Supplementary Material

Adrian Heil, Christel M. Marian∗ Institute of Theoretical and Computational Chemistry, Heinrich-Heine-University Düsseldorf, Universitätsstraÿe 1, 40225 Düsseldorf, Germany

∗ [email protected] S2

N O NH O S

N+ O- Pyridine Nitromethane Pyrrole Furan Thiophene Cyclopentadiene O

N+ -O

s-trans Butadiene Acrolein Nitrobenzene Styrene Benzene Naphthalene •+ N O - • +O C- H H O O H2C Cl S Carbon monoxide Water Nitrogen dioxide Chloromethyl Thioformaldehyde Ethylene N N O

O N N O O O H2N O s-trans Glyoxal Formaldehyde Acetone Acetaldehyde Formamide s-tetrazine

S C O C Thioketene Fulvene o-Xylene Benzocyclobutene Ketene p-Xylylene S HN

O N O H Dibenzothiophene o-Xylylene Thymine

Figure S1. Molecules used in the tting set S3

S HO O S

S OH• p-Benzosemiquinone Phenanthrene Fluorene Tetrathiafulvalene Hydroxyl

•HC all-trans-1,3,5,7-octatetraene Styrene o-Xylylene Ethylenyl

NH2

O H N F N

N N H Be Dibenzofuran Perylene Adenine Fluorobenzene Beryllium monohydride F

F F

FF F :N O Tetracene 1,2,4,5-Tetrafluorobenzene 1,2,3,4-Tetrafluorobenzene Nitric oxide

H N S •+ S N - F O O Terthiophene Carbazole 1,5-Hexadiene-3-yne Ethylfluoride Nitrogen dioxide

O•

Acenaphthylene Acenaphthene 2,3-Benzofluorene Phenoxyl Cl S F Cl S F Pentacene Azulene Bithiophene Dichlorodifluoromethane

C•

Ovalene Hexacene Phenyl

Figure S2. Molecules with doublet states used for assessment S4

O C O S C S S C O O S O Carbon dioxide Carbon disulfide Carbonyl sulfide sulfur dioxide Ethylene Propene

F Isobutene cis-2-butene trans-2-butene Trimethylethylene Tetramethylethylene Fluoroethylene F F

F F F F F Cl F F F FF F F F F 1,1-difluoroethylene cis-1,2-difluoroethylene trans-1,2-difluoroethylene Trifluoroethylene Tetrafluoroethylene Chlorotrifluoroethylene F

Cl F Chloroethylene Acetylene Propyne 1-butyne 3,3,3-trifluoropropyne 1,3-butadiene

C trans-1,3-pentadiene cis-2-trans-4-hexadiene 1,3-cyclohexadiene 1,5-hexadiene 1,4-cyclohexadiene Propadiene F F F

F F F F

F F

F F F FF Benzene Fluorobenzene o-difluorobenzene 1,3,5-trifluorobenzene 1,2,3,4-Tetrafluorobenzene 1,2,4,5-Tetrafluorobenzene F

F F F

F F O S NH F F N F F F N Pentafluorobenzene Hexafluorobenzene Furan Thiophene Pyrrole Azomethane

S N O N N N N+ N O- Cl Cl Azo-tert-butane Nitromethane Thiophosgene 1,3-cyclopentadiene Pyridine Pyrazine O

N+ S N O NN O -O HN

S N N NH2 Pyrimidine s-Triazine Acetone Acetamide Nitrobenzene Dithiosuccinimide

Figure S3. Molecules with singlet and/ or triplet states used for assessment S5

Table S1. Basis sets used in the tting of the doublet states Molecule Basis set Benzocyclobutene SV(P)a s-trans Butadiene cation TZVPb Methylidine QZVPPc Chloromethyl def2-TZVPDd Dibenzothiophene SV(P)a Ethylene TZVPPb Fulvene SV(P)a Ketene TZVPb Thioketene def2-TZVPDd Formaldehyde aug-cc-pVTZe Water TZVPPb Naphthalene cation SV(P)a Nitrogen dioxide aug-cc-pVTZe o-Benzoquinone SV(P)a o-Xylene SV(P)a o-Xylylene cation SV(P)a p-Xylylene SV(P)a Thymine SV(P)a a Ref. [1], b Ref. [2], c Ref. [3], d Ref. [4],e Ref. [5]

Table S2: Vertical excitation energies and molecular states with singlet, doublet and triplet multiplicities used for parameter optimization. Note that the molecule name corresponds to the geometry used in doublet case, o-Xylylene+ refers to electronic absorption of the cation, o-Xylene refers to photoelectron spectroscopy from the neutral molecule, therefore the neutral ground state geometry was used. All energies are given in eV. The energies of Lyskov's parameterization are given as reference for closed shell system State Type Experiment DFTMRCI-R DFT/MRCI-A DFT/MRCI-A tight

Pyridine C5H5N 1 ∗ a b 1 B2 n → π 4.44 , 4.45 4.86 4.84 4.84 1 ∗ a c 1 B1 n → π 4.99 , 4.99 5.13 5.11 5.09 1 ∗ a a c d 2 A1 π → π 6.38 , 6.30 , 6.38 , 6.32 6.31 6.26 6.22

Nitromethane H3C − NO2 21A0 π → π∗ 6.25e, 6.23e, 6.23f 6.34 6.32 6.34 13A” n → π∗ 3.80f 3.79 3.75 3.72

Pyrrole C4H4NH 3 g h i 1 B1 π → Ry 4.21 , 4.21 , 4.2 4.21 4.20 4.16

Furan C4H4O 1 ∗ g h j 1 B1 π → π 6.04 , 6.06 , 6.04 6.09 6.06 6.06 1 ∗ h j 3 A1 π → π 7.82 , 7.8 7.90 7.90 7.92 3 ∗ i h 1 B1 π → π 4.0 , 3.99 3.94 3.92 3.89 3 ∗ i h 1 A1 π → π 5.2 , 5.22 5.15 5.14 5.10

Thiophene C4H4S 1 ∗ h g k 2 A1 π → π 5.48 , 5.43 ,5.52 5.48 5.47 5.46 3 ∗ i g h 1 B1 π → π 3.7 , 3.74 , 3.75 3.77 3.64 3.65 3 ∗ h i k 1 A1 π → π 4.62 , 4.6 , 4.7 4.58 4.46 4.47

Cyclopentadiene C5H5 1 ∗ g l m 1 B2 π → π 5.22 , 5.26 , 5.33 5.29 5.25 5.24 3 ∗ g l 1 B2 π → π 3.15 , 3.10 3.16 3.13 3.11 Continued on next page S6

s-trans Butadiene C4H6 1 ∗ g h o 1 Bu π → π 5.91 , 5.9 , 5.92 5.75 5.71 5.70 1 ∗ p 2 Ag π → π 6.27 6.32 6.31 6.28 3 ∗ g o n 1 Bu π → π 3.24 , 3.22 , 3.2 3.18 3.15 3.13 3 ∗ g o n 1 Ag π → π 4.92 , 4.91 , 4.95 4.99 4.98 4.97 s-trans Butadiene cation + C4H6 2 ∗ ah 1 Au π → π 2.32 - 2.68 2.65 2 ∗ ah 2 Au π → π 4.20 - 4.49 4.50

Acrolein C3H4O 11A” n → π∗ 3.76q, 3.75r, 3.71s 3.58 3.57 3.56 13A” n → π∗ 3.08s, 3.05t 3.33

Nitrobenzene C6H5NO2 1 ∗ u v 2 A1 π → π 5.17 , 5.11 4.78 4.75 4.74

Styrene H5C6CH = CH2 21A0 π → π∗ 4.43w, 4.43x 4.53 4.51 4.52 13A0 π → π∗ 3.40w 3.19 3.16 3.18

Benzene C6H6 1 ∗ y z A 1 B3u π → π 4.80 , 4.90 , 4.89 5.00 4.98 4.96 1 ∗ y z 1 B2u π → π 6.25 , 6.03 6.12 6.07 6.04 1 ∗ y,A z 2 B3u π → π 6.95 , 6.87 6.92 6.91 6.92 3 ∗ y A 1 B2u π → π 3.90 , 3.89 4.13 4.10 4.10 3 ∗ y A 2 B3u π → π 5.59 , 5.69 5.49 5.44 5.42

Naphthalene C10H8 1 ∗ B C 1 B3u π → π 4.0 , 3.97 4.18 4.17 4.52 1 ∗ B C 1 B2u π → π 4.45 , 4.45 4.55 4.52 4.52 1 ∗ B C 2 B3u π → π 5.89 , 5.89 5.76 5.74 5.77 1 ∗ C 2 B2u π → π 6.14 6.09 6.07 6.09 Naphthalene cation + C10H8 2 ∗ ad 1 B3g π → π 1.84 - 2.00 1.99 2 ∗ ad 1 B2g π → π 2.69 - 2.75 2.73 2 ∗ ad 2 B2g π → π 3.25 - 3.39 3.38 Carbon monoxide CO 21A n → π∗ 8.39D 8.16 8.13

Water H2O 1 E F 1 B2 n → Ry 7.5 , 7.4 7.99 7.98 7.98 2 ∗ aa 1 A1 σ → n 2.11 - 1.96 1.94 2 ∗ aa 1 B1 n → n 5.93 - 6.02 6.01

Nitrogen dioxide NO2 2 ∗ ab 1 B2 n → n 2.81 − 2.85 - 2.73 2.73 2 ∗ ab 1 B1 n → n 3.1 - 3.27 3.24

Chloromethyl CH2Cl 2 ac 1 A1 π → Ry 4.99 - 4.66 4.65 1 ∗ ac 2 B2 π → π 6.20 - 6.45 6.47 2 ac 3 B2 π → Ry 6.59 6.83 6.85

Thioformaldehyde H2C = S 1 ∗ G 1 A2 n → π 2.03 2.22 2.18 2.15 3 ∗ G 1 A2 n → π 1.80 1.95 1.91 1.89 Continued on next page S7

Ethylene H2C = CH2 1 H I 1 B1u π → Ry 7.11 , 7.11 7.20 7.19 7.17 2 ∗ ai 1 B1g σ → π 2.3 - 2.31 2.30 2 ∗ ai 1 Ag σ → π 4.2 - 4.26 4.27 2 ∗ ai 1 B3u σ → π 5.3 - 5.28 5.25 2 ∗ ai 1 B2u σ → π 8.6 - 8.42 8.42 3 ∗ J 1 B2u π → π 4.36 4.36 4.32 4.28

Ethylene dimer 2x[H2C = CH2] 1 ∗ ∗ 3 2 A ππ → π π 2xE(1 B2u) 8.71 8.64 8.67 3 ∗ ∗ 3 3 A ππ → π π 2xE(1 B2u) 8.71 8.64 8.67 s-trans Glyoxal HOC − COH 1 ∗ K 1 Au n → π 2.8 2.71 2.68 2.67 1 ∗ K 1 Bg n → π 4.2 3.97 3.96 3.94 3 ∗ K 1 Au n → π 2.5 2.37 2.35 2.34

Formaldehyde H2C = O 1 ∗ L M 1 A2 n → π 3.79 , 3.94 3.83 3.81 3.79 1 N O P 1 B1 n → Ry 7.09 , 7.09 , 7.10 7.11 7.13 7.14 1 N O P 2 B1 n → Ry 7.97 , 7.98 , 7.98 7.93 7.93 7.94 2 ∗ ae 1 B2 π → π 3.22 - 3.32 3.30 2 ∗ ae 1 A1 n → π 4.97 - 5.01 5.00 3 ∗ L M 1 A2 n → π 3.50 , 3.50 3.49 3.47 3.44

Formaldehyde dimer 2x[H2C = O] 1 ∗ ∗ 3 2 A nn → π π 2xE(1 A2) 7.09 7.03 7.08 1 ∗ ∗ 1 5 A nn → π π 2xE(1 A2) 7.80 7.73 7.83 3 ∗ ∗ 3 1 A nn → π π 2xE(1 A2) 7.09 7.02 7.08 3 ∗ ∗ 3 1 4 A nn → π π E(1 A1) + E(1 A2) 7.44 7.37 7.45 3 ∗ ∗ 3 1 5 A nn → π π E(1 A2) + E(1 A2) 7.44 7.37 7.45

Acetone C3H6O 1 ∗ L Q 1 A2 n → π 4.38 , 4.37 4.26 4.27 4.24 1 L Q R 1 B2 n → Ry 6.36 , 6.35 , 6.36 6.47 6.51 6.53 3 ∗ L Q 1 A2 n → π 4.18 , 4.16 3.97 3.97 3.94

Acetaldehyde C2H4O 11A” n → π∗ 4.27L 4.09 4.07 4.05 13A” n → π∗ 3.97L, 3.91R 3.78 3.76 3.74

Formamide HCONH2 11A” n → π∗ 5.65S 5.38 5.38 5.37 13A” n → π∗ 5.30T 5.14 5.13 5.11

s-Tetrazine C2H2N4 1 ∗ U X W 1 B1u n → π 2.35 , 2.25 , 2.34 2.36 2.32 2.32 1 ∗ U V 1 Au n → π 3.60 , 3.42 3.62 3.59 3.59 1 ∗ U W 1 B3u π → π 4.92 , 5.0 5.11 5.09 5.07 3 ∗ U,X Y 1 B1u n → π 1.69 , 1.70 1.85 1.81 1.80 3 ∗ X 1 Au n → π 2.95 3.37 3.34 3.33

Thioketene H2CCS 2 ∗ af 1 B1 n → π 2.43 - 2.50 2.50 2 ∗ af 2 B2 π → π 3.25 - 3.37 3.34 2 ∗ af 1 A1 n → π 5.66 - 5.84 5.86 Continued on next page S8

Fulvene C5H4 = CH2 2 ∗ ag 1 B2 π → π 1.18 - 1.03 1.03 2 ∗ ag 1 B1 σ → π 3.74 - 3.53 3.53 2 ∗ ag 3 B2 π → π 4.44 - 4.29 4.28

o-Xylene C6H4(CH3)2 2 ∗ aj 1 A2 π → π 0.52 - 0.26 0.26 2 ∗ aj 1 A1 σ → π 2.44 - 2.80 2.77 2 ∗ aj 2 B1 σ → π 2.64 - 2.95 2.96 2 ∗ aj 1 B2 π → π 3.10 - 2.93 2.94

Benzocyclobutene C8H8 2 ∗ aj 1 A2 π → π 0.58 - 0.31 0.31 2 ∗ aj 1 A1 σ → π 2.12 - 2.30 2.31 2 ∗ aj 1 B2 σ → π 2.76 - 2.62 2.63 2 ∗ aj 2 B1 π → π 3.08 - 2.87 2.84

Ketene H2CCO 2 ∗ ak 1 B1 n → π 4.21 - 4.23 4.21 2 ∗ ak 2 B2 π → π 4.97 - 5.24 4.97 2 ∗ ak 3 B1 n → π 6.45 - 6.39 6.45 2 ∗ ak 1 A1 n → π 7.07 - 6.77 6.77

p-Xylylene C6H4(CH2)2 2 ∗ al 1 B2g π → π 1.83 - 1.86 1.86

Dibenzothiophene C12H8S 2 ∗ am 1 A2 π → π 0.41 - 0.22 0.22 2 ∗ am 2 A2 π → π 1.33 - 1.30 1.30 2 ∗ am 3 B2 π → π 2.03 - 1.88 1.88 2 ∗ am 4 B2 π → π 2.72 - 2.67 2.66 2 ∗ am 1 A1 n/σ → π 3.45 - 3.42 3.43 + o-Xylylene cation C6H4(CH2)2 2 ∗ aj 1 B2 π → π 1.44 - 1.74 1.72 2 ∗ aj 2 A2 π → π 2.36 - 2.53 2.52 2 ∗ aj 2 B2 π → π 2.82 - 2.90 2.91

Thymine C5H6N2O2 12A0 n/σ → π∗ 0.95an - 0.63 0.64 22A” π → π∗ 1.26an - 1.24 1.25 22A0 n/σ → π∗ 1.70an - 1.42 1.44 32A” π → π∗ 3.08an - 3.17 3.18 Methylidyne radical CH• 2 2 2 1 2 ap A ∆ (1σ) (2σ) (3σ) (1πx) 2.88 - 2.42 2.41 2 2 2 1 2 ap A ∆ (1σ) (2σ) (3σ) (1πy) 2.88 - 2.75 2.73 2 + 2 2 1 2 2 2 1 2 ap C Σ (1σ) (2σ) (3σ) (1πx) ,(1σ) (2σ) (3σ) (1πy) 3.94 - 3.64 3.65 a Ref. [6], b Ref. [7], c Ref. [8], d Ref. [9], e Ref. [10], f Ref. [11], g Ref. [12], h Ref. [13], i Ref. [14],j Ref. [15] k Ref. [16], l Ref. [17], m Ref. [18], n Ref. [19], o Ref. [20], p Ref. [21], q Ref. [22], r Ref. [23], s Ref. [24], t Ref. [25] u Ref. [26], v Ref. [27], w Ref. [28], x Ref. [29], y Ref. [30], z Ref. [31], A Ref. [32], B Ref. [33], C Ref. [34], D Ref. [35] E Ref. [36], F Ref. [37], G Ref. [38], H Ref. [39], I Ref. [40], J Ref. [41], K Ref. [42], L Ref. [43], M Ref. [44] N Ref. [45] O Ref. [46], P Ref. [47], Q Ref. [48], R Ref. [49], S Ref. [50], T Ref. [51], U Ref. [52], V Ref. [53], W Ref. [54], X Ref. [55] aa Ref. [56], ab Ref. [57], ac Ref. [58], ad Ref. [59], ae Ref. [60], af Ref. [56], ag Ref. [60], ah Ref. [61], ai Ref. [62], aj Ref. [63] ak Ref. [64], al Ref. [65], am Ref. [66], an Ref. [67] S9

Table S3: Vertical excitation energies of selected doublet states (in eV), comparison between experimental data and our new all-multiplicitiy

Hamiltonian with δEsel = 1.0Eh and δEsel = 0.8Eh (tight) parameters. Neutral molecules were calculated at their neutral singlet ground state geometry and are experimentally measured by photoelectronspec- troscopy. Charged molecules (cations and anions) are calculated at their corresponding cation or anion geometry of their doublet ground state, experimentally measured by electronic absorption spectroscopy. State Exp. DFT/MRCI-A tight DFT/MRCI-A p-Benzosemiquionone aniona 2 ∗ 1 B1u(π → π ) 2.87 2.69 2.68 2 ∗ 1 Au(π → π ) 3.22 3.09 3.10 2 2 B1u(π → 3pz) 3.92 3.81 3.84 Perylene cationb 2 ∗ 1 B3g(π → π ) 1.56 1.68 1.70 2 ∗ 1 B2g(π → π ) 1.69 1.64 1.66 2 ∗ 2 B3g(π → π ) 1.93 1.85 1.87 2 ∗ 2 B2g(π → π ) 2.32 2.22 2.25 2 ∗ 4 B3g(π → π ) 3.73 3.75 3.80 Fluorene cationc 2 ∗ 1 B1(π → π ) 0.86 0.88 0.88 2 ∗ 2 A2(π → π ) 1.18 1.07 1.06 2 ∗ 2 B1(π → π ) 1.93 1.95 1.95 2 ∗ 1 B2(σ → π ) 3.13 3.23 3.24 2 ∗ 3 B1(π → π ) 3.64 3.83 3.81 2 ∗ 4 B1(π → π ) 4.08 3.63 3.63 Tetrathiafulvalened 2 ∗ 1 B2g(π → π ) 2.14 2.22 2.22 2 ∗ 1 B3g(π → π ) 2.51 2.93 2.93 2 ∗ 2 B2g(π → π ) 2.86 3.01 3.01 2 ∗ 2 B3g(π → π ) 3.67 3.71 3.71 1,5-Hexadiene-3-ynee 2 ∗ 1 Bu(n → π ) 1.16 1.32 1.31 2 ∗ 1 Bg(π → π ) 2.19 2.06 2.05 2 ∗ 2 Au(π → π ) 3.20 3.25 3.22 2 ∗ 1 Ag(n → π ) 4.14 4.14 4.13 2 ∗ 2 Bu(π → π ) 4.44 4.44 4.45 2 ∗ 5 Ag(π → π ) 6.14 6.30 6.33 2 ∗ 7 Ag(π → π ) 7.19 7.11 7.13 all-trans 1,3,5,7-octatetraene cationf 2 ∗ 1 Au(π → π ) 1.67 1.75 1.73 2 ∗ 2 Au(π → π ) 2.77 2.77 2.78 2 ∗ 2 Bg(π → π ) 2.97 2.84 2.83 all-trans 1,3,5,7-octatetraeneg 2 ∗ 1 Au(π → π ) 1.82 1.76 1.75 2 ∗ 2 Bg(π → π ) 3.10 2.89 2.87 2 ∗ 1 Ag(n → π ) 3.93 4.16 4.16 o-Xylyleneh 12B(π → π∗) 1.90 1.96 1.94 22A(π → π∗) 2.35 2.42 2.41 Continued on next page S10

State Exp. DFT/MRCI-A tight DFT/MRCI-A 22B(π → π∗) 2.79 2.87 2.88 32A(π → π∗) 3.74 4.04 4.05 52B(π → π∗) 4.40 4.62 4.63 Styreneh 22A”(π → π∗) 0.80 0.75 0.74 32A”(π → π∗) 2.09 2.18 2.17 12A0(n/σ → π∗) 3.04 3.40 3.40 22A0(n/σ → π∗) 3.70 3.58 3.59 32A0(n/σ → π∗) 4.38 4.31 4.32 Carbazolei 2 ∗ 1 A2(π → π ) 0.39 0.26 0.25 2 ∗ 2 A2(π → π ) 1.46 1.43 1.42 2 ∗ 2 B2(π → π ) 2.15 2.03 2.02 2 ∗ 3 B2(π → π ) 3.19 3.21 3.20 Dibenzofuranj 2 ∗ 1 B2(π → π ) 0.25 0.20 0.20 2 ∗ 2 A2(π → π ) 1.26 1.01 1.00 2 ∗ 2 B2(π → π ) 1.97 1.84 1.83 2 ∗ 1 A1(n/σ → π ) 3.12 3.51 3.53 Dichlorodiuoromethanek 2 ∗ 1 A2(n → n ) 0.3 0.19 0.18 2 ∗ 1 B2(n → n ) 0.9 0.61 0.60 2 ∗ 1 A1(n → n ) 1.2 1.35 1.34 Adeninel 0 12A (n/σ → π∗) 0.98 0.95 0.96 22A”(π → π∗) 1.07 1.21 1.21 0 22A (n/σ → π∗) 1.98 1.87 1.88 32A”(π → π∗) 2.04 2.11 2.12 0 32A (n/σ → π∗) 2.88 2.72 2.74 Fluorobenzenem 12A”(π → π∗) 0.4 0.38 0.38 2 ∗ 1 B1(π → π ) 2.9 3.10 3.10 2 ∗ 2 B2(n/σ → π ) 2.9 2.95 2.92 2 ∗ 1 A1(n/σ → π ) 3.6 3.70 3.71 2 ∗ 2 B1(n/σ → π ) 4.5 4.78 4.80 2 ∗ 3 B1(n/σ → π ) 5.2 5.28 5.29 2 ∗ 2 A1(n/σ → π ) 5.8 5.68 5.69 2,3-Benzouorene cationn 22A”(π → π∗) 0.80 0.70 0.70 42A”(π → π∗) 1.72 1.79 1.79 62A”(π → π∗) 2.70 2.83 2.84 82A”(π → π∗) 3.06 3.39 3.42 112A”(π → π∗) 3.49 3.90 3.94 Tetracene cationj 2 ∗ 1 B3g(π → π ) 1.43 1.49 1.50 2 ∗ 1 B2g(π → π ) 1.65 1.61 1.61 2 ∗ 4 B3g(π → π ) 3.14 3.41 3.46 Continued on next page S11

State Exp. DFT/MRCI-A tight DFT/MRCI-A 1,2,4,5-Tetrauorobenzene 2 ∗ 1 B3g(π → π ) 0.7 0.61 0.61 2 ∗ 1 B1u(π → π ) 3.0 2.85 2.83 2 ∗ 1 B3u(n → π ) 4.2 4.30 4.32 2 ∗ 1 Ag(n → π ) 4.2 4.13 4.14 2 ∗ 1 B1g(n → π ) 5.1 5.19 5.21 1,2,3,4-Tetrauorobenzenem 2 ∗ 1 B2(π → π ) 0.0 0.32 0.32 2 ∗ 2 B2(π → π ) 2.7 2.81 2.78 2 ∗ 1 A1(n → π ) 3.9 3.96 4.13 2 ∗ 1 B1(n → π ) 3.9 4.13 3.95 2 ∗ 2 A1(n → π ) 4.8 4.95 4.95 Ethyluoridem 0 12A (n/σ → π∗) 0.5 0.06 0.05 0 22A (n/σ → π∗) 1.6 1.48 1.49 22A”(π → π∗) 2.1 2.12 2.13 0 32A (n/σ → π∗) 3.6 3.17 3.18 32A”(π → π∗) 4.7 4.88 4.89 0 42A (n/σ → π∗) 4.7 4.42 4.43 0 52A (n/σ → π∗) 8.6 8.51 8.52 Hydroxyl radicalo A2Σ+(σ → π∗) 4.09 4.05 4.05 B2Σ+(σ → 3s) 8.65 8.45 8.46 Acenaphthylene cationp 2 ∗ 1 A2(π → π ) 0.80 0.70 0.71 2 ∗ 2 B2(π → π ) 1.15 1.12 1.12 2 ∗ 2 A2(π → π ) 2.53 2.51 2.51 Acenaphthene cationj 2 ∗ 2 B1(π → π ) 1.88 2.02 2.01 2 ∗ 2 A2(π → π ) 2.74 2.75 2.73 Bithiophene cationq 2 ∗ 2 Au(π → π ) 2.10 2.13 2.12 2 ∗ 3 Au(π → π ) 2.92 2.93 2.94 Phenanthrene cationq 2 ∗ 2 A2(π → π ) 1.38 1.49 1.49 2 ∗ 2 B2(π → π ) 1.95 2.01 2.02 2 ∗ 3 B2(π → π ) 2.63 2.60 2.61 2 ∗ 3 A2(π → π ) 2.91 3.15 3.18 2 ∗ 4 B2(π → π ) 3.13 3.29 3.33 2 ∗ 4 A2(π → π ) 3.59 3.45 3.48 Beryllium monohydrideo A2Π(n → π∗) 2.48 2.40 2.39 2 B Π(n → 3px) 6.32 6.33 6.34 Ethylenylr 12A”(π → n∗) 3.08 3.04 3.02 0 62A (σ → n∗) 7.37 7.25 7.23 0 72A (σ → n∗) 7.53 7.60 7.59 Continued on next page S12

State Exp. DFT/MRCI-A tight DFT/MRCI-A Nitric oxidej A2Σ+(π → 3s) 5.92 6.06 6.11 2 + D Σ (π → 3pz 7.03 7.11 7.15 Nitrogen dioxides 2 ∗ 2 B1(π → π ) 5.22 4.83 4.80 2 2 A1(n/σ → 3s) 7.50 7.37 7.38 2 5 B1(n/σ → 3px) 8.60 8.63 8.66 2 3 B2(n/σ → 3py) 8.60 8.74 8.78 2 5 A1(n/σ → 3pz) 8.60 8.68 8.72 2 7 B1(n/σ → 3s) 9.66 9.31 9.34 Azulene cationj 2 ∗ 2 A2(π → π ) 2.58 2.77 2.76 2 ∗ 3 B2(π → π ) 3.37 3.38 3.36 Azulenet 2 ∗ 1 B2(π → π ) 1.07 1.11 1.10 2 ∗ 2 A2(π → π ) 2.64 2.64 2.63 2 ∗ 3 B2(π → π ) 3.42 3.28 3.27 Pentacene cationj 2 ∗ 1 B1u(π → π ) 1.26 1.17 1.16 2 ∗ 1 Au(π → π ) 1.30 1.27 1.27 2 ∗ 3 Au(π → π ) 2.91 3.05 3.08 Tertiophene cationq 2 ∗ 1 B2(π → π ) 1.46 1.52 1.52 2 ∗ 3 B2(π → π ) 2.25 2.24 2.25 Styrene cationh 32A”(π → π∗) 2.14 2.26 2.25 52A”(π → π∗) 3.75 3.75 3.73 Ovalene cationj 2 ∗ 1 Au(π → π ) 1.0 1.25 1.26 2 ∗ 1 B1u(π → π ) 1.27 1.49 1.49 2 ∗ 3 B1u(π → π ) 2.21 2.17 2.19 2 ∗ 4 Au(π → π ) 2.68 2.88 2.93 Hexacenet 2 ∗ 1 B3g(π → π ) 1.11 1.07 1.07 2 ∗ 1 B1u(π → π ) 1.7 1.68 1.69 2 ∗ 2 Au(π → π ) 2.12 1.97 1.99 2 ∗ 5 B2g(π → π ) 2.92 2.91 2.96 2 ∗ 3 B3g(π → π ) 2.92 2.77 2.81 2 ∗ 5 Au(π → π ) 3.51 3.35 3.42 2 ∗ 9 B1u(π → π ) 3.51 3.79 3.88 2 ∗ 7 B3g(π → π ) 3.86 3.79 3.86 Phenoxylu 2 ∗ 1 B1(π → π ) 1.10 0.90 0.89 2 ∗ 1 A2(π → π ) 1.98 2.29 2.27 2 ∗ 2 B2(π → π ) 3.12 3.33 3.31 2 ∗ 2 A2(π → π ) 4.20 4.40 4.41 2 ∗ 4 B2(π → π ) 5.18 5.17 5.19 Continued on next page S13

State Exp. DFT/MRCI-A tight DFT/MRCI-A 2 ∗ 3 A2(π → π ) 5.95 5.72 5.75 Phenylv 2 ∗ 1 B2(π → π ) 2.43 2.65 2.63 2 ∗ 3 A1(π → π ) 5.27 4.94 4.93 2 ∗ 3 B1(π → π ) 5.86 5.69 5.67 a Ref. [68], b Ref. [59], c Ref. [69], d Ref. [70], e Ref. [60], f Ref. [61], g Ref. [71], h Ref. [63], i Ref. [72],j Ref. [66] k Ref. [73], l Ref. [67], m Ref. [74], n Ref. [75], o Ref. [76], p Ref. [77], q Ref. [78], r Ref. [79], s Ref. [80], t Ref. [81] u Ref. [82], v Ref. [83]

Table S4: Vertical excitation energies of selected singlet and triplet states (in eV) compared between experimental data, the original parameterization (DFT/MRCI-S), Lyskov's redesign parameters and our new all- multiplicity-operator (DFT/MRCI-M) State Exp. DFT/MRCI-S DFT/MRCI-R DFT/MRCI-A DFT/MRCI-A (tight) Carbon dioxidea 1 ∗ ∆u(π → π ) 8.6 8.75 8.80 8.80 8.83 Carbon disuldea 3 ∗ ∆u(π → π ) 3.36 3.39 3.37 3.33 3.38 1 ∗ ∆u(π → π ) 3.91 4.02 4.02 3.99 4.00 1 ∗ Πg(π → π ) 6.79 6.71 6.69 6.69 6.77 Carbonyl suldea 3 + ∗ Σu (π → π ) 4.94 4.86 4.95 4.90 4.91 1 ∗ ∆u(π → π ) 5.53 5.57 5.59 5.54 5.55 1 Πg(π → 3s) 7.36 7.29 7.31 7.28 7.29 1 + ∗ ? Σu (π → π ) 8.02 8.26 8.11 8.06 8.08 Sulfur dioxidea 3 ∗ B2(n → π ) 3.40 3.23 3.27 3.23 3.23 1 ∗ A2(n → π ) 4.31 4.28 4.29 4.26 4.28 Ethylenea 3 ∗ B3u(π → π ) 4.32 4.25 4.35 4.32 4.32 1 ? B1u(π → 3s) 7.28 7.23 7.20 7.19 7.19 1 ∗ B3u(π → π ) 7.6 7.64 7.51 7.46 7.46 1 ∗ B3g(σ → π ) 8.25 8.21 8.25 8.22 8.23 1 ? B1u(π → 3d) 8.91 8.94 8.89 8.87 8.87 Propenea 3A0(π → π∗) 4.28 4.14 4.35 4.10 4.11 1A00(π → 3s) 6.6 6.61 6.64 6.63 6.64 1A0(π → π∗) 7.17 7.22 7.16 7.13 7.13 Isobutenea 3 ∗ A1(π → π ) 4.22 4.04 4.30 4.29 4.29 1 B1(π → 3s) 6.1 6.26 6.29 6.30 6.30 1 ∗ ? A1(π → π ) 6.71 6.69 6.67 6.66 6.66 1 ∗ A1(π → π ) 7.78 7.88 7.88 7.87 7.87 cis-2-Butenea 3 ∗ B2(π → π ) 4.21 4.27 4.41 4.38 4.38 1 ∗ B2(π → π ) 7.10 7.40 7.32 7.29 7.29 trans-2-Butenea 3 ∗ Bu(π → π ) 4.24 4.18 4.37 4.35 4.35 Continued on next page S14

State Exp. DFT/MRCI-S DFT/MRCI-R DFT/MRCI-A DFT/MRCI-A (tight) 1 Au(π → 3s) 6.3 6.31 6.33 6.33 6.33 1 ∗ Bu(π → π ) 6.95 7.12 7.06 7.03 7.03 Trimethylethylenea 3A0(π → π∗) 4.16 4.02 4.31 4.30 4.30 1A00(π → 3s) 5.76 5.87 5.91 5.91 5.91 1A0(π → π∗) 6.47 6.59 6.60 6.59 6.60 1A0(π → π∗) 6.97? 7.20 7.19 7.18 7.19 Tetramethylethylenea 3A(π → π∗) 4.10 4.07 4.27 4.25 4.25 1A(π → 3s) 5.55 5.67 5.70 5.70 5.70 1A(π → π∗) 6.57 6.65 6.64 6.62 6.62 Fluoroethylenea 3A0(π → π∗) 4.40 4.34 4.46 4.43 4.43 1A00(π → 3s) 7.02 7.09 7.09 7.07 7.07 1A0(π → π∗) 7.50 7.66 7.53 7.49 7.49 1A00(π → 3p) 8.08 7.88 7.89 7.88 7.88 1A00(π → 3d) 8.87 9.02 8.98 8.97 8.97 1,1-diuoroethylenea 3 ∗ A1(π → π ) 4.63 4.47 4.68 4.66 4.66 1 B2(π → 3s) 6.95 6.98 6.99 6.98 6.99 1 ∗ A1(π → π ) 7.50 7.67 7.59 7.57 7.57 1 A2(π → 3p) 8.23 7.93 7.98 7.97 7.98 cis-1,2-diuoroethylenea 3 ∗ B1(π → π ) 4.43 4.43 4.53 4.49 4.50 1 B2(π → 3s) 6.52 6.43 6.48 6.47 6.47 1 ∗ B1(π → π ) 7.82 7.96 7.80 7.76 7.76 1 A1(π → 3p) 8.38 8.29 8.24 8.22 8.22 1 B2(π → 3d) 9.01 8.81 8.79 8.78 8.78 trans-1,2-diuoroethylenea 3 ∗ Bu(π → π ) 4.18 4.27 4.37 4.34 4.34 1 Bg(π → 3s) 6.44 6.60 6.70 6.68 6.69 1 ∗ Bu(π → π ) 7.39 7.68 7.53 7.48 7.48 Triuoroethylenea 3A0(π → π∗) 4.43 4.39 4.61 4.60 4.60 1A00(π → 3s) 6.56 6.47 6.56 6.55 6.55 1A0(π → π∗) 7.65 7.85 7.77 7.74 7.74 1A00(π → 3p) 7.98 7.79 7.78 7.77 7.77 1A00(π → 3d) 8.74 8.77 8.78 8.76 8.77 Tetrauoroethylenea 3 ∗ B2u(π → π ) 4.68 4.85 4.84 4.80 4.80 1 B1u(π → 3s) 6.62 6.81 6.80 6.78 6.78 1 ∗ B2u(π → π ) 8.84 8.79 8.58 8.52 8.52 Chlorotriuoroethylenea 3A0(π → π∗) 4.43 4.41 4.51 4.48 4.48 1A00(π → 3s) 6.51 6.54 6.59 6.58 6.58 Chloroethylenea 3A0(π → π∗) 4.08 4.08 4.22 4.19 4.20 Continued on next page S15

State Exp. DFT/MRCI-S DFT/MRCI-R DFT/MRCI-A DFT/MRCI-A (tight) 1A0(π → π∗) 6.72? 6.82 6.73 6.70 6.71 Acetylenea 3 + ∗ Σu (π → π ) 5.2 5.23 5.43 5.41 5.42 3 ∗ ∆u(π → π ) 6.0 5.72 5.87 5.85 5.85 1 ? Πu(π → 3s) 8.16 7.96 7.91 7.90 7.90 Propynea 3A0(π → π∗) 5.2 5.16 5.48 5.48 5.49 3A0(π → π∗) 5.8 5.62 5.89 5.89 5.89 1A0(π → 3s) 7.18 6.91 6.94 6.94 6.95 1-butynea 3A0(π → π∗) 5.2 5.11 5.45 5.45 5.46 3A0(π → π∗) 5.8 5.57 5.85 5.85 5.86 3,3,3-triuoropropynea 3A0(π → π∗) 5.0 5.27 5.39 5.36 5.36 3A0(π → π∗) 5.8 5.75 5.81 5.78 5.78 1A0(π → 3s) 8.80 8.55 8.52 8.50 8.51 1,3-butadienea 3 ∗ Bu(π → π ) 3.22 3.13 3.18 3.15 3.17 3 ∗ Ag(π → π ) 4.91 4.84 4.99 4.98 5.03 1 ∗ ? Bu(π → π ) 5.92 5.88 5.75 5.70 5.71 trans-1,3-pentadienea 3A0(π → π∗) 3.14 3.14 3.19 3.16 3.18 3A0(π → π∗) 4.87 4.83 4.99 4.98 5.03 1A0(π → π∗) 5.80 5.83 5.70 5.66 5.66 cis-2-trans-4-hexadienea 3A0(π → π∗) 3.11 3.12 3.17 3.14 3.15 3A0(π → π∗) 4.8 4.93 5.02 5.01 5.05 1A0(π → π∗) 5.69 5.71 5.61 5.58 5.59 1,3-cyclohexadienea 3B(π → π∗) 2.94 2.89 2.95 2.92 2.93 1B(π → π∗) 4.94 5.06 4.96 4.92 4.92 1,5-hexadienea 3A(π → π∗) 4.25 4.00 4.29 4.29 4.33 1,4-cyclohexadienea 3 ∗ B2g(π → π ) 4.29 4.15 4.35 4.34 4.35 1 ∗ B3g(π → π ) 6.15 6.27 6.30 6.30 6.31 1 ∗ B3g(π → π ) 7.95 7.88 7.89 7.89 7.90 Propadiene (allene)a 3 ∗ A1(π → π ) 4.28 4.38 4.61 4.60 4.61 1 ∗ A1(π → π ) 7.24 7.16 7.14 7.13 7.14 Benzenea 3 ∗ B2u(π → π ) 3.90 4.12 4.13 4.11 4.14 3 ∗ B3u(π → π ) 5.59 5.51 5.49 5.44 5.45 1 ∗ B3u(π → π ) 4.80 5.04 5.00 4.97 5.01 1 ∗ B2u(π → π ) 6.25 6.23 6.12 6.07 6.08 Fluorobenzenea 3 ∗ A1(π → π ) 3.90 4.16 4.19 4.16 4.17 Continued on next page S16

State Exp. DFT/MRCI-S DFT/MRCI-R DFT/MRCI-A DFT/MRCI-A (tight) 3 ∗ B1(π → π ) 5.72 5.63 5.61 5.56 5.57 1 ∗ B1(π → π ) 4.78 5.02 5.00 4.98 5.01 1 ∗ A1(π → π ) 6.23 6.26 6.15 6.10 6.11 o-diuorobenzenea 3 ∗ B1(π → π ) 3.92 4.17 4.19 4.16 4.17 3 ∗ A1(π → π ) 5.67 5.65 5.63 5.58 5.58 1 ∗ A1(π → π ) 4.76 5.04 5.01 4.99 5.02 1 ∗ B1(π → π ) 6.22 6.30 6.19 6.14 6.15 1,3,5-triuorobenzenea 3 ∗ A1(π → π ) 3.95 4.23 4.25 4.22 4.23 3 ∗ B2(π → π ) 5.62 5.61 5.59 5.54 5.54 1 ∗ B2(π → π ) 4.87 5.13 5.12 5.10 5.13 1 ∗ A1(π → π ) 6.20 6.34 6.24 6.19 6.19 1,2,3,4-tetrauorobenzenea 3 ∗ A1(π → π ) 3.95 4.19 4.20 4.17 4.17 1 ∗ A1(π → π ) 4.85 5.07 5.04 5.02 5.05 1 ∗ B1(π → π ) 6.43 6.37 6.27 6.21 6.22 1,2,4,5-tetrauorobenzenea 3 ∗ B2u(π → π ) 4.0 4.18 4.18 4.16 4.19 1 ∗ B3u(π → π ) 4.69 4.97 4.95 4.93 4.95 1 ∗ B2u(π → π ) 6.3 6.40 6.28 6.23 6.24 Pentauorobenzenea 3 ∗ A1(π → π ) 3.90 4.26 4.22 4.19 4.20 1 ∗ B1(π → π ) 4.79 5.09 5.07 5.06 5.08 1 ∗ A1(π → π ) 6.36 6.44 6.33 6.27 6.28 Hexauorobenzenea 3 ∗ B2u(π → π ) 3.86 4.18 4.15 4.11 4.16 1 ∗ B3u(π → π ) 4.80 5.15 5.10 5.07 5.10 1 ∗ B2u(π → π ) 6.36 6.51 6.38 6.32 6.33 Furana 3 ∗ B1(π → π ) 3.99 3.82 3.94 3.92 3.93 3 ∗ A1(π → π ) 5.22 4.99 5.15 5.14 5.16 1 ∗ B1(π → π ) 6.06 6.15 6.09 6.06 6.07 1 ∗ A1(π → π ) 7.82 7.95 7.90 7.89 7.90 Thiophenea 3 ∗ ? B1(π → π ) 3.66 3.75 3.78 3.75 3.76 3 ∗ A1(π → π ) 4.62 4.55 4.56 4.53 4.55 1 ∗ A1(π → π ) 5.48 5.50 5.47 5.45 5.47 1 ∗ A1(π → π ) 7.05 7.10 7.11 7.11 7.13 Pyrrolea 3 ∗ B1(π → π ) 4.21 4.04 4.21 4.20 4.21 1 A2(π → 3s) 5.22 5.11 5.07 5.06 5.07 1 ∗ B1(π → π ) 5.98 6.01 5.98 5.97 5.98 Azomethanea 3 ∗ Bg(n → π ) 2.75 2.62 2.79 2.75 2.76 1 ∗ Bg(n → π ) 3.50 3.44 3.48 3.44 3.45 1 Bu(n → 3p) 6.71 6.92 6.97 6.65 6.66 Continued on next page S17

State Exp. DFT/MRCI-S DFT/MRCI-R DFT/MRCI-A DFT/MRCI-A (tight) 1 ∗ Bu(π → π ) 7.8 8.07 8.02 8.00 8.01 Azo-tert-butanea 3 ∗ Bg(n → π ) 2.67 2.30 2.61 2.60 2.61 3 ∗ Bu(π → π ) 4.9 4.70 5.08 5.09 5.10 1 ∗ Bg(n → π ) 3.37 3.10 3.28 3.28 3.28 1 Bu(n → 3p) 7.3 7.39 7.53 7.56 7.58 Nitromethanea 3A00(σ → π∗) 3.8 3.69 3.79 3.75 3.77 1A00(σ → π∗) 4.45 4.35 4.36 4.31 4.32 1A0(π → π∗) 6.23 6.31 6.34 6.32 6.34 Thiophosgenea 3 ∗ A1(π → π ) 3.1 3.08 3.10 3.06 3.06 1 ∗ A2(σ → π ) 2.61 2.65 2.68 2.64 2.64 1 ∗ A1(π → π ) 4.89 4.91 4.88 4.86 4.87 1,3-cyclopentadienea 3 ∗ B2(π → π ) 3.1 3.11 3.16 3.13 3.14 1 ∗ B2(π → π ) 5.26 5.39 5.29 5.25 5.25 Pyridineb 1 ∗ A2(n → π ) 5.43 5.39 5.43 5.41 5.46 Pyrazineb 3 ∗ B1u(n → π ) 3.33 3.55 3.61 3.56 3.60 3 ∗ B2g(n → π ) 4.59 4.87 4.96 4.94 4.97 1 ∗ B1u(n → π ) 3.83 4.03 4.04 4.01 4.05 1 ∗ B2g(n → π ) 5.19 5.33 5.45 5.43 5.48 Pyrimidinec 1 ∗ A2(n → π ) 4.62 4.83 4.86 4.85 4.88 s-Triazinec 1 ∗ B2(n → π ) 4.59 4.62 4.66 4.64 4.66 Acetoned 3 ∗ A2(n → π ) 4.16 3.70 3.97 3.97 3.98 1 ∗ A2(n → π ) 4.37 4.11 4.26 4.27 4.27 Acetamidee 1A”(n → π∗) 5.44 5.27 5.43 5.43 5.45 Nitrobenzenef 1 ∗ A2(n → π ) 3.65 3.32 3.52 3.49 3.53 Dithiosuccinimideg 3 ∗ B1(n → π ) 2.63 2.44 2.58 2.56 2.59 1 ∗ B1(n → π ) 2.77 2.65 2.73 2.71 2.73 1 ∗ A2(n → π ) 3.04 2.84 2.93 2.91 2.93 a For experimental energies see Ref. [84] and references therein. b Ref. [85], c Ref. [8], d Ref. [48], e Ref. [86], f Ref. [26], g Ref. [87] ? Band maximum. S18

ADDITIONAL DETAILS FOR IMPLEMENTATION

Besides the diagonal matrix elements (same space part, same spin coupling) discussed in the main part of this article, three o-diagonal classes exist: a) same space part, dierent spin coupling, b) one-electron dierence in space part and c) two-electron dierence in space part. [88] For DFT/MRCI one therefore needs the equations for the CI matrix elements [89] as well as the corresponding DFT/MRCI corrections. Case a): The same equation as in the diagonal case with same spin coupling is used.

X 1X 1X 1 1 H = E + F ∆w + V ∆w ∆w + V − ∆w ∆w + w w − w + ηji nn SCF ii i 2 ijij i j 2 ijji 2 i j 2 i j i ij i i6=j i6=j

1X 1 1 + V ∆w ∆w + w w − w (1) 2 iiii 2 i j 2 i i i i

Only the part depending on the exchange-like integral in conjunction with ji contributes to this o-diagonal element. ηij For dierent spin coupling, we arrive at the correction

DF T/MRCI (2) Hnn = (1 − pF )Hnn Case b): For matrix elements between two congurations diering in one electron occupation

j 0 j 1 0 j 1 0 j j kj Hnn0 = Fijηi + Σ Vikjk∆wkηi + Σ Vikkj(− ∆wkηi + wkηi − ni + ηik ) k6=i,j k6=i,j 2 2 1 0 1 0 j 1 0 1 0 j (3) + Viiij( 2 ∆wi + 2 wi)ηi − Vijjj( 2 ∆wj + 2 wi − 1)ηi we use a damping function to avoid double counting of the electron correlation

DF T/MRCI p1 0 (4) Hnn0 = 5 5 Hnn 1 + (p2 · δ) arctan(p2 · δ) Case c): For matrix elements between two congurations diering in a two electron occupation

jl lj −1 (5) Hnn” = (Vikjlηik + Vikljηik)[(1 + δik)(1 + δjl)] the same damping function as in the one electron dierence is used

DF T/MRCI p1 (6) Hnn” = 5 5 Hnn” 1 + (p2 · δ) arctan(p2 · δ) S19

References

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