The electronic spectrum of C60

Journal Pre-proofs Research paper The Electronic Spectrum of C60 Kerstin Andersson PII: DOI: Reference:

S0009-2614(19)30957-1 https://doi.org/10.1016/j.cplett.2019.136976 CPLETT 136976

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Chemical Physics Letters

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3 September 2019 12 November 2019 19 November 2019

Please cite this article as: K. Andersson, The Electronic Spectrum of C60, Chemical Physics Letters (2019), doi: https://doi.org/10.1016/j.cplett.2019.136976

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The Electronic Spectrum of C60 Kerstin Andersson Department of Mathematics and Computer Science, Karlstad University, SE-651 88 Karlstad, Sweden. Telephone: +46 54 700 1873

Abstract Using the CASSCF/CASPT2 methodology the electronic transitions HOMO → LUMO, HOMO → LUMO+1, HOMO-1 → LUMO and HOMO-2 → LUMO are determined for C60 . Comparison to experiment suggests an accuracy better than 0.3 eV. Some illustrative examples are (with experimental data within parentheses) the first excited state, 3 T2g , at 1.54 eV (1.60 eV), the two lowest-lying 1 T1u states (for spin- and symmetry-allowed transitions) at 3.09 eV (3.08 eV) and 3.19 eV (3.30 eV) and the lowest singlet excited states (1 Gg , 1 T1g , 1 T2g , 1 Hg ) at [1.84, 1.95] eV (1.90 eV with mainly 1 Gg and 1 T1g and minor 1 T2g character). Keywords: CASSCF, CASPT2, C60 , electronic spectrum 1. Introduction Since the speculations of its existence around 1970 [1], the theoretical determination of its stability in the 1970s [2], and its actual discovery in 1985 [3], the molecule C60 continues to fascinate. During the years experimental data of C60 has accumulated and it is the purpose of this study to present theoretically obtained data, mainly electronic transition energies, and compare those to experimental values. Right after its discovery, measurements on C60 were performed. In 1987 Heath et al. reported an isolated absorption band near 3.21 eV in a spectrum of C60 van der Waals complexes [4]. They assigned it to the first spin- and symmetry-allowed electronic transition of C60 . This and several other experimental results have been used in this study to either verify the theoretically Email address: [email protected] (Kerstin Andersson)

Preprint submitted to Chemical Physics Letters

November 22, 2019

obtained data or propose alternative assignments. Though fascinating in its own right, Buckminster fullerene (C60 ) and its derivatives are also useful, e.g. in organic solar cells [5], biomedicin [6] and water systems [7], and it is the hope of the author that the calculated data presented here may serve the research community in future studies of C60 . In Section 2 the methodology used for obtaining the properties of C60 is described. The quality of the results are discussed in Section 3 together with comparison to experimental data, followed by conclusions in Section 4. 2. Methodology When discussing the methodology, the high symmetry of C60 and its consequences have to be considered. These issues are analyzed in Section 2.1. This analysis is useful in setting up the calculations and is addressed in the following two subsections (see Sections 2.2 and 2.3). 2.1. Symmetry of C60 As has been discussed previously (see e.g. Ref. [8]), the point group of C60 , Ih , can be approximated by the rotation group SO(3). The 60 π-orbitals may then be classified according to an angular momentum, l, and filled according to the Aufbau principle (low l-values first). By using representation theoretical results, it can be determined which irreducible representations of Ih each l-value contains (see Table 1). In Table 1 the irreps and energies of the π-orbitals with l = 0, . . . , 6 are presented. The molecular orbitals (MOs) and energies are obtained from a self-consistent field (SCF) calculation using the software MOLCAS [9] (version 8.0). An atomic natural orbital (ANO) type basis set (ANO-S) of size 3s2p1d was applied and the geometry was given by experimental data (see Refs. [10] and [11]). In Table 1 the highest occupied (HO) and the lowest unoccupied (LU) MOs of C60 also are present. In the SO(3) approximation of Ih the π-electrons are moving on a sphere with constant potential and the energy levels are classified according to the angular momentum l with degeneracy 2l + 1 [8]. As is illustrated in Table 1 the energy levels with l = 0, . . . , 4 are all full (including in total 50 electrons) and degenerate (l = 0, 1, 2) or nearly degenerate (l = 3, 4). For l = 5 the hu orbitals are full (including in total 10 electrons) and the t1u and t2u orbitals are empty. These three sets of energy levels are non-degenerate. Further, according to the SO(3) approximation, the energy difference between level l and l − 1 should be proportional to l [8], and indeed the energy differences 2

Table 1: HOMOs and LUMOs of C60 (π-orbitals).

la 6 6 6 5 6 5 5 4 4 3 3 2 1 0 a

MO LUMO+8 LUMO+7 LUMO+3 LUMO+2 LUMO+1 LUMO HOMO HOMO-1 HOMO-2 HOMO-3 HOMO-4 HOMO-7 HOMO-11 HOMO-14

Irrep (Ih ) gg ag hg t2u t1g t1u hu hg gg gu t2u hg t1u ag

Energy (H) 0.198 0.196 0.077 0.071 0.029 -0.036 -0.294 -0.359 -0.369 -0.468 -0.484 -0.567 -0.628 -0.660

Angular momentum in spherical symmetry (see Ref. [8]).

between the five lowest levels are 0.032, 0.061, 0.091 and 0.112 (if the avarage value is used for levels l = 3 and 4), which clearly is almost proportional to l. Finally, at low and high energies the π-orbitals are interleaved with σorbitals, therefore the non-consecutive numbering of orbitals. + − 2.2. States of C60 , C60 and C60 − The ground states of C60 , C+ 60 and C60 are considered in this study. In the ground state of C60 orbitals up to the HOMO are all filled leading to a 1 Ag state. The ground state of C+ 60 is obtained when one electron is removed from the HOMO (hu ), leading to a five-fold degenerate 2 Hu state. Likewise, the ground state of C− 60 is obtained when one electron is added to the LUMO (t1u ), leading to a three-fold degenerate 2 T1u state. Excited states are only considered for the neutral molecule. The lowest excited states are expected to be derived from single electronic excitations from the HOMO (hu ) to the LUMO (t1u ), giving the singlet and triplet states T1g , T2g , Gg and Hg (determined from the direct product of hu and t1u ). The next lowest set of excited states are expected to be derived from excitations from the HOMO to the LUMO+1 or from the HOMO-1 and HOMO-2 (because of near degeneracy) to the LUMO. All these excitations

3

are treated in this study and they are summerized in Table 2. In addition, excitations from the HOMO-1 and HOMO-2 to the LUMO+1 are considered in order to facilitate the identification of states resulting from excitation from the HOMO-1 and HOMO-2 to the LUMO (see Section 3.1). Table 2: Singlet and triplet excited electronic states of C60 .

Excitation from HOMO (hu ) HOMO-1 (hg ) HOMO-2 (gg )

Excitation to LUMO (t1u ) LUMO+1 (t1g ) T1g , T2g , Gg , Hg T1u , T2u , Gu , Hu T1u , T2u , Gu , Hu T1g , T2g , Gg , Hg T2u , Gu , Hu T2g , Gg , Hg

2.3. Computational details It is the intention of this study to use the MOLCAS software [9] (version 8.0) and the complete active space (CAS) SCF and second order perturbation theory (CASPT2) methodologies for determining properties of C60 . Since only the point group D2h and its subgroups are allowed in MOLCAS, the point group D2h is used in the calculations instead of the proper point group Ih . One consequence of this is that components of an irreducible representation of Ih may span different irreducible representations of D2h (see Table 3) resulting in non-degenerate components. For example, the ground state of C+ 60 span the irreducible representation Hu of Ih and the four irreducible representations Au (two components), B1u , B2u and B3u of D2h . Table 3: The state energy average CASSCF/CASPT2 calculations of ground states.

Molecule C60 C+ 60 C− 60 a

States D2h a 1 1 Ag 2 1–2 Au 2 1 Biu 2 1 Biu

Ih b 1 Ag 2 Hu (2) 2 Hu 2 T1u

i = 1, 2, 3. b The number of components, if different from 1, within parentheses.

Another consequence of using the wrong point group is that an irreducible representation of D2h may contain components of several irreducible representations of Ih , which in practice (due to near degeneracy) results in 4

CASSCF calculations optimizing the average energy of several states (see Table 4). For example, one component of 1 Gg and two components of 1 Hg , from the HOMO → LUMO electronic excitation of C60 , span the irreducible representation Ag of D2h . Since they are close in energy they are obtained by optimizing the average energy of the second, third and fourth states (the first state is the ground state, which can be separately optimized). Table 4: State energy average CASSCF/CASPT2 calculations of excited states of C60 with HOMO, LUMO and LUMO+1 as active orbitals.

Electronic excitation HOMO → LUMO HOMO → LUMO+1 a

States D2h a 1 2–4 Ag 3 1–3 Ag j 1–4 Big j 1–3 Au j 1–4 Biu

Ih b j G +j H (2) g g j T +j T +j G +j H 1g 2g g g j G +j H (2) u u j T +j T +j G +j H 1u 2u u u

i = 1, 2, 3; j = 1, 3. b The number of components, if different from 1, within parentheses.

The states in Tables 3 and 4 are considered in a first set of calculations, where HOMO, LUMO and LUMO+1 constitute the active orbital space. In a second set of calculations HOMO-2, HOMO-1, LUMO and LUMO+1 constitute the active orbital space (with HOMO inactive) and the excited states of C60 in Table 5 are calculated (together with the ground state). Table 5: State energy average CASSCF/CASPT2 calculations of excited states of C60 with HOMO-2, HOMO-1, LUMO and LUMO+1 as active orbitals.

Electronic excitation HOMO-1 HOMO-2

)

HOMO-1 HOMO-2

)

→ LUMO → LUMO+1

Statesa 1–6 1–7 2–7 1–6 1–7

D2h b j Au j Biu 1 Ag 3 Ag j Big

Ih c 2j Gu +2j Hu (2) j T +2j T +2j G +2j H 1u 2u u u 2j Gg +2j Hg (2) j T +2j T +2j G +2j H g 1g 2g g

a

HOMO inactive. b i = 1, 2, 3; j = 1, 3. c The number of components, if different from 1, within parentheses.

Other computational details concern geometry and core orbitals. For all molecules studied the experimental geometry of C60 in the ground state is used (with bond lengths 1.391 ˚ A and 1.455 ˚ A) [10, 11]. Therefore vertical 5

energies are calculated throughout. The core orbitals are optimized in an SCF calculation (see Section 2.1) and are kept frozen in subsequent CASSCF and CASPT2 calculations. Otherwise, the default setting of parameters is used in the SCF, CASSCF and CASPT2 programs. 3. Results In this section the results from the calculations using the two active spaces described in Section 2.3 are discussed. First, in Section 3.1, the quality of the results, considering using the wrong point group, is analyzed. Secondly, in Section 3.2 the calculated values are discussed and compared to experimental data. 3.1. Quality of results In the first set of calculations, the HOMO, LUMO and LUMO+1 constitute the active orbital space, and the first ionization potential (IP) and the first electron affinity (EA) (given by the states in Table 3) are determined together with the transition energies for the neutral molecule from the ground state to the excited electronic states HOMO → LUMO and HOMO → LUMO+1 (given by Table 4). These calculations allow the computation of low-lying electronic states including a 1 T1u -state, which gives rise to one of the lowest-lying spin- and symmetry-allowed electronic transitions of C60 . The results are presented in Table 6. The usage of the wrong point group leads to mixing of states. Nevertheless, the states can be labelled with an irreducible representation of Ih . Components of Gg and Hg (or Gu and Hu ) can be identified by inspecting energies, since the two Hg (or Hu ) components, in the irreducible representation Ag (or Au ) of D2h , are supposed to be degenerate. Components of T1 and T2 1 can be distinguished by calculating dipole oscillator strengths of transitions between the HOMO → LUMO states and the HOMO → LUMO+1 states. Of the two direct products T1 ×G and T2 ×G only the latter contains T1 , and T1u is the irreducible representation spanned by the dipole operator. Dipole oscillator strengths are calculated using the RASSI program in MOLCAS at CASSCF level of theory. In the calculations the strongest T2 × G transition has an oscillator strength that is typically an 1

Gerade/ungerade subscripts are omitted in some places.

6

Table 6: Properties of C60 using ANO-S basis set of size 3s2p1d at CASPT2 level of theory (the energies are given relative the ground electronic state (1 Ag )). The geometry is given by experimental data.a The active space consists of the HOMO, LUMO and LUMO+1.

Property

State 1 T1u 1 G HOMO → LUMO+1 1 u Hu 1 T2u 3 Hu 3 T HOMO → LUMO+1 3 2u T1u 3 Gu 1 Hg 1 T2g HOMO → LUMO 1 T1g 1 Gg 3 Gg 3 Hg HOMO → LUMO 3 T1g 3 T2g 2 First IP Hu 2 First EA T1u

Calc.b (eV) 3.09d 2.96j 2.88±0.01j 2.80 2.92±0.02 2.79 2.67 2.55±0.09 1.95±0.03 1.93 1.91 1.84±0.06 2.01±0.06 1.85±0.01 1.84 1.54 7.49±0.10 2.64

a

Exptl.c (eV) 3.08e,f , 3.05g , 3.04h , 3.21i 2.99h , (3.06f ) (2.36h ) 3.02k , (2.26h , 2.07l )

2.07–2.21h 1.92 or 1.94k , 2.00h 1.90m , 1.94l , 1.78g , 1.98n (2.02–2.27h ) (1.97h ) 1.82h 1.60o , 1.56±0.03p , 1.55q 7.61±0.02r 2.689±0.008s

See Refs. [10] and [11]. b The error bars indicate the degree of non-degeneracy of components. c Unclear assignments within parentheses. d The oscillator strength is 2 at CASSCF level of theory for transition to the ground state. e In gas phase [12]. f In Ne matrix at 4 K [13, 14]. g In benzene solution [15]. h In n-hexane solution [16]. i Cold van der Waals complexes of C60 [4]. j Only components in 1 Au (D2h ) was considered due to mixing of 1 Gu and 1 Hu in 1 B1u , 1 B2u and 1 B3u (D2h ). k In Ar matrix at 5 K [17]. l In Ne matrix at 4 K [18]. m In a decaline/cyclohexane glass [19]. n In cyclohexane [20]. o In Xe matrix at 30 K (corrected for the gas-to-matrix shift) [21]. p In toluene solution [22]. q Thin films of C60 deposited on Si(100) [23]. r Vertical energy in gas phase [24]. s Adiabatic energy in gas phase [25].

7

order of magnitude larger than that of the corresonding T1 × G transition. For example, the oscillator strength of the strongest 1 T2g → 1 Gu transition is 0.09, while that of the strongest 1 T1g → 1 Gu transition is 0.008. The 1 T1u state, on the other hand, is easily identified because of the large oscillator strength of 2 for the transition to/from the ground state. Observe that the oscillator strength is not a precise tool for identifying states. The oscillator strength may be small even though the transition is allowed. In fact, the transitions from the ground state to the two lowest 1 T1u states give large oscillator strengths at the CASSCF level of theory (2 and 4, respectively), while the experimental values are rather small (0.015 ± 0.005 for the lowest state [16]). Another consequence of not using the correct point group is that the degeneracy of components of a state is broken, which is demonstrated in many states (with a peak for the 2 Hu state of the kation, where the five components differ by at most 0.20 eV). In spite of the problems, of not using the correct point group, the resulting properties that can be compared to experimental data reach an accuracy better than 0.30 eV. When comparing to experimental values, though, one has to be aware of minor contributions of vibrational energies. In the second set of calculations the active orbital space consists of the HOMO-2, HOMO-1, LUMO and LUMO+1. The excited electronic states HOMO-1,2 → LUMO2 and HOMO-1,2 → LUMO+1 (given by Table 5) are determined and the results are presented in Table 7. The comments above regarding the first set of calculations also apply to the second set of calculations. Worth mentioning is the expected overlap between the states HOMO1,2 → LUMO and HOMO → LUMO+1. Both groups of states appear in the energy range 2.4–3.2 eV. Another remark concerns the near degeneracy of some of the states, in particular singlet states, resulting from the near degeneracy of HOMO-1 and HOMO-2. For example, the two 1 Hu states differ by less than 0.1 eV. The same applies to the two 1 T2u states, the two 1 Gg states and the two 1 Hg states. A final remark is about the gap of about 1 eV between the HOMO1,2 → LUMO and HOMO-1,2 → LUMO+1 states. This gap is certainly not empty, but may contain the states HOMO → LUMO+2 and possibly the states HOMO-3,4 → LUMO. The states HOMO-1,2 → LUMO+1 were 2

HOMO-1,2 is a shorthand notion for HOMO-1 and HOMO-2.

8

mainly determined in order to facilitate the determination of irreps for the HOMO-1,2 → LUMO states. 3.2. Comparison to experimental data The first results that will be discussed are the EA end IP. The calculated EA value of 2.64 eV is in agreement with the experimental adiabatic value of 2.689 ± 0.008 eV [25]. The calculated IP value of 7.49 eV is in agreement with the experimental vertical value of 7.61 ± 0.02 eV [24]. The lowest transition energies are expected from the HOMO → LUMO excitations, which is clear form Table 6. The first excited state, 3 T2g , is calculated at 1.54 eV and is in agreement with several experimental data: 1.60 eV (in Xe matrix at 30 K and corrected for the gas-to-matrix shift) [21], 1.56 ± 0.03 eV (in toluene solution) [22] and 1.55 eV (using thin films of C60 deposited on Si(100)) [23]. The second excited state, 3 T1g , is calculated at 1.84 eV and is in agreement with the experimental value of 1.82 eV (in n-hexane solution) by Leach et al. [16]. Leach et al. also provide evidence of other triplet states below the excited singlet states in the 1.97 eV and 2.02–2.27 eV regions [16]. The 1.97 eV region may correspond to the 3 Hg state calculated at 1.85 eV and the 2.02–2.27 eV region may correspond to the 3 Gg state calculated at 2.01 eV. Vibronic analyses by Gasyna et al. position a 1 T1g state at 1.92 or 1.94 eV [17]. The calculated value for the first 1 T1g state is 1.91 eV, in agreement with the experimental value. Further, Gasyna et al. position a 1 T2u state at 3.02 eV [17]. The three calculated 1 T2u states are close in energy: 2.80, 2.97 and 3.03 eV, and close to the experimental value, which makes them all possible candidates to the measured state. Apart from assigning the first 3 T1g state, Leach et al. have made assignments (guided by calculations) of several other states of C60 , e.g., the first 1 T1g and 1 T2g states. The calculated transition energies are 1.91 and 1.93 eV, respectively, while the experimental values are 2.00 and in the range 2.07–2.21 eV, respectively [16]. Further, the transitions at 2.90 and 2.99 eV measured by Leach et al. are assigned by them to the 21 Hu and 21 Gu states [16]. The calculations position 21 Hu at 3.06 eV (almost degenerate with 31 Hu ) and 21 Gu at 2.96 eV, in fair agreement with experiment. Finally, according to Leach et al., there are states at 2.26 and 2.36 eV [16]. They propose singlet states (1 T2u and 1 Hu , respectively). However, in this region there are no states according to the calculations and the difference between

9

Table 7: Properties of C60 using ANO-S basis set of size 3s2p1d at CASPT2 level of theory (the energies are given relative the ground electronic state (1 Ag )). The geometry is given by experimental data.a The active space consists of the HOMO-2, HOMO-1, LUMO and LUMO+1.

Property

State 1 T1g 1 T2g 1 ) T2g HOMO-1 → LUMO+1 1 Gg HOMO-2 1 Gg 1 Hg 1 Hg 3 T1g 3 Gg 3 ) Hg HOMO-1 3 → LUMO+1 T2g HOMO-2 3 T2g 3 Hg 3 Gg 1 T1u 1 Gu 1 ) Hu HOMO-1 1 → LUMO Hu HOMO-2 1 T2u 1 T2u 1 Gu 3 Gu 3 Hu 3 ) T2u HOMO-1 3 → LUMO Hu HOMO-2 3 T1u 3 T2u 3 Gu

a

Calc.b (eV) 4.55 4.48 4.28 4.26±0.01 4.25±0.02 4.22±0.04 4.16±0.07 4.68 4.48±0.01 4.38±0.04 4.38 4.21 4.12±0.03 4.00±0.10 3.19d 3.09g 3.08±0.02g 3.06g 3.03 2.97 2.91g 3.10±0.02 3.09±0.05 3.06 2.92±0.03 2.69 2.49 2.47±0.01

Exptl.c (eV)

3.30e , 3.29f (3.31h ) 2.90f (3.02i ) (3.06h )

See Refs. [10] and [11]. b The error bars indicate the degree of non-degeneracy of components. c Unclear assignments within parentheses. d The oscillator strength is 4 at CASSCF level of theory for transition to the ground state. e In Ne matrix at 4 K [13, 14]. f In n-hexane solution [16]. g Only components in 1 Au (D2h ) was considered due to mixing of 1 Gu and 1 Hu in 1 B1u , 1 B2u and 1 B3u (D2h ). h In Ne matrix at 4 K [14]. i In Ar matrix at 5 K [17].

10

the calculated and measured values for the suggested states is 0.5 eV. A more probable candidate is the first 1 Hg state calculated at 1.95 eV. One of the HOMO → LUMO+1 excitations is the spin- and symmetryallowed electronic transition to the first 1 T1u state. The calculated value of 3.09 eV is in agreement with the absorption band near 3.21 eV measured by Heath et al. in 1987 and also with several other experimental data: 3.08 eV (in gas phase) [12], 3.08 eV (in Ne matrix at 4 K) [13, 14], 3.04 eV (in nhexane solution) [16] and 3.05 eV (in benzene solution) [15]. In the latter experiment by Fujitsuka et al. also the fluorescence was measured [15], and assuming that the excited singlet state in the emission is the first singlet excited state then the transition energy to the first 1 Gg state is 1.78 eV. This value is in agreement with the calculated value of 1.84 eV. The fluorescence was also measured by Catal´an and Elguero, giving a value of 1.98 eV for C60 in cyclohexane [20]. The corresponding transition during absorption is 1.99 eV, making Catal´an and Elguero conclude that C60 has a negligible Stokes’ shift [20]. Both these values are in agreement with the calculated value of 1.84 eV for the first 1 Gg state. Although the first 1 Gg state is assumed to be the lowest singlet state in the discussion in the two previous paragraphs, it might not be the case. All four singlet states resulting from the HOMO → LUMO transitions are close in energy (1.84–1.95 eV) according to the calculations and any of the four states is a plausible candidate for the lowest singlet state. Through a vibronic analysis of the fluorescence of C60 van den Heuvel et al. concluded that the lowest singlet state has ”not only T1g character” (as many authors have believed) ”but also significant Gg and possibly minor T2g character” [19]. The measured transition to the first singlet state is 1.90 eV [19], right in the middle of the calculated singlet manifold. A detailed analysis of the excitations to the lowest singlet manifold has been performed by Sassara et al. [18], giving 1.94 eV for the lowest transition. In this work also a suggestion of assignment of the lowest 1 T2u state at 2.07 eV is given. The calculated value is 2.80 eV, which is far off the suggested value. Instead the measured transition at 2.07 eV could be due to the 1 Hg state calculated at 1.95 eV. Further evidence of transitions around 2 eV is given by the gas phase measurements by Haufler et al., who recorded a complex pattern of sharp lines in the 2.00–2.08 eV region [12]. One of the HOMO-1 → LUMO excitations is the spin- and symmetryallowed electronic transition to the second 1 T1u state. The calculated value of 3.19 eV is in agreement with experimental data: 3.30 eV (in Ne matrix 11

at 4 K) by Sassara et al. and 3.29 eV (in n-hexane solution) by Leach et al. [16]. Sassara et al. also identified two transitions at 3.06 and 3.31 eV, which they assigned to 1 Gu states. The three calculated 1 Gu states are positioned at 2.91, 2.96 and 3.09 eV. Possibly the transition at 3.06 eV is due to the two lowest 1 Gu states (calculated at 2.91 and 2.96 eV) and the transition at 3.31 eV is due to the third 1 Gu state (calculated at 3.09 eV). 4. Conclusions The CASSCF/CASPT2 methodology (using an ANO-S basis set of size 3s2p1d and point group D2h ) [9] enables the calculation of the low-energetic states of the electronic spectrum of C60 . Electronic transitions originating from HOMO → LUMO, HOMO → LUMO+1, HOMO-1 → LUMO and HOMO-2 → LUMO have been determined and comparison to some experimental data suggests an accuracy better than 0.3 eV. The degree of nondegeneracy of components of an irreducible representation, because of using the wrong point group (D2h instead of Ih ), is better than 0.1 eV. The lowest excited states are the HOMO → LUMO excitations, where the triplet state manifold is lowest in energy and is somewhat overlapping with the singlet manifold. The first excited state, 3 T2g is calculated at 1.54 eV and is in agreement with experimental data (1.60 eV [21]). The elusive identity of the lowest-lying excited singlet state is confirmed by the calculations, where the four lowest-lying excited singlet states (1 Gg , 1 T1g , 1 T2g and 1 Hg ) are close in energy (1.84–1.95 eV) and any of them is therefore a plausible candidate. According to experiment the lowest vibrational level in this singlet manifold has mainly Gg and T1g character and minor T2g character [19]. The measured transition is 1.90 eV [19], right in the middle of the calculated singlet manifold. The remaining singlet state, 1 Hg , calculated at 1.95 eV, may be the cause of the transition at 2.07 eV measured by Sassara et al. [18] and the complex pattern of sharp lines in the 2.00–2.08 eV region measured by Haufler et al. [12]. It could also be the cause of the transitions at 2.26 and 2.36 eV measured by Leach et al. [16]. From the calculations it is clear that the HOMO → LUMO+1 and HOMO1,2 → LUMO excitations are overlapping. The HOMO → LUMO+1 and HOMO-1,2 → LUMO triplet states are in the energy ranges [2.55, 2.92] eV and [2.47, 3.10] eV, respectively. The corresponding energy ranges for the singlet states are [2.80, 3.09] eV and [2.91, 3.19] eV, respectively. Among the latter states are two 1 T1u states, that give spin- and symmetry-allowed 12

electronic transitions to/from the ground state. They are calculated at 3.09 and 3.19 eV and agree with experimental data (3.08 eV [12] for the first state and 3.30 eV [13, 14] for the second state). Finally, apart from electronic transition energies, the first ionization potential and the first electron affinity are calculated to 7.49 and 2.64 eV, respectively, in agreement with experiment (7.61 eV [24] and 2.69 eV [25], respectively). Acknowledgments The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at National Supercomputer Centre (NSC) at Link¨oping University. The author also gratefully acknowledges the comments and suggestions of reviewers. References [1] E. Osawa, Philos. Trans. R. Soc. London, Ser. A 343 (1993) 1. [2] I. V. Stankevich, M. V. Nikerov, D. A. Bochvar, Russ. Chem. Rev. 53 (1984) 640. [3] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R. E. Smalley, Nature 318 (1985) 162. [4] J. R. Heath, R. F. Curl, R. E. Smalley, J. Chem. Phys. 87 (1987) 4236. [5] G. Garcia-Belmonte, P. P. Boix, J. Bisquert, M. Sessolo, H. J. Bolink, Solar Energy Materials & Solar Cells 94 (2010) 366. [6] Q. Liu, Q. Cui, X. J. Li, L. Jin, Connect Tissue Res. 55 (2014) 71. [7] S.-R. Chae, S. Wang, Z. D. Hendren, M. R. Wiesner, Y. Watanabe, C. K. Gunsch, J. Membr. Sci. 329 (2009) 68. [8] F. Rioux, J. Chem. Educ. 71 (1994) 464. [9] F. Aquilante, L. de Vico, N. Ferr´e, G. Ghigo, P.-˚ A. Malmqvist, P. Neogr´ady, T. B. Pedersen, M. Pito˘ na´k, M. Reiher, B. O. Roos, L. Serrano-Andr´es, M. Urban, V. Veryazov, R. Lindh, J. Comput. Chem. 31 (2010) 224. 13

[10] P. Senn, J. Chem. Educ. 72 (1995) 302. [11] W. I. F. David, R. M. Ibberson, J. C. Matthewman, K. Prassides, T. J. S. Dennis, J. P. Hare, H. W. Kroto, R. Taylor, D. R. M. Walton, Nature 353 (1991) 147. [12] R. E. Haufler, Y. Chai, L. P. F. Chibante, M. R. Fraelich, R. B. Weisman, R. F. Curl, R. E. Smalley, J. Chem. Phys. 95 (1991) 2197. [13] A. Sassara, G. Zerza, M. Chergui, S. Leach, Astr. J. Suppl. Series 135 (2001) 263. [14] A. Sassara, G. Zerza, M. Chergui, Phys. Chem. Comm. 28 (2002) 1. [15] M. Fujitsuka, O. Ito, Y. Maeda, M. Kako, T. Wakahara, T. Akasaka, Phys. Chem. Chem. Phys. 1 (1999) 3527. [16] S. Leach, M. Vervloet, A. Despr`es, E. Br´eheret, J. P. Hare, T. J. Dennis, H. W. Kroto, R. Taylor, D. R. M. Walton, Chem. Phys. 160 (1992) 451. [17] Z. Gasyna, P. N. Schatz, J. P. Hare, T. J. Dennis, H. W. Kroto, R. Taylor, D. R. M. Walton, Chem. Phys. Lett. 183 (1991) 283. [18] A. Sassara, G. Zerza, M. Chergui, F. Negri, G. Orlandi, J. Chem. Phys. 107 (1997) 8731. [19] D. J. van den Heuvel, G. J. B. van den Berg, E. J. J. Groenen, J. Schmidt, I. Holleman, G. Meijer, J. Phys. Chem. 99 (1995) 11644. [20] J. Catal´an, J. Elguero, J. Am. Chem. Soc. 115 (1993) 9249. [21] A. Sassara, G. Zerza, M. Chergui, Chem. Phys. Lett. 261 (1996) 213. [22] R. R. Hung, J. J. Grabowski, J. Phys. Chem. 95 (1991) 6073. [23] A. Lucas, G. Gensterblum, J. J. Pireaux, P. A. Thiry, R. Caudano, J. P. Vigneron, P. Lambin, W. Kr¨atschmer, Phys. Rev. B 45 (1992) 13694. [24] D. L. Lichtenberger, M. E. Jatcko, K. W. Nebesny, C. D. Ray, D. R. Huffman, L. D. Lamb, MRS Proceedings 206 (1990) 673. [25] X.-B. Wang, C.-F. Ding, L.-S. Wang, J. Chem. Phys. 110 (1999) 8217. 14

Highlights The following three points I consider as highlights of the article ”The Electronic Spectrum of C60 ”: • Orbitals: The resemblance of C60 with a sphere has consequences on the degeneracy of orbitals. This is not a new insight, but it is demonstrated in the article. • Active spaces: The above point has consequences on the choices of active orbital spaces for calculations of excited electronic states of C60 . • Comparison with experimental data: The accuracy of calculated values with many experimental data is striking. Further, the near degeneracy of the states in the lowest singlet excited state manifold is in accordance with experiment.