Delocalized Character of Mixed-Valence

Article pubs.acs.org/JPCA

Predicting the Localized/Delocalized Character of Mixed-Valence Diquinone Radical Anions. Toward the Right Answer for the Right Reason Manuel Renz and Martin Kaupp* Theoretische Chemie, Institut für Chemie, Technische Universität Berlin, Sekr. C7, Straße des 17. Juni 135, 10623 Berlin, Germany S Supporting Information *

ABSTRACT: The Robin−Day class II/III mixed-valence character is established quantum-chemically for a series of mixed-valence diquinone radical anions. Particular emphasis is placed on the radical anion of tetrathiafulvalenedibenzoquinone, Q-TTF-Q, which has recently been used to evaluate constrained density functional approaches (CDFT) and new range hybrid functionals. Using a computational protocol based on hybrid functionals with 35−42% exact-exchange admixture and inclusion of solvent models during the structure optimization, it is demonstrated that a) Q-TTF-Q•−, 1, and the related diquinone radical anions 2−4 are all delocalized class III species in the gas phase and in nonpolar solvents, in contrast to previous assumptions; b) 1,4,5,8-anthracenetetraone radical anion, 2, remains class III in polar aprotic solvents, c) systems 1, 3 and 4 become class II, providing excellent agreement between computed and experimental intervalence charge-transfer excitations, thermal electron-transfer (ET) barriers and ESR hyperfine couplings. The direct conductor-like screening model for real solvents (D-COSMO-RS) allows the inclusion of specific hydrogen-bonding effects without the computational effort of molecular dynamics simulations and provides increased ET barriers, as well as a predicted incipient symmetry breaking for 2, due to hydrogen bonding in alcohol solvents. For the first time D-COSMO-RS optimizations in solvent mixtures have been evaluated. As previous computational studies of Q-TTF-Q•− neglected solvent effects during structure optimizations and obtained charge localization in gas-phase optimizations by CDFT or by exaggerated exact-exchange admixtures, they provided at best the right answer for the wrong reason.

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INTRODUCTION A mixed-valence (MV) system contains one particular element or molecular fragment in two different formal oxidation states, coupled via a bridge. The Robin−Day classification1 of MV compounds (Figure 1) distinguishes three classes, based on the strength of the electronic coupling 2Hab between the redox centers A and B via the bridge: class I, where the coupling is negligible, is the least interesting case. It is described by two diabatic potential curves for the two possible, identical, isolated distributions of the two oxidation states onto the sites A and B (see Figure 1a). Class II has non-negligible coupling which, however, is less than the Marcus reorganization energy (2Hab < λ).2−4 This leads to partly localized oxidation states with an adiabatic double-well potential along the reaction coordinate for transfer of the electron from A to B or vice versa (Figure 1b), separated by a transition state for thermal electron transfer (ET). When 2Hab > λ, the adiabatic ET barrier vanishes, and the ground-state potential has a single, delocalized structure (with symmetrical structure parameters when A = B, see Figure 1c). The importance of both transition-metal-based and organic MV compounds as genuine intramolecular models for ET processes can hardly be overrated. Moreover MV compounds are of interest in many fields of application, ranging from © 2012 American Chemical Society

molecular electronics to biocatalysis, and they have thus been in the focus of thousands of experimental and theoretical studies.5−10 Different applications may be favored by either a class II or a class III character of the system. During the past decade, substantial interest has focused furthermore on systems close to the class II/III borderline.11−23 Then the ET barrier becomes small, and different experimental techniques for characterization may come to different conclusions on the localized/delocalized character, depending on the time and energy scales involved. In these cases, solvent relaxation may become a dominant contribution to the ET process. A first-principles quantum-chemical prediction of the character of systems relatively close to the class II/III transition is highly desirable but faces substantial challenges: (i) sophisticated post-Hartree−Fock approaches which include the appropriate (dynamical and nondynamical) electron correlation contributions are typically too demanding to be applied to realistically sized MV compounds of chemical interest;24−29 (ii) Hartree−Fock theory overlocalizes and overpolarizes such compounds due to the lack of correlaReceived: August 21, 2012 Revised: October 1, 2012 Published: October 2, 2012 10629

dx.doi.org/10.1021/jp308294r | J. Phys. Chem. A 2012, 116, 10629−10637

The Journal of Physical Chemistry A

Article

Figure 1. Robin−Day classification for MV systems: (a) class I, diabatic states, no coupling, fully localized; (b) class II, adiabatic states, weak coupling, partly localized; (c) class III, adiabatic states, strong coupling, fully delocalized.

tion;28,30,31 (iii) density functional theory (DFT) methods with standard functionals tend to provide too delocalized molecular and electronic structures due to inherent self-interaction errors;17,32−34 (iv) almost all experimental studies are performed in more or less polar solvents, and these favor inherently a localized situation. A reliable quantum-chemical approach thus needs to provide a reasonable compromise between computational effort, treatment of electron correlation, minimal self-interaction errors, and it needs to account for solvent effects. We have recently provided such an approach based on nonstandard global hybrid functionals with about 35% or 42% exact-exchange admixture (BLYP35 and BMK functionals) and, initially, continuum solvent models. 35−37 Validation for MV bis-triarylamine radical cations,36,37 for neutral triarylamine-triarylmethyl radicals,37 and most recently for dinitroaryl radical anions35 have shown that the approach allows a successful Robin-Day classification, even relatively close to the class II/III borderline. We also demonstrated that it is possible to go beyond continuum solvents with moderate computational effort and include realistically specific solvent− solute interactions like hydrogen bonding in alcoholic solvents without the need for explicit molecular dynamics simulations, by using the direct conductor-like screening model for real solvents (D-COSMO-RS).35 The latter allowed us to describe the transition from class III to class II behavior of 1,4dinitrobenzene radical anion when going from aprotic to alcohol solvents, as well as the increased ET barriers for other radical anions in protic solvents. In general, the solvent environment turned out to be even more important than implied by many of the discussions in previous work. In the gas phase and in nonpolar solvents, many of the species studied are delocalized, and they exhibit symmetry breaking only in polar solvents. As most previous studies (but note refs 38−40) had neglected the solvent environment in ground-state structure optimizations, they inevitably gave delocalized structures even for cases, where experimental evidence in solution points to class II behavior.17 In very few cases, localization was obtained in gas-phase calculations when special functionals were used or when the system was forced to exhibit a class II electronic structure by adding artificially terms to the Hamiltonian (this has been termed “constrained DFT”, CDFT).27,41 In this context, the radical anion 1 of tetrathiafulvalenediquinone (QTTF-Q; Chart 1) has recently received particular attention in two computational studies.42,43 Wu and Van Voorhis used 1 to

Chart 1. Bis-Quinones Investigated

demonstrate gradient optimizations within the CDFT approach. At the gas-phase CDFT/B3LYP/6-31+G(d) level, a localized class II structure was obtained, and an ET barrier was reported. In a subsequent study, Vydrov and Scuseria used unconstrained DFT and advocated the use of the LC-ωPBE range-separated hybrid functional, as this provided a localized double-well potential in gas-phase optimization, in contrast to several other functionals studied.44 Both of these computational studies implied that Q-TTF-Q•− is a class II system not only in solution, as found experimentally, but also in the gas phase. In view of our extensive recent experience with a wide variety of cationic, neutral and anionic organic MV compounds (see 10630



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above),35−37 we doubted this interpretation. Here we report a systematic computational study of Q-TTF-Q•−, 1, and of the related three MV diquinone radical anions 2−4 (Chart 1) in the gas phase, in solution using continuum solvent models, and using the D-COSMO-RS model for both aprotic and protic solvents. A variety of exchange-correlation functionals is evaluated, including LC-ωPBE. We find that with appropriate exact-exchange admixtures, all of these systems exhibit class III character in the gas phase and become class II only in solution, depending on both solvent and nature of the bridge. A variety of ground-state properties, ET barriers, and intervalence charge-transfer (IV-CT) excitation energies and transitiondipole moments are provided, and general recommendations for the quantum-chemical treatment of these and related MV systems are given.

Arrhenius plots provided an estimated ET barrier of ca. 26 kJ mol−1.

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COMPUTATIONAL DETAILS Functionals. The previously validated BLYP35 global hybrid functional35−37 corresponds to EXC = (1 − a)(EXLSDA + ΔEXB88) + aEXHF + ECLYP

with a = 0.35. This turned out to be near optimum for groundstate properties, ET barriers, and IV-CT excitation energies of the previously studied organic MV systems.35−37 Recently we found that the BMK meta-GGA hybrid functional (a = 0.42),54 which (in contrast to BLYP35) is simultaneously accurate for general main-group thermochemistry, performs similarly well.35 In addition to these two best previous performers for organic MV systems, here we also evaluate Truhlar’s “double exactexchange variant” of the M05 functional, M05-2X (a = 0.56)55 and the range-separated hybrid LC-ωPBE.44,56−58 These two functionals have recently been applied to gas-phase calculations of 1.44 The comparisons of functionals have been done with Gaussian 09,59 with full structure optimization at each level, using the CPCM solvent model with appropriate dielectric constants,60,61 and TZVP basis sets62 for all atoms. Where necessary, both localized and delocalized starting structures were employed. Symmetry restrictions were applied to locate the symmetric, delocalized transition state for adiabatic electron transfer in case of localized minima. The reported computational thermal ET barriers, ΔH‡, include neither zero-point vibrational corrections nor thermal corrections. Note, however, that the various solvent models do include solvent thermal effects and even some entropic contributions. The underlying approximations cause uncertainties in the computed activation enthalpies of at least 5 kJ mol−1. Dipole moments are always given with respect to the center of charge (they serve to indicate localization/delocalization and are not compared to any experimental data). IV-CT excitation energies were computed at time-dependent DFT (TDDFT) level, both for minima and transition states, with the same functional, basisset, and solvent (including nonequilibrium solvation). In some cases (for the highest exact-exchange admixtures, see below), negative excitation energies at the symmetrical transition-state structures indicated instabilities of the Kohn−Sham groundstate wave functions with respect to symmetry breaking. In such cases, the “stable=opt” keyword in Gaussian 09 led to energy lowering (and thus lower ET barriers), symmetry-broken wave functions, and only positive excitation energies in the TDDFT calculations. Spin-density isosurface plots were obtained with the Molekel program.63 Subsequent calculations of HFCs at the optimized ground-state structures used IGLO-II basis sets (H (3s1p)/[5s1p], C, N, O (5s4p1d)/[9s5p1d], S (7s6p2d)/ [11s7p2s]).64 D-COSMO-RS Calculations. To go beyond the limitations of continuum solvent models, the COSMO-RS approach65 has been applied. D-COSMO-RS allows a self-consistent treatment of the solute in the potential exerted by the effective chemical potential (σ-potential) of a solvent or solvent mixture, including energy gradients needed for structure optimization, as well as linear response TDDFT calculations. The initial implementation of D-COSMO-RS by Neese, Klamt and coworkers within the ORCA code66 had been validated for gtensors of nitroxide radicals. We have recently validated35 the D-COSMO-RS implementation in a development version of

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EXPERIMENTAL INFORMATION ABOUT THE DIQUINONE RADICAL ANION TITLE SYSTEMS Tetrathiafulvalene−Diquinone Radical Anion, 1. Interest in radical anion 1 arises mainly from the fact that tetrathiafulvalene (TTF) has been used as a strong π-donor bridge in a variety of organic materials studies.45 Observation of a broad IV-CT band and the temperature dependence of the ESR spectra in solvents like DCM, ethyl acetate (EtOAc), tertbutanol (t-BuOH), and a 10:1 mixture of the latter two solvents indicate a class II situation.45 Because of solubility problems, detailed studies of the ET process by ESR were only possible in the (10:1) mixture, giving a broad IV-CT band peaking around 8000 cm−1, an ET barrier of about 30 kJ mol−1, and 1H-HFC constants of 2.47 G at 260 K and 1.23 G at 340 K.45 Estimated ET rates in DCM and ethyl acetate are somewhat higher, whereas the one in pure t-BuOH is lower. The previous theoretical work on this system will be compared to our own results further below.42−44,46,47 1,4,5,8-Anthracenetetraone Radical Anion, 2. Both the hyperfine couplings (HFCs) found in ESR (aH = 0.9 G, four signals) and the sharp bands near 6410 cm−1 (1560 nm) with vibrational fine structure in the NIR spectra indicate 2 (and its substituted analogues) to be a symmetrically delocalized classIII case in aprotic solvents like dimethylformamide (DMF) or dimethyl sulfoxide (DMSO).48,49 1,4,8,11-Pentacenetetraone Radical Anion, 3a (3b). In contrast, 3a shows a broad and nearly flat absorption band in the NIR from 600 to 2100 nm in DMF, indicating a class II system.50 Temperature dependent ESR measurements on the tetramethyl-substituted analogue 3b (and on its 6,12-dihexyl substituted analogue) in DMF and DCM indicated localization at 210 K (aH = 2.8 G), whereas at 294 K HFCs to both quinone moieties were observed (aH = 1.3 G), consistent with a fast equilibrium on the ESR time scale. ESR-based Arrhenius plots over relatively narrow temperature ranges suggested adiabatic electron-transfer (ET) barriers, ΔH‡, on the order of about 15−25 kJ mol−1 in DMF, depending on the substitution pattern and the concentration of counterions.51 In the less polar MTHF,52 3b exhibits averaged HFCs down to 160 K, suggesting significantly lower barriers or even a class III behavior. Triptycene-Bis-Quinone Radical Anion, 4. While the bridge pathway of 4 is shorter than that of 3, interruption of the delocalized π-framework is expected to also reduce electronic coupling between the two quinone moieties. Indeed, ESR in acetonitrile (MeCN) indicated localization of the spin density around 218 K and an averaged spin density at 298 K.53 10631



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Table 1. Computeda Ground State Dipole Moments μ (in Debye), ET Barriers ΔH‡ (in kJ mol−1), C−O Bond Lengths (d1, d2 in Å), Excitation Energies (E1 and 2Hab in cm−1),b and 1H-HFC Constants aH (in G)c 1

2

3a

3b

4

environment

μ

ΔH‡

d1(C−O)

d2(C−O)

E1

2Hab

aH (C1)

aH (C2/i)

gas phase DCM DMF expt gas phase DCM DMF exptf DMSO gas phase DCM DMF gas phase THF DCM DMF MeCN exptg gas phase DCM DMF MeCN expth DMSO

0.04 17.81 18.27

0.0 23.2 27.0 31 −0.1 −0.1 0.0 −f 0.0 −0.1 7.8 10.3 −0.3 7.4 8.0 10.1 10.1 16−20 0.0 12.3 14.6 14.6 25.9 14.7

1.230 1.213 1.213

1.230 1.257 1.257

1924 894 879

1.232 1.235 1.235

1.232 1.235 1.237 1.237 1.233 1.255 1.256 1.235 1.257 1.258 1.259 1.259

1.232 1.216 1.216 1.216

1.232 1.260 1.261 1.261

1.216

1.261

7459

1.3 (4) −2.5 (2) −2.5 (2) 2.5 (2)e −0.8 (4) −0.7(4) −0.7 (4) 0.9 (4) −0.7 (4) −1.4 (4) −3.5 (2) −3.6 (2) 0.8 (12) 2.5 (6) 2.5 (6) 2.6 (6) 2.6 (6) 2.9 (6) −1.0 (4) −2.2 (2) −2.3 (2) −2.3 (2) −h −2.3 (2)

1.3 (4) 1.2 (4) 1.2 (4) 1.2 (4)e −0.8(4) −0.7 (4) −0.7 (4)

1.235 1.233 1.220 1.220 1.235 1.222 1.222 1.221 1.221

1924 5730 6318 ∼8000d 8225 7495 7547 6410 7577 5135 7645 8033 4862 6776 6866 7198 7208 −g 2946 6600 7399 7394

0.01 0.02 0.31 0.39 0.03 16.12 17.12 27.94 43.76 44.02 44.93 44.92 4.61 10.94 11.30 11.30 11.32

8226 7494 7532 7561 5137 3767 3780 4862 3864 5013 3824 4078 2942 2072 2082 2257 2110

−0.7 −1.4 −1.4 −1.4 0.8 0.8 0.8 0.7 0.7 1.3 −1.0 −0.9 −0.9 −0.9 1.1 −0.9

(4) (4) (4) (4) (12) (12) (12) (12) (12) (12) (4) (4) (4) (4) (4) (4)

a

Gaussian 09 BLYP35/TZVP/CPCM results. bExcitation energies obtained at symmetry broken (E1) and at C2/i-symmetric structure (2Hab). Further excitation energies and corresponding transition dipole moments are given in Tables S1 and S2, Supporting Information. cHFCs of the quinone units. Other HFCs are provided in Tables S1−S5, Supporting Information. Number of signals is given in parentheses. dBroad IV-CT band at 7700 cm−1.45 eExperimental HFCs for EtOAc/t-BuOH (10:1) solvent mixture: 2.5 G (2 protons) at 260 K, 1.2 G (4 protons) at 340 K.45 fClass III case, IV-CT energy and aH in DMF from ref 48. gBroad IV-CT band from 4750 to 16700 cm−1 in DMF for 3a.50 ET barriers and aH from temperature-dependent ESR measurments in DMF51 and MTHF52 for 3b and in DCM for analogous diquinones.51 hData in MeCN from temperature-dependent ESR measurements, pentet with aH = 1.1 G obtained at 298 K, triplet when cooling down to 218 K.53

TURBOMOLE 6.367 for the question of the class II/III behavior of dinitroaromatic radical anions. The influence of hydrogen bonding in alcohol solvents on symmetry breaking and ET barriers could be described faithfully using D-COSMORS, in contrast to continuum solvent models. We will thus also evaluate D-COSMO-RS for the title systems, including for the first time a solvent mixture (see below). All D-COSMO-RS and COSMO68 results are reported at the BLYP35/TZVP level with a local development version of TURBOMOLE 6.3. For the relevant solvents, BP86/TZVP pregenerated σ-potentials have been obtained from the COSMOtherm program package69,70 and have been used for structure optimizations and TDDFT calculations. The following dielectric constants have been used in COSMO calculations (both those presented and those underlying the generation of the σ-potentials): ethyl acetate (EtOAc, ε = 5.9867), dichloromethane (DCM, ε = 8.93), t-BuOH (ε = 12.47), N,N-dimethylformamide (DMF, ε = 36.7), acetonitrile (MeCN, ε = 36.64) and a weighted value for the mixture of EtOAc with t-BuOH (10:1, ε = 6.576).

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1−4 also seem presently out of reach. We note, however, that for smaller dinitroaryl radical anions, where post-HF gas-phase calculations had been attempted, the very small computed ET barriers most likely would disappear upon convergence to the basis-set limit.35 Together with the excellent performance of the BLYP35/TZVP/CPCM approach in reproducing the characteristics of those dinitroaryl35 and the present diquinone radical anion systems (see below) in solution, we regard this result as reliable. A delocalized situation at BLYP35/TZVP level pertains also to all systems in a nonpolar solvent like hexane or ethyl acetate (see also D-COSMO-RS results below). The underlying assumptions of previous computational studies that 1 is a class II case also in the gas phase (see discussion below), thus seem clearly unwarranted. Only in polar solvents, 1, 3a, 3b, and 4 localize to a class II situation, whereas 2 remains delocalized in all aprotic solvents investigated (see below for a discussion of protic solvents). These conclusions are supported (Table 1) by the computed ground-state dipole moments, by the adiabatic ET barriers ΔH‡, and by the C−O bond lengths of both quinone units. Experimentally, the ESR-based ET barriers have been obtained in different solvents for 1, 3a, 3b, and 4. Starting with the DCM-based value for 1 (see below for a more detailed discussion of the solvent dependence for 1), we see that the roughly estimated ESR-value in this solvent is underestimated only slightly at the BLYP35/TZVP/CPCM level. The ESRbased ET barrier of 3b in DMF is also underestimated somewhat, and the ESR-based barrier of 4 in MeCN is also somewhat higher than the computed BLYP35/TZVP/CPCM

RESULTS

General Evaluation of BLYP35/TZVP/CPCM Results. Gas phase calculations at the BLYP35/TZVP level give delocalized class III situations for all systems 1−4 of the present study. No experimental gas-phase results are available, and converged high-level post-Hartree−Fock calculations for 10632



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Figure 2. Spin-density distributions (isovalue ±0.001 au) of 1, calculated with BLYP35/TZVP in the gas phase (left) and in DCM (right).

Table 2. Dependence of Computeda Ground State Dipole Moments μ (in Debye), ET barriers ΔH‡ (in kJ mol−1), C−O Bond Lengths (d1, d2 in Å), Excitation Energies (E1 and 2Hab in cm−1),b and 1H-HFC Constants aH (in G)c for 1 on ExchangeCorrelation Functional functional BMK

M05-2X

LC-ωPBE

expt45

environment

μ

ΔH‡

d1(C−O)

d2(C−O)

E1

2Hab

gas phase DCM DMF gas phase DCM DMF gas phase DCM DMF EtOAc−t-BuOH (10:1)

0.52 17.98 18.44 14.13 17.86 18.31 14.67 18.13 18.55

0.5 29.0 32.9 12.0 43.3 47.2 48.1 81.6 85.6 31

1.225 1.209 1.209 1.211 1.211 1.212 1.208 1.209 1.209

1.226 1.251 1.251 1.251 1.257 1.258 1.248 1.254 1.254

1674 8189 8791 3447 8393 8958 5267 3642 4179 ∼8000

1657 −491d −521d 657 −1855d −1868d −2931d −3650d −3663d

aH (C1)

aH (C2)

−1.4 −2.8 −2.8 −2.5 −2.5 −2.5 −2.3 −2.3 −2.3 2.5

−1.4 −1.3 −1.3 −1.3 −1.2 −1.2 −1.1 −0.9 −0.9 1.2

(4) (2) (2) (2) (2) (2) (2) (2) (2) (2)

(4) (4) (4) (4) (4) (4) (4) (4) (4) (4)

a Gaussian 09 results. Cf. Table 1 for BLYP35 data. bExcitation energies are obtained at symmetry-broken minimum (E1) and at symmetric transition state structure (2Hab). c1H-HFC constants aH for localized minima and symmetrical transition-state structures. dBecause of ground-state instabilities, negative excitation energies have been obtained, cf. text.

1530 cm−1) for the radical anion in DMSO.48 This is well reproduced at the BLYP35/TZVP/CPCM level (neutral 1676 cm−1, anion 1571 cm−1) after uniform frequency scaling by 0.95 (as done previously for dinitroaromatic radical anions).35 This holds also for 4, where the experimental IR carbonyl stretching frequencies in DMSO (1650 cm−1 for the quinone side, 1505 cm−1 for the semiquinone side)50 are well reproduced by the scaled computed frequencies (1652 cm−1 for the quinone side, 1466 cm−1 for the semiquinone side). Table 1 shows also computed and experimental 1H-hyperfine couplings (HFC, aH) on the quinone moieties (see Tables S1− 2, Supporting Information for other 1H-HFCs). Calculations for the localized class II minima of 1 give values near 2.5 G on the localized semiquinone moiety and negligible HFCs on the quinone side, consistent with experimental data at low temperatures.45 The symmetrical transition-state structures give half this value on both sides, consistent with full spin delocalization and (averaged) experimental ESR data at higher temperatures.45 Consistent with these HFCs, Figure 2 shows how the computed spin density distribution in 1 is delocalized in the gas-phase optimization but localized after optimization in DCM. Delocalized 2 gives four signals both computationally and experimentally.48 For 3b, HFCs of 1.34 G to all 12 methyl protons have been measured at room temperature in DMF,51 due to a fast equilibrium between two localized minima. Our delocalized gas-phase calculations or the calculations at the symmetrical transition state structures are consistent with these values (after taking into account rotational averaging of the methyl groups). At lower temperatures, solubility problems hamper the measurements, but the situation is consistent with HFCs of ca. 2.9 G to six methyl protons and of 0.64 G to two

value (Table 1). We see thus a general moderate underestimate of the barriers at this level. Previous work, e.g., on dinitroaryl radical anions, suggested rather good agreement.35 In any case, the class II/III behavior of all four systems in polar aprotic solvents is well characterized. Most notably, we do not need any artificial CDFT constraints to simulate the class II character of 1, 3a, 3b, and 4 in such environments. This provides additional support to our conclusion (see above) that all systems are delocalized class III cases in the gas phase or in nonpolar solvents. The computed barrier for 3b in the moderately polar solvent THF is only about 7 kJ mol−1, indicating a class II/III borderline case. This is consistent with the fact that ESR measurements in methyltetrahydrofuran or in dimethoxyethane (with essentially the same dielectric constant as THF) did not give evidence for any localization down to 160 K.52 In the absence of further spectroscopic (e.g., UV/vis or NIR) data, no clearer classification is possible in these less polar solvents. Because of solubility problems for the present diquinone radical anions, almost no reliable UV/vis or NIR studies of the ET parameters are available. For 1, NIR spectra have been obtained, which give a broad IV-CT band near ca. 8000 cm−1,45 but even here solubility problems prevented a more quantitative study.71 The computed IV-CT excitation energy in DCM is about 2000 cm−1 too low, which is a bit more than expected at the BLYP35/TZVP/CPCM level for a class II system.35 A sharp IV-CT band at 6410 cm−1 has been found for the class III case 2 in DMF.48 This is overestimated by ca. 1100 cm−1 in the calculations, consistent with previous results for class III systems.35−37 IR spectra for 2 exhibit a decrease of the carbonyl stretch frequency from 1672 cm−1 for the neutral compound to 1581 cm−1 (with two more bands at 1540 and 10633



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Table 3. Comparison of COSMO and D-COSMO-RS Results (BLYP35/TZVP)a for Ground-State Dipole Moments μ (in Debye), ET Barriers ΔH‡ (in kJ mol−1), C−O Bond Lengths (d1, d2 in Å), and Excitation Energies (E1 and 2Hab in cm−1)a of 1, 2, and 3a environment 1

EtOAc EtOAc − t-BuOH (10:1) expt45 DCM t-BuOH

2

t-BuOH

3a

DCM expt51 DMF expt51

μ

ΔH‡

d1(C−O)

d2(C−O)

E1

2Hab

17.55 (17.37) 17.91 (17.47)

20.2 (20.9) 23.9 (21.8) 30.8 24.0 (24.3) 37.2 (26.4) 3.0 (0.0) 7.7 (8.3) 16.7b 12.3 (12.1) 17.6b

1.213 (1.213) 1.213 (1.213)

1.256 (1.257) 1.258 (1.257)

1379 (1377) 1381 (1378)

1.214 (1.214) 1.213 (1.214) 1.222 (1.232) 1.223 (1.222)

1.259 (1.258) 1.264 (1.258) 1.257 (1.242) 1.258 (1.256)

5381 (5061) 5626 (5200) ∼8000 5111 (5585) 6517 (5867) 10 314 (8125) 8049 (7924)

1.221 (1.222)

1.254 (1.258)

8124 (8511)

4463 (4551)

17.62 (17.75) 18.82 (18.01) 6.78 (2.00) 15.59 (15.94) 17.56 (17.37)

1385 (1380) 1389 (1381 8115 (8006) 4540 (4489

a Values in parentheses obtained by BLYP35/TZVP/COSMO. Further calculated data is available in Tables S7−11, Supporting Information. bExp. ET barriers for the 2,3,9,10-tetramethyl-6,12-dihexyl substituted analogue in DCM and DMF respectively.51

in solution. Indeed, when adding a polar solvent model, completely unrealistic barriers of more than 80 kJ mol−1 are obtained (Table 2). This indicates far too high exact-exchange admixture and dramatic overlocalization. Similar behavior has been found in our study on dinitroaryl radical anions when using the LC-BLYP and ωB97-X range hybrids, whereas the CAM-B3LYP range hybrid was much closer to the BLYP35 and BMK results.35 It appears that the introduction of full exact exchange at long-range in some range hybrids may lead to an overlocalization. We maintain that 1 is most likely a class III system in the gas phase. Scuseria and Vydrov also noted Kohn− Sham wave function stability problems (see above) at the symmetrical transition state, another indication that too much Hartree−Fock exchange is involved.44 In that study, M05-2X was also found to give (erroneous) symmetry breaking in the gas phase, but less instability. This is in line with our recent results for dinitroaryl radical anions, where functionals like M05-2X or M06-2X also overlocalized, but not to the same extent as LC-BLYP or ωB97-X.35 Indeed, our own calculations at M05-2X level confirm this notion: symmetry breaking occurs already in the gas phase, and the ET barriers in solution are overestimated appreciably, albeit not as much as for LC-ωPBE. Triplet instabilities at the symmetrical transition-state structures are more pronounced than for BMK (see above), but less than for LC-ωPBE, in contrast to BLYP35, where we do not find such instabilities. BMK, LC-ωPBE, and M05-2X results for 2−4 are provided in Tables S3−6 in Supporting Information. We only note here that 2 exhibits (possibly erroneous) incipient symmetry breaking in DMF or DMSO, when using the BMK functional. The barriers remain very small, however. In contrast, M05-2X or LC-ωPBE give a class II structure and significant barriers even in DCM. The latter functional provides also vastly overestimated barriers for 3a, 3b, and 4. In some cases, unphysically large S2 expectation values (>0.9) at the symmetrical transition-state structures are observed. Together, these findings corroborate the too large exact-exchange admixture and concomitant overpolarization and overlocaliza-

anthracene protons on one side. The calculations (Table 1) again agree (after rotational averaging). Our computations are also consistent with experimental results for 4 (Table 1): Delocalized structures reproduce the room temperature ESR spectra with HFCs of 1.1 G to four quinone protons,53 whereas localized structures produce HFCs to two protons (2.3 G; these could not be observed experimentally, but localization is consistent with the line broadening observed at lower temperatures). Evaluation of BMK, M05-2X, and LC-ωPBE Density Functionals. In our previous study on MV dinitroaryl radical anions, the BMK meta-GGA global hybrid with 42% Exx admixture exhibited comparable accuracy as BLYP35. It has the advantage over the latter of good performance for general main-group thermochemistry. Therefore, BMK/TZVP results for 1 in some solvents are compared to those of two other functionals in Table 2. We note in passing that gas-phase BMK results agree with the BLYP35 data in predicting class III behavior for all four systems (see Table 2 and Tables S3−6, Supporting Information). In agreement with the BLYP35 results, BMK gives localization for 1 in all polar aprotic solvents. Structural symmetry breaking is slightly more pronounced, the computed BMK and BLYP35 dipole moments for a given solvent dielectric constant are very similar. The ET barriers are a few kJ mol−1 higher than the BLYP35 results, providing slightly better agreement with experiment. However, due to the higher exact-exchange admixture (42%), UKS instabilities at some transition-state structures are found, leading to negative excitation energies in subsequent TDDFT calculations. Wave function optimization using the “stable=opt” in Gaussian provides a symmetry-broken spin density and a dipole moment of 17 D, in spite of the symmetrical nuclear framework. We also evaluated the LC-ωPBE range hybrid, as Scuseria and Vydrov advocated its use and found a class II localized structure for 1 in the gas phase.44 This is confirmed by our LCωPBE results reported in Table 2: the computed ET barrier of ca. 48 kJ mol−1 is almost twice as large as the ESR-based barrier 10634



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treatment was possible here due to the limited solubility). Quantitatively, the increase compared to DCM or 10:1 EtOAc/ t-BuOH appears to be too large (for example, the experimental rate in t-BuOH is only by a factor 10 lower than that in DCM).71 Interestingly, the D-COSMO-RS data (not the COSMO data!) for 2 in t-BuOH suggest incipient symmetry breaking for this radical anion, which is clearly class III in all aprotic solvents (see above). Experimental studies of 2 in alcohol solvents would thus be very interesting. Note that for some dinitroaryl radical anions, where more data in alcohols are available, the increased symmetry breaking caused by hydrogen bonding has been faithfully reproduced by D-COSMO-RS (in contrast to COSMO).35

tion with M05-2X, and particularly with LC-ωPBE. We note in passing that HFCs exhibit only small direct dependence on the functional (Table 2) and are influenced mainly by the localized or delocalized structure. As mentioned in the introduction, Wu and Van Voorhis used the CDFT approach to constrain the Fock matrix to a localized state during B3LYP/6-31+G(d) gas-phase structure optimizations. This gave an ET barrier of about 14 kJ mol−1,42 which is lower than the experimental value in polar solvents (cf. above). While such a CDFT gas-phase calculation provides the desired class II behavior, in view of our gas-phase results it has to be considered at best the right answer for the wrong reason, akin to the LC-ωPBE and M05-2X gas-phase results above. We also note that the CDFT calculations of ref 42 gave an overall more nonplanar structure than, e.g., our BLYP35/TZVP/CPCM result in MeCN: the BLYP35/TZVP/MeCN dihedral angle between the two quinone planes is 170°, similar to the unconstrained B3LYP results of ref 42, in spite of the class II structure. The CDFT dihedral angle was ca. 160°, which is thus probably an artifact of the constraints applied. An additional advantage of the present unconstrained calculations is that the transition state structure is the truly appropriate one for the adiabatic ET, without further approximation. We emphasize, however, the usefulness of the CDFT approach in creating and studying electronic situations that may be difficult to reach without constraints, or to create diabatic potential curves. CDFT wave functions may furthermore be useful as starting point (initial guess) to converge to solutions that may be difficult to obtain otherwise, e.g., for broken-symmetry wave functions of some antiferromagnetically coupled systems. D-COSMO-RS Results. Because of solubility problems, the more reliable ESR and UV/vis data for 1 had been obtained in a 10:1 mixture of EtOAc and t-BuOH45 (in particular, temperature-dependent ESR spectra, see above). The description of solvation by such a mixture of aprotic and protic solvents is clearly outside the range of applicability of continuum solvent models. Even solvation by a pure alcohol solvent could not be described. As we recently showed that the D-COSMO-RS approach gave a remarkably good simulation of the effects of hydrogen bonding in alcohol solvents on the structures and ET barriers of MV dinitroaryl radical anions,35 and as COSMO-RS applies also to solvent mixtures, we now report D-COSMO-RS results for 1 (data for 2-4 have also been obtained but will be mentioned only briefly). To our knowledge these are the first D-COSMO-RS results for a solvent mixture. Table 3 compares COSMO and D-COSMORS data (obtained at BLYP35/TZVP level with TURBOMOLE 6.3, cf. Computational Details) for 1, 2, and 3a. COSMO calculations in the 10:1 EtOAc/t-BuOH mixture used a weighted averaged dielectric constant (but differences compared to the results in the two pure solvents are almost negligible). Starting with pure aprotic solvents (EtOAc and DCM in the case of 1), we find essentially negligible differences between the COSMO and D-COSMO-RS results (only the changes in the IV-CT excitation energies are somewhat more notable). In the 10:1 solvent mixture, D-COSMO-RS gives a slightly more distorted structure, slightly larger dipole moment, and slightly larger ET barrier than the COSMO data, consistent with a small influence of the protic minority solvent component. The increased D-COSMO-RS barrier in pure tBuOH, due to the effects of hydrogen bonding, is consistent with the slower experimental rate constant (no Arrhenius

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CONCLUSIONS We have shown that our recently suggested computational protocol35−37 is able to predict the localized/delocalized character of Q-TTF-Q•− and of related mixed-valence diquinone radical anions. In contrast to other studies, the combination of a hybrid functional with 35% (BLYP35) or 42% (BMK) exact-exchange admixture in combination with a suitable solvent model can give the right answer, essentially for the right reason. Use of a continuum solvent model (CPCM, COSMO) provides a good description in aprotic solvents (e.g., for experimental ET barriers), and D-COSMORS allows us to extend the description also to protic solvents or to protic/aprotic solvent mixtures. When using an adequate modeling of the environment, global hybrids with intermediate exact-exchange admixtures such as BLYP35 or BMK describe these mixed-valence systems adequately, without the need to apply artificial constraints. We currently evaluate the predictive power of the present approach also for mixed-valence transition-metal complexes, with promising initial results.72 Ongoing developments of the computational protocol concentrate on improved functionals (e.g., local hybrids with positiondependent exact-exchange admixture) and on yet more realistic treatments of environmental effects (e.g., including solvent dynamics).

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ASSOCIATED CONTENT

S Supporting Information *

Further tabulated BLYP35/TZVP excitation energies, transition dipole moments and 1H-HFC constants as well as ground and excited state properties calculated with other functionals, and further D-COSMO-RS results. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Prof. Andreas Klamt and Dr. Michael Diedenhofen for helpful discussions on the D-COSMO-RS approach.This work has been supported by the Deutsche Forschungsgemeinschaft within the Graduiertenkolleg 1221 “Control of electronic 10635



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properties of π-conjugated molecules”. M.R. is grateful to “Bayerisches Eliteförderungsgesetz” for a Ph.D. scholarship. Further support from the DFG excellence cluster “Unifying Concepts in Catalysis” (unicat) and from TU Berlin is gratefully acknowledged.

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Delocalized Character of Mixed-Valence

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