Wrap-around errors - Revision history

Huebsch at 09:00, 24 October 2025

2025-10-24T09:00:09Z

← Older revision		Revision as of 09:00, 24 October 2025
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	~~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		sufficiently large. It can be shown that no errors exist
	if the FFT meshes contain all <math>\mathbf{G}</math> 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>]]		[[File:Wrap errors spheres.png\|350px\|thumb\|Fig. 1: Sphere intersections for <math>G_{\mathrm{cut}}</math>]]
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	The wavefunction is defined as		The wavefunction is defined as

	<math>		::<math>
	\| \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G} C_{\mathbf{G}n\mathbf{k}} \| \mathbf{k}+\mathbf{G}\rangle,		\| \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G} C_{\mathbf{G}n\mathbf{k}} \| \mathbf{k}+\mathbf{G}\rangle,
	</math>		</math>
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	and in real space it is given by		and in real space it is given by

	<math>		::<math>
	\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		\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}}.		= \frac{1}{\Omega^{1/2}} \sum_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}} C_{\mathbf{G}n\mathbf{k}}.
	</math>		</math>

	Using ~~Fast Fourier transformations~~ one can define		Using FFTs one can define

	<math>		::<math>
	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{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}}.		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}}.
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	Therefore the wavefunction can be written in real space as		Therefore the wavefunction can be written in real space as

	<math>		::<math>
	\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}}.		\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>		</math>
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	The charge density is simply given by		The charge density is simply given by

	<math>		::<math>
	\rho^{\mathrm{ps}}_{\mathbf{r}} \equiv \langle \mathbf{r} \|\rho^{\mathrm{ps}} \| \mathbf{r} \rangle =		\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) ,		\sum_\mathbf{k} w_{\mathbf{k}} \sum_n f_{n\mathbf{k}} \phi_{n\mathbf{k}}(r) \phi^{*}_{n\mathbf{k}}(r) ,
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	and in the reciprocal mesh it can be written as		and in the reciprocal mesh it can be written as

	<math>		::<math>
	\rho^{\mathrm{ps}}_\mathbf{G} \equiv		\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}}.		\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}}.
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	calculated. The action of the local-potential is given by		calculated. The action of the local-potential is given by

	<math>		::<math>
	a_{\mathbf{r}} = V_{\mathbf{r}} C_{\mathbf{r}n\mathbf{k}}.		a_{\mathbf{r}} = V_{\mathbf{r}} C_{\mathbf{r}n\mathbf{k}}.
	</math>		</math>
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	<math>2 G_{\mathrm{cut}}</math>).		<math>2 G_{\mathrm{cut}}</math>).

	If the FFT-mesh contains all components up to <math>2 G_{\mathrm cut}</math>		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. 1. Here we see that the small sphere contains all plane waves included in the basis set <math>G<G_{\mathrm{cut}}</math>.		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 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
	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 <math>a_{\mathbf{G}}</math> with <math>\| \mathbf{G}\| < G_{\mathrm{cut}}</math> are correct i.e.		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}		the small sphere does not intersect with the third large sphere}
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	{{TAG\|NGX}},		{{TAG\|NGX}},
	{{TAG\|NGY}},		{{TAG\|NGY}},
	{{TAG\|NGZ}},		{{TAG\|NGZ}}
	[[~~Energy_cut_off_and_FFT_mesh\|~~Energy cutoff and FFT ~~mesh~~]]
			[[Energy cutoff and FFT meshes]]


	~~----~~
	[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]

Ftran at 15:52, 14 October 2025

2025-10-14T15:52:59Z

← Older revision		Revision as of 15:52, 14 October 2025
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	the components <math>a_{\mathbf{G}}</math> with <math>\| \mathbf{G}\| < G_{\mathrm{cut}}</math> are correct i.e.		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}		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_cut_off_and_FFT_mesh\|Energy cutoff and FFT mesh]]

	----		----
	[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]

Ftran at 08:36, 14 October 2025

2025-10-14T08:36:14Z

← Older revision		Revision as of 08:36, 14 October 2025
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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 <math>\~~bold~~{G}</math> 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>]]		[[File:Wrap errors spheres.png\|350px\|thumb\|Fig. 1: Sphere intersections for <math>G_{\mathrm{cut}}</math>]]
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	<math>		<math>
	\| \phi_{n\~~bold~~{k}} \rangle = \sum_\~~bold~~{G} C_{\~~bold~~{G}n\~~bold~~{k}} \| \~~bold~~{k}+\~~bold~~{G}\rangle,		\| \phi_{n\mathbf{k}} \rangle = \sum_\mathbf{G} C_{\mathbf{G}n\mathbf{k}} \| \mathbf{k}+\mathbf{G}\rangle,
	</math>		</math>

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	<math>		<math>
	\langle \~~bold~~{r}\| \phi_{n\~~bold~~{k}} \rangle = \sum_\~~bold~~{G} \langle \~~bold~~{r}\| \~~bold~~{k}+\~~bold~~{G}\rangle \langle \~~bold~~{k}+\~~bold~~{G}\|\phi_{n\~~bold~~{k}} \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_\~~bold~~{G} e^{i(\~~bold~~{k}+\~~bold~~{G})\~~bold~~{r}} C_{\~~bold~~{G}n\~~bold~~{k}}.		= \frac{1}{\Omega^{1/2}} \sum_\mathbf{G} e^{i(\mathbf{k}+\mathbf{G})\mathbf{r}} C_{\mathbf{G}n\mathbf{k}}.
	</math>		</math>

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	<math>		<math>
	C_{\~~bold~~{r}n\~~bold~~{k}}= \sum_{\~~bold~~{G}} C_{\~~bold~~{G}n\~~bold~~{k}} e^{i\~~bold~~{G} \~~bold~~{r}} \qquad \qquad \qquad		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_{\~~bold~~{G}n\~~bold~~{k}}= \frac{1}{N_{\mathrm{FFT}}} \sum_{\~~bold~~{r}} C_{\~~bold~~{r}n\~~bold~~{k}} e^{-i\~~bold~~{G} \~~bold~~{r}}.		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>		</math>

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	<math>		<math>
	\langle\~~bold~~{r}\| \phi_{n\~~bold~~{k}} \rangle = \phi_{n\~~bold~~{k}}(r) = \frac{1}{\Omega^{1/2}} C_{\~~bold~~{r}n\~~bold~~{k}} e^{i\~~bold~~{k}\~~bold~~{r}}.		\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>		</math>

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

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

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

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	<math>		<math>
	a_{\~~bold~~{r}} = V_{\~~bold~~{r}} C_{\~~bold~~{r}n\~~bold~~{k}}.		a_{\mathbf{r}} = V_{\mathbf{r}} C_{\mathbf{r}n\mathbf{k}}.
	</math>		</math>

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

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	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
	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 <math>a_{\~~bold~~{G}}</math> with <math>\| \~~bold~~{G}\| < G_{\mathrm{cut}}</math> are correct i.e.		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}		the small sphere does not intersect with the third large sphere}

	----		----
	[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]		[[Category:Electronic minimization]][[Category:Charge density]][[Category:Theory]]

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
Line 86:		Line 86:

	----		----
	[[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
Line 2:		Line 2:
	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
Line 86:		Line 86:

	----		----
	[[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
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\|~~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
Line 3:		Line 3:
	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]]