  A detailed first-principle study has been performed to evaluate the
electronic and optical properties of single-layer (SL) transition metal
dichalcogenides (TMDCs) (MX${}_2$; M= transition metal such as Mo, W and X= S,
Se, Te), in the presence of vacancy defects (VDs). Defects usually play an
important role in tailoring electronic, optical, and magnetic properties of
semiconductors. We consider three types of VDs in SL TMDCs i) $X$-vacancy,
$X_{2}$-vacancy, and iii) $M$-vacancy. We show that VDs lead to localized
defect states (LDS) in the band structure, which in turn give rise to sharp
transitions in in-plane and out-of-plane optical susceptibilities,
$\chi_{\parallel}$ and $\chi_{\perp}$. The effects of spin orbit coupling (SOC)
are also considered. We find that SOC splitting in LDS is directly related to
the atomic number of the transition metal atoms. Apart from electronic and
optical properties we also find magnetic signatures (local magnetic moment of
$\sim\mu_{B}$) in MoSe$_{2}$ in the presence of Mo vacancy, which breaks the
time reversal symmetry and therefore lifts the Kramers degeneracy. We show that
a simple qualitative tight binding model (TBM), involving only the hopping
between atoms surrounding the vacancy with an on-site SOC term, is sufficient
to capture the essential features of LDS. In addition, the existence of the LDS
can be understood from the solution of the 2D Dirac Hamiltonian by employing
infinite mass boundary conditions. In order to provide a clear description of
the optical absorption spectra, we use group theory to derive the optical
selection rules between LDS for both $\chi_{\parallel}$ and $\chi_{\perp}$.
