# How can the exchange interaction make electron spins parallel?

Electrons repel by Coulomb interaction. When they get too close, Pauli exclusion principle ("Exchange interaction") becomes important. If their spins are parallel, they are further "pseudo-repelled". If they are anti-parallel they are "pseudo-attracted".

In an Hydrogen-2 molecule, both electrons are the "bonding glue" that lowers the energy of the molecule. They may repel each other, but overall staying close together between both protons lowers the total energy. So having anti-parallel spin lowers their repulsion, helping them stay together as "glue".

So like with Hydrogen-2, I would expect the exchange interaction in all materials to favor anti-parallel spins in close electrons to lower their repulsion. But this is not what happens in reality.

In ferromagnetic materials, the opposite happens. This I don't understand. When their 'atomic' & 'free' electrons get close, instead of becoming anti-parallel (which would help reduce the unstability of their repulsive proximity), they become parallel. In my very limited understanding, this should make the material have higher energy and be more unstable.

I thought that maybe, because of some complicated molecular orbital structure of the ferro-material, those particular electrons are not "bonding glue", so when they do get close together their repulsion actually destabilizes the material. But then, making them parallel will just make that repulsion even worse! So I still don't understand how the exchange interaction could ever make them parallel. It always seems like the more unstable state.

Could anyone show me how is my crude reasoning wrong?

• Sorry about my earlier answer, which was just wrong. These slides were pretty interesting: tcd.ie/Physics/research/groups/magnetism/files/lectures/5006/…. Slide 17 explains the H2 case by saying that anti-aligned spin state has a smaller energy than the aligned-spin state. It also gives some examples of $J$ in slides 20-28, including some empirical rules. I'm not an expert, but based on the complicated empirical rules it seems like the sign of $J$ is something that is difficult to explain in a simple way based on first principles, in general. Nov 27, 2021 at 21:12
• Yes, after some more searching it seems that it is the result of a lot of complicated structures of the material, probably that's why I couldn't find any clear answer anywhere. Thanks for the slides, I'll look into them Nov 27, 2021 at 22:45

In ferromagnetic materials the magnetism does not come from valence electrons but from unpaired d-electrons. These are spin aligned in agreement with Hund's rules. The valence electrons in such a material are generally paramagnetic.

If their spins are parallel, they are further "pseudo-repelled". If they are anti-parallel they are "pseudo-attracted".
In ferromagnetic materials, the opposite happens.
How can the exchange interaction make electron spins parallel?