Adiabatic Perturbation Theory in Molecular Spectroscopy

Abstract

The Born–Oppenheimer method or adiabatic perturbation theory has long provided a formal basis for separation of the electronic and nuclear motions in molecules. According to it solutions of the molecular Schrödinger equation can be obtained using expansions in power series in the parameter κ = (m/M)1/4 where m is the electronic mass and M is the average mass of the nuclei; the zeroth-order term in the eigenfunction expansion was shown to be a product of the electronic, vibrational, rotational, and translational eigenfunctions. However, this zeroth-order separation of motions was established only qualitatively because the Hamiltonian expansion was not obtained in explicit form. Therefore the adiabatic perturbation theory (to be referred to as APT) appeared to be inconvenient in applications, and practical investigations in molecular spectroscopy were based on simplified models and approximations rather than on the APT. The present review attempts to clarify and remove difficulties impeding wider applications of the APT and to give examples of its practical use. For this purpose a molecule-fixed moving coordinate system (to be referred to as mcs) is introduced by means of the Eckart conditions, and the Hamiltonian is transformed to the electronic and nuclear coordinates defined with the help of the mcs. For the transformed Hamiltonian the Born–Oppenheimer expansion is written explicitly and order-by-order calculations of the APT are carried out up to the fourth-order equation for normal and linear molecules. The results are used to discuss the stability of normal molecules and the Renner effect in linear molecules. The stability is considered for the cases of degenerate and near degenerate electronic levels. Lowering of vibrational frequencies caused by the near degeneracy is also studied, and the results are applied to infrared spectra of hexafluorides. The Renner equations for the bending vibrations of linear molecules are derived from the second-order equation of the APT. The four-atomic Renner equations are solved approximately using expansions in power series in the Renner parameters; the results are applied to vibronic spectra of C2H2 and C2D2 molecules. The influence of the Renner effect on the rotational spectra of triatomic molecules is considered with the help of the sixth-order energy correction of the APT. Higher order APT calculations are discussed briefly in the last section.