Steinhardt order parameter and radial distribution function Is there a relation between the probability distribution of Steinhardt order parameter for BCC and FCC and their radial distribution function?

In the picture we see the probability distributions of  ̄q4(solid lines) and q4(dashed lines) for the FCC and BCC. 
(source: https://arxiv.org/pdf/0806.3345.pdf)
 A: Sorry I'm late to this answer, hopefully it's useful to someone who ends up here later. No, there is in general no relationship between the Steinhardt order parameters and the radial distribution function. Both of these are useful calculations for identifying crystal structures, but they work very differently. Also, I think most people would argue that (as a correlation function with direct statistical mechanical interpretations, and as the Fourier transform of the structure factor), the RDF is a more fundamental quantity, whereas the Steinhardt order parameters are more of a convenient heuristic method for crystal structure identification.
The radial distribution function is (up to some normalization constants and averaging) a measure of the number of pairs of particles found at a certain distance from each other. Peaks in the RDF of crystal structures like FCC and BCC correspond to actual distances that you can see in a unit cell. For instance, in BCC the first peak corresponds to the distance from any corner to the body-centered particle: $\sqrt{3*0.5^2} = 0.866$ (assuming a lattice constant of 1), while the second peak is just the edge-edge length of $1$, so you get the characteristic ratio of $1/0.866=1.15$ for the position of the second peak to the first. 
The Steinhardt order parameters, on the other hand, are computed based on the angles of nearest neighbor bonds, not distances. The actual calculation is a bit involved, it involves finding invariant combinations of the spherical harmonics computed from the vectors joining particles to their nearest neighbors. The metrics are useful because, like with the RDF, certain distributions are known fingerprints for certain structures. However, they aren't really looking at characteristic distances, but rather characteristic angles (more precisely, they're looking for symmetries by looking at something like a Fourier series generalized to 3D using spherical harmonics and then computed common frequencies in the resulting frequency space, which is what the 6 means in $Q_6$, for example). If you're interested in greater detail, here's a link to the original paper by Steinhardt. This webpage also seems to have some useful resources.
