A generic Hamiltonian, which incorporates the effect of the orbital contraction on the hopping amplitude between nearest sites, is studied both analytically at the weak coupling limit and numerically at the intermediate and strong coupling regimes for a finite atomic cluster. The effect of the orbital contraction due to hole localization at atomic sites is specified with two coupling parameters V and W (multiplicative and additive contraction terms). The singularity of the vertex part of the two-particle Green's function determines the critical temperature Tc and the relaxation rate Γ(T) of the order parameter at temperature above Tc . Unlike the case in conventional BCS superconductors, Γ has a non-zero imaginary part which may influence the fluctuation conductivity of the superconductor above Tc . We compute the ground state energy as a function of the particle number and magnetic flux through the cluster, and show the existence of the parity gap A appearing at the range of system parameters consistent with the appearance of the Cooper instability. Numeric calculation of the Hubbard model (with U > 0) at arbitrary occupation does not show any sign of superconductivity in a small cluster.