Adiabatic theory for the doubly excited asymmetric states of the helium atom

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Abstract
In the asymmetric (or planetary) doubly excited states of the helium atom, one of the electrons is excited much more than the other. The motion of the electrons is strongly correlated: both of them reside mostly on the same side of the atomic nucleus. An adiabatic theory for such states is based on the approximate separation of rapid and slow motion. The rapid motion is that of the inner electron along its elliptic orbit. The parameters of the orbit (eccentricity and the aphelion vector) slowly evolve in time. The other slow motion is the radial vibrations of the outer electron. The effective Hamiltonian is constructed as the average of the exact Hamiltonian over the rapid motion. In the quadratic approximation two types of slow motion are separated and reduced to two harmonic oscillators. The unexpected feature is that the ratio of the related frequencies is very simple: 1:2. The ratio is changed when the outer electron is replaced by the particle with an arbitrary mass. The slow motions are quantized and the series of the energy levels are obtained. In the case of infinite mass of the outer particle, the potential curves of the quasimolecule are calculated. The present purely analytic results are compared with the numerical data.

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