Abstract: Strong light-matter coupling in optical cavities can alter the dynamics of molecular and material systems resulting in polaritonic excitation spectra and modified reaction pathways. For strongly coupled photon modes close in energy to nuclear vibrations the Cavity Born Oppenheimer Approximation (CBOA) in the context of quantum-electrodynamical density functional theory (QEDFT) has been demonstrated to be an appropriate description of the coupled light-matter system. In this study, we present a theory based on CBOA to study the modification of vibrational modes in molecular and insulating solid systems by coupling to low frequency photon modes in optical cavities. Using a mapping of the CBOA energy functional (U) to a finite field enthalpy (F) based on the modern theory of polarization, we can utilize existing ab initio finite electric field methods to calculate the cavity modified phonon spectra in solids. Phonon-polariton modes are then obtained from the nuclear and photonic Hessian matrix. We show that this formalism can be generalized to multiple photon modes.
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