The Mystery of Covalent Bonding in H2

The first question a curious student in a chemistry class may pose to the teacher is the following:

• What binds two neutral H atoms together into a H2 molecule?   

The answer will be covalent chemical bonding as somehow formed by "sharing of electrons". The student would certainly expect the teacher to be able to give a theoretical explanation of how the bond is established in more detail. Or at least chatGPT.

ChatGPT informs that the state of the art explanation of the covalent bonding of two H atoms into one H2 molecule is attributed to Heitler-London (1927) →Hellmann (1933) → Ruedenberg (1962) → Nordholm (2020): 

• Kinetic-energy lowering by delocalization.
• Electrons delocalize across both nuclei → wavefunction spreads out → kinetic energy drops, stabilizing the molecule.
• Bond is inherently a quantum delocalization effect.
• Conceptually deep and physically realistic.
• Most physically fundamental explanation in modern quantum chemistry.

Nordholm gives a different picture:

• No general and unique understanding of the mechanism of covalent bonding in physical terms is provided by current computational methods or by a consensus among experts. 
• The interatomic electron motion is directly related to the delocalization taking place when atomic basis functions are combined into molecular orbitals. Calculations can then identify ground states ...with delocalization. 
• ....bond energy is a sum of repulsive localization and more strongly attractive delocalization energies. 

We see that the rosy picture of chatGPT is not really followed up by Nordholm. Apparently delocalisation of electrons over both nuclei is presented as being central, but the student would probably be puzzled since localisation of electrons between nuclei would be understood as central in apparent contradiction with delocalisation.

RealQM gives a different explanation described in article and book chap 31 with iconic picture below showing two electronic wave functions with separate support in two half-spaces meeting at a separating plane midway between nuclei. The essence is localisation of charge density between nuclei without increase of kinetic energy because charge densities are not forced to approach zero between nuclei. A student would meet this information with: Aha! And you?