Example: Franck-Condon Factors: NO2¶

#!/bin/sh

# As an example of a Franck-Condon calculation, lets look at the transition of
# NO2 to NO2 - . NO2 is a small molecule with only three vibrational modes.
# Putting an extra electron on the molecule will cause a big displacement,
# resulting in large electron-phonon couplings.

# In general, the larger the molecule, the smaller the displacement and hence
# the electron-phonon couplings and Franck-Condon factors. Moreover, larger
# molecules have more vibrational modes, meaning that the already smaller
# displacement will generally be smeared out over more modes, resulting in an
# additional decrease in electron-phonon couplings. This is fortunate, since the
# number of Franck-Condon factors increases factorially with the number of
# vibrational modes, making it prohibitively expensive to take more than a few
# vibrational quanta into account for most molecules.

# In order to calculate the Franck-Condon factors for Nitrite and Nitrogen
# dioxide, the equilibrium positions of the nuclei and the vibrational modes
# have to be obtained:


AMS_JOBNAME=NO2 $AMSBIN/ams <<eor
System
  atoms
     N         0.000000    0.000000   -0.016179
     O         0.000000    1.098646   -0.492918
     O         0.000000   -1.098646   -0.492918
  end
end

Task GeometryOptimization
GeometryOptimization
  Convergence
    Gradients 1.0e-5
  End
End
Properties
    NormalModes True
End

Engine ADF
  basis
    core NONE
    type DZP
  end
  spinpolarization 1
  title Nitrogen dioxide
  unrestricted
  xc
    lda SCF VWN
  end
EndEngine
eor

# We are using an already optimized geometry for the second calculation but in a
# real experiment one should run geometry optimization first

AMS_JOBNAME=NO2_minus $AMSBIN/ams <<eor
System
  atoms
     N         0.000000    0.000000    0.126041
     O         0.000000    1.070642   -0.555172
     O         0.000000   -1.070642   -0.555172
  end
  charge -1.0
end

Task SinglePoint
Properties
    NormalModes True
End

Engine ADF
  basis
    core NONE
    type DZP
  end
  title Nitrite
  xc
    lda SCF VWN
  end
EndEngine

eor


# This runscript produces two adf.rkf files containing the frequencies and the
# normal modes for both charge states. Lets first look at the ground state to
# ground state overlap:


$AMSBIN/fcf <<eor
  STATES NO2.results/adf.rkf NO2_minus.results/adf.rkf
  QUANTA 0 0
  TRANSLATE
  ROTATE
eor

rm TAPE61 logfile

# Here, zero vibrational quanta are specified for both charge states, which
# corresponds to the vibrational ground state. Looking at the standard output,
# we see for NO2 :

# ================================================
# | Frequency (cm^-1 ) |  lambda (dimensionless) |
# | 756                |  1.979                  |
# | 1380               |  1.489                  |
# | 1716               |  0.000                  |
# ================================================

# And for NO_2^- :

# ================================================
# | Frequency (cm^-1 ) |  lambda (dimensionless) |
# | 785                |  1.552                  |
# | 1265               |  0.000                  |
# | 1338               |  2.231                  |
# ================================================

# Both states have two vibrational modes with a significant electron-phonon
# coupling. The ground state to ground state Franck-Condon factor is therefore
# expected to be quite small. And indeed, looking at the output, we see that it
# is 0.349*10^-1 , less than four percent of the total.

# Since NO2 has only three vibrational modes, many quanta can be included, and
# this indeed turns out to be necessary. Setting the maximum number of quanta at
# 20 results in 1771 permutations for both states and a total of 3136441 Franck-
# Condon factors. Even with so many factors, the average sum is still only
# 0.463. Including one extra vibrational quanta results in an additional 960135
# Franck-Condon factors, but an average sum of only 0.473, i.e. a percent more.
# This one percent is smeared out over so many factors that their individual
# contributions become negligible.


$AMSBIN/fcf <<eor
  STATES NO2.results/adf.rkf NO2_minus.results/adf.rkf
  QUANTA 20 20
  TRANSLATE
  ROTATE
eor

rm TAPE61 logfile

$AMSBIN/fcf <<eor
  STATES NO2.results/adf.rkf NO2_minus.results/adf.rkf
  QUANTA 21 21
  TRANSLATE
  ROTATE
eor

rm TAPE61 logfile

$AMSBIN/fcf <<eor
  STATES NO2.results/adf.rkf NO2_minus.results/adf.rkf
  QUANTA 0 20
  TRANSLATE
  ROTATE
  SPECTRUM 0e3 20e3 1001
eor

rm TAPE61 logfile