Quantum mechanical microcanonical rate constants from direct calculations of the Green’s function for reactive scattering

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Abstract
It has been shown previously [Miller, Schwartz, and Tromp, J. Chem. Phys. 79, 4889 (1983)] that the cumulative reaction probability, N(E), is given by a quantal trace, N(E)=2π2ℏ2 tr[Fδ(E−H) Fδ E−H)], where F is a symmetrized flux operator for flux through a surface dividing reactants from products, and δ(E−H) is the density operator related to the full Green’s operator, G+(E), by δ(E−H)=−Im G+(E)/π. Discretizing the coordinate space representation of the Schrödinger equation for the Green’s function leads to a set of linear matrix equations, and it is shown that these can be solved by an efficient recursive technique requiring little computer storage. Several simplifications are possible in this application because the Green’s function is only required near the dividing surface. A complete procedure is outlined which includes the exact solution of the recursion equations outside the interaction region. Two other approaches are explored: a discretized complex coordinate technique and the coordinate space solution of the Lippmann–Schwinger equation for the Green’s function. Results and an analysis of the numerical behavior of these procedures are given for model one-dimensional systems. The essential features of the extension of the recursive method to multidimensional systems are given.