Proton displacement

In this Section, we study some dynamical properties of the protonated water hexamer, fo-cusing our attention on the localization of the hydrated proton. We thus need to find clear indicators of the presence or the absence of PT processes occurring in the Zundel-like or Eigen-like core of the cluster.

The first intuitive observable that comes in mind is theproton displacement ddefined as d=d_1/2= ˜d_O

1/2H⁺ −d_O1O2

2 , (6.1)

withd˜_O

1/2H⁺ the distance projected onto the O₁O₂ segment. In the above equation, theO_1/2 notation suggests that the choice of the reference oxygen atom is arbitrary. Indeed, if the generated MD trajectory is long enough, the hydrated proton dynamics is ergodic and its residence time near the left O₁ or right O₂oxygen atoms isa priori the same. Unfortunately, due to the computational cost of such calculations, our CMPC-LD trajectories are too short

T = 200 K

Figure 6.8 – Bidimensional oxygen-oxygen and oxygen-proton distributions obtained by QMC-driven CMPC-LD simulations at low temperature T = 200 K (upper left panel), room temperature T = 300 K (upper right panel) and high temperatureT = 400K (bottom panel). The black circles correspond to the zero temperature equilibrium geometries of the protonated hexamer.

to ensure the full ergodicity of proton dynamics. Consequently, the obtained distributions with the simple application of the Eq. (6.1) would be asymmetric, which is not in agreement with the expected behavior of the cluster.

To solve that issue, we apply Eq. (6.1) taking each central oxygen atom as reference and we symmetrize the distribution by binning both thed₁ andd₂distances. As we can see in the left panel of Figure6.9, the obtained distributions are symmetric but not smooth because of the limited number of sampled classical configurations. The dark green curve, corresponding to a low-temperature (T= 200K) simulation, displays two sharp peaks that confirm the trap-ping of the hydrated proton around its minimum energy Eigen-like configuration. When one increases the temperature, the position of the two peaks migrates to the larger values of|d|, which confirms the larger amplitude of the proton fluctuations. We also notice that one can directly relate the height of the central part (|d|<0.05 Å) of the histograms to the number of times the protonated cluster has adopted its Zundel-like configurations, suggesting that spontaneous PT happened. It is clear that the higher the temperature, the more numerous are the Zundel-like configurations. The greater number of visited Zundel-like configurations when the cluster temperature increases is in agreement with Figure6.8. The proton hopping is more frequent at higher temperatures.

We present in the right panel of Figure6.9another observable that can help to understand the nature of the configurations adopted by the H⁺(H₂O)₆ cluster: thesharing proton coordinate, denotedδ. It corresponds to

δ=|d_O

1H⁺−d_H+O2|, (6.2)

and has been used in previous studies of aqueous proton defects in condensed phase[246,252,253]. This observable, strongly correlated to the proton displacementd, gives us precious informa-tion about the symmetry of the cluster core. Indeed, when the considered configurainforma-tion is symmetric or Zundel-like, the proton sharing displacementδ vanishes² because it is equally shared between its two neighboring water molecules. On the contrary, when the hydrated proton forms a covalent bond with its closest neighboring oxygen atom,δ takes finite values (up toδ ∼0.8 Å) which are the signature of Eigen-like states. Inspecting the right panel of Figure6.9. we can notice that the larger the temperature, the larger the average sharing pro-ton coordinateδ. This confirms that thermal effects enhance the amplitude of the hydrated proton fluctuations. In the meantime, the high-temperature distributions (T = 350−400 K) exhibit a plateau in the δ ∼ 0 region, corresponding to Zundel-like configurations. This further proves that the hydrated proton is less trapped at these temperatures and can more easily jump from one oxygen atom to another.

To pursue this configurational analysis, we would like to estimate, as quantitatively as possible, the weights of Zundel- and Eigen-like configurations from the distributions of the proton displacementd. To that purpose, we decided to fit the probability distributionP(d)according

2as the proton displacementddoes.

0

−0.40 −0.30 −0.20 −0.10 0.00 0.10 0.20 0.30 0.40

Normalized histograms

0.00 0.20 0.40 0.60 0.80 1.00

Normalized histograms

Sharing proton coordinate δ (Å)

200 K 250 K 300 K 350 K 400 K

Figure 6.9 – Distributions of the proton displacementdwith respect to the midpoint of the oxygen-oxygen distance (left panel) and the proton sharing coordinate δ (right panel), as a function of the temperature, for classical particules.

the following 3-gaussian, or 2-species model:

P(d) = (1−λE) built to fulfill the normalization condition R

P(d) = 1 and to reproduce at best the signal observed in Figure6.9. At least2of the4fitting parameters have a meaningful physical inter-pretation: λE simply represents the weight of Eigen-like configurations within the 2-species model, whereas d_E quantifies the distortion of the hydrated proton along the oxygen-oxygen distance.

The results of the fitting procedure are reported in Figure 6.10 and Table 6.5 for the 5 studied temperatures, ranging from T = 200 K to T = 400 K. The fitting parameters are given in TableF.1 of the AppendixF. We first note that, despite the noisy shape of the pro-ton displacement distributions extracted from CMPC-LD simulations, due to the shortness of their trajectory, a good agreement with the fitting function P(d) is found. The relevance of the 2-species model is further validated by the stable behavior of the 4 fitting parameters when one increasing the temperature. In particular, the parameters related to the Eigen-like configuration, namely λ_E, d_E and σ_E tend to increase with the temperature. This further proves that thermal effects tend to enhance the asymmetry of the protonated hexamer core, localizing the hydrated proton on a side water molecule.

It is also clear from Table 6.5, that the number of spontaneous proton jumps and charge rearrangements occurring inside the water hexamer core increases with the cluster tempera-ture. Indeed, the weight of the symmetric Zundel-like transition states is multiplied by a factor

3 between the low- (T= 200K) and the high- (T= 400K) temperature regimes. This implies that the proton diffusion is enhanced by thermal effects, due to a greater proton mobility.

This point will be later discussed in the Section6.4. Let us also notice that, within a classical picture, the hydrated proton remains extremely localized since the Eigen-like structure still dominates the Zundel-like one at ambient conditions.

0

Figure 6.10 – Comparison of theP(d)distributions obtained from QMC-driven CMPC-LD simu-lations (blue) and fitted using Eq. (6.3) for T= 200, 250,300,350,400 K.

To conclude, we have elucidated the impact of thermal effects on the hydrated proton dynam-ics in the protonated water cluster. Indeed, in the low-temperature regime (up to T= 200 K), the proton is chemically inert since it is trapped around its minimum energy

configura-T (K) % Zundel % Eigen

200 13 87

250 20 80

300 20 80

350 18 82

400 38 62

Table 6.5 – Species proportions obtained by fitting distributions of the proton displacement with the 3-gaussian model of Eq. (6.3) in the classical particles case.

tion, which is Eigen-like. At room conditions, the proton trapping is found to remain quite important and the predicted majority configuration of the H₁₃O⁺₆ is still Eigen-like, which is in contradiction with recent experimental data^[17,247]. Indeed, these works emphasize the paramount importance of the Zundel complex as transition state for the hydrated proton dynamics in water and aqueous solutions. As we have seen in Chapter 4, NQE dominate over thermal effects, especially at ambient conditions: their essential role must be taken into account, as we will do in the following Section.