Bifurcation of reaction pathways: the set of valley ridge inflection points of a simple three-dimensional potential energy surface

Abstract

This paper serves for the better understanding of the branching phenomenon of reaction paths of potential energy hypersurfaces in more than two dimensions. We apply the recently proposed reduced gradient following (RGF) method for the analysis of potential energy hypersurfaces having valley-ridge in¯ection (VRI) points. VRI points indicate the region of possible reaction path bifurcation. The relation between RGF and the so-called global Newton search for stationary points (Branin method) is shown. Using a 3D polynomial test surface, a whole 1D manifold of VRI points is obtained. Its relation to RGF curves, steepest descent and gradient extremals is discussed as well as the relation of the VRI manifold to bifurcation points of these curves.

Comment on “Exploring the potential energy landscape of the Thomson problem via Newton homotopies” [J. Chem. Phys. 142, 194113 (2015)]

“…This behavior is inherent to the Newton trajectory (NT) or RGF curves. 4,5,[15][16][17][18] Thus any curve of a Newton homotopy of Ref. 14 may be parameterised as an NT.…”

“…A singular NT crosses a valley-ridge inflection (VRI) point. 5 In the textbook of Jongen, Jonker, and Twilt, 39 we find the property of VRI points to build manifolds of dimension N − 2, if N is the dimension of the PES. Pieces of the manifold of VRI points were already often found on the PES of small molecules.…”

“…29 The index theorem for NTs 22 is coupled with the manifold character of the VRIs for Newton trajectories. 5 The VRI manifold forms, together with the singular NTs, a partitioning of the configuration space. The situation is similar to the Mezey-partitioning of the configuration space into catchment regions for steepest descent curves.…”

A Valley Following Method

“…However, the latter is not always the valley floor [18,19]. Newton trajectories [21,22] can be used only in certain special cases for the valley floor. The gradient extremal [3], see also [12], appeared to represent a suitable ''ansatz'' for a valley floor.…”

Finding reaction paths using the potential energy as reaction coordinate

“…16 The RGF is the new version of the distinguished coordinate method, that is widely used to locate transition states. [17][18][19][20][21] In the present study we attend to formulate the MEP or RP path, considered as IRC curve, as a function of the values of the PES. Based in this formulation we proposed techniques to locate this type of MEP in a PES.…”