Molecular orbital theory & predicting the stability of a molecule?
You need to mix the orbitals, populate them with the electrons and see if you have net bonding.

Eg:  H + H two 1s orbitals mix to form sigma and sigma*.  Two electrons total, both occupy the sigma orbital, two more electrons in bonding than antibonding orbitals, the compound is stable.



Eg:  He + He; same mixing as above.  Four electrons, two in the sigma, two in the sigma*.  Since there are as many bonding electrons as as antibonding, there is no net bond.  He2 is not possible.



Eg:  He + H; same mixing as above.  Three electrons, two in sigma, one in sigma*.  One more electron in bonding than antibonding.  He-H forms a very weak bond.  Please note the diagram is for He2+ but the He-H is very similar



Eg: Li + H; Li has 1s + 2s, while H has 1s.  This mix to form a sigma orbital from H1s+Li2s, a sigma* orbital and H1s-Li2s, and a non bonding orbital from Li1s (lower in energy than the sigma).  Four total electrons.  Nonbonding sigma is occupied, and then the sigma orbital is occupied.  Net effect: Li-H forms a stable bond.



You might have noticed that the first three diagrams had identical structure, even if their energy levels were different.  This is a property of quantum mechanical symmetry, and solving the problem only requires knowing a few diagram types and filling the electrons in.  Consider the second row.  The diagram is as :



You simply count the valence electrons and fill in the diagram.  Then you subtract the number of bonding orbital electrons from antibonding, and if you have more, you will have a bond.

A couple of webpages are very good for further explanation:

Source of the diagrams:
http://www.chem1.com/acad/webtext/chembond/cb08.html

and
http://www.meta-synthesis.com/webbook/39_diatomics/diatomics.html