Approximations of density matrices in N-electron valence state second-order perturbation theory (NEVPT2). II. The full rank NEVPT2 (FR-NEVPT2) formulation.

In Paper I, the performances of pre-screening (PS), extended PS (EPS), and cumulant (CU) approximations to the fourth-order density matrix were examined in the context of second-order N-electron valence state perturbation theory (NEVPT2). It has been found that the CU, PS, and even EPS approximations with loose thresholds may introduce intruder states. In the present work, the origin of these "false intruder" states introduced by approximated density matrices is discussed. Canonical NEVPT2 implementations employ a rank reduction trick. By analyzing its residual error, we find that the omission of the rank reduction leads to a more stable multireference perturbation theory for incomplete active space reference wave functions. Such a full rank (FR)-NEVPT2 formulation is equivalent to the conventional NEVPT2 method for the complete active space self-consistent field/complete active space configuration interaction reference wave function. A major drawback of the FR-NEVPT2 formulation is the necessity of the fifth-order density matrix. To avoid the construction of the high-order density matrices, the combination of the FR-NEVPT2 with the CU approximation is studied. However, we find that the CU approximation remains problematic as it still introduces intruder states. The question of how to robustly and efficiently perform internally contracted multireference perturbation theories with approximate densities remains a challenging field of investigation.