GW approximation of Hedin's equations - Revision history

Huebsch at 14:45, 20 February 2024

2024-02-20T14:45:23Z

← Older revision		Revision as of 14:45, 20 February 2024
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	[[Category:Many-body perturbation theory]][[Category:Theory~~]][[Category:VASP6~~]][[Category:Low-scaling GW and RPA]]		[[Category:Many-body perturbation theory]][[Category:Theory]][[Category:Low-scaling GW and RPA]]

Kaltakm at 12:48, 18 October 2022

2022-10-18T12:48:15Z

← Older revision		Revision as of 12:48, 18 October 2022
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	<ref name="Hedin">[https://journals.aps.org/pr/abstract/10.1103/PhysRev.139.A796 L. Hedin, Phys. Rev. 139, A796 (1965)]</ref>		<ref name="Hedin">[https://journals.aps.org/pr/abstract/10.1103/PhysRev.139.A796 L. Hedin, Phys. Rev. 139, A796 (1965)]</ref>
	<ref name="HybertsenLouie">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.34.5390 M. S. Hybertsen, S. G. Louie Phys. Ref. B 34, 5390 (1986)]</ref>		<ref name="HybertsenLouie">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.34.5390 M. S. Hybertsen, S. G. Louie Phys. Ref. B 34, 5390 (1986)]</ref>
	~~<ref name="rojas">[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.1827 H. N. Rojas, R. W. Godby and R. J. Needs, Phys. Rev. Lett. 74, 1827 (1995)]</ref>~~
	<ref name="shishkin-PRB74">[http://link.aps.org/doi/10.1103/PhysRevB.74.035101 M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).]</ref>		<ref name="shishkin-PRB74">[http://link.aps.org/doi/10.1103/PhysRevB.74.035101 M. Shishkin and G. Kresse, Phys. Rev. B 74, 035101 (2006).]</ref>
	<ref name="shishkin-PRB75">[http://link.aps.org/doi/10.1103/PhysRevB.75.235102 M. Shishkin and G. Kresse, Phys. Rev. B 75, 235102 (2007).]</ref>		<ref name="shishkin-PRB75">[http://link.aps.org/doi/10.1103/PhysRevB.75.235102 M. Shishkin and G. Kresse, Phys. Rev. B 75, 235102 (2007).]</ref>
	<ref name="Faleev">[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.93.126406 S. V. Faleev, M. Schilfgaarde and T. Kotani, Phys. Rev. Lett. 93, 126406 (2004).]</ref>		<ref name="Faleev">[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.93.126406 S. V. Faleev, M. Schilfgaarde and T. Kotani, Phys. Rev. Lett. 93, 126406 (2004).]</ref>
	<ref name="shishkin:prl:07">[http://link.aps.org/doi/10.1103/PhysRevLett.99.246403 M. Shishkin, M. Marsman, and G. Kresse, Phys. Rev. Lett. 99, 246403 (2007).]</ref>		<ref name="shishkin:prl:07">[http://link.aps.org/doi/10.1103/PhysRevLett.99.246403 M. Shishkin, M. Marsman, and G. Kresse, Phys. Rev. Lett. 99, 246403 (2007).]</ref>
	~~<ref name="liu">[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.94.165109 P. Liu, M. Kaltak, J. Klimes and G. Kresse, Phys. Rev. B 94, 165109 (2016).]</ref>~~
	~~<ref name="kaltak">[http://pubs.acs.org/doi/abs/10.1021/ct5001268 M. Kaltak, J. Klimeš, and G. Kresse, Journal of Chemical Theory and Computation 10, 2498-2507 (2014). ]</ref>~~
	<ref name="grumet">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.155143 M. Grumet, P. Liu, M. Kaltak, J. Klimeš and Georg Kresse, Phys. Ref. B 98, 155143 (2018). ]</ref>		<ref name="grumet">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.155143 M. Grumet, P. Liu, M. Kaltak, J. Klimeš and Georg Kresse, Phys. Ref. B 98, 155143 (2018). ]</ref>
	</references>		</references>

Huebsch at 10:04, 19 July 2022

2022-07-19T10:04:14Z

← Older revision		Revision as of 10:04, 19 July 2022
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	[[Category:Many-~~Body Perturbation Theory~~]][[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]]		[[Category:Many-body perturbation theory]][[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]]

Ftran at 07:05, 12 April 2022

2022-04-12T07:05:18Z

← Older revision		Revision as of 07:05, 12 April 2022
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	</math>		</math>

	The G<sub>0</sub>W<sub>0</sub> method avoids the direct computation of the Green's function and neglects self-consistency in <math>G</math> completely. In fact, only the Kohn-Sham energies are updated from <math>\epsilon_{n\bf k}\to E^{(1)}_{n\bf k}</math>, while the orbitals remain unchanged. This is the reason why the G<sub>0</sub>W<sub>0</sub> method is internally selected as of VASP6 with {{TAG\|ALGO}} =EVGW0 ("eigenvalue GW") in combination with {{TAG\|NELM}}=1 to indicate one single iteration, even though the method is commonly known as the G<sub>0</sub>W<sub>0</sub> approach. To keep backwards-compatibility, however, {{TAG\|ALGO}}=G0W0 is still supported in ~~VAS6~~.		The G<sub>0</sub>W<sub>0</sub> method avoids the direct computation of the Green's function and neglects self-consistency in <math>G</math> completely. In fact, only the Kohn-Sham energies are updated from <math>\epsilon_{n\bf k}\to E^{(1)}_{n\bf k}</math>, while the orbitals remain unchanged. This is the reason why the G<sub>0</sub>W<sub>0</sub> method is internally selected as of VASP6 with {{TAG\|ALGO}} =EVGW0 ("eigenvalue GW") in combination with {{TAG\|NELM}}=1 to indicate one single iteration, even though the method is commonly known as the G<sub>0</sub>W<sub>0</sub> approach. To keep backwards-compatibility, however, {{TAG\|ALGO}}=G0W0 is still supported in VASP6.

	Note that avoiding self-consistency might seem a drastic step at first sight. However, the G<sub>0</sub>W<sub>0</sub> method often yields satisfactory results with band-gaps close to experimental measurements and is often employed for realistic band gap calculations.<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/>		Note that avoiding self-consistency might seem a drastic step at first sight. However, the G<sub>0</sub>W<sub>0</sub> method often yields satisfactory results with band-gaps close to experimental measurements and is often employed for realistic band gap calculations.<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/>

Kaltakm: /* Low-scaling GW: The Space-time Formalism */

2019-07-25T10:59:43Z

Low-scaling GW: The Space-time Formalism

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 <span id="lowGW">
 == Low-scaling GW: The Space-time Formalism ==
-The GW implementations in VASP described in the papers of Shishkin ''et al.''<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/> avoid storage of the Green's function <math>G</math> as well as Fourier transformations between time and frequency domain entirely. That is, all calculations are performed solely on the real frequency axis using Kramers-Kronig transformations for convolutions in the equation of <math>\chi</math> and <math>\Sigma</math> in reciprocal space and results in a relatively high computational cost that scales with <math>N^4</math> (number of electrons).
+{{:Low-scaling GW: The space-time formalism}}
 </span>
 == Limitations of GW ==
 From a physical point of view, the scGW method yields mostly unsatisfactory results compared to experiment. Notably, the band gaps are significantly overestimated compared to experiment, and plasmonic satellites are entirely missing in the spectral

Kaltakm at 09:04, 25 July 2019

2019-07-25T09:04:13Z

← Older revision		Revision as of 09:04, 25 July 2019
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	----		----
	[[Category:Many-Body Perturbation Theory]][[Category:Theory]][[Category:VASP6]]		[[Category:Many-Body Perturbation Theory]][[Category:Theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]]

Kaltakm at 17:01, 24 July 2019

2019-07-24T17:01:49Z

← Older revision		Revision as of 17:01, 24 July 2019
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	The method can be selected in VASP with {{TAG\|ALGO}}=QPGW0. See [[GW calculations#qpgw\|here]] for more information.		The method can be selected in VASP with {{TAG\|ALGO}}=QPGW0. See [[GW calculations#qpgw\|here]] for more information.

			<span id="lowGW">
	== Low-scaling GW: The Space-time Formalism ==		== Low-scaling GW: The Space-time Formalism ==
	The GW implementations in VASP described in the papers of Shishkin ''et al.''<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/> avoid storage of the Green's function <math>G</math> as well as Fourier transformations between time and frequency domain entirely. That is, all calculations are performed solely on the real frequency axis using Kramers-Kronig transformations for convolutions in the equation of <math>\chi</math> and <math>\Sigma</math> in reciprocal space and results in a relatively high computational cost that scales with <math>N^4</math> (number of electrons).		The GW implementations in VASP described in the papers of Shishkin ''et al.''<ref name="shishkin-PRB74"/><ref name="shishkin-PRB75"/> avoid storage of the Green's function <math>G</math> as well as Fourier transformations between time and frequency domain entirely. That is, all calculations are performed solely on the real frequency axis using Kramers-Kronig transformations for convolutions in the equation of <math>\chi</math> and <math>\Sigma</math> in reciprocal space and results in a relatively high computational cost that scales with <math>N^4</math> (number of electrons).
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	In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.		In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.
			</span>

	<span id="GWLimitations">		<span id="GWLimitations">

Kaltakm: Kaltakm moved page The GW approximation of Hedin's equations to GW approximation of Hedin's equations

2019-07-24T16:56:44Z

Kaltakm moved page The GW approximation of Hedin's equations to GW approximation of Hedin's equations

← Older revision	Revision as of 16:56, 24 July 2019
(No difference)

Kaltakm at 16:49, 24 July 2019

2019-07-24T16:49:53Z

← Older revision		Revision as of 16:49, 24 July 2019
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	In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.		In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.

			<span id="GWLimitations">
			== Limitations of GW ==
			From a physical point of view, the scGW method yields mostly unsatisfactory results compared to experiment. Notably, the band gaps are significantly overestimated compared to experiment, and plasmonic satellites are entirely missing in the spectral
			function.<ref name="grumet"/>

			The fact that "sloppier" GW flavours, such as EVGW<sub>0</sub> or even the single-shot approach yield more accurate results is due to fortuitous error cancelling and can be understood in terms of the band-gap <math>\Delta</math> of a system. The DFT gap is typically smaller than the GW band gap and yields, therefore, a larger dielectric function <math>\epsilon(\omega)=1-\chi(\omega)V</math> (the polarizability is inverse proportional to the band gap of the system). Although the band gap is corrected by GW, at the same time the screening of the Coulomb interaction is weakened. Forcing self-consistency only increases the effect and deteriorates the agreement with experimental band gaps.
			The rather disappointing results of the self-consistent GW approximation shows the general limitations of Hedin's equations in the absence of vertex corrections. It can be shown that inclusion of vertex corrections yields band gaps that are again in agreement with experiment.<ref name="shishkin:prl:07"/>
			</span>

	== References ==		== References ==

Kaltakm: /* Low-scaling GW: The Space-time Formalism */

2019-07-24T16:47:38Z

Low-scaling GW: The Space-time Formalism

← Older revision		Revision as of 16:47, 24 July 2019
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	Because, preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation <math>\Sigma(z)</math> of the self-energy can be determined with high accuracy. The analytical continuation typically yields energies that differ less than 20 meV from quasi-particle energies obtained from the real-frequency calculation.<ref name="liu"/>		Because, preceding Fourier transformations have been carried out with exponentially suppressed errors, the analytical continuation <math>\Sigma(z)</math> of the self-energy can be determined with high accuracy. The analytical continuation typically yields energies that differ less than 20 meV from quasi-particle energies obtained from the real-frequency calculation.<ref name="liu"/>

	In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.		In addition, the space-time formulation allows to solve the full Dyson equation for <math>G({\bf r,r'},i\tau)</math> with decent computational cost.<ref name="grumet"/> This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.
	From a physical point of view, the scGW method yields mostly unsatisfactory results compared to experiment. Notably, the band gaps are significantly overestimated compared to experiment, and plasmonic satellites are entirely missing in the spectral
	~~function.<ref name="grumet"/>~~

	The fact that "sloppier" GW flavours, such as EVGW<sub>0</sub> or even the single-shot approach yield more accurate results is due to fortuitous error cancelling and can be understood in terms of the band-gap <math>\Delta</math> of a system. The DFT gap is typically smaller than the GW band gap and yields, therefore, a larger dielectric function <math>\epsilon(\omega)=1-\chi(\omega)V</math> (the polarizability is inverse proportional to the band gap of the system). Although the band gap is corrected by GW, at the same time the screening of the Coulomb interaction is weakened. Forcing self-consistency only increases the effect and deteriorates the agreement with experimental band gaps.
	The rather disappointing results of the self-consistent GW approximation shows the general limitations of Hedin's equations in the absence of vertex corrections. It can be shown that inclusion of vertex corrections yields band gaps that are again in agreement with experiment.<ref name="shishkin:prl:07"/>

	== References ==		== References ==