The PI index of phenylenes

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., PI(G) = ∑ _e∈E(G)[n _eu (e|G) + n _ev (e|G)], 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 phenylenes is given, and a simple relation is established between the PI index of a phenylene and of the corresponding hexagonal squeeze.     

On unicyclic conjugated molecules with minimal energies

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let U(k) be the set of all unicyclic graphs with a perfect matching. Let C _g(G) be the unique cycle of G with length g(G), and M(G) be a perfect matching of G. Let U ⁰(k) be the subset of U(k) such that g(G)≡ 0 (mod 4), there are just g/2 independence edges of M(G) in C _g(G) and there are some edges of E(G)\ M(G) in G\ C _g(G) for any G∈U ⁰(k). In this paper, we discuss the graphs with minimal and second minimal energies in U ^*(k) = U(k)\ U ⁰(k), the graph with minimal energy in U ⁰(k), and propose a conjecture on the graph with minimal energy in U(k).      

Development of dissociative force field for all-atomistic molecular dynamics calculation of fracture of polymers

A dissociative force field for all-atomistic molecular dynamics calculations has been developed to investigate impact fracture of polymers accompanying dissociation of chemical bonds of polymer main chain. Energy of dimer molecules was evaluated as a function of both bond-length b and bond-angle θ by CASPT2 calculations, whose quality is enough to describe dissociation of chemical bonds. Because we found that the bond dissociation energy D decreases with increasing bond-angle, we employed the Morse-type function V_Bond(b, θ) = {D − V_Angle(θ)}[1 − exp{−α(b − b₀) − β(b − b₀)²}] where a quartic function V_Angle(θ) = k₁(θ − θ₀) + k₂(θ − θ₀)² + k₃(θ − θ₀)³ + k₄(θ − θ₀)⁴ . This function reproduced well the CASPT2 potential energy surface in a wide range of b and θ. The parameters have been obtained for four popular glassy polymers, polyethylene, poly(methyl methacrylate), poly(styrene), and polycarbonate. © 2019 Wiley Periodicals, Inc.     

Mono-Heteroatom Effects on Singlet–Triplet Energy Gaps of Divalent Five-Membered Ring XC3H3C (X = CH,N, P,and As)

Heteroatom effects were investigated on singlet–triplet energy gaps of divalent five-membered ring XC ₃ H ₃ C, 1 _X and 2 _X (X = CH, N, P, and As) at B3LYP/6-311++G?? level. All triplet states of 1 _X and 2 _X are more stable than the related singlet state. Δ G _{s ? t} between singlet and triplet states of 1 _X and 2 _X were changed in the order (in kcal/mol): 1 _As (14.43) > 1 _P (10.07) > 1 _CH (9.60) > 1 _N (8.50) and 2 _N (11.56) > 2 _CH (9.60) > 2 _P (3.07) > 2 _As (2.13).     

Radial electron-pair densities in momentum space and shell structure of atoms

 The radial electron-pair intracule (relative motion) H(u) and extracule (center-of-mass motion) D(R) densities in position space were known to reveal four types of maxima which are related to the four inner electron shells, K, L, M, and N, of atoms. The corresponding radial electron-pair intracule Hˉ(v) and extracule Dˉ(P) densities in momentum space are studied for the 102 atoms from He (atomic number Z=2) to Lr (Z=103). The densities Hˉ(v) and Dˉ(P) are found to have either one maximum or two maxima, and the numbers of maxima in Hˉ(v) and Dˉ(P) are the same for 98 atoms. For these atoms, the locations υ _max and P _max and the heights Hˉ _max and Dˉ _max of the corresponding maxima satisfy the approximate relations υ _max ≅ 2P _max and Hˉ _max ≅Dˉ _max /2. On the basis of their Z-dependence, the maxima in Hˉ(v) and Dˉ(P) of the 102 atoms are classified into five types. Shell-pair decompositions of the radial densities show that these maxima reflect five outer electron shells of atoms. Received: 24 January 2001 / Accepted: 12 March 2001 / Published online: 13 June 2001     

The stability condition of phase equilibrium for various concentration variables

The stability conditions of phase equilibrium for various concentration variables are educed according to thermodynamic principle. When a system with k components arrives at stable equilibrium, if the mole number n_i or the mole fraction y_i(=n_i/n_k) or molality m_i[= n_i/(n_kM_k)] of component i(i = 1,2,...,k − 1) are elected as concentration variables, thermodynamic theory is able to confirm that the sign of every order determinant composed of the second-order partial differential of chemical potential with respect to these concentration variables is positive; if the mole fraction x_i(= n_i/n) or mass fraction w_i(= n_iM_i/W are elected as the concentration variables, thermodynamic theory is only able to confirm that the sign of (k−1) order determinant is positive; if molarity c_i(= n_i/V are elected as the concentration variables, thermodynamic theory is not able to confirm the sign of every order determinant.     

Rationalization of the ratio of excimer-to-monomer fluorescence emission intensity in bichromophoric diesters of 1-pyrenoic acid and mono-, di-, tri-, and tetraethylene glycols

The ratio of excimer to monomer emission intensities, denoted by I_D/I_M, was measured for Py–COO(CH₂CH₂O)_mCO ? Py, where Py denotes the 1-pyrenyl group and m = 1–4, in solvents of different viscosity, η. Three different systems were used to change the viscosity of the medium: (a) Mixtures of methanol and ethylene glycol at 25°C, (b) linear aliphatic alcohols, H(CH₂)_nOH, where n =1–6, also at 25°C, and (c) ethylene glycol over the range 6.6–35°C. The ratio I_D/I_M decreases sharply as η increases, and the rate of the decrease in I_D/I_M is a function of m. Quantitatively, the dependence of I_D/I_M on η at high viscosity, i.e., the slope [d(I_D/I_M)/d(1/η)]_∞, is larger in the present work than in another series of 1-pyrenyl diesters in which the flexible spacer is an oligomer of polyethylene, instead of an oligomer of polyoxyethylene. In the limit where η → ∞, the ratio I_D/I_M assumes its largest value in the bichromophoric compound with m = 2. However, as η decreases the compound with m = 3 becomes the one with the largest I_D/I_M. A complete rotational isomeric state analysis (for the compounds with m = 1–3) and a Monte Carlo simulation (for the compound m = 4) of the conformations of the diesters can rationalize the behavior of I_D/I_M in the high viscosity limit. ©1995 John Wiley & Sons, Inc.     

SYNTHESIS OF METAL- AND PENTEL-CONTAINING HETEROCYCLES USING NITRILES AND ISONITRILES

The reaction of benzylnitrile with InMe ₃ in a molar ratio of 3:1 leads to 2-amino-N-[Me ₂ In(TMEDA)]-4-amino-3,5-diphenyl-6-benzyl-pyridine 2. CsF accelerates the reaction. The treatment of Ph ₂ CHCN with InMe ₃ in the presence of CsF gives the metalated ketenimine [(THF)Me ₂ InNCCPh ₂ ] ₂ 3. 3 and the corresponding Ga compound also can be obtained by the metathesis reaction of Me ₂ MCl with LiNu═C═CPh ₂ . The attack of the Lewis bases ^t BuELi ₂ (E = P, As) on nitriles and isonitriles also leads to oligomerizations. ^t BuAsLi ₂ reacts with two molecules of PhCN to the aromatic heterocycle [AsNC(Ph)NC(Ph)]^? by forming the salt [Li(diglyme) ₂ ] [(TMEDA)Li([AsNC(Ph)NC(Ph)]) ₂ ] 4, while the reaction of ^t BuAsLi ₂ with three equivalents of ^c HexNC leads to the dilithium compound [{Li ₂ (diglyme)} ₂ { ^t BuAs(CN ^c Hex) ₃ } ₂ ] 5. Six molecules ^c HexNC were consumed when Li ₂ P ^t Bu was added to the isonitrile to give [{Li(DME)} ₂ { ^t BuP(CN ^c Hex) ₅ (CH)}] 6.     

Solid-state 13C nuclear magnetic resonance characterization of MDI-based polyurethanes

¹³C solid-state nuclear magnetic resonance (NMR) experiments on linear polyurethanes and poly(ether-urethane) block copolymers demonstrate that ¹³C spin-lattice relaxation experiments in the laboratory [T₁(C)] and rotating [T_1p(C)] frames provide the most information about domain morphology in these microphase-separated polymer systems. T₁(H) TCH, and T_1p(H) data are less useful in a 4,4′-methylene bis(p-phenyl isocyanate)-1,4-butanediol (MDI/BD) hard-segment material, the MDI bridging methylene and the MDI urethane carbonyl T₁(C and T_1p(C) times fall in characteristic ranges for crystalline, amorphous, interfacial, and dissolved species. BD methylene carbons have short T_1p(C) for crystalline and long T_1p(C) for amorphous hard-segment aggregates. The distinct T_1p(C) and T₁(C) fractins observed are attributed to the presence of several crystalline polymorphs. Both T₁(C) results and DSC endotherms indicate that the crystalline polymorphs present in the poly(ether-urethane) are less ordered than the types seen in the pure hard-segment material. © 1993 John Wiley & Sons, Inc.     

Preparation of C2‐Symmetric Bis[2‐(diphenylphosphino)ferrocen‐1‐yl]‐ methane and Its Use in Rhodium‐ and Ruthenium‐Catalyzed Hydrogenation

The two diphosphine ligands (R_p,R_p)‐ and (S_p,S_p)‐bis[2‐(diphenylphospino)ferrocenyl]methane, (R_p,R_p)‐ and (S_p,S_p)‐ 1 , resp., were prepared in six steps from (S)‐ and (R)‐ferrocenyl tolyl sulfoxide, respectively (Scheme). In the solid state, both the diborane complex (R_p,R_p)‐ 1 ? (BH₃)₂ and the palladium dichloride complex [PdCl₂((R_p,R_p)‐ 1 )] were found to adopt C₂‐pseudosymmetric structures according to X‐ray analyses (Figs. 2 and 3). In the Rh‐ and Ru‐catalyzed hydrogenation of selected alkenes and ketones in the presence of the new ligands, enantioselectivities of up to 55% ee were obtained.     

Nine New Sesquiterpenes from Dendrobium nobile

Nine new sesquiterpenes, i.e., dendronobilins A–I ( 1 – 9 ), with copacamphane‐type ( 1 ), picrotoxane‐type ( 2 – 6 ), muurolene‐type ( 7 ), alloaromadendrane‐type ( 8 ), and cyclocopacamphane‐type ( 9 ) skeletons, were isolated from the 60% EtOH extract of the stems of Dendrobium nobile. Their structures were established as (1R,2R,4S,5S,6S,8S,9R)‐2,8‐dihydroxycopacamphan‐15‐one ( 1 ), (2β,3β,4β,5β)‐2,4,11‐trihydroxypicrotoxano‐3(15)‐lactone ( 2 ), (2β,3β,5β,9α,11β)‐2,11‐epoxy‐9,11,13‐trihydroxypicrotoxano‐3(15)‐lactone ( 3 ), (2β,3β,5β,12R*)‐2,11,13‐trihydroxypicrotoxano‐3(15)‐lactone ( 4 ), (2β,3β,5β,12S*)‐2,11,13‐trihydroxypicrotoxano‐3(15)‐lactone ( 5 ), (2β,3β,5β,9α)‐9,10‐cyclo‐2,11,13‐trihydroxypicrotoxano‐3(15)‐lactone ( 6 ), (9β,10α)‐muurol‐4‐ene‐9,10,11‐triol ( 7 ), (10α)‐alloaromadendrane‐10,12,14‐triol ( 8 ), and (5β)‐cyclocopacamphane‐5,12,15‐triol ( 9 ) on the basis of spectroscopic analysis. The absolute configuration of compound 1 was tentatively assigned as (1R,2R,4S,5S,6S,8S,9R) according to its CD spectrum and the octant rule. Compounds 1 and 4 – 9 were inactive in our preliminary in vitro immunomodulatory bioassay.     

Freezing-point of mixtures of H 2 16 O and H 2 18 O

Freezing-point depression of mixtures of H ₂ ¹⁶ O and H ₂ ¹⁸ O were measured. The results showed that the freezing point of the mixture rose linearly with an increase in the molal concentration of H ₂ ¹⁸ O. The results suggested the formation of a solid solution of H ₂ ¹⁶ O and H ₂ ¹⁸ O by freezing, similar to that formed by H ₂ O–D ₂ O, and that H ₂ ¹⁸ O behaves as a different molecule than H ₂ ¹⁶ O.     

On the connectivity index of trees

The connectivity index χ₁(G) of a graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. Let T(n, r) be the set of trees on n vertices with diameter r. In this paper, we determine all trees in T(n, r) with the largest and the second largest connectivity index. Also, the trees in T(n, r) with the largest and the second largest connectivity index are characterized. Mei Lu is partially supported by NNSFC (No. 10571105).     

The forcing number of toroidal polyhexes

The forcing number, denoted by f(G), of a graph G with a perfect matching is the minimum number of independent edges that completely determine the perfect matching of G. In this paper, we consider the forcing number of a toroidal polyhex H(p,q,t) with a torsion t, a cubic graph embedded on torus with every face being a hexagon. We obtain that f(H(p,q,t)) ≥ min{p,q}, and equality holds for p ≤ q or p > q and t∈{ 0,p−q,p−q + 1,..., p−1}. In general, we show that f(H(p,q,t)) is equal to the side length of a maximum triangle on H(p,q,t). Based on this result, we design a linear algorithm to compute the forcing number of H(p,q,t).     

Heat capacities of molten salts with polyatomic anions

The molar heat capacities at constant pressure, C_P, of molten salts with polyatomic anions, obtained from the literature, are examined. As a rule, the C_P values are independent of the temperature T, but the molar heat capacities at constant volume, C_V, derived from them, depend on T. The latter were obtained, as far as the required density, expansibility and compressibility data are available, for 1.1T_m, presumed to be the corresponding state, T_m being the melting temperature. Their ratio γ = C_P/C_V is linear with the cation–anion distance in the molten salt, d_C–A. The communal, quasi-lattice, heat capacity ΔC_P = C_P − C_P(i.g.) is obtained by subtraction of the sum of published ideal gas heat capacities of the constituent ions at 1.1T_m, C_P(i.g.). This communal heat capacity ΔC_P is proportional to the packing fraction of the ions in the melt, y = πN_Aνd_C−A³/6V. Here N_A is Avogadro's number, ν the number of ions per formula unit, and V the molar volume at 1.1T_m. Some models for the heat capacities of molten salts are shown not to be well applicable to the set of salts discussed here, but no alternative could be suggested.     

Maximum Randić index on Trees with <Emphasis Type="Italic">k</Emphasis>-pendant Vertices

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertices u and v in G. Let T be a tree with n vertices and k pendant vertices. In this paper, we give a sharp upper bound on Randić index of T.     

On topological indices of carbon nanocones and nanotori

Let G be a graph with n vertices and d _i is the degree of its ith vertex (d _i is the degree of v _i). In this article, we compute the redefined first Zagreb index, redefined second Zagreb index, redefined third Zagreb index, augmented Zagreb index of graphs carbon nanocones CNC _k[n], and nanotori [C₄C₆C₈(p,q)]. Also, compute the multiplicative redefined first Zagreb index, multiplicative redefined second Zagreb index, multiplicative redefined third Zagreb index, multiplicative augmented Zagreb index of carbon nanocones CNC _k[n], and nanotori [C₄C₆C₈(p,q)].     

Donnan potential of dilute colloidal dispersions: Monte Carlo simulations

Donnan equilibrium between a salt-free colloidal dispersion and an electrolyte solution has been investigated by Monte Carlo simulations. The Donnan potential is directly calculated by considering two compartments separated by a semipermeable membrane. In order to understand the role played by colloid–ion interactions, the influences of colloidal characteristics, including particle size R, intrinsic particle charge Z, counterion valency z_c, and concentration c_p, on Donnan potential Ψ_D and effective charge Z_eff are examined. Our simulations show that the electroneutrality condition is not followed in both compartments and the Donnan potential arises due to the net charge density. The Donnan potential grows by increasing c_p and Z_eff and by decreasing dielectric constant ε_r, i.e., Ψ_DZ_effc_p/ε_r approximately. Note that the effective charge varies with R,Z,c_p,ε_r and z_c as well. When the salt concentration is increased, the net charge density is lowered and thus the Donnan potential decays accordingly. The validity of the classical theory based on the Nernst equation and the electroneutrality assumption is also examined. In general, the simulation results at high-salt condition can be well represented by such mean-field theory.     

Adsorption of gas mixtures on heterogeneous solid surfaces: I. Extension of Tóth isotherm on adsorption from gas mixtures

Summary It was proved that the partial adsorbed quantities can be calculated from the individual adsorption isotherms. This calculation can be carried out by equations which take into' account the energetical heterogeneity of the surface of adsorbent as well. Thus, the partial isotherms equations which can be applied are as followsV _A (p _A ,p _B ) =p _A /[b _a + (p _A +b _AB p _B ) ^m ] ^1/m andV _B (p _A ,p _B ) =p _B /[b _B + (p _B +b _BA p _A ) ^m ] ^1/m wherep _A andp _B are the partial equilibrium pressures, whileb _A ,b _B andm are constants which can be determined on the basis of the individual isotherms.

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Zusammenfassung Es wurde nachgewiesen, daß die partialen Adsorbatmengen aus den individuellen Isothermen berechnet werden können. Die Berechnung kann mit Hilfe von Gleichungen durchgeführt werden, die auch die energetische Inhomogenität der adsorbierenden Oberfläche berücksichtigen. Benutzt werden die Gleichungen: V_A(p_A,p_B) = p_A/ [b_A + (PA + b_ABp_B)^m]^1/m und V_B(p_A,p_B) p_B/[b_B + (p_B + b_BAp_A)^m]^1/m , wop _A undp _B die Gleichgewichtspartialdrucke undb _A ,b _B undm die aus den individuellen Isothermen zu bestimmenden Konstanten darstellen.

     

The Synthesis of Hexaphenylcyclotriphosphazene in Improved Yield

The reaction between phenylmagnesium bromide and hexachlorocyclotriphophazene (N ₃ P ₃ Cl ₆ ) ( 1 ) was reinvestigated for the synthesis of hexaphenylcyclotriphosphazene (N ₃ P ₃ Ph ₆ ) ( 2 ) in high yield. The reaction is complete within two weeks at room temperature using toluene as solvent. When the reactants (PhMgBr and N ₃ P ₃ Cl ₆ ) were employed in a 6:1, 36:1 and 72:1 molar ratio, compound 2 was obtained in 2.6%, 14%, and 33.4% yield, respectively. The formation of N ₃ P ₃ Cl ₆ ( 2 ) during the reaction was followed by thin-layer chromatography. Compound 2 was characterized by elemental analysis, IR, UV-VIS, ¹ H, ¹³ C, and ³¹ P NMR spectroscopy as well as by mass spectrometry.