Proton transfer by tunnelling

Proton transfer

A proton hopping between two heavy atoms in an intramolecular hydrogen bond: two equivalent configurations = double well, with a measured tunnelling splitting (malonaldehyde 21.6 cm⁻¹, tropolone 0.97). The proton's small mass makes tunnelling observable. Effective coordinate solved by finite differences in the Python backend (via gw2py).

The same double well as inversion and the same tunnelling doublet as pear-shaped nuclei: here it is the proton, in place of the nitrogen atom or the nuclear shape, that tunnels between two equivalent minima.

In an intramolecular hydrogen bond, the proton can sit near one or the other heavy atom: two minima, hence tunnelling.

The proton's double well

The coordinate is the proton's position along the O···H···O bond. The two equivalent configurations (O₁–H···O₂ and O₁···H–O₂) are the two minima of a symmetric double well; the light proton tunnels between them.

\[ -B\,\frac{d^2\psi}{dq^2} + V_b\Big[(q/q_0)^2-1\Big]^2\psi = E\,\psi,\qquad B=\frac{16.86}{\mu}\ \text{cm}^{-1}\text{Å}^2. \]

Measurable splitting

Malonaldehyde has a ground-state tunnelling splitting of 21.6 cm⁻¹, measured in microwave/IR; tropolone 0.97 cm⁻¹ (higher barrier), the formic acid dimer ~0.016 cm⁻¹ (double proton, very high barrier). The FD solver reproduces these values by calibrating the effective 1D barrier.

The thread

The same double well as inversion and the same tunnelling doublet as pear-shaped nuclei: the proton in place of the nitrogen atom or the nuclear shape.

Molecule

Select

The proton in an intramolecular H-bond sees a double well. The effective 1D barrier is calibrated on the measured splitting.

State to represent

State type Level \(n_1\)

Level \(n_2\)

The \((0,1)\) doublet is the proton «oscillating» between the two oxygens.

Small mass, large tunnelling

The proton is ~1 u: with the same barrier a heavy nucleus would not tunnel. This is why the splitting is observable (21.6 cm⁻¹ in malonaldehyde) and collapses dramatically upon deuterium substitution — the experimental signature of proton tunnelling.

An effective coordinate, not just the proton

The proton hop is accompanied by rearrangement of the heavy skeleton (the O···O distance shortens at the transit): the effective coordinate and the reduced mass \(\mu>1\) incorporate this concerted motion. This is why the effective 1D barrier differs from the pure electronic one.

What is missing (honesty)

Transfer is intrinsically multidimensional (proton + skeleton); the 1D model with calibrated \(\mu,V_b\) reproduces the splitting but not its dependence on the promoting modes. The accurate treatment is a multidimensional-surface Hamiltonian with tunnelling effects (instantons/backend).

References

S. L. Baughcum et al., «Microwave spectroscopic study of malonaldehyde», J. Am. Chem. Soc. 103, 6296 (1981). doi.
R. L. Redington, «Tunneling in tropolone», J. Chem. Phys. 92, 6447 (1990). doi.

WebNIR · CNR-IFAC | demo interface — numerical work is provided by the Python backend (gw2py).

Keywords: proton transfer, hydrogen bond, tunnelling, double well, malonaldehyde, tropolone, splitting, effective coordinate