Band Theory: At Last the Good Stuff
Now that the background has been derived for molecular orbital theory, let's look at how to derive it for bands. If we look at solids, we discover that they can be treated as an indefinitely large molecule with an infinite number of valence orbitals overlapping. The molecular orbitals of these solids are such that they occur in bands. These bands contain an infinite number of molecular orbitals, with areas of no molecular orbitals separating the bands. This separation is called the band gap. So how do we get from a simple diatomic to these bands? Lets find out! The basic idea is diagrammed out below.
As observed for hydrogen, the number of molecular orbitals for which solutions were obtained from the Schrodinger equation are equal to the number of atomic orbitals put into the equation. Now lets look at a triatomic linear molecule in which three atomic orbitals lead to three molecular orbitals. The first observation is that there is now an orbital between the antibonding and the bonding orbitals. This is a non-bonding orbital which contributes nothing to overall bonding within the molecule. The second observation is that the energy difference between the bonding and anti-bonding orbital have increased. Thirdly, the distance between any two orbitals has decreased overall. This is shown below.
As the number of atomic orbitals increases, the number of molecular orbitals will increase as well. This results in the energy difference between each individual molecular orbital decreases, while the energy difference between the lowest bonding orbital and the highest anti-bonding orbital increases, but only increases to a maximum known as 4B, where B is the overlap integral. Once 4B is achieved, and as the number of atomic orbitals increase, the resulting molecular orbitals become close together and start to blur due to an increase in the amount of overlap of the molecular orbitals.
Now if molecular orbital theory is expanded to look at a solid, some very interesting results occur. Lets choose a basis set where there is a chain of atoms, each contributing a p-orbital, to form the molecular orbitals.. For this example, the number of atomic orbitals is considered to be infinite. Therefore, upon solving the Schrodinger equation, we would expect an infinite number of molecular orbitals as the result. These molecular orbitals would all fit into an energy separation of 4B, the difference in enrgy between the higheat and lowest energy molecular orbitals. Therefore, the energy difference between each individual molecular orbital is such that it cannot be resolved, so they blur into bands. The final result is that we now have the electronic structure of the solid described by a series of bands made up of overlapping molecular orbitals, separated by band gaps, in which there are no molecular orbitals.
The final image of molecular orbitals as bands is shown below
Just for fun, here is an animation showing the origin of the bands starting from 1 molecular orbital and increasing in number until the molecular orbitals blur together. So without furhter adieu, here is the dancing band
Now, if the s and the px, py and pz orbitals are chosen as the basis set, then gaps between the bands can occur if the s and p orbital bands don't overlap. However, in some solids, overlap of these bands results and we get sp hybridized bands. The reason for the overlap stems from the interaction of the atoms. If the interact stongly, this leads to overlap. If not a band gap is formed. Normally, the bands for a soild are displayed as in the diagram below:
Follow the link to the discussion on occupation of these moleuclar orbital bands by electrons
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- Occupation of the Molecular Orbital Bands
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