Density-matrix formulation

In DFT, the $N$-particle Kohn-Sham system can be represented by:

EITHER	a set of $N$ single-particle wavefunctions $\{ \psi_i({\bf r}) \}$
OR	the single-particle density-matrix $\rho({\bf r},{\bf r'})$

	Wavefunctions $\{ \textcolor{green}{\psi_i({\bf r})}\}$	Density-matrix $\rho({\bf r},{\bf r'})$
Density	$${\sum_i^{\rm {occ}} \left\vert \textcolor{green}{\psi_i({\bf r})}\right\vert^2}$$	$\rho({\bf r},{\bf r})$
Kinetic energy	$${-{1 \over 2} \sum_i^{\rm {occ}} \int \textcolor{green}{\psi_i^{\ast}({\bf r})}\nabla_{\bf r}^2 \textcolor{green}{\psi_i({\bf r})}{\rm d}{\bf r}}$$	$${-{1 \over 2} \int \left[ \nabla_{\bf r'}^2 \textcolor{magenta}{\rho({\bf r},{\bf r'})}\right]_{{\bf r'} = {\bf r}} {\rm d}{\bf r}}$$
Constraints	$${\int \textcolor{green}{\psi_i^{\ast}({\bf r}) \psi_j({\bf r})}{\rm d}{\bf r} = \delta_{ij}}$$	${\rm Tr}(\textcolor{magenta}{\rho}) = N; \qquad \textcolor{magenta}{\rho^2}= \textcolor{magenta}{\rho}$