# Radial distribution functions¶

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

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. 52 C).

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 :ref:80 is given in Fig. 53

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. 52 B.

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. 52 D.