Vibrational scaling factors follow a power law

Harmonic vibrational scaling factors are shown to follow an empirical power law decay curve that significantly improves the accuracy of predicted molecular frequencies. While conventional scaling relies on constant or linear functions, these findings demonstrate that the underlying relationship is non-linear and monotonic. When tested across a range of molecules using B3LYP and B97-1 density functional theory, mean errors are reduced by up to 35% compared to standard constant scaling and 19% over wavenumber linear scaling. This approach provides a more physically consistent treatment of anharmonicity across the full spectral range and identifies specific pathological deviations in triple-bond bending modes. The recommendation is that B3LYP/aug-cc-pVTZ frequencies are scaled to ν_i = 1.21000014 × ω_i^0.97129120, and B97-1/aug-cc-pVTZ frequencies are scaled to ν_i = 1.18641962 × ω_i^0.97417536.