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Even though I use some Quantum ESPRESSO specific terminology here, this question applies to any plane-wave based pseudopotential DFT code. We know that before running an actual DFT calculation, a few parameters must be set following convergence test of energy. For my calculations, I usually converge these parameters in the following order:

(1) Kinetic energy cutoff for wavefunctions:

The total energy per atom must converge within a certain convergence threshold (usually around 0.1 mRy or 1.36 meV) for increasing energy cutoff. ecutwfc in QE or ENCUT in VASP.

(2) Kinetic energy cutoff for charge density:

Same as above but for charge density. ecutrho in QE. Usually 4 to 8 times larger than ecutwfc.

(3) K-point grid/k-mesh:

The total energy per atom must converge for increasingly dens k-mesh.

(4) smearing type and broadening:

The total energy per atom must converge for decreasing broadening value. degauss in QE.

When I perform a DFT+U calculation, I do not repeat the convergence tests. Also, if I do defect calculations such as atomic substitution or vacancy, then I relax the system again but do not perform a new set of convergence tests. Are the order of these convergence correct? Should I include any more convergence tests before proceeding to the actual calculation? For example, should I include a convergence test by increasing the size of the supercell to see if the total energy per atom is converging?

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The order of convergence tests you provided is a very good starting point which I believe will give very accurate numerical results in most cases. However, in my opinion and based on some prior experience with VASP, the inclusion of the onsite Coulomb interaction term (U) in DFT+U introduces an additional parameter that needs to be properly converged. I think the following steps might give you some insights:

  1. U value convergence: The total energy per atom should be examined as you vary the value of the onsite Coulomb interaction parameter (U). Typically, a range of U values is tested to ensure the convergence of the total energy and other relevant properties. It's important to choose a U value that accurately describes the electronic correlations in your system.
  2. Hubbard parameter convergence: In addition to the U value, some DFT+U methods include a Hubbard parameter (J) to account for the exchange interaction. Similar to U, the Hubbard parameter should be tested for convergence to ensure the accuracy of your results.
  3. Kinetic energy cutoff convergence: The kinetic energy cutoff for wavefunctions (ecutwfc) and charge density (ecutrho) should be re-evaluated to ensure convergence in the presence of the onsite Coulomb interaction. It is possible that different U values may require adjustments to these cutoffs for accurate calculations.
  4. K-point grid convergence: The convergence of the k-point grid or k-mesh should also be rechecked for DFT+U calculations. The inclusion of the onsite Coulomb interaction can affect the electronic structure, and it's important to ensure that the properties of interest converge with respect to the k-point sampling.

Regarding your question about increasing the size of the supercell, it is not directly related to the convergence of the DFT+U calculations. However, if you are studying defect calculations such as atomic substitution or vacancy, it can be beneficial to include a convergence test by increasing the size of the supercell. For instance, in relatively large defects such as trefoil-like or rotational defects, one needs to have a relatively large supercell to prevent the interaction between repeated images of the introduced defect. Thus, this test helps identify if the total energy per atom is converging and if the chosen supercell size is sufficient to obtain reliable defect properties.

I believe this is a very important question, and I hope we could get some answers from the developers of Electronic Stucture codes.

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    $\begingroup$ I disagree with a few points. The literatures I have come across so far refer U as the Hubbard parameter (on-site) and J as the Hund exchange. The U value itself needs to be converged but my question was whether we need to converge other parameters after applying a certain U value. DFT+U is just a linearly added constant correction term to the total Hamiltonian. Also, E is not variational with respect to U values. So, I don't see the point of repeating convergence tests. However, I see why one might want to repeat convergence after doping since doping introduces new pseudopotentials. $\endgroup$ Dec 11, 2023 at 13:49
  • $\begingroup$ any response to my previous comment? I am pretty sure why I do not need to repeat convergence test after applying DFT+U corrections but I am not 100% sure. If you could state your reasoning on why you suggest otherwise, it would be a great. In any case, I will award the bounty to your answer in a few hours. For my reasoning, see this excellent answer by Kevin J. May. $\endgroup$ Dec 18, 2023 at 15:19
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    $\begingroup$ I do agree with you that the correction term is added to the total Hamiltonian as a constant and you would not need to check its convergence, but to be honest I am not very sure that after determining a suitable Hubbard U value, the focus should shift to the specific effects of the DFT+U correction on the electronic structure rather than the convergence of the standard DFT parameters?! For example, if you are studying magnetic properties or a property that will be affected by the U value, does one need to ensure that the magnetic moments have converged with respect to the Hubbard U value? $\endgroup$ Dec 19, 2023 at 0:06
  • $\begingroup$ @AbdulMuhaymin That was my concern and as I said I am not quite sure. Other than that I would say I agree with you. $\endgroup$ Dec 19, 2023 at 0:07
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The order of convergence seems to be correct.
As far as I have seen - In supercell calculations the convergence is not required. Only the K-mesh values are needed to be decreased from the unit cell. One can check for different k-mesh sizes.
However while introduction of new doping atom one can check minimum cutoff of wavefunction and charge density from the pseudopotentials to see if they are not exceeding the chosen cutoffs.

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