  The infrared (IR) emission of polycyclic aromatic hydrocarbons (PAHs)
permeates our universe; astronomers have detected the IR signatures of PAHs
around many interstellar objects. The IR emission of interstellar PAHs differs
from their emission as seen under conditions on Earth, as they emit through a
collisionless cascade down through their excited vibrational states from high
internal energies. The difficulty in reproducing interstellar conditions in the
laboratory results in a reliance on theoretical techniques. However, the size
and complexity of PAHs requires careful consideration when producing the
theoretical spectra. In this work we outline the theoretical methods necessary
to lead to a fully theoretical IR cascade spectra of PAHs including: an
anharmonic second order vibrational perturbation theory (VPT2) treatment; the
inclusion of Fermi resonances through polyads; and the calculation of
anharmonic temperature band shifts and broadenings (including resonances)
through a Wang--Landau approach. We also suggest a simplified scheme to
calculate vibrational emission spectra that retains the essential
characteristics of the full IR cascade treatment and can directly transform low
temperature absorption spectra in IR cascade spectra. Additionally we show that
past astronomical models were in error in assuming a 15 cm$^{-1}$ correction
was needed to account for anharmonic emission effects.
