Benchmark Quantum Mechanical Calculations of Vibrationally

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Benchmark quantum mechanical calculations of vibrationally resolved cross sections and rate constants on ab initio potential energy surfaces for the F+HD reaction: comparisons with experiments Dario De Fazio, Simonetta Cavalli, and Vincenzo Aquilanti J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b01471 • Publication Date (Web): 17 May 2016 Downloaded from http://pubs.acs.org on May 19, 2016

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The Journal of Physical Chemistry

Benchmark Quantum Mechanical Calculations of Vibrationally Resolved Cross Sections and Rate Constants on ab initio Potential Energy Surfaces for the F+HD Reaction: Comparisons with Experiments Dario De Fazio

∗

Istituto di Struttura della materia - C.N.R., 00016 Roma, Italy

Simonetta Cavalli, Vincenzo Aquilanti Dipartimento di Chimica, Biologia e Biotecnologie - Universit` a di Perugia, 06123 Perugia, Italy

May 17, 2016

Abstract Quantum scattering calculations within the time-independent approach in an extended interval of energies have been carried out for the title reaction on four ab initio potential energy surfaces. The calculated integral cross sections, vibrational branching ratios and rate constants are compared with scattering experiments as well as with chemical kinetics rate data available for this system for both the HF and DF channels. The calculations on the CSZ (J. Chem. Phys. 2015, 142, 024303) and LWAL (J. Chem. Phys. 2007, 127, 174302) surfaces are in close agreement between them and reproduce satisfactorily the experimental measurements. The agreement with the experiments is improved with respect to calculations on the earlier SW (J. Chem. Phys. 1996, 104, 6515) and FXZ (J. Chem. Phys. 2008, 129, 011103) surfaces. The results presented here witness the remarkable progress made by quantum chemistry calculations in describing the inter-atomic interactions governing the dynamics and kinetics of this reaction. They also suggest that comparison with translationally and rotationally averaged experimental observables is ∗

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not sufficient to assess the relative accuracy of highly accurate potential energy surfaces. The dynamics and kinetics calculations show that temperatures lower than 50 K or molecular beam spreads below 1 meV have to be reached in order to discriminate the accuracy of the LWAL and the CSZ surfaces.

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Introduction

The F+H2 reaction and its isotopic variants have served as a prototype system for testing many experimental and theoretical approaches used in reaction dynamics. This is because with its eleven electrons this system can be accurately treated in order to exhibit many aspects of relevance also in more complex triatomic reactions. It has therefore a long history in the literature and the rate of appearance of papers on different aspects of its dynamics remains nowadays still very high. Regarding on the F+HD isotopic variant, the first complete investigation illustrating its rich dynamics was published at the beginning of the new millennium 1;2 , when it was shown that differences between the two reactive exit channels, were sufficiently well described by quantum reactive scattering calculations using the ab initio potential energy surface (PES) obtained by Stark and Werner (SW) 3 . Interestingly, this also permitted the clear observation of the resonance phenomenon in chemical reactions and made it possible to study it by reliable theoretical tools. Additional theoretical efforts have been therefore dedicated to deeply understand the resonance structure of the F+H2 and F+HD systems 4;5;6;7;8;9;10;11;12 and their stereo-dynamics 13;14;15 . To further motivate these studies, new molecular beam experiments with higher resolution, provided additional resonance effects 16;17;18;19 very well described by the theoretical simulations on a new family of ab initio PES 20;21 , obtained by the Dalian group. It was in fact well known, that various deficiencies of the SW PES affected in particular the description of both the entrance and the exit channels. The suggested semi-empirical or ab initio corrections 22;23;24;25 , although solving some specific aspects, were unable to give a completely satisfactory account of the effective reaction dynamics 26;27 , and thus improvements on the PES were needed. These deficiencies became more evident when the open-shell structure of the Fluorine atom was taken into account 28 by adding the spin-orbit coupling term in the electronic Hamiltonian to obtain the ground adiabatic PES of the system 29 . A complete set of ground and excited PESs was supplied 30 permitting to test the effect of the non-

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adiabatic couplings (mostly induced by the spin-orbit term) in the nuclear dynamics. The first multisurface calculations of the F+HD reaction were presented in Ref. 31 where the wave-packet reaction dynamics of the ground spin-orbit state reaction was compared with Born-Oppenheimer (BO) calculations performed on the spin-orbit corrected PES of Ref. 29 . The comparison showed small differences between the two calculations probably due more to the dynamical approximations used 31 rather than to significant non-adiabatic effects on the nuclear dynamics, corroborating the accuracy of the spin-orbit corrections to describe the BO ground state dynamics. More accurate multisurface calculations were presented in Ref. 32;33 where a time-independent method without any dynamical approximation was used. Although a direct comparison with BO calculations was not provided in Refs. 32;33 the main effect discussed in these works was the small reactivity of the upper spin-orbit state (neglected in BO calculations) and its relevance of this reactivity on the theoretical simulation of some experimental data. To eliminate the deficiencies of the SW PES and its family, more efforts were dedicated by the Werner group to the high accurate determination of the reaction barrier 34 and to new sets of global PESs for the ground and excited PESs 35 . From these sets a ground state PES (Li-Werner-AlexanderLique, LWAL PES hereafter) with the spin-orbit correction was also extracted and made available in Ref. 36 . However, few quantum reactive scattering calculations 35;36;37;38 have been published with the LWAL PES especially for the F+HD isotopic variant, and it is of interest to probe the improvements provided by this PES as far as the agreement with the previous experimental data is concerned. More recently, the possibility to detect experimentally resonance features also for vibrationally excited states of the F+HD reaction 39 motivated the production of a new PES ( Chen-Sun-Zhang, CSZ PES hereafter) made available by the Dalian group 40 . The CSZ PES was suggested as presenting improvements with respect to the previous Fu-Xu-Zhang (FXZ) PES 21 , featuring a more stable behavior at high energy and a lower barrier height. Extensive comparison of dynamical calculations on the CSZ PES with experiments 1;16;17;18;19;39;41 were also provided 40;42;43 , showing slight improvements with respect to the simulations obtained by the FXZ PES. In particular, the comparison with Ref. 41 , reported in 26 , failed to show the better agreement expected by the lower reaction barrier of the CSZ PES. The authors address this deficiency to possible inaccuracies of the

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entrance channel region coming from the difficulties of the employed ab initio method to properly deal with the entrance channel of the reaction and in particular with the crossing between diabatic electronic states near the collinear geometry. The result was a lower anisotropy of the PES in the van der Waals region of the entrance channel. Still more recently, CSZ and LWAL PESs have been used to simulate the electron photodetach43;44 . In these papers some effort has been dedicated to understand ment spectrum of the FH− 2 anion

which of the two PESs should be considered more reliable to reproduce the interaction with the neutral species. There is also another more fundamental reason to understand how close the new PESs reproduce the experimental observables. In fact, it is well known 45;46 that for the F+H2 system important Renner-Teller and Jahn-Teller effects could be operative near the conical intersections located in three different regions along the minimum energy path of the entrance channel. Spin-orbit may not therefore be the only non-adiabatic coupling affecting the reactive observables. Therefore a complete comparison between high accuracy BO treatments and experimental data is required for indicating the relevance of finer dynamical treatments 47 , including more extensively non-adiabatic and/or geometric phase effects. In this work, rigorous quantum scattering calculations have been performed with the CSZ and LWAL PESs and the results for both the H-atom and D-atom transfer reactions are compared with previous available experimental data 1;41;48;49;50;51;52 . Since some of the simulations (especially for the DF channel) required values of the input parameters of the employed reactive scattering code considerably stricter than previously, we also repeated SW and FXZ calculations to check the dependence on the PESs of the simulations and to test the accuracy of the data previously published 26;53;54 . In the next Sec. II, we describe the details of the PESs and of the quantum reactive scattering calculations. In Sec. III we present the theoretical results and then discuss the comparison with the experimental data. Finally, our conclusions are drawn in Sec. IV.

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Calculational details

2.1

Potential Energy Surfaces

In this section we give a brief summary of the quantum chemistry methods used to build the four ab initio PESs employed referring the reader to Refs. 3;21;35;40 for a more detailed description of the works done. The SW PES is a fit of 714 energy points calculated by the internally contracted Multi Reference Configuration Interaction (MRCI) method plus multireference Davidson corrections (+Q) 55 . Complete active space self-consistent field (CASSCF) reference functions (622/2 in the notation of Ref. 3 ) and a one electron basis set extension similar to a quintuple-zeta Dunning basis set (cc-pV5Z) 56 were employed. The ab initio energies were fitted to a many-body expansion with the simplest Aguado-Paniagua 57 functionalities with a global root mean square deviations (rmsd) of about 11 meV. On a similar ab initio method (internally contracted MRCI+Q) is based also the LWAL PES. However larger CASSFC reference functions (722/1 in the notation of Ref. 3 ) and a more extended one electron basis set (augmented (aug-) ccpV5Z 58 ) were here employed. Also the number of ab initio geometries was larger (1178). Three different fits in three different internuclear geometries regions were obtained and merged into a single set of diabatic PESs. The rmsd of the three fits were of about 5 meV in the strong interaction region (78 points) and less of 1 meV in the product channel region where most of the energies (820) were calculated. In the entrance channel the diabatic PESs were coupled by the spin-orbit and non-adiabatic terms of Ref. 30 , so that the adiabatic PES is obtained by diagonalization of a six-by-six matrix. The ab initio energies as well as the spin-orbit couplings, were corrected by using scaling factors to reproduce experimentally known features of the reaction. In particular in Ref. 35 , two different scaling factors were proposed, one equal to 1.05 reproducing well the barrier properties as found in the benchmark calculations of Ref. 34 and the other equal to 1.078 giving an excellent agreement with the experimentally determined value of the exoergiticy of the F+H2 reaction. In Ref. 36 , the LWAL PES has been obtained merging the exit channel of the PES with the correct exoergicity to the PES matching well the entrance and transition state regions.

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Much more different is the quantum chemistry theory used to build the FXZ and CSZ PESs. For these PESs a spin unrestricted coupled cluster (UCCS) method was employed. To enhance the accuracy of the quantum chemistry calculations saving the computational load, a hierarchical construction scheme 21 was used, employing different levels of UCCS calculations and different extensions of the one electron basis set. In particular in the FXZ PES all single and double excitations and the perturbative account of triple excitations (UCCSSD(T)) were treated on the augmented quadruple zeta (aug-ccpVQZ) and quintuple zeta (aug-ccpV5Z) basis sets of Kendall et al 58 . In the CSZ PES all triple and perturbative quadruples excitations were computed and aug-ccpVQZ(ccpVQZ) basis set were used for Flourine (hydrogen) atom respectively. UCCSSD(T) calculations were also performed on a aug-sextuple zeta (aug-ccpV6Z) basis set. This last set of calculations were required to obtain an extrapolation to the complete basis set (CBS) limit of 1184 energies. The total number of internuclear geometries investigated changes markedly with the different accuracy of the quantum chemistry method employed: about 15000 geometries were calculated at the lowest degree of accuracy (UCCSSD(T) with aug-ccpVQZ basis set) while for just 775 of them all triple and perturbative quadruple excitations were considered. Another important difference between CSZ and the other PESs is that the CSZ PES is the only one that does not use any scale factor of the ab-initio energies: in fact also in FXZ PES the ab-initio energies was scaled by a factor of 1.01 in order to get the correct exothermicity of the F+H2 reaction. Also very different with respect to the previous PESs was the fitting method employed. The FXZ PES fit is based on a spline interpolation method. The advantage of this method is that it avoids the inaccuracies produced by the fit of the ab-initio energies with analytical expressions preserving the accuracy of the quantum chemistry calculations. The disadvantage is that it requires regular and rather dense grid points to avoid unphysical behavior of the PES. This is hard to reach in the region of the PES well away from the minimum energy path, so that for example regions of the PES high in energy can present spurious behaviors. We will see later that these aspect can affect the convergence of dynamical results in particular conditions (cold, ultracold or very hot regimes). Different neural-network 59 schemes were instead employed to fit the various terms of the CSZ PES. At variance with the interpolation method, the neural network introduces a small fitting error but it produces much more stable PESs at internuclear geometries far from the minimum energy path.

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In the CSZ PES the rsmd were lower than 0.1 meV. We warn here that although the stability of the neural network is larger than in the spline method, its use does not insure the continuity of the first derivatives of the PES. Both the CSZ and FXZ PESs were corrected in the entrance channel in order to take into account of the spin-orbit interaction, taking the correction from the PES of Ref. 20 . In Table 1 we report a summary of some characteristic features of the four ab initio PESs employed. The comparison among the SW PES and the other ones shows the effects of the spin-orbit corrections. As discussed in previous works 35;53 the most important effects are the enhancement of the reaction barrier by about 17 meV (1/3 of the Fluorine spin-orbit splitting) and the reduction of the entrance channel van der Waal well depth, that moves at larger atom-diatom distances increasing the width of the barrier. From the numbers in Table 1 we can see that the SW barrier (without spin-orbit) is just about 5 meV lower than that for the LWAL PES explaining the success of the SW PES to describe the main reactive properties of the system. More relevant is the difference for the Entrance channel van der Waals depth (in the perpendicular configuration) that is about half in the spin-orbit corrected PESs. LWAL, FXZ and CSZ PESs agree very well for these features, except for the reaction barrier of the FXZ PES, about 5 meV higher than LWAL PES. The LWAL value is also in perfect agreement (within about 0.1 meV) to the benchmark value obtained in Ref. 34 . Although all the spin-orbit corrected PESs agree very well (within 0.3 meV) about the entrance channel van der Waals well depth, its anisotropy is slightly different. To put into evidence this feature, in Table 1 we report also the well depth of the entrance channel collinear cut. We can observe that the well is slightly deeper (of about 1 meV) in the LWAL PES, that shows therefore the lowest anisotropy. Actually, we must warn here that the collinear well lies at a larger atom-diatom distance with respect to the perpendicular van der Waals well, so that the differences between the two well depths is not exactly equivalent to the van der Waals anisotropy. However an angular cut at the atom-diatom distance of the stationary points shows the same behavior described above. More important differences are evident in the exit channel van der Waal well where the FXZ PES shows the largest depth (13.6 meV) and the LWAL PES the smallest one (9.6 meV). As shown later this will lead to some difference in the reaction dynamics. The well depth of the SW PES is very similar to the one of the CSZ PES. However we must remark that the exothermicity (without

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zero point energy) of the SW PES is about 16 meV lower than the one of the CSZ PES, so that the well lies at higher energies. Moreover the SW PES exotermicity error is also larger if the spinorbit interaction is taken in account. Excluding the SW PES the other PESs show similar values for the exothermicity (within 1-2 meV). In particular the values of the FXZ PES are supposed to be very accurate because, as reported above, a scaling factor of the ab-initio energies was applied to reproduce accurately the experimental vibrational state-specified exothermicities of the F+H2 reaction (see Tab. II of Ref. 40 ).

2.2

Quantum Reactive Scattering Method

Rigorous quantum reactive scattering calculations have been carried out using a parallelized variant of the ABC code 60 . In this variant a different algorithm (namely the Enhanced Renormalized Numerov method of Ref. 61 ), replacing the log derivative propagator 62 implemented in the original code, was employed to propagate the hyperradial wavefunctions. This makes the calculation more efficient (from one order of magnitude down to a factor two 63 ) reducing the number of the required hyperradial sectors required and making more stable the adaptation of the sector width according to the investigated collision energy range. In addition to the Open-MP/MPI structure discussed elsewhere 26;64 a new MPI parallelization in the hyperradial grid has been implemented. This permits to partially overlap the building matrix operations and the propagation of the hyperradial functions with the diagonalization step required to build the hyperspherical eigenfunctions in the middle of the hyperradial sectors. The performances of this parallelization actually are not so high (a maximum scalability between 2 and 3 has been observed) and have been found to be strongly dependent on the details of the input: however it permitted to fully exploit simultaneously also other implemented parallelizations 26;64 so that the gain in achieved elapsed time increases linearly the already high total scalability of the code. More interestingly, from the technological point of view, the code has been fully adapted to exploit the Many Integrated Core (MIC) architecture available, for example, in Xeon Phi Intel processors. In these machines the low performance of each physical core (about 1.2 GHz) is superseded by the large number of computational unit available (about 260 in the Xeon Phi Intel processors). The large Open-MP scalabilities (about 30) of the linear algebra operations in this architecture and the relevant role that these operations cover in quantum reactive scattering

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calculations, make the calculations presented very inexpensive from the computational point of view. For example, all the ’production runs’ defined later in this section, needed from 1 to 6 hours using from 1 to 10 Xeon Phi Intel processors (depending from the collision energy range investigated, see below). Because of these high computational performances, we were able to monitor comfortably the behavior of the convergence parameters of the ABC code for all the four PESs employed in the full collision energy range investigated. This is relevant in particular for the DF channel where, as discussed in 26 , the inputs previously used 26;40;53;54 , were unable to yield convergent results for the state selective observables of the DF channel in the high (above 120 meV) collision energy range. To find the convergence parameters, the full energy range (300 meV) was divided in eight intervals. The border of each interval was chosen at the energy threshold of each reactant rotational state (jin from 0 to 6). Scattering calculations in the neighborhood of reactive thresholds (within about 2 meV) require a particular choice of the convergence parameters: because of the increasing of the reactants de Broglie wavelength, larger values of the maximum hyperradius (rmax) must be employed and strong resonance features affect the integral cross section (ics). However, the threshold behavior of higher rotational reactant states has little effect on the rate constants, so that, except for j=0, we neglected it in the calculations. Inside each interval, a grid of ten evenly spaced energies was chosen to check the ics convergence of all the open reactants states. The parameter was considered convergent if the product vibrational ics of all the initial states involved changes smaller than 0.2-0.3 %. For the ’production runs’ an opportunely fine energy grid (dE parameter) was chosen in each interval in order to resolve the main resonance features in the total ics. This criterium leads to a convergence of the total ics and of the rate constant of the order of 0.1 %, that is about one order of magnitude less than for the previous calculations performed for this system 26;40;53 . In Table 2 the convergence parameters found for the SW PES are given. We can note, that with respect to the previous calculations on this system, a larger values (8 instead of 5) of the maximum projection (kmax) of the total angular momentum (J) is required to give convergent ics values at high collision energies. In particular branching ratios for the lower vibrational quantum numbers (0 and 1) of the DF channel are highly sensitive to the value of the kmax parameter employed and the value of 8 is absolutely required to give their correct energy behaviour. Also, we

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increased the collision energy range and the value of the maximum total angular momentum used (Jtotmax) to obtain better convergence for the rate constants at the highest temperature reported (350 K). Convergence parameters for the other PESs are similar (within about 10 %) except for the LWAL PES where emax (the maximum energy of the rotovibrational states included in the basis set) and jmax (the maximum rotational quantum number included in the basis set) parameters are substantially higher. For instance, a minimum value of 3.0 eV (and a maximum of 3.5 eV) has been used in the ’production runs’ with the LWAL PES for the emax parameter. This makes these calculations considerably more expensive especially at high energy, where a large dimension of the basis set must be used. The dimension of the basis set is a good indication of how ’heavy’ are the linear algebra operations performed by the code (essentially diagonalization, linear system solver and matrix-multiplications) and therefore it is a good index of the computational load spent. For example in the highest energy range the dimension of the basis set with the parameters given in Table 2 is about 7000. Convergent parameters for FXZ and CSZ PESs are similar to the ones of the SW PES. However, with the FXZ PES we were not able to obtain convergence with emax 2.5 eV so we did not perform calculations in the highest energy range. Above this value, in fact, apart from lack of convergence, affecting ics calculations, spurious deep wells were found monitoring the behavior of the lowest hyperradial eigenvalue as a function of the hyperradius in the region of strong triatomic interaction. This anomalous behavior (not present in the newer CSZ PES) is likely due to the instabilities of the spline interpolation method (used by the FXZ PES) in the regions of low ab initio points density discussed in the previous section. Similar problems (in this case for both the FXZ and CSZ PESs) are present at low collision energies where larger values of the hyperradius must be used to obtain convergent ics. In this case, unphysical behaviours of the PESs at large interatomic distances probably due to the lack of ab initio points, does not permit to check the convergence of the rmax parameter (the maximum value of the hyperradius used in the calculation) so that the parameter values found for the LWAL PES were used instead. More comments about these limitations of the new PESs will be given the next sections.

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3

Results and Discussion

In the following, the results of quantum scattering calculations carried out on the LWAL, CSZ and FXZ PESs are compared with kinetic data 41;48 and molecular beam experiments 1;49;50;51;52 , in order to probe the dynamical features of the title reaction.

3.1

Comparison with molecular-beams experiments

In Figure 1 we show the simulations of the experiment of Ref. 1 where the total integral cross sections (ics) of the two reactive channels of the F+HD reaction were measured in a wide collision energies range (5-210 meV). To simulate the finite translational energy resolution of the beam, all the theoretical data have been convoluted by a Gaussian function with full-width-half-maximum (FWHM) of 3 meV. However, simulations with different FWHM values (1 and 6 meV, not reported here) show that the main result of the convolution is to slightly alter the resonance features and the threshold energy positions, with small effect on the global agreement. As discussed later, larger changes on the simulations, especially for the HF channel, come from the rotational distribution of the reactants, so that a correct estimate of the rotational temperature of the HD beam is relevant for the comparison with the theoretical data. In this work, as in our previous simulations 26;27;53 we use a rotational temperature of about 50 K, giving a distribution of about 81 and 19% for j = 0 and 1, respectively. However, to understand the impact of this choice on the comparison with theoretical calculations, we also simulated (not shown) this experiment (and the others of the same group shown later), using 10 (about all j=0) and 70 K (about 66, 32 and 2 % for j=0,1 and 2 respectively). Since no absolute values were supplied in Ref. 1 we re-scale ( with a single normalization factor for both HF and DF channels) the experimental data optimizing the agreement with the simulation of the most recent PESs (CSZ and LWAL). The scale factor applied here is therefore slightly different (within 10 %) from the one used in our previous simulations 26;53 . As discussed in previous articles 1;26;53 the excitation functions of the two channels show a remarkably different behavior. The ics of the DF channel increases monotonically with collision energy and is completely smooth, showing the typical behaviour of an exothermic reaction with small activation energy. Otherwise, the ics of the HF channel shows a marked bump around 30 meV and increases rapidly until about 90 meV leveling off at higher energies, where an oscillatory behaviour 11



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is shown. This is due to the higher relevance of the tunneling effect 74 for the lighter isotopic variant permitting significant reactivity in the under barrier region enhanced by resonance effects. From data in the figure we can conclude that all the PESs, except the old SW PES, reproduce satisfactorily the experimental data of both reactive channels. Minor discrepancies can be noted, such as the change of slope of the excitation functions around 60 meV leading to a slight underestimate of the ics for both channels and the experimental broad oscillations of the HF ics in the high collision energy range not reproduced by the theory. Surprisingly, perhaps, is the excellent agreement between CSZ and LWAL PESs, nearly indistinguishable at the scale of the figure, notwithstanding the wide differences in the quantum chemistry methods employed discussed in Sec. 2.1 and variance of the characterizing features listed in Table 1. Such a perfect agreement between the reactive observables suggests that the theoretical approaches are near to convergence at the BO level of theory, so that differences in reactivity can be detected only employing more advanced dynamical treatments. As shown already in Ref. 40 , the differences between the FXZ and CSZ simulations are slight and the lower barrier (see Table 1) affects more the HF than the DF channel in the high collision energy range. However, this experiment is weakly sensitive to this difference, so that only small improvements can be noted between the CSZ-LWAL simulations here presented and the comparison shown in Figure 4 of Ref. 26 where the experimental data were scaled to optimize the agreement with the FXZ data. Also, we can note a slightly better description in the FXZ simulations of the broad resonance feature around to 30 meV. However, we must consider that the ratio between the intensity of this resonance peak and the behavior of the ics at high energy, is markedly dependent on the rotational distribution of the beam, so that a small enhancement (10-20 K) of the rotational temperature clearly improves the experimental agreement of the CSZ-LWAL simulations. This is due to the j-selectivity of the broad resonance peak and to the relevant enhancement of the reactivity of higher rotational states of the HF channel 53 , so that the exact choice of the rotational temperature markedly shape the behavior of the excitation function. Also we must consider that a small fraction (estimated around 16 % 32 ) of the Fluorine atom is in its excited spin-orbit state in the beam of Ref. 1 . As discussed in Ref. 32 , although its reactivity is small and moderately energy dependent, it changes slightly the intensity of the peak essentially because the spin-orbit excited ics are not affected by the resonance feature. A finer simulation including the reactivity of both the spin-orbit

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channels should lower slightly the intensity of the peak making closer the experimental agreement of the new PESs. Finally, we can note that with the beam energy spread employed in the simulation, the HF longlived resonance effects 65 are completely smeared out and cannot be observed by the experiment. However, the Regge oscillations 10 of the transition state resonance can still be distinguished in the HF ics around 90 meV. Such oscillations in fact have been observed at the state-to-state level in the experiment of Ref. 18 . Actually, as noted above, oscillations in the high energy range (above 130 meV) are present in the experimental HF behavior but they are too broad and with much wider amplitude with respect to the ones obtained by the theory. It is possible that non-adiabatic effects play some role on these features amplifying the oscillations. However, the non-adiabatic calculations 31;32 performed with the PES of Ref. 30 do not support this hypothesis. In Figure 2 the vibrational branching ratios of the HF channel are compared with the experiments. In the upper panel the Dalian experiment 50 is simulated, while in the lower panel the theoretical results are compared to the experiment of Ref. 49 . The reactants’ rotational distribution, in the Dalian molecular beam 50 is cold enough to be considered involving only ground rotational state, while the Taipei experiment 49 is simulated as in Figure 1. A comparison between the two simulations (not reported) shows that the small amount of excited rotational states present in the Taipei beam 49 produces small effects on measured quantities, so that the output of the two experiments shows mutual agreement. As analyzed in Ref. 26 the branching behaviours are strongly characterized by the wide maximum of the HF(v ′ =3) ics as soon as this vibrational manifold become available. The reaction dynamics in this vibrational channel presents peculiar characteristics different by the other vibrational channels. In particular the reaction products are mainly scattered in the forward hemisphere and most of the reactivity comes from higher partial waves suggesting the prevalence of a spectator stripping mechanism 66 at variance with the other channels affected more by a mechanism of re-bound type. Observing the upper panel, we can see that the theoretical simulations with the LWAL PES perfectly reproduce the experimental finding, with the only exception of the ratio between v ′ = 1 and v ′ = 2 at high energy, that is significantly underestimated in the theoretical predictions. The experiment is weakly sensitive to the accuracy of the PESs employed except for v ′ = 3 where the FXZ anc CSZ results show a maximum (and consequently a minimum for v ′ =

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2) much more pronounced than in the experiment. The theoretical results simulate correctly the Taipei behavior 49 while showing a much more pronounced maximum for v ′ = 3 in closer agreement with FXZ predictions. As discussed in Ref. 26 the intensity of this feature is slightly affected by the entrance channel region of the PES while it is strongly affected by the exit channel characteristics. We note here that the intensity of the peak is strictly correlated to the exit channel van der Waals well depth (see Table 1). The influence of the van der Waals well in the stripping mechanism is well documented in literature 67;68 . Also, we can note that the HF(v ′ =3) maximum is completely absent in the oldest SW PES notwithstanding the relevant well depth. However as well documented in various articles (e.g. in Refs. 6;40 ) the SW PES presents a high (around 10 meV) spurious barrier in the exit channel just after the van der Waals well, that could markedly hinder the stripping mechanism. In addition the lower exothermicity of this PES makes higher the threshold of HF(v ′ =3) (see Table 1) so that in the energy region where the mechanism is more efficient the v ′ =3 manifold is closed. Moreover, stereodynamical calculations 14 show that the HF(v ′ =3) product is mainly formed by a collinear configuration attach, where the conical intersections appear. It is therefore possible that non adiabatic effects could play some role in the comparison with the experiments. Finally, it is worthy to note that in the single feature where LWAL and CSZ PESs show significant differences, the two experiments disagree, not allowing an unambiguous discrimination of the accuracy of the two PESs. In Figure 3 the vibrational branching ratios of the DF channel are reported. In the upper panel, the first eight experimental points in the lowest energy range are extracted from Ref. 51 while the remaining measurements were published later in Ref. 52 . The lower panel compares the results with the experiment of Ref. 49 . The rotational distribution of the two beams was simulated in the theoretical calculations as described above. We can note that the vibrational distribution of the product is very hot at low collision energy where the highest vibrational levels (v ′ = 3 and 4) are the most populated. This is in agreement with the Polanyi’s rules for exothermic reaction with early transition state 69 . Increasing the collision energy the vibrational distribution becomes colder. A decrease of the validity of the Polanyi’s rule at high collision energies, characterizing the behavior of branching ratios of Figs. 2 and 3, has been documented and interpreted for a long list of reactive systems in Ref. 70 . As in the previous cases, LWAL and CSZ simulations agree

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perfectly between each other and it is very difficult to distinguish the two curves in the figure. The theoretical simulations describe very well the experiments, especially the most recent one. Also the broad maximum in the v ′ = 4 ratio around to 30-40 meV, found in the Dalian 2009 experiment 51 , seems to be reproduced, although at a slightly higher collision energy. The only relevant difference with the theoretical simulations is the similar v ′ =3 and v ′ =4 experimental ratios found at the first energy of Ref. 49 . However, we note that also the Dalian 2009 experiment of Ref. 51 shows a much higher reactivity in v ′ =3 in the same energy range and that the theoretical simulations with higher rotational temperature of the beam do not influence this feature. The FXZ and SW PESs also reproduce satisfactorily the experiments, although the agreement is significantly worse in some cases (e.g. for the v ′ =4 FXZ simulations of the Dalian 2011 experiment 50 ). Finally, we remark that at variance with the results presented in Fig. 1, no scaling factor has been applied to the experimental results in Figs. 2 and 3, since branching ratios involve relative measurements of the ics.

3.2

Comparison with experimental rate constants

Total ics are then Boltzmann averaged as described in Refs. 24;54 to obtain thermal rate constants in the range 10-350 K. As discussed in Sec. II in the present calculations more stringent values of the ics convergence parameters with respect to our previous calculations 26;53 were employed, so that we also calculated rate constants for SW and FXZ PESs to analyze the impact of the new parameters and code on the kinetic results. The new results for the four PESs are given in tabular form in Tables S1 and S2 71 for the HF and DF channel respectively. These numbers are considered as convergent within about 0.1 %. Where available we also give our previous results 53;26 , obtained with different reactive hyperspherical methods and codes 26;72;73 and the wave packet calculations of Ref. 54 , the first accurate quantum mechanical rates for the system already compared in Ref. 53 . In the comparison of the SW PES we can observe that the results in 53 and 54 agree effectively within 1 %, as claimed in the respective papers. Moreover, where differences between the two calculations can be detected, the present results are mainly in the middle for HF. However, for DF channel (where the results of Refs. 53 and 54 agree perfectly) the new results above 200 K are slightly larger than the previous ones. This is due to higher relevance for the DF dynamics of total angular momentum

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projections larger than 5 and initial j=6 for the first time included in the present calculations (see Sec. II). The same effect at the highest temperatures can be also seen in the FXZ rates where results are compared 26 . Although these differences are very hard to be seen in logarithmic plots, they can affect more markedly the quantities dependent on the temperature derivatives of the rates, such as the apparent activation energies studied in 74 . The flat decreasing behavior at high temperatures of the DF results shown in Ref. 74 , not described by the model, is in fact removed when apparent activation energies are obtained employing the new rates. Comparing CSZ and LWAL rates, we can see that they are within 1 % in the temperature range (180-350 K) covered by the experiments available 41 , much less than the experimental error bars. Theoretical rate constants and experiments 41;48 are compared in Figure 4. However we can note that the theoretical results don’t match yet the experimental error bars. The lower barrier of the CSZ PES with respect to FXZ (see Table 1) makes closer the agreement for the DF channel but is not sufficient to enter within the error bars. Also the increased HF rate makes worse the global agreement with experimental data. In Ref. 40 it was suggested that the incorrect prediction of the experimental ratio between the two channels could be due to the low efficiency of the nonadiabatic couplings that does not permit to the system to relax in the lowest adiabatic curve near to the collinear configuration of the three atoms after the conical intersection between the Σ and Π diabatic PESs. As a consequence the incoming Fluorine atom ’feels’ a stronger anisotropy with respect to the one predicted by BO calculations. Indeed, it is well known 75;76 that the anisotropy can largely affect the isotopic ratios. Although this hypothesis is meaningful, the excellent agreement with the LWAL rates shown in the present article in spite of the different anisotropy discussed in Table 1, suggests that this effect (if exists) is likely too weak in the present case to fill the gap with the experimental data. Moreover, if the non-adiabatic dynamics plays a so important role in the intramolecular isotopic effect some trace should be found in the results of Refs. 31;32 . We also remark that a comparison (not shown) of the long range behavior of the LWAL PES shows that it well reproduces the semi-empirical long range behavior found in 22;24 , indicating that it is likely rather accurate in the entrance channel region. The remarkable agreement between CSZ and LWAL rates indicates therefore that the anisotropy and the long range potential, although probably not properly described by the CSZ PES, do not have relevant effects on the reactivity at least in the

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range of temperatures covered by the experiments. The relevant effect in the rates found 26;53 when the semi-empirical PESs were used to modify the ab initio PESs, is probably due to the modification in the short range region (a region poorly described by the model employed in 22 ) affecting too much the barrier width and the consequent tunneling effect. To understand better a possible reason for the inverted accuracy hierarchy shown in Figure 4, we can try to go deeper on the origin of the results of Ref. 41 . In 2005, A. Persky published 41 absolute values of the rate constants for the two isotopic channels of the F+HD reaction reported in Figure 4. In this study, the sum of rate constants kHF + kDF was determined experimentally over a wide temperature range (193-300 K) by a comparative method, using the reaction of F atoms with D2 as a reference. This ratio of rate constants was combined with available experimental data for the F + D2 reaction 77 and with the ratio of rate constants kHF /kDF 48 , to calculate absolute values for the individual rates kHF and kDF . Arrhenius expressions were used to obtain the rates at the temperatures reported in Figure 4 where experimental data of the intramolecular isotope effect were available 48 (in the temperature range 159-298 K). In Figure 5 the two experiments 41;48 are compared to the theoretical results independently from each other. In the upper panel we can observe that, in comparison with the newest kinetic data 41 , the expected accuracy hierarchy of the four PESs is rigorously respected and that the LWALCSZ results are inside the experimental error bars. To make a more direct comparison with the experimental output, rate constants calculated with the new PESs for the F + D2 reaction should be also be provided. However, the Arrhenius expression given in 77 seems to be accurate enough in the range of examined temperature considering also the low relevance of the tunneling effect expected 74 for this isotopic variant. More questionable is the validity of the Arrhenius’s law for F+H2 and for F+HD (in particular for the HF channel), where the strong tunneling effect affects markedly the rates at room temperatures 74;78 . In the lower panel of Figure 5, we can observe that the experiment of Ref. 48 is in poor agreement with the theoretical simulations. In particular, it shows an intramolecular isotope effect almost independent from temperature, while the theoretical results seems to increase exponentially as the temperature goes down. In the inset, the behavior of the theoretical ratio below 150 K is shown to have a huge maximum at about 50 K, where the HF yield is about 27 times larger than that of DF. This is clearly due to the transition state resonance

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shown in Figure 1, affecting almost exclusively the HF channel. The increase of the intramolecular isotope effect at low temperature observed in the lower panel of Figure 5 is therefore due to the tail of the resonance maximum, a feature well consolidated from both the theoretical and experimental points of view. We found therefore surprising that no trace of this feature was found in 48 , at least at 159 K where a large amount of the reactants (about 50 %) is still in its ground rotational state, where the resonance is manifested. We conclude that the experiments of Ref. 48 should be complemented by new kinetic measurements. Doubts about this experiment come also from early work on the hydrogen fluoride chemical laser 79 , where a ratio of about 2.5 was found a value nearly double that of Ref. 48 .

4

Final Remarks and Conclusions

In a recent article 38 , the CRESU (Cinetique de Reaction en Ecoulement Supersonique Uniforme) technique 80 was sussesfully employed to study the kinetics of the F+H2 reaction at very low temperatures (10-100 K) confirming previous theoretical predictions 24 . Because these measurements are in principle feasible also for the F+HD reactions, we find it interesting to study the behavior of the rates at very low temperatures, to predict possible experimental outputs also in this case. In Figure 6 the Arrhenius plots in the full range of temperatures studied (10-350 K) are shown for the two isotopic product channels of the F+HD reaction. As evident from the figure large deviations from the linearity of the plots are shown below 100 K. For the HF channel a marked bump is present between 30 and 100 K. This is due to the resonance feature in the total ics shown in the upper panel of Figure 1 smeared out on the rate constants by the Boltzmann averaging. Under cold conditions the rates approach their Wigner law behavior 81 becoming nearly independent of the temperature. As shown elsewhere 74 , this behavior can be accurately described by Bell’s analytical models 82;83 . However, the description of the resonance bump is more demanding because the models include the effect of the tunneling effect but not that of resonances. Extension of these models are therefore required to accurately describe the rate constants of the F+HD reactions in the deep tunneling regime. Comparing the results for the different PESs we can also observe that a temperature lower than 50 K (or 30 K) must be reached to distinguish a difference between the rates obtained for the DF (or HF) channel with the CSZ and the LWAL PESs. As demonstrated in 74 , the Wigner 18


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threshold limit is a complicated function of the barrier width and height. The lower reactivity of the LWAL PES, notwithstanding its lower barrier height (see Table 1) is therefore an indication that the effective LWAL barrier is likely slightly thinner than that for the CSZ PES. However, an exact Wigner threshold limit value cannot be calculated with none of these PESs. In fact, our calculations failed to obtain the ground and excited reactants’ energetics with a resolution finer than 0.1 meV, excluding therefore the possibility to generate cold and ultra-cold observables with these PESs. This is because the numerical fits for the diatomic curves used in the FXZ,CSZ and LWAL PESs exhibit nonphysical behaviour at high energies (above 5 eV) in the repulsive diatomic wall preventing stabilization of the convergence on the accuracy required for the rotovibrational levels. In these regimes, quantum scattering is strongly affected by the entrance channel resonances 24;7 , which can be therefore calculated exclusively with the SW PES and its phenomenologically modified variants 24;22 . This work is in progress. In the lower panel of Figure 7, the ics behaviour for the HF(v ′ =3) manifold is shown as a function of the collision energy for the four PESs studied. The ics have been translationally averaged as described in Sec. 3.1 (FWHM = 3 meV). In the upper panel we show a blow up in the energy range near the reactive v ′ =3 thresholds. In this case we show the results without averaging in order to completely show the resonance features on the four PESs. An energy grid of 0.1 meV is used in the calculations. The results for the FXZ PES have been recently investigated 65 by a Regge Pole Complex-Angular-Momentum Theory (CAM) 10 . The complicated resonance energy pattern has been shown to be given by the overlap among several poles associated to long lived metastable states corresponding to stretching and bending excited van der Waal states of the exit channel. A similar physical poles structure (studied by the Smith Q-matrix method 84;6;7 in the Complex Energy Plane) was also analyzed 12 for a PES with the same exit channel of the SW PES. As shown in the upper panel of Fig. 7 the resonance features change dramatically among the four PESs reflecting the differences in the exit Van der Waals wells and in the endothermicities reported in Table 1. In particular we note that the resonance patterns in the Dalian PESs affect more the ics that for the LWAL PES case with the van der Waal state have the minimum depth well (see Table 1). The energetic problem of the SW PES (see Sec.1) shifts the resonance pattern at higher collision energies (above 75 meV). Only some features fall in the energy range of the upper panel of figure

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7. As shown elsewhere 40;35 , J=0 reaction probabilities indicate that all the four PESs support the same number of metastable states although the couplings (or residues in 65 ) with the reactants are different changing markedly the resonance patterns. As reported in Ref. 43 the Dalian PESs (CSZ and FXZ) should produce more accurate resonance features than the LWAL PES, simulating better the experiment of Ref. 16 where the interaction between ground and first excited bending states 9 amplify the resonance effects 8 . The resonance features observed in 16 are to be considered a stringent proof of the existence of these long lived resonances and give strong indications on the accuracy of the PES in the exit Van der Waals region. In particular the results of the FXZ PES should reproduce better an hypothetical molecular beam experiment because of the adaptation of ab-initio data (see Sec. 2.1) to reproduce the energetics of the exit vibrational channels of the F+H2 reaction. However, as the lower panel demonstrate, these resonance are too narrow to survive to the translational effect averaging. The only features that should be observed at the state-of-the-art of the molecular beam technique, are the Regge Pole Oscillations 10 above 80 meV given by the same pole (the transition state resonance state) affecting the total ics of Fig. 1 and decaying in the v ′ =3 channel as soon as this vibrational manifold becomes energetically accessible. Further simulations (not shown) indicate that an energy resolution higher than 1 meV is required to put into evidence the longer lifetimes effects shown in the upper panel of Fig. 7. In conclusion, in this paper highly accurate reactive scattering calculations have been performed with four ab initio PESs: integral cross sections, vibrational branching ratios and rate constants have been compared among them and against available experimental data. From the comparison among the theoretical data no relevant differences emerges between the results obtained with the most recent LWAL and CSZ PESs in the energy range probed by the experiments. Very small is also the improvement with respect to the earlier FXZ PES, suggesting that the long road for obtaining reliable theoretical data for this system has probably reached convergence at least at the Born-Oppenheimer level of theory. The experiments are reproduced quite well by the theoretical simulations, especially the ones of the Dalian groups (see upper panels of Figures 2 and 3). Minor differences remain with the Tapei experiment (see Figure 1 and lower panels of Figures. 2 and 3). Curiously, the larger differences in the molecular beam experimental features are also the only features where CSZ and LWAL PESs results differ, not permitting an unambiguous distinction

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between the accuracy of the two PESs. More serious discrepancies between theory and experiments can be observed when theoretical rate constants are compared to the kinetic data 41 (see Figure 4) where a closer agreement should be expected from the newer PESs. Actually, as shown in Figure 5, improvements there are when the sum of the rates for the two channels effectively measured in 41 is compared. The origin of the discrepancy is due to a lower intramolecular isotopic effects measured in Ref. 48 , casting doubts on the accuracy of this old experiment. Except for this discrepancy, the general good agreement with the experimental data of the Born-Oppenheimer dynamics here presented, leaves little room for non-adiabatic effects coming from the conical intersections of this system 45;46 . More relevant, from the theoretical point of view, is probably the extension of the accuracy of the Born-Oppenheimer PESs employed in the cold and ultra-cold regimes, where most of the PESs here employed fail to give stable ics behaviors (see comments above and in Sec. II). This is particularly important now that experimental techniques 80 have appeared to be able to cover low temperature and likely new experiments will be soon available for this reaction and its isotopic variants. Temperatures below 50 K (Figure 6) or translational energy resolutions better than 1 meV (Figure 7) must be reached in the experiments to discriminate among the accuracies of the LWAL, CSZ and FXZ PESs.

Acknowledgements The authors thank Z. Sun and D.H. Zhang (Dalian) for the supply of a subroutine of the CSZ PES. Also acknowledged is the High Performance Computing center at CINECA for computer time awarded via the ’Progetto ISCRA C:HP10CXWOPW and HP10C7RZR8’ and the Ministero per l’Universit´a e la Ricerca Scientifica of Italy for the PRIN 2010/2011 grant N. 2010ERFKXLPRIN.

Supporting Information Comparison of the calculated rate constants for the HF and DF formation on the LWAL, CSZ, FXZ and SW PESs as a function of temperature: tables.

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[19] Wang, T.; Yang, T.; Xiao, C.; Sun, Z.; Huang. L.; Dai, D.; Yang, X.; Zhang, D.H. Isotopedependent rotational states distributions enhanced by dynamic resonance states: a comparison study of the F + HD → HF(vH F =2) + D and F + H2 → HF(vH F =2) + H reaction. J. Phys. Chem. Letters 2014, 5, 3049-3055. [20] Xu, C. X.; Xie, D. Q.; Zhang, D. H. A global ab initio potential energy surface for F + H2 → HF + H. Chin. J. Chem. Phys. 2006, 19, 96-98. [21] Fu, B.; Xu, X.; Zhang, D.H. A hierarchical construction scheme for accurate potential energy surface generation: An application to the F+H2 reaction. J. Chem. Phys. 2008,129, 011103. [22] Aquilanti,V.; Cavalli, S.; Pirani, F.; Volpi, A.; Cappelletti, D. Potential Energy Surfaces for F-H2 and Cl-H2 : Long-Range Interactions and Nonadiabatic Couplings. J. Phys. Chem. 2001, 105, 2401-2409. [23] Aquilanti, V.; Cavalli, S.; De Fazio, D.; Volpi, A.; Aguilar, A.; Gimenez, X.; Lucas, J. M. Exact reaction dynamics by the hyperquantization algorithm: integral and differential cross sections for F+H2 , including long-range and spin-orbit effects. Phys. Chem. Chem. Phys. 2002, 4, 401-415. [24] Aquilanti, V.; Cavalli, S.; De Fazio, D.; Volpi, A.; Aguilar, A.; Lucas, J.M. Benchmark rate constants by the hyperquantization algorithm. The F + H2 reaction for various potential energy surfaces: features of the entrance channel and of the transition state, and low temperature reactivity. Chem. Phys. 2005, 308, 237-253. [25] Hayes, M.; Gustafsson, M. ; Mebel, A.M.; Skodje, R.T. An improved potential energy surface for the F+H2 reaction. Chem. Phys. 2005, 308, 259-266. [26] De Fazio, D.; Lucas, J. M.; Aquilanti , V.; Cavalli, S. Exploring the accuracy level of new potential energy surfaces for the F + HD reactions: from exact quantum rate constants to the state-to-state reaction dynamics. Phys. Chem. Chem. Phys. 2011, 13, 8571-8582. [27] De Fazio, D.; Aquilanti, V.; Cavalli, S.; Aguilar, A.; Lucas, J. M. Exact state-to-state quantum dynamics of the F+HD reaction on model potential energy surfaces. J. Chem. Phys. 2008, 129, 064303. 24


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[28] Aoiz, F.J.; Banares, L.; Castillo, J.F. Spin-orbit effects in quantum mechanical rate constant calculations for the F+H-2 -¿ HF+H reaction. J. Chem. Phys. 1999, 111, 4013-4024. [29] Hartke, B.; Werner, H.J. Time-dependent quantum simulations of FH2- photoelectron spectra on new ab initio potential energy surfaces for the anionic and the neutral species. Chem. Phys. Lett. 1997, 280 430-438. [30] Alexander, M.H.; Manolopoulos, D.E.; Werner, H.-J. An investigation of the F+H2 reaction based on a full ab initio description of the open-shell character of the F(2 P) atom. J. Chem. Phys. 2000, 133, 11084-11100 [31] Zhang, Y.; Xie, T.X.; Han, K.L.; Zhang, J.Z.H. The investigation of spin-orbit effect for the F(P-2) plus HD reaction. J. Chem. Phys. 2004, 120, 6000-6004. [32] Tzeng, Y.R.; Alexander, M.H. Role of the F spin-orbit excited state in the F+HD reaction: Contributions to the dynamical resonance. J. Chem. Phys. 2004, 121, 5183-5190. [33] Tzengz, Y.R.; Alexander, M.H. Reactivity of the F spinorbit excited state in the F + HD reaction: Product translational and rotational energy distributions. Phys.Chem.Chem.Phys. 2004 , 6 , 5018-5025. [34] Werner, H.-J.; Kallay, M.; Gauss, J. The barrier height of the F+H2 reaction revisited: Coupled-cluster and multireference configuration-interaction benchmark calculations. J. Chem. Phys. 2008, 128, 034305. [35] Li, G.; Werner, H.-J.; Lique, F.; Alexander, M. H. New ab initio potential energy surfaces for the F + H2 reaction. J. Chem. Phys. 2007, 127, 174302-174313. [36] Lique, F.; Li, G.; Werner, H.-J.; Alexander, M.H. Non-adiabatic coupling and resonances in the F + H2 reaction at low energies. J. Chem. Phys. 2011, 134, 231101 [37] Che, L.; Ren, Z. F.; Wang, X. G.; Dong, W. R.; Dai, D. X.; Wang,X. Y.; Zhang, D. H.; Yang, X. M.; Sheng L.; Li G; et al. Breakdown of the Born-Oppenheimer approximation in the F+ o-D2 → DF + D reaction. Science 2007, 317, 1061-1064.

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[38] Tizniti, M.; Le Picard, S.D.; Lique, F.; Berteloite, C.; Canosa, A.; Alexander, M.H.; Sims, I.R. The rate of the F+H2 reaction at very low temperatures. Nature Chem. 2014, 6, 141-150. [39] Wang, T.; Chen, J.; Yang, T.; Xiao, C.; Sun, Z.; Huang, L.; Dai,D.; Yang, X.; Zhang, D. H. Dynamical resonances accessible only by reagent vibrational excitation in the F + HD → HF + D reaction. Science 2013, 342, 1499-1502. [40] Chen, J.; Sun, Z.; Zhang, D. H. An accurate potential energy surface for the F + H2 → HF + H reaction by the coupled-cluster method. J. Chem. Phys. 2015, 142, 024303-024313. [41] Persky, A. The rate constants of the two channels of the reaction of F atoms with HD in the temperature range 193-300 K. Chem. Phys. Lett. 2005, 401, 455 . [42] Sun, Z.; Zhang, D.H. Development of the potential energy surface and current stage of the quantum dynamics studies of the F+H2 /HD reaction Int. Jour. Quantum Chem. 2015, 115, 689-699. [43] Yu, D; Chen, J.; Cong, S.; Sun, Z. Theoretical Study of FH− 2 Electron Photodetachment Spectra on New Ab Initio Potential Energy Surfaces, J. Phys. Chem. A 2015, 119, 1219312208. [44] Kim, J. B.; Weichman, M. L.; Sjolander, T. F.; Neumark, D. M.; Klos, J.; Alexander, M. H.; Manolopoulos, D.E. Spectroscopic Observation of Resonances in the F + H2 Reaction. Science 2015, 349, 510-513. [45] Das, A.; Sahoo, T.; Mukhopadhyay, D.; Adhikari, S.; Baer, M. Dressed adiabatic and diabatic potentials to study conical intersections for F + H2 . J. Chem. Phys. 2012, 136, 054104 [46] Csehi, A; Bende, A; Hallasz, GJ; Vibok, A; Das, A.; Mukhopadhyay, D.; Mukherjee, S.; Adhikari, S.; Baer, M. dressed adiabatic and diabatic potentials to study topological effects for F + H2 . J. Phys. Chem. A 2014, 118,6361-6366 [47] Gamallo, P.; Akpinar, S.; Defazio, P.; Petrongolo, C. Born-Oppenheimer and Renner-Teller quantum dynamics of CH(X-2 Pi) + D(S-2) reactions on three chd potential surfaces. J. Phys. Chem. A 2015, 119, 11254-11264. 26


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[48] Persky, A. Kinetic isotope effects in the reaction of fluorine atoms with molecular hydrogen. II. The F + HD/DH intramolecular isotope effect. J. Chem. Phys. 1973, 59, 5578. [49] Dong, F.; Lee, S.-H.; Liu, K. Reactive excitation functions for F+ p-H2 /n-H2 /D2 and the vibrational branching for F+HD. J. Chem. Phys. 2000, 113, 3633. [50] Dong, W.; Xiao, C.; Wang, T.; Dai, D.; Wang, X.; Yang, X. High Resolution Crossed Molecular Beams Study on the F+HD→HF+D Reaction at Collision Energy of 5.43-18.73 kJ/mol. Chin. J. Chem. Phys. 2011, 24, 507. [51] Wang, X.; Che, L.; Ren, Z.; Qiu, M.; Dai, D.; Wang, X.; Yang,X. High Resolution Crossed Molecular Beams Scattering Study of the F+HD→DF+H Reaction. Chin. J. Chem. Phys. 2009, 22, 551. [52] Dong, W.; Xiao, C.; Wang, T.; Dai, D.; Wang, X.; Yang, X. High resolution crossed molecular beams study on the F+HD→DF+H reaction at collision energy of 8.19-18.98 kJ/mol. Chin. J. Chem. Phys. 2011, 24, 521. [53] De Fazio, D.; Aquilanti, V.; Cavalli, S.; Aguilar, A.; Lucas, J. M. Exact quantum calculations of the kinetic isotope effect: Cross sections and rate constants for the F + HD reaction and role of tunnelling. J. Chem. Phys. 2006,125, 133109. [54] Zhang, D.H.; Lee, S.-Y.; Baer, M. Quantum mechanical integral cross sections and rate constants for the F+HD reactions. J. Chem. Phys. 2000, 112, 9802-9809. [55] Langhoff, S.R.; Davidson, E.R. Configuration interaction calculations on the nitrogen molecule. Int. J. Quantum Chem. 1974, 8, 61-72. [56] Dunning, T.H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007-1023. [57] Aguado, A.; Paniagua, M. A new functional form to obtain analytical potentials of triatomicmolecules. J. Chem. Phys. 1992, 96, 1265-1275. [58] Kendall, R.A.; Dunning, T.H.; Harrison, R.J. Electron-Affinities of the 1st-row atoms revisited - Systematic basis-sets and wave-functions. J. Chem. Phys. 1992, 96, 6796-6806. 27



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[59] Behler, J. Neural network potential-energy surfaces in chemistry: a tool for large-scale simulations. Phys. Chem. Chem. Phys. 2011, 13, 17930-17955. [60] Skouteris, D.; Castillo, J.F.; Manolopoulos, D. E. ABC: a quantum reactive scattering program. Comput. Phys. Commun. 2000,133, 128-135. [61] Colavecchia, F.D.; Mrugala, F.; Parker, G.A.; Pack, R.T. Accurate quantum calculations on three-body collisions in recombination and collision-induced dissociation. II. The smooth variable discretization enhanced renormalized Numerov propagator. J. Chem. Phys. 2003, 118, 10387-10398. [62] Manolopoulos, D. E. An improved log derivative method for inelastic scattering, J. Chem. Phys. 1986, 85, 6425-6429. [63] De Fazio D. The H + HeH+ → He + H+ 2 reaction from the ultra-cold regime to the three-body breakup: exact quantum mechanical integral cross sections and rate constants. Phys. Chem. Chem. Phys. 2014, 16, 11662-11672. [64] De Fazio, D.; de Castro-Vitores, M.; Aguado, A.; Aquilanti, V; Cavalli, S; The He + H+ 2 → HeH+ + H reaction: Ab initio studies of the potential energy surface, benchmark timeindependent quantum dynamics in an extended energy range and comparison with experiments. J. Chem. Phys. 2012, 137, 244306. [65] Sokolovski, D.; Akhmatskaya, E.; Echeverria-Arrondo, C.; De Fazio, D. Complex angular momentum theory of state-to-state integral cross sections: resonance effects in the F plus HD → HF(v ’=3) + D reaction. Phys. Chem. Chem. Phys. 2015, 17, 18577-18589. [66] Bennun, M.; Brouard, M.; Simons, J.P.; Levine, R.D. Peripheral chemical-reactions. Chem. Phys. Lett. 1993, 210, 423-431. [67] Kornweitz, H.; Persky, A.; Levine, R.D. The exoergic F+CH4 reaction as an example of peripheral dynamics. Chem. Phys. Lett. 1998, 289, 125-131. [68] Aoiz, F.J.; Banares, L. Effect of reagent vibrational excitation on the dynamics of the Cl+HD → HCl(DCl)+D(H) reaction. Chem. Phys. Lett. 1995 247 232-242. 28


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[69] Polanyi, J.C. Some Concepts in Reaction Dynamics. Acc. Chem. Res. 1972, 5, 161-168 [70] Jiang, B.; Guo, H. Relative efficacy of vibrational vs. translational excitation in promoting atom-diatom reactivity: Rigorous examination of Polanyi’s rules and proposition of sudden vector projection (SVP) model. J. Chem. Phys. 2013, 138, 234104. [71] See the Supporting Information for a comparison among the rate constants calculated for the HF (Table S1) and for the DF channel (Table S2) with previous reference data. [72] Aquilanti, V; Cavalli, S.; De Fazio, D. Hyperquantization algorithm. I.Theory for triatomic systems. J. Chem. Phys. 1998, 109, 3792-3804. [73] Aquilanti, V; Cavalli, S; De Fazio, D.; Volpi A., The A + BC reaction by the hyperquantization algorithm: the symmetric hyperspherical parametrization for J

0. Advances in Quantum

Chemistry 2001, 39, 103-121. [74] Cavalli, S.; Aquilanti, V.; Mundim, K. C.; De Fazio, D. Theoretical reaction kinetics astride the transition between moderate and deep tunneling regimes: the F + HD Case. J. Phys. Chem. A 2014, 118, 6632-6641. [75] Johnston, G.W.; Kornweits, H.; Schechter, I.; Persky, A.; Katz, B.; Bersohn, R.; Levine, R.D. The branching ratio in the F + HD reaction - an experimental and computational study. J.Chem.Phys. 1991, 94, 2749-2757. [76] Skouteris, D.; Manolopoulos, D.E.; Bian, W.S.; Werner, H.-J.; Lai, L.H.; Liu, K.P. van der Waals interactions in the Cl+HD reaction. Science 1999, 286, 1713-1716. [77] Persky, A.; Kornweitz, H. The kinetics of the reaction F + H2 → HF + H. A critical review of literature data. Int. J. Chem. Kin. 1997, 29, 67. [78] Aquilanti, V.; Mundim, K. C.; Cavalli, S.; De Fazio, D.; Aguilar, A.; Lucas, J.M. Exact activation energies and phenomenological description of quantum tunneling for model potential energy surfaces. The F + H2 reaction at low temperature. Chem. Phys. 2012,398, 186-191. [79] Kompa, K.L.; Parker, J.H.; Pimentel, G.C. UF6 -H2 Hydrogen fluoride chemical laser: operation and chemistry. J.Chem.Phys. 1968, 49, 4257-4264. 29



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[80] Sims, I.R.; Queffelec, J.-L.; Defrance, A.; Rebrion-Rowe, C.; Travers, D.; Bocherel, P.; Rowe, B.R.; Smith, I.W.M. Ultralow temperature kinetics of neutral-neutral reactions - the technique and results for the reactions CN + O2 down to 13 K and CN + NH3 down to 25 K. J. Chem. Phys. 1994, 100, 4229-4241. [81] Wigner, E.P. On the behavior of cross sections near thresholds. Phys. Rev. 1948, 73, 10021009. [82] Bell, R.P. Quantum mechanical effects in reactions involving hydrogen. Proc. R. Soc.A 1935, 148, 241-250. [83] Bell, R.P. The tunnel effect correction for parabolic potential barriers. Trans. Faraday Soc. 1959, 54, 1-4. [84] Smith, F.T. Lifetime matrix in collision theory. Phys. Rev. 1960, 118, 349-356.

30


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1 - Characteristic features of the LWAL PES, the CSZ PES, the FXZ PES and the SW PES employed in scattering calculations.

LWAL

CSZ

FXZ

SW†

Bent barrier height∗ /meV

71.2

72.9

76.8

66.6

Linear barrier height∗ /meV

93.1

94.8

96.2

83.3

Exoergicity∗ /eV

-1.3731

-1.3745

-1.3727

-1.3581

EHF (v=0,j=0) - EHD (v=0,j=0)/eV

-1.3534

-1.3543

-1.3526

-1.3373

58.8

56.2

58.2

71.0

-1.4229

-1.4238

-1.4221

-1.4067

Entrance Channel VdW well depth/meV

8.0

8.2

8.3

15.9

Entrance Channel Coll. Cut well depth/meV

5.9

4.7

4.6

3.0

Exit Channel VdW well depth/meV

9.6

12.3

13.6

12.2

Feature

EHF (v=3,j=0) - EHD (v=0,j=0)/meV EDF (v=0,j=0) - EHD (v=0,j=0)/eV

∗

Without zero point energies.

†

Without spin-orbit coupling.

31



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Page 32 of 42

Table 2. Values of the input parameters used in the ’production runs’ for the SW PES. For the definition of the parameters, see text. Total Collision

a

jin

Energya /eV

Jtotmax

kmax

emaxa /eV

jmax

drho/a0

rmax/a0

dE/meV

0

0.2327-0.2347

9

2

3.5

16

0.09

30

0.02

0

0.2348-0.2435

11

2

2.0

14

0.09

20

0.1

1

0.2450-0.2656

14

3

2.0

16

0.09

20

0.2

2

0.2660-0.2980

18

4

2.0

18

0.09

15

0.5

3

0.2990-0.3410

23

5

2.3

20

0.09

15

1.0

4

0.344-0.394

26

6

2.3

20

0.10

15

5.0

5

0.398-0.458

32

7

2.5

23

0.12

15

6.0

6

0.460-0.530

39

8

3.0

25

0.13

15

7.0

zero energy is located in the bottom reactant valley.

32


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Captions to figures Figure 1. Comparison of experimental and theoretical integral cross sections for the production of HF and DF molecules (upper and lower panels respectively) on three ab initio PESs. Full and open dots are the experimental results of Ref. 1 ; full, dashed, dot-dashed and dotted lines are the ics obtained from quantum scattering calculations (properly averaged on translational energy in order to simulate the experimental conditions of the molecular beam, see the main text for more details) on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively. Figure 2. Comparison of experimental and theoretical vibrational branching ratios for the production of the HF molecule in the vibrational states v ′ = 0, 1, 2, 3 and 4. The symbols are the experimental results of Ref. 50 (upper panel) and of Ref. 49 (lower panel): squares (v ′ =0); circles (v ′ =1); triangles up (v ′ = 2); triangles down (v ′ =3) and diamonds (v ′ =4). For each vibrational state, full, dashed, dot-dashed and dotted lines are the theoretical results on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively. For the simulation of the experimental conditions of the molecular beam 49 (lower panel), the results obtained from quantum scattering calculations have been properly averaged on translational energy taking into account the contributions of the ics for the ground and first excited rotational states of the HD molecule (j = 0 and j = 1, which are experimentally given as 82% and 18% respectively) . Figure 3. Comparison of experimental and theoretical vibrational branching ratios for the production of the DF molecule in the vibrational states v ′ = 0, 1, 2, 3 and 4. The symbols are the experimental results of Ref. 51;52 (upper panel) and of Ref. 49 (lower panel): squares (v ′ =0); circles (v ′ =1); triangles up (v ′ = 2); triangles down (v ′ =3) and diamonds (v ′ =4). For each selected vibrational state, full, dashed, dot-dashed and dotted lines are the theoretical results on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively. As in the case of the HF channel, the theoretical results have been translationally averaged, see the caption of Fig. 2. Figure 4. Semi-logarithmic plot of the rate constants for the HF channel (upper panel) and the DF channel (lower panel) versus inverse temperature in the range 350-150 K. Dots with error bars are from experiments by Persky (2005) 41 while full, dashed, dot-dashed and dotted lines are the theoretical results on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively.

33



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Page 34 of 42

Figure 5. The sum (kHF + kDF ) and the ratio (kHF /kDF ) of the rate constants for the HF and the DF channel as a function of temperature are shown in the upper and lower panels respectively. The dots with error bars (upper panel) and full dots (lower panel) are the experimental data from Persky 41;48 while full, dashed, dot-dashed and dotted lines are the theoretical results on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively. The intra-molecular isotopic effect (kHF /kDF ) calculated at low temperature is plotted in the inset of the lower panel of the figure. Figure 6. Logarithmic plot of rate constants as a function of inverse of temperature for the F + HD reaction in a temperature range extended down to 100 K. Dots with error bars are the experimental data from Persky (2005) 41 while full, dashed, dot-dashed and dotted lines are the theoretical results on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively. Figure 7. Integral cross sections (ics) for the HF(v ′ =3) manifold as a function of the collision energy for the four PESs studied: full, dashed, dot-dashed and dotted lines are the theoretical results on the LWAL, the CSZ, the FXZ and the SW surfaces, respectively. The ics shown in the lower panel have been translationally averaged by a Gaussian function with a window of 3 meV (see Sec. 3.1 for details). In the upper panel, a magnification of the ics, showing the resonance structure, is reported for the energy interval near the threshold for the formation of HF in v ′ =3 indicated by black dot-dashed lines in the lower panel.

34


Page 35 of 42

4

LWAL CSZ SW FXZ

3 2

Integral Cross Section (10 nm )

2

-2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

F + HD -->HF + D

0 4 3 2 1

F + HD --> DF + H 0 0

30

60

90 120 150 180 210

Collision Energy (meV)

Figure 1:

35



1 F + HD (j=0) -->HF (v') + D

0.8 v' = 2

0.6

Vibrational Branching Ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

LWAL CSZ SW FXZ

0.4

v' = 1 v' = 3

0.2

v' = 0

0 F + HD (j=0,1) -->HF (v') + D

0.8 v' = 2

0.6 0.4

v' = 3

v' = 1

0.2 0 0

v' = 0

30

60

90 120 150 180 210


Figure 2:

36


Page 36 of 42

Page 37 of 42

0.6

F + HD (j=0) -->DF (v') + H v' = 3

0.4

Vibrational Branching Ratio

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


v' = 2 v' = 4

0.2 v' = 1

v' = 0

0 0.6

F + HD (j=0,1) -->DF (v') + H v' = 3

0.4 v' = 4 v' = 2

0.2 v' = 1

v' = 0

0 0

30

60

90 120 150 180 210


Figure 3:

37



-11

10

SW CSZ

-1

Rate constant (cm molecule sec )

LWAL

-1

FXZ

F+HD → FH+D -12

10

3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

F+HD → DF+H

-11

10

SW

LWAL

CSZ FXZ

-12

10

3

4 5 -1 1000/T (K )

6

Figure 4:

38


Page 38 of 42

Rate Costants Sum x 10 Rate Costants Ratio (HF/DF)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


13

Page 39 of 42

200 SW

CSZ

150

LWAL FXZ

100

50

3.5

20

3

FXZ LWAL

10

CSZ

2.5 0

SW

100

2

200

300

1.5 150

180

210

240

270

300

Temperature (K)

Figure 5:

39



-11

10

-12

10

SW FXZ CSZ

-13

10

F+HD → FH+D

LWAL

-14

10

3

-1 -1

Rate constant (cm molecule s )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

-11

10

F+HD → DF+H

-12

10

-13

SW

10

CSZ

-14

10

FXZ LWAL

-15

10

10 -1 1000/T (K )

100

Figure 6:

40


Page 40 of 42

0.4

2 -2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


HF (v'=3) Integral Cross Sections (10 nm )

Page 41 of 42

0.3

CSZ FXZ LWAL

0.2 0.1

SW

0 55

60

0.2

FXZ

LWAL

0.1 0

75

FWHM = 3 meV

CSZ

0.3

70

65

SW

60

80

100

120

140


Figure 7:

41



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Page 42 of 42

Benchmark Quantum Mechanical Calculations of Vibrationally

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