Two new algorithms are presented for determining the minimum energy reaction path (MEP) on the reduced potential energy surface (RPES) starting with only the reactant. These approaches are based on concepts from the fast marching method (FMM), which expands points outward as a wavefront on a multidimensional grid from the reactant until the product is reached. The MEP is then traced backward to the reactant. Since the number of possible grid points that must be considered grows exponentially with increasing dimensionality of the RPES, interpolation is important for maintaining manageable computational costs. In this work, we use Shepard interpolation, which we have modified to resolve problems in overfitting. In contrast to FMM, which accurately locates the MEP, the new algorithms focus on locating the single rate-limiting transition state and provide only a rough estimate of the MEP. They do this by mapping out the RPES on a coarse grid and then refining a least action path on a finer grid. This is done so that the majority of the interpolation is done on the finer grid, which minimizes the amount of extrapolation inherent in an outward searching algorithm. The first method scans the entire PES before iteratively locating the transition state (TS) for the MEP on the lower bound estimate of the fine PES. The second method explores the coarse grid in a similar manner to FMM and then iteratively locates the rate-limiting TS in the same manner as the first method. Both methods are shown to be capable of rapidly obtaining (in less than 30 constrained optimization cycles) an approximation to the MEP and the rate limiting TS for three example systems: the 4-well potential, the molecule N-hydroxymethyl-methylnitrosaminee (HMMN), and a cluster model of DNA-uracil glycosylase.
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