Wrap-around errors: Difference between revisions

Latest revision as of 09:00, 24 October 2025

Wrap-around errors arise if the Fast Fourier transformation (FFT) meshes are not sufficiently large. It can be shown that no errors exist if the FFT meshes contain all [math]\displaystyle{ \mathbf{G} }[/math] vectors up to [math]\displaystyle{ 2 G_{\rm cut} }[/math].

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

The wavefunction is defined as

[math]\displaystyle{ | \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G} C_{\mathbf{G}n\mathbf{k}} | \mathbf{k}+\mathbf{G}\rangle, }[/math]

and in real space it is given by

[math]\displaystyle{ \langle \mathbf{r}| \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G} \langle \mathbf{r}| \mathbf{k}+\mathbf{G}\rangle \langle \mathbf{k}+\mathbf{G}|\phi_{n\mathbf{k}} \rangle = \frac{1}{\Omega^{1/2}} \sum_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}} C_{\mathbf{G}n\mathbf{k}}. }[/math]

Using FFTs one can define

[math]\displaystyle{ C_{\mathbf{r}n\mathbf{k}}= \sum_{\mathbf{G}} C_{\mathbf{G}n\mathbf{k}} e^{i\mathbf{G} \mathbf{r}} \qquad \qquad \qquad C_{\mathbf{G}n\mathbf{k}}= \frac{1}{N_{\mathrm{FFT}}} \sum_{\mathbf{r}} C_{\mathbf{r}n\mathbf{k}} e^{-i\mathbf{G} \mathbf{r}}. }[/math]

Therefore the wavefunction can be written in real space as

[math]\displaystyle{ \langle\mathbf{r}| \phi_{n\mathbf{k}} \rangle = \phi_{n\mathbf{k}}(r) = \frac{1}{\Omega^{1/2}} C_{\mathbf{r}n\mathbf{k}} e^{i\mathbf{k}\mathbf{r}}. }[/math]

The charge density is simply given by

[math]\displaystyle{ \rho^{\mathrm{ps}}_{\mathbf{r}} \equiv \langle \mathbf{r} |\rho^{\mathrm{ps}} | \mathbf{r} \rangle = \sum_\mathbf{k} w_{\mathbf{k}} \sum_n f_{n\mathbf{k}} \phi_{n\mathbf{k}}(r) \phi^{*}_{n\mathbf{k}}(r) , }[/math]

and in the reciprocal mesh it can be written as

[math]\displaystyle{ \rho^{\mathrm{ps}}_\mathbf{G} \equiv \frac{1}{\Omega} \int \langle\mathbf{r} | \rho^{\mathrm{ps}}| \mathbf{r}\rangle e^{-i \mathbf{G}\mathbf{r}}\, d \mathbf{r} \to \frac{1}{N_{\mathrm{FFT}}} \sum_{\mathbf{r}} \rho^{\mathrm{ps}}_{\mathbf{r}} e^{-i \mathbf{G}\mathbf{r}}. }[/math]

Using the above equations for [math]\displaystyle{ \rho^{\mathrm{ps}}_{\mathbf{r}} }[/math] and [math]\displaystyle{ C_{\mathbf{r}n\mathbf{k}} }[/math] it is very easy to show that [math]\displaystyle{ \rho^{\mathrm{ps}}_{\mathbf{r}} }[/math] contains Fourier-components up to [math]\displaystyle{ 2 G_{\mathrm{cut}} }[/math].

Generally it can be shown that a the convolution [math]\displaystyle{ f_r=f^1_r f^2_r }[/math] of two functions [math]\displaystyle{ f^1_r }[/math] with Fourier-components up to [math]\displaystyle{ G_1 }[/math] and [math]\displaystyle{ f^2_r }[/math] with Fourier-components up to [math]\displaystyle{ G_2 }[/math] contains Fourier-components up to [math]\displaystyle{ G_1+G_2 }[/math].

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

[math]\displaystyle{ a_{\mathbf{r}} = V_{\mathbf{r}} C_{\mathbf{r}n\mathbf{k}}. }[/math]

Only the components [math]\displaystyle{ a_{\mathbf{G}} }[/math] with [math]\displaystyle{ |\mathbf{G}| \lt G_{\mathrm{cut}} }[/math] are taken into account (see section ALGO: [math]\displaystyle{ a_{\mathbf{G}} }[/math] is added to the wavefunction during the iterative refinement of the wavefunctions [math]\displaystyle{ C_{\mathbf{G}n\mathbf{k}} }[/math], and [math]\displaystyle{ C_{\mathbf{G}n\mathbf{k}} }[/math] contains only components up to [math]\displaystyle{ G_{\mathrm{cut}} }[/math]). From the previous theorem we see that [math]\displaystyle{ a_{\mathbf{r}} }[/math] contains components up to [math]\displaystyle{ 3 G_{\mathrm{cut}} }[/math] ([math]\displaystyle{ V_{\mathbf{r}} }[/math] contains components up to [math]\displaystyle{ 2 G_{\mathrm{cut}} }[/math]).

If the FFT mesh contains all components up to [math]\displaystyle{ 2 G_{\mathrm cut} }[/math] 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 [math]\displaystyle{ G\lt G_{\mathrm{cut}} }[/math]. The charge density contains components up to [math]\displaystyle{ 2 G_{\mathrm{cut}} }[/math] (second sphere), and the acceleration [math]\displaystyle{ a }[/math] components up to [math]\displaystyle{ 3 G_{\mathrm{cut}} }[/math], which are reflected in (third sphere) because of the finite size of the FFT mesh. Nevertheless the components [math]\displaystyle{ a_{\mathbf{G}} }[/math] with [math]\displaystyle{ | \mathbf{G}| \lt G_{\mathrm{cut}} }[/math] are correct i.e. the small sphere does not intersect with the third large sphere}

@@ Line 1: / Line 1: @@
-In this section we will discuss wrap around errors.
+'''Wrap-around errors''' arise if the [[FFT meshes| Fast Fourier transformation (FFT) meshes]] are not
-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>.
+if the [[FFT meshes]] contain all <math>\mathbf{G}</math> vectors up to <math>2 G_{\rm cut}</math>.
+[[File:Wrap errors spheres.png|350px|thumb|Fig. 1: Sphere intersections for <math>G_{\mathrm{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
+components up to <math>2 G_{\mathrm{cut}}</math>, where <math>2 G_{\mathrm{cut}}</math> is
 the "longest" plane wave in the basis set:
 The wavefunction is defined as
-<math>
+::<math>
-  | \phi_\nk \rangle = \sum_G C_\Gnk  | \bk+\bG\rangle,
+  | \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G} C_{\mathbf{G}n\mathbf{k}}  | \mathbf{k}+\mathbf{G}\rangle,
 </math>
 and in real space it is given by
-<math>
+::<math>
-  \langle\br| \phi_\nk \rangle = \sum_\bG  \langle\br| \bk+\bG\rangle \langle \bk+\bG|\phi_\nk \rangle
+  \langle \mathbf{r}| \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G}  \langle \mathbf{r}| \mathbf{k}+\mathbf{G}\rangle \langle \mathbf{k}+\mathbf{G}|\phi_{n\mathbf{k}} \rangle
-      =   \frac{1}{\Omega^{1/2}} \sum_\bG e^{i(\bk+\bG)\br}  C_\Gnk.
+      =   \frac{1}{\Omega^{1/2}} \sum_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}}  C_{\mathbf{G}n\mathbf{k}}.
 </math>
-Using Fast Fourier transformations one can define
+Using FFTs one can define
-<math>
+::<math>
-  C_\rnk= \sum_{\bG} C_\Gnk  e^{i\bG \br}  \hspace{2cm}
+  C_{\mathbf{r}n\mathbf{k}}= \sum_{\mathbf{G}} C_{\mathbf{G}n\mathbf{k}}  e^{i\mathbf{G} \mathbf{r}} \qquad \qquad \qquad
-  C_\Gnk= \frac{1}{N_{\rm FFT}} \sum_{\br} C_\rnk  e^{-i\bG \br}.
+  C_{\mathbf{G}n\mathbf{k}}= \frac{1}{N_{\mathrm{FFT}}} \sum_{\mathbf{r}} C_{\mathbf{r}n\mathbf{k}}  e^{-i\mathbf{G} \mathbf{r}}.
 </math>
 Therefore the wavefunction can be written in real space as
-<math>
+::<math>
-  \langle\br| \phi_\nk \rangle = \phi_\nk(r) = \frac{1}{\Omega^{1/2}} C_\rnk e^{i\bk\br}.
+  \langle\mathbf{r}| \phi_{n\mathbf{k}} \rangle = \phi_{n\mathbf{k}}(r) = \frac{1}{\Omega^{1/2}} C_{\mathbf{r}n\mathbf{k}} e^{i\mathbf{k}\mathbf{r}}.
 </math>
 The charge density is simply given by
-<math>
+::<math>
-  \rhops_\br \equiv \langle\br | \rhops| \br\rangle =
+  \rho^{\mathrm{ps}}_{\mathbf{r}} \equiv \langle \mathbf{r} |\rho^{\mathrm{ps}} | \mathbf{r} \rangle =
-  \sum_\bk w_\bk \sum_n f_\nk \phi_\nk(r) \phi^*_\nk(r) ,
+  \sum_\mathbf{k} w_{\mathbf{k}} \sum_n f_{n\mathbf{k}} \phi_{n\mathbf{k}}(r) \phi^{*}_{n\mathbf{k}}(r) ,
 </math>
 and in the reciprocal mesh it can be written as
-<math>
+::<math>
-  \rhops_\bG  \equiv
+  \rho^{\mathrm{ps}}_\mathbf{G}  \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}.
+     \frac{1}{\Omega} \int \langle\mathbf{r} | \rho^{\mathrm{ps}}| \mathbf{r}\rangle e^{-i \mathbf{G}\mathbf{r}}\, d \mathbf{r} \to \frac{1}{N_{\mathrm{FFT}}} \sum_{\mathbf{r}} \rho^{\mathrm{ps}}_{\mathbf{r}} e^{-i \mathbf{G}\mathbf{r}}.
 </math>
-[[template:NumBlk]]
+Using the above equations for <math>\rho^{\mathrm{ps}}_{\mathbf{r}}</math> and <math>C_{\mathbf{r}n\mathbf{k}}</math> it
+is very  easy to show that <math>\rho^{\mathrm{ps}}_{\mathbf{r}}</math> contains Fourier-components up
-Inserting <math>\rhops</math> from equation (\ref{rps}) and $ C_\rnk$ from (\ref{pl2}) it
+to <math>2 G_{\mathrm{cut}}</math>.
-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 $f_r=f^1_r f^2_r$
+a the convolution <math>f_r=f^1_r f^2_r</math>
-of two 'functions' $f^1_r$ with  Fourier-components
+of two functions <math>f^1_r</math> with  Fourier-components
-up to $G_1$ and $f^2_r$ with Fourier-components
+up to <math>G_1</math> and <math>f^2_r</math> with Fourier-components
-up to $G_2$ contains Fourier-components up to  $G_1+G_2$.
+up to <math>G_2</math> contains Fourier-components up to  <math>G_1+G_2</math>.
 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
-$2 G_{\rm cut}$).
+::<math>
-\begin{figure}  \unitlength1cm
+ a_{\mathbf{r}} =  V_{\mathbf{r}}   C_{\mathbf{r}n\mathbf{k}}.
-\epsffile{algo1.eps}
+</math>
-\caption{ \label{algo-fig1}
-The small sphere contains all plane waves included in the basis set  $G<G_{\rm cut}$.
+Only the components <math>a_{\mathbf{G}}</math> with <math>|\mathbf{G}| < G_{\mathrm{cut}}</math> are taken into
-The charge density contains components up to  $2 G_{\rm cut}$ (second sphere), and
+account (see section {{TAG|ALGO}}: <math>a_{\mathbf{G}}</math> is added to the wavefunction
-the acceleration $a$ components up to  $3 G_{\rm cut}$, which are reflected
+during the iterative refinement of the wavefunctions <math>C_{\mathbf{G}n\mathbf{k}}</math>,
-in (third sphere) because of the finite size of the FFT-mesh. Nevertheless
+and  <math>C_{\mathbf{G}n\mathbf{k}}</math> contains only components up to <math>G_{\mathrm{cut}}</math>).
-the components $a_\bG$ with $| \bG| < G_{\rm cut}$ are correct i.e.
+From the previous theorem we see that <math>a_{\mathbf{r}}</math> contains
-the small sphere does not intersect with the third large sphere
+components up to <math>3 G_{\mathrm{cut}}</math> (<math>V_{\mathbf{r}}</math> contains components up to
-}
+<math>2 G_{\mathrm{cut}}</math>).
-\end{figure}
-If the FFT-mesh contains all components up to  $2 G_{\rm cut}$
+If the FFT mesh contains all components up to  <math>2 G_{\mathrm cut}</math>
 the resulting wrap-around error is once again 0. This can
-be easily seen in Fig. \ref{algo-fig1}.
+be easily seen in Fig. 1. Here we see that the small sphere contains all plane waves included in the basis set  <math>G<G_{\mathrm{cut}}</math>.
+The charge density contains components up to  <math>2 G_{\mathrm{cut}}</math> (second sphere), and
+the acceleration <math>a</math> components up to <math>3 G_{\mathrm{cut}}</math>, which are reflected
+in (third sphere) because of the finite size of the FFT mesh. Nevertheless
+the components <math>a_{\mathbf{G}}</math> with <math>| \mathbf{G}| < G_{\mathrm{cut}}</math> are correct i.e.
+the small sphere does not intersect with the third large sphere}
+== Related tags and articles ==
+{{TAG|PREC}},
+{{TAG|ENCUT}},
+{{TAG|NGX}},
+{{TAG|NGY}},
+{{TAG|NGZ}}
+[[Energy cutoff and FFT meshes]]
-----
+[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]
-[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:Theory]]