Mass-Scaling of Quantum Tunnelling in Hydrogen Bonds: Analytical Model and Comparison with Multidimensional Potentials

Abstract

Quantum tunnelling plays a central role in the structure and spectroscopy of hydrogen-bonded systems, and its sensitivity to isotopic substitution provides a stringent probe of the underlying potential-energy landscape. Despite extensive numerical studies, many high-level approaches tend to obscure the simple physical relationships linking effective mass, barrier geometry, and tunnelling amplitudes. Here, we develop a Cornell-type analytical--numerical framework to describe proton and deuteron tunnelling, combining a semi-analytical localized wavefunction ansatz with numerical solutions of the one-dimensional Schrödinger equation. The resulting tunnelling splittings exhibit an exponential dependence on the square root of the effective isotope mass,$\ln(ΔE)\propto -\sqrt{μ_{\mathrm{eff}}}$, in agreement with semiclassical Wentzel--Kramers--Brillouin (WKB) theory. Comparison with multidimensional reaction-space calculations for the formic acid dimer shows that this scaling persists in fully coupled 3D and 5D quantum models, yielding an empirical relation $\ln(ΔE)= -1.75\sqrt{μ_{\mathrm{eff}}}+2.60$. The present framework provides a transparent and computationally efficient approach for quantifying mass-scaling and tunnelling dynamics in hydrogen-bonded and other double-well systems.