Reviews in Computational Chemistry Volume 18

102 Polarizability in Computer Simulations
The electrostatic interaction between independent polarizable atoms is
simply the sum of the charge–charge interactions between the four charge
sites,
Umm ¼ 1
2
X N
X
i ¼ 1 j 6¼ i
qiqj
1
jrijj
1
jrij djj
1
jrij þ dij þ
1
jrij dj þ dij
Typically, no Coulombic interactions are included between the core and shell
charges on a single site. Note that the electrostatic interaction in this model is
implemented using only the charge–charge terms already present in Eq. [1].
No new interaction types, such as the dipole field tensor Tij of Eq. [7], are
required. The computational advantage of avoiding dipole–dipole interactions
is almost exactly nullified by the necessity of calculating four times as many
charge–charge interactions, however.
The interaction of the induced dipoles with the static field is the sum of
the terms for each individual charge site,
Ustat ¼ XN
i ¼ 1
½28Š
qi½ri E 0 i ðri þ diÞ E 00
i Š ½29Š
where E0 i is the static field at the location of the core charge, ri, and E00 i is the
static field at the location of the shell charge, ri þ di. Note that E0 i 6¼ E00 i ,in
general.
Equations [28] and [29] correspond directly to Eqs. [7] and [6], but for
the case of dipoles with finite extent. In that sense, models based on point
dipoles can be seen as idealized versions of the shell model, in the limit of infinitely
small dipoles. That is, the magnitude of the charges qi and spring constants
ki approach infinity in such a way as to keep the atomic polarizabilities
ai constant. Indeed, in that limit, the displacements will approach zero in the
shell model, and the two models will be entirely equivalent.
To the extent that the polarization of physical atoms results in dipole
moments of finite length, it can be argued that the shell model is more physically
realistic (the section on Applications will examine this argument in more
detail). Of course, both models include additional approximations that may be
even more severe than ignoring the finite electronic displacement upon polarization.
Among these approximations are (1) the representation of the electronic
charge density with point charges and/or dipoles, (2) the assumption of an
isotropic electrostatic polarizability, and (3) the assumption that the electrostatic
interactions can be terminated after the dipole–dipole term.
In describing the shell model, the charge q was described as an effective
valence charge of the atom. In some applications of the shell model, the shell
charge is indeed interpreted physically in this manner, and q is assigned based