Rajnak and Wybourne's effective operators are generalized to degenerate ground configurations ℓN and ℓ'4ℓ' ℓN+2 as well as by removing the assumption of a Hartree-Fock basis. Explicit calculation of discrete radial integrals and energy denominators permits calculation of the discrete part of E₂in Layzer's Z-expansion. Specific application is made to the carbon isoelectronic sequence. Ground state energies obtained are not as low as Hartree-Fock values, but term structure prediction is better. Continuum contributions are estimated to be fifteen percent of E₂. First and second-order fine structure calculations are too large. Reasons why third-order is expected to be important are given. The possibility of z-expansions with non-hydrogenic basis is considered.