Diffraction of molecular hydrogen from metal surfaces

Progress in Surface Science 86 (2011) 222–254

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Review

Diffraction of molecular hydrogen from metal surfaces Daniel Farías a,⇑, Rodolfo Miranda a,b a

Departamento de Física de la Materia Condensada and Instituto Nicolás Cabrera, Universidad Autónoma de Madrid, Cantoblanco, 28049 Madrid, Spain b Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), Cantoblanco, 28049 Madrid, Spain

a r t i c l e

i n f o

Commissioning Editor: Aart Kleijn Keywords: Hydrogen dissociation Surface diffraction Gas-surface dynamics Molecule-surface scattering Molecular beam experiments

a b s t r a c t The dissociative chemisorption of hydrogen at metal surfaces is the first step in the surface chemistry of heterogeneous catalysis. Up to now, most of our understanding of this process has been obtained from sticking probability measurements. Recent experiments have shown that more detailed information on the potential energy surface (PES) governing the dissociative chemisorption of hydrogen can be obtained by employing a different technique, namely diffraction of monochromatic beams of molecular hydrogen and deuterium. In this paper, we review recent progress made by using this technique to characterize the corresponding PES for hydrogen dissociative chemisorption at metal surfaces. Elastic and rotationally inelastic diffraction (RID) peaks were observed in experiments performed on different single-crystal metal surfaces, ranging from nonreactive to very reactive ones, at incident energies between 20 and 200 meV. Extrapolation of data points by using the Debye–Waller attenuation model makes comparison with theory possible. It is shown that an analysis of both H2 diffraction and RID intensities as a function of incident energy provides a very sensitive way to test the quality of ab initio determined six-dimensional PESs. This review provides an overview of the experimental procedures as well as on the theoretical tools presently being used. A comparison between theory and experiment is discussed for several illustrative examples. Perspectives for future experiments are discussed. Ó 2011 Elsevier Ltd. All rights reserved.

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 1.1. Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

⇑ Corresponding author. Tel.: +34 91 497 5550; fax: +34 91 497 3961. E-mail address: [email protected] (D. Farías). 0079-6816/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.progsurf.2011.08.002

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2.

3.

4.

5.

Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. General requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Scattering geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Molecular hydrogen beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of diffraction probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Correction for Debye–Waller attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Absolute and normalized diffraction probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Theoretical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. General trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. H2/Pd(1 1 0): Evidence of dynamic trapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. H2/Pt(1 1 1): Validity of the Born–Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . . . 4.4. H2/Ru(0 0 0 1): Performance of different functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. D2/NiAl(1 1 0): Accuracy of RIDs description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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227 227 228 230 231 231 233 234 237 237 241 242 245 247 250 251 251

1. Introduction Reactions of molecules at solid surfaces are of the utmost practical importance, since they constitute the basis for heterogeneous catalysis. This phenomenon is relevant for many large scale chemical processes in industry, like ammonia production, oil refining and natural gas conversion, as well as for a number of environmental protection processes such as the exhaust gas cleaning in automobile industry. The importance of these reactions can hardly be overestimated: About 90% of the chemical manufacturing processes employed worldwide use heterogeneous catalysis in one form or another [35]. Because these reactions are often rather complex, the usual procedure to get further insight into them consists of breaking down these complicated processes into more elementary reaction steps. Among these steps, dissociative chemisorption is perhaps the most crucial from the viewpoint of heterogeneous catalysis. In the dissociative chemisorption process, a bond in a molecule hitting the surface is broken, and the resulting fragments are bonded to the surface. If we consider the large energies required to break bonds even for diatomic molecules (e.g. 4.5 eV for H2, or 9.8 eV for N2), the importance of this special ability of a catalyst becomes apparent [61,62,124]. It seems obvious that a detailed knowledge of the underlying elementary reaction steps could greatly improve our understanding of how to make more efficient catalysts, as it was already shown by a few case studies [16,117,19]. This is particulary true for the mechanism leading to dissociation of H2 on single-crystal metal surfaces, which has been considered since the beginning of surface science as a model system by the gas-surface community. Therefore, the interaction of hydrogen with metal surfaces has been extensively studied, both experimentally [153,188,156,94] and theoretically [47,81,109,111]. From the viewpoint of the calculations, a major advantage of the H2 + metal system is that, to a good approximation, the inelastic channels related to the surface, i.e. phonons and electron–hole pair excitations, can be neglected. Under these assumptions, if we consider a H2 molecule approaching a surface with a kinetic energy of 75 meV (corresponding to a room-temperature H2 beam, and de Broglie wavelength of 0.74 Å) at a well-defined angle of incidence, there are only two possible outcomes1: (i) to undergo diffraction, or (ii) to dissociate on the surface. This situation is schematically shown in Fig. 1. Because of the quantum mechanical nature of the problem, these two channels compete with well-defined probabilities, which depend on the molecules incident conditions. So in some sense, we can speak of diffraction as the complementary channel of dissociation. Since these are the two only possible channels for the incident H2 molecule, we can write: 1 Physisorption can be excluded, due to the high temperatures (400 K) at which the diffraction experiments are performed. See Section 2.1.

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Fig. 1. Two competing paths are open for an H2 molecule approaching a solid surface with a well-defined energy and angle of incidence: (i) the diffraction channel, in which the molecule returns to the gas phase at a different final angle but with the same total energy, and (ii) the reaction channel, in which the H2 ? 2H occurs, whereby the two hydrogen atoms remain chemisorbed on the surface. Courtesy by E. Pijper.

PDiss þ PDiff ¼ 1;

ð1Þ

whereby PDiss and PDiff are the H2 dissociative sticking and diffraction probabilities, respectively. Because of the low energies employed in diffraction experiments (20–200 meV), molecular vibrations cannot be excited in the scattering process, and therefore only rotational transitions are observed. In this case, the incident molecules convert part of their translational energy into excitation of a rotational quantum level when colliding with the surface, changing the kinetic energy of the H2 molecule. This leads to the appearance of additional peaks in the diffraction spectra, the so called rotationally inelastic diffraction (RID) peaks, whereby ‘‘inelastic’’ refers to energy exchange between the hydrogen degrees of freedom, and not with respect to the substrate. The position of RID peaks within an angular distribution are easily obtained by combining the Bragg condition for surface diffraction with conservation of energy [130]. Thus, strictly speaking, diffraction is in general accompanied by the appearance of RID peaks. For this reason it has been included as part of the diffraction channel in Fig. 1. Regarding the reactivity channel, a quantitative measure is given by the initial dissociative sticking probability (S0) of H2 [151]. This probability might vary from zero (for non-reactive systems, like noble metals) to a value very close to one for highly reactive systems (also called non-activated systems) like most transition metal surfaces. As a consequence, on most reactive surfaces a hydrogen overlayer builds up quickly if the experiments are not performed at a surface temperature higher than the desorption temperature of H2 (350 K), altering the measurements. These high temperatures will increase the inelastic background (produced by the lattice vibrations), causing a strong signal attenuation of diffracted molecules, in a way similar to the well-known case of He-diffraction [127,87]. This comes on top of the fact that we are dealing with quite reactive surfaces (i.e. surfaces with a very low

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reflectivity, see Eq. (1)), so we end up with total diffraction intensities which represent just 1% of the incoming beam signal. These low intensities set an important boundary condition to diffraction experiments. As a consequence, most experimental studies reported so far for H2 + metal systems were focused on direct measurements of the H2 dissociative channel. A wealth of molecular beam and associative desorption experiments have been performed, in which sticking probability curves as a function of incident conditions were measured. These experiments provided detailed information concerning the influence of incident energy and angle, vibrational state, the molecule’s incident rotational state and even molecular alignment on reaction [126,153,151,155,156,94,173]. This is certainly a very important piece of information which can be used to test the quantitative accuracy of current state-of-the art potential energy surfaces (PES), but usually the data contain information averaged over the whole surface unit cell. A different point of view is provided by diffraction experiments. As first pointed out by Halstead and Holloway [89], hydrogen diffraction measurements over a wide incident energy range should provide precise information regarding the distribution of activation barriers within the surface unit cell. In particular, their model calculations showed that quite different H2 diffraction patterns are obtained by placing the activation barrier at different sites within the surface unit cell, whereby essentially the same sticking curves were obtained in all cases. These results resemble the well know case of He-atom diffraction from solid surfaces [65], in which the relative intensities of the peaks are quite sensitive to the smallest variations in the particle-surface PES. Thus, H2 diffraction has been proposed as a promising, and maybe unique, experimental technique to gauge the H2-surface PES [89,46]. However, these expectations have not been satisfied until recently, due to practical limitations in both theory and experiments. In the early 1980s, the first H2 and D2 diffraction experiments from low-index metal surfaces were reported. Most of them were performed on non-reactive systems, on which the repulsive part of the PES is tested, but there were also a few studies on reactive systems. For instance, H2 diffraction from Ni(1 1 0) was clearly resolved, even with the surface at 400 K [160]. However, this line of work was soon abandoned by the experimental groups involved, presumably because of the realization that the link between the experimental H2 diffraction spectra and the PES can only be established by performing accurate dynamical calculations, which were impossible to handle at that time. To a large extent, the renewed interest in H2 diffraction experiments experienced over the last decade has been driven by several breakthroughs in the theoretical study of dissociative chemisorption of H2 on metal surfaces [90,86,45,113]. Though evaluation of realistic PESs for H2 + metal surface systems is not trivial, an exact theoretical description of H2 scattering from first principles (if we neglect nonadiabatic effects and phonons) is now possible, in which the remaining six degrees of freedom of the H2 molecule are treated quantum mechanically. These six-dimensional (6D) quantum dynamical calculations are based on potential energy surfaces obtained from density functional theory (DFT), using the generalized gradient approximation and a slab representation of the metal surface. The first of these studies was reported by Gross and Scheffler [82] for H2 diffraction from Pd(1 0 0). This pioneering work already contained a few intriguing predictions like, for instance, the existence of pronounced out-of-plane diffraction, which were also present in similar calculations performed for H2/Pt(1 1 1) in the group of Kroes [141,143,144], and which were not observed in experiment until a few years later, for H2/Pd(1 1 1) [69]. As the result of work done by several groups on this subject, combined with the increasing computational power available nowadays, calculations which were once thought intractable have become a standard theoretical tool to accurately describe both dissociative chemisorption and elastic diffraction of H2 on metal surfaces [47,81,109,114,111]. It is worth noting that the whole H2 diffraction approach might seem in some sense counterintuitive, since the goal is to get information on the dissociative channel by looking into the small amount (1%) of diffracted H2 molecules. In other words, the main idea behind diffraction studies is that the small amount of H2 molecules surviving dissociation are carrying detailed information on the PES topology. This expectation has been confirmed by many experimental and theoretical studies of H2 diffraction from metal surfaces. Some of them are presented in Section 4 of this review. Since the classical turning points of diffracted molecules are located approximately 2 Å away from the

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surface atom cores, H2 diffraction experiments sample the PES in a region difficult to access with other experimental techniques. In this review, we summarize H2 diffraction studies on single-crystal metal surfaces. The emphasis will be on work where experimental data have been compared to 6D state-of-the-art quantum dynamics calculations. In order to make our presentation accessible to a broader readership, we give in Sections 2 and 3 a brief description of the main experimental requirements for high-resolution H2 diffraction studies, as well as of the basic concepts relevant to the interpretation of the experimental results. The main trends observed in experiment are summarized in Section 4, which also presents several case studies. An outlook and some concluding remarks are given in Section 5. 1.1. Historical overview In the early 1930s, experiments performed by Stern in Hamburg provided the first successful observation of He and H2 diffraction from single crystal surfaces, demonstrating the wave nature of atoms and molecules [176,104,63]. They scattered H2 and He thermal energy beams from LiF, NaCl and NaF(1 0 0)-surfaces, and measured angular scattering distributions which were in full agreement with the de Broglie relation [63,64]. Later on, experiments by Frisch and Stern [75] led to the discovery of selective adsorption, which allows the incident molecules to become trapped in the particle-surface potential well [119]. These and other major experimental breakthroughs, like the accurate measurement of the magnetic moment of the proton, were performed during Sterns very productive Hamburg years, which ended abruptly in 1933 with Sterns forced emigration [170]. In spite of these early reports, the development of He-atom scattering as a surface investigation technique has been extremely slow for decades. The main reason was the lack of adequate He-beam sources. The effusive (Knudsen) cells used for many years produce low intensity beams with a Maxwell velocity distribution. The situation changed dramatically in the 1970s with the development of nozzle-beams, which allowed improvements in both intensity and monochromaticity by orders of magnitude. An excellent review on the historical development of molecular beams since Knudsen first studies is given by Comsa [38]. These developments, combined with the progress made in ultrahigh vacuum techniques, lead eventually to a broad range of applications of elastic and inelastic Hescattering in surface science [60,177,178,146,98]. This situation brought about new studies using hydrogen beams. The first H2 and D2 diffraction experiments from metal surfaces were reported in the early 1980s. First order diffraction peaks were observed from several low-index Miller surfaces, like Ag(1 1 1) [97], Cu(1 0 0) and Cu(1 1 0) [115], Ni(1 1 1) [93], Ni(1 1 0) [160] and Ag(1 1 0) [130,32]. From all of them, it is worth emphasizing the study performed in the group of Ertl on H2/Ni(1 1 0), since it deals with a non-activated system, i.e. a system which exhibits a high dissociative sticking probability even for thermal H2 beams. This puts high demands on the H2 diffraction experiments, which must be carried out at high surface temperatures (400 K), as it was actually done by Robota et al. [160]. For this reason, this study can be considered in some sense as the forerunner of current H2 diffraction experiments. The first observation of rotationally inelastic diffraction (RID) peaks came from scattering experiments of H2 from LiF(1 0 0) [20,21], and soon after for scattering of H2, HD and D2 from MgO(1 0 0) [162,163] as well as for D2 scattering from NaF(1 0 0) [24]. The coupling of RID to phonon excitations in the solid has been studied by [1] for the scattering of H2 and D2 from LiF(1 0 0). With respect to metal surfaces, RID peaks have been clearly resolved for the strongly anisotropic HD molecules scattered from Pt(1 1 1) [39], W(1 1 0) [154], Ni(1 0 0) [9], Cu(1 0 0) and Pd(1 1 1) [171], and Cu(1 0 0) [77]. In the case of H2 and D2 scattering from low-index metal surfaces, RID peaks are more difficult to observe (due to the low corrugation of the PES), and manifest themselves usually as weak shoulders in the flanks of the elastic diffraction peaks. Examples of this are given by H2 scattering experiments on Ag(1 1 1) [22], Ni(1 1 1) [93], Ag(1 1 0) [130], and for D2/Cu(1 0 0) [116]. As far as we know, the first experiments showing clearly resolved RID peaks have been performed in the group of Sibener for H2 scattering from Ag(1 1 1) [193,186]. It was not until almost a decade later that elastic diffraction as well as RID peaks were investigated in high-resolution D2 scattering experiments performed over a wide range of incident energies on Rh(1 1 0) [44], Ni(1 1 0) [11] and Cu(1 0 0) [13].

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2. Experimental 2.1. General requirements H2 diffraction experiments from reactive surfaces are very demanding, because they require a high sensitivity (to detect reflectivities that are of the order of a few percent) combined with the possibility of recording both in-plane and out-of-plane spectra, since pronounced out-of-plane scattering occurs for these systems. The requirement of a high sensitivity is a consequence of the need to keep the solid surface clean during the measurements. To illustrate this issue, let us take a look at the thermal desorption data for H2/Ru(0 0 0 1) shown in Fig. 2. As we can see, hydrogen atoms desorb from this surface in the temperature range between 150 and 450 K. A simple calculation will convince us that H2 diffraction experiments from clean Ru(0 0 0 1) cannot be performed below 400 K. In effect, the incident flux onto the target is 1014 molecules cm2 s1, corresponding to a rate of 0.065 ML s1 (a coverage H = 1 ML corresponds to the Ru(0 0 0 1) site density, N = 1.57  1015 atoms cm2). Since H2 sticks on most transition metal surfaces with a very high probability (in the range 0.5–1), it is clear that a surface kept at 100 K will be saturated with adsorbed hydrogen within 20 s. Needless to say, the measurement of a H2 diffraction spectrum takes much longer, typically 10– 30 min, depending on the desired resolution. As a consequence, H2 diffraction experiments must be performed with the crystal temperature kept slightly above the highest temperature desorption tail of H2. Under such conditions, the steady-state hydrogen coverage can be estimated by equating the incident flux of 0.065 ML s1 with the hydrogen desorption rate rdes [36,78]:

  Edes rdes ¼ mdes exp  H2 N; kT s

ð2Þ

where H is the instantaneous coverage on the surface, k the Boltzmann constant, Edes the desorption energy, and mdes the pre-exponential factor. For H2/Ru(0 0 0 1), these values have been determined from thermal desorption experiments to be mdes = 3 cm2 s1 and Edes = 1.3 eV [74]. With these numbers, we expect (assuming the sticking coefficient to be one) a steady-state coverage of less than 0.02 ML at Ts = 500 K, which can be considered a clean surface for all practical purposes. This analysis is valid for most H2/transition metal systems, which exhibit thermal desorption curves very similar to that shown in Fig. 2. The fact that H2 diffraction experiments must be performed at surface temperatures of 450–500 K is the cause of the high sensitivity required in these measurements. In effect, it is well known that

Fig. 2. Thermal desorption spectra of different hydrogen coverages (up to saturation) on Ru(0 0 0 1), prepared by saturation at 170 K and partial desorption by heating. Used by permission from [108].

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diffraction intensities decrease exponentially with surface temperature, as described by the Debye– Waller model (see Section 3.1). This reduces the total diffraction signal to 1% of the incoming beam intensity for very reactive systems, like Pd(1 1 1) or Pd(1 1 0). It is also for this reason that the measurements are actually performed as close as possible to the high-temperature thermal desorption tail, or in other words, the reason why even higher surface temperatures are avoided. With respect to the use of H2 or D2 beams, a few considerations are in order. First of all, from the viewpoint of diffraction it is clear that larger intensities are expected for H2 beams. This is just a consequence of reducing the mass of the beams particle by a factor of two in the Debye–Waller factor (described in detail in Section 3.1). Very often this effect is so strong, that diffraction is observed with H2 but not with D2 beams; a good example is provided by the Pd(1 1 1) surface [69]. On the other hand, D2 beams are often preferred when the study of RID transitions is pursued. The reason for this must be found in the different rotational energy levels, which are reduced by a factor of two compared to those of H2, making the corresponding RID transition probabilities much larger at the same incidence energy. Translational to rotational energy transfer during the scattering process is also very efficient for HD. A theoretical study reported by Ramírez and Busnengo [149] shows that the primary reason for that is not the asymmetric mass distribution of HD, but the larger number of open channels available for rotational excitations. At this point, it is worth mentioning that the situation for H2 or D2 beams is quite different from the theory point of view, due to the different population of the rotational levels in the incident beam (see Section 2.3). Therefore, at incident energies in the range employed for diffraction studies (20–200 meV), calculations for rotational states ji = 0, 1, 2, 3 are mandatory for D2, whereas only the states ji = 0, 1 must be considered for H2 beams. Finally, a word on the quality of the samples used. H2 diffraction apparatuses offer the possibility of switching from He to H2 beams within a few minutes, so usually the sample quality is checked in situ by He-diffraction just before performing H2 diffraction measurements. The single-crystal surfaces employed in the experiments reviewed in Section 4 were cleaned in situ in UHV by ion sputtering and annealing. For further details on sample preparation see [69] for Pd(1 1 1), [66,67] for NiAl(1 1 0), [134] for Pt(1 1 1), and [4] for Pd(1 1 0).

2.2. Scattering geometries An excellent description of the experimental demands for atomic beam diffractive scattering from surfaces can be found in the two books edited by Scoles [168,169]. An overview of the use of molecular beams to study chemical dynamics at surfaces, with a focus on heavy particle scattering, has been reviewed by Kleyn [103]. Generally speaking, the apparatuses used for He and H2 diffraction experiments can be classified according to their scattering geometries in two different groups: (a) ‘‘Fixed angle’’ systems, in which the angle between incident and outgoing beams is fixed and (b) ‘‘Rotary detector’’ systems, which allow to rotate the detector about two axes independently of the incident conditions. With the latter, all diffraction intensities can be recorded for a given scattering geometry, allowing an easier comparison with calculations. The major disadvantage of this configuration is that it hardly allows for differential pumping of the detector, resulting in general in a smaller dynamical range of measured intensities as compared to the differentially pumped ‘‘fixed angle’’ systems used in time-of-flight experiments. In the Surface Science Laboratory at the Universidad Autónoma de Madrid there are available one of each kind of these machines. For a detailed description of both fixed angle and rotary detector systems the reader is referred to [7] and [136], respectively. In the following we provide a brief description of these two scattering geometries. A schematic representation of the two setups is shown in Fig. 3. In the first one, the detector can be rotated 200° in the scattering plane (defined by the incident beam direction and the normal to the surface) as well as ±15° from the scattering plane for a fixed angle of incidence. Another advantage of this setup is that it allows measurement of the direct incident beam intensity, making it possible very accurate determination of absolute diffraction probabilities. On the other hand, since the detector lies very close to the sample, the angular resolution is usually limited to 1.5°. Also, the background pressure at the detectors position is very high (the H2/D2 base pressure in the scattering chamber is

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Fig. 3. Setups used in atomic and molecular beam diffraction experiments. In the one shown on top, the detector can be rotated in and out of the scattering plane while keeping the angle of incidence fixed. In the time-of-flight set-up (bottom), the angle of incidence is changed continuously during a measurement by rotating the sample. These two setups are available in the authors laboratory at the Universidad Autónoma de Madrid. Reproduced by permission from [71].

109 mbar), which limits the dynamical range in this type of apparatus to 3  103 of the incoming beam intensity. Measurements of the angular locations of the diffraction peaks allow the determination of the dimensions of the surface unit cell as well as its orientation relative to the incoming beam. If we consider a two-dimensional, rectangular unit cell, defined by the vectors a1 and a2, and assuming that the incoming wavevector ki is parallel to a2, the following formulae for the beam (mn), located at hmn and /mn can be derived:

m a1   1 n ¼ sin hi þ k cos /mn a2

sin /mn ¼ k

ð3Þ

sin hmn

ð4Þ

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where hmn is measured in the scattering plane (which is spanned by the wave vector ki and the surface normal) and /mn is measured away from this plane (the so-called ‘‘out-of-plane’’ angle), as shown schematically in Fig. 3. The corresponding equations for arbitrary scattering conditions can be found in [60] (p. 74). The second apparatus is a high-resolution time-of-flight (TOF) spectrometer with a fixed angle of hSD = 105.4° between incident and outgoing beam. Angular distributions are measured by rotating the crystal about an axis normal to the plane of the incident and outgoing beams. This means that the incidence angle hi is continuously varied during a measurement, and this angle is related to the corresponding final angle hf through hi + hf = hSD. After scattering at the surface, the particles are detected by a home-made magnetic mass spectrometer detector located at 1.698 m from the target surface. This allows detection of diffraction intensities as low as 104 of the incoming beam, while the angular resolution is determined by the detector acceptance angle, which is about 0.2°. 2.3. Molecular hydrogen beams A monochromatic H2 beam can be produced through an adiabatic expansion into vacuum, i.e. using the same technique as for He beams [132]. However, since the pumping systems are less efficient to evacuate H2 than He, the inlet pressures applied in the nozzle are never as high as in the case of He beams, for which pressure values up to 150 bar are often employed. In the case of H2 beams, the working pressures are usually below 70 bar, and the purity of H2 used is >99.9990%. The beam energy can be varied in the range of 20–200 meV by changing the nozzle temperature between 100 and 1000 K. The velocity spread of the incident beam is typically <1% for He and  8% for H2 and D2 beams. Detection of H2 beams is commonly carried out with a quadrupole mass spectrometer, although time-of-flight systems very often use a homemade detector, which presents a much higher sensitivity [164]. In rotary detector apparatuses, and owing to the continuous background produced by the incoming H2/D2 beams, the signal is usually recovered from the background by means of a lock-in system. It is well known that molecular hydrogen occurs in two forms [72]: one with its two proton spins aligned parallel, i.e. the total nuclear spin quantum number of the molecule is I = 1 (triplet state), and the other with its two proton spins aligned antiparallel, i.e. the total nuclear spin quantum number of the molecule is I = 0 (singlet state). The triplet state is called ortho hydrogen, o-H2, and the singlet state is termed para hydrogen, p-H2. Therefore, normal hydrogen is a well-defined mixture of p-H2 and o-H2 in the ratio 1:3. When hydrogen is cooled, only p-H2 is stable. However, the spontaneous conversion of o-H2 into pH2 by flipping of a nuclear spin occurs very slowly, over a time of years [88]. Ortho–para conversion can be accelerated using paramagnetic salts as catalysts, so that pure p-H2 can be prepared at low temperatures [129]. For normal D2, since the spin quantum number of each nuclei is I = 1, we get a ground state mixture of p-D2 (J odd) and o-D2 (J even) in the ratio 1:2 [172]. Between these two types of H2/D2, transitions are rather strictly forbidden. This means that only transitions within the term system with even J and within that with odd J are possible. As a consequence, only states with J = 1, 3, 5, . . . are allowed for o-H2, whereas only states with rotational quantum number J = 0, 2, 4, 6, . . . are allowed for pH2 [88]. High resolution H2 diffraction experiments performed in the group of Toennies showed that also large differences are observed in the diffraction intensities of normal H2 and p-H2 from the LiF(1 0 0) surface [14]. Calculations performed by Kroes and co-workers demonstrated unambiguously that the observed differences are a consequence of the electrostatic coupling between the quadrupole moment of H2 and the ionic lattice, which, in a zeroth order approximation, is averaged out in the case of p-H2 [110]. The calculations also show that, in general, the propensity rule Dmj = 0 does not hold for rotational scattering of H2 from surfaces [140]. This result suggests a technique for producing rotationally polarized H2 beams, by selecting the appropriate diffraction channel in the scattering of unpolarized H2 beams [140]. As already mentioned, in the diffraction process of H2 and D2 rotationally inelastic transitions can occur, making the determination of diffraction probabilities for molecules more complicated than for atoms. One main complication is the need to determine the occupation probability of the rotational

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levels in the incident beam. Although this can be measured very precisely by Raman spectroscopy [133], and by using the REMPI technique for both incident and scattered beams [183,184], most of the molecular beam diffraction systems available do not offer this possibility. Therefore, an alternative approach must be found. A number of previous investigations [148,101,73,132] have shown that the rotational populations of the lowest rotational states of highly expanded supersonic molecular beams follow a nearly Boltzmann distribution which can be characterized by an effective rotational temperature TR, which is somewhat smaller than the source temperature T0. The relation between TR and T0 is complex and depends not only on the parameter P0d (where P0 is the stagnation pressure and d is the nozzle orifice diameter) but also on T0 [128]. To determine rotational temperatures in the incident beam over the very wide range of nozzle temperatures used in diffraction experiments, the procedure described by Faubel et al. [73] is usually applied. Rotational populations of supersonic molecular beams can be interpolated by a logarithmically linearized correlation function, which, for H2 expansion at T0 = 293 K, is given by:

logðT R =T 0 Þ ¼ 0:21  logðP0 dÞ þ 0:043;

ð5Þ

and, for D2, by:

logðT R =T 0 Þ ¼ 0:40  logðP0 dÞ þ 0:16;

ð6Þ

when P0d is expressed in units of Torr cm. Eqs. (5) and (6) scale with the inverse Knudsen number (Kn)1 = P0d/T0 for nozzle temperatures different from room temperature [73]. Once the rotational temperature TR is determined, rotational populations nj can be obtained from a Boltzmann distribution. Table 1 gives the fractional populations of rotational levels for n-D2 beams. For a n-H2 beam, the corresponding population fractions (for stagnation condition P0d = 38 Torr cm) in states with angular momentum quantum number ji = 0, 1, 2, 3 are 0.21, 0.74, 0.04 and 0.01 at Ei = 75 meV (corresponding to T0 = 300 K), and 0.11, 0.61, 0.14 and 0.13 at Ei = 150 meV (T0 = 600 K), respectively. Because of the large spacing between the vibrational levels of the H2 molecule, the occupation of the excited vibrational states is always less than 1% for incident energies Ei 6250 meV and can be safely ignored in the analysis of the scattering data [10]. 3. Determination of diffraction probabilities 3.1. Correction for Debye–Waller attenuation The theoretical methods used to obtain diffraction intensities assume that the atoms of the lattice are at rest. However, both zero-point motion and thermal vibrations of the surface atoms lead to inelastic scattering of the incoming molecules. The main observable effect of this is a thermal attenuation of the diffraction intensities. This problem is well known in neutron and X-ray diffraction from crystals, where the Debye–Waller factor relates the intensity I(T) of a diffraction peak with the intensity I0 from a lattice at rest by [127,87]:

IðTÞ ¼ I0 e2WðTÞ ;

ð7Þ

Table 1 Fractional populations (in %) of the rotational levels ji for n-D2 beams for stagnation condition P0d = 41.25 Torr cm and source temperature T0, estimated from effective beam rotational temperatures TR as reported by [73]. Ei (meV)

T0 (K)

TR (K)

ji = 0

ji = 1

ji = 2

ji = 3

85 95 108 120 133 140 149

300 334 380 422 468 492 525

99 115 138 159 184 198 216

48.7 43.5 37.7 33.4 29.8 28.3 26.5

32.3 31.6 30.2 28.8 27.1 26.3 25.2

18.0 23.1 29.0 33.2 36.8 38.4 40.2

1.0 1.8 3.1 4.6 6.2 7.0 8.1

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where exp[2W(T)]is the Debye–Waller factor. This expression has been shown to describe many He and H2 diffraction experiments rather well [65]. For the scattering of thermal atoms from surfaces, the effect of the attractive well near the surface is usually taken into consideration by the so-called ‘Beeby correction’ [8]. For the specular beam, W(T) can be easily expressed as a function of the incident beam energy Ei and the angle of incidence hi [65]:

WðTÞ ¼

12mðEi cos2 hi þ DÞT MkB H2D

;

ð8Þ

where M is the mass of a surface atom, m is the mass of the incoming particle, kB is the Boltzmann constant, HD is the surface Debye temperature, and D is the potential well depth. Eq. (8) indicates that the strength of the diffraction intensities should be highest for grazing incidence, low incidence energy and low surface temperatures, in agreement with experiment. The usual procedure to experimentally determine the Debye–Waller attenuation consists of measuring the intensity of the specular peak at different incident conditions as a function of surface temperature. An example of such measurements obtained for the system H2/Ru(0 0 0 1) is shown in Fig. 4. A similar temperature dependence of the specular peak has been reported for the scattering of H2 and D2 from Ag(1 1 1) [193], as well as for D2/Ni(1 1 0) [11], D2/Cu(1 0 0) [13], H2/Pt(1 1 1) [39,134], and D2/ NiAl(1 1 0) [67,7]. Thus, before making a comparison with calculations, diffraction intensities are usually extrapolated to a 0 K surface assuming a Debye–Waller attenuation model. One might wonder wether the same attenuation with surface temperature is obeyed by RID peaks. The answer to this question is provided by the data for D2/NiAl(1 1 0) shown in Fig. 5. We see that diffraction intensities for different elastic and RID channels are equally attenuated with increasing surface temperature. This behavior also extends for the whole incident energy range explored, between 20 and 150 meV, where no significant broadening or shifting of the peaks was measured. Since a similar slope is obtained for both diffraction and RID peaks, the Debye–Waller correction can be avoided in cases where the peaks intensities are normalized within the same angular distribution, like for fixed-angle setups. If this is done, it can be estimated that the maximum error made for the intensity of peaks in the [15°; +15°] range around the specular peak is lower than 20% [7]. This observation is very important for practical purposes, since suggests that, to a good approximation, phonons can be neglected in theoretical models of diffraction from metal surfaces. We close this Section by mentioning that rotational excitation experiments performed in the group of Sitz provided clear evidence for the existence of important H2-surface energy exchange effects. For H2/Pd(1 1 1), Sitz and co-workers showed that state-to-state rotational excitation probabilities are

 0 symmetry Fig. 4. Temperature dependence of the (0 0) peak for H2/Ru(0 0 0 1) measured for incidence along the ½1 0 1 direction, for different incidence angles and energies. Solid lines show the temperature dependence obtained using Eqs. (7) and (8), with D = 60 ± 5 meV and HD = 473 ± 2 K. Reproduced by permission from [135].

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 0 Fig. 5. Surface temperature dependence of the intensities for the most intense RID channels in D2/NiAl(1 1 0) along the ½1 1 azimuth. The incidence energy was Ei = 86 meV. Reproduced by permission from [7].

strongly dependent on the surface temperature, and rotational excitation was observed even when the incident H2 translational energy was lower than the energy level spacing between the initial and the final rotational states [183]. Obviously, in this case part of the excitation energy must come from the surface and not only from energy transfer between translational and rotational molecular degrees of freedom. These results could be accounted for by classical dynamics simulations of the rotational excitation, including the possibility of energy exchange with surface phonons through a 3D surface oscillator model [27]. 3.2. Absolute and normalized diffraction probabilities The diffraction peak intensities are usually determined by fitting Gaussian profiles to the peaks of angular distributions measured for a given energy, Ei, and by averaging their integrated intensity over several measurements. The absolute transition probability from an initial state ji to a final rotational state jf coupled to the reciprocal lattice vector Gmn (corresponding to the diffraction beam (mn)) can be calculated from the following expression [59]:

Iðmn : ji jf Þ Pðmn : ji jf Þ ¼ Ii nðji ÞgðHi Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ei þ DErot 2WðT S Þ ; e Ei

ð9Þ

where I(mn:jijf) is the integrated intensity for RID peaks measured at a surface temperature TS, Ii is the intensity of the incident beam, n(ji) is the fractional population of the rotational level ji in the incident beam and the geometrical factor g(Hi) represents the area of the sample illuminated by the incident beam with respect to the total incident intensity. The square root factor corrects for the detector efficiency dependence on the final translational energy (Ei + DErot) of the scattered molecules. The exponential factor takes into account the thermal attenuation of intensities expected from the Debye– Waller model, which is determined experimentally in the form discussed in the previous section. Note that Eq. (9) gives the experimental probability averaged over the magnetic quantum number mj, which cannot be measured in the experiment. When the incident beam intensity can be measured, like for instance in apparatuses with rotary detector geometry, the elastic diffraction probabilities are obtained from Eq. (9) averaging over the rotational state distribution in the incident beam. Thus, absolute elastic diffraction probabilities are given by:

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PðmnÞ ¼

X

Pðmn : ji ji Þ  nðji Þ:

ð10Þ

ji

However, most high-resolution measurements are obtained using fixed-angle geometry machines, which do not allow determination of incident beam intensities. In these cases, the probabilities are expressed as a normalized probability ratio, relative to a diffraction peak appearing in the same angular distribution. The same procedure is applied for RID channels. 3.3. Theoretical methods It is beyond the aim of this review to give a detailed description of the theoretical methods employed. However, for the sake of completeness we include a brief description of the main theoretical approaches employed. The scattering dynamics of H2 on metal surfaces is usually described in terms of a single 6D molecule-surface PES obtained from state-of-the-art electronic structure calculations [81,109]. These calculations are performed within the Born–Oppenheimer approximation, and both electronic and phononic excitations of the surface are not considered. Inclusion of such inelastic effects in a quantum calculation modelling all six degrees of freedom of the problem is computationally too expensive, and currently not possible. A few attempts have been made to include inelastic phonon processes, both in the context of quantum wave-packet [43,58,181,182] and classical calculations [180,27,145]. In the case of H2 diffraction, as already discussed in Section 3.1, increasing the surface temperature leads to smaller diffraction intensities, but maintaining the relative importance of the diffraction intensities. This justifies to a large extent the frozen surface approximation employed in the dynamical calculations. The approximation of neglecting electron–hole pair excitations can be justified on the basis of recent experimental results for H2/Pt(1 1 1) [134], which are described in more detail in Section 4.3. A description of the theoretical treatments of dynamics at metal surfaces which incorporate the effects of energy dissipation by both phonon and electronic mechanisms can be found in the review by Tully [179]. Experimental evidence for electronically nonadiabatic molecule-surface interactions has been recently reviewed by Wodtke and co-workers [190,191]. The 6D PES is determined by DFT calculations performed within the generalized gradient approximation (GGA). In applying the GGA, the two most widely used exchange-correlation functionals are the PW91 and the RPBE functionals, these functionals typically providing PESs of semi-quantitative accuracy. Very recently, a semi-empirical version of DFT was demonstrated to provide chemical accuracy in the description of reactive and rotationally inelastic scattering of H2 from Cu(1 1 1) [53]. One serious problem that arises in the use of ab initio PESs is that, in general, to solve the Schrödinger equation one needs a continuous description of the potential. The DFT calculations, however, just provide total energies for discrete configurations of the nuclei. This problem is solved by applying an interpolation method able to reproduce accurately the whole PES from a limited set of DFT data (typically, a few thousand points). The interpolation methods usually employed are the corrugation reducing procedure (CRP) [26] and the modified Shepard (MS) method [41,42]. A detailed comparison of the performance of these two methods to interpolate a PW91 DFT data set for H2/Cu(1 1 1) has been recently reported by Díaz et al. [54]. The CRP has been shown to provide a precision better than 30 meV in the dynamically relevant regions for several H2-metal systems [29,138,158]. Another successful interpolation methods include a neural network approach [121,122], which has been recently employed to study the dissociation of O2 on Al(1 1 1) [33], as well as the use of suitable analytical functions to get a continuous representation of the PES [85]. A combination of the CRP with neural networks has been also reported [123]. The PESs employed in the calculations presented in Section 4 have been interpolated using the CRP method. For further details, see [29] for H2/Pd(1 1 1), [138] for H2/Pt(1 1 1), [158] for H2/NiAl(1 1 0), and [56] for H2/Pd(1 1 0). As an example, we show in Fig. 6 two dimensional cuts of the full 6D PES of H2/Pd(1 1 1), for different values of H, which measures the orientation of the molecules axis with respect to the surface normal. This figure nicely illustrates one of the most intriguing predictions of 6D DFT calculations, namely the strong dependence of the dissociation barrier height on H. Whereas no barrier is presented for molecules impinging at H = 90 deg, we see that a quite large barrier is seen by molecules with

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Fig. 6. 2D cuts of the full 6D PES of H2/Pd(1 1 1) with different values of H. Full lines correspond to positive values of the energy and dashed lines to negative values referred to the isolated H2 potential energy minimum. Contour spacing is 100 meV. Adapted from [26].

H < 45 deg. The molecular axis orientation has a dramatic effect, and low activation barriers are only met over a small range of H values around H = 90 deg [26]. In view of the large initial sticking probability of H2 on Pd(1 1 1) observed experimentally (S0  1), these 2D plots could appear counter-intuitive or even non-realistic. In effect, it looks like the very reactive Pd(1 1 1) surface would behave as a noble metal surface for molecules impinging with a low H value. However, owing to the excellent agreement obtained with a wealth of molecular beam, associative desorption, state-resolved scattering and diffraction experiments, we can trust the microscopic description of the PES given by these calculations. A nice overview of the current level of agreement between first principles theory with detailed dynamical experimental results for H2 and N2 dissociative adsorption on single crystal metal surfaces is given by Luntz [125]. Once the 6D PES is computed, there are two ways to perform the 6D quantum dynamics simulations: by solving the time-independent or the time-dependent Schrödinger equation. Both approaches are in principle equivalent and should produce the same results. The first approach applies a timeindependent coupled channel formalism, which is based in the use of reaction path coordinates [84,81,109]. In the second approach, a time-dependent wave packet (TDWP) method [107] is used [111]. The quantum dynamics calculations presented in Section 4 have been carried out using the TDWP method. For a detailed discussion of the pros and cons of these two methods, the reader is referred to the reviews by Gross [81] and Kroes [109]. We mention for completeness that Díaz et al. [52] made use of a discretization method to estimate H2 diffraction intensities using six-dimensional classical trajectory calculations. The method has been applied to the D2/NiAl(1 1 0) and H2/Pd(1 1 1) systems, (which are models for activated and non-activated dissociative chemisorption, respectively) using realistic potential energy surfaces obtained from

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Fig. 7. In-plane and out-of-plane diffraction spectra for D2/NiAl(1 1 0) (left) and H2/Pd(1 1 1) (right). Black curves: experiment; red lines: 6D quantum dynamical calculation. Theoretical calculations have been convoluted with a Gaussian function of width r = 0.7° to take into account the limited angular resolution of the measurements. Intensities have been normalized to the height of the specular peak obtained in the quantum calculations [69]. (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)

DFT–GGA calculations. Comparisons with experimental results and 6D quantum dynamical calculations showed that, in general, the method is able to predict the relative intensity of the most important diffraction peaks. Fig. 7 shows a comparison between experiment and 6D quantum calculations for both Pd(1 1 1) and NiAl(1 1 0) surfaces. The very good agreement obtained for these systems suggests that quantum dynamics calculations treating all molecular degrees of freedom can accurately predict diffraction patterns for hydrogen scattering from reactive metal surfaces. Inclusion of all molecular degrees of freedom is essential to account for the competition between dissociative and non dissociative channels. This supports the use of 6D DFT to properly describe the H2-metal surface interaction even in regions far away from the surface. Finally, it shows that out-of-plane diffraction measurements are crucial to test the details of the PES in a wide region of space (see also Section 4.3). In the quest for building better functionals for describing the H2-metal surface system, the issue of including the Van der Waals interaction is often brought up [135]. As it is well know, the generalised gradient approximations (GGAs) used to model the molecule-surface interaction with DFT do not provide a proper description of the Van der Waals dispersion interaction [83,111]. Here we would like to make a few comments on this issue, which are based on the most recent experimental results. First of all, accounting for this interaction turned out to be unimportant in the detailed comparison between theory and experiment for diffractive scattering and dissociative chemisorption for the H2 + Pt(1 1 1) system, as discussed in detail in Section 4.3 [134]. The presence of the Van der Waals well may affect the scattering in three different ways. First, it affects the thermal attenuation of the measured diffraction intensities in the way described by the Debye–Waller model. This is taken into account in the comparison between theory and experiment through the Beeby correction (see Eq. 8 above). Secondly, the Van der Waals well may have a large influence on the scattering from a 0 K or finite temperature

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surface at very low collision energies due to selective adsorption resonances [95,165]. However, this phenomenon is unimportant for incidence energies above 70 meV and normal collision energies exceeding 30 meV, which is usually the case in the majority of H2 diffraction experiments reported. Finally, the neglect of the Van der Waals attractive interaction could lead to a lower corrugation amplitude, which can in turn lead to less first order diffraction relative to specular scattering. However, the results for H2 + Pt(1 1 1) [134], and previous less detailed results for H2 + Pd(1 1 1) [69], suggest that for reactive surfaces the Van der Waals interaction is either unimportant compared to chemical interactions, or the DFT overestimates the chemical interaction somewhat, thereby compensating for the lack of the Van der Waals attractive interaction in DFT at the GGA level. To investigate this further, new DFT functionals are needed that incorporate the Van der Waals interaction in a seamless way. For a more detailed account of the above arguments, the reader is referred to the supporting online material provided by Nieto et al. [134]. 4. Results 4.1. General trends The diffraction of H2 and D2 molecular beams from surfaces is in principle quite similar to He diffraction [98,65], the two major differences being: (i) the possibility of rotational-state transitions in the case of molecular scattering and (ii) the fact that diffraction competes with the reactivity channel. Concerning (i), rotationally inelastic diffraction (RID) peaks can be observed in the form of additional diffraction peaks in the angular distributions. In this process, the incident molecules convert part of their translational energy into excitation of a rotational quantum level when colliding with the surface. The position of RID peaks within an angular distribution can be obtained by combining the Bragg condition for surface diffraction with conservation of energy:

DK ¼ Kf  Ki ¼ Gmn Ef  Ei ¼ DErot :

ð11Þ

Here, Kf and Ki are the parallel components of the outgoing and incident wave vectors, respectively, Gmn is a surface reciprocal lattice vector, Ef and Ei the final and incident beam energies and DErot is the rotational transition energy. For H2 (D2) this energy is jDErotj = 44.6 (22.2) meV for the lowest transitions j = 0  2, and jDErotj = 74.3 (36.88) meV for j = 1  3 transitions [130]. In general, the diffraction intensities measured for H2 are larger than those for He, and a twodimensional corrugation is detected, even on surfaces which look quasi one dimensional with Hediffraction, like the fcc(1 1 0) ones. The large qualitative difference in the behavior of H2 as compared to He is a consequence of its larger polarizability, which gives rise to a stronger attractive interaction [120]. As a result, larger potential well depths D are usually measured by H2 beams as compared to He beams in selective adsorption resonances experiments. Typical D values reported for H2 beams are in the range 30–40 meV [192,34,2,3], compared with 5–10 meV for He beams [65]. Evidence for more pronounced diffraction intensities when using H2 beams was already realized in the first comparisons of He and H2 diffraction data reported, for instance on Ag(1 1 1) [97], Cu(1 0 0) and Cu(1 1 0) [115], Ni(1 1 1) [93], Ni(1 1 0) [160], MgO(1 0 0) [106], as well as by the more recent results on D2 diffraction from Rh(1 1 0) [44], Ni(1 1 0) and Cu(1 0 0) [11,13], and NiAl(1 1 0) [67]. Fig. 8 shows experimental H2/D2 diffraction spectra measured from NiAl(1 1 0), Pt(1 1 1), and Pd(1 1 1) under otherwise similar incidence conditions [67,134,69,50]. These data, obtained using a rotary detector setup, illustrate a very significant result: out-of-plane diffraction is important, and its importance increases with increasing surface reactivity, i.e. when going from NiAl(1 1 0) and Pt(1 1 1) to Pd(1 1 1). This observation seems to be a general trend for H2 diffraction experiments, and has been confirmed also by measurements in other systems, like H2/Ru(0 0 0 1), which is discussed in more detail in Section 4.4. As we can see in Fig. 8, for NiAl(1 1 0), RID peaks corresponding to the 0 ? 2 and 2 ? 0 transitions are observed besides first order in-plane and out-of-plane (/ = 9°) diffraction peaks. However, outof-plane peaks are less intense than in-plane ones for this system, which has a sticking probability

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Fig. 8. Comparison of in-plane and out-of-plane H2/D2 diffraction spectra for NiAl(1 1 0), Pt(1 1 1) and Pd(1 1 1) obtained under similar incident conditions. See text for further details. Reproduced by permission from [71].

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of zero at Ei=100 meV (see data of Fig. 9). Very different is the situation for the more reactive Pt(1 1 1) surface, which exhibits clear out-of-plane diffraction peaks (/ = 7° and / = 14°) whereas in-plane ones are not observed. For other incidence conditions, small in-plane peaks were observed, but still the general trend is that out-of-plane diffraction is more important than in-plane diffraction [134]. If we consider the much more reactive Pd(1 1 1) surface, a remarkable feature of these spectra is the appearance of pronounced out-of-plane diffraction (/ = 15°), while no diffraction peaks are observed in the scattering plane [70]. Indeed, the intensity of the (0 1) peak is almost 30% of the specular one at Ei=100 meV. These results show clearly that out-of-plane diffraction is much more important than previously assumed in most H2 diffraction experiments. At this point, it is worth emphasizing again that the first evidence showing that out-of-plane diffraction is more efficient than in-plane diffraction came actually from H2 diffraction calculations from Pd(100) [82] and Pt(1 1 1) [143,144]. This also illustrates very nicely how powerful are state-of-the-art six-dimensional quantum calculations, which were able to predict a very important effect never observed experimentally. In the case of the H2/Pt(1 1 1) system, these calculations clarified a contradiction presented by molecular beam experiments on sticking of H2 and D2 [126] and rotationally inelastic diffraction of HD [39] from Pt(1 1 1). The latter experiment showed a small in-plane diffraction intensity, suggesting that the corrugation of the HD/Pt(1 1 1) PES was small. However, the experiments by Luntz et al. showed the sticking to depend on the initial momentum of D2 parallel to the surface, which indicates a relatively large corrugation [126]. These contradictory conclusions were conciliated in the light of the calculations performed by Pijper et al. [143,144], which showed that small in-plane diffraction peaks cannot be considered a definite proof of a small corrugation, and put in evidence the need of measuring also out-of-plane diffraction to asses the true corrugation of the H2-surface interaction potential [102]. The relative strength of the ratio of the out-of-plane/in-plane diffraction intensities depends on impact energy and incidence angle. This is the consequence of a dynamical effect associated with grazing incidence and not of a purely static one such as surface corrugation (as it is for instance the case in He-diffraction, [65]. A model has been proposed by Salin, based on the periodicity of the potential along the incidence direction. According to this model, momentum change along the incidence direction is of second order, while momentum change along the transverse direction is of first order.

Fig. 9. Left: Experimental (symbols) and classical calculations (solid lines) of the H2 dissociation probability as a function of incident beam energy on NiAl(1 1 0), Pt(1 1 1), Pd(1 1 0) and Pd(1 1 1). Right: Closest approach distance for H2 molecules scattered off these surfaces at normal incidence and Ei ¼ 100 meV. See text for further details. Reproduced by permission from [71].

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This implies that, along the incidence direction, any increase (decrease) of the molecule linear momentum is compensated by a decrease (increase), while the variation of the transverse linear momentum is cumulative [70]. This model demonstrates that out-of-plane diffraction should dominate in the scattering of particles under a grazing angle of incidence. It also explains very nicely the recent observation of pronounced out-of-plane diffraction in the grazing scattering of fast atoms (incident energies up to 25 keV) and H2/D2 molecules from different surfaces [161,167,166]. Finally, we mention that the existence of pronounced out-of-plane diffraction could be also reproduced by recent model calculations of diffraction intensities performed using a three-dimensional soft potential [136]. These simulations show that out-of-plane diffraction should dominate already for angles of incidence above 50°, as it is the case of the H2/Pd(1 1 1) diffraction data shown on the bottom of Fig. 8. According to these calculations, pronounced out-of-plane intensities are also expected in the case of He-diffraction from an hexagonal lattice [136]. As already mentioned, the counterpart of molecules undergoing diffraction are those following dissociative adsorption on the surface (see Fig. 1). We show for completeness in Fig. 9 the dissociative sticking probability data of H2 on NiAl(1 1 0), Pt(1 1 1), Pd(1 1 0) and Pd(1 1 1) as a function of incident beam energy. Symbols correspond to supersonic molecular beam experiments [17,18,126] and solid lines are the results of classical trajectory and quantum dynamics calculations [158,142,29]. We see that, for the four surfaces considered, classical results reproduce quite well the experimental trends. This strongly supports the use of the present theoretical approach (i.e. the Born–Oppenheimer approximation + DFT/GGA + classical molecular dynamics) to investigate adsorption mechanisms and their connection with experiments. Fig. 9 clearly shows an increasing reactivity when going from NiAl(1 1 0) to Pt(1 1 1), Pd(1 1 0) and Pd(1 1 1). On NiAl(1 1 0) and Pt(1 1 1), dissociative adsorption is an activated process, whereas it is nonactivated on Pd(1 1 1) and Pd(1 1 0). For H2/NiAl(1 1 0) and H2/Pt(1 1 1), the classical results present an energy threshold for the dissociation probability, which lie close to the minimum activation barrier obtained from DFT calculations (300 meV and 50 meV, respectively). Above these thresholds, the dissociation probability increases monotonously with Ei. This behavior is characteristic of systems on which a single direct mechanism dominates adsorption. Classical calculations show that this is the case for both Pt(1 1 1) and NiAl(1 1 0), on which molecules approach the surface and directly dissociate or are scattered back to the vacuum after a single rebound on the surface [158]. For H2/ Pd(1 1 1) and H2/Pd(1 1 0), in contrast, such a direct dissociation mechanism only dominates for energies above 100 meV. At low energies, an indirect mechanism called dynamic trapping dominates [29,56]. In this case, the molecules remain trapped during a long time due to energy exchange from motion normal to the surface toward other degrees of freedom, which prevents the molecules from escaping the surface attraction. Therefore, dynamic trapping enhances dissociative adsorption at very low energies, but its role decreases with increasing incident energy. This leads to the initial decrease of the sticking probability as a function of Ei observed for H2/Pd(1 1 1) [40,28] and H2/Pd(1 1 0). [56,31]. In the latter case, dynamic trapping has a dramatic effect on diffraction, as it will be discussed in more detail in Section 4.2. We close this Section showing in Fig. 9 the distribution of the classical turning points for H2 scattered off NiAl(1 1 0), Pt(1 1 1), Pd(1 1 1) and Pd(1 1 0). The results correspond to classical trajectory calculations performed at Ei = 100 meV and normal incidence. As we can see, a clear trend is observed in which H2 molecules come closer to the surface with increasing reactivity. Let us consider for instance H2/NiAl(1 1 0), i.e. the least reactive system. Here the molecules are reflected far from the surface, between Z = 2.5 Å and 2.8 Å. The corresponding PES presents a repulsive behavior in the entrance channel, and molecules impinging the surface with a perpendicular energy of 100 meV find a slightly corrugated hard-wall-like potential in that region [157]. As a consequence, specular reflection is dominant on this surface (similar to the case of He-scattering), and diffraction is relatively small. For H2/ Pt(1 1 1), the closest approach distance of reflected molecules is smaller than in the case of NiAl(1 1 0) (between 2.25 Å and 2.5 Å) and is slightly larger than Zb  2.1 Å which corresponds to the minimum activation barrier for dissociation of 50 meV [137,138]. Molecules reflected closer to the surface will sample a more corrugated region of the PES and, therefore, diffraction peaks become more important as compared to the specular one, in agreement with the experimental data shown in Fig. 8. This effect is even more clear in the case of Pd surfaces. On Pd(1 1 1), for instance, H2 molecules are reflected

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between 1.5 Å and 2 Å where the PES corrugation is very strong. As a consequence, the specular channel is strongly suppressed, and first-order diffraction becomes relatively more important. It is worth mentioning that there might be exceptions to this general behavior, depending on the details of the PES involved. A good example is provided by the H2/W(1 0 0) system, which presents a dissociative sticking curve similar to that of Pd(1 1 0). In spite of being a quite reactive system, classical calculations performed by Busnengo and Martínez [25] show that, at Ei=100 meV and normal incidence, H2 molecules are reflected relatively far from the surface, at 2.5 Å. The topology of the PES is such that all molecules reaching a critical distance Z = 2.25 Å eventually dissociate. A similar situation is presented by the Cu(1 0 0) and Cu(1 1 1) surfaces, which have the lowest barrier for reaction rather close to the surface, at 1 Å, even though the activation barrier for dissociation is quite large (0.5–0.7 eV) [131,174,175]. Thus, for incidence energies of 0.4 eV, the H2 molecules can come very close to the surface before being diffracted. 4.2. H2/Pd(1 1 0): Evidence of dynamic trapping The interaction of H2 with Pd(1 1 0) is an excellent example of a system where both diffraction data and classical trajectory calculations are essential to understand the H2 scattering dynamics. As we can see in Fig. 10 (left panel), the sticking coefficients of H2 on Pd(1 1 1) and Pd(1 1 0) are very similar. However, the corresponding diffraction spectra look very different. Fig. 10 also shows an in-plane H2 diffraction spectrum recorded along the [001] azimuth of Pd(1 1 0) (red curve).2 The comparison with a typical H2 diffraction spectrum on Pd(1 1 1) (shown in Fig. 8c) is surprising: whereas intense specular reflection is observed from Pd(1 1 1), specular diffraction is completely suppressed from Pd(1 1 0). The same behavior was observed at different incident energies between 25 and 150 meV and angles of incidence from 40° to 70°. In all cases, out-of-plane spectra were also measured, since – as it was discussed in Section 4.1 – it is known that this channel is important for H2 diffraction. No evidence for H2 diffrac azimuth. tion from Pd(1 1 0) was found. Similar results have been obtained along the [1 0 1] The absence of H2 diffraction peaks in Pd(1 1 0) cannot be attributed to inelastic processes, since specular and first-order out-of-plane diffraction is clearly observed for H2/Pd(1 1 1) at the same surface temperature, Ts = 435 K (see spectra in Fig. 8c). Neither a bad quality Pd(1 1 0) surface could be the reason, since several diffraction peaks are clearly resolved with He diffraction (black spectrum in Fig. 10). These results are very surprising and demonstrate that H2 reflected peak intensities from Pd(1 1 0) must be, in the best case, below 3  103 of the incoming beam intensity, which is the resolution of the apparatus employed in the experiments. This limit could be reduced to less than 104 of the incoming beam intensity by performing the same experiment with a high-sensitivity, fixed-angle geometry TOF instrument [4]. It is worth emphasizing that, using a similar TOF machine, pronounced specular and first-order D2 diffraction was observed along the same azimuth from Rh(1 1 0) at Ts 400 K [44] and Ni(1 1 0) at Ts  700 K [11], which exhibit an even lower reflectivity (S0  0.9, [36]) than Pd(1 1 0). To shed some light on these surprising observations, classical trajectory calculations have been performed at the same initial conditions as in the current experiments [49,50]. The calculations made use of the ab initio six dimensional H2/Pd(1 1 0) PES reported by Di Cesare et al. [56] for a rigid surface. These simulations show that even a quite small difference in reactivity, like the one found between H2/Pd(1 1 1) and H2/Pd(1 1 0), can still entail very different behaviors in the dynamics of scattered molecules. At this point it is worth recalling the concept of dynamic trapping, already introduced in connection with Fig. 9. In the dynamic trapping mechanism, part of the normal incidence energy is transferred to other degrees of freedom, so that the molecule interacts repeatedly with the surface. In the classical calculations, dynamic trapping is characterized by a large number (n P 10) of rebounds. Fig. 11 shows the ratio of scattered molecules that have been temporarily trapped with respect to the total number of unreactive scattered molecules as a function of Ei (for normal incidence and H2(m = 0, J = 0)). On Pd(1 1 0), at low energies a large fraction of scattered molecules have been trapped, whereas on Pd(1 1 1) scattering is essentially a direct process. However, it does not mean that trapping does not take place on Pd(1 1 1). Both surfaces attract H2 molecules in the entrance channel 2

For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.

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Fig. 10. Left: Sticking coefficient curves of H2 on Pd(1 1 1) and Pd(1 1 0). Solid lines correspond to classical trajectory calculations [56], and symbols to experimental data points [152,18]. Right: In-plane angular distributions from Pd(1 1 0) along [0 0 1] recorded with H2 (red curve) and He beams (black curve). The latter has been shifted upwards for clarity. Reproduced by permission from [4] and [71].

(i.e. above 2 Å) and the variation of the PES with molecular orientation and position on the unit cell produces energy exchange between the different degrees of freedom that results in molecules becoming trapped. However, the larger number of energetically (and dynamically) accessible dissociation pathways makes that on Pd(1 1 1), almost all trapped molecules will dissociate, whereas on Pd(1 1 0) trapped molecules still have a non-negligible probability to be scattered back to vacuum. This results in a very different angular distribution of molecules scattered from Pd(1 1 1) and Pd(1 1 0) at low energies, as can be seen on the right panel of Fig. 11. For H2/Pd(1 1 0), a cosine-like distribution is observed, which is a consequence of the memory loss effect experienced by a large fraction of scattered molecules. In contrast, for H2/Pd(1 1 1) a direct scattering mechanism dominates, leading to the appearance of a pronounced peak of reflected molecules in the specular direction [48,49]. The absence of the specular peak for H2 and D2 diffraction from Pd(1 1 0) is reminiscent of the selective adsorption resonance processes well known in the case of He-diffraction [95,165]. Note that in the current case, the attraction felt by H2 molecules in the entrance channel is much stronger than in the case of He atoms interacting with metal surfaces, since the well depth for H2/Pd is about one order of magnitude larger than the physiorption well for He/metal surfaces [139,99]. Thus, for H2/Pd(1 1 0), dynamic trapping is possible and efficient for a wide range of initial impact conditions preventing the observation of the specular diffraction peak in experiments. 4.3. H2/Pt(1 1 1): Validity of the Born–Oppenheimer approximation As mentioned in Section 3.3, the scattering dynamics of H2 on metal surfaces is usually described in terms of a single six-dimensional molecule-surface PES, taking advantage of the Born–Oppenheimer approximation [23]. This approximation is at the heart of the standard model of reactivity, and requires that electrons not to be excited to higher quantum states by the motion of the atoms as they react [37,189]. An advantage of the Born–Oppenheimer approximation is that it allows dynamics results for reactive scattering to be computed in a straightforward way. However, the ability of electronically adiabatic theory to describe reactive scattering of molecules from metal surfaces can be questioned. In effect, electron–hole pair excitations can take place with infinitesimally small excitation energies on metals. These could act as an energy sink or source, and thereby affect the molecule’s reaction on or scattering from a metal surface. Direct evidence that electron–hole pair excitations can

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Fig. 11. Classical trajectory calculations for H2/Pd(1 1 1) and H2/Pd(1 1 0). Left: Fraction of reflected molecules after trapping as a function of Ei (at normal incidence) for H2/Pd(1 1 1) and H2/Pd(1 1 0). Right: Angular distribution of scattered H2 molecules (hi = 45° and Ei = 50 meV) from both surfaces [49]. Reproduced by permission from [71].

accompany molecule-surface scattering were provided by the observation of electronic excitation following highly exothermic chemisorption of atoms and molecules [76], and of electron emission following collisions of highly vibrationally excited NO molecules with a low work function metal surface [187]. For a review on recent experiments quantifying the significance of non-adiabatic effects in surface chemical processes see work by Hasselbrink [92] and Wodtke et al. [190,191]. One possible way of testing the validity of the Born–Oppenheimer approximation for a moleculemetal surface reaction is by comparison of electronically adiabatic theory to detailed experiments on both reaction and diffraction. The comparison for reaction probes the presence of energy dissipation to electron–hole pairs that accompanies the molecule’s motion towards the reaction barrier, whereas the comparison for diffraction will also test the presence of energy dissipation on the way back from the barrier. Even if one could argue that an electronically adiabatic theory could reproduce experiment for reaction due to fortuitous cancellation of errors in the PES and errors resulting from the neglect of non-adiabatic effects, it seems extremely unlikely that a good description of both reaction and diffraction experiments could be achieved in the presence of significant non-adiabatic energy dissipation. With these ideas in mind, the work reported by [134] shows unambiguously that theory can accurately describe both reaction and diffractive scattering of H2 from Pt(1 1 1) within an electronically adiabatic picture. The quantum dynamical calculations reported included the motion in all six degrees of freedom of H2, and were based on DFT PES. The diffraction experiments were performed for fixed angles of incidence, measuring both H2 in-plane and out-of-plane diffraction for incidence along the two main symmetry directions of the Pt(1 1 1) surface. The incident beam’s intensity was measured and used to normalize scattered beam intensities with respect to the incident beam, thereby yielding absolute diffraction probabilities. The diffraction experiments cover the range of incident energies (up to 15 kJ/mol) relevant to heterogeneous catalysis: the average collision energy of H2 with a Pt-surface is 2 kT = 12.5 kJ/mol for the most important catalytic process involving H2 and Pt, reforming of gasoline, which proceeds at 750 K [35]. Fig. 12 shows the comparison of both experimental absolute diffraction and reaction probabilities with the theoretical results for the energy range explored in experiment. The agreement between theory and experiment is very good. The energy dependence and the relative values of the diffraction  incidence direction probabilities are well reproduced by the theory. The agreement along the [1 0 1] (not shown) is likewise very good. At this point, it is worth emphasizing once more the importance

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of measuring out-of-plane diffraction, which represents roughly 50% of the total diffraction intensity. It is only in this way that the full extent of agreement between experiment and theory can be established. In view of the reaction probabilities, excellent agreement is obtained for both normal and off-normal incidence. In particular, the variation of the reaction threshold with Epar = Ei sin2 hi is well described and there is good overall quantitative agreement. The conclusions of this study are certainly good news for the theoretical treatment of these systems, since it shows that application of electronically adiabatic theory to scattering of H2 from metal surfaces should allow the calculation of accurate reaction and diffraction probabilities. Similar conclusions were drawn from 6D DFT calculations which incorporate electronic friction in the dissociative adsorption of H2 on Cu(1 1 0) [100].

Fig. 12. Top: Experimentally determined H2 diffraction probabilities (symbols) are compared with computed diffraction probabilities (curves), for specular scattering (black) and several first order out-of-plane (dark blue and red) and in-plane (light  azimuth. Probabilities for blue and pink) diffractive scattering transitions. The results are for incidence along the [1 1 2] symmetry equivalent transitions were summed. Error bars represent 68 % confidence intervals. Bottom: Theoretical reaction  direction compared to experimental results probabilities computed for normal and off-normal incidence along the [1 1 2] (squares) by [126]. Reprinted with permission from [134]. (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)

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4.4. H2/Ru(0 0 0 1): Performance of different functionals The results summarized in the previous Section suggest that electron–hole pair excitation does not affect H2 diffraction from a metal surface in a major way, so that the Born–Oppenheimer approximation can be used. At this point, one might try to address the issue of the performance of different functionals for reactive and diffractive scattering from a metal surface. This issue has been investigated in a recent work [135], in which experimental diffraction data for H2/Ru(0 0 0 1) were compared with quantum dynamics calculations performed by using DFT-based, 6D PESs calculated using two different functionals employing the generalized gradient approximation, namely the PW91 and RPBE functionals. 3 Even though one might expect the RPBE PES to give better agreement with experiments, as it usually provides more accurate molecule-surface interaction energies than PW91 [91], a much better agreement between theory and experiment has been found for PW91, at least in the case of H2 diffraction. The H2 diffraction experiments on Ru(0 0 0 1) were carried out in a variable angle setup at Epar = Ei sin 2hi = 35 meV, along the two main symmetry directions. The intensities extracted from angular distributions were firstly normalized with respect to the incident beam, and then extrapolated to 0 K applying the Debye–Waller model, as described in Section 3.1. Experimental intensities were then compared with theoretical diffraction probabilities computed using PESs based on the PW91 and RPBE functionals for H2 molecules with j = 0,1. These results were weighted to simulate a normal hydrogen cold beam, i.e. 25% with j = 0 (p-H2) and 75% with j = 1 (o-H2). The comparison of experimental with  0] symmetry direction is shown in Fig. 13 (similar results theoretical results corresponding to the [1 0 1  0] symmetry direction). were obtained along the [1 1 2 The first thing to point out in this figure is that the comparison between experimental results and theoretical calculations using the RPBE functional is rather unsatisfactory. First of all, the ratio of first order to specular diffraction intensities from simulations is much smaller than for experiments. For instance, the ratio I01/I00 from the experiments is 25% for Ei = 78 meV and 40% for Ei = 150 meV, while the values from the calculations are 3% and 27%, respectively. This points to a too low corrugation amplitude for the PES calculated using the RPBE functional. Even more important is the fact that the behavior of experimental intensities as a function of incident energy is not reproduced by the calculations. While all diffraction probabilities at first increase with Ei in the calculations, an overall decreasing or constant behavior with increasing incident energy is observed in the experiment. For instance, in the experiments the (0 1) diffraction peak is the most intense one (besides the specular one), but in the calculations the (1 0) diffraction peak is the most intense one, except for high Ei. From this comparison we can conclude that the PES calculated using RPBE functional does not give a good description of the H2 diffraction experiments from Ru(0 0 0 1). If we consider now the quantum calculations with the PW91 PES, we see that the agreement between theory and experiment is quite good in both absolute value and intensity trends as a function of incident energy. However, it is worth pointing out that the agreement obtained here with a PW91 PES for H2/Ru(0 0 0 1) is not yet as good as the agreement obtained using a Becke–Perdew PES for H2/ Pt(1 1 1) [134]. The results presented in Fig. 13 suggest that the PW91 functional describes the H2/ Ru(0 0 0 1) interaction more accurately than the RPBE functional. Although the PW91 functional overestimates the reaction probability (as discussed below), it yields a fairly good description of diffractive scattering of H2 from Ru(0 0 0 1). Since diffraction intensities are mainly determined by the geometrical corrugation along the unit cell, we can conclude that a much better description of the geometrical corrugation of the surface unit cell is given by the PW91 PES than by the RPBE PES, for the energies explored in the experiments. In order to give the whole picture of the performance of both functionals, we show in Fig. 14 the corresponding results for reaction probabilities. To arrive at a reliable comparison, the computed mono-energetic reaction probabilities were convoluted with the velocity distributions characterizing the experimental beams [79]. The PW91 reaction probabilities are too large compared to the experimental sticking probabilities for the entire range of collision energies studied, whereas the RPBE 3 Calculations were also performed using the so called MIX functional, which is obtained from a weighted averaged of both PW91 and RPBE functionals. For further details see [135]

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Fig. 13. Total diffraction probabilities calculated with the RPBE and PW91 PES, compared with experimental results measured  0] symmetry direction. Both theory and experiment correspond to a fixed initial Epar = 35 meV. Adapted from along the [1 0 1 [135].

reaction probabilities are too low in the energy range explored by diffraction experiments (i.e. below 0.20 eV). The results summarized above show that, although a preference for one or the other functional cannot be given for the reaction probabilities, the diffraction data tend to favor the PW91 functional, at least at energies below 0.15 eV. However, it is also clear that neither the PW91 nor the RPBE functionals accurately describe reaction of H2 on Ru(0 0 0 1) over the entire energy range investigated (see Fig. 14). In comparison, a much better description of reaction was obtained using the a Becke–Perdew

Fig. 14. Reaction probabilities obtained from the PW91 and RPBE PES are compared to experiment [79] for normal incident (m = 0, j = 0) H2 on Ru(0 0 0 1) as a function of the average collision energy. Adapted from [135].

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PES for H2 + Pt(1 1 1) [134]. These results suggest that the PES used for H2 + Pt(1 1 1) accurately describes the variation of the reaction barrier height with surface site, i.e., the energetic corrugation [83,109]. Since the corrugation of the PES governs diffraction, it is therefore not surprising that the PES used for H2 + Pt(1 1 1) also gave a better description of diffractive scattering of H2 from Pt(1 1 1) than the one obtained using PW91 for diffractive scattering from Ru(0 0 0 1). Clearly, a better PES is needed for H2 + Ru(0 0 0 1) to get the same high quality description of diffraction in this system as observed earlier for H2 + Pt(1 1 1). 4.5. D2/NiAl(1 1 0): Accuracy of RIDs description In Section 4.3 we have seen that quantum dynamics calculations performed by using a six-dimensional PES, constructed upon interpolation of a set of DFT data, provides an excellent description of diffraction spectra. One may wonder how good are these calculations to reproduce rotationally inelastic diffraction (RID) transitions, which usually represent a small amount of the total diffracted intensity (ca. 10–20%, depending on the incident conditions). This issue has been recently addressed by a combined theoretical and experimental work on the scattering of D2 from NiAl(1 1 0) [118] which we summarize in this section. As we mentioned in Section 1.1, RID peaks were observed for the first time in the scattering of H2 from LiF(1 0 0) [20,21], and later on for H2, HD and D2 scattering from MgO(1 0 0) [162,163]. In the case of metal surfaces, H2 scattering experiments are far more challenging, since rotationally inelastic probabilities are usually a small fraction of the total diffracted flux. Clearly resolved RID peaks using H2 and D2 beams have been reported for just a few systems, like Ag(1 1 1) [193,186], Rh(1 1 0) [44], Ni(1 1 0) [11], Cu(1 0 0) [12,13,15] and NiAl(1 1 0) [67,68]. These studies have suggested that, to answer several questions regarding rotational steering effects in the scattering process, a theoretical analysis based on six-dimensional quantum dynamical calculations is needed. Such quantum calculations on rotationally inelastic diffraction have been performed for H2/Pd(1 0 0) [80], H2/Pt(1 1 1) [144], HD/Pt(1 1 1) [102], and H2/Pd(1 1 1) [51]. However, there is a lack of a systematic comparison between these 6D quantum calculations and experiment, performed for the same system and the same incident conditions. The first detailed study along this direction is given by the experiments on the scattering of D2 from NiAl(1 1 0) summarized here. The experiments have been carried out using a time-of-flight setup, with fixed angle geometry. The high angular resolution of this machine, combined with its large dynamical range, make it the ideal tool to resolve the different RID peaks, even those overlapping in the diffraction spectra. It is worth emphasizing that this setup complicates significantly the theoretical analysis, since the final scattering angle cannot be varied independently of the incidence angle and, therefore, different incidence angles must be used for each observed diffraction peak at a given incidence energy. For this reason, such calculations have been prohibitively expensive until very recently due to computational limitations. The theoretical work carried out by Díaz and co-workers provides the first example of a systematic study performed for this kind of setup [118]. The choice of the D2/NiAl(1 1 0) system is justified because, owing to its minimum reaction barrier of 300 meV [157], the H2/D2 dissociation probabilities can be considered negligible in the energy range covered in the experiment (20–150 meV). Thus, low temperature targets can be used in the experiments, which minimizes Debye–Waller attenuation. In addition, previous classical and quantum dynamics calculations for this system have led to a good description of both dissociative adsorption and elastic diffraction of D2 [69,159], which gave confidence on the quality of the available PES to perform dynamical calculations. Fig. 15 shows an angular distribution of D2 molecules scattered from the NiAl(1 1 0) surface along  0] azimuth. The incidence energy is Ei = 86 meV, and the surface temperature was set to 100 K the [1 1 in order to reduce the multiphonon background in the scattering experiments. Several RID and diffraction peaks are visible in the spectrum. The RID peaks are labelled as (mn):jijf, whereby (mn) denote the diffraction peak involved in the rotational transition, and ji and jf the initial and final rotational state, respectively. RID peaks associated with the (ji = 0 ? jf = 2), (ji = 1 ? jf = 3) and (ji = 2 ? jf = 0) rotational transitions are resolved in the spectrum. Note that inelastic peaks for excitation to higher rotational levels lie at higher final scattering angles DHf with respect to their associated elastic peaks and at

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lower angles for deexcitations. In order to perform a systematic study of RID peaks, a series of angular distributions of D2 scattered from NiAl(1 1 0) at incidence energies between 80 and 146 meV was recorded. The intensity of each diffraction channel has been determined by fitting Gaussian profiles to the peaks of the angular distributions for a given incidence energy, Ei, and by averaging their integrated intensity over several measurements, applying the general procedure described in detail in Section 3.2. The experimentally normalized (with respect to the first order diffraction peak) determined diffraction and RID probabilities are shown in Fig. 16. Also shown are the results of quantum dynamics calculations performed using a 6D PES. Due to the experimental fixed-angle geometry used and to the ji population of the D2 beam, a total of 78 wave packet propagation had to be carried out. We can see an excellent agreement between experiment and theory for the elastic channel. From the top panel of Fig. 16, it can be seen that both the relative intensities as well as their behavior as a function of incident energy is well reproduced by the calculations. In the case of the RID peak intensities, however, significant quantitative and qualitative discrepancies can be observed between theory and experiment (bottom panel of Fig. 16). Experimental measurements show that the intensity of the (0 0):02 peak increases with energy by a factor of ten in the explored energy range, while the intensity of the other peaks remains more or less constant. This behavior of the (0 0):02 peak is not well reproduced by the theoretical calculations. Furthermore, the relative intensity of the RID peaks associated with transitions from rotational states with ji > 0 are not correctly described by quantum calculations. For instance, theory predicts that the (1 0):20 peak is less intense than the (0 0):02 one in the whole range of incidence energies explored, while the opposite is observed in experiment below 130 meV.  0Þ:02 peak is more intense The same holds for less intense features: while theory predicts that the ð2  0Þ:13 one for all incidence energies, the opposite ratio is observed in the experiment. In than the ð2 addition, although the relative RID peak intensities associated with excitations from ji = 0 are correctly reproduced by the theory, the quantitative agreement with the experimental measurements is clearly less satisfactory than for the elastic diffraction channels. At this stage, it is worth recalling the level of agreement obtained in previous comparisons of theory to experiment for rotationally inelastic scattering of H2 and HD from metal surfaces. Poor agreement with experiment was obtained from 6D quantum dynamics calculations reported for HD/ Pt(1 1 1) [102] and for rotationally and rovibrationally inelastic scattering of H2 from Cu(1 0 0) [185]. For a more detailed discussion on the possible reasons for these discrepancies see [111]. As discussed

 0] azimuth at a surface temperature Fig. 15. Angular distribution of D2 scattered from the NiAl(1 1 0) surface along the [1 1 TS = 100 K. The incidence beam energy is Ei = 86 meV. Reprinted with permission from [7].

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Fig. 16. Top: Experimentally determined normalized diffraction probabilities (solid symbols) compared with quantum diffraction probabilities (open symbols) for specular scattering (black), first order (red) and second order diffraction (blue). Bottom: Experimentally determined normalized RID probabilities (solid symbols) compared with quantum RID probabilities (open symbols). Results are for incidence along the [110] symmetry direction. Error bars represent 68 % confidence intervals. Dashed lines are only a guide to the eye. Used by permission from [118]. (For interpretation of the references in colour in this figure legend, the reader is referred to the web version of this article.)

at the end of Section 3.1, experiments for H2/Pd(1 1 1) yielded non-zero values of rotational excitation probabilities even at energies below the threshold of the corresponding rotational excitation [183]. These results could be reproduced by a model including the possibility of energy exchange with surface phonons [27]. For energies above the threshold, a reasonable good description of rotationally inelastic scattering of H2 from Pd(1 1 1) was obtained from 6D quantum [30] and classical [149] dynamics calculations. In the most recent comparison reported, a quite good agreement was obtained for rotationally inelastic scattering data for H2/Cu(1 1 1) by calculations based on the specific reaction parameter (SRP) approach to DFT [53,55]. Coming back to the D2/NiAl(1 1 0) results shown in Fig. 16, the observed discrepancies between experimental and theoretical results may be caused by any of the approximations made in the calculations, namely: (i) the frozen surface approximation, (ii) the Born–Oppenheimer approximation, and (iii) inaccuracies in the PES employed in the dynamics calculations. The first point can be ruled out, since the same attenuation with surface temperature is observed experimentally for both elastic and RID peaks intensities (see data shown in Fig. 5). A violation of the Born–Oppenheimer approximation is also

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discarded by the authors of this work, mainly because previous calculations performed within the Born– Oppenheimer approximation were able to successfully account for dissociative adsorption [159] and elastic diffraction [69] in the same system. In addition, there is actually no evidence suggesting that RID processes should be more influenced by non-adiabatic effects than elastic diffraction processes. Therefore, the discrepancies between experimental and theoretical results shown in Fig. 16 were attributed to inaccuracies in the PES employed in the calculations. These inaccuracies can be due either to the interpolation process or to the DFT functional used in the calculations. The fact that the PES used in this study has led to a very good description of dissociative adsorption [159] and elastic diffraction [69] strongly suggests that RID peaks are far more sensitive to subtle details of the DFTbased PES. Therefore, the present study suggests that an accurate evaluation of RID intensities requires PESs that are more accurate than the existing ones. If this can be done within the framework of DFT is something that requires a thorough systematic theoretical investigations.

5. Outlook We have reviewed recent experimental and theoretical work on diffraction of H2 and D2 from single-crystal metal surfaces. Experiments performed on Pd(1 1 1), Pt(1 1 1), NiAl(1 1 0) and Ru(0 0 0 1) demonstrate that out-of-plane diffraction is important for H2 + metal systems, and its importance increases with increasing reactivity. State-of-the-art 6D quantum dynamics calculations were performed within the rigid surface model of electronically adiabatic molecule-surface scattering. In this approach, the ground state electronic PES is based on DFT calculations using the GGA to the exchange-correlation energy [81,109,111]. The systematic studies reported in recent years show that such calculations provide, generally speaking, a very good description of diffraction of H2 and D2 from metal surfaces. In spite of the progress made, there are still some open questions that deserve further investigations. A good example is provided by the rotationally inelastic diffraction (RID) peaks. The detailed study performed for D2/NiAl(1 1 0) [118] clearly shows that present theory does not yield an as yet accurate description of RID transitions. The discrepancies between experimental and theoretical results are very likely due to inaccuracies in the PES employed in the calculations. Similarly, the results reported for H2/Ru(0 0 0 1) show that, although the PW91 functional gives a better description of diffractive scattering than the RPBE one, the agreement obtained is not as good as the one obtained for H2/Pt(1 1 1) using a Becke–Perdew PES. These results are expected to stimulate further developments of the theory in order to get a more accurate description of reaction, elastic diffraction and RID transition from metal surfaces. Overall, the results summarized in this review suggest that the level of accuracy of the 6D PES required to get a good description of the experimental results increases when moving from sticking to diffraction data [54]. RID transitions, however, still represent a challenge for theorists. The combined theoretical and experimental work of D2 diffraction from NiAl(1 1 0) reported by Laurent et al. [118] suggests that current 6D quantum dynamics calculations on a DFT-based PES are not accurate enough to describe RID transitions. Whether such a description can be given within the framework of the Born–Oppenheimer approximation remains still an open question. Certainly a very promising development is the specific reaction parameter (SRP) approach to DFT, which allows studies of reactive and non reactive scattering of H2 from metal surfaces using a chemically accurate PES [53,55]. Dynamics calculations performed using a SRP-DFT 6D PES have been able to reproduce dissociative adsorption probability and rotationally inelastic scattering data for the H2/ Cu(1 1 1) system with chemical accuracy, i.e. with errors 64.2 kJ mol1 [53,55]. What has not yet been done by theorists is to fit an SRP-DFT functional to molecular beam sticking data, and then to perform calculations using that functional on diffractive scattering and RID transitions in the specific system considered. From the experimental point of view, perhaps one of the most challenging future developments would be to combine H2 diffraction with laser techniques in the same apparatus. It would allow, for instance, the measurement of diffraction spectra for vibrationally excited molecules, making it possible the comparison with calculations performed with H2 molecules beyond the ground state. This

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would allow testing some interesting predictions of the theory, like for instance that m = 0 and m = 1 H2 react at different surface sites on Cu(1 0 0) [112,175], which should produce a different dependence of diffraction on incident energy in the two cases. Another interesting direction for the future concerns the study of more complex systems, like the surfaces of thin films or multicomponent alloys, with the aim of getting surfaces with dissociation properties different from those of transition metals. In particular, it would be interesting to see how good the description given by 6D DFT-based quantum dynamics calculations is for reactive and diffractive scattering from these systems. In recent years, a great deal of effort has been devoted to the development of a Scanning Helium Atom Microscope (SHeAM), in which a focused beam of neutral He atoms is used as imaging probe [96,57,150]. In fact, the first SHeAM micrograph has been recently reported [105]. The potential lateral resolution of SHeAM is 50 nm, and would allow the study in real-space of delicate biological materials, weak polymers, and insulating glass surfaces, among other samples which are difficult to be examined by electron microscopy techniques, due to the appearance of sample charging or electron excitation effects. Recent work has shown that the same mirrors used to focus He-atoms exhibit a large reflectivity for H2-beams [5,6,147]. This opens up a very interesting extension of SHeAM, namely the development of a H2-microscope technique suitable for lithography applications at the nanoscale. Although in principle this technique is mainly concerned with specular reflection, theory work will be needed, for instance, to characterize the etching and contrast mechanisms. In this sense, the insight gained from 6D DFT-based quantum dynamics calculations on single-crystal metal surfaces is certainly going to be very helpful. In summary, H2 diffraction measurements represent a quite sensitive technique to gauge the 6D H2-surface PES within the surface unit cell. It samples the PES in a region around 2 Å away from the surface atom nuclei, which corresponds to the classical turning points of diffracted H2 molecules. Therefore, the accuracy of PESs determined by state-of-the-art calculations can be tested in a region of interest also for other molecule/surface interactions, which is a difficult piece of information to get with other experimental techniques. Acknowledgements We gratefully acknowledge K.H. Rieder and J.P. Toennies for the donation of the scattering apparatuses used in our experiments. Without their generosity, most of the experimental work reviewed here would never have been performed. We would also like to thank H.F. Busnengo and F. Martı´n, who contributed many ideas in the process of writing a topical review article on which they were co-authors. We are also indebted to H.F. Busnengo, C.Díaz, A. Gross, G.-J. Kroes and M.F. Somers for their critical reading of the manuscript. Special thanks go to S. Montero for helpful comments and suggestions, and to E. Hulpke for technical assistance during the setting up of the TOF machine in Madrid. The authors appreciate support from the Ministerio de Educación y Ciencia through Projects ‘‘CONSOLIDER en Nanociencia Molecular’’ (CSD 2007-00010) and FIS 2010-18847, and from Comunidad de Madrid through the Program NANOBIOMAGNET. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

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