Electronic Quenching of N(2D) by N2: Theoretical Predictions

Letter pubs.acs.org/JPCL

Electronic Quenching of N(2D) by N2: Theoretical Predictions, Comparison with Experimental Rate Constants, and Impact on Atmospheric Modeling B. R. L. Galvaõ ,† A. J. C. Varandas,‡,§ J. P. Braga,† and J. C. Belchior*,† †

Departamento de Quı ́mica-ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Pampulha, (31.270-901) Belo Horizonte, Minas Gerais, Brazil ‡ Departamento de Quı ́mica, and Centro de Quı ́mica, Universidade de Coimbra, 3004-535 Coimbra, Portugal ABSTRACT: Rate constants for the electronic quenching reaction N(2D) + N2 → N(4S) + N2 are calculated for temperatures over the range of 240 ≤ T/K ≤ 1000 using an accurate set of three global electronic potential energy surfaces for the N3 system (4A″,2A′, and 2A″). The nuclear motion is treated by running quasiclassical trajectories, incorporating spin-forbidden transitions with the trajectory surface hopping method. The exclusively theoretical results are compared with available experimental data for the reaction and contribute to clarify the discrepancies among them. The rate constants at higher temperatures achieved in the atmosphere, for which no experiments have been performed, are presented for the first time. The impact of the results in atmospheric modeling is analyzed, predicting at what altitudes this reaction will play an important role.

SECTION: Kinetics and Dynamics

T

effective of the above deactivation reactions is believed7,8 to be eq 3, but as noted by Slanger and Black,7 above altitudes of 120 km, the temperature rises fast, and if reaction 4 has a significant impact, it would lead to a further decrease in NO concentration at higher altitudes. Given the importance of such reactions to atmospheric modeling, several experimental measurements have been performed at different temperatures.9 Specifically, for the electronic quenching of N(2D) by molecular nitrogen, eq 4, multiple measurements have been carried out,7,10−15 but the results are rather incompatible, as written by Herron9 in his review on the subject, “The disagreement on temperature coefficients is so great that no recommendation is provided”. Such discrepancies cause a lack of knowledge on the impact that this reaction may have in atmospheric mechanisms such as reaction 2, and therefore, theoretical studies seem necessary to elucidate this issue. The present work aims at contributing to this topic by providing the first theoretical analysis of reaction 4 and recommending the most accurate values for the temperature dependence of the rate constant up to now. This has not been pursued yet due to the high complexity of the potential energy surfaces (PESs) involved, at least two doublet and one quartet state PESs that show several crossings, transition states (TSs), and wells being necessary.16 However, as will be shown, the

he presence of excited and ionized species, such as N(2D), NO+, and N+2 , plays a major role in the ionosphere, where their concentration is known to be significant. Particularly, the first excited state of atomic nitrogen is formed by electronic impact reactions such as1,2 N2 + e− → N(2 D) + N + e− N+2 + e− → N(2 D) + N NO+ + e− → N(2 D) + O

(1) 3

and has a relatively long lifetime. The reaction of this species with oxygen is known to be the major source of nitric oxide in the atmosphere,4 which in turn plays a fundamental role in the cooling down mechanism of the atmosphere N(2 D) + O2 → NO + O

(2)

The amount of N(2D) available for such a reaction is regulated by the deactivation to its ground state, which may occur either by spontaneous emission of a photon or through collisions with other atmospheric species, such as5 N(2 D) + O → N( 4S) + O

(3)

N(2 D) + N2 → N( 4S) + N2

(4) 2

Besides competing for the available N( D), the above reactions also generate N(4S), which is a sink for atmospheric NO2,6 via N(4S) + NO → N2 + O(3P), and thus are important for controlling the amount of nitric oxide available. The most © XXXX American Chemical Society

Received: May 15, 2013 Accepted: June 25, 2013

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The Journal of Physical Chemistry Letters

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electronic problem within the Born−Oppenheimer approximation has recently been accurately solved for five PESs,17−19 and if the methodology used for solving the nuclear motion problem is accurate enough, a quantitative description of the reaction can be achieved and compared to all available experimental information for the specific reaction defined by eq 4. In the framework of this study, a PES is the eigenvalue of the nonrelativistic electronic Hamiltonian and does not include spin−orbit (SO) effects. In this picture, the quartet and doublet PESs are independent, and the approach of the N2 molecule toward N(2D) splits the degenerate levels of the atom into five doublet PESs,18 1 and 2 of 2A′ symmetry and 1−3 of 2A″. The ground-state N(4S) atom, on the other hand, is nondegenerate, and only the N3(4A″) PES describes its interaction with N2. Because the lowest doublet states are attractive and the quartet is repulsive, they show crossings between them for different arrangements of the molecule, such points being the most probable for the electronic transitions in which we are interested. The first PES for the system was a LEPS form20 to study the N(4S) + N2 collisions, and the rising interest on this topic has led to other studies,21−24 including our analytic PES,17 which was modeled using the double many-body expansion (DMBE) method25−27 for accurate ab initio data. The first set of PESs able to describe the N(2D) + N2 interaction was recently obtained18,19 as double-sheeted representations for the 1,22A″ states and for the 1,22A′ ones, which were modeled using accurate energies of multireference configuration interaction quality with the Davidson correction (MRCI+Q).28,29 Such PESs describe all known features of the N3 system and are represented in a relaxed unidimensional plot in Figure 1, where

Of particular relevance to the present work is the crossing between the quartet and doublet sheets highlighted in Figure 1 (crossing I). Although the surfaces also cross for different molecular arrangements, this one lies closely to the minimumenergy path for the reaction and also shows a stronger coupling16 (and thus the transition probability is expected to be higher). Morokuma and co-workers16 have studied the doublet/quartet crossings and found that the minimum on the crossing seam of Figure 1 slightly deviates from linearity, and the energy height calculated is in very good agreement with the one predicted by our set of PESs. In their work, the norm of the SO coupling matrix elements were also calculated for a given geometry as VSO = ⟨D|H SO|Q⟩

(5)

where |D⟩ and |Q⟩ stand for the doublet and quartet electronic wave funtions. Crossings for nonlinear geometries also exist (here denoted as crossings II, III, and IV), as seen in the relaxed C2v plot of Figure 2.

Figure 2. Potential energy profiles for optimized C2v geometries. The crossing between surfaces of the same symmetry (conical intersections, CIs) are marked with a ◊ symbol, while the doublet/quartet crossings16 are marked with ⊙ and assigned by roman numerals.

For a nonlinear attack, the barrier for reaction is higher, and therefore, a T-shaped N(2D) + N2 attack will have less probability of entering the covalent region, the barrier in this case being 6.8 kcal mol−1, as shown in Figure 2. Other interesting aspects of this plot are the presence of CIs between PESs of same symmetry and the minimum corresponding to the cyclic-N3, which is a new isomer of N3 recently detected experimentally30 but not yet isolated. Given the large masses of the nitrogen atoms, it has been shown31 for the N(4S) + N2 case that quasiclassical trajectories (QCTs) are capable of achieving the same accuracy as quantum mechanical treatments if one is interested only in averaged properties such as rate constants. The present work involves however spin-forbidden transitions, and one further approximation is introduced, the trajectory surface hopping32−34 (TSH) method as implemented in ref 35, in which a trajectory may be allowed to hop to another electronic state when reaching a crossing point, with the probability given by the Landau−Zenner formula. It was shown in ref 19 that electronic transitions to excited states of the same symmetry do not play a relevant role in the N(2D) + N2 exchange reaction, and for this reason, we focus

Figure 1. Optimized linear path for a N atom approaching a N2 molecule, showing the ground and excited 2A″ states (black lines), the 2 A′ states (gray), and the quartet one (black dotted line). The ⊙ symbol corresponds to the crossing (I) between the quartet and doublet PESs.

the optimized linear path for the atom−diatom attack is shown. As can be seen, the two lowest doublet states (12A′ and 12A″) are degenerate at the global minimum corresponding to the N3 molecule in its 2Πg state, which lies 59 kcal mol−1 under the N(2D) + N2 limit. The TS shown in the graph corresponds to the lowest-energy barrier for reaching the covalent region from this asymptote and is also degenerate for both doublet states, lying 2.8 kcal mol−1 above the reactants.16 2293



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Figure 3. Comparison of the present results for the rate constants with various experimental measurements at different temperatures. The inset gives a larger view of the area around room temperature.

here only on 12A′ → 14A″ and 12A″ → 14A″ transitions. This means that it is not possible for a trajectory starting at the 2A′ state to end up at the 2A″ one. In fact, it is seen from Figures 1 and 2 that after hopping to the quartet state, the system will experience a strong gradient toward dissociation and quickly form the products, and thus, secondary hops are not observed. For fixed temperatures, we have sampled the initial conditions (rovibrational state of N2, relative translational energy, impact parameter, and orientation of the reactants) according to their probability distribution following the QCT method.36,37 The calculation starts on a doublet PES with an atom−diatom separation of 17 a0 and the integration of the Hamilton’s equations of motion performed with a time step of 0.1 fs. At each point of the trajectory, we monitor the difference between the current doublet PES and the quartet one to check for crossings; if the sign of this difference is changed, we rewind one step and start the integration again with a time step 200 times smaller to find more precisely the crossing point. There, the Landau−Zenner transition probability for diabatic surfaces is calculated according to35,38 ⎛ −2πV 2 ⎞ SO ⎟ PLZ = 1 − exp⎜ ⎝ ℏ|ΔFv| ⎠

calculated by ab initio in ref 16. The only relevant crossing for which no coupling is available is the one between 2A′/4A″ shown in Figure 2 with a bond angle of around 120°, which we approximate using the value of the similar crossing II. After running a sufficient number of trajectories N, the Monte Carlo integrated rate constant is given by ⎛ 8kBT ⎞1/2 2 N r k x(T ) = ge(T )⎜ ⎟ πbmax N ⎝ πμ ⎠

while the total result is the sum k(T) = k2A′→4A″(T) + k2A″→4A″(T). In the above equation, kB is the Boltzmann constant, μ is the reduced mass of the reactants, and bmax is the maximum value of the impact parameter, optimized by trial and error such as to guarantee that no trajectory will be reactive with a larger value. The number of trajectories that hopped to the quartet state and finished with the quenched nitrogen N(4S) atom is given by Nr, while N is the total number of integrated trajectories, which varies from 6 × 105 for the lowest temperature to 5 × 104 for the highest. The electronic degeneracy factor is calculated using its analytic form ge(T) = qN3/qN(2D)qN2, and the included partition functions are given by qN3 = 2, qN2 = 1, and qN(2D) = 6 + 4 exp(−12.53 K/T). Finally, the associated 68% error bars are given by Δkx = kx[(N − Nr)/ (NNr)]1/2. At room temperature, the average relative translational energy is 0.6 kcal mol−1, and a reactive trajectory, which involves overcoming the potential barrier of 2.8 kcal mol−1, is a very rare event under thermal equilibrium. This makes it necessary to integrate a huge number of trajectories to achieve a significant statistical population. To speed up the calculations, we have employed the quantum mechanical threshold41 (QMT) method, where only trajectories that show enough total energy to reach the zero-point energy (ZPE) of the TS are integrated, all others being considered as nonreactive. We have tested this approach against the normal scheme (integrating every trajectory) for T = 298 and 1000K and found no reactive trajectory disobeying the above energy criterion. Therefore, the results should be essentially the same without using the QMT method, while this approach decreases considerably the computational time for lower temperatures.

(6)

In this equation, ΔF is the difference in forces (between the two surfaces) and v the velocity, both evaluated at the crossing point and in the direction normal to the crossing seam. This product can be expressed in Cartesian coordinates (via the chain rule) as35 v ·ΔF =

dQ n ∂ΔV dt ∂Q n

9

=

∑ i=1

∂ΔV q̇ ∂qi i

(8)

(7)

where Qn is the direction normal to the seam of crossing and qi are Cartesian coordinates. The derivatives are obtained numerically. Ideally, the magnitude of the SO coupling at the geometry of the crossing should be evaluated as a function of the coordinates.39,40 In this work, whenever a crossing is reached by the trajectory, we determine its “region” according to the doublet/quartet crossing seams assigned in Figures 1 and 2. The magnitude of VSO is then given by the corresponding value 2294



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Table 1. Rate Constants for the N(2D) Quenching

There is a considerable amount of experimental measurements for the title reaction rate constant,7,9−15 especially around room temperature, as summarized in Figure 3. The temperature dependence of the rate constant was given by Slanger and Black7 and by Suzuki et al.15 via an Arrhenius fit to five temperatures, but the results are rather discrepant, especially on the prediction of the rate constant for higher temperatures. Without resorting to the present results, one could judge the two experimental curves considering the recent accurate electronic structure calculations16,18 (not available at the time of the experiments) as follows. If the real curve may indeed be described by an Arrhenius law, one could correlate the fitted experimental activation energy with the potential barrier for the N(2D) + N2 approach because the system can only reach a doublet/quartet crossing after overcoming this barrier (see Figure 1). In this perspective, the results of Slanger and Black7 predict a barrier of 1.0 kcal mol−1, while those of Suzuki et al.15 give 3.2 kcal mol−1. If compared to the high-level MRCI+Q calculations of Morokuma and co-workers16 of 3.15 kcal mol−1 (including ZPE), the latter is seen to be in perfect agreement. In fact, it has been shown18 that even after extrapolation to the complete basis set limit, the barrier height predicted in ref 16 is not considerably changed. Thus, the experimental slope of Suzuki et al.15 lies close to estimates of refs 16 and 18 and should therefore be trusted. The results of the quenching rate constants calculated in the present work (with no empirical input whatsoever) are also compared in Figure 3. Because the DMBE PESs used in the present work employed the same level of ab initio calculations as those of ref 16 (thus giving the correct barrier height), the slopes are in good agreement with the experimental results of Suzuki et al.15 but predict a somewhat lower reactivity. At room temperature, our results agree within the error bars with most of the available experimental measurements except, surprisingly, with the results of Suzuki et al.15 Although we believe that the latter shows the correct slope, it is not possible to state that it is more accurate than all other experimental data at room temperature, especially due to the discrepancy with Sugawara et al.,14 which was performed in the same laboratory and using the same technique.9 The value recommended by Herron9 is in perfect agreement with our result at room temperature, but this can hardly be indicative of accuracy because the recommendation is simply an average among the experimental results. Obviously, our calculations are not exact, and a series of well established-approximations has been used, but we stress that the agreement with the experimental values at room temperature is remarkable. The temperatures within the range of experimental measurements (213 ≤ T/K ≤ 372) are quite low and show the worst case scenario for the QCTs. Our predictions should improve for higher temperatures and were obtained (Table 1) to cover most of the temperatures achieved in the ionosphere, where no experiments have been carried out. We have found that the simple Arrhenius law (suitable for the limited temperature range of the experimental measurements) is not flexible enough to model the whole curve. A fit to a modified Arrhenius expression k(T) = ATm exp(−B/T) yields A = 4.52 × 10−14 K−m cm3 s−1, m = 0.678, and B = 1437.7 K. Note that the slope is not constant, with our curve crossing that of Suzuki et al.15 for higher temperatures. As pointed out in the introductory paragraph, the main quencher of N(2D) is believed to be its reaction with oxygen

T/K 240 270 298 330 360 500 750 1000

k(T)/cm3s−1 (4.5 (1.0 (1.7 (3.1 (4.4 (1.7 (6.0 (11.5

± ± ± ± ± ± ± ±

0.6) 0.1) 0.1) 0.3) 0.4) 0.2) 0.3) 0.5)

× × × × × × × ×

10−15 10−14 10−14 10−14 10−14 10−13 10−13 10−13

log10[k(T)] −14.34 −14.00 −13.78 −13.51 −13.36 −12.78 −12.22 −11.94

atoms (reaction 3), playing a fundamental role in the regulation of its concentration and consequently in the production of NO in the atmosphere. In the following discussion, we try to analyze the relative importance of the title reaction as another competitor for quenching N(2D). It is clear that the quenching rate for a given partner will depend on the associated rate constant and on the quencher concentration via ΓX = kX[X][N(2D)], with X = O, N2. The recommended rate constant9 for collisions with O is much larger than our calculated rate constant with N2 for any temperature, but the concentration of atomic oxygen is negligible at low altitudes, while that of molecular nitrogen is very high. The situation changes for altitudes higher than 200 km, where [O] surpasses [N2], and hence, the rate of depletion (and the relative importance of each quencher) will vary with altitude. To compare ΓO with ΓN2, we have used the rate constant for quenching with oxygen atoms recommended by Herron9 and our modified Arrhenius law for the quenching with N2. The concentration profiles of O and N2 as a function of altitude were taken from the MSIS-E-90 model42 for a typical set of parameters (date and geographic coordinate), which is shown in Figure 4a. This model also gives the temperature as a function of altitude, which allows expression of the rate constants also as a function of altitude. Although we do not have data for [N(2D)], we can compare the relative importance of each quencher as ΓO/ΓN2. As shown in Figure 4b, the role played by reaction with O is vanishingly small for low altitudes because its concentration is negligible there (this region is probably not interesting because [N(2D)] is likely to be also very small). As the altitude increases, [O] reaches a maximum at around 100 km and so does its relative importance, as shown in Figure 4. After this peak, [O] declines but at a shorter pace than [N2]. After 200 km, we have [O] > [N2] and ΓO largely favored from there upward. However, in the important range between 125 and 200 km, ΓO is less than 5 times larger than ΓN2, with the ratio ΓO/ΓN2 reaching a minimum of about 3 at 150 km. We may therefore conclude that for the above interval of temperatures, the title reaction cannot be considered to be negligible in atmospheric models involving the reactions of N(2D). In summary, a series of systematic and accurate theoretical studies on the N3 system has been shown to culminate with the prediction of accurate rate constant data. For this, five accurate ab-initio-based DMBE PESs17−19 have been employed and quasiclassical dynamics studies carried out with TSH. A comparison with the available experiments has been performed, and predictions for higher temperatures have been done for the first time. A discussion of the impact of the results in modeling the ionosphere has further been presented and highlighted the 2295



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(8) Davenport, E.; Slanger, T. G.; Black, G. The Quenching of N(2D) by O(3P). J. Geophys. Res. 1976, 81, 12−16. (9) Herron, J. T. Evaluated Chemical Kinetics Data for Reactions of N(2D), N(2P), and N2(A3Σ+u ) in the Gas Phase. J. Phys. Chem. Ref. Data 1999, 28, 1453−1483. (10) Black, G.; Slanger, T. G.; John, G. A. S.; Young, R. A. VacuumUltraviolet Photolysis of N2O. IV. Deactivation of N(2D). J. Chem. Phys. 1969, 51, 116−121. (11) Lin, C.-L.; Kaufman, F. Reactions of Metastable Nitrogen Atoms. J. Chem. Phys. 1971, 55, 3760−3770. (12) Husain, D.; Kirsch, L. J.; Wiesenfeld, J. R. Collisional Quenching of Electronically Excited Nitrogen Atoms, N(22DJ, 22PJ) by TimeResolved Atomic Absorption Spectroscopy. Faraday Discuss. Chem. Soc. 1972, 53, 201−210. (13) Husain, D.; Mitra, S. K.; Young, A. N. Kinetic Study of Electronically Excited Nitrogen Atoms, N(22DJ, 22PJ), by Attenuation of Atomic Resonance Radiation in the Vacuum Ultra-violet. J. Chem. Soc., Faraday Trans. 1974, 70, 1721−1731. (14) Sugawara, K.; Ishikawa, Y.; Sato, S. The Rate Constants of the Reactions of the Metastable Nitrogen-Atoms, 2D and 2P, and the Reactions of N(4S) + NO → N2+O(3P) and O(3P) + NO + M → NO2 + M. Bull. Chem. Soc. Jpn. 1980, 53, 3159−3164. (15) Suzuki, T.; Shihira, Y.; Sato, T.; Umemoto, H.; Tsunashima, S. Reactions of N(2D) and N(2P) with H2 and D2. J. Chem. Soc., Faraday Trans. 1993, 89, 995−999. (16) Zhang, P.; Morokuma, K.; Wodtke, A. M. High-Level Ab Initio Studies of Unimolecular Dissociation of the Ground-State N3 Radical. J. Chem. Phys. 2005, 122, 014106. (17) Galvão, B. R. L.; Varandas, A. J. C. Accurate Double Many-Body Expansion Potential Energy Surface for N3 from Correlation Scaled Ab Initio Energies with Extrapolation to the Complete Basis Set Limit. J. Phys. Chem. A 2009, 113, 14424−14430. (18) Galvão, B. R. L.; Varandas, A. J. C. Ab Initio Based DoubleSheeted DMBE Potential Energy Surface for N3(2A″) and Exploratory Dynamics Calculations. J. Phys. Chem. A 2011, 115, 12390−12398. (19) Galvão, B. R. L.; Caridade, P. J. S. B.; Varandas, A. J. C. N(4S/2D) + N2: Accurate Ab Initio-Based DMBE Potential Energy Surfaces and Surface-Hopping Dynamics. J. Chem. Phys. 2012, 137, 22A515. (20) Laganà, A.; Garcia, E.; Ciccarelli, L. Deactivation of Vibrationally Excited Nitrogen Molecules by Collision with Nitrogen Atoms. J. Phys. Chem. 1987, 91, 312−314. (21) Esposito, F.; Capitelli, M.; Gorse, C. Quasi-Classical Dynamics and Vibrational Kinetics of N + N2(v) system. Chem. Phys. 2000, 257, 193−202. (22) Wang, D.; Stallcop, J. R.; Huo, W. M.; Dateo, C. E.; Schwenke, D. W.; Partridge, H. Quantal Study of the Exchange Reaction for N + N2 Using an Ab Initio Potential Energy Surface. J. Chem. Phys. 2003, 118, 2186−2189. (23) Wang, D.; Huo, W. M.; Dateo, C. E.; Schwenke, D. W.; Stallcop, J. R. Quantum Study of the N + N2 Exchange Reaction: State-to-State Reaction Probabilities, Initial State Selected Probabilities, Feshbach Resonances, and Product Distributions. J. Chem. Phys. 2004, 120, 6041−6050. (24) Rampino, S.; Skouteris, D.; Laganà, A.; Garcia, E.; Saracibar, A. A Comparison of the Quantum State-Specific Efficiency of N + N2 Reaction Computed on Different Potential Energy Surfaces. Phys. Chem. Chem. Phys. 2009, 11, 1752−1757. (25) Varandas, A. J. C. A General Approach to the Potential Energy Function of Small Polyatomic Systems: Molecules and vdW Molecules. J. Mol. Struct. Theochem. 1985, 21, 401−424 (The article has been published as part of Vol. 120 of the same journal).. (26) Varandas, A. J. C. Intermolecular and Intramolecular Potentials: Topographical Aspects, Calculation, and Functional Representation via a DMBE Expansion Method. Adv. Chem. Phys. 1988, 74, 255−338. (27) Varandas, A. J. C. In Lecture Notes in Chemistry; Laganà, A., Riganelli, A., Eds.; Springer: Berlin, Germany, 2000; Vol. 75; p 33.

Figure 4. Dependence on the altitude of selected properties. (a) Concentrations of atmospheric O, N2, and O2. (b) Relative importance of O and N2 for the quenching of N(2D).

regions of the atmosphere where the title reaction is expected to play a role.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. § E-mail: [email protected] (A.J.C.V.).

■

ACKNOWLEDGMENTS B.R.L.G. thanks the Conselho Nacional de Desenvolvimento Cientı ́fico e Tecnológico (CNPq) for the Grant 150071/20132. The authors also acknowledge the financial support from CNPq and Fundaçaõ de Amparo à Pesquisa do estado de Minas Gerais (FAPEMIG). The support from Fundaçaõ para a Ciência e Tecnologia, Portugal, to one of us (A.J.C.V.) is also gratefully acknowledged.

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REFERENCES

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Electronic Quenching of N(2D) by N2: Theoretical Predictions

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