Two-channel semiclassical S-matrix for the E × E Jahn-Teller problem including spin-orbit coupling

An exact analytical solution of the time-dependent semiclassical equations is obtained for the relativistic E × E Jahn-Teller problem, which includes linear vibronic coupling and spin-orbit splitting of the degenerate electronic state in a molecular system with three-fold symmetry axis. The configurational displacements from the point of intersection are assumed to be linear functions of time. The analytical solution of the nonstationary problem is obtained with the use of the ordered exponential operator method. The asymptotic analysis of the exact solution in the limits t → ± ∞ yields the two-channel S-matrix, which gives the transition amplitude of the problem. The results derived in this work reveal explicitly the dependence of the nonadiabatic transition amplitude on the Jahn-Teller coupling parameter and the spin-orbit splitting.