Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM

The concepts of normalized irreducible tensorial matrices (NITM ) are extended to all finite and compact unitary groups by a development that clarifies their relationship to group theory and matrix algebra. NITM for a unitary group G are shown to be elements of a basis obtained by symmetry adapting to G the matrix basis of a matrix space M (α₁ × α₂). Elements [X] ∈ M (α₁ α₂) transform under G_a ∈ G according to [G_a] [X][G^?1_a], where [G_a] and [G^?1_a] belong to irreducible representations of G . The usual properties of NITM and the Wigner–Eckart theorem follow from these results, which are valid for both finite and compact unitary groups. The NITM span M (α₁ × α₂) are orthonormal under the trace and transform irreducibly with respect to G . This NITM basis of M (α₁ × α₂) is said to be simple. A compound NITM basis of a matrix space results when the space is partitioned into two or more subspaces, each spanned by a simple NITM basis. NITM determined from Griffith's V coefficients for the octahedral group are tabulated and used to construct a six-coordinate superposition Hamiltonian.     

Subgroup‐chain symmetry‐adapted irreducible matrices and CG coefficients of point groups

The irreducible matrices and Clebsch–Gordan coefficients of any crystallographic point group adapted to all possible canonical subgroup chains are calculated ab initio for both single‐valued and double‐valued representations and tabulated with exact values in the form of or and with components labeled by the irrep labels of the group chain in Koster notation. The phases and ordering of the components of irreducible bases for the cubic point groups are properly chosen so that irreducible matrices for all subgroup chains of G=T_d, O, O_h obey the associated relations D(G)=D(G)D(G), i=4, 6, and the complex conjugation relation for the group T, D(T)=D(T)*. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 75: 67–80, 1999     

One‐range Addition Theorems for Complete Sets of Modified Exponential Type Orbitals and Noninteger n Slater Functions in Standard Convention

Using the L ‐generalized Laguerre polynomials L ‐GLPs) and φ ‐generalized exponential type orbitals φ ‐GETOs) introduced by the author in standard convention, the one‐ and two‐center onerange addition theorems are established for the complete sets of Ψ^(α*) ‐modified exponential type orbitals (Ψ^(α*) ‐METOs) and noninteger n χ‐Slater type orbitals (χ‐NISTOs), where p_l* = 2l + 2 ‐ α* and α* is the integer (α* = α, ?∞ < α ≤2) or noninteger (α* ≠ α, ?∞ < α* < 3) self‐frictional quantum number. It should be noted that the origin of the L ‐GLPs, φ ‐GETOs and Ψ^(α*) ‐METOs, therefore, of the one‐range addition theorems presented in this work is the Lorentz damping or self‐frictional field produced by the particle itself.     

Kinetics of the reaction O(3P) + SO2 + M → SO3 + M over the temperature range of 299°–440°K

Rate constants for the reaction O(³P) + SO₂ + M have been determined over the temperature range of 299°–440°K, using a flash photolysis–NO₂ chemiluminescence technique. For M?Ar, the Arrhenius expression was obtained. At room temperature k₂^Ar = (1.05 ± 0.21) × 10^?33 cm⁶/molec²·sec. In addition, the rate constants k₂ = (1.37 + 0.27) × 10^?33 cm⁶/molec²·sec, k₂ = (9.5 ± 3.0) ± 10^?33 cm⁶/molec²·sec, k₃ = (1.1 ± 0.2) ± 10^?31 cm⁶/molec²·sec, and k₃ = (2.6 ? 0.9) ± 10^?31 cm⁶/molec²·sec were obtained at room temperature where k₃^M is the rate constant for the reaction O + NO + M → NO₂ + M. The rate data are compared and discussed with literature values.     

New RuII(arene) Complexes with Halogen‐Substituted Bis‐ and Tris(pyrazol‐1‐yl)borate Ligands

[RuCl(arene)(μ‐Cl)]₂ dimers were treated in a 1:2 molar ratio with sodium or thallium salts of bis‐ and tris(pyrazolyl)borate ligands [Na(Bp)], [Tl(Tp)], and [Tl(Tp^{iPr, 4Br})]. Mononuclear neutral complexes [RuCl(arene)(κ²‐Bp)] ( 1 : arene=p‐cymene (cym); 2 : arene=hexamethylbenzene (hmb); 3 : arene=benzene (bz)), [RuCl(arene)(κ²‐Tp)] ( 4 : arene=cym; 6 : arene=bz), and [RuCl(arene)(κ²‐Tp^{iPr, 4Br})] ( 7 : arene=cym, 8 : arene=hmb, 9 : arene=bz) have been always obtained with the exception of the ionic [Ru₂(hmb)₂(μ‐Cl)₃][Tp] ( 5′ ), which formed independently of the ratio of reactants and reaction conditions employed. The ionic [Ru(CH₃OH)(cym)(κ²‐Bp)][X] ( 10 : X=PF₆, 12 : X=O₃SCF₃) and the neutral [Ru(O₂CCF₃)(cym)(κ²‐Bp)] ( 11 ) have been obtained by a metathesis reaction with corresponding silver salts. All complexes 1 – 12 have been characterized by analytical and spectroscopic data (IR, ESI‐MS, ¹H and ¹³C NMR spectroscopy). The structures of the thallium and calcium derivatives of ligand Tp, [Tl(Tp)] and [Ca(dmso)₆][Tp]₂ ? 2 DMSO, of the complexes 1 , 4 , 5′ , 6 , 11 , and of the decomposition product [RuCl(cym)(Hpz^{iPr, 4Br})₂][Cl] ( 7′ ) have been confirmed by using single‐crystal X‐ray diffraction. Electrochemical studies showed that 1 – 9 and 11 undergo a single‐electron Ru^II→Ru^III oxidation at a potential, measured by cyclic voltammetry, which allows comparison of the electron‐donor characters of the bis‐ and tris(pyrazol‐1‐yl)borate and arene ligands, and to estimate, for the first time, the values of the Lever E_L ligand parameter for Bp, Tp, and Tp^{iPr, 4Br}. Theoretical calculations at the DFT level indicated that both oxidation and reduction of the Ru complexes under study are mostly metal‐centered with some involvement of the chloride ligand in the former case, and also demonstrated that the experimental isolation of the μ³‐binuclear complex 5′ (instead of the mononuclear 5 ) is accounted for by the low thermodynamic stability of the latter species due to steric reasons.     

On the ν1/2 ν2 Resonance in the Complex of Carbon Dioxide with Dimethyl Ether

Infrared and Raman spectra of solutions in liquid argon and krypton containing dimethyl ether or its fully deuterated isotopomer and ¹²CO₂ or ¹³CO₂ are investigated. The spectra lead to new data on the ν₁/2 ν₂ resonances appearing in the complex of CO₂ with the ether. The experimental data, and their interpretation, is supported by MP2/6‐311++G(2d,2p) calculations of the cubic and quartic force constants and of the first and higher order dipole moment derivatives required for the modelling of the Fermi and Darling‐Dennison resonances observed.     

29Si NMR spectra of methylphenylcyclotri-and -tetrasilazanes

²⁹Si NMR spectra of various methylphenyl-substituted cyclotri- and -tetrasilazanes, composed of different combinations of the units d = Me₂SiNH, d^ph = MePhSiNH and d = Ph₂SiNH, have been investigated. For most of the compounds having more than one asymmetric centre, the spectra of both the individual isomers and their mixtures have been studied, and the signals of the individual isomers have been assigned. The effect of a different arrangement of phenyl groups in the rings, and the influence of the stereochemistry of the isomers on the ²⁹Si chemical shifts in cyclotri- and -tetrasilazanes have been studied. The ²⁹Si chemical shifts in cyclotrisilazanes are additive and can be represented as the sum of increments corresponding to the number and positions of the phenyl groups in the rings. In these compounds, the spatial structure of the isomers does not exert any noticeable effect on the spectra. On the contrary, chemical shifts are not simply additive in cyclotetrasilazanes: a pronounced stereochemical dependence is apparent.     

Activation parameters and kinetic isotope effect in the system tropaeolin O-OH−

The kinetics of the deprotonation of tropaeolin O by OH^? ions was investigated between 9° and 30°C, and by OD^? ions at 24.7°C. The pH range was 10.7–12.5, and the ionic strength 0.1M throughout. All results were obtained by the temperature jump method. On the basis of a mechanism suggested earlier, rate constants k₃₁ for the reaction between OL^? and the internally bonded weak acid and k₃₂ for the opening of the internal hydrogen bond were evaluated. The activation energies in ordinary water were found to be ΔH≠₃₁ = 3.6 kcal/mol, ΔS≠₃₁ = –19 eu, and ΔH≠₃₂ = 27 kcal/mol, ΔS≠₃₂ = 46 eu. The kinetic isotope effect was k₃₁/k₃₁ ～ 1.5 and k₃₂/k₃₂ ～ 0.9. The unusual results for reaction path are discussed in terms of solvent participation.     

Pyrazolato Ligands towards Gold(I) Derivatives with Aurophilic Interactions: X‐Ray Structure of Bis[3,5‐(dibutoxyphenyl)‐1H‐pyrazolato‐κN1]bis(triphenylphosphine)digold(Au⋅⋅⋅Au)([Au(pz)(PPh3)]2; RC4H9?O?C6H4)

A crossed ‘torch' structure and a short Au⋅⋅⋅Au contact was established by X‐ray analysis for the dimeric complex [Au(pz)(PPh₃)]₂ (pz=3,5‐disubstituted pyrazolato, RC_nH_2n+1 O C₆H₄, n=4; 1 ). The complex is a representative member of a new well‐characterized family of derivatives containing the pyrazolato ligand in an uncommon monodentate coordination form. In addition, 1 is luminescent in the solid state at 77 K.     

Convergence accelerator approach for the evaluation of some three‐electron integrals containing explicit rij factors

A simplified analysis is presented for the evaluation of the three‐electron one‐center integrals of the form ∫rrrrrred r ₁d r ₂d r ₃, for the cases i, j, k, ≥−2, l=−2, m≥−1, n≥−1. These integrals arise in the calculation of lower bounds for energy levels and certain relativistic corrections to the energy when Hylleraas‐type basis sets are employed. Convergence accelerator techniques are employed to obtain a reasonable number of digits of precision, without excessive CPU requirements. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 72: 93–99, 1999     

Structural analysis of certain linear operators representing chemical network systems via the existence and uniqueness theorems of spectral resolution VII

The present Part VII of this series of articles is a direct extension of Part VI, where (1) fundamental methodology in the repeat space theory (RST) and (2) frontier electron theory of reactivity indices were theoretically linked. This part presents an estimate of the size of the regular index set, which was a central notion in Part VI, and two new theorems that are simpler and more powerful than the main theorem in Part VI. The main theorem in this part enables one to globally contextualize the Generalized Alpha Existence Theorem (a theorem essential in the RST and proved in Part V) and the μ Existence Theorem (derived from the main theorem in Part VI) into the star algebra structure of the generalized repeat space ??_r(q, d). © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

Aldehyde‐Assisted Hydrogen Transfer during the Formation of Hydride–Iridafurans from Alkynes and Aldehydes

The bis(ethylene) Ir^I complex [TpIr(C₂H₄)₂] ( 1 ; Tp=hydrotris(3,5‐dimethylpyrazolyl)borate) reacts with two equivalents of aromatic or aliphatic aldehydes in the presence of one equivalent of dimethyl acetylenedicarboxylate (DMAD) with ultimate formation of hydride iridafurans of the formula [TpIr(H){C(R¹)?C(R²)C(R³)O }] (R¹=R²=CO₂Me; R³=alkyl, aryl; 3 ). Several intermediates have been observed in the course of the reaction. It is proposed that the key step of metallacycle formation is a C? C coupling process in the undetected Ir^I species [TpIr{η¹‐O‐R³C(?O)H}(DMAD)] ( A ) to give the trigonal‐bipyramidal 16 e^? Ir^III intermediates [TpIr{C(CO₂Me)?C(CO₂Me)C(R³)(H)O }] ( C ), which have been trapped by NCMe to afford the adducts 11 (R³=Ar). If a second aldehyde acts as the trapping reagent for these species, this ligand acts as a shuttle in transfering a hydrogen atom from the γ‐ to the α‐carbon atom of the iridacycle through the formation of an alkoxide group. Methyl propiolate (MP) can be used instead of DMAD to regioselectively afford the related iridafurans. These reactions have also been studied by DFT calculations.     

1H, 13C and 15N nuclear magnetic resonance coordination shifts in Au(III), Pd(II) and Pt(II) chloride complexes with phenylpyridines

¹H, ¹³C and ¹⁵N nuclear magnetic resonance studies of gold(III), palladium(II) and platinum(II) chloride complexes with phenylpyridines (PPY: 4‐phenylpyridine, 4ppy; 3‐phenylpyridine, 3ppy; and 2‐phenylpyridine, 2ppy) having the general formulae [Au(PPY)Cl₃], trans‐/cis‐[Pd(PPY)₂Cl₂] and trans‐/cis‐[Pt(PPY)₂Cl₂] were performed and the respective chemical shifts (δ, δ and δ) reported. ¹H, ¹³C and ¹⁵N coordination shifts (i.e. differences between chemical shifts of the same atom in the complex and ligand molecules: , , ) were discussed in relation to the type of the central atom (Au(III), Pd(II) and Pt(II)), geometry (trans‐/cis‐) and the position of a phenyl group in the pyridine ring system. Copyright © 2009 John Wiley & Sons, Ltd.     

Addition of Ethylene or Hydrogen to a Main‐Group Metal Cluster under Mild Conditions

Reaction of the tin cluster Sn₈(Ar)₄ (Ar=C₆H₂‐2,6‐(C₆H₃‐2,4,6‐Me₃)₂) with excess ethylene or dihydrogen at 25 °C/1 atmosphere yielded two new clusters that incorporated ethylene or hydrogen. The reaction with ethylene yielded Sn₄(Ar)₄(C₂H₂)₅ that contained five ethylene moieties bridging four aryl substituted tin atoms and one tin–tin bond. Reaction with H₂ produced a cyclic tin species of formula (Sn(H)Ar)₄, which could also be synthesized by the reaction of {(Ar)Sn(μ‐Cl)}₂ with DIBAL‐H. These reactions represent the first instances of direct reactions of isolable main‐group clusters with ethylene or hydrogen under mild conditions. The products were characterized in the solid state by X‐ray diffraction and IR spectroscopy and in solution by multinuclear NMR and UV/Vis spectroscopies. Density functional theory calculations were performed to explain the reactivity of the cluster.     

Functionalization of Non‐activated C?H Bonds of Alkanes: An Effective and Recyclable Catalytic System Based on Fluorinated Silver Catalysts and Solvents

The complexes F_n‐TpAg(L) (F_n‐Tp=a perfluorinated hydrotris(indazolyl) borate ligand; L=acetone or tetrahydrofuran) efficiently catalyze the functionalization of non‐activated alkanes such as hexane, 2,3‐dimethylbutane, or 2‐methylpentane by insertion of CHCO₂Et units (from N₂CHCO₂Et, ethyl diazoacetate, EDA) into their C? H bonds. The reactions are quantitative (EDA‐based), with no byproducts derived from diazo coupling being formed. In the case of hexane, the functionalization of the methyl C? H bonds has been achieved with the highest regioselectivity known to date with this diazo compound. This catalytic system also operates under biphasic conditions by using fluorous solvents such as Fomblin or perfluorophenanthrene. Several cycles of catalyst recovery and reuse have been performed, with identical chemo‐ and regioselectivities.     

Reduction of CO2 by Pyridine Monoimine Molybdenum Carbonyl Complexes: Cooperative Metal–Ligand Binding of CO2

[(^ArPMI)Mo(CO)₄] complexes (PMI=pyridine monoimine; Ar=Ph, 2,6‐di‐iso‐propylphenyl) were synthesized and their electrochemical properties were probed with cyclic voltammetry and infrared spectroelectrochemistry (IR‐SEC). The complexes undergo a reduction at more positive potentials than the related [(bipyridine)Mo(CO)₄] complex, which is ligand based according to IR‐SEC and DFT data. To probe the reaction product in more detail, stoichiometric chemical reduction and subsequent treatment with CO₂ resulted in the formation of a new product that is assigned as a ligand‐bound carboxylate, [(PMI)Mo(CO)₃(CO₂)]^2?, by NMR spectroscopic methods. The CO₂ adduct [(PMI)Mo(CO)₃(CO₂)]^2? could not be isolated and fully characterized. However, the C?C coupling between the CO₂ molecule and the PDI ligand was confirmed by X‐ray crystallographic characterization of one of the decomposition products of [(PMI)Mo(CO)₃(CO₂)]^2?.     

33S chemical shifts and line widths of selected sulphones,sulphoximides, sulphimides,sulphides and sulphonium ions

The ³³S chemical shifts, δ, and the line widths of selected sulphones, sulphoximides, sulphimides, sulphides and sulphonium ions were measured at 23.009 MHz. δ increased with decreasing electronic charge on sulphur. The line width varied from 3–10 ppm for sulphones to 400 ppm for sulphonium ions. The higher geometrical flexibility of five-membered ring sulphones with respect to the six-membered ring compounds was reflected in the variations of δ with methyl substitution. Open-chain sulphoximides are shielded with respect to the corresponding sulphones, whereas open-chain sulphimides are deshielded with respect to the corresponding sulphoximides.     

Dipole induced photodetachment dynamics of halide anions

In this article the dynamics of photodetachment from closed shell anion in the presence of two color (bichromatic) laser field have been explored in the context of polar environmental situation. 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 energy of the halide ion concerned. The potential so generated are shown to support only one bound state. 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 two ways i) 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 (strength of fluctuation) is fixed and ii) in the presence of perturbation produced by a neighboring solvent dipole, which changes the effective potential. The detachment rate constant is studied as a function of important system parameters of the used light field. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008