Highly Efficient and Scalable Compound Decomposition of Two-Electron Integral Tensor and Its Application in Coupled Cluster Calculations

Abstract

The representation and storage of two-electron integral tensors are vital in large-scale applications of accurate electronic structure methods. Low-rank representation and efficient storage strategy of integral tensors can significantly reduce the numerical overhead and consequently time-to-solution of these methods. In this work, by combining pivoted incomplete Cholesky decomposition (CD) with a follow-up truncated singular vector decomposition (SVD), we develop a decomposition strategy to approximately represent the two-electron integral tensor in terms of low-rank vectors. A systematic benchmark test on a series of 1-D, 2-D, and 3-D carbon-hydrogen systems demonstrates high efficiency and scalability of the compound two-step decomposition of the two-electron integral tensor in our implementation. For the size of the atomic basis set, N, ranging from ∼100 up to ∼2,000, the observed numerical scaling of our implementation shows [Formula: see text] versus [Formula: see text] cost of performing single CD on the two-electron integral tensor in most of the other implementations. More importantly, this decomposition strategy can significantly reduce the storage requirement of the atomic orbital (AO) two-electron integral tensor from [Formula: see text] to [Formula: see text] with moderate decomposition thresholds. The accuracy tests have been performed using ground- and excited-state formulations of coupled cluster formalism employing single and double excitations (CCSD) on several benchmark systems including the C molecule described by nearly 1,400 basis functions. The results show that the decomposition thresholds can be generally set to 10 to 10 to give acceptable compromise between efficiency and accuracy.

Downfolding of many-body Hamiltonians using active-space models: Extension of the sub-system embedding sub-algebras approach to unitary coupled cluster formalisms

“…Similar reduction can also be achieved for Gaussian basis sets when combined Cholesky and singular value decompositions are employed to represent twoelectron integrals (see Refs. [27,35] for details) From the point of quantum computing applications, the net effect of the number of basis set functions and the number of non-vanishing terms in Hamiltonian define the circuit depth that determines the efficiency of quantum algorithms. The reduction in the number of non-negligible terms may also be achieved by employing localization techniques for Gaussian basis sets [36][37][38].…”

Quantum Filter Diagonalization with Double-Factorized Hamiltonians

“…A key technical element of this manuscript is the "compressed" double factorization (C-DF) approximate representation of the electronic Hamiltonian, which provides for reduced gate count requirements in quantum circuits for QFD time propagation and reduced measurement requirements for Hamiltonian expectation values. Below, we review the representation of the electronic Hamiltonian in the double-factorized representation as previously discussed by several authors in the literature [12][13][14][15][16][17][18][19], including the popular "explicit" double factorization procedure [14,16,17,19] (X-DF) for numerically finding the tensor factors. We then develop a new "compressed" double factorization (C-DF) procedure for numerically finding the tensor factors with enhanced compression and accuracy.…”

From NWChem to NWChemEx: Evolving with the Computational Chemistry Landscape

“…7) without a significant loss in accuracy for ground-state energies, excitation energies, and non-linear optical properties (see Ref. 368 and Fig. 8 for details) -where is the number of basis functions.…”