For example, should a computation be considered to converge when the differences in energy per atom are smaller than a certain value?


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From the VASP wiki, ENCUT specifies the cutoff energy for the plane-wave-basis set in eV. This is the same as ecutwfc of Quantum ESPRESSO (which has a unit of Ry). In principle (according to the variational principle), one would obtain the lowest total energy when the energy cutoff is infinite. Since this is impossible in practice, we use as much value as we can afford for ENCUT.

Answer to the question

I haven't seen any specific recommended value for the convergence threshold on the total energy with respect to the kinetic energy cutoff. However, I have generally seen people doing it at $0.1 \;\mathrm{mRy}\; (\approx1.36 \;\mathrm{meV})$. The rationale behind this is the following.

At room temperature $(\mathrm{T}=300 \mathrm{K})$, the value of $\mathrm{k_BT} \approx 25 \; \mathrm{meV}$. So, $1.36 \;\mathrm{meV}$ is much smaller than that. Let's say you used ENCUT as $400 \;\mathrm{eV} $ and found the energy to be $E_1$. And for ENCUT = $600 \;\mathrm{eV} $, you found the energy to be $E_2$. Now, if $\Delta E = (E_2-E_1)/200 < 1.36 \;\mathrm{meV}$, you can safely use $600 \;\mathrm{eV} $ as your ENCUT. In other words, you have to divide the total energy differences by the difference in the kinetic energy cutoff and this should be less than your convergence criterion.

A little caution if you want to proceed to the vibrational properties: I am not sure if this is the case in VASP too, but in Quantum ESPRESSO one needs to use a larger ecutwfc in lattice vibrational calculation than the one used in the SCF calculation. This is because the phonon frequencies in Quantum ESPRESSO have the unit of $\mathrm{cm}^{-1}$ which roughly corresponds to $0.12 \;\mathrm{meV}$. So it requires a larger cutoff (to ensure a smaller $\Delta E$).


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