Anomalously Strong Effect of the Ion Sign on the Thermochemistry of Hydrogen Bonded Aqueous Clusters of Identical Chemical Composition

The sign preference of hydrogen bonded aqueous ionic clusters X±(H2O)i (n =1–5, X = F; Cl; Br) has been investigated using the Density Functional Theory and ab initio MP2 method. The present study indicates the anomalously large difference in formation free energies between cations and anions of identical chemical composition. The effect of vibrational anharmonicity on stepwise Gibbs free energy changes has been investigated, and possible uncertainties associated with the harmonic treatment of vibrational spectra have been discussed.


Introduction
The importance of a clear and insight understanding of ion-induced nucleation phenomena for a number of issues related to the Earth climate, air quality, public health and various technologies is well established [1][2][3][4][5][6][7][8]. Although the importance of the ion-induced nucleation became recognized long time ago, the pronounced ion sign effect on nucleation rates observed in Wilson's pioneering experiments OPEN ACCESS [1] in the cloud chamber and known as a sign preference has remained a mystery up until now. Castleman and Tang [6] stated over 30 years ago that ''understanding the effect of ions would be very difficult, or even impossible, if the ion's specific chemical characteristics had a significant effect on their nucleating efficiency". Recently, Nadykto et al. [5] pointed out that the strong effect of ion properties on nucleation rates is essentially quantum in nature, and is controlled by the electronic structure of the core ion through the influence on the intermolecular bonding energies during the initial steps of cluster formation. While core ions considered in [5] differ in both sign and chemical composition, species presented in this study differ in sign only. This allows separating the "sign" and "composition" effects and permits explicit treatment of the "pure" sign preference in aqueous systems. In the present Communication, the thermochemical properties of aqueous clusters Χ ± (H 2 O) i (n = 1-5, Χ = F; Cl; Br) have been studied using the quantum theory at DFT-PW91PW91/6-311++G(3df,3pd) level. The main goals of the present Communication are to quantify the effect of the ion sign on the thermochemical properties of aqueous clusters of identical chemical composition and to estimate the effect of vibrational anharmonicity on the computed free energies.

Methods
Initial generated structures were treated initially by semi-empirical PM3 method and then by PW91PW91/6-31+G*. Finally, the most stable (within ~4 kcal/mole from the lowest energy isomer) structures obtained at PW91PW91/6-31+G* level have been optimized at PW91PW91/6-311++G(3df.3pd) level. PW91PW91/6-311++G(3df,3pd) has been used to obtain both equilibrium geometries and thermochemical properties from computed vibrational spectrums. The PW91PW91 density functional has been used in the combination with the largest Pople basis set 6-311++G(3df,3pd) that provides quite small basis set superposition error (BSSE). In order to ensure the quality of the obtained DFT results, additional MP2/6-311++G(3df,3pd) calculations, both harmonic and anharmonic, have been carried out.

Difference in Formation Free Energies Between Cations and Anions
The interest to stepwise Gibbs free energy changes as standalone quantities is related directly to very high sensitivity of nucleation rates to the thermochemistry of initial cluster growth steps. Figure 1 presents the geometries of most stable isomers of Cl + (H 2 O) i and Cl -(H 2 O) i . In addition to most stable isomers, a number of less stable conformers/local minima located within ~ 5 kcal mol -1 of the most stable isomer/global minimum (1-5 for each n) have been identified for all the species The difference in the cluster structure has direct impact on the stepwise Gibbs free energy changes associated with the addition of water molecules to the ionic clusters. Figure 2 presents the stepwise Gibbs free energies associated with As seen from Figure 2, cations have great growth advantage over anions. The difference in the Gibbs free energy is largely associated with the formation enthalpy and it decreases dramatically with the number of molecules in the clusters. The sign preference at the first growth steps is very strong, and the difference in the stepwise Gibbs free energy changes ΔG 0,   reactions (a, b, c), and the difference in the Gibbs free energy δ(j) between j-mers formed over core ions of opposite sign (d). Curves and symbols of refer to theoretical results and experimental data, respectively. Experimental data and theoretical data for F -(H 2 O) n , Cl -(H 2 O) and Br -(H 2 O) n , were adopted from [11] , [5] and [10], respectively. The calculations were performed at the ambient temperature of 298.15 K and ambient pressure of 101.3 KPa. It is important to note that the sign preference of Cl ± (H 2 O) n obtained in molecular-based studies using empirical interaction potential (TI4P) has a different sign. Moreover, the difference between quantum results and Monte-Carlo model predictions using empirical TIP4P water model is excessively large. While the Monte Carlo model [12][13] predicts the thermochemistry of Cl -(H 2 O) n in a good

(d) (c)
agreement with the present results and experimental data, its deviation in Gibbs free energy for Cl + (H 2 O) n [13] from the present study exceeds several tens of kcal mole -1 .

Vibrational Anharmonicity
Ongoing discussion [7][8] about the validity of the commonly used harmonic approximation motivated us to perform additional analysis of harmonic and anharmonic cluster spectra and validate them against available experimental data. Tables 1, 2 and 3 present the comparison of the theoretical PW91PW91/6-311++G(3df,3pd) and MP2/6-311++G(3df,3pd) results obtained using the harmonic approximation with anharmonic results and experimental data.     Tables 1, 2 and 3 show clearly that the performance of the harmonic approximation implemented in the framework of the PW91PW91/6-311++G(3df,3pd) method is well beyond satisfactory in all the cases studied here. It is important to note that the application of the anharmonic correction [19] does not lead to any substantial improvement in the theoretical predictions.
Another important indication of the reasonable performance of the harmonic approximation implemented in the frame of DFT PW91PW91/6-311++G(3df,3pd) method has been obtained from the comparison of Zero Point Energies (ZPE) computed using the harmonic and anharmonic approximation. As seen from Figure 3 and Table 4, the difference between the harmonic and anharmonic ZPE typically does not exceed 2-3 %. This finding is in excellent agreement with the recent MP2 study [14] of pure water clusters. Another important detail is that DFT results are in good agreement with ab initio MP2 predictions. The contribution of the vibrational anharmonicity to the computed Gibbs free energies does not exceed 0.03-0.2 kcal mol -1 . This leads us to conclude that anharmonicity is unlikely a source of large uncertainties in the computed Gibbs free energies in all the cases studied here.

Conclusions
The present study leads us to the following conclusions: It is important to note that the positive sign preference found for unary water vapours does not contradict with the opposite (negative) sign preference observed in recent experiments in binary sulfuric acid-water vapours [20][21][22]. In contrast to unary clusters, whose stability is controlled by the affinity of water monomers to ions, the stability of more complex binary clusters studied by Froyd and Lovejoy [20,21] and Sorokin et al. [22] is controlled by two somewhat competing factors : affinities of H 2 O and H 2 SO 4 to the ions. While the affinity of water to cations is stronger than that to anions, the affinity of H 2 SO 4 , the key binary nucleation precursor, to ions exhibits the opposite behavior. This leads, due to the very large difference in the affinity of H 2 SO 4 between anions and cations, to the negative sign preference observed in the experiments [23].
(b) The harmonic approximation implemented in the framework of the DFT works well in the case of aqueous ionic clusters. Both DFT and ab initio MP2 studies show that the effect of vibrational anharmonicity is mild, and is unlikely a source of large uncertainties in computed free energies.