# Double-counting correction in a LDA + DMFT calculation

To theoretically study correlated materials, one usually has to consult to the LDA + DMFT calculations, in which the two-particle interaction is usually double-counted. A general recipe for the double-counting correction is given as $$E_{DC}=\frac{1}{2}U_{avg}N^{l}(N^{l}-1)-\frac{1}{2}J_{avg}\sum_{\sigma}N_{\sigma}^{l}(N_{\sigma}^{l}-1),$$ where $N^{l}=\sum_{m\sigma}\langle n_{m\sigma}\rangle$ is the number of interacting electrons per site.

My question is the following:

(1) is $N^{l}$ evaluated only from the LDA bands, which remains unchanged in the DMFT loop?

(2) if $N^{l}$ is calculated from the LDA bands once for all, may I understand it as a constant shift to the chemical potential in the DMFT, which actually does not mean anything in a fixed density calculation.

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First a remark to clarify the terms: LDA (local density approximation) is an approximation used within the density functional formalism, which yields the exact solution of our many-body problem as long as we would know the exact functional. In contrast, DMFT (dynamical mean-field theory) is a concept to solve periodic (lattice) Hamiltonians with local interactions. So in fact, the double counting depends on the local interaction (in DMFT) as well as on the employed density functional.

Second: double counting is not specifically arising for DFT+DMFT but it is a general problem if one tries to combine DFT with a many-body correction (i.e. DFT+U ...). The double counting correction you posted looks like the "atomic limit" approximation (see M.T. Czyzyk, G.A. Sawatzky, Phys. Rev. B 49 (1994) 14211–14228).

(1) + (2) Usually "correlated" orbitals are "defined" (either by projection or by a suitable basis set) prior to the calculation, so that the correction of the DF functional by +U/DMFT is limited to a subspace (e.g. 3d or 4f-type electrons) of the original Hilbert space. This subspace is defined by projection operators which are applied to the DF orbital set. Your double-counting correction is derived from the local Hamiltonian (I suspect it is the (mean-field?) Kanamori Hamiltonian, otherwise the average of U and J has no meaning) and therefore it is also written in the basis of "correlated" orbitals. So in the end, $N^l$ as well as $N^l_\sigma$ are evaluated after the DMFT cycles (self-consistent DMFT calculation) with the basis set used in the DMFT cycles (projected orbitals).

Due to the projection operator, you have a well-defined relationship between your DFT and +U/DMFT basis set and thus you can evaluate the total energy functional in the original DFT basis.

Further information (reviews on DFT+DMFT):

• G. Kotliar et. al., Review of Modern Physics Vol. 78, 866-951
• A. Georges, arxiv:cond-mat/0403123

p.s. a short remark at the end: those functionals are not anymore sole density functionals, but orbital-dependent density functionals

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Thank you for your detailed reply. I agree with all what you wrote here. However, I do not really understand the logic of updating $N^{l}$ and $N_{\sigma}^{l}$ in each DMFT cycle. I would believe that $N^{l}$ and $N_{\sigma}^{l}$ calculated from the LDA band approximates the interaction considered in the DFT calculation, thus, after subtracting this double-counting term from the DFT relevant bands, we have a well-defined (subject to the ansatz above) tight-binding (TB) model. Otherwise, if I update $N^{l}$ and $N^{l}_{\sigma}$, I have a different TB model in each DMFT cycle. –  Katuru Jan 28 '13 at 16:55
Hello Katuru. Perhaps, my formulation above was misleading: you are correct in that case, that for a self-consistent run of DMFT, the double-counting is kept fixed. However in a charge self-consistent calculation regarding the total energy of the system, the value of the double counting term may change from DFT iteration to DFT iteration (because in each iteration a TB model is constructed which is solved self-consistently in +U/DMFT). If you do a one-shot calculation (no charge self-consistency), than the double-counting correction may be used in construction of the TB model. –  supermarche Feb 12 '13 at 9:33