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 =e2∙u , 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.