Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 447
Limitations due to the non-convergence of these expansion series are reduced by the
use of multicenter expansions: in such a way it is possible to reach a satisfactory representation
even for polyatomic molecules at short distances using the first expansion term only.
In general, the expansion sites are the (heavy) atoms of the dimer, using a simpler formulation
due to London. The London formulation uses the averaged (isotropic) value of the
static dipole polarizabilities, α j , of the atoms and their first ionization potentials I i and I j:
with
6 ( ) ∑ ∑ 6(
)
− −
DIS( 6) ≈ C6 AB R = C i, j Rij 3
C6( i, j)
=
2
α α II
I + I
i j i j
i j
A
i
B
j
6
[8.56]
[8.57]
This formula derives from the standard RS-PT with the expansion over the excited state
truncated at the first term. For isolated atoms i and j, the coefficients C6(i,j) can be drawn
from experimental data, while for atomic fragments of molecules only from computations.
The exchange (or repulsion) term
The exchange terms are related to the introduction of the intermonomer antisymmetrizer
AAB in the expression giving the electrostatic contribution. The correction factors introduced
in the SAPT methods 15 may be related to a renormalization factor that may be written
in the following form when AAB is replaced with 1+P for simplicity:
1 1 1
= =
Φ| PΦΨΨ| PΨΨ P
0 0
Α
B A B
( 1+
)
[8.58]
This factor is different from 1, because contains all the multiple overlap values between
MOs of A and B. may be then expanded into terms containing an increasing number of
multiple overlaps (or exchanges of orbitals):
P = P + P + P + P + � [8.59]
1 2 3 4
The exchange energy may be expanded into increasing powers of MO overlap
integrals S:
with
( ) ( ) ( )
EX = EX S + EX S + EX S + � [8.60]
( )
2 3 4
EX S = VP − P
[8.61]
2 2 2
This first term is sufficient for modeling intermolecular potentials. The overlaps depend
on the monomer separation roughly as exp(-αR) so the following terms give small
contribution at large-medium distances. At short distances the positive exchange term rapidly
increases. Reasons of uniformity with the other terms of the interaction potential sug-