Radial distribution functions#

The radial distribution function (RDF) or pair correlation function gAB(r) between particles of type A and B is defined in the following way:
(435)#gAB(r)=ρB(r)ρBlocal=1ρBlocal1NAiANAjBNBδ(rijr)4πr2

with ρB(r) the particle density of type B at a distance r around particles A, and ρBlocal the particle density of type B averaged over all spheres around particles A with radius rmax (see Fig. 52 C).

../../_images/rdf.png

Fig. 52 Definition of slices in gmx rdf: A. gAB(r). B. gAB(r,θ). The slices are colored gray. C. Normalization ρBlocal. D. Normalization ρBlocal,θ. Normalization volumes are colored gray.#

Usually the value of rmax 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 δ-function. An example of the RDF of oxygen-oxygen in SPC water 80 is given in Fig. 53

../../_images/rdfO-O.png

Fig. 53 gOO(r) for Oxygen-Oxygen of SPC-water.#

With gmx rdf it is also possible to calculate an angle dependent rdf gAB(r,θ), where the angle θ is defined with respect to a certain laboratory axis e, see Fig. 52 B.

(436)#gAB(r,θ)=1ρBlocal,θ1NAiANAjBNBδ(rijr)δ(θijθ)2πr2sin(θ)
(437)#cos(θij)=rijerije

This gAB(r,θ) is useful for analyzing anisotropic systems. Note that in this case the normalization ρBlocal,θ is the average density in all angle slices from θ to θ+dθ up to rmax, so angle dependent, see Fig. 52 D.