Concrete barriers to quantifier elimination in finite dimensional C*‐algebras

Work of Eagle, Farah, Goldbring, Kirchberg, and Vignati shows that the only separable C*‐algebras that admit quantifier elimination in continuous logic are $\mathbb{C}$, ${\mathbb{C}}^2$, $M_2(\mathbb{C})$, and the continuous functions on the Cantor set. We show that, among finite dimensional C*‐algebras, quantifier elimination does hold if the language is expanded to include two new predicate symbols: One for minimal projections, and one for pairs of unitarily conjugate elements. Both of these predicates are definable, but not quantifier‐free definable, in the usual language of C*‐algebras. We also show that adding just the predicate for minimal projections is sufficient in the case of full matrix algebras, but that in general both new predicate symbols are required.     

Paramagnetic oxotungsten(V) complexes containing the hydrotris(3,5-dimethylpyrazol-1-yl)borate ligand

Sky-blue Tp*WOCl(2) has been synthesized from the high-yielding reaction of Tp*WO(2)Cl with boron trichloride in refluxing toluene. Dark-red Tp*WOI(2) was prepared via thermal decarbonylation followed by aerial oxidation of Tp*WI(CO)(3) in acetonitrile. From these precursors, an extensive series of mononuclear tungstenyl complexes, Tp*WOXY [X = Cl(-), Y = OPh(-), SPh(-); X = Y = OPh(-), 2-(n-propyl)phenolate (PP(-)), SPh(-), SePh(-); XY = toluene-3,4-dithiolate (tdt(2-)), quinoxaline-2,3-dithiolate (qdt(2-)), benzene-1,2-diselenolate (bds(2-)); Tp* = hydrotris(3,5-dimethylpyrazol-1-yl)borate], was prepared by metathesis with the respective alkali-metal salt of X(-)/XY(2-) or (NHEt(3))(2)(qdt). The complexes were characterized by microanalysis, mass spectrometry, electrochemistry, IR, electron paramagnetic resonance (EPR), and electronic absorption spectroscopies, and X-ray crystallography (for X = Y = OPh(-), PP(-), SPh(-); XY = bds(2-)). The six-coordinate, distorted-octahedral tungsten centers are coordinated by terminal oxo [W≡O = 1.689(6)-1.704(3) ?], tridentate Tp*, and monodentate or bidentate O/S/Se-donor ligands. Spin Hamiltonian parameters derived from the simulation of fluid-solution X-band EPR spectra revealed that the soft-donor S/Se ligand complexes had larger g values and smaller (183)W hyperfine coupling constants than the less covalent hard-donor O/Cl species. The former showed low-energy ligand-to-metal charge-transfer bands in the near-IR region of their electronic absorption spectra. These oxotungsten(V) complexes display lower reduction potentials than their molybdenum counterparts, underscoring the preference of tungsten for higher oxidation states. Furthermore, the protonation of the pyrazine nitrogen atoms of the qdt(2-) ligand has been examined by spectroelectrochemistry; the product of the one-electron reduction of [Tp*WO(qdtH)](+) revealed usually intense low-energy bands.     

Design, synthesis, and structure of novel cesium receptors

The podand, bis(2-methoxyphenoxy)m-xylene (1), was designed and synthesized as a potential host for the cesium ion. Its structure was determined by single crystal X-ray diffraction. It crystallizes in P2₁/c with cell dimensions: a = 8.5723(7) Å, b = 8.3931(14) Å, c = 25.2879(26) Å, = 96.224(9), and V = 1808.7(4) ų. A crown derivative, tribenzo-21-crown-6 (2), was also prepared and structurally characterized. It also crystallizes in P2₁/c with cell dimensions: a = 15.5825(13) Å, b = 15.8648(16) Å, c = 8.8266(7) Å, = 95.247(6), and V = 2172.9(3) ų. The structures exhibit hydrogen bonding, and are evaluated in terms of complementarity and preorganization for cesium binding.     

Redox interplay of oxo-thio-tungsten centers with sulfur-donor co-ligands

The oxo-thio-W(VI) complexes TpWOS(S(2)PR(2)-S) and TpWOS(pyS-S) (Tp = hydrotris(3,5-dimethylpyrazol-1-yl)borate, R = OEt, Ph; pyS = pyridine-2-thiolate) have been prepared and characterized by microanalytical, spectroscopic, and structural techniques. Crystals of the 1,2-dichloroethane hemisolvate of TpWOS(S(2)PPh(2)-S) belong to the triclinic space group Ponemacr; with a = 10.732(6) A, b = 16.91(1) A, c = 10.021(4) A, alpha = 104.40(4) degrees, beta = 107.52(3) degrees, gamma = 96.09(5) degrees, V = 1647(1) A(3) for Z = 2. The complex exhibits a distorted octahedral structure featuring a facial tridentate Tp ligand and mutually cis terminal oxo (W-O(1) = 1.712(7) A), terminal thio (W-S(1) = 2.162(3) A), and monodentate dithiophosphinate ligands. X-ray absorption and extended X-ray absorption fine structure results support a related oxo-thio formulation for TpWOS(pyS-S). The complexes are reduced to the corresponding oxo-thio-W(V) anions, [TpWOS(S(2)PR(2)-S)](-) and [TpWOS(pyS-S)](-), which exhibit highly anisotropic EPR spectra. They are oxidized to form the EPR-active (dithio)oxo-W(V) cations, [TpWO(S(3)PR(2)-S,S')](+) and [TpWO(pyS(2)-N,S)](+) (pyS(2) = pyridine-2-dithio). Green [TpWO(pyS(2)-N,S)]BF(4), formed in the reaction of TpWOS(pyS) and NOBF(4), has been isolated and spectroscopically and structurally characterized. Crystals of [TpWO(pyS(2)-N,S)]BF(4) belong to the monoclinic space group Cc with a = 16.007(5) A, b = 14.091(4) A, c = 13.608(4) A, beta = 124.525(4) degrees, V = 2528.8(13) A(3) for Z = 4. The cation exhibits a distorted octahedral structure featuring facial tridentate Tp, terminal oxo (W-O(1) = 1.632(12) A), and bidentate pyridine-2-dithio-N,S (W-S(1) = 2.317(7) A, S(1)-S(2) = 2.037(9) A) ligands. The structures and redox behavior of the complexes are compared and contrasted with those of the related molybdenum complexes, TpMo(VI)OS(S(2)PR(2)-S) and TpMo(IV)O(pyS(2)-N,S) (Hill, J. P.; Laughlin, L. J.; Gable, R. W.; Young, C. G. Inorg. Chem. 1996, 35, 3447).     

Synthesis, characterization, and biomimetic chemistry of cis-oxosulfidomolybdenum(VI) complexes stabilized by an intramolecular Mo(O)=S...S interaction

The reactions of jade-green Tp*MoIVO(S2PR2) [Tp* = hydrotris(3,5-dimethylpyrazol-1-yl)borate; R = Et, Pri, Ph] with propylene sulfide produce ochre-red Tp*MoVIOS{SP(S)R2}. The complexes have been characterized by microanalysis, mass spectrometry, cyclic voltammetry, spectroscopy (IR, NMR, UV-vis, and X-ray absorption), and X-ray crystallography. The distorted-octahedral isopropyl and phenyl derivatives feature a tridentate fac-Tp* ligand, a terminal oxo ligand, and a unique five-membered Mo(=S){SP(=S)R2 ring moiety formed by a weak, intramolecular, bonding interaction between the Mo=S1 and (uncoordinated) S3=P moieties. The Mo=S1 [2.227(2) A (R = Pri) and 2.200(2) A (R = Ph)] and S1...S3 distances [2.396(3) A (R = Pri) and 2.383(2) A (R = Ph)] are indicative of a pi-bonded Mo=S1 unit and a weak (bond order ca. 1/3) S1...S3 interaction; the solid-state structures are maintained in solution according to S K-edge X-ray absorption data. The complexes react with excess cyanide to form thiocyanate and Tp*MoO(S2PR2), under anaerobic conditions, or Tp*MoO2(S2PR2), under aerobic conditions; the latter models the production of thiocyanate and desulfo molybdenum hydroxylases upon cyanolysis of molybdenum hydroxylases. The complexes react with triphenylphosphine to give Tp*MoO(S2PR2) and SPPh3, with cobaltocene or hydrosulfide ion to produce [Tp*MoVOS(S2PR2)]-, and with ferrocenium salts to yield [Tp*MoVO(S3PR2)]+; in the last two reactions, Mo(V) is produced by direct or induced internal redox reactions, respectively. The presence of the Mo(O)=S...S interaction does not radically lengthen the Mo=S bond in the complexes or preclude them from reactions typical of unperturbed oxosulfidomolybdenum(VI) complexes.     

Nonpreorganized neutral acyclic anion receptors: Structural investigations of N,N′-diphenyl-1, 3-benzenedisulfonamide and N,N′bis(4-t-butylphenyl)-1, 3-benzenedisulfonamide

The structure of N,N-diphenyl-1,3-benzenedisulfonamide (1) was determined by single crystal X-ray diffraction. It crystallizes in P2₁/n with cell dimensions: a = 11.8390(6) Å, b = 12.3950(10) Å, c = 12.1184(10) Å, = 94.388(6)°, and V = 1773.1(2) ų. Its di-t-butyl derivative, N,N-bis(4-t-butylphenyl)-1,3-benzenedisulfonamide (2), was prepared and structurally characterized as two solvated structures. Both crystallize in P with cell dimensions: 2 CF₃CH₂OH, a = 9.469(2) Å, b = 10.0039(18) Å, c = 16.385(3) Å, = 85.561(16)°, = 83.035(18)°, = 72.459(16), and V = 1467.7(5) ų; 2 ClCH₂CH₂Cl, a = 9.559(2) Å, b = 9.8125(12) Å, c = 17.100(6) Å, = 82.495(19)°, = 83.47(2)°, = 70.100(15), and V = 1491.1(6) ų. The structures exhibit hydrogen-bonding, and are evaluated in terms of preorganization for anion binding.     

Contact angle calculations from the contact/maximum diameter of sessile drops

This paper presents two algorithms for computing the contact angle of sessile liquid drops given data about the drops. The first yields the contact angle given the volume, surface tension and maximum diameter (or contact diameter) of a single drop. This algorithm is an extension of existing algorithms based on knowledge of the maximum diameter or of the contact diameter of a drop. A sensitivity analysis is included for this algorithm, allowing estimates to be made of the error in computed contact angle caused by errors in the measurement of the volume and/or diameter. The second algorithm requires only the volume and maximum or contact diameter of two different drops as input, and it produces both the contact angle and surface tension as output. Both algorithms are based on Newton's method applied to a function whose value is computed by solving a system of ordinary differential equations obtained from the Laplace equation of capillarity. The techniques are applicable to both hydrophobic and hydrophilic surfaces. Copyright © 2000 John Wiley & Sons, Ltd.     

Kinetic and electrochemical study of nitrile adducts of tetrachloro-bis (1,2-bis(diphenylphosphine)methane)dirhenium(II)

Summary The addition of two nitrile ligands to the complex Re₂Cl₄-(dppm)₂ (dppm= 1,2-bis(diphenylphosphine)methane)in CH₂Cl₂ solution has been investigated electrochemically. Upon addition of one equivalent of nitrile NCR (R = aromatic or aliphatic group) to the CH₂Cl₂/0.1m tetra-N-butylammonium hexafluorophosphate (TBAH) solution, Re₂Cl₄(dppm)₂(NCR) is formed immediately, without dissociation of chloride; electrochemical investigation indicates this nitrile addition is reversible upon oxidation of the dirhenium complex. On addition of two or more equivalents of nitrile, a slow ligand substitution takes place with addition of a second nitrile and concomitant loss of a chloride ion to form [Re₂Cl₃-(dppm)₂(NCR)₂]⁺. The rate of addition of nitrile to Re₂Cl₄(dppm)₂(NCR) appears to depend on the electrondonating or electron-withdrawing abilities of the ligand. The change from monoadduct to diadduct was followed with differential pulse voltammetry for various concentrations of added nitrile. The addition was found to be first order in nitrile.