  The effective independent-particle (mean-field) approximation of the Hubbard
Hamiltonian is described in a many-body basis to develop a formal comparison
with the exact diagonalization of the full Hubbard model, using small atomic
chain as test systems. This allows for the development of an intuitive
understanding of the shortcomings of the mean-field approximation and of how
critical correlation effects are missed in this popular approach. The
description in the many-body basis highlights a potential ambiguity related to
the definition of the density of states. Specifically, satellite peaks are
shown to emerge in the mean-field approximation, in departure from the common
belief that they characterize correlation effects. The scheme emphasizes the
importance of correlation and how different many-body corrections can improve
the mean-field description. The pedagogical treatment is expected to make it
possible for researchers to acquire an improved understanding of many-body
effects as found in various areas related to electronic properties of molecules
and solids, which is highly relevant to current efforts in quantum information
and quantum computing.
