PHYSICAL REVIEW B VOLUME 15, NUMBER 10 15 MA Y 1977

Self-consistent numerical-basis-set linear-colnbination-of-atomic-orbitals model

for the study of solids in the local density formalism

Alex Zunger and A. J. FreemanPhysics Department and Materials Research Center, Northwestern University, Evanston, Illinois 60201

(Received 26 July 1976)

A new approach to the fully self-consistent solution of the one-particle equations in a periodic solid within theHohenberg-Kohn-Sham local-density-functional formalism is presented. The method is based on systematicextensions of non-self-consistent real-space techniques of Ellis, Painter, and collaborators and the self-

consistent reciprocal-space methodologies of Chancy, Lin, Lafon, and co-workers. Specifically, our approachcombines a discrete variational treatment of all potential terms (Coulomb, exchange, and correlation) arisingfrom the superposition of spherical atomiclike overlapping charge densities, with a rapidly convergent three-

dimensional Fourier series representation of all the multicenter potential terms that are not expressible by asuperposition model. The basis set consists of the exact numerical valence orbitals obtained from a directsolution of the local-density atomic one-particle equations and (for increased variational freedom) virtual

numerical atomic orbitals, charge-transfer (ion-pair) orbitals, and "free" Slater one-site functions. The initial

crystal potential consists of a non-muffin-tin superposition potential, including nongradient free-electroncorrelation terms calculated beyond the random-phase approximation. The usual multicenter integrationsencountered in the linear-combination-of-atomic-orbitals tight-binding formalism are avoided by calculating all

the Hamiltonian and other matrix elements between Bloch states by three-dimensional numerical Diophantineintegration. In the first stage of self-consistency, the atomic superposition potential and the correspondingnumerical basis orbitals are modified simultaneously and nonlinearly by varying (iteratively) the atomicoccupation numbers (on the basis of computed Brillouin-zone averaged band populations) so as to minimizethe deviation, hp(r), between the band charge density and the superposition charge density. This stepproduces the "best" atomic configuration within the superposition model for the crystal charge density andtends to remove all the sharp "localized" features in the function hp(r) by allowing for intra-atomic chargeredistribution to take place. In the second stage, the three-dimensional multicenter Poisson equation associatedwith Ap(r) through a Fourier series representation of b, p(r) is solved and solutions of the band problem arefound using a self-consistent criterion on the Fourier coefficients of d p(r). The calculated observables includethe total crystal ground-state energy, equilibrium lattice constants, electronic pressure, x-ray scattering factors,and directional Compton profile. The efficiency and reliability of the method is illustrated by means of resultsobtained for some ground-state properties of diamond; comparisons are made with the predictions of othermethods.

I. INTRODUCTION

The current popularity of energy-band theorystems from its successful application to the studyof increasingly diverse problems in solid-statephysics. Recent new sophisticated experimentson both tra, ditional materials and those havingcomplex crystallographic structures have de-manded, however, not only theoretical descrip-tions of phenomena related to the one-electroneigenvalue but also detailed and precise uavefunctions with which to determine the expectationvalues of different observable operators. Such ademanding test of the predictions of one-electrontheory has the additional virtue in permitting, bytheir comparison with experiment, accurate de-terminations of the relative magnitude and impor-tance of many-body effects in real solids. Thus,there has developed considerable interest in ap-plying the Hohenberg- Kohn- Sham" local- density-functional (LDF) formalism to the investigation ofvarious ground- state properties of solids, despitethe usual difficulties of solving the associated one-

particle equation characterized by a multicenternonspherical potential. A variety of well-knownapproximations have been introduced in order toreduce the complexity of the problem. For exam-ple, the various forms of orthogonalized-plane-wave (OPW) techniques3'~ suffer from serious con-vergence difficulties when applied to solids con-taining first-row atoms with no P states in theatomic cores. Recent studies with extended setsof orthogonalized plane waves' have indicated er-rors in the eigenvalues of the order of up to a fewvolts due to poor convergence in calculations em-ploying several hundreds of OPW's. The muffin-tin approximations applied to augmented-plane-wave (APW) calculations" have been recentlycriticized as introducing errors of up to a rydbergin the potential in diamond' and other covalentstructures, ' and overestimating the binding ener-gy in some covalently bonded molecules by asmuch as (100-300)/~. " Recent LCAO (linear orbit-als)-type calculations" "have overcome the diff i-culty of treating non-muffin-tin potentials and havedemonstrated that efficient convergence with respect

4716

SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . . 4717

to the size of the basis set" can be obtained. How-

ever, the problem of carrying this type of calculationtowards self-consistency still remains a formid-able task within the framework of present tech-niques. It thus seems that although it is possibleto obtain reasonable results for the eigenvaluesof the local-density one-particle equations withinthe above-mentioned approximations both forsome molecules"" and solids (i.e. , bandstructures), and accurate evaluation of the pre-dictions of this theory for ground-state functionalsof the electron density is still a nontrivial taskfor state- of- the- art methods in one- electron theo-ry.

In this paper, we describe a new approach to thefully self-consistent (SC) solutions of the one-particle equations in a periodic solid within thelocal- density- functional formalism. It is designedand developed to incorporate special featureswith which to overcome difficulties encountered byother methods. Specifically, as will be shown in

detail, the method combines a discrete variationaltreatment of all potential terms (Coulomb, ex-change, and correlation) arising from the super-position of spherical atomiclike overlappingcharge densities, with a rapidly convergent three-dimensional Fourier-series representation of allthe multicenter potential terms that are not ex-pressible by a superposition model. The basisset consists of the accurate numerical valenceorbitals obtained from a direct solution of thelocal-density atomic one-particle equations. Toobtain increased va, riational freedom, this basisset is then augmented by virtual (numerical)atomic orbitals, charge-transfer (ion pair) or-bitals, and "free" Slater one- site functions. Theinitial crystal potential consists of a non-muffin-tin superposition potential, including nongradientfree-electron correlation terms ca1.culated beyondthe random-phase approximation. The Hamilton-ian matrix elements between Bloch states a,recalculated by the three- dimensional Diophantineintegration scheme of Haselgrove, "and Ellis andPainter, ' thereby avoiding the usual multicenterintegrations encountered in the LCAO tight-bindingformalism. Self-consistency is obtained in twostages: in the first state ("charge and configura-tion self-consistency" ), the atomic superpositionpotential and the corresponding numerical basisorbitals are modified simultaneously and non-linearly by varying (iteratively) the atomic occupa-tion numbers (on the basis of the computedBrillouin-zone averaged band population) so as tominimize the deviation Ap(r) between the bandcharge density and the superposition charge den-sity. This step produces the "best" atomic con-figuration (for the employed numerical basis

orbitals) within the superposition model for the

crystal charge density and tends to remove allthe sharp "localized" features in the function 4p(r)by allowing for intra-atomic charge redistribu-tion to take place. Having obtained a low- ampli-tude smooth function 4p(r) that contains zerocharge, we proceed in the second state of self-consistency to solve the three-dimensiona, l multi-center Poisson equation associated with &p(r)through a Fourier- series representation of &p(r ).The solution of the band problem is repeated untilthe changes in the Fourier coefficients of 4p(r) insuccessive iterations are lower than a prescribedtolerance. The calculated quantities include thetotal crystal ground-state energy, equilibriumlattice constants, electronic pressure, x- rayscattering factors, and directional Compton pro-file in addition to the one-electron band structure.

Section II reviews those elements of the local-density-functional formalism which are the basisof this work. Section III describes some basicelements (crystal potential, basis functions, andHamiltonian matrix elements) needed to obtainfully self-consistent solutions by the method dis-cussed in Sec. IV. Of particular concern to usis the question of the role of the usual approxima-tions used to solve the LDF problem —muffin-tinpotential, ' ' non-self-consistency' on the ground-state properties of solids. In Sec. V, we pre-sent illustrative results of energy-band studiesof diamond which are compared with the predict-ions of other theoretical calculations.

II. LOCALDENSITY-FUNCTIONAL FORMALISM

The Hohenberg- Kohn- Sham" local- densityformalism is based on the fundamental theoremthat the ground-state properties of an inhomogen-eous interacting electron system are functionalsof the electron density p(r) and that in the presenceof an external potential V,„,(r) the total ground-state energy in its lowest variational state can bewritten

Z(p(v))= f V.„())(t(v)dV+G(p(v)),

where G(p(r)) is a universal functional of p(r) andis independent of the external potential V,„,(r).This theorem forms the basis of our approachto the electronic structure problem in that it pro-vides an effective one-particle equation relatingself- consistently the ground- state wave functionsto the energy functionals (i.e. , potential) of theelectronic system. Identifying the external po-tential for a polyatomic system as the electron-nuclear and internuclear interactions and varyingE(t)(r)) with respect to p(r), one obtains aneffective one-particle equation of the form

4718 ALEX ZUNGER AND A. J. FREEMAN 15

~ ~

+, &fr'+ "'iI, (r) = «,g, (r),

Z p(r'), 5E„,( p(r))(2)

I R —r I I r —r'I 5p r

where the exact quantum-mechanical Laplacian operator replaces the noninteracting kinetic energy. HereZ denotes the nuclear charge of the particle at site R and E„(p(r))the total exchange and correlationenergy of the interacting (inhomogeneous) electron system (bold-face parentheses are used to denote func-tional dependence). The eigenfunctions )I&~(r} are simply related to the total ground-state charge densityof the o occupied one-particle states by

p(r) = Q y,. (r)iI,.(r}, (2)

which, in turn, determines self-consistently the local-density functional in E&l. (2). The total ground-state energy is then given by

+ Q»» + E~~( p(r)),m

n, m R„—R(4)

where the first term represents the kinetic en-ergy, the second and third terms are the totalelectrostatic potential energy, and the last termis the exchange and correlation energy.

No satisfactory formulation of E„(p(r))has beenobtained so far for a general p(r). In the limitof slowly varying density, gradient expansions ofE„,(p(r)) have been suggested '" A. lthough thereseem to be no compelling evidence for the possiblesuitability of such expansions to realistic modelsof polyatomic systems (nor is there any convincingargument as to the convergence rate of this ex-pansion}, there still seems to be some interest in

applying the LDF formalism with the presentlyavailable first-term expansion of E„(pgr))as afirst step towards a more complete electronicstructure theory based on accurate local-densityfunctionals. Note that the LDF formalism in theform described above makes no claim on thephysical significance of the eigenvalues && in Eq.(2); hence we concentrate only on ground-statecrystal properties.

Retaining only the nongradient terms in theexpansion of E„,(p(r)), the exchange and correla-tion potential becomes

5E„(p(r))/5P(r) —=F,„(p(r))+F„„(p(r)),(5)

where the exchange potential has the well-knownform

F..(P(r)) = '- «.(P(r))=- —[(2/v)p(r)]"', (6)

where «, (p(r)) is the free-electron exchange po-tential for the local density p(r). The correlationenergy of a uniform electron gas with local den-sity p(r) has been calculated by many authors usingdifferent techniques. ' ~ The agreement betweenthe most recent results lies within 5-8 mRy in the

metallic density range. Here we use the resultsof Singwi eI; al. ' fitted to a convenient analyticalform"

F„„(p(r)=-A ln[1+Bp'~'(r}],

where A = 0.0899 and B= 33.8518, in a.u. andsimilarly

«,(p(r})=—C[(1+x ) ln(1+1/x)+ 2 x —x2 —,' ],

(8)where «,( p(r)) is the free-electron correlation po-tential for local density p(r), with

x=r, /A, ~4vr', = p '(r), C=0.045.

The total exchange and correlation energy isgiven at this level of approximation by

EG&& ))=f.&")&~.,&~&~)) +—~.&p&))&&~.

The LDF formalism has been used in the past withthe functionals F,„(p(r))and F„„(p($)) to calcu-late charge densities, total ground- state energies,and ionization potentials for some atoms" andmolecules" and has yielded reasonably good re-sults. The situation in the area of applying theLDF formalism to realistic solid-state modelsis rather different. Most of the existing local-exchange one-electron band-structure calculationson solids are based on retaining only the first termin the gradient expansion"'" [namely, the free-electron exchange «„(p(r))]. Also, due to addition-al approximations invoked in order to simplify thesolution of the one-particle equation, '" a mean-ingful comparison of the predictions of the LDFdetermined ground-state observables (i.e. , totalenergy, lattice constants, brompton profiles,Fermi-surface data, structure factors, etc.) with

15 SELF-CON SISTENT N UMERICA L-BASIS-SET LCAO. . .

experiment has been hindered. Recent work in

the field has demonstrated some dramatic changesin the calculated cohensive energy, "Fermisurface '""band structure, '"'" and charge den-sity when the simplifying muffin- tin approxima-tion or the lack of self-consistency are relaxed.Of particular interest to us also in the comparisonof the predictions of LDF formalism results on

some ground-state observables with those ob-tained by the restricted Hartree-Fock model byEuwema and co-workers. "" In view of thissituation, we have undertaken the problem of de-vising an efficient and reliable method for solvingthe local-density one-particle equation [Eq. (2)]using the restricted form of the exchange and cor-relation functional [Eqs. (5)-(8)].

III. BASIC METHODOLOGY

In this section we describe the main features ofour method for solving self-consistently the LDFone-particle equation (2) with the functional definedin Eqs. (5)-(9). We will hence discuss the methodsused to generate the initial crystal potential, thebasis functions, and outline the procedure usedto construct the Hamiltonian matrix elementsneeded to obtain fully self-consistent solutions inSec. IV.

A. Crystal potential

To construct an initial crystal potential for aniterative SC solution of Eq. (2), one has to guessa charge density p(r). The most convenient choiceseems to be the conventional overlapping-atommodel in which a superposition density p,„,(r)is constructed from one-center (spherical) chargedensities p (r, Q„,Q }) in the form

N h

p'"'(r) = g g p (r- R —d, Q„,Q }), (10)

where R and d denote the position vector of themth unit cell and nth atomic sublattice site, re-spectively. N is the number of unit cells and h isthe number of atoms in each cell. p (r, {f„„Q"})is the charge density of atom o calculated fromsome independent theory (Hartree-Fock, LDFformalism, Thomas-Fermi, etc. ) assuming the(generally noninteger) atomic occupation numbersf„,for the central field quantum numbers n andl and a possible net ionic charge of Q .

Several possibilities exist as to the choice of theone-site charge densities p (r) to be used as aninitial guess for p'"'(r). Because of the SC re-quirement, the choice is rather arbitrary andshould be made by considering the convergencerate of the SC cycles. We tried a series of pro-cedures for selecting p„(r)before choosing the one

that seems to work best; we choose p„(r)as thedensity obtained from the SC solution of the LDFone-particle equation [Eq. (2)] for an isolatedatom:

(~p

r )r- r'I

+&,.„(p,(r)) V'„,((~) = &„',rV.', r(&&

VP„'c(r)= P P v (r- R„d),

where the Coulomb potential associated with sitea is given by

(13a)

v (r)=, dr'p.(r'), Z, QI r- r'l r (13b)

Q denotes the residual charge that remains un-screened by the electronic one- site charge den-sity p (r) when the summation in Eq. (13a) iscarried out to a finite range 8 ~; its magnitudeis defined as limrv (r) = Q' as r R~-

In homopolar systems, where the discretelattice sum in Eq. (13a) is carried out to about20-25 a.u. , Q' becomes very small (due to therapid fall-off of the local-density atomic wavefunctions) and V~s~~ (r) becomes the only signifi-cant contribution to the crystal Coulomb potential.When Q is larger than some prescribed toler-ance (- 10 'e), the long-range Coulomb potentialV~"„'c(r)is calculated using standard Ewald sum-mation techniques. '"" Note that all asphericalcontributions originating from the overlappingtails of the one-site potentials v (r), as well asthose contributed by the point-ion electrostatic

where E,„(p(r)) and F„,„(p(r)) are the exchangeand correlation functionals, respectively [Eqs.(6)-(7)], acting on the one-center density p (r).Omitting the indices f„,and Q from p (r), thelatter quantity is simply given by

Noc

p.(r)= g f:, , IV., ,(~)l', (12)ny l

where the sum extends to all assumed occupied(core + valence) atomic levels N Equ.ations (11)and (12) are solved self-consistently (using aHerman-Skillman" type program) for each typeof atom appearing in the crystal unit cell assuminga given set of atomic-orbital occupation numbers

f„,(generally nonintegers) and ionic charge Q .With this choice of p'"'(r), the initial electro-static potential [second and third terms in Eq.(2)] can be written as a sum of a short-rangeCoulomb Vsgc(r) and a long-range CoulombVz"„'gr) pa.rts. Because of the linearity of Pois-son's equation, the short-range Coulomb crystalpotential is expressible as a linear superposition

4720 ALEX ZUNGKR AND A. J. FREEMAN 15

crystal field VLa'c(r), are not approximated bytheir spherically projected part in our initialcrystal potential. The superposition exchange andcorrelation potentials V„'"'(r)and V'"gQr, re-spectively, are defined by applying the functionalsgiven in Eqs. (6) and (7) to p'"'(r). Owing to thenonlinearity of the local exchange and correlationfunctional with respect to the individual single-sitedensities p, (r), the initial "superposition" potential V'"'(r) given by

V'"'(r) = Vs~'„(r)+Vf,"gc (r)+ V„'"'(r)+ V,'"„',(r)(14)

is not representable as a lattice sum of one-centeratomic potentials and hence even at this "zerothiteration" stage, Eq. (2) constitutes a multicenterproblem. Many LCAO techniques"" as well asthe multiple-scattering Xa (MS Xa) method" arepossible only if the potential is representable asa sum of single-site terms, in which case eitherfunctionals like E„(p(r)}are arbitrarily linear-ized'" [e.g. , V„(r)is set to Z Z, E,

„

x(p, (r- R„-Z }]or the superposition potential isspherically averaged around each atomic site. "'The consequences of these rather drastic approxi-mations on some calculated observables have beendiscussed previously. '' ' ' As an alternative tothese approximations, one might project numeri-cally each of the term in V'"' and p'"'(r) onto ananalytic single- site basis"" using least- squarestechniques. Although possible, such an approachrequires a separate auxi11iary basis set to be de-vised for fitting each of the terms in V'"', andthis becomes almost intractable when the functionto be fitted contains high central-field l compo-nents. In previous attempts, " it was also demon-strated that when complicated quantities like thetotal energy are to be calculated, small fitting er-rors [mainly in the steep part of V„'"'(r)]producelarge numerical instabilities in the calculated ob-servables. Since no use is made in this work ofa potential function that is linear in the one-sitecontributions, none of these approximations isnecessary. In particular, the potential is allowedto have the full site symmetry required by thecrystal structure in question [i.e. , in the languageof perturbative corrections to muffin-tin meth-od,"V'"'(r) includes both "exact" warping andallows for nonspherical components inside "atom-ic spheres"]. After the convergence of the directlattice sums [Eqs. (10) and (13a)] is maintained(to a tolerance of 10 8 a.u. in the potential), ourinitial V'"'(r} is fully defined by specifying &,the postulated crystal structure, and the assumed[f„,) and charges [Q ) used to construct p (r).The latter two will be treated as essentially free

parameters in the SC procedure to produce the"best" superposition potential.

Figure 1 displays three-dimensional plots of thesuperposition Coulomb, exchange, and correlationpotentials for diamond in the (1TO) plane4', theyall follow the essential pattern of the input chargedensity. Nonconstant regions in the "intersphere"Coulomb, exchange and correlation parts areeasily visible (especially in the exchange andcorrelation parts). A lattice constant of 6.742a.u. was used and an input 1s 2s 2P2 electronicconfiguration is assumed. The exchange potentialconstitutes some 55% of the Coulomb potential inthe interstitial region while the correlation poten-tial has smaller values (5%) in that region.

B. Basis functions

As is usual in Roothaan-type variational treat-ments, the crystal eigenfunction P&(k, r)(band index

j and translational representation k) is expandedin terms of h7) Bloch functions, C „(k,r), in astandard fashion:

g~(k, r) = g P C„J(k)4„(k,r) .of=1 g=l

(15)

O

y -750-RLalzla/

O

g -O.OI-lalzLa)

-)500-Vc (r)

-0 I-

0-

0-O

K -0.0)-UJz4J

-0 I-

o -7sO-KLaJztd

- l500.Vtot (r)

FIG. 1. Three-dimensional plots of the various com-ponents of the superposition potential in the (1TO) plane.Singular potentials are denoted by vertical dashed linesat the atomic site.

Here 4„(k,r), defined in terms of the pth basisfunction on the nth sublattice, y„(r)is given by

N

C„(k,r) =N ' ' g e""~y~(r- R —Z ), (16)

and [)f„(r),p=1, . . . , q; a= 1, . . . , h] constitutesthe elementary basis set. A central problem isthe choice of an optimum basis. Qne desires alarge l.inearly independent basis set so as to main-

SELF-CONSISTENT N UMERICAL-BASIS-SET LCAO. . . 4721

tain sufficient varia. tional freedom throughout theBrillouin zone (BZ). A minimal basis set couldlead to the expansion coefficients C„&(k)beingcompletely determined by symmetry for somehigh-symmetry states in the BZ. However, sincethe number of Hamiltonian matrix elements in-creases rapidly with the size of the basis set, an

unduly large set is to be avoided. Obviously, thebasis set convergence rate of calculated observ-ables can be made to be rapid when the basis func-tions are carefully chosen to be compatible withthe type of Hamiltonian to be solved. One would

require that the basis set be flexible enough toadequately describe the various regions in thereal space unit cell (oscillatory nodal behaviornear the atomic sites with the appropriate Coulombcusps at the position of the nuclei and spatiallysmooth in the bonding interatomic region) and

various energy eigenvalue ranges (deep corestates, valence, and excited states). [Note thatwhile in nonperiodic polyatomic systems one mayneed both short and long-range basis functions todescribe the various regions in the molecule, inperiodic systems the latter are both numericallyinconvenient (due to the need to perform long-range lattice sums), and, in many cases, varia-tionally unimportant (since Bloch functions formedfrom short- or medium-range basis functions al-ready span an arbitrarily long range). ]

While band structure methods (e.g. , "conven-tional" APW, Korringa-Kohn-Rostoker and cellu-lar schemes) that rely heavily on approximatingthe multicenter crystal potential with its aspheri-cal site symmetry by sums of one-center termshaving a one-dimensional radial site symmetrycan utilize a conveniently convergent representa-tion for the eigenfunctions (one-site partial waves,plane waves, etc.}, band methods that do not re-strict the form of the crystal Hamiltonian areusually forced to resort to somewhat artificialanalytic functions to ease the difficulty of com-puting multicenter integrals. Past experiencewith these basis functions (Slater orbitals,Gaussians, etc. ) in quantum chemical molecularcomputations" has frequently revealed, alongwith a reasonable conveT'gence of one-electroneigenvalues with respect to basis set size, arather slow convergence of calculated many-elec-tron observables like the total energy. Since nouse is made in the method described in this paperof any analytical or semianalytical algorithm forcalculating the Hamiltonian matrix elements, weare free to choose I general basis set with whichto meet the considerations for an optimum set dis-cussed above. The criteria used for selecting agiven expansion set were its compactness andvariational efficiency in lowering the ground- state

total energy. Our basis set contains functionsthat are convently classified into three types:(a) "Exact" self-consistent LDF numerical orbi-tals for the occupied and lowest virtual levels ofthe atoms appearing in the crystal unit cell. (b)Additional "free" uncontracted Slater-type orbi-tals spanning the virtual space. (c) "ion-pair"charge transfer functions, e.g. , numerical atomicorbitals associated with a negatively and with apostively charged atom.

The "exact" self-consistent (LDF) numericalorbitals are directly calculated by numerical in-tegration of the central-field atomic one-particleequation in the LDF formalism:

[-aV'+g (r)]X., g(r(f:( Q ))

=&.'gX., ((r(f. &, Q™)).(17)

In this equation, E„,are the atomiclike eigen-values and g, (r, (f„,,„Qj) denote the atomiclikeradial orbitals for level n, l (for atom of type a)that depend parametrically on the prescribed pop-ulation numbers f„,of all N occupied levels inthe atom (the m degeneracy in central fields beingaveraged) and on the net atomic charge Q'. Thenet atomic charge is simply related to the nuclea, rcharge Z and the electronic populations f„,by

Noc

(18)

g, (r) = V c,„,(r)+ V,„(r}+V„„(r)+A(r), (19)

where Vc,„,(r), V,„(r),and V„„(r)are derivedfrom the total single-site ground-state chargedensity

Noc

~.(.&= g f:, ,h:, ,(., (f:.,„Q.))]' (20)

f., )*ny

The one-site potential g (r) can be any generalizedatomiclike potential, as long as it generates viaEq. (17) eigenfunctions that form a convenient and

rapidly convergent basis set for expanding thecrystal orbitals. We mould like them to be atomic-like near the atomic site, so as to be compatiblewith the crystal potential in these regions; how-ever, the long-range nature of the free-atomeigenfunctions should be controlled in order toavoid linear dependence between the Bloch func-tions generated from them. The oscillations gen-erated by the nodal character of the full (core+valence) atomic functions a.s well as the charac-teristic cusp of an exact atomic solution at thenucleus are desirable in order to partially cancelthe Coulomb singularity in the crystal potential bythe appropriate repulsive kinetic- energy terms.The one-site potential g (r) is accordingly chosenas a slightly modified atomic local-density poten-tial:

4722 ALEX ZUNGER AND A. J. FREEMAN

through the usual eth-site Poisson equation andthe local density exchange and correlation func-tionals [Eqs. (6) and (7)].

The additional term A(r) is a prescribed exter-nal potential chosen to tailor the y„,for their useas basis functions in the crystalline variationalcalculation. Again, several choices are reason-able: in order to obtain basis functions that areless diffuse than are "free" atomic orbitals, onemight choose for A(r) a Latter tail correction4~with a suitable coefficient that has the effect ofcompressing the long-range orbitals tails, an ex-ternal potential well "' ' or to adopt the Adams-Gilbert term, "modified for local potentials, togenerate more localized (and self-consistent) cen-tral-field single-site orbitals. Note that we attachno physical meaning to the c'„,and that Eq. (17),which is used only to generate basis functions, issolved numerically in a SC manner after pre-scribing a fixed set of (noninteger) f„,and Q .Thus, the single site potential g, (r) used to gen-erate the numerical basis orbitals is not re-stricted to be equal to the potential used to gener-ate the model superposition density [Eqs. (11) and(12)]. No attempt is made to fit the resulting or-bitals to any analytical form (e.g. , Slater orGaussian orbitals); instead, they are used directlyin tabular form in the crystalline variational cal-culation. Such fittings (of double-zeta quality, orso) frequently fail to accurately reproduce thecusps of the exact wave functions and result inconsiderable errors in the calculation of the largekinetic-energy terms near the nuclei.

It is perhaps worth noting that the initial super-position potential employed in our scheme is al-ready approximately self-consistent with respectto the variational numerical basis set used in thecrystal calculation, since the latter solves theone- site eigenvalue equation. In other LCAO- typecalculations" " ' the superposition potential isgenerally unrelated to the basis set, but ratherto an independent atomic model (e.g. , Hartree-Fock model for the basis functions and a local-density potential).

The numerical basis set is subsequently opti-mized nonlinearly in the variational calculation inthe solid, by recomputing these orbitals from Eq.(17) on the basis of a different {f„„Q) set chosenfrom a charge analysis of the crystalline wavefunctions throughout the BZ. The basis functionsare thus allowed to relax nonlinearly (and itera-tively) to the crystal potential, thereby increasingtheir variational flexibility. Formerly virtual at-omic states are thus allowed to become fractional-ly populated according to their variational partici-pation in the ground-state crystal eigenfunctionsand subsequently become much more localized.

The minimal numerical atomiclike basis set formsa very compact and flexible basis, whose quality,as estimated from previous molecular calcula-tions" and from our own experience, is com-parable at least to that of a double-zeta Slater set.In this paper, we use both a minimal (ls, 2s, 2Pnumerical orbital for second-row atoms) and anextended numerical set (with the addition of 3s,3P, and 3d numerical orbitals). In each case, theBloch functions formed from the basis functionsare Lowdin orthogonalized4'; linearly dependentfunctions that result in eigenvalues of the overlapmatrix (in the Bloch representation) which aresmaller than a prescribed tolerance of 10 ', areautomatically discarded (i.e. , a canonicallyorthogonalized set" is used). Whenever inclusionof higher virtual stat;es seems desirable, but be-comes impossible due to linear dependence, thenumerical set is augmented by type-(1) basisfunctions whose exponents are adjusted so that themaximum of each orbital falls within the Wigner-Seitz cell. These orbitals are then treated in anidentical manner as the numerical orbitals exceptfor the analytic evaluation of the kinetic-energymatrix elements.

The combination of basis functions of type (a)a.nd (b) seems to constitute an excellent set as faras one-electron eigenvalues are concerned. Es-sentially no further changes are observed in thevalence and low-lying conduction-state eigen-values (only states below Ez+ 15 eV are examined)in the materials studied so far (diamond, BN,LiF, Ne, TiS,) after a minimal numeric set isaugmented by the first two or three virtual states.The situation is, however, different when the totalenergy is considered where, e.g. , for diamond,a major improvement is obtained by adding anion-pair basis set [type (c)] to the usual numeri-cal set. In this case we solve the atomiclike one-site equation [Eq. (17)] not only for neutral carbon(Q' = 0) but also for the ionic configurations1s'2s'2p ' having a net charge of Q and -Q, anduse this triple-size set to construct our Blochfunctions. Experimentation with this type of setrevealed that the results are not too sensitive tothe value of ~Q ~

around 0.8 —1.0 and that the besttotal energy seems to occur for Q =+0.85.

Figure 2 displays three-dimensional plots of theoptimized numeric Bloch functions in the (ITO)plane in diamond [converged to within one part in104 with respect to the lattice sums and the ac-curacy in the solution of Eq. (17) is 1 part in 10'].These are obtained by generating the numericalone- site functions from a self- consistent solutionof Eq. (17) with a Latter tail taken as A(r) andusing an electronic configuration of1s'2s'2P'3s 3P 3d and superposing the contri-

15 SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . . 4723

25

3s

FIG. 2. Three-dimensional plots of the numericalBloch functions at the I' point in the BZ (110).

butions from various unit cells to yield 4 (%, r)as indicated in Eq. (16).

h

Q Q [H„~„~(k)—S„~„q(k)e;(k)]C„~~(k)= 0a=1

(21)

is solved, where the Hamiltonian matrix elementsare

H „q(k)=(4 (karl —&+'+V(r) I@ s(" r))

(22)

and the overlap matrix elements

S„„~(k)= (4„(k,r) l 4„~(kr)).

The eigenvalue spectrum e&(k) yields the bandstructure defined throughout the BZ; however,since they are found strictly from ground-statelocal-density operators defining V(r), they bearno rigorous relation to elementary excitations inthe system.

In order to achieve maximum flexibility in theconstruction of V(r) [e.g. , by including a.rbitrary

(23)

C. Matrix elements

Once the model initial potential V'"'(r) and theexpansion basis set are defined, the usual linear-variation secular matrix

forms for the correlation part and avoiding linear-ization of F„(p'"'(r))and F„„(p'"'(r)),etc. ] and

the 4 (k, r) (e.g. , by allowing for nonanalytic and

possible off-center basis functions) one has tohave a general form-independent algorithm forcalculating the three- dimensional multicenter in-

tegra. ls appearing in Eqs. (22) and (23).Figures (1) and (2), showing the shapes of V(r)

and 4, (k, r), reveal the fact that any integrationscheme with this type of integrands must be ableto cope with the Coulomb singularities in V(j) the

nodal character of 4„(k,r) near the nuclei and

their diffuse shape in the interstitial region. Con-

trary to the situation met in problems of inte-grating the three-dimensional corrections to themuffin-tin charge density, ' the integrands en-countered in our problem are continuous through-out the integration space. Also, unlike the situa-tion occuring in molecular three-dimensional in-

tegrations, '~'" we do not have to treat both the in-

teratomic and the "outer-molecular" tails of the

integrands, since our integration is confined to asingle unit cell and the integrands do not fall offto very small numerical values. Guided by theseconsiderations we have adopted the three-dimen-sional Diophantine numerical integration scheme,developed by Haselgrove" and Conroy" and

adapted to molecular'~ and solid state" problemsby Ellis and co-workers. The method defines aset of pseudorandom integration sampling points{r,j inside a unit cell and an associated set ofweight function {w(r,.)j together with a tabular formof the integrand, to yield

-=g w;(r, )4 (k, r, ) V(r, )4„&(k,r, ) .

(24)

The convergence properties of this scheme hasbeen subject to numerous discussions in the liter-ature" and will not be repeated here. With nu-merical Bloch functions and a multicenter crystalpotential, the sampling that seems to work bestis based on a linear superposition of functions ofthe form

(25)

centered on each atomic site a, a Gaussian dis-tribution of sampling points centered at the middleof each bond, and a constant density of pointseverywhere else. This importance samplingtreats the Coulomb singularities efficiently bymaking the function w(r,.)H(r, ) relatively smootheven at the position of the nuclei and takes care

ALEX ZUNGER AND A. J. FREEMAN

of the buildup of charge in the bond region. Themethod is very simple and straightforward to useand provides a typical convergence, using 2000sampling points, of 1 mRy or less for the valence-band structure in diamond and 2 mRy for the firstfour conduction bands. (Similar convergence rateshave been obtained .or Li. cubic boron nitride,I.iF, and Ne. ) For a typical transition-metalcompound like TiS„around 5000 integration pointswere required to achieve an over-all 1-3-mRyconvergence in the region below eF+ 15 eV. Theconvergence of the total energy and the Comptonprofile is somewhat slower. and will be discussedlater.

The direct integration of Eqs. (22) and (23)avoids the difficult multicenter integrals appearingin Roothaan- type LCAO expansions ii-z3

a carefully chosen m, (r, ) in Eq. (24) to smooth outthe Coulomb singularities, all energy integralsare treated accurately in a real-space represen-tation and the usual difficulties in treating theslowly convergent Fourier expansion of such po-tentials in reciprocal-space techniques'~52 (wheresome 20000 reciprocal-lattice vectors were typi-cally needed to obtain reasonable convergence")

are avoided. We note that the method is capableof treating general terms in V(r) (i.e. , variousfunctional forms of local exchange and correlation)without having to introduce additional fit functions(such as are used in Gaussian-transform tech-niques""" to project the multicenter potentialinto a form that is linear in the contributions orig-inating from each center) —procedures which notonly require a different projection set for eachdensity functional appearing in the Hamiltonian,but a.iso tend to require a large auxiliary fittingbasis set when the ground-state density includeslarge l components.

The calculation of both the Hamiltonian matrixelements and the total crystal energy in modelsthat consider the full (core + valence) electroncontribution to the Hamiltonian is fraught with thewell-known difficulty of having to deal with smalldifferences between the large repulsive kinetic-energy terms and attractive potential terms inthe neighborhood of the atomic cores. The use ofexact numerical atomic orbitals provides astraightforward simplification of that problem.

The (initial) crystal Hamiltonian operating on

4, ,(k, r) yields [using Eqs. (16) and (17)]

HC„(k,r) = V(F)C„„(k,r)+ T4 (k, r) = [Ve'"gc(P)+ V,'"'(r)+ V'„"gc(r)+V'„"'„(r)]4„,(k, r)

+ g e'" "~)(„(r—R d )[c', -g (r- R —d )] . (26)

If TC, (k, r) is combined numerically with-,'

V(r)C „(k,r) (the factor of ~ originates from thedetailed consideration of the real and imaginaryparts of the appropriate functions) before integration of 4„~(k, r) [HC, (k, r)] is attempted, theattractive Coulomb singularity in Ve"„'c(P)nearthe nth core is effectively cancelled by the kinetic-energy term having a repulsive —g (r —R —d )dependence. Thus, to the extent that the chargedensity arising from orbital y„contributes to theground-state Coulomb potential in V'"'(~) [Eqs. (12)and (13)], it would act to cancel the positivekinetic energy arising from the same center.This cancellation is expected to be effective onlyif the 4 (k, r) are formed from exact eigen-vectors of the operator [g (P) —~ '7'] and if theCoulomb singularities in V'"'(r) are formed fromthe superimposed Coulomb parts in g (r)."'This is so both at the initial step in our SC cycle,where V'"'(r) contains a superposition of theCoulomb part of g (r), and at an arbitrary SCiteration where the basis orbitals g~™(r)are non-linearly optimized along with identical changesbeing carried for g (r) and V»c(r). Obviously,

I

orbitals which are used to construct Bloch statesthat are not related to the one-site potential g„(r)(e.g. , Slater orbitals, virtual states of g (r),etc. ) do not contribute to the cancellation betweenV(r)4 (k, r) and TC (k, r). It is usually suf-ficient, however, to have the low-lying corestates (Is, 2s) participate ing, (r) and V(r) andto use the appropriate exact orbitals in theBloch basis to obtain a satisfactory cancellationin the integrands. The numerical Diophantineintegration is then applied to integrate4 (k, r, )[ 4H„(ek,r)] after the cancellation inthe integrand was allowed to occur algebraically.

Figure 3 shows three-dimensional plots of thefunctions TC „(k,r), and V'"'(r)C (k, r) in the(110) plane of diamon at the I' point (k=0) forsome representative Bloch functions. The largecancellation between the kinetic and potentialterms near the cores (note the different scales),especially for the 1s Bloch state, is apparent.Note that a similar, but usually arbitrary, cancella-tion in the basis of the pseudopotential method andis the cause of its success.

SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . .

+ 5500-

Eplx:

$4)s (O, r)

0).

K& -5000-

-)0,000v Ci{s(o,r)

(

hT42p (t), r)

V (r) 42p(t) p r )

FIG. 3. Three-dimensional plots of T @„(0,r) andV ~(r)@„(0,r) for the Blooh states 1s and zp, in the(1TO) plane. Singular functions are denoted by the verti-cal dashed lines at the atomic site.

IV. SELF-CONSISTENCY

A. General coasiderations

The solution of the LDF problem for an assumedmodel p'"s(r) [Eq. (10)] provides us with (r(~(k, F)that define the crystal density p,~(r) as

ty

p. (e =((2 ~ 2 f ( g(&) p; (k, et(, (& e 1d(,

os z

when the sum extends over all 0 bands spanningthe occupied volume of the Brillouin zone (OBZ)and 0 denotes the unit-cell volume. The occupa-tion numbers nz(k) are chosen according tothe Fermi statistics (for states lying on the Fermisurface, a fractional population is possible). Theintegration in Eq. (27) is performed by choosing adiscrete number of points in k space and an as-sociated set of weight functions &o&(k) chosen ac-cording to the nearest volume algorithm. We haveused 4, 6, and 19 points inthe ~—,' irreducible sec-tion of the BZ (or 14, 32, and 256 in the full zone,respectively) and have examined the convergenceof some of the calculated observables with respectto the number of sampling points. The resultspresented in Sec. V 8 indicate the contrary to ob-jections raised'+" previously, the dispersion ofthe bracket terms in Eq. (27) is usually smallenough for good results to be obtained even withthe six nearest-volume points. Hence, we use

0

p {r)=N, g (((&(k )nj=1jf~

x(k ){{(j'(k,r) {{(z(k,r),(28)

where (d~(k ) are weights that become j indepen-

r(p(~) = p, (r) —po(r) (3o)

that is not amenable to fitting by single-site basisfunctions located on atomic sites. Although theform of Eq. (29) makes the problem of calculatingthe refined crystal potential rather trivial [e.g. ,when the Poisson equation associated with theindividual p;(r) is solvable analytically of semi-analytically], a SC cycle based on the restrictionEq. (29) may lead in the general case to conver-

dent in systems having no Fermi surface and arechosen according to the Chadi and Cohen" algo-rithm as generalized by Monkhorst and Pack."In this way p„,(r) is calculated in tabular formfor 2000-3000 points in the unit cell.

Since Eq. (2) yields variational solutions onlyif the eigenvectors g& are consistent with thecharge density determining the value of the func-tional in this equation, one has obviously to per-form a SC calculation in order to obtain meaning-ful results. Since the term "self- consistency"has been used rather loosely in some previousband-structure calculations (in that severe re-strictions were imposed on the SC path) we dis-cuss various possible degrees of SC along withour particular method for obtaining "full" SC.We will thereby distinguish between charge andconfiguration self-consistency (CCSC is stage1 in SC) and full self-consistency (FSC is stage2 in SC).

The major difficulty in obtaining FSC with p, (r)given in Eq. (28) is related to the problem ofhaving to obtain the Coulomb part of the potentialby solving the Poisson equation and to generatethe refined exchange and correlation potentialsfor the next iteration. While the model densityp'"'(r) is expressed as a lattice sum of one-center contributions for which the associatedCoulomb potential is readily found, the outputcrystal density p, ~(r) is a multicenter functionwhich is not expressible in terms of one-centeratomic densities located on atomic sites.

In what follows, we define CCSC as a consistentrelation between V„„(r)and p„(r)in which welimit the crystal density to sums of one center-terms located on existing atomic sites. In thisapproach

p, (r)= p, (r)=—g g g a,.p,.(r —0 —B.)m=1 e =1

(29)

at each iteration step where a; and p; denote theexpansion coefficients and the fitting functions, re-spectively. It is to be understood that dependingon the degree of approximation involved in simula-ting p,~(r) by p, (r), there will always exist aresidual density

4726 ALEX ZUNGER AND A. J. F REE MAN

gence to an arbitrary limit, depending on theapproximations inherent in Eq. (29).

Since a direct real- space point-by-point numer-ical integration of the multicenter three-dimension-al Poisson equation related to p,~(~) is too labor-ious to be of practical use, various methods havebeen devised to circumvent this problem. Theseare broadly divided into LCAO techniques treatingthe potential in a reciprocal space' "or in a real-space representation i4, i5. 58-60 The real- spacetechniques are usually based on various modelassumption for po(r) in Eq. (29) [and analogous ex-pressions for p,'~'(r)]. Thus, in the multiple-scattering Xn (MS Xn) method" (and associatedKorringa- Kohn- Ro stoker techniques) one pro-jects p, (r) onto a nonoverlapping muffin-tinsingle-site spherical basis. Although the non-spherical components can be later restored to dif-ferent degrees of accuracy by using variouswarping techniques, "perturbative multipole ex-pansion corrections, "~point-ion corrections~ orby applying a direct perturbation in terms of thenon-muffin-tin charge, '+" these procedures tendto become intractable, especially when iteratedtowards SC. Various choices of single-site (over-lapping) p, (r) functions have been made in LCAO-Xn types of calculations, e.g. , contracted Gaus-sians, "single Slater orbitals, "angular-depen-dent functions of the form C, (r)F, (8, 4)," regularpower series in r,"etc. In the molecular dis-crete variational method of Ellis et al ."and invarious forms of the iterative-extended Huckelmethod~ the a,. 's are chosen as Mulliken grosspopulations, "or Lowdin charges, "and the p, (P)are simply the atomic densities. Since in allthese methods &p(r) is neglected and the con-vergence to SC is judged by an internal criterionwithin the specific superposition model (e.g. ,convergence of the Mulliken charges betweeniterations) it seems difficult to estimate the truedegree of SC in these methods.

In the reciprocal space representations, '~"one expands p,~(r) in a Fourier series and ob-tains the Coulomb potential in real space as a sumof reciprocal-lattice vectors (RLV) of the plane-wave modulated Fourier components of p, (r).Thus, the difficulties of solving the Poisson equa-tion in a real-space representation are replacedby the problems of having (a) to Fourier transformp,~(r) [and similarly p,'~'(r)], and (b) to convergethe reciprocal-lattice sum on p„„(r).In the firsttask, one is again faced with the difficulty ofhaving to perform integrations on the three-dimen-sional multicenter function p, (r) and is resolvedeither by resorting to a one-center superpositionmodel for "b'~

p, (r) or by restricting the basisfunctions to Gaussians'~" [in which case the three-

center integrals associated with the Fouriertransform of p, (r) are computed semianalytically].In the latter case, the iterated exchange potentialis either expanded in a (highly truncated) geo-metrical series, ' or the lattice sum of one-sitedensities p, gr) is spherically averaged over theunit cell to reduce the Fourier transform ofp', ~'(r) to a. one-dimensional integral. "~ Thevalidity of these approximations is difficult toassess, especially in systems such as covalentlybonded solids which maintain a considerablecharge buildup in the interatomic region. All themethods that use reciprocal-space techniques tocalculate the iterated crystal potential and in-corporate directly both the core and the valenceelectron contributions on the same footing'~"face the common difficulty (b) of having to treat aslowly convergent Fourier sum. Owing to theCoulomb singularities in the potential and thesharp features of the core charge density, the con-vergence of such sums is known to be extremelypoor and, unless special procedures are used toaccelerate the convergence, "some 10000-30000RLV have been required for reasonable conver-gence. "

In view of these considerations, it seems reason-able to construct a SC scheme in which the ad-vantages offered by a real-space superposition approach (in avoiding the slowly convergent Fouriersums and facilitating the solution of the Poissonequation for the crystal density) are coupled withthe efficiency in which three-dimensional multi-center contributions to the potential (that are notexpressible within the superposition model) aretreated by reciprocal space techniques.

B. Charge and configuration self-consistency

Since there exist an unlimited number of waysof partitioning p, (r) into lattice sums of single-site terms [the goal of CCSC being to minimize4p(r) over space], the relevant criteria for choos-ing a particular partitioning set are efficient inrepresenting p, (r), a possible physical signifi-cance of the projection set and the degree to whichthe particular choice facilitates the following stageof "full" SC. Our choice of the partitioning schemeis the superposition model in which the projectionset is simply the one-site population-dependentcharge densities

We thus vary iteratively the "population numbers"f 7 to minimize, in the least- squares sense, thedeviation

SE LF-CONSISTTENT N U MERIC AL-BASIS-SET LCAO. . . 4727

1o = — Ap(r)~dr. (32)

0.4

0.3—I I i I I

CHANGE IN b, I (&) DDURING STEP I OF SC

Using this choiceoice, we retain the "s„l,t;..b,~...th ry a pote ial an te crystala asis set; this not onl

' ed 1 b '

Q

kinraic cancellation betw rg

gy erms discusseden ial-ener tso avoids the use of

op o~ . ' ' Thisch '

o introduce nonlini es

'b'1't, ' to th1' ' ' ' ebasis set b ep g„,, ", „„., e I)-

yn orm of the iter

e ax

thus vary nonlinn inearly the nume'

i crated potential. We

onthebasis of themerical Bloch set

e populations demaximum simil

er ' 'mx arity between

s it turnedvariation of the basis

out, a nonlinear

onlob ye asis set proved to

tionsia ractional ele

1 1 to oft}1much more c tcon racted orbital

C orbitals yield d

i tu1 t t th us renderin tcry

crystalline variational calcul

'g hem useful in th e

t 1 btis set when ].inear-d ro

lem y eir diffuse np p

a co ece such charge r

i eration, it is nec0

ion, it is necessary to re-y afimized numer'

ions. Thus thier

h 1 oe possibility of ch '

goosing among

)bl

e as

calculationsure non- self- cour - - onsistent

In a typical CCSC iteration w

y. few f-.dti it bi

y Sttpp i

„»Qy and use the cror an assumed set

e crystal densitp vxous iteration to corn an

unc ions. The sep ange and

p oo points in

m isg

trixine in Sec. IIB for

1 t Th pp g p rpo

ave ages. The new ~cn r

F' g a con tru t dvariation in 4

ce.p'r along the

0.2—I st

O. I

O

0

-0.I—0

-0.2

-0.3

-0.4

I

0.5 I.O 1.5 2.0 2.5-0.5 0DISTANCE ALONGONG III (o.u. )

FIG. 4. Plot of Ap r = ~ )—~ . V(r) = jo.~(r)—don uring various

se -consisten cy iterations.

[111]bond directioa~

' ec ion in diamonddu'

t dfk points and a m'

e rom the six nnearest- volume

1~ ~ ~

minimal numer'

P configuration. Tth 1Md " i

relat've to the ass mede some covalent

th h I tud

eatures near the ato h' h th'

f tcrystal densit ."y. age,

e which is to be ca of 12 electrons or 8 v

compared with

i ) is obvious that t ' er-.--.--"..„R- ' ""'""'"-""'position model one h

'd tlfnume

a unctions. When o

squares procedurse or diamoond, the least-

deure reduces to h

egrees of freedwoo andling only tw

j.

tion ratio") and bo s population "","1 ~pop

omes 1s 2s' " '~ ...;.'n~.' '

~

The X, po ul

lowa ue of 2.000 durin it ce'

g erations (sin thce eg o goo

a nonoverlat d t' 1

inomic 1s orbitals so '

nf 8 & 2pp

' e e

basiartym thminimization of 0

x eration) yields the resid 1i ual

=0 098 f th

aga ion in the interatom'mic region

4728 ALEX ZUXGER AND A. J. F REEMAN

along with a structure near the nuclei due to in-terpenetration of charges from neighboring atomicsites. This value of 0 „andthe deviations of theresidual &p(r) from zero probably indicate the"best" level of SC attainable within a superpositionmodel (CCSC) of a minimal numeric set.

Although the minimal value of 0 at the conver-gence limit of CCSC is still quite large, it isremarkable that &p(r) becomes a rather smoothspatial function. Thus, while one can effectivelyaccount for all the "bonding" effects that are de-scribable by in' a-atomic charge redistribution in

a linear superposition overlapping densities mod-

el, the inter-atomic rearrangement of charge re-mains largely undescribable by a simple superposi-tion of (nonlinearly varied) spherical atomic densi-ties. In systems in which ground- state charge redis-tribution effects are very small, one might expect todo rather well with a one-center superpositionmodel, in which case, the results of CCSC arepresumably already very close to FSC results.For example, when a minimal numerical basisset ca1.culation is done for solid neon in the usual1s'2s'2p configuration [lattice constant of 4.52

A, 2000 integration points and neglecting thecorrelation functional in Eqs. (3) and (6)] the firstiteration already produces a cr value as low as0.003e (compared with a total of 10 electrons perceH)and no further changes are observed duringsuccessive iterations. On, the other hand, how-

ever, for strongly covalent systems like diamondCCSC alone is not satisfactory and one should alsoincorporate the residual np(r) in a more completetreatment. Before proceeding to the next stage

3.2—2p

—28—4JC2K

XO

1.2—„pm/2s

0.8—

in SC, some illustrative examples of the possiblearbitrariness in stage 1 need to be discussed.

A rather popular choice for a superpositionmodel to be used in a CCSC procedure is onebased on identifying the expansion coefficients a,in Eq. (29) with either Mulliken or Lowdin

atomic charge populations. "'" One constructsat each iteration step a new set of populationsalong with the resulting superposition density and

repeats the process until the populations obtainedin successive iterations agree to within a pre-scribed tolerance. The degree to which the final

po(r) simulates the charge density derived di-rectly from the wave functions remains unspecifiedthroughout the calculation. It is thus not surpris-ing that in previous investigations in molecularsystems, different prescriptions (e.g. , Mullikenor Lowdin) for partitioning the charge density intosums of atomic densities were found to lead todifferent convergence limits, as indicated bywildly different dipole moments, eigenvalues,etc. ' ' It is still of some interest to find whetherone can effectively approach the limit of a freeminimization of o [Eq. (32)] by using one of thecommonly accepted population analysis tech-niques.

%e have performed two independent CCSCcalculations for diamond using Mulliken andLowdin population analysis with a minimal numericbasis set, 3000 integration points, and carryingall lattice sums to eight shells of atomic neigh-bors. The Mulliken population~ amounts toarbitrarily dividing equally between the relevantsites all the cross terms in the charge-densityexpression between basis functions located on dif-ferent inequivalent sites. In the Lowdin pro-cedure" one orthogonalizes the Bloch functionbasis first. In this case the populations are de-fined to be proportional to the squares of thetransformed expansion coefficients. The Mullikenand Lowdin populations are calculated at a set ofpoints in the BZ and averaged to yield the totalground-state population of each of the 1s, 2s,and 2 p orbitals in the crystal. To illustrate thetype and range of possible errors made by sampl-ing only a few points in the BZ, we plot on anenlarged scale in Fig. 5 the dispersion of theMulliken and Lowdin charges throughout the zone,

TABLE I. Brillouin-zone averages of the orbitalcharges in diamond (Lowdin's definition).

I h, X W K Z l' L Q 'lN

FIG. 5. Dispersion of the Mulliken (full lines) andLowdin (dashed lines) 2s and 2P charges in diamondacross the Brillouin zone. Upper figure: 2p charges.Lower figure: 2s charges. The 1s charge, not shownin the figure, has a dispersion of less than 10 e around2.0e

Orbital

1s2$2p

6-k 10-k 19-k Exact

1.9910 1.9792 1.9793 1.9793 1.98010.9452 0.9427 0.9426 0.9425 0.94203.0638 3.0781 3.0782 3.0782 3.0779

SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . . 4729

PropertyMulliken

SCLowdin

SCFree

minimization

Final deviation cr

Eigenvalue sr25, vr„„r~, c1-2,v

L3, v

L3,cX), „

X4„

X) cX4, c

HybridizationRatio (2p/2s)

Total energyper atom (eV)

Form factorsf i'll.

f»2fr2o

0.1595 0.1714 0.098

20.41926.81934.5475.268

17.63530.8008.276

14.35029.12737.222

20.42626.83834.5625.282

17.64830.8198.276

14.36229.13837.184

20.40326.77634.5305.264

17.62430.7678.269

14.32929.08637.179

2.773 3.265 1.701

-1013.113 -1011.235 -1013.3990

3.2060.11401.986

3.2010.11101.975

3.210O. 11701.990

as computed at 24 points in the ~ irreducible BZsection. It is seen that the charge dispersion isquite low (typically of the order of 0.15e for both2s and 2P states), and that the Lowdin 2s(2P)charge is systematically lower (higher) than thecorresponding Mulliken charges. The disper-sion of the 1s charges in both schemes is smallerthan 1 part in 10' and is not shown in Fig. 5. TableI shows the BZ averaged cha.rges, using 4, 6, 10,and 19 nearest-volume points in the irreducibleBZ section. It is seen that the 6-tt sampling pro-duces charges that are accurate to within 0.005e.At the end of each iteration, p, ('Fl is used to con-struct a new potential, and the iteration cycle isrepeated until SC in populations is better than0.005e. A nonlinear va, riation of the basis set isperformed at each iteration. A simple dampingtechnique of the iteration cycle'~ (with a damp-ing coefficient of 0.3) combined with the use ofthe "Pratt improvement scheme"" near conver-gence enables self-consistency to be obtainedtypically in three to six iterations. The final wavefunctions are used to compute the total energy peratom and the x ray form factors (Appendix A).

Table II summarizes the results obtained at a

TABLE II. Comparison between some calculated prop-erties obtained in three different CCSC models. Mulliken

SC and Lowdin SC refer to iterative calculations based on

the Mulliken and Lowdin charges, respectively; "freeminimization" refers to an iterative least-squares mini-mization of the charge deviation a. Energies are givenin eV and populations in electron units. Eigenvalues aregiven with reference to the r& „point at the bottom of thevalence band.

convergence limit of these two calculations, alongwith those obtained in a calculation in which thearbitra. ry f„,f„,and f» populations were variediteratively so as to minimize the deviation cr be-tween p, (r) and p, (r). It is evident that althoughthe internal consistency in the populations is con-strained to be good, the Lowdin and Mulliken popu-lation schemes tend to converge the result todistinctly different limits. Whereas the bandstructure obtained from the Mulliken schemesagrees with that obtained by the Lowdin scheme towithin 0.01-0.04 eV over the valence and firstconduction bands, the wave- function- dependentquantities like charges and total energy aremarkedly different. The difference in the Fourier-transformed total density (form factors) are quitelow, while the difference in the individual orbitaldensities (as reflected by the 2P 2s hybridiza-tion ratio) and total energies (which depend onhigher orders of the charge density) are substan-tial. It is also to be noted thatalthough neither theLowdin nor the Mulliken schemes seem to reducesubstantially 4p(r) towards the limit obtained inthe exact CCSC procedure ("full minimization"),the Mulliken prescription does a little better thanthe Lowdin scheme. It is expected that in hetero-polar systems, where the difference in the chargesobtained by the various definitions is muchlarger, "' the observables calculated from thefirst- order density matrices within these schemeswould differ even more than in diamond. Although0'M

y yseems to offer only a small improvement

over a»„,properties that are sensitive to the de-tails of the charge density (e.g. , those nonlinearin the density like the total energy) can come outsubstantially different. Thus, due to the fact thatthe Mulliken &p(r) happened to be much smoothernear the core regions than the Lowdin &p(r) (al-though they are rather similar in the valence re-gion), the total energy obtained in the formerprocedure is closer to that obtained in the "fullminimization. " By contrast, the x- ray form fac-tors that sample mainly the interatomic region(e.g. , the "forbidden" [222] reflection) are quitesimilar in all three calculations. Note that sincethe "exact" np(r) has a null zeroth moment (byconstruction) and is rather smooth throughoutspace, the errors associated with using thissuperposition model would probably be distributedrather uniformly in space and not cause too severea problem for such properties as atomic scatter-ing factors. We conclude that although SC schemeswithin a superposition model can be useful in ob-taining "simple" quantities such a,s eigenvalues,they should be treated with great care as to thebehavior of the residual densities when morecomplicated properties are investigated. In parti-

ALEX ZUNGER AND A. J. FREEMAN 15

TABLE III. Comparison of basis-set effects in different non-self-consistent calculations in

diamond with exchange coefficient e =1 and a lattice constant a=6.728 a.u. Energies are given

in eV. The number of plane waves (PW) or atomic orbitals used in each case, is indicated.

LevelOP%90-P%

OPW'331-PW

LCAO20 Slaterorbitals

LCBO"10 Slaterorbitals

Present work'minimal set

Present workextended set

r,„

~~sv

~isc~~', c

Li, u

L3', v

L3', cL~cL2cX)

„

X4„X(~X4c

0.022.9728.6036.575.808.52

17.6927.9131.27

8.8615.8427.10

0.022, 328.834.35.58.9

19.731.730.439.78.8

16.227.441.1

0.020.2226.4533.585.177.92

17.5030.2330.2337.988.08

14.5828.2736.46

0.020.15

8.6617.56

8.4914.59

0.020.4727.6533.525.198.41

17.7630.2830.3137.628.26

14.5128.3836.84

0.020.1227.3333.425.198.39

17.7430.2030.2837.518.26

14.4928.3136.72

~Reference (3).Reference (73).

'Reference (11).

Reference (12a).'Minimal numeric set of is, 2s, and 2p.

Same as e plus 3s, 3p, and 3d Slater orbitals.

cular, when &p(r) is non-negligible the SC cyclecan converge to an arbitra. ry limit. "

C. Full self-consistency

The procedure used to obtain FSC is based on theproperties of the residual density b p(F) as obtainedat the convergence limit of the CCS, namely: (i) 4p(r)is constructed to integrate exactly to zero over theunit-ceil volume; (ii) ithasa rather smallnumericalvalue over the unit cell space due to the use of a least-squares minimization in the CCSC; (iii) It is a rathersmooth function of space. More specifically,both the characteristic cusps at the positionof the nuclei and the steep variation near thecore regions characterizing the full charge density[and the unoptimized 4p(r)] are completely elim-inated by carrying stage 1 in SC successfully toconvergence. (iv) It has the full symmetry of thecrystal These .properties suggest that 4p(r)would lend itself to a rapidly convergent Fourier-series representation.

Fourier components of 4p(r) are determinedfor a list of RLV (K, 's) shorter than some pre-scribed length, using a, direct three-dimensionalintegration over the unit cell (uc):

The sum over the RLV is expected to convergerapidly due both to the smooth form of np(r) and

to the K,' fa,ctor. The exchange and correla-tion potentials used in the iteration cycle are sim-

ply constructed in real sPace using the full den-

sity p,~(r) and the functional in Eqs. (8) and (7)thus avoiding the difficulties encountered in

Fourier expanding these nonlinear functions. ""The solution to the band problem is now repeatedwith the refined potential using the fixed numeri-cal basis set and Va"„'c+V~ac(r) obtained in thelast step of the CCSC. The resulting crystaleigenfunctions are used to construct a refinedcrystal density via Eq. (28) and a new 4p(r) and

hp(K, ). The calculation is repeated iterativelyuntil the relative change in the most sensitiveFourier components between successive iterationsis lower than 10 '. If stage 1 in SC is carried pro-perly to convergence, only a few iterations (twoto six in diamond) are required in stage 2 and onlythe first few stars of the RLV (five to ten in diamond) are required. The hp(K, ) values for largeK, reflect the core contribution to &p(r) and are,hence, negligibly small since they have been ab-sorbed in p, (r) in the CCSC stage.

1np(K, )=— e ' ~ 4p(r)dr.

Q (33)V. ENERGY-BAND STUDIES

(34)

They are employed to construct the iterated cor-rection to the electronic Coulomb potential,

nV (r)=g —4v ' ' e'".'~!K)&s

A. Basis-set convergence

Although a meaningful study of basis-set effectsin a variational theory should be based preferablyon considerations of the total energy, none of the

15 SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . . 4731

previously published calculations have attemptedsuch a task. Thus, we are forced to refer mainlyto the resulting eigenvalues and use the total en-ergy only as an internal criterion for examiningthe quality of our various basis functions. Sincepreviously published LCAO studies on diamondwere not carried out to SC nor calculated with thecorrelation functional [Eq. (7)], we will makecomparisons with our calculations obtained atthis level of approximation. Whenever a differentlattice constant or exchange parameter was usedin the literature, we have repeated our calculationto match these values as closely as possible.Since one object of this particular study is to com-pare the accuracy of various basis sets in realis-tic ab initio studies, no reference is made to em-pirical pseudopotential results, empirical tightbinding, empirically adjusted ab initio results ormuffin- tin calculations.

Among the previously published ab initio LDFmodels for diamond in the LCAO representationwe refer to the Slater basis set calculation ofChancy, Lin, and Lafon, " the discrete vari3tionalstudy of Painter, Ellis, and Lubinsky, ' theGaussian basis calculation of Simmons et al. ,

"and the linear-combination of bond orbitals (LCBO)of Kervin and Lafon. "' Several first-principlesOPW studies have been made in the past on dia-mond, e.g. , the plane-wave calculation of Herman"and Bassani and Yoshimine, ' the 302-331 plane-wave calculation of Budge" and the 965 plane-wavestudy of Euwema and Stukel. ' These calculationsare non-self-consistent, and slightly different adhoc potentials have been assumed by the variousauthors.

Table III compares our results with energyeigenvalues of the four valence and two lowestconduction bands at the high-symmetry points, I',X, and L in diamond obtained by various OP% andLCAO methods with an exchange coefficient of 1.0("Slater exchange") and a lattice constant of 6.728a.u. Our non- self- consistent results are presentedusing a minimal numerical basis set of 1s, 2s, and2P orbitals and the extended set in which 3s, 3PSlater-type orbitals (with exponents of 1.4 and2.64, respectively) were added. In the states thatcontain no P character the internal consistency be-tween the various OPW studies is about -1 eV anda similar agreement is obtained with our minimalbasis-set calculations. For states that contain Pcharacter, our results are some 2-3 eV lowerthan the best OPW result signaling serious con-vergence errors in the OPW results. "

Qur minimal nuznerical basis-set results andthe minimal LCAQ basis-set results of Chancy etal." (and Kervin and Lafon"') agree to within 0.15eV/state (i.e. , a simple average on the four va-

TABLE lV. Comparison of basis-set effects betweenvarious LCAO expansion sets. 0. =0.75829; a=6.720a.u. Energy in eV.

LCAO18 Slater Present work Present work

Level orbitals minimal numeric extended set

I"1, v

~2S, vr„12',c~2, v

Li, v

L3', v

L3', cLi, cL~,cX(

„

X4, v

X~,cL4, c

0.020.8926.8335.245.657.38

17.9130.7531.8138.79

8 ~ 3314.3628.5137.81

0.020.4726.5434.765.357.21

17.6830.4131.1738.158.32

14.3628.2137.34

0.020.3826.4534.685.347.20

17.6530.3831.1538.098.31

14.3228.2037.30

lence and two conduction bands at the I', X, andI points), and to within 0.10 eV/state for theirextended basis set. Part of the remaining dis-crepancy can well be due to the linearization ofthe exchange functional in the LCAO studies" andto the slightly different superposition potentialsused.

Qur extended basis set produces eigenvaluesthat are on the average 0.05 eV lower than thoseobtained with the minimal numerical set. We havestudied a series of extended basis sets, aug-menting the minimal set by nunzerical 3s, 3P, and3d states and various Slater-type 3d and 3P states.(In neither case do the eigenvalues presented in

Table III change by more than an average of 0.07eV relative to the minimal numerical set results. )We thus conclude that a minimal numerical setproduces eigenvalues for the valence and lowesttwo conduction bands that are roughly equivalentto those obtained by an extended analytical Slaterset. As a further check, we have compared theseeigenvalues to those obtained by Painter et al. 'with a double-zeta set, a lattice constant of 6.720a.u. and an exchange parameter a = 0.758("Schwarz value'"'). Since only a few of theireigenvalues are quoted in their paper, we repeatedtheir calculation with their Slater basis set andparameters. These results are compared inTable IV with our minimal and extended basis-setresults. The agreement with our minimal numericset is again 0.10 eV/state on the average, our(relative) eigenvalues lying consistently belowthose obtained with the double-zeta set.

In a comparative study, Painter et al. ' have

ALEX ZUNGER AND A. J. FREEMAN

TABLE V. Total ground-state energies per atom (ineV) for diamond as obtained in a non-self-consistent cal-culation, for various choices of basis sets. An exchangeparameter of 3 was used and a lattice constant of 6.74032

a.u. Correlation is omitted throughout and 6 k points areused to sample the charge density in the BZ.

Basis set Total energy

Minimal numer icalExtended set I"Extended set II'Ion-pair d

Chopped extended set I'

—i Oi 2.728—10i3.4i8—i Oi 3.792—iOi5.2i8—i Oi 3.028

is, 2s, 2p numerical local density orbitals.Set a plus Slater orbitals 3s, 3P (see text).

'Set a plus numerical 3s, 3p orbitals.Set a plus two comparable sets for a positively and a

negatively charged carbon atoms (see text).'Set a plus truncated 3s, 3p Slater orbitals.

used their double-zeta Slater basis set to computethe band structure of diamond using the muffin-tinpotential previously employed by Keown in hishighly converged APW study. ' Painter et al. found

an average agreement of -0.05 eV/state with the"exact" muffin-tin results, their values beingusually above the APW results. This tends to in-dicate that our minimal basis set results are asgood as, or slightly better than, the APW resultsand of roughly the same quality as an extendedSlater basis set. We believe our extended set tobe superio~ to all other basis sets discussed.

In order to study the variational quality of someof the basis sets employed, we have calculatedthe total ground-state energy per atom using theexperimental lattice constant" of 6.7403 a.u. andn = 3 (neglecting correlation) (see Appendix A fora description of the total energy-calculationscheme). The results are presented in Table Valong with those obtained by augmenting a minimalnumerical. set with two sets of identical sizes,corresponding to positively and negatively chargedcarbon atoms (consisting of a charge-transfer ion-pair basis set)."

A further test of the variational quality of theI CAO basis set was performed by examiningtruncated analytical orbitals. It has been previ-ously suggested" that one may significantly reducethe computational effort associated with perform-ing lattice sums over the rather long-range atom-iclike analytical orbitals with little loss in the ac-curacy of the eigenvalues, by projecting out theselong-range tails. Following Simmons et al."wehave curve fitted the 3s and 3P Slater orbitalswith several short-range Slater orbitals, and havediscarded the long- range components. The re-sulting 3s and 3P orbitals were then virtually iden-

tical to the original ones up to a distance of 1.8a.u. from the nucleus and then decayed almosttwice as fast as the original orbitals. The resultsfor this basis set are shown in the last line ofTable V.

The following conclusions may be drawn fromthis study:

(i) Augmenting the minimal numerical set by3s- and 3P-type orbitals has the effect of stabili-zing the crystal. ground-state energy by about0.9-1.1 eV/atom. This stabilization is similarfor both atomic (untruncated) Slater-type or nu-

merical orbitals. The addition of these orbitalschanges the valence-band eigenvalue spectrumonly an average of by 0.02 eV/state. The detailsof the two lowest conduction bands are changedon the average by as much as 0.06 eV/state.Hence, even a rather small mixing of the virtualatomic states into the occupied manifold in thesolid, produces a substantial total energy stabili-zation, although the accompanying change in theeigenvalues is small.

(ii) A very significant lowering of the total ener-gy, by 2.5 eV/atom, is produced by adding theion-pair basis set to the initial minimal numericalset. Again, the average change in the eigenvaluesis rather small (0.03 eV/state in the valence bandsand 0.05 eV/state in the two lowest conductionbands). Examination of the crystal density showsthat the ion-pair basis set changes only negligiblythe charge in the interatomic bonding region andthat the substantial penetration of the ion-pairorbitals into the core region of the neighboringsites is responsible for most of the variationalstabilization. We were unable as yet to optimizethe total energy as a function of the assumedionicity of the ion-pair set, due to the substantialcost of such calculations. We have instead per-formed the calculation using three discrete valuesfor the ionicity, namely +0.50e, +0.80e, and+1.00e. The results indicate that the choice of+0.80e for the ionicity (a "best" value of +0.85ewas obtained by interpolation) produces the besttotal energy, while the results obtained with achoice of +0.50e are poorer by 0.25 eV/atom.The results obtained for the choice of +1.0e aresubstantially worse than those obtained with noion-pair set (this is probably due to the high in-terelectronic repulsion in the neighboring coreregion caused by the tails of the Q = + 1.0e basisfunctions).

(iii) The use of truncated Slater orbitals alongwith the minimal numeric set introduces a seriousdestabilization of the system so that over 60k ofthe improvement in energy offered by the Slaterorbitals is lost. This result is probably quitesensitive to the exact form of the truncation pro-

SE LF-CON SISTENT N U MERICA L-BASIS-SET LCAO. . .

cedure and no general statement can be madeabout the "optimum" truncation without extensiveexperimentation. It seems, however, that the rela-tive insensitivity of the band structure to a parti-cular truncation (0.09 eV/state on the average, inour case) could hardly be used as an indication ofits adequacy, especially in covalently bonded sys-tems in which penetration of the tails coming fromneighboring atoms into the core region of the cen-tral atom contribute significantly to the stabilityof the system.

8. Convergence of reciprocal and direct lattice sums

As stated in Sec. IV, we use a special k-pointalgorithm"" to calculate the iterated crystaldensity [Eq. (28)]. Several objections have beenraised recently as to the adequacy of a small ksampling set in SC band calculations. ' '" Nat-urally, the key problem in obtaining a good BZaverage of a wave-vector-dependent property isits degree of dispersion. Figure 5 showed thatthe Mulliken and Lowdin orbital populations de-rived from the charge density have a remarkablylow dispersion even in a wide-band material likediamond. Table I shows the BZ average of theseorbital charges as calculated by the 4, 6, 10, and19 special k points used in our SC cycle to deter-mine the crystal density, together with an accuratedetermination of those quantities calculated byusing a numerical integration of the orbital chargedispersion curves of Fig. 5. It is seen that a6-k-point integration scheme is already capable ofproducing extremely accurate results for theseaverages. Figure 6 shows the total ground-state

0.8

0.6—

D

)-U) 04—LLfC3

& 02-

charge density along the bonding [111]directionas calculated by the 6-k averaging schemes. The19-k scheme produced results that are indistin-guishable from the 10-k results on the scale ofFig. 6. It is to be noted that although the band-by-band charge density shows non-negligible dis-persion, the total ground-state density of theclosed-shell system is quite constant throughoutthe BZ. We also note that the BZ averages of theband eigenvalues needed for the total energy cal-culation (Appendix A) cannot be estimated accu-rately by a special k-point algorithm (with a. few

sampling points, as used for the charge density)due to the substantial width of the occupied bandsin diamond, and that this quantity is calculated bynumerically integrating the occupied part of theone-electron band structure (Appendix A).

In examining the effect of the BZ samplingscheme on the calculated x-ray scattering factors,the results of the 4-, 6-, and 19-k-point samplings(Table VI) demonstrate the small dispersion of theFourier-transformed charge density throughoutthe BZ. However, we have found" that in hetero-Polax III-V compounds such as boron nitride,where hybridization between the orbitals of thedifferent atoms in the unit cell is important, thecharge density exhibits much larger variations inthe BZ due to the change of hybridization ratioand degree of interatomic charge transfer with thesymmetry of the state in the zone. In such cases,BZ averages needed for the SC cycle should bedone with more sampling points.

Since space lattice sums are encountered inconstructing the superposition charge density [Eq.(10)], the short-range potential [Eq. (18a)] and thebasis Bloch functions [Eq. (16)], we have examinedthe convergence of various calculated properties.When only a minimal numeric basis set was used,a cutoff distance of 14 a.u. produced energy bandsthat are stable to within 0.15 eV towards any fur-ther increase in the summation range, while in-clusion of long-range virtual (numerical) orbitalsnecessitates a much longer range. We have found,however, that if one obtains these orbitals fromthe solution of Eq. (19) with a localizing potentialA(r) in the form of an external potential well, 4'"

00

I

0.5 I.O

TABLE VI. Brillouin-zone averages of the x-rayscattering factors in diamond, as obtained by differentk-point sampling schemes.

DISTANCE ALONG [I I IJ (UNITS OF ~3/4 p)FIG. 6. Average of the valence charge along the [111]

direction in diamond; full line: 6-nearest-volume kpoints; dashed lines: 10-nearest-volume k points. Theresults for 19-nearest-volume k points are indistin-guishable from the 10-points density, on this scale.

heal

iii2203ii222

3.28ii.995i.6920.i 39

3.273i.992i.720O. i 37

i9 k

3.27 ii.990i.720O. i 35

ALEX ZUNGER AND A. J. FREEMAN

it is possible to reduce the long-range characterof these orbitals without any significant loss inthe accuracy of the resulting eigenvalues and totalenergy. The distance from the atomic origin atwhich this well was constructed was fixed by therequirement that this external potential will notchange the spatial behavior of the low-lying coreplus valence atomic orbitals. With this choice, itwas possible to converge the lattice sums of theBloch orbitals [Eq. (16)] by using a cutoff distanceof 18 a.u. ; an accurate total-energy evaluationrequired a larger cutoff distance of about 23 a.u.It thus seems that the numerical basis set in thiswork offers a natural and convenient way of local-izing the long-range free space atomic orbitals,thereby avoiding the difficulties associated withslowly convergent lattice sums. """

VI. SUMMARY

We have presented a combined Fourier-trans-form and discrete variational approach to theproblem of obtaining self-consistent solutions tothe local-density one-particle equations for solids.The method is based on systematic extensions ofnon- self- consistent real- space techniques ofEllis, Painter, and co-workers" as well as SCreciprocal space methodologies of Chancy, Lin,Lafon, and co-workers. "' Qur method over-comes the usual difficulties in "standard" APW orKorringa- Kohn- Rostoker techniques by avoidingcompletely the muffin-tin approximation to the po-tential and charge density and greatly simplifiesthe problems of basis- set convergence encounteredin tight-binding Gaussian expansion techniques.Complete self-consistency is conveniently ob-tained by combining the treatment of the iteratedsuperposition potential (with its large nonconstantspatial terms in a real-space numerical integra-tion approach) with a Fourier decomposition andreciprocal space sums of the smooth cha.rge-density difference that are not amenable to aCCSC analysis. All the many-center integrationsappearing in standard tight-binding approachesare avoided by the use of a direct Diophantinenumerical integration scheme. We believe thatwith the use of an efficient numerical basis andthe techniques outlines above for obtaining self-consistent solutions, we have come quite close to

the point where the predictions of the local densityformalism for real solids can be critically com-pared with other theoretical approaches to theelectronic structure problem.

ACKNOWLEDGMENTS

We are grateful to D. E. Ellis for making avail-able his non-self-consistent DVM program and forhelpful collaboration during an early phase of thiswork. We thank F. Averill, R. N. Euwema, J. L.Calais, and A. Seth for stimulating discussions.

APPENDIX A: CALCULATION OF OBSERVABLES

+ V„„(r)]dr,(Al)

where the various potential terms arez

V((e(r) = Q(

p(r')V„(r)=, d r',

f r- r'I

(A2)

and V„(r)and V„„(r)are given by Eqs. (6) and('1). The total potential energy is given by

Eps= p,~( )[~V, (r)+2 V„(r)

+ V„(r)+V„„)]dr, v

where V„„is the internuclear repulsion term.Since V,„/N, V„/N, and V„„/Nare divergent asN goes to infinity, we rearrange the terms in thepotential energy to yield

The wave functions, charge density, and poten-tial are used at the end of the SC cycles to com-pute the total ground-state energy and the x-rayscattering factors. We outline here briefly thecomputational schemes used to calculate theseobservable s.

1. Total energy. The total ground- state energyis computed for a fixed set of nuclear coordinatesand lattice vectors from Eq. (4). Using the factthat the band eigenvectors t/r(k, r), are solutions ofour one-particle equation, the kinetic energy isgiven by

z„,= F ~, (%) - f p,„(l I('„.td+ ('..((~ —', (', ( (

pE=& p,~ r V„,r +V„r dr+ & p, r V„,r dr+V„„+ p, r V„r+V„„rdr. (A4)

In this form, the divergences in the individual terms are grouped to yield the three finite terms in (A4),each linear in N, so that a total energy per unit cell can be defined. The first two terms in (A4) repre-sent the total electrostatic energy while the last term is the total exchange and correlation energy. Whenthe total electrostatic energy in Eq. (A4) is rearranged, one obtains the expression

SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . .

E„„=zp,~ r V„,r +V„r dr+g Z V„,a +V„aa=1

(A5)

where the "vacancy potential" V,~(n) is definedas the total electrostatic potential at the 0.th sitedue to all charges except those of the ath siteitself, and V„(a)is the value of the electronicpotential energy at this site. To calculate thetotal energy we thus need to perform three in-tegrals over the unit cell volume, namely, overp, (r)[V„,(r) + V„(r)],p, (r) V,(r), and p, (r)V

„(r)

and two sums [in Eqs. (A1) and (A5)]. The ex-change and correlation integrals were found toconverge readily using about 2000 Diophantinesample points, while the Coulomb integration wasfound to converge more slowly. The termsp, (r)V„,(r) and p, (r)V,„(r)are combined beforeintegration is attempted. It was found, however,that no further simplications of the Coulomb in-tegral are required in order to obtain good ac-curacy in its evaluation; about 4000-7000 integra-tion points, combined with a highly peaked sam-ple point distribution in the core region, weresufficient to obtain good accuracy in the cohesiveenergy.

The sum over the band eigenvalues in Eq. (Al)was performed using 32 equally spaced points inthe

,—,irreducible BZ section. Sufficient accuracy inthe crystal core eigenvalues is assured by employ-ing some 10-30 integration points inside thenuclear volume and about 300 points within the isorbital sphere.

The total energy of the free atom is computedfrom a spin-polarized version of the Herman-Skillman program using the correlation functionalfrom Eq. (l).

2. X ray scattering factors. The x-ray scat-tering factors f (K) are computed from the SC

p,~(r) by a direct three-dimensional Diophantiveintegration

(A6)

for a, set of (h, k, l) reflections K. Although theconvergence of this integral as a function of thenumber of sampling points is rather good (0.270error for 2000 points in diamond), it was foundsomewhat advantageous to combine a radial (onedimensional) integration of the spherical com-ponents of p, (r) [Eq. (31)] with a Diophantine in-tegration of [p„,(F) —p, (r)]e'"'.

*Supported by the NSF (DMR Grant No. 72-03101 and72-03019) and the AFOSR (Grant No. 76-2948).

~P. Hohenberg and W. Kohn, Phys. Rev. B 136, 864(1g64) .

2W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1965).3F. Bassani and M. Yoshimine, Phys. Rev. 130, 20

(1963).4L. Kleinman and J. C. Phillips, Phys. Bev. 125, 819

(1962).5R. N. Euwema and D. J. Stukel, Phys. Bev. B 1, 4692

(1970).~R. Keown, Phys. Rev. 150, 568 {1966).J. R. P. Neto and L. G. Ferreira, J. Phys. C 6, 3430(1973).

G. S. Painter, D. E. Ellis, and A. R. Lubinsky, Phys.Rev. B 4, 3610 {1971);D. E. Ellis and G. S. Painter,ibid. 2, 2887 {1970).

~A. R. Lubinsky, D. E. Ellis, and G. S. Painter, Phys.Rev. B 11, 1537 (1975).

' J. B. Danese, J. Chem. Phys. 61, 3071 (1974); J. B.Danese and J. W. D. Connolly, Int. J. Quantum. Chem.Symp. 7, 279 (1973).R. C. Chancy, C. C. Lin, and E. E. Lafon, Phys. Rev.B 3, 459 (1971).(a) P. W. Kervin and E. E. Lafon, J. Chem. Phys. 58,1535 (1973); {b) E. E. Lafon, MIT Solid State and Mo-lecular Theory Group Quarterly Progress Report No.

69, p. 66 (1968) {unpublished).J. E. Simmons, C. C. Lin, D. F. Fouquet, E. E. Lafon,and R. C. Chancy, J. Phys. C 8, 1549 (1975).E. J. Baerends, D. E. Ellis, and P. Bos, Chem. Phys.2, 41 (lg73).

' H. Sambe and B. H. Felton, J. Chem. Phys. 62, 1122(1975).C. B. Haselgrove, Math. Compt. 15, 373 (1961).' D. E. Ellis and G. S. Painter, Phys. Bev. B 2, 7887(1970).F. W. Averill, Phys. Rev. B 6, 3637 (1972).

'~E. C. Snow, Phys. Bev. B 8, 5391 (1973).J. F. Janak, V. L. Moruzzi, and A. R. Williams, Phys.Rev. B 12, 1257 (1975).

'M. Rasolt and D. J. W. Geldart, Phys. Bev. Lett. 35,1234 (1975).P. Nozibres and D. Pines, Phys. Bev. 14, 442 {1958).

23D. Pines, Elementary Excitations in Solids {Benjamin,New York, 1963).

24K. S. Singwi, A. Sjolander, P. M. Tosi, and R. H. Land,Phys. Bev. B 1, 1044 (1970).L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).B. Y. Tong and L. J. Sham, Phys. Rev. 144, 1 (1966).

70. Gunnarsson, P. Johansson, S. Lundqvist, and B. I.Lundqvist, Int. J. Quantum Chem. Symp. 9, 83 (1975);O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13,4274 (1970).

4736 ALEX ZUNGER AND A. J. FRKKMAN

G. S. Painter, J. S. Faulkner, and G. M. Stocks, Phys.Rev. B 9, 2448 (1974).

~ N. Elyashar and D. D. Koelling, Phys. Rev. B 13, 5362(1976).

3 A. Zunger and A. J. Freeman, Int. J. Quantum Chem.(to be published).R. N. Euwema, and D. L. Wilhite, and G. T. Surratt,Phys. Rev. 7, 818 (1973).

32G. T. Surratt, R. N. Euwema, and D. L. Wilhite, Phys.Bev. B 8, 4019 (1973).G. G. Wepfer, H. N. Euwema, G. T. Surratt, andD. L. Wilhite, Phys. Rev. B 9, 2670 (1974).

34R. N. Euwema and R. L. Green, J. Chem. Phys. 62,4455 (1975).F. Herman and S. Skillman, Atomic Structure Calcula-tions (Prentice-Hall, Englewood Cliffs, ¹J., 1963).

3 P. P. Ewald, Ann. Phys. (Leipz. ) 64, 753 (1971).3 M. P. Tosi, in Solid State Physics, edited by F. Seitz

and D. Turnbull (Academic, New York, 1964), Vol. 16,p. l.

3 K. H. Johnson and F. C. Smith, in Computational Meth-ods in Band Theory, edited by P. M. Marcus, J. F.Janak, and A. R. Williams (Plenum, New York, 1971),p. 377.P. D. DeCicco, Phys. Bev. 153, 931 (1967).All three-dimensional plots shown are produced by theperspective hidden-line subroutine PICTURE of M. L.Prueitt. The authors are indebt to Dr. Prueitt forsupplying this routine. All plots are generated fromcalculated data without interpolation.

~W. G. Richards, T. E. Walker, and R. K. Hinkley, ABibliography of Ab Initio Molecular 8'ave Eunctions(Clarendon, Oxford, 1971).

2In other calculations (e.g. , Refs. 11 and 13) the lower-ing of the one-electron eigenvalues (band structure) asa result of augmenting or changing a basis set wastaken as a criteria for the variational superiority ofthe augmented set. We avoid using a criteria of thisnature. Calculations on molecules (e.g. , Ref. 41) andsolids (e.g. , Hef. 43) have demonstrated that due to thefact that the sum over eigenvalues forms only a part ofthe total energy (usually of kinetic nature), changes inthe basis set or redistribution of the electrons amongdifferent orbitals can result in lowering of the eigen-values and a simultaneous destabilization of the totalenergy.

43A. Zunger, J. Chem. Phys. 63, 4854 (1975).44R. Latter, Phys. Bev. 99, 510 (1955).45B. E. Watson, Phys. Bev. 111, 1108 (1958).

F. W. Averill and D. E. Ellis, J. Chem. Phys. 59, 6412(1973).

YW. H. Adams, J. Chem. Phys. 34, 89 (1961); 37, 20009(1967); T. L. Gilbert, in Molecular Orbitals in Chem-istry, Physics and Biology, edited by P. O. Lowdin(Academic, New York, 1964).

4 P. O. Lowdin, Int. J. Quantum Chem. Symp. 1, Sll(1967); Adv. Phys. 5, 1 (1956).H. J. Conroy, J. Chem. Phys. 47, 5307 (1967).D. M. Drost and J. L. Fry, Phys. Rev. B 5, 684 (1972).

'(a) J. Callaway and J. L. Fry, in ComPutational Meth-ods in Band Theory, edited by P. M. Marcus, J. F.Janak, and A. H. Williams (Plenum, New York, 1971),p. 512; (b) J. Callaway and C. S. Wong, Phys. Hev. B 7,1096 (1973).

52R. C. Chancy, E. E. Lafon, and C. C. Lin, Phys. Rev.B 4, 2734 (1971).

~It is noted that this type of cancellation between kineticand potential terms is effective only for Bloch statesof the type defined in Eq. (16). The alternative form@N(k, r) =e' ~Z„)(„(r—R„—d~) used by Harris andMonkhorst (Hef. 54) in which the sum of atomic orbitalsis plane-wave modulated& does not provide effectivecancellation since T 4'~ (k, r) does not contain a positive(atomiclike) singularity. Using the form 4 ~(k, r) in-stead of Eq. (16) with a small basis set we obtainedgood accuracy in the virtual band energies but pooraccuracy in the core states.F. E. Harris and H. J. Monkhorst, in ComputationalMethods in Band Theory, edited by P. M. Marcus,J. F. Janak, and A. R. Williams (Plenum, New York,1971), p. 517.

5~N. E. Brener, Phys. Rev. B 7, 1721 (1973).+D. J. Chadi and M. L. Cohen, Phys. Rev. 8 8, 5747

(1973).~TH. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188

(1976).MK. H. Johnson and F. C. Smith, in Computational Meth-

ods in Band Theory, edited by P. M. Marcus, J. F.Janak, and A. R. Williams (Plenum, New York, 1971),p. 377.E. J. Baerends, D. E. Ellis, and P. Hos, Theor. Chim.Acta 27, 339 (1972).(a) E. W. Stout and P. Politzer, Theor. Chim. Acta 12,379 (1968); {b) B. J. Duke, ibid. 9, 760 (1968); (c)H. Rein, H. Fukuda, H, Win, G. A. Clark, and F. E.Harris, J. Chem. Phys. 45, 4743 (1966).

O'D. D. Koelling, A. J. Freeman, and F. M. Mueller,Phys. Rev. B 1, 1318 (1970); H. Schlosser and P. M.Marcus, Phys. Hev. 131, 2529 (1963); D. D. Koelling,ibid. 188, 1049 (1969).

6~W. E. Budge, Phys. Rev. 181, 1020 (1969); 181, 1024(1969).

3J. O. Slater and P. DeCicco, MIT Solid State and Mo-lecular Theory Group Quarterly Progress Report No.50, p. 46 (1963) (unpublished).

4J. E. Falk and R. J. Fleming, J. Phys. C 6, 2954 (1973).5H. S. Mulliken, J. Chem. Phys. 23, 1833 (1955).

66P. O. Lowdin, J. Chem. Phys. 1S, 365 (1950).R. C. Chancy, T. K. Tung, C. C. Lin, and E. E. Lafon,J. Chem. Phys. 52, 361 (1970).

68Similar sharp features in the initial 6p(r) have beenpreviously observed in a non-self-consistent calcula-tion on solid SiC {Ref. 9) and in Hartree- Fock calcula-tions on simple diatomic molecules (Ref. 69).

6 H. F. W. Bader and A. D. Bandrauk, J. Chem. Phys.49, 1653 (1968).G. W. Pratt, Phys. Rev. 88, 1217 (1952).

7~Similar effects have been previously indicated in astudy of the muffin-tin superposition model in mole-cules (Ref. 10) in which the eigenvalues and orbitalsobtained in the superposition approximation were indi-cated to be reasonable approximations to the "exact"results, while the binding energy in that model coulddiffer by as much as (100-300)% relative to the "full"model including the residual charge ~ (r).F. Herman, Phys. Bev. 88, 1210 (1952); 93, 1214(1954).

73W. E. Budge (unpublished); results of band structure

15 SELF-CONSISTENT NUMERICAL-BASIS-SET LCAO. . . 4737

cited by J. R. Leite, B. I. Bennett, and F. Herman,Phys. Rev. B 12, 1466 (1975).

~4ln the LCAO calculations of Chancy, Lin, and Lafon(Ref. 11) as well as in the LCBO study of Kelvin andLafon (Ref. 12a), and the OPW study of Bassani andYoshimine (Ref. 3), the potential is generated from an

overlapping atomic model using the Hartree- Fockcharge density calculated by A. Jucy [Proc. R. Soc.Lond. A 173, 59 (1939)] for the 1g 28 2p ground-stateconfiguration of the carbon atom, and the crystal ex-change potential is linearized to a lattice sum of atomiccharge potentials. The lattice constant is taken as6.728 a.u. In the discrete variational study of Painteret al. (Ref. 8), the atomic Hartree-Fock density ofClementi was used with no linearization of the exchangepotential. In this work a lattice constant of 6.7406 a.u.was used.As is well known, the OPW results for the states that

contain p character (e.g. , I')gal' I'pc I'fg X4 X$ )are probably very poorly converged since no p statesappear in the carbon core and hence at these states theOPW series reduces to a pure plane-wave series. In arecent convergence study by Euwema and Stukel (Ref.5) it was argued that some 5000 plane waves are re-quired for complete convergence.

' K. Schwarz, Phys. Rev. B 5, 2466 {1972).~'J. Thewlis and A. R. J3avey, Philos. Mag. 1, 409 (1958).~ Similar ion-pair charge-transfer set were used in

early quantum-mechanical calculations on simple di-atomics where the Heitler-London representation inan orthogonalized basis set was reformulated to in-troduce ionization in the bond, e.g. , J. C. Slater,J. Chem. Phys. 19, 220 (1951). This formulation wasshown to yield substantial improvement over the simplemolecular- orbital model.

'~A. Zunger and A. J. Freeman (unpublished).