electronic charge distribution in crystalline diamond

PHYSICAL REVIEW B VOLUME 47, NUMBER 15 15 APRIL 1993-I

Electronic charge distribution in crystalline diamond, silicon, and germanium

Z. W. Lu and Alex ZungerNational Rene&sable Energy Laboratory, Golden, Colorado 80/01

Moshe DeutschDepartment of Physics, Bar Ilan University, Ramat Gan -M900, Israel

(Received 3 November 1992)

Recent refinement studies of a consolidated set of Si structure factors have produced informationon the Si charge density with an unprecedented level of accuracy, unmatched by any other crystal-lographic study to date. In this work we examine the extent to which an accurate implementationof the local-density formalism can describe the charge distribution in silicon, as well as that of theexperimentally less-refined data on diamond and germanium. Results of a refinement study of recentgermanium and diamond measurements are presented and compared. with the ab initio calculations.Our ab initio calculated structure factors for Si show a twofold to fivefold improvement in the Rfactor over previous local-density calculations. We describe in detail, total, valence, and deformationcharge-density maps for C, Si, and Ge. We analyze the effects of high-momentum components (cur-rently outside the range of the high-precision measurements) as well as dynamic structure factorson the ensuing charge-density maps.

I. INTRODUCTION

Recent advances in measurements and analyses havenow produced extremely accurate data on the charge dis-tribution in crystalline silicon: 7 Cummings and Hartihave recently consolidated Ave data sets of high-precisionSi structure factors obtained in three independent exper-iments by Aldred and Hart, z Teworte and Bonse, s andSaka and Kato. The data were corrected for anoma-lous dispersion (using measured, wavelength-dependentdispersion factors) and nuclear scattering, and care-ful consistency checks and error estimates were con-ducted. More recently, Deutsch s fitted this consoli-dated and corrected data set of silicon structure factors toa parametrized model in which the harmonic and anhar-monic thermal effects were deconvoluted from the staticcharge density. As a result of these recent developments,(i) the structure factors of Si are now known to a millielectron level of accuracy, better by one order of magni-tude or more than any other crystal studied to dates and(ii) the static charge-density map of Si (Refs. 5 and 6) isnow known with an unprecedented level of detail. In viewof these developments we have decided to (i) examinethe extent to which the local-density formalism, im-plemented with the highest computational precision pos-sible to date, is able to capture the details of the highlyaccurate measured Si structure factors and charge den-sity, and (ii) perform a similar data refinement as donefor Si by Deutsch for the (far less complete and accurate)data sets on diamond and germanium, so that chemicaltrends along the C~Si—+Ge sequence can be assessed.

There are many previous calculations of the Si chargedensity. These include the empirical pseudopoten-tial calculations by Walter and Cohen, Chelikowskyand Cohen, Bertoni et al. , and Balderschi et ajI,'. ;

the local semiempirical pseudopotential calculation ofHamann; and the Erst-principles nonlocal pseudopo-tential calculations by Zunger and Cohen, is Zunger, s

Ihm and Cohen, Yin and Cohen, is and Nielsen andMartin. is All-electron calculations of the Si charge den-sity include the orthogonalized-plane-waves (OPW) cal-culation by Stukel and Euwemazo and by Raccah etal. ,

i the linear combination of Gauss orbital (LCGO)by Wang and Klein, zz and Heaton and Lafon, 2s andthe linear muffin-tin orbital (LMTO) calculations byWeyrich, 2 Methfessel, Rodriguez, and Andersen, zs andPolatoglou and Methfessel. In addition to the calcula-tions that use the local-density approximation, thereexists the Hartree-Fock calculation by Dovesi, Causa, andAngonoa 7 and by Pisani, Dovesi, and Orlando. Fur-thermore, Balbas et aI,. have calculated the structurefactors of Si using model bond-charge density examin-ing the role of different approximations to the exchange-correlation potential. The reasons that we undertook arecalculation of the Si charge density despite these manyprevious works are fourfold.

First, all of the previous calculations were publishedbefore the completion of the analysis of the high-precisionSi data by Cummings and Harti and by Deutsch, s s sonone includes a comparison with these data. Further-more, none of the previous authors have published bothstructure factors, as well as valence and deformationcharge-density maps.

Second, many of the previous calculations have in-corporated computational approximations whose effectson the charge density remain, in many cases, untested.These include the pseudopotential approximation,use of limited basis sets, 2z s'z7 or perturbation theory. iSpackmans has recently compared the predictions ofmany of these calculations with a few measured (pre-

47 9385 1993 The American Physical Society

9386 Z. W. LU, ALEX ZUNGER, AND MOSHE DEUTSCH 47

Cummings and Hart) structure factors, finding signif-icant discrepancies for all except the OPW results ofStukel and Euwema. We wish to perform an analogouscomparison with the current data, using calculated re-sults in which the convergence parameters of the theoryare controllably pushed to the limit where the resultsreHect as much as possible the predictions of the under-lying (local-density) Hamiltonian, unobscured by compu-tational uncertainties.

Third, we wish to examine a number of assumptionsthat have traditionally been made in the transformationof structure factors to charge-density plots. These in-clude (i) the assumption that high-momentum Fouriercomponents (which are difficult to measure with high pre-cision) refiect merely core contributions and are thereforeinconsequential for describing the valence charge den-sity or the solid versus atom density differences. Thereare, however, indications ' that the nodal structurein the latter quantities may also require high-momentumcomponents, currently outside the reach of high-precisionmeasurements. Also (ii), it is often assumed (e.g. , Zuo,Spence, and O'Keefess) that even if the high-Fouriercomponents in the Static charge-density difference arenon-negligible, the Debye-Wal. ler factor will attenuatethem, so that the dynamic density difference map willnot require high-momentum components. We wish to ex-amine this hypothesis by comparing static and dynamiccharge-density difference maps for a range of increasingmomentum cutoff values.

Finally, while the measured x-ray structure factorsof diamond and germanium 3 are significantlyless accurate ' s than those available now for Si, globaltrends in the measurement and model-refined structurefactors can nevertheless be established. We note thatprevious analysis of the Si data s revealed a vanishinganharmonic temperature term (P = 0) and a differentDebye-Wailer factor for core and valence electrons. Itwould be interesting to examine these points also for di-amond and germanium. We will hence subject the ex-perimental data to the same analysis done for Si andto examine the trends in the ensuing charge densities inthe C+Si—+Ge series. Comparison with previous calcu-lations on C and Ge will then be discussed.

The plan for the balance of this paper is as follows.Section II defines the measured quantities to be discussedbelow and analyzes these quantities for C and Ge thusproducing a separation of static versus dynamic densi-ties. Section III describes the calculated counterparts.Section IV defines the core, valence, and total charge den-sities and their respective deformation charge densities.Section V examines the convergence of the current cal-culations. Section VI presents the comparison of theorywith experiment. Finally, Sec. VII provides a summaryof the findings of the present study.

respectively, and label Fourier transforms for a momen-tum G = —(h, A:, l) as F(G) or p(C), while real-spacedensities will be denoted as F(r) and p(r).

A. Dynamic structure factors and densities

Dawson has shown that if the "rigid-atom ap-proximation" is invoked, the dynamic structure factorsF,„pt(G) can be represented as a convolution of the nthsite static structure factor p (C) and the dynamic tem-perature "smearing function" T (C) as

M

F- t(G) = ).p (G)e' '-T-(G),

where p (C) is the Gth Fourier component of the staticcharge density contributed by atomic site n (whose posi-tion vector in the unit cell is ~ ), and T (G) is the a' ssite dynamic (temperature) smearing function containingharmonic and anharmonic terms. T (G) is often approx-irnated by the Debye-Wailer factor exp( —G2B /167r ).Note that the approximation of Eq. (1) implies a linearpartitioning of the continuous three-dimensional chargedensity into subregions associated with identifiable scat-tering centers o, . This partitioning is referred to asthe rigid-atom approximation. Within this approxima-tion, it is possible to fit the data either with a singleDebye-Wailer factor or by assigning different B's to dif-ferent atomic shells ("the independent shell vibrationalmodel" ).

The dynamic real-space charge density can be syn-thesized from the Fourier components of Eq. (1) bysumming them (after subtracting anomalous scatteringcorrections) to a maximum momentum G ~„accessiblefrom diffraction experiments. This gives

&max

Fexpt(r~ Gmax) = ) Fexpt(G)eG

where the result naturally depends on the highest mo-mentum (C „)included in this sum. Fourier truncationeffects underlying Eq. (2) could affect the shape of theensuing charge-density map.

B. Static structure factors and Dawson's madel

For purposes of comparison with the results ofquantum-mechanical calculations it is often desirable todeconvolute Eq. (1) thus finding the static (purely elec-tronic) structure factor

p- (G) = ).p-(G) e' '-

II. MEASURED QUANTITIES AND THEIR.MODELING

In this section we define all of the quantities shownlater in our tables and figures. We will consistently de-note dynamic and static charge densities as E and p,

pexpt (G) = Fexpt (G )e (4a)

The resulting p,„pt(G) are then summed giving the static

For monatomic crystals in the harmonic rigid-atom ap-proximation, this deconvolution could be done by multi-plying F,„pt(G) by the inverse Debye-Wailer factor

47 ELECTRONIC CHARGE DISTRIBUTION IN CRYSTALLINE. . . 9387

Fourier-synthesized total density map

G'max

pexpt(&~ Gmax) = ) pexpt(G)eG,

(4b)

approach used originally by Dawson and subsequentlyby other practitioners of this model, ' ' one super-

poses the real spa-ce atomic model density p d,&(r) overall lattice sites R~,

The precision of this map is naturally limited bythe largest momentum component accessible to high-precision measurements as well as the validity of theaforementioned rigid-atom approximation. An alterna-tive method whose precision depends to a lesser extent onthe explicit convergence of Eq. (4b) (still using the rigid-atom approximation) was outlined by Dawson4s and re-fined by Stewart, Coppens et al. , and Deutsch.The approach is based on the fact that any ground-statecrystalline properties such as p (r) of Eq. (1) can be rig-orously expanded in an infinite set of orthonormal Kubicharmonics KP (r) of angular momentum l belonging tothe totally symmetric (ai) representation of the ath sitegroup

p ~de~(r) = ) ) @(r—R&)K~(r —"R&)l=0,3,4

This density contains a contribution correspondingto arbitrary large momentum components despite thefact that only a limited set of structure factors(F, pt(G); G & G „)are used to fit the model param-eters. This approach is hence e2;trapolative: the nonzerovalues of p ~~,~(G ) C „) are implicitly determinedby the choice of the form and number of model functionsR~(r). The second alternative is to construct a trun-cated Fourier series from the model density, analogous toEq. (4b)

p~,l~„(r) = ) R((r)KP'(r)I,=O

(5)

&max

pmodel(+~ Gmax) = ) ) p~~de](G)e«™o,=1

Here r and v" are the modulus and direction, respectively,of r and R~ (r) are the ctth site radial functions defined bythe convolution of the exact p (r) with KP (r). For theTd, site symmetry of the diamond lattice, the symmetry-allowed l values are l = 0, 3, 4, 6, 7, 8, ... . The l = 0term is the spherical contribution whereas l = odd andl = even are antisymmetric and centrosymmetric con-tributions, respectively. Conventional x-rsy refinementtechniquesso retain but the spherical term in Eq. (5) (us-ing the superposition of spherical atom densities). Daw-son's method consists of truncating Eq. (5) to includeonly the leading terms l = 0, 3, and 4 and suggestingconvenient analytic guesses for the atom-localized radialfunctions Ri(r). Deutschs and Spackman44 used

R~ 0(r) = 4~) r„&n„i(K„~r)nl

R(,,4(r) = Air"'e i", (6b)

where K„~, A~, A~, and ( are adjustable parameters andn„~ are the fixed ground-state atomic (usually Hartree-Fock) charge densities i s2 for orbitals nl The para. m-eters z„~ are intended to model monopole expansion orcontraction of the spherical atomic charge densities dueto bonding. One further parametrizes T (G) in terms ofthe Debye-Wailer factor B„~ for orbital nl and a small an-harmonic potential of magnitude P. Inserting into Eq. (1)the Fourier transform p ~,i(G) of the model density ofEqs. (5) and (6) and the model dynamic temperaturefactors T (G) then gives an analytic Fourier-space rep-resentation of F,d,~(G) in terms of the parameter set(r~t, A~, A&, (, B„i,p). These parameters are then deter-mined by a least-squares fit of the fF ~d,~(G)j to the setiFe„pt(C)). Using Eqs. (5) and (6) this gives the nth sitereal-space model density p &,&(r) and its Fourier trans-

form p ld &(G). One can then proceed and obtain thetotal crystalline model density in two ways. In the first

(7b)

In this approach all unmeasured structure factors (G )G „) are taken to be zero.

Finally, in addition to the adjustable parameters inDawson's model, one has to choose the "fixed" quantities,e.g. , the orbital densities n„~(r). These could be atomicHartree-Fock orbitals or local-density orbitals, relativis-tic or nonrelativistic orbitals, as well as "canonical" ver-sus rotated orbitals. We wil1 examine below the implica-tions and consequences of these choices.

C. Fitting Damson's model to experiment

The consolidated data set of Cummings and Hart forSi was fit to Dawson's model by Deutsch. His best fit("model p" in Ref. 6) produces a remarkably low R factorof 0.036%%uo and a goodness of fit (GoF) of 1.20. This rep-resents the most accurately determined crystalline struc-ture to date. Table I gives the model parameters obtainedin this fit (the error in the exponent ( of Ref. 6 is cor-rected in Table I and the remainder of this paper. Thecorrect value is g = 2.285 a.u. i). The results show anexpansion of 6%%uo of the 3sp valence shell and of 0.5%%uo

of the core 2' shell. The best fit is obtained when thecrystal-bound silicon atom is permitted to have a shell-dependent Debye-Wailer parameters B ~

& 0.11 A. andB, „=0.4585 A2. No evidence is found for an anhar-monic term in the e8'ective potential of the atom, i.e. ,

P = 0. Further details are given in Refs. 5 and 6. Wenext describe our analogous fits for diamond and germa-nium.

Fits for diamond

Two data sets are available here: the nine single-crystalstructure factors of Takama, Tsuchiya, Kobayashi, andSato [(TTKS), Ref. 39] measured by the Pendellosung


TABLE I. Parameters obtained by fitting Dawson's model [Eqs. (5) and (6)] to the observed structure factors of C, Si,and Ge. For C and Si we use Clementi s (Ref. 51) nonrelativistic orbital densities n„I, , while for Ge we use both Clementi snonrelativistic (NR) orbital densities as well as the relativistic LDA results. In all cases we use A~ = 4 for all l values. Anasterisk indicates that quantity was held fixed during the fit. The value ( = 2.435 for Si in Ref. 6 was in error; the correct valueis given here. We also give R value and goodness of fit (GoF) for the fit.

DiamondTTKS dataGW databSilicon'GermaniumNR, ClementiRela, tivistic'

Core

1'1'

0.9949

Valence

10.990.9382

1*0.9553

(a.u. ')

3.5413.2592.285

1.8771.913

A3(e)

0.3340.3430.4484

0.5910.583

A4

(e)

0+

—0.275—0.1270

—0.571—0.510

Bcore

0.13790.23030.4585

0.54740.5654

B

0.13790.23030.0

0.54740.5654

(eV/A )

p+

p+

p+

1.8'0.9*

0.7060.7930.036

0.1880.189

GoF

4.253.031.20

1.031.07

9 G values from Ref. 39, plus F222 taken from Ref. 37.24 C values from Ref. 36. We use Eq. (8) where the scale factor is S = 1.021 and Ebs ——0.039.

'18 G values from the CH data of Ref. 6; note that for the core states, K =q ——1.0 and e„—q——0.9949.

12 C values from MK (Ref. 40) and DHC (Ref. 43). We use Eq. (8) where the scale factors are Si = 1.015 for MK andSq ——1.003 for DHC and Fb~ = 0.'l2 C values from MK (Ref. 40) and DHC (Ref. 43). We use Eq. (8) where the scale factors are Si = 1.010 for MK andS2 ——0.999 for DHC, and Fb~ = 0.

method, and the older (1959) powder data of Gottlicherand Wolfel [(GW), Ref. 36] obtained for 24 reflections.Dispersion corrections and nuclear scattering are negli-gible for both data sets. Both data sets have problems:possible crystal imperfections observable in topographyof the crystal used (TTKS), and a powder technique re-quiring many corrections (GW). The structure factorsmeasured by TTKS are consistently larger than thoseof GW except for the (111) reflection. The measuredF(311) structure factor of TTKS of 1.648e/atom com-pares well with a previous single-crystal measurement ofLang and Mai giving 1.630e/atom, while GW found aconsiderably lower F(311)= 1.592e/atom.

Spackman44 fitted the TTKS data [adding to these the(222) reflection from Ref. 37] to Dawson's model, findingR = 0.85% and the goodness of fit equal to 4.26. We haverefitted these data for two reasons. First, Spackman usedthe rotated atomic-orbital densities n„i(r) [Eq. (6a)] ofStewart rather than the canonical Clementi results.While these sets are related by a unitary transforma-tion (hence, they produce the same total density), thevalence-only results differ significantly: Stewart's orbitalsproduce a local minimum on the atomic sites, whileClementi's orbitals produce a maximum on the atomicsites (see below) outside the atomic regions; the tworepresentations are similar. Since the Si data were fit-ted with Clementi orbitals, for reasons of consistency wewill use these also for diamond and germanium. Sec-ond, Spackman used in Eq. (6b) the exponents As = 3and A4 ——4 for diamond and A3 ——6 for germanium.Since A3 ——A4 ——4 gave the best results for Si, we willuse these fixed values also for diamond and germanium.While other choices for A~ were also tried, these seem toyield the best overall fits.

Our best-fit results for the ten points TTKS set fordiamond are given in Table I. The quality of the fit (R

factor) is similar to that obtained by Spackman. Table Ialso gives the results of the fit to the 24 points GW dataset. In this case, it is necessary to introduce a fixed scal-ing factor (S) and a constant background contribution(Fbs) to the measured data F,b, (G), i.e. ,

F pt (G):SF b (Gr) Fbg (8)

Table I shows that while both data sets give the best fitfor a zero anharmonic factor P = 0 and a single Debye-Waller factor, they differ in detail: the fit to the TTKSset produces a 50Fo smaller Debye-Wailer factor and givesno indication of the l = 4 contribution to the density. Incontrast, the fit to the GW data set gives a significantl = 4 component and a small expansion of the valenceshell. We will compare below both results to ab initiocalculations.

8. Fits for ges=ir1anium

Three recent data sets are available here: Takamaand Sato [(TS), Ref. 42], Matsushita and Kohra [(MK),Ref. 40], and Deutsch, Hart, and Cummings [(DHC),Ref. 43]. All other available structure factors were mea-sured on powders and have lower accuracies. Here, un-fortunately, there is no consolidated set of structure fac-tors. Furthermore, all data sets were measured at dif-ferent wavelengths, so dispersion corrections f' need tobe applied before comparison or averaging can be done.There are no measured f' values for any of the three sets.The Cromer-Liberman calculated valuesss's4 are highlyuncertain, e.g. , in Si, they were off by 30—40Fo from di-rectly measured f' values using Mo and Ag radiations. ss

For Ge with Cu radiation (MK measurements), which isvery close to the edge, the f' value is large, and any errorin it represents a large error in the striped F value. We


find that when the f' values are subtracted, using thebest estimate of the Debye-Wailer factor, there are fairlylarge deviations between the three sets for the common(hkt) 's.

One approach to the problem taken by Brown andSpackman4s is to combine the sets in a somewhat ar-bitrary fashion. They use the MK set but replace the(111)and add the (511) values by those of TS, scaled bythe average ratios of the common reflections in the twosets. The analysis of Brown and Spackman did not con-sider the DHC set, which was not yet published at thattime.

We have adopted a different approach. We fit thestructure factors of TS and MK separately with the same"standard" model (B,(, As, A4, Sj, where S is a scalefactor [Eq. (8)t, common to all I" 's of a set as in Ref. 45.The residuals for MK were 1.5—2 times smaller than forTS. Consequently, we adopted the set of MK as it is withno replacement or additions from scaled values of TS.However, we added to this set the DHC set, which hasa few (hkt) in common with MK and a few higher-orderreflections, which are important for the determination ofB. Where common (hkl) exist, both were included inthe set. The fit of the standard model to this combinedset (using of course two S's; one for MK and one forDHC) did not show any systematic trends in the resid-uals, which remained all of the same average magnitudeas for the MK set alone.

The number of I' 's in the measured set is 14, with somehkt having two measured values. The f' values used arethose of Creaghss for the MK data (f' = —1.089), andthose of DHC (Ref. 43) for their data (f' = 0.09). These

were fixed in all fits.Since Ge is a rather heavy element (Z = 32), one might

expect relativistic effects to be non-negligible in the cal-culated charge densities and structure factors. This isindeed born out by the comparison of the local-density-approximation (LDA) nonrelativistic (NR) atomic struc-ture factors with their relativistic (R) counterparts shownin Table II. Both sets of structure factors were calculatedfrom numerical integration of the local-density equa-tions using the spin-polarized Ceperley-Alder e~change-correlations and the s p2 configuration. The corre-sponding relativistic corrections for carbon and siliconare negligible. Table II shows also the Hartree-Fock (HF)atomic results. s~ si We see that whereas the relativisticcorrection n~(q) —nNR(q) is similar in the LDA and HFresults, there are differences between the absolute valuesof n(q) as calculated by HF and the LDA. These reflectthe absence of correlation in the HF calculation and thedifferent nature of the exchange potential (local versusnonlocal in the LDA and HF results, respectively). Sincethere are no tabulations of orbital-by-orbital Hartree-Fock relativistic densities n„i(r) for Ge, the relativisticfits were done using LDA orbitals.

Table I gives the results of the fit to the Ge data us-ing relativistic LDA orbitals. The table also gives theresults obtained with the conventional (nonrelativistic)Clementi Hartree-Fock orbitals. We see that the statis-tical quality of both Bts are nearly the same. However,as shown below, the relativistic fit produces much bet-ter agreement with (i) the measured forbidden (442) and(622) reflections and (ii) the calculated crystal structurefactors. Thus, we will use below mostly the results of

TABLE II. Calculated free-atom germanium structure factors n(q) for momentum q = sine/Aas obtained in the local-density approximation (LDA) and the Hartree-Fock (HF) approximation.We list the nonrelativistic (NR) values, as well as the relativistic (R) correction np(q) —nNp, (q).

0.050.100.150.200.250.300.350.400.500.600.700.800.901.001.101.201.301.401.50

LDAnNR(q)

31.27829.54027.49825.53223.72222.04020.45018.93416.13013.69911.69510.1148.9067.9977.3136.7896.3716.0205.710

HFnNR(q)

31.27829.52727.47725.53223.75922.10920.53619.02116.18813.71611.68110.0838.8707.9637.2866.7706.3616.0185.714

LDAn~ (q) —nNR (q)

0.0060.0190.0290.0310.0280.0240.0230.0260.0400.0540.0640.0680.0670.0650.0620.0590.0590.0600.061

HFnR (q) —nNR(q)

—0.0020.0070.0270.0350.0320.0270.0240.0260.0390.0540.0640.0680.0670.0650.0620.0600.0580.0580.060

Reference 51.Reference 57.


relativistic fits, although we will comment on the com-parison with the nonrelativistic fits.

We find that Ge exhibits a clear anharmonic term P;its value (0.9) is comparable to the measured values ofTischler and Batterman of (1.27+0.25) (Ref. 58) andRoberto, Batterman, and Keating of (1.50+0.22).ss Thenonrelativistic fit also indicates a nonzero P, which, al-though twice as large, is still in reasonable accord withexperiment. The relativistic fit shows a 4.5%%uo expansionof the valence shell compared with the 6%%uc expansionfound for silicon. No evidence was found in either fit fora shell-dependent Debye-Wailer factor of the sort foundfor silicon. However, as these conclusions are based on aset of measured structure factors of germanium which isboth smaller and less accurate than that of silicon, fur-ther higher accuracy measurements are needed to confirmthe validity of this conclusion.

III. CALCULATED QUANTITIES

A. Static structure factors and densities

While diffraction experiments produce discrete Fouriercomponents of the charge density, electronic structurecalculations produce the total static electron charge den-sity p,«, (r) directly in coordinate space. This is ob-tained by summing the squares of the one-electron crys-talline wave function over all occupied band indices i andBrillouin-zone wave vectors k enclosed within the Fermienergy e~

culcted three-dimensional density pca~c(r) of Eq. (9) intoatomic-centered quantities. One common practice isto replace T (G) by some average (T(G)) over the dif-ferent atomic species o. in the unit cell, then [in analogywith Eq. (4a)] factor (T) out of Eq. (1). While this proce-dure seems reasonable when the bonded atoms have sim-ilar vibrational properties (hence, similar Debye-Wailerfactors), it is not obvious how accurate it is otherwise.Given that any partitioning of p(r) into atomically cen-tered quantities is arbitrary, we will choose a physicallyappealing but, much like Eq. (1), nonunique scheme: hav-ing calculated a unique and continuous density p, ~,(r)from Eq. (9), we decompose it into (i) "mufBn-tin" (MT)spheres around each atom n and (ii) the remaining, inter-stitial volume between them. Denoting as pMT(G) theFourier transform of the MT charge density in the n'smuffin-tin sphere (minus the extrapolation of the inter-stitial charge density into this sphere) and by pI(G) theFourier transform of the interstitial (I) charge densityover all space, the calculated dynamic structure factorbecomes

(G)&—(HlG /isa.

M

+ ) pMT(G)e BG —1/Ger (12 )

where B are the measured Debye-Wailer factors and (B)is their atomic average. The terms of Eq. (12a) are cal-culated in a way that avoids taking Fourier transforms ofabruptly truncated functions. Note that for a monatomiccrystal such as C, Si, and Ge we have (B) = B andM=1, so

p, (,(r) = ) N, (k)g;(k, r)g, (k, r)i,k

(9) F. .(G) = p.«.(G) (12b)

where N, (k) is the occupation numbers of band i at mo-mentum k. The Fourier components of the static chargedensity can then be computed from

j-pc«c(G) = — pcalc(r)e ' 'dr,0

where 0 is the unit-cell volume. To compare withthe experimental static charge density p,„~i(r, G „) ofEq. (4b) one can then filter out all Fourier componentsabove a given momentum value of G „, finding

Gmax

pcalc(r& Gmax) = ) pcalc(G)&G

Note that Eqs. (9) and (11) differ by Fourier truncationeffects, which will be examined below.

B. Dynamic structure factors and densities

Comparison of calculated quantities with the measuredF,„~t,(G) of Eq. (1) requires the introduction of the dy-namic factor T (G) into the calculation. The obviousdifEculty here is that while the measured structure factorsanalyzed through the rigid-atom approximation [Eq. (1)]represent linear contributions from discrete scatteringcenters o. , there is no unique way of partitioning the cal-

which can be compared with F, pi(r, G a„) of Eq. (2).

IV. PARTITIONING THE CHARGE DENSITY

In order to assess bonding effects, it has beencustomaryso to subtract from p(r) a superposition (sup)of calculated ground-state spherical charge densities ofthe neutral free atoms. The Fourier components of thisreference charge density are

M

p,„p(G) = ) n (G)e' (14)

where n (G) is the Fourier transform of the o.th free-atom charge density [not to be confused with the crys-talline quantity p (G) of Eq. (1) which pertains tobonded atoms]. The dynamic superposition structurefactors are then

F,„p(G) = ) n (G)e' -T (G)

and the partitioning of Eq. (12a) is not needed. Thecorresponding calculated dynamic charge-density map isthen

Gmax

47 ELECTRONIC CHARGE DISTRIBUTION IN CRYSTALLINE. . . 939I

Pcalc(r) = Pcore(r) + Pval(r) r (i6)

where

Note that if bonding effects modify the core density rela-tive to that in the free atom, even the high Fourier com-ponents of p,„~(G) will difFer from pcalc(G). In order toisolate such efFects it is further customary to decomposep into "core" and "valence" terms. The sum over states(i, k) in Eq. (9) can be partitioned into

&P-d.l(r) = P .d.l(r) —P. .(r)= ) AP(r —R,,), (20)

(21)

&&(r) = ) .[K'.l p l(K «) —p l(r)]+ Rs(r)Ks(r)nl,

+R4(r) K4(r)

p l(r) = ) ) N;(k)@;(k, r)g;(k, r)i+core k

(17a) V. DETAILS OF AB INITIO CALCULATIONSAND CONVERGENCE

and core is 1s for diamond, ls, 2s, 2p for Si, and 1s, 2s,2p, 3s, 3p, 3d for Ge. This partitioning is based on thefact that there is a significant energy separation in theband structure between core and valence states. Usingthis partitioning, it is possible to assign different Debye-Waller factors to pco„(C) and p al(G), thus facilitatingthe comparison with the model structure factors. Thetruncated form of Eq. (17a) reads

Gmsx

pval(r~ Gmax) = ) pval(G)eG

(17b)

Gmax

+p~(r G ~ ) = ).[p~(G) —p ~,~(G)]e' 'G

where p stands for "total, " "core," or "valence. "The corresponding dynamic deformation density mapsAF(r, C „)are obtained by using the dynamic form ofF,„p(G) [Eq. (15)] and of F„l,(G) [Eq. (12a)].

Equations (17b) and (18) depict p„al and b,p in aFourier truncated form. One can use instead the model-density approach of Eq. (7a) to calculate a different ap-proximation to these quantities. 5 s M For the valence den-sity one calculates

Pmodel, v(r) —) W(r Rj) (i9a)

where

p(r) = ) r.„&p„l(~„lr)+ Rs(r)Ks(r) + R4(r)K4(r)

where pval(G) is the Fourier transform of p al(r) ofEq. (16).

In analogy with Eq. (16) one can partition the free-atom charge densities n (r) into core and valence, so thatpe„~(G) in Eq. (14) can be separated accordingly. ThedifFerence between the crystalline charge density p, l, (r)and the reference charge density p,„~(r) then yields thestatic deformation density maps

We use the all-electron linearized augmented-plane-wavess (LAPW) implementation of the local-densityformalism. No shape approximation is used in describ-ing the potential and valence charge density, while thecore wave functions are obtained self-consistently fromthe j = 0 (spherical) piece of the crystalline potential.We use the Ceperley-Alder5s exchange correlation func-tional as parametrized by Perdew and Zunger. The lat-tice constants used are taken from experiment at thetemperature where the x-ray structure-factor measure-ments were done. They are 3.5670 A (T = 298 K), 4

5.4307 A. (T = 295 K),so and 5.6579 A. (T = 298 K)(Ref. 45) for C, Si, and Ge, respectively. All calcula-tions are carried out relativistically except for the omis-sion of spin-orbit effects. The latter approximation israther good: comparison of fully relativistic (with spinorbit) and scalar relativistic (retaining mass-velocity andDarwin effects but omitting spin orbit) calculations forGe show that the structure factor changed by less than1 millielectron (me) per atom [or 0.003% for the (ill)reHection] in Ge. However, mass-velocity and Darwineffects are important for Ge, as illustrated in Table II.

The great care with which the F,„~t(G) have beenpreviously measured and analyzed calls for an equiva-lent assessment of the errors in the calculated counter-parts. There are Bve convergence parameters that con-trol the precision of the LAPW solutionss to the local-density Hamiltonian: (i) the number Nb „,of basis func-tions in which g;(r) are expanded, (ii) the number ofNl, of special k points used in the Brillouin-zone sum-mation of Eq. (9), (iii) the maximum angular momentuml „used in the Kubic harmonic expansion [analogous toEq. (5)], (iv) the radius RMT of the atomic spheres insidewhich the Kubic harmonic expansion is taken, and (v) thenumber Ng, „ofFourier components used to expand thecharge density in the interstitial region. We have variedthese parameters until an internal convergence to betterthan a millielectron was obtained.

VI. COMPARISON OF THEORY'WITH EXPERIMENT

nl=val

(19b)A. Dynamic and static structure factors

of C, Si, and Geand the sum over j in Eq. (19a) extends over unit-cellvectors R~, while the sum over nt in Eq. (19b) includesonly valence orbitals. For the model deformation densitymap6~

Dynamic structure factors for silicon

Table III gives the measured dynamic Si structure fac-tors Fe„~q(G) and their estimated standard deviation o..


TABLE III. Dynamic (F) structure factors for Si in units of e/atom corresponding to the most converged set (column 1) inTable I. The experimental data (corrected for anomalous dispersion and nuclear scattering), including the estimated standarddeviations o (in me/atom), are taken from Cummings and Hart (Ref. 1), except the (222) result taken from Alkire, Yelon, andSchneider (Ref. 7). The difference bFr is F,~~, (G) —Fe»t(G) (me/atom), while bF2 = F,„r(G) —F,„~t(C) (me/atom). Theroot-mean-square deviation for hF& is 12 me/atom and an unweighted R factor is 0.21%%uo. The dynamic F, &,(G) is obtainedfrom the static p, ~, (G) using B = 0.4632 (Ref. 30).

111220311222400331422333511440444551642800660555844880

(G)[Eq. (12a)]

10.6008.3977.6940.1616,9986.7066.0945,7605.7815.3184.1153.9313.6493.2532.9172.8022.1651.543

Dynamic, solidFe&cpt (&)[Eq (1)]10.60258.38817,68140.18206.99586.72646.11235.78065.79065.33244.12393.93493.65583.24852.91432.80092.15061.5325

2.92.21.91.01.22.02.22.12.72.01.83.45,43.41.62.12.42.6

—39

13—21

2—20—18—21—10—14

9—4—7

531

1411

10.4558.4507.8140.0007.0336.6466.0775.7695.7695.3024.1073.9253.6443.2512.9152.8022.1631.542

—14862

133—182

37—80—35—12—22—30—17—10—12

311

1210

Dynamic, atomsF-,(G)[Eq. (15)]

This is compared with the local-density calculated struc-ture factors F, &, (G) from Eqs. 10) and (12b) [using thebest single B value B = 0.4632 ~ (Ref. 30)]. The differ-ence bFr = F, ~, (G) F,„~&(G) is—also given. We see thatwhile

IhFr

I

exceeds a, the largest ]6FrI

is 21 me/atom andthe root-mean-square (rms) deviation over 18 reflectionsis only 12 me/atom. For some reflections [e.g. , (311),(222), (331), and (333)] the difference 6Fr exceeds bothcalculated and the experimental errors. We suspect thatthis might reflect a combination of imperfect knowledgeof the exchange-correlation functional and uncertaintiesin the deconvolution of p(G) from F(G) [the rigid-atomapproximation of Eq. (1)]. The extent of agreement withthe data can be measured by the (unweighted) R factor

).Ilp. ~~(G) I

—I p. ~.(G) II

B= ).Ip. ~(G)I(22)

The present calculation gives for 1'F(G)} R = 0.21%.This represents the best agreement achieved to date be-tween ab initio LDA calculated structure factors and ex-periment. For comparison, Table III also shows the dy-namic superposition structure factors F,„~(G) [Eq. (15)],where the free-atom densities n~ are calculated also fromthe local-density formalism using the same exchange cor-relation as in the crystalline calculation. The deviationfrom experiment SF' = F,„~(G) —F«'~q(G) is signifi-cantly larger than IbFr] for G ( (440), hence, solid-stateeffects are important in this momentum range. While theindividual deviations at high C are small, the difference

F,' &, (G) —F,„~(G) contributed collectively by all high-momentum components does add up to affect the shapeof AF(r) and Ap(r), as will be shown below.

We have also calculated F, t, (G) using different B val-ues for core and valence densities obtained in the modelanalysis [Table I (Ref. 6)]. For the first ll reflections theuse of two B values gives an R factor of 0.16'%%uo, slightlybetter than R = 0.21% obtained with a single B value(Table III). However, there is a systematic deviation asC increases. The B factor for the full set of 18 refIectionsis 0.35%, larger than that obtained with a single B value,which remains 0.21%. We conclude that using more thana single B value does not improve the agreement of the-ory with experiment.

8. Static structure factors for silicon

Table IU compares the experimentally deduced staticstructure factors with theory. We use two different ex-perimentally deduced static structure factors: (a) p, (G),obtained from Eq. (4a) by dividing F,„~t,(G) by a single Debye-Wailer factor with B = 0.4632 A2 of Ref. 30and (b) the model structure factors p (G) which is theFourier transform of Eq. (7a) (using two B values ob-tained in the model fits in Table I). The results arecompared to different calculations using various forms ofexchange and correlation: the Xo, , Wigner, 7 Hedin-Lunqvist, 8 Singwi et al. , Ceperley and Alder, andthe weighted-density approximation. 7 7 The table alsogives the unweighted R factors of Eq. (22). We seethat the agreement between the various theories and


p, (G) is similar to the agreement with p (C) for thefirst ll G values, for which theoretical results are avail-able. Hence, the use of two diferent Debye-Wailer factorsin the model does not significantly affect this compari-son. There are, however, systematic deviations between

p (G) and p, (G) for higher G values: while our R,

value remains 0.21'%%uo using either the first ll G or all 18G values, our R increase from 0.16%%uo (11 G values) to0.35% (18 C values).

Our values R, = 0.2l%%uo and R = 0.16'%%uo show im-provement over all previous LDA calculations. Note inparticular the improvement over the pseudopotential re-

TABLE IV. Comparison between calculated and experimentally deduced static structure factors p(C) for Si in units ofe/atom. We give two sets of experimental p(G): (i) The value p, (G) is extracted from F,„~t(G) using in Eq. (4a) a singleB = 0.4632 from Spackman (Ref. 30). The experimental F „„t(G) (corrected for anomalous dispersion and nuclear scattering)are taken from Cummings and Hart (Ref. 1) except the (222) result taken from Alkire, Yelon, and Schneider (Ref. 7). (ii) Thevalues p~(G) are the Fourier transforms of the static model-density fit by Deutsch (Ref. 6) to the same set of F,„~t,(G). Thisuses difFerent B values for core and valence (Table I). In the theoretical values we include only those pseudopotential results inwhich the core contribution has been added by the authors. We also give the R factors [Eq. (22)] and the number of data points(N) used in comparison. The calculations shown use a variety of approximations for the exchange-correlation potential, i.e. , thelocal-density approximation (LDA) (including the Xn), the Hartree-Fock (HF) approximation, or the nonlocal weighted-densityapproximation (WDA). These Hamiltonians were approximately solved by a variety of band-structure methods: LAPW is thelinearized augmented plane wave, OPW is the orthogonalized plane wave, LCAO is the linear combination of atomic orbitals,LCGO is the linear combination of Gaussian orbital, and PP is pseudopotential.

111220311222400331422333511440444551642800660555844880R, 'Fp

&m~0N

Expt,pe(G)10.7288.6568.0200.1917.4497.2476.7166.4276.4386.0464.9794.8074.5554.1763.8663.7603.1352.533

Expt.pm(G)

10.7138.6558.0270.1817.4547.2466.7126.4206.4326.0334,9524.7834.5214.1493.8343.7283.1142,509

LAPW(LDA)

10.7268.6658.0330.1687.4527.2256.6966.4046.4286.0304.9684.8024.5464.1823.8703.7613.1552.5510.210.16

11

LCAOb(HF)

10.7558.6408.0040.2177.4657.2696.7306.4266.4596.0604.9834.8154.5564.1873.8713.7583.1472.5360.240.36

11

OPW'(Xn)

10.708.678.050.177.497.266.736.416.456.075.04

0.380,39

11

LCAO~(LDA)

10.6598.6568.068

7.4607.1806.6836.4166.417

0.420.358

LCGO'(LDA)

10.6848.6308.0400.1257,4607.1916.6856.4136.4216.020

0.460.36

10

PP'(LDA)

10.6918.6307.9970.1717.4157.1916.6556.3576.385

0.620.569

PP~(LDA)

10.6998.6157.9760.1707.3807.1496.6106.3076.3365.940

1,091,02

10

PPh(LDA)

10.708.577.790.197.357.036.656.316.356.00

1.441.409

LCAO'(WDA)

10.5228.4458, 1880.1757.5057.3046.7606.534

5.970

1.531.47

Present LAPW results, using the Perdew-Zunger parametrized Ceperley-Alder LDA exchange-correlation (XC) potential and60 special k points in the Brillouin-zone integrations.

Reference 28, using the self-consistent LCAO Hartree-Fock method and a Gaussian-type of orbitals. The results quoted herewere calculated using an 8—41G * basis set, which yields the lowest total energy.'References 20 and 21, using a six k points nonrelativistic self-consistent OPW calculated valence charge density with the Xe(n = s) potential (Ref. 66), to which a core charge density was added from relativistic free-atom Hartree-Fock data.Reference 23, using the self-consistent LCAO method and the Xn (o. = 3) potential, with 19 k points in the Brillouin-zone

integrations.'Reference 22, using the self-consistent LCGO method and the Wigner XC potential (Ref. 67) with ten special k points in theBrillouin-zone integrations.Reference 19, using the self-consistent plane-wave pseudopotential method with a basis-set cutofF of 24 Ry and the Wigner

potential with ten special k points in the Brillouin-zone summation.Reference 18, using the plane-wave pseudopotential method with a basis-set cutofF of 11.5 Ry and Wigner XC potential with

ten special k points in the Brillouin-zone summation.Reference 15, using the mixed-basis pseudopotential method and exchange (n = s) plus the Singwi et aL correlation potential

(Ref. 69), with ten special k points in the Brillouin-zone integrations.'Reference 29, using a LCAO description of the bond charge and a nonlocal weighted-density approximation (WDA) (Ref. 70)to exchange correlation (Ref. 71), with six high-symmetry k points in the Brillouin-zone integrations.


suits, e.g. , Yin and Cohen~s giving R, = 1.09% andR~ = 1.02%, Zunger~s yielding R, = 1.44% and R1.40%, and the improvement over the recent weighted-density calculation of Balbas et aL s R, = 1.53% andR~ = 1.47%. The recent Hartree-Fock calculationof Pisani, Dovesi, and Orlando2s yields a rather goodR, = 0.24% and R~ = 0.36%.

The improvement in the present calculation over pre-vious pseudopotentiat results reflects our use of an all-electron representation whereby core and valence statesare treated self-consistently as solid-state wave functionson equal footing. The improvement in the present cal-culation over previous aitl-electron results stems primar-ily from a better formulation of the expansions of wavefunctions and potentials, permitting more effective con-vergence. That the improvement is not due to the choiceof the correlation functional can be seen by repeatingour LAPW calculations with different correlation func-tionals. We find for the 11 reHections of Table IVthe (R„R ) error factors of (0.21,0.16); (0.22, 0.17);(0.26,0.17), and (0.27,0.18) using the Ceperley-Alder, ss

Hedin-Lunqvist, ss Wigner, s7 and the Xo. (with o. = s)

(Ref. 66) exchange correlation, respectively. Also, theimprovement of the present calculation over the oldercalculation cannot be attributed to the increased speedand storage of present-day computers, since a completeSi calculation with better than 1 millielectron error takesonly 3 min on a GRAY YMP; even with a 20-fold slowermachine this calculation is still affordable.

8. Dynamic structure factors for diamond

Table V compares the calculated dynamic F,~~, (G)with the experimental F,»t(G) values. We have cal-culated F«~, (G) from Eq. (12b) using the appropriateDebye-Wailer factors (Table I: B = 0.1379 A.2 for theTTKS data and B = 0.2303 ~ for comparing with theGW data. The differences bFq = F«~, (G) —Fe»t(G)are larger than those seen in Si: the rms deviation is 17me/atom and the R factor for (F(G)j is 0.94% for theTTKS data. For the GW data, the rms deviation for bFqis 31 me/atom and R = 2.25%. Clearly, ab initio theoryagrees with the TTKS data much better than with the

TABLE V. Dynamic (F) structure factors for diamond in units of e/atom. We show two sets of experimental data: thesingle-crystal results of Takama et al. (TTKS) (Ref. 39) except the (222) reflection which are taken from Weiss and Middleton(Ref. 37), and the powder data of Gottlicher and Wolfel (GW) (Ref. 36). o denotes the estimated standard deviation in thedata (in me/atom). The calculated values F, ~,(G) are obtained from Eq. (4a) using the Debye-Wailer factor B appropriateto each experimental data set: B = 0.1379 A for TTKS and B = 0.2303 A for GW (see Table I). The difFerence bFq is

F, ~, (G) —F „pt(G) (me/atom). The root-mean-square deviation for 6' of TTKS is 17 me/atom and the unweighted R factoris 0.94%. For GW, we find the rms for 6' is 31 me/atom and R = 2.25%.

ill220311222400331422333511440531620533444551711642553731800733660822555751753911

F. ).(G)3.2561.9341.6500.1081.4981.4801.3441.2681.2861.2071.1551.0861.0530.9840.9500.9460.8920.8580.8610.8100.7820.7390.7380.7150.7130.6530.652

TTKS dataFexpt (&)

3.2471.9201.6480.1441.4911.4831.3631.2871.3101.198

1154

1067

10523

9142

—367

—3—19—19—24

9

F. (,(G)3.2381.9061.6180.1051.4551.4291.2861.2081.2251.1391.0841.0100.9740.9020.8660.8630.8060.7710.7730.7210.6920.6480.6480.6240.6230.5620.561

GW dataF- (&)

3,3111.9121.5860.1081.3791.4561.3011.2631.2631.1201.0901.0270.989

0.8760.8760.8240.7880.788

0.6740.6740.6390.6390.5820.582

797

10952

9355

—73—632—376

—27—15—55—38

19—6

—17—15

—10—13—18—17—15

—26—26—15—16—20—21


GW data. We also notice that the scaled GW structurefactors are systematically larger than the calculated val-ues by an average of 17 me/atom for C & (531).

Static structure factors for diamond

There are many previous calculations of the staticstructure factors 7 and charge-density mapsof diamond. Table VI compares the experimental staticp,»r, (G) deduced from the TTKS and the GW data withvarious previous theoretical calculated values.Since for diamond B«„= B ~~,„«(Table I), there isno ambiguity in the choice of p, (G) vs p (G) as dis-cussed for Si. Hence, the experimental p,»r, (G) wereobtained from Eq. (4a) by using a single Debye-Wailerfactor from Table I. As was the case for the dynamicstructure factors, we find that all calculations agree muchbetter with the TTKS data than with the GW data.The Xn calculation of Ivey"2 produces the best R fac-tor (Rq = 0.80%%uo and R2 = 1.91%), while our calculationyields Rq = 0 95%%uo and R2 = 2.36%%uo. Our R factor forthe static structure factors in diamond is 5 times largerthan our R factor for Si (0.21%%uo). This is partly due tothe fact that the denominator of B for diamond is only

s that of Si, even though the rms value of bFq is onlyslightly larger than that of Si (for the TTKS data).

$. Dynamic structure factors for germanium

Table VII compares the experimental F,»r, (G) of Gewith the calculated values using B = 0.5654 A2 obtainedin the relativistic fit (Table I). The rms error betweentheory and experiment is 170 me/atom and the R factoris 0.85%%uo. Solid-state effects are apparent by noting thatthe R factor of F,»t(G) relative to the superposition offree atom density is 1.10%%uo.

6. Static structure factors for germanium

Table VIII compares various theoretically calculatedstatic p,~~, (G) with experiment. We use two experi-mental sets of p,„~t,. one derived from a model fit usingClementi's nonretativistic Hartree-Pock free-atom struc-ture factors, and one using retatinistic LDA orbitals inthe fit (see Table I). We see that the relativistic modelfit produces consistently better R values: R~ = 0.43'%%up

and RNR = 0.58%. Both values are relatively small, ow-ing to the larger denominator [P ~p,»t(G) ~] for Ge andthe two scaling factors in the fit (Table I), even thoughthe rms of ~AFq

~(170 meV/atom) is more than 14 times

larger than that of Si (12 rneV/atom). Our results andthe OPW calculation of Raccah et at. produce the low-est B factors.

TABLE VI. Comparison between calculated and experimentally deduced static structure factors p(G) for diamond in unitsof e/atom. We show two sets of experimental p(C): The value pq(G) is extracted from F,„~t(C) of TTKS using in Eq. (4a)B = 0.1379 A . The value p2 (G) is extracted from Fe„~t(G) of GW using in Eq. (4a) B = 0.2303 A . We show the correspondingR factors of two data sets relative to different calculations. N is the number of rejections used to calculate R. See the captionto Table IV for the definitions of the calculation methods used.

111220311222400331422333511440

&a%R2%

N

Expt.c~(G)3.2741.9621.6970.1491.5571.5621.4541.3851.4091.306

3.3571.9821.6670.1141.4831.5871.4511.4271.4271.294

LAPWLDA

3.2821.9761.7000.1111.5641.5581.4341.3641.3841.3160.952.36

10

SCPWGXe3.2901.9661.6900.1171.5701,5701.4511.3981.3981.3250.801.91

10

LCAO'Xn3.2731.9921.7200.1371.4941.6001.4231.3811.385

1.562.149

LCGOHF

3.2491.9601.6930.0701.5431.5261.4271.3761.381

1.553.059

LCAO'HF

3.2741.9251.6590.0881.5351.5331.4431.3821.386

1.552.589

OPW'Xa3.331.971.660.141.531.551.421.341.371.311.722.20

10

LCAO~Xa3.3492.0231.7480.0751.5901.5641.4521.3941.401

2.182.489

'Present LAPW results, using the Perdew-Zunger parametrized Ceperley-Alder exchange-correlation (XC) potential with tenspecial k points in the Brillouin-zone integrations.

Reference 72, using the self-consistent mixed-basis plane wave plus Gaussian (SCPWG) method and the An (n = 0.75847)potential, with six high-symmetry points in the Brillouin-zone integrations.Reference 74, using a self-consistent LCAO method and exchange-only potential with ten special k points in the Brillouin-zone

integrations.Reference 75, using a minimum basis-set Gaussian-orbital Hartree-Fock method.Reference 76, using the self-consistent LCAO Hartree-Fock method with a minimum basis set of Slater-type orbitals.Reference 21, using a self-consistent OPW method and the An (n = 1) potential with four k points in the Brillouin-zone

integrations.sReference 77, the results were obtained using a self-consistent LCAO method and An (n = 1) potential with 19 k points inthe Brillouin-zone integrations.


7. "Forbidden" reflections in C, Si, and Ge

The model of superposition of spherically symmetricatomic charge densities [Eq. (15)] produces F,„p(G) = 0for the reflections G for which 6+A:+ l = 4n +2, where nis an integer. This defines the (pseudo) "forbidden" reBec-tions, e.g. , (222) and (442) and (622), (644), and (842).(Note, however, that if h+ A:+ l = 4n + 2 and any one ofthe three indices 6, k, or l is zero, then the reHection isstrictly forbidden for the diamond structure. ) Of course,the actual charge density is deformed from sphericallysymmetric densities due to bonding effects, so, in reality,the pseudoforbidden reflections carry a finite (althoughweak) intensity. Within the multipole expansion frame-work of our model, the finite structure factors for thequasiforbidden reflections are due to two effects. The firstis due to the existence of an antisymmetric, odd-l, bond-ing component in the static charge density in Eq. (5).The second is due to anharmonic thermal motion (finiteP), which exists in the centrosymmetric charge density

[all even l components in Eq. (5)].Since the intensities of the forbidden reflections are

smaller by two orders of magnitude or more than the al-lowed ones their accurate measurement is an extremelydemanding task. Thus, only the lowest-order reflections,(222), (442), and (622), were ever measured. ss ss Theimportance of these reflections stems from the fact thattheir intensity is determined solely by the small effects ofanharmonicity and bond deformation, whereas for theallowed reflections these small contributions are com-pletely masked by the much larger spherical charge con-tributions. Furthermore, since the deformation densitiesare part of the valence shell, they can provide informa-tion on postulated nonrigid thermal motion of the atom,where the valence shell moves with a different Debye-Waller factor than the core. ' Their measured valueshave been therefore employed repeatedly to test the qual-ity of model fits and ab initio calculations of the crys-talline charge distribution. In particular, they providestringent tests for the derived values of P, valence-shell

TABLE VII. Dynamic (F) structure factors for Ge in units of e/atom. The experimental data Fe„pq(G) (corrected for thethermal motion and anomalous dispersion) are from the MK (Ref. 40) and DHC (Ref. 43) data sets. The dynamic F, ~, (G)are obtained from static p, ~, (G) by using B = 0.5654 (see Table I). The difference 6Fi is F„&,'(G) —F,„pq(G) (me/atom),while OF2 = F,„'p(G) —F,„pq(G) (me/atom). The root-mean-square deviation for bFi is 170 me/atom and the unweighted Rfactor is 0.86%.

111111220311222400331422333333511440531620533444551711642553731800733660822555751777

F-~ (G)[Eq. (12a)]

27.15627.15622.86121.1210.114

18.93217.86816.20415.33315.33315.33214.05413.36312.32511.76110.91610.45110.4539.7499.3669,3628.7738.4497.9497.9497.6727.6733.942

Dynamic, solid+expt(G)[Eq (1)]27.29227.54122.68820.9190.131

18.66717.82316.04115.19215.383

13,855

10.977

7.695

3.930

60525060105371539015

53

10

bFy

—136—385

173202—17265

45163141—50

—61

12

Dynamic, atomsF. p(G)[Eq (»)]

27.04227.04222.92721.234

018.97617.83916.19415.32715.32715.32714.04113.35212.31711.75710.91010.44810.4489.7459.3609.3608.7718.4477.9487.9487.6727.6723.946

—250—499

239315

—131309

16153135—56

186



TABLE VIII. Comparison between calculated and the experimentally deduced static structure factors p(C) for Ge in unitsof e/atom. The experimental data p,„i,t(C) (corrected for the thermal motion and the anomalous dispersion) are the Fouriertransforms of the static model-density fit to the observed MK and DHC data sets. We give both the fit to the scalar-relativisticLDA (SR-LDA) atomic data and to the nonrelativistic Hartree-Fock (NR-HF) atomic data. The R factor for the LAPW (forall 12 measured data) are RsR = 0.37% and RNR = 0.62'%%uo.

111220311222400331422333511440444555777

Rsa, %RNR%

N

SR-LDA

27.89423.76622.1420.152

20.23519.48218.04017.300

16.19813.49810.6807.547

Expt.NR-HF

28.00923.83922.2080.143

20.28619.53118.07217.323

16.19913.45010.5937.461

LAPWLDA

27.51923.68322.1720.120

20.31819.43218.01617.27517.27316.18713.49310.6847.5430.430.589

OPWb2Co,

27.5423.6322.070.24

20.2719.4718.0017.2117.2416.16

0.520.739

LCGO'LDA

27.45723.61522.1250.110

20.28019.36817.95517.21817.21816.115

0.640.839

LCAOWDA

27.18123.61522.1890.185

20.39519.62218.14517.352

16.155

0.870.889

Present LAPW results, using the Perdew-Zunger parametrized Ceperley-Alder exchange-correlation (XC) potential with tenspecial k points in the Brillouin-zone integration.

Reference 21, using a self-consistent OPW method and the Xn (n = 3) potential with four k points in the Brillouin-zoneintegration.'Reference 22, using the LCGO method and Wigner potential with ten special k points in the Brillouin-zone integration.

Reference 29, using a LCAO description of the bond-charge model and weighted-density approximation (WDA) potential withsix high-symmetry k points in the Brillouin-zone. integration.

Debye-Wailer parameter, and, to a lesser extent, the de-formation density.

The measured, model-fitted, and LDA-calculatedstructure factors for the first three forbidden refiectionsare given in Table IX. Note that while the measured F222were included in the model fits (so their model valueis not a prediction), F442 and Fager were not, and areused only to test the accuracy of the model and theab initio calculations. The table shows a good over-all agreement of both model and LDA values with themeasurements. The model values areall within 10 me at most of the measured ones. The abinitio values deviate somewhat more, the maximal devia-tion being ~ 35 me for F2sz of diamond. The deviationsfor most reflections, however, are similar to those of themodel. The LDA results are consistently smaller thanthe model fitted values. For germanium, the only casefor which the fit yields p p 0, this is not surprising, sincethe LDA calculations, unlike the model, use P = 0. Toexamine these effects we calculate the forbidden reflec-tions of Ge assuming in the fit P = 0. The P = 0 modelvalues listed in Table IX for germanium indeed show amuch better agreement with the LDA results, although,as expected, they deviate more from the measured re-sults, thus indicating that anharmonic effects are signifi-cant in this case. For germanium, relativistic eKects maybe expected to be large enough to be observed. Indeed,

the use of relativistic free-atom scattering amplitudes inthe model fits improves the agreement with experimentsignificantly. However, an accurate assessment of theserelativistic efFects will have to await a more accurate, andmore complete body of measured structure factors thanavailable now.

Finally, the good agreement of the model and mea-sured forbidden structure factors lends credence to theconclusions drawn from the model fits here and in Ref. 6,i.e. , (i) P —0 and 1 eV/A. for silicon and germanium,respectively, (ii) shell-dependent thermal motion in sili-con having Bo», g B„~i,„«, (iii) the 2sp and 3' shellexpansion in silicon, and (iv) the valence-shell expansionin germanium.

B. Untruncated core, valence, and total staticcharge densities

Having established the level of accuracy of the calcu-lated F(G) and p(C) over the limited set of mornentaaccessible to high-precision measurements, we next con-sider the calculated real-space charge densities urith-out any Fourier truncation. Recall that the basis setused in our LAPW calculation includes real-space or-bitals that contain arbitrarily large momentum compo-nents. Figure 1 depicts the calculated p„ i(r) [Eq. (9)j,


while Figs. 2—4 show the deformation charge densitiesEp«„(r), Ap ~(r), and Ap«(r), respectively [Eq. (18)for G —+ oo]. In this and in the subsequent figures weshow side-by-side line plots in the (ill) direction andcontour plots in the (110)plane. The solid squares denoteatomic positions, empty triangles denote the (empty)tetrahedral interstitial sites, while the empty half dia-mond symbols at the left and right margins denote the(empty) hexagonal interstitial sites. Distances are givenin terms of the fraction of ~3a, where a is the latticeconstant. The origin is on the left atom. The bond cen-ter is at 2: = 0.125. Contour step values are given in theinsets. The basic features are as follows.

(i) As expected intuitively, the core charge densityp, „,(r) is localized around the nuclei. However, thecore is not entirely inert: the core deformation densityAp, „(r) (Fig. 2) is slightly negative in the core region.

This effect is naturally missed by all "frozen-core" andby pseudopotential calculations.

(ii) The valence charge density p ~(r) [Fig. 1] exhibitsthe following features: (a) maxima on the atomic sites,(b) the lobes on the bonds are oriented parallel to the(111)bond direction, (c) there is a local minimum at thebond center [point 6' in Fig. 1(a)], flanked by two localmaxima (point p), and (d) there is charge accumulationin the the back-band region (point a). These characteris-tics are reversed when one considers the valence deformation charge density Ap ~(r) for Si and Ge (Fig. 3): Nowthere are minima on the atomic sites, the lobes on thebonds are oriented perpendicular to the bond direction(in Si and Ge but not in C), there is a maximum at thebond center [point b in Fig. 3(a)], and charge depletionin the back-bond region o, . Table X gives the positionsR, (in units of ~3a with origin on the leftmost atom)

TABLE IX. Measured, model, and LAPW-calculated F(C) values for the "forbidden" reflec-tions h, + k+ l = 4n + 2. All values are in millielectron units. The model fits are defined in Table I.The LAPW results assume P = 0.

Model:TTKS fitGW fitLAPWB = 0.1379B = 0.2303Expt.

0.00.0

0.00.00.0

Diamond

136.8107.9

107.6105.3143.8

F442

8.37.7

F622

2.72.5

Model:LAPW:Expt. :AYsbTB1'MBTB2'

0.00.0

3.453.383.38

Silicon180.8160.5

182.0

11.27.9

4.385.254.63

3.11.7

0.631.10

Model:NR fit'NRfit P=0R fit~

RfitP=OLAPW: B = 0.5654Expt. :

MBTB2

1.810.00.880.00.0

1.321,27

Germanium

141.3136.1146.6143.8113.5

133.0

17.14.2

11.64.82.6

7.8812.29

9.41.05.61.22.0

6.988.25


'Reference 88.Reference 89.

'Reference 58. For P see Ref. 45.Using nonrelativistic free-atom form factors of Ref. 51.

~Using relativistic LDA free-atom form factors calculated by us."Reference 40.


s s

Is s s s

Is s s s

Is s s s

Is s

2 -(o)N

0 20

s si

s s s s

0 4 (o)

0.2-

s s s sI

s s

05

0 y----a

0s s

I

0.6 -(b)

I 0.4-

s s s sI ~

s ~ s sI

s s s sI

s

ISi

[

0.05

—0.o~ 0.

2-I

-(b)s s s I s s s s I

s s s'

Is s s s

s s s s I s s

s s s s

''=: Si '. -- ' --', Si '--

Si ', =,.

s

0.050.4-

0.2-

~ -0. 1—s s I s s s s

s si

s s s s

(c)0. 1-

-0. 1—

s s s s

s s s s s s s

is s

0.025 '' '

Ge ', '

', Ge, '.

0 s ~I

s s s sI

s s s sI

s s

[&s.(~ ")

[

0:-0. 1

0.2

0. 1

I s» s I «s s I s

s sI

s s s s

-(b)I s s ~

Is s

s s I s s

s sI

s s

0

cj -0. 1

0.2s s I

s s ) I s s s

(c}s s s s

is s

~ I s

s ~ I s s

0.1-

0-

-0.1-~ s I s s-0.25

Distance

nI s s s I s s I s

0 0.25 0.50along (111) (d/dc)

FIG. 2. Calculated untruncated (G ~ oo) static core de-formation charge densities Ap, „(r) [Eq. (18)] for (a) C (di-amond), (b) Si, and (c) Ge. Note that the core is not fullyinert but that the fluctuation Ap, „is localized.

0./' s s-0.25 0 0.25 0.50Distance along (111) (d/dc)

FIG. 1. Calculated untruncated (G —s oo) valence chargedensities p,~(r) [Eq. (17a)] for (a) diamond, (b) Si, and (c)Ge. In this and in the subsequent figures we show side-by-sideline plots in the (111)direction and contour plots in the (110)plane. The solid squares denote atomic positions, empty tri-angles denote the (empty) tetrahedral interstitial sites, whilethe empty half-diamond symbols at the left and right marginsdenote the (empty) hexagonal interstitial sites. Distances aregiven in terms of the fraction of v 3a, where a is the latticeconstant. The origin is on the leftmost atom. The bond centeris at 2: = 0.125. The insets denote contour steps e/A . Theoutmost contour has the same values as the contour steps.The values of the prominent features (the points a, P, p, 6,and e) are listed in Table X.

s s I s s s s i I s s s s I-0.25 0 0.25 0.50Distance along (111) (d/dc)

FIG. 3. Calculated untruncated (C —s oo) static valencedeformation charge densities &p &(r) [Eq. (18)] for (a) di-amond, (b) Si, and (c) Ge. The thick outer solid contournext to the dashed line denotes Ap = 0. Dashed contoursdenote negative Ap. The insets denote contour steps in units

3of e/A . The Dp values of the prominent features (the pointso. , P, and 6) are listed in Table XI. Note that the bond chargeis elongated parallel to the bond direction in diamond whilein Si and Ge it is perpendicular to the bond direction.

and amplitude p I(R,) of these features i = a. , p, p, b,and c for C, Si, and Ge, while Table XI gives similar re-sults for the deformation valence density Ap s„I(R;). Bothtables compare the ab initio values with those deducedthrough our model from experiments. We see that theamplitude of the local extremum of p ~I and Ap ~I at thebond-center point (6) decreases rapidly in the sequenceC~Si sGe (largely a volume efFect), as does the chargedensity at the tetrahedral interstitial site (point s). Thereare interesting differences between diamond on the onehand and Si and Ge on the other: both p,~I and Ap, I

are bond oriented in C, whereas Apv I in Si and Ge areoriented perpendicular to the bond direction. Further-more, Si and Ge exhibit deep minima on either side ofthe "camel's back" bond charge [point P in Fig. 1(a)],whereas these minima are considerably shallower in dia-mond (Table X). The significant difference between pv I

and Ap ~I (including a factor of 3 reduction in overallamplitude) suggests that many of the features of p ~I aredominated by contributions from the free atoms.

(iii) Since most of the peaked amplitude of pcc„(r) islocalized inside the core region, Ap„~I (Fig. 3) and Epact(Fig. 4) are similar over most of the unit-cell space. Thisis seen also from the last four lines of Table X, where thelocations and amplitudes of the main features of Lp ~~

and Epact are compared.(iv) The total charge density pt, t is dominated by the

core contribution and, as is expected, lacks structure.The amplitude pt t(Rp) at the bond-center point b is

Z. W. LU, ALEX ZUNGER, AND MOSHE DEUTSCH

TABLE X. Values of the valence and total charge density [p,~(R) and pq, q(R) in e/A ] and ~R~ (in units of v 3a, origin onthe left atom) at various points (o., P, p, 6, and e) along the (ill) direction as indicated in Figs. 2 and 3. The second linefor each entry lists the corresponding value obtained from the model fits to the measured structure factors. The negative p„~value obtained for Ge from the model at point e gives an idea of the uncertainty in the model fit. We only list the model fit ofTTKS data for diamond.

Calc.Expt.

Calc.Expt.

Calc.Expt.

Calc.Expt.

Calc.Expt.

Calc.Expt.

1.551.47

1.381.08

1.941.93

1.591.61

0.090.10

1.591.61

iRi

0.0560.060

0.0290.030

0.0640.069

0.1250.125

0.500.50

0.1250.125

P

p, )(R )0.310.30

p- ~(Rs)0.010.13

p-~(R~)0.570.58

p-~(R~)0.560.58

p~a(R. )0.020.02

pgog(Rp)0.560.58

Si[Rf

0.0810.083

0.0400.042

0.0960.103

0.1250.125

0.500.50

0.1250.125

0.360.40

0.020.16

0.490.60

0.460.58

0.02—0.01

0.490.61

GefRf

0.0770.079

0.0440.042

0.0910.092

0.1250.125

0.500.50

0.1250.125

TABLE XI. Values of the b, p ~~(R) and Dpq, q(R) (e/A. ) and ~R~ (in units of v 3a, origin on the left atom) at various points(n, P, and 6) along the (111)direction as indicated in Figs. 5 and 6. The second line for each entry lists the corresponding valuesobtained from the model fits to the measured structure factors. Entries left blank indicate the absence of the correspondingfeatures in the model. We use the TTKS data for the diamond model.

Si

Calc.Expt.

Calc.Expt.

Calc.Expt.

Gale.Expt.

Calc.Expt.

Calc.Expt.

Calc.Expt.

Calc.Expt.

—0.13—0.21

—0.13—0.21

0.32

0.32

0.450.45

0.450.45

—0.10—0.07

—0.10—0.07

/Rf

0.0970.098

0.0970.098

0.062

0.063

0.1250.125

0.1250.125

0.500.50

0.500.50

Ap

ap..((R )—0.10—0.10

&pro~(R~)—0.10—0.10

&p .~(Rs)—0,06

0.05&p~o~(Rp)—0.09

0.06ap..)(Rp)

0.180.20

&p~.~ (Ra)0.180.20

Ap, )(R,)—0.02—0.02

&p~o~(R )—0.02—0.02

0.0710.071

0.0720.075

0.0270.056

0.0240.061

0.1250.125

0.1250.125

0.500.50

0.500.50

—0.05—0.01

—0.05—0.01

—0.050.09

—0.060.09

0.130.25

0.130.25

—0.02—0.05

—0.02—0.05

IRI

0.0770.068

0.0760.068

0.0530.057

0.0510.057

0.1250.125

0.1250.125

0.500.50

0.500.50


close to the amplitude of p»i(R~), indicating that thecore contribution is negligible there.

(v) The deformation density maps Epics (Fig. 4) and&Fi . 3~ have sharp nodal features near the core

)& /and sharp minima in the "inner-bond region point ~~ inFigs. 3 and 4), both reflecting the fact that upon forma-tion of the solid from the atoms the nodes in the crystavalence wave functions are shifted with respect to thosein the free-atom valence orbitals. As will be shown be-low, the representation of these sharp features will requirerelatively high-momentum components in a Fourier de-scription.

C. Comparison of all-electron and pseudopotentialvalence charge density

I II

I I I II

I I

0.2-

-0.2 —. .

(b)0. 1—

s i I s

I I I I

I1 I

l II

I ~ I II

I I

~n.(~.")—05

s I i i & i I

I I ii I I l II

f I

0.025

/

/I

/I /

C ~ C

/

I

o o/

/

'=: Si '=-' ' =- Si '-="

II/

I

..-=; Si;=-../

/

I

o~ i s

I I I I

I I & i s s I s s

I I I I I I fI

F I

0.1-Ge[

---~---------&III-0

0.025 /', „Ge „,

'

Ge ''/

/

-0. 1—I I i s-0.25 0 0.25 0.50

Distance along (111) (d/dc)

FIG. 4. Calculated untruncated (C —+ oo) static totaldeformation charge densities Aptot(r) [Eq. (18)] for (a) dia-mond, (b) Si, and (c) Ge. The thick outer solid contour nextto the dashed line denotes Ap = 0. Dashed contours denotenegative Dp. The insets denote contour steps in units of e/The values at the prominent features (the points o. , P, anb) are listed in Table XI. Note that like in Fig. 3 the bondcharge is e onga e paral t d arallel to the bond direction in diamondwhile in Si and Ge it is perpendicular to the bond direction.

The pseudopotential approximation has proven to beextremely successful in many solid-state calculations asit enables the replacement of the quantum-mechanicalinterference e6'ects of core wave functions with an ef-fective, semiclassical external potential, which is com-

utationally far more expedient. The resulting pseudo-pu a ionwave-functions can be made nodeless and smoot yan appropriate choice of this external (pseudo)potential.Comparison with measured structure factors requires,however, the reintroduction of (i) core orbitals and (ii)core-valence orthogonality corrections. Current proce-dures for implementing this are often based on adding to

FIG. 5. Pseudopotential valence charge densities of (a)diamond, (b) silicon, and (c) germanium from Martins andZunger (Ref. Ql). The density is given in units of e/cell (note/A ). Contour steps is 2e/cell. Note the close similarity tothe all-electron results (contour plots in Fig. 1) outside thecore regions.

the calculated solid-state valence charge density the freeatom core densities (hence, neglecting Ap„„of Fig. 2).This includes approximately effect (i) but neglects effect(ii) (although core orthogonalization is possible for cer-tain classes of pseudopotentials, see Ref. 15 for details).Table IV indeed shows that the agreement of the cor-rected pseudopotential p, ~, (G) values with experimentis mediocre. However, the valence pseudo-charge-densityis generally very close to the valence density componentf ll-electron calculations. This is illustrated in Fig. 5,

ndwhich shows the pseudopotential results of Martins anZunger and can be compared to the current all-electronresults shown in Fig. 1. There is a remarkable agreementbetween the two sets of calculations outside the core re-gion. Of course, the maxima on the atoms and the dip Pare absent in the pseudopotential description.

D. Fourier-truncated charge-density maps for Si

Given that even deformation charge-density maps ex-hibit rather sharp features in r space, we next ex-amine the convergence of their Fourier representation.Figure 6 shows the calculated Si deformation densityApi q(r, G „) of Eq. (18) for three sets of t vectors.First [Fig. 6(a)] we use calculated p«ic(C) at the seof G vectors of Cummings and Hart (Table III), em-


0.2

0. 1

0

-0.1-

0Q) 0L

+-o+-~ -0.1-

0.2 -(c)

0.1-0y

I ' I

Apt, ~ calc)80)

aI

b,pt,~(c ale)12,12)-

I'

I' I ' I

I

I

o ''O' 0

C)

CI O

"='- Si '=- Si '=-

I

Si;-;.

eluded in the Fourier series of the type of Eq. (11).One sees a transition from a smooth behavior at low

Gm» [Fig. 7(a)] to an oscillatory function at intermedi-ate G „values [Figs. 7(c)—7(e)] and finally to a smoothfunction at high C» values [Fig. 7(f)]. The latterFourier-synthesized function is nearly indistinguishablefrom the real-space representation p I(r, oo) of Fig. 1(b).Liki e Fig. 6, Fig. 7 also shows the insuKciency of theCummings-Hart-Deutsch set in capturing the full struc-ture of the converged charge density if it is described bya Fourier series. Hence, despite the high precision of theindividual structure factors of Cummings and Hart, theset of C vectors included in them is insuKcient to repre-sent the full details of p I(r) or Ap, „(r) when used toFourier synthesize the electron density.

-0.1—-2 0 2 4

Distance along (111) (A)

FIG. 6. Convergence of the ab initio calculated static de-formation charge density Ap(r, G „) [Eq. (18)] of Si withrespect to the maximum momentum C included in theFourier series (denoted in the inset): (a) The Cummings-Hart-Deutsch set of C vectors (Table III); (b) all reflections up toCm» = (880), and (c) all reflections up to CbIs ——(12, 12, 12)(288 G components). Contour step = 0.025e/A. . Dashedcontours denote negative Ap. Note that (c) is practicallyidentical to the untruncated Ap(r, oo) of Fig. 4(b). Note fur-ther that the set of reflections used in (a) is able to capture thedetails of the converged charge in the bond region but missesthe localized features P and the amplitude of the charge den-sity on the atomic sites.

ployed also by Deutsch in his analysis. While this set ex-tends to G ( (880), it includes only 18 of the 52 allowedrefiections contained in this range. Second [Fig. 6(b)]we show Apt t calculated from all 52 reflections t

m» = (880). Finally [Fig. 6(c)] we depict Ap«t forGb;s = (12, 12, 12) (288 G components), outside therange of current high-precision measurements. The latterLp plot closely resembles the untruncated Lp ~~r j f

ig. ( ) (except for the inner-core region). The evolu-tion of 4 ( G )ptot yr& m»g with Gmax seen in Fig. 6 clearly

~ ~

exhibits robustness of the amplitude near the bond cen-ter. At the same time, the amplitudes on the (a) atomicsite, (b) inner-bond minima P, (c) back-bond miniman, and (d) the tetrahedral interstitial sites are far fromconvergence using Deutsch's set of 18 C vectors. Thisslow convergence of Ap(r, C „) with G „reflects themodification of the core of bonded atoms relative to freeatoms [see Eq. (18)]. Interestingly, Fig. 6 shows thatAp(r, Gm») does not contain noisy undulation for anylevel of truncation Gm». This disproves the assertionof Zuo, Spence, and O'Keefe33 that adding terms in aFourier series for Lp beyond about the first ten only addsnoise.

Figure 7 shows similar information for the valencedensity: it depicts the evolution of p ~r G ~j'ththe h' he ig est value of G» (indicated in the insets) in-

I

6 Ca) p„„(r,G„„)G„,=(111)

0.4

0.2

I i I s I e I

t I

0 6 gb) (311)0.4

0.2

0~ I i I g I

I' I ' I ' I

0.6

0.4-~~ 0.2-

C0

I

u 0.6-(d)=:0.4-

0.2-0

s I i I s I

II

I I I

(66&)—

I s I g I i I

I I

o 6-(e) (660)—0.4-0.2

0I i I ~ I s I

I' I '

I I ' I0.6-(f) (12,12,12)0.4-0.2

0I i I s I r I

-2 0 2 4Distance along (111) (A)

FIG, 7. Convergence of the ab initio calculated static va-lence density p»I(r Gm») [Eq. (17b)] of Si with respect tothe highest momentum included in the Fourier sum (denotedin the insets). Note that (f) is practically identical to the un-truncated p»I(r, oo) of Fig. 1(b). Note further that as Gm~„increases p I changes from a smooth function [in (a)] to anoscillatory function [(c)—(e)] and finally to a smooth function(f). Even the set G ( (880) (the best that can be mea-sured to date) is insufficient to produce a smooth p„ I.


E. Comparison of ab initio and modelcharge densities: p„~ & -(a) p„(cate)-

The text surrounding Eqs. (4b)—(7a) highlights thefundamental difFerence between a Fourier-synthesis ap-proach [Eq. (4b), illustrated in Figs. 6 and 7) and themodel-density approach [Eqs. (5)—(7a)] of Dawson,Stewart, 4 Coppens, Spackman, 4 and Deutsch:the latter method is guaranteed, by construction, to yielda smooth function despite the use of a limited set of struc-ture factors. We will next compare the results of themodel density to our fully converged results.

0

Q. p

I I I

-(b)

I I

p„(Madel')—

p~~] 2n silicon

Figure 8 compares our calculated p i(r, oo) [Eq. (17a)]for silicon with Deutsch's model valence density[Eq. (19a), see parameters in Table I (Ref. 92)]. Theagreement between theory and the experimentally de-rived function is excellent. Table X gives a quantitativecomparison of p„(calc) and p„(model) at locations n, P,p, 6, and e of Fig. 8. The only significant discrepancyexists in the inner-bond minima (point P), where our re-sult shows a significantly lower amplitude than Deutsch'sfit. Figure 6 demonstrated, however, that this featureis highly dependent on Fourier truncation. Our resultsand those of Deutsch are very difFerent from the elliptic,single-peaked density obtained by Yang and Coppensusing a truncated Fourier series.

0+,-2 0 2Distance along (111) (A)

FIG. 9. Comparison of the ab initio calculated static va-lence charge density (a) p i [Eq. (17a)] of diamond with (b)the model density from fitting the TTKS data (solid line) andGW data (dashed line). The corresponding contour plot is forthe TTKS data. Part (c) shows Spackman's results (Ref. 44)in which Stewart's orbital for 28 and Clementi's orbital for2p are used. The contour step is 0.20e/A . See the text forquantitative comparisons.

8. p ~ in diamond

0.6 -(a)

0.4-

I I I I I

] p„(canc) i-

~ 0

p~I

O. 6 Xb)~0.~-

]p„(mcdci)]-

0.2

0 Si SiI I I I I I I I


FIG. 8. Comparison of the ab initio calculated static va-lence charge density p„ i of Si [Eq. (17a)] with Deutsch's fitto the data of Cummings and Hart (b). The contour step is0.05e/A. . See text for quantitative comparisons.

Figure 9 compares the ab initio calculated valencedensity map [part (a)] with the model-density result[Eq. (19)] using the fits to the TTKS and the GW data[solid and dashed lines in part (b)]. The agreement oftheory with experiment is fair. Table X gives a quanti-tative comparison at some points. Our ab initio p~~~ atpoints n, p, and b [Fig. 9(a)] are 1.55, 1.94, and 1.59e/A,

which are close to the values 1.47, 1.93, and 1.61, respec-tively obtained from fitting the TTKS data and values1.66, 2.06, and 1.61, respectively, obtained from fittingthe GW data.

It can be seen from Eq. (19) that the shape of pat the nuclear sites depends sensitively on the choice ofatomic-orbital densities n„i since Ri~o(r = 0) = 0, seeEq. (6). We have consistently used for these quantitiesClementi'ssi Hartree-Fock data. Spackman used for nz,Stewart's density-localized valuesz while for nz„he usedClementi's values. The quality of the fit of F,„~t(C) isthe same whether one uses Clementi's orbitals or Stew-art's unitarily rotated orbitals. However, the compo-nents of the charge density depend on the choice of or-bital basis in the fit. Figure 9(c) shows the model den-sity p'

&using Stewarts orbitals in the fit. Comparison

vrith Fig. 9(b) shovrs significant difFerences betvreen thesemodel valence densities near the atomic sites: Spack-man's fit [Fig. 9(c)] gives local minima at the atomic sites,while ours [Fig. 9(b)] produces maxima on the atomicsites. Except at the atomic sites, which are expected todepend strongly on the nature of the orbitals employed,the two models are in fairly good agreement. The theo-retical calculations in Fig. 9(a) clearly agree better vrith

our model in Fig. 9(b).

pval $n gcpfAafw'lcm

Figure 10 compares our ab initio valence density of Ge[part (a)] with the result from the model fit using rela-

Z. W. LU, ALEX ZUNGER, AND MOSHE DEUTSCH

0.6 -(a)0.4-0.2

0

~ 0.6-(b)

0.4-0.2-

I~

& I

y p„{Calc)—

I ' I

p„{Model)

the local minima in the valence density, e.g. , Martinsand Zunger (see Fig. 5), Yin and Cohen, van18{b)

Camp, van Doren, and Devreese, and Nielsen andMartin. All-electron calculations have also observed

28the local minimum, e.g. , Pisani, Dovesi, and Orlan o,Methfessel, Rodriguez, and Andersen, Polatoglou andMethfessel and Weyrich. None of the previous cal-

r91culation on Ge, except that of Martins and Zunger(Fig. 5), seem to have found a local minimum in thevalence density.

0.6-(c)0.4-0.2-

I I I II

I I

p„{Model)

NR

F. Comparison of ab initio and model chargedensities: b ptot.

1. Deformation maps for silicon

e G

0 2 4Distance along (111&

6(A)

FIG. 10. Comparison of the ab initio calculated static va-lence charge density (a) p» [Eq. (17a)] of germanium withthe (b) and (c) model density. Part (b) shows the model fitusing relativistic (R) orbital, while part (c) shows the modelfit using nonrelativistic (NR) orbitals. The contour step is0.05e/A. .

The bond-centered local minimum in p„~~

Figures 8, 9, and 10 show that the valence chargedensities of Si, C, and Ge all exhibit local minimumalong the (ill) bond axes. This feature is very pro-nounced in diamond; however, its subtlety in Si andGe led to much debate in the theoretical literature.We now know from experiment that this feature isreal. Many early pseudopotential calculations on sili-con seem to have missed this feature, e.g. , Walter andCohen, 10 Chelikowsky and Cohen, 1 Ihm and Cohen, 7

Yin and Cohen, { ) Hamann, Zunger and Cohen,81and Denteneer and van Haeringen. More recent pseu-

dopotential calculations (using more accurate pseudopo-tential and better converged calculations) have predicted

tivistic [part (b)] and nonrelativistic orbitals [part (c)] inthe fit. Table X shows a quantitative comparison at somepoints. The overall shapes of p„(calc) and p„(model) arevery similar. The small negative portions of p„(model)near the interstitial sites (see also Table X) are unp ys-ical. They probably reflect a too high F, p&(ill) value.These negative regions have, however, a much smalleramplitude (by a factor of 2) in the relativistic fit whencompared with the nonrelativistic fit. As a result of thenegative segments of p„(model) in the interstitial sites,the normalization of p, (model) in the other regions mustbe too high. Indeed, the model values are higher thanthe ab initio value at the bond regions (points p andb, see Table X). Clearly, better experimental data areneeded for Ge before a definitive model can be adoptedand compared with theory.

0.2 -(a)0. 1

I' ' I I I

Apt{Calc} C)

1

Si '=-' =-' SII

.-=:, Si;=-..I

0.2C)

0. 1

-(b)

a PI I I (

hpt{ M odel)

I i I i I i I


FIG. 11. Comparison of the experimentally deduced staticd nsity deformation plots Ap~ ~(r) [Eq. (18)] of Si (b) withensi your ab initio calculation (a). The contour step is 0.025eDashed contours denote negative Ap.

Figure ll(a) compares our ab initio calculated defor-mation charge density of Ap«[Eq. (18) for G —+ oo] forSi with the model density. Table XI gives a quantitativecomparison at some points. The overall shapes are simi-lar: pt q is perpendicular to the bond, has a bond-centermaximum (point b) and a minimum at the back-bond

osition o. and a shallow negative value in the intersti-tial sites. The shape near the atoms (point P) [best seenin the line plot of Fig. 11(b)] are, however, significantlydifferent. The model shows sharp peaks at the P pointswith pronounced minima on the atomic sites. These fea-tures are unmatched by any Fourier synthesis (Fig. 6) andmay reHect the difhculty in reproducing the complexityof a realistic Apt & within the arbitrary restricted rep-resentation for RI(r) used in Dawson's model [Eqs. (6a)and (6b)]. It is remarkable, however, that despite theclear insuKciency of the Cummings-Hart-Deutsch set ofrnomenta to describe p I or Apr, q in a Fourier representation (see Figs. 6 and 7), the model density -approachof Eqs. (5)—(7a) mimics very well the overall results ob-tained from a highly converged Fourier series.


8. Deformation mays for diamond

Figure 12 compares the ab initio calculated static den-sity deformation map Apq t(r) [Eq. (18)] of diamond(a) with two experimentally deduced static density de-formation plots: that obtained from the TTKS data[Fig. 12(b)] and from the GW data [Fig. 12(c)]. Ta-ble XI gives a quantitative comparison between the abinitio result and model Ap extracted from the TTKSdata. The overall agreement is considerably better withthe TTKS data, as noticed also in comparing calculatedand measured structure factors (Sec. VI A3). Notice inparticular that the amplitude of the GW results are toohigh at the bond center, and that Ap in this model hasthe wrong slope in the interstitial region. The shapesnear the atomic sites are very different in the two modeldensities. This deserves further comment: we see fromEq. (21) that if the orbital deformation parameter r„I is1, AP is given just by the t g 0 terms RsK3 + B4K4.The conventional functional choice of AI(r) is such thatBIyp(r = 0) = 0 [Eq. (6)]. Hence, this choice forcesAIb(r) to have zero amplitude on the nuclei. On super-posing AP over unit cells [Eq. (20)) one finds some am-plitude on the atomic sites. These amplitudes result onlyfrom the penetration of the tails of RI(r) from sites i g jonto site j. This choice of a restrictive functional fIexi-bility of RIgp(r) limits its ability to capture fiuctuationsat the nuclear sites. Figure 6 further illustrates that theamplitude of Ap(r) on the nuclear sites develops onlywhen high G components are included. The GW dataset, having higher G components than the TTKS set,shows indeed more structure near the atomic sites. We

conclude that the currently used model-density approach[with r„I = 1 and RIyp(r = 0) = 0] lacks sufficient func-tional fIexibility to resolve the structure of Lp near thenuclei.

8. Deformation mays for germanium

Figure 13 compares our ab initio calculated Ap(r) ofGe with the results of the model fit using relativistic [part(b)] and nonrelativistic [part (c)] orbitals in the fit. Wesee that relativistic effects change the bond-elongatedAp [Fig. 13(c)] to a more spherically shaped bond Ap[Fig. 13(b)] in much better agreement with the theory[Fig. 13(a)]. Furthermore, the dips denoted by n and P inthe theoretical density plot along the bond in Fig. 13(a)show up only in the relativistic model fit, but not in thenonrelativistic one. Their positions also agree rather wellwith those of the model. All these indicate that relativis-tic effects have significant contributions to the electronicdensity of crystalline germanium.

G. Subjecting the Si ab initio results to model fits

Another way of comparing the ab initio results withthe experimentally deduced model values is to subjectboth sets of structure factors to the same model analysis.One then fits the set of calculated static structure factors(p, I,(G)) by the model (naturally, with p = B = 0) andobtains the fit parameters (r„I,AI, () of Eqs. (5) and (6),which can be compared with those obtained by Btting the

0.6 -(a)0.40.2

hpt(ca I c)

-0.20.6

0.40.2

~ -0.2

aI

hp (Model)-

0.6

0.40.2

0-0.2

=(c)

+---&------ --9 I t---------& I I-

C CI I

-2 0 2Distance along (111) (A)

hpt(Model)-

GW

FIG. 12. Comparison of the ab initio calculated staticdensity deformation plots b pt, t(r) [Eq. (18)] of diamond (a)and the experimentally deduced static density deformationplots of TTKS (b) and GW (c). The contour step is 0.05e/A. .Dashed contours denote negative Ap.

0.30.2

0. 1

-0. t

0.30.2

0. 1

L

0

~~ -0. 1

0.30.2

0. 1

II

I

-(a)

I

-(b)

~ I

— (c)

h pt(ca lc)

I' I ' I

hp (Model)

Ge

I

-2 0 2 4 6Distance along (111) (A)

I IG. 13. Comparison of the ab initio calculated static de-formation charge density (a) p & [Eq. (17a)] of germaniumwith the (b) and (c) model density. Part (b) shows the modelfit using the relativistic (R) orbital, while part (c) shows themodel fit using nonrelativistic (NR) orbitals. The contourstep is 0.025e/A .


(Fe„&t(C)) to the same model. The interest in this pro-cedure stems from the fact that it subjects both sets ofstructure factors —calculated and measured —to the sameanalysis. Any restriction in the model (e.g. , limited func-tional flexibility near the nuclei) will be equally reflectedin both fits. Note that p, I,(C) has no C-dependent sys-tematic deviations from the measured values where thoseare available. We will assume that the p«Ic(G) continueto be accurate also for the high-order structure factorswhich are at present outside the reach of accurate exper-iment. (This is reasonable since the higher-order termsincreasingly reflect core contributions. ) One can, there-fore, monitor the results of the fit of p«Ic (C) as a functionof C „. This allows one to separate model-dependentfeatures from those depending on the accuracy of the"measured" set of structure factors.

Figure 14 shows how the parameters g, As, A4, and[Eq. (6)], deduced from fitting p„I,(G) of Si to

the model, depend on the number of calculated struc-ture factors used in the fit. The curves in Fig. 14 con-verge after about 120 terms to their limiting values of( = 2.806 a.u. , As = 0.266e, and A4 = —0.062e. Wesee that as Gme„ is increased (i) the exponent g increases(i.e. , the radial charge density becomes more localized),(ii) the coefficient As decreases (i.e. , the antisymmet-ric component of the charge density diminishes), and(iii) A4 becomes less negative (i.e. , a lower centrosym-metric charge component). Similarly, (iv) r,I, startingout at small Gme„as rveI ( 1 (orbital expansion) in-CreaSeS, and turnS tO r„eI ) 1 (Orbital COntraCtiOn) athigher G~~„values. The converged orbital deforma-tion parameters using a large number of p, I,(G) val-

ues are nco«(n = 2) = 1.0003, and v ei = 1.0124, ascompared to the experimental values deduced from the18 known structure factors r, „(n = 2) = 0.9949, and

~——0.9382. It is possible, therefore, that a model Bt

to a more complete set of measured structure factors willyield a vanishing expansion of the 2sp shell, and a small

( 1%) contraction of the valence shell, rather than thevalues found in the model, fits to a set of 18 reflections.(v) Limiting the fit to the 11 structure factors for whichtheory versus experiment comparison exists (Table IV),we find an R factor of 0.079 jo, i.e., about half of whatwas found for LAPW versus the experimental model fit.

(vi) Finally, we made also a series of fits to the calcu-lated structure factors including in the model two moreterms of the multipole expansion, i.e. , l = 0, 3, 4, 6, 7 inEq. (7a). No improvement was observed in the statisti-cal indices of the fits, and no significant variations weredetected in the derived parameter values, as comparedwith the t = 0, 3, 4 term fits. This is in good agreementwith identical observations for fits to the canonical set ofthe 18 measured structure factors.

Figure 15 shows the model static valence density alongthe (ill) direction for (a) the fit to the 18 experimen-taL F,„~t(G) values, (b) the fit to the set of 18 calcu-lated p«Ic(G) values, and (c) the fit to the 288 caLcuLated

pceI, (G) values. In all three cases, one first fits p(G) tothe model, then uses the fitted model parameters to con-struct the crystal density [Eq. (7) and Fig. 15]. One seesremarkable agreement between Figs. 15(a) and 15(b) aswell as with the density shown in Fig. 7(f) obtained with

I I 1 II

I I I II

I I I I4(a)

0.6 -(a)

0.4-0.2

II

I I I I I

(p„(model) (-

Exp'.18 G

I

COC)

-8 I I I l I I I I

I I I II

I I I I

(b)-

0I

0.6 Cb)~0.4-

II

'I

' I

[p„(model) [-

LAP%18 G

2.8I

2e6

2.4

2-20.4

I

I I I I

I ' I ' II ' I0.6 -(c)(p„(model) (-

LAP'ItIIt

0.2-Q)

0

—0.2-0 100 200 300

Number of G vectors

Aq

FIG. 14. The variation of the Si model fitting parameters:(a) valence expansion or contraction coefficient e, (b) coeffi-cient ( of Eq. (6b), and (c) octupole and hexadecapole termsA.3 and A4 mith the number of C values included in the fit.The structure factors fitted are static LAP' values.

0.2-

-2 0 2 4DIstance aIong (111) (A)

FIG. 15. This figure gives the model valence charge den-sity along the (111) direction by subjecting both the experi-mental and LAPW structure factors to the same model. (a)The fit to the 18 experimental F oq(G), (b) the fit to aset of 18 calculated p, l, (G) values, and (c) the fit to theGb;g ——(12, 12, 12) (288 G components) calculated p,», (G)values.


a Fourier series (but no fit). In contrast, Fig. 15(c) issignificantly difFerent. This means that: (i) if one usesa large Set of structure factors, the accurate result ob-tained by Fourier summation [Fig. 7(f)] is difFerent fromthat obtained from the model fit [Fig. 15(c)]. In otherwords, there is a loss of information in fitting a structurefactor set with high-C components to Dawson's model.This is related to the limited functional flexibility of themodel near the core, discussed above, and is evidencedby the large fitting error we find in Fig. 15(c). (ii) If oneuses a stnaller set of structure factors, a direct Fourierseries produces noise [Fig. 7(e)], while fitting it to themodel [Fig. 15(b)] filters out the noise. This reflects thefact that Dawson's model extrapolates the high-G com-ponents using a spatially smooth form. (iii) It followsfrom (i) and (ii) above that when a limited set of struc-ture factors is available experimentally, the best strategyis to fit it to Dawson's model (rather than Fourier sumit directly). To compare the resulting p„(r) with the-ory, one should either subject an equivalently small set ofp, I,(G) to a model fit [Figs. 15(a) and 15(b)) or Fouriersum a converged set of p«i, (C) [Fig. 7(f)]. Subjecting alarge set of p(C) to a model fit or subjecting a small setof p(G) to Fourier summation both produce significanterrors.

H. Static versus dynamic deformation densities

~ 2 1 I I I II

I I I0

0. 1

0

-0. 1—0.2

I s I

I

I i I s

I ' I

0. 1

0

~ -0. 1—0.2

-(c)0.1-

I I I I

lI I I I II

C

Gblg)—

-0.1-0.2

-(d)0. 1—

IC

Gblg)

-0. 1—I i I i I i I


0

o o

I I

o

O

0 0

The difFerences near the nuclei between the model-density approach and ah initio calculations for thestatic deformation maps (Figs. 11—13) suggest that per-haps dynarni c effects need to be considered. Zuo,Spence, and O'Keeffes conjectured that while high-momentum Fourier components could affect the shapeof the static deformation densities Apt~t, (r, G~~„), theywould be inconsequential for the dynamic deformationdensity AFi,q(r, G „), since the Debye-Wailer factorswould efFectively attenuate such high-G components.This suggested to them the pessimistic conclusion 3

that one cannot recover any information on high-order p,„~t(G) terms from a deconvolution [Eq. (4a)]of F,„~i(C). To test this hypothesis we plotted inFig. 16 the static Apt i(r, C „) [Eq. (18)] and the dy-namic AFt~t(r, G „) [from Eq. (13)] for two trunca-tions: G „=(331) [Figs. 16(a) and 16(b)] and Gb;s ——

(12, 12, 12) (288 G components) [Figs. 16(c) and 16(d)).This shows that (i) the shapes of Ap(r, C~~„) depend onG „ in this range. Both Ap(r) and EF(r) are affectedby this truncation, and (ii) Api i = AFt t at any of thesetruncations except that a reduction of the amplitude onthe nuclei is evident. This invalidates the conjecture ofZuo, Spence, and O'Keefe. 3 Clearly, the Debye-Wailerfactor exp( —GzB) does not decay fast enough to rendera high-order contribution to Ap(C) negligible.

The similarity of Ap«I, (r, G~~„) to KF«I, (r, G~~„)over much of the unit-cell volume implies that (i) thedifficulty in calculating ab initio dynamic charge densi-ties can be efFectively circumvented for many purposesby using the far simpler static densities. (ii) Since high-momentum Fourier components F(Gb;s) clearly affect

FIG. 16. Comparison of static (b.p) and dynamic (DF)total deformation density maps of Si for two Fourier trunca-tions: G = (331) in (a) and (b), and Gb;s ——(12, 12, 12)(288 G components) in (c) and (d). The contour step is0.025e/A3. Dashed contours denote negative values. Note thesimilarity of the static and dynamic maps for the same Cand the lack of convergence of Ep and AF for G „((330).

charge-density deformation plots (viz. , Fig. 6) and va-lence charge-density plots (viz. , Fig. 7), the current in-ability to measure F(Gb;s) with high precision poses areal limitation to our ability to accurately characterize&pi.i(r)

VII. SUMMARY AND CONCLUSIONS

(i) Model fits. The multipole expansion formalismwith the lowest-order centrosymmetric and antisymmet-ric terms (I = 0, 3, 4) accounts well for the measuredstructure factors. Fits of the model to both silicon andgermanium yield a goodness of fit of 1, exhausting theaccuracy of the available data. For diamond, the onlyhigh-accuracy set available to date yields a goodness offit of 4. Further progress awaits more accurate mea-surements for germanium, and the extension of the smallset of structure factors available for both diamond andgermanium. The much better and more complete setof structure factors available for silicon could neverthe-less be improved by accurate measurements of high-orderstructure factors.

(ii) Features observed in the fits. The charge distribu-tion in silicon and germanium are similar in general, and


differ from that of diamond. The following features areobserved. (a) A valence-shell expansion of a few percentis found in silicon and germanium, but not in diamond.(b) An 0.5% expansion of the core 2sp shell of sili-con. (c) Both the octupole (l = 3) and hexadecapole(l = 4) terms are populated in silicon and germanium,but only the octupole is populated in diamond. (d) Non-rigid thermal motion with different Debye-Wailer factorswas found in silicon, but not in the other two crystals.The silicon valence charge exhibits a much reduced ther-mal vibrational amplitude, as compared to the core. (e)Relativistic effects are noticeable for germanium even atthe present 20—30-millielectron level of experimental ac-curacy. (f) No evidence for an anharmonic term in theeffective potential was found for silicon and diamond. Forgermanium, by contrast, the fits indicate the existence ofa small but non-negligible anharmonic term.

(iii) Structure factors. Precise implementation of thelocal-density theory is able to reproduce all accuratelymeasured Si structure factors with a maximum error of

20 me/atom and often considerably better. R factorsfor 18 reflections are as small as 0.21% and the rms devi-ation is 12 me/atom. The agreement with the currentlyavailable data for C and Ge is poorer (an R factor of0.94% for C and 1.05% for Ge, respectively, and the rmsdeviations are 17 me/atom and 170 me/atom for C andGe, respectively). This reflects largely the inferior qual-ity of the C and Ge data relative to the Si data.

(iv) Valence and deformation densities. The valencecharge density p ~i(r) extracted from experiment are ac-curately reproduced by the local-density theory (Figs. 8—10 and Table X). While the features of the deforma-tion densities are well reproduced in the bond regions(Figs. 11—13 and Table XI), the shapes near the atomicsites are not. We suspect that this refl.ects a limitationin the choice of a simple radial function [Eqs. (6a) and(6b)] in the model fit to experiment.

(v) Trends in calculated p i and Aptot. Valence den-sity maps p„(r) (Fig. 1) exhibit a camel's back double-maximum structure on the bond in C, Si, and Ge. Itis more pronounced in C and Ge than in Si (note thenonmonotonic trend here relative to the position in thePeriodic Table). The bond charge in p i is oriented paratlel to the borid direction. In contrast, the deformationcharge density maps Aptot(r) and Ap»i(r) (Figs. 3 and4) are parallel to the bond direction in C but per@endicular in Si and Ge. p ~i(r) and Bpt~t(r) exhibit lo-calized features reHecting changes in the atomic nodalstructures by the crystalline environment (Figs. 4). Thisleads to a slow convergence of the respective Fourier se-ries (Figs. 6 and 7). Even the state-of-art set of reflectionsof Cummings and Hart is insufhcient for capturing suchfeatures in a Fourier synthesis. A similar situation existsfor GaAs, where the measured reHections are insuK-

cient to capture many of the features of Lp.(vi) Dazuson's modeL versus Fourier summation. When

a limited set of p(G) is available, a fit to Dawson's modelproduces a smooth density. This reflects the fact thatthe model extrapolates smoothly the high-G componentsoutside the set. A direct Fourier summation of a limitedset of p(G) can produce noisy p(G). On the other hand,when a large set of p(G) is available, Dawson's modelproduces a poor flt due to its limited flexibility to rep-resent rapidly varying densities near the core. A directFourier summation of a large (converged) set of p(G)produces a smooth p(r).

(vii) Static versus dynamic densities. The conjecture ofZuo, Spence, and O'Keeffe that high-momentum com-ponents will not significantly modify dynamic maps isnot supported by our calculations. We find that high-momentum components afFect significantly both staticand dynamic deformation maps (Fig. 16, although dy-namic maps show some broadening of the localized fea-tures near the nuclei). In view of the fact that 4p doexhibit localized features, this implies that the currentinability to accurately measure high Fourier componentsdoes affect the accuracy of the ensuing density maps.

(viii) Relativistic effects in Ge Usi.ng relativistic free-atom structure factors in the Ge fits improves the agree-ment between the model and experimental forbiddenstructure factors (Table IX), reduces the unphysical neg-ative valence density in the model (Fig. 10), and improvesthe comparison of structure factors with the ab initioresults (Table VIII). Our predictions of the structurefactors and density maps of Ge await more accurate ex-perimental testing.

(ix) "Forbidden" reflections Contrib. uted solely by thedeformations and anharmonic thermal motion, the mag-nitudes of these structure factors are sensitive measuresfor the quality of any calculation or model fit. Here,the LDA-calculated and, in particular, the model valuesare in very good agreement with experiment. For thehighest-order measured reflections, (442) and (622), theagreement is within a few millielectrons. This lends fur-ther support and credibility to both the model and theLDA calculations.

ACKNOWLEDGMENTS

We are grateful to j. Spence and S. Froyen for stim-ulating discussions on the subject, to M. Spackman foruseful comments, and for supplying Stewart's atomic or-bital for carbon and to Gila Zipori at Bar-Ilan for ex-pert programming assistance. The work of Z.W.L. andA.Z. was supported by the U.S. Department of Energy,OKce of Energy Research, Basic Energy Science, GrantNo. DE-AC02-83-CH10093.

S. Cummings and M. Hart, Aust. J. Phys. 41, 423 (1988).P. J. E. Aldred and M. Hart, Proc. R. Soc. London Ser. A332, 223 (1973).R. Teworte and U. Bonse, Phys. Rev. B 29, 2102 (1984).

T. Saka and N. Kato, Acta Crystallogr. Sec. A 42, 469(1986).M. Deutsch, Phys. Lett. A 153, 368 (1991).M. Deutsch, Phys. Rev. B 45, 646 (1992); see also 46,


607(E) (1992).R. W. Alkire, W. B. Yelon, and J. R. Schneider, Phys. Rev.B 26, 3097 (1982).P. Hohenberg and W. Kohn, Phys. Rev. 136, 8864 (1964).W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).' G. P. Walter and M. L. Cohen, Phys. Rev. B 4, 1877 (1971).J. R. Chelikowsky and M. L. Cohen, Phys, Rev. B 10, 5095(1974).

C. M. Bertoni, V. Bortolani, C. Calandra, and F. Nizzoli,J. Phys. C 6, 3612 (1973).A. Baldereschi, K. Maschke, A. Milchev, R. Pickenhain, andK. Unger, Phys. Status Solidi B 108, 511 (1981).D. R. Hamann, Phys. Rev. Lett. 42, 662 (1979).A. Zunger and M. L. Cohen, Phys. Rev. B 20, 4082 (1979).A. Zunger, Phys. Rev. B 21, 4785 (1980).' J. Ihm and M. L. Cohen, Phys. Rev. B 21, 1527 (1980).(a) M. T. Yin and M. L. Cohen, Phys. Rev. B 26, 5668(1982); (b) Phys. Rev. Lett. 50, 1172 (1983).H. Nielsen and R. M. Martin, Phys. Rev. B 32, 3792 (1985).D. J. Stukel and R. N. Euwema, Phys. Rev. B 1, 1635(1970).P. M. Raccah, R. N. Euwema, D. J, Stukel, and T. C.Collins, Phys. Rev. B 1, 756 (1970).C. S. Wang and D. M. Klein, Phys. Rev. B 24, 3393 (1981).R. Heaton and E. Lafon, J. Phys. C 14, 347 (1981).K. H. Weyrich, Phys. Rev. B 37, 10269 (1988).M. Methfessel, C. O. Rodriguez, and O. K. Andersen, Phys.Rev. B 40, 2009 (1989).H. M. Polatoglou and M. Methfessel, Phys. Rev. B 41, 5898(1990).R. Dovesi, M. Causa, and G. Angonoa, Phys. Rev. B 24,4177 (1981).C. Pisani, R. Dovesi, and R. Orlando, Int. J. QuantumChem. 42, 5 (1992).L. C. Balbas, A. Rubio, J. A. Alonso, M. H. March, and G.Borstel, J. Phys. Chem. Solids 49, 1013 (1988).M. A. Spackman, Acta Crystallogr. Sec. A 42, 271 (1986).

'J. E. Bernard and A. Zunger, Phys. Rev. Lett. 62, 2328(1989).Z. W. Lu, S.-H. Wei, and A. Zunger, Acta Metall. Mater.40, 2155 (1992).J. M. Zuo, J. C. H. Spence, and M. O'Keeffe, Phys. Rev.Lett, 62, 2329 (1989).M. Renninger, Z. Kristallogr. 97, 107 (1937); Acta Crystal-logr. Sec. A 36, 957 (1955).R. Brill, H. G, Grimm, C. Hermann, and C. Peters, Ann.Phys. (Leipzig) 34, 393 (1939).S. Gottlicher and E. Wolfel, Z. Elektrochem. 63, 891 (1959).

s R. J. Weiss and R. Middleton (private communication).A. R. Lang and Z.-H. Mai, Proc. R. Soc. London Ser. A368, 313 (1979).T. Takama, K. Tsuchiya, K. Kobayashi, and S. Sato, ActaCrystallogr. Sec. A 46, 514 (1990).T. Matsushita and K. Kohra, Phys. Status Solidi 24, 531(1974).

'S. L. Mair and Z. Barnea, J. Phys. Soc. Jpn. 38, 866 (1975).T. Takama and S. Sato, Jpn. J. Appl. Phys. 20, 1183 (1981).M. Deutsch, M. Hart, and S. Cummings, Phys. Rev. 8 42,1248 (1990).M. A. Spackman, Acta Crystallogr. Sec. A 47, 420 (1991).A. S. Brown and M. A. Spackman, Acta Crystallogr. Sec.A 46, 381 (1990).B. Dawson, Proc. R. Soc. London Ser. A 298, 264 (1967);298, 379 (1967).

R. F. Stewart, J. Chem. Phys. 58, 1668 (1973).4sR. F. Stewart, Acta Crystallogr. Sec. A 32, 565 (1976).

P, Coppens, T. N. Row GuRo, P. Leung, E. D. Stevens, P.J. Becker, and Y. W. Yang, Acta Crystallogr. Sec. A 35,63 (1979).Electron and Magnetization Densities in Molecules andCrystals, edited by P. Becker (Plenum, New York, 1980).E. Clementi, IBM J. Res. Dev. 9, Suppl. 2 (1965).R. I . Stewart, in Electron and Magnetization Densities inMolecules and Crystals (Ref. 50), p. 427.D. T. Cromer and D. J. Liberman, J. Chem. Phys. 53, 1891(1970).D. T. Cromer, Acta Crystallogr. 18, 17 (1965).

ssD. C. Creagh, Aust. J. Phys. 41, 487 (1988); M. Deutschand M. Hart, Phys. Rev. B 30, 640 (1984); 37, 2701 (1988).D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566(1980).International Tables for X Ray Cr-ystallography (Interna-tional Union of Crystallography, Kynoch, Birmingham,England, 1974), Vol. IV, pp. 71—98.J. Z. Tischler and B.W. Batterman, Phys. Rev. B 30, 7060(1984).J, B. Roberto, B.W. Batterman, and D. T. Keating, Phys.Rev. B 9, 2590 (1974).J. E. Jaffe and A. Zunger, Phys. Rev. B 28, 5822 (1983).

'J. M. Zuo, J. C. H. Spence, and M. O'Keeffe, Phys. Rev.Lett. 61, 353 (1988).In a previous paper (Ref. 6), Deutsch used another defini-tion for the deformation density, which included only thenonspherical components of the deformation, [the last twoterms in Eq. (21)].While the two definitions become identi-cal for K„~ = 1, we will use here only the standard definitionof Eqs. (20) and (21). Note that the forms (19)—(21) difFerfrom (17b) and (18) since the former avoid explicit Fouriertruncation effects.S,-H. Wei and H. Krakauer, Phys. Rev. Lett. 55, 1200(1985), and references therein.J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13, 5188(1976).J. C. Slater, Quantum Theory of Molecules and Solids(McGraw-Hill, New York, 1974), Vol 4.E. Wigner, Phys. Rev. 46, 1002 (1934).L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).K. S. Singwi, A. Sjolander, M. P. Tosi, and R. H. Land,Phys. Rev. B 1, 1044 (1970).O. Gunnarsson, O. Jonson, and B. I. Lundqvist, Phys. Rev.B 20, 673 (1979).L. C. Balbas, G. Borstel, and J. A. Alonso, Phys. Lett.114A, 236 (1986); L. C. Balbas, J. A. Alonso, and G. Bors-tel, Z. Phys. D 6, 219 (1987); L. C. Balbas, A. Rubio, J. A.Alonso, N. H. March, and G. Hostel, J. Phys. Chem. Solids49, 1013 (1988).J. L. Ivey, Int. J. Quantum Chem. Symp. 8, 117 (1974).R. N. Euwema, D. L. Wilhite, and G. T. Sturratt, Phys.Rev. B 7, 818 (1973).A. Zunger and A. J. Freeman, Phys. Rev. B 15, 5049 (1977).W. von der Linden, P. Fulde, and K.-P. Bohnen, Phys. Rev.B 34, 1063 (1986).R. Dovesi, C. Pisani, F, Ricca, and C. Roetti, Phys. Rev.B 22, 5936 (1980).

7rR. Heaton and E. Lafon, Phys. Rev. B 17, 1958 (1978).R. Orlando, R, Dovesi, C. Roetti, and V. R. Saunders, J.Phys. Condens. Matter 2, 7769 (1990).

9410 Z. W. LU, ALEX ZUNCzER, AND MOSHE DEUTSCH 47

M. T. Yin and M. L. Cohen, Phys. Rev. B 24, 6121 (1981).R. Jones and M. W. Lewis, Philos. Mag. B 49, 95 (1984).P. J, H, Denteneer and W. van Haeringen, J. Phys. C 18,4127 (1985).P. E. van Camp, V. E. van Doren, and J. T. Devreese, Phys.Rev. B 34, 1314 (1986).C. O. Rodriguez, R. A. Casali, Y. Peltzer, E. L. Blanca, andO. M. Cappannini, Phys. Status Solidi B 148, 539 (1987).J, R. Chelikowsky and S. G. Louie, Phys. Rev. B 29, 3470(1984) .

G. B. Bachelet, H, S. Greenside, G. A. Baraff, and M.Schliiter, Phys. Rev. B 24, 4745 (1981).H. Aourag, G. Merad, B.Khelifa, and A. Mahmoudi, Mater,Chem. Phys. 28, 431 (1991).T. Kotani, T. Yamanaka, and J. Hama, Phys. Rev. B 44,6131 (1991).

P. Trucano and B. W. Batterman, Phys. Rev. B 6, 3659(1972).D. Mills and B. W. Batterman, Phys. Rev. B 22, 2887(1979).M. L. Cohen and J. R. Chelikowsky, Electronic Structureand Optica/ Properties of Semiconductors (Springer-Verlag,Berlin, 1988).J. L. Martins and A. Zunger, J. Mater. Res. 1, 523 (1986).

2ln constructing this plot we have summed in Eq. (19a) over

as many lattice vectors R~ as needed to converge the sum.The corresponding plot in the Deutsch paper (Ref. 5) re-

stricted the sum to only 14 atoms so there are small dif-

ferences relative to Fig. 8(b) here. These underconvergenceproblems have been corrected in the later paper (Ref. 6).Y. W. Yang and P. Coppens, Solid State Commun. 15, 1555(1974).

electronic charge distribution in crystalline diamond

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