Molecular liquids

We shall now generalize the results of the previous section to molecular liquids . We view the liquid at first as being a mixture of atomic liquids, containing as many components as there are atoms in the molecule. Eq. (2.3) then simply generalizes to

$$\rho _{\alpha \beta }(r)=G_{\alpha \beta }(r)\rho _{\beta }$$ (2.23)

where $\rho _{\alpha \beta }$ is the density of $\beta$-atoms at a distance r from a given $\alpha$-atom. We use capital $G_{\alpha \beta }(r)$ here for reasons to become clear in a minute. Defining

$$H_{\alpha \beta }(r)=G_{\alpha \beta }(r)-1$$ (2.24)

the OZ equation becomes

$$H_{\alpha \beta }(r)\rho _{\beta }=C_{\alpha \beta }(r)\rho _... ...a \beta }(\vert \vec{r}-\vec{r}^{\prime }\vert)\rho _{\beta }.$$ (2.25)

Dividing by $\rho _{\beta }$ and Fourier transforming we get in matrix notation

$$\hat{H}(k)=\hat{C}(k)+\hat{C}(k)\rho \hat{H}(k).$$ (2.26)

Here $\rho$ is the diagonal matrix with elements $\rho _{\alpha \beta }=\delta _{\alpha \beta }\rho _{\alpha }$. The transition to $\vec{k}$-space is only for reason of simpler notation.

We of course know that the correlations between the atoms are partly due to the fact that they form molecules. Therefore we write

 

$$\hat{H}(k)\rho [\hat{H}(k)\rho ]_{\mathrm{intra}}+[\hat{H}(k)\rho ]_{ \mathrm{inter}}$$ =

$$[\hat{\Omega}(k)-1]+\hat{h}(k)\rho$$ = (2.27)

where $\hat{\Omega}(k)-1$ stems from the intramolecular correlations  and $\hat{h}(k)\rho$ from the intermolecular correlations .

In the ideal gas limit we may approximate

$$\hat{H}(k)\rho = \hat{\Omega}(k) -1$$ (2.28)

which, when introduced into the OZ equation (2.26) yields

$$\hat{C}(k)\rho = 1 - \hat{\Omega}(k)^{-1} .$$ (2.29)

We therefore write in general

$$\hat{C}(k)\rho = 1 - \hat{\Omega}(k)^{-1} + \hat{c}(k)\rho$$ (2.30)

where $\hat{c} (k)$ is called the intermolecular part of the direct correlation function matrix. Introducing Eqs. (2.27) and ( 2.30) into the OZ equation we get after some algebra

$$\hat{h}(k) = \hat{\Omega}(k)\hat{c}(k)\hat{\Omega}(k) + \hat{\Omega}(k)\hat{c }(k)\rho\hat{h}(k)$$ (2.31)

where we have used the fact that $\rho$ and $\hat{\Omega}(k)$ commute. For pure molecular liquids this follows from the fact that $\rho$ is equal to the molecular density times the unit matrix. A proof for molecular mixtures is a little bit more involved. Once we know the intramolecular correlation $\hat{\Omega}(k)$ we may calculate the intermolecular correlation from Eq. ( 2.31) by using one of the closure relations

$$c_{\alpha\beta}(r) g_{\alpha\beta}(r) - y_{\alpha\beta}(r) \mbox{(PY)}$$ = (2.32)

$$y_{\alpha\beta}(r) \exp\{h_{\alpha\beta}(r)-c_{\alpha\beta}(r)\} \mbox{(HNC)}$$ = (2.33)

which we simply try in analogy with similar calculations on simple liquids. Of course this may turn out to be a very crude approximation. Experience so far shows that it works well as long as one is interested in describing the local structure in the liquid. For very small values of the wavevector k, or equivalently at large values of r problems occur which make thermodynamic predictions unreliable.

We end this section with some remarks about the intramolecular correlation  function. We Fouriertransform Eq. (2.27), and next add $\rho _{\beta }$ to each of its elements, obtaining

$$G_{\alpha \beta }(r)= \frac{1}{\rho _{\beta }}\{\Omega _{\alp... ...}(r)-\delta _{\alpha \beta }\delta (r)\}+g_{\alpha \beta }(r).$$ (2.34)

Here $\rho _{\beta }G_{\alpha \beta }(r)$ is the density of $\beta$-atoms at a distance r from an $\alpha$-atom; when $\alpha =\beta$ the atom at the origin does, by definition, not contribute to the density. $\rho _{\beta }g_{\alpha \beta }(r)$ is defined in exactly the same way, with the proviso that the $\beta$-atom should be on another molecule than the $\alpha$-atom. Using these definitions we see that $\Omega _{\alpha \beta }(r)-\delta _{\alpha \beta }\delta (r)$ is the density of $\beta$-atoms at a distance r from an $\alpha$-atom, where $\alpha$ and $\beta$ should be on the same molecule, and where, in the case $\alpha =\beta$, the atom at the origin does not contribute to the density. So finally we conclude that $\Omega _{\alpha \beta }(r)$ is the density of $\beta$-atoms at a distance r from an $\alpha$-atom, with the $\delta$-peak included when $\alpha =\beta$. In the latter case $\Omega _{\alpha \alpha }=\delta (r)$, because the total number of $\alpha$-atoms in each molecule equals one.