Convergence behavior of multireference perturbation theory: Forced degeneracy and optimization partitioning applied to the beryllium atom

High-order multireference perturbation theory is applied to the S-1 states of the beryllium atom using a reference (model) space composed of the \1s(2)2s(2)] and the \1s(2)2p(2)] configuration-state functions (CSF's), a system that is known to yield diver ent expansions using Moller-Plesset and Epstein-Nesbet partitioning a methods. Computations of the eigenvalues are made through 40th order using forced degeneracy (FD) partitioning and the recently introduced optimization (OPT) partitioning. The former forces the 2s and 2p orbitals to be degenerate in zeroth order, while the latter chooses optimal zeroth-order energies of the (few) most important states. Our methodology employs simple models for understanding and suggesting remedies for unsuitable choices of reference spaces and partitioning methods. By examining a two-state model composed of only the \1s(2)2p(2)] and \1s(2)2s3s] states of the beryllium atom, it is demonstrated that the full computation with 1323 CSF's can converge only if the zeroth-order energy of the \1s(2)2s3s] Rydberg state from the orthogonal space lies below the zeroth-order energy of the \1s(2)2p(2)] CSF from the reference space. Thus convergence in this case requires a zeroth-order spectral overlap between the orthogonal and reference spaces. The FD partitioning is not capable of generating this type of spectral overlap and thus yields a divergent expansion. However, the expansion is actually asymptotically convergent, with divergent behavior not displayed until the 11th order because the \1s(2)2s3s] Rydberg state is only weakly coupled with the \1s(2)2p(2)] CSF and because these states are energetically well separated in zeroth order. The OPT partitioning chooses the correct zeroth-order energy ordering and thus yields a convergent expansion that is also very accurate in low orders compared to the exact solution within the basis.