Extended random phase approximation method for atomic excitation energies from correlated and variationally optimized second-order density matrices

Density matrix methods are typically ground state methods. They cannot describe excited states with the same symmetry as the ground state because they rely on energy minimization. The Random Phase Approximation (RPA) is a simple method to derive excitation energies from idempotent first-order density matrices, but the quality of the resulting excitation energies is poor. The quality of the excitation spectrum may be improved by extending the RPA to correlated states. Such an ‘extended RPA’ (ERPA) method depends only on the second-order density matrix (2DM). This work studies the main differences between the ERPA and the RPA – the influence of electron correlation, variational optimality, the ensemble nature of the density matrix and N-representability errors in the input 2DM – by applying the ERPA to exact 2DM’s and variationally optimized 2DM’s. Our findings are relevant for all methods similar to ERPA that determine excitation spectra from low-order density matrices. The inclusion of correlation makes it possible to describe the low-energy excitation spectra of the atoms He–Ne adequately, and the ERPA is thus a good starting point for further refinements, as higher-order excitations should be included to obtain chemical accuracy for many-electron systems. However, the ERPA fails for ensemble density matrices and requires a positive-definite double commutator matrix Ψ0ak†al,H,aj†aiΨ0 to guarantee that the excitation spectrum is real.