J. Phys. Chem. 1995,99, 17107-17112
17107
Monte Carlo Study of Small Benzene Clusters. 2. Transition from Rigid to Fluxional Forms Lawrence S. Bartell" and Frederic J. Dulles Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109 Received: May 22, 1995; In Final Form: August 1, 1995@
Simulations were camed out on clusters composed of 12, 13, and 14 molecules interacting with realistic intermolecular forces. Cold, nearly rigid clusters were heated until they began to fragment. According to indices of melting, including the Lindemann 6 , heat capacity, coordination number, and self-diffusion, clusters began to undergo a melting-like transition at about 140 K. The 13-membered cluster remained in its lowenergy configuration somewhat longer than the others, consistent-with its closed-shell icosahedral structure. The transition was gradual, taking place over a 50 deg range, as indicated by criteria depending upon translational and out-of-plane tilting motions. Rotational diffusion about the 6-fold axis began at far lower temperatures. The activation energy for this molecular rotation to take place on the surface of the solid-like cluster is an order of magnitude lower than it is in the bulk crystal. Results of the present Monte Carlo computations are in qualitative agreement with spectroscopic observations of ( C ~ H ~ ) ( C ~ Dclusters ~ ) I Z undergoing a transition from a liquid-like to a solid-like structure in a cooling supersonic jet. Our results provide additional information about the temperatures of clusters during the transition and about the mechanism underlying the cooling which induces the transition. Factors responsible for the isotope effect strongly favoring the migration of protiated species to the cluster center during solidification are also discussed briefly.
Introduction As part of a program to investigate the effects of size and temperature on cluster properties, a series of Monte Carlo (MC) runs on small clusters of benzene were initiated. Results for the clusters (C&)N, including N = 2, 12, 13, and 14, have been described in the first paper in this series' at temperatures increasing from low through the onset of rapid isomerization among equivalent (lowest energy) structures. The present paper is concerned with effects of heating the clusters until they transform to higher energy structures, Le., until they undergo melting-like transitions. A large body of analogous work about sizeltemperatureeffects in atomic clusters has been and many of the results reported for atomic clusters resemble those found in the present investigation. On the other hand, there are marked differences between clusters of polyatomic molecules and of atoms, most notably in the size required before clusters are large enough to adopt spontaneously the structures identifiable with those of bulk matte^-.'^-^^ For polyatomic molecules this threshold size tends to be considerably smaller than for atoms. In the case of benzene, however, the gross structures of clusters with 12-14 molecules are not strikingly different from those of argon clusters of the same size. Nevertheless, the existence of orientational as well as translational degrees of freedom of molecules contributes added elements of interest. A recent spectroscopic study25of the 13-cluster heightens the interest. Clusters with the isotopic composition (C6H6)(C6D6)12 formed during free jet expansion were observed to undergo a transition from a liquid-like to a rigid structure. Results represent the first unequivocal experimental evidence for such a transition in a size-selected ensemble of molecular clusters. The investigators conjectured that the temperature during the transition was in the vicinity of 60 K. In view of the attention theorists are devoting to molecular details of how such transformations take p l a ~ e , ~it -seemed ~ ~ worthwhile to carry @
Abstract published in Advance ACS Abstracts, November 1, 1995.
out MC simulations of benzene clusters with a realistic potential function. Such simulations can elucidate the nature of the transformation and yield an alternative estimate of the transition temperature. Results of research in this direction are described in the following.
Procedure Computational procedures were described in part 1.' Several of the diagnostic procedures we will apply have been outlined in detail elsewhere.22.26These include the Lindemann 6 (an index of melting)6~7.'3.'4~'7.22 and an'index RD of self-diffusion.22 A few comments need to be made about these indices. In the published literature the Lindemann 6 has been defined in several different ways. We introduce the two most frequently seen because the one closest to Lindemann's original i n t e n t i ~ n ~ ~ , ~ ~ (which we refer to as 61)is perhaps a more fundamental but less distinct indicator of melting than the other (which we refer to as 62). In the work to follow the first is defined as
where ad2 is the variance
in the distance rij between centers of mass of adjacent benzene molecules, with averages taken over the course of -lo6 trial moves and over all adjacent pairs. The other index 62 is defined similarly except that distances between all molecules in the cluster, adjacent or otherwise, are included in the averages. Because we take r to represent the distance between centers of mass of molecules instead of between atomic centers, our indices deviate in spirit from the original intention of Lindemann (who considered only interatomic distances). The present indices, then, are simply empirical gauges of transition, gauges which cannot be expected to behave the same way for all molecular systems.
0022-365419512099-17107$09.0010 0 1995 American Chemical Society
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TIK Figure 1. Temperature dependence of configurational energies per molecule for 12-(0), 13-(W), and 14-(A)molecule benzene clusters. At temperatures above those of the last plotted points, fragmentation of clusters aborted the runs.
In an earlier paper2>we introduced the index of self-diffusion, RD,to help characterize diffusion in Monte Carlo runs for which no true element of time enters and for which the total diffusion depends upon the arbitrary step size. Index RI, sidestepped the arbitrariness by computing a ratio reflecting the ease with which molecules diffuse. It compares (a) the mean-square displacement that a molecule (a biased walker) experiences after many steps when the fraction of steps accepted is ArA) with (b) the mean-square displacement expected for an random walker with the same number of accepted steps. A virtue of this approach is that, even though the mean-square diffusion per step and the fraction f ( r A ) depend upon the maximum MC step size r A , the ratio turns out to be essentially independent of r A . Unfortunately, one of the defining relations for this index suffered a typographical error in ref 22, and this error was repeated in a thesis.26 Therefore, we point out here that the factor r in the integrand of eq 5 of ref 22 should be replaced by 3, as is required for the expression to be dimensionally correct. Another simple index of utility is the effective coordination number, nc,the average number of direct intermolecular contacts around a molecule in a cluster. Neighbors are considered to be in contact when they are closer than the position of the first minimum in the center-of-mass pair correlation function (taken to be 7.6 8, for benzene at all temperatures).
Figure 2. (left) Evolution of configurational energy with trial step number for a 13-molecule benzene cluster in a I O million step MC simulation at 150 K. (right) Histogram of the same energy values. Energies are of periodically stored configurations and not averages taken from short segments of the simulation 12,
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TIK Figure 3. Temperature dependence of heat capacities derived from fluctuations in configurational energies of 12-(0), 13-(W), and 14-(a)molecule benzene clusters. Note that the heat capacities include no contribution from kinetic energies or from intramolecular vibrations.
“isomeric” form to structures of higher energy. After the onset of excitation the cluster energy suffers irregular jumpsdeexcitations and reexcitations during its walk through configuration space at a constant temperature-as illustrated in Figure 2. The rapid fluctuations so prominent in Figure 2 are related to the cluster heat capacity by the relation30
Results Caloric curves for the 12-, 13-, and 14-molecule clusters plotted in Figure 1 show that smaller clusters are stabilized less than larger ones, at a given temperature. The fact that the curve for the 13-cluster is conspicuously closer to that of the 14-cluster than to that of the 12-membered example at low temperatures is, of course, a consequence of its closed-shell icosahedral configuration.’2.29 At higher temperatures, where clusters lose their rigidity and become fluxional, this special stability disappears. To save words, we shall refer to this melting-like transition simply as “melting.” Beyond 225-235 K, cluster fragmentation became too frequent for the resetting method’ of part 1 to remedy the problem, and runs were discontinued. In the vicinity of 140 K the curve for the 13-molecule cluster in Figure 1 can be seen to undergo a modest change in slope to a new level. For large clusters, such a behavior would be more marked and would signify melting (or some other phase change).24 For the 13-molecule case such a rise indicates that the cluster is beginning to be excited from its most stable
(3) The heat capacity so defined includes no contribution from intramolecular vibrational energy or kinetic energy, of course, but does reflect excitation to less stable configurations. Rapid fluctuations in the record of energy are punctuated by the aforementioned jumps. The histogram of energies at the right of figure 2 is broader for what amounts to an ensemble of clusters in transition (whose members coexist in each form) than are histograms at lower or higher temperatures when the cluster is in a single phase-like state. As pointed out elsewhere, autocorrelation functions of energies reveal much the same information as heat capacities.22.26Heat capacities of the clusters calculated from the fluctuations in potential energy are plotted in Figure 3. At low temperatures the heat capacity is very nearly 3R, the value expected for three-dimensional translational and librational oscillations of particles bound by Hookes’ law forces. At higher temperatures the heat capacities rise to pay for the
Monte Carlo Study of Small Benzene Clusters
J. Phys. Chem., Vol. 99, No. 47, 1995 17109
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TIK Figure 4. Temperature dependence of Lindemann index 61 for 12(O), 13-(.), and 14-(A)moleculeclusters. This index is based on nearest neighbor contacts only.
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Figure 7. Temperature dependence of translational diffusion ratios for 12-(0), 13-(B), and 14-(A)mOkCUk clusters.
Figure 5. Temperature dependence of Lindemann index 62 for 12(O), 13-(B), and 14-(A)molecule clusters. This index is based on all intermolecular distances in the clusters.
increasing separation between the attracting molecules as the clusters melt. Also indicating a melting of clusters as they are heated are the Lindemann indices 61 and 62, as shown in Figures 4 and 5. Index 61, based only on pairs of adjacent molecules, rises through the threshold of approximately 0.08 previously suggested to signify melting.22.26The second index, based on the variance of all of the intemuclear distances, exhibits a much more conspicuous rise as the clusters undergo a change in their states. In addition to the true vibrations between contacting atoms (molecules) proposed by Lindemann to destroy crystalline integrity when they become sufficiently violent,27 the second index is augmented by elements of molecular diffusion. The temperature dependence of the coordination number, nc, also shows clear signs of transition between structural states of clusters. Illustrated in Figure 6 is the rapid drop in n, after the onset of melting. What uniquely distinguishes liquids from solids is their ability to flow, a property intimately associated with self-diffusion of the constituent molecules. In Monte Carlo runs the ratios RD afford reasonable measures of self-diffusion. Plots of diffusion ratios in Figures 7-9 show that the translational and the inplane (#) and out-of-plane (e) rotational motions all undergo major increases in self-diffusion but at different temperatures. An alternative, graphic representation of the self-diffusion of in-plane rotations can be found in the evolution of histograms of the # rotations. An example is shown in Figure 10.
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Discussion Evidence for Transition. Of the various indicators of melting in clusters, the caloric curves, heat capacities, Lindemann indices, and coordination numbers are overall gauges, whereas the self-diffusion can be resolved into individual types of molecular motion. This greater discrimination is of potential value in finding whether clusters behave like certain materials in the bulk which “melt rotationally” before they become molten in the ordinary ~ e n s e . ~ ’ , ~ * The four general indicators fail to suggest any strong, abrupt transition from solid-like to liquid-like forms, although the nontraditional Lindemann index 62, which is augmented by aspects of diffusion, gives a more noticeable signal than the
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1000 T" I K ' Figure 11. Arrhenius plot of diffusion frequency k, per step derived from a series of histograms (cf. Figure IO) acquired in MC runs on 13-molecule clusters. The barrier to rotation implied is 1.7 kJimol.
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@degrees Figure 10. Histogram of in-plane (4) rotational displacements after 250 000 trial steps, from an 80 K MC simulation of a 13-molecule cluster.
others. When the change on heating is envisaged as an isomerization between a low-energy conformer and a limited set of less stable conformers, it is clear that no abrupt transition would be expected for an equilibrium ensemble. All of the general indicators, however, are consistent with a change spread over a wide temperature range beginning at about 140 K, a temperature considerably higher than the 85 K temperature at which transitions occur rapidly between equivalent low-energy isomers.' During heating, all of the indicators including the diffusion ratios show that the 13-molecule cluster resists softening for somewhat longer than the 12- and 14-homologues, but the heat capacities and Lindemann curves for the 12-, 13-, and 14-molecule clusters are by and large rather similar, overall. On the other hand, the caloric curves are appreciably different, and only that of the most rigid cluster, the completed icosahedron, has an appearance at all suggestive of a normal (but weak and diffuse) transition. The 12-molecule cluster with a mobile vacancy in its surface layer gives a less distinct, noisier indication, whereas the 14-molecule entity gives an anomalous, as yet uninterpreted appearance that may signify nothing more than statistical chance in slowly developing oscillations between configurations. With respect to the physical meaning of the temperatures mentioned above in classical simulations for such small clusters, several points arise. First, even if there is no meaning to the temperature of an individual 13-molecule particle, results are meaningful when applied to a large ensemble. The large fluctuations in Figure 2 give a graphic illustration of the lack of a unique energy of a single 13-molecule particle at a given temperature (similar to a lack of a unique temperature at a given
energy in a microcanonical ensemble). Of greater concern are the effects of zero-point energy that are absent in classical simulations. In crystalline benzene, the zero-point energy corresponds to the thermal energy of a classically vibrating crystal at -70 K but the classical and quantum energies converge well before 140 K.33 Therefore, the 85 K isomerization temperature is too high, but the 140 K melting temperature is, presumably, realistic. Self-diffusion ratios tell similar stories for the three different clusters but indicate different behaviors among the different types of molecular motion. These ratios would all be unity if motions were free walks unbiased by molecular collisions. Experience has shown that the translational ratio RD,,is very low for true crystals but still far from unity for liquids. Liquids give values of perhaps -1/6 because molecular "walks" are severely constrained, even in liquids, by the cages formed by neighboring molecules.22 Rotational motions, on the other hand, are much less constrained in liquids. With this information as a guide we infer that the development of melting is similar for translation and 8 rotations and takes place over the range of temperatures suggested by the other indicators. The behavior of rotation about the 6-fold axis is entirely different. Its onset of diffusion occurs at a considerably lower temperature. Although 4 executes librations of markedly lower amplitude at low temperatures than does 8 (attributed to a gear-tooth effect in part l), it diffuses much more rapidly (as intuition would lead one to expect for a flattened disk). Activation Energy for In-Plane Rotations. It is of some interest to compare the activation energy for the 4 rotational diffusion in the solid-like cluster with that in the bulk crystal. This can be done from the evolution of histograms (cf. Figure 10) by observing the rate at which populations of the various potential wells develop. If it is assumed that the number of diffusion steps is comparable with diffusion time as long as step size is adjusted to maintain a constant acceptance rate (our criterion for adjusting ?A), a temperature dependent rate constant can be derived, as shown in detail in ref 26. An Arrhenius plot (Figure 11) yields an activation energy of 1.7 kUmol. A simpler way is to treat the diffusion ratio R D .(Figure ~ 8) as proportional to a rate constant when it is small but to represent the rate constant by the function tan(n R ~ , d 2when ) R D .becomes ~ larger. This simple substitution naturally maps the rate constant onto a scale from zero to infinity instead of the original RD., scale from zero to unity. An Arrhenius plot of this function for the
Monte Carlo Study of Small Benzene Clusters solid-like state nearly reproduces the slope of Figure 11, giving an activation energy of 1.8 kI/mol. By contrast, values of the activation energy derived from NMR and neutron diffraction studies of crystalline benzene range from 15.5 to 19 Clearly, the barrier to rotation in the crystal, where all molecules are intimately surroundedby neighbors, is an order of magnitude higher than it is for the surface molecules in the 13-cluster. An attempt to determine the activation energy for the central molecule in the 13-cluster failed because too little rotation took place in the solid-like region to yield acceptable statistics. Qualitatively, the central molecule behaves more like molecules in the bulk. In fact, in MD treatments of molecular motions in larger clusters of SeF640and teff-butyl chloride:’ we have found that molecules in the cluster cores behave very nearly like those in the bulk at the same temperature. Relation to Spectroscopic Investigation. In a study of 13molecule clusters produced in free jet expansions (each cluster with 12 C6D6’s plus a single dopant C6H6 molecule), Easter et ~ 1 observed . ~ ~two noteworthy phenomena. First, shortly after their formation by condensation of vapor seeded into helium, the clusters were liquid-like. Their spectra displayed an inhomogeneously broadened peak attributable to the protiated molecule randomly distributed among all possible sites in the cluster. During the 25 ps of flight over which spectra were acquired, the broad peak began to be replaced by the sharper features expected for a more rigid structure. Secondly, the c&6 “dopant” molecule moved almost unerringly to the center of the cluster during freezing. Our Monte Carlo results offer a framework for interpreting the spectroscopic observations, including the probable range of temperature over which the transformation occurred, but have nothing to say about the site selectivity of the protiated species. The present simulations of clusters containing 12-14 benzene molecules leave no doubt about the existence of both rigid “solid” and nonrigid “liquid” forms (to use the notation of Easter et al.) of sufficient stability for spectroscopic observation. At temperatures above about 140 K, the liquid-like form exists in our simulations, characterized by a rapid self-diffusion of its molecules. At lower temperatures the clusters exist in each of several rather rigid, equivalent structural isomers. Below about 85 K (classical system, or lower in a quantum system), these isomers no longer interconvert at appreciable rates.’ An important point needs to be made. For our small 13-molecule clusters, heating and cooling curves (not illustrated) are virtually superposable upon each other. This is in sharp contrast to the behavior in our molecular dynamics simulations for larger clusters (1OO-5OO molecules) undergoing structural change^.^^.^ Unlike the case of 13-clusters, the transitions observed in the larger clusters are unmistakably identifiable with recognizable phase changes in the bulk. This being the case, nucleation is required to make clusters change phase when they are being cooled, whereas nucleation over and above that naturally furnished by disordered surface molecules makes specific nucleation unnecessary for phase changes in heating stages. Therefore, substantial supercooling is encountered before large clusters change phase while being cooled, and cooling curves, depending upon the element of chance, are not reproducible. Heating curves give reproducible,more or less equilibrium plots. From this information we draw the conclusion that small benzene clusters do not have to undergo a major nucleation step to go from nonrigid to rigid structures, and they therefore transform without needing to be particularly supercooled. If our modeling of temperatures is correct, then the clusters solidify by the time they reach 140 K, and the time available in the supersonic jet is more than ample.
J. Phys. Chem., Vol. 99, No. 47, 1995 17111 Implications of the present study are in excellent qualitative accord with the interpretations of the authors of ref 25 but differ in some quantitative details. The authors conjectured that the solidification took place in the vicinity of 60 K, a temperature too low for evaporation to occur on the time scale of the experiment. This estimate seems to have been based partly on the fact that the experimental expansion conditions were similar to those in an earlier study of argon clusters forming around a benzene molecule at about 40 K.42 In our own treatment of cluster formation, the cluster temperature is generally substantially warmer than the temperature of the surrounding carrier gas and tends to be proportional to the heat of vaporization of the condensing material, once clusters are well past the site of their c o n d e n s a t i ~ n ~(as ~ -also ~ ~ expected from the GspannKlots rule of t h ~ m b ~ ~ , We ~ ’ )have . found that the temperature of large benzene clusters is approximately 3 times higher than the temperature estimated by Easter et al. for their very small clusters. Our own rough modeling of the evaporatioxdcondensation dynamics for 12- to 14-clusters makes them considerably warmer than 60 K and not seriously out of line with the MC results alluded to above. Therefore, in the spectroscopic study, liquid-like clusters are probably warm enough for a molecule or two to evaporate during the 25-50 ps of flight they experience before they receive an ionizing laser pulse. This evaporation would provide an alternative and effective way for clusters to cool and solidify without requiring much assistance from the eight collisions with helium atoms suggested by Easter et al. to effect the transformation. Evaporation of a single benzene molecule would remove roughly 20 times as much energy as eight collisions with helium if clusters were at 140 K and even more if clusters were cooler.43 Why the protiated benzene molecules have such a decided preference to migrate to the cluster centers remains unanswered, but new MC or MD simulations might be enlightening. Unfortunately, our computations had to be terminated before the results of Easter et al. were published. Referring to the site selectivity, these authors stated that “the energetics responsible for this phenomenon remain a mystery”.25 It is known that the principal isotope effects on properties of liquid benzene arise from the intramolecular zero-point vibrations of the C-W C-D bonds.48 These have the effect of making C6H6 larger than its perdeutero homologue (i.e., the molar volume of C6H6 is larger) and enhance its intermolecular forces (its surface tension is larger). Simply changing the mass of rigid molecules in simulations would miss the major isotope effect. The effective size and intermolecular force (including that due to the quadrupole moment49)must also be adjusted if simple MC/ MD simulations are to have any hope of reproducing experimental findings. The somewhat greater dispersion and electrostatic forces of the protiated species would tend to drive it to the cluster center, but a larger size would oppose such a preference in an icosahedral packing of soft spheres. A very favorable specific interaction for the aspherical benzene molecules would have to be involved. The foregoing results provide information about small clusters of benzene that differ significantly from those found by potential energy minimizations with simpler intermolecular interactions. They also provide results of utility in interpreting experiments. Structures and phase-like transitions of such small clusters do not closely resemble those of bulk matter. It is of interest, then, to find how the properties evolve toward those of the bulk as clusters increase in size. The next paper in this series will address this problem.
Acknowledgment. This research was supported by a grant from the National Science Foundation. We thank the Computer
17112 J. Phys. Chem., Vol. 99, No. 47, 1995 Center of the University of Michigan for a generous allocation of computing time and gratefully acknowledge the awarding of a Regents’ Fellowship to F.J.D. We are particularly indebted to Professor W. Jorgensen for his help in initiating this study.
References and Notes (1) Dulles, F. J. ; Bartell, L. S. J. Phys. Chem. 1995, 99, 17100, part 1 in this series. (2) McGinty. D. J. J. Chem. Phys. 1973, 58, 4733.
(3) Lee, J. K.: Barker, J. A,; Abraham. F. F. J. Chem. Phgs. 1973, 58, 3166. (4) Kristensen, W. D.; Jensen, E. J.; Cotterill, R. M. J. J. Chem. Phys. 1974, 60, 4161. ( 5 ) Briant, C. L.: Burton, J. J. J . Chem. Phps. 1975, 63, 2045. (6) Kaelberer, J. B.; Etters, R. D. J. Chern. Phys. 1977, 66, 3233. (7) Etters, R. D.: Kaelberer, J. B. Phgs. Rev. A 1975, 11, 1068. (8) Etters, R. D.: Kaelberer, J. B. J. Chem. Phys. 1977, 66, 5112. (9) Nauchetel. V. V.: Pertsin, A. J. Mol. Phys. 1980, 40, 1341. (10) Quirke. N.; Sheng, P. Chem. Phys. Left. 1984, 110, 63. (11) Jellinek. J.: Beck, T. L.; Berry. R. S. J. Chem. Phys. 1986, 83. 2783. (12) Honeycutt. J. D.; Andersen, H. C. J. Phys. Chem. 1987, 91, 4950. (13) Davis. H. L.; Jellinek, J.: Berry. R. S. J. Chem. Phgs. 1987, 86, 6456. (14) Beck. 7. L.: Jellinek. J.: Berry, R. S. J. Chem. Phgs. 1987, 87. 545. (15) Quirke, N.Mol. Simul. 1988, I , 249. (16) Berry, R. S.; Beck, T. L.; Davis, H. L.: Jellinek, J. In Evolution of Size Effects in Chemical Dynamics, Part 2: Prigogine, I.. Rice, S . A,. Eds. (Adv. Chem. Phys. 1988, 70, part 2, 75). (17) Stillinger. F. H.; Stillinger, D. K. J. Chem. Phys. 1990, 93, 6013. (18) Cheng. H.; Berry, R. S. Phys. Rev. B 1992, 45, 7969. (19) Banell, L. S.: Xu, S. J. Phys. Chem. 1991, 95, 8939. (20) Boyer, L. L.; Pawley, G. S, J. Compur. Phys. 1988, 78, 405. (21) Torchet, G.: de Feraudy, M.-F.; Raoult, B.; Farges. J.: Fuchs. A. H.: Pawley, G. S. J. Chem. Phys. 1990, 92, 6768. (22) Bartell, L. S.: Dulles, F. J.: Chuko, B. J. Phys. Chem. 1991, 95, 6481. (23) Rousseau, B.; Boutin, A.: Fuchs, A. H.: Craven. C. J. Mol. Phys. 1992, 76, 1079.
Bartell and Dulles (24) Xu, S.: Bartell, L. S. J. Phys. Chem., 1993, 97, 13550. (25) Easter. D. C.: Baronavski, A. P.; Hawley, M. J. Chem. Phys. 1993, 99, 4942. (26) Dulles. F. J. Ph.D. Thesis, University of Michigan, Ann Arbor, MI. 1993. (27) Lindemann, F. A. Phys. 2. 1910, 11, 609. (28) Lindemann, F. A. Engineering 1912, 94, 515. (29) Beck. T. L.; Machioro, T. L. J. Chem. Phys. 1990, 93, 2347. (30) Steele. W. A. Adv. Chem. Phys. 1976, 34, 1. (31) Boden, N. In The Plasrically Ctytalline State: Sherwood. J . N., Ed.: Wiley: New York, 1979. (32) Xu. S.: Bartell, L. S. 2. Phys. D 1994, 31, 117. (33) Shi, X.: Bartell, L. S. J. Phgs. Chem. 1988, 92, 5667. (34) Andrew. E. R.; Eades, R. G. Proc. R. Soc. A 1953,218, 537. (35) Wendt, J.; Noack. F. Z. Naturjiorsch. 1974, 29a, 1660. (36) Noack. F.: Weithase. M.: yon Schutz, J. Z. Nururjiorsh. 1975, 30u, 1707. 137) Boden. N.; Clark, L. D.: Hanlon, S. M.; Mortimer, M. Faraday Sgmp. Chem. Soc. 1978, 109. 109. (38) Fujara, F.: Petry, W.; Schnauss. W.; Sillescu, H. J. Chem. P h y . 1988, 89, 1801. (39) Ok, J. H.; Vold, R. R.: Vold, R. L.; Etter. M. C. J. Phys. Chem. 1989, 93. 7618. (40) Bartell, L. S.: Xu, S. J. Phys. Chem., in press. (41) Chen, J.; Bartell, L. S. J. Phys. Chem. 1995, 99, 3918. (42) Hahn, M. Y.; Whetten, R. L. Phps. Rev. Lerr. 1988, 61. 1190. (43) Bartell, L. S. J. Phys. Chem. 1990, 94, 5102. (44) Bartell. L. S.: Machonkin. R. A. J. Phps. Chem. 1990, 94, 6468. (45) Bartell, L. S. Unpublished research extending the formalism of ref 43 to flow beyond the nozzle. (46) Gspann, J. In Physic:, of Elecrronic and Atomic Collisions: Datz, S.. Ed.; North-Holland: New York, 1982. 147) Klots, C. E. Phys. Rev. A 1989, 39, 339. (48) Bartell. L. S.: Roskos, R. R. J. Chem. Phys. 1966, 44, 457. (49) It is not widely recognized that the quadrupole moment of C& is comparable with that of CbF6 (Battaglia, M. R.; Buckingham, A. D.; Williams. J. H. Chem. Phgs. Left. 1981, 78.421). Accordingly, the shift of charge that makes the el, mode infrared active will contribute an isotope effect to the quadrupole moment as C-D bonds are replaced by the longer C-H bonds.
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