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Rate Coefficient for the 4Heμ + CH4 Reaction at 500 K: Comparison between Theory and Experiment Donald J. Arseneau† and Donald G. Fleming* †

TRIUMF and Department of Chemistry, University of British Columbia, Vancouver, BC V6T 2Z1, Canada

Yongle Li Department of Physics and International Center of Quantum and Molecular Structures, Shanghai University, Shanghai 200444, China

Jun Li School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China

Yury V. Suleimanov Computation-Based Science and Technology Research Center, Cyprus Institute, 20 Kavafi Str., Nicosia 2121, Cyprus Department of Chemical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States

Hua Guo* Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States ABSTRACT: The rate constant for the H atom abstraction reaction from methane by the muonic helium atom, Heμ + CH4 → HeμH + CH3, is reported at 500 K and compared with theory, providing an important test of both the potential energy surface (PES) and reaction rate theory for the prototypical polyatomic CH5 reaction system. The theory used to characterize this reaction includes both variational transition-state (CVT/μOMT) theory (VTST) and ring polymer molecular dynamics (RPMD) calculations on a recently developed PES, which are compared as well with earlier calculations on different PESs for the H, D, and Mu + CH4 reactions, the latter, in particular, providing for a variation in atomic mass by a factor of 36. Though rigorous quantum calculations have been carried out for the H + CH4 reaction, these have not yet been extended to the isotopologues of this reaction (in contrast to H3), so it is important to provide tests of less rigorous theories in comparison with kinetic isotope effects measured by experiment. In this regard, the agreement between the VTST and RPMD calculations and experiment for the rate constant of the Heμ + CH4 reaction at 500 K is excellent, within 10% in both cases, which overlaps with experimental error.

1. INTRODUCTORY REMARKS

theoretical calculations of the effects of a change in isotopic mass on reaction rates.4,8,9 Thus, it is only from muon science that it has been practical to extend the experimental H atom isotopic mass scale beyond comparisons of H and D atom reaction rates. The lightest isotope “muonium”, consisting of a positive muon and an electron (Mu = μ+e−), with a mass of

The study of mass effects on chemical reaction rates has played a central role in comparisons between theory and experiment since the discovery of deuterium (D) in 1932,1 given that the most sensitive effects of this nature are naturally found for the isotopes of the H atom. However, with a mass of only 2 amu, comparisons of the rates of H and D atom reactions, while certainly remaining important,2−6 provide a limited test of rate theory. Though tritium, with a mass of 3 amu, would be important in this regard, it is dangerously radioactive and has been mainly utilized in the field of hot atom reactivity7,8 or in © XXXX American Chemical Society

Special Issue: Bruce C. Garrett Festschrift Received: August 27, 2015 Revised: October 19, 2015

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Mu reactions, ∼13.5 kcal/mol for Mu + H243 and ∼24 kcal/mol for Mu + CH4,35 the latter being the highest yet measured in Mu reactivity in the gas phase and which has yet to be wellaccounted for by theory,4,5,29,33,34 where all such calculations give considerably lower values. Due to the much heavier mass of the muonic He atom, 4.11 amu, the effect of ZPE in the barrier region can be expected to be much less for the Heμ + CH4 reaction, leading to a much lower ZPE-corrected barrier in comparison with Mu + CH4. This was also seen previously in similar comparisons for reactions of Mu and Heμ with H2, where quantum theory and experiment are essentially in perfect accord, both on the CCI/ BH surface17 and on the BKMP2 surface,37 the latter employing the RPMD method. The effect of this huge change in mass also translates into a much earlier (and narrower) barrier for both the slightly exoergic (ΔH ≲ −1 kcal/mol) Heμ + H2 and Heμ + CH4 reactions, even giving rise to an apparent contribution to the rate from quantum tunneling near 300 K in our previous study of the Heμ + H2 reaction,17 despite the much heavier muonic-He atom mass. The RPMD method fully takes into account the effect of ZPE and also generally accounts well for quantum tunneling and is thus rapidly gaining popularity as a much more computer-efficient alternative to fully rigorous quantum calculations,13,14,17 for which the recent RPMD calculations of the isotopomers of the H3 reaction system36,37 provide impressive testimony. It is of interest, then, to see if a similar level of agreement can be found from this method for the Mu5 and Heμ + CH4 reactions, both in comparison with experiment and with VTST calculations, since, unlike H3, fully rigorous quantum calculations for the isotopologues of the CH5 system are not yet at hand. Accordingly, we report herein on the first measurement of the rate constant for the Heμ + CH4 reaction

0.114 amu, has traditionally provided a uniquely sensitive probe of quantum mass effects in chemical reactivity.10−15 More recently, “muonic He”, consisting of a negative muon screening one proton charge of the He nucleus plus one electron (Heμ = [4Heμ−]+e−), with an atomic mass of 4.11 amu,16−18 has been used to extend the isotopic mass range to even heavier than tritium. Via muon science, we are now able to compare isotopic mass effects on reaction rates over a remarkable range of a factor of 36 in atomic mass. Prior to the present study, this had been exploited mainly for the H3 reaction system,16−18 though the temperature dependence of the rate constants for the Heμ + NH3 reaction has also been reported.19 This initial focus on muon isotopic mass effects for the H3 reaction system was motivated by the fact that it is still, 80 years after the pioneering work of London, Eyring, and Polanyi, the only potential energy surface (PES) that is known with such a high degree of accuracy, ∼0.3 meV at the barrier on the CCI/BH surface of Mielke et al.,2,3,12,13,20 that truly definitive comparisons of quantum rate theory with experiment have been demonstrated.2,11,13,16,17 Similar high-quality quantum calculations of the H3 system have also been carried out by others,14,18,21 though on the earlier, less accurate, BKMP2 surface of Boothroyd et al.22 Polyatomics are much more difficult to treat due to the large increase in numbers of degrees of freedom necessary to describe the PES. Highly important in this regard is the simplest prototypical H atom abstraction reaction on a polyatomic, the H + CH4 reaction on the CH5 surface,5,23−32 and its isotopic comparisons.4−6,9,25,29,33,34 The most accurate quantum calculation of the rate coefficients for CH5 to date, although it is based rigorously on J = 0 with J-shifting for higher rotational states and restricted to the H + CH4 reaction only, is that of Manthe.32 Experimental results for the Mu + CH4 → MuH + CH3 reaction were reported in 199535 and compared later with theory, both variational transition-state theory (VTST) calculations,29,33 and reduced dimensionality quantum calculations.4,5,34 A recent quantum calculation of this nature for the rate coefficients of the isotopomers of the CH5 reaction system, that of Li et al.,5 utilized the method of ring polymer molecular dynamics (RPMD)31,36−39 on the analytical PES of ref 25, since comparisons with the rate coefficients of the H + CH4 reaction on this surface with other more rigorous calculations31,32,40 indicated that this surface was quite adequate. However, the aforementioned calculations of the Mu + CH4 reaction rates tend to significantly underestimate experiment.35 Very recently, Meng et al.41 have reported an RPMD calculation of the rate coefficients for the H + CH4 reaction using the new and more accurate PES of ref 26, showing excellent agreement with experiment and thereby strongly suggesting that this PES is, indeed, highly accurate. There are two key quantum mass effects that rate theory must address in comparison with experiment: zero-point energy (ZPE) shifts for the reactants and at the TS, particularly important for a late-barrier highly endoergic reaction like Mu + CH4,5,26,29,33 and quantum tunneling due to the light Mu atom mass, though, due to the high and wide VTST barrier for this reaction, this is expected to be relatively unimportant. The Mu + CH4 and Mu + H2 reactions are quite similar in this regard, both being endoergic by ∼7.5 kcal/mol, so that shifts in ZPE dramatically increase the height of the VTST barrier, which dominates the kinetics in both cases.5,13,21,29,36,37,42 This is reflected as well in the experimental activation energies for both

Heμ + CH4 → [Heμ ··· H ··· CH3]‡ → (Heμ)H + CH3

at a temperature of 500 K, in comparison with both RPMD and VTST theory on a recently developed PES.26

2. EXPERIMENTAL BACKGROUND AND RESULTS As described in detail in ref 17, both the Mu and Heμ atoms are formed during the slowing processes of muon (μ±) beams in matter, from initial kinetic energies of several MeV to kBT,8,17,44,45 on a time scale of 10−20 ns in a gas at ∼1 atm pressure44 and proportionately faster at higher densities. Muons are produced 100% spin-polarized in the weak interaction decay of the pion, and this polarization is maintained during Bethe−Bloch ionization processes down to ∼100 keV. Around this energy, the μ− undergoes atomic capture, principally by He but also by C and H of CH4, initially entering high muonic orbits, followed by cascade down to the muonic 1s state, 400 times smaller than the electronic 1s state in H atoms, occurring on a time scale of ∼0.1 ns.17 During this capture and cascade process, both of the electrons in the He atom are lost due to Auger emission, leaving the muonic helium ion, [4He2+μ−]+, which then thermalizes in the gas by elastic collision processes. Charge exchange near thermal energies with CH4 (ionization energy 12.5 vs 13.6 eV for H) gives the chargeneutral muonic helium atom, Heμ = [4He2+μ−]e−, with only ∼5% residual μ− polarization, due to these capture, cascade, and CE processes. NH3 was used for the purpose of neutralizing Heμ in our earlier work,16,17 but CH4 itself was later found to be more efficient.19 B

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neutral atom), as can be seen in Figure 1, where the higher CH4 partial pressure can necessitate very long run times of ∼24 h. The central feature of interest in the experiment is the measured relaxation rate, λHeμ, the damping of the signal seen in Figure 1, which increases with increasing CH4 concentration, as plotted in Figure 2 at a temperature of 500 K. Methane concentrations (CCH4) were calculated from a simple equation of state47 after verifying against tabulated values,48 but at this temperature and these partial pressures, methane behaves within 1% of ideal. The data are then fit to a simple linear dependence, characteristic of the pseudo-first-order nature of the kinetics λHeμ = λ 0 + kHeμCCH4

where λ0 is a background relaxation rate (largely determined by magnetic field inhomogeneity) and here kHeμ = (6.9 ± 0.9) × 10−16 cm3 s−1 is the bimolecular rate constant of interest, at 500 K, seen as the slope of the line in Figure 2.

Figure 1. Asymmetry plots for mixtures of CH4 in He at a total pressure of 500 bar in a weak TF of 6 G and a temperature of 500 K. The small vertical lines represent 1σ Poisson statistics. The solid lines are fits of the μ−SR data to the oscillatory dependence in eq 1 plus a similar term for slowly precessing bare muons. See discussion in the text and Figure 1 in ref 17. The top spectrum (a) is for a methane partial pressure of 70 bar; the bottom spectrum (b) is for 14 bar methane. The initial amplitude, AHeμ, is higher at the lower methane pressure since these conditions give less probability for μ− capture on carbon during the slowing-down process. The relaxations seen, λHeμ, are due to the chemical reaction of interest, Heμ + CH4 → HeμH + CH3, and are faster at higher methane concentrations (a).

It is the muon polarization that is essential to the μSR (muon spin rotation, relaxation, resonance) technique employed in the measurement.10,17,45,46 When a negative muon decays, via μ− → e− + νμ + νe, producing an electron, a neutrino, and an antineutrino, the e− is emitted preferentially opposite to the instantaneous muon spin direction (see Figure 1 in ref 17), as a consequence of the parity-violating weak interaction.46 In this experiment, the muon spin of Heμ in its triplet (S = 1) spin state precesses in the applied weak transverse magnetic field (TF). The detection of an incident muon starts a timer, which is stopped when an e− is detected by a counter at a fixed direction in the plane of precession, and the data are collected in histograms of the number of electron decay events detected for different time intervals. The histograms display an oscillatory pattern (asymmetry) superimposed on a radioactive decay curve.17,46 The asymmetry describes the experimental signal for Heμ precession, as shown in Figure 1, and is defined by AHeμ(t ) = AHeμ e−λHeμt cos(ωHeμt + ϕHeμ)

(2)

Figure 2. Plot of the relaxation rates λHeμ at 500 K at a total He + CH4 pressure of 500 bar vs CH4 concentration (number density) calculated from a parametrized equation of state. See discussion in the text. Each data point represents about 24 h of run time and has been determined from Heμ precession curves, examples of which are shown in Figure 1. The lower densities were determined from pressures accurately measured using a capacitance Baratron manometer, whereas those at the higher densties were found from a conventional needle pressure gauge, with errors set at ±1 bar. The larger error on the density point at 4 (in the units shown) is due to additional uncertainty for this specific loading. From eq 2, the slope of the fitted line gives the rate constant of interest, kHeμ = (6.9 ± 0.9) × 10−16 cm3 s−1.

3. THEORY BACKGROUND 3.1. Potential Energy Surface. The PES employed in the present work has been described in our recent paper.26 Briefly, a total of 63 041 points calculated at the level of UCCSD(T)F12a/AVTZ were fitted by the permutational invariant polynomial-neural network (PIP-NN) approach.49,50 The final PIP-NN PES employed a NN with two hidden layers each with 4 and 100 neurons, resulting in 3997 parameters with a final overall root-mean-squared error (RMSE) from an analytical fit to the ab initio points of 5.1 meV. This PIP-NN PES is very similar to the earlier Xu−Chen−Zhang (XCZ) PES,30 as the ab initio calculations are based on the same protocol and their calculated reaction probabilities are quite similar.26 3.2. Variational Transition-State Theory. The VTST rate coefficients for the hydrogen abstraction reaction were

(1)

where AHeμ is the initial amplitude of the precession and λHeμ is its relaxation rate, the principal fitted parameter of interest here, with ωHeμ and ϕHeμ being the Larmor frequency and initial phase of Heμ precession in the weak (∼6 G) TF applied, respectively. Precession of bare μ− is also observed, having a much lower frequency, no relaxation, and no relevance to chemical kinetics. As outlined above, the amplitudes of the precessing μ−SR signals are very weak, due to the muon depolarization mechanisms that affect the μ− during its capture and thermalization processes. Moreover, in the presence of CH4, this amplitude can be reduced further due to competitive μ− capture on the carbon (in competition with CE forming the C

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formed at only two temperatures, 500 and 700 K. For comparison, the RPMD rate coefficients from the earlier PES of Espinosa-Garcı ́a and co-workers (PES-2008)25 are also included in our discussion below. The convergence tests of the RPMD calculations were mostly done with the PES-2008 PES, as it is much less computationally intensive. Given the similarities of the two PESs, we are confident that the results are fully converged. The theoretical rate coefficients for the Heμ + CH4 reaction obtained from different PESs using these methods are compared with the experimental one at 500 K in Table 1. Both the RPMD and VTST rate coefficients computed on the new PIP-NN PES of ref 26 are in excellent agreement with the experimental value, somewhat lower and higher, respectively, but both within experimental error. On the other hand, the RPMD and VTST results are much closer together on the PES2008 surface and are also both about 35% lower than the experimental value. From this comparison with experiment for the Heμ + CH4 reaction at 500 K, and particularly from the RPMD calculations, which we feel are inherently more accurate than the VTST ones,61−63 we can conclude that the PIP-NN PES is slightly more accurate than the PES-2008 in generating accurate rate coefficients for this reaction.

computed using canonical variational transition-state theory (CVTST, or CVT),51 as implemented in POLYRATE.52 Quantum effects for motions transverse to the reaction path were included by using quantum mechanical vibrational partition functions, assuming the harmonic approximation, whereas quantum effects in the reaction coordinate were included by using the microcanonical optimized multidimensional tunneling (CVT/μOMT) approach,51 in which, at each total energy, the larger of the small-curvature (SCT) and largecurvature (LCT) tunneling probabilities was taken as the best estimate. The rotational partition functions were calculated classically. 3.3. Ring Polymer Molecular Dynamics. The RPMD is a recently developed quantum real-time dynamics theory that exploits the classical isomorphism between the statistical properties of a quantum system and a fictitious necklace of harmonically connected classical copies of the original quantum system in order to approximate its real-time dynamics.53 The RPMD method provides an attractive practical alternative to TST for simulating quantum dynamics, as demonstrated by its consistent and reliable performance across a variety of bimolecular chemical reactions,54 wherein accurate quantum rate coefficients are well-reproduced,5,31,36−39,54−64 including those of the X + CH4 type.5,31,56,60−63 Importantly, it provides an accurate and reliable account of quantum mass effects, zeropoint energies, and tunneling below the crossover temperature, TC = ℏωb/2πkB, where iωb is the imaginary frequency at the barrier maximum, ℏ = h/2π is Planck’s constant, and kB is Boltzmann’s constant,36,38,62 and is even more reliable above TC.31,36,37,55,56,62 The present calculations were performed at two temperatures, 500 and 700 K, both well above the crossover temperature of 329 K for the PIP-NN PES. All calculations reported here were performed using the RPMDrate code developed by one of us (Y.V.S.).65 The computational strategy of RPMD calculations of chemical reaction rates is well-documented in the RPMDrate manual.65 In brief, it is based on the Bennett−Chandler factorization, in which the rate coefficient is represented as a product of the quantum transition-state theory rate coefficient (kQTST) and transmission coefficients (κ)31,55 kRPMD = k QTST(T ;ξ ‡)κ(t → ∞ ;ξ ‡)

Table 1. Rate Coefficients (in cm3 s−1) at 500 K on Different PESs and Comparison with Experiment for the Heμ + CH4 Reaction PES, method

k (500 K)

PES-2008, VTST PES-2008, RPMD PIP-NN, VTST PIP-NN, RPMD experiment (Figure 2)

4.35 × 10−16 4.61 × 10−16 7.58 × 10−16 6.10 × 10−16 (6.9 ± 0.9) × 10−16

The results from Table 1 are also plotted in Figure 3, along with the calculated point at 700 K from Table 2. The level of

(3)

These two factors represent the static and dynamical contributions, respectively. Since the theoretical method has been discussed in detail in the literature, the reader is referred to our previous publications.31,65 The free-energy difference at the highest point of the profile (ΔG(ξ⧧)) determines the QTST rate. The transmission coefficient is subsequently computed near the barrier height (ξ⧧) to account for recrossing. The reaction coordinate ξ for the reaction X + CH4 is determined using the two standard dividing surfaces used in all of our previous studies [see eqs 12, 13, 33, and 34 in ref 31]. We note that, despite being our standard choice, these dividing surfaces may not be optimal for this particular class of reactions. However, these have no effect on the final thermal RPMD rates, which are immune to this choice.66

Figure 3. Arrhenius plots of the calculated rate coefficients for the Heμ + CH4 reaction obtained from both RPMD and VTST calculations on both the PES-2008 and PIP-NN PESs, compared as well with the experimental result at 500 K. The lines are meant to guide the eye. The current RPMD calculations on the PIP-NN surface have been carried out at only two temperatures, 500 and 700 K. Nevertheless, the excellent level of agreement between theory and experiment at 500 K on this surface is noteworthy.

4. THEORY RESULTS AND COMPARISONS WITH EXPERIMENT The present RPMD calculations on the PIP-NN PES are computationally time-consuming and thus have been perD

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at lower temperatures for the Heμ + CH4 reaction, even for the heavy muonic He atom, as was apparent for Heμ + H2 near 300 K,17 though this was on a lower-barrier PES. For reference, the classical barrier height for CH5 on the PIP-NN surface of Li et al.26 is 14.69 kcal/mol, vs 9.608 ± 0.010 kcal/mol on the CCI/ BH surface of Mielke et al. for H3.3 On both PESs, the transmission coefficients (Figure 4b) are noticeably smaller than unity, indicating a significant contribution from recrossing at the dividing surface. It is remarked that, in RPMD, tunneling is included in both kQTST and κ in eq 3, so it is impossible to separate its contribution; hence, it cannot be unambiguously claimed that κ accounts only for dynamical recrossing processes in this classical definition. As in Table 1, our previous studies also found good agreement between the RPMD and VTST calculations at high temperatures for the corresponding reactions involving the H and D isotopes.5 Consequently, we have listed in Table 3 the VTST rate coefficients at 500 K for the isotopic reactions of the X + CH4 system obtained using the same VTST protocol as described above. As pointed out by us earlier,5,62 the VTST rate coefficients for Mu might not be accurate because of the large anharmonicities that are not taken into consideration in the VTST calculations. At present, the RPMD calculations of the rate coefficients for both the Heμ and Mu + CH4 reactions on the PIP-NN surface are a work in progress, due to the large number of beads needed for convergence, particularly for the very light Mu isotope. Additional results will be published in due course. It is interesting to note from Table 3 that the calculated kinetic isotope effects (KIEs), kH/kD, are about 0.6 at 500 K on both surfaces, indicating that they are relatively insensitive to differences in barrier heights on these largely early barrier surfaces. A similar ratio was also seen over a range of temperatures in the Arrhenius plots of ref 5, consistent with the earlier findings of Pu and Truhlar on different PESs (their Table 8)33 as well. One might have expected this KIE to be >1 at all temperatures, reflecting the lighter mass of the H atom on what are essentially symmetric barrier surfaces for both reactions. That kH/kD is