Constrained–random-phase–approximation formalism - Revision history

Huebsch: Consistent hyphenation: [descriptor]-cRPA method in all headings and body prose (style guide: multi-word modifiers must be hyphenated)

2026-03-20T09:09:47Z

Consistent hyphenation: [descriptor]-cRPA method in all headings and body prose (style guide: multi-word modifiers must be hyphenated)

Huebsch: Add overview paragraph; center band-structure images; fix: typo "thuis", unclosed NB tip box, spectral heading annotation, s-cRPA self-ref, {{Available|6.6.0}}

2026-03-20T09:04:08Z

Add overview paragraph; center band-structure images; fix: typo "thuis", unclosed NB tip box, spectral heading annotation, s-cRPA self-ref, {{Available|6.6.0}}

← Older revision		Revision as of 09:04, 20 March 2026
Line 3:		Line 3:
	The resulting partially screened Coulomb interaction is often evaluated in a localized basis that spans the target space and is described by the model Hamiltonian.		The resulting partially screened Coulomb interaction is often evaluated in a localized basis that spans the target space and is described by the model Hamiltonian.
	Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher level theory, such as [[DFT+DMFT calculations\|dynamical mean-field theory]] (DMFT).		Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher level theory, such as [[DFT+DMFT calculations\|dynamical mean-field theory]] (DMFT).

			This page introduces the theoretical foundations of cRPA. The [[#Tight-binding model Hamiltonian\|first section]] presents the tight-binding model Hamiltonian and the [[Wannier basis]] used to represent the target states. The [[#Effective Coulomb kernel in constrained random-phase approximation\|second section]] describes four methods for computing the effective Coulomb kernel by excluding screening contributions from the target space. The [[#Off-center interactions\|final section]] covers off-center Coulomb integrals for extended models.

	== Tight-binding model Hamiltonian ==		== Tight-binding model Hamiltonian ==
Line 18:		Line 20:
	\| w_{i\bf R}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} e^{-i {\bf k R}} T_{i n}^{\sigma({\bf k})} \| \psi_{n\bf k}^\sigma \rangle		\| w_{i\bf R}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} e^{-i {\bf k R}} T_{i n}^{\sigma({\bf k})} \| \psi_{n\bf k}^\sigma \rangle
	</math>		</math>
	Usually, the target states and ~~thuis~~ the appropriate basis set is localized such that the interaction between periodic images can be neglected. This allows to work with Wannier functions in the unit cell at <math>\bf R=0</math>:		Usually, the target states and thus the appropriate basis set is localized such that the interaction between periodic images can be neglected. This allows to work with Wannier functions in the unit cell at <math>\bf R=0</math>:
	::<math>		::<math>
	\| w_{i}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} T_{i n}^{\sigma({\bf k})} \| \psi_{n\bf k}^\sigma \rangle		\| w_{i}^\sigma \rangle = \frac{1}{N_k}\sum_{n\bf k} T_{i n}^{\sigma({\bf k})} \| \psi_{n\bf k}^\sigma \rangle
Line 24:		Line 26:
	{{NB\|mind\|Here, <math>T</math> instead of the widely used <math>U</math> notation for the transformation matrix is used to avoid confusion with the Hubbard parameter.}}		{{NB\|mind\|Here, <math>T</math> instead of the widely used <math>U</math> notation for the transformation matrix is used to avoid confusion with the Hubbard parameter.}}
	In practice, one builds a model Hamiltonian only for a small subset of Bloch functions. These '''target states''' are typically centered around the chemical potential (or Fermi energy see {{TAG\|EFERMI}}) and are strongly localized around ions. The model Hamiltonian can be solved successfully only if the target states are well-represented by the [[Wannier basis]].		In practice, one builds a model Hamiltonian only for a small subset of Bloch functions. These '''target states''' are typically centered around the chemical potential (or Fermi energy see {{TAG\|EFERMI}}) and are strongly localized around ions. The model Hamiltonian can be solved successfully only if the target states are well-represented by the [[Wannier basis]].
	{{NB\|tip\|As an indicator of the quality of the Wannier representation, compare the [[Band-structure calculation using density-functional theory\|original band structure]] with the Wannier interpolated one.		{{NB\|tip\|As an indicator of the quality of the Wannier representation, compare the [[Band-structure calculation using density-functional theory\|original band structure]] with the Wannier interpolated one.}}

	In the following example considering SrVO3, the target space consists of three Bloch bands (red bands) that may be represented well by three Wannier states:		In the following example considering SrVO3, the target space consists of three Bloch bands (red bands) that may be represented well by three Wannier states:

	[[File:SrVO3_t2g_bands.png\|250px]]		[[File:SrVO3_t2g_bands.png\|250px\|center]]

	The complete target space is spanned by a certain number of Bloch bands. These bands can be selected with the {{TAG\|NCRPA_BANDS}} tag. Refer to the [[CRPA_of_SrVO3\|full SrVO3 cRPA calculation]] for more details of the setup.		The complete target space is spanned by a certain number of Bloch bands. These bands can be selected with the {{TAG\|NCRPA_BANDS}} tag. Refer to the [[CRPA_of_SrVO3\|full SrVO3 cRPA calculation]] for more details of the setup.
Line 36:		Line 38:
	Below is an example (face-centered-cubic Ni), where the delocalized s-band (blue) crosses the five target d-states (red):		Below is an example (face-centered-cubic Ni), where the delocalized s-band (blue) crosses the five target d-states (red):

	[[File:Ni_d_s_bands.png\|250px]]		[[File:Ni_d_s_bands.png\|250px\|center]]

	Due to the hybridization of s and d electrons, the system requires at least six [[Wannier states]] to properly represent the electronic structure of five target states.		Due to the hybridization of s and d electrons, the system requires at least six [[Wannier states]] to properly represent the electronic structure of five target states.
Line 43:		Line 45:
	If a modification of the band structure is acceptable within an energy window, these five target states might be '''disentangled''' from the remaining ones, and one arrives at the following picture:		If a modification of the band structure is acceptable within an energy window, these five target states might be '''disentangled''' from the remaining ones, and one arrives at the following picture:

	[[File:Ni_d_bands_decoupled.png\|250px]]		[[File:Ni_d_bands_decoupled.png\|250px\|center]]

	Here, the original Bloch bands (gray lines) are projected to five non-crossing Wannier states.		Here, the original Bloch bands (gray lines) are projected to five non-crossing Wannier states.
Line 216:		Line 218:
	For such cases, the {{FILE\|WAVEDER}} file should be deleted before the cRPA step. Other cRPA methods do not suffer from this problem and the usage of the long-wave limit is strongly encouraged.		For such cases, the {{FILE\|WAVEDER}} file should be deleted before the cRPA step. Other cRPA methods do not suffer from this problem and the usage of the long-wave limit is strongly encouraged.

	=== Spectral cRPA method (s-cRPA) ~~''recommended''~~===		=== Spectral cRPA method (s-cRPA) ===
	The s-cRPA method (s-cRPA) is a robust approach that remedies the main drawbacks of both w-cRPA and p-cRPA.{{cite\|kaltak:prb:2025}}		The spectral cRPA method (s-cRPA) is a robust approach that remedies the main drawbacks of both w-cRPA and p-cRPA.{{cite\|kaltak:prb:2025}}
	This method uses the eigenspectrum of the original projectors		This method uses the eigenspectrum of the original projectors
	::<math>		::<math>
Line 253:		Line 255:

	The basis orbitals can be selected with {{TAG\|LOCALIZED_BASIS}}.		The basis orbitals can be selected with {{TAG\|LOCALIZED_BASIS}}.
	{{Available\|~~VASP~~ 6.6.0}}		{{Available\|6.6.0}}

	== Additional resources ==		== Additional resources ==

Huebsch at 08:55, 20 March 2026

2026-03-20T08:55:36Z

Show changes

Huebsch: Style fixes: section spacing, TAG→FILE, TAG|value form, Available template, American English (off-center), prose fixes (comma, typo, Coulomb cap), {{Video}} for YouTube link, add Additional resources section

2026-03-20T07:56:21Z

Style fixes: section spacing, TAG→FILE, TAG|value form, Available template, American English (off-center), prose fixes (comma, typo, Coulomb cap), {{Video}} for YouTube link, add Additional resources section

← Older revision		Revision as of 07:56, 20 March 2026
Line 3:		Line 3:
	The resulting partially screened Coulomb interaction is usually evaluated in a localized basis that spans the target space and is described by the model Hamiltonian.		The resulting partially screened Coulomb interaction is usually evaluated in a localized basis that spans the target space and is described by the model Hamiltonian.
	Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher level theory, such as dynamical mean field theory (DMFT).		Usually, the target space is low-dimensional (up to 5 states) and therefore allows for the application of a higher level theory, such as dynamical mean field theory (DMFT).
	~~In a lecture on the [https://youtu.be/6F_WNIh6V7I optical gap] on our YouTube channel, cRPA is introduced.~~

	==Model Hamiltonians==
			== Model Hamiltonians ==
	A model Hamiltonian describes a small subset of electrons around the chemical potential and has, in second quantization, the following form		A model Hamiltonian describes a small subset of electrons around the chemical potential and has, in second quantization, the following form
	::<math>		::<math>
Line 73:		Line 73:
	In most applications, however, one considers the static limit <math>U=U(\omega\to 0)</math>.		In most applications, however, one considers the static limit <math>U=U(\omega\to 0)</math>.

	In practice one often, simplifies the model Hamiltonian further and works with the Hubbard-Kanamori parameters:{{cite\|vaugier:prb:86}}		In practice, one often simplifies the model Hamiltonian further and works with the Hubbard-Kanamori parameters:{{cite\|vaugier:prb:86}}
	::<math>		::<math>
	{\cal U }^{\sigma\sigma'} = \frac 1 N \sum_{i\in \cal T} U_{iiii}^{\sigma\sigma'}		{\cal U }^{\sigma\sigma'} = \frac 1 N \sum_{i\in \cal T} U_{iiii}^{\sigma\sigma'}
Line 86:		Line 86:

	== Effective Coulomb kernel in constrained random-phase approximation ==		== Effective Coulomb kernel in constrained random-phase approximation ==
	In analogy to the [[GW approximation of Hedin's equations#W_reci\|screened Coulomb kernel in GW]], the effective ~~coulomb~~ kernel is calculated as		In analogy to the [[GW approximation of Hedin's equations#W_reci\|screened Coulomb kernel in GW]], the effective Coulomb kernel is calculated as
	::<math>		::<math>
	U^{\sigma\sigma'}{{\bf G}{\bf G}'}({\bf q},i\omega)=\left[\delta{{\bf G}{\bf G}'}-(\chi^{\sigma\sigma'}{{\bf G}{\bf G}'}({\bf q},i\omega) - \tilde\chi^{\sigma\sigma'}{{\bf G}{\bf G}'}({\bf q},i\omega) ) \cdot V_{{\bf G}{\bf G}'}({\bf q})\right]^{-1}V_{{\bf G}{\bf G}'}({\bf q})		U^{\sigma\sigma'}{{\bf G}{\bf G}'}({\bf q},i\omega)=\left[\delta{{\bf G}{\bf G}'}-(\chi^{\sigma\sigma'}{{\bf G}{\bf G}'}({\bf q},i\omega) - \tilde\chi^{\sigma\sigma'}{{\bf G}{\bf G}'}({\bf q},i\omega) ) \cdot V_{{\bf G}{\bf G}'}({\bf q})\right]^{-1}V_{{\bf G}{\bf G}'}({\bf q})
Line 121:		Line 121:
	=== Disentanglement method (d-cRPA) ===		=== Disentanglement method (d-cRPA) ===
	Miyake, Aryasetiawan, and Imada{{cite\|miyake:prb:80}} propose to disentangle the target space from the full space by diagonalizing the Hamiltonian in both spaces independently.		Miyake, Aryasetiawan, and Imada{{cite\|miyake:prb:80}} propose to disentangle the target space from the full space by diagonalizing the Hamiltonian in both spaces independently.
	This yields a ~~band~~ disentangled band ~~structures~~ similar to the one shown in the third example above.		This yields a disentangled band structure similar to the one shown in the third example above.
	The corresponding target polarizability reduces to a constrained Adler and Wiser expression{{cite\|adler:1962}}{{cite\|wiser:1963}}		The corresponding target polarizability reduces to a constrained Adler and Wiser expression{{cite\|adler:1962}}{{cite\|wiser:1963}}
	::<math>\tilde \chi^\sigma_{{\bf G,G}'}({\bf q},i\omega)=		::<math>\tilde \chi^\sigma_{{\bf G,G}'}({\bf q},i\omega)=
Line 232:		Line 232:
	{{NB\|tip\|Recommended for all cRPA calculations as of VASP version 6.6.0}}		{{NB\|tip\|Recommended for all cRPA calculations as of VASP version 6.6.0}}

	== Off-~~centre~~ interactions ==		== Off-center interactions ==
	Off-~~centre~~ Coulomb integrals can be evaluated using {{TAG\|ALGO}}~~=2e4wa~~ or by adding {{TAG\|LTWO_CENTER}}=T.		Off-center Coulomb integrals can be evaluated using {{TAG\|ALGO\|2e4wa}} or by adding {{TAG\|LTWO_CENTER\|T}}.

	When chosen, the system calculates two types of integrals:		When chosen, the system calculates two types of integrals:
Line 251:		Line 251:
	The extent of displacement vectors <math>{\bf R}</math> is automatically determined based on the selected k-point grid.		The extent of displacement vectors <math>{\bf R}</math> is automatically determined based on the selected k-point grid.

	The calculation of these integrals occurs as a post-processing step. For bare, off-~~centre~~ Coulomb integrals ({{FILE\|VRijkl}}), a valid {{FILE\|WAVECAR}} file must be present in the working directory. For effectively screened Coulomb integrals ({{FILE\|URijkl}}), both a valid {{FILE\|WAVECAR}} file and {{~~TAG~~\|WFULLxxxx.tmp}} files are required. The {{~~TAG~~\|WFULLxxxx.tmp}} files are automatically generated following a successful {{TAG\|ALGO}}~~=CRPA~~ job.		The calculation of these integrals occurs as a post-processing step. For bare, off-center Coulomb integrals ({{FILE\|VRijkl}}), a valid {{FILE\|WAVECAR}} file must be present in the working directory. For effectively screened Coulomb integrals ({{FILE\|URijkl}}), both a valid {{FILE\|WAVECAR}} file and {{FILE\|WFULLxxxx.tmp}} files are required. The {{FILE\|WFULLxxxx.tmp}} files are automatically generated following a successful {{TAG\|ALGO\|CRPA}} job.

	The basis orbitals can be selected with {{TAG\|LOCALIZED_BASIS}}.		The basis orbitals can be selected with {{TAG\|LOCALIZED_BASIS}}.
	{{NB\|~~mind\|Avaliable as of~~ VASP.6.6.0.}}		{{Available\|VASP 6.6.0}}


			== Additional resources ==
			=== Lectures ===
			* {{Video\|optical:merzuk:2025\|Optical gap and the constrained random-phase approximation}}

	== Related tags and articles==		== Related tags and articles ==
	{{TAG\|ALGO}},		{{TAG\|ALGO}},
	{{TAG\|NTARGET_STATES}},		{{TAG\|NTARGET_STATES}},
Line 272:		Line 277:
	{{TAG\|VRijkl}},		{{TAG\|VRijkl}},
	{{TAG\|LTWO_CENTER}},		{{TAG\|LTWO_CENTER}},
	{{~~TAG~~\|WFULLxxxx.tmp}},		{{FILE\|WFULLxxxx.tmp}},
	[[CRPA_of_SrVO3\|cRPA of SrVO3]]		[[CRPA_of_SrVO3\|cRPA of SrVO3]]

	==References==		== References ==
	<references/>		<references/>
	----		----

	[[Category:Theory]][[Category:Many-body perturbation theory]][[Category:Constrained-random-phase approximation]][[Category:Wannier functions]]		[[Category:Theory]][[Category:Many-body perturbation theory]][[Category:Constrained-random-phase approximation]][[Category:Wannier functions]]

Kaltakm: /* Off-centre interactions */

2026-03-17T14:06:16Z

Off-centre interactions

← Older revision		Revision as of 14:06, 17 March 2026
Line 253:		Line 253:
	The calculation of these integrals occurs as a post-processing step. For bare, off-centre Coulomb integrals ({{FILE\|VRijkl}}), a valid {{FILE\|WAVECAR}} file must be present in the working directory. For effectively screened Coulomb integrals ({{FILE\|URijkl}}), both a valid {{FILE\|WAVECAR}} file and {{TAG\|WFULLxxxx.tmp}} files are required. The {{TAG\|WFULLxxxx.tmp}} files are automatically generated following a successful {{TAG\|ALGO}}=CRPA job.		The calculation of these integrals occurs as a post-processing step. For bare, off-centre Coulomb integrals ({{FILE\|VRijkl}}), a valid {{FILE\|WAVECAR}} file must be present in the working directory. For effectively screened Coulomb integrals ({{FILE\|URijkl}}), both a valid {{FILE\|WAVECAR}} file and {{TAG\|WFULLxxxx.tmp}} files are required. The {{TAG\|WFULLxxxx.tmp}} files are automatically generated following a successful {{TAG\|ALGO}}=CRPA job.

	The basis orbitals can be selected with {{TAG\|~~DMFT_BASIS~~}}.		The basis orbitals can be selected with {{TAG\|LOCALIZED_BASIS}}.
	{{NB\|mind\|Avaliable as of VASP.6.6.0.}}		{{NB\|mind\|Avaliable as of VASP.6.6.0.}}

Kaltakm: /* Related tags and articles */

2026-03-17T14:05:58Z

Kaltakm: /* Parameter definitions */

2026-03-16T11:18:32Z

Parameter definitions

← Older revision		Revision as of 11:18, 16 March 2026
Line 66:		Line 66:
	\langle u^\sigma_{n_1\bf k}\| e^{-i({\bf q + G})\cdot {\bf r}} \|u^\sigma_{n_2\bf k-q}\rangle		\langle u^\sigma_{n_1\bf k}\| e^{-i({\bf q + G})\cdot {\bf r}} \|u^\sigma_{n_2\bf k-q}\rangle
	U^{\sigma\sigma'}_{{\bf G G}'}({\bf q})		U^{\sigma\sigma'}_{{\bf G G}'}({\bf q})
	\langle u^{\sigma'}{n_3\bf k'-q}\| e^{i({\bf q - G'})\cdot {\bf r'} }\|u^{\sigma'}{n_4\bf k'}\rangle		\langle u^{\sigma'}{n_3\bf k'-q}\| e^{i({\bf q - G'})\cdot {\bf r'} }\|u^{\sigma'}{n_4\bf k'}\rangle
	T{kn_3}^{\sigma'({\bf k'-q})}		T_{kn_3}^{*\sigma'({\bf k'-q})}
	T_{ln_4}^{\sigma'({\bf k'})}		T_{ln_4}^{\sigma'({\bf k'})}
	</math>		</math>

Kaltakm: /* Parameter definitions */

2026-03-16T11:17:03Z

Parameter definitions

← Older revision		Revision as of 11:17, 16 March 2026
Line 65:		Line 65:
	T_{jn_2}^{\sigma({\bf k-q})}		T_{jn_2}^{\sigma({\bf k-q})}
	\langle u^\sigma_{n_1\bf k}\| e^{-i({\bf q + G})\cdot {\bf r}} \|u^\sigma_{n_2\bf k-q}\rangle		\langle u^\sigma_{n_1\bf k}\| e^{-i({\bf q + G})\cdot {\bf r}} \|u^\sigma_{n_2\bf k-q}\rangle
	U^{\sigma\sigma'}*_{{\bf G G}'}({\bf q})		U^{\sigma\sigma'}_{{\bf G G}'}({\bf q})
	\langle u^{\sigma'}{n_3\bf k'-q}\| e^{i({\bf q - G'})\cdot {\bf r'} }\|u^{\sigma'}{n_4\bf k'}\rangle		\langle u^{\sigma'}{n_3\bf k'-q}\| e^{i({\bf q - G'})\cdot {\bf r'} }\|u^{\sigma'}{n_4\bf k'}\rangle
	T{kn_3}^{\sigma'({\bf k'-q})}		T{kn_3}^{\sigma'({\bf k'-q})}

Kaltakm: /* Parameter definitions */

2026-03-16T11:16:00Z

Parameter definitions

← Older revision		Revision as of 11:16, 16 March 2026
Line 65:		Line 65:
	T_{jn_2}^{\sigma({\bf k-q})}		T_{jn_2}^{\sigma({\bf k-q})}
	\langle u^\sigma_{n_1\bf k}\| e^{-i({\bf q + G})\cdot {\bf r}} \|u^\sigma_{n_2\bf k-q}\rangle		\langle u^\sigma_{n_1\bf k}\| e^{-i({\bf q + G})\cdot {\bf r}} \|u^\sigma_{n_2\bf k-q}\rangle
	U^{\sigma\sigma'}*{{\bf G G}'}({\bf q})		U^{\sigma\sigma'}*_{{\bf G G}'}({\bf q})
	\langle u^{\sigma'}{n_3\bf k'-q}\| e^{i({\bf q - G'})\cdot {\bf r'} }\|u^{\sigma'}{n_4\bf k'}\rangle		\langle u^{\sigma'}{n_3\bf k'-q}\| e^{i({\bf q - G'})\cdot {\bf r'} }\|u^{\sigma'}{n_4\bf k'}\rangle
	T{kn_3}^{\sigma'({\bf k'-q})}		T{kn_3}^{\sigma'({\bf k'-q})}

Kaltakm: /* Spectral method (s-cRPA) */

2026-03-13T15:00:44Z

Spectral method (s-cRPA)

← Older revision		Revision as of 15:00, 13 March 2026
Line 229:		Line 229:
	* '''Positive interactions''': The electron number is always conserved (in contrast to p-cRPA)		* '''Positive interactions''': The electron number is always conserved (in contrast to p-cRPA)
	* '''Largest interaction values''': removes most intra-d screening effects (in contrast to w-cRPA)		* '''Largest interaction values''': removes most intra-d screening effects (in contrast to w-cRPA)
	{{NB\|~~tip~~\|s-cRPA ~~is the recommended~~ method for all cRPA calculations as of VASP version 6.6.0~~. Select this method with {{TAG\|LSCRPA}}.~~}}		{{NB\|mind\|Select the s-cRPA method with {{TAG\|LSCRPA}}.}}
			{{NB\|tip\|Recommended for all cRPA calculations as of VASP version 6.6.0}}

	== Off-centre interactions ==		== Off-centre interactions ==

← Older revision		Revision as of 14:06, 17 March 2026
Line 253:		Line 253:
	The calculation of these integrals occurs as a post-processing step. For bare, off-centre Coulomb integrals ({{FILE\|VRijkl}}), a valid {{FILE\|WAVECAR}} file must be present in the working directory. For effectively screened Coulomb integrals ({{FILE\|URijkl}}), both a valid {{FILE\|WAVECAR}} file and {{TAG\|WFULLxxxx.tmp}} files are required. The {{TAG\|WFULLxxxx.tmp}} files are automatically generated following a successful {{TAG\|ALGO}}=CRPA job.		The calculation of these integrals occurs as a post-processing step. For bare, off-centre Coulomb integrals ({{FILE\|VRijkl}}), a valid {{FILE\|WAVECAR}} file must be present in the working directory. For effectively screened Coulomb integrals ({{FILE\|URijkl}}), both a valid {{FILE\|WAVECAR}} file and {{TAG\|WFULLxxxx.tmp}} files are required. The {{TAG\|WFULLxxxx.tmp}} files are automatically generated following a successful {{TAG\|ALGO}}=CRPA job.

	The basis orbitals can be selected with {{TAG\|~~DMFT_BASIS~~}}.		The basis orbitals can be selected with {{TAG\|LOCALIZED_BASIS}}.
	{{NB\|mind\|Avaliable as of VASP.6.6.0.}}		{{NB\|mind\|Avaliable as of VASP.6.6.0.}}

← Older revision		Revision as of 14:05, 17 March 2026
Line 260:		Line 260:
	{{TAG\|NTARGET_STATES}},		{{TAG\|NTARGET_STATES}},
	{{TAG\|NCRPA_BANDS}},		{{TAG\|NCRPA_BANDS}},
	{{TAG\|~~DMFT_BASIS~~}},		{{TAG\|LOCALIZED_BASIS}},
	{{TAG\|LDISENTANGLED}},		{{TAG\|LDISENTANGLED}},
	{{TAG\|LWEIGHTED}},		{{TAG\|LWEIGHTED}},

← Older revision		Revision as of 09:09, 20 March 2026
Line 120:		Line 120:
	{{NB\|warning\|This method is not recommended for systems with entangled target states.}}		{{NB\|warning\|This method is not recommended for systems with entangled target states.}}

	=== Disentanglement cRPA method (d-cRPA) ===		=== Disentanglement-cRPA method (d-cRPA) ===
	Miyake, Aryasetiawan, and Imada{{cite\|miyake:prb:80}} propose to disentangle the target space from the full space by diagonalizing the Hamiltonian in both spaces independently.		Miyake, Aryasetiawan, and Imada{{cite\|miyake:prb:80}} propose to disentangle the target space from the full space by diagonalizing the Hamiltonian in both spaces independently.
	This yields a disentangled band structure similar to the one shown in the third example above.		This yields a disentangled band structure similar to the one shown in the third example above.
Line 146:		Line 146:
	{{NB\|mind\|Typically, this method yields the largest Hubbard-Kanamori interactions and is selected with {{TAG\|LDISENTANGLED}} tag.}}		{{NB\|mind\|Typically, this method yields the largest Hubbard-Kanamori interactions and is selected with {{TAG\|LDISENTANGLED}} tag.}}

	=== Weighted cRPA method (w-cRPA) ===		=== Weighted-cRPA method (w-cRPA) ===
	Sasioglu, Friedrich and Blügel propose an alternative approach.{{cite\|sasioglu:prb:83}}		Sasioglu, Friedrich and Blügel propose an alternative approach.{{cite\|sasioglu:prb:83}}
	They calculate the screening effects within the target space as follows:		They calculate the screening effects within the target space as follows:
Line 174:		Line 174:
	measure the probability for <math>\|\psi_{n\bf k}^\sigma\rangle</math> being in the target space.		measure the probability for <math>\|\psi_{n\bf k}^\sigma\rangle</math> being in the target space.
	This method does not alter the band structure, however, neglects screening effects within the target space as shown in next section.		This method does not alter the band structure, however, neglects screening effects within the target space as shown in next section.
	Typically, the weighted cRPA method yields the smallest Hubbard-Kanamori interactions.		Typically, the weighted-cRPA method yields the smallest Hubbard-Kanamori interactions.
	{{NB\|mind\|This method is selected with the {{TAG\|LWEIGHTED}} tag.}}		{{NB\|mind\|This method is selected with the {{TAG\|LWEIGHTED}} tag.}}

	=== Projector cRPA method (p-cRPA) ===		=== Projector-cRPA method (p-cRPA) ===
	A consistent expression for the target polarizability gives rise to the projector-cRPA (p-cRPA) method.		A consistent expression for the target polarizability gives rise to the projector-cRPA (p-cRPA) method.
	The main goal is to subdivide the target from the full Fock space in a Wannier basis.{{cite\|kaltak:prb:2025:2}}		The main goal is to subdivide the target from the full Fock space in a Wannier basis.{{cite\|kaltak:prb:2025:2}}
Line 210:		Line 210:
	P_{m_1' n }^{ \sigma ({\bf k })}		P_{m_1' n }^{ \sigma ({\bf k })}
	</math>		</math>
	The projector method usually results in larger (smaller) Hubbard-Kanamori interactions than the weighted (disentanglement) method.		The projector method usually results in larger (smaller) Hubbard-Kanamori interactions than the weighted-cRPA (disentanglement-cRPA) method.
	{{NB\|mind\|The projector method is the default cRPA.}}		{{NB\|mind\|The projector method is the default cRPA.}}

Line 218:		Line 218:
	For such cases, the {{FILE\|WAVEDER}} file should be deleted before the cRPA step. Other cRPA methods do not suffer from this problem and the usage of the long-wave limit is strongly encouraged.		For such cases, the {{FILE\|WAVEDER}} file should be deleted before the cRPA step. Other cRPA methods do not suffer from this problem and the usage of the long-wave limit is strongly encouraged.

	=== Spectral cRPA method (s-cRPA) ===		=== Spectral-cRPA method (s-cRPA) ===
	The spectral cRPA method (s-cRPA) is a robust approach that remedies the main drawbacks of both w-cRPA and p-cRPA.{{cite\|kaltak:prb:2025}}		The spectral-cRPA method (s-cRPA) is a robust approach that remedies the main drawbacks of both w-cRPA and p-cRPA.{{cite\|kaltak:prb:2025}}
	This method uses the eigenspectrum of the original projectors		This method uses the eigenspectrum of the original projectors
	::<math>		::<math>