Six-dimensional and seven-dimensional quantum dynamics study of the OH + CH4 → H2O + CH3 reaction Hongwei Song, Soo-Ying Lee, Minghui Yang, and Yunpeng Lu Citation: The Journal of Chemical Physics 139, 154310 (2013); doi: 10.1063/1.4825100 View online: http://dx.doi.org/10.1063/1.4825100 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/15?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A seven-degree-of-freedom, time-dependent quantum dynamics study on the energy efficiency in surmounting the central energy barrier of the OH + CH3 → O + CH4 reaction J. Chem. Phys. 142, 164303 (2015); 10.1063/1.4918981 Quantum dynamics study on the CHIPR potential energy surface for the hydroperoxyl radical: The reactions O + OH⇋O2 + H J. Chem. Phys. 142, 014309 (2015); 10.1063/1.4905292 Quasiclassical trajectory study of the effect of antisymmetric stretch mode excitation on the O(3P) + CH4(ν3 = 1) → OH + CH3 reaction on an analytical potential energy surface. Comparison with experiment J. Chem. Phys. 141, 094307 (2014); 10.1063/1.4893988 Effects of reactant rotation on the dynamics of the OH + CH4 → H2O + CH3 reaction: A six-dimensional study J. Chem. Phys. 140, 084307 (2014); 10.1063/1.4866426 Communication: Full dimensional quantum rate coefficients and kinetic isotope effects from ring polymer molecular dynamics for a seven-atom reaction OH + CH4 → CH3 + H2O J. Chem. Phys. 138, 221103 (2013); 10.1063/1.4811329

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THE JOURNAL OF CHEMICAL PHYSICS 139, 154310 (2013)

Six-dimensional and seven-dimensional quantum dynamics study of the OH + CH4 → H2 O + CH3 reaction Hongwei Song,1 Soo-Ying Lee,1 Minghui Yang,2,a) and Yunpeng Lu1,a) 1

Division of Chemistry and Biological Chemistry, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore 2 Key Laboratory of Magnetic Resonance in Biological Systems, State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Centre for Magnetic Resonance, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, People’s Republic of China

(Received 29 April 2013; accepted 30 September 2013; published online 17 October 2013) The reaction dynamics of hydroxyl radical with methane has been investigated using time-dependent wave packet approach within reduced six- and seven-dimensional models. Initial state-selected total reaction probabilities and integral cross sections for the hydrogen abstraction reaction have been computed on the empirical potential energy surface developed by Espinosa-García et al. [J. Chem. Phys. 112, 5731 (2000)]. Excitations of the CH stretching mode and/or the CH3 umbrella mode enhance the reaction. They are, however, both less efficient than translational energy in promoting the reaction, at least at low collision energies. Also, we studied the accuracy of two approximations: centrifugal sudden (CS) and J-shifting (JS), in the calculations of the integral cross sections by a comparison to coupled-channel (CC) calculations. The integral cross sections obtained indicated that the CS approximation works well over the whole energy range studied, and the JS approximation gives accurate cross sections at low collision energies, while noticeably overestimates them at relatively high collision energies. In addition, the OH radical acts as a good spectator as it has a negligible effect on the reaction. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4825100] I. INTRODUCTION

Hydrogen abstraction between hydroxyl radical and methane is a key step in the combustion of hydrocarbon fuels.1, 2 Methane reacts directly with OH radical to produce H2 O and CH3 , and CH3 is subsequently oxidized to produce CO and CO2 through chain propagation steps.3 In the troposphere, this reaction is the major process for the removal of methane,4 a greenhouse gas with relatively high abundance. The process accounts for 90% ± 5% of the total sink of methane.5 Numerous measurements on the rate constants for the OH + CH4 → H2 O + CH3 reaction have been reported over a wide range of temperatures.4, 6–8 The temperature dependence of the rate constant was shown to be of non-Arrhenius type, and an approximate activation energy of 3.6 kcal/mol was deduced. These measurements also indicated that vibrational activation of OH is effective in promoting reaction with CH4 , contrary to a OH role as a spectator in the early dynamics calculations of OH + CH4 .9 Lester et al.10, 11 investigated the vibrational spectroscopy and decay dynamics of CH4 –OH complexes through an infrared-ultraviolet double-resonance technique. Recently, Liu et al.12–14 studied the product correlated state-to-state dynamics for the OH + CD4 reaction using the three-dimensional ion-velocity-imaging technique in combination with (2 + 1) resonance-enhanced multiphoton ionization (REMPI) detection of the CD3 product.15 Most theoretical studies were concerned about the transition state, classical activation barrier, rate constants, and kia) Electronic addresses: [email protected] and [email protected]

0021-9606/2013/139(15)/154310/7/$30.00

netic isotope effect of the reaction.16–27 There are, however, relatively few dynamics calculations on the OH + CH4 reaction. The high dimensionality (15 dimensions) makes it a big challenge to build a global potential energy surface (PES). Till now, the only available global PES was reported by Espinosa-García and Corchado.28 Based on the PES, Yu29 has performed five-dimensional quantum scattering calculations on the OH + CH4 reaction. In pioneering studies, Nyman et al.9, 30, 31 investigated the mode selectivity of this reaction using an empirical model potential and including two degrees of freedom in the rotating line approximation and three degrees of freedom in the rotating bond approximation. The cumulative reaction probabilities from these studies showed that the reaction occurs via a direct mechanism and vibrational excitations of CH4 can enhance substantially the reaction while exciting OH stretching has a negligible effect. Actually, for the systems containing more than three atoms, the majority of quantum mechanical dynamics calculations performed nowadays are usually made under some approximations.32–42 The approximations are typically the Jshifting43, 44 (JS) or centrifugal sudden45–47 (CS) approximations, which provide tremendous conceptual simplicity and computational advantage over the coupled-channel (CC) theory. They are both based on sound physical ideas, but their applicability is not always straightforward. It is thus important to investigate the validity of the these approximations on different polyatomic systems.38 Although the accuracy of the JS or CS approximation has been tested in some tetraatomic reactions,32, 33, 35–38 their validity has never been studied in pentaatmoic or larger polyatomic systems.

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Motivated by the attempt to understand the reaction dynamics using more accurate models and to assess the accuracy of the aforementioned approximations, we apply the reduceddimensional methodology to study the reaction with higher dimensionality. In fact, reduced-dimensional wave packet approach including important modes of motion has been successfully applied to study polyatomic reaction dynamics, such as H + CH4 ,48–51 O + CH4 ,52 and OH + CH4 .9, 29–31 In this work, we employ the model Hamiltonian developed by Clary et al.,53 in which the main assumption is that nonreactive CH3 group maintains C3v symmetry, with simplification by the removal of CH3 internal rotation leading to a seven-dimensional (7D) model. This model is expected to be reasonable as ab initio calculations showed that the methyl group has a very small barrier to internal rotation.19 For clarity, we use YCZ3 below to denote CH4 . Y refers to the reactive hydrogen atom and Z refers to the non-reactive hydrogen atom. This model can be further simplified by fixing the bond length of OH, which brings about a six-dimensional (6D) model. The aims of the calculations here are: (a) to study the mode selectivity in the hydrogen abstraction reaction within the current models; (b) to assess the accuracy of the CS and JS approximations in the dynamics simulation of the OH + CH4 reaction; and (c) to clarify to what extent the OH radical can be treated as a spectator in the reaction. This paper is organized as follows. Section II outlines the theoretical methodology of the initial state selected wave packet (ISSWP) method and the results and discussion are presented in Sec. III. We conclude in Sec. IV. II. THEORY

The ISSWP method employed in this work is similar to that in dealing with tetraatomic systems and has been well documented in Refs. 54 and 55. Here, we only briefly outline the theoretical aspects of the time-dependent wave packet approach. The reactant Jacobi coordinates used in the calculation are shown in Fig. 1. R is the vector from the center of mass (COM) of OH to the COM of Y-CZ3 , r1 is the vector from

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O to H, r2 is the vector from the COM of CZ3 to Y, and ru is the bond length of CZ in the CZ3 group and is fixed at its equilibrium value of 2.067 a0 . Neglecting the CZ3 internal rotation, the C3v symmetry axis of CZ3 group, i.e., vector s, coincides with the vector r2 . χ is the angle between a CH bond and the C3v symmetry axis to describe the umbrella motion of CZ3 group. θ 1 is the bending angle between the vectors R and r1 , θ 2 is the bending angle between the vectors R and r2 , and ϕ is the azimuth angle of the rotation of Y-CZ3 around vector R. The 7D Hamiltonian for the OH + YCZ3 system is given by 1 ∂2 1 ∂2 (Jˆtot − jˆ12 )2 1 ∂2 − − + Hˆ = − 2 2 2 2μR ∂R 2μr1 ∂r1 2μr2 ∂r2 2μR R 2 +

jˆ12 jˆ22 vib + + Kˆ CH + V (R, r1 , r2 , χ , θ1 , θ2 , ϕ), 3 2 2IYCZ3 2μr1 r1 (1) c mz + m3m r 2 cos2 χ . μR is c +3mz u Y-CZ3 , μr1 is the reduced

+ where IYCZ3 = the reduced mass between OH and mass of OH, and μr2 is the reduced mass of Y-CZ3 . The current form of IYCZ3 is the result after the consideration of CZ3 umbrella mode in this model. Jˆtot is the total angular momentum operator of the system, jˆ1 is the rotational angular momentum operator of OH, and jˆ2 is the rotational angular momentum operator of YCZ3 with respect to the axis perpendicular to the C3v symmetry axis of CZ3 group. jˆ12 is coupled vib is the vibrational kinetic energy operator by jˆ1 and jˆ2 . Kˆ CZ 3 of CZ3 , which is defined as53   ∂2 ¯2 cos2 χ sin2 χ vib ˆ KCZ3 = − 2 + 2ru μx μs ∂χ 2   ∂ ¯2 1 1 sin χ cos χ − 2 − , (2) ru μs μx ∂χ μr2 r22

3 m r 2 sin2 χ 2 z u

where μx = 3mZ and μs = 3mC mZ /(mC + 3mZ ). Since the bond length of C-Z is fixed at 2.067 a0 , the kinetic operator describes the umbrella motion of CZ3 only. To construct PODVR (potential-optimized discrete variable representation) basis set56 for the umbrella angle χ , one can first vib solve the one-dimensional (1D) bound states of Hˆ χ = Kˆ CZ 3 + Vref , with the reference potential Vref determined only by the change of χ and the rest of the coordinates fixed at the equilibrium geometric values of methane in the asymptotic region, and then performs the standard DVR construction procedure. The time-dependent wave function is expanded in terms of the body-fixed (BF) rovibrational eigenfunctions as  J M ν2 Fnνj ψνJ0M j0 K0 (R, r1 , r2 , χ , t) = K,v0 j0 K0 (t)un (R)φν1 (r1 ) n,ν,j,K

ˆ rˆ1 , rˆ2 ), × φν2 (r2 )φνCZ3 (χ )YjJKM (R, (3) FIG. 1. The seven-dimensional Jacobi coordinates for OH + YCZ3 system in the reactant channel.

where n is the translational basis label, ν denotes (ν 1 ,ν 2 ,νCZ3 ), j denotes (j1 ,j2 ,j12 ), (v0 ,j0 ) denotes the initial rovibrational states, and  is the parity of the system defined as

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 = (−1)j1 +j2 +L with L being the orbital angular momentum quantum number. The YJj KM in Eq. (3) is the coupled BF total angular momentum eigenfunctions which can be written as   J j K J M −1/2 2J + 1 DK,M Yj112j2 Yj K = (1 + δK0 ) 8π j −K J (4) + (−1)j1 +j2 +j12 +J D−K,M Yj112j2 , j K

where DJK,M is the Wigner rotation matrix, and Yj12 is the 1 j2 angular momentum eigenfunction of jˆ12 defined as  j K j1 m1 j2 K − m1 |j12 Kyj1 m1 (θ1 , 0)yj2 K−m1 (θ2 , φ) Yj112j2 = m1

(5) and yjm are spherical harmonics. Note the restriction (−1)j1 +j2 +j12 +J = 1 for K = 0. The centrifugal potential, i.e., the (J − j12 )2 term in the Hamiltonian, which gives rise to the coupling between different K blocks in the BF representation, is given by57 M

J M Yj K (J − j12 )2 YjJ K  = δjj  {δKK  [J (J + 1) + j12 (j12 + 1) − 2K 2 ] + 1/2 −δK+1,K  λ+ J K λj12 K (1 + δK0 ) − 1/2 −δK−1,K  λ− } J K λj12 K (1 + δK1 )

(6)

and the quantity λ is defined as 1/2 . λ± AB = [A(A + 1) − B(B ± 1)]

(7)

In the CS approximation, we neglect the coupling between different K blocks, thus K becomes a good quantum number and is conserved in reaction. In the calculations, we construct Gaussian wave packets and propagate them using the split-operator method.58 In order to obtain the reaction probabilities PνJ1ν2 νCZ j1 j2 j12 K (E), 3 we calculate the time-independent scattering wave function, ψi+ (E), on the dividing surface S[r2F ] at r2 = r2F . The integral cross section from a specific initial state is obtained by summing the reaction probabilities over all the partial waves (total angular momentum J), σν1 ν2 j1 j2 (E) =

1 (2j1 + 1)(2j2 + 1) 

 π  J × (2J + 1)Pν1 ν2 νCZ j1 j2 j12 K (E) 3 k 2 J ≥K j K 12

=

 j K 1 σ 12 (E), (2j1 + 1)(2j2 + 1) j K ν1 ν2 νCZ3 j1 j2

(8)

12

j K

where σν112ν2 νCZ j1 j2 (E) is defined as the j12 , K, and  specified 3 cross section. In this study, we will take K0 = 0 and  = +1. Hence, we drop them from now on. III. RESULTS AND DISCUSSION

The numerical parameters used in the calculations on an L-shaped grid54 are listed in Table I. The total basis/grid number for R, NRtot is 120 within the range of 3.5–12.0 a0 , and

TABLE I. Numerical parameters used in the wave packet calculations. (Atomic units are used unless stated otherwise.) OH + CH4 Grid/basis range R ∈ [3.5, 12.0], NRtot = 120, NRint = 60 and size: r1 ∈ [0.5, 5.0], Nrtot =3 1 asy = 40, Nr2 = 10 r2 ∈ [0.5, 5.5], Nrtot 2 tot χ ∈ [0.0, π ], Nχ = 6 j1max = 42, j2max = 43  1/4 2 2 e−(R−R0 ) /2δ e−ik0 R Initial wave ϕk0 (R) = π1δ 2 packet: R0 = 11.0, k0 = 0.5, δ = 0.15 x−xa n Absorbing Fabs = exp[−t · α( xmax f or xa ≤ x ≤ xmax −xa ) ], potential: Ra = 9.5, α R = 0.05, nR = 2.5 r2a = 4.0, αr2 = 0.025, nr2 = 1.5 Flux position: r2F = 3.5

the total basis/grid used in the interaction region, NRint is 60. Three PODVR basis/points are used for r1 within the range of 0.5–5.0 a0 in the 7D calculations. For the dissociating C– H bond in the methane r2 , 40 PODVR basis/points are used in the range of 0.5–5.5 a0 in the interaction region, and 10 PODVR basis/points are used in the asymptotic region. To cater to both ground and excited vibrational states of the umbrella mode of the methyl group, 6 basis/PODVR points are used for χ in the range of 0–π . The angular basis used in this work are 42 for the OH group, j1max , and 43 for CH4 , j2max , respectively, and numerical tests show the convergence about the use of these angular basis in the calculations. The initial wave packet was prepared with its center at 11.0 a0 and propagated for 18 000 a.u. with a time increment of 15 a.u. And the flux dividing surface is positioned at r2F = 3.5 a0 to collect reactive flux. At the edges of R and r2 , absorption potentials were applied to prevent the wave packet from reflecting back from the boundaries. In order to converge the integral cross sections over the whole energy range considered, we calculated the reaction probabilities with total angular momentum J from 0 to 180 with an interval of 10 because of the extremely huge computational cost. The interpolation scheme applied in this work is the interpolation polynomial in the Lagrange form to obtain reaction probabilities for other J within the range. In the CC calculations, the number of K blocks used is min(3, J + 1) (i.e., from 0 to min(3, J + 1) − 1). As a convergence test, we calculated the reaction probabilities with J = 60 and 120 by increasing the number of K blocks from 3 to 5. The relative error (the probability difference between 3 K and 5 K blocks divided by the probability for the 5 K blocks) is no more than 0.5% for the hydrogen abstraction reaction. The calculations are based on Espinosa-García and Corchado’s PES.28 This PES is constructed by augmenting the previous potential function for H + CH4 59 with two new terms: a Morse function to describe the O–H bond and a harmonic bending term to describe the H–O–H bending mode. It is then calibrated by considering all the properties (geometry, vibrational frequency, energy, rate constant, and kinetic isotope effect) as a whole and using experimental and ab initio data. The resultant PES provides a linear saddle point with a barrier height of 0.34 eV relative to the entrance valley. The energy difference between products and reactants is

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−0.577 eV. In addition, there exists a very shallow van der Waals well with a depth of 0.018 eV in the entrance valley, which is unable to support any bound state. At the saddle point, the length of the broken C–H bond increases by 20%, and the length of the O–H bond formed is 33% larger than in the products, indicating that the hydrogen abstraction reaction proceeds via an “early” transition state. A. Accuracy of the CS approximation, the J S approximation, and the spectator-bond approximation of OH

The CS approximation can reduce the complexity of calculations and is shown to be valid for quantum scattering problems where the main contribution comes from the small J region.45, 46 Here, we first test the accuracy of the CS approximation in the polyatomic OH + CH4 reaction. In order to reduce the computing effort and make the CC calculation feasible, the bond distance of OH is fixed in the calculations, leading to a 6D model. We believe that this model is appropriate for testing the accuracy of the CS approximation for the OH + CH4 reaction. We also made a test to calculate the 6D reaction probabilities for J = 0 using two methods: including only one finite basis representation (FBR) function in the description of r1 coordinate and fixing the OH bond. They gave almost the same results over the energy range from the threshold to 1.0 eV. Figure 2 shows the 6D CC and CS reaction probabilities for the OH + CH4 reaction from the ground rovibrational state with the total angular momentum J = 30, 60, 90, and 120. The probability curve with J = 0 is also presented for comparison. Obviously, the CS probability is quite similar to the CC probability for J = 30 over the whole energy range. As J increases, the discrepancy between the CS and CC probabilities becomes clearly visible. This is consistent with the fact that the CS approximation is generally valid in the small

FIG. 2. The 6D CC and CS reaction probabilities for the OH + CH4 reaction from the ground rovibrational state with the total angular momentum J = 0, 30, 60, 90, and 120 as a function of translation energy.

FIG. 3. The 6D CC, CS, and JS integral cross sections for the OH + CH4 reaction from the ground rovibrational state as a function of translation energy.

J region. By comparing the CC and CS probabilities with J > 30, it can be found that the CS probability resembles the CC probability well at low collision energies just above the corresponding energy threshold, while the CS probability oscillates obviously around the CC probability at higher energies. Hence, we would expect that the CS approximation works very well in the low collision energy range but moderately well in the relatively high collision energy range. In Fig. 3, we present the 6D CC and CS integral cross sections from the ground rovibrational state. Clearly, the CS integral cross section resembles the CC integral cross section well over the whole energy range. At relatively high collision energies, the CS integral cross section is slightly larger than the CC cross section. The relative error is, however, no more than 10%. Thus, the CS approximation is a good approximation in the OH + CH4 reaction, especially at low collision energies. In fact, the reaction is mainly dominated by low impact parameter (small J) collisions at low collision energies, while large impact parameter collisions will take place at high energies. From Fig. 2, it can be seen that the CS probability deviates from the CC probability more clearly as the total angular momentum J becomes larger. This will undoubtedly bring about the good behavior of the CS approximation at low collision energies and slightly poor but acceptable results at relatively high energies. An even simpler approach is to use the JS approximation43, 44 in which the reaction probabilities at nonzero J values are calculated by simply shifting the collision energy in the probability for J = 0, i.e., ‡ ‡ P J (E) ≈ P J =0 (E − EJ ), where EJ = B ‡ J (J + 1) is the rotational energy at the transition state. The JS method depends upon the ability to identify a unique bottleneck geometry, such as a transition state, and is particularly well suited to reactions which have a barrier in the entrance channel. Since the OH + CH4 reaction is of an “early barrier” type and has a well-defined saddle point with a barrier height

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FIG. 4. The 6D CC and JS reaction probabilities for the OH + CH4 reaction from the ground rovibrational state with the total angular momentum J = 0, 30, 60, 90, and 120 as a function of translation energy.

of 0.34 eV,28 we expect the JS approximation to work well for calculating the reaction probabilities at nonzero J values. As the PES employed in this work provides a linear saddle point with the C–H –O angle being 180◦ (H is the hydrogen being abstracted), the rotational kinetic energy can be ‡ approximately calculated by the formula EJ ≈ B ‡ J (J + 1). The rotational constant B‡ is taken as 0.28126 cm−1 , which is obtained from the geometry of the saddle point of the PES (taken from Table II of Ref. 28). Figure 4 shows the 6D CC and JS reaction probabilities for the OH + CH4 reaction from the ground rovibrational state with the total angular momentum J = 30, 60, 90, and 120. The probability curve with J = 0 is also presented for reference. As can be seen from this figure, the JS probability resembles the CC probability well for J = 30. As J increases, the discrepancy between the CC and JS results becomes more and more noticeable. In addition, for all the J values presented, the JS approximation gives a good estimation of the energy threshold. This is easily understood considering that the OH + CH4 reaction basically satisfies the physical condition of the JS approximation. The calculated JS integral cross sections are presented in Fig. 3. Clearly, the JS approximation gives very accurate cross sections at low collision energies, while it overestimates the cross sections at relatively high collision energies. The relative error is less than 20% over the whole energy range studied. As it is based on a very simple physical model, the JS approximation works at an acceptable level of accuracy for this reaction. Also, we carried out dynamics calculations within the 6D and 7D models under the CS approximation. The 6D model is reduced from the 7D model by fixing the OH bond length. The calculated 6D and 7D CS integral cross sections are shown in Fig. 5. Obviously, the 6D integral cross section agrees very well with the 7D cross section over the energy range considered. This behavior can be understood based on the fact that

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FIG. 5. The 6D and 7D CS integral cross sections for the OH + CH4 reaction from the ground rovibrational state as a function of translational energy.

the OH bond distance is almost unchanged between reactant OH and the transition state geometry.18, 19 Similar spectator behavior of OH has been verified in OH + H2 → H + H2 O reaction.60, 61 Thus, the spectator behavior of OH is further confirmed by our direct dynamics calculations, at least at collision energies covered by this study. B. Mode selectivity

For a polyatomic system, specific vibration mode excitations on reactants are more effective than others to promote reaction because they make the reaction to approach the transition state easier. In order to examine the mode selectivity, the reactant CH4 were initially excited to different vibrational states. The CS approximation works very well over the whole energy range studied, and the OH radical can be treated as a spectator because of the negligible effect on the reaction dynamics. Therefore, it is reliable and physically meaningful to perform quantum scattering calculations within the 6D model under the CS approximation. The calculated integral cross sections for the OH (ν 1 = 0, j1 = 0) + CH4 (ν 2 = 0–1, νCZ3 = 0–2, j2 = 0) reaction as a function of translational energy and total energy are plotted in Figs. 6(a) and 6(b), respectively. As shown in Fig. 6(a), excitations of the stretching and/or the umbrella modes clearly enhance the reactivity, with the former being more efficient. The fundamental and the first overtone of the umbrella mode of CH3 is 0.17 eV and 0.34 eV, respectively, and the fundamental of the stretching mode of CH is 0.375 eV. Thus, the reduction of energy threshold is all smaller than the corresponding excitation energy, i.e., the excitation energy initially deposited in the CH stretching mode and/or the CH3 umbrella mode is not entirely used to enhance the reaction. For the umbrella mode, the higher the excitation state, the less the excitation energy initially deposited (with respect to the total energy) is used to enhancing the reaction. From

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FIG. 6. The 6D CS integral cross sections for the OH (ν 1 = 0, j1 = 0) + CH4 (ν 2 = 0,1, νCZ3 = 0–2, j2 = 0) reaction as a function of (a) translational energy and (b) total energy. Energies are referred to the bottom of the OH + CH4 entrance valley. (ν 2 , νCZ3 ) denotes quantum numbers for the CH stretching mode and the CH3 umbrella mode.

Fig. 6(b), where the integral cross section is plotted as a function of total energy, we find that excitations in the stretching mode and/or the umbrella mode do not enhance the reaction more than translational energy, i.e., translation energy is more effective than vibrational energy, at least at low energies. At high energies above about 0.8 eV, the vibrational energy of the stretching mode of CH4 becomes more effective than the translational energy, while the vibrational energy of the umbrella mode has almost no effect on promoting the reaction. In addition, the cross section profile from the state (ν 2 = 1, νCZ3 = 0) increases sharply after the threshold and then oscillates clearly over the whole energy range. The oscillation behavior can generally be found in heavy-light-heavy atom transfer reaction.62 Overall, the vibrational excitation of CH4 has a significant effect on not only the energy threshold but also the profile. This can be explained by the reaction path curvature. Espinosa-García and Corchado28 found two sharp peaks from the reaction path curvature as a function of reaction coordinate. The first one, on the reactant side, is due to the strong coupling of the reaction path to the C–H stretching and to the umbrella mode. The other one, on the product side, results from the coupling of the reaction path to the product vibrations. The first peak might predict the significant effect of the stretching and umbrella modes of CH4 on the reaction.

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The well-known Polanyi rules63 state that vibrational energy is more effective in activating a late-barrier reaction than translation energy, whereas the reverse is true for an earlybarrier reaction. The validity of the rules has been proved not only on the atom-diatom reactions but also on several atomtriatom reactions. However, exceptions have been found recently for the F/Cl + CHD3 reactions by Liu et al.64, 65 The latest theoretical studies suggested that the violation is just restricted to low collision energies, where the entrance-channel van der Waals well plays an important role.41, 66, 67 As the OH radical is isoelectronic with the F atom their interaction with methane, i.e., the PESs, have similar properties, such as very shallow entrance-channel van der Waals well, “early” transition state and exothermicity.68 It is expected that the OH + CH4 reaction should present similar reaction dynamics process to the F + CH4 reaction. However, from Fig. 6(a), it can be found that excitations in the stretching and the umbrella modes clearly enhance the reaction over the energy range studied. This is quite different from the F + CHD3 reaction that the vibration hinders the reaction rate at low collision energies. In addition, the translation energy is more effective than vibrational energy in activating the early-barrier OH + CH4 reaction at low collision energies. Obviously, the Polanyi rules is hold in OH + CH4 . But, we should point out that the conclusion is based on the reduced-dimensional models and a relatively crude PES. The conclusiveness is largely dominated by the accuracy of the models and PES employed. Some attempts to use more accurate models for calculation are in progress. As the O atom has similar mass to the OH radical, it is meaningful to compare the reaction dynamics of OH + CH4 with that of the O + CH4 reaction. The O + CH4 reaction is nearly a thermoneutral reaction with a central barrier. There exists a shallow van der Waals well in the entrance channel and a complex in the exit channel. Previous experimental and theoretical works indicated that excitations in the stretching mode of CH4 significantly enhance the reaction, while excitations in the bending and umbrella modes have a minor effect, and none of them are more efficient than translation energy in promoting the reaction.42 The vibrational enhancement originates from that the CH-stretching excitation enlarges the reactive cone of acceptance.69, 70 Obviously, these dynamical characters are quite similar to the OH + CH4 reaction, at least at low energies. Nevertheless, excitations in the stretching mode promote the OH + CH4 reaction more than translational energies when the total energy increases up to about 0.8 eV. In contrast, they do not enhance the O + CH4 reaction more than translational energy in energy range from the threshold to 1.0 eV.42 IV. CONCLUSIONS

The ISSWP method has been employed to study the reaction dynamics for the OH + CH4 reaction with and without the CS approximation in reduced seven-dimensional and sixdimensional models. The reaction probabilities and integral cross sections from ground and several excited vibrational states of CH4 have been computed based on the PES of the Espinosa-Garciá et al.28

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Song et al.

The CS approximation works well over the whole energy range considered. The JS approximation gives accurate cross sections at low collision energies while overestimating them at relatively high collision energies. In addition, the spectator model of the OH radical is a very good approximation. The calculated 6D CS integral cross sections showed that excitations of the stretching mode and/or the umbrella mode enhance the reaction, with the former being more efficient. However, both of them are less efficient than translational energy in enhancing the reaction, at least at low collision energies. ACKNOWLEDGMENTS

H. Song, S. Y. Lee, and Y. Lu were supported by a Ministry of Education, Singapore, Grant No. MOE2011T2-2-087. M. Yang was supported by the National Science Foundation of China (Project Nos. 21373266, 21073229, and 21221064). 1 I.

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Six-dimensional and seven-dimensional quantum dynamics study of the OH + CH4 → H2O + CH3 reaction.

The reaction dynamics of hydroxyl radical with methane has been investigated using time-dependent wave packet approach within reduced six- and seven-d...
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