Giant polarisability of low-barrier hydrogen bonds

An outstanding puzzle concerning simple acids and bases is their very broad infrared absorption, as highlighted in this earlier post. The first to highlight this problem was Georg Zundel. His solution involved two important new ideas:

• the stability of H5O2+ in liquid water, [the Zundel cation]
• that such complexes involving shared protons via hydrogen bonding have a giant electric polarisability, several orders of magnitude larger than typical molecules.

Both ideas remain controversial. A consequence of the second is that the coupling of electric field fluctuations associated with the solvent of the complex will result in a large range of vibrational energies, leading to the continuous absorption. 

Later I will discuss the relative merits of Zundel's explanation. Here I just want to focus on understanding the essential physics behind the claimed giant polarisability. The key paper appears to be a 1972 JACS

Extremely high polarizability of hydrogen bonds
R. Janoschek , E. G. Weidemann , H. Pfeiffer , G. Zundel

[The essential physics seems to be in a 1970 paper I don't have electronic access to].
If one considers the one-dimensional potential for proton transfer within a Zundel cation with O-O distance of 2.5 Angstroms it looks like the double well potential below.

The two lowest vibrational eigenstates are separated in energy by a small tunnel splitting omega of the order of 10-100 cm-1. These two states can be viewed as symmetric and anti-symmetric combinations of oscillator states approximately localised in the two wells. The transition dipole moment p between these two states is approximately the well separation [here roughly 0.5 Angstroms (times of the proton charge)].

At zero temperature the electric polarisability is approximately, p^2/omega. Omega is at least an order of magnitude smaller than a typical bond stretching frequency and p^2 can be an order of magnitude larger than for a typical covalent bond.
Hence, the polarisability can be orders of magnitude larger than that for a typical molecule.

A few consequences of this picture are that the polarisability will vary significantly with

• isotopic [H/D] substitution
• temperature (on the scale of the tunnel splitting)
• the donor acceptor distance.