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)∝ −√(μeff), 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√(μeff)+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.
Versions
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BibTeX
@misc{pathak2026massscaling,
title = {{Mass-Scaling of Quantum Tunnelling in Hydrogen Bonds: Analytical Model and Comparison with Multidimensional Potentials}},
author = {Krishna Kingkar Pathak},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25074},
eprint = {2609.25074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25074},
}
RIS
TY - UNPB
TI - Mass-Scaling of Quantum Tunnelling in Hydrogen Bonds: Analytical Model and Comparison with Multidimensional Potentials
AU - Krishna Kingkar Pathak
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.25074
UR - https://arxiv.org/abs/2609.25074
AB - 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)∝ −√(μ_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√(μ_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.
ER -