Resummation of divergent perturbation series: Application to the vibrational states of H2CO molecule.

Abstract

Large-order Rayleigh-Schrödinger perturbation theory (RSPT) is applied to the calculation of anharmonic vibrational energy levels of H2CO molecule. We use the model of harmonic oscillators perturbed by anharmonic terms of potential energy. Since the perturbation series typically diverge due to strong couplings, we apply the algebraic approximation technique because of its effectiveness shown earlier by Goodson and Sergeev [J. Chem. Phys. 110, 8205 (1999); ibid. 124, 094111 (2006)] and in our previous articles [A. D. Bykov et al. Opt. Spectrosc. 114, 396 (2013); ibid. 116, 598 (2014)]. To facilitate the resummation of terms contributing to perturbed states, when resonance mixing between states is especially strong and perturbation series diverge very quick, we used repartition of the Hamiltonian by shifting the normal mode frequencies. Energy levels obtained by algebraic approximants were compared with the results of variational calculation. It was found that for low energy states (up to ∼5000 cm(-1)), algebraic approximants gave accurate values of energy levels, which were in excellent agreement with the variational method. For highly excited states, strong and multiple resonances complicate series resummation, but a suitable change of normal mode frequencies allows one to reduce the resonance mixing and to get accurate energy levels. The theoretical background of the problem of RSPT series divergence is discussed along with its numerical analysis. For these purposes, the vibrational energy is considered as a function of a complex perturbation parameter. Layout and classification of its singularities allow us to model the asymptotic behavior of the perturbation series and prove the robustness of the algorithm.