## Abstract

We report a joint experimental–theoretical study of the F^{–} + HCl → HF + Cl^{−} reaction kinetics. The experimental measurement of the rate coefficient at several temperatures was made using the selected ion flow tube method. Theoretical rate coefficients are calculated using the quasi-classical trajectory method on a newly developed global potential energy surface, obtained by fitting a large number of high-level *ab initio* points with augmentation of long-range electrostatic terms. In addition to good agreement between experiment and theory, analyses suggest that the ion–molecule reaction rate is significantly affected by shorter-range interactions, in addition to the traditionally recognized ion–dipole and ion–induced dipole terms. Furthermore, the statistical nature of the reaction is assessed by comparing the measured and calculated HF product vibrational state distributions to that predicted by the phase space theory.

This article is part of the theme issue ‘Modern theoretical chemistry’.

## 1. Introduction

Simple proton-transfer reactions in an exothermic ion–molecule system are, as a rule, fast [1–3]. This is largely due to the fact that the corresponding potential energy surface (PES) is typically barrierless and dominated by attractive long-range electrostatic interactions. As a result, the rate coefficients can be readily determined using capture models based on the dominant long-range interactions, most commonly the ion–dipole and ion–induced dipole terms [4]. Exceptions to this rule exist for reactions that are not appreciably exothermic, are impeded by spin constraints [5], or involve reactants with delocalized electrons [6]. Therefore, it is surprising that the prototypical reaction of F^{−} with HCl,
1.1
has been reported [7] to occur with a room-temperature rate coefficient close to 85% of the ion–dipole collisional value and well (approx. 40%) below the expected collisional value accounting for higher-order terms [4,8]. The previous measurement [7] of the rate coefficient for reaction (1.1) was made in a flowing afterglow (FA), a method which is known to be prone to small errors for a variety of reasons [9]. Additionally, kinetics was not the primary focus of that work, but rather it was to measure the HF product vibrational distribution, which was shown to be non-statistical. However, quasi-classical trajectory (QCT) calculations using an analytical PES based on *ab initio* calculations at the MP4/6-31G(d,p) level of theory agree well with that experiment [10]. The calculations predicted the rate coefficient to approach the ion–dipole collisional value at low temperature, but to deviate sharply at elevated temperatures. The steep temperature dependence is attributed to lowered reaction probability at elevated translational energies, but neither a detailed understanding of the effect nor an explanation for the distinction between this system and other, rapidly occurring, proton-transfer reactions can be extracted from the published results. It is noted that the level of theory used to construct the earlier PES is not sufficiently accurate. Although benchmark *ab initio* calculations of the system have been reported [11], no global PES at the highest possible level of theory has yet been constructed. As a result, there is a need to re-examine the problem from both the experimental and theoretical perspectives, in order to gain a deeper understanding of this prototypical reaction.

In this work, we attempt to answer these questions by a combined experimental–theoretical study. On the experimental side, we repeat the measurement using our selected ion flow tube (SIFT) apparatus [12] over an extended temperature range, by removing the complicating presence of the F_{2} source gas and electrons in the FA measurements. The kinetics measured over the thermal energies using the SIFT apparatus provide a sensitive probe of the rate-limiting portions of the PES, which has been shown to provide greater support for predictions of subtle non-statistical effects [13,14]. Theoretically, we report here an accurate global PES based on a large number of high-level *ab initio* calculations and perform QCT calculations on the new PES. The comparison of the theoretical results with experimental measurement sheds light on the importance of various types of interaction in determining the reactivity of this reaction, as well as the statistical nature of the reaction.

## 2. Materials and methods

### (a) Theoretical

In this work, all electronic structure calculations for the reaction F^{−} + HCl → HF + Cl^{−} were carried out with an explicitly correlated (F12b) version of the coupled cluster method with singles, doubles and perturbative triples (CCSD(T)-F12b) [15], as implemented in MOLPRO [16]. The augmented correlation-consistent polarized valence triple-zeta basis set (aug-cc-pVTZ) [17] was used. This level of theory has been shown in recent studies to provide a globally accurate description of PESs within chemical accuracy (less than 1 kcal mol^{−1}) [18–21]. *Ab initio* points were firstly sampled in the normal mode space near the reactants, products and stationary points. Based on these points, a primitive PES was first constructed. Then, additional sets of geometries were generated by running QCT calculations on this PES with different initial geometries in the interaction and asymptotic regions. Points were added only if they are not too close to the existing points in the dataset, which was judged by the Euclidean distance (less than 0.05 Å). The sampling range is spanned by and . A new PES was then generated using the enlarged dataset. This process was iterated until a satisfactory PES is obtained.

Representing multi-dimensional PESs from *ab initio* data has been a challenge [22]. Here, the PES in the short range was fitted using the permutation-invariant polynomial–neural network (PIP-NN) method [23–25]. The Morse-like variables *p _{ij }*= exp(−α

*r*) with

_{ij}*α*= 4.0

^{−1}Å

^{−1}of the internuclear distances

*r*were used in the input layer of the NN. The NN with two hidden layers with 30 and 20 neurons was trained using the Levenberg–Marquardt algorithm [26] and the root mean square error (RMSE), defined as 2.1 where and are the

_{ij}*ab initio*and fitted energies, respectively, was used as the target function. To avoid overfitting, the data are randomly divided into three sets, namely, the training (90% of the data points), validation (5%) and testing (5%) sets.

The long-range parts of the PES are assumed to be dominated by the ion–induced dipole and the ion–permanent dipole interactions, following the earlier work of Su and co-workers [10]. In the F^{− }+ HCl asymptote, the long-range potential is given by
2.2
and in the Cl^{−} + HF asymptote, it is has a similar form,
2.3

In equations (2.2) and (2.3), *q* is the charge on the ion (−1e) and and are the polarizability and the dipole moment, respectively, of the corresponding neutral species. *R*_{1} is the distance from the F atom to the midpoint between the H and Cl atoms, and *R*_{2} is the distance from the Cl atom to the midpoint between the H and F atoms. The polarizabilities and dipole moments are: *α*_{HCl }= 2.63 Å^{3}, *α*_{HF }= 2.46 Å^{3}; *μ*_{HCl }= 1.08 D, *μ*_{HF }= 1.2 D [27]. *V*(F^{−}) and *V*(Cl^{−}) are ion energies and *V*(HCl) and *V*(HF) are the corresponding one-dimensional diatomic molecular potential energies, respectively.

Finally, an analytical global PES for the system of interest can be written as
2.4
where *V*_{LR1} is the long-range potential between F^{−} and HCl, *V*_{LR2} is the long-range potential between HF and Cl^{−} and *V*_{fit} represents the short-range potential fit to *ab initio* data. *S*_{1} and *S*_{2} are switching functions, ranging from 0.0 to 1.0, which smoothly connect the long-range PESs to the short-range fitted *ab initio* PES:
2.5
and
2.6
in which the distance is in bohr.

QCT calculations in this work were implemented in VENUS [28]. The trajectories were initiated with a sufficiently large separation (greater than 26.0 Å) between reactants, due to the long-range interaction between the reactants F^{−} and HCl. The trajectories were terminated when the products reached a separation of 7.0 Å. The propagation time step was chosen as 0.1 fs. The ro-vibrational state of the reactant HCl and the translation energy of reactants were sampled from the Boltzmann distribution at a specific temperature. To determine the maximal impact parameter (*b*_{max}) for each temperature, small batches of trajectories were run to determine the largest impact parameters for reaction. The temperature involved in QCT calculations ranged from 100 to 600 K, and the *b*_{max} varied from 25.0 to 35.0 Å. The scattering parameters (impact parameter, vibrational phases and spatial orientation of the initial reactants) were selected with a Monte Carlo approach. The total reaction probability (*P*) for the title reaction is
2.7
where *N*_{r} and *N*_{total} are the number of reactive trajectories and total number of trajectories, respectively; the standard error is given by .

The thermal rate coefficients can be obtained as follows:
2.8
where *μ* is the reduced mass of the reactants, *k*_{B} the Boltzmann factor, and *T* the temperature in kelvin.

To gain some insight into the reaction intermediate, we have also computed the lifetime distribution of the FHCl^{−} complex. To this end, the time the system spends in the well, defined by *R*_{FH }< 2.2 Å and *R*_{ClH }< 3.6 Å, is recorded for each trajectory.

### (b) Experimental

All measurements were performed on the Air Force Research Laboratory's variable temperature selected ion flow tube (SIFT) instrument, which has been described in detail elsewhere [29]. Briefly, F^{−} ions are created using an electron impact source in the presence of a 10% mixture of NF_{3} in He. Ions are extracted and mass-selected using a quadrupole mass filter. The ions are focused before introduction to a laminar flow tube via a Venturi inlet, where approximately 10^{4}–10^{5} collisions with a He buffer gas thermalize the ions and carry them downstream. The temperature of the flow tube is variable using resistive heating devices (300–600 K). The neutral reagent, HCl is added 59 cm upstream of the end of the flow tube, with typical reaction times of the order of 3 ms, depending on helium buffer flow (varied from 10 to 13 std. l min^{−1}), and temperature. After travelling the length of the flow tube, the core of the flow is sampled through a truncated nose-cone with a 3 mm aperture. The remainder of the flow is pumped away using a Roots pump through a throttled gate valve that acts to maintain the desired pressure (typically 0.35 Torr at 300 K) within the flow tube. After the nose-cone, the primary ions and product ions are guided using a rectilinear quadrupole into an orthogonally accelerated time-of-flight (TOF) mass spectrometer for analysis. Rate coefficients are derived by monitoring the decay of the primary ion as a function of the neutral reagent flow. Errors in the rate coefficients are estimated to be ±25% absolute and ±15% relative to each other [30].

## 3. Results

A total of 18 666 points below 4.5 eV relative to the global minimum on the PES were selected and the best PIP-NN fit has RMSE for training, validation, testing sets and the maximum deviation of 0.95, 1.03, 1.19 and 16.16 meV, respectively. The overall RMSE of the PIP-NN PES for the F^{−} + HCl system is 0.97 meV. The distribution of the fitting error, which is defined by the difference between the fitted and original potential energies at the sampling points, is shown in figure 1. As can be seen, the randomly selected points covering the dynamically relevant configuration space are distributed in a wide energy range. As shown by figure 1, the fitting error is typically within the ±0.02 eV boundaries and increases slightly with energy. The *ab initio* equilibrium geometries, harmonic frequencies and energies of the relevant stationary points on the PES are well reproduced, as shown in table 1.

In figure 2, various cuts of the PES are shown. In panels (*a*–*c*), the PES is displayed as a function of the two bond lengths at several bond angles. In the collinear F–H–Cl configuration (panel (*a*)), the reaction is feasible via an ion–molecule potential well and the exothermicity of the reaction, which are clearly seen. There is no reaction barrier in either the entrance or exit channel. In the collinear F–Cl–H configuration (panel (*c*)), on the other hand, the PES is repulsive, although a shallow van der Waals well exists. In the perpendicular geometry (panel (*b*)), the entrance and exit channels are separated by a barrier. The anisotropy of the PES is also shown in panel (*d*), in which the HCl moiety with its bond length fixed at its equilibrium geometry is placed at the origin. It is clear from figure 2 that the ion–molecule reaction has a large cone of acceptance, as F^{−} approaches HCl from the H side. Within this cone of acceptance, the PES is barrierless. However, the other angles of approach lead to repulsion.

The topography of the PES sheds light on the reaction mechanism. As the anionic atom approaches, the neutral molecule is enticed to reorient itself in order to take advantage of the attractive part of the PES in order to form the ion–molecule complex. The reorientation is particularly effective at low collision energies, making the reactivity largest at lowest energies. As the collision energy increases, it becomes harder to reorient, leading to less reactivity.

In the QCT calculations, 20 000 trajectories were run for each temperature ranging from 100 to 600 K, and the statistical errors were all found to be less than 0.2%. The total energy conservation is better than 0.05 kcal mol^{−1}. The opacity functions are plotted in figure 3 for the temperatures studied. It is seen clearly that this reaction has extremely large maximal impact parameters, due apparently to the long-range attractive interactions. The maximal impact parameter is the largest at the lowest temperature, as discussed above.

Figure 4 displays the newly measured rate coefficients at four temperatures (circles), and their values are also reported in table 2. The experimental rate coefficients decrease as a function of temperature, as expected for a barrierless ion–molecule reaction. In the same figure, QCT rate coefficients computed on the new PES are also included at six temperatures (triangles). The agreement between the new experiment and theory is quite good, with the latter within the experimental error bars. For comparison, the previous experimental [7] and theoretical [10] results are also displayed in the same figure. The newly measured SIFT rate coefficient at room temperature is slightly larger than the FA value (square), but both are within each other's error bars. These rate coefficients appear to be consistent with reaction (1.1) proceeding with nearly every collision between the species. This is indeed the case in the QCT calculations, where the capture rate, defined as the rate for entering the ion–molecule well, is essentially the same as the overall reaction rate. In fact, the SIFT rate coefficients reported here are slightly larger than the parametrized collision rate by Su & Chesnavich [4], which takes into account only the polarizability and permanent dipole moment of the neutral species.

Despite the fact that our PES has the same long-range terms as in Wei *et al*. [10], our calculated rate coefficients are much larger than the earlier QCT results. The difference can thus be attributed to the different level of the *ab initio* theory. The current CCSD(T)-F12b method is considered as the gold standard in *ab initio* calculations because it captures electron correlation due to high-order excitations that are necessary for describing the dispersion interactions. While such interactions are at shorter range than the ion–dipole interaction, they might still affect the reactivity at large impact parameters. To illustrate this point, we plot in figure 5 the minimal energy path including the centrifugal potential *J*_{tot}(*J*_{tot}* _{ }*+ 1)/2

*μR*

^{2}, where

*μ*is the reduced mass of reactants and

*R*is the distance between F

^{−}and the centre of mass of HCl, for two values of the total rotational momentum

*J*

_{tot}on our

*ab initio*PES and the long-range PES. These two values correspond to impact parameters of 14.26 and 8.28 Å, respectively, at a collision energy of 1.2 kcal mol

^{−1}. At the lower

*J*

_{tot}value, the centrifugal barrier is dominated by the long-range terms, as expected. This is in the range where the bulk of reaction takes place. However, as

*J*

_{tot}increases, the short-range part of the PES becomes more important. As shown in figure 5, the long-range PES is less attractive than our PES. This is consistent with our rate coefficients being larger than the results of the capture model of Su & Chesnavich [4], which is completely based on the long-range ion–dipole and ion–induced dipole terms. Higher-order terms, such as the ion–quadrupole attraction, may play a prominent role in this system, particularly at large impact parameters. This is as evidenced in figure 4 by the better agreement of the measured rate coefficient with the collision rate calculated using the capture model of Ervin [8], which also considers anisotropic polarizability and quadrupole moment. We note that a further parametrization by Bhowmik & Su [33] aimed to include effects of the quadrupole moment, and agrees reasonably with the model by Ervin down to temperatures of about 100 K.

Finally, figure 6*a* shows the product HF vibrational distribution at *T *= 300 K calculated by the QCT method compared with the experimental result [7]. It can be readily seen that our calculated distribution is in good agreement with experiment. It should be noted that, in the experiment, the relative intensity for *v*_{HF }= 0 was obtained by extrapolation of measured data (distributions for *v*_{HF }= 1, 2 and 3), thus it has larger error.

To understand the statistical nature of the reaction or lack thereof, we have computed the HF vibrational state distribution using the phase space theory (PST) [34,35]. The basic idea of PST is to assume that at a given energy and total angular momentum all product states have the same probabilities in the dissociation of the FHCl^{−} complex. This offers the statistical limit for a complex-forming reaction where the energy is fully randomized in the intermediate complex. For our purpose, the microcanonical cross sections obtained by counting the number of open channels of the product are averaged over the Boltzmann distribution at a given temperature, giving rise to the canonical product state distributions. As shown in figure 6*a*, the measured and QCT product state distributions are both found to deviate from the statistical limit provided by the PST results, suggesting that complete energy randomization is not achieved in the FHCl^{−} complex. Specifically, there is more population in the vibrationally excited states in both the calculated and measured distributions. This non-statistical behaviour can be attributed to the short lifetime of the intermediate complex, as shown in figure 6*b*. Most of the trajectories have a lifetime of less than 100 fs, which is not sufficient for energy randomization, especially for a complex involving a hydrogen where the intramolecular vibrational energy redistribution is not expected to be efficient.

## 4. Conclusion

Ion–molecule reactivity is traditionally thought to be controlled by the leading long-range interaction terms, namely the ion–dipole and ion–induced dipole interactions. Previous studies seem to suggest this picture might not be quantitatively valid for the title reaction. In this publication, we report a joint experimental–theoretical study to improve our understanding of this prototypical ion–molecule reaction. Theoretically, a new global PES is developed by fitting a large number of high-level *ab initio* data points, augmented by long-range interaction terms. QCT calculations on the new PES yielded rate coefficients at several temperatures as well as the product vibrational state distribution. Experimentally, the rate coefficients have been measured at several temperatures using the SIFT set-up. The agreement between theory and experiment is good. As indicated by the new experimental rate coefficients for this prototypical ion–molecule reaction and the accompanying theoretical analysis, the higher-order multipole terms play an important role. These higher-order terms affect the reactivity at large impact parameters because they become more pronounced at shorter ranges. For an accurate characterization of the kinetics for ion–molecule reactions, it is important to include all electrostatic and dispersion terms.

In addition, it is shown that this reaction is not statistical, as reflected by the HF product state distribution. There are more populations in the vibrationally excited HF states than predicted by a statistical model. The deviation from the statistical limit is attributed to the non-complete energy randomization in the short-lived reaction intermediate.

## Data accessibility

This article has no additional data.

## Competing interests

We have no competing interests.

## Funding

This work was supported by the Air Force Office of Scientific Research (FA9550-15-1-0305 to H.G. and AFOSR-16RVCOR2 to A.A.V. and N.S.S.). S.G.A. acknowledges the support of Boston College Institute of Scientific Research. X.L. thanks National Natural Science Foundation of China (grant no. 11274205) and Shandong Provincial Governmental Program to Study Abroad.

## Acknowledgements

H.G. thanks Prof. Bill Hase for some discussion on the phase space theory. This article is dedicated to Professor John N. Murrell, FRS, in his memory.

## Footnotes

One contribution of 14 to a theme issue ‘Modern theoretical chemistry’.

- Accepted October 31, 2017.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.