Plane wave expansion of the wavefunction for periodic material modelling

Question

The aim is to do an electronic calculation with static nucleus with Plane wave (PW) basis functions for an infinite periodic material. The number of functions is given by the pairs of reciprocal lattice vectors $G$ and momentum vector $k$ in the first Brillouin zone that respect the inequality $|k + G|^2 < E_{cutoff}$. Hence this inequality gives the number of PW basis functions as a function of $E_{cutoff}$.

Does this inequality also gives the size of the simulation?

By that, I mean the number of basis functions and the number of cells neighboring a 'reference cell' in a radius of $|G_{max}|$, where $|G_{max}|$ is the norm of the cells which are the most far from the 'reference' cell in the reciprocal space. Hence, giving the neighboring cells to compute the electron-electron and electron-proton potentials by solving the Poisson equation $$-\Delta V_{tot}= -4\pi (\sum_{\mathcal{T} \in \mathcal{R}} m(.+\mathcal{T}) - \rho)$$ where

• $\mathcal{T}$ is a translation in the real space $\mathcal{R}$
• $m$ the nuclei in the primitive unit cell
• $V_{tot}$ the electron-electron and electron-proton potentials
• $\rho$ is the electron density in the real/physical space this time.

Answer 1

No, $G_{max}$ is a vector in reciprocal space, so a large $|G_{max}|$ corresponds to a small real-space distance. As you increase your cut-off energy, $E_{cutoff}$, you are not extending the simulation to larger distances, you are resolving variations of the wavefunction over smaller distances.

The Fourier basis set is infinite in extent, so in some sense it can represent the entirety of the infinite material system, provided the assumption of perfect periodicity is valid. Bloch's theorem tells us that this is reliable for the particle density and the magnitude of the wavefunction, but not for the phase of the wavefunction -- this is where the phase $\vec{k}$ enters the discussion. There are solutions for any particular choice of phase, so the most general solution is to integrate over all possible choices of $\vec{k}$, which is commonly done by using a regular 3D sampling grid ("Monkhorst-Pack grid"), albeit symmetry-reduced.

For a particular rational choice of phase, e.g. in fractional coordinates $$ \vec{k} = \left(\frac{p}{q},0,0\right), $$ where $p$ and $q$ are integers, we can see that the phase does not change in the real-space $y$ and $z$ directions (I am assuming a Cartesian coordinate set for simplicity, but the argument holds in the general case as well). In the $x$ direction, the phase changes by $\frac{2\pi p}{q}$ every time we move one unit cell in that direction. After we have moved $q$ unit cells, the phase has changed by $2\pi p$ which means we are back in phase with the original cell (recall that $p$ is an integer).

Thus, any $k$-point component with an integer denominator, $q$, means that the phase is periodic with a periodicity of $q$ unit cells in the corresponding real-space direction. In this sense, the $k$-points control the "size" of the simulated system (in units of the reference cell), since they control the real-space range over which the phase is allowed to vary non-periodically. (This also shows why supercells require fewer k-points for the same accuracy as the reference cell.)