Stationary points and dynamics in high-dimensional systems

Abstract

We present some new theoretical and computational results for the stationary points of bulk systems. First we demonstrate how the potential energy surface can be partitioned into catchment basins associated with every stationary point using a combination of Newton–Raphson and eigenvector-following techniques. Numerical results are presented for a 256-atom supercell representation of a binary Lennard-Jones system. We then derive analytical formulae for the number of stationary points as a function of both system size and the Hessian index, using a framework based upon weakly interacting subsystems. This analysis reveals a simple relation between the total number of stationary points, the number of local minima, and the number of transition states connected on average to each minimum. Finally we calculate two measures of localization for the displacements corresponding to Hessian eigenvectors in samples of stationary points obtained from the Newton–Raphson-based geometry optimization scheme. Systematic differences are found between the properties of eigenvectors corresponding to positive and negative Hessian eigenvalues, and localized character is most pronounced for stationary points with low values of the Hessian index.