Transformed Dirac equation for the hydrogen atom,comparison with previous approaches in momentum space,and the anomalous Zeeman effect in momentum representation

The solution of a unitarily transformed Dirac equation for the hydrogenic electron in zero magnetic field is investigated here. The momentum‐space representation is adopted as a natural recourse. The spinor part of the transformed wavefunction in momentum space can be easily prescribed for a central potential. Hence, for the Coulomb potential, a pair of equations is obtained for the radial components in momentum space. It is shown that starting from these radial equations, one can recover the equations previously derived by Rubinowicz, Lévy, and Lombardi for the problem of the Dirac hydrogen atom in momentum space. This establishes equivalence among different approaches based on the momentum representation, including the current treatment. The recovery of the equations due to Rubinowicz permits the exact eigenvalues to be written down and exact expressions to be derived for the radial components of the transformed wavefunction in momentum space. A new approach is adopted to carry out a reduction to the nonrelativistic regime and the nonrelativistic limit. At first the transformed momentum‐space equation for the hydrogen atom is rewritten in terms of the hyperspherical coordinates. The zeroth‐order solutions of the new equation are recovered in the limit c → ∞ where c is the speed of light. These are manifestly separable into positive‐ and negative‐energy forms. For positive energy, these solutions have nonvanishing upper components that are two‐component spinors. The latter exactly correspond to the single‐component, nonrelativistic, momentum‐space solutions derived by Fock. It is shown that when the upper component is corrected through first order in v²/c² but the separability is still maintained for the transformed wavefunction, one retrieves the Pauli equation in momentum space. It is also shown that for a hydrogen atom placed in a uniform magnetic field, the nonvanishing momentum‐space matrix elements representing the anomalous Zeeman effect have a simple form, namely, the product of a radial integral and an angular integral. These integrals are equal to the well‐known radial and angular integrals in coordinate representation. The matrix elements can be easily evaluated. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

Acid-Catalyzed Hydrolysis of Some N,N-Dibenzylalkanesulfinamides in 50% Acetonitrile–Water

Abstract

The hydrolysis of some N,N-dibenzylalkanesulfinamides (RSONH(CH₂Ph)₂; 1, R = Me; 2, R = ⁱPr; 3, R = ^tBu; 4, R = 1-adamantyl) has been studied in 50% (v:v) acetonitrile–water solutions of hydrobromic and hydrochloric acids, mainly at 44.8 °C, using ultraviolet (UV) spectrophotometry to determine pseudo first-order rate constants. The compounds were found to hydrolyze by concurrent bimolecular neutral, acid-catalyzed, and acid-dependent nucleophilic (halide ion) catalysis pathways. The last-named is predominant in reactions in HBr solutions, but in HCl solutions, the acid-catalyzed pathway is predominant. The results indicate that both steric and electronic effects are important in these reactions. There appears to be no mechanistic switchover in the series 1→4.     

Near-Dirac–Hartree–Fock results for first-row atoms calculated with GTO basis sets

Relativistic basis sets for first-row atoms have been constructed by using the near-Hartree–Fock (nonrelativistic) eigenvectors calculated by Partridge. These bases generate results of near-Dirac–Hartree–Fock quality. Relativistic total and orbital energies, relativistic corrections to the total energy, and magnetic interaction energies for the first-row atoms have been presented. The smallest Gaussian expansions (13s8 p expansions) yield Dirac–Hartree–Fock total energies accurate through six significant digits, while the largest expansions (18s13p expansions) give these energies accurate through seven significant digits. These results are more accurate than some of the results reported earlier, particularly for the open-shell atoms, indicating that the basis employed is reasonably economical for relativistic calculations. © 1995 John Wiley & Sons, Inc.     

Determination of the acid dissociation constants of p-benzohydroquinone by the INDO method

The stepwise acid dissociation constants for p-benzohydroquinone (QH₂) in aqueous media have been explicitly calculated for the first time, with the INDO parametrized SCF –MO method. We have optimized the geometries of QH₂, QH^?, and Q^2? and of the QH₂ · 6H₂O, QH^? · (H₃O⁺) · 5H₂O, and Q^2? · (H₃O⁺)₂ · 4H₂O systems that model the solvated species. The presence of the associated water molecules (and hydronium ions) account for the stabilization due to hydrogen bonding as well as for a part of the effect of interaction of these molecules with the respective reaction fields in an aqueous medium. To simulate the first solvation shell in a more complete manner, four more water molecules have been considered to be placed above and below the quinonoid ring and the optimized geometries of the resulting hydrated species, QH₂ · 10H₂O, QH^? · (H₃O⁺) · 9H₂O, and QH^? · (H₃O⁺) · 8H₂O, have been determined. The standard free-energy changes calculated for the dissociation of QH₂ into QH^? and H⁺ is 0.0251 Hartree (65.9 kJ mol^?1) and that of QH^? into Q^2? and H⁺ is 0.0285 Hartree (74.8 kJ mol^?1). Experimentally observed dissociation constants for these two steps correspond to free-energy changes of 0.0214 Hartree (56.2 kJ mol^?1) and 0.0248 Hartree (65.1 kJ mol^?1), respectively. © 1995 John Wiley & Sons, Inc.     

Selective recognition of fluoride and acetate by a newly designed ruthenium framework: experimental and theoretical investigations

An effective anion sensor, [Ru(II)(bpy)(2)(H(2)L(-))](+) (1(+)), based on a redox and photoactive {Ru(II)(bpy)(2)} moiety and a new ligand (H(3)L = 5-(1H-benzo[d]imidazol-2-yl)-1H-imidazole-4-carboxylic acid), has been developed for selective recognition of fluoride (F(-)) and acetate (OAc(-)) ions. Crystal structures of the free ligand, H(3)L and [1](ClO(4)) reveal the existence of strong intramolecular and intermolecular hydrogen bonding interactions. The structure of [1](ClO(4)) shows that the benzimidazole N-H of H(2)L(-) is hydrogen bonded with the pendant carboxylate oxygen while the imidazole N-H remains free for possible hydrogen bonding interaction with the anions. The potential anion sensing features of 1(+) have been studied by different experimental and theoretical (DFT) investigations using a wide variety of anions, such as F(-), Cl(-), Br(-), I(-), HSO(4)(-), H(2)PO(4)(-), OAc(-) and SCN(-). Cyclic voltammetry and differential pulse voltammetry established that 1(+) is an excellent electrochemical sensor for the selective recognition of F(-) and OAc(-) anions. 1(+) is also found to be a selective colorimetric sensor for F(-) or OAc(-) anions where the MLCT band of the receptor at 498 nm is red shifted to 538 nm in the presence of one equivalent of F(-) or OAc(-) with a distinct change in colour from reddish-orange to pink. The binding constant between 1(+) and F(-) or OAc(-) has been determined to be logK = 7.61 or 7.88, respectively, based on spectrophotometric titration in CH(3)CN. The quenching of the emission band of 1(+) at 716 nm (λ(ex) = 440 nm, Φ = 0.01 at 298 K in CH(3)CN) in the presence of one equivalent of F(-) or OAc(-), as well as two distinct lifetimes of the quenched and unquenched forms of the receptor 1(+), makes it also a suitable fluorescence-based sensor. All the above experiments, in combination with (1)H NMR, suggest the formation of a 1:1 adduct between the receptor (1(+)) and the anion (F(-) or OAc(-)). The formation of 1:1 adduct {[1(+)·F(-)] or [1(+)·OAc(-)]} has been further evidenced by in situ ESI-MS(+) in CH(3)CN. Though the receptor, 1(+), is comprised of two N-H protons associated with the coordinated H(2)L(-) ligand, only the free imidazole N-H proton participates in the hydrogen bonding interactions with the incoming anions, while the intramolecularly hydrogen bonded benzimidazole N-H proton remains intact as evidenced by the crystal structure of the final product (1). The hydrogen bond mediated anion sensing mechanism, over the direct deprotonation pathway, in 1(+) has been further justified by a DFT study and subsequent NBO analysis.     

Octahedral CuII and NiII complexes manifesting with N′-[1-(pyridin-2-yl)ethylidene] acetohydrazide: Structural outlooks and spectral characteristics

A tridentate hydrazone precursor, N′-[1-(pyridin-2-yl)ethylidene]acetohydrazide (L) (1:1 refluxed product of acetichydrazide and 2-acetylpyridine), produced two octahedral Cu^II and Ni^II derivatives, [CuL₂]·NO₃ (1) and [NiL₂]·ClO₄·H₂O (2). Both are subjected to X-ray diffraction system, and structural investigation shows that the central metal atom (Cu^II or Ni^II) adopts a distorted octahedral geometry with N₄O₂ donor sets by coordination of a pair of independent hydrazone precursors. Besides X-ray study, IR and UV-vis spectra, thermal analysis and room temperature magnetic moments are utilized for establishing significant characteristics of both complexes. It is apparent that the M-N_pyridine bonds are slightly longer than the M-N_imino bonds, Cu1-N1 and Cu1-N4 [2.300(2) and 2.038(2) ?] for 1 and Ni1-N1 and Ni1-N4 [2.075(2) and 2.084(1) ?] for 2, Cu1-N2 and Cu1-N5 [2.062(1) and 1.932(1) ?] for 1 and Ni1-N2 and Ni1-N5 [2.008(2) and 1.975(2) ?] for 2, respectively. As per our observation, the effective magnetic moment value (μ_eff) is found to be 1.77 B.M. for 1 and 3.06 BM for 2, respectively.     

A Simple,Efficient and Green Procedure for the Knoevenagel Condensation of Aldehydes with N-Methylpiperazine at Room Temperature under Solvent-Free Conditions

Knoevenagel condensation of aromatic, aliphatic, and heteroaromatic aldehydes with active methylene compounds such as ethylcyanoacetate, malononitrile, and cyanoacetamide proceed very smoothly at room temperature by simply mixing the ingredients together under solvent-free conditions in the presence of N-methylpiperazine in excellent yields of the E-configured products.     

Reductive Cleavage of tert-Butyldimethylsilyl Ether via Intramolecular Transfer of Hydride

The cleavage of α-hydroxy tert-butyldimethylsilyl ether to diol takes place efficiently with LAH. It has been proposed that the reaction proceeds via intramolecular hydride transfer from the alkoxy aluminium hydride. In order to substantiate this, reduction of TBDMS ether with LAH in a variety of substrates was studied.     

Vibrational dephasing and hydrodynamic effects on vibrational relaxation rates in acetophenone: Raman bandshape analysis

The isotropic component of Raman band for C=O stretching mode of acetophenone in solution was analyzed by estimating the correlation coefficient with reference to Lorentzian lineshape. In the intermediate region of solute/solvent concentration there is a sharp decrease in the correlation coefficient and there appears to be a transition from non-Lorentzian to Lorentzian lineshape. The vibrational relaxation rates have been estimated from the isotropic component of Raman band in different solvents. The rate is shown to be dependent on several macroscopic as well as microscopic properties of the solute-solvent system and intermolecular interactions. The hydrodynamic and dispersion forces appear to play a major role in determining the vibrational relaxation rate and the broadening of the bands.     

Butyrate production from carbon monoxide byButyribacterium methylotrophicum

Applied Biochemistry and Biotechnology - Carbon monoxide is produced in high concentrations by gasification of coal or biomass, and is a potentially inexpensive feedstock for biological processes....