  Deuterium (D) is one of the light elements created in the big bang. As the
Galaxy evolves, the D/H abundance in the interstellar medium (ISM) decreases
from its primordial value due to "astration". However, the observed gas-phase
D/H abundances of some interstellar sightlines are substantially lower than the
expected reduction by astration alone. The missing D could have been depleted
onto polycyclic aromatic hydrocarbon (PAH) molecules which are ubiquitous and
abundant in interstellar regions. To quantitatively explore the hypothesis of
PAHs as a possible reservoir of interstellar D, we compute quantum-chemically
the infrared vibrational spectra of mono-deuterated PAHs (and their cations) of
various sizes. We find that, as expected, when H in PAHs is replaced by D, the
C-H stretching and bending modes at 3.3, 8.6 and 11.3 $\mu$m respectively shift
to longer wavelengths at $\sim$4.4, 11.4 and 15.4 $\mu$m by a factor of
$\sim$$\sqrt{13/7}$, the difference in reduced mass between the C-H and C-D
oscillators. We derive from the computed spectra the mean intrinsic strengths
of the 3.3 $\mu$m C-H stretch and 4.4 $\mu$m C--D stretch to be $\langle
A_{3.3} \rangle \sim 13.4$ km/mol and $\langle A_{4.4} \rangle \sim 7.4$ km/mol
for neutral deuterated PAHs which would dominate the interstellar 3.3 and 4.4
$\mu$m emission. By comparing the computationally-derived mean ratio of
$\langle A_{4.4}/A_{3.3} \rangle \sim 0.56$ for neutral PAHs with the mean
ratio of the observed intensities of $\langle (I_{4.4}/I_{3.3})_{\rm obs}
\rangle \sim 0.019$, we estimate the degree of deuteration (i.e., the fraction
of peripheral atoms attached to C in the form of D) to be ~2.4\%, corresponding
to a D-enrichment of a factor of ~1200 with respect to the interstellar D/H
abundance.
