The principle objective of the current work is to test the hypothesis advanced in Werdecker et al. and discussed in the Introduction of the current work. We remind the reader that this hypothesis asserts a positive correlation between the efficacy of ν 3 → ν 1 conversion induced in a methane–surface collision and the catalytic activity of the surface for the dissociative chemisorption of methane. To quantify the efficacy, we compute a branching ratio b r ν 1 / ν 3 using the following procedure. For each tagged rovibrational state η , we integrate the measured scattering intensity s η θ f over the range 11.5 ∘ = θ − < θ f < θ + = 47 ∘ of angles probed and then sum the resulting integrated fluxes S η = ∫ θ − θ + s η θ f d θ f over the different tagged rotational states of each stretch. We obtain a measure of the total scattered flux S ¯ ν a = ∑ η ∈ ν a S η of each stretch fundamental ν a . From S ¯ ν 1 and S ¯ ν 3 , we obtain the branching ratio b r ν 1 / ν 3 = S ¯ ν 1 / S ¯ ν 3 .

The branching ratios b r ν 1 / ν 3 obtained from the state-resolved angular distributions are shown in Table 1, along with the branching ratio of zero implied from the results of Werdecker et al. in their fixed scattering angle measurements for the graphene-covered nickel surface. Tabulated along with the branching ratios are the associated barrier heights for dissociative chemisorption of CH₄ as determined by density functional theory (DFT) calculations. We note that the barrier height quoted for Gr/Ni(111) was in fact computed for free-standing graphene, although the addition of a Ni(111) substrate is not expected to alter this figure significantly ¹ . Taking a low barrier height as an indication of high catalytic activity, it is clear that the order of the branching ratios and barrier heights for these three surfaces agrees with the stated hypothesis.

To assess the repeatability of the branching ratio measurements the full set of measurements for the Au(111) surface was performed twice. The data from the second run showed a somewhat better signal-to-noise ratio and so we present the branching ratio of 6.6% obtained from this run in Table 1. The first run yielded a branching ratio of 7.9%. The percentage deviation of 100 % × 7.9 % − 6.6 % 7.9 % + 6.6 % = 9 % between runs is of the same order of magnitude as those measured for other quantities derived from the same runs (e.g., the mean rotational energies E r o t . ν 1 and E r o t . ν 3 presented in Section 3.3). As there is a lack of necessary data for a more rigorous statistical analysis, we take ± 10 % as a reasonable estimate of our branching ratio uncertainty for the Ni(111) surface as well as the Au(111) surface, with the experimental procedures and data quality for the two surfaces being largely identical. From our own analysis of the signal-to-noise ratio of the measurements presented in Werdecker et al. on Gr/Ni(111), we arrive at a rough estimate of the upper limit of the branching ratio for this surface of 2%. We therefore consider the branching ratio order presented in Table 1 to be reliable.

In computing the branching ratios, we integrate over the scattering angle and sum over rotational states and thus discard any dynamical information contained in the angular and rotational state dependence of the scattering intensities. To explore in finer detail the dynamics of the molecule–surface collision, we return in this section to the state-resolved angular distributions. In the following section (Section 3.3), we narrow our attention to the gross rotational dynamics via the scattering angle-integrated state fluxes S η .

We have found it useful to summarize individual state-resolved angular distributions by three parameters, namely, m , θ max ⁡ . , and s max ⁡ . , which characterize a distribution’s width, angle of peak intensity, and peak amplitude, respectively. The parameters are determined by performing a non-linear least squares fit of the measured data using a fit function of the following form: s θ f = s max ⁡ . ⋅ cos m θ f − θ max ⁡ . (1)

That the chosen fit function gives a good description of our observations can be verified by inspection of Figure 2 which shows measured data overlaid with their associated best-fit curves (see also the Supplementary Materials for full set of angular distributions and fits).

Figure 3 shows the best-fit parameter values m , θ max ⁡ . , and s max ⁡ . for all 28 state-resolved measurements (although we include in Figure 3 the peak amplitude parameter s max ⁡ . for completeness, we omit discussion of this parameter, deferring discussion of the related scattering angle-integrated state populations S η for Section 3.3). From the figure, one can readily deduce that for both surfaces and for all lines tagged the scattering mechanism appears to be “direct” or “impulsive” in that the measured distributions strongly deviate from the m = 1 , θ max ⁡ . = 0 ∘ behavior expected of a “trapping/desorption” scattering process characterized by an extended molecular residence time at the surface (Rettner and Auerbach, 1994). Such a lack of equilibration between the surface and the scattering methane molecule has been seen before both in experiment and calculations for Ni(111) (Milot et al., 2001; Al Taleb and Farías, 2017) and Pt (111) (Kondo et al., 2018).

In addition to our classification of the scattering as direct/impulsive, we also identify three distinct trends from the analysis of the angular distributions. First, the exponents m (Figure 3, top row) are systematically higher for scattering from Ni(111) than those from Au(111), reflecting a lower degree of angular dispersion caused by the former. Second, for both Ni(111) and Au(111), the widths of the scattering peaks increase with the rotational quantum number J , suggesting a correlation between rotational excitation and momentum transfer either between the molecule and the surface or between the parallel and perpendicular components of the molecular momentum.

Third, the peak angles θ max ⁡ . (Figure 3, middle row) for scattering from Au(111) are shifted further away from surface normal compared to those for scattering from Ni(111). In a recent publication, we reported (Reilly et al., 2023) the same finding for CH₄/Ni(111) scattering with CH₄ prepared in the vibrational ground state and noted in addition a positive trend between θ max ⁡ . and total angular momentum quantum number J . From Figure 3, it is clear that this correlation holds for the vibrationally excited molecules as well, the trend being somewhat more pronounced for Au(111). Note that a positive θ max ⁡ . – J correlation is what one anticipates for the scattering of a rigid body from a flat and rigid surface based on simple considerations of conservation of energy and parallel momentum (see the red curve of Figure 3). One might therefore conclude that the CH₄/Au (111) system more closely approaches this ideal than the CH₄/Ni(111) system. However, in the limit of perfect flatness and rigidity, one also expects a unique scattering angle for each final state (i.e., m → ∞ ) . On this account, the broader peaks (characterized by smaller exponents m ) observed for scattering from Au(111) would then imply the opposite conclusion. Our results do not therefore admit a conclusive determination of which surface appears more “mirror-like” to a scattering methane molecule. Of course, a vibrating molecule cannot in the first place be said to be a rigid body, and certainly for the ν 3 → ν 1 scattering channel, the situation is further complicated by the change in the vibrational mode structure and accompanying energy release.

Going beyond the trends just discussed, we can also make a few additional qualitative observations. We first point out the anomalously large exponent m for the elastic ν 3 , J = 1 → ν 3 , J = 1 channel on Ni(111) is 38% larger than the next-largest ν 3 channel exponent and 61% larger than the largest ν 1 exponent for the same surface. In a practical sense, this fact, combined with the fact that the elastic channel on Ni(111) is relatively intense, establishes that the angular distributions for the vibrationally elastic and inelastic channels can in fact differ and that it is, therefore, essential to measure overall scattered angles to obtain the most accurate estimate of the total ν 3 → ν 1 scattering probability. Indeed, when we compute “fixed angle branching ratios” ∑ η ∈ ν 1 s η θ f / ∑ η ′ ∈ ν 3 s η ′ θ f from our Ni(111) measurements, we find that their values vary by more than 50% over the range of scattering angles θ f probed (Supplementary Section S2).

In another sense, that the vibrationally and rotationally elastic scattering peak is so much narrower than the others and should imply something about the molecule–surface interaction potential and associated scattering dynamics. We offer the following simple microscopic explanation for a strong and anomalously narrow elastic scattering distribution peaking near specular ( θ max ⁡ . − θ i < 1.5 ∘ ). Of the full range of values of impact parameters characterizing the possible initial conditions of the scattering molecule (e.g., impact site, initial molecular orientation, and internal/external axis of rotation), there is a significant fraction which results in trajectories where the molecule is turned around at distances relatively far from the surface. At these distances, the interaction between the molecule and the surface manifests primarily as a mutually repulsive force directed along the surface normal. That is, for these trajectories, the molecules do not penetrate far enough for the molecule–surface interaction to either exhibit significant lateral surface corrugation or effect significant disruption of the molecule’s or surface’s internal structure. Thus, the various mechanisms available to induce a change in the rovibrational state or deflect the scattering molecule away from specular scattering (parallel-to-perpendicular momentum transfer, surface phonon creation/annihilation, and rovibrational inelasticity) would act only weakly, leading to a narrow elastic distribution peaking near specular scattering. On the other hand, trajectories permitting deep enough penetration for the interaction to effect change in the rovibrational state will also tend to disperse angularly due to the additional effects of lateral corrugation and phonon creation/annihilation, resulting in broader scattering distributions for the rovibrationally inelastic channels. Interestingly, the strength of these dispersive forces seems to be no greater for the trajectories undergoing vibrational state change than those simply changing rotational state.

To qualitatively summarize the explanation just presented, our observations can be interpreted as implying a certain order in which the different aspects of the molecule–surface interaction “turn on” as the molecule approaches the surface, with a two-body repulsive force preceding the onset of dispersive forces, which in turn precedes the onset of forces acting to distort or rotate the molecule.

We close the section with a discussion regarding the differences in peak widths ∝ 1 / m and peak angles θ max ⁡ . , particularly for the Ni(111) data, among the levels of equal J in the ν 3 stretch. These levels have the same nominal rotational energy (within ± 2 % ²⁰) but are distinguished by their associated Coriolis stack F b and wavefunction symmetry A c . The Coriolis stack of a level indicates, loosely speaking, the relative orientation of its vibrational angular momentum and its total angular momentum (parallel, perpendicular, or antiparallel for F − , F 0 , and F + , respectively) (di Lauro, 2020). Centrifugal coupling between rotation and vibration splits a Coriolis level ( ± 0.3 % worst case (Hecht, 1961a)) into “sublevels” of definite symmetry with respect to the T d point group of the CH₄ molecules (Hecht, 1961b), of which those of A 1 and A 2 symmetries belong to the meta-CH₄ nuclear spin isomer. The sublevels of a Coriolis stack are distinguished by the internal alignment of their total angular momentum, i.e., by the different molecule-fixed axes about which the molecule is more and less likely to be found rotating (again, loosely speaking).

Compare for example the best-fit m and θ max ⁡ . values for the ( ν 3 , J = 4 , F − , A 1 ) and ( ν 3 , J = 4 , F 0 , A 2 ) levels scattering from Ni(111). The peak of the F 0 level has a 25% larger best-fit exponent m and a peak angle θ max ⁡ . 1.5 ∘ closer to normal than that of the F − level. Similar differences occur between the A 1 and A 2 sublevels in the same ( ν 3 , J = 5 , F + Coriolis stack. These rather subtle details of the internal molecular motion have a significant impact on the scattering distributions and highlight the “dynamic” nature of the CH₄/Ni(111) interaction in that the form of the state-resolved distributions are not solely dictated by the state’s energy. In a recent publication published by the authors on surface scattering of CH₄ in the vibrational ground state (Reilly et al., 2023), we found strong evidence for a more extensive molecule–surface collisional energy exchange (i.e., equilibration) after oxidation of a Ni(111) surface than before oxidation. We also found that the forms (and amplitudes) of the sublevel angular distributions (the ground vibrational state has only F 0 Coriolis stacks) produced upon scattering from the oxidized surface were essentially indistinguishable, while scattering from the unoxidized surface produced differences in the peak widths and peak angles for the different sublevels comparable to those observed in the ν 3 vibrationally excited state. We argued that the lack of imbalances observed in the collisions with the oxidized surface was consistent with the other reported observations in that they reflected an equilibration among the sublevels caused by a prolonged molecule–surface energetic coupling acting to scramble any delicate internal rotational motion.

Concerning the Au(111) surface data, while the form of the distributions among ν 3 levels of equal J is similar, we will see in the following section (Section 3.3) that the level populations S η (related indirectly to the peak amplitudes s max ⁡ . ) differ dramatically depending on the internal alignment of the molecule’s total and vibrational angular momentum, so that the scattering for this system is, in this sense, no less “dynamic.”

Figure 4 shows the scattering angle-integrated rovibrational state populations S η . In addition to illustrating what is already evident from Table 1—namely, that the overall ν 3 → ν 1 conversion is much weaker for the Au(111) surface—the plotted distributions also reveal important additional features of the rotational dynamics of the molecular–surface collisions for the systems studied here.

One feature that is directly observable is the differences in the forms of the distributions of the vibrationally elastic ( ν 3 → ν 3 ) channels for the two surfaces, with the Au(111) distribution having a relatively narrow distribution concentrated at low J , suggesting a “colder” rotational population. To quantify the degree of rotational excitation in the different stretch fundamentals, we use two different measures. The first measure is the mean rotational energy E r o t . ν a = ∑ η ∈ ν a S η ε η / S ¯ ν a obtained by taking the S η -weighted average of the rotational energies ε η of the different tagged levels η of the associated stretch ν a . The rotational energy ε η of a level η = ν a , J , F b , A c is calculated by taking the energy difference Δ ε η o between the level η and the rovibrational ground state as determined by high resolution spectroscopy (Gordon et al., 2022) and subtracting the theoretically determined energy gap Δ ε ν a separating the ν a vibrationally excited state from the ground vibrational state of a hypothetical non-rotating methane molecule (Boudon et al., 2006). The second measure is the effective rotational temperature T r o t . ν a obtained by fitting the S η of the associated stretch ν a to a thermal (Boltzmann) distribution of the form S η ∝ g η e − ε η k T r o t . ν a , where k is the Boltzmann constant and g η = 2 J + 1 is the level degeneracy.

The extracted mean rotational energies E r o t . ν a and effective rotational temperatures T r o t . ν a (Figure 5 for fits) are presented in Table 2. The differences in the ν 3 distributions mentioned previously are reflected in the mean energies, with the E r o t . ν 3 value for Ni(111) being 39% larger than that of Au(111). The difference in effective temperatures T r o t . ν 3 is less pronounced, due to the greater influence of the enhanced J = 7 level population scattering from the Au(111) surface (level populations contribute linearly to E r o t . ν a , while the T r o t . ν a fitting procedure minimizes the sum of the squared logarithmic population deviations). That the effective rotational temperatures for all four combinations of surfaces and stretching modes lie well below the surface temperature of 473   K and implies a lack of equilibration between molecular rotation and the surface degrees of freedom, offering further evidence against a trapping-desorption scattering mechanism.

Perhaps the most interesting result emerging from this analysis is that the ratio of the degree of rotational excitation for molecules scattered into the ν 1 and ν 3 states differs dramatically for scattering from Au(111) versus Ni(111). The ratio of the mean rotational energies E r o t ν 1 / E r o t ν 3 is 71% higher for Au(111) than for Ni(111). Similarly, the ratio T r o t ν 1 / T r o t ν 3 is 72% higher for Au(111) than for Ni(111).

Why do the ratios E r o t ν 1 / E r o t ν 3 and T r o t ν 1 / T r o t ν 3 differ so dramatically for the two surfaces? We begin by pointing out that the ν 3 fundamental lies 103   c m − 1 in energy above the ν 1 fundamental (Boudon et al., 2006). A more general question thus poses itself: where does this extra vibrational energy go during an SIVR process? Since we cannot measure the speed of the scattered molecules or directly detect energy transfer to the surface, a complete experimental accounting of the energy balance is not possible. Nonetheless, our observations indicate that a greater portion of this excess energy is transferred to rotation in a collision with Au(111) E r o t ν 1 − E r o t ν 3 = 52   c m − 1 ) than is transferred in a collision with Ni(111) ( E r o t ν 1 − E r o t ν 3 = 14   c m − 1 ).

In the reduced-dimensional CH₄/Ni(111) SIVR calculations presented in Werdecker et al., the surface was held frozen and the molecule was forced to assume at all times the minimum energy orientation so that energy released in the ν 3 → ν 1 conversion was constrained to be transferred completely into translational kinetic energy of the scattering molecule. In light of our results, it would be of keen interest to learn how this energy is partitioned in a dynamical simulation that includes both molecular rotation and surface vibrations as degrees of freedom. It would be equally interesting to learn to what extent the catalytic C-H bond elongation discussed in Introduction is also operative on Au(111). Perhaps the enhanced ν 1 rotational excitation on Au(111) reflects a mechanism for SIVR on Au(111) that is fundamentally different from the catalytic effect thought to be operating on Ni(111). SIVR on Au(111) might, for example, be a purely “mechanical” effect arising from the violent recoil of the molecule experiences upon rebounding from the much more massive gold surface atoms. An influence of the surface atom mass on CH₄/metal scattering inelasticity has been observed in recent dynamical calculations (Gerrits et al., 2019; Jackson, 2022) which finds Baule-like molecule–surface energy transfer in CH₄/metal surface collisions. A measurement of CH₄/Cu (111) SIVR efficiency might shed light on the role of a mechanical mechanism. The catalytic activity of Cu(111) with a calculated barrier height of 1.32 eV (Bhati et al., 2022) falls in between that of the Ni(111) and the Au(111) surface. In light of both this and the ability of the lighter copper surface atoms to better absorb the shock of impact, a Cu(111) branching ratio weaker than that of Au(111) would stand as strong evidence for a mechanical SIVR mechanism operating on Au(111).

The last observation we wish to make about the rotational distributions concerns the ν 3 levels of equal J , a subject we began discussing at the end of the previous section (Section 3.2) in the context of angular distributions. Figures 4, 5 clearly show that scattering from the Au(111) produces a highly imbalanced distribution of populations among ν 3 sublevels of equal J . These imbalances imply a scattered molecular flux with a high degree of both internal alignment of the total angular momentum (indicated by the imbalance of the A 1 and A 2 sublevel populations within the ν 3 , J = 5 , F + Coriolis stack) and relative alignment of the total and vibrational angular momentum (indicated by imbalances among the Coriolis stacks of equal J for J = 3 , 4 , 5 ). From the restricted subset of levels tagged in this experiment, there appears to be a strong tendency for preservation in the CH₄/Au(111) scattering event of the parallel (i.e., F − ) alignment of total and vibrational angular momentum prepared by the pump laser. Although the ν 3 population imbalances within a given J are not as pronounced in the CH₄/Ni(111) scattering data, it is nonetheless interesting to note that the order of the imbalances for the two surfaces are identical for the levels probed.

At the present time, the authors have no explanation for the form and extent of this internal alignment of molecular rotation observed in the scattering. We plan to present more extensive results on these interesting alignment phenomena in CH₄ surface scattering in a future publication.