Scheme 1. The molecular graph of [4]cumulene,
S-1,5-dimethyl-[4]cumulene and the (-)S(-), (+)S(-) and (+)S(+)
conformations of S-1,5-diamino-[4]cumulene, where X = [H,
CH3, (-)S(-)NH2,
(+)S(-)NH2, (+)S(+)NH2]. The magenta
circled atoms define the geometric dihedral angle ϕ specified in
the order listed. The undecorated green spheres indicate the locations
of the bond critical points (BCP s).
Helical characteristics of stereoisomers were first proposed as the
origin of chirality by Fresnel13 in 1851, later this
was demonstrated by optical experiments that show materials having
different refractive indices for right (R) and left (S) circularly
polarized light, known as circular dichroism14. These
experiments were consistent with theories of optical activity that
correlate the inherent helical identities with direction of rotation of
the circularly polarized light15–17. A ‘helix theory’
was much later hypothesized by Wang for molecular chirality and chiral
interaction who realized that evidence for this helical character was
not provided by molecular geometries18, or solely
attributable to steric hindrance. Wang referred to the association
between helical characteristics and chirality as the ‘chirality-helicity
equivalence’. Recent experiments by Beaulieu et al. on neutral
molecules19, that utilize coherent helical motion of
bound electrons, consistent with our previous work20demonstrated the need for a better understanding of the behavior of the
charge density redistribution. Consistent with Wang18,
Banerjee-Ghosh et al. also recently demonstrated that charge
density redistribution in chiral molecules, not spatial
effects21, were responsible for an enantiospecific
preference in electron spin orientation. Some of the current authors
located, but did not quantify, for chiral compounds, the unknown
chirality-helicity equivalence20 to enable the chiral
discrimination of the S and R stereoisomers consistent with the naming
schemes from optical experiments. The insufficiency of conventional
(scalar) QTAIM22 (quantum theory of atoms in
molecules) was also demonstrated for the chiral discrimination of S and
R stereoisomers. If a helical response of the electronic charge density
distribution is present on applying a bond torsion, then both non-axial
(perpendicular to the bond-path) and axial (parallel to the
bond-path) displacements of the torsional C1-C2 bond critical point
(BCP ) will be found. Recently, the unknown chirality-helicity
function, previously only located, was quantified consistent with
photoexcitation circular dichroism experiments. We then used the
chirality-helicity function to differentiate between steric effects due
to eclipsed conformations and chiral behaviors in formally achiral
species23.
In this investigation we will use NG-QTAIM (next generation QTAIM) to
investigate each of the C-C BCP s in terms of the ‘orbital-like’
{q ,q’ } path-packets and the
quantification of their wrapping around the bond-paths, the wrapping of
these bond and the chirality and helicity properties using the stress
tensor trajectory Tσ(s ).
We use Bader’s formulation of the stress tensor24 and
NG-QTAIM due to the superior performance of the stress tensor compared
with vector-based QTAIM in terms of clearer chiral discrimination of the
S and R stereoisomers of lactic acid25. The NG-QTAIM
chirality-helicity function will be applied to a range of
[4]cumulenes to determine the presence of chirality and or helicity
within the NG-QTAIM formalism.