Abstract
Quantum tunnelling reactions play an important role in chemistry when classical pathways are energetically forbidden1, be it in gas-phase reactions, surface diffusion or liquid-phase chemistry. In general, such tunnelling reactions are challenging to calculate theoretically, given the high dimensionality of the quantum dynamics, and also very difficult to identify experimentally2,3,4. Hydrogenic systems, however, allow for accurate first-principles calculations. In this way the rate of the gas-phase proton-transfer tunnelling reaction of hydrogen molecules with deuterium anions, H2â+âDââââHââ+âHD, has been calculated5, but has so far lacked experimental verification. Here we present high-sensitivity measurements of the reaction rate carried out in a cryogenic 22-pole ion trap. We observe an extremely low rate constant of (5.2â±â1.6)âĂâ10â20âcm3âsâ1. This measured value agrees with quantum tunnelling calculations, serving as a benchmark for molecular theory and advancing the understanding of fundamental collision processes. A deviation of the reaction rate from linear scaling, which is observed at high H2 densities, can be traced back to previously unobserved heating dynamics in radiofrequency ion traps.
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Data availability
The datasets used for this study are available on zenodo.org at https://doi.org/10.5281/zenodo.7148592. Source data are provided with this paper.
Code availability
The code used during this study is available from the corresponding author on reasonable request.
References
McMahon, R. J. Chemical reactions involving quantum tunneling. Science 299, 833â834 (2003).
Shannon, R. J., Blitz, M. A., Goddard, A. & Heard, D. E. Accelerated chemistry in the reaction between the hydroxyl radical and methanol at interstellar temperatures facilitated by tunnelling. Nat. Chem. 5, 745â749 (2013).
Tizniti, M. et al. The rate of the F + H2 reaction at very low temperatures. Nat. Chem. 6, 141 (2014).
Yang, T. et al. Enhanced reactivity of fluorine with para-hydrogen in cold interstellar clouds by resonance-induced quantum tunnelling. Nat. Chem. 11, 744â749 (2019).
Yuen, C. H. et al. Quantum-tunneling isotope-exchange reaction H2 + Dâ â HD + Hâ. Phys. Rev. A 97, 022705 (2018).
FerriĂšre, K. M. The interstellar environment of our galaxy. Rev. Mod. Phys. 73, 1031â1066 (2001).
Gerlich, D., Herbst, E. & Roueff, E. \({\text{H}}_{3}^{+}\)â+âHDâââH2D+â+âH2: low-temperature laboratory measurements and interstellar implications. Planet. Space Sci. 50, 1275 (2002).
Tielens, A. G. G. M. The molecular universe. Rev. Mod. Phys. 85, 1021 (2013).
Kitsopoulos, T. N., Buntine, M. A., Baldwin, D. P., Zare, R. N. & Chandler, D. W. Reaction product imaging: the Hâ+âD2 reaction. Science 260, 1605 (1993).
Harich, S. A. et al. Forward scattering due to slow-down of the intermediate in the Hâ+âHDâââDâ+âH2 reaction. Nature 419, 281 (2002).
Stevenson, D. & Hirschfelder, J. The structure of H3, \({\text{H}}_{3}^{+}\), and of \({\text{H}}_{3}^{+}\). iv. J. Chem. Phys. 5, 933â940 (1937).
StĂ€rck, J. & Meyer, W. Ab initio potential energy surface for the collisional system Hââ+âH2 and properties of its van der Waals complex. Chem. Phys. 176, 83â95 (1993).
Wang, W. et al. Observations of \({\text{H}}_{3}^{-}\) and \({\text{H}}_{3}^{-}\) from dielectric barrier discharge plasmas. Chem. Phys. Lett. 377, 512â518 (2003).
Ayouz, M., Lopes, R., Raoult, M., Dulieu, O. & Kokoouline, V. Formation of \({\text{H}}_{3}^{-}\) by radiative association of H2 and Hâ in the interstellar medium. Phys. Rev. A 83, 052712 (2011).
Zimmer, M. & Linder, F. Crossed-beam study of the Hââ+âD2âââHD(vâ)â+âDâ rearrangement reaction in the collision energy range 0.3â3âeV. J. Phys. B 28, 2671 (1995).
Belyaev, A. K., Colbert, D. T., Groenenboom, G. C. & Miller, W. H. State-to-state reaction probabilities for Hââ+âH2, D2 collisions. Chem. Phys. Lett. 209, 309â314 (1993).
Haufler, E., Schlemmer, S. & Gerlich, D. Absolute integral and differential cross sections for the reactive scattering of Hââ+âD2 and Dââ+âH2. J. Phys. Chem. A 101, 6441â6447 (1997).
Giri, K. & Sathyamurthy, N. Influence of reagent rotation on (Hâ, D2) and (Dâ, H2) collisions: a quantum mechanical study. J. Phys. Chem. A 110, 13843â13849 (2006).
Zhang, W., Liu, Y. & He, X. Effect of reagent rotation on the integral cross-sections and isotopic branching of the reactions Hââ+âHD and Dââ+âHD. Chem. Phys. Lett. 489, 237â241 (2010).
Wang, D. & Jaquet, R. Reactive scattering for different isotopologues of the \({\text{H}}_{3}^{-}\) system: comparison of different potential energy surfaces. J. Phys. Chem. A 117, 7492â7501 (2013).
Mikosch, J. et al. Inverse temperature dependent lifetimes of transient SN2 ionâdipole complexes. J. Phys. Chem. A 112, 10448â10452 (2008).
Herbst, E. et al. Calculations on the rate of the ionâmolecule reaction between N\({\text{H}}_{3}^{+}\) and H2. J. Chem. Phys. 94, 7842â7849 (1991).
Markus, C. R. et al. Vibrational excitation hindering an ionâmolecule reaction: the c-C3\({\text{H}}_{2}^{+}\) â H2 collision complex. Phys. Rev. Lett. 124, 233401 (2020).
Endres, E. S., Lakhmanskaya, O., Simpson, M., Spieler, S. & Wester, R. Upper limit of a tunneling reaction rate for Dââ+âH2âââHDâ+âHâ. Phys. Rev. A 95, 022706 (2017).
Gerlich, D. Ion-neutral collisions in a 22-pole trap at very low energies. Phys. Scr. T59, 256 (1995).
Wester, R. Radiofrequency multipole traps: tools for spectroscopy and dynamics of cold molecular ions. J. Phys. B 42, 154001 (2009).
Asvany, O. & Schlemmer, S. Numerical simulations of kinetic ion temperature in a cryogenic linear multipole trap. Int. J. Mass spectrom. 279, 147 (2009).
Wakelam, V. et al. A KInetic Database for Astrochemistry (KIDA). Astrophys. J. Suppl. Ser. 199, 21 (2012).
Luo, H., Wu, Y. & Ju, L. Variational transition-state theory study of the Dâ+H2 â HD+Hâ reaction and the Hâ+D2 â HD+Dâ reaction. Comput. Theor. Chem. 963, 475â478 (2011).
Tsallis, C. Possible generalization of BoltzmannâGibbs statistics. J. Stat. Phys. 52, 479â487 (1988).
Best, T. et al. Absolute photodetachment cross-section measurements for hydrocarbon chain anions. Astrophys. J. 742, 63 (2011).
Endres, E. S. et al. Incomplete rotational cooling in a 22-pole ion trap. J. Mol. Spectrosc. 332, 134 (2017).
Jusko, P., Asvany, O., Wallerstein, A.-C., BrĂŒnken, S. & Schlemmer, S. Two-photon rotational action spectroscopy of cold OHâ at 1âppb accuracy. Phys. Rev. Lett. 112, 253005 (2014).
Silva, R. Jr, Plastino, A. R. & Lima, J. A. S. A Maxwellian path to the q-nonextensive velocity distribution function. Phys. Lett. A 249, 401â408 (1998).
Jiulin, D. The nonextensive parameter and Tsallis distribution for self-gravitating systems. Europhys. Lett. 67, 893â899 (2004).
Rouse, I. & Willitsch, S. Superstatistical energy distributions of an ion in an ultracold buffer gas. Phys. Rev. Lett. 118, 143401 (2017).
Acknowledgements
We thank V. Kokoouline, C. H. Yuen and M. Ayouz for fruitful discussions. This work has been supported by the Austrian Science Fund (FWF) through Project I2920-N27 and through the Doctoral Programme Atoms, Light, and Molecules, Project No. W1259-N27.
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R. Wester conceived the experiment and supervised the project. R. Wild, M.N., M.S. and T.D.T. carried out the measurements. R. Wild, with support from M.N. and R. Wester, carried out the simulations. R. Wild and R. Wester wrote the manuscript, which was discussed and approved by all authors.
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Extended data figures and tables
Extended Data Fig. 1 Ion trap simulations for different densities.
Distributions of ion number and energy as a function of the radial position in the trap, at different collision rates and with a buffer gas temperature of 20âK. a With higher collision rates, ions move, on average, slightly closer to the centre, but also increase in number at large radii. Mean free path lengths are shown for reference. b When the mean free path becomes small, energies close to the trap rods increase substantially, indicating multiple heating collisions before moving away from the trap edges. The collision rates correspond to H2 densities of 4.8âĂâ1013, 1014 and 1015âcmâ3, respectively.
Extended Data Fig. 2 Ion trap simulations including stray electric fields.
Numerical simulations of the expected reaction rates at a hydrogen gas density of 1.2âĂâ1015âcmâ3, with error bars as in Fig. 3a. The electric fields were chosen to be homogeneous and oriented in one direction perpendicular to the RF trapping rods, which has the effect of slightly pushing the ions towards the RF electrodes on one side. The simulations show that an extra field on the order of 10âmVâmmâ1 could account for the higher measured reaction rates at this density.
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Wild, R., Nötzold, M., Simpson, M. et al. Tunnelling measured in a very slow ionâmolecule reaction. Nature 615, 425â429 (2023). https://doi.org/10.1038/s41586-023-05727-z
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DOI: https://doi.org/10.1038/s41586-023-05727-z
Bhibuthi bhusan Patel
Gas equation at ideal conditions of pressure P,volume V and temperature T is given by PĂV = RĂ T,where R is the universal gas constant for one mole of gas.
Thus we can get the natural parameter for tunnelling involving four atoms putting unit values for P,V and T in (PĂV) / (RĂT).
Thus,in this experiment for 4 moles of gas(2 moles of H2 + 2 moles of deuterium ion) calculated result equals to a value 5 Ă 10 to the power _20,confirminng the assumption of sole quantum nature of atom and subatomic particles.Thanks to the authors for experimental demonstration of this aspect.
Constantino Tsallis
Equation (3) [for further details, see for instance C. Tsallis, Introduction to Nonextensive Statistical Mechanics â Approaching a Complex World â Second Edition (Springer, 2023)] is illustrated in Fig. 3 (c). Prof. Roland Webster kindly shared with me the index q corresponding to the values of the hydrogen density n indicated in Fig. 3 (c). With this information, it is possible to construct the attached figure. The ideal gas limit (n=0) corresponds, as expected, to q=1, i.e., to Boltzmann-Gibbs statistical mechanics. Further experimental validation and/or theoretical approaches of the new connection (q-1) proportional n^{1/4} are naturally very welcome. https://uploads.disquscdn.c...