Application of coset representations to the construction of symmetry adapted functions

Summary A coset representation (G(/G _i )), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupG _i , is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/G _i ) orbit belongs to theG _i site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.     

Graphical Models of Characters of Groups

A graphical method of generating one- and (some) two-dimensional characters () has been developed on the basis of a reduced homomer set, which has been derived from a new concept of negative graphs. Thus, a homomer set H[G(/G _i )]={h₁,...,h _d–1,h _d } (d=|G|/|G _i |) has been generated from a regular body of G so that it has been governed by the coset representation G(/G _i ). The homomer set has been reduced into a reduced homomer set []={h₁,...,h _d–1}, where we have placed h _d –(h₁++h _d–1) in terms of negative graphs. The action of the symmetry operations of G on the reduced homomer set [] has graphically generated a one- or (some) two-dimensional character (). The versatility of the graphical method has been tested by using C _3v , D _2h , C _2h , C _2v , D _3h , and C _3h as examples. The graphical method has been compared with an alternative algebraic generation using marks (or markaracters), i.e., =G(/G _i )–G(/G).     

On p -nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S ₄-free and every minimal subgroup of P ∩ G ^N is c-supplemented in N _G(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

Synthesis and Structure of Two Keggin-Type Heteropolyanions: [VMo12O40]3n− n (1) and [H3PMoVMoVI 11O40]1− (2)

Two Keggin-type heteropolyanions were synthesized and characterized by X-ray crystal structure and elemental analysis as well as infrared spectroscopy. Both K₃[VMo₁₂O₄₀]19H₂O (1) and [N _i (H₂O)₆][H₃PMo^VMo^VI ₁₁O₄₀]₂30 H₂O (2) were prepared in aqueous solution. Compound 1 crystallized in the space group Pm-3m, a=10.6513(1) Å, V=1208.4(3) ų, Z=1. Compound 2 crystallized in the space group R-3 with a=b=13.9669(2) Å, c=42.0075(5) Å, V=7096.71(2) ų, Z=3. The compound 1 contains a {K₆VMo₁₂O₄₀} group in which six potassium ions form a regular {K₆} octahedron. The heteropolyanion [VMo₁₂O₄₀]^3– was capped by six potassium ions and enclosed by {K₆} octahedron. A three-dimensional structure was formed by the buildup of {K₃[VMo₁₂O₄₀]} _n . Compound 2 contains a one-electron reduced heteropolyanion [H₃PMo^VMo^VI ₁₁O₄₀]^1–. Ni²⁺ coordinated by six water molecules as the counter cation balances the negative charge of the molecule.     

Theoretical refinements of theendo and theexo structures of tetraborane(8) carbonyl

The molecular structures of theendo (1a) andexo (1b) isomers of B₄H₈CO have been optimized at the ab initio MP2(Full)/6-31G^* level of theory. The agreement of the computed geometrical parameters with the recently published electron-diffraction (GED) data is very good, even though a number of geometrical constraints were applied in the experimental determination. The IGLO (individual gauge for localized orbitals)¹¹B NMR chemical shifts, calculated at the II//MP2/6-31G^* level, are also in accord with experiment. The formation of1a and1b by association of B₄H₈ and CO is computed to be exothermic by 22.8 and 22.2 kcal/mol, respectively, at the MP2(Full)/6-31G^*//MP2(Full)/6-31G^* + ZPE(6-31G^*) level of theory. The Lewis acid strength of B₄H₈ toward CO is comparable to that of BH₃.     

Syntheses,structures and photochromic properties of two tetrazolylazo-based K+ and Cd2+ complexes

A 5,5′-azotetrazole (H₂AT)-based K⁺ complex of [K₃(AT)_1.5(H₂O)₄] (1) and another 5-azotetrazolyl salicylic acid (H₃ASA)-based Cd²⁺ complex [Cd(H₂ASA)₂(H₂O)₂] (2) have been synthesized and characterized by single-crystal X-ray diffraction analysis. Complex 1 shows a 3-D structure formed by two μ_7- and μ_8-bridging trans-AT^2? ligands and μ_2-bridging waters linking three kinds of K⁺ ions. Complex 2 features a 1-D chain structure in which each trans-enol H₂ASA^? anion is a μ_2-bridging ligand with one N(tetrazole) atom and one O(carboxyl) coordinating to two different Cd²⁺ ions. The photochromic and photoluminescent properties for aqueous solutions of 1 and 2 were investigated.     

A model developed for acid dissolution thermodynamics of a Turkish bentonite

A model was proposed to calculate some thermodynamic parameters for the acid dissolution process of a bentonite containing a calcium-rich smectite as clay mineral along with quartz, opal and feldspar as impurities. The bentonite sample was treated with H₂SO₄ by applying dry method in the temperature range 50–150°C for 24 h. The acid content in the dry bentonite-sulphuric acid mixture was 45 mass%. The total content (x) of Al₂O₃, Fe₂O₃ and MgO remained in the undissolved sample after treatment was taken as an equilibrium parameter. An apparent equilibrium constant, K _a, was calculated for each temperature by assuming K _a=(x _m−x)/x where x _m is the total oxide content of the natural bentonite. Also, an apparent change in Gibbs free energy, ΔG _a^o, was calculated for each temperature by using the K _a value. The graphs of lnK _a vs. 1/T and ΔG _a^o vs. T were drawn and then the real change in both the enthalpy, ΔH ^o and the entropy, ΔS ^o, values were calculated from the slopes of the straight lines, respectively. Inversely, real ΔG ^o and K values were calculated from the real ΔH ^o and ΔS ^o values through ΔG ^o = −RT ln K = ΔH ^o − TΔS ^o equation. The best ΔH ^o and ΔS ^o fittings to this relation were found to be 65687 J mol⁻¹ and 164 J mol⁻¹K⁻¹, respectively.     

Optimal use of the recurrence relations for the evaluation of molecular integrals over Cartesian Gaussian basis functions

We consider the tree search problem for the recurrence relation that appears in the evaluation of molecular integrals over Cartesian Gaussian basis functions. A systematic way of performing tree search is shown. By applying the result of tree searching to the LRL2 method of Lindh, Ryu, and Liu (LRL) (J. Chem. Phys., 95 , 5889 1991), which is an auxiliary function-based method, we obtain significant reductions of the floating point operations (FLOPS) counts in the K⁴ region. The resulting FLOPS counts in the K⁴ region are comparable up to [dd|dd] angular momentum cases to the LRL1 method of LRL, currently the method requiring least FLOPS for [dd|dd] and higher angular momentum basis functions. For [ff|ff], [gg|gg], [hh|hh], and [ii|ii] cases, the required FLOPS are 24, 40, 51, and 59%, respectively, less than the LRL1 method in the K⁴ region. These are the best FLOPS counts available in the literature for high angular momentum cases. Also, there will be no overhead in either the K² or K⁰ region in implementing the present scheme. This should lead to more efficient codes of integral evaluations for higher angular momentum cases than any other existing codes. © 1993 John Wiley & Sons, Inc.     

Ionization Constants of o-Phthalic Acid and Standard pH Values of Potassium Hydrogen Phthalate Buffer Solutions in Glycerol + Water Solvent Mixtures at Normal and Subzero Temperatures

Determination of primary standards for pH measurements in glycerol + water solventmixtures has been carried out based on reversible emf measurements of the cellPt|H₂|KHPh (m _PS) + KCl (m _Cl)|AgCl|Ag|Ptwhere KHPh denotes the potassium hydrogen phthalate buffer solution of molalitym _PS = 0.05 mol-kg^–1, at glycerol mass fractions w _G = 0.2 and 0.4, within thetemperature range –10 to 40°C. A multilinear regression procedure as a functionof electrolyte molality, glycerol mass fraction w _G, and temperature T has beenapplied for the data processing leading to the values of primary standards pH_PS.These can be represented by the following regression equationpH_PS = (4.007037±0.001113) + (3.55844±0.01776)x _G+(0.39622±0.01410)z + (4.3084±0.3377)z ²– (50.66±10.53)x _G z ² + (457.10±78.48)x _G ² z ²where z = (T – 298.15)/298.15. Parallel values of the first ionization constantof o-phthalic acid (H₂Ph; benzene-1,2-dicarboxylic acid, the parent acid of KHPh),which are essential for the above calculations, have been determined fromreversible emf measurements of the cellPt|H₂|H₂Ph (m ₁) + KHPh (m ₂) + KCl (m ₃)|AgCl|Ag|Ptover the range of solvent composition and temperatures mentioned above.     

Effects of sizes of nano-copper oxide on the equilibrium constant and thermodynamic properties for the reaction in nanosystem

Thermodynamic properties and equilibrium constant of reaction in nanosystems were analyzed theoretically. The effects of sizes of nano-CuO on thermodynamic properties and equilibrium constant were studied using the reaction of nano-copper oxide and sodium bisulfate as a system. The experimental results indicate that with the sizes of reactant decreasing, the molar Gibbs free energy (Δ_rG_m), the molar enthalpy (Δ_rH_m) and the molar entropy (Δ_rS_m) decrease, but the equilibrium constant (K) increases and there are linear trends between the reciprocal of sizes for nano-CuO and the values of Δ_rG_m, Δ_rH_m, Δ_rS_m and Ln K, which are in agreement with the theoretical analysis.     

Synthesis,Structural, and Physicochemical Study of First- and Second-Generation Melamine-Based Dendrimers

Abstract

First-tier (G₁) 2,4,6-tridiethymalonate-triazine (2,4,6-TDEMTA) and second-tier (G₂) 2,4,6-hexadiethylmalonate-triazine (2,4,6-HDEMTA) dendrimers were prepared with melamine (1,3,5-triazine) as core (G₀) and sodium malonate ester for bifurcation of chains. The trichlorotriazine (TCT) and sodium diethyl malonate (SDEM) aqueous solutions were mixed, and a creamy white precipitate of G₁ was obtained. The G₁ was hydrolyzed in alkaline medium to prepare 2,4,6-tridisodiummalonate-triazine (2,4,6-TDSMTA) by replacing ?C₂H₅ groups of COOC₂H₅ by Na, which was further neutralized by dilute HCl to obtain 2,4,6-trimalonicacid-triazine (2,4,6-TMATA). The TMATA was treated with PCl₅(s) to form –COCl, which was treated with SDEM to form the G₂.     

PI indices of pericondensed benzenoid graphs

The Padmakar–Ivan (PI) index is a graph invariant defined as the summation of the sums of n _eu (e|G) and n _ev (e|G) over all the edges e = uv of a connected graph G, i.e., , where n _eu (e|G) is the number of edges of G lying closer to u than to v and n _ev (e|G) is the number of edges of G lying closer to v than to u. An efficient formula for calculating the PI index of a class of pericondensed benzenoid graphs consisting of three rows of hexagonal of various lengths.     

An evaluation of coal fly ash as an adsorbent for the removal of methylene blue from aqueous solutions: kinetic and thermodynamic studies

Abstract

In this study the effect of the dose and particle size of the adsorbent, initial dye concentration, initial pH, contact time and temperature were investigated for the removal of by means of fly ash (FA) methylene blue (MB) from an aqueous solution. The FA dose was found to be 2.0?g and the under 270 mesh sized particles were found to be effective particles for adsorption. The adsorption process reached its maximum value at 0.5?mg/L dye concentration and attained equilibrium within 10?minutes. The adsorption isotherm was found to follow the Langmuir model. The estimated adsorption free energy (ΔG^o), enthalpy change (ΔH^o), and entropy change (ΔS^o) for the adsorption process were ?37.77?kJ mol^?1, ?13.44?kJ mol^?1 and 122 J mol^?1 K^?1 respectively at 298 K. The maximum adsorption capacity is 0,12?mg g^?1 at 298 K and 0,07?mg g^?1 at 398 K. The adsorption process was exothermic, feasible and spontaneous. The positive value of ΔS^o shows the affinity of FA for MB while the low value of ΔG^o suggests a physical adsorption process.     

Homo- and heterometallic complexes based on polytopic carboxylic acid: synthesis,characterization, and property

Two heterometallic [K₄M₄(HL)₄(H₂O)₁₂] (M=Co (1), Ni (2)) and two homometallic [M₂L(H₂O)₇]?·?2H₂O ((M=Co (3), Ni (4)) (H₄L?=?(2-(bis(carboxymethyl)amino) terephthalic acid) have been synthesized and characterized by elemental analysis, FT-IR spectrum, and single-crystal X-ray diffraction. The isomorphous 1 and 2 contain K⁺ and M²⁺, in which K⁺ were bridged with M²⁺ through μ-HL^3? and μ-H₂O, leading to 2-D layer structures. The isomorphous 3 and 4 show homometallic binuclear complexes with μ-HL^3? as the bridging ligand. Various H-bonds including different H-bond helical chains form, by which 3 and 4 assemble into 3-D supramolecular frameworks. TG analysis indicates that the decomposition temperatures are [K₄M₄(HL)₄(H₂O)₁₂] (1)?>?[M₂L(H₂O)₇]?·?2H₂O (3)?>?H₄L.     

含2,6-二甲酰胺基吡啶单元的大环合成、结构及其对四面体构型阴离子的络合性质

采用前体二胺N,N’-(3-胺基苯基)-2,6-二甲酰亚胺吡啶(1)和5,5’-亚甲基双水杨醛(2)进行缩合反应得到含酚羟基的 [1+1] 席夫碱大环L1, 将其中席夫碱C=N双键还原得到环状骨架更具柔性的饱和大环L2. 通过1H NMR、FABMS和元素分析等对大环组成进行了表征, 并通过X射线单晶衍射解析了两个大环的晶体结构. 采用UV-Vis 光谱滴定技术对大环与系列阴离子的络合作用进行了考察, 结果表明席夫碱大环L1对四面体构型的阴离子H2PO4-、 HP2O73-和H2P2O72-有明显的选择性识别作用, 进一步通过UV-Vis 光谱、核磁滴定等技术获得了识别反应的配位比及平衡常数(K), L1对3个磷酸阴离子的络合能力依H2PO4-、HP2O73-和H2P2O72-顺序递增。     

Experimental and computational investigations on the high binding-selectivity of pyrimidine derivatives by a pillar[5]arene

The binding selectivity of an adenine-monofunctionalized pillar[5]arene (H) with a series of pyrimidine derivatives were investigated through ¹H NMR experiments and density functional theory (DFT) study. High binding-selectivity was demonstrated. Typically, H displayed very strong binding strength with 6-(2,4-dioxo-3, 4-dihydropyrimidin-1 (2H)-yl)hexanenitrile (G1) [K_a >10⁵ M^?1], up to about 3000-fold as compared with 1-hexylpyrimidine-2,4(1H, 3H)-dione (G5) [K_a = 31 M^?1]. The strong binding ability of H with G1 was due to the cooperative multiple hydrogen bond, dipole-dipole, C-H···π and π···π interactions. The high binding-selectivity was also verified by calculation results. The calculated interaction energy (ΔE_i) of G1?H was ?12.92 Kcal·mol^?1 while that of G5?H was ?2.85 Kcal·mol^?1.     

Packing in honeycomb networks

Molecules arranging themselves into predictable patterns on silicon chips could lead to microprocessors with much smaller circuit elements. Mathematically, assembling in predictable patterns is equivalent to packing in graphs. An H-packing of a graph G is a set of vertex disjoint subgraphs of G, each of which is isomorphic to a fixed graph H. If H is the complete graph K ₂, the maximum H-packing problem becomes the familiar maximum matching problem. In this paper we give algorithms to find a perfect packing of HC(n) with P ₆ and K _1,3 when n is even and thus determines their packing numbers. Further we also study the packing of HC(n) with 1, 3-dimethyl cyclohexane.     

Aqua substitution from mononuclear Pt(II) complexes with 2-(pyrazol-1-ylmethyl)quinoline non-leaving ligands

The rates of aqua substitution from [Pt{2-(pyrazol-1-ylmethyl)quinoline}(H₂O)₂](ClO₄)₂, [Pt(H₂Qn)], [Pt{2-(3,5-dimethylpyrazol-1-ylmethyl)quinoline}(H₂O)₂](ClO₄)₂, [Pt(dCH₃Qn)], [Pt{2-[(3,5-bis(trifluoromethyl)pyrazol-1-ylmethyl]quinoline}(H₂O)₂](ClO₄)₂, [Pt(dCF₃Qn)], and [Pt{2-[(3,5-bis(trifluoromethyl)pyrazol-1-ylmethyl]pyridine}(H₂O)₂](ClO₄)₂, [Pt(dCF₃Py)], with three sulfur donor nucleophiles were studied. The reactions were followed under pseudo-first-order conditions as a function of nucleophile concentration and temperature using a stopped-flow analyzer and UV/visible spectrophotometry. The substitution reactions proceeded sequentially. The second-order rate constants for substituting the aqua ligands in the first substitution step increased in the order Pt(dCH₃Qn) < Pt(dCF₃Qn) < Pt(H₂Qn) < Pt(dCF₃Py), while that of the second substitution step was Pt(dCH₃Qn) < Pt(dCF₃Qn) < Pt(dCF₃Py) < Pt(H₂Qn). The reactivity trends confirm that the quinoline substructure in the (pyrazolylmethyl)quinoline ligands acts as an apparent donor of electron density toward the metal center rather than being a π-acceptor. Measured pK_a values from spectrophotometric acid–base titrations were Pt(H₂Qn) (pK_a1 = 4.56; pK_a2 = 6.32), Pt(dCH₃Qn) (pK_a1 = 4.88; pK_a2 = 6.31), Pt(dCF₃Qn) (pK_a1 = 4.07; pK_a2 = 6.35), and Pt(dCF₃Py) (pK_a1 = 4.76; pK_a2 = 6.27). The activation parameters from the temperature dependence of the second-order rate constants support an associative mechanism of substitution.     

Adsorption of phenols from contaminated water through titania-silica mixed imidazolium based ionic liquid: Equilibrium,kinetic and thermodynamic modeling studies

The present study describes the synthesis and characterization of titania-silica mixed imidazolium based ionic liquid (Ti-Si-IL) as well as evaluation of its adsorption behavior towards the 2,4-dinitrophenol (2,4-DNP) and 2,4,6-trichlorophenol (2,4,6-TCP). Synthesized Ti-Si-IL adsorbent was characterized by Fourier transform infrared spectroscopy (FT-IR), field emission scanning electron microscopy (FESEM), BET surface area Brunauer-Emmett-Teller (BET), thermogravimetric analysis (TGA) and elemental analysis (CHN). The adsorption of 2,4-DNP and 2,4,6-TCP on Ti-Si-IL was investigated systematically by evaluating the effects of adsorbent dosage, initial pH, contact time and temperature. Satisfactory adsorption 95% and 65% for 2,4-DNP and 2,4,6-TCP was observed at pH 4 and 6, respectively. The kinetic results for 2,4-DNP and 2,4,6-TCP on Ti-Si-IL indicated that the kinetic data follows pseudo-second-order model (R² = 0.9985 and 0.9750, respectively). Adsorption isotherms were fitted well by the Langmuir model for 2,4-DNP (q_m = 44.64 mg g^?1 at 318 K) and Freundlich model for 2,4,6-TCP (K_F = 0.63 mg g^?1 at 318 K). The +ΔH° and -ΔG° values demonstrated that the adsorption of 2,4-DNP was endothermic and spontaneous in nature. While the -ΔH° and +ΔG° values for 2,4,6-TCP adsorption demonstrated exothermic and comparatively nonspontaneous. During the removal process, the role of different functional groups, cyclic structure was monitored and found that the ionic property as well as π-π interactions of host molecules played important role in the extent of adsorption.     

Lennard-Jones Elastic Moduli by Liquid Structure Integral Equations and Molecular Dynamics Computer Simulations

Abstract

The infinite frequency shear modulus, G∞, and compressional modulus, K∞, of the Lennard-Jones, LJ, fluid have been determined over essentially the whole phase diagram at densities below the solid-fluid coexistence line using PY, HNC, and Rogers and Young (RY) closures of the Ornstein-Zernike relation. At low density PY is best at reproducing simulation G∞, and K∞, whereas close to the coexistence line, above the critical temperature, the RY closure is best and is remarkably accurate. Agreement is poorest for all three closures below T_c in the liquid phase.