2. Theoretical Background
The interested reader is referred to the complete mathematical
expressions and explanations in the Supplementary Materials S1 .
QTAIM analysis39–45 is used to identifycritical points in the total electronic charge density
distribution ρ (r ) where the gradient vector field
∇ρ (r ) = 0. There are four distinct categories of
critical points according to the set of ordered eigenvalues
λ1 < λ2 <
λ3, with the associated set of eigenvectors
(e1 , e2 ,e3 ), of the Hessian matrix of the
electronic charge density, ρ (r ), defined as the matrix
of partial second derivatives with respect to the spatial coordinates,
∇:∇ρ (r ). Critical points are labeled using the notation
(R , ω) where R is the rank of the Hessian matrix and ω is
the signature; the (3, -3) [nuclear critical point (NCP ), a
local maximum generally corresponding to a nuclear location], (3, -1)
and (3, 1) [saddle points, called bond critical points (BCP )
and ring critical points (RCP ), respectively] and (3, 3) [the
cage critical points (CCP )]. In this investigation we will only
be considering bond critical points (BCP s).
The ellipticity ε, quantifies the relative accumulation ofρ (r b) in the two directions(e1 ande2) perpendicular to the bond-path at
rb. For ellipticity values > 0, the
associated λ1 and λ2 Hessian eigenvalues
correspond to the shortest and longest axes of the elliptical
distribution of ρ (rb ), respectively.
Bond-flexing distortions involve the stretching of a bond (bond-path) so
that the bond-path length (BPL) exceeds the bonded inter-nuclear
geometric separation distance. A shift of a BCP position along
the containing bond-path due to changes to bonded inter-nuclear
separations results in the presence of BCP sliding. As a
consequence of this BCP sliding the chemical nature of the bond
is dependent on the relative position of the BCP , i.e. we can
quantify a degree of bond-axiality35. The construction
of the stress tensor trajectories Tσ(s )
involves the required additional symmetry breaking to identify chirality
in the form of the e3σ eigenvector. This
enables the Tσ(s ) corresponding to the
counterclockwise (CCW) and clockwise (CW) directions of torsion to be
distinguished even for the highly symmetrically positioned torsional
C1-C2 BCP , see Scheme 1 . To be consistent with optical
experiments as previously undertaken46 we defineSσ (left-handed) character to be dominant overR σ character (right-handed) for values of the
chirality Cσ (CCW) > (CW) since CCW and CW
represent left and right handed directions of torsion respectively. Note
the use of the subscript “σ” because we are using the
stress tensor Tσ(s ) in the stress tensor
Uσ-space. The chirality Cσ of a torsion
bond (in this work the torsional C1-C2 BCP and torsional C1-N7BCP ) is defined by the difference in the maximum
Tσ(s ) projections (the dot product of the stress
tensor e1σ 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(1)
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. circularmotion, since e1σ is the most preferred
direction of charge density ρ (rb )
accumulation.
The response of the C-H/D/T bonds to the CCW vs. CW torsions uses
equation 1(a) but does not define a chirality
Cσ associated with a torsional bond of a molecule.
Instead the response is referred to as the bond-twist Tσ
The least preferred (e2σ ) direction ofρ (rb ) corresponds to a more ‘difficult’
bond distortion than bond torsion, that we refer to as the bond-flexing
Fσ that is defined as:
Fσ =
[(e2σ∙dr)max ]CCW-
[(e2σ∙dr)max ]CW