Density-functional calculations

The Kohn-Sham (KS) equation for an -electron system
is[20,21]

(7) |

We use the real spherical-wave basis set
to expand the -th KS eigenstate

where the overlap matrix is given by

(10) |

(11) |

It should be emphasized that when a large system is studied, and will be sparse. In this case it is more efficient to use an iterative method based on preconditioned conjugate gradient minimization[22] to find the lowest few eigenvalues and corresponding eigenvectors of Eq. (11) than to use a direct matrix diagonalization method [23,24] in which all eigenvalue-eigenvector pairs are found.

The completeness of the basis set depends on several parameters
such as the radius of the basis sphere, ; the maximum angular
momentum component,
; and the number of values for
each angular momentum component, which we will take here to be the
same for all and is denoted by . The number of basis
functions in a basis sphere is
. For a fixed number
of , we note that the number of basis functions increases very
rapidly with respect to
. However, we will demonstrate that
most physical properties can be deduced using only a small
,
which is typically 2. The cutoff energy
for a basis
sphere is roughly given by

(12) |