Electronic structures of the tetradymites, Bi$_2$Te$_3$, Bi$_2$Te$_2$Se and Bi$_2$Se$_3$, containing various dopants and vacancies, are studied using the first principles calculations methods. We focus on the possibility of formation of the resonant levels (RL), confirming the formation of the RL by Sn in Bi$_2$Te$_3$, and predicting similar behavior of Sn in Bi$_2$Te$_2$Se and Bi$_2$Se$_3$. Vacancies, which are likely present on the chalcogen atoms in the real samples of Bi$_2$Te$_2$Se and Bi$_2$Se$_3$, are also studied and their charged donor and resonant behavior is discussed. Doping of the vacancy-containing materials with regular acceptors, like Ca or Mg, is shown to compensate the donor effect of vacancies, and $n-p$ crossover, while increasing the dopant concentration, is observed. We verify, that RL on Sn is not disturbed by the chalcogen vacancies in Bi$_2$Te$_2$Se and Bi$_2$Se$_3$, and for the Sn-doped materials with Se or Te vacancies, double-doping, instead of heavy doping with Sn, is suggested as an effective way of reaching the resonant level. This should help to avoid the smearing of the RL, which was a possible reason for an earlier unsuccessful experimental observation of the influence of the RL on thermoelectric properties of Sn doped Bi$_2$Te$_2$Se. Finally we show, that Al and Ga are possible new resonant impurities in the tetradymites, hoping that it will stimulate further experimental studies.
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