It is highly desirable for a numerical approximation of a stationary pointfor a potential energy landscape to lie in the quadratic convergence basin ofthat stationary point. However, it is possible that an approximation may lieonly in the linear convergence basin, or even in a chaotic region, and hencenot converge to the actual stationary point when further optimization isattempted. Proving that a numerical approximation will quadratically convergeto the associated stationary point is termed certifying the numericalapproximation. We employ Smale's \alpha-theory to stationary points, providinga certification that serves as a mathematical proof that the numericalapproximation does indeed correspond to an actual stationary point, independentof the precision employed. As a practical example, employing recently developedcertification algorithms, we show how the \alpha-theory can be used to certifyall the known minima and transition states of Lennard-Jones LJ$_{N}$ atomicclusters for N = 7, ...,14.
展开▼
