Three-body Coulomb problem in the dipole approximation

The doubly excited states of the two-electron atom (or ion) are considered in the case when one electron is excited much more than the other. The characteristic separation of the strongly excited electron from the atomic nucleus significantly exceeds that for the weakly excited electron. The electron-electron interaction can be approximated by the dipole term in its multipole expansion. The inner electron velocity is assumed to be much larger than that of the outer electron, which permits assigning a fixed principal quantum number for the inner electron. The same approach is extended to the general Coulomb three-body problem with the particles of arbitrary mass and charge. For the three-body states with symmetry S and ${\mathit{P}}^{\mathit{e}}$, the quantum problem reduces to the three-term recursion relations, which means that the system is effectively one dimensional. The slow evolution of the classical particle trajectories under the influence of interaction is described. It is shown that, depending on the parameters of the system, two types of librations or rotation can be realized. The semiclassical quantization rules (analogous to the well-known Bohr-Sommerfeld rules) are deduced. For S and ${\mathit{P}}^{\mathit{e}}$ states they differ only by the substitution of a half-integer quantum number by an integer. Particularly simple results are obtained in the harmonic approximation, which is valid in the vicinity of the effective-potential curve extreme.