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

A vector-based representation of the chemical bond is introduced that we refer 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 familiar quantum theory of atoms in molecules (QTAIM) bond-path length. The eigenvector-following path lengths $\mathbb{H}^{*}$ and $\mathbb{H}$ are constructed along the bond-path from the $\underline{\mathbf{\mathit{e}}}_{1}$ and $\underline{\mathbf{\mathit{e}}}_{2}$ Hessian eigenvectors respectively, which correspond to the least and most preferred directions of charge density accumulation. In particular, the paths $p$ and $q$ provide a vector representation of the scalar QTAIM ellipticity {\epsilon}. The bond-path frame-work set $\mathbb{B}$ is applied to the excited state deactivation of fulvene that involves distortions along various intramolecular degrees of freedom, 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 excited electronic states, in contrast to the QTAIM bond-path length. In particular, the eigenvector-following path lengths $\mathbb{H}^{*}$ and $\mathbb{H}$ are found to be shorter for the excited state than the ground state for both the BLA and bond torsion distortions indicating that distortions resulting in lower $\mathbb{H}^{*}$ and $\mathbb{H}$ values are easier to perform.