Chirality polynomials

Chirality algebra uses ideas from permutation group theory and group representation theory to derive chirality polynomials having appropriate transformation properties for estimation of the magnitude and sign of a given pseudoscalar property (e.g. optical rotation, circular dichroism) for a given skeleton using parameters which depend only upon the ligands located at the specific sites of the skeleton, the particular skeleton, and the particular pseudoscalar property. For all but the simplest skeletons, a qualitatively complete chirality polynomial describing all chirality phenomena associated with the skeleton contains more than one component and thus requires more than one set of ligand parameters. Qualitatively complete chirality polynomials are reviewed for the most important transitive skeletons (i.e. skeletons in which all sites are equivalent), including the polarized triangle, tetrahedron, disphenoid (allene), polarized square, polarized rectangle, polarized pentagon, octahedron, trigonal prism (cyclopropane), and polarized heptagon skeletons.     

Chemical applications of topology and group theory. XXI: Chirality in transitive skeletons and qualitative completeness [1]

This paper unifies the following ideas for the study of chirality polynomials in transitive skeletons: (1) Generalization of chirality to permutation groups not corresponding to three-dimensional symmetry point groups leading to the concepts of signed permutation groups and their signed subgroups; (2) Determination of the total dimension of the chiral ligand partitions through the Frobenius reciprocity theorem; (3) Determination of signed permutation groups, not necessarily corresponding to three-dimensional point groups, of which a given ligand partition is a maximum symmetry chiral ligand partition by the Ruch-Schönhofer partial ordering, thereby allowing the determination of corresponding chirality polynomials depending only upon differences between ligand parameters; such permutation groups having the point group as a signed subgroup relate to qualitative completeness. In the case of transitive permutation groups on four sites, the tetrahedron and polarized square each have only one chiral ligand partition, but the allene and polarized rectangle skeletons each have two chiral ligand partitions related to their being signed subgroups of the tetrahedron and polarized square, respectively. The single transitive permutation group on five sites, the polarized pentagon, has a degenerate chiral ligand partition related to its being a signed subgroup of a metacyclic group with 20 elements. The octahedron has two chiral ligand partitions, both of degree six; a qualitatively complete chirality polynomial is therefore homogeneous of degree six. The cyclopropane (or trigonal prism or trigonal antiprism) skeleton is a signed subgroup of both the octahedron and a twist group of order 36; two of its six chiral ligand partitions come from the octahedron and two more from the twist group. The polarized hexagon is a signed subgroup of the same twist group but not of the octahedron and thus has a different set of six chiral ligand partitions than the cyclopropane skeleton. Two of its six chiral ligand partitions come from the above twist group of order 36 and two more from a signed permutation group of order 48 derived from the P₃[P ₂] wreath product group with a different assignment of positive and negative operations than the octahedron.     

Geometric chirality products

Developed from Guye's produit d'asymétrie 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.On leave from the Department of Chemistry, University of the Western Cape, Bellville 7530, South Africa.     

On Quantifying Chirality

Since Pasteur's epochal discoveries a century and a half ago, the concept of chirality has continued to play a central role in chemistry and biochemistry. Can chirality be measured? It has long been known that molecular chirality can be given a quantitative meaning through functions specifically parametrized to match the magnitude of pseudoscalar observables. However, chirality is a property that is independent of its physical and chemical manifestations : for a system to be chiral, all that is required is the absence of improper rotations in the symmetry group of the system. This being the case, how can chirality be measured if the “system” is an abstract geometric figure, for example, a scalene triangle in the plane or an asymmetric tetrahedron in three-dimensional space? How does chirality vary as a function of pure shape? In this review we describe recent efforts designed to answer these and related questions.     

Chiral assembly of organic luminogens with aggregation-induced emission

Chirality is important to chemistry, biology and optoelectronic materials. The study on chirality has lasted for more than 170 years since its discovery. Recently, chiral materials with aggregation-induced emission (AIE) have attracted increasing interest because of their fascinating photophysical properties. In this review, we discussed the recent development of chiral materials with AIE properties, including their molecular structures, self-assembly and functions. Generally, the most effective strategy to design a chiral AIE luminogen (AIEgen) is to attach a chiral scaffold to an AIE-active fluorophore through covalent bonds. Moreover, some propeller-like or shell-like AIEgens without chiral units exhibit latent chirality upon mirror image symmetry breaking. The chirality of achiral AIEgens can also be induced by some optically active molecules through non-covalent interactions. The introduction of an AIE unit into chiral materials can enhance the efficiency of their circularly polarized luminescence (CPL) in the solid state and the dissymmetric factors of their helical architectures formed through self-assembly. Thus, highly efficient circularly polarized organic light-emitting diodes (CPOLEDs) with AIE characteristics are developed and show great potential in 3D displays. Chiral AIEgens are also widely utilized as “turn on” sensors for rapid enantioselective determination of chiral reagents. It is anticipated that the present review can entice readers to realize the importance of chirality and attract much more chemists to contribute their efforts to chirality and AIE study.

This review highlights the recent development of chiral materials with aggregation-induced emission properties, including their molecular structures, self-assembly and functions.     

Is There a “Most Chiral Tetrahedron”?

A degree of chirality is a function that purports to measure the amount of chirality of an object: it is equal for enantiomers, vanishes only for achiral or degenerate objects and is similarity invariant, dimensionless and normalisable to the interval [0,1]. For a tetrahedron of non-zero three-dimensional volume, achirality is synonymous with the presence of a mirror plane containing one edge and bisecting its opposite, and hence it is easy to design degree-of-chirality functions based on edge length that incorporate all constraints. It is shown that such functions can have largest maxima at widely different points in the tetrahedral shape space, and by incorporation of appropriate factors, the maxima can be pushed to any point in the space. Thus the phrase "most chiral tetrahedron" has no general meaning: any chiral tetrahedron is the most chiral for some legitimate choice of degree of chirality.     

Optisch aktive,aromatische Spirane, 9. Mitt.: Überprüfung eines Näherungsansatzes für Chiralitätsfunktionen bei 5,5′,6′-trisubstituierten 2,2′-Spirobiindanen

In order to test the semiempirical value of chirality functions in their mathematically most simple form 23 new optically active 5,5-disubstituted 2,2-spirobiindanes1 of known chirality and enantiomeric purity were prepared. Thus a set of about hundred compounds is now available with altogether sixteen types of ligands (i.e. substituents including hydrogen); the experimental molar rotation of fifteen compounds is used to determine the value of a ligandspecific parameter occuring in the used chirality polynomial. According to theory this polynomial is an approximation for the total rotation of derivatives with ligands of three different types and it approximates an experimentally separable part of the rotation as far as compounds with four different ligands are concerned.The additional chirality component, occurring exclusively in the case of derivatives with four different types of ligands turns out to be relatively small but not vanishing. Accordingly, the molar rotation predicted by our method is very good for disubstituted spirobiindanes of type1 and rather good for others with three different types of ligands but is distinctly worse for those with four different ligands. The numerical trend, however, is clearly represented even in cases where our calculation in principle refers to a part of the phenomenon only and the predicted absolute configuration is in call cases in agreement with the experiment.An adequate criterion to judge the quality of our approximation is introduced.

8. Mitt.:H. Neudeck undK. Schlögl, Mh. Chem.110, 541 (1979).     

Theorie der ChiralitÄtsfunktionen

The pseudoscalar properties of chiral molecules are object of an algebraic theory, provided a proper definition of molecule-classes is given. The analysis of the phenomenon of chirality on those classes leads to some specific features of the representation theory for chiralty functions, to a new lattice-structure of partions, to properties of permutation groups connected therewith and, finally, gives an insight into the structure of approximate points of view for chirality functions. Thus the present paper includes pure mathematical aspects. Mathematical theorems which will be stated and proved without essential reference to physics will be found in the appendix. The paper itself presents the physical phenomenon in the first place and concepts which are offered by the mathematical formalism like chirality-order, chirality-index, chirality-numbers, qualitative completeness, shortened AnsÄtze, active and inactive partitions of ligands etc. give rise to a systematicism, to an insight and to answering questions concerning the measurement of properties related to chirality of molecules. The co- and contravariant point of view for the transformation behaviour of functions, for instance, gives us two possible interpretations in the case of chirality functions. We may understand components of functions belonging to irreducible representations of \(\mathfrak{S}_n \) as chirality functions for component mixtures of isomers. Thereby projection operators get a physical interpretation as ensembleoperators for mixtures of isomers. Chap. 12 drafts applications of the theory given in this paper. First convincing comparisons of experimental data for rotatory power of allene derivatives with theoretical values on the basis of approximations according to methods given in 8–11 are available and their publication is under preparation [4]. Also, mathematical consequences from the definition of the partition-lattice given in Chapter 6 will be pursued as far as they are not be found in the present paper.     

Theory of chirality functions,generalized for molecules with chiral ligands

The theory of chirality functions described in a previous publication is generalized to allow for chiral ligands. In the earlier theory, all symmetry operations of the molecular frame could be thought of as permutations of the ligands among the sites; in the present work, improper rotations not only permute the ligands, but convert them into mirror images. The group that generates all isomers from a given ordered molecule belonging to a frame with n sites is now the hyperoctahedral group of order 2ⁿ n! consisting of all possible combinations of permutations and site reflections. The representation theory of these groups is described, and applied to the problem of constructing qualitatively complete chirality functions, and of deciding which ligand partitions, and which isomer mixtures, are chiral. It is found useful to classify chiral representations of the covering group as ligand specific and class specific. The ligand specific representations describe chiral properties which are common to all frames and arise purely from the chirality of the ligands, while the class specific representations describe the chiral properties of the frame. A number of examples are explicitly worked out.     

Laser-operated chiral molecular switch: quantum simulations for the controlled transformation between achiral and chiral atropisomers

We report quantum dynamical simulations for the laser controlled isomerization of 1-(2-cis-fluoroethenyl)-2-fluorobenzene based on one-dimensional electronic ground and excited state potentials obtained from (TD)DFT calculations. 1-(2-cis-fluoroethenyl)-2-fluorobenzene supports two chiral and one achiral atropisomers, the latter being the most stable isomer at room temperature. Using a linearly polarized IR laser pulse the molecule is excited to an internal rotation around its chiral axis, i.e. around the C-C single bond between phenyl ring and ethenyl group, changing the molecular chirality. A second linearly polarized laser pulse stops the torsion to prepare the desired enantiomeric form of the molecule. This laser control allows the selective switching between the achiral and either the left- or right-handed form of the molecule. Once the chirality is "switched on" linearly polarized UV laser pulses allow the selective change of the chirality using the electronic excited state as intermediate state.     

Chiral molecular nanosilicas

Molecular nanoparticles including polyoxometalates, proteins, fullerenes and polyhedral oligosiloxane (POSS) are nanosized objects with atomic precision, among which POSS derivatives are the smallest nanosilicas. Incorporation of molecular nanoparticles into chiral aggregates either by chiral matrices or self-assembly allows for the transfer of supramolecular chirality, yet the construction of intrinsic chirality with atomic precision in discrete molecules remains a great challenge. In this work, we present a molecular folding strategy to construct giant POSS molecules with inherent chirality. Ferrocenyl diamino acids are conjugated by two or four POSS segments. Hydrogen bonding-driven folding of diamino acid arms into parallel β-sheets facilitates the chirality transfer from amino acids to ferrocene and POSS respectively, disregarding the flexible alkyl spacers. Single crystal X-ray structures, density functional theory (DFT) calculations, circular dichroism and vibrational circular dichroism spectroscopy clearly verify the preferential formation of one enantiomer containing chiral molecular nanosilicas. The chiral orientation and chiroptical properties of POSS show pronounced dependence on the substituents of α-amino acids, affording an alternative way to control the folding behavior and POSS chirality in addition to the absolute configuration of amino acids. Through the kinetic nanoprecipitation protocol, one-dimensional aggregation enables chirality transfer from the molecular scale to the micrometer scale, self-assembling into helices in accordance with the packing propensity of POSS in a crystal phase. This work, by illustrating the construction of chiral molecular nanosilicas, paves a new way to obtain discrete chiral molecular nanoparticles for potential chiroptical applications.

A molecular folding strategy is developed to construct ferrocenyl diamino acid conjugated polyhedral oligosiloxane molecules. Hydrogen bonding-driven folding facilitates the chirality transfer from the molecular scale to the micrometer scale.     

Chemical applications of topology and group theory 13. Chirality and framework groups [1]

Pople has recently introduced the concept of a framework group to specify the full symmetry properties of a molecular structure. Furthermore, Pople has developed powerful algorithms for the use of framework groups to generate all distinguishable skeletons with a given number of sites. This paper studies the systematics of chirality arising from different framework groups. In this connection framework groups can be classified into four different types: linear, planar, achiral, and chiral. Chiral framework groups lead to chiral systems for any ligand partition including that with all ligands equivalent. Linear framework groups are never chiral even for the ligand partition with all ligands different. Planar framework groups are also never chiral since all sites are in the same plane, which therefore remains a symmetry plane for any ligand partition. However, the mirror symmetry of the molecular plane of a planar framework group can be destroyed by a process called polarization; this process can be viewed as the mathematical analogue of complexing a planar aromatic hydrocarbon to a transition metal. The chirality of four-, five-, and six-site framework groups is discussed in terms of the maximum symmetry ligand partitions resulting in removal of all of the symmetry elements corresponding to improper rotations S _n (including reflections S ₁ and inversions S ₂) from achiral and polarized planar framework groups. The Ruch-Schönhofer group theoretical algorithms for the calculation of chiral ligand partitions and pseudoscalar polynomials of lowest degree (“chirality functions”) are adapted for use with these framework groups. Other properties of framework groups relevant to a study of their chirality are also discussed: these include their transitivity (i.e. whether all sites are equivalent or not), their normality (i.e. whether proper rotations correspond to even permutations and improper rotations correspond to odd permutations), and the number of sites in their symmetry planes.     

Amplified Chirality Transfer to Aromatic Molecules through Non-specific Inclusion by Amorphous,Hyperbranched Poly(fluorenevinylene) Derivatives

Optically active, hyperbranched, poly(fluorene-2,4,7-triylethene-1,2-diyl) [poly(fluorenevinylene)] derivatives bearing a neomenthyl group and a pentyl group at the 9-position of the fluorene backbone at various ratios acted as a chirality donor (host polymers) efficiently included naphthalene, anthracene, pyrene, 9-phenylanthracene, and 9,10-diphenyanthracene as a chirality acceptor (guest molecules) in their interior space in film as well as in solution, with the guest molecules exhibiting intense circular dichroism through chirality transfer with chirality amplification. The efficiency of the chirality transfer was much higher with higher-molar-mass polymers than lower-molar-mass ones as well as with hyperbranched polymers compared to the analogous linear ones. The hyperbranched polymers include the small molecules in their complex structure without any specific interactions at various stoichiometries. The included molecules may have ordered intermolecular arrangement that may be somewhat similar to those of liquid crystals. Naphthalene, anthracene, and pyrene included in the polymer exhibited efficient circularly polarized luminescence, where the chirality was remarkably amplified in excited states, and anthracene exhibited especially high anisotropies in the emission on the order of 10⁻².     

Simple One-electron Invariants of Molecular Chirality

Pseudoscalar measures of electronic chirality for molecular systems are derived using the spectral moment theory applied to the frequency-dependent rotational susceptibility. In this scheme a one-electron chirality operator naturally emerges as a quantum counterpart of the triple scalar product, involving velocity, acceleration and second acceleration. Averaging over an electronic state vector gives rise to an additive chirality invariant (κ-index), considered as a quantitative measure of chirality. A simple computational technique for quick calculation of the κ-index is developed and various structural classes (cyclic hydrocarbons, cage-shaped systems, etc.) are studied. Reasonable behaviour of the chirality index is demonstrated. The chirality changes during the β-turn formation in Leu-Enkephalin is presented as a useful example of the chirality analysis for conformational transitions.     

Development of Ar‐BINMOL‐Derived Atropisomeric Ligands with Matched Axial and sp3 Central Chirality for Catalytic Asymmetric Transformations

Recently, academic chemists have renewed their interest in the development of 1,1′‐binaphthalene‐2,2′‐diol (BINOL)‐derived chiral ligands. Six years ago, a working hypothesis, that the chirality matching of hybrid chirality on a ligand could probably lead to high levels of stereoselective induction, prompted us to use the axial chirality of BINOL derivatives to generate new stereogenic centers within the same molecule with high stereoselectivity, obtaining as a result sterically favorable ligands for applications in asymmetric catalysis. This Personal Account describes our laboratory's efforts toward the development of a novel class of BINOL‐derived atropisomers bearing both axial and sp³ central chirality, the so‐called Ar‐BINMOLs, for asymmetric synthesis. Furthermore, on the basis of the successful application of Ar‐BINMOLs and their derivatives in asymmetric catalysis, the search for highly efficient and enantioselective processes also compelled us to give special attention to the BINOL‐derived multifunctional ligands with multiple stereogenic centers for use in catalytic asymmetric reactions.

     

三角形法和四面体法优化选择毛细管区带电泳背景电解质

建立了两种高效、快速的毛细管区带电泳背景电解质(BGE)的优化方法三角形优化法和四面体优化法。以色谱指纹图谱指数F和色谱指纹图谱相对指数Fr作为评价毛细管电泳分析系统的目标函数,以雪莲药材水提取液为样品,考察一定浓度的硼砂、硼酸、磷酸氢二钠和磷酸二氢钠溶液按三角形优化法和四面体优化法构成背景电解质时对样品的分离情况,通过添加有机改性剂和调节pH进行再优化。用三角形法优化出以50 mmol/L硼砂-含3%乙腈的150 mmol/L磷酸二氢钠(体积比为1∶1)作为BGE时分离效果最佳,用四面体法优化出以50 mmol/L硼砂-150 mmol/L磷酸二氢钠-200 mmol/L硼酸(体积比为1∶1∶2,用0.1 mol/L氢氧化钠调pH 8.55)作为BGE时分离效果最佳,分别获得28个和25个电泳峰。所建立的方法操作简捷,适用于中药材水提取液或醇提取液的毛细管区带电泳BGE的选择。     

Chiral exciplex dyes showing circularly polarized luminescence: extension of the excimer chirality rule

A series of axially chiral binaphthyls and quaternaphthyls possessing two kinds of aromatic fluorophores, such as pyrenyl, perylenyl, and 4-(dimethylamino)phenyl groups, arranged alternately were synthesized by a divergent method. In the excited state, the fluorophores selectively formed a unidirectionally twisted exciplex (excited heterodimer) by a cumulative steric effect and exhibited circularly polarized luminescence (CPL). They are the first examples of a monomolecular exciplex CPL dye. This versatile method for producing exciplex CPL dyes also improved fluorescence intensity, and the CPL properties were not very sensitive to the solvent or to the temperature owing to the conformationally rigid exciplex. This systematic study allowed us to confirm that the excimer chirality rule can be applied to the exciplex dyes: left- and right-handed exciplexes with a twist angle of less than 90° exhibit (−)- and (+)-CPL, respectively.

Axially chiral binaphthyls and quaternaphthyls possessing two kinds of fluorophores were synthesized. In the excited state, the fluorophores formed a twisted exciplex and exhibited CPL. This study gave us named the exciplex chirality rule.