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Xavier Andrade edited Sternheimer.tex
over 9 years ago
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The $\vec{k} \cdot \vec{p}$ wavefunctions can be used to compute response to electric fields in periodic systems.
In finite systems, a homogeneous electric field can be represented simply via the position operator $\vec{x}$.
However, this operator is not well defined in a periodic system and cannot be used. According to the quantum theory of polarization,
the solution is to replace
$\vec{x} $\vec{r} \varphi$ with $-i d/dk \varphi$. % cite Baroni, also Gonze papers
The response to this perturbation can be used to compute the
polarizability: polarizability from eq.~\cite{eq:sternheimer_polarizability}
\begin{align}
\alpha_{ij} \left( \omega \right) = i \sum_k f_k \left[ \left. \left< \frac{\partial \varphi_k}{\partial k_i} \right| \frac{\partial \varphi_k}{\partial \mathcal{E}_{j, \omega}} \right> + \left. \left< \frac{\partial \varphi_k}{\partial \mathcal{E}_{j, -\omega}} \right| \frac{\partial \varphi_k}{\partial k_j} \right> \right]
\end{align}