Scheme 2. The construction of the {p,p’ } precession K and {q,q’ } precession K’ corresponding to the unsubstituted cumulene molecular graph. The unit vectoru (blue arrow) is defined by thee1 eigenvector (red arrow) of the C1-C2BCP , the e2 eigenvector (purple arrow) is orthogonal to e1 . The pale magenta line indicates the interatomic surface paths (IAS) that originate at the BCP . The undecorated green spheres indicate the locations of the BCP s.
For the precession of the {q ,q } path-packet, defined by the e2eigenvector, about the bond-path, β = (π/2 – α) and α is defined by equation (3) see Scheme 1 , we can write an expression Kʹ:
Kʹ = 1 – cos2β , where cosβ =e2u , β = (π/2 – α) and 0 ≤ Kʹ ≤ 1 (3)
Note, for the general case the e3eigenvector is defined along the bond-path and is not perpendicular to the reference direction u , see Scheme 1 . For Kʹ = 0 we have a maximum degree of facile character and for K = 1 we have the minimum degree of facile character.
The presence of values of the precession K’ in the range 0 ≤ K’ ≤ 1 indicates polarization of the electron density ρ (r ) associated with the bond-path in terms of the changing orientation of the e2 eigenvectors, from parallel to perpendicular, including intermediate orientations of thee2 eigenvector. Consequently, there will be a range of ‘mixed’ bond types within the limits of the rigid shared-shell character K’ = 1, characteristic of sigma bonds and flexible closed-shell character K’ = 0, characteristic of hydrogen bonding.