What happens to the position coordinates of the nearest neighbors in supercell studies?

Question

Can anyone tell what happens to the position coordinates of the nearest neighbors when we choose to study supercell of the same material. That remains intact or any changes will be there?

For example if we have a cubic lattice then its lattice parameters are:

 [1.0, 0.0, 0.0],[0.0, 1.0, 0.0],[0.0, 0.0, 1.0] 

and nearest neighbour coordinates are:

 [1,0,0],[0,1,0],[0,0,1],[-1,0,0],[0,-1,0],[0,0,-1]. 

If we take 222 supercell of it then the position coordinates will be changed or remains same?

Answer 1

DFT calculations based on a supercell method utilize the periodicity of a perfect, infinite lattice by imposing periodic boundary conditions onto the wavefunction of the system (Bloch's theorem). In this approach all calculations must be performed on a periodic system, even when the periodicity is superficial. Thus, for example, a crystal surface must be represented by a finite-length slab. Similarly, to study molecules it is necessary to assume that they are in a box and treat them as periodic systems.

The main advantage of imposing periodic boundary conditions relates to Bloch's theorem, which states that in a periodic system each electronic wavefunction $\psi_{n, k} (\pmb{r})$ can be written as a product of a cell-periodic part $u(\pmb{r})$ and a wavelike phase factor $e^{ik\cdot\pmb{r}}$:

$$\psi_{n, k} (\pmb{r}) = u(\pmb{r}) \:e^{ik\cdot\pmb{r}}$$

Thus there are always an infinite array of periodic units in the background of such simulations. When simulating a single unit cell, calculations are explicitly performed on the electrons and ions contained within this unit cell. Nontheless, the periodic images identical to your cell are always accounted for implicitly. That is why in a study of a single defect, for example, it is essential to introduce enough separation between artificial images (i.e., build a big supercell) of such nonperiodic objects to ensure that there is no appreciable interaction between them. The situation is similar for surface calculations that are carried out in a slab geometry. The vacuum layer should be sufficiently thick to eliminate artificial interactions between the slabs.

To answer your question in short, physical properties calculated from a unit cell and any supercell size built from this unit cell are exactly the same. The position coordinates remain the same. Of course, this statement is no longer correct as soon as you introduce a defect (e.g. Doping) into your supercell.

Answer 2

In solid-state calculations, DFT code does not know whether you give it a crystalline unit cell or a supercell. Instead, it treats every input structure as a "unit cell".

So my answer is "remains same".