Wrap-around errors - Revision history

Huebsch at 16:07, 6 April 2022

2022-04-06T16:07:49Z

← Older revision		Revision as of 16:07, 6 April 2022
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	----		----
	[[Category:Electronic ~~Minimization]][[Category:Electronic Minimization Methods~~]][[Category:Charge ~~Density]][[Category:Common Pitfalls~~]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]

Vaspmaster at 11:42, 3 November 2020

2020-11-03T11:42:58Z

← Older revision		Revision as of 11:42, 3 November 2020
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	----		----
	[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:~~Calculational issues~~]][[Category:Theory]]		[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:Common Pitfalls]][[Category:Theory]]

Schlipf at 09:30, 5 April 2019

2019-04-05T09:30:29Z

← Older revision		Revision as of 09:30, 5 April 2019
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	Wrap around errors arise if the FFT meshes are not		Wrap around errors arise if the FFT meshes are not
	sufficiently large. It can be shown that no errors exist		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>\bold{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>]]		[[File:Wrap errors spheres.png\|350px\|thumb\|Fig. 1: Sphere intersections for <math>G_{\mathrm{cut}}</math>]]

Karsai at 14:50, 28 March 2019

2019-03-28T14:50:30Z

← Older revision		Revision as of 14:50, 28 March 2019
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	----		----
	[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:Theory]]		[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:Calculational issues]][[Category:Theory]]

Karsai at 15:13, 18 March 2019

2019-03-18T15:13:25Z

← Older revision		Revision as of 15:13, 18 March 2019
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	if the FFT meshes contain all $G$ vectors up to <math>2 G_{\rm cut}</math>.		if the FFT meshes contain all $G$ vectors up to <math>2 G_{\rm cut}</math>.

	[[File:Wrap errors spheres.png\|~~450px~~\|thumb\|Fig. 1: Sphere intersections for <math>G_{\mathrm{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		It can be shown that the charge density contains

Karsai at 15:12, 18 March 2019

2019-03-18T15:12:56Z

← Older revision		Revision as of 15:12, 18 March 2019
Line 4:		Line 4:
	if the FFT meshes contain all $G$ vectors up to <math>2 G_{\rm cut}</math>.		if the FFT meshes contain all $G$ 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>]]		[[File:Wrap errors spheres.png\|450px\|thumb\|Fig. 1: Sphere intersections for <math>G_{\mathrm{cut}}</math>]]

	It can be shown that the charge density contains		It can be shown that the charge density contains

Karsai at 15:12, 18 March 2019

2019-03-18T15:12:34Z

← Older revision		Revision as of 15:12, 18 March 2019
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	sufficiently large. It can be shown that no errors exist		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 $G$ 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		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 "longest" plane wave in the basis set:

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	<math>2 G_{\mathrm{cut}}</math>).		<math>2 G_{\mathrm{cut}}</math>).

	~~\begin~~{~~figure}~~ \~~unitlength1cm~~		If the FFT-mesh contains all components up to <math>2 G_{\mathrm cut}</math>
	~~\epsffile{algo1.eps~~}		the resulting wrap-around error is once again 0. This can
	~~\caption{ \label{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~~ 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 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		the acceleration <math>a</math> components up to <math>3 G_{\mathrm{cut}}</math>, which are reflected
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	the components <math>a_{\bold{G}}</math> with <math>\| \bold{G}\| < G_{\mathrm{cut}}</math> are correct i.e.		the components <math>a_{\bold{G}}</math> with <math>\| \bold{G}\| < G_{\mathrm{cut}}</math> are correct i.e.
	the small sphere does not intersect with the third large sphere}		the small sphere does not intersect with the third large sphere}
	~~\end{figure}~~
	~~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}.~~

	----		----
	[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:Theory]]		[[Category:Electronic Minimization]][[Category:Electronic Minimization Methods]][[Category:Charge Density]][[Category:Theory]]

Karsai at 15:01, 18 March 2019

2019-03-18T15:01:17Z

← Older revision		Revision as of 15:01, 18 March 2019
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	<math>		<math>
	a_\br = V_\br C_\~~rnk~~.		a_{\bold{r}} = V_{\bold{r}} C_{\bold{r}n\bold{k}}.
	</math>		</math>

Karsai at 15:00, 18 March 2019

2019-03-18T15:00:32Z

← Older revision		Revision as of 15:00, 18 March 2019
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	</math>		</math>

	~~{{NumBlk\|:\|~~<math>~~x^2 + y^2 + z~~^~~2 = 1</math>\|~~{{~~EquationRef\|1~~}}}}		Using the above equations for <math>\rho^{\mathrm{ps}}_{\bold{r}}</math> and <math>C_{\bold{r}n\bold{k}}</math> it
			is very easy to show that <math>\rho^{\mathrm{ps}}_{\bold{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		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,		The property of the convolution comes once again into play,
	when the action of the Hamiltonian onto a wavefunction is		when the action of the Hamiltonian onto a wavefunction is
	calculated. The action of the local-potential is given by		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>
			a_\br = V_\br C_\rnk.
			</math>

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

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

Karsai at 13:47, 18 March 2019

2019-03-18T13:47:17Z

← Older revision		Revision as of 13:47, 18 March 2019
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	</math>		</math>

	{{NumBlk\|:\|<math>y=~~ax+b~~</math>\|~~Eq. 3~~}}		{{NumBlk\|:\|<math>x^2 + y^2 + z^2 = 1</math>\|{{EquationRef\|1}}}}

	Inserting <math>\rhops</math> from equation (\ref{rps}) and $ C_\rnk$ from (\ref{pl2}) it		Inserting <math>\rhops</math> from equation (\ref{rps}) and $ C_\rnk$ from (\ref{pl2}) it