  We present the exact solution of the one-dimensional extended Hubbard model
in the atomic limit within the Green's function and equation of motion
formalism. We provide a comprehensive and systematic analysis of the model by
considering all the relevant response and correlation functions as well as
thermodynamic quantities in the whole parameter space. At zero temperature we
identify four phases in the plane (U,n) [U is the onsite potential and n is the
filling] and relative phase transitions as well as different types of charge
ordering. These features are endorsed by investigating at T=0 the chemical
potential and pertinent local correlators, the particle and double occupancy
correlation functions, the entropy, and by studying the behavior in the limit T
going to zero of the charge and spin susceptibilities. A detailed study of the
thermodynamic quantities is also presented at finite temperature. This study
evidences that a finite-range order persists for a wide range of the
temperature, as shown by the behavior of the correlation functions and by the
two-peak structure exhibited by the charge susceptibility and by the entropy.
Moreover, the equation of motion formalism, together with the use of composite
operators, allows us to exactly determine the set of elementary excitations. As
a result, the density of states can be determined exactly and a detailed
analysis of the specific heat allows for identifying the excitations and for
ascribing its two-peak structure to a redistribution of the charge density.
