Certifying the Potential Energy Landscape

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.