A New ab Initio Potential Energy Surface for Studying Vibrational

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A New ab Initio Potential Energy Surface for Studying Vibrational Relaxation in NO(v) þ NO Collisions Pedro Pajon-Suarez,†,‡ Jesus Rubayo-Soneira,† and Ramon Hernandez-Lamoneda*,‡ † ‡

Instituto Superior de Tecnologías y Ciencias Aplicadas, Av. Salvador Allende y Luaces, Quinta de Los Molinos, La Habana 10600, Cuba Centro de Investigaciones Químicas, Universidad Autonoma del Estado de Morelos, Av. Universidad 1001, Cuernavaca, Morelos., 62210, Mexico ABSTRACT: A new ab initio potential energy surface for the ground state of the NO-NO system has been calculated within a reduced dimensionality model. We find an unusually large vibrational dependence of the interaction potential which explains previous spectroscopic observations. The potential can be used to model vibrational energy transfer, and here we perform quantum scattering calculations of the vibrational relaxation of NO(v). We show that the vibrational relaxation for v = 1 is 4 orders of magnitude larger than that for the related O2(v) þ O2 system without having to invoke nonadiabatic mechanisms as had been suggested in the past. For highly vibrationally excited states, we predict a strong dependence of the rates on the vibrational quantum number as has been observed experimentally, although there remain important quantitative differences. The importance of a chemically bound isomer on the relaxation mechanism is analyzed, and we conclude it does not play a role for the values of v considered in the experiment. Finally, the intriguing negative temperature dependence of the vibrational relaxation rate constants observed in experiments was studied using an statistical model to include the presence of many asymptotically degenerate spin-orbit states.

I. INTRODUCTION The properties of the NO molecule have been and continue to be of interest from the point of view both of fundamental studies and for applications which range from atmospheric science to the biological sciences. In particular the nature of the chemical bond in the NO-NO dimer and the precise determination of its spectroscopic properties continue to challenge both experiment and theory.1-10 NO is an open-shell species, and this suggests that exchange interactions significantly different from those present in closed-shell dimers will lead to a stronger binding than that associated with typical van der Waals systems (compared to the N2 dimer it is roughly an order of magnitude larger). In this regard it is important to point out that the accurate experimental determination of its binding energy was performed relatively recently10 and much less has been accomplished regarding the precise determination of its van der Waals bound levels,1,11-13 indicating the difficulty inherent in dealing with open-shell systems. From the theoretical side, even the best calculations have been unable to reproduce with high accuracy its equilibrium properties,6,7 the dissociation energy being particularly difficult.9 For example the CCSD(T)/AVQZ estimate of the binding energy yields only 36% of the experimental value.9 As far as we know all previous theoretical efforts have concentrated on the properties of the equilibrium structures and there have been no calculations of the potential energy surface and therefore no studies on the bound states. From the point of view of collision dynamics the NO-NO system also shows some unique properties: the vibrational relaxation is several orders of magnitude r 2011 American Chemical Society

larger than those for similar dimer systems such as CO, N2, and even the open-shell O2.14-17 Also it has been known for many years that the temperature dependence of the rate for ::v = 1 has a minimum around 300 K and therefore a negative temperature dependence at lower temperatures.14,15,18,19 More recently Smith et al. has performed measurements at temperatures down to 7 K for v = 3 and found a strong negative temperature dependence.20 Furthermore a surprising result shown by the low temperature data is the fact that vibrational to translational (VT) relaxation is faster than the expected near-resonant vibration to vibration (VV) mechanism. The highly vibrationally excited states of NO were studied by the Wodtke group using the stimulated emission pumping technique.21,22 As the vibrational energy of the initial state is increased, the relaxation rates show different regions of enhancement with a pronounced increase for the higher vibrational levels.16 On the basis of ab initio calculations that showed the presence of several high energy chemically bound dimers,23 it was speculated that the different regions of vibrational enhancement could be related to them, but no dynamics calculations testing this hypothesis have been performed. Another possibility to explain the vibrational enhancement could be the sampling of the transition state region for the reaction NO(v) þ NO(v = 0) f N2O þ O, a channel that becomes open for v g 15. Again, no experimental or theoretical Received: January 7, 2011 Revised: February 12, 2011 Published: March 16, 2011 2892

dx.doi.org/10.1021/jp200199y | J. Phys. Chem. A 2011, 115, 2892–2899

The Journal of Physical Chemistry A calculations supporting this hypothesis have been performed. On the other hand for the case of the related system O2(v) þ O2 (v = 0), which was also studied by the Wodtke group,24 an analogous vibrational enhancement of the relaxation rates was observed for v g 25 and reduced dimensionality quantum scattering calculations25 showed that indeed the transition state region of the reaction O2(v) þ O2 (v = 0) f O3 þ O leads to enhancements of the relaxation rates. A more complete picture of the relaxation paths for the oxygen system was acquired recently, in reduced dimensionality quantum scattering studies26,27 which showed the relevance of vibrational to electronic energy transfer mechanisms mediated by spin-orbit coupling, showing the complexity and richness of relevant relaxation channels for these small but challenging systems. The purpose of the present work is to present a reduced dimensionality potential energy surface for the NO-NO system which can be used to model vibrational energy transfer processes. We also perform quantum scattering calculations of the VT mechanism which represent the first ab initio based approach to date for the collision dynamics and shed light on the intriguing properties of this system and on previous hypothetical explanations. Still our work has to be taken only as a first step in explaining the rich variety of experimental findings for this system.

II. METHODOLOGY

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Figure 1. Geometry and active coordinates. The angular coordinates correspond to the minimum of the van der Waals complex. The inactive NO is fixed at its equilibrium distance.

equilibrium values for the van der Waals complex and were taken from ref 1. For the active degrees of freedom, we choose the usual Jacobi coordinates (R,r): where R is the center of mass intermolecular distance and r is the intramolecular distance in the active NO, the other molecule being fixed at its equilibrium value r0 = 1.151 Å. The final grid uses 53 points for R in the interval [1.0,10.0] Å and 22 points for r in the interval [0.9,2.5] Å. With this density of points the cubic-spline fit leads to a smooth potential and dynamical results are well converged. B. Dynamical Calculations. We briefly review the methodology already applied in the case of the O2 dimer.36 The Hamiltonian in our reduced dimensionality model is 2 2 2 2 ^ ¼ - p D - p D þ V ðR, rÞ H 2μR DR 2 2μr Dr 2

A. Ab Initio Potential Energy Surface. The calculation of the

interaction potential via ab initio methods for the NO-NO system is a challenging task even when models of reduced dimensionality are applied. In previous paragraphs we have documented the difficulty of obtaining an accurate description of the van der Waals complex and clearly further complications arise when describing highly stretched geometries and regions where the interaction goes from van der Waals to chemical forces as are necessary to describe highly vibrationally excited states as well as the chemically bound dimers. Additionally the NO molecule has a degenerate open-shell 2π electronic ground state, leading to eight electronic states (four singlets and four triplets) for the interaction of two NO molecules in their ground states (for a previous ab initio study on this issue see ref 8). In order to tackle all the complexities involved, it is necessary to use multiconfigurational methods. From experience with the related O2 dimer system, we know that the CASPT2 method28,29 compares favorably with the more computationally demanding MRCIþQ or MRACPF methods,30,31 especially in properly describing the dispersion forces,32 and has been our choice. We calculate the ground state of the dimer which has 1A0 symmetry in the Cs point group. The reference function keeps the 1s and 2s orbitals doubly occupied and distributes the remaining 14 electrons in the full valence space of 12 orbitals leading to 85020 configuration state functions. Such a large active space is required to describe the highly vibrationally excited states and the transition from a weakly bound to a chemically bound complex. We use the 5s4p3d2f atomic natural orbital basis set and include the basis set superposition correction to the interaction potential using the counterpoise method.33,34 All calculations were peformed with the MOLCAS suite of programs.35 We found convenient to apply a level shift of 0.1 au throughout in order to avoid convergence problems and ensure a smooth potential energy surface. The geometry and active coordinates are displayed in Figure 1. The angular degrees of freedom have been fixed to their

where the masses involved are mN mO μr ¼ and mN þ mO

ð1Þ

mN þ mO 2

μR ¼

with mN and mO the atomic masses of the most abundant isotopes of nitrogen and oxygen respectively and V(R,r) the computed ab initio PES. The total wave function ψ(R, r) solution of the corresponding Schr€odinger equation (SE) is expanded in the vibrational eigenstates of the NO molecule ψðR, rÞ ¼ where

∑v jv vðRÞχv ðrÞ 0

0

0

! p 2 D2 þ V ðrÞ χv0 ðrÞ ¼ εv0 χv0 ðrÞ 2μr Dr 2

ð2Þ

ð3Þ

is the diatomic Schr€odinger equation and V(r) the diatomic potential. Substituting (2) in (1) leads to the coupled-channel equations -

p2 D2 j 0 v ðRÞ þ ½Vv0 v0 ðRÞ þ εv0 - Ejv0 v ðRÞ 2μR DR 2 v ¼

∑ Vvv ðRÞjv v ðRÞ

v6¼ v0

0

0

ð4Þ

where the coupling matrix elements are defined as: Vvv0 ðRÞ ¼ Æχv ðrÞjV ðR, rÞ - V ðrÞjχv0 ðrÞæ

ð5Þ

and Vint (R,r) = V(R,r) - V(r) is the interaction potential driving the dynamics.The scattering matrix, S, is obtained by applying the 2893

dx.doi.org/10.1021/jp200199y |J. Phys. Chem. A 2011, 115, 2892–2899

The Journal of Physical Chemistry A

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Figure 2. Intermolecular potentials for the NO-NO and O2-O2 systems. The monomers are at their equilibrium intramolecular distances.

asymptotic boundary conditions lim jv0 ðRÞ ¼ δvv0 l v

-ikv R

Rf¥

with

kv Svv0 l ikv R kv0

ð6Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2μR ðE - εv Þ kv ¼ p2

We are mainly interested in the total inelastic probabilities for a given vibrational state Pv ine ðRÞ ¼

∑ Pv f v ðRÞ

v6¼ v0

0

2

where Pvfv0 (R) = |Svv0 | are the state-to-state probabilities. In order to make qualitative comparisons with rate constant experimental data, we use the Boltzmann averaged transition probabilities defined as Z ¥ 1 l -Ec =kB T Pν ine ðEc Þ dEc Ivine ðTÞ ¼ ð7Þ kB T 0 where Ec = E - εv is the collision energy. The R-matrix method has been applied to solve the coupledchannel equations.37 In order to ensure convergence of the dynamical properties, the scattering coordinate was divided into 4000 sectors with an initial value of 2.0 bohr and final value of 80.0 bohr. After several tests with the vibrational basis set, we have included for a given vibrational state, v, eight lower and eight higher channels. The Boltzmann-averaged probabilities should be converged within 5%.

III. RESULTS AND DISCUSSION It is instructive to compare the properties of the NO dimer with those of the O2 dimer, another open-shell dimer. In Figure 2 we show the intermolecular potential for both species where the monomers are at their equilibrium intramolecular distances. Both potentials have been calculated with the CASPT2 method and similar basis sets, the results for oxygen come from ref 36 and those of NO from the current work. The binding energy for NO-NO is more than five times larger than that of O2-O2, and

Figure 3. Vibrational dependence of the interaction potential Vint(R,r) = V(R,r) - V(r): (a) region of the van der Waals complex; (b) region of the chemically bound complex.

the intermolecular equilibrium distance is significantly shortened, as expected. Comparing our results for the electronic binding energy (De) and the N-N interatomic distance with the most reliable experimental information to date10,1 we obtain De = 93 meV, RN-N = 2.40 Å vs De = 124 meV, RN-N = 2.26 Å, respectively. It is important to clarify how we estimate the experimental De: we use the experimental value of D0 obtained in ref 10 and the zero-point energy correction from ref 7 as suggested by a previous theoretical study.9 Our calculations recover roughly 75% of the binding energy and predict a slightly longer equilibrium N-N interatomic distance. As has been discussed in previous works9 the quantitative calculation of the binding energy is a challenge for ab initio methods. For the purposes of the present work, it is more important to use a method which is in practice almost size-consistent38 and which can properly describe the stretching of the NO molecule up to its dissociation yielding a smooth PES, even if it lacks high accuracy for the binding energy of the weakly bound complex. The vibrational dependence of the interaction potential, Vint, crucial in this study, is shown in Figure 3. In the first figure we show the region of intermolecular distances relevant for the van der Waals complex. The results are quite surprising: there is a very strong dependence of the intermolecular potential on the degree of stretching of the NO molecule, for example for r(NO) = 1.75 Å 2894



Figure 4. State-selected total inelastic probabilities as a function of the total energy. The arrows show the opening of the indicated vibrational channels.

which corresponds to the distance for the last maximum of the vibrational wave function for v = 25, the binding is roughly four times larger than for the unstretched molecule and the intermolecular distance shrinks by 0.2 Å. For typical van der Waals systems, such as those of halogens with noble gases or even NO with noble gases the vibrational dependence of the interaction is orders of magnitude smaller39,40 and even the more analogous case of the oxygen dimer, though it shows a qualitatively similar dependence,41 is much weaker. Evidence for such a strong vibrational dependence was observed many years ago in an infrared spectroscopical study of the NO dimer42 where the normal mode associated with antisymmetric stretching (ν4) was excited and both a significant red shift and a shrinking of the intermolecular distance relative to the ground state were observed. Our study gives support to the conclusion drawn in that work regarding a deeper van der Waals well for the effective potential of the vibrationally excited ν4 as well as a shortening of the N-N bond.42 In Figure 3b we concentrate on much shorter intermolecular distances and therefore significantly higher energies where a second minimum appears corresponding to a chemically bound species as shown both by the short N-N distances and by stronger binding energies. We recall that such chemically bound species had been studied before,23 and given our fixed angular geometry it is clear our PES is representing the isomer 2 of C2v symmetry in that study, which is the lowest in energy. In order to reach such structure, it is necessary to surmount an energy barrier whose height and position are strongly dependent on the stretching of the NO molecule as shown in the figure. At a first glance this strongly suggests a possible role of the complex on the vibrational relaxation dynamics as postulated in previous works;22,23 we defer that discussion for the dynamics results below. The state-selected probabilities for vibrational relaxation are shown in Figure 4. In general it can be seen that the probabilities tend to increase both with the vibrational quantum number and with the collision energy, as expected. On the other hand the behavior at low collision energies is more complex and not only is the dependence with collision energy not monotonous but for the lower vibrational levels they present sharp oscillations that can be of 1 or 2 orders of magnitude. A similar behavior has been

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Figure 5. Boltzmann-averaged probabilities (Iv) as a function of temperature for the NO-NO and O2-O2 systems. Notice the presence of a minimum in the NO case.

observed in the case of the oxygen dimer, and it can be traced to the strongly oscillatory nature of the scattering wave functions and the corresponding interference effects which depend strongly on the collision energy.36 Analyzing the partial contributions to the total vibrational relaxation probabilities, it is seen that the dominant mechanism is the single quantum deactivation and single quantum excitation(v f v þ 1) for collision energies where it opens. Still for some of the lower vibrational states studied there are collision energies for which two quantum deactivation can become important. One of the unique and debated properties of NO is its extremely fast vibrational relaxation compared with similar systems such as N2 or CO, for example. Due to its open-shell nature, it has even been speculated that nonadiabatic processes could be responsible for the unusual behavior.17 In Figure 5 we compare the Boltzmann averaged transition probabilities (eq 7) for NO and O2 in v = 1 as a function of temperature. The methodology applied in both cases is very similar: CASPT2 to obtain the PES and exactly the same dynamical treatment. The present model predicts a 4 orders of magnitude faster Iv (T) for NO than for O2. This relative difference is in agreement with the experimental estimates15,19,43 of the rate constants. Although a more quantitative treatment is required to completely resolve this issue, we believe the high vibrational relaxation rate of NO can be explained on the basis of a simple vibrational relaxation mechanism without having to invoke any other special mechanisms. In this regard we coincide with a qualitative analysis presented by Yang and Wodtke16,22 on the basis of the Schwartz-Slawsky-Herzfeld theory where the deeper well depth of the NO dimer implies a significantly steeper repulsive potential which induces a much higher vibrational relaxation. Another unique property of the vibrational self-relaxation of NO(v = 1) is the presence of a minimum in the rate constant as a function of temperature and therefore a negative temperature dependence for lower temperatures.14,15,18,19 From Figure 5 we can see that compared with the oxygen case, the temperature dependence of the NO self-relaxation has a convex shape with a weak negative dependence below 180 K whereas for oxygen it has a concave shape and shows the usual positive dependence. This observation is in agreement with the mechanisms proposed by Smith and co-workers20,44 regarding the role played by the 2895



Figure 6. Intermolecular potentials for the lowest electronic states correlating with NO þ NO in their ground states. The A00 states are similar to the ground state of O2-O2 shown in Figure2.

formation of collision complexes in vibrational relaxation, where a more strongly bound complex can be expected to lead to a negative temperature dependence of the rate constant. On the other hand our calculations lead to a much lower value for the minimum temperature, T = 180 K, whereas the experimental value19 is T = 300 K. Within our reduced dimensionality treatment there is another mechanism that can lead to negative temperature dependence that can be explored. In the case of open-shell species, the fact that several electronic states converge to the same asymptotic limit (different multiplets when spin-orbit is included) implies that more than one PES can be important to describe the collision dynamics.45 In Figure 6 we show the intermolecular potentials for the lowest electronic states that correlate with two NO in their ground states; they have been calculated at the same level of theory as the ground state keeping both NO molecules at their equilibrium distance. It can be seen that the 1A00 and 3A00 surfaces are similar and much less weakly bound than the 1A0 ground state with well depths of 24 and 20 meV and the same intermolecular equilibrium distance of 3.5 Å. In fact they are quite similar to the ground state of the oxygen dimer which has a well depth of 17 meV and intermolecular distance of 3.2 Å depicted in Figure 2. The 3A0 state behaves as an antibonding partner for the ground state and is more weakly bound with a well depth of 15 meV and distance of 4.0 Å. We apply a statistical approximation to estimate the role of fine structure effects on the dynamics as described in previous works.45,46 Briefly, it consists of expressing the total rate constant into an average over individual fine-structure level constants obtained in independent quantum scattering calculations. The weighting factors used to perform the average reflect the probability of initiating the collision on a given surface at temperature T and are assumed to follow a Boltzmann distribution. In the case of NO inclusion of spin-orbit effects leads to a ground state with Ω = (3/2 and an excited state with Ω = (1/2, therefore for the interaction of two NO molecules there are 16 possible spin-orbit states. The complete calculation of all the relevant PES as well as their spin-orbit couplings is beyond the scope of this work. In order to obtain a rough estimate of the role of fine-structure effects on the rates, we make the following assumption: we approximate all the excited PES with the reduced dimensionality intermolecular

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Figure 7. Boltzmann-averaged probability obtained with the statistical model (see eq ) to include fine-structure effects. Compared with Figure5 the minimum is now much more pronounced and displaced to higher temperature.

potential for the O2 dimer.36 The approximation should be quite reasonable for the A00 surfaces given the similarity in well depths and equilibrium intermolecular distance and less so for the 3A0 state. With these assumptions the final expression for the Boltzmann averaged total probability is ITot ðTÞ ¼

Ig ðTÞ þ ð3 þ 8l -ΔESOC =kT þ 4l -2ΔESOC =kT ÞIe ðTÞ ð2 þ 2l -ΔESOC =kT Þð2 þ 2l -ΔESOC =kT Þ

where Ig(T) and Ie(T) refer to the Boltzmann averaged probabilities in the ground and excited states, respectively, and ΔEsoc is the spin-orbit splitting of NO. In Figure 7 we present the results of the statistical approximation to the fine structure effects. We see that the rate constant now presents a minimum roughly around 220 K and an enhanced negative temperature dependence compared with the single surface case, indicating the possible relevance of the fine-structure effects. On the other hand our prediction for the minimum is still too low compared with experiment. Although we have made a rough approximation to the excited PES involved and therefore the role of fine-structure effects cannot be ruled out, this result seems to indicate that a different mechanism could be mainly responsible for the negative temperature dependence. Further evidence of this point of view comes from more recent and much lower temperature measurements performed by the group of Smith20 for NO(v = 3) which show a stronger negative temperature dependence, between T = 85 K and T = 7 K there is a 6-fold increase in the rate which again cannot be reproduced by the current statistical model. Further analysis of the role of complex formation and including additional degrees of freedom will have to wait until a higher dimensional PES is constructed. In particular the rotational motion of the monomers could be crucial as has been predicted for chemical reactions with strong negative temperature dependence.47 The vibrational relaxation rates for highly vibrationally excited states of NO have been measured by the Wodtke group21,22 covering the range v = 8-25 and thus providing the most complete set to date. They found that for v < 14 the relaxation rate increases approximately linearly followed by sudden jumps which show a stronger dependence on v and finally for the higher 2896



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Figure 9. Vibrational adiabats for v = 20, 25, 28. Notice the decrease in the barrier height for the formation of the chemicaly bound complex as v increases, but even for v = 28 there remains a large barrier compared with thermal collision energies.

Figure 8. Boltzmann-averaged probabilities per v (Iv/v) as a function of vibrational quantum number for T = 300 K: (a) Lower end of vibrational ladder. Notice the linear dependence up to v = 12. (b) Higher end of vibrational ladder. Notice the sharp increase after v = 21. The insets are the experimental results from ref 16.

end a stable sharp increase in the rates. In Figure 8 we show the vibrational dependence of the relaxation rates within our reduced dimensionality model. To stress the region of linear dependence, we plot Iv/v and in Figure 8b we show an inset of the experimental results taken from ref 16. For the lower vibrational levels shown in Figure 8a we see that the linear dependence is followed at the beginning but starts to increase faster after v = 12. For the higher vibrational levels shown in Figure 8b first there is an stabilization of the increase that had started at v = 12 but then after v = 20 there is a sharp increase in the relaxation rates such that for v = 25 the rate is over an order of magnitude larger than that of v = 15. Qualitatively there are many similarities with the experimental observations: a linear dependence with v for the lower vibrational states, short regions (two quanta) where the rates change slowly followed by larger jumps and for the higher vibrationally excited states a sharp increase with v. Still there are significant quantitative differences, namely, the precise pattern of sudden jumps and a shift toward higher values of v in our predictions of where the rates start to increase sharply. On the basis of our previous discussion of the interaction potential vibrational dependence (see Figure3) one is inclined to think that the final sharp increase must be due to the presence of the chemically bound species. As a matter of fact this is what we expected when we finished the PES calculations and were ready

to start the collision dynamics. It turns out that the chemically bound species does not play any role in the collision dynamics for the vibrational levels and temperatures reported. This can be easily understood by looking at Figure 9 where we show the adiabats corresponding to v = 20, 25, 28. The collision energy has been subtracted so that the value of zero for the adiabat corresponds with the classical turning point, the collision energy used is typical of the end of the Boltzmann tail for T = 300 K. We can see that the barrier to chemically bound complex formation diminishes with vibrational excitation but even for v = 28 the barrier is still too high to play a role in the dynamics. The sharp increase in rate has to be traced to the increase in well depth and shortening of the van der Waals intermolecular distance both of which become large for the higher vibrational states. This in turn leads to a shift of the classical turning point toward smaller intermolecular distances leading to increased vibrational matrix coupling elements. One can argue that our description of the chemically bound complex and, more relevant, the properties of the barrier connecting it with reactants is approximate within our reduced dimensionality treatment. In the previous ab initio work by the Gordon group,23 the calculated transition state has both NO molecules stretched whereas we have kept fixed one of them. One important consequence of this is that our energy barrier estimate is roughly 30% too high. In order to test the relevance of this point on our dynamical predictions, we constructed a scaled potential energy surface where now the energy barrier value relative to reactants corresponds to the one reported in ref 23. The scaled potential energy surface was obtained using a switching function which decreases the barrier height but leaves without change the van der Waals region which we believe is crucial for the dynamics. After repeating the collision dynamics calculations with the scaled surface, we observe that although the barriers for the different adiabats go down significantly (exactly how much depends on v) they remain too large to play a role in the dynamics as evidenced by a very small change in the rate constants obtained. As a further test of this point, we have repeated the dynamics calculations starting the propagation after the saddle point obtaining identical results with the original propagation which started on the repulsive wall of the chemically bound complex. 2897


The Journal of Physical Chemistry A After some consideration it is not surprising that the chemically bound complex does not play a role: in order to be of relevance for the dynamics is it important to consider not only the energetics but also the geometrical properties of the transition state. Although stretching the NO molecule is required to reach the transition state, it is not sufficient: one must also surmount a barrier in the intermolecular distance coordinate and this turns out to be too large for thermal energies. In light of the present results previous claims about the relevance of chemically bound complexes for the observed enhancement in vibrational relaxation should be taken with caution. Our study has showed that the lowest energy isomer cannot be of relevance due to the geometry of the saddle point involved. Looking at the saddle-point geometry of another low-lying isomer(isomer 4 in ref 23) it seems more promising since it involves one highly stretched NO and one NO close to its equilibrium geometry which makes it appear accessible from reactants side. On the other hand the two NO molecules have to be brought very close to each other(the N-N distance is 1.2 Å) and this could lead to an unsurmountable barrier at thermal energies. In order to clarify the possible role of this chemically bound isomer and also of the transition state for chemical reaction leading to N2O, it will be important to consider the geometrical properties of the saddle point and the final word will be given by the scattering calculations. Our calculations have shown that the strong vibrational dependence of the interaction potential in the region of the van der Waals complex leads to large enhancements of the vibrational relaxation rates.

IV. SUMMARY AND CONCLUSION We have calculated a two-dimensional PES for the NO-NO system which includes the vibrational and the intermolecular dependence of the interaction. This is a challenging task even in reduced dimensions due to the complexity of the system’s openshell electronic structure: electronic degeneracy of the 2Π ground state of NO correlating with eight electronic states which are degenerate asymptotically. Additionally we have to describe highly stretched geometries for NO and the transformation from a weak interaction region corresponding to the van der Waals complex to a strong interaction region leading to a chemically bound complex. Multiconfigurational approaches are strictly necessary, and we have applied the CASPT2 method to obtain the first ab initio based PES that should be useful for modeling vibrational energy transfer processes. We find an unusually large vibrational dependence of the interaction compared with typical van der Waals systems.39,40 Usually the variation in binding energy is of a few percent whereas for NO dimer it will be more than 100% for the higher vibrational states studied in this work. This confirms the observation which had been made by infrared spectroscopy when exciting the antisymmetric stretch of the dimer and where a large red shift and contraction of the intermolecular distance were measured.42 Quantum scattering calculations of the vibrationally inelastic collision process NO(v) þ NO(v = 0) have been performed. As expected the main relaxation mechanism is loss of one vibrational quantum although two quantum transitions become important for some of the lower vibrational levels. Our calculations predict for v = 1 relaxation a 4 orders of magnitude larger rate constant for the NO-NO than for the O2-O2 system in agreement with experiment.15,43 This is the first time that the unusually large vibrational relaxation rate for NO has been predicted on the basis of first principles calculations. Our study shows there is no need to invoke special mechanisms to explain this surprising behavior.

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The negative temperature dependence of the VT relaxation rate is another intriguing property known in the case of v = 1 for a number of years14,15,18,19 and studied with more detail and at lower temperatures for v = 3 more recently.20 Our calculations show a markedly different temperature dependence between the NO dimer and the O2 dimer, where the first one shows a convex shape with a weak negative temperature dependence whereas the second one follows the usual positive dependence, indicating the role of the more strongly bound NO dimer complex as suggested by Smith.44 We applied a statistical approximation to take into account the presence of many potential energy surfaces in the asymptotic region on the relaxation process. The negative temperature dependence is clearly enhanced and the temperature value for the minimum in the rate constant is shifted toward better agreement with experiment, indicating its possible relevance. Still the quantitative differences are large and we believe it is necessary to perform a more detailed study of the role of complex formation including the rotational degrees of freedom to clarify this issue. Our calculations for the relaxation rates of highly vibrationally excited NO(v = 8-25) are in qualitative agreement with experiment: for the lower vibrational states (v e 12) we find a linear dependence with the vibrational quantum number, regions where the rates change slowly followed by larger jumps and for the higher vibrationally excited states a sharp increase with v. An important finding is that, contrary to what we initially expected, the chemically bound complex does not play a role in the vibrational relaxation dynamics due to the geometry of the saddle point which generates unsurmountable barriers in the adiabats even for the highest vibrational states studied (v = 28). In this sense the possible role of other chemically bound complexes and the transition state region for the chemical reaction leading to N2O are still to be tested by performing quantum scattering calculations as those reported here, as the purely static energetic considerations invoked in the past22,23 are not enough as the present study has shown.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT We acknowledge financial support from the bilateral collaborative project Citma-Conacyt (J110.315) and SESIC-SEPFOMES2000 for unlimited time on the IBM p690 supercomputer at UAEM. ’ REFERENCES (1) McKellar, A. R. W.; Watson, J. K. G.; Howard, B. J. Mol. Phys. 1995, 86, 273. (2) Hetzler, J. R.; Casassa, M. P.; King, D. S. J. Phys. Chem. 1991, 95, 8086. (3) Dkhissi, A.; Soulard, P.; Perrin, A.; Lacome, N. J. Mol. Spectrosc. 1997, 183, 12. (4) Blanchet, V.; Stolow, A. J. Chem. Phys. 1998, 108, 4371. (5) Tsubouchi, M.; de Lange, C. A.; Suzuki, T. J. Chem. Phys. 2003, 119, 11728. (6) Lee, T. J.; Rendell, A. P.; Taylor, P. R. J. Phys. Chem. 1990, 94, 5463. (7) Gonzalez-Luque, R.; Merchan, M.; Roos, B. O. Theor. Chim. Acta 1994, 88, 425. (8) East, A. L. L. J. Chem. Phys. 1998, 109, 2185. 2898



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(9) Tobita, M.; Perera, S. A.; Musial, M.; Bartlett, R. J.; Nooijen, M.; Lee, J. S. J. Chem. Phys. 2003, 119, 10713. (10) Wade, E. A.; Cline, J. I.; Lorenz, K. T.; Hayden, C.; Chandler, D. W. J. Chem. Phys. 2002, 116, 4755. (11) Brookes, M. D.; McKellar, A. R. W.; Amano, T. J. Mol. Spectrosc. 1997, 185, 153. (12) East, A. L. L.; McKellar, A. R. W.; Watson, J. K. G. J. Chem. Phys. 1998, 109, 4378. (13) Fernandez, J. M.; Tejeda, G.; Ramos, A.; Howard, B. J.; Montero, S. J. Mol. Spectrosc. 1999, 194, 278. (14) Breshears, W. D.; Bird, P. F. Nature 1969, 224, 268. (15) Stephenson, J. C. J. Chem. Phys. 1974, 60, 4289. (16) Yang, X.; Price, J. M.; Mack, J. A.; Morgan, C. G.; Rogaski, C. A.; McGuire, D.; Kim, E. H.; Wodtke, A. M. J. Phys. Chem. 1993, 97, 3944. (17) Nikitin, E. E. Theory of elementary atomic and molecular processes in gases; Clarendon Press: Oxford, 1974. (18) Billingsley, J.; Callear, A. B. Trans. Faraday Soc. 1971, 67, 257. (19) Rebelo Da Silva, M.; Vasconcelos, M. H.; Lepoutre, F.; Perrin, M. Y. Chem. Phys. Lett. 1982, 91, 135. (20) James, P. L.; Sims, I. R.; Smith, I. W. M. Chem. Phys. Lett. 1997, 276, 423. (21) Yang, X.; Kim, E. H.; Wodtke, A. M. J. Chem. Phys. 1990, 93, 4483. (22) Yang, X.; Kim, E. H.; Wodtke, A. M. J. Chem. Phys. 1992, 96, 5111. (23) Chaban, G.; Gordon, M. S.; Nguyen, K. A. J. Phys. Chem. A 1997, 101, 4283. (24) Price, J. M.; Mack, J. A.; Rogaski, C. A.; Wodtke, A. M. Chem. Phys. 1993, 175, 83. (25) Campos-Martínez, J.; Carmona-Novillo, E.; Echave, J.; Hernandez, M. I.; Hernandez-Lamoneda, R.; Palma, J. Chem. Phys. Lett. 1998, 289, 150. (26) Dayou, F.; Campos-Martínez, J.; Hernandez, M. I.; HernandezLamoneda, R. J. Chem. Phys. 2004, 120, 10355. (27) Dayou, F.; Hernandez, M. I.; Campos-Martínez, J.; HernandezLamoneda, R. J. Chem. Phys. 2007, 126, No. 194309. (28) Andersson, K.; Malmqvist, P. A.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem. 1990, 94, 5483. (29) Andersson, K.; Malmqvist, P. A.; Roos, B. O. J. Chem. Phys. 1992, 96, 1218. (30) Werner, H.-J.; Knowles, P. J. Chem. Phys. 1988, 89, 5803. (31) Gdanitz, R. J.; Ahlrichs, R. Chem. Phys. Lett. 1988, 143, 413. (32) Bartolomei, M.; Hernandez, M. I.; Campos-Martínez, J.; Carmona, E.; Hernandez-Lamoneda, R. Phys. Chem. Chem. Phys. 2008, 10, 1. (33) Boys, S. F.; Bernardi, F. Mol. Phys. 1970, 19, 553. (34) van Lenthe, J. H.; van Duijneveldt-van de Rijdt, J. G. C. M.; van Duijneveldt, F. B. Adv. Chem. Phys. 1987, 69, 521. (35) Andersson, K. et al. MOLCAS, Version 4.0; Lund University, Sweden, 1997. (36) Hernandez-Lamoneda, R.; Hernandez, M. I.; Campos-Martínez, J. Chem. Phys. Lett. 2003, 368, 709. (37) Stechel, E. B.; Walker, R. B.; Light, J. C. J. Chem. Phys. 1978, 69, 3518. (38) Rintelman, S. V. J. M.; Adamovic, I.; Gordon, M. S. J. Chem. Phys. 2005, 122, No. 044105. (39) Swartz, B. A.; Brinza, D. E.; Western, C. M.; Janda, K. C. J. Phys. Chem. 1984, 88, 6272. (40) Jahn, D. G.; Clement, S. G.; Janda, K. C. J. Chem. Phys. 1994, 101, 283. (41) Dayou, F.; Bartolomei, M.; Campos-Martínez, J.; Hernandez, M.; Hernandez-Lamoneda, R. Mol. Phys. 2004, 102, 2323. (42) Brechignac, Ph.; De Benedictis, S.; Halberstadt, N.; Whitaker, B. J.; Avrillier, S. J. Chem. Phys. 1985, 83, 2064. (43) Ormonde, S. Rev. Mod. Phys. 1975, 47, 193. (44) Smith, I. W. M. Faraday Trans. 1997, 93, 3741. (45) Graff, M. M.; Wagner, A. F. J. Chem. Phys. 1990, 92, 2423. (46) Dayou, F.; Spielfiedel, A. J. Chem. Phys. 2003, 119, 4237. (47) Clary, D. C.; Nyman, G.; Hernandez, R. J. Chem. Phys. 1994, 101, 3704. 2899


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