# Are all molecules under the same peak in g(r) always found in the same solvation shell?

For a liquid, the peaks in $$g(r)$$ are typically associated with the solvation shells: nearest neighbors, next-nearest neighbors, and so on. However, the $$g(r)$$ is averaging over many configurations, so are all sub-populations (molecules under the same peak) necessarily in the same solvation shell? If not, how do you determine the fraction of molecules in each shell for each peak?

EDIT: This article appears to suggest that a simple way to define the solvation shells can be to use a cutoff based on the minima in $$g(r)$$, calculate the corresponding nearest, next-nearest, etc. neighbors for each molecule in a configuration, and then build up a histogram. But I don't see how that would tell you if peak 1 in $$g(r)$$ corresponds to $$x$$ fraction of molecules from shell 1, $$y$$ fraction from shell 2, etc. I would appreciate any insight.

• If solvation shell 1 is defined as molecules under peak 1 of the RDF, then all molecules under peak 1 of the RDF are part of shell 1, by definition! This question only really makes sense if you have some working definition of "solvation shell" that is completely independent of the RDF, and I for one couldn't begin to answer this question until I know what your precise definition is. Dec 21, 2022 at 3:47
• @ShernRenTee Yes, that is what I am saying: the definition of the RDF implies peak = solvation shell. I am wondering if there is another way of looking at it. The link in my edit suggests there is, but I don't understand how you get that from a cutoff.
– user2026
Dec 21, 2022 at 6:39

Yes, it is possible for configurations which contribute to a particular peak in $$g(r)$$ to be outside of that solvation shell.

As pointed out in the comments, this is only possible if you have a definition of solvation shells that makes more sense than just defining the minima of $$g(r)$$ as the solvation shells. For things like Lennard-Jones fluids that don't have strongly directional interactions, the $$g(r)$$ definition probably always makes the most sense.

For water, however, where hydrogen bonds are well-understood to be the dominant stabilizing interaction within the liquid, the $$g(r)$$ definition may not always be the best definition. For instance, in ref. [1], the authors report the average number of hydrogen bonds to each water under various definitions and find it to be around 3.6. On the other hand, if you integrate the area until the first minimum in $$g(r)$$, you find a coordination number of around 4.3. If you compare the configurations you get from each of these, some of them clearly are not hydrogen-bonded configurations and hence would not be considered as meaningfully contributing to the solvation of that water molecule.

So, there are situations where one might reasonably define the solvation shells differently than corresponding to the minima in $$g(r)$$, but this is the exception. Even so, integrating that area contains useful information. For instance, if you didn't count up the hydrogen bonds in water, one might not expect to have a meaningful number of non-hydrogen bonded interactions at such short distance in the liquid.

Also, I am not aware of any other system where this distinction between coordination number and solvation environment is regularly made. One could of course do this for any solutes in water or probably for systems like ammonia, but I don't think it's common in other contexts.

[1]: Kumar, R., Schmidt, J. R., & Skinner, J. L. (2007). Hydrogen bonding definitions and dynamics in liquid water. The Journal of chemical physics, 126(20), 05B611.

• +10 I was hoping someone would come along and answer some of our unanswered g(r) questions! Dec 27, 2022 at 19:17