2.2 The stress tensor trajectory Tσ(s) and the Chirality-helicity function Chelicity
The procedure to generate the stress tensor trajectories Tσ(s ) is provided in the Supplementary Materials S1 . In this investigation we will use Bader’s formulation of the stress tensor24 within the QTAIM partitioning which is a standard option in the QTAIM AIMAll45suite. Earlier, we demonstrated that the most preferred direction for bond displacement, corresponding to most preferred direction of electronic charge density displacement, is thee eigenvector of the stress tensor31. Recently, we established the stress tensor trajectory Tσ(s ) classifications for the S and R stereoisomers of lactic acid and alanine based on the counterclockwise (CCW) vs. clockwise (CW) torsions for thee.dr components of Tσ(s )20. The calculation of the stress tensor trajectory Tσ(s ) for the torsionalBCP is undertaken the frame of reference defined by the mutually perpendicular stress tensor eigenvectors {±e1 σe2 σe } at the torsional BCP , corresponding to the geometric dihedral angle ϕ = 0.0º. This frame of reference is referred to as the stress tensor trajectory space (also named Uσ-space).All the subsequent points along the Tσ(s ) for dihedral torsion angles in the range -180.0º ≤ θ ≤+180.0º, use this frame of reference. We adopt the convention that CW circular rotations correspond to the range -180.0° ≤θ ≤ 0.0° and CCW circular rotations to the range0.0° ≤ θ ≤+180.0°. Consistent with optical experiments, we defined from the stress tensor trajectory Tσ(s ) that S (left-handed) character is dominant over R character (right-handed) for values of (CCW) > (CW) components of the stress tensor trajectory Tσ(s ). The stress tensor trajectory Tσ(s ) are constructed using the change in position of the BCP , referred to as dr , for all displacement steps dr of the calculation. Each finiteBCP shift vector dr is mapped to a point {(e1 σ∙dr ), (e2 σ∙dr ), (e3 σ∙dr )} in sequence, forming the stress tensor trajectory Tσ(s ), constructed from the vector dot products (the dot product is a projection, or a measure of vectors being parallel to each other) of the stress tensor trajectory Tσ(s ) evaluated at the BCP. The projections of dr in Uσ-space are associated with the bond torsion: e.dr →bond-twist, e.dr → bond-flexing ande.dr →bond-axiality20,46–51. Note previously we referred toe.dr as bond-axiality or bond-anharmonicity.
The bond-twist Tσ is defined by the difference in the maximum projections (the dot product of the stress tensore eigenvector and the BCP shiftdr ) of the stress tensor trajectory Tσ(s ) values between the CCW and CW torsions Tσ = [(e∙dr)max ]CCW-[(e∙dr)max ]CW . The bond-twist Tσ quantifies the bond torsion direction CCW vs. CW, i.e. circular displacement, wheree corresponds to the most preferred direction of charge density accumulation. Note, previously we referred to bond-twist Tσ as the chirality Cσbecause a single dominant torsion bond for the molecule was being used to determine the chirality Cσ properties of that molecule. In this investigation therefore, we refer to bond-twist Tσ instead of chirality Cσ. Each of the C-C BCP s analyzed however, will still be assignedSσ or Rσ character.
The least preferred displacement of a BCP in the Uσ-space distortion set {Tσ,Fσ,Aσ} is the bond-flexing Fσ, defined as Fσ = [(e∙dr)max]CCW- [(e∙dr)max]CW . The bond-flexing Fσ provides a measure of the ‘flexing-strain’ that a bond-path is under when, for instance, subjected to an external force such as an E -field.
Previously we used the term helicity Bσ, defined as Bσ = [(e∙dr)max ]CCW-[(e∙dr)max ]CWquantifies the direction of axial displacement of the bond critical point (BCP ) in response to the bond torsion (CCW vs. CW), i.e. the sliding of the BCP along the bond-path51. In this investigation to avoid confusion with work on the helical orbitals of [n ]cumulenes we will use the term axiality Aσ≡ Bσ. The sign of the chirality determines the dominance of Sσ (Tσ > 0) andRσ (Tσ < 0) character, see Tables 2 . The axiality Aσdetermines the dominance of Sσ orRσ character with respect to the BCPsliding along the bond-path as a consequence of the bond-torsion. Note the use of the subscript “σ ” used for theSσ or Rσ assignments to denote calculation by the stress tensor trajectory Tσ(s ).
Aσ > 0 indicates dominantSσ character and the converse is true for Aσ < 0. The reason for calculating the Tσ(s ) by varying the torsion θ is to detect values of the axiality Aσ ≠ 0, i.e. BCP sliding.
The chirality-helicity function Chelicity is formed from the simple arithmetic product of the bond-twist Tσ and the axiality Aσ. The presence of a helical or chiral response, is determined by Aσ ≠ 0 or Tσ ≠ 0 of the torsional BCP to the applied torsion θ coinciding with a helical stress tensor trajectory Tσ(s ), for the conventionally chiral molecules such as lactic acid and alanine. In this investigation we will determine the Chelicity for all four of the C-C BCP s along each of the [4]cumulene variant molecular graphs.