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1.
Tetranuclear Cu(II) complexes of N-(2-hydroxymethylphenyl)salicylideneimine (H2L1-H) and its homologues (5-CH3: H2L1-Me, 5-Cl: H2L1-Cl), [Cu(L1-H)]4 · 3H2O (1), [Cu(L1-Me)]4 · 2CH2Cl2 (2), and [Cu(L1-Cl)]4 · 2CH2Cl2 (3), have been characterized by X-ray crystal structure analyses and magnetic measurements. The structure analyses revealed that the complexes 1-3 have a defective double-cubane tetra copper(II) core connected by μ3-alkoxo bridges. The intramolecular Cu?Cu distances are in the range from 5.251(2)-5.256(3) Å for the longest to 3.0518(9)-3.092(2) Å for the shortest. Each Cu(II) ion has a square-pyramidal geometry and the dihedral angles between adjacent Cu(II) basal planes are almost right angles. Magnetic measurements of the present complexes indicate that weak antiferromagnetic interactions (J=−15 to −19 cm−1) between neighboring copper(II) ions are dominant in these tetracopper cores.  相似文献   

2.
A tetranuclear Cu(II) complex [Cu4L4(H2O)4](ClO4)4 has been synthesized using the terdentate Schiff base 2-(pyridine-2-yliminomethyl)-phenol (HL) (the condensation product of salicylaldehyde and 2-aminopyridine) and copper perchlorate. Chemical characterizations such as IR and UV/Vis of the complex have been carried out. A single-crystal diffraction study shows that the complex contains a nearly planar tetranuclear core containing four copper atoms, which occupy four equivalent five-coordinate sites with a square pyramidal environment. Magnetic measurements have been carried out over the temperature range 2-300 K and with 100 Oe field strengths. Analysis of magnetic susceptibility data indicates a strong antiferromagnetic (J1 = −638 cm−1) exchange interaction between diphenoxo-bridged Cu(II) centers and a moderate antiferromagnetic (J2 = −34 cm−1) interaction between N-C-N bridged Cu(II) centers. Magnetic exchange interactions (J’s) are also discussed on the basis of a computational study using DFT methodology. The spin density distribution (singlet ground state) is calculated to visualize the effect of delocalization of spin density through bridging groups.  相似文献   

3.
A novel binuclear copper(II) complex [Cu2L(μ-SO4)](PF6)2 (1) (L = 3,5-bis (bis(pyridine-2-ylmethyl)amino)methyl)-4H-1,2,4-triazol-4-amine) has been synthesized and structurally characterized. X-ray structure shows that the two copper(II) atoms are bridged by one bidentate sulfate ion and the 1,2,4-triazole ring of L with Cu1?Cu2 distance of 4.404 Å. Each copper(II) center has a distorted trigonal-bipyramidal configuration. Variable-temperature magnetic susceptibility studies (2-300 K) indicate the existence of weak antiferromagnetic coupling between the copper(II) ions in complex 1. The interaction of complex 1 with calf thymus DNA (CT-DNA) has been studied by UV absorption, fluorescence spectroscopy, circular dichroism spectroscopy, viscosity and cyclic voltammetry. Furthermore, complex 1 was able to promote single and double strand DNA cleavage in both aerobic and anaerobic conditions, the pseudo-Michaelis-Menten kinetic parameters kcat = 2.58 h−1 and Km = 1.2 × 10−4 M were obtained for 1. The hydrolytic cleavage of DNA by the complex was supported by the evidence from free radical quenching, anaerobic experiment, thiobarbituric acid-reactive substances (TBARS) assay.  相似文献   

4.
Reaction of Cu(ClO4)2 · 6H2O and pyrazine 2,3-dicarboxylate (pzdc) in aqueous ammonia medium results [Cu(pyrazine 2,3-dicarboxylate)(H2O)2] · H2O (1). The X-ray single crystal structure reveals that the compound is a 1D polymeric sinusoidal infinite chain which through intra- and inter-molecular hydrogen bonding interactions, involving lattice and coordinated water molecules with dicarboxylate oxygens and pyrazine nitrogens, gives rise to a 3D architecture. The variable temperature magnetic measurements show weak antiferromagnetic interactions between the Cu(II) centers. The best fit parameters through the typical equation for a uniform copper (II) chain are: J=−0.25 cm−1, g=2.17, R=1.3×10−6. The EPR spectrum does not alter with temperature (from r.t. to 4 K). The spectra are typical for square-pyramidal geometry of copper(II) ions, g=2.24 and g=2.10 (average g=2.15, in good agreement to the value obtained by susceptibility fit).  相似文献   

5.
A 2D layer complex 1 and a linear trinuclear complex 2 with mixed ligands have been synthesized and characterized by elemental analyses, IR and single-crystal X-ray diffraction. In 1, the Mn(II) ions are six-coordinated and lie in distorted octahedron coordination environments. Complex 1 is connected into a 2D layer structure based on a linear trinuclear Mn3(admtrz)4(N3)6 (admtrz = 4-amino-3,5-dimethyl-1,2,4-triazole) building unit with either (6,3) topology when Mn1 cations as three-connected nodes or (4,4) network when the coordination trinuclear units being regarded as four connected nodes. In 2, the Co(II) ions are in slightly distorted octahedron coordination geometries. The magnetic behaviors are investigated in the temperature range 1.8-300 K. The magnetic susceptibility measurements show that the Mn(II) ions of complex 1 are weakly antiferromagnetically coupled with g = 1.98(1), J1 = −6.31(5) cm−1 and J2 = −1.88(1) cm−1. There is dominant zero field splitting (ZFS) effects with g values, g// = 2.38(2) and g = 4.96(4), indicated a significant presence of the spin-orbit coupling and magnetization experiment reveals large, uniaxial zero-field splitting parameters of D = −29.55 cm−1 for 2.  相似文献   

6.
A tetranuclear copper(II) complex [Cu4L2(CH3COO)2(OH)2]·6H2O, in which L stands for the dianion of N-(3-carboxylsalicylidene)-4-(2-iminoethyl)morpholine, was synthesized and characterized by elemental analysis, IR, UV-Vis, TGA and X-ray single crystal diffraction. The crystal structure shows that the coordination unit is centrosymmetric with all the Cu(II) ions in square pyramidal coordination geometry. The coordination unit consists of two equivalent parts [Cu2L(CH3COO)(OH)], each containing two Cu(II) ions, a tetradentate N2O2 Schiff base dianion L2−, a CH3COO, and a OH anion. In [Cu2L(CH3COO)(OH)], the six coordination atoms (N2O4) are nearly coplanar, with Cu(1) and Cu(2) enchased in between; the phenolate oxygen and the OH oxygen as bridging atoms bind the two Cu(II) ions in close proximity; both O4 around Cu(1) and N2O2 around Cu(2) form the basal plane of the coordination square pyramids. The two parts are connected by sharing two μ3-OH oxygens and two μ2-CH3COO oxygens from each other, forming four edge-sharing coordination square pyramids around the four Cu(II) ions. A 3D network is formed through hydrogen bonding along a and c axis, and π-π interaction along b axis.  相似文献   

7.
Combined pH-metric, UV-Vis, 1H NMR and EPR spectral investigations on the complex formation of M(II) ions (M=Co, Ni, Cu and Zn) with N-(2-benzimidazolyl)methyliminodiacetic acid (H2bzimida, hereafter H2L) in aqueous solution at a fixed ionic strength, I=10−1 mol dm−3, at 25 ± 1 °C indicate the formation of M(L), M(H−1L) and M2(H−1L)+ complexes. Proton-ligand and metal-ligand constants and the complex formation equilibria have been elucidated. Solid complexes, [M(L)(H2O)2] · nH2O (n=1 for M = Co and Zn, n=2 for M = Ni) and {Cu (μ-L) · 4H2O}n, have been isolated and characterized by elemental analysis, spectral, conductance and magnetic measurements and thermal studies. Structures of [Ni(L)(H2O)2] · 2H2O and {Cu(μ-L) · 4H2O}n have been determined by single crystal X-ray diffraction. The nickel(II) complex exists in a distorted octahedral environment in which the metal ion is coordinated by the two carboxylate O atoms, the amino-N atom of the iminodiacetate moiety and the pyridine type N-atom of the benzimidazole moiety. Two aqua O atoms function as fifth and sixth donor atoms. The copper(II) complex is made up of interpenetrating polymeric chains of antiferromagnetically coupled Cu(II) ions linked by carboxylato bridges in syn-anti (apical-equatorial) bonding mode and stabilized via interchain hydrogen bonds and π-π stacking interactions.  相似文献   

8.
The reaction of Cu(ClO4)2 · 6H2O with bis(3-aminopropyl)methylamine and sodium dicyanamide in aqueous medium results in the formation of a dimeric dicyanamide complex of Cu(II), [Cu2(medpt)2(dca)2](ClO4)2. The single crystal X-ray structure reveals that the dinuclear entities are extended to form a supramolecular 1D ladder by H-bonding. Each dinuclear entity is joined to the adjacent unit via the perchlorate anion. Variable temperature magnetic study was performed and the best-fit parameters are J = −1.20 ± 0.02 cm−1, g = 2.08 ± 0.01 with R = 2 × 10−5. These clearly indicate the antiferromagnetic interaction between the Cu(II) center.  相似文献   

9.
The macrocyclic symmetrical and a series of unsymmetrical binuclear copper(II) complexes have been synthesized by using mononuclear complex [CuL] [3,3′-((1E,7E)-3,6-dioxa-2,7-diazaocta-1,7-diene-1,8-diyl)bis(3-formyl-5-methyl-2-diolato)copper(II)]. Another compartment of the [CuL] have been condensed with various diamines like 1,2-bis(aminooxy)ethane (L1), 1,2-diamino ethane(L2a), 1,3-diamino propane(L2b), 1,4-diamino butane(L2c), 1,2-diamino benzene(L2d), 1,8-diamino naphthalene(L2e) and characterized by elemental, spectroscopic, and X-ray crystallographic methods. The influence of the coordination geometry and the ring size of the binucleating ligands on the electronic, redox, magnetic, catecholase activity, DNA binding and cleavage properties have been studied. The molecular structures of the symmetrical binuclear complex [Cu2L1(H2O)2](ClO4)2 (1) and unsymmetrical binuclear complex [Cu2L2b(ClO4)(H2O)]ClO4 (2b) were determined by X-ray crystallography. Both of them were discrete binuclear species in which each Cu(II) ions are in distorted square pyramid. The Cu?Cu distances vary from 3.0308 (2b) to 3.0361 Å (1). Electrochemical studies evidenced that two quasi-reversible one electron-transfer reduction waves −0.91 to −1.01 V, −1.26 to −1.55 V) for binuclear complexes are obtained in the cathodic region. Cryomagnetic investigation of the binuclear complexes reveals a weak antiferromagnetic spin exchange interaction between the Cu(II) ions within the complexes (−2J = 104.4-127.5 cm−1). The initial rate (Vin) for the oxidation of 3,5-di-tert-butylcatechol to o-quinone by the binuclear Cu(II)complexes are in the range 3.6 × 10−5 to 7.3 × 10−5 Ms−1. The binuclear Cu(II) complexes are avid binders to calf thymus DNA. The complexes display significant oxidative cleavage of circular plasmid pBR322 DNA in the presence of mercaptoethanol using the singlet oxygen as a reactive species. The aromatic diamine condensed macrocyclic ligands of copper(II) complexes display better DNA interaction and significant chemical nuclease activity than the aliphatic diamine condensed macrocyclic Cu(II) complexes.  相似文献   

10.
Reaction of the five-coordinate trigonal-bipyramidal platinum(II) complex, [Pt(pt)(pp3)](BF4) (pt = 1-propanethiolate, pp3 = tris[2-(diphenylphosphino)ethyl]phosphine), with I in chloroform gave the five-coordinate square-pyramidal complex with a dissociated terminal phosphino group and an apically coordinated iodide ion in equilibrium. The thermodynamic parameters for the equilibrium between the trigonal-bipyramidal and square-pyramidal geometries, [Pt(pt)(pp3)]+ + I ? [PtI(pt) (pp3)], and the kinetic parameters for the chemical exchange were obtained as follows: , ΔH0 = − 10 ± 2.4 kJ mol−1, ΔS0 = − 36 ± 10 J K−1 mol−1, , ΔH = 34 ± 4.7 kJ mol−1, ΔS = − 50 ± 21 J K−1 mol−1. The square-planar trinuclear platinum(II) complex was formed by bridging reaction of one of the terminal phosphino groups of trigonal-bipyramidal [PtCl(pp3)]Cl with trans-[PtCl2(NCC6H5)2] in chloroform. From these facts, ligand substitution reactions of [PtX(pp3)]+ (X = monodentate anion) are expected to proceed via an intermediate with a dissociated phosphino group. The rate constants for the chloro-ligand substitution reactions of [PtCl(pp3)]+ with Br and I in chloroform approached the respective limiting values as concentrations of the entering halide ions are increased. These kinetic results confirmed the preassociation mechanism in which the square pyramidal intermediate with a dissociated phosphino group and an apically coordinated halide ion is present in the rapid pre-equilibrium.  相似文献   

11.
A metal-organic coordination chain of Cu(II), viz {[Cu(Hbtc)(NH3)4](H2O)}n (1) (btc = benzenetricarboxylate) has been synthesized and structurally characterized. The sky blue single crystals of 1 were grown by the dissolution of the as-synthesized insoluble product obtained from the reaction of Cu(II), H3btc and 4,4′-bipy in aqueous NH3 solution. Structural determination reveals that 1 crystallizes chiral P212121 space group and 1D coordination chain of Cu(II) bridged by the Hbtc linker. 1D chains are H-bonded via the guest water molecules forming a 2D sheet like structure and 2D sheets through π-π and N-H···O H-bonding interactions produce a 3D supramolecular framework with 1D water filled channels along the a-axis. Temperature dependent magnetic measurement exhibits weak antiferromagnetic intrachain interaction between Cu(II) centers through the Hbtc linker with J = −0.17 cm−1 and g = 2.01 and at low temperature interchain ferromagnetic interaction is predominates. The dominant antiferromagnetic interaction is supported by the DFT calculations.  相似文献   

12.
Using the pyridine dicarboxamide derivative N,N′-bis(1,3,4-thiadiazol-2-yl)-2,6-pyridinedi-carboxamide (H2-btapca) as ligand, two novel polynuclear complexes: dimeric {[Cu2(μ2-O2H)(btapca)2]·DMF·H2O} (1) and tetrameric {[Ni4((μ2-O2H)2(btapca)4)]·DMF·MeOH·3.5H2O} (2) were obtained. In complex 1, two center Cu(II) ions are bounded by two btapca ligands and one aqueous molecule acting as a μ2-H2O bridge connect them together. Complex 2 is a tetrameric complex, in which the based backbone is an assumed Ni4 tetrahedron with two μ2-O2H bridges existing inside the tetrahedron forming a basic [Ni2(μ2-O2H)]2 core, which are surrounded by four btapca ligands. The magnetic properties of the two polynuclear complexes were determined, the results show that for both of the two complexes, the overall weak ferromagnetic exchange interactions between central metal ions are evident, the best fitting parameters are: J = 7.47 cm−1 (g = 2.21) for dimeric Cu(II) complex 1, and 2J1 = 4.8 cm−1, 2J2 = −0.00204 cm−1(g = 2.14, zJ′ = 0.00077 cm−1) for tetrameric Ni(II) complex 2.  相似文献   

13.
Two new binuclear copper complexes, [Cu2(oxpn)(bpy)(pic)(H2O)](pic) (1) and [Cu2(oxpn)(Me2bpy)(pic)](pic) (2) [H2oxpn = N,N′-bis(3-aminopropyl)oxamide; Hpic = 2,4,6-trinitrophenol; bpy = 2,2′-bipyridine; Me2bpy = 4,4′-dimethyl-2,2′-bipyridine], have been synthesized and characterized by elemental analyses, conductivity measurements, IR, UV-visible spectroscopy and single crystal X-ray analyses. Both complexes have similar molecular structures. In complex 1, the central two Cu(II) atoms are bridged by cis-oxpn2− with the Cu1-Cu2 separation of 5.221 Å and the polyhedron of each copper atom is a square-pyramid. Similarly, complex 2 is a cis-oxpn2−-bridged binuclear complex with the Cu1-Cu2 separation of 5.196 Å. Cu1(II) central atom situated in a tetrahedral geometry is four-coordinated and Cu(II) atom situated in a square-pyramidal geometry is five-coordinated. Hydrogen bonding interactions and π-π stacking interactions link the binuclear copper complex 1 or 2 into a 2D infinite network. The antibacterial assays indicate that the two complexes showed better activities than their ligands. The interactions of the two binuclear complexes with herring sperm DNA (HS-DNA) have been studied by UV absorption titration, fluorescence titration and viscosity measurements. The results suggest that the two binuclear complexes bind to HS-DNA via an intercalative mode.  相似文献   

14.
A new ligand, N,N′-dibenzylethane-1,2-diamine (L) and its four transition metal(II) complexes, ML2(OAc)2 · 2H2O (M = Cu, Ni, Zn, Co), have been synthesized and characterized by elemental analysis, mass spectra, molar conductivity, NMR and IR. Moreover, the crystals structure of Cu(II) and Ni(II) complexes characterized by single crystal X-ray diffraction showed that the complexes have a similar molecular structure. Ni(II) has an regular octahedral coordination environment complexes, but typical Jahn Teller effect influenced Cu(II) in an elongated octahedral environment. The interaction between complexes and calf thymus DNA were studied by UV and fluorescence spectra measure, which showed that the binding mode of complexes with DNA is intercalation. Under physiological pH condition, the effects of Cu(OAc)2L2 · 2H2O and Ni(OAc)2L2 · 2H2O on human serum albumin were examined by fluorescence. The results of spectroscopic measurements suggested that the hydrophobic interaction is the predominant intermolecular force. The enthalpy change ΔH0 and the entropy change ΔS0 of Cu(OAc)2L2 · 2H2O and Ni(OAc)2L2 · 2H2O were calculated to be −11.533 kJ mol−1 and 46.339 J mol−1 K−1, −11.026 kJ mol−1 and 46.396 J mol−1 K−1, respectively, according to the Scatchard’s equation. The quenching mechanism and the number of binding site (n ≈ 1) were also obtained from fluorescence titration data.  相似文献   

15.
An asymmetric single EO azido bridged dinuclear copper(II) complex, [Cu2(dmterpy)2(μ-1,1-N3)(N3)2] · NO3 · (H2O)21 [dmterpy = 5,5″-dimethyl-2,2′:6′,2″-terpyridine], and a double EO azido bridged dinuclear nickel(II) complex, [Ni2(pbdiim)4(μ-1,1-N3)2] · 2(N3) · 6(H2O) 2 [pbdiim = 2-(2′-pyridyl)benzo[1,2-d:4,5-d′]diimidazole], have been synthesized and characterized structurally and magnetically. Compound 1 consists of a single EO azido bridged CuII dimer in which each CuII ion is five-coordinated in the form of a distorted square-based pyramid. The N(μ−1,1) atom holds on the apical position of one CuII pyramid with an elongated bond length of 2.305 Å and on the basal plane of another distorted CuII pyramid with a bond length of 1.991 Å. The Cu-N(μ−1,1)-Cu angle is 117.4 (2)°. The copper(II) dimer forms a 1 D zig-zag chain via hydrogen bondings between azide ions, water molecules and the nitrate anion. Compound 2 consists of a double EO azido bridged NiII dimer with the Ni-N(μ−1,1)-Ni bond angle of 102.96 (13)°. The coordination geometry of NiII is octahedral. Their magnetic properties have been measured in the range from 300 to 2 K and correlated with the molecular structures. Compound 1 shows weak ferromagnetic interactions within the copper(II) dimer (J = 2.88 cm−1), despite the large EO azide bridge angle (117.4 (2)°). The intramolecular coupling between the NiII (S = 1) ions in compound 2 was found to be ferromagnetic (J = 27.87 cm−1).  相似文献   

16.
A trinuclear copper(II) complex, [Cu3(2,5-pydc)2(Me5dien)2(BF4)2(H2O)2] · H2O 1, has been constructed from 2,5-pyridine-dicarboxylato bridges (2,5-pydc2−) and N,N,N′,N″,N″-pentamethyl-diethylenetriamine (Me5dien) acting as a blocking ligand. The copper ions, within the centrosymmetric trinuclear cations, are connected by two 2,5-pydc2− bridges, with an intramolecular Cu···Cu separation of 8.432 Å. The central copper ion exhibits an elongated octahedral geometry, with semicoordinated ions, while the terminal ones are pentacoordinated (distorted square-pyramidal geometry). The cryomagnetic investigation of 1 reveals an antiferromagnetic coupling of the copper(II) ions (J = −5.9 cm−1, H = −JSCu1SCu2 − JSCu2SCu1a).  相似文献   

17.
《Inorganica chimica acta》2004,357(12):3574-3582
The copper(II) complexes [Cu(PyTT)2(H2O)](NO3)2 (A) and [CuCl2(μ-PyTT)2CuCl(H2O)]Cl · 3H2O (B) were synthesized and characterized by single crystal X-ray diffraction, IR spectroscopy, UV-Vis-NIR diffuse reflectance and magnetic susceptibility measurements. In the mononuclear compound A the copper ion is in a distorted square pyramidal geometry, with the equatorial plane formed by two thiazoline nitrogen atoms, one imino nitrogen atom and one water molecule, whereas the axial site is occupied by one imino nitrogen atom. The compound B is dinuclear and both Cu(II) centres present environments that can be described as slightly distorted square pyramidal geometries. The observed molar magnetic susceptibility for A (μ=2.13 BM) allows to exclude metal-metal interactions, supporting a monomeric structural formulation for this compound. In compound B, magnetic susceptibility measurements in the temperature range 6.2-288 K show an intradimer antiferromagnetic interaction (J=−11.8 cm−1).  相似文献   

18.
Crystallographic and magnetic studies have been performed on the complex, [{CuCl}2(μ-tppz)][PF6]2, where tppz is 2,3,5,6-tetra-2-pyridinylpyrazine. The crystal structure revealed an infinite, ionic chain wherein Cu(II) ions are respectively above and below the plane of the pyrazine moiety of the bridging tppz ligand with the pyridine moieties moving out of the pyrazine plane in order to coordinate to Cu(II). Each chloride ligand bonds equatorially to Cu(II) in a [{CuCl}2(μ-tppz)]2+ ion and axially to a neighboring [{CuCl}2(μ-tppz)]2+ ion so as to form a one-dimensional chain in the solid state. The temperature-dependent magnetic susceptibilitity could be satisfactorily fitted by using a modified Bleaney-Bowers expression (for H = −JSa · SbJ = −5.6 cm1 and g = 2.16) where the exchange interaction is suggested to involve the orbitals of the tppz ligand.  相似文献   

19.
Two Cu(II)-containing complexes, [Cu(pbbt)Cl2]2 · CH3OH (1) and [Cu(bbbt)1.5Cl2]n (2) (pbbt = 1,1′-(1,3-propylene)bis-1H-benzotriazole, bbbt = 1,1′-(1,4-butanediyl)bis-1H-benzotriazole), have been synthesized and characterized. Single crystal X-ray diffraction shows that 1 exhibits discrete binuclear structure, in which two Cu(II) ions are bridged together through two Cl anions and two pbbt ligands. Whereas 2 displays infinitely extended two-dimensional reticulate grid structure with hexagon units in which six Cu(II) ions act as corners and six bbbt ligands serve as sides. The electrochemical studies show that their redox processes in the potential range of 0.1-0.9 V are quasi-reversible and controlled by diffusion. The diffusion coefficients decrease with increase in the molecular weight of the complexes and the size of the molecules. Further investigation exhibits that their electrochemical properties obtained from experiments are consistent with their crystal data and the computed parameters. For example, the observed one-pair well-defined redox waves in the cyclic voltammetry (CV) diagrams correspond to the X-ray diffraction results that all of the Cu(II) ions in the title complexes are equivalent, respectively; the cathodic shifts (as compared with CuCl2) of the reduction potentials of 1 and 2 are in agreement with the computed results: the LUMO energies of the complexes (−3.789 eV for 1 and −4.330 eV for 2) are higher than that of CuCl2 (−6.942 eV).  相似文献   

20.
The malonato-bridged copper(II) complex [Cu(mal)(H2O)(azpy)1/2] · H2O (1) (mal = malonate, azpy = 4,4′-azobispyridine) has been synthesized and characterized by X-ray diffraction. The structure of 1 consists of malonato-bridged uniform copper(II) chains which are covalent connected through azpy to form two-dimensional wavelike network. The magnetic pathway of complex 1 is through a single syn-anti carboxylate bridge connecting equatorial and equatorial positions of adjacent copper(II) atoms, and have the value of the intrachain ferromagnetic coupling (J = 8.73(3) cm−1) and interchain antiferromagnetic coupling (zJ′ = − 1.31(1) cm−1) through a numerical expression for a ferromagnetic uniform chain.  相似文献   

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