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Absorption spectra by Fermi golden Rule

There is a possibility to evaluate rough estimation of UV-VIS absorption spectra by Fermi golden rule between Kohn-Sham orbitals. For a given range of photon energy are selected all pairs of occupied and unoccupied ground-state Kohn-Sham orbital with appropriate energy difference. This two orbitals are than projected onto real space grid and transition dipole moment is computed. The Oscillator strength (or probability of transition) is than proportional to square of length of transition dipole moment. The Absorption spectra can be than plotted by summing up contributions of oscillator strength for occupied and unoccupied orbital pairs with energy difference falling into particular energy bins (like weighted histogram).

Note that this method ( Fermi golden rule between ground state single particle states obtained from LDA-DFT ) is a very crude approximation of the true optical transitions. No beyond-DFT nor beyond-Hratree-Fock correction (such as RPA, TD-DFT, GW, BSE … ) are used. For this reason the transition energy ( position of peak ) might not be realistic. The method is, however, useful to explore the dependence of electronic transition probability on symmetry breaking and the changes in shape and localization of electronic states.


this is implemented only in Prokop's personal versions of Fireball located in



The computation of optical transitions is activated by setting keyword iwrtexcit = 1 in However, currently it is also necessary to activate iwrtewf = 1 in order to initialize a grid, even thought .xsf files written by iwrtewf option are not used at all and have nothing to do with the computation of transitions ( this should be corrected in future). So the typical looks like:

basisfile  = answer.bas 
nstepf     = 1
icluster   = 1
ifixcharge = 1
iwrtewf     = 1
iewform = 3
npbands = 1
pbands = 1

Than excitations.optional file should be provided to

 0        ! write_out_transition_map ( default 0, moslty just for debuging )
 0.0      ! dEmin ... minimum energy difference ( mostly 0)
 3.5      ! dEmax ... maximim energy difference

if write_out_transition_map is set to '1', an '.xsf' file containing spatial distribution of transition dipole moment is outputted for each pair of orbitals. This may be useful to identify parts of molecule which contribute mostly to the electronic transition. However, the number of transitions grow quadratically with the energy range and each .xsf file needs typically more than 100MB of disk space. So in most of cases you would quickly consume HUGE amount of disk space ( like 1 terrabyte ) if you switch this option on.

It is also recommended to chose reasonable small energy interval (dEmin,dEmax) otherwise the computation can take very long time.


Excitations.dat file is written out from the calculation. The file contains several columns of following meaning:

  1. j,j index of occupied and unoccupied orbital
  2. delta_Eij energy difference
  3. |D| length of transition dipole moment
  4. D_x,D_y,D_z cartesian components of transition dipole moment
  5. Ei,Ej energy eigenvalues corresponding to the Khon-Sham orbitals
   i      j         delta_Eij               |D|                 D_x                 D_y                   D_z               Ei                     Ej
  749    879        3.4862376070        0.9824594517        0.2759973643       -0.9336816482        0.1314937604       -3.0970434024       -6.5832810094
  750    879        3.4791174668        0.9753364513       -0.4413174043        0.2404091902        0.8358968616       -3.0970434024       -6.5761608692
  751    879        3.4582785159        0.5954896260        0.3594301771       -0.1688715641       -0.4437344220       -3.0970434024       -6.5553219183
  752    879        3.4453548553        0.8057311791       -0.1487142116       -0.4625862731       -0.6427291469       -3.0970434024       -6.5423982578
  753    879        3.4410779875        0.7684644226        0.0080712081       -0.6875508088       -0.3431418216       -3.0970434024       -6.5381213900
  754    879        3.4257564875        0.4202213606        0.0892087920       -0.1778280360       -0.3701418282       -3.0970434024       -6.5227998899

Making spectra

For transformation of pairwise table of transition dipole moment into optical spectra for plotting can be used for example following bash/awk scripts.

First script accumulate (sum up) pairwise contributions of transition dipole moment into bins of given size and then smear them with gaussian filter (to make nice smooth spectra). The most important parameter is step size. For example with step size 0.001 eV: 0 3.5 0.001 code:

awk -v min=$min -v max=$max -v step=$step '
BEGIN{for(i=0;min+i*step<max;i++){hist[i]=0}; minx=10000000; maxx=-10000000  }
x=$3; y=$4; yy=y*y;
#print maxx,minx
if(((min+i*step)>minx) && ((min+i*step)<maxx) ) {print min+i*step,hist[i]}}
' Excitations.dat > spectranice.dat

in some cases you are not interested in the sum of all contributions ( this would correspond to absorption spectra) but rather in the transition with highest oscillator strength in given interval of energy (energy bin). For this purpose script ' can be used. 0.001
#! /bin/bash
min=`awk '
END{print Emin}
' Excitations.dat`
max=`awk '
END{print Emax}
' Excitations.dat`
echo $min $max
awk -v min=$min -v max=$max -v step=$step '
END{for(i=0;min+i*step<max;i++){print min+i*step,hist[i]}}
' Excitations.dat > spectramax.dat
absorbtion_spectra_by_fermi_golden_rule.txt · Last modified: 2014/07/24 14:30 (external edit)