Synthesis,structure, and properties of M<Subscript>3</Subscript>VO<Subscript>2</Subscript>(SO<Subscript>4</Subscript>)<Subscript>2</Subscript> (M = Rb,Cs)

Vanadium(V) complexes of general composition M₃VO₂(SO₄)₂ (M = Rb, Cs) were synthesized by a solid-state route. The individuality of the synthesized compounds was proved by X-ray and neutron diffraction, vibrational spectroscopy, and microscopic analysis. The X-ray diffraction patterns of M₃VO₂(SO₄)₂ were indexed to fit the monoclinic system (space group P2/c, Z = 4) with the following unit cell parameters: a = 11.6487(2) Å, b = 8.4469(2) Å, c = 12.1110(2) Å, β = 109.483(1)°, V = 1123.43 ų (Rb); a = 12.0546(3) Å b = 8.7706(2) Å, c = 12.6496(3) Å, β = 109.843(2)°, V = 1257.99 ų (Cs). In the crystal structure of M₃VO₂(SO₄)₂, [VO₂(SO₄)₂]^3? complex anions can be discerned in which the vanadium atom is surrounded by five oxygen atoms: two oxygen atoms form short terminal V–O bonds, and three oxygen atoms are from the two sulfato groups, one of which acts as a monodentate ligand and the other acts as a bidentate chelating ligand.     

Synthesis,structure, and properties of V<Subscript>2</Subscript>O<Subscript>3</Subscript>(XO<Subscript>4</Subscript>)<Subscript>2</Subscript> (X = S,Se)

Compounds described as V₂O₃(XO₄)₂, where X = S or Se, were prepared from vanadium(V) oxide mixtures with concentrated sulfuric and selenic acids. The physicochemical properties of the products were studied; for V₂O₃(SeO₄)₂, the crystal structure was determined by powder X-ray diffraction and neutron diffraction, and its key differences from the structure of V₂O₃(SO₄)₂ were identified. V₂O₃(SeO₄)₂ crystallizes in the monoclinic system with the unit cell parameters a = 15.3831(2)Å, b = 5.54096(5)Å, c = 9.71644(7)Å, β = 111.886(1)°, V = 768.51ų, space group C2/c (no. 15).     

Synthesis,characterization and very strong luminescence of a new 3 D europium sulfate Eu<Subscript>2</Subscript>(H<Subscript>2</Subscript>O)<Subscript>4</Subscript>(SO<Subscript>4</Subscript>)<Subscript>3</Subscript>

A new 3 D europium sulfate Eu₂(H₂O)₄(SO₄)₃ 1 is synthesized solvothermally and characterized by single crystal X-ray diffraction, IR spectrum, TG, powder XRD, and ultraviolet excitation and emission spectrum. This compound crystallizes in the triclinic system with the space group P-1, a = 6.7520(9) Å, b = 9.1077(12) Å, c = 10.5910(14) Å, α = 94.432(2)°, β = 107.1170(10)°, γ = 99.160(2)°, V = 609.17(14) ų, Z = 2. The open framework of the title compound is an eight-membered ring channel along the crystallographic a-axis, which is built up by EuO₈ polyhedra and SO₄ tetrahedra.     

Synthesis,crystal structure,and vibrational spectra of M<Subscript>4</Subscript>V<Subscript>2</Subscript>O<Subscript>3</Subscript>(SO<Subscript>4</Subscript>)<Subscript>4</Subscript> (M = K,Rb, Cs)

Synthesis was performed and physicochemical properties were studied for the M₄V₂O₃(SO₄)₄ complexes, where M = K, Rb, or Cs. Their crystal structures were determined using the set of data from X-ray diffraction and neutron diffraction studies. All compounds crystallize in a triclinic lattice (space group \(P\bar 1\), Z = 2) with the parameters: a = 7.7688(2), 7.8487(1), 8.1234(1) Å; b = 10.4918(3), 10.8750(2), 11.1065(1) Å; c = 11.9783(4), 12.1336(2), and 11.8039(1) Å; α = 76.600(2)°, 77.910(1)°, 79.589(1)°; β = 75.133(2)°, 75.718(1)°, 87.939(1)°; γ = 71.285(2)°, 72.189(1)°, 75.567(1)°; V = 881.78(5), 945.42(3), 1014.34(2) ų for K, Rb, Cs, respectively. The structure of M₄V₂O₃(SO₄)₄ was found to be formed by discrete complex anions V₂O₃(SO₄) ₄ ^4? incorporating two oxygen-bridged vanadium atoms in a distorted octahedral oxygen environment. The sulfate groups are coordinated by the vanadium atoms in the chelating mode with a large scatter of S-O interatomic distances and OSO angles. Every VO₆ octahedron has a short terminal vanadium-oxygen bond with a length of about 1.6Å. The V₂O₃(SO₄) ₄ ^4? complex anions in potassium and rubidium compounds differ from that in Cs₄V₂O₃(SO₄)₄ in the type of symmetry and mutual spatial orientation. The vibrational spectra were presented and interpreted in line with the structural analysis data.     

Crystal structure of KPb<Subscript>2</Subscript>Cl<Subscript>5</Subscript> and KPb<Subscript>2</Subscript>Br<Subscript>5</Subscript>

The KPb₂Cl₅ and KPb₂Br₅ crystals are monoclinic (P2₁/c) with a microtwinned structure. X-ray analysis of chloride resulted in the parameters a = 8.854(2) Å, b = 7.927(2) Å, c = 12.485(3) Å; β = 90.05(3)°, d_calc = 4.78(1) g/cm³ (STOE STADI4, MoK_α, 2θ_max = 80°), R₁ = 0.0702 for 4094 F ≥ 4 σ(F) reflections. For bromide, a = 9.256(2) Å, b = 8.365(2) Å, c = 13.025(3) Å; β = 90.00(3)°, d_calc = 5.62(1) g/cm³ (Bruker P4, MoK_α, 2θ_max = 70°), R₁ = 0.0692 for 3076 F ≥ 4 (F) reflections.     

Crystal structure and properties of [M(NH<Subscript>3</Subscript>)<Subscript>5</Subscript>Cl](NO<Subscript>3</Subscript>)<Subscript>2</Subscript>, (M = Ir,Rh, Ru)

The crystal structures of compounds from the series [M(NH₃)₅Cl](NO₃)₂, (M = Ir, Rh, Ru) were described. The compounds crystallized in the tetragonal crystal system, space group I4, Z = 2. Crystal data for [Ir(NH₃)₅Cl](NO₃)₂ (I): a = 7.6061(1) Å, b = 7.6061(1) Å, c = 10.4039(2) Å, V = 601.894(16) ų, ρ_calc = 2.410 g/cm³, R = 0.0087; [Rh(NH₃)₅Cl](NO₃)₂ (II): a = 7.5858(5) Å, b = 7.5858(5) Å, c = 10.41357(7) Å, V = 599.24(7) ų, ρ_calc = 1.926 g/cm³, R = 0.0255; [Ru(NH₃)₅Cl](NO₃)₂ (III): a = 7.5811(6) Å, b = 7.5811(6) Å, c = 10.5352(14) Å, V = 605.49(11) ų, ρ_calc = 1.896 g/cm³, R = 0.0266. The compounds were defined by IR spectroscopy and XRPA and thermal analyses.     

Crystal structures of <Emphasis Type="Italic">trans</Emphasis>-[Re<Subscript>6</Subscript>S<Subscript>8</Subscript>(CN)<Subscript>2</Subscript>L<Subscript>4</Subscript>] complexes,L = pyridine or 4-methylpyridine

The structures of three novel octahedral rhenium cluster compounds [Re₆S₈(CN)₂(py)₄]·H₂O (1), [Re₆S₈(CN)₂(4-Mepy)₄] (2), [Re₆S₈(CN)₂(4-Mepy)₄]·4-Mepy (3) (py = pyridine, 4-Mepy = 4-methylpyridine) are determined by X-ray crystallography. Crystal data are: C2/m space group, a = 14.813(1) Å, b = 14.772(1) Å, c = 9.2122(6) Å, β = 119.085(2)°, V = 1761.7(2) ų, d _x = 3.318 g/cm³, R = 0.0585 (1); I4₁/amd space group, a = 16.0018(3) Å, c = 14.7186(5) Å, V = 3768.81(16) ų, d _x = 3.169 g/cm³, R = 0.0489 (2); P2₁/c space group, a = 9.0452(4) Å, b = 15.8065(7) Å, c = 15.2951(6) Å, β = 103.700(2)°, V = 2124.57(16) ų, d _x = 2.957 g/cm³, R = 0.0245 (3). Molecular cluster complexes interact via π-π stacking affording 3D frameworks in 1 and 2 and chains in 3.     

Synthesis,structural, spectral,thermal and comparative biological activity studies of [V<Subscript>2</Subscript>O<Subscript>2</Subscript>(SO<Subscript>4</Subscript>)<Subscript>2</Subscript>(H<Subscript>2</Subscript>O)<Subscript>6</Subscript>]

The hydrothermal reaction of a mixture of V₂O₅, VCl₃, 2,5-pyridinedicarboxylic acid and diluted H₂SO₄ for 68 h at 180°C gives a blue colored solution which yields prismatic blue crystals of IV ₂ ^IV O₂(SO₄)₂(H₂O)₆] (1) in 32% yield (based on V). Complex 1 was investigated by means of elemental analysis (C, H and S), TGA, FT-IR, manganometric titration, Single Crystal X-ray Diffraction Methods and also comparative antimicrobial activities. Crystal data for the compound: monoclinic space group P2₁/c and unit cell parameters are a = 7.3850(12) Å, b = 7.3990(7) Å, c = 12.229(2) Å, β = 108.976(12)° and Z = 2. Although structure of 1 as a natural mineral has been previously determined, this work covers new preparation method and full characterization of 1 along with comparison of antibacterial activity between 1 and the commercial vanadium(IV) oxide sulfate hydrate compounds, VOSO₄ · xH₂O (Riedel-de Haën and Alfa Aesar brand names). 1 was evaluated for the antimicrobial activity against gram-positive, gram-negative bacteria, yeasts and mould compared with the commercial VOSO₄ · xH₂O compounds. 1 showed weak activity against bacteria Bacillus cereus, Nocardia asteroides and yeast Candida albicans. A good antimicrobial activity was recorded against Cirtobacter freundii (15 mm). There are only a few reproducible well-defined vanadium(IV) starting materials to use for exploring the synthesis of new materials. VCl₄, VO(acac)₂, VOSO₄ · xH₂O and [V(IV)OSO₄(H₂O)₄] · SO₄ · [H₂N(C₂H₄)₂NH₂] are common starting materials for such applications. In addition to these compounds, 1 can be used as an oxovanadium precursor.     

Structure of A Coordination Polymer [Cu(En)<Subscript>2</Subscript>CrO<Subscript>4</Subscript>]<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The crystal structure of [Cu(En)₂CrO₄]_n (En is ethylenediamine) is determined: a = 14.7359(4) Å, b = 9.8083(3) Å, c = 14.2664(4) Å, V = 2061.98(10) ų, space group Cmce, Z = 8, d_x = 1.931 g/cm³. It is demonstrated that the studied phase is isostructural with [Сu(Еn)₂SO₄]_n. A pseudotetragonal copper atom coordination (Cu–N 2.0204 Å and 2.0244 Å, ∠N–Cu–N 84.73°) is completed to distorted octahedral by two oxygen atoms of chromate anions (Cu–O 2.433 Å and 2.380 Å).     

Phase formation in the Ag<Subscript>2</Subscript>MoO<Subscript>4</Subscript>-MgMoO<Subscript>4</Subscript>-Al<Subscript>2</Subscript>(MoO<Subscript>4</Subscript>)<Subscript>3</Subscript> system

The subsolidus region of the Ag₂MoO₄-MgMoO₄-Al₂(MoO₄)₃ ternary salt system has been studied by X-ray phase analysis. The formation of new compounds Ag_{1 ? x} Mg_{1 ? x} Al_{1 + x} (MoO₄)₃ (0 ≤ x ≤ 0.4) and AgMg₃Al(MoO₄)₅ has been determined. The Ag_{1 ? x} Mg_{1 ? x} Al_{1 + x} (MoO₄)₃ variable-composition phase is related to the NASICON type structure (space group R \(\bar 3\) c). AgMg₃Al(MoO₄)₅ is isostructural to sodium magnesium indium molybdate of the same formula unit and crystallizes in triclinic system (space group P \(\bar 1\), Z = 2) with the following unit cell parameters: a = 9.295(7) Å, b = 17.619(2) Å, c = 6.8570(7) Å, α = 87.420(9)°, β = 101.109(9)°, γ = 91.847(9)°. The compounds Ag_{1 ? x} Mg_{1 ? x} Al_{1 + x} (MoO₄)₃ and AgMg₃Al(MoO₄)₅ are thermally stable up to 790 and 820°C, respectively.     

Crystal structure of a new ternary molybdate in the Rb<Subscript>2</Subscript>MoO<Subscript>4</Subscript>-Eu<Subscript>2</Subscript>(MoO<Subscript>4</Subscript>)<Subscript>3</Subscript>-Hf(MoO<Subscript>4</Subscript>)<Subscript>2</Subscript> system

A ternary salt system Rb₂MoO₄-Eu₂(MoO₄)₃-Hf(MoO₄)₂ was studied in the subsolidus area by X-ray phase analysis. A novel ternary molybdate, Rb_4.98Eu_0.86Hf_1.11(MoO₄)₆, formed in the system. The Rb_4.98Eu_0.86Hf_1.11(MoO₄)₆ rubidium-europium-hafnium molybdate crystals were grown by solution-melt crystallization under the spontaneous nucleation conditions. The structure and composition of this compound were refined by single crystal X-ray diffraction (X8 APEX automated diffractometer, MoK _α radiation, 1753 F(hkl), R = 0.0183). The crystals are trigonal, a = b = 10.7264(1) Å, c = 38.6130(8) Å, V = 3847.44(9) ų, Z = 6, space group R \(\bar 3\) c. The three-dimensional mixed framework of the structure comprises Mo tetrahedra and two types of octahedra, (Eu,Hf)O₆ and HfO₆. The large cavities of the framework include two types of the rubidium atom. The distribution of the Eu³⁺ and Hf⁴⁺ cations over two crystallographic positions was refined.     

Structural investigation of [Au(en)<Subscript>2</Subscript>]Cl(ReO<Subscript>4</Subscript>)<Subscript>2</Subscript> and [Au(en)<Subscript>2</Subscript>](ReO<Subscript>4</Subscript>)<Subscript>3</Subscript> complexes

Novel complex salts [Au(en)₂]Cl(ReO₄)₂ (I) and [Au(en)₂](ReO₄)₃ (II), en = ethylenediamine, are obtained. Their crystal structures are determined by single crystal X-ray diffraction. Complex I crystallizes in the triclinic crystal system: a = 6.2172(7) Å, b = 7.1644(8) Å, c = 8.8829(8) Å, α = 96.605(4)°, β = 110.000(4)°, γ = 97.802(4)°, P-1 space group, Z = 1, d _x = 3.905 g/cm³; complex II crystallizes in the monoclinic crystal system: a = 15.244(2) Å, b = 7.6809(8) Å, c = 9.3476(12) Å, β = 127.004(3)°, C2 space group, Z = 4, d _x = 4.057 g/cm³.     

Crystal Structure of [UO<Subscript>2</Subscript>SO<Subscript>4</Subscript>{(CH<Subscript>3</Subscript>)HNCONH(CH<Subscript>3</Subscript>)}<Subscript>2</Subscript>]

The single crystals of [UO₂SO₄{(CH₃)HNCONH(CH₃)}₂] (I) were synthesized and studied by X-ray diffraction. The crystals are monoclinic, a = 6.847(1) Å, b = 14.259(3) Å, c = 14.297(3) Å, β = 93.451(4)°, space group P2₁/n, Z = 4. The main structural units of crystals I are ribbons whose composition coincides with the composition of the compound. The crystal chemical formula of the complex is AT³M ₂ ¹ (A = UO ₂ ²⁺ ).     

Synthesis and crystal structure of ternary molybdate compound K<Subscript>5</Subscript>Pb<Subscript>0.5</Subscript>Hf<Subscript>1.5</Subscript>(MoO<Subscript>4</Subscript>)<Subscript>6</Subscript>

Single crystals of triple molybdate of composition 5:1:3 K₅Pb_0.5Hf_1.5(MoO₄)₆ have been grown and their crystal structure has been solved from X-ray diffraction data (an automated diffractometer X8 APEX, MoK _α -radiation, 2173 F(hkl), R = 0.0321). Trigonal unit cell parameters are: a = b = 10.739(2) Å, c = 37.933(9) Å; V = 3789(1) ų, Z = 6, ρ_calc = 4.014 g/cm³, space group \(R\bar 3\). Three-dimensional mixed framework of the structure is formed by two types of MoO₄ tetrahedra and Pb and Hf octahedra linking via common O-vertices. Potassium atoms of three types occupy large vacancies in the framework.     

Synthesis and crystal structure of a 1D coordination polymer [Cd(NITpPy)<Subscript>2</Subscript>(N(CN)<Subscript>2</Subscript>)<Subscript>2</Subscript>)]<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

A novel one-dimensional chain complex [Cd(NITpPy)₂(N(CN)₂)₂)] _n (NITpPy = 2-(4′-pyridyl)-4,4,5,5-tetramethylimidazoline-1-oxyl-3-oxide) has been synthesized and characterized structurally. It crystallizes in the triclinic space group P \(\bar 1\) with a = 7.1742(13), b = 9.4913(17), c = 13.208(2) Å, α = 71.020(2)°, β=87.308(2)°, γ = 70.503(2)°, V = 799.8(3) ų, C₂₈H₃₂CdN₁₂O₄, Mr = 713.06, Z = 1, ρ _c = 1.48 g/cm³, μ(MoK _α) = 0.736 mm^?1, F(000) = 364, R = 0.0275 and wR = 0.0605 for 2702 observed reflections with I > 2σ(I). The crystal structure consists of infinite chains of [Cd(NITpPy)₂(N(CN)₂)₂)] units linked by dicyanamide anions [N(CN)₂]^?. Each Cd²⁺ ion is six-coordinated with the geometry of a distorted octahedron.     

Synthesis and X-ray investigation of Rb<Subscript>2</Subscript>[Pd(NO<Subscript>3</Subscript>)<Subscript>4</Subscript>] and Cs<Subscript>2</Subscript>[Pd(NO<Subscript>3</Subscript>)<Subscript>4</Subscript>]

Powder and single crystal X-ray diffraction studies have been performed for anhydrous nitrate complexes Rb₂[Pd(NO₃)₄] (I) and Cs₂[Pd(NO₃)₄] (II). Crystal data for I: a = 7.843(1) Å, b = 7.970(1) Å, c = 9.725(1) Å; β = 100.39(1)°, V = 597.9(1) Å ³, space group P2₁/c, Z = 2, d _calc = 2.918 g/cm³; for II: a = 10.309(2) Å, b = 10.426(2) Å, c = 11.839(2) Å; β = 108.17(3)°, V = 1209.0(4) ų, space group P2₁/c, Z = 4, d ^calc = 3.408 g/cm³. The structures are formed by isolated [Pd(NO₃)₄]^2? complex anions and alkali metal cations. The plane-square environment of the Pd atom is formed from the oxygen atoms of the monodentate nitrate groups. The geometrical characteristics of the complex anions are analyzed. Compound II has a short contact Pd...Cs 3.252 Å.     

Supramolecular 2D structure of [Co(2-Me-Pyz)<Subscript>2</Subscript>(H<Subscript>2</Subscript>O)<Subscript>4</Subscript>](NO<Subscript>3</Subscript>)<Subscript>2</Subscript>

The complex [Co(2-Me-Pyz)₂(H₂O)₄](NO₃)₂ is synthesized and its structure is determined. The crystals are monoclinic: space group P2₁/n, a = 10.685(2) Å, b = 6.837(1), c = 12.515(3) Å, β = 91.84(3)°, V = 913.8(3) ų, ρ_calcd = 1.042 g/cm ³, Z = 2. The Co²⁺ ion (in the inversion center) is coordinated at the vertices of the distorted octahedron by two nitrogen atoms of methylpyrazine and four oxygen atoms of the water molecules (Co(1)–N(1) 2.180(3), average Co(1)–O(w) 2.079(3) Å, angles at the Co atom 87.9(1)–92.1(1)°). Supramolecular pseudometallocycles are formed in the structure through the O(w)–H…N(1) hydrogen bonds between the coordinated H₂O molecules and the terminal nitrogen atoms of the 2-methylpyrazine molecules. Their interaction results in the formation of supramolecular layers joined by the NO₃ groups into a three-dimensional framework.     

High-temperature heat capacity and thermodynamic properties of pyrovanadate Pb<Subscript>2</Subscript>V<Subscript>2</Subscript>O<Subscript>7</Subscript> and orthovanadate Pb<Subscript>3</Subscript>(VO<Subscript>4</Subscript>)<Subscript>2</Subscript>

The heat capacities of Pb₂V₂O₇ and Pb₃(VO₄)₂ as a function of temperature in the range 350–965 K have been studied by the differential scanning calorimetry method. The C_P = f(T) curve for Pb₂V₂O₇ is described by the equation C_p = (230.76 ± 0.51) + (73.60 ± 0.50)×10^-3T ? (18.38 ± 0.54)×10⁵T^-2 in the entire temperature range. For Pb₃(VO₄)₂, there is a well-pronounced extreme point in the C_P = f(T) curve at T = 371.5 K, which is caused by the existence of a structural phase transition. The thermodynamic properties of the oxide compounds have been calculated.     

Crystal structures of Ti(NH<Subscript>4</Subscript>)<Subscript>2</Subscript>P<Subscript>4</Subscript>O<Subscript>13</Subscript> and Sn(NH<Subscript>4</Subscript>)<Subscript>2</Subscript>P<Subscript>4</Subscript>O<Subscript>13</Subscript>

The characteristics of crystal structures of the titanium(IV) diammonium (Ti(NH₄)₂P₄O₁₃) and tin(IV) diammonium (Sn(NH₄)₂P₄O₁₃) tetraphosphates, which are isostructural with similar silicon(IV) and germanium(IV) salts, have been obtained by the Rietveld method using X-ray powder diffraction data. The compounds crystallize in the triclinic system, space group P \(\overline 1 \), Z = 2, a = 15.0291(7) Å, b = 7.9236(4) Å, c = 5.0754(3) Å, α = 99.168(3)°, β = 97.059(3)°, γ = 83.459(3)° for Ti(NH₄)₂P₄O₁₃ and a = 15.1454(7) Å, b = 8.0103(5) Å, c = 5.1053(3) Å, α = 99.898(6)°, β = 96.806(3)°, γ = 83.881(4)° for Sn(NH₄)₂P₄O₁₃. The structure is refined in the isotropic approximation using the pseudo-Voigt function: R _p = 0.077, R _Bragg = 0.045, R _F = 0.057 for Ti(NH₄)₂P₄O₁₃; R _p = 0.082, R _Bragg = 0.044, R _F = 0.046 for Sn(NH₄)₂P₄O₁₃. The hydrogen atoms of the ammonium cations are placed in the calculated positions. A comparative analysis of the structures of the compounds of the M^IV(NH₄)₂P₄O₁₃ (M^IV = Si, Ge, Ti, Sn) series has been carried out.     

Three- and four-coordinated silver(I) ions in the coordination polymers [Ag(Me<Subscript>4</Subscript>Pyz)<Subscript>2</Subscript>]PF<Subscript>6</Subscript> and [Ag(2,3-Et<Subscript>2</Subscript>Pyz)<Subscript>2</Subscript>]PF<Subscript>6</Subscript>

The coordination polymers [AgPF₆(Me₄Pyz)₂] (I) and [AgPF₆(2,3-Et₂Pyz)₂] (II) were synthesized, and their structures were determined. Crystals of I are monoclinic, space group C2/c, a = 10.213(2) Å, b = 16.267(3) Å, c = 12.663(3) Å, β = 92.90(3)°, V = 2102.1(7) ų, ρ_calcd = 1.660 g/cm³, Z = 4. The structure of I is built of polymeric zigzag [Ag(C₈H₁₂N₂)] _∞ ⁺ chains and octahedral [PF₆] anions. The coordination polyhedron of the Ag⁺ ion is a flat triangle. Crystals of II are tetragonal, space group P \(\bar 4\)2(1)/c,a = b = 10.641(1) Å, c = 18.942(1) Å, V = 2144.6(2) ų, ρ_calcd = 1.627 g/cm³, Z = 4. In the structure of II, 2D cationic layers of fused square rings exist; the rings consist of four Ag⁺ cations linked by four bridging ligands of diethylpyrazine Et₂Pyz. The coordination polyhedron of the Ag⁺ ion is an irregular four-vertex polyhedron.