Competition between electron transfer and reactive capture in ion–molecule reactions at low collision energies: isotopic and stereodynamic effects in the reactions of CH₃F with H₂⁺, HD⁺ and D₂⁺

Abstract

The bimolecular reactions between CH₃F and H₂⁺, HD⁺ and D₂⁺ have been studied in the range of collision energies between ∼0 and k_B × 30 K using a merged-beam approach. The ion–molecule reactions were investigated following photoexciting of H₂ (HD, D₂) to high Rydberg states in a supersonic beam, merging the Rydberg-molecule beam with a cold supersonic beam of CH₃F using a surface-electrode Rydberg–Stark deflector and monitoring the CH₃⁺, CH₂F⁺ and CH₃F⁺ ions generated by the reactions of H₂⁺ (HD⁺, D₂⁺) with CH₃F within the distant orbit of the Rydberg electron. In all three reaction systems, a strong increase of the rate coefficients was observed at collision energies below k_B × 4 K. Branching ratios for the formation of CH₃⁺, CH₂F⁺ and CH₃F⁺ were measured for all three reactions as a function of the collision energy. The branching ratio for the formation of CH₃⁺ was found to decrease with increasing deuteration of the hydrogen molecular ion and to increase at collision energies below k_B × 4 K. The experimental results were interpreted using model calculations based on a rotationally adiabatic capture model as well as using classical trajectory simulations. The reaction products are shown to be generated in two distinct mechanisms: electron transfer leading to a dominant CH₂F⁺ and a weaker CH₃F⁺ product channel, and short-range complex formation leading predominantly to CH₃⁺ by F⁻ transfer, with a weaker contribution of CH₂F⁺ by H⁻ transfer. The model calculations highlight the role played by quantum-statistical and stereodynamical effects associated with the J = 1, |K| = 1 ground state of para-CH₃F and by the reduced mass of the colliding partners: the orientation of CH₃F molecules induced by the electric field of the ion favours the production of CH₃⁺ by F⁻ transfer at low collision energies and the slower approach of the reaction partners with increasing reduced mass favours electron transfer at intermediate distances.

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1 Introduction

Ion–molecule reactions are a driving force in the chemistry of low-density gaseous environments and play an important role in the chemistry of planetary atmospheres, interstellar molecular clouds, and dilute plasmas.^1–5 The electric field emanating from the ions can polarize neighbouring molecules, which induces a long-range attractive interaction and can initiate reactive collisions even between distant reaction partners. In experiments, the motion and internal states of molecular ions can be precisely controlled in the gas phase and ion–molecule reactions are excellent systems to investigate fundamental aspects of chemical reactivity, particularly at low temperatures.^6–12

Ion–molecule reactions can be broadly classified in two main categories:^13–16 (i) charge-transfer (or electron-transfer) reactions A⁺ + B → A + B⁺, sometimes followed by the fragmentation of B⁺, which can be regarded as nonadiabatic processes starting on the potential-energy surface of the reactants with asymptote A⁺ + B and ending on the potential-energy surface of the products with asymptote A + B⁺; and (ii) adiabatic reactions A⁺ + B → C⁺ + D taking place on a single Born–Oppenheimer potential-energy surface. Over the years, a broad range of concepts and methods has been elaborated to predict the rates of ion–molecule reactions and the branching ratios for the formation of different products. Whereas adiabatic reactions can be treated within the framework of transition-state theory, the description of electron-transfer processes requires a nonadiabatic treatment. In typical ion–molecule reaction systems, electron-transfer and adiabatic reactions are in kinetic competition and understanding the factors that may favour one type of reaction over the other is important. Full-dimensional quantum-mechanical treatments yield quantitative information in this regard and have brought considerable insights in the case of reactions involving atoms and diatomic molecules (see, e.g., ref. 17–21). However, these treatments rapidly become prohibitively complex and expensive with increasing number of atoms.

For many applications in astrophysics and atmospheric chemistry, the total reaction rate coefficients can be estimated using ion–molecule capture models.^1,22 In such models, thermal or state-specific capture rate coefficients k_cap are determined corresponding to ion–molecule trajectories representing what is known as the fall of a particle towards the centre.^23,24 This fall takes place whenever the collision energy is large enough to overcome barriers, including centrifugal barriers, in the long-range potential of the ion–molecule collision system. The fall results in the formation of a capture complex, which for exothermic barrier-free adiabatic reactions leads to the formation of products with unit probability. The theory of ion–molecule capture is fully developed and various approximations can be used, depending on the type of molecular systems and the temperature range of interest. The simplest model, introduced by Langevin,²⁵ considers only the long-range interaction between the charge of the ion and the induced dipole moment of the molecule and is successful in predicting the total rate of key reactions such as H₂⁺ + H₂ → H₃⁺ + H.^18,26–28 More refined capture models consider the electric multipole moments of the molecules, and parametrized rate constants extracted from trajectory calculations^13,29,30 are broadly used. Rotationally adiabatic capture models^22,31–36 are particularly successful in predicting rate constants at low temperatures and low collision energies, as verified in an increasing number of experimental studies.^37–46 At the lowest temperatures and collision energies, capture models can be extended to include the effects of quantum scattering.^47–50 However, being restricted to the treatment of long-range interactions, capture models are usually considered to be unsuitable for predicting branching ratios for competing processes taking place at short range^22,51 although there are notable exceptions: for neutral–neutral reactions capture theory has been combined with statistical treatments as in ref. 52 to enable prediction of branching ratios. In the case of the O⁺(⁴S) + HD reaction, Dateo and Clary have shown that the displacement of the centre of mass from the centre of polarization introduces a torque which preferentially exposes the H side of HD to the O⁺ ion and favours the OH⁺ product over the OD⁺ product at low collision energies.⁵³

In the present article, we report on the investigation of the competition between electron-transfer and adiabatic reactions at low collision energies in the range below k_B × 30 K, using as examples the reactions between rotationally cold (T_rot ≈ 5 K) CH₃F and H₂⁺ (as well as HD⁺ and D₂⁺), forming CH₃F⁺, CH₂F⁺, and CH₃⁺. In this range, which corresponds to the conditions encountered in interstellar molecular clouds, hardly any kinetic energy is available to drive endothermic reactions or to overcome potential barriers separating reactants from products. The reactions taking place under these conditions are therefore almost exclusively exothermic barrier-free adiabatic reactions as well as electron-transfer reactions A⁺ + B → A + B⁺ in which the adiabatic ionization energy of B is less than that of A. The observation in the present work of significant energy-dependent isotopic effects in the branching ratios for the formation of the three ion products leads to the conclusion that, at low collision energies, electron transfer competes with F⁻ and H⁻ atom transfer. This reaction system thus provides an ideal opportunity to investigate this competition and to link the observed branching ratios and energy-dependent rate coefficients to state-specific stereodynamical effects resulting from the orientation of the dipolar CH₃F molecule induced by the field of the ion. To anticipate the main conclusions of this investigation, the importance of electron transfer is found to increase with the degree of deuteration of the molecular hydrogen ion and with increasing collision energy. Somewhat unexpectedly, the main trends observed experimentally can be explained, at least in part, by capture theory, in contrast to the general belief that capture theories cannot be used for that purpose.

Several previous studies of ion–molecule reactions involving CH₃F have been reported and helped us rationalizing our findings: Chau and Bowers have reported total rate constants for the reaction of CH₃F with the rare gas ions and several molecular ions and found charge transfer to be the dominant process in most cases. They interpreted their results in terms of the Franck–Condon factors for ionization of the fluoromethane molecules.⁵⁴ Tsuji et al. have used a flowing-afterglow apparatus coupled with a low-pressure chamber to measure product-ion distributions and rate constants in the charge-transfer reactions between Ar⁺ and CH₃F. The dominant product was found to be CH₂F⁺, although CH₃⁺, CH₂⁺ and CHF⁺ were also observed as minor products.⁵⁵ Willitsch et al. reported collision-energy-dependent rate coefficients for the reaction between Ca⁺ and CH₃F in the range between k_B × 2.6 K and 3.2 K and observed a decrease of the rate coefficient with increasing collision energy.⁵⁶ Zhelyazkova et al. measured the collision-energy dependence of the total rate coefficient of the He⁺ + CH₃F reaction between ∼0 and k_B × 10 K observing the CH₂⁺, CF⁺ and CHF⁺ products, and reported a strong increase below k_B × 1 K.⁴⁰ Okada et al. reported a decrease of the capture rate constant of the Ca⁺ + CH₃F reactions with increasing rotational temperature of CH₃F at collision energies down to k_B × 30 K.⁵⁷ In a related study, Jankunas et al. have investigated the Penning-ionization reaction between metastable neon and CH₃F at low collision energies in a merged-beam apparatus and observed that the branching ratios for the production of the CH₃F⁺, CH₃⁺ and CH₂F⁺ ions are independent of the collision energy down to about k_B × 120 mK.⁵⁸ Finally, the fragmentation dynamics of CH₃F⁺ following threshold photoionization of CH₃F has been characterized in a series of detailed studies using synchrotron radiation and coincidence methods,^59–61 extending earlier experimental^62–66 and theoretical^67,68 studies of the fragmentation dynamics of CH₃F⁺.

2 Experimental setup and methods

The reactions between CH₃F and H₂⁺ (HD⁺, D₂⁺) were investigated at low collision energies using the Rydberg-merged-beam approach described in detail in earlier publications.^38,45 In this approach, one observes the ion–molecule reactions between the H₂⁺ (HD⁺, D₂⁺) ion core of Rydberg states of principal quantum number n ≈ 30 and the CH₃F molecules in their ground vibronic state. At these n values, the ion–molecule reaction takes place within the large Rydberg-electron orbit and is not affected by the distant Rydberg electron, as demonstrated experimentally.^38,69–71 The Rydberg electron, however, protects the reacting cations from being accelerated by stray electric fields, which is essential to reach low collision energies. The measurement of the collision-energy dependence of the ion–molecule reaction rate coefficient relies on the use of two pulsed skimmed supersonic beams, initially propagating at a 10° angle, generated by home-built short-pulse valves (valve-opening duration of ∼20 μs, repetition rate 25 Hz), one for the neutral precursor of the ion (here H₂, HD or D₂) and one for the CH₃F molecule. The hydrogen molecules are photoexcited to low-field-seeking Rydberg–Stark states in an electric field of about 15 V cm⁻¹ with the ion core in a selected rovibrational level (v⁺, N⁺) using the resonant three-photon excitation sequences

H₂ X ¹Σ⁺_g(v = 0, N = 1) → B ¹Σ_u⁺(3, 2) → I^{− 1}Π⁻_g(0, 2) → n = 30 [H₂⁺ X^{+ 2}Σ⁺_g(0, 1)] (1)

HD X ¹Σ⁺_g(v = 0, N = 0) → B ¹Σ_u⁺(4, 1) → I^{+ 1}Π⁺_g(0, 2) → n = 31 [HD⁺ X^{+ 2}Σ⁺_g(0, 0)] (2)

D₂ X ¹Σ⁺_g(v = 0, N = 0) → B ¹Σ_u⁺(4, 1) → I^{+ 1}Π⁺_g(0, 2) → n = 31 [D₂⁺ X^{+ 2}Σ⁺_g(0, 0)]. (3)

The ground states of ortho-H₂ and -D₂ are chosen because of the larger populations resulting from the spin-statistical factors. The electric field is generated between two parallel planar electrodes located near the entrance of a curved surface-electrode Rydberg–Stark deflector and accelerator.⁷² This deflector is used to merge the beam of H₂(n) [HD(n), D₂(n)] Rydberg molecules with the CH₃F beam and to precisely set their mean relative velocity v⁰_rel.

The collision-energy dependence of the rate coefficient is measured in the reaction zone, which consists of a Wiley–McLaren-type time-of-flight mass spectrometer with its ion-extraction axis perpendicular to the merged-beam-propagation axis, by monitoring the yield of ionic reaction products as a function of v⁰_rel. The measurements are carried out under conditions where the CH₃F molecule density is much larger than the density of H₂(n) [HD(n), D₂(n)] molecules and the reaction probability of the H₂(n) molecules is less than 1%. Consequently, the relative rate coefficients can be determined by normalising the product-ion signal by the densities of the H₂(n) and CH₃F molecules. In our experiments, we do not measure absolute but only relative densities and therefore we can only determine relative, not absolute, values of the collision-energy-dependent rate coefficients. The relative densities are extracted from the integrated signal measured with the fast-ionization gauges for the neutral CH₃F molecules and from the integrated ion signal in the time-of-flight mass spectrum for H₂⁺ (HD⁺, D₂⁺).

To precisely control the reaction-observation time τ = t₂ − t₁, we use a sequence of two electric-field pulses (see top-right inset in Fig. 1) and only monitor the product ions generated in the field-free interval between the two pulses. The first pulse, with falling edge at t₁, rejects all ions produced before t₁ and the second pulse, with rising edge at t₂, extracts the product ions formed during the interval τ for detection. Although the product ions are initially formed within the Rydberg-electron orbit, their internal energy is high so that autoionization takes place rapidly at the n values studied (n ≈ 30). To ensure that no product ions leave the reaction zone prior to the extraction pulse, we verify that the product-ion yield increases linearly with τ. Under these conditions (τ ≤ 15 μs), the slope of the increase of a given product ion with τ is directly proportional to the rate coefficient of the corresponding reaction channel. In the case of the HD⁺ + CH₃F reaction, a constant (τ-independent) offset of the CH₃⁺ product yield was observed, probably caused by a small misalignment of the two molecular beams and leading to the production of CH₃⁺ during the ion extraction after t₂.

Fig. 1 Schematic representation of the experimental setup comprising supersonic-beam source chambers, a Rydberg–Stark surface-electrode deflector to merge the beams and adjust their relative velocity, and the reaction zone consisting of an ion-time-of-flight mass spectrometer. MCP = microchannel plates, FIG = fast ionization gauge, GND = ground. All dimensions are given in mm. The inset at the top right displays the electric-potential pulse sequence that is applied to the electrodes 1 and 2 of the electrode stack to extract the ions for detection. The blue, pink and red ellipsoidal shapes schematically indicate the molecular clouds of CH₃F, ground-state hydrogen and hydrogen Rydberg molecules at different positions in the merged-beam apparatus, respectively.

The mean collision energy 〈E_coll〉_i and its distribution f_i(E_coll) are determined at each experimental cycle (index i) from (a) time-of-flight and imaging measurements of the three-dimensional velocity distribution of the cloud of deflected H₂(n) [HD(n), D₂(n)] molecules, and (b) time-of-flight measurements of the longitudinal velocity distribution of the CH₃F molecular beam, as detailed in ref. 45. Because the size of the Rydberg-molecule cloud in the reaction zone is much smaller (ϕ ≈ 1 mm) than that of the CH₃F molecule cloud, the distribution of relative velocities between the CH₃F and the Rydberg molecules is primarily given by the transverse (x, y) velocity distributions f_x(v_Rx) and f_y(v_Ry) of the Rydberg molecules and by the distribution of longitudinal (z) relative velocities f_z(v_relz) at the measured mean forward velocities of the two beams. Indeed, the longitudinal dispersion of the short CH₃F gas pulse over the long (0.9 m) propagation distance between the CH₃F valve orifice and the reaction zone guarantees that only molecules within a very narrow longitudinal velocity class (∼±15 m s⁻¹) overlap with the Rydberg-molecule cloud during the reaction-observation time. Moreover, the transverse-velocity distribution of the CH₃F molecules is limited geometrically by the size of the H₂(n) cloud and the skimmers to less than ±4 m s⁻¹ at the typical forward velocities of the CH₃F beam (∼1200 m s⁻¹).

Overall, the experimentally determined velocity distributions are well represented by Gaussian distributions, which enables us to convert the measured three-dimensional distribution of relative velocities f(v_Rx, v_Ry, v_relz) into a (one-dimensional) distribution of collision energies f(E_coll). The procedure is detailed in the appendix and includes a transformation from Cartesian coordinates (v_Rx, v_Ry, v_relz) to spherical coordinates (v_rel, θ, ϕ), as illustrated in Fig. 11. This transformation reliably describes the transition between the limiting cases of v⁰_rel = 0, where the collision energy is given by the relative motion in all three spatial directions (x, y, z), and v⁰_rel → ∞, where the collision energy is only given by the relative motion in the merged-beam-propagation direction (z). The resulting collision-energy distributions are well described by v⁰_rel and their standard deviations s_Ecoll, which are represented graphically as a function of v⁰_rel in Fig. 2. The solid lines in Fig. 2 represent fits of the collision-energy resolution s_Ecoll according to the empirical expression⁴⁰

with adjustable parameters T_r and a.

Fig. 2 Standard deviation of the collision-energy distributions as a function of the collision energy μ(v⁰_rel)²/2 for H₂, HD and D₂. The error bars correspond to indicative uncertainties of ±20 m s⁻¹ and ±40 m s⁻¹ in the mean value and the width of the relative velocity distribution, respectively. The inset shows the data within the red frame at low collision energies on an enlarged scale.

The different widths of the observed collision-energy distributions for the three reaction systems reflect the different reduced masses, differences in the intersection volumes of the three lasers used to excite H₂, HD and D₂ to the Rydberg–Stark states, and the relative positions of these volumes with respect to the deflector, as well as differences in the dynamics of the molecules in the moving electric traps at the surface of the deflector. In particular, the collision-energy resolution in the experiments involving HD(n) Rydberg states is slightly worse than for the experiments involving H₂(n) and D₂(n), which we attribute to a less efficient excitation of the HD(n) molecules that made it more difficult to optimize the experimental conditions.

Because our measurements primarily target the low-collision-energy range between 0 and k_B × 30 K, the experiments are carried out under conditions where the initial velocities of both supersonic beams are similar (∼1200 m s⁻¹). We operate the valves at a stagnation pressure of ∼3 bar. For the CH₃F beam, we use a CH₃F : Ne : He mixture with a 1 : 7 : 13 pressure ratio and heat the valve to about 380 K. Under these conditions, the vibrational motion of the CH₃F molecules in the supersonic expansion is cooled to the ground state and the rotational motion to a temperature T_rot of ∼5 K (see also Section 3). For the H₂ (HD, D₂) beams, we expand the pure gas and cool the valve to temperatures between 45 and 140 K, depending on the desired final velocity. The collision energy is then adjusted by changing the settings of the surface-electrode deflector to precisely control the mean velocity of the H₂(n) [HD(n), D₂(n)] beams.

Fig. 3 shows a typical ion-time-of-flight (TOF) mass spectrum of the D₂⁺ + CH₃F reaction system recorded after a reaction-observation time of 15 μs. The experiments are carried out under single-collision conditions and the CH₃⁺, CH₂F⁺ and CH₃F⁺ products are the only observed products for this reaction system. The remaining ions observed in the TOF spectrum, H₂O⁺ and H₂DO⁺, are products of the reaction of D₂⁺ with H₂O molecules in the background gas. They are also observed when the CH₃F valve is either turned off or triggered so that the CH₃F gas pulse does not overlap with the D₂(n) beam in the reaction zone, whereas the CH₃⁺, CH₂F⁺ and CH₃F⁺ signals disappear in these cases. The fact that the CH₃⁺, CH₂F⁺ and CH₃F⁺ peaks are well separated from the other masses enables us to derive the branching ratios for the formation of the different product ions from the integrated signal intensities, as discussed in more detail in Section 3. In experiments in which the magnitudes of the two electric-field pulses were varied, we verified that the product ions are generated by autoionization of the CH₃F(n), CH₂F(n) or CH₃(n) products during the reaction-observation time, releasing the corresponding ions (CH₃F⁺, CH₂F⁺, or CH₃⁺) and an electron.

Fig. 3 Typical ion-time-of-flight mass spectrum observed for the D₂⁺ + CH₃F reaction. The vertical lines represent the time-of-flight intervals over which the ion signals were integrated. The ion products labelled in black stem from reactions involving the background gas.

3 Results

The relative rate coefficients and product-ion branching ratios obtained from our measurements of the reactions of CH₃F with H₂⁺, HD⁺ and D₂⁺ are presented in the upper part of Fig. 4. The red dots with error bars represent the total product-ion yield, i.e., the sum of the integrated CH₃⁺, CH₂F⁺ and CH₃F⁺ ion signals, as a function of the collision energy, which is obtained as the expectation value of the distributions of collision energies f(E_coll). Each data point corresponds to the average of a large number of measurements binned according to their collision energy. As explained in Section 2, these ion yields are proportional to the total reaction rate constant. The horizontal error bars represent the standard deviation s_Ecoll of the collision-energy distribution f(E_coll) as shown in Fig. 2 and the vertical error bars represent one standard deviation of all measurements contributing to a given data point.

Fig. 4 Top panels: Scaled experimentally observed (red data points with error bars) collision-energy dependence of the total rate coefficient of the reactions between CH₃F and H₂⁺ (left), HD⁺ (middle) and D₂⁺ (right) and corresponding capture rate coefficients calculated with a rotationally adiabatic capture model (black circles, dotted black line) and obtained from classical trajectory simulations (blue diamonds, dotted blue line). The capture rates determined from the rotationally adiabatic capture model and the classical trajectory calculations are convoluted with the experimental collision-energy distribution, see Section 4.3 for details. All model calculations are for T_rot = 5 K. Bottom panels: Experimental branching ratios of the individual channels. The vertical error bars represent the statistical uncertainties and the horizontal error bars the square root of the second central moment of the collision-energy distribution.

The experimental data are compared with absolute capture rate constants k_cap obtained from classical trajectory calculations (blue diamonds, blue dotted line) as discussed in Section 4.3 and using the rotationally adiabatic capture model (black circles and dotted line) outlined in Section 4.2. Unlike the classical trajectory calculations, the adiabatic-capture-model calculations are performed for a quantum Boltzmann population distribution of the rotational states of CH₃F at a rotational temperature of 5 K, assuming that no ortho-to-para conversion takes place in the expansion.⁷³ The calculated rate constants are averaged over the experimentally determined collision-energy distributions f(E_coll) to enable comparison. For the comparison, the experimental product-ion yields were multiplied by a global scaling factor (one for each of the reaction systems involving H₂⁺, HD⁺ and D₂⁺) to minimize the respective sums of squared residual deviations from the calculated capture rate coefficients. In all three reaction systems, an increase of the total product-ion yield is observed as the collision energy is reduced towards zero and the overall trend is well described by the calculations, although the classical trajectory calculations underestimate the increase of the rate constants at low collision energies, as discussed in detail in Section 4.3.

The branching ratios corresponding to the formation of CH₃F⁺, CH₂F⁺ and CH₃⁺ are displayed in violet, yellow and blue in the lower panels of Fig. 4. These branching ratios were determined from the integrated intensities of the corresponding mass peaks in the ion-time-of-flight spectra, as explained in Section 2. In all three reaction systems, the dominant product ion is CH₂F⁺. The most striking trend in the branching ratios is the decrease of the importance of the CH₃⁺ products when H₂⁺ (CH₃⁺ branching ratio of about 38% at E_coll ≈ k_B × 25 K) is successively deuterated to HD⁺ (29%) and D₂⁺ (22%). In the D₂⁺ + CH₃F reaction system, the CH₃⁺ product even becomes as weak as the CH₃F⁺ product at collision energies beyond k_B × 20 K. The branching ratios also reveal a weak collision-energy dependence, slightly favouring the production of CH₃⁺ over the other two product ions at the lowest collision energies. This trend is barely noticeable in the H₂⁺ + CH₃F reaction system but increases with increasing degree of deuteration. It thus appears that the smaller the CH₃⁺ branching ratio is, the more it increases as the collision energy approaches zero.

All three reaction products (CH₃F⁺, CH₂F⁺, CH₃⁺) can be produced following an initial electron transfer between CH₃F and H₂⁺ (HD⁺, D₂⁺), according to

H₂⁺ + CH₃F → H₂ + CH₃F⁺, (5)

H₂⁺ + CH₃F → H₂ + [CH₃F⁺]* → H₂ + CH₂F⁺ + H, (6)

and

H₂⁺ + CH₃F → H₂ + [CH₃F⁺]* → H₂ + CH₃⁺ + F. (7)

However, the evolution of the branching ratios with increasing degree of deuteration as well as the opposite trend observed in the collision-energy dependence of the CH₃⁺ product branching ratio compared to the CH₂F⁺ and CH₃F⁺ product branching ratios suggest that electron transfer is not the only mechanism contributing to the formation of CH₃⁺. Indeed, to a first approximation, electron transfer is expected to result in very similar distributions of internal energy in CH₃F⁺, and thus in very similar fragmentation patterns, for all three reaction systems. Moreover, it seems unlikely that the collision energy would have any influence at all on the product branching ratios of an electron-transfer reaction over the very narrow range (E_coll < k_B × 30 K) studied experimentally. A second mechanism that might contribute to the CH₃⁺ product is an F⁻ atom transfer

H₂⁺ + CH₃F → [H₂⋯F⋯CH₃]⁺ → H + HF + CH₃⁺ (8)

via a short-range complex. Similarly, an H⁻ transfer might lead to the production of CH₂F⁺

H₂⁺ + CH₃F → [H₂⋯H⋯CH₂F]⁺ → H + H₂ + CH₂F⁺. (9)

We discuss in the next section how the competition between electron transfer and F⁻ or H⁻ atom transfer may be invoked to explain the trends observed in Fig. 4.

4 Discussion

The experimental results presented in Fig. 4 indicate that the CH₃⁺, CH₂F⁺ and CH₃F⁺ products of the CH₃F + H₂⁺ (HD⁺, D₂⁺) reaction systems are formed in competing processes, an electron-transfer reaction occurring at intermediate distances (∼1 nm, see below) leading to all three products and a short-range reaction that generates either CH₃⁺ by F⁻ transfer or CH₂F⁺ by H⁻ transfer. The corresponding branching ratios η_j for the formation of the different products can therefore be written aswhere η_ET and (1 − η_ET) represent branching ratios for the electron transfer and the short-range reactions, respectively. p_CH3⁺, p_CH2F⁺ and p_CH3F⁺ are the conditional probabilities that the electron transfer leads to the product ions CH₃⁺, CH₂F⁺ and CH₃F⁺, respectively, and and are the conditional probabilities that F⁻ or H⁻ is transferred at short range. CH₃F⁺ ions can only be produced by electron transfer so that and thus .

4.1 Branching ratios for the formation of CH₃F⁺, CH₂F⁺ and CH₃⁺ upon electron transfer

To estimate p_CH3⁺, p_CH2F⁺ and p_CH3F⁺, we model the electron transfer as a sudden, resonant process in which the electron hops from CH₃F molecules in their ground vibronic state and attaches to ortho-H₂⁺ X^{+ 2}Σ⁺_g(v⁺ = 0, N⁺ = 1) (HD⁺, ortho-D₂⁺ X^{+ 2}Σ⁺_g(v⁺ = 0, N⁺ = 0)) molecules to form neutral H₂ (HD, D₂) in the X ¹Σ⁺_g electronic ground state without transfer of kinetic energy. In this sudden approximation, which also assumes that the CH₃F structure is not strongly perturbed by the ion, the resulting distribution of H₂ (HD, D₂) vibrational levels is given by the Franck–Condon factors |〈v⁺ = 0|v〉|² of the e⁻ + H₂⁺ X^{+ 2}Σ⁺_g(v⁺ = 0) ↔ X ¹Σ⁺_g(v) transition and the energies transferred to CH₃F are given by the corresponding threshold-ionization energies E_I,v⁺=0←v = hc[small nu, Greek, tilde]_I,0←v of CH₃F, as summarized in the upper part of Table 1, see also Fig. 5.

Table 1 Probabilities of forming CH₃F⁺, CH₂F⁺ and CH₃⁺ after a sudden, resonant electron transfer from H₂⁺ X^{+ 2}Σ⁺_g(v⁺ = 0) forming H₂ X ¹Σ⁺_g(v). See text for details

v	CH3F^+		CH2F^+			CH3^+ and CH2F^+
	6	5	4	3	2	1	0
	0.001	0.013	0.071	0.209	0.337	0.278	0.091
[small nu, Greek, tilde] I,0←v(H2)/cm^−1	102 764	105 772	109 008	112 471	116 160	120 080	124 236
T int(CH3F^+)/cm^−1	1664	4672	7908	11 371	15 060	18 980	23 136
	0.271	0.670	0.736	0.387	0.135	0.084	0.091
p 0←v	0.001	0.039	0.239	0.370	0.207	0.107	0.038
p CH3F^+(v)	1.00	1.00	0.04	0.04	0.10	0.09	0.06
p CH2F^+(v)	0.00	0.00	0.96	0.96	0.89	0.74	0.48
p CH3^+(v)	0.00	0.00	0.00	0.00	0.00	0.17	0.46

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Fig. 5 Mass-resolved threshold photoelectron spectrum of CH₃F reproduced from ref. 60. The black solid line indicates the total intensity and is the sum of the contributions of CH₃F⁺ (violet), CH₂F⁺ (yellow) and CH₃⁺ (blue). The vertical black lines at 13.358 eV and 14.51 eV show the D₀(CH₂F⁺ + H) and D₀(CH₃⁺ + F) dissociation energies, respectively (upper horizontal scale). The assignment bars above the spectrum show the different threshold ionization energies hc[small nu, Greek, tilde]_I,0←v for H₂(v) (red), HD(v) (violet), D₂(v) (green).

The adiabatic ionization wavenumber of CH₃F is 101 100(10) cm⁻¹ (ref. 74 and 75) and, consequently, a sudden, resonant electron transfer forms CH₃F⁺ ions with an internal energy T_int(CH₃F⁺) that rapidly decreases with increasing vibrational excitation of the H₂ product, see third row of the body of Table 1 and Fig. 5. In the sudden approximation, the probability of ionizing CH₃F at [small nu, Greek, tilde]_I,0←v(H₂) corresponds to the intensity of the threshold photoelectron spectrum of CH₃F at the threshold energy corresponding to the E_I,v⁺=0←v ionization energy of H₂ (HD, D₂). The situation is illustrated in Fig. 5, which reproduces the slow photoelectron spectrum (SPES) of CH₃F reported in ref. 60 and gives the positions of the ionization energies of H₂(v), HD(v) and D₂(v) along the upper horizontal bars. The numerical values of obtained after digitalisation of the SPES⁶⁰ and normalisation, are listed in the fifth line of Table 1. The probability p_0←v of the resonant electron transfer

H₂⁺(v⁺ = 0) + CH₃F → H₂(v) + CH₃F⁺(T_int) (13)

can be determined as the normalised product and is given in the fifth row of the body of Table 1.

The dissociation energies leading to the fragments CH₂F⁺ + H and CH₃⁺ + F are 6640(40) cm⁻¹ (ref. 62, 66, 74 and 75) and 15 900(160) cm⁻¹,^60,62,74,75 respectively, and the corresponding dissociative-ionization thresholds for the formation of CH₂F⁺ + H and CH₃⁺ + F (at 13.358 eV and 14.51 eV, respectively⁶⁰) are marked by vertical lines in Fig. 5. Below 13.358 eV, the energy is insufficient for the CH₃F⁺ ions generated by threshold ionization to fragment. The probability p_CH3F⁺ for the H₂⁺ + CH₃F reaction system can thus be estimated by summing the p_0←v values in Table 1 for v = 5 and v = 6 (see Fig. 5; the energy transferred upon production of H₂(v > 6) states is insufficient to ionize CH₃F). When H₂ is formed in the v = 2–4 vibrational levels, the product of the electron transfer is mainly CH₂F⁺, with only a very minor contribution from CH₃F⁺ (see violet trace in Fig. 5). Both CH₃⁺ and CH₂F⁺ can result from electron-transfer processes forming H₂(v = 0, 1). In these cases, their relative importance corresponds to the ratio of the CH₃⁺ and CH₂F⁺ ion signals in the mass-selected slow-photoelectron spectra of CH₃F to the total signal , as measured by Tang et al.⁶⁰ and given in the bottom rows of Table 1. Summing the probabilities for the production of j = CH₃F⁺, CH₂F⁺, or CH₃⁺ by the electron-transfer mechanism yields the values listed in the first row of the body of Table 2, where they are compared with the p_j values predicted for the CH₃F + HD⁺ and CH₃F + D₂⁺ reaction systems. Tables similar to Table 1 but for the reactions involving HD⁺ and D₂⁺ are provided in the appendix (Tables 5 and 6). This analysis predicts that electron transfer should lead to a dominant CH₂F⁺ product, as observed experimentally, and to a close-to-negligible yield of CH₃⁺, which implies that the CH₃⁺ ions observed in the ion–molecule reactions are essentially only produced through F⁻ atom transfer.

Table 2 Probabilities for the formation of the three different product ions of the CH₃F + H₂⁺ (HD⁺, D₂⁺) reactions produced via electron transfer as determined from the slow photoelectron spectrum of CH₃F,⁶⁰ see text for details

	p CH3F^+	p CH2F^+	p CH3^+
CH3F + H2^+	0.10	0.87	0.03
CH3F + HD^+	0.10	0.86	0.04
CH3F + D2^+	0.13	0.85	0.02

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A deficiency of this analysis is the underestimation of the yield of CH₃F⁺. Indeed, the observed branching ratio for the production of CH₃F⁺ is around 20% whereas the predicted branching ratio of ∼10% represents an upper limit for η_ET = 1. This underestimation of the CH₃F⁺ yield is the result of the assumption, in the sudden approximation, that no kinetic energy is released upon electron transfer. A non-zero kinetic-energy release upon electron transfer would increase the amount of CH₃F⁺ and reduce the amount of CH₃⁺ predicted by the sudden approximation even further. When modelling the branching ratios in Sections 4.2.2 and 4.3.2, we therefore assume that p_CH3⁺ = 0. Eqn (10)–(12) can then be simplified to

η_CH3F⁺ = η_ETp_CH3F⁺. (16)

4.2 Analysis with the rotationally adiabatic capture model

4.2.1 Capture rate coefficients and their collision-energy dependence. For comparison with the experimental data presented in Fig. 4, we calculate capture rate coefficients using the rotationally adiabatic capture model originally developed by Clary³² and Troe.^34,36 We follow the procedure described in ref. 41 and determine adiabatic interaction potentialsbetween the ions (H₂⁺, HD⁺ and D₂⁺, treated as point charges) and CH₃F molecules in the rotational states |JKM〉. In eqn (17), μ is the reduced mass of the collision, L² = ħ²l(l + 1) is the square of the angular momentum of the collision system, R the distance between the H₂⁺ (HD⁺, D₂⁺) ion and CH₃F, α′ the polarizability volume of CH₃F (2.54 ų)^76,77 and ΔE^JKM_Stark(R) is the rotational-state-specific Stark shift of the |JKM〉 rotational level of CH₃F in the field of the ion. At the low rotational temperature of CH₃F in the seeded supersonic beam (∼5 K), only rotational levels with J ≤ 2 and K = 0, 1 are significantly populated. Moreover, because the nuclear-spin symmetry is conserved during the supersonic expansion, the Boltzmann factors of excited rotational levels are given with respect to the J = 0, K = 0 and J = 1, K = 1 levels for ortho- and para-CH₃F, respectively. At T_rot = 5 K, the most populated levels are the |000〉 (11.3%), |10M〉 (20.7%) and the |1±1M〉 (26.5%) levels. The population of the |20M〉, |2±1M〉 and |2±2M〉 levels are 13.0%, 16.6% and 0.4%, respectively.

The Stark shifts of the CH₃F rotational states and the corresponding adiabatic interaction potentials for the He⁺ + CH₃F reaction were reported by Zhelyazkova et al. (see Fig. 3 of ref. 40). These interaction potentials are practically identical to those for the D₂⁺ + CH₃F reaction presented in Fig. 7 because the two reaction systems have almost the same reduced mass. They are easily adapted to obtain the interaction potentials for the H₂⁺ + CH₃F and HD⁺ + CH₃F system by changing the value of the reduced mass μ in the first term on the right-hand-side of eqn (17).

Within the rotationally adiabatic capture model, the state-specific rate coefficients k^JKM are determined from the expression

where L^JKM_max is the maximum value of L for which the barrier [eqn (17)] remains below the collision energy E_coll. The state-averaged reaction-rate coefficient is obtained from the state-specific rate coefficients as a weighted sumover the populated rotational levels of CH₃F, with weights w^JKM corresponding to their quantum Boltzmann occupation probability at a rotational temperature T_rot = 5 K of the CH₃F molecules. As explained in ref. 40, the rate coefficients at low collision energies are dominated by the contributions of the high-field-seeking rotational states, which are subject to negative Stark shifts (see Fig. 3 of ref. 40), e.g., the state |000〉 and the J ≥ 1 states with KM = +J².

The calculated capture rate coefficient k(E_coll) is convoluted with the experimentally determined collision-energy distribution f(E_coll), leading to the black circles connected by dotted lines in the upper panels of Fig. 4, as already mentioned in Section 3. The results are compared with the experimental data after multiplying these with a global scaling factor. Overall, the calculated capture rate constants are in good agreement with the measurements, confirming the conclusions from studies of an increasing number of ion–molecule reactions that the rotationally adiabatic capture model describes barrier-free exothermic ion–molecule reactions well in the collision-energy range between ∼k_B × 1 K and k_B × 50 K.^41–45

4.2.2 Collision-energy dependence of the branching ratios. Calculations based on eqn (18) do not provide information on the branching ratio of different product channels. To describe how the branching ratios depend on the collision energy E_coll and on the degree of deuteration of the molecular hydrogen ions, we introduce an expression for the electron-transfer branching ratio η_ET within the rotationally adiabatic capture model. We assume that the electron transfer only proceeds in a distance interval R_int = [R_<, R_>] centred below 1 nm with a constant probability across the interval. η_ET thus increases with increasing time spent in this interval by the collision partners. This time can be computed, for each value of E_coll, from the rotationally adiabatic potentials V^JKM_L(R) as the time τ^JKM needed for the collision to proceed from the outer limit R_>, where the velocity isto the inner limit R_<. We determine τ^JKM by numerical evaluation of Newton's equations of motion in the potential V^JKM_L(R). Assuming an effective R-independent and rotational-state-independent electron-transfer rate constant κ_eff in the electron-transfer interval, one obtains the expression

P^JKM(L, R_<, R_>, E_coll) = 1 − e^{−κ_eff(R_<,R_>)τJKM(L,R_<,R_>,E_coll)}, (21)

for the electron-transfer probability. The value of κ_eff is taken to be the same for H₂⁺, HD⁺ and D₂⁺ and is adjusted to best represent the experimental data. With this probability, we compute the electron-transfer cross section σ_ET by modifying eqn (18):where σ^JKM_ET depends on E_coll, R_<, and R_>. Averaging the contributions σ^JKM_ET over the populated |JKM〉 states yields σ_ET. The branching ratio η^RACM_ET for the electron transfer within the rotationally adiabatic capture model is determined asand the rate coefficient for electron transfer is k_ET = σ_ETv_rel. In sets of calculations performed for different values of R_> and R_<, we found that neither the width nor the location of the interval have a significant impact on the resulting value of η_ET, because the different values of τ^JKM (see eqn (21)) resulting from specific choices of R_> and R_< are compensated by the value of κ_eff when κ_eff is adjusted to best match the experimental data. For the calculations presented below, we selected R_< = 0.5 nm and R_> = 0.7 nm, for which we obtained a value of κ_eff = 9.3 × 10¹² s⁻¹.

The full circles in Fig. 6 represent the values obtained for the branching ratio η^RACM_ET(E_coll) determined in this way as a function of the collision energy. In all three reaction systems, the branching ratio decreases with decreasing collision energy. The reason for this behaviour is that, at low collision energies, the reaction rate is dominated by contributions from high-field-seeking states, and in particular |(J = 1, 2)KM〉 states with KM = +J². The attractive internuclear potential accelerates the particles and reduces the time available for the electron transfer and thus the electron-transfer branching ratio decreases.

Fig. 6 Electron-transfer branching ratio η_ET determined from the rotationally adiabatic capture model (RACM, dots) and the classical trajectory calculations (cl, diamonds) as a function of the collision energy. See text for details.

To compare the experimentally determined values with η^RACM_ET(E_coll) resulting from eqn (23), we follow the same procedure as used for the total rate coefficients in Section 4.2.1 and convolute η^RACM_ET(E_coll) with the experimentally determined collision-energy distributions f(E_coll). Because p_CH3⁺ is assumed to be zero, as explained in Section 4.1, the conditional probabilities of forming CH₃F⁺ and CH₂F⁺ upon electron transfer are p_CH3F⁺ and (1 − p_CH3F⁺), respectively. To best represent the experimental data using eqn (14)–(16) we therefore only adjust κ_eff, p_CH3F⁺ and . The results of this analysis are presented in Fig. 9 below (see Section 4.3.2) and the optimal parameter values are listed in Table 3 below.

Table 3 Parameters obtained from fitting eqn (14)–(16) to the experimentally obtained branching ratios, see Sections 4.2.2 and 4.3.2 for details. RACM stands for rotationally adiabatic capture model

	κ eff	p CH3F^+	p CH2F^+	p CH3^+
a Adjustable parameter. b Fixed to 0 based on the analysis of the electron transfer process presented in Section 4.1. c Fixed to 0 because CH3F^+ can only be formed by charge transfer.
RACM	9.3 × 10^12 s^−1	0.45	0.55	0	0	0.41	0.59
Classical	7.4 × 10^12 s^−1	0.39	0.61	0	0	0.30	0.70

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4.3 Classical trajectory calculations

4.3.1 Capture rate coefficients and their collision-energy dependence. As an alternative interpretation of the experimental observations, we also carried out classical trajectory simulations based on several simplifying assumptions. The classical trajectories were calculated for a point-like singly positively charged ion approaching a “diatomic” model of CH₃F consisting of point-like CH₃ and F moieties but featuring the electric dipole moment (μ_el = 1.857 D)⁷⁸ and electric polarizability volume (α′ = 2.54 ų)^76,77 of CH₃F. The bond between CH₃ and F was modelled as a harmonic oscillator with a wavenumber of 1049 cm⁻¹ (ref. 76 and 79) and an equilibrium bond length chosen to reproduce the observed moment of inertia of CH₃F (I_b = 3.286 × 10⁻⁴⁶ kg m² (ref. 76 and 80)). The initial orientations of CH₃F were sampled isotropically in its equilibrium geometry with rotational angular momenta corresponding to a classical Boltzmann distribution (i.e., disregarding quantum-statistical effects) of vibrational and rotational energy at 5 K. The ion was initialized with an asymptotic velocity determined by the collision energy at a distance of 10 nm with an impact parameter b. The trajectories were computed by solving Newton's equations of motion with a constant time step considering only long-range electrostatic interactions (i.e., disregarding short-range repulsion). A trajectory was considered to have led to capture as soon as the change in the ion–molecule separation at any given step became larger than the ion–molecule separation itself. The proportion of captured trajectories ω(b) was extracted as a function of the sampled impact parameter b, which enabled the determination of the capture rate aswhere ω(b) depends on E_coll.

For comparison with the measured capture rate coefficients, the k^cl_cap values calculated at different collision energies were interpolated with a modified Akima method^81,82 as implemented in MATLAB⁸³ and the resulting function k^cl_cap(E_coll) was convoluted with the experimentally determined collision-energy distributions, following the same convolution procedure as for the capture rates calculated using the rotationally adiabatic capture model. The results are plotted as blue diamonds in the top panels of Fig. 4. They are in excellent agreement with both the experimental values and the values calculated with the rotationally adiabatic capture model above E_coll ≈ k_B × 4 K. In this range, the classical trajectory and the rotationally adiabatic capture model calculations do not only reproduce the observed trend in the capture rate coefficient but they also give the same quantitative absolute values for k_cap(E_coll), which in itself is a remarkable result, given that the classical trajectory simulations do not rely on the assumption of rotationally adiabatic channels. At lower collision energies, however, the classical-trajectory calculations, unlike the rotationally adiabatic capture model, do not reproduce the sharp increase observed experimentally. This difference can unambiguously be attributed to a quantum effect arising from CH₃F molecules in |JKM〉 rotational levels with KM = +J² because the diatomic model of CH₃F implies that there is no rotational angular momentum along the symmetric-top axis and quantum-statistical effects are disregarded.

To illustrate this point, the top panels of Fig. 7 depict the adiabatic interaction potentials V^JKM_L(R) (see eqn (17)) of the |JKM〉 states with J = 0 and J = 1 for the D₂⁺ + CH₃F reaction as a function of the ion–molecule separation R for l = 0 (full lines) and l = 70 (dashed lines). The corresponding state-specific rate coefficients calculated based on the rotationally adiabatic capture model are displayed in the bottom panels of Fig. 7, where they are compared with the rates obtained from the classical trajectory calculations (brown dots). The rate coefficients obtained with the two methods agree for the K = 0 case, which is the consequence of representing CH₃F as a diatomic molecule in the classical trajectory calculations, i.e., as a molecule with zero angular momentum (K = 0) along the internuclear axis. The interaction potentials and state-specific rate coefficients of the |JKM〉 levels with KM = ±J², which have a nonzero angular-momentum component along the F–CH₃ axis, exhibit a very different behaviour, as already pointed out in ref. 40. The KM=+J² (−J²) states are associated with strong attractive (repulsive) potentials (see Fig. 7) and their rate coefficients sharply increase (becomes zero) below E_coll = k_B × 4 K. These states are not represented in the classical trajectory calculations and are responsible for the deviations between the rate coefficients calculated with the two methods below E_coll = k_B × 4 K. This difference would persist even at T_rot = 0 because the K = 0 and |K| = 1 rotational levels of CH₃F have different nuclear-spin symmetry and represent the absolute ground states of ortho- and para-CH₃F, respectively. The simplifying diatomic description of CH₃F in the classical-trajectory simulations is crucial in revealing this effect.

Fig. 7 Top panels: Ion–molecule interaction potentials for the D₂⁺ + CH₃F reaction system as defined in eqn (17) for K = 0 (left) and |K| = 1 (right) for l = 0 (full lines) and l = 70 (dashed lines). Bottom panels: Rotational-state-specific reaction rate coefficients for the same system and states. The brown full circles represent the rate coefficients obtained from the classical trajectory calculations and the solid brown line represents the rotational-state-averaged rate coefficient calculated using the rotationally adiabatic capture model for all K = 0 (K ≠ 0) in the left (right) panel at T_rot = 5 K. The ripples in the rate coefficients correspond to the onset of contributions from individual partial waves.

4.3.2 Collision-energy dependence of the branching ratio. To model the collision-energy dependence of the branching ratios using the classical trajectory simulations, we follow the same procedure as in Section 4.2.2. At every time step along the trajectories, the distance of the centre of mass of CH₃F from the centre of mass of the ion is computed and plotted in a histogram with constant bin width, see Fig. 8. The ordinate values of the histograms in Fig. 8 are obtained from all captured trajectories and thus represent the number of time steps in the entire ensemble of captured trajectories for which the ion–molecule distance lies in an interval corresponding to the relevant bin. Because the duration of a time step is constant, the histograms represent the average time spent in this bin interval by the colliding partners. The branching ratio η^cl_ET could therefore in principle be computed as in eqn (21)–(23) using the histogram counts h(L, R_<, R_>, E_coll) instead of τ^JKM(L, R_<, R_>, E_coll). However, the calculations are carried out by sampling the impact parameter b, and therefore also L = μbR for every collision energy. The ensemble of trajectories used to generate the histograms thus contains contributions from different L values. Rather than sorting the trajectories according to L when we compute the electron-transfer probability P^JKM(L, R_<, R_>, E_coll) (see eqn (21)), it is simpler to first expand P^JKM(L, R_<, R_>, E_coll) in a Taylor series and truncate the expression after the linear term

Fig. 8 Histograms showing the average time h̃ spent at the corresponding distance for all captured trajectories. The grey shaded area indicates the R_int = [0.5, 0.7] nm range over which h̃ is considered.

The branching ratio η^cl_ET is then obtained as

where h̃(R_<, R_>, E_coll) represents the average of h over L and κ^cl_eff is an effective rate constant that is adjusted to best reproduce the experimental data. Fig. 8 depicts the L-averaged histograms corresponding solely to trajectories having led to capture. The grey area in the figure corresponds to the range of R values over which electron transfer was assumed to take place, as in the rotationally-adiabatic-capture-model calculations presented in Section 4.2.2. We verified that the linearisation of P^JKM(L, R_<, R_>, E_coll) does not significantly modify the results in equivalent calculations performed using the rotationally adiabatic capture model.

The data presented in Fig. 8 follow three main trends: (i) the time spent in the different intervals decreases with decreasing R value because the CH₃F dipole orients toward the charge, which accelerates the motion of the collision partners towards each other. (ii) The time spent by the collision partners in any given R range increases with the degree of deuteration, as expected from the increasing value of the reduced mass and hence the propensity to be accelerated. (iii) The increase with R of the time spent in the different intervals is weaker at low collision energies than it is at high collision energies. This effect primarily stems from the fact that, at low collision energies, the dipoles start orienting at larger distances, which reduces the time spent at R values larger than 0.3 nm compared to the high-collision-energy situation.

The values of the electron-transfer branching ratio η^cl_ET obtained from the classical trajectories after interpolation and convolution with the experimental collision-energy distributions are presented as diamonds in Fig. 6, where they can be compared with the corresponding values (dots) determined with the rotationally adiabatic capture model. The electron-transfer branching ratios obtained with both models reveal the same decreasing trend with decreasing collision energies resulting from the dominance, at low collision energies, of contributions from strongly oriented dipoles, as already discussed in Section 4.2.1. The fall of these branching ratios occurs in the range of collision energies where the experimental branching ratios start to change. This region also corresponds to the range where the total rate coefficient increases (see Fig. 4). At the lowest collision energies, the η_ET values calculated with the rotationally adiabatic capture model decrease more rapidly than is the case for the values extracted from the classical trajectory calculations. This difference can be attributed to the contribution of the |JKM〉 states with KM = +J² which are not included in the classical trajectory calculations, as already discussed in Section 4.3.1. Finally, the η^cl_ET values are overall slightly larger than the η^RACM_ET values, and this difference increases with the degree of deuteration.

Fig. 9 compares the CH₃⁺, CH₂F⁺ and CH₃F⁺ branching ratios obtained experimentally (full symbols with error bars) with the branching ratios derived from the rotationally adiabatic capture model (framed stars) and the classical trajectory simulations (framed diamonds), based on the assumption that CH₃⁺ is not formed by electron transfer (p_CH3⁺ = 0) and that CH₃F⁺ can only be formed by electron transfer . Only three parameters were adjusted for each model: p_CH3F⁺, and κ_eff and their optimal values are listed in Table 3. The value of κ_eff reflects the interval (chosen somewhat arbitrarily) to evaluate the charge-transfer branching ratio, as explained in Section 4.2.2. κ_eff is slightly smaller for the classical trajectory simulations than for the rotationally adiabatic capture model. This difference can be explained by the fact that the contribution of the attractive KM = J² potentials in the latter case reduces the time spent by the collision in the interval [R_<, R_>], which needs to be compensated by a higher κ_eff value.

Fig. 9 Experimentally observed collision-energy dependence of the branching ratios of the reactions between CH₃F and H₂⁺ (left), HD⁺ (middle) and D₂⁺ (right). The markers with error bars show the same experimental data as the bottom panels of Fig. 4. The filled stars indicate results from the rotationally adiabatic capture model, and the filled diamonds from the classical trajectory calculations. All model calculations are for T_rot = 5 K. See text for details.

Overall the calculated branching ratios provide a satisfactory description of the observed branching ratios, in particular they qualitatively reproduce the decreasing trend of the CH₂F⁺ and CH₃F⁺ product branching ratios and the increasing trend of the CH₃⁺ branching ratio as the collision energy is reduced below ∼k_B × 4 K. However, the trends predicted by the rotationally adiabatic capture model are stronger than observed experimentally, particularly for the D₂⁺ + CH₃F reaction system, indicating possible shortcomings of the capture model calculations. The predictions based on the classical trajectory simulations are closer to the experimental results. However, we demonstrated above that this better agreement is the result of the neglect, in the classical trajectory simulations, of rotational angular momentum along the symmetric-top axis and is not physical.

5 Conclusions

The H₂⁺(v⁺ = 0) + CH₃F reaction and the corresponding reactions of HD⁺(v⁺ = 0) and D₂⁺(v⁺ = 0) have been studied at collision energies in the range between 0 and k_B × 30 K and a CH₃F rotational temperature of ∼5 K, at which only the lowest rotational levels with J ≤ 2 are significantly populated. The measurements focused on the determination of the dependence on the collision energy of the total rate coefficient and the branching ratios for the formation of the different product ions (see Fig. 4). The total rate coefficients of all three reactions were found to increase with decreasing collision energies, particularly below E_coll ≈ k_B × 4 K. Three product ions were observed: CH₃F⁺, CH₂F⁺ and CH₃⁺. The branching ratios for their formation were found to depend on the collision energy and on the degree of deuteration of the molecular-hydrogen ion. The dominant product is CH₂F⁺ but its importance decreases with decreasing collision energy and decreasing degree of deuteration. The second most abundant product is CH₃⁺ and its branching ratio follows the exact opposite trends: it increases with decreasing collision energy and degree of deuteration. CH₃F⁺ is the least abundant product and its branching ratio is only ∼20% and hardly depends on the degree of deuteration and the collision energy.

To interpret these experimental observations, the ion–molecule capture process was modelled by classical trajectory calculations and by using a rotationally adiabatic capture model inspired by the work of Clary³² and Troe,^34,36 as described in detail in ref. 40 and 41. The rotationally adiabatic capture model quantitatively reproduces the observed collision-energy dependence of the total rate coefficient in all three reaction systems and demonstrates that the sharp rate-coefficient increase below E_coll = k_B × 4 K can be attributed to the reactions involving the high-field-seeking rotational states of CH₃F with J = 1, KM = +1 (8.8% of the CH₃F population). The classical trajectory calculations were performed for a diatomic-molecule model of CH₃F, i.e., with no angular momentum along the symmetric-top axis to highlight quantum-statistical effects and the effect of the high-field-seeking K ≥ 1 states by comparison. They quantitatively reproduced the collision-energy dependence of the total rate coefficient down to E_coll = k_B × 4 K but failed to account for the sharp rate-coefficient increase observed at the lowest collision energies, thus indirectly confirming the attribution of this increase to CH₃F molecules in J > 0, KM = +J² rotational states made on the basis of the rotationally adiabatic capture model.

All three product ions CH₃F⁺, CH₂F⁺ and CH₃⁺ can be produced following electron transfer from CH₃F to H₂⁺, HD⁺ or D₂⁺. However, modelling the electron transfer as a resonant, sudden process could not explain the observed branching ratios. CH₂F⁺ was predicted to be the dominant electron-transfer product, as observed, but the model did not predict any energy dependence of the branching ratios and led to the conclusion that electron transfer hardly produces any CH₃⁺ ions at all.

A satisfactory interpretation of the experimentally observed branching ratios and their collision-energy dependence could only be reached by invoking competitive kinetics between an electron-transfer pathway yielding only CH₃F⁺ and CH₂F⁺ and formation of a short-range complex yielding CH₃⁺ (and XF + X, with X = H, D) by F⁻ atom transfer and CH₂F⁺ (and XH + X, with X = H, D) by H⁻ atom transfer. The corresponding overall reaction scheme is presented in Fig. 10 for the H₂⁺ + CH₃F reaction system with analogous schemes for HD⁺ + CH₃F and D₂⁺ + CH₃F.

Fig. 10 Overall reaction scheme for the H₂⁺ + CH₃F reaction system at low collision energies for the three dominant product channels CH₃F⁺, CH₂F⁺ and CH₃⁺.

Because the total rate coefficient can be quantitatively accounted for by the rotationally adiabatic capture model, the electron transfer must take place at ion–molecule separations smaller than the capture radius (which is on the order of a few nm at E_coll = k_B × 4 K). The electron transfer is a nonadiabatic process connecting the potential-energy surface associated with the H₂⁺(v⁺ = 0) + CH₃F reactants with that associated with the CH₃F⁺ + H₂(v) products. Assuming that the electron transfer takes place stochastically with a rate κ_eff over a selected range of ion–molecule distances, we could explain the dependence of the product-ion branching ratios on the degree of deuteration and on the collision energies. With increasing degree of deuteration, the reduced mass increases and the reactants need more time to reach the short-range complex, thus favouring the longer-range electron-transfer reaction pathway. Similarly, low collision energies favour the reaction of high-field-seeking rotational states of CH₃F (J > 0, KM = +J²). These states are associated with strongly attractive ion–molecule interaction potentials and thus with shorter approach times to reach the short-range complex. Consequently, low collision energies favour the F⁻ and H⁻ atom-transfer reactions in a short-range complex.

Based on a simplistic stereodynamical argument, with the C–F bond pointing toward the ion, one would expect the ion–molecule approach at short range to favour the formation of CH₃⁺ by F⁻ transfer and to inhibit the formation of CH₂F⁺ by H⁻ transfer. However, at short distances, the C–F bond is not expected to exactly point toward the ion but instead to perform a pendular motion with the classical turning points at a nonzero deviation angle from the internuclear axis (see schematic structure in the lower part of Fig. 10), making H⁻ transfer possible, though not as likely as the F⁻ transfer. Considering that at long range the CH₃F molecules in J = 1 K = 1 states precess around the collision axis at an angle of 45°, it is reasonable to take this angle as representative of the amplitude of the pendular motion at short range. The angle of the C–H bond, i.e., 109° − 45° = 64° suggests that the H⁻ transfer should be possible, although less likely than the F⁻ transfer (see Fig. 10).

The observed energy-dependent branching ratios could be reproduced semi-quantitatively by fitting only three parameters, the probability p_CH3F⁺ of forming CH₃F⁺ upon electron transfer (∼45%), the probability of forming CH₃⁺ by short-range F⁻ transfer (∼60%) and an effective rate constant κ_eff for a charge transfer to occur in the range of internuclear separation between 0.5 and 0.7 nm (9.3 × 10¹² s⁻¹ in Table 3), see Fig. 9. The optimal values of the parameters p_CH3F⁺ and are both greater than 50%, which confirms the expectations from the resonant, sudden electron-transfer model and from the stereodynamical argument concerning the orientation of CH₃F during the approach towards the short-range complex. A full quantitative understanding of the experimental observations would require a careful treatment of the interactions at short range. In this context, progress has been made recently in the case of the electron-transfer reaction between He⁺ and CH₃CN⁸⁴ through the treatment of the short-range nonadiabatic dynamics. Our results are compatible with the results obtained in ref. 84 in that they show that the approach of the ion and the molecule at low collision energies is restricted to a narrow cone of orientation angles of the symmetric-top axis with respect to the collision axis, and that the charge-transfer rate coefficient can be much smaller than the capture rate coefficient.

We expect that several important aspects of the observed behaviour should be of general validity whenever electron transfer is in competition with adiabatic chemical reactions: (i) the substitution of lighter by heavier isotopes in the reacting ion or molecule should favour electron transfer. (ii) The increasing importance of the high-field-seeking states of the neutral molecule should reduce the importance of electron transfer at low collision energies. (iii) The orientation of the molecule in the field of the ion should finally lead to specific stereodynamical preferences for short-range atom transfer, particularly for low-energy collisions and for reactions involving dipolar molecules, as illustrated here with the example of CH₃F.

Conflicts of interest

There are no conflicts to declare.

Data availability

All relevant data are presented in the article. They can be obtained from the authors upon reasonable request.

Appendices

A Derivation of the collision-energy distribution

This Appendix describes how we determine the collision-energy distribution from measurements of (i) the forward velocity distribution f_Nz(v_N) of the neutral CH₃F molecules using fast-ionization gauges and (ii) the three-dimensional velocity distribution f_Rx(v_Rx)f_y(v_Ry)f_z(v_Rz) of the Rydberg H₂(n) [HD(n), D₂(n)] molecules using imaging and time-of-flight methods. All velocity distributions are well represented by Gaussian distributions and the contributions of the CH₃F molecules to the relative transverse-velocity distribution (x, y) are negligible. We define the probability density functions f as followswhere μ_i are the mean velocities with corresponding standard deviations σ_i and μ_N − μ_Rz = v⁰_rel.

The relative-velocity distribution in the forward (z) direction f_z is given by

We compute the full three-dimensional velocity distribution f as the product

f(v_Rx, v_Ry, v_relz) = f_x(v_Rx)f_y(v_Ry)f_z(v_relz) (32)

To convert the three-dimensional distribution of relative velocities into a (one-dimensional) distribution of the collision energy E_coll, we first transform to spherical coordinates (v_rel, θ, ϕ)see also Fig. 11.

Fig. 11 Example for a point in the three-dimensional relative-velocity space in both Cartesian (v_Rx, v_Ry, v_relz) and spherical (v_rel, θ, ϕ) coordinates.

Next, we exploit the relation

Expressing the collision energy aswhere μ is the reduced mass, the distribution of collision energy isThe integrals over the angular variables θ and ϕ are evaluated numerically. For the sample values listed in Table 4, this procedure yields the collision-energy distributions plotted in Fig. 12.

Table 4 Values used to compute the collision-energy distributions displayed in Fig. 12

	S1	S2	S3
μ/amu	3.6	3.6	3.6
v ^0rel/m s^−1	13.7	140.3	253.0
σ N/m s^−1	7.9	9.3	11.7
σ Rz/m s^−1	41.9	42.8	42.7
σ Ry/m s^−1	48.8	54.6	54.9
σ Rx/m s^−1	12.0	13.6	14.5

---

Table 5 Probabilities of forming CH₃F⁺, CH₂F⁺ and CH₃⁺ after a sudden, resonant electron transfer from HD⁺ X^{+ 2}Σ⁺_g(v⁺ = 0) forming HD X ¹Σ⁺_g(v). See text for details

v	CH3F^+		CH2F^+			CH3^+ and CH2F^+
	7	6	5	4	3	2	1	0
	0.0003	0.004	0.029	0.111	0.248	0.321	0.222	0.063
[small nu, Greek, tilde] I,0←v(HD)/cm^−1	102 671	105 269	108 038	110 977	114 087	117 367	120 822	124 454
T int(CH3F^+)/cm^−1	1571	4169	6938	9877	12 987	16 267	19 722	23 354
	0.222	0.901	0.759	0.571	0.228	0.101	0.087	0.088
p 0←v	0.000	0.019	0.110	0.312	0.277	0.159	0.095	0.027
p CH3F^+(v)	1.00	1.00	0.18	0.04	0.08	0.12	0.08	0.06
p CH2F^+(v)	0.00	0.00	0.82	0.96	0.92	0.86	0.66	0.48
p CH3^+(v)	0.00	0.00	0.00	0.00	0.00	0.02	0.26	0.46

---

Table 6 Probabilities of forming CH₃F⁺, CH₂F⁺ and CH₃⁺ after a sudden, resonant electron transfer from D₂⁺ X^{+ 2}Σ⁺_g(v⁺ = 0) forming D₂ X ¹Σ⁺_g(v). See text for details

v	CH3F^+		CH2F^+			CH3^+ and CH2F^+
	7	6	5	4	3	2	1	0
	0.003	0.019	0.072	0.176	0.277	0.270	0.148	0.035
[small nu, Greek, tilde] I,0←v(p − D2)/cm^−1	102 848	105 446	108 216	111 155	114 264	117 545	121 000	124 632
T int(CH3F^+)/cm^−1	1748	4346	7116	10 055	13 164	16 445	19 900	23 532
	0.345	0.873	0.707	0.539	0.222	0.101	0.090	0.090
p 0←v	0.004	0.061	0.189	0.354	0.229	0.102	0.050	0.012
p CH3F^+(v)	1.00	1.00	0.07	0.05	0.07	0.12	0.07	0.04
p CH2F^+(v)	0.00	0.00	0.93	0.95	0.93	0.86	0.66	0.48
p CH3^+(v)	0.00	0.00	0.00	0.00	0.00	0.02	0.27	0.48

---

Fig. 12 Collision-energy distributions f(E_coll) for the parameters listed in Table 4.

B Tables for the branching-ratio calculation of the reactions of CH₃F with HD⁺ and D₂⁺

Acknowledgements

We thank Hansjürg Schmutz and Josef A. Agner for their technical assistance, Fernanda B. V. Martins and Dr Valentina Zhelyazkova for their contributions to the development of the codes used to calculate the state-specific rate coefficients and for fruitful discussions and Ioana Doran for the Franck–Condon factors of the X⁺–X ionizing transitions of hydrogen. This work is supported by the Swiss National Science Foundation (Excellence Grant No. 200020B 200478). TPS acknowledges support from the Leverhulme Trust (EM2022-027).

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Footnotes

† Present address: Laboratoire de Physique des Lasers, Université Sorbonne Paris Nord, 93430 Villetaneuse, France.

‡ These authors contributed equally to this work.

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