Comment on the recent controversy over the theory of chirality functions

Derflinger,et al., have criticized the concept of qualitative completeness on the grounds that a chirality function may be qualitatively complete and still vanish identically for certain nonracemic, non-isomeric mixtures. It is pointed out that the functions considered by them are ones in which the full chirality function splits into parts which can be thought of as contributions from “effective fragments” of the molecule, and that their mixtures are indeed racemic in these fragments. The function can be augmented in a very simple way so as not to vanish for such mixtures, and the idea of qualitative completeness is in no way disturbed by this process.     

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.     

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.     

Information theoretical study of chirality: enantiomers with one and two asymmetric centra

In this work, the Kullback-Leibler information deficiency is probed as a chirality measure. It is argued that the information deficiency, calculated using the shape functions of the R and S enantiomers, considering one as reference for the other, gives an information theory based expression useful for quantifying chirality. The measure is evaluated for five chiral halomethanes possessing one asymmetric carbon atom with hydrogen, fluorine, chlorine, bromine, and iodine as substituents. To demonstrate the general applicability, a study of two halogen-substituted ethanes possessing two asymmetric carbon atoms has been included as well. The basic expression of the sum of the local information deficiency over all atoms can be decomposed into separate summations over coinciding and noncoinciding atoms, or into a global and a mixing entropy term, or into a local entropy contribution for each atom individually based on the Hirshfeld partitioning. Avnir's continuous chirality measure (CCM) has been computed and confronted with the information deficiency. Finally, the relationship between chirality and optical rotation is used to study the proposed measure. The results illustrate Mezey's holographic electron density theorem with an intuitively appealing division of the strength of propagation of the atomic chirality from an asymmetric carbon atom throughout the molecule. The local information deficiency of the carbon atom is proposed as a measure of chirality; more precisely, the difference in information between the R and the S enantiomer turns out to be a quantitative measure of the chirality of the system. It may be evaluated as the arithmetic mean of the different alignments, or considering only the alignment resulting in the highest similarity value, or using the QSSA alignment.     

Investigation of the chirality dependence of thermal properties in chiral-racemic mixtures of the cholesteric ester CE6

A scanning adiabatic calorimetric technique has been used to study the thermal properties of the chiral and racemic liquid crystal CE6 and the phase diagram covering the cholesteric phase, the three blue phases and the isotropic phase. The purpose of this investigation is to study thermal properties of liquid crystals as a function of chirality, while all other parameters remain constant. Results for the temperature and the chirality dependence of the enthalpy and of the heat capacity are reported. The latent heats between the cholesteric phase and BPI and between the different blue phases change slightly as a function of the chirality. The total heat of transition at the isotropic phase boundary is independent of the chirality, but with decreasing chirality, we observe a large increase in the latent heat and, correspondingly, a decrease in the pretransitional contribution. These experimental facts are in qualitative agreement with the predictions of a Landau-de Gennes theory for blue phases.     

Orbital views of molecular conductance perturbed by anchor units

Site-specific electron transport phenomena through benzene and benzenedithiol derivatives are discussed on the basis of a qualitative Hu?ckel molecular orbital analysis for better understanding of the effect of anchoring sulfur atoms. A recent work for the orbital control of electron transport through aromatic hydrocarbons provided an important concept for the design of high-conductance connections of a molecule with anchoring atoms. In this work the origin of the frontier orbitals of benzenedithiol derivatives, the effect of the sulfur atoms on the orbitals and on the electron transport properties, and the applicability of the theoretical concept on aromatic hydrocarbons with the anchoring units are studied. The results demonstrate that the orbital view predictions are applicable to molecules perturbed by the anchoring units. The electron transport properties of benzene are found to be qualitatively consistent with those of benzenedithiol with respect to the site dependence. To verify the result of the Hu?ckel molecular orbital calculations, fragment molecular orbital analyses with the extended Hu?ckel molecular orbital theory and electron transport calculations with density functional theory are performed. Calculated results are in good agreement with the orbital interaction analysis. The phase, amplitude, and spatial distribution of the frontier orbitals play an essential role in the design of the electron transport properties through aromatic hydrocarbons.     

Investigation of the chirality dependence of thermal properties in chiral-racemic mixtures of the cholesteric ester CE6

Abstract

A scanning adiabatic calorimetric technique has been used to study the thermal properties of the chiral and racemic liquid crystal CE6 and the phase diagram covering the cholesteric phase, the three blue phases and the isotropic phase. The purpose of this investigation is to study thermal properties of liquid crystals as a function of chirality, while all other parameters remain constant. Results for the temperature and the chirality dependence of the enthalpy and of the heat capacity are reported. The latent heats between the cholesteric phase and BPI and between the different blue phases change slightly as a function of the chirality. The total heat of transition at the isotropic phase boundary is independent of the chirality, but with decreasing chirality, we observe a large increase in the latent heat and, correspondingly, a decrease in the pretransitional contribution. These experimental facts are in qualitative agreement with the predictions of a Landau–de Gennes theory for blue phases.     

Vibrational Circular Dichroism Spectroscopy of Chiral Molecular Crystals: Insights from Theory

Chirality is a curious phenomenon that appears in various forms. While the concept of molecular (RS-)chirality is ubiquitous in chemistry, there are also more intricate forms of structural chirality. One of them is the enantiomorphism of crystals, especially molecular crystals, that describes the lack of mirror symmetry in the unit cell. Its relation to molecular chirality is not obvious, but still an open question, which can be addressed with chiroptical tools. Vibrational circular dichroism (VCD) denotes chiral infrared (IR) spectroscopy that is susceptible to both, the molecular as well as the intermolecular space by means of vibrational transitions. When carried out in the solid state, VCD delivers a very rich set of non-local contributions that are determined by crystal packing and collective motion. Since its discovery in the 1970s, VCD has become the method of choice for the determination of absolute configurations, but its applicability reaches beyond towards the study of different crystal forms and polymorphism. This brief review summarises the theoretical concepts of crystal chirality and how computations of solid-state VCD can shed light into the intimate connection of chiral structure and vibrational optical activity.     

Concerning the theory of chirality functions : Part I. Construction and discussion of chirality functions by means of [2.2] metacyclophanes

By means of [2.2]metacyclophane (skeleton symmetry C_2h) the construction of qualitatively-complete chirality functions via the first and second “Naherungsansatze” is demonstrated. In this context, a detailed comment concerning the algebraic theory of chirality functions is given. As the first term of the chirality function (“abridged Ansatz”), both methods furnish a simple superposition rule (“quadrant rule”). The second component as deduced via the method of polynomials is of third degree in ligand specific parameters. A particular property of the second approximation method is derived. The consequences with respect to observables of the chirality phenomenon, in the case of the approximations being valid, are discussed.     

Tilt and temperature dependence of the pitch in the chiral smectic C phase

The chirality of the constituent molecules in the chiral smectic phase induces a helical structure with a pitch, p₀. Because of the tilt and chirality there is a spontaneous polarization and a bend deformation which act upon the induced helix. The resulting pitch is described as a function of p₀ using the phenomenological theory of a chiral smectic C phase. The pitch, p₀, is then calculated using a molecular theory of the cholesteric phase. The results obtained explain the experimental observations, at least qualitatively.     

Pillararene Host–Guest Complexation Induced Chirality Amplification: A New Way to Detect Cryptochiral Compounds

The contradiction between the rising demands of optical chirality sensing and the failure in chiral detection of cryptochiral compounds encourages researchers to find new methods for chirality amplification. Inspired by planar chirality and the host–guest recognition of pillararenes, we establish a new concept for amplifying CD signals of cryptochiral molecules by pillararene host–guest complexation induced chirality amplification. The planar chirality of pillararenes is induced and stabilized in the presence of the chiral guest, which makes the cryptochiral molecule detectable by CD spectroscopy. Several chiral guests are selected in these experiments and the mechanism of chiral amplification is studied with a non-rotatable pillararene derivative and density functional theory calculations. We believe this work affords deeper understanding of chirality and provides a new perspective for chiral sensing.     

Pillararene Host–Guest Complexation Induced Chirality Amplification: A New Way to Detect Cryptochiral Compounds

The contradiction between the rising demands of optical chirality sensing and the failure in chiral detection of cryptochiral compounds encourages researchers to find new methods for chirality amplification. Inspired by planar chirality and the host–guest recognition of pillararenes, we establish a new concept for amplifying CD signals of cryptochiral molecules by pillararene host–guest complexation induced chirality amplification. The planar chirality of pillararenes is induced and stabilized in the presence of the chiral guest, which makes the cryptochiral molecule detectable by CD spectroscopy. Several chiral guests are selected in these experiments and the mechanism of chiral amplification is studied with a non‐rotatable pillararene derivative and density functional theory calculations. We believe this work affords deeper understanding of chirality and provides a new perspective for chiral sensing.     

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.     

Helfrich's concept of intrinsic force and its molecular origin in bilayers and monolayers

Bilayers and monolayers are excellent models of biological membranes. The constituents of the biological membranes such as lipids, cholesterols and proteins are chiral. Chiral molecules are abundant in nature (protein, nucleic acid and lipid). It is obvious that relationship between chirality and morphology (as well as function) of biological membrane is of interest for its fundamental importance and has technological implication regarding various membrane functions. The recent years have witnessed that a number of experimental studies in biomimetic systems have shown fascinating morphologies where chirality of the constituent molecule has decisive influence. Significant progress is made towards the understanding of these systems from the theoretical and computational studies. Helfrich's concept of intrinsic force arising from chirality is a milestone in understanding the biomimetic system such as bilayer and the related concepts, further progresses in molecular understanding made in recent years and experimental studies revealing the influence of chirality on morphology are the focus of the present review. Helfrich's concept of intrinsic force arising due to chirality is useful in understanding two-dimensional bilayers and one-dimensional monolayers and related mimetic systems. Various experimental techniques are used, which can probe the molecular architecture of these mimetic systems at different length scales and both macroscopic (thermodynamic) as well as microscopic (molecular) theories are developed. These studies are aimed to understand the role of chirality in the molecular interaction when the corresponding molecule is present in an aggregate. When one looks into the variety of morphologies exhibited by three-dimensional bilayer and two-dimensional monolayer, the later types of systems are more exotic in the sense that they show more diversity and interesting chiral discrimination. Helfrich's concept of intrinsic force may be considered useful in both cases. The intrinsic force due to chirality is the decisive factor in determining morphology which is explained by molecular approaches. Finally, biological and technological implications of such morphological variations are briefly mentioned.     

On the representation of atoms and molecules as self-interacting field with internal structure

The nonlinear Schrödinger equation with Gaussian convolution kernel K₂ induces the group SU₃ with reference to the classification of the multiplet structure of the eigenstates. Such a field can be used to describe some atoms (where the outermost electrons are related tos-orbitals) as a self-interacting, extended particle with an internal structure. In the case of those atoms, where the valence electrons are described byp-orbitals, and almost all molecules the Gaussian kernel K₂ has to be generalized by Hermite polynomials. By that, we can formulate a nonlinear field theory, establishing the spatial symmetry of a system via basis structure functions. Thus the symmetry represents the most essential starting-point for treating molecules as quasi-particles with an internal structure. It will be shown that there is some connection with the concept of chirality functions and the Ginzburg — Landau theory of super-conductivity. The latter theory indicates that we can consider the nonlinear Schrödinger equation and its generalizations as a classical field theory being associated with phase transitions.     

On quantifying chirality — Obstacles and problems towards unification

Recently there has been an ever growing interest and activity in the attempts on quantifying chirality which is causing this concept to become a diverse and uncorrelated entity. Possible reasons for this complication are presently discussed. It is shown that it becomes necessary to distinguish between geometric and physical chiralities. For geometrical chiral sets it is necessary to distinguish between equi- and sub-dimensional sets where the metrization of their chirality can be generalized and unified only for equi-dimensional sets. This is accomplished by the method of overlap. For sub-dimensional sets there exists no general and unique mode of quantifying chiralities, except for discrete and finite sets of points such as the comers of polyhedron, for which the approach of Hausdorff distances proves to be an efficient method of quantifying the chirality presented by their distribution. The domain of physical chiralities, although being of natural significance, is still in a premature state of development. Each physical property may have a different chiral measure so that there is no sense in a claim of unification. Equi- and sub-dimensionality exist also for physical chiralities and they can be quantified by the overlap method for equi-dimensional sets.     

Tilt and temperature dependence of the pitch in the chiral smectic C phase

Abstract

The chirality of the constituent molecules in the chiral smectic phase induces a helical structure with a pitch, p ₀. Because of the tilt and chirality there is a spontaneous polarization and a bend deformation which act upon the induced helix. The resulting pitch is described as a function of p ₀ using the phenomenological theory of a chiral smectic C phase. The pitch, p ₀, is then calculated using a molecular theory of the cholesteric phase. The results obtained explain the experimental observations, at least qualitatively.