Category:Dielectric properties - Revision history

Pmelo: /* Electron energy loss spectroscopy (EELS) */

2025-03-03T12:44:54Z

Electron energy loss spectroscopy (EELS)

← Older revision		Revision as of 12:44, 3 March 2025
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	==Other dielectric properties==		==Other dielectric properties==
	===Electron energy loss spectroscopy (EELS)===		===Electron energy loss spectroscopy (EELS)===
	In EELS experiments a narrow beam of electrons with a well defined energy is shot at the sample. These electrons then lose energy to the sample by exciting plasmons, electron-hole pairs, or other higher-order quasiparticles. The loss function can then be expressed as		In [[Electron-energy-loss spectrum\|EELS]] experiments a narrow beam of electrons with a well defined energy is shot at the sample. These electrons then lose energy to the sample by exciting plasmons, electron-hole pairs, or other higher-order quasiparticles. The loss function can then be expressed as

	::<math>		::<math>

Schlipf: /* Dielectric function */

2024-12-19T13:23:41Z

Dielectric function

← Older revision		Revision as of 13:23, 19 December 2024
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	==Dielectric function==		==Dielectric function==
	[[File:exciton.png\|thumbnail\|400px\|When light excites an electron from the valence to the conduction band~~. The~~ created hole can bind with the electron to form an exciton.]]		[[File:exciton.png\|thumbnail\|400px\|When light excites an electron from the valence to the conduction band, the created hole can bind with the electron to form an exciton.]]

	When an external electric field <math>\mathbf E</math> acts on a medium, both the electronic and ionic charges will react to the perturbing field. For dielectric materials, in a very simplistic approach, one can think that the bound charges will create dipoles inside the medium leading to an induced polarization, <math>\mathbf P</math>. The combined effects of both fields are expressed in the electric displacement field <math>\mathbf D</math>, given by		When an external electric field <math>\mathbf E</math> acts on a medium, both the electronic and ionic charges will react to the perturbing field. For dielectric materials, in a very simplistic approach, one can think that the bound charges will create dipoles inside the medium leading to an induced polarization, <math>\mathbf P</math>. The combined effects of both fields are expressed in the electric displacement field <math>\mathbf D</math>, given by

Schlipf: /* Dielectric function */

2024-12-19T13:22:44Z

Dielectric function

← Older revision		Revision as of 13:22, 19 December 2024
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	==Dielectric function==		==Dielectric function==
	[[File:exciton.png\|~~300px~~\|~~right~~\|When light excites an electron from the valence to the conduction band. The created hole can bind with the electron to form an exciton.]]		[[File:exciton.png\|thumbnail\|400px\|When light excites an electron from the valence to the conduction band. The created hole can bind with the electron to form an exciton.]]

	When an external electric field <math>\mathbf E</math> acts on a medium, both the electronic and ionic charges will react to the perturbing field. For dielectric materials, in a very simplistic approach, one can think that the bound charges will create dipoles inside the medium leading to an induced polarization, <math>\mathbf P</math>. The combined effects of both fields are expressed in the electric displacement field <math>\mathbf D</math>, given by		When an external electric field <math>\mathbf E</math> acts on a medium, both the electronic and ionic charges will react to the perturbing field. For dielectric materials, in a very simplistic approach, one can think that the bound charges will create dipoles inside the medium leading to an induced polarization, <math>\mathbf P</math>. The combined effects of both fields are expressed in the electric displacement field <math>\mathbf D</math>, given by

Schlipf at 13:17, 19 December 2024

2024-12-19T13:17:45Z

← Older revision		Revision as of 13:17, 19 December 2024
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	==Dielectric function==		==Dielectric function==
			[[File:exciton.png\|300px\|right\|When light excites an electron from the valence to the conduction band. The created hole can bind with the electron to form an exciton.]]

	When an external electric field <math>\mathbf E</math> acts on a medium, both the electronic and ionic charges will react to the perturbing field. For dielectric materials, in a very simplistic approach, one can think that the bound charges will create dipoles inside the medium leading to an induced polarization, <math>\mathbf P</math>. The combined effects of both fields are expressed in the electric displacement field <math>\mathbf D</math>, given by		When an external electric field <math>\mathbf E</math> acts on a medium, both the electronic and ionic charges will react to the perturbing field. For dielectric materials, in a very simplistic approach, one can think that the bound charges will create dipoles inside the medium leading to an induced polarization, <math>\mathbf P</math>. The combined effects of both fields are expressed in the electric displacement field <math>\mathbf D</math>, given by

Huebsch: /* Finite momentum dielectric function */

2024-10-08T12:50:01Z

Finite momentum dielectric function

← Older revision		Revision as of 12:50, 8 October 2024
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	::<math>		::<math>
	\lim_{\mathbf q\to0}\frac{\langle c\mathbf k + \mathbf q\|e^{\mathrm i\mathbf q\cdot\mathbf r}\|v\mathbf k\rangle}{q} \approx \lim_{\mathbf q\to0}\frac{\langle c\mathbf k+\mathbf q\|1 + \mathrm i\mathbf q\cdot\mathbf r\|v\mathbf k\rangle}{q} = \hat\mathbf q\cdot \langle c\mathbf k+\mathbf q\|\mathbf r\|v\mathbf k\rangle.		\lim_{\mathbf q\to0}\frac{\langle c\mathbf k + \mathbf q\|e^{\mathrm i\mathbf q\cdot\mathbf r}\|v\mathbf k\rangle}{q} \approx \lim_{\mathbf q\to0}\frac{\langle c\mathbf k+\mathbf q\|1 + \mathrm i\mathbf q\cdot\mathbf r\|v\mathbf k\rangle}{q} = \hat{\mathbf{q}}\cdot \langle c\mathbf k+\mathbf q\|\mathbf r\|v\mathbf k\rangle.
	</math>		</math>

	VASP can also analyze the effects of finite momentum excitons. This is important, e.g., in the case of the optical absorption of bulk hexagonal BN. To calculate the absorption spectrum at finite momentum, set {{TAG\|KPOINT_BSE}} to the index of the desired '''q''' point.		VASP can also analyze the effects of finite momentum excitons. This is important, e.g., in the case of the optical absorption of bulk hexagonal BN. To calculate the absorption spectrum at finite momentum, set {{TAG\|KPOINT_BSE}} to the index of the desired '''q''' point.

	===Local fields in the Hamiltonian===		===Local fields in the Hamiltonian===

Pmelo: /* Other dielectric properties */

2024-07-17T15:43:35Z

Other dielectric properties

← Older revision		Revision as of 15:43, 17 July 2024
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	==Other dielectric properties==		==Other dielectric properties==
			===Electron energy loss spectroscopy (EELS)===
			In EELS experiments a narrow beam of electrons with a well defined energy is shot at the sample. These electrons then lose energy to the sample by exciting plasmons, electron-hole pairs, or other higher-order quasiparticles. The loss function can then be expressed as

			::<math>
			\mathrm{EELS} = -\mathrm{Im}\left[\epsilon^{-1}(\omega)\right].
			</math>

	===Optical conductivity===		===Optical conductivity===
	From Maxwell's equations and the microscopic form of Ohm's law it is possible to arrive at the following relation between the tensorial dielectric function and the optical conductivity <math>\sigma(\omega)</math>		From Maxwell's equations and the microscopic form of Ohm's law it is possible to arrive at the following relation between the tensorial dielectric function and the optical conductivity <math>\sigma(\omega)</math>

Huebsch: /* Dynamical response: Green-Kubo and many-body perturbation theory */

2024-06-19T08:54:07Z

Dynamical response: Green-Kubo and many-body perturbation theory

Huebsch: /* Static response */

2024-06-19T08:52:16Z

Static response

Huebsch at 05:37, 14 June 2024

2024-06-14T05:37:49Z

← Older revision		Revision as of 05:37, 14 June 2024
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	</math>		</math>

	to compute the dielectric function. Here, <math>v</math> is the bare Coulomb potential describing electron-electron interaction. This requires increasing {{TAG\|NBANDS}} to include unoccupied states as generally the case for [[GW_and_dielectric_matrix\|GW calculations]]~~. Effects coming from local fields in the random-phase approximation (RPA) can be activated by setting {{TAG\|LRPA}}=.True.~~.		to compute the dielectric function. Here, <math>v</math> is the bare Coulomb potential describing electron-electron interaction. This requires increasing {{TAG\|NBANDS}} to include unoccupied states as generally the case for [[GW_and_dielectric_matrix\|GW calculations]].

	Two methods for computing the polarizability are available: For {{TAG\|LSPECTRAL}}=.True., VASP will avoid direct computation of <math>\chi</math> and use a fast matrix-vector product. However, this can introduce spurious peaks at low frequencies for some values of {{TAG\|CSHIFT}} and {{TAG\|NOMEGA}}. The second method computes <math>\chi</math> directly and is activated by setting {{TAG\|LSPECTRAL}}=.False.. However, it is much slower than the former method.		Two methods for computing the polarizability are available: For {{TAG\|LSPECTRAL}}=.True., VASP will avoid direct computation of <math>\chi</math> and use a fast matrix-vector product. However, this can introduce spurious peaks at low frequencies for some values of {{TAG\|CSHIFT}} and {{TAG\|NOMEGA}}. The second method computes <math>\chi</math> directly and is activated by setting {{TAG\|LSPECTRAL}}=.False.. However, it is much slower than the former method.

Pmelo: /* Low-frequency corrections from atomic displacements */

2024-05-24T14:36:14Z

Low-frequency corrections from atomic displacements

← Older revision		Revision as of 14:36, 24 May 2024
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	The corrections from ionic motion to the low frequency regime can be added to <math>\epsilon_{\alpha\beta}^\infty</math> following{{cite\|gonze:prb:1997}}		The corrections from ionic motion to the low frequency regime can be added to <math>\epsilon_{\alpha\beta}^\infty</math> following{{cite\|gonze:prb:1997}}
	::<math>		::<math>
	\epsilon_{\alpha\beta}(\omega) = \epsilon_{\alpha\beta}^\infty + \frac{4\pi}{\Omega_0}\sum_\nu\frac{S_{~~\nu,~~\alpha\beta}}{\omega_\nu^2 - (\omega+\mathrm i\eta)^2},		\epsilon_{\alpha\beta}(\omega) = \epsilon_{\alpha\beta}^\infty + \frac{4\pi e^2}{\Omega_0}\sum_\nu\frac{S_{\alpha\beta,\nu}}{\omega_\nu^2 - (\omega+\mathrm i\eta)^2},
	</math>		</math>
	where <math>\omega_\nu</math> is the phonon frequency of mode <math>\nu</math>, and <math>S_{~~\nu,~~\alpha\beta}</math> is the mode-oscillator strength, defined by		where <math>\omega_\nu</math> is the phonon frequency of mode <math>\nu</math>, and <math>S_{\alpha\beta,\nu}</math> is the mode-oscillator strength, defined by
	::<math>		::<math>
	S_{m,\alpha\beta} = \left(\sum_{I,\delta}Z^_{I,\alpha\delta}U^_{\nu\mathbf{q = 0}~~}(I\delta~~)\right)\left(\sum_{J,\delta'}Z^*_{J,\beta\delta'}U_{\nu\mathbf{q = 0}~~}(J\delta'~~)\right).		S_{\alpha\beta,\nu} = \left(\sum_{I,\delta}Z^_{I\alpha\delta}\varepsilon^_{I\delta,\nu}(\mathbf{q = 0})\right)\left(\sum_{J,\delta'}Z^*_{J\beta\delta'}\varepsilon_{J\delta',\nu}(\mathbf{q = 0})\right).
	</math>		</math>
	Here <math>Z^*_{J,\beta\delta'}</math> are the Born effective charges and <math>U_{\nu\mathbf{q = 0}~~}(J\delta'~~)</math> are the eigendisplacements associated with the vibration mode <math>\nu</math> for atom <math>J</math> along direction <math>\delta'</math>. More information on the theory and methods behind the computation of phonon frequencies and eigendisplacements can be found in the [[Phonons:_Theory\|phonons]] dedicated page.		Here <math>Z^*_{J\beta\delta'}</math> are the Born effective charges and <math>\varepsilon_{J\delta',\nu}(\mathbf{q = 0})</math> are the eigendisplacements associated with the vibration mode <math>\nu</math> for atom <math>J</math> along direction <math>\delta'</math>. More information on the theory and methods behind the computation of phonon frequencies and eigendisplacements can be found in the [[Phonons:_Theory\|phonons]] dedicated page.

	Inclusion of the low-frequency corrections can be activated in the {{TAG\|INCAR}} file with either {{TAG\|IBRION}} = 5,6 (DFPT) or 7,8 (finite differences), and by setting to .True. the variables {{TAG\|LEPSILON}} or {{TAG\|LCALCEPS}}.		Inclusion of the low-frequency corrections can be activated in the {{TAG\|INCAR}} file with either {{TAG\|IBRION}} = 5,6 (DFPT) or 7,8 (finite differences), and by setting to .True. the variables {{TAG\|LEPSILON}} or {{TAG\|LCALCEPS}}.

← Older revision		Revision as of 08:54, 19 June 2024
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	===Dynamical response: Green-Kubo and many-body perturbation theory===		===Dynamical response: Green-Kubo and many-body perturbation theory===
	===={{TAG\|LOPTICS}}: Green-Kubo formula====		===={{TAG\|LOPTICS}}: Green-Kubo formula====
	{{TAG\|LOPTICS}} allows for the evaluation of the frequency-dependent dielectric function once the ground state is computed. It uses the explicit expression to evaluate the imaginary part of <math>\epsilon</math>:		:{{TAG\|LOPTICS}} allows for the evaluation of the frequency-dependent dielectric function once the ground state is computed. It uses the explicit expression to evaluate the imaginary part of <math>\epsilon</math>:
	:<math>		::<math>
	\epsilon^{(2)}_{\alpha \beta}\left(\omega\right) = \frac{4\pi^2 e^2}{\Omega}		\epsilon^{(2)}_{\alpha \beta}\left(\omega\right) = \frac{4\pi^2 e^2}{\Omega}
	\mathrm{lim}_{q \rightarrow 0} \frac{1}{q^2} \sum_{c,v,\mathbf{k}} 2 w_\mathbf{k} \delta( \epsilon_{c\mathbf{k}} - \epsilon_{v\mathbf{k}} - \omega)		\mathrm{lim}_{q \rightarrow 0} \frac{1}{q^2} \sum_{c,v,\mathbf{k}} 2 w_\mathbf{k} \delta( \epsilon_{c\mathbf{k}} - \epsilon_{v\mathbf{k}} - \omega)
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	\langle u_{v\mathbf{k}} \| u_{c\mathbf{k}+\mathbf{e}_\beta q} \rangle,		\langle u_{v\mathbf{k}} \| u_{c\mathbf{k}+\mathbf{e}_\beta q} \rangle,
	</math>		</math>
	while the real part is evaluated using the Kramers-Kroing relation. At this level, there are no effects coming from local fields.		:while the real part is evaluated using the Kramers-Kroing relation. At this level, there are no effects coming from local fields.

	This method requires two steps: First, obtain the electronic ground state. Secondly, increase the value of {{TAG\|NBANDS}} in the {{TAG\|INCAR}} file to include unoccupied states. Always check for convergence w.r.t. the number of unoccupied states.		:This method requires two steps: First, obtain the electronic ground state. Secondly, increase the value of {{TAG\|NBANDS}} in the {{TAG\|INCAR}} file to include unoccupied states. Always check for convergence w.r.t. the number of unoccupied states.

	Furthermore, the {{TAG\|INCAR}} should also include values for {{TAG\|CSHIFT}} (the broadening applied to the Lorentzian function which replaces the <math>\delta</math>-function), and {{TAG\|NEDOS}} (the frequency grid for <math>\omega</math>).		:Furthermore, the {{TAG\|INCAR}} should also include values for {{TAG\|CSHIFT}} (the broadening applied to the Lorentzian function which replaces the <math>\delta</math>-function), and {{TAG\|NEDOS}} (the frequency grid for <math>\omega</math>).

	===={{TAG\|ALGO}} = TDHF: Casida equation====		===={{TAG\|ALGO}} = TDHF: Casida equation====
	This option performs a [[Bethe-Salpeter-equations calculations#Time-dependent Hartree-Fock calculation\|time-dependent Hartree-Fock]] or [[Bethe-Salpeter-equations calculations#Time-dependent DFT calculation\|time-dependent density-functional-theory (TDDFT)]] calculation. It follows the Casida equation and uses a Fourier transform of the time-evolving dipoles to compute <math>\epsilon</math>.		:This option performs a [[Bethe-Salpeter-equations calculations#Time-dependent Hartree-Fock calculation\|time-dependent Hartree-Fock]] or [[Bethe-Salpeter-equations calculations#Time-dependent DFT calculation\|time-dependent density-functional-theory (TDDFT)]] calculation. It follows the Casida equation and uses a Fourier transform of the time-evolving dipoles to compute <math>\epsilon</math>.

	The number of {{TAG\|NBANDS}} controls how many bands are present in the time evolution. This can be fewer empty states compared to {{TAG\|LOPTICS}}.		:The number of {{TAG\|NBANDS}} controls how many bands are present in the time evolution. This can be fewer empty states compared to {{TAG\|LOPTICS}}.

	The choice of time-dependent kernel is controlled by {{TAG\|AEXX}}, {{TAG\|HFSCREEN}}, and {{TAG\|LFXC}} tags. For calculations using hybrid functionals, {{TAG\|AEXX}} controls the fraction of exact exchange used in the exchange-correlation potential, while {{TAG\|HFSCREEN}} specifies the range-separation parameter. For a pure TDDFT calculation, {{TAG\|LFXC}} uses the local exchange-correlation kernel in the time-evolution equations.		:The choice of time-dependent kernel is controlled by {{TAG\|AEXX}}, {{TAG\|HFSCREEN}}, and {{TAG\|LFXC}} tags. For calculations using hybrid functionals, {{TAG\|AEXX}} controls the fraction of exact exchange used in the exchange-correlation potential, while {{TAG\|HFSCREEN}} specifies the range-separation parameter. For a pure TDDFT calculation, {{TAG\|LFXC}} uses the local exchange-correlation kernel in the time-evolution equations.

	===={{TAG\|ALGO}} = TIMEEV: delta-pulse electric field====		===={{TAG\|ALGO}} = TIMEEV: delta-pulse electric field====
	Uses a delta-pulse electric field to probe all transitions and calculate the dielectric function by following the [[Time_Evolution\|evolution in time of the dipole momenta]]. This algorithm is able to fully reproduce the absorption spectra from standard [[Bethe-Salpeter equations\|Bethe-Salpeter calculations]] by setting the correct time-dependent kernel with {{TAG\|LHARTREE}}=.True. and {{TAG\|LFXC}}=.True.		:Uses a delta-pulse electric field to probe all transitions and calculate the dielectric function by following the [[Time_Evolution\|evolution in time of the dipole momenta]]. This algorithm is able to fully reproduce the absorption spectra from standard [[Bethe-Salpeter equations\|Bethe-Salpeter calculations]] by setting the correct time-dependent kernel with {{TAG\|LHARTREE}}=.True. and {{TAG\|LFXC}}=.True.

	The time step is controlled automatically by the {{TAG\|CSHIFT}} and {{TAG\|PREC}}. This means that the smaller the value of {{TAG\|CSHIFT}} and the more accurate the level of precision chosen by the user, the higher the number of time steps that VASP will perform, and the higher the cost of the calculation.		:The time step is controlled automatically by the {{TAG\|CSHIFT}} and {{TAG\|PREC}}. This means that the smaller the value of {{TAG\|CSHIFT}} and the more accurate the level of precision chosen by the user, the higher the number of time steps that VASP will perform, and the higher the cost of the calculation.

	The number of valence and conduction bands involved in the time propagation is set by the {{TAG\|NBANDSO}} and {{TAG\|NBANDSV}} tags, respectively. Choose a small number of bands near the band gap to reproduce optical measurements.		:The number of valence and conduction bands involved in the time propagation is set by the {{TAG\|NBANDSO}} and {{TAG\|NBANDSV}} tags, respectively. Choose a small number of bands near the band gap to reproduce optical measurements.

	Finally, the maximum energy used in both the Fourier transform and in calculating the frequency-dependent dielectric function is set by {{TAG\|OMEGAMAX}}, and the sampling of the frequency grid is controlled by {{TAG\|NEDOS}}.		:Finally, the maximum energy used in both the Fourier transform and in calculating the frequency-dependent dielectric function is set by {{TAG\|OMEGAMAX}}, and the sampling of the frequency grid is controlled by {{TAG\|NEDOS}}.

	===={{TAG\|ALGO}} = CHI: polarizability within RPA approximation====		===={{TAG\|ALGO}} = CHI: polarizability within RPA approximation====
	Here, the frequency dielectric function is computed within the Random-Phase approximation. VASP will compute the polarizability <math>\chi</math> by setting {{TAG\|ALGO}}=Chi in the {{TAG\|INCAR}} file and then use		:Here, the frequency dielectric function is computed within the Random-Phase approximation. VASP will compute the polarizability <math>\chi</math> by setting {{TAG\|ALGO}}=Chi in the {{TAG\|INCAR}} file and then use

	::<math>		::<math>
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	</math>		</math>

	to compute the dielectric function. Here, <math>v</math> is the bare Coulomb potential describing electron-electron interaction. This requires increasing {{TAG\|NBANDS}} to include unoccupied states as generally the case for [[GW_and_dielectric_matrix\|GW calculations]].		:to compute the dielectric function. Here, <math>v</math> is the bare Coulomb potential describing electron-electron interaction. This requires increasing {{TAG\|NBANDS}} to include unoccupied states as generally the case for [[GW_and_dielectric_matrix\|GW calculations]].

	Two methods for computing the polarizability are available: For {{TAG\|LSPECTRAL}}=.True., VASP will avoid direct computation of <math>\chi</math> and use a fast matrix-vector product. However, this can introduce spurious peaks at low frequencies for some values of {{TAG\|CSHIFT}} and {{TAG\|NOMEGA}}. The second method computes <math>\chi</math> directly and is activated by setting {{TAG\|LSPECTRAL}}=.False.. However, it is much slower than the former method.		:Two methods for computing the polarizability are available: For {{TAG\|LSPECTRAL}}=.True., VASP will avoid direct computation of <math>\chi</math> and use a fast matrix-vector product. However, this can introduce spurious peaks at low frequencies for some values of {{TAG\|CSHIFT}} and {{TAG\|NOMEGA}}. The second method computes <math>\chi</math> directly and is activated by setting {{TAG\|LSPECTRAL}}=.False.. However, it is much slower than the former method.

	===={{TAG\|ALGO}} = BSE: macroscopic dielectric function including excitons====		===={{TAG\|ALGO}} = BSE: macroscopic dielectric function including excitons====
	Setting {{TAG\|ALGO}}=BSE computes the macroscopic dielectric function <math>\epsilon_M</math> by solving the [[Bethe-Salpeter equations\|Bethe-Salpeter equations]]. The electron-hole pairs are treated as a new quasi-particle called exciton, and the dielectric function is built using the eigenvectors (<math>X_{\lambda}^{cv\mathbf k}</math>) and eigenvalues (<math>\omega_\lambda</math>):		:Setting {{TAG\|ALGO}}=BSE computes the macroscopic dielectric function <math>\epsilon_M</math> by solving the [[Bethe-Salpeter equations\|Bethe-Salpeter equations]]. The electron-hole pairs are treated as a new quasi-particle called exciton, and the dielectric function is built using the eigenvectors (<math>X_{\lambda}^{cv\mathbf k}</math>) and eigenvalues (<math>\omega_\lambda</math>):

	::<math>		::<math>
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	</math>		</math>

	Here, <math>S_{\lambda\lambda'}</math> is the overlap between exciton states of indices <math>\lambda</math> and <math>\lambda'</math> (in general, the BSE Hamiltonian is not hermitian, so eigenstates associated to different eigenvalues are not necessarily orthogonal).		:Here, <math>S_{\lambda\lambda'}</math> is the overlap between exciton states of indices <math>\lambda</math> and <math>\lambda'</math> (in general, the BSE Hamiltonian is not hermitian, so eigenstates associated to different eigenvalues are not necessarily orthogonal).

	The number of occupied and unoccupied states that are included in the BSE Hamiltonian is controlled by the {{TAG\|NBANDSO}} and {{TAG\|NBANDSV}}, respectively. Normally only a few bands above and below the band gap are required to converge the optical spectrum, and the memory requirements increase quickly with the number of bands. Thus, be careful in setting up these two tags.		:The number of occupied and unoccupied states that are included in the BSE Hamiltonian is controlled by the {{TAG\|NBANDSO}} and {{TAG\|NBANDSV}}, respectively. Normally only a few bands above and below the band gap are required to converge the optical spectrum, and the memory requirements increase quickly with the number of bands. Thus, be careful in setting up these two tags.

	Regarding the comparison with optical experiments, (e.g., absorption, MOKE, reflectance), <math>q</math> is the photon momentum. Often, the <math>\mathbf q \to 0</math> limit is considered. Furthermore, the coupling between the resonant and anti-resonant terms can be switched off, in what is called the [[Bethe-Salpeter equations#Theory#Tamm-Dancoff approximation\|Tamm-Dancoff approximation]]. This approximation can be activated with the variable {{TAG\|ANTIRES}} set to 0. Setting this variable to 1 or 2 will include the coupling, but increase the computational cost.		:Regarding the comparison with optical experiments, (e.g., absorption, MOKE, reflectance), <math>q</math> is the photon momentum. Often, the <math>\mathbf q \to 0</math> limit is considered. Furthermore, the coupling between the resonant and anti-resonant terms can be switched off, in what is called the [[Bethe-Salpeter equations#Theory#Tamm-Dancoff approximation\|Tamm-Dancoff approximation]]. This approximation can be activated with the variable {{TAG\|ANTIRES}} set to 0. Setting this variable to 1 or 2 will include the coupling, but increase the computational cost.

	==Level of approximation==		==Level of approximation==

← Older revision		Revision as of 08:52, 19 June 2024
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	==== {{TAG\|LEPSILON}}: density-functional-perturbation theory (DFPT) ====		==== {{TAG\|LEPSILON}}: density-functional-perturbation theory (DFPT) ====

	By setting {{TAG\|LEPSILON}}=.True., VASP uses DFPT to compute the static ion-clamped dielectric matrix with or without local field effects ({{TAG\|LRPA}}). Derivatives are evaluated using Sternheimer equations, avoiding the explicit computation of derivatives of the periodic part of the wave function. This method does not require the inclusion of empty states via {{TAG\|NBANDS}}.		:By setting {{TAG\|LEPSILON}}=.True., VASP uses DFPT to compute the static ion-clamped dielectric matrix with or without local field effects ({{TAG\|LRPA}}). Derivatives are evaluated using Sternheimer equations, avoiding the explicit computation of derivatives of the periodic part of the wave function. This method does not require the inclusion of empty states via {{TAG\|NBANDS}}.

	At the end of the calculation, both the values of <math>\epsilon</math> including ({{TAG\|LRPA}}=.True.) or excluding ({{TAG\|LRPA}}=.False.) local-field effects are printed in the {{TAG\|OUTCAR}} file. Perform a consistency check by comparing the values excluding local-field effects and static limit of <math>\epsilon</math> obtained with {{TAG\|LOPTICS}}=.True., i.e., <math>\lim_{\omega\to0}\epsilon(\omega)</math>.		:At the end of the calculation, both the values of <math>\epsilon</math> including ({{TAG\|LRPA}}=.True.) or excluding ({{TAG\|LRPA}}=.False.) local-field effects are printed in the {{TAG\|OUTCAR}} file. Perform a consistency check by comparing the values excluding local-field effects and static limit of <math>\epsilon</math> obtained with {{TAG\|LOPTICS}}=.True., i.e., <math>\lim_{\omega\to0}\epsilon(\omega)</math>.

	==== {{TAG\|LCALCEPS}}: finite differences approach====		==== {{TAG\|LCALCEPS}}: finite differences approach====
	With {{TAG\|LCALCEPS}}=.True., the dielectric tensor is computed from the derivative of the polarization, using		:With {{TAG\|LCALCEPS}}=.True., the dielectric tensor is computed from the derivative of the polarization, using
	:<math>		::<math>
	\epsilon^\infty_{ij}=\delta_{ij}+		\epsilon^\infty_{ij}=\delta_{ij}+
	\frac{4\pi}{\epsilon_0}\frac{\partial P_i}{\partial \mathcal{E}_j}		\frac{4\pi}{\epsilon_0}\frac{\partial P_i}{\partial \mathcal{E}_j}
Line 31:		Line 31:
	{i,j=x,y,z}.		{i,j=x,y,z}.
	</math>		</math>
	However, here the derivative is evaluated explicitly by employing finite differences. The direction and intensity of the perturbing electric field have to be specified in the {{FILE\|INCAR}} file using the {{TAG\|EFIELD_PEAD}} tag. As with DFPT, at the end of the calculation, VASP will write the dielectric tensor in the {{TAG\|OUTCAR}} file. Control over the inclusion of local-field effects is done with the variable {{TAG\|LRPA}}.		:However, here the derivative is evaluated explicitly by employing finite differences. The direction and intensity of the perturbing electric field have to be specified in the {{FILE\|INCAR}} file using the {{TAG\|EFIELD_PEAD}} tag. As with DFPT, at the end of the calculation, VASP will write the dielectric tensor in the {{TAG\|OUTCAR}} file. Control over the inclusion of local-field effects is done with the variable {{TAG\|LRPA}}.

	===Dynamical response: Green-Kubo and many-body perturbation theory===		===Dynamical response: Green-Kubo and many-body perturbation theory===