Semiclassical calculation of cumulative reaction probabilities

It is shown how the rigorous quantum mechanical expression for the cumulative reaction probability (CRP) obtained by Seideman and Miller [J. Chem. Phys. 96, 4412; 97, 2499 (1992)], N(E)=4 tr[ε̂r⋅Ĝ*(E)⋅ε̂p⋅Ĝ(E)], which has been the basis for quantum calculations of the CRP for simple chemical reactions, can also be utilized with a semiclassical approximation for the Green’s function, Ĝ(E)≡(E+iε̂−Ĥ)−1=(iℏ)−1∫∞0 exp(iEt/ℏ)exp(−i(Ĥ−iε̂)t/ℏ). Specifically, a modified Filinov transformation of an initial value representation of the semiclassical propagator has been used to approximate the Green’s function. Numerical application of this trajectory-based semiclassical approximation to a simple one-dimensional (barrier transmission) test problem shows the approach to be an accurate description of the reaction probability, even some ways into the tunneling regime.