ELSEVIER Journal of Molecular Structure (Theochem) 311 (1994) 55~68

THEOCHEM

Electronic excitations in fullerenes: Jahn-Teller distortedstructures of C60

Peter R. Surjan'v":", Laszlo Udvardi", Karoly Nemeth"

"Department of Theoretical Chemistry, Eotvos University, H-1518 P.O. Box 32, Budapest 112, Hungary"Quanium Theory Group, Institute of Physics, Technical University, Budafoki ut 8, H-1521 Budapest, Hungary

(Received 19 July 1993; accepted 14 September 1993)

Abstract

The ground state wavefunction of the neutral icosahedral C60 molecule belongs to the total symmetric one-dimen­sional Ag representation. However, the degeneracy of the HOMO is five-fold while the LUMO is triply degenerate. Thismeans that the lowest excited and ionized many-electron states will also be degenerate, and thus they are subjected toJahn-Teller distortions. In this work we used a simple model Hamiltonian to study the extent and energy of thedistortions arising due to excitations. For the -r-electrons we used a Pariser-Parr-Pople type wavefunction augmentedby an empirical potential to describe the (J cores. The zr-electron part of the Hamiltonian depends on the bond lengthswhich can be optimized by means of simple gradient techniques. The excited states are described by the Tamm - Dancoffapproximation (all single CI). In order to locate the distorted states on the energy hypersurface, the degenerate excitedstates were reduced according to those subgroups of the Ih group that contain one-dimensional (that is, Jahn--Tellerinactive) irreducible representations in an excitation subspace. Distorted structures of D 2h , D 3d , D Sd and Th symmetrieswere determined. The extent of the distortions is small, the largest change in bond lengths being 0.02 A. The Jahn Tellerdistortion energies were found to be typically I~3 kcal mol": Singlet and triplet excited state spectra were computedusing the CNDO/S-CI method. Comparison of the calculated spectra with experimental results shows evidence of Jahn~Teller distortions.

1. Introduction

In spite of the thousands of papers that arepresently being published on fullerenes, thesehighly symmetric novel materials still continue toattract investigations in basic research (1-4]. Adetailed theoretical description of the electronicand geometrical structure of individual groundstate C60 molecules has already been elucidatedat various levels, including semiempirical and

*Corresponding author.

more sophisticated (ab initio) quantum chemicalmethods [5-9], including the approximatetreatment of electron correlation [10, 11]. Animportant feature of C60 is that it belongs tothe icosahedral point group lh' which is thelargest finite subgroup of the rotation group03; thus this molecule exhibits the highest possiblefinite symmetry in nature. The ground statewavefunction is of symmetry A 1g (the dominantconfiguration comes from completely filled levels)which is one-dimensional. The lh group possessesthree-, four- and five-dimensional representations

0166-1280/94/$07.00 © 1994 Elsevier Science B.V. All rights reservedSSDI 0166-1280(93)03516-A

56 P.R. Surjan et al.jJ. Mol. Struct, (Theochem) 311 (1994) 55-68

2. Models and methods

standing of experimental results, however, assumesa theoretical analysis. A step in this direction is thelocalization of the distorted structures on theexcited energy hypersurfaces. This was done inthe present study, using a simple modelHamiltonian (xHUGE model). The methodologyoutlined here can easily be extended to moresophisticated levels. For the quantitative calcula­tion of the excitation spectra at the geometriesobtained by xHUGE, we used the CNDOjS-CImethod.

The first attempts to describe the electronicstructure of fullerenes by quantum chemical meth­ods used a simple Huckel-type picture [28-30]. Atthis level, bond length optimization can be per­formed by the Coulson-Golebiewski method [31]through a bond order-bond length relationship.The method is then physically equivalent to theSu-Schrieffer-Heeger (SSH) model [32] and wasused formerly by Longuet-Higgins and Salem(LHS) [33]. The latter model, with careful repara­metrization, has been successfully applied in ourlaboratory to describe conjugated systems [34-39].

Excited states of fullerenes, however, cannot beproperly described by a one-electron model. Calcu­lations done using the Pariser-Parr-Pople (PPP)method [40,41] or all-valence electron modelsturned out to be useful [15,16,18,42-44]. TheLHS (or SSH) method has also been extended tothe PPP level [45]; the extended method is closelyrelated to the QCFFjPl method [46] used by Negriet al. [15,16].

In the present study we used the model calledxHUGE (extended Hubbard with geometry opti­mization). The model is simple enough to allow onelarge scale calculation to be made on huge clusters.The Hamiltonian is specified as [47]:

H = L afJ- L a;"afJ-" + L 'YfJ-(n~/2 - nfJ-)fJ-" fJ-

as well. These are realized in ionized and excitedstates.

Ionized states of the C60 have received particularattention [12,13] because of their importance in thedoping process. The specific feature of the ionizedstates is that they exhibit Jahn-Teller distortions,as they correspond to degenerate many-electronstates. The distorted states of C60 have recentlybeen determined by Koga and Morokuma [14]using ab initio UHF calculations. These authorsfound that typical distortions are approximately0.01 A in bond lengths and approximately 0.01 eVIn energy.

Much less is known about the energy and extentof distortions in the excited states [15-17]. Simplegroup theory tells us [18] that the low-lying excitedstates of C60 are all degenerate, as they result fromh., ---> t lu type transitions in a one-electron picture.The possible symmetries of the lowest excitedmany-electron wavefunctions are given by thedecomposition of the h., ® t lu direct productwhich reads

hu®tlu=Gg+Hg+Tlg+T2g (1)

These states are four-, five- and three-dimensional,respectively; thus, upon excitation, a Jahn-Tellerdistortion takes place. (This is a remarkabledifference from the electronic structure ofbenzene, where a transition between the degener­ate HOMO-LUMO levels contains a one­dimensional representation.)

According to the Jahn-Teller theorem [19] nostable states of a system with a degenerate elec­tronic wave function may exist (with the exceptionof diatomic molecules) or, more precisely, if theelectronic state is degenerate the energy expansionin terms of normal coordinates will contain linearterms. This means that the system will distort intoa nuclear configuration which splits the degener­acy. The theorem does not tell us anything aboutthe extent of the distortion, although the Jahn­Teller active modes can be obtained perturbatively(15,16]. (A detailed group theoretical analysis oficosahedral molecules can be found in Ref. 20.)

Experimental evidence for the existence ofdistorted excited states is provided by ESR[21,22], triplet-triplet [23-27] or singlet-singlet[23] excited state spectra. Interpretation and under-

L f3(r;) L(at"ai2"+ He); "

L 'Y(r;)n;1 n;2 + Lf(r;)i

(2)

P.R . Surjan CI ot.il . Mol. S truct. t Theochem ) 311 ( 1994) 55-68 57

Table IEmpirical parameters of the xHU G E mod el

A = -179.7003eV( = 0.32628838 A, =3.536eV( = 2.24 (eV A) - lro = 1.54AI( = 0.21 A

where HC is the hermitian conjugate. The indices /lrun over all atomic sites (71' orbitals), i labels thebonds (il and iz are the two sites of bond i), anda is the spin label. The creation and annihilationoperators are identified by a~a and a (1.a for the site /lwith spin a, and the particle number operators aredefined as:

Table 2Test calc ula tions done using the xHUG E model

Molecule Ob servabl e" xHUG E Exp .

Benzene r 1.40 1.406.£ e B2u) 4.9 4.96.£ e Btu) 3.7 3.8

Butad iene 16.£ 5.2 5.6

Naphthalene 16.£ 4.0 4.0ra 1.426 1.421rb 1.378 1.36 1r, 1.432 1.425rd 1.419 1.410

C60 gro und sta te rl 1.4023 1.40r2 1.4463 1.45

"Bond length s are in angstro ms and energies in electronvolts.

be equivalent with the fulfillment of Coulson'srelationship between the bond lengths r, and the-r-electron bond orders P,.:

(3(r) = - Aexp(-rl ( ) (4)

where A and ( are the two new parameters. Thefirst-neighbour electron repulsion integrals arepar ameterized as:

The Q:(1. and ' (1. terms are parameters of the model(Coulomb and on-site repulsion integrals, respec­tively). The bond lengths are denoted by r., andf3 (r), f er) and f (r) are unique functions of the dis­tance. The f3 integrals are assumed to have anexponential form:

where ro is the length of a single bond and (ro - ,, )is that of a double bond. The bo nd orders in theelectronic sta te III are defined by:

Pi = L (llilataai,al Ill) (8)a

where q (1. is the net charge on atom u-It is clear from the form of the Hamiltonian that

it accounts for first-neighbour interactions only("extended Hubbard model") .

2.1. Parameterization

In dealing with C60 , we took all sites as beingequivalent. Accordingly, all Q: (1. va lues were set tozero and all on- site repulsion integrals wereassigned to a uniform value, " Thus, the six inde­pendent parameters in the model are A, (, f , " "oand n, These were determined by fitting the experi­mental energy gap of polyacetylene , (CH).n to1.5 eV, its bandwidth to 10eV, and its equilibriumbond lengths to 1.455 A and 1.363 A. The lowestsinglet and triplet transition energies of eth ylene(7.7 and 4.5eV) were cho sen as two other condi­tions. The resulting parameter values are listed in

Equati on s (6) and (7) represent a system of differ­ential equ ation s for f (r), the solut ion of which is:

f (ri) = ,(ri)(PT12- qj,% )

+ 2(3(rj)(( / "- - P,. ) + frolog (r,. ) - r;]/ ( f ,,2)

(9)

(3)

(5)

(6)

(7)

f er) = I / (a )

oE = 0Br,

r, = ro - ,,-P,.

where e is an effective dielectric constant.The above Hamiltonian handles only the 71' elec­

trons explicitly, while j'(r.) is an empirical functiondescribing the effect of the a core. It can be deter­mined as follows. If we define the energy of thesystem as the expectation value of H , we candem and that the full geometry optimization ofthe model , that is the conditions

58 P.R. Surjdn el al.II. Mol. Stru ct. ( Theochem) 311 ( 1994) 55-68

2.2. Wavefunctions and densities

Table 1. The reliability of the parameter set can bechecked against the results given in Table 2.

j\lJo) = '0t'0i · · · '0t;2Ivac) (10)

11,3\lJJ) = ~?= C~ ('0t<> '0i<> ± '0ttj'0itjl\lJo) (11 )

V L., ->k

Accordingly, the excited state bond orders (densitymatrix elements) are evaluated as :

J 0 o: VJ (12)P J-L/I=PJ-L /I - J-L/I+ J-L/I

where p2/1 = 2:fccCiJ-LCi /l is the ground state bond

order with the LCAO coefficients CiJ-L'

2.3. Symmetry forcing

It is evident from Eqs. (12)-(14) that the excitedstate bond orders are not unique if state J is degen­erate. As p J enters the formula for the a core(Eq . (9)) it has a crucial role in the optimization ofthe excited state geometry (see also Eq . (7)). Inorder to locate various Jahn-Teller distortedgeometries on the hypersurface, we have to specifyunambiguously the degenerate excited states. Forthis , we used the following procedure.

(i) The symmetry loss upon distortion is assumedto be the smallest which is sufficient to split thedegeneracy of the corresponding state. Thismeans that various subgroups of I h have to beconsidered. The chain of subgroups is shown inFig . 1.

(ii) The multidimensional irreducible representa­tions of the I h group, which give rise to Jahn-Tellerdistortions, are reduced according to the subgroupsymmetries. The reductions are summarized inTable 3.

(iii) Of the different subgroup representations,the one-dimensional ones have to be pickedup; these are no longer Jahn- Teller active. Itis sufficient to consider the first such one­dimensional representation along the chain of sub­groups shown in Fig. 1.

(13)occ vir!

O~/I = :L:L C~CtCiJ-LCj/lij k

Exact diagonalization of the 1l"-electron part ofthe Hamiltonian (Eq . (2)) is very demandingcomputationally [48]. For this reason we describedthe excitations of the system by using the Tamm­Dankoff approximation. This means that theground state wavefunction is determined at theHartree-Fock level, while the excited statesare written as linear combinations of all singleexcitations:

and

(14)

The model outlined above is certainly very crudefrom a quantitative point of view. However, it isthe highest level of topological description wherethe system is determined merely by bonds. Anyfurther refinement of the model would destroy itsconceptual and practical simplicity. For conju­gated systems we can expect reliable results forbond lengths and, in most cases , for the energiesof the lowest electronic transitions (see Table 2).Transition probabilities or higher transition ener­gies cannot be evaluated. Therefore, to comparethe calculated spectra of the distorted systemswith experimental results we performed CNDOjScalculations with a CI including 2100 singly excitedconfigurations. Fig. I. Chains of subgroups of the icosahedral group lh '

P.R . Surjdn et al.l.J. Mol . Stru ct. ( Theochem } 311 ( / 994) 55-68 59

Table 3Reduction of the multidimensional irreducible representations ofh acc ording to subgro up symmetries for g sta tes (a simi lar table can becompiled for the u states)

Point group Irreducible representations

Eig + A2gE I +A2E) +A2E l +ATgEg + Ag

TEg + A2gBig + B2g + B3gE + A2

E+A2Ag + 2Bg

BI + B2 + B,E +AA+2BA' + 2A"3Ag

E2g + A2gE2 +A2

E2 +A2E2 + ATg

Eg + Ag

TEg + A2gBig + B2g + B,sE + A2E+A2

Ag + 2Bg

BI + B2 + B,E+AA+2BA' +2A"3Ag

Eig + E2gEt + E2

EI +E2E l + E2

Tg + Ag

s, + 2Ag

T +AEg + A ig +A2gAg + Big + B2g + B,gE + AI +A2E+A I +A,2Ag + 2BgA + BI + B2 + B,E +2A2A+2B2A' +2A"4Ag

E ig + E2g + A igE 1 +E2 +AIE I + E2 +A IE I + E2 + AEg + Tg

2Eg +AgE+T2Eg +Al g

2Ag + Big + B2g + B,s2E+A I

2E+A I3Ag + 2Bg2A + B I + B2 + B,2E +A3A +2B3A' + 2A"5A,

The energies at various excited states and theJahn- Teller distortion energies of C60 are given

3. Results

where E~ is the energy of the icosahedral verticallyexcited degenerate "parent" representation, Ejis the icosahedral relaxation energy, and EJ isthe energy of the distorted state. The parentrepresentation is the lowest irreducible repre ­sentation of I h which contains the selected sub ­group symmetry upon reduction (see Table 3).The icosahedral relaxation energy EJ is obtainedby exciting the system electronically but pre­serving the icosahedral symmetry of the nuclearconfiguration.

We emphasize that the above procedure doesnot give information on the physical dis­tortion process. It is merely a recipe by whichthe critical points of various symmetry typescan be localized in a systematic manner. Investi­gation .of the distortion process is much morecomplicated and, as will be seen in the nextsection, the critical points are very close to eachother, which makes dynamical Jahn-Tellereffects probable.

(16)

(15)

f:j.EJ - T = £1- (E) - EJ)

(iv) Collecting all relevant one-dimensional sub­group representations, the degenerate icosahedralstate should be projected by:

_ A I"", A

I}}J = PI}}J = - ~ X(R)RI}} Jn R

where n is the order of the subgroup, R is thesymmetry operation, and X(R) is the character ofthe selected irreducible representation. Equation(15) ensures that the projected wavefunction willbe of the desired symmetry, and thus the geometryoptimization will converge to the desired state.

Actual application ofEq. (15) implies the knowl­edge and coding the effect of all subgroup symmetryoperations on the wavefunction. In practice, inmany cases a simpler procedure will do the samejob. One can construct the matrices of someselected symmetry operations within the degener­ate subspace, and find a symmetry-adapted linearcombination of the degenerate wavefunctions bydiagonalizing these low-order matrices.

(v) Having obtained the optimum geometryin the desired state, one evaluates the Jahn­Teller distortion energy f:j.EJ _ T according to thedefinition:

60 P.R . Surjdn et al.jJ . Mol. Struct. ( Theochem ) 311 (/994) 55-68

Tabl e 4Total energies and Jahn- Teller distorsions of the optimized singlet and triplet excited states of C60

Subgroup Irreducible Parent Total energy of Jahn-Tellerrepresentation repre sentation the excited state energy

(eV) (meV)

SingletDSd A1g Hg -1 371.920 - 31

A2g T ig -1371.959 - 22A lu Hu -1371.393 - 2A2u T zu -\371.400 -34

Th Ag Gg -1371.967 - 3Au G u -1371.369 -3

D 3d A ig Gg -1 371.025 -61A2g Gg -1371.986 -22A lu Hu -1371.443 -52A zu G u -\ 371.420 -54

D Zh Au H u -1371.5\\ -120Big Gg -1 372.088 -\24Blu Hu -\371.463 -72

TripletDSd A1g Hg - 1372.075 -28

A2g Tzg -\372.352 - 185A lu Hu - 1371.46 \ -3A2u r ., -\371.573 - 20

Th Ag Gg -\371.982 - 3Au Gu -\371.6\\ -5

D3d A ig Hg -1372.057 -\0A2g Tzg -\372.226 - 59A lu G u -\371.617 -11A2u Gu -1371.649 -43

DZh Ag Hg -1372.059 -12Au Hu -1371.64 \ - 183BIg Tzg -\372.287 -120Blu Gu -\371.668 -62

in Table 4. For spatially equivalent representations(e.g. BIg, Bzg and B3g), we have indicated only one.Many critical points have been found so far (andmany more exist). These are quite close to eachother in energy, as the distortion energies are typi­cally smaller than 0.1 eV.

The optimized bond lengths are listed in Table 5(see also Fig . 2). These are also only slightly differ­ent from the icosahedral bond lengths predicted bythe xHUG E model (see the last two rows in Table2). Th is suggests that the barrier separating differ­ent distorted structures cannot be too large and,consequently, the so-called " dynamical Jahn-

Teller effect" must be taken into account even atlow temperatures.

Table 4 shows also that the lowest energy tripletdistorted structure is of D Sd synunetry. This agreeswith experimental results only if the measurementsare made in solution. In the solid phase the indi­vidual C60 molecules (unit cells) have Th symmetryas a con sequence of the perturbation due to thecrystal field at higher temperatures. Figure Ishows that Th does not contain D Sd as a sub­group; thus this latter representation cannot bereduced in the crystal phase. The next lowest tri­plet has symmetry D Zh (see Table 4). Accordingly,

P.R . Surjan et al .jl. M ol. Stru ct. ( Theochem } 31I ( 1994) 55-68 61

Table 5Distorted bond lengths in C6Q in va rio us states

Bond 'A\g IA ju IA2gIA2u

3A Ig 3A Iu 1A 2g 1A2"

D Sd point group'a 1.4457 1.4452 1.4457 1.4466 1.4457 1.4453 1.4449 1.4465b 1.4080 1.4108 1.4087 1.4024 1.4091 1.4104 1.4049 1.4028c 1.4431 1.4428 1.4429 1.4463 1.4427 1.4427 1.4451 1.4461d 1.4153 1.4106 1.4148 1.4094 1.4147 1.4111 1.3997 1.4090e 1.4400 1.4447 1.4403 1.4437 1.4406 1.4448 1.4523 1.4436f 1.4061 1.4090 1.4059 1.4183 1.4056 1.4089 1.4248 1.4179g 1.4502 1.4435 1.4499 1.4375 1.4497 1.4435 1.4230 1.4385

Dl d point group"a 1.4187 1.4080 1.4180 1.4022 1.4168 1.4205 1.4190 1.4027b 1.4418 1.4432 1.4428 1.4463 1.4448 1.4398 1.4424 1.4456c 1.4383 1.4454 1.4380 1.4482 1.4376 1.4424 1.4384 1.4480d 1.4479 1.4412 1.4483 1.4444 1.4487 1.4455 1.4481 1.4448e 1.4098 1.4098 1.4098 1.4109 1.4094 1.4091 1.4095 1.4100f 1.4508 1.4361 1.4507 1.4504 1.4494 1.4453 1.4505 1.4506g 1.4005 1.4183 1.4004 1.4052 1.4014 1.4054 1.4005 1.4052h 1.4484 1.4408 1.4481 1.4445 1.4468 1.4463 1.4482 1.4447

1.4104 1.4046 1.4112 1.4212 1.4122 1.4044 1.4103 1.42141.4373 1.4526 1.4371 1.4343 1.4381 1.4448 1.4376 1.4354

D2h point group"

IAu IBlg IBI" l Ag l Au 3Blg l B \u

a 1.4191 1.4153 1.4040 1.4042 1.4039 1.3974 1.4039b 1.4414 1.4402 1.4453 1.4456 1.4456 1.4506 1.4456c 1.4101 1.4053 l.41 25 1.4099 1.4062 1.4154 1.4062d 1.4593 1.4508 1.4461 1.4417 1.4490 1.4346 1.4490e 1.4440 1.4504 1.4458 1.4408 1.4501 1.4310 1.4501f 1.4486 1.4384 1.4469 1.4453 1.4434 1.4589 1.4434g 1.4034 1.4084 1.4027 1.4110 1.4031 1.4108 1.4031h 1.4412 1.4517 1.4407 1.4443 1.4419 1.4349 1.4419

1.4156 1.3957 1.4165 1.4108 1.4123 1.4355 1.4 123j 1.4511 1.4363 1.4556 1.4410 1.4506 1.4488 1.4506k 1.4167 1.4154 1.4164 1.4079 1.42 16 1.4019 1.42161 1.431 7 1.4398 1.4373 1.4444 1.4397 1.4465 1.4397m 1.4504 1.4403 1.4442 1.4422 1.4434 1.4470 1.4434n 1.3976 1.4207 1.4055 1.4169 1.4074 1.40 12 1.40790 1.43 10 1.4479 1.4280 1.4513 1.4282 1.4455 1.4282

Th point groupe

lAg lAu l Ag lAu

a 1.4108 1.4112 1.4107 1.4070b 1.4440 1.4443 1.4440 1.4456c 1.4096 1.4098 1.4096 1.4105d 1.4461 1.4465 1.4461 1.4465e 1.4420 1.4423 1.4420 1.4423

a Fig . 2(b) .b Fig. 2(a) .C Fig . 2(c).

62 P.R . Surj an et al.L), Mol . Str uct, ( Theochem) 311 ( 1994) 55-68

Fig. 2. Inequivalent bonds ofC60 in (a) D 3d , (b) DSd ' and (c) D2h or Th symmetries.

the lowest triplet state of crystalline C60 can bepredicted to be D2h .

Ground state spectra of fullerenes can bebasically interpreted by vertical transitions.The excited state singlet -singlet and triplet­triplet spectra, however, represent a quite differentsitua tion, as here the system spends more timein an excited state and thus the Jahn-Tellerdistortion ma y become more visible. In fact,the excited state spectra have the characteristicfeature that they do not show separate bandsbut a broad absorption curve over the range

400-600nm (2-3eV) [23-27] (see e.g. Figs I and 2in Ref. 23).

The results of the CNDO/S calculations of theexcited spectra are shown in Figs. 3 and 4. It isapparent that the icosahedral spectra (i.e. verticaltransitions) are quite sparse. Using the distortedgeometries in Table 5, we obtain the spectrashown in Fig . 4. These spectra show a significantsplitting of the bands and thus they are closer to theexperimental situation. This is clear evidence ofJahn-Teller distortions in the excited spectra. Wedo not claim, however, that the results of our

P.R . Surja n et al.lJ. Mol. Struct. ( Theochem ) 311 (19 94) 55-68 63

Singlet-singlet transit ions

BO DO700060005000energy (Al

40003000

Ih -

III III J IIII IJ111 I ,I II II I

4

7

3

2

5

6

(a) 02000

+

c.....

o(fJ

o....,.oo....

Triplet-trip let transitions

I

I h - I

I I II, I

u(fJ

o....,.oo....+

c.....

6

5

4

3

2

(b) 02000 3000 4000 5000

energy (Al6000 7000 BOOO

Fig. 3. CNDOjS-CI spectra of C6Q for vertical excitat ions at icosahedral geometry. (a) Singlet -singlet excited spectrum; the transit ionsstart from the lowest excited state. (b) Triplet-triplet spectrum; the transitions start from the lowest triplet.

BOOO

BODO

7000

7000

6000

6000

5000energy (AI

5000energy (AI

4000

Tr iplet-t r iplet trans itions

Tr- iplet-trip let trans i t i ons

4000

3000

3000

05d lAlg) -

ti ll ill, II II, I

05d IA2g) -

~ ll /L III I

a2000

6

5

a2000

. 5

3 . 5

4 . 5

(b)

'7u~

0 . 5.0

~2

+

~ 1. 5

u~

0

;-0

~

s:s: 2

(d )BOOO

BOOO

700 0

7000

6000

600 0

5000energy (AI

5000energy (A)

4000

Singlet -s inglet transit ions

Triplet-triplet transitions

4000

30 00

30 00

05d IAlU) -

,~ 1.1 I I

02h IB2g ) -

l.LJ , .l il JI, ~l "'11.1 ,Io2000

o2000

4 .5

3 .5

o 5

0 .5

4 .5

(c)

'7um0 2 .5•0~

2

::s 1.5

3 .5

3'7u~0 2 . 5;-0

~2

::~ 1.5

(a)

Tripl et-tri pl et t rans i t io ns Tr i p l e t - t n pl et t rans it i ons

05d (A2u ) -

I,~ i~ I I II

D3d (Alu) -

I

lli'l I1

I I J

1

1

8000

80 00

03d (A2g) -

7000

70006000

60 005000e ne rgy (Al

5 000en er gy IA)

Tr i pl e t - t ri Pl e t transiti ons

4000

40003000

3000

03d(Al gl -

IllU ~ I,l Ij ,I I, ,IIo2000

' . 5

. 5

u~

0

;0 2 .~

2

s1.5

4 .5

3 .

c~

::'.g 2 .5

2~

;:.1. 5

0 .5

(f)

(h)

8000

8000

7000

7000

6000

6000

5000enecgy (Al

5000enecgy (AI

4000

4000

Tr ip l e t -t r ip le t tr ansiti ons

3000

3000o2000

o2000

4.5

3 . 5

--:c~

0 2.0

~2

;:. 1. 5

(e)

(g)

o 3d (A2u ) -

JII I] 1.1, ,j II III l

Tri pl et-t r i pl e t trans it i ons• . 5

•3 .5

-:u

"0 2.5.0

~2

~

~ 1.5

0 .5

J~ ~ ~ II II, II ~ 1. ,1a Ij ~ "2000 3000 .000 5000 6000

(k) energy (AI

Triplet- tr i plet trans it ions

80 00

8000

7000

7000

6000

6000

5000energy (AI

5000energy (Al

.000

.000

Tl"ipl et -tripl e t trans iti ons

Trl plet-tri Pl et transiti ons

3000

3000

D2 h IB2g ) -

,~,l IJ III Illll. I ,Ill

o 2 h lAg) -

II ~ l~ ~ I j II. II IIIa2000

5

2 .5

3 .5

• . 5

a2000

• . 5

(I)

(j)

~ 1.5

3 . 5

um0 2 .5.0

~2

~

~ 1.5

~o.o~

8000

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P.R. Surja n et al.lJ . Mol. Struci. ( Theochem ) 3 /l (1 994) 55-68

Tr-Lp I e t r-t.r In Le t t r-an s i t r cn s4 . 5 r--- ----,- - ----.-- - - ,-- - --,- ------,-- - ---,

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Fig . 4. CNDO/S-C I spectra ofC60 at various distorted geometries. (a) Singlet - singlet excited spectrum. (b) - (m) Trip let - tri plet spectra .

CNDOjS calculations would agree quantitativelywith experimental results.

The agreement between the CNDOjS cal­culations and the singlet- singlet experimentalspectra given in Ref. 23 is quite satisfactory. Aninteresting feature of the triplet-triplet spectra[23- 27] is the strong absorption band at 740 nm .In our calculations we were not able to findan allowed transition in thi s region. In fact , inthe computed spectra, with the exception ofsome symmetries where we can see very weaktransitions at about 700 nm, this region is com­pletely empty (Figs 3(b) and 4(b)-4(m)). It is inter­esting to note that there is a strong transition atthe same place on the singlet-singlet spectrum(Fig. 4(a)). We do not understand the originof this strong triplet-triplet band; its absencecan either be due to the drawbacks of theCNDOjS-CI method, or the band ma y be due tosome unknown interaction, e.g. with the oxygen inthe solution .

4. Supplementary material

We have not included in this paper the lengthytables of transition energies and oscillatorystrengths for all symmetries. These are availablefrom the authors upon request.

Acknowledgements

Discussions with J . Kurti (Buda pest) are grate­fully acknowledged. This work was partly sup­ported by grant No. OTKA-517jl 990 from theHungarian Research Fund.

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