Reaction Rate Constants of CH4(ads) CH3(ads) + H(ads) on Ni(111

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Reaction Rate Constants of CH(ads)-CH(ads)+H(ads) on Ni(111): the Effect of Lattice Motion Wenji Wang, and Yi Zhao J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.5b08787 • Publication Date (Web): 09 Dec 2015 Downloaded from http://pubs.acs.org on December 15, 2015

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The Journal of Physical Chemistry

Reaction Rate Constants of CH4(ads) CH3(ads)+H(ads) on Ni(111): the Effect of Lattice Motion Wenji Wang1∗and Yi Zhao2 1

College of Science, Northwest A&F University, Yangling, 712100, Shaanxi Province, P. R. China, 2 State Key Laboratory for Physical Chemistry of Solid Surfaces and Fujian Provincial Key Lab of Theoretical and Computational Chemistry, College of Chemistry and Chemical Engineering, Xiamen University, Xiamen, 361005, P. R. China

Abstract The methane dissociation on metal surfaces is of great commercial importance. The dissociation and recombination rate constants of CH4 on Ni(111) are calculated using the quantum instanton approach with path integral Monte Carlo method. The Ni(111) lattice is treated rigidly, classically and quantum mechanically, so as to reveal the effects of lattice motion and quantum tunneling. For the dissociation of CH4 , the rates have the smallest value on the rigid lattice, while they possess the largest one on the quantum lattice. For instance, at 300 K, the rates on the classical and quantum lattices are 5 and 12 times as large as that on the rigid lattice, respectively. The curve of the Arrhenius plot for the dissociation rates on the rigid lattice demonstrates that the quantum tunneling effect of the ruptured H atom is remarkable, while the nearly invariable dissociation rates at low temperatures on the quantum lattice confirm that the thermally assisted tunneling should be dominant at low temperature. For the recombination of CH4 , the quantum lattice still has much larger rates than the rigid ∗

To whom correspondence should be addressed. E-mail: [email protected]

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lattice. For instance, the ratio of the recombination rates on the quantum and rigid lattices is 12 at 300 K. The quantum tunneling effect seems to play a minor role in the recombination rates on the rigid lattice, however, the thermally assisted tunneling is still very significant for the recombination process. Keywords: Quantum instanton, Path integral Monte Carlo, Quantum tunneling, Zero-point energy

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Introduction

The dissociative chemisorption of methane into CH3 and H fragments on transition metals is of great importance. It is considered to be the rate limiting step in the industrially important steam reforming reaction, CH4 +H2 O → CO+3H2 , which is usually the first step in many important large scale chemical processes, such as the synthesis of ammonia and the production of methanol. It is also a prototype for catalytic C-H activation and a benchmark for theoretical approaches of gas-surface interaction.1, 2 In recent years, with the development of various experimental and theoretical techniques, the dissociative chemisorption of methane on the low index surfaces of nickel has been extensively studied. Experimentally, molecular beam technique can not only provide the activation energy, but also reveal the roles of the translational, rotational and vibrational energies of CH4 in the cleavage of C-H bond. Beebe et al.3 exhibited that the corresponding activation energies on Ni(111), Ni(110) and Ni(100) surfaces were 12.6, 13.3 and 6.4 kcal/mol, respectively, while Egeberg et al.4 reported a larger activation barrier of 18 kcal/mol on a Ni(111) surface. Lee et al.5 studied the activated dissociative chemisorption of CH4 on Ni(111), they observed that the methane dissociation probability increased exponentially with the translational energy of the incident molecule. The rotation energy may play a minor role in the dissociation of CH4 on nickel surfaces, as Navin et al.6 have shown that the rotation of CH4 seems to be a spectator degree of freedom on a Pt(111) surface. The methane has been excited to specific vibrational quantum states by laser, the results show that methane dissociation can also be activated by vibrational excitation.7–12 The vibrational excitation is 2


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suggested to be more efficient in promoting methane dissociation than the translational energy.13 Bisson et al.14 showed that 2ν3 (the antisymmetric stretch) excitation of CH4 increased its reactivity by more than 4 orders of magnitude on Ni(111). Surface temperature is another important factor to affect the CH4 dissociation probability, since surface temperature may simultaneously increase the available reaction energy and lower the threshold energy of reaction. For CH4 dissociation on Ni(111), Killelea15 reported a strong surface temperature dependence near the translational energy threshold. Several experiments16, 17 also demonstrate that the reaction probability of methane is not only controlled by the available energy, but also sensitive to C-H stretch alignment. An in-depth insight into these experimental observations requires theoretical interpretations, fortunately, advances in quantum chemistry can provide electronic and atomic level information. The earliest, Yang and Whitten18 used a many-electron embedding theory to investigate the methane dissociation on a Ni(111) surface, they found that the methane was located at an atop site and product CH3 and H were coadsorbed at separated threefold sites. Later, the Density Functional Theory (DFT) is extensively used to provide accurate geometries and energies. With DFT, Kratzer et al.19 demonstrated that the rupture of the C-H bond occurred preferentially on the top of a Ni atom. On a perfect Ni(111) surface, Haroun et al.20 studied the adsorption of CH4 , they revealed that CH4 stabilized weakly on the flat Ni surface, the adsorption energy was about 1.2 kcal/mol, and the most stable configuration was that 2 H atoms were directed to the surface. Although Ni(111) is not the most preferred surface21, 22 for CH4 dissociation, it is the most widely investigated surface, lots of DFT calculations show that the dissociation barrier of CH4 on Ni(111) is more than 23 kcal/mol,23–27 which is larger than the experimental values. Dynamical calculations can further provide the dissociation probability and rate constant, which usually depend on accurate potential energy surfaces except the time consuming Ab Initio Molecular Dynamics (AIMD) simulation.28–30 Based on different kinds of potential energy surfaces, the classical trajectory calculation,31, 32 the quasi-classical trajectory calculation,33 the mixed quantum-classical simulation34, 35 and rather complex quantum dynamical

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approaches, such as the time-dependent wave-packet method,36, 37 the multi-configurational time-dependent Hartree method,38, 39 the microcanonical unimolecular rate theory40, 41 and the reaction path Hamiltonian method,42, 43 have been used to investigate the dissociation of methane on nickel surfaces. Numerical results demonstrate that the methane dissociation probability on Ni(111) is an increasing function of kinetic energy,44 and the vibrational excitation is proved to be highly efficient in promoting reactivity45 whereas the rotational excitation has a counterproductive effect.46 Quantum dynamical simulations also show that the dissociation of methane is a bond- and mode-selective process.47–49 Milot et al.50 and Prasanna et al.51 predicted that the excitation of the symmetric stretch vibration was more efficient in promoting reaction than the excitation of the asymmetric stretch vibration on Ni(111). Exploring the effect of lattice motion on the dissociation probability of methane is one of the most active research areas in recent years. However, due to the large degrees of freedom of CH4 /Ni(111) system, a rigorous quantum dynamical calculation is forbidden, indeed, all of the previous studies of lattice motion are based on low dimensional models. Milot et al.31 found that more than half of the translational energy was transferred to the Ni(111) surface and the rest was mostly obtained by the rotational motion. Nave and Jackson52–54 showed that during the reaction, the Ni(111) lattice could reconstruct and had time to pucker, which lowered the reaction barrier and increased the reactivity relative to the rigid lattice. Kumar et al.55 and Tiwari et al.35 showed that the thermal vibrations of the lattice enhanced the reactivity, but itself was relatively unperturbed by the methane. Teixidor and Larranage46 reported that the vibration of the nickel substrate lowered significantly the reaction energy threshold. Jackson and Nave56 found that the dissociative sticking was enhanced by the substrate temperature, particularly when the reaction had insufficient energy to surmount the barrier. Although these studies have successfully revealed some characteristics of the lattice, they are qualitative. Our previous works of H diffusion57, 58 and H2 dissociation59–61 demonstrate that an explicit treatment of the degrees of freedom of lattice atoms can provide improved results and important conclusions. For the title reaction, such a quantitative

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research has not yet been achieved. There is also a remarkable quantum tunneling effect for the title reaction. German and Sheintuch62 calculated the rate constants for methane dissociation on metal surfaces with a tunneling model, they found that the tunneling effect was remarkable on the Ni(111) surface. Luntz and Harris63 demonstrated that the dissociation mechanism of CH4 on metal surfaces was dominated by quantum mechanical tunneling, and showed that the dependence on surface temperature was caused by an enhancement of the tunneling probability that resulted from the thermal energy of the lattice, which was referred to as thermally assisted tunneling. However, due to the difficulty of accurately estimating the tunneling effect, the consistency of available values is not well. This situation motivates us to go further and explore the tunneling effect on the rates. The dynamical calculations of CH4 /Ni(111) system in the present work focus on two specific goals. One is to investigate the effect of lattice motion on the rate constants, the other is to make a systematic analysis of the tunneling effect on the rate constants. The rate constants are calculated with the Quantum Instanton (QI) method proposed by Miller et al.64–67 Together with path integral method, the QI approach can be applied to rather complex reactive system with a high degree of freedom, the full dimensional dynamical calculations with Cartesian coordinates can incorporate the effects of vibrational-rotational coupling and anharmonicity.68, 69 Moreover, the QI method has a feature that it considers all tunneling paths when calculating the rate constants, so it is especially good at predicting the tunneling effect.70, 71 The present paper is organized as follows: In Sec. 2, we describe how to calculate the rate constant with the quantum instanton method. In Sec. 3, we outline the computational details. In Sec. 4, we give the rate constants and discussions. Sec. 5 is the conclusions.

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2

Quantum Instanton Rate Constant

2.1

Potential energy surface and lattice model

Our dynamical calculations need a reasonable CH4 /Ni(111) Potential Energy Surface (PES), especially this PES must include the degrees of freedom of nickel atoms. Fortunately, Wonchoba and Truhlar72 have constructed one this kind of PES. In this PES, the authors used the Embedded Diatomics-In-Molecules (EDIM)73, 74 to describe the gas-surface interaction. The EDIM was derived from the Embedded Atom Method (EAM)75, 76 and the DiatomicsIn-Molecules (DIM)77 method. This potential energy surface can be downloaded from the POTLIB.78 There is a physisorbed state in this PES, similar to methane activation on Ir(111)79 and Pt(111),40 the classical potential energy barriers from the physisorbed state to transition state and from the chemisorbed state to transition state are 13.52 and 7.76 kcal/mol, respectively. The corresponding zero-point energy corrected potential energy barriers are 8.55 and 8.20 kcal/mol. Although this PES has a too low potential energy barrier when compared to the values (12.63 and 184 kcal/mol) of experiments and the results (more than 23 kcal/mol23–27 ) of DFT calculations, it reasonably describes the interaction of CH4 and Ni(111), and makes the dynamical calculation with an explicit treatment of the degrees of freedom of nickel atoms practicable. We will use it in our calculations. In order to properly describe the interaction between CH4 and Ni(111), we construct a large lattice model, which consists of 397 nickel atoms. This lattice model has six layers with a lattice constant of 3.52˚ A. In our quantum lattice simulation, most of these nickel atoms (372) are fixed, only 5 ones, which interact with CH4 directly, are treated quantum mechanically, the rest (20) close to the quantized ones is treated classically. In Fig. 1, we draw the geometries of CH4 at physisorbed state (PS, the C atom and the ruptured H atom are adsorbed 3.79 and 2.19 bohr above the surface, respectively, and the bond length of the breaking C-H bond is 2.24 bohr), transition state (TS, the C atom is located 3.80 bohr over a bridge site, the ruptured H atom is located 1.78 bohr above the three-fold site, and the bond length of the breaking C-H bond is 2.61 bohr) and chemisorbed state (CS, the C atom and the ruptured H atom are adsorbed 3.34 and 1.82 bohr above the surface, respectively, and 6


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the bond length of the breaking C-H bond is 3.19 bohr) with the five quantized Ni atoms. It is noted that the average distance of the C and ruptured H atoms above the surface gradually decreases, while the bond length of the breaking C-H bond increases monotonically, from PS, TS, to CS.

2.2

Rate constant

The quantum instanton theory64, 65 gives the following formula to calculate the thermal rate constant

√ k=

π~ Cf f (0) . 2Qr ∆H

(1)

Here, Qr is the reactant partition function per unit volume, Cf f (0) is zero time value of the flux-flux correlation function, and ∆H is a specific type of energy variance. In order to calculate the rate constant k with the imaginary time Path Integral Monte Carlo (PIMC)80 method, we multiply Eq. (1) by the term of Cdd (0)/Cdd (0), where Cdd (0) is zero time value of the delta-delta correlation function. Now, the rate constant k becomes the product of several ratios, that are Cdd (0)/Qr , Cf f (0)/Cdd (0) and 1/∆H. The value of Cdd (0)/Qr is evaluated by the adaptive umbrella sampling technique,81 since Cdd (0) is the quantity at the transition state, while Qr is the quantity at the reactant region. The terms of Cf f (0)/Cdd (0) and ∆H are directly calculated as a constrained average over the same ensemble of paths66, 67 ⟨ ⟩ ∑N ∑N i i i i ∇ s (r ) · (r − r ) ∇ s (r ) · (r − r ) i b P/2 −P 2 Cf f (0) i a 0 i=1 P/2+1 P/2−1 1 P −1 i=1 √ √∑ = , ∑N Cdd (0) 4~2 β 2 N −1 −1 2 2 i=1 mi (∇i sb (rP/2 )) i=1 mi (∇i sa (r0 ))

(2)

‡

∆H 2 =

⟩ 1⟨ 2 F + G ‡, 2

(3)

where } N ∑ [ ] 1 1 i )2 − V (rk , Rσ[k] ) + V (rk−1 , Rσ[k−1] ) ak F = mi (rki − rk−1 2 2 2 2~ ∆β i=1 k=1 } P∑ bath { M 2 + (Rσ − Rσ−1 ) bσ , 2~2 ∆βbath 2 σ=1 P ∑

{

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(4)


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G =

P ∑

{

N ∑

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}

fr 1 i mi (rki − rk−1 )2 ak 2 2 − 2 3 2∆β ~ ∆β i=1 k=1 } P∑ bath { fR M 2 + bσ 2 . 2 − 2 3 (Rσ − Rσ−1 ) 2∆β ~ ∆β bath bath σ=1

The ⟨· · · ⟩‡ in Eq. (2) and Eq. (3) represents the constrained path average ∫ ∫ ∫ ∫ dr1 · · · drP dR1 · · · dRPbath exp{−βΦ}∆(sa (r0 ))∆(sb (rP/2 ))(· · · ) ∫ ∫ ∫ ⟨· · · ⟩‡ = ∫ , dr1 · · · drP dR1 · · · dRPbath exp{−βΦ}∆(sa (r0 ))∆(sb (rP/2 ))

(5)

(6)

with Pbath P N M Pbath ∑ P ∑∑ i 2 Φ = (Rσ − Rσ−1 ) + 2 2 mi (rki − rk−1 )2 2~2 β 2 σ=1 2~ β k=1 i=1

+

P 1 ∑ V (rk , Rσ[k] ), P k=1

(7)

v u N u∑ 1 ∆(sγ (r)) = δ(sγ (r))t (∇i sγ (r))2 (γ = a, b). m i i=1

(8)

Here, N is the number of atoms of CH4 , mi denotes the mass of the ith atom of CH4 , while M corresponds to the mass of Ni atom. rk and rki represent the Cartesian coordinates of CH4 and the ith atom of CH4 for the kth time slice, respectively, while Rσ corresponds to the Cartesian coordinates of Ni atoms for the σth time slice. P and Pbath are the numbers of time slices for CH4 and quantized Ni atoms, respectively. The subscript σ[k] denotes a transformation from k to σ. fr and fR are the numbers of degrees of freedom for CH4 and quantized Ni atoms, respectively. ak and bσ are coefficients. β is the inverse temperature (1/(kB T )), and ∆β and ∆βbath are defined by ∆β = β/P , ∆βbath = β/Pbath . sa (r) and sb (r) define two separate dividing surfaces. The free energy surface is given by F (sa , sb ) = −kB T log[Cdd (0; sa , sb )].

(9)

For the CH4 /Ni(111) system, a special reaction coordinate s(r;ξ) is adopted, where ξ is an adjustable parameter that shifts the location of the dividing surface. In this condition, sa (r) and sb (r) in the above can be expressed as s(r;ξa ) and s(r;ξb ), respectively. Concretely speaking, s(r;ξ) is defined by a linear interpolation between two constituent reaction coordinates 8


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s0 (r) and s1 (r) through the parameter ξ, s(r; ξ) = ξs1 (r) + (1 − ξ)s0 (r).

(10)

diss For the methane dissociation, sdiss 1 (r) defines a dividing surface (s1 (r)=0) on the top

of the classical potential energy barrier, and it is determined by the length of the breaking C-H bond and the distances of the C and ruptured H atoms above the surface. sdiss 1 (r) =

ZT S (C) + ZT S (H) Z(C) + Z(H) − − [RT S (C − H) − R(C − H)], 2 2

(11)

where Z(C) (Z(H)) is the distance of the carbon atom (the ruptured hydrogen atom) above the surface. R(X-Y) denotes the interatomic distance between atoms X and Y. The subscript T S stands for the values at the transition state. diss sdiss 0 (r) in Eq. (10) describes a dividing surface (s0 (r)=0) that is located at the ph-

ysisorbed state. It is given by sdiss 0 (r) =

ZP S (C) + ZP S (H) Z(C) + Z(H) − − [RP S (C − H) − R(C − H)], 2 2

(12)

where the subscript P S denotes the values at the physisorbed state. Similarly, the reaction coordinates sreco (r) and sreco (r) for methane recombination are 1 0 given by sreco (r) = −sdiss 1 1 (r), sreco (r) = 0

Z(C) + Z(H) ZCS (C) + ZCS (H) − − [R(C − H) − RCS (C − H)], 2 2

(13) (14)

where the subscript CS represents the values at the chemisorbed state. The advantage of the above definition is that s(r;ξ) can switch smoothly from the physisorbed state (chemisorbed state), s(r; 0) = s0 (r), to the transition state, s(r; 1) = s1 (r), as ξ changes from 0 to 1.

3

Computational Details

All the quantities in quantum instanton formula are evaluated by the path integral Monte Carlo method. The numbers of time slices (P) for methane are set to 40-80 in the temperature 9



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range of 300-600 K. With these large values of P, converged rate constants are obtained on the rigid lattice. On the quantum lattice, the minimal number of time slices needed to account for the quantum effects of Ni atoms can be obtained from the formula of Markland and Manolopoulos.82 Our adopted numbers of time slices (Pbath ) are 4, 5, 6 and 8 at 600, 500, 400 and 300 K, respectively, which are much larger than the minimal values. However, the convergence of the rate constants on the quantum lattice is not tested, since it is very time consuming. The discrete paths are sampled with the Monte Carlo method, which contains three kinds of movements, that are the movements of methane, quantized Ni atoms and classical Ni atoms. The number of the Monte Carlo is about (4-8)×106 for computing a single ensemble average. The statistical errors for various factors in quantum instanton formula are estimated, and most of them are within 10%. With these statistical errors, the statistical errors of the rate constants are also calculated.

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Results and Discussions

4.1

Dissociation rate constants of CH4 on Ni(111)

We calculate the dissociation rate constants (CH4 (ads)→CH3 (ads)+H(ads), where ‘ads’ denotes that the species is adsorbed on the Ni(111) surface) of CH4 on the rigid Ni(111) surface rigid cl (labelled as kdiss in the following paragraphs). The classical rates (kdiss ) on the classical quant lattice and the quantum rates (kdiss ) on the quantum lattice are also calculated, so as to

reveal the effect of lattice motion. The rigid lattice means that all of 397 nickel atoms are fixed. The details of the quantum lattice have been given in section 2.1. For the classical lattice, all the 25 movable nickels are treated as classical particles. It should be mentioned that CH4 is always treated quantum mechanically. These three kinds of dissociation rates in the temperature range of 300-600 K are listed in Table 1, where the ‘rigid’, ‘cl’ and ‘quant’ stand for the rates on the rigid, classical and quantum lattices, respectively. Fig. 2 is the corresponding Arrhenius plots. rigid In Table 1, on the rigid lattice, we can see that the rate kdiss increases with the increasing

of temperature over the tested temperature range 300-600 K, however, the differences of these

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rates are not very large, and the Arrhenius plot of these rates is not a straight line, which has an upthrow at 300 K. The underlying mechanism of this phenomenon is the quantum tunneling effect of the ruptured H atom, which can enhance the dissociation rates of CH4 at low temperatures. Our present theoretical prediction of the quantum tunneling effect is in good agreement with the reaction mechanism suggested by Hamza and Madix,83 who have attributed the sticking probabilities of H2 at low kinetic energies to the quantum tunneling. cl On the classical lattice, the rate kdiss still increases with the increasing of temperature, cl but the rates are very close to each other, especially at low temperatures. Compared kdiss rigid with kdiss , the former is always larger than the latter, as seen in Fig. 2. For instance, rigid cl to kdiss is 2.25 at 600 K, while this ratio is increased to 4.89 at 300 K. the ratio of kdiss

This phenomenon can be explained by the fact that the classical relaxation of nickel atoms stabilizes the transition state more than the physisorbed state and lowers the potential energy barrier remarkably. We have calculated the free energy surfaces on the rigid and classical lattices at 300 K, the corresponding free energy profiles along the reaction path are displayed in Fig. 3. It is clear that the free energy barrier on the rigid lattice is higher than that on the classical one by 0.8 kcal/mol. This situation is consistent with the conclusion of Nave and Jackson,52–54 that the pucker of the Ni(111) lattice can lower the reaction barrier and increase the reactivity relative to the static surface. quant On the quantum lattice, the rate kdiss exhibits a nonmonotonic behavior, it first de-

creases and then increases with the increasing of temperature. In order to confirm this nonmonotonic behavior, we calculate the statistical errors of the rates because the variation of the rates with temperature is not large. The statistical errors of the rates are obtained from the statistical errors of various factors in quantum instanton formula, the corresponding values are tabulated in Table 1, and the error bars are displayed in Fig. 2. We can see that the quantum lattice has a larger statistical error than the rigid lattice. All of these statistical errors increase with the decrease of temperature, and they are always smaller than the variation of the rate with temperature in our tested temperature range. However, it becomes close to the variation of the rate at 300 K on the quantum lattice. Considering that the

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correlation of Monte Carlo simulation usually underestimates the statistical error, we think that the actual variations of the rates at low temperatures should be small, and the rates probably reach a plateau at low temperatures. quant rigid quant rigid Compared kdiss with kdiss , we find kdiss is much larger than kdiss , and the difference

becomes more and more remarkable as the temperature decrease. For instance, at 600 K, quant rigid rigid kdiss is 2.66 times larger than kdiss , while it is 11.55 times as large as kdiss at 300 K.

Our result is consistent with the sudden model,45 which predicts that the difference between the quantum and rigid lattices becomes more and more remarkable with the decrease of quant quant cl the substrate temperature. Compared kdiss with kdiss , we can observe that kdiss is also cl larger than kdiss over the whole tested temperature range, and the difference is slight at high quant cl temperatures, but is very large at low temperatures, for instance, the value of kdiss /kdiss is

2.36 at 300 K. In order to explain this phenomenon, we also display the free energy profile on the quantum lattice in Fig. 3. It is clear that the quantum lattice has the lowest free energy barrier, and the free energy barrier on the quantum lattice is lower than that on the classical lattice by 0.83 kcal/mol at 300 K. This situation is in line with the finding of Teixidor and Larranage,46 that the vibration of the nickel substrate lowers the reaction energy threshold significantly. Indeed, our normal mode analysis reveals that the zero-point energy of quantized nickel atoms affects the potential energy barrier little, so the role of the quantum lattice to lower the free energy barrier should mainly come from the quantum delocalization of nickel atoms. The quantization of the movable nickel atoms (nickel atom is treated as a chain with more than one bead) makes the nickel atoms easier to deviate from the equilibrium position, indeed, our estimations show that the vertical displacements of the two bridge nickel atoms (seen in Fig. 1) are in the range of -0.44 ∼ 0.40 bohr on the quantum lattice at 300 K, while it is -0.40 ∼ 0.25 bohr on the classical lattice. The larger displacement on the quantum lattice makes the barrier lower than that on the classical lattice, as Nave et al.45 have showed that the upward lattice displacement decreases the reaction barrier. Luntz and Harris63 have proposed that the tunneling probability is enhanced by the surface atom which moves towards CH4 , and described this process as the thermally assisted tunneling.

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We think that this thermally assisted tunneling contributes a lot to our rates on the quantum lattice, since the quantum rates display a tunneling mechanism that the rates are no longer decrease at low temperatures. Compared to others, German and Sheintuch62 have reported a dissociation rate constant of 9.82×103 s−1 for CH4 on Ni(111) at 500 K, this value is much lower than ours. This discrepancy is due to the difference of the potential energy surfaces, our present PES has a zero-point energy corrected potential energy barrier of 8.55 kcal/mol, while German’s PES has an activation energy of 17.39 kcal/mol, the higher the barrier is, the smaller the rate will be. Indeed, the geometries of the transition state are also different, ours corresponds to the breaking of the C-H bond with the carbon over a bridge site formed by two adjacent Ni atoms, while that of German and Sheintuch is on the top of a Ni atom. Although most of the DFT calculations show that the carbon is on the top of a Ni atom at the transition state, the theoretical dissociation barriers (more than 23 kcal/mol23–27 ) are larger than the experimental ones (12.63 and 184 kcal/mol), this inconsistency is unresolved even now.

4.2

Recombination rate constants of CH4 on Ni(111)

We also calculate the rate constants for the recombination process (CH3 (ads)+H(ads)→CH4 (ads)) of CH4 on Ni(111). The Ni(111) lattice is also treated rigidly and quantum mechanically, so rigid quant as to investigate the effect of lattice motion. The corresponding rates, kreco and kreco on the

rigid and quantum lattices, respectively, are listed in Table 2, the corresponding Arrhenius plots are displayed in Fig. 4. rigid increases monotonically with the increase of temperature, and its In Table 2, the rate kreco

Arrhenius plot gives nearly a straight line (as seen in Fig. 4), which reveals that the quantum tunneling effect of the ruptured H atom should not be remarkable in the temperature range quant changes dramatically with respect of 300-600 K on the rigid lattice. However, the rate kreco

to temperature, its Arrhenius plot first decreases obviously with the decrease of temperature, and then it reaches a plateau, where the rates at 300 and 400 K are very close to each other. We also calculate the statistical errors of the recombination rates, as tabulated in Table 2, and display the error bars of the rates in Fig. 4. It is clear that the statistical error still 13



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increases with the decrease of temperature, and these statistical errors further confirm that the rates will reach a plateau at low temperatures on the quantum lattice. The Arrhenius quant plot of kreco demonstrates that the quantum tunneling effect is very remarkable at 300 K,

and this dominant quantum tunneling effect on the quantum lattice should come from the thermally assisted tunneling, since the quantum tunneling effect is not important on the rigid lattice. rigid Detailed observation also shows that kreco changes by nearly two orders of magnitude quant in the whole tested temperature range of 300-600 K, while kreco is in the same order of quant rigid magnitude. Compared kreco with kreco , we find that the former is much larger than the quant rigid latter, for instance, kreco exceeds kreco by as much as a factor of 12 at 300 K. This situation

can be explained by the free energy profiles along the reaction path, as seen in Fig. 5, the free energy barrier on the rigid lattice is higher than that on the quantum lattice by 1.55 kcal/mol. This free energy difference is still caused by the classical relaxation and the quantum delocalization of the nickel atoms (as discussed in Section 4.1), which can lower the free energy barrier by puckering the surface. The zero-point energy of quantized nickel atoms plays a negligible role, indeed, our normal mode analysis reveals that the zero-point energy of quantized nickel atoms raises the potential energy barrier by only 0.02 kcal/mol.

5

Concluding Remarks

We have calculated the dissociation and recombination rate constants of CH4 on Ni(111) using the quantum instanton method with the potential energy surface constructed by Wonchoba and Truhlar. The effect of lattice motion is investigated by treating the nickel atoms rigidly, classically and quantum mechanically. The results show that the lattice motion plays dominant roles in both dissociation and recombination processes. Compared to the rigid lattice, the classical relaxation of the lattice enhances the rates by puckering the surface, which can lower the free energy barrier. Compared to the classical lattice, the quantum delocalization of the nickel atoms increases the rates by increasing the displacements of nickel atoms vertical to the

14


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surface. However, the zero-point energy of the quantized nickel atoms has a little influence on the rates. The quantum tunneling effect is extremely significant for the title reaction. We have obtained two kinds of quantum tunneling effects. One is the quantum tunneling of the ruptured H atom, which can enhance the rates at low temperatures. The other is the thermally assisted tunneling by the lattice, which further increases the rates and results in nearly invariable rates at low temperatures on the quantum lattice.

Acknowledgements The authors are grateful for financial support by the National Natural Science Foundation of China (Grant Nos. 21203151 and 21143007), National Basic Research Program of China (973 Program) (2013CB834602), and Chinese Universities Scientific Fund (2452015084). The computational resources utilized in this research were provided by Shanghai Supercomputer Center.

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Figure 1: The geometries of CH4 at the physisorbed state (PS), transition state (TS) and chemisorbed state (CS) on Ni(111).

Figure 2: Arrhenius plots of the dissociation rate constants for CH4 on Ni(111). The solid line with filled squares, dashed line with filled circles and dotted line with filled triangles are the rates on the rigid, classical and quantum lattices, respectively. The error bars of these rates are also displayed.

Figure 3: Free energy profiles of the dissociation for CH4 on Ni(111) at 300 K. The solid, dashed and dotted lines are the results on the rigid, classical and quantum lattices, respectively.

Figure 4: Arrhenius plots of the recombination rate constants for CH4 on Ni(111). The solid (with filled squares) and dotted (with filled triangles) lines are the rates on the rigid and quantum lattices, respectively. The error bars of these rates are also displayed.

Figure 5: Free energy profiles of the recombination for CH4 on Ni(111) at 300 K. The solid and dotted lines are the results on the rigid and quantum lattices, respectively.

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Table 1: Dissociation rate constants of CH4 on Ni(111) T (K) rigida cl quant 300 9.96±0.79(10)∗ 4.87±0.85(11) 1.15±0.17(12) 400 1.73±0.09(11) 5.18±0.44(11) 7.56±0.59(11) 500 4.25±0.18(11) 8.55±0.52(11) 1.20±0.07(12) 600 6.39±0.16(11) 1.44±0.05(12) 1.70±0.06(12) ∗ −1 Unit: s , powers of 10 are in parentheses. a ‘rigid’, ‘cl’ and ‘quant’ stand for the rates on the rigid, classical and quantum Ni(111) lattices, respectively.

Table 2: Recombination rate constants of CH4 on Ni(111) T (K) rigida quant ∗ 300 1.09±0.08(10) 1.27±0.24(11) 400 5.10±0.28(10) 1.32±0.15(11) 500 1.49±0.05(11) 3.44±0.34(11) 600 2.14±0.08(11) 4.11±0.26(11) ∗ Unit: s−1 , powers of 10 are in parentheses. a ‘rigid’ and ‘quant’ stand for the rates on the rigid and quantum Ni(111) lattices, respectively.

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4

3 0 0 K

F re e e n e rg y (k c a l/m o l)

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


rig id la ttic e q u a n tu m la ttic e 2

0

C H - 0 .5

3

(a d s) + H (a d s) 0 .0

C H 0 .5

ξ


4

(a d s) 1 .0


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

Page 32 of 37

Table of Contents


Page 33 of 37


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



Page 34 of 37

F ig . 2

rig id la ttic e c la s s ic a l la ttic e q u a n tu m la ttic e

(s )

1 . 0 ×1 0

1 2

k

d is s

-1

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

1 . 0 ×1 0

1 1

C H

1 .5

4

(a d s)

2 .0

C H 3

(a d s) + H (a d s)

2 .5 ACS Paragon Plus Environment

1 0 0 0 /T (K )

3 .0

3 .5

F ig . 3

Page 35 of 37

3 F re e e n e rg y (k c a l/m o l)

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


3 0 0 K rig id la ttic e c la s s ic a l la ttic e q u a n tu m la ttic e

2

1

0 C H 0 .0

4

(a d s)

C H 0 .5 ξ


3

(a d s) + H (a d s) 1 .0


Page 36 of 37

F ig . 4

rig id la ttic e q u a n tu m la ttic e

(s )

1 . 0 ×1 0

1 2

k

re c o

-1

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

1 . 0 ×1 0

1 1

1 . 0 ×1 0

1 0

1 .5

C H 3

(a d s) + H (a d s) 2 .0

C H 2 .5


1 0 0 0 /T (K )

4

(a d s) 3 .0

3 .5

F ig . 5

Page 37 of 37

4

3 0 0 K

F re e e n e rg y (k c a l/m o l)

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


rig id la ttic e q u a n tu m la ttic e 2

0

C H - 0 .5

3

(a d s) + H (a d s) 0 .0

C H 0 .5

ξ


4

(a d s) 1 .0

Reaction Rate Constants of CH4(ads) CH3(ads) + H(ads) on Ni(111

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