Why do RDF plots have the probability at a particular distance >1. How can probability be more than one?

I am asking this because I want to understand what do the peaks of different length but at the same position on the x axis mean

The below is an example of a RDF plot: enter image description here

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    $\begingroup$ This is not probability. $\endgroup$ Jul 12, 2020 at 8:20

1 Answer 1


The radial distribution function isn't a probability or a probability density. The usual definition is $$g(r)=\frac{\rho(r)}{\rho_0}$$ where $\rho(r)$ is the number density at a particular distance $r$ away from some specified atom and $\rho_0$ is the bulk density of the material.

It can be larger than one if the local density is greater than the bulk (which you see in the first peak of your graph), so it can't be a probability. It also can't be a probability density, as integrating the RDF from $0$ to $r$ should give you the number of particles (divided by the bulk density) in that spherical volume, which should continue to increase even up to macroscopic distances (in contrast to a probability density which should integrate to $1$).

So what does the RDF tell you? Well if you integrate it over a spherical shell and multiply by the bulk density, you obtain the number of atoms in that shell. One particular shell we are often interested in is the first coordination sphere. Using the RDF, we can define the number of particles in the first coordination sphere as $$n(r')=4\pi\rho_0\int_0^{r'}g(r)r^2dr$$ where $r'$ is the location of the first minimum of the RDF. We can similarly determine the number of atoms in the $n^{\text{th}}$ coordination sphere by integrating from the$(n-1)^{\text{th}}$ minimum to the $n^{\text{th}}$ minimum.

  • $\begingroup$ chem.libretexts.org/Bookshelves/… defines RDF for bound electrons. $\endgroup$ Jul 13, 2020 at 10:13
  • $\begingroup$ This is not about electrons $\endgroup$
    – Ian Bush
    Jul 14, 2020 at 9:03
  • $\begingroup$ Why at longer distances converges to 1, that is it behaves like bulk density? Is it because when we have larger spherical shells we approximate better the bulk? $\endgroup$
    – Anton
    May 13 at 15:50
  • $\begingroup$ @Anton considering very long (i.e. macroscopic) distance, we know it has to converge to the bulk density because eventually we are including the whole solution. For moderate microscopic distances, it should already start to approach 1, as you start to include more of the bulk structure rather than just individual local clusters. $\endgroup$
    – Tyberius
    May 13 at 16:20
  • $\begingroup$ @Tyberius When you say we are including the "whole solution" you mean that the shell dimensions are so big that it mimics the bulk right? Because $ρ(r)$ refers to density between $r$ and $r+dr$ and not $0$ and $r$ (the whole solution can be thought as the limit of $r$ going to infinity). $\endgroup$
    – Anton
    May 13 at 18:10

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