Diffusion Equation and Distance Scaling Methods of Global Optimization: Applications to Crystal Structure Prediction

Abstract

Two methods of global minimization, the diffusion equation method and the distance scaling method, are applied to predict the crystal structures of the hexasulfur and benzene molecules. No knowledge about the systems other than the geometry of the molecules and the pairwise potentials is assumed; i.e., no assumptions are made about the space groups, cell dimensions, or number of molecules in the unit cell. Both methods are based on smoothing transformations of the original potential energy surface, which remove all insignificant local minima; the surviving minima are traced back to the original potential energy surface during the so-called reversing procedure, in which the transformations are gradually removed. The crystal structures, known from experiment, were predicted correctly. To verify the power of the methods, the problem of global minimization of the potential energy of crystals of both molecules was intentionally increased considerably in complexity: viz., the numbers of molecules in the unit cell were doubled (from three to six in the case of hexasulfur and from four to eight in the case of benzene), and the search for the global minimum was repeated; the method again located the global minimum for each molecule. Additionally, local minimizations starting from the lowest-energy structures were carried out with a pressure term included, leading to the observed high-pressure structure of benzene.