  We present a new vector-based representation of the chemical bond referred to
as the bond-path frame-work set $\mathbb{B} = {p, q, r}$, where $p$, $q$ and
$r$ represent three paths with corresponding eigenvector-following path lengths
$\mathbb{H}^{*},\mathbb{H}$ and the bond-path length from the quantum theory of
atoms in molecules (QTAIM). We find that longer path lengths $\mathbb{H}$ of
the ring-opening bonds predict the preference for the transition state inward
(\textbf{TSIC}) or transition state outward (\textbf{TSOC}) ring opening
reactions in agreement with experiment for all five reactions \textbf{R1-R5}.
Competitiveness and non-competitiveness have traditionally been considered
using activation energies. The activation energy however, for \textbf{R3} does
not satisfactorily determine competitiveness or provide consistent agreement
with experimental yields. We choose a selection of five competitive and
non-competitive reactions; methyl-cyclobutene (\textbf{R1}),
ethyl-methyl-cyclobutene (\textbf{R2}), iso-propyl-methyl-cyclobutene
(\textbf{R3}), ter-butyl-methyl-cyclobutene (\textbf{R4}) and
phenyl-methyl-cyclobutene (R5). Therefore, in this investigation we provide a
new criterion, within the QTAIM framework, to determine whether the reactions
\textbf{R1-R5} are competitive or non-competitive. We that find \textbf{R2},
\textbf{R3} and \textbf{R5} are competitive and \textbf{R1} and \textbf{R4} are
non-competitive reactions in contrast to the results from the activation
energies, calling into question the reliability of activation energies.
