Lowering of the Complexity of Quantum Chemistry Methods by Choice of Representation Narbe Mardirossian, James D

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Lowering of the complexity of quantum chemistry methods by choice of representation Narbe Mardirossian, James D. McClain, and Garnet Kin-Lic Chan

Citation: The Journal of Chemical Physics 148, 044106 (2018); View online: https://doi.org/10.1063/1.5007779 View Table of Contents: http://aip.scitation.org/toc/jcp/148/4 Published by the American Institute of Physics

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Lowering of the complexity of quantum chemistry methods by choice of representation Narbe Mardirossian,a) James D. McClain, and Garnet Kin-Lic Chanb) Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA (Received 2 October 2017; accepted 2 January 2018; published online 23 January 2018)

The complexity of the standard hierarchy of quantum chemistry methods is not invariant to the choice of representation. This work explores how the scaling of common quantum chemistry methods can be reduced using real-space, momentum-space, and time-dependent intermediate representations without introducing approximations. Weﬁnd the scalings of exact Gaussian basis Hartree–Fock theory, second- order Møller-Plesset perturbation theory, and coupled cluster theory (speciﬁcally, linearized coupled cluster doubles and the distinguishable cluster approximation with doubles) to be O(N3), O(N3), and O(N5), respectively, where N denotes the system size. These scalings are not asymptotic and hold over all ranges of N. Published by AIP Publishing. https://doi.org/10.1063/1.5007779

I. INTRODUCTION in this sense, keeps the methods exact. To illustrate succinctly how representations yield a change in complexity while pre- A great deal of progress in quantum chemistry comes serving exactness, consider the electronic Hamiltonian in three from introducing approximations, for instance, to the structure different bases: a general orbital basis, a plane-wave basis, and of the wavefunction. For the conventional ladder of quantum a real-space basis such as a grid, chemistry methods (i.e., mean-field theory, perturbation theory, coupled cluster theory, etc.), such approximations lead X † X † † to significant reductions in cost relative to the formal scal- H = tijai aj + vijklai aj akal, (1) ing of the methods. For example, within a Gaussian basis, ij ijkl X † X † † the exact scaling of Hartree–Fock theory (HF), second-order H = tk k a ak + vk k k K a a ak aK 1 2 k1 2 1 2 3 k1 k2 3 Møller-Plesset perturbation theory (MP2), and coupled clus- k1k2 k1k2k3 ter theory with (singles and) doubles (CC(S)D) is commonly (K + k3 = k1 + k2), (2) accepted to be O(N4), O(N5), and O(N6), respectively, as X † X a function of system size, N. However, by assuming local- H = trr0 ar ar0 + Vrr0 nrnr0 . (3) ity in the wavefunction solutions, one can reduce the scaling rr0 rr0 of these methods to O(N).1–8 Similarly, tensor factorization Each representation is exact in the sense that no system- (i.e., density fitting, Cholesky decomposition, orbital-specific specific structure in the matrix elements is assumed, but the corrections and pair natural orbitals, tensor hypercontraction, number of elements is O(N4), O(N3), and O(N2), respectively, etc.)9–16 and stochastic methods17–20 can yield reduced costs without further approximations. under different sets of assumptions and guarantees. For exam- Using similar ideas, we will explore how a choice of repre- ple, factorization methods exploit low-rank in either the solu- sentation affects the standard hierarchy of electronic structure tions or the Hamiltonian, while stochastic methods exchange a methods. Assuming Coulombic interactions between particles, deterministic guarantee of error for a probabilistic guarantee of we find that the exact scaling of common Gaussian basis meth- variance. ods is O(N3) for Hartree–Fock, O(N3) for MP2, O(N5) for In this short note, we will be concerned with an alternate linearized coupled cluster doubles21,22 (LCCD), and O(N5) strategy to reduce the cost of quantum chemistry methods. In for the distinguishable cluster approximation with doubles23 particular, we will examine how we can change the complexity (DCD). These scalings are not asymptotic but hold over any of a method simply by changing the underlying intermedi- range of the system size, N. To reveal these scalings, we employ ate representations. While the choices of representations and real-space, momentum-space, and time-dependent intermedi- approximations are commonly considered together, here we ate representations. None of these intermediate representations draw a distinction between the complexity lowering achieved are new. Indeed, elements of our argument resemble those through representation and that achieved through approxima- in the literature dating back to the earliest days of quantum tion. This is because changing the representation does not itself chemistry.24 However, we will cleanly draw a line between introduce assumptions into the structure of the solutions, and the mathematical operations that retain exactness of the methods, and those that introduce assumptions into the solutions. a)Electronic mail: [email protected] In this way, the scalings we derive are clearly free from b)Electronic mail: [email protected] approximation.

0021-9606/2018/148(4)/044106/6/$30.00 148, 044106-1 Published by AIP Publishing. 044106-2 Mardirossian, McClain, and Chan J. Chem. Phys. 148, 044106 (2018)

II. HARTREE–FOCK THEORY to quadrature33). However, our emphasis here will be on the complexity of MP2 in an “exact” (i.e., polylogarithmic in error) As a warmup exercise to see how our results arise, consider formulation. The recent work by Schafer¨ et al. found that the Hartree–Fock exchange energy. The conventional O(N4) the MP2 algorithm could be exactly reformulated through a scaling of exact Hartree–Fock arises from the evaluation of all choice of representation to have only quartic scaling.34 Here O(N4) electron repulsion integrals, which are subsequently we find that the scaling of exact MP2 can be further reduced contracted into the one-particle density matrix, γ. However, to O(N3). (After submission, we were informed of the work of at a more basic level, the Hartree–Fock exchange energy is Moussa35 that presents a related analysis of the cost of exact simply a double integral, MP2.) 2 |γ (r1, r2)| As a single space-time integral, the two components of the EHF-X = − dr1 dr2. (4) |r1 − r2 | MP2 energy, termed direct (MP2-J) and exchange (MP2-K), Given the integrand, this integral can be “exactly” evaluated are by quadrature with O(N2) cost, regardless of the form of γ. To E = 2 g (r , r0 , τ)g (r , r0 , τ)g (r0 , r , τ) be a little more precise, we use the term “exact” for quadrature MP2-J o 1 1 o 2 2 v 1 1 because so long as singularities are appropriately handled, the × g (r0 , r , τ)v(|r − r |)v(|r0 − r0 |) dR dτ, (5) cost to obtain a desired accuracy is polylogarithmic in , v 2 2 1 2 1 2 i.e., ∼log()α. To obtain the integrand, we must evaluate γ 0 0 0 EMP2-K = − go(r1, r2, τ)go(r2, r1, τ)gv(r1, r1, τ) (here expanded in a Gaussian basis) at the coordinates (r1, P ∗ 0 0 0 r2). From γ(r1, r2) = i φi (r1)φi(r2), we see that this carries × gv(r2, r2, τ)v(|r1 − r2 |)v(|r1 − r2 |) dR dτ, (6) O(N3) cost; thus the full cost of evaluating the exchange energy 3 where dR denotes an integration over all spatial coordinates, is O(N ). 0 0 v(|r r |) is the Coulomb operator, and go(r, r , τ) and gv(r, We can also consider the cost of obtaining the Hartree– 0 Fock solution. Hartree–Fock theory is a variational the- r , τ) are occupied and virtual Green’s functions, respectively, ory, and we can use the cost of evaluating the Lagrangian defined as derivative as a proxy for the cost of solving the equa- 0 X ∗ 0 − g (r, r , τ) = φ (r)φ (r )e iτ, (7) tions. Since the Lagrangian is an algebraic function of the o i i i variational parameters in the density matrix, the rules of X 0 ∗ 0 aτ adjoint differentiation25 dictate that the cost of the deriva- gv(r, r , τ) = φa(r)φa(r )e . (8) tive is also O(N3). Thus solving the Hartree–Fock equa- a tions (for a fixed number of derivative steps) is also O(N3) To obtain the appropriate scaling of the algorithm, it is neces- cost. sary to treat the convolution integrals with the Coulomb oper- The O(N3) scaling of Hartree–Fock is certainly not a new ator in special way. Within the Fourier representation, using a result: it was already well known from work on pseudo-spectral uniform mesh in real and momentum space, the well-known (PS) methods.26–28 The above exercise merely emphasizes that result is that the Coulomb potential, J(r0), corresponding to a the O(N3) scaling does not arise from any approximations but charge distribution, ρ(r), only from the intermediate representation. Thus it describes ρ(r) 4π 0 the complexity of the exact method. J(r0) = dr = (2π)−3 ρ(G)e−iG·r dG, (9) |r − r0| G2 III. SECOND-ORDER MØLLER–PLESSET can be computed using the fast Fourier transform with PERTURBATION THEORY O(N log N) cost, which we will consider O(N) for simplicity. Errors due to periodic images can either be thought of as Second-order Møller–Plesset perturbation theory (MP2) arising from the limits of integration in the quadrature or can be is perhaps the simplest route to including electron correlation eliminated by truncating the Coulomb operator.36,37 Alterna- effects. The conventional O(N5) formal scaling arises from tively, one can compute the Coulomb potential on unstructured the atomic orbital to molecular orbital integral transformation grids by solving the real-space Poisson equation with O(N) necessary to evaluate the MP2 energy in a canonical basis. Our cost.38 treatment of MP2 resembles that of HF in Sec.II in that we In either case, assuming that the Coulomb operator can first express the MP2 energy as a multi-dimensional integral. be applied at O(N) cost and assuming that Green’s func- We employ the time-dependent (i.e., Laplace transform29) rep- tions have been formed (which requires the same O(N3) resentation of the MP2 energy, which rewrites the energy as operation for both MP2 components), we can break down an integral over space and imaginary time. The Laplace trans- the evaluation of the MP2-J expression into the following form is a common component of reduced scaling MP2 methods steps:39,40 especially in conjunction with local approximations,4 but here we emphasize that by itself it only corresponds to a choice of 0 0 0 f (r, r , τ) = go(r, r , τ)gv(r , r, τ), (10) representation. We also employ a Fourier representation of the Coulomb operator to enable its fast application. All of these F(r0, r00, τ) = f (r, r0, τ)v(|r − r00|) dr, (11) elements can be found in various earlier studies, such as in real-space versions of MP230,31 and from the tensor hypercon- 0 00 00 0 EMP2-J = 2 F(r , r , τ)F(r , r , τ) dR dτ. (12) traction literature16,32 (related to the former via the connection 044106-3 Mardirossian, McClain, and Chan J. Chem. Phys. 148, 044106 (2018)

In the first step, the two pairs of occupied and virtual Green’s functions that depend on the same real-space indices are combined at O(N2) cost, while the second step scales as O(N2) because it involves the application of the Coulomb operator at every point r0. Finally, the energy evaluation is a double integral and is thus of O(N2) cost. Note that the latter result indicates that certain variants of MP2, such as scaled opposite-spin MP2,41 have an exact complexity of O(N2) (aside from the formation of Green’s functions). The complexity of the MP2-K expression can be determined in a similar way. We group the expressions as follows: FIG. 2. Percent errors for MP2-J (blue squares) and MP2-K (green diamonds) within our pilot O(N3 log N) MP2 implementation. The test system is a G(r0 , r0 , τ) = dr dr g (r , r0 , τ)g (r0 , r , τ) 1 2 2 1 o 1 2 v 1 1 diamond cubic primitive cell at a lattice constant of 6.74 Bohr, using the f GTH-SZV basis set and GTH LDA pseudopotential. 0 0 × v(|r1 − r2 |) go(r2, r1, τ)gv(r2, r2, τ) , (13) g MP2-J algorithm scales close to quadratically with the num- E = − G(r , r , τ)v(|r − r |) dR dτ. (14) MP2-K 1 2 1 2 ber of grid points [its formal scaling in the implementation 2 With MP2-K, the four Green’s functions have unique pairs of is O(N log N)], while the MP2-K algorithm scales close to indices and cannot be straightforwardly combined as in MP2- cubically with the number of grid points. The cubic scaling J. The first step above is the most expensive, as the convolution and conventional evaluation of the Laplace transform MP2 integral (O(N) cost) is carried out for the O(N2) pairs of grid energy agree to 12 significant figures. points. Thus the entire MP2 energy can be determined at O(N3) cost. IV. COUPLED CLUSTER THEORY As a simple numerical demonstration of this algorithm, The above general arguments can be repeated to derive we have implemented an elementary cubic-scaling MP2 using lower formal complexities for a variety of different quantum the PySCF programming framework.42 We start from the inte- chemistry methods. Here we will briefly outline how they can gral expressions in Eqs. (5) and (6) and build the intermediates be extended to several coupled cluster approximations. Unlike in Eqs. (7)–(14) on a uniform cubic grid. The Coulomb opera- in MP2, the coupled cluster amplitudes are not known explic- tor is applied using a three-dimensional fast Fourier transform. itly but must be determined by solving the amplitude equations. Instead of the scaling with system size, we here carry out the For simplicity, we will discuss only the case of CCD (the sin- simpler test of scaling with respect to the number of cubic grid gles contribution is subleading in complexity), where the t points, which for fixed accuracy is proportional to the system 2 amplitude is the four index tensor tab. Conventionally, the cost size. For the diamond cubic primitive cell (lattice constant of ij 6 6.74 Bohr, GTH-SZV basis set43 and GTH LDA pseudopoten- of CCD is considered to be O(N ). Here we show that cer- 6 tial44), the timings and scalings are shown in Fig.1 for a single tain subsets of diagrams that have O(N ) cost (the LCCD and 5 Laplace point evaluation. The percent errors with respect to DCD subsets) can be reduced to O(N ) cost without assuming the large grid limit are given in Fig.2. We see clearly that the any structure in the amplitudes. A similar asymptotic scaling in a plane-wave basis, using a tensor hypercontraction approximation for the integrals but also without assuming structure in the amplitudes, has recently been reported in Ref. 45. Our analysis is related to that in Ref. 45 but illustrates that the O(N5) scaling is an exact, rather than asymptotic, result. The coupled cluster doubles correlation energy is given by the trace of the amplitudes with the integrals in Eq. (1) (assuming spin orbitals), 1 X E = tabv CCD 4 ij ijab ijab 1 = t(r , r , r , r )v(|r − r |) dr dr , (15) 4 1 2 1 2 1 2 1 2

FIG. 1. Scaling for MP2-J (bottom) and MP2-K (top) within our pilot where the real-space amplitude is deﬁned as 3 O(N log N) MP2 implementation. The circles correspond to cubic grids with 0 0 X ab ∗ ∗ 0 0 a length that is a factor of 2, 3, 5, 7, 11, or 13, while the triangles correspond t(r1, r2, r1, r2) = tij φi (r1)φj (r2)φa(r1)φb(r2) to prime numbers or non-ideal lengths. The test system is a diamond cubic ijab primitive cell at a lattice constant of 6.74 Bohr, using the GTH-SZV basis set − φ∗(r )φ∗(r )φ (r0 )φ (r0 ). (16) and GTH LDA pseudopotential. i 1 j 2 b 1 a 2 044106-4 Mardirossian, McClain, and Chan J. Chem. Phys. 148, 044106 (2018)

The coupled cluster energy is a double integral and thus The nine diagrams with reduced complexity can be grouped 2 given the real-space amplitudes, requires O(N ) cost. How- into three separate types: (1) D2c,D2d,D2ex2, and D2ex3, ever, the amplitude equations do not define the amplitudes (2) D2e and D2ex1, and (3) D3b,D3bx1, and D3bx2. The type in this form, and the transformation from the orbital basis 1 terms contain a single t2 amplitude, with the contraction to real-space is of O(N5) cost. Thus the exact cost to eval- indices corresponding to different electron coordinates (i.e., uate the coupled cluster energy, assuming the use of the r1 and r2), while the type 2 terms contain a single t2 ampli- orbital-based amplitude equations to determine the amplitude, with the contraction indices corresponding to the same 5 tudes, is O(N ), without further approximations. (If the ampli- electron coordinate (i.e., either both r1 or both r2). The tudes can be defined in a different way, for example, as in type 3 terms, despite containing a pair of t2 amplitudes, can the random-phase approximation to CCD, this cost can be be evaluated in O(N5) time because the contraction indices reduced.) contained in each amplitude correspond to the same elec- The CCD amplitude equations46 are conveniently pre- tron coordinate. To illustrate the scaling reduction for the sented in a diagrammatic form in Fig.3. Above each dia- three aforementioned types, we take a single term from each gram, we give the scaling of each term. Like the above case and define appropriate intermediates in Eqs. (17)–(19), argument for the energy, we can transform the indices of where Eq. (17) corresponds to diagram D2c, Eq. (18) corre- each amplitude into the real-space representation as needed sponds to diagram D2e, and Eq. (19) corresponds to diagram to apply the Coulomb operator, before transforming back into D3b, the orbital basis. This change of representation reduces the X 2 hKB|vˆ|CJitAC complexity of nine of the 20 terms from O(N6) to O(N5). IK KC These reductions are also indicated in Fig.3. According X 2 K(r )B(r )v(r , r )C(r )J(r )tAC dr dr to the diagrams, LCCD corresponds to the first nine terms, 1 2 1 2 1 2 IK 1 2 KC while DCD corresponds to LCCD, plus the three other terms AC X AC A X AC 6 5 t (r1) = K(r1)t t (r1) = C(r1)t (r1) whose scaling is reduced from O(N ) to O(N ) (D3b,D3bx1, I IK I I (18) 5 K C and D3bx2), plus the last four O(N ) terms. Thus the exact A A cost to determine the amplitudes in either LCCD or DCD T (r2) = t (r1)v(r1, r2) dr1 I I is O(N5). AB A T = 2 B(r2)J(r2)T (r2) dr2. 1 X IJ I hAB|vˆ|CDitCD 2 IJ In order to clarify the reduction in scaling, we will walk through CD 1 X CD the derivation for the D2c term. In a Gaussian basis, it is evident A(r )B(r )v(r , r )C(r )D(r )t dr dr 6 2 1 2 1 2 1 2 IJ 1 2 that this term scales as O(N ). After rewriting the integral in CD X X its real-space form, the contractions over C and D each require t D(r ) = C(r )tCD t (r , r ) = D(r )t D(r ) IJ 1 1 IJ IJ 1 2 2 IJ 1 O(N5) time since the former involves four orbital indices and C D one real-space index and the latter involves three orbital indices TIJ (r1, r2) = tIJ (r1, r2)v(r1, r2) and two real-space indices. Then, the result is multiplied by T A (r ) = A(r )T (r , r ) dr 4 IJ 2 1 IJ 1 2 1 the Coulomb operator in real space, at O(N ) cost. The next step is similar to that shown in Eq. (9) and scales as O(N4), T AB = 1 B(r )T A (r ) dr . IJ 2 2 IJ 2 2 while the final step is again O(N5). Thus, the scaling for (17) a term that is conventionally O(N6) can be exactly reduced

FIG. 3. Spin-summed CCD amplitude equation diagrams. The cost of diagrams associated with a single scaling is unchanged when using a real-space intermediate representation, while those with arrows experience scaling reduction [i.e., the scaling of term D2c is reduced from O(N6) to O(N5)]. 044106-5 Mardirossian, McClain, and Chan J. Chem. Phys. 148, 044106 (2018) to O(N5). with the atomic orbital Green’s functions [that require O(N3) X AC DB time to compute] deﬁned as 2 hKL|vˆ|CDitIK tLJ X − τ KLCD g 0 (τ) = C 0 C e i , (21) X µ µ µ i µi K r L r v r r C r D r tACtDB dr dr i 2 ( 1) ( 2) ( 1, 2) ( 1) ( 2) IK LJ 1 2 X aτ KLCD g¯ν0ν(τ) = Cν0aCνae , (22) AC X AC A X AC tI (r1) = K(r1)tIK tI (r1) = C(r1)tI (r1) a K C X X (19) it is possible to formulate a series of steps to evaluate Eq. (20) DB DB B DB 3 t J (r2) = L(r2)tLJ tJ (r2) = D(r2)t J (r2) where the cost is no greater than O(N ). The ﬁrst three L D intermediates require O(N3) cost, A A TI (r2) = tI (r1)v(r1, r2) dr1 X 0 ρµ(r, τ) = µ (r)gµ0µ(τ), (23) T AB = 2 tB(r )T A(r ) dr . µ0 IJ J 2 I 2 2 h (r, τ) = ρ (r0, τ)ρ (r0, τ)v(|r − r0|) dr0, (24) hµνi µ ν

0 0 V. ALTERNATE REPRESENTATIONS h 0 0 (τ) = h (r, τ)hσ λ i(r) dr, (25) hµνihσ λ i hµνi In our above arguments, we reduced the exact scalings and the final energy evaluation, of quantum chemistry methods by combining several different representations. In all three methods (HF, MP2, and X E = 2 h 0 0 (τ)h 0 0 (τ) dτ, (26) CC), we used a real-space intermediate representation. We MP2-J hµνihσ λ i hλ σ ihνµi hσ0λ0i,hµνi obtained additional cost reductions from the Fourier representation of the Coulomb operator, while the MP2 algorithm is O(N2) cost. Note that there are additional cubic steps in also used a time-dependent representation. These intermedi- the evaluation of the MP2-J term compared to the quadra- ate representations are not the only ones that lead to reduced ture implementation because in the case of quadrature, the scalings, and other choices may lead to lower computational cubic cost is confined to the formation of Green’s functions prefactors. For example, if we allow for a polylogarithmic at the beginning of the algorithm, while here the cubic cost is dependence of computational cost on the threshold error , delayed until quantities are placed on the grid. However, it is then we can regard atomic orbital bases as a form of expo- clear that by using a mixture of atomic orbitals and quadra- nentially localized real-space basis. This is the standard argu- ture, the overall prefactor is greatly reduced, as typically the ment for atomic orbital screening, but here we are interested number of atomic orbitals required in the “function” quadra- only in the reduction in complexity that can be achieved ture is much less than the number of grid points required without assuming locality in the wavefunction or by cutting for numerical quadrature. Although the above algorithm will off algebraically decaying quantities (such as R 6 contribu- exhibit cubic scaling on length scales determined only by the tions to the correlation energy47,48) that destroy the poly- atomic orbitals rather than the locality of the wavefunction, logarithmic computational cost. Within this sense of retain- the use of diffuse functions will prevent the onset of this ing the exactness of the method, as long as we also use scaling until larger systems. In such a case, alternative rep- a scheme to apply the Coulomb operator with O(N) cost, resentations may prove useful, and this is a topic of future one recovers the same complexities we have derived above work. for systems on length scales larger than the atomic orbital size. One way to apply the Coulomb operator in a fast scheme VI. CONCLUSION is to use a mixed basis and grid representation, as is com- In summary, the present work re-examines the exact scal- monly done in mixed Gaussian and plane-wave implementa- ing of several traditional quantum chemistry methods. We find 36,49–51 tions where the Coulomb operator is applied, as above, that the freedom of choice of intermediates means that HF and in the Fourier representation. As an explicit example, we out- MP2 can be reduced to cubic scaling, and variants of coupled line how to evaluate the MP2-J term using this idea as well cluster such as linearized coupled cluster doubles (LCCD) and as an atomic orbital representation. Here we use the standard the distinguishable cluster approximation with doubles (DCD) Roman and Greek symbols for molecular orbitals and atomic may be reduced to O(N5) scaling. These are scalings of the orbitals, respectively, with the molecular orbitals expanded as exact methods in the sense that no assumptions are made about P φp(r) = µ Cµp µ(r). Contributions of atomic orbital prod- the forms of the solutions. The exact scalings that we describe ucts µ(r)ν(r) will be assumed screened if ||µν|| < , and encourage a modified perspective on several topics. For exam- screened pairs will be indicated by the symbol hµνi. Start- ple, they lead to a different organization of the correlation hier- ing with the atomic orbital Laplace transform expression for archy, where the complexity gap between density functional MP2-J, methods52 and traditional wavefunction methods is eliminated.

X X 0 0 0 0 They also suggest a new way to classify diagrams in coupled EMP2-J = 2 (µ ν |σ λ )(νµ|λσ) cluster theory53 that may lead to new correlation approxima- 0 0 0 0 µ ν σ λ µνσλ tions. Finally, given that the exact scalings are lower than × gµ0µgσ0σg¯ν0νg¯λ0λ dτ, (20) that of many current approximate methods, substantial further 044106-6 Mardirossian, McClain, and Chan J. Chem. Phys. 148, 044106 (2018) reductions in cost can be obtained in practice by combining Computation, Advances in Quantum Chemistry series, Vol. 2 (Academic the ideas here with the rich existing set of techniques used to Press, 1966), pp. 1–24. 25 deﬁne approximate quantum chemistry methods. L. B. Rall, Automatic Differentiation: Techniques and Applications (Springer, 1981). 26R. A. Friesner, Chem. Phys. Lett. 116, 39 (1985). ACKNOWLEDGMENTS 27M. N. Ringnalda, M. Belhadj, and R. A. Friesner, J. Chem. Phys. 93, 3397 (1990). This work was supported by the US National Science 28R. B. Murphy, M. D. Beachy, R. A. Friesner, and M. N. Ringnalda, J. Chem. Foundation through NSF: Nos. CHE 1665333 and NSF:SSI Phys. 103, 1481 (1995). 29M. Haser¨ and J. Almlof,¨ J. Chem. Phys. 96, 489 (1992). 1657286. G.K.-L.C. is a Simons Investigator in Theoretical 30F. A. Bischoff, R. J. Harrison, and E. F. Valeev, J. Chem. Phys. 137, 104103 Physics. (2012). 31S. Hirata, T. Shiozaki, C. M. Johnson, and J. D. Talman, Mol. Phys. 115, 1E. Schwegler and M. Challacombe, J. Chem. Phys. 105, 2726 (1996). 510 (2017). 2M. Challacombe and E. Schwegler, J. Chem. Phys. 106, 5526 (1997). 32C. Song and T. J. Mart´ınez, J. Chem. Phys. 144, 174111 (2016). 3C. Ochsenfeld, C. A. White, and M. Head-Gordon, J. Chem. Phys. 109, 33R. M. Parrish, E. G. Hohenstein, N. F. Schunck, C. D. Sherrill, and T. J. 1663 (1998). Mart´ınez, Phys. Rev. Lett. 111, 132505 (2013). 4M. Schutz,¨ G. Hetzer, and H.-J. Werner, J. Chem. Phys. 111, 5691 (1999). 34T. Schafer,¨ B. Ramberger, and G. Kresse, J. Chem. Phys. 146, 104101 5G. E. Scuseria and P. Y. Ayala, J. Chem. Phys. 111, 8330 (1999). (2017). 6M. Schutz¨ and H.-J. Werner, J. Chem. Phys. 114, 661 (2001). 35J. E. Moussa, J. Chem. Phys. 140, 014107 (2014). 7S. Saebø and P. Pulay, J. Chem. Phys. 115, 3975 (2001). 36L. Fusti-Molnar¨ and P. Pulay, J. Chem. Phys. 116, 7795 (2002). 8P. Pinski, C. Riplinger, E. F. Valeev, and F. Neese, J. Chem. Phys. 143, 37L. Fusti-Moln¨ ar´ and P. Pulay, J. Mol. Struct.: THEOCHEM 666, 25 (2003). 034108 (2015). 38S. R. White, J. W. Wilkins, and M. P. Teter, Phys. Rev. B 39, 5819 (1989). 9W. Meyer, J. Chem. Phys. 58, 1017 (1973). 39H. N. Rojas, R. W. Godby, and R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995). 10M. Sierka, A. Hogekamp, and R. Ahlrichs, J. Chem. Phys. 118, 9136 (2003). 40M. Kaltak, J. Klimes,ˇ and G. Kresse, Phys. Rev. B 90, 054115 (2014). 11H.-J. Werner, F. R. Manby, and P. J. Knowles, J. Chem. Phys. 118, 8149 41Y. Jung, R. C. Lochan, A. D. Dutoi, and M. Head-Gordon, J. Chem. Phys. (2003). 121, 9793 (2004). 12H. Koch, A. S. de Meras,´ and T. B. Pedersen, J. Chem. Phys. 118, 9481 42Q. Sun, T. C. Berkelbach, N. S. Blunt, G. H. Booth, S. Guo, Z. Li, J. Liu, (2003). J. D. McClain, E. R. Sayfutyarova, S. Sharma, S. Wouters, and G. K.-L. 13F. Aquilante, T. B. Pedersen, and R. Lindh, J. Chem. Phys. 126, 194106 Chan, Wiley Interdiscip. Rev.: Comput. Mol. Sci. 8, e1340 (2018). (2007). 43J. VandeVondele and J. Hutter, J. Chem. Phys. 127, 114105 (2007). 14F. Neese, F. Wennmohs, and A. Hansen, J. Chem. Phys. 130, 114108 (2009). 44C. Hartwigsen, S. Goedecker, and J. Hutter, Phys. Rev. B 58, 3641 (1998). 15J. Yang, Y. Kurashige, F. R. Manby, and G. K. L. Chan, J. Chem. Phys. 134, 45F. Hummel, T. Tsatsoulis, and A. Gruneis,¨ J. Chem. Phys. 146, 124105 044123 (2011). (2017). 16E. G. Hohenstein, R. M. Parrish, and T. J. Mart´ınez, J. Chem. Phys. 137, 46I. Shavitt and R. J. Bartlett, Many-Body Methods in Chemistry and Physics: 044103 (2012). MBPT and Coupled-Cluster Theory (Cambridge University Press, 2009). 17A. J. W. Thom and A. Alavi, Phys. Rev. Lett. 99, 143001 (2007). 47A. K. Wilson and J. Almlof,¨ Theor. Chim. Acta 95, 49 (1997). 18S. Y. Willow, K. S. Kim, and S. Hirata, J. Chem. Phys. 137, 204122 (2012). 48B. Doser, D. S. Lambrecht, J. Kussmann, and C. Ochsenfeld, J. Chem. Phys. 19D. Neuhauser, E. Rabani, and R. Baer, J. Chem. Theory Comput. 9, 24 130, 064107 (2009). (2013). 49J. Hutter, M. Iannuzzi, F. Schiffmann, and J. VandeVondele, Wiley Interdis- 20Q. Ge, Y. Gao, R. Baer, E. Rabani, and D. Neuhauser, J. Phys. Chem. Lett. cip. Rev.: Comput. Mol. Sci. 4, 15 (2014). 5, 185 (2014). 50J. McClain, Q. Sun, G. K.-L. Chan, and T. C. Berkelbach, J. Chem. Theory 21R. J. Bartlett, Annu. Rev. Phys. Chem. 32, 359 (1981). Comput. 13, 1209 (2017). 22A. G. Taube and R. J. Bartlett, J. Chem. Phys. 130, 144112 (2009). 51Q. Sun, T. C. Berkelbach, J. D. McClain, and G. K. Chan, J. Chem. Phys. 23D. Kats and F. R. Manby, J. Chem. Phys. 139, 021102 (2013). 147, 164119 (2017). 24S. Boys and P. Rajagopal, Quantum Calculations: Which Are Accumula- 52N. Mardirossian and M. Head-Gordon, Mol. Phys. 115, 2315 (2017). tive in Accuracy, Unrestricted in Expansion Functions, and Economical in 53R. J. Bartlett and M. Musiał, Rev. Mod. Phys. 79, 291 (2007).