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The psinc Basis

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

$\displaystyle s^{\ }_{ij}$ $\textstyle =$ $\displaystyle \int D^{\ast}_{i}(\mathbf{r}) D^{\ }_{j}(\mathbf{r})
\mathrm{d} \mathbf{r}$  
  $\textstyle =$ $\displaystyle \frac{1}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot
\math...
...{i ( \mathbf{k}_{q} - \mathbf{k}_{p} ) \cdot \mathbf{r} }
\mathrm{d} \mathbf{r}$  
  $\textstyle =$ $\displaystyle \frac{V}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot
\mathbf{r}_{i} - i\mathbf{k}_{q} \cdot \mathbf{r}_{j} }
\delta_{pq}$  
  $\textstyle =$ $\displaystyle \frac{V}{N^{2}} \sum_{p} \mathrm{e}^{i\mathbf{k}_{p} \cdot
( \mathbf{r}_{i} - \mathbf{r}_{j} ) }$  
  $\textstyle =$ $\displaystyle w \delta_{ij} ,$ (45)

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

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

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

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

$\displaystyle t^{\ }_{ij}$ $\textstyle =$ $\displaystyle - \int
D^{\ast}_{i}(\mathbf{r}) \nabla^{2} D^{\ }_{j}(\mathbf{r}) \mathrm{d}
\mathbf{r}$  
  $\textstyle =$ $\displaystyle - \frac{1}{N^{2}}
\sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \ma...
...nabla^{2} \mathrm{e}^{i \mathbf{k}_{q} \cdot \mathbf{r} } \mathrm{d}
\mathbf{r}$  
  $\textstyle =$ $\displaystyle \frac{1}{N^{2}} \sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot
\math...
...{i ( \mathbf{k}_{q} - \mathbf{k}_{p} ) \cdot \mathbf{r} }
\mathrm{d} \mathbf{r}$  
  $\textstyle =$ $\displaystyle \frac{V}{N^{2}}
\sum_{pq} \mathrm{e}^{i\mathbf{k}_{p} \cdot \math...
...i\mathbf{k}_{q}
\cdot \mathbf{r}_{j} } \vert\mathbf{k}_{q}\vert^{2} \delta_{pq}$  
  $\textstyle =$ $\displaystyle \frac{w}{N}
\sum_{p} \vert\mathbf{k}_{p}\vert^{2} \mathrm{e}^{i\mathbf{k}_{p} \cdot
(\mathbf{r}_{i}-\mathbf{r}_{j})}.$ (48)


next up previous
Next: Bibliography Up: Preconditioned iterative minimization Previous: Acknowledgments
Arash Mostofi 2003-10-28