charged clusters, respectively. The kernels Kij
and Kij describe the neutral-neutral and neutral-positive

aggregation probabilities. Analogous terms are introduced

due to the presence of the negatively charged clusters (15).

During cluster growth, there is also the possibility of

charge transfer between the charged and the neutral clusters

without the accompanying cluster fusion. The probability of

electron transfer between the negatively charged and the


neutral clusters is much greater than the probability of

electron transfer between the neutral and the positively

charged species since electron affinities are much smaller

than the ionization potentials for small and medium sized

clusters. Therefore, only the electron transfer terms from

the negatively charged to the neutral clusters are included

.0 .0 0 T 0 -
(I-4) X i J= Tij Xi Xj + j Tji Xj Xi

S- 0 0-
(1-5) Xi = Xi + = Ti X X Tji Xj X
J=J ij i j ji

where Tij is the nonsymmetric charge transfer kernel. The

coupled rate equations I-(2 through 5) can then be solved

simultaneously for the concentrations of the neutral and the

charged clusters (15).

Classically, the aggregation probability for clusters I

and j with diffusion coefficients Di and Dj is proportional

to DijRij, where Dij = Di + Dj is the joint coefficient for

the two clusters, and Rij is the catching radius within which

the clusters will stick with unit probability. In a reactive

aggregation, one has to consider the reaction probability

within the interaction radius. The expression for reactive

aggregation becomes

Kiu = 4nDijRij[Pij/(Pij+PD)]



where P.j is the reaction probability per unit time and PD is

the probability of the reactants to diffuse away. One has PD

= 1/tij, where rij is the average time in which the clusters

remain within the reaction distance Rij. From the diffusion


(1-7) D = (kBT)3/2/(apmO.5

one can easily show that 1/rij= 6Dij/(Rj)2 leading to

(1-8) Kij = 4DIRijRj[Pij/(Pij+(6Dij/Rij2))]

The limiting forms of equation I-8 are of particular

interest. If PIj>PD, Kij z 4xDijRj and the distribution of

cluster sizes is governed by the classical aggregation

kinetics. No reaction-induced magic numbers will arise in

this case. The true solution to equation I-1 can be

approximated in this case by the ones corresponding to the

exactly solvable simple kernels. Since the variation of the

kernels with cluster size is not very strong in the classical

limit, a constant kernel solution may be used where Kij = 2C

and xl(t=O) = x0 may be used for qualitative purposes.

(I-9) Xi(t) = X0(CXot)i-1/(l+CXOt)1+1

A more accurate solution can be obtained provided one

includes the variation of the diffusion coefficients and the


catching radii with cluster size. Since cluster reactivity

varies with its structure, the uniqueness of the structure

explains the reproducibility of the reactivity data for small

transition metal clusters (15).

The application of classical kinetics is best

exemplified by transition metal clusters. The mass

distribution spectra of these clusters are essentially

featureless. This is what is predicted by classical

kinetics. However, although the distribution of transition

metal clusters is classical, the small and medium size

clusters will probably have unique or nearly unique shapes.

This will result in a certain amount of nonclassical behavior

which will cause certain cluster sizes to be favored (15).

In the other limit of equation I-8, where PD =

6Dij/(Rij)2 > Pij, the growth of clusters is reaction

limited. In this case, one can neglect Pij in the

denominator. Since the diffusion constants cancel, the

equation becomes

(I-10) Ku = 4x/[6(Rij)3Pij].

Thus the aggregation probability in the reaction limited

regime is dependent only on the reaction probability and does

not depend on the value of the diffusion coefficient.

Therefore, the aggregating clusters will undergo several

collisions before fusion. Significant variations in cluster


size are likely to occur since the reaction probabilities

depend on structure, symmetry, and the stability of the

reacting clusters. This is the reason why magic numbers are

observed. The reproducibility of the measured magic numbers

under a wide variety of experimental conditions is due to the

independence of the rate of fusion on the diffusion

coefficient (15).

The knowledge of reaction probabilities for each pair

of reacting clusters, including their charge dependence, is

required to calculate the cluster distribution in the

reaction limited regime. Since this is computationally

prohibitive, one is forced to make several approximations in

order to make the calculation feasible. The final expression

for the aggregation kernel becomes

(I3 e-(AGI+AGJ)/KBTav
(1-11) Kj = GRije v

after using the Polanyl-Bronsted relationship to estimate the

relative differences between transition state energies.

Also, since the reactants are probably going to undergo

considerable structural rearrangement after initial

attachment, scaled derivatives are used to describe the

energy gained upon addition of a single atom. Gibbs free

energies can be used to account for the possible temperature

dependent structures which can arise due to the dependence of

cluster entropy with structure (15).

The charge transfer kernel is approximated by

3 e-(Ai+Aj)/KBTav
(I-12) Tij= oR1e

Since the electronic wave function of a negatively charged

cluster has a relatively large radius, the Polanyi-Bronsted

proportionality factor, C, in the charge transfer kernel is

significantly larger than the corresponding factor in the

aggregation kernel (15).

The final two equations require electronic structure

calculations for only the end products (those observed in a

mass spectrum). The average temperature, Tay, is not known

at the outset, but the analysis of experimental spectra

provides an upper bound. The spectra of positively charged

clusters are thus determined by two adjustable parameters,

where as those of negatively charged clusters require an

additional parameter (C) (15).

Ziff and co-workers (16) have studied the validity of

using the Smoluchowski equation for cluster-cluster

aggregation kinetics. They investigated the validity of the

mean-field assumption by looking at the concentrations of the

cluster species and also by investigating the asymptotic

behavior of the equations. They found the mean-field

Smoluchowski equation to be appropriate in describing the


aggregation of particles which form fractal clusters. The

only problem was in determining the fractal properties of the

kernel. Even though these properties are difficult to

determine, once they are known the entire description of the

kinetics follows Smoluchowski, as presented by Bernholc and

Phillips (15).

The kinetic theory for clustering as presented by

Bernholc and Phillips (15) has been able to model the cluster

distributions found for carbon by Smalley and co-workers

(23). Bernholc and Phillips used the calculated formation

energies with a semiempirical estimate of the entropy

difference between chains and rings as input for the kinetic

energy calculations. They found that the cluster

distributions were in good agreement with the experimental

work of Smalley. This includes the data for both the

positive and negative ions produced directly in the source.

The magic numbers in the range of n equal 10 to 25 were well

reproduced. They also found that electron transfer effects

have a strong effect on the measured distributions of small

and medium clusters of the negative ions. For the positive

ions produced from photolonization of neutral clusters, the

calculated cluster distributions show that photofragmentation

and/or photothreshold and photolonization cross section

dependence on cluster size have a major effect on the

measured spectra up to about n equals 25. This was not found

to be true for larger clusters.


Even though these are just beginnings in the

understanding of what is involved in cluster formation, it

is essential to realize that the clusters which are seen by

experimentalists are the products of a complicated set of

circumstances which may possibly be at the control of the

experimentalist. With this type of background it may be

possible in the future to produce a desired cluster size by

finely tuning the experimental conditions. To be able to do

this, it will be necessary to understand what the critical

factors are in the formation of clusters. Is it the overall

flux of metal in the carrier gas? Can the amount of

ionization be controlled in order to produce the desired

cluster sizes? Or will the inherent stabilities of certain

clusters override these factors and limit the variation of

cluster size which can be easily produced? These are

questions which will only be answered through a close

synergic relationship between experiment and theory.

ESR Theory

Electron Spin Resonance (ESR) spectroscopy is concerned

with the analysis of paramagnetic substances containing

permanent magnetic moments of atomic or nuclear magnitude.

The theory of ESR spectroscopy has been dealt with by many

authors, and if desired a more in depth treatment can be

found there (24-29). In the absence of an external field

such dipoles are randomly oriented, but application of a

field results in a redistribution over the various

orientations in such a way that the substance acquires a net

magnetic moment. If an electron or nucleus possesses a

resultant angular momentum or spin, a permanent magnetic

dipole results and the two are related by

(1-13) _= TP_

where g is the magnetic dipole moment vector, pis the

angular momentum (an integral or half-integral multiple of

h/2x = f, where h is Planck's constant), and T 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

(1-14) w= -rH.

The component of u along H remains fixed in magnitude, so the

energy of the dipole in the field (the Zeeman energy)

(1-15) W= -f*H

is a constant of the motion.

The relationship between the angular momentum and the


magnetic moment is expressed by the magnetogyric ratio in

equation 1-13 and is defined by

(I-16) r = -g[e/(2mc)]

where e and m are the electronic charge and mass,

respectively, and c is the speed of light. The g factor is

equal to one for orbital angular momentum and is equal to

2.0023 (ge) for spin angular momentum. Defining the Bohr

magneton as 3=ef/2mc and combining the g factor with equation

1-13 we have (along the field direction)

(1-17) PS = -germs-

Only 2p+l orientations are allowed along the magnetic field

and are given by mSh where mS is the magnetic quantum number

taking the values

(1-18) mS = s, s-1,..., -8

because the angle of the vector g_is space quantized with

respect to the applied field H. This accounts for the

appearance in Eq. (1-17) of the mS factor for spin angular


In the case of an atom in a 2S1/2 state where only spin


angular momentum arises, the 2S+1 energy levels separate in a

magnetic field. Each level will have an energy of

(1-19) EM= gem"SH

which will be separated by geSH. The g factor is an

experimental value and mS an "effective" spin quantum number

because the angular momentum does not usually enter into the

experiment as purely spin, i.e. some orbital angular momentum

usually enters into the observed transitions. For orbitally

degenerate states described by strong coupling scheme

(Russell-Saunders), J=L+S, L+S-1, ..., IL-S) and

(1-20) Ej = gjpmjH


(1-21) gj = 1 + [S(S+1)+ J(J+1)- L(L+1)]/[2J(J+1)]

is the Lande splitting factor. This reduces to the free

electron value for L=O.

The simplest case of a free spin where mj = mS = +1/2

will give two energy levels. The equation for the resonance

condition follows:

(1-22) hv = geAHo


where HO is the static external field and v is the frequency

of the oscillating magnetic field associated with the

microwave radiation. In this research a frequency of about

9.3 GHz (X-band) was employed. The transitions observed can

be induced by application of magnetic dipole radiation

obtained from a second magnetic field at right angles to the

fixed field which has the correct frequency to cause the spin

to flip.

The Hyperfine Splitting Effect

As described above, an ESR spectrum would consist of

only one line. This would allow one to determine only a

value for the g factor for the species. Fortunately, this is

not the only interaction which can be observed via ESR

spectroscopy. These other interactions tend to greatly

increase the observed number of lines. One of the most

important of these interactions is the nuclear hyperfine

interaction. ESR experiments are usually designed so that at

least one nucleus in the species under investigation has a

non-zero magnetic moment. The magnetic moment of the odd

electron can interact with this nuclear moment and split the

single ESR line into hyperfine structure.

In the simplest case of a nucleus having a spin 1=1/2

interacting with a single electron, the magnetic field sensed

by the electron is the sum of the applied fields (external

and local). A local field would be one caused by the moment

of the magnetic nucleus. This local field is controlled by

the nuclear spin state (I=1/2, in this case). Because there

are two nuclear levels (21+1), the electron will find itself

in one of two local fields due to the nucleus. This allows

two values of the external field to satisfy the resonance

condition, which is

(1-23) Hr = (H' + (A/2)) = (H' AMI)

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=O.

A good example of an ESR spectrum is that of the

hydrogen atom with the Zeeman energy levels shown in Figure

I-5. Hydrogen has one unpaired electron for which a

transition at about ge should be observed. Because of the

spin angular momentum of the electron interacting with the

spin angular momentum of the nucleus (1=1/2), two lines are

observed. The lines are split around the "g" value for a

free electron which is ge = 2.0023 and occurs at about 3,400

Gauss in an X-Band experiment. The magnitude of the

splitting hyperfinee interaction) of the two lines about the

free electron position at ge is due to the interaction of the

free electron with the nuclear moment of the hydrogen atom.

The spin angular momentum of the unpaired electron can also

be split by several nuclei that have spins, as is the case


with CH3. The carbon nuclei (99%) are 12C which has zero

spin (1=0). The hyperfine interaction in this case arises

from the three equivalent hydrogen nuclei (each with 1=1/2)

which gives an overall 1=3/2, and four lines are observed


Several interactions are involved when a paramagnetic

species with a non-zero nuclear spin interacts with a

magnetic field. The obvious one is the direct interaction of

the magnetic moment with the external field. The precession

of the nuclear magnetic moment in the external field results

in a similar term. The equation

(1-24) gi1l= --/9N

relates the nuclear magnetic moment Iy to the nuclear g

factor (gy). In the equation the nuclear magneton, ON, is

defined as ei/2Mc where M is the proton mass and is about

1/2000th of the Bohr magneton.

The Hamiltonian can be written as

(1-25) {H) = gjeH__*{J} + hA{Ij*{Jj gPNHL*{ji

where { } indicates that the term is an operator. Small

effects such as the nuclear electric quadrupole interaction,

as well as the interaction of the nuclear moment with the


external magnetic field (Nuclear Zeeman term), which is the

last term in equation 1-25, are small and will be neglected.

The Zeeman effect in weak fields is characterized by an

external field splitting which is small compared to the

natural hyperfine splitting (hA{IJ)*({ > gSH*()J) in equation

1-25. The orbital electrons and the nuclear magnet remain

strongly coupled. The total angular momentum F = I+J orients

itself with the external field and can take the values I+J,

I+J-1,..., I-Jj. The component of F_along the field

direction, mF, has 2F+1 allowed values. In a weak field the

individual hyperfine levels can split into 2F+1 equidistant

levels which gives a total of (2J+1)(2I+1) Zeeman levels.

(Not all levels are degenerate even at zero field.)

The splitting becomes large compared to the natural

hyperfine splitting in the strong field (Paschen-Back)

region. Decoupling of I and J occurs because of strong

interaction with the external field. Therefore F is no

longer a good quantum number. Since J and I have components

along the field direction, the Zeeman level of the multiple

characterized by a fixed mj is split into as many Zeeman

hyperfine lines as there are possible values of mi (21+1).

The total energy states are still given by (2J+1)(2I+1) since

there are still (2J+1) levels for a given J. The levels in

this case form a completely symmetric pattern around the

energy center of gravity of the hyperfine multiple.


Intermediate fields are somewhat more difficult to

treat. The transition between the two limiting cases takes

place in such a way that the magnetic quantum number, m, is

preserved (In a strong field m = mi + mj, in a weak field m =

mF). The Zeeman splitting is of the order of the zero field

hyperfine splitting in this region.

With so many possible levels, the observed ESR spectrum

needs to be explained in terms of selection rules. The

transition between Zeeman levels involves changes in magnetic

moments so it is necessary to consider magnetic dipole

transitions and the selection rules pertaining to them. A

single line is observed for the mS = 1/2 <-> -1/2 transition

in the pure spin system (I=0). A change in spin angular

momentum of +f is necessary. This corresponds to selection

rule of Amj = +1. A photon has an intrinsic angular momentum

equal to f. Conservation of angular momentum therefore

dictates that only one spin can flip (electronic or nuclear)

upon absorption of a photon. The transitions usually

observed with fields and frequencies employed in the standard

ESR experiment are limited to the selection rules Amj = +1,

and AmI = 0 (The opposite of NMR work).

These interactions can be categorized as isotropic and

anisotropic, and are related to the kind interactions of the

electron with the nucleus, and can be deduced from the ESR

spectrum. The isotropic interaction is the energy of the

nuclear moment in the magnetic field produced at the nucleus


by electric currents associated with the spinning electron.

This interaction only occurs with s electrons because they

have a finite electron density at the nucleus. The isotropic

hyperfine coupling term is given by

(1-26) as = (86/3)geSgNON1((0O)2

where the final term represents the electron density at the

nucleus. There is no classical analog to this term. The as

value, also known as the Fermi contact term, is proportional

to the magnetic field, and can be of the order of 105 gauss.

It is obvious then very large hyperfine splitting can arise

from unpaired s electrons interacting with the nucleus.

Classical dipolar interactions between two magnetic

moments are the basis for describing the anisotropic

interaction. This interaction can be described by

(1-27) E = (pe*jU)/r3 -[3 (je*r) ("*rj_)]/r5

where r_ is the radius vector from the moment ge to an, and r

is the distance between them. Substituting the operators,

-g{S_) and gNN{I), for .e and Aj respectively, gives the

quantum mechanical version of equation 1-27 as

(I-28) Hdip = -gSgNPN[{I~*({L)-{S})/r3

-3(({I)*r ({SJ*r)/r5].

Then a dipolar term arises

(1-29) a = gepgIPN[(3cos20-1)/r3]

where 0 is the angle between the line connecting the two

dipoles and the direction of the magnetic field. The angular

term found in Eq. (1-29) needs to be averaged over the

electron probability distribution function because the

electron is not localized. The average of cos20 over all 0

vanishes for an s orbital because of the spherical symmetry

of the orbital.

Doublet Sigma Molecules

The spin Hamiltonian

The full spin Hamiltonian involves all the interactions

of the unpaired spin within the molecule, not just the ones

directly affected by the magnetic field. The full

Hamiltonian contains the terms below,

(1-30) H = HF + HZe + HLS + Hhf + HZn

the magnitude of the terms on the right side of Eq. (1-30)

tend to decrease going from left ot right. The first term in

equation 1-30 is the total kinetic energy of the electrons.

The "Ze" and "Zn" terms describe the electronic and nuclear

Zeeman interactions, respectively. The energy, HLS, is due


to the spin-orbit coupling interaction. The term, Hhf,

accounts for the hyperfine interaction due to the electronic

angular momentum and magnetic moment interacting with a

nearby nuclear magnetic moment. These terms have been

adequately described in detail by several authors (24-27).

This full Hamiltonian is rather complicated and difficult

to use in calculations, and the higher order terms which

could be observed in crystals have not been included. Using

a spin Hamiltonian in a simplified manner, it is possible to

interpret experimental ESR data. This was first done by

Abragam and Pryce (32). The ESR data are usually of the

lowest-lying spin resonance levels which are commonly

separated by a few cm-1. All other states lie considerably

higher in energy and are generally not observed. The

behavior of this smaller group of levels in the spin system

can be described by a simplified Hamiltonian. The splitting

are 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.

Since (S_ cannot represent a true spin, it represents an

"effective" spin. This is related to the anisotropy found in

the g factor which does not necessarily equal ge. By

convention, the "effective" g factor is defined so that the

observed number of levels equals 2S+1, just like the real

spin multiple. Therefore all the magnetic properties of a

system can be related to this effective spin by the spin

Hamiltonian. This is possible because the spin Hamiltonian

combines all of the terms in the full Hamltonian that are

effected by spin. Nuclear spins can be treated in a similar

fashion, so that the spin Hamiltonian which corresponds to

Eq. (1-30) can be written as

(1-31) HSpin = AHO*g*{S. + (I)*A*{S)

where g and A are tensor quantities and the nuclear Zeeman

term has been neglected.

The g tensor

The anisotropy of the g-tensor arises from the orbital

angular momentum of the electron through spin-orbit coupling.

The anisotropy occurs even in the sigma states which

nominally have zero orbital angular momentum. Apparently the

pure spin ground state interacts with low-lying excited

states which add a small amount of orbital angular momentum

to the ground state. This small amount is enough to change

the values of the g tensor. The interaction is generally

inversely proportional to the energy separation between the

states. This spin-orbit interaction is given by

(1-32) {H)LS = = A(L}x{S}x + {L}y(S)y + (L)z{S)z)

This term is added to the Zeeman term in the spin Hamiltonian

(1-33) {H)= BH*({L)+g({S)) + (L)*{S).

For an orbitally nondegenerate ground state represented by

IG,Ms>, the first order energy is given by the diagonal

matrix element

(1-34) WG =


where the first term is the spin-only electronic Zeeman

effect. The term, , vanishes since the ground

state is orbitally non-degenerate. The second order

correction to each element in the Hamiltonian is given by

(1-35) (H) MSM = [(l
+ geH*SnM )/n W(0)

where the prime designates summation over all states except

the ground state. The matrix elements of ge.HJSj will vanish

because = 0.

Expanding this, it is possible to factor out a quantity

(0) (0)
(1-36) A= (-E )/(Wn WG)
nn G


which is a second rank tensor. The ijth element of this

tensor is given by

(0) (0)
(1-37) Aij = (-E )/(Wn W )

where Li and Lj are orbital angular momentum operators

appropriate to the x, y, or z directions. Substituting this

tensor into HMSM yields

(1-38) HMS,MS =
+ 22{S)*{A}*{S})M'>

The first operator does not need to be considered any further

since it represents a constant contribution to the

paramagnetism. The second and third terms constitute a

Hamiltonian which operates only on spin variables. The spin

Hamiltonian results when the operator ge${H)*{S} is combined

with the last two terms of Eq. (1-38). The spin Hamiltonian

takes the form of

(I-39) HSpin = {(H}*(ge{l + 2A{A})*{S} + A2{S}*{A)*{S}

= P{H}*{g}*{S} + (S)*(D}*{S}


(1-40) {g)= g ({1) + 2A{A}


(1-41) (D} = 2(A}.

The final term in equation 1-39 Is effective only in systems

with S>l. The first term is then the spin Hamiltonian for a

2E molecule. The anisotropy of the g-tensor arises from the

spin-orbit interaction due to the orbital angular momentum of

the electron which is evident from the derivation.

The g-tensor would be isotropic and equal to 2.0023 if

the angular momentum of the system is due solely to spin

angular momentum. Deviation (anisotropy) from this value

results from the mixing in of orbital angular momentum from

excited states which is expressed through the {A} tensor.

If a molecule has axes of symmetry, they need to

coincide with the principal axes of the g tensor. Three

cases of interest can be outlined. The simplest case is one

in which g is equal to gg. This is a spin only system for

which g is isotropic. For a system containing an n-fold axis

of symmetry (n>3) there are two equivalent axes. The axis

designated z is the unique axis and the g value for the field

(H) perpendicular to z is g- and gl is the value for g when H

is parallel to z. The spin Hamiltonian therefore becomes

(1-42) HSpin = $(giHx{S)} + g-Hy)yS} + gHz{S}z).

The third case deals with the situation where the molecule



contains no equivalent axes (orthorhombic symmetry), where

gxx' gyy, and gzz are not equal and

(1-43) {H)Spin = (gxxHS)x + gyyHy(S)y + gzHz(S)}).

The A tensor

The hyperfine tensor takes into account three types of

interactions. The first term involves the interaction

between the magnetic field produced by the orbital momentum

and the nuclear moment, L*I which is usually small. More

important terms involve the interactions due to the amount of

s character of the wavefunction (the Fermi contact term) and

to the non-s character of the wavefunction also need to be

accounted for.

The isotropic interaction due to the s character is

called Aiso. Fermi (33) has shown that for systems with one

electron the isotropic interaction energy is approximately

given by

(1-44) Wiso = -(8x/3)|T(0)2 PeANPN

where W(0) represents the wave function evaluated at the


The interaction arising from the dipole-dipole

interaction of the nucleus and electron (non-s character) is

called Adip. The dipolar interaction gives rise to the


anisotropic component of hyperfine coupling in the rigid

matrix environment. The expression for the dipolar

interaction energy between an electron and nucleus separated

by a distance r is

(1-45) Wdipolar = (e*N)/r3 [3(Pe*r)(PN*r)]/r5

The term, Hhf, can now be written as

(1-46) Hhf = Hiso + Hdip

= [Aiso + Hdip]l*S

where Aiso has been given in Eq. (1-26) and Adip can be

expressed by equation 1-29. The brackets indicate the

average of the expressed operator over the wave function W.

In tensor notation the term become

(1-47) Hhf = I*A*S

where A_ = Asol + T. Here 1 is the unit tensor and T is the

tensor representing the dipolar interaction. The components

of the A tensor becomes

Aj = Aisol + Tij.



For a completely isotropic system the components of the A

tensor (Ax, Ayy, Agz) will equal Aiso. A system with axial

symmetry is treated in a manner similar to that of the g

tensor where Axx and Ayy are equal to A-. The term, Al, is

given by

(1-49) A = Also + Txx

and Azz is equal to Ag which is given by

(1-50) A, = Azz + Tzz

And finally for a system which exhibits a completely

anisotropic A tensor Ax, A yy Azz are not equal to each


In matrix isolation experiments only the absolute values

of the hyperfine parameters can be determined. In most

matrix isolation experiments, it is found that for the most

part, the signs of A_ and A, are positive. There are two

general exceptions to this. First, this may not generally be

true for very small hyperfine interactions, such as the

hyperfine interaction in CN where the splitting due to 14N is

only 5 to 10 gauss. Second, if gI is negative, the A values

will usually be negative, also (34).

Randomly oriented molecules

There is a very distinct difference between samples

held in a single crystal and those trapped in matrices. In

the case of a single crystal, the sample can be aligned to

the external field and spectra recorded at various angles of

the molecular axes to the field. Matrix isolated samples are

usually randomly oriented within the field and the observed

spectra will contain contributions from molecules at various

angles. This was first considered by Bleaney (35,36), and

later by others (37-43)

In the orthorhombic case the spin Hamiltonian can be

solved (assuming the g tensor to be diagonal), and the energy

levels can be given by

(1-51) E = OSHH(g12sin20cos2 + g22sinsin20

+ g32cos20)0.5

where S1 is the component of the spin vector S along H, gH is

the g value in the direction of H, 8 is the angle between the

molecular z axis and the field direction, and 0 is the angle

from the x axis to the projection of the field vector in the

xy plane. Returning to axial symmetry

(1-52) gH = (g2sin2 + g 2cos20)0.5

and the energy of the levels is given by


(1-53) E = PSHH(gL2sin20 + g12cos20).

It is obvious that the splitting between the energy levels

are angularly dependent. This makes the transitions between

the energy levels also angularly dependent.

The absorption intensity as a function of angle is

proportional to the number of molecules lying between 0 and

O+dO, assuming the transition probability is independent of

orientation. Since g is a function of 0 for a fixed

frequency v, the resonant magnetic field is

(1-54) H = (hv/A)(g 2cos29 + g 2sin2)-0*.5

and from this

(1-55) sin20 = (g0HO/H)2 gs2)/(ge2 g12)

where go equals (g, + 2g.)/3 and Ho equals hv/go0. From the

above equations we have

(1-56) H = hv/gl = go0H/g, at 0 = 00


(1-57) H = hu/g#3 = g0gH/g. at 0 = 90.

The absorption intensity varies from 0* to 900 and when

plotted against magnetic field, takes the appearance of


Figure I-6a, for g >g_. In a typical ESR experiment one

usually measures the first derivative of the absorption

signal. This spectrum appears in part b of Figure 1-6. The

perpendicular component is generally easily determined from

such a powder pattern. It is usually the strongest signal

observed. The parallel component is typically much weaker

and usually more difficult to detect. The values of g. and

g- can be determined as indicated assuming that the g tensor

is not very anisotropic.

Hyperfine interaction with spin containing nuclei can

split the pattern shown in Figure I-6b into (21+1) such

patterns. A simple case would be that of a molecule

containing an 1=1/2 nucleus. This is presented in Figure

I-6c. One important point is that the orientation of the

mi=l/2 pattern is opposite to that of the mi=-1/2 pattern.

This is because g, is approximately equal to g-, and Aj
Another common situation is that of the hyperfine splitting

for both parallel and perpendicular orientations are almost

equal and gi is shifted up-field from g,. In this case the

spectrum would contain two features like Figure I-6b

separated by the hyperfine splitting, A.

Molecular parameters and the observed spectrum

With all of this theory, the question now becomes what

can be learned from an ESR spectrum? To answer this, let us

begin with the solution of the spin Hamiltonian in axial

hvz=g/3Ho9,500 MHz

g =2.0023





-1/2 1/2


Figure Figure 1-5. Zeeman energy levels of an electron
interacting with a spin 1/2 nucleus.





Figure I-6. (a) Absorption and (b) first derivative
lineshapes of randomly oriented molecules with axial
symmetry and gS randomly oriented, axially symmetric molecules g . including hyperfine interaction with a spin 1/2 nucleus

( b)


symmetry including second order perturbations. This will

then show what molecular parameters can be uncovered from an

ESR spectrum.

Several authors (24,26,27,38) have given detailed

discussions of the spin Hamiltonian

(1-58) H})Spin = gjHz{S)z + g(3(Hx{S)x + Hy{S}y)

+ AI(I)z{S)z + Az({I)x{S}x + {I)y{S)y).

Considering the Zeeman term first, a transformation of axes

is performed to generate a new coordinate system x', y', and

z', with z' parallel to the field. 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'. Therefore only x and z need to be transformed.

The Hamiltonian is transformed to

(I-59) {H} = gAH{S}z, + K{I}z,(S)z, + (AtA./K)(I)x,{S}x,

+ [(A 2-A1 2)/K](glg_/g2)sin0cosO{I}x,{S)I ,

+ A. I} )y,{S})y

n n
where 12 = AglcosO/Kg, 1x = AlgLsinx/Kg, and K22 =

A 2g2cos 2 + A.22g 2sin2e. Dropping the primes and using

ladder operators {S)+ = {S)x + i{S}y and {S)- = {S)x i{S)y,

this can be rewritten in the final form

(1-60) HSpin = g"H{S}z + K(S)zI)z

+ [((A 2 Al2)/K)((gig-)/g2)

cos9sine(({S)+ + {S)-)/2){})z]

+ [((AIAA )/4K) + AI/4]((S)+{I)+ + {S)-{I)-

+ [((A[A-)/4K) + A./4]({S)+(I)- + {S)- I)+).

This Hamiltonian can be solved for the energies at any angle

by letting the Hamiltonian matrix operate on the spin kets

IMS, MI>. The solution of the spin Hamiltonian is difficult

to solve at all angles except at 0=00 and 90'. Elimination

of some of the off-diagonal elements results in some

simplification and is usually adequate. The solution is then

correct to second order, and can be used when g#H >> A, and

A-, as is typically the case. The general second-order

solution is given by Rollman and Chan (44) and by Bleaney

(36). The energy levels are given by

(I-61) AE(M,m) = g3H + Km + (A.2/8G)[(A 2 + K2)/K2]

S[I(I+1) m2] + (A H2)(AI/K)(2M 1)

where K is AI and A- at 0=00 and 90, respectively, and

G=gSH/2. Also, M is the electron spin quantum number of the

lower level in the transition, and m is the nuclear spin

quantum number. The first two terms on the right result from

the diagonal matrix elements and yield equidistant hyperfine

lines. The last two terms cause spacing of the hyperfine


lines at higher field to increase, which is referred to as a

second-order effect. This solution is routinely applied

because the hyperfine energy is usually small and not

comparable to the Zeeman energy.

As described above, the hyperfine coupling constant

consists of both an isotropic and anisotropic part. The

isotropic part (Aiso) can be written as

(1-62) Aiso = (A, + 2AL)/3 = (8x/3)ge sgNN I(0)12

The isotropic hyperfine parameter can be used to determine

the amount of unpaired s spin density. The dipolar component

can be written as

(1-63) Adip = (A + AL)/3 = ge2gNN<(3cos20 1)/2r3>

These then relate the fundamental quantities W((0) 2 and

<(3cos20-1)/(2r3)> to the observed ESR spectrum. Approximate

spin densities in the molecule can also be obtained from Ais

and Adip.

Spin densities

The electron spin density, px at a nucleus X is the

unpaired electron probability density at the nucleus. In the

case of a single unpaired electron it is the fraction of that

electron/cm3 at a particular nucleus. The spin density of

the unpaired electron is generally split among s, p and d


orbitals. The spin density at nucleus X for an s electron is

given by psXlXsX(0)l2 and for electrons in a pa orbital the

spin density is given by Ppox.

Similar expressions can be given for px and da, etc.

orbitals. The terms psX and ppaX represent the contributions

of the s and pa orbitals to the spin density at nucleus X.

The isotropic and anisotropic hyperfine parameters can be

written as

(1-64) Aiso (molecule) = (8x/3)ge egiPNPsXIXsX(0)I

(1-65) Adip (molecule) = ge egIONPpoX


Since the equations above are characteristic of atom X, it is

possible to rewrite them for Aiso and Adip as given below

(I-66) Also (molecule) = PsXAlso (atom)

(1-67) Adip (molecule) = PpoXAdip (atom).

From Eqs. (1-66,67) one can easily obtain an expression

relating the unpaired spin density to the isotropic and

anisotropic hyperfine parameters,

(1-68) PsX = Aiso (molecule)/Aiso (atom)

(1-69) P2poX = Adip (molecule)/Adip (atom).

The hyperfine parameters for the molecule are obtained from

the ESR spectra. The hyperfine parameters for the atoms can

be obtained from tables (see Weltner (24), Appendix B) and

multiplied by the appropriate correction factors. The

correct value of Adip is calculated by taking the free atom

value of P = gepgNAN and multiplying it by an angular factor

a/2 = <(3cos2a-l)/2>. The factor equals 2/5 for a p

electron, 2/7 for a d electron and 4/15 for an f electron.

The value for Aiso (atom) can be found in Table B1 (column 5)

of Weltner (24) and the uncorrected value for Adip (atom) can

be found in column 7 of the same Table.

Quartet Sigma Molecules (S=3/2)

These high spin molecules (S>1) often contain transition

metals. The metal atom will generally have a large zero

field splitting (D) value due to its large spin-orbit

coupling constant ((})). If there are only a few ligands

attached to the metal atom, the unpaired electrons will be

confined to a small volume which will cause a sizable

spin-spin interaction. A large D value will cause many

predicted lines to be unobservable.

The spin Hamiltonian

A 4E molecule will exhibit a fine structure spectrum. A

theorem due to Kramer states that in the absence of an

external magnetic field the electronic states of any molecule

with an odd number of electrons will be at least doubly

degenerate. In the case of a quartet molecule the zero

field splitting produces two Kramer's doublets, or degenerate

pairs of states, with MS values of +1/2 and +3/2.

The spin Hamiltonian for a quartet sigma molecule with

axial symmetry can be written as

(1-70) (H}spin = gS Hz{S}z + g.Hx{S)x + D(({S}))2-5/4).

This equation does not take into account hyperfine structure.

A 4X4 spin matrix can be calculated which upon

diagonalization yields four eigenvalues

(1-71) W(+3/2) = D + (3/2)gSH

(1-72) W(+1/2) = D + (1/2)g,3H

and at zero field the +3/2 level and the +1/2 level are

separated by 2D. With H parallel to the molecular axis, the

energy levels will vary linearly with the magnetic field.

For the applied field perpendicular to the principal axis

(Haz) with Hx = H and Hz = 0, the eigenvalues are more

difficult to calculate because the off-diagonal terms are no


longer zero. The eigenvalue matrix can be expanded to yield

a quartic equation

(1-73) E4 1/2(1 + 15x2)E2 + 3x2E

+ (1/16)(1 + 6x2 + 81x4) = 0,

where E = W/2D and x = g-H/(2(3) O5D). Singer (45) has

developed a more general form of the equation which can be

applied to any angle. The eigenvalues for Hz can be

expressed as

(1-74) W(+3/2) = D + (3/8D)(g-SH)2 +

(1-75) W(+1/2) = -D + g- H (3/8D)(gj H )2 + ...

when H/D or x is small. By expanding E it can then be given

as E = a + bx + cx2 +... with the levels indexed by the low

field quantum numbers.

When D > gPH*({S all the matrix elements of the type

<+3/2|(H)Spinl1/2> = <+1/2{H})Spinl+3/2> vanish. This

approach yields the eigenvalues below,

(1-76) W(+3/2) = D + (3/2)gS3Hcose

(1-77) W(+1/2) = -D + (1/2)OzH(g12cos20 + 4gL2sin28)0.5

remembering that Hz = HcosO and Hx = Hsin6 and that the angle

between the molecular axis and the applied field is 9. This


is used to introduce the "effective" or apparent g value.

The effective g value generally indicates where the

transition occurs and is defined by assuming that the

resonance is occurring within the doublet, that is between MS

= +1/2 levels with g = ge. The g values of the observable

transitions 1+3/2> <-> 1-3/2> and 1+1/2> <-> |-1/2> become

(I-78) MS = +3/2 g = 3g, = 6.0 g = 0.0

(1-79) MS = +1/2 g- = go = 2.0 f. = 2gi = 4.0

for a large zero field splitting. The underlines indicate

the effective g value. The derivative signal for the +3/2

transition is usually undetectable because of the low

population of that level. There is not a significant

population of the +3/2 level unless D is very small. Also,

the absorption pattern corresponding to the g values for this

transition would be very broad. Finally, assuming that H/D

is large implies that the transition is forbidden. The

transitions usually observed for this spin state are those

between the +1/2 levels (the lower Kramer's doublet).

Kasal (46) and Brom et al. (47) have analyzed 4E molecules

and found the following spin Hamiltonian,

(I-80) (H}Spin = g$3Hz{S}z + g-p(Hx{S}x + Hy{S)y)

+ AI(I)z(S)z + A.({I}x(S}x + (I)y{S}y)

+ D[(S}z)2 (1/3)S(S+1)]

and rewrote it as an effective spin Hamiltonian

(I-81) (H)Spin = giHz({S}) + 2gpB(Hx(S)x + Hy{S}y)

+ A ({I)z(S)z + 2AL((I}x S})x + (I})y S)y)

for the +1/2 transition. The D term vanishes and S is taken

to be 1/2. The effective spin Hamiltonian can be rearranged

to be diagonal. The Zeeman terms become

(1-82) {H}spin = gOH{S}z + A(I)}{S)z

+ ((4(A12) (Ap)2)/A)


+ (1/2)A.[(AI + A)/A]( I})+(S}- + {I)-(S}+)

+ (1/2)A.[(Al + A)/A]((I)+(S)+ + (I)-(S}-)

where g2 = (g)2cos2O + 4(g )2sin20, and A2 = ((A )2

*(g1)2/g2) cos20 + (16(A.)2(g._)2/g2)sin2 This equation can

be solved analytically at 0 = 0 and by a continued fraction

method at 0 = 90". A computer program is usually used to

match the observed lines with those calculated by the

iterative procedure in order to come up with the values for g

and A.

The observed transitions and therefore the energy levels

are typically very dependent on 0 and D. Figures I-7 and I-8

indicate the levels as a function of field for the

perpendicular and parallel orientations, respectively. Two

Figure 1-7. Energy levels for a 4Z molecule in a
magnetic field; field perpendicular to molecular axis.

LP ,3.220
0. I1

LI 0.2-

0.3 -

04 ----------------------
0 I 2 3 4 5 6 7 8 9 10

H (Kilogauss)

Figure 1-8. Energy levels for a 4E molecule in a
magnetic field; field parallel to molecular axis.


0.9 -









6 7

8 9 10

H (Kilogauss)

Figure I-9. Resonant fields of a 4' molecule as a
function of the zero field splitting.



transitions are indicated between the same two levels at 690

and 1840 G. The reason for this can be seen in Figure 1-9.

The xy2 line is shown as an arc which reaches a maximum at

about 1000 G.

Sextet Sigma Molecules

The molecules considered here will have S=5/2, axial

symmetry (at least a three-fold symmetry axis), and a large

D. Ions such as Fe3 and Mn2+ fall in this category in some

coordination complexes.

The spin Hamiltonian

The spin Hamiltonian for a 6E molecule with axial

symmetry can be given as

(1-83) (H)Spin = geiHz(S)z + g--SHx{(S)xsin

+ D(((S})2 35/12)

including all angles. For 0 = 00 all of the off diagonal

elements are zero and the eigenvalues of the 6X6 matrix are

given below;

(1-84) E(+5/2) = (10/3)D + (5/2)g,~H

(1-85) E(+3/2) = -(2/3)D + (3/2)glOH

(I-86) E(+1/2) = -(8/3)D + (1/2)glpH.


Three levels appear at zero field which are separated by 2D

and 4D. Applying the magnetic field will split these into

three Kramers' doublets, which diverge linearly with field at

high fields and with slopes proportional to Ms.

A mixing of states occurs in the perpendicular case and

no simple solution is possible. A direct solution is

possible using a computer. This type of a solution has been

done by Aasa (48), Sweeney and coworkers (49), and by Dowsing

and Gibson (50). The eigenvalues of the 6X6 matrix

calculated by computer are shown in Figure I-10. The

diagonalization of the secular determinant was done at many

fields and at four angles. It is evident that the resonant

field for some transitions is very dependent on the angle.

The plot of zero field splitting versus the resonant field is

given in Figure I-11. It was prepared by solution of the

Hamiltonian matrix at many fields and D values for 0 equal to

0 and 90". For the situation were D >> hv the xy, line at

g=6 and the z3 line at g=2 will be most easily observed. As

can be seen in Figure I-10, these correspond to +1/2


Infrared Spectroscopy

Sir William Herschel discovered infrared radiation in

1800, but it was not until the turn of the century that

infrared absorption investigations of molecules began (51).

1.6 d 0" -
300 -.... .. 6
60o --
1.4 90 -

1.0 .------ =-


0.6 -


-0. 2 .. .. .... ... ... ..
----- ---


1.0 = ge -----
D = 1.32cm'''
-1.2 = 9.4 GHz
0 2 4 6 8 10 12 14 16 18 20

Figure 1-10. Energy levels for a 6Z molecule in a
magnetic field for = 0, 30, 60, and 90.


I 2 3 4 5 6 7 8 9 10
H (Kilogauss)

Figure 1-11. Resonant fields of a 6E molecule as a
function of the zero field splitting.


The typical IR source is a Nernst glower which is heated by

passing electricity through it. The radiation, which is

emitted over a continuous range by the source, is dispersedby

using a prism, such as KBr, which is transparent over the

range of interest. Various types of detectors ranging from

thermocouples to photodetectors are used to analyze the light

which has passed through the sample.

When dealing with infrared spectroscopy, one usually

deals with three specific regions of the spectrum. The

region from 800 to 2500 nm is the near infrared region and

adjoins the visible region of the spectrum. The infrared

region is found between 2500 to 50,000 nm. And the far

infrared region borders the microwave region of the spectrum

and starts at 50,000 nm and extends to about 1,000,000 nm.

The far infrared region is used to analyze vibrational

transitions of molecules containing metal-metal bonds, as

well as the pure rotational transitions of light molecules.

Most spectrometers are used in the mid-infrared region.

This is where most molecular rotational and vibrational

transitions occur (51).


Since infrared spectra are due to the vibration and

rotations of molecules, a brief review of the theory may be

useful. When a particle is held by springs between two fixed

points and moved in the direction of one of the fixed points,

it is constrained to move linearly. A restoring force

develops as the particle is moved farther from its

equilibrium position. The springs want to return to their

equilibrium position. Hooke's law states that the restoring

force is proportional to the displacement.

(I-87) f = kx

where k is a constant of proportionality and called the force

constant. The displacement is given by x, and the restoring

force is f. The force constant is used as a measure of the

stiffness of the springs. When the particle is released

after the displacement, it undergoes vibrational motion. The

frequency of oscillation can be written as

(1-88) v = (1/2x)(k/m)0.5

where m is the mass of the particle. The frequency can also

be expressed in wavenumbers (cm-1) by dividing the right side

of the equation with the speed of light. Because we are

dealing with small particles (atoms in this case), it is

necessary to enter into a quantum mechanical description of

the oscillation. The allowed energy values are given by

Ev = (v + (1/2))hy,



where v is found in Eq.(I-88) and v is the vibrational

quantum number. The equation above tells us that the energy

of the harmonic oscillator can have values only of positive

half-integral multiples of hy. The energy levels are evenly

spaced, and the lowest possible energy is (1/2)hv even at

absolute zero (52).

If one analyzes the vibrational behavior of a simple

molecular system, such as a diatomic molecule, the system's

oscillatory motion will be nearly harmonic and the frequency

of the motion can be described by

(1-90) v(cm-1) = (1/2xc)(k/g)0.5

where p is the reduced mass of the particles and is defined


(1-91) p = (ml"2)/(m1 + m2)

Since the motion of the atoms is not completely harmonic, we

must look at the energy levels of an anharmonic oscillator.

This is given by

(1-92) Ev = (v + (1/2))hv (v + (1/2))2huvx

+ (v + (1/2))3hvye .

where the constants Xe, Ye, ... are anharmonicity constants.

These are small and typically positive and usually of the

magnitude IXel > |Yel > Izel >... (53).

Anharmonicity in a molecule allows transitions to be

observed which are called overtones. These are transitions

between v=0 and v=2 or v=3 which are designated the first and

second overtones, respectively. The first overtone is

usually found at a frequency which is a little less than

twice the fundamental frequency. Combination bands can also

arise. These are caused by the sum or difference of two or

more fundamentals.

The force constant for a molecule is related to the bond

strength between the atoms. The force constant for a

molecule containing a multiple bond is expected to be larger

than the force constant of a single bond. A large force

constant is also usually indicative of a strong bond. For

diatomic molecules there is a good correlation between (k)0.5

and u(cm-1). This relationship unfortunately does not hold

for polyatomic molecules. In this case force constants must

be calculated by a normal coordinate analysis of the

molecule. Several authors have presented detailed

descriptions of this method (51-54).

The vibrations of a molecule depend on the motions of

all of the atoms in the molecule. To describe the location

of the atoms relative to each other one looks at the degrees

of freedom of the molecule. In a molecule with N atoms, 3N

coordinates are required to describe the location of all of

the atoms (3 coordinates for each atom). The position of the

entire molecule in space (its center of gravity) is

determined by 3 coordinates. Three more degrees of freedom

are needed to define the orientation of the molecule. Two

angles are needed to locate the principal axis and 1 to

define the rotational position about this axis. For a linear

molecule the rotation about the molecular axis is not an

observable process. The number of vibrations of a polyatomic

molecule is given then by

(1-93) number of vibrations = 3N 6

for a nonlinear molecule and

(1-94) number of vibrations = 3N 5

for a linear molecule.

What is observed in an infrared spectrum is usually a

series of absorptions. These correspond to various

stretching and bending frequencies of the sample molecule. In

order for an infrared transition to be observed, there needs

to be a change in the dipole moment of the molecule when it

undergoes a stretching or bending motion. The strongest

bands are those corresponding to the selection rule

(1-95) Avk = +1


and Avi =

where j does not equal k and k equals 1, ..., 3N-6.

The most intense absorptions are those from the ground

vibrational level since that is typically the most populated

level. These type of transitions are called fundamental

frequencies. These frequencies differ from the equilibrium

vibrational frequencies, Vl,e, v2,e ..... .The fundamental

frequencies are the ones generally used in force constant

calculations because the available information is typically

not sufficient to allow the calculation of anharmonicity

constants. The fundamental frequencies of the molecule need

not be the most intense absorptions. This can happen if the

change in dipole moment, (6d/6Qk), is small or zero (51).

The phase or environment that the molecule is in will

affect the appearance of the IR spectrum. With gas phase

samples it is often possible to resolve the rotational fine

structure of the compound using high resolution instruments.

On the other hand, when dealing with a matrix isolated

sample, there usually is not much, if any, rotational fine

structure even under high resolution. This is because the

molecule is rigidly held (small molecules such as HC1 exhibit

a rotational spectrum due to a hindered rotation within the

matrix site) within the lattice of the matrix, and is not

able to rotate freely as it is able to do in the gas phase.

The elimination of the rotational fine structure simplifies

the spectrum and enables the analysis of more complicated


vibrational spectra which arise when studying larger


Fourier Transform IR Spectroscopy

The basic components of an FTIR instrument are an

infrared source, a moving mirror, a stationary mirror, and a

beamsplitter. The source in a typical FTIR spectrometer is a

glower which is heated to about 1100 *C by passing an

electrical current through it. The beam from the glower is

directed to a Michelson interferometer where the intensity of

each wavelength component is converted into an ac modulated

audio frequency waveform. Assuming that the source is truly

monochromatic, a single frequency, A/c, hits the

beamsplitter, where half is transmitted to the moving mirror

and half to the fixed mirror. The two components of the

light will return in phase only when the two mirrors are

equidistant from the beamsplitter. In this case constructive

interference occurs and they reinforce each other.

Destructive interference occurs when the moving mirror has

moved a distance of A/4 from the zero position. This means

that the radiation which goes to the moving mirror will have

to travel A/2 further than the radiation that went to the

fixed mirror, and the two will be 180* out of phase. As the

components go in and out of phase, the sample and the

detector will experience light and dark fields as a function

of the mirror traveling +x or -x from its zero position. The

intensity at the detector can be expressed as


I(x) = B(v)cos(2nxv),

where I(x) is the intensity, B(v) is the amplitude of

frequency v, and x is the mirror distance from the zero

position. For a broadband source the signal at the detector

will be the summation of Eq.(I-96) over all frequencies, and

the output, as a function of mirror movement x, is called an

interferogram (53).

The interferogram can then be converted into the

typical intensity versus frequency spectrum by performing a

Fourier transformation. The signal is transformed from a

time domain signal, which arises from the motion of the

mirror, to a frequency domain signal which is observed in the

typical IR spectrum. This can be done mathematically by


(1-97) C(y) = fI(x)cos(2xxv)dx,

where C(v) is the intensity as a function of frequency.

There are several advantages in using an FTIR

instrument. The detector in a Fourier transform instrument

gets the full intensity of the source without an entrance

slit. This yields a 100 fold improvement over the typical

prism and grating instrument. The signal to noise ratio is


theoretically improved by a factor of M1/2, where M is the

number of resolution elements. This has been termed

Fellgett's advantage since it results mathematically from one

of his derivations. A direct result of Fellgett's advantage

is that a dispersive instrument requires 3000 seconds to

collect a spectrum, whereas an interferometer needs only

about 60 seconds to collect an IR spectrum with the same

signal to noise ratio. (M equals about 3000 and the

observation time is about 1 sec/element) (53).

A complete IR investigation, when possible, can enable

one to determine the structure of the molecule of interest.

The IR spectrum allows one to determine force constants of

the various bonds and from that information the bond

strengths can be determined. The bending frequencies even

enable one to determine the bond angle between the atoms

involved in the bending motion. The shift in both stretching

frequencies and bending frequencies caused by the

substitution of isotopes into a molecule is very useful

towards this purpose since the amount of the shift is

dependent on the change in mass when the isotope is

substituted into the molecule.


ESR of VCOn Molecules


Transition-metal carbonyl molecules continue to be of

great interest, partially because of their relevance to

catalysis. The simplest molecules, those containing only one

metal atom, have been studied spectroscopically, and electron

spin resonance (ESR) has been applied successfully in some

cases, specifically to V(CO)4, V(CO)5 (55), V(CO)6 (56-59),

Mn(CO)5 (60), Co(CO)3. Co(CO)4 (61.62), CuCO, Cu(CO)3

(63,64), and AgCO, Ag(CO)3 (65,66). (Ionic carbonyls have

also been observed via ESR (67.68) but will not be explicitly

discussed here.) Theoretical discussions of the geometries,

ground states, and bonding in these types of molecules have

been given by several authors beginning perhaps with Kettle

(69) and then by DeKock (70), Burdett (71,72), Elian and

Hoffmann (73), and Hanlan, Huber, and Ozin (74). Although a

number of ab initio calculations have been made on such

carbonyls, the vanadium molecules considered here apparently

have not been treated in detail.

The background for the present investigation was

provided by the matrix work of Hanlan, Huber, and Ozin (74)

who observed the infrared spectra of V(CO)n where n equals 1


to 5, in the solid rare gases. Most notably, those authors

concluded, from experiment and theory, that [1] VCO is

nonlinear, [2] V(CO)2 exists in linear, cis, and trans forms

in all three matrices, argon, krypton, and xenon, [3] V(CO)3

is probably of D3h trigonal planar geometry. It should be

emphasized that the supporting theory usually assumed

low-spin ground states.

Morton and Preston have prepared V(CO)4 and V(CO)5 in

krypton matrices by irradiation of trapped V(CO)6. From ESR

they assign V(CO)4 as a high-spin 6A1 in tetrahedral (Td)

symmetry and V(CO)5 as 2B2 with distorted trigonal bipyramid

(C2v) symmetry. The V(CO)6 molecule is a well known stable

free radical which has been rather thoroughly researched by

infrared (75), MCD (76), ultraviolet (77), electron and X-ray

diffraction (78), and ESR. It is presumably a Jahn-Teller

distorted octahedral (2T2g) molecule at low temperatures

leading to a 2B2g ground state.

Our ESR findings are only for V(CO)n, where n equals 1

to 3, and are not always in agreement with conclusions from

optical work and semiempirical theory. The most explicit

departure is in finding that VCO and V(CO)2 are high-spin



The vanadium carbonyls synthesized in this work were

made in situ by co-condensing neon (Airco, 99.996% pure),

argon (Airco, 99.999% pure), or krypton (Airco, 99.995% pure)


doped with 0.1-5 mol% 12CO (Airco, 99.3% pure) or 13C0

(Merck, 99.8% pure) with vanadium metal [99% pure, 99.8%

51V(I=7/2)] onto a flat sapphire rod maintained at 4-6 K but

capable of being annealed to higher temperatures.

The furnace, Hell-Tran, and IBM/Bruker X-band ESR

spectrometer have been previously described (79). Vanadium

was vaporized from a tungsten cell at 1975 *C, as measured

with an optical pyrometer (uncorrected for emissivity).

ESR Spectra


Two ESR spectra of the VCO molecule were observed in

matrices prepared by condensing vanadium into CO/argon

mixtures at 4 K. We designate these two forms of VCO below

as (A) and (a). This symbolism is derived from one of their

distinguishing features: one has a considerably larger 51V

hyperfine splitting (hfs) than the other. Only the (a) form

survived after annealing the argon matrices and only it

appeared in a krypton matrix. Only (A) was observed in solid


51VCO(A) and 51VCO(a) in argon

Upon depositing vanadium metal into an argon matrix

doped with 1.0 mol% 12CO, we obtained the 4 K ESR spectrum

shown in Figure II-1. The two sets of eight strong, sharp

lines centered near 1200 G could be attributed to separate


species since upon annealing one set [designated by (A)]

disappeared. The hyperfine splitting (hfs) in the

perpendicular xy1 and xy3 lines of the (A) species due to

51V(I=7/2) is approximately 100G, whereas that in the (a)

species is about 60 G. The line centered at about 8100 G has

been observed with that intensity only once, but its

appearance, and disappearance upon annealing, correlates best

with the (A) molecule. Its complex hfs is indicative of an

off-principal axis line where forbidden AmI not equal to zero

transitions can also occur. The observed lines of both (a)

and (A) are listed in Tables II-1 and 11-2.

Annealing to 16 K and quenching to 4 K converted the

VCO (A) species into (a) which has the spectrum in argon in

Figure 11-2. Again the xy1 and xy3 lines have the same hfs,

now about 60 G, and an "extra" line appears but centered at

about 6700 G.

51V13CO (A) and 51V13CO (a) in argon

These same spectra can be observed when 13CO replaces

12CO and the effect upon the xy1 line, which is the same

effect for (A) and (a), is shown in Figure 11-3. Each line

is split into a doublet separated by about 6 G, indicating

most importantly that there is only one CO in each species.

51VCO (A) in neon

In neon only one VCO molecule appears to be trapped,

the one designated as (A) in argon with the hfs of about

Table II-1. Observed and calculated line positions
(in G) for VCO (X6W) in conformation (A)
in argon at 4 K. (v = 9.5596 GHz)

MI(51V)a xy1 xy3 Extra lines
0 = 100
Obs. Calc. Obs. Calc. Obs. Calc.

7/2 797 789 5065 5082 7906 7897
5/2 882 877 5154 5170 8015 8012
3/2 974 971 5254 5262 8119 8124
1/2 1072 1072 5364 5360 8252 8234
-1/2 1174 1176 5473 5463 8346 8344
-3/2 1282 1285 5584 5573 8467 8453
-5/2 1396 1400 5692 5689 -- 8562
-7/2 1511 1519 5819 5814 -- 8671
AMI=l+ transitions
Derived Parameters Obs. Calc.

g| 2.002(37) 7944 7938
g- 1.989(5) 7976 7970
IAi(51V)I 247(28)MHz 8054 8052
IA(51V)I 288(6) MHz 8083 8083
|DI 0.603(2)cm-1 8161 8164
Aiso(51V)a 274(13)MHz 8192 8194
Adip(55V)a -14(11)MHz 8252 8274
AL(13C)O 17(3) MHz 8304 8304
8380 8384
8423 8413
-- 8493
8514 8521
-- 8603
-- 8630

a Assuming A, and A- are positive

() Error of the reported value

Table II-2. Observed and calculated line positions
(in G) for VCO (X6W ) in conformation (a)
in argon at 4 K. (v = 9.5596 GHz)

MI(51V)a xy1 xy3 Extra lines
0 = 12*
Obs. Calc. Obs. Calc. Obs. Calc.

7/2 940 940 4460 4461 6449 6448
5/2 1000 999 4520 4520 6530 6526
3/2 1061 1060 4581 4580 6603 6603
1/2 1124 1124 4645 4644 6684 6680
-1/2 1191 1190 4710 4709 6756 6755
-3/2 1258 1257 4777 4777 6828 6831
-5/2 1326 1327 4847 4847 6906 6907
-7/2 1396 1398 4918 4921 -- 6983

AMI=l transitions


IA (51V)
fA (51V)

|D I

IAL(13C) I


183(1) MHz
177(5) MHz
-6(5) MHz
17(3) MHz



a Assuming A, and A-. are positive.

() Error in the indicated value



I I I I 1 I I 1 I




7.7 79 8.1 8.3 8.5

5.0 5.2 5.4 5.6 5.8

1 1, I 1, IX Y,


_I H(KG)

Figure II-1. ESR spectrum of an unannealed matrix at
4 K containing 51VCO(A). with hfs of about 100 G, and
51VCO(a), with hfs of about 60 G. For the conformation
(A) two perpendicular lines and an off principle axis
line are shown. v = 9.5585 GHz.






Figure 11-2. ESR spectrum of an annealed argon matrix
at 4 K containing only 51VCO in conformation (a). Two
perpendicular lines and off principal axis line are
shown. v = 9.5585 GHz.