Tensor-structured coupled cluster theory.

We derive and implement a new way of solving coupled cluster equations with lower computational scaling. Our method is based on the decomposition of both amplitudes and two electron integrals, using a combination of tensor hypercontraction and canonical polyadic decomposition. While the original theory scales as O(N6) with respect to the number of basis functions, we demonstrate numerically that we achieve sub-millihartree difference from the original theory with O(N4) scaling. This is accomplished by solving directly for the factors that decompose the cluster operator. The proposed scheme is quite general and can be easily extended to other many-body methods.

[1]  Gustavo E. Scuseria,et al.  Projected Hartree-Fock theory as a polynomial of particle-hole excitations and its combination with variational coupled cluster theory , 2017, 1702.08578.

[2]  Yang Yang,et al.  The tensor hypercontracted parametric reduced density matrix algorithm: coupled-cluster accuracy with O(r(4)) scaling. , 2013, The Journal of chemical physics.

[3]  F. L. Hitchcock.  The Expression of a Tensor or a Polyadic as a Sum of Products , 1927 .

[4]  C. David Sherrill,et al.  Communication: Acceleration of coupled cluster singles and doubles via orbital-weighted least-squares tensor hypercontraction. , 2014, The Journal of chemical physics.

[5]  Jun Li,et al.  Basis Set Exchange: A Community Database for Computational Sciences , 2007, J. Chem. Inf. Model..

[6]  Dietrich Braess,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Approximation of 1/x by Exponential Sums in [1, ∞) , 2022 .

[7]  T. Martínez,et al.  Tensor hypercontraction equation-of-motion second-order approximate coupled cluster: electronic excitation energies in O(N4) time. , 2013, The journal of physical chemistry. B.

[8]  P. Comon,et al.  Tensor decompositions, alternating least squares and other tales , 2009 .

[9]  Dimitrios G Liakos,et al.  Efficient and accurate approximations to the local coupled cluster singles doubles method using a truncated pair natural orbital basis. , 2009, The Journal of chemical physics.

[10]  Marek Sierka,et al.  Fast evaluation of the Coulomb potential for electron densities using multipole accelerated resolution of identity approximation , 2003 .

[11]  G. Scuseria,et al.  Biorthogonal projected energies of a Gutzwiller similarity transformed Hamiltonian , 2016, Journal of physics. Condensed matter : an Institute of Physics journal.

[12]  E. Neuscamman,et al.  Communication: a Jastrow factor coupled cluster theory for weak and strong electron correlation. , 2013, The Journal of chemical physics.

[13]  Frederick R. Manby,et al.  Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations , 2003 .

[14]  H. Harbrecht,et al.  On the low-rank approximation by the pivoted Cholesky decomposition , 2012 .

[15]  C. Eckart,et al.  The approximation of one matrix by another of lower rank , 1936 .

[16]  J. Almlöf,et al.  Integral approximations for LCAO-SCF calculations , 1993 .

[17]  Robert M Parrish,et al.  Discrete variable representation in electronic structure theory: quadrature grids for least-squares tensor hypercontraction. , 2013, The Journal of chemical physics.

[18]  R. Bartlett,et al.  A full coupled‐cluster singles and doubles model: The inclusion of disconnected triples , 1982 .

[19]  Robert M Parrish,et al.  Tensor hypercontraction. II. Least-squares renormalization. , 2012, The Journal of chemical physics.

[20]  Lieven De Lathauwer,et al.  A Link between the Canonical Decomposition in Multilinear Algebra and Simultaneous Matrix Diagonalization , 2006, SIAM J. Matrix Anal. Appl..

[21]  Lexing Ying,et al.  Tensor Network Skeletonization , 2016, Multiscale Model. Simul..

[22]  Robert M Parrish,et al.  Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order Møller-Plesset perturbation theory. , 2012, The Journal of chemical physics.

[23]  Henrik Koch,et al.  Method specific Cholesky decomposition: coulomb and exchange energies. , 2008, The Journal of chemical physics.

[24]  Lieven De Lathauwer,et al.  Optimization-Based Algorithms for Tensor Decompositions: Canonical Polyadic Decomposition, Decomposition in Rank-(Lr, Lr, 1) Terms, and a New Generalization , 2013, SIAM J. Optim..

[25]  Philippe Y. Ayala,et al.  Linear scaling second-order Moller–Plesset theory in the atomic orbital basis for large molecular systems , 1999 .

[26]  Gustavo E. Scuseria,et al.  Polynomial Similarity Transformation Theory: A smooth interpolation between coupled cluster doubles and projected BCS applied to the reduced BCS Hamiltonian , 2015, 1512.06111.

[27]  Lieven De Lathauwer,et al.  Unconstrained Optimization of Real Functions in Complex Variables , 2012, SIAM J. Optim..

[28]  A. Auer,et al.  Tensor decomposition in post-Hartree-Fock methods. II. CCD implementation. , 2013, The Journal of chemical physics.

[29]  Andreas Grüneis,et al.  Low rank factorization of the Coulomb integrals for periodic coupled cluster theory. , 2016, The Journal of chemical physics.

[30]  G. Scuseria,et al.  Lie algebraic similarity transformed Hamiltonians for lattice model systems , 2014, 1409.2203.

[31]  T. Martínez,et al.  Tensor Hypercontraction Second-Order Møller-Plesset Perturbation Theory: Grid Optimization and Reaction Energies. , 2015, Journal of chemical theory and computation.

[32]  Wolfgang Hackbusch,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig Tensor Decomposition in Post-hartree Fock Methods. I. Two-electron Integrals and Mp2 , 2022 .

[33]  Curtis L. Janssen,et al.  An efficient reformulation of the closed‐shell coupled cluster single and double excitation (CCSD) equations , 1988 .

[34]  Jorge J. Moré,et al.  Benchmarking optimization software with performance profiles , 2001, Math. Program..

[35]  G. Scuseria,et al.  Projected coupled cluster theory. , 2017, The Journal of chemical physics.

[36]  André Uschmajew,et al.  Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation , 2012, SIAM J. Matrix Anal. Appl..

[37]  Nikos D. Sidiropoulos,et al.  Tensor Decomposition for Signal Processing and Machine Learning , 2016, IEEE Transactions on Signal Processing.

[38]  J. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , 1977 .

[39]  Gustavo E. Scuseria,et al.  Attenuated coupled cluster: a heuristic polynomial similarity transformation incorporating spin symmetry projection into traditional coupled cluster theory , 2017, 1702.00906.

[40]  Florian Weigend,et al.  Approximated electron repulsion integrals: Cholesky decomposition versus resolution of the identity methods. , 2009, The Journal of chemical physics.

[41]  Artur F Izmaylov,et al.  Resolution of the identity atomic orbital Laplace transformed second order Møller-Plesset theory for nonconducting periodic systems. , 2008, Physical chemistry chemical physics : PCCP.

[42]  J. C. A. Barata,et al.  The Moore–Penrose Pseudoinverse: A Tutorial Review of the Theory , 2011, 1110.6882.

[43]  Robert M Parrish,et al.  Quartic scaling second-order approximate coupled cluster singles and doubles via tensor hypercontraction: THC-CC2. , 2013, The Journal of chemical physics.

[44]  Jan Almlöf,et al.  Elimination of energy denominators in Møller—Plesset perturbation theory by a Laplace transform approach , 1991 .

[45]  Jan Almlöf,et al.  THE COULOMB OPERATOR IN A GAUSSIAN PRODUCT BASIS , 1995 .

[46]  Tamara G. Kolda,et al.  Tensor Decompositions and Applications , 2009, SIAM Rev..

[47]  N. H. Beebe,et al.  Simplifications in the generation and transformation of two‐electron integrals in molecular calculations , 1977 .

[48]  Frank Neese,et al.  An efficient and near linear scaling pair natural orbital based local coupled cluster method. , 2013, The Journal of chemical physics.

[49]  Robert M Parrish,et al.  Communication: Tensor hypercontraction. III. Least-squares tensor hypercontraction for the determination of correlated wavefunctions. , 2012, The Journal of chemical physics.

[50]  Thomas Bondo Pedersen,et al.  Reduced scaling in electronic structure calculations using Cholesky decompositions , 2003 .