Matrix transformations for computing rightmost eigenvalues of large sparse non-symmetric eigenvalue problems

This paper gives an overview of matrix transformations for findingrightmost eigenvalues of Ax = x and Ax = Bx with A and B realnon-symmetric and B possibly singular. The aim is not to presentnew material. but to introduce the reader to the applicationof matrix transformations to the solution of large-scale eigenvalueproblems. The paper explains and discusses the use of Chebyshevpolynomials and the shift–invert and Cayley transformsas matrix transformations for problems that arise from the discretizationof partial differential equations. A few other techniques aredescribed. The reliability of iterative methods is also dealtwith by introducing the concept of domain of confidence or trustregion. This overview gives the reader an idea of the benefitsand the drawbacks of several transformation techniques. We alsobriefly discuss the current software situation.     

Numerical self-consistent field method for polyatomic molecules

A method for non-relativistic self-consistent field (SCF) electronic structure calculations for polyatomic molecules is described, which retains the linear combination of atomic orbitals ansatz for molecular orbitals (MO-LCAO), but replaces the usual algebraic expansion of atom-centred radial parts in terms of basis functions (usually some kind of Gauss-type functions) by a numerical representation on a set of radial grid points around each centre. The radial parts are optimized, according to the variation principle, until self-consistency is achieved. Even though Fourier integral transform techniques are used the method works completely in ordinary space. Intermediate quantities defined in momentum space are evaluated in closed form.     

Test of the stability of a three-phase contact line with negative line tension

The equilibrium structure of a three-phase contact line with negative line tension using a mean-field free-energy functional is calculated in the square-gradient approximation. The equilibrium density profiles are found by solving the Euler—Lagrange equations on a square grid of N ² points covering an area of L ². The fluctuations about the equilibrium structure are analysed via the spectrum of the free energy's second functional derivative. The equilibrium configuration is found to be stable with respect to fluctuations in the structure of the three-phase line and of the interfaces that meet at this line. In addition, the behaviour is investigated numerically of the lowest eigenvalue as the area of the grid is increased. The lowest eigenvalue is always positive and vanishes as 1/L ².     

Feedback-controlled electro-kinetic traps for single-molecule spectroscopy

A principal limitation of single-molecule spectroscopy in solution is the diffusion-limited residence time of a given molecule within the detection volume. A common solution to this problem is to immobilize molecules of interest on a passivated glass surface for extending the observation time to obtain reliable data statistics. However, surface tethering of molecules often introduces artifacts, particularly when studying the structural dynamics of biomolecules. To circumvent this limitation, we investigated alternative ways to extend single-molecule observation times in solution without surface immobilization. Among various possibilities, the so-called anti-Brownian electro-kinetic trap (or ABEL trap) seems best suited to achieve this goal. The essential part of that trap is a feedback-controlled electro-kinetic steering of a molecule’s position in reaction to its diffusive Brownian motion which is monitored by fluorescence, thus keeping the molecule within a sub-micron sized detection volume. Fluorescence trace recordings of over thousands of milliseconds duration on individual dye molecules within an ABEL trap have been reported. In this short review, we shall briefly discuss the principle and some results of ABEL trapping of individual molecules with possible extensions to future works.     

Core-halogenated,helical-chiral triphenylene-based columnar liquid crystals

It is shown that chlorination and even bromination of the mesogenic hexahexyloxytriphenylene 1 can take place at the sterically congested tetracyclic core delivering the respective monohalogenated, sevenfold substituted enantiotropic mesogens 3 and 4 in 32 and 17% yields, respectively, giving mesophases over greater temperature ranges. Miscibility tests among all mesogenic compounds of this study prove formation of the columnar hexagonal type of phase. The steric and electronic effects of those substituents, different from the six equal alkoxy groups, on the mesophase are discussed in detail. In our hands and to our surprise, the sterically hindered acceptor (nitro or halogeno) substituents of 2, 3, or 4 could not be replaced by an alkylthio group in nucleophilic aromatic substitution reactions using the aprotic polar solvent DMEU. Semi-empirical calculations on numerous hexaethers with one or two substituents, e.g. halogens or others, in one of the three bay regions of the triphenylene core reveal that these molecules are helically deformed due to the crowding in the molecule. In this new case of atropisomerism, the 1,12-disubstituted derivatives should offer the possibility of resolution into enantiomers and, therefore, of studying disc-like mesomorphic compounds possessing for the first time the origin of chirality within their molecular centre.