  The Self Consistent Random Phase Approximation (SCRPA) and a Direct
Analytical (DA) method are proposed to solve the Extended Hubbard Model in 1D.
We have considered an Extended Hubbard Model (EHM) including on-site and
off-site interactions for closed chains in one dimension with periodic boundary
conditions. The comparison of the SCRPA results with ones obtained by a Direct
Analytical approach shows that the SCRPA treats the problem of these closed
chains with a rigorous manner. The analysis of the nearest-neighbour repulsion
effect on the dynamics of our closed chains shows that this repulsive
interaction between the electrons of the neighbouring atoms induces
supplementary conductivity, since, the SCRPA energy gap vanishes when these
closed chains are governed by a strong repulsive on-site interaction and
intermediate nearest-neighbour repulsion.
