# Linear Multigrid Techniques in Self-consistent Electronic Structure Calculations

## Abstract

Ab initio DFT electronic structure calculations involve an iterative process to solve the Kohn-Sham equations for an Hamiltonian depending on the electronic density. We discretize these equations on a grid by finite differences. Trial eigenfunctions are improved at each step of the algorithm using multigrid techniques to efficiently reduce the error at all length scale, until self-consistency is achieved. In this paper we focus on an iterative eigensolver based on the idea of inexact inverse iteration, using multigrid as a preconditioner. We also discuss how this technique can be used for electrons described by general non-orthogonal wave functions, and how that leads to a linear scaling with the system size for the computational cost of the most expensive parts of the algorithm.

- Authors:

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab., CA (US)

- Sponsoring Org.:
- USDOE Office of Defense Programs (DP) (US)

- OSTI Identifier:
- 791582

- Report Number(s):
- UCRL-JC-138186

TRN: US200304%%513

- DOE Contract Number:
- W-7405-Eng-48

- Resource Type:
- Conference

- Resource Relation:
- Conference: NATO Advanced Research Workshop on Multiscale Computational Methods in Chemistry and Biology, Eilat (IL), 04/05/2000--04/07/2000; Other Information: PBD: 23 May 2000

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ALGORITHMS; BIOLOGY; CHEMISTRY; EIGENFUNCTIONS; ELECTRONIC STRUCTURE; ELECTRONS; HAMILTONIANS; NATO; WAVE FUNCTIONS

### Citation Formats

```
Fattebert, J-L.
```*Linear Multigrid Techniques in Self-consistent Electronic Structure Calculations*. United States: N. p., 2000.
Web.

```
Fattebert, J-L.
```*Linear Multigrid Techniques in Self-consistent Electronic Structure Calculations*. United States.

```
Fattebert, J-L. Tue .
"Linear Multigrid Techniques in Self-consistent Electronic Structure Calculations". United States. https://www.osti.gov/servlets/purl/791582.
```

```
@article{osti_791582,
```

title = {Linear Multigrid Techniques in Self-consistent Electronic Structure Calculations},

author = {Fattebert, J-L},

abstractNote = {Ab initio DFT electronic structure calculations involve an iterative process to solve the Kohn-Sham equations for an Hamiltonian depending on the electronic density. We discretize these equations on a grid by finite differences. Trial eigenfunctions are improved at each step of the algorithm using multigrid techniques to efficiently reduce the error at all length scale, until self-consistency is achieved. In this paper we focus on an iterative eigensolver based on the idea of inexact inverse iteration, using multigrid as a preconditioner. We also discuss how this technique can be used for electrons described by general non-orthogonal wave functions, and how that leads to a linear scaling with the system size for the computational cost of the most expensive parts of the algorithm.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2000},

month = {5}

}