On the emergence of molecular structure from atomic shape in the 1/r2 harmonium model

The formal similarity of the three-body Hamiltonians for helium and the hydrogen molecule ion is used to demonstrate the unfolding of a rotating dumbbell-like proton distribution from a (1s)2-type electron distribution by smooth variation of the particles' masses in the 1/r2 harmonium model. The 1/r2 harmonium is an exactly solvable modification of the harmonium model (also known as Hooke's law atom) where the attraction between different particles is harmonic and the repulsion between the two equal particles is given by a 1/r2 potential. The dumbbell-like molecular structure appears as an expression of increasing spatial correlation due to increasing mass. It gradually appears in the one-density distribution of the two equal particles if their mass exceeds a critical value depending on the mass of the third particle. For large mass of the equal particles, their one-density distribution approaches an asymptotic form derived from the Born-Oppenheimer treatment of H2+ in the 1/r2 harmonium model. Below the critical value, the one density is a spherical, Gaussian-type atomic density distribution with a maximum at the center of mass. The topological transition at the critical value separates molecular structure and atomic shape as two qualitatively different manifestations of spatial structure.