Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.4 Two-body interaction energy 449
In such a way it is possible to combine sites corresponding only to a point charge with
sites having more terms in the expansion. For two sites r-t both described by a point charge
(1) -1
the only interaction term is Crt Rrt
, while for a couple r-t described by a point charge and a
dipole, the contribution to the interaction is limited to the R -2 term, and, if it is described by
two dipoles, to the R -3 term. The last example shows that eq. [8.62] is cryptic or not well developed.
In fact, if we examine eq. [8.12], which reports the interaction energy between two
dipoles, it turns out that the dependence on the orientation angle θ apparently is missing in
(n)
eq. [8.62]. The coefficients Crt must be specified in more detail as given in equations [8.50]
and [8.52] by adding the opportune l, l� and m indexes (or the corresponding combination of
Cartesian coordinates), or must be reduced to the isotropic form. This means to replace the
three components of a dipole, and the five components of a quadrupole, and so on, by an average
over all the orientations (i.e., over the m values). There is a loss of accuracy, very large
in the case of two dipoles, and of decreasing importance in passing to higher multipoles.
The loss in accuracy is greatly decreased when a multi-site development is employed. Many
(n)
analytical expressions of interaction potentials use this choice. In such cases the Crt coefficients
are just numbers.
(n)
The coefficients Crt can be drawn from experimental values (but this is limited to the
cases in which there is one site only for both A and B), from ad hoc calculations, or by a fitting
of ΔE values. In the past, large use has been made of “experimental” ΔE values, derived,
e.g., from crystal packing energies, but now the main sources are the variationally computed
QM values, followed by a fit.
The use of expression [8.62] for the fitting of QM values represents a remarkable improvement
with respect to the past strategy of reaching a good fitting with complete freedom
of the analytical form of the expression. There are in the literature interaction
potentials using, e.g., non-integer values of n and/or other analytical expressions without
physical meaning. This strategy leads to potentials that cannot be extended to similar systems,
and that cannot be compared with the potential obtained by others for the same system.
Expression [8.62] has several merits, but also several defects. Among the latter we
note that given powers of R may collect terms of different origin. For example, the term R -6
describes both the induction dipole-induced dipole interaction and the dispersion dipole-dipole
interaction: two contributions with a different sensitivity with respect to changes in the
(6)
molecular system. Theory and computational methods both permit getting two separate Crt coefficients for the two contributions. This is done in a limited number of potentials (the
most important cases are the NEMOn 26 and the ASP-Wn 27 families of models). The most
popular choice of a unique term for contributions of different origin is motivated by the
need of reducing computational efforts, both in the derivation and in the use of the potential.
(6)
To fit a unique Crt coefficient means to use ΔE values only: for two coefficients there is the
need of separately fitting IND and DIS contributions.
8.4.5 THE RELAXATION OF THE RIGID MONOMER CONSTRAINT
We have so far examined decompositions of two-body interaction potentials, keeping fixed
the internal geometry of both partners. This constraint is clearly unphysical, and does not
correspond to the naïve model we have considered in the introduction, because molecules
always exhibit internal motions, even when isolated, and because molecular collisions in a
liquid (as well as in a cluster) lead to exchanges of energy between internal as well as external
degrees of freedom.