Chiral Triangle Research Articles

Chiral Redox-Active Isosceles Triangles.

Designing small-molecule organic redox-active materials, with potential applications in energy storage, has received considerable interest of late. Herein, we report on the synthesis, characterization, and application of two rigid chiral triangles, each of which consist of non-identical pyromellitic diimide (PMDI) and naphthalene diimide (NDI)-based redox-active units. (1)H and (13)C NMR spectroscopic investigations in solution confirm the lower symmetry (C2 point group) associated with these two isosceles triangles. Single-crystal X-ray diffraction analyses reveal their rigid triangular prism-like geometries. Unlike previously investigated equilateral triangle containing three identical NDI subunits, both isosceles triangles do not choose to form one-dimensional supramolecular nanotubes by dint of [C-H···O] interaction-driven columnar stacking. The rigid isosceles triangle, composed of one NDI and two PMDI subunits, forms-in the presence of N,N-dimethylformamide-two different types of intermolecular NDI-NDI and NDI-PMDI π-π stacked dimers with opposite helicities in the solid state. Cyclic voltammetry reveals that both isosceles triangles can accept reversibly up to six electrons. Continuous-wave electron paramagnetic resonance and electron-nuclear double-resonance spectroscopic investigations, supported by density functional theory calculations, on the single-electron reduced radical anions of the isosceles triangles confirm the selective sharing of unpaired electrons among adjacent redox-active NDI subunit(s) within both molecules. The isosceles triangles have been employed as electrode-active materials in organic rechargeable lithium-ion batteries. The evaluation of the structure-performance relationships of this series of diimide-based triangles reveals that the increase in the number of NDI subunits, replacing PMDI ones, within the molecules improves the electrochemical cell performance of the batteries.

Unexpected Self-Assembly of Chiral Triangles from 90° Chiral Di-Pt(II) Acceptors

Two unexpected chiral organometallic triangles rather than squares from newly designed 90° chiral di-Pt(II) acceptors were obtained through coordination-driven self-assembly. Their structures were well characterized by multinuclear NMR ((1)H and (31)P) and variable-temperature NMR experiments, ESI-TOF-MS, and elemental analysis. The PM6 semiempirical molecular simulation was employed for the interpretation of the formation and stability of such chiral triangles.

A chiral screw propeller-like scalene triangle manganese(iii) cluster

The screw propeller-like cluster complex [Mn(3)(OMe)(L(Schiff))(3)(H(2)O)Cl(2)].2H(2)O {H(2)L(Schiff) = 2-[(E)-(R)-1-hydroxypropan-2-ylimino]methyl-6-methoxyphenol} is the first chiral triangle manganese cluster compound. It is also a rare nonsymmetrical triangle compound with an unequilateral arrangement of three manganese(iii) atoms. Both ferromagnetic and antiferromagnetic interactions exist between the Mn(3+) ions, exhibiting a magnetic anisotropy.

Chiral Organometallic Triangles with Rh−Rh Bonds. 2. Compounds Prepared from Enantiopure cis-Rh2(C6H4PPh2)2(OAc)2(HOAc)2 and Their Catalytic Potentials

Enantiomers of the orthometalated dirhodium compound cis-Rh2(C6H4PPh2)2(OAc)2(HOAc)2 (R-1 and S-1) were prepared from carboxylate exchange reactions of the resolved diasteroisomers of cis-Rh2(C6H4PPh2)2(protos)2(H2O)2 (protos = N-4-methylphenylsulfonyl-l-proline anion) and acetic acid. These compounds react with excess Me3OBF4 in CH3CN, producing the enantiomers of [cis-Rh2(C6H4PPh2)2(CH3CN)6](BF4)2 (R-2 and S-2) which have six labile and replaceable CH3CN ligands in equatorial and axial positions. Reactions of R-2 and S-2 with tetraethylammonium salts of the linear dicarboxylic acids, terephthalic acid (HO2CC6H4CO2H), oxalic acid (HO2CCO2H), and 4,4'-diphenyl-dicarboxylic acid (HO2CC6H4C6H4CO2H) afford the enantiopure triangular supramolecules [cis-Rh2(C6H4PPh2)2(O2CC6H4CO2)(py)2]3, RRR-3 and SSS-3, Rh6(cis-C6H4PPh2)6(O2CCO2)3(py)5(CH2Cl2), RRR-4 and SSS-4, and Rh6(cis-C6H4PPh2)6(O2CC6H4C6H4CO2)3(py)4(CH2Cl2)2, RRR-5 and SSS-5, respectively. The absolute structures of each of the enantiomers of 1, 3, 4, and 5 were determined by X-ray diffraction analyses. The enantiomers of 3, 4, and 5 were found to be enantiomorphically isostructural, whereas R-1 and S-1 crystallized in different space groups. In 4 and 5, CH2Cl2 molecules coordinate to rhodium atoms in the axial positions. The 1H and 31P[1H] NMR spectra of all compounds are reported. The triangular compounds are redox-active, and their electrochemistry is also discussed. An assay of the catalytic activity/selectivity performance of the triangles for typical metal carbene transformation, using the model intermolecular cyclopropanation of styrene with ethyl diazoacetate in both homogeneous and heterogeneous phases, show that these chiral triangles are very active and have remarkable selectivity when compared with simple Rh2 paddle-wheel catalysts with chiral amidate ligands.

Any Scalene Triangle Is the Most Chiral Triangle

AbstractThe shape space of all possible triangles is represented by a triangular diagram, an analogue of the phase diagram for ternary mixtures. Each point of its interior corresponds to the angle set of a pair of (2D) enantiomeric physical triangles and, with appropriate conventions, to just one member of the pair. Points on median lines represent achiral triangles, those on sides represent degenerate chiral triangles, and those on vertices achiral degenerate linear triangles. A chirality index for triangles must vanish on these lines, but nowhere else within the six compartments of the diagram, and should alternate in sign between them. The archetype is the lowest A2‐symmetric eigenfunction of the Schrödinger equation for the particle confined to an equilateral triangular box. Within the constraints, the extrema of an acceptable function may be pushed onto any D3h‐symmetric hexagonal set of points in the diagram, thereby verifying a conjecture of Dunitz that any scalene triangle is the most‐chiral triangle for some legal 2D‐chirality index.

Exact consequences of the trace anomaly in four dimensions

The general form of the stress-tensor three-point function in four dimensions is obtained by solving the Ward identities for the diffeomorphism and Weyl symmetries. Several properties of this correlator are discussed, such as the renormalization and scheme independence and the analogies with the anomalous chiral triangle. At the critical point, the coefficients a and c of the four-dimensional trace anomaly are related to two finite, scheme-independent amplitudes of the three-point function. Off-criticality, the imaginary parts of these amplitudes satisfy sum rules which express the total renormalization-group flow of a and c between pairs of critical points. Although these sum rules are similar to that satisfied by the two-dimensional central charge, the monotonicity of the flow, i.e., the four-dimensional analogue of the c-theorem, remains to be proven.

Quantization of massive chiral electrodynamics reexamined.

We show that the models considered by Andrianov et al. [Phys. Rev. Lett. 63, 1554 (1989); and Phys. Rev. D 44, 2602 (1991)] are equivalent to other models where it is easily proved that the anomaly decouples and consequently the value of the chiral triangles amplitude is irrelvant for the unitarity of the S matrix.

Distance functions as generators of chirality measures

A non-numerical analysis is presented of chirality measures associated with a set of topologically equivalent distance functions. A chirality measure is defined as the minimum distance that separates a chiral and an achiral object (first kind) or two enantiomorphs (second kind). On the basis of this analysis, as applied to triangles in the Euclidean plane, results of an earlier computational study of the Hausdorff chirality measure are now fully understood. Analytical proof has been provided for an earlier conjecture, based on a numerical analysis, that the union of enantiomorphous triangles is achiral under conditions of maximal overlap. Geometric parameters for the most chiral triangle, as determined by a family of three measures of the first kind, are found to differ substantially from those determined by the corresponding measures of the second kind; none of these extremal triangles is degenerate.

A Comprehensive Coordinate Space Renormalization of Quantum Electrodynamics to Two-Loop Order

We develop a coordinate space renormalization of massless Quantum Electrodynamics using the powerful method of differential renormalization. Bare one-loop amplitudes are finite at non-coincident external points, but do not accept a Fourier transform into momentum space. The method provides a systematic procedure to obtain one-loop renormalized amplitudes with finite Fourier transforms in strictly four dimensions without the appearance of integrals or the use of a regulator. Higher loops are solved similarly by renormalizing from the inner singularities outwards to the global one. We compute all 1- and 2-loop 1PI diagrams, run renormalization group equations on them and check Ward identities. The method furthermore allows us to discern a particular pattern of renormalization under which certain amplitudes are seen not to contain higher-loop leading logarithms. We finally present the computation of the chiral triangle showing that differential renormalization emerges as a natural scheme to tackle $\gamma_5$ problems.

Geometric chirality products

Developed from Guye's “produit d'asymetrie” and formally similar to Ruch's chirality products, geometric chirality products are functions purely of molecular shape, without reference to chemical characteristics. In their normalized versions, geometric chirality products have all the attributes' of a chirality measure, i.e. they are similarity invariant and dimensionless in the interval [−1, 1]. An application to Boys' model of the tetrahedron is presented, and a detailed study of the results for triangular domains in E2 is reported. According to this measure, the most chiral triangle is infinitely flat and infinitely skewed. The analysis leads to the paradoxical conclusion that the most chiral triangle is infinitesimally close to an achiral one, The results are compared with those obtained for an overlap measure of chirality, and the relationship between molecular models and measures of chirality is briefly discussed.

Desymmetrization and degree of chirality

Chiral objects, viewed as distorted derivatives of achiral ones, may be represented by points in a configuration space that is spanned by a set of symmetry coordinates derived for the symmetry group of the achiral object of highest symmetry. We propose a measure (d) that quantifies the displacement of the representative point for a chiral object away from thenearest point representing an achiral object in such a multi-dimensional configuration space. If the symmetry coordinates are chosen so as to yield a similarity invariant measure, then the valuesd; obtained for a series ofi chiral objects can serve as a basis for comparing the degrees of chirality of these objects. The chirality of triangles inE 2 is studied by this method, and it is shown that the most chiral triangle in terms of this measure corresponds to one that is infinitely flat, and that may be approached but is never attained. This result is compared to others obtained previously for the same system by the use of different measures of chirality.

Further remarks on quantization of massive chiral electrodynamics in four dimensions.

We show that our conclusions about the consistency of massive chiral electrodynamics in four dimensions were based on the use of a peculiar, although popular, expression of the chiral triangle amplitude for massless fermions, viz., a Bose-symmetric sum of Dolgov-Zakharov poles. If the Feynman triangular amplitude is instead computed by means of standard ultraviolet regulators, no decoupling of the unphysical degree of freedom occurs, thereby jeopardizing perturbative unitarity. Our present analysis raises severe doubts about the possibility of a consistent treatment of anomalous theories in a perturbative context.