of [4]cumulene and variants
The bond-twist Tσ quantifies the non-axialdisplacement of the BCP and is defined as the difference (CCW minus CW) of the bond-twistmax component of the Tσ(s )max. The bond-flexing Fσ provides a measure of the ‘flexing-strain’ and is defined as the difference (CCW minus CW) of the bond-flexingmax component of the stress tensor trajectory Tσ(s )max. The chiral asymmetry (axiality) Aσ quantifies the degree ofaxial displacement BCP along the bond-path and is defined as the difference (CCW minus CW) of the bond-axialitymaxcomponent of the stress tensor trajectory Tσ(s )max, see Table 3and Figure 3 . The axiality Aσ is the response to the bond torsion, i.e. the sliding of the BCP along the bond-path51. The chirality-helicity function Chelicity = TσAσ is defined as the simple arithmetic product of the non-axial (Tσ) and axial (Aσ) contributions of the BCP displacement in response to the torsion θ. For comparison purposes, we examined conventionally chiral (R)-1-chloro-1-phenylethane58and found the chirality Cσ = -0.9859[Rσ] for the chiral carbon atom, the axiality Aσ (previously referred to as the helicity Bσ) = -0.0110[Rσ] and the Chelicity = 0.0108.
Note the use of the term “chirality Cσ” which is reserved for use for a single chiral atom in a molecule which is not the case for the cumulenes where we are investigating the C1-C2-C3-C4-C5 chain of BCP s, however the method of calculation for the bond-twist Tσ is the same. For [4]cumulene the values of the bond-twist Tσ and bond-flexing Fσ are both insignificant for the C1-C2-C3-C4-C5 chain however, the axiality Aσ = 0.366 is very large compared with that of the previously calculated conventionally chiral (R)-1-chloro-1-phenylethane. Examination of the Tσ(s ) along the bond-axiality axis, used to calculate the axiality Aσ, demonstrates the dominance of the axiality Aσ and the lack of bond-twist Tσ, see Figure 3 . The S-1,5-dimethyl-[4]cumulene possesses rather low values of Tσ and Chelicity for the C1-C2-C3-C4-C5 chain, however the values of Tσ and therefore Chelicity are an order of magnitude higher for the C1-C2 BCP and the C2-C3 BCP . We note that the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene all possess greater Chelicity values for the terminal C1-C2BCP and C4-C5 BCP than the central C2-C3 BCP and C3-C4 BCP . The maximum stress tensor projections{ bond-twist max, bond-flexingmax, bond-axialitymax} and the bond-flexing Fσ of the cumulene variants are presented in the Supplementary Materials S5 .
Table 3. The values of the bond-twist Tσ,bond-flexing Fσ and the axiality Aσ and of the chirality-helicity function Chelicity = (bond-twist Tσ)(axiality Aσ) for the C-CBCPs of the unsubstituted and substituted cumulenes, seeScheme 1 and the Table S5(a) of theSupplementary Materials S5 .
{Tσ, Fσ, Aσ} Chelicity[Tσ,Aσ][4]cumulene C1-C2 BCP {0.00000[Sσ], 0.00001[Sσ], 0.36621[Sσ]} 0.00000[Sσ,Sσ]C2-C3 BCP {0.00001[Sσ], -0.00002[Rσ], 0.58335[Sσ]} 0.00001[Sσ,Sσ]C3-C4 BCP {-0.00002[Rσ], 0.00001[Sσ], 0.58336[Sσ]} -0.00001[Rσ,Sσ]C4-C5 BCP {-0.00006[Rσ], 0.00000[Sσ], 0.36633[Sσ]} -0.00002[Rσ,Sσ]S-1,5-dimethyl-[4]cumuleneC1-C2 BCP {-0.02677[Rσ], -0.04781[Rσ], 0.18036[Sσ]} -0.00483[Rσ,Sσ]C2-C3 BCP {-0.00858[Rσ], -0.01873[Rσ], 0.33567[Sσ]} -0.00288[Rσ,Sσ]C3-C4 BCP {0.00052[Sσ], -0.00093[Rσ], 0.34030[Sσ]} 0.00018[Sσ,Sσ]C4-C5 BCP {-0.00282[Rσ], 0.00050[Sσ], 0.17749[Sσ]} -0.00050[Rσ,Sσ](-)S(-) S-1,5-diamino-[4]cumuleneC1-C2 BCP {0.24690[Sσ], -0.21197[Rσ], 1.28073[Sσ]} 0.31621[Sσ,Sσ]C2-C3 BCP {-0.14710[Rσ], 0.04266[Sσ], -0.45718[Rσ]} 0.06725[Rσ,Rσ]C3-C4 BCP {0.14262[Sσ], -0.04699[Rσ], -0.45292[Rσ]} -0.06460[Sσ,Rσ]C4-C5 BCP {0.31485[Sσ], -0.19825[Rσ], 1.25469[Sσ]} 0.39561[Sσ,Sσ](+)S(-) S-1,5-diamino-[4]cumuleneC1-C2 BCP {0.17738[Sσ], -0.66454[Rσ], -1.80066[Rσ]} -0.31940[Sσ,Rσ]C2-C3 BCP {-0.01920[Rσ], -0.02399[Rσ], -1.03702[Rσ]} 0.01991[Rσ,Rσ]C3-C4 BCP {-0.08852[Rσ], -0.06836[Rσ], -1.01975[Rσ]} 0.09027[Rσ,Rσ]C4-C5 BCP {0.08485[Sσ], -0.17413[Rσ], -1.86474[Rσ]} -0.15822[Sσ,Rσ](+)S(+) S-1,5-diamino-[4]cumulene C1-C2 BCP {0.86363[Sσ], -0.21694[Rσ], 6.74465[Sσ]} 5.82488[Sσ,Sσ]C2-C3 BCP {-0.16848[Rσ], -0.23924[Rσ], -7.77013[Rσ]} 1.30911[Rσ,Rσ]C3-C4 BCP {-0.18147[Rσ], -0.34965[Rσ], -7.77102[Rσ] -1.41021[Rσ,Rσ]C4-C5 BCP {0.98399[Sσ], 0.98398[Sσ], 6.70333[Sσ]} 6.59601[Sσ,Sσ]
The linear stress tensor trajectories Tσ(s ) plot for [4]cumulene reflects the presence pure axial motion, i.e. only values of axiality |Aσ| > 0 are present and are created by BCP sliding along the bond-path of the BCP , is a consequence of the value of the bond-twist Tσ = 0.0, see the Supplementary Materials S5and Table 3 . Significant values of the Chelicity are evident from twisting of stress tensor trajectories Tσ(s ) plots for S-1,5- dimethyl-[4]cumulene, see Figure 3(a) . The very large values of the chirality-helicity function Chelicity are evident from the large open spiral form of the stress tensor trajectories Tσ(s ) plots for the (-)S(-), (+)S(-) and (+)S(+) conformations of S-1,5-diamino-[4]cumulene, see Figure 4(a-c) respectively.