Tapping the implications from the functional responses of CAM-B3LYP, LC-(\omega\)-HPBE, (\omega\)-B97XD in the redistribution of the electronic cloud of the available normal modes belonging to anthracene , tetracene, phenanthrene, and pyrene

This study introduces a symmetry-guided Reduction Model to classify and interpret the vibrational normal modes of four prototypical polycyclic aromatic hydrocarbons—anthracene, tetracene, phenanthrene, and pyrene. By collapsing the full atomic structure into simplified pseudo-oscillators located at the centers of mass (COMs) of benzenoid rings, we construct a compact vibrational representation rooted in group theory. This approach not only reproduces the number and symmetry of vibrational modes obtained from conventional analysis but also provides a clear geometrical and electronic interpretation of the motions through one-dimensional (\(f(x)\)) and two-dimensional (\(f(y,z)\)) COM trajectories. When applied in conjunction with range-separated density functional theory (DFT) using CAM-B3LYP, LC-\(\omega\)HPBE, and \(\omega\)B97XD functionals, the model reveals persistent, highly regular functional-dependent signatures in vibrational descriptors—force constants, reduced masses, and spatial spans—across all systems. Two derived metrics, \(\Sigma\) and \(\Pi\), quantify the extent of electronic reorganization associated with each mode and serve as a basis to distinguish between predominantly \(\sigma\)- and \(\pi\)-type vibrations. Notably, \(f(x)\) modes show near-harmonic behavior, while \(f(y,z)\) modes exhibit pronounced oscillations and sign inversions, capturing the enhanced coupling between in-plane distortions and electron density redistribution. Zero-point energy (ZPE) trends across functionals demonstrate a smooth compensation between stiffness and mass, even as the electronic contributions vary non-monotonically. Functional-specific deviations in ZPE predictions highlight the sensitivity of vibrational properties to the exchange-correlation treatment and underscore the need for internal standards in benchmarking. Altogether, the Reduction Model provides a physically transparent, symmetry-adapted framework for analyzing vibrational dynamics, revealing transferable fingerprints of functional behavior. This compact approach has strong potential for scaling to larger molecular systems, guiding functional development, and extending to anharmonic and environment-sensitive vibrational analyses in complex systems