Ab-initio calculations of the 6D potential energy surfaces for the dissociative adsorption of H2 on the (100) surfaces of Rh, Pd and Ag

surface science ELSEVIER

Surface Science 397 (1998) 116 136

Ab-initio calculations of the 6D potential energy surfaces for the dissociative adsorption of H2 on the (100) surfaces of Rh, Pd and Ag A. Eichler, G. Kresse, J. Hafner * lnsmutJut Theoretzs~he Phystk, Techms~he Umversltat Wwn, Wtednet Hauptstrasse 8-10, A-1040 Wwn, Austria

Recmved 12 June 1997, accepted for pubhcatlon 1 September I997

Abstract Detailed investigations of the six-dimensional potential energy surface (PES) for the dlssoclat]ve adsorptaon of a hydrogen molecule on the (100) surface of Rh, Pd and Ag are presented The calculations are based on local density functional theory with generahzed gradient corrections to the exchange-correlation functional, and have been performed usmg the Vienna ab-mnltm simulation package VASP VASPworks m a plane-wave basis and uses ultrasoft pseudopotentlals We show that adsorption on Rh (100) and Pd(100) is m general non-activated, but barriers ex]st along certain reaction channels The "adiabatic" minimum-energy channel has been determined by a five-d]menslonal mlmmlzatlon of the total energy at a fixed height of the molecule The variation of the covalent hydrogen-metal bond along this channel is studied using crystal orbltal overlap populanons and the electron locallzauon funcuon On Ag(100), H2 adsorptmn Is strongly activated, with a pronounced variation of the barrier over the surface cell © 1998 Elsevier Science B V Adsorption kinetics, Chem]sorpt]on, Density functlonaI calculaUons, Hydrogen, Low index single crystal surfaces, Palladmm, Physical adsorption, Rhodmm, Salver, Single crystal surfaces

Keywords

1. Introduction T h e m o t i v a t i o n to investigate h y d r o g e n - s u r f a c e i n t e r a c t i o n p h e n o m e n a is m a n i f o l d : first, the vast field o f h e t e r o g e n o u s catalysis should be m e n tloned, where h y d r o g e n is i n v o l v e d m m a n y reactions either as a r e a c t a n t or a p r o d u c t molecule. T h e synthesis o f a m m o n i a , h y d r o c a r b o n s or alcohols by the F l s c h e r - T r o p s c h r e a c t i o n is just o n e i m p o r t a n t example. H e r e the dissociative a d s o r p tion o f m o l e c u l a r h y d r o g e n on the surface o f the

* Corresponding author Fax (+43) 1 5867760, e-marl jhafner@tph tuw]en ac at 0039-6028/98/$19 oo © 1998 Elsev]er Science B V All rights reserved PH S0039-6028 (97) 00724-3

catalyst is an i n t e r m e d i a t e step o f f u n d a m e n t a l I m p o r t a n c e [1]. Second, there is an increasing interest in using alloys or intermetallic c o m p o u n d s as a m e d i u m for storing gaseous h y d r o g en , triggered by the d e v e l o p m e n t o f fuel-cell technology. H e r e again, the a d s o r p U o n / d e s o r p t i o n o f h y d r o g e n is always the first and decisive step in the reaction. A third i m p o r t a n t aspect co m es f r o m the field o f m at er i al s science an d metallurgy, where h y d r o g e n was identified some time ago as being one o f the m a j o r reasons for e m b r i t t l e m e n t and fracture [2] In principle, h y d r o g e n should be, b o t h in its a t o m i c an d m o l e c u l a r forms, the simplest chemically r e a c t a n t adsorbate. H o w e v e r , the wealth o f e x p e r i m e n t a l d a t a a c c u m u l a t e d t h r o u g h decades

A Ezchle~ et al / Surface Scwnce 397 (1998) 116-136

of intensive investigations documents that this is still a very rich field [3]. Although the static properties are now fairly well known (at least for the close-packed low-index surfaces), much less is known about dynamics (i.e the motion of the molecule or atoms on the potential-energy hypersurface [4,5]). However, on one hand, molecularbeam adsorption experiments [6,7, 9,10] and stateresolved time-of-flight experiments [8,11,12] contribute to a deeper knowledge of the dynamical aspects of hydrogen adsorption, and on the other hand the increased efficiency of ab-initio local density functional total-energy calculations offers the possibility of constructing the potential energy surface (PES) from first principles. However, even for the simplest case of a diatomic molecule above a rigid substrate (where the PES is six-dimensional) to map the hypersurface with an accuracy sufficient to use It for classical or quantum simulations of the adsorption dynamics remains a formidable task. Only very recently was a PES for the H2/Pd(100) system derived from density functional calculations [13,14] and used by Gross et al. [15] to perform the first six-dimensional quantum calculation for the dynamical behaviour of hydrogen atoms during the adsorption/desorption process from a metal surface. The combined results of these studies are of multiple interest. (i) The PES shows both non-activated and activated pathways for adsorption. This explains the experimental observation of the initial decrease in the sticking coefficient with increasing beam energies [8,9]. At low kinetic energy, molecules with orientations unfavourable for adsorption may reorient and follow the energetically more favourable non-activated reaction channel, whereas at higher kinetic energies this steering effect becomes ineffective and an increasing number of molecules are driven along activated pathways. (ii) The large favonzation of parallel-adsorbing molecules compared to upright-impinging molecules leads to a prediction of different sticking probabilities for molecules rotating around axes parallel ("hehcopter" rotations) and perpendicular to the surface normal ("cartwheel" rotations), since the latter have a high probability of hitting

117

the surface in an upright position. For the system H2 over Pd (100), the increased sticking probability for helicoptering molecules has recently been confirmed experimentally [16]. (iil) The precise form of the PES can be used to explain the fact that the occupancy of the vibrational eigenstates of the molecule is significantly higher, whereas that of the rotational eIgenstates is lower than for a gas of hydrogen molecules at equilibrium at the temperature of the surface ("vibrational heating" [11,12] and "rotational cooling" [17]) It is clear that the modelling of the high-dimensional PES has to be based on the total energies calculated for a finite number of configurations of the molecule in front of the substrate (250 configurations were used by Wllke and Schemer [13,14]). Most of these configurations represent 2D cuts through the 6D-PES The selection of these 2D-PES's is guided by the ideas underlying our picture of the process of dissociative adsorption. One is that the favoured reaction channel is dominated by the final positions of the adsorbed atoms. For the dissociation of H2 on the (100) surface of an fcc metal, where the stable adsorption sites are the four-fold hollows, one would hence expect that the preferred geometry for the dissociation of H2 would be with the molecular axis stretching between two hollow sites, transverse to a bridge site ( " h - b - h " geometry). The second generally accepted notion is that the optimal reaction pathway must involve high symmetry, in this case C2~. This idea would allow the t - b - t as well as the h - b - h geometry. Recently, these arguments have been challenged by Feibelman [18], who emphasized the dominant role of the energetics of bond-breaking and bond-formation: the energy spent in stretching the molecular H - H bond must be gained by forming strong H-metal bonds. In this respect, a direct dissociation into the hollows is rather unfavourable. The length of a H 2 molecule is only 0.74 A, whereas the nearest-neighbour separation on Rh(100), for example, is 2.79 A. This means that even before strong H-metal bonds in the hollows can be formed, almost all the binding energy of the H2 must be spent to tear the H atoms apart. Felbelman proposed that at intermediate distances from the surface, optimal bonding

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A Ezchler et al / Surface Sctence 397 (1998) 116-136

occurs in a b - b geometry where the molecule is located between two neighbouring bridge sites (which are only 1.97 A, apart). Motivated by these arguments, we have recently performed an ab-inltio investigation of the optimal adiabatic reaction path for the dissociation of H 2 on Rh(100) [19]. The results confirm, albeit with some important modifications, the predominant influence of the quantum mechanics of bondbreaking and bond formation. In the first stage of the adsorption process, the molecule is first drawn to the on-top site and oriented along the b - t - b position Here the dominant effect is the interaction with the d3z2_r2 orbltals extending farthest from the surface. In the second stage, dissociation is driven (within the same geometry) by the H-dtzg interaction allowing the largest overlap Only in the final stage do the dissociated H atoms drift into the hollows, because the H-dx2_y-, interaction leads to a slightly higher adsorption energy. This result demonstrates that the symmetry of the atomic orbmtals of the interacting particles is important for determining the shape of the PES. In addition, this shows the advantage of using the Hellmann-Feynman forces acting between substrate and adsorbate as a guideline for exploring the 6D-PES. In the present paper, we extend these preliminary studies in the following directions (1) We analyze in detail the dependence of the PES on the helicopter and cartwheel rotations, and the variation of the molecular eigenfrequencies along the various reaction channels. This leads to a very detailed picture of the 6D-PES. (ii) We extend our studies to the (100) surfaces of Pd and Ag. The comparison of Rh and Pd adsorption is interesting because of significant differences in the presence/absence and heights of barriers in various reaction channels and the relation of these differences to the different degrees of filling of the d-band. The Ag(100) surface is studied as an example of a surface where only activated adsorption processes are possible and where the dissociative adsorption is an endothermic process. (ili) We study in detail the variation of the adsorbate-substrate bond by analyzing the crystal orbital overlap populations (coop/COOPs) and the electron localization function ( E L F ) . (iv) We analyze in detail the variation

of the local surface reactivity in the series of R h - P d Ag(100) surfaces.

2. Theory In our calculations we have used the Vienna ab-inltlO simulation program VASe [20-23]. VASP performs an iterative solution of the K o h n - S h a m equations of local density functional ( L D F ) theory in a plane-wave basis, using preconditioned conJugate-gradient and residuum-minimization techniques. Exchange and correlation are described within the Ceperley Alder local density functional [24,25] and the generalized gradient corrections (GGCs) of Perdew et al [26]. We emphasize that the GGCs have been applied self-consistently, beginning with the construction of the pseudopotentials. Optimization of the atomic geometry can be performed dynamically using molecular dynamics in a canonical ensemble, or statically using conjugate-gradient minimlzatlons The electron-ion interaction is described by optimized ultrasoft pseudopotentials [27,28], using a cut-off energy of 200 eV to limit the plane-wave basis set For the construction and tests of the ultrasoft pseudopotentials, see Refs. [29-31]. All calculations are scalar-relativistic. Brillouin-zone integrations are performed for a grid of 4 x 4 x 1 special points, using second-order Methfessel Paxton smearing [32] with a width of o-= 0.2 eV. For a much finer mesh, the adsorption energies change by at most 0.06 eV for the bridge site and 0.03 eV for the hollow site. The metal surfaces were represented by periodically repeated slabs consisting of three or five substrate layers separated by six vacuum layers, and using c(2 x 2) and p(2 x 2) surface cells. The hydrogen molecules were adsorbed on both sides of the slab. Careful tests for Rh(100) have shown that the adsorption geometries and energies concerge well for thinner slabs, and vary only slightly with coverage (by almost 0 17 eV from 0 = 0 . 5 to 0 = 1.0). The energy zero for the adsorption energies is the stun of the energies of the clean surface and a free molecule (calculated in the same supercell ) Substrate relaxation has been neglected. This is

A Etchler et al / SurJace Scwnce 397 (1998) 116-136

justified because (i) the tamescales of the substrate relaxation and the H2 adsorption process differ by too much due to the large mismatch in the masses of hydrogen and of the substrate atoms, and (li) the substrate relaxation (although it can vary substantially due to hydrogen adsorption) alters the characteristic parameters of the adsorption (energy, adsorption height) only marginally [31]. All calculations have been performed at the equilibrium lattice constants (Rh: 3.85A, Pd' 3.96 A, Ag. 4.17 A) and a bulk-terminated geometry of the surface. For results concerning bulk properties, properties of the clean surfaces and for the adsorption of atomic hydrogen, we refer the reader to previous pubhcations [29, 31,33]

3.

H 2 on

Rh(100)

3 1 Modelhng o f potential energy surfaces (PES)

The potential energy surface for the &ssoclative adsorption of a diatomlc molecule over a surface (when neglecting relaxation effects) is six-dimensional. One possible set of coordinates (see Fig. 1) consists of the bond length (d) and the height (Z) of the molecule above the surface, two lateral coordinates which describe the posmon of the centre of mass (x, y) and two rotational degrees of freedom, the so-called "helicopter" rotation (4) with the axis of rotation perpendicular to the surface, and the "cartwheel" rotation (O), with the rotational axis parallel to the surface To map this energy surface, the common strategy is to compute two-dimensional cuts through the energy surface (so-called "elbow" plots). The two angles and the two lateral coordinates are fixed, and only the bond length of the molecule and its height above the surface are varied. In the next step, from various different cuts it is possible to paramemze the six-dimensional PES by an appropriate fit [15] 3.2. Results

To obtain global information about the adsorption behaviour, we performed this procedure for (1) varaous adsorption pathways with the molecule

119

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Fig 1 Sketchofapossiblecoordinatesystemforthedescrlptlon of the posxtlon of a &atomic molecule above a surface Shown is a pnmxtive surface cell for the fcc (100) surface (a/~_x a / ~ ) and a &atomic molecule tilted in a position approximately over the four-fold hollow x, y and z are the carteslc coordinates of the centre of mass of the molecule, d is the bond length, and O and 4) describe the two rotational degrees of freedom (cartwheel and helicopter rotations) Also marked are the hollow (h), bridge (b) and top (t) posmons

parallel to the surface (O=90°), (ii) cuts with an upright geometry of the molecule ( O = 0 ° ) , and (111) for one low-symmetry position we studied also geometries with 0 = 4 5 and 135 °. In ad&tion, we attempted to determine the energetically most favourable reaction channel: starting from a large distance from the surface, at fixed Z the length of the molecule and its lateral position and orientation are relaxed. The distance from the surface is then gradually reduced. 3 2 1 2D cuts through the P E S f o r 0 = 90 ° To distinguish between the different geometries with the molecule parallel to the surface, we will characterize them by the positions where the hydrogen atoms will be adsorbed, and by that over which the centre of mass of the molecule is situated. For example, " ~ b - h ' " refers to a geometry with the centre of mass over a bridge (b) position and the atoms oriented towards the four-fold hollows (h) in which they will finally be adsorbed. " h - t - h " leads to the same final positions of the H atoms, but the molecule enters the reaction path over an on-top (t) position.

A EIchler et al / Surface Scwnce 397 (1998) 116 136

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~l,~l Fig 2 2 D section through the PES for the dissociative adsorption of H 2 l y i n g p a r a l l e l t o the surface over R h ( 1 0 0 ) In various orientations with respect to the surface according to the Inset (see text for nomenclature) Contour lines are drawn at Intervals of 0 1 eV b e t w e e n -- 1 2 a n d 1 5 eV Lines indicating negative potential energies are dashed, while thicker lines correspond to integer values

To make a comparison of the energetics along different pathways more easy, we have chosen in addition to the elbow plot (Fig. 2) a representation where the energy along the most interesting part o f the energy surface (i.e. along the bottom of the reaction channel) is plotted as a function of the height o f the molecule above the surface (Fig. 3). Energies for barriers and minima on the PES are compiled in Table 1, together with the corresponding bond lengths and heights. Another quantity we will trace along the reaction paths are the eagenfrequencies o f the H 2 molecule

Fig 3 Variation of the potential energy for H z over R h ( 1 0 0 ) along the bottom of various straight perpendicular reaction channels as a function of the height of the molecule above the surface In (a), the molecule is a l w a y s p a r a l l e l to the surface ( 0 = 9 0 ° ) T h e t h i c k g r e y line represents the energy along the " a d i a b a t i c " reaction channel described in the text In ( b ) , the thicker grey lines show the energy variation for an approach of a n u p r i g h t H2 m o l e c u l e ( O = 0 °) onto the hollow, bridge and on-top posmons of the surface, whereas the thinner black lines refer to a molecule with its centre of mass halfway between a bridge and a top position as a function of a cartwheel rotation w i t h i n a plane through the bridge and on-top positions ( v a r i o u s angles O, ~b=45 °)

(Fig. 4). In Fig. 4 the frequencies during the adsorption process are plotted as a function of the reaction coordinate, which runs along the valley of the elbow plot. This coordinate is set equal to the height Z for Z = 3 and decreases until the adsorption process is finished. The frequencies have been obtained by a parabolic fit along the reaction path in the elbow plots At some places (e.g. close to the saddle points), the accuracy o f the fit may not be better than 10%, but for qualitative analysis the frequencies are very helpful. Furthermore, they are necessary for a parametrlzatlon o f the six-dimensional PES. In the plots, we distinguish three different frequencies' (1) a frequency in the d-direction (cod), which is the vibration frequency of the H 2 molecule in the entrance channel and a frequency corresponding to a frustrated translation of the atoms in the

A Ezchler et a l / Surface Sczence 397 (1998) 116-136 Table 1 Adsorption energy E(eV), height above the surface Z and bond length d(A) of Hz over Rh(100) at characteristic positions In the 2D cuts through the PES for fixed lateral coordinates of the centre of mass (.~, y) of the H2 molecule and its orientation

(o, ~)~ Pathway

Type

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d (A)

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Barraer Minimum Barrier Mlmmum Barrier Minimum Local m i n i m u m Bamer Minimum Local m i m m u m Barrier Minimum Barrier Mmamum Barraer Local m l m m u m

0 02 - 0 92 0 13 --0 88 0 01 - 0 80 - 0 29 --0 19 --0 93 - 0 55 --0.52 --0 88 0.03 - 0 25 -0.11 -0.15

2 40 0 57 1 90 1 13 2 47 1 10 1 70 1 48 0 53 1.65 1 52 1 15 2 47 1 53 1 56 1 58

0 77 2 71 0 77 2.81 0.77 1 93 0 87 1 31 3 87 0 92 1 17 2 71 0 78 0 94 1 69 2 72

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~For the nomenclature for the pathway and the definitions of the adsorption energy, see the text

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4A] F]g 4 VariaUon of the elgenfi'equencles of an H 2 molecule (dotted hne vlbraUons in the d-dlrectaon ~0d, dashed line vibrations in the Z-direction coz, sohd hne vibrations perpendicular to the reacUon path co,) along the reaction channels w]th the molecular axis kept parallel to the R h ( 1 0 0 ) surface as a functlon of the reaction path coordinate (see text) Negauve frequencies indicate a negative curvature of the PES (2 e a locally unstable position)

121

adsorbed geometry, (li) a frequency in the Zdirection (coz), which describes the attraction of the potential in the entrance channel, and is therefore imaginary (in the adsorbed geometry this is the symmetric stretch frequency of the adsorbed H atoms), and (iii) the frequency perpendicular to the reaction path (co,), which enters a parametrization of the PES. This frequency is equivalent to cod in the entrance and equivalent to coz in the exit channel. The path which has received most attention when the stable adsorption sites are the hollows is the h - b - h path. The molecule enters the reaction channel with the bond length and frequency of the free molecule (d=0.75 A, COz=co. = 132 THz; corresponding experimental values of d = 0 . 7 4 A and co = 131.8 THz) high above the surface. Then there comes a rather long phase (compared to the other pathways discussed below) where energy and bond length remain approximately constant. There is even a very small increase in energy of about 20 meV at a height of about 2.4 A The only effect in this region is a softening of co,, which means that the reaction channel becomes wider. At a height of about 1.2 A (which is approximately the equilibrium height for atomic hydrogen adsorbed in a bridge postlon [31 ]) a significant stretching of the molecule occurs, which can be traced also by the slight kink in the E(Z) curves and in the separation of co, from coz Here, the so-called "exit" channel begins: the part of the reaction channel where the bond length is still similar to that of the free molecule is called the "entrance" channel. Finally, at a height of Z = 0 . 5 7 ,~, the dissociated H atoms are adsorbed in the four-fold hollows of the surface. The symmetric stretch frequency (coz-- 19 THz) agrees very well with the experimental value of co = 19.83 T H z [34] and an L A P W result of co=22.3 T H z [35]. A second possibility to reach the hollows as final positions as the dissociation over a substrate atom along an h - t - h path. In this case, we find a different behaviour during dissociation. The energy decreases significantly for large distances, but runs into some kind of physisorbed local minimum and has to overcome a barrier of about 100 meV before adsorption occurs. Also for this path, bond-break-

122

A Elchler et al / Surjace Sctence 397 (1998) 116 136

lng occurs approximately at the height for atomic on-top adsorption of hydrogen [31]. I f this is a general rule, we should expect a rather strange behaviour along those pathways where the centre of mass of the molecule is over a position with a lower atomic adsorption height than the final adsorption sites. Indeed, for the b - h - b and t - b t reaction channels we find that a slightly elongated molecule comes closer to the surface than the dissociated atoms adsorbed in the high-symmetry sites. For the t - b - t channel, molecular adsorption is even energetically more favourable than the atomic on-top adsorption, although this configuration is also a local minimum in the PES, but with a very low frequency for the frustrated translation (~o~). The b - h - b path is the only path with a pronounced barrier (E=0.13 eV) in the entrance channel. An important feature of this path is the soft translation frequency in the adsorbed geometry, indicating that the bridge positions are only saddle points on the PES. Sliding of the atoms into the hollow positions is possible without over-coming a barrier. A pathway proposed by Feibelman [18] to be the most favourable for distances far from the surfce is b - b . This means that the final adsorption sites are the bridges, but the centre of the molecule is over a diagonal in the surface cell, halfway between a top and a hollow position. Feibelman compared the energies of the t - b - t , h - b - h and b - b geometries at distances of 1.58 and 2.11 above the surface within an L D A - L A P W calculation, and found that at both heights b - b is more favourable than t - b - t , which is again lower in energy than h - b - h . We find exactly the same ranking in our calculation, although the absolute energies are rather different as a consequence of the different description of the exchangecorrelation potential ( L D A / G G A ) . ( F o r h - b - h Felbelman also extrapolated the height of the point where the forces vanish to be about 2.6 A above the surface, which correlates well with our barrier at 2 4 A . ) The reason why the energy minima in the b - b and b - h - b channels differ by about 70 meV IS that the two reaction pathways lead to different adsorption patterns: after b - b adsorption neighbouring

bridges are populated, which makes this configuration less favourable. The last parallel pathway which has been investigated is b t - b . At larger distances from the surface, this is by far the energetically most favourable position of the molecule. As along the h - t h channel, a shallow local minimum is found at Z ~ l . 4 5 A . Altogether, we have to note that adsorption in the bridge sites occurs at larger distances, but at almost the same energy than in the hollows (see Table 1). The discussion of the reaction channels shows that dissociation occurs most easily over the top site, with the molecule oriented towards the bridges. This suggests a more complex scenario for dissociative adsorption, i.e. first b - t b, followed by a transition from b to h. 3.2.2

2 D cuts through the P E S f o r 0 = 0 °

The elbow plots for a H 2 molecule approaching the surface in an upright position (molecular axis perpendicular to the surface) are shown in Fig. 5 for the t, b and h positions. In this orientation, no dissociation occurs. In all configurations the adsorbate-substrate interaction is repulsive Above the top position, a closer approach to the surface costs more energy than above the h and b positions, as a consequence of the strong repulsion between the substrate orbitals protruding from the surface (d3=2_r-,, p~) and the lower H atom in the molecule I f the molecule is pushed closer to the surface, the molecular bond is strongly compressed because the Pauli repulsion acts more strongly on the lower atom, Another interesting feature is that compressing the molecule favours an approach A similar result has been obtained by White et al. [36], for H 2 o v e r a tungsten (100) surface. Above the b and h positions the molecule behaves differently: a closer approach to the surface leads to a stretching of the molecule. This shows that while the lower H atom experiences a weak bonding to the substrate, for the upper H a t o m the Pauli repulsion of the overlapping orbitals dominates. The stretching effect is weak over the bridge site, but much stronger over the hollow site. In Fig. 3b the minimum energies for the optimal bond length are plotted as a function of Z. Comparing these curves, we find a scenario similar to that for planar adsorption. Far from the surface

A Etchler et al / S u t f a c e Sclence 397 (1998) 116-136

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dial Fig. 5.2D section through the PES for an upright H 2 molecule approaching the on-top (a), bridge (b) and hollow (c) posmons of a Rh(100) surface Contour lines are drawn every 0 1 eV between 0 and 3 5 eV Thlcker lmes correspond to integer values The stralght solid hnes mark the positions where the lower H atom is exactly at the height for atomic H adsorpuon at the corresponding surface position The dashed (dotted) lines indicate the optimal bond length (height of the molecule above the surface) for a fixed height of the molecule above the surface (fixed bond length)

the on-top position is the lowest in energy, but very soon the energy increases dramatically because we have to press the lower a t o m under the equilibrium height. Above the bridge and hollow positions the energy increases much less dramatically, indicating that especially above the hollows, the largest part of the Pauli repulsion is compensated by the chemical bond developing between the substrate and the lower a t o m of the molecule. In this context it is interesting to compare the dotted lines in Figs. 5a-5c representing the optimal positions of the molecule above the

123

surface at a fixed bond length d (i.e. the points where (rE/~z)ld=const.=O)), with the full lines indicating the positions of the molecule with the lower a t o m in the optimal distance for atomic adsorption. Above the top position the dotted line approaches the full line asymptotically for large d, indicatmg that only for d ~ o o (i.e. for a free hydrogen atom plus one adsorbed at the surface) can the lower part of a molecule approach the surface as closely as a free a t o m at no cost of energy. Above the bridge and hollow sites both curves cross for a certain length of the molecule. This indicates a cooperative effect between both atoms in the molecule and the substrate, reducing the dominant Pauli repulsion. Above the bridge position a rather pronounced stretching of the molecule is required, but above the hollow even a modest stretching is sufficient to allow a very close approach of the molecule to the surface. At the position on the reaction path where the potential begins to increase substantially (at about Z = 0.52 A; see Fig. 3b), the bond length of d = 1.35 (see Fig. 5c) shows that the lower H a t o m is already slightly below the surface. However, dissociation into a subsurface site along this straight reaction path is prohibited by the repulsion by the substrate atoms in the subsurface layer.

3.2.3. 2D cuts through the P E S variation with 0 We have seen that a canting of the molecule by 90 ° leads to a substantial increase in energy at any point of the surface. For two positions in the entrance channel of a H2 molecule impinging onto a bridge position of a Pd(100) surface, Wilke and Scheffer [ 14] have shown that the potential energy increases upon canting like cos 2 0 . However, this result is a consequence of the high symmetry of the bridge position. For this reason we have chosen a position with lower symmetry for our investigations, halfway between the bridge position and the on-top position, with the molecule oriented along the b - t direction. In this case a variation of O moves either the upper or the lower hydrogen a t o m towards a Rh atom. For this geometry, a c o s 2 0 dependence cannot be expected, because this would imply that configurations with O = 4 5 ° and O = 1 3 5 ° have the same energy despite very different R h - H distances. In Fig. 6, the elbow plots

124

A Ezchlet et al / Surjace Scwnce 397 (1998) 116-136

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Fig, 6 2D section through the PES for a H 2 molecule with its centre of mass halfway between a bridge and a top posmon at four stages along a cartwheel rotauon within a plane through the bridge and on-top posmons (various angles O, ff=45 ~) Contour lines are drawn every 0 1 eV between - 0 3 and 3 5 eV. Lines indicating negative adsorpnon energies are dashed, while thicker hnes correspond to integer values The four black points mdmate the posmons for whmh the O-dependence of the potentml energy is shown m Fig 7

equipotentlal lines are closer to the surface for the elongated molecule, because a stretching increases the shortest H - R h distance. To analyse the energy variation with O in m o r e detail, we have chosen four points along the reaction path and calculated the potential energy for eight configurations between O = 0 ° and O = 180 °. The results are plotted in Fig. 7. We find that for this less symmetric position the energy also varms similar to the cos 2 0 behavlour p r o p o s e d by Wflke and Scheflter [14]. The m a j o r difference is that m our case the m a x i m a and minima o f the energy are f o u n d not for the upright and the perpendicular position o f the molecule, but are shifted by 15-20 ° This means that for the m o s t unfavourable position the lower h y d r o g e n a t o m does not point simply downwards, but towards a substrate atom, and for the energy m i n i m u m the h y d r o g e n a t o m closer to the substrate a t o m hes higher above the surface than the a t o m pointing towards the bridge posinon. This also reflects the different adsorption heights at these two sites. In general the energy dependence can be fitted by Icos"(O-Oo)[, where

12

.

.

.

.

.

~"

8

for four different angles O are shown. The plot for the upright molecule is similar to the corresponding plot over an o n - t o p site, and also for 0 = 4 5 ° the energy surface does not look too different. For b o t h configurations, compressing the molecule makes an a p p r o a c h easier. However, an energy gain for impinging molecules is only possible for a molecule with greater O. For the planar g e o m e t r y (O = 90°), the elbow plot should be compared with those for the t - b - t and b - t - b geometries given in Fig. 2b and Fig. 2d. As for the t - b - t configuration, we find a local m i n i m u m o f - 0 . 3 1 eV for a molecularly adsorbed state Immediately at the beginning o f the exit channel, extruding rather far for an elongated molecule. A stretching o f the molecule (even b e y o n d the top positions) is possible without m u c h effort. In the Z-direction, on the other hand, the potentml increases dramatically, a behavlour which is typical for the o n - t o p site. In the O = 135 ° orientation, the

j~..,~.~..

D

6 42 ~

.

~

j

~

~

0 -

E[eV] 04 06 ~

.............

02

:--"~ ' J l ...~ffff.,~'~"

B

1

-0 4 -0e

/ "

0

,

02

. . . . . . . . . . . . .

04

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06

08

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Fig 7 Potennal energms at the four points along the reaction path marked in Fig 6 (A, Z=2 34A and d=0 77 A,, B, Z = 1 90A and d=0 082,~, C, Z = I 61A and d=0 90A, D Z= 1 53 A and d= 1 20 A,) as a function of Icos"(O-Oo)l The symbols mdmate the calculated potentlal energy, while the hnes connecting them are only gmdehnes for the eye The fitted parameters for (n, O0) are A (2, 28°), B (2 5, 18°), C (3, 21°), D (4, 22 °)

125

A Ezchleret al / SurJaceSczenre397 (1998) 116 136

00 is the angle describing the position with the m a x i m u m potential energy, and the exponent is n ~ 2 for the points In the entrance channel. For points in the exit channel, where the rotation causes energy variations of up to 10 eV, exponents of n ~ 3 and 4 lead to better results. The optimal fits are shown in Fig. 7, together with the fitted values for n and O 0.

:

I

o

3 2 4 "Adiabatic" reactton channel To complete our studies, we have used a different strategy to find not only the various two-dimensional pathways, but the energetically most favourable pathway within the 6D-PES. We placed the hydrogen molecule far from the surface ( Z = 4 A) in an off-symmetry position, and optimized the lateral coordinates of the atoms while fixing the height of the molecule. Then we pulled the molecule 0.1 A down and relaxed it again. This procedure was repeated until the molecule reached the surface ( Z = 0 A). At each step we monitored the forces which could lead to a canting of the molecule, but we found these forces to be too small to lead to a significant inclination of the molecule relative to the position parallel to the surface. Hence, apart from the fixed height Z, all degrees of freedom are relaxed. Assuming that along the most favourable pathway the height decreases gradually and that the resulting positions describe a continuous pathway for an adiabatic approach, this channel runs along the b o t t o m of the 6D-PES This does not imply that an impinging molecule with very low kinetic energy would choose exactly this pathway for dissociation, but we expect that this channel represents the minimum-energy geometry at each distance (which does not necessamly coincide with one of the symmetric configurations for which the 2D-PES has been calculated) Applying this strategy leads to the pathway Illustrated m Fig. 8. We find that at a distance of Z ~ 3 . 2 A the molecule rotates first such that it is parallel to the b - t - b configuration. In the next step, at a height between 2.8 and 3.0 A the molecule is drawn to the on-top site. The energies derived from the elbow plots suggest such a behaviour: while the pathways over bridge and hollow are for these heights hardly attractive (with minimum changes for helicopter rotations), the pathways

//

/

..@// Fig 8 "Adiabatic" reaction path for dissociative adsorption of a H2 molecule above Rh(100), calculated by relaxmg the bond length, orientation and center-of-massposition of the molecule at fixed heights The dots mark the positions of the H atoms centered over the on-top position are clearly favoured, especially the path with the atoms oriented towards the bridges. As the molecule is lowered further, dissociation occurs Into the bridge positions, from where the atoms finally slide sidewards into the hollows. Tracing the energy along this pathway leads to the thick line in Fig. 3a. The resulting curve is a lower envelope of the curves for the various 2-D PESs. Note also that sliding from the bridge sites into hollow positions is possible without any barrier (i e. the bridge positions are only saddle points on the 6D-PES). Pathways like this become especially important in the low-energy regime, where the movement of the molecule is sufficiently slow that the acting forces can drive it along the most attractive pathway. This steering effect has been shown to be responsible for the increase in the sticking probability for low energies [15 37] which has been observed experimentally for some transition metals like such as Pd(100) [9] and W(100) [38]. 3.3 Changes in chemical bonding along the reactton path 3 3 1 Crystal orbital overlap populatton (coop~COOP) To examine the steering effect m more detail, we analyzed the interaction of the molecular orbitals

126

A Ezchler et al / SurJace Sctence 397 (1998) 116-136

(a, o-*) of the adsorbate with the substrate orbitals using the crystal orbital overlap population (coop) introduced by Hoffman [39,40]. The coop is an overlap-population-weighted density of states, which can be calculated as the difference between the densities of states (DOSs) projected onto bondmg and antibonding hnear combinations of orbitals. Energy intervals with a negative coop reflect anti-bonding, while those with a positive value reflect a bonding regime. Integrated up to the Fermi level, the integrated coop (the C O O P ) represents a measure for the strength of the covalent bond coupling the respecUve orbitals. We used this tool to analyse the interaction between the molecule and the surface along the energetically most favourable reactmn path discussed above. We calculated the c o o p / C O O P s for linear combinations of the molecular orbltals of the adsorbate and the orbltals of a substrate atom, taking the sum over all substrate atoms with a non-neghble overlap to the adorbate. The COOPs in Fig. 9 explain the origin of the lowest-energy channel on the 6D-PES: at large distances from the surface, the molecule interacts with the surface orbitals farthest from the surface, i.e. the e g ( d 3 z 2 _ r 2 ) , p: and s orbltals. This interaction weakens the molecular orbital bond only slightly, as shown by the C O O P of the molecular

"'-hol'~'w',

"~x

rtd~ e

b-t-b

a \

a-bond plotted m the lower panel of Fig 9. At distances of Z ~ 1.5 A the interaction with the t2g orbitals reaching out towards the bridge positions begin to dominate, while the Interaction with the d3x2_,~ orbital is strongly reduced. The t2g-H-~r* interaction drives the &ssoclation. Around Z ~ 1 2 A, the eg C O O P increases steeply. In this geometry the interacUon with the dx2_y2 orbitals extending into the hollows begins to dominate. This explains why the bridge is only a saddle point on the 6D-PES, while the final adsorption site is the hollow position. While this tool enables us to classify the interaction between the molecule and the surface, we used a different method to visualize the bonding.

3 3 2 Electron locahzationfunction The electron localization function ( E L F ) [41,42] is a dimensionless quantity between 0 and 1, defined at every point in space, which is related to the local kinetic energy and hence to the strength of the Pauli repulsion acting at a given point in space. For fully localized electrons ( E L F = 1) (i.e lone electrons, electrons with different spin, etc.) the Pauli principle is active (the probability of finding a second electron with parallel spin near the reference point is zero), and therefore the kinetic energy is at its minimum (equal to the kinetic energy of a boson gas with the same density). For a free electron gas (Fermi gas), the value is by definition E L F = 0 . 5 . This leads to the following expression [41]:

2 \ "~

1- [email protected]' . i"

0 0

.

. 7~'>

..

.

\

....

o'.s

1

1 ' 5

,

1 q_(T--Db~

0

o ....

1

ELF =

:; ....

£.5

'

z[a] Fig 9 V a n a n o n of the integrated crystal orbital overlap populations (COOP) for sums over the three-centre orbltals of a H2 molecule over Rh (100) along the "'adaabatm" reaction path as a function of the hmght of the molecule above the surface Vertmal grid hnes m a r k the change of the onentaUon of the molecule at different stages of dlssoclatmn The lower part of the figure shows the COOP for the a molecular orbital of the H 2 molecule alone (the scale is reduced and shifted downwards)

Dh

( 1)

2

/

with the local kinetic energy T (the argument r has been omitted), the kinetic energy of a non-interacting Bose gas with the same density D b , and the kinetic energy of a homogenous Fermi gas with the same density Dh. To analyze the E L F there are in principle two possibilities for the representation. (i) one can plot the profile E L F within a certain plane, or (ii) one can plot the isosurface for a constant E L F value with the whole cell. In Fig. 10, lsosurfaces ( E L F = 0.25) for different steps of the dissociation/

A Etchlet et al / SurJace Science 397 (1998) 116-136

127

only outside the augmentation radii. At large distances from the surface, one recognizes the closest envelope representing the localization of the H-s electrons in the molecular a bond, and in the background a characteristic feature located in the bridge positions representing a covalent nearest-neighbour bond between the surface atoms (Figs. 10a-10d). The dissociation of the molecule is reflected by a gradual elongation of the envelope (Figs 10c and 10d) and finally its separation into two parts (Figs. 10e and 10f). On the back side, the atomic H-s E L F envelopes overlap with the substrate orbltals and one clearly recognizes the transfer of electrons from the R h - R h to the H - R h bonds The strengthening of the H - R h bonds is also a driving mechanism for the b - h transition of the adsorbed atoms. 3 3 3. Combined E L F / C O O P analysis

i•°I J

Fig 10 Isosurfaces of the electron locahzataon function for ELF=0 25 for the H 2 molecule at heights of (a) Z=3 2 A, (b) Z=2 8,~, (c) Z=24A, (d) Z = l 8A. (e) Z = I 4A, (f) Z= 1 2 A. (g) Z=0 8 A, (hi Z=0.4 A along the "'adlabatxc"reactmn path (Fig. 8 ) adsorption process are shown. In the calculations of the E L F we have considered only the contributions from the smooth pseudocharge densities, omitting for the m o m e n t the contribution from the augmentation charges (see Ref.[23] for details) Hence the E L F functions are accurate

Combining the information from the C O O P analysis with that of the E L F leads to a rather complete description of the adsorption process, which we will divide into several phases (characterized by the height of the molecule). At each stage the dominant mechanisms are as follows 3.0-1.7 A: Paull-repulsion 0.e. the energy costs for the orthogonalizatlon of the molecular orbital Og and the metal s-electrons [43], maximal over the hollows and bridges), and the possibility of an interaction with the p= and the d3z2_,2 orbital driving the molecule over the top position. Another effect postulated by Harris and Anderson [43] can be seen m the E L F figures To reduce the Pauli repulsion, s-electrons are transferred into the d3z2_r2 orbitals, which leads to the little caps over the substrate atoms. These caps appear approximately at the atomic adsorption height of hydrogen over the top position. 1 6 1.4 A' The molecular orbitals become more and more delocalized so that some of the d-electrons can be transferred back into the s-orbital, which leads to the pronounced increase in the C O O P for the s-orbital. 1.4-1 1 A: While the overlap with the d3z2_,2 orbital decreases, the t2g complex together with the Px, Py and s orbItals become more important. These orbltals form structures at the bridge positions pointing towards the H atoms and pull them

A Ezchler et al I St#face S~wnce 397 (1998) 116 136

128

from each other, while similar structures act in the bond region between the H atoms. This leads to the separation of the atoms As a further consequence, the caps over the substrate atoms have been deformed. 1.1-0.7A. The structures described above, together with an increasing influence of the eg complex (dx2_y2, the r 2 part of d3=2_~2, and d3z2_r2 of the subsurface orbitals) cause a sliding of the atoms towards the hollow position 0.7-0.0 A: The atoms are fixed at their final positions, where the dominant contribution comes from the eg complex The equilibrium adsorption height ( Z ~ 0 . 6 A) is determined by the maximal overlap with the substrate orbltals.

o 5o

ooo

-0 25 /

Palladium, with only one electron more, has very similar properties to rhodium. Detailed studies concerning the atomic adsorption of H on the (100) surfaces of rhodium and palladium [31,33,44,45] show that while the adsorption of hydrogen in the hollow- or brtdge-positions of the surface are very similar (on Pd(100) the hydrogen sinks somewhat deeper into the substrate because of the larger lattice constants), dissociative adsorption in an on-top geometry on Pd(100) (in contrast with the Rh surface) would cost energy. This difference in the adsorption energy is induced by a substantial weakening of the metal-metal bond after H adsorption (the first interlayer distance changes by 3.7% of the bulk interlayer distance for palladium, while the adsorption on rhodium leads only to a change of 0.8% [31,33]). This different adsorption behaviour will mfluence those adsorption pathways which are governed by on-top positions. Because of the similarity of Rh and Pd, we will restrict our discussions only to those features of the PES which are different 4.1

Results

4.1 1. 2 D cut through the P E S f o r 0 = 90 °

For planar adsorption, the same pathways as for Rh have been Investigated. The results are represented graphically in Figs. l la and 12, and

/b-h-b

(a)

.... l .............

/

-0 50 -0 75

.

~/ J

/

.

-

"

"x

% -t-b

/"~,, f ' ~ - b

-1 O0 -1 25

E[ov]

........

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i .....

5°

\

02, 0 00

\A,oo

"., 135

~ ~._::-:'~-...

-0 25

bridge O5

/b; '

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hollow,

-0 50

4. Hz on Pd(lO0)

t-b-t.......~ _ h-t- .

0 25

10

15

Z[al

2O

90° 25

0

Fag 11 Variation of the potentml energy for H2 over Pd(100) along the bottom of varaous straight perpendicular reaction channels as a function of the heaght of the molecule above the surface In (a), the molecule as always parallel to the surface ( 0 = 9 0 °) The thack grey hne represents the energy along the "admbatlc" reactmn channel described m the text In (b), the thicker grey lanes show the energy vanaUon for an approach of an upright H 2 molecule onto the hollow, bridge and on-top posmons of the surface, while the thinner black lanes refer to a molecule wath its centre of mass over the maddle between a bridge and a top p o s m o n as a functmn of a cartwheel rotatmn within a plane through bridge and on-top posmons (various angles O, ~b=45 °, see text and Fig 3)

energies for characteristic points along the pathways are compiled in Table 2. Comparmg the whole set of curves in Fig. 11 and Fig. 3 we can distinguish three regions. (i) The entrance channel down to about 1.7 A, depending on the pathway considered. In thts region the two metals are almost undistinguishable, except that on Pd all features are located about 0.25 A closer to the surface. (ii) The region where the bond breaking occurs. For pathways connected to the on-top position, this area exhibits the most pronounced differences. The barriers which are dominated by the on-top adsorption (and therefore are not significantly influenced by the lattice constant) increase, and as a consequence the ordering of the pathways changes For the t b - t pathway, the local minimum exactly over the substrate atoms has also vanished, showing that the top site is now an

A Ezchle~ et al / Surface Sctence 397 (1998) 116 136 150

h-b-h (a)

t-b-t (b)

h-i~-h ( c )

b+b (d)

b-h-b (e)

b-b (f)

100 50 0 150

f

IO0 50 0

150

50

O0 0'5 1 0 1'5 2 0 2 5

30

O0 05 1 0 1'5 20 2'5 30

Fig 12 Vanatlon of the elgenfrequencles of a n H 2 molecule (dotted lane vabranons m the d-dxrectlon me, dashed lane vibrations in the Z-darectlon COz,solid lane vlbrataons perpendicular to the reaction path co,) along the reaction channels wath the molecular axis kept parallel to the Pd(100) surface as a functmn of the reaction path coordinate (see text) Negatave frequencies indicate a negatave curvature of the PES 0 e a locally unstable posmon)

Table 2 Adsorption energy E(eV ), height above the surface Z and bond length d(A) of H2 over Pd(100) at charactensUc positions in the 2D cuts through the PES for fixed lateral coordinates of the centre of mass (x, y) of the H2 molecule and its onentatlon (o, ~) Pathway

Type

E (eV)

Z (A)

d (A)

h b~

Barrier Minimum Bamer Minimum Barner Mlmmum Local minimum Barner Minimum Local mmamum Barrier Minimum Barner Minimum Saddle point

0 01 --0 93 0 09 - 0 79 0 00 - 0 76 --0 12 0 25 --0 96 --0 27 - 0 10 --0 79 0 01 - 0 21 0 14

2 16 0 25 1 90 1 03 2.26 0 99 1 89 1 42 0 23 1 73 1 48 1 03 2.26 1 46 1 55

0 77 2 80 0 78 2 75 0 75 1 98 0 80 1 41 3 96 0 84 1 31 2 81 0 78 0 94 1 85

b~ b b-b h t-h

b-t-b t-b-t

129

u n s t a b l e p o s i t i o n for the H atom. We also note that the barrier in the b - t - b c h a n n e l following the local m i m m u m is n o w m u c h m o r e p r o n o u n c e d , so that at these i n t e r m e d i a t e distances the energy is lower a l o n g the b - b a n d h - b - h channels. A t slightly lower distances, however, the b - t - b channel is again m o r e favourable. (iii) The exit channel, which is solely d e t e r m i n e d by the c o r r e s p o n d i n g atomic values a n d therefore again is very similar. F o r the hollow p o s i t i o n ( ~ b - h ) , we can again c o m p a r e the frequencies (symmetric stretch m o d e coz = 13.9 T H z , parallel m o d e ~oe = 16.7 T H z ) to the experimental values of COz= 15.23 [46] (15.35 [47]) a n d c%= 18,38 T H z , [46], respectively. 4.1.2. 2 D cuts through the P E S f o r 0 5 0 ° F o r the u p r i g h t a p p r o a c h of the h y d r o g e n molecule, the PES is almost identical to that for rhod i u m The only difference is the fact that the atomic a d s o r p t i o n heights differ, which leads to a c o r r e s p o n d i n g shift for the energy variataon (see Fig 1 l b ) . T h e different energy for o n - t o p adsorption has n o influence o n the energetics of the vertically i m p i n g i n g molecule. The slightly larger lattice c o n s t a n t also induces a deeper sinking into the hollows. T h e energy stays rather c o n s t a n t between Z = 1 0 ~t a n d Z = 0 . 5 A, a n d i m m e d i a t e l y before the energy starts to increase (at (Z, d ) = (0.43, 1 . 3 0 ) A the lower a t o m is already 0.22 ]~ below the surface, i n d i c a t i n g that a similar pathway, with cartwheel r o t a t i o n s allowed, m a y lead to a s i t u a t i o n with one occupied hollow a n d one occupied subsurface p o s i t i o n ) The same scenario as for the u p r i g h t a p p r o a c h is f o u n d for the v a r i a t i o n of the energy with O at the p o s i t i o n halfway between the bridge a n d top sites. F o r the u p r i g h t a n d 45°-tilted molecule the energies a l o n g the p a t h w a y s are a b o u t equal. For the p l a n a r a n d 135°-tilted molecule, the adsorpUon energy also becomes i m p o r t a n t , a n d a stretching of the molecule at the end of the reaction p a t h is less f a v o u r a b l e t h a n for R h 4.1.3 "'Adiabatic" reactton channel A l t h o u g h the differences in the p a r t i c u l a r 2 D - P E S s are very small, we have seen (for the p l a n a r a d s o r p t i o n ) that their energetic ordering changes. Thas indicates that the most f a v o u r a b l e

A Ewhler et al / SwJace Scwnce 397 (1998) 116-136

130

pathway m a y differ from what was found for rhodium. Applying the same technique as described for rhodium leads to the reaction channel sketched in Fig. 13. Again the molecule is first driven into a b - t - b configuration, but because of the barrier for dissociation in this channel (see Fig 11 a) it rotates further into the b - b geometry, where dissociation into the bridges takes place Finally, the separated atoms drift towards the final hollow positions. The energy along this reaction channel is again always at the b o t t o m or below that of the other (planar) pathways (Fig. 11), and the barrier for dissociation IS very small (40 meV). 4.1.4 Comparison to other studies In this section we compare our results to an F L A P W study on Pd(100) by Wllke and Scheflter [14,15] and to a calculation for H 2 o n the (100) surface of bcc tungsten by White et al. [36]. The overall agreement between our study and R e f [ 14] is very good. All the features of the PES found in Ref. [14] are also present in our PES. Slight energy differences, especially for the adsorbed speoes, can be explamed by the different treatment of relativistic effects (scalar-relativisnc m our work [ 31 ], non-relativistic in Refs. [ 14,15]). Concerning the energencs for cartwheel rotation (O), we expect for palladium the same behawour

i

/

/

i!

z[X] "---...

/

Fig 13 "Adiabatic'" reaction path for the dissociative adsorption of a Hz molecule above Pd(100), calculated by relaxing the bond length, orientation and centre-of-mass position of the molecule at fixed heights The dots mark the positions of the H atoms

as already described for rhodium, which is just a generalization of the c o s 2 0 behaviour observed by Wilke and Scheffter. For the hehcopter rotation, Wilke and Scheffler state that the rotation has no effect on the energencs until the H and Pd atoms get closer than a typical H - P d bond length (1.7 A), which agrees well with our results over bridge sites ( h - b - h , t - b t ) . However, on-top sites we have at this distance an energy difference between h - t - h and b - t b of ~ 1 9 0 m e V . This dependence of the potential energy on the helicopter rotanon for on-top dissociation has also been neglected in the parametrizatlon of the 6D-PES in Ref. [15]. In this p a r a m e m z a t i o n , the frequencies perpendicular to the reaction path were also set equal for all pathways at corresponding reacnon path coordinates s, which is not valid (e.g. at s ~ 1 5 A the frequencies vary between 50 and 100 THz, see Fig. 12). Concerning the local minima in the b - t - b and h t - h pathways (which also appear in a very similar form in the PES of H2 over W(100) [36]) Wilke and Schemer find that sliding of the molecule from h - t - h via b - b towards t - h - t is purely attractive, so that the local m i m m a appear only in the 2D representation. This finding is illustrated by a graph (Fig. 4 in Ref. [14]) showing the energy variation for sliding at a height of Z = 1.6 A, which is about 0.2 A below the minimum of the h - t - h path, and therefore about 150meV higher in energy than the minimum (cf Fig. 11). At this height, b - b is clearly favoured in energy compared to h - t - h , whereas m our calculation, at Z ~ 1.9 only b ~ - b is lower m energy, and therefore we have true local minima for pathways w~th on-top dissociation. This fact has also been proved by a full 6D relaxation of the molecule starting in the local minima. These local minima are not the only c o m m o n property of the PES of H 2 o v e r W(100) [36] and our results for Pd(100). For h - b - h the PES is very flat m the entrance channel, whereas for b - h - b there is a barrier around 1 8 A. Also, the elbow plots for upright incidence look very similar. The only pathway which is really different is t - b - t , where White et al [36] found a barrier of about

131

A Ez~hlet et al / Surjace Sctence 397 (1998) 116-136

0.25 eV, whereas for palladium we found a local minimum at the corresponding position. 4.2. Changes in chemical bonding along the reaction path

For palladium we restricted our analysis to the calculation of the C O O P along the "adiabatic" reaction path, with the result shown in Fig. 14. 3.0-1 6 A: D o w n to a height of about 1.6 A, the mechanism for adsorption is completely equal to that on rhodium. The molecule is driven into the b - t - b position and the H o--bond is substantially weakened. 1 6-1.2 A: The on-top position is less favourable and the larger lattice constant would require a higher bond length to achieve an optimal overlap with the structures at the bridges (compare the E L F figures for rhodium) in this position. The C O O P for the t2g complex is substantially smaller compared to rhodium at a height of about 1.6 A. So, the molecule rotates sideways to the b - b geometry, where an optimal overlap with the tZg complex as possible 1.1 A: At this height the atoms have arrived In neighbouring bridge positions and the molecular hydrogen bond is broken 1.0-0.5 A: The reaction with the eg complex ho'l'lo~ 3

"6 0

1

~x \

b-b F-, "~x

b-t-b

; i °_(, j

--x¢,....

" px py ~ i-::-'~

I

0

"~"

z

":" "-" "-" :':"""~ ~

-1 0

~

~":': . . . . . . .

1' 05

1

15

2

2.5

3

z[A] Fig 14 Variation of the integrated crystal orbital overlap pop-

ulatmns (COOP) for sums over three-centre orbltals of a H2 molecule over Pd(100) along the °'ad~abatlc" reactmn path as a function of the height of the molecule above the surface Vemcal grid hnes mark the change of the orientation of the molecule at different stages of the dissociation The lower part of the figure shows the COOP for the a molecular orbxtal of the H 2 molecule alone (the scale as reduced and shifted downwards)

(dx2_y2, the r 2 part of d3=2_,2 and d3:2_r2 of the subsurface orbitals) and with the structures at the empty bridges built up by s, Px, P~ and t2g orbitals becomes more and more important and leads to a transition downwards into the hollows 0.4-0.0 A: The atoms are fixed at their final position (the four-fold hollow), with a maximal (total) overlap at the equilibrium position of about Z = 0.25 A.

5. Ha on A g ( 1 0 0 ) I

For the noble metal silver we expect a totally different potential energy surface. !Although the stable places for atomic adsorption are at a comparable height [33], dissociative adsorption is always an endothermic process Therefore, both the minima (as far as they exist) and the barriers will be rather high in energy, but because of the geometracal similarity of atomic hydrogens on different surface positions, the shape of the pathways (e.g. d as a function of Z along the energy minimum within a particular elbow plot) will not chan)e too much. 5 1. Results 5 1 1. 2D cuts through the P E S f o r 0 = 90 ° Elbow plots for various reaction [pathways have been computed. For the pathways with the highest ( h - t - h ) and the lowest (b h - b ) ~nergy barriers, the elbow plots are shown in Fig ~5. The energies along the b o t t o m of the 2D chanpels are shown in Fig. 16. The corresponding vibrational frequencies are shown in Fig. 17 and the! parameters of some characteristic points of thel pathways are compiled in Table 3. In the entra~lce channel an approach over on-top sites requl~es the smallest energy In contrast to Rh and Pd,] an orientation of the atoms towards the hollows 1~ more favourable than towards the bridges. However, as soon as the molecule reaches the height (or the atomic adsorption over on-top sites (2~=1.66 A), the energy increases dramatically and the pathways lead to very high barriers. Also, thee pathway with the atoms directed towards on-top positions ( t - b - t ) runs against a repulsive barrier, so that along this pathway even the local physisorbed

132

A Ewhlei et al / Surface Scwnce 397 (1998) 116 136

3

llllll!llI

Of, I'

. . . .

i,,

} ,,rrr, (a)hlllf,'ll,;l ,]

h-t-h

2 5 IIVII,~.I~PI,', ' '

150

b-h-b (b)

h-b-h (a)

t-b-t (b)

t f, "!q

','" ';,,

i'[7/",' 50

....................

%1}~\ ,,. oO 70/ ~,~.. ~i'~/,~\ ~ ~!~'i,~-'-~;l i *-i 5202530354045 0 15 2'0 25 30 35

....

o0

1 01

o

t-b-t ('" b - b ~ h ' t " ~h~"~'- I h-b-kk . ~ / " "\~ ~ " - * ~ - t - b

150

125 1

=

•

~

"~N.%

,

. . . . . . .

L X

oo

~)° \135~

,

....... ......... (b)

b-b (f)

501 O0 05 10 15 20 25 30 O0 05 10 15 20 25 30

Table 3 Adsorption energy E(eV ), height above the surface Z and bond length d(A) of H 2 over Ag(100) at characteristic posmons In the 2D cuts through the PES for fixed lateral COOl-dmates of the centre of mass (x, y) of the H2 molecule and its onentation (O, ¢)

\~--45 o

",, ~'9o°\

075

b-h-b (e)

,o0

Fig 17 Variation of the eigenfrequencies of an H 2 molecule (dotted hne vibrations in the d-direction Od, dashed hne vibrations in the Z-direction coz, solid line vibrations perpendicular to the reacUon path co,) along the reaction channels with the molecular axis kept parallel to the Ag(100) surface as a function of the reaction path coordinate (see text) Negative frequencies indicate a negative curvature of the PES (i e a locally unstable position)

(a)

025 00I ...... " " "X

,. . . . .

0

Fig 15 2D cuts through the PES for the dmsocmtive adsorption of H2 on Ag(100) for the pathways with the highest ( h - t - h ) and the lowest (b h - b ) b a m e r for dissociation The molecular axis lS always parallel to the surface Contour lines are drawn every 0 1 eV between 0 and 2 eV Thicker lines correspond to integer values

e[eV]150

; ~ ; & - .--.

i

100 ~[THz] 50

d[A]

2 O0 175 1 50 112500

r~

~\~\

Pathway

Type

E (eV)

Z (,~)

d (~.)

h b-h

Barrier Minimum Barrier Minimum Saddle point Barmer Minimum Barrier Minimum Barrier Minimum Saddle point

1 29 0 72 1 10 0 79 0 79 1 24 1 16 1 63 0 70 1 44 0 79 2 07

0 95 0 41 0 97 1 02 1 00 1 17 1 05 1 37 0 38 1 45 1 03 1 67

1 50 2 95 1 37 2 74 2 95 1 54 2 08 1 90 4 17 1 66 2 94 2 95

000 '

'0'5

....

1'0 . . . .

1'5 . . . .

z[A]

2'0 . . . .

2'5 . . . .

0

Fig 16 Variation of the potential energy of H 2 over Ag(100) along the bottom of various straight perpendicular reaction channels as a function of the height of the molecule above the surface In (a), the molecule is always parallel to the surface ( 0 = 9 0 °) In (b), the thicker grey lines show the energy vamation for an approach of an upright H2-molecule ( O = 0 °) onto the hollow, bridge and on-top positions of the surface, whereas the thinner black lines refer to a molecule with its centre of mass over the middle between a bridge and a top posmon as a function of a cartwheel rotation within a plane through the bridge and on-top potations (various angles O, ¢ - 4 5 °)

minimum disappears. The elbow plot for this scenario looks similar to those for the upright molecule over rhodium or palladium For b-b the e n e r g y b a r r i e r is r a t h e r l o w ( 1.24 e V ) a s c o m p a r e d

b-h b

b b h~ h b~ b t b t

to the other pathways, but the energy gain after t h e b a r r i e r is v e r y s m a l l ( 8 0 m e V ) because of repulsive interactions The final b-b configuration is 370 m e V h i g h e r i n e n e r g y t h a n f o r t h e b - t b and b-h-b configuration, where the occupied

133

A Ezchler et al / Surface Scwnce 397 (1998) 116-136

bridge sites are further apart. This shows that the a d a t o m interactions on silver are long-ranged as compared to r h o d m m or palladium, where these configurations differ only marginally (80-90 meV). The b - h - b channel, where we found the highest barriers in the entrance channels of rhodium and palladium, is also for silver the least favourable in the region far from the surface. But in the exit channel, this pathway has the lowest energy barrier (1.10 eV; cf. Fig. 15) at a rather constant height, where bond-breaking occurs. This means that the pathways which are less repulsave for larger distances very soon lead towards high barriers, whereas the lowest barriers can only be reached over pathways which are initially higher in energy. This could lead to a "negative steering effect", which makes adsorption even less probable. The very soft frequency cod (see Fig. 17) in the adsorbed position, which describes a motion towards the hollow positions, shows, similarly to R h and Pd, that shding from bridge sites Into hollow sites is possible without investing energy. Finally, in the h - b - h channel the barrier is also in the region where bond-breaking occurs (approximately at the height of bridge-adsorbed atomic hydrogen). The corresponding features over rhodium and palladium are the kinks m the E ( Z ) curves, which are now shifted above the zero line and become a barrier. Concerning the vibrational properties of the H2 molecule, a notable difference to the vibration spectra for Rh or Pd is that all barriers coa becomes soft, which means that for these points, positions at the same height with slightly smaller and larger bond lengths are lower in energy. This reflects the fact that all barriers are now located In the exit channel.

5.2. Vartatzon o f the dissociation barrier across the surface

In order to study the variation of the minimum b a m e r all over the surface, we calculated two additional cuts through the PES: (i) for the centre of mass over a position halfway between the top and bridge p o s m o n s and the molecule planar and perpendicular to t - b - t , and (ii) the centre of mass over the middle between hollow and bridge positions and shghtly tilted (with respect to O) (h-bt) so that the molecular axis points in the final position towards the atomic adsorption positions over bridge and hollow sites. In Fig. 18 the variation of the barrier height (E), the height of the molecule at the barrier above the surface (Z) and the bond length of the molecule at this position ar shown as a function of the position over the surface. There is an obvious correlation between the graphs: the earlier the

16 11 E[eV] 16

Despite an increase in energy, the elbow plots for upright molecules differ only marginally from what we have already seen for rhodium and palladium. For the "rotating" molecule between bridge and top site, the planar and the O = 136 ° configuration became less favourable, so that the vanaUon of the energy with the cartwheel rotation has decreased (see Fig 16).

T~~'L~

(b)

z[A]

18 5 1.2 2 D cuts through the P E S f o r 0 ~ 0 °

T ~ ( a )

T'~~f~!~(C)

11 a[A]

Fig 18 Variation of (a) the energy E, (b) the height of the centre of the mass of the molecule Z and (c) the bond length d at the dlssoclatlon barner ruth the posmon of the molecular centre of mass on the surface

134

A Emhler et al / Surface Scwnce 397 (1998) 116 136

barrier is (high Z ) , the higher 1s it in energy (high E) and the weaker is the H - H bond (large d). A very similar result has been obtained by White et al. [48] for the Cu(100) surface. They obtained a m a x i m u m b a m e r of 1.21 eV at Z = 1 40 A over the on-top position (compared to a value for Ag(100) of 1.44 eV at Z = 1.45 A) and a minimum barrier (at h0-bt) of 0.93 eV at a hetght of 1 08 A. At the same position ( h - b t ) w e obtained a barrier of 1.18 eV at a height of Z = 0 . 9 1 ?t, which is only 80 meV higher than the minimum barrier over hollow positrons. This means that although the energy corrugation on Ag(100) is approximately of the same magnitude ( 2 8 0 m e V for Cu and 340 meV for Ag), the corrugation corresponding to the position of the molecule at the barrier is much higher (0.32 A for Cu and 0.54 A for Ag). According to dynamical simulations [49] such a variation of the energy barrier leads to m o m e n t u m parallel to the surface tending to inhibit dissociation, which for Cu(100) would be in contradiction to experiment [50] (i.e. a normal energy scaling), which means that m o m e n t u m parallel to the surface does not couple to the reactmn channel. However, Darling and Holloway [49] showed that variation of the distance of the molecule at the barrier can induce an opposite effect 0 e. parallel m o m e n t u m enhancing dissocmtion) So, if the positions with a high energy barrier are correlated to a transition state high above the surface, then these two effects can cancel each other out [49], leading to the normal energy scaling observed for Cu(lO0) [50]. As the variation of the position of the barrier is much higher for Ag(100) than that observed for Cu(100), the height effect may dominate and induce a parallel m o m e n t u m enhancing dissociation over an Ag(100) surface

6. Surface chemical reactivity Fig. 19 summarizes our results for the final (atomic) adsorption energies in the hollow, bridge, and top positions and for the barriers (or energies at which bond-breaking occurs) along different adsorption paths For the adsorption energies, we find only small differences between R h and Pd,

30 25 20 15 10 05 O0 -05 -1

E[~V]

3 2 2 1 1 0 0 -0 -1

(a) b-t-b

\

HN-model _~(d) . , f L

h-t-i~

\

)<

(b)

HN-model top bridge

i e~"~-~"~ Rh

J~ :

,,*~

~.'~ i*

(b-b ) ~ ~ - ;

hollow P'd

Ag

Fig 19 Black hnes illustrate the variation of the barrier/bondbreaking energies (a) and adsorption energies (b) (in eV) along the series R h - P d - A g Also included in the plot (grey lines) are the p o s m o n s of the centre of the d-band and the adsorption energies according to the model of H a m m e r and Norskov [51,52] (For a better comparison, these energies have been shifted upwards by 2 and 3 eV ) For more details, see the text

with the exception that adsorption on the top-sites is exothermic for Rh but endothermic for Pd. On Ag, all adsorption energies are endothermic The differences between the various adsorption sites are about the same for all three metals. A more significant difference appears m the distribution of the barriers or bond-breaking energies registered along the reaction channels. The bond-breaking energy is defined as the potential energy where a substantial elongation of the molecule begins, manifested as a kink in the E ( Z ) curves in Figs. 3 and 11. For Rh, only In the h - t - h channel is a barrier between a weakly phystsorbed and a dissociated state found However, the total energy at this point is still attractive, and relative to the local minimum representing the physisorbed state, the barrier height is only minimal (see Fig. 3). Going from Rh to Pd, most barriers and bond-breaking energies are only slightly affected, except the barrier in the h - t - h path where the energy of the transition state is now repulsive, and a stronger barrier m the b - t - b channel which is only very weak for Rh. However, our calculations have shown that this barrier can be lowered by rotating and translating into a b - b positron. It IS

A Elchler et al / Surface Scwnce 397 (1998) 116-136

interesting to point out that the barriers for on-top dissociation occur at distances only slightly below the equihbrium height for atomic adsorption. For Ag, the barriers show a large variation across the surface, indicating the strong corrugation of the surface for adsorptive reactaons. The variation of the adsorption and bondbrekamng (or barrier) energies follows quite well the predictions of the H a m m e r - N o r s k o v model [51,52] measuring surface reactivity in terms of contributions arising from (1) the energy gain due to the hybrldizat~on of the d-states of the substrate and the anti-bonding a* molecular orbital of the molecule, (ii) a correspondmg term describing the interaction with the bonding % states, and (fii) a repulsive interaction arising from the orthogonalization of the a* and ag orbitals to the substrate d-states. (i) and (ii) depend on the posmon of the centre of the d-band and the degree of filling, while (i) and (iil) scale with the square of the coupling matrix element between the adsorbate and substrate states. Our ab-initio results agree rather well with the predictions of the H N model. The most important parameter is the position of the d-band, which determines the trend in the series R h - P d - A g (see Fig 19). In addition to this overall variation of the surface reactivity, we find important variations leadmg to significant differences in the PESs The appearance of a local barrier in the b - t - b channel, for example, which leads to the quite different scenarios for the adiabatic reaction path, is due to the large lnteratomic distances on Pd(100) as compared to Rh(100). The distance between the t2g-type charges m two bridge positions related across the top is simply too large: significant overlaps with a* orbitals can be achieved only after the molecular bond has actually been broken.

7. Summary We have performed detailed calculations of the 6D-PES for the dissociative adsorption of hydrogen molecules on the Rh, Pd and Ag(100) surfaces, based on ab-initio pseudopotential calculations. Our investigations are similar to those of

135

Wilke and Scheffler for Pd(100), but with some important differences. (1) Wilke and Schemer used an all-electron approach, while our calculations are based on a computatlonally more efficient plane-wave technique and ultrasoft pseudopotentlals. Hence the comparison is interesting from a methodological point of view. (i0 We have attempted to model the 6D-PES in more detail We use a larger number of reaction channels, also considering low-symmetry configurations for the adsorbing molecule, and study the variation of the vibrational frequencies of the dimer along the reaction paths. (iii) Via a 6D mimmization (fixing only the height of the molecule above the surface) we have studied the "adiabatic" reatlon channel representing the minimum potential energy as a function of the distance from the surface For Pd (100), comparison of the all-electron [ 13 ] and the present pseudopotentlal calculations firmly establishes the accuracy of a peudopotential-based approach even for transmon metals. Differences in the PESs appear only where we have allowed for a more extended exploration of the PES This leads, for example, to a more complex (but also more realistic) dependence of the potential energy on the angle O for cartwheel rotations We have also shown that the shallow local minimum in the b t b channel at Z ~ 1.65 A corresponds to a real minimum, and hence to a weakly physlsorbed state Our results also demonstrate that the vibrational frequencies depend on the value of the reaction coordinate. It remains to be seen how far these differences affect the dynamics of the adsorptlon/desorptlon process. We have also analyzed in detail the variation of the 6D-PES with the degree of filling of the d-band, from Rh(100) where only a small activation barher exists along certain reaction channels, to Pd(100) where a rather substantial barrier exists along some paths, to Ag(100) where only activated endothermlc desorptlon processes are possible. A detailed analysis of the adsorbate-substrate interactions has been performed by calculating the coop, the integrated COOP and the electron localizatlon function. This analysis leads to a very clear

136

A Elchler et al / Surface Scwnce 397 (1998) 116-136

picture of the formation of covalent hydrogenmetal bonds governing the topology of the PES T h e n e x t step in t h e a n a l y s i s will c o n s i s t o f classical a n d q u a n t u m d y n a m i c a l s i m u l a t i o n s f o r a m o l e c u l e i m p i n g i n g o n a m e t a l surface.

Acknowledgements This work was supported by the Austrian Science Foundation under project No. Pl1353-PHYS. The help of Dr. Andreas Savin ( U n i v e r s i t 6 P i e r r e et M a r i e C u r i e , P a n s ) m p e r forming the ELF calculations is g r a t e f u l l y acknowledged. This cooperation was supported by the Groupement de R e c h e r c h e E u r o p 6 e n " D y n a m i q u e m o l 6 c u l a i r e q u a n t i q u e a p p l i q u 6 e ~t la c a t a l y s e , /t l ' a d s o r p t i o n et /t l ' a b s o r p t i o n " s p o n sored by the CNRS and the Institut Frangais du P6trole. W e t h a n k A. G r o s s , S. W i l k e , B. H a m m e r a n d P. S a u t e t f o r u s e f u l discussions.

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