Rotational effects in the dissociative adsorption of H2on Cu(111)

摘要

Faraday Discuss., 1993,96,43-54 Rotational Effects in the Dissociative Adsorption of H2on Cu(ll1) George R. Darling and Stephen Holloway"? Surface Science Research Centre, University of Liverpool, PO Box 147, Liverpool, UK L69 3BX Experimental studies show that the dissociation probability of H, molecules impinging on a Cu(111) surface is strongly affected by the initial rotational state. Increasing the angular momentum slightly suppresses the dissociation at low J, but strongly enhances it at high J. We show that this is due to two competing effects; one is essentially orientational, and results in the decrease in dissociation probability, while the increased dissociation is due to the transfer of energy from the rotational coordinates into the reaction coordi- nate (R-T transfer).Quantum-mechanical wavepacket calculations are used to illustrate these effects, focussing in particular on the close connection between R-T and vibrational-translational (V-T) coupling. Vibrational excitation is now well known to promote the dissociation of H, on Cu surfaces.' This is due to the occurrence of a strong curvature in the interaction potential as the bonding switches from intramolecular to H-surface bonding before the disso- ciation barrier, and a sharp decrease in the energies of the effective vibrational levels, or, in the jargon of the field, the barrier is late., The vibrations and translations are strongly coupled in this curved interaction region resulting in enhanced dissociation, and also, if there is sufficient curvature before the barrier, in vibrational excitation of the scattered fraction3 such as observed in the H2/Cu( 11 1)system4.' More recent desorption experiments by Rettner et al.have examined the role of initial rotation on the dissociation of D, and H,.6*7 Molecules that recombine and desorb from Cu( 11 1) are state-selectively detected and, by appealing to microscopic re~ersibility,~.'dissociation probabilities are determined as a function of initial rotation- al state, J. Fitting these with a functional form which is roughly sigmoidal, they observe a slight upwards shift of the midpoint of the rise with increasing J until J z 5 or 6, when there is a monotonic decrease with increasing J. This is shown in Fig. 1.In this paper, we shall present theoretical models of these phenomena. Proper simu- lations of the rotational effects in dissociative adsorption require four dimensions; the H, centre of mass-surface distance, 2, the molecular bond length, x, and the two rota- tional degrees of freedom, 8 and @. This major project is underway and completed results will be presented elsewhere. Following from the theoretical studies of vibra- tionally enhanced dissociation, however, we can illustrate the effect of placing energy into the various coordinates by performing lower-dimensional calculations on appropri- ate model potential-energy surfaces (PESs). The results obtained supply us with an intu- itive picture which can then be used to unravel the more complicated results of higher-dimensional simulations.The models used here are two-dimensional, illustrating the competing effects of rotational-energy transfer and orientational hindering on the -f And Department of Chemistry. 43 44 Rotational Eflects in H, Dissociation 0.6 (a) h C 0.- c. n ///,o "." 0.0 0.2 0.4 0.6 0.8 1.o I I Ill I Fig. 1 (a) Experimental J-dependent adsorption probabilities (arbitrarily scaled) for D, in the vibrational ground state. Numbers on the curves indicate the value of J; 0,points of inflection. (b) Plot of the point of inflection, E,, us. J for three initial vibrational states n indicated by the numbers on the curves. From the experimental results of Michelsen et aL7 dissociation reaction.The next two sections contain model calculations for helicopter and cartwheel dynamics. The effect of translational, vibrational and rotational energy on dissociative adsorption is discussed. Finally a summary with some conclusions are pre- sented at the end of the paper. Releasing the Rotational energy For a diatomic molecule, the involvement of vibrational energy in dissociation has a clear origin in the geometry of the PES, the coordinate initially corresponding to vibra- tion evolves into the reaction coordinate as the molecule dissociates. The location of the barrier on the, now familiar, elbow potential (x and Z) then determines whether the vibrational energy is available for promotion of the reaction. The contribution from rotations is, however, less clear. In the free molecule there is some coupling between rotation and vibration, due simply to the centrifugal stretching of the bond.The size of G. R. Darling and S. Holloway Table 1 Internal energy and expectation value of the bond length as a function of rotational state for a free Morse rotor corresponding to H, in the vibrational ground state J e,/eV xla, 0 0.269 1.453 4 0.413 1.474 8 0.768 1.529 12 1.293 1.611 16 1.935 1.720 20 2.645 1.855 this is simple to estimate from the moment of inertia and the bond force constant. For the vibrational ground state the bond length, x, as a function of J, is given in Table 1. In order to place this in context, theoretical estimates hitherto have placed dissociation barriers at extensions of Ax z 1 a, (x 2.4 a,) and therefore R-V coupling at this level certainly could not be responsible for shifting the dissociation probabilities by the amount shown in experiment.Rather, in this section, we shall show that the vibrational and rotational couplings are intimately connected via the surface, i.e. it is the strong change in the bond length before the activation barrier which is responsible for the release of both vibrational and rotational energy into the reaction coordinate. The model that we employ derives from a close-coupling wavepacket (CCWP) treat-ment of the problem." If we write the wavefunction as an expansion in time-independent basis functions in the rotational coordinate, with time-dependent wavepackets for the molecular centre of mass-surface distance and the molecular bond length i.e. where, for simplicity, we consider only one rotational dimension, then substitution into the Schrodinger equation gives .av a$j(x, 2,t)I -= i 1yj(e)at at where KxJK0)are the kinetic energy operators in the x,Z(0)coordinates. The equation of motion for a particular channel wavepacket can now be obtained by projecting into the functions yit The first term on the right-hand side represents a 'rotationally adiabatic' contribution to the dynamics, while the second term represents the coupling between different rota- tional states. If we set these off-diagonal matrix elements of the potential to zero, then Rotational Eflects in H, Dissociation 4.0 03 3.0 .-+ !! 8 g 2.0 Q) m 't Tm 1.0 r 0.0 I I 0.0 1.o 2.0 ,3.0 H- H separationla, Fig.2 The late-barrier PES used in the calculations. This is based on the results of ab initio calculations by Muller," however, the barrier has been altered to remove the many sharp reson- ances, and a realistic physisorption well has been added. The zero in the x coordinate is the equilibrium bond length of H, in the gas-phase (ca. 1.4 ao). we obtain a new two-dimensional model of the sticking in which the rotational state appears in the J,2/2px2 term, which we can simply add to the potential, (V)yi.This model represents the dissociation of 'helicopter' rotors from a structureless surface and will be the subject of our first study.? Our PES is based on the ab initio calculations of Miiller,ll although we have moved the barrier to make it somewhat less late to avoid the complication of strong vibrational resonances.2*13 In addition, we have included a realistic physisorption well. This changes the shape of the barrier in the entrance channel in such a way as to fix the relative positions of the dissociation thresholds for the different vibrational states. l4 The PES is shown in Fig. 2, and the detailed form is given in the Appendix. Quantum- dynamical simulations have been performed with the split-operator method' ' using a projection grid-cutting technique to extract final, state-resolved probabilities.I6 The eigenfunctions of the initial state are those of the rotating Morse oscillator correspond- ing to the free molecule plus the centrifugal term.They (and the eigenvalues) were deter- mined using the relaxation method of Kosloff and Ta1-E~er.l~ The dynamics on the two-dimensional PES shown in Fig. 2 can be usefully (but qualitatively) described in terms of the vibrationally adiabatic potentials.12 These are the energy eigenvalues of the one-dimensional potentials obtained by taking slices of the PES orthogonal to the reaction path. By analogy, adding in the centrifugal term we obtain a set of J-dependent potentials for each vibrational state. These are shown in Fig. 3. The barrier for both n = 0 and TI = 1 clearly decreases with increasing J. This is due simply to the barrier occurring at an extension in excess of the gas-phase equilibrium bond length, xeq.Thus the barrier decreases relative to the energy of the vibrational ground state of the free molecule, because the contribution there from the centrifugal term is correspondingly smaller than at xeq.t Rotational motions with the angular momentum, J, perpendicular to the surface plane are commonlyreferred to as helicopters, while those with J parallel are cartwheels. G. R. Darling and S. Holloway 0.6 (a)t 0.4 1 0.2 } fl '-u.z bsol;' -0.4 . bsol; -0.6 0 1 2 3 4 I' -0.4 1 0 1 2 3 4 reaction path/a, Fig. 3 Vibrationally adiabatic potentials for the PES shown in Fig. 2, with the centrifugal term added, thus supplying a J-dependence. The reaction path is taken to be the line of steepest descent from the barrier maximum, with the zero occurring in the physisorption well, which has been set to 0 eV for convenience.(a) Vibrational ground state, (b)first excited state. Numbers on the curves indicate the J values. We can express this in terms of energy exchange. In the gas phase, the molecule has vibrational energy Erot= J2/21 (for a plane rotor, moment of inertia, I, and angular momentum, J). If the angular momentum remains fixed throughout the dissociation (i.e. the potential is independent of 4),then due to the increase in I = p2as the bond length stretches, Erotmust decrease. At least part of the excess energy is then available to assist in the dissociation, further stretching the bond, and further reducing the rotational energy.This analysis is borne out in the full dynamical results for this model, shown in Fig. 4. The oscillations in the dissociation probabilities for energies above the dissociation threshold are due to the onset of vibrational excitation in the refle~tivities,~ and to vibrational resonances remaining at high energies. l2 Note that, as in previous studies, ,the dissociation reaches 50 of the saturation value at an energy, E~ approximately equal to the adiabatic barrier height. Rotational Efects in H, Dissociation 1.o . 0.8 c..- D a9 0.6 P C .-0 c. $ 0.40 .--0 0.2 0.0 0.2 0.4 0.6 0.8 1.o 1.2 translational energy/eV Fig. 4 Dissociation probabilities for perfect helicopters in the vibrational ground state on a struc-tureless surface as a function of initial translational energy, and for a range of initial rotational states.The PES is that in Fig. 2. Numbers on the curves indicate the J values. We can quantify the influence of rotational energy by plotting E~(J)~~(0)-as a function of the internal energy of the initial state (note that this is less than the sum of vibrational energy and rotational energy of a rigid rotor because of the centrifugal coup-ling in the gas phase) and this is shown in Fig. 5. Included in this figure are the results obtained by assuming vibrational adiabaticity for this PES, that is, the adiabatic barrier heights relative to the J = 0 molecule shown in Fig.3. As noted above, this is a good description for the vibrational ground state in Fig. 4, but it is a poor description for n = 1. Although the approximation appears worse for high J, it is actually at low J b(J)-(O) 0.0 0.2 0.4 0.6 I I I I 0.0 h1-0.1 W I h3 w -0.2 -0.3 Fig. 5 Midpoint, cH, of the rise in dissociation probability us. internal energy of ro-vibrational state, for two different initial vibrational states, n,and two choices of PES topology: (---) early and (-) late barrier. The late barrier PESs shown in Fig. 2, and the parameters for the other case are given in the appendix. 0,Indicate the results obtained by assuming vibrational adia-baticity for n= 0 on the late barrier PES, i.e.the difference between the adiabatic barrier heights calculated from Fig. 3, while +,show the same for n = 1. G. R. Darling and S. Holloway where the sticking has substantial non-adiabatic effects due to the strong T-V coupling before the barrier (plotted in the figure are the differences between J and J = 0). This becomes less apparent for increasing J because the centrifugal term begins to dominate the potential, causing the barrier to move from late to early, as can be seen in Fig. 3. This movement of the barrier also alters the vibrational efficacy to dissociation. The energetic separation of the dissociation curves for n = 1 and n = 0 decreases from 0.3 eV at J = 0 to 0.19 eV at J = 10, a far greater amount than the decrease in the 0 -,1 vibrational quantum, which is only ca.30 meV. To demonstrate the strong interplay between vibrational and rotational motion and their effects on dissociation, we have also computed the dissociation probabilities for a middle and an early barrier, We have chosen the parameters to give the same barrier height in all cases, although we would not expect an early barrier to be so high.18-19 This allows us to observe effects due only to the barrier position. The middle barrier shows a reduced vibrational effect compared to a late barrier, while the early one shows no vibrational effect,' in agreement with the Polanyi rules2* for gas-phase scattering. It ,can be seen that this is also mirrored in the variation of E~ with J, i.e. it is weaker for the middle barrier (not shown for clarity) and essentially zero for the early barrier (there are slight variations in this case, due primarily to a combination of interpolation errors and the existence of threshold oscillations), as shown in Fig.5. In other words, when the vibrational enhancement of the sticking disappears, so also does the rotational enhance- ment. This clearly illustrates that it is the large extension of the bond before the disso- ciation barrier which is responsible for the observed rotational enhancement. Orientational Hindering The previous section dealt with molecules confined to remain in the broadside orienta- tion with respect to the surface which is believed to be the most favourable geometry for dissociation, In a realsitic beam, however, there will be molecules striking the surface with all orientations, and rotating molecules will sample the potential for a range of these during the collision.This is the origin of the decrease in dissociation with increas- ing J at low J. We can illustrate this by using an approximate PES based on the vibrationally adia- batic potential obtained for the ground-state molecule shown in Fig. 3. From the enlargement in Fig. 6 it can be seen that this is approximately Gaussian along the reaction path for the broadside geometry. For other orientations, there is very little information from which to construct a potential, however, it is clear that it must increase to a very high value for a molecule approaching end-on, i.e.at the energies of interest molecules striking end-on will simply reflect. To treat this, we add another Gaussian, with a much greater magnitude than the first. Thus, our second model involves the coordinates s, the position on the reaction path, and the polar angle, 8, coupled by the potential v(s, 8) = vb exp -Pb(s -sb)2 + c(e)K exp -Pe(s -sel2 (4) where C(8) is a corrugation function in the polar angle whose form is sinusoidal. The parameters are vb = 0.58 eV, sb = 2.9 a,, fib = 1.3 ai2, = 2.58 eV, s, = 3.7 a,, Pe = 0.9 ao2. In this example, we are simulating the scattering of perfect cartwheel rotors. If the bond length (ie. moment of inertia, which is equivalent to the 'mass' in the 8 coordinate) does not change, then the model represents an early barrier system.2' Fig.7 shows the sticking for a number of J states as a function of translational energy (the initial state is a plane wave in the 8 coordinate and a Gaussian wavepacket in the s coordinate). Clearly, the rotational motion strongly suppresses the sticking. The reason for this has been noted bef~re.~~,~~ As molecules rotate ever faster, they have a greater chance of Rotational Efects in H, Dissociation 5.0 0.5 4.0 0* 0.4 a (D 3.0 5,1-5. 0.3 m 4--aa c a 2.0 g0.2 II I I I 1.o 0.1 0.o 0.0 0.0 1.o 2.0 3.0 4.0 reaction path/a, Fig. 6 Vibrationally adiabatic potential for n = 0, J = 0 taken from Fig. 3 and the change in the molecular bond length along the reaction path 0.4 0.6 0.8 1.o 1.2 1.4 translational energy/eV Fig.7 Dissociation probability of cartwheel rotors on an early barrier PES with strong corruga- tion in the rotational coordinate as a function of initial translational energy and for various initial rotational states (J-values indicated on the curves.) Only one rotational dimension is included, so the states correspond to plane rotors. The step-like behaviour in the lower J states is caused by the onset of strong reflectivity into higher rotational states. G. R. Darling and S. Holloway 0.8 0.6 0.2 0.0 translational energy/eV Fig. 8 Dissociation probability as a function of initial translational energy, for several initial J states (values indicated on curves), for a model simulating planar cartwheels incident on a PES with a late barrier. The change in bond length has been approximated by varying the moment of inertia using the results of Fig.7. The result for J = 4 has been omitted for clarity, since it overlays those for J = 0 and J = 2. rotating out of the broadside configuration before reaching the transition state to disso- ciation, and they then scatter back from the higher barriers at less-favourable orienta- tions. In other words, they are moving so quickly in the rotational coordinate that they fly by the opening which leads to dissociation at 8 = 42. Following the quantum study of and the classical one of Beauregard and Mayne,2s we can extend this model to include a late barrier potential by making the moment of inertia vary along the reaction path.? Fig.6 shows the molecular bond length as a function of s for the PES in Fig. 2. To a good approximation, this can be represented by a linear function with a gradient of 2 switching on at s = 2.2 a, (the gradient is 2 rather than 1 because the reaction path must properly be calculated on a mass-scaled potential, i.e. with the x coordinate shrunk by a factor of 212). With this modification, it is no longer possible to use the split-operator method for propagation, because the kinetic-energy operator in the amp;coordinate depends on s, and so we must instead use the Chebychev method.26 With an initial wavepacket incident from the left in Fig. 6 and C(Q)= $(l + cos 20), we obtain the results shown in Fig.8. Clearly, the change in moment of inertia has reduced the orientational hindering at low J, and completely overwhelmed it at high J. This is exactly the trend required for agreement with experiment, cJ: Fig. 1. However, it is instructive to consider the extent to which the parameters for this PES must be changed for the agreement with experiment to be lost. If we make the surface more open, i.0. allow molecules with less favourable bond orientations to stick, by changing C(Q)to C(0) = $(1 + cos 28)3, then we find that orientational hindering is no longer in evidence. In other words, the corrugation must also be felt by molecules only slightly out of the broadside configuration for the suppression of sticking at low J to occur.t A similar model is also being investigated by Thomas Brunner. Rotational Eflects in H, Dissociation Alternatively, we can have C(0) = $(l + cos 20), but alter the strength of the corrugation by changing s, to 3.2 a,. In this case, we find that it is the R-T coupling which is lost, although the surface may now be too strongly corrugated to produce the smooth physi- sorption well seen in e~periment.rsquo;~ Conclusions We have demonstrated that coupling of the rotational coordinate with the reaction coordinate is intimately bound with the observation of vibration-translation coupling. Both arise from the large change in the molecular bond length induced by the PES. This causes an increase in the moment of inertia of the molecule with a corresponding decrease in the rotational eigenvalues, and, hence, the R-T coupling.For an early barrier, there is no bond extension at the transition state, and so also no vibrational and no rotational enhancement of dissociation. Rapid rotation can also inhibit sticking, however, owing to the strong anisotropy in the potential as a function of the orientational coordinates. This effect would be expected to occur for all early barrier systems. For a late barrier, we have demonstrated that it can be overcome at high J by the R-T coupling, giving rise to the initial slight drop followed by a strong rise in the dissociation probability as a function of J observed in the experiment. Slight variations to the two-dimensional PES employed indicate that observation of both could be a sensitive indicator to the precise details in and near the interaction region of the potential.We are grateful to Charlie Rettner and Dan Auerbach for helpful discussions on this subject. G.R.D. would like to thank IBM Almaden Research Laboratory for their hospi- tality during the course of this work. Appendix We have constructed our PES by defining equipotentials for a basic lsquo;elbowrsquo;, onto which we can add barriers and physisorption wells. The equipotentials are based on the function For each (x, 2)we evaluate f(x,z)= i/d(i+ 1/x4)(1+ 1/z4)-11 the bond length at 2 = on curve y. This is clearly singular for x, 2 = 0, so we must offset the curves by some amount, p. To change the angle between the exit and entrance channels, this is further twisted with a step function z+2, = z-Z,,,s(x -rn, , wl) (A3) where s(a, b) = i(1 + tanh ab) Using these elements , we then obtain an equivalent bond length, x,(x, 2)=f(xrsquo;, 2rsquo;)+fxrsquo; -f(xrsquo;, Zlsquo;),2rsquo; -f(xrsquo;, Zrsquo;)-p (A5) where xrsquo; = x + p, 2rsquo; = 2, + p, and the second term on the right-hand side forces the equipotentials to be almost identical in shape rather than softening at large x and 2.G. R. Darling and S. Holloway With x, we calculate the bare elbow potential K(x, Z) = D(1 + exp -ax,(x, 2)}2 The dissociation barrier and physisorption well are treated as separate components, with another step function to switch between them. The former is a Gaussian in both x and 2, amp;(x, 2)= VO exp(-Px(x -xb)2 -pZ(zf -zb)2)s(x -2, -m2 7 w2) (A7) (note that 2, replaces 2 on the right-hand side).The latter is composed of an exponen- tial repulsion term and a Van der Waals attraction wheref, is the usual cut-off function defined by fc(x)= 1 -2x(1 + x) + lexp( -2x) (A101 which are combined to give where 2, is again used in place of 2 on the right-hand side. The second step is included to force the replusion to die off at small 2 where it is replaced by the repulsive part of V, going around the elbow. The parameters have been chosen to give a well of depth ca. 30 meV outside of the reaction zone, in agreement with experimental results.28 The full potential is V(x,z, = I/e(x, z,+ Vb(x, z,+ Khys(X, 2) (A12) with the parameter values for the late barrier shown in Table Al.Although this form may seem impossibly cumbersome, the step functions actually make it possible to manipulate one region of the PES with relatively little disturbance to other regions. Thus to change to a middle barrier, we make the following parameter adjustments; Zoff= 0.18 a,, Vo = 0.72 eV, Px = 0.07 ai2, xb = 0.5 a,, P, = 1.0 ai2, yb = 0.5 a,, m2 = -0.7 a,, and to get an early barrier; Zoff= 0.18 a,, V, = 0.77 eV, xb = 0.01 a,,Liz = 1.0 ai2,yb = 1.2 a,, m2 = 1.6 a,. Table 1A The parameters for the model PES shown in Fig. 2. Ener- gies (D, V,, V, and Cvw)are given in eV, while units for other param- eters are given in the text elbow potential dissociation barrier physisorption potential D = 4.76 Vo = 0.7 Vl = 0.85 O! = 1.028 p, = 0.05 p, = 1.8 p = 2.0 Xb = 1.0 Z, = 0.25 Zoff= 0.3 p, = 0.0 C,, = 5.306 m, = 1.3 y, = 0.0 k, = 0.5 w1 = 1.5 m2 = 0.0 z,,= -1.9 w2 = 1.5 m3 = -0.1 w3 = 2.0 Rotational Eflects in H, Dissociation References 1 B.E. Hayden, in Dynamics of Gas-Surface Interactions, ed. C. T. Rettner and M. N. R. Ashfold, Royal Society of Chemistry, London, 1991, p. 137. 2 S. Holloway, in Dynamics of Gas-Surface Interactions, ed. C. T. Rettner and M. N. R. Ashfold, Royal Society of Chemistry, London, 1991, p. 88. 3 G. R. Darling and S. Holloway, J. Chem. Phys., 1992,97, 734. 4 A. Hodgson, J. Moryl, P. Traversaro and H. Zhao, Nature (London), 1992,356,501. 5 C. T. Rettner, D. J. Auerbach and H. Michelsen, Phys.Rev. Lett., 1992,68,2547. 6 H. A. Michelsen, C. T. Rettner and D. J. Auerbach, Phys. Rev. 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Holloway, J. Phys.: Condens. Matter, 1991, 3, S43. 23 A. Cruz and B. Jackson, J. Chem. Phys., 1991,94, 5715. 24 B. Jackson, J. Phys. Chem., 1989,93,7699. 25 J. N. Beauregard and H. R. Mayne, Chem. Phys. Lett., 1993,205,515. 26 H. Tal-Ezer and R. Kosloff, J. Chem. Phys., 1984,81,3967. 27 S. Andersson, L. Wilzen, M. Persson and J. Harris, Phys. Rev. B, 1989,40, 8146. 28 S. Andersson, L. Wilzen and M. Persson, Phys. Rev. B, 1988,38,2967. Paper 3/03507G; Received 16th June, 1993