Exchange energy for two closed shells generated by a bare Coulomb potential energy −<Emphasis Type="Italic">Ze</Emphasis><Superscript>2</Superscript>/<Emphasis Type="Italic">r</Emphasis> in the limit of large <Emphasis Type="Italic">Z</Emphasis>, in two dimensions

Two-dimensional (2D) inhomogeneous electron assemblies are becoming increasingly important in Condensed Matter and associated technologies. Here, therefore, we contribute to the Density Functional Theory of such 2D electronic systems by calculating, analytically, (i) the idempotent Dirac density matrix γ(r, r′) generated by two closed shells for the bare Coulomb potential −Ze ²/r and (ii) the exchange energy density e_x(r){\varepsilon_x({\bf r})} . Some progress is also possible concerning the exchange potential V _x (r), one non-local approximation being the Slater potential 2e_x(r)/n(r){2\varepsilon_x(r)/n(r)} , with n(r) the ground state electron density. However, to complete the theory of V _x (r), the functional derivative of the single-particle kinetic energy per unit area δt(s)/δn(r) is still required.     

Hosoya polynomials of TUC4C8(R) nanotubes

For a connected graph G, the Hosoya polynomial of G is defined as H(G, x) = ∑_{u,v}?V(G)x^{d(u, v)}, where V(G) is the set of all vertices of G and d(u,v) is the distance between vertices u and v. In this article, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(R) nanotubes. Furthermore, the Wiener index and the hyper‐Wiener index can be calculated. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009     

A conjecture for some partial differential operators on L 2( R n)

OnX =L ²(R ⁿ), letQ = (Q ₁,Q ₂,…,Q _n) andP = (P ₁,P ₂, …,P _n) be the operators given by (Q _jf) (x) =x _jf(x),P _j = - i∂/∂x _j. For anyC ^∞ functionh:R ⁿ →R putH ₀ =h(P) andH =H ₀ + (1 +Q ²)^-δ, where δ > 1/2. By the method of scattering theory we prove thatH _ac, the absolutely continuous part ofH is unitarily equivalent toH ₀ when (a)n = 1 and (b) forn ≥ 2, whenh is in a large class of polynomials. It is conjectured that the results are true for any polynomialh. We use the techniques of Enss’ method and the idea of bound states for momentum.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Vibronic spectra of the allyl radical at 6–8 eV with resonance-enhanced multiphoton ionization technique

The allyl radical was produced in molecular beam by pyrolysis of allyl iodide. The vibronic spectra from ground state to six new electronic states of the allyl radical at 6–8 eV, π →3d^xz, π →3d^xy and π →ns (n=4, 6, 7, 8) were observed firstly with the aid of time-of-flight mass spectros copy and resonance-enhanced multiphoton ionization technique. Vibrational progression ofv ₇(C₃ bend) with gross spacing of about 430 cm⁻¹ was observed inns Rydberg states. The adiabatic ionization potential of the allyl radical was obtained to be (65641 ± 20) cm⁻¹ ((8.138 ± 0.002) eV) by fitting the term values ofns (n=4,6,7,8) Rydberg states with Rydberg formula.     

Theoretical treatment of predissociation of the CO (3sσ) B and (3pσ) C1Σ+ Rydberg states based on a rigorous adiabatic representation

Ab initio multireference configuration interaction calculations for adiabatic potential curves, nonadiabatic couplings 〈φ _i (R,r)|d/dR|φ _j (R,r)〉 and 〈φ _i (R,r)|d²/dR ²|φ _j (R,r)〉, and nuclear kinetic energy corrections 〈dφ _i (R,r)/dR|dφ _i (R,r)/dR〉 for the (3sσ) B and (3pσ) C¹Σ⁺ Rydberg states of the CO molecule have been carried out. The energy positions and predissociation linewidths for the observed vibrational levels of these two states have been determined in a rigorous adiabatic representation by the complex scaling method employing a basis of complex scaled harmonic vibrational functions in conjunction with the Gauss-Hermite quadrature method to evaluate the complex Hamiltonian matrix elements. The present treatment correctly reproduces the observed trends in energies and line broadening for vibrational levels of the B¹Σ⁺ state and represents an improvement over the previous treatment in literature. The errors in the determined spacings of the v = 0–4 vibrational levels of the C¹Σ⁺ state are less than 2% compared with measured data. The predissociation linewidths for the v=3,4 levels of the C¹Σ⁺ state are found to be 4.9 and 8.9 cm⁻¹, respectively, in good agreement with the observed values. Received: 23 March 1998 / Accepted: 27 July 1998 / Published online: 9 October 1998     

Thermal decomposition of lighter lanthanide selenate hydrates and their solid solutions

The thermal dehydration-decomposition of Ln₂(SeO₄)₃·nH₂O (wheren=12 forLn=Pr, Nd andn=8 forLn=Sm) and Pr_xLn_2−x(SeO₄)₃·nH₂O (wheren=12 forx=1.0 andLn=Nd;n=8 forx=0.2 and 1.0 in case ofLn=Sm) have been reported.

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Zusammenfassung Die thermische Dehydratation-Zersetzung von Ln₂(SeO₄)₃·nH₂O (mitn=12 fürLn=Pr, Nd undn=8 fürLn=Sm) und Pr_xLn_2−x(SeO₄)₃·nH₂O (mitn=12 fürx=1.0 undLn=Nd;n=8 fürx=0.2 und 1.0 in Falle vonLn=Sm) wurde beschrieben.

     

Nano-composites SnO(VO<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>) as anodes for lithium ion batteries

Nano-composites of SnO(V₂O₃) _x (x = 0, 0.25, and 0.5) and SnO(VO)_0.5 are prepared from SnO and V₂O₃/VO by high-energy ball milling (HEB) and are characterized by X-ray diffraction (XRD), scanning electron microscopy, and high-resolution transmission electron microscopy techniques. Interestingly, SnO and SnO(VO)_0.5 are unstable to HEB and disproportionate to Sn and SnO₂, whereas HEB of SnO(V₂O₃) _x gives rise to SnO₂.VO _x . Galvanostatic cycling of the phases is carried out at 60 mA g⁻¹ (0.12 C) in the voltage range 0.005–0.8 V vs. Li. The nano-SnO(V₂O₃)_0.5 showed a first-charge capacity of 435 (±5) mAh g⁻¹ which stabilized to 380 (±5) mAh g⁻¹ with no noticeable fading in the range of 10–60 cycles. Under similar cycling conditions, nano-SnO (x = 0), nano-SnO(V₂O₃)_0.25, and nano-SnO(VO)_0.5 showed initial reversible capacities between 630 and 390 (±5) mAh g⁻¹. Between 10 and 50 cycles, nano-SnO showed a capacity fade as high as 59%, whereas the above two VO _x -containing composites showed capacity fade ranging from 10% to 28%. In all the nano-composites, the average discharge potential is 0.2–0.3 V and average charge potential is 0.5–0.6 V vs. Li, and the coulombic efficiency is 96–98% after 10 cycles. The observed galvanostatic cycling, cyclic voltammetry, and ex situ XRD data are interpreted in terms of the alloying–de-alloying reaction of Sn in the nano-composite “Sn-VO _x -Li₂O” with VO _x acting as an electronically conducting matrix.     

An atom-in-molecules and electron-localization-function study of the interaction between O2 and VxOy+/VxOy (x = 1, 2, y = 1–5) clusters

 The most stable structures of V _x O _y ⁺/V _x O _y (x=1, 2, y=1–5) clusters and their interaction with O₂ are determined by density functional calculations, the B3LYP functional with the 6-31G* basis set. The nature of the bonding of these clusters and the interaction with O₂ have been studied by topological analysis in the framework of both the atoms-in-molecules theory of Bader and the Becke–Edgecombe electron localization function. Bond critical points are localized by means of the analysis of the electron density gradient field, ∇ρ(r), and the electron localization function gradient field, ∇η(r). The values of the electron density properties, i.e., electron density, ρ(r), Laplacian of the electron density, ∇²ρ(r), and electron localization function, η(r), allow the nature of the bonds to be characterized, and linear correlation is found for the results obtained in both gradient fields. Vanadium-oxygen interactions are characterized as unshared-electron interactions, and linear correlation is observed between the electron density properties and the V–O bond length. In contrast, O₂ units involve typical shared-electron interactions, as for the dioxygen molecule. Four different vanadium–oxygen interactions are found and characterized: a molecular O₂ interaction, a peroxo O₂ ²⁻ interaction, a superoxo O₂ ⁻ interaction and a side-on O₂ ⁻ interaction. Received: 15 October 2001 / Accepted: 30 January 2002 / Published online: 24 June 2002     

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).     

Crystal structure of [Pd(P(i-Pr)3)2(acac)]BF4

A single crystal X-ray diffraction study is carried out for [Pd(P(i-Pr)₃)₂(acac)]BF₄, T = 150(2) K. Crystal data: a = 10.2935(4) ?, b = 11.3591(5) ?, c = 13.8728(6) ?, α = 89.154(2)°, β = 68.448(1)°, γ = 85.032(1)°, P-1 space group, V = 1502.75(11) ?³, Z = 2, d _x = 1.354 g/cm³.     

Preparation of a zwitterionic trichloroplatinum(II) complex with an alkylbipyridinum ligand and its reaction in dmso-<Emphasis Type="BoldItalic">d</Emphasis><Subscript>6</Subscript> solution

A zwitterionic trichloroplatinum(II) complex PtCl₃(4,4′-bpy-N-ⁿBu) (1) was prepared by an aqueous reaction of [4,4′-bpy-N-ⁿBu]Cl with K₂PtCl₄, and was characterized by ¹H-n.m.r. and IR spectroscopy as well as elemental analysis. Dissolving (1) into dmso-d ₆ at 25 °C yields a mixture of the complexes, cis-PtCl₂(4,4′-bpy-N-ⁿBu)(dmso-d ₆) (2-cis), trans-PtCl₂(4,4′-bpy-N-ⁿBu)(dmso-d ₆) (2-trans), [4,4′-bpy-N-ⁿBu]Cl and PtCl₂(dmso-d ₆)₂. Ratio of the products in the dmso-d ₆ solution changed depending on the temperature and the total concentration of the complexes. These compounds are in equilibrium via isomerization reaction between (2-cis) and (2-trans) and via displacement reaction of the alkylbipyridinium ligand of (2-cis) and (2-trans) with dmso-d ₆ to form [4,4′-bpy-N-ⁿBu]Cl and PtCl₂(dmso-d ₆)₂.     

The exchange-correlation potential of DFT obtained from a semiempirically fine-tuned Hartree–Fock density for inhomogeneous electron liquids

The present authors have given an exact theory of the exchange-correlation potential V _xc (r) in terms of (i) the exact ground-state electron density n(r) and (ii) the idempotent Dirac density matrix γ(r,?r′) generated by the DFT one-body potential V(r), having n(r) as its diagonal element. Here, we display two approximate consequences: (a) a form of V _xc (r) generated by the semiempirically fine-tuned HF density of Cordero et al. (N.A. Cordero, N.H. March, and J.A. Alonso, Phys. Rev. A 75, 052502 (2007)) and (b) the exchange-only potential V _x (r) determined solely by the HF ground state density for the Be atom.     

Effect of free valence on the characteristics of internal rotation in <Emphasis Type="Italic">n</Emphasis>-alkyl radicals

A study of internal rotation in the radicals n-C _n H_2n+1C^·H₂, (2 ≤ n ≤ 7) was carried out for the case of rotation around the bonds not including the radical center. 21 potential functions of internal rotation V(φ) were calculated. The coefficients in V(φ) were shown to depend only on the immediate environment at the bond of rotation. Characteristics of internal rotation in n-alkyl radicals were compared with related parameters of the corresponding alkane molecules. The generalized function V _av(φ) with the coefficients defined only by the position of the bond of rotation in the hydrocarbon chain and possessing the transposition property were proposed. The functions V _av(φ) were recommended for the simulation of the structure and properties of large molecules containing hydrocarbon fragments. This work continues a systematic study on the characteristics of the internal rotation in the n-alkane molecules and the groups containing free valence in n-alkyl radicals.     

D<Subscript>2</Subscript>O — H<Subscript>2</Subscript>O Solvent Isotope Effects on the Apparent Molar Volumes of Tetramethyl-<Emphasis Type="Italic">bis</Emphasis>-urea (Mebicarum) Solutions

Densities of solutions of tetramethyl-bis-urea (TMbU) or “Mebicarum” in H₂O and D₂O, with solute mole fraction concentrations (x ₂) ranging up to 3.2 × 10⁻³, have been measured at 288.15, 298.15, 308.15 and 318.15 K using a precision vibrating-tube densimeter. The limiting apparent molar volumes, V _φ,2 ^∞, and expansibilities, E _{p, φ, 2} ^∞, of the solute have been calculated. The isotope effect δ V _φ,2 ^∞(H₂O → D₂O;T) is negative, monotonously decreases in magnitude with temperature and reverses sign at T ≈ 318 K. Water (H₂O, D₂O) and TMbU molecules in infinitely- and highly-dilute aqueous solutions form H(D)-bonded hydration complexes with a high packing density. The hydration of TMbU should be treated as a superposition of two mechanisms, hydrophobic and hydrophilic, with the latter one predominating.     

Topological structures of tetrahedral (Td and T) fullerenes with hexagonal and triangular faces and their vibration and NMR spectra

On analyzing the topological structures of the three types of tetrahedral fullerenes (which consist only of triangles and hexagons), (1) C _n (T _d ,n=12h ²; h=1,2,…), (2) C _n (T _d ,n=4h ²;h=1,2,…), and (3) C _n (T,n=4(h ²+hk+k ²);h>k,h,k=1,2,…), we have obtained theoretically the Infrared and Raman active modes by means of the derived formulas for the decomposition of their nuclear motions into irreducible representations, and the ¹³C NMR spectra with natural abundance for ¹³C by using the distribution functions for all of the tetrahedral (T _d and T) fullerenes, respectively. Received: 25 May 1998 / Accepted: 30 July 1998 / Published online: 23 November 1998     

Kinetics of the ene reactions of 4‐phenyl‐1,2,4‐triazoline‐3,5‐dione with β‐pinene and 2‐carene: Temperature,high pressure,and solvent effects

The data on temperature, solvent, and high hydrostatic pressure influence on the rate of the ene reactions of 4‐phenyl‐1,2,4‐triazoline‐3,5‐dione ( 1 ) with 2‐carene ( 2 ), and β‐pinene ( 4 ) have been obtained. Ene reactions 1 + 2 and 1 + 4 have high heat effects: ∆H_r‐n ( 1 + 2 ) −158.4, ∆H_r‐n( 1 + 4 ) −159.2 kJ mol⁻¹, 25°C, 1,2‐dichloroethane. The comparison of the activation volume (∆V^≠( 1 + 2 ) −29.9 cm³ mol⁻¹, toluene; ∆V^≠( 1 + 4 ) −36.0 cm³ mol⁻¹, ethyl acetate) and reaction volume values (∆V_r‐n( 1 + 2 ) −24.0 cm³ mol⁻¹, toluene; ∆V_r‐n( 1 + 4 ) −30.4 cm³ mol⁻¹, ethyl acetate) reveals more compact cyclic transition states in comparison with the acyclic reaction products 3 and 5 . In the series of nine solvents, the reaction rate of 1+2 increases 260‐fold and 1+4 increases 200‐fold, respectively, but not due to the solvent polarity.     

Heteropoly Complexes Na0.5Cs2 ? x[H0.5 ? xM

Crystals of novel heterepoly complexes (HPC) Na_0.5Cs_{2 − x} [H_{0.5 − x} M _x ^II X^III(OH)₆Mo₆O₁₈] · 7−8H₂O (M^II = Fe, Mn; X^III = Cr, Al) are synthesized. Crystal structures of the complexes Na_0.5Cs_{2 − x} [H_{0.5 − x} Fe_xCr(OH)₆Mo₆O₁₈] · 7H₂O (I) (x = 0.19) and Na_0.5Cs_{2 − x} [H_{0.5 − x} Mn_xAl(OH)₆Mo₆O₁₈] · 8H₂O (II) (x = 0.22) are determined (space group Pbcn, Z = 8, a = 23.023(4) Å, b = 22.064(4) Å, c = 11.606(3) Å, V = 5895.66 ų for I and a = 22.972(9) Å, b = 22.002(8) Å, c = 11.543(5) Å, V = 5834.18 ų for II, respectively). The [X^III(OH)₆Mo₆O₁₈]³⁻ ligands were found to be coordinated in monodentate fashion to M atoms due to the participation of a terminal O atom of the cis-MoO₂ group in coordination with the Fe and Mn atoms, which was confirmed by IR data. __________ Translated from Koordinatsionnaya Khimiya, Vol. 31, No. 9, 2005, pp. 663–676. Original Russian Text Copyright ? 2005 by Gavrilova, Molchanov.     

Kinetics of heat release during the reaction ofn-decane with nitrogen dioxide in the liquid phase

The rates of heat release in the nitrogen dioxide—n-decane system at a molar ratio of nitrogen oxides ton-decane (β) from 2.4·10⁻³ to 3.1 and gaseous volumes per mole ofn-decane (V(g)) equal to 0.05–4.5 were studied in the 55.2–92.8 °C temperature range. The initial rate of the process is determined by the interaction of NO₂ withn-decane. The equilibrium constants of dissociation of N₂O₄ inn-decane and Henry's constants of NO₂ and N₂O₄ in ann-decane solution were determined by complex analysis of the thermodynamic equilibrium in the NO₂—n-decane system and dependences of the initial rates onV(g) and β. The experimentally observed self-acceleration of the process in the region of high β and lowT values was suggested to be due to the reaction of N₂O₄ with intermediate oxidation products. The rate constants of the reaction of NO₂ withn-decane were compared with analogous values determined in its mixtures with HNO₃ solutions. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 10, pp. 1789–1794, October, 1997.