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# Density-functional calculations

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

 (6)

where are the KS eigenfunctions with corresponding eigenvalues . The effective potential consists of the classical electrostatic potential, the ionic potential due to the nuclei and the exchange-correlation potential. The effective potential depends on the electron density, , which is formed from the lowest eigenstates
 (7)

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

 (8)

where is a collective label for in Eq. (5). Substituting Eq. (10) into Eq. (8), and taking inner products with the , we obtain the generalized eigenvalue problem
 (9)

where the overlap matrix is given by
 (10)

and the Hamiltonian matrix is given by
 (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)

Periodicity of the system under study is assumed and the point only is used for the Brillouin zone sampling. We have used the local density approximation (LDA) for the exchange and correlation term. Norm-conserving Troullier-Martins[25] pseudopotentials in the Kleinman-Bylander[19] form are used.

Next: Results of the calculations Up: First-principles density-functional calculations using Previous: Origin of the basis
Peter D. Haynes 2002-10-31