Exploring Model Energy and Geometry Surfaces Using Sum of Squares Decompositions.

The difficulty in exploring potential energy surfaces, which are nonconvex, stems from the presence of many local minima, typically separated by high barriers and often disconnected in configurational space. We obtain the global minimum on model potential energy surfaces without sampling any minima a priori. Instead, a different problem is derived, which is convex and hence easy to solve, but which is guaranteed to either have the same solution or to be a lower bound to the true solution. A systematic way for improving the latter solutions is also given. Because many nonconvex problems are projections of higher dimensional convex problems, Parrilo has recently shown that by obtaining a sum of squares decomposition of the original problem, which can be subsequently transformed to a semidefinite program, a large class of nonconvex problems can be solved efficiently. The semidefinite duality formulation also provides a proof that the global minimum of the energy surface has either been found exactly or has been bounded from below. It additionally provides physical insight into the problem through a geometric interpretation. The sum of squares polynomial representation of the potential energy surface may further reveal information about the nature of the potential energy surface. We demonstrate the applicability of this approach to low-dimensional potential energy landscapes and discuss its merits and shortcomings. We further show how to apply it to geometric problems by obtaining the exact distance of closest approach of anisotropic particles. Efficient molecular dynamics simulations of mixtures of ellipsoids are illustrated.