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
(
), volume
, and
grid points along
direction , where the are integers. Our basis set
is composed of periodic bandwidthlimited
deltafunctions [20], from here
on referred to as periodic sinc or psinc functions, defined as
the following finite sum of plane waves:
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 cutoff
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 cellperiodic, namely
, where is any lattice
vector. Fig. 1 shows a onedimensional analogue
of a single psinc function.
Figure 1:
Onedimensional 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

(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:
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
20031028