(e)
Figure 2. The variation of the C1-C2 BCP , C2-C3BCP , C3-C4 BCP and C4-C5 BCP bond-path precessions K’ along bond-path for geometric dihedral angles ϕ = 0.0º, 15.0º, 75.0º and 90.0ºare presented end-to-endC 1-C 2-C 3-C 4-C 5 for [4]cumulene, S-1,5-dimethyl-[4]cumulene and the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene in sub-figures(a-e) respectively.
Note, the precession K’ provides a directional interpretation of chemical bonding, see Figure 2 . The corresponding K for the wrapping of the {p ,p’ } path-packets around the C-C BCP bond-paths r are provided in the Supplementary Materials S4 . The presence of values of K’ = 1 for the entire C1-C2-C3-C4-C5 chain indicates the presence of bonding that possesses the deformation characteristics of rigid shared-shellBCP bonding and a Qrigidity = 1.000. Conversely, values of K’ = 0 for the entire C1-C2-C3-C4-C5 chain indicates the presence of bonding that possesses the deformation characteristics of flexible closed-shell BCP bonding with Qrigidity = 0.000.
For a geometric dihedral angle ϕ = 0.0º there is no variation of the degree of the wrapping of the {q,}path-packets, as measured by the precession K’ around the C-C BCPbond-paths r for the [4]cumulene and S-1,5-dimethyl-[4]cumulene molecular graphs, see Figure 2(a) , Figure 2(b) . The absence of wrapping of the{q,} path-packets around the C-C BCPbond-paths r along the C1-C2-C3-C4-C5 chain for ϕ= 0.0º is apparent from the identical values of Qrigidity = 1.000, see Table 2 . This is not the case however, for the corresponding values of the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene molecular graph at ϕ = 0.0º as demonstrated by the values Qrigidity < 1.000, see Table 2and Figure 2(c-e) . For values of ϕ = 15.0º and 75.0º the bonding becomes increasingly polarized as demonstrated by the alternating high-low values of Qrigidity .
At ϕ = 90.0º the [4]cumulene, S-1,5-dimethyl-[4]cumulene and the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene possess the most polarized character bonding with the vast majority of values of K’ = 1 or K’ = 0, indicating a switch between strong and weak chemical character, this effect being most apparent for the [4]cumulene and S-1,5-dimethyl-[4]cumulene molecular graphs, see Figure 2 and Table 2 . To summarize, precession K’ clearly identifies symmetry-breaking properties, from the asymmetrical form of K’ at ϕ = 15.0º for [4]cumulene and S-1,5-dimethyl-[4]cumulene and at ϕ = 0.0º 15.0º and 75.0º for the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene. The precession K’ does notidentify symmetry-breaking for all values of ϕ or identify any helical characteristic previously identified by the helical frontier molecular orbital55. In the next section we will therefore use the stress tensor trajectory Tσ(s )max to investigate the presence of helical character as well as chirality.
Table 2. The values of Qrigidity = Q/Qmaximum for [4]cumulene, S-1,5-dimethyl-[4]cumulene and the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene and geometric angleϕ , Q is the area enclosed by K’ for each C-C BCP bond-path and Qmaximum corresponds to the area under K’ = 1 for each C-C BCP bond-path.
Qrigidityϕ = 0.0º ϕ = 15.0º ϕ = 75.0º ϕ = 90.0º[4]cumulene C1-C2 1.0000 0.8703 0.8289 0.8334 C2-C3 1.0000 0.4975 0.0736 0.0282 C3-C4 1.0000 0.9107 0.9658 0.9801 C4-C5 1.0000 0.7413 0.2080 0.1646S-1,5-dimethyl-[4]cumulene C1-C2 0.9997 0.8640 0.8475 0.8498 C2-C3 1.0000 0.3892 0.0861 0.0640 C3-C4 1.0000 0.8762 0.9334 0.9335 C4-C5 0.9998 0.6580 0.1823 0.1522(-)S(-) S-1,5-diamino-[4]cumulene C1-C2 0.6017 0.7208 0.8660 0.8841 C2-C3 0.0911 0.0686 0.0776 0.0916 C3-C4 0.2880 0.4874 0.8441 0.9043 C4-C5 0.2622 0.2520 0.1480 0.1196(+)S(-) S-1,5-diamino-[4]cumulene C1-C2 0.6288 0.7498 0.8817 0.8838 C2-C3 0.1077 0.0776 0.0845 0.0838 C3-C4 0.3013 0.5099 0.8859 0.9111 C4-C5 0.2629 0.2444 0.1438 0.1174(+)S(+) S-1,5-diamino-[4]cumulene C1-C2 0.6110 0.7482 0.8827 0.8053 C2-C3 0.1077 0.0748 0.0794 0.0909 C3-C4 0.2819 0.5061 0.8816 0.9047
C4-C5 0.2229 0.1101 0.1096 0.1220
4.2 The chirality-helicity function Chelicity