2.1 The QTAIM ellipticity ε, the bond-path framework set B{q,q’,r} and the precession K
QTAIM is used to obtain critical points in the total electronic charge density distribution ρ (r ) by analyzing the gradient vector field ∇ρ (r )22 where the complete set critical points are ordered according to the set of ordered eigenvalues λ1 < λ2< λ3, with corresponding eigenvectorse1 , e2 ,e3 of the Hessian matrix of the total charge density ρ (r ). The complete set of critical points along with the bond-paths of a molecule or cluster is referred to as the molecular graph26. The most and least preferred directions of electron accumulation aree2 ande1 , respectively27–29; the eigenvectore 3 define the direction of the bond-path at the bond critical point (BCP ). The bond-path length (BPL) is defined as the length of the path traced out by thee3 eigenvector of the Hessian of the total charge density ρ (r ), passing through theBCP , along which ρ (r ) is locally maximal with respect to any neighboring paths30. The ellipticity ε = |λ1|/|λ2| 1, where λ1 and λ2 are negative eigenvalues of the correspondinge1 and e2 eigenvectors, respectively. The ellipticity ε quantifies the relative accumulation of ρ (rb ) in thee2 and e1 directions perpendicular to the bond-path at a BCP . The deviation from linearity, i.e. the shortest line separation, of a bond-path separating two bonded nuclei is defined as the dimensionless ratio (BPL - GBL)/GBL of the difference between the BPL and the geometric bond length (GBL) and the GBL.
The bond-path framework setB{p ,q ,r } is comprised of three ‘linkages’; p , q and r associated with thee1 , e2and e3 eigenvectors, respectively is the NG-QTAIM interpretation of the chemical bond. The p and q are the 3-D paths that are constructed from the values of the least (e1 ) and most (e2 ) preferred directions of electronic charge density accumulation along the bond-path, referred to as r . For further discussions on the construction of bond-path framework set B{p ,q ,r }31–41, see the Supplementary Materials S2 .
Two paths (q and q ’) are defined as being associated with the most preferred directione2 eigenvector sincee2 ≡ -e2lies in the same plane for the same point on the bond-path (r ); correspondingly there are two paths associated with the e1 (p and p ’), see equation (1 ). The pair of q - and q ’-paths form packetshapes, that resemble orbitals, along the bond-path are referred to as a {q ,q } path-packet. The orientation, size and location of the {q ,q’ } path-packets relative to the associated BCP indicate how much, which direction and whether a bond-path is twisted or linear. By definition, the {q ,q } path-packets are always orthogonal to the {p ,p } path-packets. The form of the constituent {q ,q’ } path-packets can be used to provide a 3-D interpretation of bonding as a mixture of the following concepts: double bond, single bond, covalent, ionic or diradical. Note that, double and single bonds for shared-shell C-C BCP s correspond to values of the C-C BCP ellipticity ε > 0.25 and ε < 0.25 respectively, also the {q ,q } path-packets have no enclosed area around the BCP for values of ellipticity ε = 0.
Using n points ri along the bond path r (associated with eigenvectore3 ) and defining εi as the ellipticity at this point, one can draw vectors q i and p i, scaled by εi, originating at this point. The tips of these vectors (q i and p i) define the paths p and q , where the form of p i and q i is defined as follows:
pi = ri+ εie1,i, qi = ri + εie2,i(1)
We will now define the extent to which the {p ,p } path-packet constructed from the e1 eigenvector wraps i.e.precesses about a C-C bond-path, see the left panel ofScheme 1 . For the {p ,p } path-packet, defined by the e1eigenvector, we wish to follow the extent to which the {p ,p } path-packet precesses about a C-C bond-path by defining the precession K for bond-path-rigidity42–44:
K = 1 – cos2α, where cosα =e1u and 0 ≤ K ≤ 1 (2)
Where u is the reference direction defined by thee1 eigenvector at the C1-C2 BCPfor each of the unsubstituted and substituted cumulenes for each choice of the geometrical dihedral angle (0.0º, 15.0º, 75.0º and 90.0º). Considering the extremes of the precession K, with α defined by equation (2 ), for K = 0, there is maximum alignment of thee1 eigenvector with the reference direction u , the least facile direction for the C1-C2BCP. When K = 1 we have the maximum degree of alignment with thee2 eigenvector, the most facile direction. In other words, K = 0 and K = 1 indicate C-C bond-paths with the lowest and highest tendencies towards C-C bond-path-flexibility, respectively. The precession K is determined relative to the BCP , in either direction along the C-C bond-path towards the nuclei at either end of the bond-path using an arbitrarily small spacing ofe1 eigenvectors. If we choose the precession K of the {p ,p } path-packet about the C-C bond-path when the ±e1eigenvector is parallel to u , the BCP will have minimum facile character, i.e. bond-path-rigidity. By following the variation of the precession K we can quantify the degree of facile character of a BCP along an entire C-C bond-path.