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Implementation of screened hybrid density functional for periodic systems with

numerical atomic orbitals: Basis function fitting and integral screening

Honghui Shang, Zhenyu Li, and Jinlong Yang

Citation: The Journal of Chemical Physics 135, 034110 (2011); doi: 10.1063/1.3610379

View online: http://dx.doi.org/10.1063/1.3610379

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THE JOURNAL OF CHEMICAL PHYSICS 135, 034110 (2011)

Implementation of screened hybrid density functional for periodic systems

with numerical atomic orbitals: Basis function ﬁtting and integral screening

Honghui Shang, Zhenyu Li, and Jinlong Yanga)

Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China,

Hefei, Anhui 230026, China

(Received 19 December 2010; accepted 23 June 2011; published online 20 July 2011)

We present an efﬁcient O(N) implementation of screened hybrid density functional for periodic sys-

tems with numerical atomic orbitals (NAOs). NAOs of valence electrons are ﬁtted with gaussian-type

orbitals, which is convenient for the calculation of electron repulsion integrals and the construc-

tion of Hartree-Fock exchange matrix elements. All other parts of Hamiltonian matrix elements

are constructed directly with NAOs. The strict locality of NAOs is adopted as an efﬁcient two-

electron integral screening technique to speed up calculations. © 2011 American Institute of Physics.

[doi:10.1063/1.3610379]

I. INTRODUCTION

Ab initio electronic structure calculations based on

density functional theory (DFT) (Refs. 1–3) or quantum-

chemical wavefunction approaches4have been widely used

to study both molecules and solids. Many-body effects in

DFT are taken into account by density functionals, which

is typically less expensive than the way to describe them in

wavefunction based post-Hartree-Fock methods, although

the latter is also becoming feasible for periodic systems.

However, widely used semilocal density functionals suffer

from self-interaction error.5A possible remedy is adding

nonlocal Hartree-Fock-type exact exchange (HFX) into

semilocal density functionals. B3LYP hybrid functional6,7

constructed this way is currently very popular in the quantum

chemistry community. However, global hybrid functionals

are typical just unfeasible to be calculated in periodic

systems8without further approximations.9At the same time,

exchange interaction at large distances is approximately

cancelled by correlation in metals and narrow-bandgap semi-

conductors,10–12 and full range hybrid density functional

calculation can thus yield qualitatively incorrect results.

As a result, screened hybrid functional (HSE03,13 HSE06

(Ref. 14)) was proposed to extend the successes of hybrid

functional into the solid state. The HSE functionals have

been implemented in some commercial softwares, such as

GAUSSIAN (Ref. 15) and VASP,16 and many results illustrate

the power of the screened exchange approximation.8

The linear combination of atomic orbitals (LCAO)

method, which uses atomic orbitals as the basis set, has been

widely used in solid state calculations. There are two kinds

of atomic orbitals that are widely used: The ﬁrst is gaussian-

type orbital (GTO), as adopted in GAUSSIAN (Ref. 15) and

CRYSTAL,17 which is designed to calculate electron repul-

sion integrals (ERIs) analytically; The second is numerical

atomic orbital (NAO), which is adopted in SIESTA,18 DMOL,19

a)Author to whom correspondence should be addressed. Electronic mail:

jlyang@ustc.edu.cn.

OPENMX,20 etc. The advantage of NAO is its strict locality,

which naturally leads to lower order scaling of computational

time versus system size. Based on it, various linear scaling

(O(N)) algorithms have been proposed to deal with systems

containing thousands of atoms.18,20–26 In contrary, Gaussian

basis function has a long tail, a simple truncation leads to a

discontinuity, which may affect the algorithm stability.

The heavy computational demand to evaluate HFX is

the main bottleneck for hybrid functional calculations. With

the GTO basis set, several approaches to reduce the com-

putational scaling for exact exchange evaluation have been

introduced.27–30 Especially, Izmaylov et al. proposed an efﬁ-

cient scheme to evaluate screened hybrid functional,31 with

a computational efﬁciency comparable with that of stan-

dard nonhybrid density functional calculations. However,

with the NAO basis set, the computation of HFX has a

big prefactor, even if the scaling can be linear.32 There-

fore, developing efﬁcient algorithm to reduce the prefactor

becomes crucial for the HFX computation with NAO basis

set.

It is interesting to take advantages of both types of atomic

orbitals, GTO, and NAO. GTO can be used for analytical

computation of ERIs in a straightforward and efﬁcient way,

while NAO can be employed to set the strict cutoff for atomic

orbitals. By this way, the computation of HFX will be much

faster, with a signiﬁcant reduction of the prefactor of O(N)

calculations. In this paper, we propose such a hybrid scheme,

denoted as NAO2GTO, and implement it in the SIESTA pack-

age. With this implementation, for the ﬁrst time, we can cal-

culate hybrid functional using NAOs for non-trivial periodic

systems.

The outline of this paper is as follows: In Sec. II, theory

of HFX and screened hybrid functionals is brieﬂy reviewed.

In Sec. III, we describe the methods we use to convert NAO

to GTO and also the screening method to decrease central

processing unit (CPU) time. In Sec. IV, benchmark results

are presented and the efﬁciency of this scheme is discussed.

Sec. Vconcludes this paper.

0021-9606/2011/135(3)/034110/7/$30.00 © 2011 American Institute of Physics135, 034110-1

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141.14.162.129 On: Fri, 23 Jan 2015 15:29:35

034110-2 Shang, Li, and Yang J. Chem. Phys. 135, 034110 (2011)

II. THEORY

A. The Hamiltonian

For periodic systems, the crystalline orbital ψi(k,r)

which is normalized in all space is a linear combination of

Bloch functions φμ(k,r), deﬁned in terms of atomic orbitals

χR

μ(r).

ψi(k,r)=

μ

cμ,i(k)φμ(k,r),(1)

φμ(k,r)=1

√N

R

χR

μ(r)eik·(R+rμ),(2)

where the Greek letter μis the index of atomic orbitals, iis the

sufﬁx for different bands, Ris the origin of a unit cell, Nis the

number of unit cells in the system. χR

μ(r)=χμ(r−R−rμ)

is the μth atomic orbital, whose center is displaced from the

origin of the unit cell at Rby rμ.cμ,i(k) is the wavefunction

coefﬁcient, which is obtained by solving the following equa-

tion:

H(k)c(k)=E(k)S(k)c(k),(3)

[H(k)]μν =

R

HR

μν eik·(R+rν−rμ),(4)

HR

μν =χ0

μ

ˆ

H

χR

ν,(5)

[S(k)]μν =

R

SR

μν eik·(R+rν−rμ),(6)

SR

μν =χ0

μ

χR

ν.(7)

In Eq. (5),HR

μν is a matrix element of the one-electron Hamil-

tonian operator ˆ

Hbetween the atomic orbital χμlocated in

the central unit cell 0 and χνlocated in the unit cell R.In

pseudopotential approximation, ˆ

His composed of the follow-

ing terms:

ˆ

H=ˆ

T+ˆ

VPS +ˆ

VH+ˆ

VXC,(8)

where ˆ

Tis the kinetic energy operator, ˆ

VPS is the pseu-

dopotential operator, ˆ

VHis the Hartree potential operator,

which is calculated on a real-space grid with the SIESTA

implementation18 by solving Poisson equation, and ˆ

VXC is

the exchange-correlation potential operator, which is local in

pure DFT, but is nonlocal in HF.

B. Hartree-Fock exchange

In semilocal DFT, VXC(r) is the same in every unit cell.

However, the HFX potential matrix element is deﬁned as

[VX]G

μλ =−1

2

νσ

N,H

PH−N

νσ χ0

μχN

ν

χG

λχH

σ,(9)

where G,N, and Hrepresent different unit cells. The density

matrix element PN

νσ is computed by an integration over the

Brillouin zone (BZ),

PN

νσ =

jBZ

c∗

ν, j(k)cσ, j(k)θ(F−j(k))eik·Ndk,(10)

where θis the step function, Fis the fermi energy, and j(k)

is the jth eigenvalue at point k.

In order to calculate the following ERI in Eq. (9):

χ0

μχN

ν

χG

λχH

σ= χ0

μ(r)χN

ν(r)χG

λ(r)χH

σ(r)

|r−r|drdr,

(11)

we use GTOs to ﬁt numerical radial functions, and then do the

integral analytically.

C. HSE06 functional

The HSE06 functional which splits the Coulomb operator

in a short-range (SR) and a long-range (LR) part

1

r=erfc(ωr)

r+erf(ωr)

r,(12)

has made hybrid functionals suitable for extend systems by

avoiding the problematic long-range HF exchange. The ex-

pression for HSE06 is given by

EHSE

xc =1

4ESR-HF

x(ω)+3

4ESR-PBE

x(ω)

+ELR-PBE

x(ω)+EPBE

c,(13)

where ω=0.11 bohr−1. All the ERI calculations in the fol-

lowing part of this paper is for the short-range part except for

the H2molecule benchmark calculation, i.e.,

χ0

μχN

ν

χG

λχH

σSR

=χ0

μ(r)χN

ν(r)erfc(ω|r−r|)χG

λ(r)χH

σ(r)

|r−r|drdr.

(14)

The short-range and long-range part of PBE functional is cal-

culated as in Ref. 13.

III. METHODS

A. Numerical atomic orbitals

A NAO is a product of a numerical radial function and a

spherical harmonic

φIlmn(r)=ϕIln(r)Ylm(ˆ

r).(15)

The radial part of the numerical atomic orbital ϕIln(r) is cal-

culated by the following equation:

−1

2

d2

dr2r+l(l+1)

2r2+V(r)+VcutϕIln(r)=lϕIln(r),

(16)

where V(r) denotes the electrostatic potential for orbital

ϕIln(r), and Vcut ensures a smooth decay of each radial func-

tion which is strictly zero outside a conﬁning radius rcut .

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141.14.162.129 On: Fri, 23 Jan 2015 15:29:35

034110-3 HFX with numerical atomic orbitals J. Chem. Phys. 135, 034110 (2011)

B. Using GTOs to ﬁt NAO

In our NAO2GTO approach, radial part of the numerical

atomic orbital ϕIln(r) is ﬁtted by the sum of several GTOs,

denoted as χ(r),

χ(r)≡

m

Dmrlexp(−αmr2).(17)

Parameters αmand Dmare determined by minimizing the

residual sum of squares of the difference

iχ(ri)/rl

i−ϕIln(ri)/rl

i2.(18)

We use the Levenberg-Marquardt algorithm to perform such a

nonlinear ﬁtting. Initial parameters are obtained from EMSL

Basis Set Library33 or from those of ﬁtted adjacent elements.

C. Using GTOs to get ERIs

The primitive GTO is deﬁned as

Ga,k(r)=(x−Ax)ax(y−Ay)ay(z−Az)az

×exp[−αk(r−A)2],(19)

where ais the label for angular momentum, Ais the center of

the atomic orbital, and kis the index of the exponent α.The

contract GTO is deﬁned as

φa(r)=

k

DakGak(r).(20)

So, the primitive ERIs is

[akbl|cmdn]= Gak(r)Gbl (r)Gcm(r)Gdn(r)

|r−r|drdr

(21)

and contract ERIs

(ab|cd)=

k

l

m

n

Dak Dbl Dcm Ddn[akbl|cmdn].

(22)

We use LIBINT (Ref. 34) package to calculate primitive

ERIs, where recursive schemes of the Obara-Saika method35

and the Head-Gordon and Pople’s variation thereof36 are im-

plemented. This scheme evaluates the integrals over functions

of non-zero angular momentum, starting with the auxiliary in-

tegrals over primitive s-functions. Evaluating the short-range

ERIs in Eq. (14) is only slightly more time consuming than

the regular ERIs since only the primitive s-functions integrals

are modiﬁed.13

D. Permutational symmetry of the ERIs in periodic

systems

To further improve the efﬁciency, we take the full permu-

tational symmetry of the ERIs into account for solids. The ﬁrst

index varies for all orbitals in the unit cell, and the other three

indexes vary for all orbitals in the extended cell. Considering

the following symmetry:

HR

μν =H−R

νμ ,(23)

we have

(μ0νH|λGσN)=(μ0νH|σNλG)

=(ν0μ−H|λG−HσN−H)=(ν0μ−H|σN−HλG−H)

=(λ0σN−G|μ−GνH−G)=(λ0σN−G|νH−Gμ−G)

=(σ0λG−N|μ−NνH−N)=(σ0λG−N|νH−Nμ−N).(24)

In this way, we save a factor of 8 in the number of in-

tegrals that have to be calculated. When the unit cell is big

enough to apply the -point approximation, it is reduced to

the molecular case.

E. Short-range ERI screening

Screening method is used to decrease the number of

ERIs that need to be calculated. For a short-range ERI

(χ0

μχN

ν|χG

λχH

σ)SR, we have used four screening rules to select

signiﬁcant ERI. For simplicity, we use (μν|λσ )SR to represent

(χ0

μχN

ν|χG

λχH

σ)SR in the following.

1. Rule-A: Schwarz screening

In the quantum chemistry community, a well-known

screening rule is the Schwarz inequality,37 substituting the SR

integrals in place of the 1/r integrals yields an upper bound of

the form

|(μν|λσ )SR |≤(μν|μν)SR(λσ |λσ )SR.(25)

Signiﬁcant shell pairs √(μν|μν ), which satisﬁes

√(μν|μν )SR Me≥tolerance 1, are preselected and stored.

Meis deﬁned as max √(μν|μν)SR. Before ERIs are cal-

culated, we use Schwarz inequality Eq. (25) to estimate a

rigorous upper bound, and only the ERIs with non-negligible

contributions are calculated. In the program, we test if

√(μν|μν )SR(λσ |λσ )SR ≥tolerance2, and (μν|λσ )SR is

calculated only if the answer is true. Using the exponential

decay of the charge distributions, the Schwarz screening

reduces the total number of ERIs to be computed from O(N4)

to O(N2). The two tolerance factors are set to 10−6and 10−4

in our program. Compared to ﬁner values (10−10,10

−10),

the bandgap difference for silicon is only 0.0038 eV with

double-zeta plus polarization (DZP) basis set.

2. Rule-B: Adjoined GTO screening

The adjoined Gaussian is a single, normalized s-type

GTO, which has the smallest exponents among all exponents

of the Gaussian functions in the contraction. The CRYSTAL06

package used the overlap of adjoint Gaussian to truncating

the extended lattice summation. Here, we adopt one of their

criteria as rule-B,

S(

R)νσ =4νσ

(ν+σ)23

4exp −νσ

ν+σ

R2>tolerance .

(26)

141.14.162.129 On: Fri, 23 Jan 2015 15:29:35

034110-4 Shang, Li, and Yang J. Chem. Phys. 135, 034110 (2011)

FIG. 1. The H 1snumerical radial function and the function ﬁtted by GTO.

We have not adopted the density matrix screening38 for νσ ,

because, in our implementation, ERIs are only calculated in

the ﬁrst SCF cycle and then stored in disk, opposite to a direct

SCF calculation, where the ERIs is calculated at every cycle.

3. Rule-C: NAO screening

The NAO is strictly truncated, so the Hamiltonian ma-

trix is sparse with semilocal density functionals. In the shore-

range ERI case, the HFX Hamiltonian is also sparse due to the

screened Coulomb potential.31 As a result, we store this HFX

Hamiltonian with a sparse matrix data structure. From Eq. (9),

we can see that ERIs whose index (μ, λ) corresponding two

atoms far away from each other is negligible.

4. Rule-D: Distance screening

Using Gaussian product theorem, the primitive ERIs can

be written as

[akbl|cmdn]= Gak(r)Gbl (r)Gcm(r)Gdn(r)

|r−r|drdr

= GP(r)GQ(r)

|r−r|drdr.(27)

TABLE I. ERIs and eigenvalues (in eV) of H2with single-zeta (SZ) basis

set using Hartree-Fock method. The H numerical basis function is ﬁtted using

six GTOs, with a resulting RSS =1.93 ×10−3.

Index NAO GTO Relative-error

(1111) 16.57689 16.5715 3.25 ×10−4

(2111) 11.58872 11.59492 5.34 ×10−4

(2121) 8.95741 8.96961 1.36 ×10−3

(2211) 13.56053 13.55606 3.29 ×10−4

(2221) 11.58872 11.59492 5.34 ×10−4

(2222) 16.57689 16.5715 3.25 ×10−4

Eigenvalue-1 –16.66648 –16.67283 3.81 ×10−4

Eigenvalue-2 9.95079 9.95189 1.10 ×10−4

TABLE II. Exponents αand contraction coefﬁcients c of the GTOs that

ﬁtted to silicon’s NAOs.

633 663

NAO2GTO αcαc

si_3s_zeta1 0.258271 9.33002 0.258271 9.33002

3.29515 −2.65134 3.29515 −2.65134

3.13540 2.99938 3.13540 2.99938

0.899316 −1.68234 0.899316 −1.68234

0.304626 −16.7716 0.304626 −16.7716

0.370524 9.04712 0.370524 9.04712

si_3s_zeta2 0.332608 1.53144 0.332608 1.53144

0.342460 1.47796 0.342460 1.47796

0.815035 4.77901 0.815035 4.77901

1.11140 −6.74352 1.11140 −6.74352

1.53810 2.70155 1.53810 2.70155

0.494049 −3.52678 0.494049 −3.52678

si_3p_zeta1 0.125764 0.165055 0.126706 0.167495

0.515463 6.22840 1.10580 2.11883

0.527350 −6.00512 0.786479 5.88454

0.627959 −5.46490

0.535605 2.83845

0.970822 −5.15502

si_3p_zeta2 0.234737 0.429051 0.234690 0.428098

0.638443 4.22164 0.584436 0.136917

0.651912 −4.15599 0.584436 0.136917

0.586068 0.135894

0.585084 0.0763346

0.711011 −0.419460

si_3d0.181544 0.117855 0.181544 0.117855

0.639086 0.146409 0.639086 0.146409

18.2322 0.0697915 18.2322 0.0697915

Distance screening accounts the 1/RPQ decay (where RPQ is

the distance between GPand GQ) between the charge distri-

butions, and it can reduce the ERI’s number to O(N). Follow-

ing Izmaylov et al.,31 before primitive ERI calculation, we

divided the ERI into near-ﬁeld and far-ﬁeld parts.29 In the

far-ﬁeld case, before calculating the primitive ERI, we per-

form distance screening.31 In screening, we use the following

approximation for (μν|λσ )SR :

(μν|λσ )SR ≈Kμν Kλσ

erfcθ1/2

ωRPQ

RPQ

,(28)

where

Kμν =√2π5/4

α+βexp −αβ

α+β(A−B)2.(29)

TABLE III. Residual sum of squares of the NAO2GTO ﬁtting for silicon’s

NAOs.

Orbital / Residual sum of squares 3g 6g

si_3s_zeta1 N/A 0.0011

si_3s_zeta2 N/A 0.0047

si_3p_zeta1 1.0 ×10−43.7 ×10−5

si_3p_zeta2 5.6 ×10−45.6 ×10−4

si_3d0.0025 N/A

141.14.162.129 On: Fri, 23 Jan 2015 15:29:35

034110-5 HFX with numerical atomic orbitals J. Chem. Phys. 135, 034110 (2011)

FIG. 2. The HSE06 band structures for Si within different basis sets in

SIESTA and VAS P.

IV. RESULTS AND DISCUSSION

In order to demonstrate the capabilities of our method,

we present benchmark calculations for both molecule and pe-

riodic system. Norm-conserving pseudopotentials generated

with the Troullier-Martins39 scheme, in fully separable form

developed by Kleinman and Bylander,40 are used to represent

interaction between core ion and valence electrons. All the

numerical atomic orbital basis functions are generated using

SIESTA’s default parameters.

A. H2molecule

We make a Hartree-Fock calculation for a H2molecule

with a H–H bond length of 0.74 Å. The H 1snumerical radial

function is ﬁtted by six GTOs, and the residual sum of squares

(RSS) is 1.93 ×10−3.AsshowedinFig.1, the GTOs ﬁt NAO

very well.

We have calculated ERIs for H2with NAO basis set

directly, using the method we proposed previously32 as the

FIG. 3. The HSE06 band structures for rutile TiO2calculated by SIESTA.

TABLE IV. The indirect and direct bandgaps of Si calculated with the

HSE06 functional. The Brillouin zone is sampled by a 11 ×11 ×11

Monkhorst-Pack (Ref. 41) k-point grid in VA SP and a 10 ×10 ×10 k-point

grid in our approach. In the VA SP case, different calculation parameters are

tried: admin (ALGO =N; NELMIN =5; IBRION =1); all (ALGO =all);

damped (ALGO =damped). All calculations were carried out on a 3.00 GHz-

Xeon(R) CPU with the fortran compiler ifort 10.1.015.

Our approach vasp

Bandgap 663 633 Admin All Damped Expt.

Indirect (eV) 1.1213 1.1149 1.1326 1.1343 1.1358 1.13a

Direct (eV) 3.2324 3.2271 3.3391 3.3400 3.3408 3.34–3.36b

aReference 43.

bReferences 43 and 44.

benchmark. A 20 Å ×20 Å ×20 Å box with a 200 Ry grid

cutoff is used, which makes the numerical accuracy be good

enough for ERI calculation with NAO basis set.32 As shown

in Table I, the ERIs’ and eigenvalues’ relative error that cal-

culated by the two kinds of methods is negligible.

B. Silicon

In silicon calculations, we set the lattice constant to

5.41 Å. Two schemes of GTO ﬁtting are used for the DZP ba-

sis set of silicon. One is named 663, where 6, 6, and 3 GTOs

areusedfors,p, and dorbitals, respectively. The other is a

smaller 633 ﬁtting. The ﬁtting exponents and contraction co-

efﬁcients are listed in Table II. The residual sum of squares of

each ﬁtted orbital are given in Table III.

With the GTO ﬁtted basis set, we are ready to calculate

the band structure of silicon. HSE06 functional is used with

DZP basis set, and the result is compared with that calculated

by plane wave code VASP.16 As shown in Table IV and Fig. 2,

the agreement is very good. The 1.12 eV bandgap calculated

by our approach also agree well with those by the Scuseria

group (1.21 eV) (Ref. 45) using Guassian basis set. As a more

stringent test, we also calculated bandgap of TiO2. Our result

is 3.00 eV, which agrees satisfactorily with the value calcu-

lated by VASP (3.38 eV) and agrees with experimental value

FIG. 4. The ERI calculation CPU time for computing solid Si in different

screening with HSE06 using SZ basis set .

141.14.162.129 On: Fri, 23 Jan 2015 15:29:35

034110-6 Shang, Li, and Yang J. Chem. Phys. 135, 034110 (2011)

FIG. 5. The band structure of silicon calculated in different screening with

HSE06 using SZ basis set.

(3.03 eV) (Ref. 46) better. The band structure of rutile TiO2

calculated with HSE06 functional is shown in Fig. 3.

Besides band structure, total energy is also a property

of interest in electronic structure calculation. However, pseu-

dopotential is used in this SIESTA based implementation to

describe interactions between ion and valence electrons, the

absolute value of total energy is not meaningful. As a test,

we calculate total energy for silicon with different supercells

including from 2 to 1000 atoms, the energy per atom agrees

well each other for different supercells.

The screening rules applied in our calculations are impor-

tant for efﬁciency (Fig. 4). With more screening rules, calcula-

tions become faster. Importantly, errors of band structure from

these screening rules are negligible (Fig. 5). Computational

scaling is an important factor for computational efﬁciency.

Previously, we have developed a series of O(N) methods

for updating density matrix and post-SCF calculations22–25,42

with NAO basis sets. It is very desirable to also calculate ERIs

at linear scaling with a small prefactor. We thus perform a

series of calculations for silicon bulk with different unit cell

sizes. As shown in Fig. 6, the CPU time for ERI calcula-

FIG. 6. The ERI calculation CPU time for computing solid Si with HSE06

using SZ basis set .

TABLE V. The ERI calculation CPU time for Si with a single-zeta basis

set. The Brillouin zone is sampled by 8 ×8×8 special k-points using the

Monkhorst-Pack scheme. Executables are complied with ifort 10.1.015, with

ﬂags “-g -O2.” All calculations were carried out on 3.00GHz-Xeon(R) CPU.

Method CPU time (s)

CRYSTAL98 (bipolar) 18.06

CRYSTAL98 (nobipolar) 63.06

Our approach 19.07

tion clearly shows a prefect linear scaling behavior. Without

screening rule-C applied, we still have a linear scaling behav-

ior but with a bigger prefactor.

For system as small as silicon, our approach is already

very efﬁcient. First, our calculations are much faster than

plane wave based HFX calculation. With all the ABCD

screening rules applied, the CPU time for ERI calculation in

our approach is even comparable with that of programs with

Gaussian basis set, such as CRYSTAL 98 (Table V). Consider-

ing the linear scaling behavior we achieved, our approach is

expected to be very comparative for large systems.

V. CONCLUSIONS

In this work, we present an implementation of HSE06

functional in SIESTA. We ﬁrst represent NAO with GTOs, and

then we calculate ERIs with GTO analytically. The full per-

mutational symmetry of ERIs for solids is considered, which

leads to a factor of 8 saving in the number of ERIs to be

calculated and stored. Additionally, four screening rules are

adopted in our implementation. We test our method for sys-

tems from molecule to solid. When DZP basis set is used,

good agreement with results obtained from plane wave pack-

ages is reached. The CPU time for ERI calculation is linear

scaling, with a favorable prefactor.

ACKNOWLEDGMENTS

We thank Wanzhen Liang and Richard M. Martin for

valuable discussions and Qiang Fu for a critical reading of

this paper. This work is partially supported by the Innovation

Fund Projects (2009) of University of Science and Technol-

ogy of China, National Natural Science Foundation of China

(Grant Nos. 20873129 and 20933006), and by the National

Key Basic Research Program (2011CB921404).

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