#
Calculation of V_{S,max} and Its Use as a Descriptor for the Theoretical Calculation of pKa Values for Carboxylic Acids

^{*}

## Abstract

**:**

_{S,max}) on the acidic hydrogen atoms of carboxylic acids to describe the H-bond interaction with water (the same descriptor that is used to characterize σ-bonded complexes) and correlate the results with experimental pKa values to obtain a predictive model for other carboxylic acids. We benchmarked six different methods, all including an implicit solvation model (water): Five density functionals and the Møller–Plesset second order perturbation theory in combination with six different basis sets for a total of thirty-six levels of theory. The ωB97X-D/cc-pVDZ level of theory stood out as the best one for consistently reproducing the reported pKa values, with a predictive power of 98% correlation in a test set of ten other carboxylic acids.

## 1. Introduction

_{a}, and its associated logarithmic quantity, pKa = −logKa, are an intrinsic characteristic of each acid. This process occurs through the abstraction of the acidic hydrogen atom using a water molecule (Equation 1). It is commonly regarded that the deprotonation reaction is mainly promoted by electrostatic interactions, given the partial positive charge on the acid hydrogen. Characterization of a protic acid through the pKa values is of practical importance and usefulness in various steps of the chemical design rationale, and therefore, it is an important quantity.

_{2}O → H

_{3}O

^{+}+ A

^{−}

_{S,max}. By assuming that the deprotonation of a carboxylic acid begins with the formation of a hydrogen bond with a water molecule, RCOOH···H

_{2}O, we propose the use of V

_{S,max}as a suitable descriptor for the strength of this interaction, which in turn correlates with the corresponding pKa values, in a similar fashion to how a σ-hole-based interaction is quantified in halogen or tetrel bonds. Previously, the nucleophilicities and electrophilicities of Lewis acids and bases, respectively, have been derived from the interpolation of the mutual dissociation energies [11].

^{2}= 0.91 [13] with the experimental pKa values. Monard and Jensen used various kinds of atomic charges of the conjugated phenolates, alkoxides, or thiolates, with the best correlations being observed for the atomic electrostatic charges from a Natural Population Analysis (NPA) calculated at the B3LYP/3-21G (R

^{2}= 0.995) and M06-2X/6-311G (R

^{2}= 0.986) levels of theory for alcohols and thiols, in implicit solvent, respectively. Other efforts include correlations on the excited states of photoacids [14] using Time Dependent DFT at the ωB97X-D/6-31G(d) level of theory for a family of hydroxyl-substituted aromatic compounds. QSPR models have yielded, for instance, a three parameters model which uses the MEP maxima, the number of carboxylic acid and amine groups for phenols, at the HF/6-31G(d,p) and B3LYP/6-31G(d,p) levels of theory (R

^{2}= 0.96) [15]. It involves a four parameters linear equation comprising the highest normal mode vibrational frequency, the partial positive and negative charges divided by the total surface area and a reactivity index, defined in terms of a population analysis on the frontier orbital HOMO (R

^{2}ca. 0.95 sic.) [16] for N-Base ligands at the semi empirical AM1 level of theory, as well as a Principal Components Analysis (PCA) for organic and inorganic acids (RMSE = 0.0195) [17]. Moreover, genetic algorithms (GA) and neural networks (NN) have employed frontier orbital energies for a chemical space of sixty commercial drugs [18] (GA, R

^{2}= 0.703; NN, R

^{2}= 0.929). Thus far, the only major commercial program capable of including the effects of molecular conformations on the estimation of pKa values is ‘Jaguar pKa′ [19,20]. For more thorough reviews on the development of pKa descriptors, please refer to References [21,22,23].

_{S,max}calculations with various DFT methods, and used MP2 as a reference (see methodology section), to the pKa values of carboxylic acids. Physically, the obtained value of V

_{S,max}on the acidic hydrogen atom reflects the attractive interaction between it and a water molecule, and thus in turn can be used to describe the deprotonation process in electrostatic terms.

## 2. Results

_{S,max}, was calculated and plotted against the experimental pKa

_{exp}value. Our model was based on simple linear regressions to obtain the best fittings. The V

_{S,max}on each acidic hydrogen atom was used for the correlations, as an example, Figure 2 depicts the location of V

_{S,max}on the acid hydrogen atom for compound 14. This value was calculated on the isodensity surface ρ = 0.001 a.u., and it was used as a descriptor for the magnitude of the attractive interaction RCOOH···H

_{2}O.

_{exp}− pKa

_{cal}. Figure 4 shows these plots for the results obtained with the functional (A), where the corresponding ΔpKa plots for the other levels of theory are collected in the Supporting Information section (Figures S4, S6, S8, S10 and S12).

^{2}= 0.9680) and the lowest ΔpKa values. Table S8 shows the pKa intervals for all levels of theory and it can be observed that all (C) models have a ΔpKa interval above 1.0 units, whereas all (A) models have ΔpKa intervals below 1.0 unit, which means an accuracy of ±0.5 pKa units. Considering these results and the calculation parameters supplied (isodensity and grid values), we proposed the following equation:

_{S,max}+ 16.1879

^{2}= 0.9801.

## 3. Discussion

#### 3.1. Computational Method: DFT or Ab Initio?

^{2}correlation coefficients (Table 1) between V

_{S,max}and pKa

_{exp}values were obtained consistently with the ab initio MP2 method. Nevertheless, the DFT functional ωB97X-D functional (A), yielded comparably similar results at a fraction of the computational cost. The lowest correlation coefficients were obtained with the B3LYP functional (B), which, despite being one of the most popular ones to model organic molecules, could be describing the surface electrostatic potential inadequately. A similar performance to that of B3LYP was observed for the PBE0 functional (E), which in turn, was slightly improved when long range corrections were included in the case of LC-ωPBE (C). The latter functional was thought to yield much better results due to this long-range correlation term; however, that was not the case.

#### 3.2. Basis Set: Is Larger Better?

_{S,max}-pKa correlations when using the relatively medium size cc-pVDZ basis set. Surprisingly, the M06-2X functional presented the largest ΔpKa deviations when combined with the largest basis set aug-cc-pVTZ (Figures S7 and S8).

#### 3.3. A Final Remark

_{S,max}of three compounds (6, 7, and 12) required an average of conformers, where the angle (D1 = O=C-O-H) was either 0.0° or 180.0°. The conformation D1 = 180.0° was stable due to strong delocalization effects from nearby п bonds to the σ*

_{O-H}orbital in the acidic hydrogen atom or intramolecular hydrogen bonding with Lewis basic motifs (Figure 7). For such kind of compounds, further improvements are required in the methodology for our linear models.

## 4. Materials and Methods

_{S,max}) calculations were performed on the wave function files with the ‘MultiWFN’ program, version 3.3.8 as in Reference [74], using an isodensity value of 0.001 a.u. All the computed values were collected in the Supporting Information (Tables S1–S6).

## 5. Conclusions

_{S,max}is a scalar quantity that characterizes a σ-hole, and according to our calculations, it has also proven to be a suitable descriptor to be correlated with the pKa value of carboxylic acids, yielding differences in pKa of high accuracy. ΔpKa = ±0.30 when the ωB97X-D/cc-pVDZ level of theory was used to calculate the associated electron density upon which the V

_{S,max}value was obtained. By means of straightforward DFT calculations with a simple implicit solvation model (CPCM), the value of the V

_{S,max}could be calculated and Equation 2 obtained herein, could be used to estimate the pKa values without the need for a full thermodynamic cycle calculation; thus, avoiding long computations of solvation free energies and other costly quantities which require high accuracy methods.

_{S,max}. Hence, we highly recommend this level of theory for geometry optimization and wave function file print. Care must be taken as the pKa value sought after should be between 0.5 and 5.0 pH units, for this is the applicability domain of our resulting equations, given the chemical space covered herein.

_{S,max}has turned out to be a powerful descriptor for predicting the pKa values of carboxylic acids as it is reflected by low, yet distinguishable differences across all methods studied herein. The presence of intramolecular non-covalent interactions, for example, hydrogen bonding, as well as highly electron-delocalizing groups within the chemical space, are key features to consider in the inclusion of an average of the V

_{S,max}for the most stable conformers. Our proposed descriptor is also dependent of the isodensity value for the definition of the surface upon which it is calculated, and it is highly recommended to keep the value suggested by Bader et al. [75] of ρ = 0.001 a.u. However, by taking these considerations into account as part of the parametrical requirements of Equation (2), then extremely accurate pKa results are obtained in a straightforward fashion.

## Supplementary Materials

_{S,max}values for carboxylic H atoms to the different levels of theory studied, Table S7: Reported pKa values for carboxylic acids studied. Figures S1, S3, S5, S7, S9 and S11: Correlation of pKa

_{exp}vs V

_{S,max}. Figures S2, S4, S6, S8, S10 and S12: Difference between the experimental and calculated pKa values (∆pKa

_{exp−cal}).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Sample Availability: Not available. |

**Figure 2.**Maximum surface electrostatic potential, V

_{S,max}, over the acidic hydrogen atom shown for compound 14 taking an isodensity value of 0.001 a.u. (isosurface not shown). Red dots represent positive V

_{S,max}values and the blue dots represent negative V

_{S,max}values.

**Figure 3.**Linear correlations between pKa

_{exp}against V

_{S,max}for DFT method (A), with the six basis sets (1) through (6).

**Table 1.**Linear regression parameters obtained for the pKa vs V

_{S,max}plots. Intercept units in kcal/mol.

Level of Theory | Slope | Intercept | R^{2} | Level of Theory | Slope | Intercept | R^{2} |
---|---|---|---|---|---|---|---|

A1 | −0.1954 | 16.1237 | 0.9626 | D1 | −0.1987 | 16.3352 | 0.9598 |

A2 | −0.1975 | 15.8213 | 0.9645 | D2 | −0.1947 | 15.5073 | 0.9598 |

A3 | −0.2185 | 16.1879 | 0.9680 | D3 | −0.2201 | 16.2212 | 0.9653 |

A4 | −0.2113 | 16.3958 | 0.9627 | D4 | −0.2082 | 16.2411 | 0.9577 |

A5 | −0.2063 | 16.3542 | 0.9594 | D5 | −0.2027 | 16.0780 | 0.9535 |

A6 | −0.2131 | 16.3967 | 0.9589 | D6 | −0.2095 | 15.9993 | 0.9534 |

B1 | −0.1902 | 15.1863 | 0.9494 | E1 | −0.1953 | 15.8840 | 0.9553 |

B2 | −0.1909 | 14.8019 | 0.9515 | E2 | −0.1953 | 15.3858 | 0.9511 |

B3 | −0.2191 | 15.5109 | 0.9570 | E3 | −0.2167 | 15.8799 | 0.9616 |

B4 | −0.2072 | 15.4614 | 0.9521 | E4 | −0.2118 | 16.1281 | 0.9536 |

B5 | −0.1983 | 15.1505 | 0.9457 | E5 | −0.2038 | 15.8232 | 0.9490 |

B6 | −0.2030 | 15.1449 | 0.9277 | E6 | −0.2125 | 15.9739 | 0.9485 |

C1 | −0.2001 | 16.4372 | 0.9654 | F1 | −0.1996 | 15.8814 | 0.9613 |

C2 | −0.1996 | 15.9700 | 0.9647 | F2 | −0.2123 | 15.9191 | 0.9625 |

C3 | −0.2272 | 16.6499 | 0.9682 | F3 | −0.2285 | 16.3778 | 0.9702 |

C4 | −0.2166 | 16.7945 | 0.9633 | F4 | −0.2198 | 16.3264 | 0.9661 |

C5 | −0.2085 | 16..4776 | 0.9597 | F5 | −0.2094 | 16.0399 | 0.9550 |

C6 | −0.2162 | 16.4972 | 0.9579 | F6 | −0.2187 | 16.1635 | 0.9616 |

**Table 2.**V

_{S,max}calculated with the A3 model. Experimental and calculated pKa values for compounds

**a**–

**j**and the differences.

V_{S,max} | pKa_{exp} * | pKa_{cal} ** | ΔpKa_{exp − cal} | |
---|---|---|---|---|

a | 70.3733 | 0.7200 | 0.8097 | −0.0897 |

b | 66.8745 | 1.3900 | 1.5743 | -0.1843 |

c | 66.2754 | 1.4700 | 1.7052 | −0.2352 |

d | 65.6112 | 1.9000 | 1.8503 | 0.0497 |

e | 62.0236 | 2.3600 | 2.6343 | −0.2743 |

f | 61.9719 | 2.9500 | 2.6456 | 0.3044 |

g | 59.0914 | 3.1600 | 3.2750 | −0.1150 |

h | 54.9561 | 3.9100 | 4.1787 | −0.2687 |

i | 54.6370 | 4.2200 | 4.2484 | 0.0284 |

j | 52.8925 | 4.3600 | 3.6296 | −0.2696 |

© 2018 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**

Caballero-García, G.; Mondragón-Solórzano, G.; Torres-Cadena, R.; Díaz-García, M.; Sandoval-Lira, J.; Barroso-Flores, J.
Calculation of *V*_{S,max} and Its Use as a Descriptor for the Theoretical Calculation of p*K*a Values for Carboxylic Acids. *Molecules* **2019**, *24*, 79.
https://doi.org/10.3390/molecules24010079

**AMA Style**

Caballero-García G, Mondragón-Solórzano G, Torres-Cadena R, Díaz-García M, Sandoval-Lira J, Barroso-Flores J.
Calculation of *V*_{S,max} and Its Use as a Descriptor for the Theoretical Calculation of p*K*a Values for Carboxylic Acids. *Molecules*. 2019; 24(1):79.
https://doi.org/10.3390/molecules24010079

**Chicago/Turabian Style**

Caballero-García, Guillermo, Gustavo Mondragón-Solórzano, Raúl Torres-Cadena, Marco Díaz-García, Jacinto Sandoval-Lira, and Joaquín Barroso-Flores.
2019. "Calculation of *V*_{S,max} and Its Use as a Descriptor for the Theoretical Calculation of p*K*a Values for Carboxylic Acids" *Molecules* 24, no. 1: 79.
https://doi.org/10.3390/molecules24010079