I think double proton transfer can be pretty amazing. The picture below shows two possible quantum states of a porphycene molecule. Note two things.
First, the two states differ by the location of the two hydrogen atoms [protons].
Second, the location of eight of the double bonds is different.
At low temperatures the ground state of the molecule is a linear superposition of the two states.
The definitive signature of this is the tunnel splitting of the vibrational states, as shown above.
This is seen experimentally, as reported here.
At higher temperatures does not see a splitting and there is a temperature activated conversion between the two tautomers.
The ground state can be written as a superposition of two Born-Oppenheimer states [products of nuclear and electronic wave functions]
Psi = |L>|A> + |R>|B>
where |L> and |R> are the two nuclear states and |A> and |B> the two electronic states.
These are approximately orthogonal to each other.
What is impressive about this?
Each electronic state involves about 28 valence electrons. Roughly each double bond can be described by a valence bond state consistent of a pair of electrons in a maximally entangled singlet state.
This is not quite a Schrodinger cat state. But it is a nanoscale kitten!
How is such a state possible? The key is that the two electronic states are very strongly coupled. They have a Hamiltonian matrix element of order of electron volts [10,000 cm^-1] . Yet the tunnel splittings are only a few cm^-1 due to the small overlap of nuclear states.
Hence, these superposition states are very fragile and will be easily destroyed at a few kelvin and/or any sort of polar solvent. So don't even start thinking about quantum biology!
Since my PhD, I have had a strong interest in the double-well model. Among others, one thing that fascinated me is the possible spectral weight transfer phenomena: there are two components of motions for a particle moving in such a potential, the intra- and inter-well motions. The former prevails at low temperatures, but the latter at high temperatures. So, raising the temperature should see a transfer between their spectral weights. There is some experimental evidence in ferroelectrics. However, a satisfactory theoretical treatment seems not existing.
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