Deuterated Polycyclic Aromatic Hydrocarbons in the Interstellar Medium: The C--D Band Strengths of Multi-Deuterated Species

Observationally, the interstellar gas-phase abundance of deuterium (D) is considerably depleted and the missing D atoms are often postulated to have been locked up into carbonaceous solids and polycyclic aromatic hydrocarbon (PAH) molecules. An accurate knowledge of the fractional amount of D (relative to H) tied up in carbon dust and PAHs has important cosmological implications since D originated exclusively from the Big Bang and the present-day D abundance, after accounting for the astration it has experienced during the Galactic evolution, provides essential clues to the primordial nucleosynthesis and the cosmological parameters. To quantitatively explore the extent to which PAHs could possibly accommodate the observed D depletion, we have previously quantum-chemically computed the infrared vibrational spectra of mono-deuterated PAHs and derived the mean intrinsic band strengths of the 3.3 $mu$m C--H stretch (A$_{3.3}$) and the 4.4 $mu$m C--D stretch (A$_{4.4}$). Here we extend our previous work to multi-deuterated PAH species of different deuterations, sizes and structures. We find that both the intrinsic band strengths A$_{3.3}$ and A$_{4.4}$ and their ratios A$_{4.4}$/A$_{3.3}$ not only show little variations among PAHs of different deuterations, sizes and structures, they are also closely similar to that of mono-deuterated PAHs. Therefore, a PAH deuteration level (i.e., the fraction of peripheral atoms attached to C atoms in the form of D) of ~2.4% previously estimated from the observed 4.4 $mu$m to 3.3 $mu$m band ratio based on the A$_{4.4}$/A$_{3.3}$ ratio of mono-deuterated PAHs is robust.