Highly accurate vibration–rotation Franck–Condon factors for high levels

Highly accurate vibration–rotation Franck–Condon factors q^ab, for a transition between two diatomic electronic states (a) and (b), are sought. When the potentials of states (a) and (b) are of the RKR type, the computation of q^ab is reduced to that of Franck–Condon integral ?^ab(i) = ∫ ψ^a(r)ψ^b(r) dr in an interval r_i, r_i+1. By using convenient interpolations for the potentials U^a and U^b in the considered interval, this integral becomes ?^ab(i) = ∑ δ (r_i+1 – r_i)ⁿ⁺¹/(n + 1), where the “coupling constants” δ depend uniquely on the eigenvalues E^a and E^b of the considered transition and on the potentials U^a and U^b (the number N of terms depends on the desired accuracy). The method used computes the Franck–Condon factors q^ab without the explicit use of the wave function and by replacing the integrals by simple summations. To test the values of q^ab obtained by this method, the orthogonality rule ∫ ψ_v′ψ_v″ dr = 0 (for v′ ≠ v″) is used for one state or the other. This test, along with other tests, show that the Franck–Condon factors computed by the present method are accurate to nine significant figures for high and low levels.     

A novel three‐dimensional copper(I) cyanide coordination polymer constructed from various bridging ligands: synthesis,crystal structure and characterization

The cyanide ligand can act as a strong σ‐donor and an effective π‐electron acceptor that exhibits versatile bridging abilities, such as terminal, μ₂‐C:N, μ₃‐C:C:N and μ₄‐C:C:N:N modes. These ligands play a key role in the formation of various copper(I) cyanide systems, including one‐dimensional (1D) chains, two‐dimensional (2D) layers and three‐dimensional (3D) frameworks. According to the literature, numerous coordination polymers based on terminal, μ₂‐C:N and μ₃‐C,C,N bridging modes have been documented so far. However, systems based on the μ₄‐C:C:N:N bridging mode are relatively rare. In this work, a novel cyanide‐bridged 3D Cu^I coordination framework, namely poly[(μ₂‐2,2′‐biimidazole‐κ²N³:N^3′)(μ₄‐cyanido‐κ⁴C:C:N:N)(μ₂‐cyanido‐κ²C:N)dicopper(I)], [Cu₂(CN)₂(C₆H₆N₄)]_n, (I), was synthesized hydrothermally by reaction of environmentally friendly K₃[Fe(CN)₆], CuCl₂·2H₂O and 2,2′‐biimidazole (H₂biim). It should be noted that cyanide ligands may act as reducing agents to reduce Cu^II to Cu^I under hydrothermal conditions. Compound (I) contains diverse types of bridging ligands, such as μ₄‐C:C:N:N‐cyanide, μ₂‐C:N‐cyanide and μ₂‐biimidazole. Interestingly, the [Cu₂] dimers are bridged by rare μ₄‐C:C:N:N‐mode cyanide ligands giving rise to the first example of a 1D dimeric {[Cu₂(μ₄‐C:C:N:N)]ⁿ⁺}_n infinite chain. Furthermore, adjacent dimer‐based chains are linked by μ₂‐C:N bridging cyanide ligands, generating a neutral 2D wave‐like (4,4) layer structure. Finally, the 2D layers are joined together via bidentate bridging H₂biim to create a 3D cuprous cyanide network. This arrangement leads to a systematic variation in dimensionality from 1D chain→2D sheet→3D framework by different types of bridging ligands. Compound (I) was further characterized by thermal analysis, solid‐state UV–Vis diffuse‐reflectance and photoluminescence studies. The solid‐state UV–Vis diffuse‐reflectance spectra show that compound (I) is a wide‐gap semiconductor with band gaps of 3.18 eV. The photoluminescence study shows a strong blue–green photoluminescence at room temperature, which may be associated with metal‐to‐ligand charge transfer.     

Melt‐state polymer chain dimensions as a function of temperature

The unperturbed chain dimensions (〈R²〉_o/M) of cis/trans‐1,4‐polyisoprene, a near‐atactic poly(methyl methacrylate), and atactic polyolefins were measured as a function of temperature in the melt state via small‐angle neutron scattering (SANS). The polyolefinic materials were derived from polydienes or polystyrene via hydrogenation or deuteration and represent structures not encountered commercially. The parent polymers were prepared via lithium‐based anionic polymerizations in cyclohexane with, in some cases, a polymer microstructure modifier present. The polyolefins retained the near‐monodisperse molecular weight distributions exhibited by the precursor materials. The melt SANS‐based chain dimension data allowed the evaluation of the temperature coefficients [dln 〈R²〉_o/dT(κ)] for these polymers. The evaluated polymers obeyed the packing length (p)‐based expressions of the plateau modulus, G = kT/np³ (MPa), and the entanglement molecular weight, M_e = ρN_anp³ (g mol^?1), where n_t denotes the number (～21) of entanglement strands in a cube with the dimensions of the reptation tube diameter (d_t) and ρ is the chain density. The product np³ is the displaced volume (V_e) of an entanglement that is also expressible as pd or kT/G. © 2002 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 40: 1768–1776, 2002     

The dependence of 10Dq upon the metal–ligand distance,R, for transition-metal complexes. What is its microscopic origin?

Experimental results on 3d O_h complexes in insulators reveal that 10Dq α R^?n, where R is the metal–ligand distance and n is close to five. This strong dependence determines the Huang–Rhys factor, S(A_1g), associated to the symmetric A_1g mode of the first excited state of complexes like MnX and CrX (X = halide) and makes it possible to measure R changes down to ∽ 10^?3 Å. This work is devoted to understanding, within a molecular orbital framework, the microscopic origin of such a dependence, which is related to the corresponding one displayed by the transferred spin densities fσ, f_s, and fπ. The analysis is focused on MnF. As a main result, it is shown that though fσ ? f_s the interaction between d(e_g) orbitals and 2s orbitals of F^? is not only primarily responsible for the R dependence of 10Dq but also makes a significant contribution to the 10Dq value itself. The present work thus shows that the significant dependence of 10Dq upon R is ultimately related to the strong dependence of f_s and the isotropic superhyperfine constant A_s upon R displayed by the experimental results of several 3d impurities. © John Wiley & Sons, Inc.     

Crystal thickness distributions from melting homopolymers or random copolymers

Differential scanning calorimetry (DSC) can be used to infer the distribution of lamellar crystal thickness l. For homopolymers, the relation between melting temperature T and thickness is described by the Gibbs relation. In this case the weight distribution function of thickness g(l) ∝ P(T)(T − T)², where P(T) is DSC power and T is the melting temperature of an infinitely thick crystal. Copolymer melting is affected by the concentration of noncrystallizable comonomer in the melt as well as lamellar thickness. Unknown melt composition in copolymers with nonequilibrium crystallinity makes determination of the correct distribution g(l) from DSC impossible. An approximate distribution g₂(l) ∝ P(T)(T − T)² is proposed, where T is based on Flory's equilibrium crystallization theory. This approximate distribution is most accurate when crystallinity is small, that is, near the upper end of the melting range. Results are reported for polyethylene homopolymer and model ethylene–butene random copolymers. Corrections were not made for distortion of the DSC endotherms by thermal lag or by melting and recrystallization; these experiments are primarily to illustrate the effect of analysis in terms of an incorrect g₃(l) ∝ P(T). Average crystal thicknesses are about 20 nm for polyethylene and 5 nm for the copolymers. Distributions are characterized by l_w /l_n ≤ 1.1 in all cases. Width of the melting range is not a reliable indicator of the breadth of the thickness distribution. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 3131–3140, 1999     

Substitutional carbon defects in silicon: A quantum mechanical characterization through the infrared and Raman spectra

The infrared (IR) and Raman spectra of eight substitutional carbon defects in silicon are computed at the quantum mechanical level by using a periodic supercell approach based on hybrid functionals, an all electron Gaussian type basis set and the CRYSTAL code. The single substitutional C _s case and its combination with a vacancy (C _sV and C _sSiV) are considered first. The progressive saturation of the four bonds of a Si atom with C is then examined. The last set of defects consists of a chain of adjacent carbon atoms C, with i = 1–3. The simple substitutional case, C _s, is the common first member of the three sets. All these defects show important, very characteristic features in their IR spectrum. One or two C related peaks dominate the spectra: at 596 cm⁻¹ for C _s (and C _sSiV, the second neighbor vacancy is not shifting the C _s peak), at 705 and 716 cm⁻¹ for C _sV, at 537 cm⁻¹ for C and C (with additional peaks at 522, 655 and 689 for the latter only), at 607 and 624 cm⁻¹, 601 and 643 cm⁻¹, and 629 cm⁻¹ for SiC, SiC, and SiC, respectively. Comparison with experiment allows to attribute many observed peaks to one of the C substitutional defects. Observed peaks above 720 cm⁻¹ must be attributed to interstitial C or more complicated defects.     

Extremal Wiener and Kirchhoff indices of globular caterpillars

The Wiener and Kirchhoff indices of a graph G are two of the most important topological indices in mathematical chemistry. A graph G is called to be a globular caterpillar if G is obtained from a complete graph K _s with vertex set {v₁,v₂,…, v _s} by attaching n _i pendent edges to each vertex v _i of K _s for some positive integers s and n₁,n₂,…,n _s, denoted by . Let be the set of globular caterpillars with n vertices (). In this article, we characterize the globular caterpillars with the minimal and maximal Wiener and Kirchhoff indices among , respectively.     

On the Sanibel coefficients in the expansion of spin-projected slater determinants

Every Slater determinant D may be uniquely analyzed in terms of spin components D_l = O_lD which are pure spin eigenfunctions, so that S²D_l = l(l+1)D. Every component D_l = O_lD may in turn be written as a sum of symmetric combinations of Slater determinants, T_k = [α^μ?kβ^k‖α^kβ^ν?k], and the coefficients c in the expansion O_lD = ∑_k c T_k are known as the “Sanibel coefficients.” By using the relation S²D_l = l(l+1)D, a recursion formula for the coefficients c is derived, which is then explicitly solved in the special case when S_z has the pure quantum number m = 0.     

Solvay type VCl3 isospecific catalysts

Solvay type S –VCl₃ catalyst has 7% of catalytically active vanadium sites ([C^*]) with k_p (rate constant of propagation) = 31 (M s)^?1 for ethylene polymerization. Addition of a comonomer, propylene of 4-methylpentene-1 (4-MP) significantly raised the ethylene polymerization activity. S –VCI₃ catalyst has very small amounts of catalytically active vanadium for propylene polymerizations: [C] = 0.19% with k_p,i = 857 (M s)^?1 and [C] = 0.45% with k_p,a = 23 (M s)^?1 for isospecific and nonspecific sites, respectively. Addition of a conomer, ethylene or 4-MP. lowered the propylene polymerization activity. S –VCI₃ is more easily reduced to the divalent ion by AIR₃ than S –TiCl₃. Methyl-p-toluate moderates the reducting power of AIR₃; it increase the productivity and stereoselectivity of the S –YiCl₃ catalyst, VCI₃ supported on MgCl₂ (CW–V catalyst) has enhanced rate constant of propylene polymerization but has the opposite effects on the S –TiCl₃ Catalyst. VCI₃ supported on MgCl₂ (CW–V catalyst) has enhances rate constant of propylene polymerization but only a minute fraction of the supported vanadiums are catalytically active: [C] = 0.019% and k_p,i = 1580 (Ms)^?1, [C] = 0.057% and k_p,i = 58 (M s)^?1. This is compared with far greater number of catalytically active titanium sites in the TiCl₃ supported on MgCl₂ catalyst: [C] = 6% and k_p,i = 200 (M s)^?1, [C] = 6% and k_p,a = 16(M s)^?1. Therefore, both the S –VCI₃ and CW–V catalysts are highly stereoselective but low in efficiency with respect to the utilization of the vanadium ion in the catalysis.     

Green's Function and Eigenvalue Representation of Time Lag for Absorptive Permeation Across a Heterogeneous Membrane

The time‐lag problem is treated for absorptive penetration across a heterogeneous membrane, where both the diffusivity D (x ) and the partition coefficient K (x ) depend on the coordinate x (0 ≦ x ≦ h ), with 0 and h being the coordinates of the upstream and downstream faces, respectively. A new concept of time‐lag distribution is introduced, and the first (time) moment and the second (time) moment over this distribution are also difined and treated in the Lapalce domain in conjuction with the Green's function G (x ,y ), and eigenvalues associated with the time‐independent diffusion equation subject to the absorbing boundary condition at both ends of the membrane. Our central results include and , where λ _i is the i th eigenvalue of the aforementioned diffusion equation. The merits of these new resprentations and comparison with the treatments of Frisch or Eyring are also discussed.     

Über Opiumalkaloide

Ohne ZusammenfassungI. Mitt.: Monatshefte für Chemie,41, 297 (1920); II.: daselbst,42, 273 (1921); III.: Ber. der Deutschen chem. Ges.,58, 200 (1925); IV.: daselbst,58. 1272 (1925); V. und VI.: daselbst,59, (1926).     

Information‐entropic measures in free and confined hydrogen atom

Shannon entropy (S), Rényi entropy (R), Tsallis entropy (T), Fisher information (I), and Onicescu energy (E) have been explored extensively in both free H atom (FHA) and confined H atom (CHA). For a given quantum state, accurate results are presented by employing respective exact analytical wave functions in r space. The p‐space wave functions are generated from respective Fourier transforms—for FHA these can be expressed analytically in terms of Gegenbauer polynomials, whereas in CHA these are computed numerically. Exact mathematical expressions of , are derived for circular states of a FHA. Pilot calculations are done taking order of entropic moments (α, β) as in r and p spaces. A detailed, systematic analysis is performed for both FHA and CHA with respect to state indices n, l, and with confinement radius (r_c) for the latter. In a CHA, at small r_c, kinetic energy increases, whereas decrease with growth of n, signifying greater localization in high‐lying states. At moderate r_c, there exists an interplay between two mutually opposing factors: (i) radial confinement (localization) and (ii) accumulation of radial nodes with growth of n (delocalization). Most of these results are reported here for the first time, revealing many new interesting features. Comparison with literature results, wherever possible, offers excellent agreement.     

Synthesis and X-ray diffraction study of complex oxides of the Zn2 ? x(ZraSnb)1 ? xFe2xO4 composition

A multicomponent system of complex refractory oxides of the composition Zn_{2 − x} (Zr _a Sn _b )_{1 − x} Fe_2x O₄ (a + b = 1; a: b = 1: 5, 1: 4, 1: 3, 1: 2, 1: 1, 2: 1, 3: 1, 4: 1; x = 0−1.0; Δx = 0.05) was studied by X-ray diffraction. The samples were prepared from oxides of appropriate metals by low-temperature plasma synthesis (hydrogen-oxygen flame). Two phases with wide homogeneity ranges were identified: α phase crystallized in the crystal system of inverse cubic spinel and β phase with the structure of tetragonal spinel. The phase boundaries were found. Structural data are presented for about 100 solid solutions.     

Deformation mechanisms of constrained polymer chains. II. Deformation energetics of tie molecules

Energy-deformation characteristics for the primary T, S, and U conformational units of tie molecules were obtained from the analysis of data generated from a constrained minimization algorithm. Energy-deformation profiles (covering the range from compact equilibrium defect structures to the fully extended chain) are reported for the S₀ and S₁ members of the S_λ family and for the U₀₀ member of the U_mn family. Estimates of the energy content V⁰ and the elastic modulus E were obtained from the computed energy-deformation data in the vicinity of the equilibrium Structure—S₀ → {60°, 180°, ?60°}, V = 1.7 kcal/mole, E = 60 kcal/cm³ [250 × 10¹⁰ dyn/cm²];S₁ → {60°, 180°, 180°, 180°, ?60°}: V = 1.7 kcal/mole, E = 25 kcal/cm³ [100 × 10¹⁰ dyn/cm²]; and U₀₀ → {60°, 180°, 60°, 180°, 60°}: V = 2.7 kcal/mole, E = 80 kcal/cm³ [340 × 10¹⁰ dyn/cm²]. Although the elastic modulus of the U₀₀ unit is comparable to the elastic modulus of the fully extended chain, the highenergy content of this unit (V₀ = 2.7 Kcal/mole) prohibits a significant population and thereby mitigates an appreciable reinforcing effect from this rigid unit. A model for a surrogate force constant is introduced to generalize the results from this study to any member of the S_λ or U_mn family as well as any combination of S_λ and U_mn units. This generalization provides a basis for estimating the deformation characteristics of tie molecules comprised of various populations of these primary conformational building blocks.     

On the separation of Co and Ni from chloride media withAliquat 336-TBP andAliquat 336-TOPO

Summary The extraction behaviour of Co and Ni chlorides withAliquat 336-TBP andAliquat 336-TOPO (Aliquat 336: tri-n-octylmethylammonium chloride,TBP: tri-n-butylphosphate,TOPO: tri-n-octylphosphine oxide) was investigated. The synergistic action ofTOPO in the extraction of Co and Ni withAliquat 336 manifested itself in an increase of the distribution ratio of Co with increasing ofTOPO concentration at constantAliquat 336 concentration without deterioration of the separation factor Co/Ni which still was above 200. Mixed complexes of the general formulaMeCl_2+mA_mB_n (Me: Co or Ni,A:Aliquat 336, A:TBP orTOPO) withm ranging from 2 to 5 andn from 1 to 3 were present in the organic phase of both systems. The coexistence of several synergistic Co/Ni species in the equilibrium organic phase is reported for the first time.

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Zur Abtrennung von Co und Ni aus chloridhaltigen Medien mitAliquat 336-TBP undAliquat 336-TOPO

Zusammenfassung Das Extraktionsverhalten von Kobalt- und Nickelchloriden mitAliquat 336-TBP undAliquat 336-TOPO (Aliquat 336: Tri-n-octylmethylammoniumchlorid,TBP: Tri-n-butylphosphat,TOPO: Tri-n-octylphosphinoxid) wurde untersucht. Die synergistische Wirkung vonTOPO bei der Extraktion von Co und Ni mitAliquat 336 manifestiert sich in einer Erhöhung des Verteilungsverhältnisses von Co mit steigenderTOPO-Konzentration, ohne daß dadurch der Co/Ni-Trennfaktor verschlechtert wird (> 200). Für beide Systeme konnten in der organischen Phase gemischte Komplexe der allgemeinen FormelMeCl_2+m A _mB_n nachgewiesen werden (Me: Co oder Ni,A:Aliquat 336,B:TBP orderTOPO,m : 2–5,n: 1–3). Zum ersten Mal wird über die Koexistenz verschiedener synergistischer Co/Ni-Spezies in der organischen Phase im Gleichgewicht berichtet.

     

Charge distributions and chemical effects. XXXIV. Concepts involved in an approximate,but accurate,description of bond energies

The concepts underlying the definition of bond energies in terms of potentials at the nuclei are outlined. The theory is rooted, first, in a definition of the energy, E_i, of “atom” i in the molecule in terms of the potential energy, V(i, mol), of nucleus Z_i in the field of all the electrons and nuclei of the molecule: E_i = K V(i, mol). The K parameter, which is not required to be a constant in the derivation of the energy expression describing the contribution of an ij bond, turns out to be virtually constant for each atomic species—a situation which is exploited in numerical applications. Second, the Hellmann—Feynman theorem is applied in the calculation of the derivative, δΔE/δZ_i, of the atomization energy, ΔE, using (i) the exact quantum-chemical definition of ΔE and (ii) the view that ΔE is the sum of bond energy contributions, ε_ij, plus a small interaction between nonbonded atoms. The individual bond energies derived in this manner necessarily depend on local charges at the bond-forming atoms. Numerical applications illustrate how this new bond-energy formula provides a simple link between typical saturated, olefinic, acetylenic, and aromatic hydrocarbons.     

Ellipsometric study of the adsorption of comb-branched polystyrene onto a metal surface

The adsorption of well-characterized comb-branched polystyrene onto a chrome plate from cyclohexane solution at the θ temperature has been studied by ellipsometry. Both the adsorbance of the polymer and the extension of the adsorbed layer are compared with values for the linear polystyrene of the same molecular mass. The adsorbance is higher than that of the linear polystyrene, whereas the extension of the adsorbed layer is smaller, reflecting the higher segment density of the branched polymer. The extension t_b of the branched polymer is given approximately t_b = t_lg, where t_l is the extension of linear polystyrene of the same molecular mass and g is the ratio of the radii of gyration of the branched and linear polymers. The ratio of the adsorbances A_b/A_l of branched and linear polymer is approximately equal to g. These results indicate that the comb-branched polymer is adsorbed as a slightly distorted randam coil with extension and adsorbance governed primarily by the experimental g_s factor.     

Specific volume of polysulfone as a function of temperature and pressure

The pressure–volume–temperature (PVT) properties of a commercial polysulfone derived from bisphenol A and 4,4′-dichlorodiphenylsulfone are studied experimentally and theoretically in the temperature range 30–370°C and for pressures to 2000 kg/cm². PVT surfaces are determined for an annealed glass, formed under zero pressure, and for the melt. Two glass-transition lines must be distinguished: T(P) which is the intersection of the glass and melt PVT surfaces, and T_g(P), which is obtained by pressurizing the melt isothermally. The application of Ehrenfest-type equations to these transitions are discussed. The Prigogine–Defay ratio r = Δ_kΔC_p/TV(Δα)² at P = 0 is found to be equal to 0.95 (±20%), using ΔC_p data determined on identical samples. The melt data is compared with the Simha–Somcynski hole theory, using the reducing parameters V^* = 0.788 cm³/g, T^* = 12,560°K, P^* = 10,875 bar. The hole fraction appearing in the theory is found to be constant along T(P), but the glass PVT relationship cannot be reproduced by using the Simha–Somcynsky theory together with the assumption that the hole fraction remains constant in the glass. At P = 0 the hole fraction must be allowed to decrease with decreasing temperature, but at a slower rate than in the melt.     

Vector coupling coefficients for calculations of transition-metal atoms and ions by the SCF coupling operator method

We derived the necessary conditions to which the vector coupling coefficients (VCC ) a and b describing atomic L,S-multiplets of the configurations d^N (1 ≤ N ≤ 9), should satisfy. Special attention is paid to the states of non-Roothaan type for which VCC depend on the choice of degenerate d-orbitals basis set determined within the accuracy up to an orthogonal transformation u. It is shown that for such states the direct sum of matrices ‖a‖ and ‖b‖ must be the non-symmetric matrix. Obtained VCC were used for the ab initio calculations (basis set (14s9p5d)/[8s4p2d] from [15]) on first-row transition atoms (from Sc to Cu) to compare to similar calculations [16], in which the Peterson's VCC have been used, and with calculations [15] carried out by the atomic SCF program [4] as well.