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We present a new vector-based representation of the chemical bond referred toas 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 ofatoms in molecules (QTAIM). We find that longer path lengths $\mathbb{H}$ ofthe ring-opening bonds predict the preference for the transition state inward(\textbf{TSIC}) or transition state outward (\textbf{TSOC}) ring openingreactions in agreement with experiment for all five reactions \textbf{R1-R5}.Competitiveness and non-competitiveness have traditionally been consideredusing activation energies. The activation energy however, for \textbf{R3} doesnot satisfactorily determine competitiveness or provide consistent agreementwith experimental yields. We choose a selection of five competitive andnon-competitive reactions; methyl-cyclobutene (\textbf{R1}),ethyl-methyl-cyclobutene (\textbf{R2}), iso-propyl-methyl-cyclobutene(\textbf{R3}), ter-butyl-methyl-cyclobutene (\textbf{R4}) andphenyl-methyl-cyclobutene (R5). Therefore, in this investigation we provide anew 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} arenon-competitive reactions in contrast to the results from the activationenergies, calling into question the reliability of activation energies.

Predicting Competitive and Non-Competitive Torquoselectivity in Ring-Opening Reactions using QTAIM and the Stress Tensor