  Recently, it has been shown that the ground-state energy of a quantum
many-body system can be written in terms of cumulants. In this paper we show
that the energies of excited states can be expressed similarly. These
representations are suitable for various approximation schemes, e.g.,
projection techniques. The explicit use of cumulants ensures size consistency
in all approximations. The theory is then applied to the computation of the
energy bands of a semiconductor with diamond structure. As we focus especially
on the effects of electron correlations, we consider a simplified model and
re-derive results obtained previously by the variational Local Ansatz method.
