Effects of Lattice Motion on Dissociative Chemisorption: Toward a

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Perspective

The Effects of Lattice Motion on Dissociative Chemisorption; Toward a Rigorous Comparison of Theory with Molecular Beam Experiments Han Guo, Azar Farjamnia, and Bret Earl Jackson J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b01948 • Publication Date (Web): 28 Oct 2016 Downloaded from http://pubs.acs.org on October 28, 2016

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The Journal of Physical Chemistry Letters

The Effects of Lattice Motion on Dissociative Chemisorption; Toward a Rigorous Comparison of Theory with Molecular Beam Experiments

Han Guo, Azar Farjamnia and Bret Jackson* Department of Chemistry, University of Massachusetts, Amherst MA 01003, USA

ABSTRACT: The dissociative chemisorption of small molecules such as methane and water on metal surfaces are key steps in many important catalyzed reactions. However, it has only very recently become possible to directly compare theory with molecular beam studies of these reactions. For most experimental conditions, such a comparison requires accurate methods for introducing the effects of lattice motion into quantum reactive scattering calculations. We examine these methods and their recent application to methane and water dissociative chemisorption. New results are presented for CO2 chemisorption and methane dissociation at step edges. The type of molecule-lattice coupling that leads to a strong variation in the dissociative sticking of methane with temperature is shown to occur for many polyatomic-metal systems. Improvements to these models are discussed. The ability to accurately compare theory with molecular beam experiments should lead to improved density functionals, and consequently more accurate thermal rate constants for these important reactions.

*

corresponding author email: [email protected]

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TOC graphic:

Keywords:

gas-surface scattering, dissociative chemisorption, reaction dynamics,

heterogeneous catalysis

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It remains a challenge to accurately model the dynamics of chemical reactions on metal surfaces.

This is unfortunate, as the industrial production of most chemicals

involves heterogeneous catalysis, often on the surface of metal particles.

In this

Perspective we examine the dissociative chemisorption of small polyatomic molecules such as CH4 and H2O on metals. These reactions, where the molecule breaks a bond as it collides with the metal, with the fragments adsorbing onto the surface, are rate-limiting steps in many key industrial processes. The dynamics of these reactions are difficult to model because the large number of electrons prohibits the use of high-level ab initio theory for the computation of barrier heights or potential energy surfaces (PESs), and in practice one typically uses methods based on DFT (Density Functional Theory). In addition, many molecules of interest contain too many degrees of freedom (DOFs) for an exact quantum dynamical treatment, and construction of a global PES can be difficult. Quasi-classical trajectory (QCT) methods can give accurate results at sufficiently high collision energies and surface temperatures. However at the energies and temperatures of most experiments, quantum effects can be important for C-H, O-H and N-H bond breaking, and QCT treatments of zero point energies can be problematic. Finally, lattice motion changes the PES for a reaction, and the reacting molecules can couple to the vibrational and electronic excitations in the metal substrate. Even with these seemingly insurmountable problems, significant advances have been made in just the past few years. Global PESs fit to tens of thousands of DFT energies have been constructed for the dissociative chemisorption of CH41-6, H2O7-16 and CO217 on rigid metal surfaces, and these have been used in several high-DOF quantum scattering calculations2, 4-9, 11-16. An approximate but full-dimensional approach based on the Reaction Path Hamiltonian18

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(RPH) has been used to describe the dissociative chemisorption of CH4 on several metal surfaces19-26, and H2O on Ni(111)27. Sudden models have been developed that introduce the effects of lattice motion into these static-surface calculations5,

28-29

. Global PESs

based on reactive force fields have been developed for methane reactions on Ni(111) and Pt(111) that include all molecular DOFs and the motion of several lattice atoms, and used in QCT studies30-31. DFT-based AIMD (ab initio molecular dynamics) has been used to model the dissociative chemisorption of CHD3 on Pt(111)32-33 and Ni(111)34. Finally, the effects of electron-hole pair excitations have been included in QCT studies of dissociative chemisorption35-37. For the first time, calculations that account for all molecular DOFs and the effects of lattice motion in a reasonable manner, and from first principles, have been directly compared with state- and energy-resolved dissociative chemisorption experiments, for CH42, 4, 19-26, 30-34 and H2O9, 12, 16, 27. The focus of this Perspective is on the last problem noted above: the effects of lattice motion on the dissociative chemisorption of polyatomic molecules. We have several important points to make. First, these effects can be very strong, often changing the probability for dissociative sticking by several orders of magnitude, and must be included in any reasonable comparison of theory with experiment. Second, these effects are more complicated than was supposed only a few years ago. Early treatments of lattice motion focused on collisional energy transfer between the molecule and the moving lattice atoms. More recently, DFT studies of methane have shown that the height of the barrier for dissociative chemisorption can vary with the vibrational displacement of the metal atom over which the methane dissociates38-42. This is responsible for the strong variation in the dissociative sticking probability with substrate temperature that has been

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observed in several experiments26,

43-48

. In this Perspective we examine the molecule-

lattice coupling for several important reactions, and show that this stronger type of coupling is more ubiquitous than might have once been expected. Third, while many theory and molecular beam studies examine reactions on the terrace sites of perfect surfaces, most reactions in real reactors likely occur at steps and other defect sites49-50. We show here that lattice motion effects can be different at defect sites, leading to a different variation in dissociative sticking with temperature. Lastly, we review some sudden approaches that have been used to incorporate the effects of lattice motion into rigid-lattice scattering calculations5, 28-29, allowing for a direct comparison of theory with experiment2, 4, 9, 12, 19-26. While these have worked well for the dissociative chemisorption of methane, the lattice coupling can be far more complex for other reactions. We discuss what advances will be necessary for there to be the type of agreement between theory and experiment for the dissociative chemisorption of molecules like H2O or CH4 that currently exists for gas-phase reactions.

Electron hole pair excitations will not be

considered here, as several recent studies have shown that they have only very minor effects on the probabilities for dissociative sticking35-37. Before proceeding, we note that these issues are also relevant to the problem of using DFT to compute PESs and thermal rate constants for reactions on metals51-53. In contrast to gas-phase reactions, there are not large databases of accurate molecular energies and geometries that one could use to test DFT results or fit exchange-correlation functionals. One could test the accuracy of DFT-based barrier heights and prefactors by using them in kinetic models and comparing with experiment, but these multi-step models can contain many parameters, and this is not optimal53-54. Perhaps a better way to

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benchmark DFT for these reactions is by comparing theory with molecular beam experiments that measure zero-coverage dissociative sticking probabilities, S0(Ei, v0, T), as a function of the translational energy of the incident molecule, Ei, its initial vibrational state, v0, and the surface temperature, T44-46. A database of chemically accurate energies (within 1 kcal/mole ≈ 43 meV) for gas-surface reactions was started within the past year, using this approach34, 51. The Kroes group was able to fit exchange-correlation functionals so that the resulting PESs, combined with quantum and QCT scattering approaches, reproduced the results of several experiments studying H2 + Cu(111)55-56 and H2 + Cu(100)57, within chemical accuracy. Chemical accuracy was also achieved in a similar fashion for CHD3 + Ni(111)34, using AIMD with a moving metal lattice. For the H2 reactions, lattice effects were unimportant because of its low mass, and for CHD3 + Ni(111) the functional was fit to special experiments performed at high Ei and T, where quantum effects were minimal. Unfortunately, most existing state and energy-resolved experiments are at lower Ei where S0 is very small and an accurate quantum treatment of the molecule is necessary43-48, and the effects of lattice motion must be included for molecules heavier than H2. Lastly, as we have noted, many molecules experience a strong coupling where the height of the reaction barrier changes with lattice motion. This implies that the lattice, if allowed to relax, will do so when the molecule is at the transition state, lowering the barrier to reaction. For systems with this type of coupling, it is thus essential to allow for lattice relaxation in the calculation of barrier heights for rate constants, and while many such calculations do so, many do not, which can lead to errors of several tenths of an eV.

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The dissociative chemisorption of methane on metal surfaces is the best studied polyatomic-surface reaction43-48, in part because it is a rate-limiting step in the steam reforming of natural gas58, which takes place over Ni-based catalysts. Molecular beam experiments find that S0 can vary dramatically with the temperature of the metal, and in order to understand this, one needs to examine how the PES, and in particular the transition state (TS) for dissociative chemisorption is modified by lattice motion. To do this, we use the DFT-based Vienna ab initio simulation package (VASP), developed at the Institut für Materialphysik of the Universität Wien59-63 to compute total energies. A supercell with periodic boundary conditions represents our system as an infinite slab, with a large vacuum space above the slab to separate it from its repeated images. The interactions between the ionic cores and the electrons are described by fully nonlocal optimized projector augmented-wave (PAW) potentials63-64, and exchange-correlation effects are treated using the Perdew-Burke-Ernzerhof (PBE) functional65-66. Our 4-layer 3x3 supercells correspond to methane coverages of 1/9 monolayer, and the Climbing Image-Nudged Elastic Band method67-68 is used to locate all transition states. Additional details can be found in our earlier studies21, 23, 42, 69. In Fig. 1 we show the transition state for CH4 dissociation on the “smooth” Ni(111) surface. The carbon atom is almost directly over one of the metal atoms (the top site), with the reacting C-H bond angled towards the surface by about 130° and stretched by several tenths of an Å from the equilibrium value of 1.1 Å. This “late” barrier is consistent with the large increases in S0 observed for vibrationally excited molecules43-48. Our results are consistent with other calculations of this nature39-40, 70-73.

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Fig.1 Total energy along the minimum energy path for dissociative chemisorption of methane on Ni(111), for three values of Q, the displacement, perpendicular to the plane of the surface, of the Ni atom over which the methane dissociates. Data are from Ref. 26

If the substrate is allowed to relax when the molecule is at the TS, the metal atom over which the molecule dissociates puckers out of the surface by a few tenths of an Å3842

. This implies that the height of the barrier to dissociation changes as this metal atom

vibrates. Other types of lattice motion do not strongly affect the TS41-42. We define Q as the displacement of this lattice atom perpendicular to the plane of the surface, with Q = 0 as the equilibrium position for a bare surface and Q > 0 away from the bulk. In Fig. 1 we illustrate how changing Q modifies the PES for reaction by plotting minimum energy paths for dissociation on Ni(111) for three fixed values of Q. As this lattice atom vibrates, the location of the transition state along Z, the distance of the carbon above the surface plane, changes by an amount αQ, where α = 0.75. This is the type of coupling typically used to model the effects of lattice motion on gas-surface scattering; atomic displacements normal to the surface change the location of the gas-surface repulsive wall. An example is the well-known Surface Oscillator Model, for which α = 174. More

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importantly, we also see that the barrier height changes by an amount –βQ, where β = 1.10 eV/Å. For Pt(111) we find similar behavior42, 75, with α = 0.83 and β = 0.95 eV/Å. Other aspects of the TS geometry do not change significantly with Q, and we find that the changes in TS location and energy are linear in Q for Q ≤ 0.2 Å42, though there can be some non-linear behavior for larger Q5. The variable Q was explicitly treated in a simple 4+1-DOF quantum model for methane dissociation on Ni41, demonstrating that the β-type coupling can lead to large variations in S0 with T. However, the large mass of the metal atoms makes such quantum treatments difficult because a large basis set is required and many initial lattice vibrational states need to be averaged over. Fortunately, because of this large mass the metal atoms move little on the timescales of reactive collisions28-29, and sudden models for lattice motion effects work well25, 28-29. We have had success using Eq. 1 to compute the dissociative sticking probability S0 from P0(Ei,v0;Q), the reaction probability on a rigid metal lattice where the atom over which the methane dissociates is displaced by Q:

S0 ( Ei ,v0 ,T ) = ∫ Plat (Q;T ) P0 ( Ei ,v0 ;Q ) dQ .

(1)

Plat(Q;T) is the probability that a surface atom is displaced by Q. We currently use a Debye model for this, with surface Debye temperatures extracted from atom-metal scattering experiments20. Assuming that the morphology of the PES is not significantly modified by changes in Q, one can make the additional approximation: P0 ( Ei ,v0 ;Q ) ≈ P0 ( Ei + β Q,v0 ;Q = 0 )

(2)

That is, we assume that the rigid-lattice reaction probability for Q≠0 is simply shifted along the energy axis relative to the Q=0 result, by the change in barrier height βQ. LowDOF studies suggested that both Eqs. 1 and 2 were good approximations28-29, and this

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was confirmed more recently in a fully quantum study of CH4 dissociation on Ni(111) that explicitly included Q in a 14-DOF PES fit to ≈ 105 DFT energies5. Lozano et al. also found Eq. 1 to be accurate in QCT studies of CH4 + Pt(111)30. The effects of the α term can be introduced using a slightly modified version25,

28-29

of the Surface Mass Model

(SMM)74. In this approach, one also averages over the momentum conjugate to Q. Changing to relative coordinates Z’ = Z-αQ, this amounts to an average over the relative collision velocity25, 28-29.

Fig. 2 Dissociative sticking probability of methane on Pt(111). Results are shown for the ground state (gs) and the 2ν3 excited state. Theory results are given for both moving (solid lines, from Ref. 20) and rigid (dotted lines) lattices. The experimental data (circles) are from Ref. 76.

Using Eqs. 1 and 2 and the modified-SMM, one can obtain S0(Ei,v0,T) at any T from a single rigid surface calculation of P0(Ei,v0;0). In Figs 2 - 4 we plot the results of such an approach for the dissociative chemisorption of CH4 on Pt(111) and Ni(111), comparing with the results of several initial-state selected molecular beam experiments. For CH4 + Pt(111), the agreement with experiment is quite reasonable. The theory results are from a RPH-based quantum mechanical scattering approach described in detail elsewhere20, 22-25, where the PES is approximated as harmonic in the normal coordinates

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about the minimum energy path, and the rotational motion is treated in the adiabatic and/or sudden limit, depending upon the collision energy20. In addition, the center of mass motion parallel to the surface, described by the coordinates X and Y, is treated using a sudden approximation similar to Eq. 1. The total molecular mass is relatively large, and for normal incidence and the high collision energies typical of these largebarrier systems, the location of the molecule in the plane of the surface unit cell should change little during the reaction. AIMD studies confirm this22, 32. It is reasonable to compute R0(Ei, v0; Q, Xj, Yj), for impact at a given surface site (Xj, Yj), and then average:

(

)

P0 ( Ei ,v0 ;Q ) = ∑ Ro Ei ,v0 ;Q, X j ,Y j . j

(3)

For the results in Figs 2-4, we have made the additional approximation:

R0 (Ei ,v0 ;Q, X j ,Y j ) ≈ R0 (Ei − ΔV,v0 ;Q, X0 ,Y0 ) ,

(4)

where ΔV(Xj,Yj) is the increase in barrier height for impact at (Xj,Yj) relative to the minimum barrier site, (X0,Y0). This last approximation appears to work reasonably well for methane on smooth surfaces19-20, 22-24, but is not expected to work in general. Our slight overestimation of S0 relative to experiment is consistent with the tendency for the PBE functional to overbind, giving barriers too low by 0.1 eV or so. Similar behavior was found in a moving-surface PBE-based AIMD study of CHD3 + Pt(111)

32

. Later

AIMD studies were able to improve agreement with experiment by including van der Waals interactions and using a mixture of PBE and RPBE, but chemical accuracy was not obtained33.

In Fig. 2 we also show the rigid surface result, P0(Ei,v0;Q=0), used to

generate S0 via Eqs. 1 and 2. Clearly, the effects of lattice motion are significant, and must be included in any comparison of theory to experiment.

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Fig.3 Dissociative sticking probability for ground state CH4 + Ni(111) as a function of incident energy, for the different levels of coupling indicated.

To better illustrate the effects of lattice motion, in Fig. 3 we consider the dissociative sticking probability of ground state CH4 on Ni(111) at different T and for various levels of coupling. The rigid lattice case corresponds to α = 0, β=0. If we only include the α-coupling (α = 0.75, β=0), we see that S0 is smaller at high collision energies than the rigid surface result. Physically, this results from recoil of the lattice atom upon molecular impact, with a transfer of energy from the molecule to the metal lattice. However, it is important to note that the modified SMM only approximates this behavior by including relative collision velocities that are less than the asymptotic molecular velocity. Molecule-metal energy transfer is not described explicitly, which may be a problem at high energies when the molecule-to-metal mass ratio is large. The α-term boosts the reactivity over the rigid surface result at lower energy, as larger relative collision velocities are also included. Including only the β-term (α = 0.0, β=1.10) boosts the reaction probability at all energies, and the effect is large, as the activation energy is actually changing. The left panel of Fig. 3 contains results for the full coupling case, α = 0.75 and β=1.10, for three temperatures. We see that the effects of energy loss and recoil 12


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dominate at low T (10 K), and S0 is actually less than the rigid surface result. As T increases, S0 can increase by orders of magnitude at lower energies, while recoil can still be important at high Ei. The effects are large, particularly at energies below the rigid lattice barrier height, and cannot be neglected.

Fig. 4 Dissociative sticking probability of methane on Ni(111). Results are shown for the ground state (gs) and 1ν3 and 2ν3 excited states. Theory results are from the RPH-based approach (solid lines, Ref. 20), and the global PES-based approach (dotted lines, Ref. 2). The experimental data (symbols) are from Refs. 76-77.

In Fig. 4, the results from the RPH-based approach show a similar agreement with experiment for reaction on Ni(111) as for Pt(111), except that the overestimation of S0 for the ground state is even larger. We note that moving surface AIMD studies of CHD3 + Ni(111) were able to achieve chemical accuracy at high energies (above 1 eV) by using a mixture of functionals containing mostly PBE65-66 with some RPBE78, and van der Waals corrections33-34. Also shown in Fig. 4 are results of quantum calculations from the Guo group, based on a flat and rigid surface single-impact-site 12-DOF global PES fit to 36,597 DFT energies2 computed using VASP with the PBE functional. Enforcing C3v symmetry of the non-reacting methyl group reduces the scattering problem to 8 DOFs, and impact site averaging and lattice motion effects were included in exactly the same

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way as in our RPH-based calculations (Eqs. 1 – 4). The agreement with experiment is similar to the RPH-based results, except that the ground state S0 was too small and not too large. This is partially due to the fact that their 2x2 supercell gives a barrier about 0.1 eV higher than for 3x3 or 4x4 supercells41.

Interestingly, for both calculations,

agreement between theory and experiment for the vibrationally excited molecules is good, and much better than for the ground state. The more reactive vibrationally excited molecules may be less sensitive to details in the PES. The dissociative chemisorption of water on metal surfaces is an important and often rate-limiting step in many important processes, including the water gas shift reaction79. Reaction on a rigid metal surface involves only 9 DOFs, making an accurate quantum treatment far more tractable than for methane. Full 9-DOF global PESs have been constructed for H2O on Cu(111)15 and Ni(111)10, 12, 16. Initially used in some 9-DOF QCT10-11 and some lower-DOF quantum studies11-14, the first fully quantum 9-DOF calculation, for H2O + Cu(111), has just appeared15. We note that electron-hole pair excitations were shown to only very weakly modify S0 for H2O + Ni(111)35. The first molecular beam study of this reaction appeared recently, measuring S0 for D2O + Ni(111) as a function of incidence energy for molecules in the ground state or with 1 or 2 quanta in the antisymmetric stretch state9. The lattice coupling is more complicated than for CH49,

27

. In Fig. 5 we show the lowest energy TS for H2O +

Ni(111). Like methane, the center of mass is nearly over a top site, and for collision at this site our modified SMM is reasonable. We find α = 0.78, similar to methane. What is different is that the motion of several atoms can modify the barrier height. The largest effect is from the motion of atom 5, mostly normal to the surface, for which β=0.63

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eV/Å, much smaller than we have observed for methane dissociation on any smooth Pt or Ni surface42. The surrounding 6 atoms have β-couplings between .16 and .29 eV/Å9, 27. Other studies have found similar results9, 80.

Fig. 5 Transition state for the dissociative chemisorption of water on Ni(111). The arrows indicate the directions and magnitudes of the largest of the β-type couplings, in units of eV/Å. Displacement in the direction of the arrow lowers the barrier.

There have been four comparisons of theory with this experiment. The first was based on a 6-DOF global PES9 and the second used the RPH-based approach27. These initial studies computed P0 over the lowest barrier impact site (Fig. 5). Both calculations used Eqs. 3 and 4 to average over other impact sites, and Eqs. 1 and 2 (with the modified SMM) to include lattice motion effects. The multiple βi couplings were treated by modifying Eq. 1 to include integrals over all the corresponding Qi21. However, after accounting for the largest one or two βi, including additional couplings did little, as the effects added both in and out of phase21. Agreement with experiment was not good, and in order to reasonably align with the data, the barrier heights had to be increased by 0.19 and 0.227 eV. Both studies used the PBE functional, with by 2x29 and 3x327 supercells. Subsequent studies10-11 have shown that the approximation of Eq. 4 is bad for this reaction. This is not surprising, as the barriers to reaction are relatively low over much of the surface unit cell, and the morphology of the PES changes from one site to another. 15



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This led to a more accurate study, where, using a 9-DOF global PES, quantum calculations were done for several 7-DOF fixed impact sites and then averaged, as in Eq. 312. This should be accurate, as site averaging has been shown to work well for the dissociative chemisorption of H281-82, HCl83-84, and H2O/Cu(111)15 where comparison with exact full-DOF rigid-surface calculations were possible. Lattice motion effects were introduced as in the earlier two studies. Unfortunately, agreement with experiment was not improved, and the DFT-PBE barriers appeared to be too small by 0.2 – 0.3 eV, a much larger error than has been observed for methane. This study was then repeated using a global PES based on the RPBE functional, resulting in higher barriers and a better, but still only qualitative, agreement with experiment16. We note that recoil effects from the α-type coupling would appear to be as or more important than the β-coupling for this reaction9,

12, 27

. Thus, the effects of lattice motion (temperature) on dissociative

chemisorption would appear to be weaker for water than for methane, though this awaits experimental confirmation. The dissociative chemisorption of CO2 on a metal surface is an important step in many heterogeneously catalyzed reactions, including recent attempts to reduce CO2 levels in the atmosphere by conversion to syngas through the dry reforming of methane8586

. The first theory study of the dynamics of direct dissociative chemisorption appeared

this year17. This QCT study used a full 9-DOF global PES for dissociation on a rigid Ni(100) surface. The computed S0 was in only qualitative agreement with molecular bean results from the Madix group87, but the effects of lattice motion were not included. We have applied our quantum RPH methods to the same reaction, with the inclusion of lattice motion, and present preliminary results here.

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Fig. 6 Transition states TS1 (left) and TS2 (right) for the dissociative chemisorption of CO2 on Ni(100). The arrows indicate the directions and magnitudes of the largest of the β-type couplings, in units of eV/Å.

The dissociative chemisorption of CO2 on many metals proceeds through a molecular precursor88, denoted CO2δ-, which is bent due to charge transfer from the metal.

We find that the barrier to forming CO2δ- is 0.20 eV, while the barrier to

dissociation of CO2δ- into chemisorbed CO and O is only 0.11 eV. ZPE corrections lower these to 0.16 and 0.04 eV, respectively. The PBE-based study by Guo and co-workers found similar results17. In Fig. 6 we show the geometries of these two transition states, denoted TS1 and TS2, and the corresponding lattice couplings. For the barrier to CO2δformation, TS1, three atoms have couplings above 0.2 eV/Å. Interestingly, the motion of atom 6 most strongly modifies the height of the barrier, though the effect is weaker than for methane. On the other hand, motion of atom 5 most strongly effects the location of the TS along Z, and we find α5 = 0.57 and α6 = 0.41. For TS2, the motion of atoms 5, 6, 8 and 9 all have a reasonably large β coupling. Note that these lattice deformations are not necessarily perpendicular to the surface. The barrier to dissociation from the precursor, TS2, is much lower than the barrier to formation of the precursor, TS1. In addition, the molecule-to-metal atom mass ratio is

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large for this reaction, and it is likely that any molecules that cross TS1 and don’t immediately dissociate will lose a significant amount of energy into lattice recoil and trap, relaxing into the precursor state. Any trapped CO2δ- molecules are far more likely to dissociate than desorb. Thus, we have approximated S0 as the probability to cross TS1, using our RPH approach with the approximations of Eqs. 1 – 4. As the molecular center of mass is roughly over atom 5, we use α5 in our modified SMM, and include sufficient βi so that additional couplings don’t change the results. We only note here, for the purposes of this Perspective, that agreement with experiment remains only qualitative, and that lattice effects are important, though the effects are weaker than for methane or water. As suggested by Guo and co-workers, the PBE functional may not adequately describe this reaction. Madix and co-workers reported almost no variation of S0 with temperature for T between 200 K and 400 K and Ei = .45 eV. While we reproduce this, we find that the variation with T can be important at energies below threshold. In addition, desorption from the CO2δ- precursor can actually cause a small decrease in S0 with increasing T at energies above threshold. Both of these predictions await confirmation. Given the large masses and the likelihood that molecular sticking in the CO2δ- state occurs, AIMD is probably the best route for further study of this reaction.

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Fig. 7 Transition state for the dissociative chemisorption of methane on Pt(211). The arrows indicate the direction and magnitude of the largest of the β-type couplings, in units of eV/Å.

Steps and defect sites are likely to dominate the kinetics in real catalytic reactors. Of interest here are how phonon couplings and lattice motion at step edges might be different from that at terrace sites. Fig. 7 shows the TS for CH4 dissociative chemisorption on the step edge of Pt(211), computed using a 4-layer 3 x 2 supercell and the PBE functional. The TS geometry is similar to that on the terrace sites, with the motion of the metal atom over which the molecule dissociates most strongly modifing the height of the barrier to dissociation. However, the magnitude of β for this atom is larger than for the terrace atoms, the direction is away from the surface normal, and the motion of several other atoms can modify the barrier height. Note that in Figs. 5 and 6 there are β-couplings that arise from metal atom motion with components both perpendicular to and lateral to the surface, but the effects of the lateral motion are small. On Pt(211), lateral motion of the step edge atoms is very important.

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Table 1. Transition State Properties for CH4 Dissociation on Pt Surfacesa.

Pt(211)

V † (eV)

α

β (eV/Å)

0.64

0.82

1.25,

Qrms(Å) 0.37, 0.071

0.37, 0.32 Pt(111)20

0.82

0.83

0.95

0.078

Pt(100)42

0.66

0.82

1.33

0.078

Pt(110)-(1x2)21

0.70

0.90

0.76,

0.67, 0.067

0.41, 0.41 a

V † is the barrier height, α and β are lattice coupling constants defined in the text, and

Qrms is the root mean square vibrational amplitude of a surface lattice atom at 475 K.

In Table 1 we compare the behavior on the (211) step sites with that on (111) and (100) Pt surfaces. Note that the (211) step has a (100) geometry, and the barrier height, α and the largest β value are in fact very similar to that on the (100) surface. As noted, the motion of several other surface atoms can also modify the barrier height at the step edge. The terrace sites on Pt(211) have a (111) geometry, and the lattice coupling on Pt(111) would appear to be weaker than on Pt(100), Overall, the variation of S0 with T should be stronger for reaction over a step edge than over a terrace site on Pt(211). Also shown in Table 1 are results for Pt(110)-(1x2), where the missing row reconstruction leaves a steplike row of exposed atoms. The lowest energy TS for methane dissociation is across one of these exposed atoms, similar to that in Fig. 721. As for the (211) step edge. several types of lattice motion can modify the barrier height, and βi > 0.3 eV/Å for 6 different atoms. While the magnitudes of these β-couplings are smaller than for the other surfaces,

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the resulting variation of S0 with T is still relatively strong, and our RPH-based methods and sudden models were able to reproduce the increase in S0 with T observed in experiment21. We expect that step edges and defect sites in general will exhibit this more complex type of coupling to lattice motion, with lateral motion of the metal atoms being important. For example, our peliminary studies of CO2 sticking on the Ni(711) step edge find that, for TS1 (compare with Fig. 6), seven different Ni atoms have β values between 0.30 and 0.54 eV/Å. The lattice relaxation required for the accurate calculation of rate constant barrier heights will involve many more atoms at defect sites than for dissociation on smooth surfaces. It is important to also consider the motion of the surface atoms. In Table 1 we list the rms value of Q based on a simple harmonic Einstein model. Here, Q describes the motion, along the surface normal, of the atom over which the methane dissociates, with the energies required to distort the atom from equilibrium computed via DFT. Qrms is actually a bit smaller at the step edges, as the metal atoms are less highly coordinated. However, this Einstein model can be misleading, as a proper treatment of collective excitations leads to larger values of Qrms. For example, we now use a Debye model for metal atom motion on the smooth (111) surfaces of Pt and Ni20, which gives a larger Qrms. Anharmonic effects can also be important, particularly at lower Ei and T24. Finally, the phonon modes that localize at step edges89 and other defect sites may result in larger amplitude vibrational motion. AIMD would be a useful tool for extracting the vibrational behavior of metal atoms at these and other sites. In conclusion, while agreement between quantum theory and molecular beam studies of the dissociative chemisorption of methane is surprisingly good, it has not

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Page 22 of 44

reached the level of chemical accuracy. However, it has not even been possible to make such comparisons until very recently. While the 15-DOF quantum calculations needed to describe these reactions at lower Ei and T remain a challenge, improvements in computer power and methodology may soon change this, and sudden treatments of X and Y (Eq. 3) should help in this regard, preferably avoiding the approximation of Eq. 46. Full 15-DOF global PESs have already been constructed1,

4, 6, 30-31

. AIMD studies combined with

special experiments conducted at high Ei, T, and nozzle temperatures, where classical methods should be accurate, have reached chemical accuracy34, suggesting that the same can be reached for quantum studies with improved functionals. However, the effects of lattice motion are clearly significant and must be included accurately.

Fortunately,

sudden treatments of lattice motion appear to work quite well for smooth metal surfaces where the coupling is relatively simple. These sudden methods can be improved. One could, for example, avoid the “energy-shifting” approximation of Eq. 2 and fit a global PES for a particular (fixed) Q, then compute P0(Ei,v0;Q)5.

The coupling is more

complicated at step edges and other defect sites, but likely tractable by lattice-sudden methods, though some work needs to be done to better understand the vibrational properties of metal atoms at these sites, perhaps by using AIMD. Given the dimensionality of the H2O/metal reaction, this would appear to be an excellent system for experiment-theory comparisons that can test both our models for lattice effects and the quality of exchange correlation functionals. We clearly need more experiments, which are difficult, but obviously not impossible. Models for including lattice motion will need to be improved. While they are probably reasonable for the TS of Fig. 5, reaction can occur at many impact sites in the surface unit cell10-11, and the

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coupling will be different at these sites. One way to do this is to implement 7-DOF fixed site calculations, and compute the β couplings for each site separately. These would then be Q-averaged for a given substrate temperature T before site averaging. Even better, with sufficient computational resources one could compute R0(Ei, v0; Q, Xj, Yj), averaging over all the slow variables. The α-term is perhaps a bit more complex. For most of the surface impact sites the molecule is not directly over a single metal atom, and it is not obvious how to approximately treat the relative velocity and recoil effects for such a collision. Our work suggests that the β-type coupling is common in gas-surface reactions. For example, while we have not yet examined the dissociative chemisorption of methanol, important in methanol fuel cells, it should exhibit strong and interesting lattice effects. Several dissociation pathways are possible, with scission of a C-H bond or O-H bond being most likely90-91. We speculate that S0 for C-H bond breaking is likely to display a stronger variation with T, similar to CH4, while the O-H scission should exhibit a weaker variation, similar to what our models predict for H2O. Given the advances of the past few years, it seems likely that quantum scattering treatments of several important gas-surface reactions will become sufficiently accurate to be able to test DFT functionals by comparison with energy and state-resolved molecular beam experiments.

While our RPH-based methods have provided semi-quantitative

results, efforts to improve the treatment of X, Y and molecular rotation may needlessly overcomplicate a model whose best feature is its simplicity. More traditional approaches using global PESs may be the best route to achieving chemical accuracy. In addition to elucidating the mechanisms of these important and interesting reactions, these studies

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will hopefully lead to the development of improved functionals, and thus more accurate rate constants for reactions that take place on the surface of metal-based catalysts.

Acknowledgments B. Jackson gratefully acknowledges support from the Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, U. S. Department of Energy, under Grant # DE-FG02-87ER13744.

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Table of Contents graphic 50x50mm (300 x 300 DPI)



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Fig. 1 82x68mm (300 x 300 DPI)


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Fig. 2 82x56mm (300 x 300 DPI)



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Fig. 3 82x61mm (300 x 300 DPI)


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Fig. 4 82x55mm (300 x 300 DPI)



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Fig. 5 82x45mm (96 x 96 DPI)


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Fig. 6 173x59mm (96 x 96 DPI)



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Fig. 7 82x46mm (300 x 300 DPI)


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Effects of Lattice Motion on Dissociative Chemisorption: Toward a

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