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Theoretical Study on Spin Alignments in Ferromagnetic Heterospin Chains with Competing Exchange Interactions: A Generalized Ferrimagnetic System Containing Organic Biradicals in the Singlet Ground State Kensuke Maekawa,† Daisuke Shiomi,*,†,‡ Tomoaki Ise,†,‡ Kazunobu Sato,† and Takeji Takui*,† Departments of Materials Science and Chemistry, Graduate School of Science, Osaka City UniVersity, Sumiyoshi-ku, Osaka 558-8585, Japan, and PRESTO, Japan Science and Technology Agency (JST), 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japan ReceiVed: December 22, 2004; In Final Form: March 11, 2005
Spin alignments in heterospin chains are examined from numerical calculations of model spin Hamiltonians. The Hamiltonians of the heterospin chains mimic an open-shell molecular assemblage composed of an organic biradical in a singlet (S ) 0) ground state and a doublet (S ) 1/2) monoradical, which are coupled by intermolecular ferromagnetic exchange interactions. It is found from numerical calculations of the spin Hamiltonians that the spin value 〈S2〉 of the ground-state singlet biradical embedded in the exchange-coupled assemblage deviates from zero and contributes to the bulk magnetization. The alternating chain is found to have two kinds of ground spin states, a high- and a low-spin state. All the spins are parallel to each other in the high-spin state, which is characterized by the spin correlation function of 〈Si‚Sj〉 ) 0.25. On the other hand, the spin alignment in the low-spin state is found to be dependent on the topology of the intermolecular exchange interactions. The energy preference of the two states depends on the relative amplitude of the exchange interactions in the chain. The intermolecular ferromagnetic couplings are competing in the ground-state singlet biradical with the intramolecular antiferromagnetic interaction. The appearance of the two kinds of ground states is attributed to a quantum spin frustration effect inherent in the triangular motif of the competing interactions. Magnetic properties of a zigzag chain complex composed of a nitronyl nitroxide biradical with a singlet ground state and Cu(hfac)2 are examined on the basis of the theoretical calculations. The vanishing magnetic moments, or the product of susceptibility and temperature χT, at low temperatures observed for the complex are consistent with those of the low-spin state predicted in the theoretical calculations.
Introduction Recent years have witnessed a great deal of interest in the construction of molecule-based magnets and other molecular functionality magnetics.1 Ferrimagnetism is one of the categories in magnetism, the physical picture of which has been initiated by Ne´el:2 An antiparallel coupling of different spin quantum numbers, e.g., S ) 1 and 1/2, gives an ordered state with bulk and net magnetization due to incomplete cancellation of magnetic moments. Ferrimagnets have been attracting revived interest in recent years as one of the new topics of lowdimensional quantum spin systems of antiferromagnetically coupled spins.3-5 Numerical calculations and theories for the systems are applicable to real heterospin compounds of transition metal ions such as Cu2+ (S ) 1/2) and Ni2+ (S ) 1).6 On the other hand, magnetic phase transition to a ferrimagnetic ordered state has not been documented so far in purely organic molecular crystalline solids. The ferrimagnetic spin ordering in purely organic molecule-based materials has been a long-standing issue in terms of materials challenge in chemistry. In addition, from a theoretical point of view, the molecule-based ferrimagnetics gives new aspects or categories in molecular science, which have been overlooked in atom- or * Address correspondence to Daisuke Shiomi at Osaka City University. Phone: +81-6-6605-3149. Fax: +81-6-6605-3137. E-mail: shiomi@ sci.osaka-cu.ac.jp. † Osaka City University. ‡ PRESTO, Japan Science and Technology Agency.
metal-based magnetics studied so far. The underlying idea of organic molecule-based ferrimagnets has been classical and crude in quantum terms,7 requiring more elaborate approaches in view of the materials development in molecule-based magnetics.8-11 One of the most important features of organic molecule-based magnetic materials is spin density delocalized over many atomic sites in an open-shell molecule. This gives a multicentered or multicontact nature to intermolecular magnetic interactions. In particular, the magnetic degree of freedom within open-shell molecules containing more than one unpaired electron should be taken into account in considering magnetic properties of molecular assemblages containing S * 1/2 molecules. This feature is negligible in metal-based ferrimagnets, in which the internal magnetic structure of S > 1/2 atoms or ions can be neglected. In our previous paper,12 we have discussed the possibility of ferrimagnetic-like spin alignment in an alternating chain consisting of biradicals in a singlet (S ) 0) ground state and a monoradical with S ) 1/2. This model system reveals the multicentered nature of organic open-shell molecules reflected in bulk magnetic properties of the molecular assemblage. The spin Hamiltonian describing the system is written as10-12 N
HA )
[-2J1Si,b1‚Si,b2 - 2J2(Si,b2‚Si,m + Si,m‚Si+1,b2) ∑ i)1 2aJ2(Si,b1‚Si,m + Si,m‚Si+1,b1)] (1)
10.1021/jp0441792 CCC: $30.25 © 2005 American Chemical Society Published on Web 04/12/2005
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Figure 1. Schematic view of the Heisenberg spin Hamiltonian of the alternating chain with N repeating units; (a) Model A and (b) Model B. The rounded rectangle represents the biradical molecule with two unpaired electron spins (Si,b1 ) 1/2 and Si,b2 ) 1/2) interacting with the adjacent monoradicals with Si,m ) 1/2. The periodic boundary condition (SN+1,I ) S1,I; I ) b1, b2, and m) is imposed. The intramolecular exchange interaction in the biradical is defined as J1. The solid and dashed lines denote the intermolecular interactions J2 and aJ2.
where Si,b1, Si,b2, and Si,m denote the spin-1/2 operators and N is the number of unit cells. The Hamiltonian (eq 1) is schematically shown in Figure 1a. The two S ) 1/2 spins, Si,b1 and Si,b2, are coupled by the intramolecular antiferromagnetic exchange interaction J1 < 0 within the biradical in the ith unit cell. These two spins are coupled with another S ) 1/2 spin, Si-1,m and Si,m, on the neighboring monoradicals by the intermolecular interactions J2 and aJ2 (0 e a e 1). We have found that the expectation value of the biradical spin deviates from zero12
〈Sb2〉 ≡ 〈(S1,b1 + S1,b2)2〉 * 0
spin states are distinguished by the 〈Sb2〉 value as well as the spin quantum number ST. In one of the ferrimagnetic-like highspin states, the biradical spins elongating from Sb ) 0 give a ferrimagnetic-like spin alignment. In the other high-spin state, the biradical spins contracting from Sb ) 1 constitute another type of ferrimagnetic-like spin alignment. The intermediate region of the coupling parameters affords the low-spin ground state. This novel class of magnetism is termed “generalized ferrimagnetism”.12 This variety of the spin alignments results from the incommutability of eq 3 based on the competing, multicentered interactions between the molecules. In this study, we extend the theoretical model of “generalized ferrimagnetism” to the case of ferromagnetic exchange interactions, J2 > 0 and aJ2 > 0, between the molecules neighboring along the chain. The topology of the exchange couplings in the chain is kept unchanged in the ferromagnetic version, as designated “Model A” in Figure 1a. The intermolecular ferromagnetic couplings are competing in the ground-state singlet biradical with the antiferromagnetic interaction J1. Thus, the ferromagnetic version of the alternating chain still has a frustrating triangular motif. The possible occurrence of ferrimagnetic-like spin alignment is elucidated from numerical calculations of the spin Hamiltonian (eq 1) of Model A. Another model system, Model B, is examined, in which the topology of the intermolecular exchange couplings is different from that of Model A: The alternation of the exchange couplings in Model B undergoes criss-cross coupling topology and twists along the chain, as depicted in Figure 1b. The spin Hamiltonian for Model B is written as N
HB )
[-2J1Si,b1‚Si,b2 - 2J2(Si,b2‚Si,m + Si,m‚Si+1,b1) ∑ i)1
(2)
2aJ2(Si,b1‚Si,m + Si,m‚Si+1,b2)] (4)
in the ground state for arbitrary values of J2 and aJ2. The nonvanishing spin on the ground-state singlet biradical is attributed to the incommutability of the biradical spin Sb and the Hamiltonian12
As a model system for the ferromagnetic chain of the groundstate singlet biradicals and monoradicals, a polymeric complex (1) composed of a nitronyl nitroxide biradical (bis[2,2′-(1-oxyl3-oxido-4,4,5,5-tetramethyl-4,5-dihydro-1H-imidazolyl)]; abbreviated as bnn) in a singlet ground state (S ) 0) and Cu(II) with S ) 1/2, [Cu(hfac)2(bnn)], is designed and synthesized.13 The biradical bnn is known to have the intramolecular exchange coupling of 2J/kB ) -448 K.14 In complex 1, the Cu(II) ion is a substitute for the organic monoradical with S ) 1/2 in the theoretical model. The crystal structure of 1 corresponds to Model B. Magnetic properties of 1 are discussed on the basis of the calculations for Model B.
[HA, Sb2] * 0
(3)
An Sb ) 0 is not a good quantum number when a biradical with a singlet (S ) 0) ground state is embedded in the exchangecoupled molecular assemblages. Even an Sb ∼ 1 (〈Sb2〉 ≈ 2) appears on the ground-state singlet biradical, when the intermolecular interaction is much larger than the intramolecular one (|J2/J1| . 1) and the spatial symmetry of the intermolecular interactions is high (a f 1). The ground-state singlet biradical should recover the contribution to the bulk magnetization:; 〈Sb2〉 * 0, although in nonquantum terms biradicals with a singlet (S ) 0) ground state would seemingly have no contribution to the magnetization at low temperatures, kBT < |J1|. Zero expectation value 〈Sb2〉 ) 0 is found only for the magnetically isolated biradical: J2 ) aJ2 ) 0, where the biradical spin Sb commutes with the Hamiltonian. An important issue for the model Hamiltonian eq 1 is exotic ground-state spin alignments resulting from quantum spin frustration inherent in the antiferromagnetic couplings in the triangular motif, as depicted in Figure 1a. From numerical diagonalization of the spin Hamiltonian, the spin chain has been found to have three kinds of ground states: two with seemingly ferrimagnetic spin alignments of the total spin ST ) N/2 and one with a spinless (diamagnetic) state. The energy preference of the three states depends on the relative magnitude of the intraand the intermolecular interactions J1, J2, and aJ2. The three
Methods of Calculation The exact numerical diagonalization of the Hamiltonian matrix was done by the Lanczos method15 and the Householder method. The largest system examined in this study consists of nine repeating units of a biradical and a monoradical (27 S ) 1/ spins) with the matrix dimension of 227 × 227 ) 134217728 2 × 134217728. The matrix was block-diagonalized according to the conservation of the z component of the total spin, giving submatrixes with the largest dimension of 27C13 × 27C13 ) 20058300 × 20058300. All the calculations were carried out on a Compaq AlphaServerGS160 at the Media Center of the Osaka City University. The periodic boundary condition SN+1,I ) S1,I (I ) b1, b2, and m) is imposed with 3N e 27 repeating units in the Hamiltonian. Results and Discussion 1. Three-Spin System in the Unit Cell of the Chain. The spin state of a pair of the biradical and the monoradical was
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Figure 2. Ground-state expectation value of the biradical spin 〈Sb2〉 ≡ 〈(Sb1 + Sb2)2〉 in the three-spin system of the biradical-monoradical pair as a function of the interaction ratios J2/|J1| and a.
examined as a minimal model system for the ferromagnetic version of the chain in Figure 1. The pair is regarded as a unit cell of the chain. The spin Hamiltonian of the pair is written as
Hpair ) -2J1Sb1‚Sb2 - 2J2Sb2‚Sm - 2aJ2Sm‚Sb1
(5)
The Hamiltonian was represented by a 23 × 23 matrix, which was analytically diagonalized to give eigenvalues and eigenvectors. The spin multiplicity for the ground state was found to be quartet (ST ) 3/2) for J2/|J1| > 1+1/a and doublet (ST ) 1/2) for J2/|J1| < 1 + 1/a. The expectation value 〈Sb2〉 ≡ 〈(Sb1 + Sb2)2〉 of the biradical spin in the ground state was calculated from the eigenvectors as
〈Sb2〉 ) 2(J2/|J1| > 1 + 1/a; ST ) 3/2) 〈Sb 〉 ) 1 + 2
(6a)
whole chain was calculated from the eigenvector of the ground state |GS〉
2 - (a + 1)(J2/J1) 2x(a2 - a + 1)(J2/J1)2 - (a + 1)(J2/J1) + 1
(J2/|J1| < 1 + 1/a; ST ) /2) (6b) 1
A three-dimensional contour plot of 〈Sb2〉 is given in Figure 2 as a function of the ratio of J2/|J1| and the parameter a. An Sb ) 0 is a good quantum number only in the limit of J2/|J1| f 0; the pure singlet (Sb ) 0) appears only when the biradical is isolated with the negligible intermolecular exchange interaction J2. Nonzero magnetic moment arising from 〈(Sb1 + Sb2)2〉 * 0 is predicted for arbitrary values of J2/|J1| and a. Even an Sb ) 1 (Sb(Sb + 1) ) 2) appears in the area of J2/|J1| g 1 + 1/a. As found for the antiferromagnetic chain,12 the ground-state singlet biradical embedded in the exchange-coupled chain has a sizable contribution to magnetization as well as the ferromagnetic chain. This result indicates the possibility of “generalized ferrimagnetism” for both ferromagnetic and antiferromagnetic intermolecular exchange interactions, J2 and aJ2. 2. Spin Alignment in the Alternating Chain of Biradical and Monoradical. (a) Model A. The Hamiltonian eq 1 was represented by a 23N × 23N matrix with the basis sets {|σ1,b1σ1,b2...σN,m〉} defined by direct product of S ) 1/2 spin functions σi,I (I ) b1, b2, and m; σ ) R or β). The matrix was numerically diagonalized to give eigenvectors and eigenvalues. All the calculations described below were made for a spin subspace with the z-component of the total spin MS 3N
(
Figure 3. Spin quantum number ST of the whole chain (a) and the expectation value of the biradical spin 〈Sb2〉 ≡ 〈(S1,b1 + S1,b2)2〉 (b) of Model A in the ground state as a function of the interaction ratio J2/ |J1| and the asymmetry parameter a. The black and red circles indicate the chain length 3N ) 15 and 18, respectively. The solid lines are to guide the eye.
∑I SIz)|σ1,b1σ1,b2...σN,m〉 ) MS|σ1,b1σ1,b2...σN,m〉
(7)
where MS ) 0 for even 3N and MS ) +1/2 for odd 3N, respectively. The ground-state spin quantum number ST of the
ST(ST + 1) ) 〈GS|ST2|GS〉
(8)
where ST2 is the total spin operator of the whole chain N
(Si,b1 + Si,b2 + Si,m)]2 ∑ i)1
ST2 ≡ [
(9)
In Figure 3a is shown the total spin quantum number ST of the ground state for the chain of length 3N ) 15 (the odd number of unit cells) as a function of the ratio J2/|J1| and the asymmetry parameter a. We find that the chain has two kinds of ground states depending on the ratios of the exchange interactions: one is a high-spin state with ST ) 15/2 and the other is a low-spin state with ST ) 5/2. The regions of the parameter space {J2/|J1|, a} corresponding to the high- and low-spin states in Model A are called “A-HS” and “A-LS”, respectively. The ground-state spin quantum number for the chain with the even number of unit cells (3N ) 18) is depicted in Figure 3a. The chain is found to have high-spin (ST ) 9) and low-spin (ST ) 3) states. The boundary of the high- and low-spin ground states for the 3N ) 18 chain coincides with that for the 3N ) 15 chain within the accuracy of the numerical calculations. The total spin of ST ) 3N/2 in region A-HS implies that all the S ) 1/2 spins in the chain are aligned in the same direction. On the other hand, the total spin of ST ) 5/2 (3N ) 15) and 3 (3N ) 18) suggests one noncompensated spin of S ) 1/2 per unit cell: ST ) N/2. The expectation value of the biradical spin Sb2 was calculated at the first cell, i.e., the origin, of the chain. Only the biradical at the “origin” is considered, because the expectation value is independent of the cell position owing to the translational symmetry of the Hamiltonian. The expectation value of Sb2 in
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Figure 4. Ground-state phase diagram for Model A (Figure 1a; eq 1) (a) and Model B (Figure 1b; eq 4) (b). The circles and squares denote the parameters {J2/|J1|, a} with the chain length of 3N ) 15 and 18, respectively, which give discontinuous changes in the total spin quantum number ST and the biradical spin 〈Sb2〉. Regions A-HS and B-HS have a high-spin ground state with ST ) 3N/2, whereas regions A-LS and B-LS correspond to low-spin ground states with ST ) N/2, ST ) 1/2, or ST ) 0. The solid lines are to guide the eye. The dashed lines represent the parameter values, where the correlation functions (Figure 5) are calculated.
the ground state is given as
〈Sb2〉 ≡ 〈GS|(S1,b1 + S1,b2)2|GS〉
(10)
The 〈Sb2〉 values are plotted as a function of the ratio J2/|J1| and the asymmetry parameter a for 3N ) 15 and 18 in Figure 3b. For both the odd and even numbers of unit cells, nonzero magnetic moment arising from 〈Sb2〉 * 0 is predicted for arbitrary values of J2/|J1| and a. Even an Sb ) 1 (Sb(Sb + 1) ) 2) appears in region A-HS. On the other hand, an Sb ) 0 is a good quantum number only in the limit of J2/|J1| f 0 for both the even and odd numbers of the unit cells: the “pure” singlet (Sb ) 0) appears only when the biradical is isolated with the negligible intermolecular exchange interaction J2. It is worth noting that discontinuous change in 〈Sb2〉 appears at the same points of {J2/|J1|, a} as the change in the ground-state spin quantum number ST as given in Figure 3. From these calculations, the ground spin state is characterized by ST and 〈Sb2〉 and is divided into two regions, as shown in Figure 4a. In region A-HS, the chain has the high-spin state with ST ) 3N/2 and 〈Sb2〉 ) 2, while the ground state in region A-LS has ST ) N/2 and 〈Sb2〉 ≈ 0. To clarify the spin structures of the two kinds of ground states, real-space spin-spin correlation functions 〈Si‚Sj〉 were examined for the chain of 3N ) 27, which is the largest for the memory capacity of our computation. The ground-state correlation functions 〈Si‚Sj〉 were calculated by using the eigenfunction of the ground state |GS〉 obtained from the exact diagonalization of the spin Hamiltonian eq 1
〈Si‚Sj〉 ≡ 〈GS|Si‚Sj|GS〉
(11)
Two points of {J2/|J1|, a} ) {2, 0.5} and {2, 0.1} are chosen
Figure 5. Real-space spin-spin correlation function 〈Si‚Sj〉 in the chain of length 3N ) 27 calculated for the spin pairs of S1,m‚SL,b1 (“m-b1”), S1,m‚SL,b2 (“m-b2”), and S1,m‚SL,m (“m-m”) as functions of the distance L between the cell at the origin (the first cell) and the Lth cell: (a) A-HS and B-HS with {J2/|J1|, a} ) {2, 0.5}; (b) region A-LS with {J2/|J1|, a} ) {2, 0.1}; and (c) region B-LS with {J2/|J1|, a} ) {2, 0.1}.
in the parameter space as representatives of the regions A-HS and A-LS, respectively. The ground-state spin quantum number is found to be ST ) 27/2 for {J2/|J1|, a} ) {2, 0.5} and ST ) 9/2 for {J2/|J1|, a} ) {2, 0.1}, which is consistent with the results for the smaller chains of 3N ) 15 and 18, as described above. The correlation function 〈Si‚Sj〉 for the regions of A-HS and A-LS is shown in Figure 5a,b as a function of the distance L between two S ) 1/2 spins Si and Sj. In region A-HS, all the spin pairs have the 〈Si‚Sj〉 value of 0.25, which is the same as 〈Si‚Sj〉 of a pure triplet (S ) 1) in a two-spin system. This indicates that all the spins in the chain retain a parallel spin alignment, as depicted Figure 6a, affording the total spin of ST ) 3N/2. In region A-LS, the sign of 〈Si‚Sj〉 is positive between the monoradical and one of the biradical spins (referred to as “m-b1”) and between the monoradicals (“m-m”), while the other pair of the biradical and the monoradical spin (“m-b2”) is negative. In the low-spin state of region A-LS, the spin correlation decays as the distance L increases. We are, however, allowed to draw schematically the short-range spin alignment of the ground state in Figure 6b on the basis of the correlation functions in Figure 5b. This is classical or Ising-like visualization of the correlation function 〈Si‚Sj〉. This spin alignment affords the total spin quantum number of ST ) N/2 in the whole chain. (b) Model B. The total spin quantum number ST 〈deleted〉 for the ground state of Model B was calculated for the chain length of 3N ) 15 and 18 in the same way as that for Model A. The results are shown in Figure 7a as functions of the ratio J2/|J1| and the asymmetry parameter a in the Hamiltonian (eq 4). As
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Figure 8. ORTEP drawing of the zigzag chain structure of the complex 1 along the a + c direction. The trifluoromethyl groups, the methyl groups, and the hydrogen atoms are omitted for clarity.
Figure 6. Schematic drawing of the spin alignments: (a) regions A-HS and B-HS; (b) region A-LS; and (c) region B-LS.
Figure 7. Spin quantum number ST of the whole chain (a) and the expectation value of the biradical spin 〈Sb2〉 ≡ 〈(S1,b1 + S1,b2)2〉 (b) of Model B in the ground state as a function of the interaction ratio J2/ |J1| and the asymmetry parameter a. The black and red circles indicate the chain length 3N ) 15 and 18, respectively. The solid lines are to guide the eye.
found in Model A, the chain of Model B has two kinds of ground states depending on the exchange interactions. The highspin state (“B-HS”) has a spin quantum number ST ) 15/2 for 3N ) 15 and ST ) 9 for 3N ) 18, whereas the low-spin state (“B-LS”) has ST ) 1/2 for 3N ) 15 and ST ) 0 for 3N ) 18. The total spin quantum number of ST ) 3N/2 for region B-HS is the same as that of A-HS in Model A. The ST value in region B-LS is rationalized below in terms of the correlation function. The expectation values of Sb2 for 3N ) 15 and for 18 of Model B are plotted in Figure 7b. The results are almost the same as those of Model A: Discontinuous change in 〈Sb2〉 occurs at the same boundary between the B-HS and B-LS as that of
the ST values. In region B-HS, 〈Sb2〉 has a constant value of 2 (Sb ) 1). The chain of Model B has a phase diagram of the ground state similar to that of Model A, as depicted in Figure 4b. The spin correlation functions are examined for the two points of {J2/|J1|, a} ) {2, 0.5} and {2, 0.1} within the B-HS and B-LS regions, respectively. The ground-state spin quantum number is found to be ST ) 27/2 for {J2/|J1|, a} ) {2, 0.5} and ST ) 1/2 for {J2/|J1|, a} ) {2, 0.1}. Panels a and c of Figure 5 depict the correlation functions 〈Si‚Sj〉 for the regions B-HS and B-LS. The correlation functions are constant, 〈Si‚Sj〉 ) 0.25, in region B-HS, indicating that all the spins are parallel. In region B-LS, on the other hand, the 〈Si‚Sj〉 values for all the pairs (“mb1”, “m-b2”, and “m-m”) exhibit rapid decaying with an alternation of signs. The spin alignment of the ground state for region B-LS is drawn in Figure 6c on the basis of the correlation functions in Figure 5c. This spin alignment underlies the minimal spin quantum number ST ) 0 (even N) or 1/2 (odd N). The total spin quantum number ST ) 1/2 in the whole chain with an odd number of unit cells gives a product of susceptibility and temperature χT ) 0.375/N emu K mol-1 for g ) 2.00, while ST ) 0 affords χT ) 0 for an even-number chain in the limit of zero temperature. The χT value of the odd-number chain should approach zero as N is larger, merging into that of the evennumber chain. Naive extrapolation to the thermodynamic limit of N f ∞ gives a nonmagnetic ground state of the whole chain. More elaborate consideration of the ground-state properties of the chain is needed to conclude the nonmagnetic ground state in the thermodynamic limit. 3. One-Dimensional Polymeric Complex of Biradical and Cu(II). As a model system for the ferromagnetic chain of the ground-state singlet biradicals and monoradicals, a polymeric complex composed of a nitronyl nitroxide biradical bnn and Cu(II) with S ) 1/2, [Cu(hfac)2(bnn)] (1), has been designed and synthesized.13 The complex has an alternating zigzag chain structure consisting of the Cu(II) ion and bnn,13,16 as depicted in Figure 8. The structure of the complex corresponds to Model B in the limit of a f 0. The temperature dependence of magnetic susceptibility χ for 1 has been measured.13 As depicted in Figure 9, the χT value decreases gradually from room temperature down to 60 K. The χT exhibits a plateau in the temperature range of 60 to 10 K, and decreases again below 10 K. From the analysis of magnetic susceptibility using eq 4 with a ) 0, the exchange parameters have been found to be 2J1/kB ) -490 ( 10 K and 2J2/kB ) 24 ( 3 K, as shown by the solid line in Figure 9. The ratio of the intermolecular interactions is J2/|J1| ) 24/490 ) 0.049 for complex 1. The ground-state expectation value 〈Sb2〉 of the biradical spin for the interaction parameters {J2/|J1|, a} ) {0.049, 0} is calculated for the chain of 3N ) 27, giving a nonzero moment of 〈Sb2〉 ) 2.9 × 10-3. This moment brings about antiparallel cancellation of adjacent S ) 1/2 spins on Cu(II), resulting in the decrease in χT at the base temperature, which is consistent with the experimental results.13 In nonquan-
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Maekawa et al. moment arising from 〈Sb2〉 * 0 is responsible for the low-spin ground state of the whole chain in 1. For realizing the highspin state predicted in the calculations, preparation of compounds with chelating bonds between the singlet biradical and a transition metal ion is in progress. Acknowledgment. This work has been supported by Grantsin-Aid for Scientific Research from the Ministry of Education, Sports, Culture, Science and Technology, Japan. Financial support from PRESTO of the Japan Science and Technology Agency (JST) is also acknowledged. References and Notes
Figure 9. Temperature dependence of χT for the randomly oriented polycrystalline sample of complex 1. The solid line represents the theoretical values of χT calculated from eq 4 with the exchange parameters of 2J1/kB ) -490 K and 2J2/kB ) 24 K and the averaged g-factor of g ) 2.18.
tum terms, the ground-state singlet biradicals have no contribution of magnetic moment in the low-temperature limit and only the S ) 1/2 spins on Cu(II) should have been left magnetically isolated, or diluted, in the chain. The decrease in χT below the plateau region observed for the chain of 1 indicates the key feature of the alternating ferromagnetic chain system composed of the ground-state singlet biradicals and the S ) 1/2 species, i.e., the appearance of nonzero magnetic moment 〈Sb2〉 ≡ 〈(Sb1 + Sb2)2〉 * 0 on the biradical molecules. Conclusion We have examined the ground spin state of an alternating chain composed of a ground-state singlet (S ) 0) biradical and a monoradical by numerical calculations of a Heisenberg spin Hamiltonian, in which the biradical is ferromagnetically coupled with the neighboring monoradical. The elongation of the biradical spin from Sb ) 0 depending on the relative magnitude of the exchange interactions has been demonstrated by the calculation of the expectation value 〈Sb2〉 for two kinds of theoretical models, Model A and Model B, with different spatial symmetry, or topology, of the intermolecular exchange interactions. The alternating chain has been found to have two kinds of ground spin states:; a high- and a low-spin state. In the highspin state, all the spins are parallel to each other in the highspin state characterized by the spin correlation function of 〈Si‚ Sj〉 ) 0.25 for both models. The spin alignment in the low-spin state is found to be dependent on the topology of the intermolecular exchange interactions of the models. The heterospin complex 1 has been synthesized and characterized as a model for the ferromagnetic version of the generalized ferrimagnetism. The low-temperature behavior of magnetic susceptibility for 1 is explained by the calculated results. The ground state of the polymeric complex 1 belongs to the low-spin state of Model B. Nonvanishing magnetic
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