Time-dependent density-functional theory - Revision history

Csheldon at 14:46, 19 June 2026

2026-06-19T14:46:38Z

← Older revision		Revision as of 14:46, 19 June 2026
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	[[Category:Theory]]		[[Category:Theory]]
	[[Category:Time-dependent density-functional theory]]		[[Category:Time-dependent density functional theory]]
	[[Category:Many-body perturbation theory]]		[[Category:Many-body perturbation theory]]
	[[Category:Linear response]]		[[Category:Linear response]]

Huebsch: Huebsch moved page Construction:Time-dependent density-functional theory to Time-dependent density-functional theory without leaving a redirect

2026-06-17T10:26:52Z

Huebsch moved page Construction:Time-dependent density-functional theory to Time-dependent density-functional theory without leaving a redirect

← Older revision		Revision as of 10:26, 17 June 2026
(No difference)

Tal: /* Two-point Dyson equation for TDDFT */

2026-06-15T15:13:24Z

Two-point Dyson equation for TDDFT

← Older revision		Revision as of 15:13, 15 June 2026
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	Here, <math>\mu_{cv\mathbf{k}}^j</math> is the dipole matrix element defined in the [[#Connection to the dielectric function\|connection to the dielectric function]] above.		Here, <math>\mu_{cv\mathbf{k}}^j</math> is the dipole matrix element defined in the [[#Connection to the dielectric function\|connection to the dielectric function]] above.

	== ~~Two-point~~ Dyson equation ~~for~~ TDDFT ==		== Dyson equation (Linear response) TDDFT ==

	Another approach to TDDFT ({{TAG\|ALGO}}=CHI) is to solve the two-point Dyson equation directly and find the density-density response function <math>\chi</math>. This approach allows one to calculate the full <math>\chi_{\mathbf q}(\mathbf G,\mathbf G',\omega)</math> and is used for finding the Coulomb interaction screened via the RPA dielectric function, ''W''. However, it requires inverting the Dyson equation at every frequency, which becomes too costly for calculating spectra with good resolution. Since this approach does not invoke the Tamm-Dancoff approximation, the RPA dielectric function obtained this way is equivalent to the Casida RPA dielectric function.		Another approach to TDDFT ({{TAG\|ALGO}}=CHI) is to solve the two-point Dyson equation directly and find the density-density response function <math>\chi</math>. This approach allows one to calculate the full <math>\chi_{\mathbf q}(\mathbf G,\mathbf G',\omega)</math> and is used for finding the Coulomb interaction screened via the RPA dielectric function, ''W''. However, it requires inverting the Dyson equation at every frequency, which becomes too costly for calculating spectra with good resolution. Since this approach does not invoke the Tamm-Dancoff approximation, the RPA dielectric function obtained this way is equivalent to the Casida RPA dielectric function.

Tal: /* Exchange-correlation kernel from exact exchange */

2026-06-15T11:43:32Z

Exchange-correlation kernel from exact exchange

← Older revision		Revision as of 11:43, 15 June 2026
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	When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form		When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form
	::<math>		::<math>
	f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = -\frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},		f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = \frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},
	</math>		</math>
	where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.		where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.

Tal: /* Related tags and articles */

2026-06-15T11:42:16Z

Tal: Adopt generalized eigenvalue form for Casida equation (consistent with category page), use X_lambda eigenvectors in dielectric function

2026-06-15T11:39:54Z

Adopt generalized eigenvalue form for Casida equation (consistent with category page), use X_lambda eigenvectors in dielectric function

← Older revision		Revision as of 11:39, 15 June 2026
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	Working within the frequency domain and using a basis set that considers transitions from valence to conduction states at the same '''k''' point (transition basis) makes it possible to recast the equation for <math>\chi(1,2)</math> into an eigenvalue problem{{cite\|casida:1995}}		Working within the frequency domain and using a basis set that considers transitions from valence to conduction states at the same '''k''' point (transition basis) makes it possible to recast the equation for <math>\chi(1,2)</math> into an eigenvalue problem{{cite\|casida:1995}}
	::<math>		::<math>
	\left(		\left(\begin{array}{cc}
	\begin{~~matrix~~}		\mathbf{A} & \mathbf{B} \\
	A & B \\		\mathbf{B}^* & \mathbf{A}^*
	-B^* & -A^*		\end{array}\right)\left(\begin{array}{l}
	\end{~~matrix~~}		\mathbf{X}_\lambda \\
	\right)		\mathbf{Y}_\lambda
	\left(		\end{array}\right)=\omega_\lambda\left(\begin{array}{cc}
	\begin{~~matrix~~}		\mathbf{1} & \mathbf{0} \\
	X \\		\mathbf{0} & -\mathbf{1}
	Y		\end{array}\right)\left(\begin{array}{l}
	\end{~~matrix~~}		\mathbf{X}_\lambda \\
	\right) = \~~omega~~ \left(		\mathbf{Y}_\lambda
	\begin{~~matrix~~}		\end{array}\right),
	X \\
	Y
	\end{~~matrix~~}
	\right),
	</math>		</math>
	which is also known as the Casida equation. The <math>A</math> and <math>B</math> matrices are given by		which is also known as the Casida equation. The <math>A</math> and <math>B</math> matrices are given by
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	=== Connection to the dielectric function ===		=== Connection to the dielectric function ===

	The macroscopic dielectric function is obtained by using the eigenvalues <math>\omega_\lambda</math> and eigenvectors <math>A^\lambda</math> of the Casida equation in the spectral representation		The macroscopic dielectric function is obtained by using the eigenvalues <math>\omega_\lambda</math> and eigenvectors <math>\mathbf{X}_\lambda</math> of the Casida equation in the spectral representation
	::<math>		::<math>
	\varepsilon_M(\omega) = 1 + \frac{4\pi}{\Omega} \sum_\lambda \left\|\sum_{cv\mathbf k} \mu_{cv\mathbf k} A_{cv\mathbf k}~~^\lambda~~\right\|^2 \left[\frac{1}{\omega + \omega_\lambda + \mathrm i\eta} - \frac{1}{\omega - \omega_\lambda + \mathrm i\eta}\right],		\varepsilon_M(\omega) = 1 + \frac{4\pi}{\Omega} \sum_\lambda \left\|\sum_{cv\mathbf k} \mu_{cv\mathbf k} X_\lambda^{cv\mathbf k}\right\|^2 \left[\frac{1}{\omega + \omega_\lambda + \mathrm i\eta} - \frac{1}{\omega - \omega_\lambda + \mathrm i\eta}\right],
	</math>		</math>
	where <math>\mu_{cv\mathbf{k}}^j=\frac{\langle c\mathbf{k}\|v_j\|v\mathbf{k}\rangle}{\varepsilon_c(\mathbf{k})-\varepsilon_v(\mathbf{k})}</math> is the dipole matrix element associated with the transition from valence state <math>v</math> to conduction state <math>c</math> at '''k'''-point <math>\mathbf k</math>, <math>v_j</math> is the velocity operator along direction <math>j</math>, and <math>\omega_\lambda</math> are the excitation energies. The eigenvalues give the peak positions and the eigenvectors determine the oscillator strengths (peak intensities) in the optical absorption spectrum <math>\mathrm{Im}\,\varepsilon_M(\omega)</math>.		where <math>\mu_{cv\mathbf{k}}^j=\frac{\langle c\mathbf{k}\|v_j\|v\mathbf{k}\rangle}{\varepsilon_c(\mathbf{k})-\varepsilon_v(\mathbf{k})}</math> is the dipole matrix element associated with the transition from valence state <math>v</math> to conduction state <math>c</math> at '''k'''-point <math>\mathbf k</math>, <math>v_j</math> is the velocity operator along direction <math>j</math>, and <math>\omega_\lambda</math> are the excitation energies. The eigenvalues give the peak positions and the eigenvectors determine the oscillator strengths (peak intensities) in the optical absorption spectrum <math>\mathrm{Im}\,\varepsilon_M(\omega)</math>.
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	When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form		When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form
	::<math>		::<math>
	f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = \frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},		f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = -\frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},
	</math>		</math>
	where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.		where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.
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	The Casida formulation of TDDFT and the [[Bethe-Salpeter equation]] (BSE) share essentially the same mathematical structure: both are non-Hermitian eigenvalue problems of the form		The Casida formulation of TDDFT and the [[Bethe-Salpeter equation]] (BSE) share essentially the same mathematical structure: both are non-Hermitian eigenvalue problems of the form
	::<math>		::<math>
	\left(\begin{~~matrix~~} A & B \\ -B^* & -A^* \end{~~matrix~~}\right) \left(\begin{~~matrix~~} X \\ Y \end{~~matrix~~}\right) = \~~omega~~ \left(\begin{~~matrix~~} X \\ Y \end{~~matrix~~}\right),		\left(\begin{array}{cc}
			\mathbf{A} & \mathbf{B} \\
			\mathbf{B}^* & \mathbf{A}^*
			\end{array}\right)\left(\begin{array}{l}
			\mathbf{X}_\lambda \\
			\mathbf{Y}_\lambda
			\end{array}\right)=\omega_\lambda\left(\begin{array}{cc}
			\mathbf{1} & \mathbf{0} \\
			\mathbf{0} & -\mathbf{1}
			\end{array}\right)\left(\begin{array}{l}
			\mathbf{X}_\lambda \\
			\mathbf{Y}_\lambda
			\end{array}\right),
	</math>		</math>
	where the matrix elements of <math>A</math> and <math>B</math> are built from valence-to-conduction transitions and contain a Hartree (Coulomb) contribution together with a term that goes beyond the random-phase approximation (RPA). The Coulomb part is identical in both methods and the difference lies entirely in how the beyond-RPA contribution is described:		where the matrix elements of <math>A</math> and <math>B</math> are built from valence-to-conduction transitions and contain a Hartree (Coulomb) contribution together with a term that goes beyond the random-phase approximation (RPA). The Coulomb part is identical in both methods and the difference lies entirely in how the beyond-RPA contribution is described:

Tal: /* Exchange-correlation kernel from exact exchange */

2026-06-15T11:31:38Z

Exchange-correlation kernel from exact exchange

← Older revision		Revision as of 11:31, 15 June 2026
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	When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form		When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form
	::<math>		::<math>
	f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = -\frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},		f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = \frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},
	</math>		</math>
	where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.		where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.

Tal: Add Tags group to Related section, add ALGO=CHI, fix nonlocal spelling, fix μ internal link

2026-06-15T11:29:05Z

Add Tags group to Related section, add ALGO=CHI, fix nonlocal spelling, fix μ internal link

← Older revision		Revision as of 11:29, 15 June 2026
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	Here, <math>\Delta t</math> is the time step chosen for the propagation. Updating the Hamiltonian with the new states is essential, as it is a functional of the time-dependent density.		Here, <math>\Delta t</math> is the time step chosen for the propagation. Updating the Hamiltonian with the new states is essential, as it is a functional of the time-dependent density.

	The same approximation hierarchy described in the [[#Approximation hierarchy\|Approximation hierarchy]] section applies to the time-evolution approach. The tags {{TAG\|LHARTREE}}, {{TAG\|LFXC}}, and {{TAG\|LADDER}} control which interaction terms are included in the time-dependent Hamiltonian: {{TAG\|LHARTREE}} controls the Coulomb (Hartree) energy, {{TAG\|LFXC}} controls the local exchange-correlation kernel, and {{TAG\|LADDER}} controls the ~~non-local~~ exchange contribution from hybrid functionals. At each time step, the corresponding energy terms in the Hamiltonian are updated based on the time-evolving density.		The same approximation hierarchy described in the [[#Approximation hierarchy\|Approximation hierarchy]] section applies to the time-evolution approach. The tags {{TAG\|LHARTREE}}, {{TAG\|LFXC}}, and {{TAG\|LADDER}} control which interaction terms are included in the time-dependent Hamiltonian: {{TAG\|LHARTREE}} controls the Coulomb (Hartree) energy, {{TAG\|LFXC}} controls the local exchange-correlation kernel, and {{TAG\|LADDER}} controls the nonlocal exchange contribution from hybrid functionals. At each time step, the corresponding energy terms in the Hamiltonian are updated based on the time-evolving density.

	=== Connection to the dielectric function ===		=== Connection to the dielectric function ===
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	\varepsilon_{ij}(\omega) = \delta_{ij} - \frac{4\pi e^2}{\Omega} \int_0^\infty \mathrm d t \sum_{cv\mathbf k} \left(\langle\mu^j_{cv\mathbf k}\|c^i_{cv\mathbf k}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega - \mathrm i\eta)t}.		\varepsilon_{ij}(\omega) = \delta_{ij} - \frac{4\pi e^2}{\Omega} \int_0^\infty \mathrm d t \sum_{cv\mathbf k} \left(\langle\mu^j_{cv\mathbf k}\|c^i_{cv\mathbf k}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega - \mathrm i\eta)t}.
	</math>		</math>
	Here, <math>\mu_{cv\mathbf{k}}^j</math> is the dipole matrix element defined in the [[#~~Casida equation formalism for TDDFT~~\|~~Casida section~~]].		Here, <math>\mu_{cv\mathbf{k}}^j</math> is the dipole matrix element defined in the [[#Connection to the dielectric function\|connection to the dielectric function]] above.

	== Two-point Dyson equation for TDDFT ==		== Two-point Dyson equation for TDDFT ==

	Another approach to TDDFT is to solve the two-point Dyson equation directly and find the density-density response function <math>\chi</math>. This approach allows one to calculate the full <math>\chi_{\mathbf q}(\mathbf G,\mathbf G',\omega)</math> and is used for finding the Coulomb interaction screened via the RPA dielectric function, ''W''. However, it requires inverting the Dyson equation at every frequency, which becomes too costly for calculating spectra with good resolution. Since this approach does not invoke the Tamm-Dancoff approximation, the RPA dielectric function obtained this way is equivalent to the Casida RPA dielectric function.		Another approach to TDDFT ({{TAG\|ALGO}}=CHI) is to solve the two-point Dyson equation directly and find the density-density response function <math>\chi</math>. This approach allows one to calculate the full <math>\chi_{\mathbf q}(\mathbf G,\mathbf G',\omega)</math> and is used for finding the Coulomb interaction screened via the RPA dielectric function, ''W''. However, it requires inverting the Dyson equation at every frequency, which becomes too costly for calculating spectra with good resolution. Since this approach does not invoke the Tamm-Dancoff approximation, the RPA dielectric function obtained this way is equivalent to the Casida RPA dielectric function.

	== Approximations for the exchange-correlation kernel ==		== Approximations for the exchange-correlation kernel ==
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	When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form		When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form
	::<math>		::<math>
	f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = \frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},		f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = -\frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},
	</math>		</math>
	where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.		where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.
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	== Related tags and articles ==		== Related tags and articles ==
			;Tags
			:{{TAG\|ALGO}}, {{TAG\|LHARTREE}}, {{TAG\|LFXC}}, {{TAG\|LADDER}}
	;How-to		;How-to
	:[[Time-dependent density-functional theory calculations]], [[Plotting exciton wavefunction]]		:[[Time-dependent density-functional theory calculations]], [[Plotting exciton wavefunction]]

Tal at 11:15, 15 June 2026

2026-06-15T11:15:55Z

← Older revision		Revision as of 11:15, 15 June 2026
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	When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form		When hybrid functionals or Hartree-Fock exchange are used in TDDFT, the exchange-correlation kernel includes a contribution from exact exchange. This nonlocal exchange contribution naturally provides the <math>-1/q^2</math> long-range behavior in the kernel, which is essential for capturing excitonic effects. The exact-exchange contribution to <math>f_\mathrm{xc}</math> takes the form
	::<math>		::<math>
	f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = -\frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},		f_{\mathrm{x}}^{\text{exact}}(\mathbf{r}, \mathbf{r'}) = \frac{\delta^2 E_{\mathrm{x}}^{\text{exact}}}{\delta^2 n(\mathbf{r},\mathbf{r'})},
	</math>		</math>
	where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.		where <math>E_{\mathrm{x}}^{\text{exact}}</math> is the exact-exchange energy.

Tal: Fix minor issues: double space, trailing whitespace, v-notation ambiguity clarification, incomplete sentence

2026-06-15T11:14:20Z

Fix minor issues: double space, trailing whitespace, v-notation ambiguity clarification, incomplete sentence

← Older revision		Revision as of 11:42, 15 June 2026
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	:[[Time-dependent density-functional theory calculations]], [[Plotting exciton wavefunction]]		:[[Time-dependent density-functional theory calculations]], [[Plotting exciton wavefunction]]
	;Theory		;Theory
	~~:[[Time-evolution algorithm]],~~ [[Bethe-Salpeter equation]], [[Dielectric properties]]		[[Bethe-Salpeter equation]], [[Dielectric properties]]

	== References ==		== References ==

← Older revision		Revision as of 11:14, 15 June 2026
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	Time-dependent density-functional theory (TDDFT) is an extension of density-functional theory (DFT) to systems with time-varying external potentials, enabling the computation of excited-state properties and response functions{{cite\|onida:revmodphys:2002}}. TDDFT calculations can be based on ground-state electronic structures obtained from DFT, [[Hybrid functionals\|hybrid functionals]], or even ''GW'' approximations.		Time-dependent density-functional theory (TDDFT) is an extension of density-functional theory (DFT) to systems with time-varying external potentials, enabling the computation of excited-state properties and response functions{{cite\|onida:revmodphys:2002}}. TDDFT calculations can be based on ground-state electronic structures obtained from DFT, [[Hybrid functionals\|hybrid functionals]], or even ''GW'' approximations.

	The theoretical foundation of TDDFT is the Runge-Gross theorem{{cite\|runge:gross:1984}}, which is the time-dependent analog of the Hohenberg-Kohn theorem of density-functional theory. It states that the time-dependent density <math>n(\mathbf r, t)</math> is uniquely determined by the time-dependent external potential <math>v_\mathrm{ext}(\mathbf r, t)</math>. As a consequence, all physical observables are functionals of the density, and the interacting many-electron problem can be mapped onto an auxiliary system of non-interacting electrons reproducing the same density (the time-dependent Kohn-Sham system).		The theoretical foundation of TDDFT is the Runge-Gross theorem{{cite\|runge:gross:1984}}, which is the time-dependent analog of the Hohenberg-Kohn theorem of density-functional theory. It states that the time-dependent density <math>n(\mathbf r, t)</math> is uniquely determined by the time-dependent external potential <math>v_\mathrm{ext}(\mathbf r, t)</math>. As a consequence, all physical observables are functionals of the density, and the interacting many-electron problem can be mapped onto an auxiliary system of non-interacting electrons reproducing the same density (the time-dependent Kohn-Sham system).

	== Linear-response formalism ==		== Linear-response formalism ==
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	B_{vc}^{v'c'} = \langle vv'\|v\|cc'\rangle - \langle vv'\|f_\mathrm{xc}\|c'c\rangle.		B_{vc}^{v'c'} = \langle vv'\|v\|cc'\rangle - \langle vv'\|f_\mathrm{xc}\|c'c\rangle.
	</math>		</math>
	Here, <math>v</math> and <math>c</math> denote valence and conduction states, respectively, ~~and~~ <math>\varepsilon_v</math> and <math>\varepsilon_c</math> are the corresponding eigenvalues. The <math>A</math> matrix describes the resonant (excitation) and anti-resonant (de-excitation) transitions, while the <math>B</math> matrix describes the coupling between them. Due to this coupling, the Casida matrix is non-Hermitian.		Here, <math>v</math> and <math>c</math> denote valence and conduction states, respectively, <math>\varepsilon_v</math> and <math>\varepsilon_c</math> are the corresponding eigenvalues, and the two-electron integrals <math>\langle \cdot\|\cdot\|\cdot\rangle</math> involve the bare Coulomb interaction or the exchange-correlation kernel as indicated. The <math>A</math> matrix describes the resonant (excitation) and anti-resonant (de-excitation) transitions, while the <math>B</math> matrix describes the coupling between them. Due to this coupling, the Casida matrix is non-Hermitian.

	=== Tamm-Dancoff approximation ===		=== Tamm-Dancoff approximation ===
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	\mathrm i \frac{\partial}{\partial t}\left\|\phi_{v\mathbf k}[n(t)]\right\rangle = \left[-\frac{\nabla^2}{2} + V_{\mathrm H}[n(t)] + V_{\mathrm{xc}}[n(t)] + V_\mathrm{ext}(t)\right]\left\|\phi_{v\mathbf k}[n(t)]\right\rangle,		\mathrm i \frac{\partial}{\partial t}\left\|\phi_{v\mathbf k}[n(t)]\right\rangle = \left[-\frac{\nabla^2}{2} + V_{\mathrm H}[n(t)] + V_{\mathrm{xc}}[n(t)] + V_\mathrm{ext}(t)\right]\left\|\phi_{v\mathbf k}[n(t)]\right\rangle,
	</math>		</math>
	where all potentials are functionals of the time-evolving density <math>n(\mathbf r, t)</math>.		where all potentials are functionals of the time-evolving density <math>n(\mathbf r, t)</math>.

	Instead of constructing and diagonalizing the full Hamiltonian in transition space as done in Casida formalism, the time-evolution method applies a delta-like perturbation		Instead of constructing and diagonalizing the full Hamiltonian in transition space as done in Casida formalism, the time-evolution method applies a delta-like perturbation
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	V_\mathrm{ext}(\mathbf r, t) = \lambda \, \mathbf r\cdot \mathbf D \, \delta(t)		V_\mathrm{ext}(\mathbf r, t) = \lambda \, \mathbf r\cdot \mathbf D \, \delta(t)
	</math>		</math>
	to the ground-state system, where <math>\lambda</math> is a small perturbation parameter and <math>\mathbf D</math> is the electric displacement field.		to the ground-state system, where <math>\lambda</math> is a small perturbation parameter and <math>\mathbf D</math> is the electric displacement field.

	=== Propagation of the time-dependent coefficients ===		=== Propagation of the time-dependent coefficients ===
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	\varepsilon_{ij}(\omega) = \delta_{ij} - \frac{4\pi e^2}{\Omega} \int_0^\infty \mathrm d t \sum_{cv\mathbf k} \left(\langle\mu^j_{cv\mathbf k}\|c^i_{cv\mathbf k}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega - \mathrm i\eta)t}.		\varepsilon_{ij}(\omega) = \delta_{ij} - \frac{4\pi e^2}{\Omega} \int_0^\infty \mathrm d t \sum_{cv\mathbf k} \left(\langle\mu^j_{cv\mathbf k}\|c^i_{cv\mathbf k}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega - \mathrm i\eta)t}.
	</math>		</math>
	Here, <math>\mu_{cv\mathbf{k}}^j</math> is the dipole matrix element defined in the [[#Casida equation formalism for TDDFT\|Casida section]].		Here, <math>\mu_{cv\mathbf{k}}^j</math> is the dipole matrix element defined in the [[#Casida equation formalism for TDDFT\|Casida section]].

	== Two-point Dyson equation for TDDFT ==		== Two-point Dyson equation for TDDFT ==
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	=== Local exchange-correlation kernel ===		=== Local exchange-correlation kernel ===

	The local exchange-correlation kernel, <math>f_\mathrm{xc}</math>, in TDDFT calculations ~~can be calculated~~		The local exchange-correlation kernel, <math>f_\mathrm{xc}</math>, in TDDFT calculations is given by
	::<math>		::<math>
	f_{\mathrm{xc}}^{\text {loc }}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\frac{\delta^2 E_{\mathrm{xc}}^{\mathrm{DFT}}}{\delta n(\mathbf{r}) \delta n\left(\mathbf{r}^{\prime}\right)},		f_{\mathrm{xc}}^{\text {loc }}\left(\mathbf{r}, \mathbf{r}^{\prime}\right)=\frac{\delta^2 E_{\mathrm{xc}}^{\mathrm{DFT}}}{\delta n(\mathbf{r}) \delta n\left(\mathbf{r}^{\prime}\right)},