Complete Duality for Quasiconvex and Convex Set-Valued Functions

This paper provides a unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set-valued functions.     

A large volume pressure cell for high temperatures

Abstract

Studies of matter under very high pressure at synchrotron radiation sources are mostly done using pressure cells with single-crystal diamond anvils. In some cases the available volume (≤ 10^?3mm³)in such cells causes problems especially at high temperature and for crystal synthesis. To ensure sufficient homogeneity of pressure and temperature, the use of cells with large sample volumes (≥ 1 mm³) is necessary.

Existing devices for such measurements are compared with a novel setup which consists of a toroidal anvil arrangement and a lightweight (50 kg) press with 250 tonnes (2.5 MN) capacity. Preliminary tests of this instrument with synchrotron radiation are reported.

Presented at the IUCr Workshop on ‘Synchrotron Radiation Instrumentation for HighPressure Crystallography’. Daresbury Laboratory 20-21 July 1991     

Intermolecular exchange-induction energies without overlap expansion

An approach to evaluate the second-order exchange-induction energies of symmetry-adapted intermolecular perturbation theory (SAPT) for single-determinant ground-state monomer wavefunctions has been derived. This approach is correct to all orders of the intermonomer overlap, that is, it takes multiple electron exchange between the monomers into account. The resulting formulae can be written in a compact way and implemented efficiently. Here, the method is employed to investigate the performance of the S ²- or single-exchange approximation at the Hartree-Fock-SAPT level. The list of test systems comprises the prototypical van der Waals- and hydrogen-bridge complexes Ne₂, Ar–HF, and (H₂O)₂, but also the systems HeCl^?, NeNa⁺ and Li⁺F^? involving closed-shell ions. It was found that the errors introduced by the S ²-approximation are more pronounced for the second-order exchange-induction energy than for the first-order exchange energy. While these errors tend to be negligible throughout the well region of complexes such as the neon dimer, they start to be significant in the repulsive part of the well regions of systems such as the water dimer, and in particular for the ionic lithium fluoride molecule. The consequences of these findings for the Hartree-Fock level estimate of higher-order induction plus exchange-induction energies, which is frequently employed in SAPT are also discussed.     

Uncovering the Hidden Landscape of Tris(4-pyridyl) Ligands: Topological Complexity Derived from the Bridgehead

Supramolecular main group chemistry is a developing field which parallels the conventional domain of metallo-organic chemistry. Little explored building blocks in this area are main group metal-based ligands which have the appropriate donor symmetry to build desired molecular or extended arrangements. Tris(pyridyl) main group ligands (E(py)₃, E=main group metal) are potentially highly versatile building blocks since shifting the N-donor arms from the 2- to the 3-positions and 4-positions provides a very simple way of changing the ligand character from mononuclear/chelating to multidentate/metal-bridging. Here, the coordination behaviour of the first main group metal tris(4-pyridyl) ligands, E(4-py)₃ (E=Sb, Bi, Ph−Sn) is explored, as well as their ability to build metal-organic frameworks (MOFs). The complicated topology of these MOFs shows a marked influence on the counter anion and on the ability of the E(4-py)₃ ligands to switch coordination mode, depending on the steric and donor character of the bridgehead. This structure-directing influence of the bridgehead provides a potential building strategy for future molecular and MOF design in this area.     

On certain arithmetic functions involving the greatest common divisor

The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ²(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.