Modeling phosphorescent lifetimes of OLED emitters

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, T₁ → S₀ ) 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.

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

1. Geometry optimization of the lowest triplet state (T₁)

2. SOC-TDDFT calculation at the optimized T₁ 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 (T₁)

Since the T₁ state is the lowest in the triplet manifold, the T₁ 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 T₁ 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

Relativistic spin-orbit coupling TDDFT calculation

To calculate the phosphorescent lifetime from the T₁ state, a SOC-TDDFT calculation is performed on the molecule in the T₁ optimized geometry. This method converges the singlet state first, and then uses response theory (TDDFT) to model the transition to and from the T₁ 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. 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 (SCF)

7. 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.

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.

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.

These SOCMEs are important parameters for designing phosphorescent OLED emitters: optimize internal conversion to T₁ 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 PBE functional was used. One can also use 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. To reduce the time in such a B3LYP calculation, without much loss of accuracy, you may use a bit coarser integration, ‘Becke Normal’ as was used in this example. 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 T₁ 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 recent 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