Next: 3. The iterative method Up: Preconditioned conjugate gradient method Previous: 1. Introduction

# 2. Formulation of the problem

We give a brief account of electronic structure calculations within density-functional theory [16] which requires the generalized eigenvalue problem to be solved (see Ref. [17] for a more comprehensive description). For a system of electrons, we need to solve self-consistently the Kohn-Sham equations which assume the following form
 (1)

where is the Kohn-Sham Hamiltonian, with energy eigenvalues and corresponding eigenstates . The effective potential consists of three terms; the classical electrostatic or Hartree potential, the exchange-correlation potential, and the external potential [17]. The electron density is formed from the lowest or occupied eigenstates
 (2)

The eigenstates satisfy the orthogonality constraints where
 (3)

for all and .

When a non-orthogonal basis set is used, the eigenstates are written as

 (4)

where labels a basis function . The right hand side of Eq. (4) has been written as a contraction between a contravariant quantity and a covariant quantity . Substituting Eq. (4) into Eq. (1), taking inner products with the , and using the definitions
 (5)

and
 (6)

we obtain the generalized eigenvalue problem
 (7)

In writing Eq. (7), we have adopted the Einstein summation notation where we sum over repeated Greek indices. The orthogonality conditions of the Kohn-Sham eigenstates in Eq. (3) translate into
 (8)

When Eq. (7) is solved, a new output electron density is obtained and a new input electron density for the next iteration can be constructed by a linear (or more sophisticated [18]) mixing scheme e.g.

 (9)

where the optimum choice for depends upon the eigenvalues of the static dielectric matrix of the system. The mixing of densities is carried out until Eq. (1) is solved self-consistently.

Next: 3. The iterative method Up: Preconditioned conjugate gradient method Previous: 1. Introduction
Peter Haynes