Reviews in Computational Chemistry Volume 18

for all i ði:e:; 8 iÞ:
@U
@qi
l ¼ 0 8 i ½40Š
Because ð@U=@qiÞ for each atom is equal to the same undetermined multiplier
l, this quantity must be identical for all atoms in the molecule, 132
@U
@qi
¼ @U
@qj
8 i; j ½41Š
Through Mulliken’s identification of @U=@q as the electronegativity, we see
that minimizing the energy with respect to the charges is equivalent to equalizing
the electronegativities,
w i
Electronegativity Equalization Models 109
@U
@qi
¼ w 0 i þ Jii qi þ X
JijðrijÞqj
for all atoms. Notice that the electronegativity of atom i in a molecule, wi, differs from the electronegativity of the isolated atom, w0 i , and now depends
on its charge, the charge of the other atoms, its hardness, and the interactions
with other atoms through JijðrijÞ. In addition, Parr et al. 132 identified the
chemical potential of an electron as the negative of the electronegativity,
m ¼ @U=@q. So electronegativity equalization is equivalent to chemical
potential equalization. Thus, this model effectively moves charge around a
molecule to minimize the energy or to equalize the electronegativity or
chemical potential. These interpretations are all equivalent (for a dissenting
opinion, see Ref. 133).
Electronegativity equalization (EE) was first proposed by Sanderson. 134
The EE model, with appropriate parameterization, has been successful in predicting
the charges of a variety of molecules. 125,135–138 The parameters w0 and
J are not typically assigned from Eqs. [33] and [34], but instead are taken as
parameters to be optimized and can be viewed as depending on the valence
state of the atom, as indicated by electronic structure calculations. 139,140
Some models 136 set JabðrijÞ ¼1=rij, and others use some type of screening.
125,135,137 In addition, some models have an expression for the energy
that is not quadratic. 125,135,141 Going beyond the quadratic term in the Taylor
expansion of Eq. [32] can possibly extend the validity of the model, but it
introduces complications in the methods available for treating the charge
dynamics, as will be discussed below.
For a collection of molecules, the overall energy is comprised of the
energy given by Eq. [36] for each molecule and an interaction between charge
j 6¼ i
½42Š