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7.4 Real-Time SCF Methods

7.4.4 Calculation of Absorption Spectra

(February 4, 2022)

The absorption cross-section σii(ω) for light polarized in the i direction (i{x,y,z}) can be obtained from the imaginary part () of the frequency-dependent polarizability, αii(ω):1292, Zhu:2021

σii(ω)=(4πωc)[αii(ω)]. (7.36)

A rotationally-averaged absorption spectrum A(ω) is then simply

A(ω)=13[σxx(ω)+σyy(ω)+σzz(ω)] (7.37)

within the electric dipole approximation. Components αij(ω) of the frequency-dependent polarizability tensor 𝜶(ω) are obtained from the Fourier transform () of the time-dependent dipole moment component μi(t), for a perturbing field j in the j direction:1292

αij(ω)=[μi(t)][j(t)]. (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 Ψn|j|Ψ0 to vanish. However, these three perturbations x, y, and 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.

   0 1
   H    0.000000   0.000000   0.000000
   H    0.000000   0.000000   0.750000

   BASIS                6-31G
   EXCHANGE             gen
   SYMMETRY             false
   TDKS                 true
   UNRESTRICTED         true
   LRC_DFT              true
   OMEGA                300
   INCFOCK              0
   PURECART             2222

   C  PBE   1.00
   X  wPBE  1.00
   X  HF    0.00

   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

View output