Analysis and Application of Potential Energy Smoothing and Search Methods for Global Optimization

Global energy optimization of a molecular system is difficult due to the well-known “multiple minimum” problem. The rugged potential energy surface (PES) characteristic of multidimensional systems can be transformed reversibly using potential smoothing to generate a new surface that is easier to search for favorable configurations. The diffusion equation method (DEM) is one example of a potential smoothing algorithm. Potential smoothing as implemented in DEM is intuitively appealing and has certain appropriate statistical mechanical properties, but often fails to identify the global minimum even for relatively small problems. In the present paper, extensions to DEM capable of correcting its empirical behavior are systematically investigated. Two types of local search (LS) procedures are applied during the reversing schedule from the smooth deformed PES to the undeformed surface. Changes needed to generate smoothable versions of standard molecular mechanics force fields such as AMBER/OPLS and MM2 are also described. The resulting methods are applied in an attempt to determine the global energy minimum for a variety of systems in different coordinate representations. The problems studied include argon clusters, cycloheptadecane, capped polyalanine, and the docking of R-helices. Depending on the specific problem, potential smoothing and search (PSS) is performed in Cartesian, torsional, or rigid body space. For example, PSS finds a very low energy structure for cycloheptadecane with much greater efficiency than a search restricted to the undeformed potential surface. It is shown that potential smoothing is characterized by three salient features. As the level of smoothing is increased, unique minima merge into a common basin, crossings can occur in the relative energies of a pair of minima, and the spatial locations of minima are shifted due to the averaging effects of smoothing. Local search procedures improve the ability of smoothing methods to locate global minima because they facilitate the post facto correction of errors due to energy crossings that may have occurred at higher levels of smoothing. PSS methods should serve as useful tools for global energy optimization on a variety of difficult problems of practical interest.