Kohn-Sham equations for multicomponent systems: The exchange and correlation energy functional

Kohn-Sham equations for multicomponent systems are derived in a rigorous way that permits the precise definition and discussion of the exchange and correlation energy, of the system as a functional of the densities of the components. In the case of a two-component electron-ion system, with $n_e{,n}_I$ the electron and ion densities, the exchange and correlation energy of the system $E_{\mathrm{xc}}\left[n_e{,n}_I\right]$ is composed of $E_{\mathrm{xc}}\left[n_e\right]$ the electron-electron exchange and correlation energy, $E_{\mathrm{xc}}^{\mathrm{II}}\left[n_I\right]$ the ion-ion exchange and correlation energy, and $E_c^{\mathrm{eI}}\left[n_e{,n}_I\right]$ the electron-ion correlation energy. $E_{\mathrm{xc}}\left[n_e\right]$ is exactly the functional encountered in the Kohn-Sham theory of electronic systems. The behavior of $E_{\mathrm{xc}}^{\mathrm{II}}\left[n_I\right]$ is investigated in the limit of a large ion mass and its relation with $E_{\mathrm{xc}}\left[n_e\right]$, for $n_e{=n}_I,$ is discussed. The structure of $E_c^{\mathrm{eI}}\left[n_e{,n}_I\right]$ is analyzed in the adiabatic approximation. In the special case of perfectly localized ion densities, $E_{\mathrm{xc}}^{\mathrm{II}}\left[n_I\right]$ results in a self-interaction correction while $E_c^{\mathrm{eI}}\left[n_e{,n}_I\right]$ vanishes.