17. Why does the LDA work for 'weakly correlated' materials then?

LDA works for normal materials because there is little orbital polarization

  • Degree of orbital polarization in any system determined by two opposing physical effects (in addition to crystal field term). U tends to polarize (as in single ion), but this is opposed by hybridization (mixing) terms arising from overlap with neighboring ions (i.e. analogues of t in Hubbard model). So itinerancy tends to homogenize the orbitals.

  • Main driving force for orbital polarization in HF is due to the non-local exchange interaction (something like -U), which is able to distinguish between the different nodal states of the various orbitals.

  • The only driving force capable of splitting orbital energies in LDA is the spin-polarization resulting from local exchange This does not split same spin states, so no orbital dependent potential. Can't separate t2g and eg with on-site interactions.

  • Spin splitting proportional to the difference in the number of up and down spin electrons. For Mn2+, (5 up spin, 0 down spin) quite a large splitting can be produced, and hence it is possible to create a small gap in MnO. The worst case scenario is Cu2+ (Ah..) where we have 5 up spin, 4 down spin, and the spin splitting will be very small.



So we get a problem when the system would like to have different occupation numbers for different d orbitals, an 'orbital polarization'. In terms of the density this will manifest itself as a marked deviation from spherical symmetry around a given ion. Now in a real physical system, the main driving force for orbital polarization like this is just U, as we saw for the single ion. That is opposed by large overlap terms which tend to homogenize the density, which are the analogue of the t terms in the Hubbard model. These terms are large in a metal, which is why the LDA works pretty well for metals.

Now in Hartree-Fock we have the exact non-local exchange operator, which is able to distinguish between the different nodal structures of the orbitals. But in LSDA there is no real mechanism for splitting d states of the same spin. We can't separate t2g and eg states, for example, apart from the small effect of the crystal field, and we certainly can't split the energies of same spin orbitals within, say, the t2g manifold.

The only mechanism for splitting in the LSDA is just spin polarization. You should note that in some ions, this actually corresponds to the correct orbital polarization - for example. Mn has five filled up spin orbitals and is more or less spherically symmetric. We want a splitting like this to be created, and that is precisely what the spin polarization does. And it will be large, because n(alpha)-n(beta) is large. So LSDA does actually work, to a certain extent, for MnO. And its only in this case that it's legitimate to say that LSDA underestimates the gap. For everything else, it tries to split the wrong sets of orbitals.

The worst case scenario will be when the difference between the number of up and down spin electrons on the TM ion is vey small, because then the spin polarization splitting mechanism will be very small... Unfortunately that means one of the worst cases is copper, where we have 5 up spin electrons and 4 down spin. Good bye high Tc superconductors.