Thermodynamics of intercalation in C60 fullerite and stability of cubic close-packed fulleride compounds

ELSEVIER

Synthetic Metals 80 (1996) 301-307

Thermodynamics of intercalation in C60 fullerite and stability of cubic close-packed fulleride compounds D. Claves a, Ph. Touzain a.b • Laboratoire Science des Surfaces et Materiaux Carbonb. ERS CNRS No. 101. ENSEEG. Domaine Universitaire. BP 75. 38402 Saint-Martin d'Heres. France b JUTl. Departement de Chimie. Universite J. Fourier. J rue F. Raoult. 38000 Grenoble. France

Received 5 May 1995; accepted 13 February 1996

Abstract Stability of known or hypothetical fullerene C60 intercalation compounds was investigated through a donor-acceptor scheme. A thermodynamic cycle was employed, requiring evaluation of energy costs related with charge transfer and formation of the intercalated so~id. Strongly exothermic reactions are predicted in several cases, especially with some elements of the rare-earth series, which should consequently fo.m stable, low dimensional phases. These latter might be interesting candidates for superconductivity, as alkali-doped C 60 • Keywords: Fullerene; Thermodynamics; Intercalation

1. Introduction Pristine C 60 forms a face-centered cubic structure, with a cell parameter ao = 14.17 A [1], in which individual molecules, held together by van der Waals forces, undergo nearly free rotation at room temperature [2,3]. This type of configuration generates tetrahedral and octahedral interstitial sites (t and 0) with sufficient sizes to accommodate a wide range of guest species, such as alkali or alkaline earth ions [4,5]. According to the size or number of intercalated species, distortion of the primitive cell may be observed, most of the time lowering the crystal symmetry, thus increasing the available space and allowing, for instance, further doping with alkali metal or intercalation of larger molecular units such as TDAE [6], sulfur [7], phosphorus [8], iodine [9], metallocene [to] or solvent molecules [ 11]. In several cases (alkali, alkaline-earth metals, TDAE) , charge transfer is observed between host-guest species [1214] and a net negative charge is ascribed to the C60 molecule making the doping process describable, using a donor-acceptor model. On steric grounds only, most of the elements of the periodic table are small enough to occupy the vacancies of the f.c.c. fullerene matrix, involving simple expansion or contraction of the lattice. In this paper, we focus on the stability of some such f.c.c. intercalated phases (known or unknown), assuming progressive and unitary filling of the t and 0 sites, yielding MxC 60-type compounds, with x= 1, 2 and 3. Starting from experimental data on the well-charac0379-6779/96/$15.00@ 1996 Elsevier Science SA All rights reserved

PllS0379-6779(96)03659-4

terized C60 and alkali-doped C60 phases, a model is proposed for the fulleride, which can easily be extrapolated to all the other elements. In some cases, stability of the compounds is expected from largely negative enthalpies offormation. However, direct determination of the phase stability relative to the doping level must be made by examining the sign of the enthalpies of the disproportionation reactions:

Mx C60 -+ ;MyC60 +( 1 -

;)c60

for y> x

This type of calculation should be useful in predicting line compound or solid solution behaviour. The influence of oxidation states, the general trends in a family of elements and a comparison with earlier works [ 15] are also discussed. As a last point, the real possibility of charge transfer, under some conditions, is estimated.

2. Thermodynamics The simplest theoretical approach consists in reasoning on the basis of a purely ionic model. The enthalpies of the intercalation reactions can then be evaluated using a classical thermodynamic cycle, decomposing the process in elementary steps. The zero-dimensional aspect of pristine C 60 allows us to use a Born-Haber cycle similar to the one accounting for the formation of simple alkali halide compounds from the elements in their standard state, as shown in Fig. 1.

D. Claves. Ph. Touzain / Synthetic Metals 80 (1996) 301-307

302

of a crystal is the sum of several terms resulting from the different types of interactions between the ions and can be expressed as

x M (gas) + C60 (gas)

i

x

U'f(solid)

1: I(q). L EA(n)

(CHARGE TRANSFER)

x &losub(M) + .1Hosub(CW> (SUBLIMATION) .1HOr =.1uoT' (x+ l).RT (FULLERIDE FORMATION)

Ix M (metal) + C6Q (solid) 1---

Fig. I. Born-Haber cycle used for the calculations.

The only steps involving calculations are the charge transfer on the C 60 molecule (total electron affinity) and the formation of the intercalated solid from the gaseous ions. Other values are tabulated [16] or can be found in the literature [17] for 6.Hsub (C 60 ), The chemical reaction writes: tHI'i

C 60 (s) +xM(s) -M xC60 (s)

= EClectroslatic + Erepulslon 101>-IOn + Evan

dcr

Waals + EVibrationaJ

Considerable simplification of the problem arises from the introduction of the below-mentioned hypothesis. Repulsion between neighbouring intercalant cations can be neglected, their size being small in front of their interionic distances. The van der Waals terms can be considered as coming mostly from the dispersive interactions between large C 60 molecules. The vibrational part of the energy, Usually small, can be neglected, to a first approximation. In the calculation of 6. CJ'?, it will be partIy compensated by the variation in internal energy of the gaseous ions on passing from 0 K to room temperature. Grouping the attractive van der Waals and repulsive anion-anion contributions, we can express A UT as 6. U'f- U'f(solid), U'f(solid) now writing: U'f(solid) = Eclectroslatic + E(C6Q/C60 )auractivefrepulsive + Erepulsion anioJ>-Cation

3.1.

EekctrostDtlC

Assuming a point charge model and complete charge transfer, the electrostatic energy per unit cell is _ 1 (qe)2 Eclectrostatic - - a - 4 - -

with

'ITEo

q

6.Hf=6.1F.ub(C60 ) +x6.1F.ub(M) +xI:I(q) n

- EEA(n) +6.HT(Mx C60 ) where 6.1F.ub is the enthalpy of sublimation of the species considered, !H(q) is the energy required for ionization of the metal to the oxidation state +q, I"EA(n) is the total electron affinity defined as the energy required for releasing n excess electrons from the negatively charged molecule, EA(n) being the one-electron affinity for the reaction: C6Qn-_C60 (n-I)-+e-. 6.HT(MxC6Q) accounts for the cohesive energy of the ionic solid and writes:

6. U'f being the variation in internal energy accompanying its formation at temperature T. Thus, 6. U'f= U'f( solid) - U'f( gas ) .

3. Cohesive energy of the solid The cohesive energy UT of an ionic solid in specified thermodynamic conditions is defined as the internal energy of the solid, relative to the state of the free ions at absolute zero. In the c1assicaltheory of the ionic solid [ 18] the cohesive energy

a

where a is the adimensional Madelung constant for the structure considered, a is the lattice parameter of the cubic cell and q the charge of the intercalant cation. Madelung constants for the different f.c.c. cases are SUIllmarized in Table 1. These were calculated stating linear rela_ tions between the well-known Madelung constants [19] of NaCI and CsCllattice types, whose superposition yields the relevant structures.

3.2. Interactions between C60 These interactions result from a London-type attractive term and a repulsive term due to overlapping of the carbon orbitals of adjacent clusters. As in the case of graphite, inter_ Table I Calculated Madelung constants according to the number and nature of OCcupied vacancies Structure

Site occupation

a

M.~

I 0 I t

3.4951 3.7829 10.7732 11.6366 22.1220

M.C60

M2C60 MZC60 M 3 C60

I t+ 1 2 t 2 t+ 1

0 0

-

D. Claves, Ph. Touzain I Synthetic Metals 80 (1996) 301-307

actions between carbon atoms can be treated under the Lennard-Jones potential form:

cPc-<:=

'Yc 8c -6'+l2 r

r

where 'Yc and 8c are constants. The high symmetry of the molecule and its dynamical properties at room temperature make it comparable to a sphere with a uniform density of carbon atoms at the surface. Integrating the Lennard-Jones potential over the surfaces of two such spheres with radius R, Girifalco [20] obtained an analytical expression for the potential energy of interaction between two C60 molecules at distance r: E(C60 /C 60 )

[ 1 + 1 --2] + B[ 1 + 1 -2]

= -A

s(s-I)3

s(s+ 1)3

S(S-1)9

s(s+I)9

S4

303

Erepulsion anion-<:ation =ZJ3± exp( - rip) the hardness parameter p taking a constant value for a family of salts. {3 ±' introduced by Pauling, accounts roughly for the valence and outer configuration of the ions and develops as

with qi the charge (including the sign) and ni the number of valence electrons on the ions (qln can be omitted in the case ofC60"- ). Assuming a constant, charge-independent 'external' radius for C60"-, we expressed Z as a function of the cation size only. At this stage, it is worth recapitulating the global analytical expression retained for the cohesive energy of the solid:

SIO

with s == r12R. Computing A and B (only interactions between first and second neighbours were considered), using for the cohesive energy of the fullerite the value -1.7 eV (Ecoh - 6. H sub ) and an equilibrium cell parameter ao = 14.17 A, we obtained A =4.745 X 10- 2 eV and B=8.990Xl0- 5 eV, in close agreement with the values reported earlier. The corresponding isothermal linear compressibility is found to be 2.2 X 10- 3kbar - I, in reasonable agreement with the mean reported value [21] of 2 X 10- 3 kbar -I. The contribution of the second neighbours to the total energy is only 3% and needs not be considered. From neutral C 60 to negatively charged Cron - , some charge effects will be taken into consideration. The number of added electrons being small in comparison with the 240 valence electrons of the molecule, A and B will be assigned to the constant values determined above. Charge effects can be simply modelled by taking into account the slight increase in the radius of the pseudo-sphere upon reduction, as determined by X-ray diffraction, thus modifying the reference surface considered for integration. This minor correction significantly increases the repulsive part of the interactions for small distances only and does not affect the global potential energy otherwise. From Refs. [ 1,4a,22,23] the recommended values according to the charge on the C60 molecule are R=3.53 (n=O), 3.54 (n=3), 3.55 (n=4), 3.57 (n=6) A. It is worth noting that such an analytical expression does not account for orientational order often observed in intercalated fullerite [4], implying a discrete sum of interactions between the carbon atoms constituting the clusters. 3.3. Anion-cation repulsion Repulsive energy of closed-shell ions was assumed by Born and Mayer to vary exponentially with their distance as

(I) where z/Zo are the coordination numbers ofthe C60n - anion towards the tlo sites (z/Zo are 016, 4/0, 8/0, 4/6, 8/6 for M1(I 0)/M 1(1 t)/M 2 (2 t)/M 2 (1 t, 1 0)/M3(2 t, 1 0) C60, respectively), Z/Zo are constants depending on the size of the cation in the t/o sites, z' is the coordination number of the C60"- anion towards its first homologous neighbours (z' = 12). Two parameters p and Z(R +) are still unknown in Eq. ( 1) requiring two independent equations. The equation of state of the cubic solid under hydrostatic pressure reduces to p=

_(aUT) av

T

which at constant temperature and under negligible pressure gives

e~Tt=0 d UT) =9ao ( 2

da2

do

4{3

(2) (3)

{3 being the volume compressibility of the solid. Rigorously, Eqs. (2) and (3) strictly apply for a static crystal only. Zhou et al. [24] reported (3=3.6±0.3X 10- 3 and 4.5 ± 0.3 X 10-3 kbar - 1for K3C60 and Rb 3C60, respectively. Resolution of Eqs. (2) and (3), taking a o = 14.25 [4b,22] and 14.40 A [23,25] for these compounds yields, within experimental error, p( K3C 60 ) = 0.23 ± 0.03 A, p(Rb3C60 ) = 0.26±0.02 A and a first value for Z(K+) and Z(Rb+). A fixed value p = 0.25 A will be employed in the following. Numerous chemically ordered binary M,.M'3-xC60 compounds, in which site selective behaviour of alkali metals was

304

D. Claves. Ph. Touzain I Synthetic Metals 80 (1996) 301-307

observed, according to the difference in size of the cations, have been reported by Tanigaki et al. [26,27]. Linear systems of equations with Z( M +) as unknown can be built and solved from Eq. (2) using couples of compounds such as K2Rb/ KRb2C 60 or Rb2Cs/RbCs2C6o. The new values of Z(M+) (M + = K +, Rb +, Cs + ) thus obtained show variations ofless than 5% from the previous ones. Eq. (2) can be easily solved using the experimentally observed lattice parameter for Na2C60 (f.c.c. C 60 packing) [28], providing a value for Z(Na +). Na3C60 is known to have an anomalous cell parameter [28] and was excluded from the present calculations. This might be due to incomplete charge transfer from the sodium in octahedral sites. Taking the mean values of Z(Rb +, Cs +) obtained so far as references, complementary values of Z(U +, Na +) were obtained from Eq. (2) applied to Li2Rb/Li2Cs/Na2RbC60 compounds. Using for alkali cations 'standard' radii such as the now common values reported by Shanon [29], it is found that an exponential law accounts very well for the evolution of Z with the cation size within the range considered. A leastsquares fit of the results gives Z (in eV) =7.595 X 107 exp[3.l334XR+ (in A)]

, e2X

1) 1

AG? (n)= -n 1-- - IOlv 2R D 4'1TEo where R is the radius of the species with charge n Ie I in the dielectric medium and D the dielectric constant ofthe solvent. In order to reduce the number of unknown parameters, an approximation A~ly(n) -n2A~oly(n= 1) is made, taking into account the major influence of the charge. Thus Eq. (4) reduces to lele l12 (n)/ref=EA(n) - (2n-l)A~lv(n= 1) + C

(5)

Most of the time, electrochemical potentials are measured versus an internal standard, such as the widely spread ferrocene/ferricinium couple, whose potential is supposed to be independent of the electrolyte, making redox potentials directly comparable between different media [32]. In aqueous solution eFc/Fc+ =0.4 V /NHE. Taking the absolute potential of the NHE as - 4.44 V [33], one obtains C= -4.84eV. A ~Iv( n = 1) remains the only unknown to be determined. This can be done by replacing the experimentally measUred [34] value of the first electron affinity of C 60, EA( 1) = 2.65 eV in Eq. (4) and neglecting the free enthalpy of solvation of neutral C 60 (AG"lOlv(C 60 ) «A~lv(C60 Successive one-electron affinities are then deduced from Eq. (5). The values of half-wave potentials were collected for six solvents (dichloroethane, chlorobenzene, pyridine dichloromethane, benz~ne-acetonitrile and benzonitrile) from Refs. [31,35], With TBA CI0 4 /PF6 as supporting electrolyte. A linear trend is observed for each medium within an 8% variation in the slope. The best correlation is obtained for EA(n)=5.54-2.89Xn (in eV). The closeness of the LUMO (t 1u ) andLUMO+ 1 (t lg ) energy levels of the cluster [36], both threefold degenerate, allows for linear extrapola_ tion up to n = 12. The limit of such a model is due to the fact that relation Eq. (4) is strictly valid in the case of dominant solvent_ solute interactions. Several factors such as solvophobic inter_ actions or ion pairing can affect the value of e l12• Our reSUlts are in good agreement with the values of EA(n) found in Ref. [30e], which seem to be the most accurate, given the value of the first EA reported (2.88 eV), with respect to the experimental one.

-».

4. Electron affinity (EA) Several theoretical models are proposed in the literature [30] for evaluating the electronic energies of the negatively charged cluster. Though good agreement is found upon linear dependence of EA with the final charge state, considerable discrepancy is observed between the numerical results. Electrochemical techniques can provide a great deal of information on the electronic properties of fullerenes, and the difference in potential between two successive reductions can be considered as an indication of the energy required to add extra electrons to the C60 molecule. It was previously reported that C60 can undergo six reversible, one-electron reductions in solution [31]. In our attempt to determine the successive electron affinities of the C 60 cluster, we chose to use the formal relationship existing between the standard electrode potential of a redox couple and EA. For the one-electron oxidation reaction: (c60n- )solv ~ (C 60(n-I) - ) lolv +

e-

one has leleO(n)/ref=AG"(n) +

In the Born model of solvation:

[AG~olv(n-l)

- A~olv(n)] + C

(4 )

where C is a constant depending on the reference electrode, AG~olv(n or n-l) is the Gibbs enthalpy of solvation of species c60(n or n-I) - , and AGO(n) the one of extraction of one electron. eO is deduced from the anodic/cathodic halfwave potential obtained reversibly in cyclic voltammetry by considering e l / 2 = ea. Neglecting entropic terms AGO(n) - EA(n).

5. Heats of formation 5.1. Alkali metals The results for alkali metals are given in Table 2. M 3C appears as the most favourable stoichiometric ratio, compare~ to lower metal concentration phases. A Hf drops from Li to Cs, due to the decreasing cohesive and ionization energies of

D. Claves. Ph. Touzain / Synthetic Metals 80 (1996) 301-307 Table 2 Heats of formation for the alkali-metal series (* indicates use of experimental lattice parameter values) Element

Fulleride Mz C60

Calculated cell param. ao (A)

!1Ht (eV ICeo )

Li Na K Rb Cs

x-3 x .. 2 x-3 x-3 x-3

13.94 14.19* 14.25* 14.40* 14.66

-0.61 -1.22 -4.83 -5.13 -5.51

305

10

(eV/C.J 7

4

·2

these elements. Our value of AH/ for K3C6Q is in reasonable agreement with local density calculations performed by Martins and TrouiIIier [13], who reported values of -1.4 to -1.7 eV per K atom, while heats of formation of nearly 8 eV per K atom are mentioned in Ref. [ 14] . CS 3C6Q was indeed shown to exist but under b.c.c. form [37] or as a mixture of b.c.1. and A15 phases [38]. These latter are probably more stable configurations than the f.c.c. packing assumed so far (see Section 5.4).

5.2. Lanthanides Cations of the rare-earth series are often observed with charge + 3, or + 2 in some cases. Extrapolation ofionic radii, when unknown, was often needed. Intercalation of most of these elements is highly likely to occur, as shown by some strongly negative values given in Table 3. Again, the most stable phase is the saturated one. AH/ increases in a nearly monotonic way for M 3 + cations from left to right in the series, with two singularities for Eu and Yb due to a strong variation in the third ionization potentials, but this oxidation state is much less favourable than + 2, except for Ce, Gd and Lu.

·5

. a +-~",-"--.--~-,-~-r~-.,-~....,.~--r~-+ 58

58

80

82

84

88

68

70

72

Fig. 2. Stability trend across the lanthanides' period: filled and open circles correspond to a metal oxidation state + 2 and + 3. respectively.

This suggests that some compounds may be obtained, exhibiting unusual oxidation states for this kind of cation. As seen in Fig. 2, no clear-cut trend is deduced for oxidation state +2, two highly positive AH/ values appearing for Gd and Lu. It is worth outlining the following points. The cohesive energy of the solid is largely dominated by the electrostatic part of the interaction potential and remains nearly constant from one element to another. The lattice parameter is governed by the charge and the number of intercalated cations. The repulsive part of the interaction between ions comes almost entirely from the repulsion of neighbouring fullerene molecules, as pointed out earlier by Yildirim et aI. [39] for small Na + cations.

Table 3 Heats of formation for the rare-earth series and elements of Group III B Element

Metal oxidation state:

Metal oxidation state:

+3 Calc. cell paramo ao (A)

Ce Pr

Nd Pm Sm Eu Gd Tb Dy Ho En Tm Yb Lu Sc y La

-3.23 -1.81 -0.55 + 1.22 +0.89 +5.03 +3.05 +3.95 +4.22 +5.28 +6.12 +6.85 +9.05 +8.02 + 17.75 +4.11 -4.82

all nearly constant and about 13.63

+2 Calc. cell param. ao (A)

-0.05 -3.33 -3.54 -2.34 -6.03 -6.50 +4.31 + 1.29 -1.17 -0.27 0.85 -1.22 -3.14 +8.11 - +6.7 - +5.9 - +1.0

13.83 13.83 13.83 13.82 13.82 13.81 13.80 13.79 13.78 13.78 13.77 13.77 13.77 13.77 -13.8 -13.8 -13.8

306

D. Claves. Ph. Touzain/ Synthetic Metals 80 (1996) 301-307

5.3. The 3d transition metals Only elements of Group III B give reasonably low values for tJ.Hf (see Table 3). Calculation for other 3d elements requires the evaluation of an additional crystal field stabilization energy. Large positive values are expected from the high ionization potentials and cohesive energies associated with these elements and an ionic model would not be appropriate. Stable coordination compounds or metallocene-like complexes, implying strong covalent interaction are rather expected and have indeed been synthesized [40,41].

5.4. Comments Close results were obtained by Tomanek et al. [ 15], based on a similar approach. The essential differences with our values mainly arise from divergences in estimating EA. We point out here the purely theoretical character, based on steric arguments only, of the phases examined, which may turn out to exist under different, but more stable forms or structures than the simple ones considered in this work. Thus, in the case of alkali metals, ionic M t C 60 compounds are predicted, according to our model, to be unstable relative to their transformation into ionic M 3C 60• Somehow, such diluted phases do exist at room temperature, but under stable polymeric [42] or metastable dimeric [43] forms. Analogous conclusions can be drawn from the non-existence of a f.c.c. CS 3C60 phase, probably less stable than the observed b.c.c. derived structures [37,38] . However, as these latter also seem metastable, spontaneous transformation must occur into more stable CS 4C 60 and polymeric Cs t C 60 compounds. Finally, such an ionic model should be regarded as an extreme form of bonding that could probably not be attained in practice and the strongly polarizing character, associated with small size cations, especially with high valency, may rather result in iono-covalent bonding. Small departure from spherical shape of the electron density, in the outer part of the sphere of influence of the Li + ion, was already reported in Li 2 CsC60 [44], revealing the partly covalent nature of the bonding, in this case.

6. Conclusions We have used an ionic model and the associated BornHaber cycle to estimate the heats of formation of some cubic close-packed C60 intercalated phases. A classical model for the fulleride reveals preponderance of the electrostatic interactions relative to all other types. The gain in electrostatic energy is balanced by the energy required for positively and negatively ionizing guest and host species. For alkali-doped phases, M 3C 60 compounds seem to be correctly described under a charge transfer formalism, at least for K and Rb elements, and their stability should justify the non-existence of M t C 60 phases under a simple f.c.c. array of

spherical ions. The size of the Cs + cation appears as a critical value, which cannot preserve the primitive fullerene lattice. From our calculations, it appears that charge transfer complexes could be very probably formed in several cases, including non-alkali metals, according to ionization potentials and/ or cohesive energies ofthe elements. These kinds offullerides may turn out to be potential novel conductors or superconductors. Particularly favourable energetics conditions are found for Sm, Eu, Yb and La, from which stable phases have been experimentally obtained [45]. References [11 P.A. Heiney, J.E. Fisher. A.R. McGhie. W.J. Romanow. A.M. Denenstein, J.P. McCauley. Jr.• A.B. Smilh III and D. Cox. Phys. Rev. Lett., 66 (1991) 2911. [2] C.S. Yannoni, R.D. Johnson, G. Meijer. D.S. Belhune and J.R. Salem. J. Phys. Chem.• 95 (1991) 9. [3] R.D. Johnson. C.S. Yannoni, H.C. Dorn, J.R. Salem and D.S. Bethune Science. 255 (1992) 1235. • [4] (a) O. Zhou and D.E. Cox. J. Phys. Chem. Solids, 53 (1992) 1373; (b) D.W. Murphy, MJ. Rosseinsky, R.M. Fleming, R. Tycko. A.P. Ramirez. R.C. Haddon, T. Siegrist, G. Dabbagh, J.C. Tully and R.E. Walstedt, J. Phys. Chem. Solids. 53 (1992) 1321. [5] AR. Kortan, N. Kopylov, E. ()zdas, A.P. Ramirez, R.M. Fleming and R.c. Haddon, Chem. Phys. Lett., 223 ( 1994) 501 and Refs. therein. [6] P.W. Stephens, D. Cox. J.W. Lauber, L. Mihaly, J.B. Wiley. P.-M. Alemand. A. Hirsch, K. Holczer, Q. Li, J.D. Thompson and F. WUdl Nature,355 (1992) 331. • [7] R.Rolhand P. Adelmann, Appl. Phys.A. 56 (1993) 169. [8] I.W. Locke, A.D. Darwish. H.W. Kroto, K. Prassides, R. Taylor and D.l.M. Walton, Chern. Phys. Lett., 225 ( 1994) 186. [9] (a) Q. Zhu. D.E. Cox, J.E. Fisher, K. Kniaz, A.R. McGhie and O. Zhou, Nature, 355 (1992) 712; (b) M. Kobayashi. Y. Akabama, H. KalJamura, H. Shinohara, H. Sato and Y. Saito, Mater. Sci. Eng. B 1911993) 100. • [10] A.L Balch. J.W. Lee, B.C. Noll and M.M. Olmstead, in K.M. Kadish and R.S. Ruoff (eds.), Recent Advances in the Chemistry and Physics of Fdlerenes and Related Materials, 1994. p. 1231. [11] B. Mlrosin. P.P. Newcomes, R.J. Baughman, BL. Venturine, D.l.oy and J.E. Schisber, Physica C, 184 (1991) 21. [12] S.C. Erwin and M.R. Pederson, Phys. Rev. Lett., 67 (1991) 1610. [13] J.1. Martins and N. Troullier, Phys. Rev. B. 46 (1992) 1766. [14] S. Saito and A. Oshiyama, Phys. Rev. B, 44 (199\) 11 536. [15] D.Tomanek, Y. WangandR. Ruoff.}, Phys. Chern.So/ids,54 (1993)

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