Gradient extremals and valley floor bifurcations on potential energy surfaces

Abstract

Gradient extremals are curves in configuration space defined by the condition that the gradient of the potential energy is an eigenvector of the Hessian matrix. Solutions of a corresponding equation go along a valley floor or along a crest of a ridge, if the norm of the gradient is a minimum, and along a cirque or a cliff or a flank of one of the two if the gradient norm is a maximum. Properties of gradient extremals are discussed for simple 2D model surfaces including the problem of valley bifurcations.

Analyses of bifurcation of reaction pathways on a global reaction route map: A case study of gold cluster Au5

“…[9][10][11][12][13][14]18,[22][23][24][25] On the other hand, when the vibrational mode with VRT is a totally symmetric mode, the reaction pathway bifurcates asymmetrically, and the products with different types are expected from the bifurcating reaction pathways. In this case, however, it is difficult to notice an occurrence of totally symmetric VRT since the IRC should terminate at one of product minima via a ridge-valley transition (RVT) point.…”

A comment to the nudged elastic band method

“…28 The solution images of task (4) may be pure GE points where the criterion (4) is a local criterion. If the GE has turning points, or bifurcations, or avoided crossings, 29 thus, if it is not always a valley-GE throughout, inclusion of parallel forces with springs may result in an averaged curve between different branches of the GE, or a bridge over the gap, and we will obtain a well adapted "MEP", cf. The SD case can overbridge the local character of operator (4), because, only if all tangents for all images are parallel to the current gradient, then we have an IRC, and then the operator of eq.…”

Trifurcation of the reaction pathway

“…If ridge character continues to the terminal point of IRC, this terminal is not a minimum but a first-order saddle point which connects two symmetrically equivalent product minima with the lower symmetry. For this type of VRI, there have been a number of theoretical studies, which include the detailed analysis of the PES near IRC, [5][6][7][8] the second-order Jahn-Teller analysis, 9 a formulation of the bifurcating reaction path, 10 an investigation of the isotope effects on the bifurcating reaction path, 11,12 and applications to organic chemical reactions. 13 Quantum wavepacket simulations have also been performed for this type of the bifurcating reaction path.…”