17100
J. Phys. Chem. 1995, 99, 17100-17106
Monte Carlo Study of Small Benzene Clusters. 1. Structure and Internal Motions Frederic J. Dulles and Lawrence S. Bartell” Department of Chemistry, Universiv of Michigan, Ann Arbor, Michigan 48109 Received: May 22, 1995; In Final Form: August 1, 1995@
To learn how the structures of clusters evolve as their sizes increase and how the results depend upon temperature, a program of Monte Carlo computations was begun. The program was initiated to elucidate the mechanisms by which one is able to control the phase of large clusters generated by the condensation of vapor in supersonic flow. In the present paper it was found for very small aggregates that 13-molecule, icosahedral clusters enjoy the special stability also observed for argon but not for TcF6 clusters. Even though the total stabilization per benzene molecule is greater for 13 molecules than for the average of 12 and 14, the stabilization per intermolecular contact is lower than in both the 12- and 14-molecule clusters. The lowtemperature structures agreed in gross architecture with the C3 structures reported by Williams and by van de Waal insofar as their icosahedral motifs are concerned. In detail, however, the molecules in our clusters assumed quite different orientations, leading to a lower overall symmetry. This difference is a consequence of the different intermolecular interactions adopted for the computations, our potential function having been developed specifically to account for benzene dimers. Published resonant two-photon ionization (R2PI) spectra of the 13-molecule cluster demand a lower symmetry than C3, evidence that supports the use of the present potential function for clusters, as well. Amplitudes of molecular oscillations were determined and compared with experimental values for benzene crystals. As the temperature increased, amplitudes increased regularly until, at 85 K, our 13-molecule cluster began to execute rapid transitions between three equivalent isomers, making the superposition of the isomers resemble the more symmetric static structure of Williams. Effects of continued heating until a melting-like transition occurred are described in the next paper of this series.
Introduction The study of clusters has become increasingly popular in the last two decades as chemists, physicists, and materials scientists find applications in such diverse areas as catalysis, homogeneous nucleation, the structure of condensed matter, and fabrication of nanodevices.’ Cluster research in this laboratory initially focused upon the structure of large liquid clusters generated by the condensation of vapor in supersonic flow.* Although this approach was successful in obtaining liquid diffraction pattems of unusually high quality, it was soon superseded because it led to the discovery of a phenomenon that was even more interesting. We observed that clusters of some systems could be generated reproducibly in either of two or, in some cases, as many as three different molecular packing arrangements, depending upon the conditions of flow at the entrance of the supersonic n ~ z z l e .It~ became compelling to understand how variations in the way vapor molecules streamed through a micronozzle could control the way the molecules organized downstream when they aggregated. According to nucleation theory, the critical nuclei formed at the extremely high supersaturations attainable in the supersonic jet consist of only a few molecule^.^ Such nuclei are so small and usually so warm from the heat of condensation that they are amorphous. As they grow by continued condensation and cool by thermal accommodation with carrier gas, however, they ultimately become large and cool enough to acquire one or another of the definite structures possible that can be identified with a bulk phase. It seemed worthwhile to investigate the properties of clusters, beginning with the very small and extending to larger aggregates to shed light on the problem. Many studies of atomic clusters of various sizes have already been p u b l i ~ h e d , ~but - ~ none answer the question posed above nor could atomic clusters have provided @
Abstract published in Aduance ACS Absrracrs, November 1, 1995.
the answer, as later studies of molecular clusters have shown. We have found that van der Waals clusters of polyatomic molecules are profoundly different from those of atoms in considerations of size/structure. Atomic clusters bound by van der Waals forces must contain thousands of atoms before they spontaneously adopt bulklike structures? whereas clusters of polyatomic molecules may need only a few Several computational studies of small molecular clusters have appeared, including those of the present subject, benzene.I3-l5 The calculations for benzene minimized potential energies of packing in vibrationless systems and were based on potential functions found to be nonoptimum for the liquid structure. Therefore, it seemed worthwhile to reinvestigate cluster structures with an updated potential functionI6 and, in addition, to study effects of thermal excitation. Our first candidates in this program of study included benzene, as stated above, and TeF6.I’ Benzene’s molecules are aspherical but nevertheless pack in bulk crystals in much the same way that argon atoms pack. Tellurium hexafluoride molecules, on the other hand, pack into several different structures in the bulk, quite unlike those of argon and benzene, despite the fact that the molecules are quasispherical. Results for our Monte Carlo (MC) simulations of 12-, 13-, and 14-molecule clusters of TeF6 and molecular dynamics (MD) simulations of larger clusters have been described elsewhere.’0,’8 Results of our MC simulations of 12-, 13-, and 14-moleculer clusters of benzene are presented in the following, with emphasis on structural aspects and molecular motions at temperatures low enough to avoid a melting-like transition. In the second paper in the series, we investigate the effects of higher temperatures.I9
Procedure Monte Carlo Computations. The program adapted for calculating properties of benzene clusters has been described
0022-3654/95/2099-17100$09.00/0 0 1995 American Chemical Society
J. Phys. Chem., Vol. 99, No. 47, 1995 17101
Monte Carlo Study of Small Benzene Clusters in detail by Jorgensen and co-workers:o*21 who applied it extensively in studies of water, organic compounds, and solutions. Certain controls and diagnostic routines have been added to the program, but the essential logic remains unchanged. Approximately spherical clusters of 12, 13, and 14 rigid benzene molecules were constructed, beginning with a molecule at the center, by carving them out of an array of molecules with the orthorhombic structure observed in bulk crystals. The initial configuration of the 13-molecule cluster, then, was approximately that of a cuboctahedron and the 12- and 14-molecule clusters were those of a cuboctahedron with one surface molecule removed or with an extra molecule resting on the surface. Although such a procedure introduces a structural bias, tests with alternative starting structures, including some from very warm clusters with rapidly diffusing molecules, confirmed that the structures attained during the course of a run converged quickly to the most stable structure. Runs began at 25 K, a temperature so low that quantum zero-point vibrational amplitudes exceed the‘ amplitudes calculated classically in the MC computations, and proceeded in 5 deg steps to temperatures at which fragmentationbecame a serious problem. Runs at a given temperature consisted of 1 OOO OOO trial moves except for a few examples of longer runs. To extend runs with intact clusters to as high a temperature as feasible, a routine was added that stopped a run when a molecule evaporated and restarted it at the last confirmed intact configuration with a new random number seed while an error message was recorded. Simulations were halted when the number of such restarts exceeded 250. Clusters were considered to be intact if all molecules had at least one neighbor within a center-of-mass distance of 8 A. Sizes of translational and rotational moves were controlled with the parameters ?-A and aA,corresponding to maximum moves along, or about, the three Cartesian axes. Each trial step consisted of translational moves along each of the axes but around only one randomly chosen axis. The ratio of the parameters ?-A and aA was fixed at the value of 0.44rad/& and the parameters were increased with temperature to maintain an acceptance rate of -35% in runs from which most of the results presented here were extracted. Additional simulations were also run at 3 and 0.1 K to provide comparisons with published structures based solely on potential energy minimization. In analyses of rotational motions of molecules it was of interest to separate rotations (At$) about the molecular 6-fold axis from rotations (A0) perpendicular to it. Our procedure to resolve the two angular components when given two arbitrary positions and orientations was first to superpose the initial and final centers of mass and then to tilt the molecule in its final orientation by A0 about its axis of intersection with the initial configuration (axis b in Figure 1) to make the molecular planes coincident. Angle At$ could then be recognized as the amount by which a reference atom had been displaced about the 6-fold axis. Included among analyses of the state of the clusters was a measure of the motions of the individual molecules. Because of the stochastic translational and rotational drift of clusters inherent in Monte Carlo computations, a procedure was developed to null out the drift to prevent it from confusing analyses of self-diffusion and molecular oscillations. Details of our computations to accomplish these ends are described exhaustively e l ~ e w h e r e . ~ ~ ? ~ ~ Potential Energy Function. An intermolecular potential function had been developed as part of an experimental investigation of the structure of liquid benzene. Diffraction patterns of large liquid clusters were poorly reproduced by the six-site models adopted by several statistical theorists.24 They
W
Figure 1. Schematic diagram illustrating the decomposition of an orientational displacement into an out-of-plane rotation (A@ and inplane rotation (A#) after center-of-mass translation has been removed. Out-of-plane rotations are defined by the angle between the vectors nu and nb, perpendicular to the initial and final molecular planes. In-plane rotations are defined as the angle between ela and C l b after superposing nu and nb.
were accounted for better, but still imperfectly, by the 12-site model found to work quite well for the observed crystal structure of b e n ~ e n e . ~The ~.~ interaction ~ potential function derived from quantum computations by Karlstrom et ~1.2’ also did not give a proper representation of the observed data. Nevertheless, a function based on Karlstrom’s computations but rescaled in size and energy and simplified by Shi16 for rapid computation was quite successful in reproducing the liquid diffraction pattern. This function was adopted for the present MC simulations with one minor modification. Shi’s cutoff of a quasielectrostatic term was found to be somewhat too abrupt. The cutoff limit was increased to 12 A. Potential parameters are listed in the Appendix.
Results Low-Temperature Structures. As had also been reported in prior studies based on potential energy minimi~ation’~.’~ and as first pointed out by van de Waal,the most stable 13-molecule cluster adopted an icosahedral configurationand the 12-molecule and 14-molecule clusters differed, respectively, by the removal of a molecule from the outer shell and by the placing of an extra molecule on a triangular face of the icosahedron. A view of the underlying organization of our clusters is given in Figure 2, where each molecular center of mass accumulated over 200 OOO moves is represented by 200 points. Nevertheless, our results differed fundamentally from those reported earlier in the orientations assumed by the molecules, as shown for the 13molecule cluster in Figure 3 and documented in detail elsewhere.23 As will be discussed in the next section, our clusters were lower in symmetry than the Williams clusters as a consequence of our different molecular interaction function. Coordinates corresponding to the Shi potential function are tabulated in the Appendix. Isomerization of Warmer Clusters. As the clusters were heated, trajectories of the atoms showed increasing agitation, but molecular orientations were preserved until the temperature reached -85 K. At this temperature, the configuration of the 13-moleculecluster, as judged from the plots of the accumulated positions averaged over the run at that temperature, began to resemble the more symmetric structures reported earlier for the vibrationless computations. On the other hand, no evidence of a “phase change” was evident in the caloric curve or in other properties. What was causing the apparent increase in symmetry was an isomerization, or interchange of the orientations of the surface molecules, without an appreciable self-diffusion among
Dulles and Bartell
17102 J. Phys. Chem., Vol. 99, No. 47, 1995
Figure 4. Superposition of three images of our low-temperature 13molecule cluster, each image differing from the others by a 120" rotation. This superposition of orientational isomers illustrates why an averaged image of our warm cluster appeared to resemble that found by Williams (Figure 3).
2 1 I .5
Figure 2. Position-density plot of the molecular centers of mass of a 13-molecule cluster at 25 K (top), a 12-molecule cluster at 75 K (bottom left), and a 14-molecule cluster at 75 K (bottom right). The points represent 200 configurations, each separated by 1000 moves. The lines in the 13-molecule image are only to aid the eye in recognizing the somewhat distorted icosahedral structure.
e
1
' 1
b ",1200
5
U
-=
no0 -
AQ)
9
400
-
0
Figure 3. Two views of the present MC structure of the 13-molecule benzene cluster at 3 K (top) and the corresponding static structure of Williams (bottom). In the left-hand panels the central molecule lies in the plane of the page. The views at the right are rotated 90" with respect to those at the left. In the top two panels members of the equatorial, axial. and diagonal orientational groups are labeled e, a, and d, respectively (see ref 23 for details).
Figure 6. Temperature dependence of mean-square amplitudes of 6 libration for 12-(0), 13-(.), and 14-(A)molecule clusters. Above -85 K transitions occur between orientational isomers and the separation of libration from diffusion increases becomes uncertain.
them. Superposed images of the three isomers of the same handedness gave a composite image resembling that of a cluster of higher symmetry. Such an image is synthesized in Figure 4 by superposing three static plots of low-temperature structures, each displaced by 120". Ultimately, at higher temperatures, all of the clusters underwent a melting-like transition. l 9 Molecular Motions in Clusters. Once the stochastic translational and rotational drifting of the clusters had been nulled, molecular motions were resolved into translational and orientational components, and the mean-square displacements were further divided into vibrationalhibrational (of molecules oscillating in the confines of the cages formed by their
neighbors) and diffusional (of molecules penetrating the cage walls). Plots of the temperature dependence of the vibrationall librational mean-square amplitudes are displayed as functions of the temperature in Figures 5-7. The quantities plotted in the figures are mean-square displacements from initial positions, not from mean positions and, hence, are larger than the latter by a factor of approximately 2 . Decomposition into vibrational and diffusional components is readily carried out for large atomic clusters, whose curves of mean-square displacements begin with a rapid upsweep as the atoms rattle about in their cages, followed by a slow, linear rise signifying diffusion. Mean-square vibrations are determined from the intercept of
0
40
no
120
I60
TIK
Monte Carlo Study of Small Benzene Clusters
Ng
J. Phys. Chem., Vol. 99, No. 47,1995 17103
1200 -
t! 8 U
3” P
.*:.&.
800-
A-
400-
OL 0
40
80
120
160
TIK Figure 7. Temperature dependence of mean-square amplitudes of 4 libration for 12-(0), 13-(W), and 14-(A)molecuie clusters. Above -85 K transitions occur between orientational isomers and the separation of libration from diffusion increases becomes uncertain.
the linear diffusion curve, and diffusion, from the slope. For molecular clusters the division is complicated by the fact that there are only 2rc radians in a complete rotation, meaning that the diffusional curve can saturate fairly quickly, introducing a decided nonlinearity. An additional complication, more troublesome for the @ than for the 8 coordinate, was that the linearity became more and more difficult to recognize as clusters became warmer. It appeared as if the cages constraining the molecular orientations were themselves diffusing. In fact, that was recognized to be essentially the case once the isomerization (see above) was identified. Therefore, the plots for 8 and @ can be seen to become noisy after the onset of isomerization, at about 85 K, due to increasing uncertainty in the decomposition of the data. The orientational blurring accompanying isomerization is graphically displayed in Figure 4. Figures 5-7 give amplitudes averaged over all of the molecules. Of course, molecules at different sites in the cluster enjoy different degrees of freedom to move. This difference is conspicuous in the representation of molecular oscillations in Figure 8.
Discussion Relative Stabilities of Clusters. Benzene clusters exhibited the same special stability at the “magic number” 13 as has been observed for rare gas cluster^.^-^ This stability is associated with the completion of the first icosahedral shell around the central molecule as manifested in the comparisons of packing energies in Table 1. It is of some interest to note that, while the stabilization per molecule is greater for 13 than for the average of 12 and 14 molecules per cluster, the stabilization per molecular contact is smaller. This decreased efficiency per contact can be understood as due to the fewer constraints on molecular orientations in the clusters with incomplete shells. In the case of clusters of argon (represented by Lennard-Jones spheres) the difference in efficiencies per contact is considerably less. Although dimers and bulk benzene both possess substantially greater interaction energies per contact than do the intermediate clusters, the reasons are different. Two molecules by themselves can choose the optimum mutual orientation and distance without the imposition of any requirements to fit closely to other adjacent molecules. Bulk benzene, on the other hand, while not quite as efficiently space filling as a small icosahedral cluster, can profit from the fact that molecular forces are longer ranged than one molecular diameter. Bulk benzene’s enhanced stability comes from the summation of interactions more distant than any existing in the small clusters.
Figure 8. Position density plot illustrating the greatly differing amplitudes of motion exhibited by the different individual molecules in a 13-molecule cluster at 35 K. Shown are the hydrogen atoms of three selected molecules. The top and bottom molecules are members of the equatorial group, e, and the axial group, a respectively, identified in Figure 3, and the middle molecule is the central molecule of the cluster.
TABLE 1: Configurational Energies of Cold Benzene, Clusters and Bulk” system
cluster energy
(Qb
contactsc
2 12 13 14 bulke
-10.582 -284.209 -324.915 -352.903
-5.291 -23.684 -24.917 -25.207 -52.3
1 36 42 45
(Q/(n,/2) 0.50 3.00 3.23 3.21 6
-10.582 -7.895 -7.712 -7.842 -8.72
Energies in kT mol-’, clusters at 3 K. bAverage energy per molecule. Number of unique intermolecular contacts in the clusters. Number of unique contacts per molecule. e Reference 25.
Amplitudes of Molecular Motions. The general increase in amplitudes as temperature increases (Figures 5-7) is, of course, expected. If oscillations were governed by harmonic restoring forces, the mean-square amplitudes would increase linearly with T. The accelerating increase evident in Figures 5-7 is an anharmonic, loosening effect associated with the increased separations of molecules as the clusters become warmer. What is somewhat counterintuitive is the striking difference between the amplitudes for 8 and @. At lower temperatures the torsional oscillations, A@, are hindered much more than the tilting oscillations, A8, probably because of the meshing C-H gear teeth. Only when the isomerization begins does the torsional oscillation become large, and that may be an artifact of the imperfect separation of libration from diffusion. This onset of isomerization occurs in the present classical simulations at about 85 K. To put the temperature into context, it should be recalled that the zero-point energy of bulk benzene crystals corresponds to the thermal energy that classical benzene would have at approximately 70 K.I6 Therefore, the quantum cluster would begin to undergo isomerization at a temperature lower than 85 K. Softer modes, however, such as those along the reaction coordinate of the isomerization, become “classical”
Dulles and Bartell
17104 J. Phys. Chem., Vol. 99, No. 47, I995 TABLE 2: Vibrational and Librational Mean-Square Amplitudes of Benzene Molecules about Their Mean Positions at 25 K: Comparison of 13- and 128-Molecule Clusters with Bulk Crystals" system
13 13 central 128 C6D6. expt
(A$) 0.022 0.010 0.028 0.01 16
@e2) 36 25 23 4.27
I
(he2) 15 1.8 10 2.27
Distances in angstroms, angles in degrees. Experimental data from neutron diffraction study of crystalline perdeuterobenzene of ref 28, extrapolated in such a way as to eliminate zero-point effects as explained in ref 23. The data in the second row refer to the central molecule of the 13-molecule cluster alone.
at lower temperatures than the coordinates overall, so the quantum correction for isomerization may be modest. The classical treatment becomes a good approximation for an entire benzene crystal at somewhat over 100 K. It is worthwhile to compare mean-square amplitudes of the different types of oscillations with those from an experimental neutron diffraction study of bulk benzene28 at a temperature well below the cluster's isomerization temperature. This is done in Table 2. The flapping of A0 is much looser for all cluster molecules than it is in the bulk, and amplitudes of all motions are larger for the surface molecules. The translational and torsional oscillations of the central molecule in the small cluster are virtually the same as in the bulk, however. Why the amplitudes for the larger 128-molecule cluster look seemingly out of line in the listing is due to the fact that over two-thirds of the molecules are in the surface layer, where they tend to be disordered. Structure. It is unremarkable that the packing of benzene molecules in small clusters is based on an icosahedral motif. Such a (van der Waals) packing is more efficient in filling space and optimizing the number of molecular contacts for soft, quasispherical particles than are the bulk close-packing arrangements, irrespective of details of intermolecular forces. Several features do depend on details of the forces, however. One is that small clusters of the quasispherical molecule TeF6, while also packing in loosely icosahedral structures, do not exhibit a special stability in the 13-molecule case. This contrary behavior has been explained Another feature, mentioned in the previous section, is that the molecular orientations, unlike the positions of the centers of mass, are sensitive to the form of the interactions. As a consequence, our structure is intrinsically different from the Williamsi3and the similar van de WaalI4 structures. Although van de Waal used a different form of atom-atom potential function than Williams, he fitted it closely to the Williams function, which had been devised to account for a variety of different crystal structures of Our functionI6 was developed specifically for benzene from quantum computations for benzene dimers at many relative orientations and distances.*' Because of the heavy computational demands imposed by the large number of configurations, the level of the computations was not high enough to ensure a faithful representation of the intermolecular forces. Indeed, Shi found it necessary to rescale distances and energies by appreciable amounts. What gives us some confidence that the resultant interaction function is plausible is that it yielded a markedly better representation of the diffraction patterns of liquid benzene than did any of the other potential functions available.'4 In portraying the equilibrium dimer configurations associated with the two potential functions, Figure 9 illustrates the essential difference between the Williams and the Shi potential functions and suggests why the corresponding cluster structures are different. The Shi function builds in a stronger
-
I
-
Figure 9. Equilibrium benzene dimer structures according to the present potential function (left) and the Williams function (right).
TABLE 3: Parameters for the Shi Potential Functionu A B
c
D K
c-c
C-H
H-H
5 930 528.0 -2862 - 1940.9 58.7 -0.407
593 629.8 -67 212.3 184.7 -58.7 0.407
-32 810.2 22 886.2 -969.1 58.7 -0.407
Potential energy in kJ mol-' when r is in angstroms. A cutoff of 12 A was used for the r-? quasielectrostatic term.
drive for adjacent molecules to orient perpendicularly into a T-shaped configuration. Such a propensity breaks the symmetry of the Williamshan der Waal structure. Still different structures of benzene clusters were proposed by Oikawa et a L i 5on the basis of energy minimization. The structures, which closely resembled those of fragments of crystalline benzene, are untenable for intermolecular interactions at all similar to the Shi/Williams/van de Waal functions. Some remarkable observations of small, mass-selected benzene clusters have been carried out spectroscopically, including clusters of C6H6 doped with a single C6H6 m o l e c ~ l e . The ~~.~~ resonant two-photon ionization (R2PI) residual splitting found in the 13-cluster demands a symmetry lower than the C3 symmetry of the Williamdvan de Waal structure. How quantitatively the spectra are accounted for by the symmetry breaking associated with the (present) Shi potential function has not yet been determined. Concluding Remarks. Although a number of properties of small clusters have been elucidated by the foregoing analyses, the results provide no special insight into the original problem stimulating this investigation. How is it possible to govern the way molecules of a condensing vapor pack together by adjusting the way the vapor flows through a nozzle? It is becoming clearer that structures of clusters depend upon their thermal history as they grow, and this history, in turn, is governed by conditions of flow through the n o z ~ l e . ~ .Therefore, ~,~' it is necessary to examine in more detail what happens to clusters as they are warmed enough to induce structural changes somewhat analogous to a phase change. Results of research in this direction are described in the next paper in this series.
Acknowledgment. This research was supported by a grant from the National Science Foundation. We thank the Computer 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 considerable help in initiating this study and acknowledge helpful comments on the manuscript by Professors B. van de Waal, R. L. Whetten, and D. C. Easter. Registry No. Benzene, 71-43-2. Appendix The intermolecular interaction function adopted in the present MC computations is a slightly modified version of Shi's 12-
Monte Carlo Study of Small Benzene Clusters
J. Phys. Chem., Vol. 99, No. 47, 1995 17105
TABLE 4: Coordinates for a 13-Molecule Benzene Cluster at 3 K According to MC Simulation X Y Z mol atom X Y 0.671 009 -1.187 54 -0.318 08 1 1 -2.796 73 -4.627 14 1 -1.315 46 0.412 662 -0.240 11 2 -2.989 2 -5.556 37 -0.258 29 0.946 224 3 1 -0.996 77 -1.012 92 -5.797 74 -0.670 89 1.185 124 4 0.319 31 1 -5.109 86 1.155 844 -0.412 54 0.237 694 1.316 693 1 1.348 317 5 -4.180 63 0.258 408 0.997 999 -0.948 64 1 -0.627 97 6 -3.939 26 1.164 156 -2.059 49 1 0.713 971 7 -0.552 32 4.326 902 -0.415 7 0.715 924 -2.282 78 1 8 1.963 55 4.724 376 -0.448 17 1 1.642 585 - 1.729 84 2.072 733 5.266 055 9 -1.164 03 2.057 076 0.932 337 1 10 0.553 55 5.410 26 0.413 285 -0.715 8 1 11 2.284 009 -0.317 24 5.012 786 1 1.731075 0.448 293 - 1.645 4.471 107 12 -0.426 42 -0.047 82 -0.765 5 -5.144 89 3.928 768 2 1 0.633 722 0.306 634 2 2 2.801 741 -0.815 51 -5.613 83 4.618 386 2 2.991 173 -0.33 82 -5.489 56 1.616 371 5.558 199 3 2 4 -4.896 35 1.853 971 0.906 791 5.808 394 1.012 587 2 -4.427 41 0.781 833 1.674 482 5.118 776 5 -1.155 43 2 1.197 179 -0.527 9 4.178 963 6 -4.551 68 - 1.344 86 2 -5.236 23 -1.728 16 4.860 194 7 -1.973 1 -0.398 63 - 1.730 58 4.259 54 2 -6.04983 0.131 998 8 - 1.296 66 2.404 392 3.154 72 2 -5.834 23 9 - 1.857 09 -0.902 46 2 -4.805 01 10 1.257 609 2.816 628 2.650 553 -3.093 96 2 -3.991 41 11 2.589 552 0.956 469 3.251 208 -3.770 39 -1.315 93 4.356 028 2 -4.207 01 12 -3.209 96 1.761431 3 1 -1.561 18 -3.978 9 -0.400 42 1.068 53 5.672 238 2 -5.321 65 -0.251 24 0.704 293 4.630 102 3 -0.387 57 -6.275 02 -1.171 72 1.15371 2.713 239 3 3 -1.359 91 1.838 5 1 3 4 -3.505 87 -5.885 65 -2.241 38 1.967363 3 5 -4.542 9 -2.390 56 2.331 6 2.880 645 -4.679 48 -3.589 53 -3.701 14 1.882 183 - 1.470 08 4.797 509 3 6 -3.278 18 0.276 132 3 7 -2.491 06 0.738 209 -3.227 68 3 -5.607 84 -3.787 08 0.106 258 -3.733 99 8 0.534 957 3 -7.261 94 -4.455 73 0.885 996 -3.609 64 - 1.062 08 9 -3.828 35 2.297 684 -2.978 98 10 -2.917 94 3 -6.586 38 -2.532 32 2.929 635 -4.256 71 -2.472 66 11 3 -3.176 76 12 3 2.149 897 -2.602 61 -2.597 01 - 1.57973 - 1.86367 4 -3.319 07 1 3.974 472 0.404 605 - 1.06022 - 1.9996 4 -4.197 53 2 5.316 45 0.258 068 -0.691 91 -4.248 21 4 -3.981 78 3 6.271 244 1.173972 -1.147 56 -5.408 31 4 -2.887 58 4 -1.971 54 5.884 059 2.236 413 -4.319 8 4 -2.009 13 4.542 08 2.382 95 -2.339 85 -2.071 19 5 4 -2.224 87 1.467 046 -0.91 1 09 3.587 287 -1.884 19 6 4 -0.725 31 7 -3.771 31 -0.268 59 3.212 132 3.272 698 4 -4.247 47 5.601 031 -0.086 29 8 -0.522 83 2.871 936 4 7.257 598 -0.876 86 9 -4.410 31 1.527017 1.066 267 4 -2.306 44 10 -4.096 98 0.522 293 6.585 832 2.909 603 4 -2.945 47 11 0.862 488 4.257 5 -3.620 82 3.163 845 4 2.207 408 2.600 932 -2.154 9 12 -3.457 98 1.574 75 1 5 -3.651 63 4.200 648 0.812 779 5.619 729 1 -0.312 68 -4.477 77 3.610 409 5.489 501 - 1.622 5 2 0.336 15 -4.760 29 1.276 973 4.892 117 -0.907 12 5 3 - 1.85767 -4.216 67 4.424 961 -1.673 76 4 5 -0.466 22 -0.784 03 -3.390 52 0.124 016 4.555 19 -1.197 13 5 0.525 294 5 2.457 45 1 -3.108 5.152 573 6 0.760 97 5 0.046 137 7 5 1.726 584 3.620 069 -0.862 21 6.058 806 -0.139 45 4.095 225 -0.526 91 5.832 86 8 -2.411 13 5 0.899 632 -1.535 24 4.796 399 9 4.406 438 - 1.257 45 5 -2.820 02 3.985 884 10 4.242 495 -2.878 88 -2.587 57 5 -0.957 25 -3.214 18 4.211 83 11 3.767 339 5 1.314422 - 1.760 62 5.248 291 12 3.456 126 5 -2.205 84 1.72332 0.396 46 1 3.391 328 -0.121 08 -4.861 82 - 1.363 54 6 1.988 811 2 4.215 724 -4.350 72 -0.943 11 6 3.221 471 0.460 669 3 4.755 677 6 - 1.590 22 -1.288 8 -3.240 82 3.775 884 4 4.471 236 6 -2.642 01 -2.657 76 -3.620 01 3.097 636 3.646 841 6 -4.201 75 1.864977 -3.153 1 -3.078 2 5 -2.452 29 3.106 887 6 -2.431 09 -4.263 6 1.310564 2.495 115 7 1.581 318 -5.677 59 -0.887 91 6 3.222 83 2.587 914 1.866 57 3.719 983 6 8 -4.790 85 -0.158 46 2.528 678 2.470 799 9 6 -2.865 17 -1.281 2 4.681 889 3.819 332 2.988 6 10 3.505 13 6 - 1.826 23 -3.133 39 4.447 878 3.623 515 6 11 1.366 465 -2.712 97 -3.862 84 3.785 769 3.740 63 12 0.404 559 6 -4.638 66 -2.740 1 2.008 465 3.308 91 -1.959 51 1.517222 1 7 -4.729 38 0.917 939 2 2.207 331 7 -2.070 45 2.804 903 -5.264 97 2.066 698 3 2.004 136 7 -0.931 38 3.605 275 -5.404 08 4.305 982 4 2.902 52 7 0.318 627 -5.00 761 3.117 965 4.004 099 5.396 508 5 0.429 563 -4.472 03 1.830 284 7 4.247 75 -0.709 51 -4.332 92 7 4.207 293 6 1.0299 12
Z
mol
atom
0.928 948 3.163 075 4.551 72 3.706 239 1.472 112 0.083 467 -1.043 38 -1.531 1 -2.816 47 -3.614 13 -3.126 41 -1.841 04 -0.098 64 -0.944 82 -3.174 94 -4.558 87 -3.712 69 - 1.48257 1.360016 2.427 664 3.078 445 2.661 577 1.593929 0.943 149 0.881 692 2.734 062 3.863 166 3.139 901 1.287532 0.158 427 -3.941 05 -3.798 -2.575 16 - 1.49537 -1.638 41 -2.861 25 -4.839 83 -4.591 65 -2.470 02 -0.596 58 -0.844 76 -2.966 39 -2.953 42 -4.223 78 -4.573 04 -3.651 93 -2.381 57 -2.032 31 -2.696 71 -4.900 8 -5.506 76 -3.908 64 - 1.704 55 - 1.098 59 2.390 49 3.662 6 4.580 641 4.226 571 2.954 461 2.036 421 1.715 731 3.922 842 5.515 642 4.901 331 2.694 22 1.101 42 3.945 893 2.869 152 1.642 211 1.492 01 2.568 75 3.795 692 4.847 695 2.979 55 0.850 806 0.590 208 2.458 353 4.587 096
7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 13 13 13 13 13 13 13 13 13
7 8 9 10 11 12 1 2
3 4 5
6 7 8 9 10 11 12 1
2 3 4 5
6 7 8
9 10 11 12 1 2 3 4 5 6 7 8
9 10 11 12 1
2 3 4 5 6 7 8 9 10 11 12 1 2
3 4 5 6 7 8 9 10 11
12 1
2 3 4 5 6 7 8
9 10 11 12
17106 J. Phys. Chem., Vol. 99, No. 47, 1995 site potential function,I6 consisting of painvise additive functions of the form
where rg represents the distance between atoms i and j and the subscript CI/~ indicates the type of atom pair involved. Following william^,^^.^^ Shi placed the hydrogen interaction sites inside the actual equilibrium nuclear positions to compensate for the asphericity of the electron density around the protons. After this shift and a subsequent minor rescaling, the C-H intersite distance was taken to be 1.031 8, and the C-C, 1.041 A. The values of the parameters AM-Kap are listed in Table 3. The r-* term was included to play the role of an electrostatic term without incurring the substantial computational expense of taking the square roots that would have been required by a true Coulomb r-' term. It was parametrized to reproduce electrostatic forces at the typical contact distances between molecules and truncated at 12 8, instead of at the original Shi limit of 5 A (requiring the inclusion of the constant term K M ) . All other interactions were calculated without any cutoff. At small values of r, the Shi H-H function became negative and, hence, physically absurd, due to the increasing dominance of the negative r-I2 term. Such interactions could result, and did initially result occasionally at high temperatures, if Monte Carlo moves happened to place hydrogens too close together. To prevent such pathology, we redefined the H-H potential energy to be a prohibitively large positive value for all r less than 1.36 A, the maximum of the Linse-Shi function. The structure of the 13-molecule cluster at low temperatures is given in Table 4.
References and Notes (1) See, for example: Berry, R. S., Ed, Proceedings of the Sixth International Symposium on Small Particles and Inorganic Clusters. in Z. Phys. D 1993, 26. (2) Bartell, L. S.; Sharkey, L. R.; Shi, X. J. Am Chem. Soc. 1988, 110, 7006. (3) Bartell. L. S.; Harsanyi. L.: Valente, E. J. J. Phys. Chem. 1989, 93, 6201.
Dulles and Bartell (4) Bartell, L. S.; Machonkin, R. A. J. Phys. Chem. 1990, 94, 6468. (5) Etters, R. D.; Kaelberer, J. B. J. Chem. Phys. 1977, 66, 3233. (6) Davis, H. L.; Jellinek, J.; Berry, R. S. J. Chem. Phys. 1987, 86, 6456. (7) Beck, T. L.; Jellinek, J.; Berry. R. S. J. Chem. Phys. 1987, 87, 545. (8) Etters, R. D.; Kaelberer, J. B. Phys. Rev. A 1975, 11, 1068. (9) Raoult. B.; Farges, J.; de Feraudy, M. F.; Torchet, G. Z. Phys. D 1989, 12, 85; Philos. Mag. B 1989, 60, 881. (10) Bartell, L. S.; Xu, S . J. Phys. Chem. 1991, 95, 8939. Xu, S.; Bartell, L. S.: J. Phys. Chem. 1993, 97. 13550. (11) Beniere, F. M.; Boutin, A.; Simon, J.-M.; Fuchs, A. H.; de Feraudy, M.-F.; Torchet, G. J. Phys. Chem. 1993, 97, 10472. (12) Bartell. L. S.; Xu, S . J. Phys. Chem., in press. (13) Williams. D. E. Acta Crysralllogr. A 1980, 36, 715. (14) van de W a d , B. W. J. Chem. Phys. 1983, 79, 3948. (15) Oikawa, S.; Minoru, T.; Hiromi, K.;Tadashi, U. Acta Crystallogr. B 1985, 41, 437. (16) Shi, X.; Bartell. L. S. J. Phys. Chem. 1988, 92, 5667. (17) A preliminary study of benzene and TeF6 clusters appeared in the following: Dulles, F. J.; Bartell, L. S.; Chuko, B.; Xu, S. In Physics and Chemistry of Finite Systems: from Clusters to Crystals; Jena, P., Rao, B. K., Khanna. S. N.. Eds.; Kluwer Academic: Dordrecht, 1992; Vol. I. p 393. (18) Chuko, B.; Bartell, L. S. J. Phys. Chem. 1993, 97, 9969. (19) Bartell, L. S.: Dulles, F. J. J. Phys. Chem. 1995, 95, 17107. (20) Jorgensen, W. L.; Binning, R. C., Jr.; Bigot, B. J. Am. Chem. Soc. 1981, 103, 4393. (21) Jorgensen. W. L.; Madura, J. D.; Seversen, C. J. J. Am. Chem. Soc. 1984, 106, 6638. (22) Bartell, L. S.: Dulles, F. J.; Chuko, B. J. Phys. Chem. 1991, 148, 271. (23) Dulles, F. J. Ph.D. Thesis, University of Michigan, Ann Arbor, MI, 1993. (24) A typical six-site function for benzene: Claessens, M.; Ferrario, M.; Ryckeaart, J.-P. Mol. Phys. 1983, 50, 217. (25) Williams, D. E. J. Chem. Phys. 1966, 45, 3770. (26) Williams, D. E.; Starr, T. L. Comput. Chem. 1977, I , 173. (27) Karlstrom, G.; Linse, P.: Wallqvist, A,; Jonsson, B. J. Am. Chem. Soc. 1983, 3777. (28) Jeffrey, G. A.; Ruble, J. R.; McMullan. R. K.; Pople, J. A. Proc. R. SOC.London A 1987, 414, 47. (29) Easter. D. C.; Whetten, R. L.; Wessel, J. E. J. Chem. Phys. 1991, 94, 3347. Easter, D. C.; Li, X.; Whetten, R. L. J. Chem. Phys. 1991, 95, 6362. Easter, D. C.: Khoury. J. T.; Whetten. R. L. J. Chem. Phys. 1992, 97, 1675. (30) Easter, D. C.: Baronavski, A. P.; Hawley, M. J. Chem. Phys. 1993, 99, 4942. (31) Banell, L. S . J. Phys. Chem. 1990, 94, 5102.
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