Time-dependent density-functional theory calculations - Revision history

Csheldon at 14:22, 24 March 2026

2026-03-24T14:22:04Z

← Older revision		Revision as of 14:22, 24 March 2026
Line 1:		Line 1:
	VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering ~~[https~~:~~//youtu.be/arTPHW28qNM~~ TDDFT theory and calculations] on our YouTube channel.		VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering {{Video\|bse:alexey:2026\|TDDFT theory and calculations}} on our YouTube channel.

	==From TDDFT (or TDHF) to Casida's equation==		==From TDDFT (or TDHF) to Casida's equation==
Line 126:		Line 126:
	Calculations at finite wavevectors can be performed in the same manner as in the [[BSE calculations#Calculations at finite wavevectors\|BSE]].		Calculations at finite wavevectors can be performed in the same manner as in the [[BSE calculations#Calculations at finite wavevectors\|BSE]].
	== References ==		== References ==
	~~<references/>~~
	~~----~~
	[[Category:Time-dependent density-functional theory]][[Category:Howto]]		[[Category:Time-dependent density-functional theory]][[Category:Howto]]

Pmelo: /* From TDDFT (or TDHF) to Casida's equation */

2026-03-18T17:17:13Z

From TDDFT (or TDHF) to Casida's equation

← Older revision		Revision as of 17:17, 18 March 2026
Line 55:		Line 55:

	This formalism is very similar to the [[Bethe-Salpeter equation\|Bethe-Salpeter equation]], only now instead of the exchange-correlation self-energy, <math>\Sigma_\mathrm{xc}</math>, and the screened interaction, <math>W</math>, it deals with the exchange-correlation potential, <math>V_\mathrm{xc}</math>, and with the exchange-correlation kernen, <math>f_\mathrm{xc}</math>.		This formalism is very similar to the [[Bethe-Salpeter equation\|Bethe-Salpeter equation]], only now instead of the exchange-correlation self-energy, <math>\Sigma_\mathrm{xc}</math>, and the screened interaction, <math>W</math>, it deals with the exchange-correlation potential, <math>V_\mathrm{xc}</math>, and with the exchange-correlation kernen, <math>f_\mathrm{xc}</math>.

			Contrary to the [[Time-evolution algorithm\|time-propagation algorithm]], here VASP requires the computation and storage in memory of the A and B matrices. So, while more robust this algorithm also requires more memory during its execution. But, it has the added advantage that it can compute not only the dielectric function, but also the eigenvalues and eigenvectors of the Casida matrix, which can provide important information on the nature of the electron-hole pairs that are involved in the absorption process.

	== Solving Casida equation ==		== Solving Casida equation ==

Pmelo: /* From TDDFT (or TDHF) to Casida's equation */

2026-03-18T17:07:00Z

From TDDFT (or TDHF) to Casida's equation

← Older revision		Revision as of 17:07, 18 March 2026
Line 22:		Line 22:
	with <math>f_\mathrm{xc}(1,2) = \frac{\delta V_\mathrm{xc}(1)}{\delta n(2)}</math> being the exchange-correlation kernel. This quantity is dependent on the choice taken to the exchange-correlation energy functional. The application of a chain rule to <math>\chi(1,2)</math> leads to a Dyson equation		with <math>f_\mathrm{xc}(1,2) = \frac{\delta V_\mathrm{xc}(1)}{\delta n(2)}</math> being the exchange-correlation kernel. This quantity is dependent on the choice taken to the exchange-correlation energy functional. The application of a chain rule to <math>\chi(1,2)</math> leads to a Dyson equation
	::<math>		::<math>
	\chi(1,2) = \chi_0(1,2) + \chi_0(1,3)\left[\frac{\delta(t_3-t_4)}{\|\mathbf r_3 - \mathbf r_4\|} + f_\mathrm{xc}(3,4)\right]\chi(4,2)		\chi(1,2) = \chi_0(1,2) + \chi_0(1,3)\left[\frac{\delta(t_3-t_4)}{\|\mathbf r_3 - \mathbf r_4\|} + f_\mathrm{xc}(3,4)\right]\chi(4,2).
	</math>		</math>
	While relatively simple, it is easier to solve this equation working within the frequency domain and in a basis set that considers transitions from valence to conduction states at the same k-point. The latter choice comes from the fact that, when dealing with optical measurements (e.g. absorption), we deal with neutral excitations and the momentum of the absorbed photons is practically zero. Once all of this is taken into account, the equation for <math>\chi(1,2)</math> is recast as an eigenvalue problem		While relatively simple, it is easier to solve this equation working within the frequency domain and in a basis set that considers transitions from valence to conduction states at the same k-point. The latter choice comes from the fact that, when dealing with optical measurements (e.g. absorption), we deal with neutral excitations and the momentum of the absorbed photons is practically zero. Once all of this is taken into account, the equation for <math>\chi(1,2)</math> is recast as an eigenvalue problem

Pmelo: /* From TDDFT (or TDHF) to Casida's equation */

2026-03-18T17:06:08Z

From TDDFT (or TDHF) to Casida's equation

← Older revision		Revision as of 17:06, 18 March 2026
Line 44:		Line 44:
	\right),		\right),
	</math>		</math>
	which is also known as the Casida equation. This formalism is very similar to the [[Bethe-Salpeter equation\|Bethe-Salpeter equation]], only now instead of the exchange-correlation self-energy, <math>\Sigma_\mathrm{xc}</math>, and the screened interaction, <math>W</math>, it deals with the exchange-correlation potential, <math>V_\mathrm{xc}</math>, and with the exchange-correlation kernen, <math>f_\mathrm{xc}</math>.		which is also known as the Casida equation. The A and B matrices are given by
			::<math>
			A_{vc}^{v'c'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'} + \langle cv'\|\frac{1}{\|\mathbf r_1-\mathbf r_2\|}\|vc'\rangle - \langle cv'\|f_\mathrm{xc}(1,2)\|c'v\rangle,
			</math>
			while the B matrix is computed via
			::<math>
			B_{vc}^{v'c'} = \langle vv'\|\frac{1}{\|\mathbf r_1-\mathbf r_2\|}\|cc'\rangle - \langle vv'\|f_\mathrm{xc}(1,2)\|c'c\rangle.
			</math>
			The A matrix describes the resonant (excitations) and anti-resonant (de-excitation) transitions, while the B matrix deals with the coupling between both. Due to the presence of this coupling, Casida's matrix is non-Hermitian.

			This formalism is very similar to the [[Bethe-Salpeter equation\|Bethe-Salpeter equation]], only now instead of the exchange-correlation self-energy, <math>\Sigma_\mathrm{xc}</math>, and the screened interaction, <math>W</math>, it deals with the exchange-correlation potential, <math>V_\mathrm{xc}</math>, and with the exchange-correlation kernen, <math>f_\mathrm{xc}</math>.

	== Solving Casida equation ==		== Solving Casida equation ==

Pmelo: /* From TDDFT to Casida's equation */

2026-03-18T16:59:42Z

From TDDFT to Casida's equation

← Older revision		Revision as of 16:59, 18 March 2026
Line 1:		Line 1:
	VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering [https://youtu.be/arTPHW28qNM TDDFT theory and calculations] on our YouTube channel.		VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering [https://youtu.be/arTPHW28qNM TDDFT theory and calculations] on our YouTube channel.

	==From TDDFT to Casida's equation==		==From TDDFT (or TDHF) to Casida's equation==
	In time-dependent density functional theory the electrons follow a time-dependent Schrödinger equation		In time-dependent density functional theory the electrons follow a time-dependent Schrödinger equation
	::<math>		::<math>
	\mathrm i \frac{\partial}{\partial t}\left\|\phi_i[n(t)]\rangle\right. = \left[-\frac{\nabla^2}{2} + V_{\mathrm H}[n(t)] + V_{\mathrm{xc}}[n(t)] + V_\mathrm{ext}(t)\right]\left\|\phi_i[n(t)]\rangle\right.,		\mathrm i \frac{\partial}{\partial t}\left\|\phi_i[n(t)]\rangle\right. = \left[-\frac{\nabla^2}{2} + V_{\mathrm H}[n(t)] + V_{\mathrm{xc}}[n(t)] + V_\mathrm{ext}(t)\right]\left\|\phi_i[n(t)]\rangle\right.,
	</math>		</math>
	where all quantities are now dependent on a time-evolving density, <math>n(t)</math>. The external potential <math>V_\mathrm{ext}(t)</math> now is explicitly dependent on time, and which is used to perturb the system away from the ground state.		where all quantities are now dependent on a time-evolving density, <math>n(t)</math>. The external potential <math>V_\mathrm{ext}(t)</math> now is explicitly dependent on time, and which is used to perturb the system away from the ground state. For time-dependent Hartree-Fock (TDHF) one would only need to replace the exchange correlation potential, <math>V_\mathrm{xc}</math>, with the Fock exchange term, <math>V_\mathrm{x}</math>.

	From linear-response theory it is possible to evaluate the change in the ground-state density due to the external potential,		From linear-response theory it is possible to evaluate the change in the ground-state density due to the external potential,
Line 24:		Line 24:
	\chi(1,2) = \chi_0(1,2) + \chi_0(1,3)\left[\frac{\delta(t_3-t_4)}{\|\mathbf r_3 - \mathbf r_4\|} + f_\mathrm{xc}(3,4)\right]\chi(4,2)		\chi(1,2) = \chi_0(1,2) + \chi_0(1,3)\left[\frac{\delta(t_3-t_4)}{\|\mathbf r_3 - \mathbf r_4\|} + f_\mathrm{xc}(3,4)\right]\chi(4,2)
	</math>		</math>
			While relatively simple, it is easier to solve this equation working within the frequency domain and in a basis set that considers transitions from valence to conduction states at the same k-point. The latter choice comes from the fact that, when dealing with optical measurements (e.g. absorption), we deal with neutral excitations and the momentum of the absorbed photons is practically zero. Once all of this is taken into account, the equation for <math>\chi(1,2)</math> is recast as an eigenvalue problem
			::<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),
			</math>
			which is also known as the Casida equation. This formalism is very similar to the [[Bethe-Salpeter equation\|Bethe-Salpeter equation]], only now instead of the exchange-correlation self-energy, <math>\Sigma_\mathrm{xc}</math>, and the screened interaction, <math>W</math>, it deals with the exchange-correlation potential, <math>V_\mathrm{xc}</math>, and with the exchange-correlation kernen, <math>f_\mathrm{xc}</math>.

	== Solving Casida equation ==		== Solving Casida equation ==

Pmelo at 16:46, 18 March 2026

2026-03-18T16:46:31Z

← Older revision		Revision as of 16:46, 18 March 2026
Line 1:		Line 1:
	VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering [https://youtu.be/arTPHW28qNM TDDFT theory and calculations] on our YouTube channel.		VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering [https://youtu.be/arTPHW28qNM TDDFT theory and calculations] on our YouTube channel.

			==From TDDFT to Casida's equation==
			In time-dependent density functional theory the electrons follow a time-dependent Schrödinger equation
			::<math>
			\mathrm i \frac{\partial}{\partial t}\left\|\phi_i[n(t)]\rangle\right. = \left[-\frac{\nabla^2}{2} + V_{\mathrm H}[n(t)] + V_{\mathrm{xc}}[n(t)] + V_\mathrm{ext}(t)\right]\left\|\phi_i[n(t)]\rangle\right.,
			</math>
			where all quantities are now dependent on a time-evolving density, <math>n(t)</math>. The external potential <math>V_\mathrm{ext}(t)</math> now is explicitly dependent on time, and which is used to perturb the system away from the ground state.

			From linear-response theory it is possible to evaluate the change in the ground-state density due to the external potential,
			::<math>
			\chi(\mathbf 1,2) = \frac{\delta n(1)}{\delta V_\mathrm{ext}(2)},
			</math>
			or due to the total Kohn-Sham potential
			::<math>
			\chi_0(\mathbf 1,2) = \frac{\delta n(1)}{\delta V_\mathrm{Ks}(2)}.
			</math>
			Both quantities are related via a Dyson equation. It is possible to show that
			::<math>
			\frac{\delta V_\mathrm{KS}(1)}{\delta V_\mathrm{ext}(2)} = \delta(1,2) + \frac{\delta(t_1-t_2)}{\|\mathbf r_1 - \mathbf r_2\|} + f_\mathrm{xc}(1,2),
			</math>
			with <math>f_\mathrm{xc}(1,2) = \frac{\delta V_\mathrm{xc}(1)}{\delta n(2)}</math> being the exchange-correlation kernel. This quantity is dependent on the choice taken to the exchange-correlation energy functional. The application of a chain rule to <math>\chi(1,2)</math> leads to a Dyson equation
			::<math>
			\chi(1,2) = \chi_0(1,2) + \chi_0(1,3)\left[\frac{\delta(t_3-t_4)}{\|\mathbf r_3 - \mathbf r_4\|} + f_\mathrm{xc}(3,4)\right]\chi(4,2)
			</math>

	== Solving Casida equation ==		== Solving Casida equation ==

Csheldon at 09:40, 6 February 2026

2026-02-06T09:40:18Z

← Older revision		Revision as of 09:40, 6 February 2026
Line 1:		Line 1:
	VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations.		VASP offers a powerful module for performing time-dependent density-functional theory (TDDFT) or time-dependent Hartree-Fock (TDHF) calculations by solving the Casida equation {{cite\|casida:jomst:2009}}. This approach can be used for obtaining the frequency-dependent dielectric function with the excitonic effects and can be based on the ground-state electronic structure in the DFT, hybrid-functional or even ''GW'' approximations. You can watch a lecture covering [https://youtu.be/arTPHW28qNM TDDFT theory and calculations] on our YouTube channel.

	== Solving Casida equation ==		== Solving Casida equation ==

Huebsch at 14:00, 28 February 2025

2025-02-28T14:00:14Z

← Older revision		Revision as of 14:00, 28 February 2025
Line 71:		Line 71:
	<references/>		<references/>
	----		----
	[[Category:Time-dependent density functional theory]][[Category:Howto]]		[[Category:Time-dependent density-functional theory]][[Category:Howto]]

Tal: Tal moved page Construction:Time-dependent density-functional theory calculations to Time-dependent density-functional theory calculations

2024-02-07T17:44:27Z

Tal moved page Construction:Time-dependent density-functional theory calculations to Time-dependent density-functional theory calculations

← Older revision	Revision as of 17:44, 7 February 2024
(No difference)

Tal: /* Time-dependent Hartree-Fock */

2024-02-07T17:42:50Z

Time-dependent Hartree-Fock

← Older revision		Revision as of 17:42, 7 February 2024
Line 11:		Line 11:
	# ground-state calculation		# ground-state calculation
	# optical absorption calculation		# optical absorption calculation
	For example, an optical absorption calculation of bulk Si can be performed using a dielectric-dependent hybrid-functional described in Refs. {{cite\|chen2018nonempirical}}{{cite\|cui2018doubly}}{{cite\|liu2019assessing}}.		For example, an optical absorption calculation of bulk Si can be performed using a dielectric-dependent hybrid-functional described in Refs.{{cite\|chen2018nonempirical}}{{cite\|cui2018doubly}}{{cite\|liu2019assessing}}.

	{{TAG\|SYSTEM}} = Si		{{TAG\|SYSTEM}} = Si