Discrete spatial symmetries: Redundant coordinates and search for stationary points in reduced spaces

Abstract

AbstractGiven the invariance of an N‐body system under discrete operations of reflection, inversion, a rotation by 2π/n, and the corresponding relations among the derivatives of energy, we have constructed through an invertible transformation a set of active and redundant coordinates. Movement along the active coordinates preserves all symmetry relations. We show that algorithms for locating stationary points or for calculating reaction paths are exactly separable in these active and redundant coordinates. We further show that this formalism is equally applicable when equations of constraints among coordinates are specified for the movement of particles. This includes geometrical constraints on bond lengths, angles, substituent group internal rotations, etc. This formalism enhances the efficiency since (laborious) cartesian derivatives need to be calculated only for the active variables and that the problem is reduced in term of m(⩽3N) variables. We apply this procedure to obtain the equilibrium geometry of H2O molecule within the subspace of C2v symmetry configurations ab initio derivatives.