7.4.4 Calculation of Absorption Spectra

The absorption cross-section ${\sigma }_{ii}(\omega )$ for light polarized in the $i$ direction ($i\in \{x,y,z\}$) can be obtained from the imaginary part ($\mathrm{\Im }$) of the frequency-dependent polarizability, ${\alpha }_{ii}(\omega )$:^{1292, Zhu:2021}

A rotationally-averaged absorption spectrum $A(\omega )$ is then simply

within the electric dipole approximation. Components ${\alpha }_{ij}(\omega )$ of the frequency-dependent polarizability tensor $𝜶(\omega )$ are obtained from the Fourier transform ($ℱ$) of the time-dependent dipole moment component ${\mu }_i(t)$, for a perturbing field ${ℰ}_j$ in the $j$ direction:¹²⁹²

To compute the spectrum in Eq. (7.37), three separate perturbations in the $x$, $y$, and $z$ directions are required, else some excitations may be missing if their transition moment is strictly perpendicular to the applied field, causing the matrix element $⟨{\mathrm{\Psi }}_n|{ℰ}_j|{\mathrm{\Psi }}_0⟩$ to vanish. However, these three perturbations ${ℰ}_x$, ${ℰ}_y$, and ${ℰ}_z$ can be applied all at once in a single calculation, in order to generate a superposition consisting of all possible excitations out of the ground state.

Two different scripts are provided to obtain the spectrum after the TDKS simulation is completed:

• •

  $QC/bin/tools/tdks_fft.py

• •

  $QC/bin/tools/tdks_pade.py

The first of these uses the Fourier transform method in Eq. (7.38) directly while the second makes use of Padé approximants to obtain comparable spectra with shorter propagation times.^Zhu:2021 The scripts can be run as follows:

$QC/bin/tools/tdks_fft.py output spectrum.txt
$QC/bin/tools/tdks_pade.py output spectrum.txt

The file spectrum.txt produced by the processing script will contain two columns: frequency (eV) and strength (arbitrary units). This data can be visualized as an $(x,y)$ plot to view the spectrum.