Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 467
8.7.1.2 Perturbation theories
In perturbation theories 44,58 one relates the properties (e.g., the distribution functions or free
energy) of the real system, for which the intermolecular potential energy is E, to those of a
reference system where the potential is E0, usually by an expansion in powers of the perturbation
potential E1=E-E0. The methods used are conveniently classified according to whether the reference system
potential is spherically symmetric or anisotropic. Theories of the first type are most appropriate
when the anisotropy in the full potential is weak and long ranged; those of the
second type have a greater physical appeal and a wider range of possible application, but
they are more difficult to implement because the calculation of the reference-system properties
poses greater problems.
In considering perturbation theory for liquids it is convenient first to discuss the historical
development for atomic liquids.
In many cases the intermolecular pair potential can be separated in a natural way into a
sharp, short-range repulsion and a smoothly varying, long range attraction. A separation of
this type is an explicit ingredient of many empirical representations of the intermolecular
forces including, for example, the Lennard-Jones potential. It is now generally accepted that
the structure of simple liquids, at least of high density, is largely determined by geometric
factors associated with the packing of the molecular hard cores. By contrast, the attractive
interactions may, in the first approximation, be regarded as giving rise to a uniform background
potential that provides the cohesive energy of the liquid but has little effect on its
structure. A further plausible approximation consists of modeling the short-range forces by
the infinitely steep repulsion of the hard-sphere potential. In this way, the properties of a
given liquid can be related to those of a hard-sphere reference system, the attractive part of
the potential being treated as a perturbation. The choice of the hard-sphere fluid as a reference
system is an obvious one, since its thermodynamic and structural properties are well
known.
The idea of representing a liquid as a system of hard spheres moving in a uniform, attractive
potential well is an old one; suffice here to recall the van der Waals equation.
Roughly one can thus regard perturbation methods as attempts to improve the theory of van
der Waals in a systematic fashion.
The basis of all the perturbation theories we shall consider is a division of the pair potential
of the form
( 12) ( 12) ( 12)
v = v + w
[8.83]
0
where v 0 is the pair potential of the reference system and w(12) is the perturbation. The following
step is to compute the effect of the perturbation on the thermodynamic properties
and pair distribution function of the reference system. This can be done systematically via
an expansion in powers either of inverse temperature (the “λ expansion”) or of a parameter
that measures the range of the perturbation (the “γ expansion”).
In spite of the fact that hard spheres are a natural choice of the reference system, for the
reasons discussed above, realistic intermolecular potentials do not have an infinitely steep
hard core, and there is no natural separation into a hard-sphere part and a weak perturbation.
A possible improvement is that to take proper account of the “softness”’ of the repulsive
part of the intermolecular potential. A method of doing this was proposed by Rowlinson, 59