On numerical methods for discrete least-squares approximation by trigonometric polynomials

Fast, efficient and reliable algorithms for discrete least-squares approximation of a real-valued function given at arbitrary distinct nodes in by trigonometric polynomials are presented. The algorithms are based on schemes for the solution of inverse unitary eigenproblems and require only arithmetic operations as compared to operations needed for algorithms that ignore the structure of the problem. An algorithm which solves this problem with real-valued data and real-valued solution using only real arithmetic is given. Numerical examples are presented that show that the proposed algorithms produce consistently accurate results that are often better than those obtained by general QR decomposition methods for the least-squares problem.

     

En route to zirconium hydrazides(2-)

The structures and properties of a series of new zirconium hydrazido(1-) complexes and the possibility of converting them to the respective hydrazido(2-) species are reported. Reaction of complex [Zr(N2TBSNpy)Cl2] (1) with the monolithiated hydrazide LiNHNMe2 gave the hydrazido(1-) complex [Zr(N2TBSNpy)(NHNMe2)Cl] (2) which exists as two isomeric forms (2a and 2b) in solution. All attempts to convert a mixture of 2a and 2b to the respective hydrazido(2-) compound by reaction with the bulky base lithium hexamethyldisilazide or via the alkyl/hydrazido(1-) complex [Zr(N2TBSNpy)(CH2SiMe3)(NHNMe2)] (3) and subsequent thermal alkane elimination failed. Reaction of 1 with LiHNNPhMe gave a mixture of stereoisomers of [Zr(N2TBSNpy)(NHNMePh)Cl] (4a and 4b), in which the hydrazido unit is end-on bound in solution and eta 2-bonded in the solid state. Reaction of this mixture with lithium hexamethyldisilazide in the presence of pyridine selectively yielded the hydrazido(2-) complex [Zr(N2TBSNpy)(NNPhMe)(py)] (5) which aggregated upon attempts to isolate it. Reaction of the insoluble precipitate with 4-dimethylaminopyridine (dmap) selectively gave the corresponding hydrazido(2-) complex [Zr(N2TBSNpy)(NNPhMe)(dmap)] (6), which could be obtained in a one-pot reaction directly from 1 and which was analytically and spectroscopically fully characterized. It appears that the isolation of stable hydrazido(2-) complexes of zirconium depends on the type of substituents at the N beta atom as well as the co-ligands coordinated to the metal centre.     

Zirconium and hafnium (1-pyridinio)imido complexes: functionalized terminal hydrazinediido analogues

Reaction of the diamidozirconium complex [Zr(N2(TBS)Npy)(NMe2)2] (1) (N2(TBS)Npy = CH3C(C5H4N)(CH2NSiMe2tBu)2) or the diamidohafnium complex [Hf(N2(TBS)Npy)(NMe2)2] (2) with one molar equiv. of 1-aminopyridinium triflate in the presence of one equiv. of pyridine gave the corresponding (1-pyridinio)imido complexes [Zr(N2(TBS)Npy)(=N-NC5H5)(OTf)(py)] (3) and [Hf(N2(TBS)Npy)(=N-NC5H5)(OTf)(py)] (4). These were converted to the acetylide complexes [Zr(N2(TBS)Npy)(=N-NC5H5)(CCPh)(py)] (5) and [Hf(N2(TBS)Npy)(=N-NC5H5)(CCPh)(py)] (6) by reaction with lithium phenylacetylide and substitution of the triflato ligand. Upon reaction of 3 and 4 with one molar equivalent of R-NC (R = tBu, Cy, 2,6-xyl), N-N bond cleavage in the (1-pyridinio)imido unit took place and the respective carbodiimido complexes [M(N2(TBS)Npy](N=C=NR)(OTf)(py)] (7-12) were formed instantaneously. A similar type of reaction with CO gave the isocyanato complex [Zr(N2(TBS)Npy](NCO)(OTf)(py)] (13). Finally, the abstraction of the pyridine ligand in compounds 3 and 4 with B(C6F5)3 led to the formation of the triflato-bridged dinuclear complexes [Zr(N2(TBS)Npy)(=N-NC5H5)(OTf)]2 (14) and [Hf(N2(TBS)Npy)(=N-NC5H5)(OTf)]2 (15).     

Crystal structure and spectroscopic properties of ammonium hydrogensquarate squaric acid monohydrate

The structure of ammonium hydrogensquarate squaric acid monohydrate has been determined by single crystal X-ray diffraction. The compound crystallizes in the monoclinic space group C2/c and exhibits a 3D network with molecules linked by intermolecular interactions with participation of the H₂Sq, HSq^?, NH₄ ⁺, and H₂O species. The HSq^? anion and the neutral H₂Sq form a strong head-to-tail dimer through O–H···O hydrogen bonding with lengths of 2.587 and 2.494 Å (protected space between numeral and unit). The layers are connected by ammonium cations and water molecules in a plane through the O···N (2.950, 2.978, 3.036 Å) and O···O (2.953, 2.781 Å) bonds. Another such layer is connected to the NH₄ ⁺ cation in the adjacent plane through bifurcated N–H···O hydrogen-bonding to form a double layer (NH···O bond lengths are 3.036, 2.978, 2.857, 2.909, 2.958, and 2.742 Å, respectively). The IR-band assignment of the compound was achieved using the polarized IR-spectroscopy of oriented colloids in a nematic host. Theoretical ab initio calculations were performed and achieved with a view to explain the IR-bands of the H₂Sq.HSq^? motif.