Perturbation of analytic hermitian matrix functions

The behaviour of real eigenvalues of selfadjoint analytic matrix valued functions under small selfadjoint analytic perturbations is studied. Attention is paid mainly to the case when the perturbation is definite (or semidefi-nite). Earlier results of the authors concerning matrix polynomials of first degree are extended to the case of analytic functions.     

The kernel of the bezoutian for operator Polynomials ∗

Based on two set partitions of the symmetric group S_n expansion theorems by diagonal elements for the permanent and the determinant are derived, for both the generic commuting and noncommuting cases. They are of the same type as the well-known Laplace expansions where either fixed rows or columns of a given matrix are chosen instead of diagonal elements.     

Ag(nic)2 (nic = nicotinate): a spin-canted quasi-2D antiferromagnet composed of square-planar S = 1/2 Ag(II) ions

Square-planar S = 1/2 Ag(II) ions in polymeric Ag(nic)(2) are linked by bridging nic monoanions to yield 2D corrugated sheets. Long-range magnetic order occurs below T(N) = 11.8(2) K due to interlayer couplings that are estimated to be about 30 times weaker than the intralayer exchange interaction.     

Experimental fingerprints for redox-active terpyridine in [Cr(tpy)2](PF6)n (n = 3-0), and the remarkable electronic structure of [Cr(tpy)2]1-

The molecular and electronic structures of the four members, [Cr(tpy)(2)](PF(6))(n) (n = 3-0; complexes 1-4; tpy = 2,2':6',2″-terpyridine), of the electron transfer series [Cr(tpy)(2)](n+) have been determined experimentally by single-crystal X-ray crystallography, by their electro- and magnetochemistry, and by the following spectroscopies: electronic absorption, X-ray absorption (XAS), and electron paramagnetic resonance (EPR). The monoanion of this series, [Cr(tpy)(2)](1-), has been prepared in situ by reduction with KC(8) and its EPR spectrum recorded. The structures of 2, 3, 4, 5, and 6, where the latter two compounds are the Mo and W analogues of neutral 4, have been determined at 100(2) K. The optimized geometries of 1-6 have been obtained from density functional theoretical (DFT) calculations using the B3LYP functional. The XAS and low-energy region of the electronic spectra have also been calculated using time-dependent (TD)-DFT. A consistent picture of the electronic structures of these octahedral complexes has been established. All one-electron transfer processes on going from 1 to 4 are ligand-based: 1 is [Cr(III)(tpy(0))(2)](PF(6))(3) (S = (3)/(2)), 2 is [Cr(III)(tpy(?))(tpy(0))](PF(6))(2) (S = 1), 3 is [Cr(III)(tpy(?))(2)](PF(6)) (S = (1)/(2)), and 4 is [Cr(III)(tpy(??))(tpy(?))](0) (S = 0), where (tpy(0)) is the neutral parent ligand, (tpy(?))(1-) represents its one-electron-reduced π radical monoanion, (tpy(2-))(2-) or (tpy(??))(2-) is the corresponding singlet or triplet dianion, and (tpy(3-))(3-) (S = (1)/(2)) is the trianion. The electronic structure of 2 cannot be described as [Cr(II)(tpy(0))(2)](PF(6))(2) (a low-spin Cr(II) (d(4); S = 1) complex). The geometrical features (C-C and C-N bond lengths) of these coordinated ligands have been elucidated computationally in the following hypothetical species: [Zn(II)Cl(2)(tpy(0))](0) (S = 0) (A), [Zn(II)(tpy(?))Cl(NH(3))](0) (S = (1)/(2)) (B), [Zn(II)(tpy(2-))(NH(3))(2)](0) (S = 0 or 1) (C), and [Al(III)(tpy(3-))(NH(3))(3)](0) (S = (1)/(2) and (3)/(2)) (D). The remarkable electronic structure of the monoanion has been calculated and experimentally verified by EPR spectroscopy to be [Cr(III)(tpy(2-))(tpy(??))](1-) (S = (1)/(2)), a complex in which the two dianionic tpy ligands differ only in the spin state. It has been clearly established that coordinated tpy ligands are redox-active and can exist in at least four oxidation levels.     

The structure and chemistry of tris(pentafluorophenyl)borane protected mononuclear nitridotitanium complexes

Treatment of TiCl(NMe(2))(3) with H(3)N·B(C(6)F(5))(3) results in N-H activation and ligand exchange to yield the structurally characterised salt [TiCl(NMe(2))(2)(NMe(2)H)(2)](+)[Ti[triple bond]NB(C(6)F(5))(3)(Cl)(2)(NMe(2)H)(2)](-). Cation exchange with [Me(4)N]Cl, [Ph(4)P]Cl and [(PhCH(2))Ph(3)P]Cl yields the respective ammonium and phosphonium salts of the [Ti[triple bond]NB(C(6)F(5))(3)(Cl)(2)(NMe(2)H)(2)](-) anion. X-ray crystallography reveals that the essential trigonal bipyramidal geometry and composition of the anion is retained in each of these salts despite some minor variations in the Ti-N-B angle and the nature of the interionic interactions. Electronic investigation by DFT calculations confirmed the Ti-N triple bond character implied by the experimentally determined bond length, with the HOMO and HOMO-1 having Ti-N π-bonding character. The dimethylamine ligands of the anion resist substitution by moderate bases but can be displaced by pyridine to give a pentacoordinate anion. In contrast, addition of 2,2'-bipyridyl gives a neutral octahedral complex. Treatment of the pyridine complex with TlCp results in the formation of a four coordinate anionic cyclopentadienyl complex.     

Dimensionality selection in a molecule-based magnet

Gaining control of the building blocks of magnetic materials and thereby achieving particular characteristics will make possible the design and growth of bespoke magnetic devices. While progress in the synthesis of molecular materials, and especially coordination polymers, represents a significant step towards this goal, the ability to tune the magnetic interactions within a particular framework remains in its infancy. Here we demonstrate a chemical method which achieves dimensionality selection via preferential inhibition of the magnetic exchange in an S=1/2 antiferromagnet along one crystal direction, switching the system from being quasi-two- to quasi-one-dimensional while effectively maintaining the nearest-neighbor coupling strength.