In this section we will discuss wrap around errors.
Wrap around errors arise if the FFT meshes are not
sufficiently large. It can be shown that no errors exist
if the FFT meshes contain all vectors up to .
Fig. 1: Sphere intersections for
It can be shown that the charge density contains
components up to , where is
the "longest" plane wave in the basis set:
The wavefunction is defined as
and in real space it is given by
Using Fast Fourier transformations one can define
Therefore the wavefunction can be written in real space as
The charge density is simply given by
and in the reciprocal mesh it can be written as
Using the above equations for and it
is very easy to show that contains Fourier-components up
Generally it can be shown that
a the convolution
of two functions with Fourier-components
up to and with Fourier-components
up to contains Fourier-components up to .
The property of the convolution comes once again into play,
when the action of the Hamiltonian onto a wavefunction is
calculated. The action of the local-potential is given by
Only the components with are taken into
account (see section ALGO: is added to the wavefunction
during the iterative refinement of the wavefunctions ,
and contains only components up to ).
From the previous theorem we see that contains
components up to ( contains components up to
If the FFT-mesh contains all components up to
the resulting wrap-around error is once again 0. This can
be easily seen in Fig. 1. Here we see that the small sphere contains all plane waves included in the basis set .
The charge density contains components up to (second sphere), and
the acceleration components up to , which are reflected
in (third sphere) because of the finite size of the FFT-mesh. Nevertheless
the components with are correct i.e.
the small sphere does not intersect with the third large sphere}