Triplet vs. Singlet diatomic molecules: Why the energy difference?

[SonOfOle]

I've been doing some research on singlet-triplet molecular states, and one things I can't perfectly settle is a rigorous demonstration of why the triplet state is a higher energy than the singlet.

One way I can qualitatively understand why the singlet state has lower energy is this: If the atoms are far apart then it does not matter much whether the spin state is triplet or singlet. But when the atoms come close together to form a molecule then we can view the two valence electrons to have almost similar position co-ordinates. It is then necessary that Pauli exclusion be strictly obeyed, and for molecule formation it's would thus be much more favorable to have spins aligned oppositely. For a triplet state with spins pointing in the same direction it is impossible for the atoms to form a molecule due to Pauli exclusion principle.

That, however, is an insufficient answer for me. It's too much "it makes sense if we want it too, but not a priori". Anyone have a good handle on this?

 

[alxm]

Well, it's not quite right either. Let's think about the simplest possible case, two hydrogen atoms come together to form a molecule.

Either hydrogen atom is either in their doublet ground state ²S. When they come together they can combine in two possible ways, forming either a singlet $$^1\Sigma$$ state or a triplet $$^3\Sigma$$ state. Which one is lower in energy?

Well, the resulting two-particle wavefunction can be written as:
$$\Psi(r_1,r_2,s_1,s_2) = \varphi(r_1,r_2)\sigma(s_1,s_2)$$
Where r are the electronic coordinates, and phi and sigma are the spatial- and spin-coordinate parts of the wavefunction.

Now, we're talking about electrons - fermions, here. So the overall wavefunction must be anti-symmetric with respect to the exchange of electrons, that is:
$$\Psi(r_1,r_2,s_1,s_2) = -\Psi(r_2,r_1,s_2,s_1)$$

In the singlet state, where the spins are antiparallel, the spin-coordinate part of the wavefunction is asymmetrical, and hence, the spatial-coordinate part is symmetrical.
In the triplet state, where the spins are parallel, then the spin-coordinate part of the wavefunction is symmetrical, and the spatial-coordinate part is asymmetrical.

What does this mean in terms of energy? Well, since the triplet spatial wavefunction is anti-symmetrical, it must have a node where r₁=r₂. Since the singlet spatial wavefunction has no node, it is lower in energy.

 

[polar5]

Hello,

I am not sur but I think the triplet is more stable than the singulet. We have S₀, with the more little energy, after T₁ and after S₁. We see for the same number of electronic state than the triplet 1 is less in energy that the singulet 1. In my course the reason is du to the spin correlation effect: the electrons have a electronic and magnetic part. If the spin are in the same direction, triplet state, they are a magnetic repulsion and the electrons are farther. This gives a less electronic repulsion, thus an lower energy state.

I am not very convinced with this theory and I such more informations about spin correlation effect, but I don't find anything...

 

[alxm]

The singlet state is usually lower in energy in diatomic molecules.

Offhand the only diatomic I can think of that has a triplet ground state (at room temperature) is oxygen.