16. Self-interaction in LSDA

  • LSDA exchange energy:

    Mean field contains all the electrons!

    Implications:

    • U (the self-exchange term) equals J (the 'different orbital' exchange term).

    • But U and J differ by and order of magnitude in real materials. LSDA therefore averages these quantities..

    • Therefore additional potential U felt by unoccupied orbitals disappears, and instead all the states are shoved up by something like the average of U and J..

    • Local exchange operator therefore not sufficient - we need an orbital dependent potential to take into account the different nodal structures of the orbitals.

    • ORBITAL POLARIZATION is the key element we re missing in DFT (e.g. we can't have different energies for t2g and eg orbitals with the same spin, other than the small difference due to the crystal field. Can't get a difference in the occupation number as in HF)



What happens in the LDA? Well, how do we define the exchange energy?

We integrate over all space the exchange energy per particle of a homogeneous electron gas with given spin-up and spin-down densities. So the mean field contains all the electrons.

Now the Coulomb self-interaction energy is mainly part of the Hartree-potential. To cancel it, we need an equivalent term in the exchange energy. We don't have one, so the self-interaction isn't cancelled.. To get that you would need a non-local exchange operator which gives a self-exchange of -U for exchange of an electron with itself and a smaller number for exchange with same-spin electrons in other orbitals, which is J. But within the LDA or GGA these are the same. So we're averaging over U and J - two numbers which as we've seen can differ by an order of magnitude. That has the effect of shoving up all the occupied bands by something like the average of these two quantities, without splitting them, which tells us why the oxygen states were so low in energy relative to the d states in the DOS plot I showed you a few overheads back. And because we no longer have an orbital dependent potential, we lose all of the Mott-Hubbard physics.