More about the stability of a molecule

You can test directly the stability of a molecule, you just have to choose Job style --> Stability. And then test with a singulet or a triplet.
But in this case what you are looking for is the stability of the wave function!
For the example of propene, we get the same, the too exist but at different energy.

For the singulet
 SCF Done:  E(UHF) =  -117.022407343     A.U. after   19 cycles
             Convg  =    0.9626D-08             -V/T =  1.9986
             S**2   =   0.1527

For the triplet
 SCF Done:  E(UHF) =  -116.910538298     A.U. after   15 cycles
             Convg  =    0.1887D-08             -V/T =  1.9942
             S**2   =   2.0203

So the results suggest that in term of energy the singulet is the more stable because it has got the lowest energy.

But with the example of O₂ you get different results.
O₂ HF 6-21G
Singulet

 The wavefunction has an RHF -> UHF instability.

Triplet

The wavefunction is stable under the perturbations considered.

So here the singulet state does not exist.

Consider the energy of the two forms (singulet and triplet) of O₂.

For the singulet
 SCF Done:  E(RHF) =  -148.686490556     A.U. after    8 cycles
             Convg  =    0.2240D-08             -V/T =  2.0032
             S**2   =   0.0000

For the triplet
 SCF Done:  E(UHF) =  -148.768525186     A.U. after    9 cycles
             Convg  =    0.4772D-08             -V/T =  2.0031
             S**2   =   2.0195

So if we just have a look at the energy we deduce that the triplet has a lower energy so it is the more stable... it is not in contradiction to our results concerning the stability of the wave function, but it is two different questions!

And if we consider the theory that we know about the orbital diagram of O₂ the filling is : $σ$ 1s² $σ$ *1s² $σ$ 2s² $σ$ *2s² $σ$ 2px² $π$ 2pz² $π$ 2py² $π$ *2pz¹ $π$ *2py¹ $σ$ *2px⁰

So it is quite logical that O₂ is more stable as a triplet because of the repartition of the electrons. Back to Calculate the energy of a molecule