Quasiclassical dynamics and kinetics of the N+NO-->N(2)+O, NO+N atmospheric reactions.

The kinetics and dynamics of the title reactions were studied using the quasiclassical trajectory (QCT) method and two ab initio analytical potential energy surfaces (PESs) developed by our group. In addition to the rate constant (T: 10-5000 K), we also considered a broad set of dynamic properties as a function of collision energy (up to 1.0 eV) and the rovibrational state of NO (v=0-2,j=1,8,12). The production of N(2)+O, reaction (1), dominates the reactivity of the N+NO system over the conditions studied, as expected from the large energy barriers associated to the NO+N exchange reaction, reaction (2). Moreover, the ground PES, which is barrierless for reaction (1), plays a dominant role. Most of the results were interpreted according to the properties of the PESs involved and the kinematics of the system. The QCT rate constants of reaction (1) are in agreement with the experimental data (T: 47-3500 K), including very recent low temperature measurements, and also with variational transition state kinetics and most of quantum dynamics calculations. In addition, the QCT average vibrational energy content of the N(2) product also agrees with the experimental and quantum data. The PESs used here could also be useful to determine equilibrium and nonequilibrium reaction rates at very high temperatures (e.g., 5000-15 000 K).


I. Introduction
The reaction of atomic nitrogen with nitric oxide has been the subject of numerous studies in the last decade. In this system two reactive processes can occur. The first one involves the production of molecular nitrogen via the highly exothermic reaction channel, 1 N( 4 S) + NO(X 2 P) → N2(X 1 Sg + ) + O( 3 P) DrH º 0 K = -3.178 eV, while the second one involves the exchange of the nitrogen atom in the nitric oxide molecule, N( 4 S) + N'O(X 2 P) → NO(X 2 P) + N'( 4 S) DrH º 0 K = 0 eV.
Reaction (1) has been used to determine NO concentrations in low-pressure discharge-flow systems and to measure NO concentrations in combustion exhaust gases. In addition to its practical implications, this reaction affects the stratosphere, since it acts as a sink for NO molecules 2,3 . The extinction of NO molecules prevents their introduction into the catalytic cycle of ozone depletion, where a single NO molecule destroys several O3 molecules, which are vital to all forms of life on Earth.
The N + NO → N2 + O reaction is also interesting from a theoretical perspective, as it proceeds via two potential energy surfaces (PESs). 4 The most important of them is the barrierless ground state PES ( 3 A'' surface), where the reaction proceeds without surmounting any energy barrier along the minimum energy path (MEP) connecting reactants and products. The other potential energy surface of interest is the first excited state PES ( 3 A' surface), which has an early barrier of 0.36 eV and only contributes significantly to the reactivity of the system at high temperatures. 4 Moreover, as this relevant system involves a small number of atoms, a strong interaction between theory and experiment is expected.
Differing from the first reaction, reaction (2) presents a very low reactivity, due to the presence of a high energy barrier (>1 eV) on both the 3 A'' and 3 A' PESs, which connects the two equivalent [N-O-N] minima involved in the N-atom exchange process. 4 The determination of the rate constant for this reaction is, however, of interest, and such data are very scarce and difficult to obtain experimentally. Measurements on reaction (2) are reported in two isotopic studies, 5,6 but only at very high collision energies, while the present study focuses on thermal conditions and lowmoderate collision energies.
Since the earlier theoretical works on the dynamics and kinetics of reaction (1) (see, e.g., Refs. 7,8,9,and 10), several theoretical methods have been applied to this system in order to better characterize its dynamics and kinetics properties, using ab initio based analytical PESs. 4 They include variational transition state theory (VTST) 4 and time-dependent real wave-packet (RWP) 11,12 approaches. These theoretical studies on reaction (1) show good agreement for the ground state PES, 3 A'', whereas the rate constants obtained with the VTST method are a factor of three larger than those obtained with RWPs for the excited state PES, 3 A'.
Here we employ the ab initio analytical representations of the ground and first excited PESs developed in our group, 4 to perform a theoretical study of the dynamics and kinetics of reactions (1) and (2), using the quasiclassical trajectory (QCT) method. The paper is organized as follows: Section II provides the computational details, Section III presents the QCT results (scalar and vector properties and microscopic reaction mechanisms) and the comparison with previous theoretical and experimental data. Finally, Section IV gives the summary and conclusions.

II. Computational method
The analytical representations (many-body expansion functions 13 ) of the 1 3 A'' and 1 3 A' PESs used in this study ( 3 A'' and 3 A' PESs hereafter) were reported in Ref. 4. They are based on an ab initio CASSCF (10,9) study, 14 in which the standard correlation-consistent cc-pVTZ basis set was used. The dynamic correlation was treated using the CASPT2 method with the G2 correction to the Fock matrix. 14 Reaction (1) is barrierless on the ground (1 3 A'') PES and has a barrier on the excited (1 3 A') PES (N-atom abstraction through a saddle point of bent geometry), while reaction (2) shows a large energy barrier on both potential energy surfaces (O-atom abstraction via a C2v minimum).
Here, we investigated the dynamics of reaction (1) on the two PESs, as a function of collision energy (ET), and the rovibrational level (v, j) of NO, using the QCT method. 15,16,17,18 The influence of temperature (T: 10-5000 K) on the dynamics and kinetics of reactions (1) and (2) has also been examined, using the same approach.
Although the QCT method was established long ago, it is still one of the most useful theoretical tools for studying chemical reaction dynamics. We verified the accuracy of the numerical integration of Hamilton's differential equations by analyzing the conservation of total energy and total angular momentum for each trajectory. The integration step size chosen (0.5´10 -16 s) was found to fulfill the conservation requirements for all trajectories. The trajectories were started at an initial distance of 12 Å between the N atom and the center of mass of the NO molecule, thus ensuring that the interaction energy was negligible with respect to the available energy. surface. As reactivity of the 3 A' surface was generally much lower than that of the 3 A'' PES, a larger number of trajectories was sampled in this case (9´10 6 at 1000 K, while 4´10 5 were sufficient at 5000 K). For reaction (1) the standard deviation of the cross sections at all temperatures studied was less than 1 % on both PESs, while for reaction (2) it varied from 17 % [ground PES (T=3000 K) and excited PES (T=2000 K)] to less than 5 % at 5000 K for both PESs. The standard deviation of reaction (1) distributions was maintained below 5 % in both PESs, whereas for reaction (2) distributions with a standard deviation below 5 % were recorded above 3000 K. Moreover, some additional calculations between 10 and 100 K were also performed on the ground PES, calculating, typically, 5´10 4 trajectories at each selected T.

III. Results and discussion
A. Rate constants Figure 1 shows the dependence of the rate constant of reaction (1), k1, on temperature for the 3  For reaction (1), the QCT results show in general a good agreement with the VTST (ICVT/µOMT-SO) 12 results. In the case of the excited PES, this agreement is almost quantitative.
The better QCT-VTST agreement found for the excited PES could be expected as this surface, unlike the barrierless ground PES, has an energy barrier along the MEP (0.379 eV; ZPE included). 4 In addition, the QCT calculations are in agreement with the experimental data over the entire temperature range, taking into account the error margins. Comparison with the quantum RWP Jshifting data 12 shows that the 3 A" QCT and RWP results are similar, but significant differences appear in the 3 A' results. The RWP J-shifting k1( 3 A') data were analyzed in Ref. 12, but the origin of the low values found remained unclear. Figure 2 shows the dependence of the rate constant of reaction (2), k2, on temperature for the 3 A'' and 3 A' PESs and for the sum of both contributions, k2 = k2( 3 A'') + k2( 3 A'). Moreover, Table   III shows the QCT results obtained together with the VTST data 4 for this reaction.
For both the ground and excited PESs, the QCT rate constant of reaction (2) increases monotonically with temperature and has very low values except for the higher temperatures studied.
These results are expected as both surfaces have a large energy barrier for the N-atom exchange reaction. 4 The variation of k2( 3 A'') and k2( 3 A') with T is quite large and, in general, k2( 3 A") is quite lower than k2( 3 A'), as the 3 A" surface has a larger energy barrier for exchange than the 3 A' surface (1.735 and 1.190 eV above reactants, including the ZPE, respectively). 4 For reaction (2), the QCT results are in quite good agreement with the VTST (ICVT/µOMT-SO) 12 results, which were obtained using the steady-state approximation. dependence for this type of systems is not observed here because energies are not large enough to make it evident. We also observe here that, as in the case of the cross section for the ground surface, s1( 3 A') presents a small dependence with the vibrational and rotational levels of the NO molecule.
These results can be rationalized taking into account that the 3 A' surface has an early barrier resulting from the high exoergicity of reaction (1) (Hammond's postulate), and in this situation it is well known (Polanyi's rules) that relative translational energy of reactants is particularly efficient to allow the system to evolve into products.
above the energy of NO(v=0) for v=1 and 2, respectively) are small in comparison with the total available energy in products (3.4 and 3.8 eV for the 3 A'' and 3 A' PESs for NO(v=1), respectively, as different ET values were considered in both surfaces). Moreover, the influence of NO vibrational excitation is more evident for the ground PES than for the excited one.
Due to the very low reactivity presented by reaction (2) (1) is exothermic. In both reaction channels the distributions become wider as temperature increases, which implies that more vibrational and rotational levels become available.

D. Two-and three-vector correlations
Having completed the study of the scalar properties of the N + NO reactive system, we then examined the stereodynamics (vector properties) of reaction (1), taking into account relevant vectors of the system. We analyzed the two-vector angular distributions kk', kj' and k'j' (where k and k' correspond to the initial and final relative velocity vectors, respectively, and j' refers to the rotational angular momentum vector of NO). The kk' angular distribution was expressed in terms of the differential cross section per unit of solid angle, d 2 s/dW (DCS hereafter), and using its dimensionless form (2p/s times the d 2 s/dW value; relative DCS hereafter), and the kj' and k'j' angular distributions were presented in terms of the probability density function [P(kj') and P(k'j'), respectively]. 26 Representative data of the kk', kj' and k'j' distributions are given in Figures 6-8, and we also analyzed the three-vector angular distribution corresponding to the kk'j' dihedral angle given in terms of the probability density function P(kk'j') 26 (Figure 9). Figure 6 shows the global kk' angular distribution for the ground and excited surfaces.
Reaction (1) takes place with some tendency towards backward scattering, this being more evident as collision energy increases. The 3 A'' PES has a lower tendency towards backward scattering than the 3 A' PES, but the relative DCS(kk') of both surfaces gradually tend to show a similar shape (backward preference) as collision energy increases. The differences observed between the 3 A'' and 3 A' surfaces can be interpreted on the basis of the absence and presence of a barrier on the minimum energy path, respectively. Moreover, The analysis of the opacity function [reaction probability (Pr(b)) vs. the initial impact parameter (b)] also helps to rationalize the behavior of the kk' angular distribution, in the case of reactions occurring through a direct reaction mode. Thus, the larger bmax values recorded for the ground PES in comparison with those for the excited PES suggest that the system will show a weaker trend towards backward scattering on the former surface, as it was really found in the calculations.
The kj' angular distributions for reaction (1) on the two PESs studied are symmetric around 90º, as they must be, due to the invariance of the distribution of the product molecule internuclear axis by reflection on the kk' plane. 26 At this angle, they exhibit a maximum and the behavior of P(kj') of both surfaces is similar (Figure 7). This trend towards a perpendicular arrangement of the k and j' vectors can be rationalized on the basis of the transformation of angular momentum vectors when evolving from reactants into products, and is fairly typical of a direct reaction.
For the 3 A'' PES and the j=8 and 12 rotational levels of NO, P(kj') evolves from a rather isotropic distribution at 0.0125 eV to a distribution with a maximum at 90º, which is already evident at 0.10 eV. 24 By the contrary, the evolution of the distribution for j=1 is not monotonic. The behavior of P(kj') is simpler for the 3 A' PES, the distribution becoming for all j values a bit wider as collision energy increases. For both surfaces, there is no tendency for j' to be aligned along k, and the condition j=1 shows a somewhat larger tendency than conditions j=8 and 12 to lead to products with j' aligned perpendicular to k.
The k'j' angular distributions of reaction (1) on the PESs studied correspond to a symmetric distribution around 90 o , as expected, due to the reflection symmetry in the scattering plane, 26 and are similar to the kj' distributions ( Figure 8). These results correlate with the dependence exhibited by the l'j' angular distribution, where l' is the orbital angular momentum vector of products, which, although it cannot be determined experimentally, is helpful in interpreting the k'j' distribution results. In fact, the trend towards parallel or anti-parallel l'j' orientation observed in the calculations leads to a tendency towards a perpendicular k'j' orientation. Two-and three-vector properties for reaction (1) for other (ET,v,j) initial conditions can be found in Ref. 24. On the overall, the results are similar to the ones described here.

E. Microscopic reaction mechanism
The reaction mode was analyzed for reactions (1) and (2)  In general, the results obtained for reaction (1) can be interpreted according to the H-H-H (heavy-heavy-heavy) kinematics of the system, and taking into account that the ground PES is barrierless, while the excited PES has a significant barrier. 4 This allows the system to evolve more rapidly into products through the first PES.
For reaction (2), the formation of collision complexes on the 3 A' PES and the absence of this reaction mode on the 3 A'' PES can also be understood if we consider the shapes of the two PESs. In fact, the NON( 3 A') minimum involved in the N-atom exchange process is energetically easier to reach from reactants (1.184 vs. 1.756 eV, along the corresponding MEP, for the 3 A' and 3 A" surfaces, respectively) and much deeper (0.585 vs. 0.0520 eV) than the NON( 3 A") minimum. 4

IV. Summary and conclusions
In this work we analyzed the kinetics of the N + NO → N2 + O reaction, using the QCT method and two ab initio analytical surfaces developed by our group for the ground (  reaction rates at very high temperatures (e.g., 5000-15000 K), which are necessary to model in a more accurate way hypersonic re-entry flows into the Earth's atmosphere. 28 This could be done for both N + NO → N2 + O and its reverse reaction, using the same PESs. Tables   Table I. QCT, VTST and RWP-J shifting rate constants for reaction (1) on the 3 A" and 3 A' PESs (in cm 3 s -1 ). 10 2    See Table III   Note: Figure 9 is suggested to be in color