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Laminar structure of bond distortion and electronic states in charged C60
Synthetic Metals, 55-57 (1993) 297%2984
2979
LAMINAR STRUCTURE OF BOND DISTORTION AND ELECTRONIC STATES IN CHARGED C60
X. SUN *t, R.T. FU t, R.L. FU§, Z. CHEN t and H.J.YE§ * International Center for Theoretical Physics, Trieste 34100, Italy. t Department of Physics, Fudan University, Shanghai 200433, China. National Laboratory of Infrared Physics, Academia Sinica, Shanghai 200083, China.
ABSTRACT For doped C60, the charge transfer distorts the bond structure of C60 and forms some self-trapping electronic states. The present theory shows that the charged C60 possesses following remarkable features: 1). Both the bond distortion and the self-trapping states have laminar structure and are localized in the equatorial area. 2). The symmetry is reduced from In of the pristine C60 to Dhd of the charged C60. 3). The carbon atoms in charged C60 are divided into eight layers with an inversion center, and there appear four groups of carbon atoms. The atoms in different groups are nonequivalent, then the spectral line in the NMR gets a fine structure consisting of four sub-lines with the strength ratio 1:1:2:2. 4). The charged C60 has two self-trapping electronic states, one is 0.06eV above HOMO with odd parity, another is 0.05eV below the LUMO with even parity.
INTRODUCTION The investigation of the carbon cluster leads to the discovery of the fullerenes [1-3]. By placing carbon atoms at each vertex of a truncated icosahedron, one generates a soccerball model of C60. In this model, all the 60 carbon atoms are equivalent, this feature has been verified experimentally by the NMR spectrum, in which there is only one line at 142.68ppm [4]. The symmetry of this model determines the HOMO level is five-fold degenerate and LUMO three-fold. The separation between HOMO and LUMO is 1.9eV [5]. Solid C60 is an insulator, its lattice has fcc structure with lattice constant a0 = 14.21 in room temperature [6]. The distance between nearest neighbor molecules in 10.01. Due to the rapid rotational motion in room temperature, the individual C60 is not oriented and a simple cube contains four unit cells. When the temperature goes below 249K, solid C60 undergoes a first-order phase transition to get an orientation order, then the four Ca0 molecules in the cube possess different orientation and lattice becomes sc [7]. Alkali-metal doped C60 M3C6o (M=K,Rb,Ce.--) are superconductors with fairly high Tc=18K for A'3C60 [8], 28K for Rb3C6o [9], 33K for RbCs2C6o [10]. Since the ions of alkali metals are quite small, e.g. the radius of K + is 1.33~, and there are vacancies in fcc structure of solid Ce0, the alkali metal atoms can enter the vacancies so that tile M3C6o retains the fcc structure [11]. In fact, the body center of the cube is a big vacancy surrounded by an octahcdron. There are four octahedrons within a cube. Meanwhile, there ~ e eight tetrahedrons, each consists of one corner and its three neighboring face 0379-6779/93/$6.00
© 1993- Elsevier Sequoia. All rights reserved
2980
centers of the solid Coo. The center of the tetrahedron is also a vacancy. Thus there are altogether 12 vacancies in a cube. When all the vacancies are filled by the Mkali metals, it is M3C6o. The ionization energy of the alkali metals are lower than other elements, the electron is easily transferred from the alkali metal to Coo. If the structure of Coo is rigid, the transferred electron should enter the LUMO of Coo. However, Cs0'S sphere has two-dimensional characteristic, its bond structure is easy to be distorted by the transferred electron. It is expected that the extra electron can be trapped by the distorted lattice to form some bound states just like what has happened in the conducting polymers [12]. Indeed, very recently, Harigaya [13] and Friedman [14] have found that the doped Coo can form a string polaron, in which the dimerization is suppressed along a meridian. This paper will further explore the bond distortion and the electronic states of doped 6'60 and suggest observable effect to demonstrate the distorted structure. Our theory shows that both the bond distortion and self-trapping electronic states possess laminar structures. THEORETICAL CALCULATION In this paper a tight-binding model is used to study the bond structure and electronic states. The hopping h~j depends on the distance I~ - 61 of atoms sitting at ~ and Vj hi3 =
h0 - a ( l ~ - 6 1 - do), nearest neighbor, 0, otherwise,
(1)
where h0 is the hopping constant, a the electron-lattice coupling, do = 1.54~ the bond length of diamond. For any configuration {~} (i=1,2,...,60), the energy spectrum of electron ¢~({~}) can be obtained by diagonalizing the matrix h O. Then the total energy E({~a}) of the system reads occ.
E({~,}) = ~ ~ ( { ~ , } ) + K ~ ( I ~ a
- 61 - do) 2,
(2)
i,j
Minimizing the total energy E({~}) with respect to the configuration {~}, the equilibrium position ~/of 60 carbon atoms can be determined. Before studying the bond charged Coo, the parameters have to be figured out. This can be done by fitting the experimental data of pristine Coo. Numerical calculation shows if to = 1.8cV,
a = 3.5eV/.4,
g = 60.OeVFt 2,
(3)
in the case of 60 electrons, the resultant long-bond length is 1.432/~, short-bond length is 1.395/~, and the LUMO is 1.83eV higher than HOMO. They are very close to the experimental d a t a 1.433/~, 1.389/~[15] and 1.9eV respectively. RESULTS AND DISCUSSION Now we are going to study what happens after one electron is transferred into Coo. By minimizing the total energy (2) for 61 electrons, a new equilibrium configuration {4} of charged Coo can be obtained. The results show that the bond distortion possesses a laminar structure. In order to demonstrate the layer structure more distinctly, it is suitable to orient the buckyball in such a way that its top and bottom are pentagons, which is shown in Fig.1. In this disposal, the Buckyball consists of eight layers, the layer numbers are indicated in the right column in the Fig.1. Meanwhile the bonds are divided into 13 layers, the numbers attached to the bonds indicate the ordinals of the bond layers.
2981
1
1
2
1 2
2
3
3
7
3
7 5 9 6 7 12
12 13
Fig.1
13
8
13
Laminar structure of C60
(i) The laminar structure of the bond distortion For the pristine C60, there are only two kinds of bonds: the long bond (between the pentagon and hexagon) and the short bond (between two adjacent hexagons). After doping, the bond structure is deformed by the transferred electron. The bond distortion possesses the following three remarkable features: 0"010 I 0.005
oooo/% -0.005
-0.010 2
3
4
5
6
7
8
9
I0
Ii
12 13
Layer Number n Fig.2 Layerdependence of changes of bond lengths
a. In each layer, all the bonds have same change in their bond lengths. The layerdependence of the bond distortion is shown in Fig.2. b. The changes of bond lengths are symmetrical with respect to the equator. c. The length changes in 6-, 7-, 8-th layers are much larger than other layers, i.e. the bond distortion is localized in the equator.
2982
Ih
Dsd
e, 12C5, 12C~, 20C3, 15C2 i, 12Sio, 12S130, 20S3, 15a k gk l mk -6.6675 Ag 1 0 -6.1251 TI~ 3 1 -5.1114
Ha
5
2
-3.9846
T2~
3
3
-3.5287
G~
4
3
-2.2669
Ga
4
4
-2.2237
Ha
5
4
-1.4074
H~
5
5
Tl~
3
5
0.9616
Tig
3
6
2.8470
Ha
5
6
3.1521
T2~
3
5
3.5869
H=
5
7
4.4464
Ga
4
6
5.7082
G~
4
7
5.8350
T2g
3
8
(HOMO)
0.4193 (LUMO)
e, i, Ek -6.6652 -6.1264 -6.1216 -5.1153 -5.1121 -5.1049 -3.9980 -3.9913 -3.5248 -3.5099 -2.2742 -2.2527 -2.2341 -2.2195 -2.2098 -1.4138 -1.4117 -1.3525 0.3663 0.4163 0.9304 0.9644 2.8461 2.8519 2.8601 3.1402 3.1676 3.5783 3.5885 3.6008 4.4438 4.4451 5.7030 5.7041 5.8296 5.8311
2Cs, 2C~, 2S~0, 2S,0, k Alg A2~ EI~ Elg Alg E2g A2~ E2~ EI~ E~ Elg E2g Aig Elg E2g EI~ E2~ Al~(q2~) A2~(k~u) EI~ Elg A2g Alg Elg E2g A2= E2~ EI~ E2~ AI= Elg E2g EI~ E2~ E2a A2a
5Q 50"d gk 1 1 2 2 1 2 1 2 2 2 2 2 1 2 2 2 2 1 1 2 2 1 1 2 2 1 2 2 2 1 2 2 2 2 2 1
l 0 1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 5 5 7 7 7 6 6 7 7 8 8
Pristine C60 possesses the symmetry Ih. The bond distortion breaks the symmetry elements Ca and $3, and the symmetry of charged Ce0 is reduced to Dsd. The representations of the groups Ih and Dsd are listed in table, where Ek is the energy level, k the representation of molecular orbit of r electron, gk degeneracy, l irreducible representation of corresponding rotation group.
2983
dd •
dd
•
0
•
-in
-m
-m
-m
•
0
•
0
~S
-S
•
d
0
•
o
-S
-S
-S
-S
•
o
o
•
-m
-S
-S
oos •
-S
I
I i
I
I
I
i
1 1
•
•
o
•
•
o
O
o
o
o
~s o • m
Fig.3
~o S
~,
s ~
S
mo
m
m
S o
s
Self-trappingstate ~
S
m
S
S
S
S
•
o
•
o
•
S •
m m •
•
o
S
S o
-m
•
o
o. S
S
-m-m
o
o
m •
•
1 -1 -1
1
1
-1
-1
•
o
•
•
o
•
o
•
o
o
;1:1
1
•
•
l i -I o
•
•
1=0.192 m=0.114 s=0.000
] In
-m •
-m o
m
-m
m
1=0.186 m=0.153 d=0.075 s=0.026
•
o
-m
o
°m
m
-
m
°
-m
m
•
• -m
s
~
Fig.4
o
..
S
S
Self-trapping state
S kI/ d
(ii) Self-trapping bound states The bond distortion partly eliminates the degeneracy and shifts the levels as shown in table. The point is that, except HOMO and LUMO, all the levels change very little. The original five-fold degenerate HOMO level H~ is split into three levels EI~, E2~ and AI~. Both E1~ and E ~ are two-fold degenerate and close to the original H~ level. But A1~, which wave function is denoted as ~d, is apparently raised up of 0.06eV from HOMO level. The original three-fold degenerate LUMO level TI~ is split into two levels EI~ and A2~. EI~ is two-fold degenerate and very close to the original T1, level. But A2~, which wave function is denoted as ~u, is apparently pulled down of 0.05eV from LUMO level. It means that the bond distortion produces only two new states ~ and Od which levels deviate apparently from the original levels. These two states possess some special properties.
2984
a. Both kg~ and ~a have laminar structure. The wave functions of ~ and ~a are shown in Fig.3 and 4, where the values of the wave functions are indicated in each site of atoms. In each layer, all the sites have the same absolute value of the wave function. Due to the symmetry Crd, tI/u and ~d have a symmetry plane through the polar axis. With respect to this symmetr.y plane, ~d is anti-symmetric and ty~ symmetric. And both ~ and tYd are anti-symmetric to the inversion center. b. ~ and k9d are localized in the equator area. It can be seen from Figs.3 and 4 that their wave functions near the equator are much larger than that near the poles. These features manifest that ~ and ~a are the self-trapping electronic bound states associated with the bond distortion induced by the charge transfer. (iii) The observable effect of the layer structure It has been mentioned in the introduction that all the 60 carbon atoms in pristine C60 are equivalent, and the spectrum of 13C-NMR of C60 has only one line at 142.68ppm. Now the bond structure of charged Cs0 is distorted, and the carbon atoms in charged C60 are no longer equivalent. Since the bond distortion has laminar structure and is symmetric with respect to the equator, the atoms in the first and eighth layers are equivalent to each other. Similarly, the atoms in second and seventh layers, third and sixth, fourth and fifth, are equivalent respectively. Hence, there are four groups of atoms in doped 6'60. The atoms in different groups are nonequivalent, then the line in the NMR of charged C60 should have a fine structure with 4 sublines, which strength ratio is 1:1:2:2. It can be roughly estimated from the screening coefficient/~i that the span of the fine structure is about 0.5ppm. This fine structure in NMR spectrum is a fingerprint of the layer structure. It suggests that the observation of the fine structure in NMR spectrum can provide the information about the bond structure in doped C60.
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation of China and the Advanced Material Committee. REFERENCES 1. E.A. Rohlfing, D.M. Cox and A. Kaldor, J. Chem. Phys. 81(1984) 3322. 2. H. Kroto, J. Heath, S. O'Brien, R. Curl and R. Smalley, Nature 318(1985) 162. 3. W. Kratschmar, L.D. Lamb, K. Fostiropoulos and D.R. Huffma, Nature ~47(1990) 354. 4. R. Taylor, J.P. Hare, A.K. Abdul-Sada and H.W. Kroto, J. Chem. Soc. Chem. Commun. (1990) 1423; H. Ajie et al., J. Phys. Chem. 94(1990) 8631. 5. J.H. Weaver, Phys. Rev. Lett. 66(1991) 1741. 6. R.M. Fleming et al., in Cluster and Cluster-Assembled Materials ed by R.S. Averback, D. Helson and J. Bernkole MRS Symposia Proc.206 (Materials Research Society, Pittsburgh 1991). 7. J.E. Fischer et a l , Science 2.5_2(1991) 1288. 8. A. Hebard et al., Nature 350.(1991) 600. 9. K. Holczer et al., Science 252.(1991) 1154. 10. K. Tanigaki et al., Nature 352.(1991) 222. 11. P.W. Stephens et al., Nature 351(1991) 632. 12. A. Heeger, S. Kivelson, J.R. Schrieffer and W.P. Su, Rev. Mod. Phys. 60(1988) 781. 13. K. Harigaya, J. Phys. Soc. Japan 60(1991) 4001. 14. B. Friedman, Phys. Rev. B45(1992) 1454. 15. J.M. Hawkins et al., Science 252(1991) 312.
2979
LAMINAR STRUCTURE OF BOND DISTORTION AND ELECTRONIC STATES IN CHARGED C60
X. SUN *t, R.T. FU t, R.L. FU§, Z. CHEN t and H.J.YE§ * International Center for Theoretical Physics, Trieste 34100, Italy. t Department of Physics, Fudan University, Shanghai 200433, China. National Laboratory of Infrared Physics, Academia Sinica, Shanghai 200083, China.
ABSTRACT For doped C60, the charge transfer distorts the bond structure of C60 and forms some self-trapping electronic states. The present theory shows that the charged C60 possesses following remarkable features: 1). Both the bond distortion and the self-trapping states have laminar structure and are localized in the equatorial area. 2). The symmetry is reduced from In of the pristine C60 to Dhd of the charged C60. 3). The carbon atoms in charged C60 are divided into eight layers with an inversion center, and there appear four groups of carbon atoms. The atoms in different groups are nonequivalent, then the spectral line in the NMR gets a fine structure consisting of four sub-lines with the strength ratio 1:1:2:2. 4). The charged C60 has two self-trapping electronic states, one is 0.06eV above HOMO with odd parity, another is 0.05eV below the LUMO with even parity.
INTRODUCTION The investigation of the carbon cluster leads to the discovery of the fullerenes [1-3]. By placing carbon atoms at each vertex of a truncated icosahedron, one generates a soccerball model of C60. In this model, all the 60 carbon atoms are equivalent, this feature has been verified experimentally by the NMR spectrum, in which there is only one line at 142.68ppm [4]. The symmetry of this model determines the HOMO level is five-fold degenerate and LUMO three-fold. The separation between HOMO and LUMO is 1.9eV [5]. Solid C60 is an insulator, its lattice has fcc structure with lattice constant a0 = 14.21 in room temperature [6]. The distance between nearest neighbor molecules in 10.01. Due to the rapid rotational motion in room temperature, the individual C60 is not oriented and a simple cube contains four unit cells. When the temperature goes below 249K, solid C60 undergoes a first-order phase transition to get an orientation order, then the four Ca0 molecules in the cube possess different orientation and lattice becomes sc [7]. Alkali-metal doped C60 M3C6o (M=K,Rb,Ce.--) are superconductors with fairly high Tc=18K for A'3C60 [8], 28K for Rb3C6o [9], 33K for RbCs2C6o [10]. Since the ions of alkali metals are quite small, e.g. the radius of K + is 1.33~, and there are vacancies in fcc structure of solid Ce0, the alkali metal atoms can enter the vacancies so that tile M3C6o retains the fcc structure [11]. In fact, the body center of the cube is a big vacancy surrounded by an octahcdron. There are four octahedrons within a cube. Meanwhile, there ~ e eight tetrahedrons, each consists of one corner and its three neighboring face 0379-6779/93/$6.00
© 1993- Elsevier Sequoia. All rights reserved
2980
centers of the solid Coo. The center of the tetrahedron is also a vacancy. Thus there are altogether 12 vacancies in a cube. When all the vacancies are filled by the Mkali metals, it is M3C6o. The ionization energy of the alkali metals are lower than other elements, the electron is easily transferred from the alkali metal to Coo. If the structure of Coo is rigid, the transferred electron should enter the LUMO of Coo. However, Cs0'S sphere has two-dimensional characteristic, its bond structure is easy to be distorted by the transferred electron. It is expected that the extra electron can be trapped by the distorted lattice to form some bound states just like what has happened in the conducting polymers [12]. Indeed, very recently, Harigaya [13] and Friedman [14] have found that the doped Coo can form a string polaron, in which the dimerization is suppressed along a meridian. This paper will further explore the bond distortion and the electronic states of doped 6'60 and suggest observable effect to demonstrate the distorted structure. Our theory shows that both the bond distortion and self-trapping electronic states possess laminar structures. THEORETICAL CALCULATION In this paper a tight-binding model is used to study the bond structure and electronic states. The hopping h~j depends on the distance I~ - 61 of atoms sitting at ~ and Vj hi3 =
h0 - a ( l ~ - 6 1 - do), nearest neighbor, 0, otherwise,
(1)
where h0 is the hopping constant, a the electron-lattice coupling, do = 1.54~ the bond length of diamond. For any configuration {~} (i=1,2,...,60), the energy spectrum of electron ¢~({~}) can be obtained by diagonalizing the matrix h O. Then the total energy E({~a}) of the system reads occ.
E({~,}) = ~ ~ ( { ~ , } ) + K ~ ( I ~ a
- 61 - do) 2,
(2)
i,j
Minimizing the total energy E({~}) with respect to the configuration {~}, the equilibrium position ~/of 60 carbon atoms can be determined. Before studying the bond charged Coo, the parameters have to be figured out. This can be done by fitting the experimental data of pristine Coo. Numerical calculation shows if to = 1.8cV,
a = 3.5eV/.4,
g = 60.OeVFt 2,
(3)
in the case of 60 electrons, the resultant long-bond length is 1.432/~, short-bond length is 1.395/~, and the LUMO is 1.83eV higher than HOMO. They are very close to the experimental d a t a 1.433/~, 1.389/~[15] and 1.9eV respectively. RESULTS AND DISCUSSION Now we are going to study what happens after one electron is transferred into Coo. By minimizing the total energy (2) for 61 electrons, a new equilibrium configuration {4} of charged Coo can be obtained. The results show that the bond distortion possesses a laminar structure. In order to demonstrate the layer structure more distinctly, it is suitable to orient the buckyball in such a way that its top and bottom are pentagons, which is shown in Fig.1. In this disposal, the Buckyball consists of eight layers, the layer numbers are indicated in the right column in the Fig.1. Meanwhile the bonds are divided into 13 layers, the numbers attached to the bonds indicate the ordinals of the bond layers.
2981
1
1
2
1 2
2
3
3
7
3
7 5 9 6 7 12
12 13
Fig.1
13
8
13
Laminar structure of C60
(i) The laminar structure of the bond distortion For the pristine C60, there are only two kinds of bonds: the long bond (between the pentagon and hexagon) and the short bond (between two adjacent hexagons). After doping, the bond structure is deformed by the transferred electron. The bond distortion possesses the following three remarkable features: 0"010 I 0.005
oooo/% -0.005
-0.010 2
3
4
5
6
7
8
9
I0
Ii
12 13
Layer Number n Fig.2 Layerdependence of changes of bond lengths
a. In each layer, all the bonds have same change in their bond lengths. The layerdependence of the bond distortion is shown in Fig.2. b. The changes of bond lengths are symmetrical with respect to the equator. c. The length changes in 6-, 7-, 8-th layers are much larger than other layers, i.e. the bond distortion is localized in the equator.
2982
Ih
Dsd
e, 12C5, 12C~, 20C3, 15C2 i, 12Sio, 12S130, 20S3, 15a k gk l mk -6.6675 Ag 1 0 -6.1251 TI~ 3 1 -5.1114
Ha
5
2
-3.9846
T2~
3
3
-3.5287
G~
4
3
-2.2669
Ga
4
4
-2.2237
Ha
5
4
-1.4074
H~
5
5
Tl~
3
5
0.9616
Tig
3
6
2.8470
Ha
5
6
3.1521
T2~
3
5
3.5869
H=
5
7
4.4464
Ga
4
6
5.7082
G~
4
7
5.8350
T2g
3
8
(HOMO)
0.4193 (LUMO)
e, i, Ek -6.6652 -6.1264 -6.1216 -5.1153 -5.1121 -5.1049 -3.9980 -3.9913 -3.5248 -3.5099 -2.2742 -2.2527 -2.2341 -2.2195 -2.2098 -1.4138 -1.4117 -1.3525 0.3663 0.4163 0.9304 0.9644 2.8461 2.8519 2.8601 3.1402 3.1676 3.5783 3.5885 3.6008 4.4438 4.4451 5.7030 5.7041 5.8296 5.8311
2Cs, 2C~, 2S~0, 2S,0, k Alg A2~ EI~ Elg Alg E2g A2~ E2~ EI~ E~ Elg E2g Aig Elg E2g EI~ E2~ Al~(q2~) A2~(k~u) EI~ Elg A2g Alg Elg E2g A2= E2~ EI~ E2~ AI= Elg E2g EI~ E2~ E2a A2a
5Q 50"d gk 1 1 2 2 1 2 1 2 2 2 2 2 1 2 2 2 2 1 1 2 2 1 1 2 2 1 2 2 2 1 2 2 2 2 2 1
l 0 1 1 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 6 5 5 7 7 7 6 6 7 7 8 8
Pristine C60 possesses the symmetry Ih. The bond distortion breaks the symmetry elements Ca and $3, and the symmetry of charged Ce0 is reduced to Dsd. The representations of the groups Ih and Dsd are listed in table, where Ek is the energy level, k the representation of molecular orbit of r electron, gk degeneracy, l irreducible representation of corresponding rotation group.
2983
dd •
dd
•
0
•
-in
-m
-m
-m
•
0
•
0
~S
-S
•
d
0
•
o
-S
-S
-S
-S
•
o
o
•
-m
-S
-S
oos •
-S
I
I i
I
I
I
i
1 1
•
•
o
•
•
o
O
o
o
o
~s o • m
Fig.3
~o S
~,
s ~
S
mo
m
m
S o
s
Self-trappingstate ~
S
m
S
S
S
S
•
o
•
o
•
S •
m m •
•
o
S
S o
-m
•
o
o. S
S
-m-m
o
o
m •
•
1 -1 -1
1
1
-1
-1
•
o
•
•
o
•
o
•
o
o
;1:1
1
•
•
l i -I o
•
•
1=0.192 m=0.114 s=0.000
] In
-m •
-m o
m
-m
m
1=0.186 m=0.153 d=0.075 s=0.026
•
o
-m
o
°m
m
-
m
°
-m
m
•
• -m
s
~
Fig.4
o
..
S
S
Self-trapping state
S kI/ d
(ii) Self-trapping bound states The bond distortion partly eliminates the degeneracy and shifts the levels as shown in table. The point is that, except HOMO and LUMO, all the levels change very little. The original five-fold degenerate HOMO level H~ is split into three levels EI~, E2~ and AI~. Both E1~ and E ~ are two-fold degenerate and close to the original H~ level. But A1~, which wave function is denoted as ~d, is apparently raised up of 0.06eV from HOMO level. The original three-fold degenerate LUMO level TI~ is split into two levels EI~ and A2~. EI~ is two-fold degenerate and very close to the original T1, level. But A2~, which wave function is denoted as ~u, is apparently pulled down of 0.05eV from LUMO level. It means that the bond distortion produces only two new states ~ and Od which levels deviate apparently from the original levels. These two states possess some special properties.
2984
a. Both kg~ and ~a have laminar structure. The wave functions of ~ and ~a are shown in Fig.3 and 4, where the values of the wave functions are indicated in each site of atoms. In each layer, all the sites have the same absolute value of the wave function. Due to the symmetry Crd, tI/u and ~d have a symmetry plane through the polar axis. With respect to this symmetr.y plane, ~d is anti-symmetric and ty~ symmetric. And both ~ and tYd are anti-symmetric to the inversion center. b. ~ and k9d are localized in the equator area. It can be seen from Figs.3 and 4 that their wave functions near the equator are much larger than that near the poles. These features manifest that ~ and ~a are the self-trapping electronic bound states associated with the bond distortion induced by the charge transfer. (iii) The observable effect of the layer structure It has been mentioned in the introduction that all the 60 carbon atoms in pristine C60 are equivalent, and the spectrum of 13C-NMR of C60 has only one line at 142.68ppm. Now the bond structure of charged Cs0 is distorted, and the carbon atoms in charged C60 are no longer equivalent. Since the bond distortion has laminar structure and is symmetric with respect to the equator, the atoms in the first and eighth layers are equivalent to each other. Similarly, the atoms in second and seventh layers, third and sixth, fourth and fifth, are equivalent respectively. Hence, there are four groups of atoms in doped 6'60. The atoms in different groups are nonequivalent, then the line in the NMR of charged C60 should have a fine structure with 4 sublines, which strength ratio is 1:1:2:2. It can be roughly estimated from the screening coefficient/~i that the span of the fine structure is about 0.5ppm. This fine structure in NMR spectrum is a fingerprint of the layer structure. It suggests that the observation of the fine structure in NMR spectrum can provide the information about the bond structure in doped C60.
ACKNOWLEDGEMENTS This work was supported by the National Science Foundation of China and the Advanced Material Committee. REFERENCES 1. E.A. Rohlfing, D.M. Cox and A. Kaldor, J. Chem. Phys. 81(1984) 3322. 2. H. Kroto, J. Heath, S. O'Brien, R. Curl and R. Smalley, Nature 318(1985) 162. 3. W. Kratschmar, L.D. Lamb, K. Fostiropoulos and D.R. Huffma, Nature ~47(1990) 354. 4. R. Taylor, J.P. Hare, A.K. Abdul-Sada and H.W. Kroto, J. Chem. Soc. Chem. Commun. (1990) 1423; H. Ajie et al., J. Phys. Chem. 94(1990) 8631. 5. J.H. Weaver, Phys. Rev. Lett. 66(1991) 1741. 6. R.M. Fleming et al., in Cluster and Cluster-Assembled Materials ed by R.S. Averback, D. Helson and J. Bernkole MRS Symposia Proc.206 (Materials Research Society, Pittsburgh 1991). 7. J.E. Fischer et a l , Science 2.5_2(1991) 1288. 8. A. Hebard et al., Nature 350.(1991) 600. 9. K. Holczer et al., Science 252.(1991) 1154. 10. K. Tanigaki et al., Nature 352.(1991) 222. 11. P.W. Stephens et al., Nature 351(1991) 632. 12. A. Heeger, S. Kivelson, J.R. Schrieffer and W.P. Su, Rev. Mod. Phys. 60(1988) 781. 13. K. Harigaya, J. Phys. Soc. Japan 60(1991) 4001. 14. B. Friedman, Phys. Rev. B45(1992) 1454. 15. J.M. Hawkins et al., Science 252(1991) 312.