Kohn-Sham Theory for Ground-State Ensembles

An electron density distribution $n\left(\mathbf{r}\right)$ which can be represented by that of a single-determinant ground state of noninteracting electrons in an external potential $v\left(\mathbf{r}\right)$ is called pure-state $v$-representable (P-VR). Most physical electronic systems are P-VR. Systems which require a weighted sum of several such determinants to represent their density are called ensemble $v$-representable (E-VR). This paper develops formal Kohn-Sham equations for E-VR physical systems, using the appropriate coupling constant integration. It also derives local density- and generalized gradient approximations, and conditions and corrections specific to ensembles.