Gradient Extremals and Their Relation to the Minimum Energy Path

Abstract

The concept of the reaction path (RP) of potential energy surfaces (PES) has gained increasing importance in theoretical chemistry [1,2]. Qualitatively, the RP is a curve in the configuration space of the atoms forming the chemical system which connects two minimizers of the PES along points of minimal energy in comparison to neighbouring points. The energy profile over the reaction path should be a “valley floor” leading via a point of highest energy, the saddle po i nt of index 1 of the PES. This point corresponds to the transition structure of the “transition state theory” . The fundamental problem in handling PES is the problem of dimensionality. Molecules with a number of atoms more than N=4 force an overwhelming number of net points in the dimension n=3N-6. The RP concept is a promising way out. It requires finding an algorithm for chemically reasonable one-dimensional curves of the PES determinable by differential properties of the PES — gradient and Hessian matrix — without knowledge of the whole, or of large parts of the PES. Slope and curvature of E(x)=E(x ¹, ..., x ⁿ) can be calculated from the gradient vector, g(x)=∇(x), and from the Hessian matrix (second derivatives, H(x)=∇∇^TE(x), of the PES), respectively. [ We shall denote geometrical vectors in the configuration space and column matrices of their Cartesian coordinates by boldface lower-case letters, second order tensors and square matrices of their components by caligraphic upper-case letters. Scalars are often denoted by greek letters.]