Direct minimisation

Express $\rho({\bf r},{\bf r}')$ in terms of:

• set $\{ \phi_i({\bf r}) \}$ of non-orthogonal localised functions - linear combinations of the $\{ \psi_i({\bf r}) \}$
• sparse Hermitian matrix $K^{ij}$ - generalised occupation numbers

$$\rho({\bf r},{\bf r}') = \sum_{ij} \phi_i({\bf r}) K^{ij} \phi_j^{\ast}({\bf r}')$$

• Find ground-state by minimising $E[\rho] = 2   {\rm Tr}[ \rho H_{\rm KS}]$ with respect to $\{ \phi_i({\bf r}) \}$ and $K^{ij}$.
• Occupation numbers are now variable parameters.

$\Longrightarrow$ need to impose constraints of normalisation and idempotency.