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# Preconditioning and periodic sinc functions

We consider a unit cell (which we shall refer to as the simulation cell) with primitive lattice vectors ( ), volume , and grid points along direction , where the 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:

 (34)

where and are integers, and the are the reciprocal lattice vectors:

 (35)

and the are the grid points of the simulation cell,

 (36)

where , and are integers: , and similarly for and . 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 , where is any lattice vector. Fig. 1 shows a one-dimensional analogue of a single psinc function.

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

 (37)

where denotes a reciprocal lattice point, denotes a grid point of the simulation cell, and is the total number of grid points in the simulation cell.

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

 (38)

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

 (39)

and the matrix elements of in the psinc basis are given by

 (40)

where , the grid point weight, and .

The operator which diagonalises is none other than the discrete Fourier transform:

 (41) (42)

where the are values on the real space grid and the 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
 (43)

Thus the eigenvalues of are given by . Substituting this into Eq. (33) gives the final expression for our preconditioned line minimization:
 (44)

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