Electronic density distribution in hydrogen bond
There are a few descriptors in 'conceptual'  DFT just to characterize hydrogen bonds. E.g.: non-covalent interaction (NCI): note that hydrogen bonds are often consider as NON-covalent bonds, Density Overlap Regions Indicator (DORI). 

One can also look at the  deformation density contributions originating from natural orbitals for chemical valence within ETS-NOCV theory.