Quantum dynamics of the N „ 4 S ... ¿ O 2 reaction on the X 2 A 8 and a 4 A 8 surfaces : Reaction probabilities , cross sections , rate constants , and product distributions

Quantum dynamics of the N „S...¿O2 reaction on the X 2A 8 and a 4A 8 surfaces: Reaction probabilities, cross sections, rate constants, and product distributions Paolo Defazio and Carlo Petrongolo Dipartimento di Chimica, Universita ` di Siena, Via A. Moro, 53100 Siena, Italy Carolina Oliva, Miguel González, and Ramón Sayós Departament de Quı ́mica Fisica i Centre de Recerca en Quı ́mica Teo `rica, Universitat de Barcelona, C/Martı́ i Franquès 1, 08028 Barcelona, Spain


I. INTRODUCTION
The reaction N( 4 S)ϩO 2 (X 3 ͚ g Ϫ )→NO(X 2 ͟)ϩO( 3 P) is involved in important atmospheric and combustion processes, and has been therefore the subject of many investigations. It is exoergic by 1.38 eV and NO products are thus formed with high vibrational and rotational quanta. The thermal rate constant has been measured between 298 and 5000 K, and the activation energy is equal to 0.28 eV. 1 Product vibrational distributions have been measured at room temperature 2 or at collision energy of about 3 eV. 3 The reactants and products correlate adiabatically to the lowest doublet X 2 AЈ and quartet a 4 AЈ electronic states of NOO in C s symmetry, neglecting spin-orbit couplings. The first theoretical work on this reaction dates back to 1987, when Walch and Jaffe 4 calculated C s ab initio potential energy surfaces ͑PES͒ of X 2 AЈ and a 4 AЈ. Later on, analytical PESs, reaction probabilities, cross sections, rate constants, and product distributions have been calculated employing different methods. Recently, we have begun an extensive theoretical study of the X 2 AЈ and a 4 AЈ surfaces, and of kinetic and dynamical properties of this reaction. We have thus obtained new ab initio PESs for describing C s abstraction and insertion mechanisms. 5 Insertions and conical intersections in C 2v symmetry were also investigated. 6 We have then calculated analytical fits to both surfaces and vibrational-transition-state-theory ͑VTST͒ rate constants. 7 Finally, we have carried out the first quantum-mechanical study of the reaction dynamics, 8 employing an earlier doublet PES ͑Ref. 9, labeled here S2͒ and a time-dependent wave packet ͑WP͒ approach. The height of the reaction barrier is 0.27 eV for the S2 surface of Ref. 9 or 0.30 eV for our new surface described in Ref. 7 ͑labeled here S3͒. However, the S3 PES improves considerably the description of the NO 2 ground electronic state, because the X 2 AЈ well depth and geometry are in good agreement with experimental data, contrary to S2 results. This feature is important in quasiclassical trajectory ͑QCT͒ calculations, since trajectories can enter into the NO 2 minimum region. 7,10 Moreover, a correct description of this region is even more important in WP calculations that depend on all details of the PESs, as we stressed in Ref. 8 and we shall show in the present work. Thus, a new WP investigation of the reaction dynamics on both doublet and quartet PESs is highly advisable, and we report here the results of such a study, employing the recent X 2 AЈ S3 surface by Sayós et al. 7 and the previous a 4 AЈ analytical PES by Duff et al., 11 whose barrier height is equal to 0.65 eV. Table I shows stationary point properties of the surfaces here employed.
Although we have recently calculated a new analytical quartet surface, 7 based on 910 ab initio points, 5 this PES has an artificial D ϱh barrier along the OϩNϩO product channel, at an OO distance of about 7.6 bohr and at about 23 eV above the NϩO 2 asymptote. This D ϱh barrier does not affect VTST calculations, 7 which depend mainly on the PES shape near the reaction barrier, or QCT results, 10 because very few trajectories sample this highly repulsive region of the surface. On the contrary, WP calculations of reaction probabilities are very sensitive to all features of the surface. Therefore, this artificial D ϱh barrier has a deep impact on WP results, reflecting back part of the WP and giving unphysical reaction probabilities at collision energies տ1.2 eV. In future work, we shall calculate more ab initio points and we shall fit better this repulsive region of the quartet PES.
We describe the theoretical method in Sec. II. Section III presents reaction probabilities and dynamics at total angular momentum quantum number Jϭ0. Section IV reports J-shifting cross sections and rate constants, and we discuss in Sec. V vibrorotational NO product distributions. We finally present our conclusions in Sec. VI.

II. METHODS
Gray and Balint-Kurti 12 ͑GBK͒ have recently developed an accurate and efficient time-dependent formalism for investigating the quantum dynamics in triatomic systems, merging the advantages of some previous approaches. Consider a reactive collision AϩBC(v, j)→AB(vЈ, jЈ)ϩC and define an initial WP, associated with an electronic state of ABC and with a vibrorotational level (v, j) of BC. The equation of motion is solved via a cos Ϫ1 mapping of a scaled Hamiltonian, with eigenvalues in ͑Ϫ1,1͒, resulting in a simple Chebyshev recursion. Only the real part of the WP is propagated, and the computational effort is thus reduced two times with respect to the propagation of a complex WP. Grid or Legendre representations of radial or angular nuclear coordinates are implemented via fast Fourier transforms or discrete variable representations, respectively. State-to-state reaction probabilities P v Ј j Ј ,v j J (E col ), at J and collision energies E col , are then obtained propagating the WP in product coordinates and performing an asymptotic analysis. 12 On the other hand, initial-state-resolved reaction probabilities P v j J (E col ) are calculated propagating the WP in reactant coordinates and performing a flux analysis. 13 The GBK method has been extended to nonadiabatic reactions 14 and to exact calculations at JϾ0. 15 We have employed the GBK approach for calculating exact reaction probabilities at Jϭ0, and we have estimated the probabilities at JϾ0 via the J-shifting ͑JS͒ approximation, 16 as implemented in Refs. 8 and 17 for bent transition states. We have used the following rotational constants of the transition states ͑TS͒: B ϭ0.31 and (A ϪB )ϭ1.94 cm Ϫ1 for X 2 AЈ, B ϭ0.34 and (A ϪB ) ϭ1.86 cm Ϫ1 for a 4 AЈ. Cross sections v Ј j Ј ,v j (E col ) and v j (E col ) and rate constants k v Ј j Ј ,v j (T), k v j (T), and k(T) at temperature T have been obtained through the usual expressions, including the electronic degeneracies 1/6 or 1/3 for X 2 AЈ or a 4 AЈ rates, respectively.
Taking into account the Pauli principle for O 2 (X 3 ͚ g Ϫ ), 18 we consider only odd j values up to jϭ11. Initial-state-resolved observables and thermal rate constants have been calculated for vϭ0 and 1, employing both X 2 AЈ and a 4 AЈ PESs and NϩO 2 Jacobi coordinates R, r, and ␥. State-to-state quantities of NO vibrorotational distributions at Tϭ300 K have been obtained for vϭ0, using the X 2 AЈ PES and NOϩO Jacobi coordinates RЈ, rЈ, and ␥Ј, and transforming the initial WP from the reactant to the product channel. Other details of the method are described in Refs. 8, 12, and 13, and Table II reports the parameters of the calculations for both reactant and product runs. The dimension of the reactant or product representation are equal to 3 240 650 or 2 429 520, respectively.

III. JÄ0 REACTION PROBABILITIES AND DYNAMICS
Because the NϩO 2 reaction is exoergic, with a barrier in the entrance channel, static or dynamical effects should dominate the mechanism at low or high E col , respectively. The analysis of the reaction probabilities at Jϭ0 gives detailed information on the dynamics, which confirm this expectation. Since the reactant-run representation of Table II and that of Ref. 8 give very similar probabilities up to E col Ϸ0.7 eV, we can contrast the present S3 results with the earlier S2 ones at low E col . The effects of the potential are more remarkable in this low-energy regime, whereas at higher energies the results depend more on the O 2 initial state and on the energy distribution between the translational and internal degrees of freedom.
As an example, Fig. 1͑a͒ shows P 0,9 0 of O 2 (0,9), the most populated level at room T, for both doublet surfaces. The S3 probabilities are larger than the S2 ones up to about 0.35 eV and the reaction has thus lower energy thresholds on the S3 PES, whereas the S2 surface is more reactive between 0.4 and 0.7 eV. Since the S3 barrier height is 0.03 eV greater than the S2 one, the enhancement of the reactivity near the threshold is a somewhat surprising result, which shows that the detailed dynamics depends on other features of the PESs that compensate the effect of the barrier height.
First note that the S3 O 2 and NO vibrorotational energies are respectively higher and smaller than the S2 energies. This implies that the reactant channel is narrower and steeper on the S3 PES than on the S2 surface, whereas the opposite is true for the product arrangement. At low E col , the S3 wave packet is thus more localized along the entrance minimum energy path ͑MEP͒ and more flux can enter into the exit channel. Both these behaviors increase the reactivity in the low energy regime.
The shape of the surfaces near their TSs is a second compensating effect. Table I shows indeed that the OO stretching frequency is smaller on S3, implying that the OO bond breaks more easily on this surface, thus increasing the S3 reactivity at low E col . Moreover the larger S3 imaginary frequency hints at a narrower barrier, and thus at a faster breaking rate of the TS and at a larger tunnel effect. By comparing indeed the static thresholds with the dynamical ones, we find an S3 tunneling at vϭ0 and for any j value, whereas the S2 tunneling occurs only at (v, j)ϭ(0,11).
Finally, we have already shown 8 that the NO 2 minimum region of the doublet PES can trap part of the wave packet. The intermediate complexes will give either the reactants or the products, thus lowering the reactivity with respect to a direct collision. Because the NO 2 well depth is equal to 7.96 or 4.71 eV in the S2 or S3 PES, respectively, this capture mechanism is less important for the S3 surface that is then more reactive.
These different features of the doublet PESs in the reactant and product channels, at the TS, and in the minimum region thus increase the S3 reaction probabilities at low E col and have a deep impact on the rate constant at low temperature, as we shall see in Sec. IV. Figure 1͑b͒ shows the O 2 (v, j) effects on the S3 dynamics, by comparing the reaction probabilities associated with the O 2 ground state ͑0,1͒ and with the most excited state ͑1,11͒ here considered. The O 2 initial excitation enhances the probability near the threshold, but slightly inhibits the reactivity at high energy. These (v, j) effects at low E col can be understood looking at the effective potentials,

͑1͒
which influence the initial wave-packet dynamics. 19 In Eq.
and R is the reduced mass associated with R. Figure 2 shows that V 0,1 and V 1,11 have a barrier at Rϭ4 bohr and a minimum at RϷ3.3 bohr, and that the V 1,11 barrier is lower than the V 0,1 one. This finding thus explains why the lowenergy reaction probabilities increase with the O 2 quanta, and implies that (v, j) effects reflect the shape of V v j (R). Figure 1͑b͒ reports also an example of a reaction probability on the a 4 AЈ surface. The quartet threshold is about 0.35 eV greater than the doublet threshold, as expected from the different barrier heights. The 4 AЈ probabilities rise steeper with E col than the 2 AЈ ones, and the quartet reactivity is thus larger than the doublet one above 1.4 eV.

IV. CROSS SECTIONS AND RATE CONSTANTS
The results of this section have been obtained using the S3 X 2 AЈ surface 7 and the a 2 AЈ surface of Ref. 11. When O 2 is in the ground level ͑0,1͒, the collision energy threshold attains the highest value, equal to 0.19 eV. The threshold is reduced at high O 2 quanta, up to the minimum value of 0.05 eV for (v, j)ϭ (1,11), which is the highest O 2 level here considered. We have estimated the reaction threshold at various temperatures, averaging the initial-state-resolved thresh-olds over a Boltzmann distribution of the O 2 initial levels. This average threshold is equal to 0.16 eV at 300 K, close to the value of 0.15 eV for the most populated O 2 (0,9) level at this temperature, and it is reduced to 0.13 eV at 3000 K. Figure 3͑a͒ shows the doublet WP and QCT cross sections for O 2 initial levels (v, j)ϭ(0,1) and ͑1,11͒. The trend of v j (E col ) with respect to E col , v, and j is similar to that of the probabilities. High O 2 vibrorotational quanta thus enhance or lower S3 v j (E col ) at low or high E col , respectively. QCT and WP cross sections on the new doublet PES are in close agreement, and also agree quite well with previous semiclassical results on other surfaces. 20,21 Figure 3͑b͒ presents the cross sections for O 2 (0,1), considering the electronic degeneracy factors 1/6 and 1/3 for X 2 AЈ and a 4 AЈ, respectively. We see that the doublet 0,1 is larger than the quartet one up to E col Ϸ1.2 eV, whereas the opposite holds at higher energies in agreement with the probability results of the previous section and with QCT calculations. 10 The variation of the degeneracy-averaged cross sections vs the O 2 initial excitation is reported in Fig. 4 at two collision energies: 0.5 eV, where only the X 2 AЈ channel is open, and 1.5 eV, where the a 4 AЈ channel dominates. The trend of the cross sections with respect to v and j confirms that of the reaction probabilities, and shows that v j oscillate with j and increase with v. At both collision energies, one O 2 vibrational quantum enhances the cross sections, whereas the most reactive O 2 rotational level depends on the collision energy. Figure 4 shows indeed that 1,j is maximum at j ϭ11 or 5 at E col ϭ0.5 or 1.5 eV, respectively.
We have calculated initial-state-resolved rate constants k v j (T) for both doublet and quartet PESs as Boltzmann averages over the collision energy and the cross section, includ- ing the electronic degeneracies, 22͑a͒ and we show in Fig. 5 these rates at Tϭ300 and 2000 K. At room temperature, k v j (T) increase nearly exponentially with j, and are enhanced more than two orders of magnitude increasing (v, j) from ͑0,1͒ to ͑1,11͒. The O 2 rotational excitation is about three times more effective than the vibrational one in increasing k v j . Of course, these (v, j) effects are strongly reduced at 2000 K, where the rates are scarcely influenced by the O 2 initial level. Thermal rate constants k(T) have been obtained as Boltzmann averages over the k v j (T) rates. To this end, we have explicit calculated k v j (vϭ2 and 3, jϭ11) for both electronic states, we have estimated k v j (vϭ2 and 3, j Ͻ11) with linear extrapolations as k 1,j (k v,11 /k 1,11 ), and we have assumed that k v j ( jϾ11)Ϸk v, 11 . Figure 5 shows that this procedure should underestimate the room-T rate constant, because k v j increases with j, but that it should be correct at high T where k v j do not vary with j. Table III and Fig. 6 show that the WP thermal rate constants are about two times smaller than the experimental ones 1 at low T, whereas the WP results are within the experi-mental error bars at TϾ1000 K. This finding can be due to an overestimation of the barrier heights of both PESs. However, the ab initio 5 value of the X 2 AЈ S3 barrier height has been increased by 0.09 eV in the analytical PES, 7 for obtaining the best agreement between VTST and experimental data at 300 K. This VTST barrier scaling seems too large for the WP k(T), because a numerical experiment has shown that a lowering of the v j thresholds by only 0.0245 eV yields WP and observed rates is nearly full agreement.  We have seen in Sec. III that the energy thresholds are smaller and the reaction probabilities at low E col are larger on the present X 2 AЈ S3 PES than on the earlier S2 surface. 8 Therefore, the S3 thermal rate constants at low T are considerably larger than the S2 ones, thus improving the agreement between WP and observed rates. S3 ͑S2͒ k͑300͒ and k͑600͒ are indeed equal to 3.52ϫ10 Ϫ17 (9.5ϫ10 Ϫ18 ) and 1.54 ϫ10 Ϫ14 (8.9ϫ10 Ϫ15 ) cm 3 s Ϫ1 , respectively, i.e., the S3 k are larger than the S2 ones by ϳ3.7 or 1.7 times at 300 or 600 K, respectively. As we have discussed in Sec. III, these S3 results are due to the PES shape in the reactant and product channels, to the TS vibrational frequencies, to a large tunneling through the barrier, and to a smaller NO 2 minimum. Moreover, the smaller OO stretching frequency of the S3 TS, with respect to the S2 one, implies that the number of available states at the TS is larger in the S3 surface. In statistical terms, this corresponds to a larger S3 partition function and therefore to a larger k.
At TϽ1000 K, S3 VTST rate constants 7 are lower than the S2 ones, in agreement with the higher S3 barrier but contrary to the present WP results. Because statistical results do not take into account quantum effects and depend mainly on the barrier region of the PES, this comparison highlights important quantum effects near the S3 reaction threshold and the role of the overall surface.

V. PRODUCT DISTRIBUTIONS
In this section we present vibrorotational distributions of NO(vЈ, jЈ) products, at E col р1.2 eV and at room temperature. These calculations were carried out by propagating wave packets in product coordinates, and employing the X 2 AЈ S3 PES, six O 2 levels with vϭ0 and jр11, and the product-run representation of Table II. Initial-state-resolved probabilities, cross sections and rate constants are very similar in both propagations, as Fig. 7 shows for 0,1 . The rate constant k 0,1 (300) is equal to 3.53ϫ10 Ϫ18 or 3.42ϫ10 Ϫ18 cm 3 s Ϫ1 , and the thermal rate k͑300͒ is equal to 3.50 ϫ10 Ϫ17 or 3.81ϫ10 Ϫ17 cm 3 s Ϫ1 in the reactant or product run, respectively. This level of agreement is actually pretty good, considering that very different coordinates and dimensions are used in the two runs.
We have compared the calculated and experimental 2 vibrational distributions of NO(vЈ) at 300 K. Recently, Caledonia et al. 3 measured a vibrational distribution at E col Ϸ3 eV. The WP calculation of this distribution at 3 eV would require product propagations on both doublet and quartet PESs and vϭ0 and 1, doubling the reactant run representation of Table II, and this is beyond our present computer capabilities. Figure 8 shows the vibrational distribution of NO(vЈ) for reactants at Tϭ300 K, where k v Ј (T) are the rate constants for the production of NO(vЈ). These rates have been calculated from the state-tostate k v Ј j Ј ,0j (T) for NϩO 2 (0,j)→NO(vЈ, jЈ)ϩO, averaging over a Boltzmann distribution at T of the j levels and summing over the jЈ levels. The quantum-mechanical distribution is inverted, peaking at v max Ј ϭ3, and is qualitatively similar to the semi-classical distributions that are also inverted, with v max Ј ϭ1, 10  We report in Fig. 9 the rotational distributions ͗ v Ј j Ј (E col ,T)͘ of NO for reactants at Tϭ300 K, for vЈ ϭ1 in panel ͑a͒, and for vЈϭ3 and 5 in panel ͑b͒, where the cross-section scales are different in the two panels. The collision energies are equal to 0.8 and 1.2 eV above and to 1.2 eV below. These results have been obtained from state-tostate cross sections v Ј j Ј ,0j (E col ), averaging over a Boltzmann distribution of the O 2 (0,j) levels. Even more than the vibrational distribution, the rotational populations depend on dynamical effects, as the opening and closing of many NO(vЈ, jЈ) channels and the energy redistribution among the product degrees of freedom. The rotational distributions are indeed strongly oscillating, different from the prior rotational distributions, 22͑b͒ and high NO rotational quanta are in general preferred. At large E col , the collision is highly selective with respect to jЈ, because only a few NO rotational levels are preferentially populated. This last finding confirms that the high-energy dynamics is predominantly direct. The distribution at vЈϭ1 and E col ϭ0.8 is maximum at j max Ј ϭ42 and 71, and the distribution at E col ϭ1.2 eV is strongly peaked at jЈϭ62. Increasing vЈ from 1 to 3, the distribution at E col ϭ1.2 is still oscillating, but the favored jЈ levels are now equal to 1 ͑strongly preferred͒, 6, and 62. Figures 8 and 9͑a͒ show that the NϩO 2 collision at room T and E col ϭ1.2 eV gives preferentially NO products with vЈϭ3 and jЈϭ1. However, the rotational distributions depend strongly on vЈ, as Fig. 9 shows for E col ϭ1.2 eV and vЈϭ1, 3, and 5. The calculated product distributions are consistent with the usual rules of an exoergic abstraction reaction, which is dynamically controlled and whose barrier is in the entrance channel. 22͑c͒ The inverted vibrational population of NO(vЈ) is also consistent with the stretched NO bond of the TS. 7 These results can be explained looking at the expectation values of the reactant radial coordinates ͗R(t)͘ and ͗r(t)͘ and at the reactant average angle ␥ (t) at time t ͑see Ref. 8 for more details͒. For NϩO 2 (0,1) and the S3 PES, Fig. 10 reports two nuclear snapshots at tϷ262 and 664 fs, and shows that the N atom approaches O 2 with a large impact parameter b, forming the NO bond with the first O atom whereas the second oxygen goes away. This large b favors high vЈ and jЈ quanta of the NO products, whose vibrational distribution is thus inverted and the rotational one is oscillating with high jЈ levels populated at any vЈ. From the comparison with the prior statistical distributions, we conclude that the detailed reaction dynamics is highly not statistical. Figure 8 shows that the calculated WP vibrational distribution of NO(vЈ) is in satisfactory agreement with that observed 2 at room T, for vЈϭ1, 2, and у4. Our population at 300 K compares also rather well with that measured at E col Ϸ3 eV, 3 except at vЈϭ3. Both experimental distributions are however oscillating and even vЈ levels are preferred, in contrast to all the theoretical results. Observed and calculated populations are mainly different at vЈϭ0 and 3, but we have not been able to reproduce the experiments, despite many long calculations. Probably, the X 2 AЈ S3 PES and the J-shifting approximation are not accurate enough to obtain a subtle quantity as the product distribution in this reaction that involves three heavy nuclei. However, Winkler et al. 2 have shown that the NO vibrational quenching in NOϩO 2 is at least 150 times faster than the NO formation in NϩO 2 , implying that this quenching can bias the results at room T of Ref.
2. This effect should indeed give a vЈϭ0 experimental population much larger than that calculated. On the other hand, Caledonia et al. 3 rule out the NO quenching at 3 eV, and therefore further theoretical and experimental works seem necessary for checking the NO(vЈ) distribution.

VI. CONCLUSIONS
This paper reports a wave packet study of several dynamical and kinetic properties of the reaction N( 4 S) ϩO 2 (X 3 ͚ g Ϫ )→NO(X 2 ͟)ϩO( 3 P) on the X 2 AЈ and a 4 AЈ PESs. We have employed the real wave packet formalism by Gray and Balint-Kurti, 12 who obtain the scattering S matrix and total probabilities solving a modified time dependent Schrödinger equation and performing asymptotic 12 or flux analyses. 13 We have propagated up to twelve initial wave packets corresponding to O 2 (v, j) vibrorotational levels with vр1 and jр11. We have employed reactant propagations and the flux method for calculating initial-stateresolved reaction probabilities, cross sections, and rate constants. State-to-state dynamical observables and product vibrorotational distributions have been obtained through product propagations and the asymptotic analysis. Large grid and Legendre representations have been used, up to 3 240 650 dimensions. We have calculated exact or J-shifting probabilities at Jϭ0 or JϾ0, respectively, and we have discussed some elementary dynamical processes by analyzing probabilities and product distributions.
We have employed the recent S3 doublet PES by Sayós et al., 7 which agrees well with all the experimental information on the stationary points of the NO 2 ground surface. This PES thus improves notably the previous surfaces of Refs. 9 and 11, which do not describe correctly the NO 2 X 2 A 1 equilibrium region. The correct S3 shape is very important in WP calculations that are sensitive to all the features of the surface. On the other hand, we have employed the earlier quartet PES by Duff et al., 11 because an artificial barrier of the new quartet PES of Ref. 7 prevents us from using the WP approach, although this repulsive region of the PES was not important in VTST ͑Ref. 7͒ or QCT ͑Ref. 10͒ studies.
The collision proceeds on the S3 surface with a lower energy threshold with respect to the S2 threshold, so that the S3 surface is more reactive despite its higher barrier. We have ascribed this result to three features of the S3 PES. ͑i͒ The shape of the reactant and product channels, which increases the localization of the WP along the entrance MEP and the flux through the exit surface. ͑ii͒ The frequency values and the narrower barrier of the S3 TS, which favor the breaking of the O 2 diatom and of the TS, and increase tunnel effects. ͑iii͒ The lower S3 hole of the NO 2 bound geometry that captures less WP with respect to that of the S2 surface, thus increasing the reactivity.
We have discussed the effects of the O 2 initial excitation in terms of the effective potential felt by the WP along its motion from the reactant channel into the interaction region. The WP cross sections are close to the QCT ͑Ref. 10͒ ones; they are enhanced by one O 2 vibrational quantum and oscillate somewhat with j. The present study improves considerably the agreement between calculated and observed rate constants at low temperatures, with respect to previous WP calculations. 8 Moreover, the accord becomes nearly quantitative provided the S3 analytical barrier height is lowered by 0.0245 eV. WP calculations also show that S3 quantum tunnel effects are very important at room temperature and are remarkable up to about 600 K.
The WP vibrational distribution of the NO(vЈ) products is inverted at 300 K, and NO is preferentially formed with vЈϭ3. The product rotational populations are strongly oscillating. These nonstatistical results agree qualitatively with previous QCT calculations. 10,11,23 We have shown that these product distributions reflect a reaction mechanism where the N atom approaches the interaction region with a large impact parameter. However, the WP and QCT inverted vibrational distributions are different from those observed, which are oscillating and prefer even vЈ quanta.
In closing, we stress that the WP formalism depends strongly on all the features of the PESs, so that it is a very useful approach for probing potential surfaces. We have shown that the recent S3 PES ͑Ref. 7͒ of the X 2 AЈ state behaves correctly nearly everywhere, whereas the a 4 AЈ PES needs some refinements. In comparing WP and observed thermal rate constants, we have found that the S3 barrier height should be probably lowered by ϳ0.0245 eV. Since this value is equal to 0.56 kcal/mol, this PES has reached a chemical accuracy within 1 kcal/mol. We are currently improving the quartet surface of Ref. 7, calculating more ab initio points in the OϩNϩO product channel, and carrying out an improved analytical fit of this repulsive region of the PES.

ACKNOWLEDGMENTS
One of the authors ͑C.P.͒ thanks Dr. S. Gray for a copy of his wave-packet codes and for many useful discussions.