Ring-Polymer Molecular Dynamics for the Prediction of Low

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Letter

Ring-Polymer Molecular Dynamics for the Prediction of LowTemperature Rates: An Investigation of the C(D) + H Reaction 1

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Kevin M Hickson, Jean-Christophe Loison, Hua Guo, and Yury V. Suleimanov J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.5b02060 • Publication Date (Web): 06 Oct 2015 Downloaded from http://pubs.acs.org on October 7, 2015

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The Journal of Physical Chemistry Letters

Ring-Polymer Molecular Dynamics for the Prediction of Low-Temperature Rates: An Investigation of the C(1D) + H2 Reaction Kevin M. Hickson,∗,†,§ Jean-Christophe Loison,†,§ Hua Guo,‡ and Yury V. Suleimanov∗,¶,k †Université de Bordeaux, Institut des Sciences Moléculaires, F-33400 Talence, France ‡Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, USA ¶Computation-based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus §CNRS, Institut des Sciences Moléculaires, F-33400 Talence, France kDepartment of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, Massachusetts 02139, United States E-mail: [email protected]; [email protected]

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Quantum chemical theory is approaching a predictive level of accuracy that is transforming our view of molecular structure and spectroscopy. Similar progress is required from quantum dynamical methods to elucidate chemical reactivity. The description of purely quantum mechanical (QM) effects of nuclear motion including zero point energy (ZPE) and tunnelling are key concerns for systems involving light atoms at low temperature, 1 particularly for triatomic insertion reactions involving H2 . These processes play important roles in Earth’s atmospheric chemistry 2 and in interstellar media, 3 being characterized by the (near) barrierless formation of long-lived complexes in deep potential wells. While these triatomic reactions are nowadays fully amenable to exact QM treatments, the large number of quantum states supported by such strongly-bound intermediates makes rigorous QM treatments particularly onerous. 4 Perhaps more importantly, investigations of larger barrierless complex-forming reactions in general lag behind those of direct reactions over activation barriers. When multiple potential energy surfaces (PESs) are involved, such studies become prohibitively expensive and alternative treatments that accurately reproduce the dynamics are highly desirable. Numerous transition-state theory (TST) methods have been developed, based on static (short-time limit) approximations to the classical real-time correlation functions describing chemical dynamics. 5 Such theories are based on the assumption that all modes are equivalent in promoting the reaction, which is often not the case due to incomplete randomization in the activated complex. Several ad hoc approximations are available to incorporate quantum tunnelling and ZPE effects into TST. While being attractive due to low computational costs, some intractable issues adversely affect TST-based methods. The main problem is an exponential sensitivity of the short-time limit of correlation functions to the choice of the dividing surface separating the reagent and product sides (the transition-state dividing surface). At low temperatures, this limits the accuracy of TST-based methods due to the multidimensional nature of quantum tunnelling even for triatomic systems, 6 while at high temperatures, recrossing (the ratio between long-time and short-time limits of the correla-

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tion functions) can be significant for complex systems, especially for heavy-light-heavy mass combinations. 7,8 These are important dynamical effects that are difficult to capture with a pure statistical approach such as TST. It is especially challenging to select the dividing surface for reactions with no or low barriers, as recrossing is prevalent near the complex-forming region. Special ad hoc approaches, such as incorporating the variable reaction coordinate model 9–11 have been conceived to tackle such problems. However, there is still no unambiguous way to treat real-time recrossing as TST is an inherently static approach. Moreover, anharmonicity in the transition-state region is difficult to treat within the TST framework leading to potentially unsatisfactory results. 8 The quasi-classical trajectory (QCT) approach represents an alternative to TST; its most advanced implementation, Gaussian binning, 12,13 estimates real-time recrossing and ZPE of reagents and products. However, QCT cannot control changes in the ZPE along the reaction coordinate. This is problematic for barrierless processes as they may violate the ZPE when exiting the potential well. 14,15 Furthermore, QCT cannot account for other quantum mechanical effects such as tunnelling. A novel alternative to these techniques is the Ring-Polymer Molecular Dynamics (RPMD) method; a full-dimensional dynamics approximation immune to many of the shortcomings of TST and QCT methods. RPMD is based on the imaginary-time path integral formalism, exploiting the classical isomorphism between the quantum system and the series of its classical copies placed in a necklace forming a ring structure with a harmonic interaction between neighbouring beads. 16 The classical evolution of this ring polymer in an extended phase-space approximates the real-time dynamics in the original quantum system by computing approximate Kubo-transformed correlation functions for many dynamical properties. The classical isomorphism provides exact mapping to various static equilibrium properties, but for real-time dynamics such approximations must also be considered as ad hoc, though its connection to the exact quantum Kubo-transformed time-correlation function via a Boltzmann conserving ”Matsubara dynamics” has been recently demonstrated (with ex-

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plicit terms which are discarded). 17 RPMD provides reliable estimates for the correlation functions responsible for thermal rate coefficients in systems with various complex reaction paths and in different temperature regimes: 6–8,14,18–31 RPMD rate coefficients are (i) exact in the high-temperature limit (ii) reliable at intermediate temperatures, (iii) more accurate than other approximate methods in the deep quantum tunnelling regime and close to the exact quantum results, (iv) able to capture ZPE effects perfectly. Unlike TST and QCT, RPMD treats the quantum Boltzmann operator accurately and automatically captures ZPE effects correctly along the entire reaction pathway; including throughout the transition-state and/or complex-formation regions. One of the main strengths of RPMD is also explained by its connection in certain situations with classical TST, instanton theory and quantum mechanical versions of TST. 16,32 RPMD is distinct from TST approaches as it is independent of the position of the transition-state dividing surface, which leads to undesirable adjustable empirical parameters in the TST case. 16 This feature is important for reactions that have a poorly defined transition state and/or if active recrossing dynamics is expected, e.g. for barrierless reactions. As RPMD is basically a classical molecular dynamics method in the extended ring-polymer phase-space, it scales favourably with the size of the system and could potentially be used to calculate the rate coefficients for systems containing many hundreds of atoms. 33 It was recently discovered that RPMD could be used to describe complex-forming reactions. 25,26 It has been employed to elucidate the kinetics of the exothermic N(2 D) + H2 and O(1 D) + H2 reactions above 270 K 25 and the kinetics of the almost thermoneutral C(1 D) + H2 and S(1 D) + H2 reactions above 200 K. 26 The results were in excellent agreement with previous QM calculations, showing the great potential of the RPMD method for determining accurate reaction rate coefficients for a wide range of barrierless complex-forming reactions. So far, RPMD theory has been mostly validated near room temperature or higher. In the present work, the RPMD approach is applied to elucidate the dynamics of the barrierless

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◦

complex-forming reaction (∆r H0 = -24.4 kJ mol−1 ) +

C(1 D) + H2 (X 1 Σg ) → CH(X 2 Π) + H(2 S),

(1)

to temperatures below 100 K for the first time, where subtle QM effects are expected to become increasingly apparent. The interaction of C(1 D) with H2 generates five singlet potential energy surfaces (PESs), two of which correlate with ground state products. While the reactivity over the 1 A′ PES has been theoretically investigated using QM and QCT methods on several occasions, 34–37 leading to rates which mostly underestimate the measured room temperature values ((2.0 – 3.7) × 10−10 cm3 s−1 ), 38–40 the role of the 1 A′′ surface is less well constrained. 41–43 QM calculations neglecting non-adiabatic interactions between these surfaces 42 yield rate coefficients that are more consistent with measurements than those including Renner-Teller and Coriolis coupling terms. 43 Indeed, recent RPMD calculations of reaction (1) over the 1 A′ PES alone also seem to underpredict the experimental rates above 200 K. 26 The present work builds on this preliminary RPMD study by explicitly including the reactivity of the 1 A′′ surface while extending the calculations down to 50 K. The RPMD calculations reported in this work used the RPMDrate code developed by one of us (YVS). 22 All the details of the RPMD methodology are well-documented in the RPMDrate manual. 22 In brief, the RPMD rate coefficient is calculated using the BennettChandler factorization 44 as a product of two factors – the static contribution, which is given by the centroid-density quantum transition-state theory (QTST) rate coefficient, 45–47 kQTST (T ; ξ ‡ ), and the long-time limit of a time-dependent ring-polymer transmission coefficient, κ(t → tp ), which accounts for recrossing of the dividing surface placed at ξ ‡ when t → tp , tp is a ”plateau” time (which is usually high for complex-forming reactions). 25,26 The second factor counterbalances kQTST (T ; ξ ‡ ), ensuring the independence of the RPMD rate coefficient kRPMD of the choice of the dividing surface. Here, we used the CH2 multi-reference configuration-interaction PESs of Ba˜ nares et al. 48 for the ground state (1 A′ ) and of Bussery-

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Table 1: The results of RPMD calculations for the C(1 D) + H2 reaction over the 1 A′ and 1 A′′ PESs at 50, 77, 128 and 300 K: centroid-density rate coefficient (kQTST ), ring polymer transmission coefficient (κ) and final RPMD rate coefficients (kRPMD ). All rate coefficients are corrected by electronic partition function Qel = 5 and are given in cm3 s−1 . T (K)

kQTST

300 128 77 50

3.4 2.6 2.1 1.6

× × × ×

300 128 77 50

5.4 7.2 7.4 7.0

× × × ×

κplateau PES 1 A′ 10−10 0.47 10−10 0.52 10−10 0.58 10−10 0.65 PES 1 A′′ 10−10 0.23 10−10 0.23 10−10 0.25 10−10 0.29

kRPMD 1.6 1.4 1.2 1.0

× × × ×

10−10 10−10 10−10 10−10

1.2 1.7 1.8 2.0

× × × ×

10−10 10−10 10−10 10−10

Honvault et al. 49 for the first excited state (1 A′′ ) which were obtained from ab initio data using similar fitting methodologies. Both PESs exhibit barrierless minimum energy paths in the entrance channel (perpendicularly constrained approach and bent approach of C towards H2 around 60◦ for 1 A′ and 1 A′′ , respectively) and both are of insertion type with deep wells (4.32 and 3.46 eV relative to the reactants for 1 A′ and 1 A′′ , respectively). Although, there are other ab initio-based PESs, 50,51 numerous dynamical studies of the title reaction have been carried out using these two surfaces, for the most part for the 1 A′ state. For both PESs we used parameters similar to those used in our previous studies of insertion reactions, 25,26 (see Table S1 in the Supporting Information (SI)). They were checked to be sufficient to converge RPMD rate coefficients to within a statistical error of ∼ 10 %. The potential of mean force along the reaction coordinate, ξ, and the time-dependent transmission coefficients κ(t) can be also found in the SI file (Figures S1 and S2, respectively). The results of the RPMD simulations are summarized in Table 1. To check the validity of the RPMD method, the C(1 D) + H2 reaction was studied experimentally over the range 50 - 296 K using the continuous supersonic flow method, 52 by 7



4.0

intensity / arb. units

3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 -50

0

50

100

150

200

250

300

350

time / microseconds

Figure 1: Exemplary temporal profiles of the H fluorescence signal recorded at 50 K. (Blue open circles) [H2 ] = 2.4 × 1014 cm−3 . (Red open circles) [H2 ] = 1.3 × 1013 cm−3 . following the kinetics of H-atom production. This technique has been described elsewhere 53,54 so only experimental details specific to this investigation are described (see the Experimental Methods section). Representative temporal profiles of the atomic hydrogen signal are shown in Figure 1. 80 70

-1

60

3

k1st / 10 s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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50 40 30 20 10 0

0

2

4

6

8

10 12 14 16 18 20 22 24 13

-3

[H2] / 10 cm

Figure 2: Pseudo-first-order rate coefficients for reaction (1) as a function of [H2 ] at 50 K. A weighted linear least-squares fit yields k1 . The error bars on the ordinate reflect the statistical uncertainties (1σ) obtained by fitting to H atom VUV LIF profiles such as those shown in Figure 1. Values of k1′ obtained from kinetic fits were plotted against [H2 ] concentration yielding the second-order rate k1 at a specified temperature from the slope, (Figure 2 and Figures

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S3-S5). The measured rate coefficients at four temperatures between 50 and 296 K are listed in Table 2 and are shown in Figure 3. Table 2: Measured rate coefficients, k1 , for the C(1 D) + H2 reaction. T (K)

[H2 ]/1014 cm−3

50±1 75±2 127±2 296

0.1-2.4 0.1-1.4 0.2-1.7 0.1-1.6

Flow density/ 1016 cm−3 25.9 14.7 12.6 16.3

Na

k1 /10−10 cm3 s−1

47 36 105 43

2.65± 2.86± 2.92± 3.20±

0.28b 0.30 0.32 0.33

a

b

Number of individual measurements. The uncertainties are quoted at the level of one standard deviation from the mean in addition to an estimated 10 % systematic uncertainty. The calculated RPMD rate coefficients for reaction (1) obtained at four temperatures

(50, 77 K, 128 K and 300 K) are also included in Figure 3 alongside the previous RPMD and QM results and earlier room temperature measurements. The measured rate coefficients are in excellent agreement with the RPMD ones, predicting a weak variation of the rate coefficient over the 50-300 K range. Theoretically, an opposite temperature effect was observed for the two surfaces: the RPMD rate for the 1 A′ surface increases with temperature, from 1.0 × 10−10 cm3 s−1 at 50 K to 1.6 × 10−10 cm3 s−1 at 300 K, while the 1 A′′ rate decreases with temperature, from 2.0 × 10−10 cm3 s−1 at 50 K to 1.2 × 10−10 cm3 s−1 (see Table 1). Consequently, the contribution of the 1 A′′ surface to the overall reactivity increases from 43 % at 300 K to 67 % at 50 K. The net effect is a very slight increase in the total rate. This is in line with previous time-dependent QM calculations performed using the real wavepacket formalism over the same two surfaces neglecting non-adiabatic couplings 42 which also demonstrated the opposite temperature effect and attributed it to different reaction mechanisms on the two surfaces due to different minimum energy paths (indirect perpendicular for 1 A′ and direct sideways insertion for 1 A′′ ). Nevertheless, the RPMD rate coefficients are closer than the previous QM ones to the present measurements, particularly at room temperature. The deviation between theoretical results (12-18 %) might be due to approximations made 9



in the QM calculations, such as a limited number of partial waves, the interpolation scheme as well as convergence issues at low collision energies. It could also be partly due to the statistical sampling uncertainties in the RPMD calculations. 4.0 A

3.5 3.0 2.5 2.0

-1

0.5

-10

cm s

1.0

3

1.5

2

katom+H / 10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.0 2.5

0

50 100 150 200 250 300 350 400 B

2.0 1.5 1.0 0.5 0.0

0

50 100 150 200 250 300 350 400

T/K Figure 3: Rate coefficients for atom + H2 reactions as a function of temperature. The C(1 D) + H2 reaction (upper panel A). Experimental values: (Pink open diamond) Fisher et al.; 39 (Green open triangle) Husain and Kirsch; 38 (Black open square) Sato et al.; 40 (Blue solid circles) This work. Theoretical values: (Blue dotted line) Defazio et al.; 42 (Blue dashed line, calculations on the 1 A′ surface) Suleimanov et al.; 26 (Blue open circles) This work. The S(1 D) + H2 reaction (lower panel B). Experimental values: (Red solid squares) Berteloite et al. 55 Theoretical values: (Red solid line) Suleimanov et al. 26 ; (Red dashed line) Berteloite et al. 55 While previous RPMD 26 and QM 34–37 calculations employing the 1 A′ surface alone lead to rate coefficients which are substantially lower than the current measurements, (see Figure 3 and Table 2), they are nonetheless in agreement with the 300 K experiments of Sato et al. 40 The discrepancy between our measurements and those of Sato et al. could have several explanations. The most likely one arises from the fitting procedure used by Sato et al. to 10


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extract rate coefficients from CH profiles. Several key secondary reaction rates required for these fits were incorrectly estimated, potentially leading to errors in the derived reaction rate. The current measurements are not hindered by similar difficulties as H-atoms are unreactive in our experiments, being mostly lost by diffusion. Moreover, secondary H-atom production from product CH reactions with H2 and CBr4 should not occur given the large endothermicity of the former process and the relative strengths of C-Br and C-H bonds for the latter one. Indeed, earlier room temperature kinetic studies of reaction (1) 38,39 are in good agreement with the present results. One other possible source of experimental error arises from a fixed 3 : 1 ratio for the ortho:para forms of H2 at all temperatures, due to inefficient gas-phase spin conversion. In contrast, the RPMD simulations inherently assume that reagent H2 obeys Boltzmann statistics, yielding an ortho:para ratio close to 1 at 77 K. Nevertheless, previous QM calculations indicate that these spin isomers react at similar rates with C(1 D) over both the 1 A′ 56 and 1

A′′ 42 surfaces, leading to negligibly small discrepancies. It is interesting to compare the present results with those obtained for the S(1 D) +

H2 reaction (Figure 3). This process occurs over a single barrierless 1 A′ PES adiabatically connecting reagents with products (the 1 A′′ PES is characterized by a barrier in the favorable approach geometry). In contrast to the RPMD rates for C(1 D) + H2 computed for a single 1

A′ surface, the rates calculated for the S(1 D) + H2 reaction 26 are only slightly smaller

than the measured ones 55 in the overlapping temperature range. The QM calculations of Berteloite et al. 55 confirm this result above 10 K. Such differences could easily arise from deficiencies in the PES or from neglecting small contributions from other low-lying surfaces. In any event, these results indicate that the RPMD method can accurately reproduce the dynamical behavior of other systems over a wide temperature range. While the RPMD rates for reaction (1) reproduce the experimental results, extending these calculations to even lower temperatures might not provide the same agreement. At 50 K the difference between the RPMD rate coefficient and the experimental one is 12

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%. Figure 3 shows that extending such comparison to lower temperatures will lead to no overlap between the RPMD and experimental results. The RPMD rate coefficient exhibits a slightly inverse temperature dependence, which accords with a typical capture regime, whereas the experiment demonstrates a positive temperature dependence. This discrepancy might be explained, firstly, by the long-lived 1 A′ resonances which were found to decrease the corresponding QM rate and thus the overall thermal rate at lower temperature 42 and which are not captured by RPMD. 25 Another important point is possible deficiencies of the present PESs in the long range region, 34,41,55 though both surfaces are isotropic and free of spurious features in the extrapolated asymptotic regions 48,49 In a regime of very low temperatures, where the total energy is close to the reactant asymptote, a precise treatment of the long-range interactions is highly desired. 55 Recently, some semiempirical attempts have been made to take more accurate account of long-range forces for the 1 A′ PES. 50,51 However, these surfaces are less well validated - different topologies were obtained which resulted in a noticeable spread of rate values. 37,57 Perhaps more importantly, a slight negative temperature dependence was observed, as opposed to the presently employed surface, which suggests that the agreement with the present experimental results would be worse. Finally, the most obvious explanation is the adiabatic limit used in the present simulations. Earlier work has shown that nonadiabatic Renner-Teller A′ –A′′ coupling for this system suppresses its reactivity at very low energies. 43 Consequently, it would be preferable to go beyond the Born-Oppenheimer approximation at lower temperatures. There are ongoing attempts to extend RPMD to nonadiabatic processes 58–60 suggesting that such calculations might be feasible in the near future. Clearly, further theory development is required in order to extend the present study to much lower temperatures. Despite these flaws, the present results demonstrate that through its unique ability to treat quantum effects, RPMD is a viable alternative to exact QM calculations to determine reaction rates at low temperature for a range of polyatomic systems, especially when multiple PESs are involved. This and other recent RPMD calculations 6–8,14,20–30,61 firmly establish the

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validity of this method, which can be expected to find wide applications in both activated and complex-forming reactions of larger sizes.

Experimental Methods All experiments were performed using a continuous supersonic flow apparatus over the range 50 - 296 K. Excess concentrations of normal-H2 (with a population ratio 3 : 1 for ortho:para spin isomers at all temperatures) were introduced into the main flow upstream of the Laval nozzle. C(1 D) atoms were generated by pulsed multiphoton dissociation of CBr4 at 266 nm. Previous work 54 showed that C(1 D) atoms are generated at the 10-15% level by this process. CBr4 was carried into the reactor by passing a small Ar flow over solid CBr4 . An upper limit of 2 × 1013 molecule cm−3 was estimated for the CBr4 concentration from its saturated vapour pressure. While C(3 P) atoms are the major photolysis product, the C(3 P) + H2 → CH + H reaction is endothermic by 95.4 kJ/mol so that it can be neglected at room temperature and below. H(2 S) atoms were detected through on-resonance VUV laser induced fluorescence (LIF) using the Lyman-α transition at 121.567 nm. VUV LIF signals were recorded as a function of delay time between photolysis and probe lasers. The evolution of the H atom signals (I H ) shown in Figure 1 is described by the expression IH = A{exp(−kL(H) t) − exp(−k1′ t)}

(2)

where k1′ = k1 [H2 ] + kL(C) , with kL(C) representing first-order losses of C(1 D), [H2 ] is the H2 concentration and t is time. The first term in expression (2) represents H-atom losses with a first-order rate kL(H) . Test experiments performed to evaluate non-reactive quenching of C(1 D) by H2 are outlined in SI.

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Supporting Information Available Supplemental experimental methodology. Supplemental figures S1-S5. Supplemental tables S1-S2. This material is available free of charge via the Internet at http://pubs.acs.org/.

Acknowledgement KMH and JCL thank the CNRS programs PCMI and PNP for their support. HG thanks the US DOE for support (DE-FG02-05ER15694). YVS thanks the European Regional Development Fund and the Republic of Cyprus for support through the Research Promotion Foundation (Project Cy-Tera NEA YΠO∆OMH/ΣTPATH/0308/31). We also thank Pascal Honvault for sending us the PESs and for several useful discussions.

References (1) Daranlot, J.; Jorfi, M.; Xie, C.; Bergeat, A.; Costes, M.; Caubet, P.; Xie, D.; Guo, H.; Honvault, P.; Hickson, K. M. Revealing Atom-Radical Reactivity at Low Temperature Through the N + OH Reaction. Science 2011, 334, 1538–1541. (2) Davidson, J. A.; Schiff, H. I.; Streit, G. E.; McAfee, J. R.; Schmeltekopf, A. L.; Howard, C. J. Temperature Dependence of O(1 D) Rate Constants for Reactions with N2 O, H2 , CH4 , HCl, and NH3 . J. Chem. Phys. 1977, 67, 5021–5025. (3) Tizniti, M.; Le Picard, S. D.; Lique, F.; Berteloite, C.; Canosa, A.; Alexander, M. H.; Sims, I. R. The Rate of the F + H2 Reaction at Very Low Temperatures. Nat. Chem. 2014, 6, 141–145. (4) Guo, H. Quantum Dynamics of Complex-Forming Bimolecular Reactions. Int. Rev. Phys. Chem. 2012, 31, 1–68.

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Dynamics be Used to Calculate Thermal Reaction Rates? J. Chem. Phys. 2015, 143, 074107.

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