Exact relation between density-matrix and density-functional theory for K plus L closed shells in a bare Coulomb field.

Abstract

The explicit form of the first-order density matrix \ensuremath{\rho}${(\mathrm{r}}_{1}$,${\mathrm{r}}_{2}$) for independent electrons moving in a bare Coulomb field is first set up for the case of K plus L closed shells. The density matrix has a simple separable form in terms of the variables ${r}_{1}$+${r}_{2}$ and \ensuremath{\Vert}${\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$\ensuremath{\Vert}, a property exclusive to the Coulomb potential. Also, the off-diagonal dependence is simply quadratic in \ensuremath{\Vert}${\mathrm{r}}_{1}$${\mathrm{\ensuremath{-}}\mathrm{r}}_{2}$\ensuremath{\Vert}. These two properties allow \ensuremath{\rho}${(\mathrm{r}}_{1}$,${\mathrm{r}}_{2}$) to be written solely in terms of electron and kinetic energy densities. Other implications for density-functional theory are briefly referred to.