Computational Chemistry

been used particularly in inorganic chemistry [ 262 ]. The most popular method of
population analysis now is probably Weinhold’s NPA, and the favored atom
charges are evidently those from NPA, and electrostatic potential charges (next
section). The methods of Mulliken, L€owdin and Weinhold are explained and
compared in more detail by Cramer [ 263 ], and those of Mulliken, L€owdin and
Mayer by Leach [ 264 ].

5.5.4.4 Electrostatic Potential

Theelectrostatic potential(ESP) is a measure of charge distribution that also
provides other useful information [ 265 ]. The electrostatic potential at a point P in
a molecule is defined as the amount of energy (work) needed to bring a unit point
positive “probe charge” (e.g. a proton) from infinity to P. The electrostatic potential
can be thought of as a measure of how positive or negative the molecule is at P: a
positive value at the point means that the net effect experienced by the probe charge
as it was brought from infinity was repulsion, while a negative value means that the
probe charge was attracted to P, i.e. energy wasreleasedas it fell from infinity to P.
The ESP at a point is the net result of the effect of the positive nuclei and the
negative electrons. The calculation of the effect of the nuclei is trivial, following
directly from the fact that the potential due to a point charge Z at a distance r away
from the unit charge is, at point P:

VðPÞ¼

Z^1

r

Z' 1

r^2

dr¼

Z

r

ð 5 : 231 Þ

Thus the ESP created by the nuclei is

VðPÞnuc¼

X

A

ZA

jrP#rAj

ð 5 : 232 Þ

where |rp#rA| is the distance from nucleus A to the point P, i.e. the absolute value
of the difference of two vectors. To obtain the expression for the ESP due to the
electrons, we can modify Eq. .(5.232) by replacing the summation over the nuclei
by an integral over infinitesimal volume elements of the electron density or charge
densityr(r) (seeSection 5.5.4.5). We get for the total ESP at P

VðPÞtot¼VðPÞnucþVðPÞelþ

X

A

ZA

jrP#rAj

Z

rðrÞ

jrP#rj

dr ð 5 : 233 Þ

The ESP at many points on the surface of the molecule can be calculated
(Section 5.5.6) and a set of atom charges then calculated to fit (by a least-squares
procedure) the ESP values, and also to sum to the net charge on the molecule (the

352 5 Ab initio Calculations