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I read this (from this website, with my emphasis):

As the lattice constant is reduced, there is an overlap of the electron wavefunctions occupying adjacent atoms. This leads to a splitting of the energy levels consistent with the Pauli exclusion principle. The splitting results in an energy band containing $2N$ states in the $2s$ band and $6N$ states in the $2p$ band, where $N$ is the number of atoms in the crystal. A further reduction of the lattice constant causes the $2s$ and $2p$ energy bands to merge and split again into two bands containing $4N$ states each. At zero Kelvin, the lower band is completely filled with electrons and labeled as the valence band.

They had taken the example of a diamond in this case.

Update: JonCuster's comment (thanks!) makes it clear that the paragraph above talking about "$2N$ states in the $2s$ band and $6N$ states in the $2p$ band" is wrong, since once atomic subshells become a part of crystal subshells, they are now governed by Bloch functions and not electron wave functions. Yet, this site has also described the reorganization of these (non-existent) $2s$ and $2p$ bands into new bands in a certain detail. With regards to this additional detail (the highlighted sentence of the quoted paragraph), my question is: are they trying to simplify certain known details? Or are they effectively creating a wrong story entirely?

My questions:

  1. Does this reorganization ("merge and split again") of energy bands occurs to lattices all elements, including metals?
  2. What exactly is the rule regarding this reorganization? (which has not been clearly mentioned there)
  3. And what do we name these new energy bands? I was told that the energy bands before reorganization are named the same as the subshell it was derived from. (A $2p$ energy band is derived by splitting the $2p$ subshells of $N$ identical atoms) Though, I am not sure how to name the new energy bands obtained after reorganization.

Regarding 2nd question, this is my thought: after a suitably large reduction of lattice constant, all the energy bands belonging to each shell of principal quantum number $n$ will reorganize with each other. For example, the $n=3$ shell had $18N$ electrons, initially organized as $2N, 6N, 10N$; but now will be reorganized as $6N$ in each new band according to me.


I hope I have clearly identified my question and provided reasonable attempts to identity the correct answers to it. Please comment for clarification. Thank you!

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    $\begingroup$ You might look at the answers (one is mine) in physics.stackexchange.com/questions/278844/… and see if that helps. Otherwise perhaps you could clarify. $\endgroup$ – Jon Custer Feb 15 '18 at 15:10
  • $\begingroup$ @JonCuster Thanks for your feedback! I read that linked answer, however, I am unable to identify much connection, except that both questions are about energy bands in semiconductors, which is a vague connection nonetheless. Could you please clarify why you think the linked question is related? Thank you! $\endgroup$ – Gaurang Tandon Feb 15 '18 at 15:23
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    $\begingroup$ The thing to realize is that Bloch functions that represent the electron wave functions in a crystal have no relation to atomic electron wave functions. Now, that doesn't prevent introductory textbooks from handwaving some connections, but in general there is no way to point to an energy level in a solid and state anything about which atomic level it came from. $\endgroup$ – Jon Custer Feb 15 '18 at 15:45
  • $\begingroup$ Re: "The thing to realize is that Bloch functions that represent the electron wave functions in a crystal have no relation to atomic electron wave functions. Now, that doesn't prevent introductory textbooks from handwaving some connections" wow, that was effectively a reset button on my introductory memory of semiconductors. Am glad it came quickly though. Anyways, what can be said about my question now? I wonder now what exactly was this site talking about. I'll update my question to reflect that. $\endgroup$ – Gaurang Tandon Feb 15 '18 at 15:51
  • $\begingroup$ @JonCuster Please see the updated question. Thanks! $\endgroup$ – Gaurang Tandon Feb 15 '18 at 15:57
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There are many papers in the literature that perform ab-initio calculations of various materials under pressure. The difficulty is often in finding one that shows variation within one crystal structure rather than the variation from structure to structure.

One that might be of interest shows the calculated band structures of bcc and fcc phases of lithium at two different volumes each (nominal density and 40% of that density). In each case, the band structure does change, although perhaps not as radically as one might think.

Note that the sloppiness of discussing $2s$ or $2p$ states is used. This is common. It also is not surprising given that many ab-initio formulations start with the atomic basis, so you can determine, for example, the $2s$-ness of some point in $E$ vs $k$. If you go back far enough, one can find papers with figures of, say, the $2p$-ness vs wave vector of a given band. Of course, since it is changing across $k$ space it should give one pause in terms of making a one-to-one linkage of a band in the solid to an atomic orbital.

A much earlier article discusses tight binding for diamond, silicon, and sodium, but doesn't do anything versus pressure. At least there the authors have the grace to be more explicit of how the atomic basis functions are used to get to the Bloch wave functions (or the 'Bloch sums' as they put it).

Now, why is there this sloppiness? It goes back to the very roots of band structure theory and calculation. If you want an orthogonal basis set to represent electrons in a solid, what better place to start then the familiar atomic orbitals. They are indeed an orthogonal basis with a ready parameterization known to any physicist worth their salt. So, one does some magic to match them up across a unit cell and turn the computational crank. Just remember that they are just a math basis to pull Bloch functions out of.

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  • $\begingroup$ Thanks for your detailed answer! I mentioned in my question that: "With regards to this additional detail (the highlighted sentence of the quoted paragraph), my question is: are they trying to simplify certain known details? Or are they effectively creating a wrong story entirely?" I've read your answer twice and I believe that your final paragraph is the summarized answer to my query. Am I correct? Thank you! $\endgroup$ – Gaurang Tandon Feb 16 '18 at 2:26

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