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