Computing mountain passes and transition states

Abstract.The mountain-pass theorem guarantees the existence of a critical point on a path that connects two points separated by a sufficiently high barrier. We propose the elastic string algorithm for computing mountain passes in finite-dimensional problems and analyze the convergence properties and numerical performance of this algorithm for benchmark problems in chemistry and discretizations of infinite-dimensional variational problems. We show that any limit point of the elastic string algorithm is a path that crosses a critical point at which the Hessian matrix is not positive definite.

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