Real-space formulation of orbital-free density functional theory...

Real-space formulation of orbital-free density functional theory using finite-element

discretization: The case for Al, Mg, and Al-Mg intermetallics

Sambit Das, Mrinal Iyer, and Vikram GaviniDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA

We propose a local real-space formulation for orbital-free DFT with density dependent kineticenergy functionals and a unified variational framework for computing the configurational forcesassociated with geometry optimization of both internal atomic positions as well as the cell geometry.The proposed real-space formulation, which involves a reformulation of the extended interactionsin electrostatic and kinetic energy functionals as local variational problems in auxiliary potentialfields, also readily extends to all-electron orbital-free DFT calculations that are employed in warmdense matter calculations. We use the local real-space formulation in conjunction with higher-orderfinite-element discretization to demonstrate the accuracy of orbital-free DFT and the proposedformalism for the Al-Mg materials system, where we obtain good agreement with Kohn-Sham DFTcalculations on a wide range of properties and benchmark calculations. Finally, we investigate thecell-size e↵ects in the electronic structure of point defects, in particular a mono-vacancy in Al.We unambiguously demonstrate that the cell-size e↵ects observed from vacancy formation energiescomputed using periodic boundary conditions underestimate the extent of the electronic structureperturbations created by the defect. On the contrary, the bulk Dirichlet boundary conditions,accessible only through the proposed real-space formulation, which correspond to an isolated defectembedded in the bulk, show cell-size e↵ects in the defect formation energy that are commensuratewith the perturbations in the electronic structure. Our studies suggest that even for a simple defectlike a vacancy in Al, we require cell-sizes of ⇠ 103 atoms for convergence in the electronic structure.

I. INTRODUCTION

Electronic structure calculations have played an im-portant role in understanding the properties of a widerange of materials systems1. In particular, the Kohn-Sham formalism of density functional theory2,3 has beenthe workhorse of ground-state electronic structure calcu-lations. However, the Kohn-Sham approach requires thecomputation of single-electron wavefunctions to computethe kinetic energy of non-interacting electrons, whosecomputational complexity typically scales as O(N3) foran N -electron system, thus, limiting standard calcula-tions to materials systems containing few hundreds ofatoms. While there has been progress in developingclose to linear-scaling algorithms for the Kohn-Sham ap-proach4,5, these are still limited to a few thousands ofatoms, especially for metallic systems6. The orbital-freeapproach to DFT7, on the other hand, models the ki-netic energy of non-interacting electrons as an explicitfunctional of the electron density, thus circumventing thecomputationally intensive step of computing the single-electron wavefunctions. Further, the computational com-plexity of orbital-free DFT scales linearly with the systemsize as the ground-state DFT problem reduces to a min-imization problem in a single field—the electron density.The past two decades has seen considerable progress inthe development of accurate models for orbital-free ki-netic energy functionals8–15, and, in particular, for sys-tems whose electronic-structure is close to a free electrongas (for e.g. Al, Mg). Also, orbital-free DFT calculationsare being increasingly used in the simulations of warmdense matter where the electronic structure is close tothat of a free electron gas at very high temperatures16–20.As the reduced computational complexity of orbital-free

DFT enables consideration of larger computational do-mains, recent studies have also focused on studying ex-tended defects in Al and Mg, and have provided impor-tant insights into the energetics of these defects21–26.

The widely used numerical implementation of orbital-free DFT is based on a Fourier space formalism using aplane-wave discretization27,28. A Fourier space formula-tion provides an e�cient computation of the extended in-teractions arising in orbital-free DFT—electrostatics andkinetic energy functionals—through Fourier transforms.Further, the plane-wave basis is a complete basis and pro-vides variational convergence in ground-state energy withexponential convergence rates. However, the Fourierspace formulations are restricted to periodic geometriesand boundary conditions that are suitable for perfectbulk materials, but not for materials systems contain-ing extended defects. Also, the extended spatial natureof the plane-wave basis a↵ects the parallel scalability ofthe numerical implementation and is also not suitable formulti-scale methods that rely on coarse-graining. In or-der to address the aforementioned limitations of Fourierspace techniques, recent e↵orts have focussed on devel-oping real-space formulations for orbital-free DFT andnumerical implementations based on finite-element29–31

and finite di↵erence discretizations32–34.

In the present work, we build on these prior e↵orts todevelop an e�cient real-space formulation of orbital-freeDFT employing the widely used non-local Wang-Govind-Carter (WGC)11 kinetic energy functional. As in priore↵orts29,30, we reformulate the extended interactions inelectrostatics and the non-local terms in the WGC ki-netic energy functionals as local variational problems inauxiliary potential fields. However, the proposed refor-mulation of electrostatic interactions is notably di↵er-

2

ent from previous works, and enables the evaluation ofvariational configurational forces corresponding to bothinternal atomic relaxations as well as external cell relax-ation under a single framework. Further, the proposedformulation naturally extends to all-electron orbital-freeDFT calculations of warm dense matter16,17. In the pro-posed real-space formulation, the ground-state orbital-free DFT problem is reformulated as an equivalent sad-dle point problem of a local functional in electron den-sity, electrostatic potential and the auxiliary potentialfields (kernel potentials) accounting for the extended in-teractions in the kinetic energy functional. We employa higher-order finite-element basis to discretize the for-mulation, and demonstrate the optimal numerical con-vergence of both the ground-state energy and configura-tional forces with respect to the discretization. Further,we propose an e�cient numerical approach to computethe saddle point problem in electron density, electrostaticpotential and kernel potentials by expressing the saddlepoint problem as a fixed point iteration problem, andusing a self-consistent field approach to solve the fixedpoint iteration problem.

We subsequently investigate the accuracy and transfer-ability of the proposed real-space formulation of orbital-free DFT for Al and Mg materials systems. To this end,we compute the bulk properties of Al, Mg and Al-Mg in-termetallics, and compare it with Kohn-Sham DFT. Asorbital-free DFT only admits local pseudopotentials, theKohn-Sham DFT calculations are conducted using bothlocal and non-local psedupotentials. Our studies indi-cates that the bulk properties computed using orbital-free DFT for Al, Mg and Al-Mg intermetallics are in goodagreement with Kohn-Sham DFT. We further investigatethe accuracy of orbital-free DFT by computing the inter-atomic forces in Al and Mg, which are also in good agree-ment with Kohn-Sham DFT calculations. Our studiesdemonstrate that orbital-free DFT is accurate and trans-ferable across a wide range of properties for Al, Mg andAl-Mg intermetallics, and can be used to study propertiesof these materials systems that require computational do-mains that are not accessible using Kohn-Sham DFT. Forinstance, in the present study we computed the forma-tion energy of �0 Al-Mg alloy containing 879 atoms in aunit cell employing the proposed real-space formulationof orbital-free DFT, but the same system was found tobe prohibitively expensive using Kohn-Sham DFT.

We finally investigate the cell-size e↵ects in the elec-tronic structure of point defects, in particular a mono-vacancy in Al. Prior studies using Fourier-based for-mulations of orbital-free DFT have suggested that theformation energy of a mono-vacancy in Al is well con-verged by 108-256 atom cell-sizes22. However, coarse-grained real-space calculations have suggested that muchlarger cell-sizes of the order of 1,000 atoms are requiredfor convergence of vacancy formation energies30, whichwas also supported by asymptotic estimates35. In or-der to understand the underpinnings of this discrepancy,we use the finite-element discretized real-space formula-

tion of orbital-free DFT and compute the vacancy for-mation energy using two boundary conditions: (i) pe-riodic boundary conditions, equivalent to Fourier-spacebased formulations; (ii) bulk Dirichlet boundary condi-tions, where the perturbations in the electronic struc-ture arising due to the vacancy vanishes on the boundaryof the computational domain. Our study suggests thatwhile the vacancy formation energy is well converged by108 atom cell-size using periodic boundary conditions,the electronic fields are not well-converged by this cell-size. On the other hand the bulk Dirichlet boundaryconditions show well converged formation energy as wellas electronic fields by cell sizes of ⇠1,000 atoms, which isconsistent with prior real-space calculations. This studyreveals that while periodic boundary conditions show asuperior convergence in formation energies due to thevariational nature of the formalism, the true cell-size ef-fects which also measure convergence of electronic fieldsare provided by the bulk Dirichlet boundary conditions.We note that the proposed real-space formulation withfinite-element discretization are crucial to employing bulkDirichlet boundary conditions, which enable the study ofisolated defects in bulk.The remainder of the paper is organized as follows.

Section II provides a description of the orbital-free DFTproblem. Section III presents the proposed real-spaceformulation of the orbital-free DFT problem, the con-figurational forces associated with structural relaxations,and the finite-element discretization of the formulation.Section IV discusses the numerical implementation of theformulation and presents an e�cient numerical approachfor the solution of the saddle point real-space variationalproblem. Section V presents the numerical convergenceresults of the finite-element discretization of the real-space formulation, the accuracy and transferability of thereal-space orbital-free DFT formalism for Al-Mg materi-als system, and the study of the role of boundary con-ditions on the cell-size e↵ects in electronic structure cal-culations of point defects. We finally conclude with asummary and outlook in Section VI.

II. ORBITAL-FREE DENSITY FUNCTIONALTHEORY

The ground-state energy of a charge neutral materialssystem containing M nuclei and N valence electrons indensity functional theory is given by1,7

E(⇢,R) = Ts(⇢)+Exc(⇢)+EH(⇢)+Eext(⇢,R)+Ezz(R) ,(1)

where ⇢ denotes the electron-density and R ={R

1

,R2

, . . . ,RM} denotes the vector containing the po-sitions of M nuclei. In the above, Ts denotes the kineticenergy of non-interacting electrons, Exc is the exchange-correlation energy, EH is the Hartree energy or classicalelectrostatic interaction energy between electrons, Eext

is the classical electrostatic interaction energy between

3

electrons and nuclei, and Ezz denotes the electrostaticrepulsion energy between nuclei. We now discuss the var-ious contributions to the ground-state energy, beginningwith the exchange-correlation energy.

The exchange-correlation energy, denoted by Exc, in-corporates all the quantum-mechanical interactions inthe ground-state energy of a materials system. Whilethe existence of a universal exchange-correlation energyas a functional of electron-density has been established byHohenberg, Kohn and Sham2,3, its exact functional formhas been elusive to date, and various models have beenproposed over the past decades. For solid-state calcula-tions, the local density approximation (LDA)36,37 andthe generalized gradient approximation38,39 have beenwidely adopted across a range of materials systems. Inparticular, the LDA exchange-correlation energy, whichis adopted in the present work, has the following func-tional form:

Exc

(⇢) =

Z

"xc

(⇢)⇢(x) dx , (2)

where "xc

(⇢) = "x(⇢) + "c(⇢), and

"x(⇢) = �3

4

✓

3

⇡

◆

1/3

⇢1/3(x) , (3)

"c(⇢) =

(

�

(1+�1

p(rs)+�2rs)

rs � 1,

A log rs +B + C rs log rs +D rs rs < 1,(4)

and rs = (3/4⇡⇢)1/3. In the present work, we use theCeperley and Alder constants37 in equation (4).

The last three terms in equation (1) represent elec-trostatic interactions between electrons and nuclei. TheHartree energy, or the electrostatic interaction energy be-tween electrons, is given by

EH(⇢) =1

2

Z Z

⇢(x)⇢(x0)

|x� x

0| dx dx0 . (5)

The interaction energy between electrons and nuclei, inthe case of local pseudopotentials that are adopted in thepresent work, is given by

Eext(⇢,R) =

Z

⇢(x)Vext(x,R) dx

=X

J

Z

⇢(x)V Jps(|x�RJ |)dx , (6)

where V Jps denotes the pseudopotential corresponding to

the J th nucleus, which, beyond a core radius is theCoulomb potential corresponding to the e↵ective nuclearcharge on the J th nucleus. The nuclear repulsive energyis given by

Ezz(R) =1

2

X

I

X

J,J 6=I

ZIZJ

|RI �RJ | , (7)

where ZI denotes the e↵ective nuclear charge on the Ith

nucleus. The above expression assumes that the core ra-dius of the pseudopotential is smaller than internucleardistances, which is often the case in most solid-state ma-terials systems. We note that in a non-periodic setting,representing a finite atomic system, all the integrals inequations (5)-(6) are over R3 and the summations inequations (6)-(7) include all the atoms. In the case ofan infinite periodic crystal, all the integrals over x inequations (5)-(6) are over the unit cell whereas the inte-grals over x0 are over R3. Similarly, in equations (6)-(7),the summation over I is on the atoms in the unit cell,and the summation over J extends over all lattice sites.Henceforth, we will adopt these notions for the domainof integration and summation.

The remainder of the contribution to the ground-stateenergy is the kinetic energy of non-interacting electrons,denoted by Ts, which is computed exactly in the Kohn-Sham formalism by computing the single-electron wave-functions (eigenfunctions) in the mean-field1. The con-ventional solution of the Kohn-Sham eigenvalue problem,which entails the computation of the lowest N eigenfunc-tions and eigenvalues of the Kohn-Sham Hamiltonian,scales as O(N3) that becomes prohibitively expensivefor materials systems containing a few thousand atoms.While e↵orts have been focused towards reducing thecomputational complexity of the Kohn-Sham eigenvalueproblem4,5, this remains a significant challenge especiallyin the case of metallic systems. In order to avoid thecomputational complexity of solving for the wavefunc-tions to compute Ts, the orbital-free approach to DFTmodels the kinetic energy of non-interacting electrons asan explicit functional of electron density7. These modelsare based on theoretically known properties of Ts for auniform electron gas, perturbations of uniform electrongas, and the linear response of uniform electron gas7–11.As the orbital-free models for the kinetic energy func-tional are based on properties of uniform electron gas,their validity is often limited to materials systems whoseelectronic structure is close to a free electron gas, in par-ticular, the alkali and alkali earth metals. Further, asthe orbital-free approach describes the ground-state en-ergy as an explicit functional of electron-density, it limitsthe pseudopotentials calculations to local pseudopoten-tials. While these restrictions constrain the applicabilityof the orbital-free approach, numerical investigations11,40

indicate that recently developed orbital-free kinetic en-ergy functionals and local pseudopotentials can providegood accuracy for Al and Mg, which comprise of tech-nologically important materials systems. Further, thereare ongoing e↵orts in developing orbital-free kinetic en-ergy models for covalently bonded systems and transitionmetals41,42.

In the present work, we restrict our focus to the Wang-Goving-Carter (WGC) density-dependent orbital-free ki-netic energy functional11, which is a widely used kineticenergy functional for ground-state calculations of mate-rials systems with an electronic structure close to a free

4

electron gas. In particular, the functional form of theWGC orbital-free kinetic energy functional is given by

Ts(⇢) = CF

Z

⇢5/3(x) dx+1

2

Z

|rp

⇢(x)|2 dx+ TK(⇢)

(8)where

TK(⇢) = CF

Z Z

⇢↵(x)K(⇠�(x,x0), |x� x

0|) ⇢�(x0) dx dx0 ,

⇠�(x,x0) =

⇣k�F (x) + k�F (x0)

2

⌘

1/�

, kF (x) =�

3⇡2⇢(x)�

1/3.

In equation (8), the first term denotes the Thomas-Fermienergy with CF = 3

10

(3⇡2)2/3, and the second term de-notes the von-Weizsacker correction7. The last term de-notes the density dependent kernel energy, TK , where thekernel K is chosen such that the linear response of a uni-form electron gas is given by the Lindhard response43.In the WGC functional11, the parameters are chosen tobe {↵,�} = {5/6 +p

5/6, 5/6�p5/6} and � = 2.7. For

materials systems whose electronic structure is close toa free-electron gas, the Taylor expansion of the densitydependent kernel about a reference electron density (⇢

0

),often considered to be the average electron density of thebulk crystal, is employed and is given by

K(⇠�(x,x0), |x� x

0|) =K0

(|x� x

0|) +K1

(|x� x

0|)��⇢(x) +�⇢(x0)�

+1

2K

11

(|x� x

0|)�(�⇢(x))2 + (�⇢(x0))2�

+K12

(|x� x

0|)�⇢(x)�⇢(x0) + . . . .(9)

In the above equation, �⇢(x) = ⇢(x)�⇢0

and the densityindependent kernels resulting from the Taylor expansionare given by

K0

(|x� x

0|) = K(⇠� , |x� x

0|)�

�

�

⇢=⇢0

K1

(|x� x

0|) = @K(⇠� , |x� x

0|)@⇢(x)

�

�

�

⇢=⇢0

K11

(|x� x

0|) = @2K(⇠� , |x� x

0|)@⇢2(x)

�

�

�

⇢=⇢0

K12

(|x� x

0|) = @2K(⇠� , |x� x

0|)@⇢(x)@⇢(x0)

�

�

�

⇢=⇢0

. . . (10)

Numerical investigations have suggested that the Taylorexpansion to second order provides a good approxima-tion of the density dependent kernel for materials systemswith electronic structure close to a free electron gas11,44.In particular, in the second order Taylor expansion, thecontribution from K

12

has been found to dominate con-tributions from K

11

. Thus, in practical implementations,often, only contributions from K

12

in the second orderterms are retained for computational e�ciency.

III. REAL-SPACE FORMULATION OFORBITAL-FREE DFT

In this section, we present the local variational real-space reformulation of orbital-free DFT, the configura-tional forces associated with internal ionic relaxationsand cell relaxation, and the finite-element discretizationof the formulation.

A. Local real-space formulation

We recall that the various components of the ground-state energy of a materials system (cf. section II) arelocal in real-space, except the electrostatic interactionenergy and the kernel energy component of the WGCorbital-free kinetic energy functional that are extendedin real-space. Conventionally, these extended interac-tions are computed in Fourier space to take advantageof the e�cient evaluation of convolution integrals usingFourier transforms. For this reason, Fourier space for-mulations have been the most popular and widely usedin orbital-free DFT calculations27,28. However, Fourierspace formulations employing the plane-wave basis re-sult in some significant limitations. Foremost of these isthe severe restriction of periodic geometries and bound-ary conditions. While this is not a limitation in the studyof bulk properties of materials, this is a significant limi-tation in the study of defects in materials. For instance,the geometry of a single isolated dislocation in bulk isnot compatible with periodic geometries, and, thus, priorelectronic structure studies have mostly been limited toartificial dipole and quadrapole arrangements of dislo-cations. Further, numerical implementations of Fourier-space formulations also su↵er from limited scalability onparallel computing platforms. Moreover, the plane-wavediscretization employed in a Fourier space formulationprovides a uniform spatial resolution, which is not suit-able for the development of coarse-graining techniques—such as the quasi-continuum method45—that rely on anadaptive spatial resolution of the basis.

We now propose a real-space formulation that is de-void of the aforementioned limitations of a Fourier spaceformulation. The proposed approach, in spirit, followsalong similar lines as recent e↵orts29,30, but the proposedformulation di↵ers importantly in the way the extendedelectrostatic interactions are treated. In particular, the

5

proposed formulation provides a unified framework tocompute the configurational forces associated with bothinternal ionic and cell relaxations discussed in III B.

We begin by considering the electrostatic interactionsthat are extended in the real-space. We denote by �(x�RI) a regularized Dirac distribution located at RI , andthe Ith nuclear charge is given by the charge distribution�ZI �(x � RI). Defining ⇢nu(x) = �PI ZI �(|x � RI |)and ⇢nu(x0) = �PJ ZJ �(|x0�RJ |), the repulsive energyEzz can subsequently be reformulated as

Ezz =1

2

Z Z

⇢nu(x)⇢nu(x0)

|x� x

0| dxdx0 � Eself , (11)

where Eself denotes the self energy of the nuclear chargesand is given by

Eself =1

2

X

I

Z Z

ZI �(|x�RI |)ZI �(|x0 �RI |)|x� x

0| dxdx0 .

(12)

We denote the electrostatic potential corresponding tothe Ith nuclear charge (�ZI �(|x0 �RI |)) as V I

˜�(x), and

is given by

V I˜�(x) = �

Z

ZI �(|x0 �RI |)|x� x

0| dx0 . (13)

The self energy, thus, can be expressed as

Eself = �1

2

X

I

Z

ZI �(|x�RI |)V I˜�(x)dx . (14)

Noting that the kernel corresponding to the extendedelectrostatic interactions in equations (12)-(13) is theGreen’s function of the Laplace operator, the electro-static potential and the electrostatic energy can be com-puted by taking recourse to the solution of a Poissonequation, or, equivalently, the following local variationalproblem:

Eself = �X

I

minV I2H1

(R3)

n 1

8⇡

Z

|rV I(x)|2dx+

Z

ZI �(|x�RI |)V I(x)dxo

, (15a)

V I˜�(x) = arg min

V I2H1(R3

)

n 1

8⇡

Z

|rV I(x)|2dx+

Z

ZI �(|x�RI |)V I(x)dxo

. (15b)

In the above, H1(R3) denotes the Hilbert space of func-tions such that the functions and their first-order deriva-tives are square integrable on R3.

We next consider the electrostatic interaction energycorresponding to both electron and nuclear charge dis-tribution. We denote this by J(⇢, ⇢nu), which is givenby

J(⇢, ⇢nu) =1

2

Z Z

�

⇢(x) + ⇢nu(x)��

⇢(x0) + ⇢nu(x0)�

|x� x

0| dxdx0 .

(16)We denote the electrostatic potential corresponding tothe total charge distribution (electron and nuclear charge

distribution) as �, which is given by

�(x) =

Z

⇢(x0) + ⇢nu(x0)

|x� x

0| dx0 . (17)

The electrostatic interaction energy of the total chargedistribution, in terms of �, is given by

J(⇢, ⇢nu) =1

2

Z

(⇢(x) + ⇢nu(x))�(x)dx . (18)

As before, the electrostatic interaction energy as well asthe potential of the total charge distribution can be re-formulated as the following local variational problem:

J(⇢, ⇢nu) = �min�2Y

n 1

8⇡

Z

|r�(x)|2dx�Z

(⇢(x) + ⇢nu(x))�(x)dxo

, (19a)

�(x) = arg min�2Y

n 1

8⇡

Z

|r�(x)|2dx�Z

(⇢(x) + ⇢nu(x))�(x)dxo

. (19b)

In the above, Y is a suitable function space corresponding to the boundary conditions of the problem. In particu-

6

lar, for non-periodic problems such as isolated cluster ofatoms Y = H1(R3). For periodic problems, Y = H1

per(Q)where Q denotes the unit cell and H1

per(Q) denotes the

space of periodic functions on Q such that the functionsand their first-order derivatives are square integrable.The electrostatic interaction energy in DFT, compris-

ing of EH , Eext and Ezz (cf. equations (5)-(7)), can berewritten in terms of J(⇢, ⇢nu) and Eself as

EH(⇢) + Eext(⇢,R) + Ezz(R) = J(⇢, ⇢nu) +X

J

Z

(V Jps(|x�RJ |)� V J

˜�(|x�RJ |))⇢(x)dx� Eself . (20)

For the sake of convenience of representation, we willdenote by V = {V 1, V 2, . . . , V M} the vector containingthe electrostatic potentials corresponding to all nuclearcharges in the simulation domain. Using the local re-

formulation of J(⇢, ⇢nu) and Eself (cf. equations (15)and (19)), the electrostatic interaction energy in DFTcan now be expressed as the following local variationalproblem:

EH + Eext + Ezz = max�2Y

minV I2H1

(R3)

Lel(�,V, ⇢,R) (21a)

Lel(�,V, ⇢,R) =� 1

8⇡

Z

|r�(x)|2dx+

Z

(⇢(x) + ⇢nu(x))�(x)dx+X

J

Z

(V Jps(|x�RJ |)� V J

˜�(|x�RJ |))⇢(x)dx

+X

I

⇢

1

8⇡

Z

|rV I(x)|2dx+

Z

ZI �(|x�RI |)V I(x)dx

�

.

(21b)

In the above, the minimization over V I represents a si-multaneous minimization over all electrostatic potentialscorresponding to I = 1, 2, . . . ,M . We note that, whilethe above reformulation of electrostatic interactions hasbeen developed for pseudopotential calculations, this canalso be extended to all-electron calculations in a straight-forward manner by using V J

ps = V J˜�and ZI to be the total

nuclear charge in the above expressions. Thus, this lo-cal reformulation provides a unified framework for bothpseudopotential as well as all-electron DFT calculations.

We now consider the local reformulation of the ex-tended interactions in the kernel energy component ofthe WGC orbital-free kinetic energy functional (cf. (9)).Here we adopt the recently developed local real-space re-formulation of the kernel energy30,31, and recall the keyideas and local reformulation for the sake of complete-ness. We present the local reformulation of K

0

and thelocal reformulations for other kernels (K

1

, K11

, K12

) fol-lows along similar lines. Consider the kernel energy cor-responding to K

0

given by

TK0(⇢) = CF

Z Z

⇢↵(x)K0

(|x� x

0|) ⇢�(x0) dx dx0 .

(22)

We define potentials v0↵ and v0� given by

v0↵(x) =

Z

K0

(|x� x

0|)⇢↵(x0)dx0 ,

v0�(x) =

Z

K0

(|x� x

0|)⇢�(x0)dx0 . (23)

Taking the Fourier transform of the above expressions weobtain

cv0↵(k) = cK0

(|k|)c⇢↵(k) ,cv0�(k) =

cK0

(|k|)c⇢�(k) . (24)

Following the ideas developed by Choly & Kaxiras44, cK0

can be approximated to very good accuracy by using asum of partial fractions of the following form

cK0

(|k|) ⇡mX

j=1

Aj |k|2|k|2 +Bj

, (25)

where Aj , Bj , j = 1 . . .m are constants, possibly com-plex, that are determined using a best fit approximation.Using this approximation and taking the inverse Fouriertransform of equation (24), the potentials in equation

7

(23) reduce to

v0↵(x) =mX

j=1

[!0

↵j(x) +Aj⇢

↵(x)] ,

v0�(x) =mX

j=1

[!0

�j(x) +Aj⇢

�(x)] . (26)

where !0

↵j(x) and !0

�j(x) for j = 1 . . .m are given by the

following Helmholtz equations:

�r2!0

↵j+Bj!

0

↵j+AjBj⇢

↵ = 0 ,

�r2!0

�j+Bj!

0

�j+AjBj⇢

� = 0 . (27)

We refer to these auxiliary potentials, !0

↵ ={!0

↵1. . .!0

↵m} and !0

� = {!0

�1. . .!0

�m} introduced in the

local reformulation of the kernel energy as kernel poten-tials. Expressing the Helmholtz equations in a variationalform, we reformulate TK0 in (22) as the following localvariational problem in kernel potentials:

TK0(⇢) = min!0

↵j2Y

max!0

�j2Y

LK0(!0

↵,!0

� , ⇢) , (28a)

LK0(!0

↵,!0

� , ⇢) =mX

j=1

CF

n

Z

⇥ 1

AjBjr!0

↵j(x) ·r!0

�j(x)

+1

Aj!0

↵j(x)!0

�j(x) + !0

�j(x)⇢↵(x) + !0

↵j(x)⇢�(x)

+Aj⇢(↵+�)(x)

⇤

dxo

.

(28b)

The variational problem in equation (28) represents asimultaneous saddle point problem on kernel potentials!0

↵jand !0

�jfor j = 1, . . . ,m. Following a similar pro-

cedure, we construct the local variational reformulationsfor the kernel energies TK1 , TK11 and TK12 correspondingto kernels K

1

, K11

and K12

, respectively. We denote byLK1(!

1

↵,!1

� , ⇢), LK11(!11

↵ ,!11

� , ⇢) and LK12(!12

↵ ,!12

� , ⇢)the Lagrangians with respective kernel potentials corre-sponding to kernel energies of K

1

, K11

and K12

, respec-tively. We refer to the supplemental material for thenumerical details of the approximations for each of thekernels used in the present work.Finally, using the local variational reformulations of

the extended electrostatic and kernel energies, the prob-lem of computing the ground-state energy for a given po-sitions of atoms is given by the following local variationalproblem in electron-density, electrostatic potentials, andkernel potentials:

E0

(R) = minp⇢2X

max�2Y

min!s

↵j2Y

max!s

�j2Y

n

CF

Z

⇢(x)5/3 dx+1

2

Z

|rp

⇢(x)|2 dx+

Z

"xc

(⇢)⇢(x) dx

+X

s

LKs(!s↵,!

s� , ⇢) + min

V I2H1(R3

)

Lel(�,V, ⇢,R)o

.(29)

In the above, s denotes the index corresponding to akernel, and X and Y are suitable function spaces cor-responding to the boundary conditions of the problem.In particular, for periodic problems, Y = H1

per(Q) andX = {p⇢|p⇢ 2 H1

per(Q),R

⇢ = N}. It is convenient to

use the substitution u(x) =p

⇢(x), and enforce the in-

tegral constraint in X using a Lagrange multiplier. Also,for the sake of notational simplicity, we will denote by !↵and !� the array of kernel potentials {!0

↵,!1

↵,!11

↵ ,!12

↵ }and {!0

� ,!1

� ,!11

� ,!12

� }, respectively. Subsequently, thevariational problem in equation (29) can be expressed as

E0

(R) = minu2Y

max�2Y

min!s

↵j2Y

max!s

�j2Y

L(u,�,!↵,!� ;R) subject to :

Z

u2(x) dx = N , (30)

L(u,�,!↵,!� ;R) = L(u) + LK(!↵,!� , u2) + Lc(u,�) + min

V I2H1(R3

)

Lel(�,V, u2,R) ,

L(u) = CF

Z

u10/3(x) dx+1

2

Z

|ru(x)|2 dx+

Z

"xc

(u2)u2(x) dx ,

LK(!↵,!� , u2) =

X

s

LKs(!s↵,!

s� , u

2) ,

Lc(u,�) = �

✓

Z

u2(x) dx�N

◆

.

8

B. Configurational forces

We now turn our attention to the configurationalforces corresponding to geometry optimization. To thisend, we employ the approach of inner variations, wherewe evaluate the generalized forces corresponding to per-turbations of underlying space, which provides a uni-fied expression for the generalized force correspondingto the geometry of the simulation cell—internal atomicpositions, as well as, the external cell domain. We con-sider infinitesimal perturbations of the underlying space ✏ : R3 ! R3 corresponding to a generator �(x) given

by � = d ✏(x)

d✏ |✏=0

such that 0

= I. We constrain thegenerator � such that it only admits rigid body deforma-tions in the compact support of the regularized nuclearcharge distribution ⇢nu in order to preserve the integralconstraint

R

�(x � RI)dx = 1. Let x denote a pointin Q, whose image in Q0 = ✏(Q) is x

0 = ✏(x). Theground-state energy on Q0 is given by

E0

( ✏) = L✏(u✏,�✏,!↵✏,!�✏;R✏) (31)

where u✏, �✏, !↵✏ and !�✏ are solutions of the sad-dle point variational problem given by equation (30)evaluated over the function space Y 0 = H1

per(Q0).

The subscript ✏ on L is used to denote that thevariational problem is solved on Q0 = ✏(Q). Forthe sake of convenience, we will represent the in-

tegrand of the Lagrangian L in equation (30) byf(u,ru,�,r�,!↵,r!↵,!� ,r!� ;Vps, V˜�,R) andg(V I

˜�,rV I

˜�;R), where f denotes the integrand whose

integrals are over Q and g denotes the integrand whoseintegrals are over R3. The ground-state energy on Q0 interms of f and g can be expressed as

E0

( ✏) =

Z

Q0f(u✏(x

0),rx

0u✏(x0),�✏(x

0),rx

0�✏(x0),!↵✏(x

0),

rx

0!↵✏(x0),!�✏(x

0),rx

0!�✏(x0);Vps(x

0), V˜�(x

0), ✏(R))dx0

+X

I

Z

R3

g(V I˜�✏(x0),r

x

0 V I˜�✏(x0); ✏(R))dx0 . (32)

Transforming the above integral to domain Q, we obtain

E0

( ✏) =

Z

Q

f(u✏( ✏(x)),rx

u✏( ✏(x)).@x

@x0 ,�✏( ✏(x)),

rx

�✏( ✏(x)).@x

@x0 ,!↵✏( ✏(x)),rx

!↵✏( ✏(x)).@x

@x0 ,!�✏( ✏(x)),

rx

!�✏( ✏(x)).@x

@x0 ;Vps( ✏(x)), V˜�( ✏(x)), ✏(R)) det(@x0

@x) dx

+X

I

Z

R3

g(V I˜�✏( ✏(x)),rx

V I˜�✏( ✏(x)).

@x

@x0 ; ✏(R)) det(@x0

@x) dx

(33)

We now evaluate the configurational force given by theGateaux derivative of E

0

( ✏):

dE0

( ✏)

d✏

�

�

�

✏=0

=

Z

Q

f(u0

(x),ru0

(x),�0

(x),r�0

(x),!↵0

(x),r!↵0

(x),!�0

(x),r!�0

(x);Vps(x), V˜�(x),R)d

d✏(det(

@x0

@x))�

�

�

✏=0

dx

+

Z

Q

@f

@ru(ru

0

)⌦ru0

+@f

@r� (r�0)⌦r�0

+X

s

⇣ @f

@r!s↵

(r!s↵0

)⌦r!s↵0

+@f

@r!s�

(r!s�0

)⌦r!s�0

⌘

!

:

✓

d

d✏

@x

@x0

�

�

�

✏=0

◆

dx

+X

J

Z

Q

u2

0

(x)�rV J

ps(|x�RJ |)�rV J˜�(|x�RJ |)

�

.

✓

d ✏(x)

d✏

�

�

�

✏=0

� d ✏(RJ)

d✏

�

�

�

✏=0

◆

dx

+X

I

Z

R3

g(V I˜�0(x),rV I

˜�0(x);R)

d

d✏(det(

@x0

@x))�

�

�

✏=0

dx+X

I

Z

R3

@g

@rV I˜�

(rV I˜�0)⌦rV I

˜�0:

✓

d

d✏

@x

@x0

�

�

�

✏=0

◆

dx . (34)

In the above, we denote by ‘⌦’ the outer product be-tween two vector, by ‘.’ the dot product between twovectors and by ‘:’ the dot product between two tensors.We note that in the above expression there are no termsinvolving the explicit derivatives of f and g with respectto R as �(|x0 � ✏(R)|) = �(|x�R|), which follows fromthe restriction that ✏ corresponds to rigid body defor-mations in the compact support of ⇢nu. We further notethat terms arising from the inner variations of E

0

( ✏)with respect to u✏, �✏, !↵✏, !�✏ and V I

˜�✏vanish as u

0

�0

,

!↵0

, !�0

and V I˜�0

are the solutions of the saddle point

variational problem corresponding to E0

( 0

). We now

note the following identities

d

d✏

(

@xi

@x0j

)

�

�

�

✏=0

=� @xi

@x0k

⇣ d

d✏

⇢

@ ✏k@xl

�

⌘ @xl

@x0j

�

�

�

✏=0

=� @�i

@xj,

(35)

d

d✏

⇢

det� @x0

l

@xm

�

�

�

�

�

✏=0

=det� @x0

l

@xm

�@xj

@x0i

⇣ d

d✏

⇢

@ ✏i@xj

�

⌘

�

�

�

✏=0

=@�j

@xj.

(36)

9

Using these identities in equation (34), and rearrangingterms, we arrive at

dE0

( ✏)

d✏

�

�

�

✏=0

=

Z

Q

E : r�(x) dx+X

I

Z

R3

E

0

I : r�(x) dx

+X

J

Z

Q

u2

0

(x)�r�V J

ps � V J˜�

��

. (�(x)� �(RJ)) dx (37)

where E and E

0 denote Eshelby tensors corresponding tof and g, respectively. The expressions for the Eshelbytensors E and E

0I explicitly in terms of u, �, !↵, !� , Vps

and V˜� are given by

E =

CFu10/3 +

1

2|ru|2 + "xc(u

2)u2 + �u2 � 1

8⇡|r�|2 + u2�+

X

J

�

V Jps � V J

˜�

�

u2 +X

s

fKs(!s↵,r!s

↵,!s� ,r!s

� , u2)

!

I

�ru⌦ru+1

4⇡r�⌦r��

X

s

@fKs

@r!s↵

⌦r!s↵ +

@fKs

@r!s�

⌦r!s�

!

(38)

E

0

I =1

8⇡|rV I

˜�|2I� 1

4⇡rV I

˜�⌦rV I

˜�(39)

In the above, for the sake of brevity, we represented byfKs the integrand corresponding to LKs . We also notethat the terms �⇢nu and V I

˜��(x �RI) do not appear in

the expressions for E and E

0

I , respectively, as r.� = 0on the compact support of ⇢nu owing to the restrictionthat � corresponds to rigid body deformations in theseregions. It may appear that evaluation of the secondterm in equation (37) is not tractable as it involves anintegral over R3. To this end, we split this integral ona bounded domain ⌦ containing the compact support of�(x � RI), and its complement. The integral on R3/⌦can be computed as a surface integral. Thus,Z

R3

E

0

I : r� dx =

Z

⌦

E

0

I : r� dx+

Z

R3/⌦

E

0

I : r� dx

=

Z

⌦

E

0

I : r� dx�Z

@⌦

E

0

I : n⌦ � ds , (40)

where n denotes the outward normal to the surface @⌦.The last equality follows from the fact that r2V I

˜�= 0 on

R3/⌦.

The configurational force in equation (37) provides thegeneralized variational force with respect to both the in-ternal positions of atoms as well as the external cell do-main. In order to compute the force on any given atom,we restrict the compact support of � to only include theatom of interest. In order to compute the stresses asso-ciated with cell relaxation (keeping the fractional coor-dinates of atoms fixed), we restrict � to a�ne deforma-tions. Thus, this provides a unified expression for geom-etry optimization corresponding to both internal ionicrelaxations as well as cell relaxation. We further notethat, while we derived the configurational force for thecase of pseudopotential calculations, the derived expres-sion is equally applicable for all-electron calculations byusing V J

ps = V J˜�.

C. Finite-element discretization

Among numerical discretization techniques, the plane-wave discretization has been the most popular and widelyused in orbital-free DFT27,28 as it naturally lends itselfto the evaluation of the extended interactions in electro-static energy and kernel kinetic energy functionals usingFourier transforms. Further, the plane wave basis o↵erssystematic convergence with exponential convergence inthe number of basis functions. However, as noted pre-viously, the plane-wave basis also su↵ers from notabledrawbacks. Importantly, plane-wave discretization is re-stricted to periodic geometries and boundary conditionswhich introduces a significant limitation, especially in thestudy of defects in bulk materials26. Further, the plane-wave basis has a uniform spatial resolution, and thus isnot amenable to adaptive coarse-graining. Moreover, theuse of plane-wave discretization involves the numericalevaluation of Fourier transforms whose scalability is lim-ited on parallel computing platforms.

In order to circumvent these limitations of the plane-wave basis, there is an increasing focus on developingreal-space discretization techniques for orbital-free DFTbased on finite-di↵erence discretization32–34 and finite-element discretization29,31. In particular, the finite-element basis46, which is a piecewise continuous poly-nomial basis, has many features of a desirable basis inelectronic structure calculations. While being a com-plete basis, the finite-element basis naturally allows forthe consideration of complex geometries and boundaryconditions, is amenable to unstructured coarse-graining,and exhibits good scalability on massively parallel com-puting platforms. Moreover, the adaptive nature of thefinite-element discretization also enables the considera-tion of all-electron orbital-free DFT calculations that arewidely used in studies of warm dense matter16,17,19. Fur-ther, recent numerical studies have shown that by using ahigher-order finite-element discretization significant com-putational savings can be realized for both orbital-free

10

DFT31 and Kohn-Sham DFT calculations6,47, e↵ectivelyovercoming the degree of freedom disadvantage of thefinite-element basis in comparison to the plane-wave ba-sis.

Let Yh denote the finite-element subspace of Y, whereh represents the finite-element mesh size. The discreteproblem of computing the ground-state energy for a givenpositions of atoms, corresponding to equation (30), isgiven by the constrained variational problem:

E0

(R) = minuh2Yh

max�h2Yh

min!s

↵j h2Yh

max!s

�j h2Yh

L(uh,�h,!↵h ,!�h ;R)

subject to :

Z

u2

h(x) dx = N . (41)

In the above, uh, �h, !↵h and !�h denote the finite-element discretized fields corresponding to square-rootelectron-density, electrostatic potential, and kernel po-tentials, respectively. We restrict our finite-element dis-cretization such that atoms are located on the nodesof the finite-element mesh. In order to compute thefinite-element discretized solution of V J

˜�, we represent

�(x � RJ) as a point charge on the finite-element nodelocated at RJ , and the finite-element discretization pro-vides a regularization for V J

˜�. Previous investigations

have suggested that such an approach provides optimalrates of convergence of the ground-state energy (cf.31,47

for a discussion).The finite-element basis functions also provide the gen-

erator of the deformations of the underlying space in theisoparametric formulation, where the same finite-elementshape functions are used to discretize both the spatialdomain as well as the fields prescribed over the domain.Thus, the configurational force associated with the loca-tion of any node in the finite-element mesh can be com-puted by substituting for �, in equation (37), the finite-element shape function associated with the node. Thus,the configurational force on any finite-element node lo-cated at an atom location corresponds to the variationalionic force, which are used to drive the internal atomicrelaxation. The forces on the finite-element nodes thatdo not correspond to an atom location represent the gen-eralized force of the energy with respect to the location ofthe finite-element nodes, and these can be used to obtainthe optimal location of the finite-element nodes—a basisadaptation technique.

We note that the local real-space variational formula-tion in section IIIA, where the extended interactions inthe electrostatic energy and kernel functionals are refor-mulated as local variational problems, is essential for thefinite-element discretization of the formulation.

IV. NUMERICAL IMPLEMENTATION

In this section, we present the details of the numeri-cal implementation of the finite-element discretization ofthe real-space formulation of orbital-free DFT discussed

in section III. Subsequently, we discuss the solution pro-cedure for the resulting discrete coupled equations insquare-root electron-density, electrostatic potential andkernel potentials.

A. Finite-element basisA finite-element discretization using linear tetrahedral

finite-elements has been the most widely used discretiza-tion technique for a wide range of partial di↵erentialequations. Linear tetrahedral elements are well suitedfor problems involving complex geometries and moder-ate levels of accuracy. However in electronic structurecalculations, where the desired accuracy is commensu-rate with chemical accuracy, linear finite elements arecomputationally ine�cient requiring of the order of hun-dred thousand basis functions per atom to achieve chem-ical accuracy. A recent study31 has demonstrated thesignificant computational savings—of the order of 1000-fold compared to linear finite-elements—that can be real-ized by using higher-order finite-element discretizations.Thus, in the present work we use higher-order hexahedralfinite elements, where the basis functions are constructedas a tensor product of basis functions in one-dimension46.

B. Solution procedureThe discrete variational problem in equation (41) in-

volves the computation of the following fields—square-root electron-density, electrostatic potential and kernelpotentials. Two solution procedures, suggested in priore↵orts31, for solving this discrete variational problem in-clude: (i) a simultaneous solution of all the discrete fieldsin the problem; (ii) a nested solution procedure, wherefor every trial square-root electron-density the discreteelectrostatic and kernel potential fields are computed.Given the non-linear nature of the problem, the simul-taneous approach is very sensitive to the starting guessand often su↵ers from lack of robust convergence, espe-cially for large-scale problems. The nested solution ap-proach, on the other hand, while constituting a robustsolution procedure, is computationally ine�cient due tothe huge computational costs incurred in computing thekernel potentials which involves the solution of a seriesof Helmholtz equations (cf. equation (27)). Thus, in thepresent work, we will recast the local variational prob-lem in equation (41) as the following fixed point iterationproblem:

{uh, �h} = arg minuh

arg max�h

L(uh,�h, !↵h , !�h ;R)

subject to :

Z

u2

h(x) dx = N. (42a)

{!↵h , !�h} = arg min!↵h

arg max!�h

L(uh, �h,!↵h ,!�h ;R) .

(42b)We solve this fixed point iteration problem using a mix-ing scheme, and, in particular, we employ the Anderson

11

mixing scheme48 with full history in this work. Our nu-merical investigations suggest that the fixed point itera-tion converges, typically, in less than ten self-consistentiterations even for large-scale problems, thus, provid-ing a numerically e�cient and robust solution procedurefor the solution of the local variational orbital-free DFTproblem. We note that this idea of fixed point iterationhas independently and simultaneously been investigatedby another group in the context of finite di↵erence dis-cretization34, and have resulted in similar findings.

In the fixed point iteration problem, we employ a si-multaneous solution procedure to solve the non-linearsaddle point variational problem in uh and �h (equa-tion (42a)). We employ an inexact Newton solver pro-vided by the PETSc package49 with field split precondi-tioning and generalized-minimal residual method (GM-RES)50 as the linear solver. The discrete Helmholtz equa-tions in equation (42b) are solved by employing blockJacobi preconditioning and using GMRES as the linearsolver. An e�cient and scalable parallel implementationof the solution procedure has been developed to take ad-vantage of the parallel computing resources for conduct-ing the large-scale simulations reported in this work.

V. RESULTS AND DISCUSSION

In this section, we discuss the numerical studies onAl, Mg and Al-Mg intermetallics to investigate theaccuracy and transferability of the real-space formu-lation of orbital-free DFT (RS-OFDFT) proposed insection III. Wherever applicable, we benchmark thereal-space orbital-free DFT calculations with plane-wavebased orbital-free DFT calculations conducted usingPROFESS27, and compare with Kohn-Sham DFT (KS-DFT) calculations conducted using the plane-wave basedABINIT code51,52. Further, we demonstrate the useful-ness of the proposed real-space formulation in studyingthe electronic structure of isolated defects.

A. General calculation details

In all the real-space orbital-free DFT calculations re-ported in this section, we use the local reformulation ofthe density-dependent WGC11 kinetic energy functionalproposed in section IIIA, the local density approxima-tion (LDA)37 for the exchange-correlation energy, andbulk derived local pseudopotentials (BLPS)40 for Al andMg. Cell stresses and ionic forces are calculated using theunified variational formulation of configurational forcesdeveloped in section III B. In the second order Taylor ex-pansion of the density-dependent WGC functional aboutthe bulk electron density (cf. Section II), we only retainthe K

12

term for the computation of bulk properties asthe contributions from K

12

dominate those of K11

forbulk materials systems. However, in the calculations in-volving mono-vacancies, where significant spatial pertur-

FIG. 1: Convergence of the finite-element approximation inthe energy of a fcc Al unit cell with lattice constant a = 7.2

Bohr.

FIG. 2: Convergence of the finite-element approximation inthe hydrostatic stress of a fcc Al unit cell with lattice

constant a = 7.2 Bohr.

bations in the electronic structure are present, we use thefull second order Taylor expansion of the density depen-dent WGC functional. We recall from section IIIA thatin order to obtain a local real-space reformulation of theextended interactions in the kinetic energy functionals,the kernels (K

0

, K1

, K11

, K12

) are approximated usinga sum of m partial fractions where the coe�cients of thepartial fractions are computed using a best fit approxima-tion (cf. equation (25)). These best fit approximationsfor m = 4, 5, 6 that are employed in the present work aregiven in the supplemental material. It has been shown inrecent studies that m = 4 su�ces for Al30,34. However,we find that m = 6 is required to obtain the desired ac-curacy in the bulk properties of Mg, and Table II shows

12

the comparison between the kernel approximation withm = 6 and plane-wave based orbital-free DFT calcula-tions conducted using PROFESS27 for Mg. Thus, we usethe best fit approximation of the kernels with m = 4 forAl, and employ the approximation with m = 6 for Mgand Al-Mg intermetallics. Henceforth, we will refer byRS-OFDFT-FE the real-space orbital-DFT calculationsconducted by employing the local formulation and finiteelement discretization proposed in section III.

The KS-DFT calculations used to assess the accuracyand transferability of the proposed real-space orbital-free DFT formalism are performed using the LDA ex-change correlation functional37. The KS-DFT calcu-lations are conducted using both local BLPS as wellas the non-local Troullier-Martins pseudopotential (TM-NLPS)53 in order to assess the accuracy and transfer-ability of both the model kinetic energy functionals inorbital-free DFT as well as the local pseudopotentials towhich the orbital-free DFT formalism is restricted to.The TM-NLPS for Al and Mg are generated using thefhi98PP code54. Within the fhi98PP code, we use the fol-lowing inputs: 3d angular momentum channel as the localpseudopotential component for both Al and Mg, defaultcore cuto↵ radii for the 3s, 3p, and 3d angular momen-tum channels, which are {1.790, 1.974, 2.124} Bohr and{2.087, 2.476, 2.476} Bohr for Al and Mg respectively,and the LDA37 exchange-correlation. For brevity, hence-forth, we refer to the KS-DFT calculations with BLPSand TM-NLPS as KS-BLPS and KS-NLPS, respectively.

In all the RS-OFDFT-FE calculations reported in thiswork, the finite-element discretization, order of the finite-elements, numerical quadrature rules and stopping tol-erances are chosen such that we obtain 1 meV/ atomaccuracy in energies, 1 ⇥ 10�7 Hartree Bohr�3 accuracyin cell stresses and 1 ⇥ 10�5 Hartree Bohr�1 accuracyin ionic forces. Similar accuracies in energies, stressesand ionic forces are achieved for KS-DFT calculationsby choosing the appropriate k-point mesh, plane-waveenergy cuto↵, and stopping tolerances within ABINIT’sframework. All calculations involving geometry opti-mization are conducted until cell stresses and ionic forcesare below threshold values of 5 ⇥ 10�7 Hartree Bohr�3

and 5⇥ 10�5 Hartree Bohr�1, respectively.

B. Convergence of finite-element discretization

We now study the convergence of energy and stresseswith respect to the finite-element discretization of theproposed real-space orbital-free DFT formulation. In aprior study on the computational e�ciency a↵orded byhigher-order finite-element discretization in orbital-freeDFT31, it was shown that second and third-order finite-elements o↵er an optimal choice between accuracy andcomputational e�ciency. Thus, in the present study,we limit our convergence studies to HEX27 and HEX64finite-elements, which correspond to second- and third-order finite-elements. As a benchmark system, we con-sider a stressed fcc Al unit cell with a lattice constant

a = 7.2 Bohr. We first construct a coarse finite-elementmesh and subsequently perform a uniform subdivision toobtain a sequence of increasingly refined meshes. We de-note by h the measure of the size of the finite-element.For these sequence of meshes, we hold the cell geometryfixed and compute the discrete ground-state energy, Eh,and hydrostatic stress, �h. The extrapolation procedureproposed in Motamarri et. al31 allows us to estimate theground-state energy and hydrostatic stress in the limitas h ! 0, denoted by E

0

and �0

. To this end, the en-ergy and hydrostatic stress computed from the sequenceof meshes using HEX64 finite-elements are fitted to ex-pressions of the form

|E0

� Eh| = Ce

✓

1

Nel

◆

qe3

,

|�0

� �h| = C�

✓

1

Nel

◆

q�3

, (43)

to determine E0

, qe, �0,& q�. In the above expression,Nel denotes the number of elements in a finite-elementmesh. We subsequently use E

0

and �0

as the exact val-ues of the ground-state energy and hydrostatic stress,respectively, for the benchmark system. Figures 1 and 2show the relative errors in energy and hydrostatic stress

plotted against⇣

1

Nel

⌘

13, which represents a measure of

h. We note that the slopes of these curves provide therates of convergence of the finite-element approximationfor energy and stresses. These results show that we ob-tain close to optimal rates of convergence in energy ofO(h2k), where k is polynomial interpolation order (k = 2for HEX27 and k = 3 for HEX64). Further, we obtainclose to O(h2k�1) convergence in the stresses, which rep-resents optimal convergence for stresses. The results alsosuggest that higher accuracies in energy and stress areobtained with HEX64 in comparison to HEX27. Thus,we will employ HEX64 finite-elements for the remainderof our study.

TABLE I: The energy di↵erence in eV between a stablephase and the most stable phase for Al and Mg computed

using RS-OFDFT-FE and KS-DFT with TM-NLPS.

Al fcc hcp bcc sc dia

RS-OFDFT-FE 0a 0.016 0.075 0.339 0.843

KS-NLPS 0 0.038 0.106 0.400 0.819

Mg hcp fcc bcc sc dia

RS-OFDFT-FE 0 0.003 0.019 0.343 0.847

KS-NLPS 0 0.014 0.030 0.400 0.822

a The zero in the first column is to indicate that these numbersare the reference against which energies of other phases aredetermined.

13

TABLE II: Bulk properties of Al and Mg: Equilibrium ground-state energy per atom (Emin in eV), volume per atom (V0 in

A3) and bulk modulus (B0 in GPa) computed using RS-OFDFT-FE, PROFESS, and KS-DFT with BLPS and TM-NLPS.

Ala RS-OFDFT-FE PROFESS KS-BLPS KS-NLPS

Emin

-57.935 -57.936 -57.954 -57.207

V0

15.68 15.68 15.62 15.55

B0

81.7 81.5 84.1 83.6

Mgb RS-OFDFT-FE PROFESS KS-BLPS KS-NLPS

Emin

-24.647 -24.647 -24.678 -24.514

V0

21.40 21.43 21.18 21.26

B0

36.8 36.6 38.5 38.6

a Cell-relaxed lattice constant for fcc Al using RS-OFDFT-FE isa0 = 7.51 Bohr.

b Cell-relaxed lattice constants for hcp Mg using RS-OFDFT-FEare a0 = 5.89 Bohr, c0 = 9.62 Bohr.

C. Bulk properties of Al, Mg and Al-Mgintermetallics

We now study the accuracy and transferability of theproposed real-space formulation of orbital-free DFT forbulk properties of Al, Mg and Al-Mg intermetallics. Tothis end, we begin with the phase stability study of Aland Mg, where we compute the di↵erence in the ground-state energy of a stable phase and the ground-state en-ergy of the most stable phase. The results for Al and Mgare shown in Table I, and are compared against those ob-tained with KS-DFT employing TM-NLPS. We note thatRS-OFDFT-FE correctly predicts the most stable phasesof Al and Mg being fcc and hcp, respectively. Further, thestability ordering of the various phases computed usingRS-OFDFT-FE is consistent with KS-DFT TM-NLPScalculations. Moreover, the energy di↵erences betweenthe various stable phases and the most stable phase com-puted using RS-OFDFT-FE are in close agreement withKS-DFT calculations.

We next consider bulk properties of Al, Mg and Al-Mgintermetallics. To this end, for each system, we first op-timize cell geometry and ionic positions to determine theequilibrium cell structure, equilibrium volume (V

0

) andground-state energy (E

min

). We subsequently computethe bulk modulus given by43

B = V@2E

@V 2

�

�

�

�

V=V0

, (44)

where E denotes the ground-state energy of a unit-cellwith volume V . To compute the bulk modulus, we varythe cell volume by applying a volumetric deformation tothe relaxed (equilibrium) unit-cell, which transforms theequilibrium cell vectors {c

1

, c2

, c3

} to {c01

, c02

, c03

} andare given by

c0ij = cij (1 + ⌘) . (45)

While keeping the cell structure fixed, we calculate the

ground-state energy for each ⌘ between �0.01 to 0.01 insteps of 0.002 and fit a cubic polynomial to the E � Vdata. We subsequently compute the bulk modulus, usingequation (44), at the equilibrium volume, V

0

. The com-puted bulk properties—ground-state energy, equilibriumvolume and bulk modulus at equilibrium—for Al and Mgare given in Table II, and those of Al-Mg intermetallics(Al

3

Mg, Mg13

Al14

, Mg17

Al12

, and Mg23

Al30

) are givenin Table III. These results suggest that the bulk proper-ties of Al, Mg and Al-Mg intermetallics computed usingRS-OFDFT-FE are in good agreement with PROFESSand KS-DFT calculations.Finally, we consider the formation energies of Al-Mg

intermetallics. In addition to the Al-Mg intermetallicsfor which we computed the bulk properties, we also com-pute the formation energy of the �0 alloy. The �0 alloyhas a disorder in 10 out of 879 sites with each site having0.5 chance of being occupied by either Al or Mg55. Inour simulations, we consider the two limits where all 10sites are occupied by either Al or Mg and refer to these as�0(Al) and �0(Mg), respectively. For these two systems,we do not provide KS-DFT results as they are computa-tionally prohibitive. The formation energies for the rangeof Al-Mg intermetallics are reported in Table IV. Ourresults suggest that the formation energies predicted byRS-OFDFT-FE are in good agreement with PROFESScalculations, and in close agreement with KS-DFT cal-culations.

D. Configurational forces and atomic displacements

As a next step in our study of the accuracy and trans-ferability of RS-OFDFT-FE, we compute the configura-tional forces on atoms that are perturbed from their equi-librium positions and compare these with Kohn-ShamDFT calculations. We investigate the accuracy of theforces in both fcc Al and hcp Mg. We begin by consider-ing the relaxed Al fcc unit cell, and the relaxed Mg hcpunit cell. In the relaxed Al fcc unit cell, we perturb the

14

TABLE III: Bulk properties of Al-Mg intermetallics: Equilibrium ground-state energy per primitive cell (Emin in eV),

volume of primitive cell (V0 in A3), and bulk modulus (B0 in GPa) computed using RS-OFDFT-FE, PROFESS, and KS-DFT

with BLPS and TM-NLPS.

Al3

Mg RS-OFDFT-FE PROFESS KS-BLPS KS-NLPS

Emin

-198.492 -198.496 -198.575 -196.162

V0

67.23 67.31 67.13 66.52

B0

69.2 67.0 67.6 71.0

Mg13

Al14

RS-OFDFT-FE PROFESS KS-BLPS KS-NLPS

Emin

-1130.083 -1130.100 -1130.972 -1117.936

V0

494.77 494.73 498.19 492.73

B0

53.1 52.1 54.7 54.8

Mg17

Al12


Emin

-1114.446 -1114.526 -1116.185 -1104.012

V0

545.32 544.85 543.67 544.21

B0

51.1 52.3 55.2 54.4

Mg23

Al30


Emin

-2306.785 -2306.762 -2307.989 -2281.082

V0

953.87 952.55 963.72 957.46

B0

64.2 60.9 60.5 60.5

TABLE IV: Formation energy per atom (eV/atom) of Al-Mg intermetallics calculated using RS-OFDFT-FE, PROFESS,and KS-DFT with TM-NLPS.

Method Al3

Mg Mg13

Al14

Mg17

Al12

Mg23

Al30

�0(Al) �0(Mg)

RS-OFDFT-FE -0.010 0.053 -0.008 -0.035 -0.026 -0.020

PROFESS -0.011 0.052 -0.011 -0.034 -0.029 -0.023

KS-NLPS -0.007 0.061 -0.027 -0.019 - -

face-centered atom with fractional coordinates 0, 1

2

, 1

2

by 0.1 Bohr in the [0 1 0] direction. In the relaxed Mg hcpunit cell, we perturb the atom with fractional coordinates2

3

, 1

3

, 1

2

by 0.1 Bohr in the [2 1 3 0] direction (directionsin hcp Mg are represented using Miller-Bravais indices).The configurational forces on the perturbed atoms arecomputed using RS-OFDFT-FE, and compared againstKS-DFT calculations. The computed restoring forces,along [0 1 0] for the Al system and along [2 1 3 0] forthe Mg system, are reported in Table V. We note thatthe computed restoring forces from RS-OFDFT-FE arein good agreement with PROFESS and KS-DFT calcu-lations.

As a more stringent test of accuracy and transferabil-ity, we consider the atomic relaxations around a mono-vacancy in fcc Al and hcp Mg. In the case of mono-vacancy in Al, we consider a supercell containing 3⇥3⇥3fcc Al unit cells and remove an atom to create a mono-vacancy. We calculate the forces on the neighboringatoms of the mono-vacancy, and their relaxation displace-ments upon ionic relaxation using both RS-OFDFT-FEand KS-DFT calculations. Periodic boundary conditions

are employed in these calculations. Table VI reports thecomputed force and relaxation displacement in Al on thenearest neighboring atom, which experiences the largestionic force and relaxation. In the case of a mono-vacancyin Mg, we consider a supercell containing 3 ⇥ 3 ⇥ 2 hcpunit cells, and Table VII reports the ionic force and relax-ation displacement on the neighboring atom that has thelargest force in the presence of the vacancy. As is evidentfrom the results, the ionic forces and relaxed displace-ments for a mono-vacancy in Al and Mg computed usingRS-OFDFT-FE are in good agreement with PROFESS,and in close agreement with KS-DFT calculations. Theseresults suggest that the proposed real-space orbital-freeDFT formulation provides a good approximation to KS-DFT for Al-Mg materials systems.

E. Cell-size studies on a mono-vacancy in Al

Prior Fourier-space calculations using OF-DFT andWGC Functional22, and KS-DFT calculations56 havesuggested that cell-sizes containing ⇠ 256 lattice sitesare su�cient to obtain a well-converged (to within 3

15

TABLE V: Restoring force (eV/Bohr) on the perturbedatom in fcc Al and hcp Mg unit cells computed usingRS-OFDFT-FE, PROFESS, and KS-DFT calculations.


Al 0.148 0.137 0.134 0.126

Mg 0.019 0.019 0.018 0.019

TABLE VI: Ionic forces (eV/Bohr) and relaxationdisplacement (Bohr) on the nearest neighboring atom to amono-vacancy in a periodic 3⇥ 3⇥ 3 fcc Al supercell,

calculated using RS-OFDFT-FE, PROFESS, and KS-DFT.f and d denote the magnitudes of ionic force and relaxationdisplacement. \f and \d denote the angles (in degrees) ofthe force and displacement vectors with respect to the

KS-NLPS force and displacement vectors.


f 0.141 0.146 0.130 0.119

d 9.90⇥10�2 9.75⇥10�2 9.47⇥10�2 8.90⇥10�2

\f 0.00 0.00 0.00 0.00

\d 0.15 0.00 0.00 0.00

meV) mono-vacancy formation energy in fcc Al. TheseFourier-space calculations, which employ periodic bound-ary conditions, compute the properties of a periodic ar-ray of vacancies. On the other hand, real-space calcu-lations on isolated mono-vacancies in bulk, computedusing the recently developed coarse-graining techniquesfor orbital-free DFT30,45, suggest that cell-size e↵ects inmono-vacancy calculations are present up to cell-sizes of⇠ 103 atoms. Although both approaches give similarconverged vacancy formation energies, this discrepancyin the cell-size e↵ects has thus far remained an open ques-tion.

In order to understand the source of this discrepancy,we conduct a cell-size study of the mono-vacancy forma-tion energy in Al using RS-OFDFT-FE with two typesof boundary conditions: (i) periodic boundary conditionson electronic fields; (ii) Dirichlet boundary conditionson electronic fields with values corresponding to that of

TABLE VII: Ionic forces (eV/Bohr) and relaxationdisplacement (Bohr) on the nearest neighboring atom to amono-vacancy in a periodic 3⇥ 3⇥ 2 hcp Mg supercell,

calculated using RS-OFDFT-FE, PROFESS, and KS-DFT.


f 0.059 0.060 0.053 0.046

d 8.26⇥10�2 8.64⇥10�2 7.00⇥10�2 5.83⇥10�2

\f 5.11 4.73 2.75 0.0

\d 5.66 5.27 3.58 0.0

a perfect crystal. These Dirichlet boundary conditions,which we refer to as bulk Dirichlet boundary conditions,correspond to the scenario where perturbations in theelectronic structure due to the mono-vacancy vanish onthe boundary of the computational domain, and the elec-tronic structure beyond the computational domain cor-responds to that of the bulk. We note that periodicboundary conditions mimic the widely used Fourier-spacecalculations on point defects, whereas the bulk Dirich-let boundary conditions correspond to simulating an iso-lated point defect embedded in bulk. We note that thelocal real-space formulation of orbital-free DFT and thefinite-element basis are key to being able to consider theseboundary conditions.

We compute the vacancy formation at constant volumeas43,57

Evf = E

✓

N � 1, 1,N � 1

N⌦

◆

� N � 1

NE (N, 0,⌦) ,

(46)where E (N, 0,⌦) denotes the energy of perfect crys-tal containing N atoms occupying a volume ⌦, andE(N � 1, 1, N�1

N ⌦) denotes energy of a computationalcell containing N � 1 atoms and one vacancy occupyinga volume N�1

N ⌦. For both periodic boundary conditionsand bulk Dirichlet boundary conditions, the lattice sitewhere the vacancy is created is chosen to be the farthestsite from the domain boundary. As we are primarily in-terested in the cell-size e↵ects of the electronic structure,we do not consider ionic relaxations in this part of ourstudy. Table VIII shows the unrelaxed mono-vacancyformation energies for di↵erent cell sizes computed usingRS-OFDFT-FE using both periodic boundary conditionsand bulk Dirichlet boundary conditions. We note thatthe mono-vacancy formation energies using both sets ofboundary conditions converge to the same value, and thisis also in good agreement with PROFESS and KS-DFTcalculations (cf. Table IX). However, it is interesting tonote that the mono-vacancy formation energies with pe-riodic boundary conditions are well converged (to within10 meV) by 3 ⇥ 3 ⇥ 3 cell-size (108 atoms), whereas werequired a 6 ⇥ 6 ⇥ 6 cell-size (864 atoms) to achieve aconverged formation energy with bulk Dirichlet bound-ary conditions.

In order to understand this boundary condition depen-dence of the cell-size e↵ects, we compute the perturba-tions in the electronic fields due to the presence of themono-vacancy by subtracting from the electronic fieldscorresponding to the mono-vacancy the electronic fieldsof a perfect crystal. To this end, we define the normal-ized perturbations in the electronic fields computed on

16

TABLE VIII: Unrelaxed mono-vacancy formation energiesfor Al computed using RS-OFDFT-FE with periodicboundary conditions (Ep

vf in eV) and bulk Dirichlet

boundary conditions (EbDvf in eV).

Cell size N EbDvf Ep

vf

2x2x2 32 -0.390 0.955

3x3x3 108 0.864 0.915

4x4x4 256 0.971 0.908

5x5x5 500 0.944 -

6x6x6 864 0.918 -

7x7x7 1372 0.914 -

TABLE IX: Unrelaxed mono-vacancy formation energies(Evf in eV) for Al computed using PROFESS27, and

KS-DFT on a 3⇥ 3⇥ 3 computational cell.

Evf

PROFESS 0.903

KS-DFT-BLPS 0.815

KS-DFT-NLPS 0.811

the finite-element mesh to be

uch =(uh � up

h) /vav (uph) ,

�ch =(�h � �ph) /vav (�ph) ,

kc↵,h =

0

@

mX

j=1

!↵j ,h �mX

j=1

!p↵j ,h

1

A /vav

0

@

mX

j=1

!p↵j ,h

1

A ,

kc�,h =

0

@

mX

j=1

!�j ,h �mX

j=1

!p�j ,h

1

A /vav

0

@

mX

j=1

!p�j ,h

1

A .

(47)

In the above, {uh,�h,!↵j ,h,!�j ,h} and{up

h,�ph,!

p↵j ,h

,!p�j ,h

} denote the electronic fields inthe computational domain with the vacancy and thosewithout the vacancy (perfect crystal), respectively.vav

(.) denotes the volume average of an electronic fieldover the computational cell. As a representative metric,in the definition of kc↵,h and kc�,h we only consider thekernel potentials corresponding to K

0

. Figures 3 and 4shows the normalized corrector fields for the mono-vacancy, computed using periodic boundary conditions,along the face-diagonal of the periodic boundary. It isinteresting to note from these results that the pertur-bations in the electronic structure due to the vacancyare significant up to 6 ⇥ 6 ⇥ 6 computational cells.Thus, although the vacancy formation energy appearsconverged by 3 ⇥ 3 ⇥ 3 computational cell while usingperiodic boundary conditions, the electronic fields arenot converged till a cell-size of 6 ⇥ 6 ⇥ 6 computational

cell. On the other hand, the cell-size convergence inmono-vacancy formation energy suggested by the bulkDirichlet boundary conditions is inline with the conver-gence of electronic fields. These results unambiguouslydemonstrate that the cell-size e↵ects in the electronicstructure of defects are larger than those suggested bya cell-size study of defect formation energies employingperiodic boundary conditions. Using bulk Dirichletboundary conditions for the cell-size study of defectformation energies provides a more accurate estimate ofthe cell-size e↵ects in the electronic structure of defects,and the extent of electronic structure perturbations dueto a defect. Further, while periodic boundary conditionsare limited to the study of point defects, bulk Dirichletboundary conditions can be used to also study defectslike isolated dislocations26, whose geometry does notadmit periodic boundary conditions.

VI. SUMMARY

We have developed a local real-space formulation oforbital-free DFT with WGC kinetic energy functionals byreformulating the extended interactions in electrostaticand kinetic energy functionals as local variational prob-lems in auxiliary potentials. The proposed real-spaceformulation readily extends to all-electron orbital-freeDFT calculations that are commonly employed in warmdense matter calculations. Building on the proposedreal-space formulation we have developed a unified vari-ational framework for computing configurational forcesassociated with both ionic and cell relaxations. Fur-ther, we also proposed a numerically e�cient approachfor the solution of ground-state orbital-free DFT prob-lem, by recasting the local saddle point problem in theelectronic fields—electron density and auxiliary potentialfields—as a fixed point iteration problem and employinga self-consistent iteration procedure. We have employeda finite-element basis for the numerical discretization ofthe proposed real-space formulation of orbital-free DFT.Our numerical convergence studies indicate that we ob-tain close to optimal rates of convergence in both ground-state energy and configurational forces with respect tothe finite-element discretization.We subsequently investigated the accuracy and trans-

ferability of the proposed real-space formulation oforbital-free DFT for Al-Mg materials system. To thisend, we conducted a wide range of studies on Al, Mgand Al-Mg intermetallics, including computation of bulkproperties for these systems, formation energies of Al-Mgintermetallics, and ionic forces in bulk and in the pres-ence of point defects. Our studies indicate that orbital-free DFT and the proposed real-space formulation is ingood agreement with Kohn-Sham DFT calculations usingboth local pseudopotentials as well as non-local pseud-potentials, thus providing an alternate linear-scaling ap-proach for electronic structure studies in Al-Mg materi-als system. We finally investigated the cell-size e↵ects

17

FIG. 3: Normalized corrector fields for a mono-vacancy, computed with periodic boundary conditions, along the facediagonal on the computational domain boundary. The abscissa d represents a normalized coordinate along the face diagonal.

Results for computational cell sizes from 2⇥ 2⇥ 2 to 4⇥ 4⇥ 4 are shown.

in the electronic structure of a mono-vacancy in Al, anddemonstrated that the cell-size convergence in the va-cancy formation energy computed by employing periodicboundary conditions is not commensurate with the con-vergence of the electronic fields. On the other hand, thetrue cell-size e↵ects in the electronic structure are re-vealed by employing the bulk Dirichlet boundary con-ditions, where the perturbations in the electronic fieldsdue to the defect vanish on the boundary of the com-putational domain. Our studies indicate that the truecell-size e↵ects are much larger than those suggested byperiodic calculations even for simple defects like pointdefects. We note that the proposed real-space formula-tion and the finite-element basis are crucial to employingthe bulk Dirichlet boundary conditions that are otherwiseinaccessible using Fourier based formulations. The pro-posed formulation, besides being amenable to complexgeometries, boundary conditions, and providing excellentscalability on parallel computing platforms, also enablescoarse-graining techniques like the quasi-continuum re-duction45,58 to conduct large-scale electronic structure

calculations on the energetics of extended defects in Al-Mg materials system, and is an important direction forfuture studies.

ACKNOWLEDGMENTS

We gratefully acknowledge the support from the U.S.Department of Energy, O�ce of Basic Energy Sciences,Division of Materials Science and Engineering underAward No. DE-SC0008637 that funds the Predictive In-tegrated Structural Materials Science (PRISMS) centerat University of Michigan, under the auspices of whichthis work was performed. V.G. also gratefully acknowl-edges the hospitality of the Division of Engineering andApplied Sciences at the California Institute of Technol-ogy while completing this work. We also acknowledgeAdvanced Research Computing at University of Michi-gan for providing the computing resources through theFlux computing platform.

18

FIG. 4: Normalized corrector fields for a mono-vacancy, computed with periodic boundary conditions, along the facediagonal on the computational domain boundary, for cell sizes ranging from 5⇥ 5⇥ 5 to 7⇥ 7⇥ 7.

19

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Supplemental Material for “Real-space formulation of orbital-free

density functional theory using finite-element discretization: The

case for Al, Mg, and Al-Mg intermetallics”

Sambit Das, Mrinal Iyer, and Vikram Gavini

Department of Mechanical Engineering,

University of Michigan, Ann Arbor, Michigan 48109, USA

1

I. PARTIAL FRACTION APPROXIMATIONS OF WGC KERNELS

We present the partial fraction approximations of the K0, K1, K11 and K12 kernel

terms arising from the second order Taylor series expansion of the WGC [1] kinetic energy

functional (cf. equations (9) and (25) in the main manuscript) that are central to the local

reformulation of the extended interactions in the kinetic energy functional. Following the

procedure by Choly and Kaxiras [2], we seek to fit with a sum of partial fractions the Fourier

transforms of the kernels given by K0(q), K1(q), K11(q) and K12(q) in a scaled Fourier space

q = k/(2kF ), where kF = (3⇡2⇢0)1/3

and ⇢0 denotes the reference electron density, generally

taken as the bulk average electron density. Taking into account the di↵erence in asymptotic

behavior of these kernels as q ! 1, two forms of partial fraction approximations are chosen,

one for the K0 kernel, and the second for all the remaining kernels. In particular, the forms

of the partial fraction approximations for the kernels of interest are given by

K0(q) =mX

j=1

q2Aj

q2 +Bj, ⇢0K1(q) =

mX

j=1

Aj

q2 +Bj,

⇢02K11(q) =

mX

j=1

Aj

q2 +Bj, ⇢0

2K12(q) =mX

j=1

Aj

q2 +Bj, (1)

where K0(q), K1(q), K11(q) and K12(q) are approximations to K0(q), K1(q), K11(q) and

K12(q), respectively. We note that the Ajs and Bjs for each of the kernels are fitted inde-

pendently, by employing a best fit approximation.

The kernels K0(q), K1(q), K11(q) and K12(q) for the second order Taylor series expansion

of the density dependent WGC functional with parameters {↵, �} = {5/6 +p5/6, 5/6 �

p5/6} and � = 2.7 are computed following the lines of Wang et. al. [1], which involves the

numerical solution of a series of ordinary di↵erential equations. Choly and Kaxiras [2] have

proposed the best fit approximation of these kernels using a sum of four partial fractions,

i.e., m = 4. However, such a fit, while satisfactory for Al is found to be inadequate for

Mg. Thus, we compute the best fit approximation of these kernels for m = 5 and m = 6.

The best approximations for m = 4, m = 5 and m = 6 employed in this work are provided

in Tables I, II and III, respectively. The errors in the best fit approximation are shown in

Figures 1, 2, 3 and 4.

2

TABLE I: Best fit approximation with m = 4 for the WGC kinetic energy functional kernels K0(q),

K1(q), K11(q), and K12(q). Only odd indices are given. The even indices j = 2, and j = 4 satisfy

the relations: A2 = A⇤1, A4 = A⇤

3, B2 = B⇤1 , and B4 = B⇤

3 , where ‘⇤’ is the complex conjugate

symbol.

j=1 j=3

K0 Aj 0.108403 + i0.079657 �0.908403 + i0.439708

Bj �0.470923� i0.465392 0.066051� i0.259678

⇢0K1 Aj �0.030515 + i0.015027 0.028915� i0.008817

Bj �0.597793� i0.294130 �0.087917� i0.164937

⇢02K11 Aj 0.008907� i0.032841 �0.034974 + i0.009116

Bj �0.537986� i0.233840 �0.041565� i0.196662

⇢02K12 Aj 0.012423� i0.034421 �0.031907 + i0.007392

Bj �0.511699� i0.266195 �0.034031� i0.188927

TABLE II: Best fit approximation with m = 5 for the WGC kinetic energy functional kernels

K0(q), K1(q), K11(q), and K12(q). Only odd indices are given. The even indices j = 2, and j = 4

satisfy the relations: A2 = A⇤1, A4 = A⇤

3, B2 = B⇤1 , and B4 = B⇤

3 .

j=1 j=3 j=5

K0 Aj �0.886 + i0.4146 0.08621� i0.09572 �0.0004198 + i0.0

Bj 0.06707� i0.2503 �0.447 + i0.4122 0.2402 + i0.0

⇢0K1 Aj �0.008928� i0.02575 0.03686 + i0.07371 �0.10395 + i0.0

Bj �0.7097 + i0.3011 �0.1313 + i0.3094 0.3323 + i0.0

⇢02K11 Aj 0.02388� i0.03564 �0.01716 + i0.004228 0.02037 + i0.0

Bj �0.2272 + i0.2447 �0.7899 + i0.1762 0.09371 + i0.0

⇢02K12 Aj 0.02387� i0.02923 �0.0172 + i0.002471 0.01633 + i0.0

Bj �0.2334 + i0.2403 �0.7962 + i0.1756 0.09016 + i0.0

3

TABLE III: Best fit approximation with m = 6 for the WGC kinetic energy functional kernels

K0(q), K1(q), K11(q), and K12(q). Only odd indices are given.The even indices j = 2, j = 4, and

j = 6 satisfy the relations: A2 = A⇤1, A4 = A⇤

3, A6 = A⇤5, B2 = B⇤

1 , B4 = B⇤3 , and B6 = B⇤

5 .

j=1 j=3 j=5

K0 Aj 0.09497 + i0.2248 0.01503 + i0.006301 �0.9100 + i0.2338

Bj �0.2711� i0.4679 �0.7524 + i0.3445 0.09512� i0.2323

⇢0K1 Aj 0.004923� i0.007041 �0.05719� i0.009509 0.04444� i0.01249

Bj �0.8584 + i0.222 �0.4359� i0.391 �0.01699 + i0.2098

⇢02K11 Aj �0.006501� i0.007079 0.02922 + i0.03002 �0.02879� i0.005261

Bj �0.9058 + i0.1325 �0.4807 + i0.3047 �0.04111 + i0.171

⇢02K12 Aj �0.006068� i0.007396 �0.02479� i0.006033 0.02405 + i0.0302

Bj �0.9081 + i0.1324 �0.04396 + i0.1684 �0.4863 + i0.3013

FIG. 1: Partial fraction approximation errors for WGC kernel, K0(q).

4



5


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[1] Y. A. Wang, N. Govind, and E. A. Carter, Phys. Rev. B 60, 16350 (1999).

[2] N. Choly and E. Kaxiras, Solid State Communications 121, 281 (2002).

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