  In most theoretical descriptions of collective strong coupling of organic
molecules to a cavity mode, the molecules are modeled as simple two-level
systems. This picture fails to describe the rich structure provided by their
internal rovibrational (nuclear) degrees of freedom. We investigate a
first-principles model that fully takes into account both electronic and
nuclear degrees of freedom, allowing an exploration of the phenomenon of strong
coupling from an entirely new perspective. First, we demonstrate the
limitations of applicability of the Born-Oppenheimer approximation in strongly
coupled molecule-cavity structures. For the case of two molecules, we also show
how dark states, which within the two-level picture are effectively decoupled
from the cavity, are indeed affected by the formation of collective strong
coupling. Finally, we discuss ground-state modifications in the ultra-strong
coupling regime and show that some molecular observables are affected by the
collective coupling strength, while others only depend on the single-molecule
coupling constant.
