# Number of k-points for unit and super cell

I am just new with solid state calculation and would like to know one thing.

Why is plane wave basis set is good for 1s and 2s orbitals, but for higher convergence is poor? Why is it better to use bigger number of k-points for case of unit cell relaxation (or any other calculations) and only one Г point for supercell?

• At least if you gave references justifying your statements I would understand, there is a misconception. Commented Jul 7, 2023 at 15:40
• Asking two questions in a single post is generally discouraged in MMSE. I have removed your second question "Why is it better to use bigger number of k-points for case of unit cell relaxation (or any other calculations) and only one Г point for supercell?" Please ask this as a separate question. Commented Jul 8, 2023 at 14:17
• @HemanthHaridas the original question has already an answer (actually two answers), so what is the point of removing part of the question now. I think this even makes the situation more confusing than helping.
– Sha
Commented Jul 8, 2023 at 21:11
• @Sha Please see this someone has already voted to close this question due to this reason. Commented Jul 9, 2023 at 2:48
• Dear @HemanthHaridas I agree with you and the rules of MSSE, I have no problem with this. But when OP has already received two answers for his "two" questions, then I do not see any good reason to remove part of the question, this even makes the answers awkward for someone reading this topic in future (part of the answer without question at the first place).
– Sha
Commented Jul 9, 2023 at 9:14

Why is plane wave basis set is good for 1s and 2s orbitals, but for higher convergence is poor?

I guess by "higher" you mean higher principle quantum number $$n$$ (i.e., 3s, 4s-orbitals). For these orbitals, the radial wavefunctions have lots of wiggles near the atomic core (so that they are orthogonal to the other s-orbitals). To represent a wave function by a set of plane waves, you are essentially expanding that wave function using the Fourier series, and a sharp peak needs a Fourier basis (in this context, the Plane waves) with much higher frequencies to be accurately described.

As the plane wave (PW) basis can be systematically improved by setting a higher kinetic energy cut-off (which is related to the highest frequency component of the PW basis) I wouldn't say the convergence is a problem, you just need to include more PWs. The problem is the cost of storing and performing matrix operations with a huge amount of PWs.

As a side note, modern DFT codes utilize pseudo potentials with frozen core approximation so that only certain frontier valence orbitals are used in the actual calculations and their "wiggly tail" behavior near the core region is replaced with a smooth one (and the corresponding wave functions are called the pseudo wave functions) so that less PW is needed to describe them. This of course comes with a certain price but I will not dive into the details here.

Why is it better to use bigger number of k-points for case of unit cell relaxation (or any other calculations) and only one $$\Gamma$$ point for supercell?

This is related to the fact that the reciprocal space is the Fourier transform of the real space, so the cell vector in reciprocal space is inversely related to the cell vector in real space. For very small cells (e.g., unit cells), the lattice vector in reciprocal space is very large, thus we need a large number of k-points to sample the crystal cell in reciprocal space (Brillouin zone). However, when the crystal cell is very large in real space (e.g., large supercell), the Brillouin zone is very small, and usually $$\Gamma$$ point is enough to describe that cell.

Hope this helps.

I will explicit my comment, the use of plane waves (PWs) for 1s and 2s is completely absurd when a linear combination of atomic orbitals gives a fast convergence. Even if they lack of significant oscillations as said in the previous answer, they are really localized, several Pws are still needed to match their structure. PWs should be used only when the potential does not vary considerably, this means for larger $$n$$ completely in desagreement with your assumption (a basic knowledge LAPW methods may help). The overlap of larger $$n$$ orbitals creates an slightly homogenous density, a few number of PW can converge quickly to a complete basis-set. A PW basis-set is a good choice compared LCAO in this case. The short-range oscillations for the orthogonality of large $$n$$ orbitals is corrected by a pseudo WF along with a pseudopotential. No matter how your cell is chosen and consequently the size of your Brillouin zone, you cannot restrict your to a single k-point $$\Gamma$$ or one another for a periodic boundary condition system. Obviously there is no need to have a large number of k-points for a large surpercell which is also associated to a computational expense.