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.