TDDFT Assessment of Functionals for Optical 0â0 Transitions in Small

Article pubs.acs.org/JPCA

TDDFT Assessment of Functionals for Optical 0−0 Transitions in Small Radicals Loïc Barnes,† Saleh Abdul-Al,‡ and Abdul-Rahman Allouche*,† †

Institut Lumière Matière, UMR5306 Université Lyon 1-CNRS, Université de Lyon, 69622 Villeurbanne Cedex, France Lebanese International University, Mouseitbeh, P.O. Box 146404 Mazraa, Beirut, Lebanon

‡

S Supporting Information *

ABSTRACT: Using LR-TDDFT, we calculated the 0−0 energies of 15 small radicals for which the experimental values in gas phase are available. We used 17 functionals. It turned out that B3LYP, M06-2X, ωB97X-D, CAM-B3LYP, and HSE06 functionals are the most effective functionals in terms of root-mean-square and average unsigned deviation. Using the standard value (0.47 a−1 0 ) of the attenuation parameter ω, the long-range-corrected LC-GGA functionals give poor results. However, the LC-PBE with ω = 0.25 a−1 0 give a performance similar to that of B3LYP. Taking into account zero-point correction improves the results, but the contribution of adiabatic correction is more important than that due to the vibration. The vertical approximation is certainly not recommended. An adiabatic calculation seems to give a good compromise between computing time (and resources) and reliability of the results for most of molecules investigated in this work.

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INTRODUCTION The high-level ab initio wave function-based methods (CASPT2, CC2, CC3, EOM-CCSD, etc.) are capable to describe, with a good accuracy, various types of excited states for a range of chemical systems.1 However, the use of these computationally expensive methods is limited to small molecules. The formalism of linear response of the timedependent density functional theory (LR-TDDFT)2,3 is an attractive alternative approach to ab initio methods for the calculation of excited electronic states of large molecules. In practice, the LR-TDDFT is used with approximate exchangecorrelation (XC) density functionals because no exact expression is known. Therefore, several extensive tests of various density functionals have been published to check their reliability to predict accurate excited states. The efficiency of LR-TDDFT was evaluated on a large number of closed-shell, small and large, organic molecules.4−7 However, for radicals, the large majority of LR-TDDFT applications are (still) performed within the so-called vertical approximation. In this framework, one estimates the excited state energies, and related properties, using the most stable ground-state geometry, without exploring the potential energy surface of excited state. This approximation has the advantage of allowing calculations on very large systems, as a single-point LRTDDFT calculation.8 This approach is certainly not wellgrounded.9,10 To obtain more meaningful theory-experiment comparisons, we should compute the 0−0 energies. That requires the optimization of excited-state geometry, the determination of its vibrational normal modes, and therefore calculation of the TDDFT Hessian, which demands very important computer resources. Due to the joint increase of computing power and implementation of TDDFT derivatives in several computational chemistry packages, a series of studies © 2014 American Chemical Society

going beyond the vertical approximation have appeared recently, especially for closed shell systems. However, to the best of our knowledge, only two studies of 0−0 energies are available in the literature for radicals.9,10 Furche et al.9 studied four radicals for which the experimental values of 0−0 energies are available. Furche et al.9 computed the zero-point vibrational corrections (ZPVC) with the B3LYP functional and adjusted the results of the other tested functionals (LSDA, PBE, BP86, TPSS, and PBE0) with the B3LYP vibrational correction. Grimme et al.10 studied 6 radicals, but the calculated values (only three functionals were used) were compared to experimental 0−0 energies for solvated radicals. (The excitation energies have been increased by 0.15 eV to take into account solvent-induced red shift.) Compared to closed-shell systems, there are mainly two reasons which make the study of openshell molecules difficult: the reference wave function in the radicals has a multideterminant structure, and the radical wave function has a more-complicated spin structure. The multireference ab initio approaches (CASPT2, MRCI,11 MRCC,12 etc.) can be used to avoid these problems, but the results of these methods are strongly dependent on the choice of active space, which requires experience and expertise. In addition, the fact that the analytical calculation of the gradient is rarely implemented for these methods makes the calculation of adiabatic effect difficult or impossible for molecules with more than 10 atoms. The calculation of harmonic frequencies for these methods is even more difficult; the second derivatives must be done numerically without taking the symmetry of the molecule into account. Compared to the XCIS calculation, Received: August 2, 2014 Revised: October 8, 2014 Published: October 28, 2014 11033

dx.doi.org/10.1021/jp5078032 | J. Phys. Chem. A 2014, 118, 11033−11046

The Journal of Physical Chemistry A

Article

Hirato et al.8 showed that the TDDFT can correctly reproduce the vertical excitation energies of valence excited states of radicals, including states with a double excitation character (at XCIS level), while the unrestricted configuration interaction with a single excitation (UCIS) method was not able to reproduce correctly the excitation energies. Therefore, TDDFT seems to be an interesting alternative to ab initio methods to study the radicals. In this work, we want to add the adiabatic and vibrational effects in the study of radicals with TDDFT. In this paper, we compile a set of 17 accurate experimental 0−0 energies for benchmark purposes, including a variety of different systems and states. Seventeen functionals have been evaluated. We also calculated the energies at ab initio UCIS and CASPT2 levels in order to study the double excitation character effects in the accuracy of results.

After extensive atomic basis set assessment on the 2 diatomic molecules included in our set of molecules (see below), we selected the 6-311++G** basis set to compute ground and excited states of polyatomic molecules of our set. As illustrated in Figure 1, the calculated vertical transition energy (Tυ) corresponds to the difference between the energy

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METHODOLOGY All calculations have been performed with the Gaussian09 D01 revision,13 Molpro 2008,14 and Gabedit.15 We used 17 functionals: BLYP, 16,17 PBE,18 B3LYP, 17,19−21 PBE0, 22 BHandH,20 BHandHLYP,20 HSE06,23 HISS,24 VSXC,25 M06L,26,27 M06,26,27 M06-2X,26,27 LC-BLYP,16,17,28 LCPBE,18,28 LC-M06L,26−28 CAM-B3LYP,29 and ωB97X-D,30 including generalized gradient approximation (GGA), hybrid, meta-GGA, and meta-hybrid functionals with or without longrange corrections. BLYP and PBE are pure GGAs without any Hartree−Fock (HF) exchange. B3LYP, PBE0, BHandH, and BHandHLYP are global hybrid GGAs containing 20%, 25%, 50%, and 50% of Hartree−Fock exchange, respectively. HSE06 and HISS are global hybrid using an error function screened Coulomb potential to calculate the exchange portion of the energy. The exact exchange is used only at short range for HSE06 and used only at medium-range for HISS. For HSE06, the HF exchange is limited to 25%, and the parameter ω used for controlling the 23 short range of the interaction is set to 0.208 a−1 The HISS 0 . functional is a multirange hybrid which partitions the Coulomb operator into three pieces rather than the two as in HSE06. The values of ω used in HISS to make the three partitions are 0.20 −1 24 a−1 0 and 0.84 a0 . VSXC and M06L are meta GGAs in which the functionals depend on the up and down spin kinetic energy densities. M06 (27% of HF) and M06-2X (54% of HF) are global hybrid meta-GGAs. LC-BLYP, LC-PBE, and LC-M06L contain 0% HF exchange at short range and 100% at long range, using 0.47 a−1 0 as a controlling separation parameter in the electron repulsion operator as

Figure 1. Transition energies calculated in this work.

(E) of the ground state (GS) and the energy of the excited state (ES) computed using the ground state optimal geometry (X). Tυ = E ES(XGS) − EGS(XGS)

The calculated adiabatic transition energy (Te) corresponds to the difference between the energy of the ground state (GS) using the optimal ground state geometry and the energy of the excited state (ES) computed using the excited state optimal geometries. Te = E ES(XES) − EGS(XGS)

The calculated 0−0 transition energy (T0) corresponds to Te + zero-point vibrational correction (ZPVC) on excited state minus that of ground state: T0 = Te + ZPVCES − ZPVCGS

The statistical analyses were made through the standard functions: average signed error (ASE), average unsigned error (AUE), root-mean-square deviation (rmsd), and standard deviation (SD) defined by N

ASE(x) =

xicalc − xiexp N

∑ i=1 N

1 − erf(ωr12) erf(ωr12) 1 = + r12 r12 r12

AUE(x) =

∑ i=1

where the erf term denotes the standard error function, r12 is the interelectronic distance between two electrons, and ω is the range-separation parameter in units of Bohr−1 (a−1 0 ). CAM-B3LYP and ωB97X-D are long-range-corrected hybrid GGAs that allow the fraction of exact exchange to vary with the interelectronic distance. The long-range-corrected functionals have been developed for overcoming the deficiencies of other functionals to reproduce correctly the transition energies of states with charge transfer. Even though, as noted below, no such states are included in this work, it remains interesting to assess how these long-range-corrected functionals differ from others when applied to valence states.

|xicalc − xiexp| N

N

rmsd(x) =

∑ i=1

(xicalc − xiexp)2 N

and SD(x) =

rmsd2(x) − ASE2(x)

Our benchmark set systems is compiled using experimental reference data only. An important selection criterion was the quality of the experimental data: only vibrationally resolved gasphase data were included and any values deduced empirically from experimental ones were avoided. The size of the 11034



Article

Table 1. Experimental Values Used as References in This Study constants

molecules

Re (Å)

cyano radical (CN)

carbon monoxide ion (CO+)

ωe (cm−1)

cyano radical (CN)

carbon monoxide ion (CO+)

T0 (cm−1)

cyano radical (CN) carbon monoxide cation (CO+) methyl radical (CH3) phosphino radical (H2P) hydroperoxy radical (HO2) HSO isocyanato radical (NCO) thiocyanato radical (SCN) nitrogen trioxide (NO3) vinoxy radical (CH2CHO) phenyl radical (C6H5) phenoxy radical (C6H5O) phenylthio radical (C6H5S) cyclohexadienyl radical (C6H7) benzyl radical (C7H7)

states

values

X2Σ+ A2Π B2Σ+ X2Σ+ A2Π B2Σ+ X2Σ+ A2Π B2Σ+ X2Σ+ A2Π B2Σ+ A2Π B2Σ+ A2Π B2Σ+ A2A1′ A2A1 A2A′ A2A′ A2Σ+ A2Π A2E′′ A2A′ A2B1 B2A2 B2A2 A2A2 B2B1

1.17182 1.2333 1.15 1.11514 1.24377 1.16877 2068.59 1812.56 2163.9 2214.24 1562.06 1734.18 9117.3 25799.7 20407.2 45636.7 46239.0 18276.6 7029.7 14367.0 22754.0 26054.56 7064.0 8036.0 18908.0 16360.0 19328.0 18191.5 22001.5

references 31 31 31 31 31 31 31

31 31 31 31 31 31 31 31 31 32,33 32,34 32,35 32,36 32,37 32,38 32,39 32,40 32,41 32,42 32,43 32,44 32,45

Table 2. Averaged Unsigned Deviation Obtained from Comparison of Calculated Values Using Several Basis Sets and that of aug-cc-pVQZ Basis Used as a Reference molecules δT0 (cm−1)

CN CO+

δRe (Å)

average CN

CO+

δωe (cm−1)

average CN

CO+

average

states

6-31G*

6-31++G**

6-311++G**

cc-pVDZ

cc-pVTZ

cc-pVQZ

aug-cc-pVDZ

aug-cc-pVTZ

A2Π B2Σ+ A2Π B2Σ+

1286 1366 262 795 927 0.013 0.009 0.010 0.012 0.012 0.029 0.014 7 22 123 9 205 261 104

1251 1302 508 1012 1018 0.013 0.009 0.009 0.012 0.012 0.034 0.015 6 53 102 4 209 284 110

470 220 137 179 251 0.003 0.002 0.005 0.005 0.003 0.013 0.005 9 40 80 4 15 142 48

1176 879 379 779 803 0.009 0.007 0.032 0.014 0.013 0.030 0.018 11 29 463 5 30 280 136

234 214 118 279 211 0.002 0.002 0.003 0.002 0.002 0.004 0.003 4 5 19 1 8 117 26

23 17 35 39 28 0.000 0.000 0.000 0.000 0.000 0.000 0.000 1 1 3 1 2 8 3

1117 1669 737 956 1120 0.010 0.008 0.033 0.013 0.014 0.034 0.019 27 22 101 14 253 301 120

227 203 106 166 175 0.002 0.002 0.003 0.001 0.002 0.003 0.002 5 6 18 2 4 108 24

X2Σ+ A2Π B2Σ+ X2Σ+ A2Π B2Σ+ X2Σ+ A2Π B2Σ+ X2Σ+ A2Π B2Σ+

molecules in our set, ranges from 2 to 14 atoms. Experimental values have been collected in Table 1.

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lengths are available for the ground and excited states of the diatomic molecules of our set. We used these values to ascertain the quality of the basis set. We used several members of Poples46,47 and Dunnings48−50 basis sets series, incorporating and not incorporating diffuse orbitals. Table S1 of the Supporting Information reports the spectroscopic constants for the two diatomic molecules of our set, namely CN and CO+

RESULTS AND DISCUSSION

Basis Set Effects. In addition to adiabatic energies, the experimental vibrational harmonic frequencies and the bond 11035



Article

Table 3. Calculated Energies for All Radicals Using 6-311++G** Basis Seta molecule

CN/A2Π

CN/B2Σ+

CO+/A2Π

CO+/B2Σ+

functional/method BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(9,8) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP wB97xD CIS CASPT2(9,8) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(9,8) BLYP PBE B3LYP PBE0

Tυ (cm−1)

Teb (cm−1)

T0b (cm−1)

−1 Tυ − Texp 0 (cm )

−1 Te − Texp 0 (cm )

−1 T0 − Texp 0 (cm )

9947 11418 9055 10900 10109 9549 10940 12750 11721 16709 13400 10145 9869 12854 11217 8854 9480 34139 12660 26417 26529 24922 25625 26656 26957 25785 27942 28525 27034 28964 31463 26962 28997 29424 24950 26645 52541 25747 23995 26530 27252 31016 33424 32570 30985 35408 22654 30579 24051 21660 30539 35105 22562 28329 28706 60662 21001 40830 43166 45452 48802

8713 10077 7550 9123 8229 7463 9195 10808 10247 14550 11271 7595 8104 11042 9538 7185 7944 33013 10566 25869 26101 24732 25622 26488 26431 25778 27140 28436 27030 27552 28448 26899 28660 29076 24933 26636 52540 25850 19928 22403 22581 26564 29130 28075 26540 31376 19796 28248 22632 21169 26681 31349 21848 23969 24687 60575 19842 40740 42766 41674 40487

8571 9931 7395 8956 8085 7292 9027 10670 10081 14300 11002 7223 7932 10892 9354 7017 7771 33600 10411 25934 26148 24749 25542 26198 25759 25681 26656 28398 26896 27532 28264 26692 28231 28952 24877 26484 52898 25867 19638 22102 22251 26256 28878 27806 26222 31142 19530 27913 22341 20960 26397 31107 21611 23653 24446 61150 19513 40561 42203 41055 39966

830 2301 −63 1782 992 432 1823 3632 2604 7591 4282 1027 751 3736 2100 −264 363 25022 3543 617 730 −878 −174 856 1158 −15 2142 2725 1234 3165 5664 1162 3198 3624 −849 845 26741 −53 3588 6123 6845 10609 13017 12163 10578 15001 2247 10172 3643 1253 10132 14698 2154 7922 8298 40255 594 −4807 −2471 −185 3166

−404 960 −1567 6 −888 −1654 78 1691 1130 5432 2154 −1522 −1014 1925 421 −1932 −1173 23896 1449 69 302 −1067 −178 688 631 −22 1341 2636 1230 1752 2648 1099 2860 3277 −866 836 26740 51 −479 1995 2174 6157 8723 7668 6133 10969 −611 7841 2224 761 6274 10942 1441 3562 4280 40167 −565 −4897 −2871 −3963 −5149

−546 814 −1722 −161 −1032 −1825 −90 1553 964 5182 1885 −1894 −1185 1775 236 −2100 −1347 24483 1293 134 348 −1051 −258 398 −41 −119 856 2599 1096 1732 2464 893 2431 3153 −922 685 27098 67 −769 1695 1844 5849 8471 7399 5815 10735 −878 7506 1934 553 5990 10700 1204 3245 4039 40743 −895 −5076 −3434 −4582 −5670

11036

T0 − Te (cm−1) −142 −146 −156 −166 −145 −171 −168 −138 −166 −250 −269 −372 −172 −150 −185 −168 −173 587 −156 65 47 16 −80 −290 −672 −97 −484 −37 −134 −20 −184 −206 −429 −124 −56 −151 358 16 −290 −300 −330 −308 −252 −269 −318 −234 −266 −335 −290 −209 −284 −242 −237 −316 −240 576 −330 −179 −562 −619 −521



Article

Table 3. continued molecule

CH3

H2P

HO2

functional/method BHandH BHandHLYP HSE0 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP wB97xD CIS CASPT2(9,8) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(7,7) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(7,6) BLYP PBE B3LYP BHandH BHandHLYP PBE0 HSE06 HISS

Tυ (cm−1)

Teb (cm−1)

T0b (cm−1)



−1 T0 − Texp 0 (cm )

T0 − Te (cm−1)

53242 52648 48673 52919 43547 55580 53446 55084 50918 54199 54277 47600 48096 65524 44014 38571 40179 42374 44677 46228 46493 44933 49595 42058 43840 36709 42931 48529 51578 55887 44473 44416 53015 45126 23160 23566 23105 23630 22079 23182 23684 23729 23249 25410 22342 20455 21655 22409 20577 22467 22088 25605 23428 9334 9672 8968 7648 8234 9279 9304 9116

41930 39180 40421 41363 41440 42962 44286 45575 44214 44241 45281 41960 41879 52099 45332 38440 40038 42216 44514 46011 46320 44758 49382 41906 43775 36646 42786 48328 51358 55098 44309 44313 52928 45691 18876 19158 18797 19250 17621 18903 19302 19383 18602 20939 18150 16186 17375 18088 15950 18178 17751 21527 19408 8903 9234 8506 7077 7648 8807 8833 8637

41577 38827 39917 41043 40820 42536 43724 45034 43637 43820 44679 41360 41371 51987 44713 38331 39929 42063 44371 45787 46158 44584 49111 41781 43972 36840 42780 48043 50992 53986 44134 44246 53075 45538 18939 19235 18864 19321 17681 18952 19374 19451 18696 20986 18165 16306 17427 18145 15936 18189 17870 21594 19475 8795 9126 8365 6904 7476 8660 8686 8469

7605 7012 3036 7282 −2090 9943 7810 9447 5281 8562 8640 1964 2459 19887 −1623 −7668 −6060 −3865 −1562 −11 254 −1306 3356 −4180 −2399 −9530 −3308 2290 5339 9648 −1766 −1823 6776 −1113 4883 5290 4828 5353 3802 4905 5408 5452 4973 7133 4066 2178 3379 4133 2300 4190 3812 7329 5151 2305 2642 1938 618 1204 2250 2274 2087

−3706 −6456 −5216 −4274 −4197 −2675 −1350 −62 −1422 −1396 −355 −3677 −3758 6463 −305 −7799 −6202 −4024 −1725 −228 81 −1481 3143 −4333 −2464 −9593 −3453 2090 5118 8859 −1930 −1926 6690 −548 600 881 520 974 −655 626 1026 1106 325 2663 −127 −2091 −902 −189 −2326 −98 −525 3251 1132 1873 2204 1476 48 618 1777 1804 1607

−4060 −6810 −5720 −4593 −4817 −3101 −1912 −603 −2000 −1816 −958 −4277 −4266 6350 −924 −7908 −6310 −4176 −1868 −452 −81 −1655 2872 −4458 −2268 −9399 −3459 1804 4753 7747 −2106 −1993 6836 −700 662 958 587 1045 −596 676 1098 1175 420 2710 −112 −1970 −850 −132 −2340 −88 −406 3317 1199 1765 2096 1336 −126 446 1630 1656 1439

−353 −353 −504 −320 −620 −426 −562 −541 −578 −420 −602 −600 −508 −112 −619 −110 −109 −153 −142 −224 −162 −174 −271 −125 196 194 −6 −285 −365 −1112 −175 −67 146 −153 63 77 67 71 59 50 72 68 94 47 15 121 52 57 −14 11 119 67 67 −108 −108 −141 −173 −172 −147 −148 −168

11037



Article


HSO

NCO

SCN

functional/method

Tυ (cm−1)

Teb (cm−1)

T0b (cm−1)



−1 T0 − Texp 0 (cm )

VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(13,9) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(13,9) BLYP PBEPBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBEPBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(15,12) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE0 HISS VSXC M06L M06 M06-2X

10501 11624 6521 3134 8380 8716 9192 8771 8634 7849 8017 15748 16162 15704 16099 15385 15373 16086 16459 16952 18244 14552 11437 16044 16491 16538 15735 15536 14082 15921 21754 21579 23471 23845 23965 25718 23780 24392 23569 24497 21533 20902 24097 23972 24597 24226 24066 31051 22967 21280 21200 25869 26942 31113 31050 26693 30501 23132 23785 24807 29230

10025 11118 6076 5362 7947 8296 8769 8310 8204 7104 7343 13505 13976 13281 13625 12331 12191 13614 13595 14739 15946 11599 9719 13354 13852 13951 13244 13070 9827 12143 20458 20196 22725 23101 23575 25326 23023 23823 22314 23168 20813 20331 23771 23570 24286 23731 23475 30478 22389 20929 20874 25816 26811 30240 29874 26578 29847 22667 23544 26526 27154

9893 10981 5961 5245 7776 8124 8583 8156 8050 6949 7203 13387 13844 13138 13469 12146 12028 13466 13427 14552 15817 11438 9753 13152 13643 13743 13079 12957 9761 12000 20805 20813 22804 23543 24089 25926 23458 24314 22674 23126 21175 20386 24063 23818 24650 24174 23861 31637 22468 21023 21037 25884 26948 30217 30097 26675 31156 22598 23745 26393 27171

3472 4594 −509 −3896 1351 1686 2162 1741 1604 819 988 1381 1795 1337 1732 1018 1006 1719 2092 2585 3877 185 −2930 1677 2124 2172 1368 1169 −285 1554 −1000 −1175 717 1091 1211 2964 1026 1638 815 1743 −1221 −1852 1343 1218 1843 1472 1312 8297 213 −4775 −4855 −186 887 5058 4995 638 4446 −2923 −2270 −1247 3175

2996 4088 −954 −1668 917 1266 1739 1280 1175 75 314 −862 −391 −1086 −742 −2036 −2176 −753 −772 372 1579 −2768 −4648 −1013 −515 −416 −1123 −1297 −4540 −2224 −2296 −2558 −29 347 821 2572 269 1069 −440 414 −1941 −2423 1017 816 1532 977 721 7724 −365 −5126 −5181 −238 756 4186 3820 523 3793 −3387 −2511 471 1099

2863 3951 −1068 −1784 746 1095 1553 1126 1021 −81 173 −980 −523 −1229 −898 −2221 −2339 −901 −940 185 1450 −2929 −4614 −1215 −724 −624 −1288 −1410 −4606 −2367 −1949 −1941 50 789 1335 3172 704 1560 −80 372 −1579 −2368 1309 1064 1896 1420 1107 8883 −286 −5032 −5017 −170 893 4162 4042 620 5102 −3457 −2309 339 1116

11038

T0 − Te (cm−1) −132 −137 −115 −117 −171 −171 −186 −154 −154 −156 −141 −118 −132 −143 −156 −185 −163 −147 −168 −187 −129 −161 33 −202 −209 −208 −165 −113 −66 −143 347 617 79 441 514 600 436 491 359 −42 362 55 292 248 364 442 386 1159 79 94 163 68 137 −24 222 97 1309 −70 202 −132 17



Article


NO3

CH2CHO

C6H5

functional/method

Tυ (cm−1)

Teb (cm−1)

T0b (cm−1)



−1 T0 − Texp 0 (cm )

T0 − Te (cm−1)

LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(15,12) PBE BLYP B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(23,15) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(13,8) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D

32640 32704 32395 30145 29387 41621 27088 13722 13154 11693 10396 5517 5809 10656 7438 14072 14648 8869 3898 6527 7149 7486 8881 9192 6316 11924 7252 7460 10096 11101 11769 13132 11090 13612 8737 10929 10652 13710 14124 13358 11748 11372 16433 10910 20964 22713 23031 25344 25546 25488 25195 27292 22609 27252 24266 24362 26495 27313 26724 24574 24333

30911 31025 30817 29701 29022 41550 26782 11887 11212 8437 8295 4736 5306 8362 6769 11988 12529 6224 3898 5630 6237 6734 6987 7341 5848 8358 5304 5314 7578 8258 8137 9231 8282 10035 6662 7434 8964 9978 10327 9972 8746 8433 10851 7461 16497 18352 18584 21062 21404 21323 20901 23355 17224 22912 19627 20203 22459 23937 22270 20223 19907

31155 31312 31116 29835 29794 41868 26850 11719 11068 8768 8684 4794 5269 8809 6795 11852 12461 6801 4736 5742 6383 6764 7339 7675 5822 8689 5838 5998 7717 8385 8280 9390 8409 10153 7102 7619 9294 10086 10424 10047 8864 8593 11071 7600 16248 18052 18340 20782 21212 21114 20619 23059 17218 22968 19768 20115 22255 21675 22241 19995 19710

6585 6649 6340 4090 3332 15566 1033 6658 6090 4629 3332 −1547 −1255 3592 374 7008 7584 1805 −3166 −537 85 422 1817 2128 −748 4860 −784 −576 2060 3065 3733 5096 3054 5576 701 2893 2616 5674 6088 5322 3712 3336 8397 2874 2056 3805 4123 6436 6638 6580 6287 8384 3701 8344 5358 5454 7587 8405 7816 5666 5425

4857 4971 4763 3646 2967 15496 727 4823 4148 1373 1231 −2328 −1758 1298 −295 4924 5465 −840 −3166 −1434 −827 −330 −77 277 −1216 1294 −2732 −2722 −458 222 101 1195 246 1999 −1374 −602 928 1942 2291 1936 710 397 2815 −575 −2411 −556 −324 2154 2496 2415 1993 4447 −1684 4004 719 1295 3551 5029 3362 1315 999

5101 5257 5061 3781 3739 15813 795 4655 4004 1704 1620 −2270 −1795 1745 −269 4788 5397 −263 −2328 −1322 −681 −300 275 611 −1242 1625 −2198 −2038 −319 349 244 1354 373 2117 −934 −417 1258 2050 2388 2011 828 557 3035 −436 −2660 −856 −568 1874 2304 2206 1711 4151 −1690 4060 860 1207 3347 2767 3333 1087 802

244 286 299 135 772 317 68 −168 −144 331 389 58 −36 447 26 −136 −68 576 837 112 146 30 352 334 −26 331 534 684 138 127 142 160 127 118 440 185 330 107 98 75 119 160 220 138 −249 −300 −244 −280 −192 −208 −282 −296 −6 56 142 −88 −204 −2262 −29 −229 −197

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C6H5O

C6H5S

C6H7

functional/method

Tυ (cm−1)

Teb (cm−1)

T0b (cm−1)



−1 T0 − Texp 0 (cm )

T0 − Te (cm−1)

CIS CASPT2(29,16) BLYP PBE B3LYP BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X PBE0 LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(5,8) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(5,8) BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(31,17) PBE BLYP

43167 24335 16354 16502 18899 22998 23553 19738 22263 17377 18271 18904 20232 19857 23706 24141 23880 21102 20492 41540 18183 15932 16478 19428 16805 23477 24252 20631 24024 17084 18094 14974 24076 26733 26884 27648 22500 21560 27161 21748 20337 20377 22075 22752 24510 25385 22715 25024 21702 22245 22030 22211 25501 25914 25177 23515 23149 42724 17667 22946 22370

42222 20006 14146 14282 16742 20799 21531 17567 20203 15072 15947 16789 17538 17696 21761 22186 21620 19055 18386 38916 15688 13662 14299 17412 14757 21252 21989 18650 22056 14845 15910 12973 21733 25801 26012 26512 20405 19545 24965 19214 18754 18754 20541 21328 23070 24362 21300 23971 20130 20684 20596 20154 24322 24829 23698 22132 21664 41439 16310 21775 21215

42315 19762 13830 14030 16528 20789 21498 17396 20189 14891 15686 16605 17484 17521 21872 22326 21814 18953 18329 39382 15474 13334 13866 17008 14438 20791 21463 18209 21614 14486 15607 12766 21246 24849 24909 25544 20007 19186 23268 18810 18041 18017 19950 20740 22512 23836 20705 23405 19427 20098 20002 19641 23860 24316 23267 21626 21163 41736 15719 21020 20489

24259 5427 −6 142 2539 6638 7193 3378 5903 1017 1911 2544 3872 3497 7346 7781 7520 4742 4132 25180 1823 −3396 −2850 100 −2523 4149 4924 1303 4696 −2244 −1234 −4354 4748 7405 7556 8320 3172 2232 7833 2420 2146 2185 3884 4561 6318 7194 4523 6833 3510 4053 3839 4020 7310 7723 6986 5324 4958 24532 −524 945 369

23314 1098 −2214 −2078 382 4439 5171 1207 3843 −1288 −413 429 1178 1336 5401 5826 5260 2695 2026 22556 −672 −5666 −5029 −1916 −4571 1924 2661 −678 2728 −4483 −3418 −6355 2405 6473 6684 7184 1077 217 5637 −114 562 562 2350 3136 4879 6171 3108 5780 1938 2492 2404 1962 6130 6638 5506 3941 3472 23247 −1881 −226 −786

23407 854 −2530 −2330 168 4429 5138 1036 3829 −1469 −674 245 1124 1161 5512 5966 5454 2593 1969 23022 −886 −5994 −5462 −2320 −4890 1463 2135 −1119 2286 −4842 −3721 −6562 1918 5521 5581 6216 679 −142 3940 −518 −151 −175 1758 2548 4321 5645 2513 5213 1236 1906 1811 1450 5668 6125 5076 3434 2972 23544 −2472 −981 −1513

93 −244 −316 −252 −214 −10 −33 −171 −14 −180 −260 −184 −53 −175 111 140 194 −102 −57 466 −214 −327 −433 −404 −319 −461 −526 −441 −442 −359 −303 −207 −487 −952 −1104 −968 −397 −359 −1697 −404 −713 −737 −591 −588 −558 −526 −595 −566 −702 −586 −593 −512 −462 −513 −431 −507 −501 297 −591 −755 −727

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Table 3. continued molecule C7H7

functional/method

Tυ (cm−1)

Teb (cm−1)

T0b (cm−1)



−1 T0 − Texp 0 (cm )

T0 − Te (cm−1)

B3LYP PBE0 BHandH BHandHLYP HSE06 VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2(35,19)

25561 27054 29842 29387 26882 24135 25046 25587 28034 29853 30088 30232 28425 28173 46323 24910

24230 25676 28487 19380 25516 22922 23652 24333 25648 15812 14814 29317 26976 26629 44974 23726

23501 24840 27526 18440 24698 22152 22958 23687 24887 14783 13698 28174 26056 25678 44986 22996

3559 5053 7841 7386 4881 2133 3044 3586 6032 7852 8086 8230 6423 6171 24321 2908

2228 3674 6485 −2622 3515 921 1651 2331 3647 −6189 −7187 7315 4974 4628 22973 1724

1499 2838 5524 −3561 2697 150 957 1686 2885 −7218 −8303 6173 4055 3677 22984 995

−729 −836 −961 −939 −818 −770 −694 −646 −761 −1029 −1116 −1142 −919 −951 11 −729

All values are given in cm−1. bGeometries optimization and zero-point vibrational corrections are calculated at the same level of theory, except for CASPT2 where we used the B3LYP zero-point correction.

a

HSE06 (2388 cm−1) clearly outperform all other functionals. The SD for those 5 functions remains lower than 2350 cm−1, while it is greater than 3000 cm−1 for LC-PBE and LC-BLYP functionals. The maximal unsigned errors (maxUE) are lower than 4700 cm−1 for B3LYP, M06-2X, ωB97X-D, and CAMB3LYP functionals. It is about 5800 cm−1 for HSE06. They are very large for all GGAs and LC-GGAs functionals. With an ASE of −423 cm−1, a rmsd of about 1940 cm−1, and a SD of less than 1900 cm−1, B3LYP is clearly the best functional to compute the 0−0 transition energies for our set of radicals. M06-2X, CAM-B3LYP, and ωB97X-D have similar performance to that of B3LYP with slight advantage for the last functional. The sign of ASE values show that B3LYP and M062X underestimate the transition energies while the HSE06, ωB97X-D, and CAM-B3LYP overestimate them. Our set of molecules can be split into two groups: group I containing the diatomic and triatomic molecules and group II containing the other molecules. Compared to results obtained for the set of all molecules, the rmsd values for group I are not very different. B3LYP (1890 cm−1), M06-2X (2249 cm−1), ωB97X-D (2477 cm−1), CAM-B3LYP (2260 cm−1), and HSE06 (2832 cm−1) always outperform all other functionals. All these functionals have almost the same performance. For group II, we obtain the same result with a rmsd = 1985 cm−1 for the B3LYP, 2115 cm−1 for M06-2X, 1986 cm−1 for ωB97XD, 2286 cm−1 for CAM-B3LYP, and 1759 cm−1 for HSE06. This behavior can be understood from the fact that the nature of transitions in group I is not very different from that of group II (see below) Adiabatic and Vibrational Correction Effects. As can be seen in Table 3, the vibrational correction (difference between T0 and Te) is generally within the 6−900 cm−1 range for most functionals. The calculation of this correction requires a lot of computing resources, especially for excited states. In order to know if the removal of this correction from the benchmarks would alter the conclusions, we report in Table 4 the statistical parameters for the difference between the adiabatic transition energies Te and the 0−0 experimental energies. These results show that the performance of the best functionals using the ZPVC remain the same without this correction. In addition, the degradation is not really significant (statistically speaking) for

radicals. Using the largest basis used here (aug-cc-pVQZ) as a reference, we calculated the statistical errors for all basis sets used in this study and they are displayed in Table 2. The averaged (on all functionals) unsigned error (AUE) between the equilibrium distances of the ground and excited states calculated using 6-311++G** and that using aug-cc-pVQZ is about 0.005 Å, which corresponds to a relative error of about 0.6%. The AUE is about 50 cm−1 (∼3%) for harmonic frequencies of ground and excited states. For 0−0 energies transitions, the AUE is about 250 cm−1 corresponding to a relative error of about 2%. One can notice that the use of double-ζ (including diffuse and polarization functions) is lesssatisfying than the triple-ζ for the calculation of 0−0 transition energies and equilibrium distances. For example, the AUE of aug-cc-pVDZ is four times larger than that of the 6-311++G** basis set. From this study, it seems that the 6-311++G** gives a good compromise between the computing time and the precision. Since there is no hydrogen atom in our selected diatomic molecules, we calculated the 0−0 transition energies for CH3 molecule, for which the experimental value is available in literature, using 6-311++G** and aug-cc-pVQZ basis sets. The AUE between the two calculations is equal to 250 cm−1, corresponding to 0.4% relative error. In conclusion, the 6-311+ +G** basis set is sufficient to estimate the properties under investigation in this work. This basis has been used to study the polyatomic molecules in our set of molecules. Comparison between Calculated T0 and Experimental Values. In Table 3, the reader can find the values of the vertical energy transitions (Tv), adiabatic energy transition (Te), and 0−0 energy transitions (T0) computed with 6-311++G** basis set for all molecules given in Table 1. In Table 4, we report the results of a statistical analysis: average signed error (ASE), average unsigned error (AUE), root-mean-square deviation (rmsd), and the standard deviation (SD), as well as the maximal and the minimal deviations obtained by comparing the transition energies of Table 3 to the experimental values. Figure 2 gives a graphical representation of the error patterns. On the basis of the consistency of the prediction as measured by rmsd, one notices that B3LYP (1935 cm−1), M06-2X (2187 cm−1), ωB97X-D (2259 cm−1), CAM-B3LYP (2272 cm−1), and 11041



Article

Table 4. Statistical Analysis Obtained from Comparison of the Theoretical and the Experimental 0-0 Energies functionnal/method

AUE (cm−1)

ASE (cm−1)

rmsd (cm−1)

SD (cm−1)

min (cm−1)

max (cm−1)

maxUE (cm−1)

2868 2791 1888 2648 3137 3496 2309 3196 2662 3186 3007 2167 3462 4065 2858 2185 2153 12328 1150

−7908 −6310 −4582 −5670 −4060 −6810 −5720 −4593 −4842 −3721 −9399 −4614 −7218 −8303 −2340 −4277 −4266 −4606 −2472

4004 4655 1844 5849 8471 7399 5815 10735 4788 7506 1934 2885 5990 10700 7747 4055 4039 40743 1625

7908 6310 4582 5849 8471 7399 5815 10735 4842 7506 9399 4614 7218 10700 7747 4277 4266 40743 2472

2893 2835 1869 2655 3238 3415 2325 3176 2642 3211 3105 2351 3389 4040 3080 2258 2160 12110 1064

−7799 −6202 −4024 −5149 −3706 −6456 −5216 −4274 −4483 −3418 −9593 −4648 −6189 −7187 −2326 −3677 −3758 −4540 −2224

4148 4823 2350 6157 8723 7668 6133 10969 4924 7841 2404 3647 6473 10942 8859 4974 4628 40167 1495

7799 6202 4024 6157 8723 7668 6133 10969 4924 7841 9593 4648 6473 10942 8859 4974 4628 40167 2224

3504 3512 2555 3020 3626 3402 2699 3392 2871 3983 3955 3791 3142 3500 2961 2565 2359 11406 2073

−7668 −6060 −3865 −2523 −1547 −1255 −1306 374 −4180 −2399 −9530 −3896 −537 85 422 −1766 −1823 −748 −1623

6090 6658 6845 10609 13017 12163 10578 15001 7008 10172 7810 9447 10132 14698 9648 7922 8298 40255 5427

7668 6658 6845 10609 13017 12163 10578 15001 7008 10172 9530 9447 10132 14698 9648 7922 8298 40255 5427

T0-T(exp) 0 BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2

2581 2331 1476 2020 2553 2863 1739 3043 2108 2916 2043 1941 3043 3621 3137 1891 1808 14081 956

−1808 −1088 −423 403 1288 927 610 2318 −554 1407 −809 −297 1421 2250 2641 623 683 13384 −160

BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2

2525 2326 1481 2008 2625 2841 1726 3053 2179 3021 2177 2056 3043 3793 3295 1951 1804 13929 871

−1672 −946 −245 553 1468 1115 768 2386 −386 1586 −708 −183 1634 2603 2892 810 783 13252 18

BLYP PBE B3LYP PBE0 BHandH BHandHLYP HSE06 HISS VSXC M06L M06 M06-2X LC-BLYP LC-PBE LC-M06L CAM-B3LYP ωB97X-D CIS CASPT2

2747 2976 2455 3357 4180 4454 3226 4931 2878 4820 3532 3802 4568 5710 5035 3321 3141 15662 2134

108 860 1846 2856 3996 4306 3070 4931 1533 4083 1548 2020 4505 5710 5035 2983 2927 15540 1744

3390 2996 1935 2679 3392 3617 2388 3948 2719 3483 3114 2187 3742 4646 3891 2272 2259 18196 1161 Te-T(exp) 0 3341 2988 1885 2712 3555 3593 2448 3973 2670 3581 3185 2358 3763 4806 4225 2399 2297 17952 1064 Tυ-T(exp) 0 3506 3615 3152 4157 5396 5488 4088 5985 3254 5703 4247 4295 5492 6697 5842 3934 3759 19277 2709

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Figure 2. Histograms of the errors computed between the 0−0 energies and the experimental ones for the 17 functionals.

Figure 3. Total root-mean-square errors (rmsd) as a function of the range-separation parameter ω in the LC-PBE functional.

the four best functionals namely B3LYP, M06-2X, CAMB3LYP, and ωB97X-D. To have an idea about the impact of the reorganization energies, we compared the calculated vertical transitions to the experimental values. Table 4 shows clearly that omitting the relaxation degrades the performance of functionals and the GGA functional, BLYP becomes the second best functional with an ASE about 108 cm−1, a rmsd of about 3500 cm−1, and a SD of about 3500 cm−1. Exact Exchange Effects. Compared to GGA functionals, an inclusion of about 20−30% of exact exchange in the functional (B3LYP, PBE0, and HSE06) improves the quality of results. However, the long-range correction using LC method provides less accurate results in most cases studied here. This inaccuracy is probably related to the very large attenuation parameter (ω = 0.47 a−1 0 ) in which the contribution of HF exchange is important at short range. However, the performance of M06-2X is similar to that of B3LYP, although the HF exchange is very important (54%). This is certainly due to a delicate balance between exchange and correlation contributions. To explore the effect of HF exchange, it is important to use the same correlation functional with a different amount of HF exchange. So, we computed the transition energies for all our set of molecules, as a function of ω ranging from 0.1 to 0.6 a−1 0 using LC-PBE functional. To explore the global exact exchange using the same correlation functional, we calculated the energies as a function of aHF, ranging from 0.1 to 0.6. aHF defines the HF exchange in the PBE0-like global hybrid functional which is given by

Figure 4. Total root-mean-square deviations (rmsd) as a function of the HF exchange fraction aHF in a PBE0-like hybrid functional.

characterize the nature of a given excitation. Λ is the spatial overlap between the occupied and virtual orbitals involved in the excitation. It is defined by Λ=

∑ia (Xia + Yia)2 Oia ∑ia (Xia + Yia)2

where Xia and Yia are the virtual-occupied and occupied-virtual transition amplitudes, respectively, and Oia is the spatial overlap integral of the modulus of the two orbitals, Oia = ∫ |ϕi(r)||ϕa(r)| dr. Λ varies from 0, for long-range excitation to 1, for very short-range transition. On the basis of the extensive benchmarks of Peach et al.,53 if Λ is less than 0.3, indicating significant long-range charge transfer character, global hybrid functionals cannot predict accurate results. In Table 5, we show our computed Λ diagnostic values using B3LYP functionals. We found that all values were above the 0.3 threshold, indicating an important overlap and no long-range charge transfer in our set of molecules, which explains the good performance for the global hybrid functionals. However, we notice the good performance for several functionals, including long-range corrections. For example, ωB97X-D and CAMB3MYP have similar performance to the best global hybrid functionals. With a rmsd of 1927 cm−1, an ASE of 458 cm−1, and a SD of 1872 cm−1, the modified long-range-corrected

ExcPBE0 − like = aHFExHF + (1 − aHF)ExPBE + EcPBE

As seen in Figure 3, the rmsd curve for LC-PBE has a minimum at ω = 0.25 a−1 0 with a rmsd error of about 1900 cm−1. This rmsd-optimized value of ω is not too far from the 0.31 a−1 0 value recommended for simultaneously describing excitation and fluorescence energies in large oligothiophenes51 and in large oligoacenes.52 The global hybrid functional (see Figure 4) has a minimum at aHF = 0.16, with a larger error of 2200 cm−1. There is practically no difference in rmsd for aHF = 0.16, 0.18, and 0.2, and 0.2 is the HF part of the exchange in B3LYP having practically the same value of our PBE0-like functional. Correlation between Error and Orbital Overlap. The Λ diagnostic introduced by Peach et al.53 can be used to 11043



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Table 5. Λ Overlap Diagnostic at TD/B3LYP/6-311++G** Level of Theory molecules cyano radical (CN) carbon monoxide cation (CO+) methyl radical (CH3) phosphino radical (H2P) hydroperoxy radical (HO2) HSO isocyanato radical (NCO) thiocyanato radical (SCN) Nitrogen trioxide (NO3) vinoxy radical (CH2CHO) phenyl radical (C6H5) phenoxy radical (C6H5O) phenylthio radical (C6H5S) cyclohexadienyl radical (C6H7) benzyl radical (C7H7)

states

Λ

A2Π B2Σ+ A2Π B2Σ+ A2A1′ A2A1 A2A′ A2A′ A2Σ+ A2Π A2E″ A2A′ A2B1 B2A2 B2A2 A2A2 B2B1

0.57 0.81 0.48 0.57 0.35 0.63 0.61 0.59 0.59 0.49 0.57 0.49 0.46 0.57 0.42 0.63 0.51

seems that the orbitals generated by KS theory should be regarded as quasi-particle states dressed with the dynamical correlation effects rather than the HF orbitals obtained without any correlation effects. To evaluate the multireference character of the ground states, we calculated the values of T1 diagnostic56 parameters at the CCSD/6-311++G** level of theory for all radicals studied in our work. Empirically, if T1 is greater than 0.02 then the ground state has a multireference character for a WFT calculation. Except for CH3 (T1 = 0.008) and H2P(T1 = 0.011) molecules, our calculated values vary from 0.028 for HO2 to 0.091 for CN, indicating a significant multireference character for these species. Despite this, the TDDFT was able to produce very good results. However, it is important to note that in the case of strong nondynamical correlation (biradical state is a known example of this case), the standard implementation of the TDDFT cannot give good results. The spin-flip approach within TDDFT was proposed for these specific cases.57

■

CONCLUSION

■

ASSOCIATED CONTENT

Using a panel of 15 radicals (17 excited states), we benchmarked 17 functionals (BLYP, PBE, B3LYP, PBE0, BHandH, BHandHLYP, HSE06, HISS, VSXC, M06L, M06, M06-2X, LC-BLYP, LC-PBE, LC-M06L, CAM-B3LYP, and ωB97X-D) in the framework of the simulation of optical 0−0 energies. The 6-311++G** basis set has been selected after the study of basis effect on 3 small molecules. For each functional and molecule, fully consistent calculations (structures, vibrational zero-point corrections, and transition energies computed with the selected approach) have been achieved. B3LYP, M062X, ωB97X-D, and CAM-B3LYP provide very similar results for the majority of radicals; with about 1900, 2167, 2153, and 2185 cm−1 mean standard deviations, respectively. Compared to GGA functionals, an inclusion of about 20−30% of exact exchange in the functional (B3LYP, PBE0, and HSE06) improves the quality of results. However, the long-range correction using the LC-GGAs method (using the standard value, 0.47 a−1 0 for ω provides less accurate results in most cases studied here). Nevertheless, the functionals CAM-B3LYP and ωB97X-D, containing also a long-range correction term, give good results. Therefore, B3LYP, M06-2X, ωB97X-D, and CAM-B3LYP functionals should be preferred when simulating 0−0 or adiabatic energies, though GGA functionals (BLYP and PBE) have a clear edge if one relies on the (faster) vertical approximation. The ZPVC certainly improves the performance of the TDDFT calculation. However, an adiabatic calculation (with relaxation of the structure of excited state) gives a good compromise between performance and computational times. Finally, it is important to note that in our set of molecules there is not any excited state with charge transfer. Nevertheless the long-range corrected functional LC-PBE with a ω = 0.25 a−1 0 has a performance very close to that of B3LYP.

functional LC-PBE (ω = 0.25 a−1 0 ) has a performance very close to that of the best hybrid functional (B3LYP). Comparison with Correlated ab initio Methods. The Kohn−Sham (KS)54 formalism was derived originally without making any assumptions about the electronic structure of the system to study. If the exact XC functional is known then closed shell systems, open shell systems, with or without multireference character would be described correctly, covering all dynamic and nondynamic correlation effects.55 Despite the fact that KS-DFT is based on a single-determinant description of the many-particle system, it will describe both singlereference and multireference systems correctly, provided the exact XC-functional is available. However, the exact XC functional is not known and therefore approximate XC functionals have to be used. It is true that the approximate exchange functionals can unintentionally mimic nondynamic electron correlation, but they do not necessarily improve the description of multireference system, mainly because of the single determinant used in KS theory. However, by the investigation of the vertical excitation energies for the first several excited states of few doublet radicals, Hirata et al.8 have shown that the TDDFT can describe excited states with substantial double excitation character at wave function theory (WFT) level, although the TDDFT is a formally exact single excitation theory based on KS orbitals of the ground state, treated as a single determinant. To investigate the double excitation character (at WFT level) and the multideterminant effect, we calculated the 0−0 transition energies for all our set of molecules, at UCIS and CASPT2 levels of theory using the 6-311++G** basis set. The calculated values are listed in Table 3. Compared to experimental values, the rmsd is about 18000 cm−1 for UCIS. By the inclusion of the dynamic and nondynamic correlation via the CASPT2 method, the rmsd is reduced to 1161 cm−1; this proves that most of the states studied here have a double excitation character. Although the CASPT2 method gives better results than TDDFT, the results of TDDFT are much better than that of UCIS. It is clear that the TDDFT includes a very important part of the correlation, neglected at the UCIS level. Although the excited states of our set of molecules have a double excitation character and are in principle not to be described by single excitation theory, it

S Supporting Information *

The calculated diatomic constants of CN and CO+ molecules using 17 functionals and nine basis sets are given in Table S1. This material is available free of charge via the Internet at http://pubs.acs.org. 11044



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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +33 (0)4 72 43 19 29. Fax: +33 (0) 4 72 43 15 07. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS In this work, we were granted access to the HPC resources of the FLMSN, “Fédération Lyonnaise de Modélisation et Sciences Numériques”, partner of EQUIPEX EQUIP@MESO.

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