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

Abstract

This theoretical study exploits the formal resemblance between three-body Hamiltonians for helium and the hydrogen molecule ion within the 1/r² harmonium framework—an exactly solvable system combining harmonic inter-particle attraction with a 1/r² repulsion between identical particles. By continuously varying particle masses, the study demonstrates how a dumbbell-shaped proton distribution unfolds from a spherically symmetric (1s)²-type electron density. The one-density distribution of the two equal-mass particles develops a dumbbell-like molecular geometry when their mass surpasses a critical threshold that depends on the third particle's mass. Below this threshold, the density remains a spherical, Gaussian-type distribution centered at the center of mass. For large equal-particle masses, the distribution converges toward an asymptotic form consistent with Born-Oppenheimer treatment of H₂⁺ in the same model. The topological transition at the critical mass value constitutes a qualitative boundary separating atomic and molecular spatial organization.

Mechanism

When the mass of the two identical particles exceeds a model-dependent critical value, increasing spatial correlation drives a topological transition in the one-density distribution from a spherically symmetric Gaussian form to a dumbbell-shaped molecular geometry.