Construction of stress tensor trajectory Tσ(s) of the torsional bond critical point
The background of QTAIM and next generation QTAIM (NG-QTAIM)[34], [50]–[55] with explanations is provided in theSupplementary Materials S1 , along with the procedure to generate the stress tensor trajectories Tσ(s ). In this investigation we will use Bader’s formulation of the stress tensor[32] within the QTAIM partitioning, which is a standard option in the QTAIM AIMAll[56] suite. The ellipticity, ε, quantifies the relative accumulation of the electronic charge densityρ (rb ) distribution in the two directions perpendicular to the bond-path at a Bond Critical Point (BCP ) with position rb . For values of the ellipticity ε > 0, the shortest and longest axes of the elliptical distribution of ρ (rb ) are associated with the λ1 and λ2 eigenvalues, respectively, and the ellipticity is defined as ε = |λ1|/|λ2| 1. We earlier 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 tensor[48]. Previously, we established the stress tensor trajectory Tσ(s ) classifications of S and R based on the counterclockwise (CCW) vs. clockwise (CW) torsions for thee.dr components of Tσ(s ) for lactic acid and alanine[30]. The calculation of the stress tensor trajectory Tσ(s ) for the torsional BCP is undertaken using the frame of reference defined by the mutually perpendicular stress tensor eigenvectors {±e1 σe2 σe } at the torsional BCP , corresponding to the minimum energy geometry ; this frame is referred to as the stress tensor trajectory space (also referred to as Uσ-space). This frame of reference is used to construct all subsequent points along the Tσ(s ) for dihedral torsion angles in the range -180.0º ≤ θ ≤+180.0º, where θ = 0.0º corresponds to the minimum energy geometry. 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°. To be consistent with optical experiments, we defined from the Tσ(s ) that S (left-handed) character is dominant over R character (right-handed) for values of (CCW) > (CW) components of the Tσ(s ). The Tσ(s ) is constructed using the change in position of the BCP , referred to as dr , for all displacement steps dr of the calculation. Each finite BCP shift vector dr is mapped to a point {(e1 σ∙dr ), (e2 σ∙dr ), (e3 σ∙dr )} in sequence, forming the 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 Tσ(s ) eigenvector components evaluated at theBCP. The projections of dr are respectively associated with the bond torsion: e.dr →bond-twist, e.dr → bond-flexing ande.dr → bond-anharmonicity[30], [53]–[55], [57]–[59].
The chirality Cσ is defined by the difference in the maximum projections (the dot product of the stress tensore eigenvector and the BCP shiftdr ) of the Tσ(s ) values between the CCW and CW torsions Cσ = [(e∙dr)max ]CCW-[(e∙dr)max ]CW . These torsions correspond to the CW (-180.0° ≤θ ≤0.0°) and CCW (0° ≤θ ≤180.0°) directions of the torsion θ. The chirality Cσquantifies the bond torsion direction CCW vs. CW, i.e. circulardisplacement, since e is the most preferred direction of charge density accumulation. The least preferred displacement of a BCP in the Uσ-space distortion set {Cσ,Fσ,Aσ} is the bond-flexing Fσ, defined as Fσ = [(e∙dr)max]CCW- [(e∙dr)max]CW . The bond-flexing Fσ therefore 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.
The chiral asymmetry that we refer to as the bond-anharmonicity Aσ, defined as Aσ = [(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-path[59]. The sign of the chirality determines the dominance of Sσ(Cσ > 0) and Rσ(Cσ < 0) character, see Tables 2-3 . The bond-anharmonicity Aσ determines the dominance ofSσ or Rσ character with respect to the BCP sliding along the bond-path as a consequence of the bond-torsion. Aσ > 0 indicates dominant Sσ 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 bond-anharmonicity Aσ ≠ 0, i.e.BCP sliding.
The stereoisomeric excess Xσ is defined as the ratio of the chirality Cσ values of S and R stereoisomers and therefore a value of Xσ > 1 demonstrates a preference for the Sσ over theRσ stereoisomer. The E -field amplification EAσ, is defined as the ratio EAσ = Cσ/Cσ|E= 0, e.g. as a consequence of the changes that occur in anE -field.