A quantitative description of complex term spectra requires accurate knowledge of the three leading coefficients in the Z expansions E = E 0 Z 2 + E 1 Z + E 2 + E 3 Z −1 + … of term energies. The screening approximation (described in a previous paper) yields exact values for E 0 and E 1 but only estimates of the remaining expansion coefficients. The present paper is concerned with the coefficient E 2, which may be split up into two physically and mathematically distinct components, E 2′ and E 2″. The intermediate states that figure in the perturbation formulae for E 2′ and E 2″ differ from the initial state in one and two principal quantum numbers respectively. The Hartree-Fock and extended Hartree-Fock (Section II) approximations to E 2 prove to be successive approximations to the component E 2′. E 2 HF, E 2 EHF and E 2′ can all be readily evaluated directly from their perturbation formulae. Examples of such calculations are given and the results are compared with the results of conventional calculations of the same quantities. The component E 2″ can be expressed as a linear combination of values of E 2″ for two-electron states in a manner first described, schematically, by Bacher and Goudsmit in 1934 ( 19). For two-electron states, the coefficient E 2 can be evaluated by existing techniques, and from E 2 one easily obtains E 2″. Using published variational calculations of E 2 for the two-electron states 1s 2 1S , 1s2s 1S and 1s2s 3S , we evaluate E 2″ for the three-electron state 1s 22s 2S . The properties of the coefficient E 2, together with considerations based on the screening approximation, enable one to understand why, in low-lying configurations, term-energy patterns appropriate to high degrees of ionization usually persist down to quite low degrees of ionization. First-order calculations of the matrix elements of one-electron operators are shown to correspond in complexity, not to complete second-order energy calculations as one might expect on general perturbation-theoretic grounds, but to calculations of E 2 EHF. They can either be based on perturbation expansions which involve single generalized sums of products of hydrogenic radial integrals, or they can be made to depend on the solution of ordinary, inhomogeneous, second-order differential equations.