The GW approximation - Revision history

Claudio at 11:45, 14 February 2020

2020-02-14T11:45:18Z

← Older revision		Revision as of 11:45, 14 February 2020
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	This runlevel calculates only the frequency dependent part of the self-energy operator, the mass operator		This runlevel calculates only the frequency dependent part of the self-energy operator, the mass operator
	[[File:PIC_doc_GW-27.png\|~~x50px~~\|center\|alt alternate text]]		[[File:PIC_doc_GW-27.png\|x150px\|center\|alt alternate text]]

	with the frequency dependent screening function defined as		with the frequency dependent screening function defined as

Claudio at 11:44, 14 February 2020

2020-02-14T11:44:49Z

← Older revision		Revision as of 11:44, 14 February 2020
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	Using the plasmon pole approximation for the screening function ([[../docs/doc_Xp.php\|Xp]]) it is possible to obtain an analytical expression for the matrix elements of the mass operator:		Using the plasmon pole approximation for the screening function ([[../docs/doc_Xp.php\|Xp]]) it is possible to obtain an analytical expression for the matrix elements of the mass operator:

	[[File:PIC_doc_GW-42.png\|~~x50px~~\|center\|alt alternate text]]		[[File:PIC_doc_GW-42.png\|x150px\|center\|alt alternate text]]

	The maximum vectors in the reciprocal lattice summation is determined by [[../vars/../vars/var_NGsBlk.php\|NGsBlk]] , while the range of the bands summation is given by [[../vars/../vars/var_GbndRnge.php\|GbndRnge]] . [[../vars/../vars/var_QPkrange.php\|QPkrange]] (or alternatively [[../vars/../vars/var_QPerange.php\|QPErange]] ) selects the matrix elements to calculate.		The maximum vectors in the reciprocal lattice summation is determined by [[../vars/../vars/var_NGsBlk.php\|NGsBlk]] , while the range of the bands summation is given by [[../vars/../vars/var_GbndRnge.php\|GbndRnge]] . [[../vars/../vars/var_QPkrange.php\|QPkrange]] (or alternatively [[../vars/../vars/var_QPerange.php\|QPErange]] ) selects the matrix elements to calculate.

Conor at 14:26, 13 September 2017

2017-09-13T14:26:50Z

← Older revision		Revision as of 14:26, 13 September 2017
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	=== The GW approximation ===		=== The GW approximation ===

	The gowo runlevel solves the Dyson equation ( [[~~The single~~ particle Green's function\|G]] ) using the ''GW approximation'' for the self-energy operator. This approximation is currently one of the most successful approximation for self-energy operator and has yielded remarkably accurate band structures for many materials. The GW approximation is obtained as first term in the expansion of Hedin equations in the expansion in the screened interaction W		The gowo runlevel solves the Dyson equation ( [[Single particle Green's function\|G]] ) using the ''GW approximation'' for the self-energy operator. This approximation is currently one of the most successful approximation for self-energy operator and has yielded remarkably accurate band structures for many materials. The GW approximation is obtained as first term in the expansion of Hedin equations in the expansion in the screened interaction W
	[[File:PIC_doc_GW-6.png\|x50px\|center\|alt alternate text]]		[[File:PIC_doc_GW-6.png\|x50px\|center\|alt alternate text]]

Conor at 14:26, 13 September 2017

2017-09-13T14:26:13Z

← Older revision		Revision as of 14:26, 13 September 2017
Line 2:		Line 2:
	=== The GW approximation ===		=== The GW approximation ===

	The gowo runlevel solves the Dyson equation ( [[~~../docs/doc_G.php~~\|G]] ) using the ''GW approximation'' for the self-energy operator. This approximation is currently one of the most successful approximation for self-energy operator and has yielded remarkably accurate band structures for many materials. The GW approximation is obtained as first term in the expansion of Hedin equations in the expansion in the screened interaction W		The gowo runlevel solves the Dyson equation ( [[The single particle Green's function\|G]] ) using the ''GW approximation'' for the self-energy operator. This approximation is currently one of the most successful approximation for self-energy operator and has yielded remarkably accurate band structures for many materials. The GW approximation is obtained as first term in the expansion of Hedin equations in the expansion in the screened interaction W
	[[File:PIC_doc_GW-6.png\|x50px\|center\|alt alternate text]]		[[File:PIC_doc_GW-6.png\|x50px\|center\|alt alternate text]]

Conor: Created page with " === The GW approximation === The gowo runlevel solves the Dyson equation ( G ) using the ''GW approximation'' for the self-energy operator. This approx..."

2017-09-13T14:20:24Z

Created page with " === The GW approximation === The gowo runlevel solves the Dyson equation ( [[../docs/doc_G.php|G]] ) using the ''GW approximation'' for the self-energy operator. This approx..."

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=== The GW approximation ===

The gowo runlevel solves the Dyson equation ( [[../docs/doc_G.php|G]] ) using the ''GW approximation'' for the self-energy operator. This approximation is currently one of the most successful approximation for self-energy operator and has yielded remarkably accurate band structures for many materials. The GW approximation is obtained as first term in the expansion of Hedin equations in the expansion in the screened interaction W
[[File:PIC_doc_GW-6.png|x50px|center|alt alternate text]]

The screening interaction is defined as

[[File:PIC_doc_GW-12.png|x50px|center|alt alternate text]]

The first term on the r.h.s. gives the exchange self-energy ([[../docs/doc_XX.php|XX]]), while the second term gives the frequency dependent part of the screening interaction. Starting from the non-interacting Green's function described in [[../docs/doc_G.php|G]] a first approximation to the (irreducible) response function is constructed

[[File:PIC_doc_GW-16.png|x50px|center|alt alternate text]]

This is used to construct the screened interaction W, and finally, the self-energy operator. At this point the Dyson equation should be solved self-consistently (to be conserving). As this scheme would be computationally too demanding, practical schemes neglects or approximates the self-consistency. Moreover approximations are introduced to make feasible the calculation of the screened interaction W.

==== The plasmon-pole approximation ====

This runlevel calculates only the frequency dependent part of the self-energy operator, the mass operator
[[File:PIC_doc_GW-27.png|x50px|center|alt alternate text]]

with the frequency dependent screening function defined as

[[File:PIC_doc_GW-32.png|x50px|center|alt alternate text]]

Using the plasmon pole approximation for the screening function ([[../docs/doc_Xp.php|Xp]]) it is possible to obtain an analytical expression for the matrix elements of the mass operator:

[[File:PIC_doc_GW-42.png|x50px|center|alt alternate text]]

The maximum vectors in the reciprocal lattice summation is determined by [[../vars/../vars/var_NGsBlk.php|NGsBlk]] , while the range of the bands summation is given by [[../vars/../vars/var_GbndRnge.php|GbndRnge]] . [[../vars/../vars/var_QPkrange.php|QPkrange]] (or alternatively [[../vars/../vars/var_QPerange.php|QPErange]] ) selects the matrix elements to calculate.

==== Dyson equation: solvers for the QP energies ====

The Dyson equation
[[File:PIC_doc_GW-50.png|x50px|center|alt alternate text]]

with

[[File:PIC_doc_GW-55.png|x50px|center|alt alternate text]]

[[File:PIC_doc_GW-58.png|x50px|center|alt alternate text]]

[[File:PIC_doc_GW-61.png|x50px|center|alt alternate text]]

can be solved by using either Newton or the secant root-finding algorithm ( [[../vars/../vars/var_DysSolver.php|DysSolver]] set to 'n' or 's'). The Newton method uses the first order of the Taylor series of a function in the vicinity of a root to evaluate the correction to the initial guess:

[[File:PIC_doc_GW-66.png|x50px|center|alt alternate text]]

This leads to

[[File:PIC_doc_GW-71.png|x50px|center|alt alternate text]]

with Z (the renormalization factor) defined

[[File:PIC_doc_GW-76.png|x50px|center|alt alternate text]]

The derivative of the self-energy with respect to the frequency is calculated from finite differences. The flag [[../vars/../vars/var_NewtDchk.php|NewtDchk]] enables the test on the convergence of this quantity. The secant method: nothing yet

==== References ====

* F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. '''61''', 237-312 (1998)
* G. Baym abd L. P. Kadanoff, Phys. Rev. '''124''', 287 (1961)
* Arno Schindlmayr, P. Garcia-Gonzalez, and R. W. Godby Phys. Rev. B '''64''', 235106 (2001)