Hierarchies of equations for the correlation energy and kinetic energy correction functionals are reviewed. Truncation of the hierarchies at different orders leads to explicit relations for the correlation energy and its kinetic component which are expanded in terms of powers of the moments of the electron density. The dependence of the correlation energy on the adiabatic constant is presented for several atoms and ions. For some two- and four-electron systems the correlation energy and its derivative are compared with the exact results, and with the local density and the generalized gradient values.