Spectrum-splitting approach for Fermi-operator expansion in all-electron Kohn-Sham

DFT calculations

Phani Motamarri and Vikram GaviniDepartment of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, USA

Kaushik Bhattacharya and Michael OrtizDivision of Engineering and Applied Science, California Institute of Technology, Pasadena, CA 91125, USA

We present a spectrum-splitting approach to conduct all-electron Kohn-Sham density functionaltheory (DFT) calculations by employing Fermi-operator expansion of the Kohn-Sham Hamiltonian.The proposed approach splits the subspace containing the occupied eigenspace into a core-subspace,spanned by the core eigenfunctions, and its complement, the valence-subspace, and thereby enablesan e�cient computation of the Fermi-operator expansion by reducing the expansion to the valence-subspace projected Kohn-Sham Hamiltonian. The key ideas used in our approach are: (i) employChebyshev filtering to compute a subspace containing the occupied states followed by a localizationprocedure to generate non-orthogonal localized functions spanning the Chebyshev-filtered subspace;(ii) compute the Kohn-Sham Hamiltonian projected onto the valence-subspace; (iii) employ Fermi-operator expansion in terms of the valence-subspace projected Hamiltonian to compute the densitymatrix, electron-density and band energy. We demonstrate the accuracy and performance of themethod on benchmark materials systems involving silicon nano-clusters up to 1330 electrons, a sin-gle gold atom and a six-atom gold nano-cluster. The benchmark studies on silicon nano-clustersrevealed a staggering five-fold reduction in the Fermi-operator expansion polynomial degree by us-ing the spectrum-splitting approach for accuracies in the ground-state energies of ⇠ 10�4Ha/atomwith respect to reference calculations. Further, numerical investigations on gold suggest that spec-trum splitting is indispensable to achieve meaningful accuracies, while employing Fermi-operatorexpansion.

I. INTRODUCTION

In the present day, quantum-mechanically informedcalculations on ground-state materials properties arereadily possible by means of electronic-structure calcu-lations -via- the Kohn-Sham density functional theory(DFT) framework1,2. Over the past few decades, DFThas been instrumental in providing significant insightsinto materials properties across a range of materials sys-tems. The approach to DFT is based on the key resultby Hohenberg and Kohn, who in their seminal work1

showed that the ground-state properties of a materialssystem can be described by a functional of electron-density, which, to date, remains unknown. This chal-lenge has however been addressed by Kohn and Sham2 inan approximate sense by reducing the many-body prob-lem of interacting electrons into an equivalent problemof non-interacting electrons in an e↵ective mean fieldthat is governed by the electron-density. This e↵ectivesingle-electron formulation accounts for the quantum-mechanical interaction between electrons by means of anunknown exchange-correlation term which is modeled inpractice, and the widely used models have been successfulin predicting a range of properties across various materi-als systems.

The self-consistent field (SCF) approach to solving theKohn-Sham DFT problem involves, in any given SCF it-eration, discretizing the Kohn-Sham eigenvalue problemusing an appropriate basis set followed by the diagonal-ization of the discrete Kohn-Sham Hamiltonian to obtainthe eigenvalues and orthonormal eigenvectors, which are

then used to compute the electron-density. The com-putational complexity of solving the Kohn-Sham DFTproblem using conventional approaches typically scalesas O(M N2) where M denotes the number of basis func-tions and N denotes the system size (number of atomsor number of electrons). As M / N , the computationalcost, which is cubic scaling with system size, becomesprohibitively expensive. This has led to numerous ef-forts focused on the development of methods3,4 aimed atreducing the computational complexity of DFT calcula-tions, which include density matrix minimization meth-ods5,6, divide and conquer methods7–9, Fermi-operatorexpansion techniques10–13, Fermi-operator approxima-tion method14,15, Fermi-operator projection method16,17,orbital minimization approach18–20 and subspace pro-jection type methods21–23. Either the locality of therepresentation of the wavefunctions, or the exponentialdecay of density matrix in real-space is generally ex-ploited in these methods. It has been demonstrated thatthese methods work well for insulating systems exhibitinglinear-scaling with system size. However, the computa-tional complexity of these approaches can deviate signif-icantly from linear-scaling, in practice, for metallic sys-tems. Further, some of the developed techniques5,16,17,21

assume the existence of a band-gap, thus restricting thesetechniques solely to insulating systems.

One reduced-order scaling technique that is equallyapplicable to both insulating and metallic systems atfinite temperatures is the Fermi-operator expansionmethod3,10,11, which computes the finite-temperaturedensity matrix through a Chebyshev polynomial approx-

2

imation of the Fermi-Dirac function of the Kohn-ShamHamiltonian. The width of the eigenspectrum (�E) ofthe discretized Hamiltonian and the smearing parame-ter (� = kB T ) in the Fermi-Dirac function determinethe accuracy of such an expansion. Numerous recent ef-forts12,13,24–26 have focused on developing methods thataim to reduce the number of terms used in the expansionto approximate the Fermi-Dirac function.

While the Fermi-operator expansion method has beenwidely employed in pseudopotential calculations, thereare major challenges in using this technique for all-electron DFT calculations. One of the challenges is thatthe discrete Hamiltonians, employing real-space basissets like finite-elements34 or wavelets27, for all-electronDFT calculations have very large spectral widths ofO(106) due to the refined discretizations that are neededto resolve the rapidly oscillating wavefunctions aroundthe nuclei. The Fermi-operator expansion method is in-accurate and impractical for such large spectral widths.This issue has recently been addressed in Motamarri etal.22 by employing the Fermi-operator expansion on asubspace projected Hamiltonian whose spectral width iscommensurate with that of the occupied eigenspectrum.A second challenge in the case of all-electron calcula-tions arises from the large width of the occupied spec-trum, especially in materials systems involving atomswith large atomic numbers, which can again render theFermi-operator expansion inaccurate. We remark thatthe magnitude of the smallest algebraic eigenvalue growsas O(Z2) where Z denotes the largest atomic numberamong all the atoms comprising the materials system.

In this paper, we overcome these challenges by propos-ing a spectrum-splitting approach for all-electron Kohn-Sham DFT calculations in which the subspace contain-ing the occupied eigenspace is split into a core-subspace,and its complement, referred to as the valence-subspace.Subsequently, the Fermi-operator expansion is e�cientlycalculated by reducing the expansion to the valence-subspace projected Hamiltonian. In this paper, the pro-posed method is discussed in the context of spectralfinite-element discretization. However, the method pre-sented is general enough to be applicable to any real-space numerical discretization employed in the solutionof the all-electron Kohn-Sham DFT problem. The mainideas constituting our approach are: (i) employ a Cheby-shev filter to compute a subspace containing the oc-cupied eigenspace, followed by a localization procedureto generate non-orthogonal localized functions spanningthe Chebyshev-filtered subspace; (ii) compute the pro-jection of the Hamiltonian onto the valence-subspace;(iii) employ the Fermi-operator expansion in terms of thevalence-subspace projected Hamiltonian to compute thedensity matrix, electron-density and band energy.

We begin by presenting an abstract mathematicalframework where the projection of the Hamiltonian ontothe valence-subspace is derived in terms of the pro-jection matrix corresponding to the core-subspace andthe Chebyshev-filtered subspace projected Hamiltonian.

These projections are expressed in a non-orthogonal lo-calized basis spanning the Chebyshev-filtered subspace,which is instrumental in realizing a reduced-order scalingnumerical implementation of DFT21,22 by taking advan-tage of the sparsity of the resulting matrices. We thenderive the expressions for the density matrix and the con-straint on the number of valence electrons in terms ofthe valence-subspace projected Hamiltonian, which aresubsequently used to develop the spectrum-splitting ap-proach for finite-element discretized all-electron Kohn-Sham DFT calculations. To this end, we first repre-sent the Kohn-Sham Hamiltonian and the correspond-ing wavefunctions in the Lowdin orthonormalized finite-element basis constructed using spectral finite-elementsin conjunction with Gauss-Lobatto-Legendre quadraturerules. The self-consistent field (SCF) iteration beginswith the action of a Chebyshev filter on a given ini-tial subspace spanned by localized single-atom wave-functions to compute an approximation to the occu-pied eigenspace of the discretized Kohn-Sham Hamilto-nian. A localization procedure is then employed to con-struct non-orthogonal localized functions spanning theChebyshev-filtered subspace. The Kohn-Sham Hamilto-nian is then projected onto the Chebyshev-filtered sub-space expressed in the localized basis. We subsequentlyemploy a second Chebyshev filtering procedure on thesubspace projected Hamiltonian to compute an approxi-mation to the core-subspace, followed by a localizationprocedure to construct non-orthogonal localized func-tions spanning the core-subspace. These localized func-tions are in turn employed to compute the projectionmatrix corresponding to the core-subspace, which is thenused to evaluate the projected Hamiltonian correspond-ing to the valence-subspace. The Fermi-operator ex-pansion of the discrete Kohn-Sham Hamiltonian, whichis used to compute the finite-temperature density ma-trix and the electron-density, is reduced to an expansionon the valence-subspace projected discrete Hamiltonianwhose spectral width is O(1).

We investigate the accuracy and performance of theproposed method on representative benchmark materialssystems involving silicon nano-clusters up to 1330 elec-trons, gold atom and a six-atom gold nano-cluster. Inthe case of silicon nano-clusters, the proposed spectrum-splitting approach resulted in a staggering five-fold reduc-tion in the Fermi-operator expansion polynomial degreefor desired accuracies of ⇠ 10�4Ha/atom in the ground-state energies. The utility of the spectrum-splitting ap-proach is even more evident in the benchmark calcula-tions on gold, which has a high atomic number (Z=79).Our results indicate that, by employing the spectrum-splitting approach, accuracies of ⇠ 10�4Ha/atom can beachieved using a Fermi-operator expansion polynomialdegree of around 1000. However, if the Fermi-operatorexpansion is employed directly on the Chebyshev-filteredsubspace projected Hamiltonian, even a polynomial de-gree of 2000 resulted in errors of O(1Ha) in ground-stateenergies per atom. This suggests that spectrum-splitting

3

is indispensable while employing Fermi-operator expan-sion in all-electron DFT calculations on materials sys-tems containing atoms with high atomic numbers.

The remainder of the paper is organized as follows.Section II describes the real-space formulation of the all-electron Kohn-Sham DFT problem. Section III presentsthe mathematical framework for spectrum-splitting, andderives the relevant expressions that will be used subse-quently. Section IV describes the various steps involvedin the spectrum-splitting algorithm within the frame-work of spectral finite-element discretization. Section Vdiscusses the numerical studies on benchmark examplesdemonstrating the accuracy and performance of the ap-proach. We finally conclude with a summary and outlookin Section VI.

II. ALL-ELECTRON KOHN-SHAM DENSITYFUNCTIONAL THEORY

Consider a materials system comprising of Na nucleiand Ne electrons. In density functional theory, the vari-ational problem of computing ground-state properties ofa given materials system is equivalent to solving the fol-lowing non-linear Kohn-Sham eigenvalue problem2:

✓�1

2r2 + Ve↵(⇢,R)

◆ i = ✏i i, i = 1, 2, · · · (1)

where ✏i and i denote the eigenvalues and correspond-ing eigenfunctions of the Hamiltonian, respectively. Wedenote by R = {R1, R2, · · ·RN

a

} the collection of allnuclear positions in the materials system. Though theproposed ideas in this work can be easily generalizedto periodic or semi-periodic materials systems and spin-dependent Hamiltonians28, for the sake of simplicity, wediscuss the formulation in the context of non-periodicsetting restricting to spin-independent Hamiltonians.

The electron-density at any spatial point x, in termsof the eigenfunctions, is given by

⇢(x) = 2X

i

f(✏i, µ)| i(x)|2 , (2)

where f(✏i, µ) is the orbital occupancy function, whoserange lies in the interval [0, 1], and µ represents theFermi-energy. In density functional theory calculations,it is fairly common to represent f by the Fermi-Diracdistribution3,29

f(✏, µ) =1

1 + exp�✏�µ�

� , (3)

where � = kBT is a smearing parameter with kB and Tdenoting the Boltzmann constant and the finite temper-ature. We note that as T & 0, the Fermi-Dirac distribu-tion tends to the Heaviside function. The Fermi-energyµ is computed from the constraint on the total number

of electrons in the system (Ne) given by

Z⇢(x) dx = 2

X

i

f(✏i, µ) = Ne . (4)

The e↵ective single-electron potential in the Kohn-ShamHamiltonian in equation (1), Ve↵(⇢,R), is given bythe sum of exchange-correlation potential (Vxc(⇢)) thataccounts for the quantum-mechanical interactions andthe classical electrostatic potentials corresponding toelectron-density (VH(⇢)) and nuclear charges (Vext(R)):

Ve↵(⇢,R) = Vxc(⇢) + VH(⇢) + Vext(R) . (5)

In the present work, the exchange-correlation interac-tions are treated using the local-density approximation(LDA)28, and in particular we employ the Slater ex-change and Perdew-Zunger30,31 form of correlation func-tional. In equation (5), the classical electrostatic po-tential corresponding to the electron-density distribution(Hartree potential), VH(⇢), is given by

VH(⇢)(x) =

Z⇢(x0)

|x� x0| dx0 . (6)

Further, electrostatic potential corresponding to the nu-clear charges, Vext(x,R), is given by

Vext(x,R) = �N

aX

I=1

ZI

|x�RI |, (7)

with ZI denoting the atomic number of the Ith nucleusin the given materials system. We note that the com-putation of electrostatic potentials VH and Vext are ex-tended in real-space. However, these quantities can bee�ciently computed by taking recourse to the solutionof the Poisson problem32–34, since the kernel correspond-ing to these extended interactions is the Green’s func-tion of the Laplace operator. Finally, for given positionsof nuclei, the system of equations corresponding to theKohn-Sham eigenvalue problem are:

✓�1

2r2 + Vxc(⇢) + VH(⇢) + Vext(R)

◆ i = ✏i i, (8a)

2X

i

f(✏i, µ) = Ne, (8b)

⇢(x) = 2X

i

f(✏i, µ)| i(x)|2. (8c)

We note that the system of equations in (8) representsa nonlinear eigenvalue problem, which has to be solvedself-consistently. Upon self-consistently solving (8), the

4

ground-state energy of the system is given by

Etot =Eband + Exc(⇢)�Z

Vxc(⇢) ⇢ dx

� 1

2

Z⇢VH(⇢) dx+ EZZ , (9)

where Exc denotes the exchange-correlation energy func-tional corresponding to Vxc, Eband denotes the band en-ergy, which is given by

Eband = 2X

i

f(✏i, µ)✏i , (10)

and EZZ denotes the nuclear-nuclear repulsive energy,which is given by

EZZ =1

2

NaX

I,J=1

I 6=J

ZIZJ

|RI �RJ |. (11)

III. MATHEMATICAL FORMULATION

In this section, we first derive the expression for theprojection of the Hamiltonian operator onto the valence-subspace of the Kohn-Sham Hamiltonian. Subsequently,we derive the expressions for the computation of electron-density and constraint on the number of valence electronsin terms of the valence-subspace projected Hamiltonian.

Let H denote the infinite-dimensional Hilbert space offunctions equipped with the inner product h.|.i and anorm k . k derived from the inner product. If the Her-mitian operator H defined on H denotes the Kohn-ShamHamiltonian of interest at a step of the self-consistent it-eration, then the projection of H onto an M -dimensionalsubspace VM

h ⇢ H is given by H ⌘ PqHPq : H ! VMh

where Pq : H ! VMh denotes the projection operator

onto VMh and is given by Pq =

PMi=1 |qii hqi|. Here,

{|qii}1iM denote an orthonormal basis for the sub-space VM

h with a real-space representation hx| qii = qi(x)for i = 1, . . . ,M . The Lowdin orthonormalized finite-element basis, which is employed in the present work,can be one choice for the basis functions.

A. Operator splitting in eigen-subspace

Let VN ⇢ VMh denote the eigen-subspace of H that

includes all the occupied states as well as the unoccupiedstates around the Fermi energy. A good approximation tothis eigen-subspace can be computed, for instance, usinga Chebyshev filtering approach35. Let {|�↵i}1↵N de-note a non-orthogonal basis spanning the eigen-subspace.The generality of using a non-orthogonal basis to repre-sent the eigen-subspace is motivated from our eventualobjective of using non-orthogonal localized basis func-tions, computed using a localization procedure22, to re-alize reduced-order scaling computational complexity forthe solution of the Kohn-Sham problem. Let the projec-tion of H onto VN be denoted by H� ⌘ P�HP� : VM

h !

VN , where P� : VMh ! VN is the projection operator

onto VN and is given by P� =PN↵,�=1 |�↵iS�1

↵� h�� |.Here, S↵� = h�↵| ��i are the matrix elements of theoverlap matrix denoted by S.

We now consider the spectral decomposition of H�

H� =NX

i=1

✏�i

��� �iED

�i

��� (12)

with ✏�i and��� �i

Edenoting the eigenvalues and the corre-

sponding eigenvectors of H�, respectively. Splitting theprojected Hamiltonian H� in (12) into a core part andits complement, referred to as the valence part, we have

H� =N

coreX

i=1

✏�i

��� �iED

�i

���+NX

i=Ncore

+1

✏�i

��� �iED

�i

���

= H�,core +H�,val (13)

In the above, H�,core and H�,val denote the projectionof H� onto the core- and valence-subspaces denoted byVN

core and VNval , respectively. Here Ncore denotes the

number of core states, while Nval = N � Ncore. Wenow derive an expression for H�,val, and, to do so, wedenote the non-orthogonal basis spanning VN

core to be{|�core↵ i}1↵N

core

. The projection operator onto VNcore ,

P�,core : VN ! VNcore , can be expressed as

P�,core =N

coreX

i=1

��� �iED

�i

��� =N

coreX

↵,�=1

|�core↵ iScore�1

↵�

⌦�core�

�� ,

(14)

where Score↵� =

D�core↵

��� �core�

Edenote the matrix elements

of the core overlap matrix denoted by Score. Further,the projection operator onto the space VN

val is given by(I � P�,core) : VN ! VN

val with I denoting the identityoperator. Thus, the projection of H� onto VN

val , H�,val :VN ! VN

val , is given by

H�,val = (I � P�,core)H�(I � P�,core). (15)

Let us now consider the single particle density operator(��) corresponding to H�, which can be expressed as

�� =NX

i=1

f(✏�i )��� �i

ED �i

��� = f(H�) , (16)

where the second equality follows from the spectral de-composition of the Hermitian operator H� in (12). Split-ting the density operator �� into core and valence parts,we have

�� =N

coreX

i=1

f(✏�i )��� �i

ED �i

���+NX

i=Ncore

+1

f(✏�i )��� �i

ED �i

���

= ��,core + ��,val . (17)

5

Using the fact that the core-states have unit occupancy,i.e. f(✏�i ) = 1 for i = 1 · · ·Ncore, equation (17) can bereduced to

�� = f(H�) = P�,core + f(H�,val) . (18)

B. Matrix representations in non-orthogonal basis

In this subsection, we derive the matrix expressions forthe operators H�,val, P�,core and ��,val expressed in thenon-orthogonal basis {|�↵i}1↵N , which are used in thespectrum-splitting algorithm described in the next sec-tion. To this end, we denote the matrices correspondingtoH� andH�,val in {|�↵i}1↵N basis asH� andH�,val,respectively. The matrix elements of H� are given by

H�↵� =NX

�=1

S�1↵�

⌦��

��H�����

↵.

The matrix elements of H�,val are given by

H�,val↵� =NX

�=1

S�1↵�

⌦��

��H�,val����

↵

=NX

�=1

S�1↵�

⌦��

�� (I � P�,core)H�(I � P�,core)����

↵.

Using the relation H� = P�H�P� in the above equation,we have

H�,val↵� =

NX

�=1

S�1↵�

⌦��

�� (I � P�,core)P�H�P�(I � P�,core)����

↵.

We now use the definition of P� to simplify the aboveequation as

H�,val↵� =NX

⌫,⌘=1

(�↵⌫ � P�,core↵⌫ )H�⌫⌘(�⌘� � P�,core⌘� ) , (19)

where P�,core↵⌫ denotes the matrix element of P�,core,which is the matrix corresponding to operator P�,core ex-pressed in {|�↵i}1↵N basis. Further, �↵⌫ denotes theKronecker delta. Thus, using matrix notation, we have

H�,val = (I�P�,core)H�(I�P�,core) . (20)

We now derive an expression for the matrix element of

P�,core as follows:

P�,core↵� =NX

�=1

S�1↵�

⌦��

��P�,core����

↵

=NX

�=1

S�1↵�

⌦��

��Pq P�,core Pq����

↵

Substituting the expressions Pq =PM

i=1 |qii hqi| and

P�,core =PN

core

↵,�=1 |�core↵ iScore�1

↵�

D�core�

��� into the above

equation, we have

P�,core↵� =NX

�=1

NcoreX

⌘,⌫=1

MX

i,j=1

S�1↵��

⇤i��

corei⌘ Score�1

⌘⌫ �core⇤

j⌫ �j� ,

(21)where �j� = hqj | ��i and �⇤

i� denotes the complex conju-gate of �i� , and the above equation can be convenientlyrecast in terms of matrices as follows

P�,core = S�1�†�coreScore�1

�core†�

= e�core

Score�1 e�core†

S , (22)

where e�core

= S�1�†�core represents the matrixwith column vectors comprising of the componentsof {|�core↵ i}1↵N

core

expressed in {|�↵i}1↵N basis.Here, � denotes the matrix whose column vectors arethe components of {|�↵i}1↵N in {|qii}1iM basis,and �† denotes the conjugate transpose of the matrix�. Furthermore, �core denotes the matrix whose col-umn vectors are the components of {|�core↵ i}1↵N

core

expressed in {|qii}1iM basis. Score, which denotes the

core overlap matrix, can be expressed in terms of e�core

as

Score = e�core†

Se�core

. (23)

Next, we derive the matrix expression for ��,val in thenon-orthogonal basis. We denote the matrix representa-tion of ��,val in {|�↵i}1↵N basis and {|qii}1iM by��,val and �q,val, respectively. We seek to express ��,val

in terms of H�,val. We have

��,val↵� =NX

�=1

S�1↵�

⌦��

����,val����

↵

=NX

�=1

S�1↵�

⌦��

��Pq ��,val Pq����

↵

=NX

�=1

MX

i,j=1

S�1↵� h�� | qii

⌦qi����,val

��qj↵hqj | ��i

=NX

�=1

MX

i,j=1

S�1↵��

⇤i��

q,valij �j� . (24)

6

We note that �q,val = f(Hq,val), where Hq,val is the ma-trix representation of H�,val in {|qii}1iM . Hence, inmatrix notation, we have,

��,val = �+f(Hq,val)� , (25)

where �+ = S�1�† denotes the Moore-Penrose pseu-doinverse of � matrix. We now seek a relation be-tween Hq,val and H�,val for which we use the relationsH�,val = P�H�,valP� and P� =

PNµ,�=1 |�µiS�1

µ� h��| toexpress Hq,val as

Hq,valij =

⌦qi��H�,val

��qj↵

=⌦qi��P�H�,valP�

��qj↵

=NX

↵,�=1

�i↵H�,val↵µ S�1

µ� �⇤j� . (26)

Using matrix notation, the above equation can be writtenas

Hq,val = �H�,val�+ . (27)

We note that the function f(✏) represents a Fermi-Diracdistribution, which is an analytic function, and, hence,f(Hq,val) admits a power series representation given by

�q,val = f(Hq,val) =1X

k=0

ak(Hq,val)k =

1X

k=0

ak(�H�,val�+)k

=1X

k=0

ak�(H�,val)k�+ = �⇣ 1X

k=0

ak(H�,val)k

⌘�+

= �f(H�,val)�+ . (28)

Substituting the above relation in equation (25), we ar-rive at

��,val = f(H�,val) . (29)

Hence, using equations (18), (22) and (29) we have

�� = f(H�) = e�core

Score�1 e�core†

S+ f(H�,val) . (30)

Finally, we derive the constraint on the number of va-lence electrons, denoted by Ne,val, using equation (30),by recalling that the constraint on the total number ofelectrons is given by

2 tr(f(H�)) = Ne

=) 2 tr(P�,core) + 2 tr(f(H�,val)) = Ne ,

=) 2 tr(f(H�,val) = Ne � 2Ncore .

Hence, the constraint on the number of valence electronsNe,val is given by

2 tr(f(H�,val)) = Ne,val = Ne � 2Ncore . (31)

IV. SPECTRUM-SPLITTING ALGORITHM

In this section, we present the various steps involvedin the spectrum-splitting approach for conducting all-electron Kohn-Sham DFT calculations within the finite-element framework. The subspace projection techniqueproposed in Motamarri et.al22 is used as the startingpoint in developing the proposed approach, which in-volves: (i) computing the discretized Hamiltonian inLowdin orthonormalized finite-element basis; (ii) com-puting the relevant eigen-subspace using Chebyshev fil-tering, followed by evaluating the non-orthogonal lo-calized basis spanning the Chebyshev filtered basis;(iii) computing the projected Hamiltonian in the non-orthogonal basis to evaluate the quantities of interestsuch as the density-matrix, electron-density and bandenergy using Fermi-operator expansion. The key ideain this work is to reformulate the Fermi-operator ex-pansion of the Kohn-Sham Hamiltonian in terms of thevalence-subspace projected Hamiltonian. The spectralwidth of the valence-subspace projected Hamiltonian,which is O(1Ha), is significantly smaller than the spec-tral widths of both the discrete Kohn-Sham Hamilto-nian in the finite-element basis as well as the Chebyshev-filtered subspace projected Hamiltonian. Further, thespectral width of the valence-subspace projected Hamil-tonian is relatively independent of the basis set discretiza-tion and the materials system. As demonstrated subse-quently, this significantly improves the accuracy of Fermi-operator expansion type techniques, which have been de-veloped to realize reduced-order scaling for the Kohn-Sham problem. A brief summary of the ideas proposedin Motamarri et.al22 is first discussed in this section,followed by the description of the proposed spectrum-splitting approach.

A. Projection of the Kohn-Sham Hamiltonian inthe Lowdin orthonormalized finite-element basis

Let VMh be theM dimensional subspace spanned by the

finite-element basis, a piecewise polynomial basis gener-ated from a finite-element discretization36 of characteris-tic mesh-size h. The various electronic fields namely thewavefunctions and the electrostatic potential expressedin the finite-element basis are given by

hi (x) =

MX

j=1

Nj(x) ji , �h(x) =

MX

j=1

Nj(x)�j , (32)

where Nj : 1 j M denotes the finite-element basisspanning VM

h . We now consider the Lowdin orthonor-malized finite-element basis qj(x) : 1 j M spanningthe finite-element space VM

h and we note the following

7

relation between qj(x) and Nj(x):

qj(x) =MX

k=1

M�1/2jk Nk(x) , (33)

where M denotes the overlap matrix associated with thebasis functions Nj : 1 j M with matrix elementsMjk = hNj | Nki. The discretization of the Kohn-Shameigenvalue problem (8) using this basis qj(x) : 1 j Mresults in the following standard eigenvalue problem:

H i = ✏hi i , (34)

where

H = M�1/2HM�1/2 (35)

with

Hjk =1

2

ZrNj(x).rNk(x) dx+

ZV he↵(x)Nj(x)Nk(x) dx ,

Mjk =

ZNj(x)Nk(x) dx .

Here i is a vector containing the expansion coe�-cients of the discretized eigenfunction h

i (x) expressedin Lowdin orthonormalized finite-element basis spanningthe finite-element space. Furthermore, the matrixM�1/2

can be evaluated with modest computational cost by us-ing a spectral finite-element basis in conjunction withGauss-Lobatto-Legendre (GLL) quadrature for the eval-uation of integrals in the overlap matrix, which rendersthe overlap matrix diagonal34.

B. Chebyshev-filtered subspace iteration andlocalization

Chebyshev-filtered subspace iteration (ChFSI) tech-nique35 is used to compute an approximation to theeigenspace of the discrete Kohn-Sham Hamiltonian,spanned by N � Ne/2 lowest eigenfunctions correspond-ing to the occupied states and a few unoccupied statesaround the Fermi energy. We refer to22,34 for the detailsof its implementation in the context of finite-element dis-cretization. The fast growth of Chebyshev polynomialsin (�1,�1) is exploited in the ChFSI technique to mag-nify the relevant eigenspectrum, and thereby providingan e�cient approach for the solution of the Kohn-Shameigenvalue problem. Further, a localized basis for thesubspace VN spanned by Chebyshev filtered vectors iscomputed by employing a localization technique as pro-posed in Garcia-Cervera et al.21. We refer to Motamarriet.al22 for the details of the numerical implementationinvolved in the computation of these localized functions.In the subsequent paragraphs, we denote by �L the ma-trix whose column vectors are the expansion coe�cientsof these compactly-supported non-orthogonal localized

functions expressed in the Lowdin orthonormalized finite-element basis.

C. Spectrum-splitting approach for computingelectron-density

We now discuss the steps involved in the spectrum-splitting approach involved in the computation of theelectron-density.a. Computation of projected Hamiltonian in

the occupied subspace: As discussed in section III,the projection of the Hamiltonian into the non-orthogonal localized basis �L is computed as

H� = S�1�TLH�L . (36)

where S�1 denotes the inverse of the overlap matrix Sresulting from the non-orthogonal localized functions andis given by S = �T

L�L.

b. Computation of the projection matrix corre-sponding to core states: As discussed in section III,the relevant subspace of the Kohn-Sham Hamiltonian,which is of dimension N , can be split into two sub-spaces VN

core and VNval spanned by the core- and valence-

subspaces, respectively. Further, let ncoreI be the num-

ber of core states associated with an isolated atom Iin the given material system, then the dimension of thesubspace VN

core ⇢ VN is given by Ncore =PN

a

I=1 ncoreI .

We now seek to compute an approximation to the core-subspace, VN

core , spanned by the core eigenfunctions inorder to evaluate the matrix corresponding to projectionoperator P�,core. To this end, we employ Chebyshev fil-tering procedure to compute the core-subspace by con-structing a filter from the projected Hamiltonian H�.We start with an initial subspace denoted by the ma-trix X� of size N ⇥Ncore, which is chosen to correspondto the core states from the localization procedure of sec-tion IVB. Subsequently, the projected Hamiltonian H�

is scaled and shifted to construct eH�such that the va-

lence spectrum of H� is mapped to [�1, 1] and the corespectrum into (�1,�1). Hence

eH�=

1

g(H��dI) where g =

✏�max � ✏�

2d =

✏� + ✏�max

2.

(37)where ✏� and ✏�max denote the upper bounds of the corespectrum and the full spectrum of H�, respectively. Es-timate for ✏�max can be computed using the Krylov-Schurmethod37. Further, ✏� is chosen to be lying in theenergy-gap between cmax and vmin, where cmax denotesthe maximum of the largest single-atom core eigenval-ues associated with each atom I in the given materialssystem, while vmin denotes the minimum of the small-est single-atom valence eigenvalues. Finally, the filter isconstructed using a Chebyshev polynomial of degree m,Tm(x), and the action of the filter on X� is recursively

8

computed as

Y � = Tm( eH�)X� =

h2 eH

�Tm�1( eH

�)� Tm�2( eH

�)iX� .

(38)The Chebyshev-filtered vectors in Y � can be expressedin terms of Lowdin orthonormalized basis by using trans-formation Y q = �LY

�.

Next, the localization procedure discussed in step Bis employed to obtain non-orthogonal localized functionsspanning the Chebyshev-filtered space Y q. Specifically,the localized functions are obtained by solving the mini-mization problem

arg min 2VN

core ,|| ||=1

Zw(x)| (x)|2 dx , (39)

where w(x) � 0 is chosen to be a smooth weighting func-tion |x � RI |2 with the position of the nuclei RI de-noting the localization center. We denote by �core

L thematrix whose column vectors are the expansion coe�-cients of these compactly supported non-orthogonal lo-calized functions spanning the core-subspace expressed inthe Lowdin orthonormalized basis. In order to computethe projection operator corresponding to core-subspace,it is computationally e�cient to express the relevant ma-trices with respect to the basis (column) vectors of �L.To this end, as discussed in section III, we express �core

L

with respect to the basis �L by means of the followingtransformation

e�core

L = S�1�†L�

coreL . (40)

Finally, the projection operator (cf. equation (22)) iscomputed as

P�,core = e�core

L Score�1 e�core†

L S , (41)

where Score in the above equation denotes the core over-lap matrix which can be evaluated as

Score = e�core†

L Se�core

L . (42)

In practice, the localized functions are truncated belowa prescribed tolerance to result in sparse matrices �L

and �coreL . If they are su�ciently sparse, P�,core can be

computed in O(N) complexity.

c. Computation of electron-density: One classof commonly employed computational methodologies toreduce the computational complexity of Kohn-ShamDFT calculations constitute the Fermi-operator expan-sion (FOE) type techniques3,4 which avoid the explicitdiagonalization of the discretized Hamiltonian in orderto compute the electron-density. The Fermi-operator ex-pansion, which approximates the Fermi-Dirac distribu-tion (cf. equation (3)) by means of Chebyshev polyno-mial expansion, is one of the most widely used techniquesto realize reduced-order computational complexity in in-

sulating as well as metallic systems. It is important tonote that the degree of polynomial required to achievea desired accuracy in the approximation depends on thespectral width of the discrete Hamiltonian, denoted by�E.

In all-electron calculations performed using the finite-element basis, the spectral width can be very large asthe largest eigenvalue of the discrete Hamiltonian isO(106) Ha even for simple systems, owing to the re-fined finite-element mesh around the nuclei to capture therapid oscillations in the wavefunctions. Fermi-operatorexpansion on such large spectral widths can be inaccu-rate and ine�cient. In a recent e↵ort22, this challengehas been addressed by projecting the problem onto a rel-evant subspace, such as a Chebyshev-filtered space con-taining the occupied states, and employing the Fermi-operator on the subspace projected Hamiltonian (H�)whose spectral width is commensurate with that of theoccupied eigenspectrum. However, one is confronted withanother challenge in the case of all-electron calculations,especially those involving heavier atoms. The spectralwidth of the corresponding Hamiltonian increases as themagnitude of the smallest eigenvalue grows as O(Z2)with Z being atomic number and this can deterioratethe accuracy of the Fermi-operator expansion. In thepresent work, we address this by reformulating the com-putation of electron-density in terms of the Fermi-Diracfunction of the Hamiltonian projected onto the valence-subspace. The spectral width of this valence-subspaceprojected Hamiltonian is O(1Ha), irrespective of the ma-terials system, and, thus, Fermi-operator expansion canbe employed to develop reduced-order scaling methodsfor all-electron Kohn-Sham DFT calculations.

We note that the electron-density is the diagonal ofthe density matrix and can be written in terms of Fermi-Dirac function of the subspace projected Hamiltonianf(H�) as (cf. equation (60) in Motamarri et al.22)

⇢(x) = 2 NT(x) �L f(H�) S�1 �T

L N(x) , (43)

with

N(x) = M�1/2 N(x),

N(x) = [N1(x) N2(x) N3(x) · · · NM (x)]T ,

and

f(H�) =1

1 + exp⇣

H��µ�

⌘ (44)

is the finite-temperature density matrix, with µ denot-ing the chemical potential and � = kB T . Using equa-tion (30), we can rewrite the above equation in terms ofthe Fermi-Dirac function of H�,val as

⇢(x) = 2 NT(x) �L

⇣P�,core + f(H�,val)

⌘S�1 �T

LN(x) ,

(45)

9

with P�,core = e�core

L Score�1 e�core†

L S.

In order to evaluate electron-density using equa-tion (45), we first computeH�,val, which we recall is givenby (cf. equation (20))

H�,val = (I�P�,core)H�(I�P�,core) .

We now use Chebyshev polynomial expansion to approx-imate f(H�,val), by first scaling and shifting H�,val toobtain H�,val

s such that its spectrum lies in [�1, 1], andthen employing a finite number of Chebyshev polynomi-als to approximate f(H�,val) as3,11

f(H�,val) =RX

n=0

an(�s, µs)Tn(H�,vals ), (46)

where

H�,vals =

H�,val � ✏

�✏; �s =

�✏

�; µs =

µ� ✏

�✏, (47)

�✏ =✏�,valmax � ✏�,valmin

2; ✏ =

✏�,valmax + ✏�,valmin

2, (48)

and

an(�s, µs) =2� �n 0

⇡

Z 1

�1

Tn(x)p1� x2

1

1 + e�s

(x�µs

)dx ,

(49)where �ij denotes the Kronecker delta. In the above,

✏�,valmax and ✏�,valmin denote the upper and lower bounds for

the spectrum of H�,val. Estimates for ✏�,valmax and ✏�,valmin arecomputed using the Krylov-Schur method37. We remarkthat H�,val is a projection onto the valence-subspace rep-resented by localized basis functions, and, hence, H�,val

is a sparse matrix. Furthermore, if H�,val is su�cientlysparse, f(H�,val) can be computed in O(N) complex-ity11. Most importantly, the degree R of the Chebyshevexpansion in (46) is proportional to the spectral width�E = ✏�,valmax � ✏�,valmin of H�,val, which is O(1Ha). We re-mark that this spectral width is much smaller than thatof H�, thus, allowing an accurate and e�cient compu-tation of Fermi-operator expansion using a Chebyshevpolynomial expansion of O(100) for all-electron calcula-tions on any materials system as subsequently demon-strated in the numerical studies.

The Fermi-energy (µ), which is required in the com-putation of the Fermi-operator expansion of f(H�,val), isevaluated using the constraint given in (31)

2 tr⇣f(H�,val)

⌘= Ne,val = Ne � 2Ncore .

Finally, the band energy (Eb), which can also be ex-

pressed in terms of H�,val, is evaluated as

Eb = 2 tr

✓e�core

L Score�1 e�core†

L S+ f(H�,val)

�H�

◆.

(50)

V. RESULTS AND DISCUSSION

The accuracy and performance of the proposedspectrum-splitting method is investigated in this section.The benchmark systems considered in this study involvenon-periodic three dimensional materials systems withmoderate to high atomic numbers. These include siliconnano-clusters of varying sizes, containing 1 ⇥ 1 ⇥ 1 (252electrons), 2⇥ 1⇥ 1 (434 electrons), 2⇥ 2⇥ 2 (1330 elec-trons) diamond-cubic unit-cells, as well as a single goldatom and a six-atom gold nano-cluster.

In order to assess the accuracy of the proposed ap-proach, we use as reference the ground-state energy of theKohn-Sham problem solved using the Chebyshev-filteredsubspace iteration method for the finite-element basis34

(ChFSI-FE). The ChFSI-FE involves the projection ofthe discrete Hamiltonian onto the Chebyshev-filteredsubspace spanned by an orthogonal basis, followed byan explicit computation of the eigenvalues and eigenvec-tors via diagonalization of the projected Hamiltonian toestimate the electron-density. In all our simulations—reference calculations, as well as, the benchmark calcu-lations discussed subsequently—we use the n-stage An-derson38 mixing scheme on the electron-density in self-consistent field iteration of the Kohn-Sham problem witha stopping criterion of 10�8 in the square of the L2 normof the change in electron-density in two successive itera-tions.

In order to demonstrate the e↵ectiveness of the pro-posed spectrum-splitting method, we compare its perfor-mance with the subspace projection algorithm (SubPJ-FE) proposed in Motamarri et.al22. In SubPJ-FE theFermi-operator expansion is employed on the Chebyshev-filtered subspace projected Hamiltonian as opposed tothe valence-subspace projected Hamiltonian in the pro-posed spectrum-splitting method, with all else beingidentical in both sets of calculations. The ground-state energies for each of these benchmark systems arecomputed for varying polynomial degrees R used inthe Chebyshev expansion of the Fermi-Dirac functionof the projected Hamlitonians, H�,val in the proposedspectrum-splitting approach and H� in SubPJ-FE. Theerror in the ground-state energies obtained in each of thetwo methods is measured with respect to the referenceenergy obtained using ChFSI-FE and is plotted againstR.

10

A. Silicon

We consider silicon nano-clusters containing 1⇥ 1⇥ 1(252 electrons), 2⇥ 1⇥ 1 (434 electrons), 2⇥ 2⇥ 2 (1330electrons) diamond-cubic unit cells with a lattice con-stant of 10.26 a.u., and conduct all-electron calculationsto test the performance of the spectrum-splitting ap-proach. Finite-element meshes with fifth-order spectralfinite-elements (HEX216SPECT) are chosen such thatthe discretization error is less than 5 mHa in the ground-state energy per atom. Identical finite-element meshesare employed in the benchmark calculations presentedbelow as well as the reference calculations.

Figures 1 and 2 show the comparison between the pro-posed spectrum-splitting method and SubPJ-FE for sil-icon nano-clusters containing 1 ⇥ 1 ⇥ 1 (252 electrons),2⇥ 1⇥ 1 (434 electrons) unit-cells respectively. A Fermi-smearing parameter of 0.003262 Ha (T=1000K) is em-ployed in these simulations. These results indicate

0 200 400 600 800 100010

−8

10−6

10−4

10−2

100

Fermi−operator Expansion Degree

Abs.

Err

or

in E

nerg

y (H

a/a

tom

)

Spectrum splitting method

SubPJ−FE

FIG. 1: Comparison between the spectrum-splitting methodand SubPJ-FE. Case study: Silicon 1x1x1 nano-cluster.

that the proposed method provides significantly betteraccuracies in the ground-state energies in comparison toSubPJ-FE for similar polynomial degrees. Accuracies inthe ground-state energies of better than 10�4 Ha/atomwith reference calculations can be obtained with just apolynomial degree R of 200 using the spectrum-splittingmethod, whereas R greater than 1000 is required toget to accuracies close to 10�4 Ha/atom with SubPJ-FE. While the error in the ground-state energy de-creases exponentially with increasing R in the spectrum-splitting method, the error is seen to stagnate at around10�6 Ha/atom. This is due to competing non-variationalerrors arising from numerical integration involved in theevaluation of coe�cients in the Chebyshev polynomialexpansion (46).

We next consider the benchmark calculations on sil-icon nano-cluster containing 2 ⇥ 2 ⇥ 2 (1330 electrons)

0 200 400 600 800 100010

−8

10−6

10−4

10−2

100

Fermi−operator Expansion Degree

Ab

s. E

rro

r in

En

erg

y (H

a/a

tom

)

Spectrum splitting method

SubPJ−FE

FIG. 2: Comparison between the spectrum-splitting methodand SubPJ-FE. Case study: Silicon 2x1x1 nano-cluster.

unit-cells for the case of two smearing parameters:0.00163 Ha(T=500K) and 0.003262 Ha (T=1000K).Figure 3 shows the comparison between the proposedmethod and SubPJ-FE for this benchmark system. The

0 500 1000 1500 200010

−8

10−6

10−4

10−2

100

Fermi−operator Expansion Degree

Ab

s. E

rro

r in

En

erg

y (H

a/a

tom

)

Spectrum splitting method: T=500 K

SubPJ−FE: T=500 K

Spectrum splitting method: T=1000 K

SubPJ−FE: T=1000 K

FIG. 3: Comparison between the spectrum-splitting methodand SubPJ-FE. Case study: Silicon 2x2x2 nano-cluster.

results from Figure 3 demonstrate that the accuracy ande↵ectiveness of the proposed method is retained for largermaterials systems. We note a five-fold reduction in thepolynomial degree for accuracies of 10�4 Ha/atom witha temperature of 1000K for the spectrum-splitting ap-proach in comparison to SubPJ-FE—a similar reduc-tion as observed in the smaller-sized nano-clusters. Fur-thermore, the simulations with a lower temperature of500K show that R ⇠ 600 results in accuracies bet-ter than 10�5 Ha/atom with the proposed approach,whereas even with R ⇠ 1500 we observe errors of ⇠

11

10�3 Ha/atom with SubPJ-FE.

B. Gold

In order to further investigate the performance of theproposed method for material systems with larger atomicnumbers, we consider gold (Atomic number 79) as ourmodel example. We consider two benchmark problems:(i) single atom gold (Au); (ii) a planar gold cluster39

containing six atoms (Au6) with Au-Au bond length of5.055 a.u. We choose finite-element meshes with fifth-order spectral finite-elements (HEX216SPECT) for thesebenchmark examples such that the discretization error isless than 5 mHa.

The error in the ground-state energies, measured withrespect to the reference energies obtained using ChFSI-FE, for the proposed spectrum-splitting method andSubPJ-FE are computed for various polynomial degreesin the Chebyshev expansion of the Fermi-operator. Fig-ures 4 and 5 show these results for case of Au single atomand Au6 nano-cluster, respectively. A Fermi-smearingparameter of 0.00163 Ha (T=500K) is used in these cal-culations.

0 500 1000 1500 200010

−6

10−4

10−2

100

102

Fermi−operator Expansion Degree

Ab

s. E

rro

r in

En

erg

y (H

a/a

tom

)

Spectrum splitting method

SubPJ−FE

FIG. 4: Comparison between the spectrum-splitting methodand SubPJ-FE. Case study: Au single atom.

These results demonstrate that the accuracy of theproposed method is far superior than SubPJ-FE. Inparticular, accuracies in the ground-state energies closeto 10�4 Ha/atom can be obtained with a polynomialdegree that is a little over 1000 using the proposedspectrum-splitting method, whereas SubPJ-FE resultedin ground-state energies with errors close to O(1Ha) peratom even with R greater than 2000. These benchmarkstudies on Au underscores the advantage of the pro-posed spectrum-splitting approach in employing Fermi-operator type techniques for material systems with heavyatomic numbers.

0 500 1000 1500 200010

−6

10−4

10−2

100

102

Fermi−operator Expansion Degree

Ab

s. E

rro

r in

En

erg

y (H

a/a

tom

)

Spectrum splitting method

SubPJ−FE

FIG. 5: Comparison between the spectrum-splitting methodand SubPJ-FE. Case study: Au6 nano-cluster.

VI. SUMMARY

In the present work, we formulated a spectrum-splitting approach for employing Fermi-operator expan-sion in all-electron Kohn-Sham DFT calculations, andpresented the approach in the framework of spectralfinite-element discretization of the Kohn-Sham DFTproblem. In the proposed method, in every itera-tion of the self-consistent field procedure, an eigen-subspace containing the occupied states of the Kohn-Sham Hamiltonian is computed using a Chebyshev fil-ter. The Kohn-Sham Hamiltonian is projected onto theChebyshev-filtered subspace spanned by localized basisfunctions which are computed using a localization pro-cedure. The core-subspace in the Chebyshev-filteredsubspace is computed using another appropriately con-structed Chebyshev filter, and the valence-subspace isextracted as its complement. The Fermi-operator ex-pansion of the Kohn-Sham Hamiltonian is subsequentlyevaluated as the sum of the projection operator corre-sponding to the core-subspace and the Fermi-operatorexpansion of the valence-subspace projected Kohn-ShamHamiltonian. As the spectral width of the valence-subspace projected Kohn-Sham Hamiltonian is O(1Ha),and is independent of the materials system and dis-cretization, the Fermi-operator expansion can be em-ployed on all-electron Kohn-Sham DFT calculations.The accuracy and performance of the proposed method

was investigated on two di↵erent materials systems: (i)Silicon with a moderate atomic number, and (ii) Goldwith a high atomic number. The benchmark systemsinvolved silicon nano-clusters up to 1330 electrons, a sin-gle gold atom, and a six-atom gold cluster. In all thecases, the proposed spectrum-splitting method providedground-state energies that are in excellent agreementwith reference calculations. In particular, in the caseof silicon nano-clusters, the proposed spectrum-splitting

12

approach resulted in a five-fold reduction in the Fermi-operator expansion polynomial degree to achieve accu-racies close to 10�4Ha/atom in the ground-state ener-gies. Further, the e↵ectiveness of the proposed approachwas even more significant in the case of gold, where thespectrum-splitting approach was found to be indispens-able in achieving chemical accuracy.

Fermi-operator expansion type techniques avoid ex-plicit diagonalization of the Kohn-Sham Hamiltonian,and o↵er a viable path to developing reduced-orderscaling methods to solve the Kohn-Sham problem forboth insulating and metallic systems. The proposedspectrum-splitting approach extends the applicability ofFermi-operator type expansion methods to all-electronDFT calculations, independent of the materials systemand discretization. The proposed spectrum-splitting ap-proach in conjunction with enriched finite-element basis,where the finite-element basis is enriched with compactlysupported single-atom Kohn-Sham orbitals, has the po-tential to enable all-electron DFT calculations on systemsizes not accessible heretofore.

ACKNOWLEDGMENTS

We gratefully acknowledge the support from the AirForce O�ce of Scientific Research under Grant No.FA9550-13-1-0113, and the support from the U.S. ArmyResearch Laboratory (ARL) through the Materials inExtreme Dynamic Environments (MEDE) Collabora-tive Research Alliance (CRA) under Award NumberW911NF-11-R-0001. V.G. also acknowledges the hos-pitality of the Division of Engineering and Applied Sci-ences at the California Institute of Technology while pur-suing this work. We also acknowledge Advanced Re-search Computing at University of Michigan throughthe Flux computing platform, and Extreme Science andEngineering Discovery Environment (XSEDE), which issupported by National Science Foundation grant numberACI-1053575, for providing the computing resources.

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