Poly[[aqua­tris­(μ4-benzene-1,2-dicarboxyl­ato)dilanthanum(III)] hemihydrate]

The asymmetric unit of the title coordination polymer, {[La2(C8H4O4)3(H2O)]·0.5H2O}n, contains two independent LaIII atoms, one of which is surrounded by eight carboxyl­ate-O atoms from six benzene-1,2-dicarboxyl­ate (BDC) anions in a bicapped trigonal–prismatic geometry. The other LaIII atom is nine-coordinated in a tricapped trigonal–prismatic geometry, formed by eight carboxyl­ate-O atoms from six BDC anions and a coordinated water mol­ecule. The BDC anions bridge the LaIII atoms, forming a two-dimensional polymeric complex parallel to (001). The crystal structure contains weak O—H⋯O and non-classical C—H⋯O hydrogen bonds. A C—H⋯π inter­action is also present in the crystal structure. The uncoordinated water molecule shows half-occupation

4,4′-Ethyl­enedipyridinium bis­(3,4,5-trihy­droxy­benzoate) sesquihydrate

The asymmetric unit of the title compound, C12H14N2 2+·2C7H5O5 −·1.5H2O, contains two 4,4′-ethyl­enedipyridinium cations, four gallate anions and three water mol­ecules. In the 4,4′-ethyl­enedipyridinium cations, the dihedral angles between the pyridinium rings are 4.3 (3) and 18.6 (3)°. Extensive classical N—H⋯O and O—H⋯O hydrogen bonding and weak C—H⋯O hydrogen bonding and C—H⋯π inter­actions are present in the crystal structure. π–π stacking is also observed, the centroid–centroid separations between the benzene and pyridine rings being 3.611 (3), 3.448 (3) and 3.536 (3) Å

4,4′-(Ethane-1,2-diyl)dipyridinium bis­(2-hy­droxy­benzoate)

In the crystal structure of the title compound, C12H14N2 2+·2C7H5O3 −, the cations and anions are linked via N—H⋯O hydrogen bonds and weak inter­molecular C—H⋯O inter­actions also occur. π–π stacking is observed between the nearly parallel benzene and pyridine rings [dihedral angle = 6.03 (8)°], the centroid–centroid separation being 3.7546 (16) Å. The 4,4′-(ethane-1,2-diyl)dipyridinium cation is centrosymmetric and the mid-point of the ethyl­ene C—C bond is located on an inversion center. An intra­molecular O—H⋯O hydrogen bond occurs in the anion

Poly[(μ4-pyridine-2,3-dicarboxyl­ato)lead(II)]

In the title coordination polymer, [Pb(C7H3NO4)]n, the PbII ion is eight-coordinated in a distorted square-anti­prismatic geometry formed by one pyridine N atom and seven carboxyl­ate O atoms from four pyridine-2,3-dicarboxyl­ate (pda) anions. In the pda anion, the dihedral angles between the pyridine ring and carboxyl­ate groups are 19.5 (6) and 73.3 (6)°. The carboxyl­ate groups of the pda anions bridge the Pb ions, forming a two-dimensional coordination polymer parallel to (100). Weak inter­molecular C—H⋯O hydrogen boning is present in the crystal structure

4-Amino­benzoic acid–4,4′-(propane-1,3-diyl)dipyridine (1/1)

In the crystal structure of the title compound, C13H14N2·C7H7NO2, the 4,4′-trimethyl­ene-dipyridine (TMDP) mol­ecule displays an approximately planar structure, the maximum atomic deviation excluding H atoms being 0.118 (2) Å and the dihedral angle between the pyridine rings 4.59 (10)°. The TMDP and 4-amino­benzoic acid (ABA) mol­ecules are linked by O—H⋯N and N—H⋯N hydrogen bonding, while ABA mol­ecules are linked by O—H⋯O hydrogen bonding. C—H⋯π interactions are also observed between the methyl­ene groups of TMDP mol­ecules and the benzene rings of ABA mol­ecules

catena-Poly[bis(μ3-3-aminobenzoato-κ4 N:O:O,O′)bis(μ2-3-aminobenzoato-κ3 O,O′:O)dilead(II)]

The PbII atom in the title compound, {[Pb2(C7H6NO2)4]}n, is chelated by two 3-aminobenzoato ligands in a distorted pentagonal-bipyramidal coordination geometry with five oxygen donors in the equatorial positions, one nitro­gen donor and one oxygen donor in the axial positions. Two mol­ecules are linked through a centre of inversion, forming a dinuclear entity. These entities are linked in a μ3-bridging mode through the amino N atom and two carboxyl­ate O atoms into a chain along the b axis. Classical inter­molecular N—H⋯O hydrogen bonding is observed in the structure. The supra­molecular structure is consolidated by π–π stacking inter­actions with centroid–centroid distances between benzene rings of 3.837 (8) Å

Poly[[tetra­aqua­bis­(μ3-pyridine-2,6-dicarboxyl­ato)(μ2-pyridine-2,6-dicarboxyl­ato)dilanthanum(III)] dihydrate]

There are two independent LaIII cations in the polymeric title compound, {[La2(C7H3NO4)3(H2O)4]·2H2O}n. One is nine-coordinated in an LaN2O7 tricapped trigonal–prismatic geometry formed by three pyridine-2,6-dicarboxyl­ate anions and two water mol­ecules, while the other is ten-coordinated in an LaNO9 bicapped square-anti­prismatic geometry formed by four pyridine-2,6-dicarboxyl­ate anions and two water mol­ecules. The two LaIII cations are separated by a non-bonding distance of 5.026 (3) Å. The pyridine-2,6-dicarboxyl­ate anions bridge the LaIII cations, forming a three-dimensional polymeric complex. The crystal structure contains extensive classical O—H⋯O hydrogen bonds and weak inter­molecular C—H⋯O hydrogen bonds. The crystal structure is further consolidated by π–π stacking between pyridine rings, the shortest centroid–centroid distance between parallel pyridine rings being 3.700 (5) Å

Poly[[hexa­aqua­(μ2-oxalato-κ4 O 1,O 2:O 1′,O 2′)bis(μ3-pyridine-2,4-dicarboxyl­ato-κ4 N,O 1:O 1′:O 4)dicerium(III)] monohydrate]

In the polymeric title compound, {[Ce2(C7H3NO4)2(C2O4)(H2O)6]·H2O}n, the Ce3+ cation is nine-coordinated in a distorted CeNO8 tricapped trigonal–prismatic geometry, formed by three pyridine-2,4-dicarboxyl­ate anions, one oxalate anion and three water mol­ecules. The mid-point of the oxalate anion is located on an inversion center. The oxalate and pyridine-2,4-dicarboxyl­ate anions bridge the Ce3+ cations, forming a two-dimensional polymeric complex parallel to (010). Inter­molecular classical O—H⋯O hydrogen bonding and weak C—H⋯O hydrogen bonding are present in the crystal structure and π–π stacking [centroid–centroid distance = 3.558 (2) Å] is observed between parallel pyridine rings of adjacent mol­ecules. The uncoordinated water mol­ecule shows an occupancy of 0.5

On the String Equation of the BKP Hierarchy

The Adler-Shiota-van Moerbeke formula is employed to derive the $W$-constraints for the $p$-reduced BKP hierarchy constrained by the string equation. We also provide the Grassmannian description of the string equation in terms of the spectral parameter.Comment: 17 page

Tunable Goos-H\"{a}nchen shift and polarization beam splitter in electro-optic crystals

We have investigated the tunable lateral shift and polarization beam splitting of the transmitted light beam through electro-optic crystals, based on the Pockels effect. The positive and negative lateral shifts could be easily controlled by adjusting the permittivity tensor, which is modulated by the external applied electric field. An alternative way to realize the polarization beam splitter was also proposed by the polarization-dependent lateral shifts. Numerical simulations for Gaussian-shaped incident beam have demonstrated the above theoretical results obtained by stationary phase method. All these phenomena have potential applications in optical devices.Comment: 5 pages, 7 figure