next up previous
Next: Results Up: Preconditioned iterative minimization Previous: Orthogonal basis

Preconditioning and periodic sinc functions

We consider a unit cell (which we shall refer to as the simulation cell) with primitive lattice vectors $\mathbf{A}^{(i)}$ ( $i \in \{
1,2,3\}$), volume $V = \vert \mathbf{A}^{(1)} \cdot (\mathbf{A}^{(2)} \times
\mathbf{A}^{(3)}) \vert$, and $N_{i} = 2J_{i} + 1$ grid points along direction $i$, where the $J_{i}$ are integers. Our basis set is composed of periodic bandwidth-limited delta-functions [20], from here on referred to as periodic sinc or psinc functions, defined as the following finite sum of plane waves:

$\displaystyle D_{klm}(\mathbf{r})$ $\textstyle =$ $\displaystyle D(\mathbf{r} - \mathbf{r}_{klm})$  
  $\textstyle =$ $\displaystyle \frac{1}{N_{1}N_{2}N_{3}}
\sum_{p=-J_{1}}^{J_{1}} \sum_{q=-J_{2}}...
... +
q\mathbf{B}^{(2)} + s\mathbf{B}^{(3)}) \cdot
(\mathbf{r}-\mathbf{r}_{klm})},$ (34)

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

\mathbf{B}^{(1)} =
\frac{2\pi}{V}(\mathbf{A}^{(2)}\times\mathbf{A}^{(3)}), \:\: \mathrm{etc.}
\end{displaymath} (35)

and the $\mathbf{r}_{klm}$ are 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)},
\end{displaymath} (36)

where $k,l$, and $m$ are integers: $k \in \{0,1,\ldots,N_{1}-1\}$, and similarly for $l$ and $m$. There is one psinc function centered on each grid point of the simulation cell.

The name ``periodic sinc'', or psinc, has been chosen to reflect the connection that this function has with the familiar ``cardinal sine'' or sinc function. The sinc function is a continuous integral of plane waves with unit coefficients up to a maximum cut-off frequency. The psinc function differs only in that this continous integral is replaced by a finite sum over the reciprocal lattice vectors of the simulation cell, as in Eq. (34). As a result, whereas the sinc function decays to zero at infinity, the psinc function is cell-periodic, namely $D(\mathbf{r}) =
D(\mathbf{r}+\mathbf{R})$, where $\mathbf{R}$ is any lattice vector. Fig. 1 shows a one-dimensional analogue of a single psinc function.

Figure 1: One-dimensional analogue of a single periodic sinc, or psinc function, centered on the origin. In this example the simulation cell is eleven grid points in length.
\scalebox{0.7}{\includegraphics*{304341JCP1.eps} }

From this point onward, for simplicity of notation, we write the psinc functions introduced in Eq. (34) as

D_{i}(\mathbf{r}) = \frac{1}{N} \sum_{p}
\mathrm{e}^{i \mathbf{k}_{p} \cdot (\mathbf{r} - \mathbf{r}_{i})},
\end{displaymath} (37)

where $\mathbf{k}_{p}$ denotes a reciprocal lattice point, $\mathbf{r}_{i}$ denotes a grid point of the simulation cell, and $N=N_{1}N_{2}N_{3}$ is the total number of grid points in the simulation cell.

Using the same model Hamiltonian $\hat{X}$ given by Eq. (14) along with the definitions presented in Eqs. (27) and (28), we write

x_{ij} = s_{ij} + k^{-2}_{0}t_{ij}.
\end{displaymath} (38)

As shown in the Appendix, the psinc functions are orthogonal,

s_{ij} = w \delta_{ij},
\end{displaymath} (39)

and the matrix elements of $-\nabla^{2}$ in the psinc basis are given by

t_{ij} = \frac{w}{N} \sum_{p}
k^{2}_{p} \mathrm{e}^{i\mathbf{k}_{p} \cdot (\mathbf{r}_{i} -
\end{displaymath} (40)

where $w=V/N$, the grid point weight, and $k_{p}=\vert\mathbf{k}_{p}\vert$.

The operator $\mathbf{F}$ which diagonalises $\mathbf{x}$ is none other than the discrete Fourier transform:

$\displaystyle \tilde{b}_{p} = \sum_{j} F_{pj} b_{j}$ $\textstyle \equiv$ $\displaystyle \frac{1}{\sqrt{N}}
\sum_{j} b_{j}
\mathrm{e}^{-i \mathbf{k}_{p} \cdot \mathbf{r}_{j}} ,$ (41)
$\displaystyle b^{\ }_{i} = \sum_{p} F^{\dagger}_{ip} \tilde{b}^{\ }_{p}$ $\textstyle \equiv$ $\displaystyle \frac{1}{\sqrt{N}}
\sum_{p} \tilde{b}_{p} \mathrm{e}^{i \mathbf{k}_{p} \cdot \mathbf{r}_{i}} ,$ (42)

where the $b_{i}$ are values on the real space grid and the $\tilde{b}_{p}$ are values on the reciprocal space grid. Using these definitions, along with Eqs. (38)-(40) and Eq. (47), it is a simple matter to show that
\tilde{x}^{\ }_{pq} = \sum_{ij} F^{\ }_{pi} x^{\ }_{ij}
...w \left( 1 + \frac{k^{2}_{p}}{k^{2}_{0}}
\right) \delta_{pq}.
\end{displaymath} (43)

Thus the eigenvalues $\xi_{p}$ of $\mathbf{x}$ are given by $\xi_{p} = w (1 + k^{2}_{p}/k^{2}_{0})$. Substituting this into Eq. (33) gives the final expression for our preconditioned line minimization:
\tilde{c}'^{\ }_{p\alpha} = \tilde{c}^{\ }_{p\alpha} -
...0} +
k^{2}_{p}}\tilde{g}_{p}^{\ \beta} S^{\ }_{\beta \alpha}.
\end{displaymath} (44)

next up previous
Next: Results Up: Preconditioned iterative minimization Previous: Orthogonal basis
Arash Mostofi 2003-10-28