A Hookean model of a three-body problem for particles with arbitrary masses and charges where two of them interact with each other through a Coulomb potential and with the third through a harmonic potential is presented. It is shown that a condition relating the masses to the harmonic coupling constants must be satisfied in order to render this problem separable. A general exact analytic solution written in terms of the relative interparticle coordinates is given as well as general expressions for the total and binding energies of this three-body system. We apply these results to examine electronic, muonic, antiprotonic, and pionic families of non-Born-Oppenheimer Hookean systems. The first contains the atoms or atomic ions: ${\mathrm{Ps}}^{-}\left(e^{+}{e}^{-}{e}^{-}\right)$, ${\mathrm{H}}^{-}\left(p^{+}{e}^{-}{e}^{-}\right)$, ${\mathrm{D}}^{-}\left(d^{+}{e}^{-}{e}^{-}\right)$, ${\mathrm{T}}^{-}\left(p^{+}{e}^{-}{e}^{-}\right)$, ${}^4\mathrm{He}\left(h{e}^{+2}{e}^{-}{e}^{-}\right)$, and the following molecular ions: ${\mathrm{Ps}}_2^{+}\left(e^{-}{e}^{+}{e}^{+}\right)$, ${\mathrm{H}}_2^{+}\left(e^{-}{p}^{+}{p}^{+}\right)$, $\mathrm{H}{\mathrm{D}}^{+}\left(e^{-}{d}^{+}{p}^{+}\right)$, ${\mathrm{HT}}^{+}\left(e^{-}{t}^{+}{p}^{+}\right)$, ${\mathrm{DT}}^{+}\left(e^{-}{d}^{+}{t}^{+}\right)$, ${\mathrm{D}}_2^{+}\left(e^{-}{d}^{+}{d}^{+}\right)$, ${\mathrm{T}}_2^{+}\left(e^{-}{t}^{+}{t}^{+}\right)$. The muonic and antiprotonic families are similar to the electronic ones except that the species are formed replacing $e^{-}$ by ${\mu }^{-}$ or $p^{-}$. The pionic family comprises exotic atoms containing at least one pion. We also apply these results to two-electron three-dimensional spherical quantum dots and for these systems we examine the effect of electronic correlation, particularly on the singlet-triplet transitions and on the collective motion of the electrons and center of mass leading to “floppy”dynamics.

• Received 13 September 2005

DOI:https://doi.org/10.1103/PhysRevA.74.042504