# Why do molecular orbitals in solids merge to bands?

Why do molecular orbitals in solids merge to bands?

For example: In silicon every atom is sp3 hybridised, but when I merge two of these orbitals then it yields a bonding and an antibonding MO. When a third atom binds to one of the $\ce{Si}$ atoms in this configuration, then again we should get two MOs, separated by the same energy. This way we we don't get so called bands. How do the bands form?

I have tried wikipedia and other sites but they answer in brief.

Let's explain this with your example of silicon that every atom is sp3 hybridised. when you merge two of these orbitals, then it yields bonding and antibonding MO. But when a third atom bounds, it yields 3 MO (not four as you mentioned) bonding, antibonding and non bonding MO. The number of MO equals the number of atomic orbitals constituting them, according to LCAO theory. The energy of the third MO is in between the nonbonding and anti-bonding MOs. When you add a forth, a fifth and so on atoms, you will have more and more MOs very close in energy and you will form a band. This is a qualitative description about the formation of a band.

I'll give a quick answer first and come back tomorrow for a more complete version.

In your question, you start with two $\ce{Si}$ atoms and then a third. As you combine these atoms into pairs and molecules, chemistry stresses that the atomic orbitals combine in different ways.

So let's take $\ce{Si2}$. We know that for the $3p$ orbitals, some will become more stable and some will become less stable. There are 3 for each $\ce{Si}$ atom, so a total of 6 molecular orbitals due to combinations of $3p$. (Actually, the $3s$ are also involved, and there's some amount of s-p mixing, but let's ignore that for now.)

Take-home message: when we combine atomic orbitals into molecules, some orbitals become more stable and some orbitals become less stable.

OK, now let's add a third $\ce{Si}$ atom as you said. I'm not going to derive the MO diagram right now. (I'll do that tomorrow and insert the picture.) The main point is that we'll have new combinations. If I take $\ce{H3}$ as another example, I should have 3 molecular orbitals.

As I grow into a linear chain, a 2D sheet, or a 3D solid, I don't have some small number of atoms like 2, 3, 4, 20, etc. Instead, I have 1,000 or 10,000 or many, many more.

Well, those atomic orbitals combine just like they do when you create molecular orbitals. Some become more stable, some become less stable. Yet, rather than having 3 molecular orbitals in $\ce{H3}$ from 3 atomic orbitals, we have 1,000 or 10,000 or many, many more.

Since there are so many orbitals, the energy spacing between them is infinitesimal. These are now called bands.

The type of material (and the atomic composition) will dictate the energy levels of the bands and the so-called "band structure" (i.e., the solid-state equivalent of molecular orbital diagrams). But simply having hundreds and thousands of atoms will give you bands.

This is explained by the Pauli exclusion principle, as applied to conductors: quantum mechanics forbids that fermions have exactly the same spin-state and energy level. In a single atom, that means, for example that H has one s electron in the "up" state and anther in thew "down". In conductors (and semiconductors), electrons from one atom can exchange with those of another, so they must have (slightly) different energy levels, if all spin-states are occupied, effectively splitting the single level into a "band" of energies. See http://en.wikipedia.org/wiki/Pauli_exclusion_principle for more detail.

BTW, this is not true for "solids" in general, e.g. sulfur or window glass, but is true for metals, alloys (e.g. brass) and conductive compounds.