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Quantum Tunnelling Across Hydrogen Bonds: Proton--Deuteron Isotope Effects from a Cornell-Type Potential Model

Krishna Kingkar Pathak

TL;DR

The paper addresses how hydrogen-bond tunnelling depends on isotope mass. It combines a Cornell-type potential with a semi-analytical wavefunction ansatz and numerical 1D Schrödinger solutions to study $ΔE$ for H and D. Key contributions include explicit scaling relations for tunnelling splittings with mass, analysis of barrier width and curvature effects, and comparisons with experimental data on representative hydrogen-bonded systems. The framework provides a general methodology for modelling tunnelling in double-well landscapes with implications for spectroscopy, enzymatic catalysis, and materials that rely on hydrogen-bond networks.

Abstract

Hydrogen bonds play a pivotal role in chemistry, biology, and condensed-matter physics, where quantum tunnelling can strongly influence structure and dynamics. Isotope substitution (H $\rightarrow$ D) provides a sensitive probe of such tunnelling, but theoretical descriptions often rely on purely numerical models or simplified potentials that obscure physical interpretation. Here we employ a Cornell-type potential combined with a double-well Schrödinger approach to investigate proton and deuteron tunnelling across hydrogen bonds. The model yields semi-analytical wavefunctions and tunnelling splittings that transparently capture isotope-dependent quantum effects. We present scaling behaviour of tunnelling splittings with isotope mass, discuss the influence of barrier width and curvature, and compare model trends with representative experimental and computational results. Beyond hydrogen bonding, the framework provides a general methodology for modelling tunnelling in double-well systems relevant to spectroscopy, enzymatic catalysis, and materials applications.

Quantum Tunnelling Across Hydrogen Bonds: Proton--Deuteron Isotope Effects from a Cornell-Type Potential Model

TL;DR

The paper addresses how hydrogen-bond tunnelling depends on isotope mass. It combines a Cornell-type potential with a semi-analytical wavefunction ansatz and numerical 1D Schrödinger solutions to study for H and D. Key contributions include explicit scaling relations for tunnelling splittings with mass, analysis of barrier width and curvature effects, and comparisons with experimental data on representative hydrogen-bonded systems. The framework provides a general methodology for modelling tunnelling in double-well landscapes with implications for spectroscopy, enzymatic catalysis, and materials that rely on hydrogen-bond networks.

Abstract

Hydrogen bonds play a pivotal role in chemistry, biology, and condensed-matter physics, where quantum tunnelling can strongly influence structure and dynamics. Isotope substitution (H D) provides a sensitive probe of such tunnelling, but theoretical descriptions often rely on purely numerical models or simplified potentials that obscure physical interpretation. Here we employ a Cornell-type potential combined with a double-well Schrödinger approach to investigate proton and deuteron tunnelling across hydrogen bonds. The model yields semi-analytical wavefunctions and tunnelling splittings that transparently capture isotope-dependent quantum effects. We present scaling behaviour of tunnelling splittings with isotope mass, discuss the influence of barrier width and curvature, and compare model trends with representative experimental and computational results. Beyond hydrogen bonding, the framework provides a general methodology for modelling tunnelling in double-well systems relevant to spectroscopy, enzymatic catalysis, and materials applications.

Paper Structure

This paper contains 4 sections, 14 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Wavefunction ansatz approach
  3. Numerical double-well Schrödinger approach
  4. Results and discussion

Figures (4)

  • Figure 1: Tunnelling splitting $\Delta E$ as a function of barrier height $V_0$ for proton (H) and deuteron (D) in a symmetric double-well potential with $d=2.7$ Å. Points are numerical Schrödinger eigenvalue differences computed on a finite-difference grid; lines are guides to the eye.
  • Figure 2: Ground- and first-excited-state wavefunctions for a symmetric quartic double-well at $V_0=0.10$ eV and $d=2.7$ Å (proton mass). The dashed curve shows the potential $V(x)$ scaled to the wavefunction amplitude range. The symmetric/antisymmetric character of $\psi_0$ and $\psi_1$ gives rise to the tunnelling splitting $\Delta E$.
  • Figure 3: Isotope mass scaling of the tunnelling splitting $\Delta E$ at $V_0=0.10$ eV and $d=2.7$ Å, comparing proton (H) and deuteron (D). The substantial reduction of $\Delta E$ for D reflects the mass dependence of barrier penetration.
  • Figure 4: Convergence of the tunnelling splitting $\Delta E$ with grid size $N$ for two domain half-widths ($L=5$ Å and $L=10$ Å) at $V_0=0.10$ eV.