Cu(I) complexes – Thermally activated delayed fluorescence. Photophysical approach and material design

Abstract Cu(I) complexes often show transitions of distinct metal-to-ligand charge transfer (MLCT) character. This can lead to small energy separations between the lowest singlet S 1 and triplet T 1 state. Hence, thermally activated delayed fluorescence (TADF) and, if applied to electroluminescent devices, singlet harvesting can become highly effective. In this contribution, we introduce the TADF mechanism and identify crucial parameters that are necessary to optimize materials’ properties, in particular, with respect to short emission decay times and high quantum yields at ambient temperature. In different case studies, we present a photophysical background for a deeper understanding of the materials’ properties. Accordingly, we elucidate strategies for obtaining high quantum yields. These are mainly based on enhancing the intrinsic rigidity of the complexes and of their environment. Efficient TADF essentially requires small energy separations Δ E (S 1 –T 1 ) with preference below about 1000 cm −1 (≈120 meV). This is achievable with complexes that exhibit small spatial HOMO–LUMO overlap. Thus, energy separations below 300 cm −1 (≈37 meV) are obtained, giving short radiative TADF decay times of less than 5 μs. In a case study, it is shown that the TADF properties may be tuned or the TADF effect can even be turned off. However, very small Δ E (S 1 –T 1 ) energy separations are related to small radiative rates or small oscillator strengths of the S 1  → S 0 transitions due to the (required) small HOMO–LUMO overlap, as discussed in a further case study. Moreover, large spin–orbit coupling (SOC) of the triplet state to higher lying singlet states can induce an additional phosphorescence decay path that leads to a luminescence consisting of TADF and phosphorescence, thus leading to a combined singlet harvesting and triplet harvesting mechanism. This gives an overall reduction of the decay time. Finally, in a strongly simplified model, the SOC efficiency is traced back to easily obtainable results from DFT calculations.

At first glance, Cu(I) complexes seem to show substantial photophysical problems with respect to OLED applications: (i) Compared to iridium, the 1st row transition metal copper induces much weaker spin-orbit coupling [108]. As a consequence, transitions between the excited triplet state and the singlet ground state are largely forbidden.
Thus, long phosphorescence decay times of several 100 s up to a few ms are frequently found [2,4,5,17,38,61,65,67,69,73,[109][110][111]. With such long decay times, OLEDs would suffer from strong saturation effects and photochemical reactions that may become significant in reducing the device stability [112][113][114][115]. (ii) After the electronic excitation, Cu(I) complexes often display flattening distortions of the molecular structure. [116][117][118][119][120][121][122][123][124][125] Such rearrangements are usually connected with an increase of non-radiative deactivation or even quenching of the emission due to a strong increase of the Franck-Condon factors of the low-lying vibrational modes of the excited state and the highly excited vibrational modes of the electronic ground state [126][127][128]. However, these shortcomings may largely be suppressed by optimizing properties of the thermally activated delayed fluorescence (TADF) (introduced in section 2 and discussed in detail in section 4) and by designing relatively rigid molecular structures, as will be addressed in section 3. We will focus in this contribution on corresponding optimization strategies being illustrated by case studies. It will further be shown that the TADF behavior, being strongly dependent on the energy separation between the lowest singlet and triplet state, can be varied over a very large range by chemical "tuning" (section 4). In this respect, we will discuss in section 5 a correlation existing for charge transfer states between this energy separation and the allowedness of the transition from the thermally activated singlet state to the electronic ground state.
By further case studies in sections 6 and 7, it will be demonstrated that SOC can strongly influence the emission behavior of the complexes. In these case studies, we will analyze corresponding SOC paths and uncover strategies to improve the material's emission properties resulting in a combined singlet harvesting and triplet harvesting mechanism.

Thermally activated delayed fluorescence and singlet harvesting
The TADF mechanism represents a molecular property that was initially described based on the emission behavior of eosin. Therefore, the observed delayed fluorescence was originally denoted as E-type (eosin-type) fluorescence. [129,130] By 6 use of Figure 1, this effect is illustrated for Cu(I) complexes. Let us first discuss an optical excitation process. For example, an energetically higher lying singlet state is excited. Subsequently, a very fast internal conversion (IC) to the S1 state of the order of 10 12 s [126,131,132] takes place (not shown in the diagram). ISC (S1→T1) is also very effective and relatively short (ISC) times, lying between approximately 3 and 30 ps depending on the individual Cu(I) complex and its local environment [81,120,123,124,133], have been reported. Accordingly, a significant prompt fluorescence (S1→S0) is not detected [4,5,44,45,61,65,[67][68][69][70][71][72], but a very bright longlived phosphorescence (T1→S0) is frequently observed at low temperature. OLEDs. [2,4] Note that the triplet state T1, consisting of three substates, is slightly split, mostly of less than 1-2 cm 1 but in a special case even 15 cm 1 . [44]  With temperature increase, the S1 state can be populated depending on the thermal energy that is available according to kBT (≈ 210 cm -1 at T = 300 K, 26 meV) and the energy separation E(S1-T1). In a situation of a fast equilibrationas usually realized in Cu(I) complexes above T ≈ 15 K [4,5,44,45,61,63,[68][69][70][71][72] population of the higher lying state is governed by a Boltzmann distribution, whereby "fast" means that the thermal equilibration is faster than all emission processes. (See also below in this section.) As a consequence of this process of up-ISC (or reverse ISC, RISC) and the (mostly) relatively short-lived S1 state, the T1 state can efficiently be depopulated at higher temperature and a strongly dominating S1 fluorescence can be observed. Since the population of the S1 state is fed from the long-lived triplet reservoir, this type of fluorescence is also long-lived. Hence, this emission is denoted as thermally activated delayed fluorescence. 1 It is remarked that the energy separation E(S1-T1) strongly governs the TADF properties. Although the E(S1-T1) value for eosin is as large as  10 kcal/mol ( 3500 cm 1 ;  0.43 eV) [130] one still detects TADF. However, for practical use of TADF compounds in OLEDs, E(S1-T1) values should be much smaller andto give the reader a rough orientation  should not exceed considerably 10 3 cm 1 (0.12 eV). A number of examples are presented below.
Herein, (S1) and (T1) represent the singlet state and the triplet state decay times, respectively, and E(S1-T1) is the energy separation between these states. At very low temperature, the exponential terms disappear and the measured decay time (T) equals the phosphorescence decay time (T1), while at high temperature (and long (T1)) the term containing (T1) can be neglected and one essentially obtains the decay time of the thermally activated delayed fluorescence (TADF). Examples (and exceptions) are discussed below.
The emission decay time as expressed by eq. (1) depends on three parameters that are determined by the individual emitter compound (and its environment). A short 9 discussion of these parameters is illustrative and, as shown below, can help in designing suitable TADF emitters: The energy separation between the lowest singlet and triplet excited state is crucial and should be as small as possible. For this situation, Cu(I) complexes are well suited, since they frequently exhibit low-lying metal-to-ligand charge transfer (MLCT) states of 3 MLCT and 1 MLCT character. In this situation, a distinct charge separation between excited and non-excited electron occurs. As a consequence, the quantum mechanical exchange interaction [142] [143, p. 86] and hence, also the singlet-triplet splitting becomes small. For example, in section 4, we will investigate the resulting properties in detail. Moreover, in section 5 we will point to a relation that exists between E(S1-T1) and the allowedness of the S1↔S0 transition or the S1 decay rate.

(S1)
The S1↔S0 allowedness, displayed, for example, by the (radiative) fluorescence decay time (S1) should be as short as possible. However, for most materials investigated so far, a short (S1) time is related to a large E(S1-T1) value. This is not a favorable situation and therefore, special efforts are required to overcome this shortcoming.
(Compare the discussion in sections 5 and 6.) The (radiative) emission decay time of the triplet state should also be "tuned" to be as short as possible. In this way, an effective radiative decay channel can be opened in addition to the TADF path. To achieve this aim, SOC of the T1 state to higher lying 10 singlet states should be enhanced. Based on case studies (sections 4.2 and 6), we will show that this can be realized. [44,45] For completeness, it is remarked that application of eq. (1) to decay time data measured at different temperatures opens experimental access to all the parameters discussed above. In particular, it becomes possible to detect energy separations that are much smaller than the spectral resolution accessible for a given compound. For example, energy separations between states being as small as only a few cm 1 can be resolved despite the MLCT emission bands being as broad as several thousand cm 1 .( Compare sections 4 and 6 and refs. [2,4,5,[139][140][141] The amount of the energy separation E(S1-T1) has a particularly strong impact on the emission properties of the TADF materials. Accordingly, this feature represents one of the main topics of this study. Thus, compounds discussed in this contribution are classified according to their E(S1-T1) values ( Figure 2). In the different case studies, we will refer to these materials and use them as examples for discussion of different aspects of their photophayical behavior. In Table 1, we summarize the compounds, their abbreviations, and the experimentally determined E(S1-T1) values.

Fig. 2:
Chemical structures of the TADF compounds discussed in this contribution. They are arranged according to the energy separations between the lowest singlet S1 and triplet T1 state E(S1-T1). These states are essentially of 1 MLCT and 3 MLCT character, respectively. Related abbreviations of the compounds and the values of E(S1-T1) are given in Table 1.  [38,61,65,72,109,122,149,153] In this section, we present two case studies discussing neutral and cationic copper(I) complexes and exemplifying the validity of this strategy for chemical engineering of strongly emissive TADF materials.

13
The charge-transfer character of the lowest excited states of the complexes can be easily elucidated by inspection of the their frontier orbitals resulting from DFT calculations, as shown in Figure 3 for are much smaller. The largest observed decrease of k r (complex 8, powder compared to PMMA) is smaller than a factor of four. This effect may be induced by small changes of the TADF properties by polarity differences of the various environments.
An MLCT transition is characterized by a significant electron density shift from the Cu(I) ion, with a d 10 configuration, to the ligand. This results in a partial oxidation of the metal.
Since for a d 9 configuration, usually a square-planar-like coordination is preferred, a four-fold coordinated Cu(I) complex has the tendency to undergo a flattening distortion. [109,116,[120][121][122][123][124]. Such a structural rearrangement results in a shift of the excited state to lower energy. Moreover, non-radiative relaxation processes to the ground state become more efficient due to enhanced vibrational coupling. [126][127][128] The geometry rearrangement in the excited state occurs most easily in fluid solution, but it is partly hindered in an organic polymer (PMMA) and, even more, in powder samples.
Obviously, high local rigidity of the environment cage significantly hinders larger distortions in polymer or crystal. However, minor geometry changes are still possible also in rigid environments (see below). ligand, as compared to the much smaller pz2BH2. Apparently, the phenyl groups serve as an "internal" rigidification of the molecule, in addition to the effects exerted by the rigid matrix.
Similar trends are observed for cationic heteroleptic Cu(I) complexes displayed in Table   3. Several of these mixed-ligand complexes exhibit intrinsic rigidity due to the steric requirements of the chelating diphosphine and diimine ligands. In particular, the diphosphine ligands pop and phanephos 3 with relatively wide P-Cu-P bite angles form rigid "semicages" for the metal ion coordinated by the second ligand. [61,[152][153][154] Moreover, as can be seen, for instance, for the molecular structure of Cu(dmp)(phanephos) + 15, small aliphatic groups in the 2 and 9 positions of phenanthroline ( Figure 5 and Table 3) and the 6 and 6' positions of bipyridine (Table 3, compound 9) proved to be crucial for "anchoring" of the diimine ligands with the framework provided by the bulky groups of the second ligand.  [61,146] a Citations refer to the compounds. b The emission decay kinetics distinctly deviate from a mono-exponential decay. Therefore, the , k r , and k nr parameters were not determined.   [150,151] level of theory. Hydrogen atoms are omitted for clarity. HOMO and LUMO exhibit distinctly different spatial distributions. The HOMO is mainly composed of the copper 3d and phosphorus sp 3 atomic orbitals, while the LUMO represents a * orbital of the dmp ligand. Adapted from [61] with permission from The Royal Society of Chemistry.

20
The lowest excitations of all cationic complexes shown in Table 3 have distinct MLCT character with significant electron density shifts from copper (HOMO) towards the diimine ligands (LUMO). (Figure 6) Thus, flattening distortions upon excitation can be anticipated which, for several complexes, result in quenching of the emission. Indeed, the complexes Cu(dmbpy)(pop) + and Cu(phen)(pop) + are only very weakly emissive in solution with quantum yields of less than 1% and non-radiative rates k nr of the order of 10 6 -10 7 s 1 . The radiationless relaxations to the ground state can be suppressed to some degree by increasing the (external) rigidity induced by the matrix. However, even for the powder material, complexes comprising diimine ligands without any substitution that induces sterical hinderences with the second ligand, represent poor emitters. For instance, the PL value of the powder sample of Cu(dmbpy)(pop) + at ambient temperature amounts to only about 9 % (Table 3), which is much below the requirements for emitter materials for efficient OLEDs.
Cu(dmbpy)(pop) + and Cu(phen)(pop) + can be modified by introducing methyl groups into the 6 and 6' positions of bipyridine and the 2 and 9 positions of phenanthroline, giving Cu(tmbpy)(pop) + 9 and Cu(dmp)(pop) + , respectively (Table 3). Such modified diimine ligands should have stronger steric interactions with the pop ligand and thus, limit excited state distortions. Consequently, the efficiency of radiationless deactivation to the ground state should be reduced and the emission red shift induced by distortions of the molecular geometry should be less pronounced. Indeed, both effects, the increase of PL (accompanied by an increase of the emission decay time ) as well as the blue shift of the emission, as compared to the unsubstituted complexes, are observed. In solution, the quantum yields of Cu(tmbpy)(pop) + 9 and Cu(dmp)(pop) + increase to 6 and 15 % (from distinctly less than 1 % in the case of Cu(dmbpy)(pop) + 9 and Cu(phen)(pop) + ), respectively. The emission spectra are blue shifted to max = 575 and 570 nm, compared to 655 and 700 nm for Cu(dmbpy)(pop) + and Cu(phen)(pop) + , respectively. [72,153] Apparently, such a ligand modification with methyl groups results in a reduction of the nonradiative rate k nr by two orders of magnitude. For Cu(dbp)(pop) + , with the 2 and 9 positions of phenanthroline substituted with n-butyl groups being sterically more demanding then the methyl group, further decrease of the nonradiative rate is expected to occur. Indeed, the effects on k nr are slightly more distinct than for Cu(dbp)(pop) + with k nr = 5.310 4 s 1 than for Cu(dmp)(pop) + with k nr = 6.110 4 s 1 (both in CH2Cl2). This strategy has also been successfully applied to homoleptic phenanthroline Cu(I) complexes. [149] Moreover, such substitutions prevent solvent molecules from coming closer to the metal center, and thus, are expected to reduce solvent-related excited-state relaxations. [160,161] It is remarked that introduction of sterically more demanding groups, such as branched aliphatic groups  (Table 3) Thus, the increase of the quantum yield of 15 is related to an increase of the radiative rate k r by a 22 factor of four as compared to Cu(dmp)(pop) + . This effect may be a consequence of a higher TADF efficiency of Cu(dmp)(phanephos) + 15 than for the other two compounds.
Similarly to the neutral complexes discussed above, the excited state distortions of complex 15 can be further reduced by increasing the matrix rigidity. Thus, the emission is blue shifted from max = 558 nm in dichloromethane to max = 535 and 530 nm in PMMA and for a powder sample, respectively. However, the value of max = 28 nm ( 950 cm 1 ) (fluid solution  powder) is relatively small. (Table 3.) In parallel, the radiationless relaxation rates in powder are only four times smaller than in solution.
Thus, both results indicate that in Cu(dmp)(phanephos) + 15, the excited state distortions are already confined at the molecular level. [61] For completeness, it is remarked that aliphatic groups (methyl, n-butyl) have electrondonating character. Thus, introduction of such groups might increase the LUMO energy and consequently lead to a blue shift of the corresponding transitions. Indeed, slight blue shifts of the MLCT absorptions are observed both from Cu(dmbpy)(pop) + towards Cu(tmbpy)(pop) + [72] and from Cu(phen)(pop) + towards Cu(dmp)(pop) + /Cu(dbp)(pop) + [153]. In emission, blue shifts of similar size would be expected. However, they are much larger, pointing to molecular rigidity effects as the main origin of these shifts, but not to electron donating effects of the alkyl groups. The dominance of the steric effects is also evident in electrochemistry of Cu(phen)(pop) + , Cu(dmp)(pop) + , and Cu(dbp)(pop) + . [153] Electron donating groups, such as methyl or n-butyl, are expected to stabilize the oxidized form of the complex, but cyclic voltammetry data reveal that the oxidation potential for the Cu(II)/Cu(I) redox couple shifts from Cu(phen)(pop) + to Cu(dmp)(pop) + by 0.15 V and to Cu(dbp)(pop) + by 0.16 V towards more positive potentials. This behavior occurs because in complexes with dmp and dbp, the ligand 23 framework resists rearrangements to a more flattened structure which would be more appropriate for the oxidized Cu(II) species. [153] Summing up, reducing the extent of the excited-state geometry distortions occurring upon MLCT excitation is essential for enhancement of the emission quantum yields. This can be achieved by optimizing the matrix material to limit such distortions of the doped emitter molecules. A promising strategy is also based on chemical modifications to achieve higher molecular rigidity by using steric requirements of the ligands. In the latter approach, excited state distortions are already hindered at the molecular level.  It is remarked that the emission of the Cu(I) complexes discussed in this chapter represents a thermally activated delayed fluorescence (TADF) 4 at ambient temperature and that the radiative and non-radiative rates, k r and k nr , calculated from the experimental PL and  values, may refer to the relaxations from the T1 and S1 states to the ground state. In this section it was not discussed which radiationless or radiative path prevails, but it is obvious that both non-radiative processes, from S1 to S0 as well as from T1 to S0, may be suppressed to a large extent in properly engineered complexes.
The occurrence of excited state geometry distortions has another important consequence. Typically, powder samples are not well suited to investigate molecular luminescence as additional inter-molecular effects, for example, energy transfer to impurities, may strongly influence the emission properties, especially the decay behavior. However, the geometry distortions are sufficient, even in the solid phase, to lower the excited state energy to such an extent that the resonance condition for energy transfer to adjacent non-excited molecules is no longer fulfilled. [4,5,65,73] Therefore, the excitation may be regarded as trapped at the initially excited emitter molecule.
Consequently, even powder samples can be used to study emission properties of these MLCT compounds. Such a self-trapping effect [162,163] is the basis for the studies of temperature dependent emission properties presented in the next sections. (Figure 8

Cu(dppb)(pz2Bph2)
The emission behavior of Cu(dppb)(pz2Bph2) 2 has been discussed in detail in refs. [67,164]. However, before presenting experimental data, it is suitable to introduce to the compound's properties by use of DFT calculations. An inspection of the frontier orbitals shows that the HOMO is largely derived from a 3d atomic orbital of the Cu(I) center with contributions from the coordinating phosphorus atoms, whereas the LUMO is mainly distributed over the o-phenylene ring of the dppb (= 1,2bis(diphenylphosphino)benzene) ligand. (Figure 9) As a consequence, the related transitions are assigned the MLCT character involving dppb as chromophoric ligand. It has been shown by TD-DFT calculations that the resulting singlet state S1 and triplet state T1 are to more than 90% of HOMO-LUMO character. Therefore, these states are assigned as 1 MLCT (S1) and 3 MLCT (T1) states, respectively. [67] Fig. 9 HOMO and LUMO of Cu(dppb)(pz2Bph2) 2 resulting from DFT calculations for the triplet state geometry at the B3LYP/def2-svp [150,151] theory level. Adapted with permission from [67]. Copyright (2015) American Chemical Society.
The spatial separation of HOMO and LUMO ( Figure 9) indicates a small exchange integral [142] and therefore, a small energy separation between 1 MLCT and 3 MLCT.
The emission properties of compound 2 were studied in ref. [67] over a wide temperature range from T = 1.5 K to 300 K. Figure 10 reproduces emission spectra and decay curves of a powder sample of 2 at different temperatures. The complex shows intense green-yellow luminescence with high emission quantum yield of PL = 70% at ambient temperature and of about 100% at T = 80 K and probably also at least down to T ≈ 30 K. The spectra are broad and unstructured even at 1.5 K. [67]   In Figure 11, the emission decay time is plotted versus temperature (together with a plot for compound 8 that is discussed below). At low temperature, i.e. in the range of the plateau up to T ≈ 50 K only a long-lived phosphorescence (T1→S0) with (T1) = 1200 s is observed. With increasing temperature, fast up-ISC (RISC) to the S1 state takes place and opens the additional radiative TADF process via the decay path from the S1 state. This leads to a drastic decrease of the emission decay time and to the observed blue shift of the emission spectrum, since the emitting S1 state lies higher in energy than the T1 state. According to fast up-and down-ISC processes, i.e. faster than the emission decay times, a thermal equilibration between the energy states is present. Thus, we can apply eq. (1) to fit the experimental data. As seen in Figure 11 the fit is excellent. Fixing the emission decay time of (T1) = 1.2 ms (experimental value in the range of the plateau), one obtains the fit parameters for the activation energy 5 of E(S1-T1) = 370 cm 1 (46 meV) and for the S1 decay time of (S1) = 180 ns. [67] It is noted that the prompt fluorescence (S1S0) was not observed even at low temperature. This is a consequence of the very fast down-ISC process of only serveral ps. [81,116,120,123,124,133,165,166] From the decay time (S1) resulting from the fit, one can determine the radiative rate for this S1↔S0 transition according to eq. (3). With PL = 70% and (S1) = 180 ns a value of k r (S1↔S0) = 3.9·10 6 s 1 is obtained. A corresponding consideration has also been carried out in Ref [61] for compound 15. For this complex, the rate was independently estimated from the absorption spectrum using the Strickler-Berg relation [167]. The rates resulting from both emission and absorption data agree fairy well with each other.
For compound 2, one obtains the energy level diagram as displayed in Figure 12.

Fig. 12
Energy level diagram for the lowest states and emission decay times as well as the rate for the S1→S0 transition for Cu(dppb)(pz2Bph2) 2 (powder). The zero-field splitting of the triplet state T1 being less than ≈ 1 cm 1 (≈ 0.1 meV) [67], is not illustrated in this diagram.
It is emphasized that Cu(dppb)(pz2Bph2) 2 has a very long decay time of the triplet state of (T1) = 1.2 ms. Nevertheless, the ambient temperature decay time amounts only to (TADF) = 3.3 s. With the measured quantum yield of PL = 70%, a radiative decay time of  r (TADF) = 4.7 s is obtained. This value belongs to the shortest ones reported for Cu(I) complexes so far. Obviously, a small energy separation of E(S1-T1) is of crucial importance to induce short-lived TADF.
Interestingly, compound 2 and its derivatives have already been applied as emitters in OLEDs (being processed by vacuum deposition). For these devices, high external quantum efficiencies of almost 18 % were reported. [34] Cu(pop)(pz2Bph2) 8 DFT calculations carried out for a gas phase show, similarly to compound 2, that the HOMO is largely located at the Cu(I) center, while the LUMO lies on the chromophoric pop ligand. (Figure 4) From TD-DFT calculations for the optimized triplet state geometry it follows that the lowest excited singlet and triplet state, 1 MLCT(S1) and 3 MLCT(T1), are largely of HOMO→LUMO character. The calculated energy separation between these states amounts to 920 cm 1 (114 meV). Again, the experimentally determined activation energy is with E(S1-T1) = 650 cm 1 (80 meV) distinctly smaller. However, both of these values are still small enough to expect occurrence of an efficient TADF.
The luminescence properties of Cu(pop)(pz2Bph2) 8 were investigated in the temperature range 1.6 ≤ T ≤ 300 K. [2,4,65] In order to minimize effects of flattening distortions occurring upon the MLCT excitation (section 3), powder samples were studied. In the whole temperature range, the emission spectra of compound 8 are broad and unstructured, as expected for MLCT transitions. At 1.6 K, the complex emits with max = 474 nm ( Figure 13). With temperature increase to T ≈ 100 K, the emission maximum does not markedly change. However, with further temperature increase, the spectra become broader and a blue shift is observed. Above T = 200 K, the emission maximum approaches the ambient temperature value of max = 464 nm. The spectral blue shift occurring with temperature increase amounts to 10 nm. It is remarked that even the T = 1.6 K spectrum shows a half-width of almost 3500 cm 1 . Accordingly, no spectral details can be resolved. However, applying the technique of measuring the temperature dependence of the emission decay time, a resolution of energy splittings even of the order of 1 cm 1 can be obtained under specific conditions. This is due to the small energy steps that can be adjusted by temperature variation. For example, the thermal energy change of kBT for T = 1 K corresponds to only 0.7 cm 1 .
In serveral cases, thermal energies of this order of magnitude can induce distinct changes of the emission decay properties due to thermally induced population changes of electronic states with different decay times.
The emission decay of compound 8 shows a very characteristic behavior at low temperature and a pronounced decrease of the decay time with temperature increase.
At T = 1.6 K, the decay profile is clearly non-exponential ( Figure 14). It can be described by a tri-exponential decay function with the individual components of 2 ms (), 600 s (, and 170 s (). A similar behavior has already been discussed for other Cu(I) complexes [65,73] as well as, for example, for Pt(II) [18,59,[168][169][170][171][172][173], Pd(II) [59,174,175], Rh(III) [176,177], Ir(III) [178] and Re(I) [179] complexes. Thus, at T = 1.6 K, the three components are assigned to originate from the three different triplet substates I, II, and III. At T = 1.6 K, these states are not thermally equilibrated due to very slow spin-lattice relaxation (SLR) processes at an energy separation between the substates of E(ZFS) ≈ 1 cm 1 or less. [18,59,[179][180][181][182] In this situation, the individual emission decay times of the three substates are much shorter than the SLR times. With temperature increase to T ≈ 20 K or ≈ 40 K, however, these SLR processes become significantly faster and a fast thermalization of the three substates takes place. As a consequence, a mono-exponential decay results and an average decay time av is observed. This average value can be expressed by [2][3][4][5]59,65,73,180,182]   1 Inserting the three decay components given above into eq. (4), an average decay time av(T1) of slightly less than 400 s is obtained. Within limits of experimental and fitting error, this value corresponds sufficiently well to the decay time of 500 s measured at T = 40 K. (Figures 11 and 14) 34 copper, being about five times smaller than that of iridium or platinum [108], but also to the less efficient SO mixing routes with higher lying 1 MLCT states (see also section 7).
With further temperature increase (above T = 100 K) and remaining high emission quantum yield (Table 2), the decay time becomes distinctly shorter and drops to (TADF) = 13 s (Figures 11 and 14). This observation, together with the blue shift of the emission spectra (Figure 13), indicates  as already studied for compound 2  that a higher lying electronic state with a distinctly greater deactivation rate becomes thermally populated. As discussed above and in analogy to a number of other investigations with Cu(I) complexes [2,4,5,44,45,61,67,68,72,73,109,110,185,186], this higher lying state is assigned to the lowest excited singlet state S1 of 1 MLCT character.
The fitting procedure applying eq. (1) to the measured decay time data of Cu(pop)(pz2Bph2) 8 ( Figure 11) gives values of E(S1-T1) = 650 cm 1 (80 meV) and of (S1) = 170 ns for the S1 decay time. These data together with the data given above are summarized in an energy level diagram. (Figure 15 However, because of the large difference in E(S1-T1) of 370 cm 1 (≈ 46 meV) (2) compared to 650 cm 1 (≈ 80 meV) (8), the temperature behavior of the decay time is rather different. (Fig. 11) With temperature decrease, TADF is frozen out below T ≈ 100 K for compound 8, while for compound 2, this is reached only below T ≈ 30 K.
Accordingly, at T = 80 K, the emission of compound 2 with a decay time of (80 K) ≈ 300 µs cannot be assigned as a phosphorescence. It represents dominantly a TADF with an intensity ratio of TADF/phosphorescence of about 75%/25% at T = 80 K. [67] Therefore, assignments with respect to the emission characteristics as phosphorescence or fluorescence (TADF) might be problematic for Cu(I) compounds, if investigated only at 300 K and 80 K, as often reported. Similar arguments hold also for the assignments of the spectral shifts between phosphorescence and TADF. ( Figure   10) (ii) The very small value of E(S1-T1) = 370 cm 1 for compound 2 is responsible for the large radiative TADF rate at T = 300 K of k r (300 K) = 2.1·10 5 s 1 being only by a factor of about three smaller than the radiative rate of the well-known Ir(ppy)3 complex. [4,60] Moreover, due to the largely forbidden nature of the T1 → S0 transition with a radiative rate of only k r (30 K) = 8.3·10 2 s 1 , the TADF effect induces a rate increase by more than a factor of 250 with temperature increase from T = 30 K to 300 K. The effect is much less pronounced for compound 8 with a rate increase by a factor of 35 from k r (80 K) = 20 10 2 s 1 to k r (300 K) = 6.9 10 4 s 1 . Compound 2 attains a much larger TADF transition rate at ambient temperature than compound 8. (iii) The rate of the radiative decay for the triplet state to the singlet ground state is dictated by the efficiency of SOC. (This is also valid for the size of ZFS, [2,4,57,183,184], section 7) Both compounds 2 and 8 exhibit relatively long triplet state decay times of 1200 s (2) and 500 s (8), respectively. Hence, only weak SOC is effective for both compounds.
In contrast, when cooling compound 10a from ambient temperature to 77 K, the emission decay time changes only slightly from 18 µs to 21 μs, while the k r change from 4.110 4 s −1 to 3.810 4 s −1 is negligible. ( Table 4) This indicates that for complex 10a the TADF mechanism is not effective and that also the emission at ambient temperature is a phosphorescence stemming from the T1 state. The slight blue shift of the high energy flank observed on heating may be explained by a small thermal broadening of the spectrum due an involvement of energetically higher lying vibronic emission components in the powder sample (compare Ref [193]). 6 This behavior is in contrast to that of complex 10 for which the entire spectrum is shifted. (Figure 17)  a thermal population of the energetically higher lying and short-lived S1 state.
Additionally, a blue shift of the emission from max = 490 nm (at 77 K) to 475 nm (300 K) occurs as the S1 state lies energetically higher than the T1 state.
The measured data, as displayed in Figure 18, can be fitted with a modified Boltzmann type function (eq. 5) [44,45].
In this equation, (T) refers to the emission decay time at a given temperature T, (I), (II), and (III) represent the individual decay times of triplet substates I, II, and III, ΔE(III−I) and ΔE(II−I) are the energy separations between the triplet substates. (S1) is the decay time of the singlet state S1, and E(S1-T1) is the energy separation between the S1 and T1 state. kB is the Boltzmann constant.  [59,180,181]. In sections 6 and 7, we will discuss effects of SOC in more detail.
The energy splitting between the triplet T1 and singlet state S1 is ΔE(S1−T1) = 740 cm −1 (92 meV). This value is in good agreement with the blue shift of the emission spectrum when heating from 77 to 300 K amounting to about 650 cm −1 . The emission decay time of the singlet state S1 is (S1) = 160 ns emphasizing the singlet nature of this state. It is remarked that in contrast to the delayed fluorescence, a prompt fluorescence was not observed for this compound as intersystem crossing from the S1 to the T1 state, probably being of the order of 10 ps, [81,123,124,133,165,166] is much faster than the prompt S1 → S0 emission.
Remarkably, the decrease of the emission decay time with increasing temperature due to an activation of the singlet S1 state by factor of three is significantly less pronounced than that for the complexes 2 and 8 discussed in section 4.1 and many other TADF Cu(I) complexes. (Compare also Table 7 below.) For example, Cu(dppb)(pz2Bph2) 2 shows a TADF induced increase of the radiative rate by a factor as large as 250. [67] However, the T1  S0 transition of the later complex with (T1) = 1.2 ms is strongly forbidden, whereas complex 10 exhibits a much shorter T1 decay time of only 34 µs, due to stronger SOC to higher lying singlet states. Therefore, complex 10 is much less susceptible to a reduction of the decay time by involving the TADF process at higher temperatures.
Complex 10a displays a similar decay behavior as compound 10 at temperatures below 100 K. The (thermalized) decay time drops drastically from about 1 ms at 1.3 K to ≈ 60 µs at 5 K. (Figure 19) Above T  30 K, the decay time is almost constant and forms a plateau with a value of   20 µs up to 300 K. In contrast to the behavior of compound 10, no distinct reduction of the decay time at T > 100 K is observed. Also the radiative rate with k r ≈ 4 10 4 s 1 is essentially constant. (Table 4) This allows us to assign the emission of complex 10a as a phosphorescence stemming from the T1 state in the entire temperature range up to 300 K. An emission via the TADF mechanism as in the case of complex 10 does not occur indicating that the energy splitting ΔE(S1−T1) is larger than 3000 cm −1 , as for such a large value no significant thermal activation is expected at T = 300 K. [45] Taking into account that the term involving the singlet state S1 in eq.(5) may be neglected, we obtain the following equation for a Boltzmann type analysis of the experimental (T) data, similar to that performed for complex 10: [2,4] The different terms are defined below eq. (5).
A fitting procedure of this equation to the experimentally determined decay times gives ΔE(ZFS) = ΔE(III−I) = 5 cm −1 , which is even slightly larger than that of complex 10. [45]  It is concluded that despite seemingly similar chemical composition of the two compounds, only complex 10, but not complex 10a, exhibits thermally activated delayed fluorescence. In fact, as summarized in Figure 20, complex 10 exhibits at ambient temperature two radiative decay paths which are thermally equilibrated: one via the S1 state as TADF and one via the T1 state as phosphorescence. Complex 10a exhibits only phosphorescence even at ambient temperature.

Tuning of TADF by variation of the torsion angle
The observation of distinctly different energy separations E(S1-T1) for the complexes 10 and 10a with "similar" chemical formulas was a surprising result. The compounds differ in two aspects: (i) Size of the -aromatic system of the N-heterocyclic carbene (NHC) ligands: simple five-member imidazole ring of IPr in 10 versus a fused benzimidazole of Bzl3,5Me in 10a. (ii) Aliphatic substituents on the pendant phenyl rings of the NHC ligands: i-propyl groups at the 2,6-positions in IPr of 10 versus methyl groups at the 3,5-positions in Bzl-3,5Me (10a). In particular, the latter difference is unlikely to have strong (electronic) impact on the singlet-triplet splitting. However, the alkyl groups can exert steric control over the mutual orientation of the two ligands towards each other. Indeed, as it can be seen in Figure 21, the IPr and py2-BMe2 ligands in molecule of 10 are almost coplanar, whereas in complex 10a, a significant twist around the Cu-C bond connecting Cu(I) with the NHC ligand occurs. This is expressed by large torsion angles between the Bzl-3,5Me and py2-BMe2 ligand planes of  NCCuN being as large as 62 to 71, as revealed by an x-ray diffraction study (two molecules in the unit cell of 10a). [66]  The difference in the HOMO distribution is due to the angular relation between the metal d and imidazole -orbitals. When the NHC and py2-BMe2 ligands are coplanar, the two sets of orbitals are orthogonal and thus do not electronically couple to each other. However, in the perpendicular orientation, the orbitals have the appropriate symmetry to conjugate and delocalize over both ligands. Consequently, spatial overlap between the HOMO and LUMO is small when the torsion angle N−C−Cu−N is 0°, whereas a significant spatial overlap exists with N−C−Cu−N of 90°. Since the lowest excited singlet S1 and triplet T1 states are largely comprised from transitions between these frontier orbitals, variations of the degree of overlap will strongly alter the exchange interaction and thus, the value of ΔE(S1−T1). According to these model calculations, the magnitude of the singlet-triplet splitting in the studied three-coordinate complexes displays a minimum for the torsion angle between the two ligands of 0 giving a small value (calculated ΔE(S1-T1) = 540 cm 1 ) and reaches a maximum (ΔE(S1-T1) = 4700 cm 1 ), when the ligands are perpendicular towards each other.  Summing up, the case study presented in this section reveals a strong dependence of ΔE(S1−T1) on the orientation of the two ligands towards each. In this regard, it should be mentioned that for a different compound (the py2-BMe2 ligand was replaced by phenanthroline) quantum mechanical calculations do not show any significant ΔE(S1−T1) dependence on the torsion angle, though the down-and up-ISC (RISC) rates vary distinctly. [199] Obviously, for engineering of emitters with optimized TADF properties, angle dependences have to be taken into account in different contexts, i.e. with respect to steric control to reduce excited state distortions that induce non-radiative relaxations (section 3) and the above mentioned effects.

Singlet-triplet splitting and fluorescence rate
As already discussed above, thermally activated delayed fluorescence represents a promising way to reduce emission decay times as the emission occurs from the thermally populated spin-allowed S1 → S0 transition in contrast to the mostly very weak T1 → S0 transitions. The TADF decay time strongly depends on the energy splitting ΔE(S1-T1) between the first excited singlet S1 and triplet T1 state. The smaller this energy separation is, the more effective is the thermal population of the S1 state at ambient temperature. A pronounced population of the S1 state leads to a shorter emission decay time. Consequently, as discussed above, many efforts have been made to minimize the singlet triplet-splitting in order to increase the TADF efficiency.
The smallest splitting reported so far is as low as 270 cm -1 (compound 1 [5]). However, for this compound, the decay time of the S1 state was as long as 570 ns [5]. This raises the question, whether it is sufficient to focus on the minimization of the energy splitting.
In fact, in this section, we will show that the transition dipole moment ⃗(S1→S0) and hence, the radiative decay rate k r (S1) of the S1 → S0 transition represents another important criterion. In particular, this rate should be as large as possible to achieve a short TADF decay time. (Compare section 2.) We will show that this is not the case, as a small singlet-triplet energy splitting is bought at the cost of a small S1 → S0 oscillator strength (radiative rate).

Theoretical considerations
In this section, we will present a qualitative model that may explain why a small splitting between the lowest singlet and triplet state ΔE(S1-T1) is related to a small radiative rate k r (S1) of the S1 → S0 transition. The model is based on the assumption that the states S1 and T1 can be approximated by a one-electron transition from HOMO to LUMO , such that further excited singlet and triplet states are lying substantially higher in energy. In this simple model, we also assume that non-radiative relaxation processes or intermolecular interactions can be neglected, and that a common set of real orbitals is chosen for the states S0, S1, and T1.
In such a situation, one can obtain both ΔE(S1-T1) and k r (S1) from integrals over the product of HOMO and LUMO . This product is called the "transition density". The radiative rate k r (S1) may be obtained from the electric dipole transition moment µ ⃗⃗(S1-S0) given approximately by µ ⃗⃗ , defined as µ ⃗⃗ , = ∫ ( ⃗) ⃗ ( ⃗) 3 = ∫ , ( ⃗) ⃗ 3 Thus, the radiative rate for spontaneous (prompt) emission is given by (see [200], p.159 and Appendix A) Here, is a numerical constant given by 16π 3 /(3ε0hc 3 ) with ε0 the vacuum permittivity, h Planck's constant, and c the velocity of light. In this equation, ν = ΔE(S1-S0)/h is the transition frequency and n is the refractive index. Thus, we see that the radiative rate is essentially related to the square of the transition density.
The energy separation ΔE(S1-T1) between S1 and T1, on the other hand, is well approximated by twice the exchange integral KH,L for HOMO and LUMO. ( [143], p. 86) Thus, we have where the exchange integral KH,L for HOMO φH and LUMO φL can be expressed in terms of the transition density ρH,L: Here, = e 2 /(4πε0) is a numerical constant.
Note that the exchange integral can be interpreted as the electrostatic interaction of the transition density ρH,L with itself. The important observation is that also the exchange integral is quadratically dependent on the transition density. (See eq. (11).) The common quadratical dependence on the transition density already explains qualitatively why small ΔE(S1-T1) and small k r (S1) are related: A small transition density implies that both ΔE(S1-T1) and k r (S1) are small.
We will discuss qualitatively several reasons, why the transition density is small. For

Small MO coefficients
The MO coefficients can be small even in the small region mentioned above in the case that the main contributions to HOMO and LUMO are well separated in space. Both reasons for small transition densities are depicted schematically in Figure 24 as "small overlap" of HOMO and LUMO.
Thus, the transition density is small in absolute value in spatial regions where at least one of the two orbitals φH and φL is small in absolute value. If the HOMO and LUMO are large in spatial regions that do not overlap, then the transition density is nothing but the product of the exponentially decaying tails of HOMO and LUMO, and this implies that the transition density is small. 8 In order to illustrate this further, we display HOMO and LUMO orbitals of complex 8 that have been computed in the T1 geometry using the B3LYP density functional. ( Figure   25) It is seen that in most spatial regions at least one of the two orbitals is vanishing which means that the transition density is only of importance in a small region where both orbitals are not vanishing. Note that in Figure 25, the product of orbitals is twenty times enlarged as compared to the contour amplitudes of HOMO and LUMO.

Fig. 24
Schematic illustration of the correlation of energy splitting ΔE(S1-T1) between first excited singlet S1 and triplet T1 state and the radiative decay rate k r (S1→S0) on the spatial overlap of HOMO and LUMO. Instead of using an LCAO-type description of HOMO and LUMO, the smallness of the transition density may also be described by representing the MOs by linear combinations of orbitals localized in different spatial regions. Within such an approach, the intensities of charge-transfer bands of ferrous, ferric, and cuprous phenanthroline complexes were explained. [202,203] In conclusion, the presented qualitative model clearly shows that the ΔE(S1-T1) value and the transition rate of the prompt fluorescence k r (S1→S0) are related and that both quantities depend quadratically on the transition density, as defined in eq. (7). A general analytical relation between these quantities could not be given. However, an experimentally based correlation, as discussed in the next section, will show that such a relation exists for many Cu(I) complexes with low lying charge transfer transitions.

Experimental correlation of ΔE(S1-T1) and S1→S0 allowedness
In order to experimentally investigate the question raised in the previous section, the decay time of the first excited singlet S1 as well as the energy separation between the singlet and triplet state have been determined for many Cu(I) compounds in the last years. These results, deduced from measurements of the emission decay time in dependence of the temperature T (see section 4), are summarized in Table 5 and reveal an interesting trend that will be discussed in the following. Table 5: Energy splitting ΔE(S1-T1) between the first excited singlet S1 and triplet T1 state, the decay time of the singlet state (S1), the photoluminescence quantum yield at ambient temperature ΦPL(300 K), and the radiative rate k r (S1→S0) = k r (S1) (calculated according to eq. (3)).

Compound ΔE(S1-T1) [cm 1 ]
(S1)   Figure 26 displays the experimentally determined data from Table 5, showing a plot of k r (S1) versus ΔE(S1-T1). Indeed, it is seen that with decreasing ΔE(S1-T1) also the allowedness of the S1→S0 transition (radiative decay rate k r (S1)) decreases. This might have an important consequence on the emission decay time of a TADF complex at ambient temperature, as can be illustrated by a very simple consideration using eq. (1).
As has been shown in a simple estimate using the experimental fit curve displayed in Figure 26 and the expression giving the (TADF) based on eq. (1), a minimum TADF decay time of several µs is obtained. [204] Fig. 26 Radiative decay rate k r (S1) of the S1→S0 transition plotted versus the energy splitting ΔE(S1-T1) between the first excited singlet S1 and triplet T1 state. The data points are taken from Table 5. The fit curve represents an exponential function as guide for the eye.
The simple model presented explains the experimental results. Thus, its basic assumption of energetically (from higher lying states) well separated S1 and T1 states derived from a HOMO-LUMO excitation seems to describe the basic physics of the systems under consideration. However, it does not contain effects induced by configuration interaction (CI). By these interactions higher lying singlets with higher oscillator strengths mixing to the S1 state can increase the radiative rate k r (S1). To investigate Cu(I) complexes exhibiting such CI effects is outside the scope of the present article but may be a fruitful topic for further research. Moreover, the role of spinorbit coupling is not taken into account in the approach of this section, but it will be shown in the following section that SOC can induce an effective reduction of the overall emission decay time at ambient temperature.

Combined TADF and phosphorescence -Importance of spin-orbit coupling
Emitter compounds that are applied in OLEDs should exhibit not only high emission quantum yields, but also emission decay times as short as possible, especially, to reduce roll-off effects [112] and to avoid bond-dissociation reactions leading to decomposition of the emitter molecules. [113][114][115]. In section 4 and in Ref [67], it was shown that the energy separation E(S1-T1) between the charge transfer states of 1 MLCT (S1) and 3 MLCT (T1) character has crucial importance for a short decay time (TADF) of the thermally activated delayed fluorescence. However, with decreasing E(S1-T1), the oscillator strength of the S0↔S1 transition decreases and thus, the radiative decay time (S1) increases. Accordingly, (TADF) cannot be diminished as preferred. (Compare eq. (1)) This restricting dependence has already been addressed in section 5. Nevertheless, a promising approach to overcome this limitation will be presented in this section. [44,45,145] Case study based on Cu2Cl2(NˆP)2 13.
An overall faster radiative decay of an emitter showing TADF can be obtained by opening an additional radiative decay path from the lowest triplet state. Such a path represents the T1→S0 phosphorescence. However, for most Cu(I) complexes, this path is ineffective mainly due to very weak spin-orbit coupling (SOC) of the T1 state to higher lying singlet states. As a consequence, the phosphorescence decay time is of the order of several hundred s and even up to ms. [2,4,5,38,61,65,69,72,73] On the other hand, development of complexes that experience large SOC with respect to the lowest triplet state is not unrealistic. Thus, in this case study, we present properties of the brightly luminescent dinuclear complex Cu2Cl2(NˆP)2 13 [44,145] (chemical formula shown in Figure 27) exhibiting efficient TADF from the S1 state and direct phosphorescence from the T1 state both being in a thermal equilibrium. [44] The phosphorescence contributes significantly to the emission even at ambient temperature due to efficient SOC that induces distinct radiative decay from the T1 state.
A first insight into the electronic structure of Cu2Cl2(NˆP)2 13 is obtained by DFT and TD-DFT calculations carried out for an optimized T1 geometry. These allow us to assign the lowest singlet and triplet state as 1 MLCT (S1) and 3 MLCT (T1) state, respectively, with an energy separation of E(S1-T1) = 940 cm 1 (0.11 eV). 9 This value is small enough to expect a TADF behavior for this compound.
The powder material exhibits a very high emission quantum yield of PL = 92% at ambient temperature. The emission spectrum with a peak maximum at max = 485 nm is very broad showing a halfwidth of about 5·10 3 cm 1 (0.62 eV). This is consistent with the MLCT character of the corresponding transition. Upon cooling to T  120 K, the emission red-shifts by = 25 nm (10 3 cm 1 ; 0.12 eV) to max = 510 nm. Further cooling to T = 1.3 K does not induce any significant spectral change apart from a slight reduction of the halfwidth. Again, no detailed spectral information can be resolved. [44] The red shift is consistent with the TADF behavior, since the TADF is frozen out at low temperature, and thus, only the energetically lower lying phosphorescence is maintained. Again, a deeper insight into the electronic structure of the complex is obtained from the analysis of the emission decay behavior.  character to the different triplet substates by SOC is extremely small to substate I, but significant to the substates II and III. The fit gives also the energy separation between the triplet and singlet state of E(S1-T1) ≈ E(S1-I) = 930 cm 1 (0.115 eV). This value corresponds very well to the value deduced from the difference of the emission maxima at T = 77 K and 300 K (1000 cm 1 ) as well as from TD-DFT calculations (940 cm 1 ). [44] The S1 decay time is determined to be (S1) = 40 ns. However, this prompt emission was not observed directly. Obviously, the ISC time is much faster than 40 ns. [81,120,123,124,133] This leads to a fast depopulation of the S1 state resulting in a population of the lower lying triplet state T1. Thus, the prompt fluorescence is (almost) quenched. The time constant of 40 ns or, more exactly, the related radiative decay rate, amounting to k r (S1) = k r (S1↔S0) = 2.3·10 7 s 1 (according to eq. (3) and PL = 92%), corresponds to the probability of the transition between the excited singlet state S1 and the electronic ground state S0.
The energy separations and decay times as obtained from the fitting procedure are summarized in the energy level diagram shown in Figure 28. The relatively short T1 decay time of (T1) ≈ 43 s, corresponding to a radiative decay rate of k r (T1) = 2.1·10 4 s 1 , allows us to conclude that the ambient temperature emission is composed of the T1 state emission as phosphorescence and the S1 state emission as TADF. Thus, two decay paths lead to a combined emission. This behavior is illustrated by discussing the temperature-dependent interplay between TADF and phosphorescence. (Compare [73].) The TADF intensity that originates from the singlet state S1 is denoted as I(S1) and the phosphorescence intensity from the triplet state T1 as I(T1), respectively, with the total emission intensity of I(total) = I(S1) + I(T1).
I(S1) is proportional to the population of the singlet state N(S1) and to the radiative rate constant k r (S1): Similarly, I(T1) is given by with a constant, being equal in both equations. The radiative rates k r (S1) and k r (T1) can be expressed in terms of quantum yields and emission decay times according to eq. (3). For this rough estimate, it is assumed that the quantum yields PL(S1) and g(S1) = 1 and g(T1) = 3 are the degeneracy factors for the singlet and the triplet states, respectively. Using I(T1) = Itot -I(S1) one obtains: By use of the eqs. (15) and (16) and the fit parameters determined for Cu2Cl2(NˆP)2 13 with E(S1-T1) = 930 cm 1 , (S1) = 40 ns, (T1) = 43 s and assuming PL(S1) = 0.92 measured at 300 K [44] and PL(T1) = 0.97 measured at 77 K [44], temperature dependent TADF and phosphorescence ratios can be visualized as plotted in Figure   29. At temperatures below T ≈ 120 K, only a T1→S0 phosphorescence occurs. With temperature increase, the phosphorescence intensity decreases, while the S1→S0 TADF grows in. At T = 300 K, about 80% of the emission is of TADF and the remaining 20% of phosphorescence character. For completeness, it is mentioned that both types of emission decay with the same decay time due to fast thermalization between the S1 and T1 states following a Boltzmann distribution.
The discussion presented above, demonstrates that the emission decay time of (300 K) = 8.3 s is not determined by the TADF process alone but additionally by a phosphorescence process. The corresponding total rate is expressed by k(300 K) = k(T1) + k(TADF) (compare compound 10 [45] and compounds described in Ref. [5]). The process of the combined emission is illustrated in Figure 30.  [108] compared to that of iridium with  = 3909 cm 1 , but essentially to the extent of mixing of adequate, energetically higher lying singlet states. In a simplified perturbational approach, the radiative rate k r (T1→S0) for the triplet state T1 can be expressed by [2,4,183,192,207]     Accordingly, the radiative rate k r of a transition from a triplet state T1 (or a triplet substate) to the singlet ground state depends on the allowedness (oscillator strength or molar extinction coefficient) of the singlet-singlet transition S0→Sn, whereby Sn is the singlet state that mixes with T1 via direct SOC. 10 Quantum mechanical considerations show that SOC between a triplet state T1 and a singlet state S1 both stemming from the same orbital configuration vanishes. [2,4,5,57,183,184,[187][188][189][190][191][192], but mixing with a higher lying state can be significant. This   Figure 31b.
If we take into account that (i) SOC is dominantly induced by the metal d-orbitals (and not by -or *-orbitals of the organic ligands), (ii) only one-center integrals on the metal contribute with large coupling constants, and (iii) a d-orbital cannot couple with itself, [2,4,5,57,183,184,[187][188][189][190][191][192], we obtain relevant SOC paths. Accordingly, since SOC between the two lowest states resulting from the same orbital transition can be ignored, only two coupling routes are relevant for the T1 state, as marked in Figure 31b. In particular, SOC between the S2 ( 1 MLCT2) state and the T1 ( 3 MLCT1) state will induce allowedness to the T1→S0 transition. As a consequence, a distinct phosphorescence decay path is opened in addition to the thermally activated path via S1 (TADF).
For example, for Cu2Cl2(PˆN)2 13, this model seems to be adequate. HOMO1 and HOMO contain contributions from different d-orbitals. The energy separation between these MOs amounts to only 0.378 eV, as determined by DFT calculations. (Table 6, below) This leads to a significant mixing of singlet character to the triplet state and thus, to a reduced overall emission decay time, as displayed in Figure 30.
For completeness, it is mentioned that the two SOC routes shown in Figure 31b govern also the zero-field splittings of the T1 state and the decay rates of the individual substates. [2,4,57,183,184] Obviously, for Cu2Cl2(PˆN)2 13 with E(ZFS) = 15 cm 1 , these SOC routes are significant. However, details concerning the coupling paths between the triplet substates and higher lying states are not in the focus of the approach presented in this section. (For example, compare [57]).
In summary, the discussions show that SOC is responsible for different, though related effects. It induces the ZFS of and the radiative decay from the lowest triplet state T1 (or from the substates) to the electronic ground state S0. A detailed discussion of this correlation is presented in Ref. [2,4,57,184].
We want to simplify the model further. To lowest order, the energy difference between singly excited states can be approximated by the energy difference of the corresponding orbitals. Thus, for a qualitative trend, it seems to be justified to replace the energy separation |E(T1) -E(Sm)| between the T1 state and the SO-coupling Sm state 12 by the coarsely related energy separation (HOMO -HOMOn). This model can easily be applied to a large number of Cu(I) compounds. In Figure 32, the experimentally determined phosphorescence decay rates k(T1) = k r (T1→S0) are plotted versus the energy separation (HOMO -HOMOn), resulting from simple DFT calculations. The data used for the plot of Figure 32 are summarized in Table 6.
Interestingly, even in this very simplified approach, a fit of the experimental data displays a quadratic dependenceas expected from eq. (18)according to k r (T1 → S0) = const [∆E(HOMO -HOMO1)] 2 k r (T1 → S0) is the experimental radiative triplet decay rate determined by use of the lowtemperature decay time in the region of the plateau. HOMOn with distinct metal-d contribution, however, different from d1. Usually n = 1.
This shows that SOC with respect to the T1 state is very weak. With a decrease of ΔE(HOMO  HOMOn) to ≈ 0.4 eV, the phosphorescence rate increases drastically, for example, for compound 1 to k r (T1→S0) = 4.8·10 4 s 1 [5]), representing an increase (relative to compound 2) by a factor of almost 60.
Obviously, the presented very simple model, being based only on DFT calculations without the need of sophisticated extensions to SOC-TD-DFT theory, treating explicitly the spin-orbit coupling [209][210][211][212][213], may be helpful in designing new materials that show efficient phosphorescence in addition to TADF at ambient temperature. This is a promising approach to reduce the overall emission decay and thus, saturation effects of an electroluminescent device.

Concluding summary
In this contribution, we present a series of Cu(I) compounds that show a large diversity of photophysical properties. These do not only open access to important applications, especially, in the field of light generation in OLEDs, but also stimulate progress in scientific research due to the possibility of chemical tuning of electronic energy states with respect to their energy positions, separations, SOC routes, oscillator strength, etc.. This is particularly interesting regarding understanding and engineering of compounds with specific luminescence properties, such as emission colors, quantum yields, or decay times. Since the focus of this study lies on understanding and improving processes of TADF materials, we present compounds that cover a wide range of TADFrelated properties as displayed in Table 7.
It is obvious that the energy separation E(S1-T1) between the lowest excited singlet and triplet states governs the radiative TADF decay time (TADF). For OLED applications, this decay should be as short as possible. E(S1-T1) is determined by the exchange interaction and accordingly by the spatial overlap of HOMO and LUMO as well as by their angular distributions. In the series discussed, E(S1-T1) varies from 270 to 1300 cm 1 and indeed, the compounds with the smallest values exhibit the shortest (TADF) decay time. (Table 7) Interestingly, this holds even though the triplet decay time belongs to the largest values observed so far with (T1) = 1200 s (compare compound 2, section 4.1). On the other hand with decreasing E(S1-T1), the decay time of the S1 fluorescence, from fitting of the (T) data, increases and reaches a value of (S1) = 570 ns for compound 1 with E(S1-T1) = 270 cm 1 . Thus, and according to eq.
(1), it may be possible that a lower limit for (TADF) might exist for "traditional" Cu(I) complexes. (Section 5) However, other mechanisms are suited to shorten the overall (radiative) decay time. In particular, "tuning-in" of an additional phosphorescence decay path is successful (section 6), since the triplet decay time (T1) can become as short as almost 20 s if SOC is efficient enough. (Table 7) A simple approach, how to understand the related SOC routes is presented in section 7. For completeness, it is remarked that also the properties of the singlet S1 state can be modified by inducing mixing with other higher lying singlet(s) by configuration interaction. These effects are not studied in this contribution, but it is indicated that the overall (radiative) decay time can efficiently be pushed down. Interestingly, values being similarly short as the triplet decay time reported for the attractive OLED emitter Ir(ppy)3 [60] may be obtained. where on the right hand side Ψ ̅ ̅ is the determinant where the spin orbital ̅ is replaced by ̅ in Ψ. Using ( [143] , p. 97)