Please ponder the Bonding & Anti-Bonding Molecular Orbitals of molecular Hydrogen, arising from the spatial overlap, of the individual atomic Hydrogen 1s orbitals:

 

The Molecular Orbital [MO] that arises from destructive interference between the atomic orbitals is called an Anti-Bonding MO b/c it does not give rise to a bond, and, in fact, increases the energy as the atoms are brought together.

 

The [atomic Hydrogen] 1s orbitals are probability waves, which can also be added with opposite sign. When the atomic orbitals are added with opposite sign, they destructively interfere... B/c the signs of the two atomic orbitals are opposite, there must be a place where the positive wave exactly cancels the negative wave. This is a node... The destructive interference, between the atomic orbitals, and the resulting node, pushes the electron density out from between the atomic nuclei. The negatively charged electrons are no longer screening the positively charged nuclei, which repel...

 

At the atomic separation that corresponds to the bond length -- that is, the separation you find in the actual molecule -- the Bonding MO is always lower in energy than the separated atoms, and the Anti-Bonding MO is always higher in energy. This is a rigorous result from quantum mechanics. It is a good approximation to say, that the energy decrease of the Bonding MO is equal to the energy increase of the Anti-Bonding MO.

 

M.D.Fayer. Absolutely Small, pp. 185-190.

So, certainly simplistically, please ponder the following (thought) experiment:

 

1. ionize isolated neutral molecular Hydrogen (4.5 eV) to create singly-ionized H₂⁺ in B-MO
   
2. bond length increases from [math]0.74 \rightarrow 1.06 \AA[/math]
   
3. since Dissociation Energy of H₂⁺ is 2.6 eV, excite the lone electron with ~5.2 eV photon, to boost it into the AB-MO
   
4. Hydrogen nuclei now repel apart, each carrying half of the electron AB-MO wave function
   
5. attract dissociating "H^+1/2" fragments towards a charged cathode detector screen
   
6. when "H^+1/2" wave functions collide with the detector screen, quantum measurement will destroy the lone electron's atomic orbital super-position state, collapsing its wave function onto one, or the other, of the Hydrogen nuclei
   
7. both the bare proton (H⁺) and neutral atom (H) will have reached the screen, in a time, corresponding to the accelerations, of particles, having half of a positive charge unit
   

Wouldn't this demonstrate the "reality", of the "smeared out" electron wave function, assuming the delicate super-position state could be maintained until measurement ?

It is a good approximation to say, that the energy decrease of the Bonding MO is equal to the energy increase of the Anti-Bonding MO.

 

Not to be a stickler, but [math] \sigma [/math] ABO have a significantly higher absolute value of enthalpy change [as compared to the non-bonding orbitals] than the corresponding bonding orbitals. I guess it just depends on how you define significant, and to what degree you're willing to approximate.

 

Sorry, just thought I should point that out so people don't get confused. People often times skip over the word "approximation". I've been guilty myself.

Differing electro-negativities, of differing atomic nuclei, in hetero-nuclear di-atomic molecules, leads to unequal sharing, of the electrons' MOs. This is also consistent, with a "realistic" interpretation of the wave function, as the "smeared-out" (charge) distribution, of the electron, which "slews", "sloughs", or "skews" towards the more eletro-negative nucleus [M.D.Fayer. Absolutely Small]