Basis set

We consider a unit cell (which we shall refer to as the simulation cell) with primitive lattice vectors $\mathbf{A}_{i}$ $(i=1,2,3)$, volume $V = \vert \mathbf{A}_{1} \cdot (\mathbf{A}_{2} \times \mathbf{A}_{3}) \vert$, and $N_{i} = 2J_{i} + 2$ grid points along direction $i$, where the $J_{i}$ are integers. We define our basis functions to be the cell periodic, bandwidth limited Dirac delta functions (Figure 1) given by,

$$D_{KLM}(\mathbf{r}) \equiv D(\mathbf{r} - \mathbf{r}_{KLM})$$

$$= \frac{1}{N_{1}N_{2}N_{3}}\sum_{p=-J_{1}}^{J_{1}+1}\sum_{q=-J_{2}}... ...bf{B}_{1}+q\mathbf{B}_{2}+s\mathbf{B}_{3})\cdot (\mathbf{r}-\mathbf{r}_{KLM})},$$ (1)

where $p,q$ and $s$ are integers, and the $\mathbf{B}_{i}$ are the reciprocal lattice vectors:

$$\mathbf{B}_{1} = \frac{2\pi(\mathbf{A}_{2}\times\mathbf{A}_{... ...ot (\mathbf{A}_{2} \times \mathbf{A}_{3})}, \:\: \mathrm{etc.}$$ (2)

Figure 1: The form of a basis function, $D(\mathbf{r})$, in two-dimensions

The NGWFs $\{ \phi_{\alpha} \}$ are expanded in terms of our basis $D(\mathbf{r})$,

$$\phi_{\alpha}(\mathbf{r}) = \sum_{KLM} C_{KLM,\alpha} D(\mathbf{r}-\mathbf{r}_{KLM}),$$ (3)

where the $C_{KLM,\alpha}$ are the expansion coefficients of $\phi_{\alpha}(\mathbf{r})$ in the basis $D(\mathbf{r})$ and the sum is over all the grid points of the simulation cell,

$$\mathbf{r}_{KLM} = \frac{K}{N_{1}}\mathbf{A}_{1} + \frac{L}{N_{2}}\mathbf{A}_{2} + \frac{M}{N_{3}}\mathbf{A}_{3},$$ (4)

where $K,L$, and $M$ are integers.

There is one basis function centered on each grid point of the simulation cell. They have the property that they are non-zero at the grid point on which they are centered and zero at all other grid points (48). This basis spans the same Hilbert space as the basis of plane-waves that can be represented by the real space grid of our simulation cell: a unitary transformation relates the two. Further properties are derived in Appendix A.

Due to the localisation of the NGWFs, the expansion coefficient, $C_{KLM,\alpha}$, of a particular NGWF, $\phi_{\alpha}$, is equal to zero if the grid point $\mathbf{r}_{KLM}$ does not fall within the LR of $\phi_{\alpha}$. Consequently, because the size of each LR is independent of system size, each NGWF is expanded in terms of a number of basis functions that is independent of system size.