PHYSICAL REVIEW B 84, 155113 (2011)

Adaptation of the projector-augmented-wave formalism to the treatment of orbital-dependentexchange-correlation functionals

Xiao Xu and N. A. W. Holzwarth*

Department of Physics, Wake Forest University, Winston-Salem, North Carolina 27109, USA(Received 12 August 2011; published 14 October 2011)

This paper presents the formulation and numerical implementation of a self-consistent treatment of orbital-dependent exchange-correlation functionals within the projector-augmented-wave method of Blochl [Phys. Rev.B 50, 17953 (1994)] for electronic structure calculations. The methodology is illustrated with binding energycurves for C in the diamond structure and LiF in the rock salt structure, by comparing results from the Hartree-Fock(HF) formalism and the optimized effective potential formalism in the so-called KLI approximation [Krieger, Li,and Iafrate, Phys. Rev. A 45, 101 (1992)] with those of the local density approximation. While the work here usespure Fock exchange only, the formalism can be extended to treat orbital-dependent functionals more generally.

DOI: 10.1103/PhysRevB.84.155113 PACS number(s): 71.15.Dx, 71.15.Mb

I. INTRODUCTION

In order to improve the physical representation of materialsbeyond that of conventional density functional theory,1,2

there has recently been renewed interest in the use oforbital-dependent exchange-correlation functionals includingthe use of hybrid functionals3–12 and the use of a combina-tion of exact-exchange and the random-phase approximation(EXX/RPA).13–19 At the moment, some of these treatmentsare treated non-self-consistently in the sense that the orbital-dependent contributions are treated by “postprocessing” wavefunctions obtained from traditional density-dependent func-tionals. Because the orbital-dependent functionals are typicallyformulated in terms of integrals of the orbitals, the refinementof these treatments to full self-consistency requires updatingthe electron orbitals by solving integral-differential equations,a process which is, in principle, outside the realm of Kohn-Sham theory.

In order to treat orbital-dependent contributions self-consistently within Kohn-Sham theory, it is necessary torepresent its effects in terms of a local potential function knownas an optimized effective potential (OEP).20–22 An interestingmeasure of the numerical effects of the various approaches canbe seen from the ground-state energies of spin-unpolarizedspherical atoms for the case of pure Fock exchange. Usingideas in the literature,23–28 we have modified our atomic code29

in order to construct Figs. 1 and 2 and Table I. Figure 1 showsthe results of calculating the ground-state total energies of spin-unpolarized spherical atoms using the exact OEP approach, anapproximate OEP approach (“KLI” described below), and thenon-self-consistent postprossessing result using local densityapproximation (LDA) or generalized gradient approximation(GGA) orbitals. In this figure, energy differences relativeto Hartree-Fock are presented. Because of the variationalproperties of the Hartree-Fock solutions, all of the energydifferences are positive.26,27 The “OEP-HF” results presentedin Fig. 1 are consistent with results found earlier by Talman.27

Also shown in Fig. 1 are the corresponding results generatedby the approximation to OEP introduced by Kreiger, Li, andIafrate (KLI).30 The corresponding “KLI-HF” energies areslightly larger than the “OEP-HF,” differing by at most 0.1 eVfor the 3d transition metals. Other approximate OEP methodshave been developed, including the localized Hartree-Fock

method31 and the common energy denominator32 and theeffective local potential33 approximations. Bulat and Levy34

have recently shown these methods to be equivalent to calcu-lating the optimized effective potential within the subspaceof occupied orbitals [vocc(r)]. We have evaluated vocc(r)for the spherical atoms in Fig. 1, finding the correspondingground-state energies to differ by at most 0.005 eV from thosecalculated within the KLI scheme.

By contrast to the self-consistent results, the results ob-tained by using wave functions from self-consistent GGA35

or LDA36 functionals and a postprocessing evaluation of theFock exchange have significantly larger values of �E. On theone hand, given that the self-consistent Hartree-Fock wavefunctions28 are very similar in shape to those obtained fromself-consistent LDA or GGA calculations, the large values ofpostprocessing energy differences shown in Fig. 1 is somewhatsurprising. On the other hand, the Hartree-Fock postprocessingenergies LDA(PP) and GGA(PP) themselves are considerablydifferent from the self-consistent ground-state LDA and GGAatomic energies as shown in Fig. 2. Interestingly, the plotof Fig. 2 shows that the LDA energy differences [LDA −LDA(PP)] have the opposite sign from the GGA energydifferences [GGA − GGA(PP)], presumably due to thedifferent correlation functional forms of LDA and GGA. Theresults of Figs. 1 and 2 suggest that postprocessing treatmentsmay introduce unintended effects into the calculations. Itis reasonable to expect that a self-consistent treatment willproduce much more reliable results, benefiting from the powerof the variational principle.

In order to have a more precise measure of the energyrelationships, some representative total energies (for rare-gasatoms) are listed in Table I. Also listed in the table are literatureresults,21,37 which are essentially identical to those generatedwith our code.

These results for atoms provide a motivation for developingmethods to calculate the orbital-dependent terms accuratelyand self-consistently in extended systems. In previous work,28

we showed how to formulate the Hartree-Fock theory withinthe projector-augmented-wave (PAW) formalism of Blochl.38

In this paper we extend this analysis to the OEP theory. Forreasons discussed in Sec. II C, it turns out that treating the fullOEP equations within the PAW formalism is computationallydemanding. As a step close to reaching that goal, the present

155113-11098-0121/2011/84(15)/155113(16) ©2011 American Physical Society

XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36Atomic Number

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

ΔE (

eV)

LDA(PP)-HFGGA(PP)-HFKLI-HFOEP-HF

H

NeAr

Sc

Cu

Kr

FIG. 1. (Color online) Plot of total energy differences �E forspin-unpolarized spherical atoms in their ground-state configurationsfor Z = 1 (H) through Z = 36 (Kr), using the self-consistentHartree-Fock energy as reference. Results for �E obtained usingself-consistent OEP or Kreiger, Li, and Iafrate (KLI) calculationsare compared with results obtained using self-consistent generalizedgradient approximation (GGA) or local density approximation (LDA)wave functions to “postprocess” the Fock exchange energies.

work focuses on the KLI approximation and its relationshipwith Hartree-Fock theory. It is assumed here that the orbital-dependent contribution is that of the full Fock exchange.More importantly, extension of this approach to other orbital-dependent functionals, including hybrid functionals, and toRPA treatments is expected to follow similar steps.

The outline of the paper is as follows. In Sec. II, we presentthe KLI formalism for spherical atoms, briefly reviewingthe all-electron formulas30 and discussing the frozen-orbitalapproximation. In Sec. II C we present the PAW formalismfor spherical atoms and derive the relations for constructingbasis and projector functions for a PAW-KLI treatment. Moredetails of this work are presented in the Ph.D. thesis of Xu.39

In Sec. III we generalize the atomic formulations of both theHartree-Fock and KLI approaches to treat periodic solids ina plane-wave representation. The methods are demonstratedin terms of binding energy curves for diamond and for LiF inSec. IV. A summary and conclusions are presented in Sec. V,where we also compare the PAW-HF and PAW-KLI approacheswith previous treatments of Fock exchange reported in theliterature. In particular, we note that a recent description of

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536

Atomic Number-50

-40

-30

-20

-10

0

10

20

30

40

50

ΔE (

eV)

LDA-LDA(PP)GGA-GGA(PP)

HNe

Ar

CuSc

Kr

FIG. 2. (Color online) Plot of energy differences �E for spin-unpolarized spherical atoms in their ground-state configurations usingLDA or GGA exchange correlation functionals, each referenced tothe corresponding energies obtained using the same wave functionsto “postprocess” the Fock exchange contribution.

the GPAW code12 which implements the PAW formalism on areal-space grid includes some PAW-HF formulations as well.

II. ELECTRONIC STRUCTURE OF SPHERICAL ATOMSWITHIN THE KLI APPROXIMATION

A. All-electron formalism

For simplicity, we discuss the formalism for spin-unpolarized, spherically averaged atoms and use the samenotation as in our previous work on developing a PAWformalism for Hartree-Fock theory.28 The total electron energytakes the same functional form as in Hartree-Fock theory,as a sum of kinetic energy (EK ), nuclear energy (EN ),electron-electron or Hartree energy (EH ), and exchange energy(Ex):

Etot = EK + EN + EH + Ex. (1)

Here the exchange energy is written in terms of radial integralsdefined by Condon and Shortley:40

Ex = −∑pq

lp+lq∑L=|lp−lq |

1

2�L

pqRLpq;qp, (2)

TABLE I. Total energies (in Ry) for some of the atoms shown in Figs. 1 and 2, comparing the self-consistent Hartree-Fock (HF), optimizedeffect potential (OEP), Kreiger-Li-Iafrate approximation (KLI), and occupied subspace (OCC) approximations with the postprocessing energiescalculated with local density approximation [LDA(PP)] and generalized gradient approximation [GGA(PP)] wave functions with the Fockexchange expression replacing the exchange-correlation contributions. Also listed are literature values for the HF, OEP, and KLI results.

He Ne Ar Kr

HF (this work) −5.7234 −257.0942 −1053.6350 −5504.1100HF (literature)a −5.7234 −257.0942 −1053.6350 −5504.1100OEP (this work) −5.7234 −257.0908 −1053.6244 −5504.0859OEP (literature)b −5.7234 −257.0908 −1053.6244 −5504.0860KLI (this work) −5.7234 −257.0897 −1053.6210 −5504.0796KLI (literature)b −5.7234 −257.0896 −1053.6210 −5504.0796OCC (this work) −5.7234 −257.0896 −1053.6211 −5504.0793LDA(PP) (this work) −5.7191 −257.0630 −1053.5940 −5504.0264GGA(PP) (this work) −5.7191 −257.0734 −1053.6125 −5504.0587

aReference 37.bReference 21.

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ADAPTATION OF THE PROJECTOR-AUGMENTED-WAVE . . . PHYSICAL REVIEW B 84, 155113 (2011)

where

RLpq;st ≡ e2

∫ ∫dr dr ′ rL

<

rL+1>

ψp(r)ψq(r)ψs(r′)ψt (r

′). (3)

Here the Fock weight factor �Lpq for the moment L for the

spherically averaged atom was derived by Refs. 23–25 and isgiven in Eq. (14) of Ref. 28. In contrast to Hartree-Fock theory,in the OEP approach, the one-electron orbitals {ψp(r)} whichappear in the energy expression are eigenstates of an effectiveHamiltonian of the form

H = K + VN (r) + VH (r) + Vx(r)(4)

with Hψp(r) = εpψp(r).

Here the expressions for the nuclear and Hartree potentials areidentical to those of Ref. 28. In the full OEP theory, the localpotential Vx(r) can be determined iteratively in terms of orbitalshift functions41 gp(r) which are solutions to inhomogeneousdifferential equations of the form

(H − εp)gp(r) = Xp(r) − Vx(r)ψp(r) − (Uxp − Vxp)ψp(r),

(5)

where

Vxp ≡ 〈�p|Vx |�p〉 and Uxp ≡ 〈�p|Xp〉. (6)

The exchange integral function Xp(r) is identical to thatdefined in Eq. (19) of Ref. 28 except that the orbitals{ψp(r)} are the self-consistent OEP orbitals instead of theself-consistent Hartree-Fock orbitals. It takes the form

Xp(r) = −∑

q

lp+lq∑L=|lp−lq |

1

Np

�Lpq WL

qp(r)ψq(r), (7)

where

WLqp(r) ≡ e2

∫dr ′ rL

<

rL+1>

ψq(r ′)ψp(r ′). (8)

For the full OEP treatment, the converged local exchangepotential Vx(r) is obtained when the combined shift functionvanishes:26,41–43 ∑

p

Npgp(r)ψp(r) = 0. (9)

The KLI approximation30 to the OEP is based on thereasonable assumption that the orbital shift functions gp(r)are numerically small so that the left-hand side of Eq. (5) canbe set to 0.44 Then a local exchange potential function V KLI

x (r)can be found which satisfies the following KLI equation:

V KLIx (r)n(r) =

∑p

Npψp(r)XKLIp (r)

+∑

p

Np|ψp(r)|2(V KLIxp − UKLI

xp

). (10)

The radial density function is defined as

n(r) ≡∑

p

Np|ψp(r)|2. (11)

In order to solve Eq. (10) it is first necessary to determinethe matrix elements V KLI

xp . The p index references each of

the S atomic shells. The boundary conditions require that forthe shell p ≡ o corresponding to the outer most orbital ψo(r),the exchange potential matrix element must satisfy21,30

V KLIxo ≡ UKLI

xo . (12)

For the S − 1 shell indices p �= o, the following linearequations must be satisfied:21,30∑

q �=o

[δpq − �pqNq]�q = p − UKLIxp , (13)

where

�q ≡ V KLIxq − UKLI

xq . (14)

Here

�pq ≡∫

dr|ψp(r)|2|ψq(r)|2

n(r)(15)

and

p ≡∫

dr|ψp(r)|2 ∑

q Nqψq(r)XKLIq (r)

n(r). (16)

Once the matrix elements V KLIxp are determined from Eqs. (12),

(13), and (14), the KLI exchange potential V KLIx (r) can be

determined from Eq. (10). Figure 3 shows three examplesof V KLI

x (r) in comparison with corresponding local potentialscalculated using the full OEP formalism. The small differencesin the potentials shown here is consistent with results on othermaterials presented in the original KLI manuscript.30

B. Frozen-core orbital approximation

The arguments favoring the frozen-core orbital treatmentover the frozen-core potential treatment for the Hartree-Fockformalism were presented in our previous work28 and apply to

r (bohr)

-2

-1

0

OEPKLI

-2

-1

0

-2

-1

0

rVx(r

) (b

ohr

Ry)

Br

Cl

F

0 1 2 3 4 5

FIG. 3. (Color online) Comparison of OEP and KLI functions ofVx(r) for Br, Cl, and F in their ground states.

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XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

the KLI formalism as well. The frozen-core orbital approxi-mation within the KLI approach is almost a trivial extension ofthe all-electron treatment. The equations have the same formas given above, with the summation over shells p includingboth valence states �v(r) which are updated and core states�c(r) which are “frozen” to the reference configuration form.45

We note that in order for the KLI exchange potential (10)to remain orbital independent and especially independent ofcore-valence orbital labels, it is essential for both valence and(frozen) core contributions to be included in the evaluationof Eq. (10). As we will see, this will be true in the PAWformulation as well. On the other hand, it is often convenient toremove the constant contributions to the energy and to definea valence electron energy from terms that involve valenceelectrons alone and terms that involve interactions betweencore and valence electrons:

Eval = EvK + Ev

N + EcvH + Evv

H + Ecvx + Evv

x . (17)

Numerical results for the frozen-core orbital approximationof KLI are comparable to those of the frozen-core orbitalapproximation of Hartree-Fock reported in our earlier work.28

For example, we considered ionization energies for sphericallyaveraged 3d atoms. The results are shown in Fig. 4, where theHartree-Fock and KLI results are compared with those ob-tained using the LDA.36 The Hartree-Fock and KLI ionizationenergies are very similar throughout the 3d series, differingfrom from each other by less than 0.08 eV. Not surprisingly,the LDA ionization energies differ from the HF ionizationenergies by roughly 2 eV. With the core states defined bythe configuration of Ar, the errors introduced by calculatingthe ionization energies within frozen-core approximations arenot visible on this scale. For the frozen-core orbital approachwithin HF, KLI, and LDA formalisms, the average error in theionization energies is 0.002, 0.002, and 0.001 eV respectively.We also calculated the ionization energies using the frozen-core potential approach described in Ref. 28 (labeled HFVon the graph). In this frozen-core potential approach, the

Sc Ti V Cr Mn Fe Co Ni Cu Zn5

6

7

8

9

10

ΔEio

n (

eV)

HFHFVKLILDA

FIG. 4. (Color online) Plot of ionization energies for sphericallyaveraged 3d atoms, assuming transitions 3dx 4s2 → 3dx 4s1, usingthe HF, KLI, and LDA exchange approximations. The resultsdesignated as HFV were calculated using the frozen-core potentialapproximation as described in Ref. 28.

average error in calculating the ionization energies is 0.05 eV,25 times larger than that of the frozen-core orbital approach.This motivates the adoption of a frozen-core orbital treatmentfor our PAW formulation, which can be accomplished in astraightforward way.

C. PAW formalism for a spherical atom

1. Motivation for various approximations

In addition to the approximations associated with the PAWformalism itself, the treatment presented here uses two majorapproximations which need justification. We first consider thereason for using the KLI approximation rather than directlysolving the full OEP equations. The reason follows from thefact that the PAW formalism is designed to represent a valenceelectron wave function �v(r) of the form

�v(r) = �v(r)

+∑ai

(a

i (r − Ra) − ai (r − Ra)

)⟨P a

i

∣∣�v

⟩, (18)

where a denotes an atomic site (which is trivial for a sphericalatom) and i denotes atom-centered functions, which include|a

i 〉, |ai 〉, and |P a

i 〉 for the all-electron basis functions,pseudo basis functions, and projector functions, respectively.Typically, the index i represents a small number of states (oneor two states per angular momentum channel) to represent theelectron wave function in typical environments found in solids.In order to adapt the PAW formalism to the OEP treatment, itis necessary to use this form to also represent the orbital shiftfunction gv(r) defined in Eq. (5). Because of the oscillatoryshape of these gv(r) functions, it is necessary to significantlyincrease the number of one-center basis functions included inthe general transformation of Eq. (18). The recent paper byBulat and Levy34 clarified this point by showing that the fulloptimized effective potential function can be determined asa sum of contributions from the space of occupied orbitalsto determine vocc

x (r) plus additional terms generated by allunoccupied states. These additional terms require additionalbasis functions. While the use of additional basis functions inthe PAW expansion is in principle possible, it is computationalexpensive. Some details are presented in Appendix A. Infact, recent work by Harl and co-workers18 describes theuse of these additional basis functions for a postprocessingtreatment of the RPA within the PAW formalism. In this case,the additional basis functions are physically motivated by theexcited state contributions to the random phase formalism,while in the exchange-only OEP treatment they are neededonly to fulfill the numerical requirements of the equationswhich represent only the ground state of the system. Onthe other hand, the minimal basis PAW formalism is wellsuited to represent the approximate OEP equations in the KLIapproach accurately and efficiently. The PAW-KLI approachpresented here can be also extended to the full occupied spaceapproximation vocc

x (r) with few changes to the formalism.The second approximation in this work is made for sim-

plicity. We have argued that the core states provide importantcontributions to the exchange interactions and in Ref. 28 wehave shown how to treat extended core states with frozen-corepseudo wave functions ψc(r) and their fully nodal counterparts

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ADAPTATION OF THE PROJECTOR-AUGMENTED-WAVE . . . PHYSICAL REVIEW B 84, 155113 (2011)

ψc(r). While including these upper core wave functions inthe calculation is straightforward, in order to simplify thepresentation in the present work, we make the assumptionthat for all core wave functions ψc(r) ≡ 0. This can be madeprecise for any system, by treating upper core states as valencestates (with additional computational cost), leaving the coredesignation to refer only to states whose orbitals are wellconfined within the augmentation sphere. What this meansis that the summations over all states in the pseudo spaceincludes only valence states. The core states then only enterthe calculations through the one-center all-electron terms.The more general case of allowing for nonzero ψc(r) forupper core wave functions can be derived by straightforwardextension.

2. PAW-KLI formalism for atoms

The purpose of deriving the PAW-KLI equations of aspherical atom is to define the consistent basis and projectorfunctions which satisfy the atomic PAW-KLI equations forthe reference configuration and to provide the basis forextending the formalism to a more general system. The PAWrepresentation of the radial electron density of a spherical atomtakes the form28,38

n(r) = n(r) + [na(r) − na(r)], (19)

where n(r) represents the radial pseudo density defined overall space and the terms with superscript a represent theatomic density confined within the augmentation sphere. Formaterials, the single index a will be replaced by a summationof one-center contributions over all augmentation spheres.

The PAW transformation allows us to approximate the left-hand side of the of KLI equality—Eq. (10)—by the form

V KLIx (r)n(r) = V KLI

x (r )n(r)

+(V aKLI

x (r)na(r) − V aKLIx (r )na(r)

). (20)

Here V KLIx (r) denotes the smooth pseudo exchange potential

in the KLI approximation, which is defined over all space,while V aKLI

x (r) and V aKLIx (r) denote the atom-centered all-

electron and pseudo exchange functions, respectively, whichwill appear in atom-centered matrix elements in the PAWformalism. In order to determine these three contributions tothe exchange potential, we assume that each of them satisfiesEq. (10) in their respective spatial and functional domains.

The pseudo-space contribution takes the form

V KLIx (r )n(r) =

∑v

Nvψv(r)XKLIv (r)

+∑

v

Nv|ψv(r)|2(V KLIxv − UKLI

xv

). (21)

Within the augmentation region, the two types of one-centercontributions can be written as

V aKLIx (r)na(r) =

∑p

Npψap(r)XaKLI

p (r)

+∑

p

Np

∣∣ψap(r)

∣∣2(V KLI

xp − UKLIxp

)(22)

and

V aKLIx (r )na(r) =

∑v

Nvψav (r)XaKLI

v (r)

+∑

v

Nv

∣∣ψav (r)

∣∣2(V KLI

xv − UKLIxv

). (23)

The V KLIxv and UKLI

xv matrix elements that appear in each ofthe three equations are determined from a sum of contributionsin the form

V KLIxp = ⟨

�p

∣∣V KLIx

∣∣�p

⟩+(⟨

�ap

∣∣V aKLIx

∣∣�ap

⟩ − ⟨�a

p

∣∣V aKLIx

∣∣�ap

⟩)(24)

and

UKLIxp = ⟨

�p

∣∣XKLIp

⟩ + (⟨�a

p

∣∣XaKLIp

⟩ − ⟨�a

p

∣∣XaKLIp

⟩). (25)

In these expressions and in others in this section, the index p

for core states is nontrivial only for the one-center all-electronterms.

In order to determine the unknown coefficients V KLIxp , a set

of linear equations similar to Eq. (13) must be solved. Thesecan be written in the form∑

q �=o

[δpq − �PAW

pq Nq

]�q = PAW

p − UKLIxq . (26)

Once the unknown matrix elements �q have been determined,they can be used in Eqs. (21), (22), and (23) to determineV KLI

x (r), V aKLIx (r), and V aKLI

x (r), respectively. In Eq. (26)the �PAW

pq matrix elements are given by

�PAWpq =

∫ ∞

0dr

|ψp(r)|2|ψq(r)|2n(r)

+∫ ra

c

0dr

[∣∣ψap(r)

∣∣2∣∣ψaq (r)

∣∣2

na(r)−

∣∣ψap(r)

∣∣2∣∣ψaq (r)

∣∣2

na(r)

].

(27)

The exchange coefficients PAWp are given by

PAWp =

∫ ∞

0dr

|ψp(r)|2 ∑q Nqψq(r)XKLI

q (r)

n(r)

+∫ ra

c

0dr

[∣∣ψap(r)

∣∣2 ∑q Nqψ

aq (r)XaKLI

q (r)

na(r)

−∣∣ψa

p(r)∣∣2 ∑

q Nqψaq (r)XaKLI

q (r)

na(r)

]. (28)

In Eqs. (27) and (28), the one-center integrands are confinedwithin the augmentation spheres (r � ra

c ) and contributions forp or q representing core states come only from the one-centerall-electron terms.

In these expressions the pseudo exchange kernel is givenby

XKLIv (r) = −

∑v′

lv+lv′∑L=|lv−lv′ |

1

Nv

�Lvv′W

Lv′v(r)ψv′(r), (29)

with the interaction function evaluated according to

WLv′v(r) = e2

∫dr ′ rL

<

rL+1>

[ψv′ (r ′)ψv(r ′) + ML

v′v(r ′)]. (30)

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XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

The summation over shells v and v′ includes contributionsfrom valence pseudo wave functions ψv(r). The momentfunction ML

v′v(r) is defined by Eq. (58) of Ref. 28.For the one-center all-electron terms (22), the sum over p

includes both valence and core contributions. The one-centerall-electron orbitals for the valence states are given by

ψav (r) =

∑i

⟨P a

i

∣∣�v

⟩φa

i (r). (31)

The PAW functions are denoted by pai (r),φa

i (r), and φai (r)

for the radial projector, all-electron basis, and pseudo basisfunctions, respectively, while the capitalized forms [for ex-ample P a

i (r)] denote the corresponding full three-dimensionalfunction. For the all-electron core contributions (p = c), theall-electron frozen-core functions ψa

c (r) are used directly.These expressions for the atom-centered radial wave func-tions are only valid in the augmentation region 0 � r � ra

c .Fortunately, in all the expressions in which they are used,it is only necessary to evaluate the one-center functionsin the augmentation regions. For evaluating the one-centerall-electron exchange kernel XaKLI

p (r) the following form forthe interaction integrals can be used for 0 � r � ra

c :

WaLqp (r) =

∑ij

⟨�q

∣∣P ai

⟩⟨P a

j

∣∣�p

⟩waL

ij (r) +(

r

rac

)L

ωaLqp , (32)

where

waLij (r) ≡ e2

∫ rac

0dr ′ rL

<

rL+1>

φai (r ′)φa

j (r ′). (33)

This expression is correct for both shell indices q and p corre-sponding to valence states. If one or both of them correspondto core states, the expressions are modified according to⟨

�ac

∣∣P ai

⟩ → δci (34)

and the replacement of φai with ψa

c (r).The one-center pseudo orbitals for the valence states are

given by

ψav (r) =

∑i

⟨P a

i

∣∣�v

⟩φa

i (r), (35)

using the same notation as above. For evaluating the one-centerpseudo exchange kernel XaKLI

v (r), the following form for theinteraction integral can be used for 0 � r � ra

c :

W aLv′v (r) =

∑ij

⟨�v′

∣∣P ai

⟩⟨P a

j

∣∣�v

⟩waL

ij (r) +(

r

rac

)L

ωaLv′v, (36)

where

waLij (r) ≡ e2

∫ rac

0dr ′ rL

<

rL+1>

[φa

i (r ′)φaj (r ′) + maL

ij (r ′)], (37)

where the augmentation moment maLij (r) has been defined in

Eq. (53) of Ref. 28. For convenience we repeat the definitionhere:

maLij (r) ≡ maL

ij gaL(r), (38)

where the charge moment coefficient maLij is given in terms of

the i and j basis functions

maLij ≡

∫ rac

0dr rL

(φa

i (r)φaj (r) − φa

i (r)φaj (r)

), (39)

and the augmentation shape function gaL(r) is localized to the

augmentation sphere, 0 � r � rac , and is normalized according

to ∫ rac

0dr rL ga

L(r) = 1. (40)

In the interaction integrals for valence states in Eqs. (32)and (36) the same constants ωaL

v′v appear. (The correspondingcontributions vanish for localized core states.) In atomiccalculations, the constants can be evaluated from the pseudopair density outside the augmentation region:

ωaLv′v ≡ (

rac

)L∫ ∞

rac

dr ′ ψv′ (r ′)ψv(r ′)r ′L+1

. (41)

In general, it is necessary to evaluate the constants ωaLv′v

within the augmentation region. This can be accomplished bymatching the boundary values of of WaL

v′v (rac ) and W aL

v′v (rac ) to

that of WLv′v(ra

c ), the full pseudo interaction integral given inEq. (30).

In terms of the given representations of orbitals and ofthe interaction integrals, the one-center all-electron exchangekernel function can be written as

XaKLIp (r) = −

∑q

lp+lq∑L=|lp−lq |

1

Np

�LpqW

aLqp (r)ψa

q (r), (42)

and the one-center pseudo exchange kernel function can bewritten as

XaKLIv (r) = −

∑v′

lv+lv′∑L=|lv−lv′ |

1

Nv

�Lvv′W

aLv′v (r)ψa

v′(r). (43)

The expressions for the exchange kernel functions givenin Eqs. (30), (42), and (43) are consistent with the PAWexchange kernel function given in Ref. 28 for the Hartree-Fock formalism, XPAW

v (r) = XHFv (r) + ∑

ai |P ai 〉XaHF

iv , withthe correspondence

XHFv (r) → XKLI

v (r) (44)

and the approximate relation

XaHFiv ≈ ⟨

ai

∣∣XaKLIv

⟩ − ⟨a

i

∣∣XaKLIv

⟩. (45)

These approximate relationships rely on the fact that withinthe augmentation spheres, ψv(r) ≈ ψa

v (r), based on the PAWexpansion given in Eq. (35) and approximate completenessrelations38 for 0 � r � ra

c such as∑i

∣∣P ai

⟩⟨a

i

∣∣ ≈ δ(r − r′) (46)

for the the pseudo space functions.The equations given above can be used to self-consistently

solve the Kohn-Sham equations within the PAW-KLI approx-imation, given a PAW dataset of basis, projector, and pseudopotential functions. For the construction of a PAW dataset in

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ADAPTATION OF THE PROJECTOR-AUGMENTED-WAVE . . . PHYSICAL REVIEW B 84, 155113 (2011)

the PAW-KLI approximation, it is necessary to evaluate theseexpressions for the reference configuration used to generatethe basis and projector functions. In this case, Eqs. (21) and(23) are identical to each other, and Eq. (22) is identical to theall-electron result, Eq. (10). By evaluating Eq. (21) or (23) todetermine the pseudo exchange potential V a

x (r), for the givenchoice of basis and projector functions, it is possible to deter-mine the unscreened local pseudo potential V a

loc(r) from thechosen screened pseudo potential V aPS(r) according to

V aloc(r) = V aPS(r) − Zava

0 (r) − V aH (r)

⌋ref − V a

x (r)⌋

ref . (47)

Here va0 (r) is defined as the Coulomb potential associated with

the L = 0 augmentation shape function gaL(r). The Hartree

contribution can be evaluated as a sum of core and valencecontributions in the form

V acH (r) ≡ Qa

coreva0 (r) and V av

H (r)⌋

ref ≡∑

v

NvW0vv(r)

⌋ref .

(48)

[The expression for V acH (r) differs from that of Ref. 28

because here we assume nacore(r) ≡ 0.] For evaluating both

the reference Hartree potential and reference pseudo exchangekernel function

Xv(r)ref = −∑v′

lv+lv′∑L=|lv−lv′ |

1

Nv

�Lvv′W

Lv′v(r)

⌋refψv′(r), (49)

the interaction function can be evaluated from

WLv′v(r)

⌋ref = e2

∫ ∞

0dr ′ rL

<

rL+1>

[ψv′ (r ′)ψv(r ′) + ML

v′v(r ′)],

(50)

For the reference configuration, the shell labels v′ and v

correspond to valence basis functions so that the momentfunctions have the simplified form

MLv′v(r)

⌋ref

= gL(r)∫ ra

c

0dr ′ r ′L[ψv′(r ′)ψv(r ′) − ψv′(r ′)ψv(r ′)]. (51)

Figure 5 shows the pseudo versions of the KLI localexchange potentials given in Fig. 3. Figure 6 shows someexamples of the unscreened local pseudo potential, V a

loc(r),showing that for the same choices of construction parametersthe shapes are quite similar to those of the LDA. These resultsfor F,46 Cl,47 and Br48 were generated using a variation of theATOMPAW code,29 using the Vanderbilt49 scheme for generatingthe pseudo basis and projector functions.

III. PLANE-WAVE REPRESENTATIONS OF PAW-HFAND PAW-KLI EQUATIONS

In order to develop the projector-augmented-wave approachfor Hartree-Fock and KLI formalisms to treat periodic systems,it is necessary to extend the equations presented in Ref. 28 andin Sec. II C to consider multiple atomic sites a and additionalangular dependence. A pseudo wave function for a Bloch state

r (bohr)

-2

-1

0

All-electronPseudo

-2

-1

0

-2

-1

0

rVx(r

) (b

ohr

Ry)

Br

Cl

F

0 1 2 3 4 5

FIG. 5. (Color online) Comparison of KLI all-electron [Vx(r)]and pseudo [Vx(r)] local exchange potentials for Br, Cl, and F in theirground states.

of band n and wavevector k can be represented in a plane-waveexpansion of the form

�nk(r) =√

1

V∑

G

Ank(G)ei(k+G)·r, (52)

where V denotes the unit cell volume, Ank(G) is an expansioncoefficient, and the summation over reciprocal lattice vectorsG includes all terms for which |k + G|2 � Ecut, for anappropriate cutoff parameter Ecut. For simplicity, we assumea spin-unpolarized system with band weight and occupancyfactors fnk.

0 0.5 1 1.5

-7

-6

-5

-4

-3

-2

-1

0

Vvl

oc(r

)

(Ry)

KLILDA

0 0.5 1 1.5r (bohr)

0 0.5 1 1.5

F Cl Br

~

FIG. 6. (Color online) Plots of V aloc(r) for F, Cl, and Br, comparing

KLI results with the corresponding LDA results.

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XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

A. Total energy expressions

The formulas for the total valence energy are the samein both the Hartree-Fock and KLI treatments. The energydifferences are only due to differences in the wave functionsused to evaluate the energies. Here we focus on the theFock exchange term only, since the other contributions areidentical to those found in earlier papers.38,50–52 The totalvalence exchange energy is a sum of smooth and one-centercontributions of the form

Exval = Evvx +

∑a

(Eavv

x + Eavcx − Eavv

x

). (53)

The fact that the core-valence interactions enter only in the one-center all-electron term is a consequence of our assumptionthat upper core states are treated as valence electrons so thatonly core states localized within the augmentation sphere aretreated as frozen-core states.

1. Smooth contributions to the Fock energy

For the purposes of evaluating the Fock energy andinteraction terms, the smooth pair density function for bandindices nk and n′k′ can be written as

ρnk,n′k′(r) ≡ �nk(r)�∗n′k′(r) + ρnk,n′k′(r), (54)

where the second term is the compensation pair charge.4,28 Fora nonspherical system, the forms of these moments must begeneralized from those presented in Eq. (58) of Ref. 28, in theform53

ρnk,n′k′(r) =∑aij

⟨P a

i

∣∣�nk⟩⟨�n′k′

∣∣P aj

⟩μa

ij (r − Ra), (55)

where the generalized moments are given by

μaij (r) ≡

∑LM

GLMlj mj limi√

4π

maLji (r)

r2YLM (r). (56)

Here all of the terms are the same as defined in Ref. 28 exceptfor the Gaunt coefficient,54 which we take to be55

GLMlj mj limi

≡√

4π

∫d�Y ∗

lj mj(r)Y ∗

LM (r)Ylimi(r). (57)

By design, this pair density function has the property

limN→∞

1

N

∫NV

d3r ρnk,n′k′(r) = δnn′ (2π )3δ(k − k′). (58)

The Fourier transform of the smooth pair density function isdefined to be

ρnk,n′k′(G) ≡∫V

d3r (ρnk,n′k′(r)e−i(k−k′)·r)e−iG·r. (59)

The compensation pair charge contribution can be evaluatedaccording to

ρnk,n′k′(G) =∑aij

⟨P a

i

∣∣�nk⟩⟨�n′k′

∣∣P aj

⟩μ

a

ij (k − k′ + G), (60)

where

μa

ij (q) ≡ e−iq·Ra∑LM

GLMlj mj limi

√4π (−i)LYLM (q)

×∫ ra

c

0drjL(qr)maL

ji (r), (61)

with jL(x) denoting a spherical Bessel function.

In terms of smooth pair densities, the corresponding Fockenergy can be written in the form

Evvx = −e2

4

∑nkn′k′

fnkfn′k′

×∫

d3r d3r ′ ρnk,n′k′(r)ρ∗nk,n′k′(r′)

|r − r′|

= −e2π

V∑

nkn′k′fnkfn′k′

∑G

|ρnk,n′k′(G)|2|k − k′ + G|2 . (62)

The evaluation of this singular integral has been the subjectof several investigations.56–60 In this work, we evaluated boththe methods of Spencer and Alavi57 and of Duchemin andGygi56 as described in more detail by Holzwarth and Xu.61

Results given in Sec. IV were obtained using the method ofSpencer and Alavi.57

2. One-center contributions to the Fock energy

The combinations of angular contributions that appear inthe one-center Coulombic contributions can be expressed interms of the four-index matrix elements

Vaij ;kl ≡

∑LM

GLMlimi lj mj

(−1)MGL−Mlkmkllml

(RaL

ij ;kl − RaLij ;kl

2L + 1

), (63)

where the generalized Condon-Shortley radial interactionintegrals RaL

ij ;kl and RaLij ;kl have been defined in Eqs. (63) and

(64) of Ref. 28.It is also convenient to define a weighted projector product

Waij ≡

∑nk

fnk⟨�nk

∣∣P ai

⟩⟨P a

j

∣∣�nk⟩. (64)

In evaluating both Vaij ;kl and Wa

ij , each basis index i,j,k,l

stands for both the radial and angular quantum numbers(nilimi , etc.).

The one-center valence-valence contribution to the Fockenergy can then be written as

Eavvx − Eavv

x = −1

4

∑ijkl

WailW

akjVa

ij ;kl . (65)

We note that in this formulation, which follows Ref. 28 andis slightly different from that of our earlier work [Eqs. (A26)and (A31) of Ref. 50], the corresponding one-center Hartreeenergy contributions take the form

EavvH − Eavv

H = 1

2

∑ijkl

Waij Wa

klVaij ;kl . (66)

The one-center valence-core contribution to the Fockenergy depends only on the Condon-Shortley interactionintegrals between valence all-electron basis functions andfrozen-core orbitals and takes the form

Eavcx = −

∑ij

Waij δli lj δmimj

Caij , (67)

where

Caij =

∑c L

Nc

2

(lc L li

0 0 0

)2

RaLic;cj . (68)

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ADAPTATION OF THE PROJECTOR-AUGMENTED-WAVE . . . PHYSICAL REVIEW B 84, 155113 (2011)

Here the sum over c is a sum over core shells with occupanciesNc = 2(2lc + 1) for each atom a.

B. Fock exchange kernel for Hartree-Fock formalism

The generalization of Eqs. (70)–(74) of Ref. 28 for a Blochstate nk can be written as

XPAWnk (r) = XHF

nk (r) +∑ai

∣∣P ai

⟩Xa

i,nk. (69)

The pseudo exchange kernel takes the form

XHFnk (r) = −1

2

∑n′k′

fn′k′Wnk,n′k′(r)�n′k′(r), (70)

where the interaction function is defined by

Wnk,n′k′(r) ≡ e2∫

d3r ′ ρnk,n′k′(r′)|r − r′|

= ei(k−k′)·r ∑G

W nk,n′k′(G)eiG·r. (71)

The Fourier transform of the interaction kernel is given by

W nk,n′k′(G) = 4πe2

Vρnk,n′k′(G)

|k − k′ + G|2 . (72)

In order to treat its singular behavior in evaluating the pseudoexchange kernel and related quantities, we used the method ofSpencer and Alavi57 which was mentioned previously in thecontext of evaluating the smooth contributions to the exchangeenergy [Eq. (62)].

The one-center matrix elements for the Hartree-Fockpseudo exchange kernel function analogous to Eq. (73) inRef. 28 (but simplified for treating only localized core orbitals)take the form

Xai,nk = −1

2

∑jkl

⟨P a

l

∣∣�nk⟩Wa

kjVaij ;kl

− 1

2

∑n′k′

fn′k′∑

j

⟨P a

j

∣∣�n′k′⟩Za

nk,n′k′;ij

−∑

j

⟨P a

j

∣∣�nk⟩δli lj δmimj

Caij , (73)

where

Zank,n′k′;ij ≡

∫d3r μa

ij (r)Wnk,n′k′(r)

=∑

G

μa∗ij (k − k′ + G)W nk,n′k′(G). (74)

C. Fock exchange potential for KLI formalism

The smooth pseudo exchange kernel XKLInk (r) in the KLI

approximation analogous to Eq. (30) takes the same formas the Hartree-Fock expression given in Eq. (70), evaluatedusing the appropriate KLI pseudo wave functions �nk(r). Thecorresponding pseudo-space potential function V KLI

x (r) can

be determined from an expression analogous to Eq. (21) inSec. II C 2:

V KLIx (r) = 1

ρ(r)

[ ∑nk

fnk�∗nk(r)XKLI

nk (r)

+∑nk

fnk|�nk(r)|2(V KLIxnk − UKLI

xnk

)]. (75)

In this expression, the valence pseudo density is given by

ρ(r) ≡∑nk

fnk|�nk(r)|2. (76)

Once the constant matrix elements V KLIxnk and UKLI

xnk are known,this expression can be evaluated most conveniently using fastFourier transform (FFT) methods.

The one-center contributions to the exchange kernel canalso be derived by extension of the atomic formalism. Theone-center all-electron full density for atom a can be writtenin the form

ρa(r) =∑ij

Waij

a∗i (r)a

j (r) +∑

c

Nc

∣∣ψac (r)

∣∣2

4πr2, (77)

were the first term represents the valence states spanned byPAW basis functions indices i and j , and the second termrepresents the localized core states with index c representingthe core shells [with occupancy Nc = 2(2lc + 1)] for that atom.In order to evaluate the summations involving the all-electronexchange kernel, it is convenient to define the followingfunctions. Analogous to the first term on the right-hand sideof Eq. (22) we define∑

p

Npψap(r)XaKLI

p (r) → ϒa cc(r) + ϒa cv(r) + ϒa vv(r).

(78)

Representing pure core contributions, we define

ϒa cc(r) ≡ −∑cc′

∑L

�Lcc′W

aLcc′ (r)

ψac (r)ψa

c′(r)

4πr2, (79)

using the all-electron notation introduced in Sec. II A. Repre-senting mixed core-valence contributions, we define

ϒa cv(r) ≡ −∑ijc

NcWaijY

∗limi

(r)Ylj mj(r)

φai (r)ψa

c (r)

r2

×∑L

(lj L lc

0 0 0

)2

WaLjc (r). (80)

In order to represent the pure valence contributions it isconvenient to define the following one-center functions

�ank(r) ≡

∑i

⟨P a

i

∣∣�nk⟩a

i (r) (81)

for the all-electron states and

�ank(r) ≡

∑i

⟨P a

i

∣∣�nk⟩a

i (r) (82)

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XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

for the pseudo states. The one-center all-electron pure valencecontributions can be written in the form

ϒa vv(r) ≡ −1

2

∑ij ;LM

a∗i (r)a

j (r)Y ∗LM (r)

×[W a ij

LM (r) +(

r

rac

)L

Ba ij

LM

]. (83)

The first term of this expression can be evaluated in termsof the interaction potentials [Eq. (33)] evaluated within theaugmentation sphere

�waLkl (r) ≡ waL

kl (r) −(

r

rac

)L

waLkl (ra

c ) (84)

with the expression

W a ij

LM (r) ≡∑kl

WailW

akj

√4π (−1)M

2L + 1GL−M

lkmkllml�waL

kl (r). (85)

The second term of Eq. (83) is related to the boundary valueconstant ωL

v′v defined in Sec. II C 2 and can be expressed in theform

Ba ij

LM ≡∑nk

fnkBa ij

LM nk

≡∑nk

fnk

∑n′k′

fn′k′⟨�nk

∣∣P ai

⟩⟨P a

j

∣∣�n′k′⟩Ja LM

nk,n′k′ . (86)

Here Ja LMnk,n′k′ is the angular component of the pseudo interaction

integral defined by Eq. (71) expanded about the atomic site a

and evaluated at the augmentation radius rac :

Ja LMnk,n′k′

= 4πiL∑

G

W nk,n′k′(G)eiqG·Ra

YLM (qG)jL

(|qG|rac

). (87)

In this expression, qG ≡ k − k′ + G, Ra denotes the atomicposition, and jL(x) denotes the spherical Bessel function.

In terms of these functions, the one-center all-electronexchange potential function can be evaluated with the relation

V aKLIx (r) = 1

ρa(r)

[ϒa cc(r) + ϒa cv(r) + ϒa vv(r)

+∑nk

fnk∣∣�a

nk(r)∣∣2(

V KLIxnk − UKLI

xnk

)+

∑c

Nc

∣∣ψac (r)

∣∣2

4πr2

(V KLI

xc − UKLIxc

)]. (88)

The one-center pseudo density, by assumption, includesonly valence contributions and can be written in the form

ρa(r) =∑ij

Waij

a∗i (r)a

j (r). (89)

The one-center pseudo exchange function can be evaluatedwith an expression similar to that of Eq. (88):

V aKLIx (r) = 1

ρa(r)

[ϒa vv(r)

+∑nk

fnk∣∣�a

nk(r)∣∣2(

V KLIxnk − UKLI

xnk

)]. (90)

The function ϒa vv(r) is evaluated using an expression similarto Eq. (83), replacing the all-electron basis functions a

i (r)with pseudo basis functions a

i (r) and the all-electron basisfunction kernel waL

kl (r) defined by Eq. (33) with the pseudobasis function kernel waL

kl (r) defined by Eq. (37). By construc-tion, the basis function kernels are equal at the augmentationradii ra

c , waLkl (ra

c ) = waLkl (ra

c ), ensuring consistency of theequations.

In order to evaluate this expressions we need to determinethe exchange integral matrix elements. Matrix elementscorresponding to core shells come only from the one-centerterms:

UKLIxc = −

∑c′;L

1

Nc

�Lcc′R

aLcc′;c′c

− 1

2

∑ij ;L

Waij δli lj δmimj

(lc L li

0 0 0

)2

RaLic;cj . (91)

In practice, each distinct core shell implies a particular sitelabel a and core-core contributions (c and c′) are restricted tothe same site. Matrix elements corresponding to the valencebands can be evaluated using the expression

UKLIxnk =

∫d3r �∗

nk(r)XKLInk (r)

− 1

2

∑a;ijkl

⟨�nk

∣∣P ai

⟩⟨P a

l

∣∣�nk⟩Wa

kjUaij ;kl

− 1

2

∑a;ij ;LM

GLMlimi lj mj√

4π

maLij(

rac

)LBa ij

LM nk

−∑a;ij

⟨�nk

∣∣P ai

⟩⟨P a

j

∣∣�nk⟩δli lj δmimj

Caij . (92)

In this expression, the coefficient Uaij ;kl is similar to the one-

center Coulomb coefficient Vaij ;kl defined in Eq. (63) and is

given by

Uaij ;kl ≡

∑LM

GLMlimi lj mj

(−1)MGL−Mlkmkllml

uaLij ;kl

2L + 1, (93)

where the radial Coulomb terms are defined according to

uaLij ;kl ≡

∫ rac

0dr

(φa

i (r)φaj (r)waL

kl (r) − φai (r)φa

j (r)waLkl (r)

)− maL

ij(rac

)LwaL

kl

(rac

). (94)

As in the atomic formulation, in order to evaluate the threecontributions to the exchange potential in Eqs. (75), (88), and(90), the potential matrix elements V KLI

xnk and V KLIxc must first

be determined by solving a set of linear equations. For thispurpose, we define

PAWnk ≡ nk +

∑a

(a

nk − ank

), (95)

where

nk ≡∫

d3r|�nk(r)|2∑

n′k′ fn′k′�∗n′k′(r)XKLI

n′k′ (r)

ρ(r), (96)

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ADAPTATION OF THE PROJECTOR-AUGMENTED-WAVE . . . PHYSICAL REVIEW B 84, 155113 (2011)

and the one-center contributions can be written as

ank − a

nk ≡∑ij

⟨�nk

∣∣P ai

⟩⟨P a

j

∣∣�nk⟩

×( ∫

r�rac

d3r a∗i (r)a

j (r)ϒa cc(r) + ϒa cv(r) + ϒa vv(r)

ρa(r)

−∫

r�rac

d3r a∗i (r)a

j (r)ϒa vv(r)

ρa(r)

). (97)

The corresponding core shell contribution can be written as

PAWc = a

c , (98)

where the notation implies that the core shell c corresponds tothe particular atomic site a. The term may be evaluated withthe expression

ac ≡

∫d3r

∣∣ψac (r)

∣∣2

4πr2

ϒa cc(r) + ϒa cv(r) + ϒa vv(r)

ρa(r). (99)

The coupling matrix elements between valence states can bewritten in the form

�PAWnk,n′k′ = �nk,n′k′ +

∑a

(�a

nk,n′k′ − �ank,n′k′

). (100)

Here, the pseudo space contribution is given by

�nk,n′k′ ≡∫

d3r|�nk(r)|2|�n′k′(r)|2

ρ(r), (101)

which can be evaluated using FFT techniques. The one-centercontributions can be evaluated from the expressions

�ank,n′k′ − �a

nk,n′k′

=∑ijkl

⟨�nk

∣∣P ai

⟩⟨P a

j

∣∣�nk⟩⟨�n′k′

∣∣P ak

⟩⟨P a

l

∣∣�n′k′⟩

×( ∫

r�rac

d3ra∗

i (r)aj (r)a∗

k (r)al (r)

ρa(r)

−∫

r�rac

d3ra∗

i (r)aj (r)a∗

k (r)al (r)

ρa(r)

). (102)

The coupling matrix elements between valence and core shellsand between core shells have only one-center contributions anddepend only on the particular atomic site a associated with thatcore shell:

�PAWnk,c = �a

nk,c and �PAWc,c′ = �a

c,c′ . (103)

The one-center matrix elements are both Hermitian and can beevaluated from the expressions

�ank,c =

∑ij

⟨�nk

∣∣P ai

⟩⟨P a

j

∣∣�nk⟩ ∫

d3ra∗

i (r)aj (r)

ρa(r)

∣∣ψac (r)

∣∣2

4πr2

(104)

and

�ac,c′ =

∫d3r

∣∣ψac (r)

∣∣2

4πr2

∣∣ψac′ (r)

∣∣2

4πr2

1

ρa(r). (105)

For evaluating the one-center integrals, it is most convenientto use a procedure found to be efficient in previous work.50 Weuse a generalized Gaussian quadrature method to sample the

angular directions rα with weight factors wα with∑

α wα =4π . For each direction rα (usually 144 points are sufficient),the radial integrals 0 � r � ra

c are evaluated usual finite-difference-based algorithms. For example,∫

d3ra∗

i (r)aj (r)

ρa(r)

∣∣ψac (r)

∣∣2

4πr2

≈∑

α

wαY ∗limi

(rα)Ylj mj(rα)

∫ rac

0dr

φai (r)φa

j (r)∣∣ψa

c (r)∣∣2

4πr2ρa(r rα).

(106)

The linear equations that must be solved can then be put in theform ∑

n′k′

(δnk,n′k′ − �PAW

nk,n′k′fn′k′)�n′k′

−∑

c

�PAWnk,c Nc�c = PAW

nk − UKLIxnk (107)

and ∑c′

(δc,c′ − �PAW

c,c′ Nc′)�c′

−∑n′k′

�PAWc,n′k′fn′k′�n′k′ = PAW

c − UKLIxc . (108)

In these expressions,

�nk ≡ V KLIxnk − UKLI

xnk and �c ≡ V KLIxc − UKLI

xc . (109)

The dimension of the the coupling matrix �PAW is equal tothe number of occupied bands nk within the k-point samplinggrid and the number of core shells c for all of the atomsof the unit cell. As in the atomic case, the coupling matrixis rank deficient, but the linear Eqs. (107) and (108) can besolved up to an arbitrary constant potential shift. We chose tofix the potential constant by setting �n0k0 = PAW

n0k0− UKLI

xn0k0,

where the index n0k0 corresponds to the highest Kohn-Shameigenvalue associated with an occupied state.

Once the potential matrix elements V KLIxnk and V KLI

xc aredetermined, the corresponding exchange potentials can becalculated. While V KLI

x (r) [Eq. (75)] contributes to the Kohn-Sham pseudo potential evaluated over all space, the one-centercontributions given in Eqs. (88) and (90) contribute to theone-center matrix elements of the Hamiltonian Da

ij in the form[V a

x

]ij

≡ ⟨a

i

∣∣V aKLIx

∣∣aj

⟩ − ⟨a

i

∣∣V aKLIx

∣∣aj

⟩. (110)

With the determination of these exchange potentials, thecalculations proceed in the same way as other Kohn-ShamPAW algorithms.

IV. RESULTS FOR DIAMOND AND LiF

In order to test the formalism, we have calculated the self-consistent electronic structure of diamond and LiF. The PAWbasis and projector functions were calculated using a modifiedversion of the ATOMPAW code29 using the parameters listedin Table II. The same parameters were used to construct theHartree-Fock, KLI, and LDA36 datasets. While the results arenot very sensitive to the details of the PAW parameters, pastexperience has shown that the choice of parameters given in

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XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

TABLE II. PAW parameters used in calculations: the pseudopotential radius ra

c and list of shell designations n1l1(rm1 )n2l2(rm2 ) . . .

of basis and projector functions used in the calculation and corre-sponding radii rmi

(in bohr) used to match the all-electron and pseudoradial wave functions. The symbol ε indicates the use of unboundbasis functions with energies (in Ry units) ε = (16.0, 10.0) and (2.0,2.0) for C and F, respectively. In each case, the local potential wasconstructed using the Troullier-Martins scheme62 for a continuumwave function of energy ε = 0 and l = 2.

Atom rac (bohr) {nili(rmi

)}Li 1.6 1s(1.4) 2s(1.6) 2p(1.6)C 1.3 2s(1.3) εs(1.3) 2p(1.3) εp(1.3)F 1.5 2s(1.5) εs(1.5) 2p(1.5) εp(1.5)

Table II and a wave function plane-wave expansion cutoff of|k + G|2 � 64 Ry are more than adequate to converge thecalculations to benchmark values. Details of the Hartree-Fockdataset construction follow Ref. 28 and extensions describedin Appendix B.

The solid calculations for the Hartree-Fock and KLI calcu-lations were performed using a modified version of the PWPAW

code.63 The LDA results were obtained using the ABINIT64

and QUANTUM-ESPRESSO65 codes as well. The Brillouin zonesampling was performed using a uniform 2 × 2 × 2 grid. Thissampling was found to be adequate for the LDA calculations;additional code development will be needed to substantiallyincrease the Brillouin zone sampling for the Hartree-Fock andKLI portions of the code. The binding energy curves shownin Figs. 7 and 8 were fit to the Murnaghan equation of state66

in order to extract the equilibrium lattice constant a and bulkmodulus B which are reported in Table III. There are manyresults of these quantities in the literature; a few of these arelisted for comparison in Table III, showing general consistencywith the present results.

To the best of our knowledge, our PAW-HF formulationis similar if not identical to that of Paier and co-workers4 asimplemented in the VASP code67 and to that of the GPAW code12

(apart from the replacement of the plane-wave expansionwith a real-space grid representation). In order to benefitfrom cancellation of systematic errors, we have been careful

3.4 3.5 3.6 3.7a (A

o)

0

0.1

0.2

0.3

0.4

0.5

E (

eV)

Diamond KLIHFLDA

FIG. 7. (Color online) Binding energy curve for C in the diamondstructure.

3.6 3.7 3.8 3.9 4 4.1 4.2 4.3a (A

o)

0

0.1

0.2

0.3

0.4

0.5

E (

eV)

LiFKLIHFLDA

FIG. 8. (Color online) Binding energy curve for LiF in the rocksalt structure.

to use atomic PAW datasets (Table II) constructed with thesame exchange-correlation or pure exchange formulation asused in the solid calculations. There is some discussion inthe literature68 that it is sometimes possible to use fixedatomic PAW datasets with the help of valence-core correctionsto perform calculations with different exchange-correlationformulations. It is our experience that the choice of ap-propriate basis function sets and augmentation radii suchas listed in Table II are rather insensitive to the exchange-correlation formulation. However, the basis and projectorfunctions themselves as well as the local potential V a

loc(r) aremore sensitive; these can easily be generated by using theATOMPAW code.29

V. SUMMARY AND CONCLUSIONS

Using Fock exchange as an example of an orbital-dependentfunctional, we have presented detailed equations neededto carry out self-consistent electronic structure calculationswithin the approximate optimized effective potential formal-ism developed by Krieger, Li, and Iafrate (KLI)30 using theprojector-augmented-wave method (PAW) of Blochl.38 Thisformalism together with the analogous development for theself-consistent Hartree-Fock formalism presented in earlierwork28 has been implemented and tested on the study ofthe ground-state properties of two well-known crystalline

TABLE III. Comparison of lattice parameters. Lattice constantsa are given in A and bulk moduli B are given in GPa.

Diamond LiF

a B a B

LDA (this work) 3.53 490 3.91 85LDA (literature) 3.54,a 3.55b 452b 3.92,a 3.96b 83b

HF (this work) 3.56 490 3.97 79HF (literature) 3.58b 480b 4.02,b 4.01c 76,b 79c

KLI (this work) 3.55 460 4.01 76Experiment 3.57d 443d 4.03d 67d

aReference 69.bReference 70.cReference 71.dReference 72.

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ADAPTATION OF THE PROJECTOR-AUGMENTED-WAVE . . . PHYSICAL REVIEW B 84, 155113 (2011)

materials—diamond and LiF. The fact that the present resultsare in general agreement with results reported in the literatureusing other computational methods provides some measureof validation of our formalism. It is interesting and notunexpected that the Hartree-Fock and KLI results for theequilibrium lattice constants are increased relative to the LDAvalues and are closer to the experimental values.

Both Hartree-Fock and KLI methods have been previouslyimplemented in all-electron and in norm-conserving pseudopotential codes by many authors, some of whom we listhere.59,60,73–78 However, the present work, as well as severalother recent studies56,79–82 show that there continue to benontrivial numerical challenges in the careful evaluation ofthe Fock exchange interaction. For the PAW method, theevaluation relies on the use of compensation pair charges[Eq. (55)] to ensure that all of the moments of chargeare correctly represented in the exchange interaction terms.This extension of ideas presented in the original PAWformalism38 was first introduced by Paier and co-workers.4

In principle, the correct treatment of the charge momentsin the PAW approach should have accuracy advantagesover norm-conserving pseudo potential methods,73,74,76,78,83,84

which explicitly control only the diagonal L = 0 pair charges.We also show that the treatment of core effects in the Fockexchange is best implemented in terms of frozen-core orbitals,which can be accomplished with high accuracy within the PAWmethod. Whether this analysis of the numerical advantages ofPAW over other pseudo-potential methods in the treatment ofthe Fock exchange interaction turns out to be numericallysignificant and/or computational efficient needs still to bequantified. Encouraging results have recently been reportedby the GPAW developers,12 who evaluate the Fock exchangeusing the PAW method on a real-space grid.

Having introduced the detailed equations needed both forthe Hartree-Fock and for the KLI implementations, we canbegin to compare the numerical and physical approximationsinvolved. While the complication of the equations presented inSec. III C does appear to be daunting, we have demonstratedit to be manageable. In fact, the added complication ofthe KLI approach results in the determination of the localexchange potential V KLI

x (r) for use in Kohn-Sham equations.While much of interest in the “exact exchange” comes fromimprovements in the band-gap energies,77,85 the few resultspresented in Sec. IV indicate that there are nontrivial effectson the structural properties as well. We expect to continue tocompare the methods in future work.

There are several ways in which the present work canbe extended. The equations here have assumed that the corestates are localized within the augmentation spheres of eachatom. This condition can be relaxed by introducing corepseudo wave functions ψa

c (r) as discussed in Ref. 28 andintroducing corresponding modifications to the PAW-KLIequations. Another extension involves improving the KLIapproximation by calculating the optimized effective potentialwithin the subspace of occupied orbitals [vocc(r)] as describedby Bulat and Levy.34 Operationally, this involves includingsome additional nondiagonal matrix elements of the localexchange potential and the exchange integral function inEq. (10). In addition, we have discussed the possibilityof extending the PAW methodogy to treat the full OEP

equations and have argued (see also Appendix A) that thiswould require a significant increase in the number of PAWone-center basis functions. A similar experience was recentlyreported by Betzinger and co-workers79 in the context ofan accurate implementation of local exact exchange poten-tials with the full-potential linearized augmented-plane-wavemethod.

The scope of the present work goes beyond the accuratetreatment of the Fock exchange interaction. By extension, thesame techniques can be used to treat other orbital-dependentexchange-correlation functionals as they are being developed.

ACKNOWLEDGMENTS

This work was supported by NSF Grant Nos. DMR-0427055 and DMR-0705239. Computations were performedon the Wake Forest University DEAC cluster, a centrallymanaged resource with support provided in part by theUniversity. We would like to thank Alan Wright and NormandModine of Sandia National Laboratory for their helpful adviceand many discussions. We would also like to thank AkbarSalam, William C. Kerr, V. Paul Pauca, and Freddie R. SalsburyJr. for serving on the Ph.D. thesis committee.

APPENDIX A: AN EXAMPLE OF ORBITALSHIFT FUNCTIONS

Figure 9 shows an an example of the shell contributionsto the combined shift function [Eq. (9)] for atomic C inthe spherically averaged 1s22s22p2 configuration. While eachcurve represents a complicated function, the sum of the curvesis zero at all radii, as required at convergence by Eq. (9).

Figure 10 shows the shape of the individual orbital shiftfunctions gp(r) for atomic C which were determined fromEq. (5). While their magnitudes are small, their shapes arevery complicated. From the forms of these functions it isevident that their accurate representation as a sum of atomicbasis functions would require the use of a large numberof basis functions. In particular, the PAW transformationEq. (18) would require that, in the augmentation region,both the all-electron valence orbital function ψv(r) and thecorresponding all-electron valence orbital shift function gv(r)

0 1 2 3 4 5 6 7r (bohr)

-0.01

-0.005

0

0.005

0.01

Npψ

p(r)g

p(r) 1s 2s

2p

C (1s22s

22p

2)

FIG. 9. (Color online) Shell contributions to the combined shiftfunction for the C atom.

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XIAO XU AND N. A. W. HOLZWARTH PHYSICAL REVIEW B 84, 155113 (2011)

0 1 2 3 4 5 6 7r (bohr)

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

g p(r) 1s

2s

2p

C (1s22s

22p

2)

FIG. 10. (Color online) Orbital shift functions gp(r) for the C atom.

be well represented by a sum of all-electron basis functions{φi(r)} for r � ra

c :

ψv(r) ≈∑

i

φi(r)⟨P a

i

∣∣�v

⟩and gv(r) ≈

∑i

φi(r)⟨P a

i

∣∣Gv

⟩.

(A1)

While the expansion of ψv(r) is well controlled, the expansionof gv(r) is computationally much more demanding. Withoutknowing the values of the expansion coefficients 〈P a

i |Gv〉,the shape of the curves in Fig. 10 suggest that accuraterepresentation of gv(r) requires many more terms than doesaccurate representation of ψv(r). Our numerical tests39 areconsistent with this analysis. This would make a full PAW-OEP treatment much more computationally demanding thanPAW-KLI.

APPENDIX B: SOME DETAILS OF HARTREE-FOCKFORMALISM

In principle, the Hartree-Fock equations are not definedfor unoccupied states. However, in order to generate a morecomplete basis set for the PAW formalism and to generateV a

loc(r), it is convenient to use continuum states. For thispurpose, we simplified a more rigorous treatment86 and defined

a continuum state at a given energy εp > 0 to satisfy theintegral-differential equation [similar to Eq. (17) of Ref. 28]

[K + VN (r) + VH (r) − εp]ψp(r) + Xp(r) = 0. (B1)

Here Xp(r) is defined by Eq. (7) with the sum over q includingonly occupied states, and where no explicit orthonormalizationconstraints are imposed. In our previous work we noted thatresults can be sensitive not only to the basis functions butalso to the magnitude of the localized pseudo potential V a

loc(r).By using Eq. (B1) and a method similar to that describedby Al-Saidi and co-workers83 to generate a norm-conservingscreened pseudo potential V aPS(r), we can then use a similarunscreening process described by Eq. (47) to determine asuitable V a

loc(r) for each atom.In order to simplify the formulation of the Hartree-Fock

PAW equations, we relaxed the orthogonality constraintsof the valence wave functions with respect to the coreorbitals. We also assumed the core states to be confinedwithin the augmentation region using the expressions given inSecs. III A 1, III A 2, and III B.

Matrix elements of the Hartree-Fock equations in theBloch basis are diagonal in wavevector k and Hermitianwith respect to band indices. In order to use the samediagonalization procedures that are used for the Kohn-Shamformulations, it is convenient to regroup terms in the evaluationof matrix elements of the exchange interaction defined inSec. III B:⟨

�n′′k∣∣XPAW

nk

⟩ = ⟨�nk

∣∣XPAWn′′k

⟩∗≡ X 1 k

n′′n +∑ail

⟨�n′′k

∣∣P ai

⟩X 2

il

⟨P a

l

∣∣�nk⟩. (B2)

Here

X 1 kn′′n ≡ −2πe2

V∑n′k′

fn′k′ρ

∗n′′k,n′k′(G)ρnk,n′k′(G)

|k − k′ + G|2 (B3)

and

X 2il ≡ −1

2

∑jk

WakjVa

ij ;kl − δli ll δmimlCa

il . (B4)

The current version of the code was written with theseequations. Further analysis will be needed to improve theefficiency of the calculations.

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