The insufficiency of scalar measures for chirality or chiral asymmetry
In this section, we first outline the scalar-based results, which are insufficient to explain chirality or chiral asymmetry due to the inherently directional nature of these effects[30]. We then present the Uσ-space distortion set {Cσ,Fσ,Aσ} using a fixed reference alpha carbon atom (C1) for lactic acid, alanine, glycine and glycine with applied electric(E )-field. The detailed analysis of the distance measures is provided in theSupplementary Materials S2 . Note that the value of theE -field = +200×10-4 a.u. causes a large structural distortion to the S and R molecular graphs in the form of a rotation of the C2-O2-H6 group about the C1-C2 BCP . Consequently, a hydrogen-bond H6–N7 BCP forms at θ = ±1.0°; for further details see the caption of Figure S3(I) of theSupplementary Materials S3 .
The variation of the (scalar) energies ∆E relative to θ = 0.0º, corresponding to the relaxed geometry of glycine, of the CW (-180.0º ≤ θ ≤ 0.0º) and CCW (0.0º ≤ θ ≤ 180.0º) torsions associated with the C1-N7BCP and C1-C2 BCP , as shown in the left and right panels respectively of Scheme 2(a) , demonstrates the greater strength of interaction of the C1-N7 BCP compared to the C1-C2 BCP . The distance measures for a torsion θ = 0.0º, demonstrate that the C1-N7BCP is closer to the C1 atom (1.198 a.u.) than the C1-C2BCP (1.393 a.u.). This asymmetry in the location of theBCP along the C1-N7 BCP bond-path leads to an asymmetry of the variation of the bond-path length (BPL) and the BCPellipticity ε with the CCW and CW torsions, see Scheme 2(b) andScheme 2(c) respectively. In particular, the lower values of the C1-N7 BCP ellipticity ε for the CCW (0.0º ≤ θ ≤ 180.0º) torsion compared to the CW (-180.0º ≤ θ ≤ 0.0º) torsion indicate a preference for the CCW torsion over the CW torsion. This is because a bond with lower ellipticity ε, e.g. single bonds, will undergo a torsion deformation more readily than bonds with a higher ellipticity ε, as is the case for double bonds. The symmetrical location of the C1-C2BCP mid-way along the associated bond-path results in symmetrical variations of the bond-path length (BPL) and the ellipticity ε with the CCW and CW torsions, and hence no preferred direction of torsion, CCW or CW, is indicated.
Considering the limitations of the scalar BCP ellipticity ε in determining preferences of CCW over CW for the C1-C2 BCP , we now proceed to examine the directional differences between the CCW and CW torsions of the torsional C1-N7 BCP and C1-C2 BCP with the vector-based stress tensor trajectories Tσ(s), seeFigure 1(a) and Figure 1(b) respectively. TheBCP ellipticity ε does not take into account the bond-path eigenvector (e ) and therefore cannot quantify torsional CCW vs. CW preferences for a symmetrically positioned C1-C2 BCP , see Scheme 2(c) .