The Secret of Covalent Chemical Bonding

Recent posts have exhibited the fact that the physics of chemical bonding still is debated as a fundamentally unresolved problem. Consensus appears to be that covalent bonding results from some form of "sharing of electrons", which decreases kernel potential energy by electron charge concentration between kernels without full compensation of increase of kinetic energy.

It is further agreed that this picture can be given support by quantum mechanics with the caveat that full solution of Schrödinger's equation for systems with several electrons is impossible. The idea is that somehow "delocalisation" of electrons over an entire molecule as a purely quantum mechanical effect, will make electron charge concentration possible without full compensation of increase of kinetic energy. But the quantitative details appear evasive.

RealQM offers a different account of the physics of covalent bonding which we here illustrate in a generic 1d molecule with two atomic kernels and two electrons. The crucial feature of RealQM is decomposition of the total electron wave function U(x) = U1(x) + U2(x) into one-electron wave functions with non-overlapping supports  meeting at a free boundary X with continuity and zero (normal) derivative. (Bernoulli condition). Running this code we get this result with kernel potential in blue and electron wave functions in red and green: 

We see a concentration of electron charge densities between the kernels meeting with non-zero joint value and zero (normal) derivates at the free boundary X. The total energy is -65.258 with kinetic energy 26.771.

We compare with a model with overlapping electron densities as in StdQM using this code:

We find a higher total energy -63.144 with substantially higher kinetic energy 35.111. 

We understand that the reason the non-overlapping wave functions above has smaller kinetic energy than the overlapping below, is that the non-overlapping meet at a free boundary with non-zero value and zero derivative allowing dU1/dx*dU1/dx and dU2/dx*dU2/dx to be smaller,  since overlapping wave functions have to meet zero end point values. The overlap region of StdQM is thus in RealQM replaced by a free boundary effectively decreasing the kinetic energy while keeping the potential energy and thus decreasing the total energy into bonding. 

Conclusion: 

RealQM offers an explanation in clear physical terms of covalent chemical bonding emerging from concentration of charge density between kernels without compensating increase of kinetic energy thus with lower total energy than separated atoms. The above experience in the basic model problem is reflected in the full molecule models reported in these posts. If this analysis is correct, it asks for major revision of the standard physics of covalent bonding.