# Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Silicon

## Abstract

We present a new efficient way to perform hybrid density functional theory (DFT)-based electronic structure calculations. The new method uses an interpolative separable density fitting (ISDF) procedure to construct a set of numerical auxiliary basis vectors and a compact approximation of the matrix consisting of products of occupied orbitals represented in a large basis set such as the planewave basis. Such an approximation allows us to reduce the number of Poisson solves from *O*($N$ $$^{2}_{ϵ}$$) to *O*( *N* _{ϵ}) when we apply the exchange operator to occupied orbitals in an iterative method for solving the Kohn–Sham equations, where Ne is the number of electrons in the system to be studied. We show that the ISDF procedure can be carried out in *O*($N$ $$^{3}_{ϵ}$$) operations, with a much smaller preconstant compared to methods used in existing approaches. When combined with the recently developed adaptively compressed exchange (ACE) operator formalism, which reduces the number of times the exchange operator needs to be updated, the resulting ACE-ISDF method significantly reduces the computational cost associated with the exchange operator by nearly 2 orders of magnitude compared to existing approaches for a large silicon system with 1000 atoms. We demonstrate that the ACE-ISDF method can produce accurate energies and forces for insulating and metallic systems and that it is possible to obtain converged hybrid functional calculation results for a 1000-atom bulk silicon within 10 min on 2000 computational cores. We also show that ACE-ISDF can scale to 8192 computational cores for a 4096-atom bulk silicon system. We use the ACE-ISDF method to geometrically optimize a 1000-atom silicon system with a vacancy defect using the HSE06 functional and computes its electronic structure. Here, we find that that the computed energy gap from the HSE06 functional is much closer to the experimental value compared to that produced by semilocal functionals in the DFT calculations.

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Univ. of California, Berkeley, CA (United States)
- Univ. of California, Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center

- 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)

- OSTI Identifier:
- 1480242

- Grant/Contract Number:
- [DMS-1652330]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Theory and Computation

- Additional Journal Information:
- [ Journal Volume: 13; Journal Issue: 11]; Journal ID: ISSN 1549-9618

- Publisher:
- American Chemical Society

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Hu, Wei, Lin, Lin, and Yang, Chao. Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Silicon. United States: N. p., 2017.
Web. doi:10.1021/acs.jctc.7b00807.
```

```
Hu, Wei, Lin, Lin, & Yang, Chao. Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Silicon. United States. doi:10.1021/acs.jctc.7b00807.
```

```
Hu, Wei, Lin, Lin, and Yang, Chao. Fri .
"Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Silicon". United States. doi:10.1021/acs.jctc.7b00807. https://www.osti.gov/servlets/purl/1480242.
```

```
@article{osti_1480242,
```

title = {Interpolative Separable Density Fitting Decomposition for Accelerating Hybrid Density Functional Calculations with Applications to Defects in Silicon},

author = {Hu, Wei and Lin, Lin and Yang, Chao},

abstractNote = {We present a new efficient way to perform hybrid density functional theory (DFT)-based electronic structure calculations. The new method uses an interpolative separable density fitting (ISDF) procedure to construct a set of numerical auxiliary basis vectors and a compact approximation of the matrix consisting of products of occupied orbitals represented in a large basis set such as the planewave basis. Such an approximation allows us to reduce the number of Poisson solves from O($N$ $^{2}_{ϵ}$) to O(Nϵ) when we apply the exchange operator to occupied orbitals in an iterative method for solving the Kohn–Sham equations, where Ne is the number of electrons in the system to be studied. We show that the ISDF procedure can be carried out in O($N$ $^{3}_{ϵ}$) operations, with a much smaller preconstant compared to methods used in existing approaches. When combined with the recently developed adaptively compressed exchange (ACE) operator formalism, which reduces the number of times the exchange operator needs to be updated, the resulting ACE-ISDF method significantly reduces the computational cost associated with the exchange operator by nearly 2 orders of magnitude compared to existing approaches for a large silicon system with 1000 atoms. We demonstrate that the ACE-ISDF method can produce accurate energies and forces for insulating and metallic systems and that it is possible to obtain converged hybrid functional calculation results for a 1000-atom bulk silicon within 10 min on 2000 computational cores. We also show that ACE-ISDF can scale to 8192 computational cores for a 4096-atom bulk silicon system. We use the ACE-ISDF method to geometrically optimize a 1000-atom silicon system with a vacancy defect using the HSE06 functional and computes its electronic structure. Here, we find that that the computed energy gap from the HSE06 functional is much closer to the experimental value compared to that produced by semilocal functionals in the DFT calculations.},

doi = {10.1021/acs.jctc.7b00807},

journal = {Journal of Chemical Theory and Computation},

number = [11],

volume = [13],

place = {United States},

year = {2017},

month = {9}

}

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