**C. K. Gan, P. D. Haynes and M. C. Payne**

**Theory of Condensed Matter, Cavendish Laboratory, Madingley
Road, Cambridge CB3 OHE, U.K.**

The use of localized basis sets is essential in linear-scaling
electronic structure calculations, and since such basis sets are
mostly non-orthogonal, it is necessary to solve the generalized
eigenvalue problem
. In this work, an
iterative method for finding the lowest few eigenvalues and
corresponding eigenvectors for the generalized eigenvalue problem
based on the conjugate gradient method is presented. The method is
applied to first-principles electronic structure calculations within
density-functional theory using a localized spherical-wave basis set,
first introduced in the context of linear-scaling methods
[Comput. Phys. Commun. 102 (1997) 17]. The method
exhibits linear convergence of
the solution, the rate of which is improved by a preconditioning
scheme using the kinetic energy matrix.

PACS numbers: 02.60; 71.15

- 1. Introduction
- 2. Formulation of the problem
- 3. The iterative method
- 4. Tests of the algorithm
- 5. Conclusions
- Acknowledgements
- Appendix A
- Appendix B
- Bibliography

Peter Haynes