>A generalized ford of a perturbation theory for a (nonrelativistic) quantum mechanical many-body Hamiltonian is given that can be useful for problems of quantum chemistry and other problems with a mcderate number of particles. A modification of Watson's t-operator allows a perturbational development of a many body problem in terms of simpler subproblems. Some examples of such cluster approximations are given. The whole theory is formulated in terms of state vectors and there are no assumptions about the degeneracy of the Hamiltonian. The given approximations to the level shift transformation are both unitary and Lie functions in every order of the development. Every step of the calcalation can be done in the domain of a Lie algebra and it is recomdended that full advantage be taken of this fact in practical calculations. The use of the diagram technique is avoided and there are no explicit partial summations of the perturbation series, but similar results are gained by the systematic use of the unitary conditions, of the Lie character, and of a modified t-operator. (auth)