Expressions for the Exchange Correlation Potential and Exchange Correlation Functional of Kohn--Sham Density Functional Theory

The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of $v_s$ and $\phi_1$, where $v_s$ is the local, one-body potential from the Kohn--Sham equations, and $\phi_1$ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|\phi_1\ran$. In other words, $|\phi_1\ran$ is the ground state eigenfunction of the noninteracting Schr\"odinger equation with the one-body potential $v_s$. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states