Analytical Derivative Approaches for Vibro-Polaritonic Structures and Properties I: Formalism and Implementation

Vibro-polaritons are hybrid light-matter states that arise from the strong coupling between the molecular vibrational transitions and the photons in an optical cavity. Developing theoretical and computational methods to describe and predict the unique properties of vibro-polaritons is of great significance for guiding the design of new materials and experiments. Here we present the \textit{ab initio} cavity Born-Oppenheimer density functional theory (CBO-DFT), and formulate the analytic energy gradient and Hessian as well as the nuclear and photonic derivatives of dipole and polarizability within the framework of CBO-DFT to efficiently calculate the harmonic vibrational frequencies, infrared absorption and Raman scattering spectra of vibro-polaritons as well as to explore the critical points on the cavity potential energy surface. The implementation of analytic derivatives into the electronic structure package is validated by comparison with the finite-difference method and with other reported computational results. By adopting appropriate exchange-correlation functionals, CBO-DFT can better describe the structure and properties of molecules in the cavity than CBO-Hartree-Fock method. It is expected that CBO-DFT is a useful tool for studying the polaritonic structures and properties.