Reduced scaling in electronic structure calculations using Cholesky decompositions

Abstract

We demonstrate that substantial computational savings are attainable in electronic structure calculations using a Cholesky decomposition of the two-electron integral matrix. In most cases, the computational effort involved calculating the Cholesky decomposition is less than the construction of one Fock matrix using a direct O(N 2 ) procedure.

Polarizability and optical rotation calculated from the approximate coupled cluster singles and doubles CC2 linear response theory using Cholesky decompositions

“…As mentioned in Ref. 1, Cholesky decomposition of the integral matrix is formally similar to the socalled resolution-of-identity ͑RI͒ ͑Refs. 3 and 4͒ or density-fitting ͑DF͒ ͑Ref.…”

“…First, M equals the minimum number of columns of the integral matrix that need to be calculated during the decomposition, although the number actually calculated usually is somewhat larger, as the integrals are calculated in shell-pair distributions. 1 Second, if sparsity is not employed, M directly determines the operation count for a subsequent electronic structure calculation. An integral-direct Cholesky decomposition algorithm is presented in Ref.…”

“…͑6͒, are constructed from the positive definite integrals (ai ͉b j) and, consequently, t i j ab form a negative definite matrix. Hence, as has been demonstrated for the MP2 energy, 1 we may employ Cholesky decomposition of either the (ai ͉b j) integral matrix or of the amplitudes themselves to reduce the operation count of the first step in the calculation of the ground state Y -intermediates. Specifically, using either…”

“…1,2 The algorithm is designed to avoid storage, in core as well as on disk, of four-index intermediates along the lines deviced by Hättig and co-workers [10][11][12][13][14] for the RI-CC2 method. Through examples involving as many as 2.8 billion coupled cluster doubles amplitudes, we have demonstrated the large-scale applicability of the implementation.…”

“…In a recent communication, 1 we have demonstrated that significant reductions in computational effort as well as storage requirements are attainable using Cholesky decompositions in electronic structure theory. Specifically, we have shown that the Cholesky representation of the two-electron integral matrix 2 can be immediately exploited for second order Møller-Plesset ͑MP2͒ perturbation theory.…”