The article information
 Jun Li
 李军
 RingPolymer Molecular Dynamics Studies of Thermal Rate Coefficients for Reaction F+H_{2}O→HF+OH
 F+H_{2}O→HF+OH反应的高精度速率常数的计算研究
 Chinese Journal of Chemical Physics, 2019, 32(3): 313318
 化学物理学报, 2019, 32(3): 313318
 http://dx.doi.org/10.1063/16740068/cjcp1808186

Article history
 Received on: August 17, 2018
 Accepted on: September 11, 2018
The rate coefficient, usually denoted as
According to the BornOppenheimer approximation, the most accurate treatment for calculating rate coefficients is to solve the nuclear Schrodinger equation on a given potential energy surface (PES), namely, the quantum mechanical (QM) approach. Although many efforts have been devoted to this field recently [58], and extremely accurate rate coefficients have been calculated by the QM approach for reactive systems of small number of atoms [9, 10], it is still very expensive, even impractical, to compute thermal rate coefficients accurately by Boltzmann averaging over initial reactant states for systems with four or more atoms, especially for those with two or more nonhydrogen atoms, as has been demonstrated in recent timedependent wave packet studies on the kinetics for the complexforming reaction OH+CO
Recently, the ringpolymer molecular dynamics approach (RPMD) was proposed and developed [1417] to provide accurate thermal rate coefficients efficiently due to the favorable scaling laws associated with classical trajectories [3]. Indeed, it has recently emerged as an efficient and reasonably accurate approach to describe quantum effects such as ZPE, anharmonicity, and tunneling, etc., for calculating thermal rate coefficients. Besides, as outlined in the recent review of RPMD rate theory, several additional desirable features are expected [3]: (ⅰ) at the high temperature limit, the ring polymer collapses to a single bead, and the resulting RPMD rate coefficient becomes exact; (ⅱ) it offers an upper bound on the RPMD rate coefficient; and (ⅲ) the RPMD rate coefficient is independent of the definition of the dividing surface, a very promising feature as an optimal dividing surface is often very difficult to be defined in the multidimensional space for polyatomic reactions. Therefore, for many bimolecular reaction systems, the RPMD rate theory has been shown to be able to provide accurate rate coefficients, compared to those calculated by exact QM approach or available experiment [3, 18, 19]. As a consequence, the results calculated by the RPMD rate theory can be considered a benchmark to provide a reliable assessment of the available experiment, or theoretical results calculated by other theories [18, 19], in particular for complicated reactive systems when rigorous QM calculation is not available.
In this work, the RPMD rate theory is used to determine the rate coefficients at temperatures ranging from 200 K to 400 K for the title reaction F+H
The most important factor in predicting kinetics is the barrier height due to its exponential contribution to the rate coefficients. Theoretically, due to the complicated electronic structure, the classical barrier height of the title reaction system ranges in 8.312.6 kcal/mol [23], 1.5 kcal/mol [25], 7.0 kcal/mol [26], 2.5 kcal/mol [27], 1.93 kcal/mol [28], at various electronic structure levels. The benchmark values are 1.534 kcal/mol by the focal point analysis (FPA) method, and 1.622 kcal/mol by the high accuracy extrapolated ab initio thermochemistry (HEAT) method [29]. In addition to the low reactantlike barriers, relatively deep prereaction complex wells [30] are found to affect the reaction dynamics and kinetics to some extent, in particular, at low collision energies or low temperatures [29, 31].
Although much has been done on the dynamics of the title system theoretically by Guo and coworkers [28, 3140], full dimensional rigorous QM approach has not been used to compute the corresponding kinetics due to the expensive calculation cost. In this work, we use the RPMD rate theory, which can provide accurate rate coefficients with relative ease, for computing thermal rate coefficients of the title reaction system, and for assessing available experimental or theoretical results.
Ⅱ. THEORY AND CALCULATION DETAILSIn 2012, the first fulldimensional global PES for the ground electronic state of the title reaction system was developed with more than 30, 000 points calculated by the multireference configuration interaction (MRCI) method [28]. In order to reproduce the correct barrier height according to benchmark calculations using the focal point analysis (FPA) method, an external correlation scaling method was employed [29], and the resulting PES is able to produce the observed reactive cross sections by QCT [32] and rate coefficients by the semiclassical transition state theory and a twodimensional master equation (SCTST/2DME) technique [29]. Besides, spinorbit (SO) corrections were determined at the complete active space selfconsistent field level [33]. Finally, the PES was fitted by using the permutation invariant polynomial neural network approach [4143] with a total root mean square error (RMSE) of 6.8 meV, or 0.157 kcal/mol. On the PES, the barrier height is 1.798 kcal/mol, and the reaction energy is 17.827 kcal/mol, which is very close to the value of 17.63 kcal/mol obtained from Ruscic's Active Thermochemical Tables (ATcT) [29].
The RPMDrate code [44] is then used to compute the rate coefficients of the title reaction. The corresponding analysis of the RPMD theory for computing bimolecular rate coefficients in the gas phase and various applications have been summarized in recent review [3]. Briefly, making use of the BennettChandler factorization [45], the RPMD rate coefficient depends on two factors,
$ \begin{eqnarray} {k_{{\rm{RPMD}}}} = {k_{{\rm{QTST}}}}\left( {T;{\xi ^ {\neq} }} \right)\kappa \left( {t \to \infty ;{\xi ^ {\neq} }} \right) \end{eqnarray} $  (1) 
$ \begin{eqnarray} {k_{{\rm{QTST}}}}(T;{\xi ^ {\neq} }) = \frac{{4\pi R_\infty ^2}}{{\sqrt {2\pi \beta {\mu _{\rm{R}}}} }}{\textrm{e}^{  \beta \left[ {W\left( {{\xi ^ {\neq} }} \right)  W\left( {\xi = 0} \right)} \right]}} \end{eqnarray} $  (2) 
where
The inclusion of the second factor
Within RPMD rate theory, it reduces to the classical limit when only one bead is used. In this limit, the static and dynamical components become identical to the classical TST rate coefficient and the classical transmission coefficient, respectively. Hence, these quantities establish the limit to which the quantum effects such as tunneling and ZPE can be captured with more beads. The following equation has been suggested to estimate the minimal number of beads needed to recover entire quantum effects [46]:
$ \begin{eqnarray} {n_{\min }} = \beta \hbar {\omega _{\max }} \end{eqnarray} $  (3) 
where
For calculating the PMF profiles, the reaction coordinate interval [0.06, 1.06] is divided into windows with an equal size d
Finally, the RPMD rate coefficients are multiplied by the following electronic factor to account for the spinorbit splitting of F(
$ \begin{eqnarray} \frac{{Q_{{\rm{elec}}}^{{\rm{TS}}}}}{{Q_{{\rm{elec}}}^{{\rm{reactants}}}}} = \frac{2}{{4 + 2\exp \left( {  \beta \Delta E} \right)}} \end{eqnarray} $  (4) 
For each temperature, 200, 300, and 400 K, we firstly perform the RPMD calculations with one bead, which provides the corresponding classical limit. Then the number of beads is increased until convergence. As shown below, most quantum effects can be captured by the first several beads. The calculation results, the PMFs along the reaction coordinate
As shown clearly, all the PMFs at 200, 300, and 400 K possess similar features. They are flat at the reactant asymptote (
Another important factor, the transmission coefficient
The final RPMD rate coefficients are plotted as a function of
Previously, the thermal rate coefficients for the title reaction were also determined by the QCT method [29]. The results are included in FIG. 4 for comparison. It can be seen that QCT underestimates rate coefficients of the title reaction within 200400 K, and the underestimation becomes more significant when the temperature is decreased. This is apparently due to the fact that the QCT method lacks of quantum effects, such as tunneling and ZPE effects. As has been reported in our previous work, the SCTST calculated tunneling corrections are 3.5 at 200 K, 2.0 at 300 K, and 1.5 at 400 K, respectively. With these tunneling corrections, the QCT calculated rate coefficients are quite close to experiment (within 20%) [29].
Ⅳ. CONCLUSIONIn this work, thermal rate coefficients at 200, 300, and 400 K for an important reaction F+H
This work was supported by the National Natural Science Foundation of China (No.21573027).
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