Capturing the directionality of hydrogen bonding

Below is a schematic picture of hydrogen bonding between a donor D and an acceptor A.

An interesting and important question is how the potential energy of the system varies with the angle phi which measures deviations from linearity. A key property of H-bonds is that they are highly directional. This leads to the four-fold co-ordination of water in liquid and solid phases.

There is a vibrational mode associated with this co-ordinate phi.
[In water this rotation is connected to the librational mode].

Previously I posted about an empirical correlation showing how this mode hardens with the increasing strength of the H-bond (decreasing R above). This is the opposite trend to the D-H stretch frequency (associated with the r co-ordinate above) which softens with decreasing R.

I was very pleased when I discovered that the model effective Hamiltonian I proposed for H-bonding can describe this correlation (see the figure below) without introducing any new parameters.

This is discussed in more detail in the final version of my paper.

Aside: I earlier posted the graph above in a post about confirmation bias. I should say that after I did get good agreement between the theory and experiment, I did go back and check all the algebra and mathematica codes.