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A vector-based representation of the chemical bond is introduced that werefer to as the bond-path frame-work set = $\mathbb{B} = \{p, q, r\}$, where$p$, $q$ and $r$ represent three paths with corresponding eigenvector-followingpath lengths $\mathbb{H}^{*},\mathbb{H}$ and the familiar quantum theory ofatoms in molecules (QTAIM) bond-path length. The eigenvector-following pathlengths $\mathbb{H}^{*}$ and $\mathbb{H}$ are constructed along the bond-pathfrom the $\underline{\mathbf{\mathit{e}}}_{1}$ and$\underline{\mathbf{\mathit{e}}}_{2}$ Hessian eigenvectors respectively, whichcorrespond to the least and most preferred directions of charge densityaccumulation. In particular, the paths $p$ and $q$ provide a vectorrepresentation of the scalar QTAIM ellipticity {\epsilon}. The bond-pathframe-work set $\mathbb{B}$ is applied to the excited state deactivation offulvene that involves distortions along various intramolecular degrees offreedom, such as the bond stretching/compression of bond length alternation(BLA) and bond torsion distortions. We find that the $\mathbb{H}^{*}$ and$\mathbb{H}$ lengths can differentiate between the ground and excitedelectronic states, in contrast to the QTAIM bond-path length. In particular,the eigenvector-following path lengths $\mathbb{H}^{*}$ and $\mathbb{H}$ arefound to be shorter for the excited state than the ground state for both theBLA and bond torsion distortions indicating that distortions resulting in lower$\mathbb{H}^{*}$ and $\mathbb{H}$ values are easier to perform.

Next-Generation Quantum Theory of Atoms in Molecules for the Ground and Excited States of Fulvene