Second-order perturbative correlation energy functional in the ensemble density-functional theory

We derive the second-order approximation (PT2) for the ensemble correlation energy functional by applying the G\"orling-Levy perturbation theory to the ensemble density-functional theory. Its performance is checked by calculating excitation energies with the direct ensemble correction method in one-dimensional (1D) model systems and three-dimensional atoms using numerically exact Kohn-Sham orbitals and potentials. Compared with the exchange-only approximation, the inclusion of the ensemble PT2 correlation improves the excitation energies in 1D model systems in most cases, including double excitations and charge-transfer excitations. However, the excitation energies for atoms are generally worse with PT2. We find that the failure of PT2 in atoms is due to the two contributions of an orbital-dependent functional to excitation energies being inconsistent in the calculations. We also analyze the convergence of PT2 excitation energies with respect to the number of unoccupied orbitals.