Collect. Czech. Chem. Commun. 1998, 63, 1368-1380

Search for the Most Stable Structures on Potential Energy Surfaces

Lucjan Piela

Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland


Smoothing techniques for global optimization in search for the most stable structures (clusters or conformers) have been a novel possibility for the last decade. The techniques turned out to be related to a variety of fundamental laws: Fick's diffusion equation, time-dependent and time-independent Schrodinger equations, Smoluchowski dynamics equation, Bloch equation of canonical ensemble evolution with temperature, Gibbs free-energy principle. The progress indicator of global optimization in those methods takes different physical meanings: time, imaginary time, Planck constant, or the inverse absolute temperature. Despite this large spectrum of physical phenomena, the resulting global optimization procedures have a remarkable common feature. In the case of the Gaussian Ansatz for the wave function or density distribution, the underlying differential equations of motion for the Gaussian position and width are similar for all these phenomena. In all techniques the smoothed potential energy function plays a central role rather than the potential energy function itself. The smoothed potential results from a Gaussian convolution or filtering out high frequency Fourier components of the original potential energy function. During the minimization, the Gaussian position moves according to the negative gradient of the smoothed potential energy function. The Gaussian width is position dependent through the curvature of the potential energy function, and evolves according to the following rule. For sufficiently positive curvatures (close to minima of the smoothed potential) the width decreases, thus leading to a smoothed potential approaching the original potential energy function, while for negative curvatures (close to maxima) the width increases leading eventually to the disappearance of humps of the original potential energy function. This allows for crossing barriers separating the energy basins. Some methods result in an additional term, which increases the width, when the potential becomes flat. This may be described as a feature allowing hunting for distant minima.

Keywords: Global optimization; Energy minimum; Stable structure; Fick's diffusion equation; Schrödinger equation; Bloch equation; Gibbs free-energy principle.