# Is total energy difference lower than 1 meV/atom good enough for k-point convergence?

How much energy difference could be considered as tight convergence? Is total energy difference lower than 1 meV/atom good enough for kpoint convergence for total energy calculation?

• This depends on the quantity you are interested in. For example, the energy difference between fcc and hcp Cu is about 8 meV/atom (with the PBE functional). Your precision nearly is on that order of magnitude. But total energies and total energy differences show different convergence behavior. The differences converge faster. The important thing therefore is to also have comparable parameter sets for different calculations. But this example also demonstrates a rather small energy difference. Oct 10, 2020 at 11:23
• Oct 10, 2020 at 12:19

## 1 Answer

When using a method like density functional theory we have to consider the accuracy of (i) the numerical solution of the problem and (ii) the physical model we use.

The convergence you are refering to is numerical (i.e. how many $$\mathbf{k}$$-points to include in the numerical approximation of replacing an integral over the Brillouin zone with a discrete sum). In this case, convergence is not something that you can decide on in absolute terms: it depends on what you are interested in. For example, if you are interested in resolving an energy of 100 meV/atom, then converging to below 1 meV/atom is definitely good enough. However, if you want to resolve an energy of 2 meV/atom, then converging below 1 meV/atom may not be sufficient.

Having said this, it is also important to keep in mind the accuracy of the physical model used. In the example of density functional theory, you can converge the numerics of your system to below 1 meV/atom, but you may have uncertainties that can be several orders of magnitude larger due to the physical approximations (e.g. choice of exchange-correlation functional).

What can we conclude from this? I would say that in most situations, converging energy differences to below 1 meV/atom is good enough. DFT is typically not accurate on the 1 meV/atom energy scale, so the physical model places a strong constraint on what you can achieve. Having said this, you should always consider your convergence on a case-by-case basis to decide what you need to confidently answer the scientific question you have.

• There are quantities on very fine energy scales that can be predicted by DFT. For example the magnetocrystalline anisotropy or spin-spiral dispersion relations. The question here is whether the inaccuracies that come with DFT are also relevant for the energy differences under investigation. Often there is an error cancellation. Oct 10, 2020 at 11:31
• @GregorMichalicek I agree with your point, and I think that my answer also makes it: in many circumstances 1 meV/atom is good enough, but one should always check depending on the quantity and level of convergence one is interested in because it may not be good enough. Oct 10, 2020 at 11:44