Dipole polarizability

$$\alpha_{\epsilon\tau}(-\omega; \omega) = - \langle\langle \hat{\mu}_{\epsilon};\hat{\mu}_{\tau} \rangle\rangle _{\omega}$$ (3.3)

For the static dipole polarizability $\alpha$(0;0)

$$\alpha(0;0) = -\left.\frac{\partial^{2}{\cal E}\left({\bf E}\right)}{\partial {\bf E}^2}\right\vert _{{\bf E} = {\bf0}}$$ (3.4)

Polarizability

$\alpha_{\mu\nu}(-\omega;~\omega)$

$\propto$

$\langle\langle$ $\hat{r}_{\mu}$; $\hat{r}_{\nu}$ $\rangle\rangle$$_{\omega}$

Symmetry

tensor, second order, symmetric

Calculation type
linear response, $\langle\langle$r;r $\rangle\rangle$ 2.Linear!$<<$r;r$>>$

@XDIPLEN XDIPLEN 7.478371663959D+00

Units
1 au of $\alpha \equiv e^2 a_0^2 E_h^{-1}$ $\equiv e^2 a_0^4 m_e \hbar^{-2}$ $\cong 1.648777251 \times 10^{-41}$ C$^2$ m$^2$ J $^{-1} \cong 1.481847 \times 10^{-25}\ 4 \pi \epsilon_0$ cm$^3$ (esu)

Comments Frequency dependence may be described using Cauchy moments (sec. 3.9). Similarly, Cauchy moments can be used to obtain long range dispersion interaction coefficients C$_n$, there is no need to compute the polarizability at imaginary frequencies.