Bath Model for N2 + C6F6 Gas-Phase Collisions. Details of the

Mar 30, 2015 - *Phone number: 806-834-3152. E-mail: [email protected]. This article is part of the Steven J. Sibener Festschrift ... data is made avai...
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Bath Model for N2 + C6F6 Gas-Phase Collisions. Details of the Intermolecular Energy Transfer Dynamics Amit K. Paul, Swapnil C. Kohale, and William L. Hase* Department of Chemistry and Biochemistry, Texas Tech University, Lubbock, Texas 79409, United States ABSTRACT: Relaxation of vibrationally excited C6F6* in a thermalized bath of N2 molecules is studied by condensed-phase chemical dynamics simulations. The average energy of C6F6 as a function of time, ⟨E(t)⟩, was determined using two different models for the N2−C6F6 intermolecular potential, and both gave statistically the same result. A simulation with a N2 bath density of 20 kg/m3 was performed to check the convergence and validate the results obtained previously with a higher bath density of 40 kg/m3. The initial ensemble of C6F6 is nearly monoenergetically excited, but the ensemble acquires as energy distribution P(E) as it is collisionally relaxed. An evaluation of P(E) and the root-mean-square deviation ⟨ΔE2⟩1/2 of P(E), versus time, shows that P(E) first broadens and then narrows. Simulations with the C6F6 vibrational excitation energy of 85.8 kcal/mol, studied experimentally, show that the width of P(E) does not affect the average collisional deactivation rate. The role of the intramolecular vibrational frequencies on the energy transfer dynamics was studied by changing the mass of the F-atoms of C6F6 to that of an H-atom. The resulting increase in the frequencies decreased the efficiency of collisional energy transfer. Simulations with classical and quantum microcanoical ensembles of the C6F6 initial vibrational energy were compared, and the initial relaxation rate was slower for the quantum ensemble.

I. INTRODUCTION Collisional intermolecular energy transfer (IET) from highly vibrationally excited molecules is important in a number of different processes, including pyrolysis, combustion, and atmospheric chemistry.1−4 The efficiency and dynamics of IET have been studied both experimentally5−7 and by atomistic simulations.8,9 This work has shown that the efficiency of IET depends upon the degree of vibrational excitation and properties of both the excited molecule and the deactivating collision partner. A number of different experimental methods have been used to study IET, i.e. infrared fluorescence detection,10,11 infrared laser transient absorption spectroscopy,12,13 diode laser spectroscopy,14 time-resolved UV absorption spectroscopy,15,16 and velocity map imaging.17,18 Some lesser used experimental approaches are kinetically controlled selective ionization detection,19 infrared multiphoton absorption coupled with time-resolved infrared fluorescence,20 and supersonic free jet expansion.21,22 The standard chemical dynamics atomistic simulation approach is to calculate classical trajectories for individual collisions between the excited molecule and the collision partner.23−25 The simulation is usually performed for a specific initial energy of the excited molecule and a specific collision energy. A large number of trajectories must be calculated to average over the random variables of the collisions, i.e. the impact parameter, orientations of the collision partners, rotational angular momenta of the colliding partners and their orientations, and phases of the vibrational modes of the excited molecule. In a previous study,26 a bath model was used to study intermolecular energy transfer for collisions of N2 with highly © 2015 American Chemical Society

excited C6F6. The excited C6F6 molecule was placed in a bath of N2 molecules, and the average energy of C6F6, ⟨E⟩, was determined versus time, t. A sufficiently large bath was used that heating of the bath was negligible, and it could be assumed that the bath remained at a constant temperature. The C6F6− bath collisions average over all the random variables of the C6F6 + N2 collisions, including impact parameter and C6F6 and N2 orientations. If the slope of the resulting ⟨E⟩ versus t, i.e., d⟨E⟩/ dt, is divided by the collision frequency, ω, the average energy transferred per collision, ⟨ΔEc⟩, is obtained. For the C6F6 + N2 simulations, excellent agreement was found between this ⟨ΔEc⟩ and that determined experimentally.27 A simulation similar to this one for C6F6 + N2 was recently reported for nitromethane in an Ar bath.28 This work reports additional details and new extensions of the previous bath model simulations of IET for C6F6 + N2. In the previous study, the sensitivity of IET to the form of the C6F6 + N2 intermolecular potential was investigated for a high N2 bath density, and in the current work this investigation is extended to a low bath density. The previous simulations were extended to a lower bath density to ensure that the single collision, low-pressure limit is attained, which includes additional approaches for analyzing the low-pressure limit. Results from three new extensions of the previous simulations are presented. The energy of C6F6 versus time is studied in Special Issue: Steven J. Sibener Festschrift Received: December 28, 2014 Revised: March 27, 2015 Published: March 30, 2015 14683

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molecules are far apart from each other, as well as from C6F6, and have a very low collision frequency. For this situation, it is a good approximation to consider a neighborhood “sphere” of solvent molecules around C6F6 outside of which interactions between C6F6 and the solvent are negligible. Such a procedure is well-known for liquid-phase simulations and commonly known as a neighbor list algorithm.34 In the current simulation with a N2 bath density of 20 kg/m3 in a cubic simulation box with sides of 132.8 Å, we applied the neighbor list algorithm in the condensed-phase version of VENUS.29,30 A sphere with a radius of 15 Å, containing N2 molecules around C6F6, is considered for the current study. This sphere is updated at every 10 integration time steps. This reduces the total number of two-body interactions calculated by a factor of 150 for a typical trajectory, with an approximate factor of 5 decrease in the computation time. Tests showed that results obtained with a 15 Å sphere are consistent with a full calculation but not those obtained for a sphere with a radius less than 15 Å.

more detail by determining: how its distribution evolves with time, i.e., P(E;t), the root mean squared values for these distributions, ⟨ΔE2⟩1/2, and the role of the C6F6 energy distribution on the ensuing P(E;t) and the average energy transferred per collision, ⟨ΔEc⟩. Also investigated are the effects on IET of increasing the intramolecular vibrational frequencies by replacing the F-atom mass of C6F6 with that of the H-atom and using different classical and quantum microcanonical models for adding the initial excitation energy to C6F6 for the simulations.

II. SIMULATION PROCEDURE Details of the simulation procedure are described in our previous publication,26 and thus only a brief description is presented here. The simulations were performed with a modified version of the chemical dynamics computer program VENUS.29,30 The system is prepared by placing 1000 N2 molecules inside a cubic box with periodic boundary conditions, with the high energy C6F6 molecule at the center of the box. The size of the box is determined by the desired N2 bath density. The initial translational and rotational energies of C6F6 are chosen from their thermal distributions at 298 K. Three random sampling methods were used to add the C6F6 initial vibrational excitation energy. They are classical microcanonical normal mode sampling,31,32 as used in our previous study;26 the same classical microcanonical normal mode sampling, but the total energy included the C6F6 zero-point energy (ZPE); and quantum microcanonical normal mode sampling,33 which explicitly includes ZPE. For the N2 bath, a Boltzmann probability distribution and a classical random phase were used to choose the bond lengths of the N2 molecules. The x, y, and z, components of the initial velocity for each N-atom are sampled from its one-dimensional velocity distribution at 298 K. Such sampling adds on average 3kBT/2 to the center of mass translation, kBT to rotation, and kBT/2 to vibrational kinetic energy to each N2 bath molecule. A MD simulation is then performed to equilibrate the N2 bath with C6F6 fixed with its initial coordinates and velocities during this process. The equilibration is a stagewise process where each stage is divided into two parts called sampling and monitoring. In the sampling part, the velocities vx, vy, and vz are resampled at each 5 integration steps of a total of 2000 steps. For the monitoring part no resampling of the velocities is done, and energy conservation is important. The temperature, radial distribution function, and vibrational, rotational, and center-ofmass translational energies of the N2 bath are monitored during this part of the equilibration. The stagewise equilibration is continued unless the properties obtained from the monitoring part gave satisfactory results. The trajectory is further integrated for 6 ps, and 8 sets of configuration and velocities are saved at random intervals. Each of these 8 configurations was selected as the solvent’s initial conditions for 12 trajectories, and for each random initial conditions were chosen for C6F6 as described above. Every trajectory is further randomized by keeping C6F6 rigid and equilibrating the N2 bath for 12 ps. Therefore, each trajectory in the ensemble has random initial conditions for C6F6 and the N2 bath. A simulation with a very low density bath is a challenge for the model used here. This is because all the trajectories have to be integrated for a sufficiently long time to obtain a considerable amount of energy transfer from the hot C6F6 molecule to the N2 bath. At a very low density the bath

III. ENERGY TRANSFER DYNAMICS A. Comparison of the Intermolecular Potentials PES1 and PES2. A detailed discussion of the complete intramolecular and intermolecular potential for the N2-bath/C6F6 system was presented in our previous publication.26 In this section we revisit the N2−C6F6 intermolecular potential to study in more detail the sensitivity of the energy transfer dynamics to this potential. It has already been shown that the N2−C2F4 system well represents the N2−C6F6 interactions without a substantial loss of accuracy.26 This reduces the electronic size necessary to model the N2−C6F6 system and decreases the ab initio computation cost. Two N2 + C6F6 intermolecular potential energy functions PES1 and PES2 were obtained by fitting ab initio potentials for four N2 + C2F4 orientations as described previously26 and shown in Figure 1. All four orientations were fit for PES2. However, for PES1 only the first three orientations were fit, and PES1 is substantially too attractive for the fourth orientation. In the previous study,26 PES1 and PES2 were found to give statistically the same average energy of C6F6 versus time, ⟨E(t)⟩, for the high N2 bath density of 324 kg/m3. This calculation was repeated, but for the lower density of 40 kg/m3, and PES1 and PES2 were again found to give statistically the same ⟨E(t)⟩. Thus, the inaccurate enhanced attractiveness for PES1 does not have a significant effect on the N2 + C6F6 intermolecular energy transfer dynamics for either low or high N2 bath densities. B. Low N2 Density Limit. If the time dependence of the C6F6 average energy is given by −d[⟨E(t)⟩]/dt = k[⟨E(t)⟩], ⟨E(t)⟩ decays exponentially and k, an energy transfer rate constant, is proportional to the N2 bath density.26 In contrast, the simulation ⟨E(t)⟩ are well-fit by the biexponential function ⟨E(t )⟩ = [E(0) − E(∞)][f1 exp( −k1t ) + f2 exp( −k 2t )] + E (∞ )

(1)

with f1 + f 2 = 1. In the low-pressure, single collision limit for C6F6 + N2 energy transfer, the proportionality constants C1 and C2 for k1 and k2, i.e.. k = C × ρ, become independent of density ρ. The previous study26 indicated that a 40 kg/m3 N2 bath density well approximates the single collision limit for the C6F6−N2 system since the C1 and C2 parameters were very similar for the 40 and 80 kg/m3 N2 bath densities. For the work reported here, an additional simulation was performed for a lower N2 bath density of 20 kg/m3 to ensure the single collision 14684

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Figure 2. Average energy of C6F6 versus time ⟨E(t)⟩ averaged over 96 trajectories, for N2 bath densities of 80, 40, and 20 kg/m3 (solid lines) and their fits to eq 1 (dashed line). There are 1000 N2 molecules in the bath. In the inset ⟨E(t)⟩ is plotted versus time × the N2 bath density.

previously.26 It is also of interest that the fitted E(∞) for the 20 kg/m3 bath density is 20.0 kcal/mol, which is very close to the total average energy of 20.6 kcal/mol for fully equilibrated C6F6 at 315 K. For 40 kg/m3, E(∞) = 22.2 kcal/mol. If we substitute the k’s in eq 1 by C × ρ, the curves of ⟨E(t)⟩ versus t × ρ for different densities should coincide if the C’s are exactly the same for the densities. This is seen from the inset of Figure 2, where the curves for 20 and 40 kg/m3 are nearly identical. As a result, and seen in Figure 3, there is hardly any

Figure 1. C2F4−N2 interaction potential energy functions PES1 and PES2 for four different orientations. The ab initio curves are calculated at the MP2/aug-cc-pVTZ level of theory, whereas the analytic curves are generated by fitting the ab initio potentials to the two-body functions in eq 3 of ref 26 for PES1 and in eq 4 of ref 26 for PES2. The right-hand side panels are for lower energies in the potential energy curves and their corresponding fits.

Figure 3. Average energy transfer per collision ⟨ΔEc⟩ versus the average energy of C6F6 for N2 bath densities of 20, 40, and 80 kg/m3 using the experimental L-J parameters.27 In the inset ⟨ΔEc⟩ is plotted versus [⟨E⟩ − E(∞)]; see eq 1.

convergence of the previous 40 kg/m3 simulation. A total of 96 trajectories were calculated for each density, with each trajectory integrated for 472 ps. The simulation with a 20 kg/m3 N2 bath density was performed with the neighbor list algorithm described in Section II. ⟨E(t)⟩ for the simulation densities of 20, 40, and 80 kg/m3 is presented in Figure 2 where the dashed lines represent the biexponential fits with eq 1. In the figure inset, the same curves are plotted as a function of time × density, i.e., t × ρ. The fitting parameters for the 40 and 80 kg/m3 simulations were given previously, and those for the 20 kg/m3 simulation are f1 = 0.240, f 2 = 0.760, k1 = 0.0138 ps−1, and k2 = 0.00409 ps−1. To within three significant figures, the proportionality constant C1 is the same for 20 and 40 kg/m3 densities, i.e., 0.00692, while the respective C2 constants are quite similar, i.e., 0.000204 and 0.000202, respectively. This analysis indicates that the 40 kg/m3 simulation is in the single collision limit as suggested

difference in the average energy transfer per collision ⟨ΔEc⟩ values for the 40 and 20 kg/m3 simulations. If the relaxation in eq 1 is a single exponential with energy transfer rate constant k, a plot of ⟨ΔEc⟩ versus [⟨E(t)⟩ − E(∞)] is linear with a slope of k/ω, where ω is the collision frequency. This slope becomes a constant when the system enters into the single collision limit. In the present case, the plot of ⟨ΔEc⟩ versus [⟨E(t)⟩ − E(∞)], as shown in the inset of Figure 3, is not exactly linear as the relaxation is biexponential but the curves obtained from the densities of 80, 40, and 20 kg/m3 coincide, indicating the system is in or near the single collision regime for these densities. The experimental27 collision frequency for the 20 kg/ m3 density is 0.21 × 10−12 s−1, and the mean free path is 25 Å. To compare with the latter value, the experimental collision diameter is 5.47 Å.26 14685

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Figure 4. Probability distribution of the C6F6 energy, for 96 trajectories, at the four trajectory integration times of 0, 24, 45, and 186 ps. The average C6F6 energies, over the 96 trajectories, for these times are 109.3, 85.8, 75, and 35.9 kcal/mol, respectively. Simulation results are for a N2 bath density of 40 kg/m3 and 1000 molecules in the bath.

trajectories and the width of the energy distribution is ∼4 kcal/ mol. For comparison, the sum of the 107.4 kcal/mol C6F6 vibrational energy and the average 298 K C6F6 rotational and translational energies is 109.2 kcal/mol. Figures 4(b), 4(c), and 4(d) give P(E) for the three additional times of 24, 45.0, and 186.0 ps, and ⟨E⟩ equals 85.8, 75.0, and 35.9 kcal/mol, respectively. The breadth of the energy distribution is a maximum at 45 ps, where the width at halfmaximum is 43 kcal/mol compared to 37 and 24 kcal/mol for the respective times of 24 and 186.0 ps. Thus, the breadth of P(E) first broadens and then narrows as C6F6 is vibrationally relaxed. The 24 kcal/mol distribution breadth at 186.0 ps is similar to the 17 kcal/mol breadth of the Boltzmann energy distribution for C6F6 at 298 K. It is of interest to determine if the breadth of P(E) affects the average energy transferred from C6F6, by the N2 collisions, and this is considered below. 2. ⟨ΔE2⟩1/2 of C6F6 versus ⟨E(t)⟩. It is quite interesting to see that the breadth of the energy distribution P(E) for C6F6 increases with time, up to a certain time, and then decreases. In order to obtain a clear understanding of this property over the entire simulation time, the root mean squared deviation of E from the average value ⟨ΔE2⟩1/2 was calculated and is ((1/ N)∑i(Ei − ⟨E⟩)2)1/2, where Ei is the energy of the ith trajectory at time t; ⟨E⟩ is the average energy over the 96 trajectories at t; and N is the total number of trajectories. The resulting plot of ⟨ΔE2⟩1/2 versus ⟨E⟩ is given in Figure 5 for the N2 bath density of 40 kg/m3. For the initial energy the value of ⟨ΔE2⟩1/2 is quite small as expected and then increases with an increase in ⟨E⟩. It reaches a maximum value at ⟨E⟩ ≈ 75 kcal/mol, which

C. Energy Distribution of C6F6 Versus Time. At the t = 0 beginning of the simulation, the vibrational excitation energy of C6F6 is distributed among its vibrational normal modes using classical microcanonical normal mode sampling.29,30 There is a narrow distribution of C6F6 energy for the 96 trajectories since the initial translational and rotational energies of C6F6 are taken from a 298 K Boltzmann distribution. The average values of the latter at 298 K are quite small (∼0.9 kcal/mol for each) compared to the initial vibrational energy of 107.4 kcal/mol. As the trajectories are integrated, each has a different profile of the total C6F6 energy versus time, E(t), and with time a distribution P(E) of C6F6 energy is formed for the 96 trajectories. The nature of this time-dependent distribution needs some attention to properly and fully understand the energy transfer dynamics. In this section, we investigate three different aspects of this C6F6 energy distribution and their effects on the value of the averaged energy transferred versus time. For understanding vibrational energy relaxation, as is done here, the first and second moments of P(E) are important35−37 (Shuler, Levine, van Kampen, 1960s to 1970s), and both are considered. 1. P(E) of C6F6 versus ⟨E⟩. The C6F6 energy distribution was determined for four different times in the simulation. Each time has a specific C6F6 average energy ⟨E(t)⟩. At the beginning of the simulation, i.e., t = 0, the energy distribution is due solely to the Boltzmann distribution of the C6F6 initial translational and rotational energies. If we only considered the initial C6F6 vibrational energy, the energy distribution would be a delta function. In Figure 4 histogram plots of the C6F6 energy distribution, P(E), are presented for the four times. Figure 4(a) displays P(E) at t = 0, where ⟨E⟩ = 109.3 kcal/mol for the 96 14686

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107.4 kcal/mol, and at this time the width of P(E) for the C6F6 molecules is substantially increased to 37 kcal/mol. Of interest is to determine if the breadth of the P(E) for C6F6 at ⟨E⟩ = 85.8 kcal/mol has any effect on the collisional energy transfer from C6F6 at this energy. To study this, a simulation was performed with a fixed C6F6 initial vibrational energy of 85.8 kcal/mol, with the C6F6 translational and rotational energies chosen from their Boltzmann distributions. The resulting initial P(E) for C6F6 is quite narrow. The result for a N2 bath density of 40 kg/m3 is presented in Figure 6, where it

Figure 5. Root mean square deviation of the trajectories’ total energy with respect to the average C6F6 energy, ⟨ΔE2⟩1/2 = ((1/N)∑i(Ei − ⟨E⟩)2)1/2, versus the C6F6 average energy. For the ensemble of 96 trajectories (N = 96), where Ei is the energy of the ith trajectory. Simulation results are for a N2 bath density of 40 kg/m3 and 1000 molecules in the bath.

corresponds to a time of 45 ps and then decreases to a value of ∼3 kcal/mol. As discussed in our earlier paper,26 the final average energy of C6F6 at the end of the 312 ps trajectories for a N2 bath density of 40 kg/m3 is 28.0 kcal/mol and does not reach the fully equilibrated thermal energy of 20.6 kcal/mol for the T = 315 K bath temperature. It is also of interest to compare the value of ⟨ΔE2⟩1/2 at the end of the simulation with the value for the C6F6 Boltzmann distribution at 315 K. To check this, only the C6F6 vibrational energy was considered since its average value is an order of magnitude larger than the translation/rotation average energy of 1.9 kcal/mol. The probability of the vibrational energy E is

( ) ρ(E)exp(− )

Figure 6. Average energy transfer per collision ⟨ΔEc⟩ versus the average energy of C6F6 minus the asymptotic fitted energy, i.e., [⟨E⟩ − E(∞)], for ensembles of 96 trajectories. The black curve is for a simulation with an initial C6F6 vibrational excitation energy of 85.8 kcal/mol, which is compared with the previous simulation26 where the initial C6F6 vibrational energy is 107.4 kcal/mol (red curve). Simulation results are for a N2 bath density of 40 kg/m3 and 1000 molecules in the bath. In the inset the average energy of C6F6 is plotted versus time. The time scale for the black curve is shifted by 24 ps.

is seen that E(t) for this simulation with an initial narrow P(E) for ⟨E⟩ = 85.8 kcal/mol is nearly identical to that for the previous simulation with an initial ⟨E⟩ of 107.4 kcal/mol and a resulting broad P(E) at ⟨E⟩ = 85.8 kcal/mol (see figure inset). A fit of the E(t) curve in the inset of Figure 6 to eq 1 gives values for the parameters E(∞), f1, f 2, k1, and k2 of 21.2 kcal/ mol, 0.168, 0.832, 0.0287 ps−1, and 0.00809 ps−1 which are close to values for the original simulation with an initial ⟨E⟩ of 107.4 kcal/mol (see Table 3 in ref 26). As a result and shown in Figure 6, the values of ⟨ΔEc⟩ are almost the same for the previous and current simulations. Similar calculations were done by Schatz and co-workers, where the simulations were done by considering both a single energy and an ensemble with the same average energy. The energy transfer quantities were found to be the same in both the cases.38−40 If one looks more closely to the curves of Figure 6 inset, there is a slight mismatch between the E(t) of the two simulations for the first ∼2 ps, which may change the value of ⟨ΔEc⟩ at t = 0. This issue is elaborated below in section E. D. Varying the Intramolecular Vibrational Frequencies: C6H6 versus C6F6. The efficiency of collisional energy transfer from a highly energetic molecule depends on its vibrational modes and their frequencies.41−44 Collisional energy transfer from a highly energized molecule to a “cold” molecule is thought to occur through gateway modes which are often low-frequency modes. It is found from experiments that the probability of transferring a quantum of energy from a lowfrequency bending mode of an excited molecule is larger than

E

P(E) =

ρ(E)exp − k T B

0

∫∞

E kBT

(2)

where ρ(E) is the C6F6 vibrational density of states at energy E. The average value of E is ⟨E⟩ = ∫ ∞ 0 EP(E)dE, and for T = 315 K and all other temperatures eq 2 gives ⟨E⟩ = 30kBT. A Monte Carlo procedure was used to determine ⟨ΔE2⟩1/2 from this P(E) distribution, and the resulting value is 2.72 kcal/mol and only slightly smaller than the ∼3 kcal/mol value obtained from the simulation. Such a difference is expected since the simulation did not reach thermal equilibrium at the end of the trajectories. However, if the C6F6 translational and rotational energies were included in the eq 2 P(E), the resulting ⟨ΔE2⟩1/2 may be ∼0.2−0.3 kcal/mol larger, and nearly exact agreement between the simulation and Boltzmann ⟨ΔE2⟩1/2 might be obtained. 3. Role of C6F6 Initial Energy. As discussed above in Sections 1 and 2, the distribution of the C6F6 energy for the 96 trajectories, which is quite narrow at t = 0 for the initial C6F6 vibrational energy of 107.4 kcal/mol, broadens as the trajectories are integrated for longer times. In comparing with experiment for a C6F6 vibrational energy of 85.8 kcal/mol, the slope d⟨E(t)⟩/dt for the simulation is evaluated at 85.8 kcal/ mol to determine the amount of energy transferred per unit time. The trajectories have been integrated for 24 ps when the average energy of C6F6 is decreased to 85.8 kcal/mol from 14687

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The Journal of Physical Chemistry C from a high-frequency stiff mode.42−44 To study this effect for our current simulation, we changed the mass of all the F-atoms of C6F6 to that for the H-atom, keeping all intramolecular and intermolecular potential parameters the same; i.e., the Morse stretch, harmonic bend, harmonic wag, and torsion parameters are the same as those for C6F6. The equilibrium C−H bond length for C6H6 is the same as C−F for C6F6.25 Thus, C6H6 considered here is not exactly benzene but a model called “model C6H6”. In this way we increased the intramolecular vibrational frequencies of the hot molecule and did not change its intermolecular potential with the N2 bath. The vibrational frequencies for C6F6 and for the “model C6H6” are listed in Table 1, where one can see that the frequencies for “model C6H6” are significantly higher.

Figure 7. Average energy transfer per collision, ⟨ΔEc⟩, versus the average C6F6/C6H6 energy minus the asymptotic fitted energy, i.e., [⟨E⟩ − E(∞)], for C6F6 (black curve) and “model C6H6” (red curve). The averages are over an ensemble of 96 trajectories. Simulation results are for a N2 bath density of 40 kg/m3 and 1000 molecules in the bath. In the inset the average energies of C6F6 and “model C6H6” are plotted versus time. The fit for the “model C6H6” curve to eq 1 is also given (blue curve).

Table 1. Intramolecular Vibrational Frequencies of C6F6 and “Model C6H6”a symmetry25

C6 F 6

model C6H6

E2u B2g A2u E2g B2u E1u E1g E2g A1g B1u E2u A2g B2g E1u E2g A1g E1u B2u B1u E2g

125(2) 172 197 271(2) 277 285(2) 379(2) 444(2) 479 592 608(2) 755 772 929(2) 1183(2) 1457 1598(2) 1611 1698 1822(2)

370(2) 557 602 715(2) 868 715(2) 900(2) 949(2) 959 1174 1130(2) 1246 1425 1567(2) 1737 1659 3505(2) 3505 3516 3511(2)

vibrational excitation energy is added to C6F6 and “model C6H6” using classical microcanonical normal mode sampling,31,32 which only includes the excitation energy and not the molecule’s zero point energy (ZPE). Details of this sampling technique are described elsewhere.31,32 For comparison we performed a simulation, for the N2 bath density of 40 kg/m3, using the same sampling model, but with the 33 kcal/ mol ZPE of C6F6 included with the excitation energy of 85.8 kcal/mol, a model identified as “classical microcanonical + ZPE”. Another sampling model considered for the 40 kg/m3 N2 bath density simulation is quantum microcanonical normal mode sampling, which considers each mode as quantum harmonic oscillator, and ZPE is explicitly included.33 In the inset of Figure 8 the results of the simulations described above for classical microcanonical, classical microcanonical + ZPE, and quantum microcanonical sampling are presented. The classical microcanonical sampling result was presented previously,26 and its initial ⟨E⟩ is smaller since it does

The vibrational frequencies are in cm−1. For “model C6H6” the masses of the F-atoms are changed to that for the H-atom without changing any of the other potential parameters.

a

A simulation was performed for “model C6H6” at a N2 bath density of 40 kg/m3, and as shown in the inset of Figure 7 the rate of energy transfer is substantially slower than for the C6F6 simulation. Fitting the E(t) curve for the “model C6H6” simulation with eq 1 gives the parameters E(∞), f1, f 2, k1, and k2 as 26.0 kcal/mol, 0.230, 0.770, 0.0103 ps−1, and 0.00442 ps−1, respectively. In Figure 7, ⟨ΔEc⟩ versus [⟨E⟩ − E(∞)] is given for the two ⟨E(t)⟩ presented in the inset of the same figure. The same experimental Lennard-Jones parameters are used to determine ⟨ΔEc⟩ values for both simulations; however, a reduced mass of C6H6−N2 is used for one, whereas a C6F6− N2 reduced mass is used for the other. At the experimental excitation energy of 85.8 kcal/mol, the ⟨ΔEc⟩ values are 1.74 and 0.84 kcal/mol for the C6F6 and “model C6H6” simulations, respectively. In comparison, for actual benzene excited with the same energy of 85.8 kcal/mol, and studied experimentally by time-resolved infrared spectroscopy,45 ⟨ΔEc⟩ = 0.126 kcal/mol for a bath of N2 molecules. E. Different Models for the Initial Intramolecular Energy. For all the simulations presented above the initial

Figure 8. Average energy transferred per collision ⟨ΔEc⟩ versus [⟨E⟩ − E(∞)] (see Figure 6) for simulations with the three different sampling models: quantum microcanonical, black curve; classical microcanonical + ZPE, red curve; and classical microcanonical (no ZPE), blue curve (see text). The averages are for ensembles of 96 trajectories. In the inset the average energy of C6F6 is plotted versus time for the three sampling models. 14688

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The Journal of Physical Chemistry C

at t = 0 for the 85.8 kcal/mol excitation energy, quantum microcanonical sampling gives a value of 2.55 kcal/mol. As seen from Figure 8, the rise of ⟨ΔEc⟩ as function of ⟨E⟩ at higher ⟨E⟩ for both the classical microcanonical and classical microcanonical + ZPE curves is exponential which may not be correct, whereas the curve for quantum microcanonical + ZPE is almost linear. Thus, initial sampling by using a quantum microcanonical ensemble may give a more accurate initial energy transfer rate as compared to classical microcanonical sampling.

not include the 33 kcal/mol ZPE. When the initial trajectory ensemble is integrated for a longer time and if the classical dynamics is ergodic,34 the system will attain a classical statistical ensemble regardless of the nature of the initial condition. This is apparently the result for the quantum microcanonical and classical microcanonical + ZPE simulations. As shown in the inset of Figure 8, their ⟨E(t)⟩ curves differ up to t = 48 ps but coincide for longer times. Also, as shown in the same plots, ⟨E(t)⟩ for the classical microcanonical (no ZPE) simulation coincides with the ⟨E(t)⟩ for the other two simulations at 270 ps, and all three curves remain the same until the trajectories are concluded. A careful inspection of the three ⟨E(t)⟩ curves in the inset of Figure 8 shows that the initial slope of ⟨E(t)⟩ is greater for the classical microcanonical and classical microcanonical + ZPE curves than for the quantum microcanonical curve. The initial rate of energy transfer is higher when the initial microcanonical ensemble is sampled classically, instead of quantum mechanically. Classical and quantum microcanonical normal mode sampling excite the modes of C6F6 differently. For the classical microcanonical ensemble each vibrational mode of the molecule has the same distribution of energy and the same average value. Including ZPE in the sampling only increases the total energy. Thus, with classical sampling the average energy of a low-frequency mode is the same as for a high-frequency mode. On the other hand, in quantum microcanonical normal mode sampling the vibrational energy levels are randomly sampled, and ZPE is included in each mode. The effect of this sampling, as compared to classical sampling, is to put more of the available energy in the higher-frequency modes. As shown in section D, this will decrease the efficiency of collisional energy transfer as compared to the classical sampling. The biexponential fitting function in eq 1 provides an excellent fit to the complete ⟨E(t)⟩ curve in the inset of Figure 8 for quantum microcanonical sampling, with parameters E(∞), f1, f 2, k1, and k2 as 22.0 kcal/mol, 0.585, 0.415, 0.0149 ps−1, and 0.00705 ps−1. However, the fittings described above in section C, for the classical microcanonical (no ZPE) ⟨E(t)⟩ curves, provide excellent fits to the overall curves but not to their 0−2 ps short-time region. The same result is found when fitting the classical microcanonical + ZPE ⟨E(t)⟩ curve in the inset of Figure 8. To ensure an accurate fit to the 0−2 ps region of the classical ⟨E(t)⟩ curves, more weight was put on this region in the fitting with eq 1. For the classical microcanonical (no ZPE) curve in the inset of Figure 8 the resulting fitting parameters are E(∞), f1, f 2, k1, and k2 as 24.4 kcal/mol, 0.066, 0.934, 0.198 ps−1, and 0.0105 ps−1 as compared to the parameters given in section C. For the classical microcanonical + ZPE curve in the inset of Figure 8 the respective fitting parameters are 23.0 kcal/mol, 0.0617, 0.938, 0.120 ps−1, 0.0103 ps−1. Plots of ⟨ΔEc⟩ versus [⟨E⟩ − E(∞)] are given in Figure 8, based on the fittings described above for the three ⟨E(t)⟩ curves in the inset of Figure 8. With the accurate short-time fitting and using the experimental collision parameters, the classical microcanonical (no ZPE) value of ⟨ΔEc⟩ at t = 0 for the 85.8 kcal/mol excitation energy is 3.13 kcal/mol, compared to the 1.74 kcal/mol value in Figure 6. For classical microcanonical + ZPE sampling, ⟨ΔEc⟩ is 3.67 kcal/mol for the experimental collision parameters using the fit to ⟨E(t)⟩ with the accurate short-time fitting. (It should be noted that linear fits of ⟨E(t)⟩ for the 0−2 ps region give the same ⟨ΔEc⟩ values as those above.) In comparison to these classical values ⟨ΔEc⟩

IV. SUMMARY In this work, features of a bath model for N2−C6F6 collisional intermolecular energy transfer (IET) are investigated in detail. Previously it was shown26 that at a N2 bath density of 324 kg/ m3 the N2−C6F6 intermolecular potential PES1 gives a similar profile of the average energy of the excited C6F6 molecules versus t, ⟨E(t)⟩, as does the more accurate PES2 potential. A N2 bath density of 40 kg/m3 is sufficiently low to model N2−C6F6 collisional energy transfer in the gas phase, and it is of interest to check the effect of the intermolecular potential on the energy transfer dynamics at this bath density. It was found that PES1 and PES2 gave similar collisional energy transfer dynamics at this density. Thus, the energy transfer dynamics are not very sensitive toward finer details of the intermolecular potential. In our previous study,26 we found that a 40 kg/m3 N2 bath density represents the low-pressure, single collision limit for the C6F6−N2 system.26 To further test this finding, an additional simulation was performed for a lower N2 bath density of 20 kg/ m3. To make this simulation more effective a neighbor list algorithm34 was implemented in the condensed-phase version of VENUS.29,30 The calculated C1 (k1/ρ) and C2 (k2/ρ) are nearly identical for the 20 and 40 kg/m3 simulations, and thus, the 40 kg/m3 simulation well approximates gas-phase collisional energy transfer for the C6F6−N2 system. Initially at t = 0, the C6F6 molecules are nearly monoenergetically excited, but during their collisional deactivation they acquire a distribution of energy. The deactivation of the C6F6 molecules is modeled by an ensemble of 96 trajectories, and the energy distribution for this ensemble was obtained at the four integration times 0, 24, 45, and 186 ps, for which the width of the C6F6 energy distribution is 4, 37, 43, and 24 kcal/mol, respectively. The time-dependent root-mean-square deviation, i.e., ⟨ΔE2⟩1/2 = ((1/N)∑i(Ei − ⟨E⟩)2)1/2; N = 96, of the energy distribution gives a clear depiction of the nature of the distribution during the complete simulation time. This deviation passes through a maximum value at an average C6F6 energy ⟨E⟩ = 75 kcal/mol, for which the trajectory is integrated to 45 ps. An analysis was made for a N2 bath density of 40 kg/m3 to determine if the width of the C6F6 energy distribution affects it collisional deactivation rate and, thus, the average energy transfer per collision, ⟨ΔEc⟩. One simulation was performed for an ensemble with an initial C6F6 excitation energy of 107.4 kcal/mol. ⟨ΔEc⟩ was then determined after the ensemble was propagated and its energy distribution broadened, with an average value of 85.8 kcal/mol. This ⟨ΔEc⟩ was found to be statistically the same as that determined when the C6F6 ensemble was excited nearly monoenergetically at 85.8 kcal/ mol. Thus, the width of the C6F6 energy distribution did not have a quantitative effect on the simulation value for ⟨ΔEc⟩. Experimentally it is known that the excited molecule’s vibrational modes play an important role in IET, with most of 14689

DOI: 10.1021/jp512931n J. Phys. Chem. C 2015, 119, 14683−14691

Article

The Journal of Physical Chemistry C the transfer occurring from low-frequency modes.42−44 To study this phenomenon, the mass of the F-atoms was changed to that of an H-atom keeping all potential parameters the same, which increased the C6F6 vibrational frequencies. A simulation for this “model C6H6” was performed for the 40 kg/m3 N2 bath density and compared with the C6F6 simulation. A significant decrease of the energy transfer rate was observed, consistent with experiment.42−44 Changing the intermolecular potential to that for C6H6−N2 instead of C6F6−N2 will further decrease the efficiency of collisional energy transfer.46,47 The current simulation also tested three different models for the initial energy sampling of C6F6; i.e., classical microcanonical,31,32 classical microcanonical + ZPE, and quantum microcanonical.33 The first is the method used in our previous study,26 while the second is the same except the C6F6 ZPE is included with the total energy. ZPE is explicitly included for the last method. For each method the C6F6 vibrational modes are treated as harmonic oscillators. The simulations for the classical microcanonical + ZPE and quantum microcanonical ensembles have the same total energy, but their average energies ⟨E(t)⟩ differ for the first 48 ps. A higher rate of IET is observed during the initial propagation of the classical microcanonical and classical microcanonical + ZPE ensembles, compared to the quantum microcanonical ensemble. The quantum microcanonical sampling puts more energy into the C6F6 highfrequency modes compared to the classical microcanonical sampling. High-frequency modes transfer energy less efficiently in intermolecular collisions, which may be the origin of the different initial IET dynamics for the classical and quantum sampling. More work needs to be done to compare these sampling methods, including comparisons with experiment. Models for IET have been proposed in which a collision complex is formed between the energized molecule and deactivating bath molecule, resulting in statistical or ergodiclike IET.48−51 It is of interest that no C6F6−N2 collision complexes were observed for the simulations reported here or in ref 26, even though the C6F6−N2 potential energy minimum has a depth of more than 1 kcal/mol. The efficiency of collision complex formation52 and ensuing collisional relaxation53 has been studied for complexes with potential energy minima deeper than 10 kcal/mol, and their dynamics are not statistical. Collision complex formation for IET is an interesting concept and may be investigated in future studies. As described above, a “model C6H6 ” molecule was considered for IET. There is a substantial amount of experiment14,38,54 and simulation25,55,56 data for benzene IET, and it clearly would be of interest to consider this work in simulations like those reported here.

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computation were also done on Robinson, a general computer cluster of Department of Chemistry and Biochemistry, Texas Tech University, purchased by the NSF CRIF-MU grant CHE0840493.

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REFERENCES

(1) Tardy, D. C.; Rabinovitch, B. S. Intermolecular Vibrational Energy Transfer in Thermal Unimolecular Systems. Chem. Rev. 1977, 77, 369−408. (2) Gilbert, R. G.; King, K. D. Gas/gas and Gas/wall Average Energy Transfer from Very Low-Pressure Pyrolysis. Chem. Phys. 1980, 49, 367−375. (3) Oref, I.; Tardy, D. C. Energy Transfer in Highly Excited Large Polyatomic Molecules. Chem. Rev. 1990, 90, 1407−1445. (4) Finlayson-Pitts, B. J.; Pitts, J. N., Jr. Atmospheric Chemistry: Fundamentals and Experimental Techniques; Wiley: New York, 1986. (5) Barker, J. R.; Toselli, B. M. Infrared Emission Studies of the Vibrational Deactivation of Benzene Derivatives. Int. Rev. Phys. Chem. 1993, 12, 305−338. (6) Flynn, G. W.; Paramenter, C. S.; Wodtke, A. M. Vibrational Energy Transfer. J. Phys. Chem. 1996, 100, 12817−12838. (7) Hippler, H.; Troe, J. Bimolecular Collisions; Ashfold M. N. R., Baggott J. E., Eds.; Royal Society of Chemistry: London, 1989. (8) Gilbert, R. G. Mechanism and Models for Collisional Energy Transfer in Highly Excited Large Polyatomic Molecules. Aust. J. Chem. 1995, 48, 1787−1817. (9) Quack, M.; Troe, J. Gas Kinetics and Energy Transfer; The Chemical Society: London, 1977 Vol: 2. (10) Rossi, M. J.; Pladziewicz, J. R.; Barker, J. R. Energy-Dependent Energy Transfer: Deactivation of Azulene (S0, Evib) by 17 Collider Gases. J. Chem. Phys. 1983, 78, 6695−6708. (11) Toselli, B. M.; Barker, J. R. Quantum Effects in Large molecule Collisional Energy Transfer? Chem. Phys. Lett. 1990, 174, 304−308. (12) Michaels, C. A.; Mullin, A. S.; Flynn, G. W. Long- and ShortRange Interactions in the Temperature Dependent Collisional Excitation of the Antisymmetric Stretching CO2(00 °1) Level by Highly Vibrationally Excited Pyrazine. J. Chem. Phys. 1995, 102, 6682− 6695. (13) Michaels, C. A.; Lin, Z.; Mullin, A. S.; Tapalian, H. C.; Flynn, G. W. Translational and Rotational Excitation of the CO2(00 °0) Vibrationless State in the Collisional Quenching of Highly Vibrationally Excited Perfluorobenzene: Evidence for Impulsive Collisions Accompanied by Large Energy Transfers. J. Chem. Phys. 1997, 106, 7055−7071. (14) Sedlacek, A. J.; Weston, R. E., Jr.; Flynn, G. W. Interrogating the Vibrational Relaxation of Highly Excited Polyatomics with TimeResolved Diode Laser Spectroscopy: C6H6, C6D6, and C6F6+CO2. J. Chem. Phys. 1991, 94, 6483−6490. (15) Hippler, H.; Troe, J.; Weldenken, H. J. Direct Observation of Collisional Deactivation of Highly Excited Toluene. Chem. Phys. Lett. 1981, 84, 257−259. (16) Nakashima, N.; Yoshihara, K. Laser Flash Photolysis of Benzene. VIII. Formation of Hot Benzene from the S2 State and Its Collisional Deactivation. J. Chem. Phys. 1983, 79, 2727−2735. (17) Liu, C.-L.; Hsu, H.-C.; Lyu, J.-J.; Ni, C.-K. Energy Transfer of Highly Vibrationally Excited Azulene: Collisions Between Azulene and Krypton. J. Chem. Phys. 2006, 124, 054302. (18) Hsu, H.-C.; Tsai, M.-T.; Dyakov, Y. A.; Ni, C.-K. Energy Transfer of Highly Vibrationally Excited Naphthalene: Collisions with CHF3, CF4, and Kr. J. Chem. Phys. 2011, 135, 054311. (19) Loehmannsroeben, H. G.; Luther, K. Selective Multiphoton Ionization in Dense Manifolds of Vibrational States. Chem. Phys. Lett. 1988, 144, 473−478. (20) Waclawik, E. R.; Lawrance, W. D.; Borg, R. A. J. State-to-State Vibrational Energy Transfer from 61 Benzene Induced by Low-Energy Collisions with Nitrogen: Temperature-Dependent Propensities. J. Phys. Chem. 1993, 97, 5798−5802.

AUTHOR INFORMATION

Corresponding Author

*Phone number: 806-834-3152. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research reported here is based upon work supported by the Air Force Office of Scientific Research (AFOSR) BRI grant FA 9550-12-1-0443 and the Robert A. Welch Foundation under Grant No. D-0005. Support was also provided by the High Performance Computing Center (HPCC) at Texas Tech University, under the direction of Philip W. Smith. Parts of the 14690

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The Journal of Physical Chemistry C (21) Zellweger, J.-M.; Brown, T. C.; Barker, J. R. Vibrationally Excited Populations from IR-Multiphoton Absorption. I. Absorbed Energy and Reaction Yield Measurements. J. Chem. Phys. 1985, 83, 6251−6260. (22) Gascooke, J. R.; Alwahabi, Z. T.; King, K. D.; Lawrance, W. D. Vibrational Deactivation of Highly Excited Hexafluorobenzene. J. Phys. Chem. A 1998, 102, 8505−8509. (23) Date, N.; Hase, W. L.; Gilbert, R. G. Collisional Deactivation of Highly Vibrationally Excited Molecules. Dynamics of the Collision Event. J. Phys. Chem. 1984, 88, 5135−5138. (24) Brown, N. J.; Miller, J. A. Collisional Energy Transfer in the Low-Pressure-Limit Unimolecular Dissociation of HO2. J. Chem. Phys. 1984, 80, 5568−5580. (25) Lenzer, T.; Luther, K.; Troe, J.; Gilbert, R. G.; Lim, K. F. Trajectory Simulations of Collisional Energy Transfer in Highly Excited Benzene and Hexafluorobenzene. J. Chem. Phys. 1995, 103, 626−641. (26) Paul, A. K.; Kohale, S. C.; Pratihar, S.; Sun, R.; North, S. W.; Hase, W. L. A Unified Model for Simulating Liquid and Gas Phase, Intermolecular Energy Transfer: N2 + C6F6 Collisions. J. Chem. Phys. 2014, 140, 194103. (27) Dumm, M.; Hippler, H.; Olschewski, H. A.; Troe, J.; Willner, J. Z. Efficient Collisional Energy Transfer of Vibrationally Highly Excited C6F6 Molecules in the Ground State. Phys. Chem. N. F. 1990, 166, 129−143. (28) Rivera-Rivera, L. A.; Wagner, A. F.; Sewell, T. D.; Thompson, D. L. Pressure Effects on the Relaxation of an Excited Nitromethane Molecule in an Argon Bath. J. Chem. Phys. 2015, 142, 014303. (29) Hase, W. L.; et al. VENUS96: A General Chemical Dynamics Computer Program; Texas Tech University, Lubbock, TX, 2005. (30) Hu, X.; Hase, W. L.; Pirraglia, T. Vectorization of the General Monte Carlo Classical Trajectory Program VENUS. J. Comput. Chem. 1991, 12, 1014−1024. (31) Hase, W. L.; Buckowski, D. G. Monte Carlo Sampling of a Microcanonical Ensemble of Classical Harmonic Oscillators. Chem. Phys. Lett. 1980, 74, 284−287. (32) Peslherbe, G. H.; Wang, H.; Hase, W. L. Monte Carlo Sampling for Classical Trajectory Simulations. Adv. Chem. Phys. 1999, 105, 171− 201. (33) Park, K.; Engelkemier, J.; Persico, M.; Manikandan, P.; Hase, W. L. Algorithms for Sampling a Quantum Microcanonical Ensemble of Harmonic Oscillators at Potential Minima and Conical Intersections. J Phys. Chem. A 2011, 115, 6603−6609. (34) Allen M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: New York, 1989. (35) Shuler, K. E.; Weiss, G. H.; Andersen, K. Studies in Nonequilibrium Rate Processes. V. The Relaxation of Moments Derived from a Master Equation. J. Math. Phys. 1962, 3, 550−556. (36) Procaccia, I.; Shimoni, Y.; Levine, R. D. Entropy and Microscopic Disequilibrium. I. Isothermal Time Evolution with Application to Vibrational Relaxation. J. Chem. Phys. 1976, 65, 3284−3301. (37) van Kampen, N. G. The Expansion of the Master Equation. Adv. Chem. Phys. 1976, 34, 245−309. (38) Bruehl, M; Schatz, G. C. Theoretical Studies of Collisional Energy Transfer in Highly Excited Molecules: Temperature and Potential Surface Dependence of Relaxation in He, Ne, Ar + CS2. J. Phys. Chem. 1988, 92, 7223−7229. (39) Lendvay, G.; Schatz, G. C. Energy Dependence of Energy Transfer in the Collisonal Relaxation of Vibrationally Highly Excited CS2. J. Phys. Chem. 1991, 95, 8748−8753. (40) Lendvay, G.; Schatz, G. C. Collisional Energy Transfer from Highly Vibrationally Excited SF6. J. Chem. Phys. 1993, 98, 1034−1041. (41) Lendvay, G. Gateway Modes in the Collisional Energy Transfer from Highly Vibrationally Excited CS2. J. Phys. Chem. A 1997, 101, 9217−9223. (42) Krajnovich, D. J.; Parmenter, C. S.; Catlett, D. L. State-to-State Vibrational Transfer in Atom-Molecule Collisions. Beams vs. Bulbs. Chem. Rev. 1987, 87, 237−288.

(43) Gordon, R. J. The Origin of Small and Large Molecule Behavior in the Vibrational Relaxation of Highly Excited Molecules. J. Chem. Phys. 1990, 92, 4632−4634. (44) Koifman, I.; Dashevskaya, E. I.; Nikitin, E. E.; Troe, J. Rotational Gateway for the Vibrational Energy Transfer from Excited Nonlinear Triatomic Molecules. J. Phys. Chem. 1995, 99, 15348−15353. (45) Yerram, M. L.; Brenner, J. D.; King, K. D.; Barker, J. R. Collisional Deactivation of Highly Vibrationally Excited Benzene Pumped at 248 nm. J. Phys. Chem. 1990, 94, 6341−6350. (46) de Sainte Claire, P.; Hase, W. L. Thresholds for the CollisionInduced Dissociation of Clusters by Rare Gas Impact. J. Phys. Chem. 1996, 100, 8190−8196. (47) Yardley, J. T. Introduction to Molecular Energy Transfer; Academic Press: London, 1980. (48) Lin, Y. N.; Rabinovitch, B. S. A Simple Quasi-Accommodation Model of Vibrational Energy Transfer. Low-Pressure Thermal Methyl Isocyanide Isomerization. J. Phys. Chem. 1970, 74, 3151−3159. (49) Nordholm, K. S. J.; Freasier, B. C.; Jolly, D. L. Ergodic Collision Theory of Intermolecular Energy Transfer. Chem. Phys. 1977, 25, 433−449. (50) Freasier, B. C.; Jolly, D. L.; Nordholm, K. S. J. Ergodic Collision Theory of Intermolecular Energy Transfer II. Quantum Effects in the Harmonic Approximation. Chem. Phys. 1978, 32, 161−168. (51) Nilsson, D.; Nordholm, S. Modeling Energy Transfer in Molecular Collisions: Statistical Theory versus Experiment for Highly Excited Toluene and Azulene. J. Chem. Phys. 2003, 119, 11212−11220. (52) Hase, W. L.; Cho, Y. J. Trajectory Studies of SN2 Nucleophilic Substitution. III. Dynamical Stereochemistry and Energy Transfer Pathways for the Cl− + CH3Cl Association and Direct Substitution Reactions. J. Chem. Phys. 1993, 98, 8626−8639. (53) Boering, K. A.; Brauman, J. I. Collisional Relaxation of Vibrational Excitation: Effects of Bath Gas Structure. J. Chem. Phys. 1992, 97, 5439−5450. (54) Toselli, B. M.; Barker, J. R. Excitation of CO2 by Energy Transfer from Highly Vibrationally Excited Benzene Derivatives. J. Chem. Phys. 1991, 95, 8108−8119. (55) Lenzer, T.; Luther, K. Intermolecular Potential Effects in Trajectory Calculations of Collisions between Large Highly Excited Molecules and Noble Gases. J. Chem. Phys. 1996, 105, 10944−10953. (56) Bernshtein, V.; Oref, I. Collisional Energy Transfer between Ar and Normal and Vibrationally and Rotationally Frozen Internally Excited Benzene-Trajectory Calculations. J. Chem. Phys. 1997, 106, 7080−7089.

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