  A necessary condition for superconductivity (SC) driven by electron
correlations is that electron-electron (e-e) interactions enhance
superconducting pair-pair correlations, relative to the noninteracting limit.
We report high-precision numerical calculations of the ground state within the
frustrated two-dimensional (2D) Hubbard Hamiltonian for a wide range of carrier
concentration $\rho$ (0 < $\rho$ <1) per site. We find that long range
superconducting pair correlations are enhanced only for $\rho$ $\simeq$ 0.5. At
all other fillings e-e interactions mostly suppress pair correlations. The
enhancement of pair correlations is driven by the strong tendency to local
singlet bond formation and spin gap (SG) in $\rho$ = 0.5, in lattices with
quantum fluctuation. We also report determinantal quantum Monte Carlo (DQMC)
calculations that are in strong agreement with our ground state results. Our
work provides a key ingredient to the mechanism of SC in the 2D organic
charge-transfer solids (CTS), and many other unconventional superconductors
with frustrated crystal lattices and $\rho$ $\simeq$ 0.5, while explaining the
absence of SC in structurally related materials with substantially different
$\rho$.
