Radial distribution function

In statistical mechanics, a radial distribution function (RDF), g(r), describes how the atomic density varies as a function of the distance from one particular atom.

Suppose, for example, that we choose an atom 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 ${\displaystyle \rho =N/V}$ is the average density, then the mean density at P given that there is an atom 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 atoms arising from the forces they exert on each other:

(mean local density at distance r from O) = ${\displaystyle \rho }$g(r) (1)

As long as the gas is dilute, the correlations in the positions of the atoms that g(r) takes into account are due to the potential ${\displaystyle \phi }$(r) that an atom at P feels owing to the presence of an atom at O. Using the Boltzmann distribution law:

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

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

Radial distribution function for the Lennard-Jones model fluid at ${\displaystyle T^{*}=0.71,\;n^{*}=0.844}$.

When the density of the gas gets higher, the so called low-density limit (2) is not applicable anymore because the atoms attracted to and repelled by the atom at O also repel and attract each other. The correction terms needed to correctly describe g(r) resemble the virial equation, which is an expansion in the density:

${\displaystyle g(r)=e^{-\phi (r)/kT}+\rho g_{1}(r)+\rho ^{2}g_{2}(r)+\ldots }$ (3)

in which additional functions ${\displaystyle g_{1}(r),\,g_{2}(r)}$ appear which may depend on temperature ${\displaystyle T}$ and distance ${\displaystyle r}$ but not on density, ${\displaystyle \rho }$.

Given a potential energy function, the radial distribution function can be found via computer simulation methods like the Monte Carlo method. It could also be calculated numerically using rigorous methods obtained from statistical mechanics like the Perckus-Yevick approximation, or the Hypernetted Chain Theory.

Importance of g(r)

g(r) is of fundamental importance in thermodynamics because macroscopic thermodynamic quantities can be calculated using g(r). A few examples:

The virial equation for the pressure:

${\displaystyle p=\rho kT-{\frac {2\pi }{3}}\rho ^{2}\int drr^{3}u^{\prime }(r)g(r,\rho ,T)}$  (4)

The energy equation:

${\displaystyle {\frac {E}{NkT}}={\frac {3}{2}}+{\frac {\rho }{2kT}}\int dr\,4\pi r^{2}u(r)g(r,\rho ,T)}$  (5)

The compressibility equation:

${\displaystyle kT\left({\frac {\partial \rho }{\partial p}}\right)=1+\rho \int dr[g(r)-1]}$  (6)

Experimental

It is possible to measure g(r) experimentally with neutron scattering or x-ray scattering diffraction data. In such an experiment, a sample is bombarded with neutrons or x-rays which then diffract in all directions. The average atomic density at each distance can be extracted according to Snells law: r=wavelength/sin(scattered angle), where r is the distance the neutron traveled during diffraction.
For an example of an RDF experiment see 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 atom 1 in ${\displaystyle d{\rm {{r}_{1}}}}$ , atom 2 in ${\displaystyle d{\rm {{r}_{2}}}}$ , etc., is given by

${\displaystyle P^{(N)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{N})d{\rm {{r}_{1}\cdots d{\rm {{r}_{N}={\frac {e^{-\beta U_{N}}d{\rm {{r}_{1}\cdots d{\rm {{r}_{N}}}}}}{Z_{N}}}\,}}}}}}}}}$  (7)

where ${\displaystyle \beta ={\frac {1}{kT}}}$  and ${\displaystyle Z_{N}}$  is the configurational integral. To obtain the probability of finding atom 1 in ${\displaystyle d{\rm {{r}_{1}}}}$  and atom n in ${\displaystyle d{\rm {{r}_{n}}}}$ , irrespective of the remaining N-n atoms, one has to integrate (7) over the coordinates of atom n + 1 through N:

${\displaystyle P^{(n)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{n})={\frac {\int \cdots \int e^{-\beta U_{N}}d{\rm {{r}_{n+1}\cdots d{\rm {{r}_{N}}}}}}{Z_{N}}}\,}}}}}$  (8)

Now the probability that any atom is in ${\displaystyle d{\rm {{r}_{1}}}}$  and any atom in ${\displaystyle d{\rm {{r}_{n}}}}$ , irrespective of the rest of the atom, is

${\displaystyle \rho ^{(n)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{n})={\frac {N!}{(N-n)!}}\cdot P^{(n)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{n})\,}}}}}}}}}$  (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:

${\displaystyle {\frac {1}{V}}\int \rho ^{(1)}({\rm {{r}_{1})d{\rm {{r}_{1}=\rho ^{(1)}={\frac {N}{V}}=\rho \,}}}}}$  (10)

It is now time to introduce a correlation function ${\displaystyle g^{(n)}}$  by

${\displaystyle \rho ^{(n)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{n})=\rho ^{n}g^{(n)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{n})\,}}}}}}}}}$  (11)

${\displaystyle g^{(n)}}$  is called a correlation function, since if the atoms are independent from each other ${\displaystyle \rho ^{(n)}}$  would simply equal ${\displaystyle \rho ^{n}}$  and therefore ${\displaystyle g^{(n)}}$  corrects for the correlation between atoms.

From (9) it can be shown that

${\displaystyle g^{(n)}({\rm {{r}_{1},\ldots ,{\rm {{r}_{n})={\frac {V^{n}N!}{N^{n}(N-n)!}}\cdot {\frac {\int \cdots \int e^{-\beta U_{N}}d{\rm {{r}_{n+1}\cdots d{\rm {{r}_{N}}}}}}{Z_{N}}}\,}}}}}$  (12)

In the theory of liquids ${\displaystyle g^{(2)}({\rm {{r}_{1},{\rm {{r}_{2})}}}}}$  is of special importance for it can be determined experimentally using X-ray diffraction. If the liquid contains spherically symmetric atoms ${\displaystyle g^{(2)}({\rm {{r}_{1},{\rm {{r}_{2})}}}}}$  depends only on the relative distance between atoms, ${\displaystyle {\rm {{r}_{12}}}}$ . People usually drop the subscripts: ${\displaystyle g(r)=g^{(2)}(r_{12})}$ . Now ${\displaystyle \rho g(r)d{\rm {r}}}$  is the probability of finding an atom at r given that there is an atom at the origin of r. Note that this probability is not normalized:

${\displaystyle \int _{0}^{\infty }\rho g(r)4\pi r^{2}dr=N-1\approx N}$  (13)

In fact, equation 13 gives us the number of atoms between r and r + d r about a central atom.

Currently, information on how to obtain the higher order distribution functions (${\displaystyle g^{(3)}({\rm {{r}_{1},{\rm {{r}_{2},{\rm {{r}_{3})}}}}}}}$ , etc.) is not available, and scientists rely on approximations based upon statistical mechanics.

References

1. B. Widom, Statistical Mechanics: A Concise Introduction for Chemists (Cambridge) 2002
2. D.A. McQuarrie, Statistical Mechanics (Harper Collins Publishers) 1976