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 thee1σ 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 thee1σ.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 σ,±e2 σ,±e 3σ}
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: e1σ.dr →bond-twist, e2σ.dr → bond-flexing ande3σ.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 tensore1σ eigenvector and the BCP shiftdr ) of the Tσ(s ) values between the CCW
and CW torsions Cσ =
[(e1σ∙dr)max ]CCW-[(e1σ∙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 e1σ 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 =
[(e2σ∙dr)max]CCW- [(e2σ∙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 =
[(e3σ∙dr)max ]CCW-[(e3σ∙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.