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.