#
Determination of pK_{a} Values via ab initio Molecular Dynamics and its Application to Transition Metal-Based Water Oxidation Catalysts

^{*}

## Abstract

**:**

## 1. Introduction

_{5}OMe(OH

_{2})]

^{2+}(where Py

_{5}OMe = 6,6${}^{\u2033}$-(methoxy(pyridin-2-yl)methylene)di-2,2${}^{\prime}$-bipyridine) [7]. The said system, and in particular the thermodynamics and kinetics of the water oxidation process, were already investigated in-depth by Luber and co-workers employing DFT simulations [7,8]. Those studies gave important insights with respect to the water oxidation mechanism and further helped to come up with some design guidelines on how to further improve those catalysts. However, the limitations imposed by the implicit solvation led to the desire for a more sophisticated description of the solvation shell. In this context, also this study serves as a mini-review and benchmark study to validate the methodology for $\mathrm{p}{K}_{\mathrm{a}}$ determination to be applied to the same or similar systems in the future.

## 2. Methodology

#### 2.1. Choice of Constraint

#### 2.2. Estimation of $p{K}_{a}$ Values from the Free Energy Differences

#### 2.2.1. Absolute $\mathrm{p}{K}_{\mathrm{a}}$

^{−}+ H

^{+}, i.e., the distance at which the covalent bond is broken [31]. The limitations with respect to the simulation cell result in an uncertainty in the $\mathrm{p}{K}_{\mathrm{a}}$ value which Davies et al. quantified as $\Delta F\left({R}_{max}\right)/\left(2.3{k}_{\mathrm{B}}T\right)$ [31].

#### 2.2.2. Relative $\mathrm{p}{K}_{\mathrm{a}}$

#### 2.2.3. Probabilistic $\mathrm{p}{K}_{\mathrm{a}}$

## 3. Computational Settings

#### 3.1. Model Systems

_{5}Me(H

_{2}O)]

^{2+}and [Ru(II)Py

_{5}OMe(H

_{2}O)]

^{2+}listed in Table 1, from a kinetic and thermodynamic point of view employing state of the art DFT simulations [7,8]. We found that both ligand frameworks Py5OMe and Py5Me were virtually identical in terms of their thermodynamics and kinetics of the water-oxidation reaction, which is a not too surprising a result as the replacement of a methyl-group by a methoxy-group at an sp${}^{3}$ carbon is not expected to remarkably alter electronics or sterics at the metal center. However, experimentally, the catalytic activity between the two ligands was found to be rather different, which was attributed to a rapid halide substitution at the catalyst bearing a Py5Me ligand. The latter leads to a deactivation of the catalyst [7]. As mechanistic studies for the more active catalyst Py5OMe are still underway, we decided to choose it as a model system even though no experimental $\mathrm{p}{K}_{\mathrm{a}}$ values are currently available. Based on our previous study we would not expect the thermodynamics of the two ligands and related $\mathrm{p}{K}_{\mathrm{a}}$ values to differ significantly [7,53]. Therefore comparing the calculated $\mathrm{p}{K}_{\mathrm{a}}$ value of the Py5OMe and Py5Me systems serves as a further validation of the method.

#### 3.2. Error Analysis

## 4. Results and Discussion

#### 4.1. Convergence of the AIMD Simulations

#### 4.2. Reference System

^{−}tend to be reprotonated by the solvent. This is also the case in our simulations. Starting from a d(O−H) of 1.5 Å, we were able to observe the reprotonation of the OH

^{−}moiety (see Figure 5). As discussed earlier, choosing a different collective variable could circumvent this issue.

#### 4.3. Overview of Calculated $p{K}_{a}$ Values

_{5}OMe(H

_{2}O)]

^{3+}is somewhat of an outliner with the largest difference compared to the experiment by about two $\mathrm{p}{K}_{\mathrm{a}}$ units. As the only difference between [Ru(II)Py

_{5}OMe(H

_{2}O)]

^{2+}and [Ru(III)Py

_{5}OMe(H

_{2}O)]

^{3+}is the charge of the system, we reasoned that the simulation cell might be too small for such highly charged species. The latter was verified by a set of simulations in a bigger simulation cell (see Table 3).

#### 4.4. Deuterated Solvent

_{2}O instead of $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ values, i.e., $\mathrm{p}{K}_{\mathrm{a}}$ values in H

_{2}O. Based on experiments, a correlation between $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ and $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ was reported already decades ago [55]. At first, the correlation was suspected to be linear only for $\mathrm{p}{K}_{\mathrm{a}}$ values > 7 and constant for more acidic species [55]. Later, Delgado et al. experimentally determined a linear relation between $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ and $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ over the whole range of $\mathrm{p}{K}_{\mathrm{a}}$ values [56]:

_{2}O solution by a pH-meter which was calibrated by H

_{2}O. Conversion to $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ is achieved by adding the empirically determined constant of 0.4 to $\mathrm{p}{K}_{\mathrm{a}}^{{\mathrm{H}}^{\ast}}$ [57]:

_{2}O respectively its $\mathrm{p}{K}_{\mathrm{w}}$ value.

_{2}O is 14.951 (25 ${}^{\circ}$C) [58]. The obtained values for ${R}_{c}$ are 1.26 Å, respectively 1.25 Å for the two simulation cells (see Supplementary Materials Figures S18 and S19). The $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ values obtained with the cut-off ${R}_{c}$ determined for D

_{2}O were converted to $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ values (see Table 4 and Table 5).

_{2}O instead of D

_{2}O. The difference when converting the $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ values either according to Equation (10) or Equation (13) is negligible.

_{2}O is acceptable. This is in particular true for the probabilistic method. The validity of this conclusion could in principle be checked by repeating all the simulations with H

_{2}O instead of D

_{2}O. With a large enough test set, it would also be possible to adjust equation of the linear relation to the employed methodology. However this is beyond the scope of the current work.

#### 4.5. Absolute and Probabilistic $p{K}_{a}$—Dependence on ${R}_{c}$

#### 4.6. Relative $p{K}_{a}$

_{5}OMe(H

_{2}O)]

^{3+}. Similar inconsistencies when applying the relative $\mathrm{p}{K}_{\mathrm{a}}$ protocol to lumiflavins have also been reported by Kiliç et al. [2].

## 5. Summary and Conclusions

_{5}Me) and Ru(Py

_{5}OMe) catalysts, for which we found, as expected, both qualitative and quantitative similar $\mathrm{p}{K}_{\mathrm{a}}$ values independent of the applied post-processing method.

_{2}O), we find that method 1 and 3 are able to quantitatively reproduce experimental values with an accuracy of about 1 $\mathrm{p}{K}_{\mathrm{a}}$ unit. In case of method 1 this is rather surprising as there is no guarantee that the free energy levels off within the scanned range of the constraint. Method 2 appears to at least qualitatively reproduce experimental $\mathrm{p}{K}_{\mathrm{a}}$ values, however there seems to be no necessity to use said approach, as it requires exactly the same set of calculations as method 3 which preformed the best within our test set. This conclusion also holds if one assumes that $\mathrm{p}{K}_{\mathrm{a}}^{D}$ instead of $\mathrm{p}{K}_{\mathrm{a}}^{H}$ values were calculated. For the sake of consistency we have only applied a very simple constraint i.e., the A−H distance. While this choice was fine for our test-set, there might be cases where more complex constraints are necessary, in particular if the conjugated base is very strong.

## Supplementary Materials

_{5}R(H

_{2}O)]

^{2+}), Figures S2–S9 (Potentials of mean force as a function of the constraint - for all systems), Figures S10–S19 (Absolute and probabilistic $\mathrm{p}{K}_{\mathrm{a}}$ values as a function of the constraint - for all systems), Tables S1–S5 (Convergence of $\mathrm{p}{K}_{\mathrm{a}}$ with respect to simulation time), Tables S7–S10 ($\mathrm{p}{K}_{\mathrm{a}}^{D}$ values converted to $\mathrm{p}{K}_{\mathrm{a}}^{H}$ values for the absolute and relative protocol), Tables S11 and S12 (Summary of simulation times per system/constraint), the following CP2K files: .inp, .ener, .xyz (only start structure), .LagrangeMultLog (forces) are available for each calculation.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

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**Figure 1.**(

**top**): Average force ($\langle \lambda \rangle $) acting on the constraint for the PhOH model system, constraining the A−H bond. (

**bottom**): Potential of mean force, i.e., free energy obtained by integrating the average forces.

**Figure 2.**Force ($\lambda $) acting on the constraint—here for PhOH at a constrained d(A−H). The vertical line at 5.0 ps marks the equilibration time in relation to the production run.

**Figure 3.**Autocorrelation function of the force acting on the constraint. From top to bottom the constraint increases by 0.1 Å, starting from 0.9 Å.

**Figure 4.**Determination of ${R}_{c}$ from simulation of water, by fitting to the experimental value. The simulations were carried out in a cubic box with a side length of 15.6406 Å.

**Figure 5.**Autodissociation of H

_{2}O (

**top**): Average force acting on the constraint. (

**bottom**): Free energy obtained by integrating the average forces. Note the drop in free energy at 1.5 Å is caused by the reprotonation of OH

^{−}.

**Figure 6.**$\mathrm{p}{K}_{\mathrm{a}}$ values calculated for PhOH using the absolute and probabilistic method. The latter is strongly dependent on the choice of ${R}_{c}$, even a change by 0.01 Å might result in change of up to 0.5 $\mathrm{p}{K}_{\mathrm{a}}$ unit.

**Figure 7.**Relative $\mathrm{p}{K}_{\mathrm{a}}$ as a function of d(A−H) according to Equation (6), the value asymptotically approaches a constant value.

**Table 1.**Model systems used in this study. ${N}_{w}$ stands for the number of water molecules in the simulation cell. The calculated $\mathrm{p}{K}_{\mathrm{a}}$ value is only given if the simulations found in the literature were obtained employing the Bluemoon methodology. The Ru complex bears a pentapyridine (Py5) ligand that is composed of two bipyridyl fragments linked to a fifth pyridyl via an sp${}^{3}$ carbon. The fourth fragment connected to the latter is either a methyl (Me) or methoxy (OMe) group resulting in Py5Me or Py5OMe (see [7,8,51] for more details). Experimental $\mathrm{p}{K}_{\mathrm{a}}$ values denoted with “*” were only available for the Py5Me ligand framework (see Supplementary Materials Figure S1 for a graphical representation of the catalysts).

Molecule | ${\mathit{N}}_{\mathit{w}}$ | Side-Length [Å] | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}$ (exp.) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}$ (calc.) |
---|---|---|---|---|

H_{2}O | 128 | 15.6 | 14.0 | – |

H_{2}O | 256 | 19.7 | 14.0 | – |

HCOOH | 126 | 15.6 | 3.8 [52] | – |

PhOH | 123 | 15.6 | 10.0 [52] | 9.7 [3] |

[Ru(II)Py_{5}Me(H_{2}O)]^{2+} | 112 | 15.6 | ∼11 [7] | – |

[Ru(II)Py_{5}OMe(H_{2}O)]^{2+} | 112 | 15.6 | ∼11 [7] * | – |

[Ru(II)Py_{5}OMe(H_{2}O)]^{2+} | 234 | 19.7 | ∼11 [7] * | – |

[Ru(II)Py_{5}OMe(H_{2}O)]^{3+} | 112 | 15.6 | ∼3 [7] * | – |

[Ru(II)Py_{5}OMe(H_{2}O)]^{3+} | 234 | 19.7 | ∼3 [7] * | – |

**Table 2.**All results presented here are calculated at 320 K, for a cubic box with side length of 15.6406 Å and a cutoff ${R}_{C}$ = 1.24 Å, over a trajectory of about 20 ps (where the first 5 ps were not included in the evaluation). The standard deviation is calculated using the block average method with a block size of 1 ps; (a) absolute, (b) relative, (c) probabilistic protocols.

Molecule | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}$ (exp.) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{(\mathbf{a})}$ | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{(\mathbf{b})}$ | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{(\mathbf{c})}$ |
---|---|---|---|---|

H_{2}O | 14.0 | – | – | |

HCOOH | 3.8 [52] | $2.7\pm 0.5$ | $6.7\pm 0.5$ | $4.2\pm 0.6$ |

PhOH | 10.0 [52] | $8.7\pm 0.3$ | $12.5\pm 0.5$ | $10.7\pm 0.4$ |

[Ru(II)Py_{5}Me(H_{2}O)]^{2+} | ∼11 [51] | $9.8\pm 0.4$ | $13.7\pm 0.3$ | $11.2\pm 0.4$ |

[Ru(II)Py_{5}OMe(H_{2}O)]^{2+} | ∼11 [51] | $9.3\pm 0.4$ | $13.3\pm 0.6$ | $11.1\pm 0.4$ |

[Ru(III)Py_{5}OMe(H_{2}O)]^{3+} | ∼2.5 [51] | $3.1\pm 0.4$ | $7.1\pm 0.4$ | $4.5\pm 0.5$ |

**Table 3.**All results presented here are calculated at 320 K, for a cubic box with a side length of 19.7340 Å and a cutoff ${R}_{C}$ = 1.24 Å over a trajectory of 10–15 ps (where the first 5 ps were not included in the evaluation). The standard deviation is calculated using the block average method with a block size of 1 ps; (a) absolute, (b) relative, (c) probability protocols.

Molecule | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}$ (exp.) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{(\mathbf{a})}$ | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{(\mathbf{b})}$ | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{(\mathbf{c})}$ |
---|---|---|---|---|

H_{2}O | 14.0 | – | – | |

[Ru(II)Py_{5}OMe(H_{2}O)]^{2+} | ∼11 [51] | $10.1\pm 0.5$ | $11.5\pm 0.5$ | $12.7\pm 0.7$ |

[Ru(III)Py_{5}OMe(H_{2}O)]^{3+} | ∼2.5 [51] | $3.1\pm 0.3$ | $4.6\pm 0.3$ | $4.3\pm 0.4$ |

**Table 4.**All results presented here are calculated at 320 K, for a cubic box with a side length of 15.6406 Å, over a trajectory of about 20 ps (where the first 5 ps were not included in the evaluation). The $\mathrm{p}{K}_{\mathrm{a}}$ values were calculated using the probabilistic method for a ${R}_{c}$ value of 1.26 Å determined for D

_{2}O. The $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ (a) was obtained referencing the calculations to H

_{2}O i.e., a ${R}_{c}$ of 1.24 Å. The $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ values were converted to $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ values according to Equation (10) (b), respectively Equation (13) (c).

Molecule | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}$ (exp.) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{H}}$(a) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{D}}$ | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{H}}$(b) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{H}}$(c) |
---|---|---|---|---|---|

H_{2}O | 14.0 | – | – | – | – |

HCOOH | 3.8 [52] | 4.2 | 4.3 | 3.8 | 4.0 |

PhOH | 10.0 [52] | 10.7 | 11.2 | 10.4 | 10.5 |

[Ru(II)Py_{5}Me(H_{2}O)]^{2+} | ∼11 [51] | 11.2 | 12.0 | 11.2 | 11.2 |

[Ru(II)Py_{5}OMe(H_{2}O)]^{2+} | ∼11 [51] | 11.1 | 11.8 | 11.0 | 11.0 |

[Ru(III)Py_{5}OMe(H_{2}O)]^{3+} | ∼2.5 [51] | 4.5 | 4.6 | 4.1 | 4.3 |

**Table 5.**All results presented here are calculated at 320 K, for a cubic box with a side length of 19.7340 Å, over a trajectory of about 20 ps (where the first 5 ps were not included in the evaluation). The $\mathrm{p}{K}_{\mathrm{a}}$ values were calculated using the probabilistic method for a ${R}_{c}$ value of 1.25 Å determined for D

_{2}O. The $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ (a) was obtained referencing the calculations to H

_{2}O i.e., a ${R}_{c}$ of 1.24 Å. The $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{D}}$ values were converted to $\mathrm{p}{K}_{\mathrm{a}}^{\mathrm{H}}$ values according to Equation 10 (b), respectively Equation (13) (c).

Molecule | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}$ (exp.) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{H}}$(a) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{D}}$ | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{H}}$(b) | $\mathbf{p}{\mathit{K}}_{\mathbf{a}}^{\mathbf{H}}$(c) |
---|---|---|---|---|---|

H_{2}O | 14.0 | - | - | - | - |

[Ru(II)Py_{5}OMe(H_{2}O)]^{2+} | ∼11 [51] | 12.7 | 13.1 | 12.2 | 12.2 |

[Ru(III)Py_{5}OMe(H_{2}O)]^{3+} | ∼2.5 [51] | 4.3 | 4.4 | 3.9 | 4.1 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Schilling, M.; Luber, S. Determination of p*K*_{a} Values via *ab initio* Molecular Dynamics and its Application to Transition Metal-Based Water Oxidation Catalysts. *Inorganics* **2019**, *7*, 73.
https://doi.org/10.3390/inorganics7060073

**AMA Style**

Schilling M, Luber S. Determination of p*K*_{a} Values via *ab initio* Molecular Dynamics and its Application to Transition Metal-Based Water Oxidation Catalysts. *Inorganics*. 2019; 7(6):73.
https://doi.org/10.3390/inorganics7060073

**Chicago/Turabian Style**

Schilling, Mauro, and Sandra Luber. 2019. "Determination of p*K*_{a} Values via *ab initio* Molecular Dynamics and its Application to Transition Metal-Based Water Oxidation Catalysts" *Inorganics* 7, no. 6: 73.
https://doi.org/10.3390/inorganics7060073