Infinite One-dimensional Chains Research Articles

A pyroxene-based quantum magnet with multiple magnetization plateaus.

Pyroxenes (AMX2O6) consisting of infinite one-dimensional edge-sharing MO6 chains and bridging XO4 tetrahedra are fertile ground for finding quantum materials. Thus, here, we have studied calcium cobalt germanate (CaCoGe2O6) and calcium cobalt silicate (CaCoSi2O6) crystals in depth. Heat capacity data show that the spins in both compounds are dominantly Ising-like, even after being manipulated by high magnetic fields. On cooling below the Néel temperatures, a sharp field-induced transition in magnetization is observed for CaCoGe2O6, while multiple magnetization plateaus beneath the full saturation moment are spotted for CaCoSi2O6. Our analysis shows that these contrasting behaviors potentially arise from the different electron configurations of germanium and silicon, in which the 3d orbitals are filled in the former but empty in the latter, enabling electron hopping. Thus, silicate tetrahedra can aid the interchain superexchange pathway between cobalt(II) ion centers, while germanate ones tend to block it during magnetization.

A europium(III) metal-organic framework as effective fluorescence sensor for detection of ofloxacin antibiotic

Antibiotic overuse, a critical global concern, poses significant threats to both the eco-environment and human health. Thus, selective and sensitive detection of antibiotics is an extremely important issue. Herein, a europium(III) metal-organic framework Eu2(FA)3(H2O)4·2DMAC (1) (FA = fumaric acid, DMAC = dimethylacetamide) was synthesized by solvothermal strategy. Single-crystal structural analyses indicate that Eu3+ ions are double-bridged by carboxylate groups of fumarate ligands to create one-dimensional (1D) infinite chains, which further form a 3D pillared-layer framework. 1 exhibits superior fluorescence detection performance toward ofloxacin (OFLX) with a high quenching efficiency Ksv of 4.1 × 104 M−1 and low detection limit of 2.4 × 10−6 M, and demonstrates excellent anti-interference properties and durability. Furthermore, the mechanism of fluorescence quenching was mainly attributed to static quenching process and internal filtration effect.

High-pressure and high-temperature modulation of one-dimensional infinite chain in SeO2

The structural evolution of lone-pair compounds under high-pressure and high-temperature conditions has been a subject of fundamental interest in revealing modulated polymorphs. As one of the archetypal lone-pair compounds, selenium dioxide (SeO2) has attracted much attention due to the pressure modulation of its one-dimensional infinite W-shaped chain arrangement. Here, through swarm intelligence algorithm in conjunction with the first-principles simulation, we propose the existence of an orthorhombic Pnma-SeO2 structure, characterized by V-shaped chains interconnected via vertex-sharing SeO3 pyramids. These V-shaped chains demonstrate reduced compressibility along their chain direction compared to the W-shaped chains. Calculations indicate that Pnma-SeO2 is a semiconductor with a large indirect bandgap of 2.39 eV. Remarkably, we synthesized the predicted Pnma-SeO2 in a laser-heated diamond anvil cell at a pressure of 48.5 or 87 GPa as identified by in situ synchrotron x-ray diffraction data. Our findings lead to a significant extension of the phase diagram and transition path of SeO2 and provide key insights into understanding the pressure modulation in lone-pair compounds.

Synthesis and crystal structure of Ba2Y0.87(1)Mn1.71(1)Te5.

We report the structural characterization of a new quaternary telluride, Ba2Y0.87(1)Mn1.71(1)Te5, which was synthesized by the direct reaction of the elements inside a vacuum-sealed fused-silica tube. The quaternary phase is the first member of the Ba-M-Mn-Te system (M= Sc and Y). The composition and structure of the phase were elucidated using SEM-EDX (scanning electron microscopy-energy dispersive X-ray spectrometry) and single-crystal X-ray diffraction (SCXRD) studies. The title phase is nonstoichiometric and crystallizes in the monoclinic system (space group C2/m) having the refined unit-cell parameters a= 15.1466 (8), b= 4.5782 (3), c= 10.6060 (7) Å and β= 116.956 (2)°, with two formula units (Z= 2). The pseudo-two-dimensional crystal structure of Ba2Y0.87(1)Mn1.71(1)Te5 consists of distorted YTe6 octahedra and MnTe4 tetrahedra as the building blocks of the structure. The YTe6 octahedra are arranged to form infinite one-dimensional chains by sharing edges along the [010] direction. These chains are further connected to the MnTe4 tetrahedra along the c axis to create layered two-dimensional polyanionic [Y0.87(1)Mn1.71(1)Te5]4- units. The stuffing of Ba2+ cations in between the layers of [Y0.87(1)Mn1.71(1)Te5]4- anions brings the charge neutrality of the structure. Each Ba atom in the structure sits at the centre of a distorted monocapped trigonal prism-like polyhedron of seven Te atoms.

Strategy for an Effective Eco-Optimized Design of Heteroleptic Cu(I) Coordination Polymers Exhibiting Thermally Activated Delayed Fluorescence.

The new series of copper(I) coordination polymers [Cu(N-N)(μ-PTA)]n[PF6]n {N-N = dmbpy (1), bpy (2), ncup (3), and phen (4)} were generated by straightforward reaction in solution or through a mechanochemical route, of [Cu(MeCN)4][PF6] with 1,3,5-triaza-7-phosphaadamantane (PTA) and the corresponding polypyridines, namely, 5,5'-dimethyl-2,2'-bipyridine (dmbpy), 2,2'-bipyridine (bpy), 2,9-dimethyl-1,10-phenanthroline (ncup), and 1,10-phenanthroline (phen). The compounds were obtained as air-stable solids and fully characterized by IR, NMR spectroscopy, and elemental analyses. The molecular structures were confirmed by single-crystal X-ray diffraction analysis (for 1, 2, and 4), revealing infinite one-dimensional (1D) linear chains driven by μ-PTA N,P-linkers. All tested Cu(I) polymeric compounds show emission at room temperature, which was attributed to thermally activated delayed fluorescence (TADF). Evidence of the involvement of the excited singlet state in the emission process is presented. Comparing the photophysical properties of 1 and 2 as well as 3 and 4, of which 1 and 3 have a stiffened structure, by introducing a methyl group to one of the ligands, we demonstrate how TADF properties depend on molecular rigidity. It is shown that stiffening of the structure reduces the flattening distortion around the Cu(I) center in the 3MLCT state. As a result, the ΔE(S1-T1) energy gap becomes smaller and the fluorescence quantum yield increases without significantly extending the emission lifetime. In particular, the ΔE(S1-T1) values for complexes 1 and 3 are among the shortest reported in the scientific literature, 253 and 337 cm-1, and the TADF lifetimes are τ(300 K) = 5.7 and 4.2 μs, respectively. The fluorescence quantum yields for these complexes are measured to be ΦPL(300 K) = 70 and 80%.

Highly efficient and selective detection of sulfur ions and picric acid through salamo-Cd(II) coordination polymer chemosensor

A salamo-Cd(II) coordination polymer [Cd(HL)2(4,4ʹ-bipy)]n (CP-Cd(II)) was synthesized by introducing a auxiliary ligand 4,4ʹ-bipy to the host ligand H2L (4-chloro-4ʹ-bromo-6-bromo-6ʹ-tert-butyl-2,2ʹ-[ethylenedioxybis(nitrilomethylidyne)]di-phenol), and used as a fluorescent chemical sensor to detect sulfur ions (S2−) and picric acid (PA). Single crystal X-ray analysis demonstrated that the Cd(II) atom has a slightly distorted hexa-coordinate octahedron configuration and forms an infinite one-dimensional chain structure due to the intervention strategy of the auxiliary ligand. In fluorescence method, the sensor CP-Cd(II) exhibits good selectivity and sensitivity for S2− and PA with fluorescence turn-on and off effects sequentially. Additionally, the quenching efficiency of PA remains unchanged even after the sensor has been used for four cycles. At the same time, the sensor CP-Cd(II) can quickly detect S2− and PA within 10 s, which make them charming, recoverable and reliable luminescent coordination polymer sensor materials for detecting S2− and PA. The sensing mechanisms were attentively explored by mass spectrometry analysis, spectroscopic studies and density functional theory (DFT) calculations. Furthermore, the potential application of the sensor for further detection of S2− ions and PA in environmental systems was studied through actual water sample detection.

Photocatalytic Activity and Thermal Stability of Hybrid Metal-Polymer-Coordinated Complexes Derived from Gallic Acid and Ethylenediamine.

Three new hybrid metal-polymer-coordinated complexes (MPCs) of copper(II), cobalt(II), and nickel(II) ions with an organic polymer derived from gallic acid and ethylenediamine (GAEtH) were synthesized. The structures of GAEtH and MPCs were characterized with FT-IR, ultraviolet (UV)-Vis, 1H and 13C NMR spectroscopy, and elemental and thermogravimetric analysis. The results reveal that the organic polymer GAEtH exhibits an infinite one-dimensional chain structure, while the hybrid MPCs have a double chain structure, with the two chains joined by metal ions. The thermodynamic and kinetic parameters of the thermal degradation stages were determined by the Coats Redfern method, and the photocatalytic behaviors of the MPCs were investigated through the decomposition of methyl orange dye under UV irradiation.

Praziquanamine enantiomers: Crystal structure, Hirshfeld surface analysis, and quantum chemical studies

In this work, we report a microwave–assisted method to perform an acid hydrolysis using 2–6 praziquanamides as substrate to produce both praziquanamine (1) enantiomers: (R)–(–)–praziquanamine (1a) or (S)–(+)–praziquanamine (1b). This method provides very good yields and short reaction times (twenty minutes). A structural characterization of praziquanamine and its enantiomers was performed by using X–ray diffraction, spectroscopic and spectrometric techniques, thermal analysis, and powder X–ray diffraction. Computational calculations were also carried out, such as density functional theory (DFT), as it yields suitable geometries for the optimized structures. We also performed frontier molecular orbital (LUMO–HOMO) and electrostatic potential (ESP) analyses. Theoretical values were calculated for electronic transitions, vibration frequencies, and chemical shifts for nuclear magnetic resonances (NMR), and they were later compared to the experimental results. Additionally, the Hirshfeld Surface Analysis was performed to evaluate intermolecular interactions. Crystal structures of both 1a and 1b enantiomers were obtained as an orthorhombic system in the non-centrosymmetric P212121 space group. Conversely, the 1 compound crystallized as a monoclinic system in the centrosymmetric P21/n space group. The following interactions define the crystal packing: N–H···O, N–H···N, C–H···O, C–H···N, C–H···π, and π–π. They all establish an infinite one-dimensional chain. The crystal architectures of praziquanamine enantiomers may be used as resolution agents to discriminate chiral topologies.

Elucidating Structure-Property Correlation in Perovskitoid and Antiperovskite Piperidinium Manganese Chloride.

In the world of semiconductors, organic-inorganic hybrid (OIH) halide perovskite is a new paradigm. Recently, a zealous effort has been made to design new lead-free perovskite-like OIH halides, such as perovskitoids and antiperovskites, for optoelectronic applications. In this context, we have synthesized a perovskitoid compound (Piperidinium)MnCl3 (compound 1) crystallizing in an orthorhombic structure with infinite one-dimensional (1D) chains of MnCl6 octahedra. Interestingly, this compound shows switchable dielectric property governed by an order-disorder structural transition. By controlling the stoichiometry of piperidine, we have synthesized an antiperovskite (Piperidinium)3Cl[MnCl4] (compound 2), the inverse analogue of a perovskite, consisting of zero-dimensional (0D) MnCl4 tetrahedra. This type of organic-inorganic hybrid antiperovskite halide is unique and scarce. Such a dissimilarity in lattice dimensionality and Mn2+ ion coordination ensues fascinating photophysical and magnetic properties. Compound 1 exhibits red emission with a photoluminescence quantum yield (PLQY) of ∼28%. On the other hand, the 0D antiperovskite compound 2 displays green emission with a higher PLQY of 54.5%, thanks to the confinement effect. In addition, the dimensionality of the compounds plays a vital role in the exchange interaction. As a result, compound 1 shows an antiferromagnetic ground state, whereas compound 2 is paramagnetic down to 1.8 K. This emerging structure-property relationship in OIH manganese halides will set the platform for designing new perovskites and antiperovskites.

Minimal model of drag in one-dimensional crystals

Using a nonperturbative classical approach, we study the dynamics of a mobile particle interacting with an infinite one-dimensional (1D) chain of harmonic oscillators. This minimal system is an effective model for many 1D transport phenomena, such as molecular motion in nanotubes and ionic conduction through solid-state materials. As expected, coupling between the mobile particle and the chain induces dissipation of the mobile particle's energy. However, both numerical and analytic results demonstrate an unconventional nonmonotonic dependence of the drag on particle speed. In addition, when this system is subjected to a constant bias, it supports multiple steady-state drift velocities.

Crystal structure and Hirshfeld surface analysis of two organic salts based on 1,3,4-thia-diazole derivatives.

During attempts to achieve inter-action between 2-amino-5-ethyl-1,3,4-thia-diazole with oxalyl chloride and 5-mercapto-3-phenyl-1,3,4-thia-diazol-2-thione with various diacid anhydrides, we obtained two co-crystals (organic salts), namely, 2-amino-5-ethyl-1,3,4-thia-diazol-3-ium hemioxalate, C4H8N3S+·0.5C2O4 2-, (I), and 4-(di-methyl-amino)-pyridin-1-ium 4-phenyl-5-sulfanyl-idene-4,5-di-hydro-1,3,4-thia-diazole-2-thiol-ate, C7H11N2 +·C8H5N2S3 -, (II). Both solids were investigated by single-crystal X-ray diffraction and by Hirshfeld surface analysis. An infinite one-dimensional chain along [100] is generated through O-H⋯O inter-actions between the oxalate anion and two 2-amino-5-ethyl-1,3,4-thia-diazol-3-ium cations in compound (I), and a three-dimensional supra-molecular framework is generated through C-H⋯O and π-π inter-actions. In compound (II), an organic salt is formed by a 4-phenyl-5-sulfanyl-idene-4,5-di-hydro-1,3,4-thia-diazole-2-thiol-ate anion and a 4-(di-methyl-amino)-pyridin-1-ium cation, which are combined by an N-H⋯S hydrogen-bonding inter-action, forming a zero-dimensional structural unit. As a result of inter-molecular π-π inter-actions, the structural units are combined into a one-dimensional chain running along the a-axis direction.

One-dimensional infinite chain Ag(i) complex with high quantum yield and TADF property: prepared by metal ion adjustment

The complex AgDpq exhibits excellent photophysical properties and TADF characteristics than Cu(i) complex, which is related to the unique one-dimensional infinite chain structure and the smaller contribution of metal to the excited states.

A one-dimensional chain of manganese(II) bridged peroxomolybdate isolated from an aqueous Mn-polymolybdate-H2O2 system.

A new manganese(II)-peroxomolybdate complex, Cs4[Mn(OH2)2(Mo7O22(O2)2)]·4.25H2O (Cs-1), was isolated from an aqueous solution containing manganese(II) sulfate, sodium heptamolybdate and hydrogen peroxide by the addition of Cs+ salt. Cs-1 was characterized by single crystal X-ray diffraction, thermogravimetry (TG), IR spectroscopy, powder X-ray diffraction (PXRD), cyclic voltammetry (CV) and ultraviolet-visible spectroscopy (UV-vis). The diperoxoheptamolybdate [Mo7O22(O2)2]6- units were linked by Mn(II) ions to form a one-dimensional infinite chain of [Mn(OH2)2(Mo7O22(O2)2)]n4n-, which represents a unique structure with the coexistence of the oxidant-reductant pair O22-/Mn2+. The interconversion between [MnII(OH2)2(Mo7O22(O2)2)]4- and [MnMo9O32]6- in the aqueous solution was monitored by UV-vis spectrophotometry. It indicates that 1 is a key intermediate during the redox cycle of Mn(II) and Mn(IV) in the Mn-polyoxometalate-H2O2 system. In the oxidation process of 3,3',5,5'-tetramethylbenzidine and ortho-phenylenediamine by H2O2, Cs-1 shows notable activity as an enzyme mimetic catalyst.

Multifunctional Catalysis by a One-Dimensional Copper(II) Metal Organic Framework Containing Pre-existing Coordinatively Unsaturated Sites: Intermolecular C-N, C-O, and C-S Cross-Coupling; Stereoselective Intramolecular C-N Coupling; and Aziridination Reactions.

The coordinatively unsaturated sites (CUS) are vital in metal-centered catalysis. Metal-organic frameworks (MOFs) provide a unique opportunity to generate and stabilize CUS due to their robust structure. Generally, the generation of CUS in MOFs needs prior activation under heat and high vacuum to remove labile molecules occupying the catalytic sites. Herein, we report a solvothermal synthesis of a ready-to-use copper MOF containing accessible pre-existing CUS that does not need activation. The single crystal X-ray diffraction structure reveals a square planar Cu(II) center with two N-methylimidazoles (Mim) and one benzenedicarboxylic acid (BDC) with the formula unit [CuII(BDC)(Mim)2]n (Cu-1D) forming an infinite one-dimensional (1D) chain along the c axis. The 1D chains are stabilized by noncovalent π-π, CH···π, and H-bonding interaction to give 2D (sheet-like) and 3D networks in the solid state. The quantification of non-covalent interaction is studied by Hirshfeld surface analysis, and the formation of a higher architecture in the solid state is confirmed by SEM analysis. The reported Cu-1D MOF acts as a solid heterogeneous catalyst and exhibits efficient catalytic activity in intermolecular and intramolecular cross-coupling reactions. Intermolecular C-heteroatom cross-coupling of a variety of N-heterocycles, aliphatic, aromatic, alicyclic amines and amides (C-N), phenols (C-O), and thiols (C-S) with aryl halides (halide = I, Br) was achieved with 70 to 95% yield, better than the state-of-the-art Cu-based homogenous system. The C-N coupling catalytic cycle is initiated by the in situ reduction of Cu(II) by KOH/DMSO to Cu(I) species. Subsequently, Cu(I) undergoes oxidative addition followed by reductive elimination to form a cross-coupled product. High stereoselectivity was found for the intramolecular C-N coupling reaction to give tetrahydroquinoxalines with an enantiomeric excess (ee) of more than 99%. For a broader application, Cu-1D was applied as the catalyst for the synthesis of a library of aziridines that gives yields of up to 99% with more than 93% recyclability for each cycle.

Exploring dithiolate-amine binary ligand systems for the supramolecular assemblies of Ni(II) coordination compounds: Crystal structures, theoretical studies, cytotoxicity studies, and molecular docking studies

Dithiolate-amine binary ligand systems based two novel supramolecular structures of Ni(II) coordination compounds [Ni(2-ap)2(i-mnt)] (1) and [Ni2(i-mnt)2(tn)2]n (2) [where i-mnt−2: 1,1-dicyanoethylene-2,2-dithiolate, 2-ap: 2-amino pyridine, tn: 1,3-diaminopropane] have been designed, synthesized and characterized by spectroscopic techniques and X-ray crystallography. In complex 1, the N2S2 coordination kernel around Ni+2 attains distorted square planar geometry (Okuniewski parameter τ/4 = 0.17) and its supramolecular array has a dominant influence of N H···N type hydrogen bonds forming significant ring geometry of graph-set-motifR428. The i-mnt−2 adopts a diverse coordination mode in 2 contrary to 1 resulting in a binuclear one-dimensional infinite polymeric chain structure of consecutive NiS4 [square planar, S4 donor set τ/4 = 0] and NiN6 (octahedral) coordination kernels linked by i-mnt−2 that runs along [1 0 0] axis wherein N H···S, N H···N, C H···S type hydrogen bonds are believed to be crystal structure stabilizers. The Hirshfeld surface analyses (HS) at the molecular and atomic levels have been extensively studied to quantify all non-covalent interactions/contacts and the nature of ligand - metal interactions. The prominent N···H/H···N [28.1 % (1), 27.3 % (2A, NiS4), 20 % (2B, NiN6)] and S···H/H···S contacts [10.4 % (1), 32.8 % (2A), 29 % (2B)] presumably play vital role in crystal packing. The non-covalent interactions were further investigated by Molecular Electrostatic Potential (MEP) surface at PBE0-D3/def2-TZVP level and QTAIM optimizations (for 1). The MTT assay study using SW620 metastatic colon cancer cells shows a significant dose-dependent decline in cell proliferation with increasing concentration for 1 but for 2, the reverse trend is followed. A molecular docking study with the APC protein has further investigated the antiproliferative activity of 1 (MTT assay).

Multistimuli-Responsive Properties of Aggregated Isocyanide Cycloplatinated(II) Complexes

Here, we describe the neutral cyclometalated tert-butylisocyanide PtII complexes, [Pt(C∧N)Cl(CNBut)] 1, the double salts [Pt(C∧N)(CNBut)2][Pt(C∧N)Cl2] 2, and the cationic complexes [Pt(C∧N)(CNBut)2]ClO43 [C∧N = difluorophenylpyridine (dfppy, a), 4-(2-pyridyl)benzaldehyde (ppy-CHO, b)]. A comparative study of the pseudopolymorphs 1a, 1a·CHCl3, 1b, 1b·0.5Toluene, 1b·0.5PhF, and 3a·0.25CH2Cl2 reveals strong aggregation through Pt···Pt and/or π···π stacking interactions to give a variety of distinctive one-dimensional (1D) infinite chains, which modulate the photoluminescent properties. This intermolecular long-range aggregate formation is the main origin of the photoluminescent behavior of 1a and 1b complexes, which exhibit highly sensitive and reversible responses to multiple external stimuli including different volatile organic compounds (VOCs), solvents, temperatures, and pressures, with distinct color and phosphorescent color switching from green to red. Furthermore, complex 1b undergoes supramolecular self-assembly via Pt···Pt and/or π···π interactions into a polymer thin polystyrene (PS) film 10 wt % in response to toluene vapors, and 3a exhibits vapochromic and vapoluminescent behavior. Theoretical simulations on the dimer, trimer, and tetramer models of 1a and 1b have been carried out to get insight into the photophysical properties in the aggregated solid state.

Crystal Structure of Chiral Drug Prenalterol and Its Precursor Prone to Spontaneous Resolution

Due to the chiral uniformity of proteins and carbohydrates, the basic building blocks of living matter, the mirror symmetry characteristics of drugs are of exceptional importance for medicinal chemistry. In this work, we present a new synthesis of the mono-enantiomeric chiral drug prenalterol 1 based on the symmetry-breaking phenomenon, namely, the spontaneous resolution of 4-hydroxyphenyl glycerol ether 2. The single crystal X-ray diffraction method was used to investigate both rac- and (S)-1 as well as (R)-2. A feature of the main crystal-forming supramolecular motif (SMM) for diol 2 is the participation of three different molecules representing different types of hydroxyl groups in the formation of its repeating unit. The type of prenalterol SMM, as in the case of the related drugs propranolol 3 and pindolol 4, appears to be a chirality driven property, and is dictated by the enantiomeric composition of the crystals. In single-enantiomeric forms, infinite one-dimensional chains are realized, organized around helical axes, while in racemates, zero-dimensional cycles are realized, organized around inversion symmetry elements. The results obtained again demonstrate the influence of the chiral polarization of a substance not only on the general (selection of a space group), but also on particular characteristics of matter crystal organization, namely selection of a specific SMM.

Superballistic and superdiffusive scaling limits of stochastic harmonic chains with long-range interactions

We consider one-dimensional infinite chains of harmonic oscillators with random exchanges of momenta and long-range interaction potentials which have polynomial decay rate |x|−θ , x → ∞, θ > 1 where is the interaction range. The dynamics conserve total momentum, total length and total energy. We prove that the systems evolve macroscopically on superballistic space-time scale when 1 < θ < 3, when θ = 3, and hyperbolic space-time scale (yɛ −1, tɛ −1) when θ > 3. Combining our results and the results in (Suda 2021 Ann. Inst. Henri Poincare B 57 2268–2314), we show the existence of two different space-time scales on which the systems evolve. In addition, we prove the fluctuations of the normal modes of the superballistic wave equation, which are analogues of the Riemann invariants and capture fluctuations along the characteristics. For the normal modes, the space-time scale is superdiffusive when 2 < θ ⩽ 4 and diffusive when θ > 4.

Preparation and Visible-Light Response of Salicylate-Stabilized Heterobimetallic Pb-Ti-Oxo Clusters Initiated via Auxiliary Quaternary Ammonium Salts and a Solvent Effect.

In this study, with the assistance of quaternary ammonium salts we have successfully prepared a new family of salicylate-stabilized heterobimetallic Pb-Ti-oxo clusters, including H(TEA)[Pb2Ti6(μ2-O)2(μ3-O)2(OiPr)4(PA)2(Sal)6(NO3)2] (PTC-321; TEA = tetraethylammonium; HOiPr = isopropanol; H2PA = phenylphosphonic acid; H2Sal = salicylic acid), {PbTi3(μ2-O)(μ3-O)(OiPr)2(PA)(Sal)3(DMF)·CH3CN}n (PTC-322; DMF = dimethylformamide), {PbTi5(μ3-O)6(Sal)3(CH3COO)2(DMF)(OiPr)2}n (PTC-323), [Pb2Ti4(Sal)6(EtO)2(OiPr)6(HOiPr)2]·CH3NH2 (PTC-324; EtOH = CH3CH2OH), H[Pb4Ti9(μ2-O)2(μ3-O)(μ4-O)6(Sal)7(OiPr)13] (PTC-325), and Pb2Ti12(μ2-O)3(μ3-O)3(μ4-O)4(Sal)4(OEt)24 (PTC-326). Single-crystal X-ray diffraction studies demonstrate that the {Ti3Pb(Sal)3} unit acts as the building block to constitute the diverse assembly of PTC-321-PTC-323. Thereinto, the clusters in PTC-322 and PTC-323 are connected into infinite one-dimensional chains. Furthermore, the solvent effects have facilitated the heterobimetallic Pb-Ti-oxo clusters into various configurations in PTC-323-PTC-326. Solid-state ultraviolet-visible spectroscopy analysis indicates that the optical absorption bands of these compounds shift effectively toward the visible-light region, and they were also employed as electrode precursors to investigate their visible-light-driven photocurrent response.

Critical temperature of one-dimensional Ising model with long-range interaction revisited

We present a generalized expression for the transfer matrix of finite and infinite one-dimensional spin chains within a magnetic field with spin pair interaction J/rp, where r∈{1,2,3,…,nv} is the distance between two spins, nv is the number of nearest neighbors reached by the interaction, and 1≤p≤2. Using this transfer matrix, we calculate the partition function, the Helmholtz free energy, and the specific heat for both finite and infinite ferromagnetic 1D Ising models within a zero external magnetic field. We focus on the temperature Tmax where the specific heat reaches its maximum, needed to compute J/(kBTmax) numerically for every value of nv∈{1,2,3,…,30}, which we interpolate and then extrapolate up to the critical temperature as a function of p, by using a novel inter-extrapolation function. We use two different procedures to reach the infinite spin chain with an infinite interaction range: increasing the chain size as well as the interaction range by the same amount and increasing the interaction range for the infinite chain. As we expected, both extrapolations lead to the same critical temperature within their uncertainties by two different concurrent curves. Critical temperatures fall within the upper and lower bounds reported in the literature, showing a better coincidence with many existing approximations for p close to 1 than the p values near 2. It is worth mentioning that the well-known cases for nearest (original Ising model) and next-nearest neighbor interactions are recovered doing nv=1 and nv=2, respectively.