Three-body Coulomb bound states

We reexamine the binding energies of three-particle systems containing two electrons and one positive particle of mass M, in an attempt to understand the approximate proportionality of the ${}_{}{}^1{}_{}{}^{}\mathrm{S}_{}^{\mathrm{e}}{}_{}{}^{}$ ground-state binding energies to the reduced masses, as pointed out by Botero and Greene [Phys. Rev. 56, 1366 (1986)]. Our investigation involves a careful evaluation of the contribution to the energy of the mass-polarization term $H_{\mathrm{mp}=\mathrm{-}(2/\mathrm{M}){\nabla }_1\mathrm{\cdot }{\nabla }_2}$ which is the only part of the Hamiltonian not scaling like the reduced mass. It does not appear that any fundamental principle is involved, since we find merely that the mass polarization decreases somewhat as the mass of the positive particle is reduced below the proton mass. In the case of the excited ${}_{}{}^3{}_{}{}^{}\mathrm{P}_{}^{\mathrm{e}}{}_{}{}^{}$ state, this reduction is not sufficient to allow binding when M approaches the electron mass. As a by-product, we compute accurately some properties of the recently observed negative muonium ion ($e^{\mathrm{-}}$${\mu }^{+}$$e^{\mathrm{-}}$).