A perturbation study of the NP states of two-electron atoms has been made. In particular, oscillator strength values for the (1*5, 2 i P) and (2 3 S, 2 3 P) transitions are obtained. The 2 i P and 2 3 P states are studied through ninth and tenth order, respectively. In addition, the N P and N Z Pstates are studied in first order through the 10P member of the series. Perturbation energy coefficients and other expectation values for several important operators have been computed. Perturbation energy coefficients for the 1*S state (through 25th order) and 2 3 S state (through 17th order) are also reported. Where comparison is possible, these results are in satisfactory agreement with the results obtained from variational calculations by C. L. Pekeris and co-workers. The variational-perturbation method for excited states requires auxiliary conditions on the perturbation wave functions. The condition on the «th-order wave function is derived here. This is a generalization of the first-order condition given by Sinanoglu.

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Oscillator strengths for electric dipole transitions between singly excited S, P, and D states are calculated via Z-dependent perturbation theory. The transition matrix elements through first order are obtained from variational perturbation first-order wave functions. Second-order expansion coefticients for the matrix elements are then estimated with the aid of several approxirnants. These matrix elements, together with accurate transition energies, yield oscillator strengths for transitions between all S-P and P-D transitions between states of two-electron ions with principal quantum number up to 9 and for values of the nuclear charge up to approximately 30.In Z-dependent perturbation theory, the nonrelativistic Hamiltonian for an X-electron atom is given, in charge-M(a, p) y ( y(a)lglq(p) ) m +I =n and where Q = r, cos8) + rzcosOz is the appropriate opera-39 4387

A multiperturbation theory for atomic systems is developed. The theory automatically decouples Rayleigh-Schrodinger perturbation theory into multiperturbation partial differential equations for the 2-through-(n+ 1)-electron components of the nth-order wave function. Variational-perturbation equations for these multiperturbation wave functions are derived together with expressions for the multiperturbation energy-expansion coeAicients. The "bare-nucleus" hydrogenic function is chosen as the zero-order wave function rather than the more customary Hartree-Fock function. With this choice the mulitperturbation wave functions are independent of the nuclear charge and of the total number of electrons in the system, and are thus completely transferable to other systems. The "cluster-expansion" wave function is analyzed in terms of the multiperturbation wave functions and the connection between cluster-type expansions and ordinary perturbation theory is shown in detail.

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