Static linear response: theory

Let’s consider three types of static perturbations

1. atomic displacements ${\textstyle u_{m}}$ with $m=I\alpha$ with $I=\{1..N_{\text{atoms}}\}$ and $\alpha =\{1..3\}$
2. homogeneous strains ${\textstyle \eta _{j}}$ with ${\textstyle j=\{1..6\}}$
3. static electric field ${\textstyle {\mathcal {E}}_{\alpha }}$ with ${\textstyle \alpha =\{1..3\}}$

By performing a Taylor expansion of the total energy in terms of these perturbations we obtain

${\begin{aligned}E(u,{\mathcal {E}},\eta )=&E_{0}+\\&{\frac {\partial E}{\partial u_{m}}}u_{m}+{\frac {\partial E}{\partial {\mathcal {E}}_{\alpha }}}{\mathcal {E}}_{\alpha }+{\frac {\partial E}{\partial \eta _{j}}}\eta _{j}+\\&{\frac {1}{2}}{\frac {\partial ^{2}E}{\partial u_{m}\partial u_{n}}}u_{m}u_{n}+{\frac {1}{2}}{\frac {\partial ^{2}E}{\partial {\mathcal {E}}_{\alpha }\partial {\mathcal {E}}_{\beta }}}{\mathcal {E}}_{\alpha }{\mathcal {E}}_{\beta }+{\frac {1}{2}}{\frac {\partial ^{2}E}{\partial \eta _{j}\partial \eta _{k}}}\eta _{j}\eta _{k}+\\&{\frac {\partial ^{2}E}{\partial u_{m}\partial {\mathcal {E}}_{\alpha }}}u_{m}{\mathcal {E}}_{\alpha }+{\frac {\partial ^{2}E}{\partial u_{m}\partial \eta _{j}}}u_{m}\eta _{j}+{\frac {\partial ^{2}E}{\partial {\mathcal {E}}_{\alpha }\partial \eta _{j}}}{\mathcal {E}}_{\alpha }\eta _{j}+{\text{terms of higher order}}\end{aligned}}$

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.

$$P_{\alpha }=-{\frac {\partial E}{\partial {\mathcal {E}}_{\alpha }}}\qquad {\text{polarization}}$$

$$F_{m}=-\Omega _{0}{\frac {\partial E}{\partial u_{m}}}\qquad {\text{forces}}$$

$$\sigma _{j}={\frac {\partial E}{\partial \eta _{j}}}\qquad {\text{stresses}}$$

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

$${\overline {\chi }}_{\alpha \beta }=-{\frac {\partial ^{2}E}{\partial {\mathcal {E}}_{\alpha }\partial {\mathcal {E}}_{\beta }}}|_{u,\eta }\qquad {\text{dielectric susceptibility}}$$

$${\overline {C}}_{jk}={\frac {\partial ^{2}E}{\partial \eta _{j}\partial \eta _{k}}}|_{u,{\mathcal {E}}}\qquad {\text{elastic tensor}}$$

$$\Phi _{mn}=\Omega _{0}{\frac {\partial ^{2}E}{\partial u_{m}\partial u_{n}}}|_{{\mathcal {E}},\eta }\qquad {\text{force-constants}}$$

$${\overline {e}}_{\alpha k}={\frac {\partial ^{2}E}{\partial {\mathcal {E}}_{\alpha }\partial \eta _{k}}}|_{u}\qquad {\text{piezoelectric tensor}}$$

$$Z_{m\alpha }=-\Omega _{0}{\frac {\partial ^{2}E}{\partial u_{m}\partial {\mathcal {E}}_{\alpha }}}|_{\eta }\qquad {\text{Born effective charges}}$$

$$\Xi _{mj}=-\Omega _{0}{\frac {\partial ^{2}E}{\partial u_{m}\partial \eta _{j}}}|_{\mathcal {E}}\qquad {\text{force response internal strain tensor}}$$