PH YSICAL REVIEW B VOLUME 15, NUMBER 10 10 MA Y 1977

Ground-state electronic properties of diamond in the local-density formalism*

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

{Received 26 August 1976)

We use our previously reported method for solving self-consistently the local-density one-particle equations in

a numerical-basis-set linear combination of atomic orbitals expansion to study the ground-state charge density,x-ray structure factors, directional Compton profile, total energy, cohesive energy, equilibrium lattice constant,

and behavior of one-electron properties under pressure of diamond. Good agreement is obtained with availableexperiment data. The results are compared with those obtained by the restricted Hartree-Fock model: therole of electron exchange and correlation on the binding mechanism, the charge density, and the momentumdensity is discussed.

I. INTRODU(. 'TION

The local density functional (LDF) formalismof Hohenberg, Kohn, and Sham, "and its recentextension as a local spin-density functional formal-ism, ' form the basis of a new approach to the study ofelectronic structure in that the effects of exchangeand correlation are incorporated directly into acharge- density- dependent potential term that isdetermined self -consistently from the solution ofan effective one-particle equation. Applications ofthe LDF formalism to atoms+' and molecules' haveyielded encouraging results. Similar applicationsfor solids are complicated by (i) the need to consider both the short-range and the long- rangemulticenter crystal potential having nonsphericalcomponents, (ii) the difficulties in obtaining fullself-consistency in a periodic system, and (iii)the need to provide a basis set with sufficientvariational flexibility. Hence, theoretical. studiesof ground- state electronic properties of solidsin the LDF formalism have been mainly l.imited tomuffin-tin models for the potential, "non-self-consistent schemes, ' treatments of simplifiedjellium models'0 or spherical cellular schemes

We have recently proposed"" a general self-consistent method for solving the LDF formalismone-particle equation for realistic solids using anumerical-basis-set LCAO (linear combination ofatomic orbitals) expansi. on and retaining all non-spherical parts of the crystal potential. We havedemonstrated a rapid convergence of the self-consistent (SC) cycle when the treatment of thefull crystal charge density is suitably apportionedbetween real- space and Fourier- transformedreciprocal-space parts and have indicated thelarge degree of variational flexibility offered bya nonlinearly optimized (exact) numerical atomic-like basis set. We have shown that all multi-center interactions as well as the nonconstant partsof the crystal potential are efficiently treated by

a three- dimensional Diophantine integrationscheme.

The purpose of this paper is to illustrate the ap-plicability of our method to real systems bystudying the ground- state electronic properties ofdiamond. Diamond has been long considered asa prototype for covalently bonded insulators" anda great deal of experimental work has been doneon its ground-state properties, including co-hensive energy, "lattice-constant studies, "x-rayscattering factors, ""charge density, "and di-rectional Compton profile. O' In addition,theoretical studies on its ground-state propertieswithin the restricted Hartree-Fock (RHF) modelare available" " so comparison with the predic-tions of the LDF formalism is possible. Althoughthe eigenvalue spectrum (band structure) of thelocal exchange Hamiltonian for diamond has beenstudied previously by a variety of first-principlestechniques [augmented planes waves (APW), '""orthogonalized plane waves (OPW),""pseudo-potential OPW, " LCAO, "" and cellular meth-ods "], the ground-state observables relatedto the ground-state crystal charge density havereceived much less attention. While our methodwas shown" to accurately reproduce the bandstructure of diamond as obtained by other tech-niques"'" (when correlation and self-consistencyis omitted so as to be compatible with the pre-viously published band- structure models and thefull nonspherical components of the potential areretained in both calculations), we do not considerthis as a stringent test since the LDF formalismin its "standard" form does not make any claimon the physical significance of the band eigen-values nor are these eigenvalues sensitive enoughto the details of the basis set and potential. " Inwhat follows we present our results for the x-rayscattering factors, charge density, directionalCompton profile, total energy, and equilibriumlattice constant and discuss the role of exchange

5049

5Q5Q ALEX ZUNGKR AND A. J. FREEMAN

and correlation in determining these observables.Comparisons are made with earlier Hartree-Fock results" "and with experiment.

II. METHOD OF CALCULATION

Since the method was described at length else-where, "we give only a brief description here inorder to place our work in proper perspective.Basically, our aim is to solve the effective one-particle equation in the local density functionalformalism, i.e. , the Bloch equation in which thepotential consists of a Coulomb and an exchangeand correlation potential V„(p{r))which is givenas a functional derivative of the total exchange andcorrelation energy E„,( p(r)) of an interacting (in-homogeneous) electron gas with respect to thecharge density. ' In our work, we use only thefree-electron exchange and correlation potential

terms" which are the first two terms in a generalexpansion'" of E„(p(r)}. For the free-electroncorrelation term we use the results of Singwiet al. as fitted to an analytic form.

One writes the ground- state charge density p(r)as

NQ &cp(r) =

(2 )' n'{kBZ

x ()(*,. (k, r) P&(k, r) dk,

where n, (k) is the Fermi occupation number of theo„occupiedcrystal eigenfunctions (t(,.(k, F) of bandj and wave vector k; and N and 0 denote the num-ber of unit cells and the unit cell volume, re-spectively. The integration is performed over theoccupied part of the Brillouin zone (BZ). The totalground-state energy is given by

&(,( = P n, (k)( 0;(k, r) ~ —2 &' ~ (t', (k, r)~ r(rl(Q; d, d, + g " r +r„.(r(q),—Z. i p(r') Z Z}r-R —d f Jr- r' ) (2)

where Z is the nuclear charge of the atomsituated at site d and R denotes the positionvector of the rnth unit cell.

It is clear that, because of the functional depen-dence of the potentials on p(r) and hence on thewave functions {i{(,.(k, r)j, the solutions must beobtained by some self-consistent (SC) procedure.As our initial guess for the crystal density (whichwill be subsequently refined in the SC iterativecycle) we chose a population-dependent overlappingatomic densities model

Ep'"'(~) = P g p. (r - R - d. , b"„'„Q )), (2)

m e

where p, (r, Q"„„„Q]) is the charge density ofatom (ion) o.'calculated numerically from an atomic(ionic) LDF one-particle equation assuming oc-cupation numbers f„, for the central-field quan-tum numbers n'l' and a possible net ionic chargeQ~. In solving these equation we use the same func-tional form of V„,(p (r)} as used for the crystalcase but replace the crystal density with p (r).The atomic eigenfunctions (p„,(r) are used todetermine (self- consistently) the one- site densityby

P (r 9 ( Q )) = Q f (V' ( (r)(p„g(r)and because of the assumed spherical symmetrythese are easily solved for self-consistently innumerical form for an assumed set of atomic popu-

lations and charge tL f"„„,, Q"j (a typical accuracy of10 ' a.u. is obtained for the eigenvalues). Thesuperposition density is then formed by carryingthe sum in Eq. (3) to convergence (a summationradius of 20 a.u. is employed). The initial guessfor the crystal potential is obtained by solvingPoisson's equation for p'"'(r) and using the ex-change and correlation functionals with this mo-del density. The Coulomb superposition potentialis separated into short-range Vss'c(r) and a long-range V'„"gc(r) parts. The first is simply given bya direct lattice sum of the short-range (over-lapping) ionic potential

RV'„"' (r)= g V (r- R —d )+

(5)

where the quantity in large parentheses denotesthe Coulomb potential associated with the neutra-lized charge of site e. This sum converges rapidlydue to the subtraction of the long-range ionic tailsin Eq. (5) and a summation radius of R ~=17-22 a.u. is sufficient to achieve an accuracyof 10 '-10 ' a.u. The remaining long-range Cou-lomb term is calculated by the Ewald4' techniqueusing the effective charges

lim rV (r).y Rmax

The exchange and correlation "superposition" po-tentials are given by applying the respective func-

15 GROUND-STATE ELECTRONIC PROPERTIES OF DIAMOND IN. . . 5051

tionals to p'"'(r) (i.e. , without linearizing thesepotentials with respect to the individual"'" p» s).Note that all aspherical contributions originatingfrom the overlapping tails of the one-site densityp (r), as well as those contributed by the point-ion electrostatic crystal field, are fully retained.However, owing to the nonlinearity of the ex-change and correlation functionals with respectto the individual single-site densities p„(r), theinitial superposition potential V'"'(r) is not rep-resentable as a lattice sum of one-center termsand hence even at this zeroth iteration stage theone-particle equation constitutes a multicenterproblem. After convergence of the direct latticesums [Eqs. (3} and (5)] is obtained, V'"'(r) iscompletely defined by specifying the atomic num-bers Z, the postulated crystal structure and theassumed populations and changes P„„Q"j. Thelatter are subsequently used as free parametersin the SC procedure to optimize the superpositionpotential (see below). By changing the relativeproportions (i.e. , hybridization ratio) of the popu-lations in the atomic LDF equations (e.g. , the car-bon 2s to 2p populations) the superposition densityand potential can be made to better simulate theoutput crystal density and potential so as to facil-itate the convergence of the SC cycle.

The crystal wave functions {{,.(R, r) are expandedin terms of ht) Bloch functions 4„,(k, r) in stand-ard form:

t/r, (k, r) = g g C„„(k)4, (k, r),n= 1 II, =1where C (k, r) is defined in terms of the pthbasis orbital y„'(r) situated on the nth site:

(8)

4 (k, r) = N ' ' g e' '"» y (r —d —H ) . (7)We use as LCAO basis functions nume~icajt atomic-like LDF orbitals. These are obtained from asolution of an atomiclike equation, similar to thatused to generate the superpostion potential

= &, X, (r [f; ~ Q ]') (8}Here, the one-site potentialg (r) can be taken asa generalized atomiclike potential as long as itgenerates through Eq. (8) orbitals that form anaccurate and rapidly convergent basis set forexpanding the crystal orbitals g,.(k, r). Such basisorbitals have to maintain the correct cusps nearthe nuclei (which are absent in a Gaussian basis)as well as appropriate nodal behavior and an~larvariation, while any long-range tail is Unwarranted(due to the possibility of formation of linear depen-dence between the Bloch functions). We hence

chose g„(r) in Eq. (8) to be the usual atomic po-tential plus an additional external potential A(r)in a form of a potential well starting from adistance R, from the nucleus. This distanceis chosen so that the low-lying occupied solutionsof Eq. (8) will be identical with the exact atomicsolution, while the virtual orbitals (3s, 3p, and3d in carbon} will have compressed tails. Ex-perimentation with this type of localizing po-tential" revealed that it is rather straightforwardto chose R, (e.g. , 17 a.u. for carbon) so that theresulting orbitals will be sufficiently localized soas to enhance the Bloch sums convergence[Eq. (7)] and still possess sufficient variationalquality in the crystal calculation (e.g. , producea total energy that is as low as that obtainedwith regular atomic orbitals). The basis setobtained in this way (for a fixed choice of (f„„Q) )is used to our work in direct tabular form withno attempt to fit it to an analytic set. Extensivestudies of the basis-set problem" (i.e. , additionof analytic Slater orbitals to a minimal basis setto better span the virtual space) have shown thata numerical basis set for carbon consisting ofls, 2s, 2p, 3s, 3p orbitals (18 orbitals per unit celldenoted here as set I) is sufficient to maintain anaccuracy of about 3 mHy in the valence and lowesttwo conduction-band eigenvalues for diamond. Inthe calculation of the total crystal energy it wasfound that the convergence of the result due toadditional virtual numerical (or Slater) orbitalswas rather slow but that augmenting the basis setby ion-pair charge-transfer orbitals [ i.e. , solvingEq. (8) for a neutral carbon atom with Q» = 0 aswell as for Q = 0.8 and Q = —0.8 and using thistriple-sized set for the crystal] produced a sub-stantial improvement in the result. We henceuse this extended set (denoted here as set II) inthis work to calculate total energies and equili-brium lattice constants and compare the resultswith the more economic set I.

Having defined the initial crystal potential andthe basis set, the usual linear variation secularequation is given as

h

Q Q [H „s(k) —S„ s (k)e,(k)] C , (k) = 0,

where the Hamiltonian and overlap matrix ele-ments in the Bloch representation are

H„„q(k) = ( 4»(k, r) I —2 +'+ V(r) I 4 s (k, r)&,(10a)

S „s(k) = ( C „,(k, r) I 4„8(k, r)) . (10b)These matrix elements, when expanded in termsof the basis orbitals y„(r} and the projected single-site potentials, give rise to a large number of two-

5052 ALEX ZUNGER AND A. J. FREEMAN 15

and three- center integrals. The computationaldifficulty in handling these integrations has led

many workers in both molecular and solid-stateelectronic structure theory to resort to somewhatartificial analytic basis functions (e.g. , Gaussians)or to simply neglect a large number of these in-tegrals. In the present work, we resolve that prob-lem by evaluating the integrals directly from Eq.(10) (using numerical forms for the potential andBloch functions) employing a three-dimensionalDiophantine integration scheme. " The details ofthe application of this scheme are given else-where. ""Here we only note that in using thisapproach all the multicenter integrals are avoidedand that any general form for the potential (variousforms for correlation, non-muffin- tin corrections,etc. ) or basis functions can be treated equally in adirect manner. The use of an exact numericalbasis set along with a superposition potential gen-erated from the same set Q„„Q ) offers anotheradvantage": the integrand appearing in Eq. (10a)exhibits an algebraic cancellation between therepulsive kinetic energy and the attractive Coulombsingularity in the vicinity of the nucl. ei in the sol.idso that the Hamiltonian elements can be calculatedto good accuracy. About 2000 Diophantine inte-gration points are required to obtain an accuracyof 2 mHy in the valence and first two eonduction-band eigenvalues in diamond while some 7000points are required for an accuracy of 0.2 eV inthe total energy calculation.

The secular equation [Eq. (9)) is solved at asetof reciprocal- lattice vectors k, ; i = 1, . . . , p,The resulting eigenfunctionsare used to constructa refined crystal density p„,(~) from Eq. (1) wherethe integration over the BZ is replaced by a sum-mation over the p,. wave vectors [k,j. We chose,for the set [kj, the first 10 (inequivalent) specialk points in the diamond BZ and their associatedweights. 4' We note that due to the low dispersionof ti,*(k, r)i(, (k, r) in diamond across the BZ" thisapproximation to the charge density does not causeany severe problems in calculating p, (r) and thetotal energy; however, the calculation of the BZaverage of the band eigenvalues (needed for thecalculation of the kinetic energy term) requiresmore k points due to the substantial width of thevalence band and is calculated in the present studyby averaging 32 inequivalent k points.

Given the output crystal density, we set a self-consistent cycle that is based on two stages. Instage 1 (charge and configuration self-consistency),we iterate over the population numbers [f„',) andcharge Q, solving at each stage for a new super-position potential and numerical basis orbitals[Eq. (8)] so as to minimize in the least-squaressense the deviation Ap(r) between the crystal den-

sity p„„(~) and the population-dependent super-position density p'"'(r). During these iter-ations electronic charge is redistributed fromthe occupied atomic orbitals to the formerlyvirtual orbitals resulting in a radial distortion ofthe ba,sis set. We thus allow our basis set torelax to the current form of the iterated crystalpotential by varying it non/ineaxly. In carryingstage 1 in SC to completion, the residual func-tion Ap(r) is minimized over space and most ofits localized features (present at the first itera-tion) are smoothed out. In other words, by varyingsimultaneously the basis set (allowing for radialdistortion, shift of nodes, and rehybridization) andcrystal potential, we select among aLL possiblesuperposition densities the one that best simulatesthe actual crystal potential. This procedure re-quires about 4-5 iterations in diamond. The resi-dual density Ap(r) obtained at the end of this stagerepresents the "misplaced" charge that is notamenable to a superposition representation usingone-center terms located on existing atomic sites.It turns out that due to the substantial penetrationof charge from nearest neighbors into the innerregion of the central. atom, the fuLL site anisotropyin diamond could not be satisfactorily representedby an optimized superposition model (a standarddeviation of 0.09e is obtained). The inclusion ofthe residual hp(r) is done in stage 2 of SC ("fullSC") by Fourier transforming numerically theresidual charge

d p(K, ) = —f e '"&'d p(r) dr,Q(the integration over the unit cell being performedby the Diophantine method) and calculating thechange in the electronic Coulomb potential dueto this density using a direct Fourier series

(12)rc, ~o I K' I

This change in the Coulomb potential is added toY~~a'c(r)+ VLgc(r) obtained in the last charge andconfiguration SC iteration. The iterated exchangeand correlation potential is computed using the fulldensity p, (r) and the solution of the eigenvalueproblem [Eq. (9)] is repeated iteratively so as todiminish 4p(r). Owing to the absorption of thecorelike cusps and localized features into p'"'(r)in stage 1 in SC, the Fourier series in Eq. (12)converges rapidly (only the first seven starsare needed) and only 3-4 iterations are required.It is noted that in performing full self-consistencywe go considerably beyond the "standard" degreesof self-consistency obtained in either muffin-tin-type treatments~ (in which the iterated redistribu-ted charge density is spherically averaged before

GROUND-STATE ELECTRONIC PROPERTIES OF DIAMOND IN. . .

the next iteration is attempted} or conventionallocal-density I.QAO schemes (in which onlycharge and configuration SC is maintained usingeither Mulliken populations" or spherical Gaus-sians ' as a superposition projection set). In whatfollows we will discuss the results of the treatmentdescribed above for the ground state electronic pro-perties of diamond using our final fully self-consistent results.

III. RESULTS AND DISCUSSION

A. Charge density and x-ray scattering factors

In order to examine the effects of exchange andcorrelation on the ground- state charge density indiamond we have performed three fully self-con-sistent calculations; the first employed only theelectrostatic electron- electron and electron-nuclear potential in the one-particle equation("electrostatic model" ), the second incorporatedalso the local exchange ("exchange model" ),while in the third calculation the correlation po-tential was also considered ("exchange and cor-relation model"}. All three calculations used anextended numerica, l set (ls, 2s, 2p, 3s, and 3porbitals per carbon) and all lattice sums wereperformed to convergence. The lattice constantwas fixed at 6.740 a.u. We arbitrarily define"exchange charge" as the total valence (i.e. , thetwo lowest Is bands are omitted) ground-statecharge-density difference between the exchangeand the electrostatic models while the "correlationcharge" is defined as the corresponding densitydifference between the exchange and correlationmodel and the exchange model. Figures 1 and 2show the contribution of some high-symmetrystates in the BZ to the exchange and correlationcharge density, respectively, along the [111] bonddirection in diamond and Fig. 3 shows the cor-responding charge-density contributions to thetotal valence charge density. Figure 4 comparesthe correlation charge along two directions inthe crystal.

The main conclusions to be drawn from thesecomparisons are the following:

(i) The exchange charge density alters the elec-trostatic charge density by as much as (10-40)%in the region around the center of the bond whilethe correlation charge density is responsible foronly a (0.1-0.3)% change. The exchange chargeis about two orders of magnitude larger than thecorrelation in most of the bond region.

(ii) Both the exchange and correlation charge den-sities show substantial k- space dispersion. Therelative changes in various k-space contributionsare larger than the corresponding variations inthe contributions to the total valence density. The

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FIG. 1. Contributions of some high-symmetxy statesin the BZ to the valence "exchange charge density" indiamond along the [111]direction. The two lowest 1s-like bands are not included. The density differenceshown results from two independent exchange-model andelectrostatic-model calculations. A carbon atom islocated at the origin while the right side of thj x axiscorresponds to the center of the C-C bond.

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DISTANCE ALONG [ I I I] ( UNITS OF p)4

FIG. 2. Contribution of some high-symmetry statesin the BZ to the valence "correlation charge density"in diamond along the [lllJ direction. The two lowest1s-like bands are not included. A carbon atom is lo-cated at the origin while the right-hand side of the ~axis corresponds to the center of the C-C bond.

5054

W

ALEX ZUNgER AN@ A. J ~ FREEMAN

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m is at the origin while the ri ht-e x axis corresponds to t

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00

about 0.2-0.4a . — . a.u. from each atom. A lan s arpenin of c

o e atomic site due to the incoof the correlatio f tion unctional in the

rve in a separate calculation onon an iso-e 2/o decrease in"

T e observed effect in the sol.id is hever much more p

o i is how-re pronounced

en into account) indicatin anh iza ion mechanism in the s

ue o the enhancement o theions rom neighboxin sit

thus seems that th hdcdb h

a e charge- dens it

huclei and around th bxc ange and correlation botho close to the

e bond center a

acting atoms.o e solid relativeive to the noninter-

Table I s lated x- ray scatter'ring

fuuil valence density (Fi . 3g. ) generally shows aer ow k-space dispersion c

tions at X 8'p Ion (compare contribu-a, TV, and L to that of the rand

1 oi tG). Th I'

an correlation char es '

space is quite small.ges in real

~ ~ ~

xiii~ The over-all effect of both exchI t'o o th ha

charge buildue c rge densit is' y is to enhance the

ui up around the centerto substant' ll

r of the bond andia y increase the char e 1

on the atoms, at therge ocaliz ation

charge density from a dou, a he expense of deletin s

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om a oughnut-shaped region at hows the calcu

TABLE I Calculated and ex riexperimental x-ray scatterin faer1ng actors for diamond

1f(K,) = — exp(iK, r)p«~(r) dr .

f(0, 0, 0) equals 6.0 Th " i 'e super osith 1 ~2 2 e atoms and a charge densiarge ensity given in Eq. (3).

Elec trostatic E hxc angemodel model

Exchange andcorrelation

model Expt. 'Haar tree SuperpositionFock '

ill220311222400331422511333

3.0621.9361.6560.0661.4701.6251.4111.3471.346

3.2731.9921.7200.1371.4941.6001.4231.3851.381

3.2811.9951.6920.1391.4931.6051.4081.3921.392

3.321.981.660.144 b1.481.581.421.421.42

3.291.931.690.081.571.551.42

3.0051.9641.7600.01.5851.5191.4321.3871.387

' Reference 18.Reference 19.

~ ReReferences 22 and 25.

GROUND-STATE ELECTRONIC PROPERTIES OF DIAMOND IN. . . 5055

factors in diamond at the three levels of localdensity approximations, together with the ex-perimental results"'" and the canonical Hartree-Fock results of Euwema et al. obtained using ans and p Gaussian basis set."'" It is apparent thatexchange acts to increase the low-angle scatteringfactors quite dramatically, reflecting the in-creased localization of charge in the interatomicregion, with the correlation effect being muchsmaller. In particular, the calculated (222) for-bidden reflection (forbidden in the approximation inwhich the charge density is given as a superpositionof spherically symmetric, possibly overlapping,atomic densities, last column in Table I) isincreased by about a factor of 2 upon introducingexchange in the potential. The value of thisscattering factor turned out to be particularly sen-sitive to the details of the self-consistency main-tained in the calculation and to the quality of thebasis set" (e.g. , a minimal basis set yieldeda value of 0.082 while a non-self-consistent ex-tended basis set yielded a value of 0.089). Wenote from Table I that a band model that allowsfor nave-function overlap produces markedlyimproved results over a spherical superpositionmodel (last column).

The general agreement between our calculatedresults and experiment is reasonable. It is ap-parent from the comparison of the atomic andcrystal values that, as expected, the scatteringfactors constitute a sensitive test of the detailsof the calculated charge density only for the firstfew reflections to which the density in the outerregions of the cell contribute. Similar conclusionscan be drawn from comparing our exchange-modelresults with those obtained in the literature" "bya number of approximations to the local exchange

problem (Table II). Although these calculationsemploy various independent approximations, theresults vary only in the first few reflections.

B. Self-consistent crystal potential and the band structure

We next consider the contributions of exchangeand correlation to the self-consistent crystal po-tential. Figure 5 shows the components of thefinal SC crystal potential, as obtained in the ex-change model and in the exchange and correlationmodel, respectively, along the bond direction indiamond. It is observed that the exchange andcorrelation potentials constitute some 50 and 5/pof the electrostatic crystal potential at the bondcenter with their relative contribution to the totalpotential decreasing rapidly as one moves towardsthe atomic sites. To further elucidate the role ofexchange and correlation in the Hamiltonian matrixrepresentation, we show in Fig. 6 the spatialbehavior of some typical Bloch functions at theI' point acted on by the kinetic-energy operator andby the SC exchange potential operator. It is seenthat the function V, (r)C'„(0, r) is largely cancellednear the core region due to the steep kinetic-energy contributions and it is only in the bondregion that the exchange potential contributessignificantly to the Hamiltonian matrix elements.The same conclusion is also valid for the correla-tion potential which has rather small and finitevalues in the core region. Thus it is the oond re-gion where the exchange and correlation potentialare expected to be important. " When the potentialis computed along the [110], [111j, and [100]directions in the crystal, it is observed the elec-trostatic, exchange, and correlationpotential exhibitmaximum anisotropies of 42%, 48%, and 32%

TABLE II. Comparison of several model calculations of the x-ray scattering factors indiamond.

SC OPW ~Pseudopoten-tial t matrix

Pseudopoten-tial OPW'

Equivalentorbi tais d

Presentstudy

111220311222400331422511333

3.231.921.640.121.521.521.401.351.32

3.211.911.580.131.51

3.321.971.660.151.48

3.311.941.690.0641.571.54

3.271.991.720.1371.491.601.421.381.38

Self-consistent OPW calculation, Ref. 48, for an exchange coefficient of &.Reference 49.' References 50 and 51.Reference 52.

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om arision wels

7 p P) For co Penvniues

occurs' T ble IG the a

.h B

at 0.7» 'ctua]. e&ge've in a

oints 1Q tresults

h-symmetry pothe

' ed at some hig - . otential fromobtain

the correlation poes almost

on removings ectra move~ a

b twtw th hgd late

entire occulmost

Vov

d'ft the ban s' l and that it

constant shil tion potentia anthe corre a isensitive to

om letely ignored icoul prth oth htion. On

odelentiona

-t eeecas expected,ff rent eigenva uepre sc

st unoccupied and isfth l bi op

lowe re d by a,bout 2.5 e r

0'4J "is-5—

-1.0

I-2.0 TlI

lI

I

0,0—3.0

S I I) DIRECTIONI I)

STA E THE II IDIS

self-consistent ex-e otential acting on a a

Bloch state T4„(0,rp mg to p= g, 2p an

longing ot sublattice 1

-20

-25

-30

h L Q wx rI' A xZw

dd orrelation banxchange an corFIG. 7. Se -c ' xc alf-consistent exc astructure for diamond.

15 GROUND-STATE ELECTRONIC PROPERTIES OF DIAMOND IN. . . 5057

crease of the direct gap by 3.5 eV. The bottomof the valence band changes by less than 0.2 eV.The effects of the removal of the exchange po-tential are manifested mainly on the states abovethe lowest valence band, again reflecting therelative insensitivity of the regions near theatomic cores to the exchange potential. We notethat although the one-particle energies of theoccupied states are stabilized by removal of theexchange potential, the total crystal energy isdestablilzed (i.e. , it is less attractive) on accountof the reduced charge buildup in the bond regionand the diminished charge localization at theatomic sites (compare I'ig. 1). This is in directcontradiction to the criteria used in the literature"for judging the variational quality of a band cal-culation according to the lowering of the bandeigenvalues.

F point X point L point

F25

F„,

0.0

20.437

26.771

X4

&mm, c

8.267

14.346

25.888

5.271

8.256

I „17.617F2 34.510 X( 29.041 L 3~ 30.824

X4 37.129 L ic 30 943

L 2~ 38.957

TABLK III. Band structure of diamond at the high-symmetry F, X, and L points, obtained at the conver-gence limit of the SC exchange and correlation model.Values are given in eV relative to the bottom of thevalence band (e&& = —30.94 eV). The two core bands lieat 271.087 eV below vacuum. b ~m ~ denotes the eigen-value at the bottom of the conduction band. v and c de-note valence and conduction, respectively.

C. Total ground state-crystal energy

Our approach for calculating the total crystalenergy is based on a direct use of Eq. (2). Wehave previously described" how the variousterms in this equation can be grouped so that thedivergencies in the individual electron- electronand electron-nuclear energies per unit cell arecancelled. We have also shown how the kineticand Coulomb terms are combined before theirintegration is attempted so as to affect a sub-stantial numerical cancellation. We are able toobtain sufficient accuracy in the core eigenvaluesby using some 10-30 Diophantine integration pointsinside the nuclear volume and about 300 integra-tion points in the 1s orbital sphere. Table IV de-scribes the results obtained with various localdensity models for diamond at the convergencelimit of the SC cycle.

Basis seteffectswere seen to be significantlyreduced in the limit of self-consistency relativeto the noniterated results: while in the non-SC

limit the ion-pair basis (set II) yields a loweringof 1.43 eV/atom relative to set I, the differencein the SC limit is reduced to 0.4 eV/atom. Boththe extended numerical set and the ion-pair setproduce results that agree to within 0.4 eV(while a minimal numerical basis set yields atotal energy that is about 1.7 eV higher). It wouldthus seem that the values obtained here with theion-pair basis (set II) are probably close to thebasis set convergence limit of our local densitymodel.

The exchange and correlation energies ~„and AE„„(defiednhere as the correspondingtotal energy differences between the results ob-tained in independent SC calculations) formabout 12 and 1%, respectively, of the total crystalenergy and somewhat smaller fractions of theatomic total energy. It is rather difficult to esti-mate the correlation energy from other sources.One can obtain an estimate of the atomic "ex-perimental" correlation energy from the work ofClementi" in which 4E„„is defined as the dif-

TABLK IV. Total (self-consistent) ground-state energies per carbon atom (values in eV).AE„and 4E'0OII are defined as the difference in total energies between column 1 and 2 and2 and 3, respectively. Lattice constant: 6.7403 a.u. The values in parenthesis indicate thevirial ratio —2T/V.

Elec tros taticmodel

Exchangemodel

Exchange andcorrelation

model &&corr

Free atom '

Crystal extendedset (set I)

Crystal ion-pairbasis set (set II)

-895.76

—887.49

—1008.23[1.0002]

—1015.10[1.0036]

—1015.52[1.0004]

—1015.72

—1027.24

-1027.64

112.47 7.49

12.12

127.61 12.14

' Values obtained from a non-spin-polarized numerical SC calculation.

5058 ALEX ZUNI'ER AN D A. J. FREEMAN

ference between the experimental total energy(with relativistic effects removed) and the bestanalytic Hartree- Fock energy. This yields some4.3 eV for carbon in its 'P ground state. Asomewhat smaller value can be deduced by com-paring the total energies obtained from a con-figuration- interaction and a single- configurationHartree-Fock (HF) study for carbon. " Our estimateof the atomic correlation energy is larger than the"experimental" values by a factor of 1.75; thisdiscrepancy probably reflects the inadequacy ofan electron-liquid correlation model for describingthe discrete atomic excitation spectrum. " Similarresults have been obtained previously by Tong andSham' and Kelly" using an interpolated Wigner"and Gell-Mann-Bruckner" correlation functional(although their results predict somewhat pooreragreement with experiment using the older correla-tion functions). Clearly, the local density func-tional is inadequate for estimating correlation en-ergies in finite systems. Only crude estimates ofthe crystal correlation energy are possible. Com-parison of HF and configuration- interaction cal-culations of the C, molecule " indicates a cor-relation energy of -5.64 eV/atom. Taking theincrease over the atomic correlation energy as anapproximation to the carbon-bond correlation en-hancement, one obtains a rough estimate of = 10eV/atom for the crystal correlation. A limitedHeitler- London type configuration- interactioncalculation on diamond indicates a stabilizationby 21+ 8 eV of the correlated state over the un-correlated state. ~'" lt would thus seem that ourestimate for the crystal correlation energy( 12 eV) constitutes a substantial improvementover the atomic value. We note that the correla-tion energy calculated as a difference between thetotal energies of the separate SC exchange and cor-relation model and the exchange model is some4%0 higher than that estimated from

where po(r) is the density calculated from an un-coryelated model (i.e. , lowest-order perturba-tive correction). This would indicate that higher-order effects brought about by the changes incharge density due to the incorporation of cor-relation effects in the potential (see Fig. 2) areas important as the direct first-order correlationenergy term. Thus, inclusion of correlation cor-rections in a solid as simple additive terms to theuncorreleted total energy'~" are bound to signi-ficantly underestimate the SC value. On the otherhand, the crystal exchange energy calculated asa total energy difference (column 5 in Table V) islower than that calculated by integrating

where V„(po(r)) is the exchange potentia, l.So far, correlation corrected HF total energy

calculations on solids have not been published.A restricted HF calculation using an s and PGaussian basis set yields a total crystal ground-state energy between~~ —1027.09 and -1030.39eV. These values are in closer agreement withour best correlated value for the crystal than isthe HF atomic total energy (- 1025.507 eV) com-pared with our atomic LDF result.

The accuracy of any calculation of the cohesiveenergy is limited by the difficulty indeterminingboth the atomic and the crystal total energies tothe same level of approximation. Since the LDFformalism seems to overestimate the correlationenergy in finite systems having no continuum oflow-lying excited states, it is difficult to assessthe validity of the atomic total energy calculation.In addition, spin polarization is expected to lowersubstantially the atomic total energy of an open-shell system like carbon relative to our spin-restricted calculation. Although spin- dependentelectron- liquid correlation functionals are avail-able, '" it is difficult to estimate the reliability ofa cohesive energy calculation based on the use ofdifferent functionals for the atom and the solid.Our exchange model predicts in the non-spin-polarized atomic limit a cohesive energy of 7.29eV while the exchange and correlation model pre-dicts in the same limit a cohesive energy of 11.92eV/atom. The experimental static binding energyof diamond (calculated by adding a Debye-modelzero-point energy to the measured value~6 usingen=2230'K) is 7.62 eV/atom. Previous studies"have indicated that spin-polarization correctionsto the atom tend to cancel the correlation en-hancement of the crystal cohesive energy so thatthe result obtained from a non-spin-polarized ex-change model comes very close to the accurateprediction. We have been informed by the refereeof this paper: "Ihave a number for the ground-stateenergy of the free carbon atom, calculated with thespin-polarized exchange and correlation potentialof Ref. 3 for the configuration

(1so.)' (1sp)' (2sn)' (2sp)' (2 pc.)'. —1021.9 eV,which would give a cohesive energy of diamond equalto 5.7 eV, much closer to the experimental result. Itshould be stressed, however, that the interpolationformulas of Befs. 3 and 39 differ slightly at r, & 1,i.e. , in the core region. "

The restricted HF value for the cohesive energy(with correlation and spin polarization neglected)is substantially lower (- 5.2 eV/atom. )"'" Thisis in line with the usual tendency of restricted

GROUND-STATE EI, ECTRONIC PROPERTIES OF DIAMOND IN. . . 5059

HF calculations to underestimate the binding en-ergy relative to the correlated limit (e.g. , 0.78eV/atom is obtained for the binding of the C,molecule in the RHF level" ~ while 5.4-6.7 eV/atom are obtained in a CI calculation. " The ex-perimental result is 6.36 eV/atom'8 comparedwith a, local-exchange result of 5.58 eV/atom").We note that in our model correlation acts tostabilize the crystal over the atom by about 4.6eV. An electrostatic-type calculation for diamond(column 2 in Table IV) seems to produce no bindingat all.

Several calculations have been published pre-viously on the cohesive energy of diamond. Theseinclude the t- matrix pseudopotential calculation ofBennemann, "thy OPW pseudopotential calculationof Goroff and Kleinman" and the Heitler-Londoncalculation of Schmidt and Dermit. We notethat whereas Gorff and Kleinman have used asmall-basis OPW calculation, a later extendedbasis OPW calculation' indicated very poor con-vergence of the eigenvalues containing P-likestates before several thousands of OPW's areincluded. The correlation energy used both byBennemann and by Goroff and Kleinman was treatedas an additive term to the total energy withoutattempting to incorporate it self-consistently intothe effective potential. The core-overlap contribu-tion was also neglected in these studies. The cal-culation of Schmidt" based on an elegant andrigorous formulation of the Heitler- London modelin an orthogonalized representation, involves ex-tensive approximations in the practical evaluationof the LCAOintegrals, resulting in large un-certainties in the cohesive energy.

In order to examine the prediction of the LDFformalism for the equilibrium lattice constant,we have repeated our total energy calculationusing our extended numerical basis (set II) atseven unit-cell parameters. Such a calculationis free from the limitations of the cohesive energyproblem in that only crystal results are con-sidered. Using the SC results corresponding tothe calculation with the closest lattice constant,only a few iterations of "state 2" in SC were re-quired for each lattice constant. The resultingtotal energy curve was interpolated to yield theequilibrium lattice constant values of a„= 3.662 Afor the exchange model and a„=3.581 A for theexchange and correlation model. The fitting errorin each of these cases is estimated to be 0.003 A.The experimental value is 3.567 A."

These results are consistent with our previousconclusion that correlation acts to stabilize thecrystal, increasing its binding energy and re-ducing the lattice constant. Similar good agree-ment with the experimental lattice constant

(a„„/a,„„-0.SS6 1.003) was obtained in the OPWmodel of Goroff and Kleinman, s' however, theapproximations made in their work make it dif-ficult to assess the accuracy of this result. Therestricted HF value for a„ is 3.545 A." It isexpected that correlation corrections would actto further reduce the HF value which is alreadysomewhat smaller than experiment. Previouslocal-density SC non-muffin-tin exchange modelcalculations on simple molecules like C„"N„CO, and H, O,"and I iF, NO, O„and N„""seem to predict equilibrium bond lengths thatare larger than experiment. Tong's" self-consistent cellular calculation on solid Na leads tothe same conclusion. Calculations within themuffin-tin approximation"' do not seem to revealany systematic behavior of the deviations from ex-periment of the computed lattice constants. Inview of the large. non-muffin- tin correctionsfound" for the cohesive energy of some covalentlybonded molecules, "it seems rather unlikely thatsuch calculations would lead to reliable resultsfor solids like diamond.

D. Behavior of one-electron properties with change in latticeconstant

In order to further elucidate the bonding mech-anism in diamond, we have studied the changesin crystal charge density with lattice constant.Figure 8 reveals the charge-density changes insome high-symmetry states in the occupied portionof the BZ as a function of lattice constant. Upondecreasing the lattice constant from a value largerthan the equilibrium value, charge density isshifted towards the bond regio~ and simultaneouslythe density at the core region is increased (atthe expense of depleting charge from an intermed-iate region).

This result confirms our earlier suggestion (cf.Figs. 1-3) that the bonding mechanism in diamondinvolves a stabilization of the crystal due to botha build up of charge density in the bond region(mainly due to exchange and correlation effects)and close to the core regions (due to penetrationof neighboring wave functions). Both of theseeffects tend to decrease the positive kinetic en-ergy, produced by the orthogonalization of the(overlapping) noninteracting atomic Bloch states,by enhancing the interaction with the attractiveelectron-nuclear, exchange, and correlation po-tentials. A minimal-basis-set description ofdiamond tends to remove charge from the coreregions into the bond region thereby producingrather realistic values for the low-angle scat-tering factors (indicating a correct charge build-up in the bond region), but still yielding small val-

5060

p(0}=2 43

FREEMANAND A. JAg, EX Z UND ER

IO—I

(I) o=6 &2 '"(2) a=6 7

LsLI

XI

= -r)s

03CI

LLIE3Kax

02

p, I

—IoI

-20

I

L3

X4

WI

LIXI

L2''0

04—

(bj

p2 p.6p40

PISTATANG E A Lp I I I (A)

I

0 = 6 52 o ti.

(2)

(3) 0=6.92 0

08-25

I

365 37035534 'ONSTANT IAI

350 'o)TTIOE CON

es in diamondelect»n energiFlQ. 9' +t The fUll d

were fittedlattice constan

d smooth lineswith a

1 ted points anactually caln'n between-

0.3O

LLI

tZ

x0.2

P(O) = I.I9

O. I )L,„

.— )x,„

I

0.8i

0.6~ M

0,40

NCE ALONG [III] IA)DISTANCE

r e- ensiity con-t'o i th aalence cha g-h-symmetry states, w efill] directio

d'constant, a g

he origin (the ratom is pe attice ons a er

, 3) while the rignoted as 1,denotes the center oand X points.

00

r . Augmenting the set byuextende a to re-

h b'd' g gy.

ag oastoret US

t thdtt, 'btotwo charge re ls

ld th o t banisms ath t seems to yjel

f the one- e electronthe sol

th va jat jonthe

ure 9 shows et. As expecte,

F gh lattice constan ~

«ding3

the bon lnen

s correspondingthe «non-

eigenvalues cob l'zed while t e'ed states are

- "virtual statessta l l

arecup

' d «ant jbondjng' . the l.attjce.

e-constant depenll ht increase jn

tice-chere, result'ng

n contracting "in only' a g

lattjce.ce bandwidth upohat the conductjot js interesting

h ge their ord I, interc ~er being

states at Li ant the crossovechant o'e of ] tt jce const'

e ssure eqe erimentaand calcula-

zero-prclose to the e~d various ban

r ofalue. Indee ~ . . '

in the order olibrium

arked sensjtjv'5 't' nal

tionsetalls o

reveal a m- f the computa othese

hus, muffin tinwhile non-

param ' . s roduce z,, 's ro-

Thus '

~ & q, wO") caicnlation P33 calculatione P

LCAO~9 and LCAO~niacin-tin (0&

The photoelec»the reverse order.l;ed pressure b

duce- ases with app '

9 whilehpld (- &r2,

&e ehpwn ininc rea

Flg.about i.2b& j.o eV.

the rang' creases 7

t be-t e

m arethe ah bandwidth in

l ttice constanIn or pthe res-bands, wehavior of the energy ban, e

suree coefficient

GROUND-STATE EI.ECTRONIG PROPERTIES OF DIAMOND IN. . . 5061

de~(k) deq(k)dP dlnV

' (13)3,5

where P is the pressure, K is the volume com-pressibility (0.18 x 10 "dyn/cm' in diamond ')and V is the unit-cell volume. Inspection of Fig.9 indicates that the one-electron energy levelscan be divided into distinct groups according totheir pressure coefficients. In the valence band,the states near the bottom of the band havingpredominantly 2s character (e.g. , 1,„X,„L,'„,etc.) have a rather large pressure coefficient[(3-5)x 10 ' eV/bar] while those near the top ofthe valence band are characterized by lower pres-sure coefficients (1.2 x 10 8 eV/bar for F»„and 1.6x 10 ' eV/bar for L,'„). In the conductionbands, most states possess an antibonding charac-ter and have a large pressure coefficient [(4-7)x 10 ' eV/bar]. The nonbonding conduction stateshowever have a very low coefficient (e.g. , 0.32x10 8 eV/bar for X„, 0.78x 10 6 eV/bar for Ls,).Transitions from the upper valence band to thelow-lying nonbonding conduction state (e.g. , I'»,-X„)would thus have a vanishingly small pressurecoefficient while zone- center transitions(F»„-I'„) are predicted to have much higher(factor of 5-8) pressure coefficients. Experi-mental data" " indeed seem to suggest an anomal-ously low-pressure coefficient for the I'»„-X„transition. A similar study of the lattice-constantdependence of the band structure was performedat the restricted HF level by Surratt et a/. " How-ever, owing to an error in this study, "we areunable to compare our results directly with theirdata.

The va.riation in the (kkl) scattering reflectionswith lattice constant is depicted in Fig. 10. It isseen that when we change from an atomic super-position model (horizontal lines on the right) toa full crystalline description, the low-angle scat-tering factors increase substantially (in absolutevalue) and ultimately start decreasing when thesolid is compressed. The high (kkl) reflectionsare rather insensitive to lattice constant varia-tion, as expected. Interestingly, the (331) and(400) scattering factors change their relativeorder in going from an atomic superposition mo-del (f», &f„,) to the crystalline model (f», &f«o).Solid-state effects introduced by wave-functionoverlap (as opposed to spherical charge-densityoverlap) are thus seen to be non-negligible.

E. Directional compton profile

Although the LDF formalism makes no definitepredictions on observables that depend on theindividual eigenfunctions of the one-particle equa-tion solved here (since they a,re not actual one-

3.0—

EO 2.5O

~l

(A

OI-o 2.0—LLI

LLJ

CC

220

3ll

40033l422333

I 0—440

0—-01—-0.2

222

3.50 3.55 3.60 3.65 3.70LATTICE CONSTANT ( A)

FIG. 10. Dependence of the x-ray scattering factorsin diamond on the lattice constant. The straight lineson the right-hand side denote the atomic scattering fac-tors as calculated by an atomic superposition exchangeand correlation model.

particle states), its resemblance to an "ef-fective" Hartree-Fock type equation makes itinteresting to examine the quality of these wavefunctions as "pseudo"-one-particle states. Sucha test is naturally provided by comparing the cal-culated directional brompton profile with experi-ment. We will thus ass~me" that the momentumdensity p(p) can be represented by the sum ofsquares of the momentum eigenfunctions"

(14)

where the Fourier- transformed Bloch function4 (p) is obtained by direct three-dimensionalDiophantine integration of

-its Ie 4 (k r)dr.1Ils(P) (2 )3/2 (15)

o(p)= gs&6 (p)4&(p)f

(16)

Since each momentum value p relates a point k;in the BZ with a given reciprocal lattice vectorG,=p, —k„we compute the crystal wave functionsg~(k, r) at a given grid g,j in the BZ and use thesefunctions to obtain the momentum eigenfunctions$J(jI) at a set of momentum points p, = k, + G, . Inall, 52 k, inequivalent points are used to obtain themomentum density

5062 ALEX ZUNGER AND A. J. FREEMAN 15

for P ~8 a.u. This function is then interpolatedby a fifth-order local interpolation to obtain p(p)for arbitrary momentum a;1d is then least-squaresfitted to a Kubic harmonic expansion '

o.i &a J [IOO] —[III]

SCHF

PY

v(P(= g, (((&, (((( (17) 0.05for I ~ 12. The Compton profile S(k, (d) in the im-pulse approximation is given by

q2.0

0.05

= —S(q, k),1

S(q k) g S(m(q) y'(~(k»&t el

we obtain the expansion coefficients S, (q) by adirect radial integration:

(19)

where q= k ~ p/m. Expressing the reduced profileS (q, k) in a harmonic expansion a,s 0 IO—

005

JK J j !00]- [ I IO]—--SCHF

PRESENT STUDY

PERIMENT

S, (q) =2n Pa, (P)P( —dP,I ql

(20)

where P, (q/p) is the Legendre polynomial. TheCompton profiles in an arbitrary direction areobtained using Eqs. (19) and (20).

We give in Fig. 11 our calcuI. ated Compton pro-file differences for the [100]-[110)and [100]-[111]directions together with the experimental resultsand those yielded by the restricted HF model ofWepfer et al." Table V gives our calculated re-sults for the [100] direction together with the ex-perimental results of Reed and Eisenberger"and Weiss and Phillips" and for comparison, thecalculated results of Seth" in the non-SC local-exchange model (with exchange coefficient of0.70) and the restricted HF results of Wepferet al. 24

The main conclusions to be drawn from this com-parison are: (i) While the RHF results for the pro-file lack sufficient high-momentum components(and hence produce too high a profile at low-mo-mentum transfer), our model predicts a slightexcess of high-momentum components. The non-self- consistent exchange model" tends to producea profile that is even higher than the RHF pro-file at low q. This result is in line with previouscalculations on atomic Compton profiles using theexchange model"" and with our results for a non-SC superposition model. " We thus conclude thatboth the inclusion of correlation and the iterationtowards self-consistency tend to redistribute thecharge so as to add some high-momentum com-ponents to it. It was previously suggested thatthe same effect can be achieved (in atomic systems)

-0.05

-O. IO

0.0I

I.Oq

2,0

FIG. 11. Directional Compton profile in diamond asobtained from the self-consistent exchange and cor-relation model. (a) AJ[100]-[111];(b) AJ[100]-[110].

by either artificially increasing the exchangeparameter or by adding a loca.l correlation func-tional to the exchange potential. " Previous super-position-model calculations for Si similarly indi-cate an overestimation. of the low-momentum pro-file in this non-SC limit. " Iteration towards self-consistency also tends to increa, se the low-anglescattering factors" (and hence to produce chargelocalization in the bond region) in accordance withits effect on the Compton profile. The occurrenceof rather strong high momentum components in ourresults partially reflects some enhanced locali-zation of the wave functions near the core regions(owing to nearest-neighbor penetration) relativeto the exchange model and superposition limit'4(see also discussion in Sec. HIA). (ii) While theRHF results agree with both experiment and with

15 GROUND-STATE ELECTRONIC PROPERTIES OF DIAMOND IN. . . 5063

TABLE V. Compton profile in diamond along the [100]direction.

SCNon-SC exchange and

Momentum exchange correlation{a.u.) model ' model

SCHF

Expt. b model c

0.00.40.81.21.62.04.0

2.2302.0751.5750.8900.4550.3000.080

2.051.931.520.960.480.330.11

2.09,2.081.91,1.941.46, 1.550.86,0.940.47, 0.45

0.31o ~ ~ 0 10

2.1802.0461.5480.8840.4600.293

~ Non-SC results of Seth {Ref. 80) using an exchange co-efficient of 0.70 and a double-zeta Slater basis set.

The first value refers to the experimental results ofReed and Eisenberger {Ref.20) while the second numbergives the data of Weiss and Phillips {Ref. 21).' Reference 24.

IV. SUMMARY

We have presented results obtained by applyinga self-consistent numerical basis set LCAOtreatment to the description of some ground-state

our results for the anisotropy along [100]-[110][Fig. 11(b)], the RHF directional profile for[100]-[111]is considerably more anisotropic thanexperiment and our data. We found that upon itera-ting our results to self-consistency (varying ourbasis set nonlinearly so as to allow more varia-tional participation of the virtual 3s and 3P atomicorbitals into the valence band) the calculated aniso-tropy decreased. It would thus seem that byallowing the formerly virtual states (having long-range tails) to participate in the bonding mani-fold, the localization of the valence-band statesdecreases and a broader nJ'(q) is produced (alongwith an increase in the binding energy —as dis-cussed in Sec. IIIC). It is possible that incorpora-tion of such "plane wave" character would alsodiminish the anisotropy in the RHF profile.

The over-all agreement of our calculated Comp-ton profile with experiment seems reasonable.It would thus seem that at least for diamond, theapparent anomaly [i.e. , J(q) too high at low q] ofthe calculated Compton profile in the exchangemodel for atoms (with an exchange coefficient of&) disappears in a SC exchange and correlationmodel. In such a covalently bonded system high-momentum components are added due to correla-tion, localization of charge in the core regions,and enhancement of the charge buildup in the bondcenter.

electronic properties of diamond in the LDFformalism. Our study of the effects ofexchange and correlation on the ground-state charge density indicated that these act toincrease the charge localization both in the"bond" region and to some extent close to thecore region —correlation effects being muchsmaller than the exchange effects, although stillconsiderably k dependent and anisotropic. Thex-ray scattering factors obtained from the SCcharge density agree quite well with experimentand seem to account for the observed "forbidden"(222) reflection. Our calculated Compton profilesimilarly indicates good agreement with experi-ment; the deficiency in the LDF formal. ismin describing the atomic profile is remedied in thesolid due to correlation and the above-mentionedcharge redistribution effects.

Our total energy studies show a substantial in-crease in stability of the solid relative to the freeatoms due to exchangeandcorrelation although thelatter is probably overestimated in the atom. Theuse of an ionpair charge-transfer basis set isshown to be rather efficient in producing a con-verged LCAO expansion, in line with the pre-viously established variational quality of theorthogonalized Heitler-London scheme for mol-ecules. '~" Our correlation model estimate forthe cohesive energy in the non- spin-polarizedatomic limit seems to overestimate the experi-mental value considerably. Since we are not ableto assess the magnitude of the corrections re-quired to account for the overestimation of thecorrelation in the atomic limit as well as thecorrect spin-polarization effects, this result istaken as a rough estimate only. Our exchangeresults for the cohesive energy in the non-spin-polarized atomic limit agree favorably with theexperimental results and predict an equilibriumlattice constant that is about 3 jo too high. Thefull exchange and correlation model reduces thecalculated lattice constant to within 0.5% ofexperiment. The HF model, on the other hand,seems to produce a lattice constant that issmaller than the experimental value even in theuncorrelated limit. Our study of the lattice-constantdependence of the band eigenvalues reveals ananomalously small pressure coefficient for theindirect I'»„-&,~, transition together with alarge pressure coefficient of the low-lying 2s-likevalence states (but in opposite senses) and theconduction bands. All in all, it seems that the dif-ferences between the predictions of a HF nonlocalexchange model and a proper self-consistent LDFmodel for the ground state properties of diamondmight be rather small compared with the differen-ces induced by an improperly balanced basis set

5Q64 ALEX ZUNGER AND J. FREEMAN 15

expansion, muffin- tin approximations and lack ofself-consistency. Further studies on other systemslike BN, LiF, and TiS, are under way and wiQhopefully further elucidate the role of local versusnonlocal exchange in determining ground-state inproperties of solids.

Note added in proof. We have been informed byDr. O. Gunnarsson that a total energy calculationof the carbon atom in the spin-polarized local-spindensity formalism (using the functional of Refs. 3and 66) yields the value of -1019.83 eV. Usingthis value and our best result for the total energy

of the solid (-1027.64 eV/atom, Table 1V), givesa binding energy of 7.81 eV/atom, which is inremarkable agreement with the observed value of7.62 eV/atom .We are grateful to Dr. Gunnarssonfor this communication.

ACKNOWLEDGMENT

The authors would like to acknowledge R.Euwema and E. Lafon for helpful discussions.We would like to thank A. Seth for making availablehis Compton routine and for helpful discussions.

*Supported by the NSF (DMR Grant Nos. 72-63101 and72-03019) and the Air Force Office of Scientific Re-search (Grant No. 76-2948).

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47It is thus possible that the poor agreement obtained in

15 GROUND-STATE ELECTRONIC PROPERTIES OF DIAMOND IN. . . 5065

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