The psinc Basis

The overlap matrix $\mathbf{s}$ of the psinc functions defined in Eq. (37) is given by

$$s^{\ }_{ij} = \int D^{\ast}_{i}(\mathbf{r}) D^{\ }_{j}(\mathbf{r}) \mathrm{d} \mathbf{r}$$

$$= \frac{1}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \math... ...{i ( \mathbf{k}_{q} - \mathbf{k}_{p} ) \cdot \mathbf{r} } \mathrm{d} \mathbf{r}$$

$$= \frac{V}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \mathbf{r}_{i} - i\mathbf{k}_{q} \cdot \mathbf{r}_{j} } \delta_{pq}$$

$$= \frac{V}{N^{2}} \sum_{p} \mathrm{e}^{i\mathbf{k}_{p} \cdot ( \mathbf{r}_{i} - \mathbf{r}_{j} ) }$$

$$= w \delta_{ij} ,$$ (45)

where $w=V/N$ is the grid point weight, and we have used the relations

$$\int \mathrm{e}^{i ( \mathbf{k}_{p} - \mathbf{k}_{q} ) \cdot \mathbf{r} } \mathrm{d} \mathbf{r} = V \delta_{pq},$$ (46)

and

$$\sum_{p} \mathrm{e}^{i\mathbf{k}_{p} \cdot ( \mathbf{r}_{i} - \mathbf{r}_{j} )} = N \delta_{ij}.$$ (47)

Furthermore, the matrix elements of $-\nabla^{2}$ in this basis are given by

$$t^{\ }_{ij} = - \int D^{\ast}_{i}(\mathbf{r}) \nabla^{2} D^{\ }_{j}(\mathbf{r}) \mathrm{d} \mathbf{r}$$

$$= - \frac{1}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \ma... ...nabla^{2} \mathrm{e}^{i \mathbf{k}_{q} \cdot \mathbf{r} } \mathrm{d} \mathbf{r}$$

$$= \frac{1}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \math... ...{i ( \mathbf{k}_{q} - \mathbf{k}_{p} ) \cdot \mathbf{r} } \mathrm{d} \mathbf{r}$$

$$= \frac{V}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \math... ...i\mathbf{k}_{q} \cdot \mathbf{r}_{j} } \vert\mathbf{k}_{q}\vert^{2} \delta_{pq}$$

$$= \frac{w}{N} \sum_{p} \vert\mathbf{k}_{p}\vert^{2} \mathrm{e}^{i\mathbf{k}_{p} \cdot (\mathbf{r}_{i}-\mathbf{r}_{j})}.$$ (48)