Occupation of the Molecular Orbitals by Electrons
When filling the molecular orbital bands with electrons, it is important to follow the following rules:
- Start filling the bands up from the lowest energy bonding molecular orbital and work your way up.
- Each individual molecular orbital can hold 2 spin paired electrons, which means with N molecular orbitals, 2N electrons can be placed within the band.
- Both the Pauli exlusion principle and Hund's rule must not be violated.
So, if each atomic orbital contributing to the molecular orbitals contains only one electron, it is observed the band is only half full, since there is only N electrons for N molecular orbitals. This leaves empty levels in the bands. If on the other hand each atomic orbital contains 2 electrons, then there is 2N electrons for N molecular orbitals. This means that the band is full. Therefore, as long as there is less than 2N electrons, then there will be empty levels within the band.
In the case where the band is only half full, the occupation of the energy levels is dictated by what is know as the Fermi-Dirac Distribution. Under this distribution, it is found that only at T=0 K does it hold true that the band is exactly half filled. The uppermost filled molecular orbital becomes known as the Fermi energy. At temperatures above 0 K, the energy rises above the Fermi energy level because the electrons start occupying higher states due to thermal excitation. The occupation follows the following equation:
The shape of the Fermi-Dirac Distribution can be seen in the image below.
In the above image, it is clearly seen that at temperatures greater than 0 K, the occupation of the molecular orbitals tails into the empty bands above the Fermi energy. This indicates that the electrons close to the Fermi level are very mobile and can move relatively freely through the solid. It should be noted that the Fermi Level is located in different positions depending on what the structure of the bands looks like. For instance, the Fermi-Level in metals is located in the middle of the half-filled band, whereas in insulators and semi-conducotrs, the Fermi-Level is calculated to be located in the band gap between the valence and conduction bands. At much higher energies, it is found that the occupation the empty levels follows a Boltzmann distribution.
Now that the Band Theory of Solids has been discussed, there is perhaps one more important property of bands to discuss: The density of states. The density of states describes the energy levels per unit energy increment. The density of states in a band is found to be non-uniform across the band. The reason for this is that the levels are packed more closely together at some energies than others as a result of overlap of the molecular orbitals. Since there are numerous ways of producing a particular linear combination of orbitals, it is possible to have more than one at a particular energy within the interior of the band. But it should be noted that there is only one way to form a fully bonding and a fully anti-bonding orbital. An example of a density of states is shown below.
Basically the image demonstrates that at the middle of the band there are more molecular orbitals than found on each end.
Well I hope you enjoyed this in depth tour of Bands to Bonds. If you have any feed back, please email me.