Hypernetted-chain equation

The direct correlation function represents the direct correlation between two particles in a system containing N − 2 other particles. It can be represented by

${\displaystyle c(r)=g_{\rm {total}}(r)-g_{\rm {indirect}}(r)\,}$

where ${\displaystyle g_{\rm {total}}(r)=g(r)=\exp[-\beta w(r)]}$ (with ${\displaystyle w(r)}$ the potential of mean force) and ${\displaystyle g_{\rm {indirect}}(r)}$ is the radial distribution function without the direct interaction between pairs ${\displaystyle u(r)}$ included; i.e. we write ${\displaystyle g_{\rm {indirect}}(r)=\exp\{-\beta [w(r)-u(r)]\}}$. Thus we approximate ${\displaystyle c(r)}$ by

${\displaystyle c(r)=e^{-\beta w(r)}-e^{-\beta [w(r)-u(r)]}.\,}$

By expanding the indirect part of ${\displaystyle g(r)}$ in the above equation and introducing the function ${\displaystyle y(r)=e^{\beta u(r)}g(r)(=g_{\rm {indirect}}(r))}$ we can approximate ${\displaystyle c(r)}$ by writing:

${\displaystyle c(r)=e^{-\beta w(r)}-1+\beta [w(r)-u(r)]\,=g(r)-1-\ln y(r)\,=f(r)y(r)+[y(r)-1-\ln y(r)]\,\,({\text{HNC}}),}$

with ${\displaystyle f(r)=e^{-\beta u(r)}-1}$.

This equation is the essence of the hypernetted chain equation. We can equivalently write

${\displaystyle h(r)-c(r)=g(r)-1-c(r)=\ln y(r).}$

If we substitute this result in the Ornstein–Zernike equation

${\displaystyle h(r_{12})-c(r_{12})=\rho \int c(r_{13})h(r_{23})d\mathbf {r} _{3},}$

one obtains the hypernetted-chain equation:

${\displaystyle \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho \int \left[h(r_{13})-\ln g(r_{13})-\beta u(r_{13})\right]h(r_{23})\,d\mathbf {r_{3}} .\,}$

• Classical-map hypernetted-chain method
• Percus–Yevick approximation – another closure relation
• Ornstein–Zernike equation