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

Abstract

The present manuscript revisits one of the earliest approaches to treating molecular systems within the Schr\"odinger 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.