Hypernetted-chain equation

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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:

where is the number density of molecules, , is the radial distribution function, is the direct interaction between pairs. with being the Thermodynamic temperature and the Boltzmann constant.


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

where (with the potential of mean force) and is the radial distribution function without the direct interaction between pairs included; i.e. we write . Thus we approximate by

By expanding the indirect part of in the above equation and introducing the function we can approximate by writing:

with .

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

If we substitute this result in the Ornstein–Zernike equation

one obtains the hypernetted-chain equation:

See also[edit]