Calculated phonon modes, infrared and Raman spectra in orthorhombic $alpha$-MoO$_3$ and monolayer MoO$_3$

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.