Theoretical Study of Catalytic CO Oxidation on (111) Metal Surfaces

9184

J. Phys. Chem. C 2007, 111, 9184-9193

Theoretical Study of Catalytic CO Oxidation on (111) Metal Surfaces: Calculating Rate Constants That Account for Tunnel Effect Ernst D. German* and Moshe Sheintuch Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel ReceiVed: July 16, 2006; In Final Form: February 26, 2007

An approximate analytical approach is suggested to calculate kinetic parameters for the catalytic oxidation of carbon monoxide on the (111) surfaces of transition metals of platinum, palladium, iridium, rhodium, and nickel. The origin of the activation barriers is discussed in terms of the decomposition analysis. Tunnel effect along the reaction coordinate is considered. An interpretation of possible non-Arrhenius behavior of rate constant of the CO oxidation reactions is suggested in terms of the tunneling. Temperature dependencies of the activation energies are calculated. Isotope effect under substitution of C(12) by C(14) is predicted. Experimental data are interpreted using the developed approach. Relationships between the activation energies and rate constants calculated accounting for quantum effects and the corresponding surface reaction energies are discussed.

1. Introduction The catalytic oxidation of CO over transition metal surfaces is of great practical and academic importance as the main reaction in automotive catalytic converters and as the simplest catalytic reaction. It is essential, therefore, to understand what is the physical origin of the reaction barrier of this reaction and what are the main factors that determine the reactivity difference between metals to improve on the present technology. Consequently, kinetics of this reaction have been extensively studied both experimentally and theoretically.1-43 Particularly, Engel and Ertl1-3 have carried out extensive molecular beam studies for CO oxidation on Pd(111) at various temperatures and pressures and have established that the oxidation follows the Langmuir-Hinshelwood (L-H) mechanism. This mechanism has been confirmed by numerous ultrahigh vacuum (UHV) studies of the coadsorption of the reactants,4,6-11,21 transient kinetic studies,12-16 and steady-state kinetics.17-19 Because oxygen can dissociate on all transition metals at room temperature, the L-H mechanism involves three elementary steps: (i) CO adsorption on the surface, (ii) O2 dissociation into O adatoms on the surface, and (iii) surface reaction, CO (ads) + O (ads) f CO2 (ads). These steps have been analyzed with the useoftheperiodicdensityfunctionaltheory(DFT)calculations,31-42,44-53 and few calculations were made by the cluster DFT method.54-58 The quantum chemical studies allowed drawing a thorough picture of the overall process; yet, the activation energies determined in various studies vary considerably. Also, despite the detailed knowledge of many features of the surface CO and O reaction, no mechanistic model exists that is fully consistent with all features of the kinetics. The numerical periodic calculations of the activation energy were found to depend on the form of the correlation functional, and there is no accepted opinion regarding which of these is the most appropriate to study the surface CO oxidation reaction. Furthermore, the numerical calculations of activation energies by the sophistical periodic DFT methods are very computer time demanding. * To whom the correspondence should be addressed: [email protected]..

E-mail:

In this work, we present an analytical approach similar to that which has been developed in ref 58 and applied in ref 59 for calculation of the kinetic parameters of O2 dissociative adsorption on transition and noble metals surfaces. Because the kinetics of CO oxidation is thermodynamically related to the kinetics of CO2 dissociation, we use here the concept of heteronuclear dissociation60 by considering the CO bond by a “frozen stick” approximation. The present technique allows us to calculate all three steps of the L-H mechanism with the same formalism. The main advantages of our approach are the following: (i) The approach is semianalytical allowing to make stronger statements concerning the results; the method provides a useful way for decomposing the various contributions of the activation energy barrier. (ii) The formalism accounts for a local reorganization of the metal surface near the adsorption sites. (iii) The method provides a simple way for calculating the transition state frequencies and the pre-exponential factor. (iv) It does not require the laborious and computer time-consuming numerical calculations of the total potential energy surface; only a quantum chemical information about the initial and final stationary points of the surface, given by the corresponding equation parameters, is sufficient for calculation of the saddle point. Therefore, the main factors that determine the different reactivity among metals and the reactivity sensitivity to physical properties of reactants and products can be discussed by performing inexpensive calculations. The main drawback of the approach is in its approximate form of the potential energy surface. Thus, at the present work we report the results of a systematic study of the activation barriers and a pre-exponential factor for CO oxidation reaction of on (111) metal surfaces of nickel, iridium, rhodium, palladium, and platinum (step (iii) in the L-H mechanism). 2. Theoretical Approach In this section, we describe our theoretical formalism developed for studying the surface reaction O(1)C (ads) + O(2) (ads) f O(1)C-O(2) (ads) (see Figure 1 for atoms notations). This formalism is based on the concept of the adiabatic potential energy surface (APES).

10.1021/jp064503o CCC: $37.00 © 2007 American Chemical Society Published on Web 06/05/2007

Catalytic CO Oxidation on (111) Metal Surfaces

J. Phys. Chem. C, Vol. 111, No. 26, 2007 9185

Figure 2. Thermodynamic cycle used to calculate the reaction surface energy: ∆Egas is the oxidation reaction energy in gas phase, ∆Esurf is the oxidation reaction energy on a surface, Eads (CO), Eads (1/2O2), and Eads (CO2) are the corresponding adsorption energies.

assumption of no interaction between vibration modes along the reaction coordinates. In this case, both potential surfaces may be written as a sum of separate contributions

Figure 1. Real (a) and model (b) reactants on a metal surface.

2.1. Potential Energy Surface and Activation Energy. The APES characterizing the reaction is constructed in the two state approximation using a multidimension diabatic PESs Ui (r, xA, xB, {rk}) (initial state) and Uf (x, r, φ,{rk}) (final state) similarly to the procedure described in ref 60. The application of the formalism60 to the surface oxidation of the CO molecule is proposed under the frozen stick assumption that is substantiated by the following arguments. In the chemisorbed CO2 molecule (reaction product), one C-O(2) bond is practically parallel or has a small angle (φ) to the metal surface (Figure 1a), and the angle between the bonds C-O(2) and C-O(1) is about 120130°.61 The vibration frequency of the C-O(1) bond is equal to ∼1800 cm-1 61 and it is close to that of the adsorbed CO molecule. The bond length of C-O(1) is slightly shorter (by ∼0.04-0.05 Å) of that in the adsorbed CO molecule. According to the quantum-classical criterion,62 the C-O(1) bond may be considered as a quantum one; this bond practically does not participate in the oxidation process on a surface. Thus, we neglect the intrinsic structure of this bond (a frozen stick approximation) while calculating the activation barrier. With this reservation, we will formally represent CO2 as a fictive two-atomic AB molecule in which A is the frozen C-O(1) fragment and B is the oxygen atom. Therefore, the initial state for the oxidation reaction is the adsorbed oxygen atom (atom B) and a CO molecule (fictive atom A) (Figure 1b). The final state is an adsorbed CO2 (or AB). In terms of A and B symbols, we consider the CO oxidation reaction as an association reaction, A + B f AB. The semiclassical approximation is used for calculation of the activation barrier. This is because the vibration frequencies [C]-metal and [CO2]-metal of the adsorbed CO and CO2 molecules and [O]-metal of the adsorbed oxygen characterizing the corresponding reactive degrees of freedom are about 360500 cm-1 (see Table E1). The vibration frequency of the fromed bond C-O(2) in the adsorbed CO2 is equal to about 900 cm-1 61 being of a quantum character. The corresponding mode will be considered as the classical one when calculating the transition configuration. The justifications for this approximation are presented in ref 58. Thus, following a broadly used procedure62 we write the lower APES in the form

U ) 1/2Ui + Uf - x(Ui - Uf)2 + 4V 2

(1)

where V is the effective resonance integral (2V is equal to the splitting of the diabatic surfaces Ui and Uf at the transition configuration).58-60 The simplest form of these diabatic PESs Ui and Uf in the two-state approximation corresponds to the

Ui(r,{rk},xA,xB) ) ui(r) + ViA(xA) + ViB(xB) + wi ({rk})

(2)

Uf (x,r,φ,{rk}) ) uf (r) + Vf (x) + zf(φ) + wf ({rk}) + ∆Esurf (3) Here, ∆Esurf is the heat of the surface oxidation reaction, which is calculated via the adsorption energies of reactants and products by the equation (see Figure 2)

∆Esurf ) ∆Egas + Eads (CO2) - Eads (1/2O2) - Eads (CO) (4) where ∆Egas ) -67.8 kcal/mol is the heat of the gas-phase reaction CO2 (gas) f 1/2O2 (gas) + CO (gas). The term Vf(x) in Uf (eq 3; x is the distance of the AB molecule center of mass from the metal surface) is the interaction between the molecularly adsorbed AB and the metal in the direction perpendicular to the surface. uf(r) is the vibrational potential of the adsorbed AB molecule (r is the A-B bond length). zf(φ) describes the hindered rotation of the AB by the angle φ around the axis parallel to the surface and passing through its center of mass in the plane perpendicular to the AB line. wf({rk}) describes the small vibrations of the metal nuclei. The first term of the initial potential Ui (eq 2) describes a repulsive interaction between separate A and B, and the fourth term, wi({rk}), has the same meaning as that for the final diabatic PES Uf. The second and third terms of Ui characterize the interaction of A and B atoms with a metal at the distances xA and xB, respectively, from the metal surface. ∆Esurf ) If - Ii, where Ii and If are the minimum values of the potential energies in the initial and final equilibrium states including electron energies. It is assumed that the origin of coordinates is at the point of minimum of the initial term. The initial coordinates, xA and xB , are related to the final coordinates x and φ by the equations given in Appendix A. Harmonic approximation is used to write wi({rk}) and wf({rk}) in an analytic form. In this approximation, an effective coordinate θ may be introduced instead of the set of the coordinates {rk}.58 In terms of this coordinate, the potential energies wi({rk}) and wf({rk}) may be written in the simple form (see Appendix B). Thus, the adiabatic surface (eq 1) is a four-dimension one. For the considered systems, we approximate the potential energies in eqs 2 and 3 by simple functions described analytically in Appendix C. The coordinates (xˆ A, xˆ B, rˆ, θˆ ) of the transition configuration on the PES U is determined by the solution of the set of eqs 5-8, which may easily be obtained using the approach described

9186 J. Phys. Chem. C, Vol. 111, No. 26, 2007

German and Sheintuch

in details in ref 58

Ui - Uf ) (1 - θ)

V(x,r)(2θ - 1)

xθ(1 - θ)

[

(5)

]

∂Uf ∂Ui ∂Ui ∂xA ∂Ui ∂xB + + ) +θ ∂r ∂xA ∂r ∂xB ∂r ∂r ∂V 2 xθ(1 - θ) (6) ∂r

(1 - θ)

[

]

∂Uf ∂Ui ∂xA ∂Ui ∂xB ∂V +θ + )2 xθ(1 - θ) ∂xA ∂x ∂xB ∂x ∂x ∂x (7)

(1 - θ)

[

]

∂Uf ∂Ui ∂xA ∂Ui ∂xB + )0 +θ ∂xA ∂φ ∂xB ∂φ ∂φ

(8)

The explicit form of these equations for the specific form of potentials is given in Appendix D. The saddle point coordinates (xˆ A, xˆ B, rˆ, θˆ ) being substituted into equation for U determine the activation barrier on the adiabatic PES for the oxidation reaction

Eaad(ox) ) Ui(rˆ,xˆ A,xˆ B,θˆ ) -

V(xˆ A,xˆ B,rˆ)

) ui(rˆ) + VAi (xˆ A) +

x(1 - θˆ )/θˆ V(x ˆ V(xˆ A,xˆ B,rˆ) ˆ B,rˆ) A,x ≡ Eanad (9) VBi (xˆ B) + θˆ 2Er x(1 - θˆ )/θˆ x(1 - θˆ )/θˆ

2.2. Pre-exponential Factor and Rate Constant. The standard transition state theory expression for a semiclassical rate constant of an adiabatic reaction is63 kscl(T) )

kBT Zˆ ′ exp[-Eascl/kBT] ≡ Ascl exp[-Eascl/kBT] h Zi

(11)

where kscl is the semiclassical rate constant calculated without allowing for the tunnel effect. The partition function of the initial state for our four-mode model, Zi, is equal to

Zi ) Zir ZixA ZixB Ziθ

(12)

and that for the transition state

Zˆ ′ ) Zˆ 2 Zˆ 3 Zˆ 4

Zir )

(13)

1 L

∫r0fL exp[-ui(r)] dr

(14)

where L is the length (formally infinite) along which there are reactants (i.e., L × L ) S is the area of a metal surface). If N is the number of particles on this length, then N/L is the linear concentration ∼4 × 107 1/cm. One can see that this integral is equal to ∼L, so that Zir ∼1. The partition functions for CO, O, and metal atoms of crystal lattice vibrations (we denoted these atom group and atoms by A, B, and M) have a standard form63

Zik )

1 1 - exp[-pωik/kBT]

(15)

where k is A, B, or M and ωik are the corresponding initial vibrational frequencies, which are usually well-known quantities. To calculate Zˆ ′, we approximate the adiabatic potential energy near the saddle point (xˆ A, xˆ B, rˆ, θˆ ) by the expansion

U≈U ˆ +

1 2

λnm(sn - sˆn)(sm - sˆm) ∑ n,m

(16)

where st (t ) n, m) are the mass-weighted coordinates correˆ are the sponding to coordinates xA, xB, r, and θ, λnm and U second derivatives and the value of U at the saddle point. The eigenvalues of the matrix equation

(10)

In eq 10, Zi is the partition function per unit area for the reactants, and Z′ is the incomplete partition function per unit area, the degree of freedom corresponding to the reaction coordinate being omitted. The energy Eascl is the energy required passing from the lowest vibration level of reactants A and B to the lowest level of the transition state. As will be seen below, the potential form along the reaction coordinate is rather sharp and is characterized by a rather high imaginary frequency. Therefore, it seems reasonable to consider the possibility of tunneling under this barrier when the system passes the transition state. The expression for the rate constant taking into account the tunneling along the reaction coordinate may be described approximately by introducing to the semiclassical rate constant expression a tunnel correction factor Qt > 163,64

kBT Zˆ ′ kq ) Qtkscl ) Qt exp(-Eascl/kBT) h Zi

First, let us consider the partition function of the initial state. Note that in the initial state we have reactants CO and O adsorbed over the surface. The first term in eq 12 is the partition function, which characterizes moving decaying potential ui(r) (see Appendix C for ui(r)). This partition function (per unit length) is equal to

2 | |ω | ˆ n δnm - λnm| ) 0

(17)

where δnm is the Kronecker symbol, give a set of frequencies ω ˆ t (t ) 1, 2, 3, and 4) characterizing the transition state; one of them has an imaginary value iω ˆ 1, whereas all others are real and equal to ω ˆ t (t * 1). One should note that vibrations of metal atoms of crystal lattice along θ coordinate are low-frequency vibrations, and the corresponding frequency may be approximately separated from other intramolecular frequencies (an approximation that is usually used in spectroscopy) under diagonalization of the frequency matrix. In this approximation, the matrix (eq 17) is really reduced to a third-order matrix, and the component of the transition state partition function corresponding to crystal lattice is cancelled by the similar component of the initial partition function. Finally, we have

kq ) Qtkscl ) Qt

kBT Zˆ 2Zˆ 3 exp(-Escl a /kBT) h ZirZxAZxB

(18)

where the partition functions Zˆ 2 and Zˆ 3 have the form given by ˆ2 eq 15, in which the frequencies ωik should be substituted by ω and ω ˆ 3. According to theory of the tunnel effect (Bell),64 the quantity Qt is equal to

Qt )

exp[Escl a /kBT] kBT

exp[-W/k T] dW

∫0∞ 1 + exp[(Escl -B W)/pωˆ ] a

(19)

1

where ω ˆ 1 ) 2πνˆ 1 is the amplitude of imaginary frequency. Equation 19 gives the exact expression for the tunnel correction.



At small values of Qt, it may be calculated approximately by the Wigner equation.65 On making the substitutions ξ ) scl ˆ 1], this becomes exp[(Escl a - W)/kBT] and γ ) exp[Ea /pω

Qt )

∫0

γ

dξ 1 + ξkBT/ pωˆ 1

TABLE 1: Reaction Heats, Activation Energies, and Rate Constants (at 500 K) Calculated in the Semiclassical Modela initial adsorption state of CO

(20)

The effect of tunneling in chemical kinetics appears more clearly when the temperature dependence of the rate constant is examined than when only reaction rates at a single temperature are considered. This dependence is usually analyzed in terms of the Arrhenius equation, which is found empirically to hold over at least a limited range of temperature a

kq ) Aq exp[-Eaq/kBT]

(21)

Thus, comparing eqs 18 and 21 we have

Aq exp[-Eq/kBT] ) Qt

kBT Zˆ ′ exp(-Escl a /kBT) h Zi

(22)

Writing eq 22 in logarithm form and differentiating the left and the right parts over temperature, one obtains

Eaq ) kBT + kBT2(dQt/dT)/Qt + Eascl

(23)

scl Introducing the denotations R ) Escl ˆ 1, and x a /kBT, β ) Ea /pω scl ) W/Ea , we rewrite Qt (eq 19) and its temperature derivative in the form

Qt )

R exp[R(1 - x)]

∫0∞ 1 + exp[β(1 - x)] dx

dQt Qt 1 )- dT T T

∫0∞

(24)

R2(1 - x) exp[R(1 - x)] dx (25) 1 + exp[β(1 - x)]

Then, substituting eqs 24 and 25 into eq 23 one obtains the relation between the activation energy calculated with taking into account tunneling, Eaq, and the semiclassical activation energy Eascl

Eaq

)

Eascl

[

]

∫0∞ dx x exp[R(1 - x)]/(1 + exp[β(1 - x)]) ∫0∞ dx exp[R(1 - x)]/(1 + exp[β(1 - x)])

(26)

The corresponding prefactor Aq may be calculated using eqs 22, 24, and 26. Equation 23 or 26 predicts that that the tunnel effect will make the activation energy Ea smaller than the semiclassical value because dln Qt/dT is always negative. The direction in which tunneling affects the pre-exponential is not so immediately obvious. A simple analysis given by Bell64 (Results and Discussion) shows that a semiclassical value of Ascl is always higher than Aq calculated with consideration of the tunnel effect. 3. Results and Discussion In this section, we present the calculated kinetic parameters for the CO oxidation reaction assuming that initially the oxygen atom is adsorbed on a three-hollow site (which is the most favorable adsorption site for it on a clear surface of all the (111) metals considered here, see Supporting Information, Tables 8-14) As for the initial adsorption state of carbon monoxide, one can see from Tables 1-7 of Supporting

metal

Ni

Ir

Rh

Pd

Pt

10.7 3.2 -5.8 -13.8 -15.3 23.4 20.8 18.1 15.1 14.3 2.75 4.0 4.94 6.46 6.79

atop

∆Esurf Eaad(scl) lgk(scl)

bridge

∆Esurf 21.2 Eaad(scl) 35.5 lgk(scl) -2.51

-4.8 25.9 1.71

3-hollow

∆Esurf 23.2 Eaad(scl) 38.3 lgk(scl) -3.75

-3.8 -10.8 -16.8 28.2 27.8 25.5 0.71 0.94 2.03

-15.8 21.3 3.84

Energies in kcal/mol, rate constants in 1/s.

Information that CO prefers to be adsorbed on a top site of some of the metals and on a hollow or bridged sites of others. For certain metals, published data do not allow us to establish unambiguously what adsorption site is the most favorable. Let us consider this problem in detail. According to refs 4 and 19 of Supporting Information (Table 1) the hollow adsorption of CO on clear (111) Ni surface is more favorable than the top one (-44 versus -36 kcal/mol4 and ∼-38 versus ∼-33 kcal/mol).19 These theoretical data, which correspond to coverages of θ ) 1/4 and 1/3, are higher than the experimental finding Eads e -30 to -31 kcal/mol at θ f 01,3,15 of Table 1; in these works there is no clear indication what kind of adsorption sites they characterize. The CO adsorption data for platinum (Table 2 of Supporting Information) are controversial. According to the periodic DFT calculations22 (Supporting Information), the atop CO adsorption on clear metal surface is slightly more preferable than the hollow CO adsorption, and the adsorption energy on a bridge site has an intermediate value. The inverse order of adsorption energies was found in theoretical works.19,38 Experimental adsorption data at low coverage listed in Table 2 of Supporting Information may be attributed to the top as well as to hollow (bridge) adsorption. Our DFT calculations performed on 18-20 atomic clusters show that hollow and bridge sites are preferable ones for the CO adsorption on clear (111) platinum surface. Analysis of theoretical and experimental data for the CO adsorption on the (111) surface of palladium (Table 3 of Supporting Information) allows us to consider the CO hollow adsorption at θ f 0 to be slightly more favorable than the atop adsorption. This conclusion is supported by our DFT cluster calculations also. Periodic and cluster DFT calculations listed in Table 4 of Supporting Information show that atop sites are more preferable for the CO adsorption on clear (111) Rh surface than hollow and bridge ones; however, the energy differences between these sites are rather small. As for the CO adsorption on the (111) surface of iridium, the comparative calculations for atop and the hollow sites were performed in ref 38 only (Table 5 of Supporting Information). The results obtained in this work show that the energy characterizing atop adsorption is higher than that for hollow sites. This conclusion is also supported by our DFT calculations on a cluster model. For obtaining more objective results, we will consider three types of initial stationary configurations of CO on (111) surfaces. This allows the comparison of these systems in terms of the same model. Our purpose is to demonstrate the possibilities of our approach and to study some details of the activation


Figure 3. A schematic diagram showing relative energies of the initial, final, and transition state on the (111) metal surfaces.

Figure 4. A part of the adiabatic potential energy U near the saddle point neighborhood constructed in coordinates xA and xB; other coordinates are fixed at the values equal to those of the saddle point.

mechanism, which were not investigated in previous publications. The reactant adsorption parameters, which correspond to the initial reactant configurations mentioned above, are listed in Table E1 along with the parameters of adsorbed CO2 molecule. 3.1. Semiclassical Model. The CO oxidation semiclassical reaction barriers and the rate constants calculated with the use of eqs 5-10 are presented in Table 1. For clarity, the reaction energetic data with the initial CO on a top site are also shown in Figure 3. In the rate constants, the frequencies characterizing the partition function of the initial reaction state were compiled from Table E1. The frequencies of the transition state were found by the diagonalization of eq 17. For all considered systems, three vibration frequencies were calculated, which include one imaginary value along the reaction coordinate. A part of a typical three-dimensional plot of U near the saddle point is shown in Figure 4. The plot was constructed in coordinates xA and xB at fixed values of other coordinates (equal to their values at the saddle point). One can see from Figure 4 that the potential energy surface along the reaction coordinate has rather sharp form. The data of Table 1 show that the calculated kinetic characteristics, Eaad(scl) and lgk(scl), correlate well with the corresponding reaction energies ∆Esurf for all the initial CO adsorption sites. One can see that when the energy ∆Esurf decreases, the activation barrier becomes lower and the rate

German and Sheintuch constant increases. The initial hollow state of adsorbed carbon monoxide leads to a higher activation energy and to a lower reaction rate constant than the atop initial state does, and the corresponding kinetic characteristics for bridge CO adsorbed state are intermediate. To understand the origin of the activation barriers, we present in Table 2 (for atop adsorbed CO initial state) the components of the activation energies using values of the saddle point and equilibrium coordinates on the potential energy surface. The decomposition analysis shows that the main contributions to the activation barriers are due to approaching reactants (∆Ea(rˆ)), which account for 55-63% of the total nonadiabatic activation energy, and to the extension of the M-O bond (∆Ea(xˆ B)) contributing from 33 to 39% depending on metal. The latter component depends very strongly on the CO initial adsorption position, because the adsorption heights, the frequencies, and the adsorption energies of atop, bridge, or hollow sites are considerably different. This explains the higher activation barrier if CO is in its initial hollow state. The contribution to the activation barrier due to the local reorganization of metal atoms is rather small; however, it is of principle importance from a physical point of view. We conclude this subsection by making some comparisons of our approximate rˆ, xˆ A, and xˆ B values of Table 2 with those obtained by periodic DFT calculations.31 The DFT-computed distance of carbon from a surface at the transition state, xˆ A, is equal to 1.88, 1.89, and 1.91 Å for CO oxidation on Pt(111), Pd(111), and Rh(111).31 These values are in remarkable agreement with our values (Table 2) of 1.916, 1.913, and 1.917 Å. The distance of oxygen from a surface at the transition state, xˆB (given31 only for Pt(111)) is equal to 1.51 Å, which agrees closely with our value 1.40 Å. The gaps in estimating rˆ (the distance C...O) are slightly larger; the values of rˆ in Table 1 are to 1.726, 1.621, and 1.710 Å compared with the values of 2.07, 2.05, and 1.89 Å of ref 31. This means that the model repulsion potential ui(r) (see eq C3) is probably slightly smoother than it follows from exact periodic calculations. 3.2. Taking into Account a Tunnel Effect. The calculated imaginary frequencies are rather high for all considered systems, being in the range of 1500-1900 cm-1. For such values of the frequencies, one can expect moderate or large tunneling along the reaction coordinate depending on reaction temperature. Therefore, it seems to be natural to perform calculations of the reaction kinetic characteristics taking into account tunneling in terms of the approximate model described in Section 2. To our knowledge, tunnel effect was not previously considered for surface CO oxidation reactions. However, it has been investigated for the surface reaction N + H f NH on Ru(0001) and for other steps of hydrogenation in ammonia synthesis on this metal using time-independent-scattering theory and variation transition state theory,66-68 and it was found that tunneling is considerable in these reactions. The tunnel correction introduces deviations of the simple Arrhenius equation. Therefore, experimental observation of nonlinear rate constant temperature dependence is thus a possible method of detecting the tunnel effect. Qualitative comparison of the classical Arrhenius plot with the rate constant temperature dependence obtained when considering the quantum tunneling is shown in Figure 5 for the specific oxidation reaction on platinum, as an example. Thus, analyzing the temperature dependence of the “quantum” rate constant kq in a wide temperature interval, one can estimate the corresponding activation energy and pre-exponential after accounting for tunneling. It is clear



TABLE 2: Saddle Point (xˆ A, xˆ B, rˆ) and Equilibrium Coordinates, and Components of the Activation Barriers of CO Oxidation Reaction on Metal Surfacesa

a

metal

Ni

Ir

Rh

Pd

Pt

∆Ea(xˆ A) ∆Ea(xˆ B) ∆Ea(rˆ) ∆Ea(θˆ ) xˆ A/xA xˆ B/xB rˆ/r

0.2 9.5 14.9 1.0 1.832/1.800 1.387/1.116 1.596/1.358

1.1 7.4 13.6 0.8 1.945/1.870 1.531/1.270 1.808/1.371

0.5 7.7 11.1 0.5 1.917/1.870 1.502/1.224 1.710/1.371

0.4 5.8 10.1 0.5 1.913/1.870 1.442/1.189 1.621/1.307

0.7 4.8 10.1 0.5 1.916/1.860 1.400/1.180 1.726/1.364

Energetic characteristics in kcal/mol, distances in Å, for the CO atop initial adsorption state.

Figure 5. Tunneling effect upon Arrhenius parameters; CO oxidation reaction on the (111) platinum surface is considered as a representative example.

from Figure 5 that the intercept of the ln kq versus 1/T plot (ln Aq) will be smaller than the semiclassical value ln Ascl. Results of calculations of the activation energies Eqa and rate constants kq taking into account the tunneling are given in Table 3 for a fixed temperature, along with published experimental data. These kinetic characteristics are compared with the corresponding semiclassical values in Figure 6a,b for the CO atop initial adsorption state. Parameters characterizing the adsorption of CO on a hollow site are not known for iridium, and those for the CO adsorption on a bridge site are not known for palladium and iridium. Therefore, the calculated kinetic data for these adsorption sites were only represented in a table form. One can see from Table 3 and Figures 6 that lgkq, as well as the similar semiclassical values, correlate linearly with the surface reaction energies, and kq are about two orders higher than the latter ones due to the tunneling. The behaviors of the Eqa and Escl a with the reaction heat are different. However, although the activation energy Eqa varies nonmonotonically in contrast to the Escl a , the corresponding rate constant changes monotonically due to a compensation effect of the prefactor. A linear relationships between the activation energies and thermodynamic characteristics of catalytic reactions have been shown previously in several works40,41,72-74 in a formal way (i.e., by correlating measured or calculated activation energies or logarithms of rate constants with the adsorption energies of reactants). In contrast to these works, the correlations in Figure 6 result immediately from eqs 3-8 of our model in which ∆Esurf (but not adsorption energies) is an evident parameter. Furthermore, this correlation may have, generally speaking, a nonlinear character. We can see from these equations that the points in the correlation plot will be scattered in a small interval if other adsorption parameters of reactants (for example, adsorption heights and frequencies, Table E1) are changed insignificantly in the set of metals. Thereby, the relationship between the rate

constants or activation energies and the corresponding reaction heats has a clear physical interpretation in our model. We discuss now temperature dependencies of the Eqa . The quantum activation energy, in contrast to the corresponding semiclassical value, depends dramatically on temperature due to the tunneling (see eq 26). We investigated these dependencies for CO oxidation on (111) platinum and rhodium surfaces, which were widely studied in experiments. The corresponding plots are shown in Figure 7a,b. We see that for the reaction on the platinum surface, the activation energy increases from ∼7 to 13 ÷ 22 kcal/mol (Figure 7a) under increasing temperature from T of ∼400 to ∼600 K depending on the model used for the CO initial adsorption state. This result is in qualitative agreement with the corresponding experimental data referred to Table 3 (see also the references cited in Table 3): they describe the increase of the activation energy with increasing temperature. For the reaction on Rh (111), the activation energy was found to be equal to 14.3 kcal/mol at T < 529 K and was equal to 25 kcal/mol at T > 529K. This result may be easily interpreted in terms of Figure 7b using the curve for the hollow model. Because of tunneling, one may expect a noticeable kinetic isotope effect (KIE) for the considered reactions. As an example, we estimate the KIE for the reaction on Pt(111) under substitution of carbon 12 in CO by its isotope C(14). The observed kinetic isotope effect is given by the relation

KIE ) kq(12)/kq(14) which may be rewritten according to eq 18 in the form

KIE ) Qt(12)/Qt(14) ‚ [k(12)/k(14)]scl

(27)

KIE is approximately determined by the ratio of the tunnel factors Qt(12)/Qt(14) since the ratio of the semiclassical rate constants is close to unit. In any case, the Qt(12)/Qt(14) term may be considered as the low estimation of the isotope effect because the term in square brackets may only increase the isotope effect. Because according to eq 20 Qt depend only on the imaginary frequency at constant temperature, we must calculate the frequency iω ˆ 1using eq 17 for the carbon mass 12 and 14. This leads to KIE(low) ) 1.08, 1.20, and 1.36 at T ) 500, 400,and 300 K. Therefore, a rather noticeable isotope effect is predicted for the considered. 4. Concluding Remarks The adiabatic potential energy surfaces for CO oxidation on the (111) surface of transition metals were constructed using the equilibrium characteristics of the reactants, CO and O, and the reaction product, CO2. The saddle point coordinates on these surfaces were calculated by solution of a set of four equations. The analysis of contributions to the activation barriers for the atop initial CO state shows that the main contributions to the



TABLE 3: Activation Energies and Rate Constants Calculated That Account for Tunneling at T ) 500Ka,b initial adsorption state of CO atop bridge 3-hollow

metal

Ni

Eqa

16.8 3.66 18.3 -0.99 20.3 -2.23

lgkq Eqa lgkq Eqa lgkq

Ir 18.9 4.47

Rh

Pd

Pt

14.6 5.64 15.7 2.89 16.5 1.96

10.1 7.31

11.2 7.42 14.2 4.81 17.9 6.9

Published experimental activation energies 22.130c 19.9 ( 224d 14.3 f 2521e 24.525f 25.422g

15.7 2.23 (14 f 27) ( 21h 28 f 2171i

17 f 404j 11.7 f 24.15k 11.320l 12.269m 9 ( 170n

b c d a Reaction heats ∆E surf as in Table 1. Energies in kcal/mol, rate constants in 1/s. T ) 425 f 625 K. T < 425 K; at T > 425 K, Ea decreases to -7 kcal/mol. eT ) 360 ÷ 779 K; polycristal.; extrapolated to θ f 0; 14.3 kcal/mol at T < 529 and 25 kcal/mol at T > 529. fT ) 500 ÷ 540 K; θ(O) corresponds to the saturation of oxygen coverage, θ(CO) f 0. gT ) 454 ÷ 625K. hT < 500 K; θ f (O) decreases from 1/3 to 1/4. iT ) 500 ÷ 575 K, apparent activation energy; at T < 500, Ea f 21 kcal/mol. jT ) 100 ÷ 700 K; for both reactants, concentration changes from ∼0 to 2.5 × 1014/cm2. kThe first value corresponds to high oxygen coverage; the second one, both reactants are at low coverage. lT is below room temperature; θ(O) ) 1/4, θ (CO) ) 1/2. mT ) 300 ÷ 400 K; θ f 0. nT ) 275 ÷ 305 K; θ(O) ) 1/4, θ(CO) ) 0.23

Figure 6. Relationship between the surface reaction energies (∆Esurf) and the kinetic characteristics (a, activation energies; b, decimal logarithms of rate constants) of the CO oxidation reaction on (111) surfaces calculated in the semiclassical model (Escl a , kscl) or while accounting for quantum effects (Eqa , kq) for atop initial-adsorbed states of carbon monoxide; points on the plots arranged from the right to the left correspond to Ni, Ir, Rh, Pd, and Pt.

Figure 7. Plots of activation energy against temperature for the CO oxidation reaction on (111) platinum (a) and rhodium (b) surfaces calculated for atop and hollow initial-adsorbed states of carbon monoxide.

activation barriers are due to approaching reactants and the extension of the M-O bond. The activation energy needed to reach the transition state is more noticeable if CO is adsorbed in a hollow site before the reaction starts. Quantum effects along the reaction coordinate were considered using an approximate approach by Bell.64 It was found that the tunneling affects dramatically the temperature dependencies of the activation energies and rate constants. At a fixed temperature the quantum

rate constant linearly decreases when the surface reaction energy ascends, and it is about two orders higher (at 500 K) than the corresponding semiclassical rate. A kinetic isotope effect upon substitution of C(12) by C(14) was predicted. Its lower limit due to the ratio of tunnel factors Qt (12)/Qt(14) was estimated to be 1.2 at T ) 400 K. We end by noting that in a future publication, we will consider the overall CO oxidation kinetics on these metals to determine



the optimal pressure and temperature. This will require the information of Tables 1 and 3. Because CO adsorption is not activated and CO desorption energy does not vary significantly from one metal to another while the trends in activation energy of reaction and oxygen adsorption are opposite in inclination, the data in the Tables suggest that the overall reaction will be limited by surface reaction at the left end of the family of metals and by oxygen adsorption at the right end. This implies that uncertainties in the value of Ea for Pt and Pd will be not affecting the results considerably. This will be a partial step toward prediction of CO oxidation kinetics in supported heterogeneous catalysis, because it does not account for the multifaceted nature of the catalyst or for the supported nature, which affects the behavior (e.g., ref 80), or the metal-oxide phase, which is known to form under certain conditions, and is the suspected to cause the oscillatory dynamics of this reaction. Yet certain aspects may be treated with similar tools to those applied here, and the overall result will be compared to experimental data. Acknowledgment. Work supported by the Israeli Science Foundation. Appendix A. The Relations between the Coordinates of Initial and Final States

mB xA ) r sin φ + x ≡ -cBr sin φ + x (A1) mA + mB xB )

mA r sin φ + x ) cAr sin φ + x mA + mB sin φ ) (xB - xA)/r

(C3)

with the parameters Bei and Ri. The components of the initial diabatic PES, VAi (xA) and B Vi (xB), characterizing interactions of the fragments CO(2) and O(2) with a metal surface, respectively, have the form 2 VA,B i (xA,B) ) DA,B{1 - exp[-βA,B(xA,B - x0A,B)]}

(C4)

where DA ≡ DM-CO and DB ≡ DM-O are the corresponding potential wells for CO and O. The anharmonicity parameters, βA and βB, of these potentials are expressed in a standard way using the perpendicular to a surface vibration frequencies ωA and ωB and the reduced masses (which are approximately equal to the masses of A and B, respectively.) The rotation potential energy of the CO bond is described in harmonic approximation

zf(φ) ) (1/2)Cf(φ - φof)2

(C5)

(A2) (A3)

Appendix D. Equations for Calculating the Saddle Point

wi({rk}) ) θ2Er

(B1)

wf({rk}) ) (1 - θ)2Er

(B2)

where Er is the reorganization energy that characterizes the local change of metal structure around adsorbed molecules.58-60,62 Appendix C. Other Components of the Initial and Final Potential Energy Surfaces The interaction between the molecularly adsorbed AB and the metal in the direction perpendicular to the surface is approximated by a Morse function

(C1)

Here Df is the depth of the potential well for chemisorption of CO2 molecule, βf ) ωxf xM/2Df is the anhamonicity parameter, M is the reduced mass for the perpendicular vibration of the CO2 (along x coordinate) relative to a surface, which is approximately equal to the mass of CO2 , and ωfx is the frequency of the perpendicular vibrations. The term uf(r) of the final state, which is identified with the C-O(2) bond of the chemisorbed CO2 molecule is approximated by a Morse function

uf(r) ) Bf{1 - exp[-Rf(r - r0f)]}2

ui(r) ) Bei exp[-2Ri(r - r0f)]

where Cf ) Iω2rock, I is the corresponding inertia moment, ωrock is the frequency of the rocking vibrations of the effective AB molecule, and φ is determined by eq A3.

Appendix B. Potential Energies of Metal Atom Vibrations

Vf(x) ) Df{1 - exp[-βf(x - x0f)]}2

CO(2) fragment, µ is the reduced mass of the vibration, and ωrf is the corresponding frequency. The distance dependence of the repulsive interaction between the separate reactants (i.e., between the fragments CO(1) (A) and O(2) (B)) is described by an exponential function

(C2)

where Rf ) ωrfxµ/2Bf, Bf is the dissociation energy of the

Equations determining the saddle point for the specific form of the above potentials have the following form -∆Esurf - (1 - 2θ)Er - Bf{1 - exp[-Rf(r - r0f)]}2 + Bei exp[-2Ri(r - r0f)] - 0.5Cf(φ - φ0f) Df(1 - exp[-βf(x - x0f)])2 + DA(1 - exp[-βA(xA - x0A)])2 + DB(1 - exp[-βB(xB - x0B)])2 - V(2θ - 1)/xθ(1 - θ) ) 0 (D1) θBfRf{exp[-Rf(r - r0f)] - exp[-2Rf(r - r0f)]} (1 - θ) Bei Ri exp[-2Ri(r - r0f)] + (1 - θ) × DAβA{exp[-βA(xA - x0A)] - exp[-2βA(xA - x0A)} cB × (xA - xB)/r + (1 - θ) DBβB{exp[-βB(xB - x0B)] exp[-2βB(xB - x0B)]} cA(xB - xA)/r - (dV/dr)xθ(1 - θ) ) 0 (D2) θDfβf{exp[βf(x - x0f)] - exp[-2βf(x - x0f)]} + (1 - θ) DAβA{exp[-βA(xA - x0A)] exp[-2βA(xA - x0A)} cB(xA - xB)/r + (1 - θ) DBβB × {exp[-βB(xB - x0B)] - exp[-2βB(xB - x0B)]} (dV/dx)xθ(1 - θ) ) 0 (D3) θCf(φ - φ0f) - (1 - θ) DAβA{exp[-βA(xA - x0A)] exp[-2βA(xA - x0A)}cBxr2 - (xB - xA)2 + (1 - θ) DBβB × {exp[-βB(xB - x0B)] - exp[-2βB(xB - x0B)]} × cAxr2 - (xB - xA)2 ) 0 (D4) Appendix E. Parameters Application of the approach described above to the calculation of the activation energies and pre-exponential factors of the of



TABLE E1: Parameters Characterizing Interaction of a Reactant with the Metal in Stationary Chemisorbed Statesa metal

Ni

Rh

Ir

Pd

Pt

1.88 505 -41

1.87 480b -33

1.86 480 -33.5

carbon monoxide atop adsorption x0A ωA Eads(CO) bridge adsorption x0A ωA Eads(CO) hollow adsorption x0A ωA Eads(CO)

1.80 400 -31.5

1.87 480 -40

1.44 380 -42

1.55 346 -41

1.33 353 -44

1.47 340 -42

1.55 380 -33 1.30 320 -36

1.37 372 -31

1.270 533 -36

1.189 490 -27

1.180 480 -25

1.902 1.432 348 656 -0.12

1.866 1.307 320 1026 -0.1

1.872 1.364 300 894 -0.11

oxygen hollow adsorption x0B ωB Eads(1/2O2)

1.116 566 -53

1.224 572 -28

carbon dioxide hollow adsorptionc x0f r0f ωrf ωxf φ0f

1.740 1.358 421 934 -0.14

1.859 1.371 390 880 -0.12

a Energetic characteristics in kcal/mol, distances in Å, angles in radian, and frequencies in cm-1. bThe value of this frequency was taken equal to that for Pt; there is no published data about this frequency for Pd. cParameters x0f and r0f are the same as the coordinates x and r in Figure 2 of Supporting Information; φ0f )A1-A2 where A1 and A2 are the angles shown in Figure 2 of Supporting Information; parameters ωrf and ωxf are the same as the frequencies ω(C-O(2)) and ω(M-CO2) in Table 15 of Supporting Information.

CO oxidation rate constant employs physical characteristics of CO2 and CO molecules and atomic oxygen in the chemisorbed state, which may be obtained from experiments or by quantum chemical calculations. These characteristics are the dissociation parameters Df, Bf , Bei , and DA,B of the adsorption Morse potentials, the adsorption heights x0f, x0A and x0B, and the corresponding vibration frequencies, ωxf , ωA, and ωB along these reactive coordinates, the equilibrium bond length r0f and the corresponding frequency ωrf, the equilibrium angle φ0f and the frequency ωrock of eq C5 By now a great number of works were published where these adsorption parameters for considered metals were measured in experiments or obtained by theoretical calculations. The summary of these characteristics for the carbon monoxide and oxygen adsorption is given in Tables 1-7 and 8-14, respectively, of Supporting Information. The values compiled from these tables and used in our kinetic calculations are listed in Table E1. In Table E1, there is also information that describes CO2 interaction with the metals; it was compiled from Table 15 of Supporting Information. One should note that data of Table E1 concern atop, bridge, and hollow CO adsorption in the initial state, and data characterizing oxygen concern hollow adsorption. The hollow adsorption of oxygen is believed to be the most favorable for all considered metals. As for CO, there is no commonly accepted opinion what site is more favorable (see Tables of Supporting Information). Our strategy consists (1) of considering the top adsorption position for all metals and (2) of comparing various adsorption sites for one metal. The parameters of the potentials C1-C5 were estimated as follows: The dissociation energy of C-O(1) bond of CO2

molecule in gas phase is equal to 127 kcal/mol.66 The parameter Bf of an adsorbed CO2 molecule (eq C2) must be lower than that for a gas phase one as the C-O(1) bond is weakened due to the interaction with the surface. To estimate this parameter, we calculated the dependence of the potential energy of a free carbon dioxide ion on the C-O bond length (under optimization of other geometric parameters) in the range 1.25-5 Å. The obtained potential curve was fitted by a Morse function. The dissociation parameter of this function was found to be equal to 88 kcal/mol. Then, following ref 58, we assume a linear correlation between this parameter and a total charge on a CO2 molecule. According to our estimations in terms of the models described in Supporting Information, the negative charges on a CO2 molecule chemisorbed on the considered metals is ∼-0.2 eu. Therefore, we can take the value of Bf for the chemisorbed carbon dioxide equal to be 117 kcal/mol. Following ref 58, we assume the parameter Bei to be equal to 0.3 × Bf and Ri to be equal to the similar parameter for the final state. The values of the dissociation parameters of the Morse curves given by eq C4 (i.e., values of DA and DB), were taken to be equal to the corresponding negative adsorption energies listed in Table E1. The parameter Df is taken to be equal to the negative adsorption energy of the CO2 molecule; the latter according to experimental estimations5,17,26,67-70 may be assumed to be ∼6 kcal/mol. The exact form of the effective electron resonance integral V (see eqs D1-D3) should be found as a result of quantum chemical calculations. In general, it depends on nuclear coordinates x and r. One may expect that this dependence will be close to an exponential. Therefore, we accept the following empirical form for the resonance integral V where

V) V0 exp[-(x - x0f)/ax](1 - exp[-(r - r0f)/ar])

(E1)

with ax and ar both being equal to 1 Å.58 Although this approach does not have a rigorous quantum mechanical basis, it should provide a plausible initial step for interpolation between two diabatic potential energy surfaces. Here, we take the value V0 ) 3 kcal/mol that was accepted in ref 58. The reorganization parameter Er (for details see ref 58) is approximately estimated as the sum of contributions of separate bonds metal-metal near an adsorption center, which changes its length under chemisorption. Our calculations on 16 atomic three-layer metal cluster modeling a metal (structure of the cluster may be seen in Supporting Information) lead to a value of Er from 4 to 8 kcal/mol and the average value of 6 kcal/mol was assumed in this work. Supporting Information Available: Tables of the adsorption characteristics of carbon monoxide, carbon dioxide, and oxygen, and figures of the cluster models, which were used for calculation of the chemisorption of CO2 on (111) metal surfaces. This material is available free of charge via the Internet at http:// pubs.asc.org. References and Notes (1) Engel, T.; Ertl, G. J. Chem. Phys. 1978, 69, 1267. (2) Engel, T.; Ertl, G. AdV. Catal. 1979, 28, 1. (3) Engel, T.; Ertl, G. In The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis; King, D. A., Woodruff, D. P., Eds.; Elsvier: New York, 1982; Vol. 4. (4) Gland, J. L.; Kollin, E. B. J. Chem. Phys. 1983, 78, 963. (5) Campbell, C. T.; Ertl, G.; Kuipers, H.; Segner, J. J. Chem. Phys. 1980, 73, 5862. (6) Matsushima, T. J. Phys. Chem. 1984, 88, 202. (7) Hagen, D. I.; Nieuwenhuys, B. E.; Rovida, J.; Somorjai, G. A. Surf. Sci. 1976, 57, 632. (8) Kuppers, J.; Plagge, A. J. Vac. Sci. Technol. 1976, 13, 256.

Catalytic CO Oxidation on (111) Metal Surfaces (9) Stuve, E. M.; Madix, R. J.; Brundle, C. R. Surf. Sci. 1984, 146, 155. (10) Conrad, H.; Ertl, G.; Kuppers, J. Surf. Sci. 1978, 76, 323. (11) Akhter, S.; White, J. M. Surf. Sci. 1986, 171, 527. (12) Matsushima, T.; White, J. M. J. Catal. 1975, 39, 265. (13) Matsushima, T. J. Catal. 1978, 55, 337. (14) Matsushima, T. Surf. Sci. 1979, 79, 337. (15) Matsushima, T. Surf. Sci. 1979, 87, 665. (16) Hori, G. K.; Schmidt, L. D. J. Catal. 1975, 38, 335. (17) Matolin, V.; Gillet, E.; Gillet, M. Surf. Sci. 1985, 162, 354. (18) Matsushima, T.; Asada, H. J. Chem. Phys. 1986, 85, 1658. (19) Campbell, C. T.; Ertl, G.; Kuippers, H.; Segner, J. J. Chem. Phys. 1980, 73, 5862. (20) Witterlin, J.; Volkening, S.; Janssens, T. V. W.; Zambelli, T.; Ertl, G. Science 1997, 278, 1931. (21) Campbell, C. T.; Shi, S.-K.; White, J. M. J. Phys. Chem. 1979, 83, 2255. (22) Peden, C. H. F.; Goodman, D. W.; Blair, D. S.; Berlowitz, P. J.; Fisher, G. B.; Oh, S. H. J. Phys. Chem. 1988, 92, 1563. (23) Goodman, D. W.; Peden, C. H. F. J. Phys. Chem. 1986, 90, 4839. (24) Schwartz, S. B.; Schmidt, L. D.; Fisher, G. B. J. Phys. Chem. 1986, 90, 6194. (25) Brown, L. S.; Siebner, S. J. J. Chem. Phys. 1988, 89, 1163. (26) Brown, L. S.; Siebner, S. J. J. Chem. Phys. 1988, 90, 2807. (27) Matsushima, T.; Matsui, T.; Hashimoto, M. J. Chem. Phys. 1984, 81, 5151. (28) Colonell, J. I.; Gibson, K. D.; Siebner, S. J. J. Chem. Phys. 1995, 103, 6677. (29) Hopstaken, M. J. P.; Niemantsverdriet, J. W. J. Chem. Phys. 2000, 113, 5457. (30) Berlowitz, P. J.; Peden, C. H. F.; Goodman, D. W. J. Phys. Chem. 1988, 92, 5213. (31) Eichler, A. Surf. Sci. 2002, 498, 314. (32) Salo, P.; Honkala, K.; Alatalo, M.; Laasonen, K. Surf. Sci. 2002, 516, 247. (33) Alavi, A.; Hu, P.; Deutsch, T.; Silvestrelli, P. L.; Hutter, J. Phys. ReV. Lett. 1998, 80, 3650. (34) Eichler, A.; Hafner, J. Phys. ReV. B. 1999, 59, 5960. (35) Liu, Z.-P.; Hu, P. J. Chem. Phys. 2001, 115, 4977. (36) Zhang, C. J.; Hu, P. J. Am. Chem. Soc. 2001, 123, 1166. (37) Zhang, C. J.; Baxter, R. J.; Hu, P.; Alavi, A.; Lee, M.-H. J. Chem. Phys. 2001, 115, 5272. (38) Zhang, C. J.; Hu, P. J. Am. Chem. Soc. 2000, 122, 2134. (39) Zhang, C. J.; Hu, P. Top. Catal. 2004, 28, 71. (40) Gong, X.-Q.; Liu, Z.-P.; Raval, R.; Hu, P. J. Am. Chem. Soc. 2004, 126, 8. (41) Michaelides, A.; Liu, Z.- P.; Zhang, C. J.; Alavi, A.; King, D. A.; Hu, P. J. Am. Chem. Soc. 2003, 125, 3704. (42) Stampfl, C.; Scheffler, M. Phys. ReV. Lett. 1997, 78, 1500. (43) Koper, M. T. M.; Jansen, A. P. J.; van Santen, R. A.; Lukkien, J. J.; Hilbers, P. A. J. J. Chem. Phys. 1998, 109, 6051. (44) Eichler, A. Surf. Sci. 2003, 526, 332. (45) Eichler, A.; Hafner, J. Phys. ReV. Lett. 1997, 79, 4481. (46) Doll, K. Surf. Sci. 2004, 573, 464. (47) Kuittinen, R.-L.; Laasonen, K. Surf. Sci. 2005, 314, 19.

J. Phys. Chem. C, Vol. 111, No. 26, 2007 9193 (48) Mittendorfer, F.; Eichler, A.; Hafner, J. Surf. Sci. 1999, 433, 75. (49) Mavrikakis, M.; Rempel, J.; Greely, J. J. Chem. Phys. 2002, 117, 6737. (50) Mavrikakis, M.; Xu, Y. Surf. Sci. 2001, 494, 131. (51) Mavrikakis, M.; Xu, Y. Surf. Sci. 2003, 538, 219. (52) Kohler, L.; Kresse, G. Phys. ReV. B. 2004, 70, 165405. (53) Honkala, K.; Laasonen, K. J. Chem. Phys. 2001, 115, 2297. (54) Jakob, T.; Muller, R. P.; Goodard, W. A., III. J. Phys. Chem. B 2003, 107, 9465. (55) Li, T.; Balbuena, P. J. Phys. Chem. B 2001, 105, 9943. (56) Koper, M. T. M.; van Santen, R. A. J. Electroanal. Chem. 1999, 472, 126. (57) Wang, Z. X.; Tlan, F. H. J. Phys. Chem. B 2003, 107, 6153. (58) German, E. D.; Kuznetsov, A. M.; Sheintuch, M. Surf. Sci. 2004, 554, 159, 170. (59) German, E. D.; Sheintuch, M. J. Phys. Chem. A 2005, 109, 7957. (60) (a) German, E. D.; Kuznetsov, A. M.; Sheintuch, M. Chem. Phys. 2006, 324, 129. (b) German, E. D.; Kuznetsov, A. M. J. Molec. Struct. (THEOCHEM) 2004, 671, 153. (61) Supporting Information to this paper. (62) Kuznetsov, A. M. Charge Transfer in Physics, Chemistry, and Biology; Gordon & Breach: Reading, U.K., 1995. (63) Glasston, S.; Laidler, K. J.; Eyring, H. The Theory of Rates Processes; McGraw-Hill Book Co, Inc: New York, 1941. (64) Bell, R. P. The Tunnel Effects in Chemistry; Chapman and Hall: London, 1980. (65) Wigner, E. Z. Phys. Chem. B. 1932, 19, 203. (66) Volpi, A.; Clary, D. C. J. Phys. Chem. B 2004, 108, 336. (67) Tauterman, C. S.; Clary, D. C. J. Chem. Phys. 2005, 122, 137702. (68) Tauterman, C. S.; Sturdy, Y. K.; Clary, D. C. J. Catal. 2006, 244, 199. (69) Kinne, M.; Fuhrmann, T.; Hzu, J. F.; Whelan, C. M.; Denecke, R.; Steinruck, H.-P. J. Chem. Phys. 2004, 120, 7113. (70) (a) Zaera, F.; Liu, J.; Xu, M. J. Chem. Phys. 1997, 106, 4204. (b) Xu, M.; Liu, J.; Zaera, F. J. Chem. Phys. 1976, 104, 8825. (71) Szanyi, J.; Kuhn, W. K.; Goodman, D. W. J. Phys. Chem. 1994, 98, 2978. (72) Bligaard, T.; Norskov, J. K.; Dahl, S.; Mattiesen, J.; Christensen, C. H.; Sehested, J. J. Catal. 2004, 224, 206. (73) Norskov, J. K.; Bligaard, T.; Logadottir, A.; Bahn, S.; Bollinger, M.; Hansen, L. B.; Bengaard, H.; Hammer, B.; Sljivancanin, Z.; Mavrikakis, M. et al. J. Catal. 2002, 209, 206. (74) Logadottir, A.; Rod, T. H.; Norskov, J. K.; Hammer, B.; Dahl, S.; Jacobsen, C. J. H. J. Catal. 2001, 197, 229. (75) Handbook of Chemistry and Physics, 81st ed.; CRC Press: Boca Raton, FL, 2000-2001, p 9.66. (76) Matsushima, T.; Matsui, T.; Hashimoto, M. J. Chem. Phys. 1984, 81, 5151. (77) Matsushima, T. J. Phys. Chem. 1987, 91, 6192. (78) Matsushima, T. Surf. Sci. 1983, 127, 403. (79) Becker, C. A.; Cowin, J. P.; Wharton, L.; Auerbach, D. J. J. Chem. Phys. 1977, 67, 3394. (80) Sheintuch, M.; Schmidt, J.; Lechtman, Y.; Yahav, G. App. Catal. 1989, 49, 55.

Theoretical Study of Catalytic CO Oxidation on (111) Metal Surfaces

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