Hartree-Fock equations for transition matrices are formulated as an extension of the ordinary theory of atomic spectra. These equations form a hierarchy which may be subjected to various truncations. Some of the truncations are identified as equivalent to different forms of many-body theories (random-phase approximation, time-dependent Hartree-Fock, many-body perturbation theory, etc.). Thereby we connect the modern many-body treatments more explicitly with the Condon-Shortley-Racah tradition of spectral theory and provide physical interpretations of various approximations. Spin and angular variables are factored out at the outset by angular-momentum techniques. The problem then takes the form of a system of integro-differential equations for radial wave functions, which affords conceptual and computational advantages. The correlations that are characteristically studied by many-body techniques are seen to be confined to short radial distances and could accordingly be treated by $R$-matrix procedures.

• Received 30 June 1975

DOI:https://doi.org/10.1103/PhysRevA.13.263