How come that Esinglet < Etriplet and Hund’s rule are both correct?

[Alex83]

Hi all, I will be most grateful if someone could help me with this:

1) It can be shown that for two electrons the singlet energy is lower
then the triplet energy.

2) According to Hund’s rule the ground spin state is symmetric.

Why (1) is not correct for atoms where Hund’s rule can be used?

Maybe the proof of (1) is wrong for more then two electrons but I can’t figure why.

With thanks,
Alex.

 

[dextercioby]

Of course. You get a singlet and a triplet when adding two spins only. Once there are more than 2 (3 or more), things are longer to calculate. Sides, an electron in an atomic shell has orbital angular momentum, too.

Daniel.

 

[Alex83]

The proof of (1) is for general symmetric and anti symmetric functions, it goes like that:

1) One can minimize the energy over all the symmetric functions and call the result Es and fs(r), then minimize over all the anti symmetric functions and call it Et and ft(r).

2) Prove that |ft(r)| yields the same mean energy as ft(r), Et.

3) |ft(r)| is obviously symmetric and so Et>Es.

I think that for more then two electrons (2) is wrong but I don’t know why.