  This work presents an in-depth investigation of the properties of complexes
composed of hydrogen, silicon or oxygen with carbon, which are major
unintentional impurities in undoped GaN. This manuscript is a complement to our
previous work on carbon--carbon and carbon-vacancy complexes. We have employed
a first-principles method using Heyd-Scuseria-Ernzerhof hybrid functionals
within the framework of generalized Kohn-Sham density functional theory. Two
H--C, four Si--C and five O--C complexes in different charge states have been
considered. After full geometry relaxations, formation energies, binding
energies and both thermal and optical transition levels were obtained. The
calculated energy levels have been systematically compared with the
experimentally observed carbon related trap levels. Furthermore, we computed
vibrational frequencies for selected defect complexes and defect concentrations
were estimated in the low, mid and high carbon doping scenarios considering two
different cases where electrically active defects: (a) only carbon and
vacancies and (b) not only carbon and vacancies but also hydrogen, silicon and
oxygen. We confirmed that $\mathrm{C_N}$ is a dominant acceptor in GaN. In
addition to it, substantial amount of $\mathrm{Si_{Ga}-C_N}$ complex exists in
a neutral form. This complex is a likely candidate for unknown form of carbon
observed in undoped $n$-type GaN.
