Rules on the changes of approximate symmetry measures along reaction paths†

Abstract

A theorem is proven, describing the differences in the gradual emergence of different symmetry elements of non-symmetric molecular structures as the nuclear geometries change along reaction paths. The gradual establishment of individual symmetry elements is described in terms of the rates of convergence of symmetry deficiency measures along steepest descent paths leading to a critical point. This critical point may be a minimum or a saddle point of any index of the potential energy hypersurface, as long as the steepest descent path leads to it. If within a quadratic neighbourhood of the critical point the local internal coordinates are given in terms of the eigenvectors of the Hessian matrix, then the theorem asserts that, for any distorted nuclear arrangement displaced along a steepest descent path within the quadratic neighbourhood of a critical point, the symmetry element that is present along the eigenvector of the lowest positive eigenvalue of the Hessian is restored the fastest, at an exponential rate relative to the restoration rate of other distorted symmetry elements. Although it is only an apparent contrast, note that, this result may appear in contrast to the known convergence behaviour of internal coordinate values, since along the same eigenvector the actual internal coordinate value is restored the slowest, while converging to the critical point. The complete set of possible nuclear configurations of a molecule can never be a vector space, however, using the manifold theoretical approach consistent sets of local coordinate systems can be introduced. Accordingly, the theorem is first proven for a symmetry deficiency measure defined in terms of the metric of the reduced nuclear configuration space (manifold M of nuclear configurations), and then for a very general family of symmetry deficiency measures with bounds of finite degree polynomial dependence on nuclear configurations. Some implications of the role of the theorem in geometry optimizations are pointed out. † This paper is dedicated to Professor Michael A. Robb.