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NO₂: Franck-Condon Factors

 

Sample directory: adf/FranckCondon_NO2/

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 (the geometry optimizations are not shown here):

$ADFBIN/adf << eor

TITLE Nitrogen dioxide

ATOMS
    N         0.000000    0.000000   -0.016179
    O         0.000000    1.098646   -0.492918
    O         0.000000   -1.098646   -0.492918
END

BASIS
    CORE NONE
    TYPE DZP
END

XC
    LDA SCF VWN
END

ANALYTICALFREQ
END

eor

mv TAPE21 NO2.t21
rm t21.* logfile

$ADFBIN/adf << eor

TITLE Nitrite

ATOMS
    N         0.000000    0.000000    0.093662
    O         0.000000    1.120366   -0.540999
    O         0.000000   -1.120366   -0.540999
END

CHARGE -1.0 1.0
UNRESTRICTED

BASIS
    CORE NONE
    TYPE DZP
END

XC
    LDA SCF VWN
END

ANALYTICALFREQ
END

eor

mv TAPE21 NO2-.t21
rm t21.* logfile

This runscript produces two TAPE21 files containing the frequencies and the normal modes for both charge states. Lets first look at the ground state to ground state overlap:

$ADFBIN/fcf << eor

STATES NO2.t21 NO2-.t21

QUANTA 0 0

eor

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)	λ (dimensionless)
1072.490460	1.216108
1434.990571	1.873915
1875.876562	0.000000

And for NO₂^-:

Frequency (cm-1)	λ (dimensionless)
816.952242	0.594643
1264.390562	2.071319
1314.362101	0.000000

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 0.7944250686*10^-2, less than one 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 0.5196635779. Including one extra vibrational quanta results in an additional 960135 Franck-Condon factors, but an average sum of only 0.5280010614, i.e. less than a percent more. This one percent is smeared out over so many factors that their individual contributions become negligible.