How to Calculate Inverse Participation Ratio

[aruth]

Hello,
I have been looking at impurity states in graphene. From looking at PARCHG files of bands around the fermi energy, many of the bands appear to be strongly localized, but I would like to follow a more systematic approach.

The method I am familiar with is called the inverse participation ratio, and it is defined as 1 divided by the norm of the wavefunction. If the state is localized, its inverse participation ration (norm) converges to a constant as the size of the supercell is increased , but there is still only 1 impurity. If the state is delocalized, the inverse participation ratio (norm) decreases (increases) proportionally to the size of the supercell. Based on the inverse participation ratio, you can define a "size" of the localized state which is either the number of electrons involved, or an equivalent volume (surface area since this is 2D).

I tried to calculate the inverse participation ratio by averaging the elements in the PARCHG file, since each element is the charge density times the volume, which is equal to the sum of the squares of the wavefunctions times the volume. The average would then be the average charge density times the volume, or the total charge. However, I always get that the average of the elements is 2, which I take to mean 2 electrons. However, I expected for a delocalized band to get 2 electrons/atom.

When I calculate the same average for a CHGCAR file, I get the expected result, the total number of valence electrons of the supercell. It seems like the PARCHG file does not have the normalization information that I need and has instead been renormalized.

Does anyone know of a way to calculate the inverse participation ratio, or of another similar method which accomplishes the same goal of determining the volume (area) of the localized state?

This information must be kept somewhere as it would be necessary when calculating matrix elements.