Estimating the Thermochemical Energies of MECP

In spin-forbidden reactions, the minimum energy crossing point (MECP) is a key structure besides stationary points. When the MECP corresponds to a new transition state, however, the reaction barrier and thermochemical energies derived from vibrational frequencies cannot be strictly defined. This study assesses the accuracy of several approximations for estimating the thermochemical energies of MECP from Hessian matrices, including the projected effective Hessian method reported in the literature, as well as additional approaches proposed here. Among them, a simple method, MSHg, based on a two-state spin-mixing model, exhibits the highest accuracy when spin-orbit coupling is not strong, roughly for elements up to xenon. Nevertheless, for spin-forbidden reactions involving 5d or heavier atoms, the MECP structure may deviate significantly from the genuine transition state, and consequently, none of these approximations can achieve thermochemical energy errors within 1.0 kcal/mol at standard temperature and pressure; in practice, the errors may reach 2.0-3.0 kcal/mol or higher, highlighting the inherent limitations of MECP calculations.