Charges Fit to Electrostatic Potentials II:  Can Atomic Charges be Unambiguously Fit to Electrostatic Potentials?

Christina Carey, Lisa Emily Chirlian, David M. Gange* and Michelle Miller Francl*

a contribution from

Department of Chemistry
Bryn Mawr College
101 N. Merion Avenue
Bryn Mawr, PA 19010-2899

and

American Cyanamid Company
Agricultural Research Division
P.O. Box 400
Princeton, NJ 08543-0400

submitted to the Journal of Computational Chemistry


Abstract:   The present work examines the conditioning of the least squares matrix for obtaining potential derived charges and presents a modification of the CHELP method for fitting atomic charges to electrostatic potentials.  Results from singular value decompositions (SVD) of the least squares matrices show that, in general, the least squares matrix for this fitting problem will be rank deficient.  Thus, statistically valid charges cannot be assigned to all the atoms in a given molecule.  We find also t
hat, contrary to popular notions, increasing the point density of the fit has little or no influence on the rank of the problem.  Improvement in the rank can best be achieved by selecting points closer to the molecular surface.  Basis set has, as expected, no effect on the number of charges that can be assigned.  Finally, a well-defined, computationally efficient algorithm (CHELP-SVD)  is presented for determining the rank of the least squares matrix in potential derived charge fitting schemes, selecting t
he appropriate subset of atoms to which charges can be assigned based on that rank estimate and then refitting the selected set of charges.

Summary of conclusions:  Singular value decomposition of the linear least squares matrices used in fitting atom based monopoles to molecular electrostatic potentials provides a tool for evaluating the integrity of the calculated charges.  Based on the SVD analysis for a selected group of molecules we note particularly the following:
	-  Increasing the molecular size reduces the fraction of charges which can be validly 	assigned.  
	-  Increasing the point density of the fit has little or no influence on the rank of the 	problem.   
	-  The symmetry problem in CHELP is due to statistical problems with the data and 	contrary to common wisdom, is not entirely a function of the point density or point 	selection algorithm.  In other words, there is generally no advantage to using 	CHELPG in place of CHELP.  Both suffer from the ill conditioning of the matrix.
We also note that improvement in the rank can be achieved by selecting points closer to the molecular surface.  Basis set has, as expected, no effect on the number of charges that can be assigned.  Finally, we show that the SVD rank estimate can be used to generate improved sets of potential derived charges.