Time-evolution algorithm - Revision history

Pmelo at 15:52, 18 March 2026

2026-03-18T15:52:24Z

← Older revision		Revision as of 15:52, 18 March 2026
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	The [[Dielectric_properties\|macroscopic dielectric function]], <math>\epsilon_{ij}(\omega)</math>, measures how a given dielectric medium reacts when subject to an external electric field. From <math>\epsilon_{ij}(\omega)</math> it is possible to extract several optical properties such as absorption, optical conductivity, and reflectance. However, it is important that the interacting electrons and holes are taken into account. This makes the evaluation of the macroscopic dielectric function more involved, since it goes beyond the single-particle level, working at the two-particle level via either the [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] (BSE) or ~~[[Time-dependent density-functional theory calculations\|~~time-dependent density-functional theory]] (TDDFT).		The [[Dielectric_properties\|macroscopic dielectric function]], <math>\epsilon_{ij}(\omega)</math>, measures how a given dielectric medium reacts when subject to an external electric field. From <math>\epsilon_{ij}(\omega)</math> it is possible to extract several optical properties such as absorption, optical conductivity, and reflectance. However, it is important that the interacting electrons and holes are taken into account. This makes the evaluation of the macroscopic dielectric function more involved, since it goes beyond the single-particle level, working at the two-particle level via either the [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] (BSE) or time-dependent density-functional theory (TDDFT).

	For both frameworks, BSE and TDDFT, users can select two different strategies to compute <math>\epsilon_{ij}(\omega)</math>. The first is based on the eigendecomposition of the electron-hole hamiltonian, <math>H^\mathrm{exc}</math>. It allows for the evaluation of <math>\epsilon_{ij}(\omega)</math> by initially obtaining the eigenvalues and eigenvectors of <math>H^\mathrm{exc}</math> and then using both to evaluate <math>\epsilon_{ij}(\omega)</math>. This strategy is based on the [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] or the [[Time-dependent density-functional theory calculations\|Casida equation]]. The second strategy transforms the mathematical expression of <math>\epsilon_{ij}(\omega)</math> into a time-dependent integral. By propagating the dipolar moments in time and then applying a Fourier transform, VASP can compute <math>\epsilon_{ij}(\omega)</math>.		For both frameworks, BSE and TDDFT, users can select two different strategies to compute <math>\epsilon_{ij}(\omega)</math>. The first is based on the eigendecomposition of the electron-hole hamiltonian, <math>H^\mathrm{exc}</math>{{cite\|sander:prb:15}}. It allows for the evaluation of <math>\epsilon_{ij}(\omega)</math> by initially obtaining the eigenvalues and eigenvectors of <math>H^\mathrm{exc}</math> and then using both to evaluate <math>\epsilon_{ij}(\omega)</math>. This strategy is based on the [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] or the [[Time-dependent density-functional theory calculations\|Casida equation]]. The second strategy transforms the mathematical expression of <math>\epsilon_{ij}(\omega)</math> into a time-dependent integral{{cite\|sander:jcp:2017}}. By propagating the dipolar moments in time and then applying a Fourier transform, VASP can compute <math>\epsilon_{ij}(\omega)</math>.

	The advantage of the later method in relation to the former is related to their cost. The time-dependent integral has a cost of the order <math>O(N^2)</math>, while the eigendecomposition has a cost of the order <math>O(N^3)</math>, where <math>N</math> is the rank of <math>H^\mathrm{exc}</math>. This means that for very large numbers of bands or k-points, the time-dependent formalism is cheaper than the eigendecomposition method.		The advantage of the later method in relation to the former is related to their cost. The time-dependent integral has a cost of the order <math>O(N^2)</math>, while the eigendecomposition has a cost of the order <math>O(N^3)</math>, where <math>N</math> is the rank of <math>H^\mathrm{exc}</math>. This means that for very large numbers of bands or k-points, the time-dependent formalism is cheaper than the eigendecomposition method.

	Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.		Below is a brief description of the time-dependent method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.
	{{NB\|warning\|In VASP < 6.6.0 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>0.}}		{{NB\|warning\|In VASP < 6.6.0 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>0.}}
			==The TDDFT Schrödinger equation==
			Much like in density-functional theory, there is an equivalent time-dependent equivalent called time-dependent density functional theory where each electron follows a time-dependent 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}[n(t)]\right]\left\|\phi_i[n(t)]\rangle\right.,
			</math>
			but now all quantities are functionally dependent on a time-evolving density, <math>n(t)</math>. Through the use of linear-response formalism this equation can be brought into a matrix equation. However, this involves the computation and storage of large matrices. To circumvent this issue it is possible to evolve the starting ground state under a perturbative potential that probes all possible excitations (see more below).

			Taking the following ansatz for the time-dependent wave-function,
			::<math>
			\left\|\phi_i(t)\right\rangle=\left\{\left\|\phi_i^0\right\rangle+\lambda\sum_{a \in \mathrm{unocc} .} c_{a i}(t)\left\|\phi_a^0\right\rangle\right\} e^{-i \varepsilon_i t},
			</math>
			one can compute the changes in an initial, occupied state <math>\|\phi_i^0\rangle</math> as time-dependent contributions from unoccupied states. In this way the original problem is recast as the propagation in time of the coefficients <math>c_{a i}(t)</math>, with the starting point being <math>\left\|\phi_i(0)\right\rangle = \left\|\phi_i^0)\right\rangle</math>.

			In essence, the time-dependent algorithm implemented in VASP works as follows:
			#Setup states at time step <math>t_n</math>
			#Update time-dependent hamiltonian <math>H[\left\|\phi_i(t)\right\rangle]</math>
			#Calculate the change in time-dependent coefficients <math>\delta c_{a i}\left(t_n\right) = \left\langle\phi_a^0\right\| H\left\{\phi_i\left(t_n\right)\right\}\left\|\phi_i\left(t_n\right)\right\rangle</math>
			#Compute the coefficients at the next time-step <math>c_{a i}\left(t_{n+1}\right)=c_{a i}\left(t_{n-1}\right)+2 i \Delta t \delta c_{a i}\left(t_n\right)</math>,
			where <math>\Delta t</math> is the time step chosen for the propagation. Updating the hamiltonian with the new states is an essential step, as it is now a functional of the time-dependent density.

			The remaining step is to prove that the computation of the dielectric function can be recast as a time-dependent problem. This is shown in the next section.
	==The macroscopic-dielectric function as a time-dependent integral==		==The macroscopic-dielectric function as a time-dependent integral==
	The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by		The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by

Csheldon at 08:59, 29 January 2026

2026-01-29T08:59:06Z

← Older revision		Revision as of 08:59, 29 January 2026
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	Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.		Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.
	{{NB\|warning\|In VASP < 6.5.2 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>0.}}		{{NB\|warning\|In VASP < 6.6.0 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>0.}}
	==The macroscopic-dielectric function as a time-dependent integral==		==The macroscopic-dielectric function as a time-dependent integral==
	The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by		The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by

Tal at 15:07, 9 April 2025

2025-04-09T15:07:41Z

← Older revision		Revision as of 15:07, 9 April 2025
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	Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.		Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.
	{{NB\|warning\|In VASP < 6.5.2 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>-1.}}		{{NB\|warning\|In VASP < 6.5.2 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>0.}}
	==The macroscopic-dielectric function as a time-dependent integral==		==The macroscopic-dielectric function as a time-dependent integral==
	The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by		The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by

Tal at 15:04, 9 April 2025

2025-04-09T15:04:46Z

← Older revision		Revision as of 15:04, 9 April 2025
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	Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.		Below is a brief description of the method, from its theoretical support to how calculations should be performed, with the relevant approximations needed in the two-particle Hamiltonian.
			{{NB\|warning\|In VASP < 6.5.2 the dielectric function is not calculated correctly with {{TAG\|ALGO}}{{=}}TIMEEV if {{TAG\|ISYM}}>-1.}}
	==The macroscopic-dielectric function as a time-dependent integral==		==The macroscopic-dielectric function as a time-dependent integral==
	The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by		The starting point is that one can re-write <math>\epsilon_{ij}(\omega)</math> as a time-dependent integral{{cite\|schmidt:prb:2003}}. It starts from its expression, given by

Huebsch at 12:31, 7 February 2025

2025-02-07T12:31:28Z

← Older revision		Revision as of 12:31, 7 February 2025
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	==References==		==References==

	~~<!--~~[[Category:Many-body perturbation theory]][[Category:Linear response]][[Category:Bethe-Salpeter equations]][[Category:Howto]]~~-->~~		[[Category:Many-body perturbation theory]][[Category:Linear response]][[Category:Bethe-Salpeter equations]][[Category:Howto]]

Huebsch: Huebsch moved page Construction:Time-evolution to Time-evolution algorithm without leaving a redirect

2025-02-07T12:30:50Z

Huebsch moved page Construction:Time-evolution to Time-evolution algorithm without leaving a redirect

← Older revision	Revision as of 12:30, 7 February 2025
(No difference)

Huebsch: /* Related tags and articles */

2025-02-05T14:30:14Z

Pmelo: /* Casida equation */

2024-10-18T09:32:16Z

Casida equation

← Older revision		Revision as of 09:32, 18 October 2024
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	Here the full [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] is employed by setting {{TAG\|ALGO}}=BSE. The interaction hamiltonian is built using the dielectric function from RPA, and has the right behaviour in the long range regime. This means that it can accurately describe bound excitons in solids and large molecules. However, it is more costly than time-evolution, scaling with <math>N_\mathrm{rank}^3</math>.		Here the full [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] is employed by setting {{TAG\|ALGO}}=BSE. The interaction hamiltonian is built using the dielectric function from RPA, and has the right behaviour in the long range regime. This means that it can accurately describe bound excitons in solids and large molecules. However, it is more costly than time-evolution, scaling with <math>N_\mathrm{rank}^3</math>.
	===Casida equation===		===Casida equation===
	Similar to the Bethe-Salpeter equation, the [[Time-dependent density-functional theory calculations\|Casida equation]] employs an eigensolver method to compute the dielectric function. This is chosen in the {{TAG\|INCAR}} with {{TAG\|ALGO}}=TDHF. The key ~~differences are~~ that the Casida method does not require a preceding GW run to compute the RPA screening and can be performed with either DFT or hybrid-functional orbitals and energies.		Similar to the Bethe-Salpeter equation, the [[Time-dependent density-functional theory calculations\|Casida equation]] employs an eigensolver method to compute the dielectric function. This is chosen in the {{TAG\|INCAR}} with {{TAG\|ALGO}}=TDHF. The key difference is that the Casida method does not require a preceding GW run to compute the RPA screening and can be performed with either DFT or hybrid-functional orbitals and energies.

	==Related tags and articles==		==Related tags and articles==

Pmelo: /* Comparison to other methods */

2024-10-18T09:31:52Z

Comparison to other methods

← Older revision		Revision as of 09:31, 18 October 2024
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	==Comparison to other methods==		==Comparison to other methods==
	VASP offers two other methods with which you can compute the macroscopic dielectric function. These are based on ~~eigen-decomposition~~ of the two particle hamiltonian, <math>H^\mathrm{exc}</math>. While more expensive than time-evolution, both these methods are able to compute eigenvalues and eigenstates of <math>H^\mathrm{exc}</math>, ~~meaning that you can have~~ direct access to the excitation energies of a system.		VASP offers two other methods with which you can compute the macroscopic dielectric function. These are based on eigendecomposition of the two particle hamiltonian, <math>H^\mathrm{exc}</math>. While more expensive than time-evolution, both these methods are able to compute eigenvalues and eigenstates of <math>H^\mathrm{exc}</math>, thus providing direct access to the excitation energies of a system.
	===Bethe-Salpeter equation===		===Bethe-Salpeter equation===
	Here the full [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] is employed by setting {{TAG\|ALGO}}=BSE. The interaction hamiltonian is built using the dielectric function from RPA, and has the right behaviour in the long range regime. This means that it can accurately describe bound excitons in solids and large molecules. However, it is more costly than time-evolution, scaling with <math>N_\mathrm{rank}^3</math>.		Here the full [[Bethe-Salpeter equations\|Bethe-Salpeter equation]] is employed by setting {{TAG\|ALGO}}=BSE. The interaction hamiltonian is built using the dielectric function from RPA, and has the right behaviour in the long range regime. This means that it can accurately describe bound excitons in solids and large molecules. However, it is more costly than time-evolution, scaling with <math>N_\mathrm{rank}^3</math>.

Pmelo: /* Choosing the direction of perturbation */

2024-10-18T09:25:53Z

Choosing the direction of perturbation

← Older revision		Revision as of 09:25, 18 October 2024
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	The dipole momentum vector <math>\mu^i</math> is direction dependent. The direction of propagation is chosen in the {{TAG\|INCAR}} with the tag {{TAG\|IEPSILON}}, which can take values of 1, 2, or 3 (corresponding to x, y, and z direction, respectively), and 4 (corresponding to all directions).		The dipole momentum vector <math>\mu^i</math> is direction dependent. The direction of propagation is chosen in the {{TAG\|INCAR}} with the tag {{TAG\|IEPSILON}}, which can take values of 1, 2, or 3 (corresponding to x, y, and z direction, respectively), and 4 (corresponding to all directions).

	While choosing a single direction of propagation decreases the computing time, it is important to pay attention to the symmetries of the material in study. For example, in the case of bulk silicon, since the material has cubic symmetry propagating along one direction (x, or y, or z) is enough. However, for a material like hexagonal boron nitride, the crystal symmetries destroy the equivalency between the x and y directions. For this system propagation should happen along both x and y, and then the dielectric function should be the average of both calculations.		While choosing a single direction of propagation decreases the computing time, it is important to pay attention to the symmetries of the material in study. For example, in the case of bulk silicon, since the material has cubic symmetry propagating along one direction (x, or y, or z) is enough. However, for a material like monolayer hexagonal boron nitride, the crystal symmetries destroy the equivalency between the x and y directions. For this system propagation should happen along both x and y, and then the dielectric function should be the average of both calculations.

	===Analysing the results===		===Analysing the results===

@@ Line 186: / Line 186: @@
 ==Related tags and articles==
-{{TAG|NBANDSO}}
+{{TAG|NBANDSO}}, {{TAG|NBANDSV}}, {{TAG|IEPSILON}}, {{TAG|NELM}}, {{TAG|LHARTREE}}, {{TAG|LADDER}}, {{TAG|LFXC}}, {{TAG|LHFCALC}}, {{TAG|LMODELHF}}, {{TAG|AEXX}}, {{TAG|HFSCREEN}}
 [[Time-dependent density-functional theory calculations]]