The two-state linear curve crossing problems revisited. III. Analytical approximations for Stokes constant and scattering matrix: Nonadiabatic tunneling case

Based on the exact solution of the linear curve crossing problems reported in the first paper of this series, approximate analytical solution is discussed here for the opposite sign of slopes of the two diabatic potentials (nonadiabatic tunneling case). Two new compact analytical formulas for reduced scattering matrix are derived for ‖b2‖≥1, where b2 represents the effective collision energy. The whole range of the two parameters a2 (effective coupling strength) and b2 is divided into five regions, in each one of which the best recommended formulas are proposed. This analysis provides a complete picture of the nonadiabatic tunneling problem, for the first time. The new formulas proposed here are simple and explicit functions of the two parameters, and thus useful for practical applications. In the case of ‖b2‖≤1, certain fitting formulas are proposed for the Stokes constant on the basis of a comparison with the exactly solvable differential equation with special quartic polynomial as coefficient.