Numerical search of discontinuities in self-consistent potential energy surfaces

Potential energy surfaces calculated with self-consistent mean-field methods are a very powerful tool, since their solutions are, in theory, global minima of the non-constrained subspace. However, this minimization leads to an incertitude concerning the saddle points, that can sometimes be no longer saddle points in larger constrained subspaces (fake saddle points), or can be missing on a trajectory (missing saddle points). These phenomena are the consequences of discontinuities of the self-consistent potential energy surfaces (SPESs). These discontinuities may have important consequences, since they can, for example, hide the real height of an energy barrier, and avoid any use of an SPES for further dynamical calculations, barrier penetrability estimations, or trajectory predictions. Discontinuities are not related to the quality of the production of an SPES, since even a perfectly converged SPES with an ideally fine mesh can be discontinuous. In this paper we explain what the discontinuities are, their consequences, and their origins. We then propose a numerical method to detect and identify discontinuities on a given SPES, and finally we discuss what the best ways are to transform a discontinuous SPES into a continuous one.