An Adiabatic Capture Theory and Quasiclassical Trajectory Study of C

Apr 23, 2012 - SINTEF Materials and Chemistry, P.O. Box 4760, 7465 Trondheim, Norway. §. Department of Chemistry, Physical Chemistry, University of ...
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An Adiabatic Capture Theory and Quasiclassical Trajectory Study of C + NO and O + CN on the 2A′, 2A″, and 4A″ Potential Energy Surfaces Terry J. Frankcombe*,† and Stefan Andersson*,‡,§ †

Research School of Chemistry, Australian National University, ACT 0200 Australia SINTEF Materials and Chemistry, P.O. Box 4760, 7465 Trondheim, Norway § Department of Chemistry, Physical Chemistry, University of Gothenburg, 41296 Gothenburg, Sweden ‡

ABSTRACT: The adiabatic capture centrifugal sudden approximation (ACCSA) has been applied to the C + NO and O + CN reactions, along with quasiclassical trajectory simulations. Existing global analytic fits to the potential energy surfaces of the CNO system in the 2A′, 2A″, and 4A″ electronic states have been used. Thermal rate constants for reaction in each of the electronic states have been calculated. In all cases a strong temperature dependence is evident in the calculated rate constants. The agreement between the calculated adiabatic capture and quasiclassical trajectory rate constants is excellent in some cases, but these rate constants differ considerably in other cases. This behavior is analyzed in terms of the anisotropy of the potential energy surfaces. On the basis of this analysis, we propose a new diagnostic for the reliability of ACCSA capture calculations.

1. INTRODUCTION The three-atom CNO system exhibits a surprisingly rich chemistry. From the reactants in their respective electronic ground states, two of the three possible atom−diatom reactions in this systemC + NO and O + CNcan barrierlessly proceed to at least one atom−diatom product that is significantly exothermic.1−3 As well as the competition between the exothermic channels producing oxygen or nitrogen atoms in the first of these reactions, both reactions are sufficiently exothermic to produce both ground state (4S) and electronically excited (2D) nitrogen atoms. Both of these reactions are thought to be important in combustion4−7 and in the interstellar medium.8−10 The theoretical description of the atom−diatom reactions in the CNO system is an ongoing effort, starting more than two decades ago. The O + CN → N + CO reaction is examined by Schmatjko and Wolfrum11 and Cobos.12 Specific aspects of the C + NO reaction are studied by Rayez and co-workers13−15 and by Abrol et al.16 Yazidi et al.17 present a detailed investigation of a large number of microscopic states in the bound triatomic regions and along paths leading to the three atom−diatom channels. A balanced treatment of all three CNO atom−diatom channels is developed in a series of papers published by Simonson, Roos, Markovic, Andersson, and co-workers.1−3,18−23 These later studies use ab initio molecular orbital theory to probe the energies of a wide range of configurations of the CNO system, primarily at the CASPT2 level of theory. The resultant energies are fit to analytic global descriptions of the adiabatic potential energy surfaces (PESs) for the 2A′, 2A″ and 4A″ electronic states of CNO. These PES fits are designed to be valid in the three atom−diatom dissociation regions as well as in the vicinity of the CNO intermediate structures. That has allowed the potential functions to be used for performing quasiclassical trajectory (QCT) studies of the dynamics of C + © 2012 American Chemical Society

NO and O + CN. These QCT studies largely agree with the available experimental data. Adiabatic capture theory has several advantages over QCT simulations, particularly in terms of computational effort, the inclusion of quantum effects, and accuracy at low temperatures. On the other hand, QCT simulations include nonadiabatic and reverse reaction effects explicitly excluded from adiabatic capture theory. For example, a recent study of the N + CN → N2 + C reaction concluded that despite the ground-state triplet exhibiting many of the features that suggest a capture theory approach, such an approach would be inappropriate due to the prevalence of recrossing.24 It is therefore useful to directly compare results from QCT simulations to adiabatic capture theory results. Thus, the approach of this work is twofold. We apply the adiabatic capture centrifugal sudden approximation (ACCSA) quantum dynamics to C + NO and O + CN on the same PESs as have been used previously for QCT simulations.2,3 Furthermore, we have extended the QCT examination of this system in order to gain deeper understanding of the strengths and weaknesses of ACCSA versus QCT for this and related systems.

2. ACCSA, ATOM−RIGID-ROTOR POTENTIALS, AND QCT CALCULATIONS The PESs used in this work are those published in ref 21 for the 2 A′′ state, in ref 1 for the 2A′′ state, and in ref 3 for the 4A″ state. All three of these states correlate with the ground states of the C (3P) + NO (2Π) and O (3P) + CN (2Σ+) atom−diatom channels. Only the 4A″ state leads to ground state N (4S) + CO Received: February 26, 2012 Revised: April 21, 2012 Published: April 23, 2012 4705

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Figure 1. The atom−rigid-rotor potentials used in this work for the C + NO (left) and O + CN (right) reactions, on the 2A′ (top) and 2A″ (middle) and 4A″ (bottom) PESs. Positive, negative, and zero contours are shown in blue, red, and black, respectively. Solid contours are spaced at intervals of 1 eV. Dashed contours around zero are spaced at intervals of 0.025 eV.

(1Σ+), with the doublet surfaces yielding excited N (2D) + CO (1Σ+). The ACCSA method is described in detail in the literature,25−28 as are the specifics of the implementation used in this work.28,29 The essence of the ACCSA method is to diagonalize the nonrelativistic atom−rigid-rotor Hamiltonian with the approximation 2

|Ĵ − j |̂ 2 ≈ J(J + 1) + j ̂ − 2Ω2

resolved step function reaction probabilities can be suitably averaged to yield state-resolved reaction cross sections and rate constants as well as thermal reaction rate constants. The ACCSA calculations use only a two-dimensional slice of the PES, keeping the diatomic fixed at its potential minimum separation as a rigid rotor. These separations are 2.175 a0 for NO and 2.215 a0 for CN. The potentials used in this work (atom−rigid-rotor slices through the potentials of refs 1, 3, and 21) are shown in Figure 1. The atom−rigid-rotor potentials for C + NO are qualitatively similar on the 2A′ and 2A″ surfaces. The potentials exhibit broad and deep wells for collinear approach at the N end on both surfaces, reflecting the fact that CNO forms by barrierless association of the reactants. A shallower reactive well exists at the O end (to form CON), protected by a small barrier that does not extend above the asymptotic energy in the 2A″ case. For approaches 50°−100° and 40°−60° from CON collinearity on the 2A′ and 2A″ surfaces, respectively, the potentials are repulsive after passing through a small physisorption well at separations of 6−7 a0. The 2A′ and 2A″ O + CN potentials are broadly similar, with deep, broad attractive wells for collinear approach of the O atom. In contrast to C + NO, for O + CN the potentials remain strongly attractive at short-range for all angles, with a repulsive island in the vicinity of the T-shaped geometry at around 5 a0 separation.

(1)

where J ̂ is the total angular momentum operator, j ̂ is the operator for the angular momentum of the rigid rotor, J is the total angular momentum quantum number, and Ω is the projection of the total angular momentum on the atom−rigidrotor center of mass vector in the asymptotic region, at a number of discrete fragment separations R. This yields a set of rotationally adiabatic potential energy curves εj,Ω(R). In turn, one can construct one-dimensional effective potentials VjJ , Ω(R ) = εj , Ω(R ) +

J(J + 1) − Bj(j + 1) 2μR2

(2)

where j and B are the rigid-rotor rotational quantum number and rotational constant and μ is the atom−rigid-rotor reduced mass. Applying a classical capture criterion on these effective potentials (in which reaction is strictly deemed to occur at energies above the highest barrier on the effective potential) results in the ACCSA method. These initial-rotational-state4706

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Figure 2. Calculated ACCSA (curves) and QCT (symbols joined by straight lines) thermal reaction rate constants for C + NO (left) and O + CN (right) on the 2A′ (top), 2A″ (middle), and 4A″ (bottom) PESs. The uncertainty in the QCT rates is smaller than the size of the symbols.

The potential for the C + NO reaction on the 4A″ surface, on the other hand, is considerably different. There are no deep reactive wells for collinear approaches on that potential. There is a reasonably deep well leading to reaction to form CNO bent at approximately 140°, but only a rather restricted angular range clearly leads to this well. There is a shallow well (that potentially leads to CON) located around 40° from collinearity, protected by a positive energy barrier. Additionally, there is a large region of weakly positive potential energy at separations greater than 7 a0 for the C atom approaching the O end of the NO diatomic. In general, the 4A″ atom−rigid-rotor potential exhibits a much less reactive character than the doublet surfaces. For the QCT calculations the same procedure has been used as in earlier studies (refs 1−3, 20, 21). One hundred thousand trajectories were run for each reaction, PES, and temperature combination. Each trajectory was characterized according to the final configuration and whether or not a collision complex had been formed during the trajectory. A collision complex was deemed to be formed if at least one minimum distance exchange occurred. That is, if the atoms giving the shortest interatomic distance in the system changed during the trajectory. This approach works well for CNO because of the similar equilibrium bond lengths involved. This procedure is able to identify in principle all trajectories where a chemically bound CNO complex is formed, whether or not a net reaction occurs. The collision complex formation rate so defined is the

QCT quantity most closely equivalent to the ACCSA capture rate. ACCSA calculations were performed from 2 to 4000 K, while QCT calculations were performed for a range of temperatures from 5 to 5000 K.

3. CALCULATED CAPTURE RATE CONSTANTS For both ACCSA and QCT calculations, the calculated rate constants are for an initial population in the entrance channel of the relevant PES. For these to be interpreted as thermal reaction rate constants they must be multiplied by temperaturedependent statistical factors to account for the thermal populations of the various entrance channel electronic states. These statistical factors are described in refs 2 and 3. The appropriate factors are C + NO f doublet (T ) =

( −23.6T K ) ⎡1 + 3 exp −23.6 K + 5 exp −62.4 K ⎤⎡2 + 2 exp −172.4 K ⎤ ( T )⎦ ( T ) ( T )⎦⎣ ⎣ 2 + 2 exp

(3)

and C + NO f quartet (T ) =

( −23.6T K ) ⎡1 + 3 exp −23.6 K + 5 exp −62.4 K ⎤⎡2 + 2 exp −172.4 K ⎤ ( T )⎦ ( T ) ( T )⎦⎣ ⎣ 4 exp

4707

(4)

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for the C + NO reaction and O + CN f doublet (T ) =

2

(

5 + 3 exp

−228 K T

) + exp( −326T K )

(5)

for the O + CN reaction. Figure 2 shows the rate coefficients for the C + NO and O + CN reactions calculated in this work. The rate constants shown in Figure 2 include the effect of the thermal factors given in eqs 3−5. The QCT calculations performed in this work are consistent with those published in refs 2 and 3. The most relevant comparison between the ACCSA and QCT results is to compare the ACCSA capture rate constants with the rate of collision complex formation in the QCT calculations. On this basis, generally the agreement between the ACCSA and QCT calculations was excellent. At temperatures up to 1000 K, both the temperature dependence and magnitude of the rate constants were reproduced well. The discrepancies between the ACCSA and QCT capture rate constants were larger for the O + CN reaction than for the C + NO reaction. With the exception of the O + CN reaction on the 2A′ PES, the ACCSA rate constants tended to be slightly larger than the QCT rate constants. Overall, this behavior is similar to the N + NH reaction, in which excellent agreement between ACCSA results and available QCT results is observed.28 Also shown in the plots of Figure 2 are the rate constants for forming atom−diatom products distinct from the reactants. For the C + NO reaction the O + CN products were strongly preferred over the N + CO products at low temperatures. At higher temperatures (around 1000 K for the reaction on the 2A″ PES, several thousand kelvin for the reaction on the 2A′ and 4A″ PESs) both the O + CN and N + CO product channels were competitive, even favoring the N + CO products at high temperatures. For the O + CN reaction the N + CO products were dominant throughout, with the rate constant for formation of C + NO negligible or zero for all but the highest temperatures. This is not surprising given that the O + CN → C + NO reaction is endothermic by more than 1.2 eV. Figure 2 shows not only the product formation rate constants, but also their sum. The difference between this “total products” rate constant and the capture rate constant gives the rate of strongly inelastic scattering events, in which the collision complex forms but the system then returns to the reactant channel. This corresponds to classical recrossing. While recrossing was evident in all the QCT simulations, it was weak for the C + NO reaction on the doublet PESs except at the highest temperatures. The rate of recrossing was most significant for the O + CN reaction, albeit not at low temperatures on the 2A′ PES. The recrossing rate in the QCT calculations is examined in more detail in Figure 3. This figure shows the proportion of trajectories that form a collision complex, then return to the reactant valley, as a function of the initial diatomic rotational state. In these calculations the initial vibrational state was sampled from the ground state and the initial translational temperature was 100 K. Clearly, in addition to being more probable, recrossing in the O + CN reaction exhibited a more varied dependence on rotation than in the C + NO reaction. This is consistent with the more variable relation between the complex formation and total products QCT rate constants for the O + CN reaction shown in Figure 2. Similar calculations

Figure 3. Proportion of complex-forming trajectories recrossing to reactants in QCT calculations at 100 K, as a function of diatomic initial rotational state. The statistical uncertainty is similar to the symbol size or smaller.

were performed at 10 and 1000 K. The opposite dependence on j for the 2A′ and 2A″ PESs was maintained at the different temperatures, but the variation with j lessened. This may be an indication that at low temperatures the lack of quantization of the rovibrational states of intermediates and products in the QCT calculations allowed kinetic energy that should be conserved as zero-point energy (ZPE) to facilitate recrossing by converting to center of mass translation (“ZPE leakage”). The recrossing rates calculated in this work could not be reliably fit to any analytic expression derived from power law potentials. The reactant thermal microscopic population factors of eqs 3−5 have a significant impact on the thermal rate constants calculated in this work. This impact is illustrated in Figure 4. Mostly, the thermal population factors served to reduce the effective reaction rate at moderate to high temperatures. In the case of the C + NO reaction on the 4A″ PES, the very low rate constant at very low temperatures was caused by the thermal population factor of eq 4 rather than the inherent properties of the PES.

4. DISCUSSION As identified above, QCT simulations of the C + NO reaction showed a clear preference to form the O + CN products. This is contrary to what one might expect given that the asymptotic potential energy in the O + CN product channel is more than 0.8 eV higher than that for N (2D) + CO (1Σ+) (on the 2A′ and 2 A″ PESs) and more than 3.3 eV higher than N (4S) + CO (1Σ+) (on the 4A″ PES). Nonetheless, the observed preference is readily rationalized by examining the minimum energy paths (MEPs) connecting the C + NO reactants to product regions. For example, a representation of the profile of the MEPs on the 2 A″ PES is given in Figure 2 of ref 1. The MEP leads barrierlessly to linear CNO, which can undergo a simple bond fission to form O + CN. Forming N + CO, on the other hand, requires that either the CNO isomerize through an intermediate well to NCO that can dissociate to N + CO (or again to O + CN), the C atom to insert into NO directly to form the NCO intermediate and dissociate, or the initial association to pass over a barrier to form the more weakly bound linear CON before dissociating over another barrier. In all cases the collision complex formation rate constants calculated from the QCT calculations increased as the temperature increased toward 5000 K. For the C + NO reaction this was mirrored in the product formation rate constants. This trend was not reproduced by the ACCSA capture rates. Clearly, this suggests that the increase in the rate 4708

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Figure 4. Effect of the statistical thermal factors. The ACCSA capture rate constants are plotted for the C + NO (left) and O + CN (right) reactions, with (thick) and without (thin) the effect of the statistical thermal factors of eqs 3−5 (c.f. Figure 2). Results for the 2A′, 2A″ and 4A″ PESs are shown in red, green and blue, respectively.

shown in Figure 2 (see also Figure 3). Recrossing is weak in the doublet C + NO reaction, making the capture rate constants calculated by either ACCSA or QCT accurate approximations to the total reaction rate constants. For C + NO on the 4A″ PES recrossing is stronger, with 28−47% of the trajectories that formed the collision complex returning to the C + NO entrance valley. Similarly, the O + CN reaction exhibited recrossing rates of up to 70%. In these latter cases the capture rate constants calculated by a capture theory such as ACCSA may well be accurate, but interpreting that capture rate constant as the rate constant for the intended bimolecular reaction fails. This is despite the MEPs exhibiting features usually interpreted as signifying capture theory as appropriate, such as substantial exothermicity and the existence of simple reaction paths. The presence of recrossing causes the ACCSA rate constants to be overestimates. Nonadiabatic effects, on the other hand, have the capability of increasing or decreasing the capture rate constant (though they usually reduce the capture rate constant through transitions to higher orbital angular momentum states). While quantum mechanical wave packet studies of some of these reactions have been published,22,23 these have not been analyzed in terms of rotational and orbital angular momentum state transitions. It is difficult to unambiguously quantify the effect of nonadiabaticities in this CNO system. However, we do offer some observations. At temperatures of 1000 K and below, in all cases except the O + CN reaction on the 2A′ PES, the capture rate constants calculated using QCT simulations were smaller than those calculated within the ACCSA. Rotationally nonadiabatic effects (as distinct from ZPE leakage in the interaction region) are principally expected to act through long-range interactions, where the orbital angular momentum states that the system can jump between are more closely spaced. Transitions between orbital angular momentum states must be accompanied by transitions between the rigid rotor rotational states, as the total angular momentum must be conserved. Thus, nonadiabatic effects can be mediated by anisotropies in the atom−rigid-rotor PES. All of the potentials shown in Figure 1 exhibit some degree of anisotropy in the long-range part. However, recalling that full three-dimensional potentials are available in this work, certain anisotropies are more evident in the potentials when examining the derivative of the energy with respect to the bond length of the rigid rotor. This derivative yields the negative of the force pulling the atoms of the rigid rotor apart when the third atom is brought close. Any motion changing the separation between the diatomic atoms corresponds to a change in the diatomic rotational

constants at high temperature is due to an effect present in the QCT calculations but not in the ACCSA. A likely explanation lies in the treatment of the diatomic as a rigid rotor in the ACCSA calculations, meaning the underlying effective potentials have no “temperature dependence”. On the other hand, in the QCT calculations the initial diatomic rovibrational state is temperature-dependent, changing the long-range PES felt by the colliding system. Higher temperatures would be accessing parts of the PES with longer diatomic bond lengths, and thus different reactivities. Furthermore, in the QCT calculations energy initially in the rovibrational motion of the diatomic can be transferred into reaction-enhancing modes. As a test, a series of ACCSA calculations was performed for the C + NO reaction at 3000 K on the 2A″ PES. The fixed NO diatomic bond length was varied to simulate high-temperature centrifugal and anharmonic stretching. The capture rate constant increased as the NO bond length was increased. When the NO bond length was increased by 8% to 2.35 a0, the capture rate constant increased by more than 20%. This suggests that improved accuracy may be obtained by using a (v,j)-dependent 2D PES derived by integrating over the diatomic bond length, weighted by the bond length distribution for the appropriate rovibrational diatomic wave function. Clearly, such an approach would increase the computational cost of ACCSA calculations. Adiabatic capture theory and QCT calculations are somewhat complementary. Recrossing and rebound effects, where the reactants capture each other to form an activated complex but then return to the reactant region, are explicitly included in QCT simulations but are absent from ACCSA calculations. QCT calculations are also free of the weak coupling, rotationally adiabatic assumption that ACCSA calculations are built on. On the other hand, the only truly quantum effect that can be reasonably considered to be incorporated in QCT calculations (beyond the quantum-classical correspondence principle) is the quantization of energy and angular momentum in the asymptotic reactant state incoming flux. Effects such as the quantization of rovibrational states in the interaction region and conservation of helicity, which are included explicitly in the ACCSA as part of the treatment of adiabaticity, are completely neglected in QCT calculations. Thus, it is sensible to compare and contrast the QCT and ACCSA results shown in Figure 2 in terms of these two principle differences between the formalisms: recrossing and nonadiabatic effects. Neglecting quantization in the interaction region, the effect of recrossing is measured by the differences between the “QCT complex formation” and “QCT total products” rate constants 4709

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Figure 5. First derivative of the potential with respect to the rigid rotor length, as a function of the interfragment coordinates, for the C + NO (left) and O + CN (right) reactions, on the 2A′ (top), 2A″ (middle), and 4A″ (bottom) PESs. Positive, negative, and zero contours are shown in blue, red and black, respectively. Solid contours are spaced at intervals of 5 eV/a0. Dashed contours around zero are spaced at intervals of 0.2 eV/a0.

the O + CN reaction. Correspondingly, for this reaction the ACCSA and QCT capture rates are in reasonable agreement.

constant and must couple to changes in the rotational state of the diatomic. Such derivatives, evaluated at the rigid rotor separation used in the ACCSA calculations, are shown in Figure 5. At longrange all of these derivatives are small and isotropic, as one would expect. However, there appears to be a critical region at separations of 4−6 a0. On the doublet PESs for the C + NO reaction (top and middle left of Figure 5), for which the ACCSA and QCT simulations agree extremely well, the potentials exhibit low anisotropy. For any separation in this range, these diatomic-altering forces differ by not much more than 1 eV/a0 across the entire angular range. For the O + CN reaction, for which the ACCSA and QCT simulations give capture rate constants that differ more widely and in a less systematic way, a much greater rotational-constant-changing force anisotropy is evident. In this region of these potentials, not only does the force differ by up to >5 eV/a0 across the angular range, but the derivative also shows substantial positive and negative regions. For the C + NO reaction on the quartet surface the potential radial derivative slice shown in Figure 5 exhibits substantial anisotropy. Whereas the O + CN potential derivatives clearly show negative derivatives in the vicinity of linear O−CN (with the derivative positive elsewhere for separations greater than 4 a0), for 4A″ C + NO the derivative alternates between positive and negative derivative regions twice from NO−C (0°) to C−NO (180°). For nonzero rigid rotor j states the average effect of this derivative would be smaller, leading to reduced nonadiabatic effects compared to in

5. CONCLUSION In this work we have performed ACCSA calculations on the C + NO and O + CN reactions on PESs for a number of electronic states. We have performed similar studies using QCT simulations, replicating previous work in greater detail. Our principle interest was in comparing and contrasting the two approaches. For the C + NO reaction there was good agreement between the ACCSA and QCT results for all three electronic states. For the O + CN reaction there was less quantitative agreement between the two approaches. We attribute this to rotational energy transfer. Apparent recrossing effects were also much stronger in the O + CN reaction. In this work we have proposed a new measure to test the likelihood of rotational nonadiabaticity. Simple anisotropy in the derivative of the potential with respect to the bond length of the diatomic qualitatively correlates with behavior we associate with rotational nonadiabaticity. While further testing is warranted, this appears to be a useful diagnostic for the appropriateness of ACCSA calculations for atom−diatom capture. 4710

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (T.J.F.) and stefan.andersson@ sintef.no (S.A.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Some of the calculations described in this paper were performed using computing resources from C3SE, Chalmers University of Technology, Sweden, and from the NCI National Facility at the Australian National University, Australia. We would like to thank Gunnar Nyman for useful discussions.

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