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.