This chapter gives a brief overview over quantum chemical methods for computing rate constants of radiative and nonradiative molecular excited‐state processes and summarizes our recent theoretical research on the photophysics of thermally activated delayed fluorescence (TADF) emitters.
8.1.1 Electroluminescent Emitters
In the first organic light‐emitting diodes (OLEDs), the electroluminescence of fluorescent dyes such as 8‐hydroxyquinoline aluminum (Alq) was exploited . Dyes of this first generation are highly fluorescent but possess slow intersystem crossing (ISC) and negligible phosphorescence rates. Therefore only the singlet excitons, that means, only about 25% of the generated excitons, can be harvested . The limited internal quantum efficiency appears to be their greatest disadvantage. Advantages are clear colors due to narrow emission bands and good operational stability due to fast radiative decay of the dyes (nanosecond regime) , .
The second generation of small‐molecule OLEDs employs phosphorescent dopants instead of fluorescent ones. Typically, the emitters are organometallic complexes with Ir or Pt cores –. Their excited singlet states undergo fast ISC to the lowest triplet state. Thus, in addition to the triplet excitons, the singlet excitons can be harvested in phosphorescent organic light‐emitting diodes (PHOLEDs), leading to an internal quantum yield of up to 100% , . The main limitation of the phosphorescent dyes is their comparatively long radiative lifetime (microsecond regime) that leads to undesirable side effects, namely, quenching processes and bleaching reactions. The latter are particularly pronounced for blue PHOLED emitters. To avoid nonradiative decay via low‐lying nonemissive metal‐centered states, complexes with strongly donating and accepting ligands were devised –, but there is still room for improvement. Because of the very limited operational stability of blue PHOLEDs, hybrid fluorescent–PHOLEDs were employed for generating white light incorporating fluorescent blue and phosphorescent green to red emitters in one device .
The third generation of OLED emitters comprises organic donor–acceptor systems  as well as transition metal (TM) complexes – with small singlet–triplet energy gap that lies within the range of thermal energy. Because of the small , reverse intersystem crossing (rISC) from the lowest triplet to the lowest singlet is reasonably fast, and therefore TADF, also called E‐type delayed fluorescence (DF) in the older literature , is possible in addition to direct fluorescence. Like PHOLEDs, TADF‐based OLEDs show an internal quantum efficiency of up to 100% . Furthermore, cheaper first‐row TMs such as Cu instead of Ir or Pt can be used. A disadvantage of many presently available TADF OLEDs that they share with the PHOLEDs is the rather low intrinsic radiative transition rate (– s) of the emitters that makes them sensitive to nonradiative decay processes such as triplet–triplet annihilation. Also, the emission from states with charge‐transfer (CT) character typically is rather broad, which is not favorable for application in displays .
The latest class of OLEDs aims to combine high internal quantum efficiency and long operational stability by using assistant dopants for the harvesting of triplet‐ and singlet excited states in addition to fluorescence emitters. The assistant dopant transfers its excitation energy nonradiatively to the fluorescent acceptor by Förster resonant energy transfer (FRET). If the donor is sufficiently phosphorescent, it is even possible to induce FRET from triplet to singlet states or vice versa , . This mechanism was exploited by Baldo et al.  for improving the efficiency of red fluorescence in OLEDs by using the green phosphor Ir(ppy) as sensitizer. Fukagawa et al.  utilized singlet‐to‐triplet FRET from TADF assistant dopants to phosphorescent Ir and Pt complexes. In this way, the amount of phosphorescent emitter could be greatly reduced. Adachi and coworkers combined purely organic, sublimable TADF assistant dopants and fluorescence emitters in one layer, thus uniting the advantages of both .
8.1.2 Thermally Activated Delayed Fluorescence
TADF is looked upon as a significant emerging technology for generating highly performant electroluminescent devices for displays and lighting systems . Despite the fact that TADF has been shown to give highly efficient OLEDs, the underlying mechanisms are still not clearly understood. Ideally, thermally stable dyes with small singlet–triplet energy gap, substantial –T (r)ISC, high fluorescence but minimal nonradiative decay to the electronic ground state are required. However, these conditions for efficient TADF emission are not easily met simultaneously. In addition to the intrinsic emitter properties, emitter–host interactions play an essential role for the luminescence properties of a device , .
The energy difference between a singlet‐ and triplet‐coupled open‐shell configuration depends on the exchange interaction of the unpaired electrons. This interaction is small when the density distributions of the orbitals involved in the excitation do not overlap substantially. Typically, this requirement is fulfilled by CT states where the unpaired electrons are far apart. Very small singlet–triplet splittings can also be achieved in molecular systems where the electron clouds in the half‐occupied orbitals are not strongly displaced with respect to each other, but where their electron density distributions peak at different atoms and hence are disjunct , . Such a situation occurs, for example, in nonalternant hydrocarbons with azulene as a well‐known representative. Unfortunately, the overlap of orbital densities between the initial and final states plays also a decisive role for the magnitude of the electronic spin–orbit coupling (SOC) and for the fluorescence rate. Electronic SOC – a further prerequisite for efficient (r)ISC – is a fairly short‐ranged interaction. Furthermore, SOC between singlet and triplet configurations with equal occupation of the spatial orbitals vanishes for symmetry reasons. As a consequence, SOC is in general very weak between singlet and triplet CT states. Owing to the near‐degeneracy of d orbitals with different magnetic moments, the situation might be more favorable in TM complexes with metal‐to‐ligand charge‐transfer (MLCT) excited states. The interplay of all the factors influencing the probability of TADF is not yet fully understood and needs further investigation. It seems to be clear, however, that a small singlet–triplet energy gap alone is not sufficient for enabling efficient TADF.
8.2 Intersystem Crossing Rate Constants
ISC is a nonradiative transition between states of different electronic spin multiplicity. Hence, a spin‐dependent interaction operator is required to mediate the transition. In most cases, electronic SOC will dominate the interaction, but in cases in which this interaction is very weak, electronic spin–spin coupling (SSC) might come into play.
Assuming the coupling of the initial and final states to be small compared with their energy difference (which will be the case in typical TADF emitters), the ISC rate can be evaluated in the framework of perturbation theory (Fermi's golden rule). The rate constant for an ISC from a manifold of thermally populated initial vibronic states to a quasi‐continuum of final vibronic states , caused by spin–orbit interaction, is then given by
where is a canonical partition function for vibrational motion in the initial electronic state, is the inverse temperature, and is the energy of the vibrational level in the initial electronic state.
In many articles relating to TADF emitters, the ISC from a singlet to a triplet state colloquially is called a downhill process, whereas the reverse transition, rISC, is called an uphill process. As may be seen from the delta distribution in Eq. 8.1, the energy is strictly conserved during the nonradiative transition, i.e. the initial and final states are isoenergetic. What people have in mind when speaking of downhill and uphill processes is the difference between the adiabatic energies of the initial and final electronic states, possibly including zero‐point vibrational energy corrections. If that energy difference is positive, the transition is dubbed a downhill process and may occur at any temperature. If that energy difference is negative, thermal energy is required in addition to bridge the gap.
The efficiency of ISC and rISC is controlled by several factors. Intrinsically molecular factors are the magnitude of the spin–orbit coupling matrix element (SOCME), the adiabatic energy difference, and the coordinate displacement of the singlet and triplet potential energy surfaces as well as further factors such as the Duschinsky rotation of the respective vibrational modes. The most important external factor – aside from environment effects – is the temperature.
8.2.1 Condon Approximation
In the Condon approximation, where it is assumed that the electronic and vibrational degrees of freedom can be separated, the ISC rate is given by a product of the electronic and vibrational parts (direct SOC):
In principle, the origin of the Taylor expansion, , can be chosen at will. It is common practice, however, to choose the minimum geometry of the initial state to determine the electronic SOCME for ISC.
In first‐order perturbation theory, each Cartesian component of the spin–orbit Hamiltonian couples the singlet state to one and only one Cartesian triplet sublevel . For this reason, phase factors do not play any role in the calculation of the total ISC rate from a given singlet state to all triplet sublevels in Condon approximation. Furthermore, a common set of vibrational wave functions is chosen for all triplet fine‐structure levels. Hence, the squared contributions from all three components can just be summed up yielding
The situation is slightly more complicated for the reverse transition from a triplet to a singlet state. In general, the fine‐structure levels of a triplet state are separated by a zero‐field splitting (ZFS). If the ZFS is large in relation to the temperature, individual rISC rate constants would have to be determined for every fine‐structure level. Fortunately, ZFSs of TADF emitters are typically very small (10 cm) compared with thermal energies (298 K cm) so that the rISC rate constants can be averaged. Hence, in first‐order perturbation theory, the total rate constant of rISC for a molecule in the triplet state is given by
where the factor of 3 in the denominator of Eq. 8.4 takes care of the degeneracy of the triplet sublevels.
126.96.36.199 Electronic Spin–Orbit Coupling Matrix Elements
Microscopic spin–orbit Hamiltonians contain vector products between the electronic momentum and the derivatives of the one‐ and two‐electron Coulomb potentials , . Because these derivatives drop off like , SOC is a fairly short‐ranged interaction. Denoting the operator for the angular momentum of electron with respect to nucleus by and the corresponding operator for the angular momentum of electron with respect to electron by , the Breit–Pauli spin–orbit Hamiltonian is given by
Herein, is the gyromagnetic factor of the electron and is the fine‐structure constant. The two‐electron terms of the spin–orbit Hamiltonian contribute roughly 50% to the SOCME in molecules composed of light elements and can therefore not be neglected. They can, however, be combined in good approximation with the true one‐electron terms to form an effective one‐electron mean‐field operator . Whether the mean‐field approximation is sufficiently accurate to compute the typically very small SOCMEs of purely organic TADF emitters is not clear at present.
The El‐Sayed rules state that ISC in organic molecules is fast between excited states of different orbital types such as , whereas ISC between states of the same orbital type such as is slow . These qualitative rules are easily understood. To this end, we consider the one‐electron spin–orbit Hamiltonian as a compound tensor operator of rank 0:
where is a system‐specific parameter and the subindices denote the tensor components of the spatial and spin angular momentum operators, respectively. Like the more familiar ladder operators, these tensor operators can shift the magnetic quantum numbers of electrons. (See Ref.  for more details.) Consider , for example. The two states are related by a single excitation from to n. While can be used to transform an out‐of‐plane orbital to an in‐plane n orbital, shifts the spin state of the electron from to . The spin–orbit operator in Eq. 8.6 does not contain any combination that changes the spin magnetic quantum number of the electron, but leaves its spatial angular momentum quantum number untouched. For that reason, will be small. Moreover, in spatially nondegenerate states (as is the case in organic compounds), angular momentum operators do not have diagonal matrix elements because they are purely imaginary. Therefore, the electronic SOCMEs between a singlet and a triplet state with equal spatial orbital occupation vanish. This means in particular that the electronic SOCMEs between two CT states of similar wave function characteristics, , which play a prominent role in TADF emitters, will be very small. However, ISC between states of equal orbital type is not always slow. In order to have appreciable SOC between states of the same orbital type, it is necessary to go beyond the Condon approximation (see also Section 8.2.2).
A further obstacle for efficient SOC in TADF emitters is the shortrangedness of the spin–orbit interaction. Because of its dependence, the largest contribution to the SOCME comes from one‐center terms. Combining this criterion with the El‐Sayed rules, one finds that in Condon approximation appreciable spin–orbit integrals may arise only if the involved molecular orbitals (MOs) exhibit electron densities at the same center and if the atomic orbitals have different magnetic quantum numbers. This excludes the typical pair of CT and CT states where the unpaired electrons typically are far apart, i.e. their spin densities have little overlap. Consequently, the two‐electron exchange integral that largely determines the singlet–triplet energy gap is very small while at the same time also their mutual spin–orbit interaction is tiny. Hypothetically, substantial SOC can be imagined even for CT states, however, namely, if more than two electronic states are involved. Consider, for example, purely organic donor–acceptor systems in which the and excitations are energetically near degenerate. Herein, and represent occupied MO and lone‐pair orbitals of the donor, respectively, and an unoccupied MO of the acceptor. Comparing configurations, it is seen that and differ from each other by a local replacement at the donor that might in turn yield large SOC. Likewise, in MLCT excited states of Cu(I) complexes, a state might be located energetically close to a and could make use of the large SOC in the 3d shell.
188.8.131.52 Overlap of Vibrational Wave Functions
When deriving qualitative rules for probabilities of radiationless transitions in large molecules, Jortner and coworkers ,  differentiated between two major cases: the weak and the strong coupling cases (Figure 8.1).
In the weak coupling case (Figure 8.1a), the coordinate displacement for each normal mode is assumed to be relatively small. In this case, the transition probability depends exponentially on the adiabatic energy difference , i.e. the smaller the energy gap, the larger the transition probability . This relation is commonly called the energy gap law. People tend to forget, however, that this qualitative rule applies only for a pair of nested states.
The strong coupling case (Figure 8.1b) is characterized by large relative displacements in some coordinates so that an intersection of the potential energy surfaces can be expected. The probability of the radiationless transition then exhibits a Gaussian dependence on the energy parameter where is the molecular rearrangement energy that corresponds to half the Stokes shift for the two electronic states under consideration . Taking, additionally, temperature effects into account by assuming a Boltzmann distribution resulted in a generalized activated rate equation similar to Marcus theory , . In agreement with this model case, occasionally an inverse relationship between the transition probability and is observed, i.e. there exist cases where the transition probability increases with increasing energy gap .
8.2.2 Beyond the Condon Approximation
In order to have appreciable SOC between states of the same orbital type, it is necessary to go beyond the Condon approximation. Henry and Siebrand  were the first who discussed various contributions of different couplings to the ISC rate. In addition to the so‐called direct spin–orbit interaction, they considered spin–orbit interaction induced by Herzberg–Teller vibronic coupling and spin–orbit interaction induced by Born–Oppenheimer vibronic coupling. In practice, the latter two types are difficult to tell apart. Similar to the Herzberg–Teller expansion of the vibronic interaction, the SOC can be expanded in a Taylor series with respect to the nuclear coordinates about an appropriately chosen reference point , for example, the equilibrium geometry of the initial state. It is plausible to use normal modes for nuclear coordinates. Then, up to the linear term in normal mode coordinates, the SOCME is 
where the vector contains normal mode coordinates of the singlet electronic state and is the adjoint of the vector comprising the first‐order derivative couplings:
If the Taylor expansion is truncated after the linear term, the ISC rate for a singlet–triplet transition is a sum of three contributions due to (i) a direct term , (ii) a mixed direct‐vibronic , and (iii) a vibronic coupling term , where the direct term is identical to the expression in Eq. 8.3 and the latter two are given by
Note that all SOCMEs between the singlet state and the Cartesian components of the triplet state are purely imaginary. This is of particular importance when computing the mixed direct‐vibronic contributions. Again, due to spin symmetry, there are no cross terms between different Cartesian components in first order. Save for a factor of 1/3 that takes account of the degeneracy of the three triplet fine‐structure levels, a similar expression is obtained for the rISC starting from the triplet state.
Numerous examples have been found in heteroaromatic molecules where El‐Sayed forbidden ISC processes have rate constants that are nearly as large as those of El‐Sayed allowed transitions –. In most cases, vibronic interaction with an energetically close‐lying excited state through out‐of‐plane molecular vibrations enhances the transition probability. Even if no n‐type orbitals are available, as, e.g. in pure hydrocarbons, pyramidalization of unsaturated carbon centers in the excited state can lead to a substantial increase of electronic SOCMEs –.
8.2.3 Computation of ISC and rISC Rate Constants
ISC and rISC rate constants are highly sensitive with respect to the relative location of the singlet and triplet states. It is, therefore, of utter importance to employ reliable electronic structure methods for the computation of the excited‐state potentials (see Section 8.3).
184.108.40.206 Classical Approach
Herein, is the adiabatic energy difference between the singlet and the triplet state and denotes the Marcus reorganization energy that can – to first approximation – be equated with the energy variation in the initial singlet excited state when switching from the singlet equilibrium geometry to the triplet equilibrium geometry , . In TADF compounds, the reorganization energy can adopt minuscule values because CT and CT states often exhibit similar equilibrium geometries.
220.127.116.11 Statical Approaches
The method for computing ISC rate constants, applied in our laboratories, is based on the generating function formalism and the multimode harmonic oscillator approximation . Herein, the triplet state mass‐weighted normal modes with frequencies are related to their singlet counterparts by a Duschinsky transformation . The Duschinsky transformation , where is the Duschinsky rotation matrix and the displacement vector, is particularly important for pairs of states with strongly displaced minimum geometries.
Rates are obtained by numerical integration of the autocorrelation function in the time domain. This approach can even be applied to molecules with a large number of normal modes or to pairs of states that exhibit a large adiabatic energy gap. In these cases, the density of states becomes enormous, and a direct summation over all final vibrational states – even in a small energy interval around the initial state – is prohibitive. The generating function formalism is also applicable to finite‐temperature conditions that are essential for uphill processes such as rISC. Herein, a Boltzmann population of vibrational levels in the initial state is assumed . The derivation of the formulas for the direct, mixed direct‐vibronic, and vibronic ISC rates in the finite‐temperature case can be found in Ref.  A similar correlation function approach for computing rates constants of direct and vibronic ISC has been pursued by Shuai and coworkers .
18.104.22.168 Dynamical Approaches
Alternatively, nonadiabatic nuclear dynamics methods have been employed for determining the kinetic constants of ISC and rISC processes –. Herein, a diabatization scheme has been used to avoid the explicit calculation of nonadiabatic coupling matrix elements , and wavepacket dynamics simulations have been carried out within the framework of the multiconfiguration time‐dependent Hartree (MCTDH) method . We refrain from going into details here because these methods are reviewed in Chapter .
8.3 Excitation Energies and Radiative Rate Constants
8.3.1 Time‐Dependent Density Functional Theory
It is well known that time‐dependent density functional theory (TDDFT) yields substantial errors for the excitation energies of CT states, when approximate standard exchange‐correlation functionals are used . The balanced description of CT and locally excited (LE) states remains to be a challenge for TDDFT methods, even if used in conjunction with modern hybrid and range‐separated density functionals. Huang et al.  systematically correlated calculated and experimental singlet‐ and triplet‐transition energies of 17 CT compounds with the aim to find a recipe for the computational prediction of these quantities. They employed TDDFT using density functionals with varying amount of Hartree–Fock (HF) exchange ranging from 0% (BLYP) to 100% (M06‐HF). They propose to determine the optimal percentage of HF exchange semiempirically from a comparison of the calculated vertical energy with the measured absorption maximum and to use this value for scaling the HF contribution to the exchange‐correlation functional. Within the chosen set of molecules, optimal HF contributions between approximately 5% and 40% were found. This semiempirical procedure seems to work well, but it becomes very involved if also LE states play a role. In that case, the authors recommend employing distinct HF contributions for the different types of states.
Moral et al.  advocate the use of TDDFT in Tamm–Dancoff approximation (TDA) instead of full linear response TDDFT. They tested the performance of these approaches on a small set of organic molecules with experimentally known singlet–triplet splitting. Among them were three typical host materials with moderately high values (0.5–0.7 eV) as well as three TADF emitters with low values (0.1–0.3 eV). In this series, TDDFT–TDA yields a smaller root‐mean‐square deviation (RMSD) than TDDFT, leading the authors to conclude that TDA is better suited for computing singlet–triplet splittings. It appears questionable, however, whether the unweighted RMSD really represents a good measure for assessing the performance of different methods on this property. Owing to their significantly larger values, the host materials dominate the error analysis. Looking at the raw data of these authors, a different picture emerges. The PBE0 functional is the only one for which both types of calculations have been carried out. Indeed, TDDFT–TDA reproduces the singlet–triplet splittings of the three host materials to a better extent than TDDFT, whereas TDDFT performs better for the three TADF emitters. If the focus is laid on vertical singlet excitation energies, the performance of TDA is very unsatisfactory. In conjunction with the B2‐PLYP functional , the excitation energies of TADF materials are underestimated by up to 0.7 eV, whereas those of the host materials are somewhat overestimated. TDDFT–TDA performs better for TADF molecules if the B2GP‐PLYP functional  is employed instead, but then the excitation energies of the host materials are largely overestimated (by up to 0.5 eV). Further investigations are necessary for a sound judgment because the set is too small for being really representative.
8.3.2 DFT‐Based Multireference Configuration Interaction
The combined density functional theory and multireference configuration interaction (DFT/MRCI) method of Grimme and Waletzke is a well‐established semiempirical quantum chemical method for efficiently computing excited‐state properties . The MRCI expansion is based on MOs from a closed‐shell Kohn–Sham DFT calculation employing the BHLYP hybrid functional . In the Hamiltonian, BHLYP orbital energies are utilized to incorporate parts of the dynamical electron correlation. Parameters that scale the Coulomb and exchange integrals and damp off‐diagonal matrix elements have been introduced in the Hamiltonian to avoid double counting of the electron correlation. These parameters were fitted to experimental data. Independent benchmark studies on a representative set of organic molecules confirmed that the mean absolute error for DFT/MRCI electronic excitation energies lies below 0.2 eV . A distributed memory parallel code facilitates the calculation of electronic spectra of larger molecules . The average deviation is somewhat larger for first‐ and second‐row TM complexes, but Escudero and Thiel found the DFT/MRCI method to be superior to the tested TDDFT approaches and thus recommended it for exploring the excited‐state properties of TM complexes . This even holds true for third‐row TM complexes, if spin–orbit interaction is included that cannot be neglected in heavy‐element compounds , . Furthermore, DFT/MRCI is one of the few electronic structure methods applicable to large systems that gives the correct order of excited states in extended polyenes and polyacenes where doubly excited configurations play an essential role , .
While the method performs very well in general, it may be problematic when treating the donor–acceptor systems with small orbital density overlap that are key components of metal‐free TADF OLEDs. Caution is advised if double excitations with four open shells contribute to the DFT/MRCI wave function with substantial weight or if singlet‐coupled CT excitations exhibit lower energies than their triplet counterparts. Recently, an alternative form of correcting the matrix elements of an MRCI Hamiltonian that is built from a Kohn–Sham set of orbitals was devised in our laboratory . The new parameterization is spin invariant and incorporates less empirism compared with the original formulation while preserving its high computational efficiency. The robustness of the original and redesigned Hamiltonians has been tested on experimentally known vertical excitation energies of organic molecules yielding similar statistics for the two parameterizations , . Besides that, the new formulation is free from artifacts related to doubly excited states with four open shells, producing qualitatively correct and consistent results for excimers and covalently linked multichromophoric systems.
Long‐range interactions are not well represented by either of the two parameterized Hamiltonians. Asymptotically, DFT/MRCI performs like the underlying BHLYP functional. For charge‐separated systems this means that the energy increases with 1/2 instead of 1/ where is the distance between the two charged subsystems. Furthermore, dispersion interactions are not properly taken care of by DFT/MRCI. The latter problem may be easily remedied, for example, by adding the semiempirical Grimme D3 dispersion correction .
8.3.3 Fluorescence and Phosphorescence Rates
Similar to ISC and rISC rate constants, also rate constants for radiative transitions can be derived in the framework of time‐dependent perturbation theory. Herein, the vector potential for the motion of the electrons in the external electromagnetic radiation field is used as perturbation operator instead of . Typically, one proceeds by a multipole expansion of the interaction, with the electric dipole operator as the leading term. The rate constant for spontaneous emission from a manifold of thermally populated initial vibronic states to a quasi‐continuum of final vibronic states due to electric dipole interaction is then given by
where is the radiation frequency, is the fine‐structure constant, is the speed of light, and the other symbols have the same meaning as in Eq. 8.1. The procedure for simplifying the expression in Eq. 8.12 further by a Taylor expansion with respect to mass‐weighted normal coordinates is similar to that described in Section 8.2. Formulas for radiative transition rates in Franck–Condon (FC) or Herzberg–Teller approximation, respectively, can readily be derived.
Once the electronic wave functions have been obtained, it is straightforward to compute the electric dipole coupling matrix elements for fluorescence emission. If wave functions are not available – which is the case in TDDFT and coupled‐cluster approaches – the electronic transition rates can be computed by means of linear response theory . Typical fluorescence rate constants for emission from LE states are of the order of 10–10 s, whereas they are several orders of magnitude smaller for states. Because of competing nonradiative processes, the latter states are optically dark in most cases. The fluorescence rates of CT states depend critically on the overlap of the electron density distributions of the orbitals involved in the transition. If that overlap is small, fluorescence rates of 10–10 s are expected at most.
In the context of spin‐forbidden transitions, the coupling matrix element in Eq. 8.12 requires a bit of attention. In this case, multiplicity‐mixed electronic wave functions need to be employed. They are complex‐valued in general. The fundamentals regarding selection rules and intensity borrowing from spin‐allowed transitions have been worked out in detail in Ref.  and need not be repeated here. Nevertheless, it is instructive to compare phosphorescence from a , a , and a CT state to the electronic ground state . According to the El‐Sayed rules (see Section 22.214.171.124), the state exhibits sizeable SOCMEs with the electronic ground state and electronically excited states. Large contributions to the electric transition dipole matrix element can originate from two terms: (i) the mutual spin–orbit interaction of and multiplied by the difference of the static dipole moments of these states and (ii) the spin–orbit interaction of with optically bright states from which intensity can be borrowed. In heteroaromatic compounds, thus phosphorescence rates of 10 s can be achieved. For a state, the direct spin–orbit interaction with is very small so that term (i) can be neglected. With regard to term (ii), it is seen that exhibits sizeable SOCMEs with states, but the latter are optically dark. Therefore, the probability of a spin‐forbidden radiative decay is much smaller for a state, with rates typically below 1 s. The same applies to CT states of purely organic donor–acceptor systems. The shortrangedness of the spin–orbit interaction makes the contributions of type (i) vanish, despite the pronounced static dipole moment difference of the CT and states. In contrast, phosphorescence from MLCT states of TM complexes can acquire substantial probability through configuration interaction. The transition dipole moment may adopt sizable values originating from the combination of and in MLCT transitions. Also the direct, first (i)‐type interaction of the MLCT state with the electronic ground state can play a role, because the SOCMEs may not be negligible. Depending on the amount of configuration mixing and the size of the SOCMEs, phosphorescence rates of 10–10 s can be reached in these complexes.
Although the concept of intensity borrowing from spin‐allowed transitions is very transparent if Rayleigh–Schrödinger perturbation theory is applied to expand the multiplicity‐mixed wave functions in terms of pure eigenfunctions, this is not the procedure followed in practice. The reason is that the (in principle) infinite perturbation sums converge very slowly with respect to the number of electronic states. In an actual calculation, it is more advantageous to use methods that avoid an explicit summation over states such as multireference spin–orbit configuration interaction (MRSOCI)  or quadratic response theory , .
8.4 Case Studies
In the following, a brief literature survey on the results of quantum chemical studies on TADF emitters will be given before turning to a review of our own recent (and still ongoing) work in more detail. For a few prototypical cases, spin‐dependent multiconfigurational electronic structure methods have been employed to describe the electronically excited‐state potentials and their couplings. These data are used as input for Fourier transform methods that enable us to determine rate constants of radiative and nonradiative transitions and thus may further the understanding of the photophysical processes in TADF emitters.
8.4.1 Copper(I) Complexes
Various organometallic complexes based upon d metal ions were found to show TADF, the most abundant ones being Cu(I) complexes –. Among them are four‐coordinated but also three‐coordinated bis‐phosphine complexes – and bridged bimetallic Cu(I) complexes of type CuX(NP) (X = Hal) , . On the basis of combined DFT and TDDFT calculations, it was concluded that restricted flexibility of four‐coordinated Cu(I)–bis‐imine–bis‐phosphine complexes leads to a reduction of nonradiative deactivation and thus an increase of emission quantum yield . In threefold‐coordinated Cu(I) complexes with a sterically demanding monodentate N‐heterocyclic carbene (NHC) ligand and a heterocyclic bidentate ligand, the relative orientation of the ligands seems to decide whether TADF or phosphorescence is observed –. In the phosphorescent complexes, the conformational analysis indicates a nearly free rotation about the C–Cu bond in solution . Conformational flexibility appears to be also the key to understanding the photophysical properties of three‐coordinate thiolate Cu(I) complexes that give bright blue emission at 77 K and orange emission at ambient temperature . Very recently, even highly efficient blue luminescence of two‐coordinated Cu(I) complexes has been reported .
Most experimental work on these complexes is accompanied by some Kohn–Sham DFT calculations that focus on the nature of the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). This bears some danger because numerous examples are known in the literature showing that the lowest electronically excited state does not necessarily originate from a HOMO–LUMO transition , . Proceeding with due caution, it is preferable to use TDDFT or – even better – approximate coupled‐cluster methods or DFT/MRCI to characterize the excited states. In several cases, TDDFT has been employed to determine singlet–triplet energy gaps of Cu(I) complexes , , , , , . Gneuß et al. found a good correlation between computed triplet emission wavelength, as obtained from TDDFT in conjunction the B3LYP functional, and measured peak maxima in a new class of luminescent mononuclear copper(I) halide complexes with tripodal ligands . Quantum chemical studies of Cu(I) complexes that explicitly take account of spin–orbit interaction are very rare, however. To our knowledge, there is only a series of papers studying the emission properties of four‐coordinated Cu(I)–bis‐phenanthroline complexes , , ,  and our own work on luminescent Cu(I)–NHC complexes . In the following, these cases will be analyzed in more detail.
126.96.36.199 Three‐Coordinated Cu(I)–NHC–Phenanthroline Complex
Using the methods described in Sections 8.2 and 8.3, recently Föller et al.  conducted a thorough quantum chemical study on the photophysical behavior of a luminescent Cu(I) complex comprising an NHC and a phenanthroline ligand (Figure 8.2). This complex had been investigated experimentally by Krylova et al.  who also performed DFT calculations and assigned the luminescence to originate from an MLCT state. The bulky isopropylphenyl substituents on the imidazol‐2‐ylidene ligand are essential for the relative orientation of the NHC and phenanthroline ligands. Substitution by methyl or even phenyl substituents in 1,3‐position of the NHC leads to a barrierless torsional relaxation of the excited triplet and singlet states yielding a perpendicular conformation of the two ligands. Dispersion, included in the calculations by means of the semiempirical Grimme D3 correction , is seen to have a small but differential effect on the torsion potentials in the ground and excited states. It preferentially lowers the coplanar arrangement of the ligands and increases the barrier between the two minima on the excited‐state potential energy surface. Qualitatively, this trend is easily understood. In the coplanar nuclear arrangement, the hydrogen atoms in positions 5 and 13 of the phenanthroline ligand (Figure 8.2a) directly point toward the aromatic system of the isopropylphenyl substituents of the NHC ligand, whereas these are far apart when the NHC and phenanthroline ligands are oriented in a perpendicular fashion.
The spin‐free vertical excitation spectra were calculated by means of the original DFT/MRCI method , . SOC was included in the calculation of absorption spectra at the level of quasi‐degenerate perturbation theory (QDPT) using the in‐house SPOCK , . Fluorescence and phosphorescence rates were obtained at the MRSOCI level . FC emission profiles and temperature‐dependent ISC and rISC rates were determined in harmonic approximation by means of a Fourier transform approach , .
The experimental absorption spectrum 90 and calculated spectra at scalar relativistic level and including SOC effects are displayed in Figure 8.3. They are seen to match perfectly, showing that the applied quantum chemical methods are very well suited for studying these complexes. Very weak bands between 400 nm and 500 nm were assigned to triplet MLCT states. In the calculated FC spectrum, the electronic excitations to the , T, and T states lie between 400 nm and 420 nm but have small oscillator strengths so that they are barely visible in Figure 8.3. At the ground‐state equilibrium geometry, and T may be characterized as (phen) single excitations where is a linear combination of a ‐like orbital of copper with in‐plane p orbitals of the phenanthroline (phen) nitrogens. At this point, T results from a (phen) excitation where is a ‐like orbital with contributions located at the phenanthroline and NHC ligands. Following the line of arguments in Section 188.8.131.52, substantial spin–orbit interaction is expected between the T and states because their coupling involves a change of orbital angular momentum. This is the case, indeed, at the FC point. However, a slight geometry distortion is sufficient to reverse the order of the two triplet states. Henceforth, we renumerate the states according to the order of their adiabatic minima. The main configurations of the relaxed and T states are (phen) excitations. The excited‐state minima with torsion angle of 0 show a T‐shaped distortion of the three‐coordinated Cu(I), as suggested by Krylova et al. .
The perpendicular arrangement constitutes a saddle‐point on the electronic ground‐state potential energy surface located approximately 0.35 eV above the minimum. In the excited states, this arrangement of the ligands yields a local minimum. It is separated from the global minimum with coplanar arrangement of the ligands only by a shallow barrier (c. 0.12 eV) that is easily overcome by thermal activation. However, comparison of the theoretical and experimental emission spectra (Figure 8.4) clearly shows that the complex emits preferentially in a coplanar arrangement of the NHC and phen ligands.
The singlet–triplet splitting changes only slightly along the path, namely, from 650 cm at 0 to 830 cm at 90, in contrast to the observations of Leitl et al.  in a related Cu(I)–NHC–dipyridyldimethylborate complex. These authors report an increase from 540 cm at 0 to 3700 cm at 70 torsion angle. The different behavior of the two complexes can be explained by the different electron density distributions of the frontier orbitals. In the Cu(I)–NHC–phen complex, the density of the HOMO is mainly located at copper and the LUMO at the phen ligand, which does not change so much during the torsion (Figure 8.5). Correspondingly, only a small increase of the energy gap is found. In contrast, the orbital overlaps of the Cu(I)–NHC–dipyridyldimethylborate increases considerably upon torsion . In that case, the LUMO is located at the NHC ligand and the HOMO of the 90 geometry has substantial additional density at the NHC ligand. Adiabatically, the singlet–triplet splitting exhibits a value of merely 0.08 eV in the Cu(I)–NHC–phen complex. As the energy gap between the and T states is so small, TADF might be possible, and therefore phosphorescence and fluorescence as well as ISC and rISC rates were investigated for both the coplanar (torsion angle 0) and perpendicular (torsion angle 90) arrangement of the ligands by Föller et al. .
Figure 8.6 provides an overview over the computed rate constants in the coplanar arrangement of the ligands at room temperature. Kirchhoff et al.  carefully analyzed the kinetics of a three‐level system relating to TADF. They concluded that the steady‐state emission properties of the three‐level system depend upon the relative values of the various rate constants as well as the relative energies of the levels and considered two limiting cases, the kinetic limit and the equilibrium limit. The basic assumption in the kinetic limit case is that the state achieves a steady‐state concentration that is negligibly affected by the rISC process. Most of the photons would then appear from prompt fluorescence or phosphorescence. The decay of the Cu(I)–NHC–phen complex reaches nearly the kinetic limit at 77 K, as is several orders of magnitude smaller than and . In contrast, the equilibrium limit case where the steady‐state populations of the and T states are determined by Boltzmann statistics appears to be more appropriate at room temperature. For this limit case to be adequate, and . We have not calculated the rate constants for the internal conversion (IC) of to () and for the ISC from T to (). Because of the substantial energy gap between the and T states on the one hand and the electronic ground state on the other hand, these processes are assumed to be much slower than the radiative decay rates and will be neglected in the following. The first condition for the equilibrium limit, namely, is certainly fulfilled at all temperatures, but the second condition is not (Figure 8.6). In such case, formula
used by many experimentalists to fit the energy gap between the and T states – and the decay times of the individual states from the temperature dependence of the total emission decay time  is not valid at low temperatures.
The respective quantum yields were calculated instead following the kinetic analysis of Hirata et al. . In deriving their expression for the TADF quantum yield in relation to the triplet quantum yield , the authors assumed that the IC from the state to the electronic ground‐state can be neglected and that the ISC from to T is much faster than the reverse process. Neglecting the nonradiative deactivation of the T sublevels, but taking account of their phosphorescence decay through the averaged high‐temperature limit of the individual rate constants, one arrives at
For 298 K our calculated quantum yield of prompt fluorescence is quite small, only 0.3%. The quantum yield for the deactivation via phosphorescence is 77.7%, and the remaining 22% are the quantum yield for TADF. This may explain why Krylova et al.  classify this complex as phosphorescent emitter.
The emission quantum yield is related to the radiative and nonradiative decay rate constants by
Using this relation, intrinsic radiative lifetimes can be deduced from the knowledge of the experimental emission decay time constant and the quantum yield . The can directly be compared with theoretical values. In the present example, Krylova et al.  had determined values of s with in CHCl and of s with in crystals at room temperature. Using Eq. 8.15, intrinsic radiative lifetimes of 80 s in CHCl at 300 K and of ≈46 s in the crystalline state can be derived. These values compare better with the calculated fluorescence lifetime of = 11 s than with the calculated phosphorescence lifetime of = 267 s .
Summarizing, it is found that the excited singlet and triplet populations of the three‐coordinated Cu(I)–NHC–phen complex have time to equilibrate before they decay radiatively to the electronic ground state. TADF is possible, but it competes with phosphorescence that is the dominating radiative decay channel. As discussed above, torsion of the ligands has only small impact on the singlet–triplet gap. However, the electronic coupling between the and T states – and hence the probability for (reverse)ISC – is seen to increase substantially when moving from a coplanar to a perpendicular arrangement of the ligands. The quantum chemical analysis by Föller et al.  suggests that a perpendicular arrangement of the ligands in a three‐coordinate NHC–Cu(I)–(NN) complex is not a hindrance per se for observing TADF (in addition to phosphorescence), provided that the electron is transferred to the (NN) ligand in the MLCT transition and not to the NHC ligand.
184.108.40.206 Four‐coordinated Cu(I)–bis‐Phenanthroline Complexes
In the 1980s, McMillin and coworkers investigated the luminescence of a series of cationic Cu(I)–bis‐phen complexes in solution . The emission intensity was found to decrease as the temperature of the solution is lowered, accompanied by a slight redshift in the positions of the emission maxima. The authors interpreted the strong temperature dependence of the solution emission of the cationic bis(2,9‐dimethyl‐l,10‐phenanthroline)copper(I) complex (Cu(dmp), Figure 8.7a) in terms of emission from two thermally equilibrated excited states separated by ca. 1800 cm whereof the lower one was ascribed triplet MLCT and the upper one singlet MLCT character. It was suggested that DF dominates the emission at room temperature.
Owing to a pseudo‐Jahn–Teller (PJT) distortion of the copper d electronic configuration, these complexes undergo a fast flattening structural change in the MLCT excited state upon photo excitation (Figure 8.7b). TDDFT calculations revealed that there are four closely spaced triplet states in energetic proximity to the lowest singlet state . Siddique et al.  carried out a combined experimental and theoretical study of the Cu(dmp) complex including a rough estimate of the spin‐forbidden transition probabilities (semiempirical spin–orbit Hamiltonian on Cu only, single‐configuration approximation of the wave functions). The dihedral angle of the two ligand planes was found to change from 90 in the electronic ground state to about 75 in the MLCT states. Siddique et al. noticed that the photophysical properties depend strongly on that dihedral angle, the flattening distortion reducing not only the magnitude of the mutual SOCMEs of the lower states but also the transition dipole moment of spin‐allowed transition from which the phosphorescence borrows intensity.
In very elaborate theoretical investigations of the photophysics of cationic Cu(I)–bis‐phen complexes, Capano et al. , ,  used methods for determining the kinetic constants of the excited‐state processes differing from those described above. They applied the vibronic coupling Hamiltonian  and the quantum dynamics within the framework of the MCTDH method  to study the primary excited‐state nonadiabatic dynamics following the photoexcitation. To this end, they identified eight important vibrational modes and determined the nonadiabatic coupling coefficients for IC in a linear coupling model ,  The results of the wavepacket dynamics show that the IC from the initially photoexcited state to the state takes place with a time constant of about 100 fs and that ultrafast ISC occurs and competes with structural dynamics associated with the PJT distortion . Approximately 80% of the wave packets are found to cross into the triplet state within 1 ps. Further analysis of the MCTDH dynamics results emphasizes the importance of vibronic interactions for the ISC. It suggests that the ISC occurs close to the FC point – despite the moderate size of the SOCMEs – from to T and T promoted by the motion along a vibrational mode that drives the state into curve crossings with T and T.
In their latest work, Capano et al.  used DFT, TDDFT, as well as classical and quantum mechanics/molecular mechanics (QM/MM) molecular dynamics (MD) simulations to investigate the influence of the geometric and electronic structure, SOC, singlet–triplet gap, and the solvent environment on the emission properties of a series of four‐coordinated cationic bis‐phen complexes. They systematically varied the parent structure by adding electron‐withdrawing or donating substituents. Furthermore, they investigated the influence of long alkyl chains at 2,9‐positions of phenanthrolines because they were said  to retard the excited‐state PJT distortion and to increase the luminescence. In agreement with earlier work by Siddique et al. , Capano et al. conclude that the magnitude of the SOC matrix elements depends critically on the dihedral angle between the ligands, thus explaining the bi‐phasic ISC observed experimentally to originate from the initially excited and relaxed structures, respectively. Furthermore, they show that the singlet–triplet gap of the MLCT states is governed by inductive effects of the substituents that also control the oscillator strength of the fluorescence.
It might be interesting to carry out similar theoretical studies for related (PP) complexes in the future. Experimentally, it was shown that the quantum efficiencies of cationic mononuclear copper(I) complexes based on phenanthroline ligands could be increased from about 1% to around 60% when exchanging one of the phenanthrolines by a diphosphineether .
8.4.2 Metal‐Free TADF Emitters
Electroluminescence from purely organic molecules was detected more than 60 years ago using acridine orange as the emitter . Despite the substantial singlet–triplet splitting of 2900 cm, acridine orange shows relatively strong DF . Most of present‐day's metal‐free TADF emitters are bi‐ or multichromophoric systems consisting of electronically weakly coupled donor (D) and acceptor (A) subunits that undergo intramolecular charge‐transfer (ICT) processes upon electronic excitation. The donor and acceptor subunits are covalently linked in such a way that the respective systems are (nearly) orthogonal, either by twisted single bonds , , – or by a spiro‐junction –.
Many TADF molecules exhibit highly twisted structures, because the energetic splitting of a singlet–triplet pair depends on the degree to which the electron density distributions of the involved half‐occupied orbitals overlap. However, in an ICT compound also the transition density, and hence the oscillator strength of the emission depends on this quantity. Recognizing that small, but nonnegligible overlap densities between the electronic wave functions of the and states may lead to a substantial increase of the fluorescence rate without significantly increasing , Shizu et al. rationally designed highly efficient green and blue TADF emitters , .
A very instructive experimental investigation of donor–acceptor–donor (D–A–D) and donor–donor–donor (D–D–D) type emitters was carried out by Dias et al . With one exception, all compounds had a disubstituted heterocyclic core, with the substituents attached either in an angular or linear fashion. In the D–D–D structures, the lowest‐lying triplet state is an LE state with characteristics. Nevertheless, all emitters show DF in ethanol solution at room temperature, though with different kinetics. Some exhibit a linear relation of the DF intensity with exciton dose, indicating thermal activation of the DF, whereas others show a quadratic dependence, implicating DF caused by triplet fusion. Particularly striking is the observation of TADF for a compound with a singlet–triplet splitting of 0.84 eV. To explain their observations, Dias et al.  postulate an intermediate state that bridges the gap between the lowest triplet and singlet states in the heteronuclear compounds. Gibson et al.  predicted second‐order SOC of the singlet and triplet CT states to an intermediate LE state to enhance the ISC and rISC rate constants of a D–A molecule, composed of a phenothiazine (PTZ) donor and a dibenzothiophene‐S‐S‐dioxide DBTO2 acceptor, by several orders of magnitude. The crucial role of vibronic SOC to an energetically close‐lying LE state for the TADF efficiency of the corresponding D–A–D compound was impressively demonstrated by Etherington et al. . These authors employed host material with temperature‐sensitive polarity to tune the LE and CT states of the PTZ‐DBTO2‐PTZ system in and out of resonance. They could show that the emission intensity goes through a maximum at the zero crossing of the energy difference. The concomitant quantum dynamics studies included spin–orbit interaction between the LE and the CT states, vibronic interaction between the two triplet states, and hyperfine interaction between the singlet and triplet CT states as possible coupling terms. The outcome of these studies suggests that hyperfine interaction between the two CT states is by far too small and that the combination of spin–orbit and vibronic interaction is required to effectuate the rISC (compare also Chapter ).
Also the theoretical study by Chen et al.  was motivated by the question why some butterfly‐shaped blue D–A–D emitters show TADF properties although their lowest excited triplet state has LE character and the singlet–triplet splitting is substantial. Adachi and coworkers explained this behavior by reverse internal conversion (RIC) from the LE to the CT state that then undergoes rISC to the CT state . Although the investigated compounds are similar to those studied by Monkman and coworkers , , , Chen et al.  come to different conclusions. They computed rISC rates between the T and states in harmonic oscillator approximation including Duschinsky effects using a Fermi's golden rule expression. Their ansatz is very similar to the one employed in our laboratory, except for the fact that they use two‐component TDDFT  for computing the electronic coupling matrix elements. Not unexpectedly, the rate of rISC from the T state with LE character to the state with ICT character is several orders of magnitude too small in Condon approximation. Scanning through the torsional angle between donor and acceptor subunits reveals an intersection of the two lowest triplet potential energy curves at 90 with a maximum for the lowest triplet state. Actually, the authors find a broken‐symmetry minimum for T in their D–A–D example. It is argued that the triplet states can approach the intersection by low‐frequency motions and that nonadiabatic interactions are expected to play a significant role. Rates for vibration‐assisted transitions have not been presented, though, in that work .
In the following, the photophysics of three metal‐free TADF emitters shall be discussed in more detail, namely, of the green TADF emitter 1,2,3,5‐tetrakis(carbazol‐9‐yl)‐4,6‐dicyanobenzene (4CzIPN) and of the assistant dopants 3‐(9,9‐dimethylacridin‐10(9H)‐yl)‐9H‐xanthen‐9‐one (ACRXTN) and 10‐phenyl‐10H,10'H‐spiro[acridine‐9,9'‐anthracen]‐10'‐one (ACRSA). 4CzIPN has very high photoluminescence efficiency in apolar solvents and films ,  and is considered a prototypical donor–acceptor multichromophoric system. ACRXTN, consisting of an acridine donor unit and a xanthone acceptor unit , is particularly interesting because of the already very involved photophysical properties of the heteroaromatic xanthone moeity . ACRSA, finally, is one of the few spiro‐compounds known to exhibit efficient TADF . These compounds were investigated recently by means of high‐level quantum chemical methods in our laboratory with the aim to shine light on some of the underlying mechanisms –. The (preliminary) results of these studies will be compared with experimental data and – where available – theoretical results from TDDFT calculations.
220.127.116.11 1,2,3,5‐Tetrakis(carbazol‐9‐yl)‐4,6‐dicyanobenzene (4CzIPN)
Attaching carbazolyl donor units to dicyano‐substituted benzene cores as acceptors, Uoyama et al.  presented a series of luminescence emitters, their color varying from turquoise to red depending on the number of carbazolyl units and the positions of the cyano substituents. The green emitter 1,2,3,5‐tetrakis(carbazol‐9‐yl)‐4,6‐dicyanobenzene (4CzIPN, Figure 8.8) turned out to be a TADF emitter with excellent internal quantum efficiency in toluene and in 4,4(bis‐carbazol‐9‐yl)biphenyl (CBP) film . OLEDs based on 4CzIPN show high luminance efficiencies and excellent operational stability .
4CzIPN is a CT system with small singlet–triplet energy gap, the magnitude of which has been estimated from Arrhenius plots assuming that only T is located energetically below (see, however, below). The estimates vary slightly depending on the conditions and the solvent. Uoyama et al.  report a value of meV in CBP film, whereas Ishimatsu et al.  give a somewhat larger value of meV in toluene. Also the measured luminescence lifetimes vary somewhat between the two studies, but the values are in the same ballpark. At room temperature, Uoyama et al. obtain time constants of 17.8 ns for prompt and 5.1 s for delayed fluorescence in toluene solution under nitrogen atmosphere, while Ishimatsu et al. report 14.2 ns and 1.82 s for these processes, respectively, in the same solvent. Uoyama et al.  also carried out quantum chemical calculations. The computed singlet–triplet energy gap depends strongly on the functional used in the TDDFT calculations, ranging from approximately 10 meV for the hybrid functionals B3LYP and PBE0 over 362 meV for M06‐2X to 700 meV for the Coulomb attenuated CAM‐B3LYP in the gas phase. Likewise, the computed vertical emission wavelengths vary substantially with the functional, ranging from 731 nm (B3LYP) to 430 nm (M06‐2X) and 420 nm (B97X‐D).
The energy gap law  states that in the weak coupling regime, that is, for small coordinate displacements, the rate of nonradiative transition between two electronic states decreases exponentially with their increasing energy separation (see also Section 18.104.22.168). This does not necessarily mean, however, that a smaller energy gap automatically leads to improved luminescence properties of a TADF emitter. A detailed experimental and theoretical study investigating the solvent effects on TADF in 4CzIPN revealed a surprising trend with regard to the efficiency of the delayed fluorescence. Ishimatsu et al.  observe larger Stokes shifts of the emission in polar solvents compared with toluene suggesting stronger ICT in the excited state. Concomitantly, the activation energy for rISC is lowered. However, despite the smaller magnitude of in polar solvents, the photoluminescence quantum yield decreases from 94% in toluene solution over 54% in dichloromethane and 18% in acetonitrile to 14% in ethanol. Using TDDFT in conjunction with the M06‐2X functional and modeling the solvent–solute interaction by a polarizable continuum model , , they also computed absorption and emission energies for 4CzIPN in these solvents. The authors consistently explain the experimentally observed trends for , the emission wavelength, and the photoluminescence quantum yields by increasing ICT character of the excited states with increasing solvent polarity, leading in turn to weaker electric dipole transitions.
Ishimatsu et al.  report torsion angles between donors and acceptor of c. 50–65 for the electronic ground state, with the largest dihedral angle for the carbazolyl donor in 1‐position and the smallest one for the donor in 3‐position. In the DFT/B3LYP optimized structure obtained for the isolated molecule in our laboratory, the dihedral angles between the carbazolyl donors in 1‐, 2‐, and 3‐positions and the isophthalonitrile core are very similar (+63 and +64), while the carbazolyl donor in 5‐position shows a stronger twist (–71) . For computing the spin‐free properties of the electronically excited states, the parallelized version of the original DFT/MRCI ,  method was employed in the preliminary calculations. The wave function of the first excited singlet state is dominated by the HOMO→LUMO ICT transition but with marked contributions from local excitations. In particular, the local HOMO‐17→LUMO excitation on the isophthalonitrile core (Figure 8.9) has a coefficient of nearly 0.1 in the wave function at the ground‐state minimum. Local excitations lead to an enhancement of the oscillator strength for . This transition is responsible for the shoulder at c. 450 nm in the absorption spectrum of 4CzIPN . The vertical DFT/MRCI excitation energy of 2.94 eV (422 nm) in the gas phase is expected to be slightly redshifted due to solvent–solute interactions. The TDDFT excitation energy obtained by Ishimatsu et al.  for the M06‐2X functional in a PCM (3.29 eV, 376 nm) is substantially higher. Interestingly, we find two triplet states below at the FC point, with the electronic structure of T corresponding to . The multiconfigurational T wave function has the HOMO‐1→LUMO configuration as leading term. T is dominated by the HOMO → LUMO configuration. Both triplet excited states exhibit larger contributions from LE on the isophthalonitrile core than their singlet counterparts.
The T nuclear arrangement was optimized with unrestricted DFT (UDFT). At the T minimum, the donors in 1‐, 2‐, and 3‐positions exhibit larger torsion angles with respect to the isophthalonitrile core (ranging between +68 and +73) than in the electronic ground state, whereas the dihedral angle is flatter () for the donor in 5‐position. This trend reflects the fact that electron density was mainly donated by the carbazolyl substituents in 1‐, 2‐, and 3‐positions upon the ICT excitation to the T state. At this point of the coordinate space, and T result predominantly from the HOMO → LUMO excitation. Still, two triplets are found below the first excited singlet state in the DFT/MRCI calculations. and T are separated by an energy gap of 86 meV here, in excellent agreement with experimental evidence. The T state is located halfway between the T and states at this geometry.
The SPOCK ,  program was used to determine SOCMEs of the DFT/MRCI wave functions. The mutual SOCMEs of the three closely spaced electronic states are small. For the and T DFT/MRCI wave functions, a sum over squared SOCMEs of cm was obtained. The electronic coupling of and T is slightly larger (sum over squared SOMCEs cm). The presence of the intermediate triplet state might therefore accelerate the ISC and rISC processes. The calculation of ISC and rISC rate constants is in progress.
Fluorescence and phosphorescence rates were obtained at the MRSOCI level . The MRSOCI calculations were performed at the UDFT‐optimized T minimum. Technically, they are at the limit of what can be handled by the current version of the SPOCK program. In the Davidson diagonalization of the MRSOCI matrix, 8 complex‐valued eigenvectors with expansion lengths of configuration state functions were determined. The sublevels of the T states are degenerate for all practical purposes. Phosphorescence is not a competitive decay mechanism in 4CzIPN. The calculated rate constants for phosphorescence are of the order of merely – s.
Although the state wave function is dominated by ICT excitations ( D), the fluorescence exhibits substantial oscillator strength. The emission from the state gains intensity from small amounts of local excitations on the isophthalonitrile core. At the T minimum geometry, a vertical emission wavelength of 482 nm is computed for the isolated molecule, in good agreement with the experimental peak maximum at 507 nm ,  measured in toluene. For a comparison of the calculated (pure) radiative lifetime of 44 ns with measured time constants, quantum yields have to be taken into account. Using Eq. 8.15 and the experimentally determined quantum yield of 21.1% for prompt fluorescence at 300 K  yields a lifetime of ns that compares well with the experimental values of 17.8 ns  and 14.2 ns  in toluene solution.
The magnitude of the local contributions to the transition density depends critically on the dihedral angles between the molecular planes of the carbazolyl and isophthalonitrile moieties. When this angle is constrained to 90, the oscillator strength drops by four orders of magnitude, thus markedly decreasing the luminescence probability. In this case, only one triplet state of B symmetry is located below the first excited singlet state, and their mutual SOCME vanishes by symmetry selection rules. This shows that the deviation from an orthogonal orientation of the donor and acceptor units is essential for the performance of the 4CzIPN TADF emitter.
22.214.171.124 Mechanism of the Triplet‐to‐Singlet Upconversion in the Assistant Dopants ACRXTN and ACRSA
Recently, one of us started investigating the photophysics of 3‐(9,9‐dimethylacridin‐10(9H)‐yl)‐9H‐xanthen‐9‐one (ACRXTN) and 10‐phenyl‐10H,10′H‐spiro[acridine‐9,9‐anthracen]‐10‐one (ACRSA) (Figure 8.10) by quantum chemical methods , . ACRXTN and ACRSA have been utilized as assistant dopants in OLEDs . The idea behind this approach is to use triplet excitons for populating the state of the assistant dopant by rISC. Instead of radiatively decaying by fluorescence, the state transfers its excitation energy by FRET to a strongly fluorescent organic emitter. Nakanotani et al.  could show that the presence of the assistant dopant substantially improved the external electroluminescence quantum efficiency of the OLED, indicating an internal exciton production efficiency of nearly 100%.
In ACRXTN, acridine and xanthone are covalently linked, with the corresponding molecular planes arranged in a perpendicular fashion (see Figure 8.10). The HOMO of ACRXTN is a ‐type orbital on the acridine moiety, whereas its LUMO is a orbital localized on xanthone. Hence, the lowest electronically excited state is expected to have ICT character. For computing electronically excited states, TDDFT  in conjunction with the B3LYP density functional, resolution‐of‐the‐identity approximated coupled‐cluster response methods (RI‐CC2) ,  as well as the redesigned DFT/MRCI‐R  quantum chemical methods were employed. All theoretical methods agree that the lowest excited triplet and singlet states originate from an ICT excitation from acridine to xanthone . Experimentally, the fluorescence (F) and phosphorescence (P) emissions in dichloromethane peak at 2.53 and 2.47 eV, respectively. The vertical DFT/MRCI‐R emission energies in vacuum are only slightly larger (F: 2.77 eV, P: 2.71 eV) in the SV(P) basis, RI‐CC2 yields an even higher value of 3.09 eV for both, whereas TDDFT/B3LYP gives 2.19 eV (F) and 2.18 eV (P). While the energetic separation between the LE and ICT states is nearly identical for DFT/MRCI‐R and RI‐CC2, this is not the case for TDDFT/B3LYP. Hence, it appears that RI‐CC2 might be better suited for the optimization of excited‐state geometries than TDDFT.
ACRXTN seems to have inherited some of the photophysical properties of the parent monochromophores . In addition to the ICT states, two low‐lying triplet states with and electronic structure as well as a state are found that correspond to local excitations of the xanthone moeity . So far, ISC and rISC rate constants have not been determined for this kind of complex. From the course of the potential energy curves and the knowledge of the coupling matrix elements, the following qualitative picture emerges.
In apolar media, the potential energy surface of ACRXTN exhibits at least two minima, the global minimum with ICT electronic structure and a local minimum originating from a local excitation on xanthone. Two or three minima are expected on the lowest triplet excited‐state surface, with the triplet ICT minimum being the global one. The second minimum on the T surface exhibits T electronic structure. It is nearly degenerate with the T minimum in apolar media. In the ICT potential well, the singlet–triplet splitting is small enough (0.06 eV) to enable, at least in principle, the thermally activated rISC from the triplet to the corresponding singlet. However, the direct SOC between the states is too small (sum of squares cm) to make this process efficient. The carbonyl stretching vibration drives the system through a crossing with the T state that mediates the coupling of the ICT states and allows for an equilibration of the singlet and triplet ICT populations. In polar media, the states are blueshifted, whereas the T and ICT states experience slight redshifts. Hence, a double minimum situation on the lowest triplet excited‐state surface can be foreseen. In contrast, only one minimum with ICT character is expected on the lowest singlet excited‐state potential energy surface. The T state continues to be the doorway state mediating the (r)ISC of the singlet and triplet ICT states in ARXCTN.
The spiro‐compound ACRSA is very similar in that respect. Within an energy interval of 0.3 eV, five electronically excited states are found in the gas phase and in apolar solvents . In ACRSA, the ICT from the phenylacridine to the anthracenone chromophores (HOMO → LUMO) is the lowest excitation in the vertical absorption region. Because of its low oscillator strength (), the singlet transition at 378 nm is barely visible in the calculated absorption spectrum. This is also true for the excitation (346 nm) and the (HOMO‐1 → LUMO) transition (318 nm). According to our calculations, the shoulder in the absorption spectrum around 310 nm stems from two local acridine transitions. The shoulder is slightly blueshifted with respect to experiment where a shoulder is observed around 320 nm in toluene solution . This is also the case for the band maximum that is found at about 280 nm compared with the experimental value at approximately 300 nm. It arises from acridine to phenyl excitations. The overall shape of the computed absorption spectrum agrees very well with the experimental spectrum in toluene. Test calculations for the isolated system suggest that the computed excitation energies are lowered by about 0.1 eV when a larger basis set of valence triple‐zeta plus polarization quality is used, thus improving the agreement with experiment.
With a relative permittivity of , toluene is an apolar solvent. Nevertheless, the solvent–solute interactions, modeled by conductor‐like solvent model (COSMO) , , preferentially stabilize the ICT state by 0.09 eV, whereas the state is destabilized by 0.10 eV with respect to the gas phase. Adiabatically, the lowest excited state does not stem from the (HOMO → LUMO) transition. At the DFT/MRCI‐R level of theory, an LE state of the anthracenone moeity, , constitutes the global minimum on the T potential energy surface. Close by, the CT and the lowest states are located. Likewise, the global minimum of the state has character that is nearly degenerate with the CT state. For this reason, strong nonadiabatic coupling is expected in addition to SOC. Similar to ACRXTN, the CO stretching mode drives the low‐lying states toward intersections of the potential energy surfaces (Figure 8.11). And indeed, rate constants for rISC of the order of 10 s in toluene have been derived from quantum dynamics calculations including vibronic coupling and SOC simultaneously . The efficient rISC in this compound is attributed to the presence of LE and states and their strong interaction with the CT states.
Vibronic interactions are also required to make fluorescence electric dipole allowed. The calculated vertical singlet emission energies of 2.93 eV (423 nm) for the ICT state and of 2.89 eV (429 nm) for the state in toluene are substantially blueshifted with respect to the photoluminescence band maximum that is found at about 500 nm in the same solvent according to Nasu et al. . At present, it is not clear where the discrepancy comes from. Two things are striking, however. Firstly, in DPEPO film the emission maximum is found experimentally at about 480 nm . Due to the polarity of DPEPO, a bathochromic shift of the peak maximum with respect to its wavelength in toluene solution would have been expected. Instead, a hypsochromic shift of at least 20 nm is found.
Secondly, ACRSA is used as an assistant dopant for the blue fluorescence emitter tetra‐ter‐butylperylene (TBPe). Herein, it is assumed that ACRSA transfers its excitation energy by FRET to TBPe. Save for the proper orientation of the transition dipole moments, FRET is only efficient, however, if the emission spectrum of the FRET donor and the absorption spectrum of the FRET acceptor have substantial overlap . As may be seen from Figure 1b of Ref. , there is barely any overlap between the emission spectrum ascribed to ACRSA and the absorption spectrum of TBPe that has its origin transition at wavelength shorter than 450 nm. As the enhancement of external quantum efficiency by the assistant dopant is undoubted, the shown emission spectrum can probably not be assigned to ACRSA.
Polar solvents such as acetonitrile () shift the states substantially toward higher excitation energies, whereas the lowest state is nearly unaffected by the solvent. In contrast, the ICT states are significantly redshifted. Because of these trends, we expect the electronic states to be turned into and out of resonance depending on the particular environment and the temperature.
8.5 Outlook and Concluding Remarks
Insight into the factors that determine the probability of TADF is a key step toward the design and optimization of third‐generation OLED emitters. Despite intensive research on this topic in the latest years, a complete and consistent rationalization of TADF is still missing. As outlined in this chapter, a small singlet–triplet splitting of the electronically excited emitter states is not sufficient for TADF to take place. Rather, the molecular parameters that steer the relative probabilities of excited‐state processes such as intramolecular charge and energy transfer, ISC, rISC, fluorescence, phosphorescence, and nonradiative deactivation have to be understood. Computational chemistry can substantially contribute to this understanding. In particular, it can provide detailed information about spectroscopically dark states and their coupling to the luminescent ones, information that is difficult or even impossible to obtain from experimental data alone. Moreover, starting from a lead structure, quantum chemistry can easily assess the effects of chemical substitution.
With regard to internal quantum yields and rate constants, experimental and theoretical information is complimentary. Experimentally, (r)ISC rate constants are often determined indirectly from the quantum yields of the prompt and delayed components. Time‐resolved spectra of such complexes, from which ISC rate constants could be retrieved directly, are still scarce. While suffering from uncertainties with regard to the underlying models and quantum chemical methods, theory can, in principle, determine rate constants and use them to derive internal TADF quantum yields that can be compared with experimental data.
The quantum chemical methods employed in our preliminary work are well suited for computing spectroscopic properties of the systems at hand. DFT/MRCI‐R is the method of choice for obtaining reliable electronic excitation energies and excited‐state properties at reasonable cost in the purely organic donor–acceptor systems. SPOCK is a powerful tool for computing electronic SOCME and phosphorescence rate constants. Less demanding single‐reference linear response methods such as RI‐CC2 or, where applicable, TDDFT can be used to obtain excited‐state minimum geometries and vibrational frequencies. Because of the comparably low nonradiative transition rates, typical for (reverse) ISC in TADF emitters, a static Fourier transform formalism seems appropriate for calculating the rate constants. What is presently missing is an efficient way to compute vibronic SOC rates for larger systems. Vibronic SOC is considered essential in donor–acceptor systems because of the small magnitude of the direct electronic SOCMEs between singlet and triplet CT states.
The latest class of hyperfluorescent OLEDs combines high internal quantum efficiency and long operational stability by using assistant dopants for the harvesting of triplet and singlet excitons in addition to fluorescence emitters. So far, quantum chemical research in that direction is missing. Modeling the excitation energy transfer from the assistant dopant to the fluorescent acceptor beyond the ideal dipole approximation is a challenging task that could be worth considering.
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