Finding reaction paths using the potential energy as reaction coordinate

Abstract

The intrinsic reaction coordinate curve (IRC), normally proposed as a representation of a reaction path, is parametrized as a function of the potential energy rather than the arc-length. This change in the parametrization of the curve implies that the values of the energy of the potential energy surface points, where the IRC curve is located, play the role of reaction coordinate. We use Caratheodory's relation to derive in a rigorous manner the proposed parametrization of the IRC path. Since this Caratheodory's relation is the basis of the theory of calculus of variations, then this fact permits to reformulate the IRC model from this mathematical theory. In this mathematical theory, the character of the variational solution (either maximum or minimum) is given through the Weierstrass E-function. As proposed by Crehuet and Bofill [J. Chem. Phys. 122, 234105 (2005)], we use the minimization of the Weierstrass E-function, as a function of the potential energy, to locate an IRC path between two minima from an arbitrary curve on the potential energy surface, and then join these two minima. We also prove, from the analysis of the Weierstrass E-function, the mathematical bases for the algorithms proposed to locate the IRC path. The proposed algorithm is applied to a set of examples. Finally, the algorithm is used to locate a discontinuous, or broken, IRC path, namely, when the path connects two first order saddle points through a valley-ridged inflection point.

Implementation of an algorithm based on the Runge‐Kutta‐Fehlberg technique and the potential energy as a reaction coordinate to locate intrinsic reaction paths

“…25 This fact implies that the point of each sub-arc with the highest potential energy should be treated in a different way than the rest of the interior points of the sub-arc, since after this point the function v(t) decreases. Because we are looking for SD curves of the type IRC, this means that the point of the sub-arc will converge to a first order saddle point or close to it at the converged curve.…”

“…DV R?q (ext). [23][24][25] The inequality of the Weierstrass E-function, E ! 0, has the next geometrical interpretation.…”

“….,q C (v j21 ), q C (v ðminÞ j ) the second sub-arc, and so on. Finally, the splines technique 25,30 is used to obtain a resulting sequence of points such that the points are equally energy spaced. The splines interpolation expression is,…”

“…The construction of each curve belonging to the sequence of iterative curves is given in the way explained in Ref. 25. Every iterative curve joints the points R:…”

“…25 This selection permits to reduce the computational effort. The basis of the algorithm is the RungeKutta-Fehlberg technique.…”