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Wraparound errors  convolutions
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
.
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
in real space it is given by
Using Fast Fourier transformations one can define

(7.3) 
Therefore the wavefunction can be written in real space as

(7.4) 
The charge density is simply given by

(7.5) 
in the reciprocal mesh it can be written as

(7.6) 
Inserting
from equation (7.5) and
from (7.3) it
is very easy to show that
contains Fouriercomponents up
to
.
Generally it can be shown that
a the convolution
of two 'functions' with Fouriercomponents
up to and with Fouriercomponents
up to contains Fouriercomponents 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 localpotential is given by
Only the components with
are taken into
account (see section 7.1: 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
).
Figure 5:
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 FFTmesh. Nevertheless
the components with
are correct i.e.
the small sphere does not intersect with the third large sphere

If the FFTmesh contains all components up to
the resulting wraparound error is once again 0. This can
be easily seen in Fig. 5.
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