Photophysics of OLED Materials with Emitters Exhibiting Thermally

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Photophysics of OLED Materials with Emitters Exhibiting Thermally Activated Delayed Fluorescence and Used in Hole/Electron Transporting Layer from Optimally Tuned Range-Separated Density Functional Theory Mojtaba Alipour* and Zahra Safari Department of Chemistry, College of Sciences, Shiraz University, Shiraz 71946-84795, Iran

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S Supporting Information *

ABSTRACT: Owing to their potential to be exploited as emitters in heavy metal-free organic light emitting diodes (OLEDs), the interest toward the molecules exhibiting thermally activated delayed fluorescence (TADF) and used in hole/electron transporting layer has been reinvigorated in recent years. In the present contribution, we take a different approach to deal with the problem where the novel optimally tuned range-separated hybrid density functionals (OT-RSHs) are developed for reliable description of the photophysical properties like absorption, emission, and singlet−triplet energies for the emitters covering a wide range of singlet−triplet energy gaps. Considering BLYP, PBE, and TPSS density functional approximations as the underlying exchange and correlation terms with different options for the short- and long-range exact-like exchange contributions as well as the range-separation parameter, we propose several OT-RSHs for accurate predictions of the photophysical properties in both gas and solution phases for a set of compounds prone to be employed in OLED materials. It is shown that newly developed OT-RSHs with correct asymptotic exchangecorrelation potential behavior provide reliable theoretical estimates of photophysics for an experimental benchmark set of compounds. We find that the proposed models not only have superior performance with respect to the standard counterparts but also outperform the earlier developed hybrids with both fixed and interelectronic distancedependent exact-like exchange. Promisingly, such good performances are also indeed the case for other emitters not included in optimally tuning processes of the range-separation parameter. As showcased by our theoretical explorations on the photophysics of OLEDs and other earlier efforts in this arena of research, it seems that a bright future lies ahead.

1. INTRODUCTION The first rationalization of the thermally activated delayed fluorescence (TADF), known also as E-type delayed fluorescence, can be traced back to the pioneering works by Perrin1 and later by Lewis et al. .2 After many years, the presence of this type of fluorescence in eosin and benzyl was reported by Parker and Hatchard3,4 and its importance to identify the delayed luminescence was also pointed out by Wilkinson and Horrocks. 5 In 2012, Uoyama et al. 6 reinvigorated the TADF mechanism by proposing the vital role of this type of delayed fluorescence as a promising mechanism for harvesting triplet excited states in organic light emitting diodes (OLEDs). In an OLED device, discovered by Tang and VanSlyke,7 upon recombination of charges injected from the electrodes the light emitters are electronically excited. Since the spins of the injected charges are uncorrelated, exciton formation gives 25% singlet excitons (MS = 0) and 75% triplet excitons (MS = −1, 0, +1). The spin statistics, however, affects such exciton formation upon the electron−hole combination. As a matter of fact, based on the molecular spectroscopy selection rules the radiative decay from triplet excitons is spin forbidden and only singlet excitons can emit light. Accordingly, the fluorescence © XXXX American Chemical Society

emission quantum yield has an upper statistical limit of 25%. Thus, although the first generation of OLEDs based on organic fluorescent emitters are able to provide high reliability, they also suffer from low electroluminescence efficiency.8 To prevail this problem, the second generation phosphorescence-based OLEDs through doping the host layer with metal−organic emitters was proposed.9 With the help of large spin−orbit coupling of heavy metal atoms such as iridium and platinum the triplet excitons can be harvested, leading to a quantum efficiency of unity. Nonetheless, not only the use of these rare elements leads to high device fabrication costs inevitably but also it is irrational to base large-scale high-volume industries on these resources. It seems that the efficient alternatives in terms of light emission, chemical stability, and cost are desirable. In this context, the molecules exhibiting the TADF and without requiring expensive and possibly toxic heavy metals have been emerged as the third generation emitters for effective harvesting of both the singlet and triplet excitons.8,10 Received: December 4, 2018 Revised: December 11, 2018

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DOI: 10.1021/acs.jpcc.8b11681 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Pragmatically, during the TADF mechanism the dark first triplet excited state (T1) indirectly fluoresces back into the singlet ground state (S0) via reverse intersystem crossing to the bright first singlet excited state (S1). Thus, the major advantage of TADF materials is a recycling of the excitons between the singlet and triplet manifolds via reverse intersystem crossing, which in turn is possible at appreciable rates when the energy gap between the lowest S1 and T1 states, ΔEST, is smaller than or comparable to kBT (kB and T stand respectively for Boltzmann constant and temperature). Moreover, the TADF provides an opportunity to reach the theoretical quantum efficiency from 25% in the first generation of OLEDs to 100%.11,12 Putting these arguments together, it seems that to having an efficient OLED in which the TADF mechanism can ascertainably be observed, the nonradiative decay should be minimized, ensuring that the triplet state lives long enough to maximize the triplet harvesting chance via reverse intersystem crossing. More importantly, and of particular relevance to our study, the condition of a small ΔEST has also to be satisfied. Inspired by the proposal of the effectiveness of the TADF mechanism for third-generation heavy metal-free OLEDs,6 immense efforts have been made for developing highly efficient TADF emitters and understanding their photophysics as a hotspot in the context of OLED technology, see some recent related reviews.13−15 In this context, the past decade has witnessed the applicability and affordability of computational tools to predict the photophysical properties of TADF-based OLEDs.16−27 Despite the usefulness and ubiquity of theoretical chemistry and its computational approaches, how to accurately predict photophysical properties of TADF emitters is still an unresolved problem. Indeed, notwithstanding the rich collection of studies in this emerging field, detailed theoretical explorations regarding the excited states of TADF emitters, which in turn can undoubtedly be helpful for a screen of potential compounds for use in OLEDs, are relatively limited. In particular, the issue of proposing the proper levels of theory which are able to present efficient and accurate photophysics of TADF-based OLEDs can also be noted as an unresolved problem in this respect. The TADF emitters as donor−acceptor molecules with charge-transfer character often consist of many atoms, limiting their investigations with high-level electron correlated multireference techniques. As an alternative approach with an affordable ratio of accuracy to computational cost, density functional theory (DFT)28−31 in time-dependent (TD) domain (TD-DFT)32−34 is of concern. Nonetheless, the presence of this kind of compounds is at odds with the limitations of TD-DFT within standard approximations for examination of such excited states, where their underestimations for the corresponding excitation energies are well-known.35−37 Besides the approximate nature of the used exchange-correlation functionals and inappropriate selections in some cases, as some responsible factors for such systematic errors the spurious interaction of an electron with itself (selfinteraction error) and the discontinuous dependence of the exchange-correlation potential on the electrons number at integer numbers of electrons (derivative discontinuity) should be mentioned.38−42 Working on these issues, it has been found that the hybrid functionals incorporation a fixed (electroncoordinate independent) amount of Hartree−Fock (HF) exchange43−45 as well as an explicit Coulomb interaction for the more strongly localized electrons46,47 can mitigate the selfinteraction error and derivative discontinuity. Plus, the

necessity of a balanced compromise between HF exchange and DFT correlation to obtain reliable results using hybrid functionals has also been pointed out. However, the hybrid density functionals cannot fully obviate these problems. As a matter of fact, the results of DFT hybrid functionals do not obey the ionization energy theorem which states that the highest occupied molecular orbital (HOMO) eigenvalues computed by DFT should be equal to the ionization energies obtained from the total energy differences.48−51 As a reason for this failure is the incorrect behavior of the electron−electron potential at asymptotically distances, where the exchange potential of a hybrid functional does not exactly recover the correct −r12−1 behavior (r12 is the interelectronic distance).52 As a physically sound solution toward overcoming the noted shortcomings, which is also of our main focus in this work, the concept of range-separated hybrid functionals has been proposed.53,54 In this context, the scheme of optimally tuned range-separated hybrid density functionals (OT-RSHs)55−58 is of concern herein. Notwithstanding earlier related efforts in the field to employ the optimally tuning approach for photophysical properties of TADF emitters,59,60 there is still room in this context. Despite these latter studies where only the effect of range-separation parameter tuning with one type of DFT approximations was taken into account, to the best knowledge of the present authors no systematic report including different views of these models in studying on TADF-based OLEDs has been appeared as of yet. In light of this scenario, in this work we present a detailed TD-DFT study based on OT-RSHs for reliable prediction of photophysical properties like absorption energies, emission energies, and singlet−triplet energy gaps for TADF-based OLEDs. More specifically, our study is not limited to assessing the density functionals through a benchmarking fashion but we propose novel OT-RSHs without any empirical parameter for the purpose. The results of different OT-RSHs in our calculations allow us to dissect not only the importance of nonempirically tuning of the range-separation parameter but also the roles played by the both short- and long-range exchange contributions and the underlying density functional approximations in describing the photophysics of OLED materials. Within such analyses, we would like to identify whether there exist the OT-RSHs having superior performance with respect to other DFT approximations for OLEDs applications.

2. THEORETICAL FOUNDATIONS As mentioned before, the framework of DFT under study herein is the RSHs and their OT scheme. In the following subsections, we provide the corresponding theoretical basics. For further details, the interested reader is directed to the original references cited in what follows. 2.A. Range-Separated Hybrid Functionals. The RSHs (long-range corrected (LC) hybrids) represent an approach within DFT which can be used for a balanced treatment of HF exchange and generalized gradient approximations (GGAs) for the exchange and correlation terms.53,54 Pragmatically, shortrange (SR) density functional exchange is mixed with longrange (LR) HF exchange through separating the electron repulsion operator into SR and LR parts using the following identity,61−63 r12−1 = r12−1{1 − [Θ(μr12)]} + r12−1[Θ(μr12)] B

(1)


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r → 0 and to (α + β)/r for r → ∞, and consequently, the right asymptotic behavior for the exchange−correlation potential is achieved upon the α + β = 1 condition. It has earlier been shown that this latter condition is necessary when studying many properties.68−74 However, it should be noted here that all the RSH functionals do not satisfy this condition, where there are also functionals not incorporating a full 100% contribution of asymptotic HF exchange, such as CAM-B3LYP in which α + β = 0.65.67 There are many interests to propose RSHs for different properties, with the differences arising mainly from the underlying density functional approximations in the energy expressions and the options for the (α, β, μ) set. Among others and just to mention some functionals, the functionals CAM-B3LYP (0.19, 0.46, 0.33),67 LC-ωPBE (0.0, 1.0, 0.47),75 ωB97XD (0.222036, 0.777964, 0.20),76 and M11 (0.428, 0.572, 0.25)77 can be named. We note here that in the RSHs with LR correction (known also as LC functionals) which are of concern in our study the HF exchange increases from SR to LR interelectronic separation in such a way that it reaches to one or a finite value. However, there also alternative types of range separation schemes, where either the HF exchange decreases from SR to LR interelectronic separation so that it vanishes at the LR part, known as screened-exchange, such as HSE0678 and MN12SX79 or HF exchange increases from SR to a maximum value at medium-range and then decreases to zero at LR, as in the HISS model.80,81 In this work, we have considered the RSHs with LR correction based on various density functional approximations with the full exchange at asymptotic distance (α + β = 1) and several choices for the (α, β, μ) set to propose new models for photophysical properties of TADF-based OLEDs. To this end, instead of using empirical parametrization against standard benchmark databases, we have considered the optimally tuning scheme. 2.B. Optimally Tuning Scheme. Optimally tuning is an approach that allows the adjustment of the two parameters from the (α, β, μ) set in RSHs, α or β and μ, to fulfill the fundamental properties that the exact functional must obey. More specifically, these parameters can be determined entirely from first-principles based on the two exact constraints: (i) enforcing the condition α + β = 1 to achieve the right asymptotic exchange-correlation potential behavior and (ii) tuning the range-separation parameter without invoking any empirical fitting so as to satisfy the ionization energy theorem.55−58 The first condition was earlier discussed and in this subsection the latter constraint of the optimally tuning scheme used in our calculations is outlined. The ionization potential (IP) and electron affinity (EA) are calculated for each value of the range-separation parameter as the differences between energies of ground state for systems with N and N ± 1 electrons,

where r12 is the interelectronic distance, μ is the rangeseparation parameter (with inverse length units) controlling how HF and DFT exchanges are mixed as a function of r12, and Θ(μr12) is a smooth range-separation function which damps the DFT exchange and complements it with HF exchange. As some representative functions for Θ, standard error function (erf)53,61 and Yukawa kernel64,65 can be mentioned. However, although the options for the splitting function are not unique, the widely used option that distinguishes the SR and LR regimes is the erf, i.e., Θ(μr12) = erf(μr12), where erf(x) = 2 2x−1/2∫ x0e−t dt.66 If the SR and LR parts are treated using DFT exchange and HF exchange terms, respectively, the expression for the exchange (x)−correlation (c) energy takes the below form LR Exc = E HF + ExSR,DFT + Ec ,DFT

(2)

This form of range-separation balances exchange and correlation contributions based on their relative importance at different interaction regimes. The DFT correlation, balanced by the DFT exchange, is primarily a SR interaction whereas the HF exchange is more pronounced at LR, enforcing the correct asymptotic potential. An extended version of splitting the electron repulsion operator, which in turn allows for different contributions of HF exchange in the SR and LR parts, has been proposed by Yanai et al.67 and denominated as the Coulomb attenuating method (CAM), r12−1 = r12−1{1 − [α + β erf(μr12)]} + r12−1[α + β erf(μr12)]

(3)

As compared to eq 1, eq 3 includes the two extra adjustable parameters, α and β, satisfying the inequalities 0 ≤ α ≤ 1, 0 ≤ β ≤ 1, and 0 ≤ α + β ≤ 1. In eqs 1 and 3, the SR effects from the first term is a SR Coulomb operator, decaying to zero on a length scale of ∼μ−1, and the LR effects from the second term dominates the first at large r. The exchange−correlation functional based on eq 3 can accordingly be expressed as follows SR LR Exc = αE HF + (1 − α)ExSR,DFT + (α + β)E HF

+ (1 − α − β)ExLR ,DFT + Ec ,DFT

(4)

Concerning eq 4 and above explanations we find a flexibility regarding the mixing of different fractions of HF exchanges at SR and LR parts, rather than the strict contributions 0 and 1, respectively, in eq 2. Moreover, the parameter α quantifies the HF exchange contribution at SR limit by a factor of α and the parameter β, which in turn scales the DFT exchange by 1 − (α + β) contribution, incorporates LR asymptotic HF exchange by α + β factor. Although setting α = 0 and β = 1 in eqs 3 and 4, respectively, eqs 1 and 2 is reproduced, respectively, the presence of the parameters α and β gives an opportunity for checking the effects of SR and LR exchanges on different properties. Moreover, in earlier works it has also been found that the SR HF exchange can lead to the improved results for different properties.68−74 On the other hand, imposing β = 1 − α in eq 3 assures the recovery of 100% of HF exchange at long interlectronic distance, constraining the correct −r12−1 asymptote for the exchange potential. In other words, α and β parameters decide the limiting behavior of the HF exchange which tend to α/r for

IP μ(N ) = Eμ(N − 1) − Eμ(N )

(5)

EAμ(N ) = Eμ(N ) − Eμ(N + 1)

(6)

Accordingly, the EA of an N-electron system is equal to the IP of the (N + 1)-electron system, EA(N) = IP (N + 1). For the exact functional, the IP of a neutral molecule is equal to the negative HOMO energy εHOMO(N), IP(N) = −εHOMO(N). However, there is no equivalent of the IP theorem that equates the EA with the negative of the lowest unoccupied molecular orbital (LUMO) energy. This is because even with the exact exchange-correlation functional, the two quantity differ by the C


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Figure 1. continued

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Figure 1. Geometrical structures of the molecules with the small (blue), medium (red), and large (green) singlet−triplet energy gaps investigated in this work. The corresponding abbreviated names are given below the structures.

conditions with the functions j(N; μ) and j(N + 1; μ),

derivative discontinuity. Nonetheless, this problem can be circumvented by considering the IP of the anion instead of the EA of the neutral molecule, leading in turn to the relation IP(N + 1) = −εHOMO(N + 1). Earlier related analyses have also been unveiled that enforcing these conditions for both neutral and anion species is equivalent to eliminating the derivative discontinuity.82 However, since density functionals have approximate nature there are differences between the two quantities. For each case, the remaining differences with respect to the exact state can be given as the separate tuning

respectively, as follows μ j(N ; μ) = εHOMO (N ) + [E(N − 1; μ) − E(N ; μ)] μ =εHOMO (N ) + IP μ(N )

(7)

μ j(N + 1; μ) = εHOMO (N + 1) + [E(N ; μ) − E(N + 1; μ)] μ =εHOMO (N + 1) + IP μ(N + 1)

(8) E


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choice of parameter α different values for the range-separation parameter μ, varying from 0.1 to 0.3 Bohr−1 in increments of 0.05, have been employed. We notice that although the performed tuning processes herein include different contributions of the HF exchange at SR, all of them recover the 100% exchange at asymptotic distance, fulfilling the condition α + β = 1. As the exchange and correlation terms during the optimally tuning processes and subsequent calculations, we utilized the widely used density functional approximations, namely the Becke−Lee−Yang−Parr (BLYP)88,89 and Perdew−Burke− Ernzerhof (PBE)90 GGAs and Tao−Perdew−Staroverov− Scuseria (TPSS)91 meta-GGA, whose reliability has extensively been confirmed before. This latter choice not only includes a correlation component free of self-interaction but also comparing its results with respect to those provided from the two formers provides a route to explore the role of the kinetic energy density for the properties under study. It has earlier been found that the computed photophysical properties, their associated errors and trend as well as the optimally tuned values of the rang-separation parameter are qualitatively consistent, where no substantial differences and visible dependence on the choice of basis sets and their extension have been observed.20,60 Plus, it has also been shown that the computed excitation energies remain somewhat close regardless of the used density functional approximations for the geometry optimizations of the ground state structures.60 Therefore, for the considered compounds collected from the recent references in the field with the recommended geometries we utilized the 6-31G(d) basis set throughout.20,60 This choice in turn provides an equal footing to straightforward comparisons between our data and previously published results in this context. The effects of the surroundings on the photophysics are difficult to account for. However, since most of the measured photophysical properties are from delayed fluorescence/ phosphorescence spectra in solution phase, besides gas phase computations the solution phase has also been included to account for the dielectric medium effects. To this end, the solvent effects with the solvent used in experiments (toluene) have been considered through the integral equation formalism variant92−94 of polarizable continuum model (IEFPCM).95,96 Pragmatically, it has repeatedly been pointed out that it is a better practice to optimally tune in vacuum and then add the solvation model afterward than to tuning in the presence of the solvent.82 Accordingly, the effect of solvation has only been included during the calculation of the photophysical properties. This is also consistent with earlier analyses where it has been found that the optimally tuning of the rang-separation parameter within a continuum solvent model may lead to the unrealistic values.69 Given some references on the using optimally tuning scheme alongside a solvent, the issue of combining the OT-RSHs with a solvation model is not a straightforward topic and deserves to be more analyzed in the future. For further details regarding the optimally tuning of the range-separation for molecules in gas, solvent, and solid phases and related issues see a most recently published review.82 From the methodological viewpoint, the Tamm−Dancoff approximation (TDA) scheme97−99 of TD-DFT has recently been shown to be reliable for absorption and emission spectra and vibrational band shapes, where especially better estimates for the singlet−triplet energy gaps have been observed.17,59,100 Accordingly, the TDA formalism has been employed in our

However, as has thoroughly been discussed in earlier works, a reliable minimization should include the computations on both neutral systems and anions.69−74 Thus, eqs 7 and 8 should be minimized simultaneously, leading to a minimization procedure using a general function J as J = j(N ; μ) + j(N + 1; μ)

(9)

Finally, although the function J can be used for the rangeseparation parameter minimization, it has been pointed out that depending on the dependencies of εHOMO and the total energy on range-separation parameter, the values of μ are very close to minima of the functions defined in eqs 7 and 8.58 Therefore, minimization of a target function J2 with the following form has been proposed69−74 J 2 = [j(N ; μ)]2 + [j(N + 1; μ)]2 1 μ =∑ [εHOMO (N + i) + IP μ(N + i)]2 i=0

(10)

We note in passing that, for those cases in which we have a negative electron affinity one should be cautious where the OT scheme should not be used; see for instance ref 83. In addition, when the words such as optimally or nonempirically are used we mean that no reference values from experiment or high level wave function based methods are employed within above equations, but there is an internal self-consistency condition between the calculated IPs and HOMO energies.

3. COMPUTATIONAL DETAILS Our calculations have been carried out on a series of donor− acceptor charge-transfer emitters with the structures in which the adjacent donor and acceptor units can be out of plane due to the steric hindrance. Thus, as expected, such structures are prone to lead to spatial separations of HOMO and LUMO, crucial for having a small value for ΔEST. More specifically, the benchmarked compounds are as follows: one class of the molecules with small ΔEST values (smaller than 0.1 eV) which can typically be employed as the TADF emitters in real materials, another category including the molecules with medium singlet−triplet energy gaps (0.1−0.5 eV), and a set of molecules with large singlet−triplet splittings (0.5−0.8 eV) which can be considered as representatives of hole/electron transporting layer.20,60 Shown in Figure 1 are the geometrical structures of the investigated molecules in this work. The corresponding experimental data for all the photophysical properties under study have been provided in Table S1 in Supporting Information. In order to compute the photophysical properties for these molecules using the developed OT-RSHs, we first carried out the range-separation parameter tuning based on minimization of eq 10. We have included different admixtures of HF exchange at SR, ranging from 0% to 25% with the intervals of 5%. Regarding the selection of the HF exchange contribution at SR, α ≠ 0, we note that the values of α can be determined from the first principles.84,85 However, this parameter can also be found by fitting processes against some benchmark databases, as has previously been shown for instance that α ≈ 0.2 is satisfactory.86,87 However, this is not indeed the case for all the purposes and since the presence of the HF exchange at SR has dominant effects on various properties, in addition to the 20% contribution for the SR HF exchange other values have also been included in the present calculations. For each F


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Figure 2. continued

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Figure 2. Plots for the rang-separation parameter tuning (the target function J2 versus the rang-separation parameter μ) for the benchmarked set of molecules in Figure 1, panel A. First row: α = 0.0, β = 1.0. Second row: α = 0.05, β = 0.95. Third row: α = 0.10, β = 0.90. Fourth row: α = 0.15, β = 0.85. Fifth row: α = 0.20, β = 0.80. Sixth row: α = 0.25, β = 0.75. In each row, the plots are respectively for the density functional approximations of BLYP (left), PBE (middle), and TPSS (right).

calculations. Note however that for the cases where the substantial charge-transfer characters come into play, negligible differences between TDA approach and full TD-DFT are expected.17,24 The TDA calculations using the OT-RSHs have been used to obtain the values of absorption and emission energies as the vertical transition energy between the S0 and S1 states, starting from the S0 and S1 equilibrium geometries, respectively. Vertical excitation energies from the ground state S0 to the first singlet excited state S1, EV(S1), and to the first triplet excited state T1, EV(T1), have been calculated as EV(S1) = [E(S1)// E(S0)] − E(S0) and EV(T1) = [E(T1)//E(S0)] − E(S0) where E(S1)//E(S0) and E(T1)//E(S0) denote the energies of the excited states at the ground state optimized geometries. The vertical singlet−triplet energy gaps are accordingly obtained through the relation ΔEST = EV(S1) − EV(T1). All the runs of our computations have been implemented in the GAUSSIAN09 suite of codes.101 As descriptors for gauging the functionals performance, we have employed five criteria as follows: mean signed deviation (MSD), mean absolute deviation (MAD), maximum absolute deviation (MaxAD), minimum absolute deviation (MinAD), and root-mean-square − deviation (RMSD). With the definition of Δi as PComputed i PReference , where PComputed and PReference refer respectively to the i i i computed and reference properties, these descriptors are defined as MSD =

1 n

n

∑i Δi , MAD =

1 n

max|Δi|, MinAD = min|Δi|, and RMSD =

1/2

( 1n ∑in Δi2)

,

where n is run over the set of studied molecules.

4. RESULTS AND DISCUSSION At first, we carried out the optimally tuning of the rangeseparation parameter of the RSH density functionals under study through the minimization of eq 10 for all the molecules displayed in Figure 1, panel A. Shown in Figure 2 are the corresponding curves for the target function J2 as a function of the rang-separation parameter μ obtained from BLYP, PBE, and TPSS density functional approximations for the six combinations of the parameters α and β as follows: α = 0.0, β = 1.0; α = 0.05, β = 0.95; α = 0.10, β = 0.90; α = 0.15, β = 0.85; α = 0.20, β = 0.80; and α = 0.25, β = 0.75. We can see from the figures that the minimization curves behave almost similar, where they have nearly flat region around minima for some molecules while there are also some other cases for which this changing pattern is less pronounced. However, in most cases a distinct minimum can be identified based on the tuning procedure through the minimization of eq 10. We also find that albeit the minima of curves do not exactly coincide, similar optimally tuned values can be derived for μ by using the three density functional approximations BLYP, PBE, and TPSS. This in turn manifests the more dependency of μ to the nature of the considered emitters. The quadratic fittings were applied for all the resulting curves in Figure 2 and the optimally tuned values of μ were subsequently determined by minimizing the derived second order polynomials. The

n

∑i |Δi |, MaxAD = H


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Table 1. Optimally Tuned Values of the Range-Separation Parameter (Bohr−1) with Various Combinations of α and β Parameters for the Benchmarked Set of Molecules in Figure 1, Panel A molecule

α = 0.0, β = 1.0

α = 0.05, β = 0.95

α = 0.10, β = 0.90

α = 0.15, β = 0.85

α = 0.20, β = 0.80

α = 0.25, β = 0.75

2CzPN ACRFLCN α-NPD NPh3 PhCz PPZ-3TPT PPZ-4TPT PPZ-DPO PXZ-DPS PXZ-TRZ

0.21 0.21 0.21 0.20 0.23 0.20 0.20 0.21 0.22 0.22

0.20 0.20 0.20 0.23 0.23 0.19 0.19 0.20 0.21 0.21

0.19 0.19 0.19 0.22 0.22 0.18 0.18 0.19 0.20 0.20

0.18 0.17 0.18 0.21 0.21 0.17 0.16 0.18 0.19 0.18

0.16 0.15 0.16 0.20 0.20 0.14 0.14 0.16 0.17 0.17

0.14 0.12 0.13 0.18 0.19 0.12 0.11 0.13 0.15 0.14

electronic spatial extent ⟨R2⟩ as an estimate of the molecular size and correlated its values with the inverse of rangeseparation values. As illustrative examples, the values of ⟨R2⟩ for the PhCz, PXZ-TRZ, and α-NPD molecules were found to be 4808, 27794, and 42627 au, respectively, using the PBE as the density functional approximation and α = 0.0, β = 1.0 combination. Moreover, we obtain the values of μ−1 as 4.35, 4.55, and 4.76 Bohr, respectively, for the same molecules. These numerical tests show that the optimally tuned values of μ can reflect the expected trend where a linear behavior with positive slope can be observed for the plot of ⟨R2⟩ against μ−1, the larger an emitter the smaller the optimally tuned value of μ. Having optimally tuned the values of the range-separation parameter they were employed for all the subsequent calculations of the photophysical properties. We have gathered in Tables 2−4 the statistical descriptors of MSD, MAD, MaxAD, MinAD, and RMSD on the performances of the proposed OT-RSHs in the calculations of absorption energies, emission energies, and singlet−triplet energy gaps of the emitters under study, respectively. All the corresponding raw data have also been collected in Tables S2−S4 in Supporting Information. At first glance, considering the values of MSDs we can see a tendency of overestimating all the photophysical properties using the OT-RSHs, as implied by the reinforced positive MSDs in the tables. On the other hand, looking at the MADs and other statistical measures for all the combinations of α and β parameters the increased accuracy of the OT-RSHs based on BLYP and PBE density functional approximations with nearly equivalent performances seems to be evident. Furthermore, scrutinizing the results it can be deduced that besides the good performances of other combinations of α and β, the proposed OT-RSHs with the first combination of the two parameter (α = 0.0, β = 1.0) yield more accurate results for the prediction of photophysical properties of the emitters against experimental reference data. To have a more general comparisons among the proposed OT-RSHs over all the computed photophysical properties and to make the models with superior performances more visible, we have also analyzed the scaled MADs (SMADs) and total SMADs (TSMADs) for the developed OT-RSHs. Considering one of the best performers as a reference, i.e., the PBE-based OT-RSH with α = 0.0, β = 1.0 combination, for each OT-RSH the values of SMAD and TSMAD have been calculated as SMADOT‑RSH = MADOT‑RSH/MADReference and the average of the SMADs, respectively. Because other OT-RSHs are compared with the reference, its SMAD is set to be 1. For a better OT-RSH, the associated SMAD is smaller than one and otherwise it is larger than one. Exhibited in Figure 3 is the corresponding graphical

corresponding optimally tuned values of the range-separation parameter which will be used for the next calculations of photophysical properties have been collected in Table 1. Overall, the optimally tuned values of μ are in the range of 0.11 to 0.23 Bohr−1, in close agreement with those determined for emitters and organic dyes in previous studies.59,60 Zooming in the numerical data we observe that for each compound the optimally tuned values of the range-separation parameter reveal a decreasing trend when going from the case of α = 0.0, β = 1.0 to α = 0.25, β = 0.75, where with increasing the HF exchange contribution at SR the values of μ decrease. These changing patterns indicate a meaningful correlation between the HF exchange contribution at SR and the values of rangeseparation parameter in OT-RSHs.69−74 As an another important point in this regard, it can clearly be seen that the optimally tuned values of μ for the considered molecules herein are smaller than, by a factor of about 2, those determined in standard LC functionals, 0.33 and 0.47 Bohr−1.62,102 In turn, these results unveil that the newly developed OT-RSHs switch from DFT exchange to HF exchange at larger r12. However, our optimally tuned values for μ are in reasonable agreement with that of the original LCωPBE0 functional, 0.2 Bohr−1.103 These findings unambiguously demonstrate that not only the optimally tuning of μ is necessary for reliable description of the considered emitters but also it can be possible to propose a more general applicable range-separation parameter for such systems, as will be shown in the following. From a different perspective, as noted before the SR term in eqs 1 and 3 decays exponentially on a length scale of ∼μ−1. On the other hand, μ−1 reflects a characteristic distance for switching between SR and LR regimes. Thus, smaller optimally tuned values of μ, which can be considered as a functional of the density of molecule, can be associated with larger systems.57,104 For instance, for π-conjugated systems there is a relationship between the optimally tuned range-separation parameter and the degree of conjugation of the system, where the μ value decreases with the increasing of chain lengths, indicating that in systems with more extended conjugation the switching to the exact-like exchange occurs at larger interelectronic distances (note that the range-separation parameter is expressed in inverse length units). From another side, such trends can also be attributed to the delocalization issue. As a matter of fact, as the orbitals delocalize with the system size increasing, the spatial ranges at which exactexchange corrections are needed become larger. As a piece of supporting evidence to this argument regarding the studied molecules herein, we have considered the I


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The Journal of Physical Chemistry C Table 2. Statistical Descriptors on the Performance of the Proposed OT-RSHs in the Calculations of Absorption Energies for the Benchmarked Set of Molecules in Figure 1, Panel Ae functionals

BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS LC-BLYPa LC-PBEa LC-TPSSa CAM-B3LYPb M11c MN12-SXd

MSD

MAD

MaxAD

OT-RSHs α = 0.0, β = 1.0 0.23 0.33 0.70 0.22 0.33 0.71 0.29 0.36 0.75 α = 0.05, β = 0.95 0.28 0.37 0.75 0.27 0.37 0.77 0.34 0.40 0.81 α = 0.10, β = 0.90 0.31 0.39 0.78 0.30 0.38 0.79 0.36 0.41 0.83 α = 0.15, β = 0.85 0.33 0.40 0.81 0.31 0.39 0.82 0.37 0.42 0.86 α = 0.20, β = 0.80 0.31 0.41 0.84 0.30 0.40 0.86 0.35 0.43 0.89 α = 0.25, β = 0.75 0.28 0.38 0.88 0.25 0.37 0.89 0.31 0.40 0.93 standard RSHs 1.20 1.20 1.71 1.20 1.20 1.75 1.23 1.23 1.75 0.49 0.49 0.86 0.84 0.84 1.23 screened-exchange functional −0.30 0.50 0.88

MinAD

Table 3. Statistical Descriptors on the Performance of the Proposed OT-RSHs in the Calculations of Emission Energies for the Benchmarked Set of Molecules in Figure 1, Panel Ae

RMSD

functionals

0.07 0.05 0.12

0.37 0.37 0.41

BLYP PBE TPSS

0.10 0.07 0.14

0.42 0.42 0.46

BLYP PBE TPSS

0.12 0.10 0.11

0.44 0.44 0.48

BLYP PBE TPSS

0.14 0.13 0.10

0.45 0.45 0.49

BLYP PBE TPSS

0.12 0.09 0.16

0.47 0.47 0.50

BLYP PBE TPSS

0.02 0.01 0.05

0.46 0.46 0.48

BLYP PBE TPSS

0.67 0.66 0.70 0.09 0.36

1.24 1.24 1.27 0.55 0.88

LC-BLYPa LC-PBEa LC-TPSSa CAM-B3LYPb M11c

0.11

0.54

MN12-SXd

MAD

MaxAD


MinAD

RMSD

0.07 0.05 0.14

0.33 0.31 0.36

0.10 0.09 0.15

0.37 0.35 0.41

0.12 0.11 0.17

0.91 0.38 0.43

0.10 0.13 0.19

0.39 0.38 0.43

0.11 0.09 0.15

0.39 0.38 0.42

0.04 0.01 0.09

0.36 0.35 0.39

1.18 1.17 1.20 0.28 0.78

1.36 1.34 1.38 0.41 0.94

0.11

0.52

α = 0.0, β = 1.0, and μ = 0.47 Bohr . α = 0.19, β = 0.46, and μ = 0.33 Bohr−1. cα = 0.428, β = 0.572, and μ = 0.25 Bohr−1. dα = 0.25, β = −0.25, and μ = 0.11 Bohr−1. eAlso provided in the table are the corresponding descriptors for other RSHs.

α = 0.0, β = 1.0, and μ = 0.47 Bohr−1. bα = 0.19, β = 0.46, and μ = 0.33 Bohr−1. cα = 0.428, β = 0.572, and μ = 0.25 Bohr−1. dα = 0.25, β = −0.25, and μ = 0.11 Bohr−1. eAlso provided in the table are the corresponding descriptors for other RSHs.

representation for the performed analysis based on SMAD and TSMAD descriptors. Not only for individual photophysical properties under study but also from the general viewpoint we can see the superiority of the OT-RSHs based on PBE (α = 0.0, β = 1.0) in comparison to others. Apart from the statistical analyses, the individual numerical data for the photophysical properties of the efficient TADF emitters for OLED applications should also be taken into account. This is more pronounced especially for the compounds with large charge-transfer character and very small singlet−triplet gaps where the first singlet and triplet excited states are locally parallel. For instance, for some molecules with an envisioned large charge-transfer character based on our recently proposed excited states index and hole− electron distributions,105 such as PPZ-DPO and PXZ-TRZ, it also been recognized that they have also small singlet−triplet energy gaps.20 The theoretically predicted values of the singlet−triplet energy gaps obtained from the OT-RSHs based on BLYP and PBE approximations with α = 0.0, β =

1.0 combination along with optimally tuned values of rangeseparation parameter are, respectively, 0.09 and 0.09 eV (PPZDPO) and 0.07 and 0.06 eV (PXZ-TRZ), in close agreement with the experimental singlet−triplet energy gaps of 0.09 and 0.06 eV for the two molecules, respectively.16 Thus, what we would like to underline here is that even very small singlet− triplet energy gaps, i.e., lower than 0.1 eV, as an essential ingredient for having an efficient TADF-based OLED, can well be predicted using the proposed OT-RSHs. Comparisons among the proposed OT-RSHs aside, from the practical standpoint it is also indispensable to evaluate the accountability of these functionals with respect to the standard RSHs and other functionals with a broad range of results. We first present the corresponding results about the comparison between our proposed OT-RSHs and other RSHs. The chosen approximations for the purpose are the LC-BLYP, LC-PBE, and LC-TPSS (α = 0.0, β = 1.0, and μ = 0.47 Bohr−1),63,106 CAM-B3LYP (α = 0.19, β = 0.46, and μ = 0.33 Bohr−1),67 M11 as a RSH Minnesota functional (α = 0.428, β = 0.46, and

a

−1 b

MSD

a

J


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0.11 Bohr−1).79 Notice that the tested functionals cover all the options of α = 0, α ≠ 0, α + β = 1, and α + β ≠ 1. The related statistical descriptors on the performances of these functionals for all the photophysical properties under study have been reported in Tables 2−4. From the results in the tables it becomes evident that in comparison to the standard RSHs and the recently proposed Minnesota functionals the newly developed OT-RSH functionals have smaller deviations with respect to the reference data. More importantly, what we can unequivocally express here is that not only imposing the correct 100% asymptotic HF exchange can lead to the improved accuracy but also an appropriate value of the range-separation parameter is necessary for well describing the photophysical properties of the emitters. As further confirmations to these findings, it has been revealed that a RSH like CAM-B3LYP without a full 100% contribution of asymptotic HF exchange (α + β = 0.65) tends to overestimate the absorption energies of TADF emitters,16 highlighting the role of imposing α + β = 1.0 condition in the construction of RSHs for the present goal. Such overestimation is also indeed the case for the LC-ωPBE (α + β = 1.0, ω = 0.47 Bohr−1) suggesting that the length scale of charge transfer for the organic molecules with TADF character is smaller than ω−1, where the positive impact of the optimally tuning of the rangeseparation parameter for these compounds are showcased. Finally, the subtle points and intricacies regarding the optimally tuning procedure unveil that the origin of larger deviations for some RSHs such as CAM-B3LYP can be related to the larger incorporation of HF exchange at SR part. Putting these perceptions together, we can conclude that a reliable description of the photophysical properties using OT-RSHs necessitates smaller admixtures of SR HF exchange than that is used in standard RSHs. It seems that the smaller the HF exchange at SR part, the better performance of the OT-RSHs for the OLEDs photophysics. Regarding the comparisons made among the performances of the newly developed OT-RSHs and other functionals from various rungs of Jacob’s ladder106 we have relied on the reliable data in the field obtained with the same geometries and basis sets that we used herein. For the benchmarked set of molecules herein, the functionals B3LYP,88,89 PBE0,107,108 and PBE90 with both diversity and literature recommendations give the MADs of 0.57, 0.45, and 1.28 eV on absorption energies, 0.59, 0.45, and 1.32 eV on emission energies, and 0.17, 0.18, and 0.22 eV on singlet−triplet energy gaps.20 These results unambiguously reveal that these functionals do not reach to the accuracy of the newly developed OT-RSHs for all the studied photophysical properties. On the other hand, considering different functionals from various rungs of Jacob’s ladder constitutes a way to further dissect the relative performances across the hierarchy of functionals. Looking at the obtained statistical descriptors on all the studied photophysical properties, we find that the performances are improved when going from the pure PBE functional (α = 0, μ = 0) to the hybrid PBE0 with a fixed amount of 25% HF exchange (α ≠ 0, μ = 0) to the OT-RSHs with the interelectronic distance-dependent HF exchange contributions (α ≠ 0, μ ≠ 0). In another comparative study, the applicability of different DFT approximations from various rungs with good performances for the absorption energies of six molecules (2CzPN, ACRFLCN, α-NPD, NPh3, PhCz, and PXZ-TRZ)16 have been compared with the best OT-RSHs developed herein. For this

Table 4. Statistical Descriptors on the Performance of the Proposed OT-RSHs in the Calculations of Singlet−Triplet Energy Gaps for the Benchmarked Set of Molecules in Figure 1, Panel Ae functionals

BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS BLYP PBE TPSS LC-BLYPa LC-PBEa LC-TPSSa CAM-B3LYPb M11c MN12-SXd

MSD

MAD

MaxAD


MinAD

RMSD

0.00 0.00 0.01

0.13 0.13 0.17

0.01 0.01 0.01

0.16 0.17 0.21

0.02 0.02 0.03

0.18 0.19 0.23

0.00 0.02 0.00

0.20 0.21 0.25

0.01 0.01 0.02

0.20 0.22 0.25

0.02 0.01 0.04

0.21 0.23 0.25

0.60 0.63 0.66 0.07 0.36

0.95 0.96 0.99 0.31 0.54

0.05

0.22

α = 0.0, β = 1.0, and μ = 0.47 Bohr−1. bα = 0.19, β = 0.46, and μ = 0.33 Bohr−1. cα = 0.428, β = 0.572, and μ = 0.25 Bohr−1.. dα = 0.25, β = −0.25, and μ = 0.11 Bohr−1. eAlso provided in the table are the corresponding descriptors for other RSHs. a

Figure 3. General performances of the proposed OT-RSHs based on SMADs and TSMSDs over all the photophysical properties under study. The PBE-based OT-RSH with α = 0.0, β = 1.0 combination, whose SMAD is set to be 1, has been considered as reference.

μ = 0.25 Bohr−1)77 and MN12-SX as a screened- exchange Minnesota density functional (α = 0.25, β = −0.25, and μ = K


Article

The Journal of Physical Chemistry C set, the hybrid functionals M06-2X,107 CAM-B3LYP,67 and ωB97XD76 provide the MADs of 0.56, 0.62, and 0.72 eV, respectively. In addition, some of the specific hybrid functionals like the reoptimized version of B3LYP (B3LYP*)109 and M06-HF110,111 with MADs of 0.46 and 1.20 eV, respectively, on this set have also been included in our comparisons. Note that, despite their different parametrizations, these latter functionals reveal different extremes of hybrid functionals with low HF exchange contribution for the former (15%) to high value for the latter (100%). For the same set of compounds, the MADs of the proposed OT-RSHs with α = 0.0, β = 1.0 combination and based on BLYP and PBE density functional approximations are 0.44 and 0.43 eV, respectively. From these comparisons, it can be concluded that the proposed OT-RSHs outperform other functionals from different categories containing both interelectronic distancedependent and fixed HF exchange contributions. Also, the results of the proposed OT-RSHs are also comparable with some other parametrized functionals. Lastly, we also observed that the parametrized M06-HF functional with full HF exchange over the entire range can corrupt the prediction of photophysical properties of the considered emitters. It seems that there is an intricate balance among the factors of DFT exchange and correlation as well as HF exchange in conventional hybrids, where a particular compromise among these terms is needed to reach a reasonable accuracy. In another analysis, we have examined the role of continuum solvent on the computed photophysical properties. More specifically, we have computed the absorption and emission energies as well as the singlet−triplet energy gaps using the OT-RSHs based on BLYP, PBE, and TPSS approximations along with α = 0.0, β = 1.0 combination in solution phase, and compared the corresponding results with those obtained in the gas phase in Table 5. The related raw data have also been

understandable. The experimental data used in our calculations as reference have often been measured in nonpolar solvents like toluene and cyclohexane with dielectric constants of 2.38 and 2.02, respectively.112 Considering this point that the solute−solvent interactions are weak with low polarity solvents, it can be anticipated that the small dielectric constants should have a small effect on photophysics.27,24 However, dissecting the numerical data and the corresponding statistical analyses we found that as compared to the absorption energies and singlet-triple energy gaps that remain largely unaffected in solution phase, the effects of solvation on photophysical properties are more pronounced for the emission energies. Finally, from the viewpoint of functionals ranking in different environments we can see that the relative ordering does not change in solvent, where the top performers are still the OTRSHs based on BLYP and PBE density functional approximations. A general conclusion in this context, however, is difficult to extricate. As the last analysis, we assessed the performance of the proposed OT-RSH approximations for predicting the photophysical properties of another set of emitters for which the range-separation parameter tuning has not been performed. It should be noted here that although the values of the rangeseparation parameter in the OT-RSHs have been shown to be system dependent, the optimally tuned parameters for similar compounds in each set may also be nearly the same. Accordingly, it would be interesting to check whether there exist good performances for these approximations to describe the photophysical properties of other emitters not included in the tuning process. To this end, we have considered another benchmark set of molecules with a wide range of singlet− triplet energy gaps (small, medium, and large) for the estimation of their photophysical properties. Panel B in Figure 1 displays the corresponding geometrical structures for these molecules. The statistical descriptors on the performance of the OT-RSHs based on BLYP, PBE, and TPSS density functional approximations with α = 0.0, β = 1.0 combination and an averaged value (μ = 0.21 Bohr−1) for the rangeseparation parameter obtained from the previous analyses in the optimally tuning process, which was in turn nearly the same for various systems, have been gathered in Table 6. Provided in the Supporting Information, Tables S8−S10, are the corresponding raw data. Similar calculations with the same ingredients have also been carried out on the first benchmarked set, where the MADs as 0.30, 0.27, and 0.10 eV were found for absorption, emission, and singlet−triplet energies, respectively. The obtained accuracy is remarkable where we can see that not only for the molecules included in Panel B but also for the compounds in the first data set the reasonable statistical data are obtained from the OT-RSHs. Upon a closer inspection of the data in Table 6, we also find that there is still the same ordering as in earlier cases for the newly proposed OT-RSHs based on various density functional approximations. We have also compared these results with those obtained from the widely used DFT functional B3LYP and the standard LC-ωPBE approximation,20 as well as the screened- exchange Minnesota functional MN12-SX. The MADs of B3LYP, LCωPBE, and MN12-SX are, respectively, 0.36, 0.74, and 0.29 eV on absorption energies, 0.66, 0.61, and 0.54 eV on emission energies, and 0.07, 0.82, and 0.09 eV on singlet−triplet energy gaps of the molecules in panel B. Comparing these data with the results obtained from the proposed OT-RSHs, it can

Table 5. Statistical Descriptors (eV) on the Performance of the OT-RSHs Based on BLYP, PBE, and TPSS Density Functional Approximations along with α = 0.0, β = 1.0 Combination for Predicting the Absorption Energies, Emission Energies, and Singlet−Triplet Energy Gaps in Solution Phase for the Benchmarked Set of Molecules in Figure 1, Panel A MSD BLYP PBE TPSS

0.24 0.23 0.30

BLYP PBE TPSS

0.32 0.30 0.37

BLYP PBE TPSS

0.10 0.10 0.15

MAD

MaxAD

absorption energies 0.32 0.67 0.32 0.68 0.36 0.72 emission energies 0.32 0.65 0.30 0.66 0.37 0.70 singlet−triplet energy gaps 0.11 0.20 0.11 0.22 0.15 0.24

MinAD

RMSD

0.12 0.10 0.13

0.36 0.36 0.40

0.10 0.10 0.16

0.35 0.34 0.40

0.01 0.01 0.05

0.12 0.13 0.17

tabulated in Tables S5−S7 in the Supporting Information. Comparing these results with those obtained at gas phase we find that, given some differences among the numerical data comparable results are obtained in the two phases where no systematic and significant improvements were found when going from gas phase to the solution. This may, however, be L


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It was revealed that the newly developed optimally tuned range-separated hybrids with right asymptotic exchangecorrelation potential behavior lead to the reliable theoretical estimates of photophysical properties. From the methodological viewpoint, the imperative roles of the range-separation parameter tuning as well as the short- and long-range HF exchanges were showcased. Comparing with the recent literatures unveils that the proposed models not only perform better than their conventional counterparts and nontuned versions but also outperform the earlier developed hybrids with both fixed and interelectronic distance-dependent HF exchange contributions. So, they can be employed as economical protocols to theoretical design of the potential molecules with low singlet−triplet energy gap for OLEDs applications. The applicability of the proposed approximations for prediction of photophysical properties in continuum solvents versus gas phase has also been explored and benchmarked, where no significant changing patterns on numerical results and functionals ranking were observed. The efficiency of the newly developed OT-RSHs for estimation of the photophysical properties of the emitters not included in the optimally tuning processes of the range-separation parameter was also highlighted. The recommendations of several models aside, the adequacy and improved accuracy of the optimally tuned range-separated approximations provide a promising path toward developing novel OT-RSHs for applications in OLEDs technology.

Table 6. Statistical Descriptors (eV) on the Performance of the OT-RSHs RSHs Based on BLYP, PBE, and TPSS Density Functional Approximations along with α = 0.0, β = 1.0 Combination (μ = 0.21 Bohr−1) Developed in This Work for Predicting the Absorption Energies, Emission Energies, and Singlet−Triplet Energy Gaps of the Molecules in the Second Benchmarked Set functional

MSD

BLYP PBE TPSS

0.32 0.32 0.38

BLYP PBE TPSS

0.18 0.17 0.24

BLYP PBE TPSS

0.17 0.18 0.21

MAD

MaxAD

absorption energies 0.32 0.47 0.32 0.46 0.38 0.53 emission energies 0.22 0.47 0.21 0.47 0.25 0.52 singlet−triplet energy gaps 0.18 0.51 0.19 0.52 0.22 0.55

MinAD

RMSD

0.09 0.06 0.15

0.34 0.34 0.40

0.00 0.02 0.01

0.26 0.26 0.30

0.02 0.02 0.07

0.23 0.24 0.27

clearly be seen that our newly developed approximations have lower or comparable deviations, though their performances are also outperformed in some cases. However, as previously discussed in the related efforts, the improved accuracy of a hybrid functional like B3LYP on singlet−triplet energy gaps can be attributed not only to a specific amount of HF exchange for the problem of concern but also to its simultaneous overestimation or underestimation of the lowest singlet and triplet excitation energies.60 To wrap up, putting all these findings together one can say that the optimally tuning procedure of the RSHs predicts the options of α = 0.0, β = 1.0, and μ = 0.21 Bohr−1 along with the considered GGAs and meta-GGAs for the underlying exchange and correlation terms as the inexpensive promising candidates for our purpose, yielding a reliable description for photophysics of OLEDs with a good compromise between accuracy and computational cost. We hope that the OLED materials based on the emitters similar to those studied in this study can benefit from the theoretical predictions of the proposed OTRSHs.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b11681.

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Reference data and computed raw data (PDF)

AUTHOR INFORMATION

Corresponding Author

*(M.A.) E-mail: [email protected]. Telephone: +98 71 36137160. Fax: +98 71 36460788. ORCID

Mojtaba Alipour: 0000-0003-3037-0232 Notes

The authors declare no competing financial interest.

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5. CONCLUDING REMARKS Summing up, through this work, we have demonstrated the applicability and accountability of optimally tuned rangeseparated density functional theory for reliable description of photophysics of OLEDs. To this end, several variants of the optimally tuned range-separated approximations were proposed and validated for predicting the photophysical properties such as absorption energies, emission energies, and singlet− triplet energy gaps of the emitters with various ranges of singlet−triplet splitting. Taking BLYP, PBE, and TPSS as the density functional approximations terms with different combinations of the related tunable parameters, short- and long-range exchange contributions as well as the rangeseparation parameter, we constructed several optimally tuned range-separated functionals for obtaining the best performing approximations for more accurate estimations of the photophysical properties in the experimental benchmark sets of molecules with OLED applications. The following is what we can draw from the findings of this work.

ACKNOWLEDGMENTS The authors wish to express their gratitude to Shiraz University for providing the computing facilities for the present project.

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REFERENCES

(1) Perrin, F. La Fluorescence Des Solutions. Ann. Phys. 1929, 10, 169−275. (2) Lewis, G. N.; Lipkin, D.; Magel, T. T. J. Reversible Photochemical Processes in Rigid media. A Study of the Phosphorescent State. J. Am. Chem. Soc. 1941, 63, 3005−3023. (3) Parker, C. A.; Hatchard, C. G. Triplet-Singlet Emission in Fluid Solutions. Phosphorescence of Eosin. Trans. Faraday Soc. 1961, 57, 1894−1904. (4) Parker, C. A.; Joyce, T. A. Activation-Controlled Delayed Fluorescence of Benzil. Chem. Commun. 1968, 22, 1421−1422. (5) Wilkinson, F.; Horrocks, A. R. Phosphorescence and Delayed Fluorescence of Organic Substances Luminescence in Chemistry; Bowen, E. J., et al., Eds., Van Nostrand: London, 1968.

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