Modeling phosphorescent lifetimes of OLED emitters
**************************************************

[image]ADF OLEDs phosphorescence

In organic light-emitting diodes (OLEDs), phosphorescent dyes can be
advantageous compared to fluorescent dyes because of the increase in
the maximum theoretical efficiency from 25% to 100%. Both can me
modeled with AMS, but to model the spin-forbidden phosphorescence
(which involves a transition from the triplet state to the singlet
ground state, T_1 → S_0 ) spin-orbit couplings must be included in
TDDFT calculation (SOC-TDDFT). This guided example explains how to use
ADF to calculate phophorescent lifetimes using SOC-TDDFT, in the case
of famous OLED emitter, the 5d transition metal complex Ir(ppy)_3 .


Example input and output files
==============================

While this example is self-contained, it is still helpful to download
the "starting geometry". Note that this starting geometry is
specifically the Λ-*fac*-Ir(ppy)_3 isomer, though other isomers can
also be investigated in the same way.

Note:

  This example may take quite some time, we recommend running the
  calculations on a HPC cluster if possible.

Essentially the procedure for predicting phosphorescent lifetimes
consists of two steps:

1. Geometry optimization of the lowest triplet state (T_1)

2. SOC-TDDFT calculation at the optimized T_1 geometry

In ADF there are two ways to include spin-orbit couplings in a
calculation: you can do a fully self-consistent variational SOC-TDDFT
calculation, or do a perturbative SOC (pSOC) calculation. A pSOC-TDDFT
calculation offers insight in the underlying singlet and triplet
contributions to the excitations, which can aid with designing ligands
which yield better emission properties. Similarly, insight in singlet
and triplet contributions may also be achieved by performing a full
SOC calculation employing a scalar-relativistic fragment of the full
molecule, as explained further below.


Optimization of the lowest triplet state (T_1)
==============================================

Since the T_1 state is the lowest in the triplet manifold, the T_1
geometry can be optimized just as a regular ground state, but with
triplet spin multiplicity. Usually the triplet state can break the
symmetry of the singlet ground state, so it is recommended to switch
off symmetry (NoSym). See also the paper by Gonzalez-Vazquez et al.
[1] where the importance of relaxing the T_1 state is discussed.

In the example we optimize the triplet state with the B3LYP
functional, a TZP basis set, scalar ZORA for relativistic effects and
we set the numerical quality to Good. Just follow these steps in the
GUI:

   1. Start AMSinput via **SCM → New Input**
   2. Build or import your structure
   3. In the **Main** panel select **Task: GeometryOptimization**
   4. Tick the **Unrestricted** box and set the Spin polarization to 2.0
   5. Select **XC functional: Hybrid:B3LYP**
   6. Select the **TZP** basis set
   7. Set **Numerical quality** to **Good**
   8. **Save** the input and **Run** the calculation

Note:

  You may ignore warnings about using hybrid functionals and frozen
  core in this case since we are not interested in the properties of
  core electrons

[image]

Tip:

  Want a faster geometry optimization? You can use a lower level of
  theory such as PBE-D3/DZP which can be orders of magnitude faster
  and only use the higher level of theory for the phosphorescence
  calculations


Relativistic spin-orbit coupling TDDFT calculation
==================================================

To calculate the phosphorescent lifetime from the T_1 state, a SOC-
TDDFT calculation is performed on the molecule in the T_1 optimized
geometry. This method converges the singlet state first, and then uses
response theory (TDDFT) to model the transition to and from the T_1
state.

Typically the lowest 3 states of the spin-orbit coupling TDDFT
calculation are important which resemble the 3 states in a triplet
state. There will be a small energy difference between the three
states because of spin-orbit coupling, the so called zero-field
splitting (ZFS). The oscillator strengths and radiative lifetimes \tau
for each transition are printed in the output, and are related
according to:

   \frac{\hbar}{\tau} = 2 \frac{(\Delta E)^2}{m_\text{e} c^2} \alpha f

Where \hbar is the reduced Planck constant, \Delta E is the energy
difference between the two states, m_\text{e} is the electron mass, c
is the speed of light, \alpha is the fine structure constant, and f is
the oscillator strength of the transition.

To set up a SOC-TDDFT for your own phosphorescent dye you can first
load the optimized triplet structure obtaine in the previous step and
follow these instructions in the GUI:

   1. In the **Main** panel select **Task: SinglePoint**
   2. Set the **Relativity** level to **Spin-Orbit**
   3. Make sure **Spin-Polarization** is set to **0** and **Unrestricted** is unticked
   4. Set the **Frozen Core** level to **None**
   5. Select the functional and basis set (could use the previous selection)
   6. Go to **Properties** and select **Excitations (UV/Vis)/CD**
   7. Set **Type of excitations: → SpinOrbit (SCF)**
   8. Set **Number of excitations: → 5**

Save then run the calculation.


Output: phosphorescent lifetimes
================================

The ouput file may be opened either with your favorite text editor or
with the ADFGUI.

If you open it with the GUI, it will take you to the output browser.
You can search for ‘tau’ to jump straight to the phosphorescent
lifetimes.

If you scroll up from the radiative lifetimes you will find the major
MO → transitions.

[image]

A cheaper alternative (ca. 5 times faster) to full spin-orbit coupling
calculations is the approximate, perturbative SOC scheme. This seems a
reasonable approximation for phosphorescence radiative lifetime
calculations, as for instance found in this highlight on OLED
phosphorescence. [6]


Perturbative SOC: spin-orbit coupling matrix (SOCME)
====================================================

The pSOC method does not include spin-orbit coupling in the ground
state at all, instead they are included purely in the response
calculation using perturbation theory. This leads to considerable
computational savings.

Setting up the calculation is simple (see instruction below). One
difference is the fact that more excitations should be converged,
since they are used in the perturbative expansion. Increasing the
number of converged states leads to better convergence, at the price
of a higher computational cost.

   1. In the **Main** panel select **Task: SinglePoint**
   2. Set the **Relativity** level to **Scalar**
   3. Set the **Frozen Core** level to **None**
   4. Select the functional and basis set (could use the previous selection)
   5. Go to **Properties** and select **Excitations (UV/Vis)/CD**
   6. Set **Type of excitations: → SpinOrbit (Perturbative)**
   7. Set **Number of excitations: → 20**

You may then save and run your calculation. The results are presented
the same as before, though be aware that in this case singlets,
triplets, and perturbative states are calculated sequentially
therefore you will have to scroll down and right before the lifetimes
you can find the spin-orbit couplings matrix elements.

These SOCMEs are important parameters for designing phosphorescent
OLED emitters:  optimize internal conversion to T_1 from the singlet
exciton and minimize non-radiative decay. Likewise, SOCMEs are useful
figure of merits to design better TADF (thermally activated delayed
fluorescence) emitters. [5]


Accuracy considerations of the phosphorescent lifetime predictions
==================================================================

The use of the XC functional is relevant for the accuracy of the
results. In this example the B3LYP functional was used. One can also
use other hybrids by chosing the appropriate fucntional in the GUI
(main panel) or setting the appropriate functional in the XC block in
your input. Hybrid functionals such as B3LYP seem to give quite good
results, however, they will take more CPU time compared to pure
functionals. You may even use frozen cores, although this is not
strictly correct for hybirds.

We find good agreement between calculated and experimental radiative
lifetimes for various phosphorescent transition metal complexes when
we use the following settings:

* Strict presets: ExactDensity, FullSCF, gradients 1e-4, SCF 1e-6,
  Becke ‘Good’ grid, DEPENDENCY bas=1e-3

* All-electron DZP basis (defaults to TZP for transition metals)

* Optimize T_1 with reduced symmetry, or NoSym

* B3LYP functional

* 20 excitations for the SR-TDDFT calculation, 4 for the STCONTRIB
  SOC-TDDFT calculation

Further accuracy considerations may be to employ a larger basis set
(TZ2P) and to include continuum solvation with COSMO. Solvation
effects may especially improve the accuracy of ZFS and lifetime
calculations [2] .

In a study by Younker and Dobbs, a good correlation is found between
calculated and observed phosphorescent rates of Ir(III) complexes when
using a pSOC-TDDFT approach with B3LYP on the BP86-optimized singlet
ground state. [6]


References
==========

[1] **Importance of geometry in triplet excited states**: J. P.
    Gonzalez-Vazquez, P. L. Burn and B. J. Powell, Interplay of Zero-
    Field Splitting and Excited State Geometry Relaxation in fac-
    Ir(ppy)3 *Inorg. Chem.* 54 (2015) 10457-10461

[2] **Comparison of calculations and experiment**: K. Mori, T. P. M.
    Goumans, E. van Lenthe, and F. Wang, Predicting phosphorescent
    lifetimes and zero-field splitting of organometallic complexes
    with time-dependent density functional theory including spin–orbit
    coupling *Phys. Chem. Chem. Phys.* 16 (2014) 14523-14530.

[3] **Excitations including spin-orbit coupling**: F. Wang, T.
    Ziegler, E. van Lenthe, S.J.A. van Gisbergen and E.J. Baerends,
    The calculation of excitation energies based on the relativistic
    two-component zeroth-order regular approximation and time-
    dependent density-functional with full use of symmetry *J. Chem.
    Phys.* 112 (2005) 204103

[4] **Perturbative approach to include spin-orbit coupling**: F. Wang
    and T. Ziegler, A simplified relativistic time-dependent density-
    functional theory formalism for the calculations of excitation
    energies including spin-orbit coupling effect , *J. Chem. Phys.*
    113 (2005) 154102

[5] **Use of SOCMEs in TADF**: P.K. Samanta, D. Kim, V. Coropceanu,
    and J.-L. Brédas, Up-Conversion Intersystem Crossing Rates in
    Organic Emitters for Thermally Activated Delayed Fluorescence:
    Impact of the Nature of Singlet vs Triplet Excited States *J. Am.
    Chem. Soc.* 139 (2017) 4042-4051

[6] **Using different method for properties geometries**: J. M.
    Younker and K. D. Dobbs, Correlating Experimental Photophysical
    Properties of Iridium(III) Complexes to Spin–Orbit Coupled TDDFT
    Predictions , *J. Phys. Chem. C* 117 (2013) 25714–25723
