PH YSICA L RK VIE% 8 VOLUME 22, NUMBER 12 15 DECEMBER 1980

Systematization of the stable crystal structure of all AB-type binary compounds:A pseudopotential orbital-radii approach

Alex ZungerSolar Energy Research Institute, Golden, Colorado 804Ã

{Received 12 June 1980}

We discuss the role of the classical crossing paints of the nonlocal density-functional atomic pseudopotentials in

systematizing the crystal structures of all binary Aft compounds {with A +8}.We show how these pseudopotentialradii (r, } can be used to "predict" the stable crystal structure of ail known {565}binary compounds. We discuss thecorrelation between [ r, ) and semiclassical scales for bonding in solids.

I. INTRODUCTION

Our experience in understanding the occurrenceof a large variety of crystal structures in naturehas been traditionally expressed in two generalframeworks: var iational quantum mechanics anda semiclassical approach. The bulk of our ex-perience in understanding the structural prop-erties of molecules and solids from the quantum-mechanical viewpoint is expressed in terms ofconstructs originating from the calculus of varia-tion: total energy minimization, optimum sub-spaces of basis functions. , etc. In this approach,one constructs a quantum-mechanical-energyfunctional representing the Born-Oppenheimersurface of a compound; its variational minimumin configuration space (RQ is then sought, usuallyby first reducing the problem to. a single-particle-like Schrodinger equation. The elementary con-structs defining this energy functional —the inter-electronic effective potential V„(r,r') and theelectron-core potential V„(r,B)—can be treatedat different levels of sophistication (e.g. , semi-empirical tight-binding, Thomas-Fermi, Hartree-Fock, density-functional, pseudopotential, etc.).Similarly, a number of choices exist for the wave-function representation (e.g. , the Hloch and mo-lecular -orbital representations or the Wannierand valence-bond models, etc.). This approachhas become increasingly refined recently, pro-ducing considerable detailed information andinsight into the electronic structure of molecules(e.g. , Refs. l and 2) and simple solids (e.g. , Ref.3).

The semiclassical approach to crystal and mo-lecular structure, on the other hand, involvesthe construction of phenomenological scales("factors") on which various aspects of bondingand structux'Rl chal Rctex'istics Rl e measured.These include chemical, crystallographic, andmetallurgical constructs, such Rs the electro-negativity, the geometry and size factors, thecoordination-number factor, the average -electron-

number factor, the orbital-promotion-energyfactor, etc.' These factors are then representedby various quantitative scales (bond-order, ele-mental work-function, ionic, metallic, and co-valent radii, electronegativity scales, etc.) thatare used to deductively systematize a variety ofstructural properties. Such intuitive and oftenheuristic scales have had enormous success inrationalizing a large body of chemical and struc-tural phenomena, often in an ingenious way. ' 'More recently, these semiclassical scales havebeen used in quantitative models, such as thesemiempirical valence force field method"" andMiedema's heat-of-formation model, ' where theremarkable predictive power of these approacheshas been demonstrated over large data bases(literally hundreds of molecules and solids).

Even before the pioneering studies of Gold-schmidt, Pauling, and others, it was known

thermodynamically that the structure -determiningenergy &E, of most ordered solids is small com-pared to the total cohesive energy &E,. Measuredheats of transformation and formation data, ""as well as quantum-mechanical calculations ofstable and hypothetical structures, indicate that&E,/bE, can be as small as 10 s-10~. This posesan acute difficulty for variational quantum-me-chanical models. The elementary constructs ofthe tluantum-mechanical approa. ch, V„(r,r') and

V„(r,R), are highly nonlinear functions of theindividual atomic orbitals that interact to formthe crystalline wave functions (due to both theoperator nonlocality of V„+V„and their self-consistent dependence on the system's wave func-tions). Conseciuently, the structural energies&E, become inseparable from the total energies&E,. One is then faced with the situation that thecomplex sneak interactions, responsible for stabi-lizing one crystal structure rather than another,are often masked by errors and physical uncer-tainties in the calculation of the s A ong Coulombicinteractions in the total interaction potentialsV„(r,r') and V„(r,R). Even though &E, can be

5839

A j.KX ZIIXGER 22

calculated using quantum mechanics with the aidof large computers (for sufficiently simple sys-tems), it is notable that the extent and complexityof the information included in V„(r,r ) and

V„(r,R) far exceeds that required to characterizea crystal structure. For example, although the12 transition meta. ls Sc, Ti, V, Cr, Fe, Y, Zr,Nb, Mo, Hf, Ta, and W have distinctly differentquantum-mechanical effective potentials and arecharacterized by systematically varying cohesiveenergies &E„,all of them appear in the same'bcc crystal form as elemental metals.

Despite this, the crystal structures of a numberof elemental solids have been successful. ly de-scribed by simple quantum-mechanical approach-es. These include the work of Deegan et a/. " inwhich the stable phase of the elemental transitionmetals was correlated with the sum of single-particle band energies of model fcc and bcc den-sities of states, as well as the density-functionalpseudopotential approach of Moriarty" in whicha pseudopotential perturbation theory and a lin-earized exchange and correlation superpositionapproximation were used to calculate binding andstructural energies of a number of non-transition-element solids. The pseudopotential formalismhas also been used by Inglesfield" to calculate thealloying behavior of a number of binary systems,emphasizing the electrochemical charge -transferenergy which is described in terms of the dif-ference in the elemental screened pseudopoten-tials.

A number of other approaches have been usedto calculate formation energies of binary alloys,neglecting the structural-energy contributions&E,. These include the tight-binding approachesof Friedel, "Cyrot and -Cyrot-Lackmann, "Hodges, "Pettifor, "and Varma" in which theshift in the alloy d-band density of states withrespect to that of its constituent elemental metalsis correlated with the heat of formation. Thevarious contributions to this energy differenceare calculated by moment approaches and charge-transfer effects are either approximated simply"'or argued to be small. " The chemical-potentialmethod of Alonso and Girifalco" focuses on thepositive contr ibution to the formation energy aris-ing from the elimination of the density mismatchat the cell boundaries of the constituent elementalmetals and the negative contribution originatingfrom a chemical-potential equalization throughcharge transfer (described by the nonlocality ofthe atomic pseudopotentials).

These approaches have successfully describedthe dominance of the d-electron contributions tothe regularities in elemental crystal structuresand cohesive energies and some of the systematics

of compound heat of formation. However, theyhave not isolated the key physical factors under-lying the structural regularities of nonelementalcomPounds (i.e. , AB with A wB) It. appears thatat present, the quantum-mechanical approachseems to lack the simple transferability of struc-tural constructs from one system to the other, aswell as the Physical t~ansPaxency required to as-sess the origin of structural regularities. Thesemiclassical approach, on the other hand, con-centrates on the construction of physically simpleand transferable coordinates that may systematizedirectly the trends underlying the structural ener-gies &E,. The major limitations of the semiclas-sical approach seem to lie in the occurrence ofinternal linear dependencies among the variousstructural factors (e.g. , orbital electronegativityand orbital promotion energy), as well as in theappearance of a large number of crystalline struc-tures placed within narrow domains of the phen-omenological structural parameters (e.g. , Moos-er-Pearson plots for nonoctet ~ compounds ordiagrams of the frequency of occurrence of a givenstructure versus average electron-per-atom ra-tio). Even so, the semielassieal approaches pro-vide valuable insight into the problem because theypoint to the underlying importance of establishingsystem-invariant energy scales (e.g. , eleetro-negativity, promotion energy) as well as lengthscales (e.g. , covalent, metallic, and ionic radii).

For the 50-60 non-transition-metal binaryoctet compounds, the problem of systematizingthe five crystal structures (NaCl, CsCl, diamond,zinc blende, and wurtzite) has been solved throughthe use of the optical dielectric electronegativityconcept of Phillips and Van Vechten. " This con-cept diagrammatically displays periodic trendswhen transferable elemental coordinates are used.Such diagrammatic Pauling-type approaches areextended here to include intermetallic transition-metal compounds (a total of 565 compounds).

In this paper, I show that the recently developedfirst-principles nonlocal atomic pseusopotentialsprovide nonempirical energy and length scales.By using a dual and transferable coordinate sys-tem derived from these scales, one is able totopologically separate the crystal structures of565 binary compounds (including simple and tran-sition-metal atoms) with a surprising accuracy.At the same time, these quantum-mechanicallyderived pseudopotentials allow one to convenientlydefine the elementary constructs, V„(r,r') andV„(r,8), and use them in detailed electronic-structure calculations for molecules, solids, andsurfaces. As such, this approach may providea step in bridging the gap between the quantum-mechanical and semiclassical approaches to elec-

S YST'KMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . .

tronic and crystal structure.The success of this approach in correctly "pre-

dicting" the structural regularities of as many as565 binary compounds using elemental coordinatesthat pertain directly only to the s and P electrons(and only indirectly to the d electrons through thescreening potential, produced by them) presentsa striking result: It suggests that the structuralpprt AE, of the cohesive energy may be do-minated by the s-p electrons.

This points to the possibility that, while therelatively localized d electrons determine bothcentral cell effects (such as octahedral ligandfiel'd and Jahn-Teller stabilizations) and the reg-ularities in the structure-insensitive cohesiveenergy &E, of crystalline and liquid alloys, thelonger range s-P wave functions are responsiblefor stabilizing one complex space-group arrange-ment rather than another. There is a strikingresemblance between this result and the semiclas-sica, l ideas indicating a correlation between thestable crystal structure of transition-metal sys-tems and the number of s and P electrons, putforward by Engel'4 "in 1939 and subsequentlygreatly refined by Brewer. " In the Engel-Brewerapproach, the d electrons play an important butindirect role in determining the energy requiredfor exciting the ground atomic configuration to onethat has available for bonding a larger number ofunpaired s and P electrons. The Engel-Brewerapproach has enabled the extension of the Hume-Bothery rules to transition-metal systems simplyby counting only the contributions of s and P elec-trons, and at the same time it has explained thestabilities of the bcc, hcp, and fcc structures ofthe 33 elemental transition metals, the effectsof alloying in multicomponent phase diagrams, aswell as pressure effects on crystal-structurestabilities, phenomena yet to be tackled by varia-tional quantum. -mechanical approaches.

These conclusions on the crucial stmctm. a/ rolesplayed by the s and P coordinates should be con-trasted with the contemporary quantum-mechanicalresonant tight-binding approaches suggested firstby Friedel" for elemental transition metals andrecently extended to compounds by Pettifor, "Varma, "and others. These approaches emphasizethe exclusive role of d electrons in determiningcohesive prop exti es.

The plan of this paper is as follows: in Sec. II,we introduce the pseudopotential concept and showhow it can be used in general to define atomicparameters that correlate with crystal structures.In Sec. III, we discuss the properties of the first-principles atomic pseudopotentials within the den-sity-functional theory of electronic structure. InSec. IV, we then show how thege atomic pseudo-

potentials can be used to define intrinsic coreradii that correlate with a large number of elec-tronic and structural properties of crystals. Theseradii are then used to systematize diagrammatical-ly the stable crystal structure of 565 binary AJ3compounds.

II. PSEUDOPOTENTIALS AND STRUCTURALSCALES

Although traditionally the inner core orbitalsand the outer valence orbitals are treated on anequal footing in variational calculations of theelectronic structure of atoms, molecules, andsolids, it was recognized as early as 1935 thata large number of bonding characteristics arerather insensitive to the details of the corestates. "'2' This relative insensitivity is a man-ifestation of the fact that the interaction energiesinvolved in chemical-bond formation (10 -10 eV),banding in solids (1-25 eV), or scattering eventsnear the Fermi energy (10 '-10 ' eV) are oftenmuch smaller than the energies associated withthe polarization or overlap of core states. Manymethods treating the quantum structure of boundelectrons, nucleons, and general fermions haveconsequently omitted any reference to the corestates, variationally treating only "valence" states[Huckel, complete neglect of differential overlap(CNDO), tight-binding, Hubbard models, opticalpotentials in nuclear physics, effective potentialsin Fermi-liquid theories, empirical valence po-tentials in atomic physics, etc. j. Clearly, how-ever, if no constraints are placed, such a varia-tional treatment will result in an unphysical lower-ing of the energy of the valence states into the"empty" core ("variational collapse" ). Much ofthe empirical parametrization characteristic ofsuch methods is implicitly directed to avoid sucha pathology. It was first recognized, however, byPhillips and Kleinman" that the price for reducingthe orbital space to valence states alone can berepresented by an additional nonlocal potentialterm (pseudopotential) in the Hamiltonian.

Although the pseudopotential concept has offeredgreat qualitative insight into the nature of bondingstates in polymers and solids (e.g. , Ref. 30), itscalculation in practical electronic-structure ap-plication has generally been avoided. "" Instead,it has been replaced by a local form with dispos-able parameters adjusted to fit selected sets ofdata (semiconductor band structures, Fermi sur-face of metals, atomic term values, etc.). Sincethe valence electronic energies near the Fermi.level are determined (to within a constant) byrelatively low-momentum transfer electron-corescattering events (~ q ~

= 2k~), it has been possiblein the past to successfully describe the one-elec-

ALEX ZUNGKR

tron optical spectra and Fermi surface of manysolids assuming pseudopotentials that are trun-cated to include only small momentum components(i.e. , smoothly varying in the core region in con-figuration space). The freedom offered by theinsensitivity of the electronic band-structure dis-persion relation &,.(k) to the variations of thepsdueopotential in the core region has beenexploited to obtain empirical potentials convergingrapidly in momentum space and hence amenableto electron-gas perturbative theories'4 and plane-w'ave -based band -structure calculations. "~'

These soft-core empirical pseudopotentials haveproduced the best fits to date for the observedsemiconductor band structures, "and their descen-dants, the soft-core self-consistent pseudopo-tentials, have yielded the most detailed informa-tion on semiconductor surface states. " The in-sensitivity of c&(k) to the high-momentum compo-nents of the pseudopotential has prompted anenormous amount of literature in w'hich differentforms for the potential have been suggested (emptycores, square wells, Gaussian-shaped, etc. ) allproducing reasonable fits to the energy levelsnear the Fermi energy. Since, how'ever, thesepseudopotentials were fitted predominantly to en-ergy levels in atoms and solids (and were not con-strained to produce physically desirable wavefunctions) they often yielded systematic discrep-ancies with experiment or all-electron calcula-tions of the bonding charge density in moleculesand solids. "" Such discrepancies result fromthe fact that higher-momentum components (e.g. ,~q~ —Gkz in crystalline silicon), not included inener gy-level-fitted soft-core pseudopotentials,are of importance in determining the directionaldistribution of the bonding charge density. Thestriking success of the empirical pseudopotentialis that it made it possible to reduce the informa-tional content of the often complex electronicspectra of semiconductors to a few (usually threeto five) nearly transferable elemental parameters(empirical pseudopotential form factors). Theimplied locality of the pseudopotential, as wellas its truncation to low-momentum components,however, has limited its chemical content to re-flect predominantly the low-energy electronicexcitation spectrum rather than explicit structuraland chemical regular ities.

Recently, Simons" and Simons and Bloch haveobserved that there exists at least one class ofs true tulsa fly significant empir ical pseudopotentialscontaining very-high-momentum components (i.e. ,

~q

~

» 2k+, or hard-core pseudopotentials). Thegeneral form of a screened pseudopotential is:

[We use a capital V(r) to denote solid-state po-tentials, while v(w) will denote atomic or ionicpotentials. ] Here V'„"(r) is the bare pseudopoten-tial acting on the lth angular momentum wave-function component, and V„,[n(r)] is the Coulomb,exchange, and correlation screening due to thepseudocharge density n(r). For the simple caseof one-electron ions, chosen by Simons and Bloch,the screening potential reduces to zero. The bareatomic pseudopotential (),',"()) was then assumedto take a simple hard-core form:

())( ) g())(/) g // gg (2)

where Z„ is the valence charge and the parameterB, is adjusted such that the negative of the orbitalenergies a„, obtained from the pseudopotentialequation:

[-2&„'+v'„",(r) ])I)„)(~) = & „,)I)„,(r )

match the observed ionization enei gies of one-electron ions such as Be", C", 0", etc. Thesehard-core pseudopotentials are characterized byan orbital-dependent crossing point r', at which

v,",I(r', ) =0. These orbital radii then possess thesame periodic trends underlying the observedsingle-electron ionization energies through thePeriodic Table. The remarkable feature of theseradii is that they form powerful structural indices,capable of systematizing ihe various crystal phasesof the octet' 8' "non-transition-metal com-pounds. " Such structural plots have been extendedby Machlin, Chow, and Phillips4' very success-fully to some 45 nonoctet (non-transition-metal)compounds.

The realization that these empirical orbitalradii are characteristic of the atomic cores, andas such are approximately transferable to atomsin various bonding situations, has led to the con-struction of a number of new phenomenologicalrelations of the form G =f(r', ), correlating phys-ical observables G in condensed Phases with theorbital radii of the constituent free ions. Someexamples of 6 are the elemental metallic workfunctions, the melting points of binary compounds,and the Miedema coordinates treated by Chelikow-sky and Phillips. " What has been realized is thatthe characteristics of an isolated atomic core,reflected in the spectroscopically determined l-dependent turning points r', contain the fundamentalconstructs describing structural regularities inpolyatomic systems. '" This ean be contrastedwith phenomenological electronegativity scalesthat are based on observables pertaining to thepolyatomic systems themselves, such as thethermochemical Pauling scale, the dielectricPhillips-Van Vechten scale, and the Walsh scale.

Since the general atomic pseudopotentail v,",I (r)

SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . .

of Eq,. (1) can be reduced to a simple form withv, =0 only for single-electron stripped ions, theempirical Simons-Bloch orbital radii can onlybe used for atoms for which str ipped-ion spectro-scopic data exist. This excludes most transitionelements, which form a wealth of interesting inter-metallic structures. Yet, even so, the extractionof a bonding scale from data on ions that lack anyvalence-valence interactions (e.g. , C" and 0",representing chemical affinities of C and 0) maydistort the underlying chemical regularities. " Inaddition, the restriction to single-electron speciesmeans that the post-transition-series atoms (e.g. ,Cu, Ag, Au or Zn, Cd, Hg) are treated as havingonly one and two valence electrons, respectively,much like the alkali and alkaline-earth elements,respectively. "~'. However, the increase in melt-ing points and heats of atomization and the de;crease in nearest-neighbor distances in goingfrom group IIB to IB metals (e.g. , Zn-Cu, Cd-Ag, and Hg -Au), as compared with the oppositetrend in going from group IIA to L4 (Ca -K,Sr-Rb, and Ba-Cs), completely eliminates anypossibility of Cu, Ag, and Au having effectivelya single -bonding electron. Similar indications onthe extensive s -d and P-d hybridization are givenby the large bulk of photoemission data on Cu andAg halides. " In keeping with the single-valence-electron restriction, one is also forced to definethe d-orbital coordinate of the post-transitionelements from the lowest unoccupied rather thanoccupied d orbital (i.e. , 4d for Cu and Zn, 5d forAg and Cd). This may be reasonably faithful tothe chemical tendencies of post-transition ele-ments with sufficiently deep occupied d orbitalsand sufficiently low unoccupied d orbitals (e.g. ,Br, Te, I), but it is questionable for the elementswith occupied semicore d shells in the vicinity ofthe upper valence band (e.g. , CdS and ZnS). Thesepathologies can be corrected by empirically ad-justing the valence charge Z„ in Eq. (2) for theseelements. " Finally, the simple pseudopotentialof Eq. (2) is not suitable for electronic-structurestudies, as indicated by Andreoni et al. ,"*"be-cause the wave functions of Eqs. (2) and (3) areseverely distorted relative to true valence orbitalsby the unphysically long-range r ' tail. This hasbeen corrected by Andreoni et al. by replacingthe long-range 8,/r' term in Eq. (2) with anA, e~'"/r' term, with the additional parametery, fixed to fit the orbital maxima. This leads toa new set of renormalized orbital radii differingconsiderably from the Simons-Bloch set.

One is hence faced with the situation that thesoft-core empirical pseudopotential", can be usedto successfully fit the low-energy electronic bandstructure of solids, but it lacks the structurally

significant turning points (i.e. , v,«(x) =0 only atr = ~); whereas the empirical Simons-Bloch" "potentials do not yield a quantitatively satisfactorydescription of the electronic structure but do yieldthe correct structural regularities. The approachthat we have taken to remedy this situation is toconstruct a pseudopotential theory from firstprinciples. The first-principles approach allowsfor the regularities of energy levels and wavefunction to be systematically built into the atomicpseudopotentials, without appealing to any ex-perimental data. Because no resort is made tosimple, single-electron models, transition ele-ments can be treated as easily as other elements,without neglecting the interactions between valenceelectrons or assuming that the highest-occupied dlevels belong to a chemically passive core. Fur-thermore, since the bare pseudopotential v' "(r)and the screening v, [n(r)] are described in termsof well-defined quantum -mechanical constructs(such as Coulomb, exchange, and correlationinteractions, Pauli forces, and orthogonalityholes) both the failures and the successes of thetheory could be analyzed.

III. FIRST-PRINCIPLES DENSITY-FUNCTIONALPSEUDOPOTENTIALS

A. Properties of the density-functional pseudopotentials

The first-principles atomic pseudopotentialsdiscussed here are derived from the spin-density-functional formalism. ' " They were first derivedby Topiol, Zunger, and Hatner""' for the firstrow atoms and further extended by Zunger andZunger and Cohen. ' " Details of the method ofconstructing these pseudopotentials, as well as ap-plications, have been previously discussed. ""Here we briefly summarize the results and es-tablish the notation to be used below.

Qne starts with the all. -electron (ae) single-particle equation for a polyatomic system in adensity-functional representation (in which coreand valence wave functions are treated on thesame footing). In this approach, the effective po-tential V;;,(r) is given as a sum of the external po-tential V,„,(r) (e.g. , the electron-nuclear attrac-tion potential) and the interelectron screeningpotential V [p, +p„]. The latter is a functionalof the core (c) and valence (v) charge densityp(r) =p, (r) +p„(r) and includes the interelectronicCoulomb V„[p(r)], exchange V„[p(r)] and correla-tion V [p(r)] potentials. "'" The eigenstates[(,(r)] of the all-electron single-particle equation:

(-—,'&'+V, ,(r)+V [p (r) +p (r)]}g~(r) =e~g~(r) (4)

are normally constrained to be orthogonal (andtherefore have nodes) and form the basis for ex-panding the self-consistent charge density

5844 ALEX ZUNGER

p(r) =+Jg~|l'y(r)

~

In the pseudopotential representation, one seeksto replace the external potential V„,(r) by a pseu-dopotential V„(r,r') such that the lowest eigen-states y&(r) have properties of valence, ratherthan core wave functions. The pseudopotentialsingle-particle equation is hence:

-', v' + V (r, r') + Vt' [n(r)]}y,(r) =~p, (r), (6)

where the valence screening V," [n(r)] has thesame functional form as V [p(r)], except thatit is a functional of the (valence) pseudo-charge-density n(r) =Z&(y~(r) ~'. Since no core statesappear below the valence states X&(r), one is freeto construct X&(r) as nodeless.

One cari either attempt to calculate the pseudo-potential V„(r,r') directly for the molecule orsolid of interest (i.e. , orthogonalized plane-wave-type pseudopotentials), or first to calculate itfor a convenient fragment (e.g. , atoms, ions) andapproximate V„(r,r') by suitable functions of suchtransferable quantities v„tr, r'). The success ofthe latter approach hinges on the degree of stateindependence that can be built into the fragmentspseudopotential v (r, r'): an atomic pseudopoten-tial is useful only to the extent that it continuesto produce chemically and physically accurateenergies and wave functions even when the atomis placed in bonding environments (e.g. , mole-

cules, solids, surfaces) which differ considerablyfrom that characteristic of the species used toconstruct the pseudopotential (e.g. , free atomsor ions). The imposition of explicit physical con-straints on the pseudo-wave-functions g&(r) leadingto an approximately energy-and state -independent(and hence transferable) atomic pseud6potentialis central. to the present approach. """ No suchconsiderations were taken in constructing theprevious empirical or semiempir ical"' "pseudopotentials.

The total pseudopotential V„(r,r') for the poly-atomic system is constructed as a superpositionof atomic pseudopotentials v'"(r)P, (where thespatial nonlocality is replaced by an angular mo-mentum dependence through the projection oper-ator P, )

V„(r,r') =Q v,',"(r -R„)P, (6)Rtf

over atoms at sites H „. The superposition ap-proximation in Eq. (6) as well as the constructionof v,',"(r) as approximately energy independentare controlled approximations: They can be testedand verified a Posteriori in the quantitative man-

.55 ~ 56

To construct v,',"(r) we specialize the all-elec-tron (Eq. 4) and pseudopotential [Eq. (6)] single-particle equations to atoms:

f- —2V'„—(Z, +Z„)/r+l(l+1)/2r'+v„[p, (r)+p„(x)]+v„[p,(x)+p„(r)]+v [p(r)+p„(x)])g„,(r)=c„,rJi„,(r) (7)

f ——,'&'„+v "(r)P, +l(l +1)/22 +v„[n(r)]+v„[n(r)]

+~ [~(~)l)X.((~)=~.,X.,(~), (8)

where v„, v„, and v denote atomic screeningpotentials and V'„ is the radial Laplacian. We nowseek to solve Eqs. (7) and (8) for the unknownpseudopotentials v'"(r), subject to a number ofphysically motivated constraints. In contrast tothe empirical pseudopotential method, "v" (r) inEq. (8) is not determined by fitting the energies6„7 to experiment, leaving the wave function X„7to be implicitly and arbitrarily fixed by such aprocess. Instead, we first construct physicallydesirable pseudo-wave-functions X„, and then solvefor the pseudopotential v,',"(r) that will producethese wave functions together with the theoretical-ly correct orbital energies &„7 =e„7 from the single-particle equation (8).

We first require that the pseudo-wave-function

X„,(r) be given as a linear combination of the"true" all-electron core and valence orbitals ofEq. (7):

- (9)n'

I

Since the pseudo-wave-functions b„,(r)fare nowthe lowest solutions to the (Hermitian) pseudo-Hamiltonian [Eq. (8)], they will be nodeless foreach of the lowest angular symmetries. The co-efficients {C'„'„].will be chosen below to satisfythis condition. In a single-determinant represen-tation, the mixing of rows and columns as givenin Eq. (9) leaves the energy invariant. We thenrequire' that the orbital energies &„7 of the pseudo-potential problem equal the "true" valence orbitalenergies z„, of Eq. (7). The first condition [Eq.(9)] assures us that the pseudo-wave-functions arecontained in the same core-plus-valence orbitalspace defined by the underlying density-functionaltheory; the second (a„,= &„,) ensures that the spec-tral properties derived from the pseudopotentialsingle-particle equation match those of the valenceelectrons as described in the all-electron problem.

Without specifying at this stage tPe choice ofthe unitary rotation coefficients (C„"„'.), Eqs. (7)-(9) can be solved to obtain the atomic pseudopo-tential v",,'(r) in terms of (C'„'„'.) and the knownquantities defining the all-electron atomic Eq. (7):

22 S YSTEMATIZATION OF THE STAB LE CRYSTAL STRUCTURE OF ALL. . .

Z& PZ,v,'."(r) = l(rr(r) — )I+]-—'+v..[p,1+v, lp. l+v Ip, ])r(v, lp. +p„]-v,lp, l

—v, fp„])

+Iv tp, +p„l-v Ip. l —v Ip.1]+Iv.,fp. l —v.,]v]I+ v. lp. ] —v„]v] + v Ip„]-v fv]),

where the "Pauli potentiairr U, (x) is given by

p„,c'„'„',0„,(~)

and the core, valence, and pseudocharge densi-ties are given as

P.(&) =g ~ g (~)~

'.

p.(r) =g-I p.",(r) I',

The atomic pseudopotential in Eq. (10) has asimple physical interpretation. The "Pauli po-tential" U, (r) is the only term in n ' (r) that de-pends on the wave function it operates on (i.e. ,"nonlocal"), whereas all other terms in Eq. (10)are common to all angular momenta (i.e., "local" ).For atomic valence orbitals that lack a matchingl component in the core, the all-electron valenceorbitals g"„,(x) are nodeless, hence X„,=g"„, and,from Eq. (11), U, (x) =0 for such states. In theseeases, the pseudopotential is local and purelyattractive due to the dominance of the all-electronterm, -(Z, +Z„)/x. In all other cases, U, (x) ispositive and strongly repulsive, but confined to theatomic-core region. U, (x) replaces the core-valence orthogonality constraint and is a realiza-tion in coordinate space of Pauli's exclusion prin-ciple. Its precise form depends on the choice ofthe mixing coefficients fC'„'„).j and is discussedbelow. We see that the pseudopotential nonlocal-ity, often neglected in the empirical pseudopo-tential approach emerges naturally in this form-ulation from the quantum shell structure of theatom. Similarly, Phillips's pseudopotential ki-netic-energy cancellation theorem" is simplyrepresented as a cancellation (or over-cancella-tion) between the nonclassical repulsive Paulipotential and the core-valence Coulomb attraction-Z„/r [Eq. (10)].

The second term in Eq. (10) represents thetotal screened potential set up by the core chargedensity p, (r). It approaches -Z,/x at small dis-tances and decays to zero exponentially at thecore radius (with a characteristic core screeninglength) due to rapid screening of the core pointcharge Z, by the core electrons. The third andfourth terms in Eq. (10) represent the nonlinearity

l

of the exchange and correlation potentia1. s, respec-tively, with respect to the interference of p, and

p„. They measure the core-valence interactionsin the system and are proportional to the pene-trability of the core by the valence electrons.

The fifth term in Eq. (10) is the Coulomb ortho-gonality hole potential. It has its origin in thecharge fluctuation &(x) =p„(r) —n(x) that resultsfrom the removal of the nodes in the pseudo-wave-functions [i.e. , the transformation in Eq. (9)].The electrostatic Poisson potential set up by &(x)is then given by the fifth term in Eq. (10). Finally,the last two terms in Eq. (10) represent, respec-tively, the exchange and correlation potentialsset up by this orthogonality hole charge density&(~).

The form of the first-principles pseudopotentialin Eqs. (10) and (11) makes it easy to establishcontact with the successfully simplified earlyempirical pseudopotentials. Hence, for example,in the Abarenkov-Heine" model potential it wasimplicitly assumed that a pseudopotential cancel-lation between a repulsive Pauli force and an at-tractive Coulomb potential -Z„/r exists, but in-stead of calculating this cancellation, its netresult was assumed to take the form of a constantv(„"(y) =A, for r smaller than some model radiusR, ( inside the core), with e,') )(x) = -Z„/r forr & R,. Abarenkov and Heine's empirical constantsA, may be identified in the present formulationwith the volume integral of [U,(r) -Z„/r] from theorigin to R, (neglecting all but the first term inEq. 9). Similarly, Ashcroft" has suggested an

empir ical "empty-core" pseudopotential, postulat-ing that the net result of the cancellation betweenU, (r) and —Z„/r inside the core region is zero.Indeed, for a sufficiently large core radius (i.e. ,of the order of Pauling's ionic radius), such asimple model well represents v(,"(r) in Eq. (10).

Up to this point, we have not yet specified theform of the transformation coefficients in Eq. (9)determining the precise relationship between thepseudo- and true wave functions. Clearly, onewould like to constrain the pseudo-wave-functionin Eq. (9) to be normalized. In addition, the re-laxation of the orthogonality constraint may beexploited. to construct y„,(x) as nodeless for eachof the lowest angular states, permitting therebya convenient expansion of the pseudo-wave-func-tions in spatially simple and smooth basis func-tions. Even so, y„,(r) is underdetermined: Thereare an infinite number of choices of (C'„"„.] leading

5846 ALEX ZUWGER 22

to normalized and nodeless g„,(r). This is amanifestation of the well-known pseudopotentialnonuniqueness. . The resolution of this nonunique-ness is precisely the point at which one appliesones physical intuition (and physical prejudices).Note, however, that in the present approach, anyof the infinite and legitimate choices of(C'„'„'.)permits a rigorous digression from the pseudo-wave-function to the true valence wave function:the choice of a linear form for y&(r) Eq. (9) allowsfor v,',"(r) to be computed from an arbitrary set(C'„'„'.] and for the resulting pseudopotential in

Eq. (10) to be used to greatly simplify the calcu-lation of the electronic structure of arbitrarymolecules or solids [Eqs. (5) and (6)]. Vpon com-pletion, one can simply recover the true wavefunction through a core orthogonalization:

q, (r) =—i~X, (r) — g,. i

y',.)q;. (r))l,( r

(15)

given the known core states g',. (r). This propertyis not shared by other pseudopotentials""'" whichare modifications of the density-functional pseudo-potential scheme. "" The choice of the transfor-mation {C„"„',] has, however, a direct bearing onthe transferability of the atomic pseudopotentialsfrom one system to another as well as on thedegree to which the true valence wave functionscan be reproduced without resort to core ortho-gonalization.

Our choice 'of wave-function transformation co-efficients" "" is based simply on maximizingthe similarity between the true and pseudo-orbitals[within the form of Eq. (9)] with a minimum coreamplitude, subject to the constraints that y„,(r)be normalized and nodeless. This simple choiceproduces highly energy-independent, and thustransferable, pseudopotentials. At the same time,the imposed wave- function similarity leads topseudo-wave-functions that retain the full chemi-cal information contained in the valence region ofthe true wave functions. Details of the numericalprocedure used to obtain {C„"„',], as well as nu-merical tests demonstrating the extremely lowenergy dependence of the associated pseudo-potentials, are given elsewhere. "~"

The general small-x expansion of the pseudo-orbital can be written as:

limy„, (r) =A,r""+A,r"""+A 0"". (14)Q

The choice of g) 2 leads to a minimum core-amplitude pseudo-wave -function with its attendantmaximum similarity to the true valence wave func-tion. Inserting (14) into (11) and (10) leads, forany g& 2, to

limv (r) =~-—+( j) @ +v~Q f J.

Hence, the Simons-Bloch~' "empirical pseudo-potential [Eq. (2)] is recovered as the small-rlimit of the first-principles pseudopotential. Wehence see that the imposition of a maximum wave-function similarity condition within the orbital sub-space spanned in Eq. (9) leads to a repulsive andshort- ranged Pauli potential U, (r) C.ombiningsuch a positive U, (r) with the negative core-attrac-tion term Z„/-r in Eq. (10) leads necessarily tocharacteristic crossing point v,',"(r', ) =0 at r =r',On the other hand, the choice g =0 leads to a soft-core pseudopotential [lim„„,v,',"(r) = const]. Theassociated pseudo-wave-function is now finiteat the origin, leading necessarily to a reducedsimilarity between the true and pseudo-wave-functions in the chemically relevant valence re-gion. In general, no crossing point r', occur atthis limit. Our choice of the wave-function trans-formation in Eq. (9) produces, therefore, unique .

pseudopotentials by going to the extreme limit ofwave-function similarity that is possible withinthe underlying density-functional orbital space.

Other possibilities for choosing pseudo-wave-functions exist and are discussed else-where. """" These procedures involve variousways of constructing pseudo-wave-functions includ-ing components lying outside the density-functionalorbital sPace [unlike Eq. (9)] and do not maintainphysically transparent analytical forms such asin Eqs. (10) and (ll). Hence, to distinguish themfrom the present density-functional pseudopoten-tials, we refer to these as "trans-density-func-tional" (TDF) pseudopotentials. " We restrictourselves in what follows to the conceptuallysimpler density-functional pseudopotentials.

The approach described above for constructingorbital-dependent pseudopotentials can easily beextended to spin- and orbital-dependent poten-tials. " This generalization is simple, and we willnot describe the details here.

Figure 1 depicts various components of the 1=0atomic pseudopotential in Eq. (10) for Sb. Thecurve labeled (1) is the Pauli term U, (r), thecurve labeled (2) shows the Coulomb attraction-Zgr, and curve (3) represents screening (terms2-6 in Eq. 10). Finally, curve (4) shows thetotal pseudopotential. First-pr inciples atomicpseudopotentials were generated for 70 atomswith 2 &Z & 57 and 72 &Z &86 (i.e. , the first fiverows).

A notable feature of these potentials is the oc-currence of a crossing point v,',"(r', ) =0 at r =r',From Eqs. (10) and (11), it is seen that, physi-cally, this point is where the repulsive Paulipotential is balanced by the Coulomb attraction

Z„/r, renormal-ized by the screened core poten-tial, exchange-correlation nonlinearity, and the

S YSTEMATIZATION OF THE STAB LE CRYSTAL STR UCTURE OF ALL'. . .

10

~ e

Q

g 4

2

0CO

2O

0'0O -4

I I I I I

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

R (a.u.)

FIG. 1. Components of the atomic pseudopotentialv~', ~(x) [Eq. (10)] for l=0 of the Sb atom: (1) Pauli po-tential U)(r), (2) the Coulomb attraction-Z„/r, (3) corescreening, and (4) the total pseudopotential. pp and pdenote the points of crossing and minimum, respectively.

Coulomb and exchange-correlation orthogonalityhole potentials.

It seems somewhat puzzling at first sight thatpseudopotentials of free -electron-like metalssuch as Na, Al, and K may have large momentumcomponents or even a hard core, because thenearly-free-electron (NFE) model seems to haveworked so well for these materials. However, thesuccesses of the NFE model may have been over-stated, in view of the fact that wave-function-reLated ProPerties of the free-electrom metals,such as the shape of the optical conductivity, "the metallic ground-state charge density and formfactors" (compare, however, with the nonlocalpseudopotential calculations of Refs. 68 and 69),as well as the properties of impurities in metals, "are poorly reproduced by local and weak pseudo-potentials. Moreover, the occurrence of rathercomplex crystal structures involving "simple"free-electron atoms (semimetals such as theB32 structure of LiAl, Laves phase materialssuch is K,Cs, the existence of the compoundNaK, but not NaK or NaK„etc. ), as weQ as theexistence of stable multiple valencies of thesesystems (e.g. , AlF vs AlF„etc.) cannot be under-stood in terms of local NFE pseudopotentials.Hence, although such weak and NFE pseudopoten-tials had to be assumed for many elements (in-cluding even groups IIIA -VIA atoms) for the verypopular low-order perturbation theories to bevalid, the underlying assumption —that the com-plex chemistry of the related compounds couldbe understood in terms of weak and isotropicperturbations of a homogeneous electron gas—

seems naive.Using the calculated atomic pseudopotentials

of Eqs. (10)-(12), we now define the crossingpoints using the ground-state screened atomicpseudopotentials v,",,'(r):

v„,(r) = v,„(r)+-- —,—+v„[n]+v„[n]+v [n](() (() I(t + I)

(16)

( t)(

Here v'„",(r) is the total effective potential ex-perienced in a ground-state pseudoatom by elec-trons with angular momentum /. These form thestructural indices {r(), which we use in connectionwith predicting the stable crystal structure ofcompounds. Table I gives the {r,j values of theVO elements for which the density-functionalpseudopotential equations have been solved. Wehave not included the heavier elements since thepresent pseudopotential theory is nonrelativistic.In what follows, we will hence not discuss thestructural stability of lanthanide and actinidecoDlpounds.

In developing the density-functional pseudopo-tentials, we have tacitly assumed a specific parti-tioning of.the atomic orbitals into core and va-lence. In the present theory, core orbitals arethose appearing as closed-shell states in the rare-gas atom of the preceding raw in the PeriodicTable. Note, however, that although w'e mayunderstand the low-energy electronic excitationspectra of a compound such as Znse by assumingthat the Zn 3d orbitals belong to a passive core,such an assumption may be invalid in intermetalliccompounds, where the Zn Sd orbitals can be innear resonance with the d orbitals of anotherelement. Given the fact that any such delineationinto core and valence is merely based on an arbi-trary assumption on the passivity of certain or-bitals to chemical interactions of interest, onemay ask whether structurally meaningful orbitalradii can be extracted froxn a pseduopotentialscheme.

In fact, the choice of the orbital radii from thescreened pseudopotential [Eq. (16)],rather thanfrom the bare pseudopotential v,',"(r) in Eq. (10)(e.g. , Refs. 39, 40, 48, and 49), is based preciselyon an attempt to avoid such a nonuniqueness. Al-though the bare pseudopotential of Eq. (10) hasthe form

v'"(r) =U((r) +f(Z„Z„,p„p„,n),the screened pseudopotentia, l can be written as

'V( )=rU, ( )+gr(Z, t .+(o„) (18)

Note that whereas U((r) [Eq. (11)]depends only

5848 AQKX ZUNGER

TABLE I. Classical crossing points of the self-consistently screened nonlocal atomic pseudopotentials (including thecentrifugal term), in a.u. The core shell is defined in each case as the rare-gas configuration of the preceding rovy.The Kohn and Sham exchange is used.

Atom yd Atom f'd

LiBeBCN

0pNe

Na

MgAlSiPSClAr

KCaScTxVCrMnFeCoNiCuZQ

GaGeAsSeBrKr

0.9850.640.480.390.330.2850.250.22

1.100.900.770.680.600.540.500.46

1.541.321.221.151.091.070.990.950.920.960.880.820.760.720.670.6150.580.56

0.6250.440.3150.250.210.180.1550.14

1.551130.9050.740.640.560.510.46

2.151.681.531.431.341.371.231.161,101.221.161.060.9350.840.7450.670.620.60

2.43

0.370.340.310.280.260.250.230.220.210.1950.1850.1750.170.160.1550.150.1430.138

0.71

SrYZrNb

MoTcBuRhPdAgCdInSnSbTeIXe

CsBaLaHfTaW

ReOsIrPtAu

HgTlPbBiPoAtRn

1.421.321.2651.231.221.161.1451.111.081.0450.9850.940.880.830.790.7550.75

1.711.5151.3751.301.251.221,191.171.161.241.211.071.0150.960.920.880.850.84

1.791.621.561.531.501.491,461.411~371.331.231.111.000.9350.880.830.81

2.601.8871.7051.611.541.5151.491.481.4681.46

1.341.221.131.0771.020.980.94

0.6330.580.540.510.490.4550.450.420.400.3850.370.360.3450.3350.3250.3150.305

0.940.8740.630.6050.590.5650.5430.5260.510.4880.4750.4630.450.4380.4250.4750.405

on orbitals with angular momentum l, the valencepseudo-charge-density n(x) [Eg. (12)] depends onall orbitals that are assigned as valence states.Consequently, for example, if the Zn 3d orbitalsare assumed to be a part of the core, the bare

pseudopotentiaf v,',"(r) for s and p electrons is dif-ferent than if the d electrons were assigned to thevalence. In contrast, it follows from Eq. (18) thatthe screened pseudopotentiaf v'„",(r) for I =0, I isinvariant under such a change in the assignmentof the d electrons. Our definition of the structuralindices r, is therefore independent of the assign-ment of orbitals g„',,(r) from other angular shellsas core or valence. Also note that the definitionof orbital radii from the screened pseudopotentialsof Eq. (16) permits a direct inclusion of electronicexchange and correlation effects in the structuralcoordinates r, (see Ref. VI), whereas the semi-

classical electron concentration-factor' is rep-resented simply by Z„.

8. Simple universal form the density-functionalpseudopotentials

The idea of atomic radii is not new in pseudo-potential theory (see Sec. II). The basic thrust ofthe pseudopotential concept is to transform thechemical picture of the existence of an orbitglsubspace of nearly chemically inert core statesinto a delineation either in configuration sPace orin nzonzenturn spgce of a core region of the poten-tial (with its attendant cancellation effects betweenorthogonality repulsion and Coulomb attraction)and a valence region (with its weaker effective po-tential). What is new in the present approach isthat whereas in the empirical pseudopotential

S YSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 5849

methods the radii were imposed extraneously,either explicitly~'6" ' ' ' or implicitly, "'"thepresent theory provides them as a natural finger-print of the internal quantum structure of the freeatom. In the empirical approa, ch, the radii are inturn transferred from various sources (Paulingionic radii, fitting energy eigenvalues to atomicterm values, optical ref lectivity of semiconductors,or the Fermi surface of metals, etc. ), such thatalthough a desired fit to selected experimental ob-servables is a.chieved, the underlying electronicand structural regularities may be obscured byfitting to different physical properties or by pos-tulating certain arbitrary analytic forms forv "'(r).

Given that the analytic form of the pseudopoten-tial in the present approach is not assumed butrather emerges as a consequence of requiring amaximum similarity between the all-electron andpsdueo-wave-functions in the tail region, it how-ever is possible to deduce g posts'ou a universalanalytic form through a fitting procedure. Such afit can be done in two different ways: either em-phasizing a high numerical accuracy for the fit(and hence using rather complicated or many fit-ting functions) or by using a physically transpar-ent fitting function, sacrificing to some extent thenumerical accuracy but obtaining the correct xeI, -ulgxities of the pseudopotentials. This has beenattempted by Lam et al. '4 using the simple form:

v i)(r) ~g-c2, r ~e-c3rC Z Z(19)

The eoeffieients (C„, C», and C,j are tabulatedby Lam et gl. Although more complicated formsthan Eq. (19) have also been used, "Eq. (19) re-veals a very important characteristic of the densityfunctional pseudopotentials: To within a reason-able approximation, the constants Cii C2i and

C, are linear functions of the atomic number, i.e. ,

C» =a, + b,Z, C» = c,+d,Z, C, = e +fZ . (20)

This constitutes a signif icant reduction in thenumber of degrees of freedom required to specifythe potential and reveals the regularities of theperiodic Table through the coordinates (Z„Z„).This can be contrasted with the empirical pseudo-potential approach in which such regularities areoften obscured by fitting certain atomic pseudo-potentials to optical data, "whereas others are fitto metallic Fermi-surface data and the resistivityof metals" or to atomic term values. ,

""'"The existence of a simple linear scaling relation-

ship in Eqs. (19) and (20) establishes a mapping ofMendeleyev's classical dual coordinates Z, andg„characterizing the digital structure of the per-iodic table, into a more refined quantum-mechan-

ical coordinate system, r,(Z„Z„), r (Z„Z„), and

r~(Z„Z„). Given the fact that Mendeleyev's dualcoordinates (Z„Z„)are already suggestive ofbroad structural trends (e.g. , the AB compoundswith Z„"=3, Z~=5 tend to form zinc-blende struc-tures for large Z,"'~ values, while compounds withg„"=1, g~ = 7 tends to form rocksalt structures,etc.), it is only reasonable to expect that withtheir present resolution into anisotxoPic orbitalcomponents, far more sensitive structural coor-dinates can be achieved.

The density- functional atomic pseudopotentialshave been previously used for self-consistentelectronic- structure calculations. These includeapplications to diatomic molecules [0, (Ref. 77),Si,, (Ref. 78)], tetrahedrally bonded semiconduc-tors such as Si (Refs. 58 and 79), Ge (Ref. 58),and GaAs (Ref. 80), elemental transition-metalsolids Mo (Refs. 81 and 82) and W (Ref. 82), therelaxed GaAs (110) surface (Ref. 80), as well asto the prediction of the cohesive energies, bulkmodulii and equilibrium lattice constant of Si (Ref.83), Mo, and W (Ref. 82). Very good agreement isobtained with the available experimental data per-taining to ground- state properties.

IV. TRENDS IN ORBITAL RADII

A. Chemical regularities

We have argued that the classical turning pointsx, of the screened density-functional ztomicpseudopotentials form a useful elemental distancescale for solids. One may then ask if indeed suchatomic quantities retain their significance in thesolid state. To answer this, we have performeda self-consistent band-structure calculation forbcc tungsten using our atomic pseudopotentials. .

This is done by assuming that the crystallinepseudopotential V"'(r) is a superposition of theatomic pseudopotentials v,',"(r) [Eq. (6)], but thescreening V," [n] is calculated from the self-con-sistent Bloch wave functions of the solid" (ratherthan from atomic orbitals). The resulting band

structure, Fermi surface, and optical spectra arein very good agreement with previously publishedresults. One can now use the self-consistent crys-talline charge density n(r), calculate the Coulomb,exchange, and correlation screening in the solid,and extract from that the screened solid-statepseudopotentials V,",~&(r) [Eq. (1)] and their classi-cal turning points. Obviously, such a solid-statescreened potential has a different form in the dif-ferent crystalline directions [h, k, l], resulting inspatially anisotropic orbital radii r, [k, k, l j. Fig-ure 2 shows the crystalline tungsten pseudopoten-tial (dashed lines) as well as the three componentsof the screening (evaluated with respect to the

ALEX ZUNGER

I.O—

0.5

Q

z'-0,5LLII-OQ IP

I I I I II

[II ]I I & I & I

I I I I I

0.8 1.2 1.6 0 0.4 0.80.4 0.8 I.2 0 0.4Dl STAN CE (UNITS OF o)

FIG. 2. Self-consistent screening poten f'otential finterelec-CO01

V (r)] and pseudopotentia$ V = &r or cc undifferent directions in the crys

COXT

stal.

/

Fermi energy~ xn e so,th solid. Although the screenedotentials show a pronounced directionalpseudopo en ia s ~

ii l in in the corecharacter, the solid-state radii, lying ~n eregion of the a oms, st show only a small directionalanisotropy: x0 =. . . 1r [lllj = 1.279 +0.002 a, .u. , ro[001

1 0 2 6 0 ~ ~ P 0 ~ ' e ~. 14 0 002 a.u. and x [110]=1.256+0.0 a.u. ,d ith the isotropic atomic value x0=

a.u. and the average crystalline value of 1.2 a.u.ith res ectThe near invariance of these radii wi resp

to the chemical environment should bbe contrastedwith the pronounced dependence of ththe classicalcrystallographic radii" on chemical factors (co-ordination number, valency, p's in state, etc.).

Inspection o eof the atomic pseudopotentials imme-diate y revel mls some clear regularities. -This maybe appreciated from Fig. , w3 which shows theradius x, 'at w xch' h the l =0 pseudopotential hasits minimum, p ol tted against the depth of the

W . The column structure of thePeriodic Table is immediately apparen .upper left corner of the figure, we see elements

low and extended pseudopotential; these elementsare indeed the least electronegative in the firstfive rows of the Periodic Table. In the lower righcorner, we find elements such as F and Q, whichare characterized by very deep and localizedpseudopoten ia s;t l . these are indeed the most elec-tronegative elements. C lea y,rl as the electro-

tivit is a measure of the power of an atom tonega ivy y int and atgain ex ra et electrons from its environmen

the same time keep its own electrons, such a pro-pensity is re ec e infl t d the potential-well structure

I ntrast with the thermochemical ordielectric electronegativity scales, however, epresent orbital radii define an gnisotropi c (or i-dependent) electronegativity scale.

%e see in F~g.i . 3 that the first-row elements areents thehat separated from the other elements, esomew se

mi ht haveformer having deeper potentials than mugbeen expecte rom ed f xtrapolating the data for otherrows. ls pTh henomenon, resulting from a weakpseudopo en ia it t' 1 kinetic-energy cancellation forfirst-row elements, is also clearly reflected inthe thermochemical stability of the corresponding

ounds. As we move from the right to the. leftcompoun s. s w

i . 3 that theof the Periodic Table, one sees in Fig. an can be well.elements belonging to a given column ca

characterized by their potential radii alone, thepotential depth being nearly constant. Thzs seemso be the basis for the success of the empty-core

pseu opo en id t ntials" postulated for simple meta1. s, inwhich v x xs assumassumed to be zero within a sp ereof radius R„. n isIn this approach only the variationin B,„within a column in the Periodic Table was

and structural data-for the corresponding me-tals. jn fact one finds that these empiricay

3.0es

t I A

II A

2.0—Na

1.5—0

So1.0—

0.5—

I

12 13I I I I0 I I I I I

7 8 9 10 ll2 30~mI.

~

and the depth of thedius r~ at which the g pseudopotential has its minimumFIG 3. The correlation between the radius ro at w ic e s pt .for the nontransition elemen s.minimum -5'0, o

SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 585l

3.0~Cs

1.0

0)~2 2.0—'el4Ko 15—0V

I~ 1.0—CL

ELLI

0.5—

0.0 I I I I I

0.0 0.5 1.0 1.5 2.0 2.5 3.0

rp (a.u. ]

FIG. 4. Correlation between Ashcroft's empty-corepseudopotential radius and the p-orbital radius of thepresent density-functional screened pseudopotential(Table I).

empty-core radii used to fit resistivity data maybe identified, within a linear scale factor, withthe y screened pseudopotential coordinate (Fig.4). Whereas the alkaii elements are characterizedpredominantly by a single coordinate (Fig. 3), inline with their free-electron properties associatedwith a shallow pseudopotential, the elements totheir right are characterized by a dual-coordinatesystem. The regularities in these dual coordinatesalso reQect well-known chemical trends: For ex-ample, the tendency towards metalization in theC —Si -Ge- Sn —and Pb series is represented bythe increased delocalization and reduced depth intheir pseudopotentials, etc.

Having discussed the periodic trends exhibitedby the atomic pseudopotentials, we now turn totheir significance in the establishment of elemen-tary distance and energy scales, which are quan-tum-mechanical extensions of similar semiclas-sical scales discussed in the Introduction.

Figures 5 and 6 display the multiplet-averageexperimental ionization energy E, of the atoms, "plotted against the reciprocal orbital radius x,'.For each group of elements, we show two lines:E, vs r,' and E~ vs x~'. The striking result is thatthe theoretical x,' is seen to form an accuratemeasure of the experimental orbital energies andhence can be used as an elementary orbital-de-pendent energy scale, much like Mulliken's electro-negativity. Indeed, since x,' is a measure of thescattering pow er of a sc reened ps eudopotentialcore towards electrons with angular momentum /,it naturally forms an electronegativity scale. Thereis an interesting relation between this picture andSlater's concept of orbital electronegativity withinthe density-functional formalism. " In his ap-proach, the spin-orbital electronegativity X,. isdefined as the orbital energy &,. of the density-functional Hamiltonian, which in turn equals the

0.5—

O.O

1.O

Ql

oUJC0tg

0.5—8OQl

0,0

1.0

Es

0.5—

0.00.0

I

0.5I

1.0 1.5r ta.u I

2.0

FIG. 5. Correlation between the observed lth orbitalmultiplet-averaged ionization energies E& and the re-ciprocal orbital radius g, (Table I) for the polyvalentelements.

derivative of the total energy E with respect tothe ith orbital occupation number: X,. = c,. = BE/Bn,In the limit where E is a quadratic function of n, ,this orbital electronegativity reduces to Mulliken'sform. This definition is based on the notion that achemical reaction takes place when electrons willflow from the highest occupited orbitals of a reac-tant to the lowest unoccupied orbitals with which a

0.40

s"- 0.30—

8 0.20—UJ'r,O

g 0.10—2

o.o

~NNaC Rb

I

0.5rI' )a.u. )

LI ES

I

1.0

FIG. 6. Correlation between the ground-state (E~) andexcited-state (E&) orbital ionization energies of the alkaliatoms and the corresponding reciprocal orbital radius

(Table I).

5852 ALEX ZUNGER 22

0.5I)I

I

0.4 +I

I

O.3 +I

I

ot2 A

~0V

5 I

u.o o.o

I

I

I~ -0.1I

tgK

II-0.3 —

y

I I I I I I I I I I

——Nb

d, lvl

r, IVI

r, INbl

maxlvl

"maxlNbl

-04—

-0.50.0

I I I I I I I I I I I I I

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8Distance(a. u.)

FIG. 7. Radial z-type all-electron wave functions for V and Nb. x ~ denotes the position of the outer orbital maxima,is the screened pseudopotential radius, and d~ is the average node position. Note that y~ is pinned inwards of the

last node and outwards of d~.

finite overlap occurs. Since the present x,' coor-dinate scales approximately with the orbital energy&„ the former coordinate is a realization ofSlater's electronegativity in a pseudopotential re-presentation.

The orbital radii r, also form an interestingdistance scale." Consider an all-electron valenceatomic wave function such as the 4s and 5s orbitalsof V and Nb, respectively, depicted in Fig. 7.These wave functions have their outer maxima atthe points denoted by z and have a number ofnodes inwards to y . An algebraic average taken

for all node positions in each wave function showsthat these average positions (denoted by d, ) arepinned at a certain distance from the pseudopoten-tial orbital radius x, . Figure 8 shows the averagenode position d, of the outer all-electron s-typevalence orbital plotted against x„and Fig. 9 showssimilar results for d orbitals (only the first andlast element of each row are denoted by the chemi-cal symbol). We find that the orbital radius r,scales linearly with the average node position,where the row-dependent scale factor increasesmonotonically with the position of the period in thetable of elements (e.g. , the scale equals 1.0, 1.5,2.0, 2.3, and 2.7 for periods 1-5, respectively).

1.0

CO~~

1.0—loIX

o0cOe~ I&

0)

Z -o.sIUltO

Rb

tgs I

'rn Dg g0.5—SZPeO-P D% e

K

0.000 0.5 1.0r, ) a.u. )

2.0

FIG. 8. Correlation between the average node positionin the valence all-electron g wave function and thescreened pseudopotential radius y, (Table I). The firstand last atom of each row are denoted by chemical sym-bols.

0.0 I

0.5rdra. u. )

1.0

FIG. 9. Correlation between the average node positionin the valence all-electron d wave function and thescreened pseudopotential radius ~„(Table I) for the4d elements.

22 SYSTEMATIZATION OF T HE STAB LE CRYSTAL STRUCTURE OF ALL. . . 5853

3.0

2.5—3.0

lO

0).2~ 2.0—tO

'oIl9

1.5—

2.0ld

ts'

1.0

0.0 '

Li

I I I I I I

Ne Na Ar K Cu Kr

1.00.0

I

0.5I I

1.0 0.5

r, ra.u.]

I

1.0

FIG. 10. Relation between Pauling's tetrahedral radiusand the s-orbital radius of the screened pseudopotential(Table I).

FIG. 12. Correlation between Pauling's univalent radiiR„and the total s -p screened pseudopotential radius8,= r,+ x&. Only the first and last elements in each roware denoted by the chemical symbol. Note the nonmono-tonic break in p„and p, occurring at the end of the Bdtransition series.

It is seen that the orbital radii x, form, therefore,an intrinsic length scale in that they carry overfrom the true wave functions the information onthe average node position. Hence, the dual coor-dinates (x„x,'j satisfy the semiclassical ideasunderlying many successful structural factors' 'in forming elementary energy and length scales.

An additional intriguing feature of these orbitalradii is their simple correlation with pauling'stetrahedral radii. ' As seen in Fig. 10, the tetra-hedral radii can be identified with the x, coordin-ate to within a row-dependent scale factor.

While the individual y, and z~ radii measure theeffective extent of the quantum cores of s and psymmetry, the sum ROA=xsA+~pA provides a mea-sure of the total size of the effective core of atom

30—

2.5—

CO 20-Y)

gg 1.5—

CO 1a-tg

C- 0.5-

00 I I I I I I I I

00 05 1 15 20 2.5 3.0 3.5 4.0

R~(8.U.)

1'IG. 11. Relation between Pauling's univalent radiiand the total s-p screened pseudopotential radius g= x&+ xs. Note that this p scale separates the univalentradii into a group of predominantly s-bonding elements(full circles) and s-p bonding elements (full triangles).

Figure 11 depicts RA versus Pauling's unival-ent radii for the first three periods of the table ofelements, whereas Fig. 12 shows their explicitcorrelation. A similar correlation exists withGordy's covalent radius. " R,"closely follows theregularities of the univalent radii, including theirdiscontinuity at the end of the transition elements.Examination of Fig. 11 reveals that the presentR,A coordinate provides a natural separation ofpauling's univalent radii into those that pertain toatoms sustaining s-p covalently bonded compoundsand those in which the s electrons largely dominatethe structural properties. It is remarkable thatthe orbital radii derived from a pseudopotentialformulation of gtomic physics provide such a closereproduction of the length scale derived experimen-tally from solid-state physics (e.g. , the empty-coreradii in Fig. 4, and Pauling's tetrahedral and uni-valent radii in Fig. 10 and Figs. 11 and 12, re-spectively).

We have concentrated in this section on revealingthe most significant correlations between the or-bital radii and some transferable (rather thancompourid-dependent) semiclassical coordinates.We will not describe correlations with compound-dependent physical properties (e.g. , melting points,deviations from ideal cia ratio in wurtzite struc-tures, elastic constants, etc. ) not only becausethis may be too excessive, but also because webelieve that many more such interesting correla-tions are likely to be discovered in the future.Such correlations between atomic fx,}values andphysical properties GA"~, GA~, etc. , may not onlyserve to systematize those properties but couldalso point to the underlying dependencies betweenthe seemingly unrelated physical observablesGA~) GAa~ etc.h)

5854 ALEX ZUÃGKH, 22

B. Screening length and orbital radii

One can view the quantity x,' as being an orbital-dependent screening constant pertinent to thescattering of valence electrons from an effectivecore. For the nontransition elements, one finds(cf. Fig. 13), as expected, that r, ' falls off mono-tonically with decreasing valence charge Z„, re-flecting a more effective screening. However, forthe 3d, 4d, and 5d transition series (Fig. 14), onefinds two distinct behaviors: Although x~' is asimple, monotonic function, both z,' and r&' showa break at the point where the d shell is filled.This is intimately related to a similar trend in theorbital shielding constants Z*, calculated by Cle-menti and Hoetti" as a rigorous extension of Sla-ter's screening rules. As seen in Fig. 15, thereciprocal screening lengths (Z,*) ' for the non-transition elements follow a regular monotonictrend, but those from the 3d transition series(Fig. 16) show a characteristic break aroundCu-Zn, much like the corresponding reciprocalradll t )

This dual behavior of x, ' separates the predom-inantly d-screening domain of the transition el.e-ments from the s-p screening domain of the post-transition elements. Note that r, ' and Z*, showuniquely this dual behavior, whereas most chemicaland physical quantities are simple monotonic func-

10I I I I I I I

1.5

1.0

0.5

0.0 I I I I I I I I I I I I I I I I I

K Ca Sc Ti V Cr Mn Fe CoNI Cu ZnGaGe As Se Br Kr

1.5

1.0

lO

CL~I or

0.5

0.0 I I I I I I I I I I I I I I I I I

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

1.5

0.5—

—10

~j

CO

—2.5

—2.0

—1.5

5.0

1.00.0 I I I I I I I I I I I I I I I

Hf Ta W Re Os Ir Pt Au Hg TI Pb Bi Po At Rn

FIG. 14. Regularities in the orbital electronegativityparameters y", (Table I) for the 3d, 4d, and 5d transitionseries. Note the linearity of the y& scale compared withthe br~ak in the r, and r& scales.

p.p l l I l I I I

Li Be B C N 0 F NeC

I I I I I I I

2.0

tions of the atomic position in these rows. It isinteresting to note that such effects are clearlymanifested by s and p coordinates rather than bythe d coordinate. This has a central role in the

1.51.0

I I I I I I I I I I I I I I I

1.0

0.5—

'

lD

M~2~@ 0.5K

P.P I I I I I I I

Na Mg Al Si P S Cl Ar

FIG. 13. Replarities in the orbital eleotronegativityparameters ~& (Table I) for the elements of the firsttwo rows.

P,P I I I I I I I I I I i I I I I

LiBeB C N 0 FNeNaMgAiSi P S CIAr

FIG. 15. The regularities in the reciprocal s-orbitalshielding constants (Ref. 88) (Z~) for the nontransitionelements, compared with the screened pseudopotentialradii y'~.

SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 5855

I I I I I I i I I I I I I I I I I

0.25

g 0.20roth

~D

ClCC 0.15

0.10

Ca Ti Cr Fe Ni Zn GeI I I I I I I I I I I I I I I I

K Sc V iiln Co Cu Ga As SeBr Kr

FIG. 16. Hegularities in the recipr ocal orbital shield-ing constants (Bef. 88) (Z*) & for the third-row ele-ments, compared with the screened pseudopotentialradii r,. Note the break at the end of the transitionseries.

structural significance of the s-p coordinates evenfor compounds containing transition elements.

C. Comparison with other orbital radii

Figure 17 compares the empirical stripped-ionradii of Simons and Bloch (SB}with the presenttheoretical values of the density-functional orbitalradii for the 41 nontransition elements calculatedby SB. The latter set has recently been corrected

I I i I -I I I I I I I I

3.0 —o e=

for the post-transition elements4' relative to theset used by Chelikowsky and Phillips ' and Mschlin,Chow, and Phillips. ~ Figure 1V includes the cor-rected values (e.g. , the 1=0 crossing point radiifor Cu, Ag, and Au are 0.38, 0.44, and 0.41 a.u. ,instead of 0.21, 0.22, and 0.13 a.u. , respective-ly~'4'}. These large corrections change quantita-tively some of the results of these previous au-thors in analyzing the nonoctet crystal structures,regu, larities of melting temperatures, as well asthe decomposition of Miedema's heat-of-formula-tion model into elemental orbital radii.

Figure 18 compares the recent orbital radiideveloped by Andreoni, Baldereschi, Biemont,and Phillips (ABBP; Refs. 48 and 49}with thepresent orbital radii, for the 27 nontransition ele-ments calculated by ABBP. The ABBP radii areobtained from a two-parameter fit of both theHartree-Fock stripped-ion ortibal energies aswell as the peak position of the orbital wave func-tions. The values for the eight 3d transition ele-ments given by ABBP are not included in Fig. 18since, as indicated by these authors, and as weconfirm, they are not as reliable.

It can be seen that although the empirical SBradii correlate overall with the present radii, thescatter is fairly large. In particular, the SBscheme predicts y, «x~ for the first-row elements,whereas the present and the ABBP scheme, whichattempt to reproduce both energies and wave func-tions, show x, &r~. The ABBP radii correlate wellwith the present radii" for the 27 nontransition

28—2.6—2.4—2I2

2.0—

b &=1

~ 1.8—Z16-It

1.4—04

1.2—1.0—0.8—

Be

80.6—

Ch ~oP.4 —Nh

Og, o

p2 F 0

bh 0

0b, o

Poh hh

h o o oALIbh o~o'boo

pp I I I I I I I I I I I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2A 2+DF

282.6—2.4—2$2

2.0—1.8—1.6—1.4—1 ~ 2

1.0—0.8—0.6—0.4—0.2—

o c=p

0o h

0ooh

0h

00

0

o b

FIG. 17. Correlation between the Simons-Blochempirical orbital radii rP and the present screenedpseudopotential density-functional radii P& (Table I)for the 4l nontransition elements given by Simons andBloch. The L = 1 coordinates of the first-row elementsand the &

= 0 coordinate of Au showing the largest spreadare denoted by their chemical symbol.

pp I I I I I I I I I I I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1A 1.6 1.8 2.0 2.2 2.4 2.6

r, " ta.u. j

FIG. 18. Correlation between the Andreoni, Balder=eschi, Biemont, and Phillips (Ref. 48) orbital radii

and the present screened pseudopotential density-functional radii y&

F (Table I) for the 27 nontransitionelements given by ABBP.

5856 ALEX ZUWGER 22

elements. Other plots (e.g. , correlation ofr-~

~

or ~r,"+x~~

between the various schemes)lead to similar conclusions.

V. APPLICATIONS TO PREDICTION OF CRYSTALSTRUCTURE OF 565 BINARY COMPOUNDS

A. Structural plots

The orbital radii (y,) derived here can be appliedto systematize the stable crystal structure of corn-

pounds in the same way as discussed by St. Johnand Bloch, ' Machlin et al. ,~ and Zunger andCohen. " Having, however, the orbital radii ofall atoms belonging to the first five rows in thePeriodic Table, the present theory can be appliedto a far larger data base of crystals (565) than hasbeen attempted previously (50-80 compounds).

Our first step was to compile a list of binary A.Bcompounds whose atoms belong to the first fiverows of the Periodic Table. We were interested

TABLE II. AB crystal structures used in the structural plots. When two entries appearfor the number of compounds, the first indicates the number of suboctet compounds and thesecond denotes the number of nontransition element superoctet compounds.

Stncktu rb eri chteor Pear son symbols

Spacegroup Unit cell Prototype

Number ofcompounds

B1B2

. B3

a4

Em3mPm3mE43mP63mcEd3m

Octet

cubiccubiccubichexagonalcubic

NaC1CsClZnSZnOdiamond

65

29114

Total

B1B2B8gB10B11B16B19B20B27B31B32B33B35B37Bacp64j'zP24

L10m C24mC32mP16mP32oC16oC16oC48oI8oP1 6tI8tI32tI32tI64

Fm3mPm3mP63/mmeI4/nmmP4/nmmPnmaPmmaP2(3PnmaPnmaFd3mCmcmP6/mmmI4/mcmP6m2P43nP63/mmeP4/mmmC2/mC2/cP3(/cP2( /nCmcmCmcaCmc2gImmmP2&2&2&

r4/mmmI4) /aI4/mcmI4&/acd

Nonoctet

cubiccubichexagonaltetragonaltetragonalorthorhombicorthorhombiccubicorthorhombicorthorhombiccubicorthorhombichexagonaltetragonalhexagonalcubichexagonaltetragonalmono clinicmono clinicmono clinicmono clinicorthorhombicorthorhombicorthorhombicorthorhombicorthorhombictetragonaltetragonaltetragonaltetragonal

NaC1CsClNiAsPbOCuTiGeSCuCdFeSiFeBMnPNaTlCrBCoSnSeTlMoPKGeLiOGuAuAsGeNaSiAsLiNS

NangKOSiPHbONaPHgC1LiGeTlTeNaPb

33+8122+ 262+ 30+2

70+7

11171630

741

30+3

63

270+4

1

0+511

0+146

0+31

0+17

453

22 SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 5857

R"s-~(

"rr"+) —(re+re) ~,

R '= (r,"-r"~

+ ~r'- r'~ . (21)

Here, R" is a measure of the difference betweenthe total effective core radii of atoms 4 and B(i.e. , size mismatch), whereas R,"s measures thesum of the orbital nonlocality of the s and p elec-trons on each site. Using the definition (Eq. 21)and the values of the orbital radii given in Table I,we construct R," vs R," maps for the binary corn™

in the most stable crystal form of each compoundand in a structure that appears in the phase dia-gram at (or close to) a 50 at. %-50 at. % composi-tion. We started the compilation by reviewingstandard tables, """as well. as a number ofbasic papers (e.g. , Befs. 96—99), which give use-ful tables for particular structures. Whenever wehave identified in this literature either a conflictin, assigning a crystal structure or expressions ofdoubt as to the identification of the structure, oc-currence of other structures at somewhat differentpressures or temperatures, substantial deviationfrom 1:1stoichiometry, etc. , we have made useof a computer-assisted literature search to findthe original papers for the compounds in question.In this way, we have surveyed some 180 refer-ences. We have identified from standard sourcesas well as from an extended computer search atotal of 565 binary AB compounds that are near-stoichiometric, ordered, and formed from atomsbelonging to the first five rows of the PeriodicTable. Their distribution among the various crys-tal structures is given in Table II. The compoundsincluded are listed in Appendix I according to theirstructural groups.

This data base of 565 binary compounds exhibitsan enormous range of physical, structural, andchemical. properties. Using the terminology of thesemiclassical structural factors, one notes thelarge range of conductivity properties spanned bythese compounds (insulators, semiconductors,semi-metals, metals, superconductors), the elec-tronegativity difference between the constitutingatoms (covalent versus highly ionic), coordinationnumbers (12 to 2), relative ionic sizes of the A andthe B atom, bonding type (covalent, ionic, metal-lic, etc.), range of heats of formation (=1-150kcal/mole), electron-per-atom ratios (=1.5 to8-9), etc. Given this distribution of the 112 octetcompounds and 453 nonoctet compounds into 5 and31 different crystal structures, respectively ex-hibiting a diverse range of properties, we now askhow well can the atomically derived orbital- radiischeme explain such a distribution.

We construct from the s and p atomic-orbitalradii the dual coordinates for an AB compound as:

4.0

3.5-

B2

3.0 -g

2.5-

2.0-

4 0lK

1.5-

1.0-

0.5-

0.0-

0.0

o +&~b

848 0(

0o~ o.88 I

op as o

bbb b

bbbA4

I

0.5

R (a.u. )

I

1.0

FIG. 19. Structural separation plot for the 112binary octet compounds A~B' ~, obtained with thedensity-functional orbital radii, with

B"s=)

(r'+ r ) (r,'+ r,')~, —

a,"'= fH-r,'f+ /r,'-r,'f.

pounds. Such maps are shown for 112 octet com-pounds in Fig. 19 and for 356 of the nonoctet com-pounds in Fig. 20. (Since the symbols in Fig. 20are crowded, if any two overlapped they wereartificially moved to show them separately).

We identify each structure by a different symboland search in the R," -R,"~ plane for the smallestnumber of straight lines, enclosing minimal areas,best separating the different structures. In somecases, there exists a unique solution to this topo-logical problem; in other cases (e.g. , B33 and thecP64 and tI64 structures), there are a number ofpermissible solutions. However, in these cases itseems to make little difference which line ischosen. While we could have lowered the numberof "misplaced" compounds by using more complexlines, we feel that the more stringent criterion ofusing straight lines provides us with a better chance

A LK X ZUNGER

3.0- Legend

0 BS(N.CtjO B2i~tiV Ba, fNiA I

Q B16[GeSID BRtMnPIV B32[NaTI)0 B33:-Bt(CrB8 oP64(KG )

& g 8 L1o{CuAu)~ mC24[AaGee mC32[NaSi8 mP16{AaLt)

OC16[NaHg)—1.S -0 tl64(NaPb)~O

1.0—

0.0—

V'Vivat, & & 0@ 0

tt)t V ~ 0 0 0

o o oooo

V V ~ QV ~O

VV V

mc 24oC 16

I

0.0 0.5R (a.u.)

I

1.0

FIG. 20. Structural separation plot for the 356 binary nonoctet coxnpounds, obtained with the density-functional orbitalradii, witli R, = ((r&+ re)-(t&+ r,)[, R, =

I tq —rs I+ I+t —+ I

'

of assessing the true success of the method.The remarkable result of these plots is that with

the same linear combination of atomic otbital-xadii, most of the structures to which the 468 com-pounds appearing in these plots belong can beseparated. The relative locations of the structuraldomains seems chemically reasonable. Hence,the 827 833 (coordina-tion number CN=7) is inter-mediate between the 81 structure (CN =6) and the82-I.le structures (CN = 8). The mostly metallicnonoctet compounds appear separate from the non-metallic regime to the right (cP64, tI64, mc32,mP16, oC16), etc. Within single structural groupsone similarly finds a chemically reasonable order-ing of compounds, e.g. , polymorphic compounds(such as SiC) appear near border lines, zinc-blende-rocksalt pairs that intertransform at lowpressure appear along the I33-B1 separating line,etc. Note that even the wurtzite-zinc-blendestructures, which only differ starting from thethird-nearest neighbors, are well separated. How-

ever, there are a number of crystallographicallyclosely related structures that overlap: I am un-able to separate the nonoctet crystal type CsC1(82) from the CuAu (I.le structure), the NiAs type(BS,) from the MnP type (831), and the CrB type(833) from the FeB type (827), etc. For clarityof display, I show the extra 81 compounds separa-tely in Fig. 21, however, using Precisely the samesePaxating lines as used fox all other nonoctetcompounds (Wg. 20).

It is not surprising that some of these structuralpairs overlap. For instance, the B27 and B33str'uctures have a common structural unit consist-ing of a row of trigonal prisms of atom g stackedside by side and centered by a zigzag chain of Batoms. "'" The structural similarity between CsC1(82) and CuAu (Llu) has been discussed by Hume-Hothery and ~ynor, ' the relation between the NiAs(88,), MnP (831), and the FeSi (820) structuresby Schubert and Eslinger, "and that between theCsC1 (82), AuCd (819), and Cu Ti (Bll) by Pear-

S YSTKMATIZATIOÃ OF THE STABLE CRYSTAL STRUCTURE OF ALL. . .

3$- p

08

84-e

e2.5-

8

Legend

Bb (QoB19 (A811 (C820 (FeB27 (FeB3? (SeB35 (Chp24 (LoC16 (Ko18 (RbmP32 (tI32 (LIoP16 (N

pounds. All the superoctet compounds (a total of34) could be separated clearly as well (Fig. 22 anddiscussion below). This illustrates the predictiveability of the present approach.

If one is to consider the pairs of related crystalstructures mentioned above as belonging to singlegeneralized structural groups, the total numberof misplaced compounds (5 octet and 32 nonoctet)forms only 7% of the total data base of binarycompounds. The present theory is hence morethan 90/o successful.

2.0—B. Discussion of "misplaced" compounds

1.5-0K

1.0-

as-

OA)-

mP32

Qo

B20B3?B35

A e

60

~hO0e% p

OOO oO

qO

819+811

I

0.5

R (a.u)

I

1.0

FIG. 21. Structural separation plot for the 81 binarynonoctet compounds, obtained with the density-functionalorbital radii, with

son. ' In fact, examination of the thermochemicaldata (Refs. 12 and 13 and references therein) in-dicates that if a certain compound exists in two ofthese related structures at somewhat differenttemperatures, the difference in their standardheats of formation is often as small as 0.3 kcal/mole. (For example, AgCd in the B19 structurehas AH=0. 094+0.004 eV, whereas the B2 struc-ture has a heat formation of 0.080 + 0.004 eV. )Also, some of these presently unseparated struc-tures indeed appear as mixtures when preparedfrom the melt (e.g. , as noted by Hohnke andParthe, ""'both the B27 and the 833 structuresare frequently found in the same arc-melted but-tons of these compounds).

Since the publication of a preliminary report ofthis work, "which included 495 compounds, we havebeen made aware of the crystal structures of 54more octet and suboctet compounds as well as 16new superoctet compounds (i.e. , a total of 565compounds). We find that the lines separating thestructural domains of the octet and suboctet com-pounds need not be changed relative to their pre-vious assignment" to incorporate the 54 new com-

The compounds that are misplaced in the pre-sent theory (i.e., their (8",s, B",s} coordinatesplace them in a differerit structural domain thanthat reported in the literature surveyed) are listedin Table III together with their {B",s, A", s} coordi-nates. In cases where a compound appears inoverlapping domains or close to a border line, weindicate all the pertinent structures. Given their(8",s, R",s} coordinates, the reader can conve-niently identify them on structural plots.

The list of misplaced compounds shows a num-ber of interesting features. At least two com-pounds, CuF (Table III, No. 1) and FeC (No. 16),reported to have the B3 (Ref. 90) and B1 (Ref. 92)structures, respectively, probably do not exist atall [for CuF see Ref. 100; for FeC, Ref. 101(a)].Their misplacement in the present theory ishence a gratifying feature. Similarly, whileOsSi (No. 11) is sometimes reported to have the82 structure" and appears in our plots in theB20 domain (No. 11 in Table III), it is known toactually have the B20 structure and appears inB2 only with impurities. ' 'b' In addition, thecompound PtB (No. 18 in Table III) which has beenreported to have the NiAs (B8,) structure" and

is placed-in our plots in an entirely unrelatedstructural domain, has been found to have ananti-¹As structure. '"+' ¹Y(No. 30) was re-ported in 1960 [Ref. 93(b), p. 678] to have theorthorombic &27 structure, in conflict with theprediction of the present scheme, while in 1964[Ref. S3(c), p. 561] it was concluded, that it isactually monoclinic with a P2,/a space group. Itseems that no clear identification for this struc-ture is yet available. While Hfpt (No. 21) isidentified in most sources as having a 833 ortho-rhombic structure, a deformed B2 modificationhas also been reported [e.g., Ref. 93(c), p. 419].Similarly, AuBe (No. 15) has been reported in1947 to have the B20 structure [e.g. , Ref. 93(b),p. 83] while in 1S62 [e.g. , Ref. 93(c), p. 64] it hasbeen identified as tetragonal. AuLa (No. 29) hasbeen reported to transform from its high-temper-

5860 ALEX ZUNGER

TABLE III. Compounds which are misplaced in thepresent theory (out of a total of 565).

ExpectedCompound structure

Structuraldomain(s) inwhich it is

found

gAB

(a.u.)

gAB

(a.u. )

1. CuF2. MgS3. BeO4. Mg Te5. MgSe

B3Bl8484Bl

Octet

BlB3Bl-B3Bl-B3Bl-B3-B4

1.6350.930.6150.360.745

0.3750.250.3050.320.285

1. CoAl. 2. FeAl3. NiAl4. CoGa5. FeGa6. NiGa7. NiIn8. MnIn9. CoPt

10. TiAl11. OsSi12. CoBe13. MBe14. NaPb15. AuBe16. FeC17. TiB18. PtB19. IrPb20. AgCa21. Hfpt22. ¹iHf23. NiLa24. NiZr25. PtLa26. RhLa27. Zr Pt28. PdLa29. AuLa30. NiY31. PtY32. LaCu

82B2B2B2B2B2B282Ll p

Ll p

828282@6482081BlBS)88(B33B33B33833B33B33833B33833B27-B33B27B27827

Nonoctet

BSg-B31-B20BSi-B31-820BS)-B31-B20BSg-B31-820BSg-B31-B20B81-B31-B20BSg-B31-B20BSi-B31-B20BSg-B31-820BS(-B31-B20BSg-B31-820BSg-B31-820B33-827B2 Llo B32Bl-B33-827Bl-BSg-B31Bl-B33-827Bl-B33-B278Si-B31-B2Bl-Llp

LloB2-LlpB2-LloB2-LlpB2-LlpB2 Ll p

B2-LlpB2-LlpB2-LlpB2-Ll

p

B2-LlpB2-Llp

0.3450.4350.5050.3250.4150.4850.130.170.680.9051.230.94i.370.561.581.471.7851.9050.5380.6250.210.730.900.6450.380.560.1250.630.420.760.241.04

0.3150.3450.3950.3550.3850.4350.43p.410.400.4150.370.380.490.52p 440.35p 4450.3850.4780.645 .

0.530.570.590.5550.550.630.5150.620.570.56'0.520.61

ature B33 form to a low-temperature $27 form(both orthorhombic) at about 660 'C, while in 1963[Ref. 93(c), p. 73] it is indicated to have thecubic B2 form. It is hence clear that for some ofthe misplaced compounds, it is not yet obviouswhether their misplacement is real. For theother compounds appearing in Table III, theirmisplacement in the present phase diagrams isreal and brings up a number of interesting ob-servations.

The octet compounds MgS and MgSe have a

NaC1 (Bl) structure but appear in our plot in theZnS (B3) domain, near the B1 border. Experi-mentally (e.g. , Ref. 102) it is found that the (nor-malized) free energy of the B3 B-1 phase transi-tion for these compounds is nearly zero.

A large number of the other misplace com-pounds have unusual properties. Two such groupsof compounds show systematic unusual proper-ties: the six Al and Ga compounds wit/ the mag-netic 3d transition elements (CoAl, CoGa, FeAl,FeGa, NiAl, and NiGa) and the group of ten CrBand three FeB structures (AgCa, HfPt, ¹Hf,¹La,MZr, PtLa, RhLa, ZrPt, PdLa, and AuLa,and NiY, PtY, and l,aCu, respectively).

The first group has the CsCl (B2) structure butappears in the present theory in the domain of the%As-MnP-FeSi structures. Their electric andmagnetic properties have been studied intensivelyin the last few years (e.g. , Refs. 103-110). Itappears that these compounds are stabilized bythe presence of defects, and they have a large,stable range of composition (45 at. %-55 at. %%uo),

wjth abrupt changes in many physical propertiesnear stoichiometric composition. Susceptibilitymeasurements as a function of magnetic fieldshow ferromagnetic impurities and antistructuredefects in such materials. More importantly, aslight nonstoichiometry often leads to the forma-tion of local magnetic moments. These resultsindicate (e.g., Ref. 110) that such slightly offstoichiometric materials are in effect spin-glasses at low temperatures. Their magnetic be-havior is intermediate between that of the inde-pendent magnetic-impurity problem (Kondo effect)and that characteristic of antiferromagnetic orferromagnetic systems having long-range orderdue to strong magnetic interactions. It is inter-esting to note that this subgroup of compoundsexhibiting stable intrinsic defects leading to mag-netic moments, are displaced in the structuralplots from the largely nonmagnetic B2 domain tothe B8, region in which many of the stoichiometxicalloys have permanent local moments. There areindications that this subgroup of compounds havecertain structural anomalies: Many authors re-port that the B2 aluminides could not be obtainedas a single phase, and in diffraction the B2 pat-tern could not be separated from other diffractionlines not belonging to this structure. " It was sug-gested (Schob and Parthe )'7 that many of thesecompounds are only metastable in the 82 struc-ture. In a recent diffraction study'" it was dis-covered that strong distortions occur around theCo sublattice sites in CoGa due to intrinsic vacan-cies. Similarly, a recent calculation'~'b' of theordering energy in FeGa using a Bragg-Williamsmodel, yielded very small interaction parameters

22 SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 5861

of 0.049 and 0.03 eV (i.e., -2 —lkT) for first andsecond neighbors, respectively. It is intriguingthat the present orbital-radii scheme has theability of identifying such unusual phenomena ina few of the 122 tabulated B2 nonoctet compounds.

Our scheme suggests that similar irregularitiesmay occur in NiIn and TiA1 and perhaps even inCoBe, but to a much smaller extent, since thesecompounds are only marginally misplaced in thepresent theory. Indeed, the absorption spectra ofMIn in the 0.7-5.5-eV range'" indicate almostno change with composition in its Drude regime aswell as above it, suggesting a constant number ofelectrons per cell due to the defect structure.This suggests that many of the unusual magneticand structural properties found in the FeAl, FeGa,CoAl, CoGa, NiAl, ~NiGa group may also be foundin NiIn.

The second large group of misplaced compounds(numbers 20-32 in Table III) form a distinctstructural group. Schob and Parthd, " Hiegerand Parthe, " and Hohnke. and Parthd""' haveindicated that all CrB (B33) and FeB (B27) com-pounds can be separated into two groups: group I,in which a transition-metal atom combines withan s-p element (B, Si, Ge, Al, Ga, Sn, or Pb,and group II, in which a transition element fromthe third or fourth group combines either withanother transition element from group 8 or fromthe Cu group. It was found that the individualtrigonal prisms in both the FeB and the CrBstructures have different relative dimensionsin groups I and II. In particular, group I of theCrB structure shows a "normal" a/c ratio greaterthan 1, but group II compounds show a compressedprism with a/c&1. Only three compounds belong-ing to group I (HfA1, ZrA1, and YAI) have a/c & l.We find that all of the CrB and FeB compoundsthat belong to group II are misplaced by our theoryinto the bordering CsCl domain, whereas thethree group-I compounds that have a/c &1 (muchlike group-II compounds) are correctly placed.Hence, the present approach is sufficiently sen-sitive to separate the true physical irregularityeven when simple structural factors such as thea/c ratio lead to the wrong conclusion. From theresults of the present approach, it would seemthat group-II compounds of the FeB and CrBstructures shouM properly be identified as aseparate group. As a result, if the errors madeby the present theory for the latter group of com-pounds, as well as the errors in the 1-8 (TableIII) local-moment materials are regarded assystematic irregularities, the remaining trueerrors amount to only 2%%uo of the total data base.

The placement of a number of B33 and B27orthorhombic compounds in the CsC1 (82) region

of the present structural plots has an interestingimplication on their electronic structure. Thecubic B2 structure of intermetallic compoundscontaining a magnetic element is stabilized by apartial ionic character and equiatomic composi-tion. The experimental proof for that is given bythe fact that the 82-type FeTi compounds has analmost vanishing atomic moment for Fe, indi-cating an ionic electron transfer which leads to anearly zero spin polarization. Any deviation fromstoichiometric composition leads to lattice dis-tortions and defect structures attempting to ac-commodate the local magnetic interactions.Hence, the normally B2 compounds FeAl, FeGa,NiAl, NiGa, CoA1, and CoGa may be distortedby off-stoichiometry and are indeed displaced inthe structural plots. Similarly, some, of the B2-displaced B33 compounds (e.g. , Table III No. 24,NiZr) do exist in a B2-type phase" +' but maybe distorted to a orthorhombic 833 phase due tooff-stoichiometry. Simultaneous measure me ntsof the composition dependence of the Curie pointand structural distortions may elucidate thispoint.

The remaining misplaced compounds may alsohave unusual properties; NaPb (No. 14) has anunusual structure with 64 atoms per cell resem-bling a molecular crystal with interacting Pb4tetrahedra"' and IrPb (No. 19), according toMiedema's model, ' has a positive heat of forma-tion of about 1 kcal/mole. Similarly, AgCa(No. 20) has been recently discovered'" to be oneof the only known glass-forming materials thatdo not contain a transition or actinide el.ement. Ithas also been noted4' that the ratio of anion-anionto cation-'anion distances in AgCa is almost anorder of magnitude smaller than in all other non-transition-metal B33 compounds, and that unlikethe latter group of compounds it has catalyticproperties in redox reactions.

CoPt (No. 9) has a tetragonal I 1, structure but

is displaced in the structural plots into the B8,domain, much like the FeAl, FeGa, CoA1, CoGa,NiAl, ¹Gagroup. By quenching and annealing itbelow its order-disorder temperature in the pres-ence of a magnetic field, it develops a uniaxialmagnetic anisotropy leading eventually to a sin-gle-ordered system having its tetragonal axis inthe direction of the applied magnetic field. '~

This material forms an excellent digital magneto-optic recording system. "' Among its otherstructural peculiarities'" it has been noted"' thatthe c/a ratio calculated from the position. of themain diffraction lines is considerably differentfrom the c/a ratio evaluated from the superlativelines, suggesting that this crystal actually con-sists of domains of different long-range order and

5862 ALEX ZUNGEB,

hence different tetragonalities.It is interesting to note that the present scheme

also predicts unusual electronic ProPerties ofcompounds belonging to the same structuralgroup. For instance, the B2 compounds CsAu andRbAu that appear in Fig. 20 as isolated from theother 147 B2+I-10 suboctet compounds have semi-conducting properties while all other suboctetcompounds belonging to these structures seem tobe metals. A recent calculation of the electronicband structure of CsAu (Ref. 119) has indicatedthat if relativistic corrections are neglected,Cshu appears to be a metal, which disagrees withexperiment, whereas the inclusion of relativisticeffects lowers the Au s valence band to form asemiconductor. It is remarkable indeed that suchcomplicated electronic-structure factors are re-quired in quantum-mechanical band-structure cal-culations to reveal the unusual semiconducting be-havior suggested here simply by the atomic-or-bital radii.

lf the predictive power of the present orbital-radii scheme in relation to unusual electronicproperties is not accidental, it would be inter-esting to speculate on its consequences. Onewould guess, for instance, that all suboctet non-transi gi on-element compounds having /"„~ largerthan roughly 0.7 a.u. are nonmetals. This includesnot only the known nonmetallic compounds belong-ing to the LiAs group (KSb, NaGe, NaSb, but notLiAs) and the KGe group (CsGe, CsSi, KGe, RbGe,and RbSi whereas KSi is a borderline case), butalso the tf64 (NaPb) group (CsPb, CsSn, KPb,KSn, RbPb, and RbSn, but not NaPb), the B2compounds LiAu and LiHg, the I1, compoundNaBi, the 0C'16 compound NaHg, and the mC32compound NaSi. In the sequence of alkali-goldcompounds LiAu, NaAu, KAu, RbAu, and CsAu,one would similarly predict that the transitionbetween metallic and insulating behavior occursbetween NaAu and KAu.

Another demonstration of the usefulness of thepresent approach in systematizing propertieswithin a given structural group is the applicationto the c/a axial ratio of the NiAs compounds. Themeasured axial ratios of the $8, compounds ofTi, V, Cr, Mn, Fe, Co, and Ni with S, Se, Te,As, Sb, and Bi show very large variations aroundthe "ideal" value of c/a = 1.633 (ranging from 1.95for TiS to 1.25 for FeSb). Defining the deviationfrom this value as 4=c/a —1.633, one immedi-ately recognizes that the known semiconductorsin this group have mostly 4&0, whereas the me-tallic NiAs compounds usually have 4&0. Onethen finds, using the ionicity coordinate B, , thatcompounds with B,"~&0.9 a.u. have 4&0 as thesize mismatch of the effective cores

) (r~+&, )

—(r~~+r, ) ~is too large to accomodate an ionic

structure. ¹S(with 4=0.073 and ft~s=1.08) isthe only exception to this rule. A similar corre-lation of 4 with 8,"~ exists for the wurzite B4compounds, ""although the range of 4 is an or~der of magnitude smaller than it is in the B8,compound s.

C. Superoctet compounds and additional structural plots

The relative orientation of the structural do-mains in Figs. 19-21 suggest that no single co-ordinate will suffice to produce a complete topo-logical separation between all structures. Since,however, the area of the 8", vs B~ plane seemsto be more or less bound (e.g. , = 3 a.u. ' in Fig.20), when extra compounds are added (i.e., com-pare the 495 compounds included in Ref. 57 withthe 559 compounds in Figs. 19-21), it is likelythat the two-dimensionality of this finite A", ~-B~~space will eventually preclude the delineation offurther structures. One may hence expect that forsome critical number of structures and com-pounds, a third coordinate may be needed. Sucha generalized multidimensional resolution ofstructural groups may also resolve some of theremaining discrepancies in the present theory.Our present approach, however, is aimed atdemonstrating the extent of structure delineationpossible with the minimum number of two coodi-nates using the. simplest possible separating lines.

One simple example for an additional coordinateis the classical~ valence-electron concentration(VEC), measuring for the binary AB system thetotal number of valence electrons Z"+ Zs [cf.,Eg. (10)] in the compound. In the semiclassicalapproaches to structure it is known that whereasthe VEC value alone does not separate differentcrystal structures, compounds with the samevalence-electron number often belong to the samebroad structural groups. One can use this addi-tional coordinate together with our orbital radiito obtain a better structural resolution of themarginally resolved structures, and at the sametime provide a clear structural delineation of allsujexoctet (i.e., Z~+Z„& 8) compounds. As withthe suboctet compounds, no previous approach hassucceeded in systematizing these complex crystalstructure s.

We find that while the definition of the structuralcoordinates R",s and 8,"s [Eq. (21)] used for theoctet (Fig. 19) and mostly suboctet (Figs. 20 and21) compounds yields an overal separation alsoof the superoctet compounds, a more sensitivedelineation is obtained with the slightly modifiedcoordinates 8, = Jr~a —r~~~ and B2=(&, ) sug-gested by Littlewood. "' Here, A, is a measure

SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 5863

SIPSUPER OCTET

16 AB Compounds (8)(VKC = 9)

SUP EROCTKT18 AS Compounds (b)

(VEC = 10)

aSIAS7- p 0

GeS Sns

ctl5

gg 4

„GePoP8

olnS

GaS hP8, hR2~GaSe ~ InSe oTIS

oInTe oTISeGaTe a

HgCIakgBrtI8

mC24TITetI32 kgI

GeSeoSnSe

PbSoGeTe SnAs

SnTeoBiTe

PbTe

oBISe TICimo

PbSeTIBr

PbPoB2

B10a SnO

aPbO

TIF

00.0

I

0,5I

0.5 1.0 0.0 1.0

~r,'- r,"~ (a.u)

FIG. 22. Structural separation plot for the 34 super-octet compounds A B + with valence electron concen-trations (VEC) equal to 9 [Fig. 22(a)] and 10 fFig. 22(b)].The structural groups are defined in Table III and in

AppendixA. The symbolR[Fig. 22(b)] denoted arhombo-hedral structure. In the 91 domain of Fig. 22(b) we havealso included the compounds with VEC= 9, 11 (see text).

of the P-orbital electronegativity difference be-tween atoms & and B, while 8, measures the s-pnonlocality on the two sites. The reason that~r~s r-~~ forms a better structural coordinate forsuperoctet compounds is that these systems in-volve relatively heavy atoms (e.g. , Pb, Sn, Bi,Tl, Hg) for which the s electrons are paired and

strongly bound relative to the p electrons. Hence,these semicore s orbitals become chemically in-active and only the contribution of the p electronsneeds to be included in the electronegativityparameter 8,.

Figures 22(a) and 22(b) shows structural plotsfor the superoctet compounds with nine and tenvalence electrons, respectively. Since there areonly a few 91 super octet compounds, we haveincluded those with VEC=9 (SnAs), VEC=10 (PbS,PbSe, PbTe, and PbPo) and VEC=11 (BiSe, BiTe)on the same plot in Fig. 22(b). Figure 22 includes18 compounds which have appeared in the previousnonoctet plots (Figs. 20 and 21): B87 (lnTe,T1Se, T1S), mC24 (GaTe, GeAs, GeP, SiAs), B16(GeS, SnS, GeSe, SnSe, InS), pseudo-B8, (GaS,

Miedema Parameters

oo00 oo oCa ~ ~

a **0 S~ 0

0 QO Z3 0

o~ QGD

o

ac ~o~ ~ Uo ~ o~

6 EB ZS 0BPIBQ~ g9 o

e) oo

LE( ENDCl- Bl'V- HeiZ - EI3&8- CP64El-, 7IB&4- e320- e20- EI33X- O~S6- i10H- NC24El- ~C32EE- DC16+- NP1EI

-0. 10 0.05 0.20 0, 35I

0. 50

R20.65 0. SO 0.95 1. 10

FIG. 23. Miedema plot for nonoctet compounds. R2 is the difference in the effective elemental workfunctions,R2=l P& —Pgl; while R~=ln~~ -nsV l, where n is the cell boundary density (Ref. 91.

5864 ALEX ZUNGEH,

GaSe, Ge Te, SnTe, InSe), and the orthorhombic-ally distorted B1 compound TlF. It is seen thatthe seven different structures of the VEC=9 com-pounds, as well as the six different VEC=10structures, are very clearly resolved, the onlyexception out of these 34 compounds being the B37compound TlS which is marginally displaced intothe neighboring hP8 hR2-domain [Fig. 22(a)].

It is important to emphasize that the ability toseparate structures shown by the present orbitalradii (Figs. 19-21) is far from being trivial oraccidental. This is demonstrated in Figs. 23-25,where structural plots are presented for some ofthe nonoctet compounds using different coordi-.nates. We use Miedema's' coordinates, 8,= ~P„*

-eel and &.= In~"'-u~s"'~ in Fig. .», ~h~~~P„* and n„*'~s are the effective elemental workfunction and chill-boundary density to the power of1s. Those coordinates were extremely success-ful in predicting the signs (and often the magni-tudes) of the heats of formation of more than 500compounds. ' In Fig. 24, we present a Mooser-Pearson4 plot, where A, is the elemental electro-negativity difference, and 8, is the average prin-cipal guantum number. In Fig. 25, we give a plot

using Shaw's parameters, "where R, is the ele-mental electronegativity difference and R, =a (Z„+ Zs)/[a (tt„+ns)]', where Z„and ri„are theatomic number and the principal quantum numberof the outer valence orbital, respectively.

In a Miedema plot (Fig. 23), one notices a roughseparation of the Bl and B8, structures (CsAu andHbAu appear, as in our case, at high AC*,dais'~'), whereas most other structures arenearly indistinguishable. This illustrates thegreat diQiculty in carrying the success of a theorythat predicts global binding energies &$0 into theprediction of st uctural energies ~, (cf. Sec. I).

The Mooser- Pearson plot for these compounds(Fig. 24) appears visually as if only 104 com-pounds (i.e., isolated points) are plotted. In fact,it includes 360 compounds belonging to 14 dif-ferent structures. This strong overlap of dif-ferent structures on the same (R„Bs) coordinatereflects the insensitivity of the scale to separatingsuch structures. A somewhat better separationis evident using Shaw's parameter (Fig. 25), butthe overlap of different structures is still ex-tremely large.

It is likely that one could construct first-prin-

I HOSER-PtFIRSON PFlRFII, ETERS

9 U C) Q O GHO

V 9 9 Q 0 Q Q Q HBO 0

9 lm II lm m) QCII QC ~ Il 0 Cl OO

0 0

QOO+OOI5+OL)GClCICl Cl

LE( ENQ0- BlV- B816- BS18- cp6&Z- TI644- BS20- BZ0- Bs~X- B168- L108- tiC24l5- t1C32EE- QC18+- NP16

-0. 1

I 1

0, 3 0. 7I

1.5

Rp2. 7

FIG. 24. Mooser-Pearson (Ref. 4) plot for 360 nonoctet compounds. Rt=l Xz —Xs l; Rt s(nz+ ns), =where Xz and

n~ are the elemental electronegativity and principal quantum number of the outer orbital, respectively.

SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL

Shaw Parameters

0 00

00

C) o03 Igl 0 cP P

oI-j

+oR

I 'INI "'LOI o & 'o

& 5 8 o%g

LEBENDQ- 81T- 881

831+- CP64H- TI60+- 8320- 820- 833X- 816- L1OIH- NC20Q- NC32g- OC16+ - NP16

-0. 1 0.5 0.8

R2

I

1.7 2.0 2. 3 2.6

FIG. 25. Structural separation plot for the nouoctet compounds using Shaw's (Ref. 72) coordinates R&= ~X&-Xs(, Rt1=a(Z~+ Zs)/[a (a&+ ns)], where Xz, Zz, and n~ are the elemental electronegativity, atomic number, and principal

quantum number of the outer orbital, respectively,

ciples atomic pseudopotential using somewhatdifferent procedures than have been used here(Sec. III A) for defining the pseudo-wave-func-tions. While this would result in a different set oforbital radii, they wi11 most likely scale withthe present set of radii. Consequently, one mayexpect that the systematization of the crystalstructures, based on such radii, will be essen-tially unchanged.

D. Discussion

As indicated in the Introduction, the success ofthe s and p orbital-radii shceme in correctly pre-dicting most of ihe structural regularities of &Bcompounds suggests that the stmctmral part ~,of the cohesive energy may be dominated by s-pelectrons, even in transition-metal systems. Theeffect of the d electrons then enters indirectly viathe screening term in Eq. (16). This points to thepossibility that while the localized d electronsdetermine central-cell effects and hence also theregularities in the structure-insensj. tive part ~,

of the total cohesive energy ~=~,+ ~, (withLREc» ~,), the longer-range s-p wave functionsare responsible for the stabilization of a certaincomplex space-group arrangement over another. "This may seem as a somewhat surprising result,in view of the fact that the resonant tight-bi. ndingmodels" '~ have explained the periodic trendsin 4$, of both elemental and alloyed transition-metal systems by considering changes in the rec-tangular distribution of the one-electron d energylevels alone. The two views need not, however,be contradictory. This point is discussed below.

The analysis of molecular and crystal bindingin variational quantum-mechanical theories interms of the predominance of a certain subset ofwave functions with a well defined (majority) ang-ular symmetry (e.g., s, P, and d), is inherentlyqualitiative. In fact, while the systematic errorsintroduced by such, approximations are oftensmall enough on the scale of ~c (so that trendsin global cohesive energies ~ may be repro-duced), they are frequently of the order of 4E,itself, or even larger. For example, the analysis

ALEX ZUN GER

of the computed formation energy ~ of NiAl interms of the differences 4, in the one-electron dorbital (l =2) energies of the compound and theelemental solids, 'shows that 4, differs by 30%from 4H. Such theories will be appropriate forpx'ed'. CQ.Q~ trends in stx"QctUrz). I, enex'gles only xn

the rare event where the structural energy ~,is much larger than about 20/g of the global bin-ding energy.

%hereas the su~:ia3ionaL total energy of a poly-atomic system is not even decomposable intoatomic site and angular momentum compounds,under suitable approximations the final totalenergy (i.e., the negative of one-half the kineticenergy) is amenable to such a partitioning. How-ever, in cases where such a decomposition hasbeen a.ttempted, ' one has obtained the somewhatcounter intuitive result that changes in the (outer)core orbitals in forming the solid from atoms aremainly responsible for the cohesion.

Such results have encouraged the utilization ofthe difference in the sum of one-electron energiesh, =Z, &~«~-Z, a", , -Z, cs, between the compound&8 and its constituents & and 8, as a measure ofthe cohesive or formabon energy of &S. Thxspermits the decomposition of binding energiesinto angular components, from which one mayhope to a,ssess the dominance of a certain term.While it was long known that Z,h, rarely shows aminimum as a function of structural coordinatesunless g, are empirically parametrized (as in theextended Huckel method (e.g., Hef. 122 and ref-erences therein), Hudenberg'" has discoveredthe remarkable result that gt equi/ib~ium, thetotal Hartree-Pock energy of a molecule equalsapproximately the sum of one-electron energies:Esp= kgZg &g, where k~ ls 811 empirical fac'tor.This energy relation is based on an earlier dis-covery by Politzer' showing that &sr=&2(V„+2V„,), where V„and V,„are, respectively, theelectron-nuclear and nuclear-nuclear potentialenergies and A, = -,'. These new energy relationshave been recently investigated in great detail inthe chemical literature. '" '" The picture thatemerges from these studies is that: (i) The er-rors made in approximating the total energy bysums of all of the occupied one-electron energiesis very large on the scale of binding energies ~,let alone molecular conformational (i.e., struc-tural) energies DE,. Even the trends (i.e., dif-ferential errors) in going from one system to theother are often obscured by such approximations,unless the constants k, and k2 are fine-tuned totake different values for atoms in different chem-ical enviroments. (ii) Whereas the Politzer-Hudenberg relations are valid only at the equilib-rium geometry, additional corrections for non-

equilibrium terms still fail to make these energyrelationships useful for finding approximate ge-ometries of molecules. Despite such gross in-accuracies, it would seem that further analysisalong these lines (e.g., Hef. 126) may be used toisolate:kn the future the major orbital contribu-tions to binding energies.

The realization that simple differences in thesums of all occupied one-electron energy levelsdo not form an adequate measure to the chemicaltrends apparent in the total energy leads to twopossible extensions. In the first approach"' onerepresents the total energy as a sum of all oc-cupied orbital energies plus an empirical classi-cal strain-energy term which is adjusted to pro-duce a minimum of the total energy at the observ-ed structural parameters. %bile this strain-energy term is small, it is crucial for physicallymeaningful predictions for the structure depen-dence of the total energy. Its inclusion, however,no longer permits the decomposition of the totalenergy into angular momentum components.

The second approach assumes that the total heatof formation scales with the difference &, betweenthe one-electron energy levels of the compoundand its constituent elemental solids, but only asingle l component (e.g. , d) is isolated as rele-vant for this scaling relation. 2' '4 The latter partof this statement is crucial: if one attempts, forinstance, to approximate the trends in the heat offormation of compounds made of a transition anda simple element by p, + &, + &, rather than by &,alone, no meaningful correlation is found'"%illiams et al.' ' have suggested an explanationfor this scaling relation in terms of a near can-cellation between the attractive Madelung inter-atomic potential and the repulsive intra-atomicCoulomb potential. If such a cancellation were tobe exact, trends in &, alone would parallel thoseobtained from variational total-energy calcula-tions. The basis of this approach is that whereasthe Madelung term V„ lowers (raises) the energyof states which occur predominantly on the neg-atively (positively) charged sublattice in an A'Bcompound, charge transfer to the negativelycharged species raises its intra-atomic Coulombenergy with respect to that of the positivelycharged species. Similar ideas where previouslytested for simple binary compounds"' by calcu-lating the sums [~„'(q)+V'„(q)] and [as(q)+ V„(Q)],where g„' and a~™ are orbital energies of specieswith formal charge Q and V„', V„are theMadelung potentials at the respective atomic site.By varying Q, it was observed that indeed a largecancellation occurs between z' and V'„, but thatthe residue of this cancellation is as large as10-209o of a„or V„, and hence significant on the

S YSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE OF ALL. . . 5867

scale of structural energy hE, . A similar resultwas recently reported'-3' for NiAl.

One would therefore conclude that from theschemes proposed so far for decomposing bindingenergies into angular momentum contributions, itis impossible to deduce the dominance of any par-ticular interaction (i.e. , d or s-P) in determingstructural re gularitie s.

The lack of contradiction between Friedel'sresonant tight-binding model and the significantstructural role assigned to the s-p coordinates inthe present model becomes obvious in consideringthe contributions entering the for, mer theory: itmay be that in binary &8 compounds with largedifferences in the constituent d-band energies,&~W) +E~(B), the s-P contribution to the struc-tural energy ~, is indeed dominant. If AE, isderived primarily from Brillouin-zone -inducedchanges in the gaps E» and these gaps can bedivided into two groups Egz and Esp (nonexis-tent in rectangular (f density of states models), ~7 2~

then the dominant structural role of s-p electronsbecomes evident if covalent hybridation leads toEs~z «E~sz~&E~(A) -E~(B). E~~ is then expectedto scale with A", much like the heteropolargaps.

The relationship between the structural stabil-ity of a polyatomic system and the degree of re-pulsiveness of the effective atomic cores of theconstituent atoms has been discussed in 1948 byPitzer" in a remarkable paper preceding allpseudopotential theories. While one might havethought then naively that the electron-core attrac-tion term Zgr w-ould lead to a strong penetra-tion by valence electrons of the core regions ofneighboring atoms, Pitzer has realized that the-

core electrons set up a repulsive potential with a,

characteristic radius inside which such a penetra-tion is discouraged. Hence, the triple-bond ener-gy of N=N is much higher than that of P-P (andthe bond length in N=-N is significantly shorterthan in P=-P) because the repulsive core size ofnitrogen is so much smaller than that of phos-phorous. Smilarly, the occurrence of multiplechemical bonds with first-row elements as com-pared with the rare occurrence of such bonds (witha small bond energy) with heavier atoms has beennaturally explained in terms of the large repul-sive core size of the latter elements. In addition,PItzer noted that whereas Pauling' has suggestedthat single bond energies (e.g. , N-N) should beroughly & of the tetrahedral bond energy (e.g. ,C-C), in fact the ratio of the two is closer to z.This discrepancy was simply explained"4 by thefact that the change from the bond angle of 90'characteristic of p-type single bonds, to a tetra-hedral angle of 109.5' minimized in the latter case

the overlap with the repulsive core.In the present orbital-radii approach, these

ideas are realized in a simple manner. To firstorder, the change in energy per atom introducedby incorporating an atom in a polyatomic systemis proportional to:

()Z- Qa, fap, (r)[((,(r)+ ~(rl]dr, (u)l

where hp, (r) is the lth component of the chargeredistribution, U, (r) is the Pauli repulsive poten-tial [Eq. (11)], 0(r) is the l-independent part ofthe atomic pseudopotential [Eq. (10)], and 4, areconstants. The first bp, (r)U, (r) term in E(l. (22)leads to a repulsive and angular-momentum-de-pendent contribution (for electron-attractingspecies) while the second term is isotropicallyattractive (for similar atoms). Neglecting forthis simple argument the nonlinear dependence ofthe charge-density redistribution 6p, (r) on U, (r),one notes that Pitzer's ideas on the destabilizingrole of large-core atoms, as well as the relativestability of structures that minimize such repul-sions through conformational changes in bondangles, are directly maninfested in Eq. (22). Onecould further note that charge-redistribution ef-fects occurring predominantly outside the pseudo-potential core [where U, (r) =0] do not contributeto such strongly directional repulsive terms. Itwould seem reasonable that the dominance of thecentrifugal barrier at small distances from theorigin will cause the charge-redistribution ef-fects in the high angular momentum componentsof the density to be confined to regions outsideU, (i.e., r&r, ). This simple picture clearly in-dicates the important structural role played bythe r, and r~ coordinates, as compared to higherangular momentum orbital coordinates.

VI. SUMMARY

It has been demonstrated that the pseudopoten-tial theory in its present nonempirical density-functional form is capable of providing transfer-able atomic pseudopotentials v(,"(r) that can beused both for performing reliable quantum-me-chanical electronic-structure calculations"" "and for defining semiclassical-like elementarylength and energy scales (r,]. The resulting radiicorrelate with a large number of classical con-structs that have been traditionally used to sys-tematize structural and chemical properties ofmany systems. At the same time, the orbitalradii derived here are capable of predicting thestable crystal structure of the 112 octet com-pounds (Fig. 19), 419 suboctet compounds (Figs.20 and 21), and 34 superoctet compounds (in Fig.

5868 ALEX ZUNGER,

22) with a remarkable success, exceeding 95 lo.The compounds for which the present theory doesnot predict the correct crystal structure are ana-lyzed and found to be largely characterized as de-fect structures with many unususal electronic,magnetic, and structural properties.

ACKNOWLEDGMENTS

I would like to thank A. ¹ Bloch, M. L. Cohen,and J. C. Phillips for inspiring and useful discus-sions. Special thanks are due to E. Parthe andF. Hulliger for reviewing the present compilationof crystal structures. I am grateful to L. Paulingfor bringing Ref. 134 to my attention and discus-sing related work.

APPENDIX

This appendix presents an alphabetic list of thestable structures of the binary AB compoundswhose atoms belong to the first five rows of thePeriodic Table. The headings include the stzuk-turberichte symbol (or Pearson's symbol, ifthe former is unavailable), the space-group nota-tion, and the crystal type, respectively. A singleasterisk indicates that there exists an uncertaintyin assigning the structure. due to off- stoichiometry,superstructures, possible metastability, existenceof a defect structure or a distorted structure. Twoasterisks indicate that there exists another com-peting phase adjacent to it in the phase diagram.Structures not plotted in Figs. 19-22 are denotedby square brackets. For nontransition- elementsystems we indicate if the compounds are super-octet (A«B" «) We do n.ot include in our list thehydrides (there is no hydrogen pseudopotential),some oxides and sulphides which do not occur intheir stable form as monomeric binary compounds(SiO, SiS, etc.), or binary systems which formcontinuous solutions or yet unknown structures.

The assignment of a compound as "octet" or"nonoctet" is sometimes arbitrary. Hence, therocksalt-type MX compounds with M =La, Sc, orY (d'S') and X=0, S, Se, Te (S'P') are sometimesassigned in the literature as nine-electron nonoctetsystems, while frequently the d' electrons of thegroup-IIIB element are considered as "chemicall. yinactive, " leading to the assignment of these com-pounds as "octet" or "pseudooctet. " We have listedthese compounds under octet, although this clas-sification does not alter the quality of separation inthe structural plots. The same considerations ap-ply to compounds such as MnS, MnSe, and MnTewhich can be considered as pseudo-octet if thefive d electrons of Mn are taken as "inactive. "

For some compounds it is difficult to establishfrom the existing literature which structure out of

B2, Pm3m, CsC1 type

CsBr, CsCl. , CsI.

B3', F43m, ZnS type

AgI, AlAs, AlP, AlSb, BAs, BN**, BP, Bepo,BeS, BeSe, BeTe, CSi**, CdPo, CdTe**, CuBr,CuC1, CuI**, GaAs, GaP, GaSb, HgTe** (alsoB9), InAs, InP, InSb**, MnSe**, ZnPo, ZnS**,ZnSe**, ZnTe**.

B4, P63mc, ZnO type

AlN, BeO, CdSe**, CdS**, GaN, HgSe** (also 89),HgS** (also 89), InN, MgTe, MnS, ZnO**.

C, Ge, Si, Sn.A4, Fd3m, diamond type

Ngnoctet compounds

Bl, Fm3m, NaC1 type, suboctet

CoO**, C rN, C rO, FeC **, FeO**, Hf8, HfC,HfN, MnO**, MoC**, NbC, NbN*, NbO*, NiO**,ScC, TaC*, TaO, TcN*, TiB, TiC*, TiN*, TiO*,VC, VN*, VO*, WC**, WN*, ZrB, ZrC, ZrN,ZO, zp**, zs**.

Bl, Fm3m, NaC1 type, superoctet:

BiSe, BiTe, PbPo, PbS, PbSe, PbTe, SnAs (alsoTlF**, an orthorhombically distorted 81 struclture).

B2, Pm3m, CsC1 type, suboctet

AgCd, AgLa, AgMg, AgSc, AgY*, AgZn, A1Sc,AuMg, AuSc, AuY, AuZn**, BaHg, CaCd, CaHg,CaIn, CaTl, CdBa, CdLa, CdSc, CdY, CoAl,CoBe, CoCa, CoGa, CoHf, CoSc, CoZr, CrPt,CsAu, CuBe, CuSc, CuY, CuZn*, FeA1, FeCa,FeCo, Feoa, FeRh~, HfOs*, HgLa, HgSc, HgY,

two competing structures is the most stable, e.g. ,AsCo in the B8, or B31 structures or RhSi in theB20 and B31 structures. In all cases where sucha "metastability" exists, we find that either thetwo structural groups are unseparable in the pre-sent theory (e.g. , BS, and B31) or that the com-pound occurs in our structural plots in the borderof the two competing structures (see text).

Octet compounds

Bl, Fm3m, NaC1 type

AgBr, AgCl, AgF, BaO, BaPo, BaS, BaSe, BaTe,CaO, CaPo, CaS, CaSe, CaTe, CdO, CsF, HgPo,KBr, KCl, KF, KI, LaAs, LaBi, LaN, LaO, LaP,LaS, LaSb, LaSe, LaTe, LiBr, LiCl, LiF, LiI,MgO, MgS**, MgSe**, NaBr, NaCl, NaF, NaI,RbBr, RbCl, RbF~ -RbI, ScAs, ScBi, ScN, ScP,ScS, ScSb, ScSe, SrO, SrPo, SrS, SrSe, SrTe,YAs, YBi, YO, YN, YP, YS, YSb, YSe, YTe.

22 SYSTEMATIZATION OF THE STABLE CRYSTAL STRUCTURE 5869

InLa, IrA1, IrQa, IrSc, LaT1, LiAg**, LiAu*,LiHg, LiPb*, LiTl, MgHg, MgLa, MgSc, MgSr*,MgT1, MgY*, MnAu*, MnHg, MnIn, MnIr* MnPd,MnPt**, MnRh*, MnZn*, NiAl, NiBe, NiQa,NiIn**, NiSc, NiZn**, OsA1, OsSi**, PdAl*,PdBe, PdCu, PdIn, PdLi, PdSc, PtSc, RbAu,ReAl, RhAl, RhQa, RhHf**, RhIn, RhMg, RhSc,RhY, RuAl, RuQa, RuHf, RuSc, RuTa*, SrCd,SrHg, SrTl', TcHf, TcTa, TiCo, TiFe**, TiIr,TiNi, TiOs, TiRe, TiRu, TiTc, TiZn, VMn, VOs,VHu*, VTc, YIn, YTl, ZnLa, ZnSc, ZnY, ZrOs,ZrPt**, ZrRu*, ZrZn.

TlBr, T1Cl

82, Pm3m, CsCI type, superoctet

810 (tP4)/nmm, PbO type, superoctet

PbO**, SnO**

Bl1, P4/nmm, CuTi type

HfAu, PdTa, TiAu, TiCd, TiCu, ZrAg, ZrCd.

816, Pnma, GeS type, superoctet

GeS, GeSe, SnS**, SnSe** (also InS, appearing inthe related OP8, Pnnnz structure. SnTe and QeTeare rhombohedral).

819,Pmma, AuCd type

AuCd**, MgCd, MoIr,' MoPt, MoRh, NbPt, TiPb,TiPd, TiPt, VPt, WIr**.

820, P2&3, FeSi type

AuBe, CoSi, CrQe, CrSi, FeSi, HfSb, HfSn,MnSi, PdGa, PtAl, PtGa, PtMg, ReSi, RhSn,RuQe, RuSi**, TcSi.

827, Pnma, FeB type

88&, P63/mmc, NiAs type

AuSn, CoS, CoSb, CoSe, CoTe, CrS*, CrSb,CrSe, CrTe*, CuQe*, CuSn**, FeS**, FeSb,FeSe**, FeSn**, FeTe**, IrPb, IrSb, IrSn, IrTe,MgPo, MnBi, MnSb, MnSn, MnTe, NiAs, NiBi,NiS**, NiSb, NiSe**, NiSn, NiTe**, PdBi~, PdSb,PdTe*, PtB, PtBi, PtPb, PtSb, PtSn, PtTe,RhBi, RhSe, RhTe, ScTe, TiAs**, TiP**, TiS**,TiSb, TiSe**, TiTe, VP, VS, VSb, VSe, VTe**,ZrAs*, ZrTe, [GaS** (hP8), GaSe** (hP8 and

hR2), InSe** (hR2) and InSn are superoctet sys-tems appearing together with CuS**, CuSe**, andTaSe** in somewhat related hexagonal structures].

831, Pnma, MnP type

AuGa, CoAs**, CoP, CrAs, Crp, peAs, Fep,IrQe, IrSi, MnAs, MnP, MoAs, NiQe, NiSi**,OsAs, OsP, PdGe, PdSi, PdSn, PtQe, ptSi, BhAs,RhQe, RhSb, RhSi**, RuAs, RuP, RuSb, VAs,WP.

832, Fd3m, NaT1 type

LiA1, LiCd, I.iQa, LiIn, LiZn**, Naln, NaT1.

B35, P6/mmm, COSn type

CoSn, FeGe, PtT1.

837, I4/mcm, SeTl type, superoctet

InTe, TlSe, TlS.

BI„P6m2, MOP type

MoP, OsC, RuC, TaN**.

cP64, P43n, KGe type

CsGe, CsSi, KGe, KSi, RbGe, RbSi.

fhPI2, P62m, NaO typei

CaAs, CaP, KS, KSe, NaO, NaS**, SrAs, SrP.

hP24, P6&/mmc, LiO type

LiO, NaS**, NaSe.

Llp, P4/mmm, CuAu type

CoPt, CuAu**, FeNi, FePd, FePt*, HfAg, I iBi,, MgIn, MnNi, NaBi, NbIr**, NbRh*, NiPt, pdCd,PdHg**, PdMg*, PdZn, PtCd*, PtHg*, Pt Zn**,TiAg, TiAl, TiQa, TiHg, TiRh, VIr*~, ZrHg.

mC24, C2/m, AsGe type, superoctet

QaTe**, GeAs, GeP, SiAs.

NaSi.

mC32, CZ/c, NaSi type

833, Cmcm, CrB type

AgCa**, AlLa**, AlY, AuLa, BaGa, BaGe, BaPb,BaSi, BaSn, CaGa, CaGe, CaSi, CaSn, CrB,QaLa, GaY, HfAl, HfPt, MoB**, NbB, NiB, NiHf,NiLa, NiZr, PdLa, PtLa, RhLa, ScGa, ScGe,ScSi, SrQe, SrPb, SrSi**, SrSn, TaB, VB, WB**,YGe, YSi, ZrAl, ZrPt. '

CoB, FeB, HfB, HfGe, HfSi, LaCu, LaQe, LaSi,MnB, NiY, PtY, TiB, TiQe, TiSi, ZrQe, . ZrSi. ** I iSn.

fmP6, P2/m, I.iSn type j

5870 ALEX ZUNt" ER

mP16, P2&/c, LiAs type

KSb, LiAs, NaQe, NaSb.

(oP16, Pbca, CdSb, superoctetg

CdAs, CdSb, ZnAs, ZnSb.

mP32, PZ&/n, NS type, superoctet

AsS**, AsSe**, NS, NSe, PS**.tI8, l4/mmm, HgC1 type, superoctet

HgBr, HgC1, HgI.

NaHg.

oC16, Cmcm, NaHg type

LiQe.

tI32„ I4&/a, LiGe type

KO.

oC16, Cmca, KO type

KC, NaC.

(tl32, I4&/acd, NaC type

SiP.

oC48, Cmc2&, SiP type, superoctet

T1Te.

tl32, I4/mcm, TlTe type, superoctet

ol8, Itnmm, RbO type

CsO, CsS, RbO, BbS.

oP16, P2&2&2&, NaP type

KAs, KP, NaAs, NaP, RbAs, RbP.

tl46, l4&/acd, NaPb type

CsPb, CsSn, KPb, KSn, NaPb, BbPb, HbSn.

((?), Cmcm, T1I type, superoctet J

InBr, InI, T1I.

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