Four-channel semiclassical S-matrix for a Σ⁺-π-Σ^--type conical intersection in triatomic systems

The time-dependent semiclassical equations are solved in analytical form for the case of a straight-line classical trajectory which passes near a point of accidental degeneracy of Σ⁺, π and Σ^- states in a linear triatomic system. The configurational displacements from the point of intersection are assumed to be linear functions of time. The asymptotic analysis of the exact solution in the limits t → ±∞ yields the four-channel S-matrix, which gives all transition amplitudes (that is, transition probabilities and phases) of the problem. The model illustrates the importance of quantum interference effects in the case of a crossing of potential-energy surfaces of more than two electronic states. The results derived in this work may prove useful for trajectory-surface-hopping calculations in the case of a potential-surface crossing of Σ⁺, π and Σ^- electronic states.