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I did a TDDFT calculation to get the emission spectra of my molecule. Here are the first five roots for the emission calculation:

 ----------------------------------------------------------------------------
  Root   1 singlet a              0.093596388 a.u.                2.5469 eV 
  ----------------------------------------------------------------------------
     Transition Moments    X -0.01267   Y  0.11330   Z -0.00853
     Transition Moments   XX -0.17432  XY -0.11698  XZ  0.08725
     Transition Moments   YY  0.50038  YZ  0.04076  ZZ -0.00276
     Dipole Oscillator Strength                    0.0008155382
     Electric Quadrupole                           0.0000000006
     Magnetic Dipole                               0.0000000172
     Total Oscillator Strength                     0.0008155561
 
     Occ.  118  a   ---  Virt.  119  a   -0.99690 X
  ----------------------------------------------------------------------------
  Root   2 singlet a              0.125536009 a.u.                3.4160 eV 
  ----------------------------------------------------------------------------
     Transition Moments    X -1.40368   Y -0.22255   Z -0.09172
     Transition Moments   XX -0.86565  XY  0.08026  XZ  0.72579
     Transition Moments   YY -4.46850  YZ -0.97403  ZZ  0.39936
     Dipole Oscillator Strength                    0.1697471134
     Electric Quadrupole                           0.0000000828
     Magnetic Dipole                               0.0000000989
     Total Oscillator Strength                     0.1697472950
 
     Occ.  118  a   ---  Virt.  120  a    0.95252 X
     Occ.  118  a   ---  Virt.  121  a    0.28877 X
  ----------------------------------------------------------------------------
  Root   3 singlet a              0.130233593 a.u.                3.5438 eV 
  ----------------------------------------------------------------------------
     Transition Moments    X  0.04368   Y  0.07607   Z  0.00812
     Transition Moments   XX  0.21272  XY -0.14124  XZ  0.00845
     Transition Moments   YY  0.21031  YZ -0.10142  ZZ -0.01092
     Dipole Oscillator Strength                    0.0006738052
     Electric Quadrupole                           0.0000000006
     Magnetic Dipole                               0.0000000089
     Total Oscillator Strength                     0.0006738146
 
     Occ.  117  a   ---  Virt.  119  a   -0.99558 X
  ----------------------------------------------------------------------------
  Root   4 singlet a              0.135112905 a.u.                3.6766 eV 
  ----------------------------------------------------------------------------
     Transition Moments    X -1.61404   Y -0.10037   Z -0.10533
     Transition Moments   XX -0.92285  XY -0.71420  XZ  0.79973
     Transition Moments   YY -3.91548  YZ -1.09555  ZZ  0.35107
     Dipole Oscillator Strength                    0.2365653107
     Electric Quadrupole                           0.0000000939
     Magnetic Dipole                               0.0000000166
     Total Oscillator Strength                     0.2365654212
 
     Occ.  118  a   ---  Virt.  120  a    0.27816 X
     Occ.  118  a   ---  Virt.  121  a   -0.94577 X
     Occ.  118  a   ---  Virt.  122  a   -0.12569 X
  ----------------------------------------------------------------------------
  Root   5 singlet a              0.144664128 a.u.                3.9365 eV 
  ----------------------------------------------------------------------------
     Transition Moments    X -1.29163   Y  1.43040   Z  0.11014
     Transition Moments   XX-13.39872  XY  1.41335  XZ -0.50657
     Transition Moments   YY 14.47237  YZ  0.64851  ZZ  0.27558
     Dipole Oscillator Strength                    0.3593917443
     Electric Quadrupole                           0.0000031743
     Magnetic Dipole                               0.0000000937
     Total Oscillator Strength                     0.3593950123
 
     Occ.  117  a   ---  Virt.  120  a   -0.06248 X
     Occ.  117  a   ---  Virt.  121  a    0.07336 X
     Occ.  118  a   ---  Virt.  121  a   -0.11547 X
     Occ.  118  a   ---  Virt.  122  a    0.97589 X
     Occ.  118  a   ---  Virt.  123  a   -0.05942 X

For my case, experimental emission is at 492 nm or 2.52 eV. In this case which root should I take to compare with the experiment? Should I always take the first root following the Kasha's rule?

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    $\begingroup$ +1 but from now on, please press the "code" button and put your output files in code blocks. I edited it for you this time (and last time), please take a look at the edit history! $\endgroup$ Feb 10 at 4:58
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In general, you should use the $S_1$ state by default, unless you have reasons to suspect that your molecule is anti-Kasha.

IMHO satisfying one of the following criteria guarantees that your molecule obeys the Kasha rule (supposing that your calculation is accurate), and you basically need not suspect otherwise:

  1. The experimental fluorescence spectrum mirrors the longest wavelength absorption peak, and the calculated $S_1$ state is a bright state. (This reasonably shows that the experimental fluorescence is from the $S_1$ state. Note that the latter criterion is important and is frequently overlooked by experimentalists, because if the $S_1$ state is dark, the longest wavelength absorption peak does not correspond to $S_1$ anymore.)
  2. The chromophore of the molecule closely resembles a system that is known to obey the Kasha rule.
  3. Optimizing the geometry of the $S_2$ state ends up at a $S_1$-$S_2$ conical intersection. (This means the $S_2$ to $S_1$ internal conversion is extremely fast, within ~ 1 ps, which prevents the $S_2$ state from fluorescing.)

Satisfying one of the following criteria, however, means that you must closely check if the fluorescence is anti-Kasha, for example by a full set of calculations of relevant internal conversion rates and fluorescence rates. Note the criteria alone are not sufficient for guaranteeing that the molecules does not obey the Kasha rule:

  1. The experimental fluorescence spectrum does not mirror the longest wavelength absorption peak, and instead mirrors a shorter wavelength absorption peak.
  2. Computations suggest that the $S_1$ state is dark (f < 0.0001) at the $S_1$ equilibrium.
  3. Optimizing the geometry of the $S_1$ state ends up at a $S_0$-$S_1$ conical intersection, however optimizing the $S_2$ state does not end up at a conical intersection.
  4. The computed $S_1$ emission wavelength is much longer than the experimental emission wavelength.
  5. The chromophore closely resembles a known anti-Kasha molecule, for example azulene.
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