Symmetry as a Tool for Solving Chemical Problems^#

Abstract

Symmetry is found all around us. It is a fundamental concept in the arts as well as in the sciences. In chemical reactions, the use of reagents and catalysts with rotational symmetry decreases the number of transition states, a situation that may lead to increased selectivity. The presence of symmetry facilitates strucure determinations, and symmetry arguments may be helpful for elucidating mechanisms and for gaining insight into dynamic molecular processes.

1. Introduction

Symmetry plays an important role in many human activities, including the arts and the sciences. The Greek philosopher Plato assumed that the world consists of elements made from the five regular platonic solids, representing the four classical elements earth, air, fire and water, and the universe. In the Elements, Euclid around 300 BC gave a mathematical description of the solids. The concept of symmetry was more than 2000 years later rationalized by the French mathematician Évariste Galois, who developed group theory,¹ a tool still indispensable not only to mathematicians, but also to chemists for relating the symmetry of a molecule to its physical properties.²

Symmetry is often connected to beauty and to harmony and proportions.³ Although asymmetry may be explored in the arts for the purpose of avoiding rigidity,⁴ symmetry is commonly used as an aesthetic element.⁵ But symmetry considerations also play a vital role in diverse areas to help understanding and for facilitating design as well as for solving scientific problems. In architecture, symmetry can be exploited for the analysis and construction of models⁶ and employed strategically in the design process.⁷ Symmetry has also found use in mathematics as a problem-solving tool,⁸ and in his lectures Richard Feynman emphasized the importance of symmetry in physical laws.⁹ In fluid physics rotational symmetry arguments can be used to reduce the number of independent variables, and symmetry can help to understand the properties of materials, such as nanoscopic magnetic clusters with complex structures.¹⁰

Symmetry considerations have been involved in chemistry ever since its beginning,¹¹ and still today symmetry arguments play a vital role in many domains of chemistry.^12,13 Symmetry considerations are essential in structural determinations, for the prediction of electronic transitions, in the interpretation of crystallographic and spectroscopic data, for the construction of molecular orbitals, and for explanations of chemical reactivity;¹⁴ the concept of the conservation of orbital symmetry emerged some 50 years ago as a powerful theory, instrumental for explaining why certain concerted reactions are allowed while others are not.¹⁵

Symmetry can also serve as a tool to enhance the selectivity in the synthesis of chemical compounds as well as to simplify their analysis. It is well recognized that the incorporation of rotational symmetry, the only kind of symmetry compatible with chirality, in ligands and metal complexes can be exploited in organic synthesis and catalysis as a means to increase the enantioselectivity in chemical reactions.¹⁶

The purpose of this review is to give some examples of how symmetry can be utilized for solving chemical problems.

2. Exploiting Symmetry in Asymmetric Catalysis and Chiral Recognition

2.1 Asymmetric Catalysis

Enantiopure chiral compounds are indispensable for use as active ingredients in e.g. pharmaceuticals and insecticides, as well as components in e.g. optical and electronic devices. An overwhelming number of methods are today known allowing the preparation of non-racemic compounds, often with complex structures. Due to the beneficial properties of reagents and catalysts with rotational symmetry in enantioselective synthesis and chiral recognition, many synthetic and separation methods are based on chiral catalysts with rotational axes. These beneficial properties are a result of a reduction of the number of competing interactions between substrates and reagents and, in metal complexes, the number of non-equivalent coordination sites as compared to reagents or catalysts lacking rotational symmetry (Figure 1). C₂-Symmetric compounds, in particular, have been widely used in synthesis and catalysis,¹⁷ and many of the so-called privileged ligands have two-fold symmetry.¹⁸ An increasing number of C₃-symmetric ligands and metal complexes have also been successfully prepared and applied in catalytic reactions.¹⁹

2.2 Chiral Recognition

Hampered by the idea that C₃ symmetry would be incompatible with chiral recognition, proofs of the efficiency of receptors with threefold symmetry were needed before the concept was fruitfully developed.²⁰ Today several examples of highly selective C₃-symmetric catalysts and receptors are known. The cage-shaped Lewis acidic boron compound 1 (Chart 1), obtained as a separable mixture of a pair of diastereomers with helical P and M chirality, exhibited chiral recognition of primary amines and a sulfoxide.²¹ The boron compound also served as an enantioselective Lewis acid catalyst for a hetero-Diels Alder reaction.

Cage compound 2 (Chart 2) is a recent example of an efficient receptor for chiral ammonium neurotransmitters, such as (1R, 2S)-ephedrine, (1R, 2S)-norephedrine, and l-adrenaline, with binding constants up to 10⁵ M⁻¹, whereas low or no binding was observed for the opposite enantiomers.²² A conformational change from a globular to an imploded structure occurred upon protonation of the three secondary nitrogen atoms, which allowed control of uptake and release of the guest.

In both cases, the threefold symmetry of the host reduces the number of possible interactions with the guest, as compared to a host lacking symmetry elements, a fact that may contribute to successful results (Figure 2).

But symmetry can not only be used for improving the selectivity in enantioselective reactions, chiral recognition, and separation of enantiomers, but also for facilitating structure elucidations, mechanistic interpretations, and for studies of dynamic molecular processes.

3. The Horeau Principle

Although powerful methods for the preparation of highly enantioenriched compounds are available today, still many catalytic enantioselective reactions do not provide the high level of enantiopurity required for a variety of applications. For particularly challenging processes, methods whereby the enantiomeric excess can be improved without requiring replacement of reagents or catalysts are therefore important. For this purpose, it is occasionally possible to take advantage of the symmetry properties of molecular structures.

In 1973 Horeau and co-workers devised a method for enhancing the enantiopurity of a scalemic mixture of a compound by linking the compound to a bifunctional reagent.^23,24 This results in a mixture of the RR, SS, RS, SR isomers, the two latter being identical or different depending on the structure of the linker (Figure 3). The heterochiral isomers can be separated from the homochiral isomers using standard separation techniques. With S being the minor enantiomer, cleavage of the homochiral compounds provides a mixture enriched in R as compared to the original mixture. From the amounts of homochiral and heterochiral compounds, the enantiomeric purity of the original mixture can be determined.²⁵

The two groups do, however, not always react independently and therefore non-statistical distributions of isomers are occasionally observed, resulting in gain or loss of selectivity. Thus, reduction of diketones 3 with LiAlH₄ did not give rise to the expected 1:2:1 ratio of isomers; instead 1,2- and 1,4-ketones gave a mixture of isomers with an excess of the heterochiral isomers, whereas the contrary was true for 1,3- and 1,5-ketones, the homo- to heterochiral ratios being 17:83, 79:21, 24:76 and 60:40 for n = 0, 1, 2 and 3, respectively,²⁶ a circumstance that may lead to increased- or decreased-selectivity in enantioselective reactions with bifunctional substrates. This nonstatistical distribution of isomers was suggested to be the result of the sterically preferred mode of hydride attack on the complexes formed after reduction of the first carbonyl group (Figure 4).

The Ugi four-component reaction converts an aldehyde, a carboxylic acid, an amine, and an isocyanide to an α-acetamidoamide.²⁷ Access to Ugi products with high enantiopurity is a challenge. In order to enhance the selectivity, bifunctional linkers equipped with one of the reaction components (either 4 or 5) were used together with the three remaining components in reactions catalyzed by chiral phosphoric acids (Scheme 1).²⁸ Products with 96–99% ee were obtained after hydrolysis of the dimers. In comparison, more modest enantioselectivities were observed in analogous reactions comprising only monomeric reagents.²⁹ In spite of the formation of heterochiral isomers, the yields obtained were still good (69–78%) – and products with high purity were obtained.

This phenomenon can also be taken advantage of in enantioselective reactions of symmetrical compounds with two or more prochiral groups residing in the same substrate, but the distribution of isomers can be difficult to predict. Reduction of the keto groups in dipyridylmethane derivatives 6a and b (Chart 3) may result in four isomers, two homochiral, S,S- and R,R, isomers, and two meso, RrS and RsS, isomers. Reduction using NaBH₄ gave close to the statistical distribution of isomers for 6a (ca 44:56 ratio of homo- and heterochiral isomers), whereas for 6b heterochiral isomers were largely favored (ca 20:80 homo- to heterochiral isomers). Enantioselective reduction of the dipyridylmethane derivatives using (−)-(S)-chlorodiisopinocampheylborane, (−)-Ipc₂BCl, should give a 90.25:4.75:4.75:0.25 ratio of isomers, assuming a 95:5 ratio of the isomers from reduction of the first keto group,³⁰ and thus a significant upgrade of the enantiopurity to ca 99.7:0.3. Experimentally, the meso isomers were not detected in the reaction mixtures.³¹ Thus, whereas the complex formed after reduction of the first keto group with NaBH₄ favored a product with opposite configuration in the second step, reduction using the chiral reagent favored the homochiral product, implying that the actual reagent for the second step was more selective than (−)-Ipc₂BCl. From the corresponding tripyridylmethane derivative, only the R,R,R- and R*,R*,S*-isomers were detected, in a ratio of 97.5:2.5, demonstrating a considerable gain in selectivity as compared to a monomeric substrate.³²

In the oxidation of C₂- and C₃-symmetric sulfides, a significant Horeau-type amplification of the selectivity was observed for several substrates (Scheme 2).³³ Instead of the anticipated 1:3 ratio of homochiral to heterochiral isomers, a ratio of 3.1:1 and an ee of 99% were experimentally observed for the sulfide 7 with Ar = 2-naphthyl, and an even higher ratio, 3.7:1 for the sulfide with Ar = 1-naphthyl.

The methodology has also been applied to a double kinetic resolution of racemic anti-1,3-diols (R,R- and S,S-8), which are important motifs in natural product synthesis in enantiopure form.³⁴ As a result of the C₂ symmetry of the substrate, a single monoacylated product, which is the sacrificial product, was obtained (Scheme 3). Since acylation of the R-center is unfavored, the R,R-diacylated product was not observed. This methodology was applied to several other anti-diols as well.

Finally, both enantiomers of a macrocyclic C₃-symmetric compound possessing helical chirality by virtue of the orientation of the tiophene rings, 9 (Chart 4), were recently prepared in enantiomerically pure form (>99% ee).³⁵

Product 9 was obtained by initial trimerization of a thiophene derivate with vinyl and triazolyl substituents, catalyzed by Rh(II) paddlewheel complex 10,³⁶ which gave the desired product along with a minor diastereomer, with one of the cyclopropane rings located on the opposite side of the triangular plane, and some undesired oligomers (Scheme 4), and final reduction. Out of a total of four stereoisomers, only two were obtained, the minor isomer being separated by standard purification.

4. Symmetry in Structure Determinations

Symmetry considerations play a vital role in structure determinations of chemical compounds. In crystallography, symmetry is used to characterize crystals, identify repeating parts in the crystal, as well as to simplify data collection and calculations.³⁷ In electron spectroscopy, the electronic transitions observed are governed by the symmetry of the orbitals involved, in IR and Raman spectroscopy selection rules help identifying allowed vibrational modes. In NMR spectroscopy, chemical and magnetic equivalence as a result of the symmetry properties of the investigated molecule serves as a valuable guide in the interpretation of spectra.

Chirally flexible, tropos, ligands and metal complexes have found widespread use in asymmetric catalysis. Ligands containing two identical, symmetrically situated, flexible units can adopt either C_s or C₂ symmetry, in the former case leading to a single meso structure, in the latter case to two enantiomeric structures. ¹H NMR spectroscopy was used to determine the structure of η³-allyl palladium complexes of this type ligands (Scheme 5).³⁸ With an achiral anion Q⁻, such as PF₆⁻, the protons in the C_s-symmetric conformer 11 are pairwise enantiotopic, except those residing in the mirror plane. Two complexes, with endo and exo allyl moieties, may in principle form, but a single major complex was observed experimentally along with a small amount of a syn,anti-allyl isomer. In contrast, due to the twofold axis in the C₂-symmetric structure 12, each enantiomer gives rise to a single complex, with all protons in each enantiomer being non-equivalent. A symmetric spectrum, with pairwise identical protons, was observed at room temperature, a situation compatible with the C_s structure.

In order to verify this conclusion, PF₆⁻ was exchanged for a chiral non-racemic anion, Δ-BINPHAT (13). In the presence of the chiral anion, enantiotopic protons become diastereotopic and are thus expected to split whereas achirotopic, i.e. those residing in the mirror plane, would not. In contrast, in the chiral C₂-symmetric structure all protons are chirotopic and thus would split. The first situation was observed, thus verifying the meso structure. This observation was of importance in connections to mechanistic studies involving chirally flexible ligands.³⁹

5. Symmetry in Mechanistic Studies

Much interest has been devoted to the isomerization of Dewar benzene (14) to benzene. The highly strained compound, first prepared in 1963, has a surprisingly high kinetic stability, with a half-life of about two days at room temperature.⁴⁰ The Woodward Hoffmann rules dictate that isomerization proceeds by a conrotatory ring-opening via the highly strained C₂-symmetric cis,cis,trans-benzene, or Möbius benzene, (15, Scheme 6) but due to the improbable existence of 15, it was for a long time universally assumed that ring-opening occurs by symmetry-forbidden disrotatory opening to provide benzene directly. Early calculations showed, however, that the transition state (TS) for disrotatory opening is indeed higher in energy than that for conroratoty opening, and a shallow minimum corresponding to 15 was identified about 100 kcal/mol above benzene.⁴¹ Cis-trans isomerization then transforms 15 to benzene, with a low barrier. Later calculations have confirmed that 14 indeed isomerizes to benzene via 15.⁴² The early calculations also supported the existence the trans isomer of Dewar benzene, 16, which has a half life time of merely 0.2 ms at 25 °C as a result of facile symmetry-allowed disrotatory ring-opening to benzene.⁴¹ That the seemingly so simple disrotatory ring-opening of 14 to benzene does not occur demonstrates the power of the orbital symmetry rules.

Analyses of the symmetry of molecular orbitals are useful also in other types of mechanisms, such as S_N2 and E2 reactions.⁴³ Using symmetry arguments and molecular orbital theory, the reactivity as well as configuration of many reactions can be explained, principles that are today used in undergraduate textbooks Scheme 7.

6. Symmetry Applied to Molecular Dynamics

Symmetry considerations can sometimes also be used to gain insight into dynamic processes. In order to determine whether tropoinversion of the biphenyl units in Pd complex 17 (Scheme 8) occurs while the ligand is coordinated to the metal or whether decoordination–recoordination is required for inversion to occur, the Pd(II) chloride complex of ligand 18, with time-averaged C_3v symmetry, was selected as a suitable model for the studies, performed by ¹H NMR spectroscopy; the different size of the chelate rings in the two complexes was considered not to affect the results.⁴⁴

Pd(II) prefers square planar coordination, and one of the three ligand arms does therefore not take part in coordination to the metal. Several dynamic processes needed to be taken into consideration. These included decoordination–recoordination, tropoinversion of coordinated and non-coordinated ligand arms, and nitrogen inversion in the coordinated ligand arms, whereas nitrogen inversion in the non-coordinated arm and conformational change in the seven-membered rings were assumed to be rapid on the NMR time-scale under the conditions of the experiment (room temperature). With these assumptions, i.e. with four dynamic processes, each of which can, on the NMR time scale under the conditions of the experiment, be rapid or slow, in principle sixteen cases needed to be considered (four processes, each of which can be slow or rapid). However, rapid metal-ligand exchange renders the three ligand arms equivalent, thus excluding different rates of tropoinversion in the three arms as well as slow nitrogen inversion at any coordination site. Furthermore, in a situation with slow metal-ligand exchange, it is not reasonable to assume that tropoinversion is rapid in the coordinated arm and slow in the free unit. This leaves eight cases, each with its expected ¹H NMR pattern, originating from a single or a mixture of complexes (for details see SI, ref. 44).

The experimentally observed signals originating from the benzylic methylene protons consisted of four doublets of equal intensity and one singlet with twice the intensity of each doublet (H_a = H_a′, H_b = H_b′, H_c = H_c′, H_d = H_d′, and H_e = H_e′ = H_f = H_f′). This result is compatible only with a situation where a single complex with time averaged C_s symmetry is observed on the NMR time-scale under the conditions of the experiment, where tropoinversion is rapid in both coordinated and non-coordinated groups but where decoordination–recoordination and ligand exchange are slow. The conclusion of this experiment is thus that decoordination of palladium is not required for tropoinversion to occur.

Triggered by the question whether it would be possible to prevent rotation around the threefold axis in hexaethylbenzene chromiun tricarbonyl, the dynamic behavior of the complex was studied with the aid of NMR spectroscopy.⁴⁵ The rate of rotation of the Cr(CO)₃ group is of interest since it has been suggested that the site of nucleophilic attack on the arene ring is controlled by the orientation of this group. Variable-temperature NMR spectroscopy was used in order to determine whether rotation around the threefold axis in the chromium compound is rapid or slow on the NMR time scale. In hexaethylbenzene alternate ethyl groups lie above and below the plane of the arene ring (19a, Figure 5), and the molecule has thus D_3d symmetry with all ethyl groups being NMR equivalent. In the chromium tricarbonyl derivative 19b, the symmetry around plane of the aromatic ring is broken, and the complex has time-averaged C_6v symmetry under conditions where rotation of the ethyl and CO groups is rapid on the NMR time scale, whereas at low temperature rotation of the ethyl groups is slow, resulting in two sets of ethyl groups, and consequently C_3v symmetry. These observations do not, however, provide information about the rotation of the Cr(CO)₃ group, since the complex has C_3v symmetry independent of the rate of rotation of the Cr(CO)₃ group. Additional desymmetrization was needed in order to solve the problem. Replacing one CO group with a CS group (19c) results in a structure with C_s symmetry. In case rotation of the Cr(CO)₃ unit is slow, the ethyl groups give rise to a 2:2:1:1 pattern of signals, which was indeed observed.

Further desymmetrization, achieved by replacing one of the remaining CO groups with NO results in a structure with C₁ symmetry with all ethyl groups different (19d). Having confirmed that the solution conformer, as well as that in the solid state, has alternating ethyl groups, the observation of a 1:2:1:2 pattern from the ring carbon atoms in the C_s-symmetric complex and an 18-line ¹³C NMR spectrum from the C₁-symmetric complex, finally solved the problem and showed that tripodal rotation is inhibited.

7. Conclusions

In this account some examples aimed at demonstrating the important role of symmetry arguments in chemistry have been collected. In addition to the benefits of rotational symmetry of reagents and catalysts, symmetry considerations can be of significant help in structure deteriminations, for insight into dynamic processes, for mechanistic studies, and for gaining selectivity in chemical reactions.

References

Christina Moberg

Christina Moberg is emeritus professor of organic chemistry at KTH Royal Institute of Technology in Stockholm. Her research interests are devoted to the development of organic synthetic methodology employing homogenous catalysis. Special interests concern the role of symmetry in asymmetric reactions, the design of self-adaptable ligands and the use of interelement compounds as synthetic tools. Her present interest is focused on recycling dissipative networks.
(Photo: Markus Scholz, copyright Leopoldina Academy)

Author notes