Theoretical investigation of CH4decomposition on Ni: electronic structure calculations and dynamics

摘要

Faraday Discuss., 1993,96, 337-347 Theoretical Investigation of CH4 Decomposition on Ni : Electronic Structure Calculations and Dynamics H. Burghgraef, A. P. J. Jansen and R. A. van Santen Laboratoryfor Inorganic Chemistry and CatalysislTheory Group, Eindhoven University of Technology, P.O. Box 513,5600 MB Eindhoven, The Netherlands The dissociative chemisorption of CH, on an Ni(ll1) surface has been studied using different cluster models. Density functional theory is used to determine the transition state and the dissociated state of CH, on the sub- strate. The transition state is explicitly determined on a one-layer Ni, cluster. We find a transition-state barrier of 210.3 kJ mol-', which is much higher than our one-atom result of 40.7 kJ mol- '. The higher barrier can be understood in terms of the intrinsic lower reactivity of the central nickel atom in the cluster and a more extended CH bond. If we use as a substrate an Nil, cluster, the barrier drops to 99.7 kJ mol-'. Vibrational frequencies are abstracted from the potential-energy surface at the transition state and the dissociated state.We have used transition-state theory to compute rate constants in terms of rotational, vibrational and translational partition func- tions. We have also determined sticking coefficients and activation energies for CH, decomposition as well as CH,-H recombination of the surface. Sticking coefficients are small, which is consistent with experimental values. At 500 K a kinetic isotope effect of 6.2 is found for CH, adsorption, while only a factor of 2.0 is found for CH,-H association at this temperature.The isotope effect on the activation energy for adsorption is 6.3 kJ mol-', while essentially no effect for the activation energy for association is found. 1. Introduction Using an ab initio density functional approach, we have investigated the dissociation of CH, on different clusters modeling an Ni( 111) surface. Specifically, we have taken Ni, , Ni7,3, Ni,,,,3 and Nil, clusters to model the substrate. We have also used an Nil, cluster, which has equivalent surface atoms. The dissociative adsorption can thus be written as: his ads CH, + Nix -HNixCH3 and the reverse reaction, a (mobile) recombination of CH, and H on the surface, as: kass des HNixCH3 -CH, + Nix In eqn.(1) and (2) x = 7, 10, 13 or 19. Using the potential-energy surfaces (PES) at the transition state (TS) and at the dissociated state (DS), we have calculated vibrational frequencies in the harmonic approximation. According to transition-state theory, the rate coefficients for dissociative adsorption and associative desorption have been calcu- lated. Technical details of the calculations are discussed in Section 2. In Section 3 the electronic structure calculations are presented and discussed. In Section 4 reaction kinetics is discussed by applying transition-state theory, and in Section 5 we summarize results and draw conclusions. 337 338 CH, Decomposition on Ni C3, axis H CH3 fragment .c Ni Fig. 1 Definition of geometric parameters.H(act) denotes the activated hydrogen, H(l) denotes the hydrogen in the mirror plane, H(2,3) denote the hydrogens before and behind the mirror plane. 8 is the tilt angle of the CH, group in the mirror plane, i.e. the angle between C,, axes, when there is no tilt as in CH, (dotted line in line with activated CH bond) and in a tilted geometry as in the TS (other dotted line). r is the NiNiC angle. 2. Computational Details We have performed quasi-relativistic calculations based on density functional theory (DFT) using the Amsterdam density functional program package (ADF) developed by Baerends and co-workers.' We have used the Vosko-Wilk-Nusair energy functional2 in combination with Becke's gradient-corrected exchange, and Stoll's correction for corre-lation., For carbon a frozen core potential is used for the 1s electrons; for nickel the electrons up to 3p are frozen. Relativistic effects were taken into account by first-order perturbation the01-y.~ The basis sets are of double quality with the exception of the nickel d orbitals which are triple cs.On all atoms polarization functions are included. We have used the Ni, cluster to optimize geometrical parameters, which are shown in Fig. 1. As other models for the substrate, we have used Ni7, ,,Ni,, ,, ,and Nil, clusters. In addition we have used a Nil, cluster which has equivalent surface atoms. This cluster does not model the (1 11) surface, but consists of a central nickel atom surrounded by its 12 nearest neighbours in a hexagonal close-packed (hcp) fashion.For all nickel clusters the bond distance of the bulk (2.49 A)was used. The CH, fragment was fixed with CH distances of 1.09 A and HCH angles of 109.48" as in CH,. The NiH bond distance was fixed at 1.49 A to restrict the number of degrees of freedom. C, symmetry was main- tained for all calculations. The TS was explicitly calculated by a four-dimensional grid in the NiC and CH distances (distance between grid points: 0.1 A)and the CH, tilt angle (0) with respect to the activated CH bond and the NiNiC angle (a) (angle between grid oints: 5.0"). Grid boundaries were 1.95 and 2.25 A for the NiC distance, 1.45 and 1.75 for the CH distance, 15.0" and 30.0" for 0 and 82.5" and 97.5" for a.Vibrational calculations were performed using the GF method.6 Surface reaction kinetics was calcu- lated according to transition-state theory.' 3. Electronic Structure Calculations Optimal parameters and binding energies of the CH,/Ni, system, the TS and the DS are shown in Table 1. They were obtained by fitting the energy calculated on grid points in the CH, gaspi,, the TS and the DS region to a second-order polynomial in the NiC H. Burghgraef, A. P. J. Jansen and R. A. van Santen 339 Table 1 Geometries and energies for CH, gas/Ni, TS and DS geometry calculation RNi,-/A RNiH/A RCH(aclJA cc/degrees eldegrees E/kJ mol -DS DFT" 2.06 1.70 co 90.0 0.0 135.8b TS DFT 2.24 1.49 1.63 90.1 26.4 210.3 CH,/Ni, DFT co Go 1.08 -0.0 0.0 DS MRCI' TS MRCT 2.12 1.48 1.51 83.5 27.0 76.9 CH,/Ni , MRCI 0 XJ 1.09 -0.0 0.0 ~~ ~ ~ a This work. Chemisorption energy H (three-fold) on Ni, 241.4 kJ mol-; chemisorption energy CH, (one-fold) on Ni,, 97.3 kJ mol-'.On Ni(lOO), ref. 11, and CH distances, the CH, tilt angle 0 and the NiNiC angle a. Although we have considered geometries outside the grid boundaries, we found large repulsive energies, especially for smaller values of a. The obtained (fitted) energy was tested by making another set of calculations with the optimal geometrical parameters. The deviation between fitted and calculated energies was negligible. All tabulated energies are binding energies with respect to CH, and Ni, at infinite distance. The enormous increase in the TS barrier height is going from a single nickel atom (barrier height: 40.7 kJ mol-') to a Ni, cluster (barrier height: 210.3 kJ mol-1)8 has two main origins.First, the nearest neighbours of the central nickel atom in the first layer are now included, resulting in a decrease in reactivity of the central nickel atom. Secondly, the activated CH bond is now 0.3 A more stretched that in the one-atom case. To investigate the energy effect of different orientations of the TS on our cluster, we have calculated three other geometries: one in which the Ni, cluster is rotated 30" (TS2) around the surface normal, one with the CH, fragment rotated 180" (TS3), and one with both the CH, fragment rotated 180" and the cluster rotated 30" (TS4).The geometry of TS1 is displayed in Fig. 2. This is the only TS that has been optimized as mentioned in the previous section. Also, the effect of cluster size and shape on the energy of TS is ,,.,..0...'.. . ... ...I.. ....1.........1,1....1.1.....(.1.......1. Fig. 2 Transition-state geometry of activated CH, at the Ni, cluster. The dotted lines show the NiC, NiH and CH bonds, which are mainly affected. CH, Decomposition on Ni Table 2 Barrier-height energies (E) for CH, dissociative adsorption for different adsorbates and different substrates, together with the number of nearest neighbours (N,,), next nearest neighbours (N,,,), and next- next nearest neighbours (N,,,,) for the nickel atom at which adsorp- tion takes place ~ ~~ cluster geometry' Eb/kJ mol-' N,: N,,,' N,,,,' Ni TS 1 210.3 Ni, TS2 213.5 Ni, Ni, Nil 9 TS3 TS4 TS1 213.5 212.2 206.6 Ni7.3 TS4 200.8 Ni7, 3.3 Nil 3 TS4 TS 1 172.3 99.7 TS1: see Fig.2; TS2: Ni, cluster rotated 30"; TS3: CH, fragment rotated 180"; TS4: Ni, fragment rotated 30" and CH3 fragment rotated 180". * Experimental CH, activation barrier Ni( 11 1) : 50.2 kJ mol -N,, for infinite Ni( 11 1): 9; N,,, for infinite Ni( 11 1): 3; N,,,, for infinite Ni( 111):9. investigated by computing the height of the barrier for some larger clusters. The results are summarized in Table 2. We see that rotation of the CH, group and/or the cluster as discussed above does not affect the energy significantly. Extending the cluster by includ- ing more nickel atoms in the top layer lowers the energy only slightly (3.7 kJ mol-l).Extending the surface by including a second layer of only three atoms has a more pro- nounced effect: (9.5 kJ mol- '); including another (third) layer of three atoms lowers the energy significantly (38.0 kJ mol-I); however, the activation barrier is still far too high. Beebe et aL9 for example, give a value of 52.7 kJ mol-' for this barrier, while Lee and co-workers" observe carbon deposition, indicative of CH, formation, on a Ni( 11 1) surface at energies above 50.2 kJ mol-l. A far lower activation barrier can be obtained if we choose as a substrate an Nil, cluster in which all surface nickel atoms form bonds with five other nickel atoms.This cluster does not have any low-coordinated boundary atoms, but it does not model an Ni( 11 1) surface. In this case the activation barrier drops to 99.7 kJ mol-'. We have added in Table 2 the number of nearest neighbours (N,,), next-nearest neighbours (N,,,) and next-next-nearest neighbours (Nnnnn).We observe that there is a correlation between these numbers and the cluster barrier height: the lowest barrier is calculated if N,, equals the value of an infinite Ni( 111) surface (N,, = 9) and the discrepancy between N,,, and N,,,, and their value at an infinite Ni(ll1) surface (N,,, = 3, N,,,, = 9) is least. The next lowest barrier is calculated when N,, is still correct, but N,,, and N,,,, deviate more strongly, etc. The only exception is the Nil3 cluster, which results in the lowest barrier.In this case N,, has its worst value (Table 2), and the cluster does not represent a (1 11) surface. All 12 surface atoms are equally unsaturated with bonds, i.e. five bonds instead of the bulk value of 12 or the surface value of nine, and will for this reason form a relatively strong bond with activat- ed CH,. For all substrates, the activation barrier will be lowered if the TS geometries would be optimized separately. We may compare our TS results to those of Swang et d.," although they have modelled a (100) surface. They calculate a barrier energy of 76.9 kJ mol-I, a less extended CH bond, a shorter NiC bond distance and a smaller NiNiC angle. This suggests a more compact, tight transition state with a significantly larger interaction energy between substrate and adsorbate as is indeed reflected in a much lower activation barrier.The experimental data for Ni(100) surfaces is not unani- H. Burghgraef, A. P. J. Jansen and R. A. van Santen 34 1 mous: Chorkendorff et a1.I2 measured an activation barrier of 52.7 kJ mol-', whereas Beebe gave a value of 26.8 kJ mol-', in which case Swang's result is a factor three too high. Many groups have computed chemisorption energies and frequencies of H and CH, and their site preference. For H we calculate a preference for the three-fold site on Ni, with a chemisorption energy of 241.4 kJ mol-', an H surface distance of 0.90 A,corre-sponding to an NiH bond length of 1.70 A,and an NiH stretch frequency of 1014.3 cm-'.CH, is chemisorbed at a one-fold site with a chemisorption energy of 97.3 kJ mol-', an NiC bond length of 2.06 A and an NiC stretch frequency of 386.8 cm-'. For H, the experimental site preference is three-fold with an H surface distance of 1.17 A,', an adsorption energy of 260.5 kJ mol-' and an NiH stretch frequency of 1137 cm-l . 13 Therefore, our calculated H surface distance is too short by 0.27 A,our chemi- sorption energy too low by only 19.1 kJ mol-1 and our frequency too low by approx- imately 123 cm-'. For CH, only an experimental CH, surface stretch frequency is available, 370 ern-','' which is in quite good agreement with our value of 386.8 cm-'. 4. Rate Constant Calculations: Application of Transition-stateTheory We have computed rate constants using transition-state theory.The transition-state formula for the rate coeficients for eqn. (1) and (2) is given by:, In eqn. (3) and (4) k, denotes Boltzmann's constant, T temperature and h Planck's constant. Qt, Q, and Qr are the translational, vibrational and rotational partition func- tions of CH,/Ni,, TS or DS. The translational partition function describes the relative translation of CH, to the Ni, cluster or, on the substrate, the two absolute and relative translations of CH, and H, implying a mobile DS. t denotes transition-state partition functions. Ecritis the minimum energy at which reaction can occur classically (critical energy) and includes the zero-point vibrational energy differences.In the Appendix we give the connection between kItTadsand kdisads and between kzzdes and k,,, des . To evalu-ate k;ftTadsand kzzzdes, we need to calculate the translational, vibrational and rotational partition functions of CH,/Ni,, the DS and the TS. However, if partition functions are the same in the DS and the TS, or in CH,/Ni, and the TS, they effectively cancel and we need only to evaluate partition functions which differ. Also, if a vibrational frequency is high (hv kB T),the partition function gives a factor of 1.0 and therefore does not con- tribute to the rate constant. We characterized the degrees of freedom at the different geometries bearing in mind that our cluster represents an (infinite) nickel surface of infinite mass and therefore overall translations and rotations of the cluster are irrelevant.Also internal lattice vibra- tions (phonons) are neglected. With these premises we have the following degrees of freedom for the TS. Three modes determine the absolute position of hydrogen: one NiH stretch, one CH stretch and one translation of hydrogen perpendicular to the mirror plane. Three modes determine the absolute position of the methyl group: one NiC stretch, one NiNiC angle (a) and one translation of CH, perpendicular to the mirror plane. Three modes determine the orientation of the methyl group: the CH, tilt angle (0) in the mirror plane, one methyl tilt out of the mirror plane, and one internal CH, rotation around the internal C,, axis. Finally, there are six internal CH, modes: three CH stretches and three internal bending modes adding up to a total of 15 degrees of CH, Decomposition on Ni freedom.For CH,/Ni, we have the following degrees of freedom: nine internal (vibrational) modes of CH, of which four are CH stretches, three are CH, internal bending modes, and two are wagging modes, one in the mirror plane and one out of the mirror plane. Together with three rotations of CH, and three relative translations of CH, with respect to the nickel surface, they make up 15 independent degrees of freedom. For the DS we have the following degrees of freedom : again the six internal CH, modes and the three modes determining the CH, orientation, one mode determining the absol- ute position of CH, perpendicular to the surface (the NiC stretch), and two (vibrational) modes determining the absolute position of CH, parallel to the surface (two translations in the mobile model).Analogously for hydrogen we have one mode determining the absolute position of hydrogen perpendicular to the surface (the NiH stretch), and two modes determining the absolute position of hydrogen parallel to the surface (again two translations in the mobile model). This choice of modes results in partition function cancellation for several modes: the six internal CH, modes and the CH, wagging mode out of the mirror plane cancel at all geometries. The CH, rotation around the C3"axes at TS and DS cancels against one CH, rotation. Thus, the following modes need explicit computation: at CH,/Ni, , three relative translations of CH,, two overall rotations of CH,, 0 and a CH stretch mode; at the TS, a, 0, one CH stretch, the NiC stretch, the NiH stretch, the CH, translation perpendicular to the mirror plane and the H trans- lation perpendicular to the mirror plane; at the DS, 8, the NiC stretch mode, the NiH stretch mode and the four translations parallel to the surface.We have not explicitly calculated the NiH stretch mode, because in the G matrix only the hydrogen mass is involved, which will result in a high frequency. As a consequence the corresponding vibrational partition function will be 1.0. For the TS the imaginary frequency is strongly dominated by a mixture of the CH stretch and a; all frequencies at this geometry are low, indicating a loose transition state.Now, according to eqn. (3) and (4) rate constants can be calculated. The results are shown in Table 3 for CH, and CD, at different tempertures. Sticking coefficients (s) are calculated in the following manner. First, we calculated the hard-sphere pre-exponential (AHS)to set an upper limit for the pre-exponential (A): This is related to the number of collisions of CH, with the substrate per unit time, dNpiJdt, via where NCH4is the number of CH, gas molecules, I/ is the volume, and S is the surface area (see also the Appendix). We calculated A and AHSand checked that AH' was indeed larger than A (Table 3). The sticking coefficient is now simply the ratio of kdis ads and AHS and is therefore effectively the reaction probability per hard-sphere collision.The last column in Table 3 reflects the effect on the sticking coefficients if we substitute for our calculated Ecritthe experimental measured values (CH,: 50.2 kJ mol-'; CD,: 58.6 kJ mol-l). Beebe gave values for the sticking coefficient at 500 K of lop8 to lo-, for (lll), (100) and (110) surfaces. We find at this temperature a value of 6.0 x lo-, and this illustrates that correct sticking coefficients can be calculated with our model and that the main remaining problem is a correct computation of the activation barrier. The isotope substitution ratio is almost constant for associative desorption (a factor of two), but varies significantly for dissociative adsorption. Our calculated value of four at T = 750 K is half the value measured in this temperature region," but tunnelling effects will decrease this discrepancy.An estimate of these tunnelling effects is not as straight- Table 3 Rate constant for dissociative adsorption (kdisads) and isotopic substitution ratio, rate constant for associative desorption (k,,, des) and isotopic substitution ratio, ratio of A to AHS,classical mechanical sticking coefficients (s)and classical mechanical sticking coefficients using the experimental activation barrier (CH,: 50.2 kJ mol- '; CD,: 58.6 kJ mol-') (sexp)for CH4 and CD4 at different temperatures A/AHS S CH4 250 3.22 x 10-40 3.27 x 10-23 0.10 225 x 10-42 3.17 x CD4 250 1.15 10-41 27.98 1.18 10-23 2.77 0.05 8.98 10-44 3.11 x 10-14 CH4 500 1.03 x 10-19 1.91 x 10-15 0.11 5.07 x 6.02 x lo-' CD4 500 1.67 x 6.18 9.38 x 2.04 0.07 9.17 10-23 5.40 x Q3CH4 750 1.04 x 9.28 x 10-13 0.15 4.18 10-15 4.69 x 10-5 L CD4 750 2.84 x 3.66 4.99 x 10-13 1.86 0.1 1 1.28 10-15 8.97 x ?cr CH4 1000 4.10 x 10-9 2.27 x lo-" 0.21 1.43 x lo-" 4.92 x 10-4 PCD4 1000 1.43 x 10--9 2.87 1.27 x lo-" 1.79 0.16 5.59 x 1.36 x 10-4 C Q3 344 CH, Decomposition on Nj 25 25 0 0 h h 7 ? -25 -25 v) N E E ---? 1 -50 -50 U 111m % ?5 C ---75 -75 -1 00 -1 00 0 1 2 3 4 5 103 KIT Fig.3 Arrhenius plots of the dissociative adsorption (kdi, ,&) and associative desorption (k,,, des) rate coefficients for CH, and CD,. Solid lines for CH,, dashed lines for CD,. A,&isads CH,; V, kdisadsCD,; a,kassdesCH?; 0,kassdesCD,.At low temperature a moderate isotopic substitution ratio is seen for the dissociative adsorption reaction; at higher temperatures curves for CH, and CD, merge smoothly. forward as in our one nickel atom study,' because now the reaction coordinate is not described by only the CH stretch mode, instead it is a mixture of all modes. As a conse- quence, we did not make an estimate of tunnelling effects. The isotope effect is more clearly seen from Fig. 3, which shows an Arrhenius plot for the adsorption/desorption of CH,/CD, . The plot data are summarized in Table 4. The electronic energy, E, denotes the energy difference between the TS, the DS and CH,/Ni, at the electronic potential-energy surface (not including zero-point energies).In the critical energy, Ecrit,the differences in zero-point energy for the TS, the DS and CH,/Ni, have been taken into account. The slope of the Arrhenius plots is reflected in the activation energy, E,, and the intercept in the pre-exponential, Aplot. We see that inclusion of zero-point energies lowers the dissociation barrier of CH, by almost 16 kJ mol -reflecting the relatively low frequencies at TS and the high stretching-mode fre- quency for CH,. The effect of deuteriation is to lower vibrational frequencies for both the TS and CH, . Therefore, the lowering effect on the barrier is smaller (10.3kJ mol-'). For the desorption barrier, we see the same trends but less pronounced. The activation energy, E,, displays the effect on the barrier, when excited rotational and vibrational levels are populated : a temperature-averaged barrier.The population of high vibrational frequency excited levels, e.g. CH stretch mode in the CH, gas/Ni or NiH stretch mode in the DS, will remain small at elevated temperatures, because of its high frequency and Table 4 Electronic energy (E), critical energy (E,,,,), activation energy (E,) and Arrhenius pre- exponential according to Fig. 4 (Aplot) for dissociative adsorption (m s-') and associative desorp- tion (m2 s-l) for CH, and CD, reaction E/kJ mol-' E,,,JkJ mol-' E,/kJ mol-' A Plot CH, dissociative adsorption 210.3 194.5 198.0 1.98 x lo2 CD, dissociative adsorption 210.3 200.0 204.3 2.04 x 102 CH, associative desorption 74.5 72.6 75.3 1.69 x 10-7 CD, associative desorption 74.5 73.5 76.5 1.11 x 10-7 H.Burghgraef, A. P. J. Jansen and R. A. van Santen 345 thereby associated high energy. In contrast, the population of relatively low vibrational frequency excited levels, as those in the TS, will increase strongly, as is reflected in the vibrational partition functions and the barrier for insertion will thus increase. The pre-exponential factor is connected with the change in entropy on going from reactant to the TS, the entropy of activation (ASS), and is therefore a measure for the gain or loss in entropy. If AS$ equals zero, the pre-exponential is approximately lo2 m s-' for an adsorption reaction and m2 s-' for a bimolecular desorption rea~tion.~ The pre-exponentials for adsorption denote therefore that there is essentially no entropy of activation.This can be understood by realizing that although three relative trans- lation are lost in going from CH4/Ni, to the TS, the TS vibrations are very loose modes and the activated CH, can rotate freely on the surface. For the desorption reaction some entropy is lost: four translation modes are lost in this case, but again because of the very loose vibrations in the TS, this loss is very limited. If we compare our results with experiment, we note the following. Ceyer and co- workers found a value of 50.2 kJ mol-I for the activation barrier of CH, and 58.6 kJ mol-' for CD, . Clearly, our calculated result on clusters modelling the (1 11) surface is still too high.We find at T = 750 K a sticking coefficient ratio of 3.3 using our calcu- lated barrier height and a value of 5.2 using the experimental value. This is not too far away from a Ni(ll1) surface sticking coeficient ratio of 8.0 in the temperature region between 640 and 830 K as found from the molecular beam experiments of Lee. Further- more, tunnelling effects will increase our calculated classical ratio. 5. Conclusions We have carried out DFT calculations to determine the TS, the DS, height and origin of the barrier of reactions (1) and (2). In addition, we calculated rate constants and sticking coefficients. On a Ni, cluster, a barrier for dissociative chemisorption of 210.3 kJ mol-' was found compared with a barrier of only 40.3 kJ mol-' for a single nickel atom.The barrier for associative desorption is 74.5 kJ mol-l. The higher dissociation barrier can be understood by realizing that the central nickel atom at which the activated CH, is chemisorbed is surrounded by neighbouring nickel atoms descreasing its intrinsic reacti- vity. In addition the CH bond is more stretched than in the one-atom case. Our kinetic analysis showed that we can reproduce experimental sticking coefficients correctly, pro- vided that we use the experimental barrier height. Therefore, the main remaining problem is the correct computation of the height of the dissociation barrier. We have shown that extending the cluster by adding more atoms in one layer or including a second or third layer does indeed lower the energy, but only for the Nil, cluster does the dissociation barrier decrease dramatically.This cluster does however not represent a crystal plane. The association barrier height cannot be compared with experimental results. However, Yang and Whitten found a value of 81.6 kJ mol-1,15 which is nearly the same as our result. This can be understood by realizing that although our Ni, cluster does not represent an infinite Ni( 11 1) surface properly, the error made in adopt- ing this cluster is the same for the TS and the DS, resulting in a correct association barrier height. Calculated pre-exponentials were connected with change in entropy for dissociation and association. They indicate a loose transition state. All calculations were performed with the ADF program on the Cray Y-MP4/464 at SARA, Amsterdam.This work has been supported by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization of Pure and Scientific Research (NWO). The computer time on the Cray Y-MP4/464 was sub- sidized by the Foundation for the use of supercomputers, National Computing Facilities (NCF). CH, Decomposition on Ni Appendix Derivation of Rate Constant Equations As a starting point, we can write down for the dissociative adsorption : Here, N:Z4 is the number of adsorbed CH, molecules. NCH4 is the number of CH, molecules in the gas phase, and k:ETads is the transition-state-theory reaction rate con- stant give by eqn.(3). As a consequence of our loose TS, the activated CH, is not bound to one specific site, and the partition functions for the two vibrational modes parallel to the surface, i.e. the CH stretch and a combination of the CH, and H modes parallel to the surface and perpendicular to the mirror plane yield an additional factor Nsites,the number of sites. Instead of changing the appropriate vibrational partition functions, we can also multiply the right-hand site of eqn. (Al) by Nsites,which results in ads -TST NCH4 -kdis ads Nsites NCH4 or equivalently, where S is the total surface area and ositethe site area, which is 5.38 x m-2 for an Ni(ll1) surface. Eqn. (A3) can be rearranged in terms of surface concentrations (left- hand side) and gas concentrations (right-hand side) to or As a starting point for the asociative desorption per site, we can write down: d ads --kTST -NCH4 -assdes NCHj NHdt Here, Ntg4 is again the number of adsorbed CH, molecules.NCHJ and N, are the number of CH, and H particles adsorbed on the surface, and kTzTdes is the transition- state theory formula for the rate constant given by eqn. (4). Taking into account the mobility of our transition state, eqn. (A6) is changed to: TST-N$Z4= -kass des -NCH3 NHdt csite or H. Burghgraef, A. P. J. Jansen and R. A. van Santen 347 which results eventually in Note that neither V in eqn. (A4) nor S in eqn. (AS) has to be known as they cancel with corresponding factors in translational partition functions.References 1 ADF program suite developed by Baerends and co-workers: E. J. Baerends, D. E. Ellis and P. Ros, Chem. Phys., 1988, 2, 41; P. M. Boerrigter, G. te Velde and E. J. Baerends, Znt. J. Quantum Chem., 1988, 33, 87; G. te Velde and E. J. Baerends, J. Comput. Phys., 1992, 99, 84; D. Post and E. J. Baerends, J. Chem. Phys., 1983,98, 5663; E. J. Baerends and A. Rozendaal, in Quantum Chemistry: the Challenge of Transition Metals and Coordination Chemistry, ed. A. Veillard, Reidel, Dordrecht, 1986, p. 159; P. J. van den Hoek, A. W. Kleyn and E. J. Baerends, Commun. Atom. Mol. Phys., 1989,23,93. 2 S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys., 1980,58, 1200. 3 A. D. Becke, Phys. Rev. A, 1988,38,3098. 4 H. Stoll, C. M. E. Pavlidou and H.Preuss, Theor. Chim. Acta, 1978, 49, 143; H. Stoll, E. Golka and H. Preuss, Theor. Chim. Acta, 1980,55,29. 5 J. G. Snijders and E. J. Baerends, Mol. Phys., 1978, 36, 1789; J. G. Snijders, E. J. Baerends and P. Ros, Mol. Phys., 1979,38, 1909. 6 E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations. Theory of Infrared and Raman Vibrational Spectra, McGraw-Hill, New York, 1955. 7 R. G. Gilbert and S. C. Smith, Theory of Unimolecular and Recombination Reactions, Blackwell, Oxford, 1990; M. Boudart and G. Djtga-Mariadassou, Kinetics of Heterogeneous Catalytic Reactions, Princeton University Press, Princeton, NJ, 1984. 8 H. Burghgraef, A. P. J. Jansen and R. A. van Santen, J. Chem. Phys., 1993, 98, 8810; Chem. Phys., accepted. 9 T. P. Beebe, D. W. Goodman, B. D. Kay and J. T. Yates, J. Chem. Phys., 1987,87,2305. 10 S. T. Ceyer, J. D. Beckerle, M. B. Lee, S. L. Tang, Q. Y. Yang and M. A. Hines, J. Vac. Sci. Technol. A, 1987,5,501; M. B. Lee, Q. Y. Yang and S. T. Ceyer, J. Chem. Phys., 1987,87,2724. 11 0.Swang, K. Faegri Jr., 0.Gropen, U. Wahlgren and P. E. M. Siegbahn, Chem. Phys., 1991,156,379. 12 I. Chorkendorff, 1. Alstrup and S. Ullmann, Surf. Sci., 1990, 227, 291. 13 R. R. Cavanagh, R. D. Kelley and J. J. Rush, J. Chem. Phys., 1982,77, 1540. 14 K. Christmann, 0.Schober, G. Ertl and M. Neumann, J. Chem. Phys., 1974,60,4528. 15 H. Yang and J. L. Whitten, J. Chem. Phys., 1992,96, 5529. Paper 31028436; Received 17th May, 1993