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Suppose I have a system with a property that is highly dependent on the density of k-points, namely the dielectric function. Using a small supercell, say with 4 atoms, I've reached the maximum allowed sampling of Quantum ESPRESSO (34x34x34), and I am not confident that I've achieved the convergence. To go further I should change the source code and recompile the program, however, this is not the case by the moment. If a make a supercell with twice the volume of the former, using the same k-sampling (the maximum one, for instance) ensures a denser mesh. Is this procedure a good workaround to increase the sampling, since I still have a manageable structure and the runtime is acceptable?

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    $\begingroup$ Using supercell will give you electronic band folding. I don't know what is worse. $\endgroup$ – Camps Aug 19 '20 at 12:31
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    $\begingroup$ +1 @Camps is correct about band folding, so an important aspect of your question is what the property of interest is. Can you please clarify this? $\endgroup$ – ProfM Aug 19 '20 at 13:54
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    $\begingroup$ @ProfM, the quantity is the dielectric function. I've edited the question. $\endgroup$ – Anibal Bezerra Aug 19 '20 at 14:45
  • $\begingroup$ @Camps, I'm aware of the band folding, but I'm not sure about its influence on the desired property, I don't think it gonna be worse. Should I? $\endgroup$ – Anibal Bezerra Aug 19 '20 at 14:48
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Short answer: Yes, doubling the volume of the simulation cell you will be able to effectively sample a finer $\mathbf{k}$-point grid to calculate the dielectric function. However, the calculation will be more expensive than simply increasing the number of $\mathbf{k}$-points directly in the primitive cell calculation.

Longer answer: In reciprocal space, the volume of the Brillouin zone is divided by two when you double the real space supercell volume. This means that, if you keep the same density of $\mathbf{k}$-points you had for your original cell, then the number of $\mathbf{k}$-points in the new BZ will only be half of the number you had in the original BZ. However, the states in the original BZ corresponding to $\mathbf{k}$-points that are now left outside the BZ of the supercell will be folded into the new BZ, such that the total number of states per $\mathbf{k}$-point will be twice as many as you originally had, so that overall you have the exact same level of sampling. What you are proposing to do is to increase the density of $\mathbf{k}$-points by sampling the same number of points in the new smaller BZ compared to the original larger BZ. In this way, you are effectively increasing your $\mathbf{k}$-point sampling.

This strategy should work OK for quantities that depend on a BZ integration, such as the calculation of the total energy or of the dielectric function. However, for quantities for which you are interested in the location of states in the original BZ (for example to locate the band gap extrema of an insulator), then band folding will complicate the analysis significantly.

A final point: although this strategy will work, I would highly recommend that you do modify Quantum Espresso to increase the hard-coded limit on the number of $\mathbf{k}$-points so that you can do the calculation with a primitive cell. With this strategy the computational cost should be smaller.

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