Quantum tunnelling in hydrogen-bonded systems is a fundamental phenomenon governing proton transfer, vibrational spectroscopy, and isotope effects across chemistry, biology, and materials science. Despite decades of study, many advanced numerical approaches—while accurate—often obscure the simple physical relationships connecting mass, barrier geometry, and tunnelling dynamics.

In this work, I develop a Cornell-type analytical–numerical framework to uncover a transparent and universal description of tunnelling in hydrogen bonds. The study is based on my recent publication:
https://doi.org/10.1016/j.chemphys.2026.113203

The central idea is to combine an analytically tractable localized wavefunction with numerical solutions of the one-dimensional Schrödinger equation. This hybrid approach allows us to retain physical insight while ensuring quantitative reliability.

A key result emerging from this framework is a simple and powerful scaling law:
the tunnelling splitting ΔE depends exponentially on the square root of the effective isotope mass,
ln(ΔE) ∝ −√μ_eff.

This behaviour is not merely a feature of simplified models—it has a clear semiclassical origin. As shown in the analysis (see Fig. 3 on page 4), increasing the mass enhances the decay rate of the wavefunction inside the barrier, leading to an exponential suppression of tunnelling. This is fully consistent with the Wentzel–Kramers–Brillouin (WKB) picture, where the tunnelling probability is controlled by the action integral.

To validate the analytical predictions, the Schrödinger equation was solved numerically for a symmetric double-well potential. The results demonstrate:
- Strong agreement with analytical scaling predictions  
- Clear isotope dependence between proton and deuteron tunnelling  
- Exponential suppression of tunnelling with increasing barrier height and mass (see Fig. 1 on page 4)  

The numerical results also reproduce experimentally observed tunnelling splittings. As summarized in Table 1 (page 3), the calculated values fall within the range of known systems such as malonaldehyde and the formic acid dimer, confirming the physical realism of the model.

A particularly important outcome of this work is the persistence of the scaling law beyond one-dimensional models. By comparing with multidimensional reaction-space calculations (3D and 5D) for the formic acid dimer, we find that the same linear relation between ln(ΔE) and √μ_eff holds (see Fig. 4 on page 5). This demonstrates that the mass-scaling behaviour is not an artifact of reduced models but a robust feature of quantum tunnelling itself.

The unified behaviour is further highlighted in Fig. 5 (page 6), where analytical, experimental, and multidimensional data collapse onto a single trend. This strongly suggests that √μ_eff acts as the primary control parameter governing tunnelling dynamics across systems.

From a broader perspective, this work establishes a bridge between:
- Simple analytical models  
- High-dimensional quantum simulations  
- Experimental observations  

The identification of a universal mass-scaling relation provides a powerful and physically transparent tool for understanding isotope effects in hydrogen-bonded systems. Beyond molecular dimers, the same framework can be applied to proton transfer in enzymes, hydrogen-bonded liquids, and quantum materials where light nuclei play a crucial role.

Overall, the present approach shows that even in complex quantum systems, simple scaling laws can emerge—offering both conceptual clarity and predictive power.

Journal link: https://doi.org/10.1016/j.chemphys.2026.113203 (Chemical Physics)