Lecture Y-6: A Computational RealQM Model of the Hydrogen Molecule (Numerical Evidence for Emergent Covalent Bonding)

A conceptually correct model for covalent bonding, suitable for teaching at the undergraduate general chemistry level, is presented. The quantum mechanical basis for the stability of the covalent bond can be derived from a qualitative discussion of the uncertainty principle, which relates the electron’s spatial confinement to its kinetic energy. The qualitative discussion is backed by quantitative data obtained from the analytical quantum mechanical solution of the H2+ molecule. In addition, the stability of molecules is used as a tangible background against which the inferences emerging from the philosophical debate between classical determinism and the Copenhagen interpretation can be tested.A conceptually correct model for covalent bonding, suitable for teaching at the undergraduate general chemistry level, is presented. The quantum mechanical basis for the stability of the covalent bond can be derived from a qualitative discussion of the uncertainty principle, which relates the electron’s spatial confinement to its kinetic energy. The qualitative discussion is backed by quantitative data obtained from the analytical quantum mechanical solution of the H2+ molecule. In addition, the stability of molecules is used as a tangible background against which the inferences emerging from the philosophical debate between classical determinism and the Copenhagen interpretation can be tested.

The Chemical EducatorMar 16, 20065 citations

Screening in the Heitler-London Model: Revisiting the Bonding and Antibonding States of the Hydrogen Molecule

Washington P. da Silva, Daniel Vieira, Jonas Maziero +1 more

The present manuscript revisits one of the earliest approaches to treating molecular systems within the Schrödinger formalism of quantum mechanics: the Heitler-London (HL) model. Originally proposed in 1927 and based on a linear combination of atomic orbitals, the HL model provided a foundational description of covalent bonds and has served as the basis for numerous variational methods. Focusing on the hydrogen molecule, we begin by revisiting the analytical calculations of the original HL model, from which the qualitative physics of bonding and antibonding states can be obtained. Subsequently, we propose including electronic screening effects directly in the original HL wave function. We then compare our proposal with variational quantum Monte Carlo (VQMC) calculations, whose trial wave function allows us to optimize the electronic screening potential as a function of the inter-proton distance. We obtain the bond length, binding energy, and vibrational frequency of the H $_2$ molecule. Beyond revisiting this foundational approach in quantum mechanics, our proposal can serve as improved input for constructing new, but still analytically simple, variational wave functions to describe dissociation or bond formation.The present manuscript revisits one of the earliest approaches to treating molecular systems within the Schrödinger formalism of quantum mechanics: the Heitler-London (HL) model. Originally proposed in 1927 and based on a linear combination of atomic orbitals, the HL model provided a foundational description of covalent bonds and has served as the basis for numerous variational methods. Focusing on the hydrogen molecule, we begin by revisiting the analytical calculations of the original HL model, from which the qualitative physics of bonding and antibonding states can be obtained. Subsequently, we propose including electronic screening effects directly in the original HL wave function. We then compare our proposal with variational quantum Monte Carlo (VQMC) calculations, whose trial wave function allows us to optimize the electronic screening potential as a function of the inter-proton distance. We obtain the bond length, binding energy, and vibrational frequency of the H$_2$ molecule. Beyond revisiting this foundational approach in quantum mechanics, our proposal can serve as improved input for constructing new, but still analytically simple, variational wave functions to describe dissociation or bond formation.

physics.chem-phCross-listedarXiv:2511.01132Nov 4, 2025

XIII. On the Model of a Triatomic Hydrogen Molecule

Niels BohrJan 1, 19810 citations

Novel binding mechanism for ultra-long range molecules

Vera Bendkowsky, Björn Butscher, J. Nipper +3 more

Molecular bonds can be divided into four primary types: ionic, covalent, van der Waals and hydrogen bonds. At ultralow temperatures a novel binding type emerges paving the way for novel molecules and ultracold chemistry [1,2]. The underlying mechanism for this new type of chemical bond is low-energy electron scattering of Rydberg electrons from polarisable ground state atoms [3]. This quantum scattering process can generate an attractive potential that is able to bind the ground state atom to the Rydberg atom at a well localized position within the Rydberg electron wave function. The resulting giant molecules can have an internuclear separation of several thousand Bohr radii, which places them among the largest known molecules to date. Their binding energies are much smaller than the Kepler frequencies of the Rydberg electrons i.e. the atomic Rydberg electron state is essentially unchanged by the bound ground state atom. Ultracold and dense samples of atoms enable the creation of these molecules via Rydberg excitation. In this paper we present spectroscopic evidence for the vibrational ground and first excited state of a Rubidium dimer Rb(5S)-Rb(nS). We apply a Born-Oppenheimer model to explain the measured binding energies for principal quantum numbers n between 34 and 40 and extract the s-wave scattering length for electron-Rb(5S) scattering in the relevant low energy regime Ekin < 100 meV. We also determine the lifetimes and the polarisabilities of these molecules. P-wave bound states [2], Trimer states [4] as well as bound states for large angular momentum of the Rydberg electron - socalled trilobite molecules [1] - are within reach in the near future and will further refine our conceptual understanding of the chemical bond.Molecular bonds can be divided into four primary types: ionic, covalent, van der Waals and hydrogen bonds. At ultralow temperatures a novel binding type emerges paving the way for novel molecules and ultracold chemistry [1,2]. The underlying mechanism for this new type of chemical bond is low-energy electron scattering of Rydberg electrons from polarisable ground state atoms [3]. This quantum scattering process can generate an attractive potential that is able to bind the ground state atom to the Rydberg atom at a well localized position within the Rydberg electron wave function. The resulting giant molecules can have an internuclear separation of several thousand Bohr radii, which places them among the largest known molecules to date. Their binding energies are much smaller than the Kepler frequencies of the Rydberg electrons i.e. the atomic Rydberg electron state is essentially unchanged by the bound ground state atom. Ultracold and dense samples of atoms enable the creation of these molecules via Rydberg excitation. In this paper we present spectroscopic evidence for the vibrational ground and first excited state of a Rubidium dimer Rb(5S)-Rb(nS). We apply a Born-Oppenheimer model to explain the measured binding energies for principal quantum numbers n between 34 and 40 and extract the s-wave scattering length for electron-Rb(5S) scattering in the relevant low energy regime Ekin < 100 meV. We also determine the lifetimes and the polarisabilities of these molecules. P-wave bound states [2], Trimer states [4] as well as bound states for large angular momentum of the Rydberg electron - socalled trilobite molecules [1] - are within reach in the near future and will further refine our conceptual understanding of the chemical bond.

arXiv:0809.2961Sep 17, 20081 citation

Simulation model for creation of the hydrogen atom

Abhijit Biswas, Krishnan R. S. Mani

Physics EssaysMar 1, 20120 citations

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molecule. Beyond revisiting this foundational approach in quantum mechanics, our proposal can serve as improved input for constructing new, but still analytically simple, variational wave functions to describe dissociation or bond formation.