Theoretical modeling of the structural, electronic, optical and dynamical properties of many-electron systems, such as organic molecules and inorganic clusters, is of utmost importance in chemistry, physics and nanoscience. In the last twenty-years an impressive development of theoretical methods and algorithms as well as computer architectures has been achieved, and nowadays first-principles simulations of systems with one hundred of atoms are possible on standard workstations. Current research in theoretical methods development aims to solve the many-body electronic problem with increasing accuracy and/or decreasing computational cost. In this work we will consider a finite system with N electrons (eg an organic molecule or an inorganic cluster) and we aim to efficiently solve the equation Tˆ þ Vˆext þ Vˆ ee