Density-functional theory

• Set $\{ \psi_i({\bf r}) \}$ of orthonormal single-particle wave-functions:
  
  $${\hat H}_{\rm KS} \psi_i({\bf r}) = \varepsilon_i \psi_i({\bf r})$$
  
  

• Set $\{ f_i \}$ of occupation numbers:
  
  $$f_i = \left\{ \begin{array}{ll} 1, & \varepsilon_i < \mu \\ 0, & \varepsilon_i > \mu \\ \end{array} \right.$$
  
  
  Chemical potential $\mu$ chosen to give correct number of electrons $N_{\rm e}$.