(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,qʹ}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,qʹ} 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