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We derive the second-order approximation (PT2) to the ensemble correlationenergy functional by applying the G\"{o}rling-Levy perturbation theory on theensemble density-functional theory (EDFT). Its performance is checked bycalculating excitation energies with the direct ensemble correction method in1D model systems and 3D atoms using numerically exact Kohn-Sham orbitals andpotentials. Comparing with the exchange-only approximation, the inclusion ofthe ensemble PT2 correlation improves the excitation energies in 1D modelsystems in most cases, including double excitations and charge-transferexcitations. However, the excitation energies for atoms are generally worsewith PT2. We find that the failure of PT2 in atoms is due to the twocontributions of an orbital-dependent functional to excitation energies beinginconsistent in the calculations. We also analyze the convergence of PT2excitation energies with respect to the number of unoccupied orbitals.

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