PHYSICAL REVIE% B VOLUME 20, 5 UMBER 10 15 NOVEMBER 1979

First-principles nonlocal-pseudopotential approach in the density-functional formalism. &L

Application to electronic and structural properties of solids

Alex Zunger* and Marvin L. CohenDepartment of Physics, University of California and Materials and Molecular Research Division,

Lawrence Berkeley Laboratory, Berkeley, California 94720

(Received 7 August 1978; revised manuscript received 21 August 1979)

We apply our previously developed first-principles nonlocal pseudopotentials (obtained for all atoms ofrows 1—5 in the Periodic Table) to study self-consistently the electronic structure of Si and Ge and the

transition metals Mo and W. For Si and Ge we find that the first-principles pseudopotentials yield valence-

band states in good agreement with the empirically adjusted pseudopotential and photoemission data,

whereas the low conduction-band states appear to be consistently lower in energy due apparently toincomplete cancellation of the self-interaction effects. The calculated x-ray scattering factors (obtained by

core orthogonalization of the pseudo-wave-functions) are in excellent agreement with experiment. The self-

consistent valence charge density shows a distinct elongation of the covalent bond along the internuclear

axis, in good agreement with the experimentally synthesized density. The systematic deviations of the

empirical pseudopotential results from the present are discussed in terms of the underlying differences in the

potentials in the high-momentum regions. Using a mixed Gaussian-plane-wave representation, we calculated

the self-consistent band structures of Mo and W, and compared them with the available augmented-plane-

wave results. We find good agreement in the internal structure of the d bands, however, the present non-

muKn-tin self-consistent calculation yields substantially different s-d and p-d splittings. The bonding

characteristics and Fermi surfaces in these materials are discussed. Finally, we show that these first-

principles potentials provide a topological separation of both the octet 2"8 ' " and the suboctet 3"8(3 & P & 6) crystal structures. It is concluded that the presently developed pseudopotentials can be

successfully used for studying both the electronic structure of wide range of materials and structural

properties, without resorting to empirical parametrization.

I. INTRODUCTION

In aprevious paper" (hereafter referred to as I),we have presented a gene ral m ethod for obtainingfirst-principles atomic nonlocal pseudopotentialsin the local-density-functional (LDF) formalism bydirect inversion of the corresponding all-electroneigenvalue equations. We have computed these po-tentials in numerical form for 68 transition andnontransition elements from the first five rows ofthe Periodic Table and discussed the chemical reg-ularities reflected in their form. We have shownthat these potentials accurately reproduce theatomic energy eigenvalues, total energy differ-ences, wave-function moments, and valence chargedensities of the underlying all-electron density-functional formalism over a wide range of excita-tion energies. These potentials contain no empir-ical data and no adjustable parameters. They areobtained in closed form in terms of the micro-scopic constructs of the all-electron problem,such as the repulsive Pauli force, orthogonalityhole potentials, and valence screening. As such,they should offer greater insight into the reasonsfor both the successes and the failures of pseudo-potential theory in interpreting observed quanti-ties, relative to the empirically adjusted modelpseudopotentials.

In the present paper we apply the first-principles

pseudopotentials to the study of electronic proper-ties as well as phase stability and structural prop-erties of solids. As the valence electronic struc-ture of solids is predominantly sensitive to the low-momentum (q =2k~, where kz is the Fermi mo-mentum) components of the effective crystal poten-tial whereas the crystal structure and total ener-gies are determined also by the high-momentumpotential segments, a combined study of theseproperties should offer the opportunity to examinethe quality of the first-principles pseudopotentialsin an extended momentum range.

II. APPLICATIONS TO BAND-STRUCTURE CALCULATIONS

A. Methodologies of construction of first-principles densityfunctional pseudopotentials

The details of our approach for the construction.of the density functional nonlocal pseudopotentialsare given in I. Here we briefly indicate the under-lying methodologies and discuss some of the im-plications of our approach.

1. Density functional pseudopotential

Let Qr„","(r)}and (e„","j indicate the occupied core(c) and valence (e) wave functions and energyeigenvalues, respectively, obtained from the all-electron (core and valence) density-functional

4082

20 FIRST-PRINCIPLES. . . . II. . . . 4083

(DF) equations for an atom in. the electronic con-figuration g. We intend to construct a suitabletotal energy expression and single-particle eigen-value problem which applies to the valence sub-space alone and is defined sofely by the DE orbitalsPace.

We first set a DF total energy expression whichdescribes the energy of a fictitious (pseudo) atomhaving only N„(valen ce) electrons with density n(r)and interacting mutually via the interelectroniccoulomb V„[n (r)] and exchange-correlationV„,[n (r)] potentials akin to the DF approach, aswell as with a yet unspecified external potentialV, (r). Using a DF variational approach we willderive from this a corresponding single-particle(pseudopotential) equation with eigenvalues e„, andorbitals y„,(r). We will impose a number of physi-cally de sir able constraints on &„,and y„,(r) via a La-grange-multiplier technique and use them to solvefor the unknown pseudopotential V, (r) ~which satis-fied them.

The constraints are (i) the one-electron spectruma„, of the pseudopotential problem equals the val-ence spectrum c"„,of the all-electronproblem, forthe chosen reference electronic configuration g;(ii) the pseudo orbitals y„,(r) be normalized andnodeless for each of the lowest angular symme-tries; (iii) the pseudo orbitals X„,(r) be given asa rotation in the all-electron "true" orbital space'Q„","(r)], i.e., X(r) =g„,C„"„g):",(r) The .coeffi-cients fC„"„',J are then chosen to minimize the ma-trix element (}t„,~P, ~X„,) of the core projectionoperator P,. This leads to a y.„,(r) which has aminimal amplitude in the core region and maxi-mum similarity, possible within the DF orbitalspace, to the corresponding "true" valence orbital$",(r) in its chemically important "tail" region.

Condition (i) assures that the one-electron spec-tral properties obtained in the pseudopotentialrepresentation match those of the underlying all-electron system. Condition (ii) will allow us (cf.Sec. II C) to expand the pseudo-wave-functions in asimple and spatially smooth basis set, relievingus from the necessity, characteristic of the all-electron approach, to accurately describe the com-plex and strong spatial variations of the mave func-tions near the core region which is usually inertto chemically and physically interesting interac-tions. Condition (iii) establishes a simple unitaryrelation of X(r) to the all-electron orbital space,which mill allow us, if so desired, to recover the"true" orbital g, (r) from the pseudo orbital, sim-ply by core orthogonalization (see Sec. IID andTable II). The maximum similarity constraint ofIt„,(r) to g, (r) in the tail region as well as the min-imum core amplitude will assure us that the chem-ical information characteristic of (t)„"~(r) will be

contained in X„,(r) and that this will continue to beso, to a good approximation, even if the pseudo-potential is to replace the core electrons in 'chemi-cal environments other than those used to deriveit. Stated formally, we are lowering the energydependence of the pseudopotential by minimizingthe orthogonality hole through a core-projectionminimization. Physically, we mean that althoughwe derive the pseudopotential from information ona certain atomic electronic state g, this potentialwill be transferable to within a good approximationto different chemical environments (e.g. , excitedatoms, molecules, solids, surfaces). This ex-pectation is confirmed by our calculations, "where we find that a DF pseudopotential derivedfrom the atomic ground-state orbitals continuesto yield very precise orbital energies (i.e., errorsof less than 0.1 eV) when applied to excited con-figurations over a 25-eV range of excitation en-ergies.

The calculation of any physical observable from

X„,(r) rather than from the "true" wave function)t)o, (r) will inevitably include errors due to theelimination of the nodal structure from }t„,(r). Thecorrections to X„,(r) are simply described in the

pr esent s chem e by the core-orthogonality term s(X

~

score)ycore.

}t„,(r)= $ C„"„',(j„;",(r), (»)n

t)" (r) = „»„,(&) —F„(» Ie-")e-"I .n, n core

Note that as X„,(r) may have different forms in thecore region when the atomic core is placed in dif-ferent chemical environments or electronic states,the error &=Z„„(x~g"") is state dePendent. Thepseudopotential derived from }t(r) is hence energydependent and may not be transferable from onesystem to the other. This is treated in the DFpseudopotential approach by directly minimizing4 via a core-projection-minimization criterion,and at the same time enforcing a maximum simi-larity constraint of }t„,(r) to tg, (r) outside the coreregion. Note that the extreme limit of this mini-mization can be achieved by zeroing X„,(r) and itsfirst derivatives at the origin, leading na,turallyto small (X

~

(I)""). Since the leading terms in thepseudopotential are of order X"(r)/X(r), the choiceof a pseudo-wave-function with a limiting behaviorof Iim„g(r) =acr~" a+,r~"" a+,r~"" '+for anyX ~ 2 leads to a "hard" pseudopotential withlim (rV)= n)rand n, &0 and an exceedingly smallenergy dependence. Choosing X= 0 one obtains a"soft" potential with limV, (r) = const, higher am-plitudes of X„, in the core region, and larger cor-rections A. One can obtain a small core overlapwith the choice of X=O if the pseudo-wave-function

4084 ALEX ZU NG EB, AND MAR V IN L. COH EN 20

is allowed to be nonmonotonic (e.g., a spike at theorigin followed by a low amplitude). Such a pseudo-wave-function would, however, require a moreextended basis set for its expansion in electronic-structure calculations. A central point in the DFpseudopotential approach is that one can generatecontinuously "softer" pseudopotentials by thechoice of {C~'„',}, compromising thereby with (i)the range ~ (between a cutoff point R, and infini-ty) at which X„,(r) is equal to the "true" wave func-tion and (ii) the energy dependence of the pseudo-potentials. Although there exist a parameter rangewhere a reasonable compromise can be achievedbetween softness and accuracy (see Sec. IIC), weusually prefer the well-defined limit of hard po-

tentialss.

Having established a set of physically motivatedconditions on e„, and )t„,(r), we apply a constrainedvariational treatment to the pseudopotential total-energy expression "and solve for V, (r) in termsof (e„",", g„",", and C„"„'.}. We refer to this as thedensity functional Pseudopotential to emphasizethat it is constructed from the DF orbital spacealone. The closed-form result (Paper I) has aphysically transparent form interpretable in termsof an analytical Pauli repulsive term replacing theorthogonality constraint, a screened core poten-tial, an exchange-correlation nonlinearity term, ,

as well as a Coulomb and exchange-correlationorthogonality hole potentials. In I, w'e have cal-culated V, (r) for 68 atoms and demonstrated quan-titatively that the desired similarity to the all-electron results (i.e., in the ground-state andexcited-state wave functions, orbital energies,and total energy differences) is achieved to withina very good accuracy. Similarly, the low pseudo-potential energy dependence and the close simula-tion of chemical trends across the periodic tableare demonstrated. Applications to diatomic mole-cules, transition metals, and their cohesive prop-erties have similarly shown very good results.

2. Trans density functional pseudopotential

The construction of &„,(r) in Eq. (1a) from a line-ar combination of the occupied DF orbital spacewill always lead to a finite difference between

X„,(r) and g„",(r) asymptotically as r goes to infini-ty. As the DF space Qi„","(r)}is orthogonal, thecoefficient ~C„"„') multiplying the P„",(x) in thelinear combination is always smaller than unity,leading to lim„„X„,(r) = C„"„'g„",(r) rather than to

rg, (r) This is .an inevitable consequence of ourchoice to remain in this unitarily rotated DF space.These large-x discrepancies show up for instancein a deviation of 1%-3% in the pseudo-orbital mo-ment (X„,~r ~II„,) from the "true" moment (g„",~x ~g„",).Although small, such deviations may be noticeable

f„,(~) = —~&„,~~"'exp(-u„, r) . (3)

The constants A„„n„,are chosen in an iterativeprocedure so that conditions (i)-(iii) above remainvalid. This leads to a simple quadratic algebraicequation for A„, as a function of a„, whose solutionsatisfies the normalization of X„,. &„, is then de-termined by requiring X„,(x) to remain nodelessand that the pseudo orbital charge accumulationfunction Qr, (R)= f ~X„,(r)~'dr match the corre-sponding all-electron valence orbital accumulationfunction (starting from R = ~) up to the smallest Rvalue possible under the above constraints. Thisleads to X„,(x), which i s numerically equal to g„",(x)to a Point R, inward to the last radial maxima oftg, (x) and is somewhat more contracted in the coreregion relative to It„,(r). This guarantees that theelectrostatic potential outside B, is identical in theall-electron and the pseudopotential representa-tion. '" The orbital moments (j.„,~x ~X„,) for p= 1, 2, 3 are now within less than I%%uo from the val-ues obtained from g„",(r).

Note that, much like in the DF approach, onecan choose in Eq. (3) A. =2, obtaining thereby avery low-amplitude pseudo-wave-function in thecore region with its associated hard-core pseudo-potential and small energy dependence or, con-versely, one may choose X= 0 and obtain a softpseudopotential. We find again that a strong pseu-dopotential produces a larger range aR (between

R, and infinity) where X„,(r) can be made identicalto P„",(r), and hence prefer the choice X=2.

The modified pseudo orbital g„,(r) is used exact-ly as before'"" to obtain the pseudopotential V, (r).We refer to this as the trans density functional(TD F)pseudopotential to emphasize that an orbitalcomponent lying outside the DF space is needed for its

in solid-state applications for crystals with an"open" structure such as Si, where the lar ge-r be-havior of the orbitals contributes to the bond

charge density. To correct for this small differ-ence, one has to go outside the DF orbital space.We chose to do that in a controlled way by addingto X„,(r) an analytic component f„,(r) which willpermit C„"„'-=1. The modified orbital )(„,(r) is

y„,(r)=p Q c„"„'.g'„",;(r)+ f, (r)I +c,(r), (2a)n'&n

with

core

Nf„,(r), (2b)

where N is a normalization constant. Any regularfunction f„,(r) which is nonorthogonal to (g,"}anddecays vapidly to zero at large t' [so that

lim„„j„,(x) =g, (r)] is adequate. We use the sim-ple choice for the cutoff function:

4085

3. Analy ticaIly continued orbita1 approach

In this approach one abandons the simple physi-cal description of the pseudo-wave-functions as alinear rotation in a core and valence orbital space,and represents it instead by an arbitrary and nu-merically convenient functional form. One possi-bility is to set the pseudo orbital y„,(r) to b'e equalto the valence all-electron orbital tg, (r) from r = ~to r =R, and then analytically continue it to r = 0 ina smooth form:

( )g„",(r); r&R,

(4a)F„,(r); r-R, .

Here E„,(r) can be

(r) a rx+l+Inf fl (4b)

where X is a constant. The coefficients a„are de-termined from continuity requirements F„(R,)

construction. We apply V, (r) (Fig. 1)for bulk Si inthis paper. It seems that the DF pseudojotential, be-ing conceptually and arialytic ally simple, will suf-fice for most applications. When, however, greaternumerical precision is required (e.g. , total energycalculations for opened-structure system), theTDF pseudopotential should be preferred. A de-tailed comparison of the DF and TDF pseudopo-tentials is given elsewhere. 'b

Both the DF and TDF pseudopotentials have sim-ple and often desirable analytic properties. Inparticular, they allow us to recover to within agood approximation the "true" wave functions sim-ply by using Eqs. (1b) or (2b) and the known coreorbitals (see Sec. IID). This permits a preciseassessment of the accuracy of the calculation ofphysical quantities from pseudo rather than from"true" wave functions. We note, however, thatthe performance of a core orthogonalization ismerely an oPtion afforded by the DF and TDF ap-proaches, and not anecessity, as both methodsexplicitly guarantee a minimal departure of theexpectation values of X from those calculated with

Our experience indicates, however, that evenif the need for core-orthogonality corrections isminimized to an extreme (possible within thephysically motivated constraints imposed), a num-ber of physically interesting properties (e.g., x-ray-structure factors) may be sensitive to suchcorrections.

The DF and TDF approaches allow fo.r a simplerecovery of the true wave functions because ofthe Iinear relations (la) and (lb) between the pseu-do-wave-functions and a canonical orbital space.If these properties can be sacrificed, yet a thirdmethod for constructing first-principles pseudo-potentials is possible.

= („,(R,); F„',(R,) = if)„',(R,); E„",(R,) = g„",(R,); F„"',(R,)= g„",(R,) as well as from the normalization condi-tion J ~X„,~'dr =1. Such an approach has been re-cently used by Christiansen et al."in the Hartree-Fock scheme. In the LDF context, we use M=4,calculate the derivatives to high numerical accu-racy by a Spline scheme, and solve the resultingset of five algebraic equations as a function of R,.We seek the smallest R, value for which a node-less, normalized, and monotonic pseudo orbitalX„,(r) can be obtained. From this, the pseudopo-tential is obtained as in the DF and TDF schemes.We refer to this as the analytically continued pseu-dopotential approach. As before, the choice ofX = 2 leads to hard-core potentials while X = 0 lead sto soft-core potentials. The form (4b) is not uniqueand was chosen since it is the natural descriptionof a central-field atomic orbital at small r andleads to simple analytic forms for the derivativesand the normalization. Exponential forms forE„,(r) are possible as well. Note that the criticalfeature of E„,(r) which determines the strength ofthe corresponding pseudopotential is the firstpower of r which is larger than 1 in its small-rexpansion. If this power is 2 the potential is hard[lim„oF"(r)/F(r) = ~], while if it is larger than 2

the potential is soft [lim„,E"(r)/F(r) = constj.We have applied the analytically continued pseu-

dopotential approach to eight atoms. We find thatthe resulting potential is numerically identical tothe TDF potential from r= to r=R, . The centralresult is that the minimum Possible value of R, ispinned to a narrow range near the outer maximumof g, (r), much like the R, value obtained in theTDF aPProach. This simply indicates that thereexist a fixed nonzero charge Q„, (R,)=—Q„, (R,) re-siding in the core region of the pseudo orbitalwhich cannot be eliminated if a normalized,smooth, and nodeless orbital is required. TheTDF and the analytically continued. orbital ap-proach directly minimize Q~~(R, ) and henceachieve the largest spatial range of similarity be-tween the pseudo and true orbitals. The DF ap-proach, restricted to an orthogonal orbital space,typically yield Q~~(R, ) values which are 5%%uo larger.Hence, as long as one attains a matching betweenthe true and pseudo-wave-functions asymptoticallyas r goes to infinity, the relevant aspect of thedescription of X(r) at small r is the degree to which- given choice minimizes Q~,s(R,). In this respectthe hard-core pseudopotential is unique in that forany X~ 2 the potential starts off at small r as n, /r'and produces a smaller core charge, lower energydependence, and a larger region AR of identity be-tween the pseudo and true wave functions, relativeto a soft-core pseudopotential. Due to its small-rform of n, /r, the hard-core pseudopotential will

4086 ALEX ZUNGER AND MARVI1V L. COHEN 20

always have a crossing point r, (i.e., where thescreened pseudopotential crosses the r axis). InSec. III we show that these quantities form a uniquescale reflecting many of the chemical regularitiesof atoms in the periodic table and, in particular,correlating with the stable crystal structure ofcompound.

In this paper, we use the TDF pseudopotentialsfor Si and Ge and the DF potential for the closelypacked metallic solids. It will seem that whilethe TDF and the analytically continued orbital ap-proaches yield numerically accurate pseudopoten-tials, the basic conceptual framework for con-structing nonempirical LDF pseudopotentials iscontained already in the DF pseudopotential meth-od. Hence, while different experimentations withthe functional choice of f„,(r) [Eq. (2)], F,(r)[Eq. (4b)], and X [Eq. (4b)] lead to families ofpseudopotentials which may be tailored to specificapplications or expansion basis sets, the unam-biguous DF pseudopotential approach lends itselfto a simple and physically transparent interpreta-tion and to sufficiently accurate results for mostproblem s.

Qualitative comparisons with semiempirical potentials

The first-principles pseudopotentials developedin I have a shape significantly different from thatof both the empirical' and semiempirical modelpotentials that have been very successful in de-scribing the band structure of many solids. Wediscuss here the implications of these differenceson the electronic structure of atoms and solids.

Figure 1 depicts the core potentials of Si as ob-tained in the present first-principles local-densityapproach'b(full lines), the first-principles Hartree-Fock (HF) approach~ (dashed lines) and the semi-empirical model potential method' (dot-dashedlines). The latter potential is local and has beendesigned to fit the overall features of the bandstructure within a smooth form. It has been usedin numerous studies of the bulk, surface, ' inter-face, and vacancy problem and is similar in shapeto the potential developed by Appelbaum andHamann' for studying surface and chemisorptionproblems. The HF potential4 has been previouslyused for studying various silicon containing mole-cules and reproduces the all-electron ab.,initio HFresults very well. Clearly, the semiempirical po-tential differs substantially from both the first-principles potentials in the core region (the formerlacking turning points) and in the valence region,close to its minima. Since Si lacks an E= 2 corestate, its d potential is purely attractive overall space. These differences in the potentials can-not be dismissed as irrelevant to the valencestates of the atom as they have a maximum at the

D

pf0

I0O.

Si—Local density--- Hartree Fock——Empirical'(local)

-6p 1.p 2.p 3.p

Distance Ia.u. )

FIG. 1. Core pseudopotentials of Si as obtained in thepresent first-principles density-functional approach(solid line), the Hartree-Fock approach (Ref. 4) (dashedline), and the semiempirical approach (Ref. 3) (dot-dashed line). The fi.rst two potentials are nonlocal whilethe third is local.

points where the 3s and 3p pseudo-wave-functionsreach values of 6F/q and 94% of their maxima, re-spectively. Direct application of the semiempiricalpotential to the self-consistent calculation of theSi atom indeed reveal discrepancies of 0.60 and0.70 eV in the 3g eigenvalues for the ground andsingly ionized configurations, respectively, rela-tive to the "exact" all-electron results obtained byusing an identical exchange functional with e= 3.The errors in the orbital moments are 6% and 8%,respectively. Hence, since the semiempirical po-tential approach attempts to fit the overall bandstructure with a soft-core or smooth form, devia-tions for atomic structure relative to "exact" all-electron results exist. The corresponding devia-tions in the first-principles potentials are smallerby more than an order of magnitude. " Similar ac-curacy is enjoyed by the first-principles HFpseudopotentials. "The comparison shown herefor Si is also characteristic of other atoms (e.g. ,C, Ge) for which similar semiempirical potentialsare available. '

Inspection of the orbital kinetic energies of the Siatom discloses part of the reason for the differ-ences in results: the semiempirical potential re-sults in extended wave functions having about 3(P/p

less kinetic energy than the corresponding first-principles-derived wave functions. Although thiscauses significant errors in describing atomiclikeconfigurations, one expects that the differences be-tween these potentials in the valence region wouldhave a smaller effect on the electronic structureof the solid. This stems from the fact that a largerpart of the wave function extends towards the bondcenter, forming a covalent charge buildup and

FIR ST-PB, IX CIPI. ES-. . . . II. . . .

thereby decreasing the relative importance of thekinetic energy in favor of the more crucial role ofthe potential energy. As all the potentials shown inFig. 1 are nearly identical at 3, distance corres-ponding to the bond center in bulk Si (2.22 a.u. ),one mould expect the corresponding valence-bandstructure to be roughly similar.

One similarly does not expect the significant dif-ferences in the potentials near the classical turningpoints to show up strikingly in the correspondingvalence-band structure as the steep portion of thepotential is sampled only by high-momentum trans-fer scattering events which do not affect the dis-persion of the bands near k~." The success of the"on the Fermi sphere" approximation" in describ-ing many of the features of the valence band struc-ture of semiconductors and the sufficiency of thefirst fem reciprocal-lattice-vector Fourier com-ponents of the potential to obtain reasonable bandstructures' "are in line with this notion. Fur-ther, when a model potential is constrained to fitonly the low-energy interband spectra, ' inclusionof high-momentum potential components often in-troduces linear dependence" and leads only tonearly rigid (k-independent) shifts in the positionsof the energy bands. One notes, how'ever, that thehigh-momentum components apparent in the first-principles potential (which are associated with itssteep portions near the classical turning points)might be of crucial importance for the study ofphase stabilities and lattice vibrations. " We willreturn to this point in Sec. III.

C. Practical considerations

The highly repulsive character of the first-prin-ciples pseudopotentials in the region of its classi-cal crossing points (cf. Fig. l) gives rise to longtails in its Fourier representation. While thisposes no practical difficulty when the electronicstructure problem is formulated in real space(e.g, using Gaussian" or exact numerical atomicorbitals" and real-space integration tech-niques" "), these long tails require a large num-ber of basis functions in reciprocal-space repre-sentations. "Although the recent advent in inte-gration techniques and orbital optimization meth-ods have made the real-space representations ofthe crystalline wave functions both-economic andaccurate, the reciprocal-space techniques arecomputationally simpler. Hence we investigatehere possible ways of suppressing the high-mo-mentum components of the pseudopotentials. "

We first inquire as to how sensitive the valenceelectronic structure is to the details of the repul-sive part of the potential. We use the behavior ofexcited atomic species as a probe to the response

of the system to modifications in the repulsive po-tential. We have used various forms of truncatingfunctions (such as Gaussian continuation of theform A,e &", parabolic continuations, and variouscombinations of hypergeometric functions) thatsmooth the effective potential with continuous firstderivative, requiring that such smoothing leave theenergy eigenvalues and orbital moments unchangedto within l%%u, over a typical excitation energy rangeof a rydberg. We have found that such smoothing ispossible with a wide range of truncating functionsprovided the classical crossing points r', and thepotential for r & 0 are untouched. Some examplesare given in Fig. 2 where the original effectivepotentials of C and Si are compared with the poten-tials smoothed in the inner core. We find, how-ever, that modifications of the turning points leadto rapid deterioration of the quality of the potentialsover the entire energy range due to the variationaltendency of the charge density to penetrate the coreregions where the Coulomb potential is attractive.These small-r smoothing procedures hence offeran approximate but practical way of employing thefirst-principles potentials in plane-wave repre-sentations. Note, homever, that the localized na-ture of these potentials outside ro (e.g. , for l =2 intransition metals) may still require a large numberof plane waves in its reciprocal-space representa-tion and is more naturally described by suitablylocalized real-space expansions (see below).

A di. fferent and equally effective method of sup-pressing the high-momentum components of thepotential is based on overmixing core characterinto the pseudo-wave-functions. "'0 If the wave-function transformation coefficients C„, „„con-structed to lead to a maximum similarity betweenthe nodeless pseudoorbital }t„,(r) and the true val-ence orbital g„,(r), are artifically scaled to includein)(„,(r) more core character than needed to satisfythis constraint, X„,(r) becomes finite atthe origin andthe repulsive part of the potential becomes smoo-ther." In the simple case of a first rom atom onehas

(lc)

e C2s, lg and C,', „are determined such thaty„(r) be nodeless and smooth (i.e., satisfy mini-mum kinetic energy) and possess the maximumsimilarity to (2,(r) possible under these con-straints. This leads to y„(0)=0 and a repulsivepotential. As C2 y is replaced by C,, „+5one ob-tains

~ X„(0)~&0 and. a resulting smoother potential.This is illustrated in Fig. 3 where the l =0 compon-ent of the screened carbon core potential is dis-played for various choices of 6. As core characteris overmixed, the node q, in V,«(q) moves outwards

4088

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i

d lfp p p

solids we avstructure of so

7P"representati. vent band calcula tions for a reptransition met

sisten

') 'll be presentdiatomic molecu

As the aim of iveloped poten-th tl dctions of the cu

orms of theclose predic i

'd economic otials rather tha n to provi ehed the effective poten-we have smoot e

~0.2 a.u. ) toe the n cleuser close tocurrentlyt n the numer

lo e exp o qf tio d g'r ies are

relative to the unsmooV respectively.

self- consistentlyW have used twoh onlocal ba p

db '" thwave an we wi o

pe method (e.g. , e .p

palence (screening ppseudopo -en '

x ansion coef icienp

= 0 1,. and 2 compo asf the wave functions is&2 character o

f interest. . The po

L&

ll in the energy

q o0bl in this sc

W'h ' "fx ansion. eof 0..1 eV or

h b

a i waves in a second-ore lus an a ision23 and-p

transition metals, with ad i ioder expansion.second-or e

20 FIRST-PRINCIPLES. . . . II. . . .

As this approach is costly due to the large num-ber of plane waves required, it was used primarilyto obtain a standard for converged results. Themixed-basis approach offers much improved con-vergence as the localized features of the wave func-tions are represented by suitably peaked Gaussiansbasis function while the remainder of the wavefunctions is expanded in plane waves. The Gaussianexponents are varied to obtain rapid convergencein the' plane-wave expansion. We obtain an accur-acy of better than 0.1 eV in the energy eigenvaluesfor W and Mo by using d Gaussian exponents of1.35 and 1.85, respectively, and 95 and 103 planewaves plus five Gaussians per atom, respectively.For Si, a s,p exponent of 1.50 a.u. ' was found tobe optimal. The mixed-basis representation is wellsuited for the pseudopotentials at hand, and since only

valence states are treated (e.g. , six electrons for W

and Mo), it constitutes a substantial simplificationover the all-electron approach to these materials(involving 42 and 74 electrons for Mo and W, re-spectively).

The calculations are carried to self-consistencyin a standard fashion, avoiding muffin-tin or othershape approximations to the potential. The chargedensity is sampled at six special k points in theBrillouin zone (I', K, I., g, E, and &) for thesemiconductors and 14 k points for the metals.For the latter we compute the density of states,Fermi. energy, and the wave-vector- and band-dependent weights at each iteration using the tetra-hedron integration scheme. '4 More details aboutthe method and its application to bulk Mo will be

given in a different publication. '

D. Si and Ge pseudopotential band structure and charge density

The self-consistent nonlocal pseudopotential band

structure of Si is given in Fig. 4. We have usedthe standard Kohn and Sham ' exchange coefficientof e = & and included the homogeneous electroncorrelation functional of Singwi et a/. ' The energyeigenvalues at high-symmetry points a.re given in

Table I where they are compared with the first-principles self- consistent orthogonalized-plane-waves (SCOPW) results of Stukel et al ,

"for n = as.

the semiempirical self-consistent local pseudo-potential" (SESCL), the empirical non-self-con-sistent nonlocal (ENSCNL) pseudopotentialm~ and

with the observed photoemission ' and optical"data. Of the four calculations compared, only theENSCNL' is not self-consistent as the effectivepotentia1. in this approach has a parametric formadjusted to reproduce the bulk interband transitionsand photoemission data and is not expressible in

terms of the charge density. It has also beenshown29 to give excellent agreement with the cyclo-

—.—2

I

—10

—14W

FIG. 4. Self-consistent exchange and correlation bandstructure of Si using the first-principles nonlocal pseudo-potential.

tron masses in Si arid the shape of the valencecharge density. Since it attempts to mimic the

geyee~ed potential in the bulk, it does not providea mechanism for charge redistribution and cannotbe used for Si-containing systems other than bulkSi (e.g. , molecules, surfaces, etc.). The semi-empirical pseudopotential SESCL has been usedmith slight modifications in the past for studies onthe Bi, -molecule, silicon surfaces, ' interfaces,and vacancies. '

Our results agree well with the experimentaldata for the occupied states and are in moderatelygood agreement with the first-principles SCQPWresults. The indirect I'» „-b, , band gap (0.5 eV)is, however, substantially smaller than the ob-served value of 1.13 eV. Although the agreementbetween the calculated gap and experiment can beimproved by treating the exchange coefficient as anadjustable parameter, '~ this approach will beavoided here. In general, the first-principles re-sults are as good as the empirically adjusted re-sults for the ground occupied states but are sys-tematically lower than both the empirical and theexperimental results for the excited states (e.g.,discrepancies of 1.65, 0.35,0.61, and 0.54 eV for the

I, „ I', „Ã, „and L„st teasrespectively). Wemill discuss this effect later on. As expected, thedifference between the results of the local pseudo-potential (SESCL) and the present nonlocal pseudo-potential are somewhat smaller than the corres-

4090 ALEX ZUNGER AND MARVIN L. COHEN 20

TABLE I. Comparison of the Si-band eigenvalues (in eV) of the present pseudopotential ex-change and correlation study (using a mixed-basis) with the SCOPW (Ref. 27) results (o = 3),the semiempir ical self-consistent local pseudopotential (o' = 3) (Ref. 28), the empirical non-self-consistent non-local (Hef. 29) and experiment (Befs. 30-33).

Level SCOP%I'resentresults SESCL ENSCNL Experimental

F( „F)5 „F2',cF15,cF(cI, „X4 „XiX41-2,vLj „&3,vLi,cL3,c

-12.040.003.312.33

-7.83-3.00

0.349.87

-9.63-7.14-1.26

1.393.12

-3.95

-12.200.002.502.487.25

-8.02-2.93

0.529,97

-9.92-7.21-1.28

1.133.36

-4.72

-12.820.003.382.705.95

-8.51-3.24

0.5111.62

-10.41-7.62-1.39

1.453.48

-4.20

-12.360.004.103.427.69

-7.69-2.86

1.17

—9.55—6 ~ 96-1.23

2.234.34

-4.47

-12.4 +0.6

4.15~ 0.05

7.6

-2.9,—2.5 +0.31.13 d

-9.3 ~0.4 '-6.4 +0.4, —6.8+ 0.2-1.2 +0.2

3.9 + 0.1-4.7 + 0.3

~ Reference 31.Reference 30.

Reference 32.Reference 33.

ponding differences in the free atom. Note, how-

ever, that the semiempirically adjusted SESCLpotential yields somewhat poorer agreement withthe observed valence-band states, as comparedwith the first-principles pseudopotential.

As the first-principles core pseudopotential isgiven as an infinite series of angular momentacomponents' Q, W, (r)P, (where P, is the projec-tion operator) and as W, (r) for the symmetrieswhich are absent in the core of the free atom issingular (e.g. , l ~ 2 in Si, cf. Fig. 1) it is impor-tant to assess the validity of the truncation of thisseries. One expects that only those angular-mo-mentum components which are present in the crys-talline wave function in an energy range of interestw'ould bg important. As the atomic central-field

symmetry is removed in the solid, states includingl» 2 componen'. s can be admixed into the occupiedvariational crystal wave functions. We have re-peated a self-consistent band-structure calculationfor Si, retaining this time only the l =0, I pseudo-potential components. We find rather smallchanges in the valence band spectra (e.g. , L, „,X, „, X~ „)along with more significant changes(0.3-0.8 eV) in some conduction band states (e.g. ,

L, „L,„I"„,andthe Z, line). It would henceseem that the higher l &2 corrections are unimpor-tant for the energy region below -5 eV above theconduction-band edge. As one moves down thePeriodic Table, one expects, however, the l ~ 2 com-ponents to become more important (i.e., As, Se)as the d states tend to be bound even in the free

24 24

FIG. 5. Total valencepseudocharge density inthe (110) plane in Si, ascalculated by the presentexchange and correlationnonlocal pseudopotential.The full dots indicate theatomic positions. The re-sults are given in units ofelectron/ (crystallographiccell).

20 FIRST-PRIN CIPLES. . . . II. . . ~ 4091

atom.The total valence pseudo-charge-density of Si in

the (110) plane is depicted in Fig. 5. The valencecharge density obtained from the present first-principles potential differs from that obtained withthe SESCL potential in a few respects: The calcu-lation depicted in Fig. 5 yields a bond charge den-sity which is elongated parallel to the bond axisand decays rather steeply towards the atomic site,whereas the semiempirical potential yields a bondcharge that is oblate and decays rather slowly to-wards the atomic site. If one is arbitrarily to de-fine the covalent bond charge density as that en-closed by the outermost contour surrounding thebond, the present calculation yields an anisotropyfactor L,/L, (where I., is the length of the bond

charge parallel to the bond axis and L, is the. lengthperpendicular to this axis) of -1.3 while the SESCLpotential yields L,/L, =0.8 and the observed bondanisotropy ratio" is about 1.4. The highest valueof the valence charge density obtained in this workis 24.0 e/cell while that obtained with the semi-empirical potential is 24.2 e/cell and the observedmaxima is 23.3 e/celL35 We note, however, thatthe experimentally synthesized valence chargedensity is not strictly comparable to pseudopoten-tial calculations as (i) the. core density is simplysubtracted in the former case from the total densitywhereas in a pseudopotential methods the coredensity is projected out (i.e., the valence wavefunctions are deorthogonalized to the core, . leavinga smooth wav'e function), and (ii) the core densityis represented in the experimentally synthesizeddensity by HF (rather than the density functional)orbitals. These are usually more contracted thanthe local density orbitals. " Note that the localpseudopotential gives rise to a single charge dens-ity maxima or to closely spaced double maxima inthe occupied m„orbital of the diatomic Si, mole-cule, '~ whereas firgt-principles LDF" calcula-tions and the present pseudopotential predicta well-separated double maxima in either side ofthe interatomic axis.

Figures 6 and 7 show the valence pseudochargedensity of Si at few high-symmetry points in thezone. Both s-type (e.g. , I, „, X, „)and p-type(e.g. , I'» „, X4 „) states show a distinct chargepolarization along the bond axis, suggesting that itis not the potential nonlocality that is responsiblefor the formation of a bond-polarized charge(L,/L2 &1). A local approximation to our potential(i.e., neglecting the I &0 components of the poten-tial) produces indeed L, /L, = 1.2. In contrast, wefind that the SESCL pseudopotential produces I.,/L, & 1 for both the s- and the p-type states. Thepolarization of the bond charge density along thebond direction is a consequence of the more local-

ti,v

lal

25,V

tbl

FIG. 6. Symmetrized pseudo charge density in the(&10) plane of Si at the I'point in the zone, as obtainedby the present exchange and correlation first-principles(nonlocal) pseudopotential. Full dots indicate the atomicpositions. (a) The bonding Bs-type I'& „state, (b) thebonding BP-type triply degenerate I'2» state, (c) theantibonding Bs*-type I'~, state.

ized nature of the first principles potential (cf.Fig. 1}. It is important to note that the differences '

in the details of the valence charge polarization be-tween the first principles and the semiempiricat.pseudopotentials do not manifest themselves sig-nificantly either in the valence band structure norin the low-angle forbidden x-ray scattering factor(the F222] reflection being 0.19 e/cell in the pres-ent calculation and 0.25 e/cell with the semiempir-ical pseudopotential). It would seem that only thehigher-momentum components (q»2k+) of thefirst-principles pseudopotential determine the de-tails of the anisotropy of the charge density. Thesemight be important in determining the total energy,phonon spectra, or chemical reactivity but arehardly material for the one-electron observablesrelated to low-momentum scattering events.

It is interesting to note in passing that the chargedensities of the top and bottom valence bands in Sichange very little across the Brillouin zone (e.g. ,

4092 ALEX. Ztj NGER AND MARVIN L. COHEN 20

FIG. 7. Symmetrized pseudocharge density in the (110)plane of Si at the X point in the zone, as obtained by thepresent exchange and correlation first-principles (non-local) pseudopotential. Full dots indicate the atomicpositions. (a) The bonding 3s-type X& „state. (b) Thebonding 3p-type E4 „state.

I',„-X,„, I'» „-X4 „). The low dispersion of theband density implies that Brillouin-zone samplingtechniques based on few representative points" areaccurate for such systems.

Inspection of the charge density of the highestoccupied p-type valence band (e.g. , I'» „}and thelowest s-type conduction band (e.g. , I', „L,,)

suggests a reason for the remarkable sensitivityof the p-s gap (e.g. , I"» „-X,„ I'» „-I'...) to scal-ing of the exchange potential. " As the I, ,-F,states are strongly antibonding states having a. den-sity which is appreciably localized near the atomicsites [cf. Fig. 6(c)], they are particularly sensitiveto the enhancement of the exchange potentialo.'p'~'(r) (via increasing the exchange coefficient u).On the other hand, the I'» „valence band is a de-Iocalized 3p state fcf. Fig. 6(b)] occupying regionsof low charge density and is consequently signifi-cantly less sensitive to exchange scaling. The dif-ferences in the degree of localization of thesestates leads to marked changes in the calculatedband gaps as the exchange coefficient is increasedfrom 0.'=-', ." One further notes that the localizednature of the lowest s-type conduction band in Simight induce important self-interaction effectswhich are spuriously included in the density func-tional potential. 4 '. These arise from the imper-fect balance between the electronic self-Coulomband self-exchange interactions and have an appreci-

able magnitude for localized states both in atomsand solids. ~ These usually produce anomalouslysmall band gaps relative both to the self-interac-tion-compensated calculations" and experiment.Hence, although the difference in the degrees ofspatial localization of the valence and conductionbands in Si enables the adjustment of the band gapvia exchange scaling, the calculated anomalouslysmall band gap appears to have a different andphysically distinct origin, As the self-inter-action corrections vanish at the limit of extendeddelocalized states, the high plane-wave-like statesof Si are likely to be correctly described by thedensity functional approach. A direct approach tothe self-interaction corrections in the spectra ofsolids has been previously developed ', however,a similar application to Si is outside the scope ofthe present paper.

Another measure of the quality of the calculatedcharge density is furnished by comparison with theobserved x-ray scattering factors. As the pseudo-potential calculation produces nodeless valencewave functions, it is not clear whether these aresufficiently accurate for comparison with the all-electron charge density or with experiment even inthe valence region, ' 'O' Since our pseudopotentialformalism represents the pseudo-wave-functionsin the atomic limit as linear combination of thetrue all-electron core and valence wave func-tions, " it is possible to recover the nodal valenceorbitals to within a good approximation by performingan appropriate core orthogonalization. ' " The ef-fects of such an orthogonalization in the atomic limitare demonstrated in Fig. 8. Here we plot the differ-ence AE, (q) between the all-electron atomic scat-tering factor fp„„(r)e'"' dr and that computedfrom the valence pseudodensiiy n(r) plus the coredensity p,(r): f[n(r)+ p,(r)]e" "dr. H'ere p, (r)is constructed from the nodal core (c) and valence(v) orbitals. As both p„„(r)and n(r }p+,(r) arenormalized to the total number of electrons in thesystem, aE, (0) =0 and similarly AE, (~) =0 as thehigh-momentum components sample only the coredensity p,(r). At intermediate momentum values, theerror LE,(q) (solid line in Fig. 8) is seen to benon-negligible relative to the experimental errorsin determining the scattering factors of bulk44 Si(vertical bars in Fig. 8 appearing at the first fewreciprocal-lattice vectors of Si). Clearly, a directuse of the pseudocharge density instead of thecore-orthogonalized density produces some errorseven at low momenta. The curve marked AE2(q)denotes the similar differences produced by the localsemiempirical pseudopotential. ' These are largerby about a factor of 3 and localized in the regionwhere the physically interesting [111]reflectionoccurs. Upon core-orthogonalizing the first-prin-

20 FIRST-PRINCIPLES. . . . II. . . .

I.o

0.8 — n,

I~ y dF (q)t0.6 — ~ &w

I] i

F~ {qI y ~04 . III

~g—Ig

I(0.2 —I.

OP f-. 0)

ppl T

c3

-0.2(q)

-0.6

-0.8 I I 1

2 4Sin 8/k (a.u. ')

ciples pseudo-wave-functions, ~,(q) for the atomvanishes identically. "b The corresponding dif-

FIG. 8. Differences between the all-electron and thepseudopotential atomic x-ray scattering factors for Si.~& (q) denotes the difference obtained with the first-principles pseudopotential; ~2(q) denotes similar dif-ferences obtained with the semiempirical pseudopoten-tial; ' 4I"3(q) denotes the difference obtained when thesemi-empirical pseudo-orbital s are orthogonalized tothe core orbitals. The corresponding difference withthe first-principles-derived orbitals is identically zero.The vertical bars on the abscissa indicate the experi-mental errors associated with measuring the crystallinescattering factors at the first few reciprocal-lattice vec-tors (Ref. 44).

ferences in the bulk are not expected to vanishidentically as the orthogonalization is performedwith respect to the atomic frozen-core orbitals (ra-ther than the variational bulk core wave functions);however, based on our experience with core ortho-gonalizations of excited state atomic wave func-tions, ""we expect to obtain accurate results for thebulk. It is interesting to note that core orthogonali. -zation of the pseudo-wave-functions derived from thesemiempiricalpseudopotential method'(in which thepseudo-wave-functions are not representable ascombinations of the true orbitals), produces evenlarger differences [curve marked LE,(j)] relativeto the unorthogonalized results.

Table II shows the calculated x-ray scatteringfactors in bulk Si obtained by core orthogonaliza-tion of the pseudo-wave-functions. '~' The ortho-gonalization has been performed using a three-dimensional Diophantine integration method. " Theresults are compared with the al)-electron SCOPWcalculation of Stukel eg gl.2' and with experiment, 4~

corrected for the Debye-Wailer factor and anoma-lous dispersion. ' The present results agree verywell with the observed data, and even the sensitivefeatures such as the "forbidden" [222) reflectionand the order of the [333] and [555] ref lectiori(degenerate in the atomic superposition limit) arecorrectly reproduced. Our scattering factors aresignificantly larger than the SCOPW and the atomicsuperposition results" at low momentum, indicat-ing a more localized valence density in the presentcalculation. At high momentum, our results areslightly lower than the SCOPW results, possiblydue to the use of somewhat different core orbitalsin the present scheme which involve homogeneouscorrelation terms in the potential. The excellentagreement obtained here between the calculatedand the observed x-ray scattering factors suggeststhat the differences between our real-space charge

TABLE II. Comparison of the observed (Ref. 44) and calculated x-ray structure factors forSi (in units of electrons per crystallographic cell). The SCOP%' results (Ref. 27) are given forthe Kohn and Sham exchange. The present results include free-electron correlation correc-tions.

ExperimentalRef. 44b Ref. 44a

Presentresults

Atomicsuperposition SCOPW

111220311222400331422333511440

10.708.487.770.177.086.816.215.875.887.65

11.128.788.050.227.407.326.726.436.406.04

10.708.577.790,197.357.036.656.316.356.00

10.528.708.150.07.477.146.656.396.396.03

10.698.648.010.177.447.216.686.386.426.02

4094 ALEX ZUNGER AND NIARVIN L. COHEN 20

TABLE III. Calculated Fourier coefficients of theself-consistent Coulomb screening [Vc,„&(G)t, ex-change screening fV„(G)t, and correlation screeningfV (G)] for Si. Hesul. ts are in rydberg per unit cellvolume. p(0) is normalized to 8.0 electrons and h, k, ldenote the index of plane waves representative of thecorresponding stars.

Vcoul (G) V„(G) V (G)

111211221222220311332331422421411431443442

—0.1 50.64-0.001 54

0.008130.008 050.006 250.001 24

—0.001 76-0.001 24-0.000 92—0.000 55—0.000 31-0.000 08

0.000 21—0.000 09

0.050 760.007420.007 260.002 990.006 920.002 790.003 470.002 360.001 890.002 120.000 130.000 240.001 230.000 60

0.003 760.000 790.000 440.000 030.000 430.000 180.000 250.000 170.000 140.000 180.000 110.000 010.000 120.000 07

density (computed from the lowest 880 scatteringfactors) and the experimentally synthesized densi-ty" (obtained by inverting the Fourier series of thelowest 21 observed scattering factors) might orig-inate from limitations of the inversionproeedure. "

Table III depicts the first few Fourier compon-ents of the self-consistent screening field. In eachcase we give the transform with respect to a singleplane wave (with index k, k, f) representative of thecorresponding star of G. A few conclusions per-taining to the screening mechanism in Si emergefrom these results. With the exception of the groupof 12 reflections belonging to the [211] star, boththe exchange (V„) and the correlation (V„„)fieldsact to screen the electron-electron Coulomb (V c,„,)interactions (i.e., they have an opposite sign). Thisscreening is very effective at short wavelengthwhere

~ V, (G)/V&-, „,(G)( is larger than unity. TheFourier components of the screening potential aredivided into a few groups with decreasing impor-tance: the first contains the strongest [111]com-ponent (eight members), the second contains thesmaller (Vc,„,-0.005 Ry) components (four starswith a total of 56 members), and finally one findsthe higher components having lower contributions(Vc,„~ -10 '-10 ' Ry). The band structure appearsto be sensitive to variations of the first two groupsof components while changes in the high-momentumcontributions affect only the details of the chargedensity (e.g. , bond anisotropy). We similarly find

that the correlation screening constitutes onlyabout 5% of the exchange screening and decay veryrapidly at high momenta. Note that the ratioV«ir (G)/V„(G) varies considerably with G for the

TABLE IV. Comparison of the present exchange andcorrelation band structure of Ge with the SCOPW (Ref.46) and empirical (non-self-consistent) pseudopotentialcalculations (Ref. 47). Results are given in eV.

Empir icalPresent results SCOPW, pseudopotential

FF25 v

~ 2', cF15.c

X(&4, v

Xi~s.c

L2, v

Lg„&s,v

L &,cL3,c1-2,c

-12.360.000.772.59

—8.40-2.85

0.958.92

-10.09-7.24-1.28

0.653.957.60

12 Q 20.01.13.0

-8.4-2.8

1.310.2

-10.1-7.1

1 Q 20.94.27.8

-12.660.000.93.1

-8.65-3.29

1.16

-10.39-7.61-1.53

0.764.20

first two groups of waves and settles to about &.05for the high-momentum parts. This suggest thatone cannot simulate these correlation terms ef-fectively for the important low-momentum compon-ents by simply linear scaling of V„. For highercomponents, an exchange coefficient of 0.70 (in-stead of the Kohn and Sham value of —', ) seems tomimic the combined effects of exchange and cor-relation reasonably well.

We have performed an analogous study of theband structure and charge density on Ge, using ourfirst-principles pseudopotentials. As the chargedensity and the corresponding conclusions born outfrom the study on Si are qualitatively similar, wepresent only a brief discussion of the resultingband structure. Table IV depicts the calculatedband structure of Ge at high-symmetry points, ascompared with the SCOPW results of Hermaneg al. ' and the empricial pseudopotential ofChelikowsky and Cohen. 4'

The agreement between the present first-princi-ples calculation and the empirically adjustedpseudopotential ' and the SCOPW ' calculation isvery good for all except the low-lying antibonding4s band (I', , A;;L, ,) -where the present study re-veals systematically lower results. The same ef-fect occurs in our calculation on Si and was dis-cussed above. Anomalously low I'» „-I'.. .andI'», -&, , gaps have been previously obtained in

first-principles OPW studies ' where it was found

that slight empirical adjustments of the calculatedFourier components of the potential are requiredto bring these results into agreement with experi-ment. Our calculated valence charge density forGe is qualitatively very similar to that of Si. The

20 FIRST-PRINCIPLES. . . . II. . . .

bond anisotropy factor L,/L, decreases from thevalue of 1.3 in Si to 1.1, indicating an increasedtendency to form a metallic bond. A local (self-consistent) semiempirical Ge pseudopotential pro-duces a consistently lower bond anisotropy (L,/L,= I.O).

E. Applications to transition metals

As a further test to the quality and generality ofthe first-principles pseudopotentials we have ap-plied them to study the electronic structure of twotransition metals —molybdenum and, tungsten. Thedetailed description of the molybdenum results willbe presented elsewhere, "and we concentrate hereon the tungsten results.

The application of pseudopotential formalism tothe study of the electronic structure of transitionmetals is difficult. This is mainly due to the inef-fectiveness of simple plane-wave expansion tech-niques to describe localized d states, the pro-nounced nonlocality of transition-metal pseudo-potentials" and the lack of sufficient experimentaldata to parametrize either the single valence-elec-tron-ionic term values or the low-energy interbandspectra. The present approach circumvents thesedifficulties by using a flexible mixed-basis repre-sentation capable of treating nonlocal potentialsand accurately describing both the localized fea-tures of the d states and the extended character ofthe s-p states. Further, no use is made of experi-mental fitting techniques in the present first-prin-ciples approach.

The establishment of the self-consistency of thecrystal potential poses special problems in metals.Whereas in semiconductors and insulators one canapproximate the variational crystal charge density

by sampling the square of the symmetrized wavefunctions at few representative points k, in theBrillouin zone (BZ) using equal weights for all theoccupied bands at a given point k, ,

' the occurrenceof a Fermi surface in a metal requires explicitconsideration of both wavevector and band depen--dent weights W&(k, ). This results from the fact thatthe BZ occupation volume depends on the details ofthe crossing of the Fermi surface by a given band.We treat this problem by sampling a grid of equallyspaced inequivalent k points in the irreducible zoneand computing at each iteration step the density ofstates (using the tetrahedron interpolationscheme24), the Fermi energy, and the fractionalvolume of each minitetrahedron that lies under theFermi surface at each band. These band-dependentweights vary at each iteration step with the shapeof the Fermi surface. They reflect both the geo-metric factor associated with each k,. point (i.e. ,nearest BZ volume occupied by k,.) and an addition-

' al factor, absent for nonmetals [in which W,.(k, )reduces to W(k, )]which measures the occupancy ofvolume under the Fermi surface. .The samplingweights obtained for 44 principal k,. points in tung-sten at the self-consistency limit are depicted inTable V.

It is seen that W&(k) is band independent for thelowest two valence bands (which are removed fromthe Fermi surface) and is a decreasing function ofthe band index for the higher bands which cross theFermi surface. The weights of the two lowestbands increase initially as one moves away fromthe I'points towards H, N, or I' due to the in-creased fractional BZ volume associated with thelatter points, while the variation of the weights ofthe highest bands with wavevector also reflect thevicinity of the band to a Fermi surface area.

TABLE V. 14-k-point band and wave-vector-dependent weights &&(k) for the self-consistenttungsteri band structure. %eights are normalized to the number of valence electrons per cell(6.0). All weights for j &4 are zero.

k point

1 (0.0, 0.0, 0.0)2 (0.25, 0.0, 0.0)3 (0.25, 0.25, 0.0)4 (0.25, 0.25, 0.25)5 (0.50, 0.0, 0.0)6 (0.50, 0.25, 0.0)7 (0.50, 0.25, 0.25)8 (0.50, 0.50, 0.0)9 (0.50, 0.50, 0.25)

10 (0.50, 0.50, 0.50)11 (0.75, 0.0, 0.0)12 (0.75, 0.25, 0.0)13 (0.75, 0.25, 0.25)14 (1.0, 0.0;0.0)

W, (k)

0.031250.125 000.125 00

. 0.187500.125 000.250 GG .

0.312 500.093 750.218 750.062 500.062 500.218 750.156250.03125

W2(k)

0.031 250.125 GQ

0.125 000.187 500.125 000.250 000.312 500.093 750.218 750.062 500.062 500.218 750.156 250.03125

S'3(k)

0..031 250.125 000.120 2910.187 500.120 2000.241 0790.302 9100.0794550.208 6700.062 5000.038 3190.184 9780.132 0690.011869

W4(k)

0.015 073 70.032 425 50.01781760.020 074 30.018401 20.020 348 40.019202 20.0000.000Q.QOO

0.001461 20.006 050 60.003 053 90.000

ALEX ZUAGER AND MARVIX L. COHEN

The self-consistent nonlocal (l =0, 1, 2) pseudo-potential band structure of tungsten is shown in

Fig. 9. Some energy eigenvalues at high-symmetrypoints are collected in Table VI, where they arealso compared with the non-self-consistentmented plane waves (APW) results of Mattheiss. ~'

The latter have been obtained within the muffin-tinapproximation to the superposition density of tung-sten atoms in the 3d'6s' configuration using an ex-change coefficient of o. =1. These results are vir-tually 'd t'cal to the similar APW calculation ofPetroff and Viswanathan, ' which were obtained bysuperposing atomic densities of the 5d'6s' config-uration. The agreement between the APW and thepresen reresent results is fair. A few noticeable discrep-ancies occur: (i) the s-d splitting (1",-H„, I, N, -is substantially smaller in the present calculation(0.18 eV) relative to the APW results (0.75 eV).We find this gap to be very sensitive to the detai. lsof the self-consistency (the I",-P» splitting bei~g0.55 eV in the zeroth iteration in which a super-position pseudocharge-density model is used) ascharge is exchanged between the s and the dshells. " As the charge density is substantiallymore localized in the d„2,2+ d, 2 state at the bot-tom of the d band (H») than in the extended s stateI", (see below), one further expects these states tovary differently under exchange scaling leading to

1 12

this gap to change by as much as 0.6 eV as the ex-

18

15

& 9

I

P 6 H G N g 1 A P DND P F H

FIG. 9. Self-consistent-exchange and correlation-bandstructure of tungsten using the nonlocal (l =0, 1, 2) first-principles pseudopotential. Dashed lines denote doublydegenerated representations.

TABLE VI. Energy-band eigenvalues (in eV) of tung-sten as obtained in the present self-consistent nonlocalpseudopotential calculation and APW (Ref. 49).

StatePresentresults APW

r,r»~&2

H(~H25,H(2- H25I

P4P3P)N,N2

N(iN)N4

N3

0.004.808.31

28.800.18

10.4410.273 329.00

15.930.562.477.898.218.92

11.03

0.005.518.38

21.370.75

11.2610.514.169.41

17.661.283.288.648.679.28

12.16

change coefficient e is varied from 1.0 to 0.7. Theearly cellular calculation of Manning andChodorow leads to results that are similar to theAPW results except that the I',-0» gap is negative.(ii) The p-d gap between N, , (6p) and iV, (5d) is closeto zero in the APW results, while in the presentcalculation it is 0.32 eV. This gap is again sensi-tive to the details of self-consistency and exchangescaling for the same reasons discussed above. Itsoccurrence constitutes a qualitative difference withrespect to molybdenum in which no p-d gap ispresent. "" This might have direct consequenceson the existence of surface states in the corres-ponding materials as a N, ,-Ã, gap would persist inthe (001)-surface projected band structure (at M).'4

(iii) The antibonding d state (second 1'», ) is muchseparated from its bonding counterpart in the pres-ent calculation (24 eV) than in the APW calcula-tion" (16 eV), suggesting stronger d-d overlap inthe present representation.

The internal structure of the d band is very sim-ilar in the two calculations as these features arerather insensitive in these materials to charge re-distribution, exchange scaling ' or muffin-tin ap-proximations. "" The d-band width is 10.27 eV inthe present calculation compared with 10.51 eV inthe APW results and similarly the distance of theFermi energy from the H~ bottom of the d band is6.20 eV in the present calculation and 6.17 eV inAPW.

The calculated density of states of tungsten isgiven in Fig. 10. The general features are verysimilar to the APW results. ~ ' Three pronouncedpeaks appear below the Fermi energy, at e~ —4.5

20 FIRST-PRINCIPLES. . . . II. . . . 4097

1 )

I

E, FI

I

Tungsten

30-

E25

O

20—C0

l5—

10—Cl

0 ) I l I I )

-0.65 -0.3 5 -0.05 0.25 0.55 0.85 I. I 5ENE, RGY (Ry)

FIG. 10. Density of states of tungsten, obtained fromthe first-principles pseudopotential calculation.

eV, e~ —3.4 eV, and e~ —1.6 eV. These peaks ap-pear at approximately -4.6, -3.3, and -1.8 eV inthe -calculation of Mattheiss" and -4.3, -3.1, and-1.V eV in the calculation of Petroff andViswanathan, "the differences prIobably reflectingthe more extensive BZ sampling used by the latterauthors than genuine differences in the band struc-ture. The three peaks at the occupied part of thedensity of states originate from predominatly s+d, 2, xy+yz+zx, and xy+yz+zx states, respec-tively. The Fermi energy appears at a region oflow density of states, followed by a broad structureof about 4 eV which constitutes the unoccupied por-tion of the d band. This structure shows two pro-nounced peaks at 1.4 eV (due to p- and d-typestates around N, , and N„respectively) and at 2.45eV (due to the d states around P, and N, +N, ).

The density of states at the Fermi energy D(e~)is 6.96 electron/(Ry atom). This is close to thevalue of -7.2 electron/(Ry atom) obtained byPetroff and Viswanathan, ' leading to an electronicspecific heat of 3.1 &10 ' cal/moleK', comparedwith the observed values of 3.10&&10 ' (Ref. 57) and

(2.5~1)X10 ' cal/moleK'. "As relativistic effects are not included in the

present calculation, a detailed comparison with themeasured Fermi surface cannot be made and onlyits gross features will be discussed. Figure 11compares the calculated (110) section of the Fermisurface (broken lines) along with the experimentaldimensions (full lines) obtained by Walsh andGrimes" using size-effect techniques, the mag-netostatic data. of Bayne et al. ,

~ and the de Haas-

H

II1

I

I

&ip

FIG. 11. Calculated (dashed line) and measured (Befs.59-61) (solid line) dimensions of the Fermi surface oftungsten in the (110) plane.

van Alphen results of Sparlin. " The agreementwith experiment is generally good. The dimensionsof the hole ellipsoid at N are sensitively deter-mined by the location of the p-type N, state. Asthe p-d gap between Ã, , and N, opens up during theself-consistency iterations, the dimensions of theellipsoid around N decrease towards their observedvalues. The hole octahedron at 0 is very accurate-ly reproduced and so is the electron jack at I'. Theabsence of spin-orbit interactions in. the presentcalculation shows up in the absence of a gap be-tween the electron jack and the hole octahedronalong 4.

The nature of the bonding in tungsten can be stud-ied from the variational charge densities. Thelowest band at I' (1',) is a pure 6s-like state where-as one progresses to the next state at I" we observethe bonding my+ye+ax state (1'», ) which has abackground of 6s character. These states are di-rected predominantly along the nearest-neighboraxis and form the metallic d bond. The lower partof the d band (e.g. , P» ) contains d, 2 and d„2 2

nonbonding character that is directed toward thenext nearest neighbors and forms a loose bond. Asone moves away from the & direction, p characteris admixed into the bond with increasing propor-tions at higher energies. The P4 state at the centerof the d band is a combination of d-like xy+yz+zxand the p-like x+y+z representation giving rise toa bond which is directed toward the nearest neigh-bors with some polarization towards the next near-est neighbors. Just above e~ at the N point (N, , )

409S ALEX ZUNGER AND MARVIN L. COHEN 20

one encounters a P state. The unoccupied portionof the conduction band is made predominanatly fromnonbonding d, 2 states (I"», P„N, ) with x'- y'components slightly admixed. The basic bondingmechanism in tungsten arises from the overlappingbond-directed xy+yz+zx "$„"-like d states whichprevail throughout the zone while s and p bondinghave somewhat lower contributions and are concen-trated predominantly at the lower and upper partsof the occupied band, respectively. The total val-ence charge density (Fig. 12) is indicative of sucha combination of localized bond directed d'-stateswith a more diffused background contributed by s-p bonding. No accurate experimental x-ray scat-tering factors are available for this material forcomparison with the present calculation.

We next consider the features of the self-consis-tent valence screening field. Table VII shows thecomponents of the valence Coulomb (Vc,„,), ex-change (V„), and correlation (V„„)potential in

tungsten. It is seen that contrary to the situationencountered in semiconductors (e.g. , Si, Sec. II D)only the first reflection is large (i.e. , Llll]), therest being usually much smaller. The rate of decayof these small components is, however, consider-ably slower, due to the localized features of thecharge density (1505 plane waves are normally re-quired to expand the total valence density effec-tively). Both exchange and correlation contribu-tions are seen to screen the Coulomb interactions,with increasing efficiency at higher momenta.As previously observed in Si, the correlationscreening is not simply proportional to the

exchange screening at low momenta (V„„/V„varying by as much as a factor of 3 over thefirst reflections) whereas at high momenta theratio V„„/V„ is roughly constant (=0.05).'This enables an approximate simulation ofcorrelation effects at high monzenta by artifical-ly increasing the exchange coefficient by 5% (i.e. ,o. = 0.70). The last column of Table VII gives forcomparison the Fourier components of the 1=0pseudopotential. The screening field is seen to beabout 1(Pjp of the core pseudopotential for the lowestreflection and of the order of 1/o for higher reflec-tions. The slow decay of the screened core pseudo-potential IVc „, + V„+V„„+W,(r)] dramatizes thedifficulties in describing the electronic propertiesof these systems within empirical procedures em-ploying truncated pseudopotential form factors.

III. APPLICATION TO CRYSTAI PHASE SEPARATIONS

A. Orbital radii and valence properties

As the valence electronic structure near theFermi energy is determined primarily by rela-tively-low-momentum scattering events, it hasbeen possible in the past successfully to describethe one-electron optical spectra and the Fermisurface of many solids assuming convenientlytruncated model pseudopotentials with q,„~3k„(i.e. , smoothly varying near the core region). Thefreedom hence offered by the insensitivity of thedispersion relation e,.(k) to the repulsive nature ofthe core potential has been exploited to obtainmodel potentials that are rapidly converged in

FIG. 12. Total valencepseudocharge density intungsten in the (110) plane.Full dots represent atomicpositions. Results aregiven in e/cell (normalizedto unity).

20 FIB, ST-PRIN CIPI. E S.. . . II. . . . 4099

TABLE VII. Fourier components (with respect to individual unsymmetrized plane waves) ofthe self-consistent pseudocharge density p (in units of e/eel. l) and the Coulomb (&c „&), ex-change (&„), and correlation (&«~) components of the screening field in tungsten. The lastCoulomb gives the Fourier components of the l =0 core pseudopotential. The origin is takenat the atomic site. The potential is given in By.

Star No. &=o(G)

12345678

15

6,00000.6905

-0.02350.0339

-0.0755-0.1755-0.0431-0.0751-0.0113

0.073 60-0.001 25-0.001 20-0.002 01-0.003 74-0.000 77-0.001 14-0.002 20

-0.741 00-0.025 00

0.004 470.002 840.003 530.006 040.001 630.002 640.000 82

-0.11760-0.001 29

0.000 350.000 140.000 230.000 330.000 110.000 160.000 07

0.44030.41570.33490.2678 .

0.21570.17600 ~ 14540.0538

reciprocal space and hence amenable to per-turbative treatments" and plariewave represen-tations. In contrast, the first-principlespseudopotentials are usually characterized bystrongly repulsive core components representedby the Pauli potential U, (r) for I present inthe core." This gives rise to zero-energy classi-cal turning points V(ro) =0 that occur as a balancebetween the repulsive Pauli term U, (r) and the nu-clear attraction -Z„/r, modified by screening,core-orthogonality, and exchange-correlation non-linearity effects."' We have seen that although U, (r)is confined primarily to the core region, it is in-timately related to the properties of the wave func-tions in the valence regions. By way of construc-tion, its repulsive nature is a direct consequenceof the constraint of maximum similarity betweenthe smooth pseudo-wave-functions g and the realwave functions P (in the valence region), imposedwithin the linear relationship of y and g.' It ishence not surprising that the l-dependent turningpoints r, tend to scale as the valence properties-in much the same way as the radii of minimum po-.

tential' r, '", their variation with the position of theatom in the Periodic Table reflects the underlyingchemical regularities of the valence states.

Recently, Simons, "Simons and Bloch~ (SB), andSt. John and Bloch ' have observed that if a "hard-core" nonlocal model potential is assumed, insteadof the more conventional smooth model semiem-pirical potentials, its classical turning points r,form powerful structural indices, capable of sep-arating the various crystal phases of the octetA ~gg' "non-transition-metal compounds. The SBeffective potential is

Here 8, is an adjustable constant and V„(r) is thepotential due to the valence field. Vs«&(r) repre-sents the repulsive potential experienced by thevalence states having the same angular momentum

l as in the core, due to the explicit relaxation ofthe core- valence orthogonality constraint. %&hen

this potential is specialized to the case of a single-valence-electron system, the complicated valence-valence interelectronic interactions vanish and

V„(r) is given in the central-field limit as

V„(r) =-Z„/r + l(l+ 1)/2r',

where Z„ is the valency. As in the early work ofFues, "they showed that the eigenvalues Q„) of thecentral field problem associated with V&,','(r} areanalytically solvable in terms of B„allowingthereby a simple determination of the latter (andthe turning points ro) in terms of the observedspectral excitation energies. These are simplygiven by

(8)

The realization that these empirical orbital radiir, =B&/Z„+ l(I+ 1)/2Z„are characteristic of theatomic core and as such are transferable to atomsin various bonding situations has led to the con-struction of a number of new phenomenologicalrelations G =f(r', ) that correlate physical observ-ables G in condensed phases with the orbital radiiof the constituent atoms. Some examples are theelemental work functions 4" given by Chelikowskyand Phillips" as

a4A ~ +y-0 rg

the melting point T"~ of binary AB compounds":

with the repulsive part chosen as

(6}

T&8 —C + C g&&+ (

4 o o 5 (10)

4100 ALEX ZUNGER AND MARVIN L. COHEN 20

where the structural coordinates R and A", aregiven by '

pAB (rA rA) + (r B rB)

pAB (rA+rA) (rB+rB)

St. John and Bloch ' have shown that a two-dimen-sional topological map of R", vs Fi!„accuratelyseparates the various structural phases of the octetbinary compounds including the wurzite-zincblendeand diamond-graphite systems. Machlin, Chow,and Phillips have subsequently' found that thesame orbital parameters provide an excellentstructural delineation of the suboctet A"B, 3&P & 6 compounds. Further, a least- squares fit-ting of the forms 9 and 10 to the available experi-mental data on CA and T"B (where a„b, and C,—C, are adjustable parameters, independent of theatom) produced very good correlations with the ob-served values of the work function and meltingpoints. ' The same authors were able to representthe two phenomenological coordinates of Meidemaet a/. "which predict the signs of the heat of form-ation of about 500 binary alloys using element-de-pendent adjustable parameters, by using two simplefunctions f,(r, ) and f2(r, ) in which the only ele-ment-dependent parameters are the spectroscopicturning points r, .

What has been realized~ ' " ""is that theessential characteristics of an isolated atomiccore, as implicit in the spectroscopically deter-mined anisotropic turning points r„contain thefundamental constructs which describe the struc-tural systematics in polyatomic systems. This canbe contrasted with other phenomenological mea-sures of the "electronegativity" parameters whichare based on various observables pertaining to thePolyatomic systems themselves, (e.g. , the thermo-chemical Pauling scale, " the dielectric Phillips-Van Vechten scale,"or the Walsh scale, ' which isrelated to diatomic vibrational force constants).

Whereas the model potential that have been suc-cessfully used for fitting the low-energy electronicband structure of solids" usually lack turning points(e.g. , Fig. 1) and hence cannot be used in the pres-ent framework to define structural parameters,the structurally significant Fues-Simons-Bloch ionpotentials ' ' do not yield a quantitatively satis-factory description of the electronic structure ofatoms" or simple polyatomic systems. ' " Thereason for that" lies both in the unphysically longrange of the repulsive Pauli field B,/r' in thesimple analytically solvable model [Eqs. (5)-(7}]and in the fact that the core potentials pertain-ing to relaxed atomic orbitals [e.g. , the ions usedinfittingEq. (8) are H~, C", N~, O~, etc. ] do notadequately describe the valence states of atoms

that occur in bonded phases as nearly neutralspecies. One is hence faced with the situationwhere the different emphasis of the Fues-Simons-Bloch potential on one hand and the band-structure-derived empirical. pseudopotentials" on the otherhand in fitting their corresponding forms in differ-ent momentum transfer regimes (the high-q and q=2k„regions, respectively), results in the lack ofa unified approach to electronic properties andstructure. We show here that the currently devel-oped first-principles pseudopotentials, which

describe well the electrohic properties of atomsand solids in the LDF framework, can alsobe used to separate all the structural phases ofboth the octet and suboctet binary compounds withan accuracy that is at least as good as that basedon the empirical Fues-Simons-Bloch scheme.

B. Construction of orbital radii and comparisons with

empirical schemes

We note that the Fues-Simons-Bloch potential(5) and (6}corresponds to the first two terms in ourfirst-principles potential [Eq. (19) in I] where U, (r)is replaced by its asymptotic form C, /r' [Eq. (21)in I] at small r Howe.ver, as U, (r) decays muchfaster than r ' in the outer core region, "the first-principles pseudopotential does not lead to unphys-ically compressed valence wave functions. "Theremaining terms in the first-principles potentialare related to interelectronic interactions in thevalence system and the orthogonality hole effects,which are absent in the single-electron representa-tion used by SB. As discussed in I, these are im-portant in producing an accurate description of theelectronic structure in nzany-electron systemssuch as atoms or solids.

In order to define generalized core radii that canbe used as nearly system-invariant transferablequantities one has to assess the importance of the(system-dependent) screening effects on the turningpoints of the potential. As the pseudopotential ap-proach replaces the all-electron potential (N„,electrons) by a sum of a static core potential W, (r)(replacing the effects of N, core electrons) and adynamic valence screening field V„(r) (representingthe interactions of N„v lean ecelectrons), theproperties of g, (r), including its turning points,would clearly depend on the choice of N, =N„, —N„.We have previously indicated' that there are fewequally plausible divisions of N« into the subsetsN, and N„; the decision is normally dictated by theassumed passivity of a certain core subspace to-wards external perturbations of interest. Hencewhile the 3d, 4d', and 5d orbitals are viewed asvalence states for atoms at the center portion ofrows 3, 4, and 5, respectively, they can be treatedas a part of the inert core towards the end of these

20 FIRST-PRINCIPLES. . . . II. . . . 4101

rows for the purpose of studying their low-energyvalence spectra. Clearly, any physically signifi-cant pseudopotential structural parameter shouldnot depend on the arbitrary assignment of N, and

p/„. We find that the screened potential V'„','(r) is,however, invariant under the redefinition ofthe core-valence subspaces (for fixed outer val-ence states). The reduction of N, in favor of N„produces deeper core potentials due to the reducedkinetic energy cancellation and this is counterbal-anced by the enhanced screening of the increasedvalence subspace. Numerically, we find thatwhereas the turning points r, and r, of the corePotential of Zn change from 0.74 and 0.86 a.u. ,respectively, when the 3d orbitals are treated ascore, to 0.44 and 0.47 a.u. when the 3d orbitals aretreated ss a valence state, the radii ro and r, of thescreened (effective) core potentials are 0.82 and1.06 a.u. , respectively, for both partitioningschemes. Further, we find that the screened po-tential radii are insensitive to the particular choiceof valence configuration (e.g. , changing from ans'p' to an s'p' configuration in column-IV atoms orfrom s'p'd' to s'P d' in Ge and As, etc. ) to withinabout 0.01 a.u. and hence can be meaningfully usedas transferable parameters.

The core radii determined from the screenedfirst-principles potentials are given in Table VIII.Self-interaction corrections" have been includedfor the single-valence-electron alkali atoms(using a so Spa' configuration). The s and Pradii of the nontransition elements are depictedgraphically in Figs. 13 and 14 according totheir position along rows. In general, r', contractswith increasing valence charge Z„ for any row inthe Periodic Table (left to right in Figs. 13 and 14)due to core-attraction effects and expands with thecore charge Z, (from the lower to the upper partof Figs. 13 and 14) due to the improved pseudo-potential cancellation associated with the increasedcore. The regularities of r, are qualitatively sim-ilar to those of the minimum potential radii r, "discussed in I and will not be repeated here.

The reciprocal classical turning points r, ' scalelinearly with some of the conventional electroneg-ativity schemes. Indeed the idea of an electroneg-ativity parameter of pure Coulombic nature datesback to Gordy, "who proposed the scale Z,«/8,where Z, ff is the effective charge experienced by avalence electron at the covalent bond radius 8,.The idea of assigning a direct 0rgitgl character tothe electronegativity was first pioneered by Mul-liken in the thirties. " The currently developedradii offer a simple generalization of these ideasto directly incorporate the anisotropy of the val-ence states (i.e. , angular-momentum dependence)into an electronegativity scale. The quantity r,

TABLE VIII. Values (in atomic units) of the classicalcrossing points ~",ff(~, ) =0 of the first-principlesscreened ground-state pseudopotential (r„r&) and thestripped-ion Simons-Bloch empirical potential (r,~

&& }(Befs. 63, 64, and 67). Interpolated values are denotedby asterisks.

Element 's

LiBeBCN

0FNe

NaMgAlSiPSClAr

KCaCuZnGaGe .

AsSeBrKr

RbSrAgCdInSnSbTeIXe

CsBaAu

HgTlPbBiPoAtRn

0.9850.640.480.390.330.2850.250.22

1.100.900.770.680.600.540.500.46

1.541.320.880.820.760.720.670.6150.580.56

1.671.421.0450.9850.940.880.830.790.7550.75

1.711.5151.211.071.010.960.920.880.850.84

1.4650.440.3150.250.210.180.1550.14

1.781.130.900.740.640.56-0.510.46

2.151.681.161.060.9350.840.745G.670.620.60

2.431.791.331.231.111.000.9350.880.830.81

2.601.891,451,341.221.131.0771.020.980.94

0.4650.3150.240.190.160.140.12

0.510.430.370,330.2950.2650.245

0.680.610.2150.320.33Q.320.310.2950.275*

0.7250.680.2250.3550.380.380.360.3450.33*

0.810.7750.130.300.3450.360.360.355*0.345*

0.94G.4650.310.2350.190.160.135

1.180.7150.540.4450.380.330.295

1 .380.9350.8050.610.520.460.4150.380.37*

1.4751.0450.830.660.5850.5350.485Q.4450.425*

1.591.170.700.630.580.5450.510.49*0.45*

forms here a direct measure of the scattering pow-er of a screened atomic core towards valence elec-trons with a given angular momentum character.It is easily seen that r, ' is directly proportional tothe energy eigenvalue e„, [cf. Eq. (8)]. We note

4102 AI, EX. ZUNGER AND MARVIN I . COHEN 20

1.4—I I I I

Nal.2—

Au

Ag1.0—

Li

Cu

irst Princi

0.8—0

Xe

0.6—Kr

0.4—Ar

0.2—

I I I I I I I I

I A II A III A IVA VA Vl A VII A VIII AI B II B

FIG. 13. I =0 classical turning points of the screenedpseudopotential along rows in the Periodic Table.

larger range for all angular momenta and showgreater separation between the various rows.

Note that whereas the present s and p radii forCu are smaller than the corresponding radii forboth Ag and Au, the empirical SB scheme yieldsthe order r", "

&r, u &r',""for all )=0, 1, and 2. Asthe empirical &=2 radii for these elements (r~c"

=2.945 a.u. , x, '=2.950 a.u. , and y2A" =2.935 a.u. )are equal to within 0.5/~, only the variations in theg and p radii along. this series are responsible inthe SB scheme for the special ordering of the cor-responding solid phase work functions [vis. thecorrelation 4 -Q, r P in Eq. (9)] and electronega-tivities: 4„„&bc„&4„~.In contrast, the presentfirst-principles scheme shows a much larger vari-ation in both the g, p and in particular in the dradii: ~, "=0.185a.u. , x,"-=0.39 a.u. , and z,"" =0.49a.u. (for Sd, 4d, and 5d, respectively). The aboveorder of C would result in the present schemefrom the variations in the nature of the val-ence d electrons in these systems. Indeed thecharacter of the valence d electrons changessubstantially in this series (Cu lacking anycore states of I =2 symmetry) as also reflected inthe corresponding d-orbital kinetic energies per

I I I I I I I I

that this is in line with the suggestion of Slater"that the energy eigenvalues of a local-density Ham-iltonian form a sensitive electronegativity scale inthat they dictate the flow of charge towards thesites characterized by a lower available unoccupiedstate. Our present treatment, based on a pseudo-potential transformation of the all-electron localdensity Hamiltonian, expresses these energy-eigenvalue electronegativities in terms of the turn-ing points r, .

Although the general trends in r, are similar tothose obtained with the empirical Simons-Bloch po-tential, ~ "(Table VIII), important quantitative dif-ferences are apparent. Our /=0 radii for the first-row atoms are about a factor of 2 larger than theSB radii whereas the /=1 radii are equa1 to withina few percent. For atoms from the second row',the present )=0 radii are still about a factor of 2larger but the correlation with the I = I radii islost. The ratio between the present radii and theSB values for the noble metals Ag and Au is 4.64and 9.31 for E=O and 1.60 and 2.07 for 1=1. As theexperimental term values for Br", I", Po", andAt" are incomplete, the corresponding SB radiihad to be obtained by extrapolation. " Similarly,no empirical radii were obtained for transitionmetals as the spectra of the corresponding singlevalence electron ions is largely unavailable. Ingeneral, the currently developed radii cover a

Na1.6—

Li

AU

Ag

1.2—Cu

1.0—DO

0.8—

Rn

Xe

0.6— Kr

0.4—Ar

0.2—Ne

I I I I I I I

I A II A III A IVA VA VI A VII A VIII AI B II B

FIG. 14. l =1 classical turning points of the screenedpseudopotential along rows in the Periodic Table.

20 FIRST-PRINCIPLES. . . . II. . . . 4103

electron: T c„=9.131 a.u. , 7« =V.813 a.u. , and &A„

=8.1VI a.u. It would seem that many of the varia-tions in the properties of the monovalent Cu, Ag,and Au compounds are better rationalized in termsof the underlying differences in the properties ofthe outer metal d electrons rather than the s and pelectrons (e.g. , the optical and photoemissionproperties of CuC1, AgC1, and AuC1 showing theinterchange of the order of the metal derived d val-ence subband with the halogen-derived p-valencesubband and the rapid decrease in the metal-d-nonmetal s, p hybridization along this series). Thefailure of the Simons-Bloch scheme to adequatelyreflect the variations in the properties of the delectrons of the Cu-Ag-Au series in r, is relatedto its confinement to treat single valence electronsystems [viz. Eq. (V)]. Consequently, the outer3d, 4d, and 5d electrons in Cu, Ag, and Au, re-spectively, are considered as a Part of the irertcore and the r, coordinate is fixed from the spec-troscopic term values pertaining to excitations intothe lowest unoccuPied 4d, 5d, and 6d levels, re-spectively. The near constancy of the empirical r,values in the Cu-Ag-Au series hence reflects theproperties of the virtuaL d orbitals that have a les-ser bearing on the d bonding in the related noble-metal systems than the vaLence d states. As thepresent first-principles scheme includes directlyvalence-valence interelectronic interactions, it is

not restricted to single-electron models and theoccupied d electrons are treated as (dynamic) val-ence states. Note also that the restriction of theSB scheme to single-valence-election systemsposes a severe problem in treating transition met-al elements.

Recently, Andreoni et al."have attempted toremedy the deficiency of the SB potential in de-scribing correctly the ionic wave functions by ex-ponentially damping the repulsive B,jr' term in(6). The new repulsive term Vs&'&(r}=Q, e""&"Irhas been fitted (varying A, and y, ) to both the ionicterm values and the HF stripped ion orbitals. Asthe latter were available only for the first-rowions, the resulting radii could not be used to ex-amine structural regularities. Their radii for thefirst-rom atoms are, however, similar to the cur-rently developed radii, their L=0 values being1.01, 0.66, 0.49, 0.39, 0.33, 0.28, 0.25, and 0.22for Li to Ne, compared with the present results of0.985, 0.64, 0.48, 0.39, 0.33, 0.285, 0.25, and0.22. For L=i their radii are 0.84, 0.41, 0.28,0.22, 0.18, 0.15, 0.13, and 0.12 compared with thepresent results of 1.46, 0.44, 0.315, 0.25, 0.21,0.155, and 0.14. The origin of these similaritiescan be understood by comparing the behavior of therepulsive potentials in the two approaches. %'eshow in Fig. 15 the variation of r'V„(r) for thealkali atoms. The regular SB model [Eq. (6)J mim-

V2—

0

~.

""~~i~+++~ + ~ + ~ Oeeyy

I I I I

I I I I

gNa

I~a

I I~~ ~ ~ ~ ~ ~ ~ +

~y~ ~ ~ ~ OO ~ ~yt ~++oeooooooot+ ~ ~ ++

LiI

2.0I

l.5I

1.0I

0.5l

2.0l

1.5I

1.0I

0.5—1

0 2.52.5 0R (a.u. )

FIG. 15. Plot of r 2 Vz(r) for the l =0 and l =1 symmetries in the alkali atoms, where Uz(r) is the first-principlespseudopotential, excluding the -Z„ /r part.

4104 ALEX ZUNGER AND MARVIN L. COHEN 20

ics this behavior by a constant line r'V„" '(r) =B,while the modification of Andreoni eI; al."yieldsr'y„"'(r) =A, e "&" for this function. As the first-principles-derived r'U, (r) function decays expon-entially in the outer core region (Fig. 15), the lat-ter approximation seems valid. Quantitative dif-ferences occur, however, both in the inner cor'eregion (where the present results are nonmonotonicdue to the shell structure of the real core) and inthe values of Q, and y, necessary to fit the presentresults in the outer core region. The latter dis-crepancy (which is more noticeable for I =1) stemsdirectly from the differences between the Z„—Iionized wave functions used by Andreoni et aI,."and the neutral ground-state wave functions usedin the present study. As we have indicated before,we feel that the latter are more appropriate to de-scribe realistic pseudopotentials for molecules and

solids. As the ground electronic state of solidsand molecules is largely determined by the config-uration interaction between the multiplets of theconstituent atoms induced by the interelectronicinteractions in the lower symmetry polyatomicsystem (e.g.; those arising from the s'p', s'p',etc. , for carbon), the highly excited '9 configura-tion of the Z„—1 ion does not characterize theground state of the polyatomic system. One wouldfurther expect that as the average of the neutralatom multiplet yields lower ionization and excita-tion energies than those obtained for the strippedion, larger equivalent orbital radii [viz. Eq. (8)]would result. This is indeed borne out by the com-parison of the SB and the Andreoni et al. radii withthe present result. However, as these multipletcorrections are similar along columns (due to thesame number of valence electrons and smoothlyvarying coupling coefficients), it would seem rea-sonable to expect that. the regularities in the SBradii would parallel those obtained for neutralatoms.

The empirical approach to orbital radii enjoysthe following advantages over the first-principlesradii: (i) the radii are determined as easily forlight and heavy elements, whereas the determina-tion of the first-principles potential is increasinglymore time consuming as the atomic number in-creases; (ii) implicit in the construction of theempirical potentials are effects that are absent in

the presently developed potentials such as relativ-istic corrections [although limited by the L-S cou-pling from underlying ('6}]and core polarizationeffects. The latter are apparent from the smallnonzero values obtained for B, even for states thathave no matching symmetry in the core (e.g. , 2pfor first-row atoms). In contrast, U, (r) is ident-ically zero for such cases due to the absence ofpseudopotential cancellation.

C. Phase separation in the orbital-radii model

3.5

3.0—

vKF

ap

v KCI

RbF

Rbclv

RbBr

Cesium chloride

t CsCI

t CsBr

2.5—

2.0—

aS v KBr

g BaSe

SrSe

iF

Rbl ~t Csl

lZnP AgCI y I.

Wurzite HgS ) ~cGaN AgBr v

AINCdS ~ lAIN ~

HgSe~ CdSeCucl QMgS

SiC ZnS 0 I L

0—InP0yiMgSe

C8~+ 0AglC CvBr

0 BPAP0GaAs0 e MgTe

BN o AIA5 0 Cvl

BeSe AISb lnSbC GaSb 0 ZnTea-0 BeS—e-~e.

Si GeSnDiamond

g 15

1.0—

0.5 0 Zinc blendeDiamondGraphite

~ Wurzite~ Rock salt& Cesium chloride

—0.5—0.5 0.5 1.0

R~ (a.u. )

1.5 2.0

FIG. 16. R~ vs R~ p&ot for the octetAB compounds,obtained with the first-principles pseudopotentials.

Following St. John and Bloch" we have con-structed a topological R „vs B, [Eq. (9)] map for77 of the octet AB compounds (Fig. 16). A similarmap has been constructed for 56 suboctet com-pounds (A "B~ " 8 ~p & 6, Fig. 17). Only the moststable forms are included.

It is seen that these coordinates separate re-markably well all the crystal phases involved, in-cluding the most sensitive wurzite and zinc-blendephases (which differ only in third-nearest neigh-bors). Among the notable exceptions we observethat CuF appears near the wurzite-rocksalt linevghi1e it was thought to crysta&li. ze in a zinc-blendeform. A recent reexamination of the data" hassuggested that this compound does not exist in factin its stable phase as a AB structure.

The general pattern of phase separation is simi-lar to that obtained by St. John and Bloch ' and

Chelikowsky and Phillips ' with the empirical ra-dii. Some of the notable differences are: (i) Thepresent scheme places the Cu and Ag halides nearthe zinc-blende- wurzite- rocksalt border in theirappropriate places, whereas the empirical scheme

FIRST-PRINCIPI KS. . . . II. . . .

3.0

2.5—

2,0—

1.5—

0b Ca

BaCd

o SrCd

S HBaHg

o CsAu

0—

o BeCuCd

Lio aBi

CaH NaPb

MgAu Naln

ZnAg LilnNaTI

MgHgB.TI

Q LITI~ LIB'

o LiCdCuZn CdAg ~- LiAu

Mg TI LiHg LiAg

~ LlOa 832o 82~ 833~ tl64i CP64

—0.5 I I

—0.5 0 1.0 1.5

R~ (a.u. j

FIG. 17. B„vsR+~~ plot for the suboctetAB com-pounds, obtained with the first-principles pseudopoten-tials.

I

0.5 2.0

places them as a nearly separated group at high 8values. (ii) The empirical R„coordinate showsonly a very small separation of the sulphides,selenides a.nd tellurides of Sr, Cd, Zn, Mg, and

Be, and the (extrapolated) R, values do not separ-ate the bromides from the iodides of I i, Na, K,Rb, A g, and Cu. 7his can be viewed from Fig.18(a) where the corresponding R„coordinates areplotted for these series. In general, the slopes ofthe corresponding curves and their separation isdistinctly higher with the present radii, indicatinglarger structural sensitivity. Similarly, the em-pirica. l radii place the silver and copper halidesbetween the alkali halides on the A scale, whilein the present scale they are more logically placedas a separated group. Both these 8, curves and

similar g curves indicate that the empirical pa-,rameters of Cu and Ag compounds are nearly de-generate. A plot of r~-r, vs r~+r, for these ele-ments shows approximate linear dependence. To alesser extent a similar effect characterizes theCd-Zn pair. (iii) The empirical R„coordinateplaces the Mg chalcogenides [Fig. 18(b)] which aremostly rocksalt, below the Zn and Cd chalcogen-ides (mostly wurzite and zinc blende). The presentscale places the Mg chalcogenides closer to therocksalt Sr salts.

I.O—

Rb

Na

Li

-Rb--K=No-Ag-Cu

=Li

Ag

Cu

O.o .F

I

Cg

I

BrI

cil

Br

"Sr

MgCd7n

{b)SrCdZn

-Mg~ Be

Z8Be

I l I I I l . l

0 S Se Te 0 S SeI

Te

F. IG. 18. Variation of the B~ coordinates of the em-pirical Simons-Bloch (right-hand. panels) and the firstprinciples pseudopotential (left-hand panels) along (a)the monovalent halides and (b) the divalent chalcogenides.The symbols A, S', and ZB indicate rocksalt, wurzite,and zincblende, respectively.

The R„coordinate measures the sum of the s-ppseudopotential nonlocalities for the pair A-B.Indeed the electronic band structure of many of thecompounds characterized by small R (e.g. , GaAs,A1As; Si, Ge, ZnSe, etc. ) has been treated suc-cessfully by local pseudopotentials. ' It is also ameasure of the g-p promotion energies in element-al semiconductors and insulators (e.g. , Ro &Rp&R~~'&R„") as the difference r, ' —r~' is proportion-al to e, —e~. A correlates successfully with theinverse of the homopolar dielectric gap E„', 'whereas A, correlates well with the ionic dielec-tric gap C."

The A~ vs A, map for the suboctet compounds(Fig. 17) separates not only the two broad group ofbcc-like structures (full symbols) and anion val-ence coordination compounds (open circles) but al-so works well for most of the individual spacegroups considered. Qf the notable exceptions,CaAg(833) and NaPb(tj64) have special proper-ties." The region of intermixing of the B32 andB2 structures which occurs in the empirical sepa-ration map of Machlin et a/. ,

" is largely eliminatedwith the present scale.

ALEX ZUNGKR AND MAR%II L. COHEIR

The & coordinate has been shown to correlatesuccessfully with the deviations ~ from the idealc/a ratio (I.633) in wurzite structures. 6' Usingthe 20 AB compounds for which crystallographicdata exist, given by Lawaetz" (Table I in Ref 7.9),we obtain a correlation coefficient of -0.702, com-pared with a correlation of -0.788 obtained fromthe empirical radii" and -0.842 obtained byLawaetz' using the empirical (e~Q/lou&~)' coordin-ate (where 8*, C, and ~, are the experimentallydeduced effective charge, Phillips's heteropolargap, and the plasma frequency). In view of thenonempirical nature of our scale and its applica-tion to systems that have phases more stable thanthe w'urzite, we view our classification as success-ful. Based on this correlation, we suggest thatHgS, . HgSe, HgTe, would have a stable wurziteform. Similar conclusions are borne by their lo-cation on the A -A map.

IV. CONCLUSIONS

We have demonstrated that the first-principlesnonlocal pseudopotentials developed in I not onlyreproduce the energies and wave-function' charac-teristics of atoms and ions very accurately, but;

they can also be used to obtain a good descrip-tion of the electronic structure of solids asdiverse as covalent semiconductors and transition.metals. In addition, they are shown to containstructural information through their characteristicelectronegativity parameters ~, . As these poten-tials are constructed in a nonempirical fashion,

their characteristic features can be convenientlyanalyzed in terms of the underlying interelectronicinteractions. This enables the systematic improve-ments in the understanding of the interaction modelby way of comparison with experiment. Althoughobtained in numerical form, the knowledge of theirlimiting behavior at small and large radius enablestheir accurate fitting to convenient analytical formssuch as

This would allow their use for a wide range ofproblems including electronic properties and crys-tal structure.

ACKNOWLEDGMENTS

The authors are grateful to J. R. Chelikowsky,L. Kahn, P. I am, and G. P. Kerker for helpfuldiscussions. %e thank J. C. Phillips for helpfuldiscussions and for sending us preliminary ver-sions of Ref. 67 prior to publication. The authorsthank D. R. Hamann for discussions on the DFpseudopotential and its application to Si, as wellas for performing an independent test on the Sipotential. This work was supported in part bythe Division of Basic Energy Sciences, U. S. De-.partment of Energy, and by NSF Grant No. DMR76-20647-A01. One of us (A.Z. ) acknowledges supportby IBM.

*Present address: Solar Energy Hesearch Institute,Golden, Colorado 80401.(a) A. Zunger and M. L. Cohen, Phys. Rev. 8 18, 5449(1978). (b) A. Zunger, J. Vac. Sci. Technol. (in press).(c) P. A. Christiansen, Y. S. Lee, and K. S. Pitzer(private communication) .

2M. L. Cohen and V. Heine, in Solid State Physics,edited by H. Ehrenreich, F. Seitz, and D. Turnbull(Academic, New York, 1970), Vol. 24, p. 38;D. Brust, in Methods in Compgtational Physics,edited by B. Alder, S. Fernbach, and M. Botenberg,(Academic, New York, 1968), Vol. 8, p. 33.

3M. L. Cohen, M. Schluter, J. R. Chelikowsky, andS. G. Louie, Phys. Rev. B 12, 5575 (1975).

4A. Bedondo, W. A. Goddard, and T. C. McGill, Phys.Rev. B 15, 5038 (1977).

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M. L. Cohen, Phys. Rev. B 13, 1654 (1976).8J. A. Appelbaum and D. R. Hamann, Phys. Rev. B 8,

1777 (1973).S, Toplol J W, Moskowltz C, F Mebus M D,Newton, and J. Jafri, ERDA Report No. C00-3077-105,New York University, 1976 (unpublished); L. R. Kahnand W. A. Goddard, Chem. Phys. Lett. 2, 667 (1968);S. Topiol, M. A. Ratner, and J. W. Moskowitz, Chem.Phys. Lett. 46, 256 (1977).

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~SL. R. Saravia and D. Brust, Phys. Rev. 176, 915 (1968);W. Saslow, T. K. Bergstresser and M. L. Cohen, Phys.Rev. Lett. 16, 354 (1966); C. Y. Fong gad M. L. Cohen,i&id 21, 22 (1968).

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20 FIRST--PRINCIPLES. . . . II. . . . 4107

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ALEX ZUNGER AND MARVIN L. COHEN 20

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