J. Chem. Phys. 151, 014110 (2019); https://doi.org/10.1063/1.5100022 151, 014110

© 2019 Author(s).

Implementation of analytic gradientsfor CCSD and EOM-CCSD using Choleskydecomposition of the electron-repulsionintegrals and their derivatives: Theory andbenchmarksCite as: J. Chem. Phys. 151, 014110 (2019); https://doi.org/10.1063/1.5100022Submitted: 14 April 2019 . Accepted: 06 June 2019 . Published Online: 03 July 2019

Xintian Feng, Evgeny Epifanovsky, Jürgen Gauss , and Anna I. Krylov

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

Implementation of analytic gradients for CCSDand EOM-CCSD using Cholesky decompositionof the electron-repulsion integrals and theirderivatives: Theory and benchmarks

Cite as: J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022Submitted: 14 April 2019 • Accepted: 6 June 2019 •Published Online: 3 July 2019

Xintian Feng,1,2 Evgeny Epifanovsky,2 Jürgen Gauss,3 and Anna I. Krylov1,3,a)

AFFILIATIONS1Department of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA2Q-Chem, Inc., 6601 Owens Drive, Suite 105, Pleasanton, California 94588, USA3Institut für Physikalische Chemie, Johannes Gutenberg-Universität Mainz, Duesbergweg 10-14, D-55128 Mainz, Germany

a)Electronic mail: krylov@usc.edu

ABSTRACTWe present a general formulation of analytic nuclear gradients for the coupled-cluster with single and double substitution (CCSD) andequation-of-motion (EOM) CCSD energies computed using Cholesky decomposition (CD) representations of the electron repulsion integrals.By rewriting the correlated energy and response equations such that the storage of the largest four-index intermediates is eliminated, CD leadsto a significant reduction in disk storage requirements, reduced I/O penalties, and an improved parallel performance. CD thus extends thescope of the systems that can be treated by (EOM-)CCSD methods, although analytic gradients in the framework of CD are needed to extendthe applicability of (EOM-)CCSD methods in the context of geometry optimizations. This paper presents a formulation of analytic (EOM-)CCSD gradient within the CD framework and reports on the salient details of the corresponding implementation. The accuracy and thecapabilities of analytic CD-based (EOM-)CCSD gradients are illustrated by benchmark calculations and several illustrative examples.

Published under license by AIP Publishing. https://doi.org/10.1063/1.5100022., s

I. INTRODUCTION

Analytic gradient techniques are of major importance in mod-ern quantum chemistry.1,2 Thus, the development of an analyticscheme for the evaluation of nuclear forces at the Hartree-Focklevel by Pulay3 opened the path for the routine computationaldetermination of molecular geometries, something which untiltoday is one of the main applications of quantum chemistry. Overthe years, essentially for all available quantum-chemical schemes,analytic-derivative techniques have been developed.4–16 Focusingon coupled-cluster (CC) theory17 as the current standard for high-accuracy computations, the first implementation of analytic gradi-ents within the CC singles and doubles (CCSD) approximation18

was reported by Scheiner et al.12 Analytic gradients for the CCSD(T)scheme,19 which augments the CCSD approximation with a

perturbative treatment of triple excitations, were first implementedby Scuseria;13 a more efficient implementation was provided some-what later by Lee and Rendell.14 Today, CC analytic gradients as wellas analytic second-derivative techniques are available for essentiallyall CC models with arbitrary excitation levels as well as additionalapproximations.15,16,20–28

CC theory can be elegantly extended in its applicability viathe equation-of-motion CC (EOM-CC) approach,29–40 which hasbecome the method of choice for the treatment of excited, ionized, orelectron-attached states. Analytic gradients for these methods havealso been reported.41–47

In recent years, significant effort has been made to extend theapplicability of existing quantum-chemical methods to larger sys-tems.48–53 Many ideas have been pursued, ranging from the useof integral-direct techniques,54–57 efficient integral screening,55,58

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-1

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

Laplace transformation,59,60 exploitation of the local character ofcorrelation,61,62 multilevel approaches,63–65 and use of pair-naturalorbitals (PNOs)51,53,66,67 up to the representation of the two-electronintegrals by means of resolution-of-identity techniques (RI, alsoreferred to as density fitting)68–70 or Cholesky decomposition(CD).52,71–77

RI schemes are nowadays routinely used in density-functionaltheory (DFT), second-order Møller-Plesset perturbation theory(MP2), and CC calculations,70,78–83 as well as multiconfigurationalmethods.84–86 These developments together with a local correlationtreatment have made it, for example, possible to carry out CC cal-culations on a protein (crambin) consisting of more than 600 atomsand with more than 6000 basis functions.66 The CD scheme is alsoroutinely used in quantum chemistry, but it was, due to its associ-ated cost for the decomposition step, until only recently applied inthe framework of multiconfigurational self-consistent field (SCF),84

CASPT2,85 or CC theory.82 Owing to the recently introduced newdecomposition algorithm,77 which removes the storage bottlenecksof the original procedure, the use of CD in quantum chemistry will,most likely, expand.

While all these developments significantly enhance the appli-cability of single-point energy computations for larger systems,somewhat less work has been done with respect to analytic gradi-ents.78,87–93 The reason for this is simply that all of the mentionedschemes that speed up computations complicate the theory and, inthis way, also the formulation of analytic gradient theory. Neverthe-less, in the context of RI, analytic gradients have been reported forDFT, MP2,78 CC methods,80,81,88,89 as well as various multireferenceschemes;90,94,95 for the CD schemes, gradient implementations havebeen described for CASSCF92 and MP2.91 Even less work has beendone so far for local correlation and PNO treatments.93,96–99 Mostof the work here has been limited to MP2 theory,97,99 although thetheory for local CC gradients has been put forth in the literature96

and an analytic scheme for the evaluation of first-order properties isavailable for the local PNO-CC approach.98

However, no analytic gradients using CD have been so farreported for CC and EOM-CC methods, despite the fact that the CDscheme significantly extends their range of applicability77,82,83 andin this way renders calculations for medium-size molecules with afew dozens of heavy atoms and 400–800 basis functions feasible ona modest hardware (e.g., midrange single computing nodes).77,100,101

The present paper addresses this issue and describes our implemen-tation of analytic CCSD and EOM-CCSD gradients when using CDfor the treatment of the two-electron integrals. Our work buildsupon Ref. 82 in which implementations for RI and CD based(EOM-)CC methods were reported. We will describe in Sec. IIthe underlying theory with a special focus on how the requiredderivatives of the Cholesky decomposed two-electron integrals areobtained together with a reformulation of (EOM-)CC gradient the-ory suitable for use with the decomposed integrals. This is followedin Sec. III by a sketch of our implementation within the Q-Chemprogram package102,103 (we note that the implementation naturallyincludes the RI gradients as a special case). Section IV demonstratesin test calculations the accuracy as well as the applicability of ourimplemented analytic gradient schemes. The latter is documented bygeometry optimizations for the ground and excited electronic statesof a dicopper single-molecule magnet (Cu2O10C8H16) with 36 atomsand 202 electrons (418 basis functions).

II. FORMALISMThe coupled-cluster (CC) wave function has the following

form:17

∣ΨCC⟩ = eT ∣Φ0⟩, (1)

where Φ0 is the reference determinant, usually the Hartree-Fock(HF) wave function, and T is the cluster operator. In the CCSDapproximation (CC with single and double substitutions), T istruncated at the double excitation level,18

T ≈ T1 + T2, (2)

withT1=∑

iatai a

†i,

T2=14∑ijab

tabij a†ib†j.

(3)

Here, we use indices i, j, k, . . . and a, b, c, . . . to denote the spinorbitals from the occupied (in Φ0) and virtual subspaces, respec-tively. To denote generic orbitals that can be either occupied orvirtual, we use letters p, q, r, s, . . .; letters µ, ν, λ, σ, . . . labelatomic orbitals. Equations for the CCSD energy (ECC) and the clusteramplitudes tai , tabij are obtained by projecting the Schrödinger equa-tion onto the subspace of the reference, singly, and doubly exciteddeterminants,

ECC = ⟨Φ0∣e−(T1+T2)He(T1+T2)∣Φ0⟩,

⟨Φai ∣e−(T1+T2)He(T1+T2)

∣Φ0⟩ = 0,

⟨Φabij ∣e−(T1+T2)He(T1+T2)

∣Φ0⟩ = 0.

(4)

EOM-CC (equation-of-motion CC) and the closely related linear-response CC theory extend the CC formalism to treat multipleelectronic states of various nature.29–40,43,46 Using general CI-likeexcitation operators R, the EOM-CCSD wave functions are given by

R∣ΨCC⟩ = ReT ∣Φ0⟩. (5)

Different types of target states can be accessed by using differentdefinitions of the operator R and different Φ0. In EOM-EE-CCSD,operator R is particle and spin conserving (1h1p and 2h2p, h = holeand p = particle), describing the excited states of the reference withN electrons,30,32,33

R ≈ R0 + R1 + R2, (6)

R0 = r01, (7)

R1 =∑iarai a

†i, (8)

R2 =14∑ijab

rabij a†ib†j. (9)

In EOM-SF, the R operator is also of 1h1p and 2h2p type, but itchanges the spin of the state (actually Sz and also, but not nec-essarily, S2).35,104 In EOM-IP-CCSD, the operator R is of 1h and2h1p type, allowing access to (N − 1)-electron states.43 In EOM-EA-CCSD, the R operator is of 1p and 1h2p type, providing access to(N + 1)-electron target states.34

By introducing the similarity-transformed Hamiltonian,H ≡ e−THeT , the EOM-CC equations can be cast in the form of a

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-2

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

non-Hermitian eigenvalue problem,

HR = RE. (10)

In the truncated space of determinants, Eq. (10) yields

⟨Φ0∣H − E∣RΦ0⟩ = 0,

⟨Φai ∣H − E∣RΦ0⟩ = 0,

⟨Φabij ∣H − E∣RΦ0⟩ = 0.

(11)

The EOM states in this formulation are just CI-like vectors and theground state is represented by Φ0 (i.e., R= 1), which is an eigen-state of H by virtue of Eq. (4). The EOM amplitudes satisfy abivariational principle,37 which is exploited in gradient and prop-erty calculations.25,42–45,105 Within this bivariational formalism, theleft eigenstates of H are defined using excitation operators L,

⟨Φ0∣(L†0 + L†

1 + L†2), (12)

L1 =∑ialai a

†i, (13)

L2 =14∑ijab

labij a†ib†j, (14)

and their amplitudes are determined by

⟨Φ0∣L†(H − E)∣Φ0⟩ = 0,

⟨Φ0∣L†(H − E)∣Φa

i ⟩ = 0,

⟨Φ0∣L†(H − E)∣Φab

ij ⟩ = 0.

(15)

For the EOM states, L0 = 0, and for the CC reference state,L0 = 1 (historically,106 the left L† operator for the reference state isdenoted by the (1 + Λ) operator). The left and right EOM states forma biorthogonal set,

⟨Φ0∣L†I RJ ∣Φ0⟩ = δIJ . (16)

The theory for analytic gradients and higher derivatives canbe conveniently derived using a Lagrangian formulation.106 TheEOM-CC Lagrangian is43–45,105,107

L(T,L,R,Z,Λ,Ω) = ⟨Φ0∣L†e−THeTR∣Φ0⟩ − E(⟨Φ0∣L†R∣Φ0⟩ − 1)

+ ⟨Φ0∣Z†e−THeT ∣Φ0⟩ +12∑pq

λpq

× ( fpq − δpq𝜖pp) +∑pq

ωpq(Spq − δpq), (17)

where the first term is the EOM-CC energy, the second term imposesthe biorthogonality constraint, the third term represents the con-straints imposed by the CC equations [Eq. (4)], Z is an operatorwhose amplitudes are the Lagrange multipliers (for the referenceCCSD state, with Z being 0), and the last two terms incorporate theHartree-Fock equations (f and S denote the Fock and overlap matri-ces, respectively, and 𝜖 denotes the orbital energies), accountingfor the orbital-response contributions to the gradient.108 The equa-tions for the amplitude response and orbital response for standardEOM-CCSD gradients, as well as the expression for the gradient interms of density matrices can be found, for example, in Refs. 43–46.

In the following, we extend the available analytic-gradient formula-tions to CD-CCSD and CD-EOM-CCSD.

The expression for CCSD/EOM-CCSD energy gradients hasthe following form (see, for example, Ref. 45):

dEdξ=∑

pqhξpqρpq +

14 ∑pqrs⟨pq∣∣rs⟩ξΠpqrs +∑

pqωpqSξpq, (18)

where hξpq and ⟨pq||rs⟩ξ are

hξpq =∂hpq∂ξ=∑

µνCµphξµνCνq, (19)

hξµν = ⟨χµ∣∂h∂ξ∣χν⟩ + ⟨

∂χµ

∂ξ∣h∣χν⟩ + ⟨χµ∣h∣

∂χν

∂ξ⟩, (20)

⟨pq∣∣rs⟩ξ =∂⟨pq∣∣rs⟩

∂ξ= ∑

µνλσCµpCνq⟨µν∣∣λσ⟩ξCλrCσs, (21)

⟨µν∣∣λσ⟩ξ = ⟨∂χµ

∂ξχν∣∣χλχσ⟩ + ⟨χµ

∂χν

∂ξ∣∣χλχσ⟩

+ ⟨χµχν∣∣∂χλ

∂ξχσ⟩ + ⟨χµχν∣∣χλ

∂χσ

∂ξ⟩, (22)

and Sξpq is

Sξpq =∑µν

CµpCνq(⟨∂χµ

∂ξ∣χν⟩ + ⟨χµ∣

∂χν

∂ξ⟩) (23)

and Cµp denotes the molecular orbital (MO) coefficients. The effec-tive density matrices ρ and Π are

ρ = γ′ + γ′′ + γ′′′, (24)

Π = Γ′ + Γ′′ + Γ′′′, (25)

with γ′ and Γ′ denoting the nonrelaxed density matrices, γ′′ andΓ′′ accounting for the amplitude-response contributions, and γ′′′and Γ′′′ representing the orbital-response contributions. Detailedexpressions for all density matrices as well as the orbital- andamplitude-response equations are found, for example, in Ref. 45.

Because the matrix II of electron-repulsion integrals (ERIs)is positive-semidefinite, it can be decomposed according to theCholesky decomposition procedure (CD).71 Using Mulliken nota-tion106 for the ERIs,

IIµν,λσ ≡ (µν∣λσ) ≈ (µν∣λσ)(CD) =M∑

P=1BPµνB

Pλσ, (26)

where M is the rank of decomposition and (µν|λσ)(CD) is referredto as CD-ERI (Cholesky-decomposed ERI) in this paper. The max-imum of M is N2 (full rank), where N is the total number of AObasis functions. When M equals the full rank, the “≈” becomes “=” inEq. (26), indicating that the ERI is fully recovered from the Choleskyvectors BP

µν. The error, which is the maximum deviation between(µν|λσ) and (µν|λσ)(CD), can be controlled by letting it be smallerthan the CD threshold 10−δ. Depending on the requested accuracy,the size of M can be much smaller than N2 (empirically, M is closeto δN, where δ is the same as in the definition of the threshold).

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-3

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

The Cholesky vectors defined by Eq. (26) can be transformed tothe MO basis,

BPpq =∑

µνCµpCνqBP

µν, (27)

so that the CD representation of the antisymmetrized ERIs (in Diracnotation106) in the MO basis becomes

⟨pq∣∣rs⟩ ≈∑QBQprB

Qqs −∑

QBQpsB

Qqr = P−rs∑

QBQprB

Qqs, (28)

where P−rs performs antisymmetrization with respect to the twoindices r and s. Expressions for the CCSD/EOM-CCSD energy andthe equations in the spin-orbital basis for the amplitudes in termsof the CD representation of the ERI can be derived by substitut-ing Eq. (28) into Eq. (4). The corresponding expressions for theamplitude equations together with those for the respective inter-mediates were given in Ref. 82. Here, we report the expressionsfor the left EOM eigenproblem (and the CCSD Λ equations), theEOM amplitude-response equations, and the density matrices. Thecorresponding expressions are given in the supplementary mate-rial. Although our spin-orbital implementation of standard and CD(EOM-)CC codes takes advantage of permutational, point-group,and spin symmetries via the libtensor library,109 additional savingsmay be achieved by using a spin-adapted implementation in atomicorbital basis (see, for example, Refs. 110–112).

A note is warranted whether the CD is used only in the corre-lation part of a CCSD/EOM-CCSD calculation or in the full compu-tation. One can consider here two definitions of CD-(EOM-)CCSDenergy: one using the HF energy computed using CD-ERIs and theother using the HF energy computed with the original ERIs. Bothschemes converge to the same result when the CD threshold is tight-ened. One can consider both options, and one needs to be only con-sistent by using an appropriate definition of the Lagrangian, Eq. (17),when deriving the orbital-response equations. Our results in Sec. IVillustrate that the differences between the two schemes are minute.In our production code, we choose to implement the approach basedon the definition of the CD-(EOM-)CCSD energy using the standardHF energy computed with original ERIs.

A. Evaluation of the derivatives of Cholesky vectorsTo simplify the expressions, in this section, we introduce the

labels bI and kJ to denote pairs of basis functions µν, i.e.,

IIµν,λσ ≡ (µν∣λσ) = IIIJ ≡ (bI ∣kJ). (29)

The indices bI and kJ run over all basis pairs µν, i.e., kJ = 1. . .N(N + 1)/2 − 1, where ≤ N(N + 1)/2 is the number of the unique µνpairs. Using this notation, (b|k) denotes the full matrix of ERIs. Theright-hand side of Eq. (26) becomes

(µν∣λσ)(CD) = (bI ∣kJ)CD =M∑

P=1BPbIB

PkJ . (30)

The canonical algorithm for the CD decomposition72,75,82,113 pro-ceeds in an iterative manner, leading to the following recurrentexpression for the Cholesky vectors, BP

b :

BJb = (

kJ ∣kJ)−

12[(b∣kJ) −

J−1

∑

K=1BKb B

KkJ], (31)

(kJ ∣kJ) = (kJ ∣kJ) −

J−1

∑

K=1BKkJB

KkJ , (32)

where (b|kJ) denotes one column in the (b|k) matrix. We notethat it is possible to reduce memory requirements and to signifi-cantly improve the efficiency of the decomposition by using the newalgorithm recently reported by Koch and co-workers.77

To obtain expressions for the nuclear derivatives of the ERIswithin the CD representation, we differentiate (b|k)(CD) with respectto the nuclear coordinate ξ,

∂(b∣k)(CD)

∂ξ=

M∑

P=1(

∂BPb

∂ξBPk + BP

b∂BP

k∂ξ)

≡∑

P(XP

bBPk + BP

bXPk ). (33)

In Eq. (34), XPb denotes the derivative of the Pth Cholesky vector.

These derivatives of the CD-ERI ultimately enter the expressions forthe derivative of the CD-(EOM-)CCSD energy. XP

b can be obtainedby differentiating Eq. (31),

XJb ≡

∂BJb

∂ξ= (kJ ∣kJ)

−12(∂(b∣kJ)∂ξ

−

J−1

∑

K=1XKb B

KkJ −

J−1

∑

K=1BKb X

KkJ)

−12(kJ ∣kJ)

−1BJb(

∂(kJ ∣kJ)∂ξ

− 2J−1

∑

I=1XIkJB

IkJ) (34)

and is evaluated recursively in a similar manner as the originalCholesky vectors. The CD algorithm needs to be modified for thispurpose such that the derivative vectors, XJ

b, are computed simul-taneously with the respective Cholesky vectors, BJ

b. Alternatively, itis possible to compute the derivatives in a separate routine, but thiswould require some additional bookkeeping. At the Jth iteration ofthe CD procedure, the following quantities are actually needed forevaluating XJ

b: the Cholesky vectors and their derivatives from theprevious steps, {BK

b ,XJb}

J−1I=1 ; a column of derivative integrals, ∂(b∣kJ)

∂ξ ;

updated diagonal ERI vector and its derivative, (kJ |kJ) and ∂(kJ ∣kJ)∂ξ .

The algorithm for computing the Cholesky vectors and theirderivatives is then as follows:

1. Compute the diagonal elements of the ERI matrix D0J = (kJ ∣kJ)

for all kJ .2. Find out the index k1 that corresponds to the maximum

element M1 = (k1|k1) of D0.From this step, the algorithm proceeds in an iterative

way until the new maximum element is smaller than thethreshold, 10−δ.

3. Compute the column of ERIs (b|ki), the new Cholesky vectorBi, and its nuclear gradient Xi according to Eqs. (31) and (34).

4. Update the diagonal vector via Di= Di−1

− (Bi)2.

5. Find out the index ki+1 that corresponds to the maximum ele-ment Mi+1 = (ki+1|ki+1) of Di. If the Mi+1 is greater than thedecomposition threshold 10−δ, go back to step 2. Otherwise,collect all B and X vectors and go to step 6.

6. Unpack B and X vectors from the basis pairs, b, to the productAO basis, µν, for subsequent CC/EOM-CC calculations.

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-4

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

B. CD orbital responseThe CCSD/EOM-CCSD orbital-response equations in terms

of one- and two-particle density matrices, γ′′′ and Γ′′′ fromEq. (25), can be found, for example, in Ref. 45. In the CDformalism, one needs to rewrite these equations by replacingthe term ∑rst⟨pr||st⟩Γqrst by 2∑s B

PpsΓPqs and ∑qs⟨pq||rs⟩λqs by

∑qs λqsP−rs∑Q BQprBQ

qs, where

ΓPpr =∑qsBPqsΓpqrs (35)

and λpq are the Lagrange multipliers from Eq. (17).

C. Evaluation of the CCSD/EOM-CCSD gradientThe CCSD/EOM-CCSD gradient is evaluated based on

Eq. (18). The equations for the nonrelaxed CCSD/EOM-CCSD den-sity matrices are the same as in the standard implementation, sothis code can be reused. However, the Λ-, amplitude, and orbitalresponse equations need to be solved using the CD integrals. Aneffective implementation involves a rewrite of the response equa-tions using the CD-ERIs, as it was already done in our CD imple-mentation for CCSD and EOM-CCSD energy calculations.82 Theresulting equations, which we derived and implemented, are givenin the supplementary material.

The final step consists in evaluating Eq. (18) via a straightfor-ward contraction of all density matrices with the derivative integralsbuilt from the Cholesky vectors and their derivatives. In the imple-mentation reported here, we rewrite these contractions as the con-tractions of the two-particle density matrix with Cholesky vectors,i.e.,

∑

pqrs⟨pq∣∣rs⟩ξΠpqrs =∑

pqrs∑

P[BP

prXPqs + XP

prBPqs − B

PpsX

Pqr − X

PpsB

Pqr]Πpqrs

= 4∑P∑

prΠP

prXPpr , (36)

whereΠP

pr =∑qsBPqsΠpqrs (37)

and XPpr is defined by Eq. (34). To obtain our final expressions, we

apply Eq. (36) separately to all blocks of the two-particle densitymatrix (i.e., Πijkl, Πabcd, Πijka, Πijab, Πiabc, and Πiajb) and collect therespective terms,

∑

pqrs⟨pq∣∣rs⟩ξΠpqrs =∑

ikPΠP

ikXPik +∑

acPΠP

acXPac +∑

jaPΠP

jaXPja, (38)

withΠP

ik = 4∑jlΠijklB

Pjl + 2∑

jaΠijkaB

Pja +∑

jlΠikabB

Pab, (39)

ΠPac = 4∑

bdΠabcdB

Pbd + 2∑

ibΠiabcB

Pib +∑

ijΠiajcBP

ij , (40)

ΠPia = 4∑

jbΠijabB

Pjb + 2∑

bcΠibacB

Pbc + 2∑

jkΠjikaB

Pjk − 2∑

jbΠibjaB

Pjb.

(41)

III. IMPLEMENTATIONOur expressions for evaluating analytic derivatives of the CD-

CCSD and CD-EOM-CCSD energies have been implemented inthe Q-Chem electronic structure program package.102,103 The imple-mentation is available both in double and in single precision ver-sions of the (EOM-)CC codes.114 In the latter, the CD decompositionand the integral transformation, however, are carried out in doubleprecision.

The main benefit of the CD or RI representation of the ERIis the reduced memory requirement. In our implementation82 ofthe CCSD and EOM-CCSD equations, we eliminated the need tocompute the full VVVV and OVVV intermediates, which are mostdemanding in this respect. However, we do compute the OOOO,OOVV, and OVOV intermediates for the right EOM vector calcu-lations. A detailed discussion of the storage requirements for thestandard and RI/CD CCSD/EOM-CCSD equations has been givenin Ref. 82. We follow the same strategy in the implementation of theequations for left EOM vectors, Λ, and amplitude response contri-butions. For the Λ-equations and left EOM vectors, we only com-pute the OOOO and OOOV intermediates (see the supplementarymaterial for the programmable equations). The reduction in mem-ory requirements is similar to those in the energy calculations. Themain benefit of using CD in the gradient calculations is that thefour-index two-particle density matrix is not needed.

IV. VALIDATION, BENCHMARKS,AND ILLUSTRATIVE APPLICATIONS

In this section, we discuss the validation of our implementationand present numerical results that quantify the errors due to the useof CD. In addition, we demonstrate with an illustrative applicationthe merits of using CD.

To validate the implementation and to quantify the errorsof CD-CCSD in comparison to standard CCSD, we consider aset of 5 small closed-shell molecules from Ref. 87: H2, N2, C2H2,CH3OH, and HCOCl (set I). To benchmark the accuracy of variousCD-EOM-CCSD schemes, we use the following examples (set II):the lowest A′ excited state of uracil (EOM-EE-CCSD), the low-est singlet state of para-benzyne (EOM-SF-CCSD), the electronicground state of the uracil cation (EOM-IP-CCSD), and the elec-tronic ground state of Na(NH3)4 (EOM-EA-CCSD). To illustratethe capabilities of the CD-(EOM-)CCSD, we consider a di-coppersingle-molecule magnet from Ref. 101.

In the analyses below, we compare the results obtained withdifferent CD thresholds against those obtained with the standardimplementation. To quantify the differences between two vectors(i.e., representing the Cartesian coordinates or forces), V and Vref ,we use the maximum absolute deviation (MD, marked by ∆),the norm of the error (NE), and the average absolute difference(AAD),

MD ≡ ∆V = max(∣V i− V i

ref ∣), i ∈ 3N, (42)

NE =

¿

ÁÁÀ

3N∑

i(V i− V i

ref )2, (43)

AAD =1

3N

3N∑

i∣V i− V i

ref ∣. (44)

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-5

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

All benchmark calculations were carried out in double pre-cision. All electrons were correlated in CCSD benchmarks (set I).In the EOM-CCSD calculations (set II), the frozen-core approxi-mation was used. The calculations for the single-molecule magnet(Sec. IV C), however, were carried out in single precision and withthe frozen-core approximation.

In the geometry optimizations, the convergence thresholdswere 3 × 10−6 a.u. for the maximum gradient component,1.2 × 10−5 a.u. for the maximum atomic displacement, and1 × 10−8 a.u. for the energy change in successive optimization cycles.All relevant geometries are given in the supplementary material.

A. ValidationWe validated the correctness of our implementation using set

I. In the first set of tests, we computed the CCSD gradients (withthe cc-pVDZ basis115,116) using CD-ERIs with thresholds of 10−3,10−4, 10−5, and 10−8 for the CD by using both the analytic proce-dure described above and numerical differentiation. In these cal-culations, the HF equations were also solved using CD-ERIs. Forselected examples, we computed in addition the numerical deriva-tives of the Cholesky vectors using two-point finite differences, totest the correctness of analytic nuclear gradient of the CD vectors.In these finite-difference calculations, the decomposition sequenceat the displaced geometries was the same as for the actual geom-etry. The convergence threshold in all of these calculations was10−11 a.u. for the SCF and CCSD energies. The step size was chosenas 10−5 Å.

Both the numerical differentiation of the Cholesky vectors andthe gradient calculations confirmed the correctness of our imple-mentation: the differences between the numerical and analytic val-ues for the individual gradient components were in all cases less than10−7 a.u. (see Table S1 in the supplementary material).

B. BenchmarksTo assess the errors introduced by CD and the impact of using

a standard HF reference (i.e., one obtained with the standard ERIs)instead of one that also uses CD, we again carried out CCSD/cc-pVDZ calculations for set I. All electrons were active in these cal-culations. These results are compiled in Tables I and II. Let us firstconsider the differences between the standard and CD gradientscomputed with thresholds of 10−3, 10−4, 10−5, and 10−8. As one cansee, the magnitude of the differences smoothly decreases with tight-ening the Cholesky threshold. Even when using a relatively loosethreshold of 10−3, the differences between the Cholesky and stan-dard gradient are small (∼10−4 a.u.) However, a more relevant quan-tity than the differences in the gradient is the differences betweenthe optimized geometries. The errors in bond lengths introduced byCD are very small: a threshold of 10−3 results in errors that are inthe largest case about 0.0004 Å, which are smaller than the intrinsicaccuracies of the underlying CCSD method. Our results are consis-tent with those reported by Lindh and co-workers87 based on densityfunctional theory calculations.

Table III reports the errors in the optimized structures andforces when computed using CD gradients but using a standard(non-CD) HF solution. The errors are essentially the same as for theproper CD calculation in which in all steps the CD-ERIs are used

TABLE I. Norm of the errors (a.u.) between analytic CD-CCSD and standard CCSDforcesa for different CD thresholds, δ.

δ

Molecule 10−3 10−4 10−5 10−8

H2 7.9× 10−5 1.3× 10−6 1.5× 10−6 1.1× 10−11

N2 8.3× 10−4 3.8× 10−4 5.3× 10−5 4.2× 10−8

C2H2 3.2× 10−4 6.9× 10−5 8.8× 10−6 1.6× 10−8

CH3OH 2.1× 10−4 1.1× 10−4 1.8× 10−5 2.0× 10−8

HCOCl 4.6× 10−4 5.8× 10−4 5.3× 10−5 1.5× 10−7

acc-pVDZ basis; all electrons correlated; at the geometries given in the supplementarymaterial.

(Table III). These results justify the use of the standard HF procedurewith regular ERIs in our production-level implementation.

To illustrate the accuracy of the CD-EOM-CCSD gradients, weconsider the examples from set II. In these calculations, we used theaug-cc-pVTZ basis115,117,118 together with the frozen-core approx-imation. The results are collected in Table IV. The errors in theoptimized geometries follow the same trend as in the CD-basedCCSD calculations. The largest errors are observed for Na(NH3)4:9 × 10−5 Å for a CD threshold of 10−3 and 8 × 10−6 Å for a CDthreshold of 10−4.

In contrast to the RI scheme with preoptimized auxiliary bases,the errors due to CD may vary along potential energy surfacesbecause the rank of the CD may change. The variations in the energyare limited by the CD threshold. The numerical tests (see, for exam-ple, Fig. 1 in Ref. 82) show that these variations are sufficiently smallnot to cause false minima and that one can consider the CD energiesas locally continuous. In the course of the geometry optimization,however, the rank of decomposition may be different at differentoptimization steps, especially when the steps are large. This can slowdown the convergence of the optimization, however, it should notaffect the quality of the optimized structures. Figures S1–S3 in thesupplementary material show the total energies and the rank of CDin the course of CD-EOM-CCSD geometry optimizations for bench-mark set II (the respective optimized geometries are summarized in

TABLE II. Errors in CD-CCSD/cc-pVDZ geometry optimizations using analytic gradi-ent with different CD thresholds, δ. The errors are computed with respect to standardCCSD values and reported as maximum deviation (MD). The MD for the optimizedbond lengths and the forces at the first optimization cyclea are denoted by |∆r |(pm = 10−2 Å) and by |∆F| (10−2 a.u.), respectively.

|∆r|, pm |∆F|, 10−2 a.u.

Molecule 10−3 10−4 10−5 10−3 10−4 10−5

H2 0.0077 0.0002 0.0001 0.0056 0.0001 0.0001N2 0.0105 0.0021 0.0003 0.0588 0.0271 0.0037C2H2 0.0216 0.0028 0.0011 0.0257 0.0048 0.0008CH3OH 0.0108 0.0057 0.0011 0.0137 0.0070 0.0011HCOCl 0.0353 0.0217 0.0007 0.0311 0.0371 0.0034

aAt the geometries given in the supplementary material.

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-6

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

TABLE III. Errors in CD-CCSD/cc-pVDZ geometry optimizations using analytic gradient with different CD thresholds, δ, anda standard HF solution with regular ERIs. The errors are computed against the standard CCSD values and reported asmaximum deviation (MD). The MDs for the optimized bond lengths and the forces at the first optimization cyclea are denotedby |∆r | (pm = 10−2 Å) and by |∆F| (10−2 a.u.), respectively.

|∆r|, pm |∆F|, 10−2 a.u.

Molecule 10−2 10−3 10−4 10−5 10−2 10−3 10−4 10−5

H2 0.0095 0.0002 0.0003 0.0001 0.0061 0.0001 0.0001 0.0001N2 0.2080 0.0121 0.0005 0.0003 0.8601 0.0318 0.0048 0.0032C2H2 0.0170 0.0112 0.0031 0.0008 0.0399 0.0298 0.0050 0.0006CH3OH 0.3176 0.0117 0.0103 0.0036 0.2919 0.0134 0.0079 0.0031HCOCl 1.5495 0.0434 0.0138 0.0010 0.9654 0.0455 0.0391 0.0036

aAt the geometries given in the supplementary material.

Table IV). As one can see, there are small variations in the rank ofdecomposition along the optimization pass and in cases when thevariations are larger, we observe that the optimization takes moresteps. Importantly, these variations do not affect the final convergedstructures.

TABLE IV. Errors in CD-EOM-CCSD/cc-pVTZ geometry optimizations using analyticgradients with different CD thresholds, δ, and a standard HF solution with regularERIs. Core electrons were frozen. The errors are computed with respect to the stan-dard EOM-CCSD values and reported as maximum deviation for bond lengths by |∆r |(pm = 10−2 Å).

|∆r|, pm

Molecule (Method) 10−3 10−4

Uracil (EE) 0.0012 0.0002Uracil cation (IP) 0.0011 0.0001para-benzyne (SF) 0.0004 0.0001Na(NH3)4 (EA) 0.0090 0.0008

FIG. 1. Structure of CUACACO2 (Cu2O10C8H16, complex 2 from Ref. 101). Cop-per atoms are shown in bronze, oxygens in red, carbons in gray, and hydrogensin white. Shown are the key distances (in Å): Cu–Cu, Cu–O, and Cu–water asdetermined from the X-ray structure (underlined), optimized CD-CCSD triplet state(plain), and optimized CD-EOM-SF-CCSD singlet state (italics).

C. Illustrative calculationTo illustrate the merit of the CD analytic gradients, we car-

ried out a geometry optimization for a single-molecule magnet fromRef. 101, i.e., CUAQUACO2 (complex 2) shown in Fig. 1. The struc-ture of this molecule has Ci symmetry, and, in the calculation, onehas to deal with 36 atoms (nuclei), 202 electrons, and 418 basisfunctions (cc-pVDZ basis set115,119). The computation also invokedthe frozen-core approximation and used a CD threshold of 10−3.We optimized the geometry of both the lowest triplet and singletstates using CD-CCSD and CD-EOM-SF-CCSD, respectively; bothcalculations used an unrestricted HF triplet reference.

In this calculation, the number of Cholesky vectors variedbetween 1440 and 1449. The decomposition procedure requiredabout 0.9 GB and 91 GB of RAM for computing the Cholesky vec-tors and their derivatives, respectively. While the Cholesky vectorswere stored on disk throughout the calculation, the derivatives ofthe Cholesky vectors were processed directly after the decomposi-tion procedure to evaluate the nuclear gradient, and the RAM wasreleased. Thus, the overall disk usage for CD is the size of Choleskyvectors, which is small compared to the high disk usage required byEOM-CCSD calculations.82

The main results are summarized in Fig. 1. As one can see, thedifferences in the key structural parameters between the X-ray andthe optimized structures are relatively small. Most importantly, theeffect of the geometry optimization on the computed singlet-tripletenergy gap is not large. At the X-ray structure, the gap betweenthe states is 180 cm−1 and the adiabatic gap between the optimizedsinglet and triplet states is 195 cm−1.

V. CONCLUSIONCD of the two-electron integrals is one of many possibili-

ties to reduce computational cost in electronic structure calcula-tions and to extend their applicability to larger systems. In thiswork, we have presented a formulation and implementation of ana-lytic nuclear gradients using CD for the CCSD and EOM-CCSDschemes. Our gradient formulation is consistent with our previouslyreported CD energy implementation by using the proper deriva-tives of the CD two-electron integrals. The errors in computedstructural parameters due to the use of CD, in comparison with

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-7

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

treatments using the regular two-electron integrals, are shown tobe more or less negligible and, furthermore, can be controlled viathe tunable Cholesky threshold. Our calculations finally documentthat the CD-based gradient scheme is indeed applicable to largersystems and render geometry optimizations for such systems on aroutine basis possible. Future work will focus on the extension ofthe current work to CCSD(T) gradients as well as the formulationof CD-based second derivatives and in this way will further enhancethe applicability of CC methods in the computation of molecularproperties.

SUPPLEMENTARY MATERIAL

See the supplementary material for programmable equations,relevant Cartesian coordinates and results of finite-difference calcu-lations.

ACKNOWLEDGMENTSThis work was supported by the U.S. Department of Energy

under the Grant No. DE-SC0018910 (A.I.K.). A.I.K. is a gratefulrecipient of the Simons Fellowship in Theoretical Physics and a vis-iting professorship of the MAINZ graduate school of excellence,which supported her sabbatical stay in Germany.

This work also benefited from scientific interactions withinthe MolSSI initiative supported by the National Science Foundation(molssi.org).

REFERENCES1P. Pulay, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 4, 169 (2014).2T. Helgaker, Encyclopedia of Computational Chemistry, Gradient Theory (Wiley& Sons, 1998).3P. Pulay, Mol. Phys. 17, 197 (1969).4S. Kato and K. Morokuma, Chem. Phys. Lett. 65, 19 (1979).5J. D. Goddard, N. C. Handy, and H. F. Schaefer III, J. Chem. Phys. 71, 1525(1979).6J. A. Pople, R. Krishnan, H. B. Schegel, and J. S. Binkley, Int. J. Quant. Chem. 13,225 (1979).7J. Gauss and D. Cremer, Chem. Phys. Lett. 138, 131 (1987).8J. Gauss and D. Cremer, Chem. Phys. Lett. 153, 303 (1988).9R. Krishnan, H. B. Schlegel, and J. A. Pople, J. Chem. Phys. 72, 4654(1980).10B. R. Brooks, W. D. Laidig, P. Saxe, J. D. Goddard, Y. Yamaguchi, and H. F.Schaefer III, J. Chem. Phys. 72, 4652 (1980).11R. Shepard, H. Lischka, P. G. Szalay, T. Kovar, and M. Ernzerhof, J. Chem. Phys.96, 2085 (1992).12A. C. Scheiner, G. E. Scuseria, J. E. Rice, T. J. Lee, and H. F. Schaefer III, J. Chem.Phys. 87, 5361 (1987).13G. E. Scuseria, J. Chem. Phys. 94, 442 (1991).14T. J. Lee and A. P. Rendell, J. Chem. Phys. 94, 6229 (1991).15J. Gauss and J. F. Stanton, J. Chem. Phys. 116, 1773 (2002).16M. Kállay, J. Gauss, and P. G. Szalay, J. Chem. Phys. 119, 2991(2003).17R. J. Bartlett and I. Shavitt, Many-Body Methods in Chemistry and Physics:MBPT and Coupled-Cluster Theory (Cambridge University Press, 2009).18G. D. Purvis and R. J. Bartlett, J. Chem. Phys. 76, 1910 (1982).19K. Raghavachari, G. W. Trucks, J. A. Pople, and M. Head-Gordon, Chem. Phys.Lett. 157, 479 (1989).20J. Gauss, J. F. Stanton, and R. J. Bartlett, J. Chem. Phys. 95, 2623 (1991).

21H. Koch, H. J. Aa. Jensen, T. Helgaker, P. Jørgensen, G. E. Scuseria, and H. F.Schaefer III, J. Chem. Phys. 92, 4924 (1990).22J. D. Watts, J. Gauss, and R. J. Bartlett, J. Chem. Phys. 98, 8718 (1993).23J. D. Watts, J. Gauss, and R. J. Bartlett, Chem. Phys. Lett. 200, 1 (1992).24J. Gauss and J. F. Stanton, Chem. Phys. Lett. 276, 70 (1997).25C. D. Sherrill, A. I. Krylov, E. F. C. Byrd, and M. Head-Gordon, J. Chem. Phys.109, 4171 (1998).26J. Gauss, J. Chem. Phys. 116, 4773 (2002).27K. Hald, A. Halkier, P. Jørgensen, S. Coriani, C. Hättig, and T. Helgaker,J. Chem. Phys. 118, 2985 (2003).28M. Kállay and J. Gauss, J. Chem. Phys. 120, 6841 (2004).29K. Emrich, Nucl. Phys. A 351, 379 (1981).30H. Sekino and R. J. Bartlett, Int. J. Quantum Chem. 26, 255 (1984).31H. Koch and P. Jørgensen, J. Chem. Phys. 93, 3333 (1990).32H. Koch, H. J. Aa. Jensen, P. Jørgensen, and T. Helgaker, J. Chem. Phys. 93, 3345(1990).33J. F. Stanton and R. J. Bartlett, J. Chem. Phys. 98, 7029 (1993).34M. Nooijen and R. J. Bartlett, J. Chem. Phys. 102, 3629 (1995).35S. V. Levchenko and A. I. Krylov, J. Chem. Phys. 120, 175 (2004).36K. W. Sattelmeyer, H. F. Schaefer, and J. F. Stanton, Chem. Phys. Lett. 378, 42(2003).37A. I. Krylov, Annu. Rev. Phys. Chem. 59, 433 (2008).38K. Sneskov and O. Christiansen, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2,566 (2012).39R. J. Bartlett, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2, 126 (2012).40A. I. Krylov, “The quantum chemistry of open-shell species,” in Reviews in Com-putational Chemistry, edited by A. L. Parrill and K. B. Lipkowitz (John Wiley &Sons, 2017), Vol. 30, Chap. 4, pp. 151–224.41J. F. Stanton, J. Chem. Phys. 99, 8840 (1993).42J. F. Stanton and J. Gauss, J. Chem. Phys. 100, 4695 (1994).43J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994).44J. F. Stanton and J. Gauss, Theor. Chim. Acta 91, 267 (1995).45S. V. Levchenko, T. Wang, and A. I. Krylov, J. Chem. Phys. 122, 224106 (2005).46P. A. Pieniazek, S. E. Bradforth, and A. I. Krylov, J. Chem. Phys. 129, 074104(2008).47M. Kállay and J. Gauss, J. Chem. Phys. 121, 9257 (2004).48Y. Jung, A. Sodt, P. M. W. Gill, and M. Head-Gordon, Proc. Natl. Acad. Sci.U. S. A. 102, 6692 (2005).49A. Dreuw and M. Head-Gordon, Chem. Rev. 105, 4009 (2005).50C. Ochsenfeld, J. Kussmann, and D. S. Lambrecht, “Linear-scaling methods inquantum chemistry,” in Reviews in Computational Chemistry (John Wiley & Sons,2007), Vol. 23, pp. 1–82.51F. Neese, A. Hansen, and D. G. Liakos, J. Chem. Phys. 131, 064103 (2009).52F. Aquilante, L. Boman, J. Boström, H. Koch, R. Lindh, A. S. de Merás, and T. B.Pedersen, “Cholesky decomposition techniques in electronic structure theory,” inLinear-Scaling Techniques in Computational Chemistry and Physics, Challengesand Advances in Computational Chemistry and Physics, edited by R. Zalesny, M.G. Papadopoulos, P. G. Mezey, and J. Leszczynski (Springer, 2011), pp. 301–343.53Q. Ma and H.-J. Werner, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 8, e1371(2018).54J. Almlöf, K. Fægri, Jr., and K. Korsell, J. Comput. Chem. 3, 385 (1982).55M. Häser and R. Ahlrichs, J. Comput. Chem. 10, 104 (1988).56H. Koch, A. Sánchez de Meráz, T. Helgaker, and O. Christiansen, J. Chem. Phys.104, 4157 (1996).57M. Schütz, R. Lindh, and H.-J. Werner, Mol. Phys. 96, 719 (1999).58D. S. Lambrecht, B. Doser, and C. Ochsenfeld, J. Chem. Phys. 123, 184102(2005).59J. Almlöf, Chem. Phys. Lett. 181, 319 (1991).60M. Häser, Theor. Chim. Acta 87, 147 (1993).61P. Pulay and S. Saebø, Theor. Chim. Acta 69, 357 (1986).62C. Hampel and H.-J. Werner, J. Chem. Phys. 104, 6286 (1996).63R. H. Myhre, A. M. J. Sanchez de Meras, and H. Koch, J. Chem. Phys. 141,224105 (2014).

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-8

Published under license by AIP Publishing

The Journalof Chemical Physics ARTICLE scitation.org/journal/jcp

64R. H. Myhre and H. Koch, J. Chem. Phys. 145, 044111 (2016).65S. Saether, T. Kjaergaard, H. Koch, and I.-M. Høyvik, J. Chem. Theory Comput.13, 5282 (2017).66C. Riplinger, B. Sandhoefer, A. Hansen, and F. Neese, J. Chem. Phys. 139,134101 (2013).67H.-J. Werner, G. Knizia, C. Krause, M. Schwilk, and M. Dornbach, J. Chem.Theory Comput. 11, 484 (2015).68J. L. Whitten, J. Chem. Phys. 58, 4496 (1973).69B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin, J. Chem. Phys. 71, 4993 (1979).70K. Eichkorn, O. Treutler, H. Öhm, M. Häser, and R. Ahlrichs, Chem. Phys. Lett.240, 283 (1995).71N. H. F. Beebe and J. Linderberg, Int. J. Quantum Chem. 12, 683 (1977).72H. Koch, A. S. de Merás, and T. B. Pedersen, J. Chem. Phys. 118, 9481 (2003).73F. Aquilante, R. Lindh, and T. B. Pedersen, J. Chem. Phys. 127, 114107 (2007).74L. Boman, H. Koch, and A. Sanchez de Meras, J. Chem. Phys. 129, 134107(2008).75F. Aquilante, T. B. Pedersen, and R. Lindh, Theor. Chem. Acc. 124, 1 (2009).76F. Aquilante, L. Gagliardi, T. B. Pedersen, and R. Lindh, J. Chem. Phys. 130,154107 (2009).77S. D. Folkestad, E. F. Kjønstad, and H. Koch, J. Chem. Phys. 150, 194112 (2019).78F. Weigend and M. Häser, Theor. Chim. Acta 97, 331 (1997).79O. Vahtras, J. Almlöf, and M. W. Feyereisen, Chem. Phys. Lett. 213, 514 (1993).80C. Hättig, J. Chem. Phys. 118, 7751 (2003).81A. Köhn and C. Hättig, J. Chem. Phys. 119, 5021 (2003).82E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y. Shao, and A. I. Krylov,J. Chem. Phys. 139, 134105 (2013).83K. D. Nanda and A. I. Krylov, J. Chem. Phys. 142, 064118 (2015).84F. Aquilante, T. B. Pedersen, R. Lindh, B. O. Roos, A. S. de Merás, and H. Koch,J. Chem. Phys. 129, 024113 (2008).85F. Aquilante, P.-Å. Malmqvist, T. B. Pedersen, A. Ghosh, and B. O. Roos,J. Theor. Comput. Chem. 4, 694 (2008).86J. Boström, M. G. Delcey, F. Aquilante, L. Serrano-Andrés, T. B. Pedersen, andR. Lindh, J. Theor. Comput. Chem. 6, 747 (2010).87F. Aquilante, R. Lindh, and T. B. Pedersen, J. Chem. Phys. 129, 034106 (2008).88U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 144, 174103 (2016).89U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 147, 044104 (2017).90W. Györffy, T. Shiozaki, G. Knizia, and H.-J. Werner, J. Chem. Phys. 138,104104 (2013).91J. Boström, V. Veryazov, F. Aquilante, T. B. Pedersen, and R. Lindh, Int. J.Quantum Chem. 114, 321 (2014).92M. G. Delcey, T. B. Pedersen, F. Aquilante, and R. Lindh, J. Chem. Phys. 143,044110 (2015).93M. Beuerle and C. Ochsenfeld, J. Chem. Phys. 149, 244111 (2018).94M. K. MacLeod and T. Shiozaki, J. Chem. Phys. 142, 051103 (2015).95B. Vlaisavljevich and T. Shiozaki, J. Chem. Theory Comput. 12, 3781 (2016).96G. Rauhut and H.-J. Werner, Phys. Chem. Chem. Phys. 3, 4853 (2001).97M. Schütz, H.-J. Werner, R. Lindh, and F. R. Manby, J. Chem. Phys. 121, 737(2004).98D. Datta, S. Kossmann, and F. Neese, J. Chem. Phys. 145, 114101 (2016).99P. Pinski and F. Neese, J. Chem. Phys. 148, 031101 (2018).100K. Nanda and A. I. Krylov, J. Chem. Phys. 149, 164109 (2018).101N. Orms and A. I. Krylov, Phys. Chem. Chem. Phys. 20, 13095 (2018).

102Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann,A. W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P. R. Horn, L. D.Jacobson, I. Kaliman, R. Z. Khaliullin, T. Kus, A. Landau, J. Liu, E. I. Proynov,Y. M. Rhee, R. M. Richard, M. A. Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L.Woodcock III, P. M. Zimmerman, D. Zuev, B. Albrecht, E. Alguires, B. Austin,G. J. O. Beran, Y. A. Bernard, E. Berquist, K. Brandhorst, K. B. Bravaya, S. T.Brown, D. Casanova, C.-M. Chang, Y. Chen, S. H. Chien, K. D. Closser, D. L.Crittenden, M. Diedenhofen, R. A. DiStasio, Jr., H. Do, A. D. Dutoi, R. G. Edgar,S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes,M. W. D. Hanson-Heine, P. H. P. Harbach, A. W. Hauser, E. G. Hohenstein, Z. C.Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R. A. King,P. Klunzinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter, K. U. Laog, A. Laurent,K. V. Lawler, S. V. Levchenko, C. Y. Lin, F. Liu, E. Livshits, R. C. Lochan,A. Luenser, P. Manohar, S. F. Manzer, S.-P. Mao, N. Mardirossian, A. V. Marenich,S. A. Maurer, N. J. Mayhall, C. M. Oana, R. Olivares-Amaya, D. P. O’Neill, J. A.Parkhill, T. M. Perrine, R. Peverati, P. A. Pieniazek, A. Prociuk, D. R. Rehn,E. Rosta, N. J. Russ, N. Sergueev, S. M. Sharada, S. Sharmaa, D. W. Small, A. Sodt,T. Stein, D. Stuck, Y.-C. Su, A. J. W. Thom, T. Tsuchimochi, L. Vogt, O. Vydrov,T. Wang, M. A. Watson, J. Wenzel, A. White, C. F. Williams, V. Vanovschi,S. Yeganeh, S. R. Yost, Z.-Q. You, I. Y. Zhang, X. Zhang, Y. Zhou, B. R. Brooks,G. K. L. Chan, D. M. Chipman, C. J. Cramer, W. A. Goddard, M. S. Gordon III,W. J. Hehre, A. Klamt, H. F. Schaefer, M. W. Schmidt III, C. D. Sherrill, D. G.Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A. T. Bell, N. A. Besley,J.-D. Chai, A. Dreuw, B. D. Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P. Hsu,Y. Jung, J. Kong, D. S. Lambrecht, W. Z. Liang, C. Ochsenfeld, V. A. Rassolov, L. V.Slipchenko, J. E. Subotnik, T. Van Voorhis, J. M. Herbert, A. I. Krylov, P. M. W.Gill, and M. Head-Gordon, Mol. Phys. 113, 184 (2015).103A. I. Krylov and P. M. W. Gill, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 3,317 (2013).104A. I. Krylov, Chem. Phys. Lett. 338, 375 (2001).105P. G. Szalay, Int. J. Quantum Chem. 55, 151 (1995).106T. Helgaker, P. Jørgensen, and J. Olsen, Molecular Electronic Structure Theory(Wiley & Sons, 2000).107K. D. Nanda and A. I. Krylov, J. Chem. Phys. 145, 204116 (2016).108The above form assumes the canonical form of the Hartree-Fock equations. Ifall orbitals are active, then this term can be reduced to∑iaλiaf ia.109E. Epifanovsky, M. Wormit, T. Kus, A. Landau, D. Zuev, K. Khistyaev, P. U.Manohar, I. Kaliman, A. Dreuw, and A. I. Krylov, J. Comput. Chem. 34, 2293(2013).110G. E. Scuseria, C. L. Janssen, and H. F. Schaefer, J. Chem. Phys. 89, 7382 (1998).111J. F. Stanton, J. Gauss, J. D. Watts, and R. J. Bartlett, J. Chem. Phys. 94, 4334(1990).112D. A. Matthews, J. Gauss, and J. F. Stanton, J. Chem. Theory Comput. 9, 2567(2013).113F. Aquilante, L. de Vico, N. Ferré, G. Ghigo, P.-Å. Malmqvist, P. Neogrády,T. B. Pedersen, M. Pitonak, M. Reiher, B. Roos, L. Serrano-Andrés, M. Urban,V. Veryazov, and R. Lindh, J. Comput. Chem. 31, 224 (2010).114P. Pokhilko, E. Epifanovskii, and A. I. Krylov, J. Chem. Theory Comput. 14,4088 (2018).115T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989).116D. E. Woon and T. H. Dunning, Jr., J. Chem. Phys. 98, 1358 (1993).117R. A. Kendall, T. H. Dunning, Jr., and R. J. Harrison, J. Chem. Phys. 96, 6796(1992).118D. Woon and T. H. Dunning (to be published), taken from EMSL.119N. B. Balabanov and K. A. Peterson, J. Chem. Phys. 123, 064107 (2005).

J. Chem. Phys. 151, 014110 (2019); doi: 10.1063/1.5100022 151, 014110-9

Published under license by AIP Publishing