BzAr LJ parameters LJ parameters

BzAr

We systematically search the PES's of BzAr with using two sets of rather different LJ parameters recommended by Ondrechen, Berkovitch-Yellin and Jortner [6] (OBJ) and Schmidt, Le Calvé and Mons [1] (SCM). In particular the SCM Ar-Ar well depth is about 20% larger than that used by OBJ and the Ar-H well is about 25% smaller. The equilibrium separations are identical for Ar-C and Ar-H and differ by about 3% for Ar-Ar, i.e. these differences are small compared with the different well depths. These parameters are summarized in Table i.

  
Table: Lennard-Jones parameters from the OBJ [6] and SCM [1] sources. is given in bohr(Å) and is given in millihartree(cm ).

We now introduce some notation which has been used, e.g. by Schmidt et al., to describe the positions of the Ar atoms with respect to the benzene molecule. Here (l|m|n) signifies that there are l Ar atoms on one side of the ring, n Ar atoms on the other side and m Ar atoms in or very close to the plane of the molecule -- the m Ar atoms are referred to as bridging atoms, and we observe such structures as local minima for BzAr with . We note that for structures with no bridging Ar atoms (i.e. m=0) the notation may be abbreviated to (l|n), where l and n are defined as above. For example a structure with three Ar atoms on either side of the benzene molecule is labeled (3|3) and if an additional Ar atom is placed in the plane of the molecule this is referred to as a (3|1|3) structure.

Following Schmidt et al. we use two additional labels c and s for describing the arrangement of the Ar atoms with respect to the axis of the benzene molecule. We use c when one of the Ar atoms lies on or close to this axis and s when this point is `shared' by more than one Ar atom. For c-type arrangements of Ar atoms those not above the axis are positioned with the benzene molecule acting as a template -- each atom lies in or near to one of the mirror planes between the H atoms. This leads to closed-packed arrangements of Ar atoms, which we would expect to see up to BzAr for one-sided structures, at which point the benzene ring becomes `saturated'. We observe s-type minima for BzAr to BzAr and for the transition state in BzAr , which connects two permutational isomers of the local minimum in a degenerate rearrangement in agreement with previous calculations by Wales [2].

The additional labels s and c are useful to describe many of the minima that we observe in the smaller BzAr clusters because of the influence that the benzene molecule has on the arrangement of the Ar atoms. For two-sided structures it is generally the case that the arrangement of Ar atoms on each side of the molecule is similar to one observed for a one-sided local minimum in a smaller cluster. For example, local minima exist for and structures and as indicated in the discussion above we also observe minima. This notation does not indicate the relative orientations of the two sets of Ar atoms and hence several local minima generally exist with the same label in such cases.

The majority of the low-lying local minima for the smaller clusters have pseudo-planar arrangements of Ar atoms on each side of the ring, i.e. each atom is approximately the same perpendicular distance from the plane of the benzene molecule. Some structures have a second plane containing one or two Ar atoms above the first. An example for which three Ar atoms are in an s-type arrangement with the fourth acting as a cap would be labelled .

This nomenclature allows us to describe many of the local minima that we observe for up to eight Ar atoms. However, for clusters as small as BzAr we observe structures not based primarily on the geometry of the benzene molecule and there are local minima with pseudo 5-fold symmetric arrangements of Ar atoms. For BzAr we see structures based on the neat Ar double icosahedron as discussed below.

Details of some low-lying minima, along with their point groups and energies, are given in Table ii. Details of sample reaction pathways are given in Table iii. It is noteworthy that one-sided minima are found for which the expected close-packed arrangement of Ar atoms does not exist, i.e. minima for which the Ar atoms are spread above the benzene molecule in patterns which do not favour the Ar-Ar interactions. We find such structures for BzAr with and these, as for the s- and c-type arrangements, are able to combine with other Ar atoms to form two-sided structures. For example, a minimum with a linear arrangement of Ar atoms above the benzene molecule exists. We label this structure and have also observed . The unfavourable arrangements of Ar atoms in these minima make them relatively weakly bound.

  
Table ii: The energies, in millihartree, and the point groups (PG) for the five lowest BzAr minima found with the OBJ and SCM LJ parameters. The labels are fully described within the text.

The geometries of the minima and the transition states, calculated without the induction energy contribution, are qualitatively unaffected in almost all cases when we change the LJ parameters from those of OBJ to SCM. Occasionally a structure characterized for one set of LJ parameters as a minimum is not so for the other set and sometimes the connectivity of the minima changes, i.e. which minima are linked by particular transition states. Although the structures generally do not change very much, we find that the energetic ordering of the local minima is rather different, and we observe that the SCM minima show a preference for one-sided structures compared to those of OBJ. The SCM lowest minima for BzAr with are all one-sided, with close-packed, pseudo-planar arrangements of Ar atoms. The size of the Ar-Ar well depth accounts for the fact that this trend was not observed for the OBJ lowest energy minima where two-sided structures have greater stability. For example, the structure is lowest in energy for BzAr for both SCM and OBJ but for BzAr and BzAr the lowest minima found with the SCM parameters are and and with the OBJ parameters are and -- see Table ii.

  
Table iii: Selected rearrangement mechanisms for some small BzAr clusters. We report the energies of the minima ( and ), the energy of the transition state ( ), the barrier heights ( and ), and the transition state point groups (PG ). All energies in millihartree.

For BzAr with we see differences in the transition states calculated using the different parameters. For BzAr we find five transition states for both OBJ and SCM. Four are degenerate rearrangements of the minimum and the other is non-degenerate, exchanging the and the minima. There is a correlation between the energies of the transition states and the number of Ar atoms crossing the benzene ring. The two highest-lying transition states involve both Ar atoms crossing the ring and mediate degenerate rearrangements. The next highest transition state is for the non-degenerate rearrangement in which one Ar atom changes sides. The remaining two degenerate rearrangements involve no ring-crossing and in the higher of the two the benzene molecule effectively rotates by , thus changing the site of one of the Ar atoms. The lower of the latter two transition states corresponds to a structure and the corresponding rearrangement changes the Ar atom lying on the axis of the molecule. The transition states permute the Ar atoms in the same way for both the OBJ and SCM parameters, and with one exception the rearrangements are equivalent. During the rotation, for the SCM parameters one Ar atom remains on the axis of the benzene molecule. However, for the OBJ parameters the rotation of the benzene molecule occurs after the structure has been reached and therefore involves a direct connection between two transtion states. If we label the minimum min, the transition state ts1 and the second transition state ts2, then we can label the two low-lying OBJ pathways as,

though the right hand side minimum is not the same permutational isomer for the two paths (Fig. 2). Branching points must exist between ts2 and ts1 on both sides of the second path. We find such rearrangements involving two transition states for the OBJ parameters in BzAr with n=2-3,5.

  
Figure 2: Two rearrangement pathways for BzAr calculated using the OBJ parameters. One includes a direct connection between two transition states. Both pathways are for degenerate rearrangements of the minimum. The arrows point to the same H atom in all cases, to clarify the relative orientations of the benzene molecule.

Click on the picture for animation: min -> ts1 -> ts2 -> ts1 -> min

We will cite one other difference in the PES's. For BzAr there exists a transition state which converges to a different minimum in one of the two downhill optimizations. For the OBJ parameters, the minima on either side of this transition state are and . For the same transition state using the SCM parameters, the minima it connects are and . If we follow the reaction pathway for these parameters we find that the structure is visited but it is not a stationary point and the optimization continues until the structure is reached.

In conclusion we find that as well as the variation in the energetic ordering of the stationary points, examples for which the OBJ and SCM PES's have other fundamental differences exist. However, these are rare, and it appears that even considerable changes in the LJ parameters leave the form of the stationary points largely unaffected. For brevity we have only tabulated a small fraction of the results we have obtained for the various stationary points.

BzAr LJ parameters LJ parameters