A basic issue is to first identify short hydrogen bonds, i.e., finding a reliable method to measure bond lengths.
I recently worked through and a nice article,
NMR studies of strong hydrogen bonds in enzymes and in a model compound
T.K. Harris, Q. Zhao, A.S. Mildvan
Surely, these bond lengths just be identified with x-ray crystallography? No.
the standard errors in distances determined by protein X-ray crystallography are 0.1–0.3 times the resolution. For a typical 2.0 Å X-ray structure of a protein, the standard errors in the distances are ±0.2–0.6 Å, precluding the distinction between short, strong and normal, weak hydrogen bonds.[Aside: I also wonder whether the fact that X-ray crystal structures are refined with classical molecular dynamics using force fields that are parametrised for weak bonds is also a problem. Such refinements will naturally bias towards weak bonds, i.e., the longer bond lengths that are common in proteins. I welcome comment on this.]
The authors then discuss how NMR can be used for bond length determinations. One of these NMR "rulers" involves isotopic fractionation, where one measures how much the relevant protons exchange with deuterium in a solvent,
is determined by the relative zero-point energy (ZPE) of a D relative to an H in the enzyme. As described in a key JACS article the ratio is given by a formula such as
If Planck's constant was zero, this ratio would always be one. It would also be one if there was no change in the vibrational frequencies of the H/D when they move from the solvent to the enzyme. Generally, as the H-bond strengthens [R gets shorter] the frequency change gets larger and so the difference between H/D gets larger [see this preprint for an extensive discussion], and phi gets smaller. However, for very short bonds the frequencies harden and phi will get larger, i.e. there will be a non-monotonic dependence on R, the distance between the donor and acceptor. This was highlighted in an extensive review which contains the following sketch.
Harris, Zhao, and Mildvan consider a particular parametrisation of the H-bond potential to connect the observed fractionation ratio with bond lengths in a range of proteins. They generally find reasonable agreement with other methods of determining the length [e.g., NMR chemical shift]. In particular the resolution is much better than from X-rays.