Complementary Eigen-Zundel Interpretation Reconciles Thermodynamics and Spectroscopy of Excess Protons in Aqueous HF Solutions

Using ab initio molecular dynamics, this study proposes a complementary Eigen-Zundel interpretation that reconciles the apparent contradiction between the distinct thermodynamic behaviors and similar vibrational spectra of excess protons in aqueous HF and HCl solutions by revealing a dynamically shared proton structure and nearly identical proton transfer free-energy profiles.

The Gist

The Great Acid Mystery: Why Two Different Acids Sound the Same

Imagine you are at a party. You have two groups of guests: Group HCl (Hydrochloric Acid) and Group HF (Hydrofluoric Acid).

According to the "textbook rules" of chemistry, these two groups should behave very differently:

• HCl is a "strong" acid. It's like a hyperactive kid who immediately lets go of their parent (the proton) and runs off to play with everyone else (water molecules). It dissociates completely.
• HF is a "weak" acid. It's like a clingy kid who refuses to let go of their parent. It stays stuck to its partner (the fluoride ion) and only occasionally lets go.

The Puzzle:
Scientists have known for a long time that if you measure the "thermodynamics" (how the chemicals react and settle down), HCl and HF are totally different. HCl is wild and free; HF is stuck and clingy.

However, when scientists shine a special light (Infrared Spectroscopy) on them to listen to their "vibrations" (like listening to the sound of a guitar string), something weird happens. They sound exactly the same.

It's as if you recorded a solo violin (HCl) and a solo cello (HF), but when you played them back, they sounded like the exact same instrument. This confused scientists for decades. How can two things that are chemically different sound identical?

The Solution: Two Different Lenses

The authors of this paper solved the mystery by looking at the problem through two different "lenses" or perspectives. They realized that the way we usually describe these acids (the Eigen picture) is great for understanding who is standing next to whom, but a different picture (the Zundel picture) is needed to understand how they move and vibrate.

Lens 1: The "Eigen" Picture (The Seating Chart)

Think of this as looking at a seating chart at a dinner party.

• In HCl: The "proton" (the guest) has let go of its original seat and is sitting in the middle of a table with three water molecules. It's a happy, shared group.
• In HF: The "proton" is still holding hands with its original partner (the fluoride). It's sitting at a table with one water molecule and one fluoride ion.

The Result: If you just look at the seating chart, the two parties look totally different. This explains why the chemistry (thermodynamics) is different. HF forms special "cliques" (called bifluoride ions) that HCl doesn't.

Lens 2: The "Zundel" Picture (The Dance Floor)

Now, imagine you stop looking at who is sitting where and start watching how they dance.

• In the Zundel picture, we care about the "dance move" where the proton jumps back and forth between two partners.
• The researchers found that even though the seating is different, the dance floor physics are identical.

The Magic Trick: Electrostatic Screening
Why do they dance the same? The paper explains that in the crowded solution, the other ions act like a crowd of people holding umbrellas.

• In the HF solution, the fluoride ion is very close to the proton. Normally, this would make the proton stick tight.
• BUT, the other ions in the solution (the "crowd") create a "shield" or "screen" around them. This shield cancels out the strong pull of the fluoride.
• Because of this shielding, the proton in HF feels just as free to jump back and forth between the water and the fluoride as the proton in HCl does between two water molecules.

The Analogy:
Imagine two people trying to talk to each other across a room.

1. HCl: They are in an empty room. They talk freely.
2. HF: They are in a crowded room, but everyone else is holding up "sound-dampening foam" (the ionic shield). Even though they are standing right next to a wall (fluoride), the foam makes it feel like they are in an open space.

Because the "sound" (the vibration) travels through the foam in the exact same way as it travels through the empty air, the spectra (the sound) are identical.

The Big Takeaway

This paper reconciles two conflicting views:

1. Thermodynamics says: "HF and HCl are different because their seating arrangements (chemical structures) are different."
2. Spectroscopy says: "HF and HCl sound the same because their dance moves (proton transfer) are identical."

The authors show that both are right.

• Use the Eigen view to understand the chemistry (who is holding hands).
• Use the Zundel view to understand the sound (how they vibrate).

The "shield" provided by the crowded solution makes the dance moves look the same, even though the partners are different. This gives us a unified picture of how excess protons behave in water, solving a puzzle that has confused scientists for nearly a century.

Technical Summary

1. Problem Statement

The paper addresses a long-standing paradox in aqueous acid chemistry regarding the behavior of excess protons in Hydrofluoric acid (HF) versus Hydrochloric acid (HCl):

• Thermodynamic Discrepancy: HF is a weak acid ($pK_a \approx 3.2$) that only partially dissociates and forms bifluoride ions ($HF_2^-$) at intermediate concentrations via the reaction $HF + F^- \rightleftharpoons HF_2^-$. In contrast, HCl is a strong acid ($pK_a \approx -7.0$) that dissociates completely. Thermodynamics predicts vastly different chemical speciation and proton environments for these two solutions.
• Spectroscopic Paradox: Despite these thermodynamic differences, experimental infrared (IR) spectra of HF and HCl solutions are nearly indistinguishable, particularly in the proton-transfer bands. Previous interpretations struggled to reconcile why HF, where protons are expected to be tightly bound to fluoride ($F^-$), exhibits spectral signatures similar to HCl, where protons are shared between water molecules.
• Simulation Conflicts: Previous ab initio molecular dynamics (AIMD) simulations yielded conflicting results: some suggested the dominance of $H_3OF$ structures (proton shared between water and fluoride) without $HF_2^-$ formation, while others observed $HF_2^-$ formation only under specific conditions, failing to explain the spectral similarity.

2. Methodology

The authors employed Ab Initio Molecular Dynamics (AIMD) simulations combined with a novel theoretical framework to analyze the solvation structure and dynamics of excess protons.

• Simulation Setup:
  • Software: CP2K using the BLYP exchange-correlation functional with D3 dispersion corrections.
  • System: 7.5 mol/kg solutions of HF and HCl in water (224 water molecules, 30 acid molecules, 30 excess protons, 30 anions).
  • Conditions: NVT ensemble at 300 K.
  • Basis Sets: DZVP-MOLOPT-SR-GTH for O/H; aug-DZVP-GTH for F/Cl.
  • Trajectory Lengths: 122 ps for HF, 84 ps for HCl.
• Analysis Frameworks:
  • Extended Eigen Picture: Used to determine chemical speciation (equilibrium populations). Protons are analyzed based on the number of hydrogen-bonded acceptors (water or anions) surrounding a central hydronium-like core.
  • Complementary Zundel Picture: Used to analyze spectroscopic signatures (vibrational dynamics). Protons are analyzed based on sharing between two proton acceptors (e.g., $H_5O_2^+$, $H_3OF$, $H_3OCl$).
  • Free Energy Landscapes: Calculated 2D free energy surfaces as a function of the heavy-atom distance ($R$) and the proton-transfer coordinate ($d$).
  • Spectral Calculation: IR absorption coefficients were derived from the Fourier transform of the polarization autocorrelation function using Wannier center localization. Difference spectra were computed to isolate proton-transfer features.

3. Key Contributions

The paper introduces a Complementary Eigen–Zundel Interpretation to resolve the thermodynamic-spectroscopic paradox:

1. Unified Speciation Model: It demonstrates that the Eigen framework correctly predicts the thermodynamic equilibrium, including the formation of bifluoride ($HF_2^-$) and the specific distribution of protonated species in HF.
2. Spectral Unification: It establishes that the Zundel framework is the appropriate descriptor for vibrational spectroscopy. The similarity in IR spectra arises not from identical structures, but from identical free-energy landscapes for proton transfer within specific Zundel-like configurations.
3. Role of Electrostatic Screening: The authors identify that electrostatic screening in concentrated solutions stabilizes the ion pair $F^- \cdots H_3O^+$, effectively lowering the free energy barrier for proton sharing in HF to match that of $H_5O_2^+$ in HCl.

4. Key Results

A. Chemical Speciation (Eigen Picture)

• HF Solutions: The excess proton is predominantly found in $H_7O_3F$ structures (61%) and $H_5O_2F_2^-$ (14%), with only 24% in the standard $H_9O_4^+$ (hydronium) form. This confirms the formation of bifluoride ($HF_2^-$) and direct contact between protons and fluoride, consistent with thermodynamic predictions.
• HCl Solutions: Dominated by the standard $H_9O_4^+$ (72%) and $H_7O_3Cl$ (24%) structures. Chloride plays a minor role in proton sharing compared to fluoride.
• Validation: The simulated speciation (14% $H^+$, 61% $HF$, 14% $HF_2^-$ at 7.5 mol/kg) aligns closely with thermodynamic estimates derived from electromotive force measurements.

B. Spectroscopic Signatures (Zundel Picture)

• Structural Differences: In HF, the dominant Zundel-like configuration is $H_3OF$ (63%), whereas in HCl, it is $H_5O_2^+$ (95%).
• Spectral Similarity: Despite these structural differences, the calculated IR difference spectra for HF and HCl are nearly identical in the Transfer-Waiting (TW) and Transfer-Path (TP) bands.
• Free Energy Landscapes:
  • The 2D free energy surfaces for $H_5O_2^+$ (in HCl), $H_5O_2^+$ (in HF), and $H_3OF$ (in HF) are remarkably similar.
  • Crucially, in concentrated HF, the free energy profile of $H_3OF$ shifts to become symmetric (double-well), resembling $H_5O_2^+$. In dilute HF, the profile is asymmetric (single-well), with the proton localized on fluoride.
  • This shift is caused by electrostatic screening from the ionic environment, which stabilizes the $F^- \cdots H_3O^+$ ion pair by approximately $1.1 k_B T$, lowering the barrier for proton sharing.

C. Mechanism of Reconciliation

The paradox is resolved because:

1. Thermodynamics is governed by the Eigen states (equilibrium populations), which differ significantly between HF and HCl (due to $HF_2^-$ formation).
2. Spectroscopy is governed by the Zundel transition states (proton transfer dynamics).
3. Electrostatic screening in concentrated HF makes the free-energy landscape for proton transfer in $H_3OF$ (HF) nearly indistinguishable from that of $H_5O_2^+$ (HCl). Consequently, the proton transfer dynamics and resulting IR signatures are identical, even though the static chemical species are different.

5. Significance

• Resolution of a Decades-Old Puzzle: The paper provides a unified microscopic picture that reconciles the contradictory thermodynamic behavior (weak vs. strong acid) with the observed spectroscopic similarity of HF and HCl.
• Methodological Advancement: It validates the use of AIMD combined with a dual-framework analysis (Eigen for equilibrium, Zundel for dynamics) for studying complex proton solvation.
• General Applicability: The findings suggest that for weak acids with high proton-affinity anions, electrostatic screening can fundamentally alter proton transfer dynamics, making them spectrally indistinguishable from strong acids under ambient conditions.
• Bifluoride Confirmation: The study provides strong computational evidence for the existence and stability of the bifluoride ion ($HF_2^-$) in concentrated aqueous solutions, resolving previous ambiguities in AIMD literature.

In conclusion, the authors demonstrate that while HF and HCl possess distinct chemical speciation (Eigen states), their proton transfer dynamics (Zundel states) are rendered identical by electrostatic screening, leading to the observed spectral convergence.