Practical approximation of the non-adiabatic coupling terms for same-symmetry interstate crossings by using adiabatic potential energies only

Abstract

A very simple equation, F_ij^App=[(∂²(V_i^a-V_j^a)/∂Q²)/(V_i^a-V_j^a)]^1/2/2, giving a reliable magnitude of non-adiabatic coupling terms (NACTs, F_ij's) based on adiabatic potential energies only (V_i^a and V_j^a) was discovered, and its reliability was tested for several prototypes of same-symmetry interstate crossings in LiF, C₂, NH₃Cl, and C₆H₅SH molecules. Our theoretical derivation starts from the analysis of the relationship between the Lorentzian dependence of NACTs along a diabatization coordinate and the well-established linear vibronic coupling scheme. This analysis results in a very simple equation, α=2κ/Δ_c, enabling the evaluation of the Lorentz function α parameter in terms of the coupling constant κ and the energy gap Δ_c (Δ_c=|V_i^a-V_j^a|_Qc ) between adiabatic states at the crossing point Q_C. Subsequently, it was shown that Q_C corresponds to the point where F_ij^App exhibit maximum values if we set the coupling parameter as κ=[(V_i^a-V_j^a)⋅(∂²(V_i^a-V_j^a)/∂Q²)]_Qc^1/2/2. Finally, we conjectured that this relation could give reasonable values of NACTs not only at the crossing point but also at other geometries near Q_C. In this final approximation, the pre-defined crossing point Q_C is not required. The results of our test demonstrate that the approximation works much better than initially expected. The present new method does not depend on the selection of an ab initio method for adiabatic electronic states but is currently limited to local non-adiabatic regions where only two electronic states are dominantly involved within a nuclear degree of freedom.