Band Theory of Solids

When Atoms come very close to each other, what happens is that the atomic orbitals of each atom overlap with each other and transform into a molecular orbital. For an instance, if you look at the 1s orbital of this sodium with another 1s orbital of sodium come together and form a 1s molecular orbital. You see atomic orbitals are unique for each atom. But now, a molecular orbital is a unique energy level for the entire molecule, and guess what? Because two atomic orbitals are all overlapping. Our molecular orbital will end up having two energy levels.

These two energy levels belong to one, as molecular orbital, we usually call them 1s and 1s star (1s*). And again, this is something that you may have already learned in chemistry molecular orbital theory, this lower energy orbital is called the bonding orbital and the higher energy orbital is called the anti-bonding orbital.

The same thing is going to happen to over 2s orbitals. They're also going to overlap and now you'll have a 2s molecular orbital which will also have two energy levels. Since the two electrons are formed in the Molecular orbital, they are no longer identical (according to Pauli’s exclusion principle). So, what do you think is going to happen if we add a third sodium atom? Well, now three atomic orbitals are overlapping. As a result, our molecular orbital conveniently ends up having three energy levels. Since the three electrons are formed in the Molecular orbital, they are no longer identical (according to Pauli’s exclusion principle).

Now, each molecular orbital will have three energy levels, there is a trend of growing each molecular orbital, the more atoms that we add, the more energy levels are molecular orbits end up having and eventually, if we have an entire solid, which is made of sodium, which has something like atoms packed together, that will our new molecular orbital of this entire solid will have now 10²³ levels.

In that scenario, can we identify individual energy (discrete energy levels) levels because the gaps between them would be extremely tiny. As a result, we like to think of these energy levels as continuous energies, so we'd like to think of this, as continuous energy. Since we do not have discrete energy levels anymore, we can’t call this an energy level, but instead, we'll call this an energy band.

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