Now for the Gory Stuff: Band Theory in Detail
The electronic structure of solids is usually described by a theory called Band Theory, a derivative of Molecular Orbital Theory. However, the best way to describe band theory is to go back and go through the process of forming molecular orbitals from the atomic orbitals for a simple diatomic like hydrogen. This will show where some of the assumptions to be used originate. Please note, it is assumed here that the reader has some knowledge of molecular orbital theory already.
To begin with, if one puts atomic orbitals into the wavefunction, and the Schrodinger equation holds true, then it is possible to make an orbital approximation where it is assumed to a reasonable first approximation that the wavefunction 
of the N electrons in the molecule can be written as the product of N one-electron wavefunctions :
The interpretation of the above equation comes down to realising that electron 1 is described by the wavefunction 
.These wavefunctions describe the molecular orbitals formed. It should be noted that the square of the wavefunction gives the probability distribution for that electron over the whole molecule(i.e. where the orbital has a large amplitude and that it will not be found at its nodes.
Now a second fundamental approximation is made. When an electron is close to the nucleus of one atom, its wavefunction closely resembles an atomic orbital of that atom. Therefore a reasonable first approximation of the molecular orbital can be achieved by adding together atomic orbitals contributed by each atom. This is the Linear Combination of Atomic Orbitals.
When constructing the molecular orbitals it is common practice to use the valence orbitals to form the molecular orbitals. Therefore, for hydrogen, by adding together two 1s orbitals, one forms a basis set for the atomic orbitals that is described by the wavefunction:
This equation describes that the molecular orbitals are built from 2 1s hydrogen atomic orbitals 
,one on atom A and the other on atom B. The coefficient c describes the extent to which each atomic orbital contributes to the molecular orbital. The higher the value of c squared, the greater the contribution. The molecular orbital that is higher in energy is described by the following equation:
These wavefunctions are arrived at by solving the Schrodinger equation for the solution that models both the highest and lowest molecular orbital. The coefficient 
and 
are both equal to positive one for the bonding molecular orbital and are opposite signs for the antibonding molecular orbital.
This leads to equal contributions of each 1s orbital into each molecular orbitals. The signs of the coefficients determine if the atomic orbitals interfere constructively(positive coefficients) or destructively(negative coefficients). This leads to an accumulation or reduction of electron density. This leads to the formation of the molecular orbitals for hydrogen below.
As can be observed, two 1s atomic orbitals of the same energy come together to form 2 molecular orbitals of different energy: one bonding and one antibonding(contains a node).
The derivation of the molecular orbitals for oxygen can be seen in the below figure as a furhter example.
This leads to the following important conclusions:
- N molecular orbitals can be constructed from a basis set of N atomic orbitals.
- By solving the Schrodinger equation, it is possible to arrive at solutions that model the molecular orbitals formed by the atomic orbitals.
- Areas of constructive interference lead to overlap of the atomic orbitals due to accumulated electron density. This results in a bonding molecular orbital that is low in energy and contributes to keeping the molecules together.
- Areas of destructive interference lead to non-overlap of the atomic orbitals due to reduction of electron density. This results in an anti-bonding molecular orbital that is high in energy and tends to tear apart the molecule.
- The are of reduced electron density resulting in a node where there is considered to be no electron density.
- In a bonding molecular orbital, the probability of finding the electrons on one, both or between the atoms is quite high.
- In an anti-bonding orbital, the probability of finding an electron in between the atoms is 0, and 1 for on one or the other atom.
So using the above assumptions and points, it will be possible to describe the molecular orbitals for metals, semiconductors, semi-metals and insulators. So now lets take a look at these materials.