Optimizing localized functions

• Optimize a set of localized functions $\{ \phi_{\alpha}({\bf r}) \}$ instead of wavefunctions $\{ \psi_i({\bf r}) \}$
• Can no longer impose orthogonality, so minimize:
  
  $$E[\{ \phi_{\alpha} \}] = {\rm Tr}\left( \sf {S^{-1}H} \right)$$
  
  
  where $\sf {S}_{\alpha \beta} = \langle \phi_{\alpha} \vert \phi_{\beta} \rangle$ and $\sf {H}_{\alpha \beta} = \langle \phi_{\alpha} \vert {\hat H} \vert \phi_{\beta} \rangle$
• Need to consider tensor properties of gradient with respect to:
  • basis set
  • localized functions