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:

(1) $$\begin{split}\begin{array}{rcl} g_{AB}(r)&=& {\displaystyle \frac{\langle \rho_B(r) \rangle}{\langle\rho_B\rangle_{local}}} \\ &=& {\displaystyle \frac{1}{\langle\rho_B\rangle_{local}}}{\displaystyle \frac{1}{N_A}} \sum_{i \in A}^{N_A} \sum_{j \in B}^{N_B} {\displaystyle \frac{\delta( r_{ij} - r )}{4 \pi r^2}} \\ \end{array}\end{split}$$

with $\langle\rho_B(r)\rangle$ the particle density of type $B$ at a distance $r$ around particles $A$, and $\langle\rho_B\rangle_{local}$ the particle density of type $B$ averaged over all spheres around particles $A$ with radius $r_{max}$ (see Fig. 51 C).

Fig. 51 Definition of slices in gmx rdf: A. $g_{AB}(r)$. B. $g_{AB}(r,\theta)$. The slices are colored gray. C. Normalization $\langle\rho_B\rangle_{local}$. D. Normalization $\langle\rho_B\rangle_{local,\:\theta }$. 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. 51 A) and makes a histogram in stead of the $\delta$-function. An example of the RDF of oxygen-oxygen in SPC water :ref:80 is given in Fig. 52

Fig. 52 $g_{OO}(r)$ 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 ${\bf e}$, see Fig. 51 B.

(2) $$g_{AB}(r,\theta) = {1 \over \langle\rho_B\rangle_{local,\:\theta }} {1 \over N_A} \sum_{i \in A}^{N_A} \sum_{j \in B}^{N_B} {\delta( r_{ij} - r ) \delta(\theta_{ij} -\theta) \over 2 \pi r^2 sin(\theta)}$$

(3) $$cos(\theta_{ij}) = {{\bf r}_{ij} \cdot {\bf e} \over \|r_{ij}\| \;\| e\| }$$

This $g_{AB}(r,\theta)$ is useful for analyzing anisotropic systems. Note that in this case the normalization $\langle\rho_B\rangle_{local,\:\theta}$ is the average density in all angle slices from $\theta$ to $\theta + d\theta$ up to $r_{max}$, so angle dependent, see Fig. 51 D.