2932
J. Phys. Chem. B 2001, 105, 2932-2938
Quantum Ferrimagnetism Based on Organic Biradicals with a Spin-0 Ground State: Numerical Calculations of Molecule-Based Ferrimagnetic Spin Chains Daisuke Shiomi,† Kazunobu Sato,‡ and Takeji Takui*,‡ Department of Materials Science and Department of Chemistry, Graduate School of Science, Osaka City UniVersity, Sumiyoshi-ku, Osaka 558-8585, Japan ReceiVed: September 18, 2000; In Final Form: December 18, 2000
The possibility of a ferrimagnetic spin alignment in a novel type of alternating heterospin chain is examined in terms of a finite-size Heisenberg spin Hamiltonian. One of the two kinds of spin sites in the chain represents an organic molecule with two S ) 1/2 spins. The two spins interact antiferromagnetically to give a biradical molecule with a singlet (S ) 0) ground state, which is coupled with adjacent S ) 1/2 molecules by the intermolecular antiferromagnetic interactions. It is shown that S ) 0 is not a good quantum number for describing the biradical embedded in the exchange-coupled chain and that the chain has a ferrimagnetic ground state with S ) N/2 (N ) the number of repeating units) or a low-spin state of S ) 1/2 or S ) 0, depending on the ratios of the intra- and intermolecular interactions. An extremely short spin-spin correlation length features in the low-spin state, indicating the occurrence of a quantum mechanical disorder resulting from a frustration effect inherent in the chain. The generalized ferrimagnetic spin alignment proposed in quantum terms uncovers a new category of molecule-based magnetic materials.
Introduction Extensive studies have been done on organic molecule-based ferromagnets and other molecular functionality magnetics in recent years.1 Together with organic ferromagnetic substances based on purely ferromagnetic intermolecular interactions, ferrimagnets have been attracting attention as one of the facile approaches to organic ferromagnets after Buchachenko’s proposal for organic ferrimagnetics in 1979.2 The concept of organic ferrimagnetics is based on the tendency for organic open-shell molecules to have antiferromagnetic intermolecular interactions. The antiferromagnetic interactions would bring about antiparallel spin alignment between neighboring molecules with different magnetic moments to result in a possible ordered state with net magnetization. Magnetic phase transition to a ferrimagnetic ordered state, however, has not been documented so far in organic molecular crystalline solids. This presents a remarkable contrast to the fact that organic molecular ferromagnets1,3 and inorganic metal-based ferrimagnets4 have been discovered. Molecule-based ferrimagnetics has been a long-standing issue in terms of a materials challenge in chemistry. In addition, from a theoretical point of view, the molecular ferrimagnetics gives new aspects or categories in molecular science, which have been overlooked in atom- or metal-based magnetics studied so far. Nevertheless, the underlying idea of organic molecule-based ferrimagnets is classical and crude in quantum terms,2 requiring more elaborate approaches in view of the materials development in molecule-based magnetics.5-7 Low-dimensional quantum systems of antiferromagnetically coupled spins have been attracting considerable interests in recent years. One of the new topics of the quantum spin systems is “mixed spin chains” or “heterospin chains”, which consist
of two kinds of spins with different spin quantum numbers S,8-10 e.g., S ) 1 and S ) 1/2. The numerical calculations and theories are applicable to real heterospin compounds of transition metal ions such as Cu2+ (S ) 1/2) and Ni2+ (S ) 1).4 The preceding theoretical studies on the heterospin systems, however, are invalidated for the molecular assemblages as models for organic ferrimagnetics. Let us point out the essential features of organic molecule-based magnetics, which make the previous theories based on the inorganic heterospin systems invalid. In organic molecule-based magnetic materials, spin density is distributed over many atomic sites in an open-shell molecule, and hence, the intermolecular interaction has a multicentered or multicontact nature. Furthermore, in most cases, intramolecular interactions in stable organic S > 1/2 molecules are in the same order of magnitude as the intermolecular interactions in crystalline solid states. In such cases, the magnetic degree of freedom within the S > 1/2 molecules and spatial symmetry of intermolecular exchange interaction should be taken into account in considering magnetic properties of molecular assemblages containing S > 1/ molecules. Both of the features are negligible in metal-based 2 ferrimagnets, in which the internal magnetic structure of S > 1/ atoms or ions can be neglected. 2 In our previous papers,5-7 we discussed the possibility of ferrimagnetic order occurring in organic molecular crystals both from numerical calculations of a Heisenberg spin Hamiltonian of an S ) 1/2 and S ) 1 alternating chain5,6 and from crucial experiments7 on only one model compound reported so far.11 The spin Hamiltonian examined in the studies is given as N
H(J1 > 0,J2 < 0) )
[-2J1Si,b1‚Si,b2 - 2J2(Si,b2‚Si,m + ∑ i)1
Si,m‚Si+1,b2) - 2aJ2(Si,b1‚Si,m + Si,m‚Si+1,b1)] (1) * To whom correspondence should be addressed. E-mail:
[email protected]. Phone: +81-6-6605-2605. Fax: +81-6-6605-3137. † Department of Materials Science. ‡ Department of Chemistry.
where Si,b1, Si,b2, and Si,m denote the spin-1/2 operators and N is the number of unit cells. In Figure 1a is schematically shown
10.1021/jp003348p CCC: $20.00 © 2001 American Chemical Society Published on Web 03/23/2001
Quantum Ferrimagnetism Based on Organic Biradicals
J. Phys. Chem. B, Vol. 105, No. 15, 2001 2933 for arbitrary parameters, J1, J2, and a. It has been found from an exact numerical diagonalization of the Hamiltonian that the expectation value of S2b for the ground state
〈S2b〉 ) 〈(S1,b1 + S1,b2)2〉
Figure 1. Schematic diagram for the Heisenberg spin Hamiltonian of the ferrimagnetic chain and the corresponding hypothetical molecular assemblage. (a) Heisenberg spin Hamiltonian of the ferrimagnetic chain with N of repeating units. The rounded rectangles are the biradical molecules with two unpaired electron spins (Si,b1 ) 1/2 and Si,b2 ) 1/2), which interact 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 thick solid line is the intramolecular interaction J1, whereas the thin solid and dashed lines are the intermolecular interactions J2 and aJ2. The Hamiltonian with the positive and negative J1 is given in eqs 1 and 1′, respectively. (b) A hypothetical molecular assemblage of a nitronyl nitroxide-based biradical 115 with a singlet (S ) 0) ground state and an S ) 1/2 monoradical 2.16 The nitroxide groups are denoted by the closed circles. (c) Model compounds 1 and 2. The methyl groups and the hydrogen atoms in the phenyl rings in 1 and 2 are omitted for clarity in (b).
the Hamiltonian (eq 1). Si,b1 and Si,b2 are coupled by the intramolecular ferromagnetic exchange interaction J1 > 0 to give a molecular site of biradical or the two-spin site. The biradical spins are coupled with a neighboring monoradical spin of Si,m ) 1/2 by the antiferromagnetic exchange interactions J2 < 0 and aJ2 < 0 (0 e a e 1). The spatial symmetry of the intermolecular exchange interactions is signified by the parameter a in the Hamiltonian. This is the simplest model Hamiltonian for molecular ferrimagnets possessing the above features; a biradical molecule with Si,b1 + Si,b2 retaining the intramolecular magnetic degree of freedom J1 * ∞ and the twofold intermolecular interactions with the definable symmetry. From a quantum Monte Carlo simulation of temperature dependence of magnetic susceptibility χ, the ground state of the Hamiltonian (eq 1) has been found to have the spin quantum number of S ) N/2.6 The spin quantum number implies the collinear ferrimagnetic spin alignment of Lieb-Mattis’s type12 in the ground state. Another important issue for the model Hamiltonian (eq 1) is incommutability of the biradical spin S2b ≡ (S1,b1 + S1,b2)2 and the Hamiltonian H(J1 > 0)6
[H(J1 > 0), S2b] * 0
(2)
(3)
takes a definite value of 2 (Sb ) 1), only when the symmetry parameter a is equal to 1.6 For arbitrary values of a, the spin magnetic moment of the biradical has been shown to be reduced by quantum mechanical fluctuation because of the incommutability. Thus, the spin contraction results from the asymmetric intermolecular antiferromagnetic interactions. The spin contraction is also found in the metal-based ferrimagnetic chains.13,14 The distinctiveness of the contraction in the organic moleculebased chain is that its physical origin is the spatial symmetry of the intermolecular magnetic interactions. An S ) 1 is not a good quantum number in describing magnetic properties of a ground-state triplet biradical embedded in the molecular assemblages of exchange-coupled heterospin systems. From the viewpoint of molecular and crystal designing for organic molecule-based magnetism, the spin contraction due to the incommutability (eq 2) deserves consideration. The biradical spin S2b is incommutable with H for J1 < 0 as well. Therefore, Sb ) 0 should not be a good quantum number, when a biradical with a singlet (S ) 0) ground state is embedded in the exchangecoupled molecular assemblages. The ground-state singlet biradical should recover the contribution to the bulk magnetization; 〈S2b〉 * 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|. The main purpose of the present paper is to explore the possibility of ferrimagnetism occurring in a molecular alternating chain of a ground-state singlet biradical and a monoradical with S ) 1/2 as shown in Figure 1b,c. The figures depict a hypothetical example of the alternating chain, which is composed of a ground-state singlet biradical, p-phenylene bis(Rnitronyl nitroxide) 1,15 and an S ) 1/2 monoradical, phenyl nitronyl nitroxide 2.16 This quantum spin chain is expressed by the spin Hamiltonian, H(J1 < 0), called eq 1′, with J1 < 0, J2 < 0, and aJ2 < 0 (0 e a e 1).17 The spin chain of H(J1 < 0), eq 1′, is distinguished from ordinary S ) 1/2 antiferromagnetic chains by the differing number of spin sites in the two sublattices of the chains; one in the sublattice composed of the monoradicals and two in the sublattice of the biradicals. The feature brings about a frustration effect in the unit cell of the chain, because all of the interactions within and between the sublattices are antiferromagnetic. It is shown that the frustration gives groundstate spin quantum numbers proportional to the size of the system or the chain length, which correspond to a diverging product of χT at T ) 0. A generalized concept of organic molecule-based ferrimagnetism is proposed in quantum terms from the calculations of the spin Hamiltonian with J1 < 0 (eq 1′). Methods of Computation The exact numerical diagonalization of the Hamiltonian matrix was done by the Lanczos method18 and the Householder method. The largest system examined in this study consists of nine repeating units of a biradical and a monoradical (twentyseven S ) 1/2 spins) with the matrix dimension of 227 × 227 ) 134 217 728 × 134 217 728. 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 ) 20 058 300 × 20 058 300. In the calculations, a
2934 J. Phys. Chem. B, Vol. 105, No. 15, 2001
Shiomi et al.
parallelization procedure is used on an NEC supercomputer SX-5 at the Computer Center of the Institute for Molecular Science and on a HITACHI supercomputer SR8000 at the Computer Center of the University of Tokyo. The periodic boundary condition SN+1,I ) S1,I (I ) b1, b2, and m) is imposed with 3N e 27 of repeating units in the Hamiltonian. Results and Discussion 1. Three-Spin System of a Biradical-Monoradical Dimer. At first, a molecular dimer of a biradical molecule with the ground-state singlet (S ) 0) and a monoradical molecule with S ) 1/2 is investigated as a minimal model system of interacting open-shell molecules with different spin quantum numbers. The dimer is regarded as a unit cell of the chain Hamiltonian (eq 1′). The spin Hamiltonian of the dimer Hdimer with three unpaired electrons is written as
Hdimer ) -2J1Sb1‚Sb2 - 2J2Sb2‚Sm - 2aJ2Sm‚Sb1
(4)
where the notation is the same as that of the chain Hamiltonian (eqs 1 and 1′). The Hamiltonian Hdimer is written in the ket space spanned by the set of direct product {|σb1σb2σm〉} defined as
Sz1|σb1σb2σm〉 ) (1/2|σb1σb2σm〉
(5)
where σI (I ) b1, b2, and m) is the spin function R or β. The Hamiltonian (eq 4) commutes with the total spin operator S2T ≡ (Sb1 + Sb2 + Sm)2 and that of the z component SzT ≡ Szb1 + Szb2 + Szm. Thus, the eigenstates of the Hamiltonian consisting of one quartet (ST ) 3/2) and two doublet (ST ) 1/2) states
|φ〉 )
∑R cR|σb1σb2σm〉R
(6)
are labeled by ST and MS
S2T|φ〉 ) ST(ST + 1)|φ〉, (ST ) 3/2, 1/2)
(7)
SzT|φ〉 ) MS|φ〉, (MS ) (3/2, (1/2)
(8)
The ground-state belongs to one of the subspaces with ST ) 1/2 for J1 < 0, J2 < 0, and 0 e a e 1. The spin operator of the biradical site S2b ≡ (Sb1 + Sb2)2 does not commute with the Hamiltonian (eq 4) owing to the intermolecular exchange interactions J2 and aJ2. The expectation value of S2b in the ground state |GS〉 is analytically given as
Pb ≡ 〈GS|S2b|GS〉 )1-
(a + 1)(J2/J1) + 2 2x(a2 - a + 1)(J2/J1)2 - (a + 1)(J2/J1) + 1
{
Region A: -3/4 e 〈Sb1‚Sb2〉 < 0, 0 e Pb < 3/2
(11a)
Region B: 0 < 〈Sb1‚Sb2〉 e 1/4, 3/2 < Pb e 2
(11b)
The boundary between A and B corresponds to apparently noncorrelating spins, 〈Sb1‚Sb2〉 ) 0 or Pb ) [1/2(1/2 + 1)]2 ) 3/ , in the biradical molecule. P approaches the limiting value 2 b of 3/2 when J2/J1 f ∞ and a f 0. The boundary is derived by putting Pb ) 3/2 in eq 9 as
J2/J1 ) 1/a
(9)
In the parameter space of {J2/J1,a}, Sb ) 0 (Pb ) 0) is a good quantum number only in the limits of J2/J1 f 0 or a f 1 as shown in Figure 2. A spin “elongation” occurs for arbitrary values of {J2/J1,a}. The parameter space of {J2/J1,a} is divided into two regions according to the sign of 〈Sb1‚Sb2〉:
(13)
where σI ) R or β (I ) 1,b1; 1,b2; ‚‚‚; N,m). All of the calculations described below are made for a spin subspace with the z component of the total spin 3N
( (10)
(12)
In the assemblage of open-shell molecules with different spin quantum numbers, a “singlet biradical” with zero magnetic moment is realized only for sufficiently large intramolecular exchange interaction (J2/J1 f 0) or completely symmetric intermolecular interactions (a f 1). Neither condition is fulfilled in the usual organic molecule-based assemblages. Biradicals with a singlet (S ) 0) ground state would seemingly have no contribution to the magnetization at low temperatures, kBT < |J1|. From the above consideration, this is found to be oversimplified for the heterospin assemblages. 2. Extended Alternating Chains of a Biradical and a Monoradical. (a) Spin Quantum Number of the Ground State. Only the ground-state properties of the extended chain are elucidated here. The total spin quantum number ST of the whole chain in the ground state is calculated for the chain Hamiltonian (eq 1′) with N of repeating units. The Hamiltonian is, as eq 5 for the three-spin system, written in the ket space spanned by the set of direct product {|σ1,b1σ1,b2‚‚‚σN,m〉} defined as
SzI |σ1,b1σ1,b2‚‚σN,m〉 ) (1/2|σ1,b1σ1,b2‚‚‚σN,m〉
The Pb value is shown in Figure 2 as a function of J2/J1 and a. Pb has a singular point at J2/J1 ) 1 along the a ) 1 line:
0 (0 e J2/J1 < 1) Pb ) 2 (J2/J1 > 1)
Figure 2. Expectation value of the biradical spin Pb ) 〈(Sb1 + Sb2)2〉 in the biradical-monoradical dimer as a function of the interaction ratio J2/J1 and the asymmetry parameter a.
∑I SzI)|σ1,b1σ1,b2‚‚‚σN,m〉 ) MS|σ1,b1σ1,b2‚‚‚σN,m〉
(14)
where MS ) 0 for even 3N and MS ) +1/2 for odd 3N. The eigenvector of the ground state
|GS〉 )
∑R cR|σ1,b1σ1,b2‚‚‚σN,m〉R
(15)
is calculated from the numerical diagonalization of the Hamil-
Quantum Ferrimagnetism Based on Organic Biradicals
J. Phys. Chem. B, Vol. 105, No. 15, 2001 2935
Figure 3. (a) Ground-state spin quantum number ST of the chain in Figure 1 calculated for the chain length 3N ) 15. (b) Ground-state expectation value of the biradical spin Pb ) 〈(S1,b1 + S1,b2)2〉 in the extended chain of 3N ) 15 as a function of the interaction ratio J2/J1 and the asymmetry parameter a. The solid lines are to guide the eye.
Figure 4. (a) Ground-state spin quantum number ST of the chain in Figure 1 calculated for the chain length 3N ) 18. (b) Ground-state expectation value of the biradical spin Pb ) 〈(S1,b1 + S1,b2)2〉 in the extended chain of 3N ) 18 as a function of the interaction ratio J2/J1 and the asymmetry parameter a. The solid lines are to guide the eye.
tonian (eq 1′). The spin quantum number ST of the whole chain is calculated from |GS〉:
ground-state spin quantum number of ST ) 3 for 3N ) 18 is indicative of the relation eq 17. These results indicate that the Hamiltonian eq 1′ has the ferrimagnetic ground state with the spin quantum number ST ) N/2, even when the intramolecular exchange interaction is antiferromagnetic. (b) Expectation Value of the Spin at the Biradical Site. The ground-state expectation value of the biradical spin at the “origin” of the chain with N of repeating units
ST(ST + 1) ) 〈GS|ST2|GS〉
(16)
In Figure 3a is shown the ground-state spin quantum number ST calculated for the chain of length 3N ) 15 as a function of the ratio J2/J1 and the asymmetry parameter a in the Hamiltonian (eq 1′). A high-spin ground state of ST ) 5/2 is found for a wide range of the parameters, whereas a low-spin ground state with ST ) 1/2 appears in an intermediate region of {J2/J1,a} as shown in Figure 3a. No intermediate spin quantum numbers 1/2 < S < 5/2 are found in the parameter space of 0 < J2/J1 e 4 and 0 e a < 1. The high-spin of ST ) 5/2 for the chain length of 3N ) 15 suggests one noncompensated spin of S ) 1/2 per unit cell:
ST ) N(1 - 1/2)
(17)
The ground-state spin quantum number ST for the chain with the even numbers of unit cells (3N ) 18) is depicted in Figure 4a, which is calculated in the same way as that for the oddnumber chain (3N ) 15). A high-spin ground state of ST ) 3 appears in a wide range of the parameters {J2/J1, a}, whereas an S ) 0 ground state is found for an intermediate region. The
Pb(N) ) 〈GS|(S1,b1 + S1,b2)2|GS〉
(18)
is calculated in order to examine the elongation effect in the extended chain. Only the biradical at the “origin” is considered, because Pb(N) is independent of the cell position owing to the translational symmetry of the Hamiltonian. The Pb(N) values are plotted as functions of the ratio J2/J1 and the asymmetry parameter a for 3N ) 15 in Figure 3b and for 3N ) 18 in Figure 4b. For both the even and odd numbers of the unit cells, the Pb(N) value approaches zero in the limit of zero-interaction between the biradical and the monoradical (J2/J1 f 0), which corresponds to a “pure” singlet appearing on the isolated biradical. On the other hand, the biradical spin behaves as an S ) 1 (Pb f 2), when the intermolecular interaction is larger than the intramolecular one and the spatial symmetry of the intermolecular interactions is high (a f 1).
2936 J. Phys. Chem. B, Vol. 105, No. 15, 2001
Figure 5. Ground-state phase diagram of the chain in Figure 1. The circles and squares denote the parameters {J2/J1,a} in the Hamiltonian (eq 1′) with the chain lengths of 3N ) 15 and 18, which give discontinuous changes in the total spin quantum number ST and the biradical spin 〈(S1,b1 + S1,b2)2〉. Regions A′ and B′ have a high-spin ground state of ST ) N/2 (3N ) 15 and 18), whereas region C has a low-spin ground state of ST ) 1/2 (3N ) 15) or ST ) 0 (3N ) 18). The solid lines are to guide the eye. The dashed line represents the boundary of the regions with 〈Sb1‚Sb2〉 > 0 and 〈Sb1‚Sb2〉 < 0, or Pb > 3/2 and Pb < 3/2, in the three-spin system of the dimer.
The appearance of Pb f 0 and Pb f 2 in the two limiting regions of the parameters {J2/J1,a} is similar to the case of the biradical-monoradical dimer as shown in Figure 2. A crucial difference between the chain and the dimer is found in the intermediate region of the parameters: a stationary behavior of Pb(N) is found between two discontinuous changes in Pb for the chain. It is worth noting that the discontinuity in Pb appears at the same points of {J2/J1,a} as the change in the groundstate spin quantum number ST as given in Figures 3 and 4. Thus the parameter space of {J2/J1,a} is divided into three regions as depicted in Figure 5: In region A′, the biradical spin elongating from Sb ) 0 gives a high-spin state with ST ) N/2. In region B′, the biradical spin contracting from Sb ) 1 gives another type of high-spin state with ST ) N/2. In the intermediate region C, the ground state has a low-spin of ST ) 0 or ST ) 1/2 depending on the chain length N.19 (c) Real-Space Spin-Spin Correlation Function. To clarify the spin structures of the two kinds of high-spin states (A′ and B′) and the low-spin state (C), real-space spin-spin correlation is examined for the ring of 3N ) 27, which is the largest for the memory capacity of our computation. Three points of {J2/ J1,a} ) {1,0.2}, {2,0.7}, and {1,0.7} are chosen in the parameter space as representatives of region A′, B′, and C, respectively. The ground-state spin quantum number is found to be ST ) 9/2 for {J2/J1,a} ) {1,0.2} (region A′) and {2,0.7} (region B′), whereas ST ) 1/2 for {J2/J1,a} ) {1,0.7} (region C). This finding is consistent with the results (eq 17) for the smaller chains of 3N ) 15 and 3N ) 18 as described above. The correlation functions 〈Si‚Sj〉 are calculated by the exact diagonalization. In Figure 6 are depicted the ground-state correlation functions 〈Si‚ Sj〉 calculated for the three points of {J2/J1,a} as functions of the distance L between two S ) 1/2 spins Si and Sj. In region A′, the correlation is positive between the monoradical and one of the biradical spins (referred to as “m-b1”) and between the monoradicals (“m-m”), whereas the other pair of the biradical spin and the monoradical spin (“m-b2”) is negative. Although long-range correlations are not expected at all in the onedimensional chain under study, the short-range spin alignment of the ground state in region A′ is schematically drawn in Figure
Shiomi et al. 7a on the basis of the correlation functions in Figure 6a. 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. A short-range scheme of spin alignment for the other high-spin region B′ is obtained in the same way as that of A′, and it is drawn in Figure 7b. Both the spin structures are regarded as a generalized spin alignment of the collinear ferrimagnetic state. The correlation function of region C is quite different from those of A′ and B′. The functions for the spin pairs of m-m, m-b1, and m-b2 exhibit extremely rapid decaying as depicted in Figure 6c.20 The short correlation length at zero temperature suggests a quantum mechanically disordered ground state because of quantum fluctuation, which underlies the minimal spin quantum number ST ) 0 or ST ) 1/2. (d) Temperature Dependence of Magnetic Susceptibility. The drastic change in the ground-state spin multiplicity depending on the ratios of the interaction parameters {J2/J1,a} can be demonstrated in the temperature dependence of magnetic susceptibility χ. χT vs T curves are simulated by diagonalizing exactly the spin Hamiltonian with J1 < 0 (eq 1′). Figure 8 shows the calculated χT as a function of the reduced temperature kBT/ |J1| for the chain length of 3N ) 9 and 12.21 The parameters {J2/J1,a} in region C afford the vanishing χT in the limit of T ) 0: the even-number chain of 3N ) 12 has χT(T f 0) ) 0, and the odd-number chain of 3N ) 9 exhibits χT(T f 0) ) 0.0417 emu K spin-1 ()0.375 emu K for the whole chain with S ) 1/2), assuming the g factor of g ) 2.00. The χT(T f 0) value of 0.375/3N emu K spin-1 for the odd-number chain should approach zero as N is larger, merging into that of the even-number chain. On the other hand, regions A′ and B′ give a minimum in χT which is similar to those observed in ordinary ferrimagnetic spin systems.4-6 The low-temperature limits of χT for A′ and B′ are explained by the Curie law for the spin quantum number S ) N/2 of the whole chain in the ground state
χT(T f 0) )
2 2 2 2 1 NAg µB S(S + 1) NAg µB (N + 1) ) (19) 3N 3kB 36kB
where NA, g, µB, and kB stand for Avogadro constant, g factor, Bohr magneton, and Boltzmann constant, respectively. From eq 19, the χT(T f 0) values are calculated to be 0.208 emu K spin-1 ()1.88 emu K in the whole chain with S ) 3/2) and 0.250 emu K spin-1 ()3.00 emu K for S ) 2). These results are fully consistent with the ground-state spin quantum numbers for the longer chains 3N ) 15, 18, and 27 as described above. Conclusion A novel type of mixed-spin chain, or heterospin chain, of antiferromagnetically coupled quantum spin systems has been examined by numerical calculations of a Heisenberg Hamiltonian. Two S ) 1/2 spins are coupled by the finite antiferromagnetic interaction to give the biradical site with the singlet (S ) 0) ground state. The biradical spin interacts with an adjacent S ) 1/2 spin of the monoradical molecule to form the alternating biradical-monoradical molecular chain. The organic assemblages composed of molecules with different spin quantum numbers are characterized by the magnetic degree of freedom in the S * 1/2 molecules and the multicentered or multicontact nature of the intermolecular interactions. These two features can be neglected in both atom- and inorganic metal-based ferrimagnets and thus have never been taken into consideration
Quantum Ferrimagnetism Based on Organic Biradicals
J. Phys. Chem. B, Vol. 105, No. 15, 2001 2937
Figure 6. Real-space spin-spin correlation function 〈Si‚Sj〉 in the chain of the 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. The interaction parameters are {J2/J1,a} ) {1,0.2} (a), {2,0.7} (b), and {1,0.7} (c), respectively.
features. The occurrence of the quantum mechanically disordered low-spin state between the two high-spin states suggests a frustration effect inherent in the unit cell of the chain. In view of the materials development in organic moleculebased magnetics, the generalized ferrimagnetic spin alignment proposed in the present calculations uncovers a new category of molecular magnetic materials, emphasizing the potential of organic molecular systems possessing quantum nature in terms of magnetism.22 Calculations on low-lying excited states and possible energy-gap formation are now in progress in order to examine the physical origin of the low-spin ground state.
Figure 7. Schematic drawings of the spin alignments of the ferrimagnetic high-spin states in regions A′ (a) and B′ (b) obtained from the correlation functions in Figure 6.
Acknowledgment. This work has been supported by Grantsin-Aid for Encouragement of Young Scientists (Nos. 07740468, 07740553, 08740462, 09740528, 10740275, and 12740385) and Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture, Japan. The authors thank the Computer Center of the Institute for Molecular Science and the Computer Center of the University of Tokyo for the use of the supercomputers and the computation facilities. References and Notes
Figure 8. Calculated products of χT as functions of the reduced temperature kBT/|J1| for the chains of 3N ) 12 and 3N ) 9. The solid, the dash-dotted, and the short-dashed lines represent the χT values calculated with the parameters {J2/J1,a} ) {1,0.2} (region A′), {2,0.7} (region B′), and {1,0.7} (region C), respectively.
so far. Thus they have been overlooked thus far in the recently developing field of molecule-based magnetics as well. The biradical-monoradical molecular chain has been found to have the two kinds of ferrimagnetic state with S ) N/2 (N ) the number of repeating units), depending on the relative ratios of the intramolecular and the intermolecular interactions. Intermediate ratios of the interactions afford a low-spin ground state, where an extremely short spin-spin correlation length
(1) For reviews of molecule-based magnetism, see: (a) Gatteschi, D., Kahn, O., Miller, J. S., Palacio, F., Eds.; Molecular Magnetic Materials; Kluwer Academic: Dordrecht, The Netherlands, 1991. (b) Iwamura, H.; Miller, J. S., Eds.; Mol. Cryst. Liq. Cryst. 1993, 232, 233. (c) Miller, J. S.; Epstein, A. J., Eds.; Mol. Cryst. Liq. Cryst. 1995, 271-274. (d) Itoh, K.; Miller, J. S.; Takui, T., Eds.; Mol. Cryst. Liq. Cryst. 1997, 305, 306. (e) Kahn, O., Eds.; Mol. Cryst. Liq. Cryst. 1999, 334, 335. (f) Lahti, P. M., Ed.: Magnetic Properties of Organic Materials; Marcel Dekker: New York, 1999. (g) Itoh, K., Kinoshita, M., Eds.; Molecular Magnetism; Gordon and Breach: Amsterdam, The Netherlands, (Kodansha, Tokyo) 2000. (2) Buchachenko, A. L. Dokl. Akad. Nauk Eng. Ed. 1979, 244, 107. (3) (a) Tamura, M.; Nakazawa, Y.; Shiomi, D.; Nozawa, K.; Hosokoshi, Y.; Ishikawa, M.; Takahashi, M.; Kinoshita, M. Chem. Phys. Lett. 1991, 186, 401. (b) Nakazawa, Y.; Tamura, M.; Shirakawa, N.; Shiomi, D.; Takahashi, M.; Kinoshita, M.; Ishikawa, M. Phys. ReV. B 1992, 46, 8906. (4) For example, see: (a) Kahn, O.; Pei, Y.; Verdaguer, M.; Renard, J. P.; Sletten, J. J. Am. Chem. Soc. 1988, 110, 782. (b) Caneschi, A.; Gatteschi, D.; Renard, J. P.; Rey, P.; Sessoli, R. Inorg. Chem. 1989, 28, 1976. (c) Stumpf, H. O.; Ouahab, L.; Pei, Y.; Bergerat, P.; Kahn, O. J. Am. Chem. Soc. 1994, 116, 3866. (d) Hagiwara, M.; Minami, K.; Narumi, Y.; Tatani, K.; Kindo, K. J. Phys. Soc. Jpn. 1998, 67, 2209. (e) Hagiwara, M.; Narumi, Y.; Minami, K.; Tatani, K.; Kindo, K. J. Phys. Soc. Jpn. 1999, 68, 2214. (5) Shiomi, D.; Nishizawa, M.; Sato, K.; Takui, T.; Itoh, K.; Sakurai, H.; Izuoka, A.; Sugawara, T.; J. Phys. Chem. B 1997, 101, 3342. (6) Shiomi, D.; Sato, K.; Takui, T. J. Phys. Chem. B. 2000, 104, 1961. (7) Nishizawa, M.; Shiomi, D.; Sato, K.; Takui, T.; Itoh, K.; Sawa, H.; Kato, R.; Sakurai, H.; Izuoka, A.; Sugawara, T. J. Phys. Chem. B 2000, 104, 503. (8) For example, see: Yamamoto, S. Phys. ReV. B 1999, 59, 1024 and references therein. (9) Pati, S. K.; Ramasesha, S.; Sen, D. Phys. ReV. B 1997, 55, 8894 and references therein.
2938 J. Phys. Chem. B, Vol. 105, No. 15, 2001 (10) Niggemann, H.; Uimin, G.; Zittartz, J. J. Phys. Condens. Matter. 1997, 9, 9031. (11) Izuoka, A.; Fukada, M.; Kumai, R.; Itakura, M.; Hikami, S.; Sugawara, T. J. Am. Chem. Soc. 1994, 116, 2609. (12) Lieb, E.; Mattis, D. J. Math. Phys. 1962, 3, 749. (13) Kolezhhuk, A. K.; Mikeska, H.-J.; Yamamoto, S. Phys. ReV. B 1997, 55, R3336. (14) Srinivasan, B.; Kahn, O.; Ramasesha, S. J. Chem. Phys. 1998, 109, 5770. (15) (a) Shiomi, D.; Tamura, M.; Sawa, H.; Kato, R.; Kinoshita, M. Synth. Met. 1993, 55-57, 3279. (b) Caneschi, A.; Chiesi, P.; David, L.; Ferraro, F.; Gatteschi, D.; Sessoli, R. Inorg. Chem. 1993, 32, 1445. (16) Ullman, E. F.; Osiecki, J. H.; Boocock, D. G. B.; Darcy, R. J. Am. Chem. Soc. 1972, 94, 7049. (17) Alternation of the intermolecular interactions along the chain is neglected in the present study. The influence of the alternation between the Si,b2‚Si,m and Si,m‚Si+1,b2 pairs and between the Si,b1‚Si,m and Si,m‚Si+1,b1 pairs in the Hamiltonian (eq 1′) on the ground-state spin structure will be discussed in a separate paper. (18) We have modified the routines of the Lanczos method in the program package TITPACK, version 2 supplied by H. Nishimori at Tokyo Institute of Technology. (19) In the thermodynamic limit of N f ∞, a phase transition between the high-spin and low-spin states is expected to occur at the boundaries
Shiomi et al. between A′ and C or C and B′, which is associated with the discontinuous change in ST and 〈S2b〉. This is not a conventional temperature-driven phase transition but an interaction-driven quantum phase transition. Alternatively, a spontaneous breakdown of symmetry in the spin system is expected in the thermodynamic limit. More elaborate consideration of the ground-state properties of the chain is needed to clarify the occurrence of the phase transition. (20) Some incommensurate order or spiral order may be present in region C, which will be clarified by exploring the spin Hamiltonian in the momentum space, not in the real space. Such numerical studies are now in progress. (21) We have calculated in ref 6 the temperature dependence of χT for the chain Hamiltonian of J1 > 0 with the chain length of up to 3N ) 60 by use of a quantum Monte Carlo simulation method. The quantum Monte Carlo method is, however, known to have a problem in frustrated spin systems, introducing numerical errors. We are allowed only to treat small spin systems of 3N e 12, for which an exact full-diagonalization of the spin Hamiltonian can be made. (22) For example, application of pressure to a molecular crystalline solid of heteromolecular assemblages may influence the ratios of the intramolecular and intermolecular interactions, resulting in a conversion of the ground-state spin states of the solid.
