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arxiv: 2603.01573 · v1 · submitted 2026-03-02 · ⚛️ physics.chem-ph · cond-mat.mtrl-sci

Weiner's theory for exactly solvable Schr\"odinger equation with symmetric double well potential

A.E. Sitnitsky This is my paper

Pith reviewed 2026-05-15 17:01 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.mtrl-sci
keywords proton transfer ratedouble-well potentialquantum tunnelingWeiner theorytrigonometric potentialammonia dimertemperature dependencehydrogen bond
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The pith

Modified Weiner's theory yields an analytic formula for proton transfer rates that tracks the crossover from thermal activation to quantum tunneling.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper adapts Weiner's theory to the exactly solvable Schrödinger equation with a symmetric trigonometric double-well potential, removing earlier approximations to produce a more accurate model. An explicit analytic expression for the proton transfer rate constant is derived and implemented using standard Mathematica functions. The method is applied to the hydrogen bond in the proton-bound ammonia dimer cation, with all input parameters taken directly from published IR spectroscopy and quantum-chemical data. The resulting temperature dependence reproduces the expected switch from Arrhenius-like exponential decay at higher temperatures to temperature-independent tunneling at low temperatures, and the same framework also accommodates vibrationally enhanced tunneling.

Core claim

The symmetric trigonometric double-well potential permits an exact solution of the Schrödinger equation that, when inserted into a modified Weiner theory, supplies a closed-form rate formula for proton transfer; when evaluated for the N2H7+ dimer with literature parameters, the formula exhibits the continuous transition from classical thermal activation to quantum tunneling as temperature is lowered.

What carries the argument

Symmetric trigonometric double-well potential (TDWP) inside the Schrödinger equation, which supplies the exact eigenfunctions and eigenvalues used to construct the modified Weiner rate expression.

If this is right

  • The same analytic expression can be evaluated for any symmetric hydrogen-bonded system once its barrier height and well separation are known.
  • The rate formula automatically incorporates the crossover between thermally activated and tunneling regimes without separate classical or quantum limits.
  • Vibrationally assisted tunneling contributions appear naturally when excited-state matrix elements are included in the expression.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be tested against path-integral or instanton calculations for the same dimer to quantify residual error from the one-dimensional model.
  • If the mapping from spectroscopic parameters proves robust, the formula offers a computationally cheap way to screen proton-transfer rates across families of related cations.
  • Extension to time-dependent driving fields would allow direct prediction of laser-enhanced tunneling rates in the same framework.

Load-bearing premise

Parameters taken from existing IR spectra and quantum-chemical calculations map directly onto the symmetric trigonometric double-well potential without further adjustment.

What would settle it

Experimental measurement of the proton-transfer rate constant in the ammonia dimer cation at several temperatures below 100 K that fails to show the predicted flattening from Arrhenius to temperature-independent tunneling.

Figures

Figures reproduced from arXiv: 2603.01573 by A.E. Sitnitsky.

Figure 1
Figure 1. Figure 1: The trigonometric double-well potential (2) at th [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The dependence of the proton transfer rate constan [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The dependence of the proton transfer rate constan [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The dependence of the probability flux and the quant [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The dependence of the proton transfer rate constan [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
read the original abstract

The Weiner's theory (WT) is developed on the basis of the exactly solvable Schr\"odinger equation with trigonometric double-well potential (TDWP). The symmetric case of TDWP is considered. This modified version of WT (mWT) enables one to eliminate some severe approximations of the original Weiner's approach and to obtain more accurate results. An analytic formula is derived which provides the calculation of the proton transfer rate with the help of elements implemented in {\sl {Mathematica}}. We exemplify the application of mWT by calculating the proton transfer rate constant in the hydrogen bond of the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ (${\rm{H_3N\cdot\cdot\cdot H^{+} \cdot\cdot\cdot NH_3}}$). The parameters of the model for this object are extracted from available literature data on IR spectroscopy and quantum chemical calculations. The approach yields the transition from the Arrhenius-like exponential temperature dependence characteristic of thermal activation to that of quantum tunneling. Besides it is well suited for describing the phenomenon of vibrationally enhanced tunnelling.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper develops a modified Weiner's theory (mWT) for the exactly solvable Schrödinger equation with a symmetric trigonometric double-well potential (TDWP). It claims to eliminate severe approximations of the original approach, derives an analytic formula for the proton transfer rate constant (implemented via Mathematica elements), and applies the method to the hydrogen bond in N2H7+. Parameters are extracted from published IR spectroscopy and quantum-chemical calculations. The results illustrate a transition from Arrhenius-like thermal activation to quantum tunneling, including suitability for vibrationally enhanced tunneling.

Significance. If the TDWP parameter mapping proves accurate and the analytic derivation is free of hidden approximations, the work supplies an efficient closed-form route to temperature-dependent proton-transfer rates that bridges classical activation and tunneling regimes. This could be useful for modeling hydrogen-bonded cations where vibrational assistance modulates tunneling probabilities, offering a lighter alternative to full numerical path-integral or instanton calculations once the potential parameters are fixed.

major comments (2)
  1. [Application to N2H7+] Application to N2H7+ (parameter extraction paragraph): The TDWP parameters (V0, a, b) are taken directly from external IR frequencies and quantum-chemical barrier heights, yet the manuscript reports no numerical diagonalization of the TDWP Hamiltonian with these same values, nor any comparison of the resulting eigenvalues or tunneling splitting against the source spectroscopic or ab-initio data. Because the rate formula depends on these parameters, the absence of this consistency check leaves the physical fidelity of the model unquantified and undermines the claim that the analytic expression yields reliable rates for the real system.
  2. [mWT derivation] Derivation of the analytic rate formula (mWT section): Although the abstract states that an analytic formula is derived and implemented in Mathematica, the manuscript supplies neither the explicit closed-form expression for the rate constant k(T) nor the key intermediate steps that eliminate the original Weiner approximations. Without these, it is impossible to verify that the claimed improvement is realized or to reproduce the temperature-dependent transition from Arrhenius to tunneling behavior.
minor comments (2)
  1. [Abstract] The abstract refers to 'elements implemented in Mathematica' without naming the specific functions or notebook structure used; adding this detail would improve reproducibility.
  2. [TDWP definition] The symmetric TDWP potential is invoked but its explicit functional form (including the precise trigonometric expression) is not restated in the main text; including it once would clarify the exactly solvable case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and indicate the changes we will incorporate in the revised version.

read point-by-point responses

  1. Referee: [Application to N2H7+] Application to N2H7+ (parameter extraction paragraph): The TDWP parameters (V0, a, b) are taken directly from external IR frequencies and quantum-chemical barrier heights, yet the manuscript reports no numerical diagonalization of the TDWP Hamiltonian with these same values, nor any comparison of the resulting eigenvalues or tunneling splitting against the source spectroscopic or ab-initio data. Because the rate formula depends on these parameters, the absence of this consistency check leaves the physical fidelity of the model unquantified and undermines the claim that the analytic expression yields reliable rates for the real system.

    Authors: We agree that a direct consistency check is important for validating the parameterization. In the revised manuscript we will add the results of numerical diagonalization of the TDWP Hamiltonian using the extracted values of V0, a, and b, together with a quantitative comparison of the computed eigenvalues and tunneling splitting against the IR spectroscopic frequencies and ab-initio barrier data cited in the paper. revision: yes

  2. Referee: [mWT derivation] Derivation of the analytic rate formula (mWT section): Although the abstract states that an analytic formula is derived and implemented in Mathematica, the manuscript supplies neither the explicit closed-form expression for the rate constant k(T) nor the key intermediate steps that eliminate the original Weiner approximations. Without these, it is impossible to verify that the claimed improvement is realized or to reproduce the temperature-dependent transition from Arrhenius to tunneling behavior.

    Authors: We acknowledge that the explicit closed-form expression for k(T) and the detailed intermediate steps were not presented with sufficient clarity. In the revised manuscript we will insert the full analytic formula for the proton-transfer rate constant together with the key algebraic steps that remove the original Weiner approximations, thereby allowing verification and reproduction of the Arrhenius-to-tunneling transition. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper derives the modified Weiner theory (mWT) from the exactly solvable Schrödinger equation with symmetric trigonometric double-well potential (TDWP), producing an analytic proton-transfer rate formula implemented via Mathematica. Parameters for the N2H7+ example are taken directly from external IR spectroscopy and quantum-chemical literature, but the derivation itself is independent of those specific values and does not reduce to them by construction. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. The result is a general analytic expression applied to external data, which is self-contained against the mathematical framework of the solvable potential.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the exact solvability of the symmetric trigonometric double-well Schrödinger equation and on external parameter values taken from spectroscopy and quantum calculations; no new entities are postulated.

free parameters (1)
  • TDWP model parameters
    Extracted from IR spectroscopy and quantum chemical calculations for the N2H7+ system; no explicit values or fitting procedure given in the abstract.
axioms (1)
  • domain assumption The trigonometric double-well potential permits an exact solution of the Schrödinger equation in the symmetric case
    This is invoked as the foundation that allows elimination of severe approximations in the original Weiner approach.

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