# The Use of Trust Regions in Kohn-Sham Total EnergyMinimization

## Abstract

The Self Consistent Field (SCF) iteration, widely used forcomputing the ground state energy and the corresponding single particlewave functions associated with a many-electronatomistic system, is viewedin this paper as an optimization procedure that minimizes the Kohn-Shamtotal energy indirectly by minimizing a sequence of quadratic surrogatefunctions. We point out the similarity and difference between the totalenergy and the surrogate, and show how the SCF iteration can fail whenthe minimizer of the surrogate produces an increase in the KS totalenergy. A trust region technique is introduced as a way to restrict theupdate of the wave functions within a small neighborhood of anapproximate solution at which the gradient of the total energy agreeswith that of the surrogate. The use of trust region in SCF is not new.However, it has been observed that directly applying a trust region basedSCF(TRSCF) to the Kohn-Sham total energy often leads to slowconvergence.We propose to use TRSCF within a direct constrainedminimization(DCM) algorithm we developed in \cite dcm. The keyingredients of theDCM algorithm involve projecting the total energyfunction into a sequence of subspaces of small dimensions and seeking theminimizerof the total energy function within each subspace. Theminimizer of a subspace energy function, which is computed by TRSCF, notonly provides amore »

- Authors:

- Publication Date:

- Research Org.:
- Ernest Orlando Lawrence Berkeley NationalLaboratory, Berkeley, CA (US)

- Sponsoring Org.:
- USDOE Director. Office of Science. Advanced ScientificComputing Research

- OSTI Identifier:
- 919759

- Report Number(s):
- LBNL-59841

Journal ID: ISSN 1064-8275; SJOCE3; R&D Project: K11116; BnR: KJ0101030; TRN: US0806483

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Journal Article

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- Journal Volume: 29; Journal Issue: 5; Related Information: Journal Publication Date: 2007; Journal ID: ISSN 1064-8275

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 36 MATERIALS SCIENCE; ALGORITHMS; CONVERGENCE; DIMENSIONS; GROUND STATES; MINIMIZATION; OPTIMIZATION; SELF-CONSISTENT FIELD; WAVE FUNCTIONS; Kohn-Sham equation Total energy minimization trust regionsconstrained optimization

### Citation Formats

```
Yang, Chao, Meza, Juan C., and Wang, Lin-wang.
```*The Use of Trust Regions in Kohn-Sham Total EnergyMinimization*. United States: N. p., 2006.
Web.

```
Yang, Chao, Meza, Juan C., & Wang, Lin-wang.
```*The Use of Trust Regions in Kohn-Sham Total EnergyMinimization*. United States.

```
Yang, Chao, Meza, Juan C., and Wang, Lin-wang. Tue .
"The Use of Trust Regions in Kohn-Sham Total EnergyMinimization". United States. https://www.osti.gov/servlets/purl/919759.
```

```
@article{osti_919759,
```

title = {The Use of Trust Regions in Kohn-Sham Total EnergyMinimization},

author = {Yang, Chao and Meza, Juan C. and Wang, Lin-wang},

abstractNote = {The Self Consistent Field (SCF) iteration, widely used forcomputing the ground state energy and the corresponding single particlewave functions associated with a many-electronatomistic system, is viewedin this paper as an optimization procedure that minimizes the Kohn-Shamtotal energy indirectly by minimizing a sequence of quadratic surrogatefunctions. We point out the similarity and difference between the totalenergy and the surrogate, and show how the SCF iteration can fail whenthe minimizer of the surrogate produces an increase in the KS totalenergy. A trust region technique is introduced as a way to restrict theupdate of the wave functions within a small neighborhood of anapproximate solution at which the gradient of the total energy agreeswith that of the surrogate. The use of trust region in SCF is not new.However, it has been observed that directly applying a trust region basedSCF(TRSCF) to the Kohn-Sham total energy often leads to slowconvergence.We propose to use TRSCF within a direct constrainedminimization(DCM) algorithm we developed in \cite dcm. The keyingredients of theDCM algorithm involve projecting the total energyfunction into a sequence of subspaces of small dimensions and seeking theminimizerof the total energy function within each subspace. Theminimizer of a subspace energy function, which is computed by TRSCF, notonly provides a search direction along which the KS total energy functiondecreases but also gives an optimal "step-length" that yields asufficient decrease in total energy. A numerical example is provided todemonstrate that the combination of TRSCF and DCM is more efficient thanSCF.},

doi = {},

journal = {SIAM Journal on Scientific Computing},

issn = {1064-8275},

number = 5,

volume = 29,

place = {United States},

year = {2006},

month = {5}

}