Category:Time-dependent density functional theory - Revision history

Tal: /* Time-evolution or real-time TDDFT ({{TAG|ALGO|TIMEEV}}) */

2026-06-17T11:40:55Z

Time-evolution or real-time TDDFT ({{TAG|ALGO|TIMEEV}})

← Older revision		Revision as of 11:40, 17 June 2026
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	\varepsilon_M(\omega) = 1 - \frac{4\pi e^2}{\Omega} \int_0^\infty \mathrm d t \sum_{cv\mathbf k} \left(\langle\mu_{cv\mathbf k}\|c_{cv\mathbf k}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega - \mathrm i\eta)t},		\varepsilon_M(\omega) = 1 - \frac{4\pi e^2}{\Omega} \int_0^\infty \mathrm d t \sum_{cv\mathbf k} \left(\langle\mu_{cv\mathbf k}\|c_{cv\mathbf k}(t)\rangle + \mathrm{c.c.}\right) e^{-\mathrm i(\omega - \mathrm i\eta)t},
	</math>		</math>
	where <math>\mu_{cv\mathbf{k}}</math> is the dipole matrix element and <math>c_{cv\mathbf k}(t)</math> are the time-dependent expansion coefficients. This approach avoids building and storing the full Hamiltonian in transition space. The detailed theory and propagation algorithm are described on the [[~~Construction:~~Time-dependent density-functional theory\|theory page]].		where <math>\mu_{cv\mathbf{k}}</math> is the dipole matrix element and <math>c_{cv\mathbf k}(t)</math> are the time-dependent expansion coefficients. This approach avoids building and storing the full Hamiltonian in transition space. The detailed theory and propagation algorithm are described on the [[Time-dependent density-functional theory\|theory page]].

	=== Scaling with the system size===		=== Scaling with the system size===

Huebsch at 10:26, 17 June 2026

2026-06-17T10:26:31Z

← Older revision		Revision as of 10:26, 17 June 2026
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	Time-dependent density-functional theory (TDDFT) extends [[:Category:Exchange-correlation functionals\|density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. The accuracy of TDDFT strongly depends on the choice of the exchange-correlation kernel <math>f_\mathrm{xc}</math> ({{TAG\|LFXC}}, {{TAG\|LADDER}}) and if the nonlocal limit of <math>f_\mathrm{xc}</math> is properly treated, excitonic effects can be captured with good accuracy{{cite\|tal:prr:2020}}. VASP provides multiple implementations of TDDFT, each with its own advantages and disadvantages, so that the best algorithm can be selected based on the problem at hand. The detailed theoretical background is given on the [[~~Construction:~~Time-dependent density-functional theory\|theory page]].		'''Time-dependent density-functional theory''' (TDDFT) extends [[:Category:Exchange-correlation functionals\|density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. The accuracy of TDDFT strongly depends on the choice of the exchange-correlation kernel <math>f_\mathrm{xc}</math> ({{TAG\|LFXC}}, {{TAG\|LADDER}}) and if the nonlocal limit of <math>f_\mathrm{xc}</math> is properly treated, excitonic effects can be captured with good accuracy{{cite\|tal:prr:2020}}. VASP provides multiple implementations of TDDFT, each with its own advantages and disadvantages, so that the best algorithm can be selected based on the problem at hand. The detailed theoretical background is given on the [[Time-dependent density-functional theory\|theory page]].

	== Casida TDDFT ({{TAG\|ALGO}}~~{{=}}TDHF~~) ==		* Lecture on {{Video\|bse:alexey:2026\|TDDFT theory and calculations}}.

			== Casida TDDFT ({{TAG\|ALGO\|TDHF}}) ==

	The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue problem		The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue problem
Line 23:		Line 25:
	VASP provides three algorithms for solving the Casida equation, selected via {{TAG\|IBSE}}:		VASP provides three algorithms for solving the Casida equation, selected via {{TAG\|IBSE}}:

	=== Exact diagonalization ({{TAG\|IBSE}}~~{{=}}2~~) ===		=== Exact diagonalization ({{TAG\|IBSE\|2}}) ===

	The Hamiltonian is diagonalized exactly. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors, which makes this approach particularly useful for analyzing individual excitons. The macroscopic dielectric function is obtained from the spectral representation:		The Hamiltonian is diagonalized exactly. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors, which makes this approach particularly useful for analyzing individual excitons. The macroscopic dielectric function is obtained from the spectral representation:
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	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, <math>X_\lambda^{cv\mathbf k}</math> are the eigenvector components, and <math>\omega_\lambda</math> are the excitation energies.		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, <math>X_\lambda^{cv\mathbf k}</math> are the eigenvector components, and <math>\omega_\lambda</math> are the excitation energies.

	=== Time-evolution algorithm ({{TAG\|IBSE}}~~{{=}}1~~) ===		=== Time-evolution algorithm ({{TAG\|IBSE\|1}}) ===

	This algorithm is based on the same time-evolution approach described in the [[#Time-evolution or real-time TDDFT (ALGO=TIMEEV)\|real-time TDDFT section]], but applied within the Casida framework. The key difference is that the Hamiltonian in transition space is built once and never updated during the propagation. Only the time-dependent dipole vector <math>\|\mu_{cv\mathbf k}(t)\rangle</math> is propagated forward in time using the fixed Hamiltonian.		This algorithm is based on the same time-evolution approach described in the [[#Time-evolution or real-time TDDFT (ALGO=TIMEEV)\|real-time TDDFT section]], but applied within the Casida framework. The key difference is that the Hamiltonian in transition space is built once and never updated during the propagation. Only the time-dependent dipole vector <math>\|\mu_{cv\mathbf k}(t)\rangle</math> is propagated forward in time using the fixed Hamiltonian.
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	where <math>\mu_{cv\mathbf k}</math> are the dipole moments and <math>\|\xi_{cv\mathbf k}(t)\rangle</math> is the time-evolved dipole vector. The solution is strictly equivalent to that of the exact diagonalization for the dielectric function.		where <math>\mu_{cv\mathbf k}</math> are the dipole moments and <math>\|\xi_{cv\mathbf k}(t)\rangle</math> is the time-evolved dipole vector. The solution is strictly equivalent to that of the exact diagonalization for the dielectric function.

	=== Lanczos algorithm ({{TAG\|IBSE}}~~{{=}}3~~) ===		=== Lanczos algorithm ({{TAG\|IBSE\|3}}) ===

	The dielectric function is expressed as a continued fraction		The dielectric function is expressed as a continued fraction
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	where <math>\|u_0\rangle</math> is an initial guess vector computed from the dipole moments. The <math>a</math> and <math>b</math> coefficients are evaluated iteratively, with the algorithm stopping once the difference between <math>\varepsilon(\omega)</math> from two consecutive iterations is below a threshold selected by {{TAG\|BSEPREC}}.		where <math>\|u_0\rangle</math> is an initial guess vector computed from the dipole moments. The <math>a</math> and <math>b</math> coefficients are evaluated iteratively, with the algorithm stopping once the difference between <math>\varepsilon(\omega)</math> from two consecutive iterations is below a threshold selected by {{TAG\|BSEPREC}}.

	=== Scaling ===		=== Scaling with the system size===

	Building the Hamiltonian scales as <math>N^4</math>–<math>N^5</math> with the system size. Solving the equation scales as <math>N^6</math> for exact diagonalization ({{TAG\|IBSE}}~~{{=}}2~~) and <math>N^4</math> for the time-evolution ({{TAG\|IBSE}}~~{{=}}1~~) and Lanczos ({{TAG\|IBSE}}~~{{=}}3~~) algorithms.		Building the Hamiltonian scales as <math>N^4</math>–<math>N^5</math> with the system size. Solving the equation scales as <math>N^6</math> for exact diagonalization ({{TAG\|IBSE\|2}}) and <math>N^4</math> for the time-evolution ({{TAG\|IBSE\|1}}) and Lanczos ({{TAG\|IBSE\|3}}) algorithms.

	== Time-evolution or real-time TDDFT ({{TAG\|ALGO}}~~{{=}}TIMEEV~~) ==		== Time-evolution or real-time TDDFT ({{TAG\|ALGO\|TIMEEV}}) ==

	An alternative to solving the Casida equation is to compute the frequency-dependent response via real-time propagation of the Kohn-Sham orbitals {{cite\|sander:jcp:2017}}. The starting point is the time-dependent Kohn-Sham equation,		An alternative to solving the Casida equation is to compute the frequency-dependent response via real-time propagation of the Kohn-Sham orbitals {{cite\|sander:jcp:2017}}. The starting point is the time-dependent Kohn-Sham equation,
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	where <math>\mu_{cv\mathbf{k}}</math> is the dipole matrix element and <math>c_{cv\mathbf k}(t)</math> are the time-dependent expansion coefficients. This approach avoids building and storing the full Hamiltonian in transition space. The detailed theory and propagation algorithm are described on the [[Construction:Time-dependent density-functional theory\|theory page]].		where <math>\mu_{cv\mathbf{k}}</math> is the dipole matrix element and <math>c_{cv\mathbf k}(t)</math> are the time-dependent expansion coefficients. This approach avoids building and storing the full Hamiltonian in transition space. The detailed theory and propagation algorithm are described on the [[Construction:Time-dependent density-functional theory\|theory page]].

	=== Scaling ===		=== Scaling with the system size===

	The compute time per time step scales as <math>N^3</math> with the system size. When nonlocal exchange is included ({{TAG\|LADDER}}{{=}}.TRUE.), an additional <math>N_{\mathbf{k}}^2 \times N_v^2 \times N_G</math> contribution arises from evaluating the exchange integrals at each time step.		The compute time per time step scales as <math>N^3</math> with the system size. When nonlocal exchange is included ({{TAG\|LADDER}}{{=}}.TRUE.), an additional <math>N_{\mathbf{k}}^2 \times N_v^2 \times N_G</math> contribution arises from evaluating the exchange integrals at each time step.

	== Dyson-equation TDDFT ({{TAG\|ALGO}}~~{{=}}CHI~~) ==		== Dyson-equation TDDFT ({{TAG\|ALGO\|CHI}}) ==

	Instead of recasting the problem as an eigenvalue equation, TDDFT can also be solved by directly evaluating the two-point Dyson equation for the density-density response function <math>\chi</math>. The macroscopic dielectric function is then obtained from		Instead of recasting the problem as an eigenvalue equation, TDDFT can also be solved by directly evaluating the two-point Dyson equation for the density-density response function <math>\chi</math>. The macroscopic dielectric function is then obtained from
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	where <math>v(\mathbf{G})</math> is the Coulomb kernel and <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> is the interacting response function in reciprocal space. This approach gives access to the full matrix <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> and is used, for instance, to compute the screened Coulomb interaction <math>W</math> needed in ''GW'' calculations. However, the Dyson equation must be inverted at every frequency point, which makes it expensive for computing spectra with fine resolution. The only nonlocal exchange-correlation kernel currently supported is the nanoquanta kernel ({{TAG\|LFXC}}).		where <math>v(\mathbf{G})</math> is the Coulomb kernel and <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> is the interacting response function in reciprocal space. This approach gives access to the full matrix <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> and is used, for instance, to compute the screened Coulomb interaction <math>W</math> needed in ''GW'' calculations. However, the Dyson equation must be inverted at every frequency point, which makes it expensive for computing spectra with fine resolution. The only nonlocal exchange-correlation kernel currently supported is the nanoquanta kernel ({{TAG\|LFXC}}).

	=== Scaling ===		=== Scaling with the system size===

	The compute time scales as <math>N^4</math> with the system size. The Dyson equation must be inverted at every frequency point ({{TAG\|NOMEGA}}), so the total cost grows linearly with the number of frequency points.		The compute time scales as <math>N^4</math> with the system size. The Dyson equation must be inverted at every frequency point ({{TAG\|NOMEGA}}), so the total cost grows linearly with the number of frequency points.

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

	~~=== Lectures ===~~
	* Lecture on {{Video\|bse:alexey:2026\|TDDFT theory and calculations}}.

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

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

2026-06-17T10:19:47Z

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

← Older revision		Revision as of 10:19, 17 June 2026
(No difference)

Tal at 15:15, 15 June 2026

2026-06-15T15:15:28Z

← Older revision		Revision as of 15:15, 15 June 2026
Line 1:		Line 1:
	Time-dependent density-functional theory (TDDFT) extends [[:Category:Exchange-correlation functionals\|density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. The accuracy of TDDFT strongly depends on the choice of the exchange-correlation kernel <math>f_\mathrm{xc}</math> ({{TAG\|LFXC}}, {{TAG\|LADDER}}); if the nonlocal limit of <math>f_\mathrm{xc}</math> is properly treated, excitonic effects can be captured with good accuracy{{cite\|tal:prr:2020}}. VASP provides multiple implementations of TDDFT, each with its own advantages and disadvantages, so that the best algorithm can be selected based on the problem at hand. The detailed theoretical background is given on the [[Construction:Time-dependent density-functional theory\|theory page]].		Time-dependent density-functional theory (TDDFT) extends [[:Category:Exchange-correlation functionals\|density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. The accuracy of TDDFT strongly depends on the choice of the exchange-correlation kernel <math>f_\mathrm{xc}</math> ({{TAG\|LFXC}}, {{TAG\|LADDER}}) and if the nonlocal limit of <math>f_\mathrm{xc}</math> is properly treated, excitonic effects can be captured with good accuracy{{cite\|tal:prr:2020}}. VASP provides multiple implementations of TDDFT, each with its own advantages and disadvantages, so that the best algorithm can be selected based on the problem at hand. The detailed theoretical background is given on the [[Construction:Time-dependent density-functional theory\|theory page]].

	== Casida TDDFT ({{TAG\|ALGO}}{{=}}TDHF) ==		== Casida TDDFT ({{TAG\|ALGO}}{{=}}TDHF) ==

Tal: /* Time-evolution or real-time TDDFT ({{TAG|ALGO}}{{=}}TIMEEV) */

2026-06-15T14:50:27Z

Time-evolution or real-time TDDFT ({{TAG|ALGO}}{{=}}TIMEEV)

← Older revision		Revision as of 14:50, 15 June 2026
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	== Time-evolution or real-time TDDFT ({{TAG\|ALGO}}{{=}}TIMEEV) ==		== Time-evolution or real-time TDDFT ({{TAG\|ALGO}}{{=}}TIMEEV) ==

	An alternative to solving the Casida equation is to compute the frequency-dependent response via real-time propagation of the Kohn-Sham orbitals ~~({{TAG\|ALGO}}{{=}}TIMEEV)~~{{cite\|sander:jcp:2017}}. The starting point is the time-dependent Kohn-Sham equation,		An alternative to solving the Casida equation is to compute the frequency-dependent response via real-time propagation of the Kohn-Sham orbitals {{cite\|sander:jcp:2017}}. The starting point is the time-dependent Kohn-Sham equation,
	::<math>		::<math>
	\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,

Tal: /* Casida TDDFT ({{TAG|ALGO}}{{=}}TDHF) */

2026-06-15T14:50:15Z

Casida TDDFT ({{TAG|ALGO}}{{=}}TDHF)

← Older revision		Revision as of 14:50, 15 June 2026
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	== Casida TDDFT ({{TAG\|ALGO}}{{=}}TDHF) ==		== Casida TDDFT ({{TAG\|ALGO}}{{=}}TDHF) ==

	The Casida formulation of TDDFT ~~({{TAG\|ALGO}}{{=}}TDHF)~~ recasts the linear-response problem as a non-Hermitian eigenvalue problem		The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue problem
	::<math>		::<math>
	\left(\begin{array}{cc}		\left(\begin{array}{cc}

Tal: /* Dyson-equation TDDFT ({{TAG|ALGO}}{{=}}CHI) */

2026-06-15T14:50:02Z

Dyson-equation TDDFT ({{TAG|ALGO}}{{=}}CHI)

← Older revision		Revision as of 14:50, 15 June 2026
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	== Dyson-equation TDDFT ({{TAG\|ALGO}}{{=}}CHI) ==		== Dyson-equation TDDFT ({{TAG\|ALGO}}{{=}}CHI) ==

	Instead of recasting the problem as an eigenvalue equation, TDDFT can also be solved by directly evaluating the two-point Dyson equation for the density-density response function <math>\chi</math> ~~({{TAG\|ALGO}}{{=}}CHI)~~. The macroscopic dielectric function is then obtained from		Instead of recasting the problem as an eigenvalue equation, TDDFT can also be solved by directly evaluating the two-point Dyson equation for the density-density response function <math>\chi</math>. The macroscopic dielectric function is then obtained from
	::<math>		::<math>
	\varepsilon_M(\omega) = 1 - v(\mathbf{G}=0) \, \chi_{\mathbf{G}=0,\mathbf{G}'=0}(\mathbf{q},\omega),		\varepsilon_M(\omega) = 1 - v(\mathbf{G}=0) \, \chi_{\mathbf{G}=0,\mathbf{G}'=0}(\mathbf{q},\omega),

Tal: /* Lanczos algorithm ({{TAG|IBSE}}{{=}}3) */

2026-06-15T14:48:28Z

Lanczos algorithm ({{TAG|IBSE}}{{=}}3)

← Older revision		Revision as of 14:48, 15 June 2026
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	- \cfrac{b_2^2}{...}}},		- \cfrac{b_2^2}{...}}},
	</math>		</math>
	where <math>\|u_0\rangle</math> is an initial guess vector computed from the dipole moments. The <math>a</math> and <math>b</math> coefficients are evaluated iteratively, with the algorithm stopping once the difference between <math>\varepsilon(\omega)</math> from two consecutive iterations is below a threshold selected by {{TAG\|BSEPREC}}. Because the starting vector is built from dipole moments, the Lanczos algorithm is sensitive only to optically active transitions and can reach convergence faster than other methods for larger matrices.		where <math>\|u_0\rangle</math> is an initial guess vector computed from the dipole moments. The <math>a</math> and <math>b</math> coefficients are evaluated iteratively, with the algorithm stopping once the difference between <math>\varepsilon(\omega)</math> from two consecutive iterations is below a threshold selected by {{TAG\|BSEPREC}}.

	=== Scaling ===		=== Scaling ===

Tal: Add NOMEGA reference in Dyson scaling; restore LFXC tag link for nanoquanta kernel

2026-06-15T13:06:14Z

Add NOMEGA reference in Dyson scaling; restore LFXC tag link for nanoquanta kernel

← Older revision		Revision as of 13:06, 15 June 2026
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	\varepsilon_M(\omega) = 1 - v(\mathbf{G}=0) \, \chi_{\mathbf{G}=0,\mathbf{G}'=0}(\mathbf{q},\omega),		\varepsilon_M(\omega) = 1 - v(\mathbf{G}=0) \, \chi_{\mathbf{G}=0,\mathbf{G}'=0}(\mathbf{q},\omega),
	</math>		</math>
	where <math>v(\mathbf{G})</math> is the Coulomb kernel and <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> is the interacting response function in reciprocal space. This approach gives access to the full matrix <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> and is used, for instance, to compute the screened Coulomb interaction <math>W</math> needed in ''GW'' calculations. However, the Dyson equation must be inverted at every frequency point, which makes it expensive for computing spectra with fine resolution. The only nonlocal exchange-correlation kernel currently supported is the nanoquanta kernel.		where <math>v(\mathbf{G})</math> is the Coulomb kernel and <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> is the interacting response function in reciprocal space. This approach gives access to the full matrix <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> and is used, for instance, to compute the screened Coulomb interaction <math>W</math> needed in ''GW'' calculations. However, the Dyson equation must be inverted at every frequency point, which makes it expensive for computing spectra with fine resolution. The only nonlocal exchange-correlation kernel currently supported is the nanoquanta kernel ({{TAG\|LFXC}}).

	=== Scaling ===		=== Scaling ===

	The compute time scales as <math>N^4</math> with the system size. The frequency ~~grid is determined via~~ {{TAG\|NOMEGA}}.		The compute time scales as <math>N^4</math> with the system size. The Dyson equation must be inverted at every frequency point ({{TAG\|NOMEGA}}), so the total cost grows linearly with the number of frequency points.

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

Tal: /* Scaling */

2026-06-15T13:04:21Z

Scaling

← Older revision		Revision as of 13:04, 15 June 2026
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	=== Scaling ===		=== Scaling ===

	The compute time scales as <math>N^4</math> with the system size. The ~~Dyson equation must be inverted at every~~ frequency ~~point, so the total cost grows linearly with the number of frequency points~~.		The compute time scales as <math>N^4</math> with the system size. The frequency grid is determined via {{TAG\|NOMEGA}}.

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

← Older revision		Revision as of 10:26, 17 June 2026
Line 1:		Line 1:
	Time-dependent density-functional theory (TDDFT) extends [[:Category:Exchange-correlation functionals\|density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. The accuracy of TDDFT strongly depends on the choice of the exchange-correlation kernel <math>f_\mathrm{xc}</math> ({{TAG\|LFXC}}, {{TAG\|LADDER}}) and if the nonlocal limit of <math>f_\mathrm{xc}</math> is properly treated, excitonic effects can be captured with good accuracy{{cite\|tal:prr:2020}}. VASP provides multiple implementations of TDDFT, each with its own advantages and disadvantages, so that the best algorithm can be selected based on the problem at hand. The detailed theoretical background is given on the [[~~Construction:~~Time-dependent density-functional theory\|theory page]].		'''Time-dependent density-functional theory''' (TDDFT) extends [[:Category:Exchange-correlation functionals\|density-functional theory]] to time-varying external potentials, enabling the computation of neutral electronic excitations and frequency-dependent response functions. The accuracy of TDDFT strongly depends on the choice of the exchange-correlation kernel <math>f_\mathrm{xc}</math> ({{TAG\|LFXC}}, {{TAG\|LADDER}}) and if the nonlocal limit of <math>f_\mathrm{xc}</math> is properly treated, excitonic effects can be captured with good accuracy{{cite\|tal:prr:2020}}. VASP provides multiple implementations of TDDFT, each with its own advantages and disadvantages, so that the best algorithm can be selected based on the problem at hand. The detailed theoretical background is given on the [[Time-dependent density-functional theory\|theory page]].

	== Casida TDDFT ({{TAG\|ALGO}}~~{{=}}TDHF~~) ==		* Lecture on {{Video\|bse:alexey:2026\|TDDFT theory and calculations}}.

			== Casida TDDFT ({{TAG\|ALGO\|TDHF}}) ==

	The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue problem		The Casida formulation of TDDFT recasts the linear-response problem as a non-Hermitian eigenvalue problem
Line 23:		Line 25:
	VASP provides three algorithms for solving the Casida equation, selected via {{TAG\|IBSE}}:		VASP provides three algorithms for solving the Casida equation, selected via {{TAG\|IBSE}}:

	=== Exact diagonalization ({{TAG\|IBSE}}~~{{=}}2~~) ===		=== Exact diagonalization ({{TAG\|IBSE\|2}}) ===

	The Hamiltonian is diagonalized exactly. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors, which makes this approach particularly useful for analyzing individual excitons. The macroscopic dielectric function is obtained from the spectral representation:		The Hamiltonian is diagonalized exactly. The excitation energies and oscillator strengths are obtained directly from the eigenvalues and eigenvectors, which makes this approach particularly useful for analyzing individual excitons. The macroscopic dielectric function is obtained from the spectral representation:
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	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, <math>X_\lambda^{cv\mathbf k}</math> are the eigenvector components, and <math>\omega_\lambda</math> are the excitation energies.		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, <math>X_\lambda^{cv\mathbf k}</math> are the eigenvector components, and <math>\omega_\lambda</math> are the excitation energies.

	=== Time-evolution algorithm ({{TAG\|IBSE}}~~{{=}}1~~) ===		=== Time-evolution algorithm ({{TAG\|IBSE\|1}}) ===

	This algorithm is based on the same time-evolution approach described in the [[#Time-evolution or real-time TDDFT (ALGO=TIMEEV)\|real-time TDDFT section]], but applied within the Casida framework. The key difference is that the Hamiltonian in transition space is built once and never updated during the propagation. Only the time-dependent dipole vector <math>\|\mu_{cv\mathbf k}(t)\rangle</math> is propagated forward in time using the fixed Hamiltonian.		This algorithm is based on the same time-evolution approach described in the [[#Time-evolution or real-time TDDFT (ALGO=TIMEEV)\|real-time TDDFT section]], but applied within the Casida framework. The key difference is that the Hamiltonian in transition space is built once and never updated during the propagation. Only the time-dependent dipole vector <math>\|\mu_{cv\mathbf k}(t)\rangle</math> is propagated forward in time using the fixed Hamiltonian.
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	where <math>\mu_{cv\mathbf k}</math> are the dipole moments and <math>\|\xi_{cv\mathbf k}(t)\rangle</math> is the time-evolved dipole vector. The solution is strictly equivalent to that of the exact diagonalization for the dielectric function.		where <math>\mu_{cv\mathbf k}</math> are the dipole moments and <math>\|\xi_{cv\mathbf k}(t)\rangle</math> is the time-evolved dipole vector. The solution is strictly equivalent to that of the exact diagonalization for the dielectric function.

	=== Lanczos algorithm ({{TAG\|IBSE}}~~{{=}}3~~) ===		=== Lanczos algorithm ({{TAG\|IBSE\|3}}) ===

	The dielectric function is expressed as a continued fraction		The dielectric function is expressed as a continued fraction
Line 51:		Line 53:
	where <math>\|u_0\rangle</math> is an initial guess vector computed from the dipole moments. The <math>a</math> and <math>b</math> coefficients are evaluated iteratively, with the algorithm stopping once the difference between <math>\varepsilon(\omega)</math> from two consecutive iterations is below a threshold selected by {{TAG\|BSEPREC}}.		where <math>\|u_0\rangle</math> is an initial guess vector computed from the dipole moments. The <math>a</math> and <math>b</math> coefficients are evaluated iteratively, with the algorithm stopping once the difference between <math>\varepsilon(\omega)</math> from two consecutive iterations is below a threshold selected by {{TAG\|BSEPREC}}.

	=== Scaling ===		=== Scaling with the system size===

	Building the Hamiltonian scales as <math>N^4</math>–<math>N^5</math> with the system size. Solving the equation scales as <math>N^6</math> for exact diagonalization ({{TAG\|IBSE}}~~{{=}}2~~) and <math>N^4</math> for the time-evolution ({{TAG\|IBSE}}~~{{=}}1~~) and Lanczos ({{TAG\|IBSE}}~~{{=}}3~~) algorithms.		Building the Hamiltonian scales as <math>N^4</math>–<math>N^5</math> with the system size. Solving the equation scales as <math>N^6</math> for exact diagonalization ({{TAG\|IBSE\|2}}) and <math>N^4</math> for the time-evolution ({{TAG\|IBSE\|1}}) and Lanczos ({{TAG\|IBSE\|3}}) algorithms.

	== Time-evolution or real-time TDDFT ({{TAG\|ALGO}}~~{{=}}TIMEEV~~) ==		== Time-evolution or real-time TDDFT ({{TAG\|ALGO\|TIMEEV}}) ==

	An alternative to solving the Casida equation is to compute the frequency-dependent response via real-time propagation of the Kohn-Sham orbitals {{cite\|sander:jcp:2017}}. The starting point is the time-dependent Kohn-Sham equation,		An alternative to solving the Casida equation is to compute the frequency-dependent response via real-time propagation of the Kohn-Sham orbitals {{cite\|sander:jcp:2017}}. The starting point is the time-dependent Kohn-Sham equation,
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	where <math>\mu_{cv\mathbf{k}}</math> is the dipole matrix element and <math>c_{cv\mathbf k}(t)</math> are the time-dependent expansion coefficients. This approach avoids building and storing the full Hamiltonian in transition space. The detailed theory and propagation algorithm are described on the [[Construction:Time-dependent density-functional theory\|theory page]].		where <math>\mu_{cv\mathbf{k}}</math> is the dipole matrix element and <math>c_{cv\mathbf k}(t)</math> are the time-dependent expansion coefficients. This approach avoids building and storing the full Hamiltonian in transition space. The detailed theory and propagation algorithm are described on the [[Construction:Time-dependent density-functional theory\|theory page]].

	=== Scaling ===		=== Scaling with the system size===

	The compute time per time step scales as <math>N^3</math> with the system size. When nonlocal exchange is included ({{TAG\|LADDER}}{{=}}.TRUE.), an additional <math>N_{\mathbf{k}}^2 \times N_v^2 \times N_G</math> contribution arises from evaluating the exchange integrals at each time step.		The compute time per time step scales as <math>N^3</math> with the system size. When nonlocal exchange is included ({{TAG\|LADDER}}{{=}}.TRUE.), an additional <math>N_{\mathbf{k}}^2 \times N_v^2 \times N_G</math> contribution arises from evaluating the exchange integrals at each time step.

	== Dyson-equation TDDFT ({{TAG\|ALGO}}~~{{=}}CHI~~) ==		== Dyson-equation TDDFT ({{TAG\|ALGO\|CHI}}) ==

	Instead of recasting the problem as an eigenvalue equation, TDDFT can also be solved by directly evaluating the two-point Dyson equation for the density-density response function <math>\chi</math>. The macroscopic dielectric function is then obtained from		Instead of recasting the problem as an eigenvalue equation, TDDFT can also be solved by directly evaluating the two-point Dyson equation for the density-density response function <math>\chi</math>. The macroscopic dielectric function is then obtained from
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	where <math>v(\mathbf{G})</math> is the Coulomb kernel and <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> is the interacting response function in reciprocal space. This approach gives access to the full matrix <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> and is used, for instance, to compute the screened Coulomb interaction <math>W</math> needed in ''GW'' calculations. However, the Dyson equation must be inverted at every frequency point, which makes it expensive for computing spectra with fine resolution. The only nonlocal exchange-correlation kernel currently supported is the nanoquanta kernel ({{TAG\|LFXC}}).		where <math>v(\mathbf{G})</math> is the Coulomb kernel and <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> is the interacting response function in reciprocal space. This approach gives access to the full matrix <math>\chi_{\mathbf{G},\mathbf{G}'}(\mathbf{q},\omega)</math> and is used, for instance, to compute the screened Coulomb interaction <math>W</math> needed in ''GW'' calculations. However, the Dyson equation must be inverted at every frequency point, which makes it expensive for computing spectra with fine resolution. The only nonlocal exchange-correlation kernel currently supported is the nanoquanta kernel ({{TAG\|LFXC}}).

	=== Scaling ===		=== Scaling with the system size===

	The compute time scales as <math>N^4</math> with the system size. The Dyson equation must be inverted at every frequency point ({{TAG\|NOMEGA}}), so the total cost grows linearly with the number of frequency points.		The compute time scales as <math>N^4</math> with the system size. The Dyson equation must be inverted at every frequency point ({{TAG\|NOMEGA}}), so the total cost grows linearly with the number of frequency points.

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

	~~=== Lectures ===~~
	* Lecture on {{Video\|bse:alexey:2026\|TDDFT theory and calculations}}.

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