In organic light-emitting diodes (OLEDs), phosphorescent dyes increase the maximum theoretical efficiency from 25% to 100% with respect to fluorescent dyes.
To model the spin-forbidden phosphorescence from the triplet state to the singlet ground state, T_{1} → S_{0}, spin-orbit coupling must be included in TDDFT calculation (SOC-TDDFT).
With ADF it is possible to calculate phophorescent lifetimes with spin-orbit coupling TDDFT, as explained in this guided example for Ir(ppy)_{3}.

You may first download the sample input files, and if you don't have access to ADF yet, request a free trial.

The procedure for predicting phosphorescent lifetimes consists of two steps:

- Optimization of the lowest triplet state (T
_{1}) - SOC-TDDFT calculation at the optimized T
_{1}geometry

Since the T_{1} state is the lowest in the triplet manifold, the T_{1} geometry can be optimized as a regular ground state, but with triplet spin multiplicity.
Usually the triplet state is lower in symmetry than the ground state, so it is recommended to switch off symmetry (NoSym).
Two sample inputs for the triplet state geometry optimization of the model OLED phosphor Ir(ppy)_{3} are included. If you want to see how this is set up in the GUI, open irppy3.geo.adf. If you want to run it from the command line, run irppy3geo.run with ADF.

Note that in the example rather large basis sets are used (all electron TZ2P for Ir and TZP for H,C,N), although one typically can get away with smaller basis sets and/or frozen-cores for such a geometry optimization.

To set up your own triplet geometry optimization for a different complex, you can copy and paste the relevant parts from the command line example, or you can follow these steps in the GUI:

Open ADFJobs SCM → New Input In the Main menu options panel: Preset → Geometry Optimization Spin-polarization → 2.0 Tick the 'yes' box next to Unrestricted: XC potential in SCF: → select the desired XC functional (e.g. BP or B3LYP) Relativity (ZORA): → Scalar Basis set: → TZP Frozen core: → None Integration accuracy: teVelde 5 Details → Basis Menu: (or you can click on the ... button next to the basis set) Select basis file for Ir → ZORA/TZ2P/Ir

Sample inputs for the SOC-TDDFT calculations at the lowest triplet state geometry are provided. From the command line, you can run the composite calculation irppy3.run, while with the GUI you will perform two consecutive calculations: irppy3.so.stcontrib.SR.adf followed by irppy3.so.stcontrib.adf which uses the results of the first calcultion.

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). Oscillator strengths and radiative lifetimes for each excitation are printed in the output, they are related according to:

1/τ = 2(ΔE)This equation is in atomic units (a.u.: e=1,i m=1, h/2pi=1). Note that in a.u. c = 137.036, and 1 a.u. of time = 2.419E-17 sec. ΔE is the excitation energy, f the oscillator strength (both in a.u.!)^{2}f/c^{3}

Note that in this example moderately-sized basis sets have been used (all electron TZP for Ir and DZ for H,C,N). For this particular system that is sufficient, but one may need larger basis sets (TZ2P, QZ4P) for other systems.

In these examples STCONTRIB is used in the spin-orbit coupled calculation, such that one can calculate the contribution to the double group excited states in terms of singlet and triplet single group excited states. Alternatively one can do a SOC-TDDFT calculation straight away.

To set up a SOC-TDDFT for your own phosphorescent dye, start from the command line example, or follow these steps in the GUI.

In the Main menu options panel of ADFInput: Preset → Single Point Spin-polarization → 0.0 Make sure the 'yes' box isNOTticked, this is a spin-restricted calculation XC potential in SCF: → select the desired XC functional (e.g. BP or B3LYP) Relativity (ZORA): → Scalar Basis set: → DZ Frozen core: → None Integration accuracy: teVelde 6 Details → Basis Menu: (or you can click on the ... button next to the basis set) Select basis file for Ir → ZORA/TZP/Ir Properties → Excitation, CD Type of excitations → SingletAndTriplet Number of excitations → 20

In the Properties → Excitation, CD pane : Type of excitations → Spin-Orbit (SCF) Number of excitations → 5

In this example we will use the scalar-relativistic calculation as a fragment to analyze the singlet and triplet contributions:

In the Main input panel change Relativity (ZORA): from Scalar to Spin-Orbit Go to the Details → User Input pane Type 'STCONTRIB' Select all atoms (Ctrl + A) Go to the Models → Regions pane Click the '+' Go to the Multilevel → Fragments pane And tick the 'Yes' box In the input box next to Region_1, enter the name of the t21 of the previous scalar relativistic calculation (or browse for it by clicking on the folder symbol), this will be used as starting point for the SOC-TDDFT calculation

To have an idea what is in the output file, without actually running the calculation, open the example output irppy3.out either with your favorite text editor or with the ADFGUI.

If you open it with ADFGUI, it will take you to the output browser, with which you can jump through sections:

Menu → Section → Excitation Energies Select 'Excitation Energies' again if this is a composite SR+SOC calculation, as in irppy3.out.Here the spin-orbit coupled excitation energies are shown. Scroll further down to find the so called 'electric dipole radiative lifetimes' for each excitated state:

All SPIN-POLARIZED excitation energies no. E/a.u. E/eV f tau/s Symmetry ------------------------------------------------------------------ 1: 0.06504 1.76990 0.1362E-05 0.5402E-02 A 2: 0.06524 1.77523 0.4479E-03 0.1633E-04 A 3: 0.06656 1.81116 0.5218E-02 0.1346E-05 A 4: 0.06746 1.83566 0.2105E-02 0.3249E-05 A 5: 0.07366 2.00428 0.1063E-04 0.5396E-03 A tau: electric dipole radiative lifetime (in seconds)

Remember that tau and k_{rad} are interrelated:

1/τ = 2(ΔE)^{2} f/c^{3}, in a.u.

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

Input files are also provided for a perturbative SOC calculation: irppy3.sopert.adf (ADFGUI) and irppy3sopert.run (command line).
Note that the key SOPERT is used in the scalar relativistic calculation of excitation energies.

The spin-orbit coupling matrix elements (SOCME) can be printed if you include PRINT SOMATRIX in your input.

The use of the XC functional is relevant for the accuracy of the results. In this example the BP 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' or 'teVelde 4', as well as reducing the calculated Number of excitations to 3.

We find good agreement (K. Mori et al., Phys. Chem. Chem. Phys. **16**, 14523-14530 (2014)) 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.

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.F. Wang, T. Ziegler, E. van Lenthe, S.J.A. vand Gisbergen and E.J. Baerends,

Perturbative approach to include spin-orbit coupling:

F. Wang and T. Ziegler,

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