Subject: partial charges

From: <[email-masked]>
Date: Fri, 30 Sep 94 19:24:54 CDT

I would like to contribute some thoughts on the subject of partial 
charges of atoms in molecules, which has received some attention on the 
CCL this week.  In particular I note that Ryszard Czerminski wrote:
	"One interesting avenue could be to see if there is a way of deriving
point charges equivalent to ESP/RESP fitted charges directly from a wave
function....  This is probably not possible exactly but maybe some further
progress is possible in this direction."
In the following I will comment directly on recent progress in this area.
	Partial charges are an important theoretical subject because they are
so useful in molecular modeling, and they are a fascinating subject because,
since they are not physical observables, their definition is--to some extent--
at our disposal.  In recent work [J. W. Storer, D. J. Giesen, C. J. Cramer, 
and D.  G. Truhlar, "Class IV Charge Models: A New Semiempirical Approach in
Quantum Chemistry," Journal of Computer-Aided Molecular Design, in 
press], we have distinguished four classes of models of partial charges.
	Class I charges are defined by simple models that make no reference
to the quantum mechanical character of electronic structure.  An example
would be obtaining the partial charges of a heteronuclear diatomic
molecule by dividing the dipole moment by the bond length.
	Class II charges are obtained by some prescription for partitioning
the density distribution corresponding to some approximate electronic 
wave function (or the exact wave function if we had it) of a molecule into 
components associated with individual atoms.  Examples include Mulliken 
or Lowdin population analysis or Richard Bader's method of partitioning 
the charge distribution.
	Class III partial charges are based on a more utilitarian approach.  As 
stated above, partial charges of atoms in molecules are interesting first of 
all because they are useful for molecular modeling.  Now there are several 
approaches to modeling the electrostatic properties of molecules based on 
placing a set of point multipoles at each of several sites in a molecule.  One 
example encountered in practice is to place partial charges both at the 
nuclei and in the lone pair regions; this approach is often used for water.  
Another example is placing a partial charge and a point dipole at each 
nuclear center.  The latter is an example of a distributed multipole 
representation.  By far the most common example of this approach is the 
nuclear-centered distributed-monopole representation in which we ignore 
the higher multipoles (dipole, quadrupole, octupole, hexadecapole, ...) and 
simply place a partial charge at each nuclear center.  The simplicity of this 
approach makes it useful for inclusion in force fields designed for 
conformational analysis, interaction potentials, solvation modeling, and 
molecular dynamics.  Class III partial charges are an attempt to find the best 
set of nuclear-centered partial charges for such modeling efforts.  Thus class 
III charges are defined such that physical observables calculated from such 
partial charges agree as well as possible (a subjective elements creeps in 
here) with the same physical observable calculated using the continuous 
psi squared charge density of an electronic wave function.  The most 
frequently used physical observable is the electrostatic potential (ESP) at 
selected points around a molecule; a special case of this would be fitting 
the dipole moment, which is equivalent to fitting the electrostatic 
potential of a polar molecule on a hypersphere of very large radius.  
Examples of partial charge methods based on ESP-fitting are the ChelpG 
method of Michelle Francl, Ken Wiberg, and coworkers and the similar 
fitting procedure of Kenny Merz and Peter Kollman.
	Class III partial charges, like any other modeling tool, have some 
deficiencies.  The first type of deficiency is numerical.  Francl herself has 
pointed out that the equations one obtains in ESP fitting are often ill-
conditioned.  In a similar vein, Bill Jorgensen has pointed out that the 
charges on buried atoms may be especially poorly determined by ESP 
fitting.  
	A second problem with class III charges is that, while they make up 
(as well as possible) for deficiencies in the replacement of a continuous 
electron density function (corresponding to some electronic structure level 
X and basis Y) by a set of nuclear-centered partial charges, they do not make 
up for the deviation of psi squared at level X/Y from the exact psi squared.  
Such deficiencies can in fact be quite serious even for popular levels X/Y 
that are considered to be high levels.  [For example, for MeSO_3H, the 
HF/6-31G* dipole moment is 3.24 D, whereas the more accurate 
MP2/cc-pVDZ dipole moment is 2.33 D.]
	Despite these deficiencies, ESP fitting is a powerful technique, and it 
is often very useful, but Class IV charges represent an attempt to make up 
for both sets of deficiencies.  Class IV charges are obtained by starting with 
class II charges and mapping them to a new set of charges (the class IV 
charges) with mapping parameters determined semiempirically such that 
the new charges reproduce experimental observables as well as possible.  
Although ESPs are in principle observable, dipole moments are more 
widely available and have been used for developing class IV charge models 
so far.
	We have so far (in the preprint mentioned above) parameterized two 
class IV charge models, which we call CM1A and CM1P.  The former begins 
the map with AM1 Mulliken charges, and the latter begins with PM3 
Mulliken charges.  The parameters in our mappings were based on 204 
neutral compounds containing a wide variety of functional groups.  
Mapping parameters are available for the following atom types: H, C, N, O. 
F, Si, S, Cl, Br, and I.  (Note the conspicuous absence of P.  We believe that
a good map for P should start with an ab initio wave function, or at least with 
something better than AM1 or PM3.  Work is "in progress" on this.)  For 23 
compounds we tested the partial charges against those obtained by ChelpG 
analysis of MP2/6-3/G* wave functions.  This chart gives the RMS errors in 
various calculated dipole moments for these 23 compounds.  The first row
is based on a continuous charge distribution; the others are based on 
partial charges:

	dipole calculated from	      RMS error (D)
	________________________      ____________
	MP2/6-31G*, psi squared		0.21
	HF/6-31G* ChelpG charges	0.33
	HF/6-31G* Mulliken charges	0.93
	AM1 Mulliken charges		0.89
	PM3 Mulliken charges		1.00
	AM1-CM1A class IV charges	0.27
	PM3-CM1P class IV charges	0.20
 
The cost of the mapping is totally negligible; thus with PM3-CM1P class IV 
charges one obtains MP2/6.31G* accuracy with NDDO cost.
	If you would like a preprint of our paper, send e-mail to
		[email-masked]
with your full mailing address and request UMSI Research Report 94/144 by 
Storer et al.
	CM1A and CM1P partial charges may be calculated for gas-phase 
molecules using AMSOL-version 4.5 [Reference: C. J. Cramer, G. D. Hawkins, 
G. C. Lynch, D. J. Giesen, D. G. Truhlar, and D. A. Liotard, QCPE Bull. 14, 55-
57 (1994)].  This program is available from QCPE at Indiana University--it is 
program 606 in their catalog.

   Don Truhlar
   Dept. of Chemistry
   University of Minnesota