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PHYSICAL REVIEW B 89, 155112 (2014) Angular momentum dependent orbital-free density functional theory: Formulation and implementation Youqi Ke, Florian Libisch, Junchao Xia, and Emily A. Carter, * Department of Mechanical and Aerospace Engineering, Program in Applied and Computational Mathematics, and Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey 08544-5263, USA (Received 3 August 2013; revised manuscript received 30 December 2013; published 10 April 2014) Orbital-free density functional theory (OFDFT) directly solves for the ground-state electron density. It scales linearly with respect to system size, providing a promising tool for large-scale material simulations. Removal of the orbitals requires use of approximate noninteracting kinetic energy density functionals. If replacing ionic cores with pseudopotentials, removal of the orbitals also requires these pseudopotentials to be local. These are two severe challenges to the capabilities of conventional OFDFT. While main group elements are often well described within conventional OFDFT, transition metals remain intractable due to their localized d electrons. To advance the accuracy and general applicability of OFDFT, we have recently reported a general angular momentum dependent formulation as a next-generation OFDFT. In this formalism, we incorporate the angular momenta of electrons by devising a hybrid scheme based on a muffin tin geometry: inside spheres centered at the ionic cores, the electron density is expanded in a set of atom-centered basis functions combined with an onsite density matrix. The explicit treatment of the angular momenta of electrons provides an important basis for accurately describing the important ionic core region, which is not possible in conventional OFDFT. In addition to the conventional OFDFT total energy functional, we introduce a nonlocal energy term containing a set of angular momentum dependent energies to correct the errors due to the approximate kinetic energy density functional and local pseudopotentials. Our approach greatly increases the accuracy of OFDFT while largely preserving its numerical simplicity. Here, we provide details of the theoretical formulation and practical implementation, including the hybrid scheme, the derivation of the nonlocal energy term, the choice of basis functions, the direct minimization of the total energy, the procedure to determine the angular momentum dependent energies, the force formula with Pulay correction, and the solution to emerging numerical instability. To test the angular momentum dependent OFDFT formalism and its numerical implementations, we calculate a diverse set of properties of the transition metal Ti and compare with different levels of DFT approximation. The results suggest that angular momentum dependent OFDFT ultimately will extend the reliable reach of OFDFT to the rest of the periodic table. DOI: 10.1103/PhysRevB.89.155112 PACS number(s): 71.15.Mb, 71.20.Be I. INTRODUCTION Based on the firm theoretical footing of the Hohenberg- Kohn theorems [1], density functional theory (DFT) has gained vast popularity as an extremely powerful tool for the first-principles simulation of electronic and structural properties of materials. The great success of DFT is attributed to the Kohn-Sham (KS) decomposition in which the intractable many-electron problem is reduced to a calculable problem of noninteracting electrons moving in an effective potential [2]. Thus far, theorists have developed two ways to implement DFT: KS orbital-based DFT [2] and orbital-free (OF) DFT [3], distinguished by the different treatments of the noninteracting kinetic energy (KE). KSDFT has been established as the workhorse for first-principles simulations due to its good compromise between accuracy and computational efficiency (as compared to more expensive correlated wave function methods) [4]. However, the required orthogonalization of the KS orbitals in standard KSDFT implementations makes the computational cost scale cubically with respect to system size, making a sample with more than a thousand atoms prohibitively costly to simulate. Several different linear scaling KSDFT algorithms have been devised (for reviews, see Refs. [5,6]), albeit with large prefactors. Notably, linear scaling * [email protected] KSDFT methods depend on approximations requiring highly localized orbitals, and their applicability is therefore limited to systems with band gaps, which excludes metallic systems. Consequently, a wide range of important applications, such as first-principles simulations of large-scale nanoelectronics [7], nanomechanics [8], or amorphous materials [9], requiring the ability to explicitly model systems with thousands of atoms (or even more), are challenging or not feasible using modern KSDFT methods. In contrast to KSDFT, OFDFT [3] features great advantages in its numerical simplicity and quasilinear scaling with system size for all types of materials. There are numerous examples of large-scale OFDFT-based simulations that highlight its numerical advantages for cases in which adequate approximate functionals are available [1019]. OFDFT employs approximate noninteracting KE density functionals (KEDFs), making OFDFT less accurate than KSDFT for almost all materials. Except for limiting cases, such as the local Thomas-Fermi (TF) KEDF [20] for the uniform electron gas and the semilocal von Weizs¨ acker (vW) KEDF [21] for single orbital systems, the exact form of the KEDF remains unknown. Recently, theorists have made considerable efforts to advance KEDFs, including two-point KEDFs based on linear response theory [2226], three- point KEDFs involving higher-order response [23,27,28], and single-point KEDFs in different forms of generalized gradient approximations [2932]. However, we are still far from a generally applicable KEDF. 1098-0121/2014/89(15)/155112(16) 155112-1 ©2014 American Physical Society
Transcript
Page 1: Angular momentum dependent orbital-free density functional ...dollywood.itp.tuwien.ac.at/~florian/PhysRevB.89.155112.pdf · Angular momentum dependent orbital-free density functional

PHYSICAL REVIEW B 89, 155112 (2014)

Angular momentum dependent orbital-free density functional theory:Formulation and implementation

Youqi Ke, Florian Libisch, Junchao Xia, and Emily A. Carter,*

Department of Mechanical and Aerospace Engineering, Program in Applied and Computational Mathematics,and Andlinger Center for Energy and the Environment, Princeton University, Princeton, New Jersey 08544-5263, USA

(Received 3 August 2013; revised manuscript received 30 December 2013; published 10 April 2014)

Orbital-free density functional theory (OFDFT) directly solves for the ground-state electron density. It scaleslinearly with respect to system size, providing a promising tool for large-scale material simulations. Removalof the orbitals requires use of approximate noninteracting kinetic energy density functionals. If replacing ioniccores with pseudopotentials, removal of the orbitals also requires these pseudopotentials to be local. These aretwo severe challenges to the capabilities of conventional OFDFT. While main group elements are often welldescribed within conventional OFDFT, transition metals remain intractable due to their localized d electrons. Toadvance the accuracy and general applicability of OFDFT, we have recently reported a general angular momentumdependent formulation as a next-generation OFDFT. In this formalism, we incorporate the angular momenta ofelectrons by devising a hybrid scheme based on a muffin tin geometry: inside spheres centered at the ionic cores,the electron density is expanded in a set of atom-centered basis functions combined with an onsite density matrix.The explicit treatment of the angular momenta of electrons provides an important basis for accurately describingthe important ionic core region, which is not possible in conventional OFDFT. In addition to the conventionalOFDFT total energy functional, we introduce a nonlocal energy term containing a set of angular momentumdependent energies to correct the errors due to the approximate kinetic energy density functional and localpseudopotentials. Our approach greatly increases the accuracy of OFDFT while largely preserving its numericalsimplicity. Here, we provide details of the theoretical formulation and practical implementation, including thehybrid scheme, the derivation of the nonlocal energy term, the choice of basis functions, the direct minimizationof the total energy, the procedure to determine the angular momentum dependent energies, the force formula withPulay correction, and the solution to emerging numerical instability. To test the angular momentum dependentOFDFT formalism and its numerical implementations, we calculate a diverse set of properties of the transitionmetal Ti and compare with different levels of DFT approximation. The results suggest that angular momentumdependent OFDFT ultimately will extend the reliable reach of OFDFT to the rest of the periodic table.

DOI: 10.1103/PhysRevB.89.155112 PACS number(s): 71.15.Mb, 71.20.Be

I. INTRODUCTION

Based on the firm theoretical footing of the Hohenberg-Kohn theorems [1], density functional theory (DFT) hasgained vast popularity as an extremely powerful tool forthe first-principles simulation of electronic and structuralproperties of materials. The great success of DFT is attributedto the Kohn-Sham (KS) decomposition in which the intractablemany-electron problem is reduced to a calculable problem ofnoninteracting electrons moving in an effective potential [2].Thus far, theorists have developed two ways to implementDFT: KS orbital-based DFT [2] and orbital-free (OF) DFT [3],distinguished by the different treatments of the noninteractingkinetic energy (KE). KSDFT has been established as theworkhorse for first-principles simulations due to its goodcompromise between accuracy and computational efficiency(as compared to more expensive correlated wave functionmethods) [4]. However, the required orthogonalization of theKS orbitals in standard KSDFT implementations makes thecomputational cost scale cubically with respect to systemsize, making a sample with more than a thousand atomsprohibitively costly to simulate. Several different linear scalingKSDFT algorithms have been devised (for reviews, seeRefs. [5,6]), albeit with large prefactors. Notably, linear scaling

*[email protected]

KSDFT methods depend on approximations requiring highlylocalized orbitals, and their applicability is therefore limitedto systems with band gaps, which excludes metallic systems.Consequently, a wide range of important applications, such asfirst-principles simulations of large-scale nanoelectronics [7],nanomechanics [8], or amorphous materials [9], requiring theability to explicitly model systems with thousands of atoms(or even more), are challenging or not feasible using modernKSDFT methods. In contrast to KSDFT, OFDFT [3] featuresgreat advantages in its numerical simplicity and quasilinearscaling with system size for all types of materials. There arenumerous examples of large-scale OFDFT-based simulationsthat highlight its numerical advantages for cases in whichadequate approximate functionals are available [10–19].

OFDFT employs approximate noninteracting KE densityfunctionals (KEDFs), making OFDFT less accurate thanKSDFT for almost all materials. Except for limiting cases,such as the local Thomas-Fermi (TF) KEDF [20] for theuniform electron gas and the semilocal von Weizsacker (vW)KEDF [21] for single orbital systems, the exact form ofthe KEDF remains unknown. Recently, theorists have madeconsiderable efforts to advance KEDFs, including two-pointKEDFs based on linear response theory [22–26], three-point KEDFs involving higher-order response [23,27,28], andsingle-point KEDFs in different forms of generalized gradientapproximations [29–32]. However, we are still far from agenerally applicable KEDF.

1098-0121/2014/89(15)/155112(16) 155112-1 ©2014 American Physical Society

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KE, LIBISCH, XIA, AND CARTER PHYSICAL REVIEW B 89, 155112 (2014)

Because conventional OFDFT typically is implementedin the context of plane-wave techniques (such as the fastFourier transform) and available KEDFs are only suitable forsystems featuring comparatively smooth electron densities,pseudopotentials are required to represent the interactionbetween the valence electrons and the ionic cores. How-ever, conventional OFDFT utilizes local pseudopotentials(LPSs). To obtain optimal LPSs, several techniques havebeen proposed [14,33–38]. Unfortunately, even elaborate LPSprocedures do not produce the high accuracy of nonlocalpseudopotentials (NLPSs) that are widely used in KSDFT.

The applicability of conventional OFDFT is thus largelyconstrained by the limited accuracy of available KEDFs andLPSs. In particular, one of the most challenging issues forOFDFT is the description of transition metals, which arecharacterized by highly localized d electrons, for which noKEDF or LPS exists. Exceptions are the coinage metals, suchas Ag, which is a special case because of their full d shells [39].

These difficulties occur because conventional OFDFTmakes exclusive use of the total electron density as the soleworking variable: electrons with different angular momentacannot be distinguished. It is therefore impossible to includethe critically important nonlocal physics in the exact KE andthe ion-electron interaction, especially in the core region.In particular, the KE potential and the pseudopotentials inOFDFT must be local quantities. By contrast, angular mo-mentum dependent (AMD) NLPSs and the exact KE operatorcontaining an AMD centrifugal potential are responsiblefor the high accuracy and excellent transferability of theKohn-Sham ansatz. Furthermore, because of the nonlineardependence of T KEDF

s on ρ, using the total electron densityin conventional OFDFT can induce unphysical interactionsbetween electrons of different angular momenta. In contrast,the exact KE depends linearly on the occupation of eachangular momentum channel. These unphysical interactions inthe KEDF can result in errors in the KE potential which, inturn, influence the electron density distribution. In addition,the modern two-point KEDFs [22–24], which are the mostaccurate KEDFs to date, reproduce the Lindhard linearresponse function in the limit of a uniform electron gas subjectto a small perturbation [22–24]. They are thus only accuratefor nearly free-electron-like systems, i.e., main group metalsand their alloys. The application of conventional OFDFT tolocalized electrons, e.g., in transition metals, can result in largeerrors in the electron density, its response to external fields,and ultimately in system properties [40].

To correct these deficiencies, we have recently developedangular momentum dependent OFDFT (AMD-OFDFT) [41],a new generation of OFDFT to advance its accuracy andgeneral applicability. The AMD-OFDFT formalism uses ageneral hybrid scheme based on a muffin tin (MT) geometry:the electron density inside MT spheres is expressed by a setof KSDFT-derived, atom-centered basis functions combinedwith an onsite density matrix, while conventional OFDFTdescribes the interstitial region where the electron density issmoother and hence more amenable to accurate descriptionby existing KEDFs. The explicit treatment of electron angularmomenta within the atom-centered spheres provides the basisfor accurately describing the important ionic core region.In addition to the total energy functionals of conventional

OFDFT, we introduce a crucial nonlocal energy term thatincludes a set of AMD energies to effectively correct errorsdue to the approximate KEDFs and LPSs in the importantcore region, resulting in improved electronic structure andsystem properties. As we have shown [41], AMD-OFDFTsubstantially improves various properties of the transitionmetal titanium over conventional OFDFT, and features goodtransferability of the AMD energies.

In this paper, we present the entire theoretical formulationand all practical implementation details of the general AMD-OFDFT. In Sec. II, we introduce a hybrid scheme based on aMT geometry, a general OFDFT total energy functional withangular momentum dependence, and a nonlocal energy termand its associated AMD energy parameters which correctsthe errors due to the use of approximate KEDFs and LPSs.Section III presents the derivation of the atom-centeredbasis functions in the MT spheres based on KSDFT-NLPScalculations of target systems. We then discuss the directminimization of AMD-OFDFT total energy functional with thenecessary constraints applied in Sec. IV. Section V describesthe modified KEDF model with a weighting function toreduce nonlinear errors of the KEDF within the MT geometry.Section VI details how to determine the AMD onsite energiesfor the MT spheres. We derive a force formula with aPulay correction (arising from the atom-centered functions)in Sec. VII. In Sec. VIII, we introduce a double-spheretechnique and a down-sampling approach for solving thenumerical instability induced by representing the MT sphereson a nonconforming three-dimensional (3D) uniform grid.We present computational details and discuss our results inSec. IX. Finally, we provide a summary in Sec. X, andadditional technical details in Appendices A, B, and C.

II. AMD-OFDFT TOTAL ENERGY

Several all-electron KSDFT methods, such as the linearizedaugmented plane-wave (LAPW) [42] and the linearized muffintin orbital (LMTO) [43] methods, are based on a MT geometry.In such a geometry, the system is partitioned into atom-centered MT spheres and an interstitial region (see Fig. 1).Here, we use a MT geometry to define a general OFDFT

FIG. 1. The muffin tin geometry partitions space into spherescentered on nuclei and an interstitial region.

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ANGULAR MOMENTUM DEPENDENT ORBITAL-FREE . . . PHYSICAL REVIEW B 89, 155112 (2014)

total energy functional with explicit angular momentumdependence.

Within the MT geometry, we rewrite the total electrondensity as [41]

ρ(�r) =∑R

ρR(�rR) + ρI (�r), (1)

where ρR(�rR) is the electron density inside the MT spherewith a radius W1R centered on site �R, ρI (�r) is the interstitialelectron density, and �rR = �r − �R. To explicitly include angularmomentum dependence, we introduce a set of fixed atom-centered basis functions ψR,lm = φR,l(rR)Ylm(�), where Ylm

are the spherical harmonics. We can now express ρR as

ρR(�rR) =∑

lm,l′m′NR,lm,l′m′ψ∗

R,lm(�rR)ψR,l′m′ (�rR). (2)

We omit the spin index throughout this paper for simplic-ity. NR denotes the onsite density matrix which containsinformation on the angular momenta of electrons. For atomswith significant hybridization between different channels, theoff-diagonal elements of NR,lm,lm are important for improvingnonsphericity of electron density inside the MT sphere. UsingEqs. (1) and (2), the OFDFT total energy functional can berewritten as

EOF[ρ] = EOF[{NR},ρI ], (3)

where the onsite density matrix {NR} and the interstitialelectron density ρI become the basic independent variables.Up to now, no physical approximation has been made to the OFtotal energy functional. Errors in the electron density inside thespheres due to use of a finite basis set can be minimized by anappropriate choice of the ψR,lm; our strategy for deriving basisfunctions from NLPS-based KSDFT calculations is presentedin Sec. III. We typically use a comparatively small numberof basis functions. Compared to conventional OFDFT, usingsuch a finite number of basis functions strongly restricts theshape of the electron density inside the MT spheres. However,this restriction results in a more accurate electron density thanin conventional OFDFT: the nonlocal effect of the KE operatorand the NLPS on the shape of the electron density are includedby using basis sets derived from KSDFT. Additionally, the totalenergy functional in Eq. (3) incorporates angular momentumdependence by using NR for the MT sphere region where theAMD physics is most important. Consequently, our hybridscheme provides the physical flexibility to include AMDcontributions not feasible in conventional implementations ofOFDFT.

The explicit form of EOF[{NR},ρI ] can be written as [41]

EOF[{NR},ρI ] = T KEDFs [ρ] + EXC[ρ] + EH[ρ]

+ELPSi-e [ρ] + ENL[{NR},ρI ], (4)

where the energies T KEDFs , EH, EXC, and ELPS

i-e are thenoninteracting KE, Hartree energy, exchange-correlation en-ergy, and LPS energy, respectively. The first four termscomprise the conventional OFDFT total energy functional.ENL describes nonlocal contributions beyond conventionalOFDFT. It corrects the errors due to the approximate T KEDF

s

and LPSs. Determining a physically sensible ENL is critical for

correct electronic structure and material properties, formingthe crux of our AMD-OFDFT formalism [41].

Ideally, ENL should contain all the differences between theKSDFT and conventional OFDFT total energy functionals,namely,

ENL = ENLPSi-e + Ts − T KEDF

s , (5)

where ENLPSi-e is only the nonlocal part of the pseudopotential

energy and Ts − T KEDFs is the KE error. Note that the nonlocal

part of the pseudopotential is not uniquely defined since itdepends on the choice of the local pseudopotential. Conse-quently, ENLPS

i-e depends on ELPSi-e ; together they define a unique

total nonlocal pseudopotential energy. The exact computationof ENL requires the exact KEDF which is unknown. We followhere our derivation for ENL in Ref. [41], which we brieflyrepeat for completeness. We choose the MT sphere radius foreach element large enough so that the nonlocal part of thepseudopotential becomes zero in the interstitial region. Theminimal MT sphere radius is thus the cutoff radius rcutoff ofthe NLPS used in solving for the basis functions (see Sec. III).

As amply demonstrated in many applications, modernKEDFs [22–24] based on the Lindhard response functionexhibit accuracy comparable to KSDFT for main group metals.Therefore, these KEDFs should be accurate enough to describethe slowly varying electron density found in the interstitialregion. We thus neglect Ts − T KEDF

s there, using conventionalOFDFT to treat the interstitial region. Unfortunately, twodifferent representations of the KE result in a discontinuousKE potential at the sphere boundary, causing unphysicalelectron occupations. We thus enforce continuity by formallyintroducing a smooth scaling function sR(r) [sR(r) = 0 forr � W1R and 0 � sR(r) � 1 for r < W1R] in the evaluationof the Ts − T KEDF

s inside the spheres, providing a rigoroushybrid KEDF model (see details in the Supplemental Materialof Ref. [41]). Then ENL becomes

ENL = ENLPSi-e +

∑R

∫MT

sR(rR)[τs(�rR) − τKEDF

s (�rR)]d�rR

= ENLPSi-e + [Ts]MT − [

T KEDFs

]MT, (6)

where [Ts]MT and [T KEDFs ]MT are the exact noninteracting KE

and the KEDF scaled by sR inside the MT spheres. Equation (6)presents a physically reasonable expression for ENL, providingan important basis for further development of AMD-OFDFT.However, determining the optimal function values of sR(rR) ona large number of radial grid points is too costly for practicaluse. Instead, we investigate the functional dependence of eachterm in Eq. (6) on the occupations NR to determine a morepractical expression containing the same physics.

We first consider ENLPSi-e and [Ts]MT in Eq. (6). They depend

linearly on the total occupation number of each l channel. Inparticular,

ENLPSi-e [NR] =

∑R,lm

NlmR 〈ψR,lm|δV l

i-e|ψR,lm〉MT

=∑R,l

N totalR,l E

l,NLPSR (7)

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KE, LIBISCH, XIA, AND CARTER PHYSICAL REVIEW B 89, 155112 (2014)

and

[Ts]MT = −1

2

∑R,lm

NlmR 〈ψR,lm|sR∇2|ψR,lm〉MT

=∑R,l

N totalR,l E

l,Ts

R , (8)

where δV li-e is the nonlocal part of the pseudopotential,

N totalR,l = ∑l

m=−l NlmR , and we have introduced the shorthand

notation NlmR = NR,lm,lm. E

l,NLPSR and E

l,Ts

R are constants thatdepend on the shape of the basis functions of each l channel.Consequently, a linear term in ENL can easily accommodatethe AMD effects of ENLPS

i-e and [Ts]MT.Next, we derive the linear and dominant nonlinear terms

in the employed KEDF to allow for their correction by ENL.We Taylor expand a general KEDF [T KEDF

s ]MT[NR] up to thirdorder around the average occupation number of each l channelN0

R,l = N totalR,l /(2l + 1) to obtain

[T KEDF

s

]MT[N ] = [

T KEDFs

]MT

[{N0

R,l

}] +∑R,lm

∂[T KEDF

s

]MT

∂NlmR

∣∣∣∣∣{N0

R,l}Nlm

R + 1

2

∑R,lm,l′m′

∂2[T KEDF

s

]MT

∂NlmR ∂Nl′m′

R

∣∣∣∣∣{N0

R,l}Nlm

R Nl′m′R

+ 1

6

∑R,lm,l′m′,l′′m′′

∂3[T KEDF

s

]MT

∂NlmR ∂Nl′m′

R ∂Nl′′m′′R

∣∣∣∣∣{N0

R,l}Nlm

R Nl′m′R Nl′′m′′

R + O(N4

R

), (9)

where NlmR = Nlm

R − N0R,l . We neglect the contribution of

off-diagonal elements of the density matrix because differentl channels hybridize minimally in the core region. Each termin the above Taylor expansion plays a different role. We thusconsider individually each term of Eq. (9) in the following.

The zeroth-order term [T KEDFs ]MT[{N0

R,l}] determines theabsolute magnitude of the KE and the total occupation of eachl channel inside the spheres. We can expand this term furtheraccording to different physical situations. In transition metals,the electron density in the core region is dominated by the totald-channel contribution, i.e., N total

R,d � {N totalR,s ,N total

R,p }. There-fore, we make another Taylor expansion of [T KEDF

s ]MT[{N0R,l}]

at N0R,s = N0

R,p = 0 because of the small contribution of s/pchannels:[T KEDF

s

]MT

[N0

R,l

] = [T KEDF

s

]MT

[N0

R,l

]∣∣N0

R,s/p=0

+∑

l′=s,p

∂[T KEDF

s

]MT

[N0

R,l

]∂N0

R,l

∣∣∣∣∣N0

R,s/p=0

N0R,l′

+ · · · . (10)

We only consider terms up to first order in this paper. It is clear

that [T KEDFs ]MT[{N0

R,l}]|N0R,s/p=0 and

∂[T KEDFs ]MT[N0

R,l ]

∂N0R,l

|N0R,s/p=0

only depend on N0R,d of the localized electrons. As shown in

the Supplemental Material of Ref. [41], for modern two-pointKEDFs [22–24], we can approximate the zeroth-order term ofEq. (10) as[T KEDF

s

]MT

[N0

R,l

] ≈ Vd,KEDFR

(N total

R,d

) 53 +

∑R,l

N totalR,l E

l,KEDFR ,

(11)

where El,KEDFR and V

d,KEDFR are constants.

For the first-order term in Eq. (9), the contributions of localand semilocal KEDFs, such as the TF and vW terms in thetwo-point KEDFs, are zero (see the Supplemental Material ofRef. [41]) because of the spherical electron density distributionwhen Nlm

R = N0R,l inside the MT sphere. However, for the two-

point KEDF, the first derivative of its nonlocal part T NLs , i.e.,

∂T NLs

∂NlmR

|{N0R,l}, depends on the quantum number m, contributing a

first-order error in Eq. (9). Since contributions of the nonlocalterm T NL

s are small in comparison to the TF and vW termsin the two-point KEDFs, we neglect the first-order term inEq. (9).

We now consider the second- and third-order terms fordifferent KEDFs in Eq. (9). Since the s channel has onlya single m value, NR,s − N0

R,s = 0, it does not contribute.We neglect contributions from the p channel since they aresmall for transition metals. For the remaining d channel, thesecond- and third-order coefficients can be written as (see theSupplemental Material of Ref. [41])

1

2

∂2[T KEDF

s

]MT

∂NdmR ∂Ndm′

R

∣∣∣∣∣{N0

R,l}= U

KEDF,dR Adm,dm′ (12)

and

1

6

∂3[T KEDF

s

]MT

∂NlmR ∂Nlm′

R ∂Nlm′′R

∣∣∣∣∣{N0

R,l}= K

KEDF,dR d,m,m′,m′′ , (13)

where Adm,dm′ = 4π∫ |YdmYdm′ |2d� and d,m,m′,m′′ =

(4π )2∫ |YdmYdm′Ydm′′ |2d� are constants independent of the

atomic species, and are evaluated numerically using a Lebedevquadrature grid [44]. The prefactors U

KEDF,dR and K

KEDF,dR ,

treated as constants in the present method, are determined bya fitting procedure (see Sec. VI). The dependence of U

KEDF,dR

and KKEDF,dR on the occupations NR could also be considered

(see Appendix A).In summary, by combining Eqs. (6)–(9) and (11)–(13), we

obtain a general form for ENL of

ENL[{NR}] =∑R,l

ElRN total

R,l −∑R,l

V lR

(N total

R,l

)5/3

−∑

R,l,mm′Ul

RAl,mm′NlmR Nlm′

R

−∑

R,l,mm′m′′Kl

Rl,mm′m′′NlmR Nlm′

R Nlm′′R .

(14)

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ANGULAR MOMENTUM DEPENDENT ORBITAL-FREE . . . PHYSICAL REVIEW B 89, 155112 (2014)

The above derivation is based on transition metals where the d

channel dominates the core region. However, our formulationcan be easily generalized to systems with s or p channeldominating the ionic core region, such as in first row elements.We have introduced the onsite AMD energies El

R , V lR , Ul

R , andKl

R . We chose V lR , Ul

R , and KlR nonzero only for l channels

featuring localized electrons, such as the d channel in transitionmetals or the p channel in main group nonmetals. At present,we treat El

R , V lR , Ul

R , and KlR as constants, neglecting their

dependence on the occupation numbers. As we show in Sec. IX(see also Ref. [41]), Eq. (14) yields a significant improvementin accuracy for an OFDFT description of transition metals.The AMD energies El

R and V lR are introduced to correct the

relative energies of different l-channel electrons inside theMT spheres, as well as of the electrons in the interstitialregion. They are thus crucial for obtaining correct occupationsinside the MT spheres. The first (linear) term of Eq. (14)includes the NLPS energy and the correction for the linearerrors in the KE inside the MT spheres. The form of N5/3

for the V lR term corrects the leading nonlinear error in the

KEDF, arising from the highly nonlinear TF KEDF term(see Appendix A). The first and second terms in Eq. (14)are important for the absolute magnitude of the total energyof the system. The last two terms in Eq. (14), especiallythe third term containing Ul

R , are introduced to correct thedelocalization error in KEDFs, which arises from unphysicalinteractions between the different m states of a given l channel.The higher-order terms proportional to Ul

R and KlR in Eq. (14)

are important for correctly distributing electrons among the2l + 1 subchannels within one l channel. The third term inour model is mathematically identical to the KSDFT + U

formalism, which corrects the self-interaction error in theXC functional due to approximate exchange energy [45]. Webenchmark our OFDFT calculations against KS calculationsusing the same XC functional without a Hubbard U . Theimprovements we observe are thus related to the correction forthe delocalization error in the kinetic energy, not the exchangeenergy. Our formalism could be trivially extended to include aKSDFT + U like contribution.

Despite the obvious importance of ENL in Eq. (14), directlyevaluating the energies El

R , V lR , Ul

R , and KlR is challenging.

Instead, we find optimal values for these AMD energies bycomparing with a small set of benchmark KSDFT properties,and then test transferability of these parameters against a largenumber of additional properties.

We have introduced the formulation of AMD-OFDFTby applying a MT-geometry-based hybrid scheme. In thepresent method, overlapping spheres are not allowed sincewe do not yet have an effective way to remove errors dueto double counting in the overlapping region. As statedabove, the lower bound for the MT sphere radius is rcutoff

of the NLPS. Since the rcutoff can be rather small and ENL

also corrects the KEDF errors inside the spheres, the actualradii of the MT spheres are essentially determined by themagnitude of acceptable errors for a given simulation andas well as the quality of the KEDF employed. If the KEDFis accurate enough to describe a larger interstitial region,the MT sphere radius can be smaller, perhaps as small asthe rcutoff .

III. DERIVING BASIS FUNCTIONS FROM BULKOR MOLECULAR KSDFT CALCULATIONS

An accurate description of the important MT sphere regioncritically depends on the quality of the basis functions, whichare fixed throughout the AMD-OFDFT calculations. In thissection, we propose a method to derive the atom-centered basisfunctions ψR,lm from a NLPS-based KSDFT calculation of atarget system. For each element, basis functions are determinedby numerical integration of the Schrodinger equation[

−∇2

2+ V (rR) + δV l

i-e,R(rR) − εlR

]ψR,lm = 0, (15)

where V (rR) is the spherical effective potential composedof the spherical part of the Hartree potential, the localpseudopotential, and the XC potential, δV l

i-e,R(rR) is thenonlocal part of the ion-electron pseudopotential, and εl

R is theenergy value at which we solve the differential equation forψR,lm. To solve Eq. (15), we have to know two basic quantities:the spherical potential V (rR) and the energy εl

R . The choice ofthese quantities is critical for the accuracy of our calculationand the transferability of the basis functions. One may obtainapproximations for both of these quantities from a single-atomKSDFT calculation. However, this solution does not includethe correct chemical environment of the solid and thus cannot yield satisfactory accuracy and transferability. Instead, weobtain V (rR) and εl

R from KSDFT calculations in a chemicalenvironment similar to the one we are interested in simulatingwith OFDFT, to minimize errors due to the use of a finite setof ψR,lm. In this way, we properly account for the effect ofthe chemical environment in solids or molecules in the basisfunctions.

We expand the bulk (or molecular) V (�rR) in terms ofspherical harmonics as

V (�rR) =∑lm

Vlm(rR)Ylm(�rR). (16)

We are only interested in the spherical component of V (�rR) foruse in Eq. (15). To obtain this component, we first calculateV (�rR) including the Hartree, local pseudopotential, and XCpotentials on a 3D uniform grid by a self-consistent KSDFTcalculation with the same NLPS as Eq. (15). Then we make afast Fourier transform (FFT) to obtain

V ( �G) = FFT[V (�rR)]. (17)

After calculating the potential in reciprocal space (i.e., on the�G grid), the spherical V (rR) on a radial grid is obtained by theintegration

V (rR) = 1

∫drR

∫V ( �G)ei�rR · �Gd �G, (18)

in which we first do an inverse Fourier transform followed by aspherical average to obtain the spherical potential. The abovedouble integration can be reduced to a Bessel transform

V (rR) =∫

V ( �G)eirR ·G sin(GrR)

GrR

d �G. (19)

To solve Eq. (15), we still need to determine the appropriateenergy values εl

R for each l channel. This is achieved by

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obtaining the local density of states nlR(E) within the MT

sphere radius centered on R from the above KSDFT-NLPScalculation, and evaluating εl

R as the average energy of eachangular momentum channel l,

εlR =

∫ EF

−∞E

nlR(E)

NlR

dE, (20)

where NlR is the total occupation number in channel l,

NlR =

∫ EF

−∞nl

R(E)dE. (21)

All KSDFT calculations are carried out using the ABINIT

package [46]. After obtaining εlR and VR(rR), the Schrodinger

equation (15) is solved by integrating from rR = 0 outward tothe atomic sphere radius using the fourth-order Runge-Kuttamethod (no boundary condition is needed since it is not aneigenstate problem) [47]. We find that the basis functionsderived in this way exhibit good transferability within similarchemical environments (see Sec. IX). Note that our AMD-OFDFT formalism would in principle allow for multiple radialbasis functions {φ(1)

R,l,φ(2)R,l, . . .} for each l channel to obtain

higher accuracy, as usually done, e.g., for Gaussian basis setsin quantum chemistry. However, we only use one radial basisfunction for each l channel in the applications presented herein.

IV. DIRECT MINIMIZATION OF THE TOTALENERGY FUNCTIONAL

Given a set of atom-centered basis functions inside the MTspheres and AMD energies El

R , V lR , Ul

R , and KlR for each

atomic sphere, the total energy functional in Eq. (4) can bedirectly minimized to obtain the ground-state electron densityfor a fixed ion configuration. To ensure that the total number ofelectrons is conserved, we perform a constrained minimizationof EOF[{NR},ρI ] by applying a Lagrange multiplier μ:

L[X] = E[X] − μ(Ntotal[X] − N0). (22)

Here, μ defines the chemical potential, the variable X ={NR},ρI , and N0 is the total number of electrons containedin the system. Ntotal[X] is the electron number functionalexpressed as

Ntotal[X] =∫

interstitialρI (r)dr +

∑R

∑lm

NR,lm,lm

×∫

MT,R

ψ∗R,lm(�rR)ψR,lm(�rR)drR.

Only the diagonal part of the onsite density matrix contributesin the above equation because all the off-diagonal elementsare eliminated by orthogonality of the different sphericalharmonics.

To satisfy the Pauli exclusion principle, we normalize thebasis functions inside the MT spheres and then constrain theonsite density matrix to ensure the occupation of each lm

channel always lies between zero and one:

0 � NR,lm,lm � 1. (23)

In addition, the same constraint has to be applied to theoccupation in each grid cell

0 � ρIdV � 1, (24)

where dV is the volume of the unit cell of a uniform grid. Tosatisfy the above constraints during the minimization process,we rewrite the density matrix N and the interstitial densityusing the McWeeny purification function [48]

NR = 3M2R − 2M3

R (25)

and

ρIdV = (3Q2

I − 2Q3I

), (26)

where MR and QI are, respectively, the auxiliary onsite densitymatrix and the auxiliary interstitial charge, which directlyyield the physical density matrix and interstitial charge. Bysetting the initial values for the auxiliary quantities MR andQI within the range [0, 1], the constraints in Eqs. (23) and (24)are satisfied automatically during the minimization [48]. Ourworking variables in Eq. (22) are changed to Y = MR,QI ,yielding

L[X[Y ]] = L[Y ] = E[Y ] − μ(Ntotal[Y ] − N0). (27)

Direct minimization of L[Y ] can be carried out using gradient-based methods, such as conjugate gradient [49] or quasi-Newton methods [50]. The total energy minimum is obtainedwhen

dL

dY= dE

dY− μ

dNtotal

dY= 0. (28)

For technical details of the minimization, see Appendix B.Finally, we briefly consider the smoothness of the ground-

state electron density at the sphere boundary. There are twomechanisms that can drive the electron density to smoothnessin the MT geometry: (i) as outlined in Sec. II, the AMD-OFDFT total energy is derived based on a rigorous hybrid KEmodel. In this hybrid KE model, the KE energy density and theKE potential are continuous at the MT sphere boundary. Sincea discontinuous total energy density (or its derivatives) caninduce discontinuities in the electron density (or its gradients),the hybrid KE model provides a basis for the continuityof the electron distribution at the minimum of total energy.(ii) Using the vW KEDF (contained in the modern two-pointKEDFs [22–24]) further drives the density and its gradientto smoothness. A large gradient in the density gives a largepositive vW KE contribution, and thus will increase thetotal energy. The total energy minimization procedure thussmoothens the electron density by optimizing its gradient.

V. A WEIGHTED KEDF FOR THE ENTIRE SYSTEMWITH REDUCED ERRORS

A sophisticated KEDF model is critical for accurately de-scribing the electron density in the interstitial region, especiallyfor systems containing localized electrons. In addition, it isalso very important to have a KEDF model with reducednonlinear errors inside the MT spheres, so that the formof ENL in Eq. (14) can be more accurate and more easilydetermined. The KEDFs currently available do not properlytreat the rapid density variations that occur in the core region.

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In this section, we therefore introduce two general strategies toimprove the KEDF model: (i) a weighting function to suppressthe (inaccurate) KEDF contributions inside the MT spheres;(ii) pseudized basis functions for the highly localized electronchannel dominant in the core region.

Among the linear-response-function-based two-pointKEDFs [22–24], the Wang-Govind-Carter (WGC99)KEDF [24] exhibits comparatively high accuracy due toits double-density-dependent response kernel. The WGC99KEDF can be written as

T KEDFs [ρ] = T TF

s [ρ] + T vWs [ρ] + T NL

s [ρ],

T TFs [ρ] = CTF

∫ρ5/3(�r)d�r,

(29)

T vWs [ρ] = 1/8

∫[∇ρ(�r)]2/ρ(�r)d�r,

T NLs [ρ] =

∫ρα(�r)k[ρ(�r),ρ(�r ′),�r,�r ′]ρβ(�r ′)d�rd�r ′,

where T TFs [ρ] is the Thomas-Fermi KEDF [20], T vW

s [ρ] isthe von Weizacker KEDF [21], and T NL

s [ρ] is a nonlocalterm introduced to yield the correct linear response of thesystem. The nonlocal kernel k[ρ(�r),ρ(�r ′),�r,�r ′] is determinedby enforcing the Lindhard response at the limit of a uniformelectron gas [22–24]. The direct application of this KEDFto inhomogeneous or strongly localized electron densitiescan result in large quantitative (or even qualitative) errorsin electronic structure and material properties compared toKSDFT [40].

To obtain a better description of localized electrons thatare tightly bound in the core region, we exploit the hybridscheme based on the MT geometry. We introduce a weightedKEDF model: we scale available KEDFs using a weightingfunction W (�r), which equals one inside the interstitial regionand smoothly decays to zero inside the MT spheres. W (�r) canbe written as

W (�r) = 1 −∑R

wR(rR). (30)

We choose the smooth function wR(rR) to be

wR(rR) = {1 − erf

[BR

(rR + rw

R

)]}/2, (31)

where erf is the error function, and 0 � w(rR) � 1. Theparameters BR and rw

R are chosen by giving function valuesat the MT sphere boundary rR = W1R of wR(W1R) ≈ 0, andat an inner radius rR,1 < W1R of wR(rR,1) � 1 [see Fig. 2 forwR(rR)]. Utilizing W (�r), we obtain a weighted KEDF modelas follows:

T TFs = CTF

∫W (�r)ρ5/3(�r)d�r,

T vWs = 1/8

∫W (�r)

[∇ρ(�r)]2

ρ(�r)d�r,

T NLs =

∫W (�r)ρα(�r)k[ρ(�r),ρ(�r ′),

�r,�r ′]ρβ(�r ′)W (�r ′)d�rd�r ′ + C, (32)

where C is introduced so that the KEDF can satisfy somelimiting conditions discussed below. The validity of this

FIG. 2. (Color online) (a) Weighting function wR(rR) inside theMT sphere with sphere radius W1R = 2.2 bohr, generated withwR(W1R) = 0.001 and wR(rR,1) = 0.75 (rR,1 = 1.4 bohr) for theKEDF model in Eq. (32). (b) The original (solid line) and pseudized(dashed line) localized d radial basis function.

weighted KEDF method within AMD-OFDDFT is guaranteedby ENL, which corrects errors caused by approximate KEDFsinside the MT region. Since we now downscale the (inaccurate)KEDF contributions from the MT sphere region, the associatederrors decrease and can be more accurately accounted for byENL. Consequently, determining the AMD energies in ENL ofEq. (14) becomes easier because of the smaller magnitude ofV l

R , UlR , and Kl

R . Moreover, the weighted KEDF method canprovide a more accurate description of the interstitial regionbecause the effect of the rapidly varying electrons in the coreregion contributes less to the KEDF.

However, the behavior of the weighted KEDFs requiresfurther consideration: in the uniform electron gas limit, thecontribution from T NL

s should be zero. To satisfy this limit, weadd an extra term C to T NL

s in Eq. (32), yielding∫ρα

CW (�r)k[ρ(�r),ρ(�r ′),�r,�r ′]W (�r ′)ρβ

Cd�rd�r ′ + C = 0, (33)

where ρC is a constant density. However, for ρ �= const, thefunctional form of C[ρ] is unknown. The above equation existsonly in the uniform electron gas limit. For the present work,we approximate C as a constant. A future careful choice ofthe functional C[ρ] might result in further improvements inmaterial properties. Note that C arises from the use of T NL

s .Thus, for local or semilocal KEDFs, there is no such concernabout the term C[ρ].

Because of the significant contribution of localized elec-trons to the total electron density, directly applying theweighted KEDFs to the whole density may still producesome errors, e.g., in the computation of the required averageelectron density ρ0 for the response kernel in T NL

s [23,24]. Asan additional strategy for improving accuracy, we thereforeintroduce a pseudized wave function for the localized electronchannel, similar to the density decomposition scheme used forthe OFDFT treatment of Ag [39]. Inside the MT spheres, wescale the basis function φR,l to obtain a pseudized φR,l :

φR,l(rR) = gR(rR)φR,l(rR), (34)

where φl(rR) is the pseudized smooth basis function, andgR(rR) is a smoothing function. gR(rR) can be chosen in many

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different ways. Here, we simply choose

gR(rR) = {1 + erf

[DR

(rR + r

g

R

)]}/2. (35)

The parameters DR and rg

R can be determined from functionvalues at the sphere boundary rR = W1R and the radialpoint having the maximum amplitude of the basis functionrR = rmax:

gR(W1R) � 1, gR(rmax) = κ,

where gR(W1R) is slightly below one, to ensure that thesmoothed wave function is quickly restored to its originalvalues outside W1R , so that the resulting error is negligibleoutside. The value κ determines the shape of the pseudizedbasis function inside the MT sphere. Its optimal value canbe determined such that ρ0 ≈ 〈ρI 〉I , where 〈. . .〉I denotes anaverage over the interstitial region. Note that smoothing isonly required for the strongly localized d channel in transitionmetals, i.e., g(r) = 1 for the s and p channels in transitionmetals.

After the above pseudization of the localized electron basisfunction, we can write the smoothed total electron density as

ρtotal =∑R

ρR + ρI , (36)

where ρR = ∑lm,l′m′ NR,lm,l′m′ψRlmψRl′m′ is the smoothed

density inside the MT sphere. Obviously, ρ = ρ in theinterstitial region.

We thus use the rescaled T KEDF[ρ] of Eq. (32) operating ona smooth electron density. By using pseudized localized basisfunctions, we can further reduce the unphysical interactionsbetween delocalized and localized channels as discussed inSecs. I and II.

VI. SEARCHING FOR THE ANGULAR MOMENTUMDEPENDENT ENERGIES

We now introduce our method for finding a set of optimalAMD energies PR = {El

R,V lR,Ul

R,KlR} for use in Eq. (14). As

our current objective is an improved treatment of transitionmetals, here we only consider V l

R , UlR , and Kl

R nonzero for thelocalized d channel, although generalization to other elementsis straightforward. We aim to find these six optimal AMDenergies, i.e., E

l=s,p,d

R , V dR , Ud

R , and KdR , by reproducing a

small set of benchmark material properties determined byKSDFT-NLPS calculations using our AMD-OFDFT ansatz.We minimize the deviation

F [PR] =N∑

i=1

Ci

[XOF

i [PR]

XKSi

− 1

]2

,

where XOF/KSi denotes the ith property value derived from

OFDFT and KSDFT, respectively. The benchmark propertiesmay include occupation numbers of each l channel within theMT spheres, the equilibrium volume, the bulk modulus, and theenergy differences between different structures. The parameterCi weights the importance of property Xi . The feasibility ofthis optimization is ensured by the high numerical efficiencyof OFDFT. In principle, the more properties included in F ,the more transferable the energies PR become. Conversely,a set of values PR should be tested for transferability against

properties not included in F . Thus we do not include propertiesof all structures but rather just fit to a minimal subset. We willdemonstrate the transferability of the AMD energies in Sec. IX.Because Ud

R and KdR account for different physics than E

l=s,p,d

R

and V dR , our searching procedure is divided into two steps: we

first search for a set of PEVR = {El

R,V dR } and then optimize

PUKR = {Ud

R,KdR} for improved properties at the fixed PEV

R

found in the first step.A good set of El

R and V dR must yield occupations compa-

rable to KSDFT for each l channel and a total energy close tothat from a KSDFT-NLPS calculation. This condition alreadyproves quite stringent, strongly restricting useful values ofEl

R and V dR . The narrow range of these energies represents a

challenge for applying global or local optimization methods,such as simulated annealing or simplex methods, withoutproviding a good starting point. To find El

R and V dR , we thus

adopt a simple adaptive-grid-based searching method that firstconsiders only the occupation numbers: we calculate F on auniform grid of E

p

R , EdR , and V d

R values. For each grid point[Ep

R,EdR,V d

R ], a change in EsR changes the occupation numbers

of all the channels. The monotonic dependence of Ns on EsR

(i.e., Ns will always increase for decreasing EsR) provides a

way to quickly determine the bounds of EsR within which

our OFDFT method gives satisfactory occupation numbersin comparison to KSDFT-NLPS calculations (see Fig. 3). Forpoints within this Es

R region, we then calculate the equilibriumvolume and bulk modulus for a simple geometry, such as bulkfcc Ti. Only if these two properties compare well to KSDFTdo we continue to calculate additional properties to fit against.We finally identify the PEV

R with the smallest error F . We findthat the AMD values determined with this algorithm alreadyreproduce the KS benchmark surprisingly well (see Sec. IX).To further refine the values PEV

R , one may carry out a localoptimization using, e.g., a simplex algorithm.

Nonzero values for UdR and Kd

R further improve the electrondistribution within the localized d channel, yielding moreaccurate properties. In a second step, we search for PUK

R on auniform grid to further improve properties, such as the energydifferences between a variety of bulk phases of titanium (seeSec. IX).

FIG. 3. Adaptive grid method for optimizing ElR and V d

R .(a) Uniform grid for E

p

R , EdR , and V d

R variables. (b) At a point [Ep

R ,Ed

R , V dR ], we determine the upper and lower bounds for Es

R using acoarse grid and then search for an appropriate Es

R value on a finergrid between the bounds.

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Once obtained for a given chemical element, the set ofAMD energies, combined with the associated basis functionsand scaling parameters for the KEDF, can be used for AMD-OFDFT simulation of materials containing that element, muchlike a pseudopotential. We envision a library of AMD energiesand basis functions for the convenient use of AMD-OFDFT.Providing well-tested, transferable parametrizations for themost common elements is ongoing work, that will ultimatelyextend the applicability of OFDFT.

VII. FORCE FORMULA WITH PULAY CORRECTIONS

Given the force on the ions, we can optimize the ionpositions to obtain an equilibrium geometry or we can carryout molecular dynamics simulations. For a general total energyfunctional Etotal[ρ,R], the force on an ion at position R iswritten as

FR = −∂Etotal[ρ,R]

∂R−

∫δEtotal[ρ,R]

δρ

dRdr. (37)

Here, the first and second terms are the Hellmann-Feynmanand Pulay forces, respectively. The Pulay force arises fromthe atom-centered basis functions that explicitly depend onR. By adding the ion-ion interaction energy, the total energyfunctional becomes

Etotal[{NR},ρI ,R] = T KEDFs [ρ] + EH[ρ] + EXC[ρ]

+ELPSi-e [ρ,R] + ENL[{NR}] + Ei-i[R].

(38)

Thus {NR}, ρI , and R are three independent variables inEtotal. The energy terms ELPS

i-e and Ei-i explicitly contain theion position R. Also, the applied KEDF model contains R

explicitly because of the weighting function W (�r) of Eq. (30)introduced in Sec. V. Note that the nonlocal energy ENL

in Eq. (14) has no explicit dependence on R because theAMD energies, once obtained, are subsequently kept constantduring the electronic structure optimization. Additionally, theNR are independent variables in Etotal. Thus, ENL yields nocontribution to the force. Consequently, the force formula inEq. (37) is explicitly rewritten as

FR = −dEi-i

dR− ∂Ei-e,local

∂R−

∫δT KEDF

s

δW

dW

dRd�r

−∫ (

VKEDF + VH + V LPSi-e + VXC

)dρR

dRdr, (39)

where we use the energy functional derivative inEq. (B8) and dρ

dR= dρR

dRsince ρI is independent of R.

According to Eq. (32), δT KEDFs

δW= CTFρ

5/3(�r) + [∇ρ(�r)]2

8ρ(�r) +∫ρα(�r) k[ρ(�r),ρ(�r ′),�r,�r ′] ρβ (�r ′) W (�r ′) d �r ′ + ∫

ρβ(�r) k[ρ(�r),ρ(�r ′),�r,�r ′]ρα(�r ′)W (�r ′)d �r ′.

As ρR is given by Eq. (2), the quantity dρR/dR can beexpressed as

dρR

dR= −

∑lm,l′m′

NR,lm,l′m′

[ψ∗

R,lm(�rR)dψR,l′m′ (�rR)

d�rR

+ψR,l′m′(�rR)dψ∗

R,lm(�rR)

d�rR

], (40)

by using ∂�rR

∂R= −1 since �rR = �r − R. Although NR is indexed

with R, it does not explicitly depend on R as mentioned above.

VIII. A DOUBLE-SPHERE TECHNIQUE

To preserve the simplicity of OFDFT in practical imple-mentations of the MT geometry, the basis functions inside theMT spheres are represented on a 3D uniform Cartesian gridby projecting from a radial mesh. From Sec. II we see thatthe basis functions are introduced inside the MT spheres butnot in the interstitial region. Special care must be taken toproperly treat the resulting discontinuity of the basis functionsat the sphere boundary. Otherwise, this nonconforming gridproblem (i.e., the mismatch between the sphere boundary andthe Cartesian interstitial grid) will cause serious numericalinstabilities for the convergence of the total energy with respectto grid size. In particular, small variations in geometry canproduce artifacts, resulting in discontinuous energy curves.This problem becomes severe for basis functions with largeamplitudes at the sphere boundary. In the following, we devisea double-sphere technique to solve this numerical problem.As shown in Fig. 4, we introduce inner and outer sphereswith respective radii W1R and W2R centered on R: the basisfunctions are used within W2R and they are reduced betweenW1R and W2R . In our double-sphere technique, the basisfunctions are modified as

φl(rR) = φl(rR)f (rR). (41)

Here, the function f (rR) is introduced to smoothly decreasethe amplitude of the basis function for rR > W1R . We choosethe form

f 2(r) = {1 − erf

[AR

(rR − r

f

R

)]}/2.0, (42)

where the parameters AR and rf

R are determined by the functionvalues f (W1R) and f (W2R). The φ(rR) are truncated outsidethe outer sphere. Both f (W2R) and W2R need to be chosen toreduce numerical noise (induced by the representation of thespheres on the uniform grid) below the accuracy requirement.

FIG. 4. (Color online) Double-sphere technique: the basis func-tions smoothly decrease in the region between W1R and W2R ,increasing numerical stability. Black curves: original basis functionsφl ; red curves: modified basis functions φl . Solid lines: s channel;dashed lines: d channel.

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We set f (W1R) � 1, so that the modified function φ(rR) isquickly restored to the original φ(rR) for rR < W1R , such thatthe difference between φ and φl is negligible within the innersphere. In the geometry with atom-centered double spheres, theelectron density inside the inner sphere is expressed only by φ,while the interstitial region starts at the inner-sphere boundaryW1R , and thus the electron density outside the inner spheresis given as a sum of the modulus squared of the reduced basisfunctions of Eq. (41) and ρI . We need to choose the sphereradii W1R and W2R so that no outer sphere overlaps with anyother inner sphere. The introduction of the double spheres doesnot change the formalism and physics we discussed above, butsignificantly increases numerical stability.

An accurate evaluation of the potential matrix elements∂E/∂NR [see Eqs. (B3), (B4), and (B8)] by numericalintegration within the double spheres can further increasenumerical stability. The rapid spatial variations of the functionf 2(r) [(obtained from the product of the basis functionsψ∗

R,lmψR,l′m′ in the expression of the density in Eq. (2)]require a fine grid to correctly capture the fast change invalue between W1R and W2R . However, we know the originalbasis functions φR and the energy potential δE/δρ are bothslowly varying functions which can be accurately representedon a coarse grid. To reduce demands on grid discretizationand maintain accuracy in the evaluation of ∂E/∂NR , we firstproject f 2(r) from a radial grid onto a 3D uniform grid,which is two times denser than the coarse working grid onwhich we calculate the total energy and the correspondingpotential. We then downsample f 2(r) from the twice densergrid to the coarse working grid by a simple 3D interpolation[see Fig. 5(a)]. As shown in Appendix C and Fig. 5(b), theaccuracy obtained can be similar to a calculation performeddirectly on the denser grid, while retaining the convergence

FIG. 5. Double-grid technique. (a) Downsampling the f (r) froma two times denser grid to a coarse grid on which we calculate thetotal energy and potentials. (b) Example for integral

∫ B

Af (x)g(x)dx:

(1) smooth function g(x); (2) fast changing f (x); (3) projecting f (x)onto a coarse grid directly; (4) projecting f (x) onto a two timesdenser grid and then downsampling to the coarse grid.

speed of optimization on the coarser grid. It should bementioned that the downsampling procedure is done only oncebefore starting the total energy optimization. Consequently, thedownsampling procedure increases numerical stability withnegligible computational cost.

IX. COMPUTATIONAL DETAILS AND RESULTS

The entire AMD-OFDFT formalism outlined above hasbeen implemented within PROFESS 2.0, a state-of-art OFDFTsoftware package [51]. For implementation details for theHartree, XC, LPS, and KEDF energies and the correspondingpotentials, please refer to Ref. [52]. Here, we apply AMD-OFDFT to calculate various properties of the transition metaltitanium and compare with other levels of approximation,to demonstrate the formalism and the associated numericalimplementation.

In all our calculations, we use the Perdew-Burke-Ernzerhof(PBE) [53] form of the generalized gradient approximation asthe exchange-correlation functional. Our KSDFT calculationsare carried out using the ABINIT [46] software package. TheTroullier-Martins (TM) form [54] of the NLPS with a nonlinearcore correction [55] is used, as generated by the FHI98 code [56]with rcutoff = 2.2 bohr for all l channels and rnlc = 1.2 bohr asthe cutoff radius for the core electron density. In KS-NLPScalculations, we use a plane-wave basis KE cutoff Ecut =1600 eV (equivalent to 6400 eV in PROFESS 2.0) for the differentTi bulk phases, and the following Monkhorst-Pack grids [57]for k-point sampling: 30 × 30 × 20 for hcp, 30 × 30 × 30for fcc and bcc, and 26 × 26 × 26 for sc, within unit cellscontaining two, one, two, and one atoms, respectively.

The bulk-derived local pseudopotential (BLPS) [34] used inKS-BLPS and conventional OF-BLPS calculations is obtainedby inverting the KS equations according to the procedureoutlined in Ref. [35]. We use the Ti bcc phase for this inversionbecause we found that the hcp and fcc phases generate veryscattered potential points. We also include the nonlinear corecorrection in the construction of the BLPS. The KS-BLPScalculations are done by ABINIT with the same plane-wavebasis KE cutoff and k-point sampling as we use for theKS-NLPS calculations. Recall that OFDFT calculations donot require k-point sampling. For the conventional OF-BLPScalculation, we use a plane-wave basis KE cutoff of 6400 eVin PROFESS 2.0. For the AMD-OFDFT calculations, we usethe s channel of the TM NLPS as the local pseudopotential;in this case, the plane-wave basis KE cutoff to convergethe total energy with an error below 0.5 meV per titaniumatom is 11 000 eV; although this cutoff sounds extreme, thecalculations are very efficient, several orders of magnitudefaster than KSDFT calculations. (We do not quote timingsin this paper as the AMD-OFDFT code has not yet beenoptimized.) The basis functions inside the MT spheres arederived from a KS-NLPS calculation of the Ti fcc phase atits equilibrium lattice structure, using the method presented inSec. III. We use W1R = 2.2 bohr (same as rcutoff of the NLPS)for the MT inner-sphere radius and W2R = 2.7 bohr for theouter-sphere radius. f 2(W1R) = 0.995 and f 2(W2R) = 0.1are used to solve for the parameters in Eq. (42) to achieve goodnumerical stability. Other choices of 0.1 � f 2(W2R) � 0.3produce negligible changes in the total energy when using

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TABLE I. Function values used to determine the scaling functionswR and gR , and resulting parameters.

wR [see Eq. (31)]

wR(W1R) 0.001 wR(W1R − 0.8) 0.75BR 3.35 rw

R (bohr) 1.53

gR [see Eq. (35)]

gR(W1R) 0.995 wR(rmax) 0.75DR 0.90 r

g

R (bohr) 0.31

the same AMD energies. A change in the shape of the basisfunctions between W1R and W2R only changes ρI during thetotal energy optimization but not the final total energy value.Values to determine the scaling functions wR and gR are listedin Table I. Changing these values is compensated by a changeof the AMD energies in ENL. Consequently, suitable parameterranges are large: the choices of wR(W1R − 0.8) � 0.55 andgR(rmax) � 0.45 all give results very close to what we presenthere.

We use the search procedures presented in Sec. VI todetermine the AMD energies E

l=s,p,d

R , V dR , Ud

R , and KdR

by considering the following small set of properties of Tidescribed within KSDFT: the fcc occupation numbers N fcc

l

for each l channel, the fcc equilibrium volume V fcc0 , the

fcc bulk modulus Bfcc0 , the phase ordering energy Efcc-hcp

between fcc and hcp, and the absolute value of the fcctotal energy. We denote use of only the first two terms ofENL in Eq. (14) with AMD-OF1 while AMD-OF2 denotesuse of all terms of ENL. We obtained the following set ofAMD energies after fitting: Es

R = −0.1133,Ep

R = 0.02,EdR =

−0.38,V dR = 0.1,Ud

R = 0.252, and KdR = 0.5 hartree. Using

these parameters within AMD-OFDFT reproduces the KSDFTbenchmark properties very well. We reported in Ref. [41] thatcalculations using this set of AMD energies deviate by 1% and5% from the KS Ti fcc equilibrium volume and bulk modulus,

FIG. 6. (Color online) Energy versus volume for the Ti hcpphase, for five different theories (see text for details). Energy shiftsof the total energy of the Ti hcp phase for the different levels ofapproximation are (in eV/atom) KS-NLPS: −268.410; KS-BLPS:−281.529; OF-BLPS: −279.672; AMD-OF1: −268.364; and AMD-OF2: −268.451.

FIG. 7. (Color online) Energy versus volume for the Ti fcc phase,for five different theories.

respectively. Moreover, the phase ordering energy betweenfcc and hcp is quantitatively reproduced by AMD-OF2. Wedemonstrated good transferability of the AMD energies byapplying them to bulk properties of Ti hcp, bcc, and sc phases,as well as the formation energies of a monovacancy in hcp Tiand of Ti hc(0001), fcc(100), and bcc(100) surfaces [41]. Here,we provide further tests of the model by investigating varioustypes of mechanical deformations and comparing differentlevels of approximation. To further demonstrate the validityof the AMD-OFDFT formalism, the deformations appliedhere are much larger than normally used for calculating bulkproperties (e.g., as presented in Ref. [41]).

Figures 6–9 show energy versus volume curves for Ti bulkhcp, fcc, bcc, and sc phases using different levels of DFTapproximation. We shift the total energy of the ground-statehcp phase to zero for each theory, as shown in Fig. 6. It is clearthat all methods produce smooth curves for all bulk phasesinvestigated (as also shown in Figs. 10–13). This smoothnessprovides a strong test for the numerical stability of the AMD-OFDFT formalism. AMD-OFDFT substantially improves allresults over the conventional OF-BLPS (see Figs. 6–9), e.g.,

FIG. 8. (Color online) Energy versus volume for the Ti bcc phase,for five different theories.

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FIG. 9. (Color online) Energy versus volume for the Ti sc phase,for five different theories.

the equilibrium volume and the energy response to systemchanges (see the curvature of the curves). In particular,conventional OFDFT volume predictions deviate about −20%from the KS-NLPS for all Ti phases, while our AMD-OFresults produce equilibrium volumes only −1% smaller thanKS benchmarks for hcp, fcc, and bcc phases and +4% for thesc phase. AMD-OFDFT produces a slowly changing energycurve in very good agreement with KS-NLPS, while OF-BLPScalculations exhibit much faster energy changes (i.e., too highbulk moduli) for all phases. For example, OF-BLPS predictsbulk moduli with about +100% error compared to AMD-OFDFT calculations that are very close to the benchmarks. Inaddition, conventional OFDFT, with its approximate KEDFand use of a BLPS, predicts a wrong phase ordering betweenfcc and bcc phases compared to KS-NLPS [41].

Although the KS-BLPS model, with its accurate KE,predicts accurate equilibrium volumes, energy versus volumecurves, and qualitatively correct energy orderings betweendifferent phases, our numerically much faster AMD-OFDFTresults are even closer to the KS-NLPS benchmarks. From theabove comparison, we see that our AMD-OFDFT formalism

FIG. 10. (Color online) Energy versus strain x in hcp Ti. Theapplied strain mode is (1 + x,0,0; 0,1 + x,0; 0,0,1).

FIG. 11. (Color online) Energy versus strain x in hcp Ti. Theapplied strain mode is (1 + x,0,0; 0,1 − x,0; 0,0,1).

provides an effective way to include the influence of anNLPS (hence the superior results compared to KS-BLPS)and to correct KE errors in KEDFs (hence the far superiorresults compared to OF-BLPS) in order to achieve an accuratedescription of system properties. The failure to describe thehypothetical sc Ti phase sufficiently well (see Fig. 9) is likelydue to its much lower coordination number (six) compared tothe other phases. The limit of transferability has been reachedin this case and motivates future refinements to the modeldiscussed earlier.

To understand the role of the AMD energies, we alsocompare the predictions of AMD-OF1 and AMD-OF2 inFigs. 6–9. It is evident that even with only the first twoterms containing El

R and V dR in ENL, AMD-OF1 reproduces

very well the equilibrium volume and energy versus volumecurves for Ti’s hcp, fcc, and bcc phases compared to KS-NLPSbenchmarks. However AMD-OF1 exhibits a +10% deviationfrom KS-NLPS for the equilibrium volume of sc Ti and awrong phase ordering between Ti hcp and fcc phases, with avery small energy difference [41]. By including Ul

R and KlR

in AMD-OF2 to correct the delocalization error in the KEDF

FIG. 12. (Color online) Energy versus strain x in hcp Ti. Theapplied strain mode is (1,0,x; 0,1,0; x,0,1).

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FIG. 13. (Color online) Energy versus strain x in hcp Ti. Theapplied strain mode is [1,0,0; 0,1,0; 0,0,1 + x].

(see Sec. II), we achieve the correct phase ordering (even theenergy differences for fcc, bcc, and sc become comparable toKS-NLPS predictions), and an improved equilibrium volumefor the Ti sc phase. Thus, corrections to the delocalizationerror are critical to reflect the effect of symmetry changes onproperties.

Up to now, we have only examined the energy response toisotropic deformations of the bulk phases. To further evaluatethe quality of the AMD-OFDFT formalism, we investigateother deformations of the ground-state hcp phase of Ti (seeFigs. 10–13). We consider four independent deformationmodes of the equilibrium hcp structure, with strain up to±6%: Figs. 10 and 11 are for two independent deformationson the basal plane while keeping the other axis constant;Fig. 12 is for a pure shear deformation; Fig. 13 is for uniaxialdeformations along the direction perpendicular to the basalplane. For details about the different types of deformations, seeRef. [19]. It is clear that our AMD-OFDFT approach agreesvery well with KS-NLPS calculations, while conventionalOFDFT significantly overestimates the energy increase withstrain. The agreement seen here provides the foundation forAMD-OFDFT simulations of the mechanical properties ofmaterials containing transition metals. AMD-OFDFT energycurves agree with KS-NLPS results for a wide range ofdeformations, providing a strong confirmation that the AMD-OFDFT formalism greatly improves the system density andthe linear response function [3].

X. SUMMARY

We have shown that AMD-OFDFT substantially improvesthe accuracy of OFDFT for describing the transition metalTi, and that the method exhibits quite good transferability. Theaccuracy and transferability of AMD-OFDFT is determined bythe quality of ENL (i.e., the AMD energies), the KEDF model,and the basis functions within the MT spheres. At present, wetreat the AMD energies as constants for a given element anddetermine them by fitting to KS-NLPS benchmarks, neglectingtheir dependence on occupation numbers that may changeas the system deforms. Nevertheless, we have demonstratedthat the constant AMD energies feature good transferability.

As we show in Sec. II, the good transferability is confirmedby a closer inspection of ENL: the AMD energies dependonly weakly on the small occupations of the delocalizedchannels (such as the s and p channels in transition metals),which are sensitive to system changes [41]. Although theydo depend on the occupation of the localized channels (suchas the d channel in transition metals), this occupation is notstrongly influenced by system changes. Treating the AMDenergies as constants is thus a good approximation. Futurework considering the dependence of the AMD energies onthe occupations may further enhance their transferability andaccuracy. The generality of our AMD-OFDFT method alsorelies on the KEDF model. Modern two-point KEDFs [22–24]should be accurate enough to describe the interstitial regionin different structures of at least any kind of metal and recentwork has shown [26] that a density decomposition permits amodern KEDF (WGC99) [24] to also describe the interstitialregions in semiconductors. The accuracy of AMD-OFDFTis also related to the quality of the basis functions employedinside the MT spheres. For example, the AMD energies dependon the basis functions (see Appendix A), whose quality in turndetermines how accurately the important ionic core region isrepresented. We derive ψR from KSDFT NLPS calculationsof a suitable target system to explicitly consider the effect ofthe surrounding electronic structure. The ψR obtained in thisway exhibit good transferability thus far, as we have shownabove and in Ref. [41].

In conclusion, we have developed a general AMD-OFDFTmethod via a hybrid scheme based on a muffin tin geometryand have written the associated computer program for first-principles OFDFT simulations of materials that can nowinclude transition metals. The AMD effects of the KE operatorand of the interaction with the ionic cores are treated by usingKSDFT-derived atom-centered basis functions inside the MTspheres, and by introducing an important nonlocal energy termto effectively correct the errors in the KEDF and the LPS.Our results for various properties of Ti show that includingangular momentum dependence is essential for improvingthe electronic structure and the system response to externalfield changes, and thus system properties: AMD-OFDFTsubstantially improves the accuracy and general applicabilityof OFDFT. Our present results are all based on the WGC99KEDF and approximations made to ENL. More sophisticatedKEDFs, or a more elaborate ansatz for ENL, should furtherimprove the accuracy and transferability of AMD-OFDFT.

ACKNOWLEDGMENTS

We gratefully acknowledge discussions with L.-W. Wang,L. Hung, C. Huang, and I. Shin. This work is supportedby the Office of Naval Research (E.A.C.) and the SfB-041ViCoM (F.L.). We thank Princeton University and the DODsupercomputing resource center for supercomputing time.

APPENDIX A: DERIVING ENL[NR]

We provide further derivation of the parameters UdR and Kd

R

in Eq. (14) by investigating their dependence on the occupation

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numbers. From Ref. [41], we know

UKEDF,dR = 4π

∫MT

δ2T KEDFs

δρ2

∣∣∣∣ρ0

R

ρ2R,dr

2dr. (A1)

To obtain the dependence of UKEDF,dR on NR , one can perform a

Taylor expansion for UKEDF,dR around N0

R,s/p = 0 for transitionmetals. We consider here the TF and vW KEDFs because oftheir dominant contributions in the modern two-point KEDFs.

For the TF KEDF, we apply the Taylor expansion to δ2T TFs

δρ2 |ρ0R

up to first order around N0R,s/p = 0. We thus obtain

UTF,dR = 4π

∫MT

δ2T TFs

δρ2

∣∣∣∣ρ0

R

ρ2R,dr

2dr

≈ 4π

∫MT

[(N total

R,d

)− 13δ2T TF

s

δρ2

∣∣∣∣ρR,d

+∑l=s,p

(N total

R,d

)− 43δ3T TF

s

δρ3

∣∣∣∣ρR,d

N totalR,l ρR,l

]ρ2

R,dr2dr

= Ntotal,− 1

3R,d BTF

R,d +∑l=s,p

Ntotal,− 4

3R,d N total

R,l DTFR,d,l . (A2)

The constants BTFR,d = 4π

∫MT

δ2T TFs

δρ2 |ρR,dρ2

R,dr2dr and DTF

R,d,l =4π

∫MT

δ3T TFs

δρ3 |ρR,dρ2

R,dρR,lr2dr only depend on the fixed basis

functions, and are thus system independent.For the case of the vW KEDF, we also apply the Taylor

expansion for δ2T vWs

δρ2 |ρ0R

up to first order around N0R,s/p = 0:

UvW,dR = 4π

∫MT

δ2T vWs

δρ2

∣∣∣∣ρ0

R

ρ2R,dr

2dr

= 4π

∫MT

[1

N totalR,d

δ2T vWs

δρ2

∣∣∣∣ρR,d

+∑l=s,p

N totalR,l(

N totalR,d

)2

δ3T vWs

δρ3

∣∣∣∣ρR,d

ρR,l

]ρ2

R,dr2dr

= 1

N totalR,d

BvWR,d +

∑l=s,p

1(N total

R,d

)2 N totalR,l DvW

R,d,l . (A3)

The constants BvWR,d = 4π

∫MT

δ2T vWs

δρ2 |ρR,dρ2

R,dr2dr and DvW

R,d,l =4π

∫MT

δ3T vWs

δρ3 |ρR,dρR,lρ

2R,dr

2dr are only dependent on the fixedbasis functions, and are thus system independent again. Thus,we see U

KEDF,dR can be approximated as

UTF+vW,dR ≈ (

N totalR,d

)− 13 BTF

R,d +∑l=s,p

(N total

R,d

)− 43 N total

R,l DTFR,d,l

+ (N total

R,d

)−1BvW

R,d +∑l=s,p

(N total

R,d

)−2N total

R,l DvWR,d,l

≈ (N total

R,d

)− 13 BTF

R,d + (N total

R,d

)−1BvW

R,d

+∑l=s,p

DTF+vWR,d,l N total

R,l . (A4)

For the parameter KKEDF,dR , we have [41]

KKEDF,dR = 4π

∫MT

δ3T KEDFs

δρ3

∣∣∣∣ρ0

R

ρ3R,dr

2dr. (A5)

For the case of TF KEDF, a Taylor expansion around N0R,s/p =

0 yields

KTF,dR = 4π

∫MT

δ3T TFs

δρ3

∣∣∣∣ρ0

R

ρ3R,dr

2dr

= 4π

∫MT

[(N total

R,d

)− 43

δ3T TFs

δρ3

∣∣∣∣ρR,d

+∑l=s,p

(N total

R,d

)− 73 N total

R,l

δ4T TFs

δρ4

∣∣∣∣ρR,d

ρR,l

]ρ3

R,dr2dr

= (N total

R,d

)− 43 GTF

R,d + (N total

R,d

)− 73 N total

R,l H TFR,d,l, (A6)

where GTFR,d = 4π

∫MT

δ3T TFs

δρ3 |ρR,dρ3

R,dr2dr and H TF

R,d,l =4π

∫MT

δ4T TFs

δρ4 |ρR,dρR,lρ

3R,dr

2dr are dependent on the radialbasis functions inside the MT spheres.

For the case of the vW KEDF, again, using a Taylorexpansion around N0

R,s/p = 0, we find

KvW,dR = 4π

∫MT

δ3T vWs

δρ3

∣∣∣∣ρ0

R

ρ3R,dr

2dr

= 4π

∫MT

⎡⎣ 1(

N totalR,d

)2

δ3T vWs

δρ3

∣∣∣∣ρR,d

+ 1(N total

R,d

)3

δ4T vWs

δρ4

∣∣∣∣ρR,d

N totalR,l ρR,l

⎤⎦ ρ3

R,dr2dr

= 1(N total

R,d

)2 GvWR,d + 1(

N totalR,d

)3 N totalR,l H vW

R,d,l, (A7)

where GvWR,d = 4π

∫MT

δ3T vWs

δρ3 |ρR,dρ3

R,dr2dr and H vW

R,d,l =4π

∫MT

δ4T vWs

δρ4 |ρR,dρR,lρ

3R,dr

2dr are only given by the radialbasis functions inside the MT spheres. Therefore, we have

KTF+vW,dR ≈ (

N totalR,d

)− 43

⎛⎝GTF

R,d −∑l=s,p

N totalR,l

N totalR,d

H TFR,d,l

⎞⎠

+ (N total

R,d

)−2GvW

R,d +∑l=s,p

(N total

R,d

)−3N total

R,l H vWR,d,l

≈ (N total

R,d

)− 43 GTF

R,d + (N total

R,d

)−2GvW

R,d

+∑l=s,p

H TF+vWR,d,l N total

R,l . (A8)

APPENDIX B: DETAILS OF THE TOTALENERGY MINIMIZATION

In each optimization step, for a given Y = {MR},QI , weneed to compute L[Y ] and the corresponding gradient dL/dY

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to calculate the line search direction for updating Y . Wecalculate dE/dY by applying the chain rule to the energyterms including T KEDF

s [ρ], EXC[ρ], EH[ρ], and ELPSi-e [ρ] in

Eq. (4) as

∂E

∂MR

= ∂E

∂NR

dNR

dMR

; (B1)

∂E

∂QI

= δE[ρ]

δρ

∂ρ

∂ρI

dρI

dQI

. (B2)

Here,

∂E

∂NR

=∫

MT

δE[ρ]

δρ

∂ρ

∂NR

d�r; (B3)

∂ρ

∂NR,lm,l′m′= ψ∗

R,lmψR,l′m′ ; (B4)

dNR

dMR

= 6MR(1 − MR); (B5)

∂ρ

∂ρI

= 1; (B6)

dρI

dQI

= 6QI (1 − QI )/dV ; (B7)

δE[ρ]

δρ= VKEDF + VH + VXC + V LPS

i-e , (B8)

where VKE, VH, VXC, and V LPSi-e are the KE potential, Hartree

potential, exchange-correlation energy potential, and localpseudopotential, respectively. dNtotal

dY= dNtotal

dXdXdY

and dENL

dY=

dENL

dXdXdY

, where X = {NR},ρI , can be calculated by using

Eqs. (B5) and (B7). During the total energy minimization, weupdate the chemical potential in Eq. (27) in each iteration by

μ =⟨dEdY

∣∣Y ⟩⟨dNtotal

dY

∣∣Y ⟩ . (B9)

In each minimization step, we note the total electron numberis not strictly conserved after the line search. Therefore, werenormalize the total number of electrons

Y = Y0 − (Ntotal[Y0] − N0)dNtotal

dY⟨dNtotal

dY

∣∣ dNtotaldY

⟩ ,where Y0 is the variable obtained directly by the line searchand the corrected Y serves as the initial variable for the nextoptimization iteration. This correction step does not changethe decrease of the total energy obtained in the line searchprocedure. As the optimization converges, this correctionbecomes smaller and smaller.

APPENDIX C: DOWNSAMPLING TECHNIQUE

We illustrate the principle using a one-dimensional functionas an example. We want to accurately calculate the integral∫ B

Af (x)g(x)dx on a coarse grid, where f (x) is a rapidly

fluctuating function while g(x) varies slowly over all space[see Fig. 5(b)]. We apply the relationship

f (xI ) = 0.5 ∗[f (x2I ) + f (x2I−1) + f (x2I+1)

2

](C1)

between the function on the dense (2I ) and coarse (I ) grids.The numerical integration on the coarse grid can then bewritten as

∫ B

A

f (x)g(x)dx ≈ x∑

I

g(xI ) ∗ f (xI ) = x∑

I

[f (x2I ) + f (x2I−1) + f (x2I+1)

2

]∗ g(xI )

= x∑

I

[f (x2I ) + f (x2I−1) + f (x2I+1)

2

]∗ g(x2I )

= x∑

I

f (x2I )g(x2I ) + f (x2I−1)g(x2I−2) + g(x2I )

2+ f (x2I+1)

g(x2I ) + g(x2I+2)

2

= x∑

I

[f (x2I−1)g(x2I−1) + f (x2I )g(x2I ) + f (x2I+1)g(x2I+1)], (C2)

where x2I = xI and x = 0.5x. Since g(x) only variesslowly, g(x) can be given in good accuracy by a simpleinterpolation. From the above derivation, we can obtain thehigh accuracy of the integral performed on a coarse grid,

comparable to the direct integration on the two times densergrid. This provides an efficient way for us to obtain accuratepotential matrix elements dE

dNinside the MT spheres with a

reasonable grid size.

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