The Mystery of the Covalent Bond of the H2 Molecule Resolved!?

The H2 molecule consisting of two Hydrogen H atoms joined by a covalent chemical bond is the simplest and most abundant molecule in the Universe, but the physical nature of the bond is still today after 100 years of quantum mechanics subject to (heated) discussion between physicists and chemists. A physicist would proclaim that the wave function of quantum mechanics describes everything there is to say, which is difficult for a chemist to embrace because the wave function lacks direct physicality. A chemist believes in the existence of molecules in space and so is not happy with only a mathematical formalism without direct physical meaning. 

In any case, both would agree that the bond is established by somehow the two electrons of the two H atoms finding minimal total energy E= -1.17 Hartree at a kernel distance = 1.4 atomic units, to be compared with E=-1 at large kernel distance, thus with a dissociation energy of 0.17 Hartree. 

The total energy is the sum of the kinetic energy Ekin and potential energy Epot of the electrons together with the repulsion energy between the kernels. Ekin increases as the volume of electron density decreases, and Epot decreases as electrons get closer to the kernels. A standard view is that these are conflicting demands as concerns decreasing total energy. 

A physicist would say that the wave function describes two overlapping electron densities which do not need to be compressed and so can overlap in the region between the kernels and so decrease Epot with both electrons profiting from closeness to both kernels. Of course overlapping electrons increase Epot but a net gain comes out, is the idea. 

For a chemist overlapping or delocalised electrons is hard to accept because a physical presence in space of both kernels and electrons is the natural concept when building models of molecules. A chemist would say that the decrease of energy in the bond is mainly coming from the accumulation of non-overlapping electrons between the kernels, but that requires electron compression and the net gain is unclear. 

Here RealQM comes in to help the chemist by opening the possibility that the two electrons can meet with positive electron density and so avoid the cost of forcing electron density to be zero on the boundary to the region occupied by the electron thus increasing the kinetic energy. You can yourself follow the formation of the bond by RealQM in this p5js-code.

Notice in particular that the two electron densities meet between the kernels with non-zero density, thus combining favorable presence between the kernels without increase of kinetic energy, thus solving the puzzle! Or?

The prevailing confusion is expressed in the introduction the Chemical Bond (eds Frenking and Shaik):

• Lowering of the energy that establishes the bond is the result of a variational competition between the kinetic energy and potential energy. 
• On the other hand, there occurs an intricate interplay between various intra-atomic and interatomic interactions. These basic agents have, moreover, to accommodate electron correlation. It emerges that, in all cases, the driving force of covalent bond formation is the lowering of the kinetic energy gained by the delocalization of electronic waves over more than one atom. 
• This observation is only superficially discordant with the virial theorem which, as mentioned earlier, requires the molecule to have a higher total kinetic energy than the separated atoms. 
• The in-depth accounting of all interconnections between the various interactions shows that the information disclosed by the actual total kinetic and potential energies per se is insufficient for drawing any inferences regarding the origin of covalent bonding.

Modern physicists overpowered chemists in the 1930s by the heavy weight of quantum mechanics (demonstrated by the atomic bomb) claiming that the final word had been said and what remained was only to ”Shut up and calculate” according to heavy weight Dirac. 

But the quest of fundamental physics to unify all forces under string theory has not delivered and maybe this gives chemists an opening to molecular realism and then why not in the form of RealQM?