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
densityfunctional 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 selfconsistently the
KohnSham equations which assume the following form

(1) 
where is the KohnSham Hamiltonian, with energy eigenvalues
and corresponding eigenstates
. The
effective potential
consists of three terms; the
classical electrostatic or Hartree potential, the exchangecorrelation
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 nonorthogonal 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 KohnSham 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 selfconsistently.
Next: 3. The iterative method
Up: Preconditioned conjugate gradient method
Previous: 1. Introduction
Peter Haynes