2003/09/02 by David J. Wales, Jonathan P. K. Doye · 4 citations
Materials Science · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #Applied mathematics #Binary number #Eigenvalues and eigenvectors #Function (biology) #Hessian matrix #Material Dynamics and Properties #Mathematical analysis #Mathematics #Maxima and minima #Physics #Quantum mechanics #Spectroscopy and Quantum Chemical Studies #Stationary point #Stationary state #Statistical physics #cond-mat
paper · pdf · doi:10.1063/1.1625644
published as J. Chem. Phys. 119, 12409-12416 (2003) · 16 pages, 2 figures
arxiv created 2003/09/02 · openalex publication_date 2003/12/04 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.