Category:Linear response: Difference between revisions

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If we restrict ourselves to the first order of the perturbation then we are in the linear regime and thus we talk about linear response.
If we restrict ourselves to the first order of the perturbation then we are in the linear regime and thus we talk about linear response.
A central quantity in linear response is the [[Dielectric properties|dielectric function]] which relates an external electric field with the internal electric displacement.
A central quantity in linear response is the [[:Category:Dielectric properties|dielectric function]] which relates an external electric field with the internal electric displacement.
The response to atomic displacements includes [[:Category:Phonons|phonons]] and [[:Category:electron-phonon interactions|electron-phonon interactions]].
The response to atomic displacements includes [[:Category:Phonons|phonons]] and [[:Category:electron-phonon interactions|electron-phonon interactions]].
The response to homogeneous strains is related to the elastic tensor and piezoelectric tensor when combined with a response to and external electric field.
The response to homogeneous strains is used to compute the elastic tensor and piezoelectric tensor when combined with a response to an external electric field.


== Polarization, berry phases, and finite electric fields ==
== Polarization, berry phases, and finite electric fields ==


The polarization in a periodic system can be computed using the [[Berry_phases_and_finite_electric_fields#Modern_Theory_of_Polarization| berry phase formulation of the polarization]] (often referred to as the modern theory of polarization). With a method to compute the polarization, we can apply a [[Berry_phases_and_finite_electric_fields#Berry_phases_and_finite_electric_fields#Self-consistent_response_to_finite_electric_fields| finite electric field]] to the system.  
The polarization in a periodic system can be computed using the [[Berry_phases_and_finite_electric_fields#Modern_Theory_of_Polarization| berry phase formulation of the polarization]] (often referred to as the modern theory of polarization). With a method to compute the polarization, we can apply a [[Berry_phases_and_finite_electric_fields#Self-consistent_response_to_finite_electric_fields| finite electric field]] to the system.  
Strictly speaking, polarization, as well as the application of a finite electric field, are ground-state properties, however, because they can be used to compute the static dielectric tensor, born effective charges, and piezoelectric tensors which are response properties, we refer to this approach here.
Strictly speaking, polarization, as well as the application of a finite electric field, are ground-state properties, however, because they can be used to compute the static dielectric tensor, born effective charges, and piezoelectric tensors which are response properties, we refer to this approach here.


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=== Dielectric tensor ===
=== Dielectric tensor ===
For the case where the external electric field is static, the static dielectric function is sufficient to determine the response. This static response can be computed by finite differences ({{TAG|LCALCEPS}}) of the polarization with respect to a finite external electric field or by using density functional perturbation theory ({{TAG|LEPSILON}}).  
For the case where the external electric field is static, the static dielectric function is sufficient to determine the response. This static response can be computed by finite differences {{TAG|LCALCEPS}} of the polarization with respect to a finite external electric field or by using density functional perturbation theory {{TAG|LEPSILON}}.
Both {{TAG|LEPSILON}} or {{TAG|LCALCEPS}} yield the same converged results for the dielectric tensor, however, the former can only be used for local or semi-local exchange-correlation functionals and applies to both semiconductors and metals while the second can be used for [[:Category: Meta-GGA|meta-GGA]] or [[:Category: Hybrid functionals|hybrid]] but only for systems with a gap.


Both {{TAG|LEPSILON}} or {{TAG|LCALCEPS}} yield the same converged results for the dielectric tensor, however, the former can only be used for local or semi-local exchange-correlation functionals and applies to both semiconductors and metals while the second can be used for [[:Category:Meta-GGA|meta-GGA]] or [[:Category:Hybrid functionals|hybrid]] but only for systems with a gap.
<!--
=== Born effective charges ===
=== Born effective charges ===
There are two approaches to compute Born effective charges implemented in VASP:
one is done by applying [[Berry_phases_and_finite_electric_fields#Self-consistent_response_to_finite_electric_fields | finite electric fields]] along the three cartesian directions and computing the forces on the atoms which is activated using {{TAG|LCALCEPS}} or by computing the derivating of the wavefunction with respect to an electric field using density functional perturbation theory (DFPT) using {{TAG|LEPSILON}}.
=== Piezolectric tensor ===
The piezoelectric tensor can be computed by finite differences with respect to a [[Berry_phases_and_finite_electric_fields#Self-consistent_response_to_finite_electric_fields |finite electric field]] using {{TAG|LCALCEPS}} or
by using DFPT with {{TAG|LEPSILON}} in combination with {{TAGO|IBRION|5,6 or 7,8}}.
=== Elastic tensor ===
=== Elastic tensor ===
=== Piezoelectric tensor ===
The elastic tensor is computed using strain finite differences using {{TAGO|IBRION|6}} in combination with {{TAGO|ISIF|3|op>=}}. It is not possible to compute it using {{TAGO|IBRION|8}} because the strain perturbation is not implemented in DFPT.
-->
 
=== Internal strain tensor ===
The internal strain tensor can be computed using finite differences using {{TAGO|IBRION|6}} or DFPT using {{TAGO|IBRION|8}}.


== Dynamic response ==
== Dynamic response ==


There are different approaches and levels of theory implemented in VASP to compute the frequency-dependent dielectric tensor. The simplest of these is done using the Green-kubo formula and is activated using {{TAGO|LOPTICS|.TRUE.}}. This however neglects local-field effects which means that it only reproduces calculations from DFPT or finite differences of a finite electric field when {{TAGO|LRPA|.TRUE.}} when the frequency is zero (static limit). To include local field effects one should use [[:Category:Dielectric_properties#ALGO_=_CHI:_polarizability_within_GW_approximation| ALGO = CHI]].
There are [[:Category:Dielectric_properties#Dynamical_response:_Green-Kubo_and_many-body_perturbation_theory|different approaches]] and levels of theory implemented in VASP to compute the frequency-dependent dielectric tensor. The simplest of these is done using the Green-kubo formula and is activated using {{TAG|LOPTICS}}. This however neglects local-field effects which means that it only reproduces calculations from DFPT or finite differences of a [[Berry_phases_and_finite_electric_fields#Self-consistent_response_to_finite_electric_fields |finite electric]] field when {{TAGO|LRPA|.TRUE.}} when the frequency is zero (static limit). To include local field effects one should use {{TAGO|ALGO|CHI}}. Additionally one can include electron-hole interaction when computing the dielectric function using the Bethe-Salpeter equation in the many-body perturbation theory framework using {{TAGO|ALGO|BSE}}.


== X-ray absorption spectroscopy ==
== X-ray absorption spectroscopy (XAS) ==


Another case of interest is the absorption of electromagnetic radiation by core electrons as measured experimentally in [[:Category:XAS| X-ray absorption near edge spectroscopy (XANES)]]. This is obtained from the frequency-dependent dielectric function with the particularity that core electrons are explicitly included in the calculation.
Another case of interest is the absorption of electromagnetic radiation by core electrons as measured experimentally in [[:Category:XAS| X-ray absorption near edge spectroscopy (XANES)]]. This is obtained from the frequency-dependent dielectric function with the particularity that core electrons are explicitly included in the calculation.
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Nuclear magnetic resonance is a widely used spectroscopy technique used to probe the structure and chemical composition of molecules and solids.
Nuclear magnetic resonance is a widely used spectroscopy technique used to probe the structure and chemical composition of molecules and solids.
VASP can compute [[LCHIMAG|chemical shielding shifts]] which is the ratio between the external and induced magnetic field compared to a reference compound, [[LHYPERFINE|hyperfine tensors]] and [[Electric_Field_Gradient|electric field gradients]].
VASP can compute [[LCHIMAG|chemical shifts]] which is the ratio between the external and induced magnetic field compared to a reference compound, [[LHYPERFINE|hyperfine tensors]] and [[Electric_Field_Gradient|electric field gradients]].


== How to ==
== How to ==

Revision as of 14:46, 8 February 2024

Apart from ground-state properties, VASP can compute how a system reacts to external perturbations. Currently, we can consider three types of perturbations:

  1. external electric field
  2. atomic displacements
  3. homogeneous strains

If we restrict ourselves to the first order of the perturbation then we are in the linear regime and thus we talk about linear response. A central quantity in linear response is the dielectric function which relates an external electric field with the internal electric displacement. The response to atomic displacements includes phonons and electron-phonon interactions. The response to homogeneous strains is used to compute the elastic tensor and piezoelectric tensor when combined with a response to an external electric field.

Polarization, berry phases, and finite electric fields

The polarization in a periodic system can be computed using the berry phase formulation of the polarization (often referred to as the modern theory of polarization). With a method to compute the polarization, we can apply a finite electric field to the system. Strictly speaking, polarization, as well as the application of a finite electric field, are ground-state properties, however, because they can be used to compute the static dielectric tensor, born effective charges, and piezoelectric tensors which are response properties, we refer to this approach here.

Static response

Dielectric tensor

For the case where the external electric field is static, the static dielectric function is sufficient to determine the response. This static response can be computed by finite differences LCALCEPS of the polarization with respect to a finite external electric field or by using density functional perturbation theory LEPSILON. Both LEPSILON or LCALCEPS yield the same converged results for the dielectric tensor, however, the former can only be used for local or semi-local exchange-correlation functionals and applies to both semiconductors and metals while the second can be used for meta-GGA or hybrid but only for systems with a gap.

Born effective charges

There are two approaches to compute Born effective charges implemented in VASP: one is done by applying finite electric fields along the three cartesian directions and computing the forces on the atoms which is activated using LCALCEPS or by computing the derivating of the wavefunction with respect to an electric field using density functional perturbation theory (DFPT) using LEPSILON.

Piezolectric tensor

The piezoelectric tensor can be computed by finite differences with respect to a finite electric field using LCALCEPS or by using DFPT with LEPSILON in combination with IBRION = 5,6 or 7,8.

Elastic tensor

The elastic tensor is computed using strain finite differences using IBRION = 6 in combination with ISIF = 3. It is not possible to compute it using IBRION = 8 because the strain perturbation is not implemented in DFPT.

Internal strain tensor

The internal strain tensor can be computed using finite differences using IBRION = 6 or DFPT using IBRION = 8.

Dynamic response

There are different approaches and levels of theory implemented in VASP to compute the frequency-dependent dielectric tensor. The simplest of these is done using the Green-kubo formula and is activated using LOPTICS. This however neglects local-field effects which means that it only reproduces calculations from DFPT or finite differences of a finite electric field when LRPA = .TRUE. when the frequency is zero (static limit). To include local field effects one should use ALGO = CHI. Additionally one can include electron-hole interaction when computing the dielectric function using the Bethe-Salpeter equation in the many-body perturbation theory framework using ALGO = BSE.

X-ray absorption spectroscopy (XAS)

Another case of interest is the absorption of electromagnetic radiation by core electrons as measured experimentally in X-ray absorption near edge spectroscopy (XANES). This is obtained from the frequency-dependent dielectric function with the particularity that core electrons are explicitly included in the calculation. The theory page describes the implementation details within the PAW method and the relevant literature.

Nuclear magnetic resonance (NMR)

Nuclear magnetic resonance is a widely used spectroscopy technique used to probe the structure and chemical composition of molecules and solids. VASP can compute chemical shifts which is the ratio between the external and induced magnetic field compared to a reference compound, hyperfine tensors and electric field gradients.

How to

Subcategories

This category has the following 5 subcategories, out of 5 total.