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It is possible to understand band formation by considering a free electron dispersion relation $E(\vec{k})=\frac{{\hbar}^2k^2}{2m}$ which is modified when a periodic potential $V(\vec{r})$ is introduced. We then have Bloch waves as the new eigenstates, characterized by a wavevector $\vec{k}$ and an index $n$ labeling a band. These eigenstates give us a new $E(\vec{k})$ with band-gap structure.

However, I often see bands being refered to as "s bands", "d bands" and so on, referencing original atomic orbitals. I can sort of understand why it is so by thinking in terms of a tight binding model, where Bloch waves are made up of superpositions of atomic states. This is closer to the picture of band formation in which bands are formed when atoms are brought close together and the original atomic orbitals form bands (something like what this video tries to represent https://youtu.be/LNsSS6Id6bM). But how would we conciliate this idea whith the "periodic potential generates gaps" one? I mean, how should the $n$ labeled states I first described be ascribed to these atomic orbitals?

For exemple, when we do band structure calculations, different sets of basis functions may be used to expand eigenstates, like pure plane waves or augmanted plane waves. Than, in the end, how are we able to talk about "d bands" or a dansity of states of "p electrons"?

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