# Low-rank factorization of electron integral tensors and its application in electronic structure theory

## Abstract

In this letter, we introduce the reverse Cuthill-McKee (RCM) algorithm, which is often used for the bandwidth reduction of sparse tensors, to transform the two-electron 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 high-dimensional two-electron integral tensors in terms of permutation matrices and low-rank Cholesky vectors. This representation facilitates the low-rank factorization of the high-dimensional tensor contractions that are usually encountered in post-Hartree-Fock calculations. In this letter, we discuss the second-order Møller-Plesset (MP2) method and linear coupled- cluster model with doubles (L-CCD) as two simple examples to demonstrate the efficiency of the RCM-CD technique in representing two-electron 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):
- PNNL-SA-122607

Journal ID: ISSN 0009-2614

- DOE Contract Number:
- AC05-76RL01830

- 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; two-electron integral tensor; tensor contraction; low-rank factorization; reverse Cuthill-McKee

### Citation Formats

```
Peng, Bo, and Kowalski, Karol.
```*Low-rank 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.
```*Low-rank 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 .
"Low-rank 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 = {Low-rank 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 Cuthill-McKee (RCM) algorithm, which is often used for the bandwidth reduction of sparse tensors, to transform the two-electron 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 high-dimensional two-electron integral tensors in terms of permutation matrices and low-rank Cholesky vectors. This representation facilitates the low-rank factorization of the high-dimensional tensor contractions that are usually encountered in post-Hartree-Fock calculations. In this letter, we discuss the second-order Møller-Plesset (MP2) method and linear coupled- cluster model with doubles (L-CCD) as two simple examples to demonstrate the efficiency of the RCM-CD technique in representing two-electron 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}

}