Perturbative Handling of the Renner-Teller Effect and Spin-Orbit Coupling in Π Electronic States of Triatomic and Tetra-atomic Molecules

Second-order perturbative formulae for handling the Renner-Teller effect combined with the spin-orbit coupling in Π electronic states of triatomic and symmetric (ABBA-type) tetra-atomic molecules with linear equilibrium geometry are derived. Two schemes for partition of the model Hamiltonian are employed: In the first the spin-orbit coupling term is treated as a perturbation, in the second it is included in the zeroth-order Hamiltonian. It is demonstrated that both approaches lead to the same results when the spin-orbit coupling constant is small compared to the bending frequency, but much larger than the splitting of potential surfaces upon bending. The perturbative formulae derived for tetra-atomic molecules are used to compute the spectrum of the X²Π_u state of the acetylene ion, employing the parameters obtained in ab initio calculations. The results are compared with those generated in corresponding variational computations.