Equilibration of Vibrationally Excited OH in Atomic and Diatomic Bath

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Equilibration of Vibrationally Excited OH in Atomic and Diatomic Bath Gases Anthony J. McCaffery,*,† Marisian Pritchard,† John F. C. Turner,† and Richard J. Marsh‡ † ‡

Department of Chemistry, University of Sussex, Brighton BN1 9QJ, U.K. Department of Physics, University College, Gower Street, London WC1E 6BT, U.K. ABSTRACT: In this work, a computational model of state-tostate energy flow in gas ensembles is used to investigate collisional relaxation of excited OH, present as a minor species in various bath gases. Rovibrational quantum state populations are computed for each component species in ensembles consisting of 8000 molecules undergoing cycles of binary collisions. Results are presented as quantum state populations and as (approximate) modal temperatures for each species after each collision cycle. Equilibration of OH is slow with Ar as the partner but much faster when N2 and/or O2 forms the bath gas. This accelerated thermalization is shown to be the result of near-resonant vibration
vibration transfer, with vibrational deexcitation in OH matched in energy by excitation in bath molecules. Successive near-resonant events result in an energy cascade. Such processes are highly dependent on molecule pair and on initial OH vibrational state. OH rotational temperatures initially increase, but at equilibration, they are lower than those of other modes. Possible reasons for this observation in molecules such as OH are suggested. There are indications of an order of precedent in the equilibration process, with vibrations taking priority over rotations, and potential explanations for this phenomenon are discussed.

1. INTRODUCTION This study continues a series of investigations into the mechanism of the evolution of gas mixtures containing species that are initially in highly non-Boltzmann thermodynamic states. We have shown that quantum state populations in the components of an ensemble of up to 10000 molecules, containing up to three different diatomic molecules, can be followed through many collisions.1
3 Studies on highly excited CO2 and N23 in various bath gases indicate that ensemble relaxation is generally a complex, multistage process, the form of which could not be predicted from single-collision measurements. Furthermore, the manner and speed of equilibration are strongly influenced by the characteristic rotational and vibrational quantum state energies of both target and bath species. Here, we examine the mechanism by which OH is collisionally relaxed in a number of different diatomic bath gases. In common with other diatomic hydrides, OH exhibits unusual collision dynamics that originate in competition between the momentum- and energy-based constraints on state-to-state transitions. Rovibrational energy levels of molecules such as HF, HCl, and OH are determined by their very low reduced masses, but the momentum they carry is related to their considerably higher molecular masses. Such species are often found to be highly resistant to collisional equilibration when formed initially in high rovibrational states, and inverted populations are frequently observed under relatively high-pressure conditions.4,5 OH is known to play a critical role in a wide range of chemical, industrial, and environmental contexts, despite constituting only r 2011 American Chemical Society

a minor component of the molecules present. An example of technological significance is the combustion of hydrocarbon fuels, where OH plays a central role in the detailed mechanism of the process.6 Laser-induced fluorescence spectroscopy from OH is an important component of the remote monitoring of the physical and chemical conditions during combustion. In Earth’s troposphere, OH is a significant contributor to the oxidation and eventual removal of hydrocarbons and sulfur- or nitrogen-containing pollutants.7 The role of OH in the complex series of elementary reactions that contribute to the balance of ozone creation and destruction is now well-established,7 and the molecule was identified as a key component of Earth’s airglow by Meinel8 in the early 1950s. Since that date, the rotational spectra of OH have been used to measure mesosphere temperatures through ground-, rocket-, and satellite-based measurements, culminating in the recent definitive determination by Cosby and Slanger9 based on the analysis of high-resolution Meinel band emission spectra using calibrated intensities. Excited OH is also found in comets, stellar atmospheres and interstellar clouds. In Earth’s mesosphere, it is thought to be formed from the reaction H þ O3 f OH þ O2 in a region around 8 km thick at ∼87 km altitude. Laboratory studies10,11 indicate the reaction to be exothermic by >27000 cm
1, with preferential population of OH vibrational states vOH = 7
9. Received: December 13, 2010 Revised: March 4, 2011 Published: April 11, 2011 4169

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The Journal of Physical Chemistry A In this work, the excited OH species in each ensemble is initially set to be in the state (vOH = 8; nOH = 6), where v and n are vibration and (nuclear) rotation quantum numbers, respectively. This choice of vOH value is to some degree a matter of computational convenience because our model at present does not extend beyond v = 10. Initial population of v = 8 permits limited observation of vibrational excitation, although, in practice, the majority of flux is toward relaxation. We have found this to be a useful starting vibrational level in cases where resonant vibration
vibration (V
V) exchange occurs with the bath species.2,3 Furthermore, the results reported here might have relevance to atmospheric studies, in view of the mesospheric reaction mentioned above. The bath gases employed are Ar, N2, O2, and a 4:1 mixture of N2 and O2. The diatomic bath gases initially are set with v = 0 and rotation values j = 10 and 12 for N2 and O2, respectively. As in our previous work, collision energy is set at 300 K. A detailed study of the temperature dependence of equilibration in these ensembles is currently underway12 and will be reported in a future publication. This model of a gas ensemble represents a thermodynamically closed system in which energy and matter are not exchanged with the surroundings. As a result, the observed behavior simulates a condition independent of experimental or environmental fluctuations and thus reflects the inherent properties of the molecular constituents of the ensemble, their initial quantum state populations, and the collision energy.

2. COMPUTATIONAL METHOD A description of the computational model can be found in recent publications,1
3 one of which3 shows a flowchart of the principal operations in the routine. In brief, an array, generally of 8000 diatomic molecules, is set up, with each molecule characterized by a specific individual rovibrational state and velocity from a thermal distribution at some preselected temperature. Three different diatomics can be specified, each initially in a welldefined (v;n) state for OH or (v;j) in the case of O2 and N2. Here, the target species is OH(8;6) in a bath gas of Ar atoms, N2(0;10), O2(0;12), or N2(0;10) and O2(0;12) in a 4:1 mixture to simulate air. In all cases, OH represents approximately 10% of the total number of species. This, relatively high OH number density is essentially determined by the total number of molecules the model can accommodate and the need, in terms of consistency of data, for a significant number of target species. The model is run with nominally 8000 molecules, or molecules and atoms, with 727 OH(8;6) molecules and 7270 bath species. Single-collision cycles commence with each molecule in the array successively picked at random and subjected to collision with another species (also picked at random). The new rovibrational states of both, as well as final molecular velocities, are computed using diatom
diatom vibration
rotation transfer (VRT) equations for momentum, angular momentum, and energy transfer.13 As molecules are chosen at random for each collision cycle, the number of collisions each molecule has actually undergone after a given number of collision cycles will be a distribution that peaks at the collision cycle number and broadens as the ensemble evolves. Here, we refer to the number of collision cycles as the collision number (CN), which represents the peak of this distribution curve. Following collision between individual pairs of diatomics, data on each molecule in its new state are recorded. The primary information following each collision cycle consists of the

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quantum state populations and molecular velocities for each of the three component species. Vibrational and/or rotational state populations of each species are displayed after each collision cycle and can be downloaded at this point. The cycles can be repeated at will with the above data available after each. Generally, this process is continued until each molecule has undergone around 1000 collision cycles. Examples of typical output data are shown below where they are discussed in detail. In addition to the quantum state populations, a valuable semiquantitative picture of ensemble evolution consists of approximate vibration, rotation, and translation temperatures (Tr, Tv, and Tt, respectively) plotted versus collision number. These temperatures are obtained by assuming that the overall populations for each mode constitute a Boltzmann distribution. The modal temperatures (Tmode) clearly are not meaningful in the early stages of ensemble development but become more reliable as the ensemble evolves. Plots of Tmode versus CN provide a highly visual overview of the equilibration process and are a useful basis for general classification purposes. However, they provide relatively little insight into the molecular mechanism of equilibration, which comes principally from detailed examination of vibration and rotation state changes as the ensemble evolves. The computational routine is based on the physical principles of the angular momentum (AM) model14 of collision-induced physical and chemical change. This approach represents an updated form of Newton’s vector mechanics in which the probability of momentum conversion is calculated directly for each collision-induced state-to-state transition within constraints set by overall and state-to-state energy conservation. For quantitative cross sections, a three-dimensional hard shape, or Newton surface, representation is used to compute momentum conversion probabilities from a large number of Monte Carlo collision trajectories. This form of mechanics was developed by the authors and co-workers following analysis of state-to-state collision dynamics experiments that indicate that the motive force for change at the molecular level originates in momentum change upon collision.15 A key element is the conversion of linear momentum of relative motion to orbital AM and, subsequently, molecular rotation. Analysis of experiment demonstrates16 that this occurs through a lever arm, or effective impact parameter, bn, of length related to molecular dimensions. For computational purposes, a three-dimensional ellipsoidal shape based on molecular bond length forms the Newton surface in combination with Monte Carlo trajectories, a procedure first adopted by Kreutz and Flynn.17 A more detailed description of the theoretical method is given in ref 3. Accurate calculation of single-collision events is a necessary requirement for a multicollision computational routine of the kind described here. The AM method is fast and reproduces quantum state distributions, giving quantitative cross sections from a wide range of processes of physical and chemical change at the molecular level. These include rotational transfer (RT) and vibration
rotation transfer (VRT) in diatomic and polyatomic species,18,19 fragment populations following dissociation of weakly bound species,20 and elementary chemical reactions.21 A review summarizes much of this earlier work and gives the principal energy and momentum equations on which the method is based.14 There are relatively few experimental data sets on state-to-state relaxation in gas ensembles against which to calibrate our predictions. However, Robinson and co-workers5 investigated the kinetics and dynamics of the OH rotational laser produced from the reaction O(1D) þ H2 f OH þ H, which is 4170

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Figure 1. Variation of modal temperatures of OH, initially in (v;n) = (8;6), with number of collision cycles for a 1:10 mixture of OH(8;6) in Ar at 300 K. The total number of molecules is (nominally) 8000. Tv (black squares) represents vibrational temperature; Tr (red circles), rotational temperature; and Tt (green triangles), translational temperature. As described in the text, the primary data are the quantum state populations. Modal temperatures are calculated assuming a Boltzmann distribution, and thus, Tv and Tr are not meaningful at the outset and in the early stages of ensemble evolution.

exothermic by 15400 cm
1. This is sufficient to populate at least the (4;11) level of product OH and higher if collision energy is included. Laser action was observed from rotational levels of vOH = 3, 2, 1, and 0 in an Ar bath at 14 Torr total pressure, with emission beginning immediately after onset of the photolysis pulse used to generate O(1D) from O3.5 Our model in its current form does not replicate the continuous OH production conditions of the laser, as radiative gain or loss does not form part of the computation at this stage. Nevertheless, calculations on an ensemble of OH(4;11) in Ar at 300 K show that the principal laser lines observed are strongly populated at some point in the ensemble’s evolution, and so, a steady-state concentration would be expected under experimental conditions.22 This suggests that our computational model constitutes a reasonably realistic representation of a gas ensemble containing atoms and diatomic molecules.

3. RESULTS 3.i. OH(8;6) þ Ar. Figure 1 shows the evolution of Tmode in a 1:10 mixture of OH(8;6) þ Ar as these species equilibrate over the course of 600 collision cycles. There are several points of note relating to this figure. First, the reduction in Tv of OH from its initial value is quite slow compared to the cases, discussed below and reported earlier,2,3 in which the bath gas is a diatomic molecule. The first-collision cross section, calculated from vibrational state population data, is 0.32 ( 0.2 Å2 for population loss from OH(8;16) in collision with Ar. This value represents the mean of 10 separate calculations, but nevertheless, the small number of OH molecules leads to errors that are quite high. The cross section is larger than that calculated for the first collision in the N2(8,10) þ Ar ensemble (0.11 ( 0.07 Å2) under the same conditions,3 and this reflects the opportunity for population transfer through quasiresonant vibration
rotation transfer (QVRT) when OH is the target species. In this phenomenon, the collision converts low-rotation
high-vibration states quasiresonantly to lower-vibration
higher-rotation states within the OH species; a more detailed discussion is provided below. Following this first, slow, initial phase, vibrational cooling then

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accelerates between 150 and 400 collision cycles. A second, and unusual, feature is the sharp rise in Tr for OH from its initial value to reach ∼2600 K at CN = 150 just prior to the onset of a lengthy period of rotational cooling. This latter phase of Tr reduction also corresponds to one of somewhat faster vibrational cooling. A third feature is that, over the range CN ≈ 150
600, Tr is reduced by nearly an order of magnitude, reaching ∼350 K by CN = 600, some 1000 K lower than the mutual equilibration temperature of Tv and Tt. This is quite surprising and, if a real phenomenon, is potentially of some significance in view of the widespread use of Tr values from OH rotational intensities to determine gas-phase kinetic temperatures. Possible reasons for this lowered value of Tr are considered in the Discussion and Conclusions section. Excited molecules in atomic bath gases lack critical fast relaxation routes such as the resonant and quasiresonant multiple V
V processes that dominate energy transfer when the bath gas is the same or some other diatomic.3 In general, molecules in high-nOH states are highly resistant to pure rotational relaxation within a vibrational manifold because the energies between successive rotational states are considerably greater than kT at 300 K. In baths of atomic partners, the principal available pathway for de-excitation of molecules in high (v;j) states is vibration
rotation transfer (VRT), often a slow process.2,3 Highly excited OH, however, because of its characteristic rovibrational energy level structure, has quasiresonant vibration
rotation transfer (QVRT) pathways in which population transfers near-resonantly to states of lower vibration, but higher rotation quanta within OH.23,24 QVRT can be very effective even at low kinetic temperatures as there are generally inter-vOH state transitions with small energy and angular momentum gaps. Within our closed system, Tt begins to increase significantly from CN ≈ 50 and exceeds 1000 K by CN = 200. From this point, Tr begins its slow decline, as relaxation within each vOH manifold begins to depopulate high-nOH states very effectively. Note that this translational warming might not be a characteristic of more open systems such as the upper regions of Earth’s atmosphere, and in such a case, high-n states could survive more collisions than our computations imply. Smith et al.24 first suggested QVRT as the mechanism responsible for the appearance of (vOH = 1, 2; nOH e 33) in measurements of Earth’s airglow at altitudes of 80
110 km using the CIRRIS 1A interferometer on the Space Shuttle. Here, we demonstrate that this mechanism does indeed lead to population of nOH > 30 in ensembles of atomic or diatomic gases. The modal temperatures present an averaged view of the equilibration process, and although they are a useful means of characterizing the evolution toward equilibrium for a particular target molecule and bath, they provide relatively little insight into the molecular mechanism of the process. For this latter aspect, the quantum state distributions, and their variation with collision number, are considerably more informative. Generally, it is the early part of an ensemble’s evolution that is the most active. Figure 2 shows the vibrational state populations (i.e., the sum of populations of all rotation states in a given vOH vibration state) of OH following 5, 10, and 20 collisions. OH is initially in its (8;6) state. Note that the population scale has been expanded in Figure 2, and subsequent plots of vibrational population, to show the transfer in both species after a small number of collisions. As a result, the vOH = 8 populations are off-scale. It is clear from Figure 2 that, over the range CN = 1
5, small amounts of population are transferred to vOH = 7, 6, and 5 and that, by CN = 20, this has extended to vOH = 0, 1, and 2. Figure 3 4171

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Figure 2. OH vibrational state population distributions in the OH(8;6) þ Ar ensemble after 5 (red columns), 10 (green columns), and 20 (blue columns) collisions. Note that the vOH = 8 populations are all off the scale. The total number of OH molecules in the ensemble of approximately 8000 molecules is 727.

Figure 3. Rotational distributions among high and low vOH levels after 5, 20, and 100 collision cycles in the OH(8;6) þ Ar ensemble. See text for discussion.

quantifies the nOH destination states for this flux over a slightly larger collision number range. From this figure, the process of downward redistribution of population in vOH and upward redistribution in nOH can be seen very clearly. This begins almost immediately to vOH = 7 and 6 and gradually spreads, as CN increases, to vOH = 0, 1, and 2. (Note that, for clarity, not all vOH states are represented in Figure 3.) The predominant rotation state change over the first 20 collisions is excitation, and this continues to CN = 100 and beyond but is now balanced by a substantial degree of rotational cooling within vOH = 8, 7, and 6. At this stage of evolution, Tt has risen to ∼700 K, and so, VRT within vOH levels is reasonably facile. However, high-nOH states persist in vOH = 0 and 1 at least up to CN = 250 3.ii. OH(8;6) þ N2(0;10). Figure 4 illustrates the increased rate of equilibration for excited OH when a diatomic molecule forms the bath gas, and with this combination, the first-collision cross section is calculated to be 1.55 ( 0.3 Å2. Vibrational temperatures of both OH and N2 have equalized within the first 100 collisions, and for both minor constituent and bath, modal equilibration is complete by CN = 400, with the exception once again of Tr for OH. Note that the Tmode scale has been expanded to show more clearly the evolution of the low-energy modes.

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Figure 4. Variation of modal temperatures of OH, initially in (v;n) = (8;6), with number of collision cycles for a 1:10 mixture of OH(8;6) in N2(0;10) at 300 K. The total number of molecules is (nominally) 8000. For OH, Tv is shown as solid black squares; Tr, as solid red circles; and Tt, as solid green triangles. Tmode for N2 follows this same color code but with open symbols.

Calculations were continued to 1000 collisions and showed little change beyond that shown at CN = 400. Tr for the target molecule once again rises markedly at the outset, although much faster than for OH(8;6) in Ar, with the peak value of Tr now occurring at CN ≈ 25. From this point onward, Tr for OH falls steadily, roughly paralleling the drop of Tv for OH as it does. By CN = 140, Tr for this species has become the lowest of the modal temperatures, leveling out at ∼500 K over the range CN = 400
1000, some 1000 K below the translational equilibration temperature of the two species. The steady-state Tr value for N2 is also significantly below the equilibrium value of Tt. An important feature in the rapid overall equilibration in this system is the steady rise of Tt in both species from the outset. Thus, by CN = 20, Tt > 600 K, a temperature not reached with Ar as the bath gas until CN > 100. Much of this kinetic energy increase originates in the rotational energy mismatch between target and bath species, which results in relatively large amounts of energy release to recoil in each intermolecular transfer process. Here, this mismatch is quite large, as Be = 1.9982 and 18.91 cm
1 for N2 and OH, respectively.25 The principal factors responsible for the rapid vibrational equilibration process shown in Figure 4 are evident from the individual vibration state populations of both target OH and bath N2 molecules early in the ensemble’s evolution. These are shown in Figure 5 for OH and N2 after 5, 10, and 20 collisions and reveal the influence of specific, near-resonant, multiquantum V
V exchanges in accelerating the vibrational de-excitation of OH. These coupled, simultaneous, intramolecular transitions involve relaxation in OH and excitation in N2. In general, the transitions in different species will involve different numbers of vibrational quanta. These fast V
V processes have been found3,26
28 to dominate when the overall energy discrepancy in the combined event in the two species is small and the fastest processes appear generally to be associated with small changes in nOH. Rotations in target and bath gases play an important role, as discussed in more detail below. In addition to rotations, translations represent an additional energy repository and, being unquantized, are able to accommodate “leftover” energies (i.e., those not in quantum states). This is likely to be the reason that Tt is generally the highest of the modal temperatures after some 1000 or so collisions. 4172

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Figure 5. Vibrational distributions in OH (red columns) and N2 (green columns) following 5, 10, and 20 collisions. These plots illustrate the preferential population of specific vOH and vN2 states by fast, nearresonant, V
V transfer.

In the ensemble containing OH(8;6) and N2(0;10) molecules, families of near-resonant V
V events are found that are able to create a population cascade, with vOH = 8 rapidly depopulating through successive collisions. This occurs because several vibrational states of OH lying below vOH = 8 are also able to undergo near-resonant V
V transfer in conjunction with N2 partners, a process that is highly state-selective. Initial and final vOH states in the cascade process strongly influence the evolution of vOH population distributions through the first and subsequent collision cycles. Thus, from vOH = 8, the initial collision cycle results in near-resonant transfer from vOH = 8 to vOH = 7 and 3, with subsequent cycles taking vOH = 7 to vOH = 6, 3, and 1. With N2 as the collision partner, the states vOH = 6, 3, and 1 undergo no further near-resonant V
V exchange. However, population that finds its way into vOH = 5 and 4 by other routes is likely to be rapidly transferred to vOH = 2 and 0, respectively, by this same mechanism. Two prominent V
V transfers can be seen from Figure 5. The first is the coupled pair [(8f7)OH;(0f1)N2], for which the overall energy discrepancy (rotations excluded) is ∼50 cm
1, and the second is [(8f3)OH;(0f6)N2], with a (vibrational) energy defect of less than 10 cm
1. This notation labels initial and final vibration quantum numbers only. In Figure 5, the vOH = 7 and vOH = 3 levels are seen to be enhanced relative to neighboring states. Furthermore, secondary processes depopulate vOH = 7 through the nearresonant [(7f6,3,1)OH;(0f1,5,8)N2] processes, the first two of these in association with nOH change, resulting in enhanced population of the destination states in OH and N2. Population of these states can be seen as the ensemble develops. The vOH = 1, 3, and 6 levels have no near-resonant outlets, and with slow VRT being the only alternative de-excitation mechanism, population is seen to build in these levels. States of this kind, in which population is partially trapped, might be of interest in gas laser design.

Figure 6. Rotational state distributions for selected vOH states in a N2 bath. The CN = 2 plot illustrates the rotational state selectivity of the near-resonant V
V processes described in the text, with vOH = 7 and vOH = 3 peaked at nOH = 6 and others close to this value. Secondary, cascading collisions have an impact on the distribution after CN = 10, and the development of the bimodal high-nOH
low-vOH populations of QVRT alongside low nOH from near-resonant V
V can be seen in these and the CN = 30 data.

Figure 7. Variation of modal temperatures of OH, initially in (v;n) = (8;6), with number of collision cycles for a 1:10 mixture of OH(8;6) in O2(0;12) at 300 K. The total number of molecules is (nominally) 8000. For OH, Tv is shown as solid black squares; Tr, as solid red circles; and Tt, as solid green triangles. Tmode for O2 follows this same color code but with open symbols.

In addition to near-resonant V
V transfers, collision-induced QVRT between vOH states also occurs and is responsible for population of the high rotational levels that are seen in the early part of the ensemble’s evolution. Thus, when a diatomic forms 4173

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Figure 8. Vibrational distributions in OH (red columns) and O2 (green columns) following 2, 5, 10, 20, and 30 collisions.

the bath gas, two competing processes are expected: one populating OH states close to the initial value nOH = 6 in the vOH levels discussed in the previous paragraph and a second group whose nOH values will be considerably higher. That this is the case is evident from Figure 6. When N2 is the bath gas, two principal groups of rotational populations are seen, and within these groups, the results of primary, secondary, and further collision cycles can be discerned. For example, in the CN = 2 distribution, the fast, near-resonant V
V transitions to vOH = 7 and 3 involve ΔnOH = 0, whereas the secondary process to vOH =1 requires ΔnOH = 1 for energy conservation. As the ensemble evolves to CN = 10, the separation of the high- from the low-nOH groups, the former populated through the QVRT mechanism, becomes evident. At this relatively early stage, high-nOH states in vOH = 1are populated with maximum values now approaching 30. This is achieved through transfers to lower and lower vOH states by successive collisions, increasing in nOH with each step. The result is the development of a bimodal distribution of overall OH rotational states, one group being centered on the initial nOH value with relatively small variations and the other more widespread with high nOH quanta associated with low vOH and vice

versa and large amounts of energy remaining within the OH molecule. As the translational temperature of the ensemble increases, nonresonant VRT becomes much faster, and the high rotational states are deactivated as the ensemble continues to evolve. When diatomics form the bath gas, this process begins much earlier than with Ar. With N2(0;10), the peak value for Tr occurs at CN ≈ 25, and it occurs at CN ≈ 30 when O2(0;12) forms the bath gas. For Ar, this peak value is close to CN = 150. Recent work12 suggests that the position of the Tr peak is also sensitive to the initial choice of ensemble temperature (300 K throughout the work reported here). Two further aspects are of particular interest. First, as with Ar bath gas, by the time most of the modes have equilibrated, Tr for OH in N2 becomes the lowest modal temperature within the ensemble by approximately 1000 K. As noted previously, these thermodynamically closed systems will equilibrate to a temperature well above that initially set (300 K) as the >23000 cm
1 initially in the form of OH vibration is redistributed within the ensemble. That Tr values might be lower than Tt is not wholly surprising for the reasons we suggest in the Discussion and Conclusions section. The second noteworthy feature is that 4174

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Figure 9. Rotational state distributions for selected vOH states in the OH(8;6) þ O2(0;12) ensemble. The effects of near-resonant V
V transfer can be seen after only a single collision cycle. In addition, population of high-nOH levels in low-vOH states is already apparent by the time five collision cycles have occurred.

there appears to be a hierarchy in the system’s strategy for equilibration, with vibrations taking precedent. Strong rotational heating accompanies the excitation and de-excitation of these major repositories of energy. Only when the vibrations are at or approaching mutual equilibration are the rotations seen to cool below the ambient translational temperature. Prior to this point, their distributions appear primarily to be driven by the needs of the dominant vibrational equilibration. This aspect of the overall relaxation process is also discussed further below. 3.ii. OH(8;6) þ O2(0;12). The evolution of modal temperatures for an ensemble consisting of OH(8;6) þ O2(0;12) in a 1:10 ratio at 300 K is shown in Figure 7. As with other work described here, the computations were carried out on ensembles of (nominally) 8000 molecules. The initial fall of Tv in OH is again very rapid. The single- (first-) collision cross section for this pair was found to be 1.6 ( 0.25 Å2. Other features of the Tmode versus CN plot are very similar to that of OH in N2, the only significant difference being the earlier equilibration, by around 100 collisions, of modal temperatures (Tr for OH excepted) with O2. This similarity suggests the operation of near-resonant V
V transitions and subsequent cascades in the case of OH(8;6) þ O2(0;12) of the kind described above. In fact, more nearresonant routes are open for fast V
V transfer in the OH(8;6) þ O2(0;10) ensemble than when N2 is the bath gas. Population is transferred down the OH vibrational ladder by near-resonant

Figure 10. Modal temperatures versus number of collision cycles for OH(8;6) in an airlike mixture [N2(0;10) þ O2(0;12) in a 4:1 ratio]. For OH, Tv is shown as solid black squares; Tr, as solid red circles; and Tt, as solid green triangles. Tmode for N2 follows the same color code but with open symbols, and Tmode for O2 follows the same shape code but with open blue symbols.

exchange with O2 initially through the three distinct routes [(8f4,5,6)OH;(0f6,5,3)O2]. As further collisions follow, [(5f4,3,2)OH;(0f2,4,6)O2], [(4f3,0)OH;(0f3,9)O2], and [(3f2,0)OH;(0f2,7)O2] processes occur with energy discrepancies requiring only small Δn transitions. Again, the above 4175

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The Journal of Physical Chemistry A notation refers to OH and/or O2 vibrations. As in the case of OH(8;6) with N2(0;10), the vibrational populations in the early part of the ensemble’s evolution reflect these near-resonant pathways for both species in the ensemble. This can be seen in Figure 8, where, after two collisions, OH population has been transferred to vOH = 4, 5, and 6, with equivalent increase in vO2 = 7, 5, and 3, respectively. Additional collisions continue the population cascade to middle and low vOH values, so that the central range of the vibrational distributions becomes fairly evenly populated. After 30 collision cycles, the low-vOH states begin to dominate, and the system resembles a high-temperature Boltzmann distribution. In addition to these near-resonant population cascades, QVRT continues, eventually populating low-vOH levels. The rotational distribution in OH(8;6) þ O2(0;12) is bimodal right from the first collision, as Figure 9 makes clear. Low-nOH states are strongly populated by the V
V mechanism, and at the same time, QVRT causes transitions to high-nOH levels of lowvOH states. The nOH states populated by QVRT rapidly migrate down to low-vOH states, with nOH g 30 in vOH = 0 after only five collision cycles. However, with this bath gas and temperature, the high-nOH states are relatively quickly relaxed. 3.iii. OH(8;6) in “Air” [N2(0;10) þ O2(0;12) in a 4:1 Ratio]. Finally, we consider the equilibration of OH(8;6) in a bath of simulated air [i.e., a 4:1 mixture of N2(0;10) þ O2(0;12)]. With this combination, it is of interest to see which is the dominant partner in the airlike mixture. Furthermore, there is potential atmospheric relevance because OH is a highly significant minor component in several regions of Earth’s atmosphere and is formed in the mesosphere in excited rovibrational states7,9 from the H þ O3 reaction. The evolution of modal temperatures as this ensemble equilibrates is shown in Figure 10. It is evident that O2, despite being the lesser component of the bath species, is at least as significant as N2 in the overall equilibration. Mutual equalization of Tv for OH with O2 in the air mixture occurs faster than for OH with N2, a reflection of the enhanced possibilities for near-resonant V
V transfer in the former case compared to the latter. However, on the whole, the overall Tmode versus CN plot could largely be predicted from those shown above for the individual bath constituents. Although not shown here, similar behavior is seen in the vibrational and rotational state populations in the critical early stages of ensemble evolution in air to that of the individual components in their interaction with OH. The same near-resonant processes for OH(8;6) with N2 and O2 are seen in the vibrational distributions with air, as is the bimodal distribution of rotations. Again, nOH > 30 states of OH are populated during the ensemble’s evolution.

’ DISCUSSION AND CONCLUSIONS Calculations on OH(8;6) in a bath of Ar demonstrate that the equilibration of highly rovibrationally excited OH is an inherently slow process at the initial temperature (300 K) used here. Direct vibration
rotation transfer for most diatomics is slow14 because large changes in rotational state are needed and the probability of Δj change falls rapidly with the magnitude of Δj. However, in certain diatomic molecules, the hydrides in particular, the magnitude of the anharmonicity and large rotational constant determine an energy level structure in which small Δn (or Δj) transitions can occur at low ΔE, resulting in hopping of population between different rovibrational levels. This quasiresonant VRT is a well-known characteristic of highly excited OH

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and begins early in the evolution process when translational temperatures are still quite low. Population can become trapped in low-vOH
high-nOH states as intra-vOH state relaxation requires kinetic energies greater than those available until quite late in the evolution. The result is a bimodal distribution of nOH states in the early stages with one nOH group centered around nOH = 6 and a second spread over a wide range of nOH values with, at certain stages, nOH > 30. Translation
vibration equilibration occurs after ∼500 collision cycles, but Tr drops well below Tv and Tt before this point is reached, finally settling to a value some 1000 K below the other modal temperatures. The presence of diatomic molecules in the bath gas introduces allows near-resonant V
V transfer of the kind conclusively demonstrated to occur in N2 by Ottinger26 and Piper27 and in O2 by Wodtke et al.28 and found to influence strongly the relaxation of vibrationally excited N2 in diatomic bath gases.3 This process can be very fast, a consequence of zero or very small Δn change in OH and Δj change in the bath diatomic. This form of energy transfer results in a group of rotations in low-vOH states that are centered on nOH = 6. Further possibilities for nearresonant V
V transfer exist from vOH states subsequent to the first collision cycle, resulting in a population cascade as further collisions also find near-resonant pathways. Simultaneously, QVRT leads to high-nOH levels in low vOH states, as found with Ar, but with diatomic molecules forming the ensemble, this process occurs much earlier in the evolution and the overall relaxation is also much advanced. It is evident that the downward cascade of population in vOH is largely responsible for the fast equilibration found when N2 and/or O2 forms the bath gas. The existence of near-resonant V
V pathways of the kind so dominant here is a property of a particular target
gas molecule pair, as energy-matching upward transitions in the bath gas form an essential element in the mechanism. Furthermore, there is a strong dependence of the primary process in the cascade on the initial vibration state that is particularly pronounced in highly anharmonic vibrators such as OH. In addition to the dominance of near-resonant exchange processes revealed here and in our previous study,3 certain other unexpected forms of modal behavior were noted as the ensembles evolve and are discussed in more detail here. The first of these is the finding that, in each case discussed above, Tr for OH is always the lowest of the modal temperatures under the steadystate conditions at the end of the equilibration process. This effect clearly needs more systematic investigation but potentially might be of some significance. OH rotational distributions are used to measure ambient temperatures in both atmospheric9 and combustion6 applications, with the assumption that the distribution of intensities in this low-energy mode accurately reflects the mode-equilibrated temperature. Our database is thus far small, but the equilibrated Tr values of the target molecule and bath diatomic are lower than Tt and Tv in ensembles containing excited CO2 and N2,3 although generally by much smaller amounts than were found here for OH. Calculations on a range of target molecule
bath combinations are currently underway to clarify the origin of this unexpected reduction of Tr and will be reported in due course. The possibility that it constitutes an artifact of our closed-system multicollision computational model must be considered, although a systematic problem of this kind might be expected to worsen as the number of collision cycles increases, which is not the case. The Tmode plot for OH
Ar in Figure 1 is truncated at CN = 400 because little change occurs beyond this point. In the 4176

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The Journal of Physical Chemistry A cases of OH with N2 and O2, calculations extend to CN = 1000 and beyond, from which it is clear that the steady-state modal temperatures persist for at least the last 500 collision cycles. A physical explanation for unexpected cooling of target-molecule rotational temperatures can be advanced that might explain why the effect is particularly marked in OH. This argument is based on the consequences of quantization and, in particular, the quadratic dependence of quantum state energy on rotational quantum number. As a result, the energy cost per unit change of rotational angular momentum will always be greater for positive ΔnOH than for negative ΔnOH. In general, at a given ambient temperature, more collision trajectories will be able to access lower-nOH states than those able to open higher-nOH channels. Thus, for example, at Tt ≈ 1500 K, the value reached for Ar close to the very end of the equilibration, access to ΔnOH g þ4 (from the initial nOH = 6) requires velocities from well into the high-energy tail of the distribution. However, access to rotational levels down to ΔnOH =
6 would become open at a much lower temperature distribution, such as one peaking at ∼1000 K. It is possible, therefore, that the apparent overcooling of Tr noted above might be a more widespread phenomenon and is particularly marked here because OH has a large rotational constant. However, we reemphasize the need for closer investigation before this effect can be regarded as substantiated. The second unexpected finding is that there appears to be an order of precedent for the thermalization process. In this regime, the requirements of vibrational modes determine which rotation states are populated and, hence, the residual energy partitioned into translation, as the system begins with the strong initial drive to achieve vibrational equilibrium. In the course of this process, Tr for OH rises to around 2000 K as rotational populations respond to the energy and angular-momentum conservation requirements of vibrational equilibration. Only when the vibrational modes are very close to equilibration does Tr for OH begin to cool. This pecking order in disposal of excess energy is somewhat reminiscent of an effect found29 in the vibrational predissociation of dimers of hydrogen halides such as HCl weakly bonded to C2H2. In that dissociation process, in which the dimer disposes of excess energy into the fragment species, rotations appear to be subordinate to the requirements of vibrational change. In that case, the priority ordering was interpreted29 as the need for the system to reduce the angular-momentum “load” that results when large amounts of energy must be disposed into rotation in molecules of small rotational constant. In the case of vibrationally excited OH equilibrating in a diatomic bath gas, a related but more general explanation is tentatively suggested here, which is that equilibration is driven by a process of force minimization. Here, in our primarily Newtonian model, force is given by F = dp/dt (where p is momentum and t is time), although the argument differs little on using the Lagrange/ Hamilton definition, F =
dV/dq (where V represents potential energy and q represents position). Vibrational motion in a diatomic represents a substantial reservoir of force in the form of reciprocating momentum, the magnitude of which increases markedly with vibrational quantum number. The forces available in vOH = 8 are considerably larger than the torques stored in rotations in their initial nOH = 6 state, and the force differential between the vOH = 8 molecules and those of the bath gas in v = 0 provides the initial impulse. Although this possibly represents a concept new to gas-phase molecular behavior, the notion of force minimization is not novel. In electronic structure calculations,

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force reduction, often through the steepest gradient, is used to find energy-minimum molecular structures. In the evolution of OH-containing ensembles, the data indicate that, once the force differentials of greatest magnitude have been minimized, those involving the rotational torques can be reduced in collisions that generate recoil orbital angular momentum, increasing Tt until translation becomes the highest-temperature mode. These effects are particularly marked in OH-containing ensembles because of the unusual rotational behavior of this species at low CN. However, the equilibration imperatives noted here can be seen in other diatomic target molecules,2,3 although the outcomes change markedly with the nature of the target and bath molecule pair.

’ AUTHOR INFORMATION Corresponding Author

*E-mail A.J.McCaff[email protected].

’ ACKNOWLEDGMENT The authors thank Dr. H. Cox for suggesting the analogy between force reduction in the ensembles described here and procedures in electronic structure calculations. ’ REFERENCES (1) Marsh, R. J.; McCaffery, A. J. J. Chem. Phys. 2002, 117, 503–506. (2) McCaffery, A. J.; Marsh, R. J. J. Chem. Phys. 2010, 132, 074304
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