J . Phys. Chem. 1986, 90, 6158-6167
6158 15
0
I
20
x 1
0.08 torr N,
x 2
0.04 torr N
x 5
0.01 torr N,
I
I
40
I
60
80
Time (nsec)
Figure 6. Fitting of Monte-Carlo results for 20 Torr of He and three nitrogen pressures, with the N2()CIIU)decay rate set equal to zero, to a first-order formation mechanism. The units of the ordinate are the same as for Figure 4 and give the N2(C311u)concentration in arbitrary units; vertical amplification factors for each curve are indicated on the drawing. The curve for 0.04 Torr of N2 has been displaced upward 2.5 units; the curve for 0.08 Torr of N2 has been displaced 5 units. The squares are the Monte-Carlo points. The solid curves are least-squares best fits to a simple exponential formation. of N, pressure. (The resulting best-fit curves are those shown in Figure 4.) Also shown in Figure 5 are the values of k f obtained from the Monte-Carlo results by analysis of the formation curves which result if k6 is set equal to zero in the calculation. It is simpler to visualize the degree to which the Monte-Carlo results can be fit to a first-order kinetic scheme from such plots. Such
Monte-Carlo curves for 20 Torr of H e are shown in Figure 6 for N2 pressures of 0.01, 0.04, and 0.08 Torr, with the least-squares "best fit" curves drawn through the Monte-Carlo points. These results confirm that a first-order formation provides a reasonable approximation to the production of N2(C3n,).The situation is similar for Ne. The formation constants given in Figure 5 may be compared with the experimental values reported by Cooper et aL2 In contrast to the qualitative agreement noted in Figure 4 between experiment and Monte-Carlo calculations, the results in Figure 5 seem to disagree with experiment. The slopes of kf vs. PN2yield values of k, = 1.9 X lOI3 dm3 mol-' s-l for He and 2.0 X lOI3 dm3 mol-' s-' for Ne. These are about four times greater than the experimentally reported values obtained from similar plots. In an effort to explain this, the experimental data have been reexamined, and we have found that the uncertainty in the analysis of the experimental data is large and that experimental values of kf more in line with the Monte-Carlo results are reasonable. Also, the much smaller intercept at PN2= 0 for Ne, in comparison with that for He, predicted by the calculation, is not evident in the original experimental plots. This may be due to deterioration of the signal quality in the experimental data at the lowest N z pressures. Further experiments are needed to determine whether this difference in intercepts is realized. The difference is expected because the loss of energy by qE-by elastic collisions is much more efficient for He than for Ne, owing to the smaller mass of the H e atom. Impurity problems could be especially troublesome at low PK2and could mask the intercept differences. Finally, Dillong has recently carried out an analytic treatment of the same problem and has obtained results close to those of the Monte-Carlo simulation. A full account of his work is being prepared for publication. Registry No. He, 7440-59-7; Ne, 7440-01-9; N,, 7727-37-9. (9) Dillon, M. A,, private communication.
Trajectory Calculations of Intermolecular Energy Transfer In SO,-Ar Collisions. 1. Method and Representative Results H. Hippler, H. W. Schranz,+and J. Troe* Institut fur Physikalische Chemie der Universitat Gottingen, D-3400 Gottingen, West Germany (Received: April 30, 1986)
The collisional energy transfer between excited SO2 and Ar has been simulated by classical trajectory calculations. The influences of the intramolecular and intermolecular potentials, of temperature, and of excitation of SO2on vibrational and rotational energy transfer are demonstrated. Average energies transferred and collisional transition probabilities are reported.
1. Introduction In spite of its importance for large classes of gas-phase reactions, collisional energy transfer of highly excited polyatomic molecules is not a well-understood process. Ultimately, one would like to know the state i-to-state if resolved energy-transfer cross-section u(i'/i;v) as a function of the relative velocity v of the collision partners or the corresponding thermally averaged rate coefficient k(i'/i;T). If the density of initial and final states of the molecule is large, averaging over groups of states becomes adequate. Then, the molecule will be characterized by the energy E and angular momentum J such that a grain-averaged k(E',J'/E,J;T) is required. So far, no "single-collision experiment" is available to Permanent address: Australian Defence Force Academy, Canberra.
0022-3654/86/2090-6158$01 .50/0
measure this quantity. Instead, in general only first and second moments of k with respect to A E = E ' - E are accessible experime~~taIly.'-'~ With the assumption that total rate coefficients (1) Troe, J. J . Chem. Phys. 1982, 77, 3485. (2) Procaccia, I.; Shimoni, Y . ;Levine, R. D. J . Chem. Phys. 1976, 65, 3284. Tabor, M.; Levine, R. D.; Ben-Shaul, A.; Steinfeld, J. I. Mol. Phys. 1979, 37, 141. (3) Van Kampen, N. G. Ado. Chem. Phys. 1976, 34, 245. (4) Tardy, D. C . ;Rabinovitch, B. S . J . Chem. Phys. 1966,45,3720; 1968,
48, 1282. ( 5 ) Troe, J.; Wagner, H. G. Recent Advances in Aerothermochemistry; AGARD: Oslo, 1966; Ber. Bunsen-Ges. Phys. Chem. 1967, 71,937. (6) Tardy, D. C.; Rabinovitch, B. S . Chem. Rev. 1977, 77, 369. (7) Quack, M.; Troe, J. In Gas Kinetics and Energy Transfer; Ashmore, P. G., Donovan, R. J., Eds.; The Chemical Society: London, 1977; Vol. 2, p 175.
0 1986 American Chemical Society
Thie Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6159
Energy Transfer in S02-Ar Collisions. 1 are approximated by Lennard-Jones rate coefficients ZLJ,such measurements lead to the average
and, in favorable cases, to 1 y-J (E’-
( U 2=)
E)2k(E’/E;T) dE’
(1.2)
(In the present article, we have chosen to write integrals instead of sums.) Only limited additional experimental information on the detailed functional dependence of k on AE has been obtained.15-18 Essentially no experimental information is available on rotational effects. Unfortunately, a complete description of collisional activation and deactivation sequences or of the competition between energy transfer and other processes like unimolecular reaction can only be given when the rotational effects in energy transfer are well understood. Only with this knowledge will master equations find their complete solutions. For unimolecular reactions, such equations have the form d[A(E,J)I = - J - J - k ( E ’ , J ’ / E , J ) [ M ] [ A ( E , J ) ] dE’dJ’+ dt
where k(E,J) denotes specific rate constants for unimolecular r e a c t i ~ n . ~ ~“Two-dimensional” .’~~~~ master equations (in E and J) have been treated formally about a decade ago.’oa31 However, in the absence of more detailed experimental information on k(E’,J’/E,J), this work apparently has found no continuation (see only the recent work in ref 22). In the absence of experimental information on the details of k(E’,J’/E,J;T), therefore, theoretical models should be inspected. Such theories may be ”statistical” or “dynamical” in nature. The present article considers dynamical theories. It describes large-scale classical trajectory calculations of the energy transfer between a highly excited triatomic molecule and an inert gas atom. The dependence of energy transfer on the excitation energy and the bath gas temperature on the role of vibrational and rotational energy is investigated. While the calculations of part 1 demonstrate the influence of the various parameters, in part 223 we derive the functional form of the two-dimensional kernel k(E’,J‘/E,J) of the integral eq 1.3 by a fit of simple two-dimensional functions to large numbers of trajectories. Such functions then will serve as the basis for solutions (8) Tardy, D. C.; Malins, R. J. J . Phys. Chem. 1979, 83, 93. (9) Tardy, D. C.; Rabinovitch, B. S.J. Phys. Chem. 1985,89,2442; 1986, 90, 1187. (10) Troe, J. J. Chem. Phys. 1977, 66, 4745. (11) Troe, J. J . Chem. Phys. 1977, 66, 4758. (1 2) Nikitin, E. E. Theory of Elementary Atomic and Molecular Processes in Gases: Clarendon: Oxford. 1974. (13) Keck, J.; Carrier, G. J. Chem. Phys. 1965, 43, 2284. (14) Snider, N . J. Chem. Phys., in press. (15) Krongauz, V. V.; Rabinovitch, B. S. Chem. Phys. 1982, 67, 201. Asakawa, R.; Kelley, D. F.; Rabinovitch, B. S.J . Chem. Phys. 1982,76,2384. KronPauz. V. V.: Rabinovitch. B. S.J. Chem. Phvs. 1983. 78, 3872. Krongauz,-V. V.;Rabinovitch, B. S.;Linkaityte-We&, E. J. Chem. Phys. 1983, 78, 5643. (16) Georgakakos, J. H.; Rabinovitch, B. S. J. Phys. Chem. 1972, 56, 5921. (17) Just, Th.; Troe, J. J . Phys. Chem. 1980, 84, 3068. (18) Eberhardt, J. E.; Knott, R. B.; Pryor, A. W.; Gilbert, R. G. Chem. Phys. 1982,69,45. Brown, T. C.; Taylor, J. A.; King, K. D.; Gilbert, R. G. J . Phys. Chem. 1983, 87, 5214. Nguyen, T. T.; King, K. D.; Gilbert, R. G. J Phys. Chem. 1983,87,494. (19) Keck, J. C. J. Chem. Phys. 1967, 46, 4211. (20) Penner, A. P.; Forst, W. Chem. Phys. 1975, 11, 243. Forst, W.; Penner, A. P. J . Chem. Phys. 1980, 72, 1435. (21) Dove, J. E.; Raynor, S.J. Phys. Chem. 1979,83, 127. (22) Borkovec, M.; Berne, B. J. J. Chem. Phys. 1986,84,4327. (23) Schranz, H.; Troe, J. J . Phys. Chem., following paper in this issue.
of the two-dimensional master eq 1.3. Trajectory calculations of energy transfer at first have been done for collinear collision c o n f @ r a t i ~ n s . ~An ~ ~early siiulation by Harter, Alterman, and Wilson% investigated collinear collisions of atoms with diatomics and triatomics, and of diatomics with triatomics, with collisional interaction described by a LennardJones 6-12 potential. It was observed that the transfer of energy was inefficient and that mass effects seem to play a role. In collinear simulations of C 0 2 Kr, Schatz and c o - ~ o r k e r s ~ ~ - ~ ~ reported no dramatic change of collisional energy transfer with transition from the regular (quasi-periodic) to the irregular (chaotic) regime but reported large differences between the behavior of harmonic and anharmonic models with variation in collision energy. Nalewajski and noted that differences in the relaxation behavior of regular and irregular molecular states depended on the strength of the intermolecular interaction. Collinear simulations suffer from the fact that they do not treat rotations. Vibration-rotation interaction can be a major source of c ~ u p l i n gand ~ ~rotations ~ ~ ~ are well-known as a major and efficient pathway for intermolecular energy Hence, full three-dimensional simulations of intermolecular energy transfer3246 should provide a more realistic picture. Early three-dimensional simulations by Bunker and J a y i ~ on h ~CH3NC ~ and by Stace and M ~ r r e l l on ’ ~ O3and H20in collision with atoms and diatomics also showed that energy transfer was inefficient and had a dependence on collider mass. In addition, rotational energy transfer was found to be more efficient than vibrational energy transfer. The efficiency of rotational energy transfer has also been noted for simulations of energy transfer of Br2 in Ar over a wide range of pressures and temperature^.^'^' Suzukawa, Wolfsberg, and Thompson38have confirmed the above trends in a simulation of low-energy C02colliding with rare gases and noted that high collision energies (short collision times) were necessary V transfer to become comparable in efficiency to T for T R transfer. With Landau-Teller this trend has been attributed to the fact that, for not too high molecular angular momentum, the rotational period is comparable or longer than the collision time, so that the rotations are generally at the efficient energy-transfer sudden limit. For low vibrational energies, the vibrational period is very short so that the vibrations are at the inefficient energy transfer adiabatic limit for low collision energies and approach the sudden limit for high vibrational energies or high collision energies. Gallucci and S ~ h a t have z ~ ~noted that
+
-
-
(24) Harters, R. J.; Alterman, E. B.; Wilson, D. J. J . Chem. Phys. 1964, 40, 2137. (25) Schatz, G. C.; Moser, M. D. J . Chem. Phys. 1978, 68, 1992. (26) Schatz, G. C. Chem. Phys. Lett. 1979, 67, 248. (27) Schatz, G. C.; Mulloney, T. J . Chem. Phys. 1979, 71, 5257. (28) Nalewajski, R. F.; Wyatt, R. E. Chem. Phys. 1983, 81, 357. (29) Nalewajski, R. F.; Wyatt, R. E. Chem. Phys. 1984,89, 385. (30) Parr, C. A.; Kuppermann, A.; Porter, R. N. J. Chem. Phys. 1977,66, 2914. (31) Frederick, J. H.; McClelland, G. M.; Brumer, P. J. Chem. Phys. 1985, 83, 190. (32) Bunker, D. L.; Jayich, S.A. Chem. Phys. 1976, 13, 129. (33) Stace, A. J.; Murrell, J. N . J . Chem. Phys. 1978, 68, 3028. (34) Jolly, D. L.; Freasier, B. C.; Nordholm, S. Chem. Phys. 1977,21,211. (35) Jolly, D. L.; Freasier, B. C.; Nordholm, S. Chem. Phys. 1977,25,361. (36) Freasier, B. C.; Jolly, D. L.; Nordholm, S. Chem. Phys. 1983,82,369. (37) Jolly, D. L.; Freasier, B. C.; Hamer, N. D.; Nordholm, S. Chem. Phys. 1984,88, 261. (38) Suzukawa, H. H.; Wolfsberg, M.; Thompson, D. L. J . Chem. Phys. 1978, 68, 455. (39) Gallucci, C. R.; Schatz, G. C. J . Phys. Chem. 1982,86, 2352. (40) Brown, N. J.; Miller, J. A. J . Chem. Phys. 1984, 80, 5568. (41) Osborn, M. K.; Smith, I. W. M. Chem. Phys. 1984, 91, 13. (42) Schlier, C. In Energy Storage and Redistribution in Molecules; Hinze, J., Ed.;Plenum: New York, 1983; p 585. (43) Date, N.; Hase, W. L.; Gilbert, R. G. J . Phys. Chem. 1984,88,5135. (44) H a s , W. L.; Date, N.; Bhuiyan, L. B.; Buckowski, D. J . Phys. Chem. 1985, 89, 2502. (45) Gelb, A. J . Phys. Chem. 1985, 89, 4189. (46) Grinchak, M. B.: Levitsky, A. A,; Polak, L. S.;Umanski, S. Y. C k m . Phys. 1984, 88, 365. (47) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics; Clarendon: Oxford, 1974.
6160 The Journal of Physical Chemistry, Vol. 90, No. 23, 1986
Hippler et al.
-
the R T pathway is more important for rotationally hot H 0 2 than for rotationally cold HOP Brown and Miller,@ in a classical trajectory study of excited H 0 2 in collision with He, found an increase in vibrational energy transfer with molecular angular momentum a t constant vibrational energy. In studies of Ar colliding with highly excited H 2 0and CH,, Hase and co-worke r ~have ~ found ~ , that ~ increasing molecular rotational energy enhances both intermolecular energy transfer and collision-induced V R transfer, the latter being more efficient. In a study of Ar and excited 03, Gelb45also finds that angular momentum plays a role in controlling energy transfer. A study of vibrational energy transfer by Osbom and Smith41 for model atom-diatom collisions has indicated that the magnitude of energy transfer is insensitive to whether collisions are complex or direct.42 TABLE I: Model SO2 Molecular Parameters When the excited molecule is large, trajectory calculations of r12c= r13c= rsoc = 1.4321 A equil bond lengths energy transfer require considerable computational effort. In this ~ 2 3 ’ = rooe = 2.4742 A case the “biased random walk” analysis of trajectories by Lim and qui1 bond angle eGS4e = 119.50 Gilbert48,49provides a means to restrict the calculations to very 012 = 813 = 8 2 3 = 2.4 A-I Morse range parameters small numbers of trajectories. By this model average energies D I 2 = D,3 = Dso = 100 kcal/mol Morse dissoc energies D23 = Doo = 30.5 kcal/mol transferred per collision ( A E ) between excited azulene and Ar dissoc energy DsW = Dso + Dw = 130.5 kcal/mol have been calculated that are in very good agreement with recent vib freq direct experimental determination^.^^*^' However, an investigation 1213 cm-I asym str of rotational effects has so far not been done. sym str 1187 cm-l Dynamical trajectory calculations of energy transfer generally bend 394 cm-‘ give results that differ considerably from the results of “statistical” moments of inertia theories. Such models, like the “strong collision model”,52 the 8.3358 g/(mol A2) G’ “scaled strong collision model”?3 “ergodic collision t h e ~ r y ” , ~ ~ , ~ he ~ 48.9133 g/(mol A2) 57.3091 g/(mol A2) and a variety of general statistical models from ref 55-58 notoICC Kc = (2B - A - c ) / ( A - c ) -0.942 riously overestimate average energies ( A E ) transferred per collision. Although such models may provide useful qualitative explanations of observed trends, they do not give the degree of TABLE 11: Collision Model Parameters detail required here. Since the extent of realistic or unrealistic predictions appears unclear, we do not consider these models here, but we refer the reader to the discussions in ref 6 and 7. Whereas SLJ 64.36 1.2612 5 statistical theories appear not to be too useful for a comparison RLJ 64.36 1.2612 5 of trajectory calculations, experimental measurements of ( A E ) provide a crucial test of the results. References 6 and 7 have given and the electronic ground state. Collisional deactivation, therefore, summaries of earlier more indirect determinations whereas more has taken place in excited electronic states coupled to the electronic direct measurements have become possible recently by laser ground state, whereas the trajectory calculations only consider techniques. Among the available data for large polyatomic the electronic ground state. More experiments on highly excited molecule^,^^*^^^^"' so far only the azulene Ar experiments have small polyatomic molecules, which do not show such complications, been compared with trajectory calculations.49 Direct measureare desirable. ments for highly excited triatomic molecules have been reported for CS262and S02.60,63 For this reason, we have chosen to 2. Model calculate classical trajectories for SO2 + Ar collision. In the The motion of the SO2 molecule and its interaction with the comparison with such experiments, one should not forget that the Ar collider is described in terms of the classical equations of highly excited molecules experimentally have been prepared by motion64 with a total Hamiltonian UV light absorption, leading to mixtures of excited electronic states
-
+
(48) Gilbert, R. G. J . Chem. Phys. 1984,80, 5501. (49) Lim, K.; Gilbert, R. G. J . Chem. Phys. 1986,84, 6129. ( 5 0 ) Rossi, M. J.; Pladziewicz, J. R.; Barker, J. R. J . Chem. Phys. 1983, 78,6695. Barker, J. R. J. Phys. Chem. 1984,88, 11. Barker, J. R.; Golden, R. E. J. Phys. Chem. 1984, 88, 1012. (51) Hippler, H.; Lindemann, L.; Troe, J. J. Chem. Phys. 1985,83,3906. (52) Oref, I.; Rabinovitch, B. S. Chem. Phys. 1977, 26, 385. Oref, I. J . Chem. Phys. 1982, 77, 5146. (53) Nordholm, S. Chem Phys. 1978,29,55. Schranz, H. W.; Nordholm, S . Chem. Phys. 1983, 74, 365. (54) Nordholm, S.; Freasier, B. C.; Jolly, D. L. Chem. Phys. 1977,25,433. Freasier, B. C.; Jolly, D. L.; Nordholm, S. Chem. Phys. 1978, 32, 161. Freasier, B. C.; Jolly, D. L.; Nordholm, S . Chem. Phys. 1978, 32, 169. Schranz, H. W.; Nordholm, S. Inr. J . Chem. Kinet. 1981, 13, 1051. (55) Keck, J.; Kalelkar, A. J. Chem. Phys. 1968, 49, 321 1. (56) Lin, Y. N.; Rabinovitch, B. S . J . Phys. Chem. 1970, 74, 3151. (57) Troe, J. Ber. Bunsen-Ges. Phys. Chem. 1973, 77, 665. (58) Bhattacharjee, R. C.; Forst, W. Chem. Phys. 1978.30, 217. (59) Fppler, H.; Troe, J.; Wendelken, H. J. J. Chem. Phys. 1983, 78, 6709. Hippler, H.; Troe, J.; Wendelken, H. J. J. Chem. Phys. 1983, 78,6718. Heymann, M.;Hippler, H.; Troe,J. J. Chem. Phys. 1984, 80, 1853. (60) Hippler, H. Ber. Bunsen-Ges. Phys. Chem. 1985, 89, 303. (61) Beck, K. M.; Ringwelski, A.; Gordon, R. J. Chem. Phys. Lett. 1985, 121, 529. Ichimura, T.; Mori, Y.; Nakashima, N.; Yoshihara, K. J. Chem. Phys. 1985, 83, 117. (62) Dove, J. E.; Hippler, H.; Troe, J. J. Chem. Phys. 1985, 82, 1907. (63) Heymann, M.; Hippler, H.; Nahr, D.; Troe, J., to be published.
ql, q2,and q3 denote the position coordinates of the S atom (atom
l), indices 4-6 characterize the first 0 atom (atom 2 ) , indices 7-9 characterize the second 0 atom (atom 3), indices 10-12 characterize the Ar atom (atom 4), and pl, ..., p12are the corresponding conjugate momenta. The masses of the atoms in the system are ms = 32.06 g/mol, mo = 16.00 g/mol, and mAr = 39.95 g/mol. The coordinate system used is illustrated in Figure 1. As usual, the total potential is separated into
...)q12)
vi”, (2.2) where Vmolis the molecular potential and vint is the collisional v(ql9
= Vmol +
interaction potential. For computational ease, we model the SO2 molecule by a potential of the central force field type.65 We assume that the molecular potential can be represented by three Morse bond potentials (64) Goldstein, H. Clussicul Mechanics; Addison Wesley: Reading, MA, 1980. (65) Wilson, E. B., Jr.; Decius, J. C.; Cross,P.C . Molecular Vibrutions; McGraw Hill: New York, 1955.
Energy Transfer in SOZ-Ar Collisions. 1 Vmol
= 412(r12)
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6161
+ 413(r13) + 423(r23)
(2.3)
where 4,(rij) is the pair potential between atoms i and j for a separation rij and the Morse potential is given by #ij(rij) = oij(1 - exp[-Oij(rij - rij?I
l2
(2.4)
The parameters of this representation are given in Table I. The Morse parameters have been chosen so that the dissociation energy, the equilibrium bond lengths, and the bond angle are reproduced. The normal-mode frequencies in Table I were obtained from power spectra66of low-energy trajectories of an isolated SOz molecule with harmonic pair potentials. While the normal-mode vibrational frequencies do not correspond exactly with those of real SO2 (1361.2, 1151.4, and 517.7 cm-’, respectively6’), they do exhibit roughly the same character of high-frequency stretches and lowfrequency bends as many other triatomic molecules. As a further approximation, the collisional interaction is modeled by pair interactions only
hl = 414(r14) + 424(r24) + 434(r34)
(2.5)
This has been shown to be a good first approximation.68 The pair interactions are described by standard Lennard-Jones 6-1 2 pair potentials
We have inspected three models for these potentials that are characterized by their Lennard-Jones parameters given in Table 11. T o a first approximation all Lennard-Jones parameters for the atom-centered pair interactions were taken identical. If one sets them equal to the Lennard-Jones parameters for the SO2-Ar interaction, obtained from the usual combining rules uso2-Ar=
so, + uAr) = 3.7835 A
1
eso2-Ar=
so,^^)''^ = 193.1 K
The method used is based on the efficient and exact microcanonical sampling procedure developed by Nordholm, Freasier, and S~hranz!~,’~ We briefly describe the essential procedure; further details are given in ref 69 and 70. A microcanonical initial state is fully specified by appropriately selected spatial coordinates q and momenta p. The.required spatial coordinates are periodically selected from a Markov chain of spatial states, successive members of which are related through a Metropolis-type microcanonical (as opposed to canonical) transition probability. In the present case, the Markov chain is generated by randomly perturbing random bond lengths in the SO2 molecule with about 50% acceptance of the new state. Spatial configurations q are then chosen every 300 Markov chain moves. The momenta p are randomly selected from a normal distribution and then uniformly scaled so that the resulting p,q state has the correct molecular energy E . In part 223 we generalize the procedure so that the molecular angular momentum is specified within a narrow range. 3.2. Molecular Orientation. The selection of the initial conditions for SO2 relative to Ar is done with crude Monte Carlo sampling as described by Hammersley and Handscomb.‘l This aspect of the selection of initial conditions can be found in more detail in the review by Porter and Raff.72 The molecular center of mass position and velocity are subtracted such that the SO2 molecule is in the center of mass frame with respect to both position and velocities. The orientation of the SO2is randomized by three consecutive rotations of the position and velocity coordinates through three random angles cp3, ‘p2, and cp, cp3
= 2rR3,
‘p2
= 2rR2,
= 2rR1
(3.1)
(where R,, R2, and R3 are uniformly random numbers between 0 and 1) about the z , y, and x axes, respectively (the 321 Euler angle sequence64). 3.3. Relative Velocity. The relative velocity, vml,between the SOz molecule and the Ar collider is selected from the Boltzmann distribution over relative velocities a t a temperature T
(2.7)
where use, = 4.102 A, uAr = 3.465 A, eso, = 328.5 K, and eAr = 113.5 K, compared to real SOZ-Ar collisions one probably overestimates the cross section and well depth. We call this first model the large Lennard-Jones (LLJ) model. A second model assigns a third of the LLJ parameters to each of the atom-centered Lennard-Jones potentials. This model is called the small Lennard-Jones (SLJ) model. It probably underestimates the cross section and well depth. A third model of the collisional interaction was provided by using the S L J parameters but leaving out the attractive r-6 term in eq 2.6. This results in a repulsive Lennard-Jones (RLJ) potential. While it is likely that none of these three models correspond exactly to the true collisional interaction, the range of behaviors they encompass yields useful qualitative information. Calculations have been performed with all three models, but the SLJ model was used in the bulk of calculations.
3. Simulation Method The simulation method used here consists of three main parts: the selection of the initial conditions, the integration of the equations of motion, and the collection and analysis of properties pertaining to a single or an ensemble of trajectories. The selection of the initial conditions concerned the internal state of the SO2 molecule and initial conditions for SO2 relative to Ar. 3.1. Microcanonical Sampling Procedure. In a first set of calculations, we prepared the SOz initially in a microcanonical state with respect to its molecular (i.e., rovibrational) energy E .
where
is the reduced mass of the SO2 and Ar system (3.3)
The vector equation in (3.2) can be split up into its independent x, y, and z components; for the x component
and analogously for the y and z components. Each of these component distributions can be scaled to the standard normal form.‘, We then proceed by generating three normally distributed random numbers, N,, N,,, and N,, using, for example, Muller’s method,71 and the relative velocity vrel is obtained by scaling v,,I
=
[ “‘ -(N>
+ N: + N:)
3’’
(3.5)
3.4. Zmpact Parameter. The impact parameter is chosen from the probability distribution 2 r b db P(b) db = -
(3.6)
4 I a x
with crude Monte Carlo selection (69) Severin, E. S.; Freasier, B. C.; Hamer, N. D.; Jolly, D. L.; Nordholm,
(66) Noid, D. W.; Koszykowski, M. L.; Marcus, R. A. J . Chem. Phys. 1977, 67, 404. (67) Herzberg, G. Infrared and Raman Spectra of Polyatomic Molecules; D. Van Nostrand: New York, 1945. (68) Redmon, M.J.; Bartlett, R. J.; Garrett, B. C.; Purvis, G. D.; Saatzer, P. M.; Schatz, G. C.; Shavitt, I. In Potential Energy Surfaces and Dynamics Calculations; Truhlar, D. G., Ed.; Plenum: New York, 1981; p 771.
S.Chem. Phys. Lett. 1978, 57, 117. (70) Schranz, H. W.; Nordholm, S.;Freasier, B. C. Chem. Phys., submitted. (71) Hammersley, J. M.; Handscomb, D. C. Monte Carlo Methods; Methuen: London, 1967. (72) Porter, R. N.; Raff, L. M. In Dynamics of Molecular Collisions; Miller, W. H., Ed.; Plenum: New York, 1976; Part B, p 1.
Hippler et al.
6162 The Journal of Physical Chemistry, Vol. 90, No. 23, 1986
b = R’/2b,,,
(3.7)
where R is a random number and b,,, is a somewhat arbitrary cutoff on impact parameter that is large enough to include all significant energy-transfer events but not too large to contain many uninteresting encounters. In this work we find b,,, = 10 8, for the LLJ collisional model and b,, = 5 8, for the S L J and RLJ models to be sufficient. It should be emphasized that the derived average energies ( A E ) transferred per collision depend on the choice of bmx. The corresponding collision frequencies per unit concentration (see also below) are given by The choice of this collision frequency, nevertheless, does not exclude a large fraction of collisions that are nearly elastic. Later on we eliminate these collision and use the Lennard-Jones collision frequency as a reference number relative to which ( AE)inelfor inelastic collisions is calculated (see section 4.7). 3.5. Specified Initial State. The initial center of mass separation of the SO2 molecule from the Ar collider is set at do = 4 q i= 15 8,for the LLJ collision model and 5 8, for the SLJ and RLJ models. This is sufficiently large that there is negligible interaction between SO2 and Ar for all initial orientations. With the SO2 center of mass at the origin and random orientation of SOz,the Ar collider can be placed anywhere within the constraint of the chosen impact parameter b and the initial center of mass separation do. It is convenient to align the relative velocity of the Ar with respect to the SO2 molecule with the negative z axis. Then velocities and coordinates of the Ar collider are (3.9)
This fully specifies the zeroth and first time derivatives for the whole system. 3.6. General Procedure. Typically 2000 initial states of the S02-Ar system are successively generated by the above procedures. The equations of motion are then integrated until the center of mass separation of SO2 and Ar again equals (or exceeds) do or until a time limit t,,, is reached. For the current calculations most trajectories are completed (that is, they have achieved a separation do) within the time limit. Those few trajectories (4 out of more than 100000) that were not completed were for the LLJ model at low temperature but are not statistically significant. It is likely that such trajectories may become more common and important for lower temperatures and deeper wells in the collisional models. The equations of motion are integrated by a Gear fifth-order predictor-corrector method with internal time step 6 t that varies with the forces present but yields results at a constant external time step At. For the present runs, the external time step is At = 48.888 fs, the minimum internal time step is 6 t = 1.19 X fs and the maximum time allowed for a trajectory is t,,, = 500At = 24.444 ps. Any necessary higher derivatives are generated by integrating forward and backward in time over very it small time steps. In general, as with previous sim~lations,’~ is the integration of the equations of motion that dominates the computational run time. The selection of the initial states takes on the order of a few percent of total run time. This can be largely attributed to the highly efficient (and exact) microcanonical sampling method that we e m p l ~ y . ~ ~ , ’ ~ For each ensemble of 2000 trajectories we record the initial SO2 molecular energy E and the temperature T . For each individual trajectory, we record the initial molecular angular momentum J , the initial impact parameter b, the initial relative velocity urcl,the final molecular energy E’, and the final molecular angular momentum J’. 3.7. Rotational and Vibrational Energies. Because our model SO2 is a good approximation to a symmetric top molecule65 ( K i= -1 in Table I), it is possible to define the “rotational energy” E j as
EJ = Be@
(3.10)
where BeCis an effective rotational constant. As noted by Brown
20.0
00
60.0
40.0
80.0
100 0
E,/(kcal/mol)
Figure 2. Distribution P(E,) 6EJ of rotational energy EJ calculated with the SLJ collision model at a molecular energy E = 100 kcal/mol (bars are the coarse-grained simulation results and the solid curve is the prediction of a rigid-rotor-harmonic oscillator treatment, eq 3.14.)
and Miller$O this only corresponds to the “adiabatic part” of the rotational energy, whereas the “active part” is included with the vibrational energy
E, = E - Ej
(3.1 1 )
Frederick, McClelland, and Brumer3’ and Brown and Miller40 have both shown that eq 3.10 is satisfied to quite good accuracy for similar approximate symmetric top molecules, sufficient to make an approximate separation between vibrational EV and rotational energy E ) In the general case of an asymmetric top molecule such a partitioning of the molecular energy E into vibrational E , and rotational E j components would not be valid. From a large number (-25) of microcanonically sampled ensembles, for molecular energies E ranging from 2.683 to 100 kcal/mol, estimates were made of Beff. Each ensemble has a maximum molecular angular momentum J,,, when all the molecular energy is rotational. So from (3.10) (3.12)
On the average, Beffwas found to be [g/(mol A2)]-’. With this value, eq 3.10 is generally obeyed to within 5% as can be seen from Figure 2, which shows the probability density P(Ej) of a rotational energy E j at a fixed molecular energy E. A rigid-rotor harmonic oscillator treatment would predict
P(Ej) 6Ej =
(s (s
+ r / 2 - I)!
- l)!(r/2
- l)!
(
1 - - :)s-l(
: ) ~ J * -I~E~ E (3.13)
where s is the number of vibrations and r is the number of rotations. If we substitute r = 2 for the number of adiabatic rotations and s = 3.5 for the number of vibrations (including the active rotation), then
The accuracy of eq 3.14 relies on the adequacy of eq 3.10 and to what degree anharmonic and nonrigid effects are important. The comparison of eq 3.14 with simulation results in Figure 2 shows that the discrepancy is of the order of a few percent except at low Ej, where it reaches 10-20%. This is typical of the agreement found at all molecular energies E ranging from 2.683 to 100 kcal/mol. Thus, the partitioning of E into Ev and EJ via eq 3.10 and 3.1 1, though approximate, is adequate for the present calculations. It should be emphasized that the partitioning of E into E , and E j is a matter of convenience only. If one prefers
Energy Transfer in S02-Ar Collisions. 1
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6163 TABLE 111: Energy Transfer Collision Models”
model
T/K
LLJ
300 1000 300 1000 300 1000
SLJ RLJ
(AE),(AE,),and (AE,) for Various
(AE)l
(kcal/mol) -1.36 -1.03 -1.89 -1.84 -1.13 -1.53
f 0.14 f 0.15 f 0.25 f 0.28 f 0.20 f 0.25
(AEJ)/
(kcallmol) -0.93 -0.60 -0.60 -0.93 -0.35 -0.81
f 0.15 f 0.15 f 0.20
h 0.24
* 0.15 f 0.22
( AEv) /
(kcallmol) -0.43 -0.43 -1.29 -0.91 -0.78 -0.72
f 0.14 f 0.14 f 0.26 f 0.26 f 0.20 f 0.23
‘Molecular energy E = 100 kcal/mol. The LLJ results are calculated with b,,, = 10 8, and the SLJ and RLJ results with b,,, = 5 A. TABLE I V Energy Transfer ( A l l ) , (AE,),and (AE,) for Morse and Harmonic Oscillator Models of SO2 (SU Model, T = 300 K)
El
D
1.0
3.0
2.0
bl
4.0
A
Figure 3. Energy transfer ( A E ) (O), (AE,) (0),and (AE,) (A) as a function of impact parameter b for the SLJ collision model at a molecular energy E = 100 kcal/mol and temperature T = 1000 K (95% confidence limits for (AE) are shown; errors for (AE,)and ( A E , ) are similar).
a representation in terms of E and J, where J corresponds to the total angular momentum exactly conserved between collisions, eq 3.10 with the given Beffallows for the conversion from E j to J. We define the molecular energy transfer gained by the molecule AE as
AE = E ‘ - E
(AE)I
(A&)/
(AEv)l
model (kcal/mol) (kcal/mol) (kcaljmol) (kcal/mol) HM 50 -0.99 f 0.15 -0.34 f 0.09 -0.65 k 0.13
I
(3.15)
and the vibrational energy transfer AEv and the rotational energy transfer AEJ are defined in an analogous manner. 3.8. Trajectory Behavior. From single trajectory runs it was found that most trajectories are not completely backward integrable, though energy and angular momentum conservation are always satisfied to high accuracy. This is due to the fact that the present model of the triatomic molecule SO2 is highly anharmonic and hence chaotic at the high energies (40-100 kcal/mol) that are covered in this study. At lower molecular energies a higher degree of backward integrability is achieved. Furthermore if, instead of Morse pair potentials, the SO, molecule is modeled with simple harmonic potentials, then the motion is highly regular and backward integrable to any required accuracy within machine error. The tolerance parameters on the predictor-corrector algorithm are set so that the system energy is conserved to better than 3 parts in lo6 and the system angular momentum is conserved to better than 1 part in 10’. Thus, the system energy is typically conserved to within 3 X lo4 kcal/mol = 0.10 cm-’. Higher accuracy could be achieved a t the expense of extended computation time but is unnecessary for present purposes. 4. Typical Results 4.1. Dependence of Energy Transfer on Collision Model and Impact Parameter. The dependence of the average molecular energy transfer ( A E ) , the average rotational energy transfer ( A E J ) and , the average vibrational energy transfer (AE,) as a function of impact parameter 6 is demonstrated in Figure 3 for the collisional interaction described by the S L J model. Energy transfer becomes negligible for 6 > 3 A. The LLJ model shows the same behavior except that the energy transfers are about 3 times larger when, instead of b, = 10 A, they are scaled to the same 6,, = 5 A. Here, energy transfer becomes negligible only for 6 > 7 A because of the much larger collision diameter. As discussed below, the larger effective energy transfer can be attributed both to the steeper wall and to the deeper well of the LLJ potential. The dependence of energy transfer upon 6 is essentially similar for the S L J and R L J models. This suggests that energy transfer is indeed dominated by the repulsive wall of the intermolecular pair potentials.
Morse
100 50 100
-2.06 f 0.31 -0.77 f 0.20 -1.29 f 0.26 -1.01 f 0.14 -0.57 f 0.12 -0.44 f 0.11 -1.89 f 0.25 -0.60 f 0.20 -1.29 f 0.26
Table I11 gives a comparison of ( A E ) , ( AE,),and ( AE,) for the LLJ, SLJ, and RLJ models. If the LLJ results are scaled up by a factor of 4 to bring them to the same 6, as the other models, it can be seen that the LLJ model has larger energy transfer by a factor of roughly 2-3 over the SLJ model. This is due to both the larger collision diameter and the steeper repulsive wall and deeper well of the LLJ model over the S L J model. Suzukawa, Wolfsberg, and Thompson3*have also noted that a steeper repulsive wall increases energy transfer as a result of the shorter collision duration. Furthermore, a deeper well also can lead to increased energy transfer since it increases the likelihood of temporary trapping of the collider and subsequent multiple collisions with the repulsive wall. Because the SOz molecule is at an energy much higher than thermal energies, multiple collisions of the Ar with the repulsive wall would increase the average amount of SO2deactivation. That the repulsive wall dominates the energy transfer at high temperatures can be seen by comparing the S L J and RLJ models. At 1000 K there is only a 10-15% difference between these two models and it is only at 300 K that the RLJ model has 40% less energy transfer than the SLJ model. This can be related to the increasing importance of the attractive well at low temperatures. It should be emphasized that the three collision models are chosen for demonstration purposes. A realistic model for S02-Ar collisions would be one between the LLJ and S L J models; see below. 4.2. Sensitivity of Energy Transfer to Anharmonicity and Chaos. A comparison of energy transfer in Morse and harmonic oscillator models of SO2 is given in Table IV. There seems to be little difference between the energy-transfer behavior of the two models despite the chaotic nature of the Morse power spectrum compared to the much less irregular (almost regular) nature s~~ of the harmonic SOzpower spectrum. Hase and c o - ~ o r k e r have speculated as to whether the form of a measured energy-transfer transition probability will depend on the chaotic or quasi-periodic dynamics of the molecule. The answers to this question will necessarily depend on the initial state that is being prepared and observed. Our results indicate that such dependence is slight for initially microcanonically excited molecular states. Nalewajski and Wyattz8s29and Schatz26 both have demonstrated similar conclusions; Le., the differences in collisional relaxation of regular and irregular molecular states are not dramatic. Of course, initially locally excited molecular states are likely to exhibit a stronger dependence of energy transfer on the degree of chaos. It should be emphasized that anharmonicity in reality does play an important role at low energies. Here, the classical trajectories have to be quantized with respect to initial and final quantum states of the molecule. Anharmonicity influences this procedure. 4.3. Temperature Dependence of Energy Transfer. The temperature dependence of the energy transfer is demonstrated in Figure 4 for the LLJ collision model. For the LLJ model, I(AE)I
6164
Hippler et al.
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986
00
10000
20000
30000
40000
50000
60000
0.0
T/K
20.0
80.0
00.0
40.0
100.0
1 1.0
E/(kcal/mol)
Figure 4. Energy transfer (AE) (0),( M J )(0),and (AE,) (A)as a function of temperature T for the LLJ collision model at a molecular energy E = 100 kcal/mol (95% confidence limits for ( A E ) are shown; errors for ( AEJ) and ( AEv) are similar).
a
x gq
= Y
g? 3.'; ?
9 1.0
.o
20.0
46.0
60.0
80.0
100.0
0
E/(kcaUmol)
Figure 5. Energy transfer ( A E ) ( O ) , ( U J(0), ) and ( A E , ) (A)as a function of molecular energy E for the LLJ collision model (T = 300 K. 95% confidence limits for (Ai?) are shown; errors for ( AEJ) and ( AE,) are similar).
decreases initially, but above T = lo00 K there is little temperature dependence. On the other hand, the SLJ model exhibits very little temperature dependence in the range T = 300-1000 K. This may be related to the fact that the well depth tij is 3 times less than in the LLJ model (see Table 11) so that the effective dimensionless "temperature" k T / t i j is 3 times larger. It is likely that at lower temperatures the SLJ model will also exhibit more temperature dependence. In general, the molecular energy transfer ( AE) is insensitive to variation in temperature. This applies to the separate rotational and vibrational contributions as well. This is not in agreement with recent direct experimental studies that suggest that the temperature effect on average energies ( AE) transferred per collision is weak or even negligible for polyatomic molecules but relatively strong for triatomic m o l e c ~ l e s . ~ ~ ~ ~ ~ 4.4. Molecular Energy Dependence of Energy Transfer. There is a marked dependence of ( AE) on molecular energy as shown in Figures 5 and 6. I(AE)I increases rapidly as E increases from thermal energies to 4 0 kcal/mol. For the LLJ model, a plateau region seems to be encountered from E = 50 to 100 kcal/mol and presumably at higher energies. However, the SLJ model exhibits strong molecular energy dependence of ( AE) from thermal energies right up to 100 kcal/mol. At average thermal energies, ( A E ) approaches zero as required by detailed balancing.
Figure 7. Energy transfer (AE) (a), ( U J(0), ) and (AE,) (A)as a function of rotational energy E, for the LLJ collision model at a molecular energy E = 100 kcal/mol and temperature T = 300 K (95% confidence limits for ( AE)are shown; errors for ( A E J ) and ( AE,) are
similar).
2
I
0.0
20.0
40.0
00.0
80.0
100.0
E,/(kcal/mol)
Figure 8. As for Figure 7 but at a temperature T = 5000 K
4.5. Rotational Energy Dependence of Energy Transfer. The dependence of the energy transfer ( A E ) , ( AEJ), and ( AEv) on the rotational energy EJ is shown in Figures 7 and 8 for various
Energy Transfer in S02-Ar Collisions. 1
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6165 0
TABLE V Estimated Inversion Energies, E," and E,"', for ( AE,) and (AE,),Respectively, as a Function of Various Collision Models, Temperaturee
T,and Molecular Energies Ep
model T/K LLJ 300
E
( W T( E I ) M C
2.683 0.596 0.596 50 11.1 100 22.2 1000 8.942 1.987 1.987
SLJ
E ': 0.6 f 0.2
E,V' 1.0 d~ 0.5
1.4 f 5.0 22 f 10
0.7 f 5.0 1.7 f 0.5 50 11.1 2.5 f 3.0 100 22.2 4.4 f 6.0 5000 50 9.936 11.1 8.0 f 5.0 100 22.2 13 f 10 300 2.683 0.596 0.596 0.6 f 0.4 25 5.5 1.9 f 2.0 50 11.1 3.4 f 5.0 100 22.2 12 f 12 1000 8.942 1.987 1.987 1.7 k 1.5 25 5.5 2.5 f 3.0 50 11.1 3.8 f 10 22.2 7.5 f 10 100
*
30 10 2.5 & 2.0 19 f 10 40 & 20 16 f 10 30 f 20 1.0 f 0.8 10 f 7 24 f 15 55 f 35 4.2 f 5.0 8.9 f 5.0 21 f 10 46 f 20
The thermal-average rotational energy (E,) and the microcanonical-average rotational energy ( E J ) M Care also given for comparison. Energies are in kcal/mol.
*
x 0.0
20.0
60.0
80.0
1 1.0
E,'/(kcal/mol)
a
temperatures T and molecular energies E. In general, the rotational energy gain (AE,)decreases with E, but increases again at high E,, whereas the vibrational energy gain (AE,)increases with E, and also decreases at high E p The decrease in the magnitude of ( A E v ) and (AE,) at high rotational energies can be explained in terms of Landau-Teller t h e ~ r y . ' ~ , ~For ' high rotational energies, the period of rotation becomes smaller than the collision time and in this adiabatic limit energy transfer is highly inefficient. The behavior at low to moderate rotational energies E, is better explained in terms of Table V. This shows simulation estimates of the inversion energies for rotational and vibrational energy transfer. The inversion energies, E," and EJn, are the rotational energies at which ( AEj) and ( AE,), respectively, change sign. Table V and Figures 7 and 8 show that the energy EJn at which ( AE ,) changes sign is always roughly equal to the thermal-average rotational energy ( E,) T,whereas (AE,) changes sign at an energy, generally much higher, roughly equal to the microcanonical average rotational energy ( E J ) M C .This suggests strongly that T rotational energy transfer occurs largely through the R pathway. For rotational energies lower than the thermal average, the rotations try to gain energy from the relative translations, whereas the roles are reversed for rotational energies greater than the thermal average. On the other hand, vibrational energy transfer seems to occur largely via the V - R pathway. For rotational energies less than the microcanonical average, the vibrations tend to lose energy, whereas energy is gained for rotational energies greater than the microcanonical average. Because ( AEv)is usually small, ( AE) roughly follows ( AE,) in behavior. The molecular energy loss - ( A E ) increases for low up to intermediate E, but ultimately decreases for high rotational energies. Similar observations on the behavior of ( AE) have been made by Gelb in a recent study of Ar-03 energy exchange.45 In fact the behavior observed here is also similar to that seen in full three-dimensional molecular dynamics simulations of excited Br2 in a thermal Ar medium by Nordholm, Freasier, and J01ly.~~-~' 4.6. Energy Flow in Collision. Figures 9 and 10 display the flow of rotational and vibrational energies for microcanonical initial states a t high and low molecular energies E, respectively. The tail of the arrow represents the initial Ev,EJ state and the head indicates the final E;, EJ' state. Transitions parallel to the EV = E - E, line are elastic (AE = 0) and all other transitions are inelastic (with respect to the molecular energy). It is apparent from both figures that there is a high proportion of almost elastic (AE = 0) transitions. In Figure 9, the molecular energy E = 100 kcal/mol is much higer than the thermal energies associated with a temperature of T = IO00 K. Thus, there are very few &activating collisions (AE > 0) but many E-deactivating collisions (AE < 0). In addition,
40.0
\
Figure 9. Flow of rotational E , and vibrational energies E , in collisions of a SO, microcanonical initial state at a molecular energy E = 100 kcal/mol with an Ar collider at a temperature T = 1000 K (SLJ collision
model).
-
0.0
4.0
8.0
12.0
1e.o
2 1.0
E;/(kcal/mol)
Figure 10. As for Figure 9 but at a molecular energy E = 8.942 kcal/
mol.
...
H
Y'I
8?
-60.0
4.0
-20.0
0.0
.
20.0
4 1.0
AE,/(kd/mol)
Figure 11. Energy transfer AEv,AE, for individual trajectories. SLJ collision model at a molecular energy E = 100 kcal/mol and temperature T = 1000 K.
Hippler et al.
6166 The Journal of Physical Chemistry, Vol. 90, No. 23, 1986
*I: 0 . .
AE/lkCOl/mOl)
I 0
-2.0
Figure 13. Collisional transition probabilities P(E’/E) for the SLJ collision model at a molecular energy E = 100 kcal/mol and temperature 7‘= 1000 K (bars are the coarsegrained simulation results, and the solid curve is the weighted mean least-squares fit of a single-exponential function, eq 4.3 and 4.4. The elastic peak (Ai7 = 0) is truncated).
”
2.0
6.0
10.0
I
AE,/(kcrWmd) Figure 12. As for Figure 11 but at a molecular energy E = 8.942 kcal/mol.
-
many large collision-induced V R transitions (AE= 0) are apparent; see also Figure 11. Figure 10 shows the behavior for a molecular energy equal to the thermal energy, E = 8.942 kcal/mol, and in this case there are substantial numbers of activating and deactivating collisions. These are roughly symmetrically distributed in the AEv vs. AEJ plane; see also Figure 12. Figures 9 and 10 directly show the relation between ( AEv)and ( AEJ). Three broad regions can be roughly identified. When E j is less than the inversion energy E,.” for ( hEJ)(see Table V) there are many AEv < 0, AEJ > 0 transitions. When Ej is above EJAbut less than EJvl,the inversion energy for ( AEv),there are many AEv < 0, AEJ < 0 transitions. When E j is above Ejvl there are many AEv > 0, AEj C 0 transitions. 4.7. Collision Frequencies and Collisional Transition Probabilities. In our presentation, energy transfer so far has been studied for randomly selected trajectories characterized by impact parameters up to b,,,. As shown in Figure 3 these collisions still contain a substantial contribution from nearly elastic collisions with AE close to zero. These contributions cancel in master equations (1.3). For many reasons, it appears desirable in practice to eliminate these elastic collision^.^^ One way of doing this is based on the definition of a total energy-transfer collision frequency (per unit concentration) that omits the elastic contributions
Zinel(E)= l m0 k ( E ’ / E )dE’
(4.1)
AE/lkcOl/mOl)
Figure 14. As for Figure 13 but at a molecular energy E = 8.942 kcal/mol.
a weighted mean least-squares fit to an exponential function of the formsJO
=
E’- E7 for) E’> E
1 exp(
(4.3)
In this fit, the central elastic peak has been eliminated in a way similar to the method of Brown and Miller.40 These authors analyzed the details of P(E’/E) with a finer graining, which leads to an even better fit by a sum of two exponential functions. For simplicity we decided to restrict ourselves to a simple exponential fit. There is, of course, the problem of the grain size, used for eliminating the elastic peak. This question is closely related to the definition of Zinel(E).We handled this by using a weighted least-squares fit minimizing M
It has been suggested6 that Zinel(E)is close to the Lennard-Jones collision frequency. Earlier trajectory calculation^^^ have confirmed this a s s ~ m p t i o n . Although ~~ the effective Lennard-Jones parameters, which would correspond to the present LLJ, SLJ, and RLJ models, have not been elaborated in detail, our present work apparently supports this point of view. For a comparison of the derived ( AE)values with measured values, therefore, our ( AE) values should be scaled (see above) to the respective Lennard-Jones frequencies; Le., they should be divided74 by the ratio bfnax/qJ2Q(232)*. Of course, this procedure would require that the experiments be interpreted with Lennard-Jones collision frequencies. Omitting the elastic contributions and dividing k(E’/E) by the one determines collisional transition collision frequencies Zinel(E), probabilities J‘(E’/E) = WE’/E)/zinel(E)
(4.2)
Figures 13 and 14 compare coarsegrained simulation results and (73) Widom, B. Adv. Chem. Phys. 1963, 5, 353. (74) Troe, J. J . Phys. Chem. 1979, 83, 114.
S = CIAEil(hi - k(AEi) 6E’)2 i= I
(4.4)
where hi is the “histogrammed” energy-transfer rate coefficient, 6E’is the energy grain, and M is the number of grains employed (usually of the order 100-500). By this fit the elastic peak AE = 0 is automatically neglected and more attention is paid to the stronger (more inelastic) collisions. The fitted exponential parameters are given in Table VI for a range of temperatures T and molecular energies E . The condition E >> CY is well satisfied such that the lower limit of integration in eq 4.1 can be put to -m. With eq 1.1, 1.2, and 4.1-4.3, the energy-transfer moments
(AE)inel = P - CY ( A E ~ ) ~=, ,2(p2 ~
+
and the normalization constant N=P+CY
(4.5) -
(4.6)
(4.7)
are obtained. The corresponding values are included in Table VI. It should be emphasized that there is a second method to derive the energy-transfer moments from the fitted exponential param-
The Journal of Physical Chemistry, Vol. 90, No. 23, 1986 6167
Energy Transfer in S02-Ar Collisions. 1 TABLE
VI: Fit of Inelastic Collisional Energy Transfer by Exponential Collision Model" TIK E a P Gle'(E) ( AE ) inel 300
1000
2.683 25 50 100 8.942 25 50 100
0.28 2.82 5.62 8.53 1.06 3.17 5.91 10.6
0.25 0.59 0.75 0.76 0.99 1.18 1.38 2.09
0.90 0.86 0.76 0.80 1.20 1.29 1.30 1.32
-0.033 -2.23 -4.87 -7.77 -0.063 -1.99 -4.53 -8.49
(AE)
( AE2)inel
0.15 13.3 55.8 134. 2.10 15.4 57.4 188.
-0.016 f 0.011 -0.57 f 0.08 -1.01 f 0.1 -1.89 f 0.26 -0.035 f 0.032 -0.45 0.08 -0.89 f 0.14 -1.84 f 0.28
*
'Equations 4.1-4.6. SLJ collision model. All energies in kcal/mol; Zine,(E)in lo-'' cm3 molecule-' s-I, Z(b,,,) = 3.99 X X cm3 molecule-' at 1000 K.
0.061 3.21 11.8 37.3 0.53 3.66 10.7 45.1 at 300 K, Z(b,,,)
= 7.29
eters. This method should be preferred, in particular, for weak collisions where l(AE)inell
