REVISTA MEXICANA DE FíSICA 44 SUI'LEMENTO J. 1&-21

Electronic and vibrational I'roperties of the e60 molecule

DICIEMBRE 1998

F. Alonso-Marroquin and J. GiraldoDepanamento de Frsica, Universidad Nacional de Colombia, Apartado aéreo 67904, Santafé de Bogotá, Colombia

A. CallesFacultad de Ciencias, Universidad Nacional Autónoma de Mlrico. 041510 México. D.F., Mexico

J.J. CastroDepartamento de Física. Centro de Investigación y Estudios Avanzados. Instituto Politécnico Nacional

Apartado postal /4740, 07000 México, D.F., Mexica

Recibido el 10 de marzo de 1998; aceptado el 14 de septiembre de 1998

We have studied the electronic and vibrational properties of (he C60 molecule by means of simplified models where only a few parametersare adjusted 10 experimental data. For (he electronic part we use ao independent electron model with a central Morse potential correctedby splitting the sphcrical degcncracy by properly including (he icosahedral symmetry given rise lO the LUMO and HOMO states. Thcvibrational spectra is obtaincd by solving t~e corresponding secular equation using a Born-Mayer type potential with two parameters adjustedlORaman and IR spectra. The results of our model are compared with sorne of the bes{ ab i!litio calculations. phenomenological models andexperimental data. We found in general a good agreemcnt for the vibrational spectrum particularly for the high frequency part. Since in ourmoJel contains only two parameters we are not ahle to reproduce accurately the whole spcctra. The main contrihution of this modcl is itssimplicity to extract most of the gross features of the C60 molccule without curnhersome computation.

Keywords: Fullerenes; C60 molecule; electron spectra; vibrational spectrum

En el presente trabajo se estudian propiedades electrónicas y vibracionales de la molécula de C60 por medio dc modelos semi+empíricossencillos que utilizan pocos parámetros para ajustar a datos experimentales. Para la parte electrónica se usa un modelo dc partícula indepen-diente. usando un potencial de Morse. con simetría esférica la cual se corrige para incluir la simetría icosaedrál y así ohtener los estadosLUMO y HOMO. Por otro lado. el espectro vibracional se ohtiene resolviendo la ecuación secular de la matriz dinámica usando potencilaestipo Bom-Mayer. entre cada par de álOmos. con s610 dos parámctros que se utilizan para ajustar a experimentos Rarnan e infrarrojo. Losrcsultados se comparan con cálculos de primeros principios. otros modelos fenomenológicos y datos experimentales. Se encuentra un buenacuerdo de nuestro modelo con datos experimentales sobre todo para la región de altas frecuencias. Sin embargo, no se reproduce con exac-Litud tooo el espectro vibracional debido a que s610 hay 2 parámetros para ajustar. La principal contribución del trabajo es que se obtienenaspectos importantes de propiedades del C60 de una manera sencilla que no implica una computación exhaustiva.

Descn'plores: Fullerenos; molécula de C60: espectro electrónico; espectro vibracional

PACS: 33.10.C5; 33.1O.Gx; 31.20Wb

1. inlroduction

The discovery of fullerenes and the method for their produc-lion in gram quantitics have stimulated a great dea! of ex-perimental and Iheoretical work. Within the whole family offullerenes, C.o is the mosl studied system mainly due lo itsinteresting physical and eheroical properties. These are par-tially a resull ofthe high symmetry ofthe molcculc. The Irun-caled icosahedral structure of lhe C60 molecule raises ¡oter-esting quc'stions regarding its electronic structure and vibra-tional properties. Therefore the study of lhe molecule ilself isa natural starting paint in arder 10 understand other membcrsof lbe fullerene family in their molecular and solid forms.

In the study of the elcctronic and vibrational propertiessorne authors have emphasized the distinction bctween the7r (delocalized) and " (Iocalized) electrons and Ihe spherical

charactcr of the molecular wave [1. 2]. Por the understand-ing ofthe vihrational spectra differenl moocls have becn usedin the !iterature ranging [rom ah initio calculations lo phc-nomenological l1lodels to determine the force conslants. Inthis work wc have taken the secand approach to study boththe eleetronic and vibrational properties of the C60 moleculc.It is based on a phenomenological mooel that allows, in avery simple ano direet way, lo extraet SOIne of the main fca-tures of lhis complex syslem. The paper is organized as fol-lows; In Sec!. 2 we present the spherical model for lhe elec-tronic structure which corresponos to the zcro order approx-imation for the electronic wave function. In Scct. 3 we dis-cuss the effcct on the electronic wavc function of breakingthe spherical syrnmetry by taking into account the icosahcdralsyrnmetry. Sect. 4 contains a summary of the mooel that weused for the study of the vibrational propcrties. The conclu-sions ano the discussion of the resuits are includcd in Sect. 5.

ELECfRONIC ANO VIBRATIONAL PROPERTlES OF THE C60 MOLECULE 19

"o .•

."

.".<0

FIGURE l. Wave functions obtained from (he Morse potential em-ployed in the prescnt work.

2. Spherical rnodel for lhe eleclronic speclra

As a slarting model for lhe eleelronie spootra of C60 we eon-sider a systcm formed by the 240 valencc electrons as movingindependenlly in a spherieal shell simulaled by a Morse po-lenlial [3]

[ ( r-ro)]' 1(1+1)V.ff(r) = D 1 + exp - -d- - D+ 2mr' . (1)

This is lhe resull of approximaling lhe whole eleelroninteraction by a sum of two lerros: ane attractive arising[rom (he intcraction betwcen each electron and the core oflhe whole moleeule (produeed by lhe 360 prolons in lhe ear-hon nuelei and lhe 120 eleelrons in lhe Is shell), and onerepulsive. The lalter is the average of the mutual repulsionbetween the valen ce eleclrons. This central field approx.ima-tion allows us lo make lhe usual separation bctwecn (he radialand angular parl of the wave function exprcssing it in lerrosof Ihe assoeiale Laguerre polynomials and lhe spherieal har-monics [4J, labeling lhe resulting wave funelions by Ihreeindexes (n, 1,m). The slales wilh n = O (no radial nodes)are oeeupied by eleelrons slrongly loealized belween atoms,corresponding thcrefore to a clectrons, whcrcas the orbital swilh n = 1 eorrespond lo " eleetrons (Fig. 1). The a slaleswbieh will be denoted by al will be oeeupied by 180 elee-lrons (1 = 9) and lhe ,,1 slales Wilh 60 deloealized eleelrons(l = 5). Tbe Morse pOlenlial paramelers were filled by im-posing the following conditions:

E(a.) - E(ao) = 30eV,

E("5) - E("o) = 7 eV,

E("5) = -5eV,E("5) = E(a.).

The aboye conditions correspond to the range of cnergies forlhe a eleetrons, for lhe" eleelrons, fit lo lbe eleelron affinily,and lbe eondilion lhat lhe highesl molecular orbilals for lhe "and C1 electrons coincide, respectively. With this we obtain

TABLE '- Encrgy levels and occupation numbcrs of the sphericalmodel.

Niv. oc. (eY) Niv. oc. (eV)

a9 18 -5.1 '" 10 -5.2a, 34 -10.6 '" 18 -7.7a, 30 -15.7 "3 14 -9.7a6 26 -20.3 ", 10 -11.2a, 22 -24.3 "1 6 -12.3a, 18 -27.7 "o 2 -12.8a3 14 -30.4a, 10 -32.5al 6 -33.9ao 2 -34.6

lhe following paramelers:

ro = 3.15 A,

D = 49.43eV,d = 0.86A,

m = rne.

The corresponding clectronic le veIs and occupation numbersare shown in Table L

3. Effecl of lhe icosahedral syrnrnelry on lheeleclronic slruclure

When we lake inlo aeeount lhe effeel of lhe ieosabedral sym-metry on the clectronic wave function wc partialIy remove thedegeneracy found in the spherical approx.imation discussed inthe last section, and the wave function is now expressed in theform:

where RnJ is the solution to the Schroedinger equation withlhe central pOlential given by Eq. (1). The angular funelionsnI,1' are now linear combinations of the spherical harmon-ies wilb lbe adapled ieosahedral symmelry, and lhe index "f

labels lhe irreducible represenlalion of lhe symmetry group.This approximalion reeovers the main lopologieal propertiesof lhe exael molecular orbilals, e.8. lhe nodal fealures (ef.Fig. 2). The splilling due lo lhe ieosahedral symmelry alsoailows lo oblain the HOMO and LUMO levels. The lasl en-ergy levels fur lhe a and " orbilals show lbe foilowing Splil-ling [1, 2J:

For lhe molecular eleelronie eonfiguration we oblain [3)

a¡'"gO --+ a.(g~)a.(h~0)"s(h~0)"5(t?u)

The Fig. 3 sbows this level splitling. Where is evidenl lbe gapbelween lhe HOMO and LUMO levels.

Rev. Mex. Fis. 44 83 (1998) 18-2t

20 F. ALONSO.MARROQUIN. A. CALLES. J.J. CASTRO. AND J. G1RALDO

HOMO HOMO( Nodal surfaee)

LUMO LUMONodal surface)

FIGURE 2. Molecular orbital s of the e60 moleculc.

FIGURE 3. Energy levels of the 1r electrons.

4. Vibrational spectra

) 1", (-2.81 eV)

-4 - -----::'-----1 1" (-4.77eV)-6 - 11, (-6.44 eV)

'" g. (-7.58 eV)--------- -----Ii" (-7.68 eV)

.8 - 'J ) ---- g,

-ID - ---~;-- '1 ---''''h- ----_.~1t1 t1u

The C.O moleeule possesses 174 vibralional modes whieh re-duces lo only 46 distinel eigenfrequeneies by lhe 1, symme-ley. Using the ieosahedral group these modes can be c1assi-fied aceording \O lhe irreducible represenlalion: r = 2A. +3T,. +4T2• +6G. +8H. +Au +4Tlu +5T2u +6Gu + 7Hu.This leads to 32 silenl modes. 10 Raman (2A. + 8H.) and 4infrared (4T,u)' There are in the lilerature a large numher oftheoretieal models predieling lhe vibrational speetra and lheassignrncnt of sorne oC the vibrational modes which are 00[-mally compared wilh experimental dala eoming from Raman.infrared. high-resolulion eleetron-energy-Ioss speelroseopy.and inclastic neutron scattering. In spitc of the high symme-lry of C.o. since it is a 60 atom moleeule. the resulting nor-mal mades of vibration are nol easy lo visualizc. Thereforea propcr construction of the eigenvectors is vcry usefu!. Inarder 10 gel good agreement between the rnadel predictionsand experimental dala il is necessary to have a correct ass:gn-ment of the experimentally observed modes and an experi-mental determination of the corresponding eigenvectors. Jt isa well known faet thal it is possible lo conslrucl models which

showa very good agrcerncnt with the whole experimental fre~queney speetra although lhe assignment of sorne of lhe modesmay be wrong. It is worth mentioning lhal even if alllhe levelassignment is made curreetly, lhe agreernent belween theoryand expcrimenl can he c1aimed only if the eigenveetors canbe verified cxperimentally; that is when the ealeulaled dis-placemenl panern can be confirmed by experimenl. This isimportanl sinee lhe displaeemenl pallern of a mode belong-ing lo a symmetry e1ass with a multiplieily bigger lhan oneis nOI delermined by lhe symmelry alone. There are in thelilerature a large numbcr of theoretical rnodels for lhr calcu-lation of the vibrational speelra of C.o whieh can be generallyc1assified as ah illitio, semiempirical, and phenornenologicalmelhods. going from a free parameler model LO others lhatuse paramelers determined by filling experimenlal dala [5-S].In this section wc present the results of a calculation forIhe vibralional speelta of C.o based on a phenomenologiealmodel formed by a Coulomb pOlenlial parlo representing lherepulsion belwcen earbon nuelei and a deeaying exponentialterrn sirnulaling lhe clcctronic c¡oud. This is a two parame~ler model filled to Ihe highesl experimentally reported vibra-tional mode. The simplieily of Ihis model potential providesan easy and quiek ealculation for bOlh Ihe normal modes ofvibralion and eigcnvectors for the whole spectra together withtheir syrnmetry c1assification. Hcre we only present the dis-cussion corresponding to the normal modes, leaving the con-struetion of the cigenvectors and the experimental detennina.tion of the displaeemenl pallern for a fulure work. In TableII we presenl the results of the ealculation of the vibrationalspeetra together wilh their symmetry assignmenl produeedby the prescnl model, and compare wilh lhose coming framdifferent models and also wilh lhe experimenlal speetra. Asit can be seen from this table, the results of this simple modelcompare quite satisfacturily wilh the results of more sophisli~caled rnodels and experimental data. In particular, we can seehow the high frequeney parl of lhe speelta is very well repro-dueed by lhis model having for sorne states an error of only2.3%. Note that in our model we did nol differentiale betweensingle and double bonds whose effeet eould be important for

-a,-12 c-

Rev. Mex. Fís. 44 53 (1998) 18--21

ELECTRONIC ANO VIBRATIONAL PROPERTIES OF THE C60 MOLECULE 21

TABLE JI. Vibrational spectra orthe Raman and IR active modes. The experimental results[6} are compared wilh previous calculations[7.to]and with our results (las1 cDlurnn).

Exp.[5] Force constant5[6] Hartree-fock melhod[7] Density functional[8] Quantum mol. dynamies[9] Bom-mayee potentialA, 491 492 0.2% 540 9.5% 483 1.6% 537 8.9% 613.1 22.1%

1470 1468 0.1% 1417 3.7% 1529 3.9% 1680 13.3% 1602.42 8.6%271 269 0.7% 263 3.0% 263 3.0% 249 8.5% 241.91 11.3%437 439 0.5% 467 6.6% 432 1.2% 413 5.6% 318.6 31.3%710 708 0.3% 739 4.0% 713 0.4% 681 4.2% 377.79 61.1%

H, 774 492 0.2% 540 9.5% 778 0.5% 845 8.8% 609.53 23.8%1099 1102 0.3% 1168 6.1% 1111 1.1% 1209 9.5% 1135.62 3.3%1250 1217 2.7% 1214 2.9% 1282 2.5% 1453 15.0% 1314.41 5.0%1428 1401 1.9% 1422 0.4% 1469 2.8% 1624 12.8% 1393.05 2.5%1570 1575 0.3% 1636 4.1% 1598 1.8% 1726 9.5% 1502.81 4.4%527 505 4.3% 556 5.4% 533 1.1% 494 6.5% 349.42 40.5%

Ti. 577 628 8.5% 540 9.5% 548 5.2% 643 10.8% 395.83 37.2%1185 1208 2.1% 1173 0.8% 1214 2.6% 1358 13.8% 1256.06 6.0%1428 1450 1.5% 1400 2.0% 1485 3.9% 1641 13.9% 1461.43 2.3%

Discrepancies 1.34% 4.56% 2.26% 10.08% 18.54%

some pan of the speetra. The main advantage of the modelproposed here is its simplieity in the ealcu1ation of the vi-brational properties of lhe Fullerene moleeule lhat permits inan easy and rather straightfOlward way lo study lhe principalphysical charactcristics related lo rhe vibrational properties.The close agreement with experimental dala and the simplie-ity of the model suggest lhe obtaintion of reliable eigenvee-lors lhat eould be used for lhe eleetron-phonon interaetion.

5. Conclusions

We have shown by using a simple phenomenologieal modelhow sorne of the main eleetronie and vibrational propertiesof the C,o moleeule can be extraeted withoUl the neeessity ofeumhersome ealculations. The use of lhe spherieal model fora zero arder electronic wave function and the brcaking of thespherieal symmetry by introducing lhe ieosahedral symmetry

1. M.R. Savina, L.L. Lohr, and A.H. Francis, Chem. Phys. Len200 (1993) 205.

2. N. Troullier and J. Manins. Phys. Re". B 46 (1992) 1754.3. F. Alonso. Eleclronic and vibrational properties 01 Ihe C60

molecule (in Spanish; lhesis work). Universidad Nacional deColombia. 1997.

4. D. Campos. Fundamelllos de jisica alómica y molecuwr, (Edi-torial Universidad Nacional de Colombia. 1997).

pro ved lo be adequate fOl reprodueing lhe main fealures ofthe elcclronic propertics. In particular it was shown how [hesyrnmetry breaking gavc cise 10 energy ¡evels foc the 11" elec-trons and the appearanee of the gap between the LUMa andHaMo slates. For the vibrational speetra we have illustratedhow lhe use of a simple phenomenologieal model for the in-teraction between carbon atoms is capable of reproducing themain characteristics of the vibrational speclra. By compar-ing our results for the vibrational speelra with those eomingfrom experimental data and ab inilio calculations it has beenshown how our model can reproduce quite satisfactorily thehigh frequeney part of the vibrational modes. The main ad-vanlage of the approaeh presented here is its simplieity andeapability in reprodueing the main physieal eharaeteristies ofthe eleelronie and vibrational states and lhe gaining of phys-kal insight without the nccessity to carry out cumbersomeeaJculalions.

5. M.S. Dressclhauss. R.A. Jishi, and R.M. Mirie, Phys. Rev. B 4S(1992) 13685.

6. J.1. Soto, Universidad Nacional Autónoma de México, (privatecomunication)

7. X.Q. Wang, C.Z. Wang. and K.M. Ho. Phys. Rev. B 48 (1993)1884.

8. G.B. Adams el al.. Phys. Re". B 44 (1992) 4052.

Re". Mex. Fís. 44 53 (1998) 18-21