Density-matrix minimization

$$\rho({\bf r},{\bf r'}) = \displaystyle{\sum_{\alpha \beta} \p... ...alpha}({\bf r}) P^{\alpha \beta} \phi_{\beta}^{\ast}({\bf r'})}$$

For a given density $n({\bf r})$ and hence Kohn-Sham potential $V_{\rm {eff}}({\bf r})$, we can:

• Write the energy as a functional of the DM: $E[\rho]$
• Minimize the energy w.r.t. $\rho({\bf r},{\bf r'})$:
  • Optimize $P^{\alpha \beta}$ and $\{ \phi_{\alpha}({\bf r}) \}$
  • Subject to the normalization and idempotency constraints

We also need to:

• Converge the energy w.r.t. the DM cutoff $r_{\rm {cut}}$ i.e.
  • the sparsity of $P^{\alpha \beta}$
  • the localization of $\{ \phi_{\alpha}({\bf r}) \}$