Supercell core-hole theory - Revision history

Tal at 08:54, 17 October 2025

2025-10-17T08:54:34Z

← Older revision		Revision as of 08:54, 17 October 2025
Line 1:		Line 1:
	In the Supercell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge, cf. {{TAG\|ICORELEVEL}}. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.		In the Supercell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge, cf. {{TAG\|ICORELEVEL}}. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

	{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}		{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}

Liebetreu at 14:32, 14 October 2025

2025-10-14T14:32:17Z

← Older revision		Revision as of 14:32, 14 October 2025
Line 16:		Line 16:

	where <math>M</math> and <math>\varepsilon</math> denote momentum matrix elements and orbital energies.		where <math>M</math> and <math>\varepsilon</math> denote momentum matrix elements and orbital energies.
	Here we consider excitations only between valence (<math>v</math>) and conduction (<math>c</math>) bands. The components of the dielectric tensor are indexed by the Cartesian indices <math>\alpha</math> and <math>\beta</math>. <math>\Omega</math>, <math>e</math> and <math>m_{e}</math> denote the unit cell volume, electron charge and mass of the electron, respectively. In the PAW method the all-electron orbitals <math>\|\psi_{n\~~bold~~{k}}\rangle</math> are given by a linear transformation of the pseudo orbitals <math>\|\tilde{\psi}_{n\~~bold~~{k}}\rangle</math>:		Here we consider excitations only between valence (<math>v</math>) and conduction (<math>c</math>) bands. The components of the dielectric tensor are indexed by the Cartesian indices <math>\alpha</math> and <math>\beta</math>. <math>\Omega</math>, <math>e</math> and <math>m_{e}</math> denote the unit cell volume, electron charge and mass of the electron, respectively. In the PAW method the all-electron orbitals <math>\|\psi_{n\mathbf{k}}\rangle</math> are given by a linear transformation of the pseudo orbitals <math>\|\tilde{\psi}_{n\mathbf{k}}\rangle</math>:

	<math>		<math>
	\|\psi_{n\~~bold~~{k}}\rangle = \|\tilde{\psi}_{n\~~bold~~{k}}\rangle + \sum\limits_{i} (\|\phi_{i}\rangle - \|\tilde{\phi}_{i}\rangle) \langle \tilde{p}_{i} \|\tilde{\psi}_{n\~~bold~~{k}}\rangle.		\|\psi_{n\mathbf{k}}\rangle = \|\tilde{\psi}_{n\mathbf{k}}\rangle + \sum\limits_{i} (\|\phi_{i}\rangle - \|\tilde{\phi}_{i}\rangle) \langle \tilde{p}_{i} \|\tilde{\psi}_{n\mathbf{k}}\rangle.
	</math>		</math>

	The pseudo orbitals depend on the band index <math>n</math> and crystal momentum <math>\~~bold~~{k}</math>. <math>\|\phi_{i}\rangle</math>, <math>\|\tilde{\phi}_{i}\rangle</math> and <math>\|\tilde{p}_{i}\rangle</math> are all-electron partial waves, pseudo partial waves		The pseudo orbitals depend on the band index <math>n</math> and crystal momentum <math>\mathbf{k}</math>. <math>\|\phi_{i}\rangle</math>, <math>\|\tilde{\phi}_{i}\rangle</math> and <math>\|\tilde{p}_{i}\rangle</math> are all-electron partial waves, pseudo partial waves
	and the projectors, respectively.		and the projectors, respectively.
	The index <math>i</math> is a shorthand for the atomic site and other indices enumerating these quantities		The index <math>i</math> is a shorthand for the atomic site and other indices enumerating these quantities

Csheldon at 09:43, 12 May 2025

2025-05-12T09:43:19Z

← Older revision		Revision as of 09:43, 12 May 2025
Line 1:		Line 1:
	In the Supercell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.		In the Supercell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge, cf. {{TAG\|ICORELEVEL}}. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

	{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}		{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}

Csheldon at 09:38, 12 May 2025

2025-05-12T09:38:36Z

← Older revision		Revision as of 09:38, 12 May 2025
Line 1:		Line 1:
	In the ~~Super-cell~~ core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.		In the Supercell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

	{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}		{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}
Line 9:		Line 9:

	== Dielectric function used in the SCH method ==		== Dielectric function used in the SCH method ==
	Since the ~~wave length~~ of the electromagnetic waves in absorption spectroscopy is usually much larger than the characteristic momentum in solids, we start from the transversal expression for the imaginary part of the dielectric function in the long wavelength limit (<math>\mathbf{q}=0</math>) which is directly proportional to the absorption spectrum		Since the wavelength of the electromagnetic waves in absorption spectroscopy is usually much larger than the characteristic momentum in solids, we start from the transversal expression for the imaginary part of the dielectric function in the long wavelength limit (<math>\mathbf{q}=0</math>) which is directly proportional to the absorption spectrum

	<math>		<math>

Csheldon at 08:03, 12 May 2025

2025-05-12T08:03:06Z

← Older revision		Revision as of 08:03, 12 May 2025
Line 1:		Line 1:
	In the Super-cell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.		In the Super-cell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

	{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within ~~this~~ methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}		{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within these methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}

	== Excited electron treatment ==		== Excited electron treatment ==

Karsai at 10:40, 26 February 2025

2025-02-26T10:40:07Z

← Older revision		Revision as of 10:40, 26 February 2025
Line 1:		Line 1:
	In the Super-cell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.		In the Super-cell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

	~~'''Very~~ important~~''' is that~~ a sufficiently large ~~super cell is used~~ to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within this methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.		{{NB\|mind\| It is important to use a sufficiently large supercell to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within this methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.}}

	== Excited electron treatment ==		== Excited electron treatment ==

Tal: /* Excited electron treatment */

2025-02-21T14:10:15Z

Excited electron treatment

← Older revision		Revision as of 14:10, 21 February 2025
Line 6:		Line 6:
	[[File:Fch_xch.png\|300px\|thumb\|right\|Schematic representation of the self-interaction effects in the XCH spectra.]]		[[File:Fch_xch.png\|300px\|thumb\|right\|Schematic representation of the self-interaction effects in the XCH spectra.]]

	Two different approaches can be used to treat the excited electron in SCH. The excited electron can be placed into the lowest conduction band in the '''excited electron and core-hole (XCH)'''{{cite\|hetenyi:jcp:2004}} approach, alternatively the excited electron can be accounted for by a negative background charge to ensure that the cell remains neutral in the '''full core-hole (FCH)''' {{cite\|Prendergasst:prl:2006}} method. These two approaches are discussed and compared in great detail in Ref. {{cite\|unzog:prb:2022}}. The main shortcoming of the XCH approach comes from the self-interaction effects of the electron exchange and correlation functional caused when the excited electron is placed in the conduction band, which is more pronounced for strongly localized conduction bands. The effects of the self-interaction error are depicted in the figure on the right, which schematically represents the excitations from the core ''s''-orbital to the lowest unoccupied ''p''-orbital, i.e., the ''K''-edge. The self-interaction error in turn causes two effects: First, it shifts up the energy level of the conduction band occupied by the excited electron from the initial energy ''C'' to the new energy ''<nowiki>C'</nowiki>''. Second, it delocalizes the orbital~~. In FCH, by placing the negative charge in the background we avoid such self-interaction effects~~. Furthermore, the conduction band electron additionally contributes to the screening of the core hole, which in turn shifts the higher-lying unoccupied states ''a~~'' to ''<nowiki>a'</nowiki>~~'' and ''b'' ~~to ''<nowiki>b'</nowiki>''~~. For systems, where the lowest conduction bands are strongly delocalized, both XCH and FCH produce very similar results. However, for systems with localized conduction states, e.g. Li-halides, the FCH has been shown to yield better agreement with the results of the [[Bethe-Salpeter equation for core excitations\|BSE+GW calculations]] as well as the experimental spectra {{cite\|unzog:prb:2022}}.		Two different approaches can be used to treat the excited electron in SCH. The excited electron can be placed into the lowest conduction band in the '''excited electron and core-hole (XCH)'''{{cite\|hetenyi:jcp:2004}} approach, alternatively the excited electron can be accounted for by a negative background charge to ensure that the cell remains neutral in the '''full core-hole (FCH)''' {{cite\|Prendergasst:prl:2006}} method. These two approaches are discussed and compared in great detail in Ref. {{cite\|unzog:prb:2022}}. The main shortcoming of the XCH approach comes from the self-interaction effects of the electron exchange and correlation functional caused when the excited electron is placed in the conduction band, which is more pronounced for strongly localized conduction bands. The effects of the self-interaction error are depicted in the figure on the right, which schematically represents the excitations from the core ''s''-orbital to the lowest unoccupied ''p''-orbital, i.e., the ''K''-edge. The self-interaction error in turn causes two effects: First, it shifts up the energy level of the conduction band occupied by the excited electron from the initial energy ''C'' to the new energy ''<nowiki>C'</nowiki>''. Second, it delocalizes the orbital. Furthermore, the conduction band electron additionally contributes to the screening of the core hole, which in turn shifts the higher-lying unoccupied states, as shown in the figure for ''a'' and ''b'' bands. In FCH, by placing the negative charge in the homogeneous background we avoid such self-interaction effects. For systems, where the lowest conduction bands are strongly delocalized, both XCH and FCH produce very similar results. However, for systems with localized conduction states, e.g. Li-halides, the FCH has been shown to yield better agreement with the results of the [[Bethe-Salpeter equation for core excitations\|BSE+GW calculations]] as well as the experimental spectra {{cite\|unzog:prb:2022}}.

	== Dielectric function used in the SCH method ==		== Dielectric function used in the SCH method ==

Tal at 12:44, 21 February 2025

2025-02-21T12:44:58Z

← Older revision		Revision as of 12:44, 21 February 2025
Line 1:		Line 1:
	In the Super-cell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.		In the Super-cell core-hole method{{cite\|karsai:prb:2018}} (SCH) a chosen core electron is removed from the core leaving behind a positive charge. Since one wants to simulate the excitation of this electron into energetically higher-lying states, one electron is added to valence/conduction bands resembling the final state of the excitation process (this is also referred to as the final state approximation). With this setup a fully self-consistent electronic minimization is carried out. The core hole is perfectly screened by the other electrons in metals so there should be no difference between core-hole calculations and regular calculations. In semiconductors and insulators, this screening is very weak and a very strong attraction between the electrons and hole occurs, which results in not only a lowering of the excited states compared to the valence states, but also a very strong change of the valence/conduction band structure. Hence the relaxation of the valence/conduction electrons is the main effect in core-hole calculations. Fortunately the relaxation of the core states in core-hole calculations is negligible. This makes the implementation into a PAW framework smooth, since no on-the fly recalculation of PAW potentials is needed in every step of the electronic self-consistent cycle.

	'''Very important''' is that a sufficiently large super cell is used to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within this methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.		'''Very important''' is that a sufficiently large super cell is used to minimize the interaction of neighboring core holes within periodic boundary conditions. Hence the computational expense within this methods comes mainly from the use of large super cells. Nevertheless this method is usually still computationally cheaper than the BSE method for core electrons.

Tal: /* Excited electron treatment */

2025-02-21T09:10:04Z

Excited electron treatment

← Older revision		Revision as of 09:10, 21 February 2025
Line 6:		Line 6:
	[[File:Fch_xch.png\|300px\|thumb\|right\|Schematic representation of the self-interaction effects in the XCH spectra.]]		[[File:Fch_xch.png\|300px\|thumb\|right\|Schematic representation of the self-interaction effects in the XCH spectra.]]

	Two different approaches can be used to treat the excited electron in SCH. The excited electron can be placed into the lowest conduction band in the '''excited electron and core-hole (XCH)'''{{cite\|hetenyi:jcp:2004}} approach, alternatively the excited electron can be accounted for by a negative background charge to ensure that the cell remains neutral in the '''full core-hole (FCH)''' {{cite\|Prendergasst:prl:2006}} method. These two approaches are discussed and compared in great detail in Ref. {{cite\|unzog:prb:2022}}. The main shortcoming of the XCH approach comes from the self-interaction effects of the electron exchange and correlation functional caused when the excited electron is placed in the conduction band, which is more pronounced for strongly localized conduction bands. The effects of the self-interaction error are ~~schematically~~ depicted in the figure on the right~~. The figure sketches a~~ s to p ~~transition in~~ the K-edge~~. The lowest conduction band at level C is represented by a p orbital~~. The self-interaction error in turn causes two effects: First, it shifts up the energy level of the ~~occupied~~ conduction band from the initial energy C to the new energy C'. Second, it delocalizes the orbital. In FCH, by placing the negative charge in the background we avoid such self-interaction effects. For systems, where the conduction bands are ~~weakly localized~~, both ~~approaches~~ produce very similar results. However, for systems with localized conduction states, e.g. Li-halides, the FCH has been shown to yield better agreement with the results of the [[Bethe-Salpeter equation for core excitations\|BSE+GW calculations]] as well as experimental spectra {{cite\|unzog:prb:2022}}.		Two different approaches can be used to treat the excited electron in SCH. The excited electron can be placed into the lowest conduction band in the '''excited electron and core-hole (XCH)'''{{cite\|hetenyi:jcp:2004}} approach, alternatively the excited electron can be accounted for by a negative background charge to ensure that the cell remains neutral in the '''full core-hole (FCH)''' {{cite\|Prendergasst:prl:2006}} method. These two approaches are discussed and compared in great detail in Ref. {{cite\|unzog:prb:2022}}. The main shortcoming of the XCH approach comes from the self-interaction effects of the electron exchange and correlation functional caused when the excited electron is placed in the conduction band, which is more pronounced for strongly localized conduction bands. The effects of the self-interaction error are depicted in the figure on the right, which schematically represents the excitations from the core ''s''-orbital to the lowest unoccupied ''p''-orbital, i.e., the ''K''-edge. The self-interaction error in turn causes two effects: First, it shifts up the energy level of the conduction band occupied by the excited electron from the initial energy ''C'' to the new energy ''<nowiki>C'</nowiki>''. Second, it delocalizes the orbital. In FCH, by placing the negative charge in the background we avoid such self-interaction effects. Furthermore, the conduction band electron additionally contributes to the screening of the core hole, which in turn shifts the higher-lying unoccupied states ''a'' to ''<nowiki>a'</nowiki>'' and ''b'' to ''<nowiki>b'</nowiki>''. For systems, where the lowest conduction bands are strongly delocalized, both XCH and FCH produce very similar results. However, for systems with localized conduction states, e.g. Li-halides, the FCH has been shown to yield better agreement with the results of the [[Bethe-Salpeter equation for core excitations\|BSE+GW calculations]] as well as the experimental spectra {{cite\|unzog:prb:2022}}.

	== Dielectric function used in the SCH method ==		== Dielectric function used in the SCH method ==

Karsai: /* Excited electron treatment */

2025-02-20T14:12:38Z

Excited electron treatment

← Older revision		Revision as of 14:12, 20 February 2025
Line 6:		Line 6:
	[[File:Fch_xch.png\|300px\|thumb\|right\|Schematic representation of the self-interaction effects in the XCH spectra.]]		[[File:Fch_xch.png\|300px\|thumb\|right\|Schematic representation of the self-interaction effects in the XCH spectra.]]

	Two different approaches can be used to treat the excited electron in SCH. The excited electron can be placed into the lowest conduction band in the '''excited electron and core-hole (XCH)'''{{cite\|hetenyi:jcp:2004}} approach, alternatively the excited electron can be accounted for by a negative background charge to ensure that the cell remains neutral in the '''full core-hole (FCH)''' {{cite\|Prendergasst:prl:2006}} method. These two approaches are discussed and compared in great detail in Ref. {{cite\|unzog:prb:2022}}. The main shortcoming of the XCH approach comes from the self-interaction effects of the electron exchange and correlation functional caused when the excited electron is placed in the conduction band, which is more pronounced for strongly localized conduction bands. The effects of the self-interaction error are schematically depicted in the figure on the right. The figure sketches a s to p transition ~~of a~~ K-edge. The lowest conduction band at level C is represented by a p orbital. The self-interaction error in turn causes two effects: First, it shifts up the energy level of the occupied conduction band from the initial energy C to the new energy C'. Second, it delocalizes the orbital. In FCH, by placing the negative charge in the background we avoid such self-interaction effects. For systems, where the conduction bands are weakly localized, both approaches produce very similar results. However, for systems with localized conduction states, e.g. Li-halides, the FCH has been shown to yield better agreement with the results of the [[Bethe-Salpeter equation for core excitations\|BSE+GW calculations]] as well as experimental spectra {{cite\|unzog:prb:2022}}.		Two different approaches can be used to treat the excited electron in SCH. The excited electron can be placed into the lowest conduction band in the '''excited electron and core-hole (XCH)'''{{cite\|hetenyi:jcp:2004}} approach, alternatively the excited electron can be accounted for by a negative background charge to ensure that the cell remains neutral in the '''full core-hole (FCH)''' {{cite\|Prendergasst:prl:2006}} method. These two approaches are discussed and compared in great detail in Ref. {{cite\|unzog:prb:2022}}. The main shortcoming of the XCH approach comes from the self-interaction effects of the electron exchange and correlation functional caused when the excited electron is placed in the conduction band, which is more pronounced for strongly localized conduction bands. The effects of the self-interaction error are schematically depicted in the figure on the right. The figure sketches a s to p transition in the K-edge. The lowest conduction band at level C is represented by a p orbital. The self-interaction error in turn causes two effects: First, it shifts up the energy level of the occupied conduction band from the initial energy C to the new energy C'. Second, it delocalizes the orbital. In FCH, by placing the negative charge in the background we avoid such self-interaction effects. For systems, where the conduction bands are weakly localized, both approaches produce very similar results. However, for systems with localized conduction states, e.g. Li-halides, the FCH has been shown to yield better agreement with the results of the [[Bethe-Salpeter equation for core excitations\|BSE+GW calculations]] as well as experimental spectra {{cite\|unzog:prb:2022}}.

	== Dielectric function used in the SCH method ==		== Dielectric function used in the SCH method ==