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Nature of the Magnetic Interaction in Organic Radical Crystals. 5. Magnetic Interaction in Mixed Radical Ion Crystals† F. Dietz,*,‡ N. Tyutyulkov,‡,§ M. Staneva,‡,§ M. Baumgarten,| and K. Mu1 llen| UniVersita¨ t Leipzig, Wilhelm-Ostwald-Institut fu¨ r Physikalische und Theoretische Chemie, D-04109 Leipzig, Johannisallee 29, Germany, UniVersity of Sofia, Faculty of Chemistry, BG-1126 Sofia, Bulgaria, and Max-Planck Institut fu¨ r Polymerforschung, D-55128 Mainz, Ackermannweg 10, Germany ReceiVed: May 16, 2001
The nature and magnitude of the spin exchange interaction within the half-filled band of 1-D stacks of mixed radical ion crystals (MRIC) consisting of two hydrocarbons H1 and H2 with different ionization potentials and electron affinities have been investigated theoretically. In a fully reduced or oxidized 1-D crystal, each elementary unit is an anion radical or a cation radical: (H1 ...H2)•- - - or - - (H1 ...H2)•+ - -. In contrast to organic radical cation (anion) crystals which consist of identical polycyclic hydrocarbons -(Hδ‚‚‚Hδ)•+- or -(Hδ‚‚‚Hδ)•-- with a metalic or semiconducting ground state, the ground state of some classes of MRICs is a magnetic one. The band theory of magnetic interaction in many electron π-systems is applied to calculate the different contributions of the effective Heisenberg exchange integral.
1. Introduction
SCHEME 1
In a recent paper1, a new type of one-dimensional (1-D) staple system with magnetic ordering has been proposed, namely 1-D molecular mixed radical crystals (MMRC) with the general formula
‚‚‚H‚‚‚R•‚‚‚H‚‚‚R•‚‚‚H‚‚‚R•‚‚‚H‚‚‚R•‚‚‚H‚‚‚ where R• is a π-conjugated radical and H is a polycyclic diamagnetic hydrocarbon with a π-conjugated system. The band theory2,3 is applied to study the nature of the spin exchange interaction within the half-filled band of the stacks. The nature and the magnitude of the exchange interaction in MMRCs are significantly different in comparison with the parent molecular radical crystals (MRC). A ferromagnetic interaction in an MRC changes into an antiferromagnetic interaction and vice versa. In this paper we want to extend the structural principle of the MMRC to a new class of 1-D mixed crystals, namely mixed radical ion crystals (MRIC). The MRICs consist of two hydrocarbons having different ionization potentials, I, and electron affinities, A. The elementary units (EU) are anion radicals or cation radicals when each EU is reduced or oxidized. The structure of the MRICs is shown in Scheme 1. In the above scheme, δ is the sum of the π-net charges, qµ, of the individual hydrocarbons (hydrocarbon subunits):
δ)
∑µ (1 - qµ)
(1)
The structure and magnetic properties of MRICs are significantly different in relation to organic cation (anion) radical crystals, whose EUs consist of two identical polycyclic aromatic
SCHEME 2
hydrocarbons (Scheme 2). Typical examples are the radical cation crystals synthesized by anodic oxidation by Wegner et al.4-6 (H are different arenes, e.g., fluoranthene), and radical cation crystals synthesized by Fritz et al.7 (H is naphthalene). The ground state of these systems is a metallic one, or by a phase transition, a semiconducting one.6 All molecules within the stacks of radical cations are crystallographically identical, i.e., the radical cation is not localized within one hydrocarbon.4 In a recent paper Kochi et al.8 have investigated the structure and energy spectra of paramagnetic monomeric and dimeric cation radicals (D)2+• (D is octamethylbiphenylene (OMB)) and the crystal structure of infinite stacks.
†
Part 4 of this series is ref 1. * To whom correspondence should be addressed. ‡ University of Leipzig. § University of Sofia. | Max-Planck Institut fu ¨ r Polymerforschung.
2. Objectives 2.1. Hydrocarbons. Here, we consider MRICs consisting of naphthalene (I ) 8.1/8.26 ( 0.1 eV9), phenanthrene (I ) 8.0
10.1021/jp011899p CCC: $20.00 © 2001 American Chemical Society Published on Web 07/27/2001
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TABLE 1: Calculated Contributions to the Effective Spin Exchange between the Electrons within the HFB of MRICs Consisting of Benzo [ghi]perylene and Naphthalene (O-1 and O-2 in Figure 1) for Two Different Distances R between the Hydrocarbonsa polymer
-Jkin
J
Jind
Jeff
∆
δ
h
O-1b O-1c O-2b O-2c
5 22 1 5
48 79 7 23
4 6 1 2
47 63 7 20
236 473 109 245
-0.387 -0.416 -0.319 -0.340
190 302 110 175
a The Coulomb integrals have been calculated with D ) 1 in eq 15. The term h is the interaction integral between the frontier (HOMO) orbitals of naphthalene and benzo [ghi]perylene (see eq 10). (All entries are in meV.) δ is the sum of the π-net charges of naphthalene. b Calculated for distances R ) 3.35 Å. c Calculated for distances R ) 3.10 Å.
TABLE 2: Calculated Values of the Different Components to the Spin Exchange between the Electrons in the HFB of 1-D Stacks, T-1, with Uniform Distances between the Hydrocarbons, and T-2 with Different Distances between the Hydrocarbons within the EU and between the Hydrocarbons of Different EUs Shown in Figure 4a polymer
-Jkin
J
Jind
Jeff
∆
T-1 T-2
14 12
164 78
9 5
159 71
401 411
a
The Coulomb integrals γ have been calculated using the MatagaNishimoto approximations with D ) 1. All entries are in meV.
Figure 1. 1-D model stacks of MRICs: face-to-face type.
3. Methods
( 0.1 eV9), and the following hydrocarbons having smaller ionization potentials:9 pyrene (I ) 7.55 eV10), perylene (I ) 6.83 ÷ 7.15 eV9), triphenylene (I ) 7.9 eV9), benzo[ghi]perylene (I ) 7.13 eV9), and pentacene (I ) 6.23 eV9). The ionization potential of fluoranthene was estimated according to the linear regression11
3.1. Spin Exchange of the Electrons in the Half-Filled Band. As in the previous papers1-3 the various contributions to the effective exchange integral Jeff in the Heisenberg Hamiltonian
I ) 6.0 + 3.2 e(HOMO) ) 7.9 eV
(ν and F denote the elementary units) can be expressed as a sum of three contributions (for the sake of simplicity the dimensionless distance parameter τ is omitted): direct, kinetic, and indirect spin exchange:
2.2. 1-D Stacks of MRICs. The stacks of MRICs are considered to be 1-D systems complying with the Born-Karman (cyclic) conditions. The hydrocarbons H are assumed to have ideal geometry with all bond lengths equal to R0 ) 1.40 Å and regular benzene rings. The intermolecular distances between the planes of two neighboring monoradicals in different MRCs varies in the range between 3.1 Å in infinite stacks of Wurster’s radicals12 up to 3.71 Å in the case of galvinoxyl stacks.13 In the case of radical ion crystals,5 the intermolecular distance between the planes of two neighboring hydrocarbons is ∼3.3 Å (see also the results in ref 8). The quantitative results obtained with various values of the interplanar distance, 3.1 Å < R < 3.4 Å, show that, qualitatively, the character of the exchange interaction does not depend on the value of R (see Table 1). Therefore, the numerical results for all of the investigated MRICs are given for the interplanar distance R ) 3.35 Å (interplanar distance of graphite) as in ref 1. Probably, as in the case of radical cation crystals,6,8 the distances between the hydrocarbons within the stacks of MRIC are not uniform as a result of Peierl’s distortions (see Table 2). Here, we consider three types of 1-D stacks of MRICs. (i) Face-to-face type. In this case, the EUs are divided by a mirror plane of symmetry, σh, perpendicular to the translation axis. The investigated systems of this type are shown in Figure 1. (ii) Slipped type. In this case, slipped-type stacks have different slip angles R shown in Figures 2 and 3. (iii) Rotated type stacks, which are characterized by a screw axis nm (Figure 3).
H)-
Jeff(ν,F)SνSF ) Jeff(|ν - F|) ) Jeff(τ) ∑ ν,F
Jeff ) J + Jkin + Jind
(2)
(3)
which are evaluated qualitatively and quantitatively for the MRICs. In eq 3, J is the Coulomb exchange integral between the localized Wannier states within the iν-th and jF-th sites. Jkin is the kinetic exchange parameter
Jkin ) -2t2/(U0 - U1) ) -2t2/U
(4)
representing the antiferromagnetic contribution to the spin exchange. U0 and U1 are the Coulomb repulsion integrals of two electrons residing in the same Wannier state and occupying adjacent Wannier states, respectively. U ) U0 - U1 is the renormalized Hubbard integral,14 and t is the transfer (hopping) parameter between adjacent Wannier functions. The term Jind expresses the indirect exchange (“superexchange”) between the magnetic electrons within the half-filled band (HFB) via delocalized π-electrons in the filled energy bands. The sign of Jind is determined by the structure of the EU and the interaction between the units. 3.2. Whangbo Condition. The Whangbo condition15
∆ < πU/4
(5)
allows a qualitative determination of the relative stability of
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Figure 3. 1-D model stacks of MRICs: slipped and rotated types, respectively.
SCHEME 3
Figure 2. 1-D model stacks of MRICs: slipped type.
the localized high-spin (magnetic), and a nonmagnetic (metallic) state of the HFB. A similar condition is derived by Hubbard:14
∆ < 2U/x3
four-orbital model) developed by Hay, Thibeault, and Hoffmann18 (see also the papers of Nesbet19 and van Kalkeren et al.20). The use of the frontier MOs of the EU for the construction of MOs of 1-D MRICs is appoximate. But, even so, it can lead to qualitatively correct results, and it can also represent the physical picture of the nature and the magnitude of magnetic interaction in MRICs in a distinct form. Here, we consider only the mixed molecular radical cation systems with nondegenerate HOMOs (see K in Scheme 3). However, the results could be immediately extended to radical anion systems (A in Scheme 3) if the LUMOs are not degenerate. If the frontier MOs (HOMO) of two hydrocarbons in the µth EU are denoted by φ1 and φ2 (see Scheme 3), then the Bloch functions of the 1-D stacks consisting of cation radicals can be expressed as follows:
(6)
2
|k> ) 1/xN The width of the HFB, ∆, and the magnitude of the electron repulsion, U, within the HFB mainly determine the character of the ground state and the spin exchange, respectively. If the HFB width ∆ is small compared with the one-site Coulomb repulsion U integral (renormalized Hubbard integral), the magnetic state is preferred. 3.3. Application of the Frontier MOs of the EU for the Study of the Magnetic Interaction of MRICs. The results for the magnetic characteristics of MRICs obtained using the methods described in Section 3.1 are numerical values. Here, we use the frontier MOs of the EU for the study of the magnetic interaction of MRICs. The use of frontier MOs (the lowest unoccupied one, LUMO, and the highest occupied one, HOMO) has been introduced by Fukui et al.16,17 for the description of the properties of complex π-electron systems. The frontier MOs are also used in the molecular theory of the superexchange interaction (four-electron
Cr(k) exp(-ikµ)|r,µ> ∑µ ∑ r)1
(7)
(k ∈[-π, π] is the wave vector, µ denotes the number of the EU, and |r,µ> is the rth MO (φ1 or φ2) within the µth EU). In the Hu¨ckel-Hubbard version of the Bloch method the MO energies e(k) of the 1-D stacks are eigenvalues of the matrix21
E(ωk) ) E0 + V exp(iωk) + V+ exp(-iωk)
(8)
In eq 8, E0 is the energy matrix of the EU, V is the interaction matrix between neighboring EUs (µth and (µ + 1)th), and V+ is the transposed matrix. In the case of 1-D stacks consisting of cation radicals (Scheme 3, K), the energies can be expressed as follows:
E+,-(k) ) (e1 + e2)/2 ( 1/2x(e1 - e2)2 + 8h2(1 + cosk) (9)
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J. Phys. Chem. B, Vol. 105, No. 33, 2001 7975
SCHEME 4
ar(τ) ) ar(µ - ν) ) δµν1/2π
∫-ππ expi(kτ)Cr dk ) δµνCr (14)
where ei (i ) 1, 2) are the orbital energies of the separate (noninteracting) frontier MOs and
h ) )
C1µC2νβµν ∑ µν
(10)
is the interaction integral between the orbitals φ1 and φ2 (h is the periodic one-electron Hamiltonian). In some cases the interaction integral h vanishes because of the symmetry. In eq 9 the sign + corresponds to the MOs of the HFB, and the sign - corresponds to the MOs of the second band. It follows that the MO energies of the HFB lie within the interval
e1(k ) π) < E+(k) < (e1 + e2)/2 + 1/2x(e1 - e2)2 + 16h2 (k ) 0) and the energies of the second band E-(k) lie within the interval
e2(k ) π) > E-(k) > (e1 + e2)/2-1/2x(e1 - e2)2 + 16h2 (k ) 0) The bandwidths (Scheme 4) are given by the expressions
∆1(HFB) ) (e1 + e2)/2 + 1/2x(e1 - e2)2 + 16h2 - e1 ) -(e1 - e2)/2 + 1/2x(e1 - e2)2 + 16h2 (11) and
∆2 ) (e1 + e2)/2 - 1/2x(e1 - e2)2 + 16h2 - e2 ) (e1 - e2)/2 - 1/2x(e1 - e2)2 + 16h2 (11a) The coefficients of the Bloch functions, eq 7, of the HFB are obtained by the following equations:
C1C*1 ) |C1|2 ) p/[p + (e1 - E+)2] |C2|2 ) 1 - |C1|2
(12)
For the calculation of the exchange integrals (as in the case described in section 3.1), it is more convenient to pass from the Bloch functions to the Wannier representation. The orbital coefficients ar(µ - ν) ) ar(τ) of the Wannier function centered on the νth unit
|ν〉 ) N-1/2
∑k
2
exp(-ikν)|k〉 )
ar(τ)|µ,r〉 ∑µ ∑ r)1
are given by the expression (N f ∞)
ar(τ) ) 1/2π
∫-ππ expi(kτ)Cr(k) dk
(13)
The Cr(k) coefficients are given by eq 12. When the orbital coefficients do not depend on k[Cr(k) ) Cr], one obtains
3.4. Parametrization. The calculations have been carried out using a standard set of parameters.1-3 The intermolecular resonance integrals between two 2p AOs have been calculated with Mulliken’s formula:22 β(R) ) β0(Sσ-σ, Sπ-π)/S(R0), taking into account the angular dependence of overlap integrals S (calculated with zC ) 3.25). A value β0(R0 ) 1.40 Å) ) -2.4 eV has been used for the resonance integrals between the 2pπ2pπ AOs of carbon. The two-center atomic Coulomb integrals γpq for the calculation of the various contributions to the effective spin exchange according to eq 3 have been evaluated using the modified Mataga-Nishimoto approximation:23
γpq ) e2/(γ/e2 + DRpq)
(15)
with a value of the one-center Coulomb integral1-3 being γ ) 10.84 eV. In eq 15, D is the screening constant. 4. Results and Discussion 4.1. Mixed Radical Ion Crystals. The numerical results in Tables 1-5 are obtained for mixed radical cation crystals, taking into account the interactions (β-integrals) only between the first neighboring π-centers of the nearest neighboring hydrocarbons. In this topological approximation the 1-D stacks are alternating systems, i.e., every starred π-center (C*) is connected with one (or two) nonstarred (Co) ones.24-26 Because of the alternating character of the 1-D stacks, the numerical results for the components of the effective exchange integral are identical for radical cation and radical anion mixed crystals, respectively (see Appendix in ref 27). This follows also from eq 10 for the interaction integral between the frontier (HOMO) orbitals φ1 and φ2 and frontier (LUMO) orbitals φ#1 and φ#2, respectively TABLE 3: Calculated Values of the Different Components to the Spin Exchange between the Electrons in the HFB of the Polymers Shown in Figure 1a polymer
-Jkin
J
Jind
Jeff
∆
h
O-1 O-2 P-1 P-2 P-3 Pe Peb F
5 1 ∼0 11 11 3 4 55
48 7 1 46 93 32 24 222
4 1 ∼0 5 8 3 3 12
47 7 1 50 90 32 23 179
236 109 32 142 352 192 192 627
190 110 0 163 247 54 54 235
a ∆ is the width of the HFB. The Coulomb integrals have been calculated using the Mataga-Nishimoto approximation (eq 15) with D ) 1 (All entries are in meV.) b D ) 3 in eq 15.
TABLE 4: Calculated Values of the Different Components to the Spin Exchange between the Electrons in the HFB of the Polymers Shown in Figure 2 (anion forms)a polymer
R(β)o
-Jkin
J
Jind
Jeff
∆
P-3r P-3r P-3r P-3r P-3β P-3β
90 67.3 50.1 38.6 70.1 -70.1
11 2 ∼0 ∼0 3 0
93 20 ∼0 7 20 8
8 2 ∼0 ∼0 2 ∼0
90 20 ∼0 7 19 8
236 177 3 9 190 97
a The Coulomb integrals have been calculated using the MatagaNishimoto approximation (eq 15) with D ) 1. (All entries are in meV.)
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TABLE 5: Calculated Values of the Different Components to the Spin Exchange between the Electrons in the HFB of the Polymers Shown in Figure 3 (anion forms)a polymer
Ro
-Jkin
J
Jind
Jeff
∆
R-1 R-2 R-3 S-1 S-2 S-3
90 70, 1 54, 1
0 0 1 2 1 0
2 3 10 9 7 1
∼0 ∼0 1 1 ∼0 0
2 3 10 8 6 1
13 22 113 161 125 17
a The Coulomb integrals have been calculated using the MatagaNishimoto approximation (eq 15) with D ) 1. (All entries are in meV.)
(see Scheme 3):
h ) 〈φ1(1)h(1)φ2(1)〉 ) 〈φ#1(1)h(1)φ#2 (1)〉 )
C*1µC2νβµν ) ∑C#1µC#*2νβµν ∑ µν µν
In accordance with the pairing theorem,24-26 C*1µ ) -C#1µ and C2ν ) -C#2ν). In the case of nonalternating systems, e.g., the fluoranthen radical ion crystals (F), the results for radical cation crystals are different in comparison with those of radical anion crystals. The effective exchange integral of the anionic form is Jeff ) 72 meV, i.e., smaller in comparison with Jeff ) 179 meV of the cationic form (see Table 3). All components of the spin exchange (Jkin,(t), J, Jind) decrease rapidly with the distance parameter τ (see eq 2) between the EUs. With τ > 1, the magnetic components are vanishingly small. For this reason, the results in Tables 1-5 are given only for adjacent EUs, i.e., for a value of the distance parameter τ ) 1. The magnitude of the exchange interaction depends on the distances R between the hydrocarbons in the stacks. This is illustrated with the results given in Tables 1 and 2. The results in Table 1 illustrate the following general rule. When the distance R decreases (O-1b and O-1c), the interaction integral h (see eq 10) between the frontier (HOMO) orbitals, the width of the HFB ∆, the sum of the π-net charges δ of the hydrocarbon with the larger ionization potential, and the effective exchange integral Jeff increase. In the case of two isomeric stack arrangements (O-1 and O-2), the effective exchange integral is larger for the stack with the larger values of δ and h, respectively. When the interaction integral h of a system is equal to zero, as in the case of P-1 (see Figure 1 and Table 3), then the effective exchange integral Jeff is zero or has a small value. The character of the exchange interaction does not depend on the distance alternation between the hydrocarbons in the stacks. Only the magnitude of the exchange interaction changes as it is shown for T-1 and T-2 in Table 2 (see Figure 4). However, if the magnitude of the magnetic interaction is small, Peierl’s distortion can change the character of the exchange interaction. In Table 3 are collected the calculated values of the exchange parameters of the MRICs shown in Figure 1. The calculations are carried out with a value of the screening constant D ) 1 in eq 15. Only the magnitude of the different components of the effective exchange integral changes with the different values of the screening constant (D > 1). This can be seen from the results for the MRICs formed by perylene (Pe) and naphthalene given in Table 3 and Figure 5. Table 4 contains the values of the contributions to the effective exchange integral and the width of the HFB calculated for various models of MRICs of the slipped type with different
Figure 4. 1-D stacks with uniform distances between the hydrocarbons, T-1, and with different distances between the hydrocarbon within the EU and between the hydrocarbons of different EUs, T-2.
slip angles R as is shown in Figures 2 and 3. The corresponding values for model MRICs of the rotated type with a screw axis nm of different order (see Figure 3) are given in Table 5. As in the case of the face-to-face type models, the effective exchange integral Jeff is proportional to the width of the HFB ∆. In all series of comparable models of the slipped and the rotated type, respectively, (P-3r, S-1 to S-3, R-1 to R-3), the effective exchange integral and the width of the HFB decrease with the decrease of the number of interacting π-centers (intermolecular 2pσ-2pσ overlap) of the different polycyclic hydrocarbons in the stacks. Qualitatively, the arising of the high-spin state with ferromagnetically coupled electrons within the HFB of the MRICs is in agreement with the Whangbo-Hubbard conditions, eqs 5 and 6. The HFB width ∆ of all MRICs (see Tables 1-5 and Figure 5) is smaller than 0.4 eV compared with the one-site Coulomb integral (renormalized Hubbard integral), which has values U > 1.3 eV. An exception is model F, formed by fluoranthen and naphthalene for which ∆ ) 0.63 eV (Table 3). However, 0.63 < 4U/π, i.e., the magnetic high-spin state is also preferred. The sum δ of the π-net charges is different for the individual hydrocarbons within the EU (see Scheme 1). For radical cation crystals, the electrons in the HFB are localized mainly in the hydrocarbons with smaller ionization potential (I is the ionization potential of the second hydrocarbon):
δ(I δ(I>) For example, in the radical cation stacks formed by perylene and naphthalene (Pe in Figure 1):
δ(perylene) ) 0.885 > 0.115 ) δ(naphthalene) For radical anion crystals, the electrons in the HFB are localized mainly in the hydrocarbons with the larger electron affinity (A>). In this case, (A< is the electron affinity of the second hydrocarbon):
|δ(A>)| > |δ(A 0.151 ) |δ(phenanthrene)| 4.2. Radical Ion Crystals Consisting of Identical Polycyclic Hydrocarbons. In the case of organic radical ion crystals consisting of identical polycyclic hydrocarbons, the frontier MOs
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J. Phys. Chem. B, Vol. 105, No. 33, 2001 7977
Figure 6. 1-D stacks of radical cations of naphthalene in the elementary units. The model is in accordance with the results in ref 6.
Figure 5. Energy dispersion of the bands of MRCI Pe (see Figure 1).
are degenerate, e1 ) e2 ) e, and eqs 9 and 11 obtain the form (see Scheme 5)
E+,-(k) ) e ( hx2(1 + cosk)
(16)
∆1(HFB) ) 2h ) -∆2
(17)
For instance, for the naphthalene radical cation crystals (see Figure 6) the HFB width is: ∆1(HFB) ) 0.416 eV. The value calculated by means of eq 8 is ∆ ) 0.427 eV. For the orbital coefficients one obtains (see eqs 9 and 12)
|C1|2 ) |C2|2 ) 1/2
(18)
and for the charge densities one obtains
q1(2) ) 1/2π
Figure 7. Energy dispersion of the bands of radical cation 1-D stacks of naphthalene (Figure 6). The bands 3-4, 5-6, 7-8, and 11-12, 13-14, 19-20 are degenerate.
∫-ππ|C1(2)|2 dk ) 1/2
i.e., the charge within the 1-D stacks of radical cations is not localized in one hydrocarbon. It follows from eqs 14 and 18 that only the Wannier coefficients are different from zero for
τ)µ-ν a1(µ - ν) ) a2(µ - ν) ) δµν1/x2
(19)
The energy gap between the HFB and the highest doubly occupied band (HDOB) of radical ion crystals consisting of naphthalene is equal to zero (see Figure 7). For that reason, the formalism in section 3.1 is not applicable to the calculation of the various contributions of the effective exchange integral Jeff
in the Heisenberg Hamiltonian (see eqs 2 and 3, and refs 2831). To obtain correct results of Jeff one must consider additionally the elementary excitation between the HFB and the HDOB. Conclusions The numerical results given in this paper illustrate the possibility of the appearance of a principally new class of 1-D pure organic π-systems with magnetic ordering, namely mixed radical ion crystals consisting of hydrocarbons with different ionization potentials or electron affinity, respectively. The
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investigations in this paper are a widening of the studies in the previous paper, ref 1, concerning mixed molecular radical crystals. The known MMRCs are the only 1-D systems with bulky ferromagnetic properties, but their critical temperature, Tc, is very low.32-34 MRICs and MMRCs are a large group of π-systems which are rarely investigated. Theoretical and experimental studies, so far, are directed mainly to extended π-electron systems (1-D polymers and oligomers) with intramolecular exchange interaction within the HFB.35 From the theoretical results in the present paper, it follows that for some classes of MRICs a stabilization of the macroscopic spin configuration (magnetic ordering) with higher critical temperature (Tc . 0 K) can be expected. Acknowledgment. This work was supported by the Deutsche Forschungsgemeinschaft (N.T., M. S.). We dedicate this work to Prof. Dr. Ju¨rgen Fabian on the occasion of his 65th birthday. References and Notes (1) Tyutyulkov, N.; Dietz, F.; Madjarova, G.; Mu¨llen, K. J. Phys. Chem. B 2000, 104, 7320. (2) Tyutyulkov, N.; Dietz, F. Band Theory of Exchange Effects in Organic Open-Shell Systems. In Magnetic Properties of Organic Materials; Lahti, P. M., Ed.; Marcel Dekker: New York, 1999; Chapter 18, p 361. (3) Dietz, F.; Tyutyulkov, N. Structure and Magnetic Properties of Nonclassical (non-Kekule) π-Conjgated Polymers. In Res. AdV. in Macromolecules; 2000, 1, 71. (4) Kro¨hnke, Ch.; Enkelmann, V.; Wegner, G. Angew. Chem. 1980, 92, 982. (5) Eichele, H.; Schwoerer, M.; Kro¨hnke, Ch.; Wegner, G. Chem. Phys. Lett. 1981, 77, 311. (6) Enkelmann, V.; Wegner, G.; Kro¨hnke, Ch. Mol. Cryst. Liq. Cryst. 1982, 86, 103.
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