Category:Many-body perturbation theory - Revision history

Csheldon: /* X-ray absorption spectra */

2025-05-12T09:39:59Z

X-ray absorption spectra

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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) ===

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

2025-05-12T08:55:43Z

Bethe-Salpeter equations (BSE)

← Older revision		Revision as of 08:55, 12 May 2025
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	* [[:Category:Bethe-Salpeter equations\|Tags and articles related to BSE calculations.]]		* [[:Category:Bethe-Salpeter equations\|Tags and articles related to BSE calculations.]]

			==== 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]].

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

Kaltakm: /* Constrained random-phase approximation */

2025-03-27T12:08:40Z

Constrained random-phase approximation

← Older revision		Revision as of 12:08, 27 March 2025
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	=== Constrained random-phase approximation ===		=== Constrained random-phase approximation ===

	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).

	* [[Constrained–random-phase–approximation formalism\|Formalism used for the ~~CRPA~~ method]]		* [[Constrained–random-phase–approximation formalism\|Formalism used for the cRPA method]]

	=== GW method ===		=== GW method ===

Huebsch at 08:21, 5 February 2025

2025-02-05T08:21:02Z

← Older revision		Revision as of 08:21, 5 February 2025
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	'''~~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 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.

			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.

	== Theory ==		== Theory ==

Huebsch at 13:42, 8 April 2022

2022-04-08T13:42:55Z

← Older revision		Revision as of 13:42, 8 April 2022
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	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).

	* [[~~:Category:~~Constrained–random-phase–approximation formalism\|Formalism used for the CRPA method]]		* [[Constrained–random-phase–approximation formalism\|Formalism used for the CRPA method]]

	=== GW method ===		=== GW method ===

Huebsch at 13:42, 8 April 2022

2022-04-08T13:42:29Z

← Older revision		Revision as of 13:42, 8 April 2022
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	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).

	* [[:Category:~~Constrained random~~-~~phase approximation~~ formalism\|Formalism used for the CRPA method]]		* [[:Category:Constrained–random-phase–approximation formalism\|Formalism used for the CRPA method]]

	=== GW method ===		=== GW method ===

Huebsch: Huebsch moved page Many-body perturbation theory to Category:Many-body perturbation theory

2022-04-08T13:29:16Z

Huebsch moved page Many-body perturbation theory to Category:Many-body perturbation theory

← Older revision	Revision as of 13:29, 8 April 2022
(No difference)

Huebsch: Created page with "'''Many-body perturbation theory''' includes screening and renormalization effects beyond the density-functional theory (DFT). It is based on the Green's-function formalism an..."

2022-04-08T13:29:05Z

Created page with "'''Many-body perturbation theory''' includes screening and renormalization effects beyond the density-functional theory (DFT). It is based on the Green's-function formalism an..."

New page

'''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.

== Theory ==

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

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

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 }}.

=== Constrained random-phase approximation ===

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).

* [[:Category:Constrained random-phase approximation formalism|Formalism used for the CRPA 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.

*{{TAG|The GW approximation of Hedin's equations}}.

=== 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}}.

* [[:Category:Bethe-Salpeter equations|Tags and articles related to BSE calculations.]]

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

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.
* '''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.
* '''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.

== 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}}.

== 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]]