3.2.2 Cooperativity in Cyclic clusters
Similar to its linear counterpart, the number of possible cyclic conformers also increases with cluster size. Therefore, it becomes imperative to choose one particular type of conformer for all the sizes, from dimer to hexamer, to study the evolution of H-bonds with increasing cluster size. We chose cyclic clusters 1, the ones with all free carbonyl groups pointing in the same direction, for the study of H-bond cooperativity. Total binding energies of CD, CTr, CTe, CP and CH were found to be -8.8, -17.9, -25.1,-31.9 and -37.9 kcal mol-1, respectively, and binding energy per interaction were -4.4, -6.0, -6.3, -6.4 and -6.3 kcal mol-1, in the same order (Table 2 and Figure 4). Here, binding energy per interaction increases monotonically with increasing cluster size up to pentamer, and then decreases marginally for hexamer. This steady increment clearly indicates that cyclic clusters enjoy cooperative stabilization and the effect gets saturated when the cluster size reaches pentamer. It may be noted here that the binding energy per interaction for linear clusters was also found to reach saturation point at pentamer.
The above values clearly show that, on average, H-bonding interaction becomes stronger with increasing size of cyclic cluster. Therefore, if one or more CHD molecules are removed from a particular cluster keeping the remaining part as it was in the cluster, the binding energy per interaction would decrease steadily with successive removal of monomeric units. For example, when one CHD molecule is removed from CH, the resulting structure would have lesser binding energy per interaction than the cluster itself. Removal of another monomer would lower the value further and so on till two monomeric units remain. Similar procedure has been followed for CP, CTe and CTr as well. The obtained values have been tabulated (Table 3) for quantitative analysis and a bar diagram has been constructed (Figure 7) for easier qualitative assessment. As monomeric units are removed from CH, the binding energy per interaction decreases from 6.3 kcal mol-1successively to 6.0, 5.6, 5.2 and 4.5 kcal mol-1. Similar trends are observed for CP, CTe and CTr as well. The above findings clearly show that upon addition of each CHD molecule, the binding energy per interaction increases steadily for all the cyclic clusters. Therefore, it could easily be inferred that the cyclic clusters enjoy cooperative stabilization.
The above discussions clearly indicate that the H-bonding interactions become stronger, and consequently H-bond accepting capability of the bound carbonyl groups increase, with the size of cyclic clusters. This increment should result in consistent increase in C=O bond length of H-bonded carbonyl groups. It was found that the C=O bond length in monomer, CD, CTr, CTe, CP and CH are 1.2041, 1.2104, 1.2130, 1.2134, 1.2134, and 1.2135 Å respectively (Table 4 and Figure 4). The lengthening of H-bonded C=O bonds with cluster size is clear evidence of positive cooperativity in cyclic clusters and the findings are in accordance with the energetic aspect discussed above.
Unlike linear clusters, where every H-bonded carbonyl group form two H-bonds, namely HB1 and HB2, number of H-bond donors attached to a single carbonyl group increases with size of cyclic clusters. It was found that one carbonyl group is attached to two C-H donors in only CD, while it is bound to three C-H donors in CTe, CTr, CP and CH forming HB3 (as defined earlier) beside HB1 and HB2. Therefore, it would be interesting to investigate how HB1 and HB2 affects each other as the size of the cluster changes and also how formation HB3 in the higher clusters affects the existing HB1 and HB2. For this purpose, all the individual H-bond lengths in cyclic clusters have been given in Table 4 and a plot against cluster size (Figure 8) has been constructed to easily analyze the trend they show. The length of HB1 in CD, CTr, CTe, CP and CH are 2.506, 2.328, 2.332, 2.332-2.345 and 2.338 Å, respectively. Similarly, HB2 bond lengths are 2.633, 2.412, 2.421, 2.412-2.446 and 2.429Å, in the same order as above. It is evident from the above values that both HB1 and HB2 becomes shorter (and therefore stronger) while moving from CD to CTr. However, they both become longer from CTe onwards. This lengthening of both HB1 and HB2 must have some connection to the behavior of HB3. The length of HB3 in CTr, CTe, CP, and CH are 2.902, 2.664, 2.568-2.582 and 2.562 Å, respectively. The above values show that there is a significant shortening, and hence strengthening, of HB3 upon going from CTr to CTe. As discussed above, HB1 and HB2 also become longer while moving from CTr to CTe. In fact, HB3 shows monotonic shortening with increasing cluster size all the way up to CH and both HB1 and HB2 behave in completely opposite manner. Therefore, for cyclic clusters, strengthening of HB3 is accompanied by concomitant weakening of HB1 and HB2, i.e. they clearly show anti-cooperative nature towards each other. Nevertheless, the three H-bonds show an overall cooperative interaction as a whole as the overall binding energy of the three H-bonds increases with increasing size of the cluster as evident from both binding energy per interaction and C=O bond length of H-bonded carbonyl groups. This behavior is in stark contrast to what was observed for the linear clusters. In those cases, the mutual anti-cooperativity existing between HB1 and HB2 showed overall anti-cooperative destabilization of higher clusters.
Finally, NBO and AIM analyses were conducted for the H-bonds in cyclic clusters in similar manner to what has been done for the same in their linear counterparts. Hyperconjugative charge transfer energies along with electron densities and their Laplacians at bond critical points were calculated (Table S2) and plotted against H-bond lengths (Figure S7). The plots clearly show a direct relationship of all the above mentioned three parameters with H-Bond length and corroborate the above findings.