Application of multiresolution analyses to electron density maps of small molecules: Critical point representations

In chemistry, the basic molecular structure information is represented in terms of molecular graphs established from X-ray diffraction experiments or mechanistic calculations. From that information, a 3D molecular property that is relevant to the study of molecular shape and interactions, i.e., the electron density (ED) distribution, can be calculated using quantum mechanical approaches. We show how working at various resolution levels may help in the identification of patterns useful to the comparison of different molecular structures.
In the present work, three methods are used in order to obtain ED images at various resolution levels. (a) In crystallography, the resolution of an ED map is defined by the ratio sin/(2 is the angle between the diffracted X-ray and the primary beam of wavelength ). Theory allows to calculate a promolecular ED map from known atomic coordinates as the Fourier transform of a set of calculated structure factors selected according to the resolution level [1]. (b) Another fast approach that is useful to obtain a promolecular ED function consists in a summation over 1s atomic Gaussian functions fitted over atomic RHF-LCAO results [2]. Smoothed images are then obtained using an analytical method derived from the multiple minima problem solving [3]. (c) The calculation of ED images at various resolution levels can be achieved using wavelet analysis, a mathematical tool in smoothing/denoising signals using convolution products between the original fonction and a low-pass filter [4]. This approach is applied to charge distributions of small molecules obtained from molecular MO-LCAO calculations.
The ED calculation stage is followed by the application of a topological analysis approach [5] in order to determine a critical point (point where the gradient of the density equals zero: local density maximum, saddle point) representation which is easily manipulated for structural interpretation and analysis. Graphs of critical points are used as representations of molecular ED distribution functions. Comparisons of graphs are carried out using a Monte Carlo/Simulated Annealing program. Multiple alignment results are obtained with programs which look for best matchings between critical points of a reference molecule and critical points of the other compounds.

[1] XTAL 3.0 User's Manual, S. R. Hall and J. M. Stewart (Eds.), Universities of Western Australia and Maryland (1990)
[2] L. Amat, R. Carbo-Dorca, J. Comput. Chem. 18, 2023 (1997)
[3] J. Kostrowicki et al., J. Phys. Chem. 95, 4113 (1991)
[4] J.-L. Stark et al., Image Processing and Data Analysis - The Multiscale Approach, Cambridge, University Press, Cambridge, UK, 1997.
[5] C. K. Johnson, Proc. Am. Crystallogr. Assoc. Meet., Asilomar CA (USA), 1977: abstract JQ6