a. ceulemans et al- electronic structure of polyhedral carbon cages consisting of hexagons and...

8/3/2019 A. Ceulemans et al- Electronic structure of polyhedral carbon cages consisting of hexagons and triangles

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Electronic structure of polyhedral carbon cages consisting of hexagons and triangles

A. Ceulemans,* S. Compernolle, A. Delabie, K. Somers, and L. F. ChibotaruDepartment of Chemistry, University of Leuven, Celestijnenlaan 200F, B-3001 Leuven, Belgium

P. W. FowlerSchool of Chemistry, University of Exeter, Stocker Road, Exeter EX4 4QD, United Kingdom

M. J. Marganska and M. SzopaInstitute of Physics, Silesian University, ul. Uniwersytecka 4, 40-007 Katowice, Poland

Received 27 December 2000; revised manuscript received 3 April 2001; published 7 March 2002

An infinite series of 3,6 cages is defined by trivalent carbon polyhedra composed of hexagonal and four

triangular rings. A zone-folding construction is applied to the graphene band structure to yield explicit expres-

sions for the -molecular orbitals, energies, and symmetries of the cages that depend only on four indices m,

n, p, and q. Leapfrog members of the series mn0 mod 3 and pq0 mod 3 have closed shells in a

neutral form with two filled nonbonding orbitals; all others have closed shells as dications. Quantum chemical

calculations on C12 , C48 , and C522 confirm this result. Embedding relationships are proved for the spectra of

3,6 cages related by inflation transformations corresponding to stretching and rotation of the polyhedral net.

DOI: 10.1103/PhysRevB.65.115412 PACS numbers: 73.22.f, 73.20.At, 36.40.c

I. INTRODUCTION

The discovery of fullerenes has revived the study of themathematics of polyhedral cages. Physicists and chemistsshow a special interest in cages that can be folded from ahoneycomb graphitic lattice by introducing pentagonal, tri-angular, or square defects. In this paper we will be concernedwith 3,6 cages, i.e., those trivalent polyhedra on v verticeswhich have four triangular faces, hv/22 hexagonal, andno other faces. The case h0 is a tetrahedron, and otherwiseh4. All polyhedra in the class have at least D2 symmetry,

1

but the most symmetrical have tetrahedral T or Td

symmetry,

2

starting from the tetrahedron itself. In a previouswork3 it was shown that the Huckel spectrum of a 3,6 cagealways contains the tetrahedral spectral roots 3, 1, 1, and1, while the remaining eigenvalues are arranged in mirrorpairs with respect to the nonbonding level, and various sub-spectrality relationships hold for cages that are related byinflation transformations. The eigenvalue spectrum of each3,6 cage was shown to be included in that of the hexagonalcovering of the torus.3 In addition, Gonzalez et al.4 presentedeigenvalue spectra of a special class of highly symmetrical3,6 cages.

In the present paper we will generalize the latter results byproviding full expressions for the spectral roots of arbitrary3,6 cages. Our method is based on an embedding of thespectrum into the highly symmetrical band structure ofgraphene, following from the embedding of the polyhedrathemselves as nets on the graphene sheet.

II. CONSTRUCTION OF A TETRAHEDRALLY

SYMMETRIC 3,6 CAGE

First we consider a tetrahedrally symmetric 3,6 cage.Such a cage can be imagined as a honeycomb lattice in-scribed on a master tetrahedron, in such a way that the ver-tices of this master polyhedron correspond to the centers of

the hexagons. Each vertex of the hexagonal lattice is onecarbon atom of the cage. The angular defect corresponding toeach vertex of the master tetrahedron is , and the four ver-tices accumulate the whole curvature of the surface of thecage and are responsible for four triangular cells in the oth-erwise hexagonal lattice. The hexagonal lattice can be in-scribed in a similar manner on two other Platonic solids: anoctahedron curvature accumulated in six squares of 2/3angular defect, giving rise to a class of 4,6 cages, and anicosahedron curvature accumulated in 12 pentagons of an-gular defect /3, corresponding to the celebrated fullereneclass of 5,6 cages.5

The primitive cell of the honeycomb lattice of graphitecontains two atoms, labeled black and white in Fig. 1a. LetT1 and T2 be the two generators of the lattice,

T1)aex , T2)

2aex

3

2aey , 1

where a is the nearest-neighbor distance. The triangle OABin Fig. 1a is the patch of hexagons forming the net of atetrahedrally symmetric 3,6 cage unfolded on the plane.Figure 1b shows a three-dimensional 3D plot of the cor-responding cage, optimized by a quantum-chemical calcula-tion vide infra, Sec. VIII.

The patch is an equilateral triangle and therefore, to definethe cage, it is sufficient to choose for its side one of thevectors A2mT12nT2 , where m and n are integers, theother at angle of 60 being B2(mn)T22nT1 . Thenumber of hexagons inside the triangle OAB is N/2, whereN4( m2n2mn) is the number of its nodes. Whenfolded, four triangles form at the corners of the tetrahedron,leaving N/22 full hexagons in the 3,6 cage itself. As thecage with nm is the mirror image of that with mn, it issufficient to assume that 0nm with m0. As a resulteach tetrahedrally symmetric 3,6 cage is uniquely deter-mined by the two integers n and m.

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The Wigner-Seitz primitive cell of the reciprocal space ofthe honeycomb lattice is the first Brillouin zone of thegraphene sheet. It is generated by two vectors P1(2/)a)ex(2/3a)ey , and P2(4/3a)ey , as shownin Fig. 2, and it takes the shape of a regular hexagon with

vertices

kx4

3)a, ky0,

kx2

3)a, ky

2

3a. 2

Of these six points, only two are linearly independent andbelong to the first Brillouin zone they are K1 and K2. Simi-

larly, only three edges of the hexagon we choose those con-taining M1 , M2 , and M3 , i.e., the thicker lines in Fig. 2belong to the zone.

III. ELECTRONIC STRUCTURE OF GRAPHENE

In this section we briefly recall the electronic structure ofthe parent graphene lattice. The wave function of a singleelectron moving in the potential of the lattice obeys the usualSchrodinger equation

2

2m2rVrrEr, 3

where V(r) is a periodic potential, i.e., for arbitrary integersi and j, V(riT1jT2)V(r). Also note that in our choiceof coordinates V(r)V(r). The solution of problem 3 inthe tight-binding approximation depends on the type of crys-tallographic lattice,6 and is well known. Here we revisit themain points of the derivation for the hexagonal lattice.7

The valence structure of graphite is formed by the bands, based on 2pz orbitals which are antisymmetric withrespect to reflection through the base plane. In second quan-tization the electron Hamiltonian of problem 3 is

Hp

ap

app ,q

ap

a q , 4

where p,q is the summation over nearest neighbors, and ap

and aq are anticommuting fermion operators for orbitals.The constant is the energy of the electron in a noninteract-ing 2pz orbital, and (0) is the nearest-neighbor matrixelement of the Schrodinger operator with the potential V.The solutions of Eq. 4 correspond to irreducible represen-tations k of the translation group. In order to prepare thesubsequent embedding of the tetrahedral lattice, we will dis-

FIG. 1. Tetrahedral 3,6 cage, unfolded on the honeycomb

plane a and in the folded state b. Indices correspond to m3

and n1, and the number of carbon atoms is 52. The folded struc-

ture was optimized by a DFT calculation on the dication C 522.

FIG. 2. Wigner-Seitz representation of the first Brillouin zone

for the honeycomb lattice. Symmetry points are found at the origin hexagonal, vertices K trigonal, and edge centers M digonal.

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tinguish the eigenstates further by their parity with respect tothe centrosymmetry. This distinction was not made in thestandard treatment of Wallace.7

The gerade state with even parity is written as

kgc k1 k1,gc k

2 k2,g , 5

where

k1,g1

N

pexp is prpkap

0 ,

k2,g1

N

pexpis prpkap

0 . 6

In Eq. 5, 1 and 2 is the band index, the summation overp in Eq. 6 is over all the N atoms at positions rp , sp1 for

black atoms and sp1 for white atoms, and ap0 is the

atomic orbital centered at atom p.The coefficients of the even wave functions Eq. 5 are

found to be

c k1

1

& k

k

1/2

, c k2

1

& kk

1/2

, 7

where

kexpikya 2 cos)2 kxa exp ikya

2 . 8

On the other hand, the ungerade state of odd parity is

kud k1 k1,ud k

2 k2,u , 9

where

k1,u1

N

psp exp is prpkap

0,

k2,u1

N

psp expisprpkap

0 . 10

For this odd wave function Eq. 9, one has

dk1 c k

1 , dk2 c k

2 . 11

For every pair of antipodal points k(kx ,ky) andk(kx ,ky) in the Brillouin zone, one may constructin each band one even eigenstate and one odd eigenstate,

which each have the same eigenenergy E k ;

E k1

14 cos2)2

kxa4 cos)

2kxa cos

3

2kya

1/2

.

12

From Eq. 12 it is clear that the two bands corresponding to1 and 2 are symmetric with respect to the energy E. The splitting of the two bands is 6 at the center ofthe zone, and then decreases toward the edges. At the points

M1 , M2 , and M3 it is 2, and at K1 and K2 and only therethe splitting vanishes. At half-filling of the band the numberof electrons is equal to the number of atoms in the cage. TheFermi energy is E, which means that in the first Brillouinzone the Fermi surface consists of just two isolated Fermipoints K1 and K2 . To simplify further discussion, we willexpress all energies with respect to the Fermi level, by put-ting 0.

Exceptional points with respect to the centrosymmetry ofthe Brillouin zone are the center of the zone and the cen-ters of the edges, M1 , M2 , and M3 , all of which coincidewith centers of symmetry. In the 1 band the point is aneven eigenstate, corresponding to the totally bonding combi-nation, while the eigenstates in the three M points are allodd. The opposite holds for the 2 band: here the pointis a single odd, totally antibonding eigenstate, and the threeM points are even combinations. The K1 and K2 points arealso special, since these are the only points for which k0, and as a result the interaction element between k1,gand k2,g vanishes, and similarly for k1,u and k2,u . AtE0 there are thus four solutions, two of even parity with

eigenvectors K11,g and K12,g , and two of odd parity witheigenvectors K11,u and K12,u . Substituting K1 with K2does not yield further solutions, since K1 and K2 are relatedto each other by centrosymmetry: K2K1P1P2 .K12,g and K21,g are thus identical within a constant phasefactor.

IV. BOUNDARY CONDITIONS

FOR THE COVERING FUNCTION

In this section we will show that the energy spectrum ofany tetrahedrally symmetric 3,6 cage can be derived fromthe spectrum of the graphene lattice by an appropriate choice

of the boundary conditions.In case of a tetrahedrally symmetric 3,6 cage, the wavefunction, solving Eq. 3, should obey the boundary condi-tions, required by matching of the function in value and gra-dient at edge points brought together in the gluing of the net,

tAAtA,

tBBtB,

AtBABtAB ,

n tAnAtA,

n tBnBtB,

nAtBAnBtAB , 13

where t is an arbitrary parameter, 0t 12 and n denotes theoutward normal unit vector on the patch boundary. The wavefunction of the cage is restricted to the area of the patchOAB. Now we can construct the covering function , whichis also a solution of Eq. 3, but defined on the wholegraphene lattice in such a way that, when restricted to thepatch, it coincides with the wave function of the unfoldedtetrahedron

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OABrr. 14

Note that, because of the identification of points O, A, and Bon the tetrahedron, (O)(A)(B). Moreover, it isno restriction on to assume that for an arbitrary vector r,

rrA and rrB, 15

i.e., that is periodic with vector periods A and B, andhence

rriAjB 16

for arbitrary integers i and j. Equation 16 extends the solu-tion from the unfolded tetrahedron to a half-tiling of theplane, represented by shaded triangles in Fig. 3. The fullcovering function of is given by assuming that, in additionto Eqs. 15, is symmetric with respect to inversion, i.e.,

rr. 17

Condition 17 extends the definition of to those tiles notaccessible by translation white triangles in Fig. 3. More-over, conditions 15 and 17 imply that the function obeys boundary conditions 13. Indeed, by the consecutive

use of Eqs. 17 and 15 we have

tAtAAtA, 18

i.e., the first equation of Eqs. 13. Similarly one can provethe fourth equation of Eq. 13

n tAntAnAtA.19

The remaining boundary conditions of Eq. 13 can be ob-tained in the same way. On the other hand, having a solution of Eq. 3, one can always construct from it a form

(1/&)(r)(r) , which is symmetric with respect toinversion 17. This is a direct consequence of the cen-trosymmetry of V(r) in our choice of coordinates. Summa-rizing, we have shown that the existence of a covering func-tion , obeying the Schrodinger equation 3 with boundaryconditions 15 and 17 on the whole plane, is a necessaryand sufficient condition for the existence of the wave func-tion that obeys the Schrodinger equation 3 with boundary

conditions 13 on the unfolded tetrahedron. One can con-clude that those orbitals of graphene which obey conditions15 and 17 will be eigenstates of the tetrahedrally symmet-ric 3,6 cage.

V. ELECTRONIC EIGENFUNCTIONS OF A 3,6 CAGE

It is not difficult to find eigenfunctions for the graphenelattice which are simultaneously eigenfunctions of the tetra-hedrally symmetric 3,6 cage. Periodic boundary conditionsin two independent directions A and B give rise to a two-dimensional grid of allowed k vectors in the Brillouin zone.As shown by Ceulemans et al.8 this zone-folding procedure

yields precisely the spectrum of a toroidal polyhex structure,based on the parallellogram patch with sides OA and OB.The allowed k vectors, which guarantee the periodicity, obey

kA2l1 ,

kB2l2 , 20

where l1 and l2 are integers counting allowed eigenstates ofthe cage and defining allowed momentum vectors

kx4

)aNmnl1nl2 ,

ky4

3aNnm l12mnl2 . 21

This result, in the particular case mn , was obtained byGonzalez et al.4 The number of momenta in the first Bril-louin zone is N, i.e., the number of atoms of the originallattice. The first Brillouin zone for the cage shown in Fig. 1,i.e., corresponding to m3 and n1, is shown in Fig. 4a.The additional condition Eq. 17 now requires that onlyeven eigenstates are selected. This effectively halves thenumber of states in the torus by selecting only those whichare symmetric with respect to inversion through the midpointof the line AB. Because of periodicity conditions 13, suchsolutions will also be symmetric with respect to the mid-points of the lines OB and OA.

In this way the gluing conditions are satisfied along all theopen edges of the unfolded tetrahedral patch, precisely asrequired for tetrahedral eigenstates. The selection of evenstates from the grid points operates in the same way as forthe full Brillouin zone. Every antipodal pair of grid pointsyields an even eigenstate in each band, while the and Mpoints yield even states in the 1 and 2 bands, respec-tively. For the K points there are two even states, owing tothe degeneracy of the two bands. The special states of thefirst Brillouin zone, , M1 , M2 , and M3 , are always al-

FIG. 3. Triangular tiling of the honeycomb plane by the patch

OAB. Black tiles are generated by translations along OA and OB.

Centrosymmetry in O maps black tiles onto white ones. The dashed

lines represent waves propagating in a direction perpendicular tothe bonds, with wavelength d . The vertical dotted line is in a

direction parallel to the bonds, with wavelength d .


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lowed momenta of the tetrahedral cage, irrespective of m andn. The point corresponds to the lowest energy state E3, and its wave function is homogeneous over the cage.The momenta M1 , M2 and M3 belong to the triply degener-ate level , and represent a wavelength 3 a .

The periodicity of the wave function in two vectors A andB is a consequence of the construction of the covering func-tion . One can, however, ask a natural question: is the wavefunction also periodic in some vectors that are parallel orperpendicular to bond directions of the hexagonal lattice?The answer is affirmative. Indeed, the three dashed lines inFig. 3 represent the periods of the triangular tiling in the

directions perpendicular to the bonds in case of the m3,n1 lattice. More generally one can show, that the period of

the tiling of the m, n cage in directions e

(1 )ex , e

(2 )

12 ex

()/2)ey and e(3 )

12 ex)/2ey , i.e., perpendicular to

the bonds, is

d2)am2n2mn

Gm ,n , 22

where G m,n is the greatest common divisor of m and n. Inthe case of the m3, n1 lattice in Fig. 3, d26)a.Note that a path which starts in an arbitrary point of the

tetrahedron and leads in one of the directing e

(i) may runseveral times around the tetrahedron before it returns to thestarting point dashed line in Fig. 3. The wavelength func-tion propagating in this direction must be commensuratewith the total length of this path. More precisely, a conditionanalogous to Eq. 20 must hold

kde i 2l , 23

where i1, 2, and 3, and l is an arbitrary integer. In the case

ofK1 and K2 vectors where the wave functions must propa-gate in directions perpendicular to the bonds Eq. 23 yieldseither

l2

3

m2n 2mn

Gm ,n

or twice this value depending on whether the directions of

K1 or K2 and e( i) are respectively at an angle of 60 or 0. It

is easy to show, that l is an integer provided that 3 is adivisor of mn and so the necessary condition for the K1and K2 vectors to belong to the first Brillouin zone is that theparameters of the lattice m,n obey this divisibility condition.

Similar considerations show that must be periodic inthe directions parallel to the bonds i.e., directions alongwhich the M waves propagate. The period for an arbitrarycage is d)d , but dd/) for a cage where 3 is adivisor of mn . In case of a cage with m3, n1, andd78a , half of the period is drawn as a vertical, dotted linein Fig. 3. The period d is always a multiple of 3a thewavelength of the M states. This is consistent with our ear-lier observation from Eq. 21, that M points are allowedmomentum vectors for all n,m.

The allowed k states form a triangular lattice like the dualof the hexagonal lattice of the tetrahedron. A graphical con-struction of the reciprocal lattice is shown in Fig. 4b. Therhombus OABA in this figure is an elementary cell inthe reciprocal lattice equivalent to the first Brillouin zonefrom Fig. 4a. It can be seen that both equilateral trianglesOAB and OAB of the rhombus are identical with thepatch of the hexagonal lattice shown in Fig. 1a, where thecenters of all hexagons are replaced by the states with al-lowed momenta. The number of these states is twice thenumber of hexagons. Note that the y axis in Fig. 4b pointsin a direction opposite to that of the y axis in Fig. 4a. Thisis because the reciprocal lattice is the mirror image of thedual to the hexagonal lattice. The above construction offers aconvenient way of building the first Brillouin zone of a tet-

FIG. 4. Allowed k states for a 3,1 cage in the first Brillouinzone. The k states form a centrosymmetric triangular grid a which

includes and M points. For all other points each pair of antipodal

points yields one allowed even eigenstate. A direct graphical con-

struction of this grid is shown in b. Starting from the original

patch OAB in direct space, we take its dual. This is the triangle

OAB, and we then apply central symmetry to form the rhombus

OABA. The mirror image of this rhombus forms an elementary

cell of reciprocal space, equivalent to the first Brillouin zone in a,

where its kx axis is taken along the longer diagonal and its ky axis

along the shorter one. b shows the reflected elementary cell, as is

clear from the opposite directions of the ky axis in a and b.


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rahedral cage without the use of Eqs. 21, by taking the dualof its unfolded hexagonal patch.

VI. BAND STRUCTURE OF THE CAGE

First consider a periodic lattice on the parent graphitesheet. The energy spectrum of this lattice in the tight-bindingapproximation is uniquely defined by the substitution of the

allowed momenta Eq. 21 in the dispersion relation ofgraphene Eq. 12. The trigonal lattice of states in the re-ciprocal space has a sixfold rotational symmetry around theorigin. On the other hand, dispersion relation 12 hasfull D6 symmetry. Indeed, E(kx ,ky)E(kx ,ky)

E(kx ,ky)E(kx ,ky)

and direct calculation shows that

E C6(kx ,ky)E(ky ,kx)

, where C6 is the rotation about /3.

All the elements of the D6 group are combinations of theabove, energy-conserving transformations. Therefore, withsome exceptions, the energies will occur in sextets or mul-tiples thereof. The only exceptions to this rule are the sixstates , M1 , M2 , M3 , K1 , and K2 marked as black pointsin Fig. 2. Note that, except for M and K points, all statesbelonging to the three upper edges of the Brillouin zone alsoform sextets. This is because the center of each side is anallowed M state and any other state occurring on that sidemust have its mirror partner on the opposite side of M bythe symmetry of trigonal lattice and two other, equal energypairs on the remaining two edges. The center is unique,and so the energy 3 is a singlet.

Now we introduce the extra tetrahedral boundary condi-tion Eq. 17 which requires that antipodal points in kspace be paired to yield even states. All sextets thus will behalved to triplets. In the tetrahedron the points M1 , M2 , andM3 , considered in Sec. III, form a gerade triplet correspond-ing to the energy . However, the degeneracy of this ei-

genvalue can be higher: all states on straight lines betweenneighboring M points and their antipodes the dashed line inFig. 2 belong to this eigenvalue. If we take, according to Eq.21, t h e (l 1 ,l2) parametrization of allowed momenta kk( l1 , l2), then the M states are M1(m ,n), M2(n ,mn), and M3(mn ,m). By simple geometricalconsiderations one can find the condition for the other statesin the first Brillouin zone to belong to the dashed hexagonfrom Fig. 2. The necessary and sufficient condition for the( l1 , l2) pair to generate a k state, other than an M point, witheigenvalue , is that

m2n 2mnmnl1nl2 , 24

where l 1 , l 2 , and l1l 2 belong to the interval (mn)l1 , l2 , l1l 2mn . One can observe that, e.g., the spec-tra for mn and n0 have exact degeneracies 3( m1) atE and 33(m1) degeneracies at E.

The degeneracy at E0, which corresponds to K points,is of special importance. These are the two points where bothbands of the graphene sheet coincide and, at the half filling,meet with the Fermi energy. The symmetry of the reciprocallattice implies that the degeneracy of the energy E0 iseither 2 or 0. The necessary and sufficient condition for theoccurrence of double degeneracy at E0 can be again de-

rived from Eq. 21, and it coincides with the condition thatmn must be a multiple of three. The relation mn0 mod 3 has a simple geometric interpretation, as it is ex-actly the condition that the 3,6 cage m,n should be aleapfrog.9 We will return to this aspect in a separate sectionon inflation relations.

Summarizing, we have shown that the spectrum of thetetrahedrally symmetric 3,6 cage always includes 3, ,

, , and always has a pseudobipartite character: twoproperties noted in Ref. 3 and proved there for a subset of the3,6 class. The spectrum has the form

13n 12 33n 23n 3 , 25

where n 1 , n 2 , and n 3 are integers. In Eq. 25, 1 correspondsto an E3 singlet, 3n1 are n 1 triplets some of them maybe multiples of 3 of energy 3E0, 33n 2 is the de-generacy ofE, 3n 3 are again n3 triplets with possiblefurther degeneracy of energy 0E3, and 2 is the de-generacy of E0 provided 3 divides mn; otherwise thisnonbonding level is absent. The pseudobipartite charactermatches the n1 and n2n 3 triplets. Examples of the Bril-

louin zones and the spectra of cages corresponding to m4 and n0, . . . ,4 are shown in Fig. 5.

The physical consequence of the above general pattern inthe eigenvalue spectrum is that a 3,6 cage of tetrahedralsymmetry has a electronic configuration of one of twotypes, entirely determined by divisibility of ( mn). Eitherthe cage is a leapfrog, with ( mn)0 mod 3, in which caseN, the number of vertices, is a multiple of 12, and the spec-trum has N/22 strictly bonding, two nonbonding and N/2strictly antibonding orbitals, and hence a closed shell con-figuration for the neutral carbon cage; or the cage is not aleapfrog, has (mn)1 mod 3, in which case N is 4 mod 12and the spectrum has N/21 strictly bonding and N/21

strictly antibonding orbitals and hence a closed shell as thedication Cn

2 . The extra four atoms of the nonleapfrogcase are to be found at the face centers of the master tetra-hedron. A typical member of the 3,6 class, if realized as astrained carbon cage, is therefore predicted to lose elec-trons easily, in contrast with the 5,6 fullerenes which areelectron-deficient in their chemistry. The electron deficien-cies of the typical fullerene are rationalised by the presencein leapfrog 5,6 cages of low-lying empty orbitals spanningthe translational and rotational symmetries, and derived fromthe bonding orbitals of the 12 pentagons.10 The degeneratepair of orbitals at the Fermi energy, which is responsible forthe opposite electron donating properties of 3,6 cages, are

related in an analogous way to the bonding orbitals of thefour triangles.

VII. IRREDUCIBLE SYMMETRY REPRESENTATIONS

OF THE TETRAHEDRAL EIGENVECTORS

Having obtained the wave functions, it is always possibleto determine their transformation under the symmetry ele-ments of the point group of the cage. General rules can begiven which relate the irreducible representations of the lev-els directly to their k values. In this section we will limitourselves to the rule which decides the symmetry labels A, E,


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and T of the tetrahedral rotation group. The threefold axes ofthis group run through the vertices and face centers of themaster tetrahedron. In Fig. 6a they are indicated on an un-folded cage. The origin of the hexagonal sheet itself coin-cides with a point of trigonal symmetry, which implies thatthere will also be a threefold axis through the center of the

Wigner-Seitz cell in reciprocal space. The point itself thuscorresponds to a totally symmetric state of A symmetry. Thedoublet at K1 and K2 is always of E symmetry. This can beshown as follows: an anticlockwise rotation of K1 in thediagram of Fig. 2 over 2 /3 turns K1 into K1P1 . Theaction on the K11,g eigenvector thus reads

C3K11,g1

Np exp isprpK1P1ap

0

exp 2i3

K11,g . 26To obtain this result it must be taken into account that theposition vectors are expressed with respect to the coordinateorigin in the center of an hexagon, i.e.,

rp13 T12T2rT1sT2 , for black points,

rp13 T1T2rT1sT2 , for white points, 27

where r and s are integers. The phase factor exp(2i /3) is

typical for a component of the twofold degenerate E repre-sentation; the other component K12,g transforms as a com-plex conjugate.

The triplets in the spectrum are symmetrically displacedaround the central point. Under a C3 rotation the threecomponents will undergo a cyclic permutation, which im-plies that the character of a triplet vanishes; (C3)0. As aresult all triplets must have either AE or T symmetry.

From the tetrahedral character table, these two possibili-ties can be distinguished by their character under any two-fold rotation axis C2 in T:

AEC23,

TC21. 28

The C2 axes run through the edge midpoints of the tetrahe-dron. On the unfolded lattice they form an interstitial trian-gulated lattice generated by the vectors A /4 and B/4 grayspots in Fig. 6b.

If we require a given wave function to be symmetricalwith respect to any C2 axis, it must obey the following con-straints:

rA/4rA/4rA/4,

rB/4rB/4rB/4. 29

In both cases the first equality expresses total symmetry withrespect to the C2 axes, while the second equality expressesthe even parity of the tetrahedral states with respect to thecenter of symmetry at the origin. By adding A/4 or B/4 in thearguments on the right and left of these equations, one ob-tains

rrA/2,

rrB/2, 30

and, by induction,

FIG. 5. Band structures and Brillouin lattice for cages with m

4 and n0, 1, 2, 3, and 4 from top to bottom. Energies are inunits of . Note the twofold-degenerate nonbonding levels in theleapfrog cages 4,1 and 4,4. All spectra include the 3,1,

1,1 eigenvalues of the primitive tetrahedron.


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rriA/2jB/2. 31

Hence totally symmetric eigenvectors with respect to all C2axes must have periods that are only half the fundamentalperiods A and B, i.e., they must fit into a torus that is fourtimes smaller than the original one. The allowed grid pointsare determined by

kA/2

2l1

,

kB/22l2 . 32

If we compare this result with the original zone-folding for-

mula in Eq. 20 one obtains l12l1 and l22l2 . Hence the

AE triplets occur only at grid points with both l 1 and l 2even. One-quarter of the total Brillouin grid is selected inthis way.

As an example consider the tetrahedron with m3 andn1. This cage contains 52 atoms, spanning the followingrepresentation of T:

5A4E13T. 33

In Fig. 6c we show the allowed ( l1 ,l2) grid points in the

Brillouin zone and also mark the (2 l1,2l2) points by heavydots.

The zone center is of course part of the heavy dots, andcorresponds to an A state. The three M points are alsopresent, but not at even grid positions; therefore they trans-form in this specific case as T. This leaves 4 (AE) and 12T triplets. We have indeed exactly 43 heavy dots and123 light dots inside the zone, which identify (AE) andTsymmetries, respectively. Note that in general the M pointsneed not be T symmetric. As an example, for a cage which istwice as large as the one which we have just considered, theM points become even grid points and thus transform

as AE.We briefly mention that one can also use similar tech-

niques to determine the symmetry if the cage has Td symme-try. Each d symmetry plane evolves into two perpendicularsymmetry planes when projecting the cage on the graphitesheet. In this way, one can distinguish between A1 and A 2representations, on the one hand, and between T1 and T2representations on the other hand.

Simple counting of orbits sets of equivalent positionsallows a complete breakdown of the eigenvector symmetriesin all T and Td 3,6 cages. Vertices of a T (m ,n), mn, n0 cage fall into sets O4 and O12 , and vertices of a Tdcage m,0 or m,m fall into sets O4 , O12 , and O24 , all

with their characteristic permutation symmetries. There arefive cases:

1 T symmetry, leapfrog cage (Nm2mnn2),

v N/3AN/3ENT. 34

2 T symmetry, nonleapfrog cage (Nm2mnn2),

v N2

3A

N1

3ENT. 35

3 Td symmetry, leapfrog cages, with m,0,

FIG. 6. Positions of the poles of the threefold a and twofold

b symmetry axes of the tetrahedral cage of Fig. 1 in the unfolded

patch. Heavy dots in c show the allowed k points with even indi-

ces l 1 and l2 in the first Brillouin zone, which correspond to A

E triplets. The point in the origin always belongs to the even

lattice, and is totally symmetric.


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v mm3

6A1

mm3

6A 2

m2

3E

mm1

2T1

mm1

2T2 . 36

4 Td symmetry, nonleapfrog cages with m,0,

v m2m1

6A 1

m2m1

6A2

m1m1

3E

mm1

2T1

mm1

2T2 . 37

5 Td symmetry, leapfrog cages with m,m,

v mm

12

A 1mm

1 2

A2m2E

m3m1

2T1

m3m1

2T2 . 38

VIII. COMPARISON WITH QUANTUM-CHEMICAL

CALCULATIONS

In order to test the validity of the simple tight-bindingapproximation, we have performed a standard density-functional theory DFT calculation on several tetrahedral 3,6 cages. Initial coordinates were generated directly from theadjacency matrix, using the method of Fowler andManolopoulos.11 The B3LYP functional was used, in combi-nation with the split-valence basis sets from Schafer et al.12

extended with a polarization function with exponent 0.80.This DFT calculation was performed with the Turbomolecode.13

The smallest tetrahedral 3,6 cage is the truncated tetra-hedron C12 (m1, n1) . This cage was optimized under Tdsymmetry constraints. Its geometry is shown in Fig. 7. Thetotal binding energy per carbon atom is only 5.37 eV, indi-cating that this structure is less stable than a cyclic alterna-tive. A frequency analysis reveals a negative Hessian eigen-

value for a normal mode with T1 symmetry. Nevertheless thecalculated frontiers orbitals are in agreement with the simplemodel predictions of the preceding sections. The highest oc-cupied molecular orbital HOMO is a fully occupied Elevel, as it should be for a leapfrog cage with mn0 mod 3. The symmetries of the lowest unoccupied mo-lecular orbital LUMO (T2) and next-LUMO ( T1) are alsoas predicted. The calculated HOMO-LUMO gap is 1.93 eV,to be compared with in the model treatment.

Next we have investigated the Td cage C48 , which isagain a leapfrog and has a closed shell as a neutral molecule.

In this case the binding energy is equal to 6.59 eV per carbonatom, while for graphite and fullerenes it is around 7.4eV.14,15 As expected, the larger value of the binding energyreflects that the strain in this larger cage is much relaxed ascompared to C12 . Arbitrary distortions of the cage, followedby optimization without symmetry restriction, always re-stored the tetrahedral structure, indicating that it is now astable minimum. Furthermore, the symmetries of the frontierorbitalsfrom HOMO2 to LUMO1are as predictedby the tight-binding model.

Finally we have also investigated the C52 cage with Tsymmetry shown in Fig. 1b. Since this cage is not a leap-frog (m3, n1) it has a closed shell as a dication, so theDFT calculation was made for C

52

2. Optimization under atetrahedral symmetry constraint yielded the folded geometryshown in Fig. 1b. In the neutral state the cage was found tobe subject to Jahn-Teller distortions, in line with a t2 ground-state configuration. However the closed-shell dication is con-firmed to remain stable with respect to symmetry-loweringdistortions. Orbital energies and symmetries were calculated,and compared to the tight-binding results. The parameters and of the -electron Hamiltonian were fitted to reproducethe calculated HOMO and LUMO levels 11.64 eVand 2.13 eV. In Table I we list the results for eight trip-lets in the vicinity of the Fermi level. The symmetries of the

FIG. 7. Structure for the truncated tetrahedron C12 as obtained

by DFT geometry optimization under Td constraints. A frequency

analysis for this structure yields imaginary frequencies for one T1mode.

FIG. 8. Arbitrary 3,6 cage, unfolded on the honeycomb plane.

Indices correspond to m3, n2, p1, and q4.


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frontier orbitals are as predicted by the tight-binding model.In addition a very satisfactory agreement of energies is

noted. Similar correspondences have frequently been notedfor fullerenes and nanotubes, and explain the success ofsimple two-parameter tight-binding models for carbon clus-ters.

IX. GENERALIZATION TO ARBITRARY 3,6 CAGES

The construction can easily be generalized to the arbitrary3,6 cage.1 The net is defined by the two vectors

A2mT12nT2 , 39

B2pT12qT2 , 40

where m, n, p, and q are integer parameters, and mqpn0. The number of nodes of such a cage is N4( mqpn). The example of a cage corresponding to ( m ,n ,p ,q)(3,2,1,4) is shown in Fig. 8. The construction of thewave function by the use of the covering function resemblesthat described in Sec. III. The only difference is that in thetiling of the plane, shown in Fig. 3, the equilateral trianglesare now replaced by arbitrary triangles defined by A and B.

To find the electronic structure of an arbitrary 3,6 cagewe again use conditions 20, assuring the periodicity of thecovering function. The solution for the allowed k vectors isnow

kx

)a

ql1nl2

mqpn ,

ky

3a2pq l 12mnl2

mqpn . 41

Such a general cage has at least D2 symmetry. The corre-sponding irreducible representations can easily be deter-mined. The and K points will always be of A type.

The neutral cage will be a closed shell with two fillednonbonding orbitals if simultanously mn0 mod 3 andpq0 mod 3 i.e., a leapfrog. Otherwise there will be a

closed shell as dication. Note that for a tetrahedral cage bothconditions coincide, since pn and qmn .

X. INFLATION RELATIONS

Consider a general transformation of the m,n indices ofa tetrahedral cage,

m,nm ,n , 42where , , , and are integers. We call such a transfor-mation an inflation if the number of atoms in the new cagewith indices (m,n) is a multiple of the number of atoms inthe original cage. For this to be true for all m and n, it isrequired that

222222.43

These relations can be turned into a quadratic equation for

as a function of and ,

20, 44

with two independent solutions and . Thefourth parameter then corresponds to and , respectively. The first solution describes the inflationof the cage as a matrix transformation of type

m,nm ,n

. 45The vectors A2mT12nT2 and B2(mn)T22nT1 , which define the cage, can easily be expressed as

linear combinations of A and B vectors of the parent cage:

A,BA,B

. 46An immediate consequence of this relationship is that thespectrum of eigenvalues of the cage OAB is entirely con-tained in the spectrum of eigenvalues of the inflated cageOAB for any inflation obeying Eq. 45. Indeed consideran eigenvector k which is periodic over OA and OB; then bycombining Eqs. 20 and 46, one immediately verifies thatit will also be periodic over OA and OB:

kA2 l1

l2

,

kB2l 1l 2 . 47

This implies that k is also an allowed eigenvector of theinflated cage, which proves the subspectrality of the parentcage. An important example of this type of inflation is theso-called leapfrog transformation, corresponding to 1and 1.

m,nLm ,n 1 11 2

. 48

TABLE I. Orbital energies and symmetries for C522 calculated

by DFT and compared to a -electron Hamiltonian; tight-binding

HOMO and LUMO were fitted to DFT results.


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The leapfrog L of a spherical trivalent polyhedron on v ver-

tices Pv

is constructed by first omnicapping Pv

and then

taking the dual of the result. L( Pv

) has 3v vertices and if

Pv

is a q,6 cage, then so is L. General results about the

spectrum ofL( Pv

) can be proved. If Pv

has at least one faceof size not divisible by 3, the spectrum of L( P

v) has

3v /2 strictly positive and 3v /2 strictly negativeeigenvalues.16 If all faces of P

vare of sizes divisible by 3,

the spectrum has 2 zero and 3v/22 strictly positiveeigenvalues.16,17 The latter result includes 3,6 leapfrogcages as a special case.

A further example is the quadrupling or chamfering18

transformation which is generated by 2 and 0:

m,nQm ,n 2 00 2

. 49This expansion corresponds to the bevelling of every edge.In this way, every edge is replaced by an hexagon. For bothexamples the spectrum of the parent 3,6 cage is containedin the spectrum of the tripled or quadrupled cage.

The second solution of Eq. 44 is of type

m,nm ,n

. 50In this case the relationship between the triangles OAB andOAB is more involved. One has

A,BA,B1

m2nmn 2 mm2nn2mn n2mnm

2n 2

n2mn m2n2 m2n 2mm2n . 51

As the coefficients in this expression are fractional numbers,subspectrality no longer holds in this case.

Inflation can also be applied to the 3,6 cages of D2symmetry. Here we will limit ourselves to cages for whichsubspectrality exists. This implies that the inflated cageOAB is obtained by a linear transformation with integercoefficients such as:

A,BA,B

. 52The multiplication factor in this case is given by .The corresponding transformation of the indices reads

m,n,p,qm ,n ,p,q 0 0

0 0

0 0

0 0

. 53Note that leapfrogging is not included in this transformation.

XI. CONCLUSIONS

By the introduction of suitable topological defects thehexagonal lattice can be wrapped up to a polyhedral cage. In

the case of four triangular defects one obtains a networkmolecule that can be laid over a tetrahedron or bisphenoid.As we have shown in this paper, the electronic structure ofsuch 3,6 cages is very close to the band structure of theparent honeycomb sheet. A straightforward projectionmethod relates this sheet to nanotubes, the nanotubes to tor-oids, and finally the toroids to tetrahedral cages. The first twosteps involve zone folding in two different directions, and thefinal step removes half of the toroidal spectrum by symmetry

adaptation to a centrosymmetric gluing condition. As a re-sult, closed-form expressions are available for the solutionsof the -electron Hamiltonian in any 3,6 cage. These con-firm and generalize previously observed subspectralityrelationships.1.

The analysis indicates that the 3,6 cages can be dividedinto two families, leapfrogs and nonleapfrogs, depending onwhether the differences between the cage indices mn andpq are divisible by 3 or not. This distinction yields asimple criterion for the electronic ground state: leapfrogshave a closed-shell ground state in the neutral form, while

nonleapfrogs are closed shells as di-cations. Furthermoreleapfrogs always have two nonbonding highest occupied mo-lecular orbitals, which can easily act as electron donor levels.So far no molecular realizations of such cages have beenreported. It is expected that triangular defects will cause a lotof strain in a carbon network. Nonetheless the parent tetra-hedral C4 cage (m1, n0) already exists in saturated formas a tetrahedrane, C4R 4 , where R represents a bulky ter-tiobutyl substituent.19 DFT calculations show that the small-est truncated tetrahedron C12 is unstable, whereas higher net-works such as C48 and C52 are stable which is in agreementwith the expectation that the strain is less in larger cages.This makes them realistic targets for chemical synthesis.

ACKNOWLEDGMENTS

The Leuven team ackowledges financial support from theFlemish National Science Foundation FWO and from theBelgian Government through the concerted action scheme.The Katowice team acknowledges support from the PolishState Committee for Scientific Research KBN under GrantNo. 5P03B03320.


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*Corresponding author.1 P. W. Fowler and J. E. Cremona, J. Chem. Soc., Faraday Trans.

93, 22552262 1997.2 M. Goldberg, Tohoku Math. J. 43, 104 1937.3 P. W. Fowler, P. E. John, and H. Sachs, 3,6 cages, Hexagonal

toroidal cages and their spectra, DIMACS Series in Discrete

Math. Theoret. Comput. Sci. 51, 139174 2000.4 J. Gonzalez, F. Guinea, and V. A. H. Vozmediano, Nucl. Phys. B

406, 771 1993.5 P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes

Clarendon Press, Oxford, 1995.6 S. L. Altmann, Band Theory of Solids: an Introduction from the

Point of View of Symmetry Clarendon Press, Oxford, 1991.7 P. R. Wallace, Phys. Rev. 71, 622 1947.8 A. Ceulemans, L. F. Chibotaru, S. Bovin, and P. W. Fowler, J.

Chem. Phys. 112, 4271 2000.9 P. W. Fowler and J. I. Steer, J. Chem. Soc. Chem. Commun. 1997,

1403.10 P. W. Fowler and A. Ceulemans, J. Phys. Chem. 99, 508 1995.

11D. E. Manolopoulos, P. W. Fowler, J. Chem. Phys. 96, 7603

1992.12 A. Schafer, H. Horn, and R. Ahlrichs, J. Chem. Phys. 97, 2571

1992.13 R. Ahlrichs, M. Bar, M. Haser, H. Horn, and C. Kolmel, Chem.

Phys. Lett. 162, 165 1989.14 W. Ebbesen, Carbon Nanotubes: Preparation and Properties

CRC Press, Boca Raton, FL, 1997.15 M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, Science of

Fullerenes and Carbon Nanotubes Academic Press, New York,

1994.16 D. E. Manolopoulos, D. R. Woodall, and P. W. Fowler, J. Chem.

Soc., Faraday Trans. 88, 2427 1992.17 P. W. Fowler and K. M. Rogers, J. Chem. Soc., Faraday Trans. 94,

2509 1998.18 M. Goldberg, Tohoku Math. J. 40, 226 1935.19 G. Maier, S. Pfriem, U. Schaefer, K.-D. Malsch, and R. Matusch,

Chem. Ber. 114, 3965 1981.


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a. ceulemans et al- electronic structure of polyhedral carbon cages consisting of hexagons and...

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