Path Optimization with Application to Tunneling

A method is presented for optimizing paths on high dimensional surfaces, i.e. scalar functions of many variables. The method involves optimizing simultaneously the end points and several intermediate points along the path and thus lends itself well to parallel computing. This is an extension of the nudged elastic band method (NEB) which is frequently used to find minimum energy paths on energy surfaces of atomic scale systems, often with several thousand variables. The method is illustrated using 2-dimensional systems and various choices of the object function, in particular (1) path length, (2) iso-contour and (3) quantum mechanical tunneling rate. The use of the tunneling paths to estimate tunneling rates within the instanton approximation is also sketched and illustrated with an application to associative desorption of hydrogen molecule from a copper surface, a system involving several hundred degrees of freedom.

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