View source for Wrap-around errors

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 $G$ vectors up to <math>2 G_{\rm cut}</math>.

It can be shown that the charge density contains
components up to <math>2 G_{\rm cut}</math>, where <math>2 G_{\rm cut}</math> is
the "longest" plane wave in the basis set:

The wavefunction is defined as

<math>
 | \phi_{n\bold{k}} \rangle = \sum_G C_{Gn\bold{k}}  | \bk+\bG\rangle,
</math>

and in real space it is given by

<math>
 \langle\br| \phi_\nk \rangle = \sum_\bG  \langle\br| \bk+\bG\rangle \langle \bk+\bG|\phi_\nk \rangle
     =   \frac{1}{\Omega^{1/2}} \sum_\bG e^{i(\bk+\bG)\br}  C_\Gnk.
</math>

Using Fast Fourier transformations one can define

<math>
 C_\rnk= \sum_{\bG} C_\Gnk  e^{i\bG \br}  \hspace{2cm}
 C_\Gnk= \frac{1}{N_{\rm FFT}} \sum_{\br} C_\rnk  e^{-i\bG \br}.
</math>

Therefore the wavefunction can be written in real space as

<math>
 \langle\br| \phi_\nk \rangle = \phi_\nk(r) = \frac{1}{\Omega^{1/2}} C_\rnk e^{i\bk\br}.
</math>

The charge density is simply given by

<math>
 \rhops_\br \equiv \langle\br | \rhops| \br\rangle =
 \sum_\bk w_\bk \sum_n f_\nk \phi_\nk(r) \phi^*_\nk(r) ,
</math>

and in the reciprocal mesh it can be written as

<math>
 \rhops_\bG  \equiv
    \frac{1}{\Omega} \int \langle\br | \rhops| \br\rangle e^{-i \bG\br}\, d \br \to \frac{1}{N_{\rm FFT}} \sum_{\br} \rhops_\br e^{-i \bG\br}.
</math>

{{template:NumBlk}}

Inserting <math>\rhops</math> from equation (\ref{rps}) and $ C_\rnk$ from (\ref{pl2}) it
is very  easy to show that $\rhops_\br $ contains Fourier-components up
to $2 G_{\rm cut}$.

Generally it can be shown that
a the convolution $f_r=f^1_r f^2_r$
of two 'functions' $f^1_r$ with  Fourier-components
up to $G_1$ and $f^2_r$ with Fourier-components
up to $G_2$ contains Fourier-components up to  $G_1+G_2$.

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
\[
 a_\br =  V_\br   C_\rnk 
\]
Only the components $a_\bG$ with $|\bG| < G_{\rm cut}$ are taken into
account (see section \ref{algo}: $a_\bG$ is added to the wavefunction
during the iterative refinement of the wavefunctions $C_\Gnk$,
and  $C_\Gnk$ contains only components up to  $G_{\rm cut}$).
From the previous theorem we see that $a_\br$ contains
components up to $3 G_{\rm cut}$ ($V_\br$ contains components up to

$2 G_{\rm cut}$).
\begin{figure}  \unitlength1cm
\epsffile{algo1.eps}
\caption{ \label{algo-fig1}
The small sphere contains all plane waves included in the basis set  $G<G_{\rm cut}$.
The charge density contains components up to  $2 G_{\rm cut}$ (second sphere), and
the acceleration $a$ components up to  $3 G_{\rm cut}$, which are reflected
in (third sphere) because of the finite size of the FFT-mesh. Nevertheless
the components $a_\bG$ with $| \bG| < G_{\rm cut}$ are correct i.e.
the small sphere does not intersect with the third large sphere
}
\end{figure}
If the FFT-mesh contains all components up to  $2 G_{\rm cut}$
the resulting wrap-around error is once again 0. This can
be easily seen in Fig. \ref{algo-fig1}.

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