ComputerPhysicsCommunications79 (1994) 447—465 Computer PhysicsNorth-Holland Communications

Simultaneouscalculationof the equilibrium atomicstructureand its electronicgroundstateusingdensity-functionaltheory

RolandStumpfand MatthiasSchefflerFritz-Haber-Institut der Max-Planck-Gesellschaft,Faradayweg4-6, D-14 195 Berlin (Dahlem),Germany

Received12 October1993

This computerprogramsolvesself-consistentlythe Kohn—Shamequationsfor the valence electronsusing the iterativemethod suggestedby Car and Parrinello.The atomic geometryis determinedsimultaneouslyusing a dampeddynamicsapproach.The computercode can handleisolated atoms,clusters,crystals,surfaces,and defects.The materials canbesemiconductorsand metals.The code is especiallyoptimized to treat systemswith hundredsof atoms,yet the hardwareneedsarejust a <15k$workstation.

PROGRAM SUMMARY

Title ofprogram: f h i 93c p No. of lines in distributed program, including test data, etc.:48230

Cataloguenumber:ACTFKeywords:density-functionaltheory, local-densityapproxima-

Program obtainable from: CPC Program Library, Queen’s tion, ab-initio pseudopotentials,plane-wavebasis, supercell,University of Belfast, N. Ireland(seeapplicationform in this chemical binding, total-energy,structure optimization,crys-issue) tals, defectsin crystals,surfaces,molecules

Licensing provisions: Personsrequestingthe program must Natureofphysicalproblemsign the standard CPC non-profit use license (see license In poly-atomicsystems,such as molecules[1,2], crystalsandagreementprintedin everyissue). defects in crystals [3,4], and surfaces [5—9], it is highly

desirableto evaluatethe electronic structureand to deter-Computerfor which the program is designedand others on mine the stableas well asmetastableatomic geometryfromwhich it has been tested: first principles and without introducing severeapproxima-Computer: IBM RS/6000 370, CONVEX C-220, CRAY tions. For a correct treatmentof the chemicalbinding it isY-MP, IBM-PC; Installation: Fritz-Haber-Institutder Max- most important to take the quantum-mechanicalkinetic-en-Planck-Gesellschaft,Berlin ergy operatoras well as the self-consistentelectronic charge

density into account.The main challengeof state-of-the-artOperatingsystem:UNIX calculations is to treat systemscomposedof 100 or more

atomswithout any restrictionsto the systemsymmetry.Programminglanguageused: FORTRAN 77 Thecomputercodedescribedbelowenablessuchcalcula-

tions,where theonly (possibly relevant)approximationis theMemoryrequired to executewith typicaldata: 2—500MB exchange-correlationenergyfunctional, which is taken in the

local-density approximation [101. We use the frozen-coreNo. of bits in a word: 32 approximation,treating the ions by ab-initio, fully separable

pseudopotentials[11,12].Memoryrequiredto executewith typical data: 4 MB

Method of solution_________ Themomentum-spacemethod [4,13] is the most efficient wayCorrespondenceto: R. Stumpf, SandiaNationalLaboratories, to calculate the Hamiltonian in a plane-wavebasis set. ToAlbuquerque,NM 87185-1111,USA (presentaddress).E-mail:rrstump@cs.sandia.gov.

0010-4655/94/$07.00© 1994 — ElsevierScienceBY. All rights reservedSSDI0010-4655(93)E0137-C

448 R. Stumpf,M Scheffler/ Simultaneouscalculationof theequilibrium atomicstructure

solve the eigenvalueproblem andto achieveself-consistency [4] WE. Pickett, Comput.Phys.Rep. 9 (1989) 117.CarandParrinello[14] proposedan iterative approach,where [5] J. Hebenstreit,M. Heinemann,and M. Scheffler, Phys.in eachiterationtheHamilton operatoris appliedto thewave Rev. Lett. 67 (1991)1031.functions.This givestheanalogof a force on eachwave-func- J. Hebenstreitand M. Scheffler, Phys.Rev. B 46 (1992)tion coefficient which points towards the electronic ground 10134.state. The program uses this iterative approach for the [6] J. NeugebauerandM. Scheffler, Phys.Rev. B 46 (1992)electronicwavefunctionsandoptimizes the total energywith 16067.respect to the atomic-structuredegreesof freedom by a [7] 0. PankratovandM. Scheffler,Phys.Rev.Lett. 70 (1983)dampeddynamics,using theHellman—Feynmanforceson the 351.ions. [8] R. Stumpf and P.M. Marcus, Phys. Rev. B 47 (1993)

16016.Restrictionson thecomplexityoftheproblem [9] R. Stumpf and M. Scheffler, Phys. Rev. Lett. 72 (1994)Only pseudopotentialswith s- or p-non-localitycanbeusedin 254.the presentversion and theyshould be given in a separable [10] D.M. CeperleyandB.J. Alder, Phys.Rev. Lett. 45 (1980)form, as for examplelisted in ref. [12]. The shapeand size of 1814.the cell maynot changeduringthe calculation. [11] L. Kleinman and D.M. Bylander, Phys. Rev. Lett. 48

(1982)1425.Typical running time: 6 mm [12] X. Gonze,R. Stumpfand M. Scheffler, Phys.Rev. B 44

(1991)8503.References R. Stumpf and M. Scheffler, Researchreport of the[1] W. Andreoni,F. Gygi andM. Parrinello,Phys.Rev. Lett. Fritz-Haber-Institut(1990).

68 (1992) 823. [13] J. Ihm, A. Zungerand ML. Cohen,J. Phys.C 12 (1979)[2] N. Troullier and J.L. Martins, Phys. Rev. B 46 (1992) 4409.

1754. [14] R. Car and M. Parrinello, Phys. Rev. Lett. 55 (1985)[3] J. Dabrowskiand M. Scheffler, Phys.Rev. Lett. 60 (1988) 2471.

2183; Phys.Rev. B 40 (1989) 10391.

LONG WRITE-UP

1. Introduction

The plane-wave, ab-initio pseudopotentialmethods in density-functional theory (DVF) for theelectronicstructurehavebeenused successfullyin recentyears to describethe electronic,structural,elastic, andvibrational propertiesof poly-atomic aggregates[1—9].

Car andParrinello[14] introducedseveraltechnicallyimportantideaswhich reducethe computationaltask to solve the Kohn—Shamequations[151of density-functionaltheory(DFT) self-consistently.Usingaplane-wavebasis set the Hamiltonian Z is representedas a M x M matrix, which is not diagonalizedexplicitly, but only has to be repeatedlyapplied to the N lowest trial eigenfunctions~ Here N isslightly larger than the numberof occupiedstatesbut clearly smallerthan M. Thismultiplication of Ztimes can be done efficiently, using the fast Fourier transformation(FFT) techniqueand fullyseparable,ab-initio pseudopotentials[4,11—13].The computercode describedbelow canbe applied tometals as well as to insulators and semiconductors.For metals, the Fermi smearingof occupationnumbersaccordingto the single-particleeigenvaluese~is introduced.In this casethe free energyF atthe electronictemperatureT~1is minimized and the energyis thenextrapolatedto the Tel —* 0 value,Ezero

By usingsingleprecisionwheneverpossiblefor datastorage(computationsarealwaysdonein doubleprecision!),andby making efficient useof virtual memory,the fastmemoryneedsarekept low (see,e.g.,our techniquefor doing the orthogonalizationof the wavefunctionsin routine g r ah am). Theevaluationof the non-localpart of the pseudopotentialof atomssitting at lattice sites is speededup dramatically,avoiding a repeatedevaluationof the sameintegrals.

Much effort was takento reducethe numberof iterations necessaryto achieve the self-consistentconvergenceand for increasing the convergencestability. As a starting charge density we use asuperpositionof contractedatomic densities.This way the starting chargedensity becomesalready

R. Stumpf,M. Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure 449

similar to the self-consistentcharge density [16]. To generatethe first set of wave-functions theKohn—ShamHamiltonian constructedfrom thischargedensityis diagonalizedin a reducedplane-wavebasis set. The resulting electrondensity is mixed with a largefraction (90%) of the startingdensity inorderto suppresslongwavelengthchargedensityoscillationsfrom the beginning(chargesloshing,1/G2instability, seeref. [4]). This mixing is reducedgradually to zero during the first seven iterations.Thechangeof the energyand thedisplacementof the wavefunctionsis monitoredduring theself-consistencyiterations.By adjustmentof the iterative timestepand the mixing coefficientand by moving the atomsonly after forcesareconverged,instabilitiesare avoided.

To get a smoothconvergencefor metallic systems, additionally the changeof Fermi occupationnumbersis damped by introducing fictitious eigenvalues[17]. These eigenvaluesfollow a dampedequationof motion wherethe driving forces point towardsthe correct eigenvalues,so that in the endbothsetsof eigenvalueswill bethe same.The fictitious eigenvaluesareusedto calculatethe occupationnumbersaccordingto a Fermidistribution.

In calculationsof slab systemswith an intrinsic dipole momenta compensatingelectric field is used.Its changeduring a calculationalso hasto bedampedto avoid oscillations.

In the following section2 we recall briefly the method. The computercodeand the input dataarethendescribedin section3. Section 4 explainshow the programshall be installedand the input andoutputof the testrun are describedin section5.

2. Self-consistentsolution of the Kohn—Shamequations

The keyquantityof density-functionaltheoryis the electrondensityn(r), which is calculatedfrom thesingle-particlewave functions~

n(r) = EfJOCC I~(r)I2, (1)1=1

wheref~°~are the occupationnumberstypically given by the Fermidistribution.The wave functionsobeythe Kohn—Shamequations[15]

(2)

Weuse Hartreeatomicunits. The exchange-correlationpotentialsis given by

~xc(n(r)) = ~[n(r) �xc(n(r))J (3)

where exc(n(r)) is the exchange-correlationenergyper electron. It is taken from the resultsof thehomogeneouselectrongasby CeperleyandAlder [10] asparametrizedby PerdewandZunger[19].V’~isthe Hartreepotential, giving the electrostaticinteractionwith the other electrons.~ representstheexternalpotentialoperator,generatedby the ionic pseudopotentialsVPS of all atoms.Foreachatomweuse the separable,fully non-localformulation afterKleinmanandBylander[11], wherethe pseudopoten-tial operatorV~ is constructedfrom the local (V~~0c)and the i-dependentparts (~V

1’~)of thesemi-localpseudopotentialsafter ref. [18] as

lmax—1 / Iz1V~~s(r)~Ps(r))K~VPs(r) I1j’~(r)I= VI~5~b04(r)+ ~ (~PS( \ ~J~pS( \\ . (4)

1—0 in——i ~ 1m~’J I ~r1 lmkTJ/

450 R.StumpfM. Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure

Here ~Pj~(r)= l’im(T/t) Rr(r) are the atomic pseudowavefunctions~, which are also used in acontractedform [16] to generatethe starting charge density. The starting wave functions 11,0 aregeneratedby a diagonalizationof the startingHamiltonian in areducedplane-wavebasis.The steepestdescentprocedurefor the self-consistencyiterationsis simply

i~t+1 = 1~Jt.t+ ~t ~~‘1J~+ orthonormalityconstraints (5)

andends,when the ~!‘J do notchangeany moreThe systemwe calculateis alwayscontainedwithin a periodicallyrepeatedsupercellspannedby the

threevectors a1, a2, and a3. The wave functions are expandedin planewaveswith wave vectors G,which are linear combinations(with integercoefficients)of the reciprocallatticevectorsb1, b2, andb3 ofthe supercell,and sampledat the ~k points k in reciprocalspace

~(r) = ~ CJG+k exp(i(G+k)r). (6)k IG+ki~Gmax

The total energyin the momentumspaceformulation [4,13] is evaluatedas

Etot = E’°’~+ ~l~s~0c + EPs,fbn_b0~ + EXC + Eion~o~~ (7)N,nk,G”~ 2G”~” 4irn(G)

~ fj°~wkIcJG+kI2(G+k)2+~I1~ n(G)

j,k,G G*0

~ N,n~,n

5

+ (1 ~ S(G) V~~l0c(G)n(G) + ~ fOCCw~PS~flOfl_lOC

+fn(r) �~w(n(r)) dr + ion-ioninteractionand G=0 terms. (8)

(1 is the volume of the supercell,~a the index of the ~a atoms.The Wk are the weightsof the k-points,and S is the structurefactor

S(G) = ~exp(iGR~). (9)

The ion—ion interaction and the terms containing n(G = 0), which divergeseparately,are calculatedpartially in G-spaceandpartially in realspaceby anEwald summation[20,21].

2.1. Efficientevaluationof the non-localpart of thepseudopotential

The mostcomplicatedandtime consumingpart of the total-energycalculationfor systemscontainingmanyatomsis the non-localpseudopotentialcontribution

ps,non—loc = I

m~-i ~ I~G~ki,~,Im(G+k) cJG+kI 2 (101 0 m i K~(r)I~V~(r)I~(r)>’

* The Yim are the sphericalharmonics.

** Insteadof the procedureof eq.(5) often the Williams—Soler[24] algorithm is usedfor the convergencyiterations.It givesforsomesystemsa fasterrateof convergency.

R. StumpfM Scheffler/ Simultaneouscalculationoftheequilibrium atomicstructure 451

with

4’n~i’ G+kAii,im(G+k) = ~~(IG~I) Yim( IG+kI ) exp(i(G+k)R~) (11)

and

~(IG+kI)=fr24VjPs(r) Rr(r) j1(IG+kI) dr. (12)

The j~are sphericalBesselfunctions.With the normalchoiceof ~ = 2, i.e. taking the d-potentialaslocal, and using a small numberof k-points, which is normally the casefor large systems,the sumsbelonging to the non-local part of the pseudopotential(fourth term in eq. (8) and sumsin eq. (10))essentiallyreduceto a triple sum of the order N X na X n~.nG is the numberof plane waveswith

a (Gm~)3a Q(Ec~1t)2/3.

Note that the evaluationof V~ff1P,,with the effective potential ~ being the sum of the Hartreepotential, the local part of the pseudopotential,andthe exchange-correlationpotential,is performedinrealspaceusingfast Fourier transforms(FFT). So thiscalculationonly scalesas nG log ~ If a foldingin G-spacewas done instead it scaledas n~. Thus, togetherwith the orthogonalizationof the wavefunctions(scalingaN2nG,but with a smallerprefactor),the evaluationof the non-localpseudopotentialconsumesthe largestshareof computertime for largesystems.

We reduce the computationaleffort to evaluate the non-local pseudopotentialcontribution byreorderingthe abovetriple sumfor thoseatoms i~f~aI which aresitting on lattice sites.The coordinatesofthose atoms can be written as r

1~e~,= ~ with the rn’s and n’s being integer. For theseatomsthe exponentialexp(i(G + k)Rj~a) of eq. (11) can only assumea small number (~scm.~ n0) ofdifferent values.nscm is the smallestcommonmultiple of n1, n2, and n3. The relationbetweenG-vectorsand thesen~mdifferentvaluesfor the exponentialexp(i(G+ k)Rji~ai) is similar for different atoms~

Appropriateordering,exploiting this similarity, thenallows for the partitioningof the sumover G of eq.(10) into nclass sub-sums,which for every atom ~ideal aremultiples of oneof the ~scmexponentials.Thesendlass sub-sums,multiplied by the accordingexponential,are thensummedup insteadof the previoussumof length nG. The number~cIass is independentof ~ and scalesroughly like n0, dependingon thetypeof the atomic lattice.For E~~t= 8 Ry the numberof operationsfor the evaluationof the non-localpseudopotentialcontributionsis loweredby abouta factor of 10.

2.2. Finite electronic temperature

In order to stabilize the convergencein the calculationof a metal [61, the electronicstatescan beoccupied according to a Fermi distribution f(e, Tel), with t~, being the eigenvalue K1If~I ZThereforethe functionalwhich is minimized with respectto n(r) is not the total energyEtot I n(r)], butthe free energy

F[n] =Etot[n] — Tdlsel, (13)

with the entropy

sdl = —2kB~[f ~ ln fOCC + (1 _fOCC) ln(1 _fOCC)1 (14)

We denotethe densityat the minimum of F[n] as ñ(r). In order to find the equilibrium position weminimize the function F~({R~}).A better treatmentwould be to minimize the function E~°((R1}) =

E~0t({R1))— TdSd/2which equalsthe total energyat zero temperaturewith an error of the order of

452 R. Stumpf,M. Scheffler/ Simultaneouscalculation oftheequilibrium atomicstructure

~9((Tel)3) [6,22]. For the small valuesof kBT~, i.e. smallerthan about0.2 eV, both functions, F~({R

1})and E,~°((R1}) give practicallythe samegeometry.Forcalculatedbindingenergies,however,oneshouldtake the resultsof E~°.

2.3. Relaxationof the atomicstructure

The minimization with respectto the atomic geometryis done by dampedNewtoniandynamicsforevery coordinateX, expressedin form of a finite differenceequation

XT+l=XT+~(XT_XT~) +6F~, (15)

with i~ and ~ controlling damping and speedof motion. For a optimum choice of ~ this dynamicscombinesfast movement towards the next local minimum with vanishing oscillations around theminimum(the aperiodiclimit). Increasingthe dampingcoefficientwill providefor a morerobustatomicrelaxation,reducingit will allow for eventualbarriersto be overcome,which canleadto another,lowerminimum. In all casesthe minimization will make implicit use of the knowledgeof the history ofdisplacementswhich is storedin the coordinates“velocity” (XT — X

T — 1)~

For fast convergencethe implementationof this algorithm is optimized for atomic displacementsaslargeas possible.After sucha largedisplacementtheelectronicsystemis distortedconsiderably.To relaxthe electronicsystemagainpurely electroniciterationsareperformed.The next atomicdisplacementisperformedwhen the atomic forces havestabilized. Important for the time consumptionis, that duringthesepurely electronic iterations the contributionsto the atomic forces by the non-local part of thepseudopotentialsare not evaluated.Thesecontributions dominate the costs for the calculationsofatomic forces.

2.4. Dipole correction and treatmentofcore-valenceexchange

The programalso allows for the compensationof an internal dipole momentas to avoid an artificialelectrostaticfield. Furthermoreit is possibleto apply an externalfield deliberately[6,25].

Implementedis also the possibility to treatthe core—valence-electronXC interactionexactlyby usingthe corechargedensityexplicitly. This non-linearinteractionis often accountedfor within the pseudopo-tential in a linearizedform only. The correctevaluationof theXC interactionof the coreandthe valenceelectronsis important if thereis a significant overlapof core andvalenceelectronslike in alkali-metals[5,23].

3. Program structure

The arrays,whosedimensionsaresystemsizedependent,areall dimensionedstatically by parametersdefinedin the file par ame t e r - h andthey areexplainedin table 1.

The control input for theprogramis readfrom 2 units, in h eade r from unit 5, which is the standardinput in UNIX, andin I n I t 1 from unit 10 from a file namedf o r t - 1 0 - Informationon theseinputvariablesis given in table2 and3.

The information aboutthe pseudopotentialsandatomicpseudowavefunctionsis read in I n I t 1 fromfort.11, fort 12,..., dependingon the numberof atomicspecies.

Typically Hartree atomic units are usedwithin the program,with some exceptions:eigenvaluesaretransformedto eV and G-vectorsaregiven in 2 IT/a lat with alat being a lengthcharacterizingthe sizeofthe supercell.

R. Stumpf,M. Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure 453

Table 1Dimensioningparametersin include file paramet er . h. After this file is changedit is necessarythat all the routines in D c p arerecompiled.

Parameter Comment

nsa numberof atomicspeciesn a x maximumnumberof atomsof anyspeciesn x numberof wavefunctionsto calculatengwx maximum numberof planewaves: ngwx ~ fl/(6~rr

2XE~”t)2”3ng w i x maximumnumberof planewavesfor startingdiagonalizationn r 1 x, n r2 x, n r 3 x dimensionsfor three-dimensionalFFT mesh.Thesamplingtheoremand

experiencetells that e.g. n r 1 x 0.8(2/~r)a1i/~~is a good choice(a1 is

oneof thethreesupercellvectors).But onehasto choosemultiplesofsmall primes,dependingon the FFT routineused.

n k p t x numberof k-pointsusedfor samplingthewavefunction

The programstartswith the routine ma I n, whichcalls h eader andthe initialization routine I n I t 1,

beforeperformingandcontrolling the iterationsfor the electronicand atomic relaxation.In I n I t 1 all the variablesneededfor the iterationsare initialized. First control andpseudopotential

information is read.Then I at g en is called,which generatesthe direct andreciprocallatticevectorsandgeneratesthe matrix s of the symmetryoperationsaccordingto pg 1 n d. I at g en also calls atp os forthe generationof the ideal atomiccoordinates,controlled by the variables I b r av and c e 11 d m. Afterthat the G and G + k vectors are calculatedin g g en and g v k. If a continuation run is done, thewave-functioncoefficients cO and positions t au 0 are read in by I — wave from unit 70. Taking theinformation aboutthe atomic coordinatesfrom unit 10, unit 70, and of the routine a t ~0 s and addingeventuallysomerandomcontributionthe final startingpositionsare generated.So in ph f ac the phasefactorsexp(i(G + k)R,) canbe calculated.Thereadditionallythe sortingof the exponentialsinto classesis doneas describedin section2. In for c s it is checked,if the atomicpositionsconformto the symmetryoperationsand the matricesfor the symmetrizationof theforces areprepared.The structurefactors S,for the different speciesarecalculatedin strucf, their “generalized”form factors arecalculatedinform f (localpart of the pseudopotentialandthe gaussianpseudocharge)andin n I sk b (non-localpartof pseudopotential).If an initial diagonalizationof the hamiltonianshall be done,now the chargedensityhas to be generated.This can be done, dependingon nrho, in formf a from contractedatomicpseudowavefunctions,in r hoof r from the readin wavefunctions,or from readingin the chargedensityon a mesh from unit 72. For every k-point the Hamiltonian matrix, developedin plane wavesup toecut I, is then generated,with the local potential,calledthe effectivepotential, calculatedinv of r ho.The lowest eigenvaluesandaccordingeigenvectorsaredeterminedin d i agon. The eigenvaluesareusedin f er m I to determinethe Fermidistribution for the occupationnumbers.

Now, backin ma I n, the iterationscanbegin:

1. Forthe first iterationsafter the initial diagonalizationthe long wavelengthchangesin the chargedensityarevery large.This would induceoscillationsin the electronicsystem.Thereforethe old chargedensityis mixedwith the new oneat a proportiongivenby r h 0_rn I x_f a c - r h 0_rn I xf ac is set in thebeginningto 0.90 (this meansthat only 10% of the chargeof the newwavefunctionsis mixed in!), andthencontinuouslyreducedto 0 in the first seventimesteps.For later iterationsit is checkedif the totalenergyandthe meansquarechangeof the wavefunction(ek I n c) decreases.If not, then the timestepd e I t is reducedfor 3 iterationsby 40%. The mixing of the chargedensityis switchedon again (withr horn i x_f ac set to 0.3), if the atomic forces do not convergewithin 10 iterations after an atomicmove. Mixing is done until the oscillationsare dampedout.

454 R. Stumpf,M Scheffler/ Simultaneouscalculationof theequilibrium atomicstructure

Table 2Input parametersfor unit 5.

Variable Type Comment

nbcg i Determinesif thecalculationshall startwith a diagonalizationin I n I t 1,if setto — 1, or if thestartingwave-functionsshall bereadin fromthewave-functionfile fort .70, ifwet to —2.

riom o r e i Gives thenumberof electronictimestepsthat shall beexecutedin a runwith purelyelectronicminimization, or, if t f o r setto true,howmanyionic movesshall bedone.

I p r i n t i Periodbetweenoutputof additionalinformationon unit 6.d eI t r The timestepin thesteepestdescentalgorithmfor theelectronicmini-

mizaton.Note thatthe factorby which theelectronicforcesaremul-tiplied is d aI t * * 2 /33 - 333 (for historicalreasons).A good valueforAl is 3.0, for thestiffer Aspseudopotentialdel t shouldbereducedtoCa. 2.0. A reductionbecauseof long wavelengthchargedensitysloshingis necessaryfor largesystems.

ied y n i The indexfor the methodusedfor minimization of theelec-tronic degreesof freedom.Up to nowonly two areimplemented:0 normalsteepstdescent;1 integrationin imaginarytime by the formula of Williams and

Soler[24], whichsometimesallowsfor a largertimestep,especiallyin high symmetrysituations.

n at ep e i Themaximumnumberof electroniciteratonsto bedoneuntil the forcesareconvergedagainafteranatomicmove. If theforcesdo not converge,the programstops.

I o r t ho i Choosesthealgorithmfor orthogonalizationof thewavefunctions.Cur-rently only theGram—Schmidtorthogonalizationis implemented,butthe routineor t ho for thelagrangianmethodof ref. [14] is includedin theprogramarchiveto beusedfor realmoleculardynamics.

t ran e, I, r Flag (t ran e) if the electronicwavefunctionsshouldberandomizedbeforeaap r e theiterations,andby which amplitude(ampre). Randomizationis useful

if somespurioussymmetryof thewavefunctionsshouldbeovercomefasterthanit is doneby normalequationsof motion.

t ran p, I, r Flag (t r anp)if the atomicpositionsshouldberandomizedbeforetheamprp iterations, andby which amplitude(amprp).t d I p0 1 1 If setto true, thedipole correctionin z-directionis done for thepoten-

tial, theenergiesandtheforces.This is usefule.g.for unsymmetricaladsorptionsystemsto decouplethe surfacesandfor a correct determi-nationof thework function.Seesection2 andref. [6].

tf I e I d, I, r An electrostaticfield in z-directionof thegivenvoltageis applied, ifvoltage tf I e Id is true.Seesection2 andref. [6].t_ad sor b at e I If setto true,thenthe atomisof thelast speciesareregardedasad-

sorbates,whichonly makesa differencefor howtheircoordinatesaregeneratedandchangedin atomicmoves.

tc 00 r d 1 Controlsfor anyofthethreecoordinatesof theadsorbatesif theyshouldad(3) bereadin from thewavefunctionfile andmovednormally(casetrue),

or if theyshouldbereadin from fort . 1 0 andkeptmixed duringthecalculation.

t I mequeue r The time in secondstherun maylast atmost.epseI r If in severalsuccessivetimestepsthechangeof total energyis smaller

thanepse I (Hartree),thanthe electronicsystemis assumedto becon-vergedandtheprogramstops.

apaf o r r Convergencecriterion for atomicrelaxation.If all forcesaresmallerthanepaf o r, theatomicsystemis assumedto berelaxed.

epaek I n c r Additional convergencycriterion for electronicconvergency.ek I n c mea-suresthemeansquarechangeof anelectroniccoordinateper timestep.

R. Stumpf,M Scheffler/ Simultaneouscalculationoftheequilibrium atomicstructure 455

Table2 (continued)

Variable Type Comment

tcondyn, ... I If t_con_dynis setto true, then n condyn atomsare movedin a smooth(calledconstraineddynamic)way to somefinal posi-tions.Theatomsandpositionsarespecifiedby thenextparameters:I s_con_dyn species;i acondyn atomindex;tau_con_dyn (3) final coordinates;ntcondyn moveto final positionin nt condyn timesteps.

2. In n I r h k b the first part of applying the non-local part of the pseudopotentialon the wavefunctions(seeeq. (4)) is done.This also gives the non-localpseudopotentialenergyEj~S,~bo~~— bc (see eq.(8)), aswell as the non-localcontributionsto the forceson the atoms.The non-localcontributionsto theforcesare costlyto compute,thereforethis calculationis only donein thoseiterationswhere the atomsare really moved.

3. r hoof r is the routinewhich calculatesthe kinetic energyE kin andthe chargedensityn(r) of thewavefunctions(exceptfor tband true). FFT is usedto transformthe wavefunctions to realspace.n(r)is symmetrizedaccordingto pg i n d.

4. In v of r ho the effectivepotential l’,ff(G) is calculatedand the routine ew a I d is calledto do theEwald summationfor the ion-ion interaction.The contributionsof both termsto the Hellman—Feynmanforceson the atomsarecalculated.vo f rho also determinesthe totalenergyEbot aswell asF and Ezero.

For tmeta I set to true (seesection2.2).

5. In d f or c e the force on thewave functions Z1I1,k is calculated,which in ma I n is usedto generatethe new wavefunctionsafter eq. (5) or the Williams—Soler [24] algorithm, and to get the eigenvalues

K~P,kI71 V’k>. The local part of the force is calculatedin real space,so FFT is usedtwice for everywavefunction.

6. In g r aham the wave functions at every k-point are orthonormalized,using Gram-Schmidtorthogonalizationin a special techniquecalled blocking. Always a set (= block) of wavefunctionsisorthogonalizedto the rest of thewavefunctionsinsteadof doingthis only for a singlewavefunction.Thiswaythe wavefunctionshaveto berun throughlessoften which savesdramaticallyon hard-diskaccessinthe casewhere not all wavefunctioncoefficientsfit into main memory.Another techniqueused in thisroutine is the flipping of the direction for running throughlarge arraysfor every run-trough. This alsoreducesdisk access.

7. f er m I is the routine to occupy the wavefunctionsaccording to the eigenvaluesand the Fermidistribution at temperatureekt. But changingof occupationnumbershas to be damped,which isdifficult to do for the occupationnumbersdirectly, as theyare constraintto give chargeneutrality andtobein the rangebetween0 and2. Thereforewe usefictitious eigenvalues,insteadof the calculatedones,and let them changein time after a dampedequationof motion with the difference to the correcteigenvaluesbeing the driving force. The occupationnumbersthen are chosenaccording to the idealfermi distribution for the fictitious eigenvalues.Obviously, if everything is converged,fictitious andcorrecteigenvaluesareequal.

8. In for c s2 the atomic forces aresymmetrizedaccordingto the symmetryoperationscalculatedinfor cs.

9. f I on s c controlsthe calculationof forcesand the relaxationof the atoms.In the first 4 timestepsafteran atomic move,no forces arecalculated.Then the local contributionsto the forcesarecalculatedfor thoseatomswith t f o r d true, until their meanchangein aniteration is less than30%. In the nextiteration also the non-localcontributionsto the forcesarecalculated,the forcesareextrapolated,taking

456 R. StumpfM. Scheffler/ Simultaneouscalculation of theequilibriumatomic structure

Table3Input parametersfor unit 10.

Variable Type Comment

i brav i Indexof Bravaislatticefor thesupercell,e.g. 1 = sc,2 = fcc, 3 = bce,4 = hexagonalandall I b r av startingwith 8 areorthorhomic.i b r avdeterminestogetherwith ccII d m theform andsize ofthe supercell,i.e. thelatticevectors a1, a2, a3, andtheatomic positionsgeneratedin atpoa.

pg 1 rid i Pointgroupindexof the symmetryoperations(seeroutine pgsym,fccsyin, shornb s,..). Every symmetryoperationkeepstheorigin un-changed.Havingsymmetryreducesthenumberof k-pointsneeded.

cc11dm(6) r cc 11dm(1) is theconventionallatticeconstant,or a multiple of it.celldm(2) and celldm(3), if notsettoO, give in theold fashionedkindof inputtheextensionsof the supercellin unitsof cc I I d rn (1). Recom-mendedis for this purposeto useccLI drn (4. . 6) which givethenumberof primitive cellsthesupercellshall containin anyof the 3 directions.

n e I i The total numberof valenceelectrons.Thenumberof conductionstatesto calculateadditionally,is givenby thedifferenceof the parametern r aand (nel+1)/2.

tine t aI I Flag that tells if Fermi occupationnumbersshall beused.Shouldbesetalwaysto true,evenfor non-metals,asthis is moresave.

ek t r The temperatureof theelectronsfor theFermi distributionin eV.t d eg en 1 If setto true,all theoccupationnumbersarereadin later on fromthis

file andfrozenin duringthecalculation(seecommentin routine I n I t 1).

e C Ut r Thekinetic energy(in Ry) of the highestwavefunctioncomponentG + kused for the plane-wave expansion.

e c u t I r Reduced cut off energy(in Ry) for thestartinghamiltonianfor theinitialdiagonalization.

m o I d I flag that indicates,if iterationsshouldbedoneafter thestartingdiagonalization.

t b an d 1 If setto true, thechargedensityandthereforethepotential is neverup-datedduringtheiterations.So e.g.a non-selfconsistentbandstructurecanbecalculatedwith non-specialk-points,onedoesnotwant to useforthecalculationof thetotal energy.The total energyfor t band=. true.normallyhasno meaning.

n rho i n r h o chooseswherethe startingchargedensityshall comefrom:1 from thepossiblycontractedatomicpseudocharges;2 from thewavefunction inputfile fort . 70;3 from the chargedensityinput file fort .72.

t psmesh I If setto true,thepseudopotentialsarereadfrom theradial logarithmicmesh,otherwiseit is calculatedfrom thefitted parametersgivenin thebeginningof thepseudopotentialinput files.

coor d wave 1 Tells theprogram,if in a continuationrun the atomiccoordinatesshallbe readfrom thewave-functioninput file f 0 r t . 70 or not.

n kpt i Thenumberof k-pointsusedto samplethewave-functions.Eventuallyn kpt is multipliedby theproductof thethreeintegersin if aCs.

If aca (3) i Givesthenumberof timesthe k-pointsshallbe folded in anyof the 3directions.This is aneasyway to makek-spacesamplingsimilar in alldimensions.

x k (3, n kpt) r Thecoordinatesof thek-points in unitsof 2Tr / a1~1or in unitsof recip-

rocallatticevectors b 1, b 2, andb 3, dependingon thechosenI b r a v.w kpt (nk pt) r Theweightsof thek-points, whichcanbedifferent in themultiplicity of

thek-points aredifferent for thesymmetryoperationsgiven.nrvacvacancy(nr_va c) i Thenumberandtheindicesof atomsto removefrom theregularatomic

latticegeneratedin theroutine at pos. thisway certainatomscanbecut outof e.g.a surfaceto generatea surfacevacancyor a step.

R. StumpfM Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure 457

Table3 (continued)

Variable Type Comment

t cor, n s cor 1, i Flagif the non-linearXC-correctionfor speciesn SCO r shouldbeper-formed.In thiscasethe accordingpseudopotentialfile mustadditionallycontainthecorechargedensity,which shouldbesmoothedat low radiifor a better k-spaceconvergency.Seesection2 and[23,5].

nsp i Thenumberof differentatomicspecies.atom(I a) a Thenameof the atomicspecies.na (I s) i Thenumberof atomsof this species.zv (I s) i Thenumberof a valenceelectronsof this species.I onfa c (1 s) r For the specialkind of dampeddynamicswe do in routinef I ons ~, thisis

thefactor to multiply the forces.It is equivalentto the factor~dt2/m in

thenormalequationof motion.The stiffer theatomicpseudopotentialis, thesmaller I on_fac hs to be.Examplesof good valuesare:for Na= 15, Al = 7, Si = 5, As = 3.

i on damp(1 s) r This is thedampingfactor,multiplying thevelocityof theatoms,inourspecialdampeddynamicsscheme.Specialmeans,that thedampingis increasedto (1—10 n darnp (1 s)), for thosecomponents,whereforceandspeedpoint in oppositedirections.This specialkind of dampeddynamics effectively inhibits oscillations,withoutreducingthespeedofconvergencytoo much.

r gauss(i a) r Theradiusof thegaussianpseudochargeof speciesI a. It shouldbechosensmallerthan~ of the smallestatom-atomdistancebut not toosmall for a low plane-wavecutoff, e.g. rgauss> =1.0 for eCut=6 .0.If smallchangesof r g auss changetheconvergedtotal energy,thanthechoicewaswrong.

110 c (I s) i The I-value(+ 1) of the componentof the pseudopotentialthat shallberegardedaslocal. This is normallythed-potentialandtherefore11 o c = 3. Settingthep-potentialto local savestime andmight work forthe ailcalines,butwith e.g. Al andSi, it giveswrong results.

t a U 0 (1 , i a, i a) r This arraycontainsthe atomiccoordinates.If a coordinateis set to0.0 in for t . 1 0, thentheaccordingcoordinateis taken from theroutineatpoS.

t fo rd (I a, 1 s) I Foreveryatomthisflag determines,if the forceon it shall becalculatedandif it shall bemoved.For atomswith t fo r d falseonly the localcontributionsto theforcesin the last iteration arecalculated.

t r an s_Iat t (1 —3) r This is thevectorwhich is addedto the atomicpositionsastheyaregeneratedin atp os. It canbeusedto makethe atomssymmetrictoa symmetryplane,which alwaysmustcontainthe origin to beusedassymmetryoperationin theprogram.

into accountthe changesin the last iterations,andthen the atomsaremovedwith the specialdampeddynamicscontrolledby i on_fa c and i on_damp(seetable3).

10. If atomshavebeenmoved, in phf aC their phasefactors exp(i(G+ k)R~)and in s t r u c f thestructurefactor haveto be recalculated.

11. Severaloutput routines,whose namesall beginningwith o —, write interesting quantitieslikeeigenvalues,positions,forces,etc. to units 80—88 in form of MATHEMATICA commands[26], so thatthe information written outcanbe easilyanalyzed.The changingpositionsandforcesarewritten to unit1, the eigenvaluesand occupationnumbersto unit 87. The momentarywave functions, the atomicpositionsandother information necessaryto continuethe calculationis written out on unit 71 in binaryform to the socalledwave-functionfile. The electronicchargedensityin binaryform is givenoutto unit73.

458 R.StumpfM Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure

12. During the run of the programthe variablesd e I t (timestep)and n o m o r e (numberof iterationsto do) canbe changedby writing newvaluesto the files d e I t and stop f I I e.

The program either stops when the iteration limit is reachedor the convergencyconditions arefulfilled. The run can be continued,usingthe wavefunctionfile as input, as describedabove.

4. Making of the program

The programis written in lower casein FORTRAN 77. Non-standardis that, no line numbersareused,commentlines can be longer than 80 characters,and somevariable namesare longer than 6characters.I M PL I C I I NONE declarationsareusedin all routines.We declareexplicitly theprecisioninbytes for all the floating point variables,choosing REAL * 4 respectivelyCOMPLEX * 8 for the longerarrays,where ever this is possiblewithout significant lossof accuracy.The file paramet e r - h is madeknown by the FORTRAN I N C LU D E statementin every routine which uses arrays with variabledimensions.The most importantCOMMON variablesaresharedamongthe routinesby including the filescommoni. h or common2- h, where the commonsare declared.There are some routines,which don’tneedtheseincludes,someBLAS routinesandthe EISPACK routinesto find the lowest eigenvectorsandeigenvaluesof ageneralhermitianmatrix. Theseroutineswe put into a library calledf_too I s - a in thesubdirectoryDf_tools with its own makefile.

For finding the lowest eigenvectorswe use in our implementation(in the subroutined i agon) thesingle precisionES SL routine C H PSV (see ref. [27]), which can be replacedby EISPACK routines inf_too I s • a if necessary.Also from ESSL the three-dimensional,complexto complex,single-precisionFFT routine SCF 13 is used,which canbe replacedin the two subroutinesf f t and f f t_f ast by theFFT routine available at the site. Information on how to do either replacementcan be found incommentsin the specific routines.

The program,the makefile, andthe datafiles arecontainedin a single tar archive,which shouldbeextractableon any UNIX machine.The makefiles possibly haveto be adaptedto the computerused.Three directoriesare generated,Dc p, D f_too I s and D run, which contain the main program, thesupportlibrary f_too I s - a,andthe set of files to run the program.Calling theUNIX makecommandin D c p generatesthe program.Runningthe shell script run_f h I 93c p generatesthe programandrunsthe testcalculation.Thisshell script is also a convenientstartingpoint for developingone’sown run timeenvironment.

5. Test run

The testsystemconsistsof a Si atom,sitting in a surfacevacancyof a 2 x 2 Al(111) slab of two layerthickness.So we havetwo species,Al andSi,whosepseudopotentialsare takenfrom ref. [12]. Wehaveadipole momentandwe get some relaxation. Four k-points and a cut off of E~~t= 8 Ry areused.Theinput from parameter- h, unit 5 and fort .10 togetherwithpartsof the outputon unit 6 is given in atthe end of this paper.The input files for t - 11 and for t - 1 2, which contain the pseudopotentialinformation for Al andSi respectively[12], and all the output files arecontainedin the tar file of theprogramdistribution.

References

[1] W. Andreoni, F. Gygi and M. Parrinello,Phys.Rev. Lett. 68 (1992)823.[2] N. Troullier andJ.L. Martins, Phys.Rev. B 46 (1992) 1754.

R. StumpfM. Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure 459

[3] J. Dabrowskiand M. Scheffler, Phys.Rev. Lett. 60 (1988) 2183; Phys.Rev.B 40 (1989) 10391.[4] W.E. Pickett, Comput.Phys.Rep.9 (1989), 117.[5] J. Hebenstreit,M. Heinemannand M. Scheffler, Phys.Rev.Lett. 67 (1991) 1031.

J. Hebenstreitand M. Scheffler, Phys.Rev. B 46 (1992) 10134.[6] J. NeugebauerandM. Scheffler, Phys.Rev. B 46 (1992) 16067.[7] 0. PankratovandM. Scheffler, Phys.Rev. Lett. 70 (1993) 351.[8] R. Stumpfand P.M. Marcus,Phys.Rev.B 47 (1993) 16016.[9] R. Stumpfand M. Scheffler, Phys.Rev. Lett. 72 (1994) 254.

[10] D.M. Ceperleyand B.J. Alder, Phys.Rev. Lett. 45 (1980) 1814.[11] L. Kleinman and D.M. Bylander, Phys. Rev. Lett. 48 (1982)1425.[12] X. Gonze,R. Stumpfand M. Scheffler, Phys.Rev.B 44 (1991)8503.

R. Stumpfand M. Scheffler, Researchreportof the Fritz-Haber-Institut(1990).[13] J. Ihm, A. Zungerand M.L. Cohen,J. Phys.C 12 (1979)4409.[14] R. Car and M. Parrinello,Phys.Rev.Lett. 55 (1985) 2471.[15] W. Kohn and L.J. Sham,Phys. Rev. A 140 (1965) 1133.[16] M.W. Finnis, J. Phys.Cond. Mat. 2 (1990) 331.[17] M.R. Pedersonand K.A. Jackson,Phys.Rev. B 43 (1991)7312.[18] G.B. Bachelet, DR. Hamann and M. Schlüter, Phys. Rev. B 26 (1982) 4199.[19] J.P. Perdew and A. Zunger, Phys. Rev. B 23 (1981) 5048.[20] A.A. Maradudin, E.W. Montroll, G.H. Weiss, and I.P. Ipatova, Solid State Physics, Suppl. 3, ed. Montroll et al. (1971).[21] G.P. SrivastavaandD. Weaire, Adv. Phys.36 (1987)463.[22] M.J. Gillan, J. Phys.Cond. Mat. 1(1989) 689.[23] S.G. Louie, S. Froyenand M.L. Cohen,Phys.Rev.B 26 (1982) 1738.[24] A. Williams andJ. Soler,Bull. Am. Phys.Soc. 32 (1952)409.[25] J. Neugebauer and M. Scheffler, Surf. Sci. 287/288 (1983) 572.[26] 5. Wolfram, MATHEMATICA, 2nd ed. (Addison—Wesley,Reading,MA 1991).[27] IBM Engineering and Scientific SubroutineLibrary Guide andReference,IBM publication(1992).

460 R. StumpfM Scheffler/ Simultaneouscalculationof theequilibrium atomicstructure

TEST RUN

Include file parameter.h

integer nsx, flax, nx, ngwx, ngx, ngwix, nrlx, nr2x, nr3x

integer nnrx, nkptx, nlmax, mrnaxx, n_fft_store

c SPECIES ATOMS STATES

parameter(nsx=2, nax=7, nx=18)

c FULL BASIS

parameter (ngwx=835, ngx=8*ngwx)

c INITIAL BASIS

parameter(ngwix30l)

c FFT: X-MESH Y-MESH Z-MESH

paraxneter(nrlx=16,nr2x=16,nr3x=36 ,nnrx(nrlx+1)*nr2x*nr3x)

c K—POINTS

parameter(nkptx4)

c Number of ffts to be stored between rhoofr and dforce

parameter(n_fft_storel)

c Number of lm-components and max length of radial mesh

parameter(nlmax4 , mmnaxx300)

Input for unit 5

-1 15 15 3600 nbeg, nomore, iprint, timequeue

3.0 1 1 delt, i_edyn, i_ortho

.false. 0.01 .false. 0.02 tra.ne, ampre, tramp, amprp

.true. .false. 18 tfor, tsdp, nstepe

.true. .true. .true. .true. tadsorbate, t_coor&ad(3)

.true. .false. 0.0 tdipol, t_field, voltage

le—5 le—3 0.18 epsel, epsfor, epsekinc

Input for unit 10

81 2 7.52 0 0 1 2 5 ibrav, pgind, celldm(1—6)

25 .true. 0.1 .false. nel, tmetal, ekt, tdegen

8.0 4.0 ecut, ecuti

.true. .false. 1 tmold, tband, nrho

2 2 .false. .true. .true. nsp, nscor, tcor, tpsmesh, coordwave

R. Stumpf,M. Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure 461

1 2 2 1 nkpt, i...facs(1—3)

0.25 0.25 0.0 1.0 xk(1—3), wkpt(ikp)

1 nr...vac

1 vacancy(i)

‘aluminum’ 7 3 7.0 atom(is), na(is), zv(is), ion_fac(is)

0.7 1.0 3 ion_damp(is), rgauss(is), l_loc(is)

0.0 0.0 0.0 .t. tauO(1—3), tford

0.0 0.0 0.0 .t. tauO(1—3), tford

0.0 0.0 0.0 .t. tauO(1—3), tford

0.0 0.0 0.0 .f. tau0(1—3), tford

0.0 0.0 0.0 .f. tau0(1—3), tford

0.0 0.0 0.0 .f. tau0(1—3), tford

0.0 0.0 0.0 .f. tau0(1—3), tford

‘silicon’ 1 4 5.0 atom(is), na(is), zv(is), ion...fac(is)

0.7 1.0 3 ion_dainp(is), rgauss(is), lloc(is)

0.0 0.0 9.0 .t. tau0(1—3), tford

0.0 0.0 0.0 trans...latt(1—3)

462 R. StumpfM. Scheffler/ Simultaneouscalculation of theequilibrium atomicstructure

Test Run Output on Unit 6

this is FHI93CPibm—RS6000 version

******* R. Stumpf, Sep 1993 ********

>>>nbeg= -1 nomore= 15 iprint= 20>>>electronic time step 3.0000 ==> electronic force—f ac= .2700>accuracy for convergency: epsel= .00001 epsfor= .00100 epsekinc= .18000

>latgen:anx,any,anz 1.0000 2.0000 5.0000>orthorhombic: ibrav,ibravh= 8 81

>celldm: 7.5200 .0000 .0000 1.0000 2.0000 5.0000>alat 9.210081 alat_bulk 7.520000 omega= 2126.295040

al 9.21008143286474912 0. 000000000000000000E+00 0 .000000000000000000E+00a2 0.000000 00000000000E+00 10.6348859890456726 0. 000000000000000000E+00a3 0. 00000000000000000E+00 0. 000000000000000000E+00 21.7083701215299278bi 1 .00000000000000022 0. 000000000000000000E+00 0. 00000000000000000E+00b2 0.000000000000 000000E+00 0.866025403784438930 0. 00000000000000000E+00b3 0. 000000000000000000E+00 0. 000000000000000000E+00 0.424264068711928566

>>nx= 1 ny= 2 nzmax= 2 nlay= 5 n_in_plane= 4 itrans= 1atpos: ineq_pos 6 4 5positions tau0 from unitlO and atpos = tau_ideal

.0000 5.3174 8.6833 4.6050 2.6587 8.6833 4.6050 7.9762 8.68333.0700 .0000 13.0250 3.0700 5.3174 13.0250 7.6751 2.6587 13.02507.6751 7.9762 13.0250

.0000 .0000 9.00001

>>ggen: ratios of FFT mesh dimensions to sampling theorem .965 .836 .921>gvk: ngwx and max nr. of plane waves: 835 820>ibrav 8 pgind 2 nrot 2 alat 9.210 omega 2126.2950 mesh= 16 16 36>ecut= 8.0 ryd gcut 68.76 ng= 6393 gcutw 17.19 nkpt= 4

k-point weight # of g—vectors1 —.12 —.11 .00 .1250 8022 .38 —.11 .00 .1250 8113 —.12 .32 .00 .1250 8074 .38 .32 .00 .1250 820

Weigthed number of plane waves npw: 809.973Ratio of actual nr. of PWs to ideal nr.: .99693

>Shell-analysis of quality of k-points after Chadi/Cohen>Number of Am0 shells N = 5>weighted sum of A_ms= .0001501 (should be small for good k—samplingof electrons= 25, # of valence states 13, * of conduction states= 5atomic data for 2 atomic speciespseudopotentialparameters for aluminum

>nr. of at.: 7, valence charge: 3.0, force fac: 7.00, speed damp: .70> lloc:3 rad. of gaussian charge:1.000pseudopotentialparameters for silicon

>nr. of at.: 1, valence charge: 4.0, force fac: 5.00, speed damp: .70> lloc:3 rad. of gaussian charge:1.000Final starting positions:

.0000 5.3174 8.6833 4.6050 2.6587 8.6833 4.6050 7.9762 8.68333.0700 .0000 13.0250 3.0700 5.3174 13.0250 7.6751 2.6587 13.02507.6751 7.9762 13.0250 .0000 .0000 9.0000

R. Stumpf, M Scheffler / Simultaneouscalculationoftheequilibrium atomicstructure 463

phfac: is, n_ideal: 1 4phfac: is, n_ideal: 2 0phfac:Lscm, n_class(is) 60 60 0>nlskb:is 1 wnl: 1 .0261286 2 .2204559 3 .2204559 4 .2204559>nlskb:is 2 wnl: 1 .0179321 2 .1301626 3 .1301626 4 .1301626ECUTI: energy cut-off for the starting wavefunctions: 4.00

starting density calculated from pseudo-atomformf a: rho, scalat and rscal for start rho of at. 2.678 5.000 4.500 1formf a: rho, scalat and rscal for start rho of at. 3.595 5.000 4.000 2>rhonor: diff between correct and actual # of el. 2.64891011

initi stores starting density for mixing in c_fft.store>>vof rho: starting dipol voltage -.137896>s(isym) in latt coord: 1 0 0 0 1 0 0 0>sym(..) in cart coord: 1.000 .000 .000 .000 1.000 .000 .000 .000 1.000>s(isym) in latt coord: 1 0 0 0 -1 0 0 0>sym(..) in cart coord: 1.000 .000 .000 .000-1.000 .000 .000 .000 1.000>Center of symmetry symO .000000 .000000 .000000Table of symmetry relations of atoms,

iasym=nr. of symmetric at., xneutauo+<sym.Op.(isym)>is ia lasym isym tauO xneu1 1 1 1 .00000 5.31744 8.68335 .00000 5.31744 8.683351 1 1 2 .00000 5.31744 8.68335 .00000 —5.31744 8.683351 2 2 1 4.60504 2.65872 8.68335 4.60504 2.65872 8.683351 2 3 2 4.60504 2.65872 8.68335 4.60504 -2.65872 8.68335

> 1 start H: k-point: —.125 -.108 .000 no. pws: 288> -9.718 —6.446 —6.091 —5.218 -4.090 -2.948 -2.624 -2.268 -2.059 —1.337 -.530

.412 .809 1.090 1.292> 2.122 2.269 2.446> 2 start H: k—point: .375 -.108 .000 no. pws: 294> -8.990 -7.436 -5.675 -5.358 -4.151 -3.921 -3.433 —2.585 -1.528 -1.369 -.568-.043 .274 1.484 1.548> 1.945 3.079 3.218

***** starting eigenvalues and occupation numbers *****

nel damp true_efermi efermi ekt seq sneq25 .700 .70048 .70048 .100 .89845 .89845

k-point —.125 -.108 .000, eigenvalues and occupation numbers:eig -9.718 -6.446 —6.091 -5.218 —4.090 —2.948 —2.624 -2.268 —2.059 -1.337eig —.530 .412 .809 1.090 1.292 2.122 2.269 2.4460CC 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.00000CC 2.0000 1.8937 .5042 .0400 .0054 .0000 .0000 .0000

(non—eq) free energy -19.927058 a.u.

(non—eq) total energy —19.923756 a.u.Harris energy = -18.170248 a.u.

kinetic energy = 6.525260 a.u.electrostatic energy = —21.688194 a.u.

real hartree energy = 24.937346 a.u.pseudopotential energy = -2.656708 a.u.

464 R. Stumpf, M. Scheffler / Simultaneouscalculation of theequilibrium atomicstructure

n-l pseudopotential energy = 4.503707 a.u.exchange-correlation energy = -6.607793 a.u.

exchange-correlation potential energy = -8.609267 au.kohn - sham orbital energy = -3.191721 a.u.

self energy = 31.516440 a.u.esr energy = .000009 a.u.

gaussian energy = 39.615255 a.u.

k-point - .125 -.108 .000, eigenvalues and occupation numbers:eig -9.913 -6.623 -6.217 —5.394 -4.260 —3.073 -2.788 -2.387 —2.207 —1.485eig -.662 .264 .661 .957 1.182 1.987 2.161 2.297occ 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.00000CC 2.0000 1.9025 .5249 .0399 .0046 .0000 .0000 .0000

>>>n_it nfl Ekinc Etot Eharr Ezero mForce mChange Seq SneqEfermi Dvolt W_a

>>> 1 0 8.7409 —19.92376 —18.17025 —19.92545 .00000 .000 .9473 .9200.5674 .036 .000

>>>main mixes old charge with rho_mix_f ac= .800>>> 2 0 5.9074 -19.71952 —18.16424 -19.72124 .00000 .000 1.0810 .9383

.2064 .459 3.245

>fionsc: mean displacement per coord = .020952 with tsdp= F>>> 11 0 .5783 -18.45440 —18.45346 -18.45644 .00774 .159 1.0990 1.1085

—.2211 —.108 4.650

>>> 95 12 .1728 —18.46113 —18.46137 —18.46306 .00007 .026 1.0475 1.0474-.0542 -.056 4.280

internal energy at zero temperature = -18.463056 a.u.non-equillibrium entropy = 1.047353 kB

equillibrium entropy = 1.047459 kBkT energy = .100 eV

(non-eq) free energy = -18.464980 a.u.(non-eq) total energy -18.461131 a.u.

Harris energy = -18.461372 a.u.kinetic energy = 6.868027 a.u.

electrostatic energy = -21.013482 a.u.real hartree energy = 23.527742 a.u.

pseudopotential energy = -2.217524 au.n-l pseudopotential energy = 4.405246 a.u.

exchange-correlation energy = -6.503399 a.u.exchange-correlation potential energy = -8.472576 a.u.

kohn - sham orbital energy = -3.930642 a.u.self energy = 31.516440 a.u.

&&s atomic positions and local+nl forces on ions:aluminum

>&&s-n .006966 5.317443 8.550038 -.000240 .000000 .000045>&&s—n 4.621964 2.700385 8.510903 .000049 .000078 .000213>&&s-n 4.621964 7.934501 8.510903 .000049 -.000078 .000213>&&s-l 3.070027 .000000 13.025022 - .004406 .000000 -.027103>&&s-l 3.070027 5.317443 13.025022 -.004209 .000000 -.036022>&&s-1 7.675068 2.658721 13.025022 .004045 .004204 -.026937

R. Stumpf,M. Scheffier/ Simultaneouscalculation of theequilibrium atomicstructure 465

>&&s-l 7.675068 7.976164 13.025022 .004045 -.004204 —.026937silicon

>&&s-n .041806 .000000 8.768970 -.001645 .000000 .001864

> 1. k-point -.125 -.108 .000, ngw 802, EWs and OCCs:>eig:-10.495 -7.394 -6.935 -5.877 -4.760 -3.903 —3.251 -2.916 -2.626 -2.436 —1.491 -.324 .024 .184 .483>eig: 1.038 1.501 1.561>occ: 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000 2.000 1.8756 .6353 .1708 .0093>0cc: .0000 .0000 .0000