Geometry optimisation on a hypersphere. Application to finding reaction paths from a conical intersection

A new method for performing constrained optimisation of critical points on the surface of an hypersphere using Newton-Raphson or rational function optimisation (RFO) steps is described. Through the parametrisation of a unitary matrix, it is possible to incorporate the constraint of the fixed distance from the centre of the hypersphere without using Lagrange multipliers. The method is applied to the problem of locating the initial relaxation directions from the vertex of a conical intersection and to the problem of following an intrinsic reaction coordinate along a ridge. An application to the [2 pi(s) + 2 pi(s)] ethylene-ethylene photochemical cycloaddition reaction path, which occurs via a rhomboid conical intersection, shows that this reaction occurs in a single step since only the reaction channel towards cyclobutane exists and any path to tetramethylene is avoided.