(c)
Figure 4. The stress tensor trajectories Tσ( s ) of the C1-C2 BCP , C4-C5 BCP (left panel) and C2-C3 BCP , C3-C4 BCP(right panel) of the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene are presented in each of sub-figures(a-c) respectively. The end of each Tσ( s ) is denoted by a cube marker.
5. Conclusions
In this investigation we visualized the bond-path framework set B{q ,q’,r } for geometric dihedral angles ϕ = 0.0º, 15.0º, 75.0º and 90.0º of [4]cumulene, S-1,5-dimethyl-[4]cumulene and the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene. There we saw a variation of the precession K for the cumulene variants and dihedral angleϕ , where K quantifies the wrapping of the {q ,q’ } path-packets around the bond-path r and the {q ,q’ } path-packets are constructed from the directions of most preferred electronic motion. The directional character of the chemical bonding is then extracted as the Qrigidity that is used to explain the lack of twisting of the {q ,q’ } path-packets of [4]cumulene and S-1,5-dimethyl-[4]cumulene around the bond-path r that occurs for the maximum possible value of Qrigidity (= 1.0) for ϕ = 0.0º. In addition, for all the cumulene variants the Qrigidity quantifies a strong trend towards polarization of the bonding between the possession of deformation characteristics of weak and strong chemical bonds as the geometric dihedral angle is increased to ϕ = 90.0º.
We have demonstrated that the presence of the chirality-helicity equivalence by observing and quantifying helical shaped stress tensor trajectories Tσ(s ) for the S-1,5-dimethyl-[4]cumulene, (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene. We used the stress tensor trajectories Tσ(s ) to determine the bond-twist Tσ, the axiality Aσ and the chirality-helicity function Chelicity . Very large values of the chirality-helicity function Chelicity are evident from the helix form of the stress tensor trajectories Tσ(s ), this finding is consistent with the helical orbitals discovered in earlier investigations into lactic acid and alanine20.
The largest bond-twist Tσ values occur for the end-point C1-C2 BCP and C4-C5 BCP of the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene and corresponds to the presence of Sσ chirality andSσ axiality. The chirality-helicity function Chelicity of S-1,5-dimethyl-[4]cumulene comprises a mix of Sσ chirality andRσ axiality and the low values of bond-twist Tσ but significant axiality Aσ results in the morphology of the stress tensor trajectories Tσ(s ) being weakly helical.
[4]Cumulene is the first molecular graph we have discovered that possesses an absence of bond-twist Tσ and a very large value of the axiality Aσ and consequently a negligible value of the chirality-helicity function Chelicity . We do not therefore expect the presence of chiral and helical properties in cumulene[4] to be discoverable by experiments such as those undertaken by Beaulieuet al. on neutral molecules19 due to the insignificant degree of the chirality-helicity function Chelicity . We do however expect, on the basis of the very large values of the chirality-helicity function Chelicity , that the chiral and helical properties of the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene would be discoverable by optical experiments.