Handbook of Solvents - George Wypych - ChemTech - Ventech!

448 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
gest replacing the exp(-αR) dependence with a highly negative power of R: R -12 in most
cases. The exchange terms describe what is often called the steric repulsion between molecules
(actually there are other short range repulsive contributions, as the electrostatic penetration
components not described by the multipole expansion).
The other terms
Charge transfer contributions (CT) have been rarely introduced in the modeling of interaction
potentials for liquids. Some attempts at modeling have been done for the study of interactions
leading to chemical reactions, but in such cases, direct calculations of the interaction
term are usually employed.
For curious readers, we add that the tentative modeling was based on an alternative
formulation of the PT (rarely used for complete studies on the intermolecular interaction,
because it presents several problems) in which the promotion of electrons of a partner on the
virtual orbital of the second is admitted. The formal expressions are similar to those shown
in eq. [8.40]. For the second order contributions we have
−
∑
K
ΨΨ| VΨΨ
a b a b
0 0 Κ 0
E − E
a a
K 0
2
The sum over K is strongly reduced (often to just one term, the HOMO-LUMO interaction),
0
with K corresponding to the replacement of the occupied MO φr of A with the virtual MO
v
φt of B (A is the donor, B the acceptor). The expression is then simplified: the numerator is
reduced to a combination of two-electron Coulomb integrals multiplied by the opportune
overlap. The CT contribution rapidly decays with increasing R.
COUP contributions, obtained as a remainder in the variational decomposition of ΔE,
are not modeled. The contributions are small and of short-range character.
A conclusive view
In this long analysis of the modeling of the separate components we have learned the following:
• All terms of the decomposition may be partitioned into short- and long-range
contributions.
• The long-range contributions present problems of convergence, but these problems
may be reduced by resorting to multicenter distributions based on suitable partitions
of the molecular systems.
• The short-range contributions are in general of repulsive character and are
dominated by the EX terms. The effect of the other short-range contributions is
strongly reduced when multicenter expansions are used.
Adapting these remarks, one may write a tentative analytical expression for ΔE(R):
∑ ∑ ∑
( n)
ΔE ≈ C R
r
t
n
rt
−n
rt
[8.62]
The expansion centers (called “sites”) are indicated by the indexes r and t for the molecule
A and B respectively. The index n indicates the behavior of the specific term with respect
to the intersite distance R rt; each couple r-t of sites has a specific set of allowed n
values.