Thomas A.
Niehaus
*a,
Thomas
Hofbeck
b and
Hartmut
Yersin
*b
aInstitut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany. E-mail: thomas.niehaus@ur.de
bInstitut für Physikalische Chemie, Universität Regensburg, 93040 Regensburg, Germany. E-mail: hartmut.yersin@ur.de
First published on 10th July 2015
Light emitting organo-transition metal complexes have found widespread use in the past. The computational modelling of such compounds is often based on time-dependent density functional theory (TDDFT), which enjoys popularity due to its numerical efficiency and simple black-box character. It is well known, however, that TDDFT notoriously underestimates energies of charge-transfer excited states which are prominent in phosphorescent metal–organic compounds. In this study, we investigate whether TDDFT is providing a reliable description of the electronic properties in these systems. To this end, we compute 0–0 triplet state energies for a series of 17 pseudo-square planar platinum(II) and pseudo-octahedral iridium(III) complexes that are known to feature quite different localization characteristics ranging from ligand-centered (LC) to metal-to-ligand charge transfer (MLCT) transitions. The calculations are performed with conventional semi-local and hybrid functionals as well as with optimally tuned range-separated functionals that were recently shown to overcome the charge transfer problem in TDDFT. We compare our results against low temperature experimental data and propose a criterion to classify excited states based on wave function localization. In addition, singlet absorption energies and singlet–triplet splittings are evaluated for a subset of the compounds and are also validated against experimental data. Our results indicate that for the investigated complexes charge-transfer is much less pronounced than previously believed.
In the past, pseudo-square planar platinum(II) and pseudo-octahedral iridium(III) complexes have shown to be excellent emitter materials for phosphorescent OLEDs and were synthesized with a variety of organic ligands.1,4,25–29 In order to obtain suitable candidate materials for future applications, a reliable computational screening procedure covering a large number of ligands and combinations thereof would be highly rewarding. As a numerically efficient quantum chemical method that provides useful accuracy for the key quantity ΔET1, time dependent density functional theory30–32 (TDDFT) has become the major tool in this regard over the last years. With recent relativistic extensions, spin–orbit interaction can be accounted for and enables the computation of triplet zero-field splittings and radiative rates for phosphorescence.33,34 These parameters can then directly be compared to experimental data. As reviewed by Escudero and Jacquemin,35 also the deactivation pathways of excited phosphors have been the subject of TDDFT simulations recently.
Although TDDFT frequently provides excited state energies with fairly systematic errors of 0.2–0.4 eV, there are also several well documented failures of the theory.36 One of these, the severe underestimation of charge-transfer excited state energies, seems to undermine the potential use of TDDFT in the field of organo-transition metal photophysics and photochemistry. This is because low lying excited states of these compounds are often characterized as MLCT states, based on experimental observations28,37 and also on more sophisticated correlated quantum chemical methods.38 This shortcoming of TDDFT is well understood39–41 by now and we summarize the main arguments below for the later discussion.
Let us assume the system of interest is a closed shell molecule and the excited state is dominated by a transition from the occupied orbital i into the unoccupied orbital a. The difference between the Kohn–Sham orbital energies associated with this transition is called ΔEia. In this case, the TDDFT eigenvalue problem that leads to the singlet (ΔES) and triplet (ΔET) excitation energies may be drastically simplified:42,43
ΔES/T = ΔEia + KS/T | (1) |
KS = 2(ia|ia) + (ia|f↑↑xc + f↑↓xc|ia) − b(ii|aa) | (2) |
KT = (ia|f↑↑xc − f↑↓xc|ia) − b(ii|aa). | (3) |
Here (ia|ia) denotes a two-electron integral in the Mulliken notation44 and is the functional derivative of the DFT exchange–correlation (xc) energy with respect to the density for spin-up (n↑) and spin-down (n↓) electrons (σ, τ ∈ {↑, ↓} are spin indices). Eqn (1)–(3) are valid both for functionals that depend on the density only (termed local in the following) or depend on the density and its gradient (i.e., semi-local functionals in the generalized-gradient approximation (GGA)). They hold as well for hybrid functionals that incorporate a fraction b of Hartree–Fock (HF) exchange. In the limit fxcστ = 0 and b = 1, the equations above reduce to the TD-HF case. For a general charge-transfer excitation, the overlap of orbitals i and a is small, such that the correction terms KS/T can be neglected for local and semi-local functionals (b = 0). The singlet–triplet splitting therefore tends to zero and excitation energies reduce to the Kohn–Sham gap. The latter differs strongly from the quasiparticle gap, defined as the difference of ionization potential (IP) and electron affinity (EA).45,46 This is because the electrons in occupied and virtual orbitals experience the same local potential in DFT. This needs to be compared with HF theory, where an additional electron in a virtual orbital feels the potential of all N electrons. Because of the exact cancellation of self-interaction, an electron in the occupied set feels the potential of N − 1 electrons in contrast. The orbital energy differences ΔEia for hybrid functionals are therefore much larger as differences from pure functionals and provide reasonable approximations for the quasiparticle gap. In fact, the excitation energy for a transition from donor (D) and acceptor (A) molecules separated by a large distance R is given by ΔE = IPD − EAA − 1/R. In this limit the last term of eqn (2) simplifies to −b/R, which shows that hybrid functionals account at least partially for the electron–hole stabilization which is completely absent for currently available LDA/GGA functionals.‡ Recently, range-separated functionals that exhibit 100% HF exchange at large electron–electron distances gained a lot of popularity.48–53 These functionals are designed to suppress the self-interaction error and provide good approximations to quasiparticle gaps.54 They were also shown to solve the erratic description of long range charge-transfer excitations.55
Given the discussion above and the fact that the excited states of phosphorescent organo-transition metal complexes have partial MLCT character, it surprising that several previous case studies show very good agreement with the experiment using simple hybrid functionals like B3LYP.56–63 Admittedly, the charge-transfer is not long range inter-molecular but intra-molecular in these systems, but questions remain how general these findings are. This information is of great importance for the rational design and computational screening of OLED emitter materials in order to assess the expected accuracy of a given calculation. In this investigation, we therefore analyze the performance of TDDFT for a comprehensive set of 17 cyclometallated platinum(II) and iridium(III) complexes with high phosphorescence quantum yields. Instead of discussing vertical absorption energies, we compute 0–0 emission energies including zero-point energy corrections to make direct contact with the experimentally relevant quantities. For a subset of the compounds, also the S1 energy and the singlet–triplet splittings (ΔEST = ΔES1 − ΔET1) are determined. Through a comparison of results for GGA, hybrid and range-separated exchange–correlation functionals, we finally attempt to answer the question, whether metal-to-ligand excitations pose significant problems for modern TDDFT.
The corresponding studies were usually carried out down to T = 1.3 K and techniques of site-selective emission and excitation spectroscopy under variation of temperature and magnetic fields as well as time-resolving methods were applied. The compounds to which it is referred in this study1,27,28,37,64,65,67–75 are summarized in Fig. 1. Table 1 displays the experimental 0–0 substrate averaged transition energies of the T1 state. Moreover, we give the matrix used and the classification as worked out for the respective T1 state.
Fig. 1 Chemical structures of compounds discussed. Complexes 11 and 13 are occasionally also abbreviated as Pt(bzq)(dpm) and Pt(bzq)2, respectively. |
# | Compound | S0 → T1 transition | PBE | B3LYP | LC-PBE (γ) | Exp. | Class [exp.] | Class [theo.] |
---|---|---|---|---|---|---|---|---|
1 | Pt(qol)2 | 5d (25%/18%) + πqol → π*qol | 1.48 | 1.85 | 1.99 (0.21) | 1.91 (ref. 65) | 3LC | 3LC |
2 | Pt(ppy2-tBu2a) | 5d (25%/19%) + πppy → π*ppy | 1.47 | 1.94 | 1.97 (0.16) | 1.99 (ref. 37 and 98) | 3LC/MLCT | 3LC |
3 | Pt(piq)(acac) | 5d (40%/33%) + πpic → π*pic | 1.90 | 2.07 | 2.18 (0.20) | 2.09 (ref. 99) | 3LC | 3LC/MLCT |
4 | Pt(2-thpy)2 | 5d (47%/34%) + π2-thpy → π*2-thpy | 1.88 | 2.11 | 2.20 (0.29) | 2.13 (ref. 64) | 3LC/MLCT | 3LC/MLCT |
5 | Pt(thpy)(acac) | 5d (34%/26%) + πthpy → 5d* (9%/9%) + π*thpy | 2.12 | 2.19 | 2.06 (0.40) | 2.23 (ref. 71) | 3LC/MLCT | 3LC |
6 | Ir(pbt)2(acac) | 5d (53%/46%) + πpbt → π*pbt | 1.93 | 2.29 | 2.28 (0.16) | 2.26 (ref. 37) | 3MLCT | 3MLCT/LC |
7 | Pt(dphpy)(CO) | 5d (27%/18%) + πdphpy → 6p* (11%/13%) + π*dphpy + π*CO | 2.06 | 2.32 | 2.46 (0.21) | 2.38 (ref. 37) | 3LC/MLCT | 3LC |
8 | Pt(ppy2-fluoren) | 5d (55%/45%) + πppy → π*ppy | 2.05 | 2.42 | 2.45 (0.18) | 2.41 (ref. 37 and 98) | 3MLCT/LC | 3MLCT/LC |
9 | Pt(ppy2-C2) | 5d (59%/48%) + πppy → π*ppy | 2.05 | 2.48 | 2.66 (0.27) | 2.42 (ref. 37 and 98) | 3MLCT/LC | 3MLCT/LC |
10 | Pt(ppy)2 | 5d (61%/50%) + πppy → π*ppy | 2.05 | 2.48 | 2.67 (0.29) | 2.43 (ref. 37) | 3MLCT/LC | 3MLCT/LC |
11 | Pt(bhq)(dpm) | 5d (33%/26%) + πbhq → π*bhq | 2.02 | 2.40 | 2.28 (0.36) | 2.44 (ref. 37 and 73) | 3LC | 3LC/MLCT |
12 | Ir(ppy)3 | 5d (60%/52%) + πppy → π*ppy | 2.05 | 2.53 | 2.47 (0.16) | 2.45 (ref. 70) | 3MLCT | 3MLCT |
13 | Pt(bhq)2 | 5d (56%/42%) + πbhq → π*bhq | 1.99 | 2.46 | 2.55 (0.28) | 2.51 (ref. 37) | 3MLCT/LC | 3MLCT/LC |
14 | Pt(4,6-dFppy)2 | 5d (55%/42%) + π4,6-dFppy → π*4,6-dFppy | 2.20 | 2.57 | 2.72 (0.26) | 2.56 (ref. 67) | 3MLCT/LC | 3MLCT/LC |
15 | Pt(ppy)(acac) | 5d (41%/35%) + πppy + πacac → π*ppy | 2.29 | 2.57 | 2.43 (0.41) | 2.56 (ref. 4 and 73) | 3LC/MLCT | 3LC/MLCT |
16 | Pt(dFpthiq)(dpm) | 5d (33%/31%) + πdFpthiq + πdpm → π*dFpthiq | 2.48 | 2.69 | 2.80 (0.21) | 2.71 (ref. 37) | 3LC/MLCT | 3LC/MLCT |
17 | Pt(ppz)2 | 5d (57%/35%) + πppz → 6p* (8%/10%) + π*ppz | 2.66 | 2.95 | 3.11 (0.28) | 2.85 (ref. 100) | 3LC/MLCT | 3LC/MLCT |
The classification of the lowest triplet states of the investigated compounds has been based on their zero-field splittings ΔE(ZFS), their radiative emission decay times, and, if available, to the (normalized) emission intensities of the vibrational satellites of metal–ligand character.28,37,72,74,75 Thus, T1 states with ΔE(ZFS) values < 2 cm−1 are assigned to a 3LC(ππ*) character. To have a course guide line, we classify states that show splitting values up to about 10 cm−1 or 20 cm−1 as 3LC(ππ*)/3MLCT(dπ*) states and for the range of about 20 cm−1 < ΔE(ZFS) < 50 cm−1, we use the notation 3MLCT(dπ*)/3LC(ππ*), while states with splittings above about 50 cm−1 are assigned as 3MLCT(dπ*) states.37,72,75 These notations are independently supported by the trends that are found for the radiative rates and the vibrational satellite structures (for details see especially ref. 28, 37, 72 and 75). Obviously, these assignments should only be used for approximate characterizations.
At the optimized geometry, excitation energies were compared for several different functionals. We consider representative functionals for each of the major classes highlighted in the introduction: the semi-local PBE,79 the often used hybrid B3LYP, and the range-separated LC-PBE functional. The latter is based on a partitioning of the Coulomb operator into a short-range and long-range component, dictated by the parameter γ, for example by utilizing the standard error-function erf(x):48,49
(4) |
The first term is a Coulomb operator decaying to zero on a length scale of ≈1/γ and is therefore short-ranged (SR), while the second term dominates at large r accounting for the long-range (LR) behavior. This gives rise to a decomposition of the xc energy functional according to
Exc = ESRx,DFT(γ) + ELRx,HF(γ) + Ec,DFT. | (5) |
In the case of LC-PBE, ESRx,DFT is the short-range form of the gradient-corrected PBE exchange functional,80ELRx,HF denotes the Hartree–Fock exchange energy evaluated with the long-range part of the interaction in eqn (4), while the correlation part Ec,DFT is left unchanged with respect to the usual form of PBE. There are different strategies to determine the value of the parameter γ. One possibility is choosing it to minimize the deviation of various molecular properties with respect to experimental or accurate first principles data on large test sets (see, e.g., ref. 81 and 82), leaving the parameter fixed in further applications. Another route is the system specific tuning proposed by Baer and co-workers,53 which we will follow here. Given that DFT with the exact exchange–correlation functional should obey Koopmans' theorem, one can find the optimal γ for the compound in question by minimizing the following error function:
ΔIP(γ) = |εHOMOγ − [Etotγ(N) − Etotγ(N − 1)]|, | (6) |
For all mentioned functionals, linear response TDDFT calculations (also known as Casida approach) were performed in the adiabatic approximation.42 Depending on the actual value of γ, the range-separated functionals may incorporate a large fraction of Hartree–Fock exchange. As shown by Tozer83,84 and Brédas,85 triplet instabilities in the ground state may lead to a significant underestimation of triplet excited state energies in such a case, especially in large π-conjugated systems. Since several of the ligands investigated in this study fall into this class, we also perform calculations in the Tamm–Dancoff approximation86 (TDA), which is known to alleviate these problems. Metal atoms were treated at the same level as in the ground state, while for the remaining atoms a split-valence basis set of triple-ζ quality plus polarization functions (6-311G*) was employed. For a subset of the studied compounds also calculations including diffuse functions for the ligands (6-311++G**) and using the triple-ζ basis LANL2TZ(f)87 for the metal have been performed. The results in the ESI (Table S4†) indicate a modest gain in accuracy (<0.1 eV).
Earlier TDDFT studies on transition metal complexes also indicate that solvent effects play a crucial role. For simulations in the gas phase, it was frequently observed that the ligand-to-ligand charge-transfer character is exaggerated, with a concomitant underestimation of excitation energies with respect to experiment.88 Here we employ the COSMO solvent model89 as implemented in NWChem with a dielectric constant of 1.95 corresponding to n-octane. This is the solvent that was used in the majority of measurements that are reported here. In order to estimate the influence of the solvent model we also tested the so-called solvation model based on density (SDM, solvent octane) put forward by Truhlar and co-workers90 and implemented in version 6.5 of NWChem. The corresponding results in the ESI (Table S5†) show only very minor changes (<0.01 eV) with respect to the COSMO model.
Based on Kasha's rule, we limit the investigation of luminescence energies to the lowest singlet and triplet state, respectively. Also, structural relaxation in the excited state is not taken into account. This reflects the experimental situation, where the matrices provide a rigid environment for the emitter molecule. Admittedly, this rigidity does not rule out the possibility of inward relaxations that reduce the occupied volume. Given the difficulties in modelling the precise morphology of the molecular environment, we discard this possibility. In order to estimate 0–0 transitions ΔE0–0, we therefore compute vertical absorption energies ΔEvert at the optimized S0 geometry which are corrected according to:
ΔE0–0 = ΔEvert + ZPET1 − ZPES0. | (7) |
Fig. 2 Potential energy surfaces for ground and excited states indicating the key quantities calculated. Depicted is a scenario with vanishing excited state relaxation. |
Here ZPE denotes the unscaled B3LYP/6-31G* zero-point energy at the minimum of the T1 and S0 potential energy surfaces, respectively. The T1 geometry optimization was carried out at the unrestricted Kohn–Sham level with the S0 minimum as initial geometry. Singlet excited state geometry optimizations based on TDDFT are computationally still very demanding. In addition, the problem of state crossing is frequently encountered during the optimization, especially for the organo-transition metal complexes discussed here with a relatively dense set of low lying states. Due to the these difficulties in obtaining ZPES1, we applied the correction protocol of eqn (7) also for singlet excited states, coarsely assuming parallelity of the T1 and S1 potential energy surfaces. Results for the correction terms and more computational details are provided in the ESI.†
(8) |
Fig. 3 Calculated versus experimentally determined triplet emission energies () for various exchange–correlation functionals. The dashed line indicates perfect agreement. The stated numbers correspond to the complexes depicted in Fig. 1. |
Strictly speaking, TDDFT does not provide direct access to the excited state wave function. As shown by Casida, eqn (8) might still be used for diagnostic purposes as an approximation to the true excited state.42 The coefficients dia are readily available from the solution of the eigenvalue equations in TDDFT. For all complexes studied here, the HOMO–LUMO transition is the dominating excitation for PBE in the expansion of eqn (8) with participation ratios of more than 93%. This allows us to estimate the two electron integral (ia|f↑↑xc − f↑↓xc|ia) according to eqn (1) from the difference of the excitation energy and Kohn–Sham energy gap. We obtain consistent values of roughly 0.1 eV which are smaller than corresponding integrals for π → π* and n → π* transition in typical organic molecules.94 This indicates a small but non-vanishing overlap between the HOMO and LUMO orbitals and hence, a partial charge-transfer character of the transition. These findings are in line with the localization of the frontier orbitals that are depicted in the ESI (Fig. S1 to S17†). The excitations may be broadly classified as a metal–ligand to ligand charge-transfer. A more quantitative measure is obtained from a Löwdin analysis95 of the atomic population (PiA) for a given molecular orbital i:
(9) |
Eqn (9) contains the square root of the overlap matrix S between atom-centered basis functions and the molecular orbital coefficients cvi. As it is well known, the division of molecular density into atomic contributions is not unique. Löwdin populations provide a better estimate for charge distributions than the more common Mulliken scheme for extended basis sets like the ones used in this study.96 They are used here as a simple and robust measure of wave function localization providing consistent results for a given basis set and functional.
In Table 1 the populations of the central metal atom are listed for the PBE and B3LYP functional. As can be seen, the HOMO is delocalized over the central metal atom and one or more ligands, while the LUMO is a π* orbital on the same ligand(s) in the majority of cases (compounds 7 and 17 show some Pt participation in the LUMO). Unexpectedly, the computed d-orbital contribution to the HOMO is larger in PBE than in B3LYP by about 10%. This is surprising as the self interaction error present in PBE (and reduced in B3LYP) gives usually rise to an overly delocalized electron density and exaggerated covalency.97 Given this fact, the charge-transfer character of the triplet states should also be larger for PBE and lead to larger deviations from experiment.
Coming to the results for the B3LYP functional, one observes in Fig. 3 an excellent agreement with the experimental data with a RMS error of only 0.05 eV, which is even smaller than the deviation found for supposedly simpler organic systems (namely hydrocarbons, heterocycles and carbonyl compounds) with this functional.91–93 Compared to the local PBE, a higher number of singly excited determinants mixes into the excited state wave function. For complex 5, as an example, the contribution of the most important determinant drops to 56%.§ This is due to the fraction of exact exchange in B3LYP which leads to larger matrix elements KT and hence also increased coupling of individual single-particle transitions. The Kohn–Sham gap ΔEia likewise increases substantially by 1.3–1.6 eV due to the exact exchange. In the computation of the final emission energies, both effects cancel to a large degree (see eqn (3), KT < 0) such that the B3LYP energies are only modestly larger than the PBE ones.
Based on the Löwdin population analysis mentioned above, we propose a simple guideline to classify the excited states as being ligand-centered, metal-to-ligand charge-transfer like or extensive mixtures of these basics types. To this end we consider the difference density Δn(r) = nS0(r) − nT1(r), which simplifies to Δn(r) ≈ |ψi(r)|2 − |ψa(r)|2 if the triplet state is well described by a single-particle transition from orbital ψi to ψa. A Löwdin projection on the central metal atom M then leads to a measure nCT = PiM − PaM of the fractional number of electrons that take part in the metal to ligand transition (cf.eqn (9)). This quantity may be used to estimate the charge-transfer character of a given excitation according to:
(10) |
As seen in Table 1, this assignment correlates rather well with the experimentally derived classification and lends further support to the quality of the B3LYP results. With adapted limits for the different excitation types, a similar procedure might also be applied at the PBE level. The agreement with experiment, however, turns out to be inferior in this case. A more refined measure that takes not only the dominant determinant into account could be based on an analysis of the difference density102 or the transition density,103,104 both of which are readily available in DFT/TDDFT. Here we use the parameter nCT as an easily accessible quantity which allows for a quick and pragmatic classification of the excited state.
Despite of the general agreement with measurements in terms of emission energies and excitation character, it should be noted that there are still remaining qualitative discrepancies. The d-orbital participation seems to be exaggerated by DFT in general. According to eqn (10), excitations with nCT up to 0.2 are still classified as 3LC, although the d-orbital participation is experimentally found to be almost neglible.27,28,37,65,75 In line with this, zero field splittings computed by TDDFT including SOC are slightly, but systematically, overestimated with respect to experiment.34 In our study, the neglect of structural relaxation in the excited state might also contribute to the magnified metal character.
Before discussing the performance of the range-separated LC-PBE functional, a short digression on the optimal values for the parameter γ seems to be appropriate. Although γ is practically found by application of the IP criterion (eqn (6)), this parameter should reflect the electronic structure of the system and is expected to depend directly on quantities like the electron density or microscopic dielectric function.105 In fact, a clear dependence of the optimal range-separation parameter with the conjugation length of polymers was found.106 Since the conjugation length strongly influences the optical gap in polymers, it is interesting to verify whether there is a direct link also in the present organo-transition metal compounds. Inspection of Table 1 reveals that there is actually no correlation between the range-separation parameter and emission energies. The values of γ vary between 0.16a0−1 and 0.41a0−1 in a rather arbitrary fashion. The LC-PBE results for the emission energies are on average slightly less accurate than the B3LYP results with a RMS deviation of 0.14 eV (cf., Table 2). Interestingly, the LC-PBE functional with a fixed value of γ = 0.3a0−1 performs only slightly worse than the tuned variant. Only for compound 2 a clear benefit of the tuning procedure is discernible.
# | Compound | PBE | B3LYP | LC-PBE (tuned) | Exp. | |||
---|---|---|---|---|---|---|---|---|
τ r | τ r | τ r | ||||||
1 | Pt(qol)2 | 1.74 | 2.6 × 10−8 | 2.39 | 8.4 × 10−9 | 2.66 | 5.5 × 10−9 | 2.33 (ref. 65) |
2 | Pt(ppy2-tBu2a) | 1.67 | 6.9 × 10−8 | 2.20 | 3.8 × 10−8 | 2.22 | 7.8 × 10−8 | 2.16 (ref. 98) |
4 | Pt(2-thpy)2 | 2.08 | 8.0 × 10−7 | 2.64 | 5.0 × 10−8 | 3.12 | 1.2 × 10−8 | 2.54 (ref. 27) |
5 | Pt(thpy)(acac) | 2.60 | 4.5 × 10−8 | 3.00 | 1.3 × 10−8 | 3.53 | 3.9 × 10−9 | 2.82 (ref. 99) |
7 | Pt(dphpy)(CO) | 2.22 | 2.7 × 10−6 | 2.63 | 8.8 × 10−7 | 2.82 | 8.9 × 10−7 | 2.58 (ref. 107) |
14 | Pt(4,6-dFppy)2 | 2.34 | 1.0 × 10−5 | 2.97 | 1.9 × 10−7 | 3.36 | 4.9 × 10−8 | 2.86 (ref. 67) |
For the theoretical analysis it should also be kept in mind that the values were approximated using the zero-point energy values of the T1 potential energy surface. Although a statistical analysis should be performed with caution for such a small number of test cases, we also report the average errors for and the singlet–triplet gap in Table 2. Note that the first excited singlet and triplet state are not necessarily strongly coupled by the spin–orbit interaction.37,72,110 Strictly speaking, the parameter ΔEST is in these cases not crucial for radiative lifetimes and OLED device performance. We include it here to obtain a broader picture of the TDDFT accuracy for organo-transition metal complexes.
Turning to the results, the gradient-corrected PBE functional strongly underestimates transition energies, but it does so consistently for both triplets and singlets. As a result, the computed singlet–triplet gap ΔEST is in surprisingly good agreement with the experiment. Considering eqn (1)–(3), this can be traced back to the cancellation of the Kohn–Sham gap, which is too small to provide accurate absolute singlet and triplet energies. The exchange integral (ia|ia) that remains after subtraction is qualitatively correct also at the PBE level. In contrast, the hybrid B3LYP yields reasonable singlet energies with an RMS deviation of 0.10 eV in good agreement with earlier more extensive benchmarks on vertical singlet excitations.111 The singlet–triplet gap is systematically overestimated by only 0.13 eV, which allows for a reliable determination of this important parameter in large scale computational screenings.
Coming to the tuned LC-PBE functional, we find a strong overestimation of singlet energies and a pronounced error for the singlet–triplet gap. These trends are likewise already documented in the literature for vertical excitations of organic compounds.92,93,111 For singlet excitations, the tuning of the LC-PBE functional turns out to be beneficial and reduces the error compared to a fixed range-separation parameter significantly. Since ΔEST is strongly dependent on the amount of HF exchange in the functional, it is not surprising that LC-PBE with 100% long-range unscreened exchange considerably overestimates this quantity with respect to the experiment. Comparison of the radiative lifetimes between the different methods reveals that PBE exhibits shorter values compared to B3LYP and LC-PBE, for 14 the difference even amounts to 2–3 orders of magnitude. In their study of TDDFT oscillator strengths for small organic molecules, Caricato et al.112 find similar trends and show that range-separated functionals (like LC-PBE) provide results close to higher level theory. In summary it can be stated that B3LYP outperforms the other studied functionals also in the description of singlet excited states.
With respect to the computational protocol, we find that vibrational contributions in the form of zero-point energies are necessary to match the experimental ΔE0–0 energies, while excited state relaxation is not crucial for emitters embedded in a rigid matrix. The B3LYP average error of only 0.05 eV for triplet emission energies is highly encouraging. This paves the way for further in silico pre-screening of possible OLED materials. We hope that the extended experimental and theoretical data set provided in this article might also be useful also in the development of novel exchange–correlation functionals which aim at a coherent description of valence, Rydberg and charge-transfer excited states in real world materials.
Our results show that charge transfer excitations in the investigated compounds do not pose a significant challenge for TDDFT. This is mainly because transitions that are often characterized as MLCT states actually involve transitions out of a strongly hybridized metal d-orbital. As mentioned in Section 4.1, there are even indications that the calculated d-orbital contribution is overestimated. This calls for further studies which connect calculated metal participations, SOC matrix elements and singlet–triplet gaps with measurements of zero-field splittings and radiative lifetimes. Such a combination of theory and experiment should be extremely helpful in the detailed understanding and further optimization of metal-organic emitters.
Footnotes |
† Electronic supplementary information (ESI) available: Listings of vertical excitation energies at different levels of theory, zero point energies and plots of relevant molecular orbitals. See DOI: 10.1039/c5ra12962a |
‡ It should be noted that even though the Kohn–Sham gap will differ from the quasiparticle gap also for the (unknown) exact Kohn–Sham exchange–correlation functional, charge-transfer excited states can be correctly described also in the framework of local functionals. To this end one has to go beyond a static xc kernel fxcστ which restores both the difference between Kohn–Sham and quasiparticle gap as well as the 1/R stabilization.47 |
§ In such a situation the analysis is better based on the concept of natural transition orbitals introduced by Martin,101 which is not yet implemented in NWChem. |
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