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7.4 Real-Time SCF Methods

7.4.4 Calculation of Absorption Spectra

(December 20, 2021)

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 Y., Herbert J. M.
J. Chem. Phys.
(2018), 148, pp. 044117.
Link

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 Zhu Y., Herbert J. M.
J. Chem. Phys.
(2018), 148, pp. 044117.
Link

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. 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