(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.