Reviews in Computational Chemistry Volume 18

of N molecules is taken as a Hartree product of the individual molecular wave
functions,
¼ YN
i ¼ 1
The Hartree product neglects exchange correlation interactions between
molecules. To include proper exchange would make these models inefficient
and impractical.
The Hamiltonian for the system
^H ¼ XN
i ¼ 1
^H 0 i
1 X
þ
2
N XN i
i ¼ 1 j 6¼ i
contains the isolated molecular Hamiltonian, ^ H 0 i
^H 0 i
¼ X2M
a ¼ 1
^Ta
X A
X 2M
Raa
a ¼ 1 a ¼ 1
^Hij
ZaðiÞ
þ X2M X
½55Š
½56Š
, given, in atomic units, by
1
r
a ¼ 1 b > a ab
where ^ T is the kinetic energy operator, ZaðiÞ is the nuclear charge of atom a on
molecule i, Raa is the distance between the nucleus of atom a and electron a,
and r ab is the distance between two electrons. The interaction Hamiltonian
between molecules i and j, ^ Hij, is
^Hij ¼ X2M X
a ¼ 1
2M
1
r
b ¼ 1 ab
þ XA
X A
a ¼ 1 b ¼ 1
ZaðiÞZbðjÞ
where Rab is the distance between atom a on molecule i and atom b on
molecule j. The interaction energy of the system is
Rab
½57Š
½58Š
E ¼h j ^ Hj i Nh 0 j ^ H 0 i j 0 i ½59Š
where 0 is the ground-state wave function of the isolated molecule and
h 0 j ^ H 0 i j 0 i is the energy of the isolated molecule. The polarization energy is
or, equivalently,
Epol ¼ XN
ðh j ^ H 0 i j i h 0 j ^ H 0 i j 0 iÞ ½60Š
i ¼ 1
Epol ¼ XN
ðh ij ^ H 0 i j ii h
i ¼ 1
Semiempirical Models 117
0
i j ^ H 0 i j 0 i iÞ ½61Š