Theoretical Model for Vibrational Relaxation in Hydrogen-Bonded Liquids from the CDUFD Framework: Calibration with Water and Outlook for Methanol and Ethanol (CDUFD Phenomenological Series III)

We present a theoretical model for vibrational relaxation rates in hydrogen‑bonded liquids, motivated by the CDUFD framework. Starting from the discrete matter–geometry coupling established in the CDUFD existence proof and employing coarse‑graining assumptions (Markovianity, local equilibrium), we derive an energy diffusion equation that expresses the relaxation rate as an integral over the radial distribution function g (r) with an exponential damping factor arising from the finite correlation length in the non‑critical regime. The final expression is\ = 4 ₃₈₅₅ \, ₍L² ₀^ r² e^-r/ g (r) \, dr, ₃₈₅₅ is the energy diffusion coefficient, n the number density, L the effective molecular diameter, and = L (1+) with =0. 01 from the CDUFD A5 axiom. Using the gold‑standard experimental ₄ₗ₏ = 3. 8510^12\, s^-1 for water (OH stretching vibration lifetime T₁=260 fs), we calibrate ₃₈₅₅ = (7. 20. 8) 10^-8 m²/s. Complete radial distribution functions for methanol and ethanol are not available; therefore quantitative predictions for these liquids cannot be made at present. The model supersedes the coarse power‑law treatment in CDUFD Phenomenological Series I and provides a direct link between microscopic liquid structure and macroscopic vibrational relaxation.