Certifying the Potential Energy Landscape

Abstract

It is highly desirable for a numerical approximation of a stationary point for a potential energy landscape to lie in the quadratic convergence basin of that stationary point. However, it is possible that an approximation may lie only in the linear convergence basin, or even in a chaotic region, and hence not converge to the actual stationary point when further optimization is attempted. Proving that a numerical approximation will quadratically converge to the associated stationary point is termed certifying the numerical approximation. We employ Smale's α-theory to stationary points, providing a certification that serves as a mathematical proof that the numerical approximation does indeed correspond to an actual stationary point, independent of the precision employed. As a practical example, employing recently developed certification algorithms, we show how the α-theory can be used to certify all the known minima and transition states of Lennard-Jones LJ${}_N$ atomic clusters for N = 7, ...,14.

4 pages

References in corpus (4)

• Phase transitions and configuration space topology
• Characterizing the network topology of the energy landscapes of atomic clusters
• Enumerating Gribov copies on the lattice
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

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• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• Energy Landscape of the Finite-Size Mean-field 2-Spin Spherical Model and Topology Trivialization
• An Inversion-Relaxation Approach for Sampling Stationary Points of Spin Model Hamiltonians
• Certification and the Potential Energy Landscape
• Statistics of Stationary Points of Random Finite Polynomial Potentials