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 thee1σ 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 thee1σ.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 σ,±e2 σ,±e 3σ}
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: e1σ.dr →bond-twist, e2σ.dr → bond-flexing ande3σ.dr →bond-axiality20,46–51. Note previously we referred toe3σ.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 tensore1σ eigenvector and the BCP shiftdr ) of the stress tensor trajectory
Tσ(s ) values between the CCW and CW torsions
Tσ =
[(e1σ∙dr)max ]CCW-[(e1σ∙dr)max ]CW .
The bond-twist Tσ quantifies the bond torsion direction
CCW vs. CW, i.e. circular displacement, wheree1σ 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 =
[(e2σ∙dr)max]CCW- [(e2σ∙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 =
[(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-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.