The Ornstein-Zernike equation and integral equations

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

$$h(r)=c(r)+\int d^{3}r^{\prime }c(r^{\prime })\rho h(\vert \vec{r}-\vec{r}^{\prime }\vert).$$ (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 more but a definition of the total correlation function.

  

Figure 2.2: Contributions to the total correlation function.

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

$$\hat{h}(k) \hat{c}(k)+\rho \hat{c}(k)\hat{h}(k)$$ = (2.16)

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

Fourier transforms are defined by

$$\widehat{f}\left( \vec{k}\right) \int d^{3}rf\left( \vec{r}\right) \exp \left\{ i\vec{k}\cdot \vec{r}\right\}$$ = (2.18)

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

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

$$g(r)=\exp \{-\beta \phi (r)\}y(r)$$ (2.20)

we see that g(r) equals y(r) outside the range of the potential. In order to obtain a short ranged function to approximate c(r), we try

c(r)=g(r)-y(r). (2.21)

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

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

$$y(r)=\exp \{h(r)-c(r)\}.$$ (2.22)

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

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