Kohn-Sham potentials and exchange and correlation energy densities from one- and two-electron density matrices forLi2,N2,andF2

Abstract

A definition of key quantities of the Kohn-Sham form of density-functional theory such as the exchange-correlation potential ${v}_{\mathrm{xc}}$ and the energy density ${\ensuremath{\varepsilon}}_{\mathrm{xc}}$ in terms of wave-function quantities (one- and two-electron density matrices) is given. This allows the construction of ${v}_{\mathrm{xc}}$ and ${\ensuremath{\varepsilon}}_{\mathrm{xc}}$ numerically as functions of r from ab initio wave functions. The behavior of the constructed exchange ${\ensuremath{\varepsilon}}_{x}$ and correlation ${\ensuremath{\varepsilon}}_{c}$ energy densities and the corresponding integrated exchange ${E}_{x}$ and correlation ${E}_{c}$ energies have been compared with those of the local-density approximation (LDA) and generalized gradient approximations (GGA) of Becke, of Perdew and Wang, and of Lee, Yang, and Parr. The comparison shows significant differences between ${\ensuremath{\varepsilon}}_{c}(\mathbf{r})$ and the ${\ensuremath{\varepsilon}}_{c}^{\mathrm{GGA}}(\mathbf{r}),$ in spite of some gratifying similarities in shape for particularly ${\ensuremath{\varepsilon}}_{c}^{\mathrm{PW}}.$ On the other hand, the local behavior of the GGA exchange energy densities is found to be very similar to the constructed ${\ensuremath{\varepsilon}}_{x}(\mathbf{r}),$ yielding integrated energies to about 1% accuracy. Still the remaining differences are a sizable fraction $(\ensuremath{\sim}25%)$ of the correlation energy, showing up in differences between the constructed and model exchange energy densities that are locally even larger than the typical correlation energy density. It is argued that nondynamical correlation, which is incorporated in ${\ensuremath{\varepsilon}}_{c}(\mathbf{r}),$ is lacking from ${\ensuremath{\varepsilon}}_{c}^{\mathrm{GGA}}(\mathbf{r}),$ while it is included in ${\ensuremath{\varepsilon}}_{x}^{\mathrm{LDA}}(\mathbf{r})$ and ${\ensuremath{\varepsilon}}_{x}^{\mathrm{GGA}}(\mathbf{r})$ but not in ${\ensuremath{\varepsilon}}_{x}(\mathbf{r}).$ This is verified almost quantitatively for the integrated energies. It also appears to hold locally in the sense that the difference ${\ensuremath{\varepsilon}}_{x}^{\mathrm{GGA}}(\mathbf{r})\ensuremath{-}{\ensuremath{\varepsilon}}_{x}(\mathbf{r})$ may be taken to represent ${\ensuremath{\varepsilon}}_{c}^{\mathrm{nondyn}}(\mathbf{r})$ and can be added to ${\ensuremath{\varepsilon}}_{c}^{\mathrm{GGA}}(\mathbf{r})$ to bring it much closer to ${\ensuremath{\varepsilon}}_{c}(\mathbf{r}).$