Superior Performance of Range-Separated Hybrid Functionals for Describing σ* ← σ UV–Vis...
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Superior Performance of Range-Separated Hybrid Functionals for Describing σ* ← σ UV−Vis Signatures of Three-Electron Two-Center Anions Ce ́ line Dupont, † E ́ lise Dumont,* ,† and Denis Jacquemin* ,‡ † Universite ́ de Lyon, Institut de Chimie de Lyon, CNRS, Ecole normale supé rieure de Lyon, 46 alle ́ ed’Italie, 69364 Lyon Cedex 07, France ‡ Chimie Et Interdisciplinarite ́ , Synthe ̀ se, Analyse, Mode ́ lisation (CEISAM), UMR CNRS 6230, Faculte ́ s des Sciences et Techniques, BP92208, Universite ́ de Nantes, 2, rue de la Houssinie ̀ re, 44322 Nantes Cedex 3, France ABSTRACT: We assess the efficiency of density functionals for the descrip- tion 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 1975 2 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 phenomena 3 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., electro- chemistry, 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 deter- mining both the stability and the optimal strength of three- electron two-center (2c−3e) species 8 (i.e., an upper bound of 0.33 for the overlap S, the optimal value being 0.17). Elemen- tary 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 wave- length of the typical σ* ← σ transition is sought together, with an understanding of its solvent dependence. An additional moti- vation for an atomic-scale understanding of these odd-number bonds is the identification of disulfide RAs within proteins. 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 predict- ing 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 Article pubs.acs.org/JPCA © 2012 American Chemical Society 3237 dx.doi.org/10.1021/jp211875r | J. Phys. Chem. A 2012, 116, 3237−3246
Transcript of Superior Performance of Range-Separated Hybrid Functionals for Describing σ* ← σ UV–Vis...
Superior Performance of Range-Separated Hybrid Functionals forDescribing σ* ← σ UV−Vis Signatures of Three-Electron Two-CenterAnionsCeline Dupont,† Elise Dumont,*,† and Denis Jacquemin*,‡
†Universite de Lyon, Institut de Chimie de Lyon, CNRS, Ecole normale superieure de Lyon, 46 allee d’Italie, 69364 Lyon Cedex 07,France‡Chimie Et Interdisciplinarite, Synthese, Analyse, Modelisation (CEISAM), UMR CNRS 6230, Facultes des Sciences et Techniques,BP92208, Universite de Nantes, 2, rue de la Houssiniere, 44322 Nantes Cedex 3, France
ABSTRACT: We assess the efficiency of density functionals for the descrip-tion of UV−vis signatures of temporary anions featuring a three-electrontwo-center bond, along a representative set of systems ranging from (pseudo)-dihalides to disulfide radical anions (dimethyl disulfide and lipoate). WhileBH&HLYP and B3LYP have been predominantly applied to perform suchsimulations so far, we outline the significantly improved performance ofseveral recently proposed functionals, including range-separated hybrids forthe computation of these specific vertical transitions.
1. INTRODUCTIONEighty years ago, Pauling sketched the existence of odd-numberelectrons bonds.1 In the lead of his seminal idea, intensiveresearch efforts have been deployed since 19752 to characterizethe formation, structure, and subsequent reactivity of thesecationic or anionic radicals (RA). They turn out to be of largeinterest in the framework of electron transfer acting as keyintermediates in redox elementary processes in biomolecules, inelectron relay phenomena3 or in the field of surface science.4
This quest of mastering the features of these species is deeplyrooted in the advent of more and more finely resolvedtechniques, which pave the way toward the detection of theseubiquitous and transient hemibonded compounds, e.g., electro-chemistry,5 X-ray,6 and electron paramagnetic resonance.7
Nevertheless, the most prominent approach is to rely ontheir hallmark UV−vis signatures characterized by flashphotolysis and/or pulse-radiolysis.2
Gill and Radom derived an elegant orbital criterion for deter-mining both the stability and the optimal strength of three-electron two-center (2c−3e) species8 (i.e., an upper bound of0.33 for the overlap S, the optimal value being 0.17). Elemen-tary molecules such as dihalides and pseudohalides from thethird or fourth rows of the periodic table verify this criterionand are therefore prone to form transient radical anions, whichcan be used as radio-protectants.9 To improve their efficiency, amost precise knowledge of the maximum absorption wave-length of the typical σ* ← σ transition is sought together, withan understanding of its solvent dependence. An additional moti-vation for an atomic-scale understanding of these odd-numberbonds is the identification of disulfide RAs within proteins.
This implies an extra layer of complexity due to the stronglyheterogeneous macromolecular environment that induces bothgeometric and electrostatic contributions significantly tuning thetwo-sulfur three-electron 2S−3e signature.10,11
As 2c−3e systems are intrinsically elusive, quantummechanics (QM) calculations are often invoked to corroboratethe spectral assignment.12 More ambitiously, it can be hoped tobuild up a predictive view of their physicochemical properties.Yet, these species are known as a pitfall for density functionaltheory (DFT), due to their large dynamical and sometimes alsostatic correlation contributions, and perhaps more importantlytheir sizable self-interaction error.13 In that framework, theBH&HLYP functional is a popular choice,14−16 due to its largeshare of exact exchange that affords valuable results, partly dueto error cancellation. More generally, DFT suffers from a seriesof well-documented deficiencies.17 For the 2c−3e systems, wehave recently shown that the ground-state intercenter distancesand electron affinities are not uniformly accurate whatever theselected functional.18 Reproducing or even reasonably predict-ing UV−vis absorption wavelengths therefore requires aprimary careful methodological inspection, all the more becauseabsorption of radical organic compounds can be more delicateto compute within a TD-DFT framework.19
An extended panel of improved density functionals havebeen proposed during the recent years. These functionalsare designed to offer a more robust and chemically sound
Received: December 9, 2011Revised: February 20, 2012Published: February 29, 2012
Article
pubs.acs.org/JPCA
© 2012 American Chemical Society 3237 dx.doi.org/10.1021/jp211875r | J. Phys. Chem. A 2012, 116, 3237−3246
computational scheme than their “old-fashioned” counterparts.We can pinpoint pure meta-GGA,20,21 hybrid meta-GGA of theMinnesota series22 that take into account the Laplacian of thedensity, double-hybrids23 that explicitly depend on the virtualorbitals, and range-separated hybrids (RSH)24−30 in which theamount of exact exchange, nE, depends on the interelectronicdistance. The relative performances of global hybrids (GH) andRSH have been recently assessed for a series of properties.27,31−34
More specifically, we have shown that both geometries andadiabatic electron affinities of halides and dimethyl disulfide maybenefit from the use of RSH functionals.18 The scope of thisstudy is to extend this preliminary benchmark toward the descrip-tion of the hallmark σ*← σ vertical transitions of a representativeset of 2c−3e systems. To the best of our knowledge, this feat hasnever been performed previously.The paper is organized as follows: first, a systematic gas-phase
investigation is performed on three small systems (gatheredon top of Figure 1) for which very accurate experimental data are
available.9 This first stage clearly establishes the superi-ority of RSH functionals as well as other specific functionals.On these small-sized systems, the basis set dependence is alsoassessed with the best performing functionals (section 3.1). Theinfluence 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 ofsurrounding water molecules, ring strain and the presence ofcharged susbtituents. Several research efforts in the 2c−3esystems have been directed toward quantifying the impact ofmicrohydration.14,35,36 To elucidate water role in a soundprediction of maximum absorption wavelength, several geo-metry optimization strategies were applied and their relativemerits discussed (section 3.3).
2. COMPUTATIONAL STRATEGY
To benchmark DFT approaches, we select a first set of twohalides and one pseudohalide (dichlorine, dibromine, anddiisocyanate), for which very accurate absorptions have beenmeasured in isolated matrix conditions.9 Independently,configuration interaction calculations including single and doubleexcitations, CISD, as well as equation of motion coupled-cluster(EOM-CCSD, with the linear response transition densities) havebeen performed to first provide a T1 diagnostic to detectpossible multiconfigurational character. These post Hartree−Fock excited-state calculations also allow a comparison betweendensity and wave function-based approaches. Nevertheless, evenEOM-CCSD cannot be used as benchmark reference to calibrateDFT results, due to the lack of perturbative corrections for tripleexcitations (T) in the implemented models.
All calculations have been performed with the Gaussian09suite of programs,37 using default options and algorithms exceptwhen noted. Our geometry optimizations have been performedwith Dunning’s augmented triple-ζ basis set (aug-cc-pVTZ) assuch an extended atomic basis set is mandatory for describinganions. The basis set dependence was estimated for dichlorineand dibromine: we verified that going from the triple-ζ to thequintuple-ζ (aug-cc-pV5Z) only induces completely triflingdifferences (1 nm at most on maximum absorption wave-lengths), and our values are therefore very close from the basisset limit. For the dianion of α-lipoic acid (lipoate, 29 atoms; seeFigure 1), a mixed basis set is used to lighten the computationalburden. It features the aug-cc-pVTZ basis set the sulfur atomsand the high-quality DZP++ basis set for all other centers. Thelatter, designed by Schaefer et al., includes both polarizationand 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 com-
puted at the second-order Møller−Plesset theory (MP2).Explicit treatment of electron correlation ensures an accuratedescription of odd-electron bonds, even at the second-order forsymmetric systems; hence, MP2 has proved its reliability for aproper description of 2S−3e bonds.39,40 Vertical absorp-tion energies were computed within the framework of linear-response time-dependent density functional theory (TD-DFT).41,42 It was verified on the three dihalides with the M06-2X functional that geometry optimization of the first excited stateleads to a dissociation of the radical anion. Therefore, onlyvertical transitions could be reported in the following, andadiabatic values cannot be computed, which is consistent withthe experimental transient nature of the excited species. Also,state-specific calculations, in which the electrostatic potential isgenerated by the excited state cannot be legitimately invokedhere as the excited-state density corresponds to a dissociatedstate ((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 readerto Gaussian’s manual for adequate literature references forthese functionals. For the records, the ⟨S2⟩ values of the RAsnever exceeded 0.77 for geometry optimization of the ground-state structure (either MP2 or DFT-based for data in Table 3).The spin contamination punctually increases and attains ∼0.80for single-point excited-state calculations; in turn, this explains⟨S2⟩ values for the σ* ← σ transition that are slightly inferior to0.75. Finally, solvent effects were taken into account by usingthe polarizable continuum model (IEF-PCM) with UA0 atomicradii43 or/and by explicitely adding a few solvent molecules(to mimick aqueous environment in section 3.3).
3. RESULTS AND DISCUSSION3.1. Gas-Phase UV−Vis Performance: (Pseudo)-
dihalides. Efficient in situ generation of (pseudo)dihalides radicalanions (RA) and subsequent UV−vis spectral measurements havebeen recently reported.9 These very accurate data allows tobenchmark a large panel of functionals on a molecular set of threecompounds, namely dichlorine, dibromine, and diisocyanate.Gas-phase vertical absorption wavelengths were computed
on the MP2/aug-cc-pVTZ geometries and compared to mea-sured λmax. Numerical data are gathered in Table 1, with func-tionals classified by categories: LDA, GGA, meta-GGA, GGAhydrids, meta-GGA hybrids, RSHs, and double hybrids (DH).Nevertheless, we emphasize that the DH implementationin Gaussian09 does not include a CIS(D)-like correction,
Figure 1. Chemical structures of three dihalides and (pseudo)halides:the first three structures form the first test subset, for which accurateexperimental data are available.
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Table
1.CalculatedGas-Phase
Maxim
umAbsorptionWavelengths
(λmax,nm
),Oscillator
Strength
(f),and⟨S
2 ⟩forThree
Different2c−3e
Bon
dedSystem
s(C
l 2,Br 2,and
(SCN) 2)
2c−3e
system
a
method
Cl 2(2.570
Å)b
Br 2(2.808
Å)b
(SCN) 2
(2.701
Å)b
family
DFc
nE(μ)d
λ max
f⟨S
2 ⟩λ m
axf
⟨S2 ⟩
λ max
f⟨S
2 ⟩
LSDA
SVWN5
0280(−
56)
0.37
0.75
300(−
72)
0.26
0.75
438(−
55)
0.15
0.75
GGA
BLY
P0
292(−
45)
0.38
0.75
311(−
61)
0.27
0.75
448(−
45)
0.15
0.76
BP8
60
288(−
48)
0.38
0.75
306(−
66)
0.31
0.75
442(−
51)
0.16
0.76
OLY
P0
289(−
47)
0.38
0.75
313(−
59)
0.30
0.75
440(−
53)
0.16
0.76
PBE
0289(−
47)
0.38
0.75
308(−
64)
0.28
0.75
441(−
52)
0.16
0.76
m-GGA
VSX
C0
298(−
38)
0.40
0.75
314(−
58)
0.37
0.75
445(−
48)
0.18
0.77
τ-HCTH
0297(−
39)
0.39
0.75
314(−
58)
0.36
0.75
448(−
45)
0.17
0.77
TPS
S0
293(−
43)
0.39
0.75
309(−
63)
0.34
0.75
441(−
52)
0.17
0.77
M06-L
0314(−
22)
0.41
0.75
309(−
63)
0.41
0.75
443(−
50)
0.20
0.76
HGGA
O3L
YP
11.61
295(−
41)
0.36
0.75
317(−
55)
0.35
0.75
430(−
63)
0.19
0.77
τ-HCTHhyb
15296(−
40)
0.37
0.75
316(−
56)
0.38
0.75
433(−
60)
0.19
0.77
B3L
YP
20302(−
34)
0.37
0.75
320(−
52)
0.37
0.75
431(−
62)
0.20
0.77
X3L
YP
21.8
304(−
32)
0.37
0.75
322(−
50)
0.37
0.75
431(−
62)
0.20
0.78
B98
21.98
298(−
38)
0.38
0.75
321(−
51)
0.38
0.75
428(−
65)
0.20
0.78
PBE0
25304(32)
0.37
0.75
322(−
50)
0.38
0.75
425(−
68)
0.21
0.78
mPW
1PW91
25304(−
32)
0.37
0.75
322(−
50)
0.38
0.75
426(−
67)
0.22
0.78
BH&HLY
P50
333(−
3)0.40
0.75
347(−
25)
0.41
0.75
440(−
53)
0.26
0.81
Hm-GGA
M06
27339(+3)
0.34
0.75
356(−
16)
0.36
0.75
454(−
39)
0.21
0.77
M05
28324(−
12)
0.36
0.75
352(−
20)
0.38
0.75
436(−
57)
0.22
0.78
BMK
42307(−
29)
0.39
0.75
339(−
33)
0.40
0.75
419(−
74)
0.23
0.78
M06-2X
54308(−
28)
0.37
0.75
357(−
15)
0.38
0.75
426(−
67)
0.23
0.77
M05-2X
56301(−
35)
0.39
0.75
345(−
27)
0.40
0.74
416(−
77)
0.24
0.78
M06-H
F100
314(−
23)
0.35
0.76
376(+4)
0.35
0.75
428(−
65)
0.24
0.78
RSH
LC-BLY
P0−
100(0.47)
343(+7)
0.40
0.75
365(−
7)0.42
0.74
441(−
52)
0.29
0.78
LC-O
LYP
345(+9)
0.40
0.75
369(−
3)0.43
0.74
443(−
50)
0.29
0.79
LC-PBE
338(+2)
0.40
0.75
361(−
11)
0.46
0.74
433(−
60)
0.29
0.79
LC-τ-H
CTH
375(+39)
0.40
0.74
389(−
17)
0.43
0.74
492(−
1)0.29
0.79
LC-TPSS
344(+8)
0.40
0.74
365(−
7)0.43
0.74
440(−
53)
0.29
0.79
ωB97
0−100(0.40)
341(+5)
0.40
0.75
357(−
15)
0.43
0.74
444(−
49)
0.28
0.78
ωB97X
15.77−
100(0.30)
327(−
9)0.40
0.75
347(−
25)
0.42
0.75
435(−
58)
0.27
0.78
ωB97X-D
22.20−
100(0.20)
312(−
24)
0.39
0.75
334(−
38)
0.41
0.75
427(−
66)
0.24
0.78
LC-ω
PBE
0−100(0.40)
332(−
4)0.40
0.75
355(−
17)
0.42
0.74
432(−
61)
0.28
0.79
CAM-B3L
YP
19−65
(0.33)
317(−
19)
0.39
0.75
337(−
35)
0.40
0.75
430(−
63)
0.24
0.78
DH
B2P
LYP(-D)
53342(+6)
0.40
0.75
355(−
17)
0.41
0.75
450(−
43)
0.26
0.82
HF&
post-H
Fmethods
HF
486(+150)
0.44
0.73
468(+96)
0.46
0.74
661(+168)
0.27
0.77
CISD
373(+37)
0.38
0.77
382(+46)
0.42
0.76
444(−
49)
0.28
0.75
EOM-CCSD
343(+7)
0.36
361(−
11)
0.40
441(−
52)
0.26
T1diagnostic
0.0106
0.007
0.0195
exp9
336
372
493
aValuesgivenin
parenthesescorrespond
toerrorswith
respectto
experim
entalvalues.bintercenterdistancesaregivenin
parentheses.c The
eightboldfacedfunctio
nalssatisfya±20
nmcriterio
non
the
twodihalides.dThe
exactexchange
percentage
(nE ,%)andtheattenuationparameters(μ)arecollatedforhybrids.
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as suggested by Grimme and co-workers44,45 and behave likeglobal hybrids for the excited state. The exact exchangepercentage (nE), included in each hybrid functional coversalmost the full range of possibilities. From a methodologicalaspect, one notes that the predicted λmax tends to increase whengoing from GGA to meta-GGA or when the exact exchangepercentage (attenuation parameter) in global (range-separated)
hybrids is increased. This is strictly the opposite to the trendfound for π* ← π transitions.32
Comparisons with experimental values are displayed ashistograms in Figure 2 for the 35 functionals (the red boldline indicates the reference values). At first sight, one clearlynotes the good behavior of RSHs. For dichlorine, which hasan experimentally determined peak at 336 nm, ten functionals
Figure 2. Calculated UV−vis maximal absorption λmax (nm) for three (pseudo)halides and along a series of functionals. Numerical data are collectedin Table 1. Reference geometries are fully optimized at the MP2/aug-cc-pVTZ level of theory.
The Journal of Physical Chemistry A Article
dx.doi.org/10.1021/jp211875r | J. Phys. Chem. A 2012, 116, 3237−32463240
lie within a ±10 nm error range: LC-PBE, BH&HLYP, M06,LC-ωPBE, ωB97, B2PLYP(D), LC-BLYP, LC-OLYP andωB97X by decreasing order of performance. Most “classical”functionals underestimate the λmax by ∼30 nm. Whengoing from GGA to hybrid meta-GGA with an increasingnE and then to RSH functionals, the agreement is continu-ously improved. Within pure meta-GGA functionals, M06-Ldominates 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 the10 nm accuracy criterion: LC-OLYP, M06-HF, LC-BLYP, LC-TPSS, and TPSS. The overlap with the ten previously selectedfunctionals is rather limited: this stresses the inherent difficultyto treat 2c−3e systems, as a subtle tuning of the exchange−correlation functional can result in a rather pronounced fluc-tuation of λmax (e.g., going from LC-PBE to LC-τ-HCTH inducesa ∼30 nm blue shift).
Table 2. Transition Characteristics of Four 2c−3e Radical Anions (Cl2, Br2, (SCN)2, and DMeDS)a
gas phase solvent
gas solvent gas solvent
method λmax f ⟨S2⟩ λmax f S2 λmax f S2 λmax f S2
Dichlorine (2.570 Å; 2.553 Å)M06 339 0.34 0.75 354 0.39 0.75 332 0.35 0.75 346 0.40 0.75LC-BLYP 343 0.40 0.74 363 0.45 0.74 335 0.41 0.75 353 0.45 0.75LC-OLYP 345 0.40 0.74 365 0.45 0.74 336 0.41 0.75 355 0.45 0.75LC-PBE 338 0.40 0.74 358 0.45 0.75 330 0.41 0.75 348 0.45 0.75LC-TPSS 344 0.41 0.74 366 0.45 0.74 335 0.41 0.75 355 0.46 0.75ωB97 341 0.40 0.74 361 0.45 0.75 332 0.41 0.75 350 0.45 0.75LC-ωPBE 332 0.40 0.75 350 0.45 0.75 324 0.40 0.75 341 0.45 0.75B2PLYP 342 0.40 0.75 360 0.45 0.75 334 0.40 0.75 351 0.45 0.75BH&HLYP 332 0.40 0.75 349 0.44 0.75 326 0.40 0.75 341 0.45 0.75exp9 336 350
Dibromine (2.808 Å; 2.792 Å)M06 356 0.36 0.75 369 0.43 0.75 350 0.36 0.75 362 0.43 0.75LC-BLYP 365 0.42 0.74 386 0.49 0.74 357 0.43 0.74 377 0.49 0.74LC-OLYP 369 0.43 0.74 391 0.49 0.74 361 0.43 0.74 382 0.49 0.74LC-PBE 361 0.43 0.74 383 0.49 0.74 353 0.43 0.74 375 0.49 0.74LC-TPSS 365 0.43 0.74 388 0.49 0.74 358 0.43 0.74 379 0.49 0.74ωB97 357 0.43 0.74 376 0.49 0.74 350 0.43 0.75 367 0.50 0.75LC-ωPBE 355 0.42 0.74 374 0.48 0.75 348 0.42 0.75 366 0.49 0.75B2PLYP 355 0.42 0.75 372 0.48 0.75 349 0.42 0.75 364 0.48 0.75BH&HLYP 347 0.41 0.75 362 0.48 0.75 340 0.41 0.75 355 0.48 0.75exp9 372 365
Diisocyanate (2.701 Å; 2.682 Å)M06 454 0.21 0.77 466 0.27 0.77 464 0.10 0.77 477 0.25 0.77LC-BLYP 441 0.29 0.78 452 0.20 0.76 450 0.28 0.78 473 0.33 0.78LC-OLYP 443 0.29 0.79 444 0.21 0.76 452 0.28 0.79 475 0.34 0.78LC-PBE 433 0.29 0.79 446 0.21 0.76 442 0.28 0.79 465 0.34 0.78LC-TPSS 440 0.30 0.79 464 0.35 0.79 449 0.28 0.79 473 0.34 0.79ωB97 444 0.29 0.78 467 0.34 0.77 454 0.27 0.78 477 0.33 0.77LC-ωPBE 432 0.28 0.79 453 0.34 0.78 441 0.27 0.79 462 0.33 0.78B2PLYP 450 0.26 0.82 467 0.32 0.81 460 0.25 0.82 478 0.31 0.81BH&HLYP 440 0.26 0.81 457 0.32 0.80 450 0.24 0.81 467 0.30 0.80exp9 493 490
DMeDS (2.702 Å; 2.699 Å)M06 382 0.12 1.05 398 0.25 0.75 381 0.17 1.03 395 0.20 0.75LC-BLYP 398 0.19 0.77 410 0.40 0.75 389 0.29 0.75 405 0.40 0.75LC-OLYP 397 0.32 0.75 412 0.40 0.74 390 0.22 0.76 407 0.40 0.75LC-PBE 386 0.35 0.75 403 0.40 0.75 383 0.36 0.75 399 0.40 0.75LC-TPSS 391 0.35 0.75 409 0.40 0.75 388 0.36 0.75 405 0.41 0.75ωB97 392 0.34 0.75 407 0.41 0.75 391 0.22 0.77 402 0.41 0.75LC-ωPBE 382 0.30 0.75 380 0.40 0.75 379 0.32 0.75 393 0.40 0.75B2PLYP 393 0.22 0.76 402 0.40 0.75 385 0.23 0.76 398 0.40 0.75BH&HLYP 385 0.18 0.76 392 0.39 0.75 378 0.28 0.75 388 0.39 0.75exp49 n/a 417−420
aFour possible cases are considered: gas-phase geometry optimization followed by gas-phase or solvent TD-DFT simulations, or geometryoptimization in the presence of solvent followed by gas or PCM absorption. Two values for the intercenter distance are reported in parentheses, inthe 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.
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One notes that the half-and-half Becke hybrid, BH&HLYP, isstill off by ∼25 nm, although remaining clearly the mostpopular functional in the 2c−3e line of research. Let us alsonote that one usually performs the DFT optimization with thesame functional, which may lead to a fortuitous cancellation oferrors. In contrast, this work underlines a clear-cut superiorityof RSHs but no clear discrimination between Hirao’s,Scuseria’s, or Head-Gordon’s schemes (respectively LC-, LC-ωPBE, and ωB97 series) could be found. Contrary to the globalhybrids (GH), RSH advantageously provides a physically sounddescription of two distant electrons. In the ground state, wehave demonstrated that the extra electron occupying the anti-bonding σ* orbital presents a larger spatial separation with res-pect to “bonding” electrons (11.5 Å46), and the descriptionof the former benefits from being treated with a larger share ofexact exchange in RSH. Indeed, accurate calculations of verticalσ* ← σ transitions require a balanced description of the(σ)2(σ*)1 vs (σ)1(σ*)2 electronic configurations. In that res-pect, RSHs offer a very significant improvement, with thenoticeable exception of CAM-B3LYP that probably suffers fromits incorrect asymptotic behavior (65% instead of 100%).Diisocyanate turns out to be a more difficult system for TD-
DFT. LC-τ-HCTH is the only functional to be within the ±20nm margin, but the same functional seriously overshot the Cl2and Br2 wavelengths (by ∼40−50 nm). All the other func-tionals, independently of their level of sophistication, under-estimate diisocyanate’s λmax by 50−70 nm. Two factors canexplain this discrepancy: the nascent multiconfigurationalcharacter (T1 diagnostic ∼0.019), which tunes both the equilib-rium geometry and the vertical estimates made with a singledeterminant scheme, and a strong dynamical electron cor-relation (triple nitrile bonds). In parallel to this TD-DFT analysis,it is noteworthy that the much more demanding EOM-CCSDprovides absorptions in the imposed accuracy criterion fordihalogens, but not for diisocyanate, which also suffers from a10% downshift. In addition, one verifies that TD-HF stronglyoverestimates 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 35-methods set, using an accuracy threshold of ±20 nm: LC-BLYP, LC-OLYP, LC-PBE, LC-ωPBE, LC-TPSS, ωB97, M06,and B2PLYP(D). In the following section, we investigate theinfluence of nonprotic solvents on the absorption wavelengthsusing this panel of hybrids, as well as the seminal BH&HLYPthat was included for comparative purposes.3.2. Influence of Nonprotic Solvents. The experimen-
tally solvatochromic shifts in dichloromethane (DCM) are +14,−7, and −3 nm for dichlorine, dibromine, and diisocynanate,respectively.9 Interestingly, these rather small incrementsqualitatively and quantitatively differ for these three systems.This is clearly a challenging case for the PCM-TD-DFT ap-proach, and it is also of interest to understand this intriguingsolvatochromism. As no specific interaction between the 2c−3esolute and the solvent molecules is expected, it is legitimate toresort to a continuum representation of the solvent. Numericaldata are gathered in Table 2, and a graphical representation canbe found in Figure 3. These wavelengths were computed onboth MP2 and PCM-MP2 geometries, to assess the direct andindirect solvent effects. The intercenter distance d(2c−3e) issystematically shrunk by ∼0.02 Å in the liquid phase, suggestingthat the solvent partly alleviates the electronic repulsion. For thetwo halides, indirect geometrical solvent effects subsequently
result in a blue shift, in line with an orbital interpretation and anear-linear dependence of λmax as a function of d.11 Comparingthe first and third columns, this decrease accounts for ca. 7 nm.But the direct (electronic) solvent effects induce a bathochromicshift of the λmax by a nearly uniform increment of ∼1520 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. Fordiisocynanate, experimental UV−vis signatures are barelyaffected by the dielectrico (−3 nm). The agreement betweencalculations and experiments is globally improved once a PCMcorrection is applied, though the solvatochromic effects are notsystematically consistent with measurements. Indeed, the PCM-TD-DFT strategy does not satisfactory render for the subtle,complex contrast between the three 2c−3e entities. Additionalcalculations were performed on difluorine (experimental absorp-tion reported to be ca. 300 nm); see first histogram of Figure 4.The intercenter equilibrium distance is considerably shorter thanfor the three other molecules (from 2.57 to 2.80 Å). In spiteof a nascent multiconfigurational character (0.022), the agree-ment is still very satisfying. This unexpected good result mightarise from error compensation with the larger dynamic electroncorrelation (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 dueto their importance and ubiquitousness in biochemistry.Indeed, among all 2c−3e systems, disulfide radical anions arethe most frequent encountered. Antioxydant redox-active enzymesor smaller agents (α-lipoic acid) have been actively characterizedand show potentialities in terms of redox protection, photo-stability, etc.47 The existence of these versatile intermediates ismost often indirectly proved by pinpointing their UV−vis sig-nature. Aliphatic entities typically absorb in the 400−450 nmrange domain, with lifetimes in water in the microsecond range.48
No wavelengths in the gas phase are available: the most accuratevalue for the prototypical dimethyl disulfide (DMeDS) is 417−420 nm in dimethyl sulfide, determined using subpicosecondlaser spectroscopy that allows a real-time probing of such tran-sient species.49 We first examine the PCM-TD-DFT performance
Figure 3. Solvatochromism of three representative 2c−3e anions: gas-phase vs DCM λmax wavelengths for a set of three (pseudo)halides.The four curves correspond to the following schemes: gas-phase andPCM TD calculations using gas-phase geometries (respectively filledand empty blue squares), as well as gas-phase and PCM TDcalculations on PCM geometries (empty and filled red circles).Corresponding numerical data are collated in Table 2. Experimentalabsorptions are denoted with black lines: solid for dimethyl sulfide,dashed for the gas phase.
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in reproducing this value (bottom of Table 2). The T1 diagnosticyields 0.013, which grounds the selection of a monodetermi-nantal approach for DMeDS. Quite interestingly, though diiso-cyanate and DMeDS present inter-sulfur distances within 0.001 Åof each other (respectively 2.701 and 2.702 Å in the gas phase),
short-range stereoelectronic contributions induce a 50 nm off-set between the two entities. For DMeDS, the evolution of theλmax as Jacob’s ladder of functionals is climbed or when nE isincreased is in the same vein as for the three (pseudo)halides.Most functionals underestimate the λmax by more than 40 nm,
Figure 4. Complementary TD-DFT calculations on difluorine and DMeDS, for the DFT functionals listed in Table 1. For the latter, a ∼400 nmvalue is reported experimentally in water. Inclusion of a PCM correction improves the agreement. As found for the first subset, RSHs outperform theother functionals.
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and only four RSHs are within 10 nm of the reference value:LC-OLYP, LC-BLYP, LC-TPSS, ωB97. They outperformBH&HLYP, which is off by 20 nm. Again, it can be argued herethat the selection of the half-and-half Becke’s functional is not tobe recommended, though it can blur its own deficiency as the2S−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. Forpractical applications, this is nonproblematic as successful micro-hydration schemes14,35,36,50 have been developed to account forwater, 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 theDMeDS UV−vis σ* ← σ transition, several strategies of in-creasing complexity have been applied. The target experimentalvalue in aqueous media is consistently claimed to be near 400 nmfor a large set of systems. For the purpose of our benchmark, itcan be taken as 390−410 nm.49 Preliminary results in the gasphase and with PCM inclusion, along the whole set of 35 func-tionals, are shown in Figure 4. The similar shape of these histo-grams, compared to the (pseudo)halides, confirms the superi-ority of RSH functionals, whereas one notices that the PCMcorrection globally improves the agreement toward a ∼400 nmvalue. Ten functionals lie in a ±10 nm error bar, namely, M06-HF, B2PLYP, LC-ωPBE, LC-PBE, M06, ωB97, ωB97X,BH&HLYP, and finally LC-TPSS, as ranked by their respectiveperformance. The above-listed functionals partly differ from theeight 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 influ-ence 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 havebeen obtained in both the gas phase and in a continuum (PCM)environment, adding n = 0, 1, or 2 explicit water molecules tothe model. Cartoon representations are given in Figure 5a,b.For DMeDS, we have reported a rapid convergence of both
geometries and other properties as a function of n (up to 6).36
We have also shown that hydrogen bond pairing with first-shellwater molecules has a significant geometrical impact that tunesthe intercenter distance and hence might indirectly affectthe absorption features.36 Assessment of the microhydrated
PCM-TD-DFT//PCM-MP2 strategies has been done systemati-cally, applying the preselected best-performing functionals. Resultsare gathered in Table 3, left side. They lie in an acceptable range oferrors, indicating that DMeDS is probably a less difficult systemto model with a QM methodology than pseudodihalides. Onceagain, let us highlight that the BH&HLYP is off by more than20 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 tooffer 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 anti-oxidant features a carboxylate group, such that after electroncapture the predominant form is a dianion. The λmax has beenreported by Sander and co-workers to be red-shifted withrespect to DMeDS, with an absorption of 410 nm (in water).47
Our PCM-TD-DFT attempts to reproduce this accurate valueare gathered in the third subblock of Table 3, left side. The totalred shift can be divided into two contributions. First, as theoptimized inter-sulfur distance is markedly increased by cycliza-tion (0.07 Å), a red shift of ∼23 nm is expected.11 Thisgeometrical contribution is counterbalanced by the negativelycharged substituent, situated at a distance of ca. 7 Å. Compar-ison between DMeDS and the lipoate dianion indicates that aPCM-TDDFT//PCM-MP2 computational scheme succeeds inpredicting the correct orientation, although the red shift tendsto be overestimated with increments of 20 nm (one watermolecule) or 30 nm (two water molecules). It is plausible that amore fine treatment of electrostatic for a dianion would help toimprove the agreement. To that aim, one would need not onlyincrease the number of water molecules but perhaps more im-portantly to go beyond a static 0 K picture and include dyna-mics contributions.Finally, beyond the TD-DFT assessment, it would be desir-
able to benefit from a coherence between the functional usedfor the excited-state calculation and for the geometryoptimization. This is tested on DMeDS and lipoate anions inthe r.h.s. of Table 3. It appears that the uniform use of a givendensity functional can severely alter the performance achievedon a reference post Hartree−Fock geometry. The capability ofa given density functional to maintain a satisfactory perfor-mance obviously depends on the evaluation of the inter-sulfurdistance d(2S−3e). Two case scenarios appear: (i) RSHs tendto underestimate this distance with respect to the MP2 refer-ence value. This is clearly illustrated by the LC-PBE approachthat induces a significant shrinkage of 0.08 Å, which impactson the λmax estimate by ca. 40 nm. (ii) Conversely, M06,B2PLYP(D), and BH&HLYP predict larger d, hence leading toan overestimation of the λmax. It should also be stated that ringstrain estimates are known to be sometimes poorly reproducedby DFT,52 hence constituting an additional difficulty for lipoatesystem. Nevertheless, several exchange−correlation functionals(e.g., LC-OLYP, LC-BLYP, and LC-ωPBE) do provide a geo-metry close to the reference MP2 structure for the DMeDSradical anion: they are found to behave correctly for the“exotic” chemistry of this prototypical 2S−3e system and con-stitute the best compromise to investigate larger structuresdifficult to model with electron-correlated wave functionschemes. We do not investigate here the possibility of combos,i.e., using two different density functionals for geometry opti-mizations and excited-state single points, as successful agree-ments will mostly reflect an inner cancellation of errors.
Figure 5. Cartoon representations of microsolvated DMeDS andlipoate radical anions. Geometries are optimized at the MP2/aug-cc-PVTZ level of theory. Key distances for d(2s−3e) and waterinteraction are specified (in Å).
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4. CONCLUSIONS
For the first time, the computational treatment of UV−vis sig-nature with TD-DFT has been assessed for three-electron two-center species. It turned out that extreme caution must be taken inthe choice of the density functional, which clearly prevents brute-force study relying on first-principle molecular simulations. Therelatively small-size of systems investigated in the presentcontribution allowed a full QM approach and offered referencevalues beyond the experimental data. In this work, we invoke thesuperiority of RSH functionals and more specifically LC-OLYP,LC-BLYP, and LC-ωPBE. They appear to provide a good com-promise between accuracy and computational effort for bothground 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 modulatedby a heterogeneous proteinic environment, in sharp constrastwith (pseudo)halides, where a subtle solvatochromism has beenevidenced. Thanks to recent methodological developments inhybrid TD-DFT/MM approaches, the question of the func-tional dependence is probably the biggest difficulty when deal-ing with 2c−3e systems.
■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected] (E.D.); [email protected] (D.J.).NotesThe authors declare no competing financial interest.
■ ACKNOWLEDGMENTSE.D. gratefully acknowledges Dr. A. Monari (Universite deLorraine) for helpful comments on this work. Ab initiocalculations were performed using the local HPC resources ofPSMN at ENS de Lyon and of GENCI (CINES/ IDRIS), projectx2011075105. D.J. acknowledges the European Research Council(ERC) and the Region des Pays de la Loire for financial support inthe framework of a Starting Grant (Marches - 278845) and arecrutement sur poste stratique, respectively.
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Table 3. Maximum Absorption Wavelengths (λmax) for DMeDS and Lipoic Acid Anions, along a Set of Best-PerformingFunctionals, Using the MP2 Geometry or after a Uniform DFT Optimization
optimized MP2 optimized DFT
method d λmax f ⟨S2⟩ d λmax f ⟨S2⟩
DMeDS−PCM + 1 Water Molecule
M06 2.686 401 0.24 0.75 2.801 454 0.29 0.75
LC-BLYP 2.686 415 0.35 0.74 2.707 423 0.36 0.74
LC-OLYP 2.686 416 0.35 0.74 2.692 416 0.36 0.74
LC-PBE 2.686 407 0.35 0.74 2.631 377 0.35 0.75
LC-TPSS 2.686 413 0.35 0.74 2.646 388 0.35 0.74
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B2PLYP 2.686 406 0.34 0.74 2.800 462 0.36 0.74
BH&HLYP 2.686 396 0.34 0.75 2.817 455 0.36 0.74DMeDS−PCM + 2 Water Molecules
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BH&HLYP 2.670 372 0.21 0.75 2.798 440 0.36 0.74
exp 400 400Lipoate−PCM + 2 Water Molecules
M06 2.758 416 0.27 0.75 2.844 447 0.30 0.75LC-BLYP 2.758 432 0.36 0.74 2.763 429 0.41 0.74LC-OLYP 2.758 434 0.35 0.75 2.749 422 0.41 0.75LC-PBE 2.758 425 0.36 0.74 2.697 383 0.42 0.75LC-TPSS 2.758 432 0.37 0.74 2.707 394 0.42 0.75ωB97 2.758 429 0.37 0.74 2.798 448 0.40 0.74LC-ωPBE 2.758 418 0.36 0.75 2.741 402 0.41 0.75B2PLYP 2.758 420 0.36 0.75 2.846 461 0.39 0.74BH&HLYP 2.758 410 0.35 0.75 2.843 446 0.38 0.75exp 410 410
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