Low-scaling GW: The space-time formalism - Revision history

Kaltakm: /* Theory */

2026-03-17T12:43:34Z

Theory

← Older revision		Revision as of 12:43, 17 March 2026
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	The GW implementations in VASP described in the papers of Shishkin ''et al.''{{cite\|shishkin:prb:2006}}{{cite\|shishkin:prb:2007}} 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.''{{cite\|shishkin:prb:2006}}{{cite\|shishkin:prb:2007}} 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 scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}}		The scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}} This rotation changes the signature of the Minkowski space-time <math>\eta=(-+++)</math> to the euclidean one <math>\eta=(++++)</math>, where correlation functions, like the Green's function do not oscillate in time.

	Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the euclidean space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space		Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the euclidean space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space

Kaltakm: /* Theory */

2026-03-17T12:39:56Z

Theory

← Older revision		Revision as of 12:39, 17 March 2026
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	The scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}}		The scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}}

	Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space		Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the euclidean space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space

	<math>G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{(0)}({\bf r}) \phi_{n{\bf k}}^{*(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right]</math>		<math>G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{(0)}({\bf r}) \phi_{n{\bf k}}^{*(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right]</math>

Csheldon at 10:18, 22 September 2025

2025-09-22T10:18:48Z

← Older revision		Revision as of 10:18, 22 September 2025
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	== References ==		== References ==
	~~</references>~~
	~~----~~

	[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:Low-scaling GW and RPA]][[Category:Theory]]		[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:Low-scaling GW and RPA]][[Category:Theory]]

Huebsch at 06:31, 21 February 2024

2024-02-21T06:31:10Z

← Older revision		Revision as of 06:31, 21 February 2024
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	----		----

	[[Category:VASP]][[Category:Many-body perturbation theory~~]][[Category:VASP6~~]][[Category:Low-scaling GW and RPA]][[Category:Theory]]		[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:Low-scaling GW and RPA]][[Category:Theory]]

Kaltakm at 12:33, 26 September 2022

2022-09-26T12:33:33Z

← Older revision		Revision as of 12:33, 26 September 2022
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	Available as of VASP.6 are low-scaling algorithms for [[RPA/ACFDT: Correlation energy in the Random Phase Approximation\|ACFDT/RPA]].{{cite\|kaltak:prb:2014}} This page describes the formalism of the corresponding low-scaling GW approach.~~<ref name="~~liu~~"/>~~		Available as of VASP.6 are low-scaling algorithms for [[RPA/ACFDT: Correlation energy in the Random Phase Approximation\|ACFDT/RPA]].{{cite\|kaltak:prb:2014}} This page describes the formalism of the corresponding low-scaling GW approach.{{cite\|liu:prb:2016}}
	A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR\|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW\|here]].		A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR\|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW\|here]].

	== Theory ==		== Theory ==

	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.''{{cite\|shishkin:prb:2006}}{{cite\|shishkin:prb:2007}} 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 scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.~~<ref name="~~rojas~~"/>~~		The scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}}

	Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space		Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space
Line 12:		Line 12:
	<math>G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{(0)}({\bf r}) \phi_{n{\bf k}}^{*(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right]</math>		<math>G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{(0)}({\bf r}) \phi_{n{\bf k}}^{*(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right]</math>

	Here <math>\Theta</math> is the step function and <math>f_{n{\bf k}}</math> the occupation number of the state <math>\phi_{n{\bf k}}^{(0)}</math>. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid <math>\tau_{m}</math>, where the number of time points can be selected in VASP via the {{TAG\|NOMEGA}} tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.~~<ref name="~~kaltak~~"/>~~		Here <math>\Theta</math> is the step function and <math>f_{n{\bf k}}</math> the occupation number of the state <math>\phi_{n{\bf k}}^{(0)}</math>. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid <math>\tau_{m}</math>, where the number of time points can be selected in VASP via the {{TAG\|NOMEGA}} tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.{{cite\|kaltak:2014}}

	Subsequently, the irreducible polarizability is calculated from a contraction of two imaginary time Green's functions		Subsequently, the irreducible polarizability is calculated from a contraction of two imaginary time Green's functions
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	</math>		</math>

	Afterwards, the same compressed Fourier transformation as for the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis <math>\chi({\bf r},{\bf r}',i\tau_m) \to \chi_{{\bf G}{\bf G}'}({\bf q},i \omega_n) </math>.~~<ref name="~~kaltak~~"/><ref name="~~liu~~"/>~~		Afterwards, the same compressed Fourier transformation as for the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] is employed to obtain the irreducible polarizability in reciprocal space on the imaginary frequency axis <math>\chi({\bf r},{\bf r}',i\tau_m) \to \chi_{{\bf G}{\bf G}'}({\bf q},i \omega_n) </math>.{{cite\|kaltak:2014}}{{cite\|liu:prb:2016}}

	The next step is the computation of the screened potential		The next step is the computation of the screened potential
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	on the real frequency axis <math>z=\omega</math>.		on the real frequency axis <math>z=\omega</math>.

	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.{{cite\|liu:prb:2016}}

	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.<noinclude>		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.{{cite\|grumet:prb:2018}} This approach is known as the self-consistent GW approach (scGW) and is available as of VASP6.<noinclude>

	=== Finite temperature formalism===		=== Finite temperature formalism===
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	== References ==		== References ==
	~~<references>~~
	~~<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="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-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="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>~~
	</references>		</references>
	----		----

	[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]		[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]

Kaltakm at 12:16, 26 September 2022

2022-09-26T12:16:59Z

← Older revision		Revision as of 12:16, 26 September 2022
Line 1:		Line 1:
	Available as of VASP.6 are low-scaling algorithms for [[RPA/ACFDT: Correlation energy in the Random Phase Approximation\|ACFDT/RPA]].~~<ref name="kaltak2"/>~~ This page describes the formalism of the corresponding low-scaling GW approach.<ref name="liu"/>		Available as of VASP.6 are low-scaling algorithms for [[RPA/ACFDT: Correlation energy in the Random Phase Approximation\|ACFDT/RPA]].{{cite\|kaltak:prb:2014}} This page describes the formalism of the corresponding low-scaling GW approach.<ref name="liu"/>
	A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR\|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW\|here]].		A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR\|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW\|here]].

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	== References ==		== References ==
	<references>		<references>
	~~<ref name="kaltak2">[https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.054115 M. Kaltak, J. Klimeš, and G. Kresse, Phys. Rev. B 90, 054115 (2014)]</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="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="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="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>

Huebsch at 10:11, 19 July 2022

2022-07-19T10:11:09Z

← Older revision		Revision as of 10:11, 19 July 2022
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	~~<noinclude>~~
	~~__TOC__~~
	Available as of VASP.6 are low-scaling algorithms for [[RPA/ACFDT: Correlation energy in the Random Phase Approximation\|ACFDT/RPA]].<ref name="kaltak2"/> This page describes the formalism of the corresponding low-scaling GW approach.<ref name="liu"/>		Available as of VASP.6 are low-scaling algorithms for [[RPA/ACFDT: Correlation energy in the Random Phase Approximation\|ACFDT/RPA]].<ref name="kaltak2"/> This page describes the formalism of the corresponding low-scaling GW approach.<ref name="liu"/>
	A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR\|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW\|here]].		A theoretical description of the ACFDT/RPA total energies is found [[ACFDT/RPA calculations#ACFDTR/RPAR\|here]]. A brief summary regarding GW theory is given below, while a practical guide can be found [[GW calculations#LowGW\|here]].

	== Theory ==		== Theory ==
	~~</noinclude>~~
	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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	on the real frequency axis <math>z=\omega</math>.		on the real frequency axis <math>z=\omega</math>.

	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.<noinclude>		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.<noinclude>
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	[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]		[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]
	~~</noinclude>~~

Huebsch at 10:05, 19 July 2022

2022-07-19T10:05:12Z

← Older revision		Revision as of 10:05, 19 July 2022
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	----		----

	[[Category:VASP]][[Category:Many-~~Body Perturbation Theory~~]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]		[[Category:VASP]][[Category:Many-body perturbation theory]][[Category:VASP6]][[Category:Low-scaling GW and RPA]][[Category:Theory]]
	</noinclude>		</noinclude>

Kaltakm: /* Theory */

2022-04-07T09:13:49Z

Theory

← Older revision		Revision as of 09:13, 7 April 2022
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	Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space		Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space

	<math>G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{*(0)}({\bf r}) \phi_{n{\bf k}}^{(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right]</math>		<math>G({\bf r},{\bf r}',i\tau)=-\sum_{n{\bf k}}\phi_{n{\bf k}}^{(0)}({\bf r}) \phi_{n{\bf k}}^{*(0)}({\bf r}') e^{-(\epsilon_{n{\bf k}}-\mu)\tau}\left[\Theta(\tau)(1-f_{n{\bf k}})-\Theta(-\tau)f_{n{\bf k}}\right]</math>

	Here <math>\Theta</math> is the step function and <math>f_{n{\bf k}}</math> the occupation number of the state <math>\phi_{n{\bf k}}^{(0)}</math>. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid <math>\tau_{m}</math>, where the number of time points can be selected in VASP via the {{TAG\|NOMEGA}} tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.<ref name="kaltak"/>		Here <math>\Theta</math> is the step function and <math>f_{n{\bf k}}</math> the occupation number of the state <math>\phi_{n{\bf k}}^{(0)}</math>. Because the Green's function is non-oscillatory on the imaginary time axis it can be represented on a coarse grid <math>\tau_{m}</math>, where the number of time points can be selected in VASP via the {{TAG\|NOMEGA}} tag. Usually 12 to 16 points are sufficient for insulators and small band gap systems.<ref name="kaltak"/>
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	=== Finite temperature formalism===		=== Finite temperature formalism===
	{{:Matsubara Formalism}}		{{:Matsubara Formalism}}

	== References ==		== References ==
	<references>		<references>

Kaltakm at 09:12, 2 March 2021

2021-03-02T09:12:39Z

← Older revision		Revision as of 09:12, 2 March 2021
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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.<noinclude>		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.<noinclude>

			=== Finite temperature formalism===
			{{:Matsubara Formalism}}
	== References ==		== References ==
	<references>		<references>

← Older revision		Revision as of 12:43, 17 March 2026
Line 6:		Line 6:
	The GW implementations in VASP described in the papers of Shishkin ''et al.''{{cite\|shishkin:prb:2006}}{{cite\|shishkin:prb:2007}} 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.''{{cite\|shishkin:prb:2006}}{{cite\|shishkin:prb:2007}} 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 scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}}		The scaling with system size can, however, be reduced to <math>N^3</math> by performing a so-called Wick-rotation to imaginary time <math>t\to i\tau</math>.{{cite\|rojas:prl:1995}} This rotation changes the signature of the Minkowski space-time <math>\eta=(-+++)</math> to the euclidean one <math>\eta=(++++)</math>, where correlation functions, like the Green's function do not oscillate in time.

	Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the euclidean space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space		Following the [[Groundstate in the Random Phase Approximation#ACFDTR/RPAR\| low scaling ACFDT/RPA algorithms]] the euclidean space-time implementation determines first, the non-interacting Green's function on the imaginary time axis in real space