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The Journal of Chemical Physics

Tensor decomposition in post-Hartree–Fock methods. II. CCD implementation

Udo Benedikt, Karl-Heinz Böhm, Alexander A. Auer

Published Dec 9, 2013• 56 citations
Publisher

Abstract

In a previous publication, we have discussed the usage of tensor decomposition in the canonical polyadic (CP) tensor format for electronic structure methods. There, we focused on two-electron integrals and second order Møller-Plesset perturbation theory (MP2). In this work, we discuss the CP format for Coupled Cluster (CC) theory and present a pilot implementation for the Coupled Cluster Doubles method. We discuss the iterative solution of the CC amplitude equations using tensors in CP representation and present a tensor contraction scheme that minimizes the effort necessary for the rank reductions during the iterations. Furthermore, several details concerning the reduction of complexity of the algorithm, convergence of the CC iterations, truncation errors, and the choice of threshold for chemical accuracy are discussed.

DOI
10.1063/1.4833565

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In a previous publication, we have discussed the usage of tensor decomposition in the canonical polyadic (CP) tensor format for electronic structure methods. There, we focused on two-electron integrals and second order Møller-Plesset perturbation theory (MP2). In this work, we discuss the CP format for Coupled Cluster (CC) theory and present a pilot implementation for the Coupled Cluster Doubles method. We discuss the iterative solution of the CC amplitude equations using tensors in CP representation and present a tensor contraction scheme that minimizes the effort necessary for the rank reductions during the iterations. Furthermore, several details concerning the reduction of complexity of the algorithm, convergence of the CC iterations, truncation errors, and the choice of threshold for chemical accuracy are discussed.
In this thesis the application of novel tensor decomposition and tensor representation techniques in highly accurate post Hartree-Fock methods is evaluated. These representation techniques can help to overcome the steep scaling behaviour of high level ab-initio calculations with increasing system size and therefore break the "curse of dimensionality". After a comparison of various tensor formats the application of the "canonical polyadic" format (CP) is described in detail. There, especially the casting of a normal, index based tensor into the CP format (tensor decomposition) and a method for a low rank approximation (rank reduction) of the two-electron integrals in the AO basis are investigated. The decisive quantity for the applicability of the CP format is the scaling of the rank with increasing system and basis set size. The memory requirements and the computational effort for tensor manipulations in the CP format are only linear in the number of dimensions but still depend on the expansion length (rank) of the approximation. Furthermore, the AO-MO transformation and a MP2 algorithm with decomposed tensors in the CP format is evaluated and the scaling with increasing system and basis set size is investigated. Finally, a Coupled-Cluster algorithm based only on low-rank CP representation of the MO integrals is developed. There, especially the successive tensor contraction during the iterative solution of the amplitude equations and the error propagation upon multiple application of the reduction procedure are discussed. In conclusion the overall complexity of a Coupled-Cluster procedure with tensors in CP format is evaluated and some possibilities for improvements of the rank reduction procedure tailored to the needs in electronic structure calculations are shown.
In this proof-of-principle study, we discuss the application of various tensor representation formats and their implications on memory requirements and computational effort for tensor manipulations as they occur in typical post-Hartree–Fock (post-HF) methods. A successive tensor decomposition/rank reduction scheme in the matrix product state (MPS) format for the two-electron integrals in the AO and MO bases and an estimate of the t 2 amplitudes as obtained from second-order many-body perturbation theory (MP2) are described. Furthermore, the AO–MO integral transformation, the calculation of the MP2 energy and the potential usage of tensors in low-rank MPS representation for the tensor contractions in coupled cluster theory are discussed in detail. We are able to show that the overall scaling of the memory requirements is reduced from the conventional N 4 scaling to approximately N 3 and the scaling of computational effort for tensor contractions in post-HF methods can be reduced to roughly N 4 while the decomposition itself scales as N 5. While efficient algorithms with low prefactor for the tensor decomposition have yet to be devised, this ansatz offers the possibility to find a robust approximation with low-scaling behaviour with system and basis-set size for post-HF ab initio methods.
;In this proof-of-principle study, we discuss the application of various tensor representation formats and their implications on memory requirements and computational eort for tensor manipulations as they occur in typical post Hartree-Fock methods. A successive tensor decomposition / rank reduction scheme in the matrix product state (MPS) format for the two electron integrals in the AO and MO basis and an estimate of the t2 amplitudes as obtained from MP2 are described. Furthermore, the AO-MO integral transformation, the calculation of the MP2 energy and the potential usage of tensors in low rank MPS representation for the tensor contractions in CC theory are discussed in detail. We are able to show that the overall scaling of the memory requirements is reduced from the conventional N 4 scaling to approximately N 3 and the scaling of computational eort for tensor contractions in post-HF methods can be reduced to roughly N4 while the decomposition itself scales as N5. While ecient algorithms with low prefactor for the tensor decomposition have yet to be devised, this Ansatz oers the possibility to nd a robust approximation with low scaling behaviour with system and basis set size for post Hartree Fock ab initio methods.
We investigate a novel approach to approximate tensor-network contraction via the exact, matrix-free decomposition of full tensor-networks. We study this method as a means to eliminate the propagation of error in the approximation of tensor-networks. Importantly, this decomposition-based approach is generic, i.e. it does not depend on a specific tensor-network, the tensor index (physical) ordering, or the choice of tensor decomposition. Careful consideration should be made to determine the best decomposition strategy. Furthermore, this method does not rely on robust cancellation of errors (i.e. the Taylor expansion). As a means to study the effectiveness of the approach, we replace the exact contraction of the particle particle ladder (PPL) tensor diagram in the popular coupled-cluster with single and double excitation (CCSD) method with a low-rank tensor decomposition, namely the canonical polyadic decomposition (CPD). With this approach, we replace an O(N6)\mathcal{O}(N^6)O(N6) tensor contractions with a potentially reduced-scaling O(N4R)\mathcal{O}(N^4R)O(N4R) optimization problem, where RRR is the CP rank, and we reduce the computational storage of the PPL tensor from O(N4)\mathcal{O}(N^4)O(N4) to O(NR)\mathcal{O}(NR)O(NR), although we do not take advantage of this compression in this study. To minimize the cost of the CPD optimization, we utilize the iterative structure of CCSD to efficiently initialize the CPD optimization. We show that accurate chemically-relevant energy values can be computed with an error of less than 1 kcal/mol using a relatively low CP rank.
A constant goal of quantum chemistry is devising accurate and computationally effective methods for molecular simulations. In this work an application of tensor decompositions in the context of highly accurate coupled cluster theory, which is often considered a "gold standard", is investigated. The scheme we develop is aimed to mitigate the steep growth of the computational cost with the system size, and hence to overcome the "curse of dimensionality" common in many potent methods in electronic structure. We show how to reduce the computational effort of the restricted coupled cluster with singles and doubles (RCCSD) by two orders of magnitude by introducing alternative parameterizations of the method using "canonical polyad decomposition" (CPD) and "tensor hypercontraction" (THC) formats. After describing CPD and THC formats in detail, we demonstrate how to cast regular index based tensors into a decomposed form. The number of parameters and the accuracy of these representations depend on the expansion length (rank) of the approximation. We investigate the dependence of rank upon the size of the system and a target accuracy and show it to be low for typical tensors in electronic structure. We then provide a generic procedure to reformulate any coupled cluster method using tensor decompositions. Two specific approximate methods, THC-RCCSD and CPD-RCCSD, are derived. We demonstrate the accuracy of these new approaches by calculating energies of a large set of organic molecules, as well as by simulations of Hubbard models. Finally, it is shown how the restriction of the number of parameters in approximate coupled cluster can improve the accuracy in the challenging strong correlation regime. We conclude by discussing a connection of our findings to other new developments in the coupled cluster theory and propose possible extensions of our approach.
We investigate a novel approach to approximate tensor-network contraction via the exact, matrix-free decomposition of full tensor-networks. We study this method as a means to eliminate the propagation of error in the approximation of tensor-networks. Importantly, this decomposition-based approach is generic, i.e., it does not depend on a specific tensor-network, the tensor index (physical) ordering, or the choice of tensor decomposition. Careful consideration should be made to determine the best decomposition strategy. Furthermore, this method does not rely on robust cancellation of errors (i.e., the Taylor expansion). As a means to study the effectiveness of the approach, we replace the exact contraction of the particle–particle ladder (PPL) tensor diagram in the popular coupled-cluster with single and double excitation (CCSD) method with a low-rank tensor decomposition, namely the canonical polyadic decomposition (CPD). With this approach, we replace an O ( N 6 ) tensor contractions with a potentially reduced-scaling O ( N 4 R ) optimization problem, where R is the CP rank, and we reduce the computational storage of the PPL tensor from O ( N 4 ) to O ( N R ), although we do not take advantage of this compression in this study. To minimize the cost of the CPD optimization, we utilize the iterative structure of CCSD to efficiently initialize the CPD optimization. We show that accurate chemically relevant energy values can be computed with an error of less than 1 kcal/mol using a relatively low CP rank.