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:

$$D_{klm}(\mathbf{r}) = D(\mathbf{r} - \mathbf{r}_{klm})$$

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

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

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

and the $\mathbf{r}_{klm}$ are the grid points of the simulation cell,

$$\mathbf{r}_{klm} = \frac{k}{N_{1}}\mathbf{A}^{(1)} + \frac{l}{N_{2}}\mathbf{A}^{(2)} + \frac{m}{N_{3}}\mathbf{A}^{(3)},$$ (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.

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

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

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

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

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

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

$$t_{ij} = \frac{w}{N} \sum_{p} k^{2}_{p} \mathrm{e}^{i\mathbf{k}_{p} \cdot (\mathbf{r}_{i} - \mathbf{r}_{j})},$$ (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:

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

$$b^{\ }_{i} = \sum_{p} F^{\dagger}_{ip} \tilde{b}^{\ }_{p} \equiv \frac{1}{\sqrt{N}} \sum_{p} \tilde{b}_{p} \mathrm{e}^{i \mathbf{k}_{p} \cdot \mathbf{r}_{i}} ,$$ (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

$$\tilde{x}^{\ }_{pq} = \sum_{ij} F^{\ }_{pi} x^{\ }_{ij} F^... ...w \left( 1 + \frac{k^{2}_{p}}{k^{2}_{0}} \right) \delta_{pq}.$$ (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:

$$\tilde{c}'^{\ }_{p\alpha} = \tilde{c}^{\ }_{p\alpha} - \fra... ...0} + k^{2}_{p}}\tilde{g}_{p}^{\ \beta} S^{\ }_{\beta \alpha}.$$ (44)