Band structure and valence electrons

[aaaa202]

The standard approach to explaining band structure is to assume that the electrons in a solid move in a potential from the ions, which is periodic leading to Blochs Theorem and the formation of band structure.
But I am a bit confused at this point. Is the approach only valid for the valence electrons in the solid? I.e. are the electrons assumed frozen out? It seems this is the case in many textbooks. If so, what justifies this approximation? In an earlier post I already touched upon the Born-Oppenheimer approximation, but this is about decoupling the ionic wavefunctions based on the big difference in their masses and not that of the core electrons.
On the other hand, if the approach were valid for all the electrons in the solid it would make sense, since the inert core electrons would then be the ones that occupy the filled up bands and would then offer an explanation for explanation for why it is valid to separate the core- and valence electrons.

 

[ZapperZ]

I don't quite understand your question. Why would you treat all the electrons in the solid the same way?

The valence electrons are treated the way they do because of a significant overlap in their wavefunctions. This is why you get the tight-binding model and all those hopping parameters. The core electrons are not treated that way because the overlap is insignificant. There's no significant hybridization of the orbitals and they essentially preserved their "atomic" identity. You can't say the same with the valence electrons.

Zz.

 

[aaaa202]

Okay so the assumption of electrons living in a periodic potential only holds for the valence electrons? Because if you think about it this is the only ingredient used to derive the band structure.

 

[ZapperZ]

Okay so the assumption of electrons living in a periodic potential only holds for the valence electrons? Because if you think about it this is the only ingredient used to derive the band structure.

Two things here:

1. Only "non-local" electrons will have that kind of a periodic potential. After all, if the electron is localized at its "mother atom", it won't see those periodic potential. So already you need a situation where the electronic wavefunction has a significant overlap.

2. The periodic potential is not the only source of a band structure. When you solve the Bloch wavefunction, you made one very important assumption: that the electrons do not interact with each other. They only interact with the periodic boundaries. While this may be OK for simple metals, this is not true in general. I've also mentioned the tight-binding band structure that I've mentioned as another example of obtaining band structure.

Zz.

 

[DrDu]

The electrons moving in a periodic potential of the nuclei is one approximation, basically the Born-Oppenheimer approximation. The electronic wavefunctions formed by the electrons moving in a periodic potential form a complete set of basis functions into which the electronic wavefunctions for other nuclear configurations than periodic can be expanded. That's what you do in the crude adiabatic approximation to describe phonons.

The valence electrons moving in the periodic potential of the core ions is another level of approximation, which is used in semi-empirical methods like tight binding, but not in ab initio methods like DFT.