# Can conventional unit cell have higher symmetry than primitive cell for the same crystal?

This question arises from the last sentence of the first paragraph (see here) of the section "Formalism". If primitive cell can have lower symmetry, why don't the DFT results using these two type of cells do not differ?

How can one decide which one to choose for DFT studies?

• Yes. For instance NaCl - take a look at lampz.tugraz.at/~hadley/ss1/crystalstructure/structures/nacl/… Commented May 1 at 10:27
• This looks like low-effort question. But yes, that's the point of conventional unit cells. Copy-pasting from Wikipedia: In some cases, the full symmetry of a crystal structure is not obvious from the primitive cell, in which cases a conventional cell may be used. A conventional cell (which may or may not be primitive) is a unit cell with the full symmetry of the lattice Commented May 1 at 10:32

A primitive and conventional cell don't have different symmetries.

The primitive cell is a Wigner-Seitz cell of the same Bravais lattice; the primitive cell is the smallest choice, but the conventional cell is an equally valid choice. The primitive and conventional cells are necessarily the same space group and the choice of unitcell is conventional (c.f. the name 'conventional cell'...).

Picking a different conventional unitcell in real-space necessarily changes the reciprocal-space unitcell (Brillouin zone, BZ) in a way that totally compensates the different choice in real space.

By picking a conventional cell (assuming sufficiently converged $$k$$-points), you get the exact same answer for energies, forces, etc. as with a primitive cell because they are all integrals over the BZ. With a non-primitive unitcell, the BZ is smaller but there are more $$k$$-points 'folded' into the 1st BZ, so that all points are still counted and results are the same.

Obviously quantities like band-structure won't appear to be identical (due to BZ folding), but with some careful thought you should be able to convince yourself that the physics is the same.

Hope this helps! Ty

• Thank you @Tyler. So, is this sentence in the link- "...the fact that unit cells often exhibit higher symmetries and simpler Brillouin zones than primitive cells (an example is face centered cubic cells)" correct and I am missing any nuances hidden there? Commented May 3 at 20:56
• Ohhhhh! Sorry. I misunderstood a little. Ignoring the basis, the unit cell itself CAN have higher symmetry. Consider e.g. the primitive FCC unit cell of silicon and the conventional simple cubic one, which is higher symmetry. The reason the physics doesnâ€™t change is still true according to the arguments above! Commented May 5 at 2:27
• Your response is appreciated, @Tyler! Can you please help me imagine or understand: When NOT ignoring the basis, the symmetry of conventional unit cell and primitive cell of FCC crystal match exactly (i.e. does not differ in any symmetry elements)?!?! Commented May 5 at 9:49
• Exactly. By definition, a unit cell must reproduce the lattice by translation. It you pick a primitive cell or conventional cell, they have to generate the same structure. The lattice vectors themselves may be different for different choices, but when combined with the set of atoms in each unit cell, the space group etc must be the same. Commented May 6 at 14:05