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 e 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 e 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(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 e 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 (e ) 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 = [(e∙dr)max ]CCW- [(e∙dr)max ]CW