Ornstein-Zernike relation

Notation used:

• $g(r)$ is the pair distribution function.
• $\mathrm{\Phi }(r)$ 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:

$h=h[c]$

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

$c=c[h]$

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)

$\gamma (\mathbf{r})\equiv h(\mathbf{r})-c(\mathbf{r})=\rho \int h({\mathbf{r}}^{\prime })c(|\mathbf{r}-{\mathbf{r}}^{\prime }|)\mathrm{d}{\mathbf{r}}^{\prime }$

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):

$h(\mathbf{r})=c(\mathbf{r})+\rho \int c(|\mathbf{r}-{\mathbf{r}}^{\prime }|)c({\mathbf{r}}^{\prime })\mathrm{d}{\mathbf{r}}^{\prime }$

$+{\rho }^2\iint c(|\mathbf{r}-{\mathbf{r}}^{\prime }|)c(|{\mathbf{r}}^{\prime }-{\mathbf{r}}^{″}|)c({\mathbf{r}}^{″})\mathrm{d}{\mathbf{r}}^{″}\mathrm{d}{\mathbf{r}}^{\prime }$

$+{\rho }^3\iiint c(|\mathbf{r}-{\mathbf{r}}^{\prime }|)c(|{\mathbf{r}}^{\prime }-{\mathbf{r}}^{″}|)c(|{\mathbf{r}}^{″}-{\mathbf{r}}^{‴}|)c({\mathbf{r}}^{‴})\mathrm{d}{\mathbf{r}}^{‴}\mathrm{d}{\mathbf{r}}^{″}\mathrm{d}{\mathbf{r}}^{\prime }$

$+{\rho }^4⨌c(|\mathbf{r}-{\mathbf{r}}^{\prime }|)c(|{\mathbf{r}}^{\prime }-{\mathbf{r}}^{″}|)c(|{\mathbf{r}}^{″}-{\mathbf{r}}^{‴}|)c(|{\mathbf{r}}^{‴}-{\mathbf{r}}^{⁗}|)h({\mathbf{r}}^{⁗})\mathrm{d}{\mathbf{r}}^{⁗}\mathrm{d}{\mathbf{r}}^{‴}\mathrm{d}{\mathbf{r}}^{″}\mathrm{d}{\mathbf{r}}^{\prime }$

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):

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

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

$\hat{\gamma }(k)=\frac{4\pi }{k}\underset{0}{\overset{\mathrm{\infty }}{\int }}r\sin (kr)\gamma (r)dr$

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

Note:

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

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

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)