J. Phys. Chem. 1988, 92,1223-7229 they will be used in subsequent work to obtain the free energies of hydration.
Acknowledgment. This work was supported by the US.-Korea Cooperative Science Program between the National Science Foundation and the Korea Science and Engineering Foundation (NSF grant INT-87-0537), by the National Science Foundation
7223
(NSF grant DMB84-0181 l ) , and by the National Institute of General Medical Sciences, National Institutes of Health ( N I H grant GM-14312). The work was carried out with the computer system at the Software Engineering Center (SEC) which was established as a partnership program between IBM Korea and the Systems Engineering Research Institute (SERI) at the Korea Advanced Institute of Science and Technology (KAIST).
Theoretical Studies of Collisional Energy Transfer in Highly Excited Molecules: Temperature and Potential Surface Dependence of Relaxation in He, Ne, Ar CS2
+
Margaret Bruehl and George C. Schatz* Department of Chemistry, Northwestern University. Evanston, Illinois 60208 (Received: July 15, 1988)
In a previous paper we presented a method for studying the collisional relaxation of highly excited triatomic molecules based on trajectory simulations of successive collisions of the molecule with bath gas atoms. The method involves the microcanonical redistribution of vibrational coordinates and momenta between each collision to approximate the aging that occurs in gas-phase experiments. This method is expected to be accurate when the molecular internal energy is high so that molecular motions are chaotic. In this paper we use this redistributed successive collision (RSC) method and a related method based on single energy collisions (SEC) to study the relaxation of CS2by He, Ne, and Ar. For He + CS2we examine the temperaturedependence of the average vibrational energy transfer ( AE) and find that, at 36 000 cm-I, ( AE) increases monotonically with T between 300 and 2000 K. The temperature dependence is similar but stronger for Ne + CS, and Ar + CS2. All of our results at 36000 cm-l are in agreement with experiment within the experimental uncertainties. The dependence of (AE)on the average vibrational energy ( E ) of CS2 is approximately linear for He + CS2and Ne + CS, at all temperatures and for Ar + CS2 at 1000 K. Except for He + CS, at 1000 K, this behavior differs from the stronger dependence that is found experimentally (often close to quadratic), and as a result the experimental values of ( A E ) at low ( E ) are smaller than those determined in the calculations. Ar + CS2 at 300 K can be fit about equally well by either linear or quadratic dependence of ( A E )on ( E ) . For this case, theory and experiment agree well at all ( E ) ’ s . We also consider the dependence of (AE)on the potential surface used in the trajectory simulations. For the CS2intramolecular potential, we consider three surfaces: harmonic, sum of Morse functions, and a many-body expansion surface. We find that ( A E )for He + CS2is insensitive to which surface we use. For the intermolecular potential we consider exponentialspline-Morse-spline-van der Waals (ESMSV) pair potentials and Lennard-Jones 6-12 and 6-20 pair potentials. We study the effect of well depth and repulsive wall steepness for the He CS2 system and find that different parametrizations of the potentials and variation of the van der Waals well depths lead to the same (AE)’s;however, an increase in steepness (as in changing from 6-12 to 6-20 potentials) leads to a significant change in ( A E ) .
+
I. Introduction The collisional relaxation of highly excited polyatomic molecules is an important but poorly understood topic in gas-phase chemical kinetics. Until recently, most estimates of collisional deactivation rates were derived indirectly through an analysis of the pressure dependence of other kinetic processes. Most of these estimates are not very precise. Recently, a number of “direct” measurements of such rates have appeared’ which provide much more detailed information about the average vibrational energy transfer per collision ( AE)than has been available previously. Some of the most extensive of these measurements concern the relaxation of CS2 by a variety of bath gases, including the rare g a ~ e s . ~These .~ measurements have characterized the dependence of ( AE) on CS2 vibrational energy E, on bath gas temperature, and on the bath gas molecules themselves. All of these functional dependencies are complex, and to date no theoretical explanation for the results has been given. In addition, lingering controversies concerning ( A E ) measurements in larger molecules such as azulene’ suggest that accurate theoretical studies of these collisional relaxation experiments are needed in order to put the results on a firm theoretical footing. In this paper we present the results of theoretical simulations of the rare gas + CS2 relaxation process, including detailed comparisons between theory and experiment that are designed (1) For a review see: Gordon, R. J. Comments A t . Mol. Phys. 1988, 21, 123. (2) Dove, J. E.; Hippler, H.; Troe, J. J . Chem. Phys. 1985, 82, 1907. (3) Heymann, M.; Hippler, H.; Plach, H. J.; Troe, J. J. Chem. Phys. 1987, 87, 3867.
0022-365418812092-7223$0 1.5010
to probe the mechanisms underlying the relaxation processes and how these mechanisms depend on energy, temperature, rare gas, and potential energy surface. The primary method that we use for the present study is classical trajectories, but within this framework we use two related methods that are specifically designed for studying collisional relaxation: the redistributed successive collision (RSC) method and the single energy collision (SEC) method. RSC, which was introduced in an earlier paper: involves calculating sequences of collisions of the excited molecule with bath gas atoms, with the assumption that the time between collisions is long enough and the molecular vibrational motions are chaotic enough to cause complete vibrational redistribution between each collision. As a result of this assumption, the initial conditions for each collision can be chosen from a microcanonical ensemble having an energy and angular momentum that are determined by the outcome of the previous collision. This approach is equivalent to simultaneously solving the collkion dynamics to determine energy-transfer rate coefficients and the master equation to determine the time evolution of the molecular energy distribution. While being computationally intensive, this method is still a valuable tool for benchmark calculations, such as the system presently studied. The RSC method was tested in ref 4 by comparison with experiment for He + CS2 at 300 K,2,3and it was found that the energy-averaged ( A E ) ’ s were in good agreement. The energy ranges used for these averages were 32 640-9030,9030-4680, and 4680-3 130 cm-’. The functional dependence of ( A E ) on E was different, ~
(4) Bruehl, M.; Schatz, G. C. J . Chem. Phys. 1988, 89, 770.
0 1988 American Chemical Society
7224 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 however, with the RSC results showing a linear dependence on E and the experimental results a more complex dependence that varied between linear and quadratic. The present manuscript will greatly extend these comparisons between RSC and experiment. One of the important conclusions of the RSC calculation in ref 4 was that CS2 rotation is in a slowly varying steady state during the relaxation process. This suggests that, as an alternative to the time-consuming RSC calculation, one can calculate (AE) at any selected energy by considering microcanonical ensembles of trajectories at that energy, locating the steady-state rotational angular momentum J and then calculating ( A E ) from an ensemble at that J. In this paper we will show that one actually can determine ( AE) from such a single energy calculation (SEC) without locating the rotational steady state very precisely. As a result, the number of trajectories needed for a SEC calculation is quite small, and this enables us to determine (AE)for systems for which the RSC calculation is computationally impractical. Since both the SEC and RSC methods derive trajectory initial conditions from microcanonical ensembles, they should generate the same ( A E ) ’ sfor a given problem. We will show that this is satisfied in our calculations. The SEC method is closely related to methods that have been used in a number of earlier studies of collisional relaxation in highly excited polyatomics.510 Some studies have considered canonical rather than microcanonical distributions in the excited molecules?*6 and others differ in how ( A E ) was determined,8v9but the calculations by Gallucci and Schatz’ and by Hase and co-workersI0 were based on methods that are functionally the same as SEC. However, the SEC method used here is different from even these calculations in two ways. First, the calculation is done at a Jvalue that is close to the steady-state value. The steady-state value is is zero. that J where the average rotational energy change (AE,) In earlier studies, either the initial J was taken from a Boltzmann distribution or it was chosen arbitrarily. Second, the maximum impact parameter b,, is chosen so that the product of ( AE) times .rrb,,: is converged. The total cross section is then used to scale the experimental results so that the comparison between theory and experiment is independent of bmx. Earlier studies have used but in all cases the a variety of procedures for assigning b, comparison between theory and experiment has depended on the value assumed. One limitation associated with both the RSC and SEC calculations is the assumption that the distribution of CS2 coordinates and momenta is microcanonical before each collision. This is the case if the CS2 internal motions are chaotic, since the time between collisions in the experiments is much longer than the slowest vibrational period, and thus randomization of the CS2 internal coordinates and momenta is expected between collisions. However, if motion is not chaotic, then the microcanonical redistribution between collisions is no longer appropriate. We studied this point in ref 4 by doing successive collision calculations with the CS2 vibrational coordinates and momenta frozen (conserved) between the end of one collision and the beginning of the next. This CSC (conserved successive collisions) calculation produced ( AE)’s that are stronger functions of E (close to quadratic) than in the RSC results, but the ( A E ) ’ s themselves are much too small at low E. The small ( AE)’s resulted from CS2molecules in which substantial amounts of energy (up to 26 000 cm-I) are “frozen” in certain modes that subsequently relax extremely slowly. This appears to be an error arising from the use of classical mechanics. The most likely quantum pathway for relaxation of such metastable states, namely, multiquantum transitions to states that are close in energy to the metastable state but which have a very different (5) Bunker, D. L.; Jayich, S. A. Chem. Phys. 1976, 13, 129. (6) Stace, A. J.; Murrell, J. N. J . Chem. Phys. 1978, 68, 3028. (7) Gallucci, C. R.; Schatz, G. C. J . Phys. Chem. 1982, 86, 2352. (8) Brown, N. J.; Miller, J. A. J . Chem. Phys. 1984, 80, 5568. (9) Hippler, H.; Schranz, H. W.; Troe, J. J . Phys. Chem. 1986, 90,6158. Schranz, H. W.; Troe, J. J . Phys. Chem. 1986, 90, 6168. (10) Hase, W . L.; Date, N.; Bhuiyan, L. B.; Buckowski, D. J. J . Phys. Chem. 1985, 89, 2502. Hu, X.; Hase, W. L., submitted for publication in J . Phys. Chem.
Bruehl and Schatz
*’
~
4
3
2
3
4 , 1 ’ 3 ’ 4 ’ rc-s 1%
Figure 1. Contour plots of CS2 potential as a function of the two CS stretch coordinates: (a) CM, (b) M, (c) H surfaces. The contour interval is 3856 cm-I so that the highest contour is at 36000 cm-l.
distribution of quanta, is a classically forbidden process. Most of the present analysis will consider internal energies that are high enough so that the assumption of chaos which underlies the RSC method is reasonable. However, it appears from our CSC calculations that even a small amount of nonchaotic behavior can have a noticeable influence on how ( A E ) depends on E, so we will view the comparison of our RSC results with experiment with this in mind. 11. Calculations A . Potential Functions. We consider the potential surfaces for He-CS2, Ne-CS2, and Ar-CS2. None of these potentials are known accurately, but a reasonable approximation can be generated by making the usual separation into intramolecular and intermolecular contributions as follows:
VM-CS~ = Vintra + Vintei
where Vintra is the isolated CS2 intramolecular potential and Vinter is the rare gas CS2 intermolecular potential. In our earlier work4 we represented Vintra by an empirically derived global potential function due to Carter and Murrell” (CM). This function describes the quadratic, cubic, and quartic terms in the equilibrium force field accurately and it also dissociates correctly, but the validity of the surface between these two limits is not known. In this paper we will use the C M potential for most of our work, but we will also examine two other potential functions which we label as Morse (M) and harmonic (H). The harmonic potential is simply the harmonic part of the C M potential. It thus has the correct normal-mode frequencies, but it does not dissociate at all. The Morse potential consists of Morse functions that describe the C-S, C-S’, and S-S’ bonds plus a damped harmonic function for the S-C-S bend angle. Parameters are chosen so that CS2 has the desired equilibrium geometry, normal-mode frequencies, and dissociation energies to S + C S and C S2. The function used is
+
+
VM,, = Decs[1 - exp(-pcs(r - rcS))l2 + D,cs[ 1 - exp(-ps(r’ - rcs))12 + D,Ss[ 1 - exp(-PSS(r”rss))12 + Xk02[)/,(1- tanh dr’’ - rss”9)l (2)
where r, r’, and r f fare the CS, CS’, and SS‘ distances and 6 is the SCS bend angle. The parameters in eq 2 (in atomic units) are Decs = 0.1407, pCs = 1.2634, rcs = 2.9347, Dess = 0.0717, pss = 0.5947, rss = 5.8694, k = 0.1324, y = 2.0, and rSSmf= 11.739.
Figure 1 presents contour diagrams of linear CS2for the CM, M, and H potentials. The highest contour shown is 36 000 cm-I, the highest energy of interest in this study. All three potentials are close at low energies, but at 36 000 cm-l, the H potential is substantially different. The C M and M potentials are similar, but the M potential is more attractive at long range. The harmonic frequencies on all three potentials are within 1 cm-l of 1532 cm-I (asymmetric stretch), 674 cm-’ (symmetric stretch), and 400 cm-’ (bend). The lowest energy dissociation threshold is at 45 OOO cm-’ (to CS(’2) S(’D)) on both the M and C M surfaces. This assumes that spin is conserved during dissociation as we have discussed previ~usly.~
+
(11 )
Carter, S.; Murrell, J. N.
Croat. Chem. Acta 1984, 57, 355.
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 7225
Energy Transfer in Highly Excited Molecules TABLE I: Intermolecular Potential Functions potential pair potential for HeS, HeS', HeC label He-1 ESMSV' He-2 LJ 6-12' He-3 LJ 6-12b He-4 LJ 12' He-5 LJ 6-20d Ne ESMSV Ar LJ 6-12f
TABLE II: Gas Kinetic Rate Constants for M
well depth, cm-' 44 47 59 0 47
130 270
'Reference 12. *Same parameters as in He-2, except well depths in pair potentials are 25% deeper. cSame parameters as in He-2, but attractive part of potential is eliminated. dParameters selected to give same well depth and equilibrium separation as He-2, but steeper repulsive wall. e Reference 13. fReference 14.
rHe-C/ao Figure 2. Comparison of the five He-CS, intermolecular potentials as a function of the He-CS, separation for perpendicular geometries. Surfaces as defined in text and labeled as follows: He-1, solid; He-2, pluses; He-3, long dash; He-4, dot-dash; He-5, short dash.
The intermolecular potential was represented as a sum of atom-atom pair potentials for all three rare gases. Table I summarizes the functions used, the literature sources for the parameters, and the van der Waals well depths for C, geometries. For H e CS2 we consider five potentials so as to determine the sensitivity of the results to the functional form used. Figure 2 compares the various potentials at perpendicular He-CS2 geometries. The He-1 and He-2 potentials are based on He-Ne and He-Ar pair potentials available from the literature.12 He-1 is composed of the exponentialspline-Morsespline-van der Waals (ESMSV) functions that we used previously,4and He-2 is the sum of Lennard-Jones 6-12 potentials that fit the same scattering data. He-3 is the same as He-2 except the 6-12 well depths have been chosen to be 25% deeper. He-4 has the same parameters as He-2; however, the attractive part of the potential has been eliminated. He-5 consists of 6-20 rather than 6-12 functions, with parameters chosen so that the well depth and equilibrium geometry are the same as in He-2, making the repulsive wall in He-5 much steeper. By comparing the Lennard-Jones surfaces (He-2 through He-5), we can determine the sensitivity of the results with respect to well depth for a given repulsive wall steepness and steepness for a given well depth. The Ne-CS2 potential in Table I was developed by using ESMSV functionsI3 analogous to He-1 . Ar-CS2, on the other hand, was represented by 6-12 functions that were taken from a potential developed for Ar-OCS.I4 The latter potential fits known structural information on the Ar-OCS van der Waals complex, and it also gives V T rates for Ar + OCS(Ol0) that are within a factor of 2 of experiment. B. Gas Kinetic Rates. As discussed in detail in ref 4,only the product of ( A E ) and the gas kinetic rate constant k is uniquely determined in our calculation. This is because classical mechanics is unable to determine the gas kinetic rate for potentials of infinite
+
-
(12) Chen, C. H.; Siska, P. E.; Lee, Y . T. J . Chem. Phys. 1973, 59,601. (13) Ng, C. Y . ;Lee,Y.T.; Barker, J. A. J . Chem. Phys. 1974,61, 1996. (14) Gibson, L. L.; Schatz, G. C. J . Chem. Phys. 1985, 83, 3433.
+ CS2 (cm3 s-')'
M T. K 300
700 1000
He
Ne
Ar
1.14 X 4.85 X lo-'' 1.74 X 6.41 X lo-'' 2.08 X 7.28 X lo-''
6.74 X lo-'' 2.93 X lo-''
8.48 X lo-'' 3.42 X lo-''
1.23 X lo4 4.17 X lo-''
1.55 X lo4 4.29 X lo-''
1500
2.55 X
2000
8.37 X lo-'' 2.94 x 10-9 9.26 X lo-''
"Top entry in each category is the rate constant based on b,, for which AE(b,,,.J < 0.05(AE). Bottom entry is that from ref 2 and 3 which is extracted from the Lennard-Jones transport cross section. range. (It diverges for infinite maximum impact parameter b-.) Fortunately, the experimental studies of CS2 also measure the product of ( A E ) and the gas kinetic rate constant (rather than ( AE) itself), so the comparison between theory and experiment can be made without having to address this point of ambiguity. It is convenient, however, to present results in terms of ( AE)rather than k ( AE) as the ( A E ) ' s relate more directly to other kinetics measurements, but to do so we need to adopt a convention for defining k. Several possible definitions were presented in ref 4, and others have been discussedsJOincluding that used by Hippler, Troe, and co-workers based on the Lennard-Jones viscosity transport collision integral Qfg)*,that obtained from the elastic cross section which can be determined from the spherically averaged potential, and that from the effective cross section which < 0.05( AE). is associated with choosing b,, such that AE(b,) Since the last of these possibilities appears naturally in our calculations, it is convenient to use this in presenting our results. Table I1 presents the gas kinetic rates used for the rare gas and temperature combinations that we consider. Also included, for reference, are the gas kinetic rates used by Hippler, Troe, and co-workers based on the transport collision integral.2J In all comparisons between our ( A E ) ' s and those from ref 2 and 3 we have multiplied the experimental results by the ratio of gas kinetic rates in Table I1 so that all results are referenced to the same gas kinetic rate. This ratio is in the range 0.28-0.45 for the entries in Table 11. C . RSC and SEC Calculations. The RSC method has been described in detail in ref 4. SEC has not been discussed previously, but many features of it are similar to RSC. The key difference is that in SEC the initial vibrational energy E and rotational angular momentum J of the CS2are the same for each trajectory in the ensemble, while in RSC the E and J values for each trajectory are determined by the previous trajectory. For both RSC and SEC the process of selecting initial conditions for a given E and J from a microcanonical ensemble is described in ref 4. Since CS2is a linear molecule, careful treatment of angular momentum is necessary, and in ref 4 we found that a reasonable way to define the microcanonical ensemble involves separately choosing J (the angular momentum component perpendicular to the molecular axis) and 1 (the vibrational angular momentum). For RSC,the initial choice of J and 1 was found to be unimportant as J rapidly comes to steady state (after a few collisions) while I, which is much smaller than J for thermal rotations, is randomized due to coupling between the bend and stretch modes. Because of this randomization, I can be redistributed between collisions, and the present calculations include I in the list of redistributed variables. The choice of J and 1 in the SEC calculation is guided by what we have learned from RSC. Since J comes to steady state in RSC, the optimum J for SEC calculations is the steady-state value, and in general this value is expected to occur when the average change in rotational energy (AE,) vanishes. Table I11 shows typical SEC results as a function of J for H e + CS2at E = 9030 cm-I and I = 0.5. The ( AE,) values are positive for low J , negative for high J, and vanish at J = 30. Thus, J = 30 is the SEC steady-state
Bruehl and Schatz
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
1226
TABLE III: J and I Dependence of SEC Energy Transfers (in cm-I) for He CSz at G = 9030 cm-I and T = 300 K
+
0 20 30 44 60 80 100
1 0.5 0.5 0.5 0.5 0.5 0.5 0.5
44 44 44 44 44 44
0.0 0.5 5.0 10.0 15.0 20.0
J
(AE) -4.8 f 1.1 -5.0 f 1.3 -5.0 f 1.4 -4.2 f 1.3 -4.2 f 1.3 -4.3 f 1.6 -3.8 f 1.8
4.6 -0.2 -5.0 -13.0 -25.6 -45.2 -66.9
f 1.4 f 1.8 f 2.2 f 2.8 f 3.1 f 5.1 f 6.6
-1.3 -4.2 -6.0 -6.7 -6.3 -6.5
-10.7 -13.0 -15.1 -15.5 -14.8 -14.4
f 2.7 f 2.8
f 1.2 f 1.3 f 2.0 f 2.3 f 2.0 f 2.1
(MI)
f 3.1 f 3.3 f 3.1 f 3.1
(AE,) 9.4 f 0.7 4.9 f 1.2 0.0 f 1.7 -8.7 f 2.5 -21.5 f 3.5 -40.9 f 4.9 -63.1 f 6.4 -9.4 -8.7 -9.1 -8.8 -8.5 -7.9
f 2.4 f 2.5
f 2.4 f 2.4 f 2.4 f 2.4
TABLE I V Parameters in E l l 3 Obtained from Fits to He + CSz Data T, K A, cm-’ B C, cm-I (AE),O cm-’ 300 31 577 f 385 (6.73 f 0.24) X lo4 834 -23.7 f 0.8 1946 -42.1 f 1.8 700 30758 f 208 (1.24 f 0.05) X lo-) 2780 -44.3 f 2.8 1000 29017 f 399 (1.33 f 0.09) X lo-) 1500 28066 f 563 (2.03 f 0.13) X lo-) 4170 -64.5 f 4.1 2000 27790 f 306 (2.49 f 0.23) X lo-) 5560 -75.9 f 7.0 “ A t 36000 cm-’
TABLE V Comparison of Calculated and Experimental (AG)’s (in em-’) for He + CSz at 300 and 1000 K 300 K 1000 K ( E ) , cm-‘ calcd exptl calcd exptl 36000 -23.7 f 0.8 -42 -44.3 f 2.8 -49 32640 -21.4 f 0.8 -40 20000 -12.9 f 0.5 -20 -22.9 f 1.5 -35 10000 -6.2 f 0.2 -5.5 -9.6 f 0.6 -12 9030 -5.5 f 0.2 -5.2 5000 -2.8 f 0.1 -1.5 -3.0 f 0.2 -2.3 4640 -2.6 f 0.1 -1.6 3130 -1.5 f 0.1 -1.0
2 . i A 2.0t W
8l
4
v, 2 1.51
7
15 5 10 Collision Number/ 1 0 2
Figure 3. RSC ensemble averaged vibrational energy (in cm-’) versus collision number n for He + CS2at T = 300,700, 1000, 1500, and 2000 K: (a) ( E ) versus n, (b) In ( E ) versus n.
value, and, in fact, this agrees well the RSC calculations. The vibrational energy transfer ( A E ) also agrees (-5.0 cm-I versus -5.5 cm-’ from RSC) as we would expect, but note that the same value of ( A E ) is obtained for all the Ss considered in Table I11 (within statistical uncertainty). From this we infer that it is not important to locate the steady-state J very precisely, and this will save substantial effort in the calculations. Table 111 also presents SEC results as a function of I for J = 44. Here we see that ( A E ) is noticeably smaller than the RSC result (-5.5 f 0.2 cm-I) for I = 0, but for I I 0.5, ( A E ) is insensitive to I within statistical uncertainty. Given the rather small range where ( AE) is sensitive to 1, it is not surprising that our fixed I and redistributed I (RSC) results are the same. Thus, it appears that 1 can be either fixed (provided that it is 10.5) or chosen from a microcanonical ensemble in our SEC calculations. We have chosen the former approach in what we report here. 111. Results
A . Temperature Dependence of ( E ) and ( AE)for He + CS2. We begin our analysis by considering the temperature dependence of the He + CS2 relaxation. We modeled this system with the RSC method using the ESMSV C M potential functions. Our results are based on calculations at 300, 700, 1O00, 1500, and 2000 K in which we calculated ensembles consisting of 9, 13, 15, 20, and 19 sequences, respectively. Except for 300 K, each sequence involved 1500 successive collisions, which is sufficient to relax the CS2’s from an initial vibrational energy of E = 32 640 cm-’ (the experimental initial energy) to a final average energy ( E ) of less than 10000 cm-’. The 300 K relaxation is slower and has been analyzed in more detail experimentally for ( E ) less than 10000 cm-I, so more collisions (4500) were considered. Figure 3a presents the dependence of ( E ) on the collision number n for the five temperatures studied. The 300 K results have been presented in ref 4 and are given again here. They exhibit an exponential falloff with n, as is most apparent from the semilog
+
i
YL-4-yI A
I
-0.1
2.5 Log T
+
Figure 4. log ( A E ) (cm-I) versus log T (K) for He CS2 at ( E ) = 36000 an-’, comparing calculated (circles) and measured (solid line from ref 3) results, with their respective error bars.
plot that is presented in Figure 3b. The higher temperature results in Figure 3b are not as linear as at 300 K, with noticeable positive curvature setting in for n > 500, but the dominant dependence at small n is still exponential, so we will analyze the results by fitting them to the function (E)= C
+ Ae-Bn
(3) where C is determined by the equipartition theorem and A and B are parameters. Table IV presents the parameters obtained from these fits, along with the value of ( AE) derived from eq 3 at ( E ) = 36000 cm-’. (This ( E ) was used in much of the analysis in ref 3 and is approximately the CS2 dissociation energy if spin conservation is ignored.) In Table IV we see that ( A E ) increases monotonically by a factor of 3.2 as the temperature increases from 300 to 2000 K. The comparison of ( AE) with experiment3 is indicated in Table V and Figure 4. Figure 4 gives ( M )as a function of temperature at 3 6 0 0 0 cm-I, and we see that the calculated ( A E ) ’ s are increasing more rapidly with T than the measured ( A E ) ’ s . The error bars on the experimental values are quite large, however, so the theoretical predictions lie mostly within those error bars. The results in Table V indicate that the comparisons between theory and experiment improve at lower energy. At 10000 cm-’, for example, the calculated ( M ) ’ s at 300 and 1000 K are within 25% of experiment. The results at 1000 K are actually in reasonable agreement with experiment at all ( E ) ’ s . B. Collisional Relaxation in Ne + CS2. We now consider relaxation in the Ne + CS, system, using the RSC method and the ESMSV C M potential function. Figures 5 and 6 present ( E ) versus n at 300 and 1000 K, respectively. Also included in both figures are graphs of In ( E ) versus n and of (E’)-’ versus n ( ( E ’ ) = ( E ) - (Ethemal), where (Ethemal) is the thermal energy
+
The Journal of Physical Chemistry, Vol. 92, No. 26, 1988 7227
Energy Transfer in Highly Excited Molecules
.
I
\
I
TABLE VI: Comparison of Calculated and Experimental ( A E ) ’ s (in cm-’) for Ne t CS, at 300 and 1000 K 300 K 1000 K ( E ) .cm-l calcd exDtl calcd exDtl I
.
I
36000 32640 20000 10000 9030 5000 4640 3130
-12.0 -10.9 -6.6 -3.1 -2.8 -1.4 -1.3 -0.8
f 0.6
2.01
j
I
’
-37
-21.6 f 1.7 -10.3 f 0.5
-16 -4.1
-4.7 f 0.1
-0.7
,
’
I
Figure 5. RSC ensemble averaged vibrational energy (in crn-’) versus collision number n for Ne + CS2 at 300 K (three sequences included in ensemble average): (a) ( E ) versus n, (b) In ( E ) versus n, (c) (E’)-I versus n. The statistical error bars associated with the calculation are given at intervals of 1000 collisions. ,
”
.
’
W
a
v,
.
2 1.0-
2 4 6 Collision Number/103
3.6~
’
-39.6 f 3.6
A 1.5-
q / , ‘
-16.1 -13.9 -5.7 -1.4 -1.3 -0.35 -0.41 -0.23
f 0.5 f 0.3 f 0.2 f 0.1 f 0.1 f 0.1 fO.l
(all
i L
0.5
3.0
2.5
l
Log T
+ CS2at 36000 cm-I comparing calculated (circles) and measured (solid line from ref 3) results, with their respective error bars.
Figure 7. log (AE)(cm-I) versus log T (K) for Ne
3.61 .
,
h
0 ’
(b)
10-
\i
98
2
,
U
1 - 47
c
31
i 5 10 15 20 Collision Number/102
Figure 6. Same as Figure 5, except T = 1000 K and seven sequences are included in ensemble average.
at temperature T from the equipartition theorem). The latter plots would be linear if ( A E ) were proportional to ( E o 2 , but we see curves linear. Instead, that at neither temperature are the (,!?’)-I the In ( E ) curves better approximate a straight line (with a small amount of negative curvature), indicating that ( A E ) is closest to being a linear function of ( E ) . This allows us to use eq 3 to define the analytical dependence of ( E ) on n, and from this we have determined (hE)’swhich are tabulated in Table VI. (The values for A, B, and C in eq 3 are 31 292 f 2 ern-', (3.42 f 0.16) X and 834 cm-’, respectively, at 300 K, and 33 074 f 32 cm-I, (1.13 f 0.12) X lo4, and 841 f 516 cm-l, respectively, at 1000 K.) Comparison of our results with experiment is excellent at the highest energies, but at low ( E )our (hE)’sare substantially higher than experiment. This indicates that the experimental ( A E ) is a stronger function of ( E ) (roughly quadratic). In ref 4 we found that because the redistribution procedure in the RSC method forces the molecules to continuously sample regions of phase space where relaxation is fast, RSC omits sequences where the CS2’smay become “frozen” in states of extra stability for long periods. If these sequences were included, then our (AE)’s would be smaller at low ( E ) where phase space is more
3
6 9 1’2 Collision Number/l03
Figure 8. RSC ensemble averaged vibrational energy (in cm-’) versus collision number n for Ar + CS2at 300 K (three sequences included in ensemble average): (a) ( E ) versus n, (b) In ( E ) versus n, (c) (E)-I versus n. The statistical error bars associated with the calculation are given at intervals of 3000 collisions.
regular, and this would lead to a stronger dependence of ( A E ) on ( E ) than the linear dependence seen in RSC. Thus, the direction of the deviation between theory and experiment in Table VI is understandable. Presumably, a conserved successive collision (CSC) calculation such as that used in ref 4 would lower the ( AE) value; at low ( E ) , but our results in ref 4 suggest that the agreement with experiment may not be improved, as CSC tends to give ( AE)’s that are too small. This occurs because the collisional relaxation of metastable molecules is dominated by classically forbidden transitions. Figure 7 compares the experimental and calculated temperature dependence of ( A E ) at 36 000 cm-l. Since the ( E ) being considered is high, the agreement between theory and experiment is excellent. Note that ( A E ) is a stronger function of temperature for Ne than for He (increasing by a factor of 2.3 for Ne between 300 and 1000 K, compared to a factor of 1.9 for He).
7228 The Journal of Physical Chemistry, Vol. 92, No. 26, 1988
Bruehl and Schatz 2.01
"
'
'
'
'
I
I
5 -0 .
i
2.5
3.0 Log T
7
+ CS2 at 36000 cm-I comparing calculated (circles) and measured (solid line from ref 3) results, with their respective error bars. Figure 10. log (AE)(ern-') versus log T (K) for Ar
TABLE VIII: Comparison of Intramolecular Potentials 2 3 Collision Numberl103
system
Figure 9. Same as Figure 8, except T = 1000 K and six sequences are included in ensemble average. TABLE VII: Comparison of Calculated and Experimental ( A s ) %(in cm-') for Ar CS, at 300 and 1000 K 300 K 1000 K
+
( E ) , cm-I
calcd
exptl
calcd
exptl
36000 32640 20000 10000 9030 5000 4640 3130
-5.6 f 0.9 -4.7 f 0.6 -2.1 f 0.5 -0.74 h 0.11 -0.65 f 0.11 -0.32 f 0.10 -0.30 f 0.10 -0.22 f 0.10
-9.7 -8.1 -3.1 -0.8 -0.7 -0.2 -0.2 -0.1
-23.2 f 2.7
-18.5
-12.0 f 1.4 -5.0 f 0.6
-6.6 -1.5
+
-1.5 f 0.2
-0.2
C. Collisional Relaxation in Ar CS2. Figures 8 and 9 present ( E ) , In ( E ) , and (E')-' versus n for Ar CS2 based on RSC calculations using the 6-12 CM potential function. At 300 K, Figure 8b,c indicates that ( A E ) is about equally well modeled as being linear or quadratic in ( E ) . The small difference between these two possibilities is in part due to the fact that a smaller range of ( E ) ' s is included in the plots due to the slower relaxation compared to He or Ne. (Note that 15000 collisions per sequence were calculated.) At 1000 K the relaxation is faster so our simulation extends over a large range of ( E ) ' s . Figure 9b,c indicates that here ( A E ) is best described as varying linearly in ( E ) . In order to determine ( A E ) ' s from our results, we have used eq 3 to fit Figure 9, and the corresponding hyperbolic formula E = C / ( A n + B) + D (4)
+
+
to fit Figure 8. The constants derived from these fits are A = (3.81 f 0.49) X C = 1.143 B = (3.088 f 0.010) X f 0.010 cm-', and D = -4926 f 905 cm-' for Figure 8 and A = 29957 f 110 cm-l, B = (6.98 f 0.80) X and C = 2780 cm-' for Figure 9. Table VI1 compares calculated and measured ( A E ) ' s that have been derived from these formulas at experimentally relevant ( E ) ' s . Note that Figure 8 only determines ( E ) down to 9000 cm-', so the (AE)'s at lower energy in Table VI1 are based on an extrapolation using eq 4 whose reliability is uncertain. Nevertheless, the agreement between theory and experiment for T = 300 K is quite good over the entire range of energies. By contrast, the agreement at 1000 K is only good at the highest ( E ) values. Figure 10 compares the experimental and calculated temperature dependence of (AE)at 36000 cm-'. The agreement between theory and experiment is not as good here as it was for Ne in Figure 7, although the results do agree within the experimental error bars. Note that, for the rare gases studied, the ( A E ) ' s for Ar + CS2 show the strongest dependence on temperature, increasing by a factor of 4.1 as temperature increases from 300 to
I. He
(a)% (in cm-') Using Different CM
CS2 surface M
H
-11.3 h 0.5 -4.3 f 0.4 -1.8 f 0.3
-11.7 f 1.1 -3.6 f 0.6 -3.7 f 1.0
-9.9 f 0.8 -4.6 f 0.8 -2.2 f 0.7
-1.6 f 0.1
-1.8 f 0.1
+ CS2 (300 K)
32640-9030 em-' 9030-4680 cm-l 4680-3 180 em-' 11. Ar CS2 (300 K) 32640-9030 cm-'
+
+
TABLE I X Comparison of ( A E ) ' s (in cm-') for He CS2 at 300 K Using Different Intermolecular Potentials potential E = 32640 cm-l E = 4640 ern-'
He- 1 He-2 He-3 He-4 He-5
-17.7 -16.3 -16.9 -16.9 -63.2
f 3.7
-1.5 -1.2 -1.5 -1.2 -4.0
f 3.7 f 3.6
f 4.9 f 11.O
f 0.6 f 0.9 f 0.9
f 0.6 f 1.8
1000 K (as opposed to 1.9 for H e and 2.3 for Ne). D. Dependence of ( L I E )on Intramolecular Potential. In Table VI11 we summarize the intramolecular potential surface comparison. For each of the surfaces discussed in section 1I.A (CM, M, and H), we report the average of ( A E ) over the experimental energy ranges referred to earlier. Each ensemble consists of three sequences except for the He CS2 CM ensemble where nine sequences were calculated. Table VI11 shows that is reasonably insensitive to the intramolecular potential, with most results in agreement within statistical uncertainty. Although our CSC calculations in ref 4 demonstrated that AE for each collision depends strongly on the distribution of energy among CS2 modes prior to the collision, the present results suggest that the microcanonical average of AE is not sensitive to how much anharmonicity there is for each mode. This is sensible in that the most probable microcanonical states are those that distribute energy evenly among the modes so that even for a strongly anharmonic potential CS2 is most likely located in a part of phase space that is reasonably harmonic. However, our result is also somewhat surprising in that previous trajectory s t ~ d i e sof ' ~collisional energy transfer at low internal energies show the magnitude of (AE)to be different for harmonic and anharmonic oscillators. E . Dependence of (AE)on Intermolecular Potential. Table IX presents a comparison of (AE)'s obtained from SEC calculations at E = 32 640 and 4640 cm-' for He CS2 at 300 K using the five intermolecular potentials defined in section 1I.A combined with the C M intramolecular potential. At both E's, the (AE) values are the same within statistical uncertainty for the first four potentials, but for He-5, the value is significantly higher. The agreement between the He-1 and He-2 results indicate that (AE) is not sensitive to the precise function used to fit the same potential
(z),
+
(z)
+
( 1 5 ) Schatz, G. C . ; Moser, M.
D.J . Chem. Phys. 1978, 68, 1992
J . Phys. Chem. 1988, 92,1229-1232 surface. Agreement between He-2, He-3, and He-4 shows that ( A E ) is not sensitive to changes in the well depth and suggests that, at 300 K, the attractive well is relatively unimportant in the collision dynamics. The large change in (AE)for the He-5 potential indicates that the repulsive inner wall of the intermolecular potential plays a significant role in determining ( A E ) . This is not a surprising result?16 as the He CS2collisions are expected to be impulsive, and is consistent with the lack of dependence on well depth in the He-2, He-3, and He-4 potentials. We have examined our H e + CS2 trajectories and find that none have more than one inner turning point in the He-CS2 separation coordinate, which is also consistent with an impulsive mechanism. Even Ne and Ar + CS2 show only a small fraction (
