Radial Distribution Function

When there are two solutes dissolved in water, the Brownian motion separates them by different distances r at different times. The radial distribution function, $g(r)$, gives the probability of finding a particle in the distance $r$ from another particle. If we count the appearance of two molecules at separation r, from $r=0$ to $r=\infty$, we can get the radial distribution function $g(r)$. The radial distribution function is a useful tool to describe the structure of a system, particularly of liquids. In a solid, the radial distribution function has an infinite number of sharp peaks whose separations and heights are characteristic of the lattice structure. Consider a spherical shell of thickness $\delta r$ at a distance $r$ from a chosen atom (Fig. 10).

Figure 10: Radial distribution functions use a spherical shell of thickness $\delta r$.

The volume of the shell is given by

$$V=\frac{4}{3}\pi(r+\delta r)^{3}-\frac{4}{3}\pi r^{3}\approx4\pi r^{2}\delta r.$$ (21)

If the number of particles per unit volume is $\rho$, then the total number in the shell is $4\pi\rho r^{2}\delta r$, and the number of atoms in the volume element varies as $r^{2}$.

The radial distribution function of a liquid is intermediate between the solid and the gas, with a small number of peaks as short distances, superimposed on a steady decay to a constant value at longer distances.

Figure 11: Radial distribution function determined from a $100$ ps molecular dynamics simulation of liquid argon at a temperature of $100$ K and a density of $1.396$ g/cm$^{3}$[15].

A typical radial distribution function calculated from a MD simulation is shown in Fig. 11. At short distances (less than atomic diameter) $g(r)$ is zero. This is due to the strong repulsive forces. The first (and large) peak occurs at $r\approx3.7$ Å, with $g(r)$ having a value of about $3$. This means that it is three times more likely that two molecules would be found at this separation. The radial distribution function then falls and passes through a minimum value around $r\approx5.4$ Å. The chances of finding two atoms with this separation are less. At long distances, $g(r)$ approaches one which indicates there is no long-rang order.

To calculate the pair distribution function from a simulation, the neighbors around each atom or molecule are sorted into distance bins. The number of neighbors in each bin is averaged over the entire simulation. For example, a count is made of the number of neighbors between $2.5$ and $2.75$, $2.75$ and $3.0$ Å and so on for every atom or molecule in the simulation. This count can be performed during the simulation itself or by analyzing the configurations that are generated.

Radial distribution function can be measured experimentally using X-ray diffraction. The regular arrangement of the atoms in a crystal gives the characteristic X-ray diffraction pattern with bright, sharp spots. For liquids, the diffraction pattern has regions of high and low intensity but no sharp spots. The X-ray diffraction pattern is analyzed to estimate an experimental distribution function, which is compared with the results obtained from the simulation.

Thermodynamic properties can be studied by calculating the radial distribution function. For example, in the calculation for the energy of a real gas, we consider the spherical shell of volume $4\pi r^{2}\delta r$ that contains $4\pi^{2}\rho g(r)\delta r$ particles. If the pair potential at a distance $r$ has a value $u(r)$, the energy of interaction between the particles in the shell and the central particle is $4\pi r^{2}\rho g(r)u(r)\delta r$. The total potential energy of the real gas is obtained by integrating $r$ from $0$ to $\infty$ and multiplied by $N/2$ (the factor $1/2$ ensures that we only count each interaction once). The total energy is

$$E=\frac{3}{2}Nk_{B}T+2\pi N\rho\int_{0}^{\infty}r^{2}u(r)g(r)dr.$$ (22)

It is usually more accurate to calculate such properties directly, partly because the radial distribution function is derived by dividing the space into small bins.

For molecule, the orientation must be taken into account if the true nature of the distribution is to be determined. The radial distribution function for molecules is usually measured between two fixed points, such as between the centers of mass. This may then be supplemented by an orientation distribution function. For linear molecules, the orientational distribution function may be calculated as the angle between the axes of the molecule, with values ranging from $-180^{\circ}$to $+180^{\circ}$. For more complex molecules one usually calculates a number of site-site distribution functions. For example, for a three-site model of water, three functions can be defined: $g_{OO},$ $g_{OH}$ and $g_{HH}$. An advantage of the site-site model is that they can be directly related to information obtained from X-ray scattering experiments. The O-O, O-H, and H-H radial distribution functions have been particularly useful for refining various potential models for simulating liquid water.