Abstract
Anisotropy is a fundamental property of particle interactions. It occupies a central role in cold and ultracold molecular processes, where orientationdependent longrange forces have been studied in ultracold polar molecule collisions^{1,2}. In the cold collisions regime, quantization of the intermolecular degrees of freedom leads to quantum scattering resonances. Although these states have been shown to be sensitive to details of the interaction potential^{3,4,5,6,7,8}, the effect of anisotropy on quantum resonances has so far eluded experimental observation. Here, we directly measure the anisotropy in atom–molecule interactions via quantum resonances by changing the quantum state of the internal molecular rotor. We observe that a quantum scattering resonance at a collision energy of k_{B} × 270 mK appears in the Penning ionization of molecular hydrogen with metastable helium only if the molecule is rotationally excited. We use stateoftheart ab initio theory to show that control over the rotational state effectively switches the anisotropy on or off, disentangling the isotropic and anisotropic parts of the interaction.
Main
Anisotropy leaves fingerprints on many atomic and molecular processes, but its contribution is difficult to evaluate and quantify at high energies due to the many quantum states needed to describe the process. For example, the effects of anisotropy can be traced to changes in the angular distribution of products in collisions and halfcollisions that are detected with advanced ion imaging techniques^{9}. Numerous experiments have reported orientation effects including the study of bimolecular processes such as the reactive scattering of H + D_{2} (ref. 10), and F + CD_{4} (ref. 11) as well as the inelastic scattering of NO by Ar (ref. 12). Spectroscopic techniques have been successfully used to identify hindered H_{2} rotation in an anisotropic potential of van der Waals molecules such as H_{2}O + H_{2} and H_{2}HF complexes^{13,14}.
In cold and ultracold molecular processes, where the number of accessible quantum states is greatly reduced, anisotropy effects have so far been studied only in fast reactions. It has been shown that the orientation of the longrange dipole–dipole interaction between molecules can be tuned via external electric fields^{15}. In KRb molecules this method enabled the observation of rethermalization^{1} and the suppression of a bimolecular chemical reaction^{2}. In addition, the reaction of N_{2}^{+} + Rb has been shown to be dominated by the anisotropic charge–quadrupole longrange interaction^{16}. Similarly, fast Penning ionization reactions have been studied, demonstrating that the rotational state of a molecule defines the type and strength of the effective longrange interaction^{17}. These fast processes are fully described by the longrange interactions alone. At short internuclear separations the Penning ionization occurs with unit probability, acting as a perfect absorber and preventing the formation of outgoing waves. In such a case there is no quantization along the internuclear coordinate that precludes the appearance of quantum scattering resonances.
Experiments that are capable of resolving quantum resonance states allow for collision studies with near spectroscopic precision. For example, isotope shifts have been observed in the Penning ionization of molecular hydrogen isotopologues^{6}. Similar effects between isotopes were observed in rotational statetostate crosssections and linked to quantum scattering resonances appearing at different energies^{18}. Feshbach resonances in ultracold anisotropic atom–atom collisions allowed the emergence of chaotic scattering in ultracold Er and Dy to be observed^{19}.
The emergence of discrete quantized metastable states formed during a cold collision provides a new way to study anisotropy. Instead of recording the angular distribution of the collision products, one can conveniently alter the way anisotropy contributes by choosing different rotational quantum states of the interacting molecules. The sensitivity of the orbiting resonances to the molecular degrees of freedom has been theoretically predicted for the reaction D + H_{2} (ref. 20), the reactive scattering of F + H_{2} (ref. 21) and inelastic collisions of CO + H_{2} (ref. 22), for example. This effect has so far eluded experimental investigation due to the insufficient resolution of individual quantum resonance states^{22}. The introduction of merged beam experiments that can reach temperatures as low as 10 mK (refs 3,6) allows for both low enough collision energies and sufficiently high resolutions due to the correlation that develops during the free propagation of a supersonic molecular beam^{23}. This merged beam configuration is used in the current collision measurements, as described in the Methods.
Here we demonstrate the dramatic effect of anisotropy on shape resonances in cold Penning ionization reactions of metastable helium in the 2^{3}S_{1} state (He^{∗}) with molecular hydrogen in the rotational ground and first excited states (Fig. 1). In the classical collision regime down to temperatures of several kelvin, H_{2} behaves similarly, independent of whether it is in the j = 0 or j = 1 internal rotational state. However, below one Kelvin we observe a striking difference. While a strong shape resonance exists for j = 1 at a collision energy of k_{B} × 270 ± 20 mK, where k_{B} is the Boltzmann constant, no resonances are observed in this energy range in the case of j = 0. Using stateoftheart ab initio theory and coupledchannel quantum molecular dynamics we find that the interaction between hydrogen, which is internally in the ground rotational state, and metastable helium is effectively isotropic. This holds for any collision where the interaction anisotropy is small compared with the rotational constant of the molecule. In such a case, the anisotropic part of the interaction potential becomes important only for molecules in an excited rotational state. The wavefunction of rotationally excited H_{2} molecules is not spherically symmetric, which causes the anisotropic components of the interaction to come into play, whereas in the rotational ground state, where the wavefunction is spherically symmetric, the contribution of anisotropic components is zero. We show that by accurately measuring the orbiting resonance positions we can independently probe the isotropic and anisotropic parts of the interaction. As the resonance wavefunctions span both the short and the longrange parts of the potentials, we are able to benchmark firstprinciples quantum dynamics calculations to within 7 × 10^{−3} cm^{−1} or 200 MHz precision. This allows for a rigorous comparison with theory and a quantitative evaluation of the different levels of treating electron correlations in electronic structure calculations.
Conveniently, merged beam experiments with molecular hydrogen can be carried out with control over the rotational state of the molecule. A normal molecular hydrogen beam cooled in supersonic expansions consists of 75% orthohydrogen and 25% parahydrogen. It follows from the Pauli principle that the lowest rotational state for parahydrogen is j = 0 whereas j = 1 is the lowest molecular rotational state for orthohydrogen. A pure parahydrogen sample can be prepared by catalytic conversion at cryogenic temperatures, allowing the determination of the stateselected rate coefficient, as has been described previously^{17}.
In Fig. 1, we compare our experimental data to the results of firstprinciples calculations. A twodimensional potential energy surface was obtained with the current gold standard of electronic structure theory, the coupledcluster method with single, double and approximate noniterative triple excitations [CCSD(T)], and further refined to include higher order correlation effects. Subsequently, the potential energy surface was used in quantum scattering calculations with reactive boundary conditions at short range^{24}. The theoretical results demonstrate that collisions with j = 0 hydrogen are not sensitive to the anisotropic part of the interaction potential, such that the hydrogen molecule effectively behaves like an atom. Even scaling the anisotropic part, V_{2}, by 50% does not introduce an observable change in the rate coefficient, as can be seen in the inset in Fig. 2. In stark contrast the j = 1 hydrogen rate coefficient is very sensitive to the anisotropy, emerging as shifts in the position of the orbiting resonance for different scaling factors of the anisotropic potential shown in Fig. 2.
The sensitivity of quantum resonance states to small changes in the interaction allows us to make a meaningful comparison of different levels of treating electron correlations in ab initio theory (Fig. 1). In fact, the interaction potential obtained with the ‘bare’ CCSD(T)/augccpV6Z + midbond method is not accurate enough to reproduce the lowenergy resonances for collisions with both j = 0 and j = 1 H_{2} (see the green curves in Fig. 1). To account for higherorder electron correlation effects, the full configuration interaction (FCI/augccpVQZ) contribution beyond CCSD(T) has been calculated and added to the potential. The FCIcorrected potential energy surface performs significantly better (blue curves in Fig. 1), in particularly for collisions with H_{2}(j = 1). However, it still erroneously predicts a resonance in the millikelvin range for H_{2}(j = 0). To further improve the theoretical model, we have uniformly scaled the correlation energy. This is motivated by the fact that incompleteness of the orbital basis set is the main source of error in the ab initio calculations. It particularly affects the correlation energy, whereas the Hartree–Fock energy is almost fully converged in the augccpV6Z basis set (see Methods for details). In fact, increasing the correlation energy in the interaction potential by merely 0.4% leads to an excellent agreement for the resonance position of H_{2}(j = 1) and confirms absence of this resonance for H_{2}(j = 0) (red curves in Fig. 1). The latter can be understood as follows: scaling the correlation energy by 0.4% corresponds to increasing the depth of the isotropic potential by around 0.1 cm^{−1}. This shifts the erroneous resonance obtained with CCSD(T) and CCSD(T) + δFCI down such that it becomes a truly bound state and does not affect the reaction rate.
A similar comparison of the different levels of theory and experiment could in principle also be performed using rovibrational spectroscopy of bound states. However, while spectroscopic methods measure the spacings between rovibrational states with high precision, these spacings are not sensitive to a global energy shift of the potential. In collisional spectroscopy the spectrum emerges above the dissociation threshold enabling the measurement of the energies of quasibound states on an absolute scale. Thus changes in the effective well depth due to the anisotropy are more pronounced in absolute scale measurements of the spectrum (Fig. 3).
To qualitatively understand the role of anisotropy, it is convenient to consider the effective adiabatic interaction potentials. To this end, the interaction potential is expanded into two terms: V_{0}(R), the isotropic term where R is the interparticle separation and V_{2}(R)P_{2}(cosθ), which describes the angular dependence to first order, where θ is the polar angle between the axis of the H_{2} molecule and the axis joining the H_{2} centre of mass with the He atom. We use an adiabatic theory^{25} inspired by previous works^{26,27} to construct effective potential curves that conserve the total angular momentum J. These potentials asymptotically correlate with a welldefined rotational state of H_{2}(j) and a welldefined collisional angular momentum, that is, partial wave l. The anisotropy, V_{2}(R), is small in our case compared with the rotational constant for all collision distances, such that mixing of the rotational states of the same parity in the H_{2} molecule is negligible. Figure 3 compares the three adiabatic potentials that support the lowerenergy resonance state that asymptotically correlates with j = 0 and j = 1 H_{2} and orbital angular momentum (partial wave) l = 3. The black curve corresponding to J = 3 and j = 0 H_{2} is identical to the isotropic part of the interaction potential and includes the centrifugal term. It is deep enough to support a single bound state just below the threshold with a binding energy of k_{B} × 0.01 K, as shown in the inset of Fig. 3. In the case of j = 1 H_{2}, the anisotropy is responsible for reducing the well depth of the corresponding J = 3 effective potential, shown in red, enough to raise the bound state above the dissociation threshold (Fig. 3 inset). This state is located below the top of the centrifugal barrier, and is accessible via tunnelling in scattering experiments at a collision energy of k_{B} × 270 mK. The grey adiabat in Fig. 3 represents two nearly degenerate effective potential curves with the total angular momentum of J = 2 and J = 4 and j = 1 H_{2}. Both potentials do not support resonances and have bound levels at k_{B} × 0.18 K below the dissociation threshold. Following a similar procedure, one can show that the higherenergy resonance, appearing for j = 0 as well as j = 1, asymptotically correlates with partial wave l = 4.
Orbiting resonances amplify minute differences in particle interactions, making them easily detectable in cold collision experiments provided that the energy resolution is sufficiently high. They are a manifestation of the quantization of the intermolecular degrees of freedom, which form a wave matter analogue of an optical cavity. Quantum resonances thus provide an ideal probe to illuminate the different contributions to particle interactions with a sensitivity that is a challenge for current stateoftheart ab initio theory. The sensitivity of quantum resonances to the internal state of a molecule can be exploited in many experiments. Changing the rotational state of a molecule effectively switches the anisotropy of the interparticle interaction on and off, shifts resonance positions and thus dramatically changes the collisional crosssection. Such control over collisional properties can be critical to the success of molecular cooling methods that rely on collisions, such as evaporative or sympathetic cooling.
Methods
The experimental setup, which has been discussed previously^{3,6,17}, consists of two supersonic beams that are generated by pulsed Even–Lavie valves^{28} placed at a relative angle of 10°. The straight beam consists of noble gas mixtures of H_{2}. The second beam consists of ^{4}He, excited to He^{∗} using a dielectric barrier discharge^{29} that is located at the exit of the valve and is merged with the straight beam using a 20 cm magnetic guide, leading to zero relative angle between the beams at the detection region. The He^{∗} beam velocity is kept constant at 870 m s^{−1} while the H_{2} beam velocity is tuned by changing the valve temperature and seeding the H_{2} in noble carrier gas mixtures. The collision energy is determined by the relative mean velocity between the beams and the velocity distribution of each beam ranging from 300 K to 10 mK (ref. 23).
The metastable He beam is characterized using an onaxis multichannel plate. The H_{2} beam is characterized using a timeofflight mass spectrometer (TOFMS)^{30}, which contains an additional ionization element. The TOFMS is positioned perpendicular to the beams’ propagation, operating in a multipulsed mode, at 4 μs intervals. The reaction product ions are detected with the TOFMS with the ionization element turned off, while the neutral molecules are characterized with the ionization element turned on. The relative rate coefficient is calculated according to the ratio between the measured signals of the products and reactants at each time interval. The rate coefficient is normalized to an absolute scale according to roomtemperature thermal rate measurement of He(2^{3}S_{1}) + H_{2} (ref. 31).
The paraH_{2} sample is prepared by liquefying normal H_{2} in liquid helium while in contact with the paramagnetic catalyst nickel sulfate. The sample is examined by resonanceenhanced multiphoton ionization followed by detection with the TOFMS to determine its purity, showing that more than 98% of the H_{2} is in the ground rotational state, as previously described in detail^{32}.
Rate coefficients were obtained by carrying out coupledchannel quantum reactive scattering calculations using a method^{24} in which only the reactant configuration is explicitly considered, and knowledge of the real value of the interaction energy is sufficient. This approach is based on the assumption that the reaction proceeds irreversibly once the reactants reach a sufficiently short interparticle distance R_{c}, the ‘capture radius’; and reactive collisions are simulated by imposing appropriate shortrange boundary conditions. Such a quantum capture approach is particularly suitable for describing a chemiionization process because of its irreversibility and strongly exoenergetic character, which implies that the total rate coefficient is exclusively determined by the dynamics in the entrance channel of the reaction. As the realvalued Born–Oppenheimer interaction potential between He^{∗} and H_{2} does not include any coupling with the ionization continuum, He + H_{2}^{+} + e^{−}, we modified the purely repulsive wall of the potential, assuming that the isotropic part V_{0}(R) is constant (V_{c}) at short intermolecular distance (R < R_{c}) and equal to the energy released in the ionization process, that is, V_{0}(R) ≡ V_{c} = −3.646 eV if R ≤ R_{c}. This results in a barrier at short range and allows for the loss of the flux into the reactive channels via quantum tunnelling.
Closecoupling quantum scattering calculations were performed separately for collisions of He^{∗} with paraH_{2}(j = 0) and with orthoH_{2}(j = 1). The angular basis functions were constructed as products of the rotational state of the H_{2} molecule and the partial wave describing the orbital motion of the whole complex, and were symmetryadapted to values of the total angular momentum J and parity. The basis set was restricted to even or odd rotational states of H_{2} for collisions with paraH_{2}, or orthoH_{2}, respectively. For collision energies up to 10 K, the basis set that yielded fully converged crosssections included all of the relevant functions with j ≤ 4 and J < 8. The scattering wave functions were propagated from 7.0a_{0} to 200a_{0} by means of a renormalized Numerov propagator. The capture radius was determined by the condition that the elastic cross sections are fully converged; choosing a smaller R_{c} only results in a homogeneous scaling of the reaction rate coefficients, as does a modification of V_{c}. Shortrange and asymptotic boundary conditions were imposed afterwards, as described in the original paper^{24}. The calculated rate coefficients are convoluted with the experimental resolution, which is 8 mK for the lowest collision energies.
The potential energy surface was obtained using the counterpoise corrected supermolecular method in which the interaction energy is obtained as the difference of the total energy of the dimer (He^{∗} + H_{2}) and the total energies of the monomers, calculated in the basis set of the dimer with basis functions placed at the position of the interacting partner with charges set to zero. As the He^{∗} + H_{2} system is not the actual triplet ground state of the system (which is He(1^{1}S) + ^{3}Σ excited H_{2}) we have employed the maximum overlap method^{33,34} with the orbitals corresponding to the isolated He^{∗} + H_{2} potential with an appropriate occupancy of 1s^{1}2s^{1}σ_{g}^{2} as the starting point. The spinrestricted coupledcluster method with single and double excitations with noniterative corrections to triple excitations [CCSD(T)] in a very large basis set, now considered the gold standard of quantum chemistry, was employed, using the MOLPRO suite of codes^{35}. The augmccpVXZ family for the hydrogen atom^{36} and the augccpVXZ family optimized for the triplet state of the metastable helium dimer^{37} were employed, where X = 4, 5, 6, 7 stands for the cardinal number, which corresponds to the highest angular momentum, equal to l = 3,4,5,6, respectively. To better account for the dispersion interaction we have also used a set of midbond functions located between the centre of mass of the H_{2} molecule and the He atom. These were obtained from polarization functions of hydrogen atom basis sets. The potential used in the subsequent scattering calculations was obtained with the augccpV6Z basis set. At large intermolecular separations, between R = 16a_{0} and R = 20a_{0}, it was smoothly connected to the analytic longrange potential, employing a switching function^{38} and the longrange coefficients (C_{6} and C_{8}) of ref. 39. The interatomic distance of the H_{2} molecule was set to 1.4487a_{0}, which corresponds to the expectation value of the interatomic separation in the ground vibrational state of the hydrogen molecule. To ensure a proper convergence of the expansion into Legendre polynomials, the potential was calculated from 0° to 90° with a step of 15°. The projection onto Legendre polynomials was performed with a published integration procedure^{38}.
The interaction energy was corrected beyond CCSD(T) with the correction obtained as the difference between the coupledcluster method with fourthfold excitations (CCSDTQ) and CCSD(T) in the same basis set. Owing to the enormous numerical cost, CCSDTQ calculations, which are equivalent to FCI for a fourelectron system and exact for a given basis set, were performed only for linear and Tshape geometries, with a smaller basis set, augccpVQZ and no midbond functions, using the CFOUR package (http://www.cfour.de) interfaced with the MRCC programme^{40}. The dominant source of uncertainty of the potential energy surface is incompleteness of the basis set, whereas relativistic effects and the diagonal Born–Oppenheimer (DBO) correction were found to be negligible: relativistic effects, estimated by calculating the scalar relativistic perturbation correction (mass velocity and oneelectron Darwin term) are of the order of 0.01 cm^{−1} in the global minimum with a well depth of 14.30 cm^{−1} at 10.5a_{0}. The DBO correction was obtained as the difference between the corrections for the dimer and infinitely separated monomers for Tshape and Lshape geometries, calculated with the augccpVQZ basis set and the Hartree–Fock wavefunction (it is not sensitive to the size of the basis set nor the quality of the wavefunction). The contribution of the DBO correction to V_{0} is 0.044 cm^{−1} in the global minimum. The incompleteness of the basis set only affects the correlation energy, whereas the Hartree–Fock interaction energy is converged to within 0.01 cm^{−1} in our calculations. Truncation of the basis set results in a shallower potential and an inner turning point that is slightly shifted towards larger intermolecular distances. The limit of the basis set is estimated assuming a convergence pattern A + BX^{−3} or A + BX^{−3} + CX^{−5} for singlet systems and A + BX^{−5} for triplet helium as the correlation energy was found to converge with the cardinal number as X^{−3} for singlet electron pairs and as X^{−5} for triplet pairs^{41,42}. The correlation energy in the H_{2} and He^{∗} monomers are very different in magnitude (9,031 ± 1 cm^{−1} compared with 214.7 ± 0.01 cm^{−1}). For the total system, the basis set limit for the CCSD(T) calculations in the global minimum extrapolated from the augccpV6Z and augccpV7Z basis sets (in the latter case we could afford for such calculations for single geometry) is −14.62 cm^{−1}, whereas extrapolation from the X = 5,6,7 basis set yields the limit −14.42 cm^{−1}. As a conservative estimate, we take the difference between the potential obtained in augccpV6Z and the basis set limit extrapolated from X = 6,7 as the uncertainty of the potential.
Data availability.
The data that support the plots within this Letter and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
The authors acknowledge financial support from the European Commission through ERC Grant EUFP7ERCCoG 1485 QuCC. Additional financial support from the German–Israeli Foundation (Grant no. 1254), the COST Action MOLIM (CM1405) and the Polish National Science Center (Grant no. DEC2012/07/B/ST2/00235) is acknowledged. This research was also supported by the ICore (the Israeli Excellence Center) ‘Circle of Light’, by the Israel Science Foundation Grant nos 298/11 and 1530/15.
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Klein, A., Shagam, Y., Skomorowski, W. et al. Directly probing anisotropy in atom–molecule collisions through quantum scattering resonances. Nature Phys 13, 35–38 (2017). https://doi.org/10.1038/nphys3904
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