Time-evolution algorithm - Revision history

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
Line 6:		Line 6:

	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===

Pmelo: /* Setting up the time-step */

2024-10-18T09:24:16Z

Setting up the time-step

← Older revision		Revision as of 09:24, 18 October 2024
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	====Setting up the time-step====		====Setting up the time-step====
	VASP is now integrating a time-dependent differential equation so the time-step used to propagate the dipole moments can be specified in the {{TAG\|INCAR}}. By default, VASP will use 20000 steps, however a different number can be set with the tag {{TAG\|~~NELMGW~~}}. Nevertheless, {{TAG\|~~NELMGW~~}} must be larger than 100, otherwise VASP will revert to the default.		VASP is now integrating a time-dependent differential equation so the time-step used to propagate the dipole moments can be specified in the {{TAG\|INCAR}}. By default, VASP will use 20000 steps, however a different number can be set with the tag {{TAG\|NELM}}. Nevertheless, {{TAG\|NELM}} must be larger than 100, otherwise VASP will revert to the default value.

	The time-step, <math>\Delta t</math>, and maximum propagation time, <math>T_\mathrm{max}</math>, are not dependent on the size of the interacting hamiltonian matrix. However they are dependent on the system in case and the input tag {{TAG\|CSHIFT}} and {{TAG\|OMEGAMAX}}. This comes from the Fourier transform used to integrate the time-dependent dipole moments, which leads to <math>T_\mathrm{max} \approx 1/\mathrm{CSHIFT}</math> and <math>\Delta t \approx 1/\mathrm{OMEGAMAX}</math>.		The time-step, <math>\Delta t</math>, and maximum propagation time, <math>T_\mathrm{max}</math>, are not dependent on the size of the interacting hamiltonian matrix. However they are dependent on the system in case and the input tag {{TAG\|CSHIFT}} and {{TAG\|OMEGAMAX}}. This comes from the Fourier transform used to integrate the time-dependent dipole moments, which leads to <math>T_\mathrm{max} \approx 1/\mathrm{CSHIFT}</math> and <math>\Delta t \approx 1/\mathrm{OMEGAMAX}</math>.

Pmelo: /* Exchange-correlation effects from time-dependent density functional theory */

2024-10-18T08:55:15Z

Exchange-correlation effects from time-dependent density functional theory

← Older revision		Revision as of 08:55, 18 October 2024
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	where <math>c_X</math> controls the fraction of the exchange energy functional that is included in the kernel (see {{TAG\|AEXX}}). This lets users perform time-dependent calculations using hybrid functionals.		where <math>c_X</math> controls the fraction of the exchange energy functional that is included in the kernel (see {{TAG\|AEXX}}). This lets users perform time-dependent calculations using hybrid functionals.

	~~As these~~ kernels often lack the long-range component (which goes as <math>-1/q^2</math>, where <math>q</math> is the momentum difference between the electron and the hole), they fail to properly reproduce the binding energies of electron-hole pairs ~~in periodic systems or very large molecules~~.		These kernels often lack the long-range component (which goes as <math>-1/q^2</math>, where <math>q</math> is the momentum difference between the electron and the hole). When using them in periodic or extended systems it is very likely that they will fail to properly reproduce the binding energies of electron-hole pairs.

	====Ladder diagrams from many-body perturbation theory====		====Ladder diagrams from many-body perturbation theory====

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 ==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]]