Handbook of Solvents - George Wypych - ChemTech - Ventech!

8.7 Theoretical and computing modeling 465
8.7.1.1 Integral equation methods
The structure of the integral equation approach for calculating the angular pair correlation
function g(r12ω1ω2) starts with the OZ integral equation [8.76] between the total (h) and the
direct (c) correlation function, which is here schematically rewritten as h=h[c] where h[c]
denotes a functional of c. Coupled to that a second relation, the so-called closure relation
c=c[h], is introduced. While the former is exact, the latter relation is approximated; the form
of this approximation is the main distinction among the various integral equation theories to
be described below.
In the OZ equation, h depends on c, and in the closure relation, c depends on h; thus the
unknown h depends on itself and must be determined self-consistently. This (self-consistency
requiring, or integral equation) structure is characteristic of all many body problems.
Two of the classic integral equation approximations for atomic liquids are the PY
(Percus-Yevick) 51 and the HNC (hypernetted chain) 52 approximations that use the following
closures
h − c = y −1 (PY) [8.77]
h − c =ln y
(HNC) [8.78]
where y is the direct correlation function defined by g(12)=exp(-β v(12))y(12) with v(12)
the pair potential and β=1/kT.
The closures [8.77-8.78] can be also written in the form
β ( 12)
( 1 )
c g e v
= −
( )
(PY) [8.79]
c = h −βv 12 −ln
g (HNC) [8.80]
For atomic liquids the PY theory is better for steep repulsive pair potentials, e.g., hard
spheres, whereas the HNC theory is better when attractive forces are present, e.g.,
Lennard-Jones, and Coulomb potentials. No stated tests are available for molecular liquids;
more details are given below.
As said before, in practice the integral equations for molecular liquids are almost always
solved using spherical harmonic expansions. This is because the basic form [8.76] of
the OZ relation contains too many variables to be handled efficiently. In addition, harmonic
expansions are necessarily truncated after a finite number of terms. The validity of the truncations
rests on the rate of convergence of the harmonic series that depends in turn on the
degree of anisotropy in the intermolecular potential.
We recall that the solution to the PY approximation for the hard sphere atomic fluid is
analytical and it also forms a basis for other theories, e.g., in molecular fluids the MSA and
RISM theories to be discussed below.
The mean spherical approximation (MSA) 52 theory for fluids originated as the extension
to continuum fluids of the spherical model for lattice gases. In practice it is usually applied
to potentials with spherical hard cores, although extensions to soft core and
non-spherical core potentials have been discussed.
The MSA is based on the OZ relation together with the closure