Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 441
E = E + E
( 2) ( 2) ( 2)
exc [8.43]
exc −dis exc − ind
The partition of this second order contribution into two terms is based on the nature of
m
the MO replacements occurring in each configuration ΨK appearing in the sums of eq.
[8.40]. These two terms give a mixing of exchange and dispersion (or induction) contributions.
Passing to higher orders E (n) the formulas are more complex. In the RS part it is possi-
( 3)
ble to define pure terms, as ERSdis , but in general they are of mixed nature. The same happens
(n)
for the Eexccontributions. The examination of these high order contributions is addressed in the studies of the
mathematical behavior of the separate components of the PT series. Little use has so far
been made of them in the actual determination of molecular interaction potentials.
8.4.4 MODELING OF THE SEPARATE COMPONENTS OF ΔE
The numerical output of variational decompositions of ΔE (supplemented by some PT decompositions)
nowadays represents the main source of information to model molecular interaction
potentials. In the past, this modeling was largely based on experimental data
(supported by PT arguments), but the difficulty of adding new experimental data, combined
with the difficulty of giving an interpretation and a decoupling of them in the cases of complex
molecules, has shifted the emphasis to theoretically computed values.
The recipes for the decomposition we have done in the preceding sections are too complex
to be used to study liquid systems, where there is the need of repeating the calculation
of ΔE(r,Ω) for a very large set of the six variables and for a large number of dimers. There is
thus the need of extracting simpler mathematical expressions from the data on model systems.
We shall examine separately the different contributions to the dimer interaction energy.
As will be shown in the following pages, the basic elements for the modeling are to a
good extent drawn from the PT approach.
The electrostatic term
ES may be written in the following form, completely equivalent to eq. [8.16]
T 1
ES = ∫∫ρAr1 ρ
r
T () ( )
12
B
r drdr
2 1 2
[8.44]
we have here introduced the total charge density function for the two separate monomers.
T
The density functionρ M ( r) is a one-electron function, which describes the distribution in the
space of both electrons and nuclei. Formula [8.44] is symmetric both in A and B, as well as
in r1 and r2. It is often convenient to decompose the double integration given in eq. [8.44] in the
following way
where
() ()
T
ES = ∫ ρ r1 V r1 dr1
[8.45]
A
B
V r ( r )
r dr
T 1
B( 1) = ∫ ρ B 2
2
[8.46]
12