The behavior of the Colle-Salvetti (CS) correlation functional based on the Hartree-Fock (HF) second-order density matrix is investigated in the case of the He atom. The analysis of the correlation hole and energy contributions show that correlation effects are not taken into account appropriately due to the missing kinetic correlation. The CS final simplified energy expression also has some problems. To address these problems, we have constructed a new correlation functional based on the HF second-order density matrix including the effects of not only electron-electron interaction but also the kinetic energy by using an adiabatic connection formula. In addition, correlation effects for opposite and parallel spins are treated independently. This functional reproduces accurate correlation energies for H-Ar atoms. Combining it with the Becke 1988 exchange functional, we get reasonable atomization energies for the G2 set.