Calculational scheme for exact exchange and correlation potentials based on the equation of motion for density matrix plus the perturbation theory

The equation of motion method is used to express the external potential in terms of the first- and second-order interacting density matrices (DMs) for a molecule. By also introducing noninteracting DMs built from the Kohn-Sham (KS) reference system orbitals, the exchange-correlation potential is derived. Applying the perturbation theory of Görling and Levy [Phys. Rev. B 47, 13 105 (1993)], this potential is separated into the exchange-only potential and various orders of the correlation terms. Each potential term can be determined self-consistently from a specific equation written in terms of both occupied and excited KS orbitals and differences of orbital energies, plus potential terms of lower orders. Naturally, the determination of these orbitals and energies requires a self-consistent solution of KS equations. Explicit expressions for calculation of the exchange potential and the leading term of the correlation potential are obtained and discussed. An approximate exchange potential written directly in terms of noninteracting DMs is also proposed.