- Radial distribution function
In

computational mechanics andstatistical mechanics , a**radial distribution function**(RDF), "g"("r"), describes how the density of surrounding matter varies as a function of the distance from a distinguished point.Suppose, for example, that we choose a molecule at some point O in the volume. What is then the average density at some point P at a distance r away from O? If $ho=N/V$ is the average density, then the mean density at P "given" that there is a molecule at O would differ from ρ by some factor g(r). One could say that the radial distribution function takes into account the correlations in the distribution of molecules arising from the forces they exert on each other:

(mean local density at distance r from O) = $ho$g(r) (1)As long as the gas is

**dilute**the correlations in the positions of the molecules that g(r) takes into account are due to the potential $phi$(r) that a molecule at P feels owing to the presence of a molecule at O. Using the Boltzmann distribution law:

$g(r)\; =\; e^\{-phi(r)/kT\}\; ,$ (2)

If $phi(r)$ was zero for all r - i.e., if the molecules did not exert any influence on each other g(r) = 1 for all r. Then from (1) the mean local density would be equal to the mean density $ho$: the presence of a molecule at O would not influence the presence or absence of any other molecule and the gas would be ideal. As long as there is a $phi(r)$ the mean local density will always be different from the mean density $ho$ due to the interactions between molecules.

When the density of the gas gets higher the so called low-density limit (2) is not applicable anymore because the molecules attracted to and repelled by the molecule at O also repel and attract each other. The correction terms needed to correctly describe g(r) resembles the

virial equation , it is an expansion in the density:

$g(r)=e^\{-phi(r)/kT\}+\; ho\; g\_\{1\}(r)+\; ho^\{2\}g\_\{2\}(r)+ldots$ (3)

in which additional functions $g\_\{1\}(r),\; ,\; g\_\{2\}(r)$ appear which may depend on temperature $T$ and distance $r$ but not on density, $ho$.

Given a

potential energy function, the radial distribution function can be found via computer simulation methods like theMonte Carlo method . It could also be calculated numerically using rigorous methods obtained fromstatistical mechanics like thePerckus-Yevick approximation .**Importance of g(r)**g(r) is of fundamental importance in thermodynamics for macroscopic thermodynamic quantities can be calculated using g(r). A few examples:

"The virial equation for the pressure:" :$p=\; ho\; kT-frac\{2pi\}\{3\}\; ho^\{2\}int\; d\; r\; r^\{3\}\; u^\{prime\}(r)\; g(r,\; ho,\; T)$ (4)

"The energy equation:" :$frac\{E\}\{NkT\}=frac\{3\}\{2\}+frac\{\; ho\}\{2kT\}int\; d\; r\; ,4pi\; r^\{2\}\; u(r)g(r,\; ho,\; T)$ (5)

"The compressibility equation:" :$kTleft(frac\{partial\; ho\}\{partial\; p\}\; ight)=1+\; ho\; int\; d\; r\; [g(r)-1]$ (6)

**Experimental**It is possible to measure g(r) experimentaly with

neutron scattering orx-ray scattering diffraction data. In such an experiment, a sample is bombarded with neutrons or x-ray which then diffract to all directions. The average molecular density at each distance can be extracted in according toSnells law : r=wavelength/sin(scattered angle), where r is the distance the neutron traveled during diffraction.

For an example of RDF experiment see [*http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JCPSA6000125000001014508000001&idtype=cvips&gifs=yes Eigen vs. Zundel structures in HCl solution, 2006*]**Formal derivation**Consider a system of "N" particles in a volume "V" and at a temperature "T". The probability of finding molecule 1 in $d\; m\{r\}\_\{1\}$, molecule 2 in $d\; m\{r\}\_\{2\}$, etc., is given by

$P^\{(N)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{N\})\; d\; m\{r\}\_\{1\},cdots,\; d\; m\{r\}\_\{N\}=frac\{e^\{-eta\; U\_\{Nd\; m\{r\}\_\{1\}\; cdots\; d\; m\{r\}\_\{N\{Z\_\{N\; ,$ (7)where $Z\_\{N\}$ is the configurational integral. To obtain the probability of finding molecule 1 in $d\; m\{r\}\_\{1\}$ and molecule "n" in $d\; m\{r\}\_\{n\}$, irrespective of the remaining "N-n" molecules, one has to integrate (7) over the coordinates of molecule "n" + 1 through "N":

$P^\{(n)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{n\})\; =frac\{int\; cdots\; int\; e^\{-eta\; U\_\{Nd\; m\{r\}\_\{n+1\}\; cdots\; d\; m\{r\}\_\{N\{Z\_\{N\; ,$ (8)Now the probability that "any" molecule is in $d\; m\{r\}\_\{1\}$ and "any" molecule in $d\; m\{r\}\_\{n\}$, irrespective of the rest of the molecules, is

$ho^\{(n)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{n\})\; =frac\{N!\}\{(N-n)!\}\; cdot\; P^\{(n)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{n\})\; ,$ (9)For "n" = 1 the one particle distribution function is obtained which, for a crystal, is a periodic function with sharp maxima at the lattice sites. For a (homogeneous) liquid:

$frac\{1\}\{V\}\; int\; ho^\{(1)\}(\; m\{r\}\_\{1\})d\; m\{r\}\_\{1\}=\; ho^\{(1)\}=frac\{N\}\{V\}=\; ho\; ,$ (10)It is now time to introduce a correlation function $g^\{(n)\}$ by

$ho^\{(n)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{n\})=\; ho^\{n\}g^\{(n)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{n\})\; ,$ (11)$g^\{(n)\}$ is called a correlation function since if the molecules are independent from each other $ho^\{(n)\}$ would simply equal $ho^\{n\}$ and therefore $g^\{(n)\}$ corrects for the correlation between molecules.

From (9) it can be shown that

$g^\{(n)\}(\; m\{r\}\_\{1\},ldots,\; m\{r\}\_\{n\})=frac\{V^\{N\}N!\}\{N^\{n\}(N-n)!\}cdotfrac\{int\; cdots\; int\; e^\{-eta\; U\_\{Nd\; m\{r\}\_\{n+1\},cdots,\; m\{r\}\_\{N\{Z\_\{N\; ,$ (12)In the theory of liquids $g^\{(2)\}(\; m\{r\}\_\{1\},\; m\{r\}\_\{2\})$ is of special importance for it can be determined experimentally using

X-ray diffraction . If the liquid contains spherically symmetric molecules $g^\{(2)\}(\; m\{r\}\_\{1\},\; m\{r\}\_\{2\})$ depends only on the relative distance between molecules, $m\{r\}\_\{12\}$. People usually drop the subscripts: $g(r)=g^\{(2)\}(r\_\{12\})$. Now $ho\; g(r)\; d\; m\{r\}$ is the probability of finding a molecule at**r**given that there is a molecule at the origin of**r**. Note that this probability is not normalized:

$int\_\{0\}^\{infty\}\; ho\; g(r)\; 4pi\; r^\{2\}\; dr\; =\; N-1\; approx\; N$ (13)In fact, equation 13 gives us the number of molecules between r and r + d r about a central molecule.

As of current, information on how to obtain the higher order distribution functions ($g^\{(3)\}(\; m\{r\}\_\{1\},\; m\{r\}\_\{2\},\; m\{r\}\_\{3\})$, etc.) is not known and scientists rely on approximations based upon

statistical mechanics .**References**#D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976

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