# “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 |

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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} | - |

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## 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