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Eight new algorithms for reaction path following are presented, ranging from third order to sixth order. Like the second-order algorithm [J. Chem. Phys. 90, 21254 (1989)] these are implicit methods, i.e., they rely on the tangent (and in some cases the curvature) at the endpoint of the step. The tangent (and the curvature, if needed) are obtained by a constrained optimization using only the gradient. At most, only one Hessian calculation is needed per step along the path. The various methods are applied to the Müller-Brown surface and to a new surface whose reaction path is known analytically to test their ability to follow the reaction path and to reproduce the curvature along the path.
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