# On the emergence of molecular structure from atomic shape in the 1/r2 harmonium model.
> 1/r²ハーモニウムモデルにおける原子形状から分子構造への出現に関する理論的考察


## 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.

## Bibliographic

- **Authors**: Müller-Herold U
- **Journal**: J Chem Phys
- **Year**: 2006 (2006-01-07)
- **PMID**: [16409022](https://pubmed.ncbi.nlm.nih.gov/16409022/)
- **DOI**: [10.1063/1.2141509](https://doi.org/10.1063/1.2141509)
- **Study type**: other
- **Delivery route**: not specified
- **Effect reported**: not assessed

## Delivery context

The delivery route is not clearly identifiable from this paper. For hydrogen intake, inhalation is the most efficient route; inhalation, however, carries explosion risk (empirical LFL of 10%; high-concentration devices are not recommended).

## Safety notes

The delivery route is not clearly identifiable from this paper. For hydrogen intake, inhalation is the most efficient route; inhalation, however, carries explosion risk (empirical LFL of 10%; high-concentration devices are not recommended).

See also:
- [Inhalation concentration and LFL / UFL](https://h2-papers.org/en/safety-notes/inhalation-concentration)
- [Consumer Affairs Agency accident cases](https://h2-papers.org/en/safety-notes/accident-cases)

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> **Cite as**: H2 Papers — PMID 16409022. https://h2-papers.org/en/papers/16409022
> **Source**: PubMed PMID [16409022](https://pubmed.ncbi.nlm.nih.gov/16409022/)