number of general aspects are discussed in detail (Sec. \[sec:sft\]). To demonstrate its usefulness, two applications will be considered for the single-band Hubbard model: In Sec. \[sec:dmft\] it is shown that the dynamical mean-field theory (DMFT) [@GKKR96] can be recovered within the self-energy-functional approach, namely by choosing a decoupled set of impurity Anderson models as a reference system. The DMFT generally requires the treatment of a quantum-impurity problem with an infinite number of degrees of freedom ($n_{\rm s} = \infty$). In Sec. \[sec:ed\] a new approximation is discussed which is based on an impurity model with a [*finite*]{} number of degrees of freedom only and which approaches the DMFT for $n_{\rm s} = \infty$. The method is closely related to the exact-diagonalization approach (ED). [@CK94; @SRKR94] Opposed to the ED, however, thermodynamical consistency is guaranteed at any stage of the approximation. New approaches beyond the mean-field level will be discussed elsewhere. The conclusions and an outlook are given in Sec. \[sec:con\].

Self-energy-functional approach {#sec:sft}
===============================

Consider a general Hamiltonian $H = H_0({\bf t}) + H_1 ({\bf U})$ with one-particle (“hopping”) parameters ${\bf t}$ and two-particle interaction parameters ${\bf U}$: $$H = \sum_{\alpha\beta} t_{\alpha\beta} 
  c_{\alpha}^\dagger c_{\beta}
  + \frac{1}{2} 
  \sum_{\alpha\beta\gamma\delta} U_{\alpha\beta\delta\gamma}
  c_{\alpha}^\dagger 