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The configuration-space Hamiltonian for a many-electron atom is derived clearing up some of the ambiguities concerning the projection operators which must occur in this Hamiltonian. In the process, numerical calculations which have been previously performed are justified. The elementary two-electron interaction in the presence of many other electrons is also discussed.

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A discussion of the problem of the construction of a configuration-space Hamiltonian describing a many-electron atom with relativistic effects included is given. It appears that the total three-body energy in such an atom must, be small if the problem of the construction of this Hamiltonian is to be simply solvable. To that end, a relativistic three-body potential is constructed for a three-electron system. It has novel terms which arise from a pathology of the two-body Hamiltonian previously noted. It is shown that a speculative generalization to the many-electron system results in a total three-body energy which may be many rydbergs but which is still a small perturbation on the total result.

The wave function of a two-level atom driven by a low-frequency laser pulse is obtained in analytical form and is used to determine the power spectrum of the dipole moment of the atom for different initial atomic states. The solution goes over some known approximate solutions and gives the energy of the two dressed states, the position of the hyper-Raman lines, the amplitudes both of the hyper-Raman lines and of the odd harmonics, and the position of the cutoff. The form of the hyper-Raman spectrum results independently of the initial atomic state and shows an onset and a cutoff of the emission.

The lowest-order configuration-space Hamiltonian {CSH) for a heavy atom is constructed from quantum electrodynamics by a variational procedure. A variational potential function 0 is introduced which in effect allows some freedom in the choice of the definition of the difference between electrons and positrons. The optimization of 0 results in a nonlinear equation from which it is shown that 0 is probably not small for relativistic electrons. The procedure results in a CSH which contains a new two-body interaction which is relativistic in origin and which is apparently of the same order of magnitude as the Breit interaction when acting between relativistic electrons.

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