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I am currently studying Diode Lasers and Photonic Integrated Circuits, second edition, by Coldren, Corzine, and Masanovic. In chapter 1.2 ENERGY LEVELS AND BANDS IN SOLIDS, the authors say the following:

On the other hand, in a covalently bonded solid like the semiconductor materials we use to make diode lasers, the uppermost energy levels of individual constituent atoms each broaden into bands of levels as the bonds are formed to make the solid. This phenomenon is illustrated in Fig. 1.4. The reason for the splitting can be realized most easily by first considering a single covalent bond. When two atoms are in close proximity, the outer valence electron of one atom can arrange itself into a low-energy bonding (symmetric) charge distribution concentrated between the two nuclei, or into a high-energy antibonding (antisymmetric) distribution devoid of charge between the two nuclei. In other words, the isolated energy level of the electron is now split into two levels due to the two ways the electron can arrange itself around the two atoms.$^1$ In a covalent bond, the electrons of the two atoms both occupy the lower energy bonding level (provided they have opposite spin), whereas the higher energy antibonding level remains empty.
$^1$The energy level splitting is often incorrectly attributed to the Pauli exclusion principle, which forbids electrons from occupying the same energy state (and thus forces the split, as the argument goes). In actuality, the splitting is a fundamental phenomenon associated with solutions to the wave equation involving two coupled systems and applies equally to probability, electromagnetic, or any other kind of waves. It has nothing to do with the Pauli exclusion principle. enter image description here If another atom is brought in line with the first two, a new charge distribution becomes possible that is neither completely bonding nor antibonding. Hence, a third energy level is formed between the two extremes. When $N$ atoms are covalently bonded into a linear chain, $N$ energy levels distributed between the lowest-energy bonding state and the highest-energy antibonding state appear, forming a band of energies. In our linear chain of atoms, spin degeneracy allows all $N$ electrons to fall into the lower half of the energy band, leaving the upper half of the band empty.

What does "spin degeneracy" mean in this context?

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  • $\begingroup$ You might want to look at calculation of the energy levels in polyenes as they get longer, familiar to chemists, and which show the same behaviour as in your figure. $\endgroup$
    – porphyrin
    Commented Mar 31, 2021 at 7:10

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I'm not a solid-state chemist, but I think the meaning seems reasonably clear from context to me. "Spin degeneracy" here means that each energy level is capable of holding a spin-up electron as well as a spin-down electron, i.e. in each orbital there are two different spin states which are degenerate (some would call these spin orbitals). Thus, $N$ atoms form $N$ different energy levels which $N$ electrons need to be distributed between. Because of this degeneracy, only the lowest-energy $N/2$ energy levels are actually populated: each energy level has $2$ electrons.

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  • $\begingroup$ Thanks for the answer. What does it mean for a spin state to be "degenerate", as you described? $\endgroup$
    – The Pointer
    Commented Mar 30, 2021 at 20:51
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    $\begingroup$ It’s not one spin state that is degenerate, it’s two spin states. Degenerate means that they have the same energy, so it doesn’t make sense to say that one spin state is degenerate. $\endgroup$
    – orthocresol
    Commented Mar 30, 2021 at 21:05
  • $\begingroup$ Hmm, I'm trying to understand this part: "... in each orbital there are two different spin states which are degenerate ...". What does this mean? The wording is confusing to me. $\endgroup$
    – The Pointer
    Commented Mar 30, 2021 at 21:10
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    $\begingroup$ Each orbital corresponds to two spin states. You must be familiar with the standard atomic orbitals, 1s, 2s, ... The 1s orbital can hold two electrons: one spin-up and one spin-down. That is the same thing as saying that the 1s orbital has two different spin states. The shape of the orbital in a solid is different, but it doesn't change the fact that each orbital can hold one spin-up and one spin-down electron. $\endgroup$
    – orthocresol
    Commented Mar 30, 2021 at 21:19
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    $\begingroup$ This is getting into the territory of semantics, and I'd rather not go too deep down that rabbit hole. My take on it is that when you say two things are degenerate, it means they have the same energy. You can't say that one thing (like an orbital) is degenerate, because effectively you are saying that "an orbital has the same energy". Same energy as what? It's an incomplete sentence, which doesn't really make sense. I'm saying that the two spin states are degenerate, i.e. the two states spin-up and spin-down have the same energy. Someone else might have a different take. $\endgroup$
    – orthocresol
    Commented Mar 30, 2021 at 21:46

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