Certifying the Potential Energy Landscape

TL;DR

This paper introduces a method using Smale's -theory to certify that numerical approximations of stationary points in potential energy landscapes are mathematically guaranteed to converge quadratically to actual stationary points, enhancing reliability in computational chemistry.

Contribution

It applies Smale's -theory to certify stationary points in potential energy landscapes, providing a rigorous proof of convergence for numerical approximations.

Findings

Successfully certifies all known minima and transition states for Lennard-Jones clusters N=7 to 14.

Demonstrates the practical use of certification algorithms in computational chemistry.

Ensures numerical approximations are mathematically verified to correspond to true stationary points.

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 \alpha-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 \alpha-theory can be used to certify all the known minima and transition states of Lennard-Jones LJ$_{N}$ atomic clusters for N = 7, ...,14.