Sunday, June 26, 2011

Aufbau principle and Hund's Rule

In the independent-particle model (i.e., the all-electron wave function is written as a Slater determinant) we have in general more spin orbitals than we have electrons. This means that we have to make a choice with which orbitals to construct the wave function. The choice is not difficult, but the procedure is often presented in separated parts, dealing with different cases. Here these parts are put together.
We use the Aufbau principle and Hund's rules to make the choice. The Aufbau principle says that you should look at the orbital energies, and take the orbitals with the lowest energies. In almost all cases this is the only rule you need. Suppose you have (spatial) orbitals with increasing energies e0, e1, e2, e3, ..., and you have six electrons. Then you can put two electrons in the orbitals with energies e0, e1, and e2 each. The orbital diagram looks as follows

Application Aufbau principle

The Aufbau principle does not suffice if the highest level in which you put electrons is degenerate and there are various ways to distribute the electrons. There is not problem with one electron in a highest non-empty degenerate level. All possibilities for this one electron yield the same energy, so there is no preference. The same holds if the level would be completely occupied if one extra electron would be added. (The level then has one hole). Hund's rules (sometimes also collective called Hund's rule) tell what to do if there are at least two electrons and at least two holes. Hund's first rule says that one should distribute the electrons as much as possible over the spatial orbitals. The reason is that in this way the electrons stay away from each other as much as possible thus reducing their repulsion. Hund's second rule says that unpaired electrons should be given the same spin. In this way the exchange interaction is maximized. As this interaction has a negative sign in the Fock operator and the expression for the electronic energy, this reduces the energy. The following diagram shows how to apply all three rules.

Application Hund's rules

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