Friday, February 20, 2015

What does the Hubbard model miss?

How is a Hubbard model related to Density Functional Theory?

Jure Kokalj and I recently wrote a paper where considered the effect of strong correlations on thermal expansion, all within the framework of a Hubbard model. This is mostly concerned with explaining anomalies in organic charge transfer salts at temperatures less than 100 K, i.e. much less than the Fermi energy.

One referee stated
“However conceptually this Hamiltonian can not capture the free energy of the relevant electrons loyally. Recall the total energy decomposition in density functional theory, the Hamiltonian corresponds only to the band energy part (which is a summation of occupied Khon-Sham states and different from the kinetic energy) plus interaction term. And the remaining Hartree part, exchange-correlated part and also ionic part, which depend on the lattice constants, are totally ignored. It is not known whether the contributions from such terms are trivial or monotonic especially when strong correlation is present. The neglect of such terms in the electronic model in use is not justified. In this sense, even though the parameters are taken from first principles estimations, it is not surprising that the results are not consistent with experimental data quantitatively and sometimes even qualitatively."
There are some subtle issues here that I would like to understand.
I am not sure I fully understand the referee's comments.
And, I am not sure I agree.

1. Do I understand that the referee is suggesting that the Hubbard model does not include the effects contained in the Hartree and exchange correlation term? Surely, this is not correct.

2. I agree that the Hubbard model will be missing all effects associated with core electrons and ionic terms. However, surely any effects associated with these will not vary significantly on energy and temperature scales of the order of 100 K?

I welcome any comments and insight.


  1. The referee's comment seems to be somewhat in a standard negative tone, and I also not sure I understand the message.
    Leaving that aside, I'd put forward two main shortcomings: 1. Absence of the energy dependence of the interaction (which would occur by integrating out phonons, for example). On the temperatures of the order of 100K that might be relevant. 2. s-orbitals, which are spherically symmetric (might be a subtle issue here, since the thermal expansion has a direction).

    1. Thanks for the comment.
      The issue is rather subtle.
      We don't totally integrate out the phonons since we allow the Hubbard model parameters to vary with lattice constant.

  2. Just a guess - but could the referee mean by "the remaining Hartree part..." that the the total Hartree is a sum of two parts, one of which is captured by the Hubbard model, and the other (long-range) part which is neglected, since you only have an on-site U? That might be why he/she says the neglected part depends on the lattice constants.

    1. Perhaps this is what the referee means.
      But it is a bit confused.
      Both DFT and the Hubbard model have Hartree terms, but they are not the same thing.

  3. I think there is a confusion here between approximation scheme and the Hamiltonian