High-Level Ab Initio Computations of the Absorption Spectra of

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High Level Ab-Initio Computations of the Absorption Spectra of Organic Iridium Complexes Felix Plasser, and Andreas Dreuw J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp5122917 • Publication Date (Web): 13 Jan 2015 Downloaded from http://pubs.acs.org on January 17, 2015

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The Journal of Physical Chemistry

High Level Ab-Initio Computations of the Absorption Spectra of Organic Iridium Complexes Felix Plasser∗ and Andreas Dreuw Interdisciplinary Center for Scientific Computing, Ruprecht-Karls-University, Im Neuenheimer Feld 368, 69120 Heidelberg, Germany E-mail: [email protected]

∗

To whom correspondence should be addressed

1

ACS Paragon Plus Environment


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Abstract The excited states of fac-tris(phenylpyridinato)iridium [Ir(ppy)3 ] and the smaller model complex Ir(C3 H4 N)3 are computed using a number of high-level ab-initio methods, including the recently implemented algebraic diagrammatic construction method to third order ADC(3). A detailed description of the states is provided through advanced analysis methods, which allow a quantification of different charge transfer and orbital relaxation effects and give extended insight into the many-body wavefunctions. Compared to the ADC(3) benchmark an unexpected striking difference of ADC(2) is found for Ir(C3 H4 N)3 , which derives from an overstabilization of charge transfer effects. Time-dependent density functional theory (TDDFT) using the B3LYP functional shows an analogous but less severe error for charge transfer states, while the ωB97 results are in good agreement with ADC(3). Multi-reference configuration interaction computations, which are in reasonable agreement with ADC(3), reveal that static correlation does not play a significant role. In the case of the larger Ir(ppy)3 complex, results at the TDDFT/B3LYP and TDDFT/ωB97 levels of theory are presented. Strong discrepancies between the two functionals, which are found with respect to the energies, characters, as well as the density of the low lying states, are discussed in detail and compared to experiment.

Introduction Organic iridium complexes have attracted a great deal of attention as dyes in organic light emitting diodes (OLED) 1,2 and electrochemical cells. 3 A wide array of complexes has been synthesized in an immense experimental effort. 3–8 However the quest for suitable molecules with all the required properties is still ongoing and in particular the construction of blue emitters remains a major challenge. 9 This endeavor is hampered by the fact that some of the properties of interest, e.g. the phosphorescence quantum yield and the photochemical stability depend on the intricate interplay of a number of singlet and triplet states. 3,8,10 A

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complete understanding of these states by experiment alone is hardly achievable, especially due to the spectroscopically dark nature of many of them. Computation provides essential complementary information needed for an effective rational design process. However, computations on organic iridium complexes are highly challenging, as well, due to their intricate electronic structure properties and the large system sizes of interest. For the most part, time-depdendent density functional theory (TDDFT) has been applied, heavily relying on the the B3LYP 11 functional, 9,12–22 and to the best of our knowledge only one wavefunction based ab-initio study 23 has been performed so far. TDDFT computations have undoubtedly moved forward our understanding of these complexes by providing essential qualitative insight, in particular concerning the shape of the orbitals and the nature of the transitions involved. Moreover, TDDFT provides predictive power through correlations between computed and measured quantities. 10,18,20 However, the situation is still far from satisfying from a methodological viewpoint. First and foremost, the reliability of TDDFT/B3LYP has not been rigorously tested despite its extensive usage. General shapes of experimental absorption spectra can indeed be reproduced quite well but this does not automatically mean that the state assignments are correct. For example, most TDDFT/B3LYP studies on various iridium complexes assigned the long wavelength shoulder in the UV absorption spectrum primarily to low energy singlet metal-to-ligand charge transfer (1 MLCT) states. 14,15,17 By contrast, the use of range separated hybrid functionals tended to increase the energy of the 1 MLCT states, leaving only triplet states in the low energy part of the spectrum. 21–23 This discrepancy highlights the well-known weakness of TDDFT in computing charge transfer states 24 and shows that the choice of functional is crucial. Another critical aspect derives from the fact that a number of factors affecting the spectra, e.g. vibrational broadening, non-equilibrium effects, spin-orbit coupling and solvation, are usually excluded from the computations. This may lead to spurious agreements between computation and experiment with limited transferability and interpretative power. A curious result in this context is, for example, the finding that TDDFT phosphorescence energies 20 and radiative

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rates 10 correlated better with experiment when these were computed at the S0 rather than T1 equilibrium geometries, which contradicts the standard view of Kasha’s rule. 25 In summary, while a lot of interesting work has been done, significant additional insight could be gained by more comprehensive computational investigations of these systems. In this work high level computational methods and sophisticated analysis strategies are applied to study the excited states of organic iridium complexes. The computations are centered around the ab-initio algebraic diagrammatic construction (ADC) scheme of the polarization propagator. 26–28 Using the variants of this method from first to third order in many-body perturbation the effects of electron correlation are analyzed. These calculations are supplemented by results at the equation of motion coupled cluster level including single and double excitations (EOM-CCSD). 29,30 Furthermore, to gauge the importance of multireference effects, complete active space self-consistent field (CASSCF) and multi-reference configuration interaction (MR-CI) computations are performed. 31,32 The excitation energies, as well as various other wavefunction properties, are compared to results at the TDDFT level. In this case the global hybrid B3LYP 11 and the range-separated ωB97 33 functionals are applied. For a discussion of the strengths and weaknesses of all these methods, see e.g. Refs 34,35. A detailed analysis of the wavefunctions produced and a systematic comparison between the various methods is achieved through recently developed analysis protocols 36,37 using specific extensions for transition metal complexes introduced here. In order to be able to use such high level methods at an acceptable computational cost a model system is constructed: fac-tris(3-iminoprop-1-en-1-ido)iridum [Ir(C3 H4 N)3 ], which represents the core structure of the well-known and characterized fac-tris(phenylpyridinato)iridium [Ir(ppy)3 ] complex 10,12,21,38,39 (Scheme 1). While a wide array of methods is applied to the smaller complex, only TDDFT computations are performed in the case of Ir(ppy)3 . Interesting parallels are drawn between the complexes and extrapolations from the smaller to the bigger complex allow us to assess the reliability of the computations on Ir(ppy)3 . The scope of this work is a high-level description of the singlet and triplet excited state

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N

N

NH

Ir

Ir N

HN

N H

Scheme 1: Iridium complexes considered in this work: Ir(ppy)3 (left) and the smaller model compound Ir(C3 H4 N)3 (right). wavefunctions of the isolated complexes at their equilibrium geometries in the scalar relativistic limit. In this way an internally consistent framework is created, which allows a direct comparison of the different computational methods. While the accurate reproduction of experimental absorption and emission spectra in solution (and ultimately the device) is of course the final goal, taking a shortcut toward this goal is dangerous as it may just lead to spurious agreements and error compensation. As mentioned previously, the comparison with experiment is only straight forward if spin-orbit coupling, 10,40–42 solvation, 18,43 and vibronic effects are consistently included, and usage of computational benchmarks is a way to avoid this problem.

Excited state wavefunction analysis for transition metal complexes A quantitative or even qualitative analysis of the excited states of transition metal complexes is a major challenge even when a suitable excited state method can be found. This is due to the fact that in many cases 13,21,23 the excited state transitions are not well represented by the canonical orbitals (created by Hartree-Fock or a DFT method) and many interacting configurations are present. To overcome this problem, a systematic quantitative approach is taken here, which does not depend on the resolution of the canonical orbitals at all. This is achieved by extending previously introduced methods 36,37 to the specific case of transition 5



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metal complexes. Different local and charge transfer contributions are quantified with a particular focus on modulations between ligand centered (LC) and MLCT character, which are in the center of attention as these affect spin-orbit coupling, radiative rates, and band shapes. 3,7,8,18 The methodology used here is based on density matrices, which are well defined quantities independent of the wavefunction model. This allows a comparison of the results between the different methodological approaches used.

Analysis of the transition density matrix The one-particle transition density matrix (1TDM) between the ground and excited states does not only code for the individual hole and electron components of the excitation but contains also information about correlations between them. 44 In the present case this means, for example, that it is possible to differentiate between a mixture of MLCT and ligand-tometal CT (LMCT) and a combination of locally excited states. The formalism presented here extends our previous work, 36,37 which is in turn inspired by other authors. 45,46 The main quantities for the 1TDM characterization are the charge transfer numbers (see also Ref. 46). For two molecular fragments A and B these are given as 37

ΩAB

i 1 X Xh 0I 0I 0I 0I D S µν SD µν + Dµν SD S µν = 2 µ∈A ν∈B

(1)

where the sums run over all atomic basis functions µ, ν on the respective fragment, D0I is the 1TDM, and S is the overlap matrix. The normalization factor

Ω=

X

ΩAB

(2)

A,B

which can be viewed as the squared norm of the 1TDM, 37 measures the single excitation character of the excitation (see also Ref. 47 for a more detailed discussion of this quantity).

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The total charge transfer character

ωCT =

1 X ΩAB Ω B6=A

(3)

is obtained by summing over all the off-diagonal elements of the Ω-matrix. To extend this work to transition metal complexes, the situation of one central metal atom ”M” and several ligands ”L1,L2,...” (indexed L) is considered. Using this arrangement, the Ω-matrix is partitioned into five parts related to metal centered (MC), ligand centered (LC), metal-to-ligand CT (MLCT), ligand-to-metal CT (LMCT), and ligand-to-ligand CT (LLCT) contributions (Scheme 2). 1 ΩMM Ω 1X ΩLL Ω L 1X ΩML Ω L 1X ΩLM Ω L 1 X ΩLL′ Ω L6=L′

ωMC = ωLC = ωMLCT = ωLMCT = ωLLCT =

(4) (5) (6) (7) (8)

Using these definitions, the sum over the individual CT contributions equals the initially defined total CT value (3):

ωCT = ωMLCT + ωLMCT + ωLLCT

(9)

Furthermore, deriving from these quantities, the total hole (Scheme 2, red) 1 ωH (M) = Ω

ΩMM +

X L

7

ΩML

!

= ωMC + ωMLCT


(10)


L1

LC

LC

LLCT

M MC M

LC

LLCT

L2

LMCT

L3

Hole

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MLCT L1

L2

L3

Electron Scheme 2: Decomposition of the transition density matrix of a complex containing a central metal ion M and three ligands L1, L2, L3 into metal centered (MC), ligand centered (LC), metal-to-ligand CT (MLCT), ligand-to-metal CT (LMCT), and ligand-to-ligand CT (LLCT) contributions. Areas contributing to the total hole charge on the metal ion ωH (M) are marked in red, while areas contributing to the electron charge ωE (M) are marked in lightblue. and electron (Scheme 2, lightblue) 1 ωE (M) = − Ω

ΩMM +

X

ΩLM

L

!

= −ωMC − ωLMCT

(11)

charges of the 1TDM with respect to the central metal atom are computed. The sum of these ω∆ (M) = ωH (M) + ωE (M)

(12)

constitutes the primary charge shift on the metal. While the 1TDM is strictly speaking only accessible for wavefunction based methods, it is possible to extend the formalism to the TDDFT framework. In the implementation 48 used here the transition density matrix is simply equated to the sum of the excitation and de-excitation components of the TDDFT response vector XI and YI :

D0I = XI + YI

(13)

There is some arbitrariness in this assignment and a different formalism suggests that, addi-

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tionally, the response vectors should be scaled by the orbital energy differences. 49,50 In spite of this formal problem, a comparison between TDDFT and wavefunction based results is certainly reasonable, at least on a semi-quantitative level.

Analysis of the difference density matrix As an alternative to the 1TDM it is also possible to analyze the 1-particle difference density matrix (1DDM). This quantity is obtained by subtracting the density matrix of the ground state D00 from the one of the excited state DII

∆0I = DII − D00

(14)

While a direct visualization of the difference density is not generally instructive, the 1DDM can be used to construct the attachment/detachment densities. 51 This procedure starts with a diagonalization WT ∆0I W = diag(κ1 , κ2 , . . .)

(15)

where it is assumed that ∆0I is given with respect to a set of orthogonal MOs {φp (r)}. The detachment density matrix is constructed by only considering the negative eigenvalues

di = min(κi , 0)

(16)

and a back-transformation to the initial orbital basis:

DD = Wdiag(d1 , d2 , . . .) WT .

(17)

The detachment density is simply obtained as

ρD (r) =

X

DD,pq φp (r)φq (r).

pq

9


(18)


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The attachment density is constructed from the positive eigenvalues in an analogous fashion

ai = max(κi , 0)

(19)

DA = Wdiag(a1 , a2 , . . .) WT X ρA (r) = DA,pq φp (r)φq (r).

(20) (21)

pq

In addition to plotting these densities it is also of interest to analyze the individual eigenvectors of the difference density matrix, 37,52,53 termed natural difference orbitals (NDOs). These orbitals do not only give a representation of the excitation process but also provide cues about higher excitation and orbital relaxation effects, which are not covered by the 1TDM. 52 The collective influence of all NDOs can be measured by the promotion number

p=−

X

di =

i

X

ai

(22)

i

which corresponds to the spatial integral over the attachment or detachment density. Its value gives the total number of rearranged electrons counting double excitation character as well as orbital relaxation. 37,52 Aside from plotting, the attachment/detachment densities can be subjected to a Mulliken population analysis. Here, the symbols pD (A)/pA (A) are used to denote the detachment/attachment population on atom (or fragment) A. Per convention pD (A) is greater or equal to zero, as it describes the positive hole, and conversely pA (A) ≤ 0. The sum of these two populations amounts to the total shift of charge on atom A during the excitation process, i.e. p∆ (A) = pD (A) + pA (A) = cI (A) − c0 (A)

(23)

where cI (A) is the Mulliken charge of atom A in state I (with I = 0 being the ground state). In the case of CIS, using unrelaxed densities, it holds that pD (A) = ωH (A) and

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pA (A) = ωE (A), see also Ref. 37. But this is not the case for general wavefunctions and it will be shown below that these values can differ significantly for many-body theories.

Computational Details ADC(1), ADC(2), and ADC(3) calculations 26–28 are carried out using a developmental version of Q-Chem 4.1. 54,55 Note that ADC(1) excitation energies and state density matrices are equivalent to configuration interaction with single excitations (CIS) while there is a small additional term in the case of the 1TDMs. Density matrices at the ADC(3) level are obtained by contracting the third order vectors with the second order intermediate state representation. 56 CASSCF calculations are performed using 12 electrons in 9 active orbitals [CASSCF(12/9)] consisting of three π, three iridium-d and three π ∗ orbitals, which were identified as the main active orbitals in the ADC(3) calculations. MR-CI 32,57,58 calculations are performed allowing single excitations out of this reference. The MR-CI energies are reported with an extensivity correction according to Pople et al., 59 denoted here +P. These computations are performed using Columbus 31,60 with integrals from the Molcas package. 61 By contrast to previous studies 23,43,62 no multireference perturbation theory computations are carried out here considering that this is by far not trivial and large active spaces are needed for accurate results. MR-CI, as applied here, should be at least more stable and free of intruder states due to its variational nature. However, it should be pointed out that due to the truncated reference space and the restriction to single excitations only semi-quantitative accuracy can be expected from these calculations. For all excited state calculations the iridium atom was described with the LANL2DZ effective core potential (ECP), in its ”small-core” version, and the corresponding basis set for the active (5s, 5p, 5d, 6s, 6p) orbital shells 63 (see also Ref. 12), while for the remaining atoms the 6-31G* basis set 64 was employed. The frozen core approximation was used in

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the wavefunction based calculations for the 1s orbitals on second row atoms while the (5s, 5p) ”semi-core” orbitals on iridium were kept active in the correlated calculations. The ground state structures were optimized at the DFT/PBE0 level 65 with the def2-TZVP basis set 66 (see also Ref. 22) using the Turbomole program system. 67 TDDFT excitation energies of Ir(C3 H4 N)3 and Ir(ppy)3 were computed using the global hybrid B3LYP 11,68 and range separated ωB97 33 functionals, as implemented in Q-Chem. 54 The extended wavefunction analysis procedures were performed using a recent implementation 37 within the ADC module of Q-Chem. For CASSCF and MR-CI analysis the newly released TheoDORE 1.0 package was applied. 36,52,69 In the TDDFT case the implementation of Richard and Herbert 48 was used. For the CCSD case only energies are reported, since no comparable implementation is available. Post processing of wavefunction analysis results was carried out using TheoDORE 1.0. 69 For the visualization of molecular structures, orbitals, and isodensities the Jmol 70 and VMD 71,72 packages were employed.

Results and discussion A number of computations were performed to address different aspects of the complexes under study. These will be presented in four different parts. Firstly, the main geometric parameters of Ir(C3 H4 N)3 and Ir(ppy)3 are discussed and the general suitability of Ir(C3 H4 N)3 as a model is established. Secondly, a detailed analysis of the excited states of this complex computed at the ADC(3) level is presented. Thirdly, an array of different methods is benchmarked against this reference considering a wide range of wavefunction properties as introduced above. Finally, the information from the previous sections is applied to Ir(ppy)3 .

Geometries of Ir(ppy)3 and Ir(C3 H4 N)3 A three-dimensional model of Ir(C3 H4 N)3 and selected structural parameters are presented in Figure 1, and the analogous values for Ir(ppy)3 are given in parentheses. For Ir(C3 H4 N)3 12


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the C-Ir covalent bond amounts to 1.99 ˚ A and the N-Ir coordinate bond to 2.12 ˚ A. These values are very similar to the analogous bond lengths in Ir(ppy)3 , which amount to 2.01 and 2.14 ˚ A. This comparison shows that roughly similar ligand field effects are expected for both complexes. An experimental reference 39 for these values (obtained in the case of Ir(tpy)3 , i.e. the same complex with an added methyl group per ligand) puts these values in close agreement at 2.02 and 2.13 ˚ A, respectively. In addition to the bond lengths also the angle between the C and N atoms in trans position is monitored to gauge the distortion from ideal octahedron symmetry. Also these values are similar for both complexes with 167◦ and 173◦ , respectively. As a rough gauge of the electronic structure we also compare the PBE0/def2-TZVP HOMO/LUMO gaps of the two complexes: 3.97 eV for Ir(ppy)3 and 4.30 eV for Ir(C3 H4 N)3 . In both cases the HOMO is of mixed Ir-d and ligand-π character while the LUMO is a ligand π ∗ orbital. These considerations show that Ir(C3 H4 N)3 is indeed a promising model for Ir(ppy)3 , and more calculations reinforcing this are shown below. 2 (2 .12 .1 4)

167° (173°) 99 ) 1. .01 (2

Figure 1: Molecular structure of Ir(C3 H4 N)3 : Values of the Ir-N and Ir-C bond lengths (˚ A) as well as the trans C-Ir-N’ angle are shown (corresponding values for Ir(ppy)3 in parentheses).

Excited states of Ir(C3 H4 N)3 computed at the ADC(3) level Having established the general suitability of Ir(C3 H4 N)3 as a model system, we will proceed to discuss its excited states in detail. For this purpose, results at the ADC(3) level of theory will be considered, which is a recently implemented method 28 of highest computational accuracy but yet of an affordable computational cost allowing an analysis of Ir(C3 H4 N)3 . To 13



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start the discussion, it is favorable to analyze the frontier orbitals. Unfortunately, the canonical Hartree-Fock orbitals are unfit for this purpose because diffuse contributions strongly mix with the valence virtual orbitals preventing a clear identification of the excited state characters. This problem is overcome by the use of state-averaged natural transition orbitals (SA-NTOs), 37 which can be interpreted in a similar sense to the Hartree-Fock orbitals only that they are specifically adapted to the set of excited states of interest. The SA-NTOs averaged over the first twelve lowest singlet and triplet excited levels are presented in Figure 2. 73 The primary hole orbitals, shown on the left side, possess predominant Ir-d character, consisting of a totally symmetric orbital (denoted da in the following text) as well as a pair of degenerate orbitals (de ). Additional smaller contributions are present in the form of a pair of degenerate πe orbitals. The dominant particle orbitals are of π ∗ character. In addition, there are non-negligible contributions of Ir-d orbitals (d∗e ), which are mixed with σ ∗ orbitals yielding an overall repulsive interaction for the Ir-C and Ir-N bonds (cf. e.g. Ref. 3). A detailed analysis of the excited states of Ir(C3 H4 N)3 is presented in Table 1. The threefold symmetry of the system leads to the fact that the states are arranged in triples. Each of these includes an exactly degenerate pair, transforming as an E irreducible representation of the C3 point group, and a third state of A symmetry. In Table 1 the twelve lowest singlet and triplet levels are presented, which are three of each 1 A, 1 E, 3 A, and 3 E symmetries. The lowest three excited states (13 E and 13 A) at around 2.82 eV are locally excited ππ ∗ states located on the three ligands (ωLC ≈ 0.80). As a representative of this first set of states, the 13 A state was chosen and the hole and particle densities plotted (Figure 3, top). This representation confirms the previous analysis: The hole density is primarily localized on the ligand π-system with a small contribution on the central iridium atom while the particle density resides only in the π-system. Interestingly, after this first triple of states there is gap of 1 eV before the next level (11 E) appears at 3.86 eV. This state has again pronounced LC character (ωLC = 0.48) but as opposed to the lowest triplet states there is also significant MLCT present (ωMLCT = 0.32). In the SA-NTO representation (Figure 2)

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Hole

Particle

da (4.51)

πe∗ (5.00)

de (4.50)

πe∗ (5.00)

de (4.50)

πa∗ (4.63)

πe (0.50)

d∗e (0.10)

πe (0.50)

d∗e (0.10)

Figure 2: State-averaged natural transition orbitals of the 12 lowest singlet and triplet excited state levels of Ir(C3 H4 N)3 computed at the ADC(3) level of theory. Summed amplitudes are given in parentheses.

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this level corresponds mostly to the da → πe∗ transition. The strong difference between the lowest singlet and triplet states may be unexpected at first sight but it conforms with the general expectation: exchange repulsion disfavors the locally excited singlet state pushing it up in energy where it acquires partial CT character. The second set of triplet states consists of the 23 A state, which lies at 3.87 eV and corresponds to the de → πe∗ transition, and the degenerate 23 E level at 4.00 eV, which is of de → πe∗ character. These are the first states with predominant MLCT character (ωMLCT = 0.44), however, also MC, LC, and LLCT contributions play a role. At 3.91 eV the 21 A (da → πa∗ ) state follows, which is related to the 11 E level discussed before. The hole and particle densities of this state are shown in Figure 3 (bottom). When analyzing this figure it should be recalled that 64% of the hole density (i.e. ωH = ωLC + ωLMCT + ωLLCT = 0.64) is in fact located on the ligands. This is not quite apparent in the graphical representation, probably because of the different radial structure of the Ir-5d orbitals as opposed to the 2p orbitals on carbon and nitrogen. Going beyond the lowest excited states, Table 1 shows that the higher states maintain MLCT character, but that also LLCT excitations play an enhanced role. To analyze the properties of the excited states presented in Table 1 in more detail, we start with the double excitation character in analogy to Ref. 23. For this purpose, the squared norm of the 1TDM Ω [Eq. (2)] is used as a wavefunction-model-independent measure of single excitation character. The case of Ω = 1 indicates a purely singly excited state while any higher excitations result in a lowered value of Ω. In agreement with computations on [Ir(ppy)2 (bpy)]+ the double excitation character is somewhat stronger for the singlet states 23 with Ω at about 0.82 while Ω is closer to one for all the triplet states. The highest single excitation character Ω = 0.89 is found for the lowest two triplet states, which are of LC character. In summary, the double excitation character and its modulations are rather modest and no significant methodological problems are expected to derive from it. Going beyond the double excitation character, a much stronger impact of many-body effects can be found in the promotion numbers p, the integrals over the attachment or

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Table 1: Excitation energies (∆E, eV), oscillator strengths (f ), and density matrix analysis of the 12 lowest singlet and triplet excited state levels of Ir(C3 H4 N)3 computed at the ADC(3) level. State ∆E 13 E 2.823 3 1A 2.824 11 E 3.862 23 A 3.873 21 A 3.908 23 E 4.003 3 3E 4.215 33 A 4.279 21 E 4.337 1 3A 4.353 31 E 4.567 41 A 4.715

f 0.047 0.067 0.018 0.085 0.010

a,b a,b a,b a,b a,b Ωa ωMC ωLC ωMLCT ωLMCT ωLLCT 0.89 0.02 0.80 0.10 0.05 0.03 0.89 0.02 0.81 0.10 0.04 0.03 0.83 0.08 0.48 0.32 0.07 0.06 0.86 0.15 0.20 0.44 0.06 0.16 0.83 0.05 0.50 0.31 0.07 0.07 0.85 0.13 0.19 0.45 0.06 0.18 0.83 0.12 0.11 0.45 0.06 0.25 0.83 0.11 0.10 0.47 0.06 0.26 0.82 0.10 0.11 0.45 0.07 0.27 0.82 0.09 0.08 0.50 0.06 0.27 0.82 0.09 0.10 0.44 0.07 0.30 0.82 0.11 0.11 0.36 0.10 0.32 a Analysis of the transition density matrix. b Values above 0.25 marked in bold. c Analysis of the difference density matrix.

Hole

pc 1.20 1.20 1.35 1.35 1.32 1.41 1.43 1.33 1.45 1.36 1.49 1.30

pA (Ir)c -0.10 -0.09 -0.29 -0.39 -0.28 -0.39 -0.40 -0.38 -0.37 -0.37 -0.38 -0.36

pD (Ir)c 0.14 0.13 0.43 0.62 0.39 0.61 0.60 0.59 0.58 0.60 0.57 0.50

Particle 13 A

23 A

21 A

Figure 3: Hole (red) and particle (blue) densities of the three lowest totally symmetric excited states of Ir(C3 H4 N)3 computed at the ADC(3) level of theory (isovalues: 0.008, 0.0016).

17



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detachment density [Eq. (22)], which count the total number of rearranged electrons. As pointed out previously, 52 a p value significantly larger than one in a singly excited state is an indication of orbital relaxation effects. In agreement with Ref. 52 it is also found here that p is large whenever significant charge transfer occurs. To shed light on this phenomenon, the 23 A state, the lowest energy state with significant CT character, is chosen for further analysis. For this purpose the natural transition orbitals (NTO, the singular vectors of the 1TDM) and the natural difference orbitals (NDO, the eigenvectors of the 1DDM) are compared in Figures 4 and 5. Starting with the NTO case (Figure 4), it is seen that there are two degenerate primary transitions of de → πe∗ type with corresponding singular values of λ1 = λ2 = 0.40. In addition, these NTOs possess small admixtures of different character: π for the hole and iridium d∗ character for the particle case. Aside from these primary NTOs, there is a third pair with a non-negligible amplitude of λ3 = 0.05, which resembles the da → πa∗ transition (cf. Figure 2). The first, second and fifth NDO sets closely correspond to the three primary NTO pairs, see Figure 5. However, there are two additional NDO contributions, which are not covered by the NTOs. The additional detachment orbitals possess the shape of hybrid orbitals on the C and N atoms pointing toward the iridium atom while the attachment orbitals resemble the d∗ orbitals. The effect of this is an electron shift toward iridium mediated through the σ-system reducing the net amount of charge transferred. This can also be observed in the attachment density [Figure 5, top)], which has a significantly stronger contribution on the iridium atom as compared to the particle density [Figure 4, top)].

Ir(C3 H4 N)3 : Comparison of electronic structure methods Having discussed the nature of the electronically excited states of Ir(C3 H4 N)3 , we will now proceed to evaluate how well these states are described by different electronic structure methods. In Table 2 the excitation energies at different levels of theory are collected: ADC from first to third order, EOM-CCSD, CASSCF, MR-CIS, TDDFT/B3LYP and TDDFT/ωB97. 18


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Hole

Particle

ω = 0.861

ω = 0.861

λ1 = 0.400

λ1 = 0.400

λ2 = 0.400

λ2 = 0.400

λ3 = 0.045

λ3 = 0.045

Figure 4: Hole and particle densities (isovalues 0.008, 0.0016) and their decomposition into NTOs (isovalue 0.05) for the 23 A state of Ir(C3 H4 N)3 computed at the ADC(3) level of theory.

19



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Detachment

Attachment

pD = −1.346

pA = 1.346

d1 = −0.445

a1 = 0.436

d2 = −0.445

a2 = 0.436

d3 = −0.069

a3 = 0.076

d4 = −0.069

a4 = 0.076

d5 = −0.061

a5 = 0.067

Figure 5: Attachment/detachment densities (isovalues 0.008, 0.0016) and their decomposition into NDOs (isovalue 0.05) for the 23 A state of Ir(C3 H4 N)3 computed at the ADC(3) level of theory.

20


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In Figure 6 a compact graphical representation of this data is given. Starting with ADC(2), the first striking observation is that except for the first two triplet levels (13 E and 13 A) the excitation energies at this level are severely underestimated giving discrepancies well above one eV for most of the states. A similar trend was reported for the [Ir(ppy)2 (bpy)]+ complex when either using the ADC(2) or the related CC2 method benchmarked against CASPT2. 23 At the ADC(1) level, the first two triplet state energies are described well while the others are severely overestimated. CCSD and MR-CIS+P perform comparatively well with the largest discrepancies to ADC(3) on the order of 0.5 eV. The same holds true for the CASSCF(12/9) calculations with the only exception of the 41 A state, which could not be described correctly and lies at significantly higher energies (6.19 eV). At the TDDFT/B3LYP level only the lowest triplet levels (13 E and 13 A) are within 0.5 eV of the ADC(3) reference while the energies of all other states are underestimated quite significantly (see also Ref. 23). But this deviation is not as severe as in the ADC(2) case. Of all the methods considered here, the range separated ωB97 functional shows the best agreement to ADC(3) with no discrepancies above 0.3 eV. The deviations between the different methods, as presented in Table 2 and Figure 6, and in particular the poor performance of ADC(2) is at first sight quite puzzling. To understand the reasons for this problem in more detail, an extended analysis of the excited state wavefunctions will be carried out in the remaining part of this section. First, the primary charge shift on iridium ω∆ (Ir) and its decomposition into hole ωH (Ir) and electron charges ωE (Ir) are analyzed (Figure 7). These values, which are computed from the 1TDM [Eq. (11) and Eq. (10)], allow a well defined quantification of the CT character across the different methods. The ADC(3) results show the trends already presented in Table 1. The lowest triplet states are almost completely ligand centered with ω∆ (Ir) ≈ 0.05 while this value goes up to about 0.5 for some of the higher lying excited states. ωE (Ir) remains rather constant around -0.15 showing that the particle orbitals are always primarily of ligand π ∗ type. By contrast, there are strong modulations in ωH (Ir) depending on whether the hole is localized on the iridium atom or the ligands. ADC(2) diverges from this reference

21



4A

21E

3E

1 A

3 2

A

33A

13E

23E

33E

2 A

(a)

6

1 3

A

1

1

5 4 3 6

(b)

11E

1

5

Excitation energy (eV)

4 3 6

(c)

3

5 4 3 6

(d)

5 4

F

CIS +P B3 LY P wB 97

M R

-

SD

CA SSC

(1 )

CC

(2 ) C

C

C A D

A D

(3 )

3

A D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6: Excitation energies of Ir(C3 H4 N)3 computed at different levels of theory, arranged according to symmetry and multiplicity: (a) singlet A, (b) singlet E, (c) triplet A, and (d) triplet E states.

22


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Table 2: Wide distribution of excitation energies of the 12 lowest singlet and triplet excited levels of Ir(C3 H4 N)3 when computed at different levels of theory. State 13 E 13 A 11 E 23 A 21 A 23 E 33 E 33 A 21 E 31 A 31 E 41 A

ADC(3) 2.823 2.824 3.862 3.873 3.908 4.003 4.215 4.279 4.337 4.353 4.567 4.715

ADC(2) 2.572 2.336 2.815 3.043 2.484 2.898 3.135 3.168 2.949 3.239 3.352 3.429

ADC(1) 2.844 2.847 4.962 4.694 5.115 4.830 5.172 5.400 5.675 5.814 5.978 5.866

CCSD 2.917 2.828 3.585 3.639 3.377 3.640 3.800 3.821 3.827 4.005 4.207 4.439

CASSCF 3.407 3.384 4.138 4.443 4.169 4.469 4.607 4.601 4.403 4.491 4.620 6.186

MR-CIS+P 3.242 3.151 3.899 3.920 3.558 3.918 4.152 4.193 4.166 4.400 4.610 4.160

B3LYP ωB97 2.592 2.595 2.510 2.571 3.307 4.000 3.239 3.801 3.104 3.948 3.237 3.895 3.355 4.202 3.385 4.286 3.569 4.410 3.677 4.426 3.973 4.714 4.110 4.912

showing a significant overestimation of MLCT character. In most cases the hole charge ωH (Ir) is above 0.7 and the total charge shift ω∆ (Ir)> 0.5. The discrepancy is particularly pronounced for the lowest triplet states, represented as black boxes in Figure 7 (c) and (d). Incidentally, as shown in Table 2, these states have the best agreement to ADC(3) in terms of the excitation energy highlighting that a simple comparison of excitation energies is not sufficient for benchmarking computational methods. At first sight, when considering only the primary charge shifts, ADC(1) appears to agree well with ADC(3). However, a more detailed look shows that the electron charges are too low for some of the states. This phenomenon derives from enhanced MC character, i.e. from d → d∗ transitions, which are completely absent for all the other methods. For technical reasons the primary charge shifts were only computed for singlet states in the case of CASSCF and MR-CI, see Figure 7. It is interesting to observe that in spite of a general agreement in energies between these two methods, the computed charge shifts differ quite strongly (see also Figure 8, discussed below). Generally CASSCF shows increased positive charges on iridium when compared to MR-CI, which are in turn still larger than in the ADC(3) case. The only exception is the 41 A state (blue triangle in Figure 7), which stands out at the CASSCF(12/9) level yielding also particularly high excitation energies (Table 2). 23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

An analogous analysis 48 is possible at the TDDFT level if the response vectors are interpreted as approximate transition density matrices and the results are shown in Figure 7, as well. Interestingly, neither of the functionals, B3LYP or ωB97, shows the same drastic overestimation of MLCT character as is the case for ADC(2). Especially the lowest triplet levels (13 A and 13 E), which also show good agreement in terms of excitation energies (Table 2) are described remarkably well, only that MLCT character is somewhat overestimated at the B3LYP level. For the higher triplet states both functionals correctly predict enhanced CT character, but as shown in Table 2 the excitation energies are severely underestimated at the TDDFT/B3LYP level while they are accurate for TDDFT/ωB97. The trends are not as clear for the singlet states. However, it can be seen that the ωB97 charges are somewhat higher but otherwise in the same overall ordering when compared to ADC(3) while at the B3LYP level there is a more random distribution. The 1TDM analysis, as described above, is concerned with the description of the primary transition processes in a one-electron picture. By contrast, the difference density matrix (1DDM) is also affected by many-particle effects like double excitations and orbitals relaxation. 52 1DDMs are computed for the ADC(1), ADC(2), ADC(3), CASSCF, and MR-CI methods and the results plotted in Figure 8. The main difference of these results as opposed to Figure 7 is that the net charge shifts from iridium are significantly reduced staying well below 0.4 e in most cases. The analysis of the individual attachment and detachment contributions, represented as small symbols connected with dotted lines in Figure 8, reveals that this change in charge transfer character is mediated by a strongly enhanced attachment component, i.e. pA (Ir)≪ ωE (Ir), while the detachment component pD (Ir) is increased only weakly over ωH (Ir) leading to an overall lowering in charge transfer: p∆ (Ir)< ω∆ (Ir). This effect derives from orbital relaxation as shown in Figure 5. It is present in similar magnitude for the ADC and the multi-reference methods except for the ADC(1) method whose wavefunction expansion is not flexible enough to allow for such relaxation. In Figure 8 also CASSCF and MR-CI results for the triplet states are shown, which were missing for technical

24


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0.8

(a)

0.4

0

0.

-0.4

0.8

1 3

A

1

2 A

4A 1

(b)

0.4

0

0.

e

Charge ( )

-0.4

0.8

11E

21E

3E

3

A

33A

3

E

3E

1

(c)

0.4

0

0.

-0.4

0.8

3

1 A

2

(d)

0.4

0

0.

3

97

F

CIS M R

(1 ) C

CA SSC

A D

(2 ) C A D

(3 ) C

2

wB

3

1 E

B3 LY P

-0.4

A D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 7: Primary charge shift ω∆ (Ir) on iridium (large symbols) and its decomposition into hole ωH (Ir) and electron ωE (Ir) components (small symbols connected by dotted lines) computed for the lowest excited states of Ir(C3 H4 N)3 at different levels of theory, arranged according to symmetry and multiplicity: (a) singlet A, (b) singlet E, (c) triplet A, and (d) triplet E states.

25



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reasons in Figure 7. In agreement to ADC(3) small charge shifts are observed for the lowest triplet states confirming their ligand centered character. At the TDDFT level relaxed state densities are available only after solving an additional response equation and were therefore not analyzed in this work. However, interesting modulations in charge shifts have indeed been found for other molecules when performing this step. 74 Finally, the promotion numbers p, i.e. the integrals over the attachment or detachment densities as defined in Eq. (22), are compared among the computational methods. While p is equal to one for CIS and ADC(1), 51 this value may be significantly larger for correlated methods 37,52 and can thus be seen as a compound measure counting all post-CIS effects deriving e.g. from orbital relaxation or double excitation character. The p values for the lowest excited states of Ir(C3 H4 N)3 evaluated at different levels of theory are presented in Figure 9. As discussed previously, these are quite large indicating orbital relaxation effects that go along with the charge transfer transitions. At the ADC(3) level p lies between 1.20 and 1.50 e while there is a significant enhancement for ADC(2) (above 1.5 e in most cases) indicating that an overestimation of orbital relaxation effects takes place. In agreement with previous experience, 52 this occurs without any rise in the double excitation character: Ω stays between 0.83 and 0.87 for ADC(2) similarly to ADC(3), and there is no increase in the doubly excited ADC amplitudes either. CASSCF(12/9) generally exhibits low promotion numbers indicating that the chosen active space is not large enough to cover all orbital relaxation effects (see also Refs 23,43,62). The MR-CIS values are in overall agreement but generally larger when compared to the ADC(3) case. While the strong discrepancies between the different methods are quite puzzling at first sight, the detailed wavefunction analysis performed here allows to delve deeper into the origin of this behavior. In fact, the previous analyses suggest that the crucial point lies in the description of CT, and in particular MLCT, states. The ADC(1) (or CIS) excitation energies for the CT states are too high owing to the lack of orbital relaxation in the excited state description. 75,76 By contrast, ADC(2) overestimates orbital relaxation as seen by the

26


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0.8

1

2 A 1 3

(a)

A 4A 1

0.4

0

0.

-0.4

0.8

(b)

11E 21E 3E 1

0.4

0

0.

e

Charge ( )

-0.4

0.8

(c)

3

1 A

3 A 33A 2

0.4

0

0.

-0.4

0.8

13E 23E 3E

(d)

3

0.4

0

0.

M R

-C

IS

F

CA SSC

(1 ) C A D

(2 ) C A D

C

(3 )

-0.4

A D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 8: Total charge shift (computed from the difference 1DM, large symbols) on iridium and its decomposition into hole pD (Ir) and electron pA (Ir) components (small symbols connected by dotted lines) computed for the lowest excited states of Ir(C3 H4 N)3 at different levels of theory, arranged according to symmetry and multiplicity: (a) singlet A, (b) singlet E, (c) triplet A, and (d) triplet E states.

27



1.8

1

(a)

2 A 1

3 A 1

1.5

4 A

1.2

0.9 1.8

(b)

1

1 E 1

2 E

1.5

Promotion number

1

3 E 1.2

0.9 1.8

(c)

3

1 A 3

2 A

1.5

3

3 A 1.2

0.9 1.8

3

(d)

1 E 3

2 E

1.5

3

3 E 1.2

-C

IS

F M R

CA SSC

(1 ) C A D

(2 ) C A D

C

(3 )

0.9

A D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 9: Electronic promotion numbers of the lowest excited states of Ir(C3 H4 N)3 computed at different levels of theory, arranged according to symmetry and multiplicity: (a) singlet A, (b) singlet E, (c) triplet A, and (d) triplet E states.

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fact that for most states the promotion number p exceeds 1.5. The CASSCF and MR-CIS methods show a somewhat enhanced MLCT character when compared to ADC(3). If even wavefunction based methods have problems describing the CT states, then it is even more important to pay careful attention to computations at the TDDFT level owing to its generally poor description of this phenomenon. 24,77 Interestingly, both functionals tested here perform better than ADC(2) with respect to the excitation energies as well as the amount of charge transfer. However, the errors at the B3LYP level, i.e. an underestimation by more than 0.5 eV for all MLCT states, are still significant. Even more, as the system size increases, the possibility for spurious long range charge transfer is enhanced and the performance may thus deteriorate further during scale-up (see e.g. Ref. 78). These results indicate, at the least, that more careful benchmarking is required to assess the reliability of the widely used TDDFT/B3LYP method in the case of transition metal complexes. The TDDFT/ωB97 energies agree remarkably well with the ADC(3) reference with the only difference that the CT character is somewhat overestimated. Of the methods investigated here, this level of theory appears to be most promising for the description of larger complexes. Aside from MLCT character and orbital relaxation no other particularly problematic property seems to be present. The system shows reasonable single reference character in the ground state: At the CASSCF(12/9) level the closed shell has a contribution of 83.9 % followed by ππ ∗ and dπ ∗ configurations with weights below 1.6 %. These numbers are similar to what is observed for typical heteroaromatic molecules at their eqilibrium geometries, which are generally assumed to be properly described by single reference methods. 79 Furthermore, as discussed above, no significant double excitation character was observed in any of the calculations.

Excited states of Ir(ppy)3 After the methodological considerations of the previous sections, we now proceed to analyzing the larger Ir(ppy)3 complex. Considering the bad performance of the ADC(2) method and 29



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the high computational cost of the other wavefunction based ab-initio methods, TDDFT is chosen for this task. We will first consider results at the TDDFT/ωB97 level, which according to the previous investigation is the most promising method considered here and later proceed to TDDFT/B3LYP. While the low energy excited states of Ir(C3 H4 N)3 could be readily apprehended, a significantly enhanced number of states are present for Ir(ppy)3 . Rather than discussing each of these states individually, as in the previous sections, we present an overview of their excitation energies, oscillator strengths, and charge shifts. The results at the TDDFT/ωB97 level are presented in Figure 10. In part (a) of this figure the oscillator strengths of the singlet states are plotted against the excitation energies representing an approximate absorption spectrum. However, this panel shows only the subset of bright singlet states. To represent all the excited states irrespective of their optical activity, we consider the primary charge shifts on the iridium atom ω∆ (Ir) and their decomposition into hole and particle charges in analogy to Figure 7 and plot them against the excitation energies. This is performed in Figure 10 (b) for the singlets and (c) for the triplets. At first sight, the complexity of the involved states becomes apparent. Considering, for example, the range below 6.0 eV, there are already 30 singlet and 51 triplet states of varying oscillator strengths and charge transfer character present. In a similar sense to Ir(C3 H4 N)3 it is observed that triplet states start at significantly lower energies when compared to their singlet counterparts. The lowest triplet state (13 A), which is located at 2.81 eV is again of almost entirely ligand centered character with virtually no contribution of the Ir atom to the excited state [ωH (Ir)=0.06, ωE (Ir)=0.02]. The hole and particle densities of this state are shown in the left part of Figure 11. The lowest singlet states follow only at 4.00 eV (11 E) and 4.02 eV (21 A). These possess mixed LC/MLCT character with ωH (Ir)=0.38 and ωE (Ir)=-0.04. With oscillator strengths of 0.07 per E state and 0.12 for the A state, these certainly produce a strong band in the UV absorption spectrum. The hole and particle densities of these states are presented in Figure 11 (middle, 21 A). The hole density shows a strong contribution on the iridium atom

30


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in the shape of a d-orbital and only some smaller participation of the ligands. However, it should be pointed out that this picture is somewhat misleading as the Ir atom does in fact contribute only 38% of the hole density while the remaining part is distributed over the ligands. This skew is probably caused by the enhanced size of the 5d-orbitals on iridium owing to their different radial structure. At higher excitation energies also states with enhanced electron components on the iridium atom ωE (Ir) are found in Figure 10. The state where this is most pronounced is the 101 A state located at 5.93 eV showing ωH (Ir)=0.67 and ωE (Ir)=-0.39. The hole and particle densities for this state are presented in Figure 11 (right) showing its metal centered d→d∗ character. For comparison, the TDDFT/B3LYP results are presented in Figure 12. These show marked differences when compared to the ωB97 case, following the trends found for Ir(C3 H4 N)3 , previously. In analogy to Table 2, the excitation energy of the first singlet excited state (21 A) is significantly lowered to 2.93 eV as opposed to 4.02 eV at the ωB97 level. The lowest triplet state (13 A) is positioned at 2.68 eV, which is similar to the ωB97 case. However, there is a strong increase in charge transfer for this state with an ωH (Ir) value of 0.34 while it is only 0.06 at the ωB97 level showing an even more pronounced difference when compared to Ir(C3 H4 N)3 (Figure 7). Comparing Figure 12 as a whole to Figure 10 illuminates the dramatically different results obtained with the two functionals. For example, the density of low energy excited states is significantly enhanced at the B3LYP level. Considering, first, an excitation energy of 4.0 eV, there are 3 singlet and 9 triplet states at or below this threshold at the ωB97 level while these numbers are 18 and 24 for B3LYP. Using a threshold of 5.0 eV these numbers are 12/21 in the ωB97 case and 50/60 for B3LYP. Furthermore, the amount of charge transfer is strongly varying showing that there are not only shifts in energy but that also different kinds of states appear. For example, there is almost no involvement of the Ir atom in the particle density at the B3LYP level for any of the states shown and ωE (Ir) never goes beyond -0.06. Moreover, a number of singlet states with no involvement of the iridium atom at all appear between 4.0 and 4.5 eV. A full decomposition of the 1TDM

31



0.5

(a)

Osc.

Singlets

strength

0.4 0.3 0.2 0.1 0.

0 (b)

0.4

Singlets

e

Charge ( )

0.6

0.2 0.

0

-0.2 -0.4 0.6

(c)

0.4

T

riplets

e

Charge ( )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.2 0.

0

-0.2 -0.4 3

4

5

6

Excitation energy (eV) Figure 10: TDDFT/ωB97 analysis of the manifold of the excited states of Ir(ppy)3 : (a) oscillator strengths, (b) primary charge shifts on the iridium atom ω∆ (Ir) (solid circles) as well as their decomposition into hole ωH (Ir) and electron ωE (Ir) components (empty triangles) for the 30 singlet, and (c) the 51 triplet excited states located below 6.0 eV. The three states shown in Figure 11 are marked with crosses.

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13 A (2.81 eV)

21 A (4.02 eV)

101 A (5.93 eV)

Figure 11: Hole (top) and electron (bottom) densities for selected excited states of Ir(ppy)3 computed at the TDDFT/ωB97 level of theory. (Scheme 2) reveals that, except for the brightest state at 4.34 eV, these are of predominant LLCT character, reflecting again the tendency of B3LYP to yield low energy CT states. In summary, there are substantial differences between the ωB97 and B3LYP descriptions over a wide energy range. Both spectra roughly agree with the experimental result 21,39 in the sense that strong absorption starts above 4 eV and that only darker states are present below that energy. However, the assignment of the bands is quite different. The B3LYP computation suggests that the low energy absorption stems from 1 MLCT states while in the ωB97 case only triplet states of mostly local character are present in the area below 4 eV. Moreover, the experimental emission maximum of 2.52 eV 39 is rather well reproduced by both methods, i.e. the lowest triplet state at the Franck-Condon point is at a somewhat higher energy than this. But there is again some discrepancy with respect to the nature of this state. At the ωB97 level, it is almost a purely ligand centered state while at the B3LYP level it contains about 30% MLCT character. A more detailed comparison with experiment favors the B3LYP picture to some extent considering the strength of the low energy absorption 33



band and the emission band shape, 39 but as stated previously it is difficult to exclude that the agreement is just spurious.

0.5

(a)

Osc.

Singlets

strength

0.4 0.3 0.2 0.1 0.

0 (b)

0.4

Singlets

e

Charge ( )

0.6

0.2 0.

0

-0.2 -0.4 0.6

(c)

0.4

T

riplets

e

Charge ( )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.2 0.

0

-0.2 -0.4 3

4

5

Excitation energy (eV) Figure 12: TDDFT/B3LYP analysis of the manifold of the excited states of Ir(ppy)3 : (a) oscillator strengths, (b) primary charge shifts on the iridium atom ω∆ (Ir) (solid circles) as well as their decomposition into hole ωH (Ir) and electron ωE (Ir) components (empty triangles) for the 50 singlet, and (c) the 60 triplet excited states located below 5.0 eV. In summary, significant differences are observed between the ωB97 and B3LYP descriptions of the excited states of Ir(ppy)3 , similarly to other studies of related complexes. 21–23,43 While the overall spectroscopic properties are similar, huge discrepancies are observed with respect to the dark states. These differences will, for example, strongly affect spin-orbit cou34


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pling and thus phosphorescence, which is determined by the density of available states and their interactions with each other (see e.g. Refs 3,8,18). Moreover, the dynamics after charge injection or photon absorption, which determines whether photoemission, non-radiative decay or a disintegration of the complex occurs, is affected by all the states present in the energy range (see e.g. Ref 35). Theoretical evidence in this work suggests that the results using the range separated ωB97 functional are more accurate than the global hybrid B3LYP. This claim is substantiated through excellent agreement of TDDFT/ωB97 to the ADC(3) reference for Ir(C3 H4 N)3 and the knowledge of the general deficiency of global hybrid functionals for charge transfer states, 24,77 of which many are observed in the present case. By contrast, the TDDFT/B3LYP results showed somewhat better agreement with experiment. While the experiment is of course the best benchmark, it is difficult to exclude that the agreement is spurious since many factors are not included in the simulation (solvation, spinorbit coupling, vibrational broadening). It is, thus, too early to give a definite answer to the question whether B3LYP, ωB97 or none of them provide a reliable description of this system. However, the importance of this issue could be highlighted and the results suggest, at the very least, that TDDFT/B3LYP should be applied with caution.

Conclusions The aim of this work was to proceed toward a more fundamental understanding of the excited states of organic iridium complexes. For this purpose high level wavefunction based ab-initio (ADC, EOM-CCSD, CASSCF, and MR-CIS), as well as TDDFT computations were performed. A new analysis formalism was applied to extract the physically relevant information out of the many-body wavefunctions and to perform a systematic comparison of the different approaches. Due to the generally large system sizes of the iridium complexes of interest, a smaller model Ir(C3 H4 N)3 was introduced, which was amenable to highest level ab-initio calculations. The excited states of this model were analyzed in detail at the

35



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ADC(3) level of theory. An interesting observation in this study was that the lowest triplet states lay about 1.0 eV below the lowest singlet states. The former possessed predominantly ligand centered character, whereas the latter showed enhanced MLCT contributions. Moreover, the importance of orbital relaxation effects and charge shifts caused by these could be illuminated. Of methodological relevance was the observation that the primary charge shift, as measured from the 1TDM, was significantly enhanced when compared to the ”relaxed” charge shift, given by the 1DDM. The ADC(3) results were used as a benchmark for a number of other computational methods. For this purpose not only the excitation energies but also a number of other properties of the wavefunctions were compared. The most surprising outcome in this context was the poor performance of the ADC(2) method, which underestimated excitation energies by more than 1 eV and overemphasized charge transfer. This phenomenon, which had been observed for the [Ir(ppy)2 (bpy)]+ complex as well, 23 was assigned to an amplification of orbital relaxation effects. CASSCF(12/9) and MR-CIS generally provided results in agreement with ADC(3) showing that multi-reference effects do not play a significant role. To perform a comparable analysis at the TDDFT level, two density functionals were tested. The global hybrid B3LYP showed a significant underestimation of the energy of charge transfer states (by at least 0.5 eV), but interestingly this failure was not as pronounced as in the case of ADC(2). By contrast, the range separated ωB97 functional exhibited an exceptional agreement with respect to ADC(3) in terms of excitation energies and provided good estimates for the amount of charge transferred. Computations on the larger Ir(ppy)3 complex were performed at the TDDFT/ωB97 and TDDFT/B3LYP levels of theory. Both of these methods roughly reproduced the shape of the absorption spectrum and provided similar energies for the lowest triplet state. In spite of these superficial agreements the two methods showed dramatic differences when a more detailed analysis was performed. As opposed to ωB97, the B3LYP treatment gave a significantly greater number of low energy states and generally emphasized their charge

36


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transfer character. A practical consequence is that at the B3LYP level the low energy absorption is assigned to 1 MLCT states while at the ωB97 level only triplet states are present in this area. The B3LYP spectrum showed a somewhat better agreement with experiment contrary to the earlier comparison to ADC(3), which explicitly suggested that the energy of 1

MLCT states is underestimated at this level. Additional in-depth investigations are needed

to resolve this issue. On the one hand, extended treatment of solvation, spin-orbit coupling, and vibrational broadening may clarify whether this agreement is in fact only spurious. On the other hand, high level computational reference data can provide a more fundamental way of testing the description of all the excited states, even the ones that are not easily accessible by spectroscopy. Until then it is advised to use the TDDFT/B3LYP method with caution. While it may indeed possess some predictive power with respect to different correlations to experiment, this method should not be applied to more detailed or mechanistic studies without further scrutiny.

Acknowledgement We thank P.H.P. Harbach for implementing and supplying the ADC(3) code, and S.A. Bäppler and J.M. Mewes for proof-reading the manuscript. F.P. acknowledges a fellowship for postdoctoral researchers by the Alexander von Humboldt-Stiftung.

Supporting Information Available Molecular geometries and total energies of Ir(C3 H4 N)3 and Ir(ppy)3 .

This material is

available free of charge via the Internet at http://pubs.acs.org/.

Notes and References (1) Adachi, C.; Baldo, M. A.; Thompson, M. E.; Forrest, S. R. Nearly 100% Internal Phosphorescence Efficiency in an Organic Light-Emitting Device. J. Appl. Phys. 2001,

37



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

90, 5048–5051. (2) Reineke, S.; Lindner, F.; Schwartz, G.; Seidler, N.; Walzer, K.; Lussem, B.; Leo, K. White Organic Light-Emitting Diodes with Fluorescent Tube Efficiency. Nature 2009, 459, 234–238. (3) Costa, R. D.; Orti, E.; Bolink, H. J.; Monti, F.; Accorsi, G.; Armaroli, N. Luminescent Ionic Transition-Metal Complexes for Light-Emitting Electrochemical Cells. Angew. Chemie-Int. Ed. 2012, 51, 8178–8211. (4) Lowry, M. S.; Bernhard, S. Synthetically Tailored Excited States: Phosphorescent, Cyclometalated Iridium(III) Complexes and their Applications. Chem. Eur. J. 2006, 12, 7970–7977. (5) Ulbricht, C.; Beyer, B.; Friebe, C.; Winter, A.; Schubert, U. S. Recent Developments in the Application of Phosphorescent Iridium(III) Complex Systems. Adv. Mater. 2009, 21, 4418–4441. (6) Chi, Y.; Chou, P.-T. Transition-metal phosphors with cyclometalating ligands: fundamentals and applications. Chem. Soc. Rev. 2010, 39, 638–655. (7) Tian, N.; Lenkeit, D.; Pelz, S.; Fischer, L. H.; Escudero, D.; Schiewek, R.; Klink, D.; Schmitz, O. J.; González, L.; Schäferling, M. et al. Structure-Property Relationship of Red- and Green-Emitting Iridium(III) Complexes with Respect to Their Temperature and Oxygen Sensitivity. Eur. J. Inorg. Chem. 2010, 2010, 4875–4885. (8) Yersin, H.; Rausch, A. F.; Czerwieniec, R.; Hofbeck, T.; Fischer, T. The Triplet State of Organo-Transition Metal Compounds. Triplet Harvesting and Singlet Harvesting for Efficient OLEDs. Coord. Chem. Rev. 2011, 255, 2622–2652. (9) Meier, S. B.; Sarfert, W.; Junquera-Hernández, J. M.; Delgado, M.; Tordera, D.; Ort´ı, E.; Bolink, H. J.; Kessler, F.; Scopelliti, R.; Grätzel, M. et al. A Deep-Blue 38


Page 38 of 48

Page 39 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Emitting Charged Bis-cyclometallated Iridium(III) Complex for Light-Emitting Electrochemical Cells. J. Mater. Chem. C 2013, 1, 58–68. (10) Younker, J. M.; Dobbs, K. D. Correlating Experimental Photophysical Properties of Iridium(III) Complexes to SpinOrbit Coupled TDDFT Predictions. J. Phys. Chem. C 2013, 117, 25714–25723. (11) Becke, A. D. A New Mixing of Hartree-Fock and Local Density-Functional Theories. J. Chem. Phys. 1993, 98, 1372–1377. (12) Hay, P. J. Theoretical Studies of the Ground and Excited Electronic States in Cyclometalated Phenylpyridine Ir(III) Complexes Using Density Functional Theory. J. Phys. Chem. A 2002, 106, 1634–1641. (13) Polson, M.; Ravaglia, M.; Fracasso, S.; Garavelli, M.; Scandola, F. Iridium Cyclometalated Complexes with Axial Symmetry: Time-Dependent Density Functional Theory Investigation of Trans-bis-cyclometalated Complexes Containing the Tridentate Ligand 2,6-Diphenylpyridine. Inorg. Chem. 2005, 44, 1282–1289. (14) Liu, T.; Xia, B.-H.; Zhou, X.; Zhang, H.-X.; Pan, Q.-J.; Gao, J.-S. Theoretical Studies on Structures and Spectroscopic Properties of Bis-Cyclometalated Iridium Complexes. Organometallics 2007, 26, 143–149. (15) De Angelis, F.; Fantacci, S.; Evans, N.; Klein, C.; Zakeeruddin, S. M.; Moser, J. E.; Kalyanasundaram, K.; Bolink, H. J.; Gratzel, M.; Nazeeruddin, M. K. Controlling Phosphorescence Color and Quantum Yields in Cationic Iridium Complexes: a Combined Experimental and Theoretical Study. Inorg. Chem. 2007, 46, 5989–6001. (16) Terki, R.; Simoneau, L. P.; Rochefort, A. Tailoring the Photoluminescence Properties of Ionic Iridium Complexes. J. Phys. Chem. A 2009, 113, 534–541.

39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(17) Beyer, B.; Ulbricht, C.; Escudero, D.; Friebe, C.; Winter, A.; Gonzlez, L.; Schubert, U. S. Phenyl-1H-[1,2,3]triazoles as New Cyclometalating Ligands for Iridium(III) Complexes. Organometallics 2009, 28, 5478–5488. (18) Fantacci, S.; De Angelis, F. A Computational Approach to the Electronic and Optical Properties of Ru(II) and Ir(III) Polypyridyl Complexes: Applications to DSC, OLED and NLO. Coord. Chem. Rev. 2011, 255, 2704–2726. (19) Himmetoglu, B.; Marchenko, A.; Dabo, I.; Cococcioni, M. Role of Electronic Localization in the Phosphorescence of Iridium Sensitizing Dyes. J. Chem. Phys. 2012, 137, 154309. (20) Tavasli, M.; Moore, T. N.; Zheng, Y.; Bryce, M. R.; Fox, M. A.; Griffiths, G. C.; Jankus, V.; Al-Attar, H. A.; Monkman, A. P. Colour Tuning from Green to Red by Substituent Effects in Phosphorescent Tris-cyclometalated Iridium(III) Complexes of Carbazole-based Ligands: Synthetic, Photophysical, Computational and High Efficiency OLED Studies. J. Mater. Chem. 2012, 22, 6419. (21) Fine, J.; Diri, K.; Krylov, A. I.; Nemirow, C.; Lu, Z.; Wittig, C. Electronic Structure of Tris(2-phenylpyridine)iridium: Electronically Excited and Ionized States. Mol. Phys. 2012, 110, 1849–1862. (22) Wang, J.; Bai, F.-Q.; Xia, B.-H.; Zhang, H.-X.; Cui, T. Accurate Simulation of Geometry, Singlet-Singlet and Triplet-Singlet Excitation of Cyclometalated Iridium(III) Complex. J. Mol. Model. 2014, 20, 2108. (23) Bokarev, S. I.; Bokareva, O. S.; Kuhn, O. Electronic Excitation Spectrum of the Photosensitizer [Ir(ppy)2(bpy)]+. J. Chem. Phys. 2012, 136, 214305. (24) Dreuw, A.; Head-Gordon, M. Failure of Time-Dependent Density Functional Theory for Long-Range Charge-Transfer Excited States: The Zincbacteriochlorin-Bacteriochlorin

40


Page 40 of 48

Page 41 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


and Bacteriochlorophyll-Spheroidene Complexes. J. Am. Chem. Soc. 2004, 126, 4007– 4016. (25) Kasha, M. Characterization of Electronic Transitions in Complex Molecules. Discuss. Faraday Soc. 1950, 9, 14–19. (26) Trofimov, A. B.; Schirmer, J. An Efficient Polarization Propagator Approach to Valence Electron-Excitation Spectra. Journal of Physics B 1995, 28, 2299–2324. (27) Wormit, M.; Rehn, D. R.; Harbach, P. H.; Wenzel, J.; Krauter, C. M.; Epifanovsky, E.; Dreuw, A. Investigating Excited Electronic States Using the Algebraic Diagrammatic Construction (ADC) Approach of the Polarisation Propagator. Mol. Phys. 2014, 112, 774–784. (28) Harbach, P. H.; Wormit, M.; Dreuw, A. The Third-Order Algebraic Diagrammatic Construction Method (ADC(3)) for the Polarization Propagator for Closed-Shell Molecules: Efficient Implementation and Benchmarking. J. Chem. Phys. 2014, 141, 064113. (29) Stanton, J. F.; Bartlett, R. J. The Equation of Motion Coupled-Cluster Method. A Systematic Biorthogonal Approach to Molecular Excitation Energies, Transition Probabilities, and Excited State Properties. J. Chem. Phys. 1993, 98, 7029–7039. (30) Krylov, A. I. Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhiker’s Guide to Fock Space. Annu. Rev. Phys. Chem. 2008, 59, 433–462. (31) Lischka, H.; Mueller, T.; Szalay, P. G.; Shavitt, I.; Pitzer, R. M.; Shepard, R. COLUMBUS-a Program System for Advanced Multireference Theory Calculations. Wiley Interdiscip. Rev. Mol. Sci. 2011, 1, 191–199. (32) Szalay, P. G.; M¨ uller, T.; Gidofalvi, G.; Lischka, H.; Shepard, R. Multiconfiguration

41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Self-Consistent Field and Multireference Configuration Interaction Methods and Applications. Chem. Rev. 2012, 112, 108–81. (33) Chai, J.-D.; Head-Gordon, M. Systematic Optimization of Long-Range Corrected Hybrid Density Functionals. J. Chem. Phys. 2008, 128, 084106. (34) Dreuw, A.; Head-Gordon, M. Single-Reference ab Initio Methods for the Calculation of Excited States of Large Molecules. Chem. Rev. 2005, 105, 4009–4037. (35) Plasser, F.; Barbatti, M.; Aquino, A. J. A.; Lischka, H. Electronically Excited States and Photodynamics: a Continuing Challenge. Theor. Chim. Acta 2012, 131, 1073. (36) Plasser, F.; Lischka, H. Analysis of Excitonic and Charge Transfer Interactions from Quantum Chemical Calculations. J. Chem. Theory Comput. 2012, 8, 2777–2789. (37) Plasser, F.; Wormit, M.; Dreuw, A. New Tools for the Systematic Analysis and Visualization of Electronic Excitations. I. Formalism. J. Chem. Phys. 2014, 141, 024106. (38) Dedeian, K.; Djurovich, P. I.; Garces, F. O.; Carlson, G.; Watts, R. J. A New Synthetic Route to the Preparation of a Series of Strong Photoreducing Agents - Fac Tris-OrthoMetalated Complexes of Iridium(III) with Substituted 2-Phenylpyridine. Inorg. Chem. 1991, 30, 1685–1687. (39) Tamayo, A. B.; Alleyne, B. D.; Djurovich, P. I.; Lamansky, S.; Tsyba, I.; Ho, N. N.; Bau, R.; Thompson, M. E. Synthesis and Characterization of Facial and Meridional Tris-Cyclometalated Iridium(III) Complexes. J. Am. Chem. Soc. 2003, 125, 7377–87. (40) Li, X.; Minaev, B.; Agren, H.; Tian, H. Theoretical Study of Phosphorescence of Iridium Complexes with Fluorine-Substituted Phenylpyridine Ligands. Eur. J. Inorg. Chem. 2011, 2517–2524. (41) Smith, A. R. G.; Riley, M. J.; Burn, P. L.; Gentle, I. R.; Lo, S.-C.; Powell, B. J. Effects of Fluorination on Iridium(III) Complex Phosphorescence: Magnetic Circular 42


Page 42 of 48

Page 43 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Dichroism and Relativistic Time-Dependent Density Functional Theory. Inorg. Chem. 2012, 51, 2821–2831. (42) Shiozaki, T. Analytical Nuclear Gradients of Density-Fitted DiracFock Theory with a 2-Spinor Basis. J. Chem. Theory Comput. 2013, 9, 4300–4303. (43) Escudero, D.; Gonzalez, L. RASPT2/RASSCF vs Range-Separated/Hybrid DFT Methods: Assessing the Excited States of a Ru(II)bipyridyl Complex. J. Chem. Theory Comput. 2012, 8, 203–213. (44) Bäppler, S. A.; Plasser, F.; Wormit, M.; Dreuw, A. Exciton Analysis of Many-Body Wave Functions: Bridging the Gap Between the Quasiparticle and Molecular Orbital Pictures. Phys. Rev. A 2014, 90, 052521. (45) Tretiak, S.; Mukamel, S. Density Matrix Analysis and Simulation of Electronic Excitations in Conjugated and Aggregated Molecules. Chem. Rev. 2002, 102, 3171–3212. (46) Luzanov, A. V.; Zhikol, O. A. Electron Invariants and Excited State Structural Analysis for Electronic Transitions Within CIS, RPA, and TDDFT Models. Int. J. Quant. Chem. 2010, 110, 902–924. (47) Matsika, S.; Feng, X.; Luzanov, A. V.; Krylov, A. I. What We Can Learn from the Norms of One-Particle Density Matrices, and What We Can’t: Some Results for Interstate Properties in Model Singlet Fission Systems. J. Phys. Chem. A 2014, DOI: 10.1021/jp506090g. (48) Richard, R.; Herbert, J. Time-Dependent Density-Functional Description of the 1La State in Polycyclic Aromatic Hydrocarbons: Charge-Transfer Character in Disguise? J. Chem. Theory Comput. 2011, 7, 1296–1306. (49) Casida, M. E. In Recent Adv. density Funct. methods, Part I ; Chong, D. P., Ed.; World Scientific: Singapore, 1995; pp 155–192. 43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(50) Curchod, B. F. E.; Rothlisberger, U.; Tavernelli, I. Trajectory-Based Nonadiabatic Dynamics with Time-Dependent Density Functional Theory. ChemPhysChem 2013, 14, 1314–40. (51) Head-Gordon, M.; Grana, A.; Maurice, D.; White, C. Analysis of Electronic Transitions as the Difference of Electron Attachment and Detachment Densities. J. Phys. Chem. 1995, 99, 14261–14270. (52) Plasser, F.; Bäppler, S. A.; Wormit, M.; Dreuw, A. New Tools for the Systematic Analysis and Visualization of Electronic Excitations. II. Applications. J. Chem. Phys. 2014, 141, 024107. (53) Kannar, D.; Szalay, P. G. Benchmarking Coupled Cluster Methods on Singlet Excited States of Nucleobases. J. Mol. Model. 2014, 20, 2503. (54) Shao, Y.; Molnar, L. F.; Jung, Y.; Kussmann, J.; Ochsenfeld, C.; Brown, S. T.; Gilbert, A. T.; Slipchenko, L. V.; Levchenko, S. V.; O’Neill, D. P. et al. Advances in Methods and Algorithms in a Modern Quantum Chemistry Program Package. Phys. Chem. Chem. Phys. 2006, 8, 3172–3191. (55) Krylov, A. I.; Gill, P. M. W. Q-Chem: an Engine for Innovation. WIREs: Comp. Mol. Sci. 2013, 3, 317–326. (56) Schirmer, J.; Trofimov, A. B. Intermediate State Representation Approach to Physical Properties of Electronically Excited Molecules. J. Chem. Phys. 2004, 120, 11449. (57) Lischka, H.; Shepard, R.; Brown, F. B.; Shavitt, I. New Implementation of the Graphical Unitary-Group Approach for Multi-Reference Direct Configuration-Interaction. Int. J. Quant. Chem. 1981, S15, 91–100. (58) Shepard, R.; Shavitt, I.; Pitzer, R. M.; Comeau, D. C.; Pepper, M.; Lischka, H.; Sza-

44


Page 44 of 48

Page 45 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


lay, P. G.; Ahlrichs, R.; Brown, F. B.; Zhao, J. G. A Progress Report on the Status of the Columbus MRCI Program System. Int. J. Quant. Chem. 1988, 34, 149–165. (59) Pople, J. A.; Seeger, R.; Krishnan, R. Variational Configuration Interaction Methods and Comparison with Perturbation Theory. Int. J. Quant. Chem. 1977, 12, 149–163. (60) Lischka, H.; Shepard, R.; Shavitt, I.; Pitzer, R. M.; Dallos, M.; Mueller, T.; Szalay, P. G.; Brown, F. B.; Ahlrichs, R.; Boehm, H. J. et al. COLUMBUS: An Ab Initio Electronic Structure Program, Release 7.0, www.univie.ac.at/columbus. 2013. (61) Aquilante, F.; De Vico, L.; Ferr, N.; Ghigo, G.; Malmqvist, P.-k.; Neogrdy, P.; Pedersen, T. B.; Pitok, M.; Reiher, M.; Roos, B. O. et al. MOLCAS 7: The Next Generation. J. Comp. Chem. 2010, 31, 224–247. (62) Vancoillie, S.; Zhao, H.; Tran, V. T.; Hendrickx, M. F. A.; Pierloot, K. Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems. J. Chem. Theory Comput. 2011, 7, 3961–3977. (63) Hay, P. J.; Wadt, W. R. Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for K to Au Including the Outermost Core Orbitals. J. Chem. Phys. 1985, 82, 299–310. (64) Hariharan, P.; Pople, J. The Influence of Polarization Functions on Molecular Orbital Hydrogenation Energies. Theoretica chimica acta 1973, 28, 213–222. (65) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods Without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158–6170. (66) Schafer, A.; Huber, C.; Ahlrichs, R. Fully Optimized Contracted Gaussian-Basis Sets of Triple Zeta Valence Quality for Atoms Li to Kr. J. Chem. Phys. 1994, 100, 5829–5835.

45



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(67) Ahlrichs, R.; Bar, M.; Haser, M.; Horn, H.; Kolmel, C. Electronic-Structure Calculations on Workstation Computers - The Program System TURBOMOLE. Chem. Phys. Lett. 1989, 162, 165–169. (68) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785–789. (69) Plasser, F. TheoDORE 1.0: A Package for Theoretical Density, Orbital Relaxation, and Exciton Analysis. http://theodore-qc.sourceforge.net/. (70) Jmol: an open-source Java viewer for chemical structures in 3D. http://www.jmol. org/. (71) Humphrey, W.; Dalke, A.; Schulten, K. VMD – Visual Molecular Dynamics. Journal of Molecular Graphics 1996, 14, 33–38. (72) Stone, J. An Efficient Library for Parallel Ray Tracing and Animation. M.Sc. thesis, Computer Science Department, University of Missouri-Rolla, 1998. (73) Note that due to the C3 symmetry of the system these orbitals either come as isolated totally symmetric orbitals or in pairs of e symmetry. (74) Ronca, E.; Angeli, C.; Belpassi, L.; De Angelis, F.; Tarantelli, F.; Pastore, M. Density Relaxation in Time-Dependent Density Functional Theory: Combining Relaxed Density Natural Orbitals and Multireference Perturbation Theories for an Improved Description of Excited States. Journal of Chemical Theory and Computation 2014, 10, 4014–4024. (75) Subotnik, J. E. Communication: Configuration Interaction Singles Has a Large Systematic Bias Against Charge-Transfer States. J. Chem. Phys. 2011, 135, 071104.

46


Page 46 of 48

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(76) Liu, X.; Subotnik, J. The Variationally Orbital-Adapted Configuration Interaction Singles (VOA-CIS) Approach to Electronically Excited States. J. Chem. Theory Comput. 2014, 10, 1004–1020. (77) Peach, M. J. G.; Benfield, P.; Helgaker, T.; Tozer, D. J. Excitation Energies in Density Functional Theory: An Evaluation and a Diagnostic Test. J. Chem. Phys. 2008, 128, 044118. (78) Casida, M. E.; Huix-Rotllant, M. Progress in Time-Dependent Density-Functional Theory. Annu. Rev. Phys. Chem. 2012, 63, 287–323. (79) Plasser, F.; Aquino, A. J. A.; Lischka, H.; Nachtigallova, D. In Photoinduced Phenomena in Nucleic Acids II ; Barbatti, M., Borin, A. C., Ullrich, S., Eds.; Topics in Current Chemistry; Springer International Publishing, 2015; Vol. 356; pp 1–37.

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High-Level Ab Initio Computations of the Absorption Spectra of

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