Some Further Results for the Stationary Points and Dynamics of Supercooled Liquids

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.

arXiv comment16 pages, 2 figures

References in corpus (30)

• The relationship between fragility, configurational entropy and the potential energy landscape of glass forming liquids
• Inherent Structure Entropy of Supercooled Liquids
• Crossover to Potential Energy Landscape Dominated Dynamics in a Model Glass-forming Liquid
• Saddles in the energy landscape probed by supercooled liquids
• Energy Barriers and Activated Dynamics in a Supercooled Lennard-Jones Liquid
• Saddle Points and Dynamics of Lennard-Jones Clusters, Solids and Supercooled Liquids
• The potential energy landscape of a model glass former: thermodynamics, anharmonicities, and finite size effects
• Energy landscape of a Lennard-Jones liquid: Statistics of stationary points
• Phase space geometry and slow dynamics
• Hopping in a Supercooled Lennard-Jones Liquid: Metabasins, Waiting Time Distribution, and Diffusion
• Geometric approach to the dynamic glass transition
• An improved Monte Carlo method for direct calculation of the density of states
• The thermodynamical liquid-glass transition in a Lennard-Jones binary mixture
• Real space origin of temperature crossovers in supercooled liquids
• Metastable states as a key to the dynamics of supercooled liquids
• Thermodynamic and structural aspects of the potential energy surface of simulated water
• Fragile vs strong liquids: a saddles ruled scenario
• Activated mechanisms in amorphous silicon: an activation-relaxation-technique study
• Non-topographic description of inherent structure dynamics in glass formers
• Role of the unstable directions in the equilibrium and aging dynamics of supercooled liquids
• Quasi-saddles as relevant points of the potential energy surface in the dynamics of supercooled liquids
• General features of the energy landscape in Lennard-Jones like model liquids
• Finite-Size Effects in a Supercooled Liquid
• Topological Description of the Aging Dynamics in Simple Glasses
• On the origine of the Boson peak
• Entropy, Dynamics and Instantaneous Normal Modes in a Random Energy Model
• Inherent-Structure Dynamics and Diffusion in Liquids
• Potential Energy Landscape in Lennard-Jones binary mixture model
• Saddles and dynamics in a solvable mean-field model
• Off-equilibrium dynamics in the energy landscape of a simple model glass

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