Category:Strongly correlated electrons - Revision history

Csheldon: /* QPGW */

2026-04-15T09:24:37Z

QPGW

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	=== QPGW ===		=== QPGW ===
	The [[:Category:GW\|GW]] approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW (quasi-particle GW), which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons {{cite\|Cunningham2023}}.		The [[:Category:GW\|GW]] approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW (quasi-particle GW), which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons {{cite\|Cunningham2023}}{{Cite\|shishkin:prl:2007}}.

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

Liebetreu at 07:52, 15 April 2026

2026-04-15T07:52:47Z

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	The total energy within DFT+U can be written as		The total energy within DFT+U can be written as

	<math>		<math display="block">
	E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right],		E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right],
	</math>		</math>
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	* {{TAG\|LDAUTYPE}}=3: Linear-response calculation of <math>U</math>. Within this approach the effective interaction <math>U</math> can be determined via the linear response approach {{cite\|cococcioni:2005}}		* {{TAG\|LDAUTYPE}}=3: Linear-response calculation of <math>U</math>. Within this approach the effective interaction <math>U</math> can be determined via the linear response approach {{cite\|cococcioni:2005}}
	:<math>		<math display="block">
	U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}.		U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}.
	</math>		</math>
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	The response function without the contribution of the target states or constrained polarizability <math>\chi_c</math> is calculated by explicitly removing the response in the target space <math>\chi_d</math> from the total response function:		The response function without the contribution of the target states or constrained polarizability <math>\chi_c</math> is calculated by explicitly removing the response in the target space <math>\chi_d</math> from the total response function:

	<math>\chi_c(\omega) = \chi(\omega) - \chi_d(\omega)</math>.		<math display="block">\chi_c(\omega) = \chi(\omega) - \chi_d(\omega)</math>.

	[[Constrained–random-phase–approximation formalism#Effective Coulomb kernel in constrained random-phase approximation\|VASP provides several approaches for calculating <math>\chi_d</math>]]		[[Constrained–random-phase–approximation formalism#Effective Coulomb kernel in constrained random-phase approximation\|VASP provides several approaches for calculating <math>\chi_d</math>]]

Liebetreu at 07:41, 15 April 2026

2026-04-15T07:41:20Z

Liebetreu at 07:38, 15 April 2026

2026-04-15T07:38:14Z

← Older revision		Revision as of 07:38, 15 April 2026
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	== DFT+U ==		== DFT+U ==
	[[:Category:DFT+U\|DFT+U]] is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.		[[:Category:DFT+U\|DFT+U]] is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.
	{{NB\|mind\|It is not currently possible to apply $U$ to both $d$ and $f$ states of the same atom}}		{{NB\|mind\|It is not currently possible to apply <math>U</math> to both <math>d</math> and <math>f</math> states of the same atom}}
	The total energy within DFT+U can be written as		The total energy within DFT+U can be written as
	~~\begin{equation}~~
			<math>
	E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right],		E=E_{DFT}+\sum_I\left[\frac{U^I}{2} \sum_{m, \sigma \neq m^{\prime}, \sigma^{\prime}} n_m^{I \sigma} n_{m^{\prime}}^{I \sigma^{\prime}}-\frac{U^I}{2} n^I\left(n^I-1\right)\right],
	~~\end{equation}~~		</math>

	where the first term is the standard DFT energy, the second term is the Hubbard on-site interaction and the third term accounts for the double counting. The on-site interaction is described by $U^I$ and $n_m^{I\sigma}$ are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.		where the first term is the standard DFT energy, the second term is the Hubbard on-site interaction and the third term accounts for the double counting. The on-site interaction is described by $U^I$ and $n_m^{I\sigma}$ are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.

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	* {{TAG\|LDAUTYPE}}=3: Linear-response calculation of $U$. Within this approach the effective interaction $U$ can be determined via the linear response approach {{cite\|cococcioni:2005}}		* {{TAG\|LDAUTYPE}}=3: Linear-response calculation of $U$. Within this approach the effective interaction $U$ can be determined via the linear response approach {{cite\|cococcioni:2005}}
	~~\begin{equation}~~		:<math>
	U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}.		U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}.
	~~\end{equation}~~		</math>
	The shortcoming of this method is that the effective interaction accounts for the response due to all electrons including the target states (localized $d$ or $f$ orbitals), thus leading to double counting		:The shortcoming of this method is that the effective interaction accounts for the response due to all electrons including the target states (localized <math>d</math> or <math>f</math> orbitals), thus leading to double counting in the derived effective potential U.
	in the derived effective potential U.

	== Constrained Random Phase Approximation (cRPA) ==		== Constrained Random Phase Approximation (cRPA) ==

Csheldon: /* Additional resource */

2026-04-14T09:27:31Z

Additional resource

← Older revision		Revision as of 09:27, 14 April 2026
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	The [[:Category:GW\|GW]] approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW (quasi-particle GW), which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons {{cite\|Cunningham2023}}.		The [[:Category:GW\|GW]] approximation in its simplest, non-self-consistent form (i.e., the one-shot approach) exhibits a strong dependence on the choice of starting point, and thus inherits the limitations of the underlying DFT description of localized states. In contrast, self-consistent GW schemes such as QPGW (quasi-particle GW), which are independent of the starting electronic structure, have been shown to provide an accurate description of correlated electrons {{cite\|Cunningham2023}}.

	== Additional ~~resource~~==		== Additional resources ==
	=== Books ===		=== Books ===
	* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.		* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.

Csheldon at 09:23, 14 April 2026

2026-04-14T09:23:51Z

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	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT {{Cite\|martin:book:2016}}. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed {{Cite\|marianetti:kotliar:2006}}{{Cite\|savrasov:kotliar:2004}}{{Cite\|turkowski:book:2021}}.		To model such systems, several extensions of DFT have been developed {{Cite\|marianetti:kotliar:2006}}{{Cite\|savrasov:kotliar:2004}}{{Cite\|turkowski:book:2021}}.

Csheldon at 09:23, 14 April 2026

2026-04-14T09:23:26Z

← Older revision		Revision as of 09:23, 14 April 2026
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	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed.		To model such systems, several extensions of DFT have been developed {{Cite\|marianetti:kotliar:2006}}{{Cite\|savrasov:kotliar:2004}}{{Cite\|turkowski:book:2021}}.

	== DFT+U ==		== DFT+U ==
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	=== Books ===		=== Books ===
	* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.		* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.
			* ''Dynamical Mean-Field Theory for Strongly Correlated Materials'' by Volodymyr Turkowski - a book about dynamical mean field theory (DMFT) {{Cite\|turkowski:book:2021}}.

	=== Tutorials ===		=== Tutorials ===

Csheldon: /* Additional resource */

2026-04-14T09:15:08Z

Additional resource

← Older revision		Revision as of 09:15, 14 April 2026
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	== Additional resource==		== Additional resource==
			=== Books ===
			* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.

	=== Tutorials ===		=== Tutorials ===
	* Tutorial for {{Tutorial\|bulk:e09\|NiO DFT+U calculations}}.		* Tutorial for {{Tutorial\|bulk:e09\|NiO DFT+U calculations}}.
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	* Tutorial for [https://vasp.at/wiki/DFT%2BDMFT_calculations NiO DFT+DMFT calculations].		* Tutorial for [https://vasp.at/wiki/DFT%2BDMFT_calculations NiO DFT+DMFT calculations].
	* Tutorial for [https://www.vasp.at/wiki/NiO_GGA%2BU NiO GGA+U calculations].		* Tutorial for [https://www.vasp.at/wiki/NiO_GGA%2BU NiO GGA+U calculations].

	~~=== Books ===~~
	* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.

	== References ==		== References ==

	[[Category:Linear response]][[Category:DFT+U]][[Category:VASP]]		[[Category:Linear response]][[Category:DFT+U]][[Category:VASP]]

Csheldon: /* Tutorials */

2026-04-13T16:04:29Z

Tutorials

← Older revision		Revision as of 16:04, 13 April 2026
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	=== Tutorials ===		=== Tutorials ===
	* Tutorial for {{Tutorial\|bulk:e09\|NiO DFT+U calculations}}.		* Tutorial for {{Tutorial\|bulk:e09\|NiO DFT+U calculations}}.
			* Tutorial for {{Tutorial\|magnetism:e03\|antiferromagnetic NiO}}.
			* Tutorial for {{Tutorial\|magnetism:e04\|LSDA+U structure relaxation of NiO}}.
			* Tutorial for {{Tutorial\|magnetism:e05\|Heisenberg model for NiO using DFT+U}}.
			* Tutorial for {{Tutorial\|magnetism:e07\|magnetic anisotropy in FeO}}.

	=== How-to's ===		=== How-to's ===

Csheldon: /* Additional resource */

2026-04-13T16:01:50Z

Additional resource

← Older revision		Revision as of 16:01, 13 April 2026
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	=== Books ===		=== Books ===
	* 'Interacting Electrons - Theory and Computational Approaches' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.		* ''Interacting Electrons - Theory and Computational Approaches'' by Richard Martin, Lucia Reining, and David Ceperley - a book about strong correlation {{Cite\|martin:book:2016}}.

	== References ==		== References ==

	[[Category:Linear response]][[Category:DFT+U]][[Category:VASP]]		[[Category:Linear response]][[Category:DFT+U]][[Category:VASP]]

← Older revision		Revision as of 07:41, 15 April 2026
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	[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]		[[File:Ni_d_s_bands.png\|200px\|thumb\|Band structure of a typical strongly correlated system - Ni]]
	Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT {{Cite\|martin:book:2016}}. These systems typically include elements with partially filled $d$ and $f$ electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.		Strongly correlated materials are systems in which electron-electron interactions play a dominant role and cannot be adequately described by independent-particle approximations such as standard DFT {{Cite\|martin:book:2016}}. These systems typically include elements with partially filled <math>d</math> and <math>f</math> electron orbitals which are localized. Correlation effects lead to phenomena such as metal-insulator transitions, magnetism, and unconventional superconductivity.
	To model such systems, several extensions of DFT have been developed {{Cite\|marianetti:kotliar:2006}}{{Cite\|savrasov:kotliar:2004}}{{Cite\|turkowski:book:2021}}.		To model such systems, several extensions of DFT have been developed {{Cite\|marianetti:kotliar:2006}}{{Cite\|savrasov:kotliar:2004}}{{Cite\|turkowski:book:2021}}.

	== DFT+U ==		== DFT+U ==
	[[:Category:DFT+U\|DFT+U]] is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term $U$ that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to $d$ or $f$ states.		[[:Category:DFT+U\|DFT+U]] is the simplest and the most computationally efficient approach to treat strong correlations within electronic structure calculations. Within this approach, standard DFT is augmented with an on-site Hubbard interaction term <math>U</math> that explicitly penalizes fractional occupation of localized orbitals. The Hubbard interaction is typically applied to <math>d</math> or <math>f</math> states.
	{{NB\|mind\|It is not currently possible to apply <math>U</math> to both <math>d</math> and <math>f</math> states of the same atom}}		{{NB\|mind\|It is not currently possible to apply <math>U</math> to both <math>d</math> and <math>f</math> states of the same atom}}
	The total energy within DFT+U can be written as		The total energy within DFT+U can be written as
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	</math>		</math>

	where the first term is the standard DFT energy, the second term is the Hubbard on-site interaction and the third term accounts for the double counting. The on-site interaction is described by $U^I$ and $n_m^{I\sigma}$ are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.		where the first term is the standard DFT energy, the second term is the Hubbard on-site interaction and the third term accounts for the double counting. The on-site interaction is described by <math>U^I</math> and <math>n_m^{I\sigma}</math> are occupation numbers that are defined as projections of occupied Kohn-Sham orbitals on the states of a localized basis set.

	VASP provides the following approaches to include the Hubbard corrections:		VASP provides the following approaches to include the Hubbard corrections:
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	* {{TAG\|LDAUTYPE}}=4: The same approach as {{TAG\|LDAUTYPE}}=1 but uses spin-averaged expression that's simpler and assumes an average spin configuration.		* {{TAG\|LDAUTYPE}}=4: The same approach as {{TAG\|LDAUTYPE}}=1 but uses spin-averaged expression that's simpler and assumes an average spin configuration.

	* {{TAG\|LDAUTYPE}}=2: The simplified approach which is also rotationally invariant but uses isotropic effective interaction $U^I_{eff}=U^I-J^I$ {{cite\|dudarev:prb:98}}. This approach neglects the anisotropy of the orbitals and thus the on-site interaction depends on the occupations but not the orbitals themselves.		* {{TAG\|LDAUTYPE}}=2: The simplified approach which is also rotationally invariant but uses isotropic effective interaction <math>U^I_{eff}=U^I-J^I</math> {{cite\|dudarev:prb:98}}. This approach neglects the anisotropy of the orbitals and thus the on-site interaction depends on the occupations but not the orbitals themselves.

	A common approach is to treat the effective on-site interaction $U$ as an adjustable parameter, tuning it to reproduce selected experimental observables such as the band gap, magnetic moments, or lattice parameters. Alternatively, fully ''ab initio'' schemes exist that determine the Hubbard interaction directly from first principles, avoiding empirical fitting.		A common approach is to treat the effective on-site interaction <math>U</math> as an adjustable parameter, tuning it to reproduce selected experimental observables such as the band gap, magnetic moments, or lattice parameters. Alternatively, fully ''ab initio'' schemes exist that determine the Hubbard interaction directly from first principles, avoiding empirical fitting.

	* {{TAG\|LDAUTYPE}}=3: Linear-response calculation of $U$. Within this approach the effective interaction $U$ can be determined via the linear response approach {{cite\|cococcioni:2005}}		* {{TAG\|LDAUTYPE}}=3: Linear-response calculation of <math>U</math>. Within this approach the effective interaction <math>U</math> can be determined via the linear response approach {{cite\|cococcioni:2005}}
	:<math>		:<math>
	U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}.		U=\chi^{-1}-\chi_0^{-1} \approx\left(\frac{\partial N_I^{\mathrm{SCF}}}{\partial V_I}\right)^{-1}-\left(\frac{\partial N_I^{\mathrm{NSCF}}}{\partial V_I}\right)^{-1}.
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	== Constrained Random Phase Approximation (cRPA) ==		== Constrained Random Phase Approximation (cRPA) ==
	[[Constrained–random-phase–approximation formalism\|cRPA]] is a first-principles method used to compute the effective interaction parameters for the DFT+U or DMFT calculations.		[[Constrained–random-phase–approximation formalism\|cRPA]] is a first-principles method used to compute the effective interaction parameters for the DFT+U or DMFT calculations.
	cRPA allows to separate the screening originating from the target states from the rest of the system and to determine the effective interaction $U$ that is free of double counting.		cRPA allows to separate the screening originating from the target states from the rest of the system and to determine the effective interaction <math>U</math> that is free of double counting.

	The response function without the contribution of the target states or constrained polarizability $\chi_c$ is calculated by explicitly removing the response in the target space $\chi_d$ from the total response function:		The response function without the contribution of the target states or constrained polarizability <math>\chi_c</math> is calculated by explicitly removing the response in the target space <math>\chi_d</math> from the total response function:
	~~$\chi_c(\omega) = \chi(\omega) - \chi_d(\omega)$.~~

	[[Constrained–random-phase–approximation formalism#Effective Coulomb kernel in constrained random-phase approximation\|VASP provides several approaches for calculating $\chi_d$]]		<math>\chi_c(\omega) = \chi(\omega) - \chi_d(\omega)</math>.

			[[Constrained–random-phase–approximation formalism#Effective Coulomb kernel in constrained random-phase approximation\|VASP provides several approaches for calculating <math>\chi_d</math>]]

	== Dynamical Mean-Field Theory (DMFT) ==		== Dynamical Mean-Field Theory (DMFT) ==
	DMFT is a dynamical extension of DFT+U that provides a more accurate treatment of strongly correlated materials where DFT+U is insufficient {{cite\|kotliar:rmp:2006}}. Like DFT+U, DMFT augments the DFT calculation with an additional local correlated subproblem — typically a specific $d$- or $f$-shell. The key idea is to map the full lattice problem onto a quantum impurity model: a single correlated site embedded in a self-consistently determined effective bath representing the rest of the lattice. The central approximation is that the self-energy is purely local — frequency-dependent but momentum-independent. This makes the problem tractable while still capturing many-body effects beyond the reach of DFT+U, such as quasiparticle mass renormalization, Hubbard bands, and the Mott metal–insulator transition. The interaction parameters entering DMFT can be determined from first principles using cRPA (see [[Constrained–random-phase–approximation formalism\|cRPA section above]]).		DMFT is a dynamical extension of DFT+U that provides a more accurate treatment of strongly correlated materials where DFT+U is insufficient {{cite\|kotliar:rmp:2006}}. Like DFT+U, DMFT augments the DFT calculation with an additional local correlated subproblem — typically a specific <math>d</math>- or <math>f</math>-shell. The key idea is to map the full lattice problem onto a quantum impurity model: a single correlated site embedded in a self-consistently determined effective bath representing the rest of the lattice. The central approximation is that the self-energy is purely local — frequency-dependent but momentum-independent. This makes the problem tractable while still capturing many-body effects beyond the reach of DFT+U, such as quasiparticle mass renormalization, Hubbard bands, and the Mott metal–insulator transition. The interaction parameters entering DMFT can be determined from first principles using cRPA (see [[Constrained–random-phase–approximation formalism\|cRPA section above]]).

	== Other methods ==		== Other methods ==