It is highly desirable for numerical approximations to stationary points for a potential energy landscape to lie in the corresponding quadratic convergence basin. 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 certification. Here, we apply Smale's α-theory to stationary points, providing a certification serving 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 α-theory can be used to certify all the known minima and transition states of Lennard-Jones LJNatomic clusters for N = 7, …, 14.
Received 08 March 2013Accepted 16 April 2013Published online 03 May 2013
D.M. was financially supported by the US Department of Energy under Contract No. DE-FG02-85ER40237. J.D.H. would like to thank the US National Science Foundation and Air Force Office of Scientific Research for their support through DMS-1262428 and FA8650-13-1-7317, respectively.