Theory and practice of reaction dynamics on twohyphen;dimensional potential energy surfaces is investigated. Nonequilibrium multidimensional barrier crossing, occurring when the initial density is located near the ridgeline separating reactants and products, is treated by solving the timehyphen;dependent Smoluchowski equation as a function of diffusion anisotropy. For a locally separable potential, and slow relaxation in the perpendicular mode, the problem reduces to a onehyphen;dimensional Smoluchowski equation with a sink term. It may be further approximated as a firsthyphen;order reaction with a timehyphen;dependent rate coefficient. These approximations are compared with exact twohyphen;dimensional propagations on a potential surface representing CO binding to agr; heme. The intermediatehyphen;time powerhyphen;law decay of the survival probability is analyzed with the aid of the above approximations. The power also shows some kind of critical behavior near the isotropic diffusion limit, where these approximations are no longer valid. For fast relaxation, a nonmonotonic survival probability is observed. The long time decay of the survival probability is governed by the equilibrium rate coefficient. We calculate its anisotropy dependence and compare it with two asymptotic expansions for the lowest eigenvalue of the Smoluchowski operatorminus;for the onehyphen;dimensional sinkhyphen;Smoluchowski operator and the fully twohyphen;dimensional operator. The difference between the fast relaxation limit of these onehyphen; and twohyphen;dimensional equations may provide a quantitative explanation for previous problems in extrapolating to high temperatures using the Agmonndash;Hopfield model. The implications of these results to heme protein dynamics are discussed and new experiments are proposed.
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