Semiclassical calculation of cumulative reaction probabilities

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Abstract
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[ε̂ r ⋅Ĝ*(E)⋅ε̂ p ⋅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.