Handbook of Solvents - George Wypych - ChemTech - Ventech!

464 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
Figure 8.5. Example of the radial distribution function
g(r) of a typical monoatomic liquid. Figure 8.6. Examples of h(r) and c(r) functions for a typical
monoatomic liquid.
a) the site-site correlation function g αβ(r αβ) proportional to the probability
density that sites α and β on different molecules are separated by distance r,
regardless of molecular orientations;
b) the total correlation function h(r 12ω 1ω 2)= g(r 12ω 1ω 2)-1;
c) the direct correlation function c(r 12ω 1ω 2).
Simple examples of h and c functions are reported in Figure 8.6.
In the list above, item (a) implies definite sites within molecules; these sites may be the
nuclei themselves or sites at arbitrary locations within the molecules.
In addition, the total correlation, h, between molecules 1 and 2 can be separated into
two parts: (i) a direct effect of 1 on 2 which is short-ranged and is characterized by c, and (ii)
and indirect effect in which 1 influences other molecules, 3, 4, etc., which in turn affect 2.
The indirect effect is the sum of all contributions from other molecules averaged over their
configurations. For an isotropic and homogeneous fluid formed by non-spherical molecules,
we have
( r ) ( r ) r ( r ) ( r )
h 12ωω 1 2 = c 12ωω 1 2 + ρ∫ d 3 c 13ωω 1 3 h 32ω3ω2 [8.76]
ω3
which is the generalization of the Ornstein-Zernike (OZ) equation 50 to non-spherical molecules.
The OZ equation is the starting point for many theories of the pair correlation function
(PY, HNC, etc.); however, numerical solutions starting from [8.76] are complicated by
the large number of variables involved. By expanding the direct and total correlation functions
in spherical harmonics, one obtains a set of algebraically coupled equations relating
the harmonic coefficients of h and c. These equations involve only one variable in the place
of many in the original OZ equation. In addition, the theories we shall describe below truncate
the infinite set of coupled equations into a finite set, thereby enabling a reasonably simple
solution to be carried out.