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

TL;DR

The paper addresses proton tunneling splittings in hydrogen-bonded systems using a one-dimensional wavefunction-based double-well model with a Cornell-type potential. It combines analytical WKB estimates with numerical finite-difference solutions to validate the approach and quantify its accuracy. As a molecular benchmark, the authors map the Formic Acid Dimer barrier to a quartic 1D double-well with $V_b \approx 2848\ \mathrm{cm^{-1}}$, obtaining a tunneling splitting of $\Delta E \approx 0.037\ \mathrm{cm^{-1}}$, in line with reduced-dimensional results by Qu and Bowman. The work demonstrates the pedagogical value and diagnostic utility of simple 1D models while highlighting their limitations compared to full multidimensional quantum treatments, and it suggests future extensions to connect with ab initio surfaces and vibrational coupling.

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.

Figures (5)

• Figure 1: Numerical ground-state (blue) and first-excited-state (red) wavefunctions in the symmetric Cornell-type double-well potential. The two states are nearly degenerate and localized in opposite wells, with a small admixture due to tunneling.
• Figure 2: Computed tunneling splitting $\Delta E$ (arbitrary units) as a function of barrier separation $d$ for representative barrier heights $V_b$ in the Cornell-type double-well potential. The exponential decrease of $\Delta E$ with $d$ highlights the sensitivity of tunneling to geometric parameters.
• Figure 3: Direct comparison of tunneling splittings $\Delta E$ computed numerically (circles) and semiclassically using the WKB approximation (triangles) as a function of barrier separation $d$ for representative barrier heights. The close agreement demonstrates that semiclassical theory captures the essential scaling of tunneling in the Cornell-type double-well model.
• Figure 4: Quartic 1D double-well model parametrized to the reduced-dimensional formic acid dimer (FAD) potential with barrier height $V_b = 2848\ \mathrm{cm^{-1}}$ and minima at $\pm 0.7236\ \text{\AA}$. Horizontal dashed lines mark the numerically computed ground and first excited states ($E_0$ and $E_1$), whose splitting reproduces the reduced-dimensional tunneling result.
• Figure 5: Ground- and first-excited-state wavefunctions for the FAD 1D quartic model. The symmetric and antisymmetric combinations of localized states produce a tunneling splitting of $\sim 0.037\ \mathrm{cm^{-1}}$, in agreement with the reduced-dimensional results of Qu and Bowman.