arX

iv:1

810.

0154

9v1

[ph

ysic

s.co

mp-

ph]

3 O

ct 2

018

Efficient ab initio auxiliary-field quantum Monte Carlo calculations

in Gaussian bases via low-rank tensor decomposition

Mario Motta,1 James Shee,2 Shiwei Zhang,3, 4 and Garnet Kin-Lic Chan1

1Division of Chemistry and Chemical Engineering,

California Institute of Technology, Pasadena, CA 91125, USA2Department of Chemistry, Columbia University, New York, NY 10027, USA

3Center for Computational Quantum Physics, Flatiron Institute, New York, NY 10010, USA4Department of Physics, College of William and Mary, Williamsburg, VA 23187-8795, USA

We describe an algorithm to reduce the cost of auxiliary-field quantum Monte Carlo (AFQMC)calculations for the electronic structure problem. The technique uses a nested low-rank factoriza-tion of the electron repulsion integral (ERI). While the cost of conventional AFQMC calculations inGaussian bases scales as O(N4) where N is the size of the basis, we demonstrate numerically thathigh accuracy can be achieved in the calculation of ground-state energies through tensor decomposi-tion with O(N3) scaling. This is the same computational scaling as observed in AFQMC algorithmsusing plane-waves and pseudo-potentials, but with a reduced prefactor, due to the compactness ofGaussian basis sets. The algorithm is applied to hydrogen chains and square grids, water clusters,and hexagonal BN.

I. INTRODUCTION

Correlated electronic structure calculations often re-quire to store and manipulate tensors, that can havehigh rank and act on vector spaces with high dimen-sion. Frequently, the input-output and algebraic opera-tions involving such high-rank tensors constitute a com-putational bottleneck of the calculations.The cost of tensor manipulations and storage can be

significantly reduced by low-rank decompositions [1–4],in which a higher-rank tensor is represented as contrac-tions of lower-rank tensors. The most common tensorappearing in Gaussian basis calculations is the rank-4electron-repulsion integral (ERI) tensor

Vprqs =

∫

drdr′ϕp(r)ϕq (r′)

1

|r− r′|ϕr(r)ϕs (r′) , (1)

where the real-valued Gaussian atomic orbitals (AOs)ϕp(r)Mp=1 form a non-orthogonal basis for the one-electron Hilbert space. Density-fitting (DF) [1, 5–7] andmodified Cholesky (CD) [2, 8, 9] are commonly appliedto obtain a low-rank decomposition of the ERI in the AObasis in terms of a rank-3 tensor Lγpr,

vprqs ≃Nγ∑

γ=1

LγprLγqs . (2)

Importantly, it is known that the error in such approxi-mations of the ERI decay exponentially with the numberof vectors Nγ , and require only M = O(N) vectors fora fixed error per atom as a function of increasing sys-tem size[10]. Using the DF or CD approximations re-duces the cost of storing the ERI from O(N4) to O(N3)[10], although the computational scaling of most elec-tronic structure methods using DF or CD integrals isnot changed.More recently, several strategies to represent the ERI

by contractions of rank-2 tensors have been introduced.

These include the tensor hyper-contraction scheme [4,11–15] and the nested matrix diagonalization introducedin [16]. Unlike CD or DF, these low-rank representa-tions can be used to obtain lower-computational scalingin many different electronic structure methods, includingcoupled-cluster [14, 17–20] and Moller-Plesset perturba-tion theory [15], and even algorithms for quantum simu-lation of the electronic structure Hamiltonian [21].

In the present work, we apply the nested matrix diag-onalization introduced in [16] to the auxiliary-field quan-tum Monte Carlo (AFQMC) method in a Gaussian basis[22, 23]. While the cost of conventional AFQMC in suchbases scales as O(N4), relying on this simple and efficientform of low-rank factorization reduces the complexity ofthe method to O(N3) [24] with a simple modificationof the original algorithm. The same computational scal-ing is observed in applications to the electronic struc-ture problem with plane-waves as the one-electron basis,where the ERI is naturally represented in a low-rank fac-torized form, and the fast Fourier transform leads to re-duced scaling [25]. However, in most scenarios, Gaussianbasis sets are more compact than plane-wave bases [26]and thus the cubic scaling Gaussian basis AFQMC algo-rithm will generally show a reduced computational pref-actor compared to plane wave implementations.

The rest of the paper is organized as follows. In SectionII we provide a brief description of the AFQMC method.In Section III, we show that the low-rank decomposi-tion can be used to accelerate the most expensive part ofAFQMC simulations, the calculation of the local energy.In Section IV, we assess the performance and accuracyof AFQMC calculations using Gaussian basis sets basedon low-rank decompositions, and conclusions are drawnin Section V.

2

II. THE AFQMC METHOD

In this Section, we introduce the AFQMC method andillustrate that the origin of its quartic cost for generalelectronic structure problems lies in the local energy cal-culation. Throughout the rest of the paper, we use lettersprqs to indicate a general orthogonal or non-orthogonalbasis function ϕp (range 1 . . .N), ijkl for particles (in-dices range from 1 . . . O), γµν for auxiliary indices asso-ciated with the low-rank decompositions (range 1 . . .Mfor γ, 1 . . . ργ for µν). Spin labels are suppressed forcompactness.The AFQMC [22, 23] is a projective quantum Monte

Carlo (QMC) method, which estimates the ground-stateproperties of a many-fermion system by statistically sam-pling the ground-state wavefunction

|Ψβ〉 =e−βH |ΦT 〉

〈ΦT |e−βH |ΦT 〉β→∞−−−→ |Ψ0〉

〈ΦT |Ψ0〉. (3)

In Equation (3), Ψ0 is the ground-state wavefunction ofthe system, ΦT is an initial wavefunction not orthogonalto Ψ0, which for simplicity we assume to be a single Slaterdeterminant, and H is the Hamiltonian of the system,which without loss of generality [23] can be written inthe form

H = E0 +∑

pq

tpqEpq +1

2

∑

prqs

VprqsEprEqs . (4)

Comprising H are a constant correction, a one-body partwritten in terms of the Choi [27] operator Epq = a†paq,and a two-body part. The underlying single-particle basisin Eq. (4) must be an orthogonal basis. Thus, whenemploying a Gaussian AO basis, the AO ERI in Eq. (1)must first be transformed to an orthogonal basis, as mustthe DF or CD vectors in the decomposition (2). Using thetransformed DF or CD vectors, the two-body part can bewritten as a sum of squares of one-body operators,∑

prqs

VprqsEprEqs =∑

γ

v2γ , vγ =∑

pr

LγprEpr . (5)

These are illustrated in Figure 1(a) and (b). For suf-ficiently large β, expectation values computed over Ψβyield ground-state averages. AFQMC projects ΨT to-wards Ψ0 iteratively, writing

e−βH =(

e−∆τH)n

, (6)

where ∆τ = βnis a small imaginary-time step. The prop-

agator is represented, through a Hubbard-Stratonovichtransformation [28, 29], as

e−∆τH =

∫

dx p(x) B(x) , (7)

where

B(x) ∝ exp

−∆τ H1 + i√∆τ

Nγ∑

γ=1

xγ vγ

(8)

is an independent-particle propagator that dependson the vector of fields x, p(x) is the standard nor-

mal Nγ-dimensional probability distribution and H1 =∑

pq tpqEpq is the one-body part of H . The representa-

tion (7) maps the original interacting many-fermion sys-tem onto an ensemble of non-interacting systems subjectto a fluctuating potential. The imaginary-time projec-tion can be realized as an open-ended random walk overpaths of auxiliary-fields x [22]. Importance sampling thetrajectories of the random walk leads to a representationof Ψβ as a stochastic weighted average of Slater determi-nants.

|Ψβ〉 ≃1

∑

wWw

∑

w

Ww

|Φw〉〈ΦT |Φw〉

. (9)

Because the phase in vγ can be complex for general two-body interactions, AFQMC suffers from a phase problem.This can be controlled using a trial state |ΦT 〉 and im-posing the phaseless approximation (Ph) and a real localenergy estimator [22, 23]; the error of these approxima-tions vanishes if the trial state is exact.The accuracy of Ph-AFQMC calculations of ground-

and excited-state energies has been extensively bench-marked both in ab initio studies [30–33] and lattice mod-els of correlated electrons [34, 35]. The random walkstake place in the over-complete manifold of Slater de-terminants, in which fermion antisymmetry is by con-struction maintained in each walker. Recently, the Ph-AFQMC has also been extended to the calculation ofgeneral ground-state properties, energy differences andinteratomic forces in realistic materials [36–38].In ab initio computations, the electron repulsion inte-

grals entering into the AFQMC calculation can be ob-tained in different computational bases, such as plane-waves and pseudo-potentials [22, 39] or Gaussian type or-bitals [40]. This choice of representation is important be-cause it affects the cost of the AFQMC algorithm. Whenplane-waves are used, the standard AFQMC methodol-ogy is known to scale as O(N3), as documented in Ap-pendix VII. When using a Gaussian basis, on the otherhand, state-of-the-art calculations feature O(N4) cost.This computational bottleneck in both cases is the localenergy calculation, which we describe below.

A. Local energy calculation

AFQMC calculations require the computation of thefollowing local energy functional for each sample,

Eloc(Φ) =〈ΦT |H|Φ〉〈ΦT |Φ〉

, (10)

from which the total energy is obtained as E =∑

iWwEloc(Φw). The local energy is also needed to de-termine the weights in Ph calculations. The most de-manding part of its calculation comes from the two-body

3

(a) V

r

p s

q

(b) L

r

p

γ

L

q

s

(c) µ

X

U

r

p

γ

γ

ν

U

X

q

s

FIG. 1. (color online) Pictorial illustrations (a) of therank-4 electron repulsion integral (ERI) tensor Vprqs, (b)of its Cholesky (CD) or density-fitting (DF) decomposition

Vprqs =∑Nγ

γ=1 LγprL

γqs, and (c) of the low-rank decomposition

Vprqs =∑Nγ

γ=1

∑ργµν=1 X

γµp Uγµ

r Xγνq Uγν

s used in the presentwork. Lines emerging from colored blocks indicate free in-dices, and lines connecting blocks, indices summed over. De-compositions (b,c) break down the original ERI into tensorsof progressively lower rank, leading to increasingly moderatememory requirements and faster evaluation of the local en-ergy.

term H2 which, from the generalizedWick’s theorem [41],can be written as

2Eloc,2(Ψ) = 2〈ΨT |H2|Ψ〉〈ΨT |Ψ〉 =

=∑

prqs

Vprqs (GprGqs −GpsGqr)(11)

where the one-body reduced density matrix (RDM1)

Gpr =〈ΦT |a†par|Φ〉

〈ΦT |Ψ〉 =[

Φ(

ΦTΦ)−1

ΦT

]

rp

=∑

i

ΘriΦT ip .(12)

appears, defined in terms of the matrices Φ (of dimen-sion N × O) and ΦT (O × N) parametrizing the Slaterdeterminant and trial wave-function respectively,

|Φ〉 =∏

i

a†ψi|∅〉 , |ψi〉 =

∑

p

Φpi|ϕp〉

〈ΦT | = 〈∅|∏

i

aϕi, 〈ϕi| =

∑

p

ΦT ip〈ϕp| .(13)

Note that the expression for the RDM1 sample resemblesthe expression for the RDM1 of the trial Slater determi-nant, with one ΦT matrix (walker independent) replacedby Θ (dependent on the walker). Explicit evaluation of(12) costs O(ON2) per sample while the summation inthe two-body local energy costs O(N4) per sample. ForN ≫ O, it is more efficient [40] to first contract the two-body matrix elements with ΦT ,

Vpiqj =∑

rs

ΦT irΦT jsVprqs, (14)

which may be carried out once and stored at the start ofthe AFQMC calculation at a cost of O(ON4 + O2N3).

The local energy then follows as the sum

2Eloc,2(Φ) =∑

piqj

VpiqjΘpiΘqj (15)

at a cost of O(O2N2) per sample. When memory is nota limitation, this is the most efficient conventional al-gorithm for local energy evaluation and is the one wecompare against in this work.

As mentioned in the introduction, the Cholesky decom-position (2) allows one to significantly reduce the storagerequirements by replacing the 4-index integrals by a trun-cated set of 3-index quantities. However, it does not re-duce the computational cost of local energy evaluation.Inserting (12) into (11) and using the CD form in (2)(after transformation to an orthogonal basis) gives

2Eloc,2(Φ) =∑

ijγ

fγiifγjj − fγijf

γij , (16)

with the intermediate fγij defined as

fγij =∑

pr

(

ΦT ipLγpr

)

Θrj. (17)

This is computed most efficiently by precomputing andstoring the quantity in brackets, Lγir =

∑

p ΦT ipLγpr, at

the beginning of the AFQMC run, at cost O(ON2M),and subsequently carrying out the second contraction foreach sample with O(O2NM) ∼ O(N4) cost. However,as M > N , the reduced memory cost afforded by CDis offset by an increased computational cost of the lo-cal energy evaluation, compared with the conventionalalgorithm in (11). The operations described so far areillustrated diagrammatically in Figure 2.

(a) 2Eloc,2(Φ) = G L

r

p

γ

GL

q

s

− L

r

p

γ

G

G

L

q

s

(b)

p

r

G =

p

i

r

ΦT

Θ

,

p

i

ΦT

r

L

γ=

i

r

L

γ,

r

i

Θ

j

L

γ=

i

r

L

γ=

j

i

fγ

(c) 2Eloc,2(Φ) = fi

γf j − f

γ

i

j

f

FIG. 2. (color online) (a) Pictorial representation of the lo-cal energy calculation based on the CD decomposition of theERI. (b) separable structure of the RDM1, of its use in pre-computing the tensors L and f . (c) Expression of the localenergy based on the f tensor.

To overcome this increased cost, we now describe howwe can exploit additional structure in the Cholesky vectorLγpr.

4

III. LOW-RANK FACTORIZATION VIA

NESTED MATRIX DIAGONALIZATION AND

ACCELERATION OF LOCAL ENERGY

EVALUATION

The Cholesky vectors in the AO basis are sparse, be-cause for a given Cholesky index γ, the elements Lγpr arenon-zero only if pr correspond to spatially close AO func-tions. This sparsity can also be revealed via a low-rankfactorization of Lγ , as first introduced in [16].To illustrate this structure, we start from the CD of

the ERI, Equation (2), where all elements of the residual

Rprqs = Vprqs −∑

γ

LγprLγqs (18)

are kept smaller in absolute value than a predefinedthreshold εCD. After transformation to an orthogonalbasis, we carry out an eigenvalue decomposition of thematrix Lγpr for each γ,

Lγpr =∑

µ

UγpµσγµU

γrµ , (19)

and only eigenvalues larger in absolute value than a pre-defined threshold εET are kept, |σγµ| ≥ εET . This eigen-value truncation (ET) leads to the approximation

Vprqs ≃∑

γ

∑

µν

(Uγpµσγµ)U

γrµ(U

γqνσ

γν )U

γsν = ,

=∑

γ

∑

µν

XγpµU

γrµX

γqνU

γsν

(20)

where ργ ≤ N is the number of retained eigenvalues forthe matrix Lγ and Xγ

pµ = Uγpµσγµ. The decomposition

(20) is diagrammatically illustrated in Figure 1(c).The key is that, due to the sparsity in Lγ , the average

number of eigenvalues of the Cholesky vectors

〈ργ〉 =1

Nγ

∑

γ

ργ , (21)

grows logarithmically with increasing system size [16],as shown in Figure 3 focussing on hydrogen chains andsquare grids. Following [16], we fit 〈ργ〉 to a function ofthe form α log(NH + β) + γ, in all cases observing log-arithmic behavior. (When only the size of the basis isincreased, for example in order to approach the completebasis set limit, we have observed that 〈ργ〉 grows lin-early withM , albeit with a prefactor smaller than 1 [21]).Thus for a fixed truncation accuracy and up to logarith-mic factors, we can consider the number of significanteigenvalues as independent of system size. In the presentwork we choose εCD = εET ≡ ε, but the two thresholdscan in principle be tuned separately [16].The low-rank structure revealed in the Cholesky vec-

tors leads directly to a reduced cost to construct the fol-

lowing intermediate fγij in the evaluation of the local en-

ergy (see also Figure 4),

fγij =∑

prγµ

(ΦT ipUγpµσ

γµ)(U

γrµΘrj)

=∑

γµ

Aγµi Bγµj (22)

where A can be evaluated at the beginning of theAFQMC run with cost O(NMργ), B is evaluated foreach sample with cost O(ONMργ), and the assemblyinto fγij is O(O2Mργ) per sample. For logarithmic ργ ,

this is then O(N3) cost for the energy evaluation.The use of the low-rank factorization not only results

in a lower-scaling local energy calculation, but also ina significant reduction in the memory needed to storethe ERI and to compute the local energy. This memoryreduction is documented in Figure 5, where the ratio be-tween the size of the tensors V ′ and A, B is shown forhydrogen chains and grids. As seen, for large systems,the size of A, B is only ≃ 5% of that of V ′.

IV. RESULTS

We now apply the formalism outlined in Section IIIto several test systems, including both molecules andcrystalline solids. In each case we show the logarithmicgrowth of 〈ργ〉 with system size, and compare the localenergy evaluation time TEloc

from conventional AFQMCand AFQMC with CD+ET, and assess the accuracy ofthe ET procedure, investigating the scaling with systemsize. Timing calculations were performed on a clusterwith nodes having 2 CPUs with 14 cores each (Intel E5-2680, 2.4 GHz).

A. Networks of H atoms

1. Timing and accuracy

We first consider the test case of hydrogen (H) chains[33, 42], at a representative bondlength R = 1.8 aB, usingthe minimal STO-6G basis and RHF trial wavefunction.We use identical thresholds for CD and eigenvalue trun-cation, εCD = εET = 10−4, 10−5, 10−6. The local en-ergy evaluation time using the conventional AFQMC for-mula, Eq. (15), and CD+ET-based AFQMC (CD+ET-AFQMC) of Section III, is shown in Figure 6. Localenergy calculations times are reproduced well by the for-mulae

t ≃ t0NαH

tCD+ET ≃ t0NβH log(NH + γ)

(23)

with exponents α = 3.91(2), 3.99(1), 3.99(1) and β =3.13(6), 3.12(9), 3.5(1) for ε = 10−4, 10−5, 10−6 au re-spectively. The two local energy calculation strategies

5

40

80

120

160

〈ργ〉

CD, ET, ε = 10−4 CD, ET, ε = 10−5 CD, ET, ε = 10−6

0 40 80 120 160 200

NH

0

40

80

120

160

〈ργ〉

35

70

105

140

175

〈ργ〉

CD, ET, ε = 10−4 CD, ET, ε = 10−5 CD, ET, ε = 10−6

0 14 28 42 56 70

NH

0

70

140

210

280

350

〈ργ〉

FIG. 3. (color online) Logarithmic increase in the average number 〈ργ〉 of eigenvalues for H chains (top) and square grids(bottom), at the representative bondlength R = 1.8aB , at STO-6G (left) and cc-pVDZ (right) level, using thresholds ε = 10−4,10−5, 10−6 a.u. (red circles, green squares, blue diamonds) are explored. Black dotted lines represent the number M of basisfunctions, providing an upper bound for 〈ργ〉. Coloured lines are the result of fit to α log(NH + β) + γ. Sub-linear growth isvisible in all cases.

display the anticipated quartic and soft-cubic complex-ities respectively. The prefactors in the two functionsdetermine the number N∗

H of H atoms required for thetwo curves to cross. We find that N∗

H ≃ 25, 35, 40 for thethree thresholds we have considered.In the insets, we compute the difference ∆Ec between

the correlation energies per atom from AFQMC andCD+ET, as function of the number of H atoms, usingthe estimator

∆Ec =1

Nw

∑

w

Eloc,c(Φw)− E′loc,c(Φw) , (24)

where Eloc,c(Φ) = Eloc(Φ) − EHF is defined in termsof the standard local energy functional (15), but us-ing integrals reconstructed from the CD vectors, whileE′loc,c(Φ) = E′

loc(Φ) − E′HF is formulated in terms of

the CD+ET expression, Section (III), for the local en-ergy. In Figure 6, ∆Ec is evaluated on 6 independentlygenerated populations Φww of walkers equilibrated forβ = 2E−1

Ha. Using all thresholds, the energies per atomagree to within 0.02% of the total correlation energy ex-trapolated to the thermodynamic limit (TDL), confirm-ing the good accuracy of the CD+ET decomposition forconservative choices of the threshold ε.

2. Asymmetric dissociation of the H chain

We next study the asymmetric dissociation of the infi-nite H chain using the STO-6G basis in Figure 7. Morespecifically, we compute the potential energy surface of anetwork of H atoms at positions Rk,± = (0, 0, zk,±) with

zk,± = ±R2+ k(R′+R), k = 0 . . . N

2− 1, for a total num-

ber of atoms between NH = 10 and NH = 50, as functionof the intra-bond and inter-bond lengths R, R′. We usethe UHF Slater determinant as trial wavefunction. Forall R,R′ in a mesh of points between 1.2 and 3.6 aB, weextrapolate the energy per atom E(R,R′, N) to the TDLusing standard procedures [33], and compute correlationenergies using AFQMC with CD+ET and the truncationthreshold ε = 10−5 au. The extrapolated potential en-ergy surface E(R,R′) = limN→∞E(R,R′, N) is shownin Figure 7, and values for R′ = 1.6, 2.4, 3.2 aB are givenin Table I.

The diagonal of Figure 7 corresponds to the symmetricdissociation of the chain, R = R′ [33], the minimum en-ergy being reached at the saddle point R = R′ ≃ 1.83 aB.For large R,R′ the potential energy surface increases to-wards the energy EH = 0.471EHa of a single H atom

6

(a) 2Eloc,2(Φ) = G µ

X

U

r

p

γ

γ

ν

U

X

q

s

G −

G

G

µ

X

U

r

p

γ

γ

ν

U

X

q

s

(b)

ΦT

µ

X

i

p

γ=

Θ

ν

U

j

p

γ=

µ

A

i

γ

ν

B

i

γ

µ

A

B

j

i

γ

γ

, =

i

j

fγ

FIG. 4. (color online) Pictorial illustrations (a) of the localenergy calculation based on the CD+ET decomposition ofthe ERI, (b) of the precomputed and intermediate tensorsinvolved in the calculation. The final expression for the localenergy coincides with the one in Figure 2(c).

R′ E(R = 1.6, R′) E(R = 2.4, R′) E(R = 3.2, R′)

1.2 -0.51517(6) -0.55578(9) -0.5652(1)

1.4 -0.52857(9) -0.5619(2) -0.5704(1)

1.6 -0.53362(6) -0.5582(2) -0.5660(3)

1.8 -0.54288(8) -0.5507(2) -0.5569(4)

2.0 -0.5498(1) -0.5411(1) -0.5454(3)

2.4 -0.5582(2) -0.5233(1) -0.5223(1)

2.8 -0.5634(2) -0.5219(1) -0.5037(1)

3.2 -0.5660(2) -0.5223(1) -0.4915(1)

3.6 -0.5672(2) -0.5228(1) -0.4902(1)

TABLE I. Energy per atom of the H chain at STO-6G level oftheory, extrapolated to the thermodynamic limit, as functionof the inter-bond length R′ for R = 1.6, 2.4, 3.2 (left to right).

in the STO-6G basis, and the global minimum of theenergy is reached for R′ → ∞, R ≃ 1.4 aB, correspond-ing to a collection of uncoupled H2 molecules, with en-ergy EH2

= −0.573EHa. This illustrates the well-knownPeierls instability of equally spaced atomic chains underlattice distorsions.

We continue our assessment of accuracy and perfor-mance by studying, in Figure 8, two-dimensional squaregrids of H atoms, where the H atoms occupy positionsRij = (0, iR, jR), i, j = 0 . . . n − 1. Here n is related tothe number NH of atoms in the grid as NH = n2, andwe work at the representative bondlength R = 1.8 aB.

The trends seen for H chains are confirmed: the stan-dard and CD+ET-based local energy calculation timesare well described by (23) with exponents α = 4.09(2),4.31(3), 4.66(7) and β = 3.06(18), 3.25(16), 3.28(15)for ε = 10−4, 10−5, 10−6 au respectively. Crossover be-

0.0

0.1

0.2

0.3

0.4

0.5

0.6

CD, ET, ε = 10−4 CD, ET, ε = 10−5 CD, ET, ε = 10−6

0 40 80 120 160 200

NH

0.0

0.2

0.4

0.6

0.8

1.0

MC

D+

ET

MC

D

FIG. 5. (color online) Ratio between the memory requiredfor local energy precomputing for AFQMC with CD (MCD)and AFQMC with CD+ET (MCD+ET ), as a function of thenumber NH of hydrogen atoms for H chains (top) and squaregrids (bottom) at the representative bondlength R = 1.8 a.u.at STO-6G level of theory. Three different truncation thresh-olds, ε = 10−4, 10−5, 10−6 a.u. are explored (red circles, greensquares, blue diamonds).

tween the two approaches is seen for N∗H ≃ 50, 120, 170

for increasingly small threshold. The discrepancy ∆Ecbetween correlation energies based on AFQMC andAFQMC with CD+ET is consistently below 0.01% ofthe correlation energy per atom extrapolated to the TDL,further confirming the accuracy of the truncation scheme.

B. Water clusters

To test larger basis sets and heavier elements, in Figure9, we investigate 38 water clusters (motivated by studiesof water clusters in the terrestrial atmosphere) containing2-10 water molecules [43], using the heavy-augmented cc-pVDZ basis (aug-cc-pVDZ for O, cc-pVDZ for H), trun-cation threshold ε = 10−4 au and RHF trial wavefunc-tion. Also in this case, the average number of retainedeigenvalues 〈ργ〉 grows logarithmically with the size of thesystem, as measured by the number of H2O molecules,leading to a local energy evaluation featuring soft-cubic

7

10−5

10−4

10−3

10−2

10−1

100

AFQMC, CD + ET, ε = 10−4

AFQMC, CD + ET, ε = 10−5

AFQMC, CD + ET, ε = 10−6

AFQMC, CD, ε = 10−4

AFQMC, CD, ε = 10−5

AFQMC, CD, ε = 10−6

10−5

10−4

10−3

10−2

10−1

100

TE

loc[s]

10 20 50 100 200

NH

10−5

10−4

10−3

10−2

10−1

100

0 60 120 180

NH

-2

-1

0

1

2

∆E

c[µ

EH

a]

-0.0128

-0.0064

0.0

0.0064

0.0128

∆E

c[%

Ec,T

DL]

0 60 120 180

NH

-2

-1

0

1

2

∆E

c[µ

EH

a]

-0.0128

-0.0064

0.0

0.0064

0.0128

∆E

c[%

Ec,T

DL]

0 60 120 180

NH

-2

-1

0

1

2

∆E

c[µ

EH

a]

-0.0128

-0.0064

0.0

0.0064

0.0128

∆E

c[%

Ec,T

DL]

FIG. 6. (color online) Main figures: local energy evaluationtime TEloc

as function of the number NH of hydrogen atomsfor H chains at the representative bondlength R = 1.8 a.u. atSTO-6G level of theory, from AFQMC with CD (empty mark-ers) and AFQMC with CD+ET (filled markers). Truncationthresholds, ε = 10−4, 10−5, 10−6 a.u. (top to bottom) are ex-plored. Solid, dashed lines are the result of fit of AFQMCwith CD, CD+ET to αN

βH , αNβ

H log(NH + γ) respectively.Insets: average difference in the correlation part of the localenergy, per atom, between AFQMC with CD and CD+ET.

scaling (upper panel). The dependence of the local en-ergy calculation time on the number of water molecules,shown in the inset of the upper panel, is again well rep-resented by the functional forms (23) with α = 4.01(1)and β = 3.21(4), so that crossover between conventionalAFQMC and CD+ET local energy calculation times isseen at NH2O ≃ 13.

1.2 1.6 2.0 2.4 2.8 3.2 3.6

R[aB]

1.2

1.6

2.0

2.4

2.8

3.2

3.6

R′ [

aB]

−0.575

−0.555

−0.535

−0.515

−0.495

−0.475

−0.455

ET

DL(R

,R′ )[E

Ha]

R R

R′

FIG. 7. (color online) Energy per atom of the H chain atSTO-6G level of theory, as function of the intra-bond andinter-bond lengths R,R′. Results are obtained for R,R′ =1.2, 1.4, 1.6, 1.8, 2.0, 2.4, 2.8, 3.2, 3.6, and the potential energysurface produced via cubic spline interpolation.

In the inset of the lower panel, we show the differ-ence ∆Ec between the correlation energies per atom fromAFQMC with CD integrals and CD+ET, as a functionof the number of monomers. ∆Ec is evaluated on 6 inde-pendently generated populations Φww of walkers equi-librated for β = 2E−1

Ha and, for a given cluster size ∆Ec isaveraged over all cluster structures with the same numberof monomers. For example, for N = 5, ∆Ec is averagedover the 6 water pentamers labelled CYC, CAA, CAB,CAC, FRA, FRB, FRC in [43]. The accuracy seen fornetworks of H atoms is also seen here.The binding energy per water molecule for the most

stable clusters, labelled 2Cs, 3UUD, 4S4, 5CYC, 6PR,7PR1, 8D2d, 9D2dDD, 10PP1 in [43], is shown in thelower panel of Figure 9. As seen, the correlation energyper molecule decreases almost monotonically with thenumber of monomers in the cluster, reaching Eb/NH2O ≃−9 kcal/mol for NH2O ≥ 8.Numerical data supplied in Table II provide a com-

parison with RHF, MP2, CCSD and CCSD(T). Energiesfrom these methodologies are computed without perform-ing any truncation on the Hamiltonian, while AFQMCenergies are estimated adding the correlation energy froma calculation with CD+ET to the RHF energy of the un-truncated Hamiltonian,

EAFQMC =(

ECD+ETAFQMC − ECD+ET

RHF

)

+ ERHF . (25)

As seen, correlated methods are in relatively good agree-ment with each other. AFQMC is in good agreementwith CCSD(T), with an average deviation of only ∆ =

8

10−5

10−4

10−3

10−2

10−1

100

101

AFQMC, CD + ET, ε = 10−4

AFQMC, CD + ET, ε = 10−5

AFQMC, CD + ET, ε = 10−6

AFQMC, CD, ε = 10−4

AFQMC, CD, ε = 10−5

AFQMC, CD, ε = 10−6

10−5

10−4

10−3

10−2

10−1

100

TE

loc[s]

10 20 50 100 200

NH

10−5

10−4

10−3

10−2

10−1

100

0 50 100 150 200

NH

-2

-1

0

1

2

∆E

c[µ

EH

a]

-0.01

-0.005

0.0

0.005

0.01

∆E

c[%

Ec,T

DL]

0 50 100 150 200

NH

-2

-1

0

1

2

∆E

c[µ

EH

a]

-0.01

-0.005

0.0

0.005

0.01

∆E

c[%

Ec,T

DL]

0 50 100 150 200

NH

-2

-1

0

1

2

∆E

c[µ

EH

a]

-0.01

-0.005

0.0

0.005

0.01

∆E

c[%

Ec,T

DL]

FIG. 8. (color online) Main figures: local energy evaluationtime as function of the number NH of hydrogen atoms, for Hsquare grids at the representative bondlength R = 1.8 a.u. atSTO-6G level of theory, from AFQMC with CD (empty mark-ers) and AFQMC with CD and ET (filled markers). Crossoverbetween the two strategies is seen for NH ≃ 50, 100, 150 forincreasingly small threshold. Insets: average difference in thecorrelation part of the local energy, per atom, between state-of-the-art AFQMC and AFQMC with eigenvalue truncation.

−0.59(29) kcal/mol. Data for the different water pen-tamers are showed in Figure 10. Binding energies fromcorrelated methods are in good agreement with eachother and display the same trends. The average devi-ation between CCSD(T) and AFQMC is ∆ = 0.06(61)kcal/mol.

0 3 6 9 12

NH2O

0

55

110

165

220

〈ργ〉

1 2 3 4 5 6 7 8 9 10

NH2O

-12

-9

-6

-3

0

3

Eb

NH

2O

[

kcal

mol

]

1 5 10 20

NH2O

10−4

10−3

10−2

10−1

100

101

102

TE

loc[s]

0 3 6 9 12

NH2O

-12

-6

0

6

12

∆E

c

[

µkcal

mol

]

FIG. 9. (color online) Top: average number 〈ργ〉 of retainedeigenvalues as function of the number of H2O molecules insmall water clusters [43], using ε = 10−4 au. Inset: localenergy calculation time from AFQMC with CD and CD+ET(empty, filled symbols). Solid, dot-dashed lines indicate fitto a power law and a power law with logarithmic correctionsrespectively. Bottom: AFQMC binding energy per monomer,for the most stable water clusters with given number NH2O ofmonomers. Inset: difference ∆Ec between correlation energyper monomer from AFQMC with CD and with CD+ET, forall clusters.

C. Two-dimensional hexagonal boron nitride

We now consider a crystalline solid, 2D hexago-nal boron nitride (BN). To perform these calcula-tions we used an underlying single-particle basis ofcrystalline Gaussian-based atomic orbitals, which aretranslational-symmetry-adapted linear combinations ofGaussian atomic orbitals [44]. Core electrons were re-placed with separable norm-conserving GTH pseudopo-tentials [45, 46], removing sharp nuclear densities. Ma-trix elements for the Hamiltonian of the system werecomputed with the PySCF [47] package, using the GTH-DZV Gaussian basis set [48]. The RHF state was usedas trial wavefunction.Size effects were removed studying increasingly large

supercells at the Γ point. Supercells were obtained re-peating the primitive, two-atom cell Nx = Ny = 1, . . . , 5times along directions ax, ay sketched in Figure 11, andwe operated at the representative bondlength RBN =

9

5-CYC 5-CAA 5-CAB 5-CAC 5-FRA 5-FRB 5-FRC

cluster

20

25

30

35

40|E

b|

[

kcal

mol

]

RHF MP2 CCSD CCSD(T) AFQMC

FIG. 10. Binding energy for water pentamers by RHF, MP2,CCSD, CCSD(T) and AFQMC(CD+ET), in kcal/mol, usingthe heavy-augmented cc-pVDZ basis.

cluster Eb,RHF Eb,MP2 Eb,CCSD Eb,CCSD(T ) Eb,AFQMC

2Cs -3.815 -5.217 -4.912 -5.179 -5.11(31)

3UUD -10.521 -15.833 -14.670 -15.619 -14.78(64)

4S4 -19.001 -28.358 -26.210 -27.865 -26.49(46)

5CYC -25.297 -37.482 -34.627 -36.776 -36.27(59)

6PR -29.917 -47.246 -43.727 -46.823 -46.25(66)

7PR1 -37.486 -59.149 -54.619 -58.470 -60.04(77)

8D2s -46.650 -74.924 -69.023 -74.044 -74.7(1.1)

9D2dDD -53.395 -84.816 -78.164 -83.739 -81.3(1.4)

10PP1 -60.449 -96.615 -89.029 -95.453 -93.7(1.4)

TABLE II. Binding energy for the most stable water clus-ters reported in [43], by RHF, MP2, CCSD, CCSD(T) andAFQMC(CD+ET), in kcal/mol, using the heavy-augmentedcc-pVDZ basis.

2.5 A to illustrate the effects of the eigenvalue trunca-tion on top of the DF approximation. The ERI was ob-tained using the Gaussian density fitting (DF) approxi-mation [49], and eigenvalue truncation was performed onthe DF operators with truncation thresholds ε = 10−4,5 · 10−4, 10−3 au.

In the upper panel of Figure 11 we illustrate the lo-cal energy evaluation time from AFQMC with DF andDF+ET as a function of supercell size Nx ·Ny. Crossoveris seen forNx ≃ 5, 6 for increasingly small thresholds. Fora wide-gap semiconductor like BN, as discussed below,supercells of this size are sufficient to converge mean-fieldand correlation energies to the thermodynamic limit. Wethus expect the DF+ET approach to be even more ben-eficial for materials with smaller or vanishing gap (e.g.metals), requiring even larger supercells or Brillouin zonemeshes to reliably converge energies to the TDL.

In the inset of the upper panel, we illustrate the dif-ference ∆Ec between the correlation energy per cell from

AFQMC with DF and DF+ET, estimated on 3 popula-tions of walkers equilibrated for β = 4E−1

Ha. As naturallyexpected, ∆Ec increases monotonically with the trun-cation threshold, though remaining consistently below0.03% of the AFQMC correlation energy extrapolatedto the TDL.In the lower panel of Figure 11, we extrapolate the

AFQMC energy to the TDL using the power-law AnsatzEc(Nx) = α + βN−1

x . The extrapolated total energy isshown in the inset of the lower panel of Figure 11.

1 4 9 16 25 36

Nx · Ny

10−5

10−4

10−3

10−2

10−1

100

101

102

TE

loc[s]

AFQMC, DF

AFQMC, DF + ET, ε = 10−4

AFQMC, DF + ET, ε = 5 · 10−4

AFQMC, DF + ET, ε = 10−3

0 2−13−14−15−1

N−1x

-0.160

-0.155

-0.150

-0.145

-0.140

Ec[E

Ha]

AFQMC, DF + ET, ε = 10−4 TDL

B N

RBN

ax ay

0 5 10 15 20 25

Nx · Ny

-75

-50

-25

0

25

50

75

∆E

c[µ

EH

a]

-0.032

-0.016

0.0

0.016

0.032

∆E

c[%

Ec,

TD

L]

0 2−13−1 1

N−1x

-12.80

-12.75

-12.70

-12.65

-12.60

E[E

Ha]

FIG. 11. (color online) Top: Local energy evaluation timeTEloc

from AFQMC with DF approximation for the ERI(purple stars), and AFQMC with DF+ET (red circles, greensquares, blue triangles for ε = 10−3, 5 · 10−4, 10−4 a.u. re-spectively), for 2D hexagonal BN at RBN = 2.5 A, as functionof supercell size. Inset: difference in the correlation part ofthe local energy per unit cell. Bottom: extrapolation to thethermodynamic of AFQMC correlation (main plot) and total(inset) energy (orange diamonds) per unit cell. Extrapola-tions to the TDL are indicated by orange crosses.

V. CONCLUSIONS

In the present work we have shown that, through asimple and efficient low-rank factorization of the ERI,it is possible to perform AFQMC calculations for elec-tronic structure problems in Gaussian bases with O(N3)scaling, introducing a minimal modification to the orig-

10

inal algorithm, and still maintaining high accuracy. Weexpect a similar scaling will be observed with other tech-niques that exploit sparse structure in the Cholesky vec-tors, such as tensor hyper-contraction, or integral screen-ing [50]. We expect our approach will be particularly use-ful in studies of larger molecules, and of crystalline solidsrequiring extrapolations to the thermodynamic limit ofinfinite system size. The algorithmic advances may alsobe used in conjunction with parallel efforts to acceler-ate AFQMC through improved hardware implementa-tions [51]. We expect such a combination will make sys-tematic AFQMC calculations on large systems a practicalpossibility.

VI. ACKNOWLEDGMENTS

Computations were carried out on facilities supportedby the National Energy Research Scientific ComputingCenter (NERSC), on facilities supported by the Scien-tific Computing Core at the Flatiron Institute, a divisionof the Simons Foundation, on the Pauling cluster at theCalifornia Institute of Technology, and on the Storm andSciClone Clusters at the College of William and Mary.S. Z. acknowledges support from DOE (Grant No. DE-SC0001303). M. M. acknowledges Narbe Mardirossianand Yuliya Gordiyenko for help and discussion in elec-tronic structure calculations for H2O clusters, and Qim-ing Sun for generous support and guidance in under-standing and using PySCF to set calculations for BN.

VII. APPENDIX: RELATIONSHIP WITH

PLANE-WAVE FORMULATIONS

Many calculations in solid-state systems are performedusing a computational plane wave basis. AFQMC sim-ulations using this computational basis [22, 25] have anO(N3) scaling. We here briefly outline the relationshipbetween the cubic scaling achieved in the plane-wave ba-sis and that achieved using the factorization techniquesin this paper.In the plane-wave basis, the Hamiltonian with pseu-

dopotentials takes the form

H = H0 +∑

GG′

tGG′EGG′ +1

2

∑

GG′q

VqEG+qGE†G′G′+q

(26)

where G is a wave-vector in the reciprocal lattice, corre-

sponding to the plane-wave state 〈r|G〉 = eiG·r

√Ω

where Ω

is the computational cell volume. The vectors q are thetransfer momenta, and, due to momentum conservation,their number is proportional to number of plane-wavesN . Thus equation (26) is a low-rank factorization of theintegrals (with q playing the role of γ) but it is not aCholesky factorization, because the analogous quantity

LqGG′ =

√

VqδG,G′+q (27)

is not a lower triangular matrix for each q.

The local energy formula can be written analogouslyto (16),

Eloc,2(Ψ) =∑

ijq

fqiif

−qjj − fq

ijf−qji , (28)

with fqij defined formally as

fqij =

∑

GG′

LqGG′ΦT iGΘG′j (29)

Unlike in the case of the atomic orbital basis, the Lq

matrices contain N , rather than O(logN), elements, anddo not display the same low-rank structure. However, Lq

encodes a periodic delta function, which means that (27)is a convolution,

fqij =

∑

G

ΦσiGΘσG+qj. (30)

Consequently, using the fast Fourier transform, comput-ing fq

ij requires only O(O2N logN) ∼ O(N3) time, andthe local energy evaluation can be computed in soft cubictime.

[1] J. L. Whitten, The Journal of Chemical Physics 58, 4496(1973), https://doi.org/10.1063/1.1679012.

[2] N. H. F. Beebe and J. Linderberg, International Journalof Quantum Chemistry 12, 683 (1977).

[3] J. Yang, Y. Kurashige, F. R. Manby, and G. K. L. Chan,The Journal of Chemical Physics 134, 044123 (2011),https://doi.org/10.1063/1.3528935.

[4] E. G. Hohenstein, R. M. Parrish, and T. J. Martınez,The Journal of Chemical Physics 137, 044103 (2012),https://doi.org/10.1063/1.4732310.

[5] B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin, Inter-national Journal of Quantum Chemistry 12, 81 (1977).

[6] B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin,The Journal of Chemical Physics 71, 4993 (1979),https://aip.scitation.org/doi/pdf/10.1063/1.438313.

[7] H.-J. Werner, F. R. Manby, and P. J. Knowles,The Journal of Chemical Physics 118, 8149 (2003),https://doi.org/10.1063/1.1564816.

[8] H. Koch, A. S. de Meras, and T. B. Ped-ersen, J. Chem. Phys. 118, 9481 (2003),http://dx.doi.org/10.1063/1.1578621.

[9] F. Aquilante, L. De Vico, N. Ferre, G. Ghigo, P.-A.Malmqvist, P. Neogrady, T. B. Pedersen, M. Pitonak,M. Reiher, B. O. Roos, L. Serrano-Andres, M. Urban,

11

V. Veryazov, and R. Lindh, Journal of ComputationalChemistry 31, 224 (2010).

[10] M. Sierka, A. Hogekamp, and R. Ahlrichs, TheJournal of Chemical Physics 118, 9136 (2003),https://doi.org/10.1063/1.1567253.

[11] R. M. Parrish, E. G. Hohenstein, T. J. Martınez, andC. D. Sherrill, The Journal of Chemical Physics 137,224106 (2012), https://doi.org/10.1063/1.4768233.

[12] R. M. Parrish, E. G. Hohenstein, N. F. Schunck, C. D.Sherrill, and T. J. Martınez, Phys. Rev. Lett. 111,132505 (2013).

[13] E. G. Hohenstein, S. I. L. Kokkila, R. M. Par-rish, and T. J. Martınez, The Journal of Physi-cal Chemistry B 117, 12972 (2013), pMID: 23964979,https://doi.org/10.1021/jp4021905.

[14] R. M. Parrish, C. D. Sherrill, E. G. Hohen-stein, S. I. L. Kokkila, and T. J. Martınez, TheJournal of Chemical Physics 140, 181102 (2014),https://doi.org/10.1063/1.4876016.

[15] S. I. L. Kokkila Schumacher, E. G. Hohen-stein, R. M. Parrish, L.-P. Wang, and T. J.Martınez, Journal of Chemical Theory and Com-putation 11, 3042 (2015), pMID: 26575741,https://doi.org/10.1021/acs.jctc.5b00272.

[16] B. Peng and K. Kowalski, Journal of Chemical The-ory and Computation 13, 4179 (2017), pMID: 28834428,https://doi.org/10.1021/acs.jctc.7b00605.

[17] U. Benedikt, A. A. Auer, M. Espig, and W. Hackbusch,The Journal of Chemical Physics 134, 054118 (2011),https://doi.org/10.1063/1.3514201.

[18] N. Shenvi, H. van Aggelen, Y. Yang, W. Yang,C. Schwerdtfeger, and D. Mazziotti, The Jour-nal of Chemical Physics 139, 054110 (2013),https://doi.org/10.1063/1.4817184.

[19] F. Hummel, T. Tsatsoulis, and A. Gruneis, TheJournal of Chemical Physics 146, 124105 (2017),https://doi.org/10.1063/1.4977994.

[20] R. Schutski, J. Zhao, T. M. Henderson, and G. E. Scuse-ria, The Journal of Chemical Physics 147, 184113 (2017),https://doi.org/10.1063/1.4996988.

[21] M. Motta, E. Ye, J. R. McClean, Z. Li, A. Minnich,R. Babbush, and G. K.-L. Chan, arXiv:1808.02625(2018).

[22] S. Zhang and H. Krakauer, Phys. Rev. Lett. 90, 136401(2003).

[23] M. Motta and S. Zhang, WIREs Comput Mol Sci e1364,1 (2018).

[24] Here, and the remainder of the present work, we rely onthe soft-O notation, well-established in complexity the-ory: g(x) = O (f(x)) if there exists an integer k such thatg(x) = O

(

f(x)logk(x))

.[25] M. Suewattana, W. Purwanto, S. Zhang, H. Krakauer,

and E. J. Walter, Phys. Rev. B 75, 245123 (2007).[26] G. H. Booth, T. Tsatsoulis, G. K.-L. Chan, and

A. Grneis, The Journal of Chemical Physics 145, 084111(2016), https://doi.org/10.1063/1.4961301.

[27] M.-D. Choi, Linear Algebra and its Applications 10, 285(1975).

[28] R. L. Stratonovich, Soviet Physics Doklady 2, 416 (1958).[29] J. Hubbard, Phys. Rev. Lett. 3, 77 (1959).[30] W. A. Al-Saidi, S. Zhang, and H. Krakauer, The Journal

of Chemical Physics 127, 144101 (2007).[31] W. Purwanto, H. Krakauer, and S. Zhang, Phys. Rev.

B 80, 214116 (2009).

[32] Y. Virgus, W. Purwanto, H. Krakauer, and S. Zhang,Phys. Rev. Lett. 113, 175502 (2014).

[33] M. Motta, D. M. Ceperley, G. K.-L. Chan, J. A. Gomez,E. Gull, S. Guo, C. A. Jimenez-Hoyos, T. N. Lan, J. Li,F. Ma, A. J. Millis, N. V. Prokof’ev, U. Ray, G. E. Scuse-ria, S. Sorella, E. M. Stoudenmire, Q. Sun, I. S. Tupitsyn,S. R. White, D. Zgid, and S. Zhang (Simons Collabo-ration on the Many-Electron Problem), Phys. Rev. X 7,031059 (2017).

[34] J. P. F. LeBlanc, A. E. Antipov, F. Becca, I. W. Bulik,G. K.-L. Chan, C.-M. Chung, Y. Deng, M. Ferrero, T. M.Henderson, C. A. Jimenez-Hoyos, E. Kozik, X.-W. Liu,A. J. Millis, N. V. Prokof’ev, M. Qin, G. E. Scuseria,H. Shi, B. V. Svistunov, L. F. Tocchio, I. S. Tupitsyn,S. R. White, S. Zhang, B.-X. Zheng, Z. Zhu, and E. Gull(Simons Collaboration on the Many-Electron Problem),Phys. Rev. X 5, 041041 (2015).

[35] M. Qin, H. Shi, and S. Zhang, Phys. Rev. B 94, 235119(2016).

[36] M. Motta and S. Zhang, Journal of Chemical Theoryand Computation 13, 5367 (2017), pMID: 29053270,https://doi.org/10.1021/acs.jctc.7b00730.

[37] M. Motta and S. Zhang, The Journal of Chemical Physics148, 181101 (2018), https://doi.org/10.1063/1.5029508.

[38] J. Shee, S. Zhang, D. R. Reichman, and R. A. Friesner,J. Chem. Theor. Comput. 13, 2667 (2017).

[39] F. Ma, S. Zhang, and H. Krakauer, Phys. Rev. B 95,165103 (2017).

[40] W. A. Al-Saidi, S. Zhang, and H. Krakauer,J. Chem. Phys. 124, 224101 (2006),http://dx.doi.org/10.1063/1.2200885.

[41] R. Balian and E. Brezin, Nuovo Cimento B 64, 37 (1969).[42] J. Hachmann, W. Cardoen, and G. K.-L. Chan,

The Journal of Chemical Physics 125, 144101 (2006),https://doi.org/10.1063/1.2345196.

[43] B. Temelso, K. A. Archer, and G. C. Shields, The Jour-nal of Physical Chemistry A 115, 12034 (2011).

[44] J. McClain, Q. Sun, G. K.-L. Chan, and T. C.Berkelbach, Journal of Chemical Theory andComputation 13, 1209 (2017), pMID: 28218843,https://doi.org/10.1021/acs.jctc.7b00049.

[45] S. Goedecker, M. Teter, and J. Hutter, Phys. Rev. B 54,1703 (1996).

[46] C. Hartwigsen, S. Goedecker, and J. Hutter, Phys. Rev.B 58, 3641 (1998).

[47] Q. Sun, T. C. Berkelbach, N. S. Blunt, G. H. Booth,S. Guo, Z. Li, J. Liu, J. D. McClain, E. R. Sayfutyarova,S. Sharma, S. Wouters, and G. K.-L. Chan, Wiley Inter-disciplinary Reviews: Computational Molecular Science8, e1340.

[48] B. Lundqvist, Physik der kondensierten Materie 6,193205 (1967).

[49] Q. Sun, T. C. Berkelbach, J. D. McClain, and G. K.-L. Chan, The Journal of Chemical Physics 147, 164119(2017), https://doi.org/10.1063/1.4998644.

[50] D. S. Lambrecht, B. Doser, and C. Ochsenfeld,The Journal of Chemical Physics 123, 184102 (2005),https://doi.org/10.1063/1.2079987.

[51] J. Shee, E. J. Arthur, S. Zhang, D. R. Reich-man, and R. A. Friesner, Journal of Chemical The-ory and Computation 14, 4109 (2018), pMID: 29897748,https://doi.org/10.1021/acs.jctc.8b00342.