  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.
