CD and CPL for spin-forbidden electronic transitions
****************************************************

ADF Spectroscopy Circularly polarized luminescence Spin-orbit coupling

Circularly polarized luminescence (CPL) dissymmetry factors and
phosphorescence lifetimes are calculated for the spin-forbidden
triplet emission of a platinahelicene complex, as is done in Ref. [1]
on which this Tutorial is based. Spin-orbit coupling is included in
the relativistic time-dependent density functional calculations.
Emission from a triplet state to a singlet state is spin forbidden,
however, due to spin-orbit coupling such transitions may occur.

For phosphorescence one needs to compute the geometry of the lowest
excited triplet state (T_1), see for example the Tutorial Vibrational
progression of an OLED phosphorescent emitter. Here the T_1 optimized
structure of platinahelicene *P*-**3 c** will be taken from Ref. [1].

   Start AMSinput.
   Obtain the platinahelicene *P*-**3 c** by copy-pasting the following coordinates into AMSinput:

   Pt 1.88557 -0.18819 -0.09184
   N  0.53248 -1.39044 -0.93333
   O -0.83139  4.50753 -0.36816
   O  3.16072  1.14825  0.73682
   O  3.42989 -1.63253 -0.16462
   C  0.38349  1.05819 -0.12840
   C  0.40946  2.43348 -0.07897
   H  1.35956  2.94635  0.08566
   C -0.78729  3.16948 -0.27172
   C -2.06683  2.51788 -0.28913
   C -3.26016  3.26301 -0.43357
   H -3.19230  4.33714 -0.61567
   C -4.48585  2.63006 -0.39893
   H -5.40156  3.20093 -0.58304
   C -4.59331  1.25914 -0.08686
   C -5.87212  0.62486 -0.02698
   H -6.75377  1.21811 -0.28983
   C -6.00026 -0.68193  0.33815
   H -6.98145 -1.16545  0.35914
   C -4.85822 -1.41901  0.79804
   C -5.01464 -2.70951  1.35136
   H -6.00988 -3.16539  1.34100
   C -3.94932 -3.37509  1.92985
   H -4.09079 -4.36918  2.36459
   C -2.69421 -2.74701  1.99285
   H -1.85786 -3.23921  2.49787
   C -2.51273 -1.49672  1.42702
   H -1.53683 -1.01466  1.51801
   C -3.56458 -0.81800  0.77077
   C -3.40060  0.49872  0.16186
   C -2.14212  1.09766 -0.13877
   C -0.89804  0.39244 -0.39652
   C -0.75395 -0.88046 -1.01165
   C -1.74669 -1.62254 -1.69388
   H -2.74468 -1.19458 -1.80010
   C -1.44205 -2.85403 -2.23570
   H -2.20476 -3.42159 -2.77596
   C -0.13832 -3.36681 -2.09214
   H  0.14108 -4.34342 -2.49279
   C  0.81493 -2.59285 -1.44363
   H  1.85025 -2.92292 -1.32054
   C  0.36036  5.25424 -0.31870
   H  0.07055  6.30544 -0.46034
   H  1.05601  4.95538 -1.12332
   H  0.86335  5.14614  0.65906
   C  5.09120  2.13040  1.60594
   H  4.57183  2.42528  2.53454
   H  5.02604  2.98776  0.91312
   H  6.14843  1.92825  1.83129
   C  4.39181  0.94494  0.99921
   C  5.11332 -0.23796  0.77878
   H  6.16563 -0.22805  1.06805
   C  4.61920 -1.43558  0.22324
   C  5.55256 -2.60481  0.05850
   H  5.14292 -3.47731  0.59695
   H  6.56771 -2.39569  0.42671
   H  5.60487 -2.88044 -1.00966

[image]


Relatively cheap settings
=========================

As the settings that are used in the paper will lead to a long
calculation, first a gas phase PBE calculation will be performed with
a DZ frozen core basis set, using spin-orbit coupled TD-DFT with TDA
to calculate rotatory strengths. This is a relatively cheap
calculation and thus does not take so long.

   Use the Main [image: ADFPanel][image] panel
   **XC functional →  GGA →  PBE**
   **Relativity →  Spin-Orbit**
   Use the panel bar **Properties →  Excitations (UV/Vis), CD**
   **Type of excitation →  Spin-Orbit (SCF)**
   Check the 'TDA' checkbox
   Enter "3" for the 'Number of excitations`
   Check the calculate 'Rotatory strengths (CD)' checkbox
 [image][image]

We are now ready to run the calculation.

   Save your input as "P-3c_pbe_dz" using **File →  Save As...**.
   Run the calculation with **File →  Run**.
   Wait for the calculation to finish.
   Open [image: SCMMenu][image] **→  Output**
   In AMSoutput, search for "Rotatory Strengths"
   In AMSoutput, search for "lifetime"
 [image][image]

The calculated results are quite sensitive to small numerical changes,
since we are looking at a small effect.

The averaged dissymmetry factor for the emission can be computed using
rotatory and dipole strengths that are Boltzmann-averaged over the
triplet components:

   g_{lum}^{av} = 4 \frac{R_1 + e^{-(\Delta E_{1,2}/k_B T)} R_2 +
   e^{-(\Delta E_{1,3}/k_B T)} R_3}{D_1 + e^{-(\Delta E_{1,2}/k_B T)}
   D_2 + e^{-(\Delta E_{1,3}/k_B T)} D_3}

In this case the energy of the triplet components are very close
(1.75762, 1.75808, and 1.75866 eV), thus we will use

   g_{lum}^{av} = 4 \frac{R_1 + R_2 + R_3}{D_1 + D_2 + D_3} = 5.5 ~
   10^{-4}

Also the averaged radiative lifetime can be computed using lifetimes
that are Boltzmann-averaged over the triplet components:

   \tau^{av} = \frac{1 + e^{-(\Delta E_{1,2}/k_B T)} + e^{-(\Delta
   E_{1,3}/k_B T)}}{1/\tau_1 + e^{-(\Delta E_{1,2}/k_B T)}/\tau_2 +
   e^{-(\Delta E_{1,3}/k_B T)}/\tau_3}

In this case we will use again that the energy of the triplet
components are very close, thus we will use

   \tau^{av} = \frac{3}{(1/\tau_1 + 1/\tau_2 + 1/\tau_3)} = 147 ~ \mu
   s

Note that here no refractive index of a solvent is included, as this
is a gas phase calculation.

Tip:

  Insert the values into the below "Python script" and use it to
  calculate the values:

     #!/usr/bin/env amspython
     import numpy as np
     from scm.plams import Units


     def get_values():
         E_eV = np.array([1.7576, 1.7581, 1.7587])
         D = np.array([4.6005, 26.599, 196.61])
         R = np.array([-0.34615e-1, -0.35678e-1, 0.10183])
         tau = np.array([0.2433e-2, 0.4205e-3, 0.5683e-4])
         refractive_index = 1
         return E_eV, R, D, tau, refractive_index


     def main():
         E_eV, R, D, tau, refractive_index = get_values()

         use_boltzmann_weights = False

         if use_boltzmann_weights:
             weights = calc_boltzmann_weights(E_eV, T=298)
         else:
             weights = np.ones(np.shape(E_eV))

         g = 4 * np.dot(R, weights) / np.dot(D, weights)
         print(f"g_lum_av: {g:.3e}")

         tau_av = 1.0 / refractive_index**2 * np.sum(weights) / np.dot(1.0 / tau, weights)
         print(f"tau_av: {tau_av:.3e} s")


     def calc_boltzmann_weights(E_eV, T=298):
         k_B = Units.constants["k_B"] * Units.convert(1, "J", "eV")  # eV/K
         boltzmann_weights = np.exp(-(E_eV - np.min(E_eV)) / (k_B * T))
         return boltzmann_weights


     if __name__ == "__main__":
         main()


Settings as in the paper
========================

Next we will try to use the same settings as are used in Ref. [1]:
spin-orbit coupled TD-DFT with TDA to calculate rotatory strengths,
the hybrid PBE0 XC functional, full XC kernel, a TZ2P basis all
electron basis set on Pt, DZ all electron on H, and DZP all electron
on all other elements, dichloromethane as COSMO solvent, and numerical
quality good. This is a much more demanding calculation than with the
previous settings.

   Use the Main [image: ADFPanel][image] panel
   **XC functional →  Hybrid →  PBE0**
   **Relativity →  Spin-Orbit**
   **Basis set →  DZP**
   **Frozen Core →  None**
   **Numerical Quality →  Good**
   Use the panel bar **Properties →  Excitations (UV/Vis), CD**
   **Type of excitation →  Spin-Orbit (SCF)**
   Check the 'TDA' checkbox
   Enter "3" for the 'Number of excitations'
   Check the calculate 'Rotatory strengths (CD)' checkbox
 [image][image]

   Use the panel bar **Model →  Solvation**
   **Solvation method →  COSMO**
   **COSMO solvent →  Dichloromethane**
   Use the panel bar **Details →  Basis**
   Enter "ZORA/TZ2P/Pt" for Pt
   Enter "ZORA/DZ/H" for H
   Use the panel bar **Details →  XC Details**
   Check the 'Use full XC kernel' checkbox
 [image][image][image]

We are now ready to run the calculation. As this may take several
hours on a cluster, and maybe even longer on a desktop, one could also
skip the calculation, and directly look at the results below.

   Save your input as "P-3c_pbe0" using **File →  Save As...**.
   Run the calculation with **File →  Run**.
   Wait for the calculation to finish
   In AMSoutput, search for "Rotatory Strengths"
   In AMSoutput, search for "lifetime"
 [image][image]

The averaged dissymmetry factor for the emission can be computed using
rotatory and dipole strengths that are Boltzmann-averaged over the
triplet components. In this case the energy of the triplet components
are very close (1.79177, 1.79194, and 1.79222 eV), thus we will use

   g_{lum}^{av} = 4 \frac{R_1 + R_2 + R_3}{D_1 + D_2 + D_3} = 1.78 ~
   10^{-3}

Also the averaged radiative lifetime is calculated without using
Boltzmann factors as

   \tau^{av} = \frac{1}{n^2} \frac{3}{(1/\tau_1 + 1/\tau_2 +
   1/\tau_3)} = 68 ~ \mu s

where n=1.42 is the refractive index of the dichloromethane solvent
used in the measurements. For comparison, experimental data reported
in Ref. [1] for *P*-**3 c** are 1.93 eV for the transition energy, a
phosphorescence lifetime of 21 \mu s, and a g_{lum} of 3 10^-3, see
also the discussion of the results in the paper.

[1] H.D. Ludowieg, M. Srebro-Hooper, J. Crassous, J. Autschbach,
    *Optical Activity of Spin-Forbidden Electronic Transitions in
    Metal Complexes from Time-Dependent Density Functional Theory with
    Spin-Orbit Coupling*, ChemistryOpen 2022, 11, e202200020
