Handbook of Solvents - George Wypych - ChemTech - Ventech!

462 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
The complexity of the problem thus requires the assumption of some approximations;
most of the formulated theories, for example, assume
(a) a rigid molecule approximation,
(b) a classical treatment of the traslational and rotational motions, and
(c) a pairwise additivity of the intermolecular forces.
In the rigid molecule approximation it is assumed that the intermolecular potential energy
V(r N ,ω N ) depends only on the positions of the centers of mass r N =r1r2...rN for the N mol-
N
ecules and on their orientations ω = ωω 1 2� ωN.
This implies that the vibrational
coordinates of the molecules are dynamically and statistically independent on the center of
mass and orientation coordinates, and that the internal rotations are either absent, or independent
of the r N and ω N coordinates. The molecules are also assumed to be in their ground
electronic states.
At the bases of the second basic assumption made, e.g., that the fluids behave classically,
there is the knowledge that the quantum effects in the thermodynamic properties are
usually small, and can be calculated readily to the first approximation. For the structural
properties (e.g., pair correlation function, structure factors), no detailed estimates are available
for molecular liquids, while for atomic liquids the relevant theoretical expressions for
the quantum corrections are available in the literature.
The third basic approximation usually introduced is that the total intermolecular potential
energy V(r N ,ω N ) is simply the sum of the intermolecular potentials for isolated pairs ij
of molecules, i.e.,
N N ( r , ω ) ∑ ( ij, ωi, ωj)
V V r
=
i< j
[8.74]
In the sum i is kept less than j in order to avoid counting any pair interaction twice.
Eq. [8.74] is exact in the low-density gas limit, since interactions involving three or
more molecules can be ignored. It is not exact for dense fluids or solids, however, because
the presence of additional molecules nearby distorts the electron charge distributions in
molecules i and j, and thus changes the intermolecular interaction between this pair from the
isolated pair value. In order to get a more reliable description, three-body (and higher
multi-body) correction terms should be introduced.
The influence of three-body terms on the physical properties has been studied in detail
for atomic fluids, 46,47 while much less is known about molecular fluids. In the latter case, the
accurate potentials are few and statistical mechanical calculations are usually done with
model potentials. A particular model may be purely empirical (e.g., atom-atom), or
semiempirical (e.g., generalized Stockmayer), where some of the terms have a theoretical
basis. The so-called generalized Stockmayer model consists of central and non-central
terms. For the central part, one assumes a two-parameter central form (the classical example
is the Lennard-Jones, LJ, form introduced in the previous sections). The long-range,
non-central part in general contains a truncated sum of multipolar, induction and dispersion
terms. In addition, a short-range, angle-dependent overlap part, representing the shape or
core of the potential, is usually introduced. The multipolar interactions are pair-wise additive,
but the induction, dispersion and overlap interactions contain three-body (and higher
multi-body) terms. Hence, three-body interactions are strongly suspected to be of large importance
also for molecular liquids.