Berry phases and finite electric fields - Revision history

Csheldon: /* Response properties */

2025-05-05T07:34:45Z

Response properties

← Older revision		Revision as of 07:34, 5 May 2025
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	Z^*_{ij}=\frac{\Omega}{e}\frac{\partial P_i}{\partial u_j}		Z^*_{ij}=\frac{\Omega}{e}\frac{\partial P_i}{\partial u_j}
	=\frac{1}{e}\frac{\partial ~~F_i~~}{\partial \mathcal{E}_j},		=\frac{1}{e}\frac{\partial F_j}{\partial \mathcal{E}_i},
	\qquad		\qquad
	{i,j=x,y,z},		{i,j=x,y,z},

Vaspmaster: /* Modern Theory of Polarization */

2024-01-19T13:55:17Z

Modern Theory of Polarization

← Older revision		Revision as of 13:55, 19 January 2024
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	:<math>		:<math>
	\Delta \mathbf{P}= \frac{2}{\Omega_{0}} \int_{\Omega_{0}} \mathbf{r} \delta		\Delta \mathbf{P}= \frac{1}{\Omega_{0}} \int_{\Omega_{0}} \mathbf{r} \delta
	\left(		\left(
	\mathbf{r} \right) d^{3}r		\mathbf{r} \right) d^{3}r

Vaspmaster: /* Modern Theory of Polarization */

2024-01-19T13:54:05Z

Modern Theory of Polarization

← Older revision		Revision as of 13:54, 19 January 2024
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	:<math>		:<math>
	\Delta \mathbf{P}= \frac{1}{\Omega_{0}} \int_{\Omega_{0}} \mathbf{r} \delta		\Delta \mathbf{P}= \frac{2}{\Omega_{0}} \int_{\Omega_{0}} \mathbf{r} \delta
	\left(		\left(
	\mathbf{r} \right) d^{3}r		\mathbf{r} \right) d^{3}r

Karsai: /* References */

2022-07-19T07:50:22Z

References

← Older revision		Revision as of 07:50, 19 July 2022
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	</references>		</references>
	----		----
	~~[[The_VASP_Manual\|Contents]]~~

	[[Category:Dielectric properties]][[Category:Berry phases]][[Category:Theory]]		[[Category:Linear response]][[Category:Dielectric properties]][[Category:Berry phases]][[Category:Theory]]

Karsai: /* References */

2022-07-19T07:48:59Z

References

← Older revision		Revision as of 07:48, 19 July 2022
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	[[The_VASP_Manual\|Contents]]		[[The_VASP_Manual\|Contents]]

	[[Category:Dielectric ~~Properties~~]][[Category:Berry phases]][[Category:Theory]]		[[Category:Dielectric properties]][[Category:Berry phases]][[Category:Theory]]

Miranda.henrique: /* Self-consistent response to finite electric fields */

2021-10-18T07:20:59Z

Self-consistent response to finite electric fields

← Older revision		Revision as of 07:20, 18 October 2021
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	\frac{\delta \mathbf{P}\left[\{\psi^{({\mathcal E})} \}\right]}{\delta \langle \psi^{({\mathcal E})}_{n\mathbf{k}}\|}=		\frac{\delta \mathbf{P}\left[\{\psi^{({\mathcal E})} \}\right]}{\delta \langle \psi^{({\mathcal E})}_{n\mathbf{k}}\|}=
	-\frac{ie}{2\Delta k} \sum^N_{m=1}		-\frac{ie}{2\Delta k} \sum^N_{m=1}
	\left[ \| u^{({\mathcal E})}_{m\mathbf{k}_{j+1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j+1})~~\rangle~~ -		\left[ \| u^{({\mathcal E})}_{m\mathbf{k}_{j+1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j+1}) -
	\| u^{({\mathcal E})}_{m\mathbf{k}_{j-1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j-1})~~\rangle~~\right]		\| u^{({\mathcal E})}_{m\mathbf{k}_{j-1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j-1})\right]
	</math>		</math>

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	\frac{\partial \|u_{n\mathbf{k}_j} \rangle}{\partial k}=		\frac{\partial \|u_{n\mathbf{k}_j} \rangle}{\partial k}=
	\frac{ie}{2\Delta k} \sum^N_{m=1}		\frac{ie}{2\Delta k} \sum^N_{m=1}
	\left[ \| u^{({\mathcal E})}_{m\mathbf{k}_{j+1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j+1})~~\rangle~~ -		\left[ \| u^{({\mathcal E})}_{m\mathbf{k}_{j+1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j+1}) -
	\| u^{({\mathcal E})}_{m\mathbf{k}_{j-1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j-1})~~\rangle~~\right]		\| u^{({\mathcal E})}_{m\mathbf{k}_{j-1}} \rangle S^{-1}_{mn}(\mathbf{k}_j,\mathbf{k}_{j-1})\right]
	</math>		</math>

Huebsch: /* Self-consistent response to finite electric fields */

2021-08-04T05:18:23Z

Self-consistent response to finite electric fields

← Older revision		Revision as of 05:18, 4 August 2021
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	As of version 5.2, VASP can calculate the ground state of an insulating system under the application of a finite homogeneous electric field. The VASP implementation closely follows the ''PEAD'' (Perturbation Expression After Discretization) approach of Nunes and Gonze<ref name="nunes:prb:01"/> and the work of Souza ''et al.''.<ref name="souza:prl:02"/>		As of version 5.2, VASP can calculate the ground state of an insulating system under the application of a finite homogeneous electric field. The VASP implementation closely follows the ''PEAD'' (Perturbation Expression After Discretization) approach of Nunes and Gonze<ref name="nunes:prb:01"/> and the work of Souza ''et al.''.<ref name="souza:prl:02"/>

	In short: to determine the ~~groundstate~~ of an insulating system under the application of a finite homogeneous electric field <span style="font-size:16pt">''ε''</span>, VASP solves for the field-polarized Bloch functions {''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup>} by minimizing the electric enthalpy functional:		In short: to determine the ground state of an insulating system under the application of a finite homogeneous electric field <span style="font-size:16pt">''ε''</span>, VASP solves for the field-polarized Bloch functions {''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup>} by minimizing the electric enthalpy functional:

	<span id="EdotP">		<span id="EdotP">

Huebsch: /* Self-consistent response to finite electric fields */

2021-08-04T05:16:17Z

Self-consistent response to finite electric fields

← Older revision		Revision as of 05:16, 4 August 2021
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	== Self-consistent response to finite electric fields ==		== Self-consistent response to finite electric fields ==
	As of version 5.2, VASP ~~is able to~~ calculate the ~~groundstate~~ of an insulating system under the application of a finite homogeneous electric field. The VASP implementation closely follows the ''PEAD'' (Perturbation Expression After Discretization) approach of Nunes and Gonze<ref name="nunes:prb:01"/> and the work of Souza ''et al.''.<ref name="souza:prl:02"/>		As of version 5.2, VASP can calculate the ground state of an insulating system under the application of a finite homogeneous electric field. The VASP implementation closely follows the ''PEAD'' (Perturbation Expression After Discretization) approach of Nunes and Gonze<ref name="nunes:prb:01"/> and the work of Souza ''et al.''.<ref name="souza:prl:02"/>

	In short: to determine the groundstate of an insulating system under the application of a finite homogeneous electric field <span style="font-size:16pt">''ε''</span>, VASP solves for the field-polarized Bloch functions {''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup>} by minimizing the electric enthalpy functional:		In short: to determine the groundstate of an insulating system under the application of a finite homogeneous electric field <span style="font-size:16pt">''ε''</span>, VASP solves for the field-polarized Bloch functions {''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup>} by minimizing the electric enthalpy functional:
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	and u<sup>(<span style="font-size:12pt">''ε''</span>)</sup><sub>n'''k'''</sub> is the cell-periodic part of ''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup><sub>n'''k'''</sub>.		and u<sup>(<span style="font-size:12pt">''ε''</span>)</sup><sub>n'''k'''</sub> is the cell-periodic part of ''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup><sub>n'''k'''</sub>.

	The second term on the right-hand-side of the [[#EdotP\|electric enthalpy functional]] introduces a corresponding additional term to the Hamiltonian		The second term on the right-hand side of the [[#EdotP\|electric enthalpy functional]] introduces a corresponding additional term to the Hamiltonian

	<span id="HdotP">		<span id="HdotP">
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	This Hamiltonian allows one to solve for {''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup>} by means of a [[direct optimization method]].		This Hamiltonian allows one to solve for {''ψ''<sup>(<span style="font-size:12pt">''ε''</span>)</sup>} by means of a [[direct optimization method]].

	'''Note''': By analogy it can be shown that		'''Note''': By analogy, it can be shown that

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

	in the sense of a first-order finite difference scheme (higher order stencils may be similarly defined).<ref name="nunes:prb:01"/>		in the sense of a first-order finite difference scheme (higher-order stencils may be similarly defined).<ref name="nunes:prb:01"/>

	'''Note''': One should be aware that when the electric field is chosen to be too large, the electric enthalpy functional will lose its minima, and VASP will not be able to find a stationary solution for the field-polarized orbitals.		'''Note''': One should be aware that when the electric field is chosen to be too large, the electric enthalpy functional will lose its minima, and VASP will not be able to find a stationary solution for the field-polarized orbitals.

Miranda.henrique at 19:36, 4 February 2021

2021-02-04T19:36:44Z

← Older revision		Revision as of 19:36, 4 February 2021
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	{{TAG\|NPPSTR}},		{{TAG\|NPPSTR}},
	{{TAG\|DIPOL}}		{{TAG\|DIPOL}}

	~~{{sc\|Berry phases and finite electric fields\|Examples\|Examples that use this tag}}~~

	== References ==		== References ==

Karsai: /* References */

2019-03-08T13:46:44Z

References

← Older revision		Revision as of 13:46, 8 March 2019
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	[[The_VASP_Manual\|Contents]]		[[The_VASP_Manual\|Contents]]

	[[Category:Berry phases]][[Category:~~Dielectric Properties~~]]		[[Category:Dielectric Properties]][[Category:Berry phases]][[Category:Theory]]