The plasma in thermodynamic equilibrium has been extensively discussed under the assumption that all particles can be classified as “bound” or “classically free”. Under this assumption simple models led to divergencies of the partition functions of bound and free particles as well as to discrepancies of the predicted level shift. On the basis of a quantum-mechanical treatment we develop an “adapted free-bound approximation” for the eigenstates and a classical approximation for the partition function of the free particles. It is a decisive feature of the analysis that it takes freebound interaction into account. The results produce values for the free-bound limit and the limit of the series continuum. They also remove the divergence difficulties.