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PHYSICAL REVIEW B 66, 035119 ~2002!

Nonorthogonal generalized Wannier function pseudopotential plane-wave method

Chris-Kriton Skylaris, Arash A. Mostofi, Peter D. Haynes,* Oswaldo Dieguez, and Mike C. PayneTheory of Condensed Matter, Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, United Kingdom

~Received 28 September 2001; revised manuscript received 12 March 2002; published 30 July 2002!

We present a reformulation of the plane-wave pseudopotential method for insulators. This new approachallows us to perform density-functional calculations by solving directly for ‘‘nonorthogonal generalized Wan-nier functions’’ rather than extended Bloch states. We outline the theory on which our method is based andpresent test calculations on a variety of systems. Comparison of our results with a standard plane-wave codeshows that they are equivalent. Apart from the usual advantages of the plane-wave approach such as theapplicability to any lattice symmetry and the high accuracy, our method also benefits from the localizationproperties of our functions in real space. The localization of all our functions greatly facilitates the futureextension of our method to linear-scaling schemes or calculations of the electric polarization of crystallineinsulators.

DOI: 10.1103/PhysRevB.66.035119 PACS number~s!: 71.15.Ap, 31.15.Ew

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I. INTRODUCTION

The pseudopotential plane-wave method for densfunctional theory~DFT! calculations has been developed aperfected over many years into a reliable tool for predictstatic and dynamic properties of molecules and solid1

Kohn–Sham DFT maps the interacting system of electrto a fictitious system of noninteracting particles,2 which canbe fully described by the single-particle density matr(r ,r 8), expressed as a sum of contributions from singparticle Bloch statescnk(r ),

r~r ,r 8!5(n

f n

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~2p!3E1BZcnk~r !cnk* ~r 8!dk. ~1!

We have assumed that we are dealing with an insulator wcompletely filled~occupation numberf n51) or empty (f n50) states andV is the volume of the simulation cell. Thk-point integration is carried out in the first Brillouin zon~1BZ!. The single-particle states are the eigenfunctions ofKohn–Sham Hamiltonian at eachk-point. They are requiredto be orthonormal and, in general, extend over the whsimulation cell. The consequence of this orthogonalityquirement is that the cost of a DFT calculation involving t$cnk% grows cubically with the system-size. The electroncharge density is equal to the diagonal part of the denmatrix multipied by a factor of 2 to take into account the spdegeneracy and is commonly abbreviated asn(r )52r(r ,r ).

The most general representation of the density matequivalent to~1!, and first applied to linear-scaling DFT caculations by Hernańdez and Gillan,3 is in terms of a set oflocalized nonorthogonal functions$faR%,

r~r ,r 8!5(ab

(R

faR~r !KabfbR* ~r 8!, ~2!

where the sum overR runs over the lattice vectors of thcrystal and the matrixKab is called the density kernel,generalization of the occupation numbers$ f n%. We will callthe $faR% ‘‘nonorthogonal generalized Wannier functions~NGWFs! as they can be derived from a subspace rotationM

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between a set of Bloch orbitals at eachk-point and a unitarytransformation of the results ink-space~Wannier transforma-tion!,

faR~r !5V

~2p!3E1BZe2 ik•RF(

ncnk~r !MnaGdk, ~3!

where the number of NGWFs at each lattice cellR can begreater than or equal to the number of occupied Bloch baat eachk-point. The density kernel is the result of applyinthe inverse of these transformations on each side on theagonal occupation number matrix diag($ f n%),

Kab5(n

Nna f n~N†!n

b , ~4!

whereN5M21.In our presentation so far, we have considered the Bl

states as the natural representation and starting point fwhich to construct the charge density and NGWFs. Thisalso the usual order which has been followed in discussiand derivations of Wannier functions4 in the literature. Gen-eralized Wannier functions, orthonormal or not, are mcommonly constructed in a postprocessing fashion5–8 afterthe end of a plane-wave band structure calculation.

Since the energy is variational with respect to the chadensity, directly varying the NGWFs and the density kernof Eq. ~2! is equivalent to varying the Bloch states of Eq.~1!.The localization properties of the NGWFs are seta priori,i.e., each NGWF is nonzero only within a predefined locization region. As a result, the computational cost ofdensity-functional calculation scales only quadratically wsystem-size rather than cubically. Furthermore, it canmade to scale linearly with one extra variational approximtion: the truncation of the density kernel in Eq.~2! when thecenters of the functionsfaR(r ) andfbR(r ) lie beyond somecutoff distance.32,33

The NGWFs are expanded in terms of a basis of periobandwidth limited delta functions~Appendix A!. These arecentered on the points of a regular real-space grid andrelated to an equivalent plane-wave basis through a uni

©2002 The American Physical Society19-1

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SKYLARIS, MOSTOFI, HAYNES, DIEGUEZ, AND PAYNE PHYSICAL REVIEW B66, 035119 ~2002!

transformation~the Fourier transform!. Hence, the methodwe present is directly equivalent to the Bloch state pseupotential plane-wave density-functional approach. The dfunction basis allows us to restrict our NGWF expansionscontain only the delta functions that are included in sospherical localization region. This should be an accuratesumption for insulators in which the NGWFs decexponentially.9 Our approach is closely related to that of Henandezet al.10 who developed a method to do calculatioby optimizing both the density kernel and the functio$faR%, which they call ‘‘support functions.’’

Real-space methods, as an alternative to pure reciprospace plane-wave methods, have been used by manyauthors for DFT calculations11–18 in the past in order to takeadvantage of the benefits of localization in real-space.particular, approaches have been developed that use ftions strictly localized in spherical regions on real-spagrids.19,20These functions play the same role as the NGWwe present here. A different approach is taken by theSIESTA

program,21–23 which uses a basis set of numerical atomorbitals. These are generated as described by Artaet al.24,25 and are not optimized during the calculations.

Even though these real-space methods have led to simportant methodological developments, it would be vedesirable if they could be directly comparable with planwave pseudopotential DFT. In other words, we wish to haa method that rigorously adheres to a basis set we canprove systematically, such as the plane-wave basis wherconvergence towards completeness is controlled by thenetic energy cutoff parameter. Our method achieves thisworking both in real- and reciprocal-space. We demonstthat our approach can actually be viewed as an alternaway of performing plane-wave DFT calculations whicheasy to turn into a linear-scaling method in the future wonly trivial modifications. It is thus also directly applicabto any Bravais lattice symmetry, in contrast to common findifference methods that are usually restricted to orthorhobic lattices.

We should note at this point that there exist othbasis sets, apart from plane-waves, that can be improvedtematically: the B-splines of Hernańdez et al.,26 and thepolynomial basis in the finite-element approach of Pasketal.,27,28 who have done some pioneering work using a tenique for solving differential equations common in engineing applications and adapting it for electronic structure cculations.

In what follows, we begin by describing the calculationthe total energy in our scheme, directly with NGWFs, in SII. In Sec. III we describe our strategy for total energy opmization, i.e., minimization of the energy with respectboth the density kernel and the NGWFs. In Sec. IVpresent the FFT box technique, an essential ingredientlowering the cost of the calculations and for eventuaachieving linear-scaling behavior. In Sec. V we present teon a variety of systems showing the accuracy and efficieof this method and finally we conclude and mention whatsee as future developments.

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II. CHARGE DENSITY AND TOTAL ELECTRONICENERGY WITH NONORTHOGONAL GENERALIZED

WANNIER FUNCTIONS

Linear-scaling DFT calculations are aimed at large stems, and in particular, large unit cells. Thus in this work wwill be concerned with calculations only at theG-point, i.e.,k50. This means that the Bloch bands and thereforeNGWFs can be chosen to be real. We can also dropdependence of the NGWFs onR, so thatfaR(r )5fa(r ).

Our basis set is the set of periodic bandwidth limited defunctions that are centered on the pointsrKLM of a regularreal-space grid,

DKLM~r !51

N1N2N3

3 (P52J1

J1

(Q52J2

J2

(R52J3

J3

ei (PB11QB21RB3)•(r2rKLM),

~5!

whereB1 is one of the reciprocal lattice vectors of the simlation cell.N1 is the number of grid points in the direction odirect lattice vectorA1, andN152J111. The delta functionbasis is equivalent to the plane-waves that can be represeby the real-space grid since it is related to them via a unittransformation. An important property of the basis set is tthe projection of a functionf (r ) on DKLM(r ) is

EVDKLM~r ! f ~r !dr5W fD~rKLM !, ~6!

whereW is the volume per grid point andf D(r ) is the resultof bandwidth limiting the functionf (r ) to the same planewave components as in~5!.

We represent the NGWFs in the delta function basis b

fa~r !5 (K50

N121

(L50

N221

(M50

N321

CKLM ,aDKLM~r !, ~7!

and in the plane-wave basis by

fa~r !51

V (P52J1

J1

(Q52J2

J2

(R52J3

J3

3fa~PB11QB21RB3!ei (PB11QB21RB3)•r.

~8!

where it is straightforward to show that the amplitudfa(PB11QB21RB3) are the result of a discrete Fourietransform on the delta function expansion coefficieCKLM ,a .

In ~7! the sum over theK, L andM indices formally goesover the grid points of a regular grid that extends overwholesimulation cell. From now on however, we will restricall NGWFs to have contributionsonly from delta functionscentered inside a predefined spherical region. This spheregion is in general different for each NGWF. Thus we impose on~7! the condition

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NONORTHOGONAL GENERALIZED WANNIER FUNCTION . . . PHYSICAL REVIEW B66, 035119 ~2002!

CKLM ,a50 if rKLM does not belong to the sphere offa .~9!

This of course does not affect the form or the applicabilityEq. ~8!.

The charge density of Eq.~2! with our NGWFs becomes~from now on we will use the summation convention frepeated Greek indices!

n~r !52r~r ,r !52fa~r !Kabfb~r !52Kabrab~r !

52 (X50

(2N121)

(Y50

(2N221)

(Z50

(2N321)

Kab

3rab~rXYZ!BXYZ~r !, ~10!

which involves the fine grid delta functionsBXYZ(r ) that aredefined in a similar way to theDKLM(r ) of Eq. ~5! but in-clude up to twice the maximum wave vector ofDKLM(r ) inevery reciprocal lattice vector direction~see also AppendixA!. This is necessary because a product of twoDKLM(r )delta functions is a linear combination of fine grid delta funtions BXYZ(r ), a result reminiscent of the Gaussian functiproduct rule.29

The expressions for the various contributions to the toelectronic energy with the NGWFs are simple to derive fro~10!. The total energy is the sum of the kinetic energyEK ,the Hartree energyEH , the local pseudopotential energEloc , the nonlocal pseudopotential energyEnl , and the ex-change and correlation energyExc ,

E@n#5EK@n#1EH@n#1Eloc@n#1Enl@n#1Exc@n#.~11!

The kinetic energy is written as a trace of the product ofdensity kernel and of the matrix elements of the kineticergy operatorT52(1/2)¹2,

EK@n#52Kab^fbuTufa&. ~12!

To compute these matrix elements we can applyT to theplane-wave representation~8! of fa(r ) and then evaluate thintegral in real-space where it is equal to a discrete sum ogrid points whereTfa(r ) obviously plays the role off D(r )of Eq. ~6!.

Calculation of the Hartree energy requires first the Hartpotential. From Eq.~10! we see that the charge density isfine grid delta function expansion, thus the same shouldtrue for the Hartree potential, which is a convolution of tcharge density with the Coulomb potential. Therefore,VH(r )can be written as a linear combination of fine grid delta futions and extends over the whole simulation cell,

VH~r !5 (X50

(2N121)

(Y50

(2N221)

(Z50

(2N321)

VH~rXYZ!BXYZ~r !.

~13!

The Hartree energy is

EH@n#51

2E VH~r !n~r !dr5Kab^fbuVHufa&. ~14!

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This quantity can be calculated as a discrete summationthe fine grid of the product ofVH(r ) with n(r ) or equiva-lently as a trace of the product of the density kernel andpotential matrix elements. The local potential matrix ements are integrals that are identically equal to discrete son the regular grid provided of course that (VHfa)D(r ) isfirst put on the regular grid.

The local pseudopotential energy is calculated in antirely analogous manner to the Hartree energy and canrepresented by Eq.~14! if we put Vloc in place ofVH andmultiply it by a factor of 2 to take into account the lack oself-interaction in this case.

The nonlocal pseudopotential energyEnl@n# is the expec-tation value of the nonlocal potential operatorVnl in theKleinman–Bylander form,30

Vnl5(A

(lm(A)

udVl(A)C lm

(A)&^C lm(A)dVl

(A)u

^C lm(A)udVl

(A)uC lm(A)&

, ~15!

where theA-summation runs over the atoms in the systeand thelm-summation runs over the pseudo-atomic orbitof a particular atom. ThedVl

(A) is an angular momentumdependent component of the nonlocal potential of a pseuHamiltonian for a particular atom and theC lm

(A) are theatomic pseudo-orbitals associated with it. In the NGWF reresentation the nonlocal potential energy is again expresas a matrix trace,

Enl@n#52Kab^fbuVnlufa&. ~16!

The ^fbuVnlufa& matrix elements require the calculationoverlap integrals faudVl

(A)C lm(A)& between the NGWFs and

the nonlocal projectorsudVl(A)C lm

(A)&. These are simple tocompute as discrete summations on the regular grid, starfrom the plane-wave representation of the nonlocal projtors which is analogous to the plane-wave representatiothe NGWFs in Eq.~8!. These integrals need only be calclated when the sphere of functionfa(r ) overlaps the core ofatomA.

The exchange-correlation energy is obtained by appromating the exchange-correlation functional expression adirect summation on the fine grid, which first involves thevaluation of a functionF(n(r )) whose particular form de-pends on our choice of exchange-correlation functional,2

Exc@n#5EVF~n~r !!dr.

V

8N1N2N3(XYZ

F~n~rXYZ!!.

~17!

This is the only approximation in integral evaluation in omethod as all direct summations described up to now wexactly equal to the analytic integrals. However, in the cof the exchange-correlation energy, the exchange-correlafunctionals usually contain highly nonlinear expressions tcan not be represented without any aliasing even whenuse the delta functions of the fine grid. The resulting errhowever will be of the same nature as in conventional plawave codes and therefore negligible.1

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III. TOTAL ENERGY OPTIMIZATION

The total energy is a functional of the charge densE@n#. From Eq.~10! we see that the charge density is epanded in fine grid delta functions whereKabrba(rXYZ) arethe expansion coefficients. Therefore the energy will havvariational dependence on these coefficients providedform an N-representable charge density. Consequently,energy should also have a variational dependence on thesity kernel Kab and the NGWF expansion coefficienCKLM ,a since theKabrba(rXYZ) are constructed from them

E@n#5E~$Kab%,$CKLM ,a%!. ~18!

It is thus sufficient to minimize the energy with respect$Kab% and $CKLM ,a%. We must however do this under twconstraints. The first is that the number of electrons cosponding to the charge density

Ne5EVn~r !dr52KabSba ~19!

should remain constant. The second is that the ground sdensity matrix should be idempotent, or in other wordseigenfunctions of the Kohn–Sham Hamiltonian have toorthonormal,

r~r ,r 8!5EVr~r ,r 9!r~r 9,r 8!dr 9

or

Kab5KagSgdKdb. ~20!

We choose to carry out the total energy minimization in tnested loops, in a fashion similar to the ensemble Dmethod of Marzariet al.31 The density kernel will play therole of the generalized occupation numbers and the NWGwill play the role of the orbitals. So we can reach the mimum energy in two constrained-search stages,

Emin 5 min$CKLM ,a%

L~$CKLM ,a%!, ~21!

with

L~$CKLM ,a%!5 min$Kab%

E~$Kab%,$CKLM ,a%!, ~22!

where the minimization with respect to the density kerneEq. ~22! ensures thatL of Eq. ~21! is a function of theNGWF coefficients only. In practice in Eq.~22! we do notjust minimize the energy with respect toKab but we alsoimpose the electron number and idempotency constra~19! and ~20!. There are a variety of efficient methods fachieving this available in the literature, derived from tneed to perform linear-scaling calculations with a localizbasis.32–36Any of these methods would ensure that the dsity kernel in ~22! adapts to the current NGWFs so thatminimizes the energy within the imposed constraints.

In the present work we have used the variant of theNunes, and Vanderbilt~LNV ! ~Ref. 32! method that was developed by Millam and Scuseria37 in calculations with

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Gaussian basis sets. We emphasize again, though, that athe other available methods could have been used as wFor simplicity of presentation, our analysis from now on wassume that the energy of Eq.~22! is minimized without anyconstraints. In order to take into account the constraints,formulas we derive will have to be modified according to tdensity kernel minimization method one chooses to use. Tis a straightforward but tedious exercise.38

The minimization of Eq.~22! can be performed iterativelywith the conjugate gradients method.39 As in the simplersteepest descents method, the essential ingredient is thedient. It is easy to show40 that this quantity is equal to twicethe matrix elements of the Kohn–Sham Hamiltonian,

]E

]Kab52^fbuHufa&. ~23!

The nonorthogonality of our NGWFs has to be taken inaccount when computing search directions with the abgradient by transforming it to a contravariant second ortensor.41,42

The minimization stage of Eq.~21! is also performed it-eratively with the conjugate gradients method. In this caone can show by using the properties of the delta functbasis set that the gradient is

]L

]CKLM ,a54WKab~Hfb!D~rKLM !, ~24!

whereW is the weight associated with each grid point. Hea contravariant-to-covariant tensor correction is needed wthis gradient is used to calculate the search direction durinconjugate gradient step.43 The (H# fb)D(r ) functions in gen-eral contain contributions from all delta functions of thsimulation cell but we wish to keepfa(r ) restricted to itsspherical region. For this reason in every minimization sof ~21! we zero all the components of~24! that correspond todelta functions outside the sphere offa(r ).

When the minimization with respect to the density kernof Eq. ~22! is carried out under the electron number aidempotency constaints, Eq.~24! contains extra terms asresult of the constraints imposed in~22!. These terms ensurthat the electron number and idempotency constraintsautomatically obeyed in~21! and as a result, the optimizatiowith respect to the support functions can be carried out inunconstrained fashion.

IV. THE FFT BOX TECHNIQUE

Our discussion so far has demonstrated how to locathe NGWFs in real-space in a manner that ensures thatare composed by a number of delta functions that is conswith system-size. On the other hand, each delta functioexpanded in the plane-waves that can be supported byregular grid of the simulation cell. The number of theplane-waves is proportional to the size of the system. Aresult, the cost of a calculation still scales cubically wsystem-size as in the traditional plane-wave case.

In order to reduce the computational cost we must rest

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the number of plane-waves that contribute to each delta fution so that it is independent of system-size. This is not sa straightforward matter as the restriction of the NGWFsreal-space. Several factors have to be taken into accounmost important of which is the Hermiticity of operatorsintegrals between NGWFs and also a representation oferators that is consistent when they are acting on diffeNGWFs. We have investigated these matters in detail incontext of the evaluation of kinetic energy integrals wNGWFs in a previous paper,44 where we have proposed th‘‘FFT box’’ technique as an accurate and efficient solutionis shown there that all the imposed conditions are satisfiethe plane-waves that are used to expand the delta funcare restricted to belong to a miniature simulation cell whwe call the ‘‘FFT box.’’ The FFT box must be large enougto contain any possible orientation of overlapping NGWFand must be a parallelepiped with a shape commensuwith the simulation cell. It should contain a regular grwhich is a subset of the simulation cell grid and thusorigin of the grid of the FFT box should coincide withparticular grid point of the simulation cell. FFT techniquusing smaller boxes have been used in the past to takevantage of localized functions: Pasquarelloet al.45,46 usethem to efficiently deal with the augmentation charges tarise when using ultrasoft pseudopotentials; Hutteret al.47

use them to FFT localized Gaussian orbitals. However, tobest of our knowledge, this is the first time that a small Fbox has been used to define a systematic basis set foentire total energy calculation.

Figure 1 demonstrates in two dimensions the Fbox inside the simulation cell as defined for two overlappNGWFs fa and fb . In the same figure the regular gridalso visible and we can observe that a portion of it iscluded in the FFT box. For the subset of grid points insthe FFT box we can define delta functions as we did forsimulation cell. We represent these FFT box delta functiby dklm(r ) and in general we follow the convention of usinlowercase letters to denote quantities associated withFFT boxanduppercase letters for quantities associated wthe simulation cell.

Based on the knowledge gained through the use ofFFT box to compute kinetic energy integrals,44 we extend

FIG. 1. The FFT box as defined for the pair of functionsfa andfb .

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here its use to the calculation of all the terms that constitthe total energy~Sec. II! and all the terms needed for itsubsequent optimization with respect to the density kerand NGWF coefficients~Sec. III!. For this purpose we neeto be able to express or ‘‘project’’ a function from the simlation cell to the FFT box and back. Assuming the functionexpressed entirely by delta functionsDKLM(r ) on grid pointscommon to the simulation cell and the FFT box, this taskstraighforward and all we have to do is to rewrite the funtion as a linear combination of the FFT box delta functiothat lie on the same centers as the simulation cell delta futions. This task is expressed formally by theP(ab) projec-tion operator. When this operator acts on a function insimulation cell, it maps it onto a function in the FFT boThis is demonstrated for the function of Eq.~7! by the fol-lowing expression which can also be considered as a detion of P(ab) @a more explicit expression forP(ab) isgiven in Appendix B#:

P~ab!fa~r !

5 (k50

n121

(l 50

n221

(m50

n321

cklm,adklm~r !

5 (k50

n121

(l 50

n221

(m50

n321

C(k1Kab)( l 1Lab)(m1Mab)dklm~r !

~25!

while the adjoint operatorP†(ab) can act on a function inthe FFT box and turn it into a function in the simulation cein an analogous way.

With this compact notation it is relatively straightforwarto devise a way of calculating the total energy by using othe delta functions~and hence the plane-waves! periodic inthe FFT box as the basis set for each NWGF. It is aequally straightforward to write formulas that represent tprocess in a concise way.

We start with the charge density of Eq.~2!, which shouldof course extend over the whole simulation cell, howevercontributionsrab(r ) from pairs of NGWFs need not, and dnot, when they are calculated with the FFT box techniqTherefore the charge density is calculated by replacing thpair contributions in~10! by the following expression:

rabbox~r !5Q†~ab!$@ P~ab!fa~r !#@ P~ab!fb~r !#%,

~26!

which involves multiplyingfa(r ) with fb(r ) after theyhave been transferred into the FFT box and as a consequtheir product is limited to extend only over the volume of tFFT box. Here we have made use of the fine grid delta fution projection operatorQ(ab) which defines a mappingbetween fine grid delta functionsBXYZ(r ) of the simulationcell and the fine grid delta functionsbxyz(r ) of the FFT box.It is defined in an analogous manner toP(ab) of Eq. ~25!.

The matrix elements of one-electron operators, suchthe kinetic energy and nonlocal potential can be easily cculated in the FFT box rather than in the simulation cell

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simply transferring the NGWFs in the FFT box before evaating the integrals. The notation for this set of operatiosimply involves ‘‘sandwiching’’ the one-electon operator btween the standard grid projector operator and its adjoThe kinetic energy expression of Eq.~12! becomes

EK@n#52Kab^fbuP†~ab!TP~ab!ufb& ~27!

and a similar expression can be written for the nonlocaltential energy.

The matrix elements of the Hartree and local potentcan be calculated with the same ease provided howevehave a way to transfer these potentials to the FFT box. Tis indeed possible as these local potentials are expressterms of the fine grid delta functions and therefore we ctransfer them if we use the fine grid delta function projecQ(ab). Therefore the Hartree energy of Eq.~14! becomes

EH@n#5Kab^fbuP†~ab!@~Q~ab!VH!~r !# P~ab!ufa&.~28!

Nothing changes in the evaluation of the exchancorrelation energy which is simply an integral over the figrid of the whole simulation cell except of course for the fathat the charge density is now calculated by summingrab

box(r )terms in Eq.~10! in place of therab(r ).

As far as the optimization of the energy with respectthe density kernel is concerned, the results of Sec. III arevalid provided the Hamiltonian matrix elements that constute the density kernel gradient~23! are calculated with theFFT box method as we have described above.

In the same way, the total energy gradient with respecthe NGWF expansion coefficients of Eq.~24! can be calcu-lated in the FFT box provided we use only the part of tKohn-Sham Hamiltonian that exists in the FFT box. Thefore all we have to do is to substituteH of Eq. ~24! by

H~ab!5 P†~ab!@ T1Vnl1~Q~ab!VHlxc!~r !# P~ab!,~29!

where VHlxc(r ) is the sum of the Hartree potential, locpseudopotential and exchange-correlation potential andcourse now this Kohn–Sham operator is in general differfor each pair of functions since it contains only the partsthe local potentials that fall inside the FFT box and therefchanges with the location of the FFT box relative to tsimulation cell.

As far as implementation is concerned, there are mmore issues and algorithmic details about the use of thebox that are beyond the scope of this paper. We descthese in another paper.48

V. THE NGWF PSEUDOPOTENTIAL PLANE-WAVEMETHOD IN PRACTICE

We have implemented our method in a new code andhave performed extensive tests on a variety of systems.have also performed comparisons with CASTEP,1 an estab-lished pseudopotential plane-wave code that we use aspoint of reference. We expect our approach to have e

03511

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-

se

isin

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-

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-

oft

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ciency comparable to the traditional cubic-scaling planwave pseudopotential method and thus it would be possto use it in its place for systems with a band gap. In that caa calculation with our method would afford a set of optimlocalized functions which could be used directly in applictions such as the calculation of polarization changes in ctalline solids.49,50 However, the most important applicatiothat we envisage is the extension of the present formalismlinear-scaling calculations on very large systems. Suchextension requires the truncation of the density kernel,issue which has been already investigated in detail.34,36,51

The resulting linear-scaling method would be directly coparable to and have the same advantages as the plane-approach.

Since we optimize the NGWFs iteratively, some initiguesses are required for them. In this work we use pseuatomic orbitals ~PAOs! that vanish outside a sphericaregion.52 These orbitals are generated for the isolated atowith the same radii, norm-conserving pseudopotentialskinetic energy cutoff as in our calculations. Even thouthese NGWF guesses are optimal for the isolated atoms,undergo large changes during our calculations so thapractice any guess that resembles an atomic orbital coulused, such as Slater or Gaussian functions.

We first demonstrate the accuracy of the FFT box tenique as compared to using the entire simulation cell asFFT grid. We define the quantity

DE[Ebox@n#2E@n#, ~30!

whereEbox is the total energy per atom calculated using tFFT box technique andE is that calculated using the entirsimulation cell. Figure 2 showsDE for the butane molecule(C4H10) for different FFT box sizes. For this test we usedcubic simulation cell of side length 50a0 and grid spacing0.5a0. The PAOs on all the atoms were confined withspherical regions of radius 6.0a0. The carbon atoms had on2s and three 2p orbitals and the hydrogen atoms had a sin1s orbital. In this case the PAOs were not optimized duri

FIG. 2. DE plotted for a butane molecule as a function of FFbox size. All PAOs were confined to atom centered localizatregions of radius 6.0a0, and the grid spacing was 0.5a0.

9-6

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ane

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thelr ttho

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n-lo-.ithHas

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the calculation. It is seen that the error associated with usthe FFT box rather than the entire simulation cell is onlythe order of 1027 Eh per atom, which is insignificant in thecontext of DFT calculations. We also note that the convgence of the total energy with FFT box size is not stricvariational, as is expected; as the FFT box size is increait is true that the basis set expands, but the smaller basnot necessarily a subset of the larger one. For a givenbox size, however, the kinetic energy cutoff of our bafunctions~and hence the grid-spacing! is a variational param-eter, just as in traditional plane-wave DFT. Further testsdiscussion of the FFT box technique are publishelsewhere.48

Our next example involves the potential energy curvethe LiH molecule inside a large cubic simulation cell of silength 40a0. In Fig. 3 the potential energy curve is showncalculated by CASTEP and by our method with the sakinetic energy cutoff of 538 eV. As we have used norconserving Troullier–Martins53 pseudopotentials, this istwo electron system which we describe by one NGWFeach atom. It can be seen that when we use NGWFsradii of 8.0a0, we have mEh agreement in total energies witthe CASTEP results. Furthermore, the equilibrium bolength and vibrational frequency for this case differ from tCASTEP results by only20.19% and 0.74%, respectivelFor the smaller radius of 6.0a0 the curve diverges from theCASTEP curve at large bond lengths. This is becauseNGWF sphere overlap, and therefore the number of dfunctions between the atoms, decreases more rapidly fosmall radii as the atoms are pulled apart. Also shown insame figure is a curve that has been generated withmethod but without optimization of the NGWFs, which wekept constant and equal to the initial PAO guesses. Thiequivalent to a tight-binding calculation with a minimal PAbasis. As can be seen, the total energies deviate significafrom the CASTEP result, as one would expect. The equirium bond length for this case differs by 3.34% froCASTEP, as compared to21.24% for the 6.0a0 NGWF cal-

FIG. 3. Potential energy curves for LiH generated with tCASTEP plane-wave pseudopotential code and with our methodNGWF radii of 6.0a0 and 8.0a0 and fors-type PAOs with a radiusof 6.0a0.

03511

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nth

d

etaheeur

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culation. Thus, optimizing the NGWFs improves the esmate of the bond length. The vibrational frequency obtainfrom the PAO case, however, differs by 3.51% froCASTEP, as compared to 5.33% for the 6.0a0 NGWF calcu-lation. This, we believe, is an artifact of the localization costraint imposed on the NGWFs and suggests that in factcalization radii greater than 6.0a0 should be used in practice

We now show convergence of the total energy wNGWF radius. For our tests we have used a silane (Si4)molecule with the same simulation cell and grid spacingdescribed above. A local Troullier–Martins53 norm-conserving pseudopotential was used on the hydrogen aand a nonlocal one on the silicon atom. The numberNGWFs on each atom was as many as in the valence sof the isolated atoms, i.e., one on hydrogen and foursilicon. Figure 4 shows total energy results calculated for tsystem as a function of NGWF sphere radii. Convergencuniform and to mEh accuracy by the time we get to a radiuof 7.0a0. Such a NGWF radius should be adequate for prtical calculations.

Here we also show that large qualitative changes occuthe shapes of the NGWFs during optimization. In Fig. 5show plots of isosurfaces of the NGWFs for an ethene mecule in a large simulation cell, before and after optimiztion. The NGWF radius was 8.0a0 for all atoms. In particu-lar, the carbon 2px orbital, which is collinear with the C–Cs bond, focuses more around this bond and gains two mlobes and nodes at the positions of the hydrogen atomsthest from its carbon center. The hydrogen functions, starfrom 1s, obtain after optimization a complicated shape thextends over the whole molecule and has nodal surfacestwen the carbons and the rest of the hydrogens. The dqualitative changes to the shapes of the NGWFs that ocduring their optimization with our method are obviously neessary for obtaining a plane-wave equivalent result. Ourtimized NGWFs in general look nothing like the atomic obitals they started from and are adjusted to their particumolecular environment. We therefore conclude that usingdelta function basis set and performing all operations contently with the plane-wave formalism is important for obtaiing the systematic convergence that plane-waves have.

or

FIG. 4. Total energy of a silane molecule calculated with omethod for various NGWF radii (r c).

9-7

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Our final example demonstrates the direct applicabilityour method to any lattice symmetry without any modifiction. This is a consequence of being consistent throughwith the plane-wave formalism. As we have shown for tcalculation of the kinetic energy in this way,44 we alsoachieve better accuracy at no additional cost compareda finite difference approach. In Fig. 6 we show a porti(Si8H16) of an infinite linear silane chain inside a hexagonsimulation cell on which we have performed a total enecalculation at a kinetic energy cutoff of 183 eV. The radiithe NGWFs were 6.0a0 on silicon and 5.0a0 on hydrogen. Atotal energy of239.097 Eh was obtained when we optimized the density kernel only~with the NGWFs kept con-stant and equal to PAOs!. When both the density kernel anthe NGWFs were optimized, the energy lowered252.216 Eh , which is another manifestation of the fact thboth the density kernel and the NGWFs should be optimiin calculations with our method.

FIG. 5. Isosurfaces of NGWFs for the ethene molecule, befand after optimization. The light gray surfaces are positive anddark gray are negative. A drawing of the ethene molecule is suimposed on each NGWF in order to show its location with respto the atoms.

03511

f-ut

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ly

d

VI. CONCLUSIONS

We have developed a new formalism where we havecast the plane-wave pseudopotential method in terms of northogonal localized functions instead of Kohn–Sham banA key ingredient for computationally efficient calculationwith our approach is the restriction of the local functioboth in real and in reciprocal space. We have written a ncode to implement and test this approach. Even thoughequivalent to plane-waves, our method performs calculatidirectly with localized functions without ever resortingKohn–Sham states. As a consequence it could be moreable for application to fields such as the theory of elecpolarization of insulators. However we anticipate that tmain use of this approach will be in density-functional cculations on insulators whose cost scales linearly withsize of the system. Its extension to linear-scaling calculatirequires the reduction of the elements of the density keby truncation. Our test calculations on a variety of systeconfirm that such a linear-scaling method should be direcomparable to traditional plane-waves. Advantages ofapproach include high accuracy, applicability to any lattsymmetry, and systematic basis set improvement controby the kinetic energy cutoff.

ACKNOWLEDGMENTS

C.-K.S. would like to thank the EPSRC~Grant No. GR/M75525! for postdoctoral research funding. A.A.M. woullike to thank the EPSRC for a Ph.D. studentship. P.Dwould like to thank Magdalene College, Cambridge forResearch Fellowship. O.D. would like to thank the Europe

eer-t

FIG. 6. A portion (Si8H16) of an infinite linear silane chain in ahexagonal simulation cell.

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Commission for a Research Training Network Fellowsh~Grant No. HPRN-CT-2000-00154!.

APPENDIX A: DELTA FUNCTIONS

In this paper, whenever we refer to ‘‘delta functions’’ wwill assume a periodic and bandwidth limited version of tDirac delta functions. These delta functions are thrdimensional versions of the ‘‘impulse functions’’ that acommon in signal processing applications of FFTs.54 In anelectronic structure, similar functions have been used‘‘mesh delta functions’’ in the ‘‘exact finite differencemethod’’ of Hoshiet al.55 and in recent studies of their possible application when we consider the limit of an infinisimulation cell.56

In our derivations we will assume that we have a simution cell of any symmetry, which in general is a paralleleped defined by its primitive lattice vectorsA1 , A2, andA3.In this simulation cell we define aregular grid with an oddnumber of pointsN152J111, N252J211, andN352J311 in every direction~the adaptation of our results to thcase of even numbers of points is straightforward!. Thereforepoint rKLM of this regular grid is defined as

rKLM5K

N1A11

L

N2A21

M

N3A3 ~A1!

with K50,1, . . . (N121), etc.Bandwidth limited delta functions centered at points

the regular grid are defined as

DKLM~r !5D000~r2rKLM !

51

N1N2N3

3 (P52J1

J1

(Q52J2

J2

(R52J3

J3

ei (PB11QB21RB3)•(r2rKLM),

~A2!

whereB1 , B2 andB3 are the primitive reciprocal lattice vectors of the simulation cell. Plane-waves whose wave vecis a linear combination of these reciprocal lattice vecthave periodicity compatible with the simulation cell antherefore so do our delta functions, or any other functexpanded in terms of these plane-waves. Theseperiodicbandwidth limiteddelta functions are our basis set. A plota two-dimensional version of one of these delta functionshown in Fig. 7. It is obvious from~A2! that the delta func-tions are real-valued everywhere in space. They are notmalized to unity but they are normalized to the grid povolume (V is the volume of the simulation cell!

W5V

N1N2N3. ~A3!

Their value at grid points is equal to one when the grid pocoincides with the center of the function and zero forother grid points,

03511

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--

f

rs

n

is

r-t

tl

DKLM~rFGH!5dKFdLGdMH . ~A4!

The delta functions act as Dirac delta functions with tadded effect of filtering out any plane-wave components tare not part of them. For example, iff (r ) is a function peri-odic with the periodicity of the simulation cell but not banwidth limited, it can be expressed in terms of its discreFourier transform~plane-wave! expansion,

f ~r !51

V (S52`

`

(T52`

`

(U52`

`

f ~SB11TB21UB3!

3ei (SB11TB21UB3)•r, ~A5!

whereV is the volume of the simulation cell.It is straightforward to show that the projection off (r )

onto DKLM(r ) is

EVDKLM~r ! f ~r !dr

51

N1N2N3(

S52J1

J1

(T52J2

J2

(U52J3

J3

f ~SB11TB21UB3!

3e2 i (SB11TB21UB3)•rKLM

5W fD~rKLM !.

We define heref D(r ) to be the bandwidth limited version othe functionf (r ), limited to the same frequency componenasDKLM(r ).

As the NGWFs are linear combinations of the delta funtions according to~7!, the result of Eq.~A6! is very impor-tant since it leads to the following relation:

EVfa~r ! f ~r !dr5W (

K50

N121

(L50

N221

(M50

N321

CKLM ,a f D~rKLM !

~A6!

which means that the integral in the left-hand side ofabove equation isexactly equalto a discrete summation ovalues on the grid, provided we use the bandwidth limitversion of f (r ).

As a corollary we observe that the delta functions areorthogonal set since

EVDFGH~r !DKLM~r !dr5WDFGH~rKLM !

5WdFKdGLdHM . ~A7!

We also need to define thefine grid delta functionsBXYZ(r ) ~here theXYZare just grid point indices for the finegrid, they arenot related to any Cartesian coordinates!.These functions are the analogs of the delta functionshave just described that would be obtained if we doubledminimum and maximum values that their wave vectors ctake. Consequently, they have the same periodicity but tcorrespond to a grid with twice the number of points in evedirection, i.e., 2N1 , 2N2, and 2N3 points. They are definedby

9-9

t.


FIG. 7. A two-dimensional version of one of the functions that constitute our basis set. Here functionD00(r ) is shown which is identicallyequal to 1 at its center~point r00) and equal to zero at the centers~shown as black dots in the picture! of all other functions in the basis se

a

a

oelel

izeofng

theor

Tsent

torit

BXYZ~r !5B000~r2rXYZ!

51

8N1N2N3(

P5(2N111)

N1

(Q5(2N211)

N2

(R5(2N311)

N3

3ei (PB11QB21RB3)•(r2rXYZ). ~A8!

As expected, the fine grid delta functions also satisfyequation similar to~A6!,

EVBXYZ~r ! f ~r !dr5

W

8f B~rXYZ!, ~A9!

wheref B(r ) is again a bandwidth limited version off (r ) butthis time it is limited to contain any of the plane-waves thconstituteBXYZ(r ) rather thanDKLM(r ). It is easy to verifythat any function that can be written as a sum of productspairs of delta functions can also be written as a fine grid dfunction expansion. We define and use the fine grid dfunctions because of this ‘‘product rule’’ property.

03511

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ftata

APPENDIX B: FROM THE SIMULATION CELLTO THE FFT BOX AND BACK

Even though the FFT box is universal in shape and sfor a given system, its position with respect to the gridthe simulation cell is determined by the pair of overlappiNGWFs, sayfa(r ) and fb(r ), we are dealing with at anygiven time. An operator therefore that would mapfa(r )from one representation to another would depend also onposition offb(r ). We therefore define such an operator fthe pair of functionsfa(r ) andfb(r ) by

P~ab!51

W (k50

(n121)

(l 50

(n221)

(m50

(n321)

udklm&

3^D (k1Kab)( l 1Lab)(m1Mab)u, ~B1!

where the numbersKab , Lab , and Mab denote the gridpoint of the simulation cell on which the origin of the FFbox is located. Here lowercase letters are used to reprequantities related to the FFT box, son1 , n2, andn3 are thenumbers of grid points in the FFT box in each lattice vecdirection. Because of the periodic boundary conditions

9-10

nels

tions

ed

ds,


should also be understood that if the indices of a delta fution of the simulation cell exceed the grid point indices, ththis function coincides with its periodic image that falwithin the simulation cell. As an example, assumeN15N25N3520. Then,

D (5)(21)(23)~r !5D (5)(1)(3)~r !. ~B2!

We also need to define an operator that projects a funcfrom the portion of the fine grid associated with functio

R

d

a

lt

BJ

lliQ

.

.

03511

c-n

n

fa(r ) andfb(r ) to the FFT box. Such an operator is definin a similar fashion toP(ab) by

Q~ab!58

W (x50

(2n121)

(y50

(2n221)

(z50

(2n321)

ubxyz&

3^B(x12Kab)(y12Lab)(z12Mab)u. ~B3!

OperatorsP†(ab) and Q†(ab) map a function from theFFT box to the simulation cell in the standard and fine grirespectively.

s.

ev.

ut.

ns

ry,-

m.

om-

M.

bilt,

bilt,

e,

*Author to whom correspondence should be addressed. Fax:144~0!1223 337356; Electronic address: [email protected]; Uhttp://www.tcm.cam.ac.uk/;pdh1001/

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