Lower bounds on the ground-state energy and necessary conditions for the existence of bound states: The few-body problem

Abstract

We consider a positive diatomic molecule consisting of N electrons and two nuclei and obtain some results that exhibit scaling properties of the adiabatic energy ${\mathit{E}}_{\mathrm{ad}}$---the ground-state energy with the nuclei fixed---under changes in the charges, and we obtain lower bounds on ${\mathit{E}}_{\mathrm{ad}}$. In particular, with the use of available numerical results for ${\mathit{E}}_{\mathrm{ad}}$ for a single electron in the Coulomb field of two fixed nuclei with charges ${\mathit{Z}}_{\mathit{A}}$e and ${\mathit{Z}}_{\mathit{B}}$e, we show that composite bound states cannot be formed for N=1 and ${\mathit{Z}}_{\mathit{A}}$+${\mathit{Z}}_{\mathit{B}}$\ensuremath{\ge}3 nor for N=2 and ${\mathit{Z}}_{\mathit{A}}$=${\mathit{Z}}_{\mathit{B}}$\ensuremath{\ge}3, results not too far from the true lower limits.