Ring-Polymer Molecular Dynamics Rate Coefficient Calculations for

Jan 31, 2014 - It is shown that the unique ability of the RPMD approach among the existing ..... they are only slightly higher than the QD results of ...
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Ring-Polymer Molecular Dynamics Rate Coefficient Calculations for Insertion Reactions: X + H2 → HX + H (X = N, O) Yongle Li,†,∥ Yury V. Suleimanov,*,‡,§,∥ and Hua Guo*,† †

Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, New Mexico 87131, United States Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States § Department of Mechanical and Aerospace Engineering, Combustion Energy Frontier Research Center, Princeton University, Olden Street, Princeton, New Jersey 08544, United States ‡

S Supporting Information *

ABSTRACT: The thermal rate constants of two prototypical insertion-type reactions, namely, N/O + H2 → NH/OH + H, are investigated with ring polymer molecular dynamics (RPMD) on full-dimensional potential energy surfaces using recently developed RPMDrate code. It is shown that the unique ability of the RPMD approach among the existing theoretical methods to capture the quantum effects, e.g., tunneling and zero-point energy, as well as recrossing dynamics quantum mechanically with ring-polymer trajectories leads to excellent agreement with rigorous quantum dynamics calculations. The present result is encouraging for future applications of the RPMD method and the RPMDrate code to complex-forming chemical reactions involving polyatomic reactants.

SECTION: Kinetics and Dynamics

R

within the TST framework,3−6,10 a rigorous description requires a multidimensional quantum mechanical treatment with an accurate PES. The second problem is anharmonicity, for which approximations also exist within the TST framework.3−6 However, an accurate calculation of partition functions for anharmonic systems remains a challenge, especially in the transition-state region. Perhaps most importantly, there is essentially no good way to treat recrossing because it is inherently a dynamical problem that is incompatible with the statistical nature of TST. Recrossing, which is particularly prevalent for barrierless reactions, is minimized in variational TST,3−6 which is based on Wigner’s observation that the classical TST provides an upper limit of the rate coefficient.11 Alternatively, quasi-classical trajectory (QCT) methods have been proposed, which are capable of estimating recrossing and zero-point energy effect of the reactants and products. However, it suffers from its complete inability to account for other quantum mechanical effects.2,12 A recently proposed ring polymer molecular dynamics (RPMD) approach13 for calculating rate coefficients14−16 offers an attractive alternative approach to the conventional methods based on TST or QCT. Taking advantage of the isomorphism between the statistical properties of a quantum particle and a fictitious ring polymer consisting of harmonically connected

ate coefficients of chemical reaction play an indispensible role in kinetic modeling of combustion, atmospheres, and interstellar media. Since experimental measurements of rate coefficients can often be difficult and expensive, there is a great need for theoretical calculations. The theoretical approach not only allows comparison to the experimental data, but also provides insights into reaction mechanism and assessment of the underlying potential energy surface (PES). For many reactions involving light atoms, a quantum mechanical treatment is necessary. While quantum scattering calculations provide a bottom-up approach to rate coefficients, the computational costs are formidable except for the smallest systems.1,2 The most popular method for estimating rate coefficients is the transition-state theory (TST) originally proposed by Wigner, Erying, Evans and Polanyi, as well as its modern variants, which are all based on statistical ansätze.3−6 The inclusion of approximate treatments of tunneling and vibrational quantization have transformed TST to an inexpensive and, in many cases, a reliable method for determining theoretical rate coefficients, particularly those with apparent activation barriers. Other advances of TST, such as incorporating the variable reaction coordinate model, have also been made to tackle reactions with no intrinsic barriers.7−9 Despite these successes, there are still issues that are difficult to address accuracy within the TST framework. The first concern is tunneling near the transition state. While approximate schemes have been proposed and implemented © 2014 American Chemical Society

Received: January 9, 2014 Accepted: January 31, 2014 Published: January 31, 2014 700

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Table 1. Comparison of RPMD Rate Coefficients (cm3 s−1) of the N + H2 and O +H2 Reactions and Previous Theoretical and Experimental Results

classical beads,17 RPMD is a full-dimensional approximate quantum mechanical method for computing Kubo transformed real-time correlation functions for many dynamical properties. Unlike TST and QCT, RPMD treats the quantum Boltzmann operator accurately and automatically captures zero-point energy effect correctly along the entire reaction pathway including the transition-state region.18 Thanks to its equivalence to the instanton theory in some certain situations,19 RPMD provides very accurate estimates in the deep tunneling regime, where conventional TST approaches are inaccurate and QCT methods inadequate. In the high-temperature limit, RPMD becomes the classical TST, which is accurate at high temperatures, when the ring polymer collapses to a single bead.15 Interestingly, its (t→0+) limit gives rise to a quantum mechanical version of the TST.20,21 Furthermore, the RPMD rate coefficient is unique in that it is independent of the position of the dividing surface,16 thanks to an important distinction of the RPMD method from the TST approaches, namely, the inclusion of real-time (t > 0) recrossing dynamics. This feature is particularly important for reactions that do not have a well-defined transition state.22 Numerically, since the main computational costs are running classical trajectories, RPMD scales well with dimensionality. The RPMD calculations are usually only by an order of magnitude more expensive than conventional QCT calculations. Recently, it has been shown22,23 that the bimolecular rate coefficients can be efficiently computed via an RPMD implementation of the Bennett−Chandler factorization.24,25 This computational strategy has a number of distinct features. It avoids the partition functions altogether by performing thermodynamic integration from the reactants to the desired location in the transition-state region. The RPMD approach implemented in the software package RPMDrate developed by

N + H2 PES of Ho et al.34 T/K kQTST κ kRPMD QD39 expt.52 PES of Zhou et al.35 T/K kQTST κ kRPMD QD39 QD36 expt.52

T/K kQTST κ kRPMD QD43 QD42 QD45 expt.

270 1.08 × 10−11 0.207 2.24 × 10−12 2.05 × 10−12 (1.67 ± 0.04) × 10−12

300 1.44 × 10−11 0.220 3.17 × 10−12 2.87 × 10−12 (2.44 ± 0.34) × 10−12

400 2.84 × 10−11 0.251 7.12 × 10−12 6.36 × 10−12 --

-300 -1.41 × 10−11 -0.227 -3.21 × 10−12 2.05 × 10−12 2.87 × 10−12 −12 1.72 × 10 2.44 × 10−12 (1.67 ± 0.04) × (2.44 ± 0.34) × 10−12 10−12 O + H2

400 2.81 × 10−11 0.249 6.98 × 10−12 6.36 × 10−12 5.63 × 10−12 --

300 2.67 × 10−10 0.414 1.11 × 10−10 9.90 × 10−11 1.00 × 10−10 1.25 × 10−10 (1.03 ± 0.04) × 10−10 46 (1.20 ± 0.18) × 10−10 49

420 2.69 × 10−10 0.424 1.14 × 10−10 1.01 × 10−10 1.25 × 10−10

Figure 1. Comparison of PMFs (upper panels) and transmission coefficients (lower panels) for the N + H2 and O + H2 reactions. 701

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(below the thermal time ∼ βℏ).32 This means that RPMD has a potential to capture short-time quantum dynamics effect, but it does not include long-time quantum effects such as interference. Both types of resonances (short- and long-lived) in the complex-forming reactions due to the reaction intermediate have been observed in numerous studies.2 However, it is interesting to note that these studies also demonstrated that resonances, although they collectively contribute to the reactivity, have essentially no manifestation in thermally averaged rate constants. To summarize, we expect RPMD not to fail for this particular class of chemical reactions. In this work, we investigate the rate coefficients of two prototypical insertion-type reactions, namely, the N/O + H2 → NH/OH + H reactions. Our choice of these reactions for benchmark calculations is motivated by two factors. First, for both reactions, accurate experimental and quantum dynamical rate coefficients exist.2 Second, they have different potential well depths and exothermicities, which affect the lifetime of the reaction intermediates. These reactions also exhibit different potential energy profiles along the reaction pathways, with (N + H2) and without (O + H2) a barrier in the entrance channel. We believe that the experience acquired in the present work will be very valuable in future studies of complex-forming polyatomic chemical reactions using the RPMD approach and the RPMDrate code. The details of the RPMD method and the RPMDrate code can be found in previous work,26 so only a brief summary of the present calculations is given here. Taking advantage of the Bennett-Chandler factorization,24,25 the RPMD rate can be conveniently obtained in the following form:22,23 kRPMD = k QTST(T ; ξ ‡)κ(t → ∞ ; ξ ‡)

(1)

The first term denotes the static contribution, while the second is the dynamical correction. In particular, kQTST(T,ξ‡) is the centroid-density quantum transition-state theory (QTST) rate coefficient. This quantity depends on the position of the dividing surface and is determined entirely by the static equilibrium properties, namely the potential of mean force (PMF). By default, RPMDrate evaluates this quantity at the maximum value of the free-energy barrier, ξ‡, along the reaction coordinate ξ(q) in order to reduce recrossings in the dynamical correction provided by the second factor (κ(t → ∞;ξ‡)) in eq 1. This factor is the long-time limit of a time-dependent ringpolymer transmission coefficient accounting for recrossing at ξ‡. It counterbalances kQTST(T,ξ‡), ensuring the independence of the RPMD rate coefficient kRPMD(T) of the choice of the dividing surface. The reaction coordinate for the X + H2 (X = N, O) reactions is defined in terms of standard two dividing surfaces used in the previous atom−diatom studies.26 Note that while this choice of dividing surfaces may be not optimal, it does not affect the final RPMD results. In this work, the thermal rate coefficients for the N + H2 reactions were calculated in the range of 270−400 K. For the O + H2 reactions, rate coefficients at two temperatures, 300 and 420 K, are calculated. The PESs used for the N + H2 reaction are from Ho et al.34 and Zhou et al.35 For the O + H2 reaction, the PES from Dobbyn and Knowles36 was used. All calculations were performed using the RPMDrate code.26 The input parameters for all calculations are summarized in the Supporting Information (SI). Note that these parameters are similar to previous ones used in the RPMDrate studies of

Figure 2. Comparison of the RPMD, QD, and experimental rate coefficients for the N + H2 and O + H2 reactions. Experimental values are taken from the work of Suzuki et al., Atkinson et al., Matsumi et al., Talukdar and Ravishankara, Vranckx et al., and Sander et al.

one of us (Y.V.S.)26 has been tested against available accurate quantum dynamical results with good agreement18,22,23,27−29 and applied to several other reactions.30−33 So far, the RPMD theory has been largely applied to reactions dominated by a significant barrier, and its applicability for barrierless reactions has not been extensively tested. Such barrierless reactions play an important role in combustion, atmospheres, and cold interstellar media.2 Due to the formation of intermediate complexes, quantum dynamical studies of such reactions are much more demanding, and hence lag significantly behind those for thermally activated reactions.2 The difficulties associated with a barrierless reaction are 2-fold. First, it is not easy to define the barrier along the free-energy profile, which is typically located in the asymptotes and depends on energy or temperature. Second, the long-lived reaction intermediate results in significant recrossing, especially for systems with slow rates for energy randomization. Because of its independence of the dividing surface in rate calculations and inclusion of recrossing, the RPMD approach is expected to offer an accurate alternative to conventional TST approaches. It is also expected to work better than the QCT approach due to more accurate treatment of quantum mechanical effects. It should also be noted that RPMD provides accurate estimates of real-time correlation functions only at short times 702

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The O(1D) + H2 → OH + H reaction is another prototypical insertion reaction with significant exothermicity of ∼45 kcal/ mol.36 Unlike the N + H2 reaction discussed above, this reaction has no barrier in the entrance channel. Initial state (ji = 0) specific QD calculations have been reported by several groups,42−45 all using the PES of Dobbyn and Knowles.36 The rate coefficients obtained by Gray et al.42 and by Pradhan et al.45 were based on approximated reaction probabilities for J > 0, while those of Lin and Guo were obtained with all partial waves explicitly calculated.43,44 Table 1 and Figure 2b compare the RPMD rate coefficients with previous theoretical and experimental results. The theoretical rate coefficients are all consistent with the weak temperature dependence in the experimental data,46−50 reflecting the barrierless nature of the reaction. Although the RPMD rate coefficients are smaller than most of the experimental data, they are only slightly higher than the QD results of Lin and Guo,43 with the other two approximate QD values either above or below. The QCT rate coefficient at 300 K is slightly larger (1.2 × 10−10 cm3 s−1),51 presumably due to violation of ZPE. Similar to N + H2, RPMD demonstrates small error, ca. +10−15%. As discussed above, the discrepancy between the RPMD and QD results might be attributed to the lack of contributions from higher rotational states in the QD rate coefficients or the overestimation of the RPMD method. Although the PES for this reaction has no potential-energy barrier in the entrance channel, there is a bottleneck, manifested by a tiny free-energy barrier. As shown in Figure 1c, the bottleneck is just before the entrance into the H2O well, and the position of the free-energy barrier is temperature dependent, as expected. The free-energy barrier provides a convenient choice for computing the transmission coefficient in eq 1, which is shown in Figure 1d. As expected, the recrossing in this system is quite significant at the both temperatures where the RPMD calculations were performed. The most striking observations is that the transmission coefficients are not converged until ∼1500 fs. This extremely long time for the convergence of the transmission coefficients is a characteristic of the complex-forming reaction mechanism.2 As in the N + H2 system discussed above, trajectories entering the deep H2O well might still return to the entrance channel bottleneck region, due to slow intramolecular vibrational energy randomization. This time frame is consistent with the lifetime of the H2O complex in the well, as reported before.52 To summarize, in this work, the RPMD method is extended to complex-forming reactions. Calculations of the rate coefficients for such reactions are challenging for two reasons. First, the potential-energy surface often has a very small or null barrier, although a bottleneck is often present. Second, there is considerable recrossing due to the lack of a significant barrier and the deep well. It is demonstrated for two complex-forming reactions (N + H2 and O + H2) that the Bennett-Chandler factorization implemented in the RPMDrate code provides a convenient and reliable computational procedure to calculate the rate coefficients for complex-forming reactions. It is also shown that the RPMD transmission coefficients converge slowly for complex-forming reactions, especially when there is no potential barrier. However, its calculations can be conveniently handled by parallel computing options implemented in the RPMDrate code. The ability of the RPMD approach to capture the recrossing dynamics quantum mechanically with ring-polymer trajectories is unique among the current theoretical methods for computing reaction rate

thermally activated chemical reactions.18,22,23,27−29 and applied to several other reactions.30−33,37 However, recrossing factor calculations required more computations as discussed below. RPMDrate provides convenient parallel computing options, which we used to speed up the recrossing factor calculations (see ref 26). Note that the final rate coefficients are corrected by electronic partition functions: for both reactions, Qel = 5 was used. The N(2D) + H2 → NH + H is a prototypical insertion reaction with exothermicity of ∼29 kcal/mol. This reaction has a small (∼2 kcal/mol) barrier in the entrance channel,38 and the reaction path is dominated by the deep NH2 well. Thanks to its small size, quantum dynamical (QD) calculations have been reported by Lin and Guo39 on the PES of Ho et al.,34 and the initial state (ji = 0) specific rate coefficients were found to be larger than the thermal rate coefficients obtained using a QCT method on the same PES.40 These authors attributed the difference to tunneling over the entrance channel barrier, a quantum effect ignored by QCT. Subsequently, a new PES was developed by Zhou et al.,35 which has a slightly higher barrier height. Quantum mechanical initial state (ji = 0) specific rate coefficients on this new PES were found to be in better agreement with experiment. The RPMD rate coefficients on both PESs are compared with the previous QD and experimental values in Table 1. Interestingly, the RPMD rate coefficients are quite similar on the two PESs, and are only slightly higher than the experimental values and the QD results on PES of Ho et al. with an error of ∼ +10−15%, close to typical convergence error in RPMD. The deviation between rate coefficients from RPMD and QD on PES of Zhou et al. is larger, about 30%. The deviation between QD and RPMD results could also be due to the fact that the QD rate coefficients did not include higher rotational states of the H2 reactant, which should increase the total QD rate. Another possibility is the known propensity of RPMD to overestimate rate coefficients for asymmetric reactions with energy barriers.29 Nonetheless, the level of accuracy for this insertion reaction with a small barrier in the entrance barrier is quite impressive. In Figure 1a, the PMFs for this reaction are plotted for three temperatures. The location of the small free-energy barrier is very close to that of the potential-energy barrier, which is defined as ξ = 1. Consequently, the behavior of this insertion reaction superficially resembles an activated reaction, evidenced by the Arrhenius type temperature dependence of the rate coefficients in Figure 2a. It is interesting to note that the slope of the RPMD rate is essentially the same as the QD counterpart, where the QCT slope (not shown)40 differs significantly. This concurs with the earlier conclusion39 that the QCT underestimates the rate coefficients due to the neglect of tunneling. As shown in both Table 1 and Figure 1b, there is large recrossing. Indeed, the transmission coefficient takes a long time (∼100 fs) to converge, while in previous studies of reactions with significant barriers it converges within several tens of femtoseconds. The transmission coefficients are in the range of 0.2−0.3, and highly oscillatory at short time. The oscillations are due to the particular choice of the dividing surface, thus having little physical significance. The large recrossing stems apparently to the deep NH2 well from which trajectories may return after a long excursion. The time taken for the transmission coefficients to converge compares well with the reaction time.41 703

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(13) Habershon, S.; Manolopoulos, D. E.; Markland, T. E.; Miller, T. F., III. Ring-Polymer Molecular Dynamics: Quantum Effects in Chemical Dynamics from Classical Trajectories in a Extended Phase Space. Annu. Rev. Phys. Chem. 2013, 64, 387−413. (14) Craig, I. R.; Manolopoulos, D. E. Quantum Statistics and Classical Mechanics: Real Time Correlation Frunction from Ring Polymer Molecular Dynamics. J. Chem. Phys. 2004, 121, 3368−3373. (15) Craig, I. R.; Manolopoulos, D. E. Chemical Reaction Rates from Ring Polymer Molecular Dynamics. J. Chem. Phys. 2005, 122, 084106. (16) Craig, I. R.; Manolopoulos, D. E. A Refined Ring Polymer Molecular Dynamics Theory of Chemical Reaction Rates. J. Chem. Phys. 2005, 123, 034102. (17) Chandler, D.; Wolynes, P. G. Exploiting the Isomorphism between Quantum Theory and Classical Statistical Mechanics of Polyatomic Fluids. J. Chem. Phys. 1981, 74, 4078−4095. (18) Pérez de Tudela, R.; Aoiz, F. J.; Suleimanov, Y. V.; Manolopoulos, D. E. Chemical Reaction Rates from Ring Polymer Molecular Dynamics: Zero Point Energy Conservation in Mu + H2 → MuH + H. J. Phys. Chem. Lett. 2012, 3, 493−497. (19) Richardson, J. O.; Althorpe, S. C. Ring-Polymer Molecular Dynamics Rate-Theory in the Deep-Tunneling Regime: Connection with Semi-Classical Instanton Theory. J. Chem. Phys. 2009, 131, 214106. (20) Hele, T. J. H.; Althorpe, S. C. Derivation of a True (t → 0+) Quantum Transition-State Theory. II. Recovery of the Exact Quatnum Rate in the Absense of Recrossing. J. Chem. Phys. 2013, 139, 084115. (21) Hele, T. J. H.; Althorpe, S. C. Derivation of a True (t → 0+) Quantum Transition-State Theory. I. Uniqueness and Equivalence to Ring-Polymer Molecular Dynamics Transition-State-Theory. J. Chem. Phys. 2013, 138, 084108. (22) Collepardo-Guevara, R.; Suleimanov, Y. V.; Manolopoulos, D. E. Bimolecular Reaction Rates from Ring Polymer Molecular Dynamics. J. Chem. Phys. 2009, 130, 174713. (23) Suleimanov, Y. V.; Collepardo-Guevara, R.; Manolopoulos, D. E. Bimolecular Reaction Rates from Ring Polymer Molecular Dynamics: Application to H + CH4 → H2 + CH3. J. Chem. Phys. 2011, 134, 044131. (24) Bennett, C. H. Molecular Dynamics and Transition State Theory: The Simulation of Infrequent Events. In Algorithms for Chemical Computations; Christofferson, R. E., Ed.; ACS Symposium Series; ACS: Washington, DC, 1977. (25) Chandler, D. Statistical Mechanics of Isomerization Dynamics in Liquids and the Transition State Approximation. J. Chem. Phys. 1978, 68, 2959−2970. (26) Suleimanov, Y. V.; Allen, J. W.; Green, W. H. RPMDrate: Bimolecular Chemical Reaction Rates from Ring Polymer Molecular Dynamics. Comput. Phys. Commun. 2013, 184, 833−840. (27) Li, Y.; Suleimanov, Y. V.; Yang, M.; Green, W. H.; Guo, H. Ring Polymer Molecular Dynamics Calculations of Thermal Rate Constants for the O(3P) + CH4 → OH + CH3 Reaction: Contributions of Quantum Effects. J. Phys. Chem. Lett. 2013, 4, 48−52. (28) Li, Y.; Suleimanov, Y. V.; Li, J.; Green, W. H.; Guo, H. Rate Coefficients and Kinetic Isotope Effects of the X + CH4 → CH3 + HX (X = H, D, Mu) Reactions from Ring Polymer Molecular Dynamics. J. Chem. Phys. 2013, 138, 094307. (29) Suleimanov, Y. V.; Pérez de Tudela, R.; Jambrina, P. G.; Castillo, J. F.; Sáez-Rábanos, V.; Manolopoulos, D. E.; Aoiz, F. J. A Ring Polymer Molecular Dynamics Study of the Isotopologues of the H + H2 Reaction. Phys. Chem. Chem. Phys. 2013, 15, 3655−3665. (30) Suleimanov, Y. V. Surface Diffusion of Hydrogen on Ni(100) from Ring Polymer Molecular Dynamics. J. Phys. Chem. C 2012, 116, 11141−11153. (31) Allen, J. W.; Green, W. H.; Li, Y.; Guo, H.; Suleimanov, Y. V. Communication: Full Dimensional Quantum Rate Coefficients and Kinetic Isotope Effects from Ring Polymer Molecular Dynamics for a Seven-Atom Reaction OH + CH4 → CH3 + H2O. J. Chem. Phys. 2013, 138, 221103.

coefficients. Finally, we note very good agreement with previous quantum dynamical calculations for these two reactions. RPMD is well-known for systematic and consistent performance with a predictable level of errors when applied to higher-dimensionality systems. The present results are certainly encouraging for future applications of the ring-polymer theory to larger reactive systems involving reaction intermediates, for which rigorous quantum dynamics calculations are currently impossible.

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ASSOCIATED CONTENT

S Supporting Information *

Parameters used in the RPMDrate calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Author Contributions ∥

Y.L. and Y.V.S. contributed equally to this work.

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Y.L. and H.G. were supported by the Department of Energy (DE-FG02-05ER15694 to HG). Y.V.S. acknowledges the support of a Combustion Energy Research Fellowship through the Combustion Energy Frontier Research Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Basic Energy Sciences under Award Number DE-SC0001198.

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REFERENCES

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The Journal of Physical Chemistry Letters

Letter

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dx.doi.org/10.1021/jz500062q | J. Phys. Chem. Lett. 2014, 5, 700−705