4.1 The bond-path framework set B{q,q’,r} and the precession K of [4]cumulene and variants
Examination of the distance measures reveals that none of the C-CBCP s are located at the mid-points of the C-C BCPbond-paths as may be expected, see Table 1 , further scalar measures, include distances and the relative energies E are provided in the Supplementary Materials S2 . The locations of the C-C BCP s are rather similar for the [4]cumulene and S-1,5-dimethyl-[4]cumulene molecular graphs, where the terminal C1-C2 BCP and C4-C5 BCP are located closest to the corresponding terminal C1 and C5 atom respectively, than to the C2 or C4 atoms. The central C2-C3 BCP and C3-C4 BCP display the opposite effect in that both these BCP s are located closest to the more central C3 atom. The effect of the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene is to reverse the positions of the BCP s where the terminal BCP s are located closest to the C2 and C4 atoms for the C1-C2 BCP and C4-C5BCP respectively. The positions of the central C2-C3 BCPand C3-C4 BCP are also located closest to the C2 and C4 atoms respectively.
Table 1. The partial bond-path lengths (in a.u.) of cumulene and the variants for values of the geometric dihedral angle ϕ = 0.0º.
(C1-BCP,BCP-C2) (C2-BCP,BCP-C3) (C3-BCP,BCP-C4) (C4-BCP,BCP-C5)[4]cumulene 0.9293,1.5379 1.4434,0.9600 0.9600, 1.4434 1.5379, 0.9293S-1,5-dimethyl-[4]cumulene 0.9381,1.5355 1.4392,0.9657 0.9657, 1.4392 1.5354, 0.9381(-)S(-) S-1,5-diamino-[4]cumulene 1.3456, 1.1454 0.9942,1.4096 1.4096,0.9943 1.1449, 1.3458(+)S(-) S-1,5-diamino-[4]cumulene 1.3514, 1.1395 0.9903,1.4134 1.4134,0.9905 1.1396, 1.3511(+)S(+) S-1,5-diamino-[4]cumulene 1.3569, 1.1340 0.9868,1.4170 1.4173,0.9866 1.1396, 1.3573
The bond-path framework set B{q ,q’ ,r } comprise the most preferred {q ,q’ } path-packets of [4]cumulene, S-1,5-dimethyl-[4]cumulene and the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene at the geometrical dihedral angle of 0.0° are presented in Figure 1 . At a geometrical dihedral angle of 0.0° the C2-C3 BCP and C3-C4BCP , on the basis of the size of the {q ,q’ } path-packets (BCPellipticity ε = 0.15), are considered to be single bond-like and C1-C2BCP and C4-C5 BCP are double bond-like (BCPellipticity ε = 0.42) for both the [4]cumulene, S-1,5-dimethyl-[4]cumulene molecular graphs, see Figure 1(a-b) . The corresponding {q ,q’ } path-packets of the C2-C3 BCP and C3-C4 BCP contain increasing degrees of polar character, indicated by the asymmetrical character of the {q ,q’ } path-packets, for the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene, see Figure 1(c-e) . A lower degree of asymmetrical character of the {q ,q’ } path-packets is apparent for geometric dihedral angles ϕ = 15.0º, 75.0º and 90.0º, see theSupplementary Materials S2 . The {p ,p’ } path-packets that correspond to the least preferred directions of charge density accumulation are provided in the Supplementary Materials S3.