Quantum bifurcation in a coulombic‐like potential

Because of the lack of nonlinear dynamics, up to now no bifurcation phenomenon in its original sense has been discovered directly in quantum mechanical systems. Based on the formalism of complex‐valued quantum mechanics, this article derives the nonlinear Hamilton equations from the Schrödinger equation to provide the necessary mathematic framework for the analysis of quantum bifurcation. This new approach makes it possible to identify quantum bifurcation by the direct evidence of the sudden change of fixed points and their surrounding trajectories. As a practical application of the proposed approach, we consider the quantum motion in a Coulombic‐like potential modeled by V(r) = A/r² ? B/r, where the first term describes the centrifugal trend and the second deals with the Coulombic attraction. As the bifurcation parameter evolves, we demonstrate how local and global bifurcations in quantum dynamics can be identified by inspecting the changes of fixed points and their surrounding trajectories. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Foundations of the relativistic theory of many-electron bound states

Most of the existing calculations of relativistic effects in many-electron atoms or molecules are based on the Dirac–Coulomb Hamiltonian H_DC. However, because the electron–electron interaction mixes positive- and negative-energy states, the operator H_DC has no normalizable eigenfunctions. This fact undermines the quantum-theoretic rationale for the Dirac–Hartree–Fock (DHF ) equations and therefore that of the relativistic configuration-interaction (RCI ) and multiconfiguration Dirac–Fock (MCDF ) methods. An approach to this problem based on quantum electrodynamics is reviewed. It leads to a configuration-space Hamilton H which involves positive-energy projection operators dependent on an external potential U; identification of U with the nuclear potential V_ext corresponds to use of the Furry bound-state interaction picture. It is shown that the RCI method can be reinterpreted as an approximation scheme for finding eigenvalues of a Hamiltonian H, with U identified as the DHF potential; the theoretical interpretation of the MCDF method needs further clarification. It is emphasized that if U differs from V_ext one must consider the effects of virtual-pair creation by the difference potential δU = V_ext ? U; an approximate formula for the level-shift arising from δU is derived. Some ideas for dealing with the technical problems introduced by the projection operators are discussed and relativistic virial theorems are given. Finally, a possible scheme for adapting current MCDF methods to Hamiltonians involving projection operators is described.     

Quantum effects in semiclassical treatment of molecular reactions

By the use of the known Coulomb wavefunctions, “quantal” Coulomb streamlines are defined for the eikonal of the system. Comparisons of such streamlines with the classical Coulomb trajectories are made. Use of such “quantum” Coulomb streamlines is proposed for reaction systems containing the Coulomb interaction U. The derived equations of motion are of particular use for cases where the residual potential V₁ (V₁ = V ? U) has either no classical forbidden region or has relatively small effects on trajectories.     

Nonadiabatic effects in the nuclear probability and flux densities through the fractional Schrödinger equation

Nonadiabatic effects in the nuclear dynamics of the H₂⁺ molecular ion, initiated by ionization of the H₂ molecule, is studied by means of the probability and flux distribution functions arising from the space fractional Schrödinger equation. In order to solve the fractional Schrödinger eigenvalue equation, it is shown that the quantum Riesz fractional derivative operator fulfills the usual properties of the quantum momentum operator acting on the bra and ket vectors of the abstract Hilbert space. Then, the fractional Fourier grid Hamiltonian method is implemented and applied to molecular vibrations. The eigenenergies and eigenfunctions of the fractional Schrödinger equation describing the vibrational motion of the H₂⁺ and D₂⁺ molecules are analyzed. In particular, it is shown that the position-momentum Heisenberg's uncertainty relationship holds independently of the fractional Schrödinger equation. Finally, the probability and flux distributions are presented, demonstrating the applicability of the fractional Schrödinger equation for taking into account nonadiabatic effects.     

Derivation and pilot application of a scheme for intermediate Hamiltonians in Fock space

A Fock space formulation of the intermediate Hamiltonian approach is derived by introducing shift operators in the equations determining effective Hamiltonians in Fock space. A non-hermitian intermediate Hamiltonian is constructed from the Fock space Bloch equation. An alternative derivation, based on a similarity transformation expression, is presented providing access to hermitian intermediate Hamiltonians. In a pilot application, the potential curves of the two lowest¹ _g ⁺ states of the H₂ molecule are calculated demonstrating the applicability of the scheme.     

Six-dimensional tunneling dynamics of internal rotation in hydrogen peroxide molecule and its isotopomers

The six-dimensional torsion-vibration Hamiltonian of the H₂O₂ molecule and its H/D- and ¹⁸O/¹⁶O-isotopomers is derived. The Hamiltonian includes the kinetic energy operator, which depends on the tunneling coordinate, and the potential energy surface represented as a quartic polynomial with respect to the small-amplitude transverse coordinates. Parameters of the Hamiltonian were obtained from DFT calculations of the equilibrium geometries, eigenvectors, and eigenfrequencies of normal vibrations at the stationary points corresponding to the ground state and both the cis- and trans-transition states, carried out with the B3LYP density functional and 6-311+G(2d,p) basis set. The quantum dynamics problem is solved using the perturbative instanton approach generalized for the excited states situated above the barrier top. Vibration-tunneling spectra are calculated for the ground state and low-lying excited states with energies below 2000 cm^–1. Strong kinematic and squeezed potential couplings between the large-amplitude torsional motion and bending modes are shown to be responsible for the vibration-assisted tunneling and for the dependence of tunneling splittings on the quantum numbers of small-amplitude transverse vibrations. Mode-specific isotope effects are predicted.     

Temperature dependence of the rate constants of cryochemical reactions

The temperature dependence of the reaction rate constant for tunneling transfer of an atomic particle in solid near absolute zero was studied. Different mechanisms describing the temperature dependence were considered: reorganization of the medium, modulation of parameters of the potential barrier, and under-barrier friction. It was established that for the rate constant (K) at low temperatures the equation InK=InK ₀+C ₄ T ⁴+C ₅ T ⁵+C ₆ T ⁶+C ₈ T ⁸ is valid. Experimental data were compared with the theory. A good agreement is achieved when the quantum nature of the hydrogen crystal is applied under the assumption of a predominant role of reorganization of the medium. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 10, pp. 1906–1914, October, 1999.     

Inner-sphere oxidation of ternary nitrilotriacetatocobalt(II) complexes involving oxalate and phthalate by periodate

The oxidation of [Co^II(nta)(ox)(H₂O)₂]³⁻ and [Co^II(nta)(ph)(H₂O)₂]³⁻ (nta = nitrilotriacetate, ox = oxalic acid and ph = phthalic acid) by periodate have been studied kinetically in aqueous solution over 20–40 °C and a variety of pH ranges. The rate of oxidation of [Co^II(nta)(ox)(H₂O)₂]³⁻ by periodate, obeys the following equation: d[Co^III]/dt = [Co^II(nta)(ox)(H₂O)₂³⁻][H₅IO₆] {k ₄ K ₅ + (k ₅ K ₆ K ₂/[H⁺]} while the reaction of [Co^II(nta)(ph)(H₂O)₂]³⁻ with periodate in aqueous acidic medium obeys the following rate law: d[Co^III]/dt = k ₆ K ₈[Co^II]_T [I^VII]_T/{1 + [H⁺]/K ₇ + K ₈[I^VII] _T }. Initial cobalt(III) products were formed and slowly converted to final products, fitting an inner-sphere mechanism. Thermodynamic activation parameters have been calculated. A common mechanism for the oxidation of ternary nitrilotriacetatocobalt(II) complexes by periodate is proposed and supported by an excellent isokinetic relationship between ΔH* and ΔS* values for these reactions.     

Thermodynamic investigation of several natural polyols (II)

The low-temperature heat capacity C _p,m of sorbitol was precisely measured in the temperature range from 80 to 390 K by means of a small sample automated adiabatic calorimeter. A solid-liquid phase transition was found at T=369.157 K from the experimental C _p-T curve. The dependence of heat capacity on the temperature was fitted to the following polynomial equations with least square method. In the temperature range of 80 to 355 K, C _p,m/J K⁻¹ mol⁻¹=170.17+157.75x+128.03x ²-146.44x ³-335.66x ⁴+177.71x ⁵+306.15x ⁶, x= [(T/K)−217.5]/137.5. In the temperature range of 375 to 390 K, C _p,m/J K⁻¹ mol⁻¹=518.13+3.2819x, x=[(T/K)-382.5]/7.5. The molar enthalpy and entropy of this transition were determined to be 30.35±0.15 kJ mol⁻¹ and 82.22±0.41 J K⁻¹ mol⁻¹ respectively. The thermodynamic functions [H _T-H _298.15] and [S _T-S 298.15], were derived from the heat capacity data in the temperature range of 80 to 390 K with an interval of 5 K. DSC and TG measurements were performed to study the thermostability of the compound. The results were in agreement with those obtained from heat capacity measurements.     

Temperature dependence of the rate constant of ionic analogs of H + H2 → H2 + H reaction by quantum instanton method

Rate constant ratios k(T)/k(1,500K) for two symmetrical reactions H^? + H₂ → H₂ + H^? and H⁺ + H₂ → H₂ + H⁺ are reported. Direct method based on quantum instanton approximation for evaluation of the temperature dependence of the quantum‐mechanical reaction rate constant is used. Implementation of the theory involves thermodynamic integration and path integral Monte Carlo method. Results of anionic case shows resemblance to neutral case, whereas cationic case is significantly different and below 1,000K rate constant shows strong deviation form linearity of Arrhenius plot due to high activation barrier. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Monohydration selectivity mechanism of alkali cations in the potassium channel of excitable biomembranes

Nonempirical quantum chemical method Hartree–Fock–Roothan LCAO SCF MO in a two-exponent Dunning basis with the use of an extended set of Gaussian functions by Huzinaga–Dunning with consideration of electron correlation according to the Meller–Plesset theory of excitations of the second order was used to study monohydrates of Li⁺, Na⁺, K⁺, and HCOO^? ions. The indicated basis was supplemented with polarization functions of d-type on the O atom and of p-type on the hydrogen atom as well as with diffusion functions of p-type on the oxygen atom. It has been found that binding energies of the water molecule with Li⁺, Na⁺ appeared to be higher and with K⁺ lower than with HCOO^? · H₂O. Potential curve shapes of K⁺ + H₂O and HCOO^? + H₂O reactions are shown to be similar. The molecular mechanism of K⁺ channel selectivity of an excitable membrane is explained on the basis of the obtained calculations.     

On the sudden approximation of cross: computational tests and factorization of cross sections and related scattering phenomena

A sudden approximation recently derived by Cross using a semiclassical treatment of the orbital motion is recast into a form which permits factorization of differential and integral degeneracy averaged cross sections, opacities as a function of final angular momentum quantum number, the scattering amplitude, and the phenomenological cross section which describes spectral line broadening. Calculations are done using an average of initial and final orbital angular momentum quantum numbers for the partial wave parameter for ArN₂, ArTIF, H⁺H₂ and Li⁺H₂. The results indicate that the method is a good approximation for integral cross sections and opacities when the energy sudden approximation is valid and when the coupling of the orbital motion is important.     

A Genuine Jahn–Teller System with Compressed Geometry and Quantum Effects Originating from Zero‐Point Motion

First‐principle calculations together with analysis of the experimental data found for 3d⁹ and 3d⁷ ions in cubic oxides proved that the center found in irradiated CaO:Ni²⁺ corresponds to Ni⁺ under a static Jahn–Teller effect displaying a compressed equilibrium geometry. It was also shown that the anomalous positive g_∥ shift (g_∥?g₀=0.065) measured at T=20 K obeys the superposition of the |3 z²?r²? and |x²?y²? states driven by quantum effects associated with the zero‐point motion, a mechanism first put forward by O'Brien for static Jahn–Teller systems and later extended by Ham to the dynamic Jahn–Teller case. To our knowledge, this is the first genuine Jahn–Teller system (i.e. in which exact degeneracy exists at the high‐symmetry configuration) exhibiting a compressed equilibrium geometry for which large quantum effects allow experimental observation of the effect predicted by O'Brien. Analysis of the calculated energy barriers for different Jahn–Teller systems allowed us to explain the origin of the compressed geometry observed for CaO:Ni⁺.     

Low-temperature heat capacities and thermodynamic properties of 2,2-dimethyl-1,3-propanediol

The molar heat capacities C _p,m of 2,2-dimethyl-1,3-propanediol were measured in the temperature range from 78 to 410 K by means of a small sample automated adiabatic calorimeter. A solid-solid and a solid-liquid phase transitions were found at T-314.304 and 402.402 K, respectively, from the experimental C _p-T curve. The molar enthalpies and entropies of these transitions were determined to be 14.78 kJ mol⁻¹, 47.01 J K⁻¹ mol⁻ for the solid-solid transition and 7.518 kJ mol⁻¹, 18.68 J K⁻¹ mol⁻¹ for the solid-liquid transition, respectively. The dependence of heat capacity on the temperature was fitted to the following polynomial equations with least square method. In the temperature range of 80 to 310 K, C _p,m/(J K⁻¹ mol⁻¹)=117.72+58.8022x+3.0964x ²+6.87363x ³−13.922x ⁴+9.8889x ⁵+16.195x ⁶; x=[(T/K)−195]/115. In the temperature range of 325 to 395 K, C _p,m/(J K⁻¹ mol⁻¹)=290.74+22.767x−0.6247x ²−0.8716x ³−4.0159x ⁴−0.2878x ⁵+1.7244x ⁶; x=[(T/K)−360]/35. The thermodynamic functions H _T−H _298.15 and S _T−S _298.15, were derived from the heat capacity data in the temperature range of 80 to 410 K with an interval of 5 K. The thermostability of the compound was further tested by DSC and TG measurements. The results were in agreement with those obtained by adiabatic calorimetry.     

Kinetic studies of the oxidation of chromium(III) by N-bromosuccinimide

Summary A kinetic study of the oxidation of chromium(III) by N-bromosuccinimide (NBS) in aqueous solutions and H₂O-MeOH solvent mixtures were performed. The kinetics in aqueous solutions obeyed the rate law: d[Cr^VI]/dt = {k ₄ K _h K ₂[NBS][Cr^III]_T}/[₊]{1 + K _h/[H⁺] + (K ₁ + K _h K ₂/[H⁺][NBS])} where K _h, K ₁ and K ₂ are the hydrolysis constant of [Cr^III(H₂O)₆]³⁺, and pre-equilibrium formation constants for the protonated and deprotonated precursor complexes, respectively. An innersphere mechanism is proposed. An argument based on isokinetic correlations among activation parameters for the oxidation of a series of cobalt(II) and chromium(III) complexes including [Cr(H₂O)₆]³⁺ is presented in support of a common mechanism for these reactions. Abstracted from the Ph.D. Thesis (Ain Shams University) of A. E.-D. M. Abdel-Hady.     

The Effect of Guest Metal Ions on the Reduction Potentials of Uranium(VI) Complexes: Experimental and Theoretical Investigations

Two mononuclear uranyl complexes, [UO₂L¹] ( 1 ) and [UO₂L²] ⋅ 0.5 CH₃CN ⋅ 0.25 CH₃OH ( 2 ), have been synthesized from two multidentate N₃O₄ donor ligands, N,N′-bis(5-methoxysalicylidene)diethylenetriamine (H₂L¹) and N,N′-bis(3-methoxysalicylidene)diethylenetriamine (H₂L²), respectively, and have been structurally characterized. Both complexes 1 and 2 showed a reversible U^VI/U^V couple at −1.571 and −1.519 V, respectively, in cyclic voltammetry. The reduction potential of the U^VI/U^V couple shifted towards more positive potential on addition of Li⁺, Na⁺, K⁺, and Ag⁺ metal ions to acetonitrile solutions of complex 2 , and the resulting potential was correlated with the Lewis acidity of the metal ions and was also justified by theoretical DFT calculations. No such shift in reduction potential was observed for complex 1 . All four bimetallic products, [UO₂L²Li_0.5](ClO₄)_0.5 ( 3 ), [UO₂L²Na(ClO₄)]₂ ( 4 ), [UO₂L²Ag(NO₃)(H₂O)] ( 5 ), and [(UO₂L²)₂K(H₂O)₂]PF₆ ( 6 ), formed on addition of the Li⁺, Na⁺, Ag⁺, and K⁺ metal ions, respectively, to acetonitrile solutions of complex 2 , were isolated in the solid state and structurally characterized by single-crystal X-ray diffraction. In all the species, the inner N₃O₂ donor set of the ligand encompasses the equatorial plane of the uranyl ion and the outer open compartment with O₂O′₂ donor sites hosts the second metal ion.     

Lie algebraic approach to the scattering system He + H2

In this article, the Hamiltonian for the scattering of the H_e + H₂ system is given by using the interaction potential V(X, Y, Θ) determined by experiments and the semiclassical method. From this Hamiltonian we find a dynamical algebra h₆. The statistical expectation of the energy and the transition probability of H₂, P_n→m, are derived; therefore, selection rules have been found easily.     

Study of equilibria and kinetics of the acid catalysed interaction of iron(III) with salicylaldehyde ando-hydroxyacetophenone

The equilibria and kinetics of the reaction of Fe^III with salicylaldehyde ando-hydroxyacetophenone, leading to 1∶1 chelate formation, have been studied at different temperatures (25–35°C) and ionic strength, I = 1.0 mol dm⁻³ (NaClO₄+HClO₄). A dual path mechanism involving both Fe _aq ³⁺ and Fe(OH) _aq ²⁺ species and undissociated free ligand (LH) is consistent with the experimental observations where [H⁺]≫[Fe]_T≫[L]_T (where [Fe]_T and [L]_T stand for total concentrations of iron and ligand respectively). The results conform to k_obs/B = k₁[H⁺]+k₂K_h where B = [Fe]_T/(K_h+[H⁺])+1/Q; K_h = hydrolysis constant of Fe _aq ³⁺ ; k₁, k₂ are the forward second order rate constants of Fe _aq ³⁺ and Fe(OH) _aq ²⁺ , respectively, and Q is the equilibrium constant of the reaction, Fe³⁺+LH⇋FeL²⁺+H⁺. Thermodynamic parameters for each of the steps have been determined. Fe(OH) _aq ²⁺ appears to react in a dissociative fashion (Eigen-Tamm mechanism), whilst Fe _aq ³⁺ appears to react through the associative inter-change (I_a) mechanism. The equilibrium constants (Q) obtained spectrophotometrically are compared with those obtained from kinetic studies. TMC 2638     

Thermal behavior of the polyoxometalates derived from H3PMo12O40 and H4PVMo11O40

The aim of this study is to investigate the influence of some monovalent counter-ions (NH₄ ⁺, K⁺ and Cs⁺) on thermal behavior of polyoxometalates derived from H₃PMo₁₂O₄₀ (HPM) and H₄PVMo₁₁O₄₀ (HPVM) by replacing the protons. The IR and UV-VIS-DRS spectra of some acid and neutral NH₄ ⁺, K⁺, Cs⁺ salts, which derived from HPM and HPVM, confirmed the preservation of Keggin units (KU) structure. The X-ray diffraction spectra clearly showed the presence of a cubic structure. The non-isothermal decomposition of studied polyoxometalates proceeds by a series of processes: the loss of crystallization water; the loss of O₂ accompanying with a reduction of V⁵⁺→V⁴⁺ and Mo⁶⁺→Mo⁵⁺; the loss of constitution water started at 360°C for HPVM salts and 420°C for HPM salts; the decomposition of ammonium ion over 420°C with NH₃, N² and H₂O elimination and simultaneous processes of reduction (V⁵⁺→ V⁴⁺ and Mo⁶⁺→ Mo⁵⁺ or Mo⁴⁺) associating with endothermic effects; reoxidation of Mo⁵⁺, Mo⁴⁺ and V⁴⁺with a strong exothermic effect; destruction of KU to the oxides: P₂O₅, MoO₃ and V₂O₅ and the crystallization of MoO₃. This revised version was published online in July 2006 with corrections to the Cover Date.     

Heat Capacities and Thermodynamic Properties of Aqueous SrCl<Subscript>2</Subscript> Solutions in the Temperature Range from 80 to 320 K

The molar heat capacities of three different concentrations of aqueous SrCl₂ solutions, 0.1212, 0.4615 and 1.878 mol⋅kg⁻¹, were measured, using a precision automated adiabatic calorimeter in the temperature range from 80 to 320 K. Solid–liquid phase transitions were observed at 272.83, 270.18 and 255.15 K, respectively, for these three solutions. The molar enthalpies and entropies of the phase transitions were evaluated. The experimental heat capacity data were fitted to polynomial equations, and based on the polynomial equations and thermodynamic relationship, the thermodynamic functions relative to 298.15 K, [H _T −H _298.15 K] and [S _T −S _298.15 K], of the three solutions were derived in the range of 80 to 320 K with an interval of 5 K.