The role of internal molecular degrees of freedom, such as rotation, has scarcely been explored experimentally in low-energy collisions despite their significance to cold and ultracold chemistry. Particularly important to astrochemistry is the case of the most abundant molecule in interstellar space, hydrogen, for which two spin isomers have been detected, one of which exists in its rotational ground state whereas the other is rotationally excited. Here we demonstrate that quantization of molecular rotation plays a key role in cold reaction dynamics, where rotationally excited ortho-hydrogen reacts faster due to a stronger long-range attraction. We observe rotational state-dependent non-Arrhenius universal scaling laws in chemi-ionization reactions of para-H2 and ortho-H2 by He(23P2), spanning three orders of magnitude in temperature. Different scaling laws serve as a sensitive gauge that enables us to directly determine the exact nature of the long-range intermolecular interactions. Our results show that the quantum state of the molecular rotor determines whether or not anisotropic long-range interactions dominate cold collisions.
The hydrogen molecule is the most abundant molecule in interstellar space and participates in many reactions of astrochemical relevance1. At low temperatures of several kelvins up to several tens of kelvins (typical to the interstellar medium), hydrogen can be found in two quantum states: nuclear spin anti-symmetric para-hydrogen, which remains in the rotational ground state, and the symmetric ortho-hydrogen, which is rotationally excited. The ratio between para- and ortho-hydrogen is an important parameter used to estimate the age of interstellar clouds2,3. It is thus critical to understand the dependence of inelastic and reactive processes on the internal rotational molecular quantum state.
At low temperatures of a few kelvins, fast barrierless reactions dominate the chemistry of interstellar space4,5,6. In the absence of an activation barrier, the rate-limiting part of the interaction potential shifts from the transition state, which is usually located at the reactants’ shortest separation, to the long-range part of the interaction potential. As a result, the conventional Arrhenius law for the dependence of the reaction rate with temperature is replaced by universal scaling power laws7 as first derived by Paul Langevin in 19058. In his pioneering work, Langevin treated ion and structureless atom collisions. For molecules, however, the long-range interaction is orientation-dependent.
The effect of molecular rotation in fast low-temperature reactions has been studied in the past, but understanding is limited. It was studied theoretically by Clary5 and Troe9 and colleagues, and the case of H2 molecule/ion reactions has been recently addressed by Dashevskaya and co-authors10. Additionally, rotational state-selective quenching rates have been experimentally studied by Olkhov and Smith11 at room temperature. However, experimental evidence of the role of the rotational quantum state of molecules in cold collisions has been entirely missing so far4.
In the experiments described here, we are able to measure the universal scaling of rotational state-selected reaction rates over a wide range of energies reaching sub-kelvin temperatures. The observed power law serves as a force gauge revealing the exact nature of the long-range interactions that govern collisions for a particular quantum level of the molecular rotor. Here we show that the rotational state of the molecule plays a central role in cold reactions by modifying the long-range forces that are responsible for the reactants’ capture at low temperatures. Contrary to the high-temperature regime, where the rotational state of the hydrogen molecule has no influence on the total measured reaction rate, at temperatures of several kelvins, rotationally ground-state and excited-state hydrogen molecules behave as distinct chemical species with different reaction rates.
Some of the prospects of controlling the long-range dominated reactions have been demonstrated by selecting the reactive channel via exchange statistics12 or electrostatic filtering of molecular conformers that have different dipole moments13. In our case, because molecular hydrogen possesses a quadrupole moment, we are able to demonstrate that the contribution of the purely anisotropic, orientation-dependent quadrupole–quadrupole interaction with an open-shell atom depends on the rotational state of the hydrogen molecule. When the molecule is in the ground rotational state, which has an isotropic spatial distribution of the nuclei, only interactions with an isotropic component such as van der Waals can contribute, while the purely anisotropic interactions average to zero. However, if the molecule is in an excited rotational state, the spatial distribution of the nuclei is no longer isotropic, and anisotropic interactions start playing a central role in the reaction dynamics. The quadrupole–quadrupole interaction scales as R–5, while the van der Waals attraction scales as R–6, so its strength overcomes the van der Waals contribution at the large separations that are most relevant to single-digit kelvin temperature range reactions. At these temperatures, the stronger long-range interaction for rotationally excited molecules leads to a dramatic increase in the reaction rate as well as a change in the universal Langevin scaling from the T1/6 that describes van der Waals long-range interactions to T1/10, corresponding to quadrupole–quadrupole interactions7. Our result is not limited to a particular collision partner of molecular hydrogen, and the reaction rates at low temperatures will be different for para- and ortho-hydrogen every time purely anisotropic forces are at play.
Langevin's theory predicts the universal scaling laws of reaction rates, kn ∝ E(n–4)/2n, as a function of the collision energy E, where n describes the scaling of the long-range interaction strength with intermolecular separation7. To date, some of these universal scaling laws have only been observed in ion-neutral reactions over a limited range of discrete energies between 8 K and 200 K in CRESU (Cinétique de Réaction en Ecoulement Supersonique Uniforme) experiments14,15. These measurements of absolute reaction rates revealed the equivalent thermal rate scaling laws k2 ∝ T–1/2 in the case of ion–dipole interactions and that k4 is independent of the collision temperature in the case of ion-induced-dipole interactions. For neutral reactants, inelastic and reactive molecular collisions with near unit probability have been observed at ultracold temperatures12,16,17,18,19, but the direct observation of how the internal molecular degrees of freedom affect the universal scaling has so far eluded experimental investigation. The difficulty has been to experimentally access both the warm and cold reaction regime below the several kelvins needed to distinguish between the low fractional power-law scalings that are characteristic of neutral reactants. Recently, in the case of permanent-induced dipole interactions, the reaction rate of He(23S1) + NH3 was found to be approximately proportional to E1/6 (ref. 20).
Only recently have experiments that span many orders of magnitude in collision energies, extending into the cold regime, been realized. Such methods, which perform collisions with merged supersonic molecular beams in the moving frame of reference, enable continuous tuning of the collision energy from hundreds of degrees kelvin down to 10 mK (refs 21,22). In a different implementation of the same idea, two beams have been merged with separate magnetic and electrostatic guides, leading to cold reactions20,23. In the method used here, one beam is magnetically aligned with a second unmanipulated beam21,22,24, as shown in Fig. 1. So far, these merged beam experiments have been used to study chemi-ionization reactions such as Penning ionization25, where one reactant is ionized upon collision with a second internally excited reactant. Examples of such reactions, which have attracted interest in astrophysical studies26, include . In these reactions, strong shape resonances have been observed, emerging as sharp peaks in the reaction rate21,24. The observation of sharp resonance structures indicates that these reactions strongly deviate from the universal limit. Indeed, in the case of He(23S1) collisions with H2, less than 1% of collisions are reactive at a collision energy of 1 K. These resonances have also been predicted to be sensitive to the rotational state of molecular hydrogen in collisions with F by Alexander and colleagues27. In Penning ionization, the electron transfer rate that is responsible for the ionization step is exponentially dependent on atom–molecule separation, as shown in Fig. 3. The repulsive short range interaction between He(23S1) and H2 leads to a separation larger than 9a0 (a0 is the Bohr radius) between the colliding particles at the position of the inner classical turning point at near-threshold energies (Fig. 2a). When the separation is so large, ionization is suppressed, explaining the observation of the low total ionization rate along with the shape resonances.
However, we can dramatically modify the short-range behaviour of the interaction potential without affecting the ionization width appreciably by exciting the 23S1 metastable helium to the 23P2 state (Fig. 3). In the 23P2 state, helium possesses a quadrupole moment, leading to a strong anisotropy in the interaction potential. Depending on the relative orientation of the H2 molecule and the 2p occupied orbital of He(23P2), the interaction may result in either a strongly repulsive ‘wall’ or an attractive well with a depth exceeding 5,000 K at short range (Fig. 2b). As the atom–molecule separation decreases, the anisotropy increases. When the anisotropy is on the order of the rotational level spacing of the molecule, it becomes strong enough to reorient the reactants to the most attractive configuration28,29 and the ‘event horizon’ or point of no return is reached (Supplementary Section, ‘Interaction details’). The reorientation shifts the classical turning point to less than 3a0, compared with more than 9a0 for the isotropic potential for H2 with both states of helium, leading to an increase of more than four orders of magnitude in the total ionization width (Fig. 3). Thus, the overall probability for a reaction to occur once the centrifugal barrier is overcome increases to unity as the reactants are inevitably pulled towards each other into a ‘black hole’ from which they will only emerge after reacting.
Results and discussion
The total He(23P2) + para/ortho-H2 reaction rates obtained experimentally, including both the Penning and associative ionization channel products, are shown in Fig. 4. The products of these channels include and HeH+, with weights of 90 and 10%, respectively for all the collision energies observed. In both cases the rate for He(23P2) is more than two orders of magnitude faster than for He(23S1), indicating the dramatic increase in reaction probability at short separations driven by the strong anisotropy21,24. In fact, the total reaction probability P is higher than 99% once the reactants are in the most attractive orientation, calculated according to the classical approximation25, , where Rc is the classical turning point, Γ(R) the ionization width, μ is the reduced mass and V(R) is the interaction potential.
Because the reaction takes place regardless of the complicated details that exist within the ‘event horizon’, only the long-range part of the potential is needed to describe the reaction. Similarly, it has been shown that inelastic rotational and spin–orbit scattering can be described using the details of ab initio calculated long-range non-adiabatic interactions30. The picture is further simplified because the anisotropy is not strong enough to reorient the reactants at long separations and an averaged, effective long-range potential emerges. Kotochigova and colleagues predicted that such averaging can give rise to an isotropic van der Waals interaction at long range in the universal limit, for ultracold Li + LiYb interactions, which are strongly anisotropic at short range31. In the case of He(23P2) + H2, the long-range interaction is characterized mainly by the van der Waals attraction, which scales as R–6, and the quadrupole–quadrupole interaction, with a different scaling of R–5. The specific type of the effective long-range potential is determined by the rotational quantum state of the molecule, which governs how anisotropy can come into play. Molecular hydrogen formed in the supersonic expansion consists of 75% ortho-H2 and 25% para-H2. Due to the fermionic nuclei statistics, the lowest rotational state of para-H2 is j = 0, where j is the molecular rotation quantum number, whereas the lowest rotational state of ortho-H2 corresponds to the first excited state, j = 1. To measure the reaction rate of para-H2 we produced pure para-H2 by liquefying normal-hydrogen in the presence of a catalyser (see Methods). The rate of ortho-H2 was determined by subtracting the para-H2 contribution from the normal-H2 measurements according to kortho = (4/3)(knormal – (1/4)kpara).
When the molecule is in the ground rotational state, j = 0, such as in the case of para-H2, the quadrupole–quadrupole interaction, which is purely anisotropic32, is averaged out at long-range because the distribution of the molecule's nuclei is isotropic. This leads to a reaction rate dominated by the higher-order van der Waals interaction, whose scaling law was first derived by Gorin7,33 and is written explicitly as
where the reaction rate is defined as , in which σ is the total cross-section and is the effective van der Waals interaction strength, which in this case equals our estimated isotropic coefficient . The observed rate of the He(23P2) + para-H2 reaction follows this theoretical universal absolute rate with the E1/6 scaling law over more than two orders of magnitude of collision energies down to 0.8 K (Fig. 4a, dashed red line). Below 0.8 K, the reaction rate deviates from the unitary limit, eventually slowing by more than 50% at the lowest collision energies.
The rate for the He(23P2) + ortho-H2 reaction down to ∼30 K is within the standard deviation of the He(23P2) + para-H2 results, consistent with the scaling law E1/6 where for ortho-H2, as shown in Fig. 4b (dashed red line). Because ortho-H2 is in the j = 1 rotational state, the contributions of the anisotropic components of the interaction are no longer zero and they emerge in the effective long-range potential. These contributions can account for the difference in effective strengths of the van der Waals interaction between ortho-H2 and para-H2. Below ∼30 K the difference in reaction rates of ortho-H2 and para-H2 becomes evident. The deviation from the van der Waals scaling law at low energies occurs because the purely anisotropic quadrupole–quadrupole interaction, which was averaged out for para-H2, where j = 0, is the leading order term in the effective long-range potential for the j = 1 state and now dominates at the lower collision energies. This leads to a reaction rate that scales as E1/10 and is explicitly given by
where is the effective quadrupole–quadrupole interaction strength7. The reaction rate with ortho-H2 follows the E1/10 scaling law with a fitted interaction strength of over more than three orders of magnitude of collision energy (Fig. 4b, dashed blue line).
To confirm that the change in the reaction rate scaling law at low collision energies is only due to the hydrogen molecule's rotational state, we performed additional measurements of the total rate of the He(23P2) + HD reaction (Fig. 4a, grey symbols). The products of the Penning and associative ionization channels include HD+, HeH+ and HeD+, with relative weights of ∼90, ∼3 and ∼7%, respectively, for all the collision energies observed. Because HD is a heteronuclear molecule, the lowest rotational state is j = 0, as in the case for para-H2, and the potential surfaces of the two molecules are nearly identical at long range, such that only the reduced mass is changed. Indeed, the resulting energy dependence of the reaction rate for the two isotopologues is nearly identical. The absolute scale of the H2 reaction rates was normalized according to the thermal rate measurement of He(23S1) + H2 by Oskam and colleagues34 in an afterglow study in helium–hydrogen mixtures at 300 K (see Methods). The systematic error in normalization, depicted by the green error bar in Fig. 4b, is much larger than the error in the theoretical . For convenience, the relative H2 rates were therefore scaled up by 22% from the green circle to match the theoretical rate prediction. The rate of He(23P2) + HD is given in arbitrary units and is scaled down by 30% relative to the H2 measurement.
Importantly, the sharp resonances, which are clearly visible in the reactions with He(23S1)21,24, disappear in the case of the much faster reactions with He(23P2). This is consistent with the behaviour predicted in the unitary limit, as was recently investigated by Jachymski and colleagues35. Below the p-wave barrier energy of 0.04 K for He(23P2) + ortho-H2, the observed rate assumes an approximately constant value. At this energy, there is no centrifugal barrier in the quantum limit as only s-wave scattering can take place and the scaling of the reaction rate becomes consistent with the prediction by the Wigner threshold laws36,37,38. The observed continuous transition from the classical Langevin to quantum behaviour can be used to test recent developments in quantum defect theory, extending it to the non-universal regime35,39. Note that our results showing stronger interaction for ortho-hydrogen are supported by the earlier spectroscopic measurements of bound van der Waals H2O–H2 complexes40. The linear scale insets of the reaction rates show the slow changes in the power-law behaviour (Fig. 4).
We have shown that universal Langevin scaling laws can be used to gauge the long-range forces responsible for reactivity in the cold regime, and have experimentally demonstrated that rotationally excited molecules experience stronger long-range attraction, leading to a faster reaction at the low energy range relevant to astrochemical processes.
The merged beam configuration used consisted of two Even–Lavie supersonic molecular beam valves41 placed at a relative angle of 10° (Fig. 1). Noble gas mixtures of para-H2/normal-H2/HD were entrained into a straight beam, which was aimed directly at the detection region in a time-of-flight mass spectrometer42 (TOF-MS). The second beam, which was composed of 4He, was excited to the metastable 23S1 state by a dielectric barrier discharge43 located directly after the valve. The 20 cm curved magnetic quadrupole guide turned the low-field-seeking metastable helium beam into the straight beam path so that the two beams had a relative angle of zero when they met in the TOF-MS. The relative mean velocity between the beams and the width of their velocity distribution, which was locally decreased by the short opening durations of the Even–Lavie valves, determined the resulting collision energy22. Tuning the mean velocities of the beams by seeding the reactants in noble gas mixtures and changing the temperature of the valve allowed the collision energy to be tuned continuously from 300 K to 10 mK.
Collisions of He(23P2) + H2/HD were realized by exciting He(23S1) in the fully stretched m = 1 state to the 23P2 state at 1,083 nm in the reaction volume of the TOF-MS (Fig. 1). A continuous-wave distributed Bragg reflected (DBR) diode laser stabilized by saturated absorption spectroscopy on a helium discharge cell was used for excitation well above the saturation intensity of the transition (0.17 mW cm–2)44, giving a 1:1 population ratio between the two helium states, forming a two-level system. The laser was σ+ polarized with the quantization, and the quantization axis was defined by a weak magnetic field perpendicular to the beam propagation direction. Excitation therefore occurred along the optical pumping transition between the He(23S1) m = 1 state and the 23P2 m = 2 state.
The H2/HD beam was characterized by operating the TOF-MS in a multi-pulsed mode, giving the beam intensity profile at four 4 μs intervals, and the He(23S1) beam was characterized with an on-axis microchannel plate (MCP). He(23P2) was estimated as half He(23S1) due to the 1:1 population ratio between the two. The reaction products were measured in the same way as the H2/HD beam was characterized, but with the ionizing element turned off. The relative Penning ionization reaction rate at each collision energy of He(23S1) + H2/HD, measured with the laser off, is given by in equation (3), where Nn is the measured signal of a species in the nth time interval. The total reaction rate He(23P2) + H2/HD was measured with the laser on and included both the Penning and associative ionization channels, depicted by , given in equation (4). In both cases the measurement took place over ten time intervals centred at the time when both beams were spatially and temporally overlapped in the TOF-MS.
The reaction rates were normalized to an absolute scale according to the thermal rate measurement of He(23S1) + H2 by Oskam and colleagues34 in an afterglow study in helium–hydrogen mixtures at 300 K. The thermal rate was calculated with our results, which were all below 300 K, and the rates at higher collision energies by Martin et al.45, which were measured in a crossed beam experiment. The rate for ortho-H2 was determined by subtracting the para-H2 contribution from the normal-H2 measurements.
Samples of para-H2 were prepared by liquefying normal-H2 with liquid helium while in contact with a catalyst of nickel(II) sulfate. The purity of the sample was determined with resonance-enhanced multiphoton ionization (REMPI) followed by detection with the TOF-MS, which indicated that more than 95% of the H2 was in the j = 0 state, compared to 25% in rotationally cold normal-H2. The 3 + 1 REMPI process makes use of the spectroscopically resolved three-photon transition along the R branch at 289.5 nm, followed by a fourth photon that ionizes the molecule46. Laser pulses (8 mJ in 10 ns) were generated by a pulsed dye laser (Sirah Cobra-Stretch, pumped by a Spectra Physics Quanta-Ray Pro) and focused by 500 mm lens. The observed rotational state abundance spectrum of H2 is shown in Supplementary Fig. 2.
Potential energy curve calculation details
The potential energy curves were calculated for the two most symmetric configurations: linear and T-shaped geometries. The potential energy curves for the interaction of He(23P) + H2 in the 3Π state for the linear geometry and the 3B2 state for the T-shaped configuration, shown in Fig. 2b, as well as He(23S1) + H2 in the linear and T-shaped geometries, were obtained with the supermolecular approach using the restricted open-shell Hartree–Fock (ROHF) method followed by spin-unrestricted coupled-cluster calculations with single, double and noniterative triple excitations [UCCSD(T)]. The 3B1 state of the T-shaped geometry was also calculated using this method and is shown in Supplementary Fig. 1. In the ROHF calculations, convergence to the desired excited state (corresponding to the interaction of H2 with He(23P)), was enforced by a suitable rotation of the occupied orbitals. A similar approach24,47 has already been successfully applied to He(23S1) + H2. The H2 bond length was kept fixed at r = 1.449a0.
For the 3Σ state of the linear geometry and the 3A1 state of the T-shaped geometry, which are shown in Supplementary Fig. 1, a different approach was used. For these symmetries we obtained the potentials in the spirit of the stabilization method. To this end, we performed standard calculations of the excited states employing linear response theory within the coupled-cluster singles, doubles and linear triple excitations (LR-CC3) and carefully identified states that correspond to the He(23P) + H2 asymptote. This procedure was repeated for a series of basis sets: d-aug-cc-pVTZ/-pVQZ/-pV5Z, yielding very similar results (for example, the well depth of the 3Σ potential differs by less than 2.5% between the d-aug-cc-pVQZ and d-aug-cc-pV5Z basis sets). Good convergence with respect to the size of the basis set allows us to assume that the calculated potential energy curves are a decent estimation of the real potentials. The presented interaction potentials are also fairly consistent with results obtained from symmetry adapted perturbation theory extended to open-shell systems48.
Calculation details for long-range interaction strengths
To compute the isotropic coefficient, the dipole polarizability for H2 at imaginary and real frequencies was taken from Bishop and Pipin49, and the polarizability for He(23P) was constructed from the sum-over-states expression. The dipole moments of the necessary states and transitions were obtained from multi-reference configuration interaction calculations with single and double excitations (MRCISD) with 27 orbitals in the active space. A doubly-augmented Dunning's d-aug-cc-pV5Z basis set was chosen for the hydrogen and helium atoms in all calculations. The potential calculations used the MOLPRO50 and DALTON51 ab initio quantum chemistry suites of codes.
Ionization width calculation details
Penning ionization width as a function of the He(23P) – H2 distance was calculated using the ab initio Fano–algebraic diagrammatic construction (Fano-ADC) method52,53,54 adapted to the decay of triplet excited states. A hole-localization-based configuration selection procedure52 was used to identify the initial state and final state subspaces within the extended second-order ADC scheme [ADC(2)x]. Ab initio calculations were performed using the fully uncontracted cc-pV5Z Gaussian-type basis55 augmented by 8s4p3d2f1g Kaufmann–Baumeister–Jungen continuum-like diffuse Gaussians56. Stieltjes imaging57 was used for renormalization and interpolation of the Penning matrix elements resulting from the L2 calculations. The Hartree–Fock solution for the ground state of the neutral system was obtained using MOLCAS 7.6 quantum chemistry package58.
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The authors thank R. Kosloff and R. Moszynski for discussions as well as O. Tal and D. Rakhmilevitch for advice and help in the generation of para-hydrogen. This research was made possible, in part, by the historic generosity of the Harold Perlman family. The authors acknowledge financial support from the European Commission through ERC grant EU-FP7-ERC-CoG 1485 QuCC (Y.S., A.K., E.N.), from the Alexander von Humboldt Foundation (W.S.), from the Lee Family Foundation (R.Y.) and from the UK's Engineering and Physical Sciences Research Council (V.A., R.Y.) through the Career Acceleration Fellowship (award EP/H003657/1) and the Programme Grant on Attosecond Dynamics (award EP/I032517), as well as from the Deutsche Forschungsgemeinschaft through Research Unit 1789 (V.A.).
The authors declare no competing financial interests.
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Shagam, Y., Klein, A., Skomorowski, W. et al. Molecular hydrogen interacts more strongly when rotationally excited at low temperatures leading to faster reactions. Nature Chem 7, 921–926 (2015). https://doi.org/10.1038/nchem.2359
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