Density-matrix formulation

Combine $\{ \psi_i({\bf r}) \}$ and $\{ f_i \}$ into $\rho({\bf r},{\bf r}')$, the single-particle density-matrix:

$$\rho({\bf r},{\bf r}') = \sum_i f_i   \psi_i({\bf r}) \psi_i^{\ast}({\bf r}')$$

• normalised:
  
  $$N[\rho] = 2   {\rm Tr}[\rho] = 2 \int {\rm d}{\bf r}   \rho({\bf r},{\bf r}) = 2 \sum_i f_i = N_{\rm e}$$
  
  

• idempotent:
  
  $$\rho^2({\bf r},{\bf r}') = \int {\rm d}{\bar {\bf r}}   \rho(... ...bf r}}) \rho({\bar {\bf r}},{\bf r}') = \rho({\bf r},{\bf r}')$$
  
  

• short-ranged:
  
  $$\rho({\bf r},{\bf r}') \rightarrow 0 {\rm  as } \left\vert {\bf r} - {\bf r}' \right\vert \rightarrow \infty$$