Hardy's Inequality for the fractional powers of Grushin operator

We prove Hardy's inequality for the fractional powers of the generalized sublaplacian and the fractional powers of the Grushin operator. We also find an integral representation and a ground state representation for the fractional powers of generalized sublaplacian

Structural and DFT Studies on the Polymorphism of a Cadmium(II) Dipicolinate Coordination Polymer

The coordination polymer [Cd₂(dipic)₂(H₂O)₃]_<i>n</i> was prepared by the reaction of cadmium­(II) chloride or bromide and dipicolinic acid (dipicH₂) at 60 °C under autogenous pressure. The <i>C</i>2/<i>c</i> polymorph (<b>1</b>) was almost exclusively obtained. However, a few crystals of the <i>P</i>2/<i>c</i> polymorph (<b>2</b>) were occasionally found in the mixture with the <i>C</i>2/<i>c</i> polymorph, thus making it a disappearing and concomitant polymorph. The polymeric chains in <b>1</b> are connected into dimers by π–π interaction and O–H···O hydrogen bonds. These dimers are in turn connected by intermolecular O–H···O hydrogen bonds into a 2D network. The polymeric chains in <b>2</b> are connected by intermolecular O–H···O hydrogen bonds into a zigzag chain along the [001] direction. According to DFT calculations, the hydrogen bonding is of similar order in both polymorphs (∼7.5 kcal mol^–1 per hydrogen bond). However, there is additional stability imparted in <b>1</b>, as shown by dispersion-corrected DFT, through π–π stacking between polymeric chains, making <b>1</b> the thermodynamically favored polymorph. Polymorph <b>1</b> was characterized by IR spectroscopy, PXRD analysis, and TGA and DSC methods. The DSC analysis did not show any sign of phase transition between <b>1</b> and <b>2</b>. This was also confirmed by variable temperature PXRD, since the pattern of <b>1</b> remained unchanged until the decomposition of <b>1</b>

Structural and DFT Studies on the Polymorphism of a Cadmium(II) Dipicolinate Coordination Polymer

The coordination polymer [Cd₂(dipic)₂(H₂O)₃]_<i>n</i> was prepared by the reaction of cadmium­(II) chloride or bromide and dipicolinic acid (dipicH₂) at 60 °C under autogenous pressure. The <i>C</i>2/<i>c</i> polymorph (<b>1</b>) was almost exclusively obtained. However, a few crystals of the <i>P</i>2/<i>c</i> polymorph (<b>2</b>) were occasionally found in the mixture with the <i>C</i>2/<i>c</i> polymorph, thus making it a disappearing and concomitant polymorph. The polymeric chains in <b>1</b> are connected into dimers by π–π interaction and O–H···O hydrogen bonds. These dimers are in turn connected by intermolecular O–H···O hydrogen bonds into a 2D network. The polymeric chains in <b>2</b> are connected by intermolecular O–H···O hydrogen bonds into a zigzag chain along the [001] direction. According to DFT calculations, the hydrogen bonding is of similar order in both polymorphs (∼7.5 kcal mol^–1 per hydrogen bond). However, there is additional stability imparted in <b>1</b>, as shown by dispersion-corrected DFT, through π–π stacking between polymeric chains, making <b>1</b> the thermodynamically favored polymorph. Polymorph <b>1</b> was characterized by IR spectroscopy, PXRD analysis, and TGA and DSC methods. The DSC analysis did not show any sign of phase transition between <b>1</b> and <b>2</b>. This was also confirmed by variable temperature PXRD, since the pattern of <b>1</b> remained unchanged until the decomposition of <b>1</b>

Directed Assembly of acac-Based Complexes by Deliberately Fine-Tuning Electrostatic Molecular-Recognition Events

A protocol for supramolecular synthesis of one-dimensional (1-D) chains comprising octahedral metal-complexes linked by predictable hydrogen bonds has been established. The synthetic process was refined by adjusting the reactants in an iterative manner to shift the magnitude of the electrostatic potential surfaces of competing hydrogen-bond acceptor sites thereby avoiding the formation of unwanted products. The synthetic space for this study was provided by a combination of reactants: Co­(II)/Ni­(II) cations and acac-based anions (dibenzoylmethanato, dbm/hexafluoracetylacetonato, hfac) combined with a series of pyridine-oxime ligands (4-pyridinealdoxime, 4-Hoxpy; methyl 4-pyridyl ketoxime, 4-Meoxpy; 3-pyridinealdoxime, 3-Hoxpy; methyl 3-pyridyl ketoxime, 3-Meoxpy). The initial self-assembly process did not produce the desired 1-D chains held together by oxime···oxime hydrogen bonds; however, through deliberate fine-tuning of the reactants, a robust synthetic protocol for the reproducible synthesis of the correct supramolecular products was obtained. The successful synthetic protocol was arrived at in much the same way as organic synthesis is systematically altered and refined in response to product yields

From Simple Palladium(II) Monomers to 2D Heterometallic Sodium–Palladium(II) Coordination Networks with 2‑Halonicotinates

The 2D heterometallic sodium–palladium(II) coordination polymers with 2-halonicotinates [2-chloropyridine-3-carboxylate (2-chloronicotinate), 2-Clnic– and 2-bromopyridine-3-carboxylate (2-bromonicotinate), 2-Brnic–], {[Na2(H2O)2(μ-H2O)4PdCl2(μ-2-Clnic-N:O′)2]}n (1), and {[Na2(H2O)2(μ-H2O)4PdBr2(μ-2-Brnic-N:O′)2]·2H2O}n (2) were prepared in aqueous solutions under the presence of NaHCO3, while palladium(II) monomers with the neutral 2-chloronicotinic and 2-bromonicotinic acid ligands, [PdCl2(2-ClnicH-N)2]·2DMF (3) and [PdCl2(2-BrnicH-N)2]·2DMF (4), were prepared in DMF/water mixtures (DMF = N,N′-dimethylformamide). The zigzag chains of water-bridged sodium ions are in turn bridged by [PdCl2(2-Clnic)2]2– moieties in 1 or by [PdBr2(2-Brnic)2]2– moieties in 2, leading to the formation of the infinite 2D coordination networks of 1 or 2. The DFT calculations showed the halosubstituents type (Cl vs Br) does not have an influence on the formation of either trans or cis isomers. The trans isomers were found in all reported compounds; being more stable for about 10 to 15 kJ mol–1. The 2D coordination networks 1 and 2 are more stabilized by the formation of Na–Ocarboxylate bonds, comparing to the stabilization of palladium(II) monomers 3 and 4 by hydrogen-bonding with DMF molecules. The difference in DFT calculated energy stabilization for 1 and 2 is ascribed to the type of halosubstituents and to the presence/absence of lattice water molecules in 1 and 2. The compounds show no antibacterial activity toward reference strains of Escherichia coli and Staphylococcus aureus bacteria and no antiproliferative activity toward bladder (T24) and lung (A549) cancer cell lines

From Simple Palladium(II) Monomers to 2D Heterometallic Sodium–Palladium(II) Coordination Networks with 2‑Halonicotinates

The 2D heterometallic sodium–palladium(II) coordination polymers with 2-halonicotinates [2-chloropyridine-3-carboxylate (2-chloronicotinate), 2-Clnic– and 2-bromopyridine-3-carboxylate (2-bromonicotinate), 2-Brnic–], {[Na2(H2O)2(μ-H2O)4PdCl2(μ-2-Clnic-N:O′)2]}n (1), and {[Na2(H2O)2(μ-H2O)4PdBr2(μ-2-Brnic-N:O′)2]·2H2O}n (2) were prepared in aqueous solutions under the presence of NaHCO3, while palladium(II) monomers with the neutral 2-chloronicotinic and 2-bromonicotinic acid ligands, [PdCl2(2-ClnicH-N)2]·2DMF (3) and [PdCl2(2-BrnicH-N)2]·2DMF (4), were prepared in DMF/water mixtures (DMF = N,N′-dimethylformamide). The zigzag chains of water-bridged sodium ions are in turn bridged by [PdCl2(2-Clnic)2]2– moieties in 1 or by [PdBr2(2-Brnic)2]2– moieties in 2, leading to the formation of the infinite 2D coordination networks of 1 or 2. The DFT calculations showed the halosubstituents type (Cl vs Br) does not have an influence on the formation of either trans or cis isomers. The trans isomers were found in all reported compounds; being more stable for about 10 to 15 kJ mol–1. The 2D coordination networks 1 and 2 are more stabilized by the formation of Na–Ocarboxylate bonds, comparing to the stabilization of palladium(II) monomers 3 and 4 by hydrogen-bonding with DMF molecules. The difference in DFT calculated energy stabilization for 1 and 2 is ascribed to the type of halosubstituents and to the presence/absence of lattice water molecules in 1 and 2. The compounds show no antibacterial activity toward reference strains of Escherichia coli and Staphylococcus aureus bacteria and no antiproliferative activity toward bladder (T24) and lung (A549) cancer cell lines

From Simple Palladium(II) Monomers to 2D Heterometallic Sodium–Palladium(II) Coordination Networks with 2‑Halonicotinates

The 2D heterometallic sodium–palladium(II) coordination polymers with 2-halonicotinates [2-chloropyridine-3-carboxylate (2-chloronicotinate), 2-Clnic– and 2-bromopyridine-3-carboxylate (2-bromonicotinate), 2-Brnic–], {[Na2(H2O)2(μ-H2O)4PdCl2(μ-2-Clnic-N:O′)2]}n (1), and {[Na2(H2O)2(μ-H2O)4PdBr2(μ-2-Brnic-N:O′)2]·2H2O}n (2) were prepared in aqueous solutions under the presence of NaHCO3, while palladium(II) monomers with the neutral 2-chloronicotinic and 2-bromonicotinic acid ligands, [PdCl2(2-ClnicH-N)2]·2DMF (3) and [PdCl2(2-BrnicH-N)2]·2DMF (4), were prepared in DMF/water mixtures (DMF = N,N′-dimethylformamide). The zigzag chains of water-bridged sodium ions are in turn bridged by [PdCl2(2-Clnic)2]2– moieties in 1 or by [PdBr2(2-Brnic)2]2– moieties in 2, leading to the formation of the infinite 2D coordination networks of 1 or 2. The DFT calculations showed the halosubstituents type (Cl vs Br) does not have an influence on the formation of either trans or cis isomers. The trans isomers were found in all reported compounds; being more stable for about 10 to 15 kJ mol–1. The 2D coordination networks 1 and 2 are more stabilized by the formation of Na–Ocarboxylate bonds, comparing to the stabilization of palladium(II) monomers 3 and 4 by hydrogen-bonding with DMF molecules. The difference in DFT calculated energy stabilization for 1 and 2 is ascribed to the type of halosubstituents and to the presence/absence of lattice water molecules in 1 and 2. The compounds show no antibacterial activity toward reference strains of Escherichia coli and Staphylococcus aureus bacteria and no antiproliferative activity toward bladder (T24) and lung (A549) cancer cell lines

From Simple Palladium(II) Monomers to 2D Heterometallic Sodium–Palladium(II) Coordination Networks with 2‑Halonicotinates

The 2D heterometallic sodium–palladium(II) coordination polymers with 2-halonicotinates [2-chloropyridine-3-carboxylate (2-chloronicotinate), 2-Clnic– and 2-bromopyridine-3-carboxylate (2-bromonicotinate), 2-Brnic–], {[Na2(H2O)2(μ-H2O)4PdCl2(μ-2-Clnic-N:O′)2]}n (1), and {[Na2(H2O)2(μ-H2O)4PdBr2(μ-2-Brnic-N:O′)2]·2H2O}n (2) were prepared in aqueous solutions under the presence of NaHCO3, while palladium(II) monomers with the neutral 2-chloronicotinic and 2-bromonicotinic acid ligands, [PdCl2(2-ClnicH-N)2]·2DMF (3) and [PdCl2(2-BrnicH-N)2]·2DMF (4), were prepared in DMF/water mixtures (DMF = N,N′-dimethylformamide). The zigzag chains of water-bridged sodium ions are in turn bridged by [PdCl2(2-Clnic)2]2– moieties in 1 or by [PdBr2(2-Brnic)2]2– moieties in 2, leading to the formation of the infinite 2D coordination networks of 1 or 2. The DFT calculations showed the halosubstituents type (Cl vs Br) does not have an influence on the formation of either trans or cis isomers. The trans isomers were found in all reported compounds; being more stable for about 10 to 15 kJ mol–1. The 2D coordination networks 1 and 2 are more stabilized by the formation of Na–Ocarboxylate bonds, comparing to the stabilization of palladium(II) monomers 3 and 4 by hydrogen-bonding with DMF molecules. The difference in DFT calculated energy stabilization for 1 and 2 is ascribed to the type of halosubstituents and to the presence/absence of lattice water molecules in 1 and 2. The compounds show no antibacterial activity toward reference strains of Escherichia coli and Staphylococcus aureus bacteria and no antiproliferative activity toward bladder (T24) and lung (A549) cancer cell lines

From Simple Palladium(II) Monomers to 2D Heterometallic Sodium–Palladium(II) Coordination Networks with 2‑Halonicotinates

The 2D heterometallic sodium–palladium(II) coordination polymers with 2-halonicotinates [2-chloropyridine-3-carboxylate (2-chloronicotinate), 2-Clnic– and 2-bromopyridine-3-carboxylate (2-bromonicotinate), 2-Brnic–], {[Na2(H2O)2(μ-H2O)4PdCl2(μ-2-Clnic-N:O′)2]}n (1), and {[Na2(H2O)2(μ-H2O)4PdBr2(μ-2-Brnic-N:O′)2]·2H2O}n (2) were prepared in aqueous solutions under the presence of NaHCO3, while palladium(II) monomers with the neutral 2-chloronicotinic and 2-bromonicotinic acid ligands, [PdCl2(2-ClnicH-N)2]·2DMF (3) and [PdCl2(2-BrnicH-N)2]·2DMF (4), were prepared in DMF/water mixtures (DMF = N,N′-dimethylformamide). The zigzag chains of water-bridged sodium ions are in turn bridged by [PdCl2(2-Clnic)2]2– moieties in 1 or by [PdBr2(2-Brnic)2]2– moieties in 2, leading to the formation of the infinite 2D coordination networks of 1 or 2. The DFT calculations showed the halosubstituents type (Cl vs Br) does not have an influence on the formation of either trans or cis isomers. The trans isomers were found in all reported compounds; being more stable for about 10 to 15 kJ mol–1. The 2D coordination networks 1 and 2 are more stabilized by the formation of Na–Ocarboxylate bonds, comparing to the stabilization of palladium(II) monomers 3 and 4 by hydrogen-bonding with DMF molecules. The difference in DFT calculated energy stabilization for 1 and 2 is ascribed to the type of halosubstituents and to the presence/absence of lattice water molecules in 1 and 2. The compounds show no antibacterial activity toward reference strains of Escherichia coli and Staphylococcus aureus bacteria and no antiproliferative activity toward bladder (T24) and lung (A549) cancer cell lines