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Tunnelling measured in a very slow ion–molecule reaction


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, H2 + D → H + HD, 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 (5.2 ± 1.6) × 10−20 cm3 s1. 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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Fig. 1: Overview of the experiment.
Fig. 2: Ion–molecule reaction kinetics.
Fig. 3: Density dependence of rates and velocity distributions.
Fig. 4: Tunnelling rate coefficient in comparison with theory.

Data availability

The datasets used for this study are available on at data are provided with this paper.

Code availability

The code used during this study is available from the corresponding author on reasonable request.


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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.

Author information

Authors and Affiliations



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.

Corresponding author

Correspondence to Roland Wester.

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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 20 K. 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 4.8 × 1013, 1014 and 1015 cm−3, respectively.

Source data

Extended Data Fig. 2 Ion trap simulations including stray electric fields.

Numerical simulations of the expected reaction rates at a hydrogen gas density of 1.2 × 1015 cm−3, 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 10 mV mm−1 could account for the higher measured reaction rates at this density.

Source data

Source data

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Wild, R., Nötzold, M., Simpson, M. et al. Tunnelling measured in a very slow ion–molecule reaction. Nature 615, 425–429 (2023).

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