Ornstein–Zernike equation

The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to the text book for the full derivation.[2]

It is convenient to define the total correlation function:

${\displaystyle h(r_{12})=g(r_{12})-1\,}$

which is a measure for the "influence" of molecule 1 on molecule 2 at a distance ${\displaystyle r_{12}}$ away with ${\displaystyle g(r_{12})}$ as the radial distribution function. In 1914 Ornstein and Zernike proposed[3] to split this influence into two contributions, a direct and indirect part. The direct contribution is defined to be given by the direct correlation function, denoted ${\displaystyle c(r_{12})}$. The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as

${\displaystyle h(r_{12})=c(r_{12})+\rho \int d\mathbf {r} _{3}c(r_{13})h(r_{32})\,}$

which is called the Ornstein–Zernike equation. Its interest is that, by eliminating the indirect influence, ${\displaystyle c(r)}$ is shorter-ranged than ${\displaystyle h(r)}$ and can be more easily described.

If we define the distance vector between two molecules ${\displaystyle \mathbf {r} _{ij}:=\mathbf {r} _{i}-\mathbf {r} _{j}}$ for ${\displaystyle i,j=1,2,3}$, the OZ equation can be rewritten using a convolution.

${\displaystyle h(\mathbf {r} _{12})=c(\mathbf {r} _{12})+\rho \int d\mathbf {r} _{3}c(\mathbf {r} _{12}-\mathbf {r} _{32})h(\mathbf {r} _{32})=c(\mathbf {r} _{12})+\rho \int d\mathbf {r} _{32}c(\mathbf {r} _{12}-\mathbf {r} _{32})h(\mathbf {r} _{32})=c(\mathbf {r} _{12})+\rho (c\,*\,h)(\mathbf {r} _{12})\,}$.

If we then denote the Fourier transforms of ${\displaystyle h(\mathbf {r} )}$ and ${\displaystyle c(\mathbf {r} )}$ by ${\displaystyle {\hat {H}}(\mathbf {k} )}$ and ${\displaystyle {\hat {C}}(\mathbf {k} )}$, respectively, and use the convolution theorem we obtain

${\displaystyle {\hat {H}}(\mathbf {k} )={\hat {C}}(\mathbf {k} )+\rho {\hat {H}}(\mathbf {k} ){\hat {C}}(\mathbf {k} ),\,}$

which yields

${\displaystyle {\hat {C}}(\mathbf {k} )={\frac {{\hat {H}}(\mathbf {k} )}{1+\rho {\hat {H}}(\mathbf {k} )}}\,\,\,\,{\text{and}}\,\,\,\,\,\,\,{\hat {H}}(\mathbf {k} )={\frac {{\hat {C}}(\mathbf {k} )}{1-\rho {\hat {C}}(\mathbf {k} )}}.\,}$

One needs to solve for both ${\displaystyle h(r)}$ and ${\displaystyle c(r)}$ (or, equivalently, their Fourier transforms). This requires an additional equation, known as a closure relation. The Ornstein–Zernike equation can be formally seen as a definition of the direct correlation function ${\displaystyle c(r)}$ in terms of the total correlation function ${\displaystyle h(r)}$. The details of the system under study (most notably, the shape of the interaction potential ${\displaystyle u(r)}$) are taken into account by the choice of the closure relation. Commonly used closures are the Percus–Yevick approximation, well adapted for particles with an impenetrable core, and the hypernetted-chain equation, widely used for "softer" potentials. More information can be found in.[4]

Closure relations are independent second equations that connect the total correlation ${\displaystyle h(r)}$ and the direct correlation ${\displaystyle c(r)}$. The Ornstein-Zernike equation and a second equation are needed in order to solve for two unknowns: the total correlation ${\displaystyle h(r)}$ and the direct correlation ${\displaystyle c(r)}$.[1] The word "closure" means that it closes or "completes" the conditions for a unique determination of ${\displaystyle h(r)}$ and ${\displaystyle c(r)}$.[1]

• Percus–Yevick approximation, a closure relation for solving the OZ equation
• Hypernetted-chain equation, a closure relation for solving the OZ equation

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