  As pioneering experiments have shown, strong vibrational coupling between
molecular vibrations and light modes in an optical cavity can significantly
alter molecular properties and even affect chemical reactivity. However, the
current theoretical description is limited and far from complete. To explore
the origin of this exciting observation, we investigate how the molecular
structure changes under strong light-matter coupling using an ab-initio method
based on the cavity Born-Oppenheimer Hartree-Fock ansatz. By optimizing H$_2$O
and H$_2$O$_2$ resonantly coupled to cavity modes, we study the importance of
reorientation and geometric relaxation. In addition, we show that the inclusion
of one or two cavity modes can change the observed results. On the basis of our
findings, we derive a simple concept to estimate the effect of the cavity
interaction on the molecular geometry using the molecular polarizability and
the dipole moments.
