Chapter 1: Introduction to Molecular Chirality Free

Let us imagine that one can pass through the mirror, as Alice did in Lewis Carroll’s novel “Through the Looking-Glass”,¹  and see how the world we know would look like on the other side of the mirror. Surprisingly, biscuits would still be sweet,^2,3  although not fattening, since a mirror image of glucose cannot be phosphorylated by the first enzyme in the glycolysis pathway, hexokinase.^4,5  The toxicity of a “red tide”, i.e., algal bloom, would be greatly reduced since the mirror counterpart of naturally occurring anatoxin-a does not fit acetylcholine receptors and consequently is more than a hundred times less neurotoxic.^6,7  Caraway would smell like mint, and mint would smell like caraway, which is a consequence of one of mirror forms of carvone being responsible for caraway’s odor, while the other is responsible for mint’s odor.⁸  Noticeable effects would be observed at the cellular and even organ level, as could be the case with a fly, Drosophila melanogaster, which has an asymmetrically twisted hindgut.^9,10  This handful of examples illustrates the dramatic impact of molecular chirality on the world around us. See Figure 1.1 for more examples of the importance of chirality in Nature. Moreover, differences in the bioactivity of a dozen chiral molecules and the resulting severe sociomedical consequences, such as the thalidomide scandal, are discussed in Section 5.2.

The fact that an entity can differ from its mirror image can have a substantial effect on its physical,^23–26  chemical,^27–29  and biological^30,31  properties, and it has been the subject of research in many fields of science,^32–35  including fields as seemingly distant from each other as the theory of knots, the theory of electroweak interactions, the mechanism of pharmacologically active ingredients and even the structure of hereditary material. These numerous examples showing the impact of chirality on the processes around us indicate not only the importance of chirality in Nature and the environment surrounding us³⁶  but also the need for a wide range of methods for studying chirality both qualitatively and quantitatively. Therefore, the question of how we can perceive chirality is fundamental, and the attempt to answer it forms the core of this book.

Given its richness, chirality has been a subject of many extensive reviews and dedicated books. Readers interested in various aspects of chirality research can find a lot of complementary information shedding light on chirality in a broader context in the following sources: Gerlach’s brief note on chirality,³⁷  Hegstrom and Kondepudi’s essay on the handedness of the Universe,³⁸  Gardner’s “The Ambidextrous Universe”³⁹  being a classical position written in popular-science style, Guijarro’s book on the origin of chiral biomolecules,⁴⁰  Flügel’s essay on the relationship between chemistry evolution and chirality,⁴¹  Barron’s “Molecular Light Scattering and Optical Activity”,⁴²  a fundamental book on chiroptical spectroscopies and related optical phenomena from the standpoint of molecular scattering of polarized light, Wenzel’s “Differentiation of Chiral Compounds Using NMR Spectroscopy” which is a basic book in the field of NMR,⁴³  Kamenetskii’s edited book on recent advances in physical studies of chirality, magnetism, and magnetoelectricity,⁴⁴  Flapan’s book presenting a mathematician’s view and tools used to study chirality,⁴⁵  and Oerter’s “The Theory of Almost Everything: The Standard Model, the Unsung Triumph of Modern Physics”⁴⁶  which is a highly accessible volume introducing the elementary particle physics theory relevant to chirality in Chapter 9. This list is undoubtedly not exhaustive, but it can serve as an initial point for the reader’s own studies on chirality. As the focus of the book is on studies within the field of both chirality and nuclear magnetic resonance, two encyclopaedias serving as comprehensive references are recommended. These are “Comprehensive Chirality” edited by Carreira and Yamamoto⁴⁷  and “Encyclopedia of NMR” edited by Harris and Wasylishen.⁴⁸ 

In some cases, a physical property of a molecular structure may depend on the handedness of the chosen coordinate system – this book is devoted to such structures. Consequently, we can attribute left- or right-handedness to any of such structures. Handedness goes even deeper than its description using a frame of reference – it is an inherent property of such molecular structures.^† It exists regardless of whether the molecular structure is placed in a given frame of reference or not.

To illustrate how handedness manifests at the level of geometry, let us take an example shown in Figure 1.2: a “right-handed” tetracube placed in a right-handed Cartesian coordinate system and a “left-handed” tetracube placed in the corresponding left-handed Cartesian coordinate system. Handedness of polyhedrons shown in Figure 1.2 is assigned arbitrarily. A tetracube (a solid figure formed by joining four equal cubes face to face) should not be confused with a tesseract (a four-dimensional analogue of the cube). An analogous example taken from everyday life is the act of using left or right-handed scissors (corresponding to the tetracube) in left or right hand (being a counterpart of the coordinate system).

The description of the tetracubes on the left and right are alike, i.e., the x-axis coincides with the edge joining the purple and yellow faces, the y-axis coincides with the edge joining the yellow and blue faces, and the z-axis coincides with the edge joining the blue and purple faces. Distances between vertices and planar angles between edges are exactly the same for the tetracube shown on the left and right sides, but it is impossible to superimpose the left tetracube on the right one. Each time the bases of the tetracubes, consisting of three cubes, superimpose, the four cubes do not overlap. Let us rotate the right-handed tetracube around the z-axis, causing three cubes in the xy-plane to overlap. Then, the fourth cube points downward for the right-handed tetracube, but the left-handed tetracube points upward.

Distances and planar angles are the same, regardless of whether we consider them for each coordinate system separately or describe them for a right-handed tetracube in a left-handed coordinate system. However, not all the quantities describing the tetracubes presented in Figure 1.2 are independent of the choice of handedness of the coordinate system. A dihedral angle given by three bolded line segments of the right-handed tetracube is +90° if measured in the right-handed coordinate system. Although the analogous dihedral angle takes exactly the same value for the left-handed tetracube placed in the left-handed coordinate system, the dihedral angle of the right-handed tetracube measured in the left-handed coordinate system is −90°. Similarly, for the dihedral angle of the left-handed tetracube in the right-handed coordinate system, one obtains −90° instead of +90°, as expected if the description of the tetracube were independent of the handedness and its coordinate system. Notice that the dependence on the coordinate system lies more in our description of the space and the object embedded in it, rather than in the choice of the particular coordinate system itself, i.e., here a Cartesian coordinate system used in this example. Other coordinate systems also possess handedness, i.e., in the case of the spherical coordinate system, one has to specify in which direction (clockwise or counter-clockwise) the polar and azimuthal angles increase.

Nature has a preference for being either right- or left-handed. In particular, while the decision to call the Cartesian system right-handed if the thumb of the right hand points to the x-axis, the index finger points to the y-axis, and the middle finger points to the z-axis, as in Figure 1.2, is arbitrary, there are physical systems that behave differently depending on their chirality. The most well-known example of such behaviour comes from particle physics studies,⁵⁰  namely, only left-chiral fermions, e.g., electrons, and right-chiral antifermions, e.g., positrons, exhibit weak interactions.^‡

The chirality of a particle determines the direction in which it gains phase in a complex plane.^§ If the phase is gained counter-clockwise about the direction of motion, then the particle is a right-handed spinor; conversely, clockwise rotation indicates that the particle is a left-handed spinor. In experimental studies, the chirality of elementary particles was observed with the aid of weak interactions – demonstrated for the first time in 1950s. A sample containing ⁶⁰Co nuclei was cooled down to cryogenic temperatures (T ≈ 3 mK), and their nuclear moments were aligned by a strong magnetic field. It was observed that the electrons emitted from the β-decay of ⁶⁰Co were emitted in the direction opposite to that of the nuclear magnetic moment.⁵⁴  This direction marks which is the “north pole” of the ⁶⁰Co nucleus and consequently carries information about preferred handedness. Therefore, the notion of handedness extends far beyond the mathematical convention or geometrical description of three-dimensional objects and is an inherent property of Nature (see “Ozma problem” stated in ref. 55 for further discussion).

Depending on the mathematical, physical, chemical, and biological context, the notion of handedness usually requires refinement and further clarification. From a mathematical perspective, the vital issue is to place the object that is the subject of our research and the types of operations that can be performed within a strict framework, revealing the inherent handedness. The basis of such a framework is illustrated in Figure 1.3, which shows a scalene triangle, i.e., a triangle whose sides are of different lengths, placed in a two-dimensional plane.

Operations that do not change the distances between the vertices of a triangle are called isometries. More precisely, let X and Y be metric spaces with metrics d_X and d_Y. Commonly, for physicists and chemists, X and Y are ordinary three-dimensional spaces, the Euclidean spaces. Metric is a more precise formulation of the intuitively understood concept of distance.

A map, i.e., one-to-one correspondence, f: X → Y is called isometry or distance preserving if any a and b that are in X satisfy that d_X(a, b) = d_Y(f(a), f(b)). The distance measured from a to b in space X is the same as that measured from f(a) to f(b) in the space Y. There are several kinds of isometries, e.g., translations, rotations, and reflections. In the case of objects with handedness, an important feature of isometry is whether it preserves the object’s handedness or reverses it. For the scalene triangle shown in Figure 1.3, the isometry f₁, a reflection, reverses the direction of arrows on the sides of the triangle from the counter-clockwise to clockwise. In contrast, in the case of the f₂ isometry, the direction of the arrows on the sides of the rotated (by 180°) triangle remains unchanged. Note that in the second case, a rotation of 180° in the plane is the same as an inversion about a point. In the case of three-dimensional space, the space inversion, i.e., the operation $(x,y,z)↦ ( − x , − y , − z )$ leads to a change in the handedness of the object, as shown for the tetracube in Figure 1.2. Therefore, the concept of isometry preserving or altering the handedness of a geometric figure is inherently related to the space in which we consider isometry and the method of measuring distance, i.e., the metric. See ref. 56–58 for discussion of chirality in this context.

What follows, is that the space in which the molecule is embedded influences whether the molecule is considered chiral or not. For instance, 1-nitronaphthalene is a planar molecule, and thus achiral as a free molecule in three-dimensional space, but when it is adsorbed onto a gold surface, its motion is restricted to two dimensions, making it chiral.⁵⁹  From a geometrical perspective, the equivalent of a tetrahedron in three-dimensional space with different vertices is a scalene triangle in two-dimensional space shown in Figure 1.3.

Since we have used the term handedness in an intuitive manner so far, let us clarify a little more precisely what handedness entails from the point of view of the considerations presented here. Generally speaking, one can assign an orientation to an orientable space or an element of it (e.g., a three-dimensional Euclidean space, a ball-and-stick model of an amino acid alanine, or a scalene triangle), i.e., being either clockwise and counter-clockwise in an abstract sense. In the case of ordinary two- or three-dimensional space, one can check whether an isometry changes orientation by checking the sign of the determinant of the matrix that corresponds to that isometry. The matrix of a space inversion in nth dimensional space is a matrix whose diagonal elements are all −1, and all other elements are zero. It follows that the space inversion leaves the orientation in two-dimensional space unchanged since the determinant is +1, and it reverses the orientation in three-dimensional space because the determinant is −1. A pictorial description of the issue of the number of reflections and parity (i.e., invariance under flip in the sign of a spatial coordinate) can also be found in the first chapter of ref. 39. In the light of the above discussion, one can therefore rephrase the question of chirality-sensitive interactions as “Would a mirror image of the world behave in the same way as our world?” or equivalently but in more technical language, “Is the parity conserved in Nature?”. This equivalent formulation is usually the starting point of discussion about molecular chirality in physics.

It may happen that the object obtained by applying an isometry that reverses orientation differs from the original object at the start of the operation. This property is called chirality. Quoting the words of Lord Kelvin from 1884, “I call any geometrical figure, or any group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realised, cannot be brought to coincide with itself”.⁶⁰  In this definition, the emphasis is placed on a specific isometry, with both the space to which it refers and the method of measuring distances being implied and intuitive. The definition is formulated with the usage of a negation – those objects that cannot be superimposed on each other are chiral. In this view, it is potentially easier to show that the molecular structure is achiral by identifying a transformation that superimposes the mirror image of the structure on itself. For complicated molecular structures, one may have to check a large number of potential transformations. Consequently, showing that the structure is chiral may be a nontrivial task, especially if we analyse flexible objects. An example of a molecule that is so complex that determining whether it is chiral requires the use of topological methods is given in Section 3.3.

Before we move on to a more detailed description of molecular chirality, let us examine the minimum requirements that a group of atoms must fulfil to make up a chiral object in three-dimensional space. Consider four different points scattered in space so that three of them span a plane, and the fourth one is out of that plane (Figure 1.4).

If points are distinguishable, one can order them in a series. In Figure 1.4, district colours are used for this purpose. The same arrangement of points may be seen conceptually from left to right in the figure as vertices of a tetrahedron, points spanning two intersecting planes, and a point on a helix. As long as the connections between points are beyond the scope of considerations, all these three descriptions of a group of points are equivalent. Frequently, depending on the entire structure of the molecule, it is convenient to adopt one of the points of view given above. Specific examples of molecules are given later in the chapter; here, we will examine the nomenclature of chiral molecules in which a tetrahedral part can be easily distinguished. On the one hand, this choice is motivated by their frequent occurrence among biomolecules and, on the other, because it provides handy terminology in the theoretical description of chiral molecules.

The most common naming system for uniquely assigning a descriptor to a molecule was proposed by Cahn, Ingold, and Prelog.⁶¹  The basic terminology of stereochemistry can be found in ref. 62, and only a summary is provided here. The alanine molecule, shown in Figure 1.5, has an α-carbon atom with four substituents that can be ordered from the heavier to the lighter atom or chemical group: NH₂, COOH, CH₃, and H. Different spatial arrangements of these substituents result in alanine occurring in two non-superimposable forms, called enantiomers (enantios, from Greek, meaning opposite). Viewed from the lightest substituent, one can observe a counter-clockwise sense from the heaviest to the lightest for (S)-alanine (sinister, left from Latin), and a clockwise ordering for (R)-alanine (rectus, right from Latin). It is common practice to write descriptors, such as the stereo descriptors R and S, in italics. In Nature, (S)-alanine is much more common than (R)-alanine. In studies of sugars and α-amino acids, commonly a Fischer projection is used in which substituents below the plane of the figure are drawn horizontally while those which are above it are shown vertically.

Based on this projection, a Fischer–Rosanoff convention was introduced, which uses the stereodescriptors L (laevus, left from Latin) and D (dexter, right from Latin) with the arbitrary assignment of d-glyceraldehyde to (R)-2,3-dihydroxypropanal. It follows that l-alanine is a synonym of (S)-alanine, and d-glucose corresponds to (2R,3S,4R,5R)-2,3,4,5,6-pentahydroxyhexanal. For brevity, in the following text, we will refer to enantiomers of a chiral molecule simply as the (R)-enantiomer and (S)-enantiomer when there is no need to refer to a specific molecule.

Flexible molecules can occur in many forms due to rotations along single bonds called rotamers; therefore, usually this kind of rotation is not considered as breaking the mirror relationship between enantiomers unless a hindered rotation is a source of chirality as it is in the case of atropisomers.⁶³  Likewise, precise indication of an electronic, vibrational, and rotational state of a molecule is usually disregarded when chirality is considered from a chemical perspective, even though from the perspective of quantum mechanics one should consider the full wave function of the molecule for its appropriate description. In addition to the (R)-enantiomer and (S)-enantiomer of a chiral molecule, quantum mechanics offers a third possibility: a superposition of the (R)-enantiomer and (S)-enantiomer. An illustrative model of such a superposition is discussed by Hund⁶⁴  and Barron.⁶⁵  Let us consider a slightly modified example. Considering the isotopologue of ammonia, the chiral molecule of ¹⁵N¹H²H³H, one can notice that there is a superposition of states that are interconverted by the umbrella motion, i.e., the electron lone pair up or down with respect to the plane of the oriented triangle formed by the apexes of ¹H, ²H, and ³H. This structure does not have an unambiguously determined sense of chirality, as its structure is formed by the superposition of the equally probable (R)-enantiomer and (S)-enantiomer configurations. More generally, if wave functions of the (R)-enantiomer and (S)-enantiomer are |ψ_R〉 and |ψ_S〉 and parity is conserved, then only their superpositions $1 2 (| ψ R 〉+| ψ S 〉)$ and $1 2 (| ψ R 〉−| ψ S 〉)$ are time-independent states of such a molecule. The reason is that these two superpositions are eigenstates of the parity operator, i.e., $P ˆ (| ψ R 〉±| ψ S 〉)=±(| ψ R 〉−| ψ S 〉)$ ⁠. Consequently, a pure (R)-enantiomer will interconvert into an (S)-enantiomer given a sufficiently long timescale. The time scale of the interconversion is proportional to the energy that the (R)-enantiomer has to overcome in order to reverse its configuration, and for rigid molecules, such a process is too slow to be observable experimentally (see also Section 8.5 and Figure 8.1).

The interconversion of enantiomers (racemisation) is an important aspect in experimental studies of chirality, especially when the time scale of the intramolecular movements is comparable to the time of the measurement. For instance, hydrogen peroxide is a chiral molecule from the point of view of time-resolved optical spectroscopy, but fast interconversion of its enantiomers, i.e., racemisation, results in no influence of its chirality on reactivity visible to the chemist. Consequently, one may view “chirality” as a term describing the behaviour of a specific subset of properties under mirror reflection that is most relevant to the particular experiment or studied system. Following a pictorial analogy, one can fairly reasonably use chirality in the context of a class of objects having a clearly defined mirror image (such as a tetrahedral molecule having four different substituents or shoes), while the term becomes meaningless or at last complicated to handle when applied to objects that formally possess sufficiently low symmetry and are thus chiral, but whose exact mirror images are not easily found in Nature, e.g., an octahedral molecule having eight different substituents or a particular potato.^66–69 

Molecules can be considered as objects consisting of nuclei and chemical bonds joining them, such that one can classify molecules of a given composition (isomers) according to the scheme shown in Figure 1.6. Frequently, their three-dimensional shape may be deduced based on theories like valence shell electron pair repulsion (VSEPR) theory.^70,71 

The molecular formula C₃H₇NO₂ is a representative of many constitutional isomers that differ in the way the atoms are connected. Most of them do not possess a defined relationship to each other since they differ in the carbon skeleton and functional groups (e.g., 1-nitropropane vs. N-(2-hydroxyethyl)formamide shown in the upper left corner of the figure). Some of the constitutional isomers can be grouped into sets of molecules that have the same atom connectivity but different spatial arrangements, i.e., called stereomers. One can classify each pair of stereomers either as enantiomers if they are non-superimposable mirror images of each other or otherwise as diastereomers. Molecules that are diastereomers may be achiral, e.g., (1E)-2-methoxyprop-1-en-1-ol vs. (1Z)-2-methoxyprop-1-en-1-ol shown in the right upper corner of the figure, or chiral. Frequently, in the context of studies of chirality, the term diastereomer is used in the latter sense, especially if there is more than one atom that renders the molecule chiral. This particular atom is known as a stereogenic centre. It is marked by an asterisk in Figure 1.6, e.g., 2-aminopropanoic acid (alanine) and 2-hydroxypropanamide (lactamide), which are shown, respectively, on the left and right sides in the bottom part of the figure have a carbon stereogenic centre called an asymmetric carbon. A sufficient condition for an atom to be a stereogenic centre is that its chemical bonds form a non-planar (commonly tetrahedral) arrangement and that its four substituents (including a lone electron pair, e.g., that of phosphorus in a molecule of hindered pyramidal inversion) are different from each other. While planar molecules, by definition, cannot be chiral, the absence of a plane of symmetry or inversion point does not ensure that a molecule is chiral. The molecules exemplifying this belong to the S_2n point group for n = 2, 4, 6… while the achiral molecule 2,3,7,8-tetramethyl-spiro[4.4]nonane lacks both a plane of symmetry and a centre of inversion (shown in Figure 1.7). The molecule 2,3,7,8-tetramethyl-spiro[4.4]nonane when rotated by a quarter of a full turn and then reflected in a horizontal plane, returns to its initial arrangement of atoms, thus rendering it achiral. However, any of the methyl groups do not transform into each other by a space inversion or any mirror reflection.

Therefore, the strict definition of chirality based on the symmetry operation of the point group of the molecule must include the requirement of a lack of any improper symmetry elements (a combination of a symmetry plane and rotation about an axis) even though examples of achiral molecules without a plane of symmetry and inversion point are extremely rare. It follows that symmetry point groups of the chiral molecules in Schönflies notation^72–75  are cyclic symmetries C₁, C₂, C₃, … , dihedral symmetries D₂, D₃, … , and the three polyhedral groups, T (chiral tetrahedral symmetry), O (chiral octahedral symmetry), and I (chiral icosahedral symmetry). The importance of the symmetry of a molecular structure in determining whether it is chiral or not was emphasised by Prelog in his Nobel lecture in 1975: “An object is chiral if it cannot be brought into congruence with its mirror image by translation and rotation. Such objects are devoid of symmetry elements, which include reflection: mirror planes, inversion centres, or improper rotational axes.”⁷⁶ 

At the level of macromolecules, chirality occurs in protein motifs such as an α-helix, where amino acids are arranged in a right-handed helical structure with a 100° turn in the helix per amino acid residue and a pitch of 1.5 Å.⁷⁷  Another widely known example is DNA: the chirality of a double helix of deoxyribonucleic acid depends on its form: forms A and B of DNA are right-handed while form Z is left-handed.

Allotropes of carbon, the fullerenes that belong to the D₂ point group, are chiral, e.g., C₇₆. Figure 1.8 shows several examples of moderate molecular weight molecules that exhibit structural elements such as a centre (point), an axis or a plane of chirality that are useful in classifying chirality. These elements are an extension of the term “stereogenic centre”. It is worth noting that the number of stereoisomers increases rapidly with the number of structural elements providing chirality to the molecule, which can be clearly exemplified by the bacteriocidal antibiotic boromycin, which has 2¹⁸ = 262 144 stereoisomers.⁷⁸ 

The point in space relative to the four substituents like NH₂, COOH, CH₃, and H in the case of alanine, does not have to be an atom that belongs to that molecule. For instance, if one replaces a carbon atom with a larger structure such as adamantane and substitutes it at three suitably chosen positions, then the point (called a chirality centre) lies roughly at the geometric centre of the molecule, which is not occupied by any atom of the molecule.

The next symmetry element that may be a key-element of the molecular structure providing chirality is a chirality axis (e.g., helical, propeller, or screw-shaped molecular entity) which, in a classical case, is oriented along either the axis of the prolate molecule or the shortest axis of an oblate molecule. These two cases illustrate chiral biphenyl derivatives and helicenes, respectively. For axially chiral molecules, the stereodescriptors used to differentiate between enantiomers are P (or plus) for a right-handed helix and M (or minus) for a left-handed helix. If the details of their structure are ignored, then both molecules resemble the shape of an open book. Similarly, hydrogen peroxide possesses a chirality axis, which coincides with the oxygen–oxygen bond.⁷⁹  Therefore, a primary reason for the difference between the axially chiral enantiomers in a broad sense is a torsional angle (either positive or negative). Just as a different dihedral angle can be a source of chirality, a planar angle may play in some sense an analogous role, which is exemplified in the case of a transoid (BF)O(BF)-quinoxalinoporphyrin, where chirality originates from a bond-angle inversion.⁸⁰  The use of such a planar angle as an indicator of chirality requires the adoption of a convention regarding the sense of rotation in three-dimensional space, e.g., by reference to the molecular skeleton (see Section 1.8). This kind of stereoisomerism is called akamptisomerism. As in the case of axial chirality, where a screw can be taken as a macroscopic object illustrating this type of chirality, in the case of planar chirality, the object to be referred to is a lid of a pot with a handle, provided that for an object to be chiral, either the handle or the lid symmetry is not too high. In particular, planar chirality occurs in molecules having a planar fragment, such as the H–C═C–H moiety of trans-cyclooctene or a substituted phenyl ring of [2.2]paracyclophane, which has a bridge that makes the molecule chiral.

It is not possible to identify a clear source of chirality for all (chiral) molecules, as is the case for twistane (bottom in the middle in Figure 1.8).⁸¹  Structures resembling a closed chain exhibit chirality if their constituents are arranged in a suitable manner.^82,83  Examples include cyclic enantiomers of alanine and molecules containing two or more intertwined rings,⁸⁴  catenanes, having several applications in constructing molecular machines.⁸⁵  Chirality noticeably appears in nanostructures. Examples include chiral nanotubes^86,87  and nanoparticles.^88–96 

Chirality is a property of the entire system under consideration. Crystals can, for instance, be chiral even though they are composed of achiral molecules and vice versa. A crystal may, therefore, be chiral due to either the periodic symmetry of the structure of the crystal lattice, the chirality of the molecules that make up the crystal, or both of these factors.

Several examples of chiral metal–organic frameworks and chiral nanoparticles illustrating this point are given in ref. 97–100. In some cases, the chiral molecules form a chiral crystal, and there is a direct correspondence between the chirality at the molecular level and the chirality at the macroscopic level of the single crystal. This is the case for tartaric acid, whose potassium salt spontaneously forms the so-called “wine diamonds” on the cork or bottom of the bottle. Crystals of the enantiomers of potassium bitartrate are enantiomorphs, i.e., they are non-superimposable mirror images of each other as macroscopic objects.¹⁰¹  In general, the determination of the absolute configuration of the constituent molecules from the absolute structure of a crystal requires a dedicated experimental strategy.¹⁰² 

Chirality, defined as the non-superimposability of the object and its mirror image, can be considered at the level of elementary particles, molecules, and the systems formed by them. In the domain of particle physics, chirality is a quantum mechanical phenomenon that describes a property of a state of particle.^103,104  From a chemical perspective, it is sufficient in almost all instances to focus only on the atomic nuclear network surrounded by electrons that bind to the nuclei. Therefore, the details of the chirality of each particle are generally not considered, and the emphasis is on the non-superimposable framework formed by the molecule’s nuclei. In order to discuss molecular chirality from the point of view of quantum chemistry, one has to translate the geometric concepts used for the mathematical description of chirality into quantum mechanical concepts. In particular, the shape of the molecule is determined unambiguously by the molecular wave function ψ. We will restrict ourselves to the non-relativistic limit. The wave function depends on the position of electrons and nuclei, i.e., ψ(r₁, r₂, … , r_n; R₁, R₂, … , R_N), where the vectors are bolded, small letters denote electrons, and capital letters refer to nuclei. The parity operator $P ˆ$ corresponds to space inversion.

Let us check the effect of the parity operator on the wave function. One could simply write, that since $P ˆ r ˆ =− r ˆ$ ⁠, the resulting wave function is ψ(−r₁, −r₂, … , −r_n; −R₁, −R₂, … , −R_N); however, a clearer insight is obtained if one uses an internal coordinate representation instead of vectors describing the position of the electrons and nuclei. An internal coordinate representation called a Z-matrix is built in a chain-like fashion. The first entity (an electron of a nucleus) is at the origin of the coordinate system, the second is at a distance r₁ from the first, the entity is at a distance r₂ from the first and the first three entities form a planar angle, φ₁. Introducing the fourth entity requires a dihedral (torsional) angle θ₁ that is spanned by the first four entities, and so on. Several examples of application of the internal coordinate representation for small, highly symmetric molecules are described in ref. 105.

The wave function parametrised using an internal coordinate representation takes the form ψ(r₁, r₂, φ₁, r₃, φ₂, θ₁, …; R₁, R₂, Φ₁, R₃, Φ₂, Θ₁, …). Application of the parity operator on this wave function yields ψ(r₁, r₂, φ₁, r₃, φ₂, −θ₁, …; R₁, R₂, Φ₁, R₃, Φ₂, −Θ₁, …). The coordinates, such as distances and planar angles, are not affected by the parity operator since they are independent of the handedness of the coordinate system. This is not the case for dihedral angles. Similar to the tetracube shown in Figure 1.2, the dihedral angles reverse their sign under the parity operator. The observation that the parity operator reverses only those coordinates of the molecular wave function that are dependent on the chosen handedness of the coordinate system applies to both chiral and achiral molecules. In the case of achiral molecules, one can recover the original wave function from the wave function obtained after applying the parity operator by choosing a different sequence of electrons and nuclei in the Z-matrix. Unlike achiral molecules, for chiral molecules, one cannot transform ψ and $P ˆ ψ$ wave functions into each other by simply labelling of constituents of the molecule differently.¹⁰⁶ 

Let us consider a hydrogen peroxide molecule, H¹–O²–O³–H⁴, and consider its dihedral angle, −180° < θ₁₂₃₄ ≤ 180°, formed by atoms numbered from 1 to 4. If the angle θ₁₂₃₄ is 0° (cis-H₂O₂) or 180° (trans-H₂O₂), the molecule is planar and achiral. The spatial inversion of trans-H₂O₂ about a midpoint of the oxygen–oxygen bond exchanges its protons H¹ ↔ H⁴ and oxygens O² ↔ O³. However, it leaves the dihedral angle unchanged since θ₁₂₃₄ = θ₄₃₂₁. In contrast, H₂O₂ is non-planar and chiral; the dihedral angle 119.8° corresponds to its equilibrium configuration.¹⁰⁷  In this case, the dihedral angle does not depend on the order of numbering, i.e., θ₁₂₃₄ = θ₄₃₂₁, but reverses upon spatial inversion.

Usually, it is assumed that the energy of the molecule is not affected by the parity operator, i.e., $〈ψ| H ˆ |ψ〉=〈 P ˆ ψ| H ˆ | P ˆ ψ〉$ ⁠, where $H ˆ$ is the Hamiltonian of the molecule. Despite a strong theoretical basis for the small energy difference between two mirror forms of a molecule, this extremely tiny effect was not confirmed experimentally.^108,109  For instance, quantum chemistry computations indicate that the difference in energy between alanine enantiomers is on the order of 10⁻¹⁶ kJ mol⁻¹;^110,111  thus, the difference in energy caused by electroweak interactions is extremely small compared to the total energy of the molecule.

The violation of parity conservation follows that essentially a “mirror image” of the molecule is not only a mirror image considering its spatial coordinates, but additionally, one has to replace all particles forming the molecules by the corresponding antiparticles.¹¹²  At the most basic level, the laws of Nature, according to the current knowledge, remain invariant under the combined symmetry operation involving the parity operator $P ˆ$ ⁠, charge conjugation $C ˆ$ (replacing all particles by their antiparticles), and time reversal $T ˆ$ (reversing all changes over time) applied consecutively.^113,114  See ref. 115 and 116 for a broader discussion on the importance of the charge, parity, and time reversal (CPT) invariance in physics.

Figure 1.9 shows the parity and charge conjugation symmetries using alanine as an example. The energy of (S)-alanine is exactly the same as the energy of (R)-alanine made of antimatter. These two molecules are CP-enantiomers. The energy of (S)-alanine is very close but not the same as the energy of (R)-alanine made from ordinary matter. This is the case of P-enantiomers (or, in short, enantiomers). Therefore, enantiomers, which are non-superimposable mirror images of each other are only approximately equivalent because they differ in energy. However, for questions concerning molecular structure and chemistry, the approximation that the enantiomers of a chiral molecule are only interconverted by a parity transformation (as opposed to the combined action of parity, charge conjugation, and time-reversal), is completely sufficient.

As mentioned in Section 1.8, in the case of molecules, the impact of parity violation on their properties is too tiny to be observed experimentally; so from the point of view of their studies, left-handedness, and right-handedness are assumed to be arbitrary. More specifically, under this condition, no fundamental physical laws force us to distinguish one of the enantiomers. Both are equivalent, and naming them as, e.g., (R)-alanine or (S)-alanine, is just a label used for identification purposes. Consequently, one needs to refer to some external physical or chemical entity that serves as a reference for the sense of chirality. This situation is similar to the one presented for the tetracubes in Figure 1.2, i.e., the molecule corresponds to the tetracube and the coordinate system is a reference entity. Based on the choice of the reference entity, one can divide methods capable of sensing chirality into those in which information regarding chirality can be obtained directly, i.e., those that allow determining the absolute configuration of the molecule, and those that can distinguish between enantiomers, but where the absolute configuration needs to be inferred. In fact, the borderline between direct and indirect methods lies in the degree of the complexity of the reference rather than a particular law of Nature. Let us look at two extreme cases. A photon is the simplest object that carries information about handedness (in physical context: helicity). Its spin $S ˆ$ can be directed parallel or antiparallel to the momentum vector p, resulting in a pseudoscalar $S ˆ ⋅p$ that is either positive or negative. The difference in the interaction between the electromagnetic fields of differently oriented spins $S ˆ$ relative to the momentum p and a chiral molecule is utilised in chiroptical spectroscopy techniques and X-ray crystallography (vide infra). The system consisting of an electromagnetic field and a particle immersed in it is simple enough to determine its absolute configuration directly. On the other hand, although larger than subatomic particles, objects that are much more complicated than a photon, e.g., macromolecules^117,118  or surfaces of substances, are more difficult to describe theoretically; however, they may prove to be more straightforward in experimental applications. In this case, we consider these methods as indirect rather than direct because determining the absolute configuration involves comparing experimental results with those obtained for chemically (structurally) similar molecules. Although this is not a strict division, as a rule of thumb, it can be said that in direct methods, chirality-sensing is manifested as a change in the phase of the measured signal, and in the case of indirect methods, it is manifested as a change in the position or amplitude of the measured signal.

Since Lord Kelvin first used the word “chiral” 130 years ago, a number of methods have been developed to study the chirality of molecules; among these, techniques using the interaction of electromagnetic radiation with matter have made a leading contribution to the knowledge of chiral molecules. The electromagnetic field depending on the frequency induces various phenomena in molecules; at high frequencies (hundreds of THz) electronic transitions occur, midrange frequencies are associated with the oscillatory structure of molecules, and at relatively low frequencies (up to about a GHz), transitions between nuclear states occur. The high and midrange frequencies cover three major branches of chiroptical spectroscopy: polarimetry, circular dichroism, and vibrational optical activity. The low-frequency response of a studied chiral sample on electromagnetic radiation manifests in NMR. Each of these approaches to seeing chirality provides insight into a different aspect of the structure and dynamics of chiral molecules. For instance, polarimetry is mainly used for the determination of the chiral purity of a sample with no structural information. While vibrational circular dichroism in conjunction with computations can be a powerful technique, it is generally limited to small molecules and those that can be observed in organic solvents. Raman optical activity can be applied in both organic and aqueous environments and for biomolecules, providing the structural information about protein motifs (alpha helices and beta-sheets) and some general insights into sugars. The latter two techniques greatly benefit from reference calculations that can be used to determine the absolute configuration of a chiral molecule. The use of very high-energy electromagnetic radiation, i.e., having an energy of hundreds of eV compared to a few eV in the optical domain, namely, X-ray crystallography, allows us to study periodic chiral materials, although the difficulty of determining the absolute configuration of investigated molecules is strongly influenced by the atomic weights of the constituent atoms.

Alternatively, information about the chirality of a molecule has also been derived from chromatographic separations with a chiral stationary phase.^119,120  Other techniques used to study molecular chirality, currently at the proof of concept stage, are piezoelectric gas sensors,¹²¹  chemical force microscopy,¹²²  and fluorescent molecular sensors.¹²³ 

Most of the direct methods are based on the interaction between an electromagnetic field, typically light, and the sample, typically a solution containing the molecules of interest. Besides the classical studies of homogenous solutions for structure determination and analytical purposes, there is a growing interest in nanophotonics of chiral microscopic objects – see, for example, ref. 124–126.

For instance, methods utilising chiroptical spectroscopy techniques, such as measurements of optical rotation and measurements of absorption (electronic as well as vibrational circular dichroism,^127–130  and rotational spectroscopy¹³¹ ), emission (circularly polarised luminescence), and scattering (Raman optical activity¹³²  and anomalous X-ray scattering) of electromagnetic radiation, belong to this group. Details about chiroptical spectroscopy techniques are given in Chapter 2. However, not all methods that permit the direct determination of molecular chirality are based on the interaction of molecules with the electromagnetic field, as exemplified by Coulomb explosion imaging (CEI)^133–138  and atomic force microscopy (AFM).¹³⁹ 

In the case of CEI, a beam of molecular ions accelerated to a fraction of the speed of light passes through a foil several tens of angstroms thick. On impact, the binding electrons of the molecules are stripped off in at least an order of magnitude less time than the rotation and vibration of the molecule so that the configuration of the atomic nuclei of the molecule remains largely unchanged after passing through the foil. Consequently, the mutual electrostatic repulsion of positively charged atomic fragments results in their dissociation. Next, the potential Coulomb energy is transferred to kinetic energy, and after a flight distance of a few meters, detectors record the positions of the fragments. The spatial distribution of the ion impacts at the detector differentiates between enantiomers, as shown by Herwig et al. for trans-2,3-dideuterooxirane.¹⁴⁰ 

Even more direct evidence of the absolute configuration of a molecule is provided by AFM. Application of a CO-functionalised tip in studies of a diamonide, [123]tetramentane, deposited on the cold Cu(111) surface (T = 15 K) permits the identification of its (P)-enantiomer and (M)-enantiomer through visual inspection (this molecule is shown in Figure 1.1).¹¹  Each of the ways that are used for direct determination of the chirality of a molecule can be, in principle, used indirectly to differentiate between enantiomers by transformation of an enantiomer to a diastereomer, i.e., by chemical reaction with a chiral agent and alternatively by introducing a chiral environment.

Each of the ways that are used for direct determination of the chirality of a molecule can, in principle, also be used indirectly to differentiate between enantiomers by transforming an enantiomer to a diastereomer, i.e., by a chemical reaction or association with a chiral agent, or alternatively by the introduction of a chiral environment. For instance, although normal (non-resonant) X-ray photon scattering is not sensitive to chirality, each enantiomer of the studied chiral molecule when mixed with other chiral molecules of fixed chirality may form crystals with a different structure. For example, the (R)-enantiomer of N-methylamphetamine, a potent central nervous system stimulant, forms crystals with (R)-mandelic acid of the P2₁ space group in the monoclinic crystal system; however, (S)-N-methylamphetamine crystalises with (R)-mandelic acid in the P2₁2₁2₁ space group in the orthorhombic system.¹⁴¹  Therefore, using a chiral agent makes it possible to differentiate between the enantiomers of N-methylamphetamine based on the different positions of reflexes. The same is true for optical and infrared spectroscopy. One could potentially make an optical-rotation strength measurement observing the effect of the presence of a different chiral compound mixed with that being studied, which is the so-called Pfeiffer effect.^142–144 

Enantiomers are routinely discriminated using chromatographic separation with a chiral stationary phase^145–151  and electrophoresis.^152,153  This method is generally empirical and requires a different stationary phase for each studied system. To some degree, one can anticipate the most suitable stationary phase from the chemical similarity between compounds, but in practice, usually, this trial-and-error-based method requires several chromatographic columns and relatively large amounts of compound. Chromatography is especially helpful in separating enantiomers, but it only provides the structure of the compound if the separation parameters are previously known for each enantiomer. Here, also the computation of the relevant molecular interactions is often not straightforward.

Despite the great importance of NMR spectroscopy as a tool in chemical studies for finding or confirming the structure of molecules,^154–157  this branch of spectroscopy, in contrast to chiroptical spectroscopy, cannot distinguish enantiomers of a chiral molecule in a direct way, and consequently, the handedness of a molecule is determined in NMR spectroscopy with empirical rules rather than on the basis of pure physical principles. However, the relationship between NMR spectroscopy and chirality in general terms is rather complicated and, when observed from the perspective of chemical interactions, it may be intractable; therefore, it is convenient to consider it from the point of view of the framework based on physical laws.

One infers about chirality from the NMR spectra of diastereomers obtained using chiral derivatising and solvating agents,^158,159  complexes with chiral molecules,^160,161  and chiral carrageenan gels.¹⁶²  This broad subject is discussed in further chapters of this book: usage of chiral liquid crystals in Section 2.5.4, the application of chiral agents in Chapters 4 and 5, and a xenon-based NMR chirality sensor in Section 6.6.

Let us briefly discuss the need to search for new methods for studying chirality using NMR spectroscopy, which, unlike classical methods that are indirectly sensitive to chirality, would allow inferring about chirality directly.

Among the various interactions between nuclear dipole magnetic moments and a magnetic field, which are the subject of NMR spectroscopic studies, chirality-sensitive are those that are affected by a mirror reflection of the system under study. The description of this dependence may vary greatly depending on the level of the theory, i.e., the assumptions we make. Such description, in full generality, lies far beyond the scope of the physicochemical studies of chirality by NMR because an exact mirror image would also reverse all nuclear properties, e.g., the proton’s magnetic moment would spin in the opposite direction – counter-clockwise in a magnetic field – instead of clockwise as it does in our world. This presumption is experimentally confirmed since the magnetic dipole moment of the antiparticle of a proton – an antiproton – has the same length as a proton but with the opposite sign (see Section 1.8).¹⁶³  Since the subjects of the studies in physical chemistry are systems composed of ordinary matter, let us limit ourselves to the cases in which chirality arises from a suitable spatial orientation of their constituents.

Even if only the spatial arrangement of nuclei that form a molecule is not the same as their mirror reflection, one can theorise the existence of a difference between the energy of a chiral molecule and its non-superimposable mirror image (enantiomer). Although parity violation would allow for direct discrimination between enantiomers in NMR, due to its low energy, all of the experimental attempts to observe the influence of parity violation on NMR spectra have failed, except for several theoretical predictions postulating small differences in chemical shifts of enantiomers.^164,165  The same is true for other experimental methods such as laser,^166,167  Mössbauer,¹⁶⁸  neutron,¹⁶⁹  and rotational¹⁷⁰  spectroscopy techniques.

Consequently, it is usually assumed in NMR that the energies of the enantiomers are the same, and the differentiation methods are based on the transformation of the enantiomers into diastereomers. Enantiomers can be transformed to diastereomers directly (forming a covalent bond) through chemical reaction with a chiral chemical compound or indirectly (utilising non-covalent interactions) by forming solvates with a chiral reagent or by placing the molecules in a chiral environment, e.g., a chiral liquid crystal. In this case, the interaction energy resulting from the presence of the chiral auxiliary substance is on the order of a few µJ, which is three orders of magnitude lower than in the case of chemical shifts observed in typical NMR measurements and is usually easily detectable. The disadvantage of this approach is the lack of rules that unambiguously allow us to determine which enantiomer has a given absolute configuration. An alternative to indirect methods is provided by a series of predicted effects described in this book in detail that are related to an electric field called NMER (nuclear magnetoelectric resonance) spectroscopy.^171–178  NMER is a branch of liquid-state NMR spectroscopy that utilises the permanent electric dipole moment of a chiral molecule and antisymmetric nuclear interactions, which cannot be used in the case of chiroptical spectroscopy since the electric dipole moment cannot follow the electromagnetic field at frequencies significantly higher than 10 GHz. The range of energies expected for NMR chirality-sensitive signals is about one-hundredth of the energy observed in the case of indirect methods; thus, these signals fall within the range of experimental detection. Unlike indirect NMR methods, the influence of chirality on NMER spectra manifests as the phase shift of the signals, like in chiroptical spectroscopy studies.

One of the arguments for introducing direct methods into NMR spectroscopy is to shorten and facilitate the determination of the absolute configuration of a molecule. To make this argument meaningful for an experimenter, one must also add that the signal directly recorded in the experiment, even in the case of chiroptic measurements, does not immediately provide information about the absolute configuration. In particular, the positive or negative phase signal is not assigned unambiguously to the (R)-enantiomer or (S)-enantiomer. Theoretical computations of at least a pseudoscalar sign characteristic of a given chiroptic method serve as a bridge, enabling the linkage of the measured quantity with the structure. Therefore, the measurement and the method of calculating the pseudoscalar must be feasible and satisfactorily simple. Achieving this level of usability is usually a challenge. Such theoretical and experimental difficulties usually increase rapidly with the size of the molecule being studied and the complexity of the system under consideration.

Another, perhaps more critical, argument for the need to advance NMR spectroscopy toward direct methods of studying chiral molecules is the expansion of the scope of information obtained. The range of information obtained from the chirality-sensing experiment matters because efficient methods for chiral discrimination are of considerable importance for medical and pharmaceutical studies, and the Food and Drug Administration now requires monitoring the handedness of a chiral drug molecule throughout the entire production process. This is particularly important in the case of in situ biomolecular studies and the quality control of medical products. Therefore, efficient methods of chiral discrimination can have some long-term effects on the quality control of industrial processes, such as food processing. Considering the large and rapidly growing scope of this field, one can conclude that new chirality-sensing methods are highly desirable for academic purposes, and their successful results will offer significant practical implications.

Moreover, the indirect methods are specific to a particular target molecule or a class of molecules. They are also time-consuming, and in the case of derivatisation, the recovery of the chiral analyte is difficult. Therefore, the exploration of other direct ways to solve the issue of chirality determination with NMR spectroscopy could be very advantageous. The introduction of these methods may permit the determination of the interatomic connectivity, the three-dimensional structure, and the chirality of a molecule and, therefore, fully characterise the structure of the molecule by using the measurements performed using only one kind of spectroscopy.

Over a century of studies on chirality have revealed many aspects beyond the often-referenced definition of molecular chirality as the property of non-superimposability of mirror images of a molecule. From the point of view of particle physics, Nature has a preference between being left-handed or right-handed when weak interactions are considered. This fact is of great importance for understanding the fundamental laws of Nature, particularly, serving as one of the foundations of particle physics. However, it has limited application in the field of spectroscopy because parity violation occurs in molecules to a minimal extent. The description of chirality using quantum chemical methods, e.g., VSEPR and related theories, enables identification of key symmetry elements of a chiral molecule. Some limitations of this perspective are that molecules are not rigid, and most of them have only local symmetries, or their symmetry is only approximate. Instead, topological methods can be used for molecules whose shapes deviate from group theory. The quantum view also provides a slightly different view of the enantiomer compared to the classical physics approach. Just as the simultaneous knowledge of the momentum and position of a particle is limited according to Heisenberg’s principle, determining the enantiomeric state of a molecule and the time interval during which it is most likely to be in such a state is not independent. On the other hand, in practical cases, the racemisation of molecules occurs in many ways, so a simple model of the ammonia isotopologue does not fully reflect the experimental situation (especially in solution), where the fraction of states with a well-defined parity is typically small.

Despite the advantages and limitations of various perspectives on chirality, the impact of chirality on the properties of molecules is clearly visible, and almost the entire electromagnetic spectrum can be used for chirality sensing. In NMR spectroscopy, indirect methods for studying chiral molecules are widely used, which are described in Chapters 4–6 and 10. However, from a general perspective, NMR spectroscopy is the only branch of spectroscopy that is not directly sensitive to chirality, which motivates the search for new methods, such as NMER, discussed in Chapters 7 and 8 of this book, and TWENMR described in Chapter 9.

I would like to acknowledge Prof. Laurence Barron (Glasgow University) for his comments regarding the physical principles of molecular chirality, Prof. Peer Fischer (Max Planck Institute for Medical Research in Heidelberg), Prof. William S. Price (Western Sydney University), Prof. Marek Orlik (Faculty of Chemistry, University of Warsaw) for their general comments, Prof. Jan Romański and Prof. Anna Piątek (Faculty of Chemistry, University of Warsaw) for proofreading structures, names, and formulas of compounds listed in the chapter, and Dr Grzegorz Łach (Faculty of Physics, University of Warsaw) for helpful discussions about chirality in the context of elementary particles interactions. Moreover, I would like to acknowledge the European Research Council for the financial support through the ERC Starting Grant (project acronym: NMER, agreement ID: 101040164).