Static linear response: theory - Revision history

Csheldon at 08:21, 13 November 2025

2025-11-13T08:21:03Z

← Older revision		Revision as of 08:21, 13 November 2025
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	== References ==		== References ==

	[[Category:Linear response]][[Category:Theory]]		[[Category:Linear response]][[Category:Theory]][[Category:Forces]][[Category:Dielectric properties]]

Csheldon at 08:08, 13 November 2025

2025-11-13T08:08:47Z

← Older revision		Revision as of 08:08, 13 November 2025
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	== References ==		== References ==

	[[Category:Linear response]]		[[Category:Linear response]][[Category:Theory]]

Csheldon at 10:26, 12 November 2025

2025-11-12T10:26:40Z

← Older revision		Revision as of 10:26, 12 November 2025
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	== References ==		== References ==

			[[Category:Linear response]]

Miranda.henrique at 08:08, 30 April 2025

2025-04-30T08:08:02Z

← Older revision		Revision as of 08:08, 30 April 2025
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	# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>		# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>

	By performing a Taylor expansion of the total energy $E$ in terms of these perturbations we obtain{{cite\|wu:prb:2005}}		By performing a Taylor expansion of the total energy <math display="inline">E</math> in terms of these perturbations we obtain{{cite\|wu:prb:2005}}
	:<math>		:<math>
	\begin{aligned}		\begin{aligned}

Miranda.henrique at 15:08, 8 February 2024

2024-02-08T15:08:49Z

← Older revision		Revision as of 15:08, 8 February 2024
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	# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>		# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>

	By performing a Taylor expansion of the total energy in terms of these perturbations we obtain{{cite\|wu:prb:2005}}		By performing a Taylor expansion of the total energy $E$ in terms of these perturbations we obtain{{cite\|wu:prb:2005}}
	:<math>		:<math>
	\begin{aligned}		\begin{aligned}
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	<math display="block">		<math display="block">
	Z_{m\alpha}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} \|_{\eta}		Z^*_{m\alpha}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} \|_{\eta}
	\qquad \text{Born effective charges}		\qquad \text{Born effective charges}
	</math>		</math>
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	\begin{aligned}		\begin{aligned}
	\chi_{\alpha\beta} &= \overline{\chi}_{\alpha\beta} +		\chi_{\alpha\beta} &= \overline{\chi}_{\alpha\beta} +
	\Omega_0^{-1} Z_{m\alpha} (\Phi)^{-1}_{mn} Z_{n\beta}		\Omega_0^{-1} Z^_{m\alpha} (\Phi)^{-1}_{mn} Z^_{n\beta}
	\qquad \text{dielectric susceptibility}\\		\qquad \text{dielectric susceptibility}\\
	C_{jk} &= \overline{C}_{jk} +		C_{jk} &= \overline{C}_{jk} +
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	\qquad \text{elastic tensor}\\		\qquad \text{elastic tensor}\\
	e_{\alpha j} &= \overline{e}_{\alpha j} +		e_{\alpha j} &= \overline{e}_{\alpha j} +
	\Omega_0^{-1}Z_{m\alpha} (\Phi)^{-1}_{mn} \Xi_{nj}		\Omega_0^{-1}Z^*_{m\alpha} (\Phi)^{-1}_{mn} \Xi_{nj}
	\qquad \text{piezoelectric tensor}		\qquad \text{piezoelectric tensor}
	\end{aligned}		\end{aligned}
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	The ionic contributions to the dielectric tensor are: <math display="block">		The ionic contributions to the dielectric tensor are: <math display="block">
	\epsilon^{\text{ion}}_{ij}=\frac{4\pi ~~e^2~~}{\Omega}		\epsilon^{\text{ion}}_{ij}=\frac{4\pi}{\Omega}
	\sum_{kl}		\sum_{kl}
	Z_{ik}		Z^*_{ik}
	\Phi^{-1}_{kl}		\Phi^{-1}_{kl}
	Z_{lj}		Z^*_{lj}
	</math>		</math>

	The ionic contributions to the elastic tensor <math display="block">		The ionic contributions to the elastic tensor <math display="block">
	C^{\text{ion}}_{ik}=e		C^{\text{ion}}_{ik}=
	\sum_{kl}		\sum_{kl}
	\Xi_{ij}		\Xi_{ij}
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	The ionic contributions to the piezoelectric tensor <math display="block">		The ionic contributions to the piezoelectric tensor <math display="block">
	e^{\text{ion}}_{ij}=e		e^{\text{ion}}_{ij}=
	\sum_{kl}		\sum_{kl}
	Z_{ij}		Z^*_{ij}
	\Phi^{-1}_{jk}		\Phi^{-1}_{jk}
	\Xi_{kl}		\Xi_{kl}

Miranda.henrique at 08:47, 2 August 2022

2022-08-02T08:47:29Z

@@ Line 82: / Line 82: @@
 \Xi_{mj}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \eta_j} |_{\mathcal{E}}
 \qquad \text{force response internal strain tensor}
 </math>
 == References ==

Miranda.henrique at 15:00, 1 August 2022

2022-08-01T15:00:59Z

← Older revision		Revision as of 15:00, 1 August 2022
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	Let’s consider three types of static perturbations		Let’s consider three types of static perturbations
	# atomic displacements <math display="inline">u_m</math>		# atomic displacements <math display="inline">u_m</math> with <math>m=I\alpha</math> with <math>I=\{1..N_\text{atoms}\}</math> and <math>\alpha=\{1..3\}</math>
	# homogeneous strains <math display="inline">\eta_j</math> with <math display="inline">j=\{1..6\}</math>		# homogeneous strains <math display="inline">\eta_j</math> with <math display="inline">j=\{1..6\}</math>
	# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>		# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>

	By performing a Taylor expansion of the total energy in terms of these perturbations we obtain		By performing a Taylor expansion of the total energy in terms of these perturbations we obtain{{cite\|wu:prb:2005}}
	:<math>		:<math>
	\begin{aligned}		\begin{aligned}
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	\qquad \text{force response internal strain tensor}		\qquad \text{force response internal strain tensor}
	</math>		</math>

			== References ==

Miranda.henrique: Created page with "Let’s consider three types of static perturbations # atomic displacements $u_m$ # homogeneous strains $\eta_j$ with..."

2022-08-01T14:48:33Z

Created page with "Let’s consider three types of static perturbations # atomic displacements <math display="inline">u_m</math> # homogeneous strains <math display="inline">\eta_j</math> with..."

New page

Let’s consider three types of static perturbations
# atomic displacements <math display="inline">u_m</math>
# homogeneous strains <math display="inline">\eta_j</math> with <math display="inline">j=\{1..6\}</math>
# static electric field <math display="inline">\mathcal{E}_\alpha</math> with <math display="inline">\alpha=\{1..3\}</math>

By performing a Taylor expansion of the total energy in terms of these perturbations we obtain
:<math>
\begin{aligned}
E(u,\mathcal{E},\eta) =
&E_0 + \\
&\frac{\partial E}{\partial u_m} u_m +
\frac{\partial E}{\partial \mathcal{E}_\alpha} \mathcal{E}_\alpha+
\frac{\partial E}{\partial \eta_j} \eta_j + \\
&\frac{1}{2} \frac{\partial^2 E}
{\partial u_m \partial u_n }
u_m u_n +
\frac{1}{2} \frac{\partial^2 E}
{\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta }
\mathcal{E}_\alpha \mathcal{E}_\beta +
\frac{1}{2} \frac{\partial^2 E}
{\partial \eta_j \partial \eta_k}
\eta_j \eta_k + \\
&\frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha}
u_m \mathcal{E}_\alpha +
\frac{\partial^2 E}{\partial u_m \partial \eta_j}
u_m \eta_j +
\frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \eta_j}
\mathcal{E}_\alpha \eta_j +
\text{terms of higher order}
\end{aligned}
</math>

The derivatives of the energy with respect to an electric field are the polarization, with respect to atomic displacements are the forces, with respect to changes in the lattice vectors are the stress tensor.

<math display="block">
P_\alpha = -\frac{\partial E}{\partial \mathcal{E}_\alpha}
\qquad \text{polarization}
</math>

<math display="block">
F_m = -\Omega_0\frac{\partial E}{\partial u_m}
\qquad \text{forces}
</math>

<math display="block">
\sigma_j = \frac{\partial E}{\partial \eta_j}
\qquad \text{stresses}
</math>

This leads to the following ‘clamped-ion’ or ‘frozen-ion’ definitions:

<math display="block">
\overline{\chi}_{\alpha\beta} =
- \frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \mathcal{E}_\beta} |_{u,\eta}
\qquad \text{dielectric susceptibility}
</math>

<math display="block">
\overline{C}_{jk} =
\frac{\partial^2 E}{\partial \eta_j \partial \eta_k} |_{u,\mathcal{E}}
\qquad \text{elastic tensor}
</math>

<math display="block">
\Phi_{mn}=\Omega_0 \frac{\partial^2 E}{\partial u_m \partial u_n} |_{\mathcal{E},\eta}
\qquad \text{force-constants}
</math>

<math display="block">
\overline{e}_{\alpha k} =
\frac{\partial^2 E}{\partial \mathcal{E}_\alpha \partial \eta_k} |_{u}
\qquad \text{piezoelectric tensor}
</math>

<math display="block">
Z_{m\alpha}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \mathcal{E}_\alpha} |_{\eta}
\qquad \text{Born effective charges}
</math>

<math display="block">
\Xi_{mj}=-\Omega_0 \frac{\partial^2 E}{\partial u_m \partial \eta_j} |_{\mathcal{E}}
\qquad \text{force response internal strain tensor}
</math>