Klassifikation der verallgemeinerten Regelflächen durch ihre Kommerellhyperflächen

In this paper we investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family of k-dimensional linear subspaces of the n-dimensional Euclidean space \(\mathbb{E}^n \) .     

The molecular structure and conformation of trichloronitromethane as determined by gas-phase electron diffraction and theoretical calculations

The molecular structure of trichloronitromethane has been studied in the gas phase using electron diffraction data. The molecules are found to undergo low barrier rotation about the CN bond with a planar CNO₂ moiety in agreement with HF/MP2/B3LYP/6-311G(d,p) calculations. The experimental data are consistent with a dynamic model using a potential function for the torsion of V = (V₆/2)(1 − cos 6τ). The major geometrical parameters (r_g and ) for the eclipsed form, obtained from least squares analysis of the data are as follows: r(NO₃) = r(NO₄) = 1.213(2) Å, r(CN) = 1.592(6) Å, r(CCl)_av = 1.749(1) Å, Cl₅CN/Cl₆CN = 109. 6°/106.3°(2), O₃NC/O₄NC = 117. 6°/114.1°(4), τCl₅C₁N₂O₃ = 0.0°, and V₆ = 0.20(25) kcal/mol.     

Determination of compositions and isomer structures in mixtures of cis- and trans-1,2-dichloroethene (ClHC=CHCl) by gas-phase electron diffraction

Using gas-phase electron diffraction it has been demonstrated that a composition of known isomer mixtures can be determined with error limits of about 5%, all relevant structural parameters being refined simultaneously by the least-squares method. If, however, corresponding bond distances and valence angles have erroneously been assigned equal values in the two isomers, a large increase in the least-squares error limits from 5% to 12% is noticed. Apparently innocent assumptions about some of the parameters can lead to incorrect isomer composition and to too small error limits as estimated by the least-squares formulae.

From the reinvestigation of pure cis-1,2-dichloroethene the following bond distances (r_a) and valence angles () were determined: r(C---H) = 1.090(29) Å, r(C=C) = 1.345(6) Å, r(C---Cl) = 1.716(4) Å, C=C---Cl = 123.8(2)°, C=C---H = 119.4(26)°. Error limits are 2σ.     

Molecular structure and conformation of 2,3-dichloro-1-propene as determined by gas-phase electron diffraction

The molecular structure and conformation of 2,3-dichloro-1-propene have been determined by gas-phase electron diffraction at nozzle temperatures of 24, 90 and 273°C. The molecules exist as a mixture of two conformers with the chlorine atoms anti (torsion angle ∠φ = 0°) or gauche (∠φ = 109°) to each other and with the anti form the more stable. The composition (mole fraction) of the vapor with uncertainties estimated at 2σ was found to be 0.55 (0.08), 0.49 (0.08) and 0.41 (0.10) at 24, 90 and 273°, respectively. These values correspond to an energy difference with estimated standard deviation ΔE° = E°_g-E°_a = 0.7 ± 0.3 kcal mol^?1 and an entropy difference ΔS° = S°_g-S°_a = 0.6 ± 0.9 cal mol^?1 K^?1. Some of the diffraction results, together with spectroscopic observations, permit the evaluation of an approximate torsional potential function of the form 2V = V₁ (1 - cos φ) + V₂ (1 - cos 2φ) + V₃ (1 - cos 3φ); the results are V₁ = 4.4 ± 0.5, V₂ = ?2.9 ± 0.5 and V₃ = 4.8 ± 0.2, all in kcal mol^?1. The results at 24°C for the distance (r_a) and angle (∠_α) parameters, with estimated uncertainties of 2σ, are: r(C_sp²-H) = 1.098(0.020)Å, r(C_sp³-H) = 1.103(0.020)Å, r(CC) = 1.334(0.009)Å, r(C-C) = 1.504(0.013)Å, r(C_sp²-Cl) = 1.752(0.021)Å, r(C_sp³-Cl) = 1.776(0.020)Å, ∠C-CC = 127.6(1.1)°, ∠C_sp³-C_sp²-Cl = 110.2(1.0), ∠C_sp²-C_sp³-Cl = 113.1(1.2)°, ∠H-C_sp³-H = 109.5° (assumed), ∠CC-H = 120.0° (assumed) and ∠φ = 108.9(3.4)°.     

From the {FeIII2Ln2} Butterfly's Perspective: the Magnetic Benefits and Challenges of Cooperativity within 3 d–4 f Based Coordination Clusters

In this Review we discuss the tuning handles which can be used to steer the magnetic properties of Fe^III-4 f “butterfly” compounds. The majority of presented compounds were produced in the context of project A3 “Di- to tetranuclear compounds incorporating highly anisotropic paramagnetic metal ions” within the SFB/TRR88 “3MET”. These contain {Fe^III₂Ln₂} cores encapsulated in ligand shells which are easy to tune in a “test-bed” system. We identify the following advantages and variables in such systems: (i) the complexes are structurally simple usually with one crystallographically independent Fe^III and Ln^III, respectively. This simplifies theory and anaylsis; (ii) choosing Fe allows ⁵⁷Fe Mössbauer spectroscopy to be used as an additional technique which can give information about oxidation levels and spin states, local moments at the iron nuclei and spin-relaxation and, more importantly, about the anisotropy not only of the studied isotope, but also of elements interacting with this isotope; (iii) isostructural analogues with all the available (i. e. not Pm) 4 f ions can be synthesised, enabling a systematic survey of the influence of the 4 f ion on the electronic structure; (iv) this cluster type is obtained by reacting [Fe^III₃O(O₂CR)₆(L)₃](X) (X=anion, L=solvent such as H₂O, py) with an ethanolamine-based ligand L′ and lanthanide salts. This allows to study analogues of [Fe^III₂Ln₂(μ₃-OH)₂(L′)₂(O₂CR)₆] using the appropriate iron trinuclear starting materials. (v) the organic main ligand can be readily functionalised, facilitating a systematic investigation of the effect of organic substituents on the ligands on the magnetic properties of the complexes. We describe and discuss 34 {M^III₂Ln₂} (M=Fe or in one case Al) butterfly compounds which have been reported up to 2020. The analysis of these gives perspectives for designing new SMM systems with specific electronic and magnetic signatures     

Photoaddressable polymers for liquid crystal alignment

We demonstrate reversible photoinduced in situ reorientation of low molecular mass liquid crystals (LCs) by means of photoaddressable polymers (PAPs). These polymers contain mesogenic azobenzene side chains optimized to reorient cooperatively and effectively upon illumination with polarized light. Various low molecular mass LCs were introduced between two PAP layers and these sandwich devices were tested with respect to stability and reversibility of photoinduced orientation. Dissolution of the PAP layer by the low molecular mass LC was observed for several material combinations and systematically investigated. Different anisotropic dyes were added as fluorescence markers in order to monitor the photoinduced LC orientation. With an optimized material combination, more than 10 reversible reorientation processes could be realized with polarized light of either 514 or 405 nm wavelength, without any reduction in alignment quality. Further, microscopic polarized fluorescence patterns could be produced and erased within short exposure times.     

Trace formulas for additive and non-additive perturbations

Trace formulas for pairs of self-adjoint, maximal dissipative and accumulative as well as other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a pair {H^′,H}$\{H^{\prime },H\}$ of maximal accumulative operators has been proved. We investigate also the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H   and H^′=H+V$H^{\prime }=H+V$ are maximal accumulative and V is trace class, we prove the existence of a summable   complex-valued spectral shift function. We also obtain trace formulas for pairs {H,H^?}$\{H,H^{?}\}$assuming only that H and  H^?$H^{?}$are resolvent comparable. In this case the determinant of the characteristic function of H is involved in trace formulas.