# “Transitivity”: A Code for Computing Kinetic and Related Parameters in Chemical Transformations and Transport Phenomena

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Phenomenology of Temperature Dependence of the Reaction Rate Constant

#### 2.2. Calculation of Reaction Rate Constant

#### 2.2.1. Deformed Transition-State Theory ($d$-TST)

#### 2.2.2. Bell35 and Bell58

#### 2.2.3. Skodje and Truhlar, ST

#### 2.3. Solvent Effect on Reaction Rate Constant

#### 2.3.1. Collins–Kimball Formulation

#### 2.3.2. Kramers’ Formulation

## 3. Handling the Transitivity Code

^{−1}and inserted in the “Solvent Type” option. The temperature dependence of viscosity of water using the experimental data [74] is expressed as $\eta \left(T\right)={\mathrm{2.7024.10}}^{-4}Poise{\left(1-213.0543/T\right)}^{-2.75634}$. When the Kramers formulation is chosen, the friction coefficient of the solvent ($\mu /{s}^{-1})$, the Kramers transmission coefficient, and the overall reaction rate constant (${k}_{Obs}/{\mathrm{cm}}^{3}\xb7{\mathrm{mol}}^{-1}\xb7{\mathrm{s}}^{-1}$or ${\mathrm{s}}^{-1}$) are provided. If the Collins–Kimball formulation is selected, the overall reaction rate constant (${k}_{Obs}/{\mathrm{cm}}^{3}\xb7{\mathrm{mol}}^{-1}.{\mathrm{s}}^{-1}$), separate diffusion coefficients (${\mathrm{cm}}^{2}\xb7{\mathrm{s}}^{-1}$) for the reactants and the Smoluchowski diffusion rate constant (${\overrightarrow{k}}_{D}/{\mathrm{cm}}^{3}\xb7{\mathrm{mol}}^{-1}\xb7{\mathrm{s}}^{-1})$ are provided.

## 4. Examples

#### 4.1. Fitting Mode—Arrhenius and Transitivity Plots

_{2}

`→`H + H

_{2}O reaction [76] using the Arrhenius and Aquilanti–Mundim formulas. The third example is centered on the investigations that have revealed the super-Arrhenius behavior for the rates of the processes promoted by enzymatic catalysis [77,78,79]. Here, we regard the reaction rate constant of the hydride transfer between the substrate and NAD

^{+}, catalyzed by F147L, which exhibits a strong convex curvature in the temperature range of 5 to 65 °C [78]: A fitting is performed with the Arrhenius, Aquilanti–Mundim and VTF formulas. The final case is aimed at showing a case of anti-Arrhenius behavior, which is characterized by the decrease of the reaction rate constant with the increase in the temperature. The OH + HBr

`→`Br + H

_{2}O reaction is prototypical in studies of an example, both from a theoretical and an experimental point of view: It exhibits negative temperature dependence of the reaction rate constant [80,81,82]. Fitting is performed with the Arrhenius and the Aquilanti–Mundim formulas.

_{2}reaction. The VTF and Aquilanti–Mundim formulas seem adequate to fit super-Arrhenius behavior. The Aquilanti–Mundim parameters that were obtained for the OH + HBr reaction indicate that they provide an excellent option for the anti-Arrhenius behavior. As expected, the Arrhenius formula is clearly inadequate to account for deviations at low temperature for all the reactions presented.

#### 4.2. Reaction Rate Constants’ Mode

`→`H

_{2}O + Cl reaction in the gas-phase was performed to validate TST and tunneling corrections, implemented in the Transitivity code. Furthermore, using the suggestion of the Eyringpy code [54], the NH

_{3}+ OH

`→`NH

_{2}+ H

_{2}O reaction was selected to demonstrate the accuracy of the Collins–Kimball and Kramers models to estimate reaction rate constants in an aqueous solution.

#### 4.2.1. The OH + HCl `→` H_{2}O + Cl Reaction

#### 4.2.2. The NH_{3} + OH `→` NH_{2} + H_{2}O Reaction

_{3}+ OH

`→`NH

_{2}+ H

_{2}O reaction permits to illustrate the accuracy of the methodology in the liquid-phase. Energies, geometries, and frequencies of stationary points were extracted at the same level of calculation used in the Eyringpy code [54].

_{3}+ OH

`→`NH

_{2}+ H

_{2}O reaction from 273 to 4000 K using Kramers and Collins–Kimball models. The Smoluchowski diffusion rate constant ${\overrightarrow{k}}_{D}$, which evaluates the diffusion limit for a bimolecular reactive process including the solvent effect, is shown in the lower right panel. The Kramers transmission correction, which evaluates the interference of the friction effect of the solvent in the reactive process as a function of temperature, is shown in the lower left panel. At 298.15 K, Kramers’ formulation gives for the reaction rate constant the value 6.73 × 10

^{11}cm

^{3}mol

^{−1}s

^{−1}, while the Collins-Kimball formulation yields 6.77 × 10

^{11}cm

^{3}mol

^{−1}s

^{−1}, (the experimental value indicates ~10

^{11}cm

^{3}mol

^{−1}s

^{−1}) [91,92,93]. The value of ${\overrightarrow{k}}_{D}$ from Smoluchowski (Collins–Kimball) is 3.73 × 10

^{12}cm

^{3}mol

^{−1}s

^{−1}, in accordance with that calculated in Reference [54], ${\overrightarrow{k}}_{D}$ = 3.60 × 10

^{12}cm

^{3}mol

^{−1}s

^{−1}.

#### 4.3. CPMD Input Files Generator

## 5. Final Remarks

- Calculation of the kinetic rate constants for chemical reactions from the potential energy surface features profile, such as the CH
_{4}+ OH [60], CH_{3}OH + H [99], OH + HCl [44], OH + HI [43], to proton rearrangement of enol forms of curcumin [100], OH + H_{2}[101], and chiral nucleophilic substitution reaction [102].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Symbols | Nomenclature |
---|---|

$k$ | Rate constant |

T | Temperature |

${k}_{B}$ | Boltzmann constant |

$\beta $ | Lagrange multiplier |

$\gamma $ | Transitivity function |

$d$ | Deformed parameter |

$h$ | Planck’s constant |

$Q$ | Partition functions |

AM | Aquilanti-Mundim |

$dH$ | Enthalpy of reaction |

ASCC | Aquilanti–Sanchez–Coutinho–Carvalho |

NTS | Nakamura–Takayanagi–Sato |

TST | Transition-State Theory |

GSA | Generalized Simulated Annealing |

$d$-TST | Deformed Transition-State Theory |

ST | Skodje and Truhlar tunneling correction |

Bell35 | Bell’s tunneling correction of 1935 |

Bell58 | Bell’s tunneling correction of 1958 |

${\epsilon}^{\u2021}$ | barrier height (Eyring’s parameter) |

${E}_{a}$ | Apparent Activation Energy |

${E}_{0}$ | Energy parameter from NTS formula |

${E}_{\upsilon}\text{}$ | Energy parameter from ASCC formula |

B | Temperature parameter from VFT formula |

${T}_{0}$ | Temperature parameter from NTS and VFT formulas. |

${T}_{c}$ | Crossover temperature |

${\overrightarrow{k}}_{D}$ | Diffusion rate constant |

${\nu}^{\u2021}$ | Imaginary frequency |

${k}_{Obs}$ | Overall reaction rate constant |

${\kappa}_{Kr}$ | Transmission factor from Kramers’ model |

$\mu $ | Friction constant |

$\eta $ | Viscosity |

DFT | Density functional theory |

BOMD | Born-Oppenheimer molecular dynamics |

CPMD | Car-Parrinello molecular dynamics |

PIMD | Path-Integral molecular dynamics |

MTD | Metadynamics |

TSH | Trajectory Surface Hopping |

## References

- Valter Henrique, C.-S.; Coutinho, N.D.; Aquilanti, V. From the Kinetic Theory of Gases to the Kinetics of Chemical Reactions: On the Verge of the Thermodynamical and the Kinetic Limits. Molecules
**2019**. to be submitted. [Google Scholar] - Aquilanti, V.; Coutinho, N.D.; Carvalho-Silva, V.H. Kinetics of Low-Temperature Transitions and Reaction Rate Theory from Non-Equilibrium Distributions. Philos. Trans. R. Soc. London A
**2017**, 375, 20160204. [Google Scholar] [CrossRef] - Aquilanti, V.; Borges, E.P.; Coutinho, N.D.; Mundim, K.C.; Carvalho-Silva, V.H. From statistical thermodynamics to molecular kinetics: the change, the chance and the choice. Rend. Lincei. Sci. Fis. e Nat.
**2018**, 28, 787–802. [Google Scholar] [CrossRef] - Gentili, P.L. The fuzziness of the molecular world and its perspectives. Molecules
**2018**, 23, 2074. [Google Scholar] [CrossRef] - Gentili, P.L. Untangling Complex Systems: A Grand Challenge for Science; CRC Press: Boca Raton, FL, USA, 2018; ISBN 9780429847547. [Google Scholar]
- Atkinson, R. Kinetics and mechanisms of the gas-phase reactions of the hydroxyl radical with organic compounds under atmospheric conditions. Chem. Rev.
**1986**, 86, 69–201. [Google Scholar] [CrossRef] - Limbach, H.-H.; Miguel Lopez, J.; Kohen, A. Arrhenius curves of hydrogen transfers: tunnel effects, isotope effects and effects of pre-equilibria. Philos. Trans. R. Soc. B Biol. Sci.
**2006**, 361, 1399–1415. [Google Scholar] [CrossRef][Green Version] - Smith, I.W.M. The temperature-dependence of elementary reaction rates: Beyond Arrhenius. Chem. Soc. Rev.
**2008**, 37, 812–826. [Google Scholar] [CrossRef] - Sims, I.R. Low-temperature reactions: Tunnelling in space. Nat. Chem.
**2013**, 5, 734–736. [Google Scholar] [CrossRef] [PubMed] - Peleg, M.; Normand, M.D.; Corradini, M.G. The Arrhenius Equation Revisited. Crit. Rev. Food Sci. Nutr.
**2012**, 52, 830–851. [Google Scholar] [CrossRef] [PubMed] - Darrington, R.T.; Jiao, J. Rapid and Accurate Prediction of Degradant Formation Rates in Pharmaceutical Formulations Using High-Performance Liquid Chromatography-Mass Spectrometry. J. Pharm. Sci.
**2004**, 93, 838–846. [Google Scholar] [CrossRef] - Giordano, D.; Russell, J.K. Towards a structural model for the viscosity of geological melts. Earth Planet. Sci. Lett.
**2018**, 501, 202–212. [Google Scholar] [CrossRef] - Klinman, J.P.; Kohen, A. Hydrogen Tunneling Links Protein Dynamics to Enzyme Catalysis. Annu. Rev. Biochem.
**2013**, 82, 471–496. [Google Scholar] [CrossRef][Green Version] - Warshel, A.; Bora, R.P. Perspective: Defining and quantifying the role of dynamics in enzyme catalysis. J. Chem. Phys.
**2016**, 144, 180901. [Google Scholar] [CrossRef] - Laidler, K.J. A Glossary of Terms Used in Chemical Kinetics, Including Reaction Dynamics. Pure Appl. Chem.
**1996**, 68, 149–192. [Google Scholar] [CrossRef] - Tolman, R.C. Statistical Mechanics Applied to Chemical Kinetics. J. Amer. Chem. Soc.
**1920**, 42, 2506–2528. [Google Scholar] [CrossRef] - Glasstone, S.; Laidler, K.J.; Eyring, H. The Theory of Rate Processes: The Kinetics of Chemical Reactions, Viscosity, Diffusion and Electrochemical Phenomena; International chemical series; McGraw-Hill: New York, NY, USA, 1941. [Google Scholar]
- Truhlar, D.G. Current Status of Transition-State Theory. J. Phys. Chem.
**1983**, 2664–2682. [Google Scholar] [CrossRef] - Kooij, D.M. Über die Zersetzung des gasförmigen Phosphorwasserstoffs. Zeitschrift für Phys. Chemie
**1893**, 12, 155–161. [Google Scholar] - Bělehrádek, J. A unified theory of cellular rate processes based upon an analysis of temperature action. Protoplasma
**1957**, 48, 53–71. [Google Scholar] [CrossRef] - Vogel, H. Das temperature-abhangigketsgesetz der viskositat von flussigkeiten. Phys. Z
**1921**, 22, 645–646. [Google Scholar] - Fulcher, G.S. Analysis of Recent Measurements of the Viscosity of Glasses. J. Am. Ceram. Soc.
**1925**, 8, 339–355. [Google Scholar] [CrossRef] - Tammann, G.; Hesse, W. Die Abhängigkeit der Viscosität von der Temperatur bie unterkühlten Flüssigkeiten. Zeitschrift für Anorg. und Allg. Chemie
**1926**, 156, 245–257. [Google Scholar] [CrossRef] - Nakamura, K.; Takayanagi, T.; Sato, S. A modified arrhenius equation. Chem. Phys. Lett.
**1989**, 160, 295–298. [Google Scholar] [CrossRef] - Aquilanti, V.; Mundim, K.C.; Elango, M.; Kleijn, S.; Kasai, T. Temperature dependence of chemical and biophysical rate processes: Phenomenological approach to deviations from Arrhenius law. Chem. Phys. Lett.
**2010**, 498, 209–213. [Google Scholar] [CrossRef] - Coutinho, N.D.; Silva, Y.S.; de Fazio, D.; Cavalli, S.; Carvalho-Silva, V.H.; Aquilanti, V. Chemical Kinetics under Extreme Conditions: Exact, Phenomenological and First-Principles Computational Approaches. In The Astrochemical Observatory: Focus on Chiral Molecules; Accademia Nazionale delle Scienze detta dei XL: Rome, Italy, 2018; pp. 1–15. [Google Scholar]
- Carvalho-Silva, V.H.; Coutinho, N.D.; Aquilanti, V. Temperature dependence of rate processes beyond Arrhenius and Eyring: Activation and Transitivity. Front. Chem.
**2019**, 7, 380. [Google Scholar] [CrossRef] [PubMed] - Fernandez-Ramos, A.; Ellingson, B.A.; Garrett, B.C.; Truhlar, D.G. Variational Transition State Theory with Multidimensional Tunneling. In Reviews in Computational Chemistry; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2007; pp. 125–232. ISBN 9780470116449. [Google Scholar]
- Marcus, R.A. Electron transfer reactions in chemistry. Theory and experiment. Rev. Mod. Phys.
**1993**, 65, 599–610. [Google Scholar] [CrossRef][Green Version] - Miller, W. Quantum mechanical transition state theory and a new semiclassical model for reaction rate constants. J. Chem. Phys.
**1974**, 61, 1823. [Google Scholar] [CrossRef] - Richardson, J.O. Ring-Polymer Approaches to Instanton Theory. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 2012. [Google Scholar]
- Zhang, Y.; Stecher, T.; Cvitaš, M.T.; Althorpe, S.C. Which Is Better at Predicting Quantum-Tunneling Rates: Quantum Transition-State Theory or Free-Energy Instanton Theory? J. Phys. Chem. Lett.
**2014**, 5, 3976–3980. [Google Scholar] [CrossRef] - Xiao, R.; Gao, L.; Wei, Z.; Spinney, R.; Luo, S.; Wang, D.; Dionysiou, D.D.; Tang, C.J.; Yang, W. Mechanistic insight into degradation of endocrine disrupting chemical by hydroxyl radical: An experimental and theoretical approach. Environ. Pollut.
**2017**, 231, 1446–1452. [Google Scholar] [CrossRef] - Luo, S.; Gao, L.; Wei, Z.; Spinney, R.; Dionysiou, D.D.; Hu, W.P.; Chai, L.; Xiao, R. Kinetic and mechanistic aspects of hydroxyl radical‒mediated degradation of naproxen and reaction intermediates. Water Res.
**2018**, 137, 233–241. [Google Scholar] [CrossRef] - Gao, Y.; Ji, Y.; Li, G.; An, T. Mechanism, kinetics and toxicity assessment of OH-initiated transformation of triclosan in aquatic environments. Water Res.
**2014**, 49, 360–370. [Google Scholar] [CrossRef] - De Sainte Claire, P. Degradation of PEO in the solid state: A theoretical kinetic model. Macromolecules
**2009**, 42, 3469–3482. [Google Scholar] [CrossRef] - Ahubelem, N.; Shah, K.; Moghtaderi, B.; Page, A.J. Formation of benzofuran and chlorobenzofuran from 1,3-dichloropropene: A quantum chemical investigation. Int. J. Quantum Chem.
**2015**, 115, 1739–1745. [Google Scholar] [CrossRef] - Zavala-Oseguera, C.; Galano, A.; Merino, G. Computational study on the kinetics and mechanism of the carbaryl + OH reaction. J. Phys. Chem. A
**2014**, 118, 7776–7781. [Google Scholar] [CrossRef] [PubMed] - Döntgen, M.; Przybylski-Freund, M.-D.; Kröger, L.C.; Kopp, W.A.; Ismail, A.E.; Leonhard, K. Automated discovery of reaction pathways, rate constants, and transition states using reactive molecular dynamics simulations. J. Chem. Theory Comput.
**2015**, 11, 2517–2524. [Google Scholar] [CrossRef] [PubMed] - Piccini, G.; McCarty, J.; Valsson, O.; Parrinell, M. Variational Flooding Study of a S2 Reaction. J. Phys. Chem. A
**2017**, 8, 580–583. [Google Scholar] - Fleming, K.L.; Tiwary, P.; Pfaendtner, J. New Approach for Investigating Reaction Dynamics and Rates with Ab Initio Calculations. J. Phys. Chem. A
**2016**, 120, 299–305. [Google Scholar] [CrossRef] [PubMed] - Lancar, I.T.; Mellouki, A.; Poulet, G. Kinetics of the reactions of hydrogen iodide with hydroxyl and nitrate radicals. Chem. Phys. Lett.
**1991**, 177, 554–558. [Google Scholar] [CrossRef] - Coutinho, N.D.; Carvalho-Silva, V.H.; de Oliveira, H.C.B.; Aquilanti, V. The HI + OH → H2O + I Reaction by First-Principles Molecular Dynamics: Stereodirectional and Anti-Arrhenius Kinetics, 2017; Volume 10408 LNCS, ISBN 9783319624037.
- Coutinho, N.D.; Sanches-Neto, F.O.; Carvalho-Silva, V.H.; de Oliveira, H.C.B.; Ribeiro, L.A.; Aquilanti, V. Kinetics of the OH+HCl→H2O+Cl reaction: Rate determining roles of stereodynamics and roaming and of quantum tunneling. J. Comput. Chem.
**2018**, 39, 2508–2516. [Google Scholar] [CrossRef] [PubMed] - Isaacson, A.D.; Truhlar, D.G.; Rai, S.N.; Steckler, R.; Hancock, G.C.; Garrett, B.C.; Redmon, M.J. POLYRATE: A general computer program for variational transition state theory and semiclassical tunneling calculations of chemical reaction rates. Comput. Phys. Commun.
**1987**, 47, 91–102. [Google Scholar] [CrossRef] - Duncan, W.T.; Bell, R.L.; Truong, T.N. TheRate: Program forab initio direct dynamics calculations of thermal and vibrational-state-selected rate constants. J. Comput. Chem.
**1998**, 19, 1039–1052. [Google Scholar] [CrossRef] - Barker, J.R. Multiple-Well, multiple-path unimolecular reaction systems. I. MultiWell computer program suite. Int. J. Chem. Kinet.
**2001**, 33, 232–245. [Google Scholar] [CrossRef][Green Version] - Ghysels, A.; Verstraelen, T.; Hemelsoet, K.; Waroquier, M.; Van Speybroeck, V. TAMkin: A versatile package for vibrational analysis and chemical kinetics. J. Chem. Inf. Model.
**2010**, 50, 1736–1750. [Google Scholar] [CrossRef] [PubMed] - Glowacki, D.R.; Liang, C.-H.; Morley, C.; Pilling, M.J.; Robertson, S.H. MESMER: An Open-Source Master Equation Solver for Multi-Energy Well Reactions. J. Phys. Chem. A
**2012**, 116, 9545–9560. [Google Scholar] [CrossRef] [PubMed] - Gao, C.W.; Allen, J.W.; Green, W.H.; West, R.H. Reaction Mechanism Generator: Automatic construction of chemical kinetic mechanisms. Comput. Phys. Commun.
**2016**, 203, 212–225. [Google Scholar] [CrossRef][Green Version] - Euclides, H.O.; Barreto, P.R. APUAMA: a software tool for reaction rate calculations. J. Mol. Model.
**2017**, 23, 176. [Google Scholar] [CrossRef] [PubMed][Green Version] - Canneaux, S.; Bohr, F.; Henon, E. KiSThelP: A program to predict thermodynamic properties and rate constants from quantum chemistry results. J. Comput. Chem.
**2014**, 35, 82–93. [Google Scholar] [CrossRef] [PubMed] - Coppola, C.M. Mher V Kazandjian; Matrix formulation of the energy exchange problem of multi-level systems and the code FRIGUS. Rend. Lincei Sci. Fis. e Nat.
**2019**, in press. [Google Scholar] - Dzib, E.; Cabellos, J.L.; Ortíz-Chi, F.; Pan, S.; Galano, A.; Merino, G. Eyringpy: A program for computing rate constants in the gas phase and in solution. Int. J. Quantum Chem.
**2018**, 119, 11–13. [Google Scholar] [CrossRef] - Mundim, K.C.; Tsallis, C. Geometry optimization and conformational analysis through generalized simulated annealing. Int. J. Quantum Chem.
**1998**, 58, 373–381. [Google Scholar] [CrossRef] - Sato, S. Tunneling in bimolecular reactions. Chem. Phys.
**2005**, 315, 65–75. [Google Scholar] [CrossRef] - Bell, R.P. Quantum Mechanical Effects in Reactions Involving Hydrogen. Proc. R. Soc. London. Ser. A, Math. Phys. Sci.
**1935**, CXLVIII.A, 241–250. [Google Scholar] - Bell, R.P. The Tunnel Effect Correction For Parabolic Potential Barriers. Faraday Soc. Contrib.
**1958**, 1–4. [Google Scholar] [CrossRef] - Skodje, R.T.; Truhlar, D.G. Parabolic tunneling calculations. J. Phys. Chem.
**1981**, 85, 624–628. [Google Scholar] [CrossRef] - Carvalho-Silva, V.H.; Aquilanti, V.; de Oliveira, H.C.B.; Mundim, K.C. Deformed transition-state theory: Deviation from Arrhenius behavior and application to bimolecular hydrogen transfer reaction rates in the tunneling regime. J. Comput. Chem.
**2017**, 38, 178–188. [Google Scholar] [CrossRef] [PubMed] - Collins, F.C.; Kimball, G.E. Diffusion-controlled reaction rates. J. Colloid Sci.
**1949**, 4, 425–437. [Google Scholar] [CrossRef] - Kramers, H.A. Brownian motion in a field of force and the diffusion model of chemical reactions. Phys.
**1940**, 7, 284–304. [Google Scholar] [CrossRef] - CPMDversion 3.17.1; CPMD, version 4.; CPMDversion 4.1; CPMDversion 3.17.1 Copyright IBM 2012.
- Claudino, D.; Gargano, R.; Carvalho-Silva, V.H.; E Silva, G.M.; Da Cunha, W.F. Investigation of the Abstraction and Dissociation Mechanism in the Nitrogen Trifluoride Channels: Combined Post-Hartree-Fock and Transition State Theory Approaches. J. Phys. Chem. A
**2016**, 120, 5464–5473. [Google Scholar] [CrossRef] - Cavalli, S.; Aquilanti, V.; Mundim, K.C.; De Fazio, D. Theoretical reaction kinetics astride the transition between moderate and deep tunneling regimes: The F + HD case. J. Phys. Chem. A
**2014**, 118, 6632–6641. [Google Scholar] [CrossRef] - Bell, R.P. The Tunnel Effect in Chemistry; Champman and Hall: London, UK, 1980. [Google Scholar]
- Christov, S.G. The Characteristic (Crossover) Temperature in the Theory of Thermally Activated Tunneling Processes. Mol. Eng.
**1997**, 7, 109–147. [Google Scholar] [CrossRef] - Onsager, L. Electric Moments of Molecules in Liquids. J. Am. Chem. Soc.
**1936**, 58, 1486–1493. [Google Scholar] [CrossRef] - Wong, M.W.; Wiberg, K.B.; Frisch, M.J. Solvent effects. 3. Tautomeric equilibria of formamide and 2-pyridone in the gas phase and solution: an ab initio SCRF study. J. Am. Chem. Soc.
**1992**, 114, 1645–1652. [Google Scholar] [CrossRef] - Henriksen, N.E.; Hansen, F.Y. Theories of Molecular Reaction Dynamics: The Microscopic Foundation of Chemical Kinetics; Oxford University Press: New York, NY, USA, 2008; ISBN 9780191708251. [Google Scholar]
- Smoluchowski, M.V. Drei Vortrage uber Diffusion, Brownsche Bewegung und Koagulation von Kolloidteilchen. Phys. Zeit.
**1916**, 17, 557–585. [Google Scholar] - Collins, F.C.; Kimball, G.E. Diffusion-Controlled Reactions in Liquid Solutions. Ind. Eng. Chem.
**1949**, 41, 2551–2553. [Google Scholar] [CrossRef] - Eigen, M. Acid-Base Catalysis, and Enzymatic Hydrolysis. Part I: Elementary Processes. Angew. Chemie Int. Ed. English
**1964**, 3, 1–19. [Google Scholar] [CrossRef] - Hallett, J. The Temperature Dependence of the Viscosity of Supercooled Water. Proc. Phys. Soc.
**1963**, 82, 1046–1050. [Google Scholar] [CrossRef] - Savitzky, A.; Golay, M.J.E. Smoothing and Differentiation of Data by Simplified Least Squares Procedures. Anal. Chem.
**1964**, 36, 1627–1639. [Google Scholar] [CrossRef] - Ravishankara, A.R.; Nicovich, J.M.; Thompson, R.L.; Tully, F.P. Kinetic study of the reaction of hydroxyl with hydrogen and deuterium from 250 to 1050 K. J. Phys. Chem.
**1981**, 85, 2498–2503. [Google Scholar] [CrossRef] - Kohen, A.; Cannio, R.; Bartolucci, S.; Klinman, J.P. Enzyme dynamics and hydrogen tunnelling in a thermophilic alcohol dehydrogenase. Nature
**1999**, 399, 496–499. [Google Scholar] [CrossRef] - Liang, Z.X.; Tsigos, I.; Bouriotis, V.; Klinman, J.P. Impact of protein flexibility on hydride-transfer parameters in thermophilic and psychrophilic alcohol dehydrogenases. J. Am. Chem. Soc.
**2004**, 126, 9500–9501. [Google Scholar] [CrossRef] - Truhlar, D.; Kohen, A. Convex Arrhenius plots and their interpretation. Proc. Nat. Acad. Sci. USA
**2001**, 98, 848–851. [Google Scholar] [CrossRef][Green Version] - Coutinho, N.D.; Aquilanti, V.; Silva, V.H.C.; Camargo, A.J.; Mundim, K.C.; De Oliveira, H.C.B. Stereodirectional Origin of anti-Arrhenius Kinetics for a Tetraatomic Hydrogen Exchange Reaction: Born-Oppenheimer Molecular Dynamics for OH + HBr. J. Phys. Chem. A
**2016**, 120, 5408–5417. [Google Scholar] [CrossRef] [PubMed] - de Oliveira-Filho, A.G.S.; Ornellas, F.R.; Bowman, J.M. Quasiclassical Trajectory Calculations of the Rate Constant of the OH + HBr → Br + H
_{2}O Reaction Using a Full-Dimensional Ab Initio Potential Energy Surface Over the Temperature Range 5 to 500 K. J. Phys. Chem. Lett.**2014**, 5, 706–712. [Google Scholar] [CrossRef] [PubMed] - Coutinho, N.D.; Silva, V.H.C.; de Oliveira, H.C.B.; Camargo, A.J.; Mundim, K.C.; Aquilanti, V. Stereodynamical Origin of Anti-Arrhenius Kinetics: Negative Activation Energy and Roaming for a Four-Atom Reaction. J. Phys. Chem. Lett.
**2015**, 6, 1553–1558. [Google Scholar] [CrossRef] [PubMed] - Zuniga-Hansen, N.; Silbert, L.E.; Calbi, M.M. Breakdown of kinetic compensation effect in physical desorption. Phys. Rev. E
**2018**, 98, 032128. [Google Scholar] [CrossRef][Green Version] - Sims, I.R.; Smith, I.W.M.; Clary, D.C.; Bocherel, P.; Rowe, B.R. Ultra-low Temperature Kinetics of Neutral-neutral Reactions - New Experimental and Theoretical Results For OH + HBr Between 295 K and 23 K. J. Chem. Phys.
**1994**, 101, 1748–1751. [Google Scholar] [CrossRef] - Souletie, J.; Tholence, J.L. Critical slowing down in spin glasses and other glasses: Fulcher versus power law. Phys. Rev. B
**1985**, 32, 516. [Google Scholar] [CrossRef] - Stickel, F.; Fischer, E.W.; Richert, R. Dynamics of glass-forming liquids. II. Detailed comparison of dielectric relaxation, de-conductivity, and viscosity data. J. Chem. Phys.
**1996**, 104, 2043. [Google Scholar] [CrossRef] - Drozd-Rzoska, A. Universal behavior of the apparent fragility in ultraslow glass forming systems. Sci. Rep.
**2019**, 9, 6816. [Google Scholar] [CrossRef] - Silva, V.H.C.; Aquilanti, V.; de Oliveira, H.C.B.; Mundim, K.C. Uniform description of non-Arrhenius temperature dependence of reaction rates, and a heuristic criterion for quantum tunneling vs classical non-extensive distribution. Chem. Phys. Lett.
**2013**, 590, 201–207. [Google Scholar] [CrossRef][Green Version] - Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G.A.; et al. Gaussian09 Revision D.01 2009; Gaussian, Inc.: Wallingford, CT, USA, 2009. [Google Scholar]
- Ravishankara, A.R.; Wine, P.H.; Wells, J.R.; Thompson, R.L. Kinetic study of the reaction of OH with HCl from 240 to 1055 K. Chem. Phys. Lett.
**1985**, 17, 1281–1297. [Google Scholar] [CrossRef] - Hickel, B.; Sehested, K. Reaction of hydroxyl radicals with ammonia in liquid water at elevated temperatures. Int. J. Radiat. Appl. Instrum. Part C. Radiat. Phys. Chem.
**1992**, 39, 355–357. [Google Scholar] [CrossRef] - Men’kin, V.B.; Makarov, I.E.; Pikaev, A.K. Pulse radiolysis study of reaction rates of OH and O-radicals with ammonia in aqueous solutions. High Energy Chem. (Engl. Transl.)
**1989**, 22, 333–336. [Google Scholar] - Neta, P.; Maruthamuthu, P.; Carton, P.M.; Fessenden, R.W. Formation and reactivity of the amino radical. J. Phys. Chem.
**1978**, 82, 1875–1878. [Google Scholar] [CrossRef] - Aquilanti, V.; Mundim, K.C.; Cavalli, S.; De Fazio, D.; Aguilar, A.; Lucas, J.M. Exact activation energies and phenomenological description of quantum tunneling for model potential energy surfaces. The F+H
_{2}reaction at low temperature. Chem. Phys.**2012**, 398, 186–191. [Google Scholar] [CrossRef] - Rampino, S.; Pastore, M.; Garcia, E.; Pacifici, L.; Laganà, A. On the temperature dependence of the rate coefficient of formation of C
_{2}^{+}from C + CH^{+}. Mon. Not. R. Astron. Soc.**2016**, 460, 2368–2375. [Google Scholar] [CrossRef] - Coutinho, N.D.; Silva, V.H.C.; Mundim, K.C.; de Oliveira, H.C.B. Description of the effect of temperature on food systems using the deformed Arrhenius rate law: deviations from linearity in logarithmic plots vs. inverse temperature. Rend. Lincei
**2015**, 26, 141–149. [Google Scholar] [CrossRef] - Capitelli, M.; Pietanza, L.D. Past and present aspects of Italian plasma chemistry. Rend. Lincei. Sci. Fis. e Nat.
**2019**. [Google Scholar] [CrossRef] - Agreda, N.J.L. Aquilanti–Mundim deformed Arrhenius model in solid-state reactions: Theoretical evaluation using DSC experimental data. J. Therm. Anal. Calorim.
**2016**, 126, 1175–1184. [Google Scholar] [CrossRef] - Sanches-Neto, F.O.; Coutinho, N.D.; Silva, V. A novel assessment of the role of the methyl radical and water formation channel in the CH3OH + H reaction. Phys. Chem. Chem. Phys.
**2017**, 19, 24467–24477. [Google Scholar] [CrossRef] [PubMed] - Santin, L.G.; Toledo, E.M.; Carvalho-Silva, V.H.; Camargo, A.J.; Gargano, R.; Oliveira, S.S. Methanol Solvation Effect on the Proton Rearrangement of Curcumin’s Enol Forms: An Ab Initio Molecular Dynamics and Electronic Structure Viewpoint. J. Phys. Chem. C
**2016**, 120, 19923–19931. [Google Scholar] [CrossRef] - Carvalho-Silva, V.H.; Vaz, E.C.; Coutinho, N.D.; Kobayashi, H.; Kobayashi, Y.; Kasai, T.; Palazzetti, F.; Lombardi, A.; Aquilanti, V. The Increase of the Reactivity of Molecular Hydrogen with Hydroxyl Radical from the Gas Phase versus an Aqueous Environment: Quantum Chemistry and Transition State-Theory Calculations. In Lecture Notes in Computer Science; Springer: Cham, Switzerland, 2019; pp. 450–459. [Google Scholar]
- Rezende, M.V.C.S.; Coutinho, N.D.; Palazzetti, F.; Lombardi, A.; Carvalho-Silva, V.H. Nucleophilic substitution vs elimination reaction of bisulfide ions with substituted methanes: exploration of chiral selectivity by stereodirectional first-principles dynamics and transition state theory. J. Mol. Model.
**2019**, 25, 227. [Google Scholar] [CrossRef] [PubMed]

Sample Availability: Not available. |

**Figure 2.**Arrhenius plots comparing the experimental reaction rate constant and fitted formulas for keto–enol tautomerization reaction (sub-Arrhenius behavior under deep tunneling), OH + H

_{2}⟶ H

_{2}O + H reaction (sub-Arrhenius behavior under moderate tunneling), hydride transfer with enzymatic catalysis (super-Arrhenius behavior) and OH + HBr ⟶ H

_{2}O + Br reaction (anti-Arrhenius behavior). NTS and ASCC formulas were of use for sub-Arrhenius behavior under deep-tunneling regime. The Aquilanti–Mundim formula was of use for sub-Arrhenius cases under moderate-tunneling regime, for super-Arrhenius and for anti-Arrhenius behaviors. VFT also was of use for super-Arrhenius situations. The references of experimental data can be found in Table 1.

**Figure 3.**The Arrhenius (upper panel) and Transitivity (lower panel) planes of the temperature dependence of relaxation time of the propylene carbonate. The diamond symbols represent the transitivity values obtained numerically and smoothing with the Savitzky–Golay filter. Red lines emphasize two regions where the temperature dependence of the transitivity is linearized, as expected by the Aquilanti–Mundim law.

**Figure 4.**Arrhenius plot obtained from the Transitivity code for the OH + HCl

`→`Cl + H

_{2}O reaction using TST with Bell35, Bell58, ST tunneling correction, and $d$-TST. Experimental data in the literature [90] are available for comparison and shown as full dots.

**Figure 5.**Upper panels present the Arrhenius plots as given by the program for the NH

_{3}+ OH

`→`NH

_{2}+ H

_{2}O reaction using Kramers’ and Collins–Kimball formulations. The lower panels show the Kramers transmission and Smoluchowski diffusion limit constant as a function of inverse temperature.

**Figure 6.**An exemplary view of the input generation function of the transitivity program. Details of input files can be found in the www.vhcsgroup.com/transitivity web page.

**Table 1.**Fitted parameters for the Arrhenius, AM, ASCC, NTS and VFT formulas, using the Transitivity code for keto–enol tautomerization [7], OH + H

_{2}[76], enzymatic catalysis [78] and OH + Br [84] reactions. Energy (${E}_{a}$, ${\epsilon}^{\u2021}$, ${E}_{\upsilon}$ and ${E}_{0}$) is in cal/mol and temperature (${T}_{0}$and $B$) in K. Pre-factor units can be identified in the references.

Formula | Chemical Processes | ||||
---|---|---|---|---|---|

Fitted Parameters | Keto-enol Tautomerization [7] Sub -Arrhenius (Deep-Tunneling) | OH + H_{2} → H + H_{2}[76] Sub-Arrhenius (Moderate Tunneling) | Enzymatic Catalysis [78] Super-Arrhenius | OH + HBr → Br + H_{2}O[84] Anti-Arrhenius | |

Arrhenius $k\left(T\right)=A\mathrm{exp}\left(-\frac{{E}_{a}}{{k}_{B}T}\right)$ | $A$ | 1.74 × 10^{3} | 2.16$\xb7$10^{-11} | 1.52 × 10^{11} | 1.66 × 10^{-11} |

${E}_{a}$ | 214 | 4891 | 14600 | −94.6 | |

$\mathsf{\chi}2$ | 1.10 × 10^{-2} | 4.20$\xb7$10^{-3} | 2.60 × 10^{-2} | 6.69 × 10^{-2} | |

Aquilanti–Mundim (AM) $k\left(T\right)=A{\left(1-d\frac{{\epsilon}^{\u2021}}{{k}_{B}T}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$d$}\right.}$ | $A$ | 3.32 × 10^{6} | 1.11$\xb7$10^{-10} | 1.91 × 10^{4} | 7.43 × 10^{-14} |

${\epsilon}^{\u2021}$ | 318.06 | 9170 | 2391 | −324.61 | |

$d$ | −0.81 | −0.086 | 0.207 | 1.24 | |

$\mathsf{\chi}2$ | 3.68 × 10^{-2} | 6.80$\xb7$10^{-4} | 2.91 × 10^{-2} | 2.78 × 10^{-3} | |

Aquilanti–Sanchez–Coutinho–Carvalho (ASCC) $k\left(T\right)=A{\left(1-d\frac{{\epsilon}^{\u2021}}{{k}_{B}T+{E}_{\nu}}\right)}^{\frac{1}{d}}$, $d=-\frac{1}{3}{\left(\frac{{E}_{\upsilon}}{2{\epsilon}^{\u2021}}\right)}^{2}$ | $A$ | 2.33 × 10^{4} | - | - | - |

${\epsilon}^{\u2021}$ | 2441 | - | - | - | |

${E}_{\upsilon}$ | 429 | - | - | - | |

$\mathsf{\chi}2$ | 2.18 × 10^{-2} | - | - | - | |

Sato–Nakamura–Takayanagi (NTS) $k\left(T\right)=A\mathrm{exp}\left[-\frac{{E}_{0}}{{k}_{B}{\left({T}^{2}+{T}_{0}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}\right]$ | $A$ | 3.12 × 10^{4} | - | - | - |

${E}_{0}$ | 1655 | - | - | - | |

${T}_{0}$ | 168 | - | - | - | |

$\chi 2$ | 7.38 × 10^{-3} | - | - | - | |

Vogel–Fulcher–Tammann (VFT) $k\left(T\right)=A\mathit{exp}\left(\frac{B}{T-{T}_{0}}\right)$ | $A$ | - | - | 1.25 × 10^{5} | - |

$B$ | - | - | −1298 | - | |

${T}_{0}$ | - | - | 175 | - | |

$\chi 2$ | - | - | 2.16 × 10^{-2} | - |

© 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/).

## Share and Cite

**MDPI and ACS Style**

Machado, H.G.; Sanches-Neto, F.O.; Coutinho, N.D.; Mundim, K.C.; Palazzetti, F.; Carvalho-Silva, V.H. “Transitivity”: A Code for Computing Kinetic and Related Parameters in Chemical Transformations and Transport Phenomena. *Molecules* **2019**, *24*, 3478.
https://doi.org/10.3390/molecules24193478

**AMA Style**

Machado HG, Sanches-Neto FO, Coutinho ND, Mundim KC, Palazzetti F, Carvalho-Silva VH. “Transitivity”: A Code for Computing Kinetic and Related Parameters in Chemical Transformations and Transport Phenomena. *Molecules*. 2019; 24(19):3478.
https://doi.org/10.3390/molecules24193478

**Chicago/Turabian Style**

Machado, Hugo G., Flávio O. Sanches-Neto, Nayara D. Coutinho, Kleber C. Mundim, Federico Palazzetti, and Valter H. Carvalho-Silva. 2019. "“Transitivity”: A Code for Computing Kinetic and Related Parameters in Chemical Transformations and Transport Phenomena" *Molecules* 24, no. 19: 3478.
https://doi.org/10.3390/molecules24193478