An empirical, variational Method of Approach to
unsymmetric Valley-Ridge Inflection Points

by Wolfgang Quapp   and   Benjamin Schmidt

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Abstract

Valley-ridge inflection points (VRIs) emerge on a potential energy surface
of a chemical reaction if the reaction pathway bifurcates.
The valley of the reaction path branches into two valleys, and a ridge in
between.  It can happen in uphill, or in downhill direction.
Newton trajectories (NT) are curves for the description of the reaction
path.
They are curves where at every point the gradient of the potential energy
surface  points into the same direction.
Singular Newton trajectories are a special case: they bifurcate
at VRI points. To find a singular Newton trajectory is quasi equivalent with
the determination of the corresponding VRI point where this NT bifurcates.

Often the bifurcation of the reaction path is governed by a symmetry of the
problem. Then the symmetry axis is usually the first branch of the singular
NT, and so its determination is easy. In case of an unsymmetric
branching,  however, such a guiding line is missing.
We name the place of such a bifurcation a skew VRI. We propose a variational
calculation of the singular NT through the VRI of interest by an empirical,
iterative method. Before, the variational theory of possible reaction
pathways is developed, and applied to the intrinsic reaction coordinate (IRC),
as well as to NTs. We have to employ the theory of NTs with its many facets,
we use especially the Branin equation.
The developed method is applied to the calculation of VRI points on the
potential energy surface of HCN, and to a VRI point of alanine dipeptide being
adjacent to the C5 minimum.