Convergence property of multireference many-body perturbation theory analyzed by the use of a norm of effective Hamiltonian

It is proposed to use a norm of a nth order effective Hamiltonian, for analyzing the convergence property of the multireference many-body perturbation theory (MR-MBPT). The utilization of the norm allows us to employ only (1) a single number for all the states that we are interested in, and (2) values which decreases from the positive side to zero as the order n of the perturbation increases. This characteristic features are in contrast to those in the usually used scheme where several numbers, namely, the eigenvalues of the target states, should be used and they may oscillate around exact eigenvalues. The present method has been applied to MR-MBPT calculations of the (H₂)₂, CH₂, and LiH molecules based on the multireference versions of Rayleigh-Schrödinger PT, Kirtman-Certain-Hirschfelder PT, and the canonical Van Vleck PT; and following features are found: (1) the above three versions of the perturbation theories have essentially the same convergence property judged from the lowering of the norm; (2) the lower order truncation of the perturbation series gives reasonable solutions; (3) the norm decreases irrespective of the perturbation expansion being convergent or divergent for the first several orders (up to about the sixth order).