  Typically, there is no guarantee that a numerical approximation obtained
using standard nonlinear equation solvers is indeed an actual solution, meaning
that it lies in the quadratic convergence basin. Instead, it may lie only in
the linear convergence basin, or even in a chaotic region, and hence not
converge to the corresponding stationary point when further optimization is
attempted. In some cases, these non-solutions could be misleading. Proving that
a numerical approximation will quadratically converge to a stationary point is
termed \textit{certification}. In this report, we provide details of how
Smale's $\alpha$-theory can be used to certify numerically obtained stationary
points of a potential energy landscape, providing a \textit{mathematical proof}
that the numerical approximation does indeed correspond to an actual stationary
point, independent of the precision employed.
