6232
J. Phys. Chem. C 2008, 112, 6232-6239
Energy Spectra and Electric and Magnetic Properties of 1D Stacks of Conjugated π-Electron Systems with Defect Surface States. I. π Systems with Tamm Surface States N. Tyutyulkov,†,‡ N. Drebov,†,‡ K. Mu1 llen,§ A. Staykov,† and F. Dietz*,† UniVersita¨t Leipzig, Wilhelm-Ostwald-Institut fu¨r Physikalische und Theoretische Chemie, D-04103 Leipzig, Johannisallee 29, Germany, UniVersity of Sofia, Faculty of Chemistry, Chair of Physical Chemistry, 1 J. Bourchier Bd, BG-1126, Sofia, Bulgaria, and Max-Plank-Institut fu¨r Polymerforschung, D-55128 Mainz, Ackermannweg 10, Germany ReceiVed: NoVember 19, 2007; In Final Form: February 11, 2008
The energy spectra and the electric and magnetic properties of one-dimensional (1D) stacks consisting of conjugated π-electron systems (polycyclic aromatic hydrocarbons and polymethines) with Tamm defects are investigated theoretically by means of the many-electron band theory. The conditions for the relative arrangement of the stacks in terms of the slip parameters for yielding different magnetic and electric groundstate properties are studied.
1. Introduction The investigations of pure organic π-electron systems with unconventional electric (organic metals and semiconductors) and magnetic (organic ferromagnets) properties are focused mainly on one-dimensional (1D) and two-dimensional (2D) π systems with intramolecular interaction of the π electrons.1-5 So far, the magnetic properties of pure organic systems with intermolecular π-π interaction have not been investigated systematically and are seldom found in the literature.1-5 Typical examples are 1D molecular radical crystals (MRC) consisting of weekly interacting stable π-electron monoradicals.6,7 The presence of different types of defects in a semiconductor leads to the existence of one or more defect states and changes its physical properties significantly.8-14 This phenomenon is investigated for 3D semiconductors. However, it is not investigated for 1D stacked π systems with large number of π centers and with different types of defects.15 There are different types of defects in a polycyclic aromatic hydrocarbon (PAH):16 Tamm,17 Schottky,18 Shockley,19 Frenkel,20 and chemisorption10,11 defects, respectively. The synthesis of large PAHs and their derivatives (see ref 21 and the references given therein) and 1D stacked systems with well-defined architectures, crystals, and columnar discotic materials22-24 determines the aim of the investigations in this series of papers. In this first communication, we consider the energy spectra and the electric and magnetic properties of 1D stacks consisting of PAH with Tamm defects. Tamm17 was the first to demonstrate the existence of these surface states by terminating the Kronig-Penney potential for representing the surface in finite crystals. For the first time, Coulson25 has emphasized the significance of the defects (in that case Schottky vacancies) for the explanation of the graphite properties. In ref 26, the energy spectra of 1D stacks consisting of PAHs with different topologies and symmetries and without defects has been studied * Corresponding author. E-mail:
[email protected]. † Universita ¨ t Leipzig. ‡ Permanent address: University of Sofia, Faculty of Chemistry, BG 1126 Sofia, Bulgaria. § Max-Planck-Institut fu ¨ r Polymerforschung.
Figure 1. Investigated homonuclear monoradicals with Tamm defects used as elementary units of the 1D stacks (here and below only one valence formula is depicted). Tamm defects with vacancies on the periphery of the hydrocarbon could also be formed when the valence state of a certain periphery atom is changed from sp2 to sp3 as is shown in the Figure.
theoretically. It has been shown that the energy gap, ∆E, of the stacks is different from zero for PAHs with N e 102 π centers; that is, the ground state of the stacks is a dielectric one. The magnetic properties of 2D graphene slabs with ribbons of different types of periphery defects have been investigated by Klein et al.,27-30 Fujita et al.,31 and Harigaya et al.32 The interlayer interactions in graphite were investigated by Yumura and Yoshizawa33 (see also refs 34-37 and references given therein). Essentially, pure organic systems with metallic ground states synthesized by Haddon et al.38,39 are 1D stacks consisting of monomers with Tamm defects. 2. Point Tamm Defects in PAH There are two types of Tamm defects that can occur in a molecule that is originally represented by a closed-shell structure:
10.1021/jp7109884 CCC: $40.75 © 2008 American Chemical Society Published on Web 03/29/2008
Energy Spectra, Electric and Magnetic Properties
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6233
Figure 3. Investigated polymethine cyanines: pentamethine streptocyanine Pm(n ) 2), heptamethine streptocyanine Hm(n ) 3), nonamethine streptocyanine Nm(n ) 4).
TABLE 1: Calculated Values of the Transitions Energies of the Cyanine Dye Aggregates with Different Values of the Slip Angle r (All Entries Are in eV) Pm
Figure 2. Weitz-type monoradicals used as elementary units of the investigated 1D stacks, and slip parameters of stacks (∆x and ∆y).
SCHEME 1
Hm
Nm
Ro
∆E(J)
∆E(Htt)|
∆E(J)
∆E(Htt)|
∆E(J)
∆E(Htt)|
90 55.3 35.8 25.7 19.8 16.1 0a 0b
1.92 2.31c 2.70 2.84
3.00 2.91 2.87 2.96
1.26 1.55 1.89c 2.15 2.22
2.34 2.30 2.26 2.23 2.32
0.84 1.07 1.35 1.59c 1.79 1.82 1.92 1.91 1.55d
1.92 1.89 1.87 1.84 1.82 1.90
3.00 3.15 2.46d
2.34 2.28 1.97d
a Calculated values of the longest-wavelength transition of noninteracting molecules (monomers). b Experimental values of the longestwavelength transition of noninteracting molecules (monomers).67,68 c Calculated transition energy of the J aggregate (best agreement with the experimental value). d Experimental values of the transition energies of J aggregates (∆E(J)).67,68
TABLE 2: Calculated Values of Transitions Energies of H Aggregates, ∆E(Htt) and ∆E(Hbb), Respectively, Consisting of Pentamethine Cyanine for Different Values of the Slip Angle r (All Entries Are in eV)
- defects that do not change the closed-shell ground state, and - defects determining an open-shell ground state. Here, we consider the following point defects which determine a doublet ground state: - Tamm defects arising by lattice vacancies in a polycyclic aromatic hydrocarbon, - Tamm states arising by substitution of one or more -CHd groups in the periphery of a PAH (or substitution of the CH2 end groups in 1D polymethines by heteroatoms). 2.1. Tamm States Arising by Lattice Vacancies. The vacancies could be created when one or more carbon atom(s) are removed from the periphery of the hydrocarbon. If the resulting system is a non-classical (non-Kekule) hydrocarbon that has {S*} starred and {Ro} non-starred π centers in accordance with the Coulson-Rushbrooke-Longuet-Higgins (CRLH) theorem40,41 the perturbed hydrocarbon has at least
N ) S* - Ro
(1)
nonbonding MOs (NBMO). The vacancies are essentially a surface perturbation that lead to a NBMO within the energy gap. This means that the NBMOs can be considered as surface Tamm states. From this point of view, the half-filled NBMO band (HFB) with intramolecular ferromagnetically coupled electrons in the well-known examples of the 1D polymers can be considered to be Tamm states. For instance, poly(1,3-phenylene methine)3,4 can be considered as a structure arising from the polyacene through Tamm defects (Scheme 1). In Figure 1, the investigated monoradicals with Tamm defects are shown: D-1, D-2, D-3, which constitute the model 1D stacks.
Ro
∆E(Htt)
∆E(Hbb)
90 55.3 35.8 25.7
3.00 2.91 2.87 2.96
3.00 3.07 3.12 3.03
2.2. Tamm States Arising by Periphery Impurities. Openshell derivatives of the PAHs with Tamm defects are Weitztype mono-radicals,42,43 for example, W-1, W-2, and W-3 shown in Figure 2. 2.3. Closed-Shell Monomers with Tamm Defects. The classical polymethine cyanines are 1D chains with an odd number of methine groups and with different atoms of groups III, V, or VI of the Periodic Table at the end (R,ω) positions. They are typical examples of closed-shell molecules with Tamm defects. In Figure 3, the investigated polymethine molecules are shown. 2.4. Models of the 1D Stacks. The stacks are considered to be 1D crystals for which the Born-von Karman cyclic conditions are fulfilled. The arrangement of the stacks is of a slipped faceto-face type with different slip parameters ∆x and ∆y, respectively. In Figure 4, the slip parameters of the stacks consisting of D-1, D-2, and D-3 radicals (Figure 1) are shown, and in Figure 2 the corresponding parameters for the stacks of the investigated Weitz-type radicals are given. The intermolecular distances between the planes of two neighboring monoradicals in different pure MRCs vary in the range between 3.1 Å in infinite stacks of Wurster’s radicals44 up to 3.71 Å in the galvinoxyl stacks.45 The calculations have been carried out with different values of the distance R between the planes of neighboring monomers. Because the results for the magnetic properties, obtained with different values of the interplanar distance R, are qualitatively identical, in Tables 1-6
6234 J. Phys. Chem. C, Vol. 112, No. 16, 2008
Tyutyulkov et al.
TABLE 3: Calculated Values of Different Contributions to the Effective Exchange Integral (see Eq 4) between Adjacent EUs of Polymer D-1 for Different Values of the Slip Parameters ∆x and ∆y (see Figure 4)a ∆y (Å)
∆x (Å)
∆
J
Jkin
Jind
Jeff
0.00 -1.40 -2.80 -4.20 -5.60 -7.00 -8.40 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.70 -0.70 -0.70 -2.10 -2.10 -2.10 -3.50 -3.50 -3.50
0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.212 2.424 3.636 4.848 6.060 7.272 8.484 9.696 1.212 3.636 6.060 1.212 3.636 6.060 1.212 3.636 6.060
1145 0 0 525 0 0 241 0 179 0 596 0 127 0 169 0 0 0 520 485 228 0 0 0
0 34 8 0 26 2 0 37 0 16 0 15 0 4 0 32 10 11 0 0 0 33 13 19
-183 0 0 -33 0 0 -5 0 -4 0 -42 0 -2 0 -2 0 0 0 -38 -28 -5 0 0 0
-4 14 4 -6 10 1 -4 19 -16 8 -17 8 -4 2 -6 14 5 6 -15 6 -13 16 6 11
-187 48 12 -39 36 3 -9 56 -20 24 -59 23 -6 6 -8 46 15 17 -53 -22 -18 49 19 30
a ∆ is the HFB width (all entries are in meV). The results are obtained considering the interaction between the first neighboring AOs only. The interplanar distance is R ) 3.35 Å as in graphite.
and Table 8 only the results for one value of R (3.35 Å) are given. In Table 7, the results for two different values of R (R ) 3.35 Å and R ) 3.5 Å) are compared. 3. Methods of Investigation 3.1. Energy Spectra. The MOs of the 1D system have the form of Bloch running waves
|k> ) N-1/2
∑µ ∑r Cr(k) exp(-ikµ)|r,µ>
(2)
(k ∈[-π, π] is the wave vector, µ denotes the number of the elementary units (EU), and |r,µ> is the rth AO within the µth EU). In the Hu¨ckel-Hubbard version of the Bloch method, the MO energies e(k) are eigenvalues of the energy matrix46
E(k) ) E0 + V exp(ik) + V+ exp(-ik)
(3)
where E0 is the energy matrix of the EU, V is the interaction matrix between neighboring EUs (µth and µ + 1th), and V + is the transposed matrix. 3.2. Spin-Exchange Interaction in the Half Filled Band. In accordance with Anderson’s theory of magnetism,47 it was shown (see ref 48 and references given therein) that the effective exchange integral, Jeff, in the Heisenberg-Dirac-Van Vleck Hamiltonian (i and j denote the EUs)
H ) -2
Jeff (i, j) Si Sj ) -2 ∑ Jeff (i, j) Si Sj ) ∑ i, j i,j -2 ∑ Jeff(τ) Si Sj i, j
(4)
Figure 4. Slip parameters of stacks consisting of D-1 radicals (a), D-2 radicals (b), and D-3 radicals (c).
J is the direct (Coulomb, Hund) exchange integral between the Wannier states localized at the ith and jth sites. Jkin(Jkin < 0) is the kinetic exchange parameter representing the antiferromagnetic contribution to the spin exchange
Jkin ) -∆2/2U ) -∆2/2(U0 - U1)
(6)
where ∆ is the NBMO bandwidth and U is the renormalized Hubbard parameter.49,50 U0 is the Coulomb repulsion integral of two electrons residing the same Wannier state. U1 is the Coulomb repulsion integral of two electrons occupying adjacent Wannier states (τ ) 1). The term Jind expresses the indirect exchange (superexchange) via delocalized π electrons in the filled energy bands. The sign of Jind is determined by the structure and the interaction between the EUs. The terms can be calculated using a formalism described in ref 51. 3.3. Parametrization. The calculations have been carried out using a standard set of parameters.46 The dependence of the resonance integrals between neighboring carbon atoms within the EU on the bond lengths, R, has been calculated with Mulliken’s formula52
β(R,Θ) ) β0 [S(R,Θ)]/S0 cos Θ
(7)
can be expressed as a sum of three contributions:
Jeff ) J + Jkin + Jind
(5)
where S are the overlap integrals between 2p-2p AOs (calculated with ZC ) 3.25, ZN ) 3.90). The intermolecular resonance
Energy Spectra, Electric and Magnetic Properties
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6235
TABLE 4: Calculated Values for the Components of the Effective Exchange Integral, Jeff, between the Electrons in the HFB, and the HFB Width, ∆E, (All Values Are in meV) of 1D Stacks Consisting of D-2 Radicals with an Interplanar Distance of R ) 3.5 Å, and Different Values of the Slip Parameters ∆y and ∆x (in Å, Figure 4) (Considering First Neighbor Interactions Only)
TABLE 6: Calculated Values of the Components of the Effective Exchange Integral, Jeff (in meV), between the Electrons in the HFB, and the HFB Width, ∆E (in meV), of 1D Stacks Consisting of Weitz-Type Radicals W-1 (Figure 2) with Interplanar Distance R ) 3.35 Å and Different Values of the Slip Parameters ∆y and ∆x (in Å), Considering First and Second Neighbor Interactions
∆y
∆x
∆
J
Jkin
Jind
Jeff
∆y
∆x
∆
J
Jkin
Jind
Jeff
0.00 1.40 2.80 4.20 5.60 7.00 0.00 0.00 0.00 0.00 0.00 0.00 0.70 0.70 0.70 1.40 1.40 2.10 2.10 2.10
0.00 0.00 0.00 0.00 0.00 0.00 1.212 2.424 3.636 4.848 6.060 7.272 1.212 3.636 6.060 2.424 4.848 1.212 3.636 6.060
855 0 0 356 0 0 0 278 0 161 0 0 0 0 0 0 0 413 114 103
0 24 9 0 9 1 9 0 5 0 3 0 9 7 2 5 14 0 0 0
-44 0 0 -7 0 0 0 -4 0 -1 0 0 0 0 0 0 0 -10 -1 0
-4 8 4 -5 2 1 5 -12 4 -4 2 -1 4 3 2 3 7 -4 -6 -2
-48 32 13 -12 11 2 14 -16 9 -5 5 -1 13 10 4 8 21 -14 -7 -2
0.00 -1.40 -2.80 -4.20 -5.60 0.00 -0.70 -2.10 -3.50
0.00 0.00 0.00 0.00 0.00 2.424 1.212 1.212 1.212
754 222 17 352 170 124 177 301 39
1 11 2 2 5 1 7 2 5
-41 -4 0 -7 -2 -1 -2 -6 0
0 2 0 -1 1 0 1 0 0
-40 9 2 -6 4 0 6 -4 5
TABLE 5: Calculated Values of the Components of the Effective Exchange Integral (in meV), Jeff, between the Electrons in the HFB, and the HFB Width, ∆E, (in meV) of 1D Stacks Consisting of D-3 Radicals with Interplanar Distance R ) 3.35 (in Å), and Different Values of the Slip Parameters ∆y and ∆x (in Å) (Considering First Neighbor Interactions Only) ∆y
∆x
∆
J
Jkin
Jind
Jeff
0.00 1.40 2.80 4.20 5.60 7.00 8.40 0.00 0.00 0.00 0.00 0.00 0.00 0.70 0.70 0.70 2.10 2.10 2.10 3.50 3.50 3.50
0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.212 2.424 3.636 4.848 6.060 7.272 2.212 3.636 6.060 1.212 3.636 6.060 1.212 3.636 6.060
1145 0 0 499 0 0 207 0 339 0 225 0 10 0 0 0 555 179 131 0 0 0
0 59 26 0 2 8 0 2 0 16 0 6 0 19 22 4 0 0 0 21 30 10
-85 0 0 -14 0 0 -2 0 -7 0 -3 0 0 0 0 0 -19 -2 -1 0 0 0
-9 20 11 -11 8 4 -5 11 -23 12 -9 4 -2 9 8 3 -8 -14 -3 8 9 6
-94 79 37 -25 34 12 -7 13 -30 28 -12 10 -2 28 30 7 -27 -16 -4 29 39 16
integrals between two 2p orbitals that belong to neighboring radicals have also been calculated with Mulliken’s relation, eq 7, taking into account the angular dependence of the overlap integrals (Scheme 2). The three contributions to the exchange parameter in eq 5 were calculated using the Mataga-Nishimoto potential53 for the two-center Coulomb atomic integrals
γrs(M) ) e /(a + DRrs) 2
(8)
where the constant a is a ) e2/γ, and D is the screening constant (with D ) 1, eq 8 corresponds to the Mataga-Nishimoto potential). The results obtained by means of the Ohno potential54
TABLE 7: Calculated Values of the Components of the Effective Exchange Integral between the Electrons in the HFB, the HFB Width, ∆E, (All Values in meV) of 2D Langmuir-Blodgett Films of W-2 Radicals with Different Values of the Slip Parameter ∆x (in Å)a ∆x
∆
J
Jkin
Jind
Jeff
0.000 0.606 1.212 1.818 2.424
933 (692) 815 (610) 520 (386) 250 (188) 286 (213)
0 (0) 0 (0) 1 (1) 1 (0) 1 (1)
-58 (-34) -48 (-26) -19 (-10) -4 (-2) -5 (-3)
-1 (0) 0 (0) 0 (0) 0 (0) 0 (0)
-59 (-34) -48 (-26) -18 (-9) -3 (-2) -4 (-2)
a The results have been obtained with an interplanar distance R ) 3.35 Å and R) 3.50 Å (values in parentheses).
TABLE 8: Calculated Values of the Components of the Effective Exchange Integral (in meV), between the Electrons in the HFB and the HFB Width (in meV), ∆E, of 1D Stacks Consisting of Weitz-Type Radicals W-3 with Interplanar Distance R ) 3.35 Å and Different Values of the Slip Parameters ∆y and ∆x (in Å) Taking into Account the Interaction between the First Neighbor Centers Only ∆y
∆x
∆
J
Jkin
Jind
Jeff
0.00 -1.40 -2.80 -4.20 -5.60 -7.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 2.424 4.848 7.272
1062 150 308 605 84 222 164 465 178
0 52 150 11 165 1 147 126 14
-186 -20 -81 -51 -4 -5 -8 -72 -3
-2 -8 6 0 5 -1 7 2 1
-188 24 75 -40 166 -5 146 56 12
SCHEME 2
do not differ substantially from the results calculated using the Mataga-Nishimoto approximation. The following standard values γC ) 10.84 eV, and γN ) 12.27 eV of the one-center Coulomb integrals were used.55 4. Numerical Results and Discussion 4.1. Scheibe Aggregates. First of all, we consider the Scheibe aggregates. The polymethine dye aggregates discovered by Scheibe56 and by Jelly57 are the first 1D stacks with intermolecular π system of conjugation whose physical properties, especially the optical spectra, have been studied systematically.56-60
6236 J. Phys. Chem. C, Vol. 112, No. 16, 2008
Tyutyulkov et al. SCHEME 3
Figure 5. Model of pentamethine streptocyanine aggregates.
Figure 6. Energy splitting of the frontier MOs (HOMO-LUMO) of the monomers forming a stack with different values of the slip angle R.
As in the case of the investigated MRC systems, the Scheibe aggregates are in fact 1D stacks of slipped face-to-face-type with different slip angles R. This angle is defined as the angle between the longest molecular axis and the translation axis of the stack. A model of stacks consisting of pentamethine streptocyanines is shown in Figure 5. It has been shown that the main factor determining the magnitude and the character of the electron transitions is the slip parameter R.58,61,62 There are two types of aggregates: J-type and H-type aggregates, respectively. J aggregates are characterized by a small angle R, and H aggregates have a larger slip angle. In accordance with the papers of Scheibe,61,62 the longer wavelength absorption of the J aggregates in relation to the monomer absorption is explained by an electron transition from the top of the highest occupied band to the bottom of the lowest unoccupied band (see Figure 6). In the H aggregates, the absorption is caused by an electron transition from the top of the highest occupied band (HOMO band) to the top of the lowest unoccupied band (LUMO band)61,62 labeled as Htt in Figure 6. However, the absorption of H aggregates can be also explained just as well by a transition from the bottom of the HOMO band to the bottom of the LUMO band marked by Hbb in Figure 6. The theoretical treatment of the relationship between the aggregates’ structure and the spectroscopic properties of aggregates can be carried out qualitatively and semiquantitatively by application of the theory of molecular excitons developed by Levinson, Simpson, and Curtis,63 and by Kasha et al.64,65 So far, in the literature there is no information about the band structure of 1D stacks consisting of polymethine molecules. Exceptions are the calculations reported in ref 66. However, in this quoted paper, ref 66, only the dependence of the transition energy of J aggregates, ∆E(J), on the slip parameter R has been discussed. In Figure 3, the polymethine cyanines forming the stacks for which the excitation energies were calculated are shown. The interplanar distance, R, between the planes of
neighboring monomers varies between 3.3 and 3.7 Å.59,67,68 The calculations have been carried out with different values of the distance R. Because the results of the energy spectra are obtained with different values of the interplanar distance R, in Table 1 the results are given only for R ) 3.5 Å. In Table 1, the calculated values of the transition energies of J aggregates, ∆E(J), and of H aggregates, ∆E(Htt), of the investigated polymethine cyanine stacks with different values of the slip angle R are summarized. By comparing the experimental values of the transition energies of J aggregates with the calculated ones (bold values in Table 1), one can estimate the most probable aggregate structure parameter R. From Table 2 one can conclude that ∆E(Htt) ) ∆E(Hbb) in the case of a full face-to-face structure (R ) 90°). In all other cases (R < 90°), the difference ∆E(Htt) - ∆E(Hbb) is quite small in value. The same results have been obtained for other polymethine cyanine aggregates. This means that both electron transition types (Htt and Hbb) have similar transition energies. 4.2. 1D Stacks Consisting of Homonuclear Alternant Radicals. Spin-Exchange Interaction in the Half-Filled Band in the Case of Full Face-to-Face Arrangement of Radicals in 1D Stacks. In ref 69, the symmetry breaking of the full faceto-face arrangement of the closed-shell π-electron molecules (hydrocarbons) in 1D stacks has been investigated. A theorem was formulated that shows that the π-π interaction in a full face-to-face arrangement is nonbonding; that is, the structure of such 1D systems tends toward a system with reduced symmetry and lowest π-electron energy. This theorem69 was proven only for closed-shell homonuclear molecules, in particular for disk-like hydrocarbons. Here, the investigations (considerations) are extended to homo- (and heteronuclear) open-shell π systems. In the case of a full faceto-face arrangement in 1D stacks, the monomers are separated by a mirror plane of symmetry, σh, perpendicular to the translation axis. By reflection through the σh plane the rth atom of the µth EU coincides with the rth atom in the (µ + 1)-th EU and with the rth atom in the (µ - 1)-th EU, respectively (see Scheme 3). If the resonance integrals between neighboring 2πσ, r - 2πσ, r AOs are equal for all pairs {µ, r - µ ( 1, r}, that is
2πσ,r - 2πσ,r ) βσ-σ
(9)
in topological approximation, that is, if we consider the interaction only between neighboring π centers then the interaction matrices in eq 3 are equal to
V ) V+ ) βσ-σ I
(10)
where I is the unity matrix. If the condition (eq 10) is fulfilled, then the matrix E(k) is equal to
E(k) ) E0 + 2βσ-σ cos kI
(11)
Energy Spectra, Electric and Magnetic Properties
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6237
Figure 7. Splitting of the HFB of a 1D stack as function of the degree of perturbation: NP,non-perturbed HFB; WP, weakly perturbed HFB, ferromagnetic ground state; SP, strongly perturbed HFB, metallic or antiferromagnetic ground state.
SCHEME 4
SCHEME 5
Let us denote with {ep} the MO energies of the isolated noninteracting molecules. It follows from eq 11 that each MO energy ep splits into a band of energies (Scheme 4)
ep(k) ) ep + 2βσ-σ cos k
(12)
The contribution of each bonding MO to the splitting energy, ∆ep, is equal to zero
∆ep ) 2βσ-σ
∫-ππ cos k dk ) 0
(13)
that is, the π-π interaction is nonbonding. If the considered system is an open-shell one, then eq 12 is also valid also for the HFB, and the HFB width is equal to
∆ ) 4|βσ-σ|
(14)
The character of the ground state of the 1D system is determined by the degree of perturbation, that is, by the HFB width ∆. If the splitting of the HFB is small, then one would deal with a high-spin system with ferromagnetically coupled electrons within the NBMO band. If the splitting of the NBMOs is large, then the ground state of the polymer is a low-spin metallic one (see Figure 7). The large bandwidth of the HFB (eq 14) causes the large value of the kinetic term Jkin (eq 6) and also the antiferromagnetic character of the spin exchange. The Coulomb (J) and the indirect (Jind) contributions are equal to zero or have small values. These qualitative results are in agreement with the quantitative results shown in Tables 3-5. The considerations are valid for an arbitrary planar homonuclear π-electron radical with Tamm (Figure 1), Schottky (S) or chemisorption (C) defects (Scheme 5). Results for Stacks with Different Slip Parameters. The energy spectra of the 1D stacks formed by PAH with defects are characterized by a large energy gap in which the HFB is situated. This can be seen in Figure 8 where the energy dispersion of the energy bands of 1D stacks of D-2 radicals is shown as a typical example.
Figure 8. Band structure of 1D stacks composed of D-2 monoradicals (Figure 1) with different values of the slip parameters ∆y and ∆x (in Å) (see Figure 4): (a) ∆y ) ∆x ) 0 (full face-to-face arrangement); b) ∆y ) -1.4 and ∆x ) 0. The HFB is designated with an asterisk.
In Tables 3-5, the calculated values for the different contributions to the effective exchange integral between adjacent EUs of the investigated polymers built up by radicals D-1, D-2, and D-3, respectively, and different values of the slip parameters ∆x and ∆y are given. The two-center Coulomb integrals were calculated by means of the Mataga-Nishimoto potential (eq 8) with a screening parameter D ) 1. The results obtained for D > 1 do not differ substantially from the results calculated with D ) 1. All of the monoradicals are characterized by one NBMO. The magnitude of the different contributions to the effective spin exchange between the unpaired electrons in the HFB depends essentially on the slip parameters ∆x and ∆y (see Figure 4), that is, on the arrangement of the radicals within the stacks. The values are determined mainly (principally) by the resonance integrals βσ-σ between the nearest π centers of the neighboring radicals in the stacks. In the case of a full face-to-face arrangement, as this follows from the results in Section 4.2, the interaction is an antiferromagnetic one. This result is also in agreement with the following considerations. In the case of the full face-to-face arrangement, the nearest π centers of the neighboring radicals are either both starred or both non-starred, respectively (Scheme 6). Such an 1D system is not alternant (CRLH theorem is not valid) and its HFB is not formed by degenerated NBMOs. The large value of the HFB width determines the dominant contribution of the antiferromagnetic term Jkin. If the nearest π centers of the neighboring radicals belong to different subsets (starred
6238 J. Phys. Chem. C, Vol. 112, No. 16, 2008 SCHEME 6
and non-starred), then the system is alternant. Such a system has a HFB of degenerate NBMOs. 4.3. 1D Stacks Consisting of Weitz-Type Radicals. In Tables 6-8 the calculated values of the components of Jeff for the studied 1D stacks consisting of Weitz-type radicals are given. The redox potentials and spectral and EPR properties of radical W-1 organized in Langmuir-Blodgett films have been studied in the paper of Schmittel et al.70 The exchange interaction in stacks of W-2 cation radicals is an antiferromagnetic one for all values of ∆x, decreasing with the increase of the slip parameter, as can be seen from the values of Jeff. For this system, the main component responsible for the nature of the magnetic interaction is the kinetic exchange term Jkin. The negative values of Jeff can lead either to an antiferromagnetic localized state or to a delocalized (low-spin) metallic one. 5. Conclusions and Outlook The ground-state properties (magnetic or electric) of 1D stacks of three types of π-conjugated systems have been investigated: (a) PAHs (originally representing closed-shell systems) with Tamm-type defects resulting in open-shell (doublet state) species as exemplified by D-1, D-2, D-3, (b) Weitz-type monoradicals (structures W-1, W-2, W-3), (c) Closed-shell monomers with Tamm defects (Scheibe streptopolymethine cyanine aggregates). The theoretical results in this paper illustrate the possibility for the existence of 1D stacks with magnetic ordering. The numerical results should be considered as a quantitative illustration of qualitatively correct results. The main factor that determines the character and magnitude of the magnetic interaction is the slip parameter (R and ∆x and ∆y, respectively). The full face-to-face stacking determines an antiferromagnetic exchange interaction between the electrons within the HFB. In the slipped face-to-face stacking (arrangement), the nature and the magnitude of the magnetic interaction is determined by the slip parameters, as in the case of Scheibe aggregates. When a 1D stack of alternate PAH radicals (PAH with Tamm defects) is by itself alternating, then the interaction of the electrons within the half-filled band is a ferromagnetic one. However, if the stack is a nonalternating system then the interaction of the π electrons within the HFB is an antiferromagnetic one. The structural variety of theoretically possible 1D stacks consisting of polycyclic aromatic hydrocarbons with Tamm defects is impressive. This gives rise to our belief that some of the experimentally accessible purely organic systems can be expected to be with magnetic ordering and with relatively high values of the critical temperature Tc. Acknowledgment. This work was supported by the Deutsche Forschungsgemeinschaft (N.T., N.D.) and by the Deutscher Akademischer Austauschdienst (A.S). References and Notes (1) Heeger, A. J. J. Phys. Chem. B 2001, 105, 8475 (Nobel Lecture). (2) Kies, H., Ed.; Conjugated Conducting Polymers, Springer Series in Solid State Physics; Springer: Berlin, 1992; Vol. 102.
Tyutyulkov et al. (3) Lahti, P. M., Ed.; Magnetic Properties of Organic Materials; Marcel Dekker Inc.: New York, 1999. (4) Itoh, K.; Kinoshita, M. Molecular Magnetism; Kodansha, Gordon, and Breach: Tokyo, 2000. (5) Makarova, T. L. Semiconductors 2004, 38, 615. (6) Kinoshita, M. Jpn. J. Appl. Phys. 1994, 33, 5718. (7) Dormann, E. Synth. Met. 1995, 71, 1781. (8) Koster, G. F.; Slater, J. C. Phys. ReV. 1954, 95, 1167. (9) Proc. 2nd Conference on Semiconductors Surface, Phys. Chem. Solids 1960, 14. (10) Rehr, J. J.; Kohn, W. Phys. ReV. B 1974, 9, 1981. (11) Koutecky, J. In AdVances in Chem. Physics; Prigogin, I., Ed.; Interscience Publishers: London, 1965; Vol. 9. (12) Davidson, G.; Levine, J. Surface States (Tamm States); Academic Press: New York, 1970 (13) Madelung, O. Festko¨rper Theorie III, Lokalisierte Zusta¨nde; Springer: Berlin, 1973. (14) Harrison, A. Electronic Structure and Properties of Solids (Chapter 10), Surface and Defects; W. H. Freemann & Co: San Fransisco, 1980. (15) Klein, D. J.; March, N. H. Int. J. Quantum Chem. 2001, 85, 327. (16) Dietz, F.; Tyutyulkov, N.; Madjarova, G.; Mu¨llen, K. J. Phys. Chem. B 2000, 104, 1746. (17) Tamm, I. E. Z. Phys. 1932, 76, 849; Tamm, I. E. Phys. Z. Sowjet. 1932, 1, 733. (18) Schottky, W.; Wagner, C. Phys. Chem. B 1930, 11, 163. (19) Shokley, W. Phys. ReV. 1939, 56, 317. (20) Frenkel, I. Z. Phys. 1926, 35, 652. (21) Grimsdale, A. C.; Wu, J.; Mu¨llen, K. Chem. Commun. 2005, 2197. (22) Berresheim, A. J.; Mu¨ller, M.; Mu¨llen, K. Chem. ReV. 1999, 99, 1747. (23) Meier, H. Synthesis 2002, 1213. (24) Boden, N.; Bushby, R. J.; Clements, J.; Movaghar, B. J. Mater. Chem. 1999, 9, 2081. (25) Coulson, C. A.; Herraez, M. A.; Leal, M.; Santos, E.; Senent, S. Proc. R. Soc. London, Ser. A 1963, 274, 461. (26) Tyutyulkov, N.; Ivanov, N.; Mu¨llen, K.; Staykov.; Dietz. F. J. Phys. Chem. B 2004, 108, 4275. (27) Klein, D. J. Chem. Phys. Lett. 1994, 217, 261. (28) Klein, D. J.; Bytautas, L. J. Phys. Chem. A 1999, 103, 5196. (29) Ivanciuc, O.; Bytautas, L.; Klein, D. J. J. Chem. Phys. 2002, 116, 4735. (30) Ivanciuc, O.; Klein, D. J.; Bytautas, L. Carbon 2002, 40, 2063. (31) Nakada, N.; Igami, M.; Wakabayashi, K.; Fujita, M. Mol. Cryst. Liq. Cryst. 1998, 310, 225. (32) Harigaya, K.; Enoki, T. Chem. Phys. Lett. 2002, 351, 128. (33) Yumura, T.; Yoshizawa, K. Chem. Phys. 2002, 279, 111. (34) Maruyama, M.; Kusakabe, K. J. Chem. Soc. Jpn. 2004, 73, 656. (35) Harigaya, K. Chem. Phys. Lett. 2001, 340, 123. (36) Harigaya, K. Chem. Phys. Lett. 2001, 399, 23. (37) Enoki, T.; Kawatsu, N.; Shibayama, Y.; Sato, H.; Kobori, R.; Maruyama, S.; Kaneko, K. Polyhedron 2001, 20, 1311. (38) Pal, S. K.; Itkis, M. E.; Reed, R. W.; Oakley, R. T.; Cordes, A. W.; Tham, F. S.; Siegrist, T.; Haddon, R. C. J. Am. Chem. Soc. 2004, 126, 1478. (39) Chi, X.; Itkis, M. E.; Tham, F. S.; Oakley, R. T.; Kordes, A. W.; Haddon, R. C. Int. J. Quantum Chem. 2003, 95, 853. (40) Coulson, C. A.; Rushbrooke, G. S. Proc. Cambridge Philos. Mag. 1940, 36 193. (41) (a) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc. London, Ser. A 1947, 191 39. (b) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc. London, Ser. A 1947, 192, 16. (c) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc. London, Ser. A 1948, 193, 447. (d) Coulson, C. A.; Longuet-Higgins, H. C. Proc. R. Soc. London, Ser. A 1948, 195 188. (42) Weitz, E.; Fischer, K. Angew. Chem. 1925, 38 1110. (43) Hu¨nig, S. In Optische Anregung Organischer Systeme; Verlag Chemie: Weinheim, 1966; p 184. (44) Tanaka, J.; Sacabe, N. Acta Crystallogr., Sect. B 1968, 24, 1345. (45) Williams, D. Mol. Phys. 1969, 16, 145. (46) Polansky, O. E.; Tyutyulkov, N. MATCH (Mathematical. Comm.) 1977, 3, 149. (47) Anderson, P. W. Phys. ReV. 1950, 79, 350; 1959, 115, 2. (48) Tyutyulkov, N.; Dietz, F. In Magnetic Properties of Organic Materials; Lahti, P. M., Ed.; Marcel Dekker, Inc.: New York, 1999; Chapter 18, pp 361-376. (49) (a) Hubbard, J. Proc. R. Soc. London, Ser. A 1963, 276, 238. (b) Hubbard, J. Proc. R. Soc. London, Ser. A 1964, 277, 401. (50) Tyutyulkov, N.; Dietz, F.; Mu¨llen, K. Chem. Phys. 2000, 255, 223. (51) Tyutyulkov, N.; Karabunarliev, S. Chem. Phys. 1987, 112, 293. (52) Mulliken, R. S. J. Chim. Phys. Phys.-Chim. Biol. 1949, 46, 497. (53) Mataga, N.; Nishimoto, K. Z. Phys. Chem. N. F. 1957, 13, 140 . (54) Ohno, K. Theor. Chim. Acta 1964, 2, 219.
Energy Spectra, Electric and Magnetic Properties (55) Dietz, F.; Tyutyulkov, N.; Lu¨ders, K.; Christen, A. Chem. Phys. 1997, 218, 43. (56) Scheibe, G. Angew. Chem. 1936, 49, 563; 1937, 50, 212. (57) Jelly, E. Nature 1936, 138, 1009; 1937, 139, 631. (58) Kuhn, H.; Mo¨bius, D.; Bu¨cher, H. In Physical Methods of Chemistry; Weissberger, A., Rossiter, B., Eds.; Wiley: New York, 1972; Vol. 1, Part 3B. (59) Tyutyulkov, N.; Fabian, J.; Mehlhorn, A.; Dietz, F.; Tadjer, A. Polymethine Dyes - Structure and Properties; University “St. Kliment Ohridski” Press: Sofia, Bulgaria, 1991. (60) Dietz, F. J. Inform. Record. Mater. 1973, 1, 157, 237. (61) Scheibe, G.; Wo¨rz, O. Angew. Chem. 1966, 78, 304. (62) Scheibe, G.; Wo¨rz, O.; Haimerl, F.; Seiffert, W.; Winkler, J. J. Chim. Phys. 1968, 65, 146.
J. Phys. Chem. C, Vol. 112, No. 16, 2008 6239 (63) Levinson, G. S.; Simpson, W. T.; Curtis, W. J. Am. Chem. Soc. 1957, 79, 4314. (64) McRae, E. G.; Kasha, M. J. Chem. Phys. 1958, 28, 721. (65) Kasha, M.; El Bayoumi, M. A.; Rhodes, W. J. Chim. Phys. 1961, 58, 916. (66) Dietz, F.; Tyutyulkov, N.; Karabunarliev, S. J. Inform. Record. Mater. 1992, 20, 161. (67) Grimm, B.; Da¨hne, S. J. Inform. Record. Mater. 1973, 1, 339. (68) Kobischke, H.; Da¨hne, S. Photogr. Sci. Eng. 1972, 16, 173. (69) Tyutyulkov, N.; Staykov, A.; Mu¨llen, K.; Dietz, F. Chem. Phys. Lett. 2003, 373, 314. (70) Schmittel, M.; Lal, L.; Graf, K.; Jenschke, G.; Suske, I.; Salbeck, J. Chem. Commun. 2005, 5650.
