Lowrank factorization of electron integral tensors and its application in electronic structure theory
Abstract
In this letter, we introduce the reverse CuthillMcKee (RCM) algorithm, which is often used for the bandwidth reduction of sparse tensors, to transform the twoelectron integral tensors to their block diagonal forms. By further applying the pivoted Cholesky decomposition (CD) on each of the diagonal blocks, we are able to represent the highdimensional twoelectron integral tensors in terms of permutation matrices and lowrank Cholesky vectors. This representation facilitates the lowrank factorization of the highdimensional tensor contractions that are usually encountered in postHartreeFock calculations. In this letter, we discuss the secondorder MøllerPlesset (MP2) method and linear coupled cluster model with doubles (LCCD) as two simple examples to demonstrate the efficiency of the RCMCD technique in representing twoelectron integrals in a compact form.
 Authors:
 Publication Date:
 Research Org.:
 Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1353329
 Report Number(s):
 PNNLSA122607
Journal ID: ISSN 00092614
 DOE Contract Number:
 AC0576RL01830
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Chemical Physics Letters; Journal Volume: 672
 Country of Publication:
 United States
 Language:
 English
 Subject:
 electronic structure theory; twoelectron integral tensor; tensor contraction; lowrank factorization; reverse CuthillMcKee
Citation Formats
Peng, Bo, and Kowalski, Karol. Lowrank factorization of electron integral tensors and its application in electronic structure theory. United States: N. p., 2017.
Web. doi:10.1016/j.cplett.2017.01.056.
Peng, Bo, & Kowalski, Karol. Lowrank factorization of electron integral tensors and its application in electronic structure theory. United States. doi:10.1016/j.cplett.2017.01.056.
Peng, Bo, and Kowalski, Karol. Wed .
"Lowrank factorization of electron integral tensors and its application in electronic structure theory". United States.
doi:10.1016/j.cplett.2017.01.056.
@article{osti_1353329,
title = {Lowrank factorization of electron integral tensors and its application in electronic structure theory},
author = {Peng, Bo and Kowalski, Karol},
abstractNote = {In this letter, we introduce the reverse CuthillMcKee (RCM) algorithm, which is often used for the bandwidth reduction of sparse tensors, to transform the twoelectron integral tensors to their block diagonal forms. By further applying the pivoted Cholesky decomposition (CD) on each of the diagonal blocks, we are able to represent the highdimensional twoelectron integral tensors in terms of permutation matrices and lowrank Cholesky vectors. This representation facilitates the lowrank factorization of the highdimensional tensor contractions that are usually encountered in postHartreeFock calculations. In this letter, we discuss the secondorder MøllerPlesset (MP2) method and linear coupled cluster model with doubles (LCCD) as two simple examples to demonstrate the efficiency of the RCMCD technique in representing twoelectron integrals in a compact form.},
doi = {10.1016/j.cplett.2017.01.056},
journal = {Chemical Physics Letters},
number = ,
volume = 672,
place = {United States},
year = {Wed Mar 01 00:00:00 EST 2017},
month = {Wed Mar 01 00:00:00 EST 2017}
}

In this paper, we apply reverse CuthillMcKee (RCM) algorithm to transform twoelectron integral tensors to their block diagonal forms. By further applying Cholesky decomposition (CD) on each of the diagonal blocks, we are able to represent the highdimensional twoelectron integral tensors in terms of permutation matrices and lowrank Cholesky vectors. This representation facilitates lowrank factorizations of highdimensional tensor contractions in postHartreeFock calculations. Finally, we discuss the secondorder MøllerPlesset (MP2) method and the linear coupledcluster model with doubles (LCCD) as examples to demonstrate the efficiency of this technique in representing the twoelectron integrals in a compact form.Cited by 1

Positive semidefinite tensor factorizations of the twoelectron integral matrix for lowscaling ab initio electronic structure
Tensor factorization of the 2electron integral matrix is a wellknown technique for reducing the computational scaling of ab initio electronic structure methods toward that of HartreeFock and density functional theories. The simplest factorization that maintains the positive semidefinite character of the 2electron integral matrix is the Cholesky factorization. In this paper, we introduce a family of positive semidefinite factorizations that generalize the Cholesky factorization. Using an implementation of the factorization within the parametric 2RDM method [D. A. Mazziotti, Phys. Rev. Lett. 101, 253002 (2008)], we study several inorganic molecules, alkane chains, and potential energy curves and find that this generalizedmore »