Classical derived functors as fully faithful embeddings

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Given associative unital algebras A and B and a complex T of B-A-bimodules, we give necessary and sufficient conditions for the total derived functors, ₐ(T,?):(A)(B) and?:(B)(A), to be fully faithful. We also give criteria for these functors to be one of the fully faithful functors appearing in a recollement of derived categories. In the case when T is just a B-A-bimodule, we connect the results with (infinite dimensional) tilting theory and show that some open question on the fully faithfulness of ₐ(T,?) is related to the classical Wakamatsu tilting problem.