  Orhorhombic $\alpha$-MoO$_3$ is a layered oxide with various applications and
with excellent potential to be exfoliated as a 2D ultrathin film or monolayer.
In this paper, we present a first-principles computational study of its
vibrational properties. Our focus is on the zone center modes which can be
measured by a combination of infared and Raman spectroscopy. The polarization
dependent spectra are simulated. Calculations are also performed for a
monolayer form in which "double layers" of Mo$_2$O$_6$ which are weakly van der
Waals bonded in the $\alpha$-structure are isolated. Shift in phonon
frequencies are analyzed.
