Category:Many-body perturbation theory - Revision history

Csheldon at 14:20, 24 March 2026

2026-03-24T14:20:25Z

← Older revision		Revision as of 14:20, 24 March 2026
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	There are three implementations available:		There are three implementations available:

	* '''MP2'''~~<ref name=marsman/>~~: this implementation is recommended for very small unit cells, very few k-points and very low plane-wave cutoffs. The system size scaling of this algorithm is N⁵.		* '''MP2''' {{Cite\|paier:2009}}: this implementation is recommended for very small unit cells, very few k-points and very low plane-wave cutoffs. The system size scaling of this algorithm is N⁵.
	* '''LTMP2'''~~<ref name=schaefer2017/>~~: for all larger systems this Laplace-transformed MP2 (LTMP2) implementation is recommended. Larger cutoffs and denser k-point meshes can be used. It possesses a lower system size scaling (N⁴) and a more efficient k-point sampling.		* '''LTMP2'''{{Cite\|schaefer:2017}}: for all larger systems this Laplace-transformed MP2 (LTMP2) implementation is recommended. Larger cutoffs and denser k-point meshes can be used. It possesses a lower system size scaling (N⁴) and a more efficient k-point sampling.
	* '''stochastic LTMP2'''~~<ref name=schaefer2018/>~~: even faster calculations at the price of statistical noise can be achieved with the stochastic MP2 algorithm. It is an optimal choice for very large systems where only relative errors per valence electron are relevant. Keeping the absolute error fixed, the algorithm exhibits a cubic scaling with the system size, N³, whereas for a fixed relative error, a linear scaling, N¹, can be achieved. Note that there is no k-point sampling and no spin polarization implemented for this algorithm.		* '''stochastic LTMP2'''{{Cite\|schaefer:2018}}: even faster calculations at the price of statistical noise can be achieved with the stochastic MP2 algorithm. It is an optimal choice for very large systems where only relative errors per valence electron are relevant. Keeping the absolute error fixed, the algorithm exhibits a cubic scaling with the system size, N³, whereas for a fixed relative error, a linear scaling, N¹, can be achieved. Note that there is no k-point sampling and no spin polarization implemented for this algorithm.

	* [[MP2 ground state calculation - Tutorial]]		* [[MP2 ground state calculation - Tutorial]]

	== References ==		== References ==
	~~<references>~~
	~~<ref name=marsman>[http://dx.doi.org/10.1063/1.3126249 M. Marsman, A. Grüneis, J. Paier, and G. Kresse, J. Chem. Phys. 130, 184103 (2009).]</ref>~~
	~~<ref name=schaefer2017>[http://dx.doi.org/10.1063/1.4976937 T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 146, 104101 (2017).]</ref>~~
	~~<ref name=schaefer2018>[https://doi.org/10.1063/1.5016100 T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 148, 064103 (2018).]</ref>~~
	~~</references>~~
	~~----~~

	[[Category:VASP\|Many-body perturbation theory]]		[[Category:VASP\|Many-body perturbation theory]]

Csheldon: /* Bethe-Salpeter equations (BSE) */

2026-03-24T14:15:17Z

Bethe-Salpeter equations (BSE)

← Older revision		Revision as of 14:15, 24 March 2026
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	* [[BSE calculations]] — practical guide		* [[BSE calculations]] — practical guide
	* Tutorial for {{Tutorial\|bse\|BSE calculations}}		* Tutorial for {{Tutorial\|bse\|BSE calculations}}
			* Lecture on {{Video\|bse:alexey:2026\|BSE}}

	==== X-ray absorption spectra ====		==== X-ray absorption spectra ====

Csheldon: /* X-ray absorption spectra */

2026-03-24T14:14:48Z

X-ray absorption spectra

← Older revision		Revision as of 14:14, 24 March 2026
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	* Tutorials for {{Tutorial\|xas\|XAS calculations}}		* Tutorials for {{Tutorial\|xas\|XAS calculations}}
			<!--* Lecture on {{Video\|xas:alexey:2026\|XAS}} -->

	=== Second-order Møller-Plesset perturbation theory (MP2) ===		=== Second-order Møller-Plesset perturbation theory (MP2) ===

Csheldon at 14:14, 24 March 2026

2026-03-24T14:14:05Z

← Older revision		Revision as of 14:14, 24 March 2026
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	* [[RPA/ACFDT: Correlation energy in the Random Phase Approximation]] — theory		* [[RPA/ACFDT: Correlation energy in the Random Phase Approximation]] — theory
	* [[ACFDT/RPA calculations]] — practical guide		* [[ACFDT/RPA calculations]] — practical guide
	* Lecture on ~~[https~~:~~//youtu~~.~~be/0hV5bTpY89M RPA]~~		* Lecture on {{Video\|rpa:merzuk:2025\|RPA}}.

	=== Constrained random-phase approximation (cRPA) ===		=== Constrained random-phase approximation (cRPA) ===
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	* [[Practical guide to GW calculations]] — practical guide		* [[Practical guide to GW calculations]] — practical guide
	* [[GW and dielectric matrix]]		* [[GW and dielectric matrix]]
	* Tutorial for ~~[https://vasp.at/tutorials/latest/~~gw/ GW calculations]		* Tutorial for {{Tutorial\|gw\|GW calculations}}
	* Lecture on ~~[https~~:~~//youtu.be/zGPqDxsD80o~~ GW]		* Lecture on {{Video\|gw:merzuk:2023\|GW}}
	* Lecture on the ~~[https~~:~~//youtu.be/6F_WNIh6V7I~~ optical bandgap], including using many-body perturbation theory (GW and RPA)		* Lecture on the {{Video\|optical:merzuk:2025\|optical bandgap}}, including using many-body perturbation theory (GW and RPA)

	=== Bethe-Salpeter equations (BSE) ===		=== Bethe-Salpeter equations (BSE) ===
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	* [[:Category:Bethe-Salpeter equations\|Bethe-Salpeter equations]] — all tags and articles		* [[:Category:Bethe-Salpeter equations\|Bethe-Salpeter equations]] — all tags and articles
	* [[BSE calculations]] — practical guide		* [[BSE calculations]] — practical guide
	* Tutorial for ~~[https://vasp.at/tutorials/latest/~~bse/ BSE calculations]		* Tutorial for {{Tutorial\|bse\|BSE calculations}}

	==== X-ray absorption spectra ====		==== X-ray absorption spectra ====
	The BSE/TDHF algorithm can also be used to model the X-ray absorption spectra (XAS), i.e., excitations from the core states into conduction bands. Detailed documentation of this method can be found in the [[:Category:XAS\|XAS category page]].		The BSE/TDHF algorithm can also be used to model the X-ray absorption spectra (XAS), i.e., excitations from the core states into conduction bands. Detailed documentation of this method can be found in the [[:Category:XAS\|XAS category page]].

			* Tutorials for {{Tutorial\|xas\|XAS calculations}}

	=== Second-order Møller-Plesset perturbation theory (MP2) ===		=== Second-order Møller-Plesset perturbation theory (MP2) ===

Huebsch: Rename Theory -> Available methods; distribute tutorials into each method; remove How to and Tutorials sections; fix {{TAG}} -> wikilinks; fix style guide issues

2026-03-19T09:12:47Z

Rename Theory -> Available methods; distribute tutorials into each method; remove How to and Tutorials sections; fix {{TAG}} -> wikilinks; fix style guide issues

← Older revision		Revision as of 09:12, 19 March 2026
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	Treating the electron-electron interaction within '''many-body perturbation theory''' includes screening and renormalization effects beyond ~~the~~ density-functional theory (DFT). It is based on the Green's-function formalism and can be derived and visualized in terms of a diagrammatic expansion of~~, e.g.,~~ the electron interacting with other electrons. Instead of describing ~~the~~ electrons by means of Kohn-Sham (KS) orbitals, the renormalized (or dressed) propagators yield quasiparticle orbitals.		Treating the electron-electron interaction within '''many-body perturbation theory''' includes screening and renormalization effects beyond density-functional theory (DFT). It is based on the Green's-function formalism and can be derived and visualized in terms of a diagrammatic expansion of the electron interacting with other electrons. Instead of describing electrons by means of Kohn-Sham (KS) orbitals, the renormalized (or dressed) propagators yield quasiparticle orbitals. Another area that can be discussed in the language of many-body perturbation theory is [[electron-phonon coupling]], which treats the interaction between electronic and ionic degrees of freedom.

	Here, we focus on treating the electron-electron interaction within '''many-body perturbation theory'''. Another area that can be discussed in the language of '''many-body perturbation theory''' is [[electron-phonon coupling]] to treat the interaction between electronic and ionic degrees of freedom.		== Available methods ==

	== ~~Theory~~ ==

	=== Random-phase approximation (RPA) ===		=== Random-phase approximation (RPA) ===

	GW and RPA are post-DFT methods used to solve the many-body problem ~~approximatively~~.		GW and RPA are post-DFT methods used to solve the many-body problem approximately.

	RPA stands for the random-phase approximation and is often used as a synonym for the adiabatic connection fluctuation-dissipation theorem (ACFDT). RPA/ACFDT provides access to the correlation energy of a system and can be understood in terms of Feynman diagrams as an infinite sum of all bubble diagrams, where excitonic effects (interactions between electrons and holes) are neglected. The RPA/ACFDT is used as a post-processing tool to determine a more accurate ground-state energy.		RPA stands for the random-phase approximation and is often used as a synonym for the adiabatic connection fluctuation-dissipation theorem (ACFDT). RPA/ACFDT provides access to the correlation energy of a system and can be understood in terms of Feynman diagrams as an infinite sum of all bubble diagrams, where excitonic effects (interactions between electrons and holes) are neglected. The RPA/ACFDT is used as a post-processing tool to determine a more accurate ground-state energy.
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	* [[RPA/ACFDT: Correlation energy in the Random Phase Approximation]] — theory		* [[RPA/ACFDT: Correlation energy in the Random Phase Approximation]] — theory
	* [[ACFDT/RPA calculations]] — practical guide		* [[ACFDT/RPA calculations]] — practical guide
			* Lecture on [https://youtu.be/0hV5bTpY89M RPA]

	=== Constrained random-phase approximation ===		=== Constrained random-phase approximation (cRPA) ===

	The constrained random-phase approximation (cRPA) is a method that allows calculating the effective interaction parameter <math>U</math>, <math>J</math> and <math>J'</math> for model Hamiltonians. The main idea is to neglect the screening effects of specific target states in the screened Coulomb interaction <math>W</math> of the <math>GW</math> method. Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher-level theory, such as dynamical-mean-field theory (DMFT).		The constrained random-phase approximation (cRPA) is a method that allows calculating the effective interaction parameter <math>U</math>, <math>J</math> and <math>J'</math> for model Hamiltonians. The main idea is to neglect the screening effects of specific target states in the screened Coulomb interaction <math>W</math> of the <math>GW</math> method. Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher-level theory, such as dynamical-mean-field theory (DMFT).
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	=== GW method ===		=== GW method ===

	The GW approximation goes hand in hand with the RPA since the very same diagrammatic contributions are taken into account in the screened Coulomb interaction of a system often denoted as W. However, in contrast to the RPA/ACFDT, the GW method provides access to the spectral properties of the system by means of determining the energies of the ~~quasi-particles of a system~~ using a screened exchange-like contribution to the self-energy. The GW approximation is currently one of the most accurate many-body methods to calculate ~~band-gaps~~.		The GW approximation goes hand in hand with the RPA since the very same diagrammatic contributions are taken into account in the screened Coulomb interaction of a system often denoted as W. However, in contrast to the RPA/ACFDT, the GW method provides access to the spectral properties of the system by means of determining the energies of the quasiparticles using a screened exchange-like contribution to the self-energy. The GW approximation is currently one of the most accurate many-body methods to calculate bandgaps.

	*~~{{TAG\|~~The GW approximation of Hedin's equations}}.		* [[The GW approximation of Hedin's equations]] — theory
			* [[Practical guide to GW calculations]] — practical guide
			* [[GW and dielectric matrix]]
			* Tutorial for [https://vasp.at/tutorials/latest/gw/ GW calculations]
			* Lecture on [https://youtu.be/zGPqDxsD80o GW]
			* Lecture on the [https://youtu.be/6F_WNIh6V7I optical bandgap], including using many-body perturbation theory (GW and RPA)

	=== Bethe-Salpeter equations (BSE) ===		=== Bethe-Salpeter equations (BSE) ===

	VASP offers a powerful module for solving time-dependent DFT (TD-DFT) and time-dependent Hartree-Fock equations (TDHF) (the Casida equation) or the Bethe-Salpeter (BSE) equation{{cite\|albrecht:prl:98}}{{cite\|rohlfing:prl:98}}. These approaches are used for obtaining the frequency-dependent dielectric function with ~~the~~ excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional, or [[Practical guide to GW calculations\|''GW'' ]] approximations. VASP also offers the TDHF and BSE calculations beyond the Tamm-Dancoff approximation (TDA){{cite\|sander:prb:15}}.		VASP offers a powerful module for solving time-dependent DFT (TD-DFT) and time-dependent Hartree-Fock equations (TDHF) (the Casida equation) or the Bethe-Salpeter (BSE) equation{{cite\|albrecht:prl:98}}{{cite\|rohlfing:prl:98}}. These approaches are used for obtaining the frequency-dependent dielectric function with excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional, or [[Practical guide to GW calculations\|GW]] approximations. VASP also offers the TDHF and BSE calculations beyond the Tamm-Dancoff approximation (TDA){{cite\|sander:prb:15}}.

	* [[:Category:Bethe-Salpeter equations\|~~Tags~~ and articles ~~related to~~ BSE calculations.]]		* [[:Category:Bethe-Salpeter equations\|Bethe-Salpeter equations]] — all tags and articles
			* [[BSE calculations]] — practical guide
			* Tutorial for [https://vasp.at/tutorials/latest/bse/ BSE calculations]

	==== X-ray absorption spectra ====		==== X-ray absorption spectra ====
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	There are three implementations available:		There are three implementations available:

	* '''MP2'''<ref name="marsman"/>: this implementation is recommended for very small unit cells, very few k-points and very low plane-wave ~~cuttofs~~. The system size scaling of this algorithm is N⁵.		* '''MP2'''<ref name=marsman/>: this implementation is recommended for very small unit cells, very few k-points and very low plane-wave cutoffs. The system size scaling of this algorithm is N⁵.
	* '''LTMP2'''<ref name="schaefer2017"/>: for all larger systems this Laplace transformed MP2 (~~LTMP~~) implementation is recommended. Larger cutoffs and denser k-point meshes can be used. It possesses a lower system size scaling (N⁴) and a more efficient k-point sampling.		* '''LTMP2'''<ref name=schaefer2017/>: for all larger systems this Laplace-transformed MP2 (LTMP2) implementation is recommended. Larger cutoffs and denser k-point meshes can be used. It possesses a lower system size scaling (N⁴) and a more efficient k-point sampling.
	* '''stochastic LTMP2'''<ref name="schaefer2018"/>: even faster calculations at the price of statistical noise can be achieved with the stochastic MP2 algorithm. It is an optimal choice for very large systems where only relative errors per valence electron are relevant. Keeping the absolute error fixed, the algorithm exhibits a cubic scaling with the system size, N³, whereas for a fixed relative error, a linear scaling, N¹, can be achieved. Note that there is no k-point sampling and no spin polarization implemented for this algorithm.		* '''stochastic LTMP2'''<ref name=schaefer2018/>: even faster calculations at the price of statistical noise can be achieved with the stochastic MP2 algorithm. It is an optimal choice for very large systems where only relative errors per valence electron are relevant. Keeping the absolute error fixed, the algorithm exhibits a cubic scaling with the system size, N³, whereas for a fixed relative error, a linear scaling, N¹, can be achieved. Note that there is no k-point sampling and no spin polarization implemented for this algorithm.
	~~<!-- == Dynamical-mean-field theory == -->~~
	~~== How to ==~~
	~~Practical guides to different diagrammatic approximations are found on following pages:~~

	*ACFDT: {{TAG\|ACFDT/RPA calculations}}.
	*GW: {{TAG\|Practical guide to GW calculations}}.
	*BSE: {{TAG\|BSE calculations}}.
	*Using the GW routines for the determination of frequency dependent dielectric matrix: {{TAG\|GW and dielectric matrix}}.
	*MP2 method: {{TAG\|MP2 ground state calculation - Tutorial}}.

	~~== Tutorials ==~~		* [[MP2 ground state calculation - Tutorial]]
	*~~Tutorial for~~ [~~https://vasp.at/tutorials/latest/gw/ GW calculations].~~
	*Tutorial ~~for [https://vasp.at/tutorials/latest/bse/ BSE calculations~~].
	*Lecture on [https://youtu.be/0hV5bTpY89M RPA].
	*Lecture on [https://youtu.be/zGPqDxsD80o GW].
	*Lecture on the [https://youtu.be/6F_WNIh6V7I optical bandgap], including using many-body perturbation theory (GW and RPA).

	== References ==		== References ==
	<references>		<references>
	<ref name="marsman">[http://dx.doi.org/10.1063/1.3126249 M. Marsman, A. Grüneis, J. Paier, and G. Kresse, J. Chem. Phys. 130, 184103 (2009).]</ref>		<ref name=marsman>[http://dx.doi.org/10.1063/1.3126249 M. Marsman, A. Grüneis, J. Paier, and G. Kresse, J. Chem. Phys. 130, 184103 (2009).]</ref>
	<ref name="schaefer2017">[http://dx.doi.org/10.1063/1.4976937 T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 146, 104101 (2017).]</ref>		<ref name=schaefer2017>[http://dx.doi.org/10.1063/1.4976937 T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 146, 104101 (2017).]</ref>
	<ref name="schaefer2018">[https://doi.org/10.1063/1.5016100 T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 148, 064103 (2018).]</ref>		<ref name=schaefer2018>[https://doi.org/10.1063/1.5016100 T. Schäfer, B. Ramberger, and G. Kresse, J. Chem. Phys. 148, 064103 (2018).]</ref>
	</references>		</references>
	----		----

	[[Category:VASP\|Many-body perturbation theory]]		[[Category:VASP\|Many-body perturbation theory]]

Huebsch: RPA subsection: show both theory and how-to links side by side

2026-03-19T08:49:34Z

RPA subsection: show both theory and how-to links side by side

← Older revision		Revision as of 08:49, 19 March 2026
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	RPA stands for the random-phase approximation and is often used as a synonym for the adiabatic connection fluctuation-dissipation theorem (ACFDT). RPA/ACFDT provides access to the correlation energy of a system and can be understood in terms of Feynman diagrams as an infinite sum of all bubble diagrams, where excitonic effects (interactions between electrons and holes) are neglected. The RPA/ACFDT is used as a post-processing tool to determine a more accurate ground-state energy.		RPA stands for the random-phase approximation and is often used as a synonym for the adiabatic connection fluctuation-dissipation theorem (ACFDT). RPA/ACFDT provides access to the correlation energy of a system and can be understood in terms of Feynman diagrams as an infinite sum of all bubble diagrams, where excitonic effects (interactions between electrons and holes) are neglected. The RPA/ACFDT is used as a post-processing tool to determine a more accurate ground-state energy.

	*~~{{TAG\|~~RPA/ACFDT: Correlation energy in the Random Phase Approximation ~~}}.~~		* [[RPA/ACFDT: Correlation energy in the Random Phase Approximation]] — theory
			* [[ACFDT/RPA calculations]] — practical guide

	=== Constrained random-phase approximation ===		=== Constrained random-phase approximation ===

Csheldon: /* Tutorials */

2025-09-22T10:39:55Z

Tutorials

← Older revision		Revision as of 10:39, 22 September 2025
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	*Lecture on [https://youtu.be/0hV5bTpY89M RPA].		*Lecture on [https://youtu.be/0hV5bTpY89M RPA].
	*Lecture on [https://youtu.be/zGPqDxsD80o GW].		*Lecture on [https://youtu.be/zGPqDxsD80o GW].
	*Lecture on the [https://youtu.be/6F_WNIh6V7I optical bandgap], including using many-body perturbation theory.		*Lecture on the [https://youtu.be/6F_WNIh6V7I optical bandgap], including using many-body perturbation theory (GW and RPA).

	== References ==		== References ==

Csheldon: /* Tutorials */

2025-09-22T10:35:28Z

Tutorials

← Older revision		Revision as of 10:35, 22 September 2025
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	*Lecture on [https://youtu.be/0hV5bTpY89M RPA].		*Lecture on [https://youtu.be/0hV5bTpY89M RPA].
	*Lecture on [https://youtu.be/zGPqDxsD80o GW].		*Lecture on [https://youtu.be/zGPqDxsD80o GW].
			*Lecture on the [https://youtu.be/6F_WNIh6V7I optical bandgap], including using many-body perturbation theory.

	== References ==		== References ==

Csheldon: /* How to */

2025-09-22T10:21:46Z

How to

← Older revision		Revision as of 10:21, 22 September 2025
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	*Using the GW routines for the determination of frequency dependent dielectric matrix: {{TAG\|GW and dielectric matrix}}.		*Using the GW routines for the determination of frequency dependent dielectric matrix: {{TAG\|GW and dielectric matrix}}.
	*MP2 method: {{TAG\|MP2 ground state calculation - Tutorial}}.		*MP2 method: {{TAG\|MP2 ground state calculation - Tutorial}}.

			== Tutorials ==
			*Tutorial for [https://vasp.at/tutorials/latest/gw/ GW calculations].
			*Tutorial for [https://vasp.at/tutorials/latest/bse/ BSE calculations].
			*Lecture on [https://youtu.be/0hV5bTpY89M RPA].
			*Lecture on [https://youtu.be/zGPqDxsD80o GW].

	== References ==		== References ==

Csheldon: /* X-ray absorption spectra */

2025-05-12T09:39:59Z

X-ray absorption spectra

← Older revision		Revision as of 09:39, 12 May 2025
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	==== X-ray absorption spectra ====		==== X-ray absorption spectra ====
	The BSE/TDHF algorithm can also be used to model the X-ray absorption spectra (XAS), i.e., excitations from the core states into conduction bands. Detailed documentation of this method can be found in [[:Category:XAS]].		The BSE/TDHF algorithm can also be used to model the X-ray absorption spectra (XAS), i.e., excitations from the core states into conduction bands. Detailed documentation of this method can be found in the [[:Category:XAS\|XAS category page]].

	=== Second-order Møller-Plesset perturbation theory (MP2) ===		=== Second-order Møller-Plesset perturbation theory (MP2) ===