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Inverse Volume Scaling of Finite-Size Error in Periodic Coupled Cluster Theory

March 2024
Physical Review X 14(1)

March 2024
14(1)

DOI:10.1103/PhysRevX.14.011059

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Coupled cluster theory is one of the most popular post-Hartree-Fock methods for ab initio molecular quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the length of the system. In this study, we review the current understanding of finite-size error in quantum chemistry methods for periodic systems. We introduce new tools that elucidate the mechanisms behind this phenomenon in the context of coupled cluster doubles calculations. This reconciles some seemingly paradoxical statements related to finite-size scaling. Our findings also show that singularity subtraction can be a powerful method to effectively reduce finite-size errors in practical quantum chemistry calculations for periodic systems.

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Inverse Volume Scaling of Finite-Size Error in Periodic Coupled Cluster Theory

Xin Xing 1and Lin Lin 1,2

1Department of Mathematics, University of California, Berkeley, California 94720, USA

2Applied Mathematics and Computational Research Division,

Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

(Received 18 April 2023; revised 17 January 2024; accepted 13 February 2024; published 28 March 2024)

Coupled cluster theory is one of the most popular post-Hartree-Fock methods for ab initio molecular

quantum chemistry. The finite-size error of the correlation energy in periodic coupled cluster calculations

for three-dimensional insulating systems has been observed to satisfy the inverse volume scaling, even in

the absence of any correction schemes. This is surprising, as simpler theories that utilize only a subset of the

coupled cluster diagrams exhibit much slower decay of the finite-size error, which scales inversely with the

length of the system. In this study, we review the current understanding of finite-size error in quantum

chemistry methods for periodic systems. We introduce new tools that elucidate the mechanisms behind this

phenomenon in the context of coupled cluster doubles calculations. This reconciles some seemingly

paradoxical statements related to finite-size scaling. Our findings also show that singularity subtraction can

be a powerful method to effectively reduce finite-size errors in practical quantum chemistry calculations for

periodic systems.

DOI: 10.1103/PhysRevX.14.011059 Subject Areas: Chemical Physics,

Computational Physics,

Quantum Physics

I. INTRODUCTION

In the past few decades, ab initio methods for quantum

many-body systems, such as density functional theory

(DFT), quantum Monte Carlo methods, and quantum

chemistry wave function methods, are becoming increas-

ingly accurate and applied to ever larger range of systems

[1,2]. Unlike molecular systems, periodic systems, includ-

ing solids and surfaces, require calculating properties in the

thermodynamic limit (TDL), a theoretical state in which the

system size approaches infinity. However, the TDL cannot

be directly accessed in practical applications. Finite-sized

computational supercells are employed to approximate this

limit, which introduces finite-size errors into the calcula-

tions. Finite-size errors can significantly affect the accuracy

of calculations, even for systems with thousands of atoms.

An extreme case is a moir´e system such as magic angle

twisted bilayer graphene (MATBG), where each computa-

tional unit cell consists of approximately 10 000 atoms, and

the supercell needs to have more than 100 000 atoms to

capture subtle correlation effects [3,4]. Directly tackling

finite-size effects by enlarging the supercell size is

very demanding, even for relatively inexpensive DFT

calculations with modern-day supercomputers. For more

accurate theories, this task is often computationally intrac-

table. Understanding the finite-size scaling, i.e., the scaling

of the finite-size error with respect to the system size, and

employing finite-size error correction schemes are, there-

fore, crucial for obtaining accurate results using moderate-

sized calculations.

The sources of finite-size errors in ab initio calculations

are multifaceted and complex [5–7]. These errors are

influenced by numerous factors, including system charac-

teristics such as whether it is insulating or metallic or

whether it is a three-dimensional bulk system versus a low-

dimensional system. Calculations of electron kinetic

energy, electron-ion interaction energy, Hartree energy,

Fock exchange energy, and electron correlation energy

can all contribute to finite-size errors. The first four types

are predominantly single particle in nature, while the

electron correlation energy is significantly more complex.

To a large extent, electronic correlation is short-ranged, and

this characteristic has spurred the development of local

correlation methods, whose computational cost may scale

linearly with system size. However, for ab initio methods to

be accurate, they must also effectively account for van der

Waals (vdW) interactions [8]. In solids, the cumulative

effect of weak van der Waals interactions can become a

significant contributor to the energy. The convergence of

vdW energy follows an inverse volume scaling, implying

that the finite-size error is inversely proportional to the

volume of the supercell. The origin of finite-size error is not

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solely confined to the characteristics of a physical system.

The finite-size scaling can vary significantly across differ-

ent theoretical frameworks even when applied to the same

physical system. Sometimes, the finite-size scaling can

manifest differently even when using the same theoretical

framework for the same system, simply by altering the

shape of the computational unit cell. All these complexities

require careful analysis under varying scenarios, and this is

important for accurately extrapolating toward the TDL or,

preferably, for devising improved correction schemes

aimed at reducing finite-size errors.

In recent years, there has been a growing interest in

employing quantum chemistry wave function methods,

such as Møller-Plesset perturbation theory and coupled

cluster (CC) theory [9], to compute ground-state and

excited-state properties for periodic systems [7,10–25].

Originally developed to study nuclear physics [26,27],

CC theory has become one of the most popular methods

in quantum chemistry that involves electronic correlation

[28]. The coupled cluster singles, doubles, and perturbative

triples [CCSD(T)] theory is often referred to as the “gold

standard”in molecular quantum chemistry. In these meth-

ods, there are two primary strategies to approximate the

TDL. The first involves expanding the computational

supercells within real space. The second strategy involves

performing calculations using a fixed unit cell and refining

discretization of the Brillouin zone within reciprocal space

using a k-point mesh, such as the Monkhorst-Pack mesh

[29]. This paper focuses on the latter approach, where the

number of kpoints is denoted by Nk. If the Monkhorst-Pack

mesh includes the Gamma point of the Brillouin zone, this

approach is equivalent to using a supercell comprised of Nk

unit cells. The convergence toward the TDL can be studied

by increasing this single parameter Nktoward infinity. The

computational expense of quantum chemistry methods can

rise sharply with respect to the system size. For example, in

coupled cluster singles and doubles (CCSD) calculations,

the computational cost scaling is OðN6

kÞin real-space

implementations [9] and OðN4

kÞin reciprocal-space imple-

mentations [30]. Therefore, even moderate-sized calcula-

tions with Nk¼2×2×2or 3×3×3can already be

computationally challenging. In CCSD(T) calculations, the

computational cost of the real-space implementation can

escalate to OðN7

kÞin the worst-case scenario. This implies

that even a moderate refinement of the Monkhorst-Pack grid

by a factor of 2 along each dimension can lead to an increase

in computational cost by ð23Þ7≈2million folds. Given this

computational challenge imposed by system size, it is

mostly impractical to estimate the finite-size scaling using

power-law fitting over calculations on increasingly large

systems and then determine the TDL value by extrapolation.

Instead, a more feasible approach is to acquire the exact

finite-size scaling through rigorous mathematical analysis

and subsequently utilize power-law extrapolation to esti-

mate the TDL value.

Although numerous empirical studies have examined the

finite-size scaling of some quantum chemistry methods,

there has been a notable lack of rigorous analysis. To the

best of our knowledge, the first rigorous analysis of finite-

size error in Hartree-Fock (HF) theory and second-order

Møller-Plesset perturbation theory (MP2) for insulating

systems has been conducted only recently [31]. The

principal findings in Ref. [31] include the following.

(1) In the absence of any corrections, the finite-size error

in HF scales as OðN−1=3

kÞ.AsNkis proportional to

the supercell volume, N1=3

kis proportional to the

length of the supercell (we always assume uniform

refinement of the k-point mesh along all three

dimensions for any shaped unit cell). This scaling

is, thus, also referred to as inverse length scaling.

(2) By applying the Madelung constant correction, the

finite-size error in HF improves to OðN−1

kÞ, i.e., the

inverse volume scaling.

(3) The finite-size error of the MP2 correlation energy

also satisfies the inverse volume scaling OðN−1

kÞ.

As can be seen from HF and MP2, the specific finite-size

scaling depends on the level of computational theory and

numerical treatment employed in the calculations. In

addition to confirming the empirically observed scalings,

this rigorous analysis offers significant additional insights.

It elucidates the nature of the singularity of the electron

repulsion integral (ERI) due to the Coulomb kernel and its

impact on finite-size scaling; it explains why the finite-size

error can depend on the shapes and symmetries of the unit

cell; and it has led to the development of new correction

schemes, such as the staggered mesh method, which can

expedite the convergence of HF, MP2, and random phase

approximation (RPA) calculations toward TDL [32,33].

An innovative theoretical observation provided in

Ref. [31] is that the finite-size error can be largely compre-

hended if assuming HF single-particle orbitals can be

acquired exactly at any given kpoint in the Monkhorst-

Pack mesh. This perspective helps to disentangle the

contribution due to the relatively manageable single-particle

effects from the collective and more complex electron

correlation effects. Based on this assumption, the finite-size

error can be rigorously examined from a numerical quad-

rature perspective. Specifically, the value of a physical

observable at the TDL can often be written as a multidi-

mensional integral over the Brillouin zone. As the

Monkhorst-Pack mesh forms a uniform grid discretizing

the Brillouin zone, the analysis simplifies to investigating

the quadrature error of certain trapezoidal rule in a periodic

region, a topic widely discussed in numerical analysis

literature. The novelty here lies in the recognition that the

associated integrands possess a unique singularity structure

that is asymptotically of a specific fractional form.

Reference [31], therefore, develops a new Euler-

MacLaurin type of analysis that facilitates the study of

XIN XING and LIN LIN PHYS. REV. X 14, 011059 (2024)

011059-2

the finite-size error associated with HF and MP2 methods,

taking into account this fractional form singularity.

Compared to HF and MP2, the finite-size error analysis

and finite-size error correction methods in CC methods

remain nascent. Early works in Refs. [12,34] focus on an

important intermediate quantity called the structure factor

and develop finite-size correction methods based on differ-

ent types of polynomial interpolation of the structure factor

near the Coulomb singularity. Subsequent studies on the

uniform electron gas system [35,36] numerically examine

the power-law scaling of the finite-size error in CC theory,

offering new analytical perspective on the structure factor.

These works also provide new error correction methods,

such as twist-angle techniques [37,38] and structure factor

interpolation [39] for these metallic systems. Very recently,

Ref. [40] replaces the Coulomb kernel in CC calculations

with an averaged Coulomb kernel in each quadrature

element to reduce the finite-size error for anisotropic

systems. Despite these developments, theoretical under-

standing of the finite-size error scaling in CC methods (with

and without error correction methods) has been lacking.

Recently, by applying a similar approach, we have

expanded the finite-size error analysis to include the

correlation energy of the third-order Møller-Plesset pertur-

bation theory (MP3) and CC theory [41]. The simplest CC

theory is the coupled cluster singles (CCS) theory.

However, due to Thouless’s theorem [42], CCS only rotates

the Slater determinant to another Slater determinant. For a

nondegenerate closed-shell Hartree-Fock reference, the

correlation energy from the CCS theory vanishes.

Therefore, we focus on the coupled cluster doubles

(CCD) theory, which is mathematically the simplest and

representative form of CC theory. From a diagrammatic

perspective, CC diagrams encompass all Møller-Plesset

perturbation diagrams. When the CCD amplitude equation

is solved iteratively with nfixed point iterations [referred to

as the CCDðnÞscheme], the MP2 and MP3 diagrams can

be generated from CCD(1) and CCD(2), respectively. More

generally, CCDðnÞconsists of a finite subset of Møller-

Plesset perturbation diagrams. It is worth pointing out that

there is a method called CCSD(2) which uses second-order

perturbation to rectify CCSD energies for multireference

and open-shell systems [43], and its meaning should,

therefore, be distinguished from our CCDðnÞ. Our analysis

uncovers that the finite-size error in MP3 and CC is

fundamentally different from that in MP2. Specifically,

theories such as MP2 and RPA incorporate only “particle-

hole”types of diagrams. The integrands corresponding to

these diagrams are singular, but the singularity is relatively

mild. However, starting from MP3, additional perturbative

terms, namely, the “particle-particle”and “hole-hole”

diagrams, must be considered. These terms introduce much

stronger singularities, necessitating the development of new

analytical tools. Our quadrature error analysis adapts the

Poisson summation formula in a new setting and aligns

with a recently developed trapezoidal quadrature analysis

for certain singular integrals [44]. This approach provides

more accurate estimates and can be more applicable than

our previous quadrature analysis based on the Euler-

MacLaurin formula for HF and MP2 in Ref. [31]. In the

absence of finite-size correction schemes and assuming

exact orbital energies at any kpoint, the study in Ref. [41]

concludes that the finite-size errors of MP3 and CCD both

satisfy the inverse length scaling OðN−1=3

kÞ.

Interestingly, for CCD, earlier numerical calculations did

not provide conclusive evidence regarding its finite-size

scaling, with different studies suggesting either an inverse

volume scaling [15,34] or inverse length scaling [17]. More

recent calculations demonstrate that the electron correlation

energy in periodic coupled cluster calculations should

follow an inverse volume scaling, even in the absence of

finite-size correction schemes. This observation points to a

significant gap in the theoretical understanding of the finite-

size error and prompts the central question we address in

this paper: How can we reconcile the following seemingly

paradoxical facts?

(1) Without finite-size corrections, the finite-size error

in CCD exhibits inverse volume scaling.

(2) Without finite-size corrections, the finite-size error

in MP3 exhibits inverse length scaling. This rate is

sharp and cannot be improved.

(3) All MP3 diagrams are encompassed within the CCD

formulation.

There are several often-cited physical justifications for

expecting that the CCD method, and CC theory more

generally, may exhibit superior behavior when applied to

periodic systems. One such reason stems from the size

extensivity of CC theory. A theory is size extensive when

the total energy of two noninteracting identical systems,

calculated as a combined system, equates to twice the

energy of one system computed independently. Unlike

theories such as truncated configuration interaction meth-

ods, which are not size extensive, truncated CC theory

(such as CCD) possesses this advantageous characteristic.

Another reason is that CC theory can be formulated in such

a way that it does not explicitly depend on orbital energies.

One practical consequence of this is that, upon the con-

vergence of the coupled cluster iterations, the Madelung

constant correction, often used to reduce finite-size effects

in many-body simulations, cancels out naturally. Therefore,

in this scenario, CC without the Madelung constant

correction is equivalent to that with the Madelung constant

correction.

We would like to clarify that neither size extensivity nor

the cancellation of the Madelung constant alone is suffi-

cient to address the aforementioned question. While size

extensivity is indeed a desirable property, many methods

such as HF, MP2, and MP3, among others, are all size

extensive. In fact, given that periodic systems are infinitely

large, size extensivity should be viewed as a necessary

INVERSE VOLUME SCALING OF FINITE-SIZE ERROR IN …PHYS. REV. X 14, 011059 (2024)

011059-3

condition for the applicability of any numerical method to

such systems. However, this property does not provide

insight into the convergence rate of the finite-size error. The

cancellation of the Madelung constant plays a pivotal role

here. Nevertheless, demonstrating that CCD calculation (or

even MP3) with the Madelung constant correction exhibits

inverse volume scaling in finite-size error is itself a

significant challenge. This requires the development of

new technical tools not currently available in existing

literature. Indeed, the development and application of these

tools are the main technical contribution of this paper.

In this paper, we elucidate the origin of the inverse

volume scaling behavior. Our analysis consists of two

steps. First, we investigate the structure of the CCD

amplitude equation. We show that the Madelung constant

correction, commonly used to reduce finite-size errors in

Fock exchange energy and orbital energies, can also be

applied to reduce the finite-size error in ERI contractions

within the CCD amplitude equation. We establish a con-

nection between the Madelung constant correction and a

quadrature error reduction technique known as the singu-

larity subtraction method [45]. By subtracting the leading

singular terms from the integrands in the numerical quad-

ratures, the Madelung constant correction reduces the

finite-size errors in both the ERI contractions and the

orbital energies from OðN−1=3

kÞto OðN−1

kÞ. Furthermore,

we demonstrate that, with the Madelung constant correc-

tion, the finite-size errors in CCDðnÞand converged CCD

calculations satisfy the desired inverse volume scaling.

In the second step of our analysis, we observe that, upon

convergence of the CCD amplitude equations, the

Madelung constant corrections to both orbital energies

and ERI contractions perfectly cancel each other out for any

finite-sized system. This cancellation ensures that the CCD

correlation energy remains the same, regardless of whether

the Madelung constant correction is applied. Combining

this result with the first step, we conclude that the finite-size

error of the correlation energy in converged CCD calcu-

lations satisfies the desired inverse volume scaling without

the need for any additional correction schemes. However,

prior to the convergence of the amplitude equations, this

perfect cancellation does not occur, and the finite-size error

of CCDðnÞcalculations remains OðN−1=3

kÞ. A similar lack

of cancellation occurs when the orbital energies take their

exact value at the TDL but the ERI contractions are not

corrected, resulting in an OðN−1=3

kÞfinite-size error for both

converged CCD and CCDðnÞcalculations studied

in Ref. [41].

To validate our theoretical analysis, we perform CCD

calculations on a 3D periodic hydrogen dimer system using

the

PYSCF

software package [46]. Our numerical results

support the conclusions drawn from the theoretical analysis

and provide further evidence for the finite-size scaling

behavior summarized in Table I.

The paper is organized as follows. Section II introduces

background knowledge and basic notations. Section III first

decomposes the finite-size error in CCDðnÞcalculations

into the errors in four basic components and then describes

how the four components contribute to the overall finite-

size error with the possible Madelung constant correction.

Section IV explains the key ideas of how the Madelung

constant correction can reduce the finite-size error in orbital

energies and ERI contractions from a numerical quadrature

perspective. Section Villustrates the numerical results that

corroborate our error estimate. Lastly, Sec. VI discusses the

implication of our theoretical study and future directions.

TABLE I. Finite-size scaling of different computational theories with and without corrections to orbital energies

and ERI contractions. The first two rows refer to the finite-size scaling of the Hartree-Fock (HF) exchange energy,

and the remaining rows refer to the finite-size scaling of the correlation energy (excluding the HF exchange). While

we focus on the Madelung constant correction in Ref. [31] and this work, any correction schemes that reduces the

finite-size errors in orbital energies and ERI contractions to OðN−1

kÞcan also be applied, and the conclusions drawn

here remain valid. In particular, exact values of single particle orbital energies at the TDL satisfy the condition

above. N/A means that Madelung constant correction does not apply within the theory.

Theory

Correction to

orbital energies

Correction to

ERI contractions

Finite-size

scaling References

HF N/A ✗N−1=3

k[31, Theorem 3.1] and [6,17,47]

HF N/A ✓N−1

k[31, Theorem 5.1] and [47]

MP2 ✓N/A N−1

k[31, Theorem 4.1] and [15,34]

RPA, SOSEX, drCCD ✓N/A N−1

k[33,34]

MP3 ✓✗

N−1=3

k[41]

MP3 ✓✓

N−1

kTheorem 1

CCDðnÞ=CCD ✓✗

N−1=3

k=N−1=3

k[41]

CCDðnÞ=CCD ✓✓

N−1

k=N−1

kTheorem 1/Theorem 2

CCDðnÞ=CCD ✗✓

N−1=3

k=N−1=3

kTheorem 1/Theorem 2

CCDðnÞ=CCD ✗✗

N−1=3

k=N−1

kTheorem 1/Corollary 3

XIN XING and LIN LIN PHYS. REV. X 14, 011059 (2024)

011059-4

II. BACKGROUND

Consider a unit cell and its Brillouin zone denoted as Ω

and Ω, respectively. Denote the associated real- and

reciprocal-space lattices as Land L, respectively. To

model such a periodic system, the Brillouin zone Ωis

commonly discretized using a uniform mesh Kof size Nk

(known as the Monkhorst-Pack grid [29]). The orbitals and

orbital energies (also called bands and band energies)

fψnk;εnkgindexed by orbital indices nand momentum

vectors k∈Kcan be solved by the HF method. As a

common practice, n∈fi; jgrefers to an occupied orbital

and n∈fa; bgrefers to an virtual orbital. Throughout this

paper, we use the normalized ERI:

hn1k1;n

2k2jn3k3;n

4k4i

¼4π

jΩjX0

G∈L

jqþGj2

ϱn1k1;n3ðk1þqÞðGÞˆ

ϱn2k2;n4ðk2−qÞð−GÞ;

ð1Þ

where q¼k3−k1is the momentum transfer vector,

the crystal momentum conservation k1þk2−k3−

k4∈Lis assumed implicitly, and ˆ

ϱn0k0;nkðGÞ¼

hψn0k0jeiðk0−k−GÞ·rjψnkiis Fourier representation of the pair

product. The primed summation over Gmeans that the

possible term with qþG¼0is excluded in the numerical

calculation. Using a finite mesh K, the HF orbital energy

without any correction is computed as

εNk

nk¼hnkj^

H0jnki

þ1

NkX

ki∈KX

ð2hiki;nkjiki;nki−hiki;nkjnk;ikiiÞ;

ð2Þ

where ^

H0refers to the single-particle component of the many

body Hamiltonian.

In this paper, we focus on three-dimensional insulating

systems with an indirect gap, i.e., εaka−εiki>0,

∀i; a; ki;ka. To simplify the analysis, we assume that

the orbitals are exact at any kpoint and the number of

virtual orbitals are truncated to a finite number. In addition,

we assume that the exact orbitals and orbital energies in the

TDL are smooth and periodic with respect to their

momentum vector index k∈Ω. This assumption is a

restriction in our current analysis. For systems free of

topological obstructions [48,49], these conditions may be

replaced by weaker ones using techniques based on Green’s

functions or Hamiltonians defined in the atomic orbital

basis instead of the band basis [50].

A. CCD theory

Based on the reference HF determinant jΦi, the CCD

theory represents the wave function as

jΨi¼eTjΦi;

T¼1

NkX

ijab X

ki;kj;ka∈K

Tijabðki;kj;kaÞa†

akaa†

bkbajkjaiki;

where a†

nkand ankare creation and annihilation operators,

respectively, for ψnk,Tijabðki;kj;kaÞ(commonly denoted

as taka;bkb

iki;jkjin the literature) is the normalized CCD double

amplitude, and kb∈Kis uniquely determined using the

crystal momentum conservation kiþkj−ka−kb∈L.

The amplitude tensor TNk

¼fTijab ðki;kj;kaÞg is defined

as the root of a nonlinear amplitude equation that consists

of constant, linear, and quadratic terms.

In practice, the amplitude equation can be solved using a

quasi-Newton method [9,51], which is equivalent to apply-

ing fixed point iteration to

T¼1

εNk

ANkðTÞ:ð3Þ

Here, εNkdenotes a diagonal operator with entries

εNk

iki;jkj;aka;bkb¼εNk

ikiþεNk

jkj−εNk

aka−εNk

bkb, and 1=εNkgives

the diagonal operator with 1=εNk

iki;jkj;aka;bkb. The operator

ANkðTÞis referred to as the ERI-contraction map (see the

definition in Appendix A). It consists of contractions

between ERIs and Tand does not involve orbital energies.

Note that both Tand ANkðTÞare tensors indexed by

ði; j; a; bÞand ðki;kj;kaÞ∈K×K×K. The CCD corre-

lation energy is then defined as

ENk

CCD ¼1

kikjka∈KX

ijab

Wijabðki;kj;kaÞTijab ðki;kj;kaÞ

≔GNkðTNk

Þ;

where Wijabðki;kj;kaÞdenotes the antisymmetrized

ERI 2hiki;jkjjaka;bkbi−hiki;jkjjbkb;akai.

In the TDL with Kreplaced by Ω,TNk

¼

fTijabðki;kj;kaÞg with ðki;kj;kaÞ∈K×K×Kshould

be replaced by t¼ftijabðki;kj;kaÞg, where each

tijabðki;kj;kaÞis a function of ðki;kj;kaÞin Ω×

Ω×Ω. The converged TDL amplitude tsatisfies a

similar amplitude equation

t¼1

εTDL ATDLðtÞ;ð4Þ

where ATDL can be obtained from ANkby taking the limit

ð1=NkÞPk∈K→ð1=jΩjÞ RΩdkwith Nk→∞.For

example, the four-hole–two-particle (4h2p)linear term

in ½ANkðTÞijabðki;kj;kaÞconverges in ATDLðtÞas

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NkX

kk∈KX

hkkk;lkljiki;jkjiTklabðkk;kl;kaÞ

⟶

K→Ω1

jΩjZΩ

dkkX

hkkk;lkljiki;jkjitklabðkk;kl;kaÞ:

ð5Þ

The image of ATDL is a set of functions of ðki;kj;kaÞ

indexed by ði; j; a; bÞ. The CCD correlation energy in the

TDL is defined in a similar way as

ETDL

CCD ¼1

jΩj3ZΩ×Ω×Ω

dkidkjdkaWijabðki;kj;kaÞ

×tijabðki;kj;kaÞ

≔GTDLðtÞ:

Applying nsteps of fixed point iteration over Eqs. (3)

and (4) with zero initial guess (i.e., t¼0,T¼0), we

obtain the CCDðnÞamplitude and the CCDðnÞenergy:

ENk

CCDðnÞ¼GNkðTNk

nÞETDL

CCDðnÞ¼GTDLðtnÞ

TNk

m¼ðεNkÞ−1ANkðTNk

m−1Þ→tm¼ðεTDL Þ−1ATDLðtm−1Þ

TNk

0¼0t0¼0:

ð6Þ

CCDðnÞis related to the perturbative description of CCD

and consists of a subset of finite order perturbation energy

terms in the Møller-Plesset perturbation theory. For exam-

ple, CCD(1) can be identified with MP2, and CCD(2)

contains all the terms in MP2 and MP3 and a subset of

terms in MP4.

One main result of this paper is the rigorous analysis of

the finite-size error in CCDðnÞcalculation with any fixed

n>0. If the fixed point iterations in both the finite and

TDL cases converge to the CCD amplitudes as n→∞, i.e.,

TNk

n→TNk

and tn→t(the technical definition of this

convergence is provided in Appendix C), the finite-size

error analysis for CCDðnÞalso applies to the converged

CCD calculation. In other words, we analyze the finite-size

error in CCD calculation using a perturbative approach

based on the analysis on CCDðnÞ.

B. Madelung constant correction

Reference [41] shows that the finite-size errors in

CCDðnÞand CCD both scale as OðN−1=3

kÞwhen assuming

that exact orbital energies are used in the amplitude

equation (3). The same finite-size scaling also appears in

Fock exchange energy and occupied orbital energy calcu-

lations. One common correction to reduce the finite-size

errors in the latter two calculations is to add a Madelung

constant shift [6,47,52] to the Ewald kernel. This shift

introduces a correction to all involved ERIs in the calcu-

lations as

hn1k1;n

2k2jn3k3;n

4k4i−δn1n3δn2n4δk1k3δk2k4Nkξ:

Such a correction is triggered only in ERIs which have fully

matched orbital indices, i.e., n1¼n3;n

2¼n4, and zero

momentum transfer, i.e., k1¼k3. The Madelung constant

ξis defined uniquely by the unit cell and the k-point mesh

Kas

ξ¼1

NkX

q∈Kq

−

jΩjZΩ

dqX0

G∈L

4π

jΩj

e−σjqþGj2

jqþGj2−

4πσ

þX0

R∈LK

erfcðσ−1=2jRj=2Þ

jRj;ð7Þ

where Kqis a uniform mesh that is of the same size as K

and contains q¼0and LKis the real-space lattice

associated with the reciprocal-space lattice qþGwith

q∈Kq;G∈L. Note that ξdoes not vary with respect to

σ>0[6] and this parameter σis commonly tuned to

control the lattice cutoffs in the summation over Land LK

in Eq. (7) when numerically computing ξ.

For finite-size orbital energy calculation in Eq. (2), the

Madelung constant correction gives

εNk;ξ

nk¼εNk

nkþξnis occupied;

εNk

nknis virtual:ð8Þ

For the ERI contractions in ANkðTÞ, this correction

should be applied to six linear amplitude terms (see

Appendix B2). For example, the 4h2plinear term in

Eq. (5) should be modified to

NkX

kk∈KX

hkkk;lkljiki;jkjiTklabðkk;kl;kaÞ

−ξTijabðki;kj;kaÞ:ð9Þ

Collecting the corrections to all six terms together, ANkðTÞ

is modified to

ANk;ξðTÞ¼ANkðTÞþ2ξT: ð10Þ

In the finite-size CCDðnÞand converged CCD calculations,

the Madelung constant correction can be applied to the

orbital energies, ERI contractions, or both in the amplitude

equation (3). As a result, we may have three correction

schemes compared to the standard calculation without any

correction.

Particularly, applying the Madelung constant correction

to both components gives the amplitude equation

XIN XING and LIN LIN PHYS. REV. X 14, 011059 (2024)

011059-6

T¼1

εNkþ2ξ½ANkðTÞþ2ξT¼ 1

εNk;ξANk;ξðTÞ:ð11Þ

It can be easily verified that this amplitude equation has the

same solution as Eq. (3) without any correction. Its CCD

solution is, thus, identical to the one without correction.

However, the associated CCDðnÞcalculation differs and

can be interpreted as solving the original amplitude

equation (3) using a quasi-Newton method that has a 2ξ-

diagonal shift to the Jacobian matrix.

III. MAIN STATEMENTS

We start our error analysis with the CCDðnÞcalculation

with a fixed number of iterations n>0and then general-

ize the analysis to the fully converged CCD calcu-

lation. In CCDðnÞ, the finite-size error is quantified by

ETDL

CCDðnÞ−ENk

CCDðnÞ. According to Eq. (6), this error can be

traced back to two sources: the difference in energy oper-

ators, GTDL versus GNk, and the difference in amplitudes, tn

versus TNk

n.Recallthattnand TNk

nare the amplitudes of the

system in TDL and finite-size cases, respectively. Let us

consider the evaluation map,denotedbyMK.Thismap

evaluates a tensor valued function, initially defined on the

product space Ω×Ω×Ω, on a finite-sized grid

K×K×K. Consequently, the values of the TDL amplitude

tnon this finite-size grid, K×K×K, are given by MKtnand

are approximated by the finite-size amplitude TNk

By applying the triangle inequality, we can decompose

the finite-size error into two distinct sources: the errors

arising from the discretized energy calculation using the

exact amplitude MKtnand the errors stemming from the

amplitude calculation itself:

jETDL

CCDðnÞ−ENk

CCDðnÞj

≤jGTDLðtnÞ−GNkðMKtnÞj þjGNkðMKtnÞ−GNkðTNk

nÞj

≤jGTDLðtnÞ−GNkðMKtnÞj

þC

ijab X

ki;kj;ka∈K

j½MKtn−TNk

nijabðki;kj;kaÞj:ð12Þ

Here, we use the fact that jWijabðki;kj;kaÞj can be upper

bounded uniformly by a constant.

To further break down the error in amplitude calculation,

we note that tnand TNk

nare recursively constructed by

Eq. (6) with initial values t0¼0and TNk

0¼0. As a result,

the error in the CCDðnÞamplitude calculation can

also be recursively decomposed using the same strategy

above as

MKtn−TNk

n¼1

εTDL ½MKATDLðtn−1Þ−ANkðMKtn−1Þ

þANkðMKtn−1Þ1

εTDL −

εNk

þ1

εNkhANkðMKtn−1Þ−ANkðTNk

n−1Þi:ð13Þ

The three error terms from this dissection can be interpreted

as the errors in ERI contractions using exact CCDðn−1Þ

amplitudes, orbital energies, and CCDðn−1Þamplitude

calculation composed with ANk, respectively. For the

last term, it can be shown that entries in ANkðMKtn−1Þ−

ANkðTNk

n−1Þhave the same scaling with respect to Nkas

those in MKtn−1−TNk

n−1. Replacing the last term by

MKtn−1−TNk

n−1and applying the dissection recursively,

we find that the error in the CCDðnÞamplitude calculation

is determined by those in the ERI contractions and orbital

energies, i.e., the first two terms in Eq. (13).

Overall, the finite-size error of CCDðnÞcalculation

can be decomposed into errors in three basic factors:

(i) energy calculation using exact CCDðnÞamplitude,

(ii) ERI contractions using exact CCDðn−1Þamplitude,

and (iii) orbital energies. This error decomposition is also

valid when applying the Madelung constant correction to

orbital energies [Eq. (8)] or ERI contractions [Eq. (10)]. By

analyzing these three error sources with or without cor-

rections separately, we can obtain the finite-size error

estimate for CCDðnÞwith various correction schemes.

The Madelung constant correction can reduce the finite-

size error in orbital energies from OðN−1=3

kÞto OðN−1

kÞ.

This correction is at the HF level. One main technical result

of this work is to show that the Madelung constant

correction can also reduce the finite-size error in most

(but not all) entries of the ERI contraction [note that

MKATDLðtÞ−ANkðMKtÞis a tensor] from OðN−1=3

kÞto

OðN−1

kÞ; see Sec. IV B. As a result, when applying the

Madelung constant correction to both orbital energies and

ERI contractions in the CCDðnÞcalculation, the overall

finite-size error can be successfully reduced to OðN−1

kÞ.

Theorem 1. In CCDðnÞcalculation, the finite-size error

in the correlation energy scales as OðN−1=3

kÞin each of the

following scenarios: (i) there is no finite-size correction,

(ii) the Madelung constant correction is applied only to the

ERI contraction ANk, and (iii) the Madelung constant

correction is applied only to the orbital energy εNk

nk.

When the Madelung constant correction is applied to

both ANkand εNk

nkin the CCDðnÞcalculation, the overall

finite-size error scales as OðN−1

kÞ.

As a special case, the same conclusion applies to MP3

calculations.

Proof. See Appendix B.▪

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011059-7

Now what happens to the finite-size error of the CCD

calculation as n→∞? For gapless and small-gap systems,

it has been observed in practice that the fixed point iteration

might not converge or the amplitude equation might have

multiple solutions. While we focus on systems with an

indirect gap, it is worth noting that, even in such favorable

scenarios, the existence and uniqueness of solutions to the

CCD amplitude equations in both finite and TDL cases

remain an open question and are beyond the scope of

this paper.

To study the finite-size error of CCD via the above

results on CCDðnÞ, we make additional technical assump-

tions (see Appendix C) that can guarantee the convergence

of CCDðnÞto CCD. Under these assumptions, we show

that the finite-size scaling of CCD calculation is upper

bounded by those of its associated converging CCDðnÞ

calculations. Numerical observations (see Sec. V) further

show that this finite-size scaling estimate through CCDðnÞ

is asymptotically sharp for CCD calculation with the

Madelung constant correction applied to orbital energies,

ERI contractions, or both.

Theorem 2 (informal). Under additional conditions on

the convergence of CCDðnÞto CCD, the finite-size error of

the CCD correlation energy scales as OðN−1=3

kÞin each of

the following scenarios: (i) the Madelung constant correc-

tion is applied only to the ERI contraction ANk, and (ii) the

Madelung constant correction is applied only to the orbital

energy εNk

nk.

When the Madelung constant correction is applied to

both ANkand εNk

nkin the CCD calculation, the overall finite-

size error scales as OðN−1

kÞ.

Compared to Theorem 1, Theorem 2 does not address the

finite-size error of the CCD calculation when no finite-size

correction is applied. The natural conclusion from Theorem

1 is that this finite-size scaling should be OðN−1=3

kÞas well.

However, this error estimate is loose and inconsistent with

the numerical observations which suggest an inverse

volume scaling. To obtain a tight estimate, we now use

the observation that the CCD calculation without any

correction is equivalent to the CCD calculation with the

Madelung constant correction applied to both ERI con-

tractions and orbital energies. Specifically, when applying

the Madelung constant corrections Eqs. (8) and (10), the

CCD amplitude equation in Eq. (11) can be formulated as

ðεNkþ2ξÞT¼ANkðTÞþ2ξT⇔εNkT¼ANkðTÞ:

With 2ξTon both sides canceling each other, this refor-

mulation is exactly reduced to the original amplitude

equation (3) without corrections. Therefore, the roots of

the two amplitude equations with and without the correc-

tions are the same, and the correlation energy in converged

CCD calculation without any corrections is the same as that

with the Madelung constant correction. In other words,

when investigating the finite-size error of CCD calculation

without corrections, we should apply Theorem 2 with the

Madelung constant correction applied to both ANkand εNk

nk.

This yields a sharp estimate OðN−1

kÞand explains the origin

of the inverse volume scaling of the finite-size error.

Corollary 3. Under the same additional conditions as in

Theorem 2, the finite-size error of the CCD correlation

energy without finite-size correction scales as OðN−1

kÞ.

We provide the proof of Theorem 2 in Appendix C, and

Corollary 3 follows directly from Theorem 2 and the

reasonings above.

IV. KEY IDEAS

As demonstrated in the previous section, the finite-size

errors in CCDðnÞand CCD calculations can be reduced to

the errors in three simpler basic calculations: energy

calculation using exact amplitudes, ERI contraction using

exact amplitudes, and orbital energies. A key observation is

that the finite-size errors in all three calculations can be

interpreted and analyzed from a numerical quadrature

perspective.

For a function gover a hypercube V, we denote a

(generalized) trapezoidal rule using a uniform mesh Xof V

and its quadrature error as

QVðg; XÞ¼jVj

jXjX

x∈X

gðxÞ;

EVðg; XÞ¼ZV

dxgðxÞ−QVðg; XÞ:

Under the assumption of exact orbitals at any kpoint in

finite-size calculation, the errors in (i) orbital energy,

(ii) energy, and (iii) ERI contraction ANkcan be, respec-

tively, formulated by their definitions as

εTDL

nk−εNk

nk¼1

jΩjEΩX

Wininðki;k;kiÞ;K;

GTDLðtÞ−GNkðMKtÞ¼ 1

jΩj3EΩ×Ω×ΩX

ijab

Wijabtijab ðki;kj;kaÞ;K×K×K;

MKATDLðtÞ−ANkðMKtÞ¼ 1

jΩjEΩX

hkkk;lkljiki;jkjitklabðkk;kl;kaÞ;Kþ:ð14Þ

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011059-8

For ERI contraction in the last line, the error is denoted by a

tensor with indices ði; j; a; b; ki;kj;kaÞ, and the error for

the 4h2plinear term Eq. (5) is detailed with kkbeing the

integration variable. Meanwhile, terms not shown account

for errors from computing other linear and quadratic

amplitude terms. From a numerical quadrature perspective,

all three finite-sized calculations approximate the corre-

sponding integrals in the TDL using the trapezoidal rule

and a finite mesh Kto discretize Ω. Therefore, their finite-

size errors can be estimated systematically by quadrature

error analysis.

In general, the quadrature error associated with a

trapezoidal rule is influenced by the integrand’s smoothness

and boundary conditions. If we take Δhas the mesh size

along each dimension, the quadrature error for a smooth

integrand is typically of order OðΔh2Þ. Interestingly, if the

integrand is also periodic, the error diminishes much more

rapidly than it does in a nonperiodic scenario. The decay

rate is faster than any finite power of Δh, showcasing a

superalgebraic decay [53]. However, for an integrand

periodic but marked by singularities, its quadrature error

tends to taper off at a slower rate.

For quadrature errors in Eq. (14), the involved integrands

are all periodic across their integration domains. However,

many of these integrands have singularities within the

domains, affecting the scaling of their quadrature errors. As

demonstrated next, these integrands include point singu-

larities arising from both ERIs and amplitudes, resulting in

low-order power-law decay of the corresponding quad-

rature errors as Nkincreases.

A. Singularity structure and quadrature

error estimate

All the quadrature errors in Eq. (14) have integrands

comprising of either ERIs or contractions between ERIs

and exact CCDðnÞamplitudes. The integration variables

are momentum vectors sampled in the Brillouin zone Ω.

Consequently, understanding the singularity structure of

ERIs and exact CCDðnÞamplitudes with respect to their

momentum vector indices is crucial in comprehending the

singularity structure of these integrands and ultimately

estimating the quadrature error in the three basic

calculations.

First, consider a generic ERI hn1k1;n

2k2jn3k3;n

4k4i

with fixed band indices ðn1;n

2;n

3;n

4Þand treat it as a

function of k1,k2, and q¼k3−k1in Ω. By its

definition, the ERI can be separated as

hn1k1;n2k2jn3k3;n4k4i

¼4π

jΩj

ϱn1k1;n3ðk1þqÞð0Þˆ

ϱn2k2;n4ðk2−qÞð0Þ

jqj2

þ4π

jΩjX

G∈Lnf0g

ρn1k1;n3ðk1þqÞðGÞˆ

ρn2k2;n4ðk2−qÞð−GÞ

jqþGj2:ð15Þ

Since we assume all orbitals ψnkperiodic and smooth with

respect to k∈Ω, this ERI has a point singularity at q¼0

only due to the first fraction term. Specifically, this term is of

fractional form fðk1;k2;qÞ=jqj2with a smooth numerator f.

As can be verified by direct calculation, such a fraction term

can have its point singularity at q¼0described by the

following general concept called algebraic singularity.

Definition 4 (informal). A function fðxÞhas algebraic

singularity of order γ∈Rat x0∈Rdif every lth-order

derivative near x0is bounded asymptotically by jx−x0jγ−l,

i.e.,



∂α

∂xαfðxÞ

≤Cjx−x0jγ−jαj;∀α≥0;

where αdenotes a non-negative d-dimensional derivative

multi-index. For brevity, fis said to be singular at x0with

the order of γ. See Definition 8 in Appendix Bfor the

rigorous mathematical definition.

A representative example of such a singular function of

order γis gðxÞ=jxj2, where gðxÞis smooth and scales as

Oðjxj2þγÞnear x¼0. Using orbital orthogonality, the

generic ERI exhibits singularities at q¼0with the order

of 0 when band indices mismatch (n1≠n3,n2≠n4), −1

when they partially match (n1¼n3;n

2≠n4or n1≠n3;

n2¼n4), and −2when they fully match (n1¼n3;n2¼n4).

We can now characterize the singularity structure of

integrand PiWininðki;k;kiÞdefined by orbital energy

εTDL

nkin Eq. (14). Fixing ðn; kÞ, the leading singularity in

each Wininðki;k;kiÞ(as a function of ki) comes from the

exchange term hiki;nkjnk;ikiiwhich has algebraic sin-

gularity at ki¼kof order 0 when i≠nand −2when

i¼n. In computing εTDL

nk, the overall integrand is, thus,

singular at ki¼kof order 0 for a virtual band nand −2for

an occupied band n.

For such periodic functions with one point of algebraic

singularity, the conventional textbook analysis of the

trapezoidal rule is overly pessimistic. A key technical tool

in this work is Lemma 5 below, which provides a rigorous

and sharp quadrature error estimate linking its error scaling

to the singularity order. (See Lemma 22 in Appendix Dfor

a more general statement.)

Lemma 5. Let fðxÞbe periodic with respect to V¼

½−1

2;1

2dand smooth everywhere except at x¼0with

singularity order γ≥−dþ1.Atx¼0,fðxÞis set to 0.

The quadrature error of a trapezoidal rule using an md-sized

uniform mesh Xthat contains x¼0can be estimated as

jEVðf; XÞj ≤Cm−ðdþγÞ:

If fð0Þis set to an Oð1Þvalue in the calculation, it

introduces additional Oðm−dÞquadrature error.

Combining this error estimate with the integrand singu-

larity structure for orbital energies (with m¼N1=3

k,d¼3,

and γ¼−2;0for occupied and virtual orbitals, respec-

tively), we obtain the finite-size error estimate for orbital

energy calculation as

INVERSE VOLUME SCALING OF FINITE-SIZE ERROR IN …PHYS. REV. X 14, 011059 (2024)

011059-9



εTDL

nk−εNk

nk

≤CN−1=3

knis occupied;

N−1

knis virtual:ð16Þ

To adapt the above approach for examining energy and

ERI-contraction calculations with exact amplitude, we need

to characterize the singularity structure of the exact

CCDðnÞamplitude entries. First, we note that the exact

CCD(1) amplitude is just the MP2 amplitude

nhaka;bkbjiki;jkjiðεTDL

iki;jkj;aka;bkbÞ−1o:

As a function of ðki;kj;kaÞ, each CCD(1) amplitude entry

indexed by ði; j; a; bÞhas the same singularity structure as

the included ERI term, which is singular at ka−ki¼0of

order 0. It turns out that the exact CCDðnÞamplitude with

any n>0all has the same singularity structure

as haka;bkbjiki;jkji.

Lemma 6 (singularity structure of the amplitude,

Lemma 4 in Ref. [41]). In CCDðnÞcalculation with

n>0, each entry of the exact double amplitude tn¼

f½tnijabðki;kj;kaÞg belongs to the following func-

tion space:

TðΩÞ¼ffðki;kj;kaÞ∶fis periodic with respect to ki;kj;ka∈Ω;

fis smooth everywhere except at ka¼kiwith algebraic singularity of order 0;

fis smooth with respect to ki;kjat the singularity ka¼kig:

Combining the above singularity structure characterizations of ERIs and exact CCDðnÞamplitudes, we are able to

analyze the integrands for energy and ERI-contraction calculations in Eq. (14). Take the CCDðnÞexchange energy term as

an example whose finite-size error can be formulated as

−

jΩj3EΩ×Ω×ΩX

ijab

hiki;jkjjbkb;akaitijabðki;kj;kaÞ;K×K×K;

with integration variables ðki;kj;kaÞ. For each set of

ði; j; a; bÞ, both the associated ERI and exact amplitude

in the integrand exhibit an algebraic singularity of order 0 at

ka−kj¼0and ka−ki¼0, respectively. For such a

product of two functions with algebraic singularities, we

also provide a rigorous quadrature error estimate similar to

Lemma 5. Specifically, for all the integrands in the energy

and ERI-contraction calculations, their quadrature errors

are determined by the most singular product components in

the integrand. The quadrature error still scales as m−ðdþγÞ

similar to Lemma 5 but with γdenoting the minimum

algebraic singularity order of all ERIs and exact ampli-

tudes.

For the exchange term above, all involved ERIs and

exact amplitudes have point singularities of order 0 and

similar for the direct term. Thus, we can get the finite-size

error estimate for the energy calculation using exact

CCDðnÞamplitude as

jGTDLðtÞ−GNkðMKtÞj ≤CN−1

k:ð17Þ

Similar analysis is also applicable to ERI contractions

using exact amplitudes. The key distinction lies in the fact

that the ERI contractions involve integrands formed by

ERIs with stronger singularities, such as those defined by

particle-particle or hole-hole diagrams. A prominent exam-

ple is the 4h2plinear term in Eq. (5), where the involved

ERI hkkk;lkljiki;jkjiexhibits an algebraic singularity of

order −2at kk¼kiwhen k¼iand l¼j. Consequently,

as per the above analysis, the finite-size error in the 4h2p

linear term calculation alone scales as OðN−1=3

kÞ. This error

turns out to dominate the overall finite-size error in the ERI-

contraction calculation, and we have

½MKATDL ðtÞ−ANkðMKtÞijabðki;kj;kaÞ

≤CN−1=3

k;∀i; j; a; b; ∀ki;kj;ka∈K:ð18Þ

B. Madelung constant correction as a quadrature

error reduction method

To reduce the quadrature error for a singular integrand,

one common numerical quadrature technique is the singu-

larity subtraction method [45]. Essentially, this method

involves constructing an auxiliary function hthat possesses

the same leading singularity as the integrand g. The integral

is then approximated as

dxgðxÞ≈jVj

jXjX

x∈X

ðg−hÞðxÞþZV

dxhðxÞ

¼jVj

jXjX

x∈X

gðxÞþZV

dx−jVj

jXjX

x∈XhðxÞ:ð19Þ

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This approximation consists of the numerical quadrature of

g−hand the exact integral of h(which can be computed

analytically or numerically with high precision). It is also

equivalent to adding a correction EVðh; XÞto the numerical

quadrature of g. By this correction, the quadrature

error changes from EVðg; XÞto EVðg−h; XÞ. Since h

removes the leading singularity of gin the subtraction

g−h,EVðg−h; XÞcan be asymptotically smaller than

EVðg; XÞ.

The Madelung constant defined in Eq. (7) can be

reformulated using an arbitrary σ>0as

ξ¼−

jΩjZΩ

dq−jΩj

NkX

q∈KqX

G∈L

4π

jΩj

e−σjqþGj2

jqþGj2

þOðN−1

kÞ:ð20Þ

Compared to Eq. (19), this representation connects ξto the

singularity subtraction correction defined by an auxiliary

function

hσðqÞ¼ X

G∈L

4π

jΩj

e−σjqþGj2

jqþGj2:

The effectiveness of the Madelung constant correction to

reduce the finite-size error can be rigorously explained by

this connection.

Taking the occupied orbital energy εnkas an example, its

exchange portion leads to the dominant finite-size error,

and the associated Madelung constant correction modifies

the calculation as (with a change of variable ki→k−q)

NkX

q∈KqX

hiðk−qÞ;nkjnkiðk−qÞi−ξ:

Comparing this calculation with Eqs. (19) and (20), the

Madelung constant correction exactly uses the auxiliary

function hσðqÞto remove the leading singularity (i.e.,

4π=jΩjjqj−2) of the target integrand and, thus, reduces

the associated finite-size error asymptotically to OðN−1

kÞ.

One major technical contribution of this paper is to

rigorously prove the effectiveness of the Madelung constant

correction for reducing the finite-size error in the ERI-

contraction calculations following the same singularity

subtraction interpretation. For instance, consider the

4h2plinear term calculation in Eq. (5) with any fixed

entry index ði; j; a; b; ki;kj;kaÞ. Using the change of

variable kk→ki−q, this term with the Madelung con-

stant correction can be detailed as

NkX

q∈KqX

hkðki−qÞ;lðkjþqÞjiki;jkjitklabðki−q;kjþq;kaÞ−ξtijabðki;kj;kaÞ:ð21Þ

The leading singularity of the integrand comes from the

product with ðk; lÞ¼ði; jÞ. In this product, the ERI is

singular at q¼0of order −2, and the amplitude is singular

at q¼ki−kaof order 0. The correction in Eq. (21)

defines a singularity subtraction with an auxiliary function

hσðqÞtijabðki;kj;kaÞ. This auxiliary function shares ex-

actly the same leading singularity as the integrand at q¼0

due to the ERI term, i.e.,

4π

jΩj

jqj2tijabðki;kj;kaÞ:

Similar to the orbital energy analysis, the finite-size error in

this 4h2plinear entry can be reduced to OðN−1

kÞ. However,

the key difference here is that this error reduction is the case

only for most but not all the amplitude entries. The

exception is when the amplitude singularity q¼ka−ki

is close or equal to the ERI singularity q¼0, i.e., when

computing an ERI-contraction entry whose momentum

vector indices kiand kaare close or identical. In the worst

case when ki¼ka, the two singularities overlap and the

finite-size error of such a 4h2plinear entry can be shown to

be still of scale OðN−1=3

kÞ. Similar analysis also applies to

other terms in the ERI-contraction calculation.

In summary, the Madelung constant correction does not

uniformly reduce the finite-size errors in the ERI-contrac-

tion tensor. This is the case for the 4h2plinear term and

also for other terms in the ERI contraction. More precisely,

we have the following technical error estimate (see Lemma

11 in Appendix Bfor the general statement and proof).

Lemma 7 (error in ERI contractions). The finite-size

error in the ERI contractions using exact CCDðnÞampli-

tude tnwith the Madelung constant correction in Eq. (10)

satisfies

½MKATDL ðtnÞ−ANk;ξðMKtnÞijabðki;kj;kaÞ

≤C(1

jqiaj2N−1

kqia ≠0;

N−1=3

kqia ¼0;

where qia ¼ka−kiþG0with G0∈Lchosen such that

qia ∈Ω.

As a result of this nonuniform error reduction, the

maximum entrywise finite-size error in the CCDðnÞ

INVERSE VOLUME SCALING OF FINITE-SIZE ERROR IN …PHYS. REV. X 14, 011059 (2024)

011059-11

amplitude calculation MKtn−TNk

nwith the Madelung

constant correction remains OðN−1=3

kÞ. However, the aver-

age entrywise error satisfies the bound

ijab X

ki;kj;ka∈K

j½MKtn−TNk

nijabðki;kj;kaÞj ≤CN−1

From the error decomposition in Eq. (12), such a refined

bound is sufficient for our finite-size error analysis.

Therefore, the Madelung constant corrections to the ERI

contractions and the orbital energies effectively reduce the

finite-size error in the overall CCDðnÞenergy calculation

to OðN−1

kÞ.

V. NUMERICAL EXAMPLES

To validate the above theoretical analysis, we conduct

CCD and CCDðnÞcalculations on a 3D periodic system of

hydrogen dimers. One hydrogen dimer is positioned at the

center of each cubic unit cell with an edge length of 6 bohr

in the xdirection and a separation distance of 1.8 bohr. For

each uniform mesh K, we perform an HF calculation on K

to obtain the orbitals and orbital energies. We then perform

CCD and CCDðnÞcalculations under four distinct settings,

each with a different combination of the Madelung constant

correction to the orbital energies and the ERI contractions

in the amplitude equation. Specifically, we compute CCD

and CCDðnÞwith both corrections, the correction to the

orbital energies alone, the correction to the ERI contrac-

tions alone, and without any corrections. All the calcu-

lations are carried out using the

PYSCF

package [46] with a

minimal basis set gth-szv.

Figure 1illustrates the numerical results of the CCD(1),

CCD(2), CCD(3), and converged CCD calculations. For

CCD(1), which is identical to MP2, we have T0¼0, and

the Madelung constant correction to the ERI contractions

has no effect. As a result, the curves for the calculations

with and without this correction are identical. For CCD(2)

and higher, the four correction settings produce distinct

curves. Only the CCDðnÞcalculation with both corrections

exhibits a convergence rate of OðN−1

kÞ, while the other

three calculations have convergence rates of OðN−1=3

kÞ.

This highlights the importance of taking into account the

Madelung constant correction to both the orbital energies

and ERI contractions in higher-level CCDðnÞcalculations.

As CCDðnÞconverges to CCD, the difference between the

calculation with both corrections and the one without any

corrections gradually diminishes to zero, and the finite-size

error satisfies the inverse volume scaling.

2345

-0.02

-0.019

-0.018

-0.017

-0.016

-0.015

-0.014

(a) CCD(1)

2345

-0.032

-0.03

-0.028

-0.026

-0.024

-0.022

-0.02

-0.018

-0.016

(b) CCD(2)

2345

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

2345

-0.05

-0.045

-0.04

-0.035

-0.03

-0.025

-0.02

-0.015

(d) CCD

2345

-0.01435

-0.0143

-0.01425

-0.0142

-0.01415

-0.0141

(e) CCD(1)

2345

-0.0205

-0.0204

-0.0203

-0.0202

-0.0201

-0.02

(f) CCD(2)

2345

-0.0231

-0.023

-0.0229

-0.0228

-0.0227

-0.0226

(g) CCD(3)

2345

-0.0251

-0.02505

-0.025

-0.02495

-0.0249

-0.02485

-0.0248

-0.02475

-0.0247

(h) CCD

FIG. 1. Convergence of the CCD(1), CCD(2), CCD(3), and CCD correlation energies for a 3D periodic system of hydrogen dimers

with increasing Nk. The settings are distinguished by the presence or absence of the Madelung constant correction to the orbital energies

(“εNk;ξ

nk”) and the ERI contraction (“ANk;ξ”). In (a)–(d), the dashed curves show the power-law fitting using C0þC1N−1=3

kand data

points N1=3

k¼3, 4, 5 for the two cases with partial Madelung constant correction. (e)–(h) plot the curve fittings using N−1=3

kand N−1

over the calculations with correction to both components, numerically corroborating the inverse volume scaling.

XIN XING and LIN LIN PHYS. REV. X 14, 011059 (2024)

011059-12

VI. DISCUSSION

Recent years have witnessed significant progresses in

leveraging wave function methods to study solids. One

important driver of this trend is the potential of these methods

to provide systematically improvable results. Among the

simplest post-HF methods, MP2 has been applied to increas-

ingly large periodic systems along with the development of

efficient implementations utilizing density fitting and paral-

lelization techniques [16,23]. Notably, the next-order

method, MP3, has also been applied to periodic systems

recently for the first time [24]. Though computationally

much more expensive, CC theory has also gained momentum

in this context. It is being increasingly employed to com-

pute ground state and band structures for a variety of

solid materials including insulators and metallic systems

[17–19,22,25]. Furthermore, related theories such as equa-

tion-of-motion coupled cluster theory, GW method, and

algebraic diagrammatic construction theory also start their

utilizations in computing excited state properties of solids

[20,21,24]. Because of the omnipresence of finite-size

effects, as wave function methods continue to be increasingly

applied to periodic systems, the importance of developing a

thorough understanding of finite-size effects becomes even

more pronounced.

In this work, we fill a gap in understanding the finite-size

error in periodic coupled cluster calculation for insulating

systems. Notably, we reveal an unexpected inverse volume

scaling of this error in CCD calculation. This behavior

manifests even in the absence of any finite-size correction

schemes, owing to an error cancellation. Our findings,

together with the methodologies employed in this study,

provide valuable insights for practitioners, method devel-

opers, and theorists.

For practitioners, when applying computational quantum

chemistry methods to periodic systems, reducing finite-size

errors using techniques such as power-law extrapolation

requires an in-depth understanding of the error scaling.

This is particularly important when calculations are con-

strained to small-sized systems due to the steep increase of

the computational cost with respect to the system size and

limited resources. Many production-level quantum chem-

istry packages that support periodic systems use certain

finite-size corrections to reduce errors in the HF exchange

energy calculation and the HF orbital energies. For in-

stance, the truncated Coulomb correction scheme [54,55]

can be applied to insulating systems, so that the finite-size

error in orbital energies decays superalgebraically with

respect to Nk. However, our analysis shows that if this

correction is applied only to the orbital energy but without

any correction to ERI contractions, the finite-size error of

the correlation energy deteriorates from inverse volume

scaling to inverse length scaling. This is also a finding

consistent with numerical observations.

For method developers, a key aspect of our result is the

connection between the Madelung constant correction and

the singularity subtraction method. This relationship serves

not just as a crucial element in our theoretical proof but also

points toward new methods for further finite-size error

reduction. The Madelung constant correction operates as a

one-shot correction, while the singularity subtraction

method can be systematically improved to reduce the

finite-size error. Our analysis and correction schemes can

also be extended to more advanced coupled cluster theories

such as CCSD, and coupled cluster singles, doubles, and

triples (CCSDT). Exploring the finite-size error in lower-

dimensional systems, such as MATBG that requires a

different form of the Coulomb kernel [4,56], can lead to

different finite-size scaling patterns and novel correction

schemes. Going beyond finite-size corrections, the singu-

larity structure of the amplitude, as outlined in Lemma 6,

may be of independent interest. The singularity structure

imposes important analytic constraints that should be taken

into account when developing efficient numerical methods

for compressing the CC amplitude tensor.

For theorists, several critical questions persist: How can

finite-size error analysis be integrated with the study of

complete basis limits to tackle basis set dependence? How

should the finite-size error behavior in metallic systems be

analyzed, especially when the orbital energy difference in

the denominator could vanish? Can the scope of finite-size

analysis be broadened to include more complex systems

like disordered systems and finite-temperature alloys?

These questions present a fertile ground for future research.

ACKNOWLEDGMENTS

This material is based upon work supported by the U.S.

Department of Energy, Office of Science, Office of

Advanced Scientific Computing Research and Office of

Basic Energy Sciences, Scientific Discovery through

Advanced Computing (SciDAC) program under Grant

No. DE–SC0022364 (X. X.). This work is also partially

supported by the Applied Mathematics Program of the

U.S. Department of Energy (DOE) Office of Advanced

Scientific Computing Research under Contract No. DE-

AC02-05CH1123 (L. L.). L. L. is a Simons Investigator in

Mathematics (Grant No. 825053). We are grateful to

Timothy Berkelbach, Garnet Chan, and Alexander

Sokolov for insightful discussions and to the anonymous

reviewers for their helpful suggestions, which have been

instrumental in enhancing the presentation and clarity of

our paper.

APPENDIX A: CCD AMPLITUDE EQUATION

In this section, we use a capital letter to denote an index

pair consisting of the orbital index and the kindex. For

instance, I¼ði; kiÞ,J¼ðj; kjÞ,A¼ða; kaÞ, etc. We use

P∈fI; J; K; Lgto refer to occupied orbitals (also known as

holes) and P∈fA; B; C; Dgto refer to unoccupied orbitals

(also known as particles). Any summation PPrefers to

INVERSE VOLUME SCALING OF FINITE-SIZE ERROR IN …PHYS. REV. X 14, 011059 (2024)

011059-13

summing over all occupied or virtual orbital indices p

and all momentum vectors kp∈K, while the crystal

momentum conservation is enforced according to the

summand. This notation is used only in this section to

simplify the notation and also to connect the equations to

those in the molecular case [9] for better readability.

Using a finite mesh Kof size Nk, the normalized

CCD amplitude TNk

¼fTijab ðki;kj;kaÞg ≔ftAB

IJ gwith

ki;kj;ka∈Kis defined as the solution of the amplitude

equation

tAB

IJ ¼1

εNk

IJAB

½ANkðTNk

ÞIJAB

¼1

εNk

IJAB hABjIJiþPX

κA

CtCB

IJ −X

κK

ItAB

KJþ1

NkX

χKL

IJ tAB

KL þ1

NkX

χAB

CDtCD

þP1

NkX

ð2χAK

IC −χAK

CI ÞtCB

KJ −χAK

IC tBC

KJ −χAK

CJ tBC

KI ;∀I; J; A; B; ðA1Þ

where εNk

IJAB ¼εNk

ikiþεNk

jkj−εNk

aka−εNk

bkb,½ANkðTÞIJAB ¼½ANkðTÞijabðki;kj;kaÞ, and Pis a permutation operator defined

as Pð  ÞAB

IJ ¼ðÞ

IJ þðÞ

JI . This reformulation of the CCD amplitude equation is derived from the CCSD amplitude

equation in Ref. [30] by removing all the terms related to single amplitudes and normalizing the involved ERIs and

amplitudes (which gives the extra 1=Nkfactor in the equation and the intermediate blocks). The intermediate blocks in the

equation are defined as

κA

C¼−

KLD

ð2hKLjCDi−hKLjDCiÞtAD

KL;

κK

I¼1

LCD

ð2hKLjCDi−hKLjDCiÞtCD

IL ;

χKL

IJ ¼hKLjIJiþ 1

NkX

hKLjCDitCD

IJ ;

χAB

CD ¼hABjCDi;

χAK

IC ¼hAKjICiþ 1

2NkX

ð2hLKjDCi−hLKjCDiÞtAD

IL −hLKjDCitDA

IL ;

χAK

CI ¼hAKjCIi−1

2NkX

hLKjCDitDA

IL ;

and their momentum vector indices also assume the crystal momentum conservation

κQ

P→kp−kq∈L;

χRS

PQ →kpþkq−kr−ks∈L:

In the TDL, the amplitude equation for the exact amplitude t¼ftijab ðki;kj;kaÞg ≔ftAB

IJ gas functions of

ki;kj;ka∈Ωcan be formulated by letting Kin Eq. (A1) converge to Ωas

tAB

IJ ¼1

εTDL

IJAB

½ATDLðtÞIJAB

¼1

εTDL

IJAB hABjIJiþPX

κA

CtCB

IJ −X

κK

ItAB

KJþ1

jΩjZΩ

dkkX

χKL

IJ tAB

KL þ1

jΩjZΩ

dkcX

χAB

CDtCD

þP1

jΩjZΩ

dkkX

ð2χAK

IC −χAK

CI ÞtCB

KJ −χAK

IC tBC

KJ −χAK

CJ tBC

KI ;∀I; J; A; B; ðA2Þ

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011059-14

where the intermediate blocks in the TDL are defined as

κA

C¼−

jΩj2ZΩ×Ω

dkkdklX

kld

ð2hKLjCDi−hKLjDCiÞtAD

KL;

κK

I¼1

jΩj2ZΩ×Ω

dkcdkdX

lcd

ð2hKLjCDi−hKLjDCiÞtCD

IL ;

χKL

IJ ¼hKLjIJiþ 1

jΩjZΩ

dkcX

hKLjCDitCD

IJ ;

χAB

CD ¼hABjCDi;

χAK

IC ¼hAKjICiþ 1

2jΩjZΩ

dklX

ð2hLKjDCi−hLKjCDiÞtAD

IL −hLKjDCitDA

IL ;

χAK

CI ¼hAKjCIi−1

2jΩjZΩ

dklX

hLKjCDitDA

IL :

APPENDIX B: PROOF OF THEOREM 1

Restatement of Theorem 1. In CCDðnÞcalculation, the

finite-size error in the correlation energy scales as

OðN−1=3

kÞin each of the following scenarios: (i) there is

no finite-size correction, (ii) the Madelung constant cor-

rection is applied only to the ERI contraction ANk, and

(iii) the Madelung constant correction is applied only to the

orbital energy εNk

nk.

When the Madelung constant correction is applied to

both ANkand εNk

nkin the CCDðnÞcalculation, the overall

finite-size error scales as OðN−1

kÞ.

As a special case, the same conclusion applies to MP3

calculations.

1. Proof outline

The main context of this paper has provided a brief

description of the main proof idea. In this proof, we recap

some of the equations and concepts discussed before to

make it self-contained. The proofs of Theorem 1 for

CCDðnÞcalculations with various types of corrections

are based on the error splitting in Eqs. (12) and (13), i.e.,



ETDL

CCDðnÞ−ENk

CCDðnÞ≤jGTDL ðtnÞ−GNkðMKtnÞj

þCkMKtn−TNk

nk1;ðB1Þ

MKtn−TNk

n¼1

εTDL ½MKATDLðtn−1Þ−ANkðMKtn−1Þ

þANkðMKtn−1Þ1

εTDL −

εNk

þ1

εNk½ANkðMKtn−1Þ−ANkðTNk

n−1Þ:ðB2Þ

Note that the amplitude error here is measured in the

average norm as

kTk1¼1

ki;kj;ka∈KX

ijab

jTijabðki;kj;kaÞj:

The finite-size error in CCDðnÞcalculation is, thus,

decomposed into the error in energy calculation using

exact amplitude, the error in ERI contractions, the error in

orbital energies, and the error accumulated from previous

iteration. The latter three errors together make up of the

error in amplitude calculation. The error in energy calcu-

lation using exact amplitude is studied previously in

Ref. [41]. For completeness, we provide a brief review

of the main results below.

a. Brief review of error in energy calculation

with exact amplitude

One basic observation is that this error in CCDðnÞ

calculation can be interpreted as the quadrature error of

a specific trapezoidal rule as

jGTDLðtnÞ−GNkðMKtnÞj

¼

jΩj3EΩ×Ω×ΩX

ijab

ðWijab½tnijab Þ;K×K×K

ðB3Þ

Since both Wijab and ½tnijab are periodic with respect to

ki;kj;ka∈Ω, the asymptotic scaling of this quadrature

error depends on the smoothness of these two components

that constitute the integrand.

A general ERI hn1k1;n

2k2jn3k3;n

4k4ican be viewed

as a function of momentum vectors k1and k2and its

momentum transfer vector q¼k3−k1. This function is

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011059-15

periodic with respect to each variable in Ω, and its

definition in Eq. (1) can be decomposed as

4π

jΩj

ϱn1k1;n3ðk1þqÞð0Þˆ

ϱn2k2;n4ðk2−qÞð0Þ

jqj2þ4π

jΩjX

G∈Lnf0g



jqþGj2:

ðB4Þ

The numerators of all the fractions are smooth with respect

to k1,k2, and q[note the assumption that ψnkðrÞis smooth

with respect to k]. Therefore, this ERI example is smooth

with respect to k1and k2and has one point singularity in

Ωwith respect to qat q¼0, which is due to the first

fraction term. For any fixed kiand kj, this point singularity

can be characterized using the concept of algebraic

singularity of certain orders.

Definition 8 (algebraic singularity for univariate func-

tions). A function fðxÞhas algebraic singularity of order

γ∈Rat x0∈Rdif there exists δ>0such that



∂α

∂xαfðxÞ

≤Cα;δjx−x0jγ−jαj;∀0<jx−x0j<δ;∀α≥0;

where the constant Cα;δdepends on δand the non-negative

d-dimensional derivative multi-index α. For brevity, fis

also said to be singular at x0of order γ.

The numerator of the first fraction in Eq. (B4) scales as

OðjqjsÞwith s∈f0;1;2gnear q¼0using the orbital

orthogonality. The value of sdepends on the relation

between orbital indices ðn1;n

2Þand ðn3;n

4Þ. As a result,

the algebraic singularity of the ERI example above at

q¼0with any fixed k1and k2has order in f−2;−1;0g.

In addition, to connect this singularity with varying

k1;k2∈Ω, we introduce the algebraic singularity with

respect to one variable for a multivariate function.

Definition 9 (algebraic singularity for multivariate

functions). A function fðx;yÞis smooth with respect to

y∈VY⊂Rdyfor any fixed xand has algebraic singularity

of order γwith respect to xat x0∈Rdxif there exists δ>0

such that



∂α

∂xα∂β

∂yβfðx;yÞ

≤Cα;β;δjx−x0jγ−jαj;

∀0<jx−x0j<δ;∀y∈VY;∀α;β≥0;

where constant Cα;β;δdepends on δ,α, and β. Compared to

the univariate case in Definition 4, the key additions are the

shared algebraic singularity of partial derivatives over yat

x¼x0of order γand the independence of Cα;β;δon y∈VY.

By this definition, the ERI hn1k1;n

2k2jn3k3;n

4k4iis

smooth everywhere with respect to ki;kj;q∈Ωexcept

at q¼0, and the singularity order γ∈f−2;−1;0g.

Specifically, γequals to −2,−1, and 0, respectively, when

the orbital indices are fully matched, i.e., n1¼n3;n

2¼n4,

partially matched, i.e., n1¼n3;n2≠n4or n1≠n3;n2¼n4,

and not matched, i.e., n1≠n3;n

2≠n4. If treating the ERI

example as a function of k1,k2,k3instead, we equiv-

alently claim that the function is singular at k1¼k3of

order γ.

One key result in Ref. [41] is the singularity structure

characterization for the exact CCDðnÞamplitude tn,

which is essential for estimating the quadrature error in

Eq. (B3). It turns out that each exact amplitude entry

½tnijabðki;kj;kaÞwith any n>0has one point of

algebraic singularity of order 0 at ka−ki¼0, sharing a

similar singularity structure as the ERI hiki;jkjjaka;

bkbior the exact MP2/CCD(1) amplitude entry

ðεTDL

iki;jkj;aka;bkbÞ−1haka;bkbjiki;jkji.

Restatement of Lemma 6 (singularity structure of the

amplitude, Lemma 4 in Ref. [41]). In CCDðnÞcalculation

with n>0, each entry of the exact double amplitude tn

belongs to the following function space:

TðΩÞ¼ffðki;kj;kaÞ∶fis periodic with respect to ki;kj;ka∈Ω;

fis smooth everywhere except at ka¼kiwith algebraic singularity of order 0;

fis smooth with respect to ki;kjat the singularity ka¼kig:

Based on the above singularity structures of ERIs and

exact amplitudes, the integrand in the energy calculation in

Eq. (B3) consists of products of periodic functions where

each has one point singularity of order 0. (Recall that Wijab

is the antisymmetrized ERI and consists of two ERIs that

can be treated separately.) We provide a sharp quadrature

error bound for trapezoidal rules over periodic functions in

such a product form, and its application to Eq. (B3) gives

the following lemma.

Lemma 10 (energy error with exact amplitude, Lemma 5

in Ref. [41]). In CCDðnÞcalculation with any n>0, the

finite-size error in the energy calculation using the exact

amplitude tncan be estimated as

jGTDLðtnÞ−GNkðMKtnÞj ≤CN−1

where the constant Cdepends on tn.

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011059-16

b. Error in amplitude calculation

Based on the error splitting in Eq. (B2), analysis of the

error in amplitude calculation is reduced to estimating the

two main error terms MKATDLðtÞ−ANkðMKtÞand

εTDL

nk−εNk

nkand understanding how ANkamplifies the

amplitude error MKtn−1−TNk

n−1from the previous iteration.

Like the energy calculation, the errors in ERI contrac-

tions and orbital energies consist of specific quadrature

errors. In both cases, we can show that the Madelung

constant correction is connected to certain singularity

subtraction methods and can significantly reduce the

corresponding dominant quadrature errors.

Lemma 11 (error in ERI contractions). In CCDðnÞ

calculation with any n>0, the finite-size error in the

entries of the ERI contractions using the exact amplitude tn

without any corrections can be bounded as

½MKATDL ðtnÞ−ANkðMKtnÞijabðki;kj;kaÞ≤CN−1=3

ðB5Þ

The Madelung constant correction reduces this error to

½MKATDL ðtnÞ−ANk;ξðMKtnÞijabðki;kj;kaÞ

≤C(1

jqj2N−1

kq≠0;

N−1=3

kq¼0;

ðB6Þ

where q¼ka−kiþG0with G0∈Lchosen such that

q∈Ω. In both cases, the constant Cdepends on tnbut not on

the entry index ði; j; a; bÞand ðki;kj;kaÞ∈K×K×K.

Remark 12. The prefactor 1=jqj2in the above estimate is

important when q∈Kqis near the origin. For example, if

ki;ka∈Kare adjacent to each other, jqjis of scale

OðN−1=3

kÞand the estimate in Lemma 11 suggests an error

bound of scale OðN−1=3

kÞ.

Lemma 13 (error in orbital energies). The finite-size

error in orbital energies without any corrections is

bounded as

εTDL

nk−εNk

nk≤CN−1=3

knis occupied;

N−1

knis virtual:

The Madelung constant correction reduces this error to

εTDL

nk−εNk;ξ

nk≤CN−1

In both cases, the constant Cis independent of the entry

index n; k∈K.

Note that there are three multipliers for the three error

terms in the amplitude error splitting Eq. (B2). These

prefactors are bounded by constants independent of Nk.

First, the orbital energy difference in Eq. (B2) satisfies

jεTDL

iki;jkj;aka;bkbj≥2εgby the assumed indirect gap

εTDL

aka−εTDL

iki≥εg>0. Second, since ANkðTÞconsists of

constant, linear, and quadratic terms of T, a straightforward

estimate shows (e.g., using Lemma 7 in Ref. [41]) that

max

ijab;ki;kj;ka∈Kj½ANkðMKtnÞijabðki;kj;kaÞj

≤Cmax

ijab k½tnijabk2

L∞ðΩ×Ω×ΩÞ;

where the L∞-norm of ½tnijab is finite according to

Lemma 6.

Based on the above estimates of these multipliers, we

have that the summation of the first two error terms in

Eq. (B2) is dominated by the error in the ERI contractions

and the orbital energies discussed in Lemmas 11 and 13. If

applying the Madelung constant correction to both orbital

energies and ERI contractions, the summation of the first

two error terms has its entries bounded asymptotically the

same as Eq. (B6). Otherwise, its entries are bounded

asymptotically as in Eq. (B5). Lastly, for the third error

term in Eq. (13), the application of ANk=ANk;ξcan be

proved to maintain the entrywise error scaling obtained in

Lemma 11.

Lemma 14. Consider two arbitrary bounded amplitude

tensors T;S ∈Cnocc ×nocc ×nvir×nvir ×Nk×Nk×Nkand assume their

entrywise upper bound independent of Nk. If the difference

between Tand Ssatisfies an estimate similar to that in

Eq. (B5), i.e.,

j½T−Sijabðki;kj;kaÞj ≤CN−1=3

the ERI-contraction map without any corrections ANk

satisfies

j½ANkðTÞ−ANkðSÞijabðki;kj;kaÞj ≤CN−1=3

If the difference between Tand Ssatisfies an estimate

similar to that in Eq. (B6), i.e.,

j½T−Sijabðki;kj;kaÞj ≤C(1

jqj2N−1

kq≠0;

N−1=3

kq¼0;

the ERI-contraction map with Madelung constant correc-

tion ANk;ξsatisfies

j½ANk;ξðTÞ−ANk;ξðSÞijabðki;kj;kaÞj ≤C(1

jqj2N−1

kq≠0;

N−1=3

kq¼0:

Combining the estimates of the three error terms in

Eq. (13) and the initial condition MKt0¼TNk

0¼0in both

cases with and without Madelung constant correction, we

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011059-17

can obtain the entrywise estimate of the error in the

CCDðnÞamplitude calculation recursively. First, for the

CCDðnÞcalculation without any corrections or with partial

Madelung constant corrections to either orbital energies or

ERI contractions, we have

j½MKtn−TNk

nijabðki;kj;kaÞj ≤CN−1=3

k:ðB7Þ

Accordingly, the average entrywise error can be esti-

mated as

kMKtn−TNk

nk1

¼1

ki;kj;ka∈KX

ijab

j½MKtn−TNk

nijabðki;kj;kaÞj

≤CN−1=3

Second, for the CCDðnÞcalculation with the Madelung

constant correction to both orbital energies and ERI

contractions, we have

j½MKtn−TNk

nijabðki;kj;kaÞj ≤C(1

jqj2N−1

kq≠0;

N−1=3

kq¼0:

ðB8Þ

Accordingly, the average entrywise error can be esti-

mated as

kMKtn−TNk

nk1

¼1

ki;kj∈KX

q∈KqX

ijab

j½MKtn−TNk

nijabðki;kj;kiþqÞj

≤C1

NkX

q∈Kqnf0g

jqj2N−1

kþC1

N−1=3

≤CN−1

Plugging the above estimate and the error estimate for

energy calculation in Lemma 10 into Eq. (12) then finishes

the proof of Theorem 1.

Remark 15. In our previous work [41], we use the

maximum entrywise norm to characterize the finite-size

error in the amplitude calculation, which loosens the

average entrywise norm used in the error splitting in

Eq. (B1) as

kTk1≤kTk∞≔max

ijab;ki;kj;ka∈KjTijabðki;kj;kaÞj:

Without the Madelung constant correction, the maximum

norm suffices since all entries in the amplitude error are of

the same scale, as shown in Eq. (B7). However, this norm is

no longer sufficient for the calculation with the Madelung

constant correction, bounded in Eq. (B8). Because the

maximum entrywise error is now of scale OðN−1=3

kÞ, while

most entries are of scale OðN−1

kÞ. In this case, the average

entrywise norm provides a necessary and tighter estimate of

amplitude error.

2. Proof of Lemma 11: Error in ERI contractions

According to the singularity structure of exact CCDðnÞ

amplitude in Lemma 6, we consider the ERI contraction

using an arbitrary exact amplitude t∈TðΩÞnocc×nocc ×nvir×nvir .

Fixing a set of entry index ði; j; a; bÞand ðki;kj;kaÞ∈

K×K×K, the error in the indexed ERI-contraction

entry can be detailed as [by comparing Eqs. (A1)

and (A2)]

½MKATDLðtÞ−ANk;ξðMKtÞijab;kikjka¼1

jΩjEΩX

hkkk;lkljiki;jkjitklabðkk;kl;kaÞ;KþjΩjξtijab ðki;kj;kaÞ

þ1

jΩj2EΩ×ΩX

klcd

hkkk;lkljckc;dkditijcdðki;kj;kcÞtklabðkk;kl;kaÞ;K×K

þ;ðB9Þ

where the constant terms cancel with each other, the first

term is the error in the 4h2plinear term calculation with the

Madelung constant correction, and the second term is the

error in the 4h2pquadratic term calculation. The neglected

ones are the errors in remaining linear and quadratic term

calculations, which can all be similarly formulated as

quadrature errors of specific trapezoidal rules.

In the error analysis for CCD calculation without

corrections in ERI contractions [41], it has been shown

that without the Madelung constant correction the error

entry in the ERI contractions is uniformly bounded as

½MKATDL ðtÞ−ANkðMKtÞijab;kikjka≤CN−1=3

k:ðB10Þ

More specifically, all the quadratic terms and part of the

linear terms that contain only ERIs with mismatched orbital

indices contribute at most OðN−1

kÞerrors in both Eqs. (B9)

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011059-18

and (B10). The dominant error in Eq. (B10) comes from the

calculation of the following six linear terms:

NkX

kk∈KX

hkkk;lkljiki;jkjitklabðkk;kl;kaÞ;

NkX

kc∈KX

haka;bkbjckc;dkditijcdðki;kj;kcÞ;

−

NkX

kk∈KX

haka;kkkjckc;ikiitkjcbðkk;kj;kcÞ;

−

NkX

kk∈KX

hbkb;kkkjckc;jkjitkicaðkk;ki;kcÞ;

−

NkX

kk∈KX

haka;kkkjckc;jkjitkibcðkk;ki;kbÞ;

−

NkX

kk∈KX

hbkb;kkkjckc;ikiitkjacðkk;kj;kaÞ;

which contain ERIs with fully or partially matched orbital

indices. The Madelung constant correction in Eq. (B9) is

only triggered in these six terms. In this proof, we focus on

the error estimate for the 4h2plinear term (the first term

above) with the correction, and similar analysis can be

applied to all the other five terms.

Denote the ERI-amplitude product in the 4h2plinear

term with orbital indices ðk; lÞas

Fklðq1Þ¼hkðki−q1Þ;lðkjþq1Þjiki;jkji

×tklabðki−q1;kjþq1;kaÞ

¼Hkl

eriðq1ÞHkl

ampðq1Þ;

where q1¼ki−kkis the momentum transfer vector of the

ERI. The 4h2plinear term calculation with the Madelung

constant correction using a finite mesh Kcan be reformu-

lated as

NkX

kk∈KX

Fklðki−kkÞ−ξHij

ampð0Þ

¼1

NkX

q1∈KqX

Fklðq1Þ−ξHij

ampð0Þ;

using the change of variable kk→ki−q1and the perio-

dicity of Fklðq1Þ. The error of this calculation compared to

its TDL value, i.e., the first error term in Eq. (B9), can be

written as

jΩjX

EΩðFklðq1Þ;KqÞþjΩjξHij

ampð0Þ:

Previously, in Ref. [41], the quadrature error for Fklðq1Þ

with varying ðk; lÞis estimated as

jEΩðFklðq1Þ;KqÞj ≤C8

N−1

kk≠i; l ≠j;

N−2=3

kk¼i; l ≠jor k≠i; l ¼j;

N−1=3

kk¼i; l ¼j;

∀i; j; a; b; ∀ki;kj;ka∈K:ðB11Þ

As demonstrated next, in the case of partially matched orbital indices (e.g., k¼i,l≠j), this error estimate turns out to be

loose when ki≠kaand can be further improved as

jEΩðFilðq1Þ;KqÞj ≤C(1

jqjN−1

kq≠0;

N−2=3

kq¼0;

∀i; j; a; b; ∀ki;kj;ka∈K;ðB12Þ

where q¼ka−kiþG0with G0∈Lchosen to make q∈Ω. In the case of fully matched orbital indices (k¼i,l¼j),

the Madelung constant correction is triggered in the ERI evaluation and can help remove the leading quadrature error when

ki≠kaas

jEΩðFijðq1Þ;KqÞþjΩjξHij

ampð0Þj ≤C(1

jqj2N−1

kq≠0;

N−1=3

kq¼0;

∀i; j; a; b; ∀ki;kj;ka∈K:ðB13Þ

In the following discussion, we prove these two error estimates Eqs. (B12) and (B13) respectively at ki≠ka.

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a. Error estimate for linear terms with partially

matched orbital indices

Consider a fixed set of ði; j; a; bÞand ki;kj;ka∈Kwith

ki≠ka. Assume k¼iand l≠j. Our target is a sharper

estimate of EΩðFilðq1Þ;KqÞ. Since Kqand Ωare both

inversely symmetric over q1¼0, the quadrature error can

be symmetrized as

EΩðFilðq1Þ;KqÞ¼1

2EΩðHil

eriðq1ÞHil

ampðq1Þ

þHil

erið−q1ÞHil

ampð−q1Þ;KqÞ:

This symmetrized integrand can be further decomposed

into two terms:

½ðHil

eriðq1ÞþHil

erið−q1ÞÞHil

ampðq1Þ

−½Hil

erið−q1ÞðHil

ampðq1Þ−Hil

ampð−q1ÞÞ:ðB14Þ

For the first term in Eq. (B14), we note that the ERI term

can be detailed as

Hil

eriðq1Þ¼ 4π

jΩj

ϱiðki−q1Þ;ikið0Þˆ

ϱlðkjþq1Þ;jkjð0Þ

jq1j2

þ4π

jΩjX

G∈Lnf0g



jq1þGj2:

The ERI nonsmoothness with q1∈Ωcomes from the first

fractional term whose numerator is smooth and scales as

Oðjq1jÞ near q1¼0using orbital orthogonality. It can be

verified directly that Hil

eriðq1Þis singular at q1¼0of order

−1, and after symmetrization Hil

eriðq1ÞþHil

erið−q1Þis

singular at q1¼0of order 0. Meanwhile, the amplitude

Hil

ampðq1Þis smooth everywhere in Ωexcept at q1¼−qof

order 0 according to Lemma 6. An error estimate lemma in

Ref. [41] (restated as Lemma 24 in Appendix D) provides a

quadrature error estimate for periodic functions in such a

product form and can show that

jEΩððHil

eriðq1ÞþHil

erið−q1ÞÞHil

ampðq1Þ;KqÞj ≤CN−1

where constant Cis independent of i,j,a,band

ki;kj;ka∈K.

For the second term in Eq. (B14), direct application of

Lemma 24 leads to error estimate of scale OðN−2=3

kÞ, since

Hil

erið−q1Þis singular at q1¼0of order −1. However, note

that Hil

ampðq1Þis smooth at q1¼0[recall that Hil

ampðq1Þis

singular only at q1¼−q≠0], and, thus, the subtraction

Hil

ampðq1Þ−Hil

ampð−q1Þscales as Oðjq1jÞ near q1¼0.

Multiplication by this extra Oðjq1jÞ term improves the

algebraic singularity of Hil

eriðq1Þat q1¼0, and the overall

product can be shown to be singular at q1¼0of order 0.

Intuitively, this improved algebraic singularity at q1¼0

can lead to asymptotically smaller quadrature errors. To

rigorously justify this statement, we generalize Lemma 24

to estimate the quadrature error for this special case.

Lemma 16. Let fðxÞ¼f1ðxÞf2ðxÞ, where f1ðxÞand

f2ðxÞare periodic with respect to V¼½−1

2;1

2dand

(i) f1ðxÞis smooth everywhere except at x¼z1¼0of

order γ≤−1,

(ii) f2ðxÞis smooth everywhere except at x¼z2≠0of

order 0,

(iii) ∂α

xf2ð0Þ¼0for any derivative order jαj≤s.

Assume γ>−dfor fðxÞto be integrable in Vand γþ

sþ1≤0so the leading algebraic singularity of fðxÞis at

x¼0. Consider an md-sized uniform mesh Xin V.

Assume that Xsatisfies that z1,z2are either on the mesh

or Θðm−1Þaway from any mesh points and mis sufficiently

large that jz1−z2j¼Ωðm−1Þ.Atx¼z1and x¼z2,fðxÞ

is set to 0. The trapezoidal rule using Xhas quadrature error

jEVðf; XÞj ≤CHdþ1

V;z1ðf1ÞHdþ1

V;z2ðf2Þjz1−z2j−ðsþ1Þ

×m−ðdþγþsþ1Þ:

Remark 17. The two factors Hdþ1

V;z1ðf1Þand Hdþ1

V;z2ðf2Þ

characterize the algebraic singularities of the two functions,

and their exact definition can be found in Appendix D.

Proof of Lemma 16 is provided in Appendix D2.

In order to utlize this result, we further decompose the

second term in Eq. (B14) into

½Hil

erið−q1ÞðHil

ampðq1Þ−Hil

ampð0ÞÞ

þ½Hil

erið−q1ÞðHil

ampð0Þ−Hil

ampð−q1ÞÞ:

Applying Lemma 16 to both terms with γ¼−1and s¼0

gives

jEΩðHil

erið−q1ÞðHil

ampðq1Þ−Hil

ampð−q1ÞÞ;KqÞj ≤C1

jqjN−1

where constant Cis independent of i,j,a,band

ki;kj;ka∈K. Combining the above quadrature error

estimates for the two terms in Eq. (B14), we prove a

tighter error estimate at q≠0shown in Eq. (B12) while the

previous result in Eq. (B11) at q¼0still holds.

b. Error estimate for linear terms with fully

matched orbital indices

Let k¼i,l¼jand ki≠kaand consider the corre-

sponding calculation in the 4h2plinear term. The

Madelung constant correction is applied in the ERI

evaluation at kk¼kior, equivalently, at q1¼0, and the

calculation can be written as

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011059-20

NkX

q1∈Kq

Fijðq1Þ−ξHij

ampð0Þ

¼1

NkX

q1∈KqHij

eriðq1ÞHij

ampðq1Þ−hσðq1ÞHij

ampð0Þ

þ1

jΩjZΩ

dq1hσðq1ÞHij

ampð0ÞþOðN−1

kÞ;

which uses the expansion of the Madelung constant with an

arbitrary fixed parameter σ>0in Eq. (20). Note that the

prefactor in the OðN−1

kÞremainder term above depends

only on σ. The right-hand side of the above reformation is

equivalent to a singularity subtraction method that decom-

poses the original integrand into two terms:

Fijðq1Þ¼Hij

eriðq1ÞHij

ampðq1Þ

¼Hij

eriðq1ÞHij

ampðq1Þ−hσðq1ÞHij

ampð0Þ

þhσðq1ÞHij

ampð0Þ;

and then computes the numerical quadrature of the first

term and the exact integral of the second term. The

Madelung-corrected calculation, thus, has quadrature error

only from the first term as

jΩjEΩðFijðq1Þ;KqÞþξHij

ampð0Þ

¼1

jΩjEΩHij

eriðq1ÞHij

ampðq1Þ−hσðq1ÞHij

ampð0Þ;Kq

þOðN−1

kÞ:

Following a similar approach as in the partially matched

case, the effective integrand after the Madelung constant

correction above can be split into two terms as

ðHij

eriðq1Þ−hσðq1ÞÞHij

ampðq1Þ

þ½hσðq1ÞðHij

ampðq1Þ−Hij

ampð0ÞÞ:ðB15Þ

For the first term in Eq. (B15), the subtraction part can be

detailed as

Hij

eriðq1Þ−hσðq1Þ

¼4π

jΩj

ϱiðki−q1Þ;ikið0Þˆ

ϱjðkjþq1Þ;jkjð0Þ−e−σjq1j2

jq1j2

þ4π

jΩjX

G∈Lnf0g



jq1þGj2;

where the numerator of the first fraction scales as Oðjq1jÞ

near q1¼0. This subtraction is, thus, singular at q1¼0of

order −1and shares a similar form as the ERI with partially

matched orbital indices. Using the earlier inversion sym-

metry analysis for partially matched case, the quadrature

error of the first term in Eq. (B15) when ki≠kacan be

estimated as

EΩððHij

eriðq1Þ−hσðq1ÞÞHij

ampðq1Þ;KqÞ≤C1

jqjN−1

For the second term in Eq. (B15), we exploit the

inversion symmetry of Kqand Ωagain, and its quadrature

error equals to that of its symmetrized version as

2hσðq1ÞðHij

ampðq1Þ−2Hij

ampð0ÞþHij

ampð−q1ÞÞ:

This formula uses hσðq1Þ¼hσð−q1Þafter symmetrization.

Note that hσðq1Þis singular at q1¼0of order −2, while the

term in the parentheses scales as Oðjq1j2Þnear q1¼0using

the smoothness of Hij

ampðq1Þat q1¼0. Therefore, the overall

function above is singular at q1¼0of order 0. To fit the

integrand form in Lemma 16, we further decompose the

symmetrized integrand above into

½hσðq1ÞðHij

ampðq1Þ−Hij

ampð0Þ−q1·∇Hij

ampð0ÞÞ

þ½hσðq1ÞðHij

ampð−q1Þ−Hij

ampð0Þ−ð−q1Þ·∇Hij

ampð0ÞÞ:

Applying Lemma 16 to these two terms separately with γ¼

−2and s¼1gives



EΩ1

2hσðq1ÞðHij

ampðq1Þ−2Hij

ampð0ÞþHij

ampð−q1ÞÞ;Kq

≤C1

jqj2N−1

Combining the above quadrature error estimates for the

two terms in Eq. (B15), we prove a tighter error estimate at

q≠0shown in Eq. (B13) while the previous result in

Eq. (B11) at q¼0still holds.

The same error estimate can be obtained for all the six

linear terms that trigger the Madelung constant correction,

and the remaining linear and quadratic terms contribute at

most OðN−1

kÞquadrature error. Gathering the error esti-

mates for all these terms together and plugging into

Eq. (B9), we finish the proof.

3. Proof of Lemma 13: Error in orbital energies

In the TDL, the orbital energy εNk

nkwith any fixed nand

k∈Kconverges to

εTDL

nk¼hnkj^

H0jnkiþ 1

jΩjZΩ

dkiX

ð2hiki;nkjiki;nki

−hiki;nkjnk;ikiiÞ:

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011059-21

Comparing this exact orbital energy with its finite-size

calculation in Eq. (2), the finite-size error without any

corrections can be written as

εTDL

nk−εNk

nk¼2

jΩjEΩX

hiki;nkjiki;nki;K

−

jΩjEΩX

hiki;nkjnk;ikii;K:ðB16Þ

For the first quadrature error in Eq. (B16), i.e., the finite-

size error in the direct term, the ERI with each ican be

specified as

hiki;nkjiki;nki¼4π

jΩjX

G∈Lnf0g

ρiki;ikiðGÞˆ

ρnk;nkð−GÞ

jGj2:

Note that the momentum transfer vector of this ERI is

always zero and the singular fraction term in this ERI is set

to 0 by definition. As a result, the integrand is periodic and

smooth with respect to ki∈Ω. Therefore, the quadrature

error of the direct term calculation, thus, decays super-

algebraically according to Lemma 20 as



EΩX

hiki;nkjiki;nki;K

≤ClN−l

k;∀l>0:

For the second quadrature error in Eq. (B16), i.e., the

finite-size error in the exchange term, the ERI with each i

can be written as

hiki;nkjnk;ikii¼4π

jΩj

ρiki;nkð0Þˆ

ρnk;ikið0Þ

jqj2

þ4π

jΩjX

G∈Lnf0g

ρiki;ikiðGÞˆ

ρnk;nkð−GÞ

jqþGj2;

which is singular at q≔k−ki¼0of order −2when n¼

iand 0 otherwise. An error estimate lemma in Ref. [41]

(restated as Lemma 22 in Appendix D) gives a tight

quadrature error estimate for such periodic functions with

one point of algebraic singularity, and its application to the

above integrand gives

jEΩðhiki;nkjnk;ikii;KÞj ≤C(N−1=3

kn¼i;

N−1

kn≠i:

Combining the estimates of the two error terms in

Eq. (B16), we obtain the overall finite-size error estimate

for orbital energies without any corrections as

εTDL

nk−εNk

nk≤C(N−1=3

knis occupied;

N−1

knis virtual:

From the above analysis, the dominant finite-size error in

an occupied orbital energy lies in the calculation of the

exchange term with i¼n. In the Madelung-corrected

orbital energy εNk;ξ

nk, the correction is applied to this

exchange term as (ignoring the prefactor −jΩj−1)

QΩðhnki;nkjnk;nkii;KÞ→QΩðhnki;nkjnk;nkii;KÞ

−jΩjξ:

Applying the change of variable ki→k−qand using the

periodicity of the ERI with respect to ki, this corrected

calculation can be reformulated as

QΩðhnki;nkjnk;nkii;KÞ−jΩjξ

¼QΩðhnðk−qÞ;nkjnk;nðk−qÞi −hσðqÞ;KqÞ

þZΩ

dq1hσðq1ÞþOðN−1

kÞ;

using the expansion of the Madelung constant with any

fixed σ>0in Eq. (2). The quadrature error after the

correction can be written as

EΩðhnki;nkjnk;nkii;KÞþjΩjξ

¼EΩðhnðk−qÞ;nkjnk;nðk−qÞi −hσðqÞ;KqÞ

þOðN−1

kÞ:

The effective integrand above after the correction can be

detailed as

hnðk−qÞ;nkjnk;nðk−qÞi −hσðqÞ

¼4π

jΩj

jˆ

ϱnðk−qÞ;nkð0Þj2−e−σjqj2

jqj2þ4π

jΩjX

G∈Lnf0g



jqþGj2:

The integrand singularity comes from the first fractional

term and is of order −1, since the ERI and hσðqÞshare the

same leading singular term. Similar to the analysis in the

ERI contraction, we can then combine this singularity

subtraction with the inverse symmetry of Ωand Kqto

show that

jEΩðhnðk−qÞ;nkjnk;nðk−qÞi −hσðqÞ;KqÞj ≤CN−1

Combining this estimate with the above estimate for all

the remaining direct and exchange terms in εNk;ξ

nk, we obtain

the finite-size error estimate for orbital energies with the

Madelung constant correction as

εTDL

nk−εNk;ξ

nk≤CN−1

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4. Proof of Lemma 14: Error from previous iteration

Fixing a set of entry index ði; j; a; bÞand ðki;kj;kaÞ∈K×K×K, the ERI-contraction entry ½ANk;ξðTÞijab;kikjkacan

be detailed as

½ANk;ξðTÞijabðkikjkaÞ¼haka;bkbjiki;jkjiþ 1

NkX

kk∈KX

hkkk;lkljiki;jkjiTklabðkk;kl;kaÞ

þ1

kkkc∈KX

klcd

hkkk;lkljckc;dkdiTijcdðki;kj;kcÞTklab ðkk;kl;kaÞ

þþ2ξTijabðki;kj;kaÞ;

where the neglected terms are all the other linear and quadratic terms and the Madelung constant corrections to different

terms are collected together at the end.

In the subtraction ANk;ξðTÞ−ANk;ξðSÞ, the constant terms cancel each other. The subtraction between the two Madelung

constant correction terms can be estimated directly as

j2ξ½T−Sijabðki;kj;kaÞj ≤CN−1=3

k(1

jqj2N−1

kq≠0;

N−1=3

kq¼0;

using the fact that ξ¼OðN−1=3

kÞ.

The subtraction between the two 4h2plinear terms can be formulated and bounded as



NkX

kk∈KX

hkkk;lkljiki;jkji½T−Sklabðkk;kl;kaÞ

≤1

NkX

kk∈Knfki;kag

þ1

δkk;kiþ1

δkk;kaX

hkkk;lkljiki;jkji½T−Sklabðkk;kl;kaÞ

:ðB17Þ

For the term with kk¼ki(i.e., δkk;ki), we have

NkX

hkki;lkjjiki;jkji½T−Sklabðki;kj;kaÞ

≤1

NkX

CN−1=3

k≤CN−4=3

where the ERI definition at zero momentum transfer skips the singular fraction and is Oð1Þ, and according to the

assumption on T−Sit always holds that

max

ijab;ki;kj;ka∈Kj½T−Sijabðki;kj;kaÞj ≤CN−1=3

k:ðB18Þ

For the term with kk¼ka(i.e., δkk;ka), its estimate is the same as the term above when ka¼ki. When ka≠ki∈K,we

have jqj≥CN−1=3

kand

NkX

hkka;lkbjiki;jkji½T−Sklabðka;kb;kaÞ

≤

NkX

jqj2N−1=3

k≤C1

jqjN−1

For the first summation term in Eq. (B17), we introduce the change of variable kk→ki−q1and write it as

NkX

q1∈Kqnf0;−qgX

hkðki−q1Þ;lðkjþq1Þjiki;jkji½T−Sklabðki−q1;kjþq1;kaÞ

When q¼0, this term can be further bounded using Eq. (B18) by

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ðÞ ≤1

NkX

q1∈Kqnf0g

jq1j2N−1=3

k≤CZΩ

dq1

jq1j2N−1=3

k≤CN−1=3

When q≠0, this term can be further bounded as

ðÞ ≤C1

NkX

q1∈Kqnf0;−qg

jq1j2X

G∈L

jq1þqþGj2N−1

≤CN−1

kZΩ

dq1X

G∈L

jq1j2

jq1þqþGj2

≤CN−1

G∈L

jqþGj2≤C1

jqj2N−1

where L

0denotes the set of 27 lattice vectors in Laround the origin. The first inequality is based on the lemma assumption

on T−Sthat

j½T−Sklabðki−q1;kjþq1;kaÞj ≤C1

jq1þqþG0j2N−1

k≤CX

G∈L

jq1þqþGj2N−1

where G0∈L

0is the unique lattice vector that makes q1þqþG0∈Ω. The third inequality can be obtained from the

nonsmoothness characterization of function

fðzÞ¼ZΩ

dq1

jq1j2

jq1þzj2:

Using Lemma 11 in Ref. [41],fðzÞis singular only at z¼0of order −2, and its value at z∈Ωnf0gis bounded by Cjzj−2.

Based on the above estimates of the first term in Eq. (B17), we obtain the estimate of the error accumulation in the 4h2p

linear term calculation as



NkX

kk∈KX

hkkk;lkljiki;jkji½T−Sklabðkk;kl;kaÞ

≤C1

jqj2N−1

kq≠0;

N−1=3

kq¼0:

The same analysis can be applied to all the similar linear terms that contain ERIs with matched orbital indices. For other

linear terms, the analysis can be done similarly, and they all contribute at most OðN−1

kÞerror to the overall subtraction.

Taking the subtraction between the following 3h3plinear terms as an example, it can be formulated and bounded as



NkX

kk∈KX

haka;kkkjiki;ckci½T−Skjbcðkk;kj;kbÞ

≤C1

NkX

kk∈KnfkbgX

þ1

δkk;kbj½T−Skjbcðkk;kj;kbÞj

≤C1

NkX

q1∈Kqnf0g

jq1j2N−1

kþ1

N−1=3

k≤CN−1

For the subtraction between quadratic terms, we consider the 4h2pquadratic term as an example. The subtraction

between the two 4h2pquadratic terms can be formulated and bounded as

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

kkkc∈KX

klcd

hkkk;lkljckc;dkdiðTijcdðki;kj;kcÞTklab ðkk;kl;kaÞ−Sijcdðki;kj;kcÞSklab ðkk;kl;kaÞÞ

≤C1

kkkc∈KX

klcd

jTijcdðki;kj;kcÞ½T−Sklab ðkk;kl;kaÞþ½T−Sijcd ðki;kj;kcÞSklabðkk;kl;kaÞj

≤C1

kkkc∈KX

klcd

j½T−Sklabðkk;kl;kaÞj þ C1

kkkc∈KX

klcd

j½T−Sijcdðki;kj;kcÞj

≤CN−1

Similar analysis can be done to all the remaining quadratic terms, and they all contribute at most OðN−1

kÞerror to the overall

subtraction. Gathering all the estimates above together, we finish the proof.

APPENDIX C: PROOF OF THEOREM 2

To guarantee the convergence and control the regularity

of CCDðnÞcalculations with n→∞, we introduce addi-

tional technical assumptions similar to those in Ref. [41].

One key difference is that Ref. [41] measures the error in

the amplitude calculation using the maximum entrywise

norm. When Madelung constant correction is used, the

error of the amplitude should be measured by the average

entrywise norm (related to the L1norm) instead, which is

denoted by

kTk1¼1

ki;kj;ka∈KX

ijab

jTijabðki;kj;kaÞj;

ktk1¼1

jΩj3ZΩ×Ω×Ω

dkidkjdkaX

ijab

jtijabðki;kj;kaÞj;

where Tand tdenote a generic amplitude computed using

mesh Kand in the TDL, respectively.

Assume that the CCD amplitude equations using a

sufficiently fine mesh K(with or without the Madelung

constant correction) and in the TDL have unique solutions

and denote the solutions as TNk

and t, respectively.

Convergence of the CCDðnÞamplitudes to the CCD

amplitude is defined in the k·k1-norm sense as

lim

n→∞kTNk

n−TNk

k1¼0and lim

n→∞ktn−tk1¼0:

We impose a sufficient condition that guarantees the

convergence of fixed point iterations by requiring the target

mapping to be contractive in a domain that contains both

the solution point and the initial guess.

Assumption 18. For Nksufficiently large, the following

statements hold.

(1) The exact CCDðnÞamplitude tnconverges to the

CCD amplitude tpointwisely as n→∞, i.e.,

lim

n→∞½tnijabðki;kj;kaÞ

¼½tijabðki;kj;kaÞ;∀i; j;a; b;ki;kj;ka:ðC1Þ

(2) ðεNkÞ−1ANkis a contraction map in some domain

BNk⊂Cnocc×nocc ×nvir×nvir ×Nk×Nk×Nkthat contains TNk



and the initial guess 0, i.e.,

ðεNkÞ−1ANkðTÞ∈BNk;∀T∈BNk;

kðεNkÞ−1ðANkðTÞ−ANkðSÞÞk1

≤LkT−Sk1;∀T; S∈BNk;ðC2Þ

with a Lipschitz constant L<1.

(3) The exact CCDðnÞamplitude tnand the domain BNk

above satisfy that

MKtn∈BNk;∀n>0:ðC3Þ

(4) For all the amplitudes ftng, there exists a constant C

such that

kMKATDLðtnÞ−ANk;ξðMKtnÞk1

≤CN−1

k;∀n>0;

kMKATDLðtnÞ−ANkðMKtnÞk1

≤CN−1=3

k;∀n>0:ðC4Þ

Note that, when we consider the finite-size calculation with

Madelung constant correction, the components εNkor ANk

in the second assumption need to be changed to εNk;ξor

ANk;ξaccordingly.

Remark 19. The second assumption guarantees that

fTNk

nglies in BNkand converges to TNk

. For the third

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assumption, Theorem 1 shows that with each fixed n>0

the amplitude TNk

nconverges to tnin the sense of

lim

Nk→∞kMKtn−TNk

nk1¼0;

suggesting that fTNk

ng⊂BNkconverges to tnwith

K→Ω. Therefore, BNkand MKtnshould be related to

each other, which leads to the third assumption.

For the last assumption, we note that the two related error

estimates in Lemma 11 have the prefactor Cdependent on

the amplitude tn. The assumption here is stronger in the

sense that Cneeds to be independent of tn.

Rigorous statement of Theorem 2. Under Assumption

18, the finite-size error of the CCD correlation energy

scales as OðN−1=3

kÞin each of the following scenarios:

(i) the Madelung constant correction is applied only to

the ERI contraction ANkand (ii) the Madelung constant

correction is applied only to the orbital energy εNk

nk. When

the Madelung constant correction is applied to both ANk

and εNk

nkin the CCD calculation, the overall finite-size error

scales as OðN−1

kÞ.

Proof. The finite-size error in the CCD energy calcu-

lation with or without the Madelung constant correction

can be estimated as

ETDL

CCD −ENk

CCD¼jGTDLðtÞ−GNkðTNk

Þj

≤jGNkðMKtÞ−GNkðTNk

Þj

þjGTDL ðtÞ−GNkðMKtÞj

≤CkMKt−TNk

k1þCN−1

k;ðC5Þ

where the last inequality uses the boundedness of jWijabjin

GNkand Lemma 10. To first estimate the above amplitude

error when the Madelung constant correction is applied to

both orbital energy and ERI contractions, we consider the

error splitting Eq. (B2) for the amplitude calculation at the

nth fixed point iteration as

kMKtn−TNk

nk1≤



εTDL ½MKATDLðtn−1Þ−ANk;ξðMKtn−1Þ



1

þ



1

εTDL −

εNk;ξANk;ξðMKtn−1Þ



1

þ



εNk;ξ½ANk;ξðMKtn−1Þ−ANk;ξðTNk

n−1Þ



1

≤CN−1

kþL



MKtn−1−TNk

n−1



1;

where the last estimate uses the assumption in Eq. (C4) for

the first term and the assumptions of contraction maps in

Eq. (C2) and MKtn−1∈BNkin Eq. (C3) for the third term.

Since the initial guesses in the finite and the TDL cases

satisfy kMKt0−TNk

0k1¼0, we can recursively derive that

kMKtn−TNk

nk1≤C1−Ln

1−LN−1

and, thus, the first assumption Eq. (C1) gives

kMKt−TNk

k1¼lim

n→∞kMKtn−TNk

nk1≤CN−1

Plugging this estimate into Eq. (C5) then finishes the

proof for the scenario when the Madelung constant cor-

rection is applied to both orbital energies and ERI con-

tractions.

For the two scenarios with partial Madelung constant

correction, a similar analysis as above gives

kMKtn−TNk

nk1≤CN−1=3

kþL



MKtn−1−TNk

n−1



1;

where the dominant CN−1=3

kerror comes from uncorrected

orbital energy or uncorrected ERI-contraction term.

Recursively, we can obtain the amplitude error estimate as

kMKt−TNk

k1≤CN−1=3

which finishes the proof for the two scenarios with partial

correction. ▪

APPENDIX D: QUADRATURE ERROR

ESTIMATE FOR PERIODIC FUNCTIONS

WITH ALGEBRAIC SINGULARITY

This section presents a collection of lemmas that provide

quadrature error estimates for trapezoidal rules over peri-

odic functions with specific algebraic singularities, which

are used in this paper. Most of the lemmas are proven in

Ref. [41] and are restated here for completeness. In

addition, we introduce and prove a new quadrature error

estimate that is critical in describing the efficacy of the

Madelung constant correction and inverse symmetry in

reducing the quadrature error in orbital energies and ERI

contractions.

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The lemmas presented in this section provide the

asymptotic scaling of the quadrature errors and also a

quantitative relationship between the prefactors in the

estimate and the algebraic singularities of the integrand.

In addition to the singularity order, we further

quantitatively characterize the algebraic singularity as

follows. For a univariate function fðxÞthat is smooth

everywhere in Vexcept at x¼x0with algebraic singu-

larity of order γ, we define a constant

V;x0ðfÞ¼min C∶j∂α

xfðxÞj ≤Cjx−x0jγ−jαj;∀jαj≤l; ∀x∈Vnfx0g

¼max

jαj≤l



ð∂α

xfðxÞÞ=jx−x0jγ−jαj



L∞ðVÞ:

For a multivariate function fðx;yÞthat is smooth everywhere in VX×VYexcept at x¼x0with algebraic singularity of

order γ, we define a constant

VX×VY;ðx0;·ÞðfÞ¼min nC∶j∂α

x∂β

yfðx;yÞj ≤Cjx−x0jγ−jαj;∀jαj;jβj≤l; ∀x∈VXnfx0g;y∈VYo

¼max

jαj≤l



ð∂α

x∂β

yfðx;yÞÞ=jx−x0jγ−jαj



L∞ðV×VÞ;

where “·”in the subscript “ðx0;·Þ”is a placeholder to indicate the smooth variable. Using these two quantities, we have the

following function estimates that are extensively used in this section:

j∂α

xfðxÞj ≤Hl

V;x0ðfÞjx−x0jγ−jαj;∀l≥jαj;∀x∈Vnfx0g;

j∂α

x∂β

yfðx;yÞj ≤Hl

VX×VY;ðx0;·ÞðfÞjx−x0jγ−jαj;∀l≥jαj;jβj;∀x∈VXnfx0g;y∈VY:

1. Existing results from Ref. [41]

Lemma 20. Let fðxÞbe smooth and periodic in

V¼½−1

2;1

2d. The quadrature error of a trapezoidal rule

using an md-sized uniform mesh Xin Vdecays super-

algebraically as

jEVðf; XÞj ≤Clm−l;∀l>0:

Remark 21. If we replace fðxÞby fðx;yÞdefined in

V×VYwhich is smooth and periodic with respect to xfor

each y∈VYand satisfies supx∈V;y∈VYj∂α

xfðx;yÞj <∞for

any α≥0, Lemma 20 can be generalized as

jEVðfð·;yÞ;XÞj ≤Clm−l;∀l>0;∀y∈VY;

where constant Clis independent of y∈VY.

Lemma 22. Let fðxÞbe periodic with respect to V¼

½−1

2;1

2dand smooth everywhere except at x¼0of order

γ≥−dþ1.Atx¼0,fðxÞis set to 0. The quadrature error

of a trapezoidal rule using an md-sized uniform mesh X

that contains x¼0can be estimated as

jEVðf; XÞj ≤CHdþmaxð1;γÞ

V;0ðfÞm−ðdþγÞ:

If fð0Þis set to an Oð1Þvalue in the calculation, it

introduces an additional Oðm−dÞquadrature error.

Remark 23. If we replace fðxÞby fðx;yÞdefined in

V×VYwhich is smooth everywhere in V×VYexcept at

x¼0of order γ, Lemma 22 can be generalized to

jEVðfð·;yÞ;XÞj ≤CHdþmaxð1;γÞ

V×VY;ð0;·ÞðfÞm−ðdþγÞ;∀y∈VY;

where the prefactor applies uniformly across all y∈VY.

Lemma 24. Let fðxÞ¼f1ðxÞf2ðxÞ, where f1ðxÞand

f2ðxÞare periodic with respect to V¼½−1

2;1

2dand

(i) f1ðxÞis smooth everywhere except at x¼z1¼0of

order γ≤0and

(ii) f2ðxÞis smooth everywhere except at x¼z2≠0of

order 0.

Consider an md-sized uniform mesh Xin V. Assume that X

satisfies that z1,z2are either on the mesh or Θðm−1Þaway

from any mesh points and mis sufficiently large that

jz1−z2j¼Ωðm−1Þ.Atx¼z1and x¼z2,fðxÞis set to

0. The trapezoidal rule using Xhas quadrature error

jEVðf; XÞj ≤CHdþ1

V;z1ðf1ÞHdþ1

V;z2ðf2Þm−ðdþγÞ:

If fðz1Þand fðz2Þare set to arbitrary Oð1Þvalues, it

introduces an additional Oðm−dÞquadrature error.

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Remark 25. If we replace fiðxÞwith i¼1,2byfiðx;yÞ

defined in V×VYwhich is smooth everywhere in V×VY

except at x¼ziof order γand 0, respectively, Lemma 24

can be generalized to

jEVðf1ð·;yÞf2ð·;yÞ;XÞj

≤CHdþ1

V×VY;ð0;·Þðf1ÞHdþ1

V×VY;ð0;·Þðf2Þm−ðdþγÞ;∀y∈VY;

where the prefactor applies uniformly across all y∈VY.

2. A new quadrature error estimate

Here, we prove Lemma 16, which is used in the

quadrature error estimate of the ERI contractions with

the Madelung constant correction in Appendix B2. This

lemma is a generalization of the existing result in Lemma

24 with an additional condition.

Restatement of Lemma 16. Let fðxÞ¼f1ðxÞf2ðxÞ,

where f1ðxÞand f2ðxÞare periodic with respect to V¼

½−1

2;1

2dand

(i) f1ðxÞis smooth everywhere except at x¼z1¼0of

order γ≤−1,

(ii) f2ðxÞis smooth everywhere except at x¼z2≠0of

order 0, and

(iii) ∂α

xf2ð0Þ¼0for any derivative order jαj≤s.

Assume γ>−dfor fðxÞto be integrable in Vand γþ

sþ1≤0so the leading algebraic singularity of fðxÞis at

x¼0. Consider an md-sized uniform mesh Xin V.

Assume that Xsatisfies that z1,z2are either on the mesh

or Θðm−1Þaway from any mesh points and mis sufficiently

large that jz1−z2j¼Ωðm−1Þ.Atx¼z1and x¼z2,fðxÞ

is set to 0. The trapezoidal rule using Xhas quadrature error

jEVðf; XÞj ≤CHdþ1

V;z1ðf1ÞHdþ1

V;z2ðf2Þjz1−z2j−ðsþ1Þ

×m−ðdþγþsþ1Þ:

Proof. For z1¼0and any z2∈V, we can introduce a

proper translation fðxÞ→fðx−x0Þto move both the

singular points z1and z2to the smaller cube ½−1

4;1

4din

V. The target quadrature error can be correspondingly

reformulated as

EVðfð·Þ;XÞ¼EVþx0ðfð·−x0Þ;X−x0Þ

¼EVðfð·−x0Þ;X−x0Þ;

which is the quadrature error of the translated function

fðx−x0Þ. Without loss of generality, we assume such a

translation has been applied to fðxÞand Xand both

singular points z1and z2lie in ½−1

4;1

4d.

Define a cutoff function ψ∈C∞

cðRnÞsatisfying

ψðxÞ¼1;jxj<1

0;jxj>1

Denote the distance between the two singular points as

δz¼jz2−z1jand define two local cutoff functions that

isolate the two singular points as

ψδz;1ðxÞ¼ψx−z1

δz;ψδz;2ðxÞ¼ψx−z2

δz;

whose supports are both inside V. The target integrand can

be split into three parts as

fðxÞ¼fðxÞψδz;1ðxÞþfðxÞψδz;2ðxÞ

þfðxÞð1−ψδz;1ðxÞ−ψδz;1ðxÞÞ:

All three terms satisfy the periodic boundary condition on

∂V. The first term is smooth everywhere except at x¼z1

of order γþsþ1, the second term is smooth everywhere

except at x¼z2of order 0, and the last term is smooth

everywhere. Application of Lemmas 20 and 22 to these

terms suggests that

jEVðfψδz;1;XÞj ≤CHdþ1

V;z1ðfψδz;1Þm−ðdþγþsþ1Þ;ðD1Þ

jEVðfψδz;2;XÞj ≤CHdþ1

V;z2ðfψδz;2Þm−d;ðD2Þ

jEVðfð1−ψδz;1−ψδz;2Þ;XÞj

≤CjαjZV

dxj∂α

xfð1−ψδz;1−ψδz;2Þjm−jαj;∀jαj>d:

ðD3Þ

Note that the last estimate means a superalgebraic decaying

error and constant Cjαjdepends only on Vand jαj(see

Eq. (H.6) in Ref. [41] for the derivation of this detailed

prefactor for Lemma 20). Since γþsþ1≤0, the three

estimates together prove that the quadrature error of fðxÞ

scales asymptotically as m−ðdþγþsþ1Þas m→∞. In order to

describe the extreme case where the two singular points are

only Oðm−1Þaway from each other for a given mesh X,

i.e., δz¼Oðm−1Þ, we provide a precise description of the

three prefactors in the estimates above using δz.

For the prefactor in Eq. (D1), it is defined by

V;z1ðfψδz;1Þ

¼max

jαj≤l



ð∂α

xfðxÞψδz;1ðxÞÞ=jx−z1jγþ1−jαj



L∞ðVÞ;

∀l>0:

For any derivative order jαj≤l,wehave

XIN XING and LIN LIN PHYS. REV. X 14, 011059 (2024)

011059-28

j∂α

xðf1ðxÞf2ðxÞψδz;1ðxÞÞj ≤CX

α1þα2þα3¼α

j∂α1

xf1ðxÞjj∂α2

xf2ðxÞjj∂α3

xψδz;1ðxÞj

≤CX

α1þα2þα3¼α

V;z1ðf1Þjx−z1jγ−jα1jj∂α2

xf2ðxÞjδz−jα3jδjx−z1j<1

2δz

≤CX

α1þα2þα3¼α

V;z1ðf1Þjx−z1jγ−jα1j−jα3jj∂α2

xf2ðxÞjδjx−z1j<1

2δz:ðD4Þ

The last inequality uses jx−z1j≤1

2δzby noting that ∂α3

xψδz;1ðxÞis zero when jx−z1j>1

2δz. Next, we estimate j∂α2

xf2ðxÞj

in the ball jx−z1j≤1

2δz. When jα2j≤s,wehave

j∂α2

xf2ðxÞj ¼ X

jβj¼sþ1−jα2j

sþ1−jα2j

β!ðx−z1ÞβZ1

ð1−tÞs−jα2j∂α2þβ

xf2ðz1þtðx−z1ÞÞdt

≤Cjx−z1jsþ1−jα2jsup

jx−z1j≤1

2δz;jβj¼sþ1

j∂β

xf2ðxÞj

≤Cjx−z1jsþ1−jα2jHsþ1

V;z2ðf2Þδz−ðsþ1Þ:

The first equality applies the Taylor expansion of ∂α2

xf2ðxÞat x¼z1with the assumption that ∂β

xf2ðz1Þ¼0for any jβj≤s.

The last inequality uses the algebraic singularity of f2ðxÞat x¼z2. When jα2j>s, we have

j∂α2

xf2ðxÞj ≤sup

jx−z1j<1

2δz

j∂α2

xf2ðxÞj ≤CHl

V;z2ðf2Þδz−jα2j≤CHl

V;z2ðf2Þjx−z1jsþ1−jα2jδz−ðsþ1Þ:

Plugging the estimates of j∂α2

xf2ðxÞj above into Eq. (D4), we obtain

j∂α

xðf1ðxÞf2ðxÞψδz;1ðxÞÞj ≤CHl

V;z1ðf1ÞHl

V;z2ðf2Þδz−ðsþ1Þjx−z1jγþsþ1−jαj;l≥dþ1;

which suggests that

V;z1ðfψδz;1Þ≤CHl

V;z1ðf1ÞHl

V;z2ðf2Þδz−ðsþ1Þ;∀l≥dþ1:

Similar analysis can also be applied to the prefactor in Eq. (D2) to obtain

V;z2ðfψδz;2Þ≤CHl

V;z1ðf1ÞHl

V;z2ðf2Þδz−ðsþ1Þ;∀l≥dþ1:

For the estimate of the third term in Eq. (D3) with any jαj¼l≥dþ1, the prefactor can be bounded as

dxj∂α

xfðxÞð1−ψδz;1ðxÞ−ψδz;2ðxÞÞj

≤CX

α1þα2þα3¼αZV

dxj∂α1

xf1ðxÞjj∂α2

xf2ðxÞjj∂α3

xð1−ψδz;1ðxÞ−ψδz;2ðxÞÞj

≤CHl

V;z1ðf1ÞHl

V;z2ðf2ÞX

α1þα2þα3¼αZV

dxjx−z1jγ−jα1jjx−z2j−jα2jj∂α3

xð1−ψδz;1ðxÞ−ψδz;2ðxÞÞj:

For the integral with each set of ðα1;α2;α3Þ, we consider two cases.

(i) α3¼0. The integral can be bounded by

ðÞ ≤ZVnðBðz1;1

4δzÞ∪Bðz2;1

4δzÞÞ

dxjx−z1jγ−jα1jjx−z2jjα1j−jαj≤Cð1þδzγþd−jαjÞ;

using the Hölder-inequality technique developed in the proof of Lemma 24 in Ref. [41].

INVERSE VOLUME SCALING OF FINITE-SIZE ERROR IN …PHYS. REV. X 14, 011059 (2024)

011059-29

(ii) α3>0. The integral can be bounded by

ðÞ ≤CZðBðz1;1

2δzÞnBðz1;1

4δzÞÞ∪ðBðz2;1

2δzÞnBðz2;1

4δzÞÞ

dxjx−z1jγ−jα1jjx−z2j−jα2jδz−jα3j≤Cδzγþd−jαj:

Collecting these two terms together, the error estimate in Eq. (D3) with any jαj¼l≥dþ1can be further bounded as

jEVðfð1−ψδz;1−ψδz;2Þ;XÞj ≤ClHl

V;z1ðf1ÞHl

V;z2ðf2Þδzγþd−lm−l

≤ClHl

V;z1ðf1ÞHl

V;z2ðf2Þδz−ðsþ1Þm−ðγþsþ1þd−lÞm−l

≤ClHl

V;z1ðf1ÞHl

V;z2ðf2Þδz−ðsþ1Þm−ðγþsþ1þdÞ;

where the second inequality uses the assumption δz¼Ωðm−1Þand γþsþ1þd−l≤0. Gathering the above prefactor

descriptions for Eqs. (D1)–(D3), we obtain

jEVðf; XÞj ≤CHdþ1

V;z1ðf1ÞHdþ1

V;z2ðf2Þδz−ðsþ1Þm−ðγþsþ1þdÞ;

which finishes the proof. ▪

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011059-31

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Finite size error is commonly removed from coupled cluster theory calculations by N-1 extrapolations over correlation energy calculations of different system sizes (N), where the N-1 scaling comes from the total energy rather than the correlation energy. However, previous studies in the quantum Monte Carlo community suggest an exchange-energy-like power law of N-2/3 should also be present in the correlation energy when using the conventional Coulomb interaction. The rationale for this is that the total energy goes as N-1 and the exchange energy goes as N-2/3; thus, the correlation energy should be a combination of these two power laws. Further, in coupled cluster theory, these power laws are related to the low G scaling of the transition structure factor, S(G), which is a property of the coupled cluster wave function calculated from the amplitudes. We show here that data from coupled cluster doubles calculations on the uniform electron gas fit a function with a low G behavior of S(G) ∼ G. The prefactor for this linear term is derived from the exchange energy to be consistent with an N-2/3 power law at large N. Incorporating the exchange structure factor into the transition structure factor results in a combined structure factor of S(G) ∼ G2, consistent with an N-1 scaling of the exchange-correlation energy. We then look for the presence of an N-2/3 power law in the energy. To do so, we first develop a plane-wave cutoff scheme with less noise than the traditional basis set used for the uniform electron gas. Then, we collect data from a wide range of electron numbers and densities to systematically test five methods using N-1 scaling, N-2/3 scaling, or combinations of both scaling behaviors. We find that power laws that incorporate both N-1 and N-2/3 scaling perform better than either alone, especially when the prefactor for N-2/3 scaling can be found from exchange energy calculations.

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Article

Aug 2022

We present the first implementation and applications of non-Dyson algebraic diagrammatic construction theory for charged excitations in three-dimensional periodic solids (EA/IP-ADC). The EA/IP-ADC approach has a computational cost similar to the ground-state Møller-Plesset perturbation theory, enabling efficient calculations of a variety of crystalline excited-state properties (e.g., band structure, band gap, density of states) sampled in the Brillouin zone. We use EA/IP-ADC to compute the quasiparticle band structures and band gaps of several materials (from large-gap atomic and ionic solids to small-gap semiconductors) and analyze the errors of EA/IP-ADC approximations up to the third order in perturbation theory. Our work also reports the first-ever calculations of ground-state properties (equation-of-state and lattice constants) of three-dimensional crystalline systems using a periodic implementation of third-order Møller-Plesset perturbation theory (MP3).

Integral-Direct Hartree-Fock and Møller-Plesset Perturbation Theory for Periodic Systems with Density Fitting: Application to the Benzene Crystal

Article

Aug 2022
J CHEM THEORY COMPUT

We present an algorithm and implementation of integral-direct, density-fitted Hartree-Fock (HF) and second-order Møller-Plesset perturbation theory (MP2) for periodic systems. The new code eliminates the formerly prohibitive storage requirements and allows us to study systems 1 order of magnitude larger than before at the periodic MP2 level. We demonstrate the significance of the development by studying the benzene crystal in both the thermodynamic limit and the complete basis set limit, for which we predict an MP2 cohesive energy of -72.8 kJ/mol, which is about 10-15 kJ/mol larger in magnitude than all previously reported MP2 calculations. Compared to the best theoretical estimate from literature, several modified MP2 models approach chemical accuracy in the predicted cohesive energy of the benzene crystal and hence may be promising cost-effective choices for future applications on molecular crystals.

Ground-State Properties of Metallic Solids from Ab Initio Coupled-Cluster Theory

Article

Aug 2022

Metallic solids are an enormously important class of materials, but they are a challenging target for accurate wave function-based electronic structure theories and have not been studied in great detail by such methods. Here, we use coupled-cluster theory with single and double excitations (CCSD) to study the structure of solid lithium and aluminum using optimized Gaussian basis sets. We calculate the equilibrium lattice constant, bulk modulus, and cohesive energy and compare them to experimental values, finding accuracy comparable to common density functionals. Because the quantum chemical "gold standard" CCSD(T) (CCSD with perturbative triple excitations) is inapplicable to metals in the thermodynamic limit, we test two approximate improvements to CCSD, which are found to improve the results.

Machine learning for a finite size correction in periodic coupled cluster theory calculations

Article

Apr 2022

We introduce a straightforward Gaussian process regression (GPR) model for the transition structure factor of metal periodic coupled cluster singles and doubles (CCSD) calculations. This is inspired by the method introduced by Liao and Gr\"uneis for interpolating over the transition structure factor to obtain a finite size correction for CCSD [J. Chem. Phys. 145, 141102 (2016)], and by our own prior work using the transition structure factor to efficiently converge CCSD for metals to the thermodynamic limit [Nat. Comput. Sci. 1, 801 (2021)]. In our CCSD-FS-GPR method to correct for finite size errors, we fit the structure factor to a 1D function in the momentum transfer, $G$.We then integrate over this function by projecting it onto a k-point mesh to obtain comparisons with extrapolated results. Results are shown for lithium, sodium, and the uniform electron gas.

Inverse Volume Scaling of Finite-Size Error in Periodic Coupled Cluster Theory

Abstract

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