Cavity Field-Driven Symmetry Breaking and Modulation of Vibrational Properties: Insights from the Analytical QED-HF Hessian

In this work, we present the analytical derivation and implementation of the quantum electrodynamics Hartree-Fock Hessian. We investigate how electronic strong coupling influences molecular vibrational properties, applying this framework to formaldehyde, <i>p</i>-nitroaniline, and adamantane. Our analysis reveals cavity-induced changes in vibrational frequencies and intensities. Additionally, we show how the quantum electromagnetic field breaks molecular symmetry, activating previously forbidden infrared transitions. Our findings highlight the potential of coupling photonic and electronic degrees of freedom to control and modulate molecular vibrational properties in the ground state.