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 Articles you may be interested in Editorial: JCP Communications—Updating a valued community resource The Journal of Chemical Physics 148, 010401 (2018); 10.1063/1.5019731 On the difference between variational and unitary coupled cluster theories The Journal of Chemical Physics 148, 044107 (2018); 10.1063/1.5011033 Communication: An improved linear scaling perturbative triples correction for the domain based local pair- natural orbital based singles and doubles coupled cluster method [DLPNO-CCSD(T)] The Journal of Chemical Physics 148, 011101 (2018); 10.1063/1.5011798 Improving the accuracy of Møller-Plesset perturbation theory with neural networks The Journal of Chemical Physics 147, 161725 (2017); 10.1063/1.4986081 Analytical energy gradients for explicitly correlated wave functions. 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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 the- ory, coupled cluster theory, etc.), such approximations lead X y X y y 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 y X y y 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 y 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 meth- ods, 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 jγ (r1, r2)j As a single space-time integral, the two components of the EHF-X = − dr1 dr2. (4) jr1 − r2 j 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(jr − r j)v(jr0 − r0 j) dR dτ, (5) cost to obtain a desired accuracy is polylogarithmic in , v 2 2 1 2 1 2 i.e., slog()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(jr1 − r2 j)v(jr1 − r2 j) 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.