[QE-users] Thermodynamics with DFT+U

Mon Dec 11 12:03:32 CET 2023

Dear Eduardo,

Your questions are tricky. There is a lot one can say. Please see my comments below. Maybe someone else can have a different viewpoint and comment as well.

  *   Should we choose one average value, or use the computed value for each system?

Both options are used in the literature. From my experience, it is better to use the second one.

  *   In DFT+U with empirical U people often use one value and compare the total energies. Why? One reason is because how would you choose different U values for different systems (e.g. FM vs AFM)? Maybe this can be done, but it is easier to use one empirical value. And it is claimed that the total energies must be compared with the same U value. But why? Is there a theorem or a proof? See below for the discussion why I would not use the same U value.
  *   In the second case, one uses different U values for different structures, provided that these U value are computed ab initio. Does this make sense? At least to me, yes. Why? Because different structures require different corrections. And, indeed, if one computes U e.g. for the Co-3d states in LiCoO2 and CoO2, the U values appear to be different. Why? Because the electronic screening is different, and the magnitude of self-interaction errors is different in LiCoO2 and CoO2. One can make an approximation and use an average U value for these two systems, but why doing so? From our experience using different ab initio U values and comparing total energies gives results in good agreement with experiments (e.g. voltages for batteries). But we do not have a (mathematical) justification for doing so, as well as we do not have a proof why one should not do it. Hence, at present there is no consensus in the literature on this topic. More investigations for various systems is needed to see trends. But for me, comparing total energies with different U values obtained from linear-response theory makes sense and it provides reasonable results.

  *   Concerning the advantage of self consistency, let me rise the example LiCoO2 that comes with the HP code. The example produces U for Co and also for O, as well as V(Co-O). U(O-2p)=8.0439 eV. Is this parameter useful? As the example is not converged w.r.t. to k-points and cutoffs the number may change, but U(O-2p) is still there. I read PRB101, 064305 (2020) by Floris et al, and it seems that U(O-2p) is discarded. I am curious why, but I couldn't find a discussion. Maybe there is another article. My point here is that using self consistent parameters for some elements and shells, and discarding others is just a partial self-consistency.

We did not apply the U correction to O-2p states. The question of whether to apply or not the U correction to O-2p is another big question. Many things can be said here, and you will possibly receive different answers from different people. A few comments from my side:

  *   We generally do not apply U to O-2p, when U is computed from linear-response theory, because it is large (8-9 eV) and from our experience the accuracy of some properties (e.g. voltages) are worsened.
  *   If you use ACBN0 to compute U, you might get 2-3 eV, and applying this correction to O-2p might improve the results. So you see that it matters which value of U to apply to O-2 states and how it was computed. If one tunes U by hand, then of course you can get whatever you want. E.g. people apply empirical U to O-2p states in ZnO to get the right band gap. But this touches on another topic: DFT+U for band gaps. U generally improves the band gaps if the correction is applied to the edge states. Have a look at this paper: https://www.mdpi.com/2076-3417/11/5/2395
  *   In some works, even U from linear-response theory is applied to O-2p to get better band gaps.
  *   Applying U to O-2p localizes these states more. Is it good or bad? It depends on the system. E.g. in systems with strong covalency, this is not good as you will kill the hybridization between TM-3d and O-2p states. E.g. in the case of BaTiO3 applying U to O-2p does exactly that and one gets the cubic phase instead of the rhombohedral one, in contradiction to experiments. While not applying U to O-2p is ok, because the inter-site hybridization is there and the DFT+U+V approach preserves the rhombohedral symmetry: https://arxiv.org/abs/2309.04348

  *   A related question is whether the forces and energies are consistent with variable U and V. That is, Let us move the Fe impurity atom inside a crystal, and recompute the U and V for each position.  Force is the gradient of energy obtained in the Hellman-Feynman way, I guess with constant U,V.

  *   Pressure is the negative of the derivative of the energy with respect to volume, which implies a variation of U and V. I guess the stress is computed with constant U, V. I think that self-consistency could be implemented, but first we must be sure that comparing energies with variable, self-consistent parameters is correct.

Another excellent question. In Quantum ESPRESSO, U is constant and its derivative dU/dR is set to zero when computing Hubbard forces (and same for Hubbard stresses): https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.235159
In order to circumvent this problem, we perform the calculation of U in a self-consistent fashion, by performing cyclic calculations (recalculation of U and structural optimization with DFT+U), thus pushing the system to the energy extremum: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.103.045141

HTH

Greetings,
Iurii

----------------------------------------------------------
Dr. Iurii TIMROV
Tenure-track scientist
Laboratory for Materials Simulations (LMS)
Paul Scherrer Institut (PSI)
CH-5232 Villigen, Switzerland
+41 56 310 62 14
https://www.psi.ch/en/lms/people/iurii-timrov
________________________________
From: users <users-bounces at lists.quantum-espresso.org> on behalf of EDUARDO ARIEL MENENDEZ PROUPIN <emenendez at us.es>
Sent: Wednesday, December 6, 2023 10:24
To: users at lists.quantum-espresso.org <users at lists.quantum-espresso.org>
Subject: Re: [QE-users] Thermodynamics with DFT+U

Hello!
I have read this thread, which is from three years ago, and I would like to know if there is any update, consensus, or a study about this issue.

The topic of the thread was how to compare the energies of two systems when there is at least one element subject to Hubbard correction, in the case that the  Hubbard parameters are computed self-consistently via the HP code, and have different values in the two systems compared.  Should we choose one average value, or use the computed value for each system?  The two systems may be either:

  1.  Two phases of a material
  2.  Two antiferromagnetic configurations
  3.  Crystal with a transition metal impurity vs clean crystal and impurity in bulk metal.

I may have a case of type (b), with certain energy order when using the self-consistent U values for each AFM configuration, and the opposite order when the same U is used for both configurations. The same U was computed for one configuration, I am waiting for the queue to finish calculations with the other U, but this is published (Naveas et al, https://doi.org/10.1016/j.isci.2023.106033).

Concerning the advantage of self consistency, let me rise the example LiCoO2 that comes with the HP code. The example produces U for Co and also for O, as well as V(Co-O). U(O-2p)=8.0439 eV. Is this parameter useful? As the example is not converged w.r.t. to k-points and cutoffs the number may change, but U(O-2p) is still there. I read PRB101, 064305 (2020) by Floris et al, and it seems that U(O-2p) is discarded. I am curious why, but I couldn't find a discussion. Maybe there is another article. My point here is that using self consistent parameters for some elements and shells, and discarding others is just a partial self-consistency.

A related question is whether the forces and energies are consistent with variable U and V. That is, Let us move the Fe impurity atom inside a crystal, and recompute the U and V for each position.  Force is the gradient of energy obtained in the Hellman-Feynman way, I guess with constant U,V.
Pressure is the negative of the derivative of the energy with respect to volume, which implies a variation of U and V. I guess the stress is computed with constant U, V. I think that self-consistency could be implemented, but first we must be sure that comparing energies with variable, self-consistent parameters is correct.

Best regards,

Eduardo A. Menéndez Proupin
Departamento de Física Aplicada I
Universidad de Sevilla
Teléfono: +34 9554 20231
https://personal.us.es/emenendez/
https://personal.us.es/emenendez/docencia/
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