Summary so far

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

$$E = \sum_{ij}\{T_{ij} + V_{ij}^{ps}\}b_{ij} + \tilde{E}_H + \tilde{E}_{xc} + E_{i-i}$$ (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, $E_\lambda$ to allow unconstrained minimisation with respect to $c_i^\lambda$, i.e.:

$$\frac{\partial E}{\partial c_i^\lambda} = \frac{\partial \{ ... ...e{E}_H + \tilde{E}_{xc} + E_{i-i} \}}{\partial c_i^\lambda} = 0$$ (32)

 

$$\sum_j \{T_{ij} + V_{ij}^{ps} + V_{ij}^H + V_{ij,s\lambda}^{xc} - E_\lambda S_{ij}\}c_j^\lambda = 0$$ (33)

 

$$\sum_j (H_{ij} - ES_{ij})c_j = 0$$ (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).