In this work, we illustrate the recently introduced concept of the cavity
Born-Oppenheimer approximation for correlated electron-nuclear-photon problems
in detail. We demonstrate how an expansion in terms of conditional electronic
and photon-nuclear wave functions accurately describes eigenstates of strongly
correlated light-matter systems. For a GaAs quantum ring model in resonance
with a photon mode we highlight how the ground-state electronic
potential-energy surface changes the usual harmonic potential of the free
photon mode to a dressed mode with a double-well structure. This change is
accompanied by a splitting of the electronic ground-state density. For a model
where the photon mode is in resonance with a vibrational transition, we observe
in the excited-state electronic potential-energy surface a splitting from a
single minimum to a double minimum. Furthermore, for a time-dependent setup, we
show how the dynamics in correlated light-matter systems can be understood in
terms of population transfer between potential energy surfaces. This work at
the interface of quantum chemistry and quantum optics paves the way for the
full ab-initio description of matter-photon systems
