Underlying theory of a model for the Renner–Teller effect in tetra-atomic molecules: X²Π_u electronic state of C₂H₂⁺

In the present study, we prove the plausibility of a simple model for the Renner–Teller effect in tetra-atomic molecules with linear equilibrium geometry by ab initio calculations of the electronic energy surfaces and non-adiabatic matrix elements for the X²Π_u state of C₂H₂⁺. This phenomenon is considered as a combination of the usual Renner–Teller effect, appearing in triatomic species, and a kind of the Jahn–Teller effect, similar to the original one arising in highly symmetric molecules. Only four parameters (plus the spin–orbit constant, if the spin effects are taken into account), which can be extracted from ab initio calculations carried out at five appropriate (planar) molecular geometries, are sufficient for building up the Hamiltonian matrix whose diagonalization results in the complete low-energy (bending) vibronic spectrum. The main result of the present study is the proof that the diabatization scheme, hidden beneath the apparent simplicity of the model, can safely be carried out, at small-amplitude bending vibrations, without cumbersome computation of non-adiabatic matrix elements at large number of molecular geometries