Dissociative Chemisorption of Methane on Pt(110)-(1Ã2): Effects of

Article pubs.acs.org/JPCA

Dissociative Chemisorption of Methane on Pt(110)-(1×2): Effects of Lattice Motion on Reactions at Step Edges Dongwon Han,† Sven Nave,‡ and Bret Jackson*,† †

Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003, United States Institut des Sciences Moléculaires d’Orsay, Université Paris-Sud 11/CNRS-UMR 8214, Bât. 351, 91405 Orsay Cedex, France

‡

ABSTRACT: The dissociative chemisorption of methane on Pt(110)-(1×2) is examined, with a focus on how the reaction dynamics are modified by the motion of the lattice atoms. The barriers to dissociation are found to be lowest at the step edges. The relaxation of the lattice in the presence of the dissociating molecule is found to be far more complicated than on the smooth surfaces of Pt and Ni. The dissociative sticking probabilities are computed using a full-dimensional treatment based on the reaction path Hamiltonian that includes all 15 molecular degrees of freedom and the effects of lattice motion. The potential energy surface and all parameters in our model are computed from first principles. The effects of lattice motion are strong, but not significantly larger than for dissociation on smoother surfaces. Vibrational excitation of the molecule can significantly enhance reactivity, though this effect varies from mode to mode. Agreement with recent experiments with regard to the variation of reactivity with translational energy and substrate temperature is good.

I. INTRODUCTION Steam re-forming, where methane and water react over a Ni catalyst, is the primary commercial source for molecular hydrogen. The rate-limiting step in this process is the dissociative chemisorption of methane, where a single C−H bond breaks as the molecule collides with the metal surface, leaving chemisorbed H and CH3 fragments. As a result, several experimental groups have studied this reaction, mostly on Ni and Pt surfaces, and molecular beams have been used to measure how the dissociative sticking probability varies with the translational and vibrational energy of the methane.1−3 Typically, the barriers for dissociation on these surfaces are relatively large,4 the dissociative sticking probabilities are small, and reactivity increases strongly with increasing collision energy and vibrational excitation of the molecule, as well as substrate temperature.1−3 This reaction also exhibits nonstatistical behavior with regard to energy in the incident molecule. For the dissociative chemisorption of CH4 on Ni(100), adding 35 kJ/mol of energy to the molecule by exciting the symmetric stretch leads to a greater increase in reactivity than putting the same amount of energy into the incident translational energy.5 Adding a similar amount of energy by exciting the antisymmetric stretch leads to a smaller increase in reactivity than putting this energy into translation.6 Such nonstatistical behavior has been observed on several surfaces, though the details vary from surface-to-surface.2,3 The dissociative chemisorption of methane on Pt(110)(1×2) has been examined by several groups.4,7−16 Pt(110) undergoes a missing row reconstruction, leaving one-dimen© 2013 American Chemical Society

sional rows of Pt atoms on the surface, separated by large troughs, two atomic layers deep.17 This highly corrugated surface is often used as a model for real metal catalysts, which have rough irregular surfaces. Molecular beam studies of this reaction have shown that there is a precursor-mediated pathway to dissociation at low incident energies,8−10 not unusual for highly corrugated surfaces where diffraction mediated trapping is efficient. At higher energies dissociative sticking is direct and activated and increases with translational energy, vibrational excitation, and surface temperature.7−10 Although there have been density functional theory (DFT)-based studies of transition states and reaction paths for dissociation on this surface,4,12−14 there have been no attempts to calculate the dissociative sticking probability. In this paper, we use an approach based on the reaction path Hamiltonian18,19 (RPH) to examine the dissociative chemisorption of CH4 on Pt(110)-(1×2). The RPH is based on the assumption that the potential energy surface is harmonic with respect to motion away from the reaction path. Using DFT, we are able to compute from first principles all of the terms defining this Hamiltonian.20 In two studies, on Ni(100)21 and Ni(111),22 we have been able to include all 15 degrees of freedom of the methane molecule. Previously developed sudden models were used to introduce the effects of thermal Special Issue: Structure and Dynamics: ESDMC, IACS-2013 Received: March 26, 2013 Revised: April 30, 2013 Published: May 1, 2013 8651

dx.doi.org/10.1021/jp402987w | J. Phys. Chem. A 2013, 117, 8651−8659

The Journal of Physical Chemistry A

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lattice vibrations.23,24 We were able to compute full-dimensional dissociative sticking probabilities as a function of collision energy, molecular vibrational state, and substrate temperature, entirely from first principles. Comparison with experiment was good, with regard to the magnitude of the reactivity, its variation with collision energy and substrate temperature, and the enhancement of this reactivity with vibrational excitation. More importantly, our RPH formulation allows us to examine how energy moves between various degrees of freedom during the reaction. One goal of this study is to elucidate the dynamical origins of the behavior observed in the experiments. Another goal is to further test our theoretical methods for computing dissociative sticking probabilities and for introducing the effects of lattice motion (substrate temperature). A third goal is to compare the reaction dynamics on this highly corrugated surface with that on flat surfaces, which have been much better studied. It is possible that on real catalysts, most of the reactions take place on step edges or defect sites. Indeed, we find that the exposed Pt atoms on the step edges are the most reactive. We are interested not only in how reaction dynamics and sticking probabilities on these edge sites compares with that on terrace sites but also in how coupling to the lattice motion differs. We have shown how lattice relaxation in the presence of the dissociating methane can lead to a large molecule−phonon coupling, which results in a strong variation in reactivity with temperature. One might expect very different behavior at steps, and we find that this lattice relaxation and the phonon coupling are very different on the Pt(110)-(1×2) surface relative to the (111) and (100) surfaces we have examined on Pt and Ni. In the next section we describe our scattering model and DFT calculations. In section III we present the results of our DFT studies of the lowest energy transition states and reaction paths, comparing with both earlier work and results on flat Pt and Ni surfaces. We then examine the variation in the dissociative sticking probability with collision energy, vibrational state, and substrate temperature, comparing with both experiment and the behavior on terrace sites. We close with some conclusions in section IV.

Perdew−Burke−Ernzerhof (PBE) functional.33,34 We begin by fully relaxing the top four layers of atoms, in the absence of any methane, with the bottom three layers held fixed at the bulk values computed by VASP. We then locate the transition state and RP for methane dissociation while keeping the lattice atoms fixed at these bare surface positions. The effects of lattice relaxation and thermal motion are introduced later in the calculation. 2 The distance along the RP is s, where (ds)2 = Σ15 i=1(dxi) , and the xi are the mass-weighted Cartesian coordinates of the CH4 nuclei. At 25 points along this path we compute the total energy, V0(s), and perform a normal-mode analysis. This gives us the normal vibrational coordinates {Qk} and corresponding frequencies {ωk(s)}, k = 1−14, describing motion orthogonal to the RP at a point s, in the harmonic approximation. To lowest order, the potential energy surface is harmonic with respect to these displacements, and the essential assumption of the RPH is to ignore higher order (anharmonic) terms. We also get the normal mode eigenvectors, Li,k(s), that define the transformation between {xi} and our RPH coordinates s and {Qk}: 14

∑ Li ,k(s)Q k

xi = ai(s) +

k=1

(1)

where ai(s) is the configuration of the molecule at s, in terms of our mass-weighted Cartesian coordinates. Changing variables, the (classical) RPH has the form19 H = H vib + V0(s) +

1 (p − πs)2 /(1 + bss)2 2 s

(2)

where 14

H vib =

⎡1 2 1 2 ⎤ Pk + ωk (s)Q k 2 ⎥ ⎦ 2 2 k=1

∑ ⎢⎣

14

πs =

(3)

14

∑ ∑ Q kPB j k ,j (s) (4)

k=1 j=1

and

II. THEORETICAL METHODS To construct our RPH, we use the climbing image-nudged elastic band method25,26 to locate the reaction path (RP), the usual minimum energy path in our 15-dimensional configuration space from the reactant to the product configurations, passing over the transition state. Total energies along this path are computed using the DFT-based Vienna ab initio simulation package (VASP), developed at the Institut für Materialphysik of the Universität Wien.27−31 A supercell with periodic boundary conditions is used to represent our system as an infinite slab, with a 20 Å vacuum space above the slab to separate it from its repeated images. The Pt(110) surface reconstructs such that every other row of surface atoms along the [11̅0] direction is missing. We model this with a seven-layer slab, where each layer contains four atoms, except the topmost layer, which contains two. This corresponds to a methane coverage of 1/4 ML (using the symmetry of the unreconstructed surface). VASP uses a plane wave basis set, and the interactions between the ionic cores and the electrons are described by fully nonlocal optimized projector augmented-wave (PAW) potentials.31,32 Our calculations are adequately converged for an 8 × 8 × 1 grid of k-points and an energy cutoff of 500 eV for the plane wave expansion. Exchange−correlation effects are treated using the

14

bss =

∑ Q kBk ,15(s) k=1

(5)

The momenta conjugate to s and {Qk} are ps and {Pk}, respectively, and the vibrationally nonadiabatic couplings in the kinetic energy operator are given by 15

Bk ,j (s) =

∑ i=1

dLi ,k ds

Li ,j(s)

(6)

The lowest energy transition state is illustrated in Figure 1. The molecular center of mass is almost directly over a Pt atom on the exposed edge, and the dissociating C−H bond is aligned along the ridge. Note the large surface corrugation resulting from the missing row reconstruction. The functions defining our potential energy surface, V0(s), and the 14 vibrational frequencies describing motion orthogonal to the RP are plotted in Figure 2 for this transition state. When the molecule is far above the surface (large negative s), 9 of the frequencies ωk(s) are nonzero: the triply degenerate antisymmetric stretch, ν3, the symmetric stretch, ν1, the doubly degenerate bend, ν2, and the triply degenerate bend, ν4. The molecule−surface interaction removes the degeneracies and leads to a softening of the 8652



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where the normal coordinates are divided into modes that are bound at large negative s, {Qk}b, and those that are unbound (ωk = 0) at large negative s, {Qk}u. The Φn are eigenfunctions of that part of Hvib corresponding to the bound modes (1′−6′ ℏωk(s)(nk + and 1″−3″ in Figure 2), with eigenvalues Σbound k 1 /2) and quantum numbers n = {nk}b. We limit our sum over n to include only the ground and 9 singly excited vibrational states of the molecule, which is reasonable for the high-energy stretches, though it is possible that some of the lower-energy bending modes could become doubly excited at higher collision energies. The operators bss and πs describe energy flow between the vibrationally adiabatic states. The term bss couples the ground state to the A′ singly excited vibrational states, allowing for energy flow between these vibrations and motion along the RP. Given our ansatz Ψ, the “two-quanta” operator πs only couples one singly excited state to another of the same symmetry. Using the quantum mechanical form of the RPH,19 we expand in a power series in these two operators, neglecting terms in H that are second order and higher in bss and πs, because they describe transitions to multiquanta excited vibrational states not included in our ansatz Ψ. The timedependent Schrö dinger equation then leads to coupled equations of motion for the ψn (s,{Qk}u;t). Two of these {Qk}u, Q8′, and Q6″, correspond to motion along X and Y, the molecular center of mass parallel to the surface. It is reasonable to assume that this motion is slow on collision time scales, given the larger relative mass for this type of motion, the large collision energies (0.2−1.0 eV) and that we consider the case of normal incidence. We thus implement our scattering calculation for fixed values X0 and Y0 and average the resulting reaction probabilities over the surface unit cell. This has worked well for similar systems,35 and experiments have shown that this reaction is relatively insensitive to parallel motion of the molecule.36 The remaining {Qk}u describe molecular rotation asymptotically. The rotational temperature in the molecular beams is about 10 K, and it is reasonable to assume that the molecule is initially in the ground rotational state. As the molecule approaches the metal, there are always one or more H atoms pointing toward the surface, and only minor angular reorientation is required to enter the transition state. As the moment of inertia is relatively small, and there is likely to be a reasonable amount of rotational steering, we assume rotational adiabaticity, writing

Figure 1. Transition state for methane dissociation on the Pt(110)(1×2) surface, denoted K1 in the text.

Figure 2. Frequencies of the 14 normal modes of methane along the reaction path s, corresponding to the K1 transition state. The 8 vibrations with A′ symmetry (solid lines) are labeled 1′−8′, and the 6 modes with A″ symmetry (dashed lines) are labeled 1″−6″. The energy along the minimum energy path, V0, is also plotted, scaled by a factor of 1/2.

symmetric stretch and some of the bends at the transition state (s = 0). We find similar mode softening on the (100) and (111) surfaces of Ni and Pt.4,21,22 One consequence of this is that zero point energy (ZPE) corrections typically lower barrier heights by 0.1 eV or more. The remaining 5 frequencies correspond to molecular rotation and translation parallel to the surface and are all zero asymptotically. As the molecule approaches the surface, they become nonzero, describing hindered rotational and translational motion. Our RP is symmetric with respect to reflection through a plane that lies perpendicular to the surface and along the ridge, and the normal modes are either symmetric (A′) or antisymmetric (A″) with respect to reflection through this plane. We label the A′ modes 1′−8′ (solid lines), from highest to lowest frequency at the transition state. The modes with A″ symmetry (dashed lines) are labeled 1″−6″ in the same fashion. The Bij only couple modes of the same symmetry, and Bk,15 = 0 for the A″ modes. Our close-coupled wave packet RPH approach to reactive scattering is described in detail elsewhere,20,21 though we provide some details here to afford readability. We write the total molecular wave function as Ψ(t ) =

∑ ψn(s ,{Q k}u;t ) Φn({Q k}b;s) n

ψn(s ,{Q k}u ;t ) = χn (s ;X 0 ,Y0 ;t ) R 0(Q 7 ′ ,Q 4 ″ ,Q 5 ″;s)

(8)

where R0 is the ground state eigenfunction of that part of the Hamiltonian containing the coordinates, Q7′, Q4″, and Q5″. Given the spherical shape of CH4, one expects that the collision will not lead to significant rotational excitation, and experiment suggests that moderate rotational excitation of the incident molecule does little to modify the reactivity.37 These approximations lead to the following coupled equations of motion: iℏ

(7)

∂χ0 (s ;X 0 ,Y0 ;t ) ∂t

=

⎛1 2 ⎞ ⎜ p + Veff,0 + ΔV ⎟χ0 ⎝2 s ⎠ 2 ⎡ ∂χ ⎤ df ∂χ 1 d fν ⎥ + ∑ ⎢fv 2ν + ν ν + χ ds ∂s 4 ds 2 ν ⎥⎦ ⎣ ∂s v ⎢ (9)

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Table 1. Properties of Four Transition States (TS) on Pt(110)-(1×2), Designated K1, K2, L1, and L2a TS

Ea (eV)

E(ZPE) (eV) a

Z†

r†

θ†

α

β1 (eV/Å)

β2 (eV/Å)

K1 K2 L1 L2 Pt(100) Pt(111)

0.698 0.718 0.717 0.710 0.658 0.928

0.596 0.607 0.642 0.641 0.554 0.822

2.23 2.24 2.15 2.16 2.26 2.26

1.55 1.52 1.62 1.59 1.46 1.49

131 132 121 121 136 133

0.90 0.90 0.95 0.95 0.83 0.83

0.67 0.63 0.83 0.79 1.33 1.00

0.76 0.76 0.82 0.83

a The bare and zero point energy-corrected barrier heights are Ea and E(ZPE) , respectively. At the TS the distance of the carbon atom above the surface a is Z†, the length of the dissociating C−H bond is r†, and the angle of the dissociating C−H bond with respect to the surface normal is θ†. The phonon coupling parameters α and β are described in the text. The data for Pt(100) and Pt(111) are from ref 4 and correspond to the transition states labeled I2 and D1, respectively, in that paper.

iℏ

∂χν (s ;X 0 ,Y0 ;t ) ∂t

=

⎛1 2 ⎞ ⎜ p + Veff,ν + ΔV ⎟χν ⎝2 s ⎠ 2 ⎡ ∂ 2χ ⎤ df ∂χ 1 d fν ⎥ χ + ⎢fv 20 + ν 0 + ds ∂s 4 ds 2 0 ⎥⎦ ⎢⎣ ∂s +

⎡

∂χν ′

⎣

∂s

∑ ⎢gvv ′ v′

+

1 dfνv ′ ⎤ χ ⎥ 2 ds ν ′ ⎦

transition states are listed in Table 1, and the K1 configuration is shown in Figure 1. In all four transition states the carbon atom is over a Pt atom in the top layer, i.e., over a ridge top site. For the K1 configuration the dissociating C−H bond is aligned along the ridge, whereas for the L1 configuration this bond is aligned perpendicular to the ridge. The K2 and L2 configurations differ from the K1 and L1 configurations, respectively, with regard to the orientation of the nonreacting methyl group, with either 1 or 2 of the H atoms pointing toward the surface. After dissociation, the methyl group remains over the ridge top site. All four configurations have very similar activation energies, Ea. The geometries of these transition states are similar to those found on the smoother (100) and (111) surfaces of Ni and Pt, where dissociation also occurs over the top sites.4 The barriers on these smooth surfaces are higher than on the stepped Pt(110)-(1×2) ridge sites, with the exception being Pt(100), where the top-to-bridge barrier is, interestingly, a bit lower.4 As the CH4 moves away from these ridge top sites, the barriers to dissociation increase rapidly. We find that the activation energy for dissociation over one of the atoms in the second layerthe exposed atoms halfway between the ridge and the valleyis about 0.8 eV higher than for dissociation on the ridge. Thus dissociative chemisorption is dominated by reaction over the ridge atoms, and as the ZPE-corrected activation energies, E(ZPE) , are similar a for all four pathways, all are likely to contribute. In measurements of the dissociative sticking on both Pt(111) and Pt(110)-(1×2), Beck and co-workers estimate that the barrier on Pt(111) is larger by about 15 kJ/mol.10 Comparing our E(ZPE) for Pt(111) with the average E(ZPE) for the K1, K2, a a L1, and L2 transition states, we find a difference of 19 kJ/mol, in reasonable agreement. Beck and co-workers10 also conclude, on the basis of angle-resolved measurements, that reactivity is dominated by reactions at the ridge sites, in agreement with our and earlier studies.14 Anghel et al. have also computed pathways for methane chemisorption on this surface,14 and their results are in reasonable agreement with ours. They identify two transition states, both located over the ridge top sites. In one, the dissociating C−H bond is oriented along the ridge, corresponding to our K1 and K2 configurations. In the other, the dissociating H atom moves into a bridge site between the ridge Pt atom and a neighboring Pt atom in the second layer, similar to our L1 and L2 configurations. Their computed transition state geometries are very similar to ours. Although they compute a barrier Ea of only 0.40 eV for both configurations, it is important to note that they relax the top three layers of the slab in the presence of the dissociating methane. It is well-known that Ni, Pt, and Ir surfaces can relax

(10)

where the subscript 0 denotes the vibrational ground state and v = 1−9 labels the excited states corresponding to one quantum of energy in one of the modes 1′−6′ or 1″−3″. The wave packets evolve on the effective potentials 14

Veff,n(s) = V0(s) +

⎛

∑ ℏωk(s)⎝nk + ⎜

k=1

1 ⎞⎟ 2⎠

(11)

where Veff,0 is the RP with ZPE corrections. The term ΔV(s;X0,Y0) modifies the RP according to the surface impact site (X0, Y0). The couplings in eqs 9 and 10 are fν (s) = ℏ2

ℏ Bν ,15(s) 2ων(s)

(12)

and gνν ′(s) =

ℏ2 ⎡⎢ ων ′(s) Bν ,ν ′(s) − 2 ⎢⎣ ων(s)

⎤ ων(s) Bν ′ ,ν (s)⎥ ⎥⎦ ων ′(s)

(13)

The parametric dependence of the Φn on s leads to terms that depend upon the momentum and kinetic energy of the molecule, and curve crossing becomes increasingly likely at higher velocities. We use standard techniques21,22 to propagate eqs 9 and 10. The initial state consists of a wave packet of Gaussian form, centered far outside the interaction region, on the initial vibrational channel of interest. As this wave packet approaches the surface, it couples to and creates wave packets on the other channels, via the nonadiabatic couplings. These wave packets are discretized on 512-point grids, and the close-coupled equations are evolved using standard FFT-based methods.38 The reactive flux at large positive s is Fourier transformed in time on each channel n, giving both state-resolved and energyresolved reaction probabilities for all incident energies Ei included in the initial wave packet.39,40

III. RESULTS AND DISCUSSION We have characterized four transition states for dissociative adsorption on Pt(110)-(1×2), which we denote as K1, K2, L1, and L2, adopting the notation of ref 4. Some properties of these 8654



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necessary to compute the transition state energy for many distorted lattice configurations to determine the βi. The βi are simply the (Hellman−Feynman) forces on the unrelaxed lattice atoms at the transition state, usually generated at a minor computational cost for purposes of geometry optimization. The mechanical coupling still only involves the motion of the top site ridge atom over which the reaction occurs, and α is a bit larger than on Pt(100) and Pt(111), where it ranges from 0.83 to 0.86. In Figure 3 we consider the effects of averaging over surface impact sites and lattice motion, for the K1 transition state, and

significantly when CH4 dissociates and CH3 bonds to the surface.4,41−44 If we relax the top four layers of our seven-layer slab, we find that Ea decreases from 0.60 to 0.53 eV for the K1 configuration, much closer to the Anghel et al. result. The remaining 0.1 eV difference in energy may result from the additional layer in our calculation, the larger energy cutoff (500 vs 340 eV), differing exchange−correlation functionals (though PBE and PW91 generally provide very similar results), or other details of the calculation. As noted, the metal lattice relaxes in the presence of the dissociating methane. Put another way, thermal motion of the lattice atoms modify the barrier to reaction, resulting in a strong molecule−lattice (phonon) coupling and a dissociative sticking probability that varies with the substrate temperature Ts. Studies of methane dissociation on Ni(100), Ni(111), Pt(100), and Pt(111) find that if the metal lattice is allowed to fully relax at the transition state, the surface metal atom over which the C−H bond breaks puckers out of the surface plane by about 0.2 Å, lowering the energy by roughly 0.2 eV.4,42−44 The lattice is otherwise unchanged, for the most part, on these smooth surfaces. We were able to characterize the molecule−lattice coupling on these surfaces in terms of Q, the displacement of this surface metal atom normal the plane of the surface, and two parameters: α and β. The usual “mechanical” coupling, which is a measure of how the location of the barrier (or the repulsive wall of the particle−surface potential) shifts along the Z-axis as the metal atom vibrates, is characterized by α. We find that this shift is equal to αQ, for reasonable values of Q, where Q > 0 for displacements away from the bulk. For α = 1, this is equivalent to the surface oscillator model (SOM).45,46 When there is lattice relaxation in the presence of the molecule, there is a second type of coupling, characterized by β. On these smooth metal surfaces the height of the barrier changes by an amount −βQ, for reasonable values of Q, where β is typically about 1 eV/Å. We have examined how these couplings modify the reactivity, using fully quantum models that include both Q and several degrees of freedom of the methane molecule.43,44 Changes in the dissociative sticking probability due to the β term are particularly strong. Although the geometries of our transition states on Pt(110)(1×2) are very similar to those on smooth Pt and Ni surfaces, the nature of the lattice relaxation and molecule-lattice coupling is very different. On these smoother surfaces, lattice relaxation was essentially limited to the puckering of a single atom. If the top four layers of our Pt(110)-(1×2) slab are allowed to relax with methane at the transition state, both the top site atom directly under the methane and the Pt atom directly below it in the third layer, pucker normal to the surface plane by about 0.12 Å. All four atoms in the second layer shift about 0.05 Å away from the methane, and the ridge atoms on either side of the methane also shift by about 0.05 Å. The remaining atoms in the third and fourth layers also relax, but by lesser amounts. For methane dissociation on Pt(111) and Pt(100), where only motion along Q changes the transition state energy,4 β ranges from 0.93 to 1.59 eV/Å. On Pt(110)-(1×2) we have several βi that must be included in our average. The largest are β1 and β2, listed in Table 1, which correspond to the ridge top site and the third-layer atom below it, respectively. The next largest are β3 = 0.41 eV/Å for two of the second layer atoms and β4 = 0.31 eV/ Å for the ridge atoms adjacent to the methane. Though these couplings are smaller than for the terrace sites on the smooth surfaces (Table 1), there are many of them and the total effect on the sticking probability may be large. We note that it is not

Figure 3. Single-site static lattice reaction probability (solid black line) and impact site-averaged static lattice reaction probability (dashed black line) as a function of incident energy, for methane initially in the ground vibrational state. The moving-lattice results are for a surface temperature of 400 K. These correspond to inclusion of only the largest (pink), the two largest (light blue), the four largest (green), the five largest (brown), and the seven largest (dark blue) coupling constants, βi.

methane in the vibrational ground state. The single site static surface result is the solution of eqs 9−13 for X0 = Y0 = 0 only, corresponding to impact at the ridge top site, where the barrier is lowest (ΔV = 0). The single site reaction probability saturates near one for energies above the ZPE-corrected barrier height, and at incident energies below this the probability drops exponentially, as the only mechanism for reaction involves tunneling. When we average over all impact sites in the surface unit cell, the saturation value decreases by about an order of magnitude, as the barriers increase rapidly with impact away from the ridge top site. We estimate that ΔV increases harmonically with X and Y, proportional to ω8′(0) and ω6″(0) squared, respectively.21 The effects of lattice motion are introduced using sudden models based on studies showing that the lattice atoms move slowly on the time scale of the collision.23,24 We sample values of Q, and its conjugate momentum P, from a Boltzmann distribution at the substrate temperature Ts. The reaction probability for each Q is approximated by shifting the static surface reaction probability, P(Ei), along the energy axis by the change in barrier height, i.e., P(Ei + βQ). The mechanical coupling is included using an approach similar to the surface mass model,24,46 where the relative collision velocity depends upon α and P. This method has been used to generate Ts-dependent dissociative sticking probabilities in good agreement with fully quantum calculations for methane dissociation on Ni(111) and Pt(111).23,24 For the case of Pt(110)-(1×2) we sample lattice displacements (independently) for all lattice atoms with a significant coupling 8655



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βi, and apply our lattice sudden model. Plotted are results for the full-dimensional finite-temperature dissociative sticking probability, S0(Ei,n,Ts), with various βi included, α = 0.9 and Ts = 400 K. Including α and only β1 for the ridge top site atom significantly increases the reaction probability relative to the static lattice case. This effect is strongest below the barrier, where tunneling would otherwise be the only reaction mechanism. At energies above the ZPE-corrected barrier height, recoil effects from the α term can lower S0 below the static lattice result, but the effect is only minor here. Adding β2 further increases the reactivity, though the effect is weaker than for the first βi. As we include the motion of the two most strongly coupled second layer atoms, then the ridge atoms adjacent to the methane, then the two more weakly coupled second layer atoms, we see S0 converging. Adding additional terms, all with βi ≤ 0.1 eV/Å, does not significantly change S0. In Figure 4 we compare the effects of the α and βi terms. The static surface results in the figure correspond to α = 0 and βi =

the many contributing oscillators for Pt(110)-(1×2) are generally out of phase and the effects cancel to a great extent. In Figure 5 we compare S0(Ei,n,Ts) for four different RPs, corresponding to the four transition states in Table 1, at two

Figure 5. Dissociative sticking probabilities for methane initially in the ground vibrational state, for the four transition states and for two substrate temperatures, Ts, as labeled.

Figure 4. Dissociative sticking probabilities for methane initially in the ground vibrational state, and a substrate temperature of 400 K. Results are shown for the full-coupling case of Figure 3, with all βi included and α = 0.9 (blue), all βi included but α = 0.0 (red), all βi = 0 and α = 0.9 (purple), and the result for a single β = 1.274 and α = 0.9 (green). Also shown for comparison is the static surface result, corresponding to all βi = 0 and α = 0 (dashed black line).

substrate temperatures, 400 and 600 K. The incident methane molecules are in the vibrational ground state. First we note that the relative increase with substrate temperature is large for low collision energies, about an order of magnitude at Ei = 25 kJ/ mol. The effect decreases with increasing Ei to roughly a factor of 2 near 45 kJ/mol, eventually going to zero at saturation. These variations are consistent with those observed in the experiments8−10 and are similar for all four reaction pathways. The difference in reaction probability between the four reaction paths is mostly just a shift along the energy axis corresponding to the differences in the ZPE-corrected barrier heights. There is some variation in α and the βi, with L1 and L2 having the strongest molecule−phonon coupling. Although this can be seen in the increased (relative) sticking probabilities at lower Ei, the effect is not large. In Figures 6 and 7 we consider the effects of vibrational excitation for the K1 reaction path. Static lattice single-site

0. If we include only the “mechanical coupling”, α = 0.9, there is significant enhancement of the reactivity below the ZPEcorrected barrier, with some lowering of S0 due to lattice recoil at higher energies, similar to the SOM.45,46 On the other hand, including only the βi terms, with α = 0, leads to an enhancement in the reactivity at all energies, due to a lowering of the barriers by thermal motion of the lattice. This effect is much stronger than for the α coupling. Adding the α coupling leads to the full-coupling result, which is very similar to the βionly result, except for some enhancement at lower energy and the addition of some recoil effects. It is interesting to compare the stepped surface behavior with that on the terrace sites of smooth surfaces, where there is only a single βi, corresponding to vibration of the top site metal atom. In Figure 4 we plot results for β = 1.247, the root-sum-square of the βi for Pt(110)(1×2). This value is close to that for the reactive Pt(100) surface, though larger than for Pt(111). We see that a single oscillator with a large coupling is more effective at promoting reaction than many oscillators with smaller βi, though the effect is not dramatic for this system This result is not unexpected, as

Figure 6. Single-site static lattice reaction probabilities as a function of incident energy, for methane initially in the ground vibrational state (black line), and several vibrationally excited states each with one quanta of energy in the 1′, 2′, 3′, 4′, 6′, or 1″ mode (using the notation and line color of Figure 2). 8656



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model. In comparing with experimental efficacies for methane dissociation on Ni(100) and Ni(111), we concluded that our model tended to underestimate η a bit, though the trends with mode were reproduced.21,22 No state-resolved experimental data exist for the singly excited states we study here. However, Beck and co-workers have measured the efficacies of some 2quanta vibrational states,10,11 and they vary between 0.40 and 0.67, larger than what we observe here, on average, but not significantly different. Walker and King also report an increase in S0 with nozzle heating,8,9 consistent with an increase in reactivity with vibrational excitation of the incident methane. Finally, we note that high dimensional treatments of the dissociative sticking of polyatomic molecules, with reasonable descriptions of the vibrational motion, are rare. However, recent studies of methane dissociation on Ni(100)21 and Ni(111),22,47 and water dissociation on Cu(111)48,49 also find varying amounts of enhancement with vibrational excitation. To compare with experiment, we average the sticking probabilities computed for the four reaction paths in Figure 5, though the effect of this averaging is not significant. In Figure 8 we plot these averaged results along with the experimental

Figure 7. Dissociative sticking probabilities as a function of incident energy, for methane initially in the ground vibrational state (black line), and several vibrationally excited states each with one quanta of energy in the 1′, 2′, 3′, 4′, 6′, or 1″ mode (using the notation and line color of Figure 2). The results are averaged over impact sites and lattice motion, at a substrate temperature of 400 K.

reaction probabilities are plotted in Figure 6 for the ground state and several excited states, each corresponding to one quanta of energy in the modes indicated. As we have observed on Ni(111) and Ni(100), the vibrational enhancement is largest for the symmetric stretch (3′), due in part to the mode softening near the transition state.21,22 Even in the vibrationally adiabatic limit (Bk,j = 0) a molecule in the 3′ state has an effective barrier (eq 11) that is lower than that in the ground state by about 15 kJ/mol, due to this softening. However, the reaction probability curve for the 3′ state is shifted to lower energies relative to the ground state curve by almost twice this amount. This arises from the couplings Bk,j and Bk,15, which allow for transitions from the 3′ state to vibrational states of lower energy, and the ground state, respectively. The excess vibrational energy from these transitions is transferred into motion along the reaction path, corresponding to bond breaking at the transition state. For the 3′ state, over-thebarrier processes are possible for Ei above 22 kJ/mol (E(ZPE) a minus the asymptotic vibrational energy) and the reaction probability drops exponentially for Ei below this energy. Although the same is true for the 1′ and 2′ antisymmetric stretches, the couplings are weaker and there is negligible mode softening, so the reaction probabilities for Ei between 22 kJ/ mol and the effective barrier height are considerably smaller than for the 3′ state. For the 1″ stretch the coupling B1″,15 = 0, by symmetry, and there is no mode softening. Thus, excitation of this mode only weakly promotes reaction. The A′ bending modes have relatively strong couplings and some mode softening and are reasonably effective at promoting reaction. In Figure 7 we plot the dissociative sticking probabilities, S0, for the same vibrational states plotted in Figure 6, for a substrate temperature of 400 K. Again, exciting the 3′ mode has the largest effect. The efficacy of the 3′ mode for promoting reaction, η3′, defined as the shift in S0(Ei,3′,Ts) along the energy axis relative to the ground state curve, divided by the asymptotic value of ℏω3, is about 0.66 over most of the energy range in Figure 7. The efficacies for the three stretching modes comprising the ν3 triplet are clearly much smaller, though η is about 0.43 for the 2′ state at the lowest energies plotted. For the 4′ bend η is about 0.39, whereas for the 6′ bend it is much smaller. The A″ modes tend to give very small efficacies in our

Figure 8. Dissociative sticking probabilities for methane initially in the ground vibrational state, as a function of incident energy and substrate temperature, Ts, as indicated. The lines correspond to the theory and the symbols are experimental data from the Beck group (ref 10). Theory results are given for two nozzle temperatures, Tn.

results of the Beck group.10 The agreement with respect to both the magnitude of the sticking and the variation with Ts is good, considering the complexity of the system. As in our studies on Ni, the model tends to overestimate the reactivity at saturation. One likely cause is our limited expansion in vibrational states. As more of these become excited at higher collision energies, energy is removed from the reaction coordinate, lowering S0. The increase in reactivity with substrate temperature is strongest at lower temperatures for both experiment and theory, and the computed magnitude of this increase is in good agreement with the data of Beck and coworkers. The experiments were performed with a nozzle temperature Tn in the range 323−373 K, and this can affect the results at low Ei. The vibrational state distribution in the beam is roughly Boltzmann at Tn, and our results correspond to Tn = 0. At low incident energies, small populations of highly reactive vibrationally excited molecules can make a significant contribution to S0. In this study we have not computed sticking probabilities for many of the excited states populated at 8657



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Energy Research, U.S. Department of Energy, under Grant No. DE-FG02-87ER13744.

these temperatures, such as the bend overtones, and experimental efficacies are not known for most of these states. However, we can estimate the Tn effects, as described in an earlier work.22 We assume a vibrational efficacy of 0.7 for every excited state and approximate the sticking probability for an excited state n, with energy En, as S0(Ei,n,Ts) ≈ S0(Ei + 0.7En,0,Ts), where S0(Ei,0,Ts) is our computed ground state sticking probability. The results for Tn = 348 K, the average experimental value, are plotted in Figure 8. We see that adding these nozzle temperature effects tends to lower the enhancement in reactivity with increased substrate temperature, as a larger percentage of the reactivity results from vibrationally excited molecules that are closer to the saturation value of S0. Agreement with experiment at lower incident energies is significantly improved. We note that Walker and King also report an increase in S0 with increasing surface temperature.8,9 It is clear that our models for including the effects of lattice motion and substrate temperature work reasonably well for this system, though many types of lattice motion need to be included.

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(1) Larsen, J. H.; Chorkendorff, I. From Fundamental Studies of Reactivity on Single Crystals to the Design of Catalysts. Surf. Sci. Rep. 1999, 35, 163−222. (2) Juurlink, L. B. F.; Killelea, D. R.; Utz, A. L. State-Resolved Probes of Methane Dissociation Dynamics. Prog. Surf. Sci. 2009, 84, 69−134. (3) Utz, A. L. Mode Selective Chemistry at Surfaces. Curr. Opin. Solid State Mater. Sci. 2009, 13, 4−12. (4) Nave, S.; Tiwari, A. K.; Jackson, B. Methane Dissociation and Adsorption on Ni(111), Pt(111), Ni(100), Pt(100), and Pt(110)(1×2): Energetic Study. J. Chem. Phys. 2010, 132, 054705. (5) Maroni, P.; Papageorgopoulos, D. C.; Sacchi, M.; Dang, T. T.; Beck, R. D.; Rizzo, T. R. State-Resolved Gas-Surface Reactivity of Methane in the Symmetric C-H Stretch Vibration on Ni(100). Phys. Rev. Lett. 2005, 94, 246104. (6) Juurlink, L. B. F.; McCabe, P. R.; Smith, R. R.; DiCologero, C. L.; Utz, A. L. Eigenstate-Resolved Studies of Gas-Surface Reactivity: CH4 (ν3) Dissociation on Ni(100). Phys. Rev. Lett. 1999, 83, 868−871. (7) McMaster, M. C.; Madix, R. J. Alkane Dissociation Dynamics on Pt(110)−(1×2). J. Chem. Phys. 1993, 98, 9963−9976. (8) Walker, A. V.; King, D. A. Dynamics of the Dissociative Adsorption of Methane on Pt{110}(1×2). Phys. Rev. Lett. 1999, 82, 5156−5159. (9) Walker, A. V.; King, D. A. Dynamics of Dissociative Methane Adsorption on Metals: CH4 on Pt{110}(1 × 2). J. Chem. Phys. 2000, 112, 4739−4748. (10) Bisson, R.; Sacchi, M.; Beck, R. D. State-Resolved Reactivity of CH4 on Pt(110)-(1×2): The Role of Surface Orientation and Impact Site. J. Chem. Phys. 2010, 132, 094702. (11) Bisson, R.; Sacchi, M.; Beck, R. D. Mode-Specific Reactivity of CH4 on Pt(110)-(1×2): The Concerted Role of Stretch and Bend Excitation. Phys. Rev. B 2010, 82, 121404. (12) Petersen, M. A.; Jenkins, S. J.; King, D. A. Theory of Methane Dehydrogenation on Pt{110}(1×2). Part 1: Chemisorption of CHx (x=0−3). J. Phys. Chem. B 2004, 108, 5909−5919. (13) Petersen, M. A.; Jenkins, S. J.; King, D. A. Theory of Methane Dehydrogenation on Pt{110}(1×2). Part II: Microscopic Reaction Pathways for CHx -> CHx‑1(x=1−3). J. Phys. Chem. B 2004, 108, 5920−5929. (14) Anghel, A.; Wales, D.; Jenkins, S.; King, D. Pathways for Dissociative Methane Chemisorption on Pt{110}-(1×2). Phys. Rev. B 2005, 71, 113410. (15) Sacchi, M.; Wales, D. J.; Jenkins, S. J. Mode-Specific Chemisorption of CH4 on Pt{110}-(1×2) Explored by First-Principles Molecular Dynamics. J. Phys. Chem. C 2011, 115, 21832−21842. (16) Sacchi, M.; Wales, D. J.; Jenkins, S. J. Bond-Selective Energy Redistribution in the Chemisorption of CH3D and CD3H on Pt{110}(1×2): A First-Principles Molecular Dynamics Study. Comput. Theor. Chem. 2012, 990, 144−151. (17) Lee, J. I.; Mannstadt, W.; Freeman, A. J. Multilayer-Relaxed Structure of the (1×2) Pt(110) Surface. Phys. Rev. B 1999, 59, 1673− 1676. (18) Marcus, R. A. On Analytical Mechanics of Chemical Reactions. Quantum Mechanics of Linear Collisions. J. Chem. Phys. 1966, 45, 4493−4499. (19) Miller, W. H.; Handy, N. C.; Adams, J. E. Reaction-Path Hamiltonian for Polyatomic-Molecules. J. Chem. Phys. 1980, 72, 99− 112. (20) Nave, S.; Jackson, B. Vibrational Mode-Selective Chemistry: Methane Dissociation on Ni(100). Phys. Rev. B 2010, 81, 233408. (21) Jackson, B.; Nave, S. The Dissociative Chemisorption of Methane on Ni(100): Reaction Path Description of Mode-Selective Chemistry. J. Chem. Phys. 2011, 135, 114701. (22) Jackson, B.; Nave, S. The Dissociative Chemisorption of Methane on Ni(111): The Effects of Molecular Vibration and Lattice Motion. J. Chem. Phys. 2013, 138, 174705.

IV. CONCLUSIONS In conclusion, we have used DFT-based electronic structure methods to locate the lowest energy reaction paths to CH4 dissociation on the highly corrugated Pt(110)-(1×2) surface. The lowest energy barriers occur for dissociation over the top of Pt atoms on the exposed ridges, in agreement with experiment10 and earlier calculations.14 Normal mode calculations are performed along the lowest energy reaction path to generate a reaction path Hamilton, which we use to perform fully quantum reactive scattering calculations. The effects of lattice motion are introduced using sudden models, and all molecule−phonon coupling constants are computed from DFT. Vibrational excitation of the molecule can strongly enhance the reaction probability, but the effect is a bit weaker than we have computed on Ni(100) and Ni(111).21,22 This trend is consistent with what has been observed experimentally.2,3,5,6,10,11 Lattice relaxation in the presence of the dissociating methane is far more complicated on Pt(110)(1×2) than on the relatively smooth (111) and (100) surfaces of Pt and involves the displacement of many lattice atoms. In spite of this, the effect of surface temperature on dissociative sticking is not any stronger on the ridge sites of this corrugated surface than it is on the smooth (111) and (100) terraces of Pt. Having said this, S0 does increase strongly with temperature, and the magnitude of the increase is in good agreement with experiment.10 The primary source of this effect is a modulation of the barrier height when the two Pt atoms beneath the dissociating molecule vibrate normal to the surface. The magnitude of our computed S0 is also in good agreement with available measurements.10

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS B.J. gratefully acknowledges support from the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of 8658



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