Communication: Importance of rotationally inelastic processes in low-energy Penning ionization of CHF₃

Scattering processes between particles in the gas phase include elastic, inelastic, and reactive collisions. In general, each of these processes has its own reaction rate, and a complete understanding of the collision dynamics requires knowledge of all these rates. Because they all have an individual dependence on the collision energy, the complete characterization of molecular scattering events is a highly complex task, both theoretically and experimentally. While in chemistry one is interested foremost in the reactive rates, studies of elastic and inelastic scattering processes are essential for understanding of molecular scattering and fundamental processes that determine chemical dynamics.^1–8 

In recent years, detailed investigation of scattering processes has attracted considerable interest in particular in the context of cold molecule formation and trapping.^9,10 Recent experimental developments have opened the route to investigations of molecular scattering in supersonic expansions at collision energies substantially below 10 K.^5,11–21 The collision energy depends on the relative velocity of the reactants. In a molecular beams experiment this is determined not just by individual beam speeds but also by the angle between them. Traditional crossed-beam measurements are performed at an angle of ninety degrees and, given the high speeds of supersonic expansions,²² cannot reach below 50 K. This domain is currently accessible only by reducing the angle between the beams.^5,16 To reach energies below 1 K, a complete superposition of the two beams is necessary which is the basis of the merged beams technique. Several studies in the past years have demonstrated the power of this approach, addressing different aspects of Penning ionization reactions in collisions of excited rare gas atoms with other atoms, diatomics, or light polyatomic molecules.^{13–15,17–21}

Only a handful of low-energy collision studies of polyatomic molecules have been performed to date.^18–21 In general, higher mass and increasingly complex internal structure of the colliding molecules open more inelastic and reactive channels. Therefore, the complete description of a cold chemical reaction must account for all possible reaction routes and also include inelastic collisions.

Here, we investigate the contribution of rotationally inelastic collisions to the total reaction cross section as a function of collision energy. Using the experimental setup shown in Fig. 1 and described in Ref. 17 we produce and superpose pulsed supersonic expansions of He(³S₁) and CHF₃. The collision energy is given as $E coll = μ 2 ( v 1 − v 2 ) 2$⁠, where μ is the reduced mass and the v_i are the beam velocities which are controlled experimentally via the choice of valve temperature and carrier gas. In the current reaction we tune the collision energy between 500 mK and 120 K, thus covering almost three orders of magnitude in energy. Because the ionization energy of CHF₃ is less than the internal energy of He(³S₁) such collisions can lead to Penning ionization, accompanied by dissociation. Specifically, the following principal reactive routes are energetically accessible and observed:²³ 

CHF$3 +$ itself is not observed, presumably because all of the accessible states readily (pre-)dissociate.^23,24 Threshold energies for the formation of CF$3 +$⁠, CHF$2 +$⁠, and CF⁺ are 13.9 eV, 14.2 eV, and 17 eV, respectively.²³ The ionized molecules are detected in a time-of-flight mass spectrometer, and product ions CF⁺, CHF$2 +$⁠, and CF$3 +$ are counted individually to monitor reaction channels separately as a function of collision energy.

In the present work we probe a wide range of energy regimes, from low collision energies where no channels for rotationally inelastic collisions are open, to energies where rotationally inelastic collisions become possible. By recording the probability of reactive collisions and comparing it with the calculated total collision cross section we obtain—in an indirect fashion—the probability for inelastic collisions.

Theoretical modeling of collisions between polyatomic molecules and electronically excited atoms is an extremely challenging task. Rigorous treatment of Penning ionization process involving a molecule would require obtaining the relevant potential energy surfaces with the precision that is currently beyond reach, even using state-of-the-art numerical procedures. However, effective models can be constructed to capture the essential features of the problem.^25–27 In the current study, the rotational structure of CF₃ is expected to have strong influence — the CF₃ rotational constants of A = B = 0.3452 cm⁻¹ and C = 0.189 25 cm⁻¹ are comparable to the experimental energy scales. As the threshold energies of different rotational states are given by E_J,K = CJ(J + 1) + (A − C) K², increasing the kinetic energy opens new channels, which makes the scattering problem even more complex. On the other hand, the channels describing the reaction products lie energetically much below the entrance channel and one can assume that there is no flux back to the initial state. Here we do not aim to do a full scattering calculation involving all the relevant channels, but only to construct an effective model to explain the observed rate of ionization events. To this end, we make use of the description based on quantum defect theory (QDT), which has been successfully applied before to describe Penning ionization of the NH₃ molecule.^19,20 The model relies on the knowledge of the interaction potential at long range and an effective parametrization of the short-range dynamics (see Refs. 19 and 28 for details). The necessary parameters of the model are the short-range reaction probability P_re and the short-range phase ϕ that sets the scattering length of the entrance channel potential and controls the positions of shape resonances. These quantities are used to construct short-range boundary conditions for the wave function in the region where the potential is still deep, but dominated by the asymptotic van der Waals terms. This boundary condition can be conveniently written in a WKB-like form with the outgoing flux modulated by a factor $1 − P re$⁠. One then performs scattering calculations with the long-range potential only. Both P_re and ϕ can in principle depend on the collision energy and partial wave quantum number, and are thus sensitive to the presence of rotationally excited states. Away from the energy thresholds, their energy dependence is usually weak and in previously studied systems assuming constant values turned out to be a good approximation.^19,28 In our simple model, we only aim to account for the impact of rotationally excited states on our QDT parameters.

The first possible effect of the rotational structure is resonances that may occur due to coupling to bound states supported by the closed channels. At a resonance, the short range phase varies rapidly. The second consequence is that multiple open exit channels, corresponding to rotational excitations, appear as the collision energy grows, so the probability for reaction should decrease. It is possible to estimate the density of states affecting the reaction just by knowing the rotational constants and using simplified potential energy curves.^29,30 For simplicity, we assume that all the possible inelastic processes that lead to rotational excitation have similar probability. If this is the case, then the probability for reactive collision can be described as

where P₀ is the reaction probability at zero energy, which was fitted to P₀ = 0.1. $N$ counts the number of energetically accessible rotational states and γ is a fitting parameter describing the branching ratio between reactive and inelastic events, assumed to be constant over the entire range of energies studied here. For each calculation we randomly choose energies at which π-shifts of the short-range phase occur, resulting in resonances with density approximately governed by the expected density of states. Naturally, the distribution of these resonances depends on the level structure of the molecule, and thus on both potential and rotational constants.

The last thing that is needed is the interaction potential at long range, which is dominated by van der Waals attraction. We have obtained the leading coefficients for dispersion $C 6 , disp 00 =350.9$ a.u. and induction $C 6 , ind 00 =132.9$ a.u. using symmetry-adapted perturbation theory (SAPT(UHF))³¹ and suitable asymptotic expressions of the multipole expansion, respectively.³² Moreover, in order to estimate the well depth of the molecular complex we also performed SAPT(UHF) calculations up to the second order in intermolecular interaction operator. The global minimum of 72.64 cm⁻¹ corresponding to the collinear He(³S₁)–HCF₃ geometry was found.

Fig. 2 shows the most important results of our study. As seen in Fig. 2(a), the branching ratio between the three accessible reaction channels is nearly constant over the entire energy range covered here. The experimental rate coefficient for formation of CF⁺ is shown in Fig. 2(b) as black dots (the experiment does not provide absolute rate coefficients; the scale is determined through the calculations). The red dashed-dotted curve in that panel is the rate coefficient from our QDT model calculations in the absence of inelastic collisions. Dashed blue line is the prediction from a pure Langevin capture model that for van der Waals interactions predicts the reaction rate constant to scale with $E coll 1 / 6$⁠. Clearly, our results do not follow any power law behavior. The impact of rotational excitations is visible above 10 K where the red curve distinctly overestimates the rates. Satisfactory agreement is achieved only when inelastic collisions are included, as shown by the black solid line in Fig. 2(b). The minimum of the reactive rate coefficient is located at the energy corresponding to the binding energy of the excited complex, and its presence is independent of inelastic collisions.^25,33 The position of this minimum obtained from QDT calculations closely matches the experimental result.

Theoretical calculations do not correctly describe the results at the lowest collision energies. This behavior has been observed also in previous studies of NH₃ Penning ionization and in the He^∗ + H₂ reaction¹⁵ and can be attributed either to a small potential barrier which is not found in our potential calculations or to enhanced coupling to another state. Further studies are needed to understand the source of disagreement.

In order to further explore the features of our model we show, in Fig. 3(a), results of calculations in which we changed the rotational constants of the molecule while keeping the interaction potential of the He(³S₁)–CHF₃ complex. The dashed-dotted green and solid black lines were computed assuming rotational constants for NH₃ and CHF₃, respectively, and the dashed red line corresponds to a molecule with rotational constants ten times smaller than CHF₃. Because of the large rotational constant of NH₃ most inelastic collision channels remain closed and the green curve in Fig. 3(a) only slightly differs from the red curve in Fig. 2(b) at high energies. Moreover, we assume that the probability for rotationally inelastic scattering is directly proportional to the number of accessible channels. In consequence, CHF₃ with rotational constants³⁴ of A = B = 0.3452 cm⁻¹ and C = 0.189 25 cm⁻¹ should experience more rapid drop in the probability for reactive scattering than NH₃ with rotational constants³⁴ of A = B = 9.4443 cm⁻¹ and C = 6.1960 cm⁻¹. This effect is shown in Fig. 3(b). Even though the direct proportionality assumed in our model is a rough approximation,⁶ we stress that it allowed to achieve satisfactory agreement with the experimental data by including the possibility for inelastic scattering. Finally, for comparison, Fig. 3(c) shows the experimental results from our study of the Ne^∗ + ND₃ reaction.¹⁹ Both experiment and theory confirm that for light molecules with large rotational constants it is not necessary to explicitly include inelastic scattering channels.

Our simplified theoretical treatment allows only to estimate the Penning ionization rates and is not suitable for calculating the rates for production of rotationally excited states. However, a full multichannel model would inevitably contain a large number of free parameters needed to describe couplings to open channels. We stress that due to experimental energy resolution and very large number of channels, the detailed behaviour of each single inelastic process would anyway be washed out and change the ionization rates only slightly. As a result, our QDT model works well despite its crudeness. A detailed theoretical study of competing reactive and inelastic processes in CHF₃ would allow to understand the system more deeply, but for now remains an open challenge.

We have demonstrated that rotationally inelastic collisions may drastically reduce Penning ionization rates, even in the low-energies regime of a few Kelvin. This effect is related to the magnitude of the rotational constant which appears to be critical. When the rotational constant is low, a considerable number of inelastic collision channels may be open and suppress reactivity. Indeed, suppressed reactivity at high energies has been observed in the case of CHF₃, but not in any of the lighter systems.^{13–15,18–21} The principal difference between CHF₃ and CH₃F or ammonia is larger mass and smaller rotational constants. This leads to a different appearance of the energy dependent reaction rate coefficient. Based on these results we suggest that this is a general property of molecular collisions: collisions of heavier molecules will be largely determined by inelastic scattering, even at the lowest temperatures. This striking result means that classical scaling laws that are known to work for barrierless reactions of light molecules cannot be applied to polyatomic molecules.

This work was supported by Foundation for Polish Science within the START program, Alexander von Humboldt Foundation, Swiss Science Foundation (Grant No. PP0022-119081), and by EPFL.