Radial distribution functions

gmx rdf

The radial distribution function (RDF) or pair correlation function $g_{AB}(r)$ between particles of type $A$ and $B$ is defined in the following way:

(437)$$\begin{array}{r}\begin{array}{rcl}{g}_{AB}(r)& =& {\displaystyle \frac{⟨{\rho }_B(r)⟩}{⟨{\rho }_B{⟩}_{local}}}\\ & =& {\displaystyle \frac{1}{⟨{\rho }_B{⟩}_{local}}}{\displaystyle \frac{1}{N_A}}\sum _{i\in A}^{N_A}\sum _{j\in B}^{N_B}{\displaystyle \frac{\delta (r_{ij}-r)}{4\pi r^2}}\end{array}\end{array}$$

with $⟨{\rho }_B(r)⟩$ the particle density of type $B$ at a distance $r$ around particles $A$, and $⟨{\rho }_B{⟩}_{local}$ the particle density of type $B$ averaged over all spheres around particles $A$ with radius $r_{max}$ (see Fig. 52 C).

Fig. 52 Definition of slices in gmx rdf: A. $g_{AB}(r)$. B. $g_{AB}(r,\theta )$. The slices are colored gray. C. Normalization $⟨{\rho }_B{⟩}_{local}$. D. Normalization $⟨{\rho }_B{⟩}_{local,\theta }$. Normalization volumes are colored gray.

Usually the value of $r_{max}$ is half of the box length. The averaging is also performed in time. In practice the analysis program gmx rdf divides the system into spherical slices (from $r$ to $r+dr$, see Fig. 52 A) and makes a histogram in stead of the $\delta$-function. An example of the RDF of oxygen-oxygen in SPC water 80 is given in Fig. 53

Fig. 53 $g_{OO}(r)$ for Oxygen-Oxygen of SPC-water.

With gmx rdf it is also possible to calculate an angle dependent rdf $g_{AB}(r,\theta )$, where the angle $\theta$ is defined with respect to a certain laboratory axis $\mathbf{e}$, see Fig. 52 B.

(438)$$g_{AB}(r,\theta )=\frac{1}{⟨{\rho }_B{⟩}_{local,\theta }}\frac{1}{N_A}\sum _{i\in A}^{N_A}\sum _{j\in B}^{N_B}\frac{\delta (r_{ij}-r)\delta ({\theta }_{ij}-\theta )}{2\pi r^{2}sin(\theta )}$$

(439)$$cos({\theta }_{ij})=\frac{{\mathbf{r}}_{ij}\cdot \mathbf{e}}{\Vert r_{ij}\Vert \Vert e\Vert }$$

This $g_{AB}(r,\theta )$ is useful for analyzing anisotropic systems. Note that in this case the normalization $⟨{\rho }_B{⟩}_{local,\theta }$ is the average density in all angle slices from $\theta$ to $\theta +d\theta$ up to $r_{max}$, so angle dependent, see Fig. 52 D.