SWKB Approach to Confined Isospectral Potentials

Recently we had formulated the supersymmetric Wentzel–Kramers–Brillouin (SWKB) quantization rule for one-dimensional confined quantum systems and applied the same to two trigonometric potentials, tangentially limited by infinite walls at x=0 and x=L, viz., V(x)=V ₀cot²(x/L) and the Pöschl–Teller potential, V(x)=V ₀₁cosec²({x/(2L))}+V ₀₂sec²(x/(2L)). Both the potentials have received quite a lot of attention by various authors because of their importance in molecular physics. Though these potentials have been studied in the framework of WKB, BS (Bohr–Sommerfeld), mBS (matrix formulation of BS) formalisms, it was observed that the supersymmetric approach not only rendered the calculations simpler and more transparent, it also reproduced the exact analytical energies in both the cases.In this study, we shall generate isospectral Hamiltonians of the above potentials with the help of a modified form of Darboux's theorem. We shall show that though the new potentials look different from the original ones, and have different eigenfunctions, they too, are confined in the same region of space, and share the same energy spectrum as their original counterparts. This may be of substantial importance in determining the energy spectrum of highly non-trivial systems.     

Slater sum for an inhomogeneous liquid of Fermions generated by a sech2 x potential in one dimension

In one dimension, the Slater sum S(x, β), which is the diagonal element of the canonical density matrix, satisfies a known partial differential equation characterised by a one-body potential V(x). Here, for the case of a sech² x potential in one dimension, it is stressed that S(x, β) is explicitly related to the limit S ₀(β) as V(x) → 0 and to V(x) itself. This is the same input information as in the Thomas–Fermi result. The relevance to density functional theory is emphasised.     

Zur Bedeutung von V2O3(OH)4(g) für den chemischen Transport von V2O5(s) in Anwesenheit von Wasser. Nachweis des Gasteilchens,thermodynamische Daten und Modellrechnungen

V₂O₃(OH)₄(g), Proof of Existence, Thermochemical Characterization, and Chemical Vapor Transport Calculations for V₂O₅(s) in the Presence of Water By use of the Knudsen-cell mass spectrometry the existence of V₂O₃(OH)₄(g) is shown. For the molecules V₂O₃(OH)₄(g), V₄O₁₀(g), and V₄O₈(g) thermodynamic properties were calculated by known Literatur data. The influence of V₂O₃(OH)₄(g) for chemical vapor transport reactions of V₂O₅(s) with water ist discussed. Δ_BH°(V₂O₃(OH)₄(g), 298) = –1920 kJ · mol^–1 and S°(V₂O₃(OH)₄(g), 298) = 557 J · K^–1 · mol^–1, Δ_BH°(V₄O₁₀(g), 298) = –2865,6 kJ · mol^–1 and S°(V₄O₁₀(g), 298) = 323.7 J · K^–1 · mol^–1, Δ_BH°(V₄O₈(g), 298) = –2465 kJ · mol^–1 and S°(V₄O₈(g), 298) = 360 J · K^–1 · mol^–1.     

Solid Solution Formation between Vanadium(V) and ­Tungs­ten(V) Oxide Phosphate

The solid solutions (V_1–xW_x)OPO₄ with β‐VOPO₄ structure type (0.0 ≤ x ≤ 0.01) and α_II‐VOPO₄ structure type (0.04 ≤ x ≤ 0.26) were obtained from mixtures of V^VOPO₄ and W^VOPO₄ by conventional solid state reactions and by solution combustion synthesis. Single crystals of up to 3 mm edge length were obtained by chemical vapor transport (CVT) (800 → 700 °C, Cl₂ as a transporting agent). Single crystal structure refinements of crystals at x = 0.10 [a = 6.0503(2) Å, c = 4.3618(4) Å, R₁ = 0.021, wR₂ = 0.058, 21 parameters, 344 independent reflections] and x = 0.26 [a = 6.0979(2) Å, c = 4.2995(1) Å, R₁ = 0.030, wR₂ = 0.081, 21 parameters, 346 independent reflections] confirm the α_II‐VOPO₄ structure type (P4/n, Z = 2) with mixed occupancy V/W for the metal site. Due to the specific redox behavior of W⁵⁺ and V⁵⁺, solid solutions (V_1–xW_x)OPO₄ should be formulated as (V^IV_xV^V_1–2xW^VI_x)OPO₄. The valence states of vanadium and tungsten are confirmed by XPS measurements. V⁴⁺ with d¹ configuration was identified by EPR spectroscopy and magnetic measurements. Electronic spectra of the solid solutions show the IVCT(V⁴⁺ → V⁵⁺) and the LMCT(O^2– → V⁵⁺). (V_0.74W_0.26)OPO₄ powders exhibit semi‐conducting behavior (E_g = 0.7 eV).     

Effects of Cr doping on the electrochemical performance of Li3V2(PO4)3 cathode material for lithium ion batteries

A series of Cr-doped Li₃V_2 − x Cr _x (PO₄)₃ (x = 0, 0.1, 0.25, and 0.5) samples are prepared by a sol–gel method. The effects of Cr doping on the physical and chemical characteristics of Li₃V₂(PO₄)₃ are investigated. Compared with the XRD pattern of the undoped sample, the XRD patterns of the Cr-doped samples have no extra reflections, which indicates that Cr enters the structure of Li₃V₂(PO₄)₃. As indicated by the charge–discharge measurements, the Cr-doped Li₃V_2 − x Cr _x (PO₄)₃ (x = 0.1, 0.25, and 0.5) samples exhibit lower initial capacities than the undoped sample at the 0.2 C rate. However, both the discharge capacity and cycling performance at high rates (e.g., 1 and 2 C) are enhanced with proper amount of Cr doping (x = 0.1). The highest discharge capacity and capacity retention at the rates of 1 and 2 C are obtained for Li₃V_1.9Cr_0.1(PO₄)₃. The improvement of the electrochemical performance can be attributed to the higher crystal stability and smaller particle size induced by Cr doping.

     

Photodetachment dynamics from closed‐shell anions in the presence of a bichromatic field

I have studied the dynamics of photodetachment from closed‐shell anions in the presence of a two‐color (bichromatic) laser field. The electronic states of halide ions are modeled by a 1‐D Hamiltonian with a potential V(x) = ?V₀e. The two parameters V₀ and σ are fixed by requiring V(x) to reproduce the experimentally observed ground‐state ionization energy of the halide ions concerned. The potentials so generated are shown to support only one bound state in each case. The time‐dependent Fourier grid Hamiltonian method is used to follow the detachment dynamics with fairly high intensities of light. The environmental effects on the dynamics are sought to be modeled by allowing the well depth (V₀) to fluctuate randomly (V₀(t) = V₀[1 + ΔVR(t)]; R(t) randomly fluctuates between +1 and ?1 with time, when ΔV is fixed). The average detachment rate constants k_av are seen to increase with increase in the intensities of used bichromatic field. An alternative model potential, V(x) = ?V₀e^?σx, is also shown to yield similar results. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

Modeling quantum dynamics of photodetachment from closed‐shell anions: Static versus fluctuating well‐depth models

The electronic states of halide ions are modeled by a one‐dimensional Hamiltonian with a potential V(x)=−V₀e. The two parameters V₀ and σ are fixed by requiring V(x) to reproduce the experimentally observed ground‐state ionization potentials of the halide ions concerned. The potentials so generated are shown to support only one bound state in each case. The time‐dependent Fourier grid Hamiltonian method is used to follow the ionization dynamics in monochromatic light of fairly high intensities. The total Hamiltonian, in the most general case, reads H(t)=P/2m−V₀e−ϵ₀s(t)ex sin(ωt). For pulsed fields [s(t)=sin²(πt/t_p)], the computed ionization rate constants are seen to increase with increase in the peak intensity (ϵ₀) of the electric field of light. The possibility of additional transient bound states being generated at the high intensities of light and its possible consequences on the observed ionization rates are explored. The environmental effects on the dynamics are sought to be modeled by allowing the well depth (V₀) to fluctuate randomly [V₀(t)=V₀+ΔVR(t); R(t) randomly fluctuates between +1 and −1 with time, ΔV is fixed]. The ionization rate constants (k_ϵ) are shown to be significantly affected by fluctuations in V₀ and pass through a well‐defined minimum in each case for a certain specified frequency of fluctuation. An alternative model potential V(x)=−V₀e^−σx is also shown to yield similar results. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 469–478, 1999     

Methane Activation Mediated by a Series of Cerium–Vanadium Bimetallic Oxide Cluster Cations: Tuning Reactivity by Doping

The reactions of cerium–vanadium cluster cations Ce_xV_yO_z⁺ with CH₄ are investigated by time‐of‐flight mass spectrometry and density functional theory calculations. (CeO₂)_m(V₂O₅)_n⁺ clusters (m=1,2, n=1–5; m=3, n=1–4) with dimensions up to nanosize can abstract one hydrogen atom from CH₄. The theoretical study indicates that there are two types of active species in (CeO₂)_m(V₂O₅)_n⁺, V[(O_t)₂]^. and [(O_b)₂CeO_t]^. (O_t and O_b represent terminal and bridging oxygen atoms, respectively); the former is less reactive than the latter. The experimentally observed size‐dependent reactivities can be rationalized by considering the different active species and mechanisms. Interestingly, the reactivity of the (CeO₂)_m(V₂O₅)_n⁺ clusters falls between those of (CeO₂)_2–4⁺ and (V₂O₅)_1–5⁺ in terms of C?H bond activation, thus the nature of the active species and the cluster reactivity can be effectively tuned by doping.     

Trigonal Prismatic Coordination of Discrete Rare Earth Ions,Enforced by the Polyoxotungstate [P4W27O99(H2O)]16–

A family of solution-stable polyanions [Na⊂{Ln^III(H₂O)}{W^VIO(H₂O)}P^V₄W^VI₂₆O₉₈]¹²⁻ (Ln=Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu and Y) represent the first examples of polyoxometalates comprising a single lanthanide(III) or yttrium(III) ion in a rare trigonal prismatic O₆ environment. Their synthesis exploits the reactivity of the organophosphonate-functionalized precursor [P₄W₂₄O₉₂(C₆H₅P^VO)₂]¹⁶⁻ with heterometal ions and yields hydrated potassium or mixed lithium/potassium salts of composition K_xLn_yH_12–x–y[Na⊂{Ln(H₂O)}{WO(H₂O)}P₄W₂₆O₉₈]⋅nH₂O⋅mLiCl (x=8.5–11; y=0–2; n=24–34; m=0–1.5). The Dy, Ho, Er and Yb derivatives are characterized by slow magnetization relaxation.     

Gas-Phase Synthesis of Singly and Multiply Charged Polyoxovanadate Anions Employing Electrospray Ionization and Collision Induced Dissociation

Electrospray ionization mass spectrometry (ESI-MS) combined with in-source fragmentation and tandem mass spectrometry (MS/MS) experiments were used to generate a wide range of singly and multiply charged vanadium oxide cluster anions including V_xO_y ^n– and V_xO_yCl^n– ions (x = 1–14, y = 2–36, n = 1–3), protonated clusters, and ligand-bound polyoxovanadate anions. The cluster anions were produced by electrospraying a solution of tetradecavanadate, V₁₄O₃₆Cl(L)₅ (L = Et₄N⁺, tetraethylammonium), in acetonitrile. Under mild source conditions, ESI-MS generates a distribution of doubly and triply charged V_xO_yCl^n– and V_xO_yCl(L)^(n–1)– clusters predominantly containing 14 vanadium atoms as well as their protonated analogs. Accurate mass measurement using a high-resolution LTQ/Orbitrap mass spectrometer (m/Δm = 60,000 at m/z 410) enabled unambiguous assignment of the elemental composition of the majority of peaks in the ESI-MS spectrum. In addition, high-sensitivity mass spectrometry allowed the charge state of the cluster ions to be assigned based on the separation of the major from the much less abundant minor isotope of vanadium. In-source fragmentation resulted in facile formation of smaller V_xO_yCl^(1–2)– and V_xO_y ^(1–2)– anions. Collision-induced dissociation (CID) experiments enabled systematic study of the gas-phase fragmentation pathways of the cluster anions originating from solution and from in-source CID. Surprisingly simple fragmentation patterns were obtained for all singly and doubly charged V_xO_yCl and V_xO_y species generated through multiple MS/MS experiments. In contrast, cluster anions originating directly from solution produced comparatively complex CID spectra. These results are consistent with the formation of more stable structures of V_xO_yCl and V_xO_y anions through low-energy CID. Furthermore, our results demonstrate that solution-phase synthesis of one precursor cluster anion combined with gas-phase CID is an efficient approach for the top-down synthesis of a wide range of singly and multiply charged gas-phase metal oxide cluster anions for subsequent investigations of structure and reactivity using mass spectrometry and ion spectroscopy techniques.      

Structural and Electrochemical Study of Vanadium‐Doped TiO2 Ramsdellite with Superior Lithium Storage Properties for Lithium‐Ion Batteries

Titanium‐oxide‐based materials are considered attractive and safe alternatives to carbonaceous anodes in Li‐ion batteries. In particular, the ramsdellite form TiO₂(R) is known for its superior lithium‐storage ability as the bulk material when compared with other titanates. In this work, we prepared V‐doped lithium titanate ramsdellites with the formula Li_0.5Ti_1?xV_xO₂ (0≤x≤0.5) by a conventional solid‐state reaction. The lithium‐free Ti_1?xV_xO₂ compounds, in which the ramsdellite framework remains virtually unaltered, are easily obtained by a simple aqueous oxidation/ion‐extraction process. Neutron powder diffraction is used to locate the Li channel site in Li_0.5Ti_1?xV_xO₂ compounds and to follow the lithium extraction by difference‐Fourier maps. Previously delithiated Ti_1?xV_xO₂ ramsdellites are able to insert up to 0.8 Li⁺ per transition‐metal atom. The initial gravimetric capacities of 270 mAh g^?1 with good cycle stability under constant current discharge conditions are among the highest reported for bulk TiO₂‐related intercalation compounds for the threshold of one e^? per formula unit.     

Schrödinger Equation Solutions for the Central Field Power Potential Energy I. V(r) = V 0(r/a 0)2ν−2, ν ≥ 1

The solution of a generalized non-relativistic Schrödinger equation with radial potential energy V(r)=V ₀(r/a ₀)^2–2 is presented. After reviewing the general properties of the radial ordinary differential equation, power series solutions are developed. The Green's function is constructed, its trace and the trace of its first iteration are calculated, and the ability of the traces to provide upper and lower bounds for the ground eigenvalue is examined. In addition, WKB-like solutions for the eigenvalues and eigenfunctions are derived. The approximation method yields valid eigenvalues for large quantum numbers (Rydberg states).     

Reactivity Control of C?H Bond Activation over Vanadium–Silver Bimetallic Oxide Cluster Cations

Vanadium–silver bimetallic oxide cluster ions (V_xAg_yO_z⁺; x=1–4, y=1–4, z=3–11) are produced by laser ablation and reacted with ethane in a fast‐flow reactor. A reflectron time of flight (Re‐TOF) mass spectrometer is used to detect the cluster distribution before and after the reactions. Hydrogen atom abstraction (HAA) reactions are identified over VAgO₃⁺, V₂Ag₂O₆⁺, V₂Ag₄O₇⁺, V₃AgO₈⁺, V₃Ag₃O₉⁺, and V₄Ag₂O₁₁⁺ ions, in which the oxygen‐centered radicals terminally bonded on V atoms are active sites for the facile HAA reactions. DFT calculations are performed to study the structures, bonding, and reactivity. The reaction mechanisms of V₂Ag₂O₆⁺+C₂H₆ are also given. The doped Ag atoms with a valence state of +1 are highly dispersed at the periphery of the V_xAg_yO_z⁺ cluster ions. The reactivity can be well‐tuned gradually by controlling the number of Ag atoms. The steric protection due to the peripherally bonded Ag atoms greatly enhances the selectivity of the V–Ag bimetallic oxide clusters with respect to the corresponding pure vanadium oxide systems.     

Theoretical studies of the concentration dependences of d–d transition band,g factors,and local distortion for V4+ in Li2O–PbO–B2O3–P2O5 glasses

In this work, the g factors, d–d transition band, local distortion, and their concentration dependences for impurity V⁴⁺ in 20Li₂O–20PbO–45B₂O₃–(15 − x)P₂O₅:V₂O₅ (0 ≤ x ≤ 2.5 mol%) glasses are theoretically investigated by using perturbation formulas of g factors for a tetragonally compressed octahedral 3d¹ cluster. In the light of the cubic polynomial concentration functions for cubic field parameter D_q, covalency factor N, and relative tetragonal compression ratio ρ, the calculated concentration dependences of d–d transition band and g factors for V⁴⁺ show good agreement with the experimental data. With increasing x, N (≈0.7682–0.8165) displays the monotonously increasing trend, whereas ρ (≈6.5–4.2%) and D_q (≈1504.9–1481.1 cm⁻¹) exhibit the decreasing tendencies. The above concentration dependences can be ascribed to the modifications of the V⁴⁺–O²⁻ bonding and orbital admixtures around the impurity V⁴⁺ due to the effects of V₂O₅ doping on the stability of the glass network, the strength of local crystal fields, and the electron cloud distribution.     

Concentration dependence of the global and anisotropic dimensions of confined macromolecules

The chain dimensions 〈R²〉 of nondilute polymer solutions confined to a slit of the width D were studied using lattice simulations. It was found that the chain compression induced in good solvents by the concentration ϕ is enhanced in a slit relative to the bulk. The global dimensions of chains also change with ϕ in confined and unconfined theta solutions. At intermediate slit widths, a region was noted where coils are squeezed along all three axes. This region is manifested as a channel on the three‐dimensional surface 〈R²〉(D,ϕ) in both good and theta solvents. The coil anisotropy, given by the ratio of the parallel and perpendicular components of the chain dimensions 〈R_y²〉/〈R_x²〉, reaches high values at strong confinements, where coils form quasi‐two‐dimensional pancakes. The concentration‐induced reduction of the global chain dimensions in good solvents is almost fully transmitted to the parallel component 〈R_y²〉. The computed effects of concentration and confinement were compared with the predictions of mean‐field and scaling theories, and implications of the results to ultrathin films and layered nanocomposites were discussed. In addition, the distribution functions of the components of the end‐to‐end distance R perpendicular and parallel to the plates, W (R_x) and W (R_y), were calculated. The function W (R_x) combined with the concentration profile ϕ (x) along the pore provided details of the chain structure close to walls. A marked difference in the pace of the filling up of the depletion layer was noticed between chains in theta and good solvents. From the distribution functions W (R_x) and W (R_y), the highly anisotropic force‐elongation relations imply the deformation of chains in confined solutions and ultrathin bulk films.     

Regenerating Spent Solutions of Vanadium‐containing Heteropoly Acids in the Presence of Additives

Vanadium‐containing heteropoly acid solutions of Keggin H_3+xPMo_12–xV_xO₄₀ and modified H_aP_zMo_yV_xO_b types (P‐Mo‐V HPAs) are promising nanosized inorganic metal‐oxygen cluster compounds with the property of reversible oxidability (V^V ↔ V^IV). The oxidation of reduced P‐Mo‐V HPAs at a temperature of 130–170 °C and an oxygen pressure of 4 atm is a convenient method for their regeneration, but results in regeneration degree of only 75–88 %. Various materials with electron transfer or oxidative properties, such as nitrogen doped carbon nanofibers (N‐CNFs), Sibunit‐4, HNO₃, and MoO₂, were investigated as additives to facilitate and accelerate the regeneration of HPA solutions. Among the studied additives HNO₃ was found to show the best efficiency, resulting in regeneration degree of higher 95 %. Rapid and efficient regeneration of spent HPA catalysts is an important criterion for achieving high productivity and sustainability of oxidative processes on their basis.     

The green's functions of the Schrödinger equation for the simplest systems

The Green's functions for the simplest quantum mechanical systems the linear harmonic oscillator, the three-dimensional isotropic oscillator, the Morse oscillator, the Kratcer potential, and the double-minimum potential V_(x) = (mw²/2)(/x/?R)² are presented in closed analytical forms.     

Two novel windmill-like tetrasupporting heteropolyoxometalates: [MoVI7MoVVIV8O40(PO4)][M(phen)2(OH)]2[M(phen)2(OEt)]2 (M=Co,Ni)

Two interesting neutral tetrasupporting heteropolyoxometalates: [Mo^VI₇Mo^VV^IV₈O₄₀(PO₄)][M(phen)₂(OH)]₂[M(phen)₂(OEt)]₂·xH₂O (phen=1,10-phenanthroline, EtOH=ethanol, M=Co, x=7, 1; M=Ni, x=6, 2) were hydrothermally prepared and structurally characterized. The mixed molybdenum–vanadium polyoxoanion [Mo^VI₇Mo^VV^IV₈O₄₀(PO₄)]⁴⁻ exist in both two complexes, which acts as a bridge to covalently link two pairs of transition metal complex fragments, generating neutral windmill-like trimetallic nanocluster polyoxometalates. Variable-temperature magnetic susceptibility measurements of complexes 1 and 2 reveal that antiferromagnetic exchange interaction exists in this type of trimetallic tetrasupporting heteropolyoxometalates.     

Atomic‐Level Alloying and De‐alloying in Doped Gold Nanoparticles

Atomically precise alloying and de‐alloying processes for the formation of Ag–Au and Cu–Au nanoparticles of 25‐metal‐atom composition (referred to as Ag_xAu_25?x(SR)₁₈ and Cu_xAu_25?x(SR)₁₈, in which R=CH₂CH₂Ph) are reported. The identities of the particles were determined by matrix‐assisted laser desorption ionization mass spectroscopy (MALDI‐MS). Their structures were probed by fragmentation analysis in MALDI‐MS and comparison with the icosahedral structure of the homogold Au₂₅(SR)₁₈ nanoparticles (an icosahedral Au₁₃ core protected by a shell of Au₁₂(SR)₁₈). The Cu and Ag atoms were found to preferentially occupy the 13‐atom icosahedral sites, instead of the exterior shell. The number of Ag atoms in Ag_xAu_25?x(SR)₁₈ (x=0–8) was dependent on the molar ratio of Ag^I/Au^III precursors in the synthesis, whereas the number of Cu atoms in Cu_xAu_25?x(SR)₁₈ (x=0–4) was independent of the molar ratio of Cu^II/Au^III precursors applied. Interestingly, the Cu_xAu_25?x(SR)₁₈ nanoparticles show a spontaneous de‐alloying process over time, and the initially formed Cu_xAu_25?x(SR)₁₈ nanoparticles were converted to pure Au₂₅(SR)₁₈. This de‐alloying process was not observed in the case of alloyed Ag_xAu_25?x(SR)₁₈ nanoparticles. This contrast can be attributed to the stability difference between Cu_xAu_25?x(SR)₁₈ and Ag_xAu_25?x(SR)₁₈ nanoparticles. These alloyed nanoparticles are promising candidates for applications such as catalysis.     

Generic hypersurface singularities

The problem considered here can be viewed as the analogue in higher dimensions of the one variable polynomial interpolation of Lagrange and Newton. Let x₁,...,x _r be closed points in general position in projective spacePn, then the linear subspaceV ofH ⁰ (⨑ⁿ,O(d)) (the space of homogeneous polynomials of degreed on ⨑ⁿ) formed by those polynomials which are singular at eachx _i, is given by r(n + 1) linear equations in the coefficients, expressing the fact that the polynomial vanishes with its first derivatives at x₁,...,x _r. As such, the “expected” value for the dimension ofV is max(0,h ⁰(O(d))−r(n+1)). We prove thatV has the “expected” dimension for d≥5 (theorem A). This theorem was first proven in [A] using a very complicated induction with many initial cases. Here we give a greatly simplified proof using techniques developed by the authors while treating the corresponding problem in lower degrees.