  Inspired by the formulation of quantum-electrodynamical time-dependent
density functional theory (QED-TDDFT) by Rubio and coworkers, we propose an
implementation that uses dimensionless amplitudes for describing the photonic
contributions to QED-TDDFT electron-photon eigenstates. The leads to a
symmetric QED-TDDFT coupling matrix, which is expected to facilitate the future
development of analytic derivatives. Through a Gaussian atomic basis
implementation of the QED-TDDFT method, we examined the effect of dipole
self-energy, rotating wave approximation, and the Tamm-Dancoff approximation on
the QED-TDDFT eigenstates of model compounds (ethene, formaldehyde, and
benzaldehyde) in an optical cavity. We highlight, in the strong coupling
regime, the role of higher-energy and off-resonance excited states with large
transition dipole moments in the direction of the photonic field, which are
automatically accounted for in our QED-TDDFT calculations and might
substantially affect the energy and composition of polaritons associated with
lower-energy electronic states.
