UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
APW CALCULATION OF THE ELECTRONPHONON 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 putup 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 11. Purpose of Calculation ......................... 1 12. Assumptions and Approximations .................. 2 13. Outline of Calculation ........................... 4 II. FORMAL THEORY ANID METHICDS OF CALCULATION ................. 7 21. BornOppenheimer Theory for Crytalline Solids .... 7 22. The Xa Hami' onian ................ .... ........ 11 23. The APW Method ...... ............................ 13 24. The FrIbJlich Hamiltorian ....................... 16 25. The Derivation of the Forn for the EPME ......... 27 III. RLLTLIS OF CALCULATIONS .. ................... ............. 34 31. API Energy Band Calculaticn of Argon and Neon ... 34 32. The SCH Phonon Spectra of Argon and Neon ........ 38 33. The Perturbed Phonon Spectra of Argon and Neon .. 41 IV. CO CLUSIOJ :S ........................... ....... ............ 43 I. SEI"COS i' ;'f T HARMONIC THEORY AND L.'NARJDJONES 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 ELECTRONFHONON INTERACTION IN SOLID ARGON AND NEON By Joseph Patrick Worth December, 1976 Chairman: Samuel B. Trickey Major Department: Physics The clectrcnphonon interaction in solid argon and neon was calculated within the selfconsiszent field au~.nted plane wave local exchange(SCF APVX:a) model. The Frihlich Hamiltonian, which uses the electronrhoron 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(ectronphonou mnatrix element (EPME) was used to compute correctionLc to ohonons iun.pecturbed by the electronphonon interaction. The unpertur'bep. phonons were calculated within the selfconsistent harmonic (SCB) appro': imation, assuming a superpo.;tion of LennardJones 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 APVX> cohecsivc cJIuergies. Explicit expressions for the EPME were derived in the AP;' formali ~, as wer; selcction rules between representations involved in the relevant interband 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 11. 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 electronphonon (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 ronpho:'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 coputi 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 muffintin 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 muffintin 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. 32. Assumptions and Approxinmations Perhaps the most crucial assumiptica for the overall calculations is that of allowing the nuclei and electrons to interact among themselves and each ctheer only through shortranged screened potentials. Such screened potentials a'.: inherent in the selfconsist n.t field (SrC) APW description of th electrons and the Fr'hlich Eailtoni:'sn which will be used to calcu late energy corrections to the phoinns ariing 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 BornOppenheimer 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 nowtraditional muffintin type (the sort originally discussed by Slater2), a oneelectron potential must be available. Even at the level of HartreeFock 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:uffintin 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 muffintin 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 multiphonon 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 LennardJones pair poten tial constants. s The fitted constants are used in a LennardJones 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. 13. 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.fconsistency is obtained through oi)n of three criteria: convergence (or more exactly, iteration to iteration stability) on the pressure, oneclectron 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 126 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 LennardJones 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 plike valence band states to the lowest slike conduction band states. The fact that these bands exd:ibit simple sy~rcietry properties throughout the zone is a great conve.nienrce for ccmirputttion. 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 21. BornOppenheimer 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) (21) k An n k V(r4R ) = V(r) (2la) 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 xn The extension to multipartic]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)) (22) j n 1 ne e j1 i where Rydberg atomic units will be employc'e throughout this work except as otherwise noted. In (22), M is the nuclear mass with the suia j over nuclei, the sum i is over electrons, V (R) i; the nuclearnuclear inter n.  action potential, V (r,.) i: :he nucle:rel"c.ron interaction potential, ne and V (r) is the eolcrronclcctron 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 (22) 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 (22) 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) (23) th with E,(R) the total electronic energy of the kt state dependent upon the nuclear coordinates R as parameters. The electronic eigenfunctions of (23) 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) (24) k  th where the expansion coefficients depend on the k electronic state and the nuclear coordinates R. Operating on (24) with the total Hamiltonian H (22) 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)7V (kR) 1/2M (_ R)2, (r,R)) (25) R k R R /2 R k By multiplying the last two terms in (25) by *, (r,R) and integrating over the electronic coordinates, these terms; may be shown1720 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 (25) (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 (25) ( H(R) + E (R) ) .(k,R) = E (R) a(k,R) (26) where, in the case of a lattice dynamical treatment, E (R) = (n + 1/2) (27) 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 (26) 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 deterination of the dynamical matrix (a matrix of selfconsis 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" rnor c Qtr, "ii'rvv prciM,m; (73) n r.',s" from, 1h''. ro'Tnybody Ppt,;re of the electronelectron interaction ccmbii.ed with the requirement that the electronic wave functions be properly antisymmletrized. Almost all treat ments have as their aim the construction of a oneelectron 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 (23) that results, with particular e  emphasis on the features needed to achieve the eventual goal of this work. 22. The Xa Parailtonian A brief outline of the Xa method for the nonspin 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 Hamilion 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 X ij T 1 i L j m j  12 12 + 1/2 fp(1) xU (1) d (210) 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) (211) with n. the occupancy of the spin orbital u.. The local exchangecorrela 1 1 tion operator in the nonspin polarized case is defined as 1/3 U (1) = 9a(3/8Tr p ())13 (212) with a a parameter to be calibrated against, for example, atomic properties. Application of thc variational principle to (28) leads to the effective oneelectron Schradi nger equation [ + Z./r + r(2Z,/r1)(2)dr, 6c(3/8. p(i)) /3]u.(1) = E.u.(1) (213) [1 13 2 The oneelectro:. eigenvalues E. of (213) 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/) = X c solid X; atom This def:iiitior is co.i:sitent 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 23. The APW Method The APW method is a wellestablished scheme for solvig a given one electron eigenvalue problem for a crystal. An excellent review by Mattheiss et al.3 discusses the selfconsistent 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 nonprimitive lattice translations will be dropped. The potential, of the functional form discussed in section (22), is assumed to be spherically symmretric inside spheres centered at each nuclear site in a solid and constant outside those spheres. This periodic muffintin (MT) potential is a physically plausible one which approxinmates the exyct 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 eecLtrcnic 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.) (215) K  with the sum K over all reciprocal lattice veccors, The APW basis function of (215) is, letting R be the MT sphere radius, D(k+K,r,E.) = ext[i(k+K)'] r>R (216a) jj (sk+KR ) 4ij k R) ,Js = t 4 4i z) Y (e k u(r,E.)Y (e,) r In (216b), 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 (217) where V1 is the muffintin 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 (216) to obtain the the SAPW basis functions a (k+K,r, E ) F* (R) R.,(kb rE.) (218) 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 operacions of the group of (Gk). Then, with these SAI'/W, the ..yci :Lri.zed Bloch functions are obtained S() Cc' )' (+, r, Ec) (219) nk Ks s n,s wi h 1in hot (?1') rn1r (979) 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 (216) is defined by 1 R ( k,r,E) = (k,R r,E.) = (Rk,r,E.) (220) 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[iRkr] r>R (221) n,s n,s  s ZREG (R )Rk"r = ,*a( r (R) n 4iV s Y(eRk,9k) u .(Er)Y (e,) r REG m u (E ,R )  k s S where on the RHS of both (221) and (222) Rk R(k+K) and k IRki R(k+K) (223) In most applications using the APW basis, the addition theorem of the spherical harmonics (A312) may be invoked to further simplify (222) ,j 2(kR ) Rk'r to (k+K,r,E ) = I 1* (R) i  P ( )u (E ,r) (224) REC 2. u(ER ) The purpose of this symInetrication is twofold: 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 electronphonon interaction considered here nmut 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 21 for the typical fee solid with one atom per unit cell. In Table 21 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 (219) is, in general, over the entire periodic macrocell; 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] (225) 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 (223) has been used in (225) and in Appendix III where the formcalae for the EPME are derived based on the SAPW in the form (222). 24. The FrJhlich Hiailtonian Though Frohlich 4' and many others9 using his approach had the principal objective of explaining metallic superconductivity, the Hamil tonian and approach to the electronphonon interaction which he formulated is of a geFereal n .?t!re (wvithvn. nF cour ', certpin anprny:imatlaions) and oc 0 44 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 4i , 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 41 i 41 C) C n G * 0) 4i G w 0 0 cd Cl CC *,I UN .4 0CI i i o; 0 iiio 0 r i Ci E r.' ? 1. cri xj z 0( x X < X r 14 C' ,1 cq (svTun lariainTqll') A)uanbyj Table 21 Phonon Eigenvectors of the Six Nonequivalent 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 pseudolongitudineal and the last is actually pseudotransverse. 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 BornOppenheimer Hamiltonian. As a convenient starting point, the Hamiltonian H of (22), 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 electronelectron H(Ro) = H (R0 + H(R) + HE n  e  = [T (RO)+Vn (P0)] + [T e(Ro)+V (RO)+V (R0)] + 6RV (V +V ) o (226) n n e nee R ne eR HEP is the electronphonon interaction to first order in the displacement and by an obvious simplification of (22), T has replaced the kinetic energy sums. The individual operators of (226) 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. (227) with the a's denoting boson quasiparticle operators, H (Ro) = C , e k k .i k  SnkE (228) k  with the c's denoting fermi quasiparticle operators, and 4 S= M(k';k)c,ck(a + a) (229) EP k' OX OX k,k' In (229), 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[iqR]]C j  1/2 (2o)1/2 (23 = (NM)1/2 (2 ), (a +a ) (230) and the electronphonon matrix element (EPME) [between electronic Bloch states (23)] is defined as M(k';k) = Tk' EP k = (NM2 ) 1/2< RV +V) ol'> (231) k.' k 1 ne I R e The energies of equations (227) and (228) have the same meaning as those of (23) and (27) and the superscript o means the static equili brium lattice (unperturbed). The totality of (227), (228), and (229) 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 ) (232) F k  k So, :e explanation of these equations is now in order. In (228) aind (229), t,i I; and k' also imply band indices. Bra and ket notation simpli fy the electronic coordinate integration in (231). The Bloch condition of (21) has been used in (231) to remove the exponential of (230) and to give the selection rule k = k' k + K (233) which governs the sum restrictions in (229) and (232). 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 (233) with k denoting the initial state (valence or "plike" band), k' the final state (conduc tion or "slike" band), and q the phonon wave vector. The eigenvectors and normal mode amplitudes cf (230) are chosen to have the following phase convention Q = Q (234a) S = c (234b) ::'RX _q~ The eigenvectors for the symmetry points used in this work (see Appendix I) were given in the previous section (Table 21) 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 (235a) L P (235b) with i,j tbhe :a, y, or z coi;:pcen':t of the eige2nvector. In waiting down .., of (232), the expli cit. lectronCelectron and e]ectronnucrlear" interim .crions are accou:,Led for by using an appropriate screened forr. of Lhe potenrtiars in the EPME e*xp .ssion (231) 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 LennardJones superposition) is used to calculate the unperturbed phonon spectra. Therefore, as used here, the bose and fermi creation and annihilation operators of (227) and (228) are formally "quasiparticle" 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 (232), 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 ylTcighSii:rdinger perturbation theory. First, we define some quantities for purposes of simipl ification Ho = H (o) + H (RO) = I Eaa + E cc (236) .i k .. k.. 9 q K n. ag I. k E = L n (237) wheri the pola i/z.tio labl 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 "quasiparticle" nature of the field operators. The total energy to second order in the perturbation 't may now be written as31 E = E + 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 21 and (24)] with nk electrons in state k and n phonons in state q. The first order energy term in (238) 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 (238) may be shown to be31 E(2) = < ( k';k) a ic (E Hc1() aE C S k ck Ck + akk ('E.~) t'11> (239) where the sum is restricted by (233). Fur clarity and illustration, FigLre 22 shows the interactions inherent in the assumed form of HiE and the processes that contribute to ticn total secicrd older energy of (239). The use of (236) and (237) in terfus o ooccLpatior nu;hrs reduces (239) to n (n +1) E( = ;k) 1 )[ +  (240) k, k' ,o Eb . , k k k K a where, again, the sum: i re strictcd by (233). '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 22. 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, (240) may be simplified to 2r, (E ,Ek) (1n (2) 11 (k' 2 k' k E2 = IM(k';k) in,[+ (241) k,k' k o 2 o02 0 0 o kk' (E,) E k  (241) is the total second order energy correction to the system and the electronphonon 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 (238) necessary to increase the h phonon occupation by n gives31 E wX in~00 2r (EE ) = + M(k';k) 12[ 2 2 (242) X ,o o 2 02 O 2 E k'  with the sum over k' still restricted by (233) 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 (242) has been programmed (see Appendix IV) to calculate the total second order energy corrections only for transitions involving all "plike" filled valence band states to all excited empty "slike" 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. 25. The Derivation of the Form. for the EPME In section (24), the existence of an EPME was established and shown to follow as a natural consequence of the BornOppenheimer 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 muffintin (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 selfconsistent description of the BornOppenheimer 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 WignerSeitz 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 displaccmueiLts (small wavenumber) which is really only pertinnt to metals or semiconductors. ?Estimates, based on Kenney's suggestion, of the mean square displacement (in Rydberg atomic units) 2 (23) (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' (244) k k' E E k k k' with the band indices implicit (as in section 24) 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 23 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 24, 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:ioi FSirrplified c"d s'..perscriven from (215), become u = C0 o k, (k' iK) (245) '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(k4K) +4 C(k,k')(k') (246) K k' k+K where the k label on the I.HS is only to indicate the initial eigenstate of the system. In (246), all possible wave vectors are needed due to the nonperi 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 (246) to first order in.the nuclear displacements gives C(k,k+K) = Co(kkK) + 6RV RC1 (247)  R c6(k+K) = (k+) + iPVR I o (248) 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 (247) and (248) in (246), and keeping 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 (249) ai L' o K R The second term of (249) 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 (250) k'k where the sum k' over the scattered states is now restricted to the excited or conduction band eigeustates. The third term of (249) is seen [through the unperturbed coefficients Co(k,kK)] 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' (251) k R o K R Comparing (244) and (249) shows that M(k' ;k) <0, E > + <'! IS > (252) o 0E k k k' k k, k' The electrolnici Schr!dingcr equaticn for the perturbed situation may be taken fror; (23) as Hiek = F: (253) e'k k k with V (r) and V (r,R) the sum of displaced muffintin potentials e  ne  V (r,R) + V (r) = 1 V,(rR) (254) ne e MT and j labeling the nuclei and their respective muffintins. Expanding (253) and (254) about the R equilibrium positions to j terms linear in the lattice displacements (6R.), one finds Ho(Ro)[S + B] + 6V E(S + B) (255) V a 3 R. o oth) +; = E0o 0 an d & V M' T_. o e' k k k R R3 (250) and (252) may be used to obtain M(k';k) = A(k';k) + <,!B > (256) E E0 1 Solving for A(k';k) by using (255) gives (Ek'E)A(k';k) = E E,)<, Bk> <,6V > (257) k k' k k k I k k' k With the use of (257) and (256), the final expression for the EPME becomes M(k; < J j R (258) k J R j In Figure 23, the "rigid ion" assumption is illustrated, showing th that the potential at ary poirn within the j muffintin is the same, with i.espect to the center of that muffintin, regardless of the displace men t 0 0 Sr. R. = r R. (259) i  3 ~ t with SR. = R. R" and that as a consequence the potential gradient j j J Equilibrium th Muffintin Displaced jth M in j Muffintin _./! / r Origin Figure 23. Demonstration of rigid ion approiximation for nuf fintin djspj cei:ent. This figure shows that the potential with respect to the r[uffintin cec'ter is the same regardless of disulacemei t. of (258) may be written as aVMT (r.) ;, (r .) V (r .) VMT j I = ' (260) 3R. o r o o 1 3 Since the MT potential is spherically symmetric within the sphere radius and constant outside the sphere radius, (260) becomes dV WVV(r) = e d for r = 0 for r>R (261) s where e is the unit radial vector. r In section 24, the nuclear displacements (IR.) were expanded in normal mode coordinates for use in the. assumnd EPME expression (231). Comparison between (231) and (25S) shows that the APW form of the EPME may be written M(k';k) = (MN2 )1/2 (r) V V i(r) dr (262) 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 (261) and (260), the final expression for the EPMfE becories o/2 yo dV T M(k';k) ( o(r)  "W(r)dr (263) lk f k th k d  j sphere where the selection rule of section 24 (233) or Appendix T.I (A23) 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 (222) form of (263). CHAPTER III RESULTS OF CALCULATION 31. 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 "2slike'" "2plike", and "3slike" bands; aid for argon, the "3slike", '3plike", and "4slike" bands. The other lower lying core bands were treated in the usual fashion as states of the crystalline potential having atomiclike character; thus they correspond to flaL (i.e. kindependent) 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 syietry inequivalent k points in the irreducible wedge (see Figure A21 and Appendix II). This corresponds to 256 possible wave vectors or a 4x4x4 periodic macrocell (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 Iterationtoiteration 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 31. In that Table, "XAlpha 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 "3plike" conduction band extends into the excited "3slike" 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 "slike" representations of Table A21, which were used in the actual selection rules between the "plike" valence (occupied) and the "slikc" conduction (unoccupied) bands. However, the additional contribution was not considered in the calculation of the phonon energy corrections and only the "slike" representations of Table A21 were used in the calculations. The argon reiults of Table 32 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 comparrtive study of oneelectron 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 ii:odel that can be obtained vill be utilized, regardless of experiImetia] comparison. Table 31 Calculated Lattice Constant 20 Atomic Mass 0Ne20 Exchange ParameterAlpha Pressure(in kilobars) Total Energy Total Atomic Energy C Cohesive Energy Zero Point Energy LennardJones Parameters E U Quantum Parameter and Observed Properties of Neon Calculated XAlpha 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 9.3644x104 7.5127x104 4.29037x10 4 4.99436556 5.12812803 5.128128x10 4 3.29028x104 5.47097073 5.572535x103 5.572535xi0 a Observed Experimental Value 8.4323 3.644308x104 0.0 0.0019186 4.9082x04 4 2.2990104 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 32 Calculated Lattice Constant 40 Atomic Mass ,Ar Exchange ParameterAlpha Pressure(in kilcbars) Total Energy Total Atomic Energy Cohesive Energy Zero Point Energy LennardJones Parameters a Quantum Parameter and Observed Properties of Argon Calculated XAlpha 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 Appendix I for definition i 32. The SCH Phcncn Spectra of Argon and Neon As mentioned in section 21, 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 quasiharmonic 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 A21 and Appendix II). Thus, the six nonequivalent 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 33 for neon and Table 34 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 LennardJones pair potential description. The Lare phonorns for neo at 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 eiergy (given in Tables 31 and 32) and the average total eacrgy per particle, 3er~ieruL .ud in the dCynamicL lattice using SCHI, hai'te b.een ne.H"e to t nle, rl,',! e th' 7"~ro pnt r Th ps,'il Table 33 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 34 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 31 and 32. 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. 33. The Perturbed Phonon Spectra of Argon and Neon The calculation of the perturbed phonons is crude in the sense that a periodic macrocell of eight cubic unit cells (2x2x2) has been used for the calculation. The APW and SCH calculations were done with larger peri odic macrocells to ensure the accurate representation of the solid based on the smaller macrocell. This smaller macrocell 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 nonequivalent 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 asociated with them twelve re presentations for the phonons and the "plike" electrons, and six repre sentations for the "slike" electrons. Since the "plike" 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 (242), it was deemed sufficient to sum over all possible k' states in the lov(:est "sJlike" 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 33 and 34 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 24) 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 electronphonon interaction in an insulating solid. The problem was formu lated in the APWXa 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 33 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 LennardJones 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 energydependent 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 nvrrce, a difEicult task. Clearly, the use of the LennardJones 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 electronphonon 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 electronphonon effects in a solid by direct numerical investigation. APPENDIX I SELFCONSISTENT HARMONIC APPROXIMATION ANN LENNARDJONES 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 onedimensional 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 (Al1) 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( (Al2) 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] (Al3) ere G2 2 where G ,'. 47 Taking the ground state expectation value of H yields E =<0IHI0> = 1/4 A2G + <0(Vl0> (Al4) where < I'VV Y'> = 1/2 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 (Al5), some ground state matri.; elements are needed. Consider (Al6) Using (Al6) to evaluate V(( 0>)2V) yields the identity (I0>2V) = 2<01xGVIO> + <0IV'1 0> (Al7) However the LHS vanishes because the wave function is well behaved at plus or minus infinity; therefore we get 1/2 G (Al8) Operating on (( 0>)2V) with VV and using (Al7) and (Al8) gives (Al9) If we let V be unity, (Al9) reduces to (Ai10) Using the identities (Al9) and (Al10) on (Al5) yields 2 ,2<0 '' V7A O> 0 1/4 L 1/4 C' <010: (Al11) (AI5) V[0> CG >O> which is satisfied provided 2 2 A G = <01 vIO> = 0 (Al12) Eqn. (Al12) is the selfconsistent 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 (Al12) is a ground state average tu be solved selfconsistently through G. G is seen to be explicit on the LHS ot (Al12) 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 (Al1) for the crystal problem is assumed most commonly to be a pairwise superposition of atomic potentials V = 1/2 v(r..) (Al13) 1J i,j A fairly standard and widely used potential (especially for rare gas solids) is the LennardJones (126) potential v(r..) = 4c[(/r..)12 (/r. )6] (Al14) 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 N1 E = N/2 v(ri) (Al15) i=l where r. is the nonzero distance of the ith atom from the origin. The stable lattices for argon and neon are facecentered cubic (fcc) with the volume per primitive unit cell U = a3/4 (Al16) 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 (Al16) and (Al17) the following relations are obtained 3 3 r. = 2 U mn dr./dU = 2/3 m3/r2 J J J 17) (Al18a) (Al 18b) Minimizing the total energy with respect to U (Al19) dE/dJU = 0 N/2 2/3 rt'/r dv/dr. j 3 J From (Al14) we have 4c f 212 13 6j6/r 7 dv/dr. 4 412. /r. + 6 /r7] J J J (Al20) CombJinig (Al16), (AIl)), and (Al20) gives the following 7/6 6 v 12 1/6 /a = 2 [ i.6 / r:, j J J j  (Al21) I The lattice sums in (Al21) have been evaluated for the fee, bcc, sc, and hep structures.s5'45'46 Equation (Al21) 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 (Al15) and (Al17) and solving for = 1/2 (E/N) ( .22) 2 (Al21) j [(a/(m.a/2))1 (c/(m.a/2))6 These two equations, (AI22) and (Al21), 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 nonequivalent 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 A21, 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 nonequivalent reduced zone vectors that were used in deriving the selection rules is given in 'Pable A21. 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 slectio: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 A21. The first Brillouin zone for the fcc structure. The points labeled are those used in the 32 point grid (which corresponds to six nonequivalent points); while those points labeled O are the 256 point grid (which corresponds to nineteen nonequivalent 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 A21 Representations and Nonequivalent Reduced Wave Vectors Reduced Wave Vector Representations k, k', q (units of r/a) Phonon/plike Valence slike 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 (A21) 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 nonsymorphic 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) (A22) 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 (A21) k + g = k' + K (A23) we can decompose the direct products between all small representations of k and i into all allowed small represeitations of k'. In this work, we are coulcerrced with the tccecentered 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 A22, w:e give the 48 group operations of this point Table A23 is tl! group i.ultiplicat ic t fcle for the Oh point group. Table A22 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 01 0 001 1 0 0 01 0 0 01 1 0 0 010 0 01 01 0 100 001 010 1 0 0 001 100 0 01 010 100 001 01 0 001 010 1 0 0 0 01 010 100 010 1 0 0 000 0 Oi 56 Table A22 coutilnued Operation Number 00 1 0 01 01. 0 0 01 100 0 01 1 0 0 010 01 0 001 1 C 0 0 01 1 0 0 0I U) Class Matrix Representation 00 1 01 0 100 1 0 0 001 010 01 0 1 0 0 0 01 0 01 01 0 100 1 0 0 1 0 0 100 010 01 0 010 001 10 0 Vector Effect Table A22 continued Class C3 Matrix Representation 0 1 0 0 01 i 0 0 Vector Effect y z x a The operations 2548 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 T1 r4 rI H r4 H i r4 CM4 il CM C C*> Sr M c rc mc o7% 1 vi CN ''n tn \C i c3. c a r H N C) ,r 4 1 rH i if 1 4 rH CH C4 (N CM CN M .4NcO 1" rM CO "I Hi cM H CM rl C4 C CO NNCN co NHN  in cy\ ot CM r.4 u OXO rql O i O N  m0t c 4H H L C)ci L .4 r1 U CA ri r4 H IC cl) in C) %Z N rHH rH ONCO O$  M r H H CM rl r4 r4 H CX) C,4 10 NO4 CM mI r4 ON rc m mrCLff\ r C14 r  ri 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 ra' ( . Ci M r1 I cn CC) I H C 1 C1l CM *r1 CM r H CM H4 rHHN CM rN rl rN rH i A N N vq H,q H T rD CD 4H r)  CM i 4. c) 0my n co ~Irf H HHn HN 1 ii E H( CJ e' L' 'Do rt C,) 01 rHl 00 14 \0 rL rC) 0v*1 rI rl rl ri U3 rl crl rl 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 rlr CM rl C4 L O CL CC 0'r i 4 c 0r! r I N r( C i. co ri rl C\C CI rl H CA C) 4 CM4 C\y r1 ri n NHC lN C N ri (z 4 Lowe H r  X; ; rl CM C4 H  CN M CO rm N C0 rN N H (r1 r N C) h Sr r HMH N H 4LC C ri N H Cl N C r C, C) M iC m rA rl CM rlo rHNc ' rl C ^0 M C mLr c ri ri MC rA CO rA i C.4 N HC ON N >hNH ci '0 0. mf c *> h IA , N  ,0 O cH \O 0H vO C H rr4 r l rq r H ,3 '. H q d C) N10 '. a% ,i C)t L'r0 O CH O rH r I r d0 Gn r0 crHl C' Cl rl r C C41 H C4C H ri N H ri cm4 CM H NM CM M r( CM CM CM C4 CM u Only the filled plike valence and the unfilled slike conduction bands have been considered and allowed to interact. The plike representa tions (associated with the vector k), the slike representations (associ ated with the vector k'), and the representations associated with the phonon wave vector q are listed in Table A21. 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 A24, 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 (A23), 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 A24 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 (A22) reducts 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 (A24) to 1/2 [ F.( + (JC (A25) which when evaluated over all Li representiations shows that only the AI, Table A24 Group Operations of the Thirtytwo 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 A24 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 (A26) 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 (A26) must be equal to the sum of the star number times the dimensionality of each representation on the IRS. Using (A26) 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 slike excited conduction band are used. From (A26) and Table A21, 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) rpr esentaticon has been con structed anri i;, given in Table A25. The Liectronphionon matrix element (EPI:E) calculation has been siumplified 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 A25 Selection Rules between Valence plike Band and Conduction slike 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 F5 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 A25 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 A25 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 A25 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 A25 conthued 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 A25 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 A23. 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 (A27) J j with R1 the inverse oDeration to R and the 1th star member of k deter mined by S= R (A28)   i I* The necessary group operations of the star members of the various reduced wave vectors are given in Table A26. This table is based on the irreducible wedge given in Figure A21, i.e. all the identity star members of the reduced wave vectors are contained in this wedge. Table A26 Star Group Operations Generated from the Six NonEquivalent 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 A26 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 25, that the EPME expression to be evaluated numerically from (263) 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 (A31) (A32) we can consider the x, y, and z components of (A31) separately. This is done by remembering that S= cossi sinetsinQo + cose$ r x y (A33) 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 (A34a) E m "k'k R 'L *, k R}^ L i mnm ,, *r5Ym,)) ( (A34b) Y*r' Rk' ,,R',r Y "', ,(0,4)cos sY (0, )dS2 (A34c) "m R'k, "R'k )Ykm (kd, RI,.) i'co , We shall derive explicitly the zcompon.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 (A35a) (A35b) 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 , ( ) [+ + (A36) m (+m1) CC m lm Cr (24(r)+1) ( Cm+1i) (Zm+2) sinOY r exp(i") OY Y (37) ra (22+1 C_ lm.L C Y+lm1 sinOY, exp (i) L Y I] (A38) in (2 I/2 C(ep 3i exp1 Ci] A9,+1 sin6 = 1/2i (exp[it] exp[i1]) (A39) cosO = 1/2 (exp[i]J + exp[ie]) rY*Y ) Ira "Y  E, rai (C. ,2 n (A310) (A311) P (cosy)  2 Y* m(0 9) ( ,,) (A312) 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' (A313) The LHS of (A313) may be simplified by using the following recursive relations of the Legendre pclynomidls dP dP (Z+1) du 2 [ uP P ] (A314a) 1u dP du 2 [ uP, (A314b) lu Therefore (A313) reduces to dP dP +, R+1 d (+!), cos cos d [ P (cosucosS. + P+ (ccs ucos )] 1 du 2 du 2 k 1 2 k+1 2 1 1u (A315) It is seen that we have an apparent singularity at u = 1 (which corres ponds to parallel or antiparallel directions).. However, by taking the appropriate limit of (A315) as u 1i, we may shove that ddPP d limit cos cosO (l) '(Z+) cos9 (A316) 1 tdu 2 du 1 By using (A36) in (A34c) and invoking (A311), we obtain Y ,( ,e ')Y ( ) Y* cos'Ym dQ k, m,ri m R Rk t km, (2m.) (9~~+! "1/2 ,Y. , S Z (21+1) (2+3) +lm 'k"Rk' mv Rk' Pk'v The of be fu r i + [ j +(1 t ),Y (t6 ^)li 'Rk' ) Rk ) (A317) The RPS of (A31i7) may be further simp ified through the use of (A313) and (A315) (,LFi) [(P (CosS .uc ,,) + P(cosRk,UcOSRk) ( Rk1 )R k (P (cos6 ucose ) + P (cos ucos ) ) (A 2 + R R 'k' Rk R'k' 1u (pE+1(coseRk'ucos6Rk)+ P(cosRkucosR'k) )] (A318) The reason that the terms are not combined in (A318) is that there are Zdependent radial parts multiplying each term in the parentheses within the square brackets. Combining (A318) and (A31) 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 iCk 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 (A319) 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 2u  ' 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 (A320) 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 lu sin, i(k'R3 )j2(kR ) r E dV (E) usinknRk u )) T7 , ( (E) r d   v s S o (P+1 (siniRk7siuR'k'usin Rksi6 R) j (k'IR)j (k ) Rs P (sinjksin kusin 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 (A32i) Equations (A319), (A320), and (A321) have been coded into the EPME program (see Appendix IV) together with the appropriate limiting expression (A316). 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 establishd 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 selfconsistent 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 (225)]. 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 recproc;!i space mesh and at any ter.perature, given the crystal st_'ucture, lattice ipran.neter, LteonnirdJones 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 slike band to plike band transitions or all plike band to slike 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 HExtended compiler an.' linedited ard overplayed with I.B.M.'s F128 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 selfconsistent APW potential. The nu:merical integration routine is the same as that used in the APW program and is based on a five point NewtonCotes 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. LIST OF REFERENCES 1. Golibersuch, D.C., Phys. Rev. 157, 532 (1967), 2. Slater, J.C., Phys. Rev. 51, 846 (1937). 3. Mattheiss, L.F.., Wood, J.H., and Switendick, A.C. in Methods in Computational Physics, edited by B. Adler, S. Femnbach, and M. Rotenberg (Academic Press, New York 1963), Vol. 8, p.63. 4. Frhhlich, H., Phys. Rev. 79, 845 (1950). 5. Sinha, S.K., Phys. Rev. 169, 477 (1968). 6. Toya, T., J. Res. Inst. Catalysis, Hokkaido University, 6, 161 (1958). 7. Pick, R.M., Cohen, M.H., and Marinrl, R.M., Phys. Rev. B 1, 910 (1970). 8. see the excellent review aticl.e and reference list of Chapter 7 in Dyaics of Perfect CJiytal', by G. Vcnkatakarmn, L.A. Feldkamp, and V.C. Sahni (ITT Press, Boston 1975). 9. Born, M. and Oppenheiirar, J.R., Anr. Physik 84,, 457 (1927). 10. Koehler, T.R. end Nesbet, R.K., Phys. RFv. 135, A638 (1964). 11. Slater, J.C., Phys. Rev. 81, 385 (19'l'). 12. Connolly, J.W.D., in NMi ni Thoret al Che:mistry, edited by G. Segal (Plenum I ress, NeTw York 1976), Vol. 7, p.105. 33. Nordhein., i,., Anur Phys. r, 607 (1933). 14. Kce.hler, T.R., Phy;s. Pe. Lett. 17, 89 (1966). 15. Le,;;irdlJones, J.;. n.r Inghac;:, A.E., Proc. Roy. Soc.(London) Al07, 612 (1925). 16. Z:iman, .J.M.. :ctrns and hobuons, (Oxford Press, London 1960). 17. Johnsor, .'o Prce:. Iv. Soc.(London), A310, 79 (1969). 18. Chester, G.V., Ad\an. PhiL' 1_, 357 (1961). 19. Sha1 L J. and ZiJ,;a i, .M., Solid State :Phy 15, 221 (1963). 20. Englman, R., Phys. Rev. 129, 551 (1963). 21. Kittel, C., Introduction to Solid State Physics, (Wiley, New York 1971). 22. see for example D. Pines, Elementary Excitations in Solids, (Benjamin, New York 1954). 23. Tricky, S.B., Green, F.R.,.Tr, and Averill, F.W., Phys. Rev. B 8, 4822 (1"73). 24. Sabin, J.R., WorLh, J.P., and Trickey, S.B., Phys. Rev. B 11, 3658 (1975). 25. Gaspar, R., Acta Phys. Hung. 3, 263 (1954); Kohn, W. and Sham, L.J., Phys. Rev. 140, A1133 (1965). 26. Schvarz, K., Phys. Rev. B 5, 2466 (1972). 27. Condcn, E.U. and Shortley, G.H., The Theory of Atomic Spectra (Cambridge Press, Cambridge 1951). 28. Fr8hlich, H., Proc. Phys. Soc. (London) 64, 129 (1952); Proc. Roy. Soc. A 215, 291 (1952). 29. Bouck.e'rt, L.P., Smoluchowski, R., and Wigner, E.P., Phys. Rev. 50, 58 (1 36). 30. Hedin, L. and Lundqvist, S., Solid State Phys. 23, 1 (1969). 31. Taylor, P.L., A Quantum Approach to the Solid State (Wiley, New York 1974). 32. Mida, A.B., Soviet Phys. JETP (English Transl.) 7, 996 (1958). 33. S,:tt, T.A.. private communication, 1975. 34. Kenrc; y, ., "The Energy ..)nU Problem for a Deformed Lattice", in Mass. InsL. Te;h. SSMITG SemiA n. Progr. Rep. ,2, 26 (1966). 35. Slater, J.C., Sy:nm. try and Enir: 36. P'llac.:, G.L., Rev. Mod. Phy.,. 36, 748 (1964). 37. En.coh, Y., ShLran.:, G., and ,kalyo, J. Jr., Phys. Rev. B 11, 1681 (1975). 38. Fuji, .. Lurie, N.A ., lynn:, '., and Shi.ane, C., Ph}ys. Rev. B 10, 3647 (1 74). 39. Earker, J.A. and PIknderson, D., Rev. ild. Phys. 48, 587 (1976). 40. V, rth, J.P., jerry C.L., Trickley, S.B., and Sabin, J.R., to be pub lished. 41. Sambe, H. and Felton, R.H., J. Chem. Phys. 62, 1122 (1975). 42. Koehler, T.R., Phys. Rev. 165, 942 (1968). 43. Koehler, T.R., in "Proceeding of the International School of Physics Enrico Fermi, Course 55", edited by S. Califano (Academic Press, New York 1975). 44. de Boer, J., Physica 14, 139 (1948). 45. Hirschfelder, J., Curtis, C.F., and Bird, R.B., Molecular Theory of Gases and Liquids (Wiley, New York 1964). 46. Kihara, T. and Koba, S., J. Phys. Soc. Japan 7, 348 (1952). 47. Zak, J., Phys. Rev. 151, 464 (1966). 48. von der Lage, F.C. and Bethe, H.A., Phys. Rev. 71, 612 (1947). 49. Lax, M.. ymimetry _Pinci lses in Solid State and Mclecular Physics (Wiley, New York 1974). 50. Tric:key, S.B., private communication, 1975. 51. Wood, J.H., Ph.D. Dissertation, Massachusetts Institute of Technology (1958). 52. Connclly, J.W.D.. Ph.D. Dissertation, University of Florida (1968). 53. Bickley, W.G., Math. Gaz. 25, 19 (1941). 54. HaKmirng, G., Numerical Methods for S dentists and Engneers (McGrawHill, N?w York 1962), p.127. 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 stardares 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 requirements for the degree of Doctor of Philosophy. December, 1976 ea:., Graduate Schoiol UNIVERSITY OF FLORIDA II 1262 08666 924 8 3 1262 08666 924 8 