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A vector‐based representation of the chemical bond is introduced, which we refer to as the bond‐path framework set B = {p, q, r}, where p, q, and r represent 3 eigenvector‐following paths with corresponding lengths H*, H, and the familiar quantum theory of atoms in molecules (QTAIM) bond‐path length (BPL). The intended application of B is for molecules subjected to various types of reactions and distortions, including photoisomerization reactions, applied torsions θ, or normal modes of vibration. The lengths H* and H of the eigenvector‐following paths are constructed using the e1 and e2 Hessian eigenvectors, respectively, along the bond path, these corresponding 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 ε. The bond‐path framework set 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 and bond torsion distortions. We find that the H* and H lengths can differentiate between the ground and excited electronic states, in contrast to the QTAIM BPL. Five unique paths were presented for B= {(p0,p1), (q0,q1), r} for the ground and first excited states where the profile of the scaling factor, the ellipticity ε, reveals a large unexpected asymmetry for the excited state. A "next‐generation" quantum theory of atoms in molecules (QTAIM) is introduced to analyze all types of bonding in molecules subjected to various types of reactions and distortions, including photoisomerization reactions, applied torsions θ, or normal modes of vibration. The double‐bond isomerization of fulvene is used as test model for this "next‐generation" QTAIM approach. [ABSTRACT FROM AUTHOR]

Variation of the precession K along the torsion C2-C6 BCP bond-path of fulvene for the S 0 (left panel) and S 1 (right panel) electronic states for a series of values of the torsion θ = 0.0° (Franck-Condon) to θ = 64.3° (conical intersection). Bond critical points (BCP s) are located at a distance = 0.0 a.u along the bond-path. • Determined the precessions K of the path-packets for the S 0 and S 1 states of fulvene. • The precessions K of the path-packets quantified the bond rigidity and flexibility. • An asymmetry is found for the S 1 state variation in the bond-path rigidity with torsion. • The S 1 state at torsion θ = 0.0° possesses maximum bond-rigidity along the entire bond. • The S 1 state possessed a higher degree of bond-anharmonicity than the S 0 state. Here we have determined the precessions K and K' of the { p , p ′} and { q , q ′} path-packets for the excited state deactivation reaction of fulvene in terms of bond-flexing, bond-torsion and bond-anharmonicity that includes the tendencies towards bond-path-rigidity and bond-path-flexibility. An asymmetry is found for the S 1 state resulting in an unpredictable variation in the direction of the bond-path-rigidity as a function of the torsion coordinate. Uniquely, the S 1 state at torsion θ = 0.0° possesses the possible maximum bond-path-rigidity, along the entire bond-path but then drops lower than for the S 0 state at the conical intersection indicating photo-excitation facilitates the torsion. [ABSTRACT FROM AUTHOR]