Superior Performance of Range-Separated Hybrid Functionals for

Article pubs.acs.org/JPCA

Superior Performance of Range-Separated Hybrid Functionals for Describing σ* ← σ UV−Vis Signatures of Three-Electron Two-Center Anions Céline Dupont,† Élise Dumont,*,† and Denis Jacquemin*,‡ †

Université de Lyon, Institut de Chimie de Lyon, CNRS, Ecole normale supérieure de Lyon, 46 allée d’Italie, 69364 Lyon Cedex 07, France ‡ Chimie Et Interdisciplinarité, Synthèse, Analyse, Modélisation (CEISAM), UMR CNRS 6230, Facultés des Sciences et Techniques, BP92208, Université de Nantes, 2, rue de la Houssinière, 44322 Nantes Cedex 3, France ABSTRACT: We assess the efficiency of density functionals for the description of UV−vis signatures of temporary anions featuring a three-electron two-center bond, along a representative set of systems ranging from (pseudo)dihalides to disulfide radical anions (dimethyl disulfide and lipoate). While BH&HLYP and B3LYP have been predominantly applied to perform such simulations so far, we outline the significantly improved performance of several recently proposed functionals, including range-separated hybrids for the computation of these specific vertical transitions.

1. INTRODUCTION Eighty years ago, Pauling sketched the existence of odd-number electrons bonds.1 In the lead of his seminal idea, intensive research efforts have been deployed since 19752 to characterize the formation, structure, and subsequent reactivity of these cationic or anionic radicals (RA). They turn out to be of large interest in the framework of electron transfer acting as key intermediates in redox elementary processes in biomolecules, in electron relay phenomena3 or in the field of surface science.4 This quest of mastering the features of these species is deeply rooted in the advent of more and more finely resolved techniques, which pave the way toward the detection of these ubiquitous and transient hemibonded compounds, e.g., electrochemistry,5 X-ray,6 and electron paramagnetic resonance.7 Nevertheless, the most prominent approach is to rely on their hallmark UV−vis signatures characterized by flash photolysis and/or pulse-radiolysis.2 Gill and Radom derived an elegant orbital criterion for determining both the stability and the optimal strength of threeelectron two-center (2c−3e) species8 (i.e., an upper bound of 0.33 for the overlap S, the optimal value being 0.17). Elementary molecules such as dihalides and pseudohalides from the third or fourth rows of the periodic table verify this criterion and are therefore prone to form transient radical anions, which can be used as radio-protectants.9 To improve their efficiency, a most precise knowledge of the maximum absorption wavelength of the typical σ* ← σ transition is sought together, with an understanding of its solvent dependence. An additional motivation for an atomic-scale understanding of these odd-number bonds is the identification of disulfide RAs within proteins. © 2012 American Chemical Society

This implies an extra layer of complexity due to the strongly heterogeneous macromolecular environment that induces both geometric and electrostatic contributions significantly tuning the two-sulfur three-electron 2S−3e signature.10,11 As 2c−3e systems are intrinsically elusive, quantum mechanics (QM) calculations are often invoked to corroborate the spectral assignment.12 More ambitiously, it can be hoped to build up a predictive view of their physicochemical properties. Yet, these species are known as a pitfall for density functional theory (DFT), due to their large dynamical and sometimes also static correlation contributions, and perhaps more importantly their sizable self-interaction error.13 In that framework, the BH&HLYP functional is a popular choice,14−16 due to its large share of exact exchange that affords valuable results, partly due to error cancellation. More generally, DFT suffers from a series of well-documented deficiencies.17 For the 2c−3e systems, we have recently shown that the ground-state intercenter distances and electron affinities are not uniformly accurate whatever the selected functional.18 Reproducing or even reasonably predicting UV−vis absorption wavelengths therefore requires a primary careful methodological inspection, all the more because absorption of radical organic compounds can be more delicate to compute within a TD-DFT framework.19 An extended panel of improved density functionals have been proposed during the recent years. These functionals are designed to offer a more robust and chemically sound Received: December 9, 2011 Revised: February 20, 2012 Published: February 29, 2012 3237

dx.doi.org/10.1021/jp211875r | J. Phys. Chem. A 2012, 116, 3237−3246

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computational scheme than their “old-fashioned” counterparts. We can pinpoint pure meta-GGA,20,21 hybrid meta-GGA of the Minnesota series22 that take into account the Laplacian of the density, double-hybrids23 that explicitly depend on the virtual orbitals, and range-separated hybrids (RSH)24−30 in which the amount of exact exchange, nE, depends on the interelectronic distance. The relative performances of global hybrids (GH) and RSH have been recently assessed for a series of properties.27,31−34 More specifically, we have shown that both geometries and adiabatic electron affinities of halides and dimethyl disulfide may benefit from the use of RSH functionals.18 The scope of this study is to extend this preliminary benchmark toward the description of the hallmark σ* ← σ vertical transitions of a representative set of 2c−3e systems. To the best of our knowledge, this feat has never been performed previously. The paper is organized as follows: first, a systematic gas-phase investigation is performed on three small systems (gathered on top of Figure 1) for which very accurate experimental data are

All calculations have been performed with the Gaussian09 suite of programs,37 using default options and algorithms except when noted. Our geometry optimizations have been performed with Dunning’s augmented triple-ζ basis set (aug-cc-pVTZ) as such an extended atomic basis set is mandatory for describing anions. The basis set dependence was estimated for dichlorine and dibromine: we verified that going from the triple-ζ to the quintuple-ζ (aug-cc-pV5Z) only induces completely trifling differences (1 nm at most on maximum absorption wavelengths), and our values are therefore very close from the basis set limit. For the dianion of α-lipoic acid (lipoate, 29 atoms; see Figure 1), a mixed basis set is used to lighten the computational burden. It features the aug-cc-pVTZ basis set the sulfur atoms and the high-quality DZP++ basis set for all other centers. The latter, designed by Schaefer et al., includes both polarization and diffuse orbitals and is popular in the present framework (see for instance ref 38 for more details). Geometries of RA species in the ground state were computed at the second-order Møller−Plesset theory (MP2). Explicit treatment of electron correlation ensures an accurate description of odd-electron bonds, even at the second-order for symmetric systems; hence, MP2 has proved its reliability for a proper description of 2S−3e bonds.39,40 Vertical absorption energies were computed within the framework of linearresponse time-dependent density functional theory (TDDFT).41,42 It was verified on the three dihalides with the M062X functional that geometry optimization of the first excited state leads to a dissociation of the radical anion. Therefore, only vertical transitions could be reported in the following, and adiabatic values cannot be computed, which is consistent with the experimental transient nature of the excited species. Also, state-specific calculations, in which the electrostatic potential is generated by the excited state cannot be legitimately invoked here as the excited-state density corresponds to a dissociated state ((σ)1(σ*)2). A large panel of DFT functionals have been applied and they are listed by category in Table 1. We refer the interested reader to Gaussian’s manual for adequate literature references for these functionals. For the records, the ⟨S2⟩ values of the RAs never exceeded 0.77 for geometry optimization of the groundstate structure (either MP2 or DFT-based for data in Table 3). The spin contamination punctually increases and attains ∼0.80 for single-point excited-state calculations; in turn, this explains ⟨S2⟩ values for the σ* ← σ transition that are slightly inferior to 0.75. Finally, solvent effects were taken into account by using the polarizable continuum model (IEF-PCM) with UA0 atomic radii43 or/and by explicitely adding a few solvent molecules (to mimick aqueous environment in section 3.3).

Figure 1. Chemical structures of three dihalides and (pseudo)halides: the first three structures form the first test subset, for which accurate experimental data are available.

available.9 This first stage clearly establishes the superiority of RSH functionals as well as other specific functionals. On these small-sized systems, the basis set dependence is also assessed with the best performing functionals (section 3.1). The influence of nonprotic solvents is then discussed (section 3.2). We eventually move on to two disulfide-linked compounds, which in a biological context calls for an adequate treatment of surrounding water molecules, ring strain and the presence of charged susbtituents. Several research efforts in the 2c−3e systems have been directed toward quantifying the impact of microhydration.14,35,36 To elucidate water role in a sound prediction of maximum absorption wavelength, several geometry optimization strategies were applied and their relative merits discussed (section 3.3).

2. COMPUTATIONAL STRATEGY To benchmark DFT approaches, we select a first set of two halides and one pseudohalide (dichlorine, dibromine, and diisocyanate), for which very accurate absorptions have been measured in isolated matrix conditions.9 Independently, configuration interaction calculations including single and double excitations, CISD, as well as equation of motion coupled-cluster (EOM-CCSD, with the linear response transition densities) have been performed to first provide a T1 diagnostic to detect possible multiconfigurational character. These post Hartree− Fock excited-state calculations also allow a comparison between density and wave function-based approaches. Nevertheless, even EOM-CCSD cannot be used as benchmark reference to calibrate DFT results, due to the lack of perturbative corrections for triple excitations (T) in the implemented models.

3. RESULTS AND DISCUSSION 3.1. Gas-Phase UV−Vis Performance: (Pseudo)dihalides. Efficient in situ generation of (pseudo)dihalides radical anions (RA) and subsequent UV−vis spectral measurements have been recently reported.9 These very accurate data allows to benchmark a large panel of functionals on a molecular set of three compounds, namely dichlorine, dibromine, and diisocyanate. Gas-phase vertical absorption wavelengths were computed on the MP2/aug-cc-pVTZ geometries and compared to measured λmax. Numerical data are gathered in Table 1, with functionals classified by categories: LDA, GGA, meta-GGA, GGA hydrids, meta-GGA hybrids, RSHs, and double hybrids (DH). Nevertheless, we emphasize that the DH implementation in Gaussian09 does not include a CIS(D)-like correction, 3238


3239

exp9

DH

RSH

H m-GGA

H GGA

m-GGA

DF

SVWN5 BLYP BP86 OLYP PBE VSXC τ-HCTH TPSS M06-L O3LYP τ-HCTHhyb B3LYP X3LYP B98 PBE0 mPW1PW91 BH&HLYP M06 M05 BMK M06-2X M05-2X M06-HF LC-BLYP LC-OLYP LC-PBE LC-τ-HCTH LC-TPSS ωB97 ωB97X ωB97X-D LC-ωPBE CAM-B3LYP B2PLYP (-D) HF & post-HF methods HF CISD EOM-CCSD T1 diagnostic

nE (μ)

0−100 (0.40) 15.77−100 (0.30) 22.20−100 (0.20) 0−100 (0.40) 19−65 (0.33) 53

0 0 0 0 0 0 0 0 0 11.61 15 20 21.8 21.98 25 25 50 27 28 42 54 56 100 0−100 (0.47)

d

0.44 0.38 0.36

486 (+150) 373 (+37) 343 (+7) 0.0106 336

f 0.37 0.38 0.38 0.38 0.38 0.40 0.39 0.39 0.41 0.36 0.37 0.37 0.37 0.38 0.37 0.37 0.40 0.34 0.36 0.39 0.37 0.39 0.35 0.40 0.40 0.40 0.40 0.40 0.40 0.40 0.39 0.40 0.39 0.40

280 (−56) 292 (−45) 288 (−48) 289 (−47) 289 (−47) 298 (−38) 297 (−39) 293 (−43) 314 (−22) 295 (−41) 296 (−40) 302 (−34) 304 (−32) 298 (−38) 304 (32) 304 (−32) 333 (−3) 339 (+3) 324 (−12) 307 (−29) 308 (−28) 301 (−35) 314 (−23) 343 (+7) 345 (+9) 338 (+2) 375 (+39) 344 (+8) 341 (+5) 327 (−9) 312 (−24) 332 (−4) 317 (−19) 342 (+6)

λmax

Cl2 (2.570 Å)b ⟨S2⟩

0.73 0.77

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.76 0.75 0.75 0.75 0.74 0.74 0.75 0.75 0.75 0.75 0.75 0.75

λmax

468 (+96) 382 (+46) 361 (−11) 0.007 372

300 (−72) 311 (−61) 306 (−66) 313 (−59) 308 (−64) 314 (−58) 314 (−58) 309 (−63) 309 (−63) 317 (−55) 316 (−56) 320 (−52) 322 (−50) 321 (−51) 322 (−50) 322 (−50) 347 (−25) 356 (−16) 352 (−20) 339 (−33) 357 (−15) 345 (−27) 376 (+4) 365 (−7) 369 (−3) 361 (−11) 389 (−17) 365 (−7) 357 (−15) 347 (−25) 334 (−38) 355 (−17) 337 (−35) 355 (−17) 0.46 0.42 0.40

0.26 0.27 0.31 0.30 0.28 0.37 0.36 0.34 0.41 0.35 0.38 0.37 0.37 0.38 0.38 0.38 0.41 0.36 0.38 0.40 0.38 0.40 0.35 0.42 0.43 0.46 0.43 0.43 0.43 0.42 0.41 0.42 0.40 0.41

f

Br2 (2.808 Å)b ⟨S2⟩

0.74 0.76

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.74 0.75 0.74 0.74 0.74 0.74 0.74 0.74 0.75 0.75 0.74 0.75 0.75

λmax

661 (+168) 444 (−49) 441 (−52) 0.0195 493

438 (−55) 448 (−45) 442 (−51) 440 (−53) 441 (−52) 445 (−48) 448 (−45) 441 (−52) 443 (−50) 430 (−63) 433 (−60) 431 (−62) 431 (−62) 428 (−65) 425 (−68) 426 (−67) 440 (−53) 454 (−39) 436 (−57) 419 (−74) 426 (−67) 416 (−77) 428 (−65) 441 (−52) 443 (−50) 433 (−60) 492 (−1) 440 (−53) 444 (−49) 435 (−58) 427 (−66) 432 (−61) 430 (−63) 450 (−43)

0.27 0.28 0.26

0.15 0.15 0.16 0.16 0.16 0.18 0.17 0.17 0.20 0.19 0.19 0.20 0.20 0.20 0.21 0.22 0.26 0.21 0.22 0.23 0.23 0.24 0.24 0.29 0.29 0.29 0.29 0.29 0.28 0.27 0.24 0.28 0.24 0.26

f

(SCN)2 (2.701 Å)b ⟨S2⟩

0.77 0.75

0.75 0.76 0.76 0.76 0.76 0.77 0.77 0.77 0.76 0.77 0.77 0.77 0.78 0.78 0.78 0.78 0.81 0.77 0.78 0.78 0.77 0.78 0.78 0.78 0.79 0.79 0.79 0.79 0.78 0.78 0.78 0.79 0.78 0.82

Values given in parentheses correspond to errors with respect to experimental values. bintercenter distances are given in parentheses. cThe eight boldfaced functionals satisfy a ±20 nm criterion on the two dihalides. dThe exact exchange percentage (nE, %) and the attenuation parameters (μ) are collated for hybrids.

a

family

LSDA GGA

c

method

2c−3e systema

Table 1. Calculated Gas-Phase Maximum Absorption Wavelengths (λmax, nm), Oscillator Strength ( f), and ⟨S2⟩ for Three Different 2c−3e Bonded Systems (Cl2, Br2, and (SCN)2)

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Figure 2. Calculated UV−vis maximal absorption λmax (nm) for three (pseudo)halides and along a series of functionals. Numerical data are collected in Table 1. Reference geometries are fully optimized at the MP2/aug-cc-pVTZ level of theory.

as suggested by Grimme and co-workers44,45 and behave like global hybrids for the excited state. The exact exchange percentage (nE), included in each hybrid functional covers almost the full range of possibilities. From a methodological aspect, one notes that the predicted λmax tends to increase when going from GGA to meta-GGA or when the exact exchange percentage (attenuation parameter) in global (range-separated)

hybrids is increased. This is strictly the opposite to the trend found for π* ← π transitions.32 Comparisons with experimental values are displayed as histograms in Figure 2 for the 35 functionals (the red bold line indicates the reference values). At first sight, one clearly notes the good behavior of RSHs. For dichlorine, which has an experimentally determined peak at 336 nm, ten functionals 3240



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Table 2. Transition Characteristics of Four 2c−3e Radical Anions (Cl2, Br2, (SCN)2, and DMeDS)a gas phase gas

solvent solvent

method

λmax

f

⟨S ⟩

λmax

M06 LC-BLYP LC-OLYP LC-PBE LC-TPSS ωB97 LC-ωPBE B2PLYP BH&HLYP exp9

339 343 345 338 344 341 332 342 332 336

0.34 0.40 0.40 0.40 0.41 0.40 0.40 0.40 0.40

0.75 0.74 0.74 0.74 0.74 0.74 0.75 0.75 0.75

354 363 365 358 366 361 350 360 349


356 365 369 361 365 357 355 355 347 372

0.36 0.42 0.43 0.43 0.43 0.43 0.42 0.42 0.41

0.75 0.74 0.74 0.74 0.74 0.74 0.74 0.75 0.75

369 386 391 383 388 376 374 372 362


454 441 443 433 440 444 432 450 440 493

0.21 0.29 0.29 0.29 0.30 0.29 0.28 0.26 0.26

0.77 0.78 0.79 0.79 0.79 0.78 0.79 0.82 0.81

466 452 444 446 464 467 453 467 457


382 398 397 386 391 392 382 393 385 n/a

0.12 0.19 0.32 0.35 0.35 0.34 0.30 0.22 0.18

1.05 0.77 0.75 0.75 0.75 0.75 0.75 0.76 0.76

398 410 412 403 409 407 380 402 392

2

f

gas λmax

solvent 2

S2

346 353 355 348 355 350 341 351 341 350

0.40 0.45 0.45 0.45 0.46 0.45 0.45 0.45 0.45

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

0.75 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75

362 377 382 375 379 367 366 364 355 365

0.43 0.49 0.49 0.49 0.49 0.50 0.49 0.48 0.48

0.75 0.74 0.74 0.74 0.74 0.75 0.75 0.75 0.75

0.10 0.28 0.28 0.28 0.28 0.27 0.27 0.25 0.24

0.77 0.78 0.79 0.79 0.79 0.78 0.79 0.82 0.81

477 473 475 465 473 477 462 478 467 490

0.25 0.33 0.34 0.34 0.34 0.33 0.33 0.31 0.30

0.77 0.78 0.78 0.78 0.79 0.77 0.78 0.81 0.80

0.17 0.29 0.22 0.36 0.36 0.22 0.32 0.23 0.28

1.03 0.75 0.76 0.75 0.75 0.77 0.75 0.76 0.75

395 405 407 399 405 402 393 398 388 417−420

0.20 0.40 0.40 0.40 0.41 0.41 0.40 0.40 0.39

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

S

Dichlorine (2.570 Å; 2.553 Å) 0.39 0.75 332 0.45 0.74 335 0.45 0.74 336 0.45 0.75 330 0.45 0.74 335 0.45 0.75 332 0.45 0.75 324 0.45 0.75 334 0.44 0.75 326

0.35 0.41 0.41 0.41 0.41 0.41 0.40 0.40 0.40

0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75 0.75

Dibromine (2.808 Å; 2.792 Å) 0.43 0.75 350 0.49 0.74 357 0.49 0.74 361 0.49 0.74 353 0.49 0.74 358 0.49 0.74 350 0.48 0.75 348 0.48 0.75 349 0.48 0.75 340

0.36 0.43 0.43 0.43 0.43 0.43 0.42 0.42 0.41

(2.701 Å; 2.682 Å) 0.77 464 0.76 450 0.76 452 0.76 442 0.79 449 0.77 454 0.78 441 0.81 460 0.80 450

DMeDS (2.702 Å; 2.699 Å) 0.25 0.75 381 0.40 0.75 389 0.40 0.74 390 0.40 0.75 383 0.40 0.75 388 0.41 0.75 391 0.40 0.75 379 0.40 0.75 385 0.39 0.75 378

λmax

f

f

Diisocyanate 0.27 0.20 0.21 0.21 0.35 0.34 0.34 0.32 0.32

S

2

a

Four possible cases are considered: gas-phase geometry optimization followed by gas-phase or solvent TD-DFT simulations, or geometry optimization in the presence of solvent followed by gas or PCM absorption. Two values for the intercenter distance are reported in parentheses, in the gas phase and in presence of a solvent, respectively. For the first three cases, the solvent is dichloromethane (Figure 3), whereas for DMeDS, dimethyl sulfide is applied.

lie within a ±10 nm error range: LC-PBE, BH&HLYP, M06, LC-ωPBE, ωB97, B2PLYP(D), LC-BLYP, LC-OLYP and ωB97X by decreasing order of performance. Most “classical” functionals underestimate the λmax by ∼30 nm. When going from GGA to hybrid meta-GGA with an increasing nE and then to RSH functionals, the agreement is continuously improved. Within pure meta-GGA functionals, M06-L dominates with an error twiced compared to VSXC, τ-HCTH, and TPSS.

Results for dibromine also confirm the superiority of RSHs, though a more limited number of functionals (five) satisfy the 10 nm accuracy criterion: LC-OLYP, M06-HF, LC-BLYP, LCTPSS, and TPSS. The overlap with the ten previously selected functionals is rather limited: this stresses the inherent difficulty to treat 2c−3e systems, as a subtle tuning of the exchange− correlation functional can result in a rather pronounced fluctuation of λmax (e.g., going from LC-PBE to LC-τ-HCTH induces a ∼30 nm blue shift). 3241



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One notes that the half-and-half Becke hybrid, BH&HLYP, is still off by ∼25 nm, although remaining clearly the most popular functional in the 2c−3e line of research. Let us also note that one usually performs the DFT optimization with the same functional, which may lead to a fortuitous cancellation of errors. In contrast, this work underlines a clear-cut superiority of RSHs but no clear discrimination between Hirao’s, Scuseria’s, or Head-Gordon’s schemes (respectively LC-, LCωPBE, and ωB97 series) could be found. Contrary to the global hybrids (GH), RSH advantageously provides a physically sound description of two distant electrons. In the ground state, we have demonstrated that the extra electron occupying the antibonding σ* orbital presents a larger spatial separation with respect to “bonding” electrons (11.5 Å46), and the description of the former benefits from being treated with a larger share of exact exchange in RSH. Indeed, accurate calculations of vertical σ* ← σ transitions require a balanced description of the (σ)2(σ*)1 vs (σ)1(σ*)2 electronic configurations. In that respect, RSHs offer a very significant improvement, with the noticeable exception of CAM-B3LYP that probably suffers from its incorrect asymptotic behavior (65% instead of 100%). Diisocyanate turns out to be a more difficult system for TDDFT. LC-τ-HCTH is the only functional to be within the ±20 nm margin, but the same functional seriously overshot the Cl2 and Br2 wavelengths (by ∼40−50 nm). All the other functionals, independently of their level of sophistication, underestimate diisocyanate’s λmax by 50−70 nm. Two factors can explain this discrepancy: the nascent multiconfigurational character (T1 diagnostic ∼0.019), which tunes both the equilibrium geometry and the vertical estimates made with a single determinant scheme, and a strong dynamical electron correlation (triple nitrile bonds). In parallel to this TD-DFT analysis, it is noteworthy that the much more demanding EOM-CCSD provides absorptions in the imposed accuracy criterion for dihalogens, but not for diisocyanate, which also suffers from a 10% downshift. In addition, one verifies that TD-HF strongly overestimates λmax (by ca. 100−170 nm), whereas CIS(D) yields a +50 nm error for dihalogens. On the basis of this first gas-phase benchmark, we can select the eight best-performing functionals out of the original 35methods set, using an accuracy threshold of ±20 nm: LCBLYP, LC-OLYP, LC-PBE, LC-ωPBE, LC-TPSS, ωB97, M06, and B2PLYP(D). In the following section, we investigate the influence of nonprotic solvents on the absorption wavelengths using this panel of hybrids, as well as the seminal BH&HLYP that was included for comparative purposes. 3.2. Influence of Nonprotic Solvents. The experimentally solvatochromic shifts in dichloromethane (DCM) are +14, −7, and −3 nm for dichlorine, dibromine, and diisocynanate, respectively.9 Interestingly, these rather small increments qualitatively and quantitatively differ for these three systems. This is clearly a challenging case for the PCM-TD-DFT approach, and it is also of interest to understand this intriguing solvatochromism. As no specific interaction between the 2c−3e solute and the solvent molecules is expected, it is legitimate to resort to a continuum representation of the solvent. Numerical data are gathered in Table 2, and a graphical representation can be found in Figure 3. These wavelengths were computed on both MP2 and PCM-MP2 geometries, to assess the direct and indirect solvent effects. The intercenter distance d(2c−3e) is systematically shrunk by ∼0.02 Å in the liquid phase, suggesting that the solvent partly alleviates the electronic repulsion. For the two halides, indirect geometrical solvent effects subsequently

Figure 3. Solvatochromism of three representative 2c−3e anions: gasphase vs DCM λmax wavelengths for a set of three (pseudo)halides. The four curves correspond to the following schemes: gas-phase and PCM TD calculations using gas-phase geometries (respectively filled and empty blue squares), as well as gas-phase and PCM TD calculations on PCM geometries (empty and filled red circles). Corresponding numerical data are collated in Table 2. Experimental absorptions are denoted with black lines: solid for dimethyl sulfide, dashed for the gas phase.

result in a blue shift, in line with an orbital interpretation and a near-linear dependence of λmax as a function of d.11 Comparing the first and third columns, this decrease accounts for ca. 7 nm. But the direct (electronic) solvent effects induce a bathochromic shift of the λmax by a nearly uniform increment of ∼1520 nm (comparing, on the one hand, the first and second columns, and, on the other hand, the third and fourth columns in Table 2), hence counterbalancing the impact of bond elongation. For diisocynanate, experimental UV−vis signatures are barely affected by the dielectrico (−3 nm). The agreement between calculations and experiments is globally improved once a PCM correction is applied, though the solvatochromic effects are not systematically consistent with measurements. Indeed, the PCMTD-DFT strategy does not satisfactory render for the subtle, complex contrast between the three 2c−3e entities. Additional calculations were performed on difluorine (experimental absorption reported to be ca. 300 nm); see first histogram of Figure 4. The intercenter equilibrium distance is considerably shorter than for the three other molecules (from 2.57 to 2.80 Å). In spite of a nascent multiconfigurational character (0.022), the agreement is still very satisfying. This unexpected good result might arise from error compensation with the larger dynamic electron correlation (LC-τ-HCTH, M05-2X, M06, LC-TPSS, LC-OLYP, LC-BLYP, M05). Beyond the experimental work by Marnicek et al. focusing on (pseudo)halides, disulfide RAs deserve a special attention due to their importance and ubiquitousness in biochemistry. Indeed, among all 2c−3e systems, disulfide radical anions are the most frequent encountered. Antioxydant redox-active enzymes or smaller agents (α-lipoic acid) have been actively characterized and show potentialities in terms of redox protection, photostability, etc.47 The existence of these versatile intermediates is most often indirectly proved by pinpointing their UV−vis signature. Aliphatic entities typically absorb in the 400−450 nm range domain, with lifetimes in water in the microsecond range.48 No wavelengths in the gas phase are available: the most accurate value for the prototypical dimethyl disulfide (DMeDS) is 417− 420 nm in dimethyl sulfide, determined using subpicosecond laser spectroscopy that allows a real-time probing of such transient species.49 We first examine the PCM-TD-DFT performance 3242



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Figure 4. Complementary TD-DFT calculations on difluorine and DMeDS, for the DFT functionals listed in Table 1. For the latter, a ∼400 nm value is reported experimentally in water. Inclusion of a PCM correction improves the agreement. As found for the first subset, RSHs outperform the other functionals.

short-range stereoelectronic contributions induce a 50 nm offset between the two entities. For DMeDS, the evolution of the λmax as Jacob’s ladder of functionals is climbed or when nE is increased is in the same vein as for the three (pseudo)halides. Most functionals underestimate the λmax by more than 40 nm,

in reproducing this value (bottom of Table 2). The T1 diagnostic yields 0.013, which grounds the selection of a monodeterminantal approach for DMeDS. Quite interestingly, though diisocyanate and DMeDS present inter-sulfur distances within 0.001 Å of each other (respectively 2.701 and 2.702 Å in the gas phase), 3243



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PCM-TD-DFT//PCM-MP2 strategies has been done systematically, applying the preselected best-performing functionals. Results are gathered in Table 3, left side. They lie in an acceptable range of errors, indicating that DMeDS is probably a less difficult system to model with a QM methodology than pseudodihalides. Once again, let us highlight that the BH&HLYP is off by more than 20 nm in the presence of two water molecules. Concomitantly, one restores the good performance of LC-OLYP and LC-BLYP: this is all the more desirable as these functionals were found to offer a robust behavior for halides. We have also considered another 2s−3e system, namely the radical dianion of α-lipoic acid.51 In aqueous media, this antioxidant features a carboxylate group, such that after electron capture the predominant form is a dianion. The λmax has been reported by Sander and co-workers to be red-shifted with respect to DMeDS, with an absorption of 410 nm (in water).47 Our PCM-TD-DFT attempts to reproduce this accurate value are gathered in the third subblock of Table 3, left side. The total red shift can be divided into two contributions. First, as the optimized inter-sulfur distance is markedly increased by cyclization (0.07 Å), a red shift of ∼23 nm is expected.11 This geometrical contribution is counterbalanced by the negatively charged substituent, situated at a distance of ca. 7 Å. Comparison between DMeDS and the lipoate dianion indicates that a PCM-TDDFT//PCM-MP2 computational scheme succeeds in predicting the correct orientation, although the red shift tends to be overestimated with increments of 20 nm (one water molecule) or 30 nm (two water molecules). It is plausible that a more fine treatment of electrostatic for a dianion would help to improve the agreement. To that aim, one would need not only increase the number of water molecules but perhaps more importantly to go beyond a static 0 K picture and include dynamics contributions. Finally, beyond the TD-DFT assessment, it would be desirable to benefit from a coherence between the functional used for the excited-state calculation and for the geometry optimization. This is tested on DMeDS and lipoate anions in the r.h.s. of Table 3. It appears that the uniform use of a given density functional can severely alter the performance achieved on a reference post Hartree−Fock geometry. The capability of a given density functional to maintain a satisfactory performance obviously depends on the evaluation of the inter-sulfur distance d(2S−3e). Two case scenarios appear: (i) RSHs tend to underestimate this distance with respect to the MP2 reference value. This is clearly illustrated by the LC-PBE approach that induces a significant shrinkage of 0.08 Å, which impacts on the λmax estimate by ca. 40 nm. (ii) Conversely, M06, B2PLYP(D), and BH&HLYP predict larger d, hence leading to an overestimation of the λmax. It should also be stated that ring strain estimates are known to be sometimes poorly reproduced by DFT,52 hence constituting an additional difficulty for lipoate system. Nevertheless, several exchange−correlation functionals (e.g., LC-OLYP, LC-BLYP, and LC-ωPBE) do provide a geometry close to the reference MP2 structure for the DMeDS radical anion: they are found to behave correctly for the “exotic” chemistry of this prototypical 2S−3e system and constitute the best compromise to investigate larger structures difficult to model with electron-correlated wave function schemes. We do not investigate here the possibility of combos, i.e., using two different density functionals for geometry optimizations and excited-state single points, as successful agreements will mostly reflect an inner cancellation of errors.

and only four RSHs are within 10 nm of the reference value: LC-OLYP, LC-BLYP, LC-TPSS, ωB97. They outperform BH&HLYP, which is off by 20 nm. Again, it can be argued here that the selection of the half-and-half Becke’s functional is not to be recommended, though it can blur its own deficiency as the 2S−3e linkage optimal BH&HLYP distance is overestimated. In short, the influence of an aprotic solvent is limited, but difficult to capture numerically with a continuum model. For practical applications, this is nonproblematic as successful microhydration schemes14,35,36,50 have been developed to account for water, the (protic and polar) solvent relevant in biochemistry. 3.3. Aqueous Solvation of Dimethyl Disulfide and Lipoate Anions. To elucidate the influence of water on the DMeDS UV−vis σ* ← σ transition, several strategies of increasing complexity have been applied. The target experimental value in aqueous media is consistently claimed to be near 400 nm for a large set of systems. For the purpose of our benchmark, it can be taken as 390−410 nm.49 Preliminary results in the gas phase and with PCM inclusion, along the whole set of 35 functionals, are shown in Figure 4. The similar shape of these histograms, compared to the (pseudo)halides, confirms the superiority of RSH functionals, whereas one notices that the PCM correction globally improves the agreement toward a ∼400 nm value. Ten functionals lie in a ±10 nm error bar, namely, M06HF, B2PLYP, LC-ωPBE, LC-PBE, M06, ωB97, ωB97X, BH&HLYP, and finally LC-TPSS, as ranked by their respective performance. The above-listed functionals partly differ from the eight best-performing functionals as established on (pseudo)halides: for instance, LC-OLYP and LC-BLYP are slightly off. At this stage, it becomes important to address the possible influence of hydrogen bonds, or at least weak interactions S···H−O. This task required the inclusion of explicit water molecules, which has been done in a second step. MP2 geometries have been obtained in both the gas phase and in a continuum (PCM) environment, adding n = 0, 1, or 2 explicit water molecules to the model. Cartoon representations are given in Figure 5a,b. For DMeDS, we have reported a rapid convergence of both

Figure 5. Cartoon representations of microsolvated DMeDS and lipoate radical anions. Geometries are optimized at the MP2/aug-ccPVTZ level of theory. Key distances for d(2s−3e) and water interaction are specified (in Å).

geometries and other properties as a function of n (up to 6).36 We have also shown that hydrogen bond pairing with first-shell water molecules has a significant geometrical impact that tunes the intercenter distance and hence might indirectly affect the absorption features.36 Assessment of the microhydrated 3244



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Table 3. Maximum Absorption Wavelengths (λmax) for DMeDS and Lipoic Acid Anions, along a Set of Best-Performing Functionals, Using the MP2 Geometry or after a Uniform DFT Optimization optimized MP2 method

d

λmax

M06 LC-BLYP LC-OLYP LC-PBE LC-TPSS ωB97 LC-ωPBE B2PLYP BH&HLYP

2.686 2.686 2.686 2.686 2.686 2.686 2.686 2.686 2.686

401 415 416 407 413 412 401 406 396

M06 LC-BLYP LC-OLYP LC-PBE LC-TPSS ωB97 LC-ωPBE B2PLYP BH&HLYP exp

2.670 2.670 2.670 2.670 2.670 2.670 2.670 2.670 2.670

379 400 401 375 380 380 372 381 372 400

M06 LC-BLYP LC-OLYP LC-PBE LC-TPSS ωB97 LC-ωPBE B2PLYP BH&HLYP exp

2.758 2.758 2.758 2.758 2.758 2.758 2.758 2.758 2.758

416 432 434 425 432 429 418 420 410 410

optimized DFT f

⟨S2⟩

d

DMeDS−PCM + 1 Water Molecule 0.24 0.75 2.801 0.35 0.74 2.707 0.35 0.74 2.692 0.35 0.74 2.631 0.35 0.74 2.646 0.35 0.74 2.741 0.34 0.74 2.696 0.34 0.74 2.800 0.34 0.75 2.817 DMeDS−PCM + 2 Water Molecules 0.26 0.75 2.784 0.34 0.74 2.691 0.34 0.74 2.676 0.29 0.74 2.616 0.30 0.74 2.630 0.30 0.75 2.736 0.29 0.75 2.680 0.19 0.76 2.779 0.21 0.75 2.798 Lipoate−PCM + 2 Water Molecules 0.27 0.75 2.844 0.36 0.74 2.763 0.35 0.75 2.749 0.36 0.74 2.697 0.37 0.74 2.707 0.37 0.74 2.798 0.36 0.75 2.741 0.36 0.75 2.846 0.35 0.75 2.843

■

4. CONCLUSIONS

λmax

f

⟨S2⟩

454 423 416 377 388 442 401 462 455

0.29 0.36 0.36 0.35 0.35 0.36 0.36 0.36 0.36

0.75 0.74 0.74 0.75 0.74 0.74 0.75 0.74 0.74

441 410 403 366 377 431 391 445 440 400

0.30 0.35 0.35 0.34 0.34 0.37 0.35 0.36 0.36

0.75 0.74 0.74 0.75 0.75 0.74 0.75 0.74 0.74

447 429 422 383 394 448 402 461 446 410

0.30 0.41 0.41 0.42 0.42 0.40 0.41 0.39 0.38

0.75 0.74 0.75 0.75 0.75 0.74 0.75 0.74 0.75

AUTHOR INFORMATION

Corresponding Author

For the first time, the computational treatment of UV−vis signature with TD-DFT has been assessed for three-electron twocenter species. It turned out that extreme caution must be taken in the choice of the density functional, which clearly prevents bruteforce study relying on first-principle molecular simulations. The relatively small-size of systems investigated in the present contribution allowed a full QM approach and offered reference values beyond the experimental data. In this work, we invoke the superiority of RSH functionals and more specifically LC-OLYP, LC-BLYP, and LC-ωPBE. They appear to provide a good compromise between accuracy and computational effort for both ground and excited states. This study opens the door to the robust, rational prediction of UV/vis spectra of more complex, biological disulfide anions, whose photochemical characteristics can be strongly modulated by a heterogeneous proteinic environment, in sharp constrast with (pseudo)halides, where a subtle solvatochromism has been evidenced. Thanks to recent methodological developments in hybrid TD-DFT/MM approaches, the question of the functional dependence is probably the biggest difficulty when dealing with 2c−3e systems.

*E-mail: [email protected] (E.D.); Denis.Jacquemin@ univ-nantes.fr (D.J.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS E.D. gratefully acknowledges Dr. A. Monari (Universite de Lorraine) for helpful comments on this work. Ab initio calculations were performed using the local HPC resources of PSMN at ENS de Lyon and of GENCI (CINES/ IDRIS), project x2011075105. D.J. acknowledges the European Research Council (ERC) and the Région des Pays de la Loire for financial support in the framework of a Starting Grant (Marches - 278845) and a recrutement sur poste stratiqué, respectively.

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