The energy band structure of ferromagnetic nickel

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Title:
The energy band structure of ferromagnetic nickel
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vii, 103 leaves : ill. ; 28 cm.
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Connolly, John William Domville, 1938-
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Ferromagnetism   ( lcsh )
Nickel -- Magnetic properties   ( lcsh )
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Thesis--University of Florida.
Bibliography:
Bibliography: leaves 100-102.
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by John William Domville Connolly.
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Manuscript copy.
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Vita.

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University of Florida
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Full Text












THE ENERGY BAND STRUCTURE

OF FERROMAGNETIC NICKEL











By
JOHN WILLIAM DOMVILLE CONNOLLY


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY











UNIVERSITY OF FLORIDA
August, 1966

























DEDICATED TO MY WIFE















ACKNOWLEDGMENTS


I would like to acknowledge the help afforded by ny super-

visory committee, and in particular by Professor J.C.Slater, who

suggested the problem upon which this dissertation is based.

Thanks are due to many members of the Quantum Theory Project

at the University of Florida,including Drs. S.O.Goscinski and K.H.

Johnson for many piquant discussions, and especially to Dr. J.B.Conklin,

Jr. for his assistance with the computer programs and constructive

criticism of the manuscript.

I would also like to express my appreciation to Prof. J.H.

Wood of M.I.T. for his generosity in providing several computer pro-

grams, and to the University of Florida Computing Center for the use

of their facilities.

Finally, my continuing gratitude goes to my wife, Charlotte,

whose patience and understanding were exceeded only by the excellence

of her typing of the manuscript.


iii
















TABLE OF CONTENTS


ACKNOWLEDGMENTS . ....

LIST OF TABLES . .. .....

LIST OF FIGURES . ..


Page
. iii

S. .. vi

. vii


Chapter


I. INTRODUCTION . ..

1.1. Purpose . .
1.2. Outline of Calculation .

II. METHODS OF CALCULATION . .

2.1. The Unrestricted Hartree-Fock Method.
2.2. Exchange Approximations ..
2.3. Superimposed-atom Potential .
2.4. The Energy Band Calculation .
2.5. The Self-Consistent Calculation .

III. RESULTS OF THE CALCULATION .


. e .


. a .


1

1
3

8

8
12
18
23
31

35


3.1. The Self-Consistent Spin-Dependent Energy Bands in
Nickel. . 35
3.2. Comparison with Previous Calculations 44
3.3. Comparison with Experimental Data . 53

IV. CONCLUSIONS. . . 63

Appendix

I. APW MATRIX ELEMENTS . ... 70

II. RECIPE FOR THE CONSTRUCTION OF A LINEARLY INDEPENDENT
BASIS SET . ... 75

III. APW CHARGE DENSITY. . 84

IV. DERIVATION OF THE COULOMB POTENTIAL FROM A GIVEN
CHARGE DENSITY.. . .. 88











V. DETERMINATION OF THE DENSITY OF STATES 94

VI. NOTATION .................... 97

BIBLIOGRAPHY .................. .100

BIOGRAPHICAL SKETCH . . .103














LIST OF TABLES

Table Page

2.1. Hartree-Fock one-electron energy parameters for
Fe (3dd)5 (3dJ)3 (4s)2 ... 17

2.2. Hartree-Fock one-electron energy parameters for
Ni-+ (3dd)5 (3dp)3 . . ... 19

3.1. Coordinate grids in reciprocal space . 37

3.2. Comparison of energy differences for various potentials. 43

3.3. Detail at an L-point . .. 46

3.4. Results of Snow, Waber and Switendick for paramagnetic
nickel . . .. 48

3.5. Charge distributions . . 52

3.6. Comparison of calculated and experimental quantities 62

A2.1. Frequency numbers for the fcc structure . 81

A5.1. A list of Kubic harmonics . .. 95
















LIST OF FIGURES


ure

1.1. Schematic diagram of energy levels in nickel .

1.2. Flow-chart for the self-consistent calculation .

2.1. Core polarization for the Ni4+ ion ....

2.2. Extrapolation of a new starting potential .

3.1. Convergence of the L-point energy values .

3.2. Self-consistent energy bands for V3 (Vx = x) .

3.3. Self-consistent energy bands for V4 (Vx ,V) .

3.4. Detail of the energy bands at an L-point for the
two self-consistent potentials .

3.5. Comparison of transition element energy band
calculations . .

3.6. The density of states curves ..

3.7. Fermi surface cross-sections for V4 ..

A4-1. The Ewald problem . .


Page

. 5

. 7

. 20

. 33

. 38

. 40

. 42


. 45


. 50

. 55

. 58

. 89


vii


Fig














CHAPTER I


INTRODUCTION


1.1. Purpose

The properties of ferromagnetic solids have interested

physicists for many years, and are of sufficient complexity that a

comprehensive theoretical treatment is not yet a reality. The pur-

pose of this dissertation is to apply the unrestricted (or spin-

polarized) Hartree-Fock method to such a solid. This scheme has had

a certain amount of success in the explanation of atomic properties,

and, as will be shown here, appears to be an acceptable model for a

ferromagnet.

Experimental information on the metallic properties of

ferromagnets, which are dependent on the electronic configuration,

has recently become available, anrd with it some analyses to explain

these properties on the basis of the energy band model. Of the

ferromagnetic materials, nickel is perhaps the best understood.

Energy band:,models have been presented, that explain empirically

the available data on the electronic and optical properties of

nickel. Also within the last five years, several papers have appear-

ed in the literature,5-10 which give energy band structures calcu-

lated from basic considerations, i.e., they present solutions for

Schrodinger' s equation in a crystal using some form of one-electron










potential. This potential, in all cases, has been derived from

an atomic calculation, corrected for the effects of placing the

atoms on a crystalline lattice.

Although these previous calculations are qualitatively

quite similar, the arbitrariness of the potential used is enough

to create differences between them which are large with respect

to the experimental effects which are to be explained. The calcula-

tion described in this dissertation attempts to eliminate this

dependence on an arbitrary potential by solving the equations self-

consistently, in the same way as the Hartree-Fock method used in

atomic calculations. In this way, it is possible to examine the

validity of the approximations which must be made in order to solve

the equations.

The case of ferromagnetic nickel turns out to be extremely

sensitive to slight changes in these approximations. In particular,

the form in which the exchange effects responsible for the ferro-

magnetic structure are inserted into the theory can radically change

the final results. This effect was not pointed out in previous

calculations on nickel or other materials, either because the solu-

tions were not carried to self-consistency or because the particu-

lar case was not sensitive enough to show a definite discrepancy

with experiment. In a sense, then, a calculation on nickel is a

test case, which examines the validity of an approximation which has

been used in many calculations of this type.

The procedures used in this work are outlined in the










following section, and described in more detail in Chapter II. The

results and comparison with the 'experimental data are presented in

Chapter III followed by the conclusions and discussion in Chapter IV.


1.2. Outline of Calculation

The one-electron model for a crystal consists of the assump-

tion that the electronic wave functions satisfy a Schrodinger equa-

tion of the form

(- v' VU ) *C/(f^U c = k(c^ (1.1)

where V(r) is a potential identical for all electrons, and includes;

(i) an attractive term due to the nuclei situated on the
lattice sites,

(ii) a repulsive term due to the Coulomb interaction with the
other electrons, and

(iii) an attractive term which similutates the exchange effects
of other electrons.

As a first approximation, V(7) can be generated from the

assumption that the electronic wave functions are unchanged from their

atomic values. The atomic potentials on the appropriate crystalline

lattice sites are then overlapped to form what has been called a

"superimposed-atom potential". This, of course, makes V(?) dependent

on the particular atomic configuration chosen. Although the most

logical choice would be the ground state configuration of the atom,

this is not always the best choice. It is true that the wave functions

do not change greatly on going from a free atom to a crystalline

environment, but the effective occupation number of each type of

orbital (s,p,d, etc.) may change. The reason for this is that the










discrete atomic levels which the electrons occupy in a free atom broad-

en into bands when these atoms come together to form a solid. If these

bands happen to overlap, as they do in many metals, then electrons in

one band will "spill over" into the other, thus reducing the effective

number of the first type in favor of the second. This occurs in the

case of nickel (figure 1.1.), in which the overlap of the 3d and 4s

bands alters the configuration such that the effective number of

"d-like" electrons is increased from the atomic ground state value of

8, to approximately 9 in the solid.

However, in a self-consistent calculation, where the potential

is regenerated after each iteration, the arbitrariness in the choice

of V(?) disappears. The superimposed-atom potential is used only as

a starting point for the self-consistent procedure.

Once the initial potential is chosen, the energy bands are

calculated for a selected number of vectors in reciprocal space by

means of the augmented plane wave (APW) method.11l13 This method has

beensufficiently developed and checked out against other methods 14-16

so that it gives a solution (for a given potential) as accurate as

desired. In its simplest form, the APW method solves equation (1.1)

for a V(f) of the "muffin-tin" type, i.e., spherically symmetric in

spheres about each lattice site-and constant between these spheres.

This is not a necessary restriction, 17 but it can be removed only at

cost of much greater complexity, without much improvement in the final

results.

The energy eigenvalues obtained in this way are then used to

find the associated wave functions. The total charge density is











Cf

I z

d)


c

4-.-- 0
- I
LJ *o


0>
SJ
lit)










generated from these wave functions, by summing over the occupied

levels. The procedure for forming the new muffin-tin potential to
17, 18
be used for the next iteration is straightforward, 1718 with a

certain amount of care necessary in the choice of the intersphere

constant. The iterations are continued, going through the above

manipulations each time, until there is no significant difference

between the energies obtained from two sucessive iterations.

Each of these steps is described in detail in Chapter II,

and the entire calculation represented by the flow chart in figure

1.2.











ASSUMED ATOMIC
CONFIGURATION



UNRESTRICTED
HARTREE-FOCK
PROGRAM FOR
ATOMS


ATOMIC
ORBITALS


SUPERIMPOSED-ATOM
POTENTIAL
(MUFFIN-TIN FORM)


NON-REIATIVIS'
APW PROGRAM



ENERGY BANDS


RECIPROCAL
LATTICE VECTORS
(DERIVED FROM
GROUP THEORY)


&
WAVE FUNCTIONS


SPHERICALLY-
AVERAGED
CHARGE DENSITY


I
NEW MUFFIN-TIN
POTENTIAL (FROM
EWALD METHOD)


SELF-
CONSISTENCY


Figure 1.2. Flow-chart for the self-consistent calculation


1













CHAPTER II


METHOD OF CALCULATION

2.1. The Unrestricted Hartree-Fock Method
The Hartree-Fock formalism consists of approximating the

wave function for an N-electron system by a single antisymmetrized
product of one electron functions, ui of space and spin coordinates

xi = ('r, si)


(2.1)


For a Hamiltonian consisting only of electrostatic interac-
tion and kinetic energy terms, the variational principle leads to
N coupled integro-differential equations of the fbrm 19


L4. J i1i' .a=l J


l/ sp-s] i(, ,,12. 11]

[I/ Sps [ i +, 2, .E. =1 J


(2.2)


where is the one-electron kinetic and nuclear potential operator.
The second summation is restricted to one-electron states (j) with
spin parallel to state (i), whereas the first is over all the elec-
trons including the ith state itself.


1E _== 4 at(x,) UL(X ) ... tLI(Xy .








These equations can also be written in the form

[- V ,() + Ve ()] U i) (2.3)




where Vn ( i) = (atom)

2?
Z, (solid)


is the nuclear potential term,

v (- F! d


is the Coulomb potential, and

S(S;,, ) A I
v/ 1 /"]
[spin sJ
is the "exchange potential". Note, that Vn and V are "local" poten-
c
tials, i.e., they appear in the equations as a multiplicative ope-
rator acting on ui, but Vx(s) is not. It is this peculiarity of the
equations which causes the greater part of the difficulty in their
solution. The exchange potential can be expressed as a local operator,
but with the result that it will have a different form for each
orbital:


[II spins]









The equations are solved by an indefinite number of iterative

steps. Some initial set of ui's is chosen, Vc and Vx are generated

from them, and the resulting differential equations are solved for

a new set of ui's. The procedure is then repeated for this new set

until the final ui's are the same as the initial set (i.e., "self-

consistency" is achieved). Fortunately, for an initial set of ui's

close enough to the self-consistent solution, this process is conver-

gent.

Because of the difficulties involved in solving the three

dimensional integro-differential equations (2.3), their solution

involves a major computational effort, and calculations involving

heavy atoms are few. In practice, the Hartree-Fock formalism is

often modified by assuming the potentials in equations (2.3) are

spherically symmetric. (Here we are referring to the case of a free

atom. The equations for a solid will be discussed in subsequent

sections of this chapter.) This allows us to separate the wave

function in the form


x = P .^ (2.5)
J-.
where Y is a spherical harmonic with angular momentum quantum

numbers t and m and (s) is a spin function with spin quantum

number, s = 1- This one assumption (known as the central field

approximation) effects enormous simplification in the equations, in

that all integrals and differential equations become one-dimensional.

In particular (2.3) reduces to

S + P) =- (2.6)
;, I^ Ci 9 L L









where VC -2Z 4 fl p() dk + 2 J,2 (rq' j .?
7 o k-

The exchange potential Vx(r) is slightly more complicated than Vc(r)

involving an extra summation over integrals which contain Pi(r) as

well as P (r). (See reference 19, page 17.)

In general, Pi will depend on 1, m, and s, as well as the

principal quantum number n which labels the eigenvalues of (2.6) in

increasing order. In the so-called "conventional" Hartree-Fock

scheme, Pi is assumed independent of m and s, resulting in a degene-

racy of 2 (21+1) for each energy eigenvalue, thus reducing the

number of equations to be solved. This has the advantage of giving

a final wave function (2.1) which is an eigenfunction of the total

angular momentum L2 and total spin operator S2. However, this wave

function may give entirely false results in interpreting results in

which spin effects are important. For example, consider a transition

element, for which the ground state has two 4s electrons and an

unfilled 3d shell. The conventional Hartree-Fock solution in this

case would have a zero spin density(i.e., the difference between the

charge densities of opposite spins) near the nucleus. But the inter-

pretation of the hyperfine structure of an atom with non-zero nuclear

spin demands that the spin density be non-zero. This sort of effect

is the justification for the use of "unrestricted" Hartree-Fock

sbheme.20'21 The term "unrestricted" refers to the relaxation of

the restriction that Pi be independent of s. The number of equations

to be solved is doubled, giving a (21+1) fold degeneracy for









each ns. For a configuration which has an unequal number of up

and down spins, such as an unfilled 3d shell, the relaxation of

the constraint will result in a non-zero spin density. This desira-

ble feature of the wave function is gained at the sacrifice of the

symmetry requirement that (2.1) be an eigenfunction of S2. However,
22
it has been shown that the unrestricted wave functions are in

substantially better agreement with experimental data. It also should

be noted here that the unrestricted solution can be made an eigen-

function of S2 by the application of projection operators, 23 but

such a complication is beyond the scope of the present calculation.

Both the conventional and the restricted forms of the Hartree-Fock

must be regarded as approximations whose error is impossible to esti-

mate without doing the actual exact calculations. Since these are

intractable at the present time for atomic systems with more than a

few electrons, appeal must be made to experiment to judge the use-

fulness of the different forms. In particular, as will be seen, the

conventional scheme is much less suitable than the unrestricted for

the case of a ferromagnetic solid.


2.2. Exchange Approximations

The exchange potential of equation (2.4) for a particular

orbital ui involves a sum over all electrons uj of spin parallel to

that of ui, of integrals involving both ui and uj. Looking at the

situation from a one-electron point of view, this means that each

electron moves in a different effective potential. For a large









number of electrons, this results in formidable computational

difficulties even when the central field approximation is used.

These difficulties are not insurmountable for a free atom, but are

virtually impossible to overcome for the case of a solid. Also, we

are interested in solving the problem within the framework of the

one-electron model, in which all electrons move in the same effective

potential. This simplification has enjoyed a great deal of success

despite its obvious limitations, and represents the present limit

of complexity to which calculations in solids can be carried.

What is clearly indicated, then, is the formation of some

kind of average exchange potential which will retain the main fea-

tures of equation (2.4) One such average has been suggested by

Slater, 24i.e.,


'C 2' u.(i-)Au (2.7)
[II sp ,s] i,/

which is simply the sum of Vx weighted by the probability factor,



There is one particular case for which the expression (2.7)

can be evaluated exactly, viz., the free-electron gas. If the free-

electron wave functions ui= exp(dii~ ) are substituted into (2.7),

the integral can be easily performed, 19 and the sum over i up to

the Fermi level gives

X )(2.8)









where is the constant electron charge density (in units of the

electronic charge).

In an atom, of course, the electrons are far from free, and

the charge density is not a constant, but if we take the extreme

step of assuming the exchange interactions between electrons in an

atom ( or a solid) depend only on the local electronic charge

density, then a function is obtained which turns out to be remarkably

similar to (2.7). Explicitly this function is

V) -Y 11] (2.9)


In at least one case, that of the ion Cu+, a detailed

comparison of the approximations (2.7) and (2.9) with the exact

Hartree-Fock expression (2.4), has been worked out. 25 The results

show that the Slater average (2.7) and the free-electron average

(2.9) are practically idential. At least, the deviations of (2.7)

from (2.9) are much less than the deviations of either from the

Hartree-Fock value (2.4). The point to be drawn here is that there

is no advantage in using (2.7) instead of (2.9), and that if one is

forced by practical considerations into using a local exchange poten-

tial, then the errors involved can be ascribed to the averaging pro-

cedure rather than the use of the free-electron approximation.

Recently, Kohn and Sham 26 by the use of a different averaging scheme

have concluded that (2.9) should be multiplied by a factor of 2/3.

Calculations on various atoms 27,28 have shown that perhaps this is

closer to the truth, in that the resultant wave functions are more

similar to the Hartree-Fock values. Lindgren 29,0 by using a









parameterized one-electron exchange potential and minimizing the

correct total Hartree-Fock energy with respect to these parameters

gets amazingly good agreement with the best Hartree-Fock values.

This "optimum" potential, he finds,31 is always smaller in magnitude

than (2.9). The consensus appears to be that, although the free-
-fe
electron approximation Ve reproduces the general features of the

Hartree-Fock scheme, it overestimates the exchange effects.

The extension of this approximation to the unrestricted

case is straightforward. In the course of the derivation of (2.8),

half of the electron gas was assumed to have up spin, and the other

half down spin. This introduces an extra factor of 2 in equation (2.9)

when we replace the total charge density by that of only one spin.

The effective exchange potential is then


X4 GP) r, )-1 (2.10)

where the summation now contains the orbitals of one spin only. This

is the approximation used in this calculation. Substitution of (2.10)

into (2.6) gives the actual equations to be solved:



where V () = 2 f(r)d/ 1 2 d
0



,^ ) = P )/ r7 s = ,
(2.11)
f'(')= ^ <^ r} + t (r)










Wnas is the number of electrons in the configuration with quantum

numbers n and f, and spin s.

These equations are exactly the same, with the exception

of the added spin subscript, as those used by Herman and Skillman

in their atomic calculations.32 Their program, which was published

with their calculations, was modified for the unrestricted case

during the course of this work. As a test case, this program was

used to find the wave function for the Fe atom in the configuration

(3dd)5 (3dl)1 (4sd)1 (4sf)1, in order to compare with the results

of Wood and Pratt,21 who did exactly this calculation. A comparison

of the one-electron energies is shown in table 2.1. The energy values

in the unrestricted case tend to bracket those of Herman and Skillman

and the splitting are larger for the n=3 electrons than for the

other shells, as would be expected. Apart from a large discrepancy

(almost certainly an error in their paper) at the ls level, the

results are comparable to those of Wood and Pratt. The results are

not identical in that their calculation was not taken out to as

great a degree of self-consistency.

At first glance, the discrepancy between the energies calcu-

lated using the free electron approximation and the exact Hartree-

Fock values seems discouragingly large. However, this does not mean

that the corresponding wave function is inaccurate to the same degree.

As Lindgren30 has pointed out, Koopman's theorem no longer holds

when the Hartree-Fock equations are solved approximately, and

therefore the one-electron energies no longer have the same














Table 2.1. Hartree-Fock one electron energy parameters for
Fe (3do)5 (3df)3 (4s)2

Exact Approx. Approx. Apprx.
CHFa CHFb UHFc UHF


Is -522.7e -515.8 -584.5 -584.2 -515.8 -515.8

2s -63.84 -60.96 -61.08 -60.40 -61.03 -60.75

2p -54.79 -53.08 -53.16 -52.65 -53.12 -52.91

3s -8.308 -7.269 -7.463 -6.930 -7.492 -6.929

3p -5.455 -4.891 -5.061 -4.540 -5.111 -4.558

3d -1.271 -0.963 -1.122 -0.664 -1.168 -0.664

4s -0.510 -0.545 -0.532 -0.428 -0.592 -0.491

a Exact conventional Hartree-Fock values, taken from Watson33.
b -fe
Approximate conventional Hartree-Fock values (Vx = Vx ) taken
from Herman and Skillman32.

c Approximate unrestricted Hartree-Fock values (Vx = V ) taken
from Wood and Pratt21.
d Approximate unrestricted Hartree-Fock values (Vx = Vf) calcu-
lated during the course of this work.
e All energies are in Rydbergs.









significance of being equal to the binding energies. By applying

the appropriate corrections, he has shown that binding energies

can be extracted from the approximate solutions which are in much

better agreement with the exact values.

To further examine the method, a calculation was done for

the ion Ni+ in the configuration (3dU)5 (3df)3 for which an exact
20
unrestricted Hartree-Fock solution is available. The results are

shown in table 2.2,both for the standard exchange and for one 2/3

as large. Note that the discrepancies with the exact energy values

are larger in the latter case, but that core polarization (figure 2.1)

is in better agreement. This serves as an illustration of the fact

that inaccuracies in the one-electron energies are not necessarily

indicative of inaccuracies in the wave function itself.


2.3. Superimposed-Atom Potential

In the preceding section we have shown how to obtain a

one-electron potential valid for electrons in an atom. We are now

faced with the problem of how to extend this method when the atoms

come together to form a solid. Suppose we make the rash assumption

that the electron charge density in the solid is unchanged from the

atomic values. If J (r) is the spherical charge density of spin s

in the atom, derived from a self-consistent calculation, then the

charge density in the solid (A ) is just a sum,


(2.12)
i
















Table 2.2. Hartree-Fock one-electron energy parameters for
Ni (3do()5 (3df)3


Exact UHFa Approximate UHFb
Vx SO Vx 3 x



Is -612.6 -612.6 -605.3 -605.3 -597.2 -597.2

2s -77.30 -77.15 -74.20 -74.04 -72.17 -72.08

2p -67.35 -67.21 -65.47 -65.35 -63.27 -63.20

3s -11.35 -10.96 -10.19 -9.901 -9.477 -9.294

3p -8.200 -7.725 -7.445 -7.158 -6.762 -6.583

3d -2.898 -2.705 -2.782 -2.519 -2.266 -2.111


a Exact unrestricted Hartree-Fock values, taken from Watson and
Freeman.20
b Approximate unrestricted Hartree-Fock values, calculated during
the course of this work.
c All energies are in Rydbergs.










































































o oN o o <

I I


0
C..










0
r4
a







U)

0~









where ai is the position vector of the ith lattice site. We consider
a particular case here, that of only one atom in a unit cell
The Coulomb potentials defined in (2.11) will also combine linearally;

v^?V C ( (2.13)

where Vc(r) is the atomic Coulomb potential and the exchange is

=' 4 (2.14)

For computational reasons, which will be made clear in the
next section, we wish to have a potential which will have the so-
called muffin-tin form, i.e., spherically symmetric in non-overlapping
spheres of radius R about each lattice site, and a constant between
these spheres. To find the spherical average <(J> of the charge
density (2.12), we integrate over all angles to get

< > -n, = 1 j 2, J si+ (L 2-,.S
i. 0
ai + (2.15)





where ni is the number of neighboring atoms at distance ai.
The spherically averaged Coulomb potential same form, and by substituting (2.15) into (2.14) we can approximate
the spherically averaged exchange potential T( .
The constant intersphere potential V0s is defined by

S /
.A
0









where 12 = J.I L R3 is that part of the unit cell which
3
lies outside the spheres. This can be evaluated in terms of the

spherically averaged quantities by splitting up the integral over

V, (r) into separate parts over. and the sphere J, i.e.,





= 4 T Jf4Ve(r)& + e' ( v'>d .
0 1-
Finally we have
o f,, (2.18)

a 0 6



00
is the charge of sin in the volume J2 and is the fraction

of electrons with spin s.

This scheme affords a straightforward method of deriving a

one-electron potential suitable for a solid from the results of a

self-consistent atomic calculation. The sum over neighbors in equa-

tion (2.15) typically converges after two or three terms, and the

integrals involved, being all one-dimensional, present no difficul-

ties. The sphere radius R is arbitrary, but better convergence in the

energy values is obtained in the solution of the energy band problem

if R is taken as large as possible, i.e., equal to one-half the

nearest neighbor distance. For the face centered cubic structure with

one atom per unit cell, R= a/242, where a is the "lattice parameter",










the distance between second nearest neighbors.

There are two basic difficulties with the potential descri-

bed here:

(1) The neglect of the non-spherical terms in the potential.

This can be remedied by adding them in as a perturbation after a

solution is found. However, if these terms are not negligible, formi-

dable complexities are introduced into the solution.

(2) The assumption that the charge density is unchanged from

its atomic value is not always valid. The orbitals will certainly be

distorted to some degree by the influence of neighboring atoms. This

would be much more noticeable in a metal, where the wave functions

have a substantial overlap, than in an insulator where the charge

density is more confined. It may be possible to choose an atomic

configuration other than the ground state which will generate a

charge density close to the actual charge density in the solid, but

this is impossible to determine non-empirically.

However, this problem will not concern us in this work,

since we will be finding a self-consistent solution. The superimposed-
/
atom potential as described here is used only as a logical starting point

for the first iteration.


2.4. The Energy Band Calculation

Within the framework of the one-electron model for a solid

containing N atoms and ZN electrons, the total electronic wave func-

tion is an ZN by ZN determinant of functions 4i, each of which

satisfy an equation of the form












[- V2 + V ( )] VL = E- 4 (2.19)


where Ei are the allowed energy levels. The potential V(P) includes

the interactions with the N nuclei, which are assumed to be in fixed

positions on the lattice sites, and simulates the interactions with

all the other electrons. Effectively, the problem of solving the

Schr8dinger equation for the (Z+1)N particles has been replaced by

an easier one, in which the ZN electrons are pictured as non-inter-

acting particles moving in a fixed potential. In this picture, the

Fermi statistics hold, so that the ground state of the system is

obtained by choosing the solutions of (2.19) corresponding to the

lowest ZN values of Ei.

For a solid with a ferromagnetic structure, the problem is

generalized slightly, in that the electrons are divided in two

separate systems, according to their spin, which interact differently

with their environment. In the one-electron model, this means solving

two equations:

[-V V )]E (2.20)


The lowest ZN eigenvalues of the combined set of solutions
will represent the ground state. Since the potentials V. and Vg

are unequal, the two sets Ei and Ej will be different. This will, in

general, imply an excess of one spin over the other in the ground

state, which is found experimentally in a ferromagnetic material.









Once this model has been chosen, we are left with the pro-

blem of solving the equations (2.20)..This can be made feasible by
the application of the symmetry properties of the Hamiltonian de =
- V+ Vs(r).
First, the translational symmetry of RL allows us to choose
the eigenfunctions i*(r) to be Bloch functions, i.e., to satisfy the
relation

-l( Jt ) = e ') (2.21)

where aj is a lattice vector, and k is a real vector which takes on
N values corresponding to the N one-dimensional representations of
the translation group. In the language of group theory, a Bloch
function is a basis function for the k-th representation of this
translation group T.
The reciprocal lattice vectors g are defined by the rela-
tion

lw n (2.22)


where the nij are integers.
If Ik> is a Bloch function belonging the k-th representation
of T, then it follows from (2.21) and (2.22) that iZ+ i> also
belongs tothe same representation, and further that I and Ik')
belong to the same representation if and only if k-k' is a reciprocal
lattice vector. We define the Brillouin zone as that set of N vectors

ko in k-space satisfying the relation:










Sj T V Tr (2.23)

Every vector in k-space is equal to the sum of a "reduced" vector
to (in the Brillouin zone) and a reciprocal lattice vector.
Suppose, we have available a complete set of Bloch functions
( > corresponding to all possible vectors in k-space, then 4i can
be expressed as a linear combination of these functions. But, if 4L.
is to satisfy (2.21), then only those functions which belong to the
th representation will contribute to it. Therefore lt. can be expres-
sed as a linear combination of that subset of the complete set which
has the same reduced vector, i.e.,

SCo (2.24)

is the most general eigenfunction corresponding to the irreducible
representation associated with the reduced vector ko.
A secular equation for a particular ko can now be set up by
substituting (2.24) into (2.20):

^ < i?,r le- E_!,T) > C j/ = 0
< kb+ + E( o0)I 0i, '> C-.+ -o (2.25)

and the energies E(ko) can be found by the solution of the determi-
nantal equation

I h E( o) Aj | 0 (2.26)

where = <









and A = o
The order of the determinant in (2.26) will be equal to the

number of lattice vectors needed to give convergent values for E(ko).

This, of course, will be dependent on the basis set chosen. For

example, the plane waves, ei"' form a complete set of Bloch functions,

but are quite unsuitable for a problem in which the potential is

atomic-like.
For a "muffin-tin" potential, as described in Section 2.3,

suitable basis functions are the augmented plane waves 1 (APW'S)

defined by

I e outside the spheres,


Z mt u -E) Ylm(g9)? (2.27)

inside the spheres,


where Ymare spherical harmonics, iU satisfy a radial equation

corresponding to energy E, and the a1 are constants defined so that

Ik)is continuous at the sphere radius r = R, (see Appendix I). These

satisfy the requisite Bloch condition (2.21), and the matrix elements

required for the secular equation (2.26) are easily derived. For

reference, these elements are listed in Appendix I, equations (AI-16,17).

The translational symmetry used to derive the secular equa-

tion (2.26) is only part of the total symmetry of a crystal. In gene-

ral, the Hamiltonian is invariant under a space group, whose elements
are combinations of translations and rotations (either proper or








improper). The translation group is always a subgroup and in
certain cases, called symmorphic, the space group can be expressed
as a semi-direct product of the translational subgroup and another
subgroup called the point group, consisting only of rotations. This
is the case for the face-centered cubic structure, for which the
point group is the full cubic group Oh.
The representations for a symmorphic space group are parti-
cularly simple, and it can be shown that a basis function ikoT
of the translation group can also be chosen to be a basis function
for that subgroup of the point group which consists of elements
transforming the vector ko into an equivalent vector. This subgroup,
known as the group of the reduced vector ko, can be defined mathemati-
cally as


(0 =I^ (2.28)

where are the rotational elements of the space group.

A basis function for this group can be manufactured from

(Tk by the application of the standard "projection" operator

oiI = T ^^z^ (2.29)

where p F) is the (I,J) matrix element of the pth representation
of f(. The projected function,

li? r > O (2.30)








will transform as the Ith row of rp, i.e.,

3tlEPt> = ^ 00^t) ( I^pIW>. (2.31)

It can be shown 35-36 that the effect of operating on a
Bloch function I1o- such as an APW, with the rotation a is to
transform it into another Bloch function corresponding to ko
Therefore the projected basis function is

S(2.32)

The projection operators (2.29) satisfy the orthogonality
relation
P Q P
QTL 0MN In P LM TN ? (2.33)

where G is the order of 1(2) and n is the dimension of r .
0 p p
Because of the relation (2.33), any operator A which is
invariant under all operations of(fk) has zero matrix elements between
projected functions belonging to different representations, or to
different rows of the same representation. It follows directly from
(2.33) that

< 'po P I3" A I AP Y --i 'pp"< i' (2.34)

The requisite overlap and energy integrals are found by
substituting A=I (the identity operator) and A=~Prespectively. The
explicit form of these matrix elements between projected AFW's is
shown in Appendix I.









If the projected functions Iko PIJ are chosen instead of

I k then the secular equation (2.26) is considerably simplified
at symmetry points, i.e., where (k)~ contains more than one element.
The orthogonality of equation (2.34) has the effect of
"splitting" up the secular equation into blocks, one for each combi-
nation of P and I, which can be solved separately. Also, since the
right hand side of (2.34) is independent of the row I, then we need
solve for only one row of each representation. Thus for each P we
have a smaller secular equation:
P P
Ole 7.;J' 3 1J'l= (2.35)

where O= K P I IE >
P 0


a '- <- o l cer -


The projected functions (2.30) at a symmetry point are not
all independent. In Appendix II is given a recipe for selecting out
of all the functions Iko + P I J >, a linearally independent set.
In practice, the energies E(ko) are calculated at a few
symmetry points, because of the saving in time afforded by the use
of projected functions, and some kind of interpolation scheme is
used to find the energy bands over the whose Brillouin zone. It is
a straightforward matter for a computer to calculate the matrix
elements (equations AI-16 to 22) and diagonalize the secular equa-
tion. There are two infinite sums involved (over the values, and









the lattice vectors 1), but empirically it is found 14 that, for

a metal, 61 values and 50 lattice vectors (at a non-symmetry point)

are enough to give convergence in the energy values to less than

.001 Rydberg. This means that one never has to diagonalize a matrix

larger than 50 by 50, and at a symmetry point, or if less accuracy

is desired, considerably less.


2.5. The Self-Consistent Calculation

In the preceding section, it has been shown how the energy
eigenvalues and eigenfunctions can be found from a given potential,

( or two potentials as in the ferromagnetic case). If a self-consis-

tent solution is desired, there remains the problem of generating,

from the calculated wave functions, a new potential to be used on

the subsequent iteration.

When an approximate exchange of the type given in equation

(2.8) is-used, the one-electron potential is a functional of the

charge density (t)


V,(?) = VC [] + V [j] (2.36)

where Vx is the particular exchange approximation to be used, and

Vc is the Coulomb potential satisfying Poisson's equation,


V Vc j- oJ (atomic units).

In the computations reported here, the charge density is assumed to

be spherically symmetric in a sphere about each lattice site, and









constant between. The departures of the actual charge density from

this simplified model can be handled as perturbations on the final

solution, but only at a cost of considerable complication.

In Appendix III it is shown how to obtain an averaged charge
density of this type from the occupied wave functions. The quantities

to be calculated are closely related to those already found in the

expression for the overlap integral used in the secular equation.

Taking advantage of this fact simplifies the work to be done.

The formation of the Coulomb part of the muffin-tin poten-
tial from the averaged charge density can be carried out, using

Ewald's technique for handling the lattice summations. This is out-

lined in Appendix IV, the result being

v, 2"(rlOrd t +i(t) dt

Sv (2.38)
\ t-= 2 c'/ ,

where Qo= Jp(r) dr is that part of the charge density found

outside the spheres, a= 22 R is the lattice parameter and c'2.41583

is a constant characteristic of the fcc structure. The superscripts

on Vc indicate the potential inside and outside the spheres.

Equations (2.38) give us a straightforward way of generating
a new potential, once the spherical average of the charge density has

been computed using the formulas of.Appendix III. This can now be

used as the input to the energy band calculation. However this pro-
cedure very often leads to a divergence, i.e., the difference between

the potentials of successive iterations increases. As in atomic calcu-









lations, it is found that convergence is usually obtained by the
use of an extrapolation procedure. The simplest of these is the

arithmetic average. If (Vin, Vout) is the initial (muffin-tin)

potential, and (VjC Vout) is the iterated potential, then we take
the new potential to be (V i V ) defined by
v.^ yrvi. + V 1
i[ de ie d(2.39)

Vo.t= y[ v.,, vo.]

Even better, empirically, is a scheme invented by Pratt 37
and used to advantage by Herman & Skillman 32 in which the results of
two iterations are used to extrapolate a new starting potential. This

is best illustrated by a diagram (figure 2.2)

ITERATED
POTENTIAL

/

II
I V

I
I I
I I



I I
I I
I I INITIAL
I POTENTIAL


Figure 2.2. Extrapolation of a new starting potential








In the figure, the abcissa represents the starting potential,
and the ordinate, the potential obtained after the iteration by
equation (2.38). Each calculation is therefore represented by a single
point on the diagram. The next starting potential in this procedure
is chosen at the intersection of the line joining the points repre-
senting two successive iterations with the 450 line. If (V1-- VI)
and (V2 -+V2) are the initial and iterated potentials for the two
iterations, then the new potential V3 is defined by the equations;


V ut potV t / out
3Jui P I J L
Poot 1





Pout = [1V '-< o /[ t V.j]
/n of 0 0 u


Pin [V -] V1 Vo t-
[V ~V t] Vi -VI ou

(2.40)

(Since the zero-point of the potential is arbitrary, the extrapola-
tion procedure uses the potentials with respect to their constant
value between the spheres.)
It was found, during this calculation, as Herman and
Skillman 32 had for atoms, that quick convergence is obtained if the
two schemes (2.39) and 2.40) are alternated.














CHAPTER III

RESULTS OF THE CALCULATION

3.1. The Self-Consistent Spin-Dependent
Energy Bands in Nickel:

The self-consistent energy eigenvalues were computed for

two values of the exchange potential: (i) the averaged free electron

value a= 6(6g/ 8r ) and (ii) the value suggested by Kohn and

Sham 26 = 2/3 *t The initial potential chosen was a superimposed-

atom potential, as defined in Section 2.3, generated from self-consis-

tent atomic orbitals corresponding to the configuration

3d= 5.0, 3dP= 4.0, 4sg= 4sf= 0.5 .

The value of the lattice parameter used in the calculation
38
is the one quoted by Wyckoff, i.e.,

a = 6.6586 atomic units.

The radius of the sphere about each lattice site corres-

ponding to this value is R = 2.3544 atomic units. The summations over

1 in the matrix elements (AI -26) were taken up to a value of 1 = 6

and the maximum reciprocal vector magnitude ( which determines the

size of the secular equation) was K = 61T/ a. By calculating the
max
energy eigenvalues for a few points for larger values, it was found

that the chosen values, of emax and Kmax were sufficient to insure

a convergence in the energy values of less than .005 Ry. The.

"d-like" states tend to converge more slowly than the "s and p like"










states, because of the large negative values of the logarithmic

derivatives for i= 2 This implies that the APW basis functions

corresponding to the d-states will have a large discontinuity in

their derivatives, and more basis functions are necessary to

"smooth out" the wave function.

The eigenvalues and the corresponding wave functions were

calculated at the 32 points of a cubic mesh in reciprocal space

(grid #1 of table 3.1) during the course of each iteration. Because

of symmetry, actually only 6 points were calculated, and the

corresponding charge densities were multiplied with the appropriate

weights to derive the total charge density. For the last two itera-

tions of each calculation, the mesh was reduced to half the former

size, for an equivalent of 256 points (grid #2 of table 3.1).

It was found that 6 or 7 iterations were sufficient to

achieve a convergence of the energy levels to within .01 Rydbergs.

More accuracy than this was deemed unnecessary, because of the

uncertainties in the one-electron potential and the approximations

inherent in the muffin-tin form of this potential. The superimposed-

atom potential used for the first iteration turned out to be quite

far from the self-consistent solution, with the result that the first

two iterations appeared to be diverging (See figure 3.1). However,

the extrapolation procedure described in Section 2.5 was sufficient

to give eventually the self-consistent solution.

In each iteration, after the energies were calculated, the

Fermi energy E was estimated by counting the states of both spins










Table 3.1. Coordinate grids in reciprocal space.


Grid #1

k*(a/7r)

000

001

002

011

012

111


(32 equivalent points)

Symmetrya Weight


Grid #2 (256

Symmetry Weight


equivalent points)

k*(a/2TT)


Symmetry Weight


a The symmetry symbols are those of Bouckaert,
Wigner.39


Smoluchowski and


k*(a/2wr)

000

001

002

003

004

011

012

013

014

022


- -----









SI I p it I


..>-


- 0







n:1



N






II
ICn


.t


o o 0 m
- 0 d o
CA 0O


0o f

103d'S3 HIM


I I I I I I I I I I I


o
(AU)


0 d
A9O3N3


- 0
o o


- >



















N









in order of increasing energy until 10 bands were filled, corres-

ponding to the 10 valence electrons of.the nickel atom. For example,

if the energies had been computed at the 32 points of grid #1, then

the lowest 320 states were chosen as the occupied levels for the

generation of the next potential. Because of this procedure, the

configuration (i.e., the effective number of s,p, and d-like states)

can change from iteration to iteration, unlike the usual atomic self-

consistent calculation, in which the configuration is held fixed.

This might be expected to lead to a "collapse" of the energy bands,

i.e., the number of occupied levels of the minority spin might tend

to increase with each iteration until finally it has been become

equal to that of the majority spin. In the course of the calculation,

it was found that the difference between the number of electrons of

either spin did decrease slightly, but levelled off at a constant

value. This may be a fortuitous result of using a finite number of

points in the Brillouin zone to calculate the density but the fact

that this constant value is close to the experimental is encouraging.

The calculation was carried out on an IBM 709 computer.

The time taken per iteration was approximately one hour for grid #1

(32 points in the Brillouin zone) and three hours for grid #2 (256

points in the Brillouin zone).

Figure 3.2 shows the final bands for the first calculation

(denoted by V3) in which the exchange potential was chosen to have
the averaged free electron value,
the averaged free electron value, x*






40



,1 --. m-ll
I %


!/ /

1 '' 1









a Ia
S' 0














.r-
-- |---- -- ^ c ^






II













/ .,
-1


I I i












m in o
So o o S







- -- I -JII Ip p
0 d d d


0.
N


I I
ID U)
0i 0 0


6"-A 01 .L dS3HI HLIM (ANL) AkO3N3


n










Here, the energy values are plotted along the three main symmetry
directions in reciprocal space:

A: r o,-oo) X x(, o)

A: P- L(


Figure 3.3 shows the same information for the second calculation

(denoted by V4), for which Vx = 2/3 V e

Both figures have the same general shape, which is essen-

tially determined by the symmetry of the face-centered cubic lattice.

Both show the feature, typical of the transition metals, of a narrow

( .3 Ry.) "d-band" imbedded in a much broader (^.8 Ry.) "s-p" or
"conduction" band. The splitting between the two spin bands is not

uniform throughout the zone, but is approximately .07 Ry for the

d-like states and zero in the conduction states.

The principal difference between the two sets of results

is in the width and position of the d-bands. These are considerably

wider (-30%) and higher ( 1.15 Ry.), with respect to the conduction

band, for V4 than for V3 (See table 3.2). The reason for this is that

reducing the exchange effects elevates the potential by an amount

proportional to ./3. Therefore, the energies of the d-type electrons,

which are mostly confined to the higher-density interior of the atom,

are raised more than the energies of the conduction band electrons,

which are found further out where the density is less. This elevation

of the d-bands tends to spread out the wave functions thus increasing

































































a, Of- -o
a s
I I I -- II I I I I I
O (0 oU) 1 N j o
o o d do d d o d
DOA 01 1J03dS3U HIM C(A) AOU3N3


rf

















0)
,..




ct












C..









Table 3.2.Comparison of energy differences for various potentials.


Potential



V1o(b

V113

V2qc
V2P

d
V3


V4e
V4
vB


v54

V5fl


d-s separation



.481a .627

.537 .687


.437
.491

.322

.403

.478

.542


.488

.535


.571

.633

.441

.531

.625

.698


.639

.695


d width

XF-X,

.320

.349

.299

.325

.251

.282


.330

.362


.347

.372


s-p width

X-/ _pr

- .837

.838

.836

.837

.833

.834


.841

.842


.865

.866


.070


.070



.070

.076


splitting



.016



.013



-.005


.060 -.004


.067


.018


a The energy differences are given in-Rydbergs.

b V =a superimposed-atom potential with Vx = V, corresponding to
the atomic configuration (3dc = 5.0, 3d = 4.4, 4sq = 4s = 0.3).

c V = a superimposed-atom potential with Vx V=x, corresponding to
the atomic configuration (3do = 5.0, 3d = 4.0, 4so(= 4s, = 0.5).

V3 = the self-consistent potential with Vx = Vx.


e V4 = the self-consistent


potential with Vx = (2/3) Ve.


V5 = Wakoh's potential of reference 10.









the d-band width.

Although the general shape of the bands is quite similar,

the change in the position of the d-bands with respect to the con-

duction band is sufficient to alter some of the details. For example,

at the L point (fT/a, '/a, T/a) there are two doubly degenerate

d-like levels Lo, and two singly degenerate levels: L ( a mixture of

s and d-like states ) and L2 (which has p-like symmetry). Figure 3.4

shows the bands in the region around this point. A is the line from

r (0,0,0) to L, and Q is the line perpendicular toAon the face of

the zone, going from L to W (ir/a,2ir/a, O ). It can be seen from this

figure that the energy bands are substantially different in the two

cases, the p-like level I4 being much lower with respect to the

d-bands for Vq than for V3. This is particularly significant in that

the position of the Fermi level with respect to the conduction band,

and consequently the Fermi surfaces, are considerably altered by the

shifting of the 12 level. Table 3.3 lists the ordering and the

resultant Fermi surfaces, and shows that the reduction of the exchange

effects has qualitatively changed these surfaces.

The experimental data (See Section 3.3) favor the second

calculation (VN), because of the multiply-connected Fermi surface

in the sixth O-band. This surface is similar to the famous copper

Fermi surface, having "necks" along the (1,1,1) direction.


3.2. Comparison with Previous Calculations:

Within the last five years, several calculations or estimates

of the energy band structure of nickel have appeared in the literature.






45






V3 v3

0.6- L2 -


A,
0.5 Ef L3 Q -



0 0.4 -


Q
a 0.3- L





SI I I I I
Li L3

I


V4 a V4

0.7 -
SCDL __ __
iL
w0.6 A3 La


A3

SL QL3, Q .
0.5- Q



0.4- -


Figure 3.4. Detail of the energy bands at an L-point
for the two aelf-consiatent potentials















Table 3.3. Detail at an L-point.


Potential Ordering at Resultant shape of Fermi
the L-point surfaces near the L-point


V3o L3 < Ef < Closed surface in the
6th band.


V3 3 Ef 5th and 6th bands, holes
in the 4th band.


V4 L3 ace in the 6th band with
a "neck" at the L-points.

V4J3 L 6th band.









The augmented plane wave method has been applied to the paramagnetic

case by (i) Hanus, 5 who used a potential generated from renormalized

atomic orbitals of the configuration (3d)8(4s)2, (ii) Mattheiss,6

who used a superimposed-atom potential corresponding to the atomic

configuration (3d)9(4s)1, (iii) Snow, Waber and Switendick, 7 who

tried several different superimposed-atom potentials with configu-

rations (3d)10-(4s)x, 0 < x < 2.

The three calculations are qualitatively similar to each

other, differing only in the position of the d-bands with respect

to the conduction band. In reference 7 it is shown, as confirmed by

the results presented in Section 3.1, that this d-band shift has a

profound effect on the topology of the Fermi surface, caused mainly

by the altered ordering of the L point levels. (See table 3.4.)

Yamashita and Wakoh, 8-10 have also done several calculations

for nickel, both in the paramagnetic and ferromagnetic states, using

the Green's function method. Since this method is formally equivalent

to the APW method, 40 their results are also very similar, showing

the same sensitivity of the d-bands to changes in the potential.

In order to compare all of these results, we note that the

results of each calculation can be fairly well described by only two

parameters:

(i) the position of the d-band with respect to the conduction

band, which can be characterized by the energy difference between the

states r2 and P, and

(ii) the d-band width, a measure of which is the energy

difference between X5 and X1.















Table 3.4.


Results of Snow, Waber and Switendick7
for paramagnetic nickel


Atomic configuration Ordering of Fermi surface in
used for the super- energies at the 6th band
imposed-atom potential the L-point

(3d)8"0 (4s)2.0 1e Ef <( Closed surface with pro-
trusions toward the X-
points.
(3d)8.5 (48)1.5 3

(3d)9.0 (4s)10 4< T < Ef Multiply-connected surf-
ace, with "necks" at the
L-points.
(3d)9'5 (4s)05 L4 trusions toward the X
and L-points.
(3d)10.0 IL









Using these two parameters as coordinates, each calculation

is plotted on the graph in figure 3.5. For comparison, a few calcu-

lations for other elements (Cu,Co,Fe) with an fcc structure are

included. We note that no matter what the potential, all the calcu-

lations tend to lie on a straight line, which suggest that the ener-

gy band structure of an fee transition metal could be described

by a single parameter, say, the position of the d-bands.

This diagram shows clearly the considerable variation between

the results of previous work. The principal differences can be traced

back to the atomic configuration used. Those calculations which

assumed a smaller number of d-electrons are found toward the lower

left hand corner of the diagram. Although all of the calculations

listed here used the averaged free-electron exchange potential V ,

the effect of reducing the exchange moves the results in the opposite

direction.(See Section 3.1.) In short, anything which makes the

potential more attractive for the d-electrons narrows the d-band

and lowers it with respect to the s-p band.

Of the previous calculations on transition elements, only

two have been carried to self-consistency, viz., that of Wakoh 10

on copper and ferromagnetic nickel, and that of Snow and Waber 41

on copper, both of which used Vx We can see by looking at figure

3.5, that there is a considerable difference between the two results

for Cu (references 10 and 41) although both lie near the straight

line. Wakoh's d-bands are displaced upward from those of Snow and

Waber by approximately .15 Ry. This is almost exactly the same as the






50





ITERATION OF THIS CALCULATION
0 PREVIOUS CALCULATIONS
m I
0
a/
INCREASING
"- NUMBER OF
SD ELECTRONS/ INCREASING/
I* EXCHANGE /
1i EFFECTS /
N /
- 0.7

n INCREASING/
z ATOMIC /
< NUMBER /
m /
9-0.6 -
. Ni. I/ NIP- WAKOH (10)
o NIp-v,
I / -CO-MATTHEISS(6)
uw NI- MATTHEISS(6) K od 0D
N NNIP-4 FE(fcc)-WOOD (14)
. 0 .5 2 o f
w NIa-V1'V- 0-- NNi WAKOH (10)
w: NNl*_V4
S CU-MATTHEISS(6)D Ni -V4
z 0-- NI-HANUS (5)
a CU-ARLINGHAUS (4) NNIV2 NI YAMASHITA & WAKOH (9)
A 0.4 CU-BURDICK
o (16) NI-SNOW ET AL(?)
z CU-WAKOH (10) NIBJ-V3
o 3

o8 CU-SEGALL (15)
0.3 -
S0.3 Nla-V3
I



0.2 0 C1 U- SNOW a WABER (41)



0.2 0.3 0.4 0.5 0.6
D-BAND WIDTH E(x5)-E(x) RYDBERGS

Figure 3.5. Comparison of transition element
energy band calculations









displacement between Wakoh's bands for ferromagnetic Ni and those

of V3 (the self-consistent bands for Ve ). The reason for this

discrepancy lies in the simplified version of self-consistency used

by Wakoh. The wave functions used to generate a new potential after

each iteration were not chosen uniformly over the Brillouin zone. His

procedure was to pick out 5 functions representative of the d-like

electrons (corresponding to local maxima in the density of states

curve) and another function representing a conduction electron,

corresponding to energy E( ?) + (3/5)EI. These functions were then

weighted to give 5.0 do-electrons, 4.4 d3 -electrons and 0.3 conduc-

tion electrons of both spins. For the copper calculation, the corres-

ponding weights were 10 d-electrons and one conduction electron. Since

the "conduction electron" function chosen here is a mixture of s and

d-like functions, this procedure tends to fix the amount of d-like

character at too high value, thus displacing the d-bands upward. His

final result is very close to that of V4 (the self-consistent bands

for 2/3 Vxe ), since the effect of overestimating the exchange tends

to balance the effect of using too many d-electrons.

For comparison, the effective number of d-like electrons

calculated for Ni from the self-consistent results, according to

the formulas of Appendix III, are listed in table 3.5. We note that,

although the effective number of d-electrons is about the same (- 9.4)

for the superimposed-atom potential V and the self-consistent poten-

tial V3, the resulting bands are displaced 1.l tRy. from each other.

This is a reflection of the distortion of the d-functions on going
















Table 3.5. Charge distributions.


Inside charge


.14

.16


.24

.26



.44

.50


Total
charge


4.72

4.02


4.47

3.79



9.45

9.33


5.33

4.67


5.31

4.69



10.97

10.95


Effective
d-charge
I a
%d


4.97

4.38


4.79

4.14


9.96

9.90


a = Qd(QtQt-Qo)). The
dix III.


quantities Qo etc. are defined in Appen-


b The results for Cu are taken from Arlinghaus2 who calculated the
bands for a superimposed-atom potential (Cu-SAP) and for the poten-
tial used by Burdicklo (Cu-B).


Potential


Outside
charge


V3c


V40o


.20

.20


.24

.24



.50

.51


Cu-B









to self-consistency.The atomic d-functions are more localized than

the self-consistent ones, which has the effect of elevating the

bands through the Coulomb repulsion term. There is a similar displa-

cement between the self-consistent copper bands of Snow and Waber 4

and those of Burdick 16 and Arlinghaus, 2 even though the number of

d-electrons is close to 10 in both cases. (See table 3.5.)


3.3. Comparison With Experimental Data

Of the wealth of experimental data that has been published

on nickel, there are several firmly established facts which should

be explained by an energy band calculation, if the model is to have

any validity at all: (1) the saturation value of the magnetization,

(2) the electronic specific heat at low temperatures, (3) the Fermi

surface topology as deduced from de Haas-van Alphen measurements,

(4) the saturation of the magnetoresistance, and (5) the negative

spin density found from neutron diffraction and positron annihilation

data.

In order to interpret the first two of these properties, we

need to calculate from the energy bands, the density of states, i.e.,

the number of allowed energy levels per unit energy. Mathematically
44
this is 4


E- 3 VI k (3.1)
(2I) 70i d f S, t

where Es( k ) is the energy of an electron with spin s and reduced

wave vector k andI2 is the volume of the unit cell. Since Es is not








known analytically from the computation, we approximate (3.1) by

Z w(e) (3.2)
XE ( ;Rf ,< *46f
where the sum is over those I points at which Es ( ) is known, and
w ( k ) is the weight for the point !. (See table 3.1.)
In this calculation, Es ( k ) was calculated at only 20
non-equivalent points of the Brillouin zone, which is not enough to
give an accurate n. (E). In Appendix V is shown a procedure for inter-
polating Es ( k ) at enough points to give sufficient accuracy.
The calculated n. (E) for both spins is shown in figure 3.6
for the two self-consistent potentials V3 and V4. It can be seen that
the narrowing of the d-bands in V3 substantially increases the height
of the peaks in n. (E).
The Fermi level E; is determined by the relation,


Z)2 (3.3)


where nv is the number of valence electrons, 10 in nickel, and EFti
is the minimum energy of the corresponding bands. The difference
in the number of electrons of either spin is then



.mis]
The measured value of corresponding to the saturation
value of the magnetization is 0.606 electrons/atom (reference 43
page 317).






55








3

V3














o
A t






0 V3



-3
0


CO
a)




H I 3








4








.1 .1 .3 .4- .5 .6
1 I I I I.5

Energy (Rydbergs) with respect to Voc(

Figure 3.6. The density of states curves









The calculated values ofu. for the superimposed-atom

potentials using Vx are slightly larger than the experimental
value, which tends to substantiate the conjecture that this appro-

ximation overestimates the exchange effects. Going to self-consis-

tenty slightly reduces the value bringing it closer to .6, i.e.,

,kA(V3) = .65, t(V4) = .62. The error in the calculation of A is
fairly large ,A- n (Ef)AEf, since np (Ef) is high (- 40 electrons/

atom-Ry.) An error of AEf '.001 Ry. therefore gives an error in /

of -.04 electrons/atom, so that the agreement here is quite satisfac-

tory.

The electronic specific heat, c,, at low temperatures is

related to the density of states at the Fermi energy by the equation

V- J [ 4(^> A (3.5)
J

(reference 45, page 150). The measured value of cv corresponds to a
46
total density of 3.1 electrons/atom-ev. The calculated values of

n(Ef) are n(V3) 4.5, n(V4) 3.0 electrons/atom-ev. The discrepancy

in the value for V3 is a further indictment of these bands, since the

method of calculating n(E) should be expected to give a density lower

than the exact result.

Both calculations show the Fermi energy occurring at a high

peak of the density of states curve which agrees with Slater's

criterion, 47 for ferromagnetism. The results are also confirmed by

the recent measurements of the magnetic susceptibility, 48 X which

satisfies a relationship of the form;










I I I
( + (3.6)
xa h(W E() 11" y
The measurements indicate a very small Xd, which would be

obtained only if no (El) is small. We note that in both V3 and V4,

since the A d-bands are filled, n 0 (Ef) < np (Ef).

Although the Fermi surface of nickel is not as completely

mapped out as that of Cu and the noble metals, at least one feature

has been firmly established. The de Haas-van Alphen measurements,49'50

indicate a small cross-sectional area of the Fermi surface in the

(1,1,1) direction. This is interpreted as a "neck" in a Fermi surface
similar to that found in copper, i.e., roughly spherical with protru-

sions that make contact with the Brillouin zone at the L-points. The

neck area is about 1/10 that found in Cu, corresponding to an angle

of 6.80 + 0.20 subtended at the V -point. 50 If the energy surfaces

are assumed to be hyperbolic,


M (3 "7)

then mt and mt are the transverse and longitudinal effective masses,

whose measured values are 50

mt = .26 + .04, mA = .65 + .10

The Fermi surface derived from the energy bands for V4 is

shown in two cross-sections in figure 3.7. It consists in this case

of four sheets for the down-spin (P) bands and one in the up-spin

(oc) bands. The i- sheets consist of small hole surfaces at the























CLU





C) 0

030




5~ C..4
02
03


0)









~d Ed






L L U



0 0
X <1




4 04


Sz


14 I2 a1 C-'









X-points in the 3rd and 4th bands, a closed surface with protrusions
along the 7 directions in the 5th band, and closed surface with
protrusions along the A -directions in the 6th band. The sole
o( sheet is a multiply-connected surface with necks at the L-points,
in qualitative agreement with experiment. The calculated angle sub-
tended at F is 100, slightly larger than the measured value. The
theoretical values of the effective masses at the neck are in much
worse agreement; mt .5, m1 = 1.1.
This discrepancy is probably due to the fact that the
energy surfaces at L2 do not conform to the simple equation (3.7),
but is more like the solution of a quadratic equation
E -- ".'[
(,5 ti-^ a-kc9a- k 4(p*k)] (3.8)

along a particular direction in r-space, where s = E(L3) + E(L2) and
p = E(3) E(I).
The principal result of this behavior is that the effective
mass mX = ( 8E/ d1k)EE varies strongly with the position of
Ef, unlike (3.7) where it is a constant. In fact, shifting the level
L2 up with respect to Ef by an amount of .02 Ry would bring the
effective masses into approximate agreement with the experimental
values. A shift of this magnitude would also bring the neck size into
agreement with the measured value.
The model of the o( Fermi surface is consistent with the
magnetoresistance date of Fawcett and Reed. 51,52 These measurements
also show that the magnetoresistance saturates at high magnetic fields.
This type of behavior is usually associated with an uncompensated









material. Nickel being an even-valenced metal, would ordinarily be

expected to behave like a compensated material. 52 This would be the

case in the ordinary energy band model in which the two spins are

degenerate. However, the unrestricted case can give rise to the

situation where a different number of bands are occupied for either

spin, the sum of the occupation numbers being an odd number.

Experimentally, 51 the number of electrons per atom must satisfy the

relation


We teQ..) -t4(0d () 10 (3.9)

where ne (s) is the number of electrons/atom on the electron Fermi

surfaces, and nh (s) is that on the hole Fermi surfaces. Equation

(3.9) is satisfied by both of the energy bands V3 and V since in

both cases five of the o( -bands and four of the -bands are filled,

so that only one of the ten valence electrons is left to be distri-

buted over the Fermi surfaces.

Recently, it has been discovered through measurements of

positron annihilation 5and neutron diffraction 54 in ferromagnetic

Ni, that the electron spin density is negative in the outer regions

of the unit cell. It is found that for both V3 and V the conduction

electrons are polarized opposite to the d-electrons, so there is a

theoretical negative spin-density outside the spheres, and just inside

the spheres. The measured 54 value of 19% of the magneton number /A

(.606) is larger than the theoretical ones for both calculations,

perhaps because of the neglect of the non-spherical terms of the










potential. As pointed out in reference 4, these non-spherical terms

are essential for the interpretationof the neutron diffraction data,

and will have to be included before quantitative agreement can be

obtained.

There are also a great deal of optical data available for

nickel. There are summarized in references 1 and 2, which analyze

the ferromagnetic Kerr effect, the dielectric constant and the

conductivity. The structure found in these data (at .3, .6 and 1.4 ev)

is interpreted as interband transitions. Because of the errors involved

in calculating a density of states curve (See Appendix V), it is

difficult to determine whether or not structure on this small a

scale is present. It can be seen from figures 3.2 and 3.3 that

transitions to empty states above the Fermi level are possible in the

regions around the X and L points. These are the most probable

since the density of states is high at these points. However, the

density of states curve calculated here is not accurate enough to

compare the theoretical energy bands with the experimental optical

data.

A summary of the experimental and calculated data is found

in table 3.6. It can be seen that the self-consistent bands for

x = 2/3 Ve (V ) are in much better agreement than those for

,= V; (V3)'













Table 3.6. Comparison of calculated and experimental quantities.


Calculated Experimental
V3 V4

Saturation moment
(Bohr magnetons) .65 .62 .606

Density of states at
the Fermi level
(electrons/atom-ev) 4.5 3.0 3.1

Neck of the 6d Fermi
surface (angle subtend- no
ed at the P -point) neck 10.2 6.804i.2

Effective masses at the mt -0.5 .26+.02
neck (in units of the
electron mass) mI 1.1 .65+.10

High field Hall
coefficient (electrons) 1.0 1.0 1.0

Negative spin-density
(percentage of the
saturation moment) 7.8% 9.4% 19%















CHAPTER IV


CONCLUSIONS


The purpose of this work has been to solve as accurately

as possible within the limits of present computational techniques,

the unrestricted Hartree-Fock (UHF) equations in a ferromagnetic

solid. Whether or not this is a valid model will not be discussed here.

To really test this method would involve a determination of the excited

state energies, which would permit an analysis of the temperature-

dependent properties in a way similar to that of the Heisenberg

model. However, this calculation has been limited to the ground state

of ferromagnetic nickel. This follows naturally from the use of the

energy-band model, which although it can explain certain temperature-

independent properties, has not yet been extended in a rigorous way.

In attempting to solve the UHF equations, we have used the

one-electron approximation. This has had considerable success for

atomic systems, and it can be shown, 30 that it can give a solution

that is very nearly as good as the exact Hartree-Fock value. There

is no reason to believe that the same should not be true for a

solid. The problem here is to find the "best" one-electron approxi-

mation, i.e., to find an effective local exchange potential that can

accurately reproduce the exact Hartree-Fock solution. One way to do

this would be to calculate the total energy of the solid in the










correct way, viz., Et == ( ft', where i is a determinant made up

of the approximate self-consistent one-electron functions and e

is the exact Hamiltonian. and then to minimize Et with respect to

the exchange approximation. Lindgren 29 has found that this method

works very well for atoms, giving better agreement with the experi-

mental binding energies than the exact Hartree-Fock method. Such a

scheme is of course much more difficult in a solid, but de Cicco 17

has recently shown that it is feasible to compute the total energy

of a solid, so that an investigation of this type might be possible

in the future.

This has not been done in this work, but is has been shown

that slightly changing the one-electron Hamiltonian critically

changes the resulting energy band structure. In particular, it turns

out that using the averaged free-electron exchange approximation
-e )1/3
Vx = -6(6 /81r)/ gives qualitative disagreement with experiment,

whereas reducing the exchange to (2/3)Vx gives agreement for most of

the experimental data. The strongest argument for this conclusion is

the Fermi surface structure deduced from the calculated energy bands.

The existence of a copper-like Fermi surface with a small "neck" in

the (1,1,1) direction, which has been firmly established 49-52 by

de Haas-van Alphen and magneto-resistance experiments, is-not predict-

ed by the V~-bands, but does occur when the exchange is reduced.

This tends to confirm the conclusion (See Section 2.2.) that others

have reached for atomic systems, i.e., that V overestimates the

exchange effects.










This sensitivity of the energy bands to changes in the

one-electron approximation was not pointed out in previous calculations

of this type, mainly because they were not carried out to self-

consistency, and therefore no conclusions could be made about the

adequacy of the Hamiltonian. There has been one other calculation

(that of Snow and Waber 41 on Cu) that was self-consistent using the

Vx exchange. The discrepancy of the Fermi surface did not occur in

this case, since the position of the Fermi level with respect to the

position of the d-bands is different for Cu than for Ni. In Cu, there

are 11 valence electrons per atom, 10 of which are "d-like", so that

the d-bands are full. The Fermi level therefore lies above the

d-bands in Cu, whereas in Ni, having one less valence electron, has

its Fermi level slightly below the top of the d-bands. In both cases

(this work and reference 41), going to self-consistency lowers the

d-bands with respect to the s-p bands. For Cu, the position of Ef

with respect to the s-p bands remains essentially the same, but for

Ni, Ef is lowered with respect to the s-p bands. As shown in Section

3.1, the position of Ef with respect to the s-p bands is what

determines the shape of the Fermi surfaces. Therefore, going to self-

consistency in Cu showed no basic change in the Fermi surface, as is

the case for Ni. This sensitivity makes Ni a better test case than
-6e
Cu for testing the validity of an exchange approximation such as Vx

Actually, before we can definitely say whether the discre-

pancy is due to the inaccuracy of V we should examine the approxi-

mations made in this computation, in order to see if there could










possibly be some other effect which would shift the bands enough to

bring the V- bands into agreement with experiment. From Figure 3.2,

it can be seen that this shift would have to be approximately .1 Ry.

in order to bring the d-bands up far enough to give the experimentally

observed Fermi surface.

There are four important approximations which have been

used in this computation:

(1) the neglect of relativistic effects. They can be handled

within the framework of the energy band model, and are certainly

important for heavy atoms. Calculations which have been made (e.g.,

reference 36) show that these effects are on the same order of magni-

tude in a solid as in a free atom. For the Ni atom, the relativistic

splitting of the one-electron energy values are 32 .01 Ry., so that

it does not appear that they could qualitatively change the Fermi

surface structure.

(2) the neglect of the non-spherical component of the

potential. In deriving a potential from a muffin-tin charge density

(See Appendix IV) we used the spherical average instead of the actual

Ewald potential. In Reference 18, it is shown how these effects can

be taken into account, by expanding the non-muffin-tin part of the

potential in terms of a Fourier series. For example, it turns out

that the L2 level can be well approximated by a single symmetrized

plane wave corresponding to k = (ir/a) ( 1,1,1). The Fourier component

of the Ewald potential corresponding to i = (2ar/a) (1,1,1) is 18

equal to -(Q/a) (.0096). Substituting in the values of Q = .26 and










a = 6.65 shows that the energy shift of the L level would be

on the order of only .001 Ry. Since this level is the one which

determines the neck of the Fermi surface, it appears that the

inclusion of these non-muffin-tin terms would change the resultant

Fermi surface only by a negligible amount.

(3) the neglect of the non-spherical components of the

charge density. The Ewald potential was derived on the assumption

of a muffin-tin charge density. However, the departures of the

actual density from this approximation are certainly significant,

as can be seen from the neutron diffraction data. 4 These effects

are much harder to estimate, 17 especially the non-spherical parts

within the spheres. It can be done by an expansion in terms of

spherical harmonics, but if more than a few a-values are important,

the number of integrals to be calculated becomes prohibitively large.

They should, of course, be investigated if the data are to be

accurately interpreted, but it is unlikely that they will seriously

affect the energy band structure as presented here.

(4) the neglect of the distortion of the core states. In

this calculation, the core state wave function (ls,2s,2p,3s and 3p)

were assumed to remain unchanged from the values found for the Ni

atom. Snow and Waber 41 in their calculation on Cu tried to estimate

the error involved in this assumption. They used two values of the

core density, one obtained from an ordinary self-consistent atomic

calculation, and the other, from an atomic calculation in which all

of the wave functions were constrained to be in the Wigner-Seitz











sphere (i.e., that sphere with volume equal to a unit cell). It was

found that constraining the core states had the effect of pushing

the d-states Mu with respect to the s-p band by approximately .03Ry.

During the course of this work, the bands corresponding to the 3s

and 3p levels were computed using the actual crystalline potential,

and a new potential derived from the resultant wave functions. The

changed potential shifted the d-bands up by an amount less than .01

Ry. Neither estimate is enough to qualitatively change the Fermi

surfaces, although a more accurate calculation should certainly

take the distortion of the core states into account.

The arguments presented here, although not conclusive,

tend to suggest that any discrepancy found can be attributed to the

approximation made in the Hamiltonian, rather than in the approxima-

tion made to find the eigenvalues of this Hamiltonian.

To the list of approximations, could possibly be added

another, i.e., the neglect of any correlation effects. These are of

course not included in the Hartree-Fock model, but they may be im-

portant in ferromagnetic nickel. However, the importance of correla-

tion cannot be determined if the exchange is not known accurately.

Indeed, it may be that the averaged free-electron approximation is

a better estimate of the Hartree-Fock exchange effects than it appears

here, and that reducing the magnitude of the exchange merely introdu-

ces some effective correlation. More study is needed on these effects,

if the energy band model is to be successful in the explanation of

ferromagnetic effects.






69



In conclusion, the main result of this work is that the

unrestricted Hartree-Fock equations can be solved in a solid, at

least in the one-electron model, and that this scheme forms a

reasonable model for a ferromagnetic solid. The accuracy obtained

is sufficient to give at least qualitative agreement with experiment,

and could probably be improved if better approximations to the

Hamiltonian were used.













APPENDIX I


APW MATRIX ELEMENTS
A single APW basis function has the form, for a crystal with


one atom per unit cell


APW
fs =


It


where E (X)


an ( d. )


is Ea step function,

is a step function,


S, ,I


- 4.,i.-jf,.)


ki~


(Al-1)
tbC ILe( 8 Y^),(J


YA: (k)


(Al-2)


The spherical harmonics Jmare chosen according to the phase
convention of Condon and Shortley, i.e.,


y, (i)#
Jim


4,j (#-)!e


I
cls'(os W-4)A-k


so that they have the following properties:


(Al-5)


(P)


= S .


(Al-6)


SA


#7r


A( I A


(Al-7)


by =-a


,) ( )= E(,-,).(A1-3)


4


A A
/" \H y "^ /


y,,, ) --


/Y a4


Y2,1,, ( ) d121


Y J (-z )










e (Al-8)

.e= o mn=-A
This last property (Al-8) makes the APW (Al-1) continuous
at the sphere radius R.
The contribution to the overlap integral (~/k'from the
volume outside the spheres is just

o t _i__-_) (Al-9)

and that from inside the spheres is

< l^ 4>!n s 2 ., .l/ (t),j (Al-10)
-= o m=-i. o
using (Al-6). This can be simplified by applying (Al-7) to derive
the relation




Similarly, for the energy matrix






(Al-13)

because of the relation (Al-3).
In addition, there is a surface contribution due to the
discontinuity of the derivative of the APW I> at r = R ,

-S S d~ S

S








where 6[f] is the discontinuity in the gradient on the sphere S.


This reduces to:




7-t = &


=-o i -.


In summary, the requisite matrix elements between two single APW's are


/where>
where




Kk.Ibjk.'>~


(r / / />


J-2. -/Ch(/ '/P)
TC- jr/


47r (2 1 h-f k ) e (k' ) J ( (At t-19)
(Al-19)


p


5- a
P () +EL


D, ()


(E) + ( E



?Ljd 01


. (Al-15)


(Al-16)


(Al-17)


(Al-18)


(Al-20)


O,(E)


S (E) (E)


0,E(
^^)


(Al-21)


(A1-22)


<,k y / -A r


tii 4 l-.lt
k,100u '>
00~~/d -',J
4- Z < /;C> '


4* k)^ E)k^








The secular equation is obtained by forming < kI/ -E I kZ
for all and k having the same reduced vector k, and finding the
zeroes of the resulting determinant:

M E ( + D (E) = 0 (Al-23)

where k! tI > = < k o( > .
The symmetrized combinations of APW functions corresponding
to the pth representation of (j, the group of the reduced wave vector
ko, are defined by:

c(A1-24)

where rP, (&) is the matrix element of the group element corres-
ponding to the Ith row and the Jth column.
The matrix element between 2 symmetrized APW's corresponding
to the same reduced vector, of any operator 7 which commutes with all
the group elements can be shown, setting k = k0 + g, = ko + g, to
reduce to
= PI Jp / l1

SSPP S < (A-25)


Kp Lt L (4)
where Go is the order of (), np is the dimension of the representation.
Thus for a particular value of P and I, the secular equation
will have elements of the following form:
( Pr I R-E I k,' P 'j -
< Zj -^ P r j +< l( A1 2 6 )



1 j iE








We note that this expression is independent of the row subscript I,
which implies an np fold degeneracy for each energy eigenvalues
of the secular equation:


IIB- EA\ + D, (E)

where




P
t = P<


S I = o,


'^ P J 7>5

PC" P T>
IjZ- J


(A-27)












APPENDIX II


RECIPE FOR THE CONSTRUCTION OF A LINEARLY INDEPENDENT BASIS SET
The problem is to choose from among the functions Qjiko +
a linearly independent set of functions for a particular row I of the
representation p of F(.), the group of the wave vector ko,

-W- i { I 30 (A2-1)

k + g> is any Bloch function, such as an APW, corresponding to
the reciprocal vector ko + g.
The projection operators ^r as defined in equation (2.29)
satisfy the orthogonality relation
TL MN P LM N (A2-2)
.TL Q^ = P, LM N (A2-2)


Consider the function,
+itb ) +;^,>


1^o+ ?


Since gi + g is also a reciprocal lattice vector, then
0 &tk+ is a member of the basis set.
Now,.+ o3 X .





L
-pf,, ^'^^ ^ Iw ^ ^ >








so that the function OBr L I L + > is linearly dependent on the
set of np functions ( >IL IItj> L = 1, 2 ...np). Therefore,
the first ingredient of the recipe is;
RULE 1: If the function Jko+ g is chosen to be in the basis set, then
no functions of the form Otik + g are to be chosen.
This takes care of the case of unequal vectors. Now consider
the functions with fixed g. Let = = o + g The number of
linearly independent functions OPI (J = 1, 2 ...n ) is equal
to the number of non-zero eigenvalues of the matrix

(A2-4)
=- I j, > (independent of I)

Suppose I is invariant under a subgroup P of 9t.)
( # is the stabilizing group of ), i.e.,

S- j [ / t e % ) ( (A2-5)
If a e then the use of (A2-3) gives

r PP( I L
IL

L LV
so that the corresponding matrix satisfies the relation,

A= rOx)A = ( a(^)
A (A2 -6)
P -





77


where and are elements of is a representa-
tion for YZ, and therefore is also a representation for the sub-
group which will in general be reducible. Therefore, there
exists a transformation 5 which will reduce rp to block form.


(D9
r 0


Let A\ = A\$ be the transformed matrix, which will have the
same eigenvalues as A From (A2-6), it follows that:

A"A ^ A\ k )A< N,(A2-7)

Consider the decomposition of A\ into the same block form as A

A,, 1I A\12

A A\t A\
A\3, A33,.

Application of (A2-7) gives the equations


A\t C R cf)A\
and A\ 1 Af I' ".
By Schur's lemma, since (A2-8) holds for all elements
and belonging to then, either A\ .= or =F
(the totally symmetric representation of ).









Thus, the number of non-zero eigenvalues of A\ is equal to
the frequency of in ( .This can be expressed in mathematical
form to give
RULE 2: The number of projections of the function = ko + that
should be included in the basis set for the representation 1p is:

^ C)' = J? ^ W (A2-9)

where is the stabilizing subgroup for @ with order G O and
p (a) are the characters of the pth representation of (koI
At first glance, this theorem would seem to be of little
use, in choosing the basis set, since it gives only the number of
projections, and not the actual columns of the representation matrix
which must be used.
However, due to the inherent symmetries of the functions
Ik + g~, ap(l)very often vanishes.
To show the usefulness of (A2-9), consider the following
three examples:
(1) rp = P the fully symmetric representation of OL.), then
Xp(ait) = 1 ;1 and therefore a1(-)= 1 for all the functions, .
From which it follows that, every function Iko + must be chosen
once and only once in the basis set for r .
(2) Suppose 3 has no intrinsic symmetry, i.e., P consists only of
the identity element ', then:
ap^ == p= .f









Therefore, a non-symmetric 4+g must be chosen np times
for rp .

(3) Suppose r has the full symmetry of 7(t ), i.e., = ()
then; ap ( # /


Therefore, a fully symmetric ko+g must be chosen once for
the representation PI but not for any other representation. This is
the case if g = o, and ko is a point inside the Brillouin zone.
In summary, we have the following recipe for constructing the
basis set. For a given reduced wave vector ko, this is to be repeated
for each irreducible representation, Pp of lk.) .
1. Form a list of vectors ko+ g.by running through all the
values of the reciprocal lattice vectors g, up to the value at which
the series is to be truncated.
2. Starting at the lowest magnitude, choose a vector k +g ,
cross off the list all those vectors k+g which are related to it by
a relation ko+g = & (k+1), JEC(JO and continue in this fashion.
3. Find the stabilizing group of each vector I left on the
list, and from the character table of 4(k ), find the frequency
number a ( ) / ^ ( ) .


4. If ap ( )= 0 cross 0 off the list. If a ( ) O 0
and np > 1, then the actual projections must be formed to determine
the linearly independent set.

5. As a final check, the total number of projections of all










the representations must be equal to the original number of vectors

on the list.

In table A2-1 is found a list of the ap (b ) for the points

of high symmetry in the Brillouin zone for the face centered cubic

structure. The notation used for the representations is that of

Bouckaert, Smoluchowski and Wigner, and are found along the top

of each table. The general form of the V 's found in the vector

list is found along the leftmost column. The small letters along the

top are the type of atomic orbitals compatible with each representa-

tion, e.g., in the first table, it can be seen that a d-orbital is

compatible at ko = (0,0,0) with the two representations P and r2.









Table A2-1. Frequency numbers for the fcc structure


Gammna 1 2 12 25' 15' 2' 1' 12' 15 25

(0,0,0) s d d f pf f

000 1 0 0 0 0 0 0 0 0 0

aO0 1 0 1 0 0 0 0 0 1 0

aa0 1 0 1 1 0 0 0 0 1 1

aaa 1 0 0 1 0 1 0 0 1 0

abO 1 1 2 1 1 0 0 0 2 2

aab 1 0 1 2 1 1 0 1 2 1

abc 1 1 2 3 3 1 1 2 3 3





X 1 2 3 4 1' 2' 3' 4' 5 5'

(0,2,0)*(r/a) sd d d f f pf d pf2


OaO 1 0 0 0 0 0 0 1 0 0

a0a 1 0 1 0 0 0 0 0 0 1

abO 1 1 0 0 0 0 1 1 1 1

aOb 1 1 1 1 0 0 0 0 0 2

aba 1 0 1 0 0 1 0 1 1 1

abc 1 1 1 1 1 1 1 1 1 2










Table A2-1 (continued)


L

(1, 1,)*(7/a)


1 2 1'

sd f


aaa 1 0 0 1 0 0

aab 1 0 0 1 1 1

abc 1 1 1 1 2 2


W

(1,2,0)*(Tr/a)


1 1'

sdf d


2'

pdf


3

pdf2


abO 1 0 0 1 1

a0b 1 0 0 1 1

abc 1 1 1 1 2


1

spdf2


3

pd2f2


Sigma

(x,x,0)


1

spd2f2


aaa 1 0 0

aba 1 0 1

abc 1 1 2


aa0 1 0 0 0

aab 1 0 1 0

abO 1 0 0 1

abc 1 1 1 1


2'

pf2


Lambda

(x,x,x)


3

pdf2


4

pdf2










Table A2-1 (continued)


Delta

(O,x,0)


2'

df


1

spdf


1' 5

pdf2


OaO 1 0 0 0 0

aba 1 0 1 0 1

Oab 1 1 0 0 1

abO 1 1 0 0 1

abc 1 1 1 1 2


z

(x, 2T/a, 0)


1

spd2f2


3

pdl2


4

pdf2


abO 1 0 1 0

a0b 1 0 0 1

abc 1 1 1 1



S 1 2 3 4

(x,21T/a,x) sd2f2 pdf p2df2 df2


a0a 1 0 0 0

aba 1 0 1 0

a0b 1 0 0 1

abc 1 1 1 1














APPENDIX III


APW CHARGE DENSITY

The electronic wave function corresponding to the ith row

of the pth irreducible representation of the group of the reduced

wave-vector k is

FL I P -a
.o N- Pr > (A3-
0 k., y,5
where Ik PIJ> are symmetrized APW's as defined in equation (Al-24),
pIT
and Cie are the elements of the eigenvector as obtained from the

secular equation (Al-26). The sum is carried over those combinations

of lattice vector f and column index J which give a linearly indepen-

dent set of symmetrized APW's, as described in Appendix II. N

is the normalization integral, which will be derived in equation

(A3-U).

To see the explicit spatial dependence of the wave function

inside the spheres we can express (A3-1) in the form


in Y E) (A3-2)

where a 03-g^j J(A3 3)


The coefficients aA are defined in equation (Al-2). The spherically-

averaged charge density inside the sphere is

S,. (A3-4)

a0E ^ ^-.
JS.SO i^-L~~









where we have used the orthogonality of the spherical harmonics (Al-6).





addition n theorem (A ), this can be reduced to








A.
rRF ( : 8+ + ltl ) Ml I + -5 1

/ lA-A






,' (A3-5)

By using the defiorthogonity properties of the min equatrix el-2) ments the
additexpression inside the curly brackets be reduced to
yrk ) &.(? 'J


1t, CR, EC


X 4U 2. r z + I p'ty I P-) j pL I*7:S'I
apelL (A3-6)

By using the orthogonality properties of the matrix elements, the
expression inside the curly brackets becomes

CT. 9^ 3'.J Cj, )A')J
"qZ P+(


A z


S< t I I* ^t PJ>r 7


(A3-7)


where the matrix is exactly the same quantity that was calculated
for the secular equation (Al-27).
Therefore, we have as a final expression for the inside


charge


r









density (spherically averaged)





The outside wave function has the form

oi (A3-9)

The average charge density outside the sphere involves the same
calculation as in the derivation of (Al-9), and turns out to be

o ? c (A3-10)

where the (, matrix was also calculated for the secular equation.
The normalization integral can now be obtained by the equation


0


kZ c fj<^ o JtjIPFJ/> (A3-l1)



0
where I(E)= t -
o
The equations (A3-8), (A3-9) and (A3-11) can now be used to
find the "muffin-tin" part of the charge density corresponding to an
APW wave function. Once the secular equation (Al-27) has been diagonal-
prJ-
ized to find the C. we have only to sum up over quantities
that are already available. The determination of these quantities for









a suitable number of occupied levels gives a total charge density
from which a new one-electron potential can be determined.
In Section 3.2, we refer to the amount of s-like, p-like
and d-like charge inside the sphere. These terms refer to the
quantities,

S-J" c I l IP TJIT> ..(E)
(A3-12)

for .= 0, 1 and 2 respectively. The total charge outside the spheres
is just



J T JJJ/














APPENDIX IV


DERIVATION OF THE COULOMB POTENTIAL FROM A GIVEN CHARGE DENSITY
The generation of an averaged potential (muffin-tin type)
from an averaged charge density is a simple electrostatics problem 18
The equation to be solved is

VV 8 IJ (A4-1)


which is Poisson's equation in atomic units, where p =f, (constant)
outside the spheres and p = y (r) (spherically symmetric) inside each
sphere, subject to the condition

4rJ r^() di ^ P'(A4-2)
0
R is the sphere radius and J2 = 11 4 r J is that part of the unit
J
cell outside the sphere. There is also a point charge Z at each
lattice site. (An electronic charge is taken as positive). Since
(A4-1) is a linear equation, we can express the solution as a sum of
two potentials
V = V, V,
Iv-V 8 (A4-3)

J) consists of a charge Z+Q at each lattice site, a spherical charge
p(r) -p, inside each sphere, and zero charge outside. Y2 consists of
a charge Q at each lattice site and a constant charge 6 elsewhere









(cf. figure A4-1). Q is defined so that the total charge in both
problems is zero, i.e.,


where QU is the charge


V


outside each sphere.


V,


Figure A4-l The Ewald problem
We now consider the two problems separately:
1) the spherically symmetric charge in each sphere produces
a zero field outside it, since the total charge inside is zero. There-
fore V1 inside each sphere has contributions only from the charge
density inside that sphere. Choosing V1 = 0 outside the spheres, we
have the solution for V1
S2 v


Vo
v0I


= 0


+ i f' l 2- 4t
2j. [o-lJL


(A4-5)


where a(r) = 4-1T p( r) To( ') 4-"lKg F I


(A4-4)










2) the charge density Y 2 produces a non-spherically symmetric
potential, whose derivation is less straightforward. Formally it is

given by:

V ()=2. + constant, (A
(AC-6)

where t are the lattice vectors. However, the sum over t diverges, and

the constant is infinite. To avoid these difficulties, we can use a

mathematical trick to extract a convergent series from the two infini-

ties in (A4-6). By using the identity;

Xe -w A (A4-7)
0
we can express the sum in (A4-6) as

V.2 O- i + e constant.(A4-8)
orr

The lattice sum in (A4-8) is periodic in r and can therefore
be expressed as Fourier series, i.e., a sum of reciprocal lattice
vectors, g, which is easily shown to be:

e e -, e (A-9)


The trick now is to notice that the lattice sum on the left hand side

of (A4-9) is rapidly convergent for large u, and the reciprocal lattice

sum on the right hand side is rapidly convergent for small u. Therefore

we divide the integral over u in (A4-7) into two parts, introducing an
arbitrary constant :

V,0)J- 477I .e e If & .(A10
J2 d- (A4-0l)
v i.A









The infinity in this expression arises from only one term,

i.e., that corresponding to 3=0 in the reciprocal sum. Subtracting
this term off leaves a finite potential, which is dependent on E
However, the derivative of (A4-l1) with respect to E is a constant
independent of r;

ae I- ~o VTo /
S7 <(A4-11)
4TQ/JSS3 ,

where we have used the relation of (A4-9). Thus, by adding the constant
2 IT Q//IE1 to V2, we will have a final potential independent of E ;

v? (^)- 7Q E' 2q e eE_ l ] Y
1 ~~~ e-t ^ ""p r~'- ?/ Q
J 3 (A4-12)
where erfc is the complementary error function. The constant E is
chosen so as to optimize the convergence of the two sums.
Equation (A4-12)-can be put in dimensionless form, by
setting g = 2 K, = aR, = ax, ` o=Ea, where a is the lattice para-
a.
meter and 2= a 3/4 for the fcc structure:

a Vt 4 el X 7 K LK 71 1' *j
S 4- 7T7r k 4L
Kjo

e I [f x-- j i ] 4 r
(A4-13)


Empirically, it is found that if the sums in (AA-12) are
truncated at the 6th nearest neighbors, then foro(= 3 the error in
a V2/2Q is less than 10-8 for all values of r. The values of this
function for the feec structure are tabulated in reference 18. These









values were recalculated during the course of this work and were found
to be in agreement up to the fifth decimal place. V2 has a singularity

at the origin due to the charge Q situated there. From the calcula-

tion the behavior near the origin was found to be,

Atm La2 7] (A4-14)


where b = 4.584862.

The spherical average of the potential inside a sphere (A4-13)
will be just the expression (A4-14) plus a term due to the constant
charge density pJ,

< v"n 2.q 2qb / ( /4-7r ).
S- a. 3 (A4-15)

Since (A4-12) is valid as an expression for V2 for all values

of 6 it holds in the limit as E -4 o

V- 2Q e
I Z (A4-16)

This is an exact expression which does not diverge, but is too slowly
convergent for actual calculation. However, it can be easily seen from
this form that the average value of V2 taken over a unit cell is zero.
This allows us to calculate the average of V2 in the region outside the
spheres, from the relation
< 0 >d
s oV,'> a*d = 0, (A4-17)
0
from which we can find

<.( L 3* /322- J ri ( .-18
2L 3a / 5a a3