In statistical mechanics the hypernetted-chain equation is a closure relation to solve the Ornstein–Zernike equation which relates the direct correlation function to the total correlation function. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function. It is given by:
ln y ( r 12 ) = ln g ( r 12 ) + β u ( r 12 ) = ρ ∫ [ h ( r 13 ) − ln g ( r 13 ) − β u ( r 13 ) ] h ( r 23 ) d r 3 , {\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}} ,\,}
where ρ = N V {\displaystyle \rho ={\frac {N}{V}}} is the number density of molecules, h ( r ) = g ( r ) − 1 {\displaystyle h(r)=g(r)-1} , g ( r ) {\displaystyle g(r)} is the radial distribution function, u ( r ) {\displaystyle u(r)} is the direct interaction between pairs. β = 1 k B T {\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}} with T {\displaystyle T} being the Thermodynamic temperature and k B {\displaystyle k_{\rm {B}}} the Boltzmann constant.
Derivation The direct correlation function represents the direct correlation between two particles in a system containing N − 2 other particles. It can be represented by
c ( r ) = g t o t a l ( r ) − g i n d i r e c t ( r ) {\displaystyle c(r)=g_{\rm {total}}(r)-g_{\rm {indirect}}(r)\,}
where g t o t a l ( r ) = g ( r ) = exp [ − β w ( r ) ] {\displaystyle g_{\rm {total}}(r)=g(r)=\exp[-\beta w(r)]} (with w ( r ) {\displaystyle w(r)} the potential of mean force) and g i n d i r e c t ( r ) {\displaystyle g_{\rm {indirect}}(r)} is the radial distribution function without the direct interaction between pairs u ( r ) {\displaystyle u(r)} included; i.e. we write g i n d i r e c t ( r ) = exp { − β [ w ( r ) − u ( r ) ] } {\displaystyle g_{\rm {indirect}}(r)=\exp\{-\beta [w(r)-u(r)]\}} . Thus we approximate c ( r ) {\displaystyle c(r)} by
c ( r ) = e − β w ( r ) − e − β [ w ( r ) − u ( r ) ] . {\displaystyle c(r)=e^{-\beta w(r)}-e^{-\beta [w(r)-u(r)]}.\,}
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