Quantum nuclear motion in proton sponges

There is a nice paper Hydrogen Motion in Proton Sponge Cations: A Theoretical Study
by Yevhen Horbatenko and Sergei Vyboishchikov

Proton sponge is a trade name for a particular compound which is a strong base, i.e. it likes to bond to protons.

The proton sponge compounds are of particular interest to me because they work by the proton forming a strong hydrogen bond between two nitrogen atoms.

In this paper the authors first use quantum chemistry to calculate the adiabatic potential energy surface for the ground state as a function of the proton position. They then calculate the proton vibrational wavefunction and energy.

The three potential energy curves above correspond to the three distinct cases of hydrogen bonding: i) strong hydrogen bond, ii) low barrier hydrogen bond, iii) weak hydrogen bond. [They arise naturally in my simple model of H-bonds.]

I found two results of the authors particularly interesting.

First, the shape of the potential energy curve [and specifically the presence or absence of a barrier] depends on the level of quantum chemistry or density functional used in the calculation.

Second, the paper has a nice physical insight for strong hydrogen bonds that I have not seen before. The vibrational energy levels have spacing similar to that for a square well potential of with width comparable to the donor-acceptor distance. Specifically, the energy of the n-th level is proportional to n^2, whereas for a harmonic potential it is proportional to (n-1). This reflects how anharmonic the potential is.