State-to-State Rate Constants for the O(³P)H₂(v) System: Quasiclassical Trajectory Calculations

Abstract

The rate constants of elementary processes in the atom–diatom system $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2\left(v\right)$, including the processes of vibrational relaxation and dissociation, were studied using the quasiclassical trajectory method. All calculations were carried out along the ground potential energy surface (PES) ³$A^{″}$ that was approximated by a neural network. Approximation data were obtained using ab initio quantum chemistry methods at the extended multi-configuration quasi-degenerate second-order perturbation theory XMCQDPT2 in a basis set limit. The calculated cross-sections of the reaction channels are in good agreement with the literature data. A complete set of state-to-state rate constants was obtained for the metathesis reaction, the dissociation and relaxation of the H₂ molecule upon collision with an O atom. According to these data, Arrhenius approximations over a wide temperature range were obtained for the thermal rate constants of considered processes. Data obtained on the dissociation constants and VT relaxation of vibrationally excited H₂ molecules can be used in constructing kinetic models describing the oxidation of hydrogen at high temperatures or highly nonequilibrium conditions.

1. Introduction

Modelling of combustion processes in the hydrogen–oxygen system is of great practical importance, since hydrogen is a promising fuel that allows us to improve both the efficiency and environmental characteristics of gas turbine engines and power plants [1,2]. For hydrogen as a fuel, there are a large number of kinetic mechanisms of various degrees of detail [3,4], but the problem of quantitative description of the hydrogen–oxygen kinetics cannot be considered fully solved, since there is no complete agreement between theoretical modelling and experiments [2,3]. In recent years, there was an active discussion in the literature about the necessity of taking into account the processes of vibrational relaxation in kinetic mechanisms to describe combustion and detonation [5,6,7,8], and the question of the role of vibrational excitation of radicals (for example, the HO₂ molecule) formed in chain reactions of oxidation was raised [3,9]. Moreover, there are still uncertainties in third body efficiencies for collision-induced dissociation, recombination, and molecular internal energy relaxation processes [10,11]. This work is devoted to the application of molecular dynamics methods to obtain new information about the probabilities of elementary physicochemical processes of hydrogen oxidation, which are important from both theoretical and practical points of view, for example, to improve the efficiency of modelling hydrogen combustion in future gas turbine engines.

Numerical modeling of molecular dynamics is an actively developing field of science, which has become the de facto main method to obtain new information about the rate constants of physicochemical processes [12,13]. The progress of molecular dynamics methods is due to the availability of powerful computers, the development of computational algorithms (in particular, the increasingly widespread use of neural networks), the ability to perform numerical simulation from first principles, and the absence in a numerical experiment of some natural limitations inherent in laboratory diagnostic methods [14]. In particular, the methods of molecular dynamics allow us to obtain the most complete description of the dependence of the probabilities of physicochemical processes on the quantum states (rotational, vibrational, electronic) of reacting particles in a wide temperature range [15]. The complete sets of state-to-state rate constants acquired in this way can be used to obtain averaged characteristics of the reacting system (for example, thermal rate constants), or used to assemble kinetic models for thermally nonequilibrium gas. Such nonequilibrium models considering vibration–chemistry coupling (state-to-state [7,16,17,18,19] and multitemperature [5,6,16,20,21,22] models) are widely used to simulate flows with shock waves [16,17,20,21,23,24,25,26,27,28] and detonation waves [5,6,7,8,9], flows in a supersonic nozzle [18,29,30], gas under discharge conditions [31,32,33], during emission/absorption of intense radiation [34,35], in the upper layers of the Earth’s atmosphere [36], to describe chemistry of the interstellar space [37,38]. Thus, the development of the methods for state-specific rate constant calculations and the generation of new data on rate constants is an important and urgent task, to which this study is devoted.

The main difficulty in estimating the state-to-state rate constants of the reactions and the energy exchange processes is that it is impossible to apply simple estimation approaches (the model of solid spheres, the capture model, the theory of the transition state, etc.) without additional assumptions and empirical corrections. To obtain state-to-state rate constants, it is necessary to use dynamic calculation methods based either on solving the equations of classical mechanics (for example, the method of quasi-classical trajectories (QCTs) [39]), or on solving the quantum dynamics equations of the system under study (for example, the real wave packets method [40]). The latter group of methods is quite difficult to implement even for simple atom–diatom systems; therefore, the main method for estimating the state-to-state rate constants used in most studies is the QCT method [21,41,42,43,44,45,46,47,48,49,50,51,52]. According to this, it should be noted that the QCT method, as any other classical method, does not take into account quantum effects, for instance, the tunneling effect and over-barrier reflection. In this paper, the implementation and application of the QCT method will be considered.

The implementation of the QCT method requires either a preliminary assignment of the entire potential energy surface (PES) of a system of colliding particles in some analytical form (PES can be obtained from ab initio calculations or based on the modelling multidimensional potentials, for example, LEPS potential [53]), or determination of the PES along each calculated trajectory “on the fly” based on calculating the energy gradient at the trajectory point (usually by ab initio methods) [54]. In this paper, the first approach was used, i.e., the PES approximation based on data from ab initio calculations and subsequent construction of the trajectory ensembles of interest, since the number of trajectories per ensemble should be large enough that the relative statistical error of the rate constant estimations is less than 10%.

A separate task is to represent the PES as a multidimensional function of the atomic nuclei coordinates, for which approximations based on the least squares method are often used, defining the PES function as a series of degrees of atomic coordinates [55]. An alternative approach is to use a neural network as a universal approximator. One such method is approximation by a neural network that accepts symmetrized coordinates as an input (PIP-NN—permutation invariant polynomials–neural network method) [50,56,57,58]. The use of such symmetrized coordinates makes it possible, with the existing indistinguishability in permutations of identical atoms in the reacting system, to reduce the number of PES points necessary for the approximation.

The purpose of the present work is, first, to implement the QCT method for calculating the state-to-state rate constants in the atom–diatom A + BC test system using the PIP-NN method to approximate the PES of the system, and, second, to obtain a complete set of state-resolved rate constants for the exchange reaction A + BC→AB + C, dissociation, and VT (vibrational–translational) relaxation of BC molecules on A atoms. The $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system was chosen as such a test system, since it was previously studied in detail by molecular dynamics methods in the works of other authors [37,59,60,61], which will allow us to validate our implementation of the QCT method. Nevertheless, in previous studies [37,59,60,61], the probabilities of dissociation and VT relaxation of H₂$\left(v\right)$ molecules on the O atom were not considered. So, the QCT method for these processes, as far as the authors know, is used for the first time.

2. Methodology

2.1. PES Approximation

All calculations of the PES of the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system along the ground state ³$A^{″}$ surface were carried out at the level of the extended multi-configuration quasi-degenerate second-order perturbation theory XMCQDPT2 [62]. We choose this method because XMCQDPT2 combines an acceptable calculation cost and sufficient accuracy in calculating the dynamic correlation energy for similar molecular systems [63,64,65,66]. As an initial approximation of the wave function, the results of the calculation by the multi-configuration self-consistent field method with a dynamic choice of weights DW-CASSCF [67] was used. Such a choice of a multi-configuration calculation method is associated with the need for a smooth construction of the PES, which is most likely provided by the DW-CASSCF method (see ref. [68] for details). The size of the active space specified in the DW-CASSCF method is 8 electrons distributed in 6 orbitals, i.e., the full valence space for the considered system. The main set of basis functions in all calculations is the family of Dunning basis sets with diffuse functions aug-cc-pVXZ (X = D, T, Q) [69]. Based on calculations in the basis sets aug-cc-pVXZ (X = D, T, Q), the energies were calculated at each point of the PES in the limit basic set aug-cc-pV∞Z; for MCSCF energies, the exponential extrapolation scheme [70,71], and for XMCQDPT2 energies, the scheme using Riemann zeta function [72], were utilized. All ab initio calculations were carried out in the Firefly QC v.8.2.0 software package [73], partially based on the GAMESS (US) [74] source code.

The obtained set of PES data was approximated using the PIP-NN method [56]. This approach to fit PES has important advantages: the universality of the method, the availability of neural network training tools (for example, TensorFlow [75], PyTorch [76], Keras [77] software packages, etc.), the ability to implement the analytical nuclear gradient at any point of the PES. At the same time, the main drawback of this approach is related to the peculiarity of the neural network device that slows down calculations comparing with, for example, LEPS potential; for a neural network with a sigmoid activation function or a hyperbolic tangent, the main calculation time of the output neuron is spent on multiple calculations of this function, as well as matrix–vector multiplication operations.

In the framework of the PIP-NN method, before constructing and training a neural network that plays the role of a universal approximator, it is necessary to transform the Cartesian coordinates of atomic nuclei into symmetrized coordinates. Since the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system corresponds to the AB₂ type, the general form of symmetrized coordinates is already known via Morse coordinates [78]:

$$\left\{\begin{array}{c}{G}_1=exp\left(-\lambda r_{{\mathrm{H}}_1{\mathrm{H}}_2}\right)\hfill \\ G_2=exp\left(-\lambda r_{\mathrm{O}{\mathrm{H}}_1}\right)+exp\left(-\lambda r_{\mathrm{O}{\mathrm{H}}_2}\right)\hfill \\ G_3=exp\left(-\lambda r_{\mathrm{O}{\mathrm{H}}_1}\right)·exp\left(-\lambda r_{\mathrm{O}{\mathrm{H}}_2}\right)\hfill \end{array}\right.$$

(1)

Here, $r_{\mathrm{XY}}$ is the distance between X and Y atoms in Å, and $\lambda$ is a dimensional coefficient equal to 1–2 Å⁻¹. In general, it is necessary to solve a combinatorial problem with respect to permutations of atoms and the corresponding permutations of interatomic distances. Software packages such as MAGMA [79] and SINGULAR [80], which are able to determine permutation groups based on a given permutation matrix, help in solving these problems. The coordinates obtained in this way within the PIP-NN approach are fed to the input of the neural network, and at the output, the neural network gives the potential energy value.

2.2. Rate Constant Estimation

The direct calculation of the rate constants by the QCT method is based on a statistical analysis of the outcomes of trajectory ensembles with specified conditions. We will briefly describe the stages of all calculations that eventually lead to estimates of the rate constants of interest, according to the methodology from [39,81,82].

First of all, it is necessary to convert the Cartesian coordinates of the atomic nuclei into Jacobi coordinates for the atom–diatomic molecule system. In the case of the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system, these are the three coordinates of the H₁ atom relative to the H₂ atom, the three coordinates of the $\mathrm{O}{(}^{3}P)$ atom relative to the mass center of the H₁ H₂ molecule, and the three coordinates of the mass center of the entire $\mathrm{O}{(}^{3}P)$H₁H₂ system. After the corresponding linear transformation, the Hamiltonian of the system will take the form:

$$H\left(\mathbf{p},\mathbf{q}\right)=\sum _{i=1}^3{\displaystyle \frac{p_i^2}{2{\mu }_{{\mathrm{H}}_1{\mathrm{H}}_2}}}+\sum _{i=4}^6{\displaystyle \frac{p_i^2}{2{\mu }_{\mathrm{O},{\mathrm{H}}_1{\mathrm{H}}_2}}}+\sum _{i=7}^9{\displaystyle \frac{p_i^2}{2M_{\mathrm{O}{\mathrm{H}}_1{\mathrm{H}}_2}}}+V\left(\mathbf{q}\right).$$

(2)

Here, $\mu$ represents appropriate reduced masses, $M_{\mathrm{O}{\mathrm{H}}_1{\mathrm{H}}_2}$ is the total mass of the entire system, $\mathbf{q}$ or $q_i$ is the Jacobi coordinates, $\mathbf{p}$ or $p_i$ is the generalized momentum corresponding to $\mathbf{q}$, and $V\left(\mathbf{q}\right)$ is the potential energy of the system at the point with the coordinates $\mathbf{q}$. The resulting Hamiltonian (2) allows us to compose the equations of motion of Hamiltonian mechanics, which were then solved numerically using an explicit one-step Runge–Kutta–Merson method of the 4th order of accuracy with a variable integration step starting from 0.02 fs.

Each ensemble of trajectories was set by initial conditions which are described in detail in [39,81,82]. Here, we only note following. The calculation of the reaction cross-section was carried out without scanning over the impact parameter; each ensemble of trajectories was set only by the vibrational and rotational numbers of the H₂ molecule and the kinetic energy of the incoming $\mathrm{O}{(}^{3}P)$ atom. Starting and final distances between colliding particle are 8 Å; the upper limit of collision energy is 9 eV. Also, we note that rotational quantum number J is estimated in the same way as described in [82] and vibrational quantum number v is chosen as the nearest number corresponding to the precalculated rovibratinal energy level of a product molecule (in this sytem, the OH or H₂ molecules).

Also, to construct the initial conditions and to analyze each trajectory, it is necessary to know the rovibrational levels of H₂ and OH molecules, but the approximation of the anharmonic Morse potential may be inaccurate when describing upper rovibrational levels. Therefore, the energy levels of diatomic molecules were determined by numerically solving the radial component of the stationary Schrödinger equation for a diatomic molecule with the LEVEL16 program [83]. However, it is worth noting that this program, alongside bound states, also calculates the energy levels of quasi-bound states [83]. Despite the fact that some of the quasi-bound states may be metastable and have a lifetime of tens of hours, such states will not be considered within the framework of the QCT method, and the trajectories that led to such an outcome will be attributed to the dissociation process.

According to the methodology described above, it becomes possible to calculate the cross-sections of the following reactions in this system:

$$\begin{array}{c}\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v_1,J_1)\to \mathrm{H}+\mathrm{OH}(v_2,J_2),\\ \mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v_1,J_1)\to \mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v_2,J_2),\\ \mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v,J)\to \mathrm{O}{(}^{3}P)+\mathrm{H}+\mathrm{H}.\end{array}$$

(3)

The reaction cross-section was calculated using the ratio of the trajectory number $N_{\mathrm{R}}$ that led to the required outcome to the total number of trajectories of the ensemble N according to the formula from [39,48,50,51,81,82]:

$$\sigma \left(v,J,E\right)\approx \pi b_{\max }^2{\displaystyle \frac{N_{\mathrm{R}}}{N}}$$

(4)

with the statistical error

$$\Delta \left(v,J,E\right)\approx \pi b_{\max }^2{\displaystyle \frac{N_{\mathrm{R}}}{N}}\sqrt{{\displaystyle \frac{N-N_{\mathrm{R}}}{N{N}_{\mathrm{R}}}}}.$$

(5)

Here, $b_{\max }$ is the maximum value of the impact parameter at which the probability of a particular process (reaction, dissociation, or VT relaxation process) is above zero, and E is the kinetic energy of the incoming $\mathrm{O}{(}^{3}P)$ atom. In all calculations, the total number of trajectories that compose one ensemble was equal to 30,000, and the highest kinetic energy of the $\mathrm{O}{(}^{3}P)$ atom was equal to 9 eV.

On the basis of the obtained cross-sections, the rate constants $k(v,J,T)$ and their error $\Delta k(v,J,T)$ were calculated by the following formulas:

$$\begin{array}{c}k(v,J,T)=N_{\mathrm{A}}{\displaystyle \frac{g_{\mathrm{PES}}^{\mathrm{e}}}{g_{\mathrm{R}}^{\mathrm{e}}}}\sqrt{{\displaystyle \frac{8k_{\mathrm{B}}T}{\pi {\mu }_{\mathrm{A},\mathrm{BC}}}}}\underset{0}{\overset{\infty }{\int }}\sigma \left(v,J,E\right)exp\left(-{\displaystyle \frac{E}{k_{\mathrm{B}}T}}\right){\displaystyle \frac{dE}{{\left(k_{\mathrm{B}}T\right)}^2}},\\ \Delta k(v,J,T)=N_{\mathrm{A}}{\displaystyle \frac{g_{\mathrm{PES}}^{\mathrm{e}}}{g_{\mathrm{R}}^{\mathrm{e}}}}\sqrt{{\displaystyle \frac{8k_{\mathrm{B}}T}{\pi {\mu }_{\mathrm{A},\mathrm{BC}}}}}\underset{0}{\overset{\infty }{\int }}\Delta \left(v,J,E\right)exp\left(-{\displaystyle \frac{E}{k_{\mathrm{B}}T}}\right){\displaystyle \frac{dE}{{\left(k_{\mathrm{B}}T\right)}^2}}.\end{array}$$

(6)

Here, $g_{\mathrm{PES}}^{\mathrm{e}}$ and $g_{\mathrm{R}}^{\mathrm{e}}$ are the electronic statistical weights of the studied PES and reagents, respectively, and $k_{\mathrm{B}}$ is the Boltzmann constant. The integrals of Formula (6) were calculated numerically with piecewise linear interpolation of the reaction cross-section $\sigma \left(v,J,E\right)$:

$$\sigma \left(v,J,E\right)=a_{i}E+b_i,$$

(7)

where $a_i={\displaystyle \frac{{\sigma }_i-{\sigma }_{i-1}}{E_i-E_{i-1}}}$ and $b_i={\displaystyle \frac{{\sigma }_{i-1}{E}_i-{\sigma }_i{E}_{i-1}}{E_i-E_{i-1}}}$. Such an approximation of the reaction cross-section makes it possible to calculate the rate constant $k\left(T\right)$ by simple summation according to the following formula (the error of such a numerical approach is many times smaller than the statistical error of trajectory calculations [81]):

$$\begin{array}{cc}\hfill k(v,J,T)& =N_{\mathrm{A}}{\displaystyle \frac{g_{\mathrm{PES}}^{\mathrm{e}}}{g_{\mathrm{R}}^{\mathrm{e}}}}\sqrt{{\displaystyle \frac{8k_{\mathrm{B}}T}{\pi {\mu }_{\mathrm{A},\mathrm{BC}}}}}\underset{0}{\overset{\infty }{\int }}\sigma \left(v,J,E\right)exp\left(-{\displaystyle \frac{E}{k_{\mathrm{B}}T}}\right){\displaystyle \frac{dE}{{\left(k_{\mathrm{B}}T\right)}^2}}\approx \hfill \\ \hfill N_{\mathrm{A}}{\displaystyle \frac{g_{\mathrm{PES}}^{\mathrm{e}}}{g_{\mathrm{R}}^{\mathrm{e}}}}& \sqrt{{\displaystyle \frac{8}{\pi {\mu }_{\mathrm{A},\mathrm{BC}}{k}_{\mathrm{B}}T}}}\times \hfill \\ \hfill \times & \sum _{i=0}^n\left[exp\left(-{\displaystyle \frac{E}{k_{\mathrm{B}}T}}\right)\left(a_i\left(E^2+2E{k}_{\mathrm{B}}T+2{\left(k_{\mathrm{B}}T\right)}^2\right)++b_i\left(E+k_{\mathrm{B}}T\right)\right)\right]{|}_{E_{i-1}}^{E_i}.\hfill \end{array}$$

(8)

From the obtained vibrational–rotational state-to-state rate constants (6), it is possible to determine the vibrational state-resolved rate constants $k_v\left(T\right)$ (assuming that the translational and rotational degrees of freedom of the reagents and the products are in equilibrium) and the thermal equilibrium rate constant $k_{\mathrm{therm}}\left(T\right)$ as follows:

$$k_v\left(T\right)=\sum _{J=0}^{M\left(v\right)}w(v,J,T)k(v,J,T),$$

(9)

$$k_{\mathrm{therm}}\left(T\right)=\sum _{v=0}^N\sum _{J=0}^{M\left(v\right)}w(v,J,T)k(v,J,T),$$

(10)

where $w(v,J,T)$ is the normalized population of the H₂ energy level with the vibrational and rotational numbers v and J, N is the number of vibrational levels of the molecule H₂, and $M\left(J\right)$ is number of rotational levels in the v-th vibrational level.

3. Results and Discussion

For the construction of the full PES of the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system along the ground state ³$A^{″}$ surface, 6650 energies for the training set and 1089 energies for the test set were calculated according to the methodology of this work. Several neural networks of the multilayer perceptron type with two hidden layers were trained. The neural network with 45 neurons in the first hidden layer and 55 in the second was chosen because it gives the smallest RMS error of about 10 meV after training. In Figure 1 the minimum energy path for reaction O(³$P$)+H₂→H+O₂ on the approximated PES is depicted. It is important to note that the result obtained within the framework of the transition state theory energy barrier of the exchange reaction turned out to be lower ($E_{\mathrm{a}}=11.5$ kcal/mole with zero point energy (ZPE) and $E_{\mathrm{a}}\approx 11$ kcal/mole without ZPE) than the corresponding values from [84] ($E_{\mathrm{a}}=13$ kcal/mole) or from [85] ($E_{\mathrm{a}}=13.6$ kcal/mole). This difference is caused by a different level of theory in ab initio calculations. In [59], the PES was used from [84], where the mainframe level of theory was the ICCI method, whereas in this work, XMCQDPT2 was used. In

Table 1. Geometry properties of critical points (linear orientation) obtained in this work on the ³$A^{″}$ PES of the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system.

Structure	${\mathit{r}}_{\mathbf{O}\mathbf{H}}$, Å	${\mathit{r}}_{\mathbf{H}\mathbf{H}}$, Å
O−HH	2.828 (2.996) 1	0.741 (0.729) 1
OH−H	0.970 ($0.953$) 1	2.691 (4.079) 1
3TS	1.239 (1.193) 1	0.882 (0.872) 1

¹ Values in brackets are taken from [85].

, distances between atoms in critical points (linear orientation) on the target PES are presented along with corresponding values from [85]. It could be seen that these geometry properties of critical points are in good agreement with the results of [85].

To validate the robustness of the methodology described above for obtaining state-to-state rate constants and assess the overall correctness of its specific implementation, it is fruitful to compare predictions of our work with the results of QCT and direct quantum dynamic calculations performed by other authors. For the considered system $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$, the data on the cross-sections and state-specific rate constants for the processes $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v,J)\to \mathrm{H}+\mathrm{OH}$ are widely presented in the literature. So, in Figure 2, the cross-sections for the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v,J=0)\to \mathrm{H}+\mathrm{OH}$ reaction from our QCT estimates are presented in comparison with the similar QCT results from [59]. One can notice a slight overestimation of cross-sections in our modelling compared to [59]. This discrepancy may be the reflection of the properties of the two PES used in different works to perform trajectory calculations, and (as was mentioned above) the most significant difference in the PES properties could be the lower energy barrier of the exchange reaction in the frame of our simulations when compared to that in [59].

Figure 3 shows a comparison of the state-to-state rate constants found by the QCT method in this work and obtained on the basis of quantum dynamics calculations in [37]. This comparison shows a good qualitative agreement of the results of the two computational studies. Nevertheless, for reaction channels with a quantum number $v>4$ and the difference in the values of the rate constants determined by different methods can be significant, and for the temperatures $T<500$ K, they can reach an order of magnitude.

A comparison of the thermal equilibrium $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2\to \mathrm{H}+\mathrm{OH}$ reaction rate constant, calculated in this work via the QCT method and obtained in [37] on the basis of quantum dynamics calculations, is shown in Figure 4. In addition, Figure 4 shows an estimate of this rate constant obtained using the non-variational transition state theory adjusted for the tunnel effect and various Arrhenius approximations of the rate constant used in the kinetic models [86,87,88]. One can notice an expected overestimation of the rate constant by the transition state theory, while the QCT method gives closer values, although somewhat higher at $T<2500$ K, relative to the known data. Such an overestimation has the same cause as in the case of the reaction cross-section—lowering the energy barrier at the XMCQDPT2/aug-cc-pV∞Z level of theory. The discrepancy in the values of the rate constant at $T>2500$ K is explained by the different temperature ranges for which the Arrhenius approximation was obtained. Note that our estimation for the thermal equilibrium rate constant under consideration can be approximated with good accuracy in the temperature range $T=100-5000$ K by the following Arrhenius dependence (cm³/(mole · s)):

$$k_{\mathrm{exch}}^0\left(T\right)=1.21·{10}^9{T}^{1.37}exp\left(-3684/T\right).$$

(11)

For the dissociation reaction and the VT relaxation process of H₂ molecules in collision with O atoms, there are no data on rate constants in the literature (as far as the authors know). Nevertheless, it is interesting to compare the estimates of the rate constants of these processes obtained by the QCT method with the available information on the rates of dissociation and VT relaxation of hydrogen molecules in collisions with other partners. For the dissociation process, such a comparison is presented in Figure 5. It turns out that the equilibrium rate constant of the dissociation of H₂ obtained in this work significantly exceeds those with the different partners available in the literature. However, at temperatures $T>1000$ K, the dissociation rate constant of hydrogen upon collision with O atoms coincides by the order of magnitude with the dissociation rate constants in collision with H and H₂O. The thermal equilibrium rate constant of H₂ dissociation obtained in this work in the temperature range $T=100-5000$ K can be described with good accuracy using the double Arrhenius dependence (cm³/(mole s)):

$$k_{\mathrm{dis}}^0\left(T\right)=1.92·{10}^9{T}^{0.63}exp\left(-39143/T\right)+1.21·{10}^{15}{T}^{0.19}exp\left(-47567/T\right).$$

(12)

Figure 6 shows a comparison of the VT relaxation time of H₂ molecules in collisions with various partners. Note that for the processes of hydrogen relaxation on H, H₂, and O, the relaxation time was determined using the VT rate constant of the transition from the first excited vibrational level to the ground state:

Figure 6. Vibrational relaxation time of H₂ molecules on various collisional partners. Dashed lines are experimental data (Dove1974 [91]), while solid lines are calculation results (Cacciatore1989 [92], Gorse1987 [41]).

Figure 6. Vibrational relaxation time of H₂ molecules on various collisional partners. Dashed lines are experimental data (Dove1974 [91]), while solid lines are calculation results (Cacciatore1989 [92], Gorse1987 [41]).

$$p\tau ={\displaystyle \frac{k_{\mathrm{B}}T}{k_{\mathrm{V},\mathrm{T}}^{1,0}\left(1-exp\left(-{\theta }_v/T\right)\right)}}.$$

(13)

Here, $k_{\mathrm{V},\mathrm{T}}^{1,0}$ is the VT relaxation rate constant, and ${\theta }_v$ is the characteristic vibrational temperature. It can be seen from Figure 6 that according to the results of calculations performed in this work, the vibrational relaxation of H₂ proceeds faster on the O atom than on the Ar atom, but slower than on the H atoms, which is consistent with the qualitative expectations of the dependence of the relaxation time of the molecule on the mass of the collisional partner, expressed, for example, in the well-known Millikan–White formula [93]. At the same time, the results of the calculation using the QCT method predict a more effective VT relaxation of H₂ in collisions with O atoms than in collisions with molecular hydrogen. It can be assumed that this effect is due to the fact that a molecule as a collisional partner, in addition to translational degree of freedom, has rotational and vibrational freedom, unlike a single atom.

The lack of data in the literature on the probabilities of dissociation and VT relaxation of H₂ molecules in collision with O atoms is due to the fact that the exchange reaction O+H₂$\to$ H+OH is considered as a more probable outcome of the collision of H₂ and O and hence is more important for kinetic models. State-resolved rate constants for these processes calculated in line with Equation (9) are shown in Figure 7. The comparison shows that for H₂ molecules in the ground vibrational state, the exchange reaction is indeed a much more probable process than dissociation. However, for vibrationally excited H₂ molecules, the rate constants of dissociation and VT relaxation upon collision with O can be comparable or even many times higher than the corresponding exchange reaction constants. Thus, the data obtained in this paper on the state-to-state rate constants for dissociation and VT relaxation of molecular hydrogen can be an important addition for the kinetic models describing hydrogen oxidation at high translational temperatures or under essentially nonequilibrium conditions. Note that the convenient approximations of vibrationally resolved rate coefficients estimated in this study are available as the Supplementary Materials.

4. Conclusions

In this study, an implementation of the QCT method was created for atom–diatom systems using the PES approximation of this reacting system by a neural network. Using this implementation, cross-sections and state-to-state rate constants for exchange reactions, dissociation, and VT relaxation processes were calculated for the $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ system. Based on the data on the state-to-state rate constants for exchange and dissociation reactions, Arrhenius approximations of thermal equilibrium rate constants over a wide temperature range were found.

It was shown that the cross-sections and rate constants obtained in this work for the exchange reactions $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2(v,J=0)\to \mathrm{H}+\mathrm{OH}$ are in good qualitative agreement with the results of the QCT and the quantum dynamics method calculations presented in the works of other authors [37,59]. In addition, our estimation of the thermal equilibrium rate constant for the exchange reaction is in good agreement with the corresponding rate constants used in various kinetic models [86,87,88].

As far as the authors are aware, there are no reliable data on the rate constants of dissociation and VT relaxation of H₂ molecules in collision with O atoms in the literature, and the study of these processes in the frame of the QCT method is the novelty of this work. And for the first time, the full set of vibrationally resolved rate constants for metathesis reaction, VT relaxation, and dissociation in the system ³$A^{″}$ $\mathrm{O}{(}^{3}P)+{\mathrm{H}}_2$ was obtained by the QCT method and tabulated in a convenient form (see Supplementary Materials). A comparative analysis of the obtained state-to-state rate constants showed that when H₂ on the lower vibrational levels ($v<5$) collides with an O atom, the probability of its dissociation is negligible compared to the probability of an exchange reaction (however, this difference decreases with increasing temperature). But for the upper vibrational levels of the hydrogen molecule, the rate constants of dissociation and VT relaxation of H₂ on O are comparable or may even exceed the corresponding state-resolved rate constants of the exchange reaction. Thus, the rate constants of dissociation and VT relaxation of H₂ on O obtained in this work can be useful in constructing kinetic models describing hydrogen oxidation under thermal nonequilibrium conditions or at high temperatures.

The results of this work make it possible to increase the efficiency of modeling processes in gas turbine power plants using promising types of fuel, such as methane–hydrogen mixtures or pure hydrogen at high-temperature conditions inherent in hydrogen fuels.

In the future, it is planned to refine the calculation of the rate constants of energy exchange processes in this system, since, in this work, the interaction of the $\mathrm{O}{(}^{3}P)$ atom and the H₂ molecule along the ³$A^{\prime }$ and the first excited ³$A^{″}$ terms of this system was not taken into account. In addition, due to the versatility of our PES approximation method, the created computational complex can be easily adapted for different triatomic systems and other configurations of collisions, for example, for a recombination reaction A+B+C.