I will now attempt to summarise the method shown so far, and draw out
the salient calculational points. Using the approximate functions for
*E*_{H} and *E*_{xc} derived in the last few sections we can rewrite the
total energy as

(31) |

where

*E* is now minimised, however an additional constraint has to be
applied in order to keep the wavefunctions orthonormal; this can be
imposed in terms of the overlap matrix, *S*,

Now all we need is Lagrange undetermined multipliers, to
allow unconstrained minimisation with respect to , *i.e.*:

(32) |

(33) |

(34) |

Equation 3.2.39 is the Kohn-Sham equations once more, and is written in terms of two generalised eigenvalue problems (one for each spin).