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The Ornstein-Zernike equation and integral equations

As we have stressed at the beginning of theprevious section, the surplus of density $\rho (r)-\rho =\rho h(r)$ may beregarded as being the convolution of a short range function and $\rho h(r)$ itself. We formalize this by introducing the direct correlation functionc(r) according to

$\begin{displaymath}h(r)=c(r)+\int d^{3}r^{\prime }c(r^{\prime })\rho h(\vert\vec{r}-\vec{r}^{\prime }\vert).\end{displaymath}$

(2.15)

The total correlation at $\vec{r}$ is the sum of a direct correlation plus an indirect contribution coming from all surrounding points:the surplus induced at $\vec{r}^{\prime }$ causes an effect at $\vec{r}$ (see Fig. (2.2)). Notice that Eq. (2.15) is nothing morebut a definition of the total correlation function.

**Figure 2.2:**Contributions to the total correlation function.
$\begin{figure}\setlength{\unitlength}{1mm}\begin{picture}(100,55)(-15,0)\pu......$ }\put(60,37){$\rho h(\vert\vec{r}-\vec{r}'\vert)$ }\end{picture}\end{figure}$

A simple relation exists betweenS(k) and the Fourier transform $\hat{c}(k)$ ofc(r). Fourier transforming Eq. (2.15) we get

$\displaystyle \hat{h}(k)$	=	$\displaystyle \hat{c}(k)+\rho \hat{c}(k)\hat{h}(k)$	(2.16)
S(k)	=	$\displaystyle 1+\rho \hat{h}(k)=\frac{1}{1-\rho \hat{c}(k)}.$	(2.17)

Fourier transforms are defined by

$\displaystyle \widehat{f}\left( \vec{k}\right)$	=	$\displaystyle \int d^{3}rf\left( \vec{r}\right) \exp\left\{ i\vec{k}\cdot \vec{r}\right\}$	(2.18)
$\displaystyle f\left( \vec{r}\right)$	=	$\displaystyle \frac{1}{\left( 2\pi \right) ^{3}}\int d^{3}k\hat{f}\left( \vec{k}\right) \exp \left\{ -i\vec{k}\cdot \vec{r}\right\}$	(2.19)

We have introduced the direct correlation function such that it is a shortrange function. Writing

$\begin{displaymath}g(r)=\exp \{-\beta \phi (r)\}y(r)\end{displaymath}$

(2.20)

we see thatg(r) equalsy(r) outside the range of the potential. Inorder to obtain a short ranged function to approximatec(r), we try

c(r)=g(r)-y(r).

(2.21)

This equation is called the Percus-Yevick closure. Together with theOrnstein-Zernike equation it constitutes the Percus-Yevick equation .

For a first principles derivation of the Percus-Yevick equation we refer tothe literature mentioned above. Using a slightly different approximationthan the one producing the PY closure, one may also derive

$\begin{displaymath}y(r)=\exp \{h(r)-c(r)\}.\end{displaymath}$

(2.22)

This result is called the hypernetted chain closure. Together with the OZequation it gives the HNC equation .

Here we restrict ourselves to mentioning that both equations have been verysuccessful in predicting correlation functions, the PY equation being themore successful one in the case of hard spheres , and the HNC being the more successful one in the caseof Lennard-Jones atoms.

Next:Molecular liquids Up:Integral equations for polymer Previous:The radial distribution function W.J. Briels

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