Dispersion models describing coupled systems in doped crystals. I. A generalized model for bound–bound and bound–free coupling

State-of-the-art descriptions of coupled phonons rely on many-body Green’s functions, which, in principle, capture both single-particle properties and mutual interactions. Although this framework offers detailed physical insight, its practical use in dispersion modelling is limited by the need for simplifying assumptions, empirical parameters, and substantial numerical complexity. As a result, the obtained response functions are typically valid only in restricted spectral regions and cannot be used directly as general dispersion formulas. In this work, we develop a generalized classical model capable of consistently describing coupling between different components of the system. We first reformulate the classical equations of motion using the factorized harmonic-oscillator approach and show how Fano-type quantum interference can be represented within a Lorentz model with complex amplitude, leading to the established asymmetric-peak formalism. We then identify a fundamental limitation of asymmetric-peak model—its inability to account for coupling between bound and free particles—and resolve it by introducing a physically consistent extension that treats both types of couplings on equal footing. Further generalizations include replacing the Lorentzian broadening with an arbitrary symmetric profile and enabling coupling between components represented by absorption bands. Together, these advances provide a versatile framework for constructing dispersion models of optical constants.