Kinetic modelling and kinetic parameters fitting

The required kinetic parameters are the activation energy and the pre-exponential factor in the modified Arrhenius equation. For their search, software that assists in construction of microkinetic mechanisms of heterogeneous catalytic reactions mech_optimiz [1] is used. The problem of minimization of the objective function is solved and the selection is based on the calculation of the sum of the squares error between the values of the model responses and the experimental conversion values:

where Yiu,exp – experimental value of the u-th substance conversion in the i-th experiment, %,
Yiu,sim – value of the u-th substance conversion in the i-th experiment from simulation, %,
Ni – the number of experiments,
Nu – the number of substances for that the conversion data is measured.

A genetic algorithm to find the minimum of the objective function is used in the software mech_optimiz . Using this optimization method, a fitness function is calculated that is inversely proportional to the objective function.

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Calculation of the kinetic parameters of the reaction using DFT-methods

Modeling the reaction and calculation of the kinetic parameters of the reaction using DFT-methods is done in several steps.

First of all, to simulate a chemical reaction, a kinetic scheme is drawn up, which takes into account the addition and detachment of molecules from the catalyst and the transformation of molecules in the state associated with the catalyst. With the help of molecular editors such as ChemCraft and Avogadro, the geometry of the molecules involved in the reaction is constructed. This geometry must be optimized, but the optimization algorithms built into molecular editors (force-field) are very approximate, and on the basis if this geometry one cannot evaluate energy differences like reaction barriers. Therefore, specialized programs for quantum-mechanical calculations are used. We use the Orca program for quantum-mechanical calculations. This is a free software for academic purposes, developed at the Max Planck Institute for Chemical Energy Transformations.

Optimization of the geometry of the molecules involved in the reaction is necessary to obtain the exact value of the total energy and calculate the energy difference and kinetic parameters. For optimization in Orca we use B3LYP hybrid functional. DFT-calculations with hybrid functional are, probably, the most popular calculations at the moment due to the combination of high accuracy and relatively high speed. We use triple zeta basis sets prepared by Ahlrichs and coworkers from Karlsruhe with polarization (def2-TZVP). Also, one should choose a model of solvation if the reaction proceeds in solution. We usually use the COSMO solvation model, which models the electrostatic interaction of a molecule with a solution. After optimization, the vibrational frequencies of the molecule are calculated, the presence of only positive frequencies indicates that the molecule is in a stable state. Before searching for a transition state, we perofrm a relaxed potential energy scan (Relaxed PES Scan), gradually moving molecules along the direction of formation / breaking of bonds and each time reoptimizing the system with the calculation of its total energy. After each step, a file with the coordinates of the optimized geometry is formed. The maximum value of energy in most cases indicates a transition state of some reaction. Having selected the necessary file with coordinates, you can start searching for a transition state.

The search for a transition state is carried out by minimizing the total energy along all directions on the surface of potential energy, except for one, which is called the reaction coordinate. Along this direction, on the contrary, one searches for the maximum energy. In other words, the transition state is a first-order saddle point on the surface of potential energy. Orca performs maximization along the direction of the strongest mode by default. For the final geometry found, the presence of the transition state can be checked by calculating the vibrational frequencies. The presence of a single saddle point is postulated if one and only one frequency is imaginary. In addition, the imaginary frequency must correspond to the vibration of atoms or to their rotation (change of angles) in the direction of formation / breaking of the necessary bonds corresponding to the reaction being studied. This complex displacement of atoms is the internal coordinate of the reaction.

To check the transition state, the IRC method is used (the internal reaction coordinate method). This method allows you to confirm the presence of a transition state and its belonging to the selected reaction. One performs a descent on the surface of potential energy from the transition state point in both directions along the reaction coordinate, i.e., to the reactant and to the product (Figure 1). To do this, one performs a simple optimization (search for a minimum of energy). If the final structures after the IRC method calculation correspond to the products and reactants of the reaction, then the transition state belongs to the selected reaction.

Potential energy curve

Figure 1. Chemical reaction potential energy curve along the reaction coordinate

Then, the activation energy of the forward and backward reactions is calculated by subtracting energy the energy of the initial substances or products from the energy of the transition state . The preexponential factor is calculated on the basis of the product of vibrational frequencies using the harmonic transition state theory.

Vasilyev M.V and Mitrichev I.I., Ph.D.

Citation: Vasilyev M.V., Mitrichev I.I. Calculation of the kinetic parameters of the reaction using DFT-methods. Virtual Catalysis Center, 2018. Access mode: http://centralcatalysis.muctr.ru/en-us/theory/18-quantum-chemical-calculations.html

Literature
1. Plekhovich S.D., Zelentsov S.V. Calculation of transition states by quantum chemistry methods. Teaching manual. Nizhny Novgorod: Nizhny Novgorod State University, 2015, 21 p.
2. Neese, F. The ORCA program system // Wiley Interdisciplinary Reviews - Computational Molecular Science, 2012, Volume 2, Issue 1, Pages 73–78.
3. Neese, F., Wennmohs, F. Orca-An ab initio, DFT and semiempirical SCFMO package. Version 3.0.3. Max-Planck-Institute for Chemical Energy Conversion. Germany, Mülheim, 2015, 595 p.