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An empirical, variational method of approach to unsymmetric valley-ridge inflection points


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.

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We thank the referees for suggestions and comments.

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Correspondence to Wolfgang Quapp.

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Quapp, W., Schmidt, B. An empirical, variational method of approach to unsymmetric valley-ridge inflection points. Theor Chem Acc 128, 47–61 (2011).

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  • Potential energy surface
  • Variation of reaction pathways
  • Singular Newton trajectory
  • Skew valley-ridge inflection point