GW and dielectric matrix - Revision history

Csheldon at 14:07, 24 March 2026

2026-03-24T14:07:17Z

← Older revision		Revision as of 14:07, 24 March 2026
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	{{NB\|mind\|[[Practical_guide_to_GW_calculations#First_step:_DFT_calculation\|All GW calculations require a preceding DFT calculation with many unoccupied states.]]}}		{{NB\|mind\|[[Practical_guide_to_GW_calculations#First_step:_DFT_calculation\|All GW calculations require a preceding DFT calculation with many unoccupied states.]]}}

	There is a lecture available on our YouTube channel on calculating the ~~[https~~:~~//youtu.be/6F_WNIh6V7I~~ optical gap] and ~~[https~~:~~//youtu.be/3YKJZHmcGhY~~ dielectric properties].		There is a lecture available on our YouTube channel on calculating the {{Video\|optical:merzuk:2025\|optical gap}} and {{Video\|dielectric:pedro:2025\|dielectric properties}}.

	== Spectral Method ==		== Spectral Method ==

Csheldon at 10:38, 22 September 2025

2025-09-22T10:38:22Z

← Older revision		Revision as of 10:38, 22 September 2025
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	If only the frequency-dependent dielectric matrix should be computed, {{TAG\|ALGO}}=''CHI'' can be used to skip the calculation of GW quasi-particle energies.		If only the frequency-dependent dielectric matrix should be computed, {{TAG\|ALGO}}=''CHI'' can be used to skip the calculation of GW quasi-particle energies.
	{{NB\|mind\|[[Practical_guide_to_GW_calculations#First_step:_DFT_calculation\|All GW calculations require a preceding DFT calculation with many unoccupied states.]]}}		{{NB\|mind\|[[Practical_guide_to_GW_calculations#First_step:_DFT_calculation\|All GW calculations require a preceding DFT calculation with many unoccupied states.]]}}

			There is a lecture available on our YouTube channel on calculating the [https://youtu.be/6F_WNIh6V7I optical gap] and [https://youtu.be/3YKJZHmcGhY dielectric properties].

	== Spectral Method ==		== Spectral Method ==

Miranda.henrique at 21:03, 27 September 2022

2022-09-27T21:03:58Z

Kaltakm: /* Direct calculation of the dielectric function */

2022-09-27T12:58:11Z

Direct calculation of the dielectric function

← Older revision		Revision as of 12:58, 27 September 2022
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	This method is selected with {{TAG\|LSPECTRAL}}=''.FALSE.'' and yields smoother dielectric functions than the spectral method described above.		This method is selected with {{TAG\|LSPECTRAL}}=''.FALSE.'' and yields smoother dielectric functions than the spectral method described above.
	Also, the direct calculation requires less memory.		Also, the direct calculation requires less memory.
	{{NB\|warning\|The direct calculation of the polarizability is much slower than the spectral method.}}		{{NB\|warning\|The direct calculation of the polarizability is much slower than the spectral method.}}
			A minimal {{FILE\|INCAR}} for such a calculation looks as follows:
			{{TAGBL\|ALGO}} = CHI # skip quasi-particle energies
			{{TAGBL\|LSPECTRAL}} = .FALSE. # direct calculation of chi
			{{TAGBL\|NOMEGA}} = 100 # number of frequency points

	== Local field effects ==		== Local field effects ==
	Both methods support the inclusion of local field effects in the dielectric function on the RPA level. These effects can be included with with {{TAG\|LRPA}}=''.TRUE.''.		Both methods support the inclusion of local field effects in the dielectric function on the RPA level. These effects can be included with with {{TAG\|LRPA}}=''.TRUE.''.

Kaltakm: /* Spectral Method */

2022-09-27T12:57:14Z

Spectral Method

← Older revision		Revision as of 12:57, 27 September 2022
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	However, reducing {{TAGBL\|CSHIFT}} in the spectral method introduces additional spikes in the dielectric function. Thus for visualization, one often performs a Lorentzian filter on the raw data, even though such a filter typically shifts peaks of the original dielectric function. There is no particular reason why the Lorentzian filter should be used, since smoothing data is essentially cosmetics; Gaussian smearing is also perfectly acceptable.		However, reducing {{TAGBL\|CSHIFT}} in the spectral method introduces additional spikes in the dielectric function. Thus for visualization, one often performs a Lorentzian filter on the raw data, even though such a filter typically shifts peaks of the original dielectric function. There is no particular reason why the Lorentzian filter should be used, since smoothing data is essentially cosmetics; Gaussian smearing is also perfectly acceptable.

			A minimal {{FILE\|INCAR}} for such a calculation looks as follows:
			{{TAGBL\|ALGO}} = CHI # skip quasi-particle energies
			{{TAGBL\|NOMEGA}} = 100 # number of frequency points
	Next an alternative approach, which already includes some sort of smoothing of the dielectric function, is presented.		Next an alternative approach, which already includes some sort of smoothing of the dielectric function, is presented.

Kaltakm at 12:51, 27 September 2022

2022-09-27T12:51:57Z

Show changes

Kaltakm at 11:42, 27 September 2022

2022-09-27T11:42:10Z

← Older revision		Revision as of 11:42, 27 September 2022
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	__toc__		__toc__

	For GW calculations the frequency dependent dielectric matrix in the Random Phase Approximation (RPA) is determined via the polarizability <math>{\bf \chi}</math> and the Coulomb potential <math>V</math> using <math>\epsilon(\omega)= 1 -V \cdot \chi(\omega)</math>.		For GW calculations the frequency dependent dielectric matrix <math>\epsilon(\omega)</math> in the Random Phase Approximation (RPA) is determined via the polarizability <math>{\bf \chi}</math> and the Coulomb potential <math>V</math> using <math>\epsilon(\omega)= 1 -V \cdot \chi(\omega)</math>.
	{{NB\|mind\|[[Practical_guide_to_GW_calculations#Low_scaling_GW_algorithms\|low-scaling GW algorithms]] determine the dielectric matrix on the imaginary frequency axis and cannot be used to calculate <math>{\bf \epsilon}</math> on the real frequency axis.}}		{{NB\|mind\|[[Practical_guide_to_GW_calculations#Low_scaling_GW_algorithms\|low-scaling GW algorithms]] determine the dielectric matrix on the imaginary frequency axis and cannot be used to calculate <math>{\bf \epsilon}</math> on the real frequency axis.}}
	The real-frequency dependent dielectric matrix can be calculated with the quartic-scaling GW implementation, which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.		The real-frequency dependent dielectric matrix can be calculated with the quartic-scaling GW implementation, which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.

Kaltakm at 16:23, 26 September 2022

2022-09-26T16:23:22Z

Show changes

Kaltakm at 13:41, 26 September 2022

2022-09-26T13:41:51Z

← Older revision		Revision as of 13:41, 26 September 2022
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	~~The GW routine also determines the frequency dependent dielectric matrix <math>{\bf \epsilon}</math>~~
	~~in the Random Phase Approximation (RPA) via the polarizability <math>{\bf \chi}</math> and the Coulomb potential <math>V</math> using <math>\epsilon(\omega)= 1 +V \cdot \chi(\omega)</math>.~~

	Note, [[Practical_guide_to_GW_calculations#Low_scaling_GW_algorithms\|low-scaling GW algorithms]] determine the dielectric matrix on the imaginary frequency axis. These algorithms cannot be used to calculate <math>{\bf \epsilon}</math> on the real frequency axis. Thus one, has to rely on the quartic-scaling GW implementation, which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.

	~~For instance, if one studies only the frequency dependent dielectric matrix, {{TAG\|ALGO}}=''CHI' can be used and the calculation of GW quasi-particle energies is skipped.~~
	To this end, VASP calculates the polarizability <math>{\bf \chi}_{{\bf G G}'}({\bf q},\omega) </math> in two steps. First the spectral density of <math>{\bf \chi}</math> is calculated with Fermi's golden rule{{cite\|shishkin:prb:2006}}
	~~<math>~~
	~~\sum_{ij}\delta(\epsilon_i - \epsilon_j -\omega) \left\langle i\right\mid {\bf r} \left\mid i\right\rangle~~
	~~</math>~~


	~~The latter algorithm is of particular interest if one studies local field effects in the dielectric matrix ({{TAG\|LRPA}}=.TRUE.). To this end, the polarizability is determined~~


	Note, the calculated microscopic frequency dependent dielectric function without local field effects		Note, the calculated microscopic frequency dependent dielectric function without local field effects
	corresponds to the same function obtained using {{TAG\|LOPTICS}}=''.TRUE.''.		corresponds to the same function obtained using {{TAG\|LOPTICS}}=''.TRUE.''.

Kaltakm at 13:41, 26 September 2022

2022-09-26T13:41:35Z

← Older revision		Revision as of 13:41, 26 September 2022
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	The GW routine also determines the frequency dependent dielectric matrix		The GW routine also determines the frequency dependent dielectric matrix <math>{\bf \epsilon}</math>
	~~without local field effects and with local field effects~~ in the ~~random~~		in the Random Phase Approximation (RPA) via the polarizability <math>{\bf \chi}</math> and the Coulomb potential <math>V</math> using <math>\epsilon(\omega)= 1 +V \cdot \chi(\omega)</math>.
	~~phase approximation~~ (RPA, {{TAG\|~~LRPA~~}}=''.~~TRUE~~.''),
	or the ~~DFT approximation~~ ({{TAG\|LRPA}}=''.~~FALSE.'', see Sec~~. ~~\ref{incar-rpa}~~).		Note, [[Practical_guide_to_GW_calculations#Low_scaling_GW_algorithms\|low-scaling GW algorithms]] determine the dielectric matrix on the imaginary frequency axis. These algorithms cannot be used to calculate <math>{\bf \epsilon}</math> on the real frequency axis. Thus one, has to rely on the quartic-scaling GW implementation, which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.
	~~The~~ calculated microscopic frequency dependent dielectric function,
	~~must match exactly those determined~~ using ~~the optical~~		For instance, if one studies only the frequency dependent dielectric matrix, {{TAG\|ALGO}}=''CHI' can be used and the calculation of GW quasi-particle energies is skipped.
	~~routine (~~{{TAG\|LOPTICS}}=''.TRUE.'', and~~, in the static limit,~~		To this end, VASP calculates the polarizability <math>{\bf \chi}_{{\bf G G}'}({\bf q},\omega) </math> in two steps. First the spectral density of <math>{\bf \chi}</math> is calculated with Fermi's golden rule{{cite\|shishkin:prb:2006}}
	the density functional perturbation routines ({{TAG\|LEPSILON}}=''.TRUE.'').		<math>
			\sum_{ij}\delta(\epsilon_i - \epsilon_j -\omega) \left\langle i\right\mid {\bf r} \left\mid i\right\rangle
			</math>


			The latter algorithm is of particular interest if one studies local field effects in the dielectric matrix ({{TAG\|LRPA}}=.TRUE.). To this end, the polarizability is determined


			Note, the calculated microscopic frequency dependent dielectric function without local field effects
			corresponds to the same function obtained using {{TAG\|LOPTICS}}=''.TRUE.''.
			and the density functional perturbation routines ({{TAG\|LEPSILON}}=''.TRUE.'').
	In fact, it is guaranteed that the results are identical to those determined		In fact, it is guaranteed that the results are identical to those determined
	using a summation over conduction band states ({{TAG\|LOPTICS}}).		using a summation over conduction band states ({{TAG\|LOPTICS}}).

← Older revision		Revision as of 12:58, 27 September 2022
Line 71:		Line 71:
	This method is selected with {{TAG\|LSPECTRAL}}=''.FALSE.'' and yields smoother dielectric functions than the spectral method described above.		This method is selected with {{TAG\|LSPECTRAL}}=''.FALSE.'' and yields smoother dielectric functions than the spectral method described above.
	Also, the direct calculation requires less memory.		Also, the direct calculation requires less memory.
	{{NB\|warning\|The direct calculation of the polarizability is much slower than the spectral method.}}		{{NB\|warning\|The direct calculation of the polarizability is much slower than the spectral method.}}
			A minimal {{FILE\|INCAR}} for such a calculation looks as follows:
			{{TAGBL\|ALGO}} = CHI # skip quasi-particle energies
			{{TAGBL\|LSPECTRAL}} = .FALSE. # direct calculation of chi
			{{TAGBL\|NOMEGA}} = 100 # number of frequency points

	== Local field effects ==		== Local field effects ==
	Both methods support the inclusion of local field effects in the dielectric function on the RPA level. These effects can be included with with {{TAG\|LRPA}}=''.TRUE.''.		Both methods support the inclusion of local field effects in the dielectric function on the RPA level. These effects can be included with with {{TAG\|LRPA}}=''.TRUE.''.

← Older revision		Revision as of 21:03, 27 September 2022
Line 3:		Line 3:
	For GW calculations the frequency dependent dielectric matrix <math>\epsilon(\omega)</math> in the Random Phase Approximation (RPA) is determined via the polarizability <math>{\bf \chi}</math> and the Coulomb potential <math>V</math> using <math>\epsilon(\omega)= 1 -V \cdot \chi(\omega)</math>.		For GW calculations the frequency dependent dielectric matrix <math>\epsilon(\omega)</math> in the Random Phase Approximation (RPA) is determined via the polarizability <math>{\bf \chi}</math> and the Coulomb potential <math>V</math> using <math>\epsilon(\omega)= 1 -V \cdot \chi(\omega)</math>.
	{{NB\|mind\|[[Practical_guide_to_GW_calculations#Low_scaling_GW_algorithms\|low-scaling GW algorithms]] determine the dielectric matrix on the imaginary frequency axis and cannot be used to calculate <math>{\bf \epsilon}</math> on the real frequency axis.}}		{{NB\|mind\|[[Practical_guide_to_GW_calculations#Low_scaling_GW_algorithms\|low-scaling GW algorithms]] determine the dielectric matrix on the imaginary frequency axis and cannot be used to calculate <math>{\bf \epsilon}</math> on the real frequency axis.}}
	The real-frequency dependent dielectric matrix can be calculated with the quartic-scaling GW implementation, which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.		The real-frequency dependent dielectric matrix can be calculated with the quartic-scaling GW implementation, in which usability is limited to relatively small unit cells containing a few dozen atoms at maximum.
	Here, two algorithms are available and can be selected via {{TAG\|LSPECTRAL}}. The methods are discussed below.		Here, two algorithms are available and can be selected via {{TAG\|LSPECTRAL}}. The methods are discussed below.

	If only the frequency dependent dielectric matrix ~~is of interest~~, {{TAG\|ALGO}}=''CHI'' can be used to skip the calculation of GW quasi-particle energies.		If only the frequency-dependent dielectric matrix should be computed, {{TAG\|ALGO}}=''CHI'' can be used to skip the calculation of GW quasi-particle energies.
	{{NB\|mind\|[[Practical_guide_to_GW_calculations#First_step:_DFT_calculation\|All GW calculations require a preceding DFT calculation with many unoccupied states.]]}}		{{NB\|mind\|[[Practical_guide_to_GW_calculations#First_step:_DFT_calculation\|All GW calculations require a preceding DFT calculation with many unoccupied states.]]}}

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	Here, VASP computes the polarizability in two steps.		Here, VASP computes the polarizability in two steps.

	First, the spectral density of the polarizability is calculated ~~with~~ Fermi's golden rule{{cite\|gajdos:prb:2006}}{{cite\|shishkin:prb:2006}}		First, the spectral density of the polarizability is calculated using Fermi's golden rule{{cite\|gajdos:prb:2006}}{{cite\|shishkin:prb:2006}}

	<math>		<math>
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	</math>		</math>

	The delta function is approximated ~~with~~ a triangular function centered at the transition energy ~~and is rather efficient~~. The real and imaginary part of the polarizability is calculated in a second step using a Hilbert transformation{{cite\|shishkin:prb:2006}}		The delta function is approximated by a triangular function centered at the transition energy. The real and imaginary part of the polarizability is calculated in a second step using a Hilbert transformation{{cite\|shishkin:prb:2006}}

	<math>		<math>
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	This integration is performed semi-analytically by restricting the integration variable ω' to a frequency grid that is determined by {{TAG\|NOMEGA}}, {{TAG\|OMEGATL}}, {{TAG\|OMEGAMIN}} and {{TAG\|OMEGAMAX}}.		This integration is performed semi-analytically by restricting the integration variable ω' to a frequency grid that is determined by {{TAG\|NOMEGA}}, {{TAG\|OMEGATL}}, {{TAG\|OMEGAMIN}} and {{TAG\|OMEGAMAX}}.

	Together with the approximation of the delta function in the spectral density (see above), the integration can be carried out analytically and one arrives essentially at a matrix-vector ~~formula~~		Together with the approximation of the delta function in the spectral density (see above), the integration can be carried out analytically and one arrives essentially at a matrix-vector product

	<math>		<math>
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	{{TAGBL\|ALGO}} = CHI # skip quasi-particle energies		{{TAGBL\|ALGO}} = CHI # skip quasi-particle energies
	{{TAGBL\|NOMEGA}} = 100 # number of frequency points		{{TAGBL\|NOMEGA}} = 100 # number of frequency points
	Next an alternative approach, which already includes some sort of smoothing of the dielectric function, is presented.		Next, an alternative approach, which already includes some sort of smoothing of the dielectric function, is presented.

	== Direct calculation of the dielectric function ==		== Direct calculation of the dielectric function ==
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	== Local field effects ==		== Local field effects ==
	Both methods support the inclusion of local field effects in the dielectric function on the RPA level. These effects can be included with with {{TAG\|LRPA}}=''.TRUE.''.		Both methods support the inclusion of local field effects in the dielectric function on the RPA level. These effects can be included with with {{TAG\|LRPA}}=''.TRUE.''.
	{{NB\|mind\|The GW routine is the only routine capable to include local field effects for the frequency dependent dielectric function.}}		{{NB\|mind\|The GW routine is the only routine capable to include local field effects for the frequency-dependent dielectric function.}}
	== Output ==		== Output ==
	The imaginary and real part of the frequency dependent dielectric function		The imaginary and real part of the frequency-dependent dielectric function
	is determined in all GW calculations. It can be conveniently grepped		is determined in all GW calculations. It can be conveniently grepped
	from the file using the command (note two ~~blanks~~ between the ~~two~~ words)		from the {{FILE\|OUTCAR}} file using the command (note the two spaces between the words)
	grep " dielectric constant" OUTCAR		grep " dielectric constant" OUTCAR
	The first value is the frequency (in eV) and the other two are		The first value is the frequency (in eV) and the other two are
	the real and imaginary ~~part~~ of the trace of the dielectric matrix.		the real and imaginary parts of the trace of the dielectric matrix.

	Furthermore, VASP writes following data into {{FILE\|OUTCAR}}.		Furthermore, VASP writes the following data into the {{FILE\|OUTCAR}} file.

	* The head of the microscopic dielectric function (without local field effects): <math>\epsilon_{\rm mic}(\omega) = \lim_{{\bf q}\to 0} \epsilon_{\bf 00}({\bf q},\omega)</math>		* The head of the microscopic dielectric function (without local field effects): <math>\epsilon_{\rm mic}(\omega) = \lim_{{\bf q}\to 0} \epsilon_{\bf 00}({\bf q},\omega)</math>
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	= \lim_{{\bf q}\to 0} \left[\epsilon^{-1}\right]_{\bf 00}({\bf q},\omega)</math>		= \lim_{{\bf q}\to 0} \left[\epsilon^{-1}\right]_{\bf 00}({\bf q},\omega)</math>

	The latter potentially includes local field effects depending on value of {{TAGBL\|LRPA}}.		The latter potentially includes local field effects depending on the value of {{TAGBL\|LRPA}}.
	A detailed explanation of these quantities is found in the [[Media:VASP_lecture_Dielectric.pdf\| lecture notes on dielectric properties]].		A detailed explanation of these quantities is found in the [[Media:VASP_lecture_Dielectric.pdf\| lecture notes on dielectric properties]].
	{{NB\|warning\|{{FILE\|OUTCAR}} contains only a subset of the complete dielectric function ( the one restricted to {{TAG\|NOMEGA}} points).}}		{{NB\|warning\|{{FILE\|OUTCAR}} contains only a subset of the complete dielectric function ( the one restricted to {{TAG\|NOMEGA}} points).}}
	The complete frequency dependence ({{TAG\|NEDOS}} frequency points) is written to {{FILE\|vasprun.xml}} and has following format:		The complete frequency dependence ({{TAG\|NEDOS}} frequency points) is written to {{FILE\|vasprun.xml}} and has following format:
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	== Comparison with k-p and density perturbation theory ==		== Comparison with k-p and density perturbation theory ==

	The calculated microscopic frequency dependent dielectric function without local field effects ~~corresponds to~~ the same function obtained using {{TAG\|LOPTICS}}=''.TRUE.'',		The calculated microscopic frequency-dependent dielectric function without local field effects is the same function obtained using {{TAG\|LOPTICS}}=''.TRUE.'',
	as well as the one obtained from density functional perturbation routines ({{TAG\|LEPSILON}}=''.TRUE.'').		as well as the one obtained from density functional perturbation routines ({{TAG\|LEPSILON}}=''.TRUE.'').
	In fact, it is guaranteed that the results are identical to those determined		In fact, it is guaranteed that the results are identical to those determined
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	differences can arise, because		differences can arise, because

	* ~~the~~ GW routine uses ~~less~~ frequency points and different frequency grids than the optics routine or		* The GW routine uses fewer frequency points and different frequency grids than the optics routine or

	* again from a different complex shift.		* again from a different complex shift.
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	the calculations, by calculating the response functions only		the calculations, by calculating the response functions only
	at the <math>\Gamma</math>-point. This can be achieved by setting the following values in the {{TAG\|INCAR}} file:		at the <math>\Gamma</math>-point. This can be achieved by setting the following values in the {{TAG\|INCAR}} file:
	NKREDX = number of k-points in direction of first lattice vector		NKREDX = number of k-points in direction of the first lattice vector
	NKREDY = number of k-points in direction of second lattice vector		NKREDY = number of k-points in direction of the second lattice vector
	NKREDZ = number of k-points in direction of third lattice vector		NKREDZ = number of k-points in direction of the third lattice vector
	The calculation of the QP shifts can be bypassed by setting		The calculation of the QP shifts can be bypassed by setting
	{{TAG\|ALGO}}=''CHI''.		{{TAG\|ALGO}}=''CHI''.