CALCULATION OF THE ELECTRONIC STRUCTURE OF THE U CENTER
AND RELATED COLOR CENTERS IN ALKALI HALIDES
BY USING THE MULTIPLE SCATTERING METHOD
By
HSILING YU
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR TEE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1975
ACKNOWLEDGEMENT S
I would like to thank Dr. J. W. D. Connolly for his
suggestion of the present topic and for his advice and
encouragement in the course of this report.
Special thanks are extended to Dr. S. B. Trickey for
his constructive criticism and his patience in correcting
errors in the manuscript. I would like also to acknowledge
Dr. J. R. Sabin and other members of the Quantum Theory
Project at University of Florida for their helpful cosrents.
Thanks are also due to Dr. M. L. De Siqueira for his
helpful discussion concerning this work.
Finally, I would like to thank, especially, my wife,
Ning, for her help in preparation of the manuscript and
for her patience and continuous encouragement throughout
this research.
The use of the facilities of the North East Regional
Data Center and the financial support from the National
Science Foundation for part of this research are gratefully
acknowledged.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . . . . . . . . ii
ABSTRACT . . . . . . . . . . iv
CHAPTER I INTRODUCTION . . . . . .. 1
CHAPTER II COLOR CENTERS IN ALKALI HALIDES . 3
2.1 Optical Absorption. . . . . .. 3
2.2 Experimental Methods . . . . . 4
2.3 Model for Color Centers . . . . 7
2.4 Theoretical Approaches . . . . 9
CHAPTER III METHODS OF CALCULATION. . . ,. 16
3.1 The Multiple Scattering Method ..... 16
3.2 Choice of a in Vxa . . . . .. 24
3.3 Transition State . . . . . . 25
3.4 Radii for Ions . . . . . . 27
3.5 Ionic Correction Potential . . 31
CHAPTER IV RESULTS AND DISCUSSION . . . . 38
4.1 The U Centers in Alkali Halides . . 38
4.2 The F Centers in Alkali Halides . . 71
4.3 The U2 Center in Potassium Chloride .86
4.4 The U1 Center in Potassium Chloride . 92
4.5 Cluster Calculation of a Pure KCi Crystal 95
CHAPTER V CONCLUSIONS . . . . .. . 109
APPENDIX A THE EVJEN METHOD. . . . .. 114
APPENDIX B SYMMETRY ORBITALS . . . .. 117
BIBLIOGRAPHY . . . . . . . . . .
BIOGRAPHICAL SKETCH . . . . . . . .
121
125
iii
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
CALCULATION OF THE ELECTRONIC STRUCTURE OF THE U CENTER
AND RELATED COLOR CENTERS IN ALKALI HALIDES
BY USING THE MULTIPLE SCATTERING METHOD
By
HSILING YU
March, 1975
Chairman: J. W. D. Connolly
Major Department: Physics and Astronomy
The spinpolarized multiple scattering Xa selfconsistent
field method has been applied to the problem of color centers
in ionic crystal. Within the muffintin approximation, the
electronic structures of the U and F centers in several
alkali halides and the U1 and U2 centers in potassium chloride
have been calculated and studied using a cluster of ions
including the first nearestneighbors. From the transition
state calculation, the U and F center optical absorption
energies in several alkali halides are found to be in good
agreement with experiment except in the case of a crystal
with a very small interionic distance. In the U2 center case,
two possible transitions are investigated and one transition
agrees well with experiment while the other, a charge
transfer process, gives an energy too small. The hyperfine
iv
interaction of the U2 center and the optical absorption of
the Ui center have also been studied.
Larger clusters including the second nearestneighbors
and the third nearestneighbors for the U and F centers
in KC1 have been calculated and the absorption energies
are found to oscillate with the size of the cluster. This
problem has been investigated and discussed by comparing
the cluster calculation of the pure KC1 crystal with the
energy band calculation and the experimental data. One of
the improvements for the large cluster calculation is a
more accurate crystalline potential. In the present calcu
lation this potential is calculated from a pointion
assumption.
CHAPTER I
INTRODUCTION
Color centers in alkali halide crystals have been
observed and studied for many years. But the calculation of
the electronic structure and the properties of the color
center or impurity in solid is difficult because of the
complexity of the system involved. Thus, most of the studies
are done by methods that are semiempirical in nature.
Recently, the Xa method (Slater, 1972; Slater and
Johnson, 1972) which makes use of the socalled Xa statisti
cal exchange, has been successfully applied to many problems.
For molecules, the multiple scattering Xa selfconsistent
field (MSXaSCF) method (Johnson, 1973) gives satisfactory
results in many cases and is also very efficient in computing
time. Thus, one of the purposes of the present dissertation
is to study the electronic structure of color centers by
direct calculation of the properties of a cluster of ions
using the MSXaSCF method. From the calculation the optical
absorption process of the color center can be studied and the
transition energy can be calculated.
Several clusters with different sizes were used in the
calculation, but only one cluster (Figure 2.1) is used for
extensive study of the U center from LiF to KBr. Other color
centers (F, U1 and U2 centers) are studied mainly in the KC1
1
2
crystal. From these investigations, the capability and
limitation of the present approach can be checked.
Another purpose of this dissertation is to compare the
present cluster calculation of a pure crystal with energy
band calculations and the experimental data. From this
comparison, the characteristics of the present cluster method
and possible improvements on the method for future applica
tions can be found.
CHAPTER II
COLOR CENTERS IN ALKALI HALIDES
2.1 Optical Absorption
The alkali halides are, in many respects, ideal crystals
for theoretical and experimental studies. They have cubic
structure and many of their properties are affected by the
strong Coulomb interactions which, in turn, are due to the
highly ionic nature of these crystals. For a pure alkali
halide crystal, the energy gap between the conduction band
and the valence band is of the order of 10 eV. This corres
ponds to a light quantum in the far ultraviolet and, hence,
the crystal appears transparent to the eye. When imperfections
develop in these crystals extra absorptions may appear in the
visible region and the crystals will appear colored. One of
the more interesting problems in solid state physics for
many years has been the study of these imperfections in
ionic crystals both experimentally and theoretically.
The color centers occurring in alkali halides come
from many varieties of defects in the crystal. They occur
when a few alkali or halogen ions are replaced by other ions
or atoms, or when ions are removed, leaving ion vacancies
in the crystal. There are also cases in which extra ions or
atoms exist in the solids, forming interstitial impurities.
4
Distortions of the lattice from the perfect configuration of
the crystal also can create color centers. In all cases, the
effect of these color centers is to allow certain energy
levels to exist in the gap between the conduction band and
valence band of the energy band picture. Thus, energies
needed to excite electrons from these color center levels
to the allowed energy levels above are small compared
to the energies of the interband transition or the energies
from the valence band to the exciton levels. In some cases,
this will happen in the visible light region, causing the
crystal to appear colored, e.g., the F centers. In other
cases, the excitation energies are outside the visible
light region; thus, the crystal will remain transparent,
e.g., the U centers.
2.2 Experimental Methods
Color centers are produced by many different kinds
of techniques. They can be produced by exposing the crystals
to Xrays or yrays, or by injecting electrons directly
into the crystals. Bombarding the sample with heavy particles
or electrons can also create imperfections in the solid.
Another method is the process of additive coloration in
which the crystal is heated in the presence of excess
alkali or halogen or other atoms. Several review articles
by Markham (1966) and Seitz (1946, 1954) have discussed
these and other methods for producing color centers.
The optical absorption spectrum of a color center
generally shows a broad band whose position and shape are
temperature dependent. For example, in the case of the F
center, it is found that the width of the absorption band
broadens with an increase of the temperature, while the
peak energy of the band decreases with the increase of the
temperature. It will be seen in section 2.4 that a very
simple model for the F center can give a qualitatively
good explanation for these characteristics of the absorption
band.
A study of the peaks of the absorption bands in various
alkali halides for a certain color center shows that there
is a relationship between these energies and the nearest
neighbor distances of the solid. This kind of relation
was first found empirically by Mollwo (1931) and later
refined by Ivey (1947). In the cases of the U center and the
F center, the relations are
20.16 d1.10 for U band
max = 1.84 (2.1)
17.64 d for F band,
where E is in Electron Volts and d (the nearestneighbor
max
distance) is in Angstroms.
From the optical absorption data alone, very little
information about the color center and its surroundings
can be obtained. There are two other methods which can
provide more insight into the structure of imperfections
in the crystal. One of these methods is the Electron Spin
Resonance method (ESR) (Hutchison, 1949) and the other is
the Electron Nuclear Double Resonance method (ENDOR)
(Feher, 1957). These methods have been reviewed by Seidel
and Wolf (1968) and Markham (1966).
In ESR spectroscopy, magnetic dipole transitions
between the spin levels of an unpaired electron of the color
center can be measured in a static magnetic field. In
ENDOR spectroscopy, the nuclear spin resonances of the
neighboring nuclei that are coupled to the unpaired electron
of the color center are measured. Usually, these two reson
ance methods are applied only to the ground states of color
centers. Measurements on excited states are difficult
due to the short lifetime of these states.
From the ESR measurements, one can determine the
nature of the center, that is, whether it is paramagnetic
or diamagnetic. Sometimes the ESR absorption line is broad
ened due to the interactions between the magnetic moment
of the unpaired electron and those of the neighboring
nuclei, and information about the neighboring ions is
difficult to obtain from the structureless absorption line
shape. With the relatively new ENDOR measurements, more
precise determination about the surroundings of the imper
fection can be obtained and this provides an accurate way
to check the ground state wave function of the paramagnetic
center from theoretical calculations.
Using the data from these optical and magnetic measure
ments, models for these centers can be established and the
electronic structure of the color centers can be studied by
various theoretical approaches.
2.3 Models for Color Centers
In the present calculations, only U, F, U, and U2
centers in alkali halides are considered. From various
experimental studies, especially ESR (Hutchison, 1949; Kip
et al., 1953) and ENDOR (Feher, 1957; Seidel, 1961; Doyle,
1962), the F center can be established to be an electron
trapped at an anion vacancy in the alkali halide crystal.
The U center is believed to be a negative hydrogen ion
substituted for an anion in the ionic crystal (Delbecq et al.,
1956). The Ul and U2 centers are interstitial defects which
occupy a position inbetween the regular sites of the
crystal. The U1 center is believed to be a negative
hydrogen ion at an interstitial position in the crystal and
the U2 center is a hydrogen atom at the same site (Delbecq
et al., 1956; Spaeth, 1966).
The clusters of ions that are used in the present
calculations for these color centers are shown in Figures
2.12.3.
Figure 2.1 shows the eightcenter cluster (including
the outer sphere) for the U center and the F center calcula
tion with Oh symmetry. For the U center, it is a negative
hydrogen ion in the center surrounded by six alkali ions. For
the F center, the cluster is an electron trapped in the
central sphere with six nearest alkali ions surrounding it.
II
Figure 2.1. The eightcenter cluster used in the
calculation. Regions I, II and III are
the atomic, intersphere and outer regions,
respectively.
Figure 2.2 shows the cluster with Td symmetry for the
U1 and U2 center. In the case of the U2 center, this cluster
is a hydrogen atom in the center with four anions and four
cations forming two interpenetrating tetrahedra. For the
U1 center, the central sphere is a negative hydrogen ion.
Figure 2.3 shows a large cluster with several shells
of ions that is used in the U and F center calculations.
This cluster has Oh symmetry, and the difference for the
U center and the F center is at the central sphere which is
occupied by a hydrogen ion in the case of the U center and
an electron in the case of the F center. Calculations with
the halogen ion in the central sphere of this cluster have
also been done in order to compare with energy band calcu
lations.
2.4 Theoretical Approaches
The F center is the simplest defect that can occur in
alkali halides. As mentioned in the previous section, this
center is an electron at the vacant anion site trapped in
the potential well due to the nearby positive ions. An
extremely simple model (Stockmann, 1952; van Doom, 1962)
in which the F center is thought to be a particle in an
infinitely deep threedimensional square well, i.e., a
particle in a box, can qualitatively explain several pro
perties of the optical absorption spectra that werementioned
before. Using this model, the excitation energy from the
ground state to the first excited state is
Figure 2.2. The cluster used in the U1 and U2 center
calculations.
Figure 2.3. The large cluster with 01 symmetry used in
the calculation. C=central ion, l=first nearest
neighbor, 2=second nearestneighbor, 3=third
nearestneighbor.
A (/ (2.2)
where d is the distance from center of the potential well
to the potential barrier. This relation shows the same form
as the empirical relation for the F center in Equation (2.1).
Thus, from this model the F band energy is found to be rela
ted directly to the spacing of the lattice. The decrease
of the peak energy of the absorption band can be explained
as the expansion of the lattice due to the increase of the
temperature, and from the temperature dependence of lattice
vibrations, the width of the absorption band can be expected
to increase with the temperature.
Because of the simplicity of the electronic structure
and the availability of the magnetic properties of an F
center in an ionic crystal, this center served as a good
testing example for the theoretical methods developed for the
general impurity problem. Thus, most of the methods in the
beginning are concentrated on the F center. Before 1957, the
techniques used are mostly semiempirical in nature. For
example, there is a continuum model (Simpson, 1949) in
which the polarizable crystalline medium is replaced by
a dielectric continuum and the influence of lattice periodi
city on the electron is considered by using an effective
mass for the electron. Modified version, the semicontinuum
model (Krumhansl and Schwartz, 1953), employs more detailed
treatment of potential for the central part of a color
center in which the previous model fails. There are other
methods using the variation techniques, e.g., the pointion
approximation (Gourary and Adrian, 1957) and molecular
orbital calculation (Inui and Uemura, 1950). A summary of
these methods up to 1960 is given by Gourary and Adrian
(1960). More extensive calculations using the extended point
ion approximation (Wood and Joy, 1964) and pseudopotential
approximation (Kubler and Friauf, 1965) have also been done.
These methods are discussed in a review article by Fowler
(1968). Since the U center has the same structure as the F
center, calculations of U centers (Spector et al., 1967;
Wood and Opik, 1967) have employed the same methods that were
developed for the F center.
As to the U1 center, there have been no theoretical
calculations, partially because it has no interesting
magnetic properties (the ground state is a closed shell)
and partially because the peak of U1 band in the absorption
spectrum is generally not well defined (Delbecq et al.,
1956; Rolfe, 1958). Contrary to the case of the U1 center,
a great deal of work has been done on the U2 center. Kerkhoff,
Martienssenand Sander (1963) examined various possibilities
of the optical process for the U2 center in a semiclassical
way, and concluded that the optical U2 band is due to a
charge transfer from the nearestneighbor halogen ions to
the hydrogen atom. The configuration of the excited state
of the U2 center is thought to be a negative hydrogen ion
plus a hole in the p orbitals of the nearestneighbor
halogen ions. Using these configurations, Cho et al. (1966)
and Cho (1967) obtain wave functions to discuss the hyper
fine interaction.
Other calculations involving the variation technique
have been done by Sammel (1969). Calculations of the transi
tion energy of the U2 band have been done semiempirically
by Hagston (1971) and Schechter (1969) and earlier by
Mimura and Uemura (1959).
In this dissertation, the optical absorption energies
of the U and F centers in alkali halides and the U1 and U2
centers in KC1 are calculated. Unlike other approaches, the
U centers, instead of the F centers, in various alkali
halides were investigated first by using the multiple
scattering method. This is because the starting potential
used in the present calculation usually is a superimposed
ionpotential and it is relatively simple to get an ion
potential for the H ion by using the HermanSkillman
atomic program (Herman and Skillman, 1963).
In all calculations, the effects of the lattice dis
tortion and polarization and the temperature dependence
mentioned before are not included. Some of these effects,
though, will be discussed in later sections. One of the
purposes of the present calculation is to investigate the
capability of the multiple scattering cluster method of
handling the problem of impurities in solid; therefore, no
adjustment of parameters to match the experimental data
15
was made. However, the effects of some of the parameters
will be discussed in the results of the calculations given
in Chapter 4.
CHAPTER III
METHODS OF CALCULATION
3.1 The Multiple Scattering Method
One practical way to solve the problem of a many
electron system is the oneelectron selfconsistent field
method in which the interactions between the electrons and
nuclei are replaced by the interactions of one electron
with an averaged electronic charge distributions and with
all nuclei plus some exchangecorrelation effects. By
solving the Schroedinger Equation in such a field, a set of
eigenfuncticns can be found and from these eigenfunctions
the charge densities can be calculated. Thus, by repeating
the procedures a selfconsistent field can be reached. This
method was first used by Hartree (1928) in his proposed
oneelectron equation.
If one startsfrom an antisymmetrized oneelectron
product function or a Slater determinant function (Slater,
1929) to express the total energy of the system, a set of
oneelectron equations can be obtained by the variation
technique. These are the HartreeFock equations (Fock, 1930)
which have been used in many problems. For a complicated
system, the exchange potential term in the HartreeFock
equation becomes extremely difficult to calculate.
An approximate method called the Xa. method (Slater,
1972; Slater and Johnson, 1972) can be used to avoid the
difficulty in the exchangecorrelation potential. In this
method, the exchange term in the HartreeFock oneelectron
equation is replaced by a local exchange potential (Slater
and Wood, 1971) (see Equation (3.2)) which gives an approxi
mate exchangecorrelation term and is easy to calculate.
A numerical technique to solve the X< equation in a
large molecule or cluster is the multiple scattering (MSXA )
method (Johnson, 1973). In its formulation, the cluster is
divided into three regions, The first is the atomic sphere
region which contains one atom or ion inside the sphere.
The second region is the intersphere region that is the
space outside all the atomic spheres but inside an outer
sphere which enclosesthe whole cluster. The third region is
the outer region which includesall space outside the outer
sphere. The problem is to solve the Schroedinger problem
of a single electron in each region and after the matching
of the wave functions and their first derivatives on all
boundaries of the cluster we get a relation in which the
eicenvalues can be determined. The oneelectron wave func
tion with spin up, u (1), satisfies the following Xo
equation (in Rydberg units).
VS* VC !4 Vx1t ( V ] VI %Litj!)= i C ILtl)t (3.1)
where V (1) is the Coulomb potential at position 1 due to
c
all electronic and nuclear charges of the cluster, and
V x(1) is the statistical exchangecorrelation potential
which is given by
L kiLi') (3.2)
where the summation is over all eigenfunctions with spin up
and n. is the occupation number of the spinorbital u.. A
discussion of the choice of a in this local exchange potential
is given in the next section. The term V represents the
Coulomb potential due to all ions outside the cluster. The
determination of VI is treated in section 3.5. Similar
expressions of Equations (3.1) and (3.2) with spin down give
the oneelectron equation satisfied by ui(l1).
In order to simplify the problem, a muffintin form
of potential has been used in the calculations. In this
approximation, the potential inside the atomic sphere
region is spherically averaged and the potential in the
intersphere region is a constant equal to the volume average
of potentials in that region. The potential in the outer
region is also spherically averaged. Because of the locally
spherically symmetric potential, we can express the solution
of the oneelectron Equation (3.1) in the ath sphere of the
first region (for simplicity, the spin subscripts are
dropped for the following discussion) as
j = z C'R" ( 1, ,0 ,(3.3)
Jm
where r, is a vector measured from the center of the Eth
sphere and m (r )'s are the real spherical harmonics where
r, represents the angular part of r, and C m' are coeffi
cients to be determined, and R(JIrl ,e) is the solution of
the oneelectron radial Schroedinger equation with a certain
energy E, namely,
r.   V ) R) ) o
where V is the spherically averaged potential in the cth
sphere.
Similarly, the solution in the outer region can be
written as
0 20 ( A
Cn R, c; r (3.4)
where r5 is a vector measured from the center of the outer
sphere. In the intersphere region, the volume average of Vc,
Vx and VI gives a constant potential VII, and the solutions
of the Schroedinger equation for the cases
expanded as
S. L( Am.
(3.5)
2L ('A Kr 2 Kr
j M J m Ot
where K=(VII ) and i k (1) are modified spherical
Bessel and Hankel function, respectively. For the cases
where E > V II the solution in the intersphere region can
be expanded as
0 A r A
Sd"k'~ (3.6)
where K=( EV I) and j is an ordinary spherical Bessel
function and r. is an ordinary spherical Neumann function.
In order to match the logarithmic derivative of the
wave functions on the boundaries of a particular atomic
sphere or the outer sphere, we have to transform the multi
centered wave function in region II in terms of only the
one center that is under consideration. Several expansion
theorems which are useful for this purpose have been
discussed by Johnson (1973).
Suppose we consider the case of < (V and try to
match u and u (and their first derivatives) on the
boundary of the th sphere. After cancelling the C m
coefficients, we get a set of linear equations in terms
of the AIm coefficients, namely, for a particular j and m
4 4 I P ( C3.7)
"A + ALM L L NA
3*cA LM
0o ,(3.7)
+1 ALM J, LM
21
where
t ) r 4L E,A K[ r ) W \N K (rI
evaluated at r= b (the radius of the oCth sphere). Here
we have used the Wronskian bracket notation
'J r adr The G (U) coefficients
SHLM
depend only on the relative positions of centers in the
cluster and are defined as
(3.8)
S1^1) for
where R is a vector from the center of the oth sphere
to the center of the (i th sphere. The term I ,, (km;LM)
is a Gaunt coefficient and is defined as anintegral of the
product of three spherical harmonics over all solid angles,
namely,
I(; L M') I((r) (3.9)
Similarly,
^ (, 4^0^2 iT ()I) Z, &.)
ICt, LM JPv0
(3.10)
From above we know that in Equation (3.7) the dependence
on the nature of a particular atom or ion in the cluster
is only from the term t (e) through its dependence on
RI ( r ,c ), the solution of the radial Schroedinger equa
tion, and the G (0) matrix element is only a structure
factor with dependence on eigenvalue parameter E through
its relation with K .
After matching the logarithmic derivatives of the wave
functions on the boundaries of all atomic spheres and the
outer sphere, we complete the set of equations in terms of
A 's. It is well known that for such a set of homogeneous
linear equations, the condition for having nonvanishing
solutions for all variables requires that the determinant
of the coefficients of these equations equals to zero. Thus,
et I L rM + 0( (3.11)
where the indices t and ^ start from the outer sphere and
run through all atomic centers. The highest I value used
in the calculation is dependent on the atoms and the
symmetry of the cluster.
In the case that the cluster possesses symmetry, we
can simplify the determinant by expanding the wave function
in each region in terms of some linear combinations of
the products of spherical harmonics and radial functions
centered on the equivalent sites of the cluster that form
bases for various irreducible representations of the
appropriate symmetry group. The size of the symmetrized
array of a particular representation that we need to solve
is equal to the number of the basis functions that we used
in that representation and is considerably smaller than that
of (3.11) especially in the cases where the cluster has a
high symmetry or many centers. In any case, the determinant
can not be solved directly, but the zeroes of the determinant
can be searched by evaluating the determinant in steps of
the energy parameter E .
For each eigenvalue EL found, the eigenfunction in
each region can be obtained. From these eigenfunctions, we
can calculate the charge densities. After all eigenvalues
F's are found, we can calculate the Coulomb potentials in
each region by using the muffintin form of the total
charge densities. Using the spherically averaged potential
in region I and region III and the volume averaged potential
in region II, the entire calculation can be started again.
In order to start the initial calculation, a super
position of atomic potentials is used. For each new iteration,
a weighted mixture of the newly generated potential and the
previous one is used as the input potential. This procedure
is repeated until the difference between the new and the
old potentials is within a preset tolerance. Thus, self
consistency is achieved.
3.2 Choice of a in V
xc
The statistical exchangecorrelation potential (Slater
and Wood, 1971) given in Equation (3.2) was originally
derived by Slater (1951) with a = 1, and later with another
approach it was derived independently by Gaspar (1954) and by
Kohn and Sham (1965) with a = 2/3. In actual cases, values of
a in between the above two values are expected to give better
exchangecorrelation effects. In the case of atoms, Schwarz
(1972) has determined two setsof a's by using two different
criteria. The first is to adjust the statistical total energy
to the HartreeFock total energy of the isolated atom and the
second is to adjust the parameter a so that the virial theorem
is satisfied when Xa orbitals are used in the RHF expressions
for the expectation value of the kinetic and potential energies
of the isolated atom.
The two sets of a's turn out to be very close to each
other and the variation of a from atom to ion is not
significant. Thus, it seems justified to use the same a for
atom or ion in the solid. All of Schwarz' determinations
were done without spin polarization. In the present
calculations, the a~HF values with the HartreeFock total
energy criterion were used for all atomic spheres except
for the hydrogen ion sphere in which the oL value is equal
to 0.77, which is appropriate for a spinpolarized calcula
tion (Slater, 1973; Singh and Smith, 1971). For other ions,
the oHF values are approximately the same in a spin
polarized calculation. For the case of KC1, the same value
of 0.72 was used both for the K ion and the Cl ion. A
weighted average of V is also used for the intersphere and
the outer sphere regions. All the oc values used for various
ions in the present calculations and the XHF and oVT
HF VI
values determined by Schwarz (1972) are given in Table 3.1.
3.3 Transition State
In order to find the absorption energies of color
centers in ionic crystals, the transition state concept
(Slater, 1972) has been used in the multiple scattering Xo(
method.
Instead of calculating the difference of the total
energies of the ground state and the excited state of the
whole system, the excitation energy can be approximated by
taking the difference of the oneelectron eigenvalues of
the two eigenstates involved in the transition with the
occupation numbers set to be halfway between the initial
and the final state of the system. In other words, one can
remove onehalf of the charge from the lower eigenstate
and add onehalf of a charge to the higher eigenstate to
TABLE 3.1
The a values used in the present calculations and the
aHF and aV values determined by Schwarz.
a a
Ion a aHFa VT
H 0.77000b
Li+ 0.78147 0.78147 0.78087
F 0.73732 0.73732 0.73651
+
Na 0.73115 0.73115 0.73044
Cl 0.72000 0.72325 0.72277
K+ 0.72000 0.72117 0.72072
Schwarz (1972).
Singh and Smith (1971)
form a "transition state" of the system, and the excitation
energy can be easily obtained from the eigenvalues.
The justification behind the transition state calcula
tion follows from the fact that the eigenvalue in the Xx
method can be expressed as a first derivative of the total
energy with respect to the occupation number of that state.
Using the power expansion of the total energy, it can be
shown that the error of the excitation energy for using
the transition state calculation is mainly from the third
order terms which can be shown to be relatively small
(Slater and Wood, 1971).
3.4 Radii for Ions
In section 3.1, the division of space into three
regions in the multiple scattering formulation has been
discussed and the atomic region for each atom or ion is
defined by a sphere. Thus, a set of radii acting as para
meters for all atomic spheres is required in the calcula
tion. Because of the muffintin approximation, it is
natural to choose the radii of atomic sphere and outer
sphere to touch one another so that the space in the inter
sphere region will not be too large, since the use of the
constant potential in this region will probably contribute
some errors to the final results.
For ionic crystals, there are a few sets of empirical
ionic radii (Goldschmidt, 1926; Pauling, 1948) which are
widely used in many books. Thus, at the beginning of this
investigation, several calculations were carried out by
using the radii of Goldschmidt (1926) for the ion spheres.
However, the basic reason for having such a set of radii
is the fact that the differences between the values of the
nearestneighbor distance in the corresponding halides of a
pair of alkali metals are approximately constant, and the
same holds for the alkali metal salts of pairs of halogens;
thus, by specifying any one radius, the whole set of ionic
radii can be determined with a small correction so that the
radii of anion and cation can add up exactly to the nearest
neighbor distance of that particular crystal. Different
methods for choosing the radii generally will lead to a
different set of ionic radii and a review of various kinds
of ionic radii is given by Tosi (1965).
In order to make best use of the muffintin approxima
tion, it is desirable to have a set of radii which are
close to the relative sizes of ions in crystals. There are
a few experiments employing the Xray diffraction techniques
to study the electron charge distributions of ionic crystal
(Witte and Wolfel, 1955; Krug et al., 1955).
From the minimum of the charge densities in between a
pair of anion and cation the crystal radii for this pair of
ions can be defined. Using the above criterion, the crystal
radii of the Na+ and Cl ions are 1.17 A and 1.64 A, respec
tively. Starting from these two radii, Gourary and Adrian
(1960) were able to deduce a set of corrected radii for the
ionic crystals. This set of radii give a larger relative
size of cation to anion than that of the previous sets.
Values of different sets of ionic radii mentioned are given
in Table 3.2.
Calculations of various color centers by using Gourary
and Adrian Xray corrected ionic radii have been done for
many alkali halides. The electronic charge in the intersphere
region was observed to be less than those of previous calcu
lations and the potential has a smaller discontinuity when
we go from one atomic region to another. Therefore, in most of
thecalculations in this dissertation the corrected radii of
Gourary and Adrian were used.
There are other ways to find the crystal radii for a
particular ionic crystal, for example, by using the charge
densities of ions or the potentials of the ions. But in
most cases the radii determined are not far from the
Xray corrected radii.
Above all, the radii of the atomic spheres are not
used as adjustable parameters in the present calculation.
However, for some color centers, calculations with different
setsof radii were available and effects of this variation
of radii will be discussed in the results of those parti
cular color centers.
TABLE 3.2
The ionic radii of Goldschmidt and the Xray
corrected radii of Gourary and Adrian.
Ion Goldschmidta Gourary and Adrian
(in A) (in a.u.) (in A) (in a.u.)
Li+ (LiF only) 0.78 1.47 0.92 1.74
Li+ 0.78 1.47 0.94 1.78
Na 0.98 1.85 1.17 2.21
K+ 1.33 2.51 1.49 2.82
F (LiF only) 1.33 2.51 1.09 2.06
F 1.33 2.51 1.16 2.19
Cl 1.81 3.42 1.64 3.10
Br 1.96 3.70 1.80 3.40
aGoldschmidt (1926).
bGourary and Adrian (1960).
3.5 Ionic Correction Potential
In all of the present calculations except the U2 center
calculations, the cluster being used is not neutral. Thus,
some form of the potential correction is needed to stabilize
it. The most common method is by adding the potential from
a charged sphere to the potential of the cluster. This
method was first used by Watson (1958) to stabilize the 0
ion. Using this technique with the charge on the Watson
sphere being the negative of the charge Q of the cluster
(see Table 3.3), the ionic correction potential VI for an
electron in Equation (3.1) is given by
R2 Ry for r R
V (r) = (3.12)
I (2Q Ry for r > R
r
where R is the radius of the Watson sphere, usually chosen
to be the same as the outer sphere, and r is measured from
the center of the cluster. With this charged sphere, the
potential for the cluster will approach zero outside the
cluster, and presumably the potential inside the cluster is
close to the actual potential of the cluster imbedded in the
solid. Several calculations were carried out with this
potential correction, and some characteristics of this
correction were found. That is, the correction inside the
cluster depends on the choice of the radii of the ion spheres
through the fact that the R in Equation (3.12) is always
TABLE 3.3
Constants in Equations (3.12) and (3.13) for
various clusters used in the U center
and F center calculations.
Clustera A Q
Eightcenter 3.96 5.00
Twentycenter 3.97 7.00
Twentyeightcenter 0.50 1.00
See Figure 2.3. The eightcenter cluster includes the
first nearestneighbors, and the twentycenter cluster
includes the second nearestneighbors, and the twenty
eightcenter cluster includes the third nearestneigh
bors.
33
equal to the sum of a constant (in the eightcenter calcu
lation, this constant is the nearestneighbor distance) and
the radius of the most distant ion of the cluster. The
situation is more serious when the cluster has a large Q.
A better potential correction,which depends only on the
nearestneighbor distance of the crystal, is needed. There
are several methods for calculating the Madelung potential
inside the ionic crystal (for a review, see Tosi (1965)),
and for the present purpose the Evjen method (Evjen, 1932)
is a simple method which can be easily programed to use in
a computer. A volume average of the potential due to the
surrounding ions is calculated inside the cluster and
details of this calculation are discussed in Appendix A.
Thus, the ionic correction potential for an electron can be
rewritten as
2A
d Ry for r < R
VI(r) = (3.13)
2Q
Ry for r > R
r
where d is the nearestneighbor distance or the anioncation
distance in an ionic crystal, A is a constant, which depends
on the size and structure of the cluster. The different
values of A for various clusters are given in Table 3.3.
In a large cluster, the volume averaged correction
potential is not sufficient to give a reasonable correction
for each ion, therefore, separate ionic correction poten
tials for ions were sought. Since the potential for each
ion in the muffintin approximation is spherically averaged,
the correction potential for each ion is particularly
simple to find.
We note that in the jth ion sphere of the cluster the
ionic correction potential at an arbitrary point, rj, is
given by
V = r (3.14)
I i r Rjij
where r. is measured from the center of the jth sphere,
the summation of i is over all ions outside the cluster
and q. is the point charge of ith ion, and R.. is a vector
measured from the center of the jth sphere to the ith ion.
It can be shown very easily that the spherically averaged
V (r.) is equal to the ionic correction potential at the
I 3
center of the jth sphere:
(r,) 41... ^ 1. ^ (3.15)
Therefore, the ionic correction potential for an electron
inside any ionic sphere is a constant and can be calculated
by using the Madelung constant c (see Appendix A ), e.g.,
35
Z% L 2 v. 2z 2A (3.16)
V,  1
where d is the nearestneighbor distance of the crystal,
the summation of k is over all ions in the cluster
except the jth, and R. is the distance from the jth center
jk
to the kth center. The ionic correction potential in the
intersphere region of the cluster can also be found by
the following method. The spherically averaged or the
volume averaged potential inside the outer sphere of a
cluster due to all ions outside will give a constant poten
tial that is the same as the potential at the origin of the
cluster, and since the potentials inside the ion spheres
are known, the ionic correction potential for the inter
sphere region can be calculated from the following equation:
o) 5 I^ Sk 2,At ^(3.17)
V  Ry
1 So  s_ )
where V (0) is the potential at the center of the cluster,
S is the volume of the outer sphere, the summation of k
0
is over all the ion spheres in the cluster, and Sk is
the volume of the kth ion sphere. The value of Aint depends
36
on the relative size of the cation and anion, and values of
Aint for KC1 and LiF along with the Aj's for the large
cluster shown in Figure 2.3 are given in Table 3.4. The
correction potentials for the outer region in all cases are
the same as defined in Equation (3.12).
TABLE 3.4
Constants in the separate ionic correction potentials
for the twentyeightcenter cluster.
Region Aj or Ainta
Central ion sphere 0.386
First nearestneighbor 0.439
Second nearestneighbor 0.235
Third nearestneighbor 0.682
Intersphere (KC1) 0.381
Intersphere (LiF) 0.477
aThe twentyeightcenter cluster is shown in Figure 2.3.
Constants are defined in Equations (3.16) and (3.17).
CHAPTER IV
RESULTS AND DISCUSSION
4.1 The U Centers in Alkali Halides
The peak energies of the optical absorption band of
the U centers in several alkali halides have been calculated
with the use of the eightcenter cluster (see Figure 2.1),
which has been discussed in section 2.3. The electronic
structure and the absorption energy of the U center in KC1
have also been calculated using the twentycenter cluster.
This has a negative hydrogen ion at its center and includes
up to the second nearestneighbors. Calculations with the
twentyeightcenter cluster shown in Figure 2.3 including
up to the third nearestneighbors have also been done.
All calculations have been carried out using the spin
polarized multiple scattering Xa method in which a separate
local exchange potential for each spin is used in the one
electron equation. The a values used for the various atomic
regions are given in Table 3.1. In the U center calculation,
the a value for the hydrogen sphereis 0.77 which is
usually used in a spinpolarized calculation of a two
electron ion (Slater, 1973; Singh and Smith, 1971).
The input data for the U center calculations using the
eightcenter cluster are given in Table 4.1. The interionic
TABLE 4.1
Radii used in the eightcenter cluster calculations
of the U centers and the F centers in alkali halides.
Crystal da R+ R Rut
LiF 3.800 1.740 2.060 5.540
NaF 4.370 2.190 2.180 6.560
NaCl 5.310 2.210 3.100 7.520
NaBr 5.630 2.210 3.420 7.840
KF 5.046 2.834 2.212 7.880
KC1 5.934 2.834 3.100 8.768
KBr 6.220 2.820 3.400 9.040
All quantities are measured in atomic
aTosi (1965).
units.
distance (d), shown in the first columnis the same as the
equilibrium nearestneighbor distances in a pure crystal.
The R 's are the radii for the alkali spheres and the R_'s
+
are the radii for the halogen spheres, which is occupied by
a negative hydrogen ion in the present case. All radii are
chosen according to Gourary and Adrian's corrected ionic
radii shown in Table 3.1), although small adjustments were
made so that the atomic spheres touch one another. Rout is
the radius of the outer sphere. The averaged Madelung
potential correction defined in Equation (3.13) was used for
all these calculations.
The electronic structure of the ground state of the U
center in KC1 calculated by using the eightcenter cluster
is shown in Figure 4.1. In this diagram, one can see that
no spinpolarization effect appeared in the eigenvalues.
This is expected from a system consisting of closed shell
ions. The hydrogen Is level appears as two alg symmetry
orbitals, one with each spin. These two hydrogen alg levels
are well localized in the central part of the cluster as the
calculated result showed that 85% of the charge of these
levels is within the central sphere and the rest of the
charge is distributed mostly in the intersphere region.
Below the hydrogen levels, the next occupied levels
are a group of seven levels for each spin that arise from
the atomic 3p levels of the six K ions. These levels are
relatively deep compared to the hydrogen alg orbital and
SPIN DOWN
O.OOF0
tlu
0.10
tlu
alg
0.20
0.30 
0.40
1.24
1.25
1.26
1 27 
1.28
1.281
alg
(U)
tig
tlu
e
t2u
t2g
tlu
alg
tig
tlu
eg
t2u
t2g
tlu
alg
Figure 4.1. The ground state electronic structure of the
U center in KC1 calculated by using the
eightcenter cluster.
SPIN UP
their charge distribution is concentrated in the six K+
spheres. Hence, only a small interaction is expected between
this group of levels and the hydrogen levels. The excitation
of the U center in this picture is expected to be an electron
excited from the hydrogen alg orbital to the next higher
allowed level, which is a tlu symmetry orbital with the same
spin. By using the transition state concept discussed before,
the excitation energy for the U center can be calculated.
The calculated transition energies of the U center for
several alkali halides are shown in Table 4.2 along with
the experimental absorption energies and other theoretical
results. The e(alg) and E(tlu) are the eigenvalues of the
two states involved in the excitation process in a "transi
tion state" calculation, and the difference of these two
energies gives the present value, AE, for the U center
absorption. Comparing with the experimental results, one
can see that AE's are in good agreement with the observed
values, especially for the two potassium halides. Two
other sets of energies are also given in Table 4.2. One of
them is calculated by Gourary and Adrian (1957) and Spector
et al. (1967) using the pointion model and their values
are given under AEpi, and the other set of values, AE vey,
is calculated by Ivey's relation given in Equation (2.1),
which is determined empirically from the experimental
data.
The absorption energies obtained both theoretically and
A4 A U
2 z
H i 4
r u
U
u
>1
.0
4
x
09
I
Q
Md
<3
!
H,
ft
09
r'r
.,I
rI
1o1
H
r0
H
CdJ
r
rHU
4)2
O U
t0
49
aC (
4J
00)
J c
4 1
44
0,
LO 4
r ro
1,
H H
0
04
0
Ei
4
*
C.,
*H
0
4l)
H 0
U) a I
oa o
CN
*o
tr>
S r4
I H
H 4C 
U 0 CN
co u ;'
0 C C O
io: U) C0 ()
experimentally are plotted against the nearestneighbor
distances of alkali halides in Figure 4.2. As one can see
from this figure, Ivey's relation is the result of averaging
experimental data, and there is in fact considerable devia
tion from it.
The absorption energies calculated by the pointion
model show a smooth curve in parallel with the Ivey's
empirical curve, and their values can be fitted to a rela
tion with the same form as the Ivey's. Unfortunately, this
simple model produces results which are approximately 15%
lower than the observed values. In the pointion model
calculation of the U center, one uses a trial wave function
for a ground state, with variation parameters to be deter
mined, in a field produced by all surrounding ions which
are treated as point charges. By minimizing the energy
expression of the Hamiltonian for this twoelectron system
using the trial wave function, one can obtain the energy
for the ground state. A similar procedure using another
trial wave function gives the energy of the excited state,
thus, the excitation energy of the U center can be found.
In this pointion model, only the nearestneighbor distance
of the crystal is directly involved in the calculation, it
is not surprising that the calculated energy changes
smoothly with the interionic distance of the crystal. On
the contrary, the results calculated from the present
cluster method do show some deviations from the smooth
IVEY'S RELATION
/"
O EXP
* MSXo
A PI
'N
'N
\:"
6\
\ /
F 1 
5.0
5.5
6.0
6.5
NEARESTNEIGHBOR DISTANCE (A.U.)
Figure 4.2. The optical absorption energies of the U center
as a function of the nearestneighbor distance
of the crystal.
0.60
0.55
0.50
0.45
0.40 
0.35
4.0
4.5
variation of an Ivey type of relation. This can be seen,
especially for the case of the NaBr crystal with an inter
ionic distance of 5.63 a.u., in Figure 4.2. In that case,
the U center energy is lower than the energy for a U center
in KC1 which has an interionic distance of 5.934 a.u. As
one can see, the experimental results also indicate the
same characteristics in that region. Therefore, the present
model, including the detailed structure of the first
nearestneighbors, does show that although the interactions
between the hydrogen ion and the six alkali ions are expected
to be small as discussed before, the effects on the energy
levels and wave function of the U center due to the structure
of the surrounding alkali ions are, nevertheless, not
negligible. Wood and Opik (1967) have also calculated the
U center absorption energies in potassium halides by a semi
empirical method. In their calculation, the electronic
structure of the first nearestneighbor ions is considered,
and the calculated transition energies, 0.409 Rydbergs in
KC1 and 0.403 Rydbergs in KBr, showed an energy somewhat too
low for KC1 and a comparable energy for the KBr case. They
also tried another calculation by including polarization
effects and taking more surrounding ions into consideration
and obtained a transition energy of 0.437 Rydbergs for KC1
which is higher than the experimental value by approximately
the same amount as the difference of their first calculated
result. In any case, the ability of a model to predict a
single transition energy is not a very satisfactory criterion
for the validity of the model. It is necessary at least to
examine several cases to obtain a better understanding of the
model that is used. Thus, the calculation of Wood and Opik
and the calculation of Hayns (1972), in which only the U
center transition energy in LiF was calculated by a semi
empirical CNDO method, will not be discussed.
Experimental data for the U center absorption energy are
available for several alkali halides with interionic distance
greater than and equal to that of the sodium chloride
crystal. Comparison of the present calculated energies with
the observed values shows that the average deviation is
about 3.5%, which is very good. For alkali halides, with
interionic distances less than that of the sodium chloride
crystal, one has to consider that the results from Ivey's
empirical relation will probably give a reasonable guess as
to the transition energy of the U center. From Figure 4.2,
the results from. the present cluster method can be seen to
deviate away from the predictions of the Ivey relation as
the nearestneighbor distance gets smaller.
For the case of the LiF crystal, which has the smallest
interionic distance of 3.8 a.u., the transition energy
calculated is 0.577 Ry which is approximately 16% lower
than Ivey's result and is comparable to the calculated value
of the pointion model. Discussion of the possible causes
for this drop in the calculated U center excitation energy
in the region of small interionic distance and of the
neglected factors, namely, the lattice distortion and
polarization effect, will be given later.
At first, it may seem that the resulting curve of the
cluster model in Figure 4.2 is quite complicated as to its
variation with the interionic distance despite the fact
that it has been shown to be consistent with the experi
mental data. With a closer examination, the calculated
energies do show a characteristic variation with the inter
ionic distance within the sodium halides and the potassium
halides (see Figure 4.3). This is reasonable, since in the
eightcenter calculation only the nearest cations are
considered directly in the calculation and the variation
within a certain family of cation only changes the space
occupied by the H ion. Therefore, the difference in energy
between the ground state and the excited state of the U
center decreases with the increase of the space occupied by
the H ion, but for the same space occupied by H ion with a
different set of cations, the lighter cations will give a
larger difference between the two levels of the U center. In
the pointion calculation, the effect of the nearest cation
is only related to the distance measured from the center of
the cation, since only the Coulomb potential of pointcharge
of that cation is considered. Thus, the interionic distance,
which is the sum of the size of H and that of the nearest
cation, will be related directly to the U center transition
energies. On the other hand, the effects of cations are
shown in the cluster calculation. This implies that the
49
0.5 
POINTION MODEL
A
0.4 /
/7 /0 '/KBr
0.2 
I 7
oC /7 KF
'0.3 /,5
LiF /
IVEY'S RELATION
0.2
0.6 0.7 0.8
LOG (d)
Figure 4.3. The logarithmic graph of the optical
absorption energies of the U center, AE
(in Ry), as a function of the nearest
neighbor distance of the crystal, d (in
a.u.).
interactions between the hydrogen ion and different cations
have different degrees of deviation from the interactions
between a hydrogen ion and the approximated pointcharges of
the cations. We can see in Figure 4.3 that the transition
energies calculated from the cluster for the sodium halides
and the potassium halides fall on different curves. The
reason for this is probably the different ionicities of the
sodium ion and the potassium ion. In other words, the
electronic charge distributions of different cations have
slightly different forms, a feature which cannot be included
in the pointion model.
In order to investigate this possible difference, the
electronic charge distribution within each region in the
transition state cluster calculation for various alkali
halides is shown in Table 4.3. Now, the transition state
is, in the exact case, a mixed state corresponding to equal
weighting of single determinant ground and excited states
(Trickey, 1973). Comparing the charge distribution of the
ground state to that of the transition state in KC1, one
finds that the charge inside the K+ sphere, Q+, almost stays
the same (difference is less than 0.001 e), and a fraction
of an electronic charge (about 0.32 e) is transferred from
the hydrogen sphere to both the intersphere region and the
outer region as the cluster goes from the ground state to
the transition state. With this in mind, one can use the
general features of the charge distributions given in Table
4.3 to discuss the problem mentioned above.
TABLE 4.3
The distribution of electronic charges within each
in the U center transition state calculation.
Crystal Q Q Qint out
KBr 17.842 1.450 1.274 0.224
KC1 17.846 1.410 1.242 0.271
KF 17.849 1.260 1.187 0.455
NaBr 9.954 1.455 0.681 0.143
NaCI 9.954 1.408 0.682 0.187
NaF 9.954 1.239 0.659 0.380
LiF 1.998 1.201 0.442 0.369
All calculations are
cluster.
carried out using the eightcenter
region
From Table 4.3, one finds that the charges in the K+,
Na+, and Li ion spheres are approximately 17,85, 9.95, and
2.00 electron charges, respectively, thus, the net charges
inside the K Na+, and Li+ ion spheres are approximately
+1.15, +1.05, and +1.00, respectively. The electronic
charges of cations outside the ion sphere are mostly in the
intersphere region, and the Coulomb potential due to this
diffused charge of the cations in the hydrogen sphere is
smaller than the Coulomb potential with all these charges
in the cation spheres. Therefore, the net Coulomb effect of
a potassium ion gives an equivalent charge of more than +1.
The sodium ion also has an effective ionicity greater than
+1 but less than that of the potassium ion. The lithium ion
has an ionicity equal to +1. Since the alg orbital of the
U center is localized in the central part of the cluster
as one can see from the general features of the charge
distribution of this orbital given in Table 4.4, the energy
of this alg orbital is expected to be lower than that in a
pointion field, and the amount of this difference for the
potassium halides is larger than that of the sodium halides.
On the other hand, the charge distribution of the excited
state (the tlu orbital) of the U center is mainly in the
intersphere region and the outer region (see Table 4.4).
Since in general there is more charge in the intersphere
region for the potassium halides than for the sodium halides,
the energy of this tlu orbital is expected to have a higher
53
TABLE 4.4
The charge distribution of the ground state and
the excited state orbitals of the U center
in a transition state calculation.
Orbital Crystal QH Q Qint Qo
H int out
KBr 0.891 0.004 0.085 0.001
KC1 0.872 0.005 0.096 0.001
KF 0.775 0.014 0.142 0.001
alg NaBr 0.883 0.003 0.099 0.000
NaC1 0.864 0.004 0.113 0.000
NaF 0.774 0.009 0.169 0.001
LiF 0.754 0.009 0.191 0.001
KBr 0.059 0.013 0.506 0.360
KC1 0.032 0.012 0.432 0.465
KF 0.002 0.006 0.163 0.801
tlu NaBr 0.119 0.013 0.541 0.263
NaCI 0.077 0.013 0.497 0.349
NaF 0.007 0.008 0.223 0.723
LiF 0.011 0.006 0.226 0.725
energy in the case of the potassium ions surrounding a H
ion than in the case with sodium ions surrounding a H ion.
Therefore, the U center transition energies for potassium
halides will be generally higher than those of the sodium
halides. Precisely this sort of behavior is exhibited in
Figure 4.3. Of course, one should not compare the U center
transition energies between these two families of alkali
halides quite so directly. For example, in the cases of NaC1
and KC1, the small interionic distance of NaC1 is still the
dominating factor which makes the transition energy in NaCI
greater than that in KC1. On the other hand, the argument
given above does explain the cause of the calculated U
center energy in NaBr being smaller than that in KC1.
Another result is that there is a larger difference
between the calculated energy and Ivey's result in an alkali
fluoride than the corresponding differences in an alkali
chloride and an alkali bromide (see Figure 4.3). This
suggests that the relatively small size of the F ion has
a strong effect on the hydrogen ion. As shown in Table 4.4,
the charge distribution of the ground state of the U center
inside the central hydrogen sphere for an alkali fluoride
is less than that of the corresponding alkali chloride or
alkali bromide and the charge in the intersphere region is
increased in the case of an alkali fluoride. Thus, the energy
of this state is expected to have an additional shift upward
as compared to the other cases. For the U center excited
state in an alkali fluoride, the charge distribution is
mostly in the outer region, thus, this is a very extensive
orbital as compared to the other cases. A more accurate
representation of this state probably needs the consideration
of more neighbors into the calculation or other semiempirical
methods, but the energy of this state is believed to be not
far from the present value. There are no experimental data
on the U center energies for the alkali fluorides. Thus, the
present calculated values cannot be evaluated directly.
Nevertheless, U center calculations in various alkali halides
which use the small cluster multiple scattering calculation
give a good description of the electronic structure and the
absorption energies. The absorption energies calculated for
KC1 and KBr are in excellent agreement with the observed
values, although this agreement is probably somewhat coin
cidental since some effects were neglected in the calcula
tions and the effects of the parameters and approximations
used in the calculation should be considered.
The most common effects discussed in the color center
calculation are lattice distortion and polarization. To
incorporate the lattice distortion into the present calcula
tion is relatively simple. A preliminary result showed that
the U center transition energy is increased if the neighboring
ions are allowed to displace inward, and in fact, an estimate
can be obtained directly by using the graph shown in Figure
4.3. At present, the determination of the amount of the
lattice distortion near the U center by fitting data to the
experimental results is certainly unwarranted since there
are other effects to be considered. Thus, no attempt was made
in this regard, though a rough estimate showed that the U
center transition energy in KCl will increase about 0.004 Ry
if the first nearestneighbor ions are allowed to displace
inward by 1% of the interionic distance of the crystal.
Furthermore, in the excited state the probability for finding
the electron outside the first nearestneighbor distance
from the center will be appreciable. Thus, the lattice
distortion effect is expected to be more important in the
excited state than the ground state of the U center. Never
theless, the lattice distortion caused by the U center is
certainly smaller than the distortion caused by the F center,
and one calculation (Wood and Joy, 1964) estimated the dis
tortion of the ground state of the F center is of the order
of 1% or less.
When an electron moves out of the vicinity of the color
center, the surroundings of the electron will be polarized,
and this polarization will follow the movement of the elec
tron. Gourary and Adrian (1960) have discussed the quasi
adiabatic approximation to consider the instantaneous field
at the position of the electron. But there are many
difficulties in this approximation. For the oneelectron
selfconsistent field method, each electron is supposed to
respond only to the averaged positions of all other electrons.
Thus, the polarization effect cannot be considered in the
present calculation. For the ground state of the U center,
the electron will stay mainly in the central region and the
polarization effect will be negligible. For the excited
state, the polarization effect probably will lower the
energy of this state, but the effect is expected to be
small.
As to the effects of the approximations and parameters
used in the present calculation, the muffintin approximation
is expected to be a good approximation, since the Xray
diffraction experiment (Witte and Wolfel, 1955) for NaCl
showed that the charge distribution is nearly spherical for
the ions and only a very small amount of charge is in
between the ions. The gradient of the charge densities in
this interior region is small. For the case of LiF (Krug et
al., 1955), the charge distribution is less spherical near
the outer part of the ions as compared to the case of NaC1.
It is thought in this case the anions are relatively soft,
so that some overlapping of charge densities from different
ions can occur. One of the problems of using this approxima
tion usually comes from the uncertainty in choosing the
radii for the atomic spheres. From the treatment of the
intersphere potential as a constant, one knows that an
appropriate choice of radius for each ion is certainly
essential. In section 3.4, the choice of the radii for the
ions has been discussed, and the Xray corrected radii of
Gourary and Adrian (X.R.) are used for all the calculations.
In order to see the effects for using different sets of
radii, a U center calculation in KC1 with the ordinary ionic
radii of Goldschmidt (O.R.) was carried cut. The results of
these two transition state calculations in KC1 are shown in
Table 4.5. In the O.R. case, the size of the K+ ion is
decreased as compared to the size in the X.R. case, and this
decrease is the direct cause for the drop in electron charge
inside the K sphere. Thus, the net charge inside the K
sphere is about +1.32 which is higher than the net charge
(about +1.15) in the X.R. case. Similarly, the charge inside
the H sphere is increased due to the increase of the size
of the hydrogen sphere. The major effects of using the O.R.
as compared to the X.R. are the increase of the electronic
charge in the intersphere region and the decrease in energies
of the ground state and the excited state of the U center, and
also a decrease of the U center transition energy. The
increase of the charge in the intersphere region indicates
that the O.R. gives a worse representation for the size of
the ions in crystal than the X.R. does. The more positive
charge of the K+ sphere in the O.R. case gives a deeper
potential for the hydrogen ion, consequently, the energy
levels of the U center are lowered. The transition energy
of the U center is lowered by 0.01 Ry in the O.R. case which
is about 2.5% of the energy.
So far all the calculations have used the ionic
TABLE 4.5
Comparison of the U center transition state calculations
in KC1 with two different sets of ionic sphere radii.
X.R.a
VI(r Rout) c
Rout
E (alg)
(tlu)
Qint
1.335
8.768
3.100
2.834
0.427
0.488
0.061
0.270
1.240
1.410
17.846
O.R.b
1.335
8.448
3.420
2.514
0.417
0.497
0.080
0.255
2.157
1.509
17.680
All energies are in Rydbergs and distances are in a. u.
aRadii of Gourary and Adrian (1960).
bRadii of Goldschmidt (1926).
CCalculated from Equation (3.13).
correction potential defined in Equation (3.13). Several
calculations with the Watson sphere correction (see Equation
(3.12)) are shown in Table 4.6. In this table two sets of
radii are used in both the KCl and the LiF calculations.
Comparing the charge distribution of the cluster in KCl
with two different kinds of radii, one finds essentially the
same differences that are found in the calculations with the
averaged correction potential (see Table 4.5). With the
Watson sphere in the calculation, the potential inside the
cluster is lower than that with the averaged correction
potential in these calculations. This shifts the energy
levels of the U center down and a small fraction of the
charge redistributes from the outer region to the inside of
the cluster. The charges inside the cation spheres stay the
same. Comparing the transition energies found for different
ionic correction potentials, one finds that these energies
vary due to the relative insensitivity of the U center
excited state as compared to the ground state. Thus, the
correction potential for a cluster embedded in the solid
is essential in finding an accurate optical absorption
energy in the U center. Further, as can be seen in Table 4.6
the differences in the transition energies between calcula
tions which use the X.R. and the O.R. are 0.02 Ry and 0.04 Ry
in the cases of KCl and LiF, respectively. The difference
for the case of KC1 is doubled as compared to that of the
previous calculations shown in Table 4.5. One important
TABLE 4.6
Comparison of the U center calculations
with different sets of radii using
the Watson sphere potential correction.
KC1 LiF
(O.R.) (X.R.) (O.R.) (X.R.)
VI(r Rout) 1.184 1.141 1.965 1.805
Rout 8.448 8.768 5.090 5.540
R 2.514 2.834 1.290 1.740
R_ 3.420 3.100 2.510 2.060
AE 0.438 0.458 0.624 0.662
E(alg) 0.628 0.653 0.766 0.865
E(tlu) 0.190 0.195 0.142 0.203
Qout 0.138 0.091 0.174 0.104
Qnt 2.200 1.350 0.860 0.652
QH 1.540 1.439 1.380 1.217
Q+ 17.680 17.850 1.928 2.004
All energies are in Rydbergs and distances are in atomic
units. O.R.=ionic radii of Goldschmidt (1926). X.R.=
Xray corrected radii of Gourary and Adrian (1960).
factor for causing these differences is the dependence of
the Watson sphere potential on Rout for the cluster (see
section 3.5). Therefore, the averaged correction potential
of Equation (3.13) is a better potential to use in the present
eightcenter cluster calculations.
For most of the calculations in various alkali halides,
the Xray corrected radii give fairly good results for the
transition energies, and the potential calculated has a
smaller discontinuity in crossing the boundary of the
neighboring ionic spheres than is the case when the ordinary
ionic radii are used. Exceptions are the cases of the alkali
fluorides, namely, KF and NaF. In these cases, preliminary
results show that the transition energies calculated by
using O.R. are somewhat higher than the previous results
and closer to Ivey's results. Above all, the variations of
the results due to different sizes of the ions are relatively
small, as one can see in the case of KC1, for which the
variation is only 2.5%.
Actually, any set of ionic radii is only an approximation
to the apparent sizes of the ions in crystals. Thus, in order
to get a more accurate result on the sizes of ions in each
crystal, one needs to find them separately by using the
criteria of charge densities or potentials. Eventually, the
removal of the muffintin restriction on the potential can
eliminate the necessity of choosing radii for the ion spheres
and the error introduced by this approximation.
In all calculations, the parameter a in the local
exchange potential for each region is not varied. The
detailed effects of choices of a have not been investigated,
but in general a higher a value will give the electron a
deeper potential. Thus, an a value greater than 0.77 in the
hydrogen sphere will probably lower the states of the U
center and will probably result in a slightly higher
transition energy for the U center.
Several effects associated with the present cluster
calculation have been discussed and most of these effects
are small. From Figure 4.2, the calculated energies can be
seen to deviate from Ivey's relation in the region of small
interionic distance. One possible cause is the assumption
of the point charges for the surrounding ions in calculating
the average correction potential. This may become improper
as the interionic distance gets too small. There are no
experimental results on a U center in the region of small
interionic distances. A similar problem will be discussed
later, in the section on F center calculations.
A calculation on the U center in KC1 using the twenty
center cluster (including the outer sphere) has also been
carried out. In this cluster the central hydrogen ion is
surrounded by six K ions and twelve Cl ions. The radii of
the atomic spheres are the same as those used in the eight
center cluster calculation. The a values used in various
regions are given in Table 3.1. At first, the correction
potential used in the calculation is chosen simply to be
an average of the ionic correction potentials at the centers
of all atomic spheres which can be calculated by using
Equation (3.16). This constant correction potential for an
electron inside the cluster equals to 8.32/d Ry, where d is
the nearestneighbor distance of the crystal. From the tran
sition state calculation the U center absorption energy is
found to be 0.34 Ry which is about 20% lower than the
observed value.
Another calculation has used the averaged correction
potential defined in Equation (3.13). This correction
potential for an electron inside the cluster equals
7.94/d Ry which is higher than the previous potential by
0.067 Ry. The results of the transition state calculation
show no significant change from the previous calculation.
The calculated U center absorption energy is equal to 0.32
Ry. The charge in the intersphere region has 10.12
electronic charges which is very large compared to that of
an eightcenter cluster calculation. This is' mainly from the
fact that only 17.19 electronic charges are within the Cl
sphere and the rest of the charge of the Cl ion is in the
intersphere region.
The possible cause for the low transition energy is
believed to be the inaccurate correction potential used in
the calculation. For a small cluster the averaged potential
defined by Equation (3.13) may be an appropriate approxima
tion. But for a larger cluster, the deviation of this correc
tion potential from the averaged value in each region may be
appreciable. This is especially important in the present
calculation of the U center absorption energy. Since the
ground state of the U center is in general concentrated in
the hydrogen sphere region and the excited state is mainly
distributed in the intersphere region, a difference of the
correction potentials in these two regions will affect the
resulting transition energy almost directly.
The ionic correction potential inside any atomic sphere
can be calculated from Equation (3.16), and this potential
inside the hydrogen sphere is 8.46/d Ry. The correction
potential in the intersphere region can be calculated by
Equation (3.17). Since the averaged correction potential
inside the cluster region has been calculated for this
twentycenter cluster (see Table 3.3), one can use this
potential to replace the Vi(o) term in Equation (3.17).
This will give a more accurate correction potential for the
intersphere region (because of the fact that the parts of
the surrounding ions inside the "outer sphere" of the
cluster are excluded in the calculation of the averaged
correction potential) and this correction potential calculated
in the present case is 7.7/d Ry.
Using the correction potential for each region discussed
above, one can see that the difference of these potentials
between the intersphere region and the hydrogen region is
about 0.13 Ry. Thus, this correction potential for the
cluster is expected to give a better U center absorption
energy than the previous two calculations. The charge inside
the Cl sphere is expected to increase also. A calculation
using this ionic correction potential for each region has
not yet been done.
The twentyeightcenter cluster model shown in Figure
2.3 has also been used for the U center calculation in KC1.
In this cluster the hydrogen ion is surrounded by three
shells of ions. The first shell consists of six K ions and
the second shell consists of twelve Cl ions and the third
shell consists of eight K+ ions. The radii used for the
atomic spheres and the a values used in various regions
are the same as those used in the smaller cluster
calculations.
A transition state calculation using this large cluster
has been carried out. In the calculation the averaged
correction potential given in Equation (3.13) is used. The
calculated U center absorption energy is 0.48 Ry which is
about 12% higher than the observed value.
In this large cluster transition state calculation, the
hydrogen alg levels of both spins are below the group of
levels which arise from the chlorine 3p states, while in the
previous twentycenter cluster calculations the hydrogen al
level with spin up (occupation number is 0.5) is above the
chlorine 3p levels and the other hydrogen alg level is below.
Furthermore, the potentials in various atomic spheres are
found to be deeper in the twentyeightcenter case than in
the twentycenter case with different magnitude. Thus, large
differences are found between the two cluster calculations
in potentials and eigenvalues. This is also shown in the
calculated U center absorption energies. The reason for the
fluctuation of the calculated energies from small cluster
calculation to the large cluster calculation is believed to
be the use of a constant correction potential inside the
entire cluster and also the use of a constant potential in
the intersphere region. As mentioned before, the ionic
correction potential for each region of the cluster can be
found using Equation (3.16) and Equation (3.17). Thus, it
is interesting to see the effects of this form of the correc
tion potential used in the calculation.
A twentyeightcenter cluster calculation of the ground
state of the U center using the above mentioned correction
potential has been done and the results show no spinpolarized
effect in the eigenvalues and potentials as would be expected
in a closed shell system. The diagram of the calculated
eigenvalues is shown in Figure 4.4. The U center ground
state is an alg orbital arising from the hydrogen Is state
with energy equal to 0.545 Ry. Below this level there is
a group of fifteen levels (see Appendix B) from the second
nearestneighbor (2nn) Cl 3p levels. These levels will
form the valence band in a band picture of the crystal when
more and more ions are added to the cluster under considera
tion. The discussion of this valence band and comparison with
other calculations and experimental data will be given in
section 4.5. Below the valence levels, the first group near
1.34 Ry consists of ten levels which arise from the 3nn
K 3p levels. The width of this group of levels is very
narrow because of the relatively small interactions with
other states outside this group. Between 1.42 Ry and 1.48
Ry there are twelve levels. Five of them are from the 2nn
Cl 3s levels and the rest are from the Inn K+ 3p levels.
Those orbitals mainly concentrated in the Inn K ions have
somewhat higher energy as compared to those concentrated in
the Cl ions. Since these levels are close in energy, they
are actually a mixture of the functions from the two groups
of ions. Several unoccupied orbitals are also shown in
Figure 4.4. The lowest of the unoccupied states is an alg
orbital. For a U center transition, the lowest unoccupied
tlu orbital is the first allowed excited state.
The calculated electronic charges in each region of
the present cluster are shown in Table 4.7. In the same
table the potential near the boundary of each atomic sphere
is also given. As one can see, these potentials are approxi
mately the same for the atomic spheres. This indicates that
the correction potentials for the atomic spheres calculated
from Equation (3.16) are adequate. For the case of KC1 the
correction potentials used in the intersphere region,
SPIN UP
SPIN DOWN
e
alg
tlu ~
alg 
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(3nn)
(Inn) 
(2nn) . . .
Figure 4.4.
The electronic structure of the U center
ground state in KC1 from the twentyeight
center cluster calculation.
e
tlu
alg
alg
(2nn) .. . .
( n
0.7
0.8
1.3
1.4
1.5
1.6
Cl 3p
K+ 3p
K+
Cl
~~~~
~
TABLE 4.7
The electronic charges and the potential in each region
from the twentyeightcenter cluster calculation
of the ground state of the U center in KC1.
Region Q Va
Intersphere 9.49 0.35
Outer 0.05
H (C) 1.74 0.67
K+ (Inn) 17.96 0.79
Cl (2nn) 17.33 0.67
K+ (3nn) 17.88 0.65
near
values given for sphere regions are the potentials
the sphere boundaries.
hydrogen sphere, Inn sphere, 2nn sphere and 3nn sphere are
0.125, 0.13, 0.148, 0.079 and 0.23 Ry, respectively. As
one can see the use of a constant correction potential for
the entire cluster is certainly inappropriate.
A transition state calculation using the twentyeight
center cluster and a separate correction potential for each
region has been tried, but the calculation has not converged
due to the difficulty in tracking the hydrogenic alg orbital
which oscillates about the chlorine valence levels. From
the correction potentials in the intersphere region and in
the hydrogen sphere, it is expected that the calculated U
center absorption energy will still be too large. Further
discussions on the present correction potential for the
cluster will be given in section 4.5 where the cluster
calculation is compared with the energy band calculation and
other results.
4.2 The F Centers in Alkali Halides
Calculations of the peak energy of the F center absorp
tion band in several alkali halides have been carried out
using the spinpolarized MSXaSCF method. In these calcula
tions, the model used for the F center is the eightcenter
cluster shown in Figure 2.1. A few calculations using the
twentycenter cluster and the twentyeightcenter cluster
have also been done in the case of KC1.
The F center calculations are similar to the U center
work discussed in the previous section. In fact, the
potential from the U center calculation is used as the
starting point for the present F center selfconsistent
calculation. For an F center, the central sphere of the
cluster is occupied by an electron. The a value used for
the local exchange terms in that region is chosen to be the
same as the a value in the surroundings. Thus, for the
eightcenter cluster, the a values in all regions are the
same as Schwarz' aHF values (see Table 3.1) for the first
nearestneighbor cation. For larger clusters (in KC1) the
a value in the central sphere is chosen to be 0.72 which is
the same value used for the rest of the cluster. The radii
used for the atomic spheres are the Xray corrected radii
(see Table 3.2). In most of the calculations the averaged
correction potential defined in Equation (3.13) is used.
The ground state of the F center in KC1 using the
eightcenter cluster has been calculated and the eigenstates
are shown in Figure 4.5. The diagram is similar to that for
the U center eigenstates diagram shown in Figure 4.1. But
the spinpolarization effect caused by the unpaired F center
electron is clearly shown in the present case. The F center
electron occupies an alg orbital with spin up at o.183 Ry.
The probability for this electron in the central sphere
region is about 0.517 from the present calculation and the
probability in the intersphere region is about 0.357. Thus,
the F center ground state orbital is localized in the anion
SPIN UP
SU
SPIN DOWN
0.00
0.10
0.20
0.30
0.40
1.26
1.27
1.28 
1.29 
(Inn)
 i K 3p
Figure 4.5. The electronic structure of the F center
ground state in KC1 calculated by using
the eightcenter cluster.
S(F)
vacancy although it is not as localized as the U center
ground state orbital in the central sphere of the cluster.
On the other hand, the first allowed excited state of the
F center (which is the tlu orbital with spin up at 0.029
Ry) is relatively extensive. Thus, the probabilities for an
electron in this orbital to be found in the intersphere
region and outer region of the present cluster are 0.464 and
0.432, respectively. The calculated potential for the spin
up electron in the anion vacancy is nearly flat. This gives
some justification for the simplified potential well model
for the F center mentioned in section 2.4. However, the
actual problem is more complicated as one notices from the
extensiveness of the excited state wave function of the F
center.
The calculated F center absorption energies, using the
transition state procedure on several alkali halides, are
given in Table 4.8 along with the experimental results and
Ivey's values from Equation (2.1). Symbols used in this
table are the same as those used in Table 4.2 and are
explained in section 4.1. As one can see, the calculated
absorption energies in NaC1, NaBr, and KC1 are in good
agreement with the observed values. In Figure 4.6 the F
center energies given in Table 4.8 are plotted against the
nearestneighbor distance of the crystal.
As discussed in the case of the U center absorption,
Ivey's relation gives the averaged experimental results and
tI
0)
*Hl
rd
.,I
H
4J
0
SU
ar
cnU
0) 4
h I
wr 0
0
0)m
4i
O
HO
W 0 Q)
Cd
0
0
EH
E4^
H1
4 (4 4
E 0 0
MMM M
H
X
rd
v
r11
&i fa
*ii rd (
il Z
4
I
0
H
4i
ro
Cd
0
4i
i 0
 H
4I
(0
H H
crd
iH
0)o
1
[f 11
(U C,
m Qa)
0.38
A MSXa
0.35
0 EXP
SIVEY'S RELATION
0.32
0.29
S0.26
z
P 0.23 
< 0.20 A
0.17
0.14
4.0 5.0 6.0
NEARESTNEIGHBOR DISTANCE (A.U.)
Figure 4.6. The optical absorption energies of the
F center as a function of the nearest
neighbor distance of the crystal.
in the present (F center) case, Ivey's relation gives a good
approximation to the observed values for all alkali halides
shown in the graph. On the other hand, although the present
calculated F center energies for most of the alkali halides
are in good agreement with the experimental values, the
energy calculated for the LiF crystal is 0.231 Ry which is
about 0.13 Ry below the experimental value. If one checks
the similar U center calculation shown in Figure 4.2, one
finds essentially the same differences (compared to Ivey's
result) near the small nearestneighbor distances of the
crystal. This indicates that the errors in both cases
probably come from the same source. The possible causes are
the neglect of ionic overlap (caused by the muffintin
assumption) in the LiF case and the inadequacy of the
correction potential calculated from point charges as the
interionic distance becomes small.
Above all, the calculations using the eightcenter
cluster give good results on the F center absorption
energies for most of the ionic crystals. There are other
effects associated with the problem that are omitted in
the present calculation. For example, the lattice distortion
effect and the polarization effect are expected to be
more important in the present case than in the U center
case. The displacement of cations toward the vacancy will
increase the F center transition energy. But the severity
of the above mentioned effects is not known. Effects
associated with the exchange parameters and other parameters
have been discussed in the previous section; that discussion
is also applicable to the present case.
From the ground state calculation of the F center in
KC1, the wave function of the F center is obtained. With
knowledge of the value of the wave function at the neighboring
nucleus, the contact interaction term can be calculated
(see, for example, Seidel and Wolf, 1968). In the present
case the calculated value is about five times greater than
the observed value. The discrepancy may be the consequence
of using a finite size cluster.
The electronic structure of the ground state of the
F center in KC1 has also been calculated in the context of
the twentycenter cluster. The averaged value of the correc
tion potential at the centers of all the atomic spheres is
used as the correction potential inside the cluster. Thus,
VI(r) equals 8.32/d Ry for all r smaller than Rout*
Other parameters used in the calculation are the same as
those in a similar U center calculation except that the a
value for the central sphere is 0.72 in the present case.
The eigenstates calculated from this twentycenter
cluster are shown in Figure 4.7. A comparison with the
eightcenter cluster result shows that the inclusion of the
second nearestneighbor Cl ions introduces a band of 15
levels at about 0.55 Ry for each spin. These two bands
arise from the atomic 3p levels of the twelve chlorine ions
and have the same width (0.047 Ry). The F center electron
SPIN DOWN
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.3
1.4
1.5
1.6
tlu
alg
1 T
alg (F)
(2nn) _
(Inn)
(2nn) 
Figure 4.7.
The electronic structure of the F center
ground state in KC1 calculated by using
the twentycenter cluster.
SPIN UP
alg
tlu
_ t^u
K 3p
  C 3s
__________ Cl 3s
occupies an alg orbital at 0.254 Rv. As one can see from
the diagram, the spin polarization effects of the unpaired
F electron on the Cl 3p band and the K+ 3p and Cl 3s
bands below are relatively small as compared to the unoccupied
levels. It is also found that the F electron level is closer
to the unoccupied levels than in the eightcenter cluster
case.
A transition state calculation shows that the F center
absorption energy is only 0.124 Ry which is about 26% below
the experimental value. This is the same sort of situation as
that which happens in the twentycenter U center calculation.
The addition of the twelve Cl ions has apparently pushed
the ground state level of the color center too high. It is
believed that a more accurate correction potential for the
large cluster is essential, especially in the intersphere
region.
The ground state of the F center in KC1 has also been
calculated using the twentyeightcenter cluster. In the
calculation, two kinds of correction potential were used.
The first is defined in Equation (3.13); it is the averaged
correction potential inside the cluster. The electronic
structure which results from this calculation is shown in
Figure 4.8. The second correction potential is defined in
Equation (3.16) and Equation (3.17); in this scheme each
region has a separate correction potential. The calculated
electronic structure is shown in Figure 4.9. A comparison
SPIN DOWN
tlu
alg
alg
F)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
1.3
1.4
1.5
1.6
alg
tlu
alg
_K Cl 3p
C 3s
K+ 3p
Figure 4.8.
The twentyeightcenter cluster calculation
of the ground state of the F center in KC1
using the averaged correction potential.
0.0 r
(2nn) ____
(2nn) E LLfwu^
(3nn) 
(Inn) .. . .
SPIN UP
SPIN DOWN
tlu
ag
alg
alg
(F)
(2nn)
ali~=i~_i~i~= 3
= ". C 3p
= Cl 3
(3nn)  
(2nn)
(Inn)
K + 3p
C1 3s
K+ 3p
Figure 4.9. The twentyeightcenter cluster calculation
of the ground state of the F center in KC1
using the separate correction potential.
0.0
0.1 
0.2
0.3
U.4 
0.5 
0.6
0.7
0.8
1.3
1.4
1.5
1.6
r
SPIN UP
of the two calculations yields an energy difference between
the F center level and the first unoccupied tlu level of
0.178 and 0.182 Ry, respectively, for the two cases. These
energies are very close to the F center absorption energy
in KC1. The gap between the first unoccupied level and the
top of the valence band with spin up is 0.397 and 0.432 Ry
respectively. The slightly higher value for the second case
is a result of the correction potential used for the 2nn
atomic sphere being smaller than those for the central
sphere and intersphere region. The effect of the separate
correction for each region in the second calculation can
also be seen from the lower bands as compared to the first
calculation. In Figure 4.8 the highest of the three bands
shown is the 2nn Cl 3s band whereas in Figure 4.9 the highest
band is found to be the 3nn K+ 3p band.
A comparison of the electronic charges inside each
region and a comparison of the calculated potentials for
a spinup electron in the intersphere region and near each
sphere boundary (which result from several F center ground
state calculations in KC1) are both shown in Table 4.9. The
electronic charge inside each ion sphere is seen to increase
with the increase of the size of the cluster. On the other
hand, the charge in the intersphere region is seen to de
crease from the twentycenter case to the twentyeightcenter
case. The twentyeightcenter cluster has a total charge of
+1, thus, a relatively small correction potential is needed
TABLE 4.9
A comparison of the electronic charges and the
in each region of the cluster between several
ground state calculations in KC1.
potential
F center
8center 20center 28centera 28centerb
Qint 1.32 10.14 9.78 9.56
Qout 0.07 0.41 0.06 0.05
QF 0.56 0.43 0.62 0.61
QK+ (Inn) 17.85 17.95 17.98 17.97
QC (2nn) 17.19 17.29 17.33
QK+ (3nn) 17.89 17.88
Vint 0.18 0.42 0.32 0.35
V F 0.59 0.58 0.72 0.76
VK+ (Inn)c 0.57 0.73 0.80 0.81
VC1(2nn)c 0.56 0.59 0.67
VK+ (3nn)c 0.71 0.65
All energies are in Rydberg.
aUse correction potential in Equation (3.13).
bUse correction potential in Equations (3.16) and (3.17).
CValues given are potentials near the sphere boundary.
in the calculation as compared to that in a smaller cluster
calculation. It seems that this large cluster should give
better results for the present problem. The calculated F
center absorption energy in KC1 turns out to be 0.222 Ry
and 0.223 Ry using the two different kinds of correction
potential mentioned before. These energies deviate from the
observed value by about the same amount but opposite direc
tion of that found in the twentycenter calculation. This
oscillation of deviation with increasing number of shells
can also be seen in the position of the F center level in
various diagrams shown in Figures 4.5, 4.7 and 4.8. The
difficulty in the present case is traceable to the fact that
the F center excited state is not localized while the F
center ground state is fairly well localized in the central
region of the cluster. Thus, a small deviation of the rela
tive magnitude between the constant potential in the inter
sphere region and the potential inside the central sphere
(which is nearly constant for the case of the F center)
will affect directly the calculated transition energy.
Clearly, a more accurate evaluation of the correction poten
tial between spheres is needed. One method is to integrate
the potential in the intersphere region directly using the
method discussed in Appendix A or to use the calculated
averaged value inside the cluster combined with Equation
(3.17). Estimates have been done and they indicate that the
calculated transition energy will be improved in both the
twentycenter and twentyeightcenter cases.
From the above discussion, we know that the eight
center cluster calculations give reasonably good results on
the F center absorption energies. But the calculation of
the contact term for the spin interaction between the F
center electron and the neighboring ions indicates that the
small size of the cluster has compressed the wave function
of F electron and gives a contact term too high compared
to the experimental value. Thus, larger clusters are needed
to see the improvement on the F center wave function. From
the ground state calculation of the large cluster, the
contact term is improved, although the value is still too
high. On the other hand, the transition energy is found to
be worse than that obtained in a small cluster calculation.
Improvement of the correction potential has been discussed
before, but there are also other difficulties which perhaps
are associated with the large cluster. Thus, calculations
with a cluster of pure crystal have been done. A study of
these calculations can give some information about the
present cluster approach. Results of these calculations will
be given in section 4.5.
4.3 The U, Center in Potassium Chloride
The calculation of the U2 center in KC1 has been done
using the cluster shown in Figure 2.2. In this cluster a
hydrogen atom is surrounded by four K ions and four Cl ions.
Since this cluster is electrically neutral, no Watson sphere
is needed in this case. The parameters used in the calcula
tion are the same as those used in the previous U center
calculations except for the radius of the interstitial
hydrogen sphere which is choose to be as large as possible
without overlapping the other spheres. By this criterion,
the radius of the hydrogen sphere is 2.04 atomic units.
The calculated orbital energies of the ground state
of this center in KC1 are shown in Figure 4.10. The hydrogen
ground state is an a, orbital with spin up in the present
calculation. This hydrogenic al orbital is mixed with the
al orbital from the neighboring chlorine ions. From the
charge distribution of the orbitals, it is found that the
most hydrogenlike al orbital with spin up is below the group
of levels from the Cl 3p levels while the unoccupied
hydrogenic al orbital with spin down is above the chlorine
levels. Thus, in this spinunrestricted calculation, we
know that the system with two electrons in the hydrogen Is
levels plus a hole in the chlorine 3p levels has a higher
energy than that which has filled chlorine 3p levels and one
electron in the hydrogen Is level. This situation was
unclear from the energy considerations of Cho et al. (1966),
although the same conclusion was reached.
In finding the excitation energy of this center, there
are two possible forms of transition. The first possibility
is an electron excited from the highest occupied a symmetry
orbital, which is a mixture of the hydrogen Is level and the
0.0
0.1
0.2
0.3
G> 0.4
z
0.5
0.6
0.7
Figure 4.10.
SPIN DOWN
3t2
3a1
2a1
2t2
Itl
it1
le
It2
3t2
3a1
2a1
2t2
it1
(Cl 3p) le
it2
1a1
lal
The electronic structure of the ground state
of the U2 center in KC1.
SPIN UP
neighboring ClI 3p levels, to the first unoccupied t2 orbital.
The second possible form of excitation is to excite an electron
from the highest occupied spin down t2 orbital, which is an
orbital concentrated on the four Cl ions, to the unoccupied
hydrogen al orbital. The excitation energies calculated from
these two kinds of transition are 0.395 and 0.213 Ry, respec
tively. Comparing these results to the experimental U2
absorption energy, 0.387 Ry (Fischer, 1967), one finds that
the first form of transition is likely to be the process.
In most of the previous calculations on optical transitions
in a U2 center (Mimura and Uemura, 1959; Hagston, 1971), the
absorption process was thought to be a charge transfer
which is equivalent to the second form of the transition in
the present calculation. There is no direct proof of this
charge transfer process in a U2 center absorption. In order
to see the effect of the possible lattice distortion around
the hydrogen atom upon the transition energy, a calculation
of the charge transfer process with the Cl ion and the K
ion displaced outward and inward, respectively, along the
line joining the ion and the hydrogen atom was carried out.
The result shows an even smaller energy for the U2 center
absorption. Thus, from the present calculations the process
of charge transfer cannot account for the observed absorption
energy, and the process with an electron excited from a
hybrid chlorinehydrogen al orbital to a higher state gives
a good interpretation for the U optical absorption.
2
The proton hyperfine interaction of the U2 center in
KC1 has been investigated by many workers in this field.
Spaeth and Seidel (1971) used the orthogonalized function
method and obtained a relative proton hyperfine constant
shift 5 = 9.27%, where 6 is defined by
aHi aH
6 = (4.1)
aH
and aHi is the proton hf constant for the U2 center while
aH is the hf constant for a free hydrogen atom. Since the
experimentally determined aHi (1378 MHz) is less than the
free hydrogen hf constant (1420 MHz), the relative proton
hf constant shift is equal to 3.01% (Spaeth and Sturm,
1970). To improve the calculated value, they tried to take
into account the van der Waals interaction between the H
atom and its neighboring ions. This has the effect of shifting
electron density away from the proton into the outer region
of the H atom resulting in a value of 6 = 9.73%. Another
model to improve the calculation of 6 is to include the
crystal field effects using the ligand field model. In this
way, Hagston (1970) obtained a value of 6 between 1% and
2% in a semiempirical calculation.
In the present calculation, the proton hf constant is
calculated from the spin density at the position of the
proton which can be calculated from the following expression:
( Z 1 r) 2 rLd I 'ir 1(4.2)
Mt blt
where n. is the occupation number of the ith state and
Yit (r) is the wave function of the ith state at position r.
The summation of i is over all states with spin up and the
summation of j is over all states with spin down. Using the
spin density at the proton, the calculated 6 is equal to
4.5% which is in fairly good agreement with the experimental
value.
The calculation of the hf contact term at the nuclei of
the Cl and K ions has also been done using the spin density
obtained from Equation (4.2). Comparing with the experimental
results the calculated hf contact term for the Cl ion is
overestimated and that for the K ion is underestimated and
is negative. In order to see the effect on the distribution
of the spinpolarization due to a particular choice of the
cluster, a different cluster for the U2 center in KCl was
used. This cluster has a K ion at its center with six
Cl ions surrounding it and also has a hydrogen atom located
at (d/2, d/2, d/2) of the cluster, where d is the nearest
neighbor distance of the crystal. The structure of the cluster
is similar to that shown in Figure 2.1 except that there is
an interstitial atom in the present case and the cluster has
C3v symmetry. The calculated hf contact term for the K+ ion
in this case is improved and has a value near zero. The hf
contact term for the Cl ion is lowered. Thus, overall the
cluster with C3v symmetry has improved results for the hf
contact term compared to the tetrahedral cluster, but the
results are not satisfactory. Since in the present calcula
tion the spin polarization is mainly within the outer sphere
of the cluster in contrast with the experimental results that
show spin polarization in the third shell of ions, a larger
cluster may be used to improve the results. On the other hand,
the hf contact term of the nearestneighbors of the inter
stitial hydrogen can also be calculated approximately using
the oneelectron wave function of the unpaired electron from
the tetrahedral cluster calculation. The hf contact term at
the Cl nucleus in the first shell is found to be 18.0 MHz
which is roughly comparable to the experimental value of
23.7 MHz. At the K nucleus, the calculated hf contact term
is 3.1 MHz and the experimental value is 1.0 MHz. These
results are better than the results obtained using Equation
(4.2). But the calculated spin density at the proton position
using the oneelectron wave function is found to be too small.
The reason for this low spin density is probably the fact
that the hydrogenlike al orbital is mixed with the al orbi
tal from the neighboring Cl ions shifting the spin density
away from the proton.
4.4 The U1 Center in Potassium Chloride
The U1 center is similar to the U2 center discussed in
the previous section except that the interstitial impurity is
a negative hydrogen ion in the present case. The study of
this center in KC1 has been done using the tetrahedral
cluster shown in Figure 2.2. A Watson sphere with charge of
+1 was used to enclose the whole cluster.
The calculated orbital energies are shown in Figure
4.11. Contrary to the case of the U2 center, both of the two
hydrogenic al orbitals are above the group of five levels
arising from the Cl 3p levels. The optical absorption
process in this case is thought to be an electron excited
from the hydrogenic al orbital to the first unoccupied t
orbital. The transition state calculation has been done
and shows a transition energy of 0.18 Ry. There are a few
experimental measurements of the absorption band of the
U1 center in KC1 (Delbecq et al., 1956; Rolfe, 1958), but
the peak of this band generally is not clearly defined.
It is estimated that the peak of the absorption band is at
about 0.32 Ry and the edge of this band is about 0.26 Ry.
In the case of this interstitial impurity, the interac
tion between the hydrogen ion and the neighboring ions is
believed to be greater than that of the previously discussed
centers. Consequently, the displacement of ions with respect
to their equilibrium positions is more important. Thus,
two more calculations with displacementswere carried out.
In these calculations, the chloride ions were allowed to
displace outward along the cubic diagonals and the potassium
ions were allowed to displace inward along the cubic diagonals.
The hydrogenlike level was found to shift down because of
the relaxation of ions surrounding it, and the resulting
transition energies were 0.22 Ry and 0.25 Ry with the ions
displaced approximately 5% and 8% of the nearestneighbor
S94
SPIN UP
SPIN DOWN
(U,)
 (Cl 3p)
Figure 4.11.
The electronic structure of the ground
state of the U1 center in KC1.
3ao
3 t2
3a
0.0i
0.1
0.2
0.3
0.4
0.5
t2
It1
le
It2
la1
2t
it1
le
it2
lal
S20,
distance from their crystal equilibrium positions, respec
tively.
Thus, from the above results, we know that the
displacement of the neighboring ions of the hydrogen played
an important role in the U1 center structure. Since the
actual amount of displacement or other distortions of the
crystal lattice are not known, we can only qualitatively
understand the problem. For more accurate calculations, the
polarization effect due to the extra electron on the hydrogen
and the more general form of potential must be considered.
4.5 Cluster Calculation of a Pure KC1 Crystal
In the previous sections of this chapter we have
discussed the calculation of several color center problems
using clusters of several different sizes. In section 3.5
we have also discussed the correction potential needed to
simulate the potential which arises from the rest of the
crystal when the cluster is embedded in the solid. From the
calculations on U centers and F centers we know that the
orbital energies of the cluster are affected by the correc
tion potentials which are calculated approximately from
point charges. It is interesting to compare the electronic
structure from a calculation on a pure alkali halide crystal
using the present cluster method with that from an energy
band calculation or experimental data, e.g., band gap, band
width, etc. of a pure crystal. In this way some insight
