Reviews in Computational Chemistry Volume 18

118 Polarizability in Computer Simulations
which is the difference between the molecular energy of wave function (or
i), which is the expectation value of the Hamiltonian in Eq. [56], and the
molecular energy of the isolated molecule wave function. This expression
for the polarization energy is comparable to Eqs. [9], [26], and [48] for the
other models.
To avoid calculating the two-electron integrals in Eq. [59], the assumption
is made that no electron density is transferred between molecules. The
interaction Hamiltonian is then
^Hijð jÞ ¼ X2M
Vað jÞþ XA
a ¼ 1
a ¼ 1
ZaðiÞVað jÞ ½62Š
where Vxð jÞ is the electrostatic potential 179 from molecule j at the position of
electron a or nuclei a of molecule i,
Vxð jÞ ¼
ð 2
j ðrÞ XA
dr þ
jrx rj
b ¼ 1
ZbðjÞ
jrx Rbj
If the Vxð jÞ coming from the electrons and nuclei of molecule j is represented
just by point charges on atomic sites, then
and
^Hijð jÞ ¼ X2M
Vxð jÞ ¼ XA
X A
a ¼ 1 b ¼ 1
b ¼ 1
qbð jÞ
jrx Rbj
qbð jÞ
þ XA
rab
X A
a ¼ 1 b ¼ 1
ZaðiÞqbð jÞ
where qbð jÞ is the partial atomic charge on atom b in molecule j derivable
from the wave function j. (Other semiempirical models have charges offset
from the atomic sites.) 172,175,176 The energy of molecule i is then changed
by the partial charges from the other molecules. Since exchange correlation
interactions are neglected as mentioned above in regard to Eq. [55], the shortrange
repulsive interactions need to be added, which can be done with a Lennard–
Jones potential. The interaction energy between molecules i and j is then
Rab
½63Š
½64Š
½65Š
Eij ¼ 1
2 ðh ij ^ Hijj iiþh jj ^ Hjij jiÞ þ ELJ ½66Š
which is used so that Eij is equal to Eji. The interactions between molecules
then consist of only Lennard–Jones and Coulombic components. Polarizability