APW calculation of the electron-phonon interaction in solid argon and neon

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APW calculation of the electron-phonon interaction in solid argon and neon
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Worth, Joseph Patrick, 1949-
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Thesis--University of Florida.
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Includes bibliographical references (leaves 80-82).
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by Joseph Patrick Worth.
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APW CALCULATION OF THE ELECTRON-PHONON INTERACTION
IN SOLID ARGON AND NEON



















By


JOSEPH PATRICK WORTH


A DISSERTATION r'-H ."'":-ED TO THE GRADUATE COUNCIL
OF TITE I *: ;T 1.;ITY OF FLORIDA
IN PARTIAL FUJLFILLENT OF THE REQLIRPJIM .I5 FOR THE
rEF' '. 1 O DOCTOR OF PHILOSOFEY





U?; \'. '. TY OF FLORIDA


1976


1






























This dissertation is dedicated to the vreiiory of Professor John C.

Slater, whose contributions to modern physics had a profound influence

on present day theories of solid state physics and quantum chemistry.

It seems only fitting that this work, which makes use both of Professor

Slater's Xa model and APW method, should be dedicated to the memory of

his rol0e in the development of the theory of condensed matter. That

Professor Slater could- not be made aware of the results of this work is

regrettable.

















ACKNOW1EDINLD TTS


Dr. Samuel B. Trickey, my dissertation advisor, is gratefully acknow-

ledged as the source for the ideas in this dissertation and for his help,

advice, and encouragement (in all forms!) throughout the course of this

work. His dedication, enthusiasm, and sense of humor have helped me in my

efforts to understand and complete this work. I particularly want to thank

him for the tremendous opportunity to accompany him on his year's sabba-

tical leave at the I.B.M. Research Lab in San Jose, California. Without

that year of uninterrupted study, concentrated work, and computational

freedom, this dissertation would not have been possible. I thank Dr. John

Connolly for bringing the Golibersuch article to my attention. I am also

grateful to the many members of the Quantum Theory Project for their help.

The International Business Machines Corporation's Research Laboratory

at San Jose, California is to be thanked for allowing a year's visit

during Dr. Trickey's sabbatical. Drs. T.R. Koehler, P. Eagus, D. Haarer,

and G. Castro and many other people of the Research Lab are to be thanked

for their hospitality and help during my visit.

This diswertion was made possible through the financial support of

the Natiooal Science Foundation, the computing support of the Northeast

Regional Data Center of the State University System of Florida, and the

computing and i;:boratory facilities of the I.IB.!. Research laboratoryy in

San Jose, California.

















PREFACE


"The whole thing is a low put-up job on our noble credulity," said Sam.





NORMAL LINDSAY, The Magic Pudding







And, so, if you can't dazzle 'emi with brilliance, ...

















TABLE OF CONTENTS
PAGE

ACKNOWLEDGEMENTS ...................... ..... .... .................. iii

PREFACE .......................................................... iv

ABSTRACT ......................... ................... ............ vii

CHAPTER

I. INTRODUCTION ................. ............................. 1

1-1. Purpose of Calculation ......................... 1

1-2. Assumptions and Approximations .................. 2

1-3. Outline of Calculation ........................... 4

II. FORMAL THEORY ANID METHICDS OF CALCULATION ................. 7

2-1. Born-Oppenheimer Theory for Crytalline Solids .... 7

2-2. The Xa Hami' onian ................ .... ........ 11

2-3. The APW Method ...... ............................ 13

2-4. The FrIbJlich Hamiltorian ....................... 16

2-5. The Derivation of the Forn for the EPME ......... 27

III. RLLTLIS OF CALCULATIONS .. ................... ............. 34

3-1. API Energy Band Calculaticn of Argon and Neon ... 34

3-2. The SCH Phonon Spectra of Argon and Neon ........ 38

3-3. The Perturbed Phonon Spectra of Argon and Neon .. 41

IV. CO CLUSIOJ :S ........................... ....... ............ 43



I. SEI"-CO-S i' ;'f T HARMONIC THEORY AND L.'-NARJD-JONES
PAI POi ALS ...................... .. ............... ...... 46

TT (C'ROU '''rETICAL EXPLN TI OF 'f TRA:S.T'1 ION SELEC' TIIN
RULLS ............ ..................................... 51










III. EXPLICIT DERIVATION AND FORMULAE FOR THE EPME ............ 72

IV. DESCRIPTION OF COMPUTER PROGRAMS AND CODES ............... 77

LIST OF REFERENCES .......................... ...................... 80

BIOGRAPHICAL SKETCH ....................................... .... 83

















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



APW CALCULATION OF THE ELECTRON-FHONON INTERACTION
IN SOLID ARGON AND NEON

By

Joseph Patrick Worth

December, 1976

Chairman: Samuel B. Trickey
Major Department: Physics

The clectrcn-phonon interaction in solid argon and neon was

calculated within the self-consiszent field au~-.nted plane wave local

exchange(SCF APV-X:a) model. The Frihlich Hamiltonian, which uses the

electron-rhoron interaction in the form of a transition probability

matrix element, iwas assunmd to be appropriate for describing corrections-

to the phonon scp ctra. Second order perturbation theory involving the

calculated (:l(ectron-phonou mnatrix element (EPME) was used to compute

correctionLc to ohonons iun.pecturbed by the electron-phonon interaction.

The unpertur'bep. phonons were calculated within the self-consistent

harmonic (SCB) appro': imation, assuming a superpo.;tion of Lennard-Jones

pair poter' tal-]. The' pair protcntial p:.ra;meters vere not determined empiri-

cally, but were fitted to th';: calculated equili bium lattice constant or

the expeir~mental lattice constant ind to the corresponding calculated

APV-X> cohecsivc cJIuergies. Explicit expressions for the EPME were derived

in the AP;' formali ~, as wer; selc-ction rules between representations










involved in the relevant inter-band transitions. To calculate the EPME

numerically, computer codes were written and tested. The perturbed pho-

nons, calculated as a test of the EPME, were found to be in poor agreement

for neon and in good agreement for argon, a distinction which can be traced

directly to the shortcomings in the original Xa calculation on neon, and

not to the method in general.


viii

















CHAPTER I


INTRODUCTION


1-1. Purpose of Calculation


It is essential to study the interaction of electrons with the lat-

tice vibrations or phonons in a crystal if the basic properties and

characteristics of solids are to be understood. Such effects as super-

conductivity in metals, phonon dispersion anomalies, and electronic speci-

fic heat enhancement may be calculated once one has detailed knowledge

of the electron-phonon (EP) interaction.

However, the aim in this work is not so ambitious as a calculation

of the above effects. Rather, the present effort is to develop a formal

computational procedure to calculate from first principles the EP inter-

action in a physically realistic model of an insulator. Once tlhe method

has been justified and demonstrated to be feasible, the application to

the effects caused by the EP interactions ray be jone.

A seemiin gly practical icand ;sounl methd)c o7 calculating the EP inter-

action waz- Ldvaonced almost 10 years ago by GolJbersuch.I His idea for

calculating the elect ron-pho:'on matrix elmtene (EPf L) vas to exploit

certain features of Slater's ougmented plane vavc (APW)2 method for the

calculation of the energy bands in s .soid.

In his article, Golial riuch jsiisifiJ a r certain specific form of the

EPME, then explicitly derived the computational form.ilae in terms of some

of Ithe qc j ii. IL o. a il l co.-vent.i.onal ,Fi co-put-i cod 3e Hol er,









no calculations based on his method have ever appeared in the literature.

The reason for this absence of calculations is probably the complicated

nature of his formulae; in particular, the occurrence of double sums over

angular moment (two Z sums and two m sums) which include some rather in-

tricate spherical harmonic transformation matrices.

An examination of Golibersuch's paper shows that his method does rot

take explicit advantage of the muffin-tin symmetry inherent in the usual

APW method.2 Also he does not consider an insulator, although his justi--

fication of the EPME is general. Therefore, a method that exploits

the muffin-tin symmetry has been developed for filled valence band solids

(not necessarily insulators), and is the subject of the present work. It

should be emphasized however, that the EPME used in this work is based on

the form justified by Golibersuch.

To test the calculation of the EPME, the effect of the electron inter-

action on the phonon spectra of solid argon and neon has been coirputed.

Though this test is rather severe (as will be explained subsequently), it

demonstrates both the power and the limitations inherent in our procedure.


3-2. Assumptions and Approxinmations


Perhaps the m-ost crucial assumiptica for the overall calculations is

that of allowing the nuclei and electrons to interact among themselves

and each ctheer only through short--ranged screened potentials. Such screened

potentials a'.: inherent in the self-consist n.t field (SrC) APW description

of th- electrons and the Fr'hlich Eai-ltoni:'sn which will be used to calcu-

late energy corrections to the phoinns ari-ing froi the EP interaction. A

truly basic approach would be to calculate first the bare nuclear inter-

action and then screen this interaction with an electron distribution









whose response is described by a dielectric function. An approach along

these lines has been derived by Sinhas for the APW method, but does not

appear to be computationally feasible. Although the formal theory of

microscopic lattice dynamics has been developed by Toya,6 Pick, and

others8 through the dielectric response concept, few, if any, practical

numerical calculations have been attempted. This work, even with its

approximations, represents an attempt at a calculation based in principle

on microscopic lattice dynamics.

The Born-Oppenheimer adiabatic theory and approximation9 are assumed

to be applicable in calculating the electronic wave functions for a static

lattice. The formal equivalence between the Fr8hlich theory and the adia-

batic theory has been pointed out previously,10 but will be derived expli-

citly in Chapter II in the present context.

In order to carry out an APW calculation of the now-traditional

muffin-tin type (the sort originally discussed by Slater2), a one-electron

potential must be available. Even at the level of Hartree-Fock theory,

the generation of such a potential for a crystal is an almost prohibitively

difficult task. A great deal of effort has gone into developing methods

which circumvent this difficulty at the cost of the introduction of

certain ad hoc physical assumptions. The most widely studied of these

schemes approximates exchange and correlation contributions to the poten-

tial along the lines of exchange (designated as X) in the electron gasI

with the added flexibility of one parameter (designated as a): the model

is therefore known as the "X' '" model.12

In this study of lattice dynamics, '.:h APW i:uffin-tin potential

associated with a particular rnucleus is .asuried to move rigidly with that

nucleus. Ihis "rigid ion" approximation alllc ws the total solid potential










to be a superposition of all muffin-tin potentials centered on the nuclear

locations.

For the nuclear motion, the potentials are assumed to be harmonic;

i.e. the effect of the force is linear in the displacement. Even the self-

consistent harmonic (SCH)14 approximation used to calculate the unperturbed

phonons has as its basis functions harmonic solutions, though the method

itself contains implicit and inherent multi-phonon processes (to infinite

order).14 The total static lattice potential is assumed to be describable

in terms of the cohesive energy and lattice constant of the solid. This

description is parameterized through a set of Lennard-Jones pair poten-

tial constants. s The fitted constants are used in a Lennard-Jones pair

potential which is superposed to yield the total lattice potential.

All these assumptions and approximations will be reviewed and examined

in the proper context in Chapter II where the formal theory is developed.

In light of the conclusions reached in Chapter IV, the severity or appro-

priateness of the approximations and assumptions will be discussed.


1-3. Outline of Calculation


The first phase of the calculation consists in obtaining the static

electronic Bloch wave functions at a given lattice constant in the APW

basis. The puinber of eigenvalues and Bloch wave functions are determined

by the type and nEcmber of atoms in the solid. Sel.f--consistency is obtained

through oi)n of three criteria: convergence (or more exactly, iteration to

iteration stability) on the pressure, one-clectron eigenvalues (i.e. band

energies or potential. Usually when one of these criteria is achieved,

the others are also fairly well satisfied.

., t0h loch fP: ctiortl CA I-.:.--L; and clf- coG:i:.istent










potential are obtained to desired accuracy, the cohesive energy of the

solid is calculated at the assumed lattice constant. The cohesive energy

and the assumed lattice constant then can be used to parameterize the

electronic contribution to the bulk, static solid in terms of Lennard-

Jones pair potential parameters.i5 This parameterization describes the

static solid by a lattice sum involving all distinguishable pairs in the

solid. The 12-6 form of the potential15 is used for computational conve-

nience in obtaining the phonon spectrum of the solid.

Next, the SCH formalism is applied to the given total lattice poten-

tial, based on the Lennard-Jones potential superposition. A phonon spec-

trum is calculated to a specified level of precision by iterative solution

of the SCH dynamical matrix eigenvalue problem (see Appendix I). This set

of phonons constitutes the unperturbed phonons (i.e. phonons unperturbed

by explicit interaction with the electrons).

In principle, the EPMYE should be calculated for all transitions be-

tween occupied bands and empty barnd.. Since, however, the transition prob-

ability is dominated by a factor inversevl proportional to the energy

difference between the states of the transition, the main contributing

transitions are take to be betL.een the higiLest single valence band states

and the lowest single conducLio:) band states. In the rare gas insulators

(e.g. argon and neon), this means only transitions from the highest p-like

valence band states to the lowest s-like conduction band states. The fact

that these bands exd:ibit simple sy~rcietry properties throughout the zone

is a great conve.nienrce for ccmirputt-tion.

The E P'ME is calculated throughout the entire first Erillouin zone

(BZ) (see Appendix I). The square of the EPME is used to find a second

"C. l-_ 'i









to the energy shifts for a pbonon of a given vector and polarization

are governed by certain selection rules that relate this phonon to the

electronic irreducible representations (which label the electron wave vec-

tor and state). These allowed electronic corrections are then summed over

the entire first BZ.

This is a basic description of the procedure involved in the EPME

and phonon correction calculation. Chapter II gives a theoretical and

formal justification for the procedures outlined here. Chapter III

includes the results, commentary, and comparison with known quantities.

Conclusions regarding the results, their implications for the assumptions,

and the method in general are presented in Chapter IV. There are four

Appendices in which are explained the potential and SCH approximations

(Appendix I), specific group theory and selection rules (Appendix II),

the actual formulae for thl EPME (Appendix III), and a description of the

computer codes developed and used (Appendix IV).

















CHAPTER II


FORMAL THEORY AND METHODS OF CALCULATION


2-1. Born-Oppenheimer Theory for Crystalline Solids


Crystalline solids, by definition, possess a definite periodic trans-

lational symmetry. Though the specific crystal structure is determined by

minimization of the free energy, the various lattice symmetries inherent

in a particular crystal structure frequently may be exploited to reduce

the difficulty in carrying through whatever formal ansatz has been elected.

Thus, the translational periodicity of a perfect crystal allows one to

reduce many problems to considerations within a single cell by application

of Bloch's theorem. Group theoretical methods also may be used to take

explicit advantage of the crystal symmetry; as, for example, in the block

diagonallzation of secular matrices commonly done in energy band and

molecular theory.

The elementary form of Bloch's theorem for a particle moving in a

periodic potential V(U) is1


V (r+R ) e.:p[ik*R ]' (r) (2-1)
k An -n k
V(r4R ) = V(r) (2-la)
-11


where '' (r) is the particle ei; :nfunction with wave vector h and energy

Sand R is any translation that tlea's the solid or lattice invariant.
hk x-n
The extension to multi-partic]ce systems is straightforward.16










The symmetries characterizing a solid are valid only for a static

crystal structure. In reality, the solid consists of moving nuclei whose

motion is centered on the lattice sites. However, because the nuclei are

bound in the crystal, they appear on the average to be at the equilibrium

(i.e. static) lattice sites. It is then intuitively plausible (because of

the smallness of the electron mass relative to the nuclear mass) to begin a

treatment of electrons in a solid by assuming that the nuclei may be taken

as fixed at the given lattice sites for a specified density. After a solu-

tion to this problem is achieved, the motion of the nuclei may be taken

into account by a perturbative calculation of the electronic response.

Just such an endeavor is the purpose of this research.

The formal basis for this kind of effort was first enunciated by Born

and Oppenheimer9 and the general theory has become known by their names.

Following the notation of references 10, 16, and 17, the total solid

Hamiltonian H is written as


H(R) H (R) + H (r,R)
n e
( p/2 + V (R) ) ( p2 + V (r,R) + V (r)) (2-2)
j n- 1 ne e
j1 i

where Rydberg atomic units will be employc'e throughout this work except

as otherwise noted. In (2-2), M is the nuclear mass with the suia j over

nuclei, the sum i is over electrons, V (R) i; the nuclear-nuclear inter-
n. -

action potential, V (r,.) i: :he nucle:r-el-"c-.ron interaction potential,
ne

and V (r) is the eolcrron-clcctron interacti.:n potential. Also R is a
e -

supervector of all nuclear position vectors and r plays a similar role

for electrons. The grouping of the terms ir (2-2) is suggestive of the

role which the nuclear coordinates play as parat:iters in the adiabatic

or ins-'tri~.n~ous resc'n)uns teorv of el:--ctronLc structure. That is, the









last three terms in (2-2) give the SchrE'dinge equation for the electronic

state as


H (R)Y (r,R) = (7 p2 + V (r,R) + V (r) )P (r,R)
e -- k i ne e k
= Ek(R){ k (r,R) (2-3)

th
with E,(R) the total electronic energy of the kt state dependent upon

the nuclear coordinates R as parameters.

The electronic eigenfunctions of (2-3) form in general a complete

basis set and therefore may be used to expand the total system (nuclear

and electronic) wave function


E(r,R) (= L (k,R)k (r,R) (2-4)
k -
th
where the expansion coefficients depend on the k electronic state and

the nuclear coordinates R. Operating on (2-4) with the total Hamiltonian

H (2-2) gives three terms


HE(r,R) = ( (': (R)[ p2/2M + V (R) + E,(R) ] (k,R)
k j n .
k -

1/M T V (r,R)7-V (kR) 1/2M (_ R)2, (r,R)) (2-5)
R k R- R /2 R k


By multiplying the last two terms in (2-5) by *, (r,R) and integrating

over the electronic coordinates, these terms; may be shown17-20 to be

related to electronic scattering from tl.h. state k to k'. Becaue this

scattering is not in accorIa.nce with the adiabatjc theory (which requires

the electronic state be unchalnged) and because tO terMrs are small due to

the factor of t h nuclear mass, the last two terns may be eliminated from

consideration.

Exa:ineation of the recs of (2-5) (after integration over the elec-

Ci c- O"nc .tt









we have achieved a quite plausible separation between the nuclear and

the electronic motions. This separation is emphasized by requiring that

the Q(k,R) be eigenfunctions of the nuclear motion determined from (2-5)


( H(R) + E (R) ) .(k,R) = E (R) a(k,R) (2-6)


where, in the case of a lattice dynamical treatment,


E (R) = (n + 1/2) (2-7)


Eq is the total energy of the nth state for the th phonon of frequency

w and X polarization.

The separation of the electronic and nuclear motion has been accom-

plished by use of the adiabatic hypothesis. The nuclear motion governed

by (2-6) may now be described as a phonon field either of the ordinary

harmonic typ. or the intrinsically quanrum mechanical type given by

SCH theory (see Appendix I). Both approaches diagonalize a force constant

or dynamical matrix to obtain the phonon eigenaulues. The only difference

is in the deter-ination of the dynamical matrix (a matrix of self-consis-

tently averaged second derivatives in the SCH theory as opposed to equili-

brium values in plain harmonic theory).

The quantcum parameter A2 introduced in Appendix I is a measure of the

quantum ierchanic;al character (or zero point motionn) of a syst;::. Of the

two rare gs solids: cnidcr' d here, the value of A for argon is fairly

saall, while that for neoin indicates soiwr':iat i:-ore substantial zero point

mot on. l.:therefore the unperturLbe. (i. adiabatic) phonon spectra of

argon and neon h~ave been c.] culated by the SC'I theory.

As is 'e.ll knoi \, t-.e priiipal diffic lty i. solving the adiabatic

ele-" r-nor c Qt-r, "ii'rvv pr-ciM,-m; (7-3) n r.',s"- from, 1h-''. ro'Tny-body Ppt,;re of










the electron-electron interaction ccmbii.ed with the requirement that the

electronic wave functions be properly anti-symmletrized. Almost all treat-

ments have as their aim the construction of a one-electron model in which

the electrons are treated as independent particles moving in some well-

defined average field due to all the other electrons. Even at this level,

rigorous treatment of Fermi statistics introduces complicated exchange

contributions to the effective Hamiltonian. The inclusion of soatial cor-

relation introduces further complications. The net effect is to make the

problem almost intractable for a crystal. For example, correlation and

exchange effects may be treated through a computational realization of the

configuration interaction scheme for atoms and molecules, but in solids

such an approach is, to this date, well beyond the limits of feasibility

and other schemes must be invoked.22 A much explored alternative is to

make some sacrifice in rigor (small, one hopes) and thereby obtain a sol--

uble model of electron behavior. The next section deals with the form of

such a mode:. and the Hamiltonian H (R) of (2-3) that results, with particular
e -

emphasis on the features needed to achieve the eventual goal of this work.


2-2. The Xa Parailtonian


A brief outline of the Xa method for the non-spin poclrlzr d case

will be given here. The method itself is well documented and has had

many useful applications in a variety of circumstances. 2 For the present

pur poses, .ite method allows a co::putationally manageable form for the

Hamili-on i.n in calculations of the electronic properties. of a solid.

For a static lattice, in the adiebatic sense, the X~ method assumes

for the total electronic energy the expression


- -
K'I X:L__X


(2--)









where


= In.u (1)Vi.(1)dr (2-9)
X ij T 1 i --L

= -'Z.Z /R. l'p(1)(2Z /r )dr +p(1)((2)!/r dr
j m j -- 12 -12

+ 1/2 fp(1) xU (1) d (2-10)

th th th
with Z. denoting the j nuclear charge and R. the j and m nuclear

separation. The charge density is


p(l) = n.u(l)u.(1) (2-11)


with n. the occupancy of the spin orbital u.. The local exchange-correla-
1 1
tion operator in the non-spin polarized case is defined as

1/3
U (1) = -9a(3/8Tr p ())13 (2-12)


with a a parameter to be calibrated against, for example, atomic properties.

Application of thc variational principle to (2-8) leads to the effective

one-electron Schradi nger equation


[- + Z./r + r(2Z,/r1)(2)dr, 6c(3/8. p(i)) /3]u.(1) = E.u.(1) (2-13)
[1 13 2

The one-electro:. eigenvalues E. of (2-13) are related to the occupancies

n. so that Ferrmi statistics are obeyed.

The static lattice cohesive energy per particle for a solid of N

constituent nuclear particles and of a specifie; density is defined as


(E/) = (2-14)
X c solid X; atom


This def:iiitior is co.i:si-tent with A)ppendixl I provided the two Xa total

energier r. ce1C'l t the F4*'' U.










The selection of a value for a has been the subject of much investi-

gation and discussion. However, as a result of previous calculations

done on argon23 and neon,24 a value of 2/3 has been used for neon and a

value of 0.72131 for argon. The neon value is that due to Gaspar, Kohn,

and Sham,25 while the argon value is such that the virial theorem is

satisfied for the energies determined from a single determinant of argon

Xa orbitals using that value cf a.26


2-3. The APW Method


The APW method is a well-established scheme for solvig- a given one-

electron eigenvalue problem for a crystal. An excellent review by Mattheiss

et al.3 discusses the self-consistent field synmmetrized form of the APW

method (SAPW). Just this form has been used in the present research.

Therefore the presentation here will follow the notation and style of that

review. As the calculation was done on two solids which have the same

synmmorphic space group (see Appendix I), the explicit reference to

non-primitive lattice translations will be dropped.

The potential, of the functional form discussed in section (2-2), is

assumed to be spherically symmretric inside spheres centered at each

nuclear site in a solid and constant outside those spheres. This periodic

muffin-tin (MT) potential is a physically plausible one which approxinmates

the exy-ct periodia-. potential while being much mr ore convenient. In' the

conEtant potenL ia1 region, the- AI'W ba';is functions are taken to be plane

w;?VL'S: while inside the sphere the APW basicc functions are expanded in

sphcrical harm:onrics- a:d energy dependenti radial functions.
Sth th e
The eecLtr-cnic Blioch fIction for tlhe j band and i- wave vector

wiL : encrg:1 _--,- E ( r ,y be ex::-eanded in ter...: f t":'- : P",'. bae e f,"nction -










Y A(r) = I C(K)k(k+X,r,E.) (2-15)
K -

with the sum K over all reciprocal lattice veccors, The APW basis function

of (2-15) is, letting R be the MT sphere radius,


D(k+K,r,E.) = ext[i(k+K)'-] r>R (2-16a)

jj (sk+KR )
4ij k R) ,J-s

= t 4 4i z) Y (e k u(r,E.)Y (e,) r u (R ,E.) ZikKp k +K 2 lj m s

In (2-16b), Ykm is a spherical harmonic (and is of the Condon and Shortley

convention 7), j is a spherical Bessel function, 6k+K and >k+K are the

angular coordinates of the vector k+K, E. is the energy eigenvalue of the
J
th th
j band and k wave vector and the u,(r,E,) are solutions to the radial
z J
Schrbdinger equation (in Rydberg units)


-1/r2 d/dr (r2 du /dr) + [((+l)/r2) + VJT] u Eu (2-17)


where V1 is the muffin-tin potential.
th th
The Bloch function of the j band and k wave vector can be sym:-e-

trized to transform irreducibly as the nt basis partner of the th repre-

sentation of the group of k. First, projection operators are applied to

the APW basis functions of (2-16) to obtain the the SAPW basis functions


a (k+K,r, E ) F* (R) R.,(kb rE.) (2-18)
n,s n, s -- -- 3
RcG
-- k
th th th
with r* the n and s e .lmeat of the m.atri:,: representation of R,
n,s
and R belonging to the operac-ions of the group of (Gk). Then, with these

SAI'/W, the ..yci :Lri.zed Bloch functions are obtained


S() Cc' )' (+, r, Ec) (2-19)
nk Ks s n,s

wi h 1in hot (?-1-') rn1r (97-9) t he en'er -v ei.e,nvalue of the o.th









representation. Strictly speaking, this eigenvalue should have a basis

partner index, but for most cases (as in the present calculation) the

eigenvalue is degenerate in the basis partner index.

The effect of a rotation operation R on the APW basis function (2-16)

is defined by

-1
R ( -k,r,E) = (k,R- r,E.)

= (Rk,r,E.) (2-20)
J

-I
with R the inverse operation to R. The final form of the SAPW becomes


a O(K.r ) = K F*a (R) exp[iRk-r] r>R (2-21)
n,s n,s -- s
ZREG






(R )Rk"r
= ,*a( r (R) n 4iV s Y(eRk,9k) u .(Er)Y (e,-) r ni~G nS a R
REG m u (E ,R ) -
k s S
where on the RHS of both (2-21) and (2-22)


Rk R(k+K) and k I-Rki R(k+K) (2-23)


In most applications using the APW basis, the addition theorem of the

spherical harmonics (A3-12) may be invoked to further simplify (2-22)

,j 2(kR ) Rk'r
to (k+K,r,E ) = I 1* (R) i ----- P ( )u (E ,r) (2-24)
-REC 2. u(ER )

The purpose of this symInetrication is two-fold: the reduction of the

size of the secular matrix to be solved, and clear identification of the

conrribut- icg syri.etry to the electronic bands. As a consequence of using

SAPWs the entire problem must be formulated in symmetrized tcrnms. This

means that the phonons of the nuclear motion for the electron-phonon

interaction considered here nmu-t be expressed in terms of the appropriate

irreducible representations and basis partner (which is equivalent to the









phonon wave vector and polarization). The results cf the symmetrization

of the phonons (in the form of their dispersion curves) are given in

Figure 2-1 for the typical fee solid with one atom per unit cell. In

Table 2-1 are given the eigenvectors of the phonons at each of the symme-

try points and at each polarization (or basis partner).

One final point of the APW method concerns normalization. Proper

normalization is required if the electronic B]och functions are to be

used outside of the actual SAPW program. The normalization integral of

(2-19) is, in general, over the entire periodic macro-cell; however, using

Bloch's theorem the integral may be reduced t) N ties the integral cover

a single unit cell (where N is the number of unit cells in the periodic

volume). Reference 3 shows that the application of standard group theore-

tical methods gives for the normalization


IY*(r)'k(r)dr = N (g/d ) 1 1 Y F*( (Q [Q6
nk -nk Ka ( RE n,n Rk" ,k+K
K REG knn

j (JRk'-(k+K).R) j (k'R) (kR ) s 2
4R5 ---4Rk--- ) 2 + 4(2+i) ) uurdr] (2-25)
s l +K) I R u* (Ea,R )u (ER ) o
k s s

where g is the order Gk, d is the dimensionality of the a h representa-

tion, and 0 is the primitive unit cell volume. The notation of (2-23) has

been used in (2-25) and in Appendix III where the formcalae for the EPME

are derived based on the SAPW in the form (2-22).


2-4. The FrJhlich H-i-ailtonian


Though Frohlich 4' and many others9 using his approach had the

principal objective of explaining metallic superconductivity, the Hamil-

tonian and approach to the electron-phonon interaction which he formulated

is of a geFereal n .?t!re (wvithvn. nF cour- ', certp-in anprny:imatlaions) and




































oc
0 4-4
11 0)



, o



,4 W

0 0
0m0
0 o 0
OH ca)
0 r- 3
cu


C J rd 4 -





P4 0
0 u C


a) 0

-o -44


0 ri


> j

Q U 41




p N
0 0*



o ao 41
o 0
0I Ci4-


0

Ca)
0 4-i ,
il4-

0 -,
0 C,'r
0 0 t)

SH C


OC)
C.)
SU ri





0 0G



-d U)
-) C )
-1 cro c0



-H u




0 C C
'0 CG





O rc 1E1
'0 0 4-
4-1 -i
4-1 C) C

n G


* 0)


4i

G w


0 0




cd



C-l
CC *,-I




UN



.4 0C-I
i- i


o; 0

i-iio



0 r i




Ci E-
r.' ?














-1.
cri
xj

z



0(





x


X









< X





r-



1-4





C'




,-1











cq


(svTun laria-inTqll') A)uanbyj









Table 2-1


Phonon Eigenvectors of the Six Non-equivalent BZ Points


Symmetry Matrix Form of
Point Eigenvector


Tl T2


0 -li

i 0
1
0 12


0 -li li

1 0 0
1 1
0 2


1 li 1 + li
2 /12 2 /T2
-1 li -1 + ii
2 /12 2' 12
li --li
3 V3


0
1 -
2
1 +
2


0 i
-1 li
2 2 0
-1 li
2 2 0


R ow denotes x, y, z componcrits; column denotes polarization
(L longitudinal; TI and T2 transverse 1 and 2); i=/-l

The first two eigenvectors of T are actually pseudo-longitudineal
and the last is actually pseudo-transverse.









will be utilized in this work. An excellent paper by Hedin and Lundqvist30

(the author is grateful to Dr. N.Y. Ohrr for bringing this reference to his

attention) gives a strictly formal justification for the assumed form of

the FrBhlich Hamiltonian that has been used here. The purpose here is to

present that assumed form within certain approximations and show the

natural connection to the Born-Oppenheimer Hamiltonian.

As a convenient starting point, the Hamiltonian H of (2-2), for N

nuclei of mass M, is expanded about the equilibrium lattice sites

(Ro) in powers of the displacements (SR). In keeping with the harmonic

model, the gradient of the total energies of the electrons and of the

nuclei taken together vanishes. The only terms left to first order in the

displacements are the electron-electron

H(Ro) = H (R0 + H(R) + HE
n -- e --


= [T (RO)+Vn (P0)] + [T e(Ro)+V (RO)+V (R0)] + 6R-V (V +V ) o (2-26)
n- n- e nee- R ne eR


HEP is the electron-phonon interaction to first order in the displacement

and by an obvious simplification of (2-2), T has replaced the kinetic

energy sums. The individual operators of (2-26) may be expressed (see,

for instance, references 22 and 31) in second quantized field operator

form as


H (Ro) = E0 a a
n L

= (n + 1/2) I. (2-27)


with the a's denoting boson quasi-particle operators,


H (Ro) = C ,
e- k k .i
k -










SnkE (2-28)
k --

with the c's denoting fermi quasi-particle operators, and

4-
S= M(k';k)c,ck(a + a) (2-29)
EP k' OX OX
k,k'

In (2-29), the sum is restricted to = k' k + K (K, a reciprocal

lattice vector), the displacements have been expressed in normal mode

coordinates


6R. = (NM)-1/2 Qexp[iq-R]]C
j -
-1/2 (2o)-1/2 (2-3
= (NM)-1/2 (2 ), (a +a ) (2-30)


and the electron-phonon matrix element (EPME) [between electronic Bloch

states (2-3)] is defined as


M(k';k) =
Tk' EP k

= (NM2 ) -1/2< RV +V) ol'> (2-31)
k.' k 1 ne I R e

The energies of equations (2-27) and (2-28) have the same meaning as

those of (2-3) and (2-7) and the superscript o means the static equili-

brium lattice (unperturbed).

The totality of (2-27), (2-28), and (2-29) comprise the assumed former

for the Fr'hlich Hamiltonian H used in this work

J. C -
1H 1 E c, + EO a + M(k' 1:)c (cX(a +a ) (2-32)
F k -- -k-

So, :e explanation of these equations is now in order. In (2-28) aind

(2-29), t,i I; and k' also imply band indices. Bra and ket notation simpli-

fy the electronic coordinate integration in (2-31). The Bloch condition









of (2-1) has been used in (2-31) to remove the exponential of (2-30) and

to give the selection rule


k = k' k + K (2-33)


which governs the sum restrictions in (2-29) and (2-32). The presence of

the reciprocal vector K allows for unklapp processes.16'19 The convention

of the wave vectors throughout this work is based on (2-33) with k denoting

the initial state (valence or "p-like" band), k' the final state (conduc-

tion or "s-like" band), and q the phonon wave vector.

The eigenvectors and normal mode amplitudes cf (2-30) are chosen to

have the following phase convention


Q = Q- (2-34a)


S = c (2-34b)
-::'RX -_q~

The eigenvectors for the symmetry points used in this work (see Appendix I)

were given in the previous section (Table 2-1) and have been adapted in.

symmetrized form to the present calculation from those vectors of refer-

ence 32. These eigenvectors ate seen to satisfy the following orthogo-

nality and closure relations


e -, 5 (2-35a)



L P (2-35b)

with i,j tbhe :a, y, or z coi;:pcen':t of the eige2nvector.

In waiting down .-., of (2-32), the expli cit. lectron-Celectron and

e]ectron-nucrlear" interim .-crions are accou:,Led for by using an appropriate

screened forr. of Lhe potenrtiars in the EPME e*xp .ssion (2-31) and by cal-









culating the "bare" or unperturbed phonons of the Hamiltonian PF using

some method that gives the correct small wavenumber behavior of the phonon

energies (i.e. proportional to the wavenumber). The screening is inherent

in the Xa potential approximation for exchange and correlation effects

and the correct behavior of the phonon energies is contained in the SCH

theory if any reasonable (though not necessarily accurate) total lattice

potential (in this work, the Lennard-Jones superposition) is used to

calculate the unperturbed phonon spectra. Therefore, as used here, the

bose and fermi creation and annihilation operators of (2-27) and (2-28)

are formally "quasi-particle" operators as opposed to the usual single-

particle electron or phonor operators. This point is emphasized and given

a strictly formal justification by Hedin and Lundqvist.30 HF in the form

of (2-32), therefore will be assumed to be applicable to the present cal-

culation. It should be mentioned, however, that energy corrections cal-

culated using Hy have beei found to affect only electrons close to the

Fermi surface, while the phonon dispersion was altered substantially. 3 C 32

These facts will be used both as an aid and as a check for this calculation.

By means of HF, the energy corrections may be calculated using second

order Ra ylTcigh-Sii:rdinger perturbation theory. First, we define some

quantities for purposes of simipl ification


Ho = H (o) + H (RO)


= I Eaa +- E cc (2-36)
.i k .. k..
9 q K


n. ag I. -k
E -= L n (2-37)


wheri the -pola i/z.tio lab-l is; im plict in a as is t1he band index in k,

aid n, d doe the. occupation u..;cr-, of tihe electronic state 4 and
.9.'-









phonon state q respectively. It should be noted that the electronic occu-

pation numbers have an implicit spin summation factor of two contained in

them and that both the phonon and electronic occupation numbers are not

necessarily integers because of the "quasi-particle" nature of the field

operators. The total energy to second order in the perturbation 't may

now be written as31


E = E + + (2-38)


where bra and ket notation indicate nuclear and electronic integration

and E is the ground state total system wave function [in the sense of

section 2-1 and (2-4)] with nk electrons in state k and n phonons in

state q. The first order energy term in (2-38) vanishes due to the harmonic

nature of the phonons (formally this is equivalent to the creation or

destruction of a phonon in the total system and the resulting orthogo-

nality of that system state to the system ground state). The total second

order energy correction of (2-38) may be shown to be31


E(2) = < ( k';k) a ic (E -Hc1() aE C
S- k -ck Ck

+ akk ('E.-~) t'11> (2-39)


where the sum is restricted by (2-33). Fur clarity and illustration, FigLre

2-2 shows the interactions inherent in the assumed form of HiE and the

processes that contribute to tic-n total seci-crd older energy of (2-39). The

use of (2-36) and (2-37) in terfus o ooccLpatior nu-;hrs reduces (2-39) to
n (n +1)
E( = ;k) 1- )[----- + --- (2-40)
k, k' ,o Eb -. ,-
k k k K a
where, again, the sum: i re strictcd by (2-33). 'ihe symmetry of the eigen-

vec'tors (?-'4b) -C = ; tb'" in :!:--'. eP2:i2j rfum, it is eYpec-
q _












k'





q = (k k' + K)


(k.' k + K)


k'




(c)


k'

k k
.^--^*-






(d)


Fig:-ie 2-2.


I1 ustratio:- of the inttcractiorts involved in the perturbation
Haw iI:Jt or'T n.in cnd oj t he resultant second order energy pro-
cesses. Thi v~ ic uS picture; arin their corresponding terms are:


EEP


a c ,c
- k' k


H
EP


t
c c a
k' k!


() 2) i- -
c) .' ..c.c... -;0


-,--.-, .... _J.- ft^ __ ---


d) (2)
(d) E









ted that n = n Therefore, (2-40) may be simplified to

2r, (E ,-Ek) (1--n
(2) 11 (k' 2 k' k
E2 = IM(k';k) in,[+ (2-41)
k,k' k o 2 o02 0 0 o
kk' (-E,) E k- --

(2-41) is the total second order energy correction to the system and the

electron-phonon effect on the phonons is in the tern: proportional to the

phonon occupation n .

Identification of the phonon perturbation with the total energy E of

(2-38) necessary to increase the h phonon occupation by n gives31

E
wX in~00
2r (E-E )
= + M(k';k) 12[ 2 2 (2-42)
X ,o o 2 02
O 2 ---E

k' -

with the sum over k' still restricted by (2-33) and the polarization

explicitly included as a reminder that the transitions are between

representations of the wave vectors. For this calculation, the sum over k'

is only over available scattered states in the conduction or excited bands.

Expression (2-42) has been programmed (see Appendix IV) to calculate the

total second order energy corrections only for transitions involving all

"p-like" filled valence band states to all excited empty "s-like" conduc-

tion band states. These transitions are based on the selection rules

calculated in Appendix IT between the electron wave vector representations

and the phonon wave vector representations.













2-5. The Derivation of the Form. for the EPME


In section (2-4), the existence of an EPME was established and shown

to follow as a natural consequence of the Born-Oppenheimer theory and was

used in the Fr8hlich Hamiltonian to calculate second order energy correc-

tions to the phonons. The purpose of this section is to derive and justify

an explicit form of the EPME specific to the APW formalism.

The muffin-tin (MT) potentials are assumed to move rigidly with the

nuclei. This assumption is similar to the "rigid ion" approximation,1

the fundamental, and ~ hysically more reasonable, distinction being that

the Mi potentials are a self-consistent description of the Born-Oppenheimer

solid rather than a simple superposition of atomic potentials. Because the

MT spheres are traditionally chosen to touch so as to maximize the frac-

tion of the cell volume they enclose, there appears to be a problem of

overlapping and mixing of adjacent potentials when the MTs are displaced

(a point originally emphasized to the author33). Kenney34 has stated that

the validity and applicability of the APW method are rather independent

of the sphere radius provided the sphere remains within the Wigner-Seitz

cell (primitive unit cell). Fe also gives a plausible argument, in the

case of metals, that the sphere radius be reduced five to ten percent from

half the nea. est neighbor distance. Zimanir devotes a considerable dis-

cussion to the effects of large displaccmue-iLts (small wavenumber) which

is really only pertinnt to metals or semi-conductors. ?Estimates, based

on Kenney's suggestion, of the mean square displacement (in Rydberg atomic

units)

2 (2-3)
(Xx)2 (2-(3)









for neon and argon show that the average displacements are about fifteen

percent for neon and about ten percent for argon. In spite of this, and

due to the fact that Ziman and Kenney consider large displacements only

for metals, the overlap effect has been neglected. Both the smallness and

the flatness (small derivative) of the potential in the overlap region

(as contrasted to metals) tend to make this approximation plausible.

Following Golibersuch, the Bloch function for the jth band at the
th
k wave vector in the perturbed lattice (i.e. perturbed by the electron-

phonon interaction) is given approximately by first order perturbation

theory as

M(k';k)
T = k' + --- k' (2-44)
k k' E -E k
k k'
with the band indices implicit (as in section 2-4) in the wave vector and

with the sumration in general over all states including core and valence

(occupied) and conduction and excited (unoccupied). The superscript o

will denote throughout this section the unperturbed static equilibrium

lattice, while its absence will denote the perturbed lattice. The explicit

r dependence and the band index j of section 2-3 will be dropped

henceforth (except where essential for clarity) with the convention that

the wave vector includes the band labeling. The square of M(k';k) is for-

mally the first order (in the wave function) transition probability be-

tween the equilibzium states k and k' where, as in section 2-4, k is the

initial state and k' the final state. It is therefore appropriat- to take

this quantrity as the- EPE. Frotm the viewpornt of scattering, the -'.i. can

be related to that part of the transition matrix ( T mLatrix) linear in

the nuclear displacement.

The static unpertuIrbed Rioch0b fn tict:io-i FSirrplified c"d s'..perscriven









from (2-15), become


u = C0 o k-, (k' iK) (2-45)
'k
-- K

where the coefficients now contain an explicit reference to the initial

state wave vector k. This explicit reference is to remind us that the per-

turbed wave function, constructed frnr. -he perturted APW basis functions,

is not necessarily a periodic function


Ck = C(k,k+K)O.5(k-4K) +4 C(k,k')(k') (2-46)
K k' k+K

where the k label on the I.HS is only to indicate the initial eigenstate

of the system.

In (2-46), all possible wave vectors are needed due to the non-peri-

odic nature of the perturbed wave function; that is, the label k no longer

denotes eigensOates of the now perturbed system. Expanding the perturbed

APW basis functions and coefficients of (2-46) to first order in.the

nuclear displacements gives


C(k,k+K) = Co(kk--K) + 6R-V RC1 (2-47)
-- R

c6(k+K) = (k+) + iPVR- I o (2-48)


where 6R ar- the displ.ce.ents, V are the gradients with respect to the

nuclear coordinte- Cl, and V re the equilibriun configuration.

Using (2-47) and (2-48) in (2-46), and ke-eping only constant and

linear terms


SC ,k+) (+K) + C k+K) + C (k+K + TVilC ok')
kY R ok, L- o
K K R K R
k' k+,--

So + V C ,'(k') + Y C(k, k4).V (2-49)
ai L' o K R









The second term of (2-49) represents scattering into an available

equilibrium eigenstate, since the sum is ever all possible k' states and

reciprocal lattice vectors K. "Available" is emphasized because all scat-

tering is considered from the ground state configuration, and the lowest

energy eigenstates are filled to the Fermi level and are NOT available as

final states into which scattering occurs. Rewriting the second term to

emphasize this fact


Sk R R'l (k')
all k' R

S A(k';k)+0 (2-50)
k'k

where the sum k' over the scattered states is now restricted to the

excited or conduction band eigeustates.

The third term of (2-49) is seen [through the unperturbed coefficients

Co(k,k-K)] to allow the unperturbed APW basis function to follow the motion
th th
of the nuclei and the rest (those not in the j band and k vector eigen-

state) of the electrons. This is also pointed out by Kenney34 in a dif-

ferent context. For convenience, this term will be rewritten as


B = C (k,k+K)V re' (2-51)
k R o
K R

Comparing (2-44) and (2-49) shows that

M(k' ;k)
<0, E > + <'! IS > (2-52)
o 0E k k k' k
k, k'

The electrolnici Schr!dingcr equaticn for the perturbed situation may

be taken fror; (2-3) as


Hiek = F: (2-53)
e'k k k









with V (r) and V (r,R) the sum of displaced muffin--tin potentials
e -- ne --


V (r,R) + V (r) = 1 V,(r-R) (2-54)
ne e MT--


and j labeling the nuclei and their respective muffin-tins.

Expanding (2-53) and (2-54) about the R equilibrium positions to
-j
terms linear in the lattice displacements (6R.), one finds


Ho(Ro)[S + B] + 6V E(S + B) (2-55)

V

a- -3 R.
o oth) +; = E0o 0 an d & V M' -T_. o
e-' k k k R R3


(2-50) and (2-52) may be used to obtain

M(k';k)
-= A(k';k) + <,!B > (2-56)
E -E0 1

Solving for A(k';k) by using (2--55) gives


(Ek'-E)A(k';k) = E -E,)<, Bk> <,6V > (2-57)
k k' k k k I k k' k


With the use of (2-57) and (2-56), the final expression for the

EPME becomes


M(k; < J -j R (2-58)
k --J R
-j
In Figure 2-3, the "rigid ion" assumption is illustrated, showing
th
that the potential at ary poirn within the j muffin-tin is the same,

with i.espect to the center of that muffin-tin, regardless of the displace-

men t

0 0
Sr. R. = r R. (2-59)
--i -- 3 ~- t-

with SR. = R. R" and that as a consequence the potential gradient
-j -j J





















Equilibrium

th Muffin-tin


Displaced

jth M in-
j Muffin-tin


_./! / r


Origin


Figure 2-3. Demonstration of rigid ion approiximation for nuf fin-tin
djspj cei:ent. This figure shows that the potential with
respect to the r[uffin-tin cec'ter is the same regardless
of disulacemei t.









of (2-58) may be written as

aVMT (r.) ;, (r .) V (r .)
VMT -j I = -' (2-60)
3R. o r o o
-1 3

Since the MT potential is spherically symmetric within the sphere radius

and constant outside the sphere radius, (2-60) becomes

dV
WVV(r) = e d for r MT r dr s

= 0 for r>R (2-61)
s


where e is the unit radial vector.
r
In section 2-4, the nuclear displacements (IR.) were expanded in

normal mode coordinates for use in the. assum-nd EPME expression (2-31).

Comparison between (2-31) and (2-5S) shows that the APW form of the EPME

may be written


M(k';k) = (MN2 )-1/2 (r) V V i(r) dr (2-62)
XA thk-' -- I, MT -2
th -3
j unit cell
where E is the eigenvector and ) is the energy or frequency of the
-RA RA
th
qth phonon of X polarization, M or. th- RHS ib -th nuclear mass, and N is

the number of nuclei or unit cells (for a Eravais lattice). Using (2-61)

and (2-60), the final expression for the EPMfE becories
o-/2 yo dV T
M(k';k) --( o(r) | "W(r)dr (2-63)
lk f k -th k d -
j sphere
where the selection rule of section 2-4 (2-33) or Appendix T.I (A2-3) must

be satisfied. Appendix II shows that this selection rule allows only cer-

tain transitions between the representations of the cleclron wave vectors

and the polarizations of the phonon wave vector. In Appendix III, explicit

formulae are derived for the SAPU basis function (2-22) form of (2-63).
















CHAPTER III


RESULTS OF CALCULATION


3-1. APW Energy Band Calculation of Argon and Neon


The Xa energy bands and wave functions in argon and neon have been

calculated for the two highest occupied valence bands and the lowest un-

occupied conduction band. For neon, this would correspond to the "2s-like'"

"2p-like", and "3s-like" bands; aid for argon, the "3s-like", '3p-like",

and "4s-like" bands. The other lower lying core bands were treated in the

usual fashion as states of the crystalline potential having atomic-like

character; thus they correspond to flaL (i.e. k-i-ndependent) bands.

As is expected for an insulator, the valence Lands of argon and neon

are completely filled with a definite energy gap between the highest

valence band and the bottom of the conduction band. By construction, the

bands also exhibit the features characteristic of -cc structures. 3

All band calculations were done at nineteen sy-ietry inequivalent k

points in the irreducible wedge (see Figure A2-1 and Appendix II). This

corresponds to 256 possible wave vectors or a 4x4x4 periodic macro-cell

(64 unit cells with 4 atoms per unit cell for the fcc structure). The

angular momentum sums were evaluated through =10 and the :aximun square
2 2
magnitude used for a reciprocal vector was 80:/a Iteration-to-iteration
-6
convergence on the total energy wa.: typically at least 106 Ryd. and

usually better. As a result of this energy cthnv.prgence, the pressure,

virial rati, and poten"-i~' rs'o were wel? converiged which ensured

numerically the best possible eigeinvalues and cigenfunctions.
34









Relevant results of the neon calculations are given in Table 3-1.

In that Table, "X-Alpha Equilibrium" refers to the static lattice constant

determined to correspond to the lowest crystalline total energy. "Experi-

mental Equilibrium" means the calculation done at the experimental

constant.

Neon at the calculated equilibrium lattice constant (7.7 a.u.) is

seen to be overbound (in comparison with experiment) by a factor of two.

This overbinding is also evident in the valence and conduction bands. The

valence band widths are expanded by nearly a factor of two as compared with

the other neon (8.4348 a.u.) calculation. The bottom of the excited "3p-like"

conduction band extends into the excited "3s-like" conduction band; in all

the other calculations (including argon), the lowest excited conduction

bands are well separated. This extension would add in principle, another

representation to the "s-like" representations of Table A2-1, which were

used in the actual selection rules between the "p-like" valence

(occupied) and the "s-likc" conduction (unoccupied) bands. However, the

additional contribution was not considered in the calculation of the

phonon energy corrections and only the "s-like" representations of Table

A2-1 were used in the calculations.

The argon rei-ults of Table 3-2 appear to be very good, particularly

the calculated cohesive energy. However, the calculated band gap is only

about sixty percent of the- expcriLnLi value.

Since this work is not a comparr-tive study of one-electron models,

but an attempt to extend a particularly familiar example of such a mode]

intc a previously unexplored area, a discussion of the relative merits

of the XKa calculations is not necessary. Rather, the best possible solu-

tions of the X i--i:odel that can be obtained v-ill be utilized, regardless

of experiImetia] com-parison.









Table 3-1


Calculated






Lattice Constant
20
Atomic Mass 0Ne20

Exchange Parameter-Alpha

Pressure(in kilobars)

Total Energy

Total Atomic Energy C

Cohesive Energy

Zero Point Energy

Lennard-Jones Parameters
E
U


Quantum Parameter


and Observed Properties of Neon

Calculated
X-Alpha Experimental
Equilibrium Equilibrium

7.7 8.4348

(3.644308x104) (3.644308x104)

2/3 2/3

-0.24 -4.59

-254.9852860 -254.9844249

-254.9815919 -254.9315919

-0.0036941 -0.002833

9.3644x10-4 7.5127x10-4


4.29037x10 4
4.99436556

5.1281280-3
5.128128x10


-4
3.29028x104
5.47097073

5.572535x10-3
5.572535xi0


a


Observed
Experimental
Value

8.4323

3.644308x104



0.0






-0.0019186

4.9082x0-4


-4
2.299010-4
5.2695

8.3457x10 ~


a All quantities are in Rydberg atomic units except where noted

b
See references 21 and 36

c See reference 24


dSee Ape for definition
See Appendix I for definition










Table 3-2


Calculated






Lattice Constant

40
Atomic Mass ,Ar

Exchange Parameter-Alpha

Pressure(in kilcbars)

Total Energy

Total Atomic Energy

Cohesive Energy

Zero Point Energy

Lennard-Jones Parameters

a


Quantum Parameter


and Observed Properties of Argon

Calculated
X-Alpha Experimental
Equilibrium Equilibrium

9.66773 10.039

4 4
(7.28457841 04) (7.2845784xl04)

0.72131 0.72131

-1.21 -3.99

-1053.603303 -1053.602791

-1053.596643 -1053.596648

-0.006655 -0.006143

6.7300x104 6.2250x104


-4
7.72920x10
6.237067

9.03364x10


-4
7.13456x104
6.51149

9.07603x104


a


Observed
Experimen al
Value

10.039

7.2845784xl04



0.0






-0.006479
Z
5.9600x10


--4
7.55570x10
6.51149

8.57550x104


All quantities are in Rydberg atomic units except where noted


See references 21 and 36

SSee reference 23


dSee A-ppendix I for definition


i









3-2. The SCH Phcncn Spectra of Argon and Neon


As mentioned in section 2-1, SCH theory, instead of conventional har-

monic theory, was used to calculate the phonon spectra of argon and neon

because of zero point motion considerations. In addition, it was felt that

SCH theory represents a conceptual improvement over quasi-harmonic theory

for the calculation of unperturbed phonons.

All phonon calculations used an extremely fine reciprocal space mesh

(large number of phonon wave vectors). Symmetry considerations allow the

calculation to be done in the irreducible wedge of the first BZ (see

Figure A2-1 and Appendix II). Thus, the six non-equivalent wave vectors

used in the selection rules and their respective frequencies represent

a small subset of the calculated frequencies.

Due to the three polarizations of the phonons and to degeneracies of

the phonons among these polarizations, the set of six points corresponds

to twelve energetically different phonons. These twelve phonons and their

respective representations are listed as "Bare Phonons" in Table 3-3 for

neon and Table 3-4 for argon.

The overbinding of neon in the Xa model is reflected in the "bare"

phonon spectra. This is expected since the static solid cohesive energy

is included in Lth Lennard--Jones pair potential description. The Lare

phonorns for neo a-t 7.7 a.u. are a factor of tvo higher than the experi-

mentally det rimined phonons. Though the 8.4348 a... calculaticoi is less

overbound, tle- "bare" phonon frequencies ere still substantially different

front the ecxp.uri J Eentai frequenciies.

The static lattice cohesive ei-ergy (given in Tables 3-1 and 3-2) and

the average total eacrgy per particle, 3er~i-eruL .ud in the dCynamicL lattice

using SCHI, hai'te b.een ne.H"e to t nle, rl,',! -e th' 7"~ro pn-t r Th -ps,'il-









Table 3-3


Comparison of the Calculated and Experimental Neon Phonons


Calculation
a=7.7 a.u. a=8.4348 a.u.


Experiment a
a=8.4395 a.u.


Representation b


r15

AI

As






El

C3

Z4

Li



W1

W3


Bare
Phonons

0.0c

2.327

1.619

3.375

2.285

2.891.

2.343

1.512

3.389

1.513

2.267

2.886


Perturbed
Phonoons

0.0

2.709

2.063

3.560

2.532

3.161

2.544

2.001

3.619

1.974

2.617

3.102


Bare
Phonons

0.0

1.873

1.301

2.715

1.837

2.326

1.884

1.216

2.726

1.216

1.823

2.321


Perturbed
Phonons

0.0

2.194

1.675

2.873

2.045

2.559

2.060

1.618

2.927

1.597

2.110

2.505


aExperimental results are from reference 37 and are at T = 6,5K

bSymetry labels of ar the f 29
Symmetry labels of phonons are those of reference 29


c requ 1012
Frequr.cy units are 10 Hz (Tf!z)


0.0

1.120

0.805

1.726

1.139

1.395

1.129

0,745

1.603

0.752










Table 3-4


Comparison of the Calculated and Experimental Argon Phonons


Calculation
a=9.66773 a.u. a=10.039 a.u.


Experiment a
a=10.040 a.u.


Representation b


Fis

At

As



Xw



C3

Z4

L2'

L



W3


Bare
Phonons

0.0c

1.599

1.124

2.333

1.588

1.994

1.624

1.054

2.337

1.055

1.573

1.999


Perturbed
Phonons

0.0

1.848

1.399

2.434

1.762

2.171

1.745

1.428

2.486

1.366

1.814

2.143


Bare
Phonons

0.0

1.480

1.040

2.159

1.469

1.845

1.503

0.976

2.163

0.976

1.456

1.849


Perturbed
Phonons

0.0

1.723

1.288

2.282

1.634

2.021

1.627

1.284

2.309

1.270

1.677

1.989


a Experimental results are from rrefencrr e 38 and are for 1Ar at T=l'0K.
Phonons have been scaled by (36/40) .

b tr labels f from rfr 29
Syr :

SFrquc s are 1012
Frequency unSits are 10 Hz (Tiz)


0.0

1.342

0.958

1.909

1.363

1.716

1.356

0.906

1.915

0.911









ting values and the estimated true -zero point energy are listed in Tables

3-1 and 3-2. The overbindirg of neonf is seen again to be reflected in this

calculated zero point energy.

The argon phonons compare more favorably with experiment, due to the

better agreement between the calculated and experimental static cohesive

energy. The calculated zero point energy for argon also compares well with

experiment.


3-3. The Perturbed Phonon Spectra of Argon and Neon


The calculation of the perturbed phonons is crude in the sense that

a periodic macro-cell of eight cubic unit cells (2x2x2) has been used for

the calculation. The APW and SCH calculations were done with larger peri-

odic macro-cells to ensure the accurate representation of the solid based

on the smaller macro-cell. This smaller macro-cell is equivalent to 32

atoms since the fee crystal structure has four atoms per cubic unit cell.

In the actual evaluation of the EPME expressions and selection rules, only

six non-equivalent wave vectors of the electrons and phonons (correspon-

ding to 32 possible wave vectors) were considered. As pointed out in

Appendix II, these six wave vectors have a-sociated with them twelve re-

presentations for the phonons and the "p-like" electrons, and six repre--

sentations for the "s-like" electrons.

Since the "p-like" valence bands are full, the transition from any k

state in th- p band is only possible to an) empty k' state in the conduc-

tion bands. In view of the energy difference denominator for the second

order energy cc.rr.ct.ions (2-42), it was deemed sufficient to sum over all

possible k' states in the lov(:est "s-Jlike" ccndu: Lion band.

As the su:a; limit of the A,'W calculation was 10, the EPiE expressions









of Appendix III could only be summed through =9 because of the presence of

terms dependent on (.+1). The same square magnitude for the reciprocal

lattice vectors was used as in the APW calculation.

The perturbed phonons for neon and argon are listed in Tables 3-3

and 3-4 respectively. Typically most bare phonons were perturbed from

approximately ten to thirty percent. This is to be expected in view of the

discussion (see also section 2-4) by Hedin and Lundqvist30 of substantial

corrections to the bare phnonos given by the FrShlich Hamiltonian. However,

phonons of both neon and argon are corrected by about the same percentage

so the method in general appears to be reasonable. That is, the phonon

corrections do not appear to reflect the overbinding situation in neon.

The neon perturbed phonons are in both cases poor because the bare

phonons reflect the overbinding. The argon results are substantially

better, probably as a consequence of the good agreement between the calcu--

lated and experimental static cohesive energy. Though the good agreement

for the argon cohesive energy may be fortuitous, the method of calculating

the second order energy corrections would seem to give good results if the

bare phonons are a reasonably decent representation of the dynamic lattice

without the electrol.-phonon interaction.
















CHAPTER IV


CONCLUSIONS


The purpose of this work has been to develop a feasible first prin-

ciples (or nearly so) approach to the calculation of the effects of the

electron-phonon interaction in an insulating solid. The problem was formu-

lated in the APW-Xa formalism as a means of obtaining some comnputatiornally

practical realization of the calculation. This utilization of the Xc mnodl,

in a way which had not been previously attempted, clearly constitutes a test

of the method. Nevertheless, it was not the purpose of this work to weigh

the merits of various density functional or local exchange schemes. Rather,

the purpose was to see to what extent the features of a prototype density

functional model, with all its admitted shortcomings, could be exploited

for unconventional purposes.

The results ani discussion in section 3-3 indicated that the method

seemed to be a reasonable approach, even though a rather crude calculation

was done. This is encouraging in view of the approximations made to

achieve the calculation. Also it seems reasonable, based on the results

of Chapter III, that better "oare" phonons and crystalline potentials

would give much tUetter agreement with exp, ri:.ent. The unrealistiic nature

(at least as co-;.-red .ith the results from potential ls krovwn to be reason-

able) of the itrc phonoiis used here can be traced, in part, to the assured

Lennard--Jones pair potential.39 At best, such a pair potential is well

known to b inaCdquate to describe a rare gas crystal. Mor eo r, h.:.










already seen that the Xa model overbinds neon severely and argon

slightly, so that the calculated equilibrium geometry is compressed with

respect to the observed lattice. An improved functional form for the

electronic Hamiltonian would be very desirable on these grounds alone.

When one considers the problems with the Xa band gap in argon, and the

consequent likelihood that the Xa static lattice conduction band states

are of less than optimum quality, the value of an improved electronic

Hamiltonian becomes even imore obvious. In spite cf these drawbacks, it

remains a somewhat remarkable fact that a model as simple as Xa can be

extended so far as to yield a respectable estimate of phonon dispersion

in a rare gas crystal.

In the present APW realization of the formalism, however, it is not

practical computationally to undertake more refined calculations. The

problem is simply that of excessive computer time (see Appendix IV). This

is a direct consequence of the energy-dependent APW basis, a problem met

in other related work.40 A change of basis functions, to say, a Gaussian

type, (once the APW method has been used to calculate the eigenvalues and

Bloch functions) would reduce substantially the computational labor of

constructing a different basis set at each eigenvalue. The analogue for

solids of Sambe': molecular work, for instaLce, (a Gaussian basis Xa

method) would be appropriate.

The results represent an attempt, within certain approxima-

tions arid assumptions, at a first principles (or nearly so) calculation

of microscopic lattice dynamics. A more rigorous approach would include

the dielectric function, and not assui.:: screening effects as this work

has done. However, this would coiiplicate the calculation by requiring the

aridd.-jonls cromp'-t :tion of the dielect ic function nd its it nvrr-ce, a









difEicult task. Clearly, the use of the Lennard-Jones form for the unper-

turbed nuclear motion represents an intuitively plausible circumvention

of these difficulties.

If the method can be refined and made computationally practical, it

could, one might hope, be applied to calculations of other effects caused

by the electron-phonon interactioL:. Based on this work, the calculation

and methcd appear, in general, to be promising. This conclusion is proba-

bly the most important result of this work: the demonstration of the com-

putational accessibility of the EPME given certain approximations. With

the calculation now known to be at least possible, one may hope to achieve

better quantitative understanding of electron-phonon effects in a solid

by direct numerical investigation.
















APPENDIX I


SELF-CONSISTENT HARMONIC APPROXIMATION ANN LENNARD-JONES PAIR POTENTIALS


Since the unperturbed or "bare" rhonons were calculated from compu-

ter codes written by T.R. Koehler, we shall follow the SCH treatment and

notation developed by Koehler in a series of articles.14'42143 Following

Koehler, we direct our attention to a one-dimensional single particle

treatment to obtain the basic ideas and concepts of the SCH framework.

This treatment is only intended to be representative.

Consider a particle with a total Hamiltonian H expressed in energy

units of and in length units of a and having a mass M,


H = T + V (Al-1)


where T = -1/2 A V2 and V is the potential energy. A2 is a quantum

parameter (= 2/Mc2E ), related to de Boer's parameter4 by a factor of (2T)2

Taking the harmonic approximation to H (expanded about x=0) and

writing it as h, we get


h = T + Vo + 1/2 x2( (Al-2)


whcre V0 = V! x and 7 = V7V x=. The quantu-~. mechanical solution of

the harmouic oscillator is well known and the normalized ground state

eigenfunction is


10> = (G/I7)14 exp[-1/2 x G] (Al-3)


ere G2 2
where G ,'.





47



Taking the ground state expectation value of H yields


E =<0IHI0> = 1/4 A2G + <0(Vl0>


(Al-4)


where < I'VV Y'> = 1/2 has bee-n used.

Minimizing E with respect to G and taking explicit account of the

normalization gives


3E 2
0 = 1/4 A-- -r< i
-G <010>


S<0jx ><0 V 0>
<0 0>2


To evaluate (Al-5), some ground state matri.; elements are needed. Consider


(Al-6)


Using (Al-6) to evaluate V(( 0>)2V) yields the identity


(I0>2V) = -2<01xGVIO> + <0IV'1 0>


(Al-7)


However the LHS vanishes because the wave function is well behaved at

plus or minus infinity; therefore we get


1/2 G- =


(Al-8)


Operating on (( 0>)2V) with VV and using (Al-7) and (Al-8) gives


=1/2 G- I + 1/4 G-2<01 VVVO>


(Al-9)


If we let V be unity, (Al-9) reduces to


= 1/2 <0oo0>


(Ai-10)


Using the identities (Al-9) and (Al-10) on (Al-5) yields

2 ,-2<0 '' V7A O>
0 1/4 L 1/4 C--'
<010-:


(Al-11)


(AI-5)


V[0> -CG >O>









which is satisfied provided


2 2
A G = <01 vIO> = 0 (Al-12)


Eqn. (Al-12) is the self-consistent requirement and shows the

fundamental difference between ordinary classical harmonic theory and SCH

lattice dynamics. The force constant is classical harmonic theory is the

second derivative of the potential at equilibrium; the force constant in

(Al-12) is a ground state average tu be solved self-consistently through

G. G is seen to be explicit on the LHS ot (Al-12) and to be implicit on the

RHS through !0>. The extensiLon of the above derivation to three dimensions

and N particles of a crystalline solid is readily done (see Koehler 3).

The transformation of the problem to reciprocal space simplifies the

actual numerical calculation, just as in ordinary lattice dynamics.

The potential corresponding to (Al-1) for the crystal problem is

assumed most commonly to be a pairwise superposition of atomic potentials


V = 1/2 v(r..) (Al-13)
1J
i,j

A fairly standard and widely used potential (especially for rare

gas solids) is the Lennard-Jones (12-6) potential


v(r..) = 4c[(/r..)12 (/r. )6] (Al-14)


This potential has a zero at r..= u and an energy minimum of -e at

. 21/6 .
13
Since we want to fit E: and C, to our API results, we need the total

energy expression and the volume derivative of the total energy expression

at equilibrium volume. By choosing an N atom solid with our coordinate

system origin centered on an atom, we tay evaluate thr total classical





49



energy as

N-1
E = N/2 v(ri) (Al-15)
i=l

where r. is the non-zero distance of the ith atom from the origin.

The stable lattices for argon and neon are face-centered cubic (fcc)

with the volume per primitive unit cell


U = a3/4


(Al-16)


where a is the cubic lattice parameter. The jth atom in the lattice

is located by


r. = a/2 [ m .e + m .e + m .e ] (Al
--3 xj x yj y zj z

2 2 2 1/2
with Ir.| = r. = a/2 m. m. = (m + in + m .) with m ., m
SJ J xJ yj] ZJ X yj
th
m a set of integers appropriate to the j atom.
zj
Using (Al-16) and (Al-17) the following relations are obtained


3 3
r. = 2 U mn


dr./dU = 2/3 m3/r2
J J J


-17)


(Al-18a)


(Al- 18b)


Minimizing the total energy with respect to U


(Al-19)


dE/dJU = 0 N/2 2/3 rt'/r dv/dr.
j 3 J


From (Al-14) we have


4c f -212 13 6j6/r 7
dv/dr. 4- 4-12. /r. + 6 /r7]
J J J


(Al-20)


CombJinig (Al-16), (AIl-)), and (Al-20) gives the following


-7/6 -6 v 12 1/6
/a = 2 [ i-.6 / r:, j
J J j -


(Al-21)


I









The lattice sums in (Al-21) have been evaluated for the fee, bcc, sc, and

hep structures.s5'45'46 Equation (Al-21) shows that a/a is independent

of the lattice constant and is determined solely by the crystal structure.

The cohesive energy or energy per particle may be found using (Al-15)

and (Al-17) and solving for


= 1/2 (E/N) ( .-22)
2 (Al-21)
j [(a/(m.a/2))1 (c/(m.a/2))6


These two equations, (AI--22) and (Al-21), together with the appro-

priate lattice sums, have been used to fit and a for a representative

pair potential that best describes the lattice as calculated by the APW

program. Tihe resulting pair potential is utilized as input by KoeKler's

SCH program and the "bare" or unperturbed phoi.uns are calculated.

















APPENDIX II


GROUP THEORETICAL EXPLANATION OF THE TRANSITION SELECTION RULES


This appendix is concerned with the specific use and description of

the group theory employed throughout this problem, particularly in deriving

the appropriate selection rules and in the labeling of the bands, phonon

dispersion curves, Brillouin zone, and wave vectors. Also the group

operations of all the non-equivalent wave vectors and of the unique vectors

in their corresponding stars are given.

The entire APW calculation was done in the irreducible wedge of the

first Brillouin zone (BZ) appropriate to the fee crystal. The reduced

zone is shown in Figure A2-1, together with the irreducible wedge end the

two k space mesh grids that were used. The labeling con-'entioni of

Bouckaert, Smoluchowski, and Wigner (BSW)'9 is used throughout this disser-

tation for the energy bands, phonon dispersion curves, and all wave vector

labels (elecLrons and phoronrs). The list of the six non-equivalent

reduced zone vectors that were used in deriving the sele-ction rules is

given in 'Pable A2-1.

The derivation of i.he selection rules is based on the theory developed

by Zak4 of small re1presenta -ion.I of the gr.,ps of the various reduced wiave-

vectors. The s-lectio:i rule is calculated ca the transition probability

beLw:en a state of wave 'ecto:: 1 to a statc of wave- vector k' by a

perturbationi or exc taticn cf ave.'e vecto .,y 1'etry _. The actual calcu-

lation js u.uallv done in thj foilowi..n integral forn



















k












IU




E Wt Z" Z / Y

K













Figure A2-1. The first Brillouin zone for the fcc structure. The points
labeled are those used in the 32 point grid (which
corresponds to six non-equivalent points); while those
points labeled O are the 256 point grid (which corresponds
to nineteen non-equivalent points). The points K and U are
equivalent and the wave vector axes are in T/a units, where
a is the cubic lattice constant. The irreducible wedge (1/48th
of the first zone) is shaded.









Table A2-1


Representations and Non-equivalent Reduced Wave Vectors


Reduced Wave Vector Representations

k, k', q (units of r/a) Phonon/p-like Valence s-like Conduction

T(O 0 0) F1s

A(0 1 0) Al As Ai

X(0 2 0) X4 X5 Xi

(1 1 0) Ei E3 14 E1

L(1 1 1) L! LA L1

W(l 2 0) W, W3 W2











fY, (r)V (r)Y (r) dr (A2-1)


where T(W(r) is the initial state, Y*, (r) is the final state, and V (r) is
k k-
the perturbation connecting the two states. This integral vanishes unless

k + a = k' + K ( where K is a reciprocal lattice vector), due to lattice

symmetry.

Zak's treatment iL done for general non-symorphic space groups and

therefore can be somewhat simplified for the present work. We follow his

notation and convention and let Gk, G Gk, stand for the groups of the

vectors k, q, k' respectively, Letting g be the number of common group

elements and G be the group of common elements among the three groups
c
Gk, G Gk' one gets for the selection rule formula


/gc I ki(R)j (R) k (R) (A2-2)
CEG
c
where ki(R), qj (R), k'l(R) are the characters of the group element R
n th .th Kth
in the i j 1 small representations of k, a, k' respectively. The

above formula actually only tells whether a certain transition is allowed.

However, by utilizing the wave vector selection rule inherent in (A2-1)


k + g = k' + K (A2-3)


we can decompose the direct products between all small representations of

k and i into all allowed small represei-tations of k'.

In this work, we are coulcerrced with the tcce--centered cubic (fcc)

crystal structure with one at om pei pr.i-.tive uni t cell (i.e. 05(Fa3 )

space group in the Schaoe:flicr a:-d Intc -rational Crystallography notation).

The point group -i, 0 an d in Table A2--2, w:e give the 48 group operations

of this point-
Table A2-3 is tl! group i.ult-iplicat i-c t fcle for the Oh point group.










Table A2-2


Oh Point Group Operations
h


Operation
Number

1



2



3



4


Vector
Effect


Class


E


9
C2
4


2
C
4


2
C
4


Matrix
Representation

1 0 0
100
010
001

-1 0 0
0-1 0
001

1 0 0
0-1 0
0 0-1

-1 0 0
010
0 0-1

0-1 0
100
001

010
-1 0 0
001

100
0 0-1
010

100
001
0-1 0

001
010
-1 0 0

0 0-1
010
100

010
1 0 0
000
0 O-i





56



Table A2-2 coutilnued


Operation
Number


00
--1 0
0-1


0-1. 0
0 0-1
100

0 0-1
-1 0 0
010

0-1 0
001
-1 C 0

0 0-1
1 0 0
0-I U)


Class


Matrix
Representation

00 1
0-1 0
100

-1 0 0
001
010

0-1 0
-1 0 0
0 0-1

0 0-1
0-1 0
100
-1 0 0

-1 0 0




100
010
0-1 0






010
001
10 0


Vector
Effect










Table A2-2 continued


Class


C3


Matrix
Representation

0 1 0
0 0-1
-i 0 0


Vector
Effect

y
-z
-x


a The operations 25-48 are obtained by reversing the sign of the first 24
operations; ..e. they are the inversion operator (denoted by the class
symbol J ) times each of the first 24 operations. As an example, consider
operation 4 times the inversion: the vector effect becomes (x -y z),
the operation number becomes 28, and the class becomes JC .
4.


Operations
Number

24 a






















_- j C4 cq -T '.0o K co r- 0 -1 "' c'1 -z- iLn 'o WX r- OC)r' cJ '- -4 '- M
T-1 r-4 r-I H r-4 H -i r-4 CM4 il CM C C-*>


Sr M c r-c mc o7%
--1


v-i CN ''n tn \C i c3. c a r- H N C) -,r
-4 -1 rH i if -1 -4 r-H CH C4 (N CM CN M


.4NcO
1" rM CO



"I Hi cM




H CM r-l C4
C CO
NNCN







co NHN -
in cy\ ot








CM r-.4

u OXO








r-ql O i
O N -



















m0t c
-4H H L






C)ci L











.4 r-1
U


CA r-i r-4 H-







IC- cl) in C) %Z
N r-HH r-H






-ONCO -O$ --
M- r- H H-


CM r-l r-4 r-4

H- CX) C,4 10 NO4

CM m-I r-4 ON r-c
m mr--CLff\ r












C14 r- -- r-i


r ( c- d M o--
r- m C 1-.4 t C)









CNM r- r--


r- 4 -4 -C H -4

Lr).n c- co0 -t C-)

SC) c -A CM) -



C) N .4 r--a' ( .
C-i M r-1 -I

cn CC-) -I H C
-1 C-1l CM

*r-1 CM NN


r- H CM H4
rHHN- CM r-N



rl rN rH i
A N N v--q

H,-q H -T rD CD


4H r-) -

CM -i 4. c) 0my n


co ~Ir-f
H

HHn HN
1- ii E


H-( C-J e' L' 'Do r-t C,) 01


rHl

00


1-4


\0
rL


rC) 0v*-1
r-I r-l


r-l r-i
U3 r-l
crl r-l


CM ool u


r--




r -1
H C


hr-


ilr


m c r inL i'D
rH HH ri

r)in o o o) O -
r- -l Hr


-M4 C NM
NNN




CM






NO
r- C
CM4





rl





-r-lr CM











r-l

C-4 L O
CL CC 0'r
i -4- c


0r!
r- I-


N
r-(


C i-.- co
r-i r-l









C\C
CI r-l

H CA C)
4 CM4


C\y r-1




r-i n


NHC

lN C N
r-i (-z








-4





Lowe


H

r -


X-; ;


r-l CM



C4 H -
CN M


CO rm -N- C0
rN N H


(r1- r-










N C) h-
Sr- r
HMH












N H
-4LC
C r-i









N H

-Cl N- C





-r C, C)

M iC m


r-A r-l
CM rlo






rHNc '
r-l C
^0 M C


mLr c r-i
r-i

MC r-A CO
r-A -i


C.4
N





HC
-ON
N

>hNH











ci '-0 0.
m-f c




*> h- IA
,- N -

,0 O
cH \O


0H v-O C H
rr-4


r- l- r-q r



H- ,-3 '. H -q
-d C) N10 '.- a%








-,i C)t L'r0 O CH O






r-H r- --I r
d0 Gn r0
crHl C' C-l rl









r- C C41 H C4C
H r-i N H r-i











cm4 CM H NM CM -M


r( CM CM CM C4 CM


u









Only the filled p-like valence and the unfilled s-like conduction

bands have been considered and allowed to interact. The p-like representa-

tions (associated with the vector k), the s-like representations (associ-

ated with the vector k'), and the representations associated with the

phonon wave vector q are listed in Table A2-1. t.1 wave vectors are in the

reduced zone scheme and are in units of T/a, where a is the lattice constant.

In Table A2-4, are listed the group operations belonging to each

of the 32 equivalent BZ points used throughout this work. This table and

the character tables of the small representations of the wave vectors2

will be used in the illustrative example that will follow.

As an example of the selection rule calculation consider an inter-

action involving the vectors k = (0 1 0) and ( = (1 1 0), and the k repre-

sentation Ai and the representation Zi. By using (A2-3), all possible

allowed k' vectors are seen to be k' = (0 1 0), (1 ] 1), and (1 2 0).

Therefore one must use the selection rule formula on all the k' represen-

tations of A (0 1 0), L (1 1 1), and W (1 2 0).

Considering first only the A representations of k', Table A2-4 shows

that the groups of operations associated with c = (] 1 0) and k =- (-1 0 0)

have only two group elements in cominon: the identity element in class E
9
and an improper rotation element in class JC(. Therefore the selection

rule (A2-2) reduct-s to


1/2 [a(E))E (E)(6 (L) + (JC 2 (j c)2 i J (A4)


The character tables in the ESW article9 are used to reduce (A2-4) to


1/2 [ F.( + (J-C (A2-5)


which when evaluated over all Li representiations shows that only the AI,









Table A2-4


Group Operations of the Thirty-two Equivalent First BZ Points


Symmetry
Point

F

A












X


Group Operations


Star Member
(Ti/a units)

(0 0 0)

(1 0 3)

(0 1 0)

(0 0 1)

(-1 0 0)

(0 -1 0)

(0 0 -1)

(2 0 0)

(0 2 0)

(0 0 2)

(1 1 0)

(1 -1 0)

(-1 1 0)

(-1 -1 0)

(1 0 1)

(1 0 -1)

(-1- 0 1)

(-1. 0 -1)



(0 1 -3)

(0 -1 1)

(0 -1 -1)


All

3 7

4 9

2 5

3 7

4 9

2 5

8

10

6


483

8

10

6

8

10

6

13 1

12 1

II 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

i 1

1 1

1 1

1 I

1 1


40

39

38

40

39

38

28 31. 32 37 40

28 33 34 36 39

28 29 30 35 38


Operations

26 28 37

26 27 36

27 28 35

26 28 37

26 27 36

27 28 35

6 25 26 27

5 25 26 27

4 25 26 27

.1 26 38

4 26 35

.4 26 35

1 26 38

2 28 39

5 28 36

5 28 36

2 28 39

3 27 40

6 27 37

6 27 37

3 27 40









Table A2-4 continued


Symmetry Star
Point (%i/a

L (1


Member
units)

1 1)


Group Operations


(-1 1 1)

(1 --i 1)

(-1 -1 1)

(1 2 0)

(2 1 0)

(0 1 2)

(0 2 1)

(1 0 2)

(2 0 1)


41 42

45 46

43 44

47 48









A2, and As representations contribute.

Applying these same techniques to the other two cases (the L (1 1 1)

point and W (1 2 0) point), gives the following decomposition between the

Ai and El representations


A, s Yi, = Ai e A2z As e L1 ( LI L3 L3 I & W1 i Wi & Ws (A2-6)


where E denotes direct product and (t denotes direct sum of the represen-

tations.49 As a check on the decomposition, the product of the star number

(48 divided by the order of the group) times the dimensionality of each

representation of the LHS of (A2-6) must be equal to the sum of the star

number times the dimensionality of each representation on the IRS. Using

(A2-6) as an example, we see that the LHS is equal to 72 [Ai dim. = 1,

star = (48/8) = 6; 1i dim. = 1. star = (8/4) = 12] and the RHS is equal

to 72 [Ai dim. 1, star = 6; A2 imu. = ., star 6; As dim. = 2, star

6; Li dim. = 1, star = 4; LZ dim. = 1, star -= 4; L3 dim. = 2, srar = 4;

LS dim. = 2, star = 4; W1 dim. = 1, star = 6; WV din. = 1, star = 6; W3

dim. = 2, star = 6]. The direct product is also seen to be sytmnetric, i.e.

A1 E lj = 1i N A1.

In the actual calculation only the rreresentations that are in the

s-like excited conduction band are used. From (A2-6) and Table A2-1,

the only representations that contribute in our example are seen to be

A!, LI, anrd i A list of all possible vajlence band to conduction band

excitatio:ns for each a (phonon wave vector) r-pr esentaticon has been con-

structed anri i;, given in Table A2-5.

The Liectron-phionon matrix element (EPI:E) calculation has been

siumplifie-d through thi- use of group theory. TSpcifically the iEPc:i is first

lcl !at.r d for a fixp ri szt;?r i;irn,iber of tie- -' vector a rd all possil le star









Table A2-5


Selection Rules between Valence p-like Band and Conduction s-like Band


k
Representation

Fis

A1

As


xA


XE
XI



Z3

E4

L2

L1

W?

Ws

Fis


AI



x

xi

ZS

E3

E4

Li

L73

W3:

V3


a1
Representation

is5

f15

F15

Fis


l15






F5

S15
PI


F-5


A,
Ai



A1



A1
A1
A1

A1


A!





A,

A1


k'
Representation

Fi

Al

A1

Xi

Xi

No Contribution

2ZI

No Contribution

Li

LI

No Contribution


WA




Li
Xi 6 Zi eC F








A, E+ Li W






C1


xl
X, ~ ir





64



Table A2-5 continued


k
Representation

I15

Ai

As

x

xi

Ez

23
EZ

L,~

LT

W1

W3

F15


A5

xAs
x4

XI

E,

Z3

Z4

Li
Li


L'-


WG


a
Representation

As

As

As

As5


A5
As


As

As

As
As

As


xT
As



xi
X4



x
XZ

xi
x 1



X

X4'

xi

Xii
xi
xI

x"

xZ L


k'
Representation

A1

-1



= i $w

ai e wj

S A e Ll e W'

Ai e L1. ID Wi

Ai e Li e W'


Z1
= 221



Xi 2C1







rl X1 e2
Xi

Axl



= Fi e xi













= No Contribution

A11
= Ai









Table A2-5 continued


k
Representation

r15
Ai

As


A 4
xl



w3

F4
LCi

LI

W

W3


Al

As

x?


"S

ZL'


xl


Li


W


I
Representation

x

X!
xv

X?

Ax

xT

xl


X5


xi
xy





xi

C1



C1












Xi

Zi

C1

Ci


k'
Representation

= xi

= W

= A e w2

= No Contribution

= rFI 2XI

= E.

2Z1

Ll

= L1

= 2LI

= No Contribution

A AI W2

No Contribution












- A 1



A! & W2

ti A1e W



A1 e L T





66



Table A2-5 continued


k
Representation

1 r


-S
Representation

Z3

:3

13

13

Z3

E3

-3


z3
E3

23
Z3

-73

Z4


1:4
E4
:4



E4



C4
2:4

Z4



2,


k'
Representation

= 221

= LI

Ai Li 9 W2
- AzeLIGWL

= El

2i1

2:1
= 2lX121




= .A1

= AI

= A e w2

= LI

S Ai e L1i Wz

= No Contribution




A= e L1E Wi








1= 1

Fi e Xi e Zi

= No Contribution

- Ai O W"



/,, .i, vQ










Table A2-5 cont-hued


k
Representation

r15

A1

A5








U3
44



x?













A\1
Z4

L21

LI
A5


















U!
4


_q
Representation

L2*










L2.


zI
L21



LA,

Lzv








I,




L 1


i.'


L)

L.1







L1


k'
Representation

Li






LI

Ll

AI, We

Al

No Contribution

rI e XI

XI

E1



L1



21I

LI

2Li

A1 e W2

A, C W2

A, e w?

No Contribution

r1 e 2X1



71









Table A2-5 continued


k
Representation

15

AI

As
xg



El
xi



Z3

Z4



Ll

wl



Fis





X,'
A5


xT
x5

El

F3

Z4

LW

Ll



W3


1-
Representation


wt
W2


wl






VTT
wl
WY




wi
wl

wl



wi



W3


Ws


23

W 3

W3
W3



W3
W.



W.,


k'
Representation

= No Contribution

= XI 1



= No Contribution

= No Contribution

= Ai & Wl

SL








= 2 e XI 2ZI

= x1


= vz


= 2XI 8 2z1


- A1



= AI LL W&W
= AI 8 LI 8Wi

= 2 e Li ( W

= 1





S Fi X 1 2Z1
= Zi

=w









members of the vector k, then this set of EPIEs is transformed to a new

set at each of the other members of the star of k'.

In the computer program, the EPME is calculated for the identity

star member of k' and all the unique star members of k. The EPME is in

general a 3x3 matrix (three components: x, y, z; and three polarizations

or alternatively basis partners) and may be operated on by the appropri-

ate rotation matrices of Table A2-3. To write down the explicit transfor-
th
nation, a group operation of the mth star member of the vector k is denoted

by and the identity star member of k' and ith star member of k of
--m*
the EPME is denoted by M1i(k' k; ). The transformation of this EPME to

the EPME of the jth star of k' and Ith star of k is accomplished by


M. j(k' kL ) = R = M 1(kl k) Ri (A2-7)
J -j

with R-1 the inverse oDeration to R and the 1th star member of k deter-

mined by


S= R (A2-8)
-- -- i I*

The necessary group operations of the star members of the various

reduced wave vectors are given in Table A2-6. This table is based on the

irreducible wedge given in Figure A2-1, i.e. all the identity star members

of the reduced wave vectors are contained in this wedge.









Table A2-6


Star Group Operations Generated from the Six Non-Equivalent BZ Points


BZ
Point

r

A












x


Star
Generation

R(O 0 0) = (0 0 0)

R(0 1 0) = (0 1 0)

= (0 0 1)

= (1 0 0)

= (0 -1 0)

= (0 0 -1)

= (-1 0 0)

R(0 2 0) = (0 2 0)

= (0 0 2)

= (2 0 0)

R(1 1 0) = (1 1 0)

= (1 -.1 0)

= (-1 1 0)

= (-i -1 0)

(0 1 1)



= (0 1 -1)

= (0 -1 -1)

= (1 0 1)

= (1 0 -1)

= (-1 0 1)

= (-1 0 )


Group
Operations

= All 46 Operations

= 1 4 9 10 26 27 36 39

= 7 13 17 21 32 40 43 47

= 6 11 18 24 29 38 44 46

= 2 3 12 15 25 28 33 34

= 8 16 19 23 31 37 41 45

= 5 14 20 22 30 35 42 48

= 1 2 3 4 9 10 12 15 25 26 27 28 33 34 36 39

= 7 8 13 16 17 19 21 24 31 32 37 40 41 43 45 47

= 5 6 11 14 18 20 22 24 29 30 35 33 42 44 46 48

= 1 11 26 38

= 3 6 28 29

= 4 5 27 30

= 2 14 25 35

= 10 17 39 43

= 12 21 33 47

= 9 23 36 45

S15 19 34 41

S7 18 40 46

= 8 24 37 '4

= 13 20 32 48

= .C 22 31 42


F,









Table A2-6 continued


BZ
Point

L


Star
Generation

R(U 1 1) = (1 1 1)

= (-1 1 1)

= (1 -1 1)

= (-1 -1 1)

R(1 2 0) = (1 2 0)

= (2 1 0)

= (0 1 2)

= (0 2 1)

=(1 0 2)

= (2 0 1)


Group
Operations

= 1 14 15 16 17 18 25 38 39 40 41 42

= 3 5 10 13 19 24 27 29 34 37 43 48

= 4 6 7 12 22 23 28 30 31 36 46 47

= 2 8 9 11 20 21 26 32 33 35 44 45

= 1 3 13 16 26 28 31 32

= 5 11 19 21 30 38 41 47

= 6 14 17 23 29 35 43 45

S10 12 22 24 33 39 42 44

= 2 4 7 8 25 27 37 40

=9 15 18 20 34 36 46 48


w
















APPENDIX III


EXPLICIT DERIVATION AND FORMULAE FOR THE EPME


We have shown in Chapter II, section 2-5, that the EPME expression

to be evaluated numerically from (2-63) using SAPW basis functions is


M( ', ;n,) -


(j 2 7T I I C*a (K')C' (K)rF, (R')rF
K' K l R ,s- n,s n ,s- n,s-
K' K R'SG REG s s



m' R'k' R'k')Y m(Rk''Rk) m'-' r imd
A xrM m -- -- P
/t2~~~~~~~~~ 'tPk''r m6R~R rZ


"' j 3(k'Rs )(kR )r s ud d
1 S-,. j j u* tE',r) V u (E,r)r dr'
Sc u*,.(EiR )u (ER )j E dr Z
k S s 0
0
By considering an arbitrary polarization vector of the form


S- = ae + b + ce
--CA x y


(A3-1)


(A3-2)


we can consider the x, y, and z components of (A3-1) separately. This

is done by remembering that


S= cossi sinetsinQo + cose$
r x y


(A3--3)


where the angular convention of the spherical harmonics is used. Taking

the dot pre'duct between '. and c gives three separate angular parts,
r
one for each conponent


.m y fR
Ym* (' ) ,i, ) v* (6.6r)cosi'sinOYY (6,0)df (A3-4a)
E m "k'k R 'L *, k R}^ L i mnm









,, *r5Ym,)) ( (A3--4b)

Y*r' Rk' ,,R',r Y "', ,(0,4)cos sY- (0, )dS2 (A3-4c)
"m R'k, "R'k )Ykm (kd, RI,.) i'co ,


We shall derive explicitly the z-compon.nt expression to demonstrate

how the final expressions were calculated. The following identities for the

spherical harmonics and Legendre polynoTials are based on unpublished notes

of Trickeyso for use in a k.E energy band method. The spherical harmonics

are of the Condon and Shortley convention27 and are defined by


Ym (6,6) = Cm P m(cose)exp(im )

Sm (2.+_)(,-m)! 1/2
k = (-1) ] -K.)
"m v / l 4W. (br.)!


where


(A3-5a)


(A3-5b)


and Pkm are the associated Legendre polynomials.

The identities that are used in the angular integrals to invoke the

orthogonality of the spherical harmonics are

( l)., +m)
cosY -, ( ) [--+ + (A3-6)
-m (+m1) CC m -lm

Cr (24(r)+-1-) ( C-m+1i) (Z-m+2)
sinOY r exp(i") OY Y -(-3-7)
ra (22-+1 C_ -lm-.L C Y+lm-1

sinOY, exp (-i) L-- Y I] (A3-8)
in (2 I/2 C(ep 3i -e-xp-1 Ci] A-9,+1

sin6 = 1/2i (exp[it] exp[-i1]) (A3-9)


cosO = 1/2 (exp[i]J + exp[-ie])


rY*Y ) Ira "Y -
E, rai (C. ,2 n


(A3-10)


(A3--11)


P (cosy) -- 2-- Y* m(0 9) ( ,,) (A3-12)
m=-P'

where cosY ucoscoso coss + in +sinO cos( -2) and P' (u) are the
1 2 1 2 1 '









Legendre polynomials. The greatest simplification of the spherical har--

monic expressions is due to the next identity which is generally not well

known, but which is proven by Trickays0 :,n his unpublished notes

dP dP k
dP+1 dP (4 ) (-m- /211 m(6 Y2(2, )
cos -- cos 4r [ ( Y* 2 )y
1 du 2 du (22+1)(22+3) m 2 +lm 1'
(A3-13)

The LHS of (A3-13) may be simplified by using the following recursive

relations of the Legendre pclynomidls

dP
dP (Z+1)
du 2 [ uP P ] (A3-14a)
1-u
dP
du 2 [ uP, (A3-14b)
l-u
Therefore (A3-13) reduces to

dP dP +,
R+1 d (+!),
cos cos d [ P (cos-ucosS. + P+ (ccs -ucos )]
1 du 2 du 2 k 1- 2 k+1 2 1
1-u
(A3-15)


It is seen that we have an apparent singularity at u = 1 (which corres-

ponds to parallel or anti-parallel directions).. However, by taking the

appropriate limit of (A3-15) as u 1i, we may shove that

ddPP d
limit cos--- cosO-- (l) '(Z+) cos9 (A3-16)
1 tdu 2 du 1

By using (A3-6) in (A3-4c) and invoking (A3-11), we obtain


Y ,( ,e ')Y ( ) Y* cos'Ym dQ
k, m,ri -m R- Rk- t km,
(2-m.) (9~~+! "1/2 ,Y. ,
S Z (2-1+-1) (2+3) +lm 'k"Rk' mv Rk' Pk'v
The of be fu r i

+ [ j +(1 t ),Y (t6 ^)li 'Rk' ) Rk ) (A3-17)


The RPS of (A3-1i7) may be further simp ified through the use of









(A3-13) and (A3-15)

(,LFi) [(P (CosS -.uc ,,) + P(cosRk,-UcOSRk)
( Rk1 )R k
(P (cos6 -ucose ) + P (cos -ucos ) ) (A-
2 + R R 'k' Rk R'k'
1-u

(pE+1(coseRk'-ucos6Rk)+ P(cosRk-ucosR'k) )] (A3-18)


The reason that the terms are not combined in (A3-18) is that there are

Z-dependent radial parts multiplying each ter-m in the parentheses within

the square brackets. Combining (A3-18) and (A3-1) gives for the final

expression of the z component


M (n',k';nk) -4i C ( (R')* (R)
2 ." S c n s --
K K R'E( RES s- s
iC-k

k +1 Ek Rk' kRk



u (E'R ),:-ER) )+1 T r


(P (cos9 *-ucosek) + P (cosO, --cosRk) )
-p






R'k' Rk + P (cOS RkUCOSe R'kI)

j (k'R )i (kR ) Rs d
_S s__ dV






s) s u (E' r)- (E,r)rdr (A3-19)
k s .u-,1 s '

For the M1 and M co7.ponents of the EPME, we have applied the above
x y
methods and have obr.aJned the following e.rpressions

(M (- ,- 4,) i y (K)? ip,')P* (R)
x ? LI
x K' K R'cG., RG s' s n ,s n,- ns

(I (P (cos Rkin0 uos' sirn9 ) +
2 (1 R k Ru '' Rk'
-R 2-u -- '-


S...k. (k )j i ,r' )., s .dV 2
P2 (co Rksin, .k''-u Rk -in Rk) u .... u E r ( )+1 )d- ( )r dr
K+1 s I sC.

(P (co ,sin0 -c; i ) +.
+1 R'k k ,k Ri.









ji ('R ) j Z+1(kR ) RsV 2
P (cos sinS -ucosj sino )) -- -, *(El)--l )(E)r dr ]
-- 'R -ER -, d -+i


(A3-20)


M ( ',k' ;,k) = 47i Y
Y K'


K R' EC ,
k


- k


s rs -- a,s n ,- u1, -
s S


+I) [ (P (sin. in9 using R' n',) +
2 +i Rk Ak .R' k
l-u


-sin, i(k'R3 )j2(kR ) r E dV (E)
-usinknRk u )) T--7- -, ( (E) r d
-- -- v s S o


-(P+1 (siniRk7siuR'k'-usin Rksi6 R)


j (k'IR)j (k ) Rs
P (sinjksin k-usin R'k'sinRk )) s "(E'
e R k R'k' Rk' R )us c
z s V s


dV 2
)--V(E)r dr
dr +1


(A3-2i)


Equations (A3-19), (A3-20), and (A3-21) have been coded into the

EPME program (see Appendix IV) together with the appropriate limiting


expression (A3-16).


P (sin Rk'sinR,'k'
















APPENDIX IV


DESCRIPTION OF COMPUTER PROGRAMS AND CODES


All numerical calculations have been done using a set of three

programs. Two of the programs have been used and tested in many different

calculations. The other program was writte,-i y the author during the

course of this calculation.

One of the establish-d programs calc.:laies the energy bands and

electronic wave functions based on the APW model, using computer codes

developed by Wood and others,51,3 and -:,iade self-consistent and modified by

Connolly and others. s52''"3 In the course of this work, it was found

necessary to modify the numerical precision (from single to double) of

the card output of the potential, eigenvalues, and basis function

coefficients. Also the basis function coefficients had to be normalized

properly for use outside of the APW program [using (2-25)].

The other established program is Koebler's SCH code' based on the

formalisri cutli ~ ed in Appendix II. This program calculates a set of phonons

on any given rec-proc;!i space mesh and at any ter.perature, given the

crystal st_'ucture, lattice ipran.neter, Lteonnird-Jones parameters, and the

temperature. YThi code lihac b,'en used ai.d modified by Trickevy' for dev:--

lopment of certain point transfornr problems.

The third pro ras: w;!s vriitten eiiti;rcly by tihe author except for

certain subroutines which vere borro:'ed from. thie AP program to perform

similia tas.ks. YTh program: calculates the UP i. ov'ir- the entire Brillouin









zone. Presently all s-like band to p-like band transitions or all p-like

band to s-like band transitions can be done, given that one of the bands is

full. The calculation of the perturbed phonon spectrum is also done within

this program.

It should also be mentioned that the APW and EPME programs use

synmetrized basis functions. Group theory and projection operators are

used to specify lists of contributing reciprocal lattice vectors at any

given reduced zone vector. This "specification deck" is used as input to

the APW and the EPME programs.

All programs are written n FORTRAN anr have been compiled with

I.B.M.'s FORTRAN H-Extended compiler an.' lin-edited ard overplayed with

I.B.M.'s F-128 linkage editor. Numerical precision is double precision

throughout all programs approximatelyy 16 decimal digits).

The EPME program has been overlayed and arranged to run in 128K bytes

of core. The program is also designed to be restartable at any point in the

actual EPME calculation. The typical Lunning time for the calculations

done in this dissertation was 70 minutes CPU (360/M195) which is approx-

imately 140 minutes CPU (370/M165) for all the p to s transitions

on a 32 point reciprocal space mesh. The APW calculations ran about 5 to

10 minutes CPU (370/M165) depending on the desired convergence. The

amount of required core depended on the basis size. The SCH calculations

ran typically Iessf than a minute and in about 150K of core.

Numerical integration an deifferentiation are done in the I,'ME

program. The po.teiiLal has to b7 differentiated, which is done by using

an eleven point Lagranglan diffcrefntiation aigorith.53 The routine was

checked against a known f'cct;cn ( f(r) 1!/r exp[-ar] ) whose exact

derivative is clc1ulable. lThI nuua.ricca and exact derivative values agreeci









to full double precision for all values of r tested. This function was

chosen as a test case because it closely resembles the numerical form of

the self-consistent APW potential. The nu:merical integration routine is

the same as that used in the APW program and is based on a five point

Newton-Cotes integration algorithm.54 The radiaJ wave function of the

Schridinger eqn. is solved by the Nuaercv method3 in a subroutine also

borrowed from the APW program.

All program labeling and numbering conventions of the energy bands

and electron wave vectors follow BSW29 (see Appendix II) and are based on

the "specs" deck used for both the APW and EPME calculation. The same

notation is used in labeling the dispersion curves and phonon wave vectors.
















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BIOGRAPHICAL SK~TCE


Joseph Patrick Worth was born on 25 July 1949 in Salzburg, Austria.

He attended various public and army base schools in Europe and the United

States. He was graduated from Cuyahoga Falls High School in Cuyahoga Falls,

Ohio on 9 June 1967. From October 1967 to June 1969, he served in the

United States Army and was then honorably discharged. He enrolled at the

University cf Akron in September 1969 and was graduated in June 1972 with

a Bachelor of Science (Physics) and in June 1973 with a Master of Science.

In September 1973, he began Ph.D. graduate study at the University

of Florida. He joined the Quantum Theory Project in June 1974 znd has been

working toward his degree since that tine.


















I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




S.B.. Trickey, Clairman
Associate Profes or of Physics






I certify that I have read this study and that in my opinion it
ccnforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.



A/ A/lt/
N.Y. Ohn
Professor of Chemistry and Physics






I certify that I have read this study and that in my opinion it
conform. to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dJss>rtation for the degree of
Doctor of Philosophy.



Z*-i_-^____----~
JE Sab in
As ociaLe Professor of Physics
end Chemistry


















I certify that I have read this study a& d that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




T.A. Scott
Professor of Physics






I certify that I have read this study and that in my opinion it
conforms to acceptable star-dares of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.




C.E. Reid
Associate Professor of Chemistry





This dissertation was submitted to the Graduate Faculty of the Department
of Physics in the Colle'g ot Arts ard Scieaces and to the Graduate Council,
and was accepted as partial fulfJile:et cf the re-quirements for the degree
of Doctor of Philosophy.

December, 1976


ea:., Graduate Schoiol




































UNIVERSITY OF FLORIDA
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