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 localisation for the displacements corresponding to Hessian eigenvectors in samples of stationary points obtained from the Newton-Raphson-based geometry optimisation scheme. Systematic differences are found between the properties of eigenvectors corresponding to positive and negative Hessian eigenvalues, and localised character is most pronounced for stationary points with low values of the Hessian index.

Reaction channels of the potential energy surface: application of Newton trajectories

“…Thus, most of the configuration space in a large chemical system must lie near the equipotential surface of the corresponding potential energy, cf. [31]. Or, if we think in terms of the singular 'walls', most of the 'space of all NTs' in S n21 will lie near the walls which partition the regular sets.…”

Characterizing Potential Surface Topographies through the Distribution of Saddles and Minima

“…In this sense, the focus here, being on relations between topography and dynamics, is complementary to the correlations that have been studied previously between different aspects of the topography of the potential landscape; see, for example, the study by Wales and Doye. 14 …”

Characterizing the network topology of the energy landscapes of atomic clusters

“…For example, the number of minima increases exponentially with system size 7-9 and the size scaling for higher-order saddle points has also been obtained. [10][11][12] It is also known that the distribution of minima should be a Gaussian function of the potential energy. 13,14 In this paper we seek to provide fundamental new insights into the structural organization, and particularly the connectivity, of an energy landscape.…”