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Hypernetted-chain equation
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<h2 id="derivation">Derivation</h2> <p>The direct correlation function represents the direct correlation between two particles in a system containing <i>N</i> − 2 other particles. It can be represented by </p> 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)\,} <p>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 <a href="/facts/Potential_of_mean_force/rSRzkSJp">potential of mean force</a>) 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 <i>approximate</i> c ( r ) {\displaystyle c(r)} by </p> 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)]}.\,} <p>By expanding the indirect part of g ( r ) {\displaystyle g(r)} in the above equation and introducing the function y ( r ) = e β u ( r ) g ( r ) ( = g i n d i r e c t ( r ) ) {\displaystyle y(r)=e^{\beta u(r)}g(r)(=g_{\rm {indirect}}(r))} we can approximate c ( r ) {\displaystyle c(r)} by writing: </p> c ( r ) = e − β w ( r ) − 1 + β [ w ( r ) − u ( r ) ] = g ( r ) − 1 − ln y ( r ) = f ( r ) y ( r ) + [ y ( r ) − 1 − ln y ( r ) ] ( HNC ) , {\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}}),} <p>with f ( r ) = e − β u ( r ) − 1 {\displaystyle f(r)=e^{-\beta u(r)}-1} . </p><p>This equation is the essence of the hypernetted chain equation. We can equivalently write </p> h ( r ) − c ( r ) = g ( r ) − 1 − c ( r ) = ln y ( r ) . {\displaystyle h(r)-c(r)=g(r)-1-c(r)=\ln y(r).} <p>If we substitute this result in the <a href="/facts/Ornstein%25E2%2580%2593Zernike_equation/FBUedwnW">Ornstein–Zernike equation</a> </p> h ( r 12 ) − c ( r 12 ) = ρ ∫ c ( r 13 ) h ( r 23 ) d r 3 , {\displaystyle h(r_{12})-c(r_{12})=\rho \int c(r_{13})h(r_{23})d\mathbf {r} _{3},} <p>one obtains the hypernetted-chain equation: </p> 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}} .\,} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Classical-map_hypernetted-chain_method/wPfAEwKu">Classical-map hypernetted-chain method</a></li> <li><a href="/facts/Percus%25E2%2580%2593Yevick_approximation/5vzRVMQI">Percus–Yevick approximation</a> – another closure relation</li> <li><a href="/facts/Ornstein%25E2%2580%2593Zernike_equation/FBUedwnW">Ornstein–Zernike equation</a></li></ul>