Optimal Site Charge Models for Molecular Electrostatic Potentials

Abstract

This paper presents a comprehensive methodology for the fitting of site charge models to molecular electrostatic fields computed using quantum mechanical calculations. Charges may be placed both at atomic positions and at “satellite” positions associated with selected atoms. The positions of these satellites is also optimised. Molecular symmetry is taken into account for the determination of physically meaningful charge magnitudes and satellite positions. Maximum tolerances may be imposed on the deviations of the total charge, dipole moment and quadrupole computed by the site charge model from the corresponding quantum mechanical values.

^†For simplicity of notation, we neglect the constant 1/4 \mgreek{p} \epsilon _{0}.

The fitting is achieved via the solution of a nonlinear least squares problem that incorporates specified upper bounds on the statistical error of the charge magnitudes; this automatically excludes satellite charge positions that lead to ill-conditioned and inaccurate charge estimates. The overall optimisation problem is non-convex, involving multiple local minima. Molecular symmetry is exploited to derive an exact reformulation of the original problem that involves a reduced number of optimisation decision variables. A multi-level single-linkage (MLSL) stochastic minimization algorithm, coupled with a quadratic programming local minimisation algorithm and analytical derivatives for both objective function and constraints, is used for the reliable determination of globally optimal solutions of the reformulated problem.

A number of examples illustrating the applicability of the methodology are presented. The use of the optimal site charge models in the area of ab initio crystal structure prediction is briefly discussed.

Keywords:

• Site charges
• Electrostatic potentials
• Global optimization
• Confidence intervals
• Cyclobutane

Acknowledgements

The authors wish to thank Professor D. N. Theodorou of the National Technical University of Athens for useful discussions during the course of this work. The financial support of the Mitsubishi Chemical Corporation and the United Kingdom's Engineering and Physical Sciences Research Council (EPSRC) under Platform Grant GR/N08636 is gratefully acknowledged.

Notes

^†For simplicity of notation, we neglect the constant 1/4 \mgreek{p} \epsilon _{0}.

^‡Alternatively, one could use the experimental values if they are available.