On a Method of Nonlinear Optimization for the Comparison of Spatial Structure of Molecules

Abstract

To compare the geometry of two or more geometric structures consisting of N ordered points, which can be considered as solids in three-dimensional space, a method based on the minimization of a certain comparison function is developed. This function is the sum of squared distances between pairs of elements of the two structures under comparison with the same indices. Distances change when changing the mutual orientation of the structures with all possible shifts and rotations of the structures as rigid bodies. The comparison function is minimized by superpositioning the centers of mass
of the two structures under comparison and minimizing this function with respect to Euler angles. The minimization of the comparison function with respect to Euler angles is carried out numerically by the Rosenbrock method. The comparison method of two geometric structures is used to solve problems in structural chemistry, that is, to compare two molecules with the same structural formula in one crystal.