Nuclear quantum effects and proton transport in equimolar water–phosphoric acid mixtures

We investigated isotope effects associated with nuclear quantum dynamics in equimolar water–phosphoric acid mixtures and their relationship to the large low-temperature isotope effect previously reported for proton conductivity. Neutron Compton scattering measurements were performed using the VESUVIO beamline at the ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, UK. Nuclear kinetic energies and momentum distributions were determined for hydrogen and deuterium. Pronounced isotope dependence was observed in the nuclear kinetic energies of H and D, together with non-Gaussian components in their nuclear momentum distributions. These signatures are consistent with an anharmonic, barrierless local effective Born–Oppenheimer potential rather than a double-well potential. The microscopic observations provide a local nuclear-dynamical counterpart to the reported isotope effect in proton conductivity. The transport behaviour is consistent with zero-point-energy-driven lowering of the effective proton-hopping barrier rather than tunnelling-dominated motion. Although smaller, site-specific tunnelling contributions cannot be excluded, the site- and isotope-averaged neutron Compton scattering observables do not require tunnelling to be the dominant transport mechanism. These findings demonstrate neutron Compton scattering as an isotope-selective diagnostic of nuclear quantum effects in an ionic-liquid-like proton-conducting medium and establish a framework for applying nuclear momentum distributions to disordered hydrogen-bonded electrolytes and glassy proton conductors.

Proton conduction in condensed-phase systems is central to many functional materials. The underlying microscopic mechanism, however, depends on hydrogen-bond topology, protonic defects and structural disorder [1–3]. A key example is Grotthuss-type transport, in which protons move through a hydrogen-bonded network by successive proton-transfer steps coupled to local hydrogen-bond rearrangements [4, 5]. In such systems, the hydrogen-bonded network is affected by the quantum nature of the light nuclei [6]. Two classes of nuclear quantum effects are particularly relevant to proton transfer: zero-point-energy contributions that lower the effective activation barrier and reshape local nuclear dynamics, and nuclear quantum tunnelling through a barrier along the proton-transfer coordinate [7].

Phosphoric acid (H 3 PO 4) is a useful reference system for studying these effects because its proton conductivity has been investigated by conductivity measurements [8], quasielastic neutron scattering [9], and ab initio molecular dynamics simulations [10–12]. In neat H 3 PO 4, simulations by [10] showed that protons in polarisable hydrogen bonds undergo a Grotthuss-like mechanism, accounting for the high proton conductivity. The corresponding mechanism in aqueous H 3 PO 4 solutions is less clear. Chemical disorder, speciation, proton exchange and glass formation make the microscopic role of nuclear quantum effects difficult to isolate experimentally or computationally.

This difficulty is highlighted by a recent study of the isotope effect in H 2 O–H 3 PO 4 and D 2 O–D 3 PO 4, which revealed a large increase in proton conductivity below 160 K upon D/H substitution, while the D/H conductivity ratio approached the value expected for a classical proton-transfer mechanism above 250 K [13]. The same study showed that D/H substitution shifts the structural relaxation to longer times, increases the viscosity and raises the glass-transition temperature. Structural proton transport in such a hydrogen-bonded liquid requires transient reorganisation of the local molecular environment; a more viscous and dynamically slower matrix therefore reduces the probability of forming favourable proton-transfer configurations and suppresses conductivity. After the viscosity contribution had been accounted for, the residual H/D difference was attributed to zero-point-energy lowering of the effective proton-transfer barrier, with no indication that tunnelling is the dominant transport channel [13]. The present neutron Compton scattering study complements those macroscopic conductivity, viscosity and structural-relaxation measurements by probing the isotope-resolved nuclear momentum distributions, kinetic energies and non-Gaussian components of the local H and D nuclear dynamics directly.

Applying neutron Compton scattering to proton-conducting liquids and glassy electrolytes is non-trivial. Chemical speciation, rapid proton exchange, orientational disorder and the coexistence of multiple hydrogen-bond motifs mean that the measured nuclear momentum distribution is an ensemble- and orientation-averaged quantity rather than the response of a single crystallographic proton site. We therefore use neutron Compton scattering here not as a unique tool for reconstructing a local effective Born–Oppenheimer potential, but as an isotope-selective diagnostic of nuclear quantum dynamics in a chemically disordered proton-conducting medium. The experimentally accessible quantities–the distribution width, excess kurtosis and mean kinetic energy–directly reveal how the local H and D nuclear dynamics depart from a harmonic, Gaussian description.

Here, we apply this strategy to equimolar H 2 O–H 3 PO 4 and D 2 O–D 3 PO 4, a model proton-conducting liquid/glassy electrolyte. This composition, equivalent to 85 wt% acid in water, was chosen because it is the composition for which the large low-temperature conductivity isotope effect and the associated changes in structural relaxation were reported [13]. This choice therefore allows the neutron Compton scattering observables to be compared directly with the established macroscopic transport behaviour, while recognising that the measured response is necessarily an average over multiple local hydrogen-bond motifs rather than a single microscopic environment. To our knowledge, this is the first application of neutron Compton scattering as an isotope-selective probe of nuclear quantum dynamics in this class of ionic-liquid-like proton-conducting media. By combining neutron-Compton-scattering-derived momentum widths, excess kurtosis values and kinetic energies with harmonic ab initio vibrational calculations on representative acid–water clusters, we establish a practical framework in which the calculations provide a harmonic reference for diagnosing isotope effects and nuclear quantum effects in disordered hydrogen-bonded electrolytes and for connecting microscopic nuclear dynamics with macroscopic proton transport.

Equimolar phosphoric acid–water samples and their fully deuterated analogues were measured on the VESUVIO beamline at Target Station 1 of the ISIS Neutron and Muon Spallation Source. Measurements were performed at 300, 160, 80 and 10 K. Details of the sample preparation and handling, data reduction and fitting procedures, and Bayesian model selection are provided in the Supplementary Material.

The Cs, AdoWpo and AdohWpo H 3 PO 4 –H 2 O and D 3 PO 4 –D 2 O clusters from the study of [14] were used to calculate harmonic nuclear momentum widths, excess-kurtosis values and isotope-dependent kinetic-energy ratios for comparison with the proton and deuteron longitudinal momentum distributions. Full computational details are given in the Supplementary Material.

The Bayesian model comparison, described in the Supplementary Material and shown in Supplementary Figures S1 and S2, indicates that the model based on the fourth-order Gram–Charlier representation (GC4), in which a Gaussian profile is augmented by a Gaussian-weighted fourth-order Hermite term, provides a lower- χ 2 description than the purely Gaussian model for both H and D at 10 K. Here, GC4 denotes the complete truncated representation, whereas c 4 denotes the coefficient multiplying H 4 and quantifies the excess kurtosis. The corresponding log-evidence differences and Bayes factors are reported in Supplementary Tables S4 and S5. This result is important because it establishes that the low-temperature nuclear momentum distributions contain measurable non-Gaussian components, rather than being adequately described by a single harmonic Gaussian response.

The proton longitudinal momentum distribution in the H 2 O–H 3 PO 4 sample at 10 K is shown in Figure 1, and the corresponding deuteron distribution in D 2 O–D 3 PO 4 is shown in Figure 2. In both cases, the fitted model reproduces the focused recoil profile and the residuals remain small across the measured momentum range. The key information extracted from these spectra is the width of the distribution, which determines the isotope-specific kinetic energy, and the fourth-order Hermite coefficient c 4, which quantifies the departure from a harmonic, Gaussian local-motion picture. Because the Hermite polynomials used in the Gram–Charlier representation are orthogonal under the Gaussian weight, σ and c 4 are fitted as independent expansion parameters. Consequently, the kinetic energies derived from σ are not determined by the magnitude of the excess-kurtosis term [15, 16]. The corresponding kinetic energies are listed in Table 1.

Three of the most stable H 3 PO 4 –H 2 O clusters identified by [14], termed Cs, AdoWpo and AdohWpo, together with their D 3 PO 4 –D 2 O analogues, are shown in Figure 3. The corresponding harmonic ab initio vibrational calculations are not intended to provide a complete structural model of the liquid. Rather, they define the range of nuclear momentum widths, excess-kurtosis values and kinetic energies expected from representative harmonic acid–water hydrogen-bond motifs. Figures 4, 5 compare these harmonic predictions with the experimentally measured H and D nuclear momentum widths and excess-kurtosis parameters. The cluster calculations are therefore used only as a local harmonic reference. They cannot represent the full liquid-state structure of the equimolar acid–water mixture, including long-range chemical disorder, speciation, proton exchange and collective matrix dynamics. A fully predictive treatment of the isotope-dependent condensed-phase nuclear quantum dynamics would require path-integral or related quantum-dynamical simulations of a statistically representative liquid/glass ensemble, which is beyond the scope of the present work. This limitation is mitigated by the local character of neutron Compton scattering: in the impulse-approximation regime, eV neutrons probe the short-time, single-particle momentum distribution of H or D nuclei, so the comparison is most sensitive to local nuclear quantum dynamics rather than to the nuclear dynamics dictated by changes in long-range structural order.

The central experimental observation is the breakdown of this harmonic Gaussian reference for the protonated system below approximately 160 K. In H 2 O–H 3 PO 4, c 4 increases sharply from 0.2 at 160 K to a plateau value of 0.9–1.0 below 80 K, while σ rises from 5.2 Å -1 at 160 K to 6.6–7.0 Å -1 below 80 K. For any spherically averaged three-dimensional harmonic anisotropic Gaussian momentum distribution, the value of c 4 cannot exceed 0.8 (see [17] and the derivation in the Supplementary Material). The physical basis of this criterion is that an isotropically averaged harmonic Gaussian momentum ellipsoid has a restricted fourth moment: the variance is controlled by the trace of the momentum-width tensor, while the excess kurtosis is controlled by the spread of its eigenvalues. Even the most anisotropic positive-semidefinite harmonic tensor can generate at most c 4 = 4 / 5. The threshold c 4 > 0.8 therefore provides a practical selection rule separating line shapes compatible with harmonic anisotropy from those requiring anharmonic local nuclear dynamics. This violation of the harmonic anisotropic Gaussian bound, c 4 ≤ 0.8, provides direct evidence of strongly anharmonic local proton dynamics. The hydrogen momentum distributions measured at 10 and 80 K therefore cannot be explained by a harmonic anisotropic Gaussian model and require an anharmonic contribution. This interpretation is consistent with potential-of-mean-force analyses of neutron Compton profiles, in which deviations of the mean force from the harmonic anisotropic reference provide a complementary way of diagnosing anharmonicity. For the GC4 representation to be physically admissible, its longitudinal momentum distribution must remain positive, which requires 0 ≤ c 4 < 4 / 3. A stationary-point analysis of the potential of mean force additionally gives c 4 ≤ c 4, P M F = 2 [ 1 − M k B T / (ℏ 2 σ 2) ] 2. These combined positivity and single-minimum conditions are summarised in Supplementary Equation S27, with the derivation given in Supplementary Equations S18-S27 of the Supplementary Material. All experimental values of the width and excess kurtosis for H and D satisfy both conditions (Supplementary Tables S6, S7), so the fitted GC4 profiles are physically admissible and the simplified potentials inferred from them remain single-minimum at every temperature. The excess kurtosis therefore diagnoses anharmonicity here without implying a multi-well potential or coherent tunnelling [18–20]. In contrast, the deuterated system shows smaller low-temperature values, approximately 0.5 for c 4 and 6.6 Å -1 for σ (Figure 5), and remains within the harmonic anisotropic Gaussian bound, c 4 ≤ 0.8, over the full temperature range.

We emphasise that this conclusion does not rest on the bound being exceeded at a single temperature. Below 160 K the protonated system departs from the harmonic anisotropic Gaussian reference simultaneously in three observables: the momentum width σ, the excess kurtosis c 4, and the proton-to-deuteron kinetic-energy ratio (Figure 6), all of which move together and in the same direction. The propagated uncertainties in Table 1 show that the low-temperature enhancement of the kinetic-energy ratio is robust, whereas deviations close to 160 K are less decisive. It is this concerted, isotope-specific departure, rather than any individual data point, that constitutes the evidence for anharmonic local proton dynamics.

To place this observation in a physically transparent context, we compare the neutron-Compton-scattering-derived isotope effect with the two-state diabatic model of [21] which describes nuclear quantum effects in an idealised hydrogen bond X–H(D)…Y as a function of donor–acceptor separation. In this model, large donor–acceptor distances are associated with double-well local effective Born–Oppenheimer potentials and bimodal proton densities, whereas shorter distances lead to flatter, strongly anharmonic and ultimately barrierless local effective Born–Oppenheimer potentials. The systematic evolution of these local effective Born–Oppenheimer potentials is correlated with the isotope effect on the OH/OD stretching frequency. In applying this model here, it is important to recognise that neutron Compton scattering measures a site- and orientation-averaged view of H/D nuclear dynamics across multiple local environments, whereas the McKenzie model refers to an isolated hydrogen bond. We therefore use it as a qualitative reference for interpreting the isotope dependence of the experimental observables, not as a one-to-one reconstruction of a local effective Born–Oppenheimer potential.

This interpretation is also consistent with neutron-scattering and ab initio studies of short, strong hydrogen bonds in urea–phosphoric-acid and related molecular crystals. In those systems, proton transfer or proton migration toward the hydrogen-bond midpoint could not be explained by a fixed one-dimensional proton potential and thermal occupation of its levels alone; instead, low-frequency phonons and collective structural fluctuations were shown to reshape the effective proton potential [22, 23]. The present liquid/glassy electrolyte is not a single-crystal short-hydrogen-bond system, so these studies should not be used as a direct microscopic model. They nevertheless support the broader view that strongly hydrogen-bonded phosphoric-acid environments can display anharmonic, environment-modulated proton dynamics rather than a simple static double-well tunnelling response. Neutron Compton scattering provides a particularly direct illustration of the isotope dependence of quantum tunnelling in potassium dihydrogen phosphate. In the paraelectric phase of K H 2 P O 4 (KDP), the proton momentum distribution projected along the O − H ⋯ O hydrogen bond was found to narrow substantially and to exhibit oscillatory structure characteristic of interference between spatially separated components of a wavefunction coherently delocalised over the two sites of an effective double-well potential [24]. The Born–Oppenheimer potential reconstructed from the measured momentum distribution yielded an estimated tunnelling splitting of approximately 90 m e V. In contrast, neutron Compton scattering measurements of the deuterated analogue K D 2 P O 4 (DKDP), performed below and above its paraelectric–ferroelectric transition, produced nearly Gaussian deuteron momentum distributions that changed only weakly with temperature and showed no corresponding oscillatory signature of coherent occupation of two sites [25]. The absence of a resolved coherence feature led to the conclusion that the deuteron tunnelling splitting in DKDP is much smaller than 20 m e V, in marked contrast to the pronounced proton-tunnelling signature observed in KDP.

The absence of an observable tunnelling signature in DKDP should not be interpreted as evidence that deuterons are universally incapable of tunnelling. In particular, the DKDP momentum distribution is also compatible with a symmetric double-well potential whose tunnelling splitting is small compared with the thermal energy, so that thermal population of the lowest states suppresses the momentum-space interference feature. Local asymmetry, reduced coherence and the temperature and resolution of the measurement may likewise obscure an otherwise finite tunnelling contribution. Furthermore, deuteration of KDP does not constitute a simple change of nuclear mass in an unchanged potential: it increases the O ⋯ O separation and modifies the coupling between hydrogen-bond motion and displacements of the heavier ions, thereby changing the effective Born–Oppenheimer potential itself [25, 26]. Deuteration therefore usually reduces the tunnelling splitting, but whether deuteron tunnelling persists or remains experimentally observable is determined by the material-specific potential-energy landscape, collective coupling, temperature and experimental timescale. For the present mixtures, by contrast, every measured profile satisfies the GC4 positivity condition and the corresponding mean-force potential remains single-minimum. This supports a physically admissible, strongly anharmonic single-minimum interpretation while leaving open smaller site-specific tunnelling contributions that may be obscured by ensemble and orientational averaging.

Beyond coherent tunnelling signatures, isotope substitution can modify proton-transfer dynamics through zero-point delocalisation, reshaping of the transfer free-energy profile and coupling to hydrogen-bond-network reorganisation. Path-integral simulations of an excess proton in liquid water show that nuclear quantum fluctuations increase proton excursions along hydrogen-bonded wires and permit transient delocalisation over several water molecules [27]. Simulations of water autoionisation likewise show that quantised nuclei substantially alter the proton-abstraction free-energy curve and generate a pronounced isotopic dependence of the acidity [28]. In a solid-acid model, ring-polymer simulations of proton hopping in H-CHA zeolite find that nuclear quantum effects modify equilibrium site populations, preferred transfer pathways and hopping rates, and produce a temperature-dependent H/D kinetic isotope effect [29]. These studies emphasise that mass-dependent proton-transfer kinetics need not arise solely from motion through a fixed one-dimensional barrier. They can emerge from the combined effects of zero-point motion, free-energy-barrier renormalisation, collective network fluctuations and dynamical recrossing. In the present mixtures, the broader and more non-Gaussian H momentum distribution below 160 K is therefore consistent with a protonated network sampling transfer-favourable configurations more readily than its deuterated analogue, although neutron Compton scattering diagnoses the local nuclear dynamics rather than measuring a proton-transfer rate directly.

Although OH and OD stretching frequencies are not measured directly by neutron Compton scattering, the nuclear momentum distribution is dominated by high-energy local vibrational motions that carry most of the nuclear kinetic energy [15, 16, 30, 31]. It is therefore meaningful to compare the isotope effect in the measured kinetic energy with the isotope effect in the harmonic OH/OD frequencies calculated for the Cs, AdoWpo and AdohWpo motifs (Figure 6). The propagated ratios are 2.21 ± 0.17 at 10 K and 2.07 ± 0.15 at 80 K, well above the harmonic cluster reference shown in Figure 6. At 160 K 1.53 ± 0.11 and 300 K 1.70 ± 0.12, the deviations from the harmonic reference are smaller and should be interpreted more cautiously. The H/D kinetic-energy ratio can be viewed in two complementary ways. In a simple mean-field picture it reflects the curvature and effective mass associated with one local effective potential and one effective vibrational frequency. More generally, it is the first moment (centre of gravity) of a Boltzmann-population-weighted, atom-projected vibrational density of states: low-frequency intermolecular modes involve larger effective masses and collective reorientations, whereas high-frequency stretching modes are closer to the bare H or D mass. A departure from the harmonic H/D ratio can therefore arise when anharmonic low-frequency intermolecular motions soften or reshape the potential of mean force underlying the site- and isotropically averaged longitudinal momentum distribution [19, 32, 33]. Within the qualitative McKenzie-model framework, this site-averaged response is most naturally associated with a population of short, strong hydrogen-bond environments on the barrierless/strongly anharmonic side of the model rather than with a tunnelling-dominated double-well local effective Born–Oppenheimer potential.

At 10 K, the vibrational zero-point energy of protons in the equimolar water–phosphoric-acid mixture is approximately twice that of deuterons in the corresponding deuterated system (see Table 1). If the dominant contribution arises from the O–H stretching coordinate, this magnitude of the isotope effect is consistent with strongly anharmonic local dynamics with steep confinement, approaching the square-well limit in the idealised model. At higher temperatures, the site-averaged response is expected to become less strongly anharmonic and to approach the harmonic limit around 160 K.

The neutron Compton scattering results therefore provide a local nuclear-dynamical counterpart to the macroscopic conductivity isotope effect. The microscopic anharmonicity observed below 160 K correlates with the temperature range where the conductivity isotope effect becomes significant in the measurements of [13]. The same temperature range in which transport becomes strongly isotope dependent is also the range in which the proton momentum distribution becomes broader, increasingly non-Gaussian and can only be explained by an underlying anharmonic local effective Born–Oppenheimer potential, while its deuteron counterpart remains within the realm of the harmonic approximation. In macroscopic terms, the lower glass-transition temperature of the protonated mixture means that, at the same low temperature, its hydrogen-bond network remains more able to undergo the local rearrangements required for structural proton transport than the deuterated analogue. The larger proton zero-point energy additionally lowers the effective transfer barrier; together these effects rationalise the large H/D conductivity difference without requiring a dominant contribution from nuclear quantum tunnelling [13]. A tunnelling-dominated transport mechanism would be expected to show different temperature dependence and momentum-distribution signatures associated with coherent delocalisation, which are not observed here. At the same time, the present liquid/glassy data are site- and isotropically averaged, so small or transient tunnelling contributions associated with a subset of hydrogen-bond motifs could be partially averaged into the observed J (y). The conclusion is therefore not that tunnelling is absent, but that the measured distributions do not display the characteristic coherent-delocalisation signatures that would make tunnelling the dominant interpretation [34].

The temperature dependence of the extracted kinetic energies may also reflect a dynamical crossover of the hydrogen-bond network. Here, the term dynamical crossover denotes a broad change in the local dynamical regime, not a liquid–liquid transition or a sharp thermodynamic glass transition. Around and above 160 K, the decrease in the measured momentum width for both isotopic samples coincides with the temperature range in which quasielastic neutron-scattering studies report the activation of local proton dynamics and structural relaxation in phosphoric-acid-based systems [9, 13]. This temperature coincides with the region where structural relaxation becomes significant in the protonated system. Activation of local reorientations of H 3 PO 4 motifs that rearrange hydrogen-bond donor and acceptor directions and can assist structural proton transport would average over, and effectively soften, the local potential experienced by the light nuclei. Such softening would narrow the longitudinal momentum distribution and reduce the apparent kinetic energy. The similar trend observed for H and D suggests that this contribution is associated with matrix dynamics, whereas the much larger low-temperature non-Gaussian response remains specific to the protonated system. Above this temperature range the H and D kinetic energies do not evolve identically. This difference is expected because isotope substitution changes both the zero-point contribution and the thermal population of local vibrational, librational and reorientational modes; consequently, the same matrix reorganisation can affect H and D longitudinal momentum distributions with different weights [19, 33, 35].

We note that the extracted mean kinetic energy is non-monotonic in temperature, passing through a minimum near 160 K for both isotopic species (Table 1). This is consistent with the dynamical-crossover picture outlined above: the activation of local librations and reorientations near 160 K motionally softens the effective potential experienced by the light nuclei and reduces the apparent kinetic energy, whereas the larger value recovered at 300 K reflects the ordinary thermal increase in vibrational kinetic energy. For the deuterated system the 160 and 300 K values agree within their combined uncertainties; for the protonated system the 300 K value lies only marginally above the 160 K minimum, so we do not attach quantitative significance to the precise depth of the minimum. During the measurements the temperature was stabilised within approximately 2 K, which is much smaller than the spacing between the nominal measurement temperatures; this uncertainty therefore does not affect the interpretation of the 160 K feature as a broad crossover between measured temperature points.

On the whole, in equimolar water–phosphoric-acid mixtures and their deuterated analogues, neutron Compton scattering identifies the microscopic isotope-dependent nuclear dynamics that accompany the low-temperature change in proton transport.

This work demonstrates the use of neutron Compton scattering as an isotope-selective probe of nuclear quantum effects in a chemically disordered proton-conducting liquid/glassy electrolyte. Applied to equimolar water–phosphoric-acid mixtures and their fully deuterated analogues, neutron Compton scattering provides direct access to H and D nuclear momentum distributions, excess-kurtosis values and mean kinetic energies across the 10–300 K range.

The main physical result is a pronounced low-temperature departure of the proton nuclear momentum distribution from the harmonic Gaussian reference. Below approximately 160 K, the protonated system shows a sharp increase in both distribution width and excess kurtosis, accompanied by a weaker response in the deuterated analogue. Interpreted together with harmonic cluster calculations and the McKenzie model as a qualitative reference for isolated hydrogen bonds, these observations are most consistent with site-averaged signatures of strongly anharmonic local nuclear dynamics dominated by zero-point-energy effects. They do not require a tunnelling-dominated interpretation. The experimental GC4 longitudinal momentum distributions for both systems at all temperatures satisfy the positivity condition, and the corresponding effective potentials of mean force satisfy the single-minimum criterion. Partial tunnelling contributions from a subset of local environments cannot be excluded, but they are not needed to account for the concerted behaviour of the measured width, excess kurtosis and H/D kinetic-energy ratio.

These findings provide a microscopic nuclear-dynamical counterpart to the large low-temperature isotope effect previously observed in proton conductivity. In macroscopic terms, the protonated liquid can be viewed as a hydrogen-bond network that remains more able to rearrange at low temperature than its deuterated analogue, while the proton zero-point energy further lowers the effective barrier for transfer. More generally, the combination of the model-independent excess-kurtosis bound with the requirement that width, kurtosis and isotope-dependent kinetic-energy ratios depart from the harmonic reference in concert provides a transferable, isotope-selective diagnostic of anharmonic nuclear quantum effects, applicable to other disordered hydrogen-bonded electrolytes and glassy proton conductors without recourse to a system-specific structural model. This provides a framework for applying nuclear momentum distributions to diagnose anharmonic nuclear quantum effects in chemically disordered hydrogen-bonded electrolytes. The same framework connects naturally to recent theoretical and experimental work showing that isotope-dependent proton-transfer dynamics can reflect not only tunnelling, but also mass-dependent zero-point delocalisation, free-energy-barrier reshaping and coupling to collective hydrogen-bond-network reorganisation [18, 27–29, 34, 35].