# Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations

## Abstract

We describe here a novel iterative strategy for Kohn–Sham density functional theory calculations aimed at large systems (>1,000 electrons), applicable to metals and insulators alike. In lieu of explicit diagonalization of the Kohn–Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ a two-level Chebyshev polynomial filter based complementary subspace strategy to (1) compute a set of vectors that span the occupied subspace of the Hamiltonian; (2) reduce subspace diagonalization to just partially occupied states; and (3) obtain those states in an efficient, scalable manner via an inner Chebyshev filter iteration. By reducing the necessary computation to just partially occupied states and obtaining these through an inner Chebyshev iteration, our approach reduces the cost of large metallic calculations significantly, while eliminating subspace diagonalization for insulating systems altogether. We describe the implementation of the method within the framework of the discontinuous Galerkin (DG) electronic structure method and show that this results in a computational scheme that can effectively tackle bulk and nano systems containing tens of thousands of electrons, with chemical accuracy, within a few minutes or less of wall clock time per SCF iteration on large-scale computing platforms. We anticipate that our method will be instrumental in pushing the envelopemore »

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
- Univ. of California, Berkeley, CA (United States). Dept. of Mathematics; Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
- Georgia Inst. of Technology, Atlanta, GA (United States). College of Engineering
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Physics Division

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); National Science Foundation (NSF)

- OSTI Identifier:
- 1512626

- Report Number(s):
- LLNL-JRNL-757223

Journal ID: ISSN 1549-9618; 943724

- Grant/Contract Number:
- AC52-07NA27344; SC0017867; 1450372

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Theory and Computation

- Additional Journal Information:
- Journal Volume: 14; Journal Issue: 6; Journal ID: ISSN 1549-9618

- Publisher:
- American Chemical Society

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Banerjee, Amartya S., Lin, Lin, Suryanarayana, Phanish, Yang, Chao, and Pask, John E..
```*Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations*. United States: N. p., 2018.
Web. doi:10.1021/acs.jctc.7b01243.

```
Banerjee, Amartya S., Lin, Lin, Suryanarayana, Phanish, Yang, Chao, & Pask, John E..
```*Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations*. United States. doi:10.1021/acs.jctc.7b01243.

```
Banerjee, Amartya S., Lin, Lin, Suryanarayana, Phanish, Yang, Chao, and Pask, John E.. Mon .
"Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations". United States. doi:10.1021/acs.jctc.7b01243. https://www.osti.gov/servlets/purl/1512626.
```

```
@article{osti_1512626,
```

title = {Two-Level Chebyshev Filter Based Complementary Subspace Method: Pushing the Envelope of Large-Scale Electronic Structure Calculations},

author = {Banerjee, Amartya S. and Lin, Lin and Suryanarayana, Phanish and Yang, Chao and Pask, John E.},

abstractNote = {We describe here a novel iterative strategy for Kohn–Sham density functional theory calculations aimed at large systems (>1,000 electrons), applicable to metals and insulators alike. In lieu of explicit diagonalization of the Kohn–Sham Hamiltonian on every self-consistent field (SCF) iteration, we employ a two-level Chebyshev polynomial filter based complementary subspace strategy to (1) compute a set of vectors that span the occupied subspace of the Hamiltonian; (2) reduce subspace diagonalization to just partially occupied states; and (3) obtain those states in an efficient, scalable manner via an inner Chebyshev filter iteration. By reducing the necessary computation to just partially occupied states and obtaining these through an inner Chebyshev iteration, our approach reduces the cost of large metallic calculations significantly, while eliminating subspace diagonalization for insulating systems altogether. We describe the implementation of the method within the framework of the discontinuous Galerkin (DG) electronic structure method and show that this results in a computational scheme that can effectively tackle bulk and nano systems containing tens of thousands of electrons, with chemical accuracy, within a few minutes or less of wall clock time per SCF iteration on large-scale computing platforms. We anticipate that our method will be instrumental in pushing the envelope of large-scale ab initio molecular dynamics. As a demonstration of this, we simulate a bulk silicon system containing 8,000 atoms at finite temperature, and obtain an average SCF step wall time of 51 s on 34,560 processors; thus allowing us to carry out 1.0 ps of ab initio molecular dynamics in approximately 28 h (of wall time).},

doi = {10.1021/acs.jctc.7b01243},

journal = {Journal of Chemical Theory and Computation},

issn = {1549-9618},

number = 6,

volume = 14,

place = {United States},

year = {2018},

month = {4}

}