Application of symmetry-adapted wavefunctions to systematictreatment of electronic structure of fullerenes

Au Chin Tang and An Yong Li*National Key L aboratory of T heoretical and Computational Chemistry and Institute ofT heoretical Chemistry, Jilin University, Changchun, Jilin 130023, China

By constructing symmetry-adapted wavefunctions and investigating the topological structure, a systematic method is proposedfor resolving the p-electron Hamiltonian of a fullerene with a certain symmetry and for treating second-neighbour hoppinganalytically. Calculations for fullerenes with di†erent symmetries show that second-neighbour hopping has almost no inÑuencefor small molecules but may play an important role in the electronic and bonding properties of large carbon clusters.

1 IntroductionIn fullerenes, four valence electrons of each carbon are dividedinto two parts, three link three neighbouring atoms and con-struct the r frame of the molecule, the remaining electronoccupies the orbital and forms the large p-conjugate2p

zsystem. The Hamiltonian can be written approximately asand we can determine the energy levels byH \H

s] H

presolving the p Hamiltonian. This model has been applied toexplain successfully the electronic properties of many largecarbon clusters.1h11

For large fullerenes, the main difficulty is how to decom-pose the Hamiltonian. Many methods have been proposed toresolve the problem. Generally, the symmetry of the moleculeis always used. However, until now, the symmetry of the mol-ecule point group has not been completely considered ; more-over, for those molecules with inversion symmetry, the parityof the energy level cannot be determined directly by resolvingthe secular equation.

In this paper, we start from the topological structure of thefullerene, discuss in detail the relationship of the symmetry-adapted wavefunctions and the molecule coordinate system,and construct generally the complete symmetry-adaptedwavefunctions. Based on these, we propose a systematicmethod to factorize completely the p-electron Hamiltonianinto irreducible representations of the maximal point group ofthe fullerene and the parity can be given directly. We alsotreat analytically second-neighbour hopping and Ðnd that itplays a role in the electronic structure of the fullerene.

2 Formulation of the methodWithin the p-electron approximation, the single-particle Ham-iltonian of the fullerene can be written as (in units + \ 1C

nand m\ 1) :

H \ [12Z2] ;i/1

n l(r [ Ri) (1)

where is the potential produced on one electron at rv(r [ Ri)

by the carbon cation at We assume that the p orbitalRi.

wavefunction (to be denoted as o iT) satisÐes the fol-s(r [ Ri)

lowing equations

[[12Z2 ] l(r [ Ri)] o iT \ e0 o iT (2)

and

Si o iT \ 1 (3)

The eigenfunctions can be constructed from linear com-bination of the functions (i \ 1, 2, . . . , n) ands(r [ R

i)

obtained by resolving the Schro� dinger equation

HW \ eW (4)

In order to simplify the calculation, we need Ðrst to constructthe symmetry-adapted wavefunctions and then form the func-tion W.

2.1 Representative patch and coordinate system

We assume that the point group of the fullerene isCn

G\. . . , with irreducible representations Da ofMg1 \ e, g2 , g

hN,

dimension The n atoms can be classiÐed into l classes,da .atoms within each class being permuted under the operationsof G. The net of the polyhedron can be divided into hC

nequivalent fragments, each containing one atom of each classand travelling over the whole surface of the polyhedron underthe operations of G. The boundaries of the fragments may besymmetry planes or axes, the atoms on the boundary may belocated on symmetry planes or three-fold axes and maybelong to two, three or six fragments. We take one of thesefragments as the representative patch of the fullerene.

If there are atoms lying on three-fold axes or symmetryplanes, we must choose one of an equivalent set of three-foldaxes or symmetry planes containing atoms as the coordinateaxis or plane of the molecule coordinate system.

It can be proved that any point group has, at most, threekinds of symmetry planes and and is perpendicu-ph , pv pd , phlar to and and that if a point group contains more thanpv pd ,one three-fold axis, all the three-fold axes are equivalent. It iswell known that there are 28 possible fullerene point groups :

I ; T ; (n \ 2, 3, 5, 6) ;Ih , Td , Th , Dnd , D

nh , Dn

S2n , Cnv , C

nh , Cn(n \ 2, 3) ; because the polyhedron of the fullereneCs , Ci ; C1,is comprised of pentagonal and hexagonal rings.12h14 For all

the fullerene groups, is also normal to a correspondingpvplane Therefore, we can always make a coordinate system,pd .let the three-fold axis (if present) be the coordinate axis andthe symmetry plane (if present) be the coordinate plane.

One exception is the group. For the fullerene withTh Cn

Thsymmetry, the atom number n satisÐes : n(h, k ; [ h,h ] k) \ 4(5h2] 8hk ] 2k2) with h and h ] 2k positive. Thenet of the polyhedron consists of 8 equilateral and 12C

n(Th)isosceles triangles12 such as OCB and OAB in Fig. 1. P, P@

and PA are the three two-fold axes, POPA, PACP@ and PBP@are the three mutual-vertical symmetry planes, is a three-P3fold axis. We Ðnd that there are always carbon atoms locatedon the symmetry planes, and when h ] 2k is not divisible by 3

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Fig. 1 The representative patch of the fullerene n(h, k ; [ h,Cn(T h) :h ] k)\ 4(5h2] 8hk ] 2k2) with h and h ] 2k positive. O is the

origin of the coordinate system XOY , and the coordinates of A and Bare, respectively, (h, k) and ([h, h ] k)

there are two atoms on each three-fold axis. For this kind ofmolecule, according to the above rule, we must make a coor-dinate system such that the symmetry planes are taken as thecoordinate planes and simultaneously, the three-fold axis isP3taken as the Z axis. However, this is impossible because, obvi-ously, the three-fold axis does not lie on any symmetry plane.Therefore, for these types of fullerenes with (mTh h ] 2k D 3mis an integer), we cannot factorize the Hamiltonian accordingto its maximal symmetry. However, we can factorize it accord-ing to T and two subgroups of the group and, thus,D2h , T hthe degeneracy and parity of the energy level can be deter-mined. For the T group, the three-fold axis is taken as the Zaxis ; for the group, the three symmetry planes are takenD2has the coordinate planes. Fig. 1 illustrates the representativepatch POPACP@BP according to the group, which is equalD2hto of the surface of the polyhedron. If h ] 2k \ 3m, there is18no atom on the three-fold axis, we can decompose the Hamil-tonian according to the group, the representative patch isThwhich is equal to of the surface of the poly-POP3BP 124hedron, the three symmetry planes are chosen as the coordi-nate planes.

2.2 Symmetry-adapted wavefunctions

If the atom j is situated inside the representative patch, it doesnot lie on any symmetry element. The number of equivalentatoms is h, their p orbital wavefunctions can act as a base toproduce a representation of G and combine linearly intoD

jbases of irreducible representations as follows :

tjlaj \Sda

h;

g | G

Da(gü~1)jl o gü jT (j, l\ 1, 2, . . . , da)(5)

The representation is decomposed intoDj

Dj\ ;

a^da Da (6)

where each irreducible representation Da appears times,dawhich is consistent with

;a

(da)2\ h (7)

The representative patch can be chosen arbitrarily from theh equivalent patches, therefore, in eqn. (5), j can be replaced by

j or by We obtaingü @~1 Da(gü~1) Da(gügü @)~1].

tjlaj \Sda

h;

g | G

Da(gü @~1gü~1)jl o gü jT (gü @ ½ G) (8)

If an atom j is located on the boundary of the patch and itssite symmetry group is or (their orders areS

jCs , C3 C3vrespectively 3, 6), the number of its equivalent atoms ism

j\ 2,

Provided that one equivalent atom j@ of j lies on thenj\ h/m

j.

coordinate axis or plane and is transferred to j by the oper-ation in eqn. (8) we obtain the symmetry-adaptedgü

j, gü @\ gü

j,

wavefunctions

tjlaj \Sda

h;

g | G

Da(güj~1gü~1)jl o gü jT (9)

Let and (i \ 1, 2, . . . , respectively, representsüji sü

j@i \ süj(0)i m

j),

the elements of the site symmetry groups of j and j@, the groupG can be expressed as

G\ Moü psüji ; p \ 1, 2, . . . , n

j; i \ 1, 2, É É É , m

jN (10)

Since eqn. (9) is rewritten assüji \ gü

jsüj(0)igü

j~1,

tjlaj \Sda

h;i, p

Da(süj(0)~igü

j~1oü ~p)jl o gü jT (11)

Because the symmetry axis or plane where the atom j@ lies ischosen as the coordinate axis or plane, the matrix sum

Da(Sj) \ ;

iDa(sü

j(0)~i) \;

iDa(sü

j(0)i) (12)

is diagonal. Provided that the number of diagonal elements isthe representation is decomposed inton

ja, D

jD

j\ ;

a^n

ja Da (13)

and

nj\ ;

anja da (14)

Based on the above discussion, we can combine linearly then p-orbital wavefunctions of the fullerene into the sym-C

nmetrical wavefunctions.

tjlaj \Sda

h;

g | G

Da(güj~1gü~1)jl o gü jT (15)

where j \ 1, 2, . . . , l, l\ 1, 2, . . . , and j can only take someda ,numbers of 1, 2, . . . , which satisfy For then

ja da Da(S

j)jjD 0.

atom j inside the patch, and For the atom jnja \ da , gü

j\ eü .

on the symmetry plane or the three-fold axis, is an oper-güj~1

ation which transfers the symmetry plane or axis where theatom j lies to the equivalent symmetry plane or axis which ischosen as the coordinate plane or axis.

2.3 Factorization of Hamiltonian

Taking the function

w\ ;ajjl

cjlaj tjlaj (16)

and substituting into eqn. (4), we derive the matrix equation :

H (a)C(a)\ eS(a)C(a) (17)

where C(a) is a column matrix of components(;j/1l n

ja)] 1

H(a) and S(a) are, respectively, the energy andcjaj( \ cjlaj ),overlap matrices (containing l ] l blocks) with the blocks hij(a)

matrix) and matrix) :(nia ] n

ja s

ij(a) (n

ia ] n

ja

hij(a)\ h

ji(a)`\ ;

g | GDa(gü

i~1gügü

j)*Si o H o gü jT (18)

sij(a) \ s

ji(a)`\ ;

g | GDa(gü

i~1gügü

j)*Si o gü jT (19)

The matrix expresses the interaction of the atomsDa(güi~1gügü

j)*

i and Once we obtain the matrices with help of thegü j. Da(gü ),representative patch, the eigenvalues and eigenfunctions canbe obtained by resolving eqn. (17).

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2.4 The Ðrst approximation

In the Ðrst approximation, we neglect the integrals Si o H o jTwhen the minimum r-bond number between the atoms i and jis greater than 1 and Si o jT when Let Si o H o iT \ a andiD j.Si o H o jT \ b if i is bonded to j. Since Si o iT \ 1, we have

and where is an unitsij(a)\ 0 (iD j) s

ii(a)\ m

iD

ia(eü ), D

ia(eü ) n

ia ] n

ia

matrix. Multiplying the left-hand side of eqn. (17) by S(a)~1,and assuming x \ (e [ a)/b, we obtain

A@(a)C(a)\ xC(a) (20)

where the block of the matrix A@(a) isAij@(a)

Aij@(a)\

1

mi

;i~ˆj/1

Da(güi~1gügü

j)* (21)

If the atom i is bonded to atom the value of satisfying thegü j, gürequirement is Only when i and j are both inside them

im

j.

representative patch, Therefore, it is not convenientmim

j\ 1.

to use eqn. (20) directly to resolve the eigenvalues. However,we can easily show that the column blocks of the matrix A@(a)express exactly the interaction of the atoms. The l blocks ofeach column (such as the i column) contain a total of threenon-zero terms each expressing the interactionDa(gü

j~1gü~1gü

i)*,

of i and bonded to i. Considering a matrix and its trans-gü jposed matrix having common eigenvalues, we take the trans-posed matrix of A@(a)

A(a)\ A@(a)T (22)

where the element of A(a) is

Aij(a)\ ;

i~ˆj/1Da(gü

i~1gügü

j) (23)

The atom i connects with three atoms, each of which can betransferred one atom (such as j) in the representative patch bysome operations of G ; from these operations we need onlytake any one to calculate according to eqn. (23). If twogü A

ij(a)

or three of the atoms bonded to i can be transferred to thesame atom j in the patch, is equal to the sum of two orA

ij(a)

three terms. Above all, for each atom bonded to i we take onlyone operation gü .

The eigenvalue x is obtained by resolving the secular equa-tion

o A(a) [ xI(a) o\ 0 (24)

2.5 Second-neighbour hopping

We assume the wavefunction in eqn. (4) to be of the form

t\ ;j/1

ncjs(r [ R

j) (25)

The coefficients are determined by minimizing the expecta-cjtion value e of H :

e \;

ijci* c

jSi o H o jT

;ij

ci* c

jSi o jT

\ e0 ]C`H@CC`SC

(26)

where C is an n ] 1 column matrix of the component C` isci,

its Hermitian conjugate, and S and H@ are n ] n matriceswhose matrix elements are, respectively,

Sij\ Si o jT (27)

Hij@ \

TiK G

;kEi, j

l(r [ Rk)] 12[l(r [ R

i)] l(r [ R

j)]H K

jU

\ [e0 Sij] Si o H o jT (28)

The integrals and are determined approximately byHij@ S

ijthe minimum number k of r bonds between the atoms i and j,

and marked by and We obtainbk

sk.

e \ e0 ] ;kw0

bkC`A

kCN

;kw0

skC`A

kC (29)

where the matrix is an n ] n matrix, whose element j)Ak

Ak(i,

is equal to 1 when the minimum r-bond number between theatoms i and j is k, otherwise j) \ 0 is the unit matrix).A

k(i, (A0The fullerene consists of pentagonal and hexagonal rings,

each atom being bonded to three Ðrst-neighbour atoms andhaving six second-neighbour atoms. Starting from any atomand walking two random steps along the r-bond chain, onewill reach the six second-neighbour atoms once and the initialatom three times. Therefore, we have

A2 \ A12[ 3A0 (30)

With the help of this equation, we can take into account thesecond-neighbour interaction. Let b

k\ 0(k P 3), s

k\ 0(k P 3),

because the eigenvalue of is equal to the Hu� ckel energy xA1(in units of b), we obtain the energy expression :

e \ e0]b0] b1x ] b2(x2 [ 3)

1 ] s1x ] s2(x2 [ 3)(31)

If we also neglect the Ðrst-neighbour and second-neighbouroverlap integrals and and let thes1 s2 g \ b2/b1 (b1 \ b),energy expression is of the form

e \ e0 ] b0 ] b[x ] g(x2[ 3)]\ a ] b[x ] g(x2[ 3)] (32)

3 Summary and DiscussionWe have proposed a systematic method to factorize the p-electron Hamiltonian according to the maximal point groupof the fullerene. The method can be applied to any fullerenewith a certain symmetry. First, one should investigate thetopological structure of the fullerene and determine its repre-sentative patch according to its maximal symmetry. Second,an appropriate coordinate system should be constructed,whose axes or planes are the symmetry axes or planes con-taining carbon atoms. Thirdly, the interaction matrix Da

of the atoms in the representative patch must be(güi~1gügü

j)

determined. Then we can resolve the secular equation (24) andobtain the energy eigenvalues, if the irreducible representation

has been obtained.Da(gü )We treat analytically second-neighbour hopping, which has

inÑuence on the energy levels of the fullerene. The Hu� ckelenergies x (in units of b) of the fullerene satisfy 3 P x [ [3.For the lowest occupied molecular orbital (LOMO), x \ 3 ;for the highest unoccupied MO (HUMO), x is slightly greaterthan [3. Because g [ 0 and b \ 0, when second-neighbourhopping is included, the frontier energy levels whose Hu� ckelenergies satisfy x2\ 3 will increase. The occupied energylevels with and the unoccupied energy levels withx [ J3

will decrease as g increases, the former do not inÑu-x \[J3ence the order of the energy levels, but the latter will. TheHUMO level decreases fastest, it will Ðrst intersect withHOMO and LUMO levels which increase fastest, and whoseHu� ckel energies satisfy 1[ x [ [1 (except for and areC20),almost equal to zero when the number of carbon atoms isvery large. Therefore, when g [ gF0 \ [1/(xHOMO ] xHUMO)Bthe frontier energy levels will be changed. Generally, the13,value of g is much less than the order of the energy levels13,and the frontier orbitals are not altered by second-neighbourhopping.

The gap between HOMO and LUMO is D\ (xHOMOFor a closed-shell molecule[ xLUMO)[1 ] g(xHOMO ] xLUMO)].with n/2 bonding molecular orbitals (BMO) within the Hu� ckelapproximation, xHOMO [ 0 P xLUMO , DB DHu? ckel\ xHOMOHowever, for a pseudo-closed-shell molecule with[ xLUMO .

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Table 1 Calculated results for the icosahedral fullerenes (n \ 60h2, h \ 1È25)Cn

n h xave xHOMO xLUMO DHu? ckel gFO gNBMO60 1 1.5527 0.6180 [0.1385 0.7565 0.5000 0.2361

240 2 1.5689 0.4368 [0.0596 0.4964 0.4087 0.1555540 3 1.5721 0.3241 [0.0336 0.3577 0.3815 0.1120960 4 1.5732 0.2550 [0.0221 0.2771 0.3686 0.0869

1500 5 1.5737 0.2096 [0.0159 0.2255 0.3610 0.07092160 6 1.5740 0.1777 [0.0121 0.1898 0.3562 0.05992940 7 1.5742 0.1541 [0.0096 0.1637 0.3527 0.05183840 8 1.5743 0.1361 [0.0079 0.1440 0.3502 0.04564860 9 1.5743 0.1218 [0.0066 0.1284 0.3482 0.04086000 10 1.5744 0.1102 [0.0056 0.1158 0.3467 0.03697260 11 1.5744 0.1007 [0.0049 0.1056 0.3454 0.03378640 12 1.5745 0.0926 [0.0043 0.0969 0.3444 0.0310

10 140 13 1.5745 0.0858 [0.0038 0.0896 0.3435 0.028711 760 14 1.5745 0.0799 [0.0034 0.0833 0.3428 0.026713 500 15 1.5745 0.0747 [0.0030 0.0777 0.3421 0.024915 360 16 1.5745 0.0702 [0.0027 0.0729 0.3416 0.023417 340 17 1.5745 0.0662 [0.0025 0.0687 0.3411 0.022119 440 18 1.5745 0.0626 [0.0023 0.0649 0.3406 0.020921 660 19 1.5745 0.0594 [0.0021 0.0615 0.3402 0.019824 000 20 1.5746 0.0565 [0.0019 0.0584 0.3399 0.018926 460 21 1.5746 0.0539 [0.0018 0.0557 0.3396 0.018029 040 22 1.5746 0.0515 [0.0017 0.0532 0.3393 0.017231 740 23 1.5746 0.0493 [0.0015 0.0508 0.3390 0.016434 560 24 1.5746 0.0473 [0.0014 0.0487 0.3388 0.015837 500 25 1.5746 0.0455 [0.0013 0.0468 0.3386 0.0152

In the simple Hu� ckel approximation, the electronic conÐguration is and are, respectively, the critical values of g changing(hu)10(tlu)0. gFO gNBMOthe frontier orbitals and the number of the bonding molecule orbitals and deÐned as andgFO\ [1/(xHOMO] xHUMO) gNBMO\xHOMO/(3[ xHOMO2 ).

more than n/2 BMO in the Hu� ckel approximation, xHOMO [If and are not too small,xLUMO[ 0, D[ DHu? ckel . xHOMO xLUMOD will deviate largely from Second-neighbour hoppingDHu? ckel .may play an important role in the electronic structure and

spectroscopy of the fullerene.We Ðnd that inclusion of the second-neighbour hopping

will cause the number of BMO (NBMO) to decrease. Theresults for a large number of fullerenes show that the energy ofthe HOMO is generally greater than zero, except for some oftetrahedral fullerenes.15 Since is small, a small value ofxHOMOg will change the sign of the[PgNBMO \ xHOMO/(3[ xHOMO2 )]HOMO level and make it become an antibonding MO. For

the critical value is g \C60 , xHOMO\ (J5 [ 1)/2 B 0.6180,In fact, g is less than this value, therefore,)5[ 2 B 0.2361.

even though second-neighbour hopping is included, NBMOof is not changed. However, if the number of carbonC60atoms is large, the result is di†erent. For the largest one of thecalculated icosahedral fullerenes theC37 500 , xHOMO B 0.0455,critical value of g is 0.0152, which may be less than the practi-cal value of g. Therefore, it seems that we can conclude thatthe large fullerene has some electrons Ðlled in the conductionband and exhibits the properties of a metal. In the simpleHu� ckel model, this character is also possessed by some tetra-hedral fullerenes with antibonding HOMO. Here, we Ðnd thatthe highest occupied molecular orbitals of some fullerenes,which are bonding orbitals in the Hu� ckel approximation, willbe antibonding orbitals when second-neighbour interaction istaken into account.

In Table 1, we list some results of calculations for the icosa-hedral fullerenes (n \ 60h2, h \ 1È25). is deÐned asC

nxave

xave \ ;i

nixi/n

where is the number of the electrons Ðlled in the occupiedniMO with the Hu� ckel energy (in units of b).x

iThe above discussion is based on eqn. (32). If we include theoverlap integrals and use eqn. (31), the results are notchanged. Because the overlap matrix S is positive deÐnite,

is always greater than zero. So the1q \ 1 ] s1x ] s2(x2 [ 3)above analysis for NBMO is retained. The energy level with

x2\ 3 increases and that with x2[ 3 decreases as g increasesbecause q is positive. Let j \ (x2 [ 3)/[1] s1x] s2(x2[ 3)].The di†erential coefficient isdj/dx \ q2[2x] (x2] 3)s1]positive for positive and small negative x, and negative for xnear Hence, the HUMO level decreases[3(s1B 0.25).16fastest and the HOMO and LUMO levels increase fastestwith increasing g. Therefore, the above discussion on the e†ectof second-neighbour hopping on the order of energy levelsholds. For the frontier energy levels, x is small, qB 1, so thediscussion about the gap between the HOMO and theLUMO is not inÑuenced by introduction of overlap integrals.Usually, we consider the frontier levels, and can use eqn. (32)to analyse the electronic properties

work is supported by the National Natural ScienceThisFoundation of China.

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Paper 8/03351J ; Received 5th May, 1998

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