Reviews in Computational Chemistry Volume 18

M i
z i + q i
d i
k i
and shell charges. For a neutral atom i with a core charge of þqi, an equal and
opposite shell charge of qi, and a shell charge that is displaced by a distance
di from the core charge, the dipole moment is
l i ¼ qidi ½24Š
As with any model involving inducible dipoles, the potential energy of
the induced dipoles contains terms representing the interaction with any static
field, the interaction with other dipoles, and the polarization energy, that is,
U ind ¼ Ustat þ Umm þ U pol
The polarization energy arises in this case from the harmonic spring separating
the two charges,
Upol ¼ 1
2
XN kid
i ¼ 1
2 i
for a collection of N polarizable atoms with spring constants ki and charge
displacements di ¼jdij. Using di from Eq. [24] and comparing it with
Eq. [10] for polarizable point dipoles, we see that the polarizability of an
isotropic shell model atom can be written as
ai ¼ q 2 i =ki
−q i
Shell Models 101
Figure 2 In the shell model, a ‘‘core’’ charge zi þ qi is attached by a harmonic spring
with spring constant ki to a ‘‘shell’’ charge qi. For a neutral atom, zi ¼ 0. The center of
mass is at or near the core charge, but the short-range interactions are centered on the
shell charge. (Not drawn to scale; the displacement di between the charges is much
smaller than the atomic radius.)
½25Š
½26Š
½27Š