Apparent failure of the Born–Oppenheimer static surface model for vibrational excitation of molecular hydrogen on copper

Abstract

The accuracy of dynamical models for reactive scattering of molecular hydrogen, H₂, from metal surfaces is relevant to the validation of first principles electronic structure methods for molecules interacting with metal surfaces. The ability to validate such methods is important to progress in modeling heterogeneous catalysis. Here, we study vibrational excitation of H₂ on Cu(111) using the Born–Oppenheimer static surface model. The potential energy surface (PES) used was validated previously by calculations that reproduced experimental data on reaction and rotationally inelastic scattering in this system with chemical accuracy to within errors ≤ 1 kcal/mol ≈ 4.2 kJ/mol [Díaz C, et al. (2009) Science 326:832–834]. Using the same PES and model, our dynamics calculations underestimate the contribution of vibrational excitation to previously measured time-of-flight spectra of H₂ scattered from Cu(111) by a factor 3. Given the accuracy of the PES for the experiments for which the Born–Oppenheimer static surface model is expected to hold, we argue that modeling the effect of the surface degrees of freedom will be necessary to describe vibrational excitation with similar high accuracy.

Reactions of molecules with metal surfaces are of tremendous importance, as the production of most man-made chemicals involves heterogeneous catalysis by a metal surface. One of the achievements recognized by the 2007 Nobel Prize in Chemistry, awarded to G. Ertl, was the detailed description of the sequence of elementary molecule-surface reactions by which ammonia is produced (1). In such reaction sequences, the dissociative chemisorption of molecules often plays a prominent role.

In the present state of the art, theoretical calculations can reproduce overall rates of heterogeneously catalyzed reactions like ammonia production to within an order of magnitude (2). Achieving greater precision has been hampered by the lack of accuracy inherent in the density functional theory (DFT) used to calculate the interactions of molecules with metal surfaces.

Because dissociative chemisorption involves stretching bonds until they rupture, reactivity of molecules on surfaces is closely related to transfer of energy to and from the vibrational degrees of freedom (3). Thus vibrationally inelastic scattering can be a sensitive probe of the barrier region of the potential energy surface (PES), and experiments (4, 5) and theory (6, 7) suggest that this process is governed by the same region of the PES as dissociative chemisorption: The picture of vibrational excitation occurring in competition with dissociative chemisorption is that it results from a stretching of the molecule as it approaches the transition state (4, 5). In principle, experiments on this competition may reveal much interesting information. For instance, comparison of data on vibrationally inelastic scattering and reaction on a surface has been suggested to provide information on whether these processes occur predominantly at similar or different surface sites (7, 8). Finally, we note that recent experiments and calculations on H + D₂ present evidence that the bond stretching mechanism for vibrational excitation in competition with reaction is also relevant for gas phase reactions (earlier, vibrational excitation in gas phase molecular collisions was usually assumed to result from bond compression only) (9, 10).

Theoretical research suggests that vibrationally inelastic scattering of molecules from surfaces may be promoted by energy exchange with surface phonons (11, 12). Vibrationally inelastic scattering may also be promoted by energy exchange with electron-hole (e-h) pairs (3, 13, 14). The H₂ + Cu system is a benchmark system for investigating the effects of e-h pair excitation (15, 16) and energy transfer to phonons (12, 17) on reaction of molecules with metal surfaces. Theoretical studies on e-h pair excitation suggest the effects to be small (15, 16), in line with research on H₂ + Pt(111) (18). Also, specific features have been identified in PESs for H₂ + Cu that promote vibrationally inelastic scattering in an electronically adiabatic mechanism (6–8). Theoretical studies on the effect of energy transfer to phonons on reaction also suggest this effect to be small at low surface temperature (T_s) (17), in agreement with experiments (19–21).

The absence of important effects of e-h pairs and phonons on reaction makes H₂ + metal surface systems ideal for testing first principles electronic structure methods for their accuracy for molecules interacting with metal surfaces, which is important to the accurate modeling of heterogeneous catalysis. Calculations with the Born–Oppenheimer (or electronically adiabatic) static surface (BOSS) model can take into account the motion in all molecular degrees of freedom of these systems while using quantum mechanics for the dynamics and nonetheless remaining computationally tractable (22, 23). However, vibrational deexcitation of H₂ on Cu(100) is accompanied by substantial energy loss to the surface (24), and there is theoretical evidence that the loss could be due to e-h pair excitation (25). This raises the question of whether experiments on vibrationally inelastic scattering of H₂ from metal surfaces can be used to validate electronic structure methods for molecule–metal surface interactions with dynamics calculations employing the BOSS model. There is thus ample reason to investigate vibrationally inelastic scattering of H₂ from Cu, and the role played by the surface degrees of freedom.

Previous quantitative comparisons between theory and experiment on vibrationally inelastic scattering of H₂ from surfaces (26–28) were hampered by the accuracy of the PESs used being too low to say whether discrepancies with experiment were due to inaccuracies in the PES or to other problems, such as the neglect of the surface degrees of freedom in the calculations. This has now changed. We recently showed (21) that specific reaction parameter DFT (SRP-DFT) (29) allows a chemically accurate description (i.e., to within 1 kcal/mol ≈ 4.2 kJ/mol) of experiments on the reaction of H₂ and D₂ in molecular beams, on the influence of the initial vibrational and rotational state of H₂ on reaction, and on rotational excitation of H₂, in scattering from Cu(111). As a result, as argued below we can now address the question whether the BOSS model affords a quantitative description of experimental results on vibrational excitation of H₂ on Cu(111) (5, 30).

Here we compute rovibrational energy transfer probabilities and, from these, time-of-flight (TOF) spectra used experimentally to probe vibrational excitation; the computed TOF spectra can be compared directly to experiments on scattering of H₂ from Cu(111) (30). The calculations use the BOSS model that, with the SRP-DFT PES also used here, accurately reproduces the experiments on scattering of H₂ and D₂ from Cu(111) referred to above (21). Comparison of the computed TOF spectrum to experiment shows that our calculations underestimate the contribution of vibrational excitation to previously measured TOF spectra of H₂ scattered from Cu(111) by a factor 3. Because the PES used has been shown to reproduce several other observables with chemical accuracy (21) (see also above), we argue that this failure can be predominantly attributed to the only remaining sources of error; i.e., to one or both of the approximations in the BOSS model (the neglect of e-h pairs and phonons).

Results

Computed TOF spectra of H₂ scattered from Cu(111) into its (v = 1, j = 3) state are compared to experimental data in Fig. 1. Here, v is the vibrational and j the rotational quantum number of H₂. The appearance of the experimental TOF spectrum is explained from the way the experiments were done in Materials and Methods, to which we refer for an easier understanding of the following. To simulate scattering at off-normal incidence (θ_i = 15°, θ_i being the angle of incidence with the surface normal) (30) and for reasons of computational efficiency, computed spectra were based on P(v,j → v^′ = 1,j^′ = 3) calculated for normal incidence (θ_i = 0°). Results are shown assuming that probabilities at off-normal incidence only depend on the total incidence energy E_i [total energy scaling (TES)], and assuming that the probabilities only depend on the translational energy normal to the surface, E_i cos²(θ_i) [normal energy scaling (NES)]. Both experimental and theoretical spectra exhibit a “loss peak” (the “loss” being relative to the incident beam) at long times (Fig. 2). This peak reflects loss of (v = 1, j = 3) H₂ due to reaction and vibrational deexcitation, and rotational redistribution within v = 1 [we explain that it appears at long times in Materials and Methods (30)]. The excellent agreement of the loss peaks shows that the BOSS model provides an accurate simultaneous description of these processes.

Fig. 1.

The experimental TOF spectrum of H₂ scattered into the (v = 1, j = 3) state (30) (black dots) is compared to spectra computed assuming NES and TES of the probabilities for scattering into (v = 1, j = 3), for f(K) = 0. For TES, computational results are also shown for f(K) = 0.3 and 0.5 (see text for explanation).

Fig. 2.

The experimental TOF spectrum of (v^′ = 1, j^′ = 3) H₂ (30) (black dots) is compared to the spectrum computed assuming TES, for f(K) = 0.3 (solid line). The experimental spectrum of (v = 1, j = 3) H₂ in the incident beam is also shown (blue dots). The computed spectrum was obtained by multiplying the calculated contribution of vibrational excitation from (v = 0, j) states (dotted line) by 2.1, while leaving the contribution of rotionally (in)elastic scattering from (v = 1, j) states (dashed line) intact.

The spectra also exhibit a “gain peak” at short times, due to vibrational excitation of H₂ (v = 0) in the beam to (v = 1, j = 3) (Figs. 1 and 2). For this peak, good agreement with experiment can only be obtained by multiplying the computed P(v = 0,j → v^′ = 1,j^′ = 3) with a factor 2.5 (3.7) assuming TES (NES). This suggests that our calculations reproduce the energy dependence of the P(v = 0,j → v^′ = 1,j^′ = 3) to within a constant multiplicative factor. Comparison of normal incidence to off-normal incidence results computed for the (v = 0, j = 3) initial state (Fig. S1) shows that the computed probabilities for vibrational excitation at off-normal incidence obey a intermediate scaling between TES and NES; taking the average we estimate that the BOSS model underestimates the contribution of vibrational excitation to the TOF spectrum by about a factor 3.1. At least three factors may contribute to this discrepancy, with the first two related to the neglect of the surface degrees of freedom.

First, state-to-state measurements on vibrational deexcitation showed that, in scattering of H₂ from Cu(100) and Cu(110), the deexcited molecule lost a fraction f(K) of the translational energy K that would be available to it in scattering from a static surface (24). Taking this effect, which arises if energy is transferred to surface motion, into account by assuming f(K) > 0 leads to higher simulated gain peaks (Fig. 1): after losing energy to the surface the molecule travels more slowly through the detection zone, increasing its detection (Eq. 1). Assuming a reasonable (24) value for f(K) (of 0.3; SI Text) increases the computed gain peak by 20%, reducing the disagreement with experiment from a factor 2.5 (3.1) to 2.1 (2.6), assuming TES (intermediate scaling) [see Fig. 2 for TES]. Assuming the factor used for f(K) to be correct and assuming intermediate scaling, the BOSS model would then underestimate vibrational excitation probabilities measured while taking energy loss to the surface into account by a factor 2.6. The factor by which the pure BOSS model underestimates the contribution of vibrational excitation to the TOF spectrum (3.1) is larger because molecules not having lost energy to the surface are detected less efficiently in the experiment.

The second factor contributing to the discrepancy between theory and experiment was found by reanalyzing the experimental data. Specifically, a scrutiny of the dependence of the experimental TOF spectrum on T_s; i.e., replotting the results of figure 4 of ref. 30 using the same baseline, shows that increasing T_s from 400 to 700 K increases the contribution of vibrational excitation by 20% (Fig. S2). This increase was not noted earlier because the experimentalists were looking for a much more dramatic dependence of vibrational excitation on T_s as exhibited in scattering of NO from Ag(111), which was attributed to a mechanism involving decay of thermally excited e-h pairs (13). The promoting effect of T_s we observe suggests that fully incorporating surface motion and, thereby, the effect of T_s in the dynamical model would further increase the contribution of vibrational excitation to the spectrum, as needed for better agreement with experiment. Linear extrapolation of the effect of T_s to 0 K would decrease the contribution of the gain peak to the experimental spectrum by 27%, further reducing the disagreement with theory from a factor 2.6 to 1.9, assuming intermediate scaling.

Third, the height of the gain peak depends strongly on the nozzle temperature, T_n (Fig. 3). In the experimental paper the value of T_n was stated (2,000 K), but not its uncertainty (30). The effect of an error in T_n could be substantial (Fig. 3), but with a realistic estimate of this error (100 K; see SI Text for a discussion of the possible range of T_n as used in Fig. 3) the statistical error of the peak height should only be about 6%. The combined effect of the first two systematic sources of error, which are both related to surface motion, is much larger.

Fig. 3.

The experimental TOF spectrum of (v^′ = 1, j^′ = 3) H₂ (30) (black dots) is compared to spectra computed assuming TES, for f(K) = 0.3, for five different values of T_n in the range 1725–2160 K.

In summary, the BOSS model underestimates the contribution of vibrational excitation to the TOF spectrum measured at T_s = 400 K by about a factor 3. The discrepancy cannot be explained by an error in T_n, nor can the model be patched up by accounting for increased detection efficiency through energy loss to the surface and linearly extrapolating the T_s-dependence of the experimental results to 0 K. [Our calculations do suggest that the disagreement between BOSS theory and experiment would be somewhat smaller (a factor of about 2.5) for vibrational excitation probabilities, if these probabilities were extracted from experiments while taking energy loss to the surface into account].

Discussion

We now speculate about what T_s dependence of vibrational excitation could lead to agreement of an experiment at T_s = 0 K with the theory, assuming f(K) = 0.3. For this we assume the probability for vibrational excitation to exhibit a pseudo-Arrhenius dependence on T_s according to P(E_i,T_s) = P(E_i,0 K){1 + A exp(-E_a/kT_s)} (3, 12). We choose this model because it gives a reasonable description of the T_s dependence of vibrational excitation probabilities of D₂ scattering from Cu(111) at values of E_i near to those relevant to the evaluation of the gain peak (12). In (12), these probabilities were computed modeling motion in a subset of the molecular degrees of freedom, and using an approximate model for the phonons and how the phonons affect the molecule–surface interaction. Here we use the same E_i dependence for the T_s dependent and independent parts because agreement between our T_s independent theoretical and the T_s dependent experimental results can be enforced by multiplying the theoretical gain peak with a constant factor (Fig. 2). Because experimental information is available for only two values of T_s and the model we fit to contains three adjustable parameters, we cannot extrapolate the experimental results to 0 K. However, we can determine the activation energy E_a needed for agreement with the theory. This value (23 meV) is close to the range of E_a characterizing the promotion of vibrational excitation of D₂ scattering from Cu(111) by phonons in the quantum dynamics calculations referred to above (30–50 meV for E_i above threshold) (12).

The E_a required in our model is much lower than the E_a required for a E → V mechanism involving energy transfer from e-h (E) pairs to vibration (V), in which E_a should be equal to (a large fraction of) the molecule’s vibrational quantum (3, 25) (50 kJ/mol for H₂). If E → V transfer were to account for the discrepancy with experiment, at the E_i corresponding to the gain peak (80 kJ/mol) the probability for this process would have to be in the range 0.10–0.15 (30). The likelihood of such a probability can be assessed by considering recent experiments on vibrational excitation of NO scattering from Au(111) (31) and Cu(110) (32). The v = 0 → 1 vibrational excitation energy of NO (23 kJ/mol) is close to the frequency of the H₂ vibration near the barrier (which may be taken as approximately one half the vibrational quantum (19); i.e., as 25 kJ/mol). Because vibrational excitation of NO occurs in the strong E - V coupling regime (31), NO v = 0 → 1 vibrational excitation probabilities may serve as upperbound estimates to the probability of vibrational excitation of H₂ due to E → V transfer. The experiments on NO, which were done for similar E_i and similar T_s (400 K) as the experiment we describe, show v = 0 → 1 probabilities of about 0.01 or smaller at T_s = 400 K (31, 32). Such low values rule out an important role of E → V transfer in H₂ + Cu(111). If e-h pairs do play a role in the H₂ + Cu experiment, it is more likely to involve accompanying energy dissipation from translation to the electrons (T → E) leading to enhanced detection of the vibrationally excited H₂.

Interestingly, for impact on top sites [collisions with this site should yield an important contribution to vibrational excitation (8)] DFT calculations show that the PES displays a similar dependence on the motion of a surface atom as in the model used in ref. 12 to describe the effect of surface motion on reaction of D₂ on Cu(111). Specifically, the model employed assumes that at the reaction barrier geometry (Z_b, r_b) (these coordinates being the molecule-surface distance and the H-H distance at the barrier) the dependence of the potential on the displacement of the surface Cu-atom perpendicular to the surface, Q, can be described using V(Q,Z_b,r_b) = V(0,Z_b - Q,r_b). The fact that this equation describes the Q-dependence of DFT calculations with the PW91 functional rather well (Table S1) suggests that phonons may indeed play an important role in promoting vibrational excitation. However, further experimental research is needed to establish whether as low a value of E_a as suggested here is realistic, and whether phonons are more important for promoting vibrational excitation than e-h pairs.

In the original interpretation of the experiments (30), vibrational excitation probabilities could only be extracted assuming the vibrational excitation to be rotationally elastic (j = j^′ = 3). However, the calculations suggest that scattering from the (v = 0, j = 5) state contributes more to the (v^′ = 1, j^′ = 3) H₂ TOF intensity than scattering from the (v = 0, j = 3) state (Fig. 4). The contribution of P(v = 0,j = 5 → v^′ = 1,j^′ = 3) is larger because, while being similar in size to P(v = 0,j = 3 → v^′ = 1,j^′ = 3), it peaks at lower E_i, where the intensity of the incident beam is higher (Fig. 2). Our findings suggest that for quantitative extraction of vibrational excitation probabilities the experiments should be extended to yield state-to-state probabilities P(v = 0,j → v^′ = 1,j^′), for instance by transferring population among (v = 0, j) states in the incident beam using stimulated Raman pumping.

Fig. 4.

Contributions from different initial (v = 0, j) states to the computed gain peak in the TOF spectrum are shown for TES and f(K) = 0.

Assuming intermediate scaling, the pure BOSS model underestimates the contribution of vibrational excitation by a factor 3.1, even though a PES was used that reproduced several other observables with chemical accuracy. Experience with theoretical studies of gas phase reaction dynamics suggests this is not necessarily proof that the BOSS model is to blame for this discrepancy. Semiempirical potentials or electronic structure methods that were calibrated by demanding an accurate description of rate constants or integral cross sections for overall reaction are not guaranteed to provide an accurate description of more detailed reactive scattering data, such as product angular distributions (33) and product energy partitioning (34). In a similar vein, one could argue that an electronic structure scheme adjusted semiempirically to give an accurate overall description of reaction is not guaranteed to give an accurate description of the scattering, for instance because reaction and scattering might be dominated by different regions of the PES. We nevertheless argue that the discrepancy should not be predominantly due to errors in the PES, but rather to the approximations inherent in the BOSS model; i.e., the neglect of e-h pair excitations and phonons. Our evidence is as follows (see SI Text).

Within the BOSS model our PES describes data on rotationally inelastic scattering with chemical accuracy, in the sense discussed in ref. 21. This suggests that the PES used accurately describes the anisotropy of the molecule–surface interaction, and it raises the question why the combination of the semiempirical PES and model used here reproduces data on rotationally inelastic scattering, but not on vibrationally inelastic scattering. Furthermore, the SRP-DFT PES we used accurately describes (21) those experiments on reactive scattering of H₂ from Cu(111) for which the BOSS model is expected to hold, based on previous theoretical research (15–18, 21) and experiments (19–21). The fact that this PES yields accurate results concerning how vibrational preexcitation of H₂ promotes reaction suggests (7) that the features of the PES responsible for promoting vibrational excitation [large curvature of the reaction path in front of a late barrier (7)] should also be accurately described, at least at the reactive sites. The fact that the PES yields an accurate description of the overall reaction probability measured in molecular beam experiments on reaction of D₂ over a lange range of energies suggests that the anisotropy and corrugation of the molecule–surface interaction are also correctly described in the PES, at least as far as the reaction barrier height is concerned (SI Text). Combining these observations suggests that an explanation of the discrepancy with experiment in terms of an incorrect description of the PES’s features promoting vibrational excitation at nonreactive impact sites/orientations is implausible. Here, it is helpful that, according to experiments, both reaction and vibrational excitation occur with large probabilities, the maximum probabilities extracted from experiments being in the range 20–30% for both processes (19, 30), making it even more unlikely that these processes occur in mutually exclusive regions of the PES. Experience with modeling gas phase reactions suggests that an empirical PES (35) may do well at modeling both reaction and vibrationally inelastic scattering if both processes occur with large probability, and if the vibrationally inelastic scattering may be viewed as “frustrated reaction,” in “collisions that almost lead to reaction but the reagents separate” (36), the picture that is also applicable to H₂ + Cu(111) (4, 5, 30).

Continuing our line of argument, the SRP-DFT method used is not just any semiempirical method. By taking the PES as a weighted average of the PW91 (37) and RPBE (38) functionals, which have been successfully applied to numerous surface science problems, we ensure that the SRP-DFT functional inherits the solid physics built into these functionals. For instance, DFT work on Bayesian error theory (39) suggests that the PW91 and RPBE functionals predict chemisorption energy differences between adsorption sites rather accurately, giving further support to the argument used above that the SRP-DFT PES should provide an accurate description of the corrugation of the molecule–surface interaction. Finally, with the family of SRP density functionals to which the SRP density functional used here and in (21) belongs (i.e., of weighted averages of the PW91 and RPBE functionals), it is not possible to provide an accurate description of the contribution of vibrational excitation to the measured TOF spectrum discussed here within the BOSS model (Fig. S3 and SI Text). On the other hand, as already noted several other observables can be reproduced with chemical accuracy with a functional of this family, using the BOSS model.

Based on the above line of reasoning, we consider it highly unlikely that errors in the PES should constitute the predominant cause of the discrepancy with experiment for vibrational excitation. This conclusion is supported by our reinvestigation of the experimental data concerning the effect of T_s, which shows a substantial effect of T_s on the contribution of vibrational excitation to the TOF spectrum going from 400 to 700 K. This constitutes direct evidence of the breakdown of the BOSS model for vibrational excitation. Further evidence comes from analyzing the effect that the energy loss to the surface, which was observed earlier in experiments on vibrational deexcitation of H₂ scattering from Cu-surfaces (24), should have on the measured contribution of vibrational excitation to the TOF spectrum simulated here. As noted above, our analysis suggests that taking the combined effects of T_s and energy loss into account decreases the discrepancy between theory and experiment for vibrational excitation from a factor 3.1 to 1.9. This admittedly primitive analysis suggests that effects related to the breakdown of the BOSS model account for about 60% of the discrepancy with experiment, in line with our conclusion that the discrepancy is predominantly due to errors associated with the dynamical model.

The research presented here shows that experiments on vibrational excitation of H₂ from metal surfaces cannot be used to validate electronic structure methods for molecule-metal surface interactions with dynamics calculations employing the BOSS model. To allow validation, the dynamical model should be improved. Improvements to the dynamical model consist of approximate ways of including the effect of phonons in quantum dynamical calculations (40) or in ab initio molecular dynamics (AIMD) simulations (41). In the latter approach it should also be possible to incorporate e-h pair excitation in an approximate way using electronic friction (25) or the independent electron surface hopping method (42). Calculations employing the latter method in combination with a classical description of molecular and phonon motion provide a qualitatively accurate description of multiquantum vibrational relaxation of highly vibrationally excited NO scattering from Au(111) (43). Fortunately, at the collision energy (80 kJ/mol) at which the contribution of vibrational excitation to the experimental TOF spectrum peaks, quasiclassical dynamics (as used in AIMD) is quite accurate for vibrational excitation of H₂ on Cu from v = 0 to 1 (44). [The quasiclassical method would be expected to overestimate the contribution of vibrational excitation to the TOF spectrum at lower collision energies (44).] Experiments can contribute to progress by determining state-to-state vibrational excitation probabilities. Progressing to the state-to-state level could allow an accurate determination of the energy lost to the surface with the TOF techniques already in place, which would increase the accuracy of the extracted state-to-state probabilities. Further improvements could consist of measuring TOF spectra exhibiting vibrational excitation for a large range of T_s and for more accurately determined T_n values.

Materials and Methods

The quantum dynamical calculations used here to compute state-to-state vibrational excitation probabilities treat the motion of H₂ in all six degrees of freedom, with a method that has been presented in detail (21, 45). The accurate SRP-DFT PES used here was computed using a weighted average of the RPBE (38) and PW91 (37) exchange correlation functionals. Its construction, and the numerical details of the quantum dynamical calculations, are described in (21). Here we focus on how the calculated state-to-state vibrationally inelastic and elastic scattering probabilities are used to compute theoretical TOF spectra for comparison to experiments (30).

The TOF scattering experiments (30) observed molecules scattered into the j^′ = 3 rotational state within the final v^′ = 1 vibrational state using resonantly enhanced multiphoton ionization (REMPI). This TOF spectrum S(t) may be simulated with theory using

[1]

Eq. 1 represents an extension of equations published earlier, which did not yet allow for changes in j (30). In Eq. 1, v_i (v_s) is the velocity of the incident (scattered) molecule, both velocities depending on the initial (v, j) state of H₂ (30),v₀ is the stream velocity of the incident molecular beam (4115 m/s), and the width parameter α is 1358 m/s (2379 m/s) for v_i < v₀ (v_i > v₀) (30). N is a normalization factor, c defines an offset, and x_i (x_s) describes the distance traveled by the incident (scattered) H₂, x_t being their sum. The weight w_vj is the Boltzmann population of the initial (v, j) state in the incident beam divided by the population of the final (v^′, j^′) state in the incident beam at the experimental value of T_n [2,000 K (30)]. For the nearly effusive (thermal) molecular beam used in the experiments to achieve a wide range of collision energies (30), the Boltzmann populations may be calculated assuming that both the rotational and the vibrational temperature of the incident molecules is equal to T_n. The crucial input from the theory are the computed state-to-state probabilities for vibrationally inelastic and elastic scattering P(v,j → v^′,j^′) already referred to above. The need to compute these for all (v, j = odd) states with significant population in the incident beam (we used states with j up to 11 for v = 0 and up to 9 for v = 1) adds considerable challenge to the calculations.

In the experiment described by Eq. 1 translational energy resolution is achieved by chopping the incident beam into short pulses far way from the surface (at large x_i) and detecting the scattered molecules close to the surface (at small x_s) with a pulsed laser, using REMPI. By taking x_i≫x_s the TOF was made to reflect the velocity of the incident molecules. A typical experimental TOF spectrum describing scattering into (v = 1, j = 3) (the black dots in Fig. 2) shows a peak at early times (large incidence energy E_i) and a peak at late times (small E_i). Because the early scattered TOF peak signal exceeds the TOF signal of the incident beam (the blue dots in Fig. 2), this peak was called the gain peak (30). The “gained” (v = 1, j = 3) population was attributed to vibrational excitation in a T → V process, which requires a high E_i because the energy transferred to vibration (V) needs to come from translation (T). Because the late scattered TOF peak signal is lower than the incident beam TOF signal (Fig. 2), this peak was called the loss peak (30) and attributed to vibrationally elastic scattering within v = 1. The loss was attributed to reaction and vibrational deexcitation to v = 0.