Quantum transport in dynamically disordered continuum tight binding systems

We consider the continuum limit of three distinct models describing tightly bound electron systems in one dimension. The first model is the usual tight binding hamiltonian for monatomic lattices with nearest-neighbour hopping between sites. The second model describes a two-subband tight binding system involving two different atoms per unit cell. Finally, the third model represents a monatomic system with two energy levels per atomic site and different nearest-neighbour hopping parameters for hopping between equivalent and non-equivalent levels. The continuum limits of these models result in field-theoretic hamiltonians showing similarities with the Dirac hamiltonian. Assuming the different types of site energies to be dynamically disordered with gaussian whitenoise spectra, we calculate exactly the quantum mechanical mean square displacement <x ²(t)>. Due to the use of Novikov's theorem for the evaluation of configuration averages our analysis for the two-band models is restricted to the degenerate case, where the average positions of the two types of atomic levels coincide. Fort we find coherent motion, <x ²(t)>t ², for the one-band model and disorder induced diffusive contributions for the two-band models. However, for the two-level atomic model the diffusive term is dominated by at ²-term describing coherent hopping between equivalent levels. These findings are discussed in relation to previous results for both discrete and continuum models.     

A direct solver for the least-squares spectral collocation system on rectangular elements for the incompressible Navier–Stokes equations

A least-squares spectral collocation scheme for the Stokes and incompressible Navier–Stokes equations is proposed. The original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least-squares sense. The collocation conditions and the interface conditions lead to overdetermined systems. These systems are directly solved by QR decomposition of the underlying matrices. By numerical simulations it is shown that the direct method leads to better results than the approach with normal equations. Furthermore, it is shown that the condition numbers can be reduced by introducing the Clenshaw–Curtis quadrature rule for imposing the average pressure to be zero. Finally, our scheme is successfully applied to the regularized and lid-driven cavity flow problems.     

Tridentate Lewis Acids Based on 1,3,5‐Trisilacyclohexane Backbones and an Example of Their Host–Guest Chemistry

Directed tridentate Lewis acids based on the 1,3,5‐trisilacyclohexane skeleton with three ethynyl groups [CH₂Si(Me)(C₂H)]₃ were synthesised and functionalised by hydroboration with HB(C₆F₅)₂, yielding the ethenylborane {CH₂Si(Me)[C₂H₂B(C₆F₅)₂]}₃, and by metalation with gallium and indium organyls affording {CH₂Si(Me)[C₂M(R)₂]}₃ (M=Ga, In, R=Me, Et). In the synthesis of the backbone the influence of substituents (MeO, EtO and iPrO groups at Si) on the orientation of the methyl group was studied with the aim to increase the abundance of the all‐cis isomer. New compounds were identified by elemental analyses, multi‐nuclear NMR spectroscopy and in some cases by IR spectroscopy. Crystal structures were obtained for cis‐trans‐[CH₂Si(Me)(Cl)]₃, all‐cis‐[CH₂Si(Me)(H)]₃, all‐cis‐[CH₂Si(Me)(C₂H)]₃, cis‐trans‐[CH₂Si(Me)(C₂H)]₃ and all‐cis‐[CH₂Si(Me)(C₂SiMe₃)]₃. A gas‐phase electron diffraction experiment for all‐cis‐[CH₂Si(Me)(C₂H)]₃ provides information on the relative stabilities of the all‐equatorial and all‐axial form; the first is preferred in both solid and gas phase. The gallium‐based Lewis acid {CH₂Si(Me)[C₂Ga(Et)₂]}₃ was reacted with a tridentate Lewis base (1,3,5‐trimethyl‐1,3,5‐triazacyclohexane) in an NMR titration experiment. The generated host–guest complexes involved in the equilibria during this reaction were identified by DOSY NMR spectroscopy by comparing measured diffusion coefficients with those of the suitable reference compounds of same size and shape.     

Dispersion of collective modes at inhomogeneous metal surfaces

A simple model in which electrons are assumed to be specularly reflecting on a plane located a distance u above the boundary of a homogeneous metallic region is proposed for studying dynamical properties of metal surfaces. This model is believed to incorporate essential effects of the inhomogeneity of a pure jellium surface in a new way. It leads to a strong reduction of hydrodynamic dispersion effects for long wavelengths surface plasmons, in agreement with experiment and with various previous calculations. The model is also applied to surface phonons and exact numerical dispersion curves for both surface phonons and plasmons at arbitrary wavelengths are presented.