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That is, why is the potential energy with the orbitals overlapping less than with the Hydrogen atoms 'independent'. Similarly, why is a noble gas configuration stabler than if an electron were to be removed or added? Is this because pairs of electrons are more stable than single electrons? If so, why?

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You may find a discussion of the H atom attraction and bonding here. The noble gas atomic configuration is really a different topic, namely the so-called Aufbau or build-up principle of the Periodic System. –  Lupercus Sep 6 '12 at 21:30
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The pairs of electrons are more stable if you see the pair as a filled orbital (which can contain maximum of 2 different spin electrons via Pauli's Exclusion Principle).

I think the simplest way of explaining the lower energy of two-nuclei orbital is that the size of the electron's 'playground' grows 2 times, and as the area over which the electron may soar increases, it's kinetic energy decreases.

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Thanks for that- but also- why is it more stable (or with less potential energy, same thing)? –  Alyosha Sep 6 '12 at 20:57
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I don't think it's possible to explain it further on an intuitive level, at least I can't think of a way :( Electrons can't do whatever they want, and ultimately bonding is what they do because of "choosing what to do" from limited set of options (due to quantum restrictions). See the link provided by @Lupercus for more in-depth discussion. –  pafau k. Sep 7 '12 at 8:53
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