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RRKM Simulation of Hydrogen Dissociation on Cu(111): Addressing Dynamical Biases, Surface Temperature, and Tunneling Scott B Donald, and Ian Harrison J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 13 Dec 2013 Downloaded from http://pubs.acs.org on December 15, 2013

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The Journal of Physical Chemistry

RRKM Simulation of Hydrogen Dissociation on Cu(111): Addressing Dynamical Biases, Surface Temperature, and Tunneling Scott B. Donald, Ian Harrison* Department of Chemistry, University of Virginia, Charlottesville, VA 22904-4319

(Submitted 8/30/2013; Revised 11/5/2013)

Abstract The effects of dynamics, surface temperature, and tunneling on the dissociative chemisorption of hydrogen on Cu(111) are explored using a dynamically-biased precursor mediated microcanonical trapping (d-PMMT) model. Transition state vibrational frequencies were taken from recent GGA-DFT electronic structure calculations and the model’s few remaining parameters were fixed by optimizing simulations to a limited number of quantum-state-resolved associative desorption experiments. The d-PMMT model reproduces a diverse variety of dissociative chemisorption and associative desorption experimental results, and, importantly, largely captures the surface temperature dependence of quantum-state-resolved dissociative sticking coefficients. Molecular translational energy parallel to the surface was treated as a spectator degree of freedom. The efficacy of molecular rotational energy to promote dissociation, relative to normal translational energy, varied monotonically from -45% to 33% as the rotational energy increased. Efficacies for molecular vibrational energy and surface phonon energy were 60%. The efficacies did not vary with isotope change from *

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H2 to D2. The thermal dissociative sticking coefficient for H2/Cu(111) is predicted to vary as S (T ) = S0 exp ( − Ea / RT ) where S 0 = 0.075 and Ea = 49.2 kJ/mol over the

300 K ≤ T ≤ 1000 K temperature range. Dynamical effects are significant and suppress S (T ) by ~2 orders of magnitude as compared to statistical expectations. For thermal

dissociative chemisorption of H2/Cu(111) at 1000 K, a temperature of catalytic interest, normal translational energy is calculated to provide 57% of the energy necessary to react, surface phonons 23%, molecular rotation 15%, and vibration 5%. Tunneling is calculated to account for 13% of S (T ) at 1000 K, and more than 50% at temperatures below 400 K. These results demonstrate that many aspects of gas-surface reactivity can be modeled using microcanonical transition state theory subject to a few dynamical constraints.

Keywords: surface reactions, dissociative chemisorption, heterogeneous catalysis, kinetics, reaction dynamics

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I. Introduction The dissociative chemisorption of hydrogen on Cu(111) is an important model system for gas-surface reactivity. Thermally-driven associative desorption (the time reversal of dissociative chemisorption) experiments, interpreted using the principle of detailed balance, have shown that molecular vibrational and rotational energies are not as efficacious in promoting dissociative chemisorption as molecular translational energy directed along the surface normal.1-2 In a recent theoretical advance, a specific reaction parameter, density functional theory, potential energy surface (SRP-DFT PES) for H2/Cu(111) was optimized to give chemically accurate agreement with some experimental data based on six-dimensional (6D) dynamical calculations that addressed all molecular degrees of freedom but assumed a static Cu surface.3-5 Although several dynamical approaches towards treating the effects of surface thermal energy on reactivity are available,6-8 none9 have quantitatively accounted for Hodgson’s surface temperature dependent quantum-state-resolved H2 & D2/Cu(111) associative desorption observations10 that have been used to argue that surface thermal energy and normal translational energy have similar efficacies in promoting dissociative chemisorption. Here, we explore an alternative, high dimensional approach for analyzing and predicting experimental data based on a dynamically-biased precursor mediated microcanonical trapping11-13 (d-PMMT) model. This model employs transition state properties derived from the SRP-DFT PES and is largely consistent with Hodgson’s observations. Its few parameters, optimized to a limited set of associative desorption experiments, predominantly describe dynamical deviations from statistical reactive behavior. The importance of dynamics to the gas-surface reactivity could be readily assessed by setting the d-PMMT model’s dynamical parameters to limiting statistical values that recover a statistical (s-) PMMT model of the reactivity. In this paper, the d-PMMT model is used to estimate the impact 3 ACS Paragon Plus Environment


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of dynamics, surface phonons, and tunneling on hydrogen/Cu(111) dissociative chemisorption, in advance of more sophisticated dynamical calculations that would require at least 7 dimensions. II. d-PMMT Model The d-PMMT model has been described elsewhere in the context of the activated dissociative chemisorption of CH4 on Pt(111).11-13 Figure 1 provides a schematic depiction of the PMMT model for H2/Cu(111). Briefly, gas-surface collision complexes comprised of an incident molecule and a few local surface oscillators are assumed to have their exchangeable energy microcanonically randomized, at least as averaged over the ensemble of collision complexes prepared,14-16 if their pooled energy is sufficient to gain access to the strongly state-mixing regions of the reactive PES near the transition state. These “precursor complexes” (PCs) are effectively trapped in the vicinity of the molecular physisorption well located between the transition states for desorption and dissociative chemisorption (reaction). At energies sufficient to react, desorption lifetimes are ultrafast which curtails energy exchange with the surrounding surface. The PCs formed are assumed to go on to desorb or react with Rice-Ramsperger-KasselMarcus (RRKM)17 rate constants,

ki ( E * ) =

Wi ( E * ) hρ ( E * )

Eq. (1)

where E * is the active exchangeable energy whose zero occurs for the reactants at rest at infinite separation, Wi ( E * ) is the sum of states for transition state i = D , R with threshold energies ED* = 0 and E R* = E0 (see Fig. 1), ρ ( E * ) is the PC density of states, and h is Planck’s constant.


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Figure 1: Schematic depiction of the kinetics and energetics of hydrogen dissociative chemisorption via precursor-mediated microcanonical trapping (PMMT). At energies sufficient to react, collisionally formed precursor complexes, PCs, comprised of a hydrogen molecule interacting with s surface oscillators in the spatial vicinity of the physisorption well, are presumed to become transiently trapped between the transition states for desorption and reaction. Zero-point energies are implicitly included within the potential energy curve along the reaction coordinate. Applying the steady state approximation to the H2(p) coverage of the Fig. 1 kinetics scheme, *

F0 f ( E ) kR ( E )  → H 2( p )  H 2( g ) ← → 2 H (c) *  *

kD ( E )

Eq. (2)



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yields an expression for the experimental dissociative sticking coefficient, ∞

S = ∫ S ( E * ) f ( E * )dE * 0

Eq. (3)

where S ( E * ) is the microcanonical sticking coefficient,

kR (E* ) WR ( E * ) S (E ) = = k R ( E * ) + k D ( E * ) WR ( E * ) + WD ( E * ) *

Eq. (4)

and f ( E * ) is the probability distribution for forming a PC with exchangeable energy E * , which is calculated by convolution over the molecular and surface energy distributions describing the particular experimental conditions of interest. The experimental sticking coefficient is the average of the microcanonical sticking coefficient over the experimental probability of forming a PC with energy E * . The microcanonical sticking coefficient is the ratio of the number of open channels to react to the total number of open channels to either react or desorb. S ( E * ) is a completely statistical quantity derived solely from the quantum structure of the transition states. Equations (3-4) define the s-PMMT model whose derivation does not rely on detailed specification of PC characteristics other than knowledge of how many surface oscillators are involved in one. At energies sufficient to react, the PCs should be microcanonically distributed over the phase space between the desorption and reaction transition states.14, 18 Under steady state conditions involving collisional preparation, desorption, and reaction of PCs, this microcanonical (random) condition at any particular E * is likely to be approximated in a collision ensemble averaged way sufficient for the RKKM formalism to apply (vide infra).14-16 Alternatively, the PC can be considered as a kinetic construct to arrive at the Eq. (4) unbiased choice of possible


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outcomes for a PC formed at energy E * that defines the microcanonical (dissociative) sticking coefficient. However, for activated dissociative chemisorption of larger molecules, such as propane on Pt(111),19 the physisorption well is appreciable ( Ead = 41.5 kJ/mol),20 desorption lifetimes are longer, energy-exchange with the surrounding surface begins to be appreciable (and can be treated in a master-equation approach19, 21), and the PCs can be more readily described as being transiently localized in the vicinity of the physisorption well. With increasing alkane size, dissociative chemisorption eventually becomes unactivated and a precursor mediated thermalized trapping (PMTT)22 model is commonly used wherein the precursor is a molecule in the physisorption well thermalized to the surface temperature. For H2/Cu(111), the s-PMMT model does not admit any energy exchange between the PCs and surrounding surface that could lead to thermalization of PCs to the surface temperature and the depth of the physisorption well should be very close to the Ead = 2.14 kJ/mol value determined by inelastic scattering for HD on Cu(111).23 In the s-PMMT model all trajectories pass through a PC and Eqs (3-4) are uniformly applied so there is no need to define a trapping coefficient (it would be 1). Furthermore, the statistical model does not distinguish special trajectories involving no bounces, multiple bounces, or snarled trajectories as being “direct” or “indirect” or otherwise labeled in any way. The model has been demonstrated to provide a reasonable qualitative, first order approximation to the H2/Cu(111) reactivity24-25 but needs to be subjected to some dynamical biasing constraints developed below in the d-PMMT model to quantitatively replicate the experiments. The H2/Cu(111) system exhibits some intrinsic dynamical, non-statistical, and non-RRKM reactive behavior26 and the PMMT modeling presents an opportunity to quantitatively assess the effects of the dynamical deviations away from statistical behavior.



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Experimental dissociative sticking coefficients for hydrogen on Cu(111) are found to scale with the translational energy directed along the surface normal,27 En = Et cos 2 ϑ , and are independent of the translational energy directed parallel to the surface, Et sin 2 ϑ . Consequently, translational motion parallel to the surface is taken to be a spectator degree of freedom. Dynamical biases are introduced11 only during formulation of the PC exchangeable energy available to surmount the reaction barrier, E * = En + ηv Ev + η r Er + η s Es

Eq. (5)

where En , Ev , Er , Es are the normal translational energy, vibrational energy, rotational energy, and surface energy, respectively, and the ηi are efficacies for different kinds of energy to promote reaction relative to En . For example, the vibrational efficacy is,

∂S ηv ≡ ∂Ev

Ei≠v

∂S ∂En

−1

Eq. (6) E j≠n

which may be measured experimentally in quantum-state-resolved dissociative sticking coefficient experiments as,28

ηv =

∆En ∆Ev

Eq. (7) ∆Sv

where ∆Sv is the change in S accompanying a ∆Ev increase in vibrational energy at some particular En , and ∆En is the change in En required to gain the same ∆Sv change in S for molecules maintained in the lower energy vibrational state. The rotational efficacy may be 8 ACS Paragon Plus Environment

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measured similarly,29 but the surface efficacy cannot. For hydrogen dissociative chemisorption on Cu(111), similar kinds of efficacy values (vide infra) have been determined from thermal associative desorption experiments interpreted on the basis of detailed balance arguments.1-2 In the d-PMMT model presented here, all parameters and efficacies are determined by optimizing theoretical simulations of quantum-state resolved product state distributions from associative desorption experiments. The flux distribution for creation of PCs with exchangeable energy E * is calculated by convolution over the active energy distributions of the incident molecules and the surface energy distribution of the local surface oscillators describing the particular experiment,

f (E∗ ) = ∫

E∗ 0

f n ( En ) ∫

(E

∗

)

− En /ηv

f v ( Ev ) ∫

0 −1 s

( E − E −η E )/η ∗

n

0

v v

r

f r ( Er )

Eq. (8)

∗

× f s (η ( E − En − ηv Ev − ηr Er )) dEr dEv dEn

The surface energy distribution, f s ( Es ) , is calculated for the s surface oscillators making up each PC under the assumption the oscillators all vibrate at the mean phonon frequency of Cu,

ν s = 34 k Bθ Debye / hc = 175 cm-1. Hydrogen vibrational and rotational energy levels are calculated as Ev (v) = hc [ vωe − v(v + 1)ωe χ e ] and Er ( J ) = hc  Be J ( J + 1) − De J 2 ( J + 1) 2  using Herzberg’s spectroscopic constants including anharmonic and centrifugal corrections.30 The BeyerSwinehart-Stern-Rabinovitch31 algorithm was used to calculate sums and densities of states as necessary for Eqs. (4) & (8). Over the ultrafast desorption lifetimes of PCs formed at reactive energies, nuclear spin flips are assumed not to occur and so the manifold of rotational states available for desorption depends on the ortho or para spin statistics of the incident molecules. For simulations of 9 ACS Paragon Plus Environment


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supersonic molecular beam experiments involving molecules with thermally populated rotational states, dissociative sticking coefficients for the ortho and para components30 of the incident gas flux were calculated separately (e.g, with para-specific f p ( E ∗ ) and S p ( E ∗ ) in Eq.(3)) and then averaged using the “high temperature” statistical ratios, appropriate to room temperature cylinders of gas, of 3:1 for H2 and 2:1 for D2 to calculate the experimental S . Internal state distribution for hydrogen incident from supersonic molecular beams were modeled assuming the beam nozzle temperature, TN , sets the molecular vibrational and rotational temperatures as Tv = TN and Tr = 0.9 TN , respectively.2 Thermal dissociative sticking coefficients were

calculated for the thermal equilibrium mixture of the hydrogen ortho and para spin components. The microcanonical sticking coefficient, S ( E * ) of Eq.(4), depends on the properties of the transition states for desorption and reaction and is formulated statistically with no dynamical biasing (i.e., as appropriate for a s-PMMT model with {ηi } = 1 ). The transition state for PC desorption is taken to occur with the hydrogen molecule freely rotating and vibrating in the gasphase far from the s surface oscillators left behind on the surface. The reaction co-ordinate for desorption is translational motion along the surface normal and so this degree of freedom, along with the two spectator translations parallel to the surface, are missing from WD ( E * ) . Transition state properties for hydrogen dissociative chemisorption on Cu(111) were generously derived from the recent SRP-DFT-PES3 by Krishnamohan, Díaz, Olsen, and Kroes and these properties appear in tabled form for reaction with H2, D2, and HD within the electronic Supplementary Information (SI). The reaction co-ordinate for hydrogen dissociative chemisorption is a mixture of molecular bond extension and motion towards the surface3 and its vibrational mode frequency is imaginary. This reaction co-ordinate mode, along with the two spectator translational modes in 10 ACS Paragon Plus Environment

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the surface plane, are missing from WR ( E * ) . Calculation of S ( E * ) requires specification of {E0 , s} . Typically, {E0 , s} are free variables of a PMMT model and chosen by optimization of PMMT simulations to experiments. The apparent threshold energy for reaction,

E0 = Ec + ( ZTS − Z Re ) , is the classical barrier, Ec , plus the difference in vibrational zero-point energies between the transition state and reactants. For the SRP-DFT-PES, Ec =60.6 kJ/mol which yields E0 = 56.5 kJ/mol, 56.8 kJ/mol, and 57.7 kJ/mol for H2, HD, and D2 on Cu(111), respectively, based on the vibrational frequencies listed in the SI. Here, we treat E0 for H2/Cu(111) as a parameter and consistently list only its value below even when discussing results for other isotopes. Threshold energies for other isotopes were calculated from the H2/Cu(111) value using the SRP-DFT-PES transition state vibrational frequencies32 and Herzberg’s expressions30 for the experimentally observed vibrational frequencies of gas-phase hydrogen. The H2/Cu(111) dissociative chemisorption SRP-DFT-PES3-4 is classically exothermic by 31.4 kJ/mol (i.e., with no zero-point corrections). To account for tunneling through the barrier to dissociative chemisorption, the RRKM rate constant for dissociative chemisorption was written in its generalized form as,33 E WR ( E ) = ∫ p(ε t )k R ( E , ε t )d ε t 0 hρ ( E ) E 1 = p(ε t ) ρ R ( E − ε t )d ε t hρ ( E ) ∫0

kR ( E ) =

Eq. (9)

where E = E * + Z Re is the classical energy above the electronic potential energy surface whose zero is set by the well-separated reactants at T =0 K, ε t is the translational energy along the reaction coordinate leading to separated products, p (ε t ) is the tunneling probability, and 11 ACS Paragon Plus Environment


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ρ R ( E − ε t ) is the density of states, excluding the reaction coordinate mode, of the reactive transition state evaluated at the energy available to populate vibrational states of the transition state complex when tunneling occurs at ε t .17-18 The barrier to chemisorption was approximated by a 1D Eckart potential34-35 whose height was Ec = E0 − ( ZTS − Z Re ) while the curvature and exothermicity were fixed by the SRP-DFT-PES (see Fig. 2(a)). The analytic expression34 for the tunneling probability through a 1D Eckart barrier relates to ε t above the classical electronic potential evaluated for the well-separated reactants, without regard to zero-point energies, and so the sum of states for the reactive transition state referenced to the E * energy scale can be written as,

WR ( E * ) = ∫

E * + Z Re

0

p (ε t ) ρ R ( E * + Z Re − ε t )d ε t

Eq. (10)

where the argument of ρ R is the energy available to populate vibrational states of the transition state complex, taking in to account their vibrational zero-point energy. Tunneling was not relevant to calculation of WD ( E * ) for molecular desorption because there is no barrier for intact molecular adsorption/desorption that rises above the E * energy floor defined by the infinitely separated molecule and surface (e.g., Fig. 1). Fig. 2(b) illustrates the effects of tunneling on the microcanonical sticking coefficient, S ( E * ) .


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Figure 2: (a) The 1D Eckart potential representing the barrier for H2/Cu(111) dissociative chemisorption derived from the Kroes’ 6D SRP-DFT-PES with the d-PMMT optimized threshold energy of E0 = 43.4 kJ/mol (c.f., SRP-DFT-PES value of E0 = 56.5 kJ/mol). The zeropoint energies of the separated reactants and the transition state, Z Re and ZTS , as well as some other energies relevant to the RRKM and tunneling Eqs. (9-13) are also labeled. (b) Microcanonical dissociative sticking coefficient, S ( E * ) , evaluated with and without tunneling.

An average relative discrepancy, ARD, between theoretical simulations of quantities and their experimental values for a particular set of experiments was defined, for example with regards to dissociative sticking coefficients, as,

ARD =

Stheory − Sexp t min( Stheory , Sexp t )

Eq. (11)

ARDs were a useful measure for evaluating how well simulations reproduced experiments.



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A statistical microcanonical transition state model is recovered in the limit that {ηi } → 1 . Earlier s-PMMT models for H2/Cu(111) have been qualitatively successful in reproducing the surface temperature dependence of the reactivity,24-25 and a 3-parameter model treating rotation as a spectator25 gave fairly close quantitative agreement with other aspects of the reactivity (e.g., associative desorption angular distributions, product state distributions, isotope effects, etc.). III. Results and Discussion The d-PMMT model parameters {E0 , s,ηv ,η s ,η r } were fixed by minimizing the ARD between simulations and experiments for the 925 K hydrogen/Cu(111) associative desorption data of Figs 3 & 4, that comprise a limited subset of the available experimental data. The simulations assumed the efficacies are independent of isotope, and threshold energies for dissociative chemisorption vary only with changes in zero-point energies, E0 = Ec + ( ZTS − Z Re ) . The H2/Cu(111) parameters are { E0 = 43.4 kJ/mol, s = 1 , ηv = 0.60 , η s = 0.60 ,

ηr = −0.45 + 0.78 Erf [0.026 Er ] } where the rotational efficacy ηr ( Er ) is a function of the rotational energy expressed in kJ/mol and Erf is the error function. The H2 threshold energy for dissociative chemisorption fixes both Ec = 47.5 kJ/mol and E0 ( D2 ) = 44.6 kJ/mol. All the dPMMT calculations in the paper employ this single set of parameters and we generally list only the H2/Cu(111) values even when discussing experiments involving other isotopes. A. Associative Desorption at 925 K 1. Rovibrationally-Resolved Translational Energy Distributions


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The thermally driven associative desorption of chemisorbed hydrogen atoms to form desorbing hydrogen molecules was studied experimentally at IBM’s Almaden laboratories1-2 by diffusing hydrogen gas through a heated Cu(111) single crystal that formed the end of a pipe such that a continuous flow of reaction products could be maintained under ultrahigh vacuum conditions. For hydrogen associatively desorbing from the Cu(111) surface at 925 K, Figure 3 provides some examples of state-resolved translational energy distributions derived from timeof-flight (TOF) spectra following state-selective resonance enhanced multi-photon ionization. Figure 4 summarizes how the mean translational energy, Et , of the desorbing molecules varied with rovibrational state for the measured TOF spectra. The principle of detailed balance applied to hydrogen gas and a Cu(111) surface at thermal equilibrium with one another requires that the associatively desorbing flux should exactly balance the dissociatively sticking flux, even when the fluxes are specified at quantum state resolved levels of detail.2, 36 With this principle, and assuming the absence of an ambient thermal gas over the experimental Cu(111) surface did not alter the thermal associative desorption dynamics, the d-PMMT model of dissociative chemisorption was used to calculate the associative desorption flux distributions. The model’s parameters were fixed by minimizing the ARD of simulations to the experimental data of Figs. 3 & 4. The state-resolved Et data of Fig. 4 are reproduced by the d-PMMT model with an ARD of 2.4 % and the full TOF spectra of Fig. 3 are also simulated quite well. Figure 4 provides evidence for translational activation of the dissociative chemisorption because the state-resolved

Et are considerably greater than 2k BTs . Under thermal equilibrium conditions, if the dissociative sticking coefficient was independent of Et then the all molecules striking the surface, including the successfully chemisorbing molecules, would have a flux–weighted



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Maxwell-Boltzmann distribution with Et = 2kBTs and by detailed balance so too must the associatively desorbing molecules. Indeed, by detailed balance, the Fig. 4 data provide the mean translational energies required to successfully react for molecules impinging on to the surface with a Maxwell-Boltzmann distribution at temperature Ts . The reduction in the Et required to react as the vibrational energy increases indicates the dissociative chemisorption is vibrationally activated. However, vibrational energy is not as efficacious as normal translational energy in promoting dissociative chemisorption1-2 and the d-PMMT modeling indicates a vibrational efficacy of ηv = 0.60 .

Figure 3: Vibrational state dependence of experimentally-derived translational energy distributions (dashed lines)1-2 for the associative desorption of (a) H2( v , J =2) and (b) D2( v , J =2) are compared to d-PMMT simulations (lines). Fig 4 shows that increasing the rotational energy of thermal molecules incident on the surface initially increases and then decreases the Et required to overcome the activation


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barrier to dissociatively chemisorb. Consequently, the rotational efficacy is a function of the rotational energy1-2 and the d-PMMT model finds ηr ( Er ) varies from -0.45 to 0.33 with increasing rotational energy as depicted in Fig. 5. The effect of ηr ( Er ) on d-PMMT simulations of translational energy distributions for associatively desorbing (or dissociatively sticking) H2(v=0) over an extended range of rotational quantum number J is shown in Fig. 6.

Figure 4: Measured rovibrational state resolved mean translational energies (points)1-2 for recombinatively desorbing (a) H2 and (b) D2 are compared to d-PMMT simulations (lines). A reference line demarking the Et = 2kBTs expectation for a system exhibiting a dissociative sticking coefficient independent of Et is also provided.



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Figure 5: The efficacy of rotation (solid line) and exchangeable rotational energy (dashed line) used in d-PMMT simulations are plotted as a function of the rotational energy of the incident molecules for hydrogen dissociation on Cu(111).


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Figure 6: Rotational state dependence of measured translational energy distributions (points)2 for the associative desorption of H2( v =0, J ) are compared to d-PMMT simulations (lines). The efficacy of surface vibrational (phonon) energy to promote reactivity influences the leading and trailing edges of the state-resolved translational energy distributions (c..f., Fig. 3 with Fig. 5 of Ref. 25) and the d-PMMT finds η s = 0.60 . Interestingly, although the molecular and surface vibrational quanta differ substantially in energy (i.e., 4159 cm-1 (H2), 2992 cm-1 (D2) vs. 175 cm-1 (phonon)), the molecular and surface vibrational efficacies share the same value of 0.60. 2. Rovibrational Product State Distributions



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Vibrationally-resolved rotational Boltzmann plots for hydrogen associatively desorbing from Cu(111) at a surface temperature of 925 K are shown in Figure 7. The experimental rovibrational product state probabilities, Pv , J , derive from analysis of the REMPI TOF spectra and are normalized to 1 when summed over the range of measured v, J states.1-2 Theoretical simulations of Pv , J were made over a wider range of vibrational ( v = 0 − 5 ) and rotational (

J = 0 − 26 ) states and were normalized accordingly. Agreement between theory and experiment is quite good (ARD= 47.3%), especially when considering that no product population information is embedded within the Figs 3-4 experimental data from which the d-PMMT parameters were fixed. An effective rotational temperature, Tr ,eff , for the associatively desorbing molecules defined as,

Tr ,eff

 ∂ ln  Pv , J / g n (2 J + 1)   = −kB     ∂Er  

−1

Eq. (12)

where g n is the nuclear spin degeneracy, depends on the slope of the Boltzmann plots. Fig. 7 shows that the effective rotational temperature varies with rotational energy and vibrational state. For vibrational ground state molecules, the calculated Tr ,eff initially increases with Er , beginning at 650 K and rising to a peak of 1750 K near Er = 40 kJ/mol, whereupon Tr ,eff monotonically diminishes with Er towards a limiting value of Tr ,eff = Ts = 925 K. The modulation of Tr ,eff ( Er ) diminishes for molecules with increasing vibrational excitation.


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Figure 7: Boltzmann plots compare the measured vibrationally-resolved rotational energy distributions (points)1-2 for associative desorption of (a) H2 and (b) D2 to d-PMMT simulations (lines). Vibrational product state distributions for associative desorption are provided in Table I. The experimental geometry led to a polar angle of acceptance range for state-resolved molecular detection of ϑ ≤ 20° measured away from the direction of the surface normal. Given the desorption angular yield distribution ∝ cos9 ϑ at Ts = 925 K,37 46% of the desorbing molecules would have passed through the ϑ ≤ 20° range for state-resolved detection. Agreement between theory and experiment1-2 for the angle-constrained vibrational product state distribution is excellent save for the populations in the lowest two vibrational states of D2. The calculated product state distribution integrated over all desorption exit angles has slightly higher levels of vibrational excitation, as might be expected by detailed balance with a dissociative sticking coefficient that scales with only the normal component of the molecular translational energy. 21 ACS Paragon Plus Environment


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H2

Pv – Expt.1-2

Pv – d-PMMT

Pv – d-PMMT

ϑ ≤ 20°

ϑ ≤ 20°

Angle integrated

v

J sum

0

0-10

96.7%

96%

93.7%

1

0-7

3.3%

3.9%

6.1%

100%

99.9%

99.8%

∑P

v, J

D2

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0

0-14

82.4%

90.2%

86.3%

1

0-11

16.8%

9.1%

13.0%

2

0-8

0.8%

0.6%

0.7%

100%

99.9%

100%

∑P

v, J

Table 1. Vibrational product state distributions for thermal associative desorption of H2 and D2 from Cu(111) at 925 K.

B. Associative Desorption: Angular Distributions Angular yield distributions for thermally desorbing hydrogen and deuterium are presented in Figure 8. The d-PMMT simulations reproduce the experimental distributions37 fairly well with an overall ARD of 33%. The simulations are narrower than the experimental distributions, particularly at low surface temperatures. At 925 K, the H2/Cu(111) associative desorption angular yield distribution varies experimentally as cos9 ϑ which compares to predictions of cos10 ϑ by d-PMMT calculation and cos 25 ϑ by 5-D quantum scattering calculations.38 22 ACS Paragon Plus Environment

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Figure 8: Measured angular distributions (points)37 for H2 and D2 recombinative desorption from Cu(111) at several surface temperatures are compared to d-PMMT simulations (lines).

For associatively desorbing D2 at Ts = 1000 K, Fig. 9 compares experimental mean translational energies39 as a function of desorption angle, to detailed balance simulations of the mean energies of the successfully reacting precursor complexes (e.g., Ei (ϑ )

R

for degree of

freedom i) formed in thermal dissociative chemisorption at this temperature. This figure also makes apparent the difference between the reactive means of the total active (i.e., non-spectator) energy, Etot =

∑

Ei , and the exchangeable energy, E * =

i = n,v ,r , s

PMMT Et (ϑ )

R

∑

ηi Ei , where ηn = 1 . The d-

i = n,v ,r , s

underpredicts the experimental values across nearly all angles, yielding an

ARD of 6%. Some underprediction is expected because the experimental Et (ϑ )

R

were

calculated from only the fast component of the desorbing D2 time-of-flight spectra. Nevertheless, the d-PMMT prediction that Et (ϑ )

R

should increase slowly with ϑ to reach a broad maximum

near 30° before decreasing sharply at higher angles towards an asymptotic limit of 23 ACS Paragon Plus Environment


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Et (ϑ → 90°)

R

Page 24 of 62

→ 2kBTs , seems in good qualitative agreement with the experiments. Other

theoretical models, such as the 1D Van Willigen model,40-41 5D quantum scattering model,38 or commonly applied error function models10, 42 predict that Et (ϑ )

R

should monotonically

increase with ϑ . For a dissociative chemisorption system obeying normal translational energy scaling, the Et (ϑ → 90°)

R

→ 2kBTs limit should rigorously apply because at ϑ = 90° there is

no influence of Et on the dissociative sticking coefficient and so the successfully reacting molecules at that angle must have the same translational energy distribution and mean as the incident molecules. The d-PMMT calculations of Fig. 9 indicate the surface and molecular internal energies become increasingly important to the gas-surface reactivity as the molecular angle of incidence increases away from the surface normal.

Figure 9: Measured mean translational energies (points)39 for the fast component of D2 recombinative desorption as a function of angle are compared to d-PMMT predictions (lines) of


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the mean energies derived from different degrees of freedom for the successfully reacting precursor complexes formed in thermal dissociative chemisorption of D2 at 1000 K.

C. Associative Desorption: Surface Temperature Dependence of Pv,J(Et; ϑ = 0°) Rovibrational eigenstate-resolved, translational energy distributions, P ( Et ,ϑ = 0o ;ν , J , Ts ) , for hydrogen associative desorption from Cu(111) were measured by Murphy and Hodgson10 at several surface temperatures and are shown in Figure 10 along with d-PMMT simulations. Hodgson used detailed balance to calculate temperature-dependent, relative dissociative sticking coefficients according to, P ( Et ,ϑ = 0o ;ν , J , Ts ) S ( Et ,ϑ = 0 ;ν , J , Ts ) ∝ f MB ( Et , Ts ) o

Eq. (13)

where f MB ( Et , Ts ) ∝ Et exp ( − Et / kbTs ) is the flux weighted Maxwell-Boltzmann distribution for molecular translational energy at temperature Ts . These relative dissociative sticking coefficients were normalized to a common absolute scale by assuming the high translational energy limit reached was always 1 (c.f., 0.25 for H2/Cu(111) by Rettner and co-workers2). Experimentally derived, state-resolved dissociative sticking coefficients calculated in this manner10 are shown in Figure 11(a) and where they are compared against direct d-PMMT calculations. The d-PMMT model underpredicts these experimentally derived dissociative sticking coefficients (ARD = 4911%), particularly at the lowest translational energies where the signal-to-noise ratio and subtraction of background gas contribution in the P ( Et ,ϑ = 0o ;ν , J , Ts ) distributions become increasing concerns in experimental determination of S ( Et , ϑ = 0o ;ν , J , Ts ) .



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Figure 10: Measured rovibrational state resolved translational energy distributions (points)10 for the associative desorption of (a) H2( v =0, J =1) and (b) D2( v =0, J =2) at several surface temperatures are compared to d-PMMT simulations (lines).

Figure 11: (a) Surface temperature dependent, rovibrational state resolved, relative dissociative sticking coefficients (points)10 derived from the measured associative desorption energy distributions of Fig. 9 are compared to d-PMMT simulations of absolute dissociative sticking 26 ACS Paragon Plus Environment

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coefficients (lines). (b) Effective activation energies derived from Arrhenius plots of the dissociative sticking coefficients shown in (a) are plotted for experiments (solid points) and dPMMT simulations (lines). Effective activations energies " Ea (Ts )" = − k B ∂ ln S / ∂Ts−1 , calculated from the slopes of the best fit lines in the Arrhenius plots of Fig. 11(a), are plotted as a function of normal translational energy in Figure 11(b). Interestingly, the effective activation energies fall as the normal translational energy increases. The experimentally determined " Ea (Ts )" fall linearly with constant slope ∂ " Ea (Ts )"/ ∂En = −0.9 . The d-PMMT predicts ∂ " Ea (Ts )"/ ∂En = −η s−1 = −1.7 initially and a falling off of the slope towards zero in the neighborhood of En

E0 . This

theoretical behavior can be explained fairly readily. The non-equilibrium Tolman expression for the effective activation energy21 is " Ea (Ts )" = −kB ∂ ln S / ∂Ts−1 = Es

R

− Es , where Es

R

and

Es are the mean surface energies for those PCs that successfully react and for all the PCs formed, respectively. Furthermore, in order for a PC to successfully react the active exchangeable energy must be greater than the reaction threshold energy such that for eigenstateresolved dissociative sticking experiments, Es

R

≥ [ Eo − ( En + ηv Ev + ηr Er ) ] / η s . When

molecular energy alone is insufficient to surmount the reaction barrier and Es

R

must be much

greater than Es to allow for reaction, the last expression effectively becomes an equality plus a Ts –dependent constant and ∂ " Ea (Ts )"/ ∂En = ∂ Es

R

/ ∂En = −ηs−1 . With increasing molecular

energy (increasing En in these experiments), the inequality constraint on Es becomes negligible such that Es

R

R

eventually

→ Es , " Ea (Ts )" → 0 , and ∂ " Ea (Ts )"/ ∂En → 0 .



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Murphy and Hodgson10 argued that thermal motion of the surface is just as efficacious in promoting reactivity as normal translational motion because their measured " Ea (Ts )" values at low translational energies are very similar to the barriers to translational activation on a cold surface. They further speculated that because ∂ " Ea (Ts )"/ ∂En = −0.9 , the effective activation energy with respect to surface temperature should eventually become negative for En > E0 whereupon surface thermal motion would inhibit dissociative chemisorption. Holloway and coworkers10 performed 5D quantum wavepacket calculations for D2/Cu(111) reactivity on a reduced dimensionality potential energy surface in which a thermal surface oscillator was coupled to the incident deuterium molecule. Their calculations of the dissociative sticking coefficient for D2( v =1, J =2) versus En found an Arrhenius dependence with surface temperature at low En . They found that " Ea (Ts )" decreased with En linearly with slope ∂ " Ea (Ts )"/ ∂En = −0.86 over the narrow energy range 29 kJ/mol ≤ En ≤ 35 kJ/mol. As En increased further ∂ " Ea (Ts )"/ ∂En rolled off considerably and this slope reduction was tentatively attributed to the increasing role of unspecified quantum resonances and threshold features not present at lower En . Unfortunately, there are no experimental S ( Et , ϑ = 0o ;ν , J , Ts ) data for D2(

v =1, J =2)/Cu(111) as a function of Ts with which to compare. The d-PMMT model discussion above provides a simpler explanation, grounded in transition state theory, for the variation of " Ea (Ts )" with En as a result of molecule-surface oscillator coupling.

D. Non-equilibrium Dissociative Sticking: Supersonic Molecular Beams with Surface at Ts = 120 K.


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Experimental dissociative sticking coefficients1-2 performed using seeded supersonic beams of H2 and D2 with a heated nozzle impinging on Cu(111) surface at Ts = 120 K beam are reported as a function of normal translational energy in Fig. 12 along with predictions of the dPMMT model and those from 6D quantum dynamics3-4 calculations. The H2/Cu(111) experiments are replicated quite well with the d-PMMT model achieving an ARD of 26% and the 6D quantum dynamics calculations an ARD of 60%. The specific reaction parameter of the SRP-PES was optimized for agreement of the 6D quantum dynamics calculations with the D2/Cu(111) experiments at TN =2100 K of Fig. 12(c). For the D2/Cu(111) experiments, the 6D quantum dynamics calculations achieved an ARD of 23% which compares to 48% for the dPMMT model which performed relatively poorly only for the TN =2100 K experiments. Unfortunately, experimental beam velocity and temperature parameters required for accurate theoretical simulations are only available in the literature3-4 for this subset of an extended range of supersonic beam measurements made at IBM.1-2 Rettner, Michelsen, and Auerbach (RMA)2 found they could reproduce all their H2/Cu(111) supersonic beam experimental results for the Ts = 120 K surface adequately well using detailed balance and the parameters derived from error function (erf) curve fits to state-resolved associative desorption experiments made at Ts = 925 K, given some assumptions about the surface temperature dependence of the erf parameters for v = 0, 1, free fitting of erf parameters for v = 2 to the Ts = 120 K experiments, and normalization of all the state-resolved dissociative sticking coefficients to a common limiting high value of 0.25.2 RMA’s assertion that detailed balance applies well to the H2/Cu(111) system, is affirmed here, in an assumption free manner, by the dissociative sticking predictions of the d-PMMT model whose relatively few parameters were determined solely on the basis of the Figs 3-4 product translational energy distributions from the thermal associative desorption of hydrogen at 925 K 29 ACS Paragon Plus Environment


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(i.e., without any information about relative yields across the desorption product states, or about dissociative sticking coefficients).

Figure 12: Non-equilibrium dissociative sticking coefficient measurements (filled points)1-2 for supersonic molecular beams of H2 and D2 impinging on Cu(111) at Ts = 120 K and with nozzle temperature TN are compared to 6D quantum dynamics3-4 and d-PMMT simulations (open points).


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E. Thermal Dissociative Sticking: Ambient Gas In Fig. 13(a) the thermal dissociative sticking coefficient, S (T ) , for ambient H2 gas above an isothermal Cu(111) surface calculated by the d-PMMT model is compared to RMA’s extrapolation2 based on their erf analysis of thermal associative desorption at Ts = 925 K, and the non-equilibrium dissociative sticking of supersonic molecular beams at Ts = 120 K discussed above. Over the 300 K ≤ T ≤ 1000 K temperature range, the best Arrhenius fit to the d-PMMT model calculations is S (T ) = S0 exp ( − Ea / RT ) where S 0 = 0.075 and Ea = 49.2 kJ/mol. The activation energy at 925 K according to the RMA extrapolation2 is 48.4 kJ/mol. To assess the impact of dynamics on S (T ) an additional calculation was made for a statistical(s) -PMMT model in which all the efficacies of the d-PMMT model were set to their limiting statistical values of one (i.e., ηi → 1 ). Fig. 13(a) indicates that over the temperature range of catalytic interest, dynamical effects are substantial and serve to reduce the thermal gas-surface reactivity by two orders of magnitude as compared to statistical expectations. Figure 13(b) extends d-PMMT calculations of S (T ) for H2 and D2 to lower temperatures where tunneling through the reaction barrier is signaled by exaggerated curvature and a kneeing of the Arrhenius plots. To make the effects of tunneling explicitly apparent, d-PMMT calculations of S (T ) without inclusion of tunneling pathways are also shown. Tunneling clearly dominates the reactivity at the lowest temperatures where it can increase the tunneling-excluded reactivity by many orders of magnitude. Interestingly, even at the high temperatures of catalytic interest, tunneling pathways remain responsible for a significant fraction of the observed reactivity. The d-PMMT calculations of Fig. 13 (c)-(d) explicitly illustrate the kinetic isotope



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Page 32 of 62

effect and indicate that tunneling pathways for H2/Cu(111) account for 13% of S (T ) at 1000 K, increasing to greater than 50% as the temperature falls below 400 K.

Figure 13: (a) Thermal dissociative sticking coefficient, S (T ) , for H2/Cu(111) calculated by several models. (b) d-PMMT S (T ) for H2 and D2 on Cu(111) are compared at lower temperatures where the effects of tunneling through the reaction barrier can be discerned (solid lines). Dashed lines give equivalent d-PMMT calculations without tunneling. (c) d-PMMT


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kinetic isotope effect. (d) d-PMMT calculation of the fraction of S (T ) that derives from trajectories that tunnel through the energetic barrier to dissociative chemisorption. Although the d-PMMT model performs well in quantitatively replicating the supersonic molecular beam dissociative sticking coefficient measurements of Fig. 12 for both H2 and D2 these relatively high energy experiments are not ideally suited for discerning the effects of tunneling. Preferable would be to compare thermal dissociative sticking coefficients for H2 and D2 at as low a temperature as possible. Unfortunately, there are no modern surface science measurements of S (T ) for hydrogen on Cu(111).43 Instead, we compare d-PMMT and RMA2 predictions of S (T ) for H2(D2)/Cu(111) with experimental measurements of S (T ) for the H2(D2)/Cu(110) system44 in Fig. 14. This is not entirely unreasonable because RMA found that sputtering their Cu(111) surface to introduce defects at 100 K did not appreciably effect the supersonic molecular beam dissociative sticking coefficient measurements of the kind made in Fig. 12 at En = 30 kJ/mol which suggests that surface structure sensitivity for this reaction may be relatively modest.2 The d-PMMT calculations for S (T ) on Cu(111) are generally less than experimental values on Cu(110) which is consistent with the typical finding45 that flat, close packed surfaces are less reactive than more open and atomically rougher surfaces. Nevertheless, the activation energy for the H2/Cu(110) data is Ea = 60±6 kJ/mol,44 slightly higher than the theoretical predictions for Cu(111). The experimental measurements on Cu(110) have some scatter but their averaged kinetic isotope effect (KIE) is 1.14 with a mean value near 500 K of 1.48. This compares to KIE predictions for Cu(111) at 500 K for the d-PMMT model of 3.1 (2.48 w/o tunneling), and 2.55 for the RMA model. The nominal effect of tunneling on the KIE



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Page 34 of 62

in the d-PMMT model is modest even at this low temperature for which modern experimental S (T ) data is available only for the related Cu(110) surface.

Figure 14: d-PMMT and RMA2 model calculations of S (T ) for H2 and D2 on Cu(111) are compared to experimental measurements44 on Cu(110). Fractional energy uptakes defined as fi = Ei

R

/ Etot

R

,where Ei

R

is the mean energy

derived from the ith degree of freedom for the successfully reacting PCs formed and Etot

R

is the

mean total active energy for the successfully reacting PCs formed, are useful for assessment of which degrees of freedom are most important in supplying the active energy required to surmount the activation barrier to thermal dissociative chemisorption. By application of detailed balance to the thermal equilibrium reactivity, these uptakes are equivalent to the fractional total active energy releases into the different degrees of freedom for the products of thermal associative desorption. Fig. 15 reports mean energies for the successfully reacting PCs formed, and fractional energy uptakes calculated for thermal dissociative chemisorption over a range of 34 ACS Paragon Plus Environment

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temperatures of catalytic interest, whereas Fig. 16 shows how the f i vary with molecular angle of incidence for reaction at 1000 K. The angle-integrated f i of Fig. 15(b) show that normal translational energy supplies more than 48% of the energy required to surmount the activation barrier, the surface contributes 12 - 26%, rotation 5 - 18%, and vibration 0.2 - 8%. The angleresolved f i of Fig. 16 for reaction at 1000 K show that when normal translational energy ( En = Et cos 2 ϑ ) is readily available near ϑ = 0° f n is particularly high but as ϑ increases and

En falls off then it is increasingly energy from the surface and molecular vibration that is used to surmount the activation barrier. Indeed, for ϑ > 40° the fractional energy uptake is greatest from the surface. An important point to note from Fig. 15 is that surface vibrational energy (phonons) always plays a substantial role in activated thermal dissociative chemisorption, and, apparently, a more substantial role than either molecular rotational or vibrational energy.

Figure 15: d-PMMT predictions of (a) mean energies derived from different degrees of freedom for the successfully reacting precursor complexes formed in thermal dissociative chemisorption, 35 ACS Paragon Plus Environment


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Page 36 of 62

and (b) the fractional energy uptakes from the different degrees of freedom (i.e.,

fi = Ei

R

/ Etot

R

) for thermal dissociative sticking, over the temperature range of interest to

catalysis.

Figure 16: d-PMMT predictions of the fractional energy uptakes as a function of molecular angle of incidence at 1000 K. RMA’s calculations of the contributions different rovibrational states made to the thermal dissociative sticking coefficient of H2 on Cu(111) led them to conclude that S (T ) is dominated by translationally hot molecules in the ground vibrational state, that contributions from vibrationally excited molecules are negligible, and that rotational motion slightly inhibits reactivity at catalytically relevant temperatures. These conclusions can be compared to application of detailed balance to the experimentally measured product state distributions from thermal associative desorption at 925 K shown in Table I and Fig. 7 which should also represent the distributions of the successfully dissociatively chemisorbing molecules under thermal


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equilibrium conditions at 925 K (albeit only for those incident at angles ϑ ≤ 20° ). The subthermal Tr ,eff of Eq.(12) observed at low J in associative desorption is consistent with the slightly inhibitory effect of rotational motion on dissociative sticking for the thermally-easilyaccessible low J states (n.b., for the higher J states Tr ,eff is superthermal). In related work,44 Campbell and Campbell’s study of the pressure dependence of hydrogen reactivity on Cu(110) led them to predict that the activation barrier to dissociative sticking is surmounted almost entirely by translational energy, with, at best, only modest assistance from rotational and/or vibrational energy. The fractional energy uptakes of Fig. 15 and 16 provide a convenient quantitative guide to what kinds of energy are used to surmount the activation barrier to thermal dissociative chemisorption and allow the importance of surface phonon energy in this process to be easily recognized. Given the fractional energy uptake from rotation is relatively modest and slowly varying, it may be reasonable to simply treat rotation as a spectator degree of freedom, especially because the rotational efficacy is low. d-PMMT simulations performed with rotation as a spectator (see figures in Supplementary Information) were found capable of reproducing the majority of the experimental data available with comparable ARDs to those demonstrated here. In that case, optimal parameters for H2/Cu(111) were { E0 = 48 kJ/mol, s = 1, ηv = 0.50, ηs = 0.60} . The finding that rotation plays a relatively modest role in the activated dissociative chemisorption of this benchmark H2/Cu(111) system with fully state-resolved experimental data is interesting because the same rotation-as-a-spectator approximation has also met with some success in simulations of the dissociative chemisorption of larger molecules, such as CO2/Rh(111)46 and CH4/Pt(111).11-13 If rotation could generally be approximated as a spectator degree of freedom in



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activated dissociative chemisorption it would simplify the modeling of these important gassurface reactions.

F. Comparison to Other Work 1. Analysis with Error Function Sticking Model Rettner and coworkers1-2 and also Murphy and Hodgson10 modeled their state-resolved associative desorption and molecular beam dissociative sticking experiments for hydrogen on Cu(111) by assuming the molecular eigenstate-resolved dissociative sticking coefficient obeyed a semi-empirically defined error function form,

S ( Et , ϑ ; v, J , Ts ) =

A(v, J )  1 + Erf 2 

 Ee − Ed (v, J )      W (v, J ; Ts )  

(14)

in which A( v , J ) , Ed ( v, J ) , and W ( v, J ; Ts ) are adjustable parameters. An effective translational energy is defined as, Ee = Et cosn ϑ

(15)

where n is an adjustable parameter. Sticking coefficients of the Eq. (14) form have a sigmoid shape as a function of Ee . Rettner and coworkers assumed normal translational energy scaling of

S with n = 2 was applicable,2 but Murphy and Hodgson10 argued that n = 1.8 or n = n( Et ) could improve agreement with experiments. The A( v , J ) parameter is the limiting value of the sticking coefficient at high Ee , Ed ( v, J ) is the value of Ee when the sticking has reached half its limiting value, and W ( v, J ; Ts ) is a width parameter. There is sufficient flexibility in the erf


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functionality that state-resolved experimental data can be fit very well. The success of the erf functionality is typically ascribed to the idea that a Gaussian distribution of reaction barriers is sampled by the ensemble of incident molecules colliding with the surface.47-50 The particular barrier an individual molecule encounters is presumed to be a function of the molecule’s state, and its experimentally-uncontrolled orientation, vibrational phase, and impact parameter across the surface unit cell. If the molecule’s Ee is greater than the particular barrier it encounters it is presumed to react with unit probability. Under the premises of the erf model, Ed ( v, J ) is the mean dynamical barrier to reaction encountered for a molecule incident in the (v, J ) rovibrational state. Rettner and coworkers1-2 defined a vibrational efficacy for reaction as,

ην ,e =

Ed (v − 1, J ) − Ed (v, J ) ∆Ed = Ev (v − 1, J ) − Ev (v, J ) ∆Ev

(16)

to characterize how efficacious vibrational energy was in lowering the translational energy requirement to surmount the mean dynamical barrier [c.f., Eqs (6-7)]. The experimentally derived Ed ( v, J ) values are numerically very similar to the rovibrationally resolved Et values of Fig. 4. For H2 and D2 on Cu(111),2 experiments found that ην ,e = 0.51 ± 0.02 for ν = 1 and 2, independent of J state and isotope. The rotational efficacy, ηr ,e , defined similarly to Eq. (16), was initially negative, beginning near -1, because Ed ( v, J ) initially increases slightly with J (c.f., Et variation in Fig. 4). With increasing J , the rotational efficacy steadily increases before leveling off to a constant value within the range 0.25-0.48, depending weakly on vibrational state and isotope. Rettner and coworkers’ efficacy findings1-2 based on erf model 39 ACS Paragon Plus Environment


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analysis of their experiments are essentially recapitulated by the d-PMMT model analysis which finds ηv = 0.60 and ηr = −0.45 + 0.78 Erf [0.026 Er ] , independent of vibrational state and isotope, such that ηr varies from -0.45 to a limiting value of 0.33 as J increases. However, the d-PMMT model extracts an additional efficacy value for surface vibrational energy of η s = 0.60 . The erf model does not allow for calculation of a surface efficacy but does find that the width parameter W ( v, J ; Ts ) , which determines the broadness of the transition region of the sigmoid shaped sticking curve [i.e., S ( Et , ϑ ; v, J , Ts ) is sigmoidal when plotted on a linear scale versus

Ee ], increases with Ts .42 The extrapolated Ts dependence of the reactivity,2, 42 based primarily on molecularly state-averaged experiments, may be treated poorly in the erf model given its inability to replicate Hodgson’s P ( Et ,ϑ = 0o ;ν , J , Ts ) of Fig. 11 (c.f., Fig. 8 of Ref. 10). The experimental vibrational efficacies determined using Eq.(16) for H2/Cu(111) are equivalent to the efficacies defined by Eq. (7) given the RMA finding that the A( v , J ) values for this system are all AH 2 = 0.25. The En required to achieve a dissociative sticking coefficient of

AH 2 / 2 for the ground vibrational state molecules is then Ed (v = 0, J ) whereas it is Ed (v = 1, J ) for the v = 1 state molecules. Consequently, beginning at En = Ed (v = 1, J ) for a ground state molecule where the dissociative sticking coefficient is negligible a change ∆Sv = AH 2 / 2 is possible by increasing En by ∆En = Ed (v = 0, J ) − Ed (v = 1, J ) or by increasing the vibrational energy by one quantum, ∆Ev , such that the ratio ∆En / ∆Ev yields the vibrational efficacy of both Eqs 7 and 16. For the D2/Cu(111) system, Rettner and coworkers1 found the limiting A( v , J ) values varied with vibrational state (by as much as a factor of 2), which makes the experimental


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vibrational efficacies reported based on Eq. (16) no longer exactly comparable to the d-PMMT vibrational efficacies of Eqs.(6-7), despite similar numeric values. There is some economy of parameterization gained when analyzing the experimental data using the d-PMMT model as compared to the erf model. Seven d-PMMT parameters were fixed by fitting the experimental state-resolved associative desorption data of Figs 3-4 alone – dynamical data which provided no information concerning desorption state relative yields or dissociative sticking coefficients. Subsequently, the full range of other desorption phenomena and dissociative sticking coefficients at different surface temperatures could be reproduced with good fidelity (i.e., Figs 6-14). The erf model requires fixing 3 parameters per quantum state to define S ( Et , ϑ ; v, J , Ts ) at a given surface temperature. To optimally describe the associative desorption of the 19 H2 and 36 D2 rovibrationally resolved states of Fig. 4 at Ts = 925 K required specification of 165 parameters. Additional state-averaged supersonic molecular beam dissociative sticking coefficient experiments at Ts = 120 K were required to fix absolute values for the A( v , J ) parameters and some associative desorption experiments at different temperatures42 were needed to fix further parameters describing the surface temperature dependence of the W ( v, J ; Ts ) parameters. The surface temperature dependence of the erf model is poorly understood and Hodgson notes that the Ts -dependence assumed by RMA to make their extrapolation and assertion that detailed balance applies at 925 K fails in its application to his state resolved P (E t ) of Fig. 10 (e.g., as simulated in his Fig. 8 of Ref. 10). Although the erf model offers sufficient flexibility to accurately fit state-resolved experimental data, there is no compelling necessity to invoke (v, J , Ts ) state-specific Gaussian distributions of dynamical barriers to theoretically explain the variation of S ( Et , ϑ ; v, J , Ts ) with Et . In contrast to the less 41 ACS Paragon Plus Environment


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parameterized d-PMMT model, which in the limit {η j } → 1 recovers conventional microcanonical transition state theory, there is no rigorous theoretical derivation48, 50 for the empirical erf model. The d-PMMT model can be considered in the context of an information theory/maximal entropy formalism51-52 to be one for which the prior distribution is a s-PMMT [maximal entropy; based on Eq. (4)] model and the efficacy values constitute the dynamical constraints on the active exchangeable energy in Eq.(8) necessary to recover the experimental results. To the extent that 5 parameters fixing the 3 efficacies of the d-PMMT model allow for adequate reproduction of the experimental data, the dynamical information content of the experiments can be summarized by 5 or less independent parameters. Outside these dynamical constraints, the H2(D2)/Cu(111) system apparently behaves in close accord with the precepts of conventional microcanonical transition state theory.

2. Dynamical Simulations Dynamical simulations of increasing dimensionality4, 38, 53 have been evolving to treat the H2/Cu(111) dissociative chemisorption. Most recently, Kroes and coworkers3-5 used sixdimensional (6D) quantum dynamical calculations, addressing all molecular degrees of freedom but assuming a static Cu surface, to optimize the specific reaction parameter of a SRP-DFT PES for H2/Cu(111) such that some experimental dissociative sticking coefficients (i.e., Fig. 12 and another similar to Fig. 4) could be reproduced sufficiently well to claim chemical accuracy (to within 4.2 kJ/mol) for the SRP-DFT PES. The threshold energy for dissociative chemisorption of H2/Cu(111) for the SRP-DFT PES is E0 =56.5 kJ/mol, somewhat higher than the E0 =43.4 kJ/mol and Ea =49.2 kJ/mol values of the d-PMMT model, or the RMA calculation of Ea =48.4 42 ACS Paragon Plus Environment

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kJ/mol for the thermal activation energy at 925 K. On the other hand, higher dimensional dynamical calculations including surface degrees of freedom might lead to a lower E0 in a SRPDFT PES optimization procedure because new non-reactive energy transfer processes involving the surface might more successfully compete with new reactive channels such that E0 would need to be lowered to match the experimental reactivity. Certainly, Hodgson’s experimental data and analysis (Figs. 10 & 11) dramatically demonstrate that surface energy can play a significant role in the gas-surface reaction dynamics. The d-PMMT model predicts a fractional energy uptake from surface phonons of f s =22% for thermal dissociative chemisorption at 925 K. These observations encourage the development of higher dimensional dynamical calculations that more fully account for surface phonons. Perhaps fortuitously, the canonical54 6D PES for H2/Cu(111) calculated by Norskov and coworkers55 in 1994 using GGA-DFT has a classical barrier for reaction of Ecl =48.2 kJ/mol, which when corrected for vibrational zero-point energies yields E0 =44.0 kJ/mol, which is in remarkable accord with the experimentally-derived d-PMMT model value of E0 =43.4 kJ/mol.

The Eq. (16) vibrational and rotational efficacies evaluated by Kroes and coworkers using 6D quantum dynamical calculations on the SRP-DFT PES are ηv ,e =0.65 (0.58) and

η r ,e [ν = 0; high J ] =0.46 (0.41) for H2(D2)/Cu(111).3 The vibrational efficacies are similar to those reported experimentally, ην ,e = 0.51 ± 0.02 ,2 and by the d-PMMT model, ηv = 0.60 . The preferential reactivity for normal translational energy as compared to vibrational energy is related to the early/late positioning of the transition state as described by the Polanyi rules developed for gas-phase atom plus diatomic molecule reactions56-57 and discussed by Holloway for H2 dissociative chemisorption.53 In this instance, the transition state is sufficiently late that 43 ACS Paragon Plus Environment


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both normal translational energy and vibrational energy can promote reactivity, but normal translational energy is more efficacious. The 6D quantum dynamics calculations yield a rotational efficacy that matches the experimental findings quite well at high J but the negative going η r ,e ( J ) observed experimentally at low J and modeled by the d-PMMT as the ηr ( Er ) reported in Fig. 5 was not replicated. Lower dimensional dynamical models have reproduced the dip and climb in reactivity as rotational energy increases,58-59 but this variance was not retained in the 6D dynamical calculations on the SRP-DFT PES. The lower dimensional modeling indicates that, because the barrier for dissociation is very much lower for molecules oriented parallel to the surface as compared to impinging end-on, the low J effect is due to orientational hindering whereas the high J effect is due to the moment of inertia coupling of rotational energy to bond extension as molecules approach the transition state.60 Polarized laser detection of the products of associative desorption for D2/Cu(111) found that molecules desorbing in the J =0 state are isotropically oriented, whereas with increasing J the average angular momentum vector of the desorbing molecules are increasingly polarized towards the surface normal (i.e., a preference for desorption of “helicoptering”, rather than “cartwheeling”, molecules develops with increasing J ).61 By detailed balance, this behavior is consistent with the orientationally anisotropic PES efficiently steering incident molecules at the lowest J towards the preferred parallel-to-the-surface orientation for dissociation but a weakening of this efficiency with increasing J . Subsequent experiments62 probing the variation of the rotational polarization of desorbing D2 in high J states [( v =1, J =6), ( v =0, J =11)] found the polarization decreased with increasing translational energy of the desorbing molecules. By detailed balance, this is opposite the expectation for steering where the longer time available for steering by the PES for slower incident molecules should lead to decreasing polarization as 44 ACS Paragon Plus Environment

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incident translational energy is decreased. Consequently, steering by the PES for these molecules, with substantial internal energies and translational energies greater than ~30 kJ/mol, is not believed to be significant. The behavior of these high J molecules presumably still points to the orientational anisotropy of the PES where the difference in the energetic requirements to dissociate through helicoptering rather than cartwheeling orientations has more impact for molecules incident at lower, rather than higher En . As the total active energy available to overcome the barriers to dissociation increases, more trajectories are able to overcome the local barriers to dissociate and effects related to any one degree of freedom become increasingly difficult to observe [c.f., effect of Ts with increasing En in Fig. 11(b)].

G. Applicability of PMMT Models The s-PMMT model described at the beginning of Section II above is predicated on the idea that for some activated gas-surface reactions a microcanonical statistical treatment of the reaction dynamics may be appropriate because there are so many states involved in the dynamics, the PES and/or intramolecular vibrational energy redistribution (IVR) may be sufficiently state-mixing at energies sufficient to access the transition state region, and there will be considerable inevitable collisional averaging over impact parameter across the surface unit cell, molecular vibrational and rotational phases, and vibrational phase and thermal energy distribution of the surface atoms.14, 21 Although the hydrogen/Cu(111) reactive system is the most dynamically biased one known in surface science, the experiments modeled above clearly show that En , Ev , Er , and Es can all be used to help surmount the activation barrier for reaction and may therefore by exchangeable to some degree on the multidimensional PES. State-resolved inelastic scattering experiments63-64 show evidence for ultrafast translational to vibrational 45 ACS Paragon Plus Environment


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energy transfer (ν =0 → ν =1) with onsets at energies sufficient to access the reactive transition state region of the H2(D2)/Cu(111) PES. Experiments61-62 measuring the rotational polarization of associatively desorbing D2 from Cu(111) are consistent with a strong orientational anisotropy of the reactive PES near the transition state. Dynamical calculations60 indicate that this orientational anisotropy will drive translational to rotational energy transfer particularly efficiently for trajectories that can access the anisotropic transition state region. These findings beg the questions of how ergodic and state-mixing is the process of a trajectory accessing the transition state region of the PES and would the Eq.(4) unbiased choice amongst possible outcomes be a reasonable way to estimate the microcanonical dissociative sticking coefficient.


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Figure 17: (a-b) s-PMMT model properties. The precursor complex density of states, ρ ( E * ) , and time constants for molecular desorption and reaction, τ i = h ρ ( E * ) / Wi ( E * ) , are also given. (c) Components of a d-PMMT calculation of a dissociative sticking coefficient, S ( En ) , for a hypothetical rovibrational state-selected beam of molecules striking a 925 K surface at a particular En . The flux distributions for the precursor complexes formed, f ( E * ) , and for the successfully reacting precursor complexes formed, P ( E * ) = S ( E * ) f ( E * ) / S ( En ) , [0.01 vertically offset for readability] are shown. (d) d-PMMT predictions of a rovibrational state-selected

S ( En ) over a range of En for several surface temperatures. 47 ACS Paragon Plus Environment


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The fundamental elements of the s-PMMT and d-PMMT models and their calculations for H2/Cu(111) are illustrated in Figure 17. Also shown in Fig. 17 (a-b) are PC properties for the typical choice of precursor location for much larger molecules, at the middle of the molecular adsorption well. Based on inelastic scattering of HD on Cu(111),23 the physisorption well depth for H2(p) was estimated as Ead = 2 kJ/mol and the vibrational frequency along the direction of the surface normal as ν z = 55 cm-1. Other than being bound in the surface normal (z) direction, the H2(p) is assumed to have the same characteristics as the gas-phase molecule. The time constant for molecular desorption at reactive energies is τ D ~600 fs which compares to τ s = 190 fs period for the surface oscillator vibrating at the mean phonon frequency of 175 cm-1. Hydrogen molecules traveling with 20-100 kJ/mol of translational energy traverse 1 Å in 1-3 ps in free space but substantive collision interaction with the reactive PES likely occurs on a timescale of 50-100 fs.65 It would seem our choice of PC has too long a desorption (interaction) lifetime, and with the molecule in the weakly coupled physisorption well it’s not so clear how states could become microcanonically randomized across the phase space between desorption and reaction other than through their initial collisional interaction with the surface. The desorption lifetime is given by τ D = h ρ ( E * ) / WD ( E * ) where the PC density of states, ρ ( E * ) , is a rapidly increasing function of the exchangeable energy above the zero-point corrected PES illustrated in Fig. 1. Consider now an alternate location for a “dynamical” PC which would be the located far to the right of the physisorption well in Fig. 1 and approaching the reactive transition state such that rotational motion is hindered and the density of states is diminished by roughly an order of magnitude. Then the timescale for desorption and substantive interactions in molecular trajectory calculations would come in to better agreement. Furthermore, the transition state for “desorption” could be taken to occur at the physisorption state immediately adjacent to the 48 ACS Paragon Plus Environment

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asymptotic fall-off region of the reactive PES where the molecule is once again able to freely rotate (e.g., where the potential of Fig. 1 first meets the E * =0 asymptote from the right). The region between these transition states for “desorption” and reaction is a more physically reasonable choice for our PC domain where, at reactive energies, state mixing may efficiently occur60, 63-64 and states are more likely to become microcanonically occupied, at least in a collision ensemble averaged sense.14-16 Because the reaction coordinate for desorption is the z vibration (normal translation) that doesn’t appear in the desorption sum of states, then WD ( E * ) for the desorption transition state chosen at the position of the inner turning point of a physisorbed molecule with E * =0 or at its outer turning point will be very similar. With an accurate PES one could calculate WD ( E * ) at the inner turning point (or other positions nearby) exactly but in the absence of such a PES it remains expedient to approximate it using the WD ( E * ) of the outer turning point. With these approximations, we can conceptualize the

dynamical PC domain as extending only over the region wherein the incident molecule interacts substantively with the reactive PES. This is exactly the domain over which the d-PMMT calculations of this paper were carried out using the Eckart potential of Fig. 2 which admits no physisorption well whatsoever. Important to keep in mind is that the exact details of the PC are largely irrelevant to the derivation of the central result of the PMMT models, the Eq.(4) expression for S ( E * ) - the PC needs only be located between the transition states for desorption and reaction, and its states be microcanonically randomized, at least in a collision ensemble averaged way.14-16 The estimated 100 fs lifetime of the dynamical PC at reactive energies is comparable to the τ s = 190 fs cycle period for the surface oscillator. Consequently, the surface oscillator atom can move considerably over the dynamical PC lifetime. Surface atom motion modulates the reactive PES, and its barrier height,9, 66-67 and so gas-surface energy coupling is to 49 ACS Paragon Plus Environment


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be expected. The s-PMMT models developed for H2/Cu(111) with active24 and inactive25 rotations have provided qualitative guides to the experimental behavior but have not quantitatively replicated all the experiments. The d-PMMT model of H2/Cu(111) employs exactly the same Eq.(4) microcanonical sticking coefficient, S ( E * ) , as for the s-PMMT model displayed in Fig. 17(b). Dynamical biases are introduced only through the efficacies, η j , that refine the active exchangeable energy in Eq. (5) wherein the statistical limit is reached when all efficacies are one. To better appreciate the degree of averaging inherent to experiments studying gas-surface reactive collisions, it is instructive to consider the idealized hypothetical experiment of Fig. 17(c) in which a beam of H2( v =0, J =0) molecules with exactly En =10 kJ/mol is made incident onto a Cu(111) surface at 925 K. The energy breadth of the flux distribution of the PCs formed is due solely to the surface oscillator’s thermal energy distribution (narrowed somewhat by ηs =0.60). More interesting is the energy breadth of the flux distribution for the successfully reacting PCs formed is about 15 kJ/mol at full width at half-maximum (fwhm). Consequently, even for this idealized eigenstate-resolved experiment at En =10 kJ/mol, the dissociative sticking coefficient,

S ( En ) , calculated as the Eq.(3) integral over S ( E * ) f ( E * ) [i.e., the average of the microcanonical dissociative sticking coefficient over the flux distribution of the PCs formed], is a substantially energy averaged quantity. This kind of energy averaging is likely to help bring a microcanonical theoretical model into better agreement with the experiments, even for intrinsically dynamical reactive systems.68-69 Additional averaging incurred by the experiments over the molecular impact position across the surface unit cell and the phases of the vibrational and rotational motions is not explicitly treated by the d-PMMT model. However, this latter kind of collisional 50 ACS Paragon Plus Environment

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averaging alone is sometimes considered sufficient to generate a collision ensemble averaged microcanonical distribution of collisional complexes sufficient for the RRKM formalism to successfully proceed.15-16 The H2/Cu(111) reactive potential has the additional demonstrated state-mixing property of supporting ultrafast energy transfer between molecular translation and vibration,63-64 and likely rotation,60 at energies that can access the transition state region. Consequently, application of microcanonical models to this reactive system is not unreasonable, but the reaction dynamics retain some intrinsically non-RRKM behavior15, 26, 57, 70 that doom the simplest s-PMMT model to quantitatively fail.

Figure 17(d) shows how the eigenstate-resolved S ( En ) should vary with En at several temperatures according to the d-PMMT model. The experimental translational energy distribution for H2( v =0, J =0) molecules associatively desorbing from a 925 K surface in Fig. 6 is well replicated by P ( Et ) = S ( En ) f MB ( En ) / S averaged over the desorption angles detected. Similar calculations closely replicated Hodgson’s Ts dependent associative desorption product translational energy distribution in Fig. 10 and so the surface temperature variation of S ( En ) in Fig. 17(d) should be quite accurately predicted.

The S ( En ) curves of Fig. 17(d) would be represented in the erf model by Eq.(14) wherein only the W ( v, J ; Ts ) width parameters2, 10, 42 are assumed to vary with Ts . However, in the erf model, the many S ( En ) parameters required for each quantum state, isotope, and temperature are always fit to experiments or the results of dynamical calculations. The 6D quantum dynamics simulations3 on the SRP-DFT-PES made the static surface approximation which precluded their direct evaluation of the effect of surface temperature on dissociative sticking. In contrast, the d-PMMT model of H2(D2)(HD)/Cu(111), whose relatively few 51 ACS Paragon Plus Environment


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parameters were fit to the thermal associative desorption product state distributions of Figs. 3-4 at the one surface temperature of 925 K, makes definite predictions about dissociative sticking coefficients for any arbitrary experiment for which the PC flux distribution of Eq. (8) can be defined.

IV. Conclusions A minimally parameterized, d-PMMT model of the dissociative chemisorption of hydrogen on Cu(111) was developed and shown capable of reproducing the diverse range of experimental results available for this benchmark system for gas-surface reaction dynamics. The model’s utility for analysis of experiments, identification of dynamical biases, prediction of experimental outcomes (e.g., at low T), and description of reactive energy uptakes was demonstrated. At temperatures of catalytic interest, dynamical biases are predicted to reduce the H2/Cu(111) thermal reactivity by several orders of magnitude as compared to the expectations of statistical transition state theory. Tunneling was calculated to account for a significant fraction of the thermal reactivity. The model predicts that the energy necessary to surmount the activation barrier to thermal dissociative chemisorption is supplied primarily by molecular translational energy directed along the surface normal, secondarily by surface phonon energy, and lagging behind in importance, by molecular rotational energy and vibrational energy. As compared to existing erf and 6D quantum dynamics models, the d-PMMT model of the hydrogen/Cu(111) is more closely grounded in transition state theory and provides easier access to predictions of dissociative sticking coefficients over extended and arbitrary experimental conditions.

Acknowledgements: This work was supported by National Science Foundation Grant # CHE1112369 and an AES Graduate Fellowship in Energy Research for S.B.D. We thank G. P.


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Krishnamohan, C. Díaz, R. A. Olsen, and G. J. Kroes for kindly calculating and providing transition state vibrational frequencies for their SRP-DFT-PES3 for hydrogen on Cu(111).

Supporting Information Available: (i) a table of transition state properties for the SRP-DFTPES3 for hydrogen dissociative chemisorption on Cu(111), and (ii) several figures representative of simulations with a reduced dimensionality d-PMMT model in which rotation was taken as a spectator degree of freedom. This information is available free of charge via the Internet at http://pubs.acs.org.



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Phys. 1995, 102, 4625-4641. 3. Diaz, C.; Pijper, E.; Olsen, R. A.; Busnengo, H. F.; Auerbach, D. J.; Kroes, G. J. Chemically Accurate Simulation of a Prototypical Surface Reaction: H2 Dissociation on Cu(111). Science

2009, 326, 832-834. 4. Diaz, C.; Olsen, R. A.; Auerbach, D. J.; Kroes, G. J. Six-Dimensional Dynamics Study of Reactive and Non Reactive Scattering of H2 From Cu(111) Using a Chemically Accurate Potential Energy Surface. Phys. Chem. Chem. Phys. 2010, 12, 6499-6519. 5. Diaz, C.; Olsen, R. A.; Busnengo, H. F.; Kroes, G. J. Dynamics on Six-Dimensional Potential Energy Surfaces for H2/Cu(111): Corrugation Reducing Procedure Versus Modified Shepard Interpolation Method and PW91 Versus RPBE. J. Phys. Chem. C 2010, 114, 1119211201. 6. Hand, M.; Harris, J. Recoil Effects in Surface Dissociation. J. Chem. Phys. 1990, 92, 76107617. 7. Dohle, M.; Saalfrank, P. Surface Oscillator Models for Dissociative Sticking of Molecular Hydrogen at Non-Rigid Surfaces. Surf. Sci. 1997, 373, 95-108. 54 ACS Paragon Plus Environment

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J. Chem. Phys. 1998, 108, 4199-4211. 11. Donald, S. B.; Harrison, I. Dynamically Biased RRKM Model of Activated Gas-Surface Reactivity: Vibrational Efficacy and Rotation as a Spectator in the Dissociative Chemisorption of CH4 on Pt(111). Phys. Chem. Chem. Phys. 2012, 14, 1784-1796. 12. Navin, J. K.; Donald, S. B.; Tinney, D. G.; Cushing, G. W.; Harrison, I. Communication: Angle-Resolved Thermal Dissociative Sticking of CH4 on Pt(111): Further Indication that Rotation is a Spectator to the Gas-Surface Reaction Dynamics. J. Chem. Phys. 2012, 136, 061101. 13. Donald, S. B.; Navin, J. K.; Harrison, I. Methane Dissociative Chemisorption and Detailed Balance on Pt(111): Dynamical Constraints and the Modest Influence of Tunneling. J. Chem.

Phys. 2013, 139, 214707. 14. Ukraintsev, V. A.; Harrison, I. A Statistical-Model for Activated Dissociative Adsorption Application to Methane Dissociation on Pt(111). J. Chem. Phys. 1994, 101, 1564-1581. 15. Bunker, D. L.; Hase, W. L. On Non-RRKM Unimolecular Kinetics: Molecules in General, and CH3NC in Particular. J. Chem. Phys. 1973, 59, 4621-4632. 16. Freed, K. General Discussion. Faraday Discuss. 1979, 67, 231-235. 55 ACS Paragon Plus Environment


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17. Forst, W. Unimolecular Reactions: A Concise Introduction. 1st ed.; Cambridge University Press: Cambridge UK, 2003. 18. Baer, T.; Hase, W. L. Unimolecular Reaction Dynamics. Oxford University Press: New York, NY, 1996. 19. Cushing, G. W.; Navin, J. K.; Donald, S. B.; Valadez, L.; Johanek, V.; Harrison, I. C-H Bond Activation of Light Alkanes on Pt(111): Dissociative Sticking Coefficients, Evans-Polanyi Relation, and Gas-Surface Energy Transfer. J. Phys. Chem. C 2010, 114, 17222-17232. 20. Tait, S. L.; Dohnalek, Z.; Campbell, C. T.; Kay, B. D. N-Alkanes on Pt(111) and on C(0001)/Pt(111): Chain Length Dependence of Kinetic Desorption Parameters. J. Chem.

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Table of Contents Graphic:


Addressing Dynamical Biases, Surface Tempera - American Chemical

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