Jahn-Teller distortion and possible electron pairing in molecular systems

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Synthetic Metals 75 (1995) 55-60

Jahn-Teller distortion and possible electron pairing in molecular systems Tokio Yamabe a,b, Kazunari Yoshizawa b, Yukihito Matsuura a, Kazuyoshi Tanaka a,. "Division of Molecular Engineering, Faculty of Engineering, Kyoto University,Sakyo-ku, Kyoto 606-01, Japan h Institutefor Fundamental Chemistry, 34-4 Takano-Nishihiraki-eho, Sakyo-ku, Kyoto 606, Japan Received 1 May 1995; revised 12 July 1995; accepted 17 July 1995

Abstract

Negative effective electron-electron interaction (U) due to Jahn-Teller distortion and its role in molecular superconductivity are discussed in degenerate molecular systems. It is demonstrated within the Htickel molecular orbital theory that the U value could become negative in the + 1 and + 3 charged states of [N] annulenes, (CH)Jv ring systems, with D6h geometry, where N---6 (2m + 1) (m = 0, 1, 2, 3, • •. ). Possible intermolecular electron pairing between two electrons with opposite discrete molecular wave vectors is proposed. Keywords: Jahn-Teller distortion; Electron pairing; Molecular systems

1. Introduction Since Little' s proposal [ 1] of high-temperature superconductivity based on the polyene (CH)N chain, there has been much interest in the study of superconductivity in molecular systems. Advances in the design and synthesis of organic materials have yielded a large number of organic superconductors [2-4] since the first observation of Tc=0.9 K for (TMTSF) 2PF6. Moreover, recent discovery of high-temperature superconductivity in the molecular solid of electrondoped fullerene A3C6o, where A represents alkali metals such as K and Rb [5-7], has greatly intensified the interest in superconductivity in molecular systems. Several theoretical models [ 8-12] have been proposed to account for this interesting solid-state property of the fullerene systems. The JahnTeller effect [ 13] in this highly degenerate system and its role in the superconductivity are of interest. On the other hand, it is well known that the diamagnetic anisotropy of aromatic hydrocarbons and annulenes can be attributed to the induced ring currents in their w-electronic systems [ 14-16]. The relationship between the ring current and the virtual superconducting state in these molecular systems has been discussed. For instance, an analytical relationship between the resonance energies and ring currents of the (4n + 2) w-electronic system has been demonstrated, and the analogy between the molecular orbital (MO) wave function of benzene and the Bardeen--Cooper-Schrieffer (BCS)-type [ 17] paired configuration has been suggested [ 18]. The con* Corresponding author. 0379-6779/95/$09.50 © 1995 Elsevier Science S.A. All rights reserved

SSD10379-6779(95)03390-6

densation of electrons into boson-like Cooper pairs has been further discussed to explain the aromatic stabilization energy [19]. One of the authors [20] has indicated the role of the vibronic interactions in the normal and superconducting states of conjugated polymers. The aim of this paper is to show a possibility that effective electron--electron interaction becomes negative in electronor hole-doped states of degenerate Jahn-Teller molecular systems. Moreover, we will discuss that electrons with opposite discrete molecular wave vectors would form an electron pair on separate molecules. For this purpose we consider [N]annulenes, i.e., finite (CH)N ring systems with N = 6 (2m + 1 ) w electrons (m = 0, 1, 2, 3, - • • ) within the framework of the Htickel method. In fact, the 6(2m + 1 ) w system is a subgroup of the ( 4 n + 2 ) w system, since 6 ( 2 m + 1) can be written as 4 ( 3 m + 1) + 2 . There has been a long history concerning the research on the structure of annulenes from the quantum chemical viewpoint [21,22]. The existence of the bond alternation has been predicted in annulene ring systems [ 23 ]. However, it is interesting to study whether there is an actual bond alternation in the [6(2m + 1) ]annulenes of medium size, because X-ray structural study [24] and semi-empirical and ab initio MO calculations including electronic correlation [25-29] have shown that [ 18] annulene, the second smallest [6(2m + 1 ) ] annulene, has O6h geometry like benzene and that the difference in the bond lengths is within 0.01 A. We will therefore assume that [ 6 ( 2 m + l ) ] a n n u l e n e s of medium size are degenerate molecular systems with D6h geometry and nearly equidistant C--C bond lengths.

T. Yamabe et al./ Synthetic Metals 75 (1995) 55-60

56

2. Electronic states of degenerate molecular systems

(9)

We can write the total Hamiltonian of general molecular systems in the form h2

02

02

V(q,Q) =-V,(q) + Ve(q,Q) + V3(Q)

h2

02

(2)

where q = {qi} represents the set of all the electronic coordinates; Q = {Qj} those of nuclei with masses {Mr; and the others have their usual meanings. V(q,Q) includes electronelectron, electron-nucleus, and nucleus-nucleus interactions, as indicated in Eq. (2). Assuming the kinetic energy of the nuclei as a small perturbation [ 30], we solve the Schr'odinger equation for the electronic motion in the field of arbitrarily fixed positions of the nuclei:

Ze]ec(a,( q,Q ) = En(Q ) qb,(q,Q )

(5)

(6)

sin(N) The energy gap between the highest occupied MO (HOMO) and the lowest unoccupied MO (LUMO) is written as Eg=41fll sin(N)

(+s)

(7)

(+s)

(-4) " 0 0 "

"00"

(+4)

(-3) " 0 0 "

-00"

(+3)

(-2) - 0 0

-00"

(+2)

(-1) - ( 3 0 -

"00"

(+1)

-O0(o)

~

(+14)

(-13)

~

(+13)

(-12)

~

(+12)

(-11)

~

(+11)

(-10)

- -

(+10)

(-9)

~

(+9)

(-e) ~

~

(+8)

(4)

where/3 is the resonance energy, and the total energy in the framework of the Htickel theory is given by 4/3

~

(-14)

(3)

where a is the one-dimensional unit vector of translational symmetry around the ring; the azimuthal quantum number is j = 0 , +__1, +2, . . . , + ( N / 2 - 1 ) , N / 2 ; Xr the ~ atomic orbital of site r; and kj the discrete wave vector (kj = 2"rrj/ Na) defined in the range [ - w / a , ar/a]. For j4=0 and j ~ N / 2 , the degenerate pair MOs ~b±tv correspond to two independent traveling waves going around the molecular ring, one clockwise and the other counter-clockwise. The one-electron energy levels of [18]- and [30]annulenes are shown in Fig. 1. The orbital energies are expressed by

Eo =

(+7)

(-6) ~

(15)

r=]

,±kj = 2/3 c o s ( ~ -~)

(+0)

~

(a)

Here we consider [N]annulenes, i.e., finite (CH)N rings with N = 6(2m + 1 ), as mentioned above. We assume here that the [ 6 (2m + 1 ) ] annulenes have planar D6n geometries with equidistant C-C bonds. The rr-MO by the simple Htickel theory [ 31 ] can be written as 1 u q~+~,=~77~ y ' exp{ +_ikjar}xr

- -

(-7) ~ (-'J) ~

,g*'(q,Q) = -~m~i-~qz+V(q,Q) h2

(-8) - -

(-7) - 0 0 -

- 0 ( 3 - (+7)

(-s) - 0 0 "

-00"

(+0)

(-s) " 0 0 -

-00-

(+s)

(-4) < 3 < ) -

-00-

(+4)

(-3) "00"

-OG (+3)

(-2) "0"0"

" 0 0 " (+2)

(-1) " 0 0 "

-00"

(b)

(+1)

"O0(o)

Fig. 1. One-electron energy levels of (a) [18]- and (b) [30]annulenes. The values in parentheses are azimuthal quantum numbers.

When electron or hole doping is performed to prepare anionic or cationic [N] annulene, respectively, the Coulomb potential energy V(q,Q) changes and therefore the molecular geometry would be slightly distorted. For such small displacements of the nuclei, V(q,Q) can be expanded around the equilibrium position as

V( q,Q) -~ V( q,Qo)

o(5_J

1_[ ~2v,

The second term of the right-hand side of Eq. (8) represents vibronic interaction, which plays an important role in JahnTeller distortion of molecular systems and in superconductivity in the solid state. According to the first-order perturbation theory [32,33], change in the orbital energy AE can be determined from the following secular equation:

AV]~(Q)-AE AV21(Q)

AV12(Q) A V22(Q) -

AE = 0

(9)

57

T. Yamabe et al. / Synthetic Metals 75 (1995) 55-60

where Kjs, stands for the effective force constant and the A Qs terms the displacements from the original D6h equilibrium positions of the nuclei.

OO O O O HOMO (elg)

3. Effective electron-electron interactions

Fig. 3 shows the schematic representation of (a) electron doping into the twofold degenerate LUMO and (b) hole doping into the twofold degenerate HOMO. Let us first consider the total energies of the electron-doped [N] annulenes. Assuming that MOs are frozen except for the degenerate LUMO of the neutral-state molecule, we can write down the total energy of the negatively charged states, which can be decomposed as the 7r-electronic energy and the distortion energy:

LUMO (e2u)

E2g 1800 cm "1

Fig. 2. Twofold degenerate HOMO and LUMO in the real form and a pair of E2g modes for [ 30] annulene.

E ( - 1 ) =Eo+21/3 ] sin

-A+Eeast

E( - 2) = Eo -t- 2 { 2 , ~, sin(N) - A ' ) -t- E~dist

The solution of this equation has the form:

(14)

A E - - ~ ( A V I I -~-m V22 )

E(-3)

1

E ( - 4 ) = E o + 81/31 sin(N)

where A Vl2 is

AV'2=fcb*~ 0(-~)

(11)

The first term of the right-hand side ofEq. (10) would vanish from symmetry considerations [ 32]. In neutral [N] annulenes with N = 6(2m + 1) ~ electrons, the HOMO and LUMO are of etg and ez,, respectively, for even m; and the HOMO and LUMO are of e2, and e~, respectively, for odd m. When an electron is added to the LUMO or an electron is removed from the HOMO, Jahn-Teller distortion occurs to remove the degeneracy. From the analysis of the direct product shown below elg X elg = e2, × e2, = Alg q- Azg + E2g

(12)

only the Ezg modes of vibration are found to lift the degeneracy, since Atg is totally symmetric and A2g is asymmetric so that they cannot couple. The HOMO and LUMO in the real form, and a pair of E2g modes of [ 30] annulene are shown in Fig. 2. According to the Jahn-Teller effect, the D6h geometry would be distorted into the D2, one through the Ezg modes of vibration. The energy AE~st required for this distortion is approximated by E al~t=-~ 1 ~.~Kjs,AQi AQ~, .~jt

=Eo+61/31sin(N)- A"+ E~ist

(13)

where Eo is given by Eq. (6) and the A terms signify the Jahn-Teller splitting energies corresponding to the second term of the right-hand side of Eq. (10). In the - 2 charged state, the triplet state probably retaining the original D6t , geometry could compete in energy with the singlet state. However, since the Hiickel model is spin independent, we assume here only the non-magnetic condition. The effective electron-electron interaction energy U(n) [34] for the n charged state is defined in terms of the total energies as

U(n)=[E(n+l)-E(n)]+[E(n-1)-E(n)]

(15)

This value can thus be written for the electron-doped states in the form ofEq. ( 16):

(a)

I 2A- I - I 2A' (o)

+2N'

(-1)

(-2)

(-3)

t.O-O

<3<)

--0-

~

(-4)

(b) (0)

(+1)

(+2)

(+3)

(+4)

Fig. 3. Schematic representation of (a) electron doping into the degenerate LUMO and (b) hole doping into the degenerate HOMO.

T. Yamabeet al. / Synthetic Metals 75 (1995) 55~0

58

U( - 1) =2(,4 - , 4 ' ) q- (E'dist-- 2Edist) U(-2) =2,4'-A

- A " + (Edist"[- E'dist-- Etdist)

U( - 3) = 2(,4 ' t - ,4 t) ..[_ (E~dist_

(16)

2E,~ist )

If ,4 = A' = '4" and Edist = E'dist = EFdist, the U value can become negative in the - 1 and - 3 charged states. These conditions may be satisfied if the extent of the Jahn-Teller distortion is not largely dependent on the charged states. On this assumption the U value would always be positive in the 0 and - 2 charged states. This result is completely the same in the positively charged states from the electron-hole symmetry within the Hfickel MO theory as shown in Fig. 3. According to Ref. [ 10], let us next assume ' 4 ' = 2,4 and ESdist = 4Edis~. We can also assume A " = '4 and E'~ist= Edist in twofold degenerate systems because, in the - 3 charged state, two electrons occupying the upper and lower levels that are originally degenerate are approximately canceled. Upon these simple assumptions, we may obtain: U ( - 1) = U ( - 3 ) = - U ( - 2 ) = - 2'4 +2E~s,

(17)

In this expression the sign of U ( - 2 ) is clearly reversed to that of U( - 1 ) and U( - 3). When ,4 is larger than Edict, U ( - 1) and U ( - 3 ) can be negative and U ( - 2 ) positive. In this way, the so-called negative U [35] possibly appears in Jahn-Teller molecular systems if certain conditions are fulfilled. Since the direct Coulomb repulsive interaction is approximated by Uc = 1/ eR, where R is the diameter of molecule, and the dielectric constant e is 2.4 for benzene ( [ 6] annulene) [36], it is greatly reduced with an increase in molecular size in such a way that Uc = 2.1 eV for N--- 6, 1.0 eV for N = 18, 0.5 eV for N = 30, and 0.4 eV for N = 42. We can therefore expect that the negative U value can surpass the direct Coulomb repulsion in these molecular systems with medium size. In fact, NMR studies [37-40] have shown that there is a disproportionation between monoanions and dianions of various annulenes in solution: [ (CH)~v] - + [ (CH)N] - ~ [ (CH)N] + [ (CH)N] 2and that the equilibrium state contains dianions in [ 10] annulene [ 37], but quite the reverse in [ 16] annulene [ 38 ]. Moreover, it has been reported that treatment of [ 18]annulene with potassium results in the formation of both the monoanion and dianion to yield a 7:3 mixture at the'equilibrium [39]. Considering the direct repulsive interactions in the dianions, such disproporfionation is rather surprising. These experimental results may suggest that these annulenes are the negative U system. Quite similar phenomenon has been observed in C6o in solution [41,42] but not in C7o, the HOMO and LUMO of the latter not being degenerate. We consider that the important point in these results is the existence of degenerate HOMO and LUMO in the molecular systems and the resultant Jahn-Teller distortions in the electron- or holedoped state.

Finally, we discuss possible electron pairing between two anions of [ 6(2m + 1)] annulenes in the solid state. The MO wave function of Eq. (4) has the discrete wave vector +kj defined in the range [ - ~ r / a , rr/a], where + kj are two independent wave vectors going around the molecule, one clockwise and the other counter-clockwise. Since the degenerate LUMO has wave vectors, +_kj = ++_(kv + av/Na), the LUMO can be written as 1

qb±=~TT~exp{q-i(kFa+N)r}x

r

(18)

r=l

where kF ( = IT/2a) represents the Fermi wave vector. In the BCS theory of superconductivity [ 17 ], we consider the electron-electron interaction arising from the exchange of a virtual phonon as a scattering process. There are two intermediate states allowed by momentum conservation. An electron with wave vector k emits a phonon of wave vector - q to scatter it into a state k + q. This phonon is then absorbed by another electron with - k to scatter it into a state - k - q. We combine this electron-electron interaction arising from the electron-phonon coupling with the Coulomb repulsive interaction between the electrons [ 43 ]. In contrast to metals and alloys, molecular solids unavoidably contain disorders, kinks, or solvent molecules, and therefore we may not define a continuum wave vector. In order to consider the electron pairing in molecular systems, we write the electron-electron interaction Vk+q.k in the form of the BCS reduced Hamiltonian [43] shown below:

V ~ VqhoJq/ N

Vk + q k = ' (Ek--Ek+q)2--(h(,Oq)

wC°ul°mb

2"~- V k + q 'k

(19)

If [ ek-- ek+q[ < hcoq, the first term of Eq. (19) becomes negative, where we consider molecular vibration with energy he% being of the order 0.1-0.2 eV. The second term is inversely proportional to the diameter of the molecule as mentioned above. Let us consider electron pairing of the molecular systems in the - 1 and - 3 charged states. A possibility of electron pairing is that between two electrons with opposite discrete molecular wave vectors + kj and opposite spins. There is an intriguing analogy between Eq. (4) and the BCS-type wave function, although +kj in Eq. (4) are discrete molecular wave vectors. Thus, we think that intermolecular electron pairing can occur between two electrons with opposite molecular wave vectors and opposite spins on separate molecules, as shown in Fig. 4. In this figure each molecule has - 1 charge on average. We consider that two electrons with + kj can be attractively coupled. Schematic representation of the LUMO level in the - 1 and - 3 charged states of [6(2m + 1 ) ] annulenes is shown in Fig. 5. These charged states are typical Jahn-Teller systems, and the energy gap A will appear through certain modes of vibration which remove the degeneracy. The Jahn-Teller splitting A may correspond to the value [ek-ek+q[ in Eq. (19). If A is of the order of the molecular vibration energy,

T. Yamabe et al. / Synthetic Metals 75 (1995) 5 5 4 0

Oi

© ©

I+kj>((~

Fig. 4. Electron pairing between two electrons with discrete + k~ on separate molecules with charge - 1, where + ky are two wave vectors going around the molecule, one clockwise and the other counter-clockwise. n = -1

59

the Jahn-Teller distortion of the degenerate LUMO and HOMO in molecular systems. The disproportionation observed in negatively charged annulenes and C6o would support the present theory. This prediction can be expanded to general molecular systems with the degenerate HOMO and LUMO, e.g., coronene, hexabenzocoronene, circumcoronene, and so on. Since the direct Coulomb repulsion is reduced with an increase in molecular size, the net negative U behavior may appear in the degenerate molecular systems of medium size. Furthermore, we have indicated that intermolecular electron pairing can occur between two electrons with the opposite discrete molecular wave vectors in the electron- or hole-doped state. The present study on the new type of electron pairing will hopefully provide a new insight into the properties of molecular superconductivity.

LUMO

Acknowledgements HOMO

¢-kj) (a)

(+kj)

This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan, and the JSPS Program for Supporting University-Industry Cooperative Research Project. The authors would like to thank the reviewer of this paper for useful comments.

n = -3

4-4 (-kj)

(b)

.OMO

(÷~j)

Fig. 5. Schematic representation of electron-doped LUMO levels with + kj in (a) - 1 and (b) - 3 charged states.

attractive electron pairing can appear between two molecules, as shown in the first term in the right-hand side of Eq. (19). We know a good example of A for C ~ is of the order 0.1 eV [44]. This could eventually make the total Vk+q.k negative, since molecular vibration energy is also of the order 0.1 eV and l-~°ul°mb ek+q,k is reduced with increasing molecular size. As estimated above, this direct Coulomb repulsion is not so large in molecules with medium size. This fact possibly reinforces the origin of the superconductivity in A3C6othrough the JahnTeller active H e modes of molecular vibration [ 5-7 ]. The existence of molecular superconducting materials with short-range structural ordering can be explained from the approach of this discrete wave vector rather than from traditional concepts of long-range crystalline ordering and continuum wave vector. We again emphasize the relationship between molecular vibration and possible superconductivity in degenerate Jahn-Teller molecular systems.

4. Conclusions We have shown within the Hiickel MO theory that the effective electron-electron interaction can become negative in electron- or hole-doped states of [ 6 ( 2 m + 1)]annulenes w i t h D6h geometry. The origin of this interaction comes from

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