Exact local exchange potential from Fock equations at vanishing coupling constant, and delta Tc/ delta n from wave-function calculations at full coupling constant.

Abstract

Consider H${\mathrm{^}}_{\ensuremath{\lambda}}$=T^+\ensuremath{\lambda}V${\mathrm{^}}_{\mathit{ee}}$+${\ensuremath{\sum}}_{\mathit{i}=1}^{\mathit{N}}$${\mathit{v}}_{\ensuremath{\lambda}}$(r${\ensuremath{\rightarrow}}_{\mathit{i}}$), where, in atomic units, T^=${\ensuremath{\sum}}_{\mathit{i}=1}^{{\mathit{N}}^{\mathrm{\ensuremath{-}}}}$1/2${\mathrm{\ensuremath{\nabla}}}_{\mathit{i}}^{2}$ and V${\mathrm{^}}_{\mathit{ee}}$=${\ensuremath{\sum}}_{\mathit{i}}$${\ensuremath{\sum}}_{\mathit{j}\mathrm{\ensuremath{\gtrsim}}\mathit{i}}$\ensuremath{\Vert}r${\ensuremath{\rightarrow}}_{\mathit{i}}$-r${\ensuremath{\rightarrow}}_{\mathit{j}}$${\mathrm{\ensuremath{\Vert}}}^{\mathrm{\ensuremath{-}}1}$ and the local-multiplicative potential ${\mathit{v}}_{\ensuremath{\lambda}}$ is constructed to keep n, the Hartree-Fock density of H${\mathrm{^}}_{\ensuremath{\lambda}}$, independent of the coupling constant \ensuremath{\lambda}. Moreover, n is simultaneously the exact physical ground-state density of interest and ${\mathit{v}}_{\ensuremath{\lambda}=0}$ is its Kohn-Sham potential. For the purpose of extracting the exact Kohn-Sham exchange potential ${\mathit{v}}_{\mathit{x}}$([n];r\ensuremath{\rightarrow}), it is shown that ${\mathit{v}}_{\mathit{x}}$([n];r\ensuremath{\rightarrow})=-\ensuremath{\partial}${\mathit{v}}_{\ensuremath{\lambda}}$([n];r\ensuremath{\rightarrow})/\ensuremath{\partial}\ensuremath{\lambda}${\mathrm{\ensuremath{\Vert}}}_{\ensuremath{\lambda}=0}$-\ensuremath{\int}[ n(r\ensuremath{\rightarrow}\ensuremath{'})/\ensuremath{\Vert}r\ensuremath{\rightarrow}-r${\ensuremath{\rightarrow}}^{\ensuremath{'}}$\ensuremath{\Vert}]${\mathit{d}}^{3}$${\mathit{r}}^{\ensuremath{'}}$. The correlation potential for n(r\ensuremath{\rightarrow}) is then obtained by subtracting the above ${\mathit{v}}_{\mathit{x}}$ from the exact exchange-correlation potential for n(r\ensuremath{\rightarrow}). Furthermore, it is shown that the potential ${\mathit{w}}_{\ensuremath{\lambda}}$(r\ensuremath{\rightarrow}), which keeps the exact ground-state density of T^+\ensuremath{\lambda}V${\mathrm{^}}_{\mathit{ee}}$+${\ensuremath{\sum}}_{\mathit{i}=1}^{\mathit{N}}$${\mathit{w}}_{\ensuremath{\lambda}}$(r${\ensuremath{\rightarrow}}_{\mathit{i}}$) independent of the value of \ensuremath{\lambda}, determines the functional derivative of ${\mathit{T}}_{\mathit{c}}$[n] with respect to the density, i.e., the kinetic contribution to the correlation potential ${\mathit{v}}_{\mathit{c}}$, \ensuremath{\delta}${\mathit{T}}_{\mathit{c}}$[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(r\ensuremath{\rightarrow})${\mathrm{\ensuremath{\Vert}}}_{\mathrm{\ensuremath{\rho}}=\mathit{n}}$=${\mathit{v}}_{\mathrm{xc}}$([n];r\ensuremath{\rightarrow})+\ensuremath{\int}n(r\ensuremath{\rightarrow}\ensuremath{'})${\mathit{d}}^{3}$${\mathit{r}}^{\ensuremath{'}}$/\ensuremath{\Vert}r\ensuremath{\rightarrow}-r${\ensuremath{\rightarrow}}^{\ensuremath{'}}$\ensuremath{\Vert}-\ensuremath{\partial}${\mathit{w}}_{\ensuremath{\lambda}}$([n];r\ensuremath{\rightarrow})/\ensuremath{\partial}\ensuremath{\lambda}${\mathrm{\ensuremath{\Vert}}}_{\ensuremath{\lambda}=1}$. \textcopyright{} 1996 The American Physical Society.