Benchmarking Proton Tunneling Splittings with a Wavefunction-Based Double-Well Model: Application to the Formic Acid Dimer

TL;DR

This paper develops a wavefunction-based double-well model to benchmark proton tunneling splittings, applying it to the formic acid dimer and comparing with multidimensional quantum calculations to assess its accuracy and limitations.

Contribution

It introduces a simplified, analytical framework for proton tunneling splittings using a double-well potential, serving as a benchmark tool for complex quantum models.

Findings

The model accurately reproduces tunneling splittings for the formic acid dimer.

Numerical and semiclassical results show consistent agreement across barrier parameters.

The approach highlights the pedagogical value and limitations of one-dimensional tunneling models.

Abstract

Proton tunneling across hydrogen bonds is a fundamental quantum effect with implications for spectroscopy, catalysis, and biomolecular stability. While state-of-the-art instanton and path-integral methods provide accurate multidimensional tunneling splittings, simplified one-dimensional models remain valuable as conceptual and benchmarking tools. Here we develop a wavefunction-based framework for tunneling splittings using a Cornell-type double-well potential and apply it as a benchmark for hydrogen-bond tunneling. Analytical WKB estimates and numerical finite-difference solutions are compared across a range of barrier parameters, showing consistent agreement. As a test case, we map the formic acid dimer (FAD) barrier onto a quartic double-well model parameterized to reproduce the reported barrier height of $V_b \\approx 2848~\\text{cm}^{-1}$. The resulting tunneling splitting of about $0.037~\\text{cm}^{-1}$ matches the reduced-dimensional calculations of Qu and Bowman. The close agreement between numerical and semiclassical results highlights the pedagogical and diagnostic value of one-dimensional models, while comparison with molecular benchmarks clarifies their limitations relative to full multidimensional quantum treatments.