# Numerical methods for the inverse problem of density functional theory

## Abstract

Here, the inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.

- Authors:

- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
- Purdue Univ., West Lafayette, IN (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- NSF Funded from Graduate School; USDOE

- OSTI Identifier:
- 1421863

- Report Number(s):
- SAND-2017-7018J

Journal ID: ISSN 0020-7608; 655000

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- International Journal of Quantum Chemistry

- Additional Journal Information:
- Journal Volume: 118; Journal Issue: 1; Journal ID: ISSN 0020-7608

- Publisher:
- Wiley

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; density functional theory; inverse problems; PDE-constrained optimization

### Citation Formats

```
Jensen, Daniel S., and Wasserman, Adam.
```*Numerical methods for the inverse problem of density functional theory*. United States: N. p., 2017.
Web. doi:10.1002/qua.25425.

```
Jensen, Daniel S., & Wasserman, Adam.
```*Numerical methods for the inverse problem of density functional theory*. United States. doi:10.1002/qua.25425.

```
Jensen, Daniel S., and Wasserman, Adam. Mon .
"Numerical methods for the inverse problem of density functional theory". United States.
doi:10.1002/qua.25425. https://www.osti.gov/servlets/purl/1421863.
```

```
@article{osti_1421863,
```

title = {Numerical methods for the inverse problem of density functional theory},

author = {Jensen, Daniel S. and Wasserman, Adam},

abstractNote = {Here, the inverse problem of Kohn–Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the numerical methods for solving each problem are substantially different. We examine both problems in this tutorial with a special emphasis on the algorithms and error analysis needed for solving the inverse problem. Two inversion methods based on partial differential equation constrained optimization and constrained variational ideas are introduced. We compare and contrast several different inversion methods applied to one-dimensional finite and periodic model systems.},

doi = {10.1002/qua.25425},

journal = {International Journal of Quantum Chemistry},

number = 1,

volume = 118,

place = {United States},

year = {Mon Jul 17 00:00:00 EDT 2017},

month = {Mon Jul 17 00:00:00 EDT 2017}

}

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