# The Ionic Product of Water in the Eye of the Quantum Cluster Equilibrium

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. QCE Theory

#### 2.2. Modified Partition Functions

^{−1}. The former is a sensitive choice in the recommended range of reasonable values between 20 and 100 cm

^{−1}and the latter represents an upper limit. Choosing both allows us to investigate the influence of this parameter in a sensible way [38].

#### 2.3. Systems Investigated

_{3}O

^{+}donating three hydrogen bonds), except for the cluster W10ip2, which contains the Zundel ion motif (one H

^{+}shared between two water molecules) [16]. The ion pair clusters range in size from a pentamer, W5ip, to decamer clusters, W10ip and W10ip2. Please note that the hydroxide anion always accepts three hydrogen bonds.

#### 2.4. Quantum-Chemical Calculations

## 3. Results and Discussion

#### 3.1. Variation in the QCE Parameters

^{−1}.

^{−1}and $1.5$ $\mathrm{K}$, respectively. This very small dependence of ${b}_{\mathrm{xv}}$ on the method is expected, as this parameter corrects the translational volume for the eigenvolume of the clusters and thus depends neither on the quantum-chemical description of the clusters nor on the precise details of the vibrational partition function.

^{3}mol

^{−2}. Again, if all other parameters were kept the same, such a shift in ${a}_{\mathrm{mf}}$ would lead to an increase in the computed temperature of phase transition of about $24.5$ $\mathrm{K}$, whereas the computed density remains almost the same, increasing only by 0.01 g cm

^{−1}. Its value depends significantly on the functional, increasing in the order PBEh-3c < PBE0/D3/gCP < B3LYP/D3/gCP. Since ${a}_{\mathrm{mf}}$ accounts for interactions between clusters, the parameter has in the past been observed to compensate for underbinding of the electronic structure method, in which case ${a}_{\mathrm{mf}}$ values become larger. Increasing the cutoffs for the mRRHO model also leads to larger values of ${a}_{\mathrm{mf}}$, indicating an indirect impact of the treatment of these internal rotations on the mean-field energy. The vibrational motions at low frequencies are usually internal rotations or translations, and hence can be related to the process of cluster association. It is therefore reasonable that the mean-field parameter ${a}_{\mathrm{mf}}$ is sensitive to the particular form of the vibrational partition function for these modes, while ${b}_{\mathrm{xv}}$ is not.

#### 3.2. Ionic Product Dependence on mRRHO

^{−1}leads to similar, but less pronounced changes, except for PBE0, where the slope of $\mathrm{p}{K}_{\mathrm{w}}$ is sightly increased again.

^{−1}while the others exhibit none or only two (see Table 2). Keeping in mind that the y-axis has a logarithmic scale, it is clear that W10ip2 is the dominating ion pair cluster, causing the overall increase in $\mathrm{p}{K}_{\mathrm{w}}$.

^{−1}, the populations of these clusters are not influenced by the modified vibrational partition function. The populations of clusters that are dominated by ring structures, be it cyclic clusters or condensed rings in spiroclusters, are reduced if the mRRHO50 model is employed.

^{−1}and their individual population is reduced. However, W5c and W8b also show these modes and obtain slightly higher or unaltered populations. For the sandwich-type decamer, the population is significantly increased if the hindered rotor model is used to treat internal rotations. Please note that this cluster shows no modes below 50 cm

^{−1}, see Table 2. While it is still not the dominant cluster in the liquid phase, its population is enhanced by a factor of around four so that within the mRRHO50 model up to 20% of all water molecules are arranged in this shape. Interestingly, the population of the cubic W8c cluster is decreased at low temperatures and increased at higher temperatures.

#### 3.3. Clausius–Clapeyron Analysis

^{−1}is observed for PBE0/D3/gCP. All mRRHO calculations lead to an overestimation of this quantity, whereas the B3LYP data are least sensitive to the mRRHO approach and yield the best overall data, which is in agreement with the observations made for the ionic product. With the exception of PBEh-3c, all methods reproduce the experimental enthalpy of vaporization to within 4 kJ mol

^{−1}, a threshold often referred to as a targeted chemical accuracy.

#### 3.4. Entropy

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Ball-and-stick representation of the neutral water clusters. Oxygen: blue; hydrogen: white.

**Figure 2.**Ball-and-stick representation of the ion pair clusters. Oxygen: blue; hydrogen: white; red shows the hydroxide anions and green shows the hydronium cations.

**Figure 3.**Temperature dependence of the negative logarithm of the ionic product, $\mathrm{p}{K}_{\mathrm{w}}$, for all selected methods as well as experimental [54] values. The standard method is represented by solid, mRRHO50 by dashed, and mRRHO100 by dotted lines.

**Figure 4.**Temperature dependence of the monomer-normalized populations for the ion pair clusters, showing B3LYP/D3/gCP data as a representative example. Solid lines: conventional QCE, dashed lines: mRRHO50.

**Figure 5.**Temperature dependence of the monomer-normalized populations for the regular clusters, showing B3LYP/D3/gCP data as a representative example. Solid lines: conventional QCE, dashed lines: mRRHO50. Note the semilogarithmic scale.

**Figure 6.**Pressure–temperature graph for B3LYP/D3/gCP (green line) and compared to experimental [55] data (red line).

**Figure 7.**Clausius–Clapeyron plot: Logarithmic plot of pressure P versus inverse temperature T for B3LYP/D3/gCP (green line) and experiment [55] (red line).

**Figure 8.**Temperature dependence of the entropy for all methods (colored lines) as well as experimental values (red marks). Standard QCE results are presented as solid, mRRHO50 results as dashed, and mRRHO100 results as dotted lines. The grey line is a linear extrapolation of the gas-phase curve. Experimental values are from Refs. [57,58,59].

**Table 1.**Optimized QCE parameters ${a}_{\mathrm{mf}}$ in $\mathrm{J}{\mathrm{m}}^{3}{\mathrm{mol}}^{-2}$ and ${b}_{\mathrm{xv}}$ (dimensionless) for different methods.

Method | Standard QCE | mRRHO50 | mRRHO100 | |||
---|---|---|---|---|---|---|

${\mathit{a}}_{\mathrm{mf}}$ | ${\mathit{b}}_{\mathrm{xv}}$ | ${\mathit{a}}_{\mathrm{mf}}$ | ${\mathit{b}}_{\mathrm{xv}}$ | ${\mathit{a}}_{\mathrm{mf}}$ | ${\mathit{b}}_{\mathrm{xv}}$ | |

B3LYP/D3/gCP | 0.201 | 1.50 | 0.213 | 1.50 | 0.216 | 1.50 |

PBE0/D3/gCP | 0.192 | 1.52 | 0.205 | 1.50 | 0.207 | 1.50 |

PBEh-3c | 0.173 | 1.50 | 0.185 | 1.52 | 0.187 | 1.52 |

Cluster | # Modes $\tilde{\mathit{\nu}}<50\phantom{\rule{3.33333pt}{0ex}}{\mathbf{cm}}^{-1}$ | # Modes $50\phantom{\rule{3.33333pt}{0ex}}{\mathbf{cm}}^{-1}<\tilde{\mathit{\nu}}<100\phantom{\rule{3.33333pt}{0ex}}{\mathbf{cm}}^{-1}$ |
---|---|---|

W5p | 1 | 2 |

W5c | 2 | 2 |

W6c | 2 | 4 |

W7 | 3 | 4 |

W8p | 4 | 5 |

W8b | 4 | 4 |

W8c | − | 5 |

W9 | 6 | 5 |

W10 | − | 7 |

W5ip | − | 1 |

W8ip | 3 | 5 |

W8cip | − | 3 |

W10ip1 | 2 | 6 |

W10ip2 | − | 6 |

**Table 3.**First line: enthalpy of vaporization $\Delta {H}_{\mathrm{vap}}$ in kJ mol

^{−1}calculated through a Clausius–Clapeyron analysis. The experimental value is based on the data shown in the SI from Ref. [56]. Next lines: entropy of vaporization in J mol

^{−1}K

^{−1}at 298 $\mathrm{K}$ and at the temperatures of phase transition (373 $\mathrm{K}$). See Section 3.4 for details. Experimental values are from Refs. [57,58,59].

B3LYP/D3/gCP | PBE0/D3/gCP | PBEh-3c | Exp | |||||||
---|---|---|---|---|---|---|---|---|---|---|

mRRHO | − | 50 | 100 | − | 50 | 100 | − | 50 | 100 | |

$\Delta {H}_{\mathrm{vap}}$ | 43.74 | 45.89 | 45.93 | 43.57 | 45.61 | 46.26 | 46.31 | 48.20 | 49.03 | 41.58 |

$\Delta {S}_{\mathrm{vap}}$(298 K) | 128.69 | 131.21 | 130.68 | 123.45 | 129.70 | 129.92 | 131.47 | 132.82 | 133.90 | 118.76 |

$\Delta {S}_{\mathrm{vap}}$(trs) | 115.78 | 121.46 | 121.73 | 115.64 | 121.21 | 122.37 | 122.97 | 128.62 | 130.51 | 109.54 |

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Kirchner, B.; Ingenmey, J.; von Domaros, M.; Perlt, E.
The Ionic Product of Water in the Eye of the Quantum Cluster Equilibrium. *Molecules* **2022**, *27*, 1286.
https://doi.org/10.3390/molecules27041286

**AMA Style**

Kirchner B, Ingenmey J, von Domaros M, Perlt E.
The Ionic Product of Water in the Eye of the Quantum Cluster Equilibrium. *Molecules*. 2022; 27(4):1286.
https://doi.org/10.3390/molecules27041286

**Chicago/Turabian Style**

Kirchner, Barbara, Johannes Ingenmey, Michael von Domaros, and Eva Perlt.
2022. "The Ionic Product of Water in the Eye of the Quantum Cluster Equilibrium" *Molecules* 27, no. 4: 1286.
https://doi.org/10.3390/molecules27041286