Projector-augmented-wave formalism - Revision history

Pmelo: /* Basics of the PAW formalism */

2025-10-17T12:06:13Z

Basics of the PAW formalism

Huebsch: /* Basics of the PAW formalism */

2024-09-06T07:00:44Z

Basics of the PAW formalism

← Older revision		Revision as of 07:00, 6 September 2024
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	proposed and implemented by Blöchl<ref name="bloechl:94"/>. The formal relationship between Vanderbilt-type		proposed and implemented by Blöchl<ref name="bloechl:94"/>. The formal relationship between Vanderbilt-type
	ultrasoft pseudopotentials and the PAW method has been derived by Kresse and		ultrasoft pseudopotentials and the PAW method has been derived by Kresse and
	Joubert<ref name="kressjoub:99"/>, and the generalization of the PAW method to ~~non collinear~~		Joubert<ref name="kressjoub:99"/>, and the generalization of the PAW method to noncollinear
	magnetism has been		magnetism has been
	discussed by Hobbs, Kresse and Hafner<ref name="hobbs:00"/>.		discussed by Hobbs, Kresse and Hafner<ref name="hobbs:00"/>.

Wolloch: /* Basics of the PAW formalism */

2023-10-19T08:32:46Z

Basics of the PAW formalism

← Older revision		Revision as of 08:32, 19 October 2023
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	\langle \widetilde{p}_{i} \|\widetilde{\psi}_{n\mathbf{k}} \rangle.		\langle \widetilde{p}_{i} \|\widetilde{\psi}_{n\mathbf{k}} \rangle.
	</math>		</math>


	The pseudo orbitals		The pseudo orbitals

Huebsch at 15:23, 6 April 2022

2022-04-06T15:23:10Z

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Huebsch: Huebsch moved page PAW formalism to Projector-augmented-wave formalism

2022-04-06T15:19:18Z

Huebsch moved page PAW formalism to Projector-augmented-wave formalism

← Older revision	Revision as of 15:19, 6 April 2022
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Huebsch: Huebsch moved page PAW method to PAW formalism without leaving a redirect

2022-04-06T15:18:35Z

Huebsch moved page PAW method to PAW formalism without leaving a redirect

← Older revision	Revision as of 15:18, 6 April 2022
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Karsai at 11:59, 1 October 2019

2019-10-01T11:59:27Z

← Older revision		Revision as of 11:59, 1 October 2019
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	== Basics of the PAW formalism ==		== Basics of the PAW formalism ==

	The PAW formalism is a generalization of ideas of both Vanderbilt-type~<ref name="vander:90"/>		The PAW formalism is a generalization of ideas of both Vanderbilt-type<ref name="vander:90"/>
	ultrasoft-pseudopotentials ~~(USPP)~~~<ref name="kresshaf:94"/> and the		ultrasoft-pseudopotentials<ref name="kresshaf:94"/> (USPP) and the
	linearized augmented-plane-wave ~~(LAPW) method~~~<ref name="andersen:75"/>. The method was first		linearized augmented-plane-wave<ref name="andersen:75"/> (LAPW) method. The method was first
	proposed and implemented by Blöchl~<ref name="bloechl:94"/>. The formal relationship between Vanderbilt-type		proposed and implemented by Blöchl<ref name="bloechl:94"/>. The formal relationship between Vanderbilt-type
	ultrasoft pseudopotentials and the PAW method has been derived by Kresse and		ultrasoft pseudopotentials and the PAW method has been derived by Kresse and
	Joubert~<ref name="kressjoub:99"/>, and the generalization of the PAW method to non collinear		Joubert<ref name="kressjoub:99"/>, and the generalization of the PAW method to non collinear
	magnetism has been		magnetism has been
	discussed by Hobbs, Kresse and Hafner~<ref name="hobbs:00"/>.		discussed by Hobbs, Kresse and Hafner<ref name="hobbs:00"/>.
	We briefly summarize the basics of the PAW method below (following Refs. <ref name="bloechl:94"/>		We briefly summarize the basics of the PAW method below (following Refs. <ref name="bloechl:94"/>
	and <ref name="kressjoub:99"/>).		and <ref name="kressjoub:99"/>).
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	</math>		</math>

	can be rewritten as (for details we refer to Ref.<ref name="bloechl:94"/>):		can be rewritten as (for details we refer to Ref. <ref name="bloechl:94"/>):

	<math>		<math>

Karsai at 08:02, 3 May 2019

2019-05-03T08:02:37Z

← Older revision		Revision as of 08:02, 3 May 2019
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	</references>		</references>
	----		----
	~~[[The_VASP_Manual\|Contents]]~~

	[[Category:Electronic Minimization]][[Category:PAW]][[Category:Theory]]		[[Category:Electronic Minimization]][[Category:PAW]][[Category:Theory]]

Karsai: /* References */

2019-03-07T10:05:25Z

References

← Older revision		Revision as of 10:05, 7 March 2019
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	[[The_VASP_Manual\|Contents]]		[[The_VASP_Manual\|Contents]]

	[[Category:PAW]][[Category:Theory]]		[[Category:Electronic Minimization]][[Category:PAW]][[Category:Theory]]

Karsai: /* References */

2019-03-07T10:04:39Z

References

← Older revision		Revision as of 10:04, 7 March 2019
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	[[The_VASP_Manual\|Contents]]		[[The_VASP_Manual\|Contents]]

	[[Category:PAW]]		[[Category:PAW]][[Category:Theory]]

← Older revision		Revision as of 12:06, 17 October 2025
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	\langle \mathbf{r} \| \widetilde{\psi}_{n\mathbf{k}} \rangle =		\langle \mathbf{r} \| \widetilde{\psi}_{n\mathbf{k}} \rangle =
	\frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{n\mathbf{kG}}		\frac{1}{\Omega^{1/2}} \sum_{\mathbf{G}} C_{n\mathbf{kG}}
	e^{i(\mathbf{G}+\mathbf{k})\cdot \mathbf{r}},		e^{i(\mathbf{G}+\mathbf{k})\cdot \mathbf{r}} = e^{i\mathbf{k}\cdot \mathbf{r}}\tilde u_{n\mathbf k}(\mathbf r),
	</math>		</math>

	where <math>\Omega</math> is the volume of the Wigner-Seitz cell. The all-electron (AE)		where <math>\Omega</math> is the volume of the Wigner-Seitz cell and <math>\tilde u_{n\mathbf k}(\mathbf r)</math> is the cell periodic part of the pseudo orbital. The all-electron (AE)
	partial waves		partial waves
	<math>\phi_{\alpha}</math> are solutions of the radial Schrödinger equation for a		<math>\phi_{\alpha}</math> are solutions of the radial Schrödinger equation for a
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	the length of the vector <math>\mathbf{r}-\mathbf{R}_\alpha</math>.		the length of the vector <math>\mathbf{r}-\mathbf{R}_\alpha</math>.
	Note that the radial component of the partial wave <math>u_{\alpha}</math> is independent of		Note that the radial component of the partial wave <math>u_{\alpha}</math> is independent of
	<math>m_\alpha</math>, since the partial waves are calculated for a spherical atom.		<math>m_\alpha</math>, since the partial waves are calculated for a spherical atom. Also, do not confuse the radial functions, <math>u_{\alpha}(\|\mathbf r\|)</math>, with the functions of the cell-periodic part, <math>\tilde u_{n\mathbf k}(\mathbf r)</math>.
	Furthermore, the spherical harmonics depend on the angular quantum numbers only		Furthermore, the spherical harmonics depend on the angular quantum numbers only
	and not on the reference energy. The pseudo partial		and not on the reference energy. The pseudo partial