Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 425
Figure 8.1. Interaction energy for a dimer with respect to
the mutual approach distance at a fixed orientation.
where necessary, with r ab and Ω ab. ΔE AB(R)
has the status of a PES, with a shift in the
reference energy, given here by the sum of
the energies of the two monomers.
A function defined in a 6-dimensional
space is hard to visualize. Many devices
have been introduced to render in graphic
form selected aspects of this function.
Some will be used in the following: here we
shall use the simplest graphical rendering,
consisting in fixing an orientation (Ω ab) and
two coordinates in the r ab set in such a way
that the remaining coordinate corresponds
to the mutual approach between molecule
A and B, along a given straight trajectory
and a fixed mutual orientation. A typical example is reported in Figure 8.1.
The energy curve may be roughly divided into three regions. Region I corresponds to
large separation between interaction partners; the interaction is feeble and the curve is relatively
flat. Region II corresponds to intermediate distances; the interactions are stronger
compared to region I, and in the case shown in the figure, the energy (negative) reaches a
minimum. This fact indicates that the interaction is binding the two molecules: we have here
a dimer with stabilization energy given, in first approximation, by the minimum value of the
curve. Passing at shorter distances we reach region III; here the interaction rapidly increases
and there it gives origin to repulsion between the two partners. Before making more comments,
a remark must be added. The shape of the interaction energy function in molecular
systems is quite complex: by selecting another path of approach and/or another orientation,
a completely different shape of the curve could be obtained (for example, a completely repulsive
curve). This is quite easy to accept: an example will suffice. The curve of Figure 8.1
could correspond to the mutual approach of two water molecules, along a path leading to the
formation of a hydrogen bond when their orientation is appropriate: by changing the orientation
bringing the oxygen atoms pointing against each other, the same path will correspond
to a continuously repulsive curve (see Figure 8.2 below for an even simpler example). We
have to consider paths of different shape, all the paths actually, and within each path we
have to consider all the three regions. To describe a liquid, we need to know weak
long-range interactions as well as strong short-range repulsion at the same degree of accuracy
as for the intermediate region. Studies limited to the stabilization energies of the dimers
are of interest in other fields of chemical interest, such as the modeling of drugs.
The ΔE AB(R) function numerically corresponds to a small fraction of the whole QM
energy of the dimeric system. It would appear to be computationally safer to compute ΔE directly
instead of obtaining it as a difference, as done in the formal definition [8.11]. This can
be done, and indeed in some cases it is done, but experience teaches us that algorithms starting
from the energies of the dimer (or of the cluster) and of the monomers are simpler and
eventually more accurate. The approaches making use of this difference can be designed as
variational approaches, the others directly aiming at ΔE are called perturbation approaches,
because use is made of the QM perturbation theory. We shall pay more attention, here be-