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# 7.4.4 Calculation of Absorption Spectra

(February 4, 2022)

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 ($\Im$) of the frequency-dependent polarizability, $\alpha_{ii}(\omega)$:1292, Zhu:2021

 $\sigma_{ii}(\omega)=\left(\frac{4\pi\omega}{c}\right)\Im\bigl{[}\alpha_{ii}(% \omega)\bigr{]}\;.$ (7.36)

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

 $A(\omega)=\tfrac{1}{3}\bigl{[}\sigma_{xx}(\omega)+\sigma_{yy}(\omega)+\sigma_{% zz}(\omega)\bigr{]}$ (7.37)

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

 $\alpha_{ij}(\omega)=\frac{\mathcal{F}\bigl{[}\mu_{i}(t)\bigr{]}}{\mathcal{F}% \bigl{[}\mathcal{E}_{j}(t)\bigr{]}}\;.$ (7.38)

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 $\langle\Psi_{n}|\mathcal{E}_{j}|\Psi_{0}\rangle$ to vanish. However, these three perturbations $\mathcal{E}_{x}$, $\mathcal{E}_{y}$, and $\mathcal{E}_{z}$ 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.

Example 7.18  TDKS job using a CW field and a CAP.

$molecule 0 1 H 0.000000 0.000000 0.000000 H 0.000000 0.000000 0.750000$end

$rem BASIS 6-31G EXCHANGE gen SYMMETRY false TDKS true UNRESTRICTED true LRC_DFT true OMEGA 300 INCFOCK 0 PURECART 2222 SCF_CONVERGENCE 9 PRINT_GENERAL_BASIS true$end

$xc_functional C PBE 1.00 X wPBE 1.00 X HF 0.00$end

$tdks DT 0.05 MAXITER 5 PROPAGATOR MMUT FIELD_VECTOR 1 1 1 FIELD_TYPE cw FIELD_FREQUENCY 1.55 FIELD_AMP 0.0001 DO_CAP true CAP_TYPE atom_centered_spherical CAP_R0 12.0 ! should be in bohr/a.u. CAP_ETA 8.0$end


View output