Ornstein-Zernike relation

Notation used:

• $g(r)$ 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 integral equation is given by:

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

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 Ornstein-Zernike relation 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 Ornstein-Zernike relation becomes (Eq. 6 of Ref. 1)

$\gamma ({\mathbf {r} })\equiv h({\mathbf {r} })-c({\mathbf {r} })=\rho \int h({\mathbf {r} '})~c(|{\mathbf {r} }-{\mathbf {r} '}|){\rm {d}}{\mathbf {r} '}$

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):

$h({\mathbf {r} })=c({\mathbf {r} })+\rho \int c(|{\mathbf {r} }-{\mathbf {r} '}|)c({\mathbf {r} '}){\rm {d}}{\mathbf {r} '}$

$+\rho ^{2}\iint c(|{\mathbf {r} }-{\mathbf {r} '}|)c(|{\mathbf {r} '}-{\mathbf {r} ''}|)c({\mathbf {r} ''}){\rm {d}}{\mathbf {r} ''}{\rm {d}}{\mathbf {r} '}$

$+\rho ^{3}\iiint c(|{\mathbf {r} }-{\mathbf {r} '}|)c(|{\mathbf {r} '}-{\mathbf {r} ''}|)c(|{\mathbf {r} ''}-{\mathbf {r} '''}|)c({\mathbf {r} '''}){\rm {d}}{\mathbf {r} '''}{\rm {d}}{\mathbf {r} ''}{\rm {d}}{\mathbf {r} '}$

$+\rho ^{4}\iiiint c(|{\mathbf {r} }-{\mathbf {r} '}|)c(|{\mathbf {r} '}-{\mathbf {r} ''}|)c(|{\mathbf {r} ''}-{\mathbf {r} '''}|)c(|{\mathbf {r} '''}-{\mathbf {r} ''''}|)h({\mathbf {r} ''''}){\rm {d}}{\mathbf {r} ''''}{\rm {d}}{\mathbf {r} '''}{\rm {d}}{\mathbf {r} ''}{\rm {d}}{\mathbf {r} '}$

etc.

Diagrammatically this expression can be written as :

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 Ornstein-Zernike relation can be derived by performing a functional differentiation of the grand canonical distribution function.

Ornstein-Zernike relation in Fourier space

The Ornstein-Zernike equation may be written in Fourier space as ( Eq. 5):

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

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

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

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

Note:

${\hat {h}}(0)=\int h(r){\rm {d}}{\mathbf {r} }$

${\hat {c}}(0)=\int c(r){\rm {d}}{\mathbf {r} }$

References

Related reading

• Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5
• Yan He, Stuart A. Rice, and Xinliang Xu "Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids", Journal of Chemical Physics 145 234508 (2016)