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 $M$ electrons, we need to solve self-consistently the Kohn-Sham equations which assume the following form

$$\hat{H} \psi_m({\bf r}) = \left[-\frac{\hbar^2}{2m_{\mathrm ... ...f r}) \right] \psi_m({\bf r}) = \varepsilon_m \psi_m({\bf r}),$$ (1)

where ${\hat H}$ is the Kohn-Sham Hamiltonian, with energy eigenvalues $\varepsilon_m$ and corresponding eigenstates $\psi_m({\bf r})$. The effective potential $V_{\mathrm{eff}}$ 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 $M$ lowest or occupied eigenstates

$$\rho({\bf r}) = \sum_{m=1}^{M} \left\vert \psi_m({\bf r}) \right\vert^2.$$ (2)

The eigenstates satisfy the orthogonality constraints where

$$\int \d {\bf r}\ \psi_m^{\ast}({\bf r}) \psi_n({\bf r}) = \delta_{mn},$$ (3)

for all $m$ and $n$.

When a non-orthogonal basis set $\{ \chi_{\alpha}({\bf r})\}$ is used, the eigenstates are written as

$$\psi_n({\bf r}) = \sum_{\alpha} {x_n}^{\alpha} \ \chi_{\alpha}({\bf r}),$$ (4)

where $\alpha$ labels a basis function $\chi_{\alpha}({\bf r})$. The right hand side of Eq. (4) has been written as a contraction between a contravariant quantity $x_n$ and a covariant quantity $\chi({\bf r})$. Substituting Eq. (4) into Eq. (1), taking inner products with the $\{ \chi_{\alpha}({\bf r})\}$, and using the definitions

$$S_{\alpha\beta} = \int \d {\bf r}\ \chi^{\ast}_{\alpha}({\bf r}) \chi_{\beta}({\bf r}) ,$$ (5)

and

$$H_{\alpha\beta} = \int \d {\bf r}\ \chi^{\ast}_{\alpha}({\bf r}) \hat{H} \chi_{\beta}({\bf r}) ,$$ (6)

we obtain the generalized eigenvalue problem

$$H_{\alpha\beta} {x_n}^{\beta} = \varepsilon_n S_{\alpha\beta} {x_n}^{\beta}.$$ (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

$${x_m^{\ast}}^{\alpha} S_{\alpha\beta} {x_n}^{\beta} = \delta_{mn}.$$ (8)

When Eq. (7) is solved, a new output electron density $\rho^{(i)}_{\mathrm{out}}$ 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.

$$\rho^{(i+1)}_{\mathrm{in}} = f \rho^{(i)}_{\mathrm{out}} + (1-f) \rho^{(i)}_{\mathrm{in}},$$ (9)

where the optimum choice for $f$ 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.