The Kohn-Sham matrix

To minimize the energy within a basis set requires construction of the Kohn-Sham matrix, the matrix representation of the operator in equation (2.51). Writing the exchange-correlation energy as
$$\begin{align}E_{xc} = \int f_{xc}(\rho,\nabla \rho) \, d{\bf r}, \end{align}$$
the exchange-correlation elements of the Kohn-Sham matrix are given by
 $$\begin{align} F^{xc}_{\mu \nu} = \int \phi_{\mu} v_{xc} \phi_{\nu} \, d{\bf r} \end{align}$$
with the potential defined by equation (2.50). By using calculus of variations [83] the potential can be expressed as
$$\begin{align}v_{xc} = \frac{\partial f_{xc}}{\partial \rho} - \nabla . \left( \frac{\partial f_{xc}}{\partial \nabla \rho} \right). \end{align}$$

However, Pople et al. [151] pointed out that the calculus of variations procedure involves integration by parts in the $\nabla \rho$ contribution. Thus the numerical integration of equation (2.101) will have an increased error. A more consistent approach is to obtain the exchange-correlation part of the Kohn-Sham matrix from the direct minimization of the energy with respect to orbital variations, that is
$$\begin{align}F_{\mu \nu}^{xc} = \frac{\partial E_{xc}}{\partial P_{\mu \nu}} = \... ...{\partial \nabla \rho} . \nabla (\phi_{\mu} \phi_{\nu}) \, d{\bf r}. \end{align}$$
Another advantage of this formulation is that the second-derivative of the density is no longer required, a major computational saving.