  A generic Hamiltonian, which incorporates the effect of the orbital
contraction on the hopping amplitude between the nearest sites, is studied both
analytically at the weak coupling limit and numerically at the intermediate and
strong coupling regimes for 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 Gamma(T) of the
order parameter at temperature above Tc. Unlike in conventional BCS
superconductors, Gamma has a non-zero imaginary part which may influence the
fluctuation conductivity of 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 Delta appearing at the range
of system parameters consistent with the appearance of Cooper instability.
Numeric calculation of the Hubbard model (with U>0) at arbitrary occupation
does not show any sign of superconductivity in small cluster.
