Ornstein-Zernike relation

Notation:

• $g(r)$ is the pair distribution function.
• $\Phi (r)$ is the pair potential acting between pairs.
• $h(1,2)$ is the total correlation function.
• $c(1,2)$ is the direct correlation function.
• $\gamma (r)$ is the indirect (or series or chain) correlation function.
• $y(r_{12})$ is the cavity correlation function.
• $B(r)$ is the bridge function.
• $\omega (r)$ is the thermal potential.
• $f(r)$ is the Mayer f-function.

The Ornstein-Zernike relation (OZ) integral equation is

$h=h\left[c\right]$

where $h[c]$ denotes a functional of $c$. This relation is exact. This is complemented by the closure relation

$c=c\left[h\right]$

Note that $h$ depends on $c$, and $c$ depends on $h$. Because of this $h$ must be determined self-consistently. This need for self-consistency is characteristic of all many-body problems. (Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the OZ has the form (5.2.7)

$h(1,2)=c(1,2)+\int \rho ^{(1)}(3)c(1,3)h(3,2)d3$

If the system is both homogeneous and isotropic, the OZ relation becomes (Ref. 1Eq. 6)

$\gamma (r)\equiv h(r)-c(r)=\rho \int h(r')~c(|r-r'|)dr'$

In words, this equation (Hansen and McDonald, section 5.2 p. 107)

``...describes the fact that the total correlation between particles 1 and 2, represented by $h(1,2)$, 
is due in part to the direct correlation between 1 and 2, represented by $c(1,2)$, but also to the indirect correlation,  
$\gamma (r)$, propagated via increasingly large numbers of intermediate particles."

Notice that this equation is basically a convolution, i.e.

$h\equiv c+\rho h\otimes c$

(Note: the convolution operation written here as $\otimes$ is more frequently written as $*$) This can be seen by expanding the integral in terms of $h(r)$ (here truncated at the fourth iteration):

$h(r)=c(r)+\rho \int c(|r-r'|)c(r')dr'+\rho ^{2}\int \int c(|r-r'|)c(|r'-r''|)c(r'')dr''dr'+\rho ^{3}\int \int \int c(|r-r'|)c(|r'-r''|)c(|r''-r'''|)c(r''')dr'''dr''dr'+\rho ^{4}\int \int \int \int c(|r-r'|)c(|r'-r''|)c(|r''-r'''|)c(|r'''-r''''|)h(r'''')dr''''dr'''dr''dr'$

etc. Diagrammatically this expression can be written as (Ref. 2):

where the bold lines connecting root points denote $c$ functions, the blobs denote $h$ functions. An arrow pointing from left to right indicates an uphill path from one root point to another. An `uphill path' is a sequence of Mayer bonds passing through increasing particle labels. The OZ relation can be derived by performing a functional differentiation of the grand canonical distribution function (HM check this).

OZ equation in Fourier space

The OZ equation may be written in Fourier space as (Eq. 5 in Ref. 3):

${\hat {\gamma }}=(I-\rho {\hat {c}})^{-1}{\hat {c}}\rho {\hat {c}}$

The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly to:

${\hat {\gamma }}(k)={\frac {4\pi }{k}}\int _{0}^{\infty }r~\sin(kr)\gamma (r)dr$

$\gamma (r)={\frac {1}{2\pi ^{2}r}}\int _{0}^{\infty }k~\sin(kr){\hat {\gamma }}(r)dk$

Note:

${\hat {h}}(0)=\int h(r)dr$

${\hat {c}}(0)=\int c(r)dr$

References

1. L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci. 17 pp. 793- (1914)
2. James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A 45 pp. 816 - 824 (1992)
3. Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics 103 pp. 2625-2633 (1995)
4. Hansen and MacDonald "Theory of Simple Liquids"