Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 463
This review will be mainly restricted to a discussion of small rigid molecules in their
ground electronic states. However, one can generalize in a number of directions by extending
the pair potential v(r ij,ω i,ω j) of eq. [8.74] so as to treat:
(a) non-rigid molecules,
(b) molecules with internal rotation,
(c) large (e.g., long chain) molecules which may be flexible,
(d) electronic excited state molecules.
In addition, the inclusion of covalence and charge transfer effects are also possible. The first
two generalizations are straightforward in principle; one lets v depend on coordinates describing
the additional degrees of freedom involved. As for (c), for very large molecules the
site-site plus charge-charge model is the only viable one. Under (d) we can quote the
so-called long-range ‘resonance’ interactions.
As the last note, we anticipate that in the following only equilibrium properties will be
considered; however, it is fundamental to recall from the beginning that dynamical (e.g.,
non-equilibrium) analyses are an important and active field of research in the theory of
physical models. 42,48,49
Before entering into more details of each theory, it is worth introducing some basic
definitions of statistical mechanics. All equilibrium properties of a system can be calculated
if both the intermolecular potential energy and the distribution functions are known. In considering
fluids in equilibrium, we can distinguish three principal cases:
(a) isotropic, homogeneous fluids, (e.g., liquid or compressed gas in the absence
of an external field),
(b) anisotropic, homogeneous fluids (e.g., a polyatomic fluid in the presence of
a uniform electric field, nematic liquid crystals), and
(c) inhomogeneous fluids (e.g., the interfacial region).
These fluid states have been listed in order of increasing complexity; thus, more independent
variables are involved in cases (b) and (c), and consequently the evaluation of the necessary
distribution functions is more difficult.
For molecular fluids, it is convenient to define different types of distribution functions,
correlation functions and related quantities. In particular, in the pair-wise additive
theory of homogeneous fluids (see eq. [8.74]), a central role is played by the angular pair
correlation function g(r 12ω 1ω 2) proportional to the probability density of finding two molecules
with position r 1 and r 2 and orientations ω 1 and ω 2 (a schematic representation of such
function is reported in Figure 8.5).
In fact, one is frequently interested in some observable property , which is (experimentally)
a time average of a function of the phase variables B(r N ,ω N ); the latter is often a
sum of pair terms b(r ij,ω i,ω j) so that is given by
1
( ) ( ) ( 1 2) ( 1 2)
N N N N N N
B = ∫dr d P r B r = N∫dr g r b r
ω ω ω ρ ωω ωω ωω
2
1 2
[8.75]
i.e., in terms of the pair correlation function g(12). Examples of such properties are the configurational
contribution to energy, pressure, mean squared torque, and the mean squared
force. In eq. [8.75] g( rωω 1 2) b(
rωω 1 2)
means the unweighted average over orienta-
ωω 1 2
tions.
In addition to g(12), it is also useful to define: