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In this work, GHz and THz complex dielectric spectra of a polyethylene glycol dimethyl ether (2000 g/mol) aqueous solution were studied. The reorientation relaxation of water in this kind of macro-amphiphilic molecule solution can be well described by three Debye models, including under-coordinated water, bulk-like water [water molecules in a tetrahedral hydrogen bond network (bulk water) and bulk water molecules affected by hydrophobic groups], and slow hydrating water (water molecules donating hydrogen bonds to hydrophilic ether groups). The reorientation relaxation timescales of bulk-like water and slow hydration water both show increases with concentration from 9.8 to 26.7 ps and from 46.9 to 100.1 ps, respectively. By estimating the ratios of the dipole moment of slow hydration water to the dipole moment of bulk-like water, we calculated the experimental Kirkwood factors of bulk-like and slow hydrating water. The experimental Kirkwood factor of bulk-like water increased from 3.17 to 3.44 with concentrations, while the experimental Kirkwood factor of slow hydrating water roughly remained constant at 4.13 from concentrations of 15%–60%. The estimated water molecule numbers of three water components around monomers also confirm our sorting for water components. [ABSTRACT FROM AUTHOR]

Magic clusters are metastable faceted nanoparticles that are thought to be important and, sometimes, observable intermediates in the nucleation of certain faceted crystallites. This work develops a broken bond model for spheres with a face-centered-cubic packing that form tetrahedral magic clusters. With just one bond strength parameter, statistical thermodynamics yield a chemical potential driving force, an interfacial free energy, and free energy vs magic cluster size. These properties exactly correspond to those from a previous model by Mule et al. [J. Am. Chem. Soc. 143, 2037 (2021)]. Interestingly, a Tolman length emerges (for both models) when the interfacial area, density, and volume are treated consistently. To describe the kinetic barriers between magic cluster sizes, Mule et al. invoked an energy parameter to penalize the two-dimensional nucleation and growth of new layers in each facet of the tetrahedra. According to the broken bond model, barriers between magic clusters are insignificant without the additional edge energy penalty. We estimate the overall nucleation rate without predicting the rates of formation for intermediate magic clusters by using the Becker–Döring equations. Our results provide a blueprint for constructing free energy models and rate theories for nucleation via magic clusters starting from only atomic-scale interactions and geometric considerations. [ABSTRACT FROM AUTHOR]

Di-tert-butyldiphosphatetrahedrane (tBuCP)2 (A) is a reactive tetrahedral molecule which may serve as a source of new phosphaorganic molecules and ligands. However, the redox chemistry of this compound has not yet been investigated. Here, we show that the reduction of A with alkali metals (AM = Li, Na, K, Rb and Cs) affords 1,2,3-triphospholides [AM(crown ether)][1,2,3-P3C2tBu2] (1–5, [AM(crown ether)] = [Li([12]crown-4)2]+, [Na([15]crown-5)2]+, [K([18]crown-6)]+, [Rb([18]crown-6)]+, and Cs+) with 1,3-diphospholides [AM(crown ether)][1,3-P2C3tBu3] (6–10) formed as by-products. The potassium salt 3 was isolated on a preparative scale, allowing for reactivity studies. Transmetalation with iron(II) and ruthenium(II) chlorides yielded the sandwich complexes [Cp*M(η5-1,2,3-P3C2tBu2)] (11, M = Fe; 12, M = Ru, Cp* = C5Me5) featuring η5-coordinated triphospholide ligands. Treatment of 3 with [Cp2Fe][BAr4F] or [H(Et2O)2BAr4F] (BAr4F = B{C6H3(CF3)2}4) afforded the polyphosphorus compound tBu4C4P6 (13), which presumably results from the dimerisation of a 1,2,3-triphospholyl radical intermediate (1,2,3-P3C2tBu2)˙ (3˙). Tetracyclic 13 is closely structurally related to an isomer of the hydrocarbon hypostrophene (C10H10). [ABSTRACT FROM AUTHOR]

Using theoretical and computational tools for predicting thermophysical properties of fluid systems and the soft matter has always been of interest to the physical, chemical, and engineering sciences. Certainly, the ultimate goal is to be able to compute these macroscopic properties from first-principles calculations beginning with the very atomic constitution of matter. In this work, Mie potential parameters were obtained through dimer interaction energy curves derived from ab initio calculations to represent methane and substituted-methane molecules in a spherical one-site coarse-grained model. Bottom-up-based Mie potential parameters of this work were compared with top-down-based ones from the statistical associating fluid theory (SAFT) models for the calculation of thermodynamic properties and critical point by molecular dynamics simulations and SAFT-VR Mie equation of state. Results demonstrated that bottom-up-based Mie potential parameters when averaging the Mie potential parameters of a representative population of conformers provide values close to the top-down-based ones from SAFT models and predict well properties of tetrahedral molecules. This shows the level of consistency embedded in the SAFT-VR Mie family of models and confers the status of a purely predictive equation of state for SAFT-VR Mie when a reasonable model is considered to represent a molecule of interest. [ABSTRACT FROM AUTHOR]

The restoration of endodontically treated teeth is one of the main challenges of restorative dentistry. The structure of the tooth is a complex assembly in which the materials that make it up, enamel and dentin, have very different mechanical behaviors. Therefore, finding alternative replacement materials for dental crowns in the area of restorative care isa highly significant challenge, since materials such as ceramic and zirconia have very different stress load resistance values. The aim of this study is to assess which material, either ceramic or zirconia, optimizes the behavior of a restored tooth under various typical clinical conditions and the masticatory load. A finite element analysis (FEA) framework is developed for this purpose. The 3D model of the restored tooth is input into the FEA software (Ansys Workbench R23)and meshed into tetrahedral elements. The presence of masticatory forces is considered: in particular, vertical, 45° inclined, and horizontal resultant forces of 280 N are applied on five contact points of the occlusal surface. The numerical results show that the maximum stress developed in the restored tooth including a ceramic crown and subject to axial load is about 39.381 MPa, which is rather close to the 62.32 MPa stress computed for the natural tooth; stresses of about 18 MPa are localized at the roots of both crown materials. In the case of the zirconia crown, the stresses are much higher than those in the ceramic crown, except for the 45° load direction, while, for the horizontal loads, the stress peak in the zirconia crown is almost three times as large as its counterpart in the ceramic crown (i.e., 163.24 MPa vs. 56.114 MPa, respectively). Therefore, the zirconia crown exhibits higher stresses than enamel and ceramic that could increase in the case of parafunctions, such as bruxism. The clinician's choice between the two materials should be evaluated based on the patient's medical condition. [ABSTRACT FROM AUTHOR]

The high-pressure behavior of simple molecular systems, devoid of strong intermolecular interactions, provides a unique avenue toward a fundamental understanding of matter. Tetrahalides of the carbon group elements (group 14), lacking all intermolecular interactions but van der Waals, are among the most elementary of molecular compounds. Here, we report the investigation of CF4 up to 46.5 GPa—the highest pressure up to which any tetrahalides of group 14 elements have been studied so far—by a combination of single-crystal x-ray diffraction (SC-XRDp), Raman spectroscopy, and ab initio calculations. These measurements reveal a pressure-induced reentrant phase transition (phase II → 2.8 G P a phase III → ∼ 20 G P a phase IIR) at room temperature and the formation of a previously unknown CF4 cubic polymorph, named phase IV, after the laser heating of CF4 at 46.5 GPa. In this work, the structures of phases IIR, III, and IV were solved and the atomic coordinates were refined on the basis of SC-XRDp. A comparison of tetrahalides of group 14 elements underlines that reducing the intermolecular halogen–halogen distances leads to a structural rearrangement from close packing of the tetrahedral molecules to close packing of the halogen atoms. [ABSTRACT FROM AUTHOR]

The bottom‐up synthesis of 3D tetrakis(hexa‐peri‐hexabenzocoronenyl)methane, "tetrahedraphene", is reported. This molecular nanographene constituted by four hexa‐peri‐hexabenzocoronene (HBC) units attached to a central sp3 carbon atom, shows a highly symmetric arrangement of the HBC units disposed in the apex of a tetrahedron. The X‐ray crystal structure reveals a tetrahedral symmetry of the molecule and the packing in the crystal is achieved mostly by CH⋅⋅⋅π interactions since the interstitial solvent molecules prevent the π⋅⋅⋅π interactions. In solution, tetrahedraphene shows the same electrochemical and photophysical properties as the hexa‐tBu‐substituted HBC (tBu‐HBC) molecule. However, upon water addition, it undergoes a fluorescence change in solution and in the precipitated solid, showing an aggregation induced emission (AIE) process, probably derived from the restriction in the rotation and/or vibration of the HBCs. Time‐Dependent Density Functional Theory (TDDFT) calculations reveal that upon aggregation, the high energy region of the emission band decreases in intensity, whereas the intensity of the red edge emission signal increases and presents a smoother decay, compared to the non‐aggregated molecule. All in all, the excellent correlation between our simulations and the experimental findings allows explaining the colour change observed in the different solutions upon increasing the water fraction. [ABSTRACT FROM AUTHOR]

The bottom‐up synthesis of 3D tetrakis(hexa‐peri‐hexabenzocoronenyl)methane, "tetrahedraphene", is reported. This molecular nanographene constituted by four hexa‐peri‐hexabenzocoronene (HBC) units attached to a central sp3 carbon atom, shows a highly symmetric arrangement of the HBC units disposed in the apex of a tetrahedron. The X‐ray crystal structure reveals a tetrahedral symmetry of the molecule and the packing in the crystal is achieved mostly by CH⋅⋅⋅π interactions since the interstitial solvent molecules prevent the π⋅⋅⋅π interactions. In solution, tetrahedraphene shows the same electrochemical and photophysical properties as the hexa‐tBu‐substituted HBC (tBu‐HBC) molecule. However, upon water addition, it undergoes a fluorescence change in solution and in the precipitated solid, showing an aggregation induced emission (AIE) process, probably derived from the restriction in the rotation and/or vibration of the HBCs. Time‐Dependent Density Functional Theory (TDDFT) calculations reveal that upon aggregation, the high energy region of the emission band decreases in intensity, whereas the intensity of the red edge emission signal increases and presents a smoother decay, compared to the non‐aggregated molecule. All in all, the excellent correlation between our simulations and the experimental findings allows explaining the colour change observed in the different solutions upon increasing the water fraction. [ABSTRACT FROM AUTHOR]

General characteristics of the crystal-chemical features of the sulfide class of minerals, including the genesis, chemical bonding, different approaches to the classification, etc., are given. The structures of the main representatives of tetrahedral sulfides, disulfides, cluster sulfides, sulfides of elements with incomplete valence shells, and sulfosalts are considered. Mineralogically possible iron sulfides in the Earth's core are presented. It is noted that the geophysical interest in sulfide Fe5S2 is related to its possible presence in the core of not only the Earth but also in the other planets in the solar system. [ABSTRACT FROM AUTHOR]

Tetrakis‐4‐(4‐pyridyl)phenylmethane (TPPM) is a tetrahedral rigid molecule that crystallizes forming a dynamically responsive supramolecular organic framework (SOF). When exposed to different stimuli, this supramolecular network can reversibly switch from an empty to a filled solvated solid phase. This article describes a novel expanded form of a TPPM‐based SOF that has been mechanochemically synthesized and whose crystal structure has been determined by 3D electron diffraction analysis using a novel electron diffractometer. [ABSTRACT FROM AUTHOR]

New solid compounds A3−xGaO4H1−y (A = Sr, Ba; x ∼0.15, y ∼0.3), which are the first oxyhydrides containing gallium ions, have been synthesized by high-pressure synthesis. Powder X-ray and neutron diffraction experiments revealed that the series adopts an anti-perovskite structure consisting of hydride-anion-centered HA6 octahedra with tetrahedral GaO4 polyanions, wherein the A- and H-sites show partial defect. Formation energy calculations from the raw materials support that stoichiometric Ba3GaO4H is thermodynamically stable with a wide band gap. Annealing the A = Ba powder under flowing Ar and O2 gas suggests topochemical H− desorption and O2−/H− exchange reactions, respectively. [ABSTRACT FROM AUTHOR]

The tetrahedral shape‐persistent molecule 14+, containing four identical pyridyl pyridinium units connected via a sp3 hybridized carbon atom, has been investigated in detail by means of steady‐state and time resolved spectroscopy. Remarkable photophysical properties are observed, particularly in comparison with protonated and methylated analogues (1H48+, 1Me48+), which exhibit substantially shorter excited state lifetimes and lower emission quantum yields. Theoretical studies have rationalized the behavior of the tetrameric molecules relative to the monomers, with DFT and TD‐DFT calculations corroborating steady‐state (absorption and emission) and transient absorption spectra. The behavior of the monomeric compounds (each consisting in one of the four identical subunits of the tetramers, i. e. 2+, 2H2+ and 2Me2+) considerably differs from that of the tetramers, indicating a strong electronic interaction between the subunits in the tetrameric species, likely promoted by the homoconjugation through the connecting sp3 C atom. 2+ is characterized by a peculiar S1‐S2 excited state inversion, whereas the short‐lived emitting S1 state of 2H2+ and 2Me2+ exhibits a partial charge‐transfer character, as substantiated by spectro‐electrochemical studies. Among the six investigated systems, only 14+ is a sizeable luminophore (Φem=0.15), which is related to the peculiar features of its singlet state. [ABSTRACT FROM AUTHOR]

In order to ensure the good stability and reliability of the vane pump, the modal characteristics of the mechanical structure were studied and analyzed through finite element simulation and impact modal testing. Based on the stiffness criterion, an optimization plan was proposed for the structure of the vane pump without reducing the natural frequency. Adopting tetrahedral adaptive mesh method, the mesh quality of the modal simulation model was verified to ensure the accuracy and efficiency of the calculation. In modal testing, force sensors and acceleration sensors were arranged reasonably. The excitation signal would be transmitted to the integrated vibration testing system for processing, and modal analysis and processing would be carried out through LMS Test Lab. Based on curve fitting, the spectrum and modal parameters of the tested structure could be obtained. The influence of different curvature radius of cover plates on natural frequencies were studied, and the results show that increasing the curvature radius within a certain range can enhance stiffness and improve processability. [ABSTRACT FROM AUTHOR]

The structure and vibrational spectroscopy of centrohexaindane, 1, was investigated. This unusual molecule has a quaternary carbon atom that is coordinated to four further such quaternary carbon atoms as its core, each pair of which is bonded to an ortho‐phenylene unit. Previous NMR studies have shown that the molecule has tetrahedral (Td) symmetry in solution. The infrared and Raman spectra of chloroform and deuterochloroform solutions of 1 are completely in agreement with this conclusion, as the only modes that are visible are those allowed for Td symmetry. This is not the case in the solid state: X‐ray powder diffraction indicates that the unit cell is triclinic or monoclinic with a volume in excess of 4000 Å3. The vibrational spectroscopy is consistent with C1 site symmetry and the presence of at least two molecules in the primitive cell. It is likely that the space group is centrosymmetric. [ABSTRACT FROM AUTHOR]

Geomagnetic depth sounding (GDS) is a geophysical electromagnetic (EM) method that studies the deep structure and composition of the Earth by using long-period EM signals from geomagnetic observatories and satellites. In this paper, a 3-D anisotropic GDS modelling algorithm is developed. The curl–curl equation is discretized using the edge-based finite-element method on unstructured tetrahedral grids. In order to solve the computationally demanding problem of EM modelling on a global scale, the complex linear system is first separated into the equivalent real linear systems and then the real system is iteratively solved by the flexible generalized minimum residual method with a block diagonal pre-conditioner. This will greatly reduce the condition number of the linear system and thus speed up the solution process. We verify the accuracy of the proposed algorithm by comparing our results with the existing methods. After that, we design a subduction zone model to simulate the EM responses under isotropic and anisotropic environments, respectively. The numerical results show the high efficiency of the proposed algorithm and the response differences between isotropic and anisotropic models. This research can provide theoretical and technical support for the high-accuracy and efficient inversion of GDS data for the geo-dynamic study. [ABSTRACT FROM AUTHOR]

A main task in cryo-electron microscopy single particle reconstruction is to find a three-dimensional model of a molecule given a set of its randomly oriented and positioned projection-images. In this work, we propose an algorithm for ab initio reconstruction for molecules with tetrahedral or octahedral symmetry. The algorithm exploits the multiple common lines between each pair of projection-images as well as self common lines within each image, and integrates the information from all images at once. The applicability of the proposed algorithm is demonstrated using simulated and experimental cryo-electron microscopy data. [ABSTRACT FROM AUTHOR]

Purpose: The ability to characterise highly conducting objects, that may also be highly magnetic, by the complex symmetric rank–2 magnetic polarizability tensor (MPT) is important for metal detection applications including discriminating between threat and non-threat objects in security screening, identifying unexploded anti-personnel landmines and ordnance and identifying metals of high commercial value in scrap sorting. Many everyday non-threat items have both a large electrical conductivity and a magnetic behaviour, which, for sufficiently weak fields and the frequencies of interest, can be modelled by a high relative magnetic permeability. This paper aims to discuss the aforementioned idea. Design/methodology/approach: The numerical simulation of the MPT for everyday non-threat highly conducting magnetic objects over a broad range of frequencies is challenging due to the resulting thin skin depths. The authors address this by employing higher order edge finite element discretisations based on unstructured meshes of tetrahedral elements with the addition of thin layers of prismatic elements. Furthermore, computer aided design (CAD) geometrical models of the non-threat and threat object are often not available and, instead, the authors extract the geometrical features of an object from an imaging procedure. Findings: The authors obtain accurate numerical MPT characterisations that are in close agreement with experimental measurements for realistic physical objects. The assessment of uncertainty shows the impact of geometrical and material parameter uncertainties on the computational results. Originality/value: The authors present novel computations and measurements of MPT characterisations of realistic objects made of magnetic materials. A novel assessment of uncertainty in the numerical predictions of MPT characterisations for uncertain geometry and material parameters is included. [ABSTRACT FROM AUTHOR]

The precise control over hierarchical self‐assembly of superstructures relying on the elaboration of multiple noncovalent interactions between basic building blocks is both elusive and highly desirable. We herein report a terpyridine‐based metallo‐cage T with a tetrahedral motif and utilized it as an efficient building block for the controlled hierarchical self‐assembly of superstructures in response to different halide ions. Initially, the hierarchical superstructure of metallo‐cage T adopted a hexagonal close‐packed structure. By adding Cl−/Br− or I−, drastically different hierarchical superstructures with highly‐tight hexagonal packing or graphite‐like packing arrangements, respectively, have been achieved. These unusual halide‐ion‐triggered hierarchical structural changes resulted in quite distinct intermolecular channels, which provided new insights into the mechanism of three‐dimensional supramolecular aggregation and crystal growth based on macromolecular construction. In addition, the chiral induction of the metallo‐cage T can be realized with the addition of chiral anions, which stereoselectively generated either PPPP‐ or MMMM‐type enantiomers. [ABSTRACT FROM AUTHOR]

The precise control over hierarchical self‐assembly of superstructures relying on the elaboration of multiple noncovalent interactions between basic building blocks is both elusive and highly desirable. We herein report a terpyridine‐based metallo‐cage T with a tetrahedral motif and utilized it as an efficient building block for the controlled hierarchical self‐assembly of superstructures in response to different halide ions. Initially, the hierarchical superstructure of metallo‐cage T adopted a hexagonal close‐packed structure. By adding Cl−/Br− or I−, drastically different hierarchical superstructures with highly‐tight hexagonal packing or graphite‐like packing arrangements, respectively, have been achieved. These unusual halide‐ion‐triggered hierarchical structural changes resulted in quite distinct intermolecular channels, which provided new insights into the mechanism of three‐dimensional supramolecular aggregation and crystal growth based on macromolecular construction. In addition, the chiral induction of the metallo‐cage T can be realized with the addition of chiral anions, which stereoselectively generated either PPPP‐ or MMMM‐type enantiomers. [ABSTRACT FROM AUTHOR]

Airborne electromagnetic (AEM) surveys using airborne mobile platforms enable rapid and efficient exploration of areas where groundwork is difficult. They have been widely used in fields such as shallow resource exploration and environmental engineering. Three-dimensional AEM inversion is the main technique used in fine structural interpretation. However, most current methods focus on separate component data inversions, which limit the kinds of structures that can be recovered in the inversion results. To address this issue, a method for the robust 3D joint inversion of multicomponent frequency-domain AEM data was developed in this study. First, a finite element method based on unstructured tetrahedral grids was used to solve the forward problem of frequency-domain AEM data for both isotropic and anisotropic media. During inversion, a limited-memory quasi-Newton (L-BFGS) method was used to reduce the memory requirements and enable the joint inversion of large-scale multicomponent AEM data. The effectiveness of our algorithm was demonstrated using synthetic models for both isotropic and anisotropic cases, with 5% Gaussian noise added to the modeling data to simulate the measured data for separate and joint inversions. The results of the synthetic models show that joint inversion has advantages over separate inversion in that it enables the recovery of finer underground structures and provides a novel approach for the fine interpretation of frequency-domain AEM data. [ABSTRACT FROM AUTHOR]

Reaction rates are a complicated function of molecular interactions, which can be selected from vast chemical design spaces. Seeking the design that optimizes a rate is a particularly challenging problem since the rate calculation for any one design is itself a difficult computation. Toward this end, we demonstrate a strategy based on transition path sampling to generate an ensemble of designs and reactive trajectories with a preference for fast reaction rates. Each step of the Monte Carlo procedure requires a measure of how a design constrains molecular configurations, expressed via the reciprocal of the partition function for the design. Although the reciprocal of the partition function would be prohibitively expensive to compute, we apply Booth's method for generating unbiased estimates of a reciprocal of an integral to sample designs without bias. A generalization with multiple trajectories introduces a stronger preference for fast rates, pushing the sampled designs closer to the optimal design. We illustrate the methodology on two toy models of increasing complexity: escape of a single particle from a Lennard-Jones potential well of tunable depth and escape from a metastable tetrahedral cluster with tunable pair potentials. [ABSTRACT FROM AUTHOR]

Humanity has always benefited from an intercapillary study in the quantification of natural occurrences in mathematics and other pure scientific fields. Graph theory was extremely helpful to other studies, particularly in the applied sciences. Specifically, in chemistry, graph theory made a significant contribution. For this, a transformation is required to create a graph representing a chemical network or structure, where the vertices of the graph represent the atoms in the chemical compound and the edges represent the bonds between the atoms. The quantity of edges that are incident to a vertex determines its valency (or degree) in a graph. The degree of uncertainty in a system is measured by the entropy of a probability. This idea is heavily grounded in statistical reasoning. It is primarily utilized for graphs that correspond to chemical structures. The development of some novel edge-weighted based entropies that correspond to valency-based topological indices is made possible by this research. Then these compositions are applied to clay mineral tetrahedral sheets. Since they have been in use for so long, corresponding indices are thought to be the most effective methods for quantifying chemical graphs. This article develops multiple edge degree-based entropies that correlate to the indices and determines how to modify them in order to assess the significance of each type. [ABSTRACT FROM AUTHOR]

Topologically interlocked materials (TIMs) are an emerging class of architectured materials that each of block is supported by their neighboring blocks without any connector, which is inspired by microstructures observed in nature. Therefore, they showed alluring attributes such as resistance to impact, toughness, high flexibility even when using brittle materials, and high crack propagation resistance. In these structures, the presence of lateral compressive force, applied on peripheral blocks, ensures the integrity and identified properties. Lateral load, however, can adversely affect TIM plates and cause instability, and initiate buckling. In the present study, to focus on the buckling behavior of TIM plates due to the lateral force, a finite element model of TIM plates, made of tetrahedral blocks, was first validated using numerical results available in the literature and then analyzed. In this regard, an appropriate procedure has been presented to simulate the possible instability of these structures. Unlike monolithic plates, geometric parameters of TIM plates are dependent and this makes their parametric study more challenging. Therefore, some case studies have been accomplished to illustrate the effect of diverse parameters. The current study indicates that increasing the assembly plate length and decreasing its thickness causes a reduction of buckling load; nevertheless, it is necessary to mention that the thickness variation is a more impressive factor. [ABSTRACT FROM AUTHOR]

The tethering dynamics of PyridoPyrrole Quinoxaline (PPQ) anticancer with gold nanoparticles (AuNPs) forming the PPQ/AuNPs complex were investigated. PPQ was synthesized from various substituted nitro-fluoroquinolone and the resulting chemical structure was validated using high-resolution 1H- and 13C-NMR. The tethering in PPQ/AuNPs was activated by irradiating the samples with light at λ ~ 525 nm (surface plasmon resonance (SPR) of AuNPs) and simultaneously with electrical pulsing applied to the electrical electrodes. FTIR and UV–Vis spectroscopy as well as Density Functional Theory (DFT) modeling were used to obtain results on the PPQ/AuNPs with different PPQ concentrations. The results elucidate the formed intermolecular bonding between AuNPs with PPQ on the sites of fluorine atoms or/and hydrogen in the hydroxyl group. The absorbance spectra of PPQ/AuNPs revealed appearing new peaks at 300 nm, 386 nm, and 711 nm. FTIR spectrum was calculated using a DFT model based on tethered a single gold nanoparticle with four PPQ molecules in the fluorine sides where the optimized structure was a distorted tetrahedral geometry of bounded four PPQ molecules with AuNP as a center. The calculated FTIR spectrum was in excellent matching with the experimental FTIR spectrum. The calculated complexation energy was equal to − 3975.5 kcal/mole which is less than the sum of the energy of the PPQ (− 959.5 kcal/mol) and the AuNP (− 135.4 kcal/mol) indicating that tethering led to a stable complex. The binding energy of the PPQ/AuNPs complex deduced from the DFT calculations proved that the highest tethering stability which leads to the minimum energy is at the site of the fluorine atoms. [ABSTRACT FROM AUTHOR]

Bidentate Schiff bases (I1–I9) are synthesized by condensation between substituted anilines, benzothiazoles, and 2-hydroxy aromatic aldehydes. The Zn(II) derivatives (1–9) are prepared by reacting the respective ligand and zinc acetate dihydrate in a 2:1 mole ratio in dry methanol using triethylamine as a base. They are characterized by FTIR, NMR, and single-crystal XRD techniques. The XRD data confirmed distorted tetrahedral molecular geometry for 6; the intermolecular interactions are investigated comprehensively by 3-D Hirschfeld surface analysis and 2-D fingerprint plots. The DFT studies for the compounds are conducted for geometrical and energy optimization by applying 3-21 G* and 6-311++G (d, p)/B3LYP basis sets. Molecular docking data show that ligands possess GOLD scores in the range 51.5–46.7, whereas 2 and 3 have shown good fitness score of 57.7 and 53.3, respectively, with a maximum score of 60.2 for 7. The antibacterial data for 7 shows zone of inhibition as 17.3 mm against Staphylococcus aureus and 16.3 mm against Staphylococcus epidermidis, whereas 2 exhibits maximum value of 15.6 mm against gram negative bacteria Escherichia coli. The maximum and comparable alpha amylase inhibition potential of 12.6 and 12.3% are found for 8 and 3, respectively. [ABSTRACT FROM AUTHOR]

The crystal structures of the new phosphorus oxide nitrides P40O31N46 and P74O59N84, which were synthesized from amorphous phosphorus oxide nitride imide, exhibit complex frameworks built up from P(O,N)4 tetrahedra. The latter form various chain‐like building units with various degrees of branching. These modular units can be combined and arranged in different ways, which leads to closely related structures and several disordered configurations in each compound. As the material was obtained by high‐pressure high‐temperature synthesis, the disorder is most likely a consequence of quenching a high‐pressure phase with P(O,N)5 trigonal bipyramids. Under ambient conditions, P atoms are expected to relax by moving to the centers of the face‐sharing tetrahedra that constitute the bipyramid. Diffraction patterns acquired with microfocused synchrotron radiation reveal that domains of both compounds are intergrown with H3P8O8N9, whose tetrahedral framework represents a cutout of the structures of both P40O31N46 and P74O59N84. Powder diffraction patterns do not indicate any further phases. [ABSTRACT FROM AUTHOR]

A double‐walled tetrahedral metal‐organic cage assembled in solution from silver(I), 2‐formyl‐1,8‐naphthyridine, halide, and a threefold‐symmetric triamine. The AgI4X clusters at its vertices each bring together six naphthyridine‐imine moieties, leading to a structure in which eight tritopic ligands bridge four clusters in an (AgI4X)4L8 arrangement. Four ligands form an inner set of tetrahedron walls that are surrounded by the outer four. The cage has significant interior volume, and was observed to bind anionic guests. The structure also possesses external binding clefts, located at the edges of the cage, which bound small aromatic guests. Halide ions bound to the silver clusters were observed to exchange in a well‐defined hierarchy, allowing modulation of the cavity volume. The principles uncovered here may allow for increasingly more sophisticated cages with silver‐cluster vertex architectures, with post‐assembly tuning of the interior cavity volume enabling targeted binding behavior. [ABSTRACT FROM AUTHOR]

A double‐walled tetrahedral metal‐organic cage assembled in solution from silver(I), 2‐formyl‐1,8‐naphthyridine, halide, and a threefold‐symmetric triamine. The AgI4X clusters at its vertices each bring together six naphthyridine‐imine moieties, leading to a structure in which eight tritopic ligands bridge four clusters in an (AgI4X)4L8 arrangement. Four ligands form an inner set of tetrahedron walls that are surrounded by the outer four. The cage has significant interior volume, and was observed to bind anionic guests. The structure also possesses external binding clefts, located at the edges of the cage, which bound small aromatic guests. Halide ions bound to the silver clusters were observed to exchange in a well‐defined hierarchy, allowing modulation of the cavity volume. The principles uncovered here may allow for increasingly more sophisticated cages with silver‐cluster vertex architectures, with post‐assembly tuning of the interior cavity volume enabling targeted binding behavior. [ABSTRACT FROM AUTHOR]

Random access memories utilize fast, reversible switching between ordered and disordered states of matter in phase change materials (PCMs) such as Ge2Sb2Te5−5x. The short-range structure in the disordered phase has been described either as (i) a network of Ge tetrahedra or (ii) Peierls distorted Ge/Sb octahedra. The PCM transition was investigated in bulk Ge2Sb2Se5xTe5−5x (GSST), in which amorphization sets in with Se doping (x ≈ 0.85) upon quenching. GSST has a hexagonal crystalline ground state with Ge/Sb octahedral coordination, but the phase change transition to the amorphous state that is only observed when the system is quenched brings a short-range structure with sharp, tetrahedrally coordinated Ge/Sb correlations and shortened bonds that are distinctly different from the expected octahedral pairing. [ABSTRACT FROM AUTHOR]

Presented herein is an example of the conversion of an aluminum oxo cluster (AlOC) to an aluminum oxo cluster organic cage (AlOCOC). We successfully synthesized the first example of an aluminum cluster-based organic cage-Al12 tetrahedral cage via an Al3 cluster. The use of 4-pyrazolecarboxylic acid plays an important role in the construction of the organic cage. Due to the presence of partially deprotonated ligands, the hydrogen-bonding interactions between the discrete tetrahedra generate porous supramolecular structures. Considering the high porosity and the abundant N–H sites, we further investigated the performance of the material towards iodine capture. [ABSTRACT FROM AUTHOR]

This paper presents a finite element method to search for equilibrium points around a heterogeneous small body, which degenerates from the full two-body problem in finite element form. The gravitational potential, acceleration, gravitational gradient matrix, and the linearized perturbation equation, which are key formulas for solving the equilibrium points and discriminating their stabilities, are interpolated by the tetrahedral model nodes of the small body. The finite element method could capture the complex internal structures of small bodies and provide a uniform and simple formula for various configurations. The method is applied to the Kuiper Belt Object (486958) Arrokoth, which is a contact binary asteroid. Three types of heterogeneous structures are hypothesized to investigate the evolutions of equilibrium points, namely density disparity of two sub-lobes, hardcore structure, and cavity structure. Under the condition of constant Arrokoth mass, the density disparity of the two sub-lobes greatly influences the local gravitational field. It thus has a significant influence on the positions of equilibrium points. The hardcore and cavity structures have less impact on the equilibrium points. These three simulation groups verified that the equilibrium points of a heterogeneous small body are quite different from those of a homogeneous small body. Such investigation of the equilibrium points may give an in-depth understanding of the dynamical environment around the heterogeneous small body, which is significant for future deep-space missions. [ABSTRACT FROM AUTHOR]

Objective. Freehand 3D ultrasound volume reconstruction has received considerable attention in medical research because it can freely perform spatial imaging at a low cost. However, the uneven distribution of the original ultrasound images in space reduces the reconstruction effect of the traditional method. Approach. An adaptive tetrahedral interpolation algorithm is proposed to reconstruct 3D ultrasound volume data. The algorithm adaptively divides the unevenly distributed images into numerous tetrahedrons and interpolates the voxel value in each tetrahedron to reconstruct 3D ultrasound volume data. Main results. Extensive experiments on simulated and clinical data confirm that the proposed method can achieve more accurate reconstruction than six benchmark methods. Specifically, the averaged interpolation error at the gray level can be reduced by 0.22–0.82, and the peak signal-to-noise ratio and the mean structure similarity can be improved by 0.32–1.83 dB and 0.01–0.05, respectively. Significance. With the parallel implementation of the algorithm, one 3D ultrasound volume data with size 279 × 279 × 276 can be reconstructed from 100 slices 2D ultrasound images with size 200 × 200 at 1.04 s. Such a quick and accurate approach has practical value in medical research. [ABSTRACT FROM AUTHOR]

The purpose of this study was to achieve a specific unlocking of the butterfly effect: nanointerface-based electrochemical biosensing of adenosine triphosphate (ATP) and alkaline phosphatase (ALP). Based on the Faraday-cage concept reported first by our group, we built a new outer Helmholtz plane (OHP)-based electrochemical biosensor by using an unique nanocomposite involving three-dimensional graphene-Au nanoparticles (3D-GO-AuNPs), tetrahedral DNA nanostructures (TDNs), and separated ATP aptamers, in which methylene blue (MB) was employed as the electrochemical signal output. In this process, the prepared nanocomposites were attached favorably onto the TDN substrate electrode surface due to the interaction of ATP and its aptamer, creating a better OHP of the electrode owing to its large enough specific surface area; then a detection limit of 0.25 pM was calculated by 3δ/slope. Whereas, the hydrolysis for ATP of ALP can hinder this binding process, therefore, the biosensor could be indirectly applied for ALP analysis with a detection limit of 0.21 mU/L (3δ/slope). Since some small changes of the two targets will set off a whole series of changes in system, the OHP-extended biosensor provides a superior electrochemical platform for complex biological processes with causal relationships in clinical diagnosis and drug development, similar to the butterfly effect. [ABSTRACT FROM AUTHOR]

We apply here a class of substructuring method to improve the performance of linear tetrahedral element used in finite element analysis (FEA). The method is novel and relied on the construction of mesh inside a tetrahedron volume which behaves as an assembly of substructures. The corresponding stiffness matrix of the mesh is assembled using a static condensation procedure which is used further to obtain strain energy from a set of particular displacement vectors. This energy is the key to obtain a so-called energy ratio that will modify the stiffness matrix of a linear tetrahedral element. In the numerical tests, we show that the method can improve the performance of the tetrahedral element to approximate displacement and stress fields from the analytical solutions for cantilever and stress concentration problems, respectively. [ABSTRACT FROM AUTHOR]

Self‐assembly FeII complexes of phenazine (Phen), quinoxaline (Qxn), and 4,4′‐trimethylenedipyridine (Tmp) with tetrahedral building blocks of [HgII(XCN)4]2− (X=S or Se) formed six new high‐dimensional frameworks with the general formula of [Fe(L)m][Hg(XCN)4]⋅solvents (L=Phen, m/X=2/S, 1; L=Qxn, m/X=2/S, 2; L=Qxn, m/X=1/S, 3; L=Qxn, m/X=1/Se, 3‐Se; L=Tmp, m/X=1/S, 4; and L=Tmp, m/X=1/Se, 4‐Se). 1, 3, and 3‐Se show an intense sub‐terahertz (sub‐THz) absorbance of around 0.60 THz due to vibrations of the solvent molecules coordinated to the FeII ions and crystallization organic molecules. In addition, crystals of 1, 4, and 4‐Se display low‐frequency Raman scattering with exceptionally low values of 0.44, 0.51, and 0.53 THz, respectively. These results indicate that heavy metal FeII−HgII systems are promising platforms to construct sub‐THz absorbers. [ABSTRACT FROM AUTHOR]

Self‐assembly FeII complexes of phenazine (Phen), quinoxaline (Qxn), and 4,4′‐trimethylenedipyridine (Tmp) with tetrahedral building blocks of [HgII(XCN)4]2− (X=S or Se) formed six new high‐dimensional frameworks with the general formula of [Fe(L)m][Hg(XCN)4]⋅solvents (L=Phen, m/X=2/S, 1; L=Qxn, m/X=2/S, 2; L=Qxn, m/X=1/S, 3; L=Qxn, m/X=1/Se, 3‐Se; L=Tmp, m/X=1/S, 4; and L=Tmp, m/X=1/Se, 4‐Se). 1, 3, and 3‐Se show an intense sub‐terahertz (sub‐THz) absorbance of around 0.60 THz due to vibrations of the solvent molecules coordinated to the FeII ions and crystallization organic molecules. In addition, crystals of 1, 4, and 4‐Se display low‐frequency Raman scattering with exceptionally low values of 0.44, 0.51, and 0.53 THz, respectively. These results indicate that heavy metal FeII−HgII systems are promising platforms to construct sub‐THz absorbers. [ABSTRACT FROM AUTHOR]

In this paper, we detect a magnetic null inside the MMS space‐craft (SC) tetrahedron using a linear interpolation that might be nearly impossible for very small inter‐SC separation like MMS (previously implemented on Cluster mission which is larger separation SC). We also report, successive or identical magnetic nulls that enter/reenter and cross the MMS spacecraft tetrahedron, and the "bifurcations" (change of the null type) of magnetic nulls are captured when it crosses one of four MMS tetrahedral surfaces during the 19 September 2015 event. During this event, we observe the eigenvalues of the radial null that enters and crosses the MMS tetrahedron changes from all real to one conjugate complex pair and one real eigenvalues, when the null exits through one of the tetrahedral boundary cells. This bifurcation is also evidenced by measuring the topological distance between two nulls before and after "bifurcation" using the Earth Mover's Distance. Key Points: First, we find a magnetic null and track it using a linear interpolation method in MMSWe identify successive magnetic nulls that enter/reenter, inside, and cross the MMS tetrahedron during a reconnection event on 19 September 2015 at 07:43:30We measure the topological distance between each successive magnetic null and try to capture the "bifurcations" or the change of magnetic topology when null crosses one of the boundary cells of the MMS tetrahedron [ABSTRACT FROM AUTHOR]

Fracture produces new mesh fragments that introduce additional degrees of freedom in the system dynamics. Existing finite element method (FEM) based solutions suffer from increasing computational cost as the system matrix size increases. We solve this problem by presenting a graph‐based FEM model for fracture simulation that is remeshing‐free and easily scales to high‐resolution meshes. Our algorithm models fracture on the graph induced in a volumetric mesh with tetrahedral elements. We relabel the edges of the graph using a computed damage variable to initialize and propagate fracture. We prove that non‐linear, hyper‐elastic strain energy density is expressible entirely in terms of the edge lengths of the induced graph. This allows us to reformulate the system dynamics for the relabelled graph without changing the size of the system dynamics matrix and thus prevents the computational cost from blowing up. The fractured surface has to be reconstructed explicitly only for visualization purposes. We simulate standard laboratory experiments from structural mechanics and compare the results with corresponding real‐world experiments. We fracture objects made of a variety of brittle and ductile materials, and show that our technique offers stability and speed that is unmatched in current literature. [ABSTRACT FROM AUTHOR]

III–V semiconductors such as GaAs are widely studied as promising candidates for high-speed integrated circuit. Despite these applications for conventional bulk structures, their fundamental physical properties in the nanoscale limit are still in scarcity, which is of great importance for the development of nanoelectronics. In this work, we demonstrate that the III–V semiconductor MX (M = Al, Ga, In; X = P, As, Sb) in its two-dimensional (2D) limit could exhibit double layer honeycomb (DLHC) configuration and distorted tetrahedral coordination, according to our first-principles calculations with HSE06 hybrid functional. It is found that surface reconstruction endows 2D III–V DLHCs with pronouncedly different electronic and magnetic properties from their bulk counterparts due to strong interlayer coupling. Mexican-hat-shape bands emerge at the top valence bands of pristine AlP, GaP, InP, AlAs, and InAs DLHCs, inducing the density of states showing a sharp van Hove singularity near the Fermi level. As a result, these DLHCs exhibit itinerant magnetism upon moderate hole doping, while the rest GaAs, AlSb, GaSb, and InSb DLHCs become magnetic under tensile strain with hole doping. With an exchange splitting of the localized pz states at the top valence bands, the hole-doped III–V DLHCs become half-metals with 100% spin-polarization. Remarkably, the InSb DLHC shows inverted band structure near the Fermi level, bringing about nontrivial topological band structures in stacked InSb DLHC due to the strong spin-orbital coupling. These III–V DLHCs expand the members of 2D material family and their exotic magnetic and topological properties may offer great potential for applications in the novel electronic and spintronic devices. [ABSTRACT FROM AUTHOR]

In2Te5 is a stoichiometric compound in the In–Te system of interest for applications in phase change electronic memories and thermoelectrics. Here, we perform a computational study of the structural, dynamical, and electronic properties of the liquid, supercooled liquid, and amorphous phases of this compound by means of density functional molecular dynamics simulations. Models of the supercooled liquid and amorphous phases have been generated by quenching from the melt. The structure of the liquid phase is characterized by a mixture of defective octahedral and tetrahedral local environments of In atoms, while the amorphous phase displays a mostly tetrahedral local geometry for In atoms with corner and edge sharing tetrahedra similar to those found in the crystalline phases of the In2Te5, InTe, and In2Te3 compounds. Comparison with our previous results on liquid and amorphous In2Te3 and further data on the structural properties of liquid In2Te3 are also discussed. The analysis of the electronic properties highlights the opening of a mobility gap in In2Te5 at about 150 K below the liquidus temperature. [ABSTRACT FROM AUTHOR]

Liquid water has anomalous liquid properties, such as its density maximum at 4 °C. An attempt at theoretical explanation proposes a liquid-liquid phase transition line in the supercooled liquid state, with coexisting low-density liquid (LDL) and high-density liquid (HDL) states. This line terminates at a critical point. It is assumed that the LDL state possesses mesoscopic tetrahedral structures that give it solidlike properties, while the HDL is a regular random liquid. But the short-lived nature of these solidlike structures makes them difficult to detect directly. We take a thermodynamic approach instead and calculate the thermodynamic Ricci curvature scalar R in the metastable liquid regime. It is believed that solidlike structures signal their presence thermodynamically by a positive sign for R, with a negative sign typically present in less organized fluid states. Using thermodynamic data from ST2 computer simulations fit to a mean field (MF) two state equation of state, we find significant regimes of positive R in the LDL state, supporting the proposal of solidlike structures in liquid water. In addition, we review the theory, compute critical exponents, demonstrate the large reach of the MF critical regime, and calculate the Widom line using R. [ABSTRACT FROM AUTHOR]

Herewith, we present fast Fourier transforms of time resolved signals, obtained by use of the femtosecond transient transmission (TT) spectroscopy, for three tetrachlorides, CCl4, SiCl4, and GeCl4, and chloroform, CHCl3. Due to coherent excitation of molecules, the isotopic splitting of their spectral bands in the range of symmetric stretching vibration can be observed with high resolution not available in spontaneous Raman scattering. The intensity distribution in the isotopic fine structure pattern appears to differ for various studied molecules, which is explained by the role of intermolecular interactions and the local order of molecules in the liquids. In particular, in SiCl4, the vibrational band exhibits anomalous ratios of the peak amplitudes, which do not agree with the natural abundance of the isotopologues. Using the simple oscillatory model of the liquid and fitting theoretical curves to the experimental results, we have been able to find the intermolecular force constants for all three liquids and to formulate the conclusion that the anomalous spectral pattern in SiCl4 results from strong interactions between the closest Cl atoms belonging to adjacent molecules. Application of the windowed Fourier transform enables us to study the dynamics of intermolecular interactions. The strength of intermolecular interactions in CCl4, SiCl4, and GeCl4, found by the TT technique, is compared with the results obtained by means of the femtosecond optical Kerr effect spectroscopy. [ABSTRACT FROM AUTHOR]

We calibrate basis sets and performance of two theoretical approaches to compute X-ray absorption spectra (XAS) of condensed water by comparison to experiments on hexagonal ice Ih. We apply both the transition-potential half-core-hole approach and the complex polarization propagator using four different models of the crystal with increasing oxygen and proton disorder but find poor agreement with experiments. We note that there are large variations in experimental spectra depending on detection mode and how the ice samples were prepared, which leads us to critically investigate what structures were actually prepared and measured in each case. This is done by using a Monte Carlo-based fitting technique which fits the spectra based on a library of precomputed spectra and assigns weights to contributions from different model structures. These are then used to generate O–O and O–H radial distribution functions and tetrahedrality parameters associated with each of the measured spectra. We find that all spectra are associated with sharp peaks at the oxygen positions in the perfect lattice, but with significant disorder around these positions. We suggest that presently available XAS of hexagonal ice are not fully representative of the perfect crystalline lattice, but contain varying amounts of defects and possible contributions from low-density amorphous ice. [ABSTRACT FROM AUTHOR]

Edge‐based and face‐based smoothed finite element methods (ES‐FEM and FS‐FEM, respectively) are modified versions of the finite element method allowing to achieve more accurate results and to reduce sensitivity to mesh distortion, at least for linear elements. These properties make the two methods very attractive. However, their implementation in a standard finite element code is nontrivial because it requires heavy and extensive modifications to the code architecture. In this article, we present an element‐based formulation of ES‐FEM and FS‐FEM methods allowing to implement the two methods in a standard finite element code with no modifications to its architecture. Moreover, the element‐based formulation permits to easily manage any type of element, especially in 3D models where, to the best of the authors' knowledge, only tetrahedral elements are used in FS‐FEM applications found in the literature. Shape functions for non‐simplex 3D elements are proposed in order to apply FS‐FEM to any standard finite element. [ABSTRACT FROM AUTHOR]

We disclose an approach to a library of novel multidimensional O‐heterocycles based on a rigid, tetrahedral core. New derivatives of tetraphenylmethane (TPM) have been synthesized by employing Claisen rearrangement (CR) and ring‐closing metathesis (RCM) as key steps. Mono‐, di‐, tri‐, and tetra‐substituted benzofurans, 2H‐chromenes, and benzoxepine derivatives are reported. Various heterocyclic cores present in these compounds are key structural elements present in several natural products and pharmaceuticals as well as valuable precursors for the construction of complex molecules with the tetrahedral framework. To the best of our knowledge, this is the first report involving the synthesis of five‐, six‐, and seven‐membered O‐heterocycles in TPM core and the basecatalyzed inexpensive synthesis of several isomerized intermediates that are useful to prepare oxacycles such as benzofuran and 2H‐chromenes. The symmetrical and unsymmetrical one, two, three, and four‐armed architecture of TPM makes it a valuable starting point for building complex molecular and supramolecular architectures. To expand the unmet chemical space of the TPM core, we have prepared twenty new derivatives by a unified approach. Additionally, we report a simple, catalyst‐free, and inexpensive recipe for double bond isomerization based on KOtBu to assemble various oxacycles derived from different trityl chloride derivatives. [ABSTRACT FROM AUTHOR]

Unstructured tetrahedral grids have been applied in magnetotelluric (MT) forward modeling using the finite element (FE) method because of their adaptability to complex anomalies. However, high-quality results require an extreme refinement of the near-surface area, which leads to excessive meshes and an increased degree of freedom (DoF) of the governing equation of the finite element system. To reduce the computational cost, we have developed a hybrid mesh based on triangular prisms and tetrahedrons. The required elements in the near-surface area are reduced because the quality of the triangular prism is not limited by the element aspect ratio. The deep area is discretized by tetrahedral elements to ensure the flexibility of the unstructured grids. The superiority of this hybrid mesh has been tested on a layered model, the DTM1 model and terrain relief models. The results show that the modeling efficiency has been improved, especially for high-frequency data. The accuracy of the modeling using the hybrid mesh is significantly higher than that of the tetrahedral mesh with a similar DoF. Usage of the hybrid mesh can be easily adapted to complex geoelectric models with strong terrain fluctuations, which requires less computational cost than using conventional unstructured elements. [ABSTRACT FROM AUTHOR]

Highly dispersed metal sites on the surface of silica, achieved from immobilization of metal precursor within hydroxyl groups, has gained increasing attention in the field of heterogeneous catalyst. However, the special role of adsorbed water derived by hydroxyl groups on the silica is generally ignored. Herein, a new understanding of adsorbed water on the formation of highly dispersed tetrahedral Co(II) (Td-cobalt(II)) sites is illustrated. It is indicated that sufficient adsorbed water induces the transformation of precursor of Co(NO3)2 into intermediate of [Co(H2O)6]2+. Subsequently, [Co(H2O)6]2+ makes the highly dispersed Td-cobalt(II) sites to be available during direct H2-reduction process. A systematic characterization and DFT calculation prove the existence of the adsorbed water and the importance of the intermediate of [Co(H2O)6]2+, respectively. The as-synthesized catalyst is attempted to the propane dehydrogenation, which shows better reactivity when compared with other reported Co based catalysts. Metal sites immobilized by hydroxyl groups gain great attention, however, the role of adsorbed water derived by hydroxyl groups on the silica is generally ignored. Herein, the authors propose a new understanding of adsorbed water on the immobilization of highly dispersed tetrahedral Co(II). [ABSTRACT FROM AUTHOR]

The Laplace-Beltrami operator is a fundamental tool for analysis on manifolds. It plays a vital role in differential geometry, geometry processing, data analysis and machine learning. Here we present a Laplace--Beltrami operator in spatial Schwarzschild manifolds, denoted by SLBO. Since spatial Schwarzschild manifold has global coordinates, we derive two global approximated schemes of the SLBO in 3-dimensional grids and tetrahedral meshes: (1) We discretize the SLBO by the seven-point stencil finite difference method. This derives a small correction of the classical finite difference scheme, and we prove the convergence for this scheme; (2) we give and prove a correctedcotan formula of the SLBO. According to the two discrete schemes, the robot path planning and tetrahedral mesh smoothing, are designed to demonstrate the effectiveness of the proposed methods using the SLBO. [ABSTRACT FROM AUTHOR]