Cartesian and Curvilinear Anharmonic Frequencies and Intensities: Perturb-then-Diagonalize Polyad Transport and Adiabatic Discrete-Variable Representation

Abstract

Anharmonic intensities reveal vibrational coupling that band positions alone cannot determine. We derive explicit second-order infrared transition moments in curvilinear normal coordinates, including kinetic metric derivatives and transport through variational resonance polyads. The vibrational perturbation-to-second-order-then-diagonalization (VP2TD) treatment covers fundamentals, overtones, three-quantum transitions, and hot bands. A general computational platform combines Extended Symmetry-Adapted Local Coordinates (ESALCs), using Cartesian (ESALC-C) or internal (ESALC-I) primitives, with interchangeable quantum-chemical programs and parallel finite differences. Independent oscillator matrices verify the intensity equations; a molecular water check reproduces second-order amplitudes within $8.4\times10^{-8}$ atomic units. Comparisons with established tools reproduce a formaldehyde Fermi polyad within 0.163 cm$^{-1}$ and 0.061 km mol$^{-1}$ and water Raman activities within $10^{-7}$ Angstrom$^6$. For 29 matched formaldehyde rotational line pairs, the predicted hot-to-fundamental intensity ratio, using the measured lower-state energy, is consistent with the nuclear-spin-corrected median: the differences are 3.9% (ESALC-I) and 3.8% (ESALC-C), compared with a 15% uncertainty estimate for individual measured line intensities. Common analytic reference derivatives ensure identical harmonic observables, while internal coordinates reduce and localize intermode anharmonic couplings. Rotation-vibration constants extend the same tensor treatment. An integrated adiabatic discrete-variable representation treats large-amplitude motions, and finite-level partition functions provide anharmonic thermochemistry.