Multivalued property of Rayleigh–Schrödinger perturbation series for vibrational energy levels of molecules

Abstract

We have calculated 200 lower vibrational energy levels of the formaldehyde molecule using ab initio quartic force field, high-order Rayleigh–Schrödinger perturbation theory (up to 200th order) and algebraic Hermit–Padé approximants (up to sixth degree) to sum up divergent perturbation series, corresponding to each vibrational state. Both energy levels and algebraic Hermit–Padé approximants are, in fact, multivalued functions and consist of several branches. We find that branches of the approximant constructed using coefficients of the perturbation series for a particular vibrational state can reproduce not only the energy of this state, but also of other states which are close in energy to the considered one and are coupled with them by anharmonic resonances. The multivalued property of algebraic approximants has been previously demonstrated by Jordan in the case of two-state avoided crossings of the potential energy curves of diatomic molecules (Jordan 1975 Int. J. Quantum Chem. 9 325) and by Fernandez and Diaz in the case of one-dimensional periodic eigenvalue problems (Fernandéz and Diaz 2001 Eur. Phys. J. D 15 41). This paper extends the study of this nontrivial property of perturbation theory to the vibrational states of polyatomic molecules, a problem with many dimensions. Additionally, we have found that this property is useful in some special cases when perturbation series diverges so fast that most high-order approximants do not reproduce the correct value of the energy with the required accuracy.