Group theoretical concepts for C60 and other fullerenes

122

Materials Science and Engineering, B19 (1993) 122-128

Group theoretical concepts for C60 and other fullerenes M. S. Dresselhaus Department of Electrical Engineering and Computer Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 (USA)

G. Dresselhaus Francis Bitter National Magnet Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139 (USA)

Riichiro Saito Department of Electronics Engineering, The University of Electro-Communications, Chofu 182, Tokyo (Japan)

Abstract

The physical properties of C60 and general fullerenes are discussed with particular regard to group theoretical concepts. We summarize the role of group theory in specifyingthe multiplet electronic structures for C60, doped C60and photoexcited neutral C60,as well as the multiplet levelsand the vibronicinteractions for the photoluminescence spectra of excitonic levels.

1. Introduction

C6o is one of few examples of a molecule with icosahedral Ih symmetry [1], i.e. the highest point group symmetry, with 120 distinct symmetry operations. This high symmetry gives rise to many degeneracies in the vibrational modes and electronic levels of C6o. The eigenstates of C6o are conveniently classified using symmetry in terms of the irreducible representations of the icosahedral point group. The splitting of these nondegenerate irreducible representations as the symmetry is lowered is useful for predicting the effects of doping and external perturbations on the eigenstates of the unperturbed molecule. Considering the large number of degrees of freedom within the unit cell, the room temperature Raman spectra for both pristine C6o and alkali-metal-doped MxC6o ( M - K , Rb, Cs) are surprisingly simple and similar to each other [2, 3] and to the solution spectra of the free molecule [4]. These observations indicate the importance of the symmetry and suggest that we can model the more complex crystalline C60 and the even more complex doped, crystalline M6C6o using the highly symmetrical C6o molecule as the unperturbed system. In this context, we consider the effect of the periodic potential of the crystalline lattice of C6o as a very weak symmetry-loweringperturbation and the effect of the alkali metal doping on the lattice modes of solid 0921-5107/93/$6.00

C60 in a similar manner to the approach taken in carrying out the lattice dynamics studies in alkali-metalintercalated graphite [5, 6]. For both (260and the graphite intercalation compounds (GICs), the alkali metal interacts only weakly with the near-neighbor carbon atoms, so that the main effect of intercalation on the lattice modes is the charge transfer associated with the alkali metal intercalation process. As for the electronic structure, the symmetries of the ground states for general (including large) icosahedral fullerenes and for C'~6d: molecular ions can be treated by considering the fullerenes to have an approximately spherical shape, allowing the use of spherical harmonics to classify their electronic configurations [7]. Hund's rule is applied to find the quantum numbers S, L and J for the ground state configurations that are allowed by the Pauli principle [7]. The icosahedral symmetry is then introduced as a symmetry-lowering effect, treating the icosahedral groups Ih and I as subgroups of the full rotational group (see Table 1 for the character table for the double group o f / ) . Two limiting cases are considered explicitly, depending on whether the icosahedral perturbation is small or large compared with the electron--electron interaction. Using this general approach, it is possible to identify fullerenes that are especially stable, because they are already stable before the icosahedral symmetry-lowering perturbation is applied. © 1993-Elsevier Sequoia. All rights reserved

123

M. S. Dresselhaus et al. / Group theory for C6o and other fullerenes TABLE 1. Character table for the double point group I 1

E

R

12C5

12C5

12(752

12C52

20C 3

20C 3

30C 2

Basis functions

FI(A) F2(Ft)

+1 +3

+1 +3

+1 +r

+1 + ~-

+1 - 1/~-

+1 - 1/~-

+1 0

+1 0

+1 - 1

Y0.0 YI,-I, Yl.o, YI.1

F3(F2)

+3

+3

- 1/r

-l/r

+r

+ ~-

0

0

- 1

( - (2/5)v2y3, -3 + (3/5)I/2y3, 2) Y3,0 ((2/5)1/2Y3 3 + (3/5)1/2Y3 -2)

F4(G)

+4

+4

Fs(H)

+5

+5

0

0

I'6

+2

-2

+~-

-~-

F7

+2

-2

-1/~-

Fa

+4

-4

+1

+6

-6

-1

-1

-1

1/~-1

+1

-1 0 -1/r

-1

+1

+1

0

0

-1

-1

+1

1/~-

+1

-1

0

((3/5)U2y3. -3 + (2/5)U2y3. 2) Y3, - 1 1 ~3' (3/5)1/2Y3.3 + (2/5)x/2y3. -2)

Y2.-2, Y2,-t, 112,0, Y2.x, Y2,2

+~-

-~-

+1

-1

0

~b~/2._ x/2, ~l/x v2 { ((7/10)1/24'7/2.-3/2- (3/10) t/2) ((7/10)1/24'7/2,3/2+ (3/10)1/2~;b7)

+1

-1

-1

+1

0

~/2.-~/2, 4'~/2.-i/2,4'3:~i/2,~/2,3

-1

+1

0

0

0

~5/2. - 5/2 ~b5/2. - 3/2 ~5/2, - i/2 ~5/2.1/2 t~5/2. 3/2 ~5/2. 5/2

In this table, ~-= (1 + 51/2)/2, - 1/~-= 1 - % ~.2= 1 + r.

Note: C5 and (75-I are in different classes, labelled 12C5 and 12C52 in the character table. R and (~i represents a double rotation. The group lh is derived from group I as the direct product group l®i=lh and the irreducible representations of Ih are classified as even and odd, respectively, using the subscripts g and u.

2. Group theory of the electronic structure of fullerenes I n c o n s i d e r i n g t h e r o l e o f g r o u p t h e o r y in simplifying t h e t h e o r e t i c a l f r a m e w o r k for d e s c r i b i n g t h e e l e c t r o n i c s t r u c t u r e o f i c o s a h e d r a l f u l l e r e n e s , w e c o n s i d e r first t h e v a r i o u s i c o s a h e d r a l f u l l e r e n e s CN,,m t h a t c a n b e c o n s t r u c t e d m a t h e m a t i c a l l y . I c o s a h e d r a l s y m m e t r y req u i r e s t h a t N , , , m = 2 0 ( n 2 + m 2 + n m ) , in which n a n d m a r e i n t e g e r s [8]. F r o m t h e (rn, n ) values, t h e s y m m e t r i e s a r e d e t e r m i n e d : Ih (for m = n o r m n = 0) a n d I (for all o t h e r (m, n ) ) ( s e e T a b l e 2) [8], w h e r e p o i n t g r o u p / d o e s n o t c o n t a i n t h e inversion o p e r a t i o n . Since e a c h v e r t e x on a f u l l e r e n e c o n n e c t s a c a r b o n a t o m (having f o u r v a l e n c e e l e c t r o n s ) to t h r e e o t h e r c a r b o n a t o m s , we can assign o n e a v a i l a b l e zr e l e c t r o n to e a c h c a r b o n a t o m o f t h e f u l l e r e n e . W e can t h e n f o r m e l e c t r o n i c c o n f i g u r a t i o n s f r o m t h e s e ,r e l e c t r o n s , as shown in T a b l e 2, by assigning 2 ( 2 / + 1 ) o f t h e s e e l e c t r o n s to e a c h a n g u l a r m o m e n t u m s t a t e a n d using t h e c o n v e n t i o n a l n o t a t i o n s, p, d, f, g, h, i, k, 1, m, n, o, q, r, t, ... for a n g u l a r m o m e n t u m s t a t e s l = 0, 1, 2, .... The/max v a l u e for e a c h f u l l e r e n e in T a b l e 2 c o r r e s p o n d s to t h e maxi m u m I v a l u e for t h e h i g h e s t o c c u p i e d m o l e c u l a r o r b i t a l ( H O M O ) for e a c h e l e c t r o n i c c o n f i g u r a t i o n . A l s o listed

in t h e t a b l e are ntot = 2(/m~x+ 1) 2, which is t h e n u m b e r o f e l e c t r o n s n e e d e d to fill t h e highest lm~x state, a n d t h e t o t a l n u m b e r nv o f v a l e n c e electrons. W e t h e n a p p l y H u n d ' s rule to g e t t h e g r o u n d state q u a n t u m n u m b e r s S, L a n d J listed in T a b l e 2. A l s o given in T a b l e 2, using t h e n o t a t i o n in p a r e n t h e s e s in T a b l e 1, a r e the i r r e d u c i b l e r e p r e s e n t a t i o n s o f the i c o s a h e d r a l group, c o r r e s p o n d i n g to e a c h J v a l u e for t h e H u n d ' s rule g r o u n d state. O f p a r t i c u l a r i n t e r e s t a r e t h e special f u l l e r e n e s C6o, Clso a n d C42o which f o r m J = 0 g r o u n d states, even w h e n w e c o n s i d e r the full s p h e r i c a l symmetry, i.e. b e f o r e t h e i c o s a h e d r a l p e r t u r b a t i o n o f t h e i c o s a h e d r a l f u l l e r e n e m o l e c u l e s is i n t r o d u c e d . It is e x p e c t e d t h a t t h e high stability o f t h e s e J = 0 m o l e c u l e s will b e f u r t h e r e n h a n c e d by t h e s y m m e t r y - l o w e r i n g effect o f t h e i c o s a h e d r a l p e r t u r b a t i o n , y i e l d i n g even m o r e stable molecules. C l o s e d shells for m o l e c u l e s which a r e topologically i c o s a h e d r a l for l =/max Occur very rarely. T h e s m a l l e s t p o s s i b l e N,,,.n value for a n i c o s a h e d r a l f u l l e r e n e CNm.n having a c l o s e d shell c o n f i g u r a t i o n (i.e. nv = 0 o r S = 0, L = 0, J = 0) is Nm. n = 275 467 380, which occurs for (m, n ) = (2192, 2093) a n d /max= 11 735. A l l t h e topologically a l l o w e d i c o s a h e d r a l f u l l e r e n e s for Nm.n < 7 8 0 a r e listed in T a b l e 2. H o w e v e r , for t h e s e f u l l e r e n e s to b e stable in i c o s a h e d r a l s y m m e t r y r e q u i r e s

124

M. S. Dresselhaus et al. / Group theory for C6o and other fuUerenes

T A B L E 2. Spherical ball approximation for the ~" molecular orbitals for icosahedral fullerenes CNm.n

(m, n)

Sym.

"/max

ntot

nv

Full configuration

S~

La

J~

Icosahedral symmetry b

C2o C60 C80 Cl40 Cl80 C240 C260 C320 C380 C420 C50o C540 C56o C620 C72o C74o CTs0

(1, (1, (2, (2, (3, (2, (3, (4, (3, (4, (5, (3, (4, (5, (6, (4, (5,

lh Ih 1h I Ih lh I lh I I Ih lh I I Ih I I

3 5 6 8 9 10 11 12 13 14 15 16 16 17 18 19 19

32 72 96 162 200 242 288 338 392 450 512 578 578 648 722 800 800

2 10 8 12 18 40 18 32 42 28 50 28 48 42 72 18 58

sZp6dl°f2 s2p6d1°f~4g18hl° sZp6dl°...h22i s s2p6dl°...k3°112 s2p6dl°...134m18 s2p6dl°...maSn 4° s2p6dl°...n42o 18 s2p6dl°.., o46q32 s2p6d1°...qS°r 4~ sZp6dl°.., rs4t 2s s2p6dl°...tSSu 5° sZp6dt°.., u6Zv28 s2p6dl°...u62v 48 s2p6dl°...v66w 42 s2p6dl°...w7°x72 sEp6dl°...x74y TM s2p6dl°...x74y "58

l 5 4 6 9 1 9 9 6 14 6 14 9 14 ]. 9 10

5 5 20 30 9 19 45 63 90 14 114 70 135 98 35 189 190

4 0 16 24 0 20 36 72 96 0 120 56 144 112 36 180 200

Gg, Hg Ag A s, 2Fig, F2g, 2Gg, 3Hg A, 2F1, 2F2, 4G, 4 H Ag A s, 2Fig, 2F2g, 2Gg, 4Hg 2,4, 4F~, 3F2, 5G, 6H ... ... A ... ... ... ... 2Ag, 4Fig , 3F2g , 5Gg, 6Hg ... ...

:

i

i

:

i

!

i

i

0) 1) 0) 1) 0) 2) 1) 0) 2) 1) 0) 3) 2) 1) 0) 3) 2)

aS, L and J values for the ground state are given according to H u n d ' s rule. tqrhe irreducible representations for icosahedral symmetry are explicitly listed for J < 5 0 .

that a non-degenerate electronic ground state exists for the molecule in question. Table 2 suggests that there are an extremely limited number of fullerene molecules which would satisfy these constraints. Open shell structures may be easily distorted by symmetry-lowering interactions when the ground states are degenerate. Non-degenerate ground states appear in a filled shell (labelled/max) whenever we have nv = 0, S = 0, L = 0 and J = 0, for example, for N = 18, 32, 50, 72, 98, 128, 162, 200, ..., 2(lmax+l)2). However, in general, these fullerenes do not have icosahedral symmetry (see Table 2). In addition, there are the Hund's rule states where L = S and J = 0, and these also give non-degenerate ground states for spherical symmetry (for example, N = 2 4 , 40, 60, 84, 110, 144, 180, 220, .., 420 . . . . . 2(/m~,+l)(lm~+2)). By these arguments, all these J = 0 states are especially stable and one might expect these configurations to give rise to particularly stable fullerenes. The relative magnitudes of the configurational interaction and the icosahedral (or other symmetry-lowering) splittings determine the most convergent approach to calculations of the electronic structure. If the icosahedral perturbation is large compared with the electron-electron interaction, then the most appropriate calculational strategy is to apply the icosahedral perturbation first to split the large l values and then to formulate the many-electron levels from the icosahedrally split states (these are designated in Table 3 as low spin states). However, the electron correlation effects are much larger than the icosahedral splittings, then one should form the multiplet states first and

then apply the weak icosahedral perturbation later (these are designated in Table 3 as high spin states). We illustrate both these approaches in Table 3, which lists the levels of various C'~6omolecular ions - 6 ~
M. S. Dresselhaus et al. / Group theory for C~o and other fullerenes

125

I

:A

c

"O t~

~

~

~1~1

~

~

I ~ 1

~

~

~

I I 1 [ I

~

t-

II

II

"'n

O

0

d~

~J ~I

~

I ~

~

I ~

I ~

~

I ~

'G

[--,

=o t~

e-.

O

A"

__o

c~

O

I

I

I

'~D tt% ',:1"

I O'5

I ¢q

I r--~ ~

+

+

+

,--.~

¢',1

to)

126

M. S. Dresselhaus et al. / Group theory for C6o and other fullerenes

TABLE 4. Decomposition of angular momenta into irreducible representations of 1 F~(A) 0 (S) (D) (F) (G) (H) (I) (K) (L) (M) (N) (O) (Q) (R) (T) (U) (V) (W) (X) (Y) (Z)

r3(F9

r,(G)

Fs(H)

1

1 (P)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

r2(el)

1

1 1 1 1 1

1 1 1 1 1 2 1 1 1 2 2 2 1 2 2

r7

1 1

1 1 1 1 1 1 1 2 1 2 1 2 2 2 2

1 1 1 2 1 1 2 2 2 2 2 2 3 2 2

fullerene, the atomic potential modulation is expanded to high I values and the energy gap decreases with increasing radius. Thus, the high spin state would be expected to be the preferred description for the large fullerenes. For the high spin state limit, 7r electrons in fullerenes are well represented by a large wave function amplitude on a ball. If we look at the entries in Table 3, we note that a number of the H u n d ' s rule ground states have A , symmetry, which is indicative of a non-degenerate, stable ground state. In other cases, the icosahedral ground state is degenerate and a splitting would be expected based on a J a h n - T e l l e r distortion argument. One example of a degenerate icosahedral ground state is that for the C6o4- ion where the icosahedral symmetry shows a fivefold degenerate Hg ground state. Interestingly, the crystal structure of M4C6o is tetragonal [12], which is consistent with a D2h symmetry-lowering distortion of the C6o4- molecular ion, which splits the five fold Hg level into a non-degenerate and two doubly degenerate levels, so that the non-degenerate level can form the ground state. Likewise, the C6o1- state is a doublet 2/, which is split by the spin-orbit interaction into a twofold and a fourfold degenerate state. Since the spin-orbit coupling for carbon is very small [13], such hyperfine structure could only be observed in accurate high resolution experiments. F r o m the standpoint of the fullerene host, Table 3 suggests that a C6o2- molecular ion would be a more satisfactory host for endohedral dopants than would C6o1-, since C6o2- has a non-

1 1 1 1 2 1 2 2 2 2 3 2 3 3 3 4 4

5/2 7/2 9/2 11/2 13/2 15/2 17/2 19/2 21/2 23/2 25/2 27/2 29/2 31/2 33/2 35/2 37/2 39/2 41/2

/'8

r,

1

3/2

1

1

/'6 1/2

1

1

J

1

1 1 1 1 1 1 1 1 2 1 1 1 1 2

1 1 1 1 1 1 1 1 1 1 2 1 1

1 1 1 1 1 1 2 2 1 2 2 2 3 2 2 3 3

1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4

degenerate ground state and C6o 1 - does not. However, to determine the stability of an endohedrally doped fullerene, the symmetry of the endohedral dopant also must be considered. Two of the common endohedral dopants, namely y3+ and La 3+, both have closed shell configurations, for which the ion potential would not lower the icosahedral symmetry of the fullerene shell. In this context, Tables 3 and 4 are useful for discussing the stability of endohedrally doped fullerenes. For example, the introduction of one La ion into Coo gives rise to La-C6o, where the La has been reported to be in a + 3 charge state, indicating a negatively charged fullerene, i.e. C6o3- [14, 15]. The Hund's rule ground state for an icosahedrally symmetric molecule in this case would have Fs- symmetry, within the I h symmetry point group, and would therefore be fourfold degenerate. Such a degenerate state would be expected to be unstable with regard to a Jahn-Teller distortion, so that a symmetry lowering of the icosahedral symmetry of the molecule would be expected. Recent measurements of the magnetic properties of Cs-doped C6o [16] show unexpected behavior in the susceptibility, which implies that the molecular ions have magnetic dipole moments. For the negatively charged fullerene ions, the maximum magnetic m o m e n t is observed experimentally at C6o4 - (x = 4 for Cs4C60), which is consistent with the molecular ion ground states listed on the right-hand side of Table 3, where the maximum J value ( J = 2 ) also occurs at C6o4-. For acceptor-doped C6o, the maximum J value is predicted (by this work) to occur for the + 3 ion, though this

M. S. Dresselhaus et al. / Group theory for C6o and other fullerenes

effect has not yet been confirmed experimentally. Although the maximum magnetic moment occurs for the C604- ion, the measured effective Bohr magnetron is much smaller than the theoretical effective Nohr magneton given by g[J(J+l)]~/2=2(6v2). This large discrepancy may arise from a thermal averaging of J (since the spin-orbit interaction for fullerenes is so small) and from an itinerant character of the electron wave functions between the balls in the face-centered cubic (f.c.c.) lattice.

3. Optical and vibronic transitions

To calculate the energies of the excited states of the neutral molecule, we must first consider the hgf~~ configuration, which comes from excitations of the l = 5 configuration of Table 2. By considering the decomposition of the direct product (H®FI), as shown in Table 5, one can form Pauli-allowed states, which are singlet wave functions with symmetries ~F~g, ~F2g, ~Gg, 1Hg as well as triplet wave functions with symmetries 3Fig, 3F2g, 3Gg, 3Hg (see Table 5) [17]. On the righthand side of the state designations in this table are given the icosahedral irreducible representations obtained by decomposing the direct product F~® Fs of the orbital F~ and spin Fs angular momenta, where F~=A for the singlet state and F~=F1 for the triplet state. TABLE 5. C6o° states allowed by Pauli principle Config.

Singlet

h l°

~A

A

-

hga t

tH

H

3H

Ft+F2+G+H Ft + F 2 + G + H F2+G+H G+H A+Ft+H

hgfl I

h9f2~

hgg I

h9h t

Triplet

IH

H

3H

IG

G

36

tF2

F2

3F2

IF 1

F1

3F 1

IH IG 'Fz

H G F2

31-1 36 3F2

IF 1

Fl

3F 1

IH 1H tG

H H G

31t 3H 3G

tF2 1FI

F2 FI

3F2 3FI

tH 1H IG IG

H H G G

3H 31t 3G 3G

IF2 tF1 1-4

F2

3F2

Ft

3F~

A

3A

F~+F2+G+H F2+G+H G+H A + F 1+ H F~+F2+G+H Ft+F2+G+H F2+G+H G+H A + FI +H Ft+Fz+G+H FI+Fz+G+H Fe+G+H Fe+G+H G+H A + Ft + H FI

127

The optical transitions in the neutral C6o molecules or in pristine C60 films can be understood from Table 5. The dorhinant optical transition is to the excited molecular state (or in the solid to the excitonic state). This excited (or excitonic) state will have one of the allowed configurations given in the table. Since the spin--orbit coupling is small, we expect the conversion from spin singlet to spin triplet to be relatively slow. Thus, optical transitions will dominantly conserve the spin angular momentum and the ground state (1As) will make optical transitions to the singlet states listed. Since the electric dipole has symmetry Flu, the allowed transitions from the ground state would be expected to have symmetry ~Flu. However, the lowest excited state configuration hugf~u1 has even parity, so that direct optical transitions are forbidden. There are several mechanisms which would contribute experimentally to these forbidden transitions. Imperfections or impurities would play an important role. In particular, 02 contamination is known to enhance the triplet-to-singlet conversion [18]. Forbidden transitions also can be made observable through a second-order process called a vibronic transition, in which a vibrational mode is also involved. This process is often the dominant absorption mechanism in molecules and is observed in the experimental luminescence spectra of fullerenes [19]. Group theoretical arguments determine the selection rules for both direct first-order allowed transitions and vibronic transitions involving a single quantum of vibrational energy. In general, the allowed matrix elements for the optical dipole transitions for zero vibrations are of the form (O(Fi)~-Al~b(Ff)). Since the momentum operatorp transforms as the irreducible representation Flu(F2-) of the fullerene point group Ih, the selection rule for an allowed zero-vibrational optical transition from an initial state i to a final state f is determined by requiring the direct product F~® Ff to contain the irreducible representation FIn(F2-). For fullerenes with inversion symmetry, this selection rule requires that the initial and final states have opposite parity. For a vibronic transition containing a vibrational mode, the selection rule requires the direct product Fi ® Ff® Fv to contain the irreducible representation Flu(F2-), where Fv is the symmetry of the normal mode vibration. The allowed vibrational mode symmetries permitting vibronic transitions that might be observed in a luminescence experiment from an excited state of symmetry Fox to the ground state of symmetry Ag are listed in Table 6. This table also gives the symmetries for the vibronic transitions that are possible in an absorption experiment. In absorption, the initial state F~ has Ag symmetry, whereas, in luminescence, the final state has Ag symmetry. In either case, the excited electronic state of symmetry Fex will show vibronic lines associated with normal modes of symmetry Fv provided

M. S. Dresselhaus et al. / Group theory for C6o and other fullerenes

128

TABLE 6. Phonon symmetries Fv coupling the Ag ground state to an excited state Fox through a vibronic transition

Ag Fig

Flu Au, Flu, Ho

Au Flu

Fig Ag, Fig, Hg

F2g Gg Hs

G., H. F2., G,, H, Flu, Fz~, G~, Hu

F2. G, Hu

Gg, Hg F2g, Gg, Hg F1g, F2s, Gg, Hg

that Fex® Fv contains the irreducible representation Flu. Since the vibrational mode symmetries in C60, as designated by their irreducible representations, are 2Ag + 3F18+ 4F28 + 6Gg + 8Hg + 1Au + 4Flu + 5F2u + 6Gu + 7Hu of the Ih group [9], we see that many vibronic transitions are possible, both in the luminescence and the absorption spectra. In summary, it is shown that the high symmetry of the icosahedral fullerenes allows group theory to simplify the analysis of the optical properties of fullerenes and leads to important new insights into the physics of fullerenes.

Acknowledgments We gratefully acknowledge NSF Grant DMR92-01878 for support of this research. One of the authors (RS) contributed to this work while visiting MIT as an Overseas Research Scholar of the Ministry of Education, Science and Culture of Japan.

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