Observation of Partial Wave Resonances in Low-Energy O2–H2 Inelastic Collisions

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Science  06 Sep 2013:
Vol. 341, Issue 6150, pp. 1094-1096
DOI: 10.1126/science.1241395

Quantum Collision Course

Our experience of a world apparently governed by classical physics is a consequence of the fact that quantum mechanical effects average out in size regimes much larger than nanometers. Even at the molecular level, the quantized nature of rotational energy distributions is often obscured by averaging effects. Chefdeville et al. (p. 1094; see the Perspective by Casavecchia and Alexander) have observed a striking manifestation of quantized rotation in the scattering trajectories of colliding H2 and O2 molecular beams. The experimentally resolved partial wave resonances show essentially complete agreement with theoretical calculations and deviate starkly from classical collision paradigms.


Partial wave resonances predicted to occur in bimolecular collision processes have proven challenging to observe experimentally. Here, we report crossed-beam experiments and quantum-scattering calculations on inelastic collisions between ground-state O2 and H2 molecules that provide state-to-state cross sections for rotational excitation of O2 (rotational state N = 1, j = 0) to O2 (N = 1, j = 1) in the vicinity of the thermodynamic threshold at 3.96 centimeter−1. The close agreement between experimental and theoretical results confirms the classically forbidden character of this collision-induced transition, which occurs exclusively in a purely quantum mechanical regime via shape and Feshbach resonances arising from partial waves with total angular momentum (J) = 2 to 4.

The accurate description of collisions between individual molecules remains a long-standing goal in physical chemistry. In a classical picture, the outcome of a collision is determined by the interaction potential and initial conditions such as relative velocity and impact parameter, that is, the distance of closest approach between the molecules in the absence of any interaction. Surely, molecules are quantum objects in nature, and the collisions must be described in terms of interfering quantum-scattering states, or partial waves, instead. Each partial wave is the quantum mechanical analog of a classical trajectory with a given impact parameter and is characterized by a definite value of total angular momentum, J, which is conserved throughout the collision (1).

The contribution of individual partial waves to the scattering cross sections constitutes the ultimate information that can be obtained about a collision event. Whereas molecular scattering processes are nowadays described successfully by quantum mechanical methods throughout, the direct observation of individual partial waves in molecular collision experiments remains highly challenging. In fact, partial waves can only be distinguished when giving rise to scattering resonances that manifest as resolved peaks in differential or integral cross sections (DCSs and ICSs) as measured in crossed-beam experiments. Such very rare findings then provide a unique insight in the most critical part of the potential energy surface (PES), the transition state region (2). However, in most cases, the collision energy spread and the participation of many overlapping partial waves corresponding to the energetically allowed values of J tend to average the individual resonance signatures and still render them elusive to experimental observation.

Partial wave resonances were first observed for a few elastic collisions that involve simple atoms (35) and then for the F + HD → HF + D system. In this textbook three-atom reaction, the resonances could be identified by comparison of experimentally determined cross sections with the outcome of high-level electronic structure and quantum-scattering calculations. A clear oscillatory structure assigned to J = 12 to 14 partial waves appears in the collision energy dependence of the state and angle-resolved DCS (6), whereas a steplike feature characterizes the resonance behavior of the total ICS (7). More recently, resonance structures have been revealed in total ICSs for Penning ionization reactions of metastable He with H2 or Ar in the cold regime (8) and in state-to-state ICSs for low-energy inelastic scattering of CO with H2 (9). Yet, the identification of the contributing partial waves suffered from imperfect agreement with theory in the latter studies. Here, we report a joint experimental and theoretical study on O2–H2 inelastic collisions for O2 (N = 1, j = 0 → N = 1, j = 1) rotational excitation, a transition which violates the semiclassical propensity rules for rotational energy transfer involving a molecule in a 3Σ state (10). The results demonstrate the purely quantum mechanical nature of the process and offer a complete characterization of fully resolved partial wave resonances with close agreement between theory and experiment.

We performed our experiments with a crossed-beam apparatus, which allowed us to tune the collision energy (the relative translational energy of colliding partners with reduced mass μ and relative velocity vr) by varying the beam intersection angle, χ, between 12.5° and 90°: ET = ½μvr2 = ½μ(vO22 + vH22 − 2vO2vH2cosχ) (9, 11). H2 and O2 beams with low and matched velocities and high velocity resolution and quantum state purity were obtained by using cryogenically cooled Even-Lavie fast-pulsed valves (12) [see table S1 (13)]. Neat beams of para-H2 and normal-H2 were used and were characterized in the crossing region by 3 + 1 resonance-enhanced multiphoton ionization (REMPI) time-of-flight mass spectrometry using three-photon (C1Πu, v = 0 ← X1Σg+, v = 0) R branch transitions near 99,150 cm–1 (14). Populations >95% for j = 0 and <5% for j = 1 were deduced when using freshly prepared samples of para–H2 and 25% j = 0, 75% j = 1 when using normal–H2. The O2 beam (0.3% O2 in He) was probed by using 2 + 1 (3Σ0, v = 2 ← X3Σg, v = 0) and (3Σ1, v = 2 ← X3Σg, v = 0) REMPI transitions around 88,900 cm–1 (15). The relative populations of the three fine-structure components of ground rotational state N = 1 were estimated to be >85% in j = 0, <15% in j = 2 at E1,2 = 2.08 cm–1, and <0.5% in j = 1 at E1,1 = 3.96 cm–1, whereas those of first excited rotational state N = 3 were negligible (13). Cross-section measurements attributed to N = 1, j = 0 → N = 1, j = 1 rotational energy transfer were performed by probing the collision-induced population in the N = 1, j = 1 level as a function of the crossing angle. The initial residual population in N = 1, j = 1 level was offset by shot-to-shot background subtraction when triggering the probe laser and the O2 beam at 10 Hz with the H2 beam at 5 Hz. The integral cross sections for O2para-H2 collisions between ET = 3.7 cm–1 and 20 cm–1 are displayed in Figs. 1A and 2. Those for O2normal-H2 collisions are shown in Fig. 2. The excitation function displays well-resolved peaks (a, b, and c), with no observable difference for the collision partners para-H2 or normal-H2. The excitation function also appears unusual for a molecular collision process. The O2 (N = 1, j = 0 → N = 1, j = 1) transition violates the propensity rules in rotationally inelastic collisions of diatomic molecules in 3Σ electronic states (10). Thus, normal scattering cannot occur. Scattering can only arise from resonances, and the three observed peaks are proof of this resonance behavior.

Fig. 1 Collisional energy dependence of the integral cross sections for O2 excitation (N = 1, j = 0) → (N = 1, j = 1).

(A) Experimental data with para-H2 (open circles, with error bars representing the statistical uncertainties at 95% of the confidence interval). Each point corresponds to 40 consecutive scans of the beam intersection angle acquired between 30° and 12.5° with –0.5° decrement and 100 laser shots per angle; theoretical ICSs were convoluted with the experimental collision energy spread (solid line). (Inset) Energy-level diagram and excitation scheme of O2 in the N = 1 state. (B) Theoretical results: partial waves J = 2, 3, and 4 (solid lines); partial waves J = 1 and J = 5 to 7(dashed lines); integral cross section (dashed-dotted line). Positions of the bound and quasi-bound states (see Fig. 3) labeled with their quantum numbers {N, j, l} (see text) are shown by vertical dashed lines. a.u., arbitrary units.

Fig. 2 Collisional energy dependence of the integral cross sections for O2 excitation (N = 1, j = 0) → (N = 1, j = 1).

Experimental data with para-H2 (open circles, data of Fig. 1A) and normal-H2 (open triangles). Error bars and scan parameters for normal-H2 as defined in Fig. 1A but with 29 consecutive scans of the beam intersection angle.

To gain insight into the resonance structure, we performed full quantum close-coupling scattering calculations by using a four-dimensional ab initio PES treating O2 and H2 as rigid rotors, which had been recently obtained by the coupled cluster method while using single and double excitation with perturbative contributions from connected triple excitations and large sets of atomic basis orbitals (16). Because the well depth of the PES strongly affects the energy, width, and intensity of the scattering resonances, we performed additional electronic structure calculations at a higher level of theory to obtain a better description of this critical parameter (fig. S1) (13). We estimated that 5% of the interaction energy was missing in the potential well. Therefore, we scaled the global PES by a factor of 1.05. A previous theoretical study on O2–H2 inelastic scattering neglecting the fine structure showed that rotation of H2 has almost no influence on the magnitude of the cross sections, including the resonances (16). We thus restricted the calculations to the j = 0 level of H2 (i.e., the PES was averaged over H2 rotations), considering that our results should also be valid for H2 (j = 1). For the scattering calculations, we solved the quantal coupled equations in the intermediate coupling scheme by using the MOLSCAT code (17) modified to take into account the fine structure of the O2 energy levels (18, 19). Cross sections were obtained up to ET = 50 cm–1 on a 0.05-cm–1 grid. The results are presented in Fig. 1B for partial waves J = 1 to 7, which contribute at the collision energies sampled in the experiment. The theoretical ICSs convoluted with the experimental collision energy spread are also reported in Fig. 1A.

The agreement between experiment and theory in Fig. 1 is very good. The convoluted theoretical curve reproduces well the position and width of the three experimental peaks. The experimental excitation function falls almost to zero at ET = 20 cm–1, which indicates a negligible contribution of collision energy transfer in the final observed state N = 1, j = 1 from N = 1, j = 2 residual population in the O2 beam (see also fig. S2) (13). Furthermore, Fig. 2 demonstrates that inelastic scattering with H2 (j = 1) behaves the same as with H2 (j = 0), which justifies the theoretical assumption made (vide supra).

A comparison of experimental and theoretical data shows that each peak in the experimental excitation function corresponds to an almost pure partial wave: peak a to J = 2, peak b to J = 3, and peak c to J = 4. The contributions from partial waves J = 1 and J > 4 and the overlaps between J = 2, 3, and 4 remain marginal. To gain insight into the nature of the peaks, we calculated adiabatic-bender potentials (20) and searched for all van der Waals stretch levels supported by these curves (21). The levels are labeled as N, j, and l, where l is the orbital angular momentum. Figure 3 shows the J = 2 potential curve, which correlates with O2 (N = 1, j = 1) + H2(j = 0). This curve supports one quasi-bound state at E1,1,2 = 4.36 cm–1, slightly above the asymptotic value at E1,1 = 3.96 cm–1 but trapped below the centrifugal barrier. Tunneling through the barrier thus gives rise to a shape (or orbiting) resonance (5, 22). Another J = 2 curve, which correlates with the asymptotically closed channel O2 (N = 3, j = 4) + H2(j = 0) at E3,4 = 16.39 cm–1, furnishes a different scenario. The O2–H2 complexes are temporally trapped in the bound state at E3,4,4 = 6.19 cm–1 before dissociating to O2 (N = 1, j = 1) + H2 (j = 0), yielding a Feshbach resonance (23). Therefore peak a can be regarded as the juxtaposition of a shape resonance in its rising edge and a Feshbach resonance in its falling edge. The J = 3 curve, which correlates with the O2 (N = 3, j = 4) + H2 (j = 0) closed channel, supports a bound state at E3,4,7 = 10.94 cm–1 in good agreement with the center of peak b. Peak b is thus a pure Feshbach resonance. Analysis of J = 4 is less straightforward, because adiabatic-bender potentials correlating with O2 (N = 3, j = 2, 3, 4) + H2 (j = 0) closed channels at E3,2 = 16.25 cm–1, E3,3 = 18.34 cm–1, and E3,4 = 16.39 cm–1 are found to support six bound and quasi-bound states situated between 15.3 and 17.1 cm–1 (fig. S3). Peak c is a composite of Feshbach resonances.

Fig. 3 Adiabatic bender curves with total angular momentum J = 2 and 3 that correlate with O2 states (N, j) = (1, 1) and (3, 4).

Quasi-bound states below (solid lines) or above (dashed lines) the dissociation limit are labeled with their quantum numbers {N,j,l} (see text).

Our results highlight the purely quantum mechanical nature of this inelastic collision process. None of the partial waves exhibits a long tail, which would indicate nonresonant rotational energy transfer. Successful O2 (N = 1, j = 0 → N = 1, j = 1) excitation exclusively occurs via shape and Feshbach resonances. Our results also validate the PES, which can now be used with confidence to calculate precise fine-structure resolved low-temperature rate coefficients for rotational (de-)excitation of O2 (X3Σg) by H2. This regime is of particular importance for astrophysics because, in the interstellar medium, O2 rotational level populations result both from radiative transitions and from inelastic collisions with overwhelmingly abundant H2. Evaluation of O2 abundance from spectral line data (24) requires the accurate knowledge of these rate coefficients.

Supplementary Materials

Materials and Methods

Figs. S1 to S3

Table S1

References (2533)

References and Notes

  1. Materials and methods are detailed in the supplementary materials on Science Online.
  2. Acknowledgments: This work extends the objectives of ANR-12-BS05-0011-02 contract with the Agence Nationale de la Recherche and contract 2007.1221 with the Conseil Régional d’Aquitaine, for which financial support is gratefully acknowledged. S.C., S.Y.T.v.d.M., C.N., and M.C. acknowledge support of Partenariat Hubert Curien Van Gogh 2013–28484TH contract. S.Y.T.v.d.M. acknowledges support from the Netherlands Organisation for Scientific Research (NWO) via a VIDI grant and from the Université de Bordeaux for a visiting professorship. Y.K. acknowledges the support of High Performance Computing SKIF-Cyberia (Tomsk State University). Y.K. and F.L. acknowledge the financial support of the CNRS national program Physique et Chimie du Milieu Interstellaire and of the Contrat de Projets Etat-Région Haute-Normandie/Centre National de Recherche Technologique/Energie, Electronique, Matériaux.
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