Appendix B: From the simulation cell to the FFT box and back

Even though the FFT box is universal in shape and size for a given system, its position with respect to the grid of the simulation cell is determined by the pair of overlapping NGWFs, say $\phi_{\alpha}(\mathbf{r})$ and $\phi_{\beta}(\mathbf{r})$, we are dealing with at any given time. An operator therefore that would map $\phi_{\alpha}(\mathbf{r})$ from one representation to another would depend also on the position of $\phi_{\beta}(\mathbf{r})$. We therefore define such an operator for the pair of functions $\phi_{\alpha}(\mathbf{r})$ and $\phi_{\beta}(\mathbf{r})$ by

$$\hat{P}(\alpha\beta)= \frac{1}{W} \sum_{k=0}^{(n_1-1)} \sum_... ...ha \beta})(l + L_{\alpha \beta})(m +M_{\alpha \beta} )} \vert$$ (40)

where the numbers $K_{\alpha \beta}$, $L_{\alpha \beta}$ and $M_{\alpha \beta}$ denote the grid point of the simulation cell on which the origin of the FFT box is located. Here lowercase letters are used to represent quantities related to the FFT box, so $n_1$, $n_2$ and $n_3$ are the numbers of grid points in the FFT box in each lattice vector direction. Because of the periodic boundary conditions it should also be understood that if the indices of a delta function of the simulation cell exceed the grid point indices, then this function coincides with its periodic image that falls within the simulation cell. As an example, assume $N_1=N_2=N_3=20$. Then

$$D_{(5)(21)(23)}(\mathbf{r}) = D_{(5)(1)(3)}(\mathbf{r}) \, .$$ (41)

We also need to define an operator that projects a function from the portion of the fine grid associated with functions $\phi_{\alpha}(\mathbf{r})$ and $\phi_{\beta}(\mathbf{r})$ to the FFT box. Such an operator is defined in a similar fashion to $\hat{P}(\alpha \beta)$ by

$$\hat{Q}(\alpha\beta)= \frac{8}{W} \sum_{x=0}^{(2n_1-1)} \su... ...}) (y+2L_{\alpha \beta}) (z + 2M_{\alpha \beta}) } \vert \,\,.$$ (42)

Operators $\hat{P}^{\dag }(\alpha \beta)$ and $\hat{Q}^{\dag }(\alpha \beta)$ map a function from the FFT box to the simulation cell in the standard and fine grids respectively.