  Experiments have demonstrated that vibrational strong coupling between
molecular vibrations and light modes can significantly change molecular
properties, such as ground-state reactivity. Theoretical studies towards the
origin of this exciting observation can roughly be divided in two categories,
with studies based on Hamiltonians that simply couple a molecule to a cavity
mode via its ground-state dipole moment on the one hand, and on the other hand
ab initio calculations that self-consistently include the effect of the cavity
mode on the electronic ground state within the cavity Born-Oppenheimer (CBO)
approximation; these approaches are not equivalent. The CBO approach is more
rigorous, but unfortunately it requires the rewriting of electronic-structure
code, and gives little physical insight. In this work, we exploit the relation
between the two approaches and demonstrate on a real molecule (hydrogen
fluoride) that for realistic coupling strengths, we can recover CBO energies
and spectra to high accuracy using only out-of-cavity quantities from standard
electronic-structure calculations. In doing so, we discover what the physical
effects underlying the CBO results are. Our methodology can aid in
incorporating more, possibly important features in models, play a pivotal role
in demystifying CBO results and provide a practical and efficient alternative
to full CBO calculations.
