On Chirality and the Universal Asymmetry: Reflections on Image and Mirror Image

On Chirality and the Universal Asymmetry Reflections on Image and Mirror Image

V HCA

On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

On Chirality and the Universal Asymmetry Reflections on Image and Mirror Image Georges H. Wagnière

V HCA Verlag Helvetica Chimica Acta · Zürich

Prof. Dr. Georges H. Wagnière Physikalisch-chemisches Institut Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich Switzerland

This book was carefully produced. Nevertheless, author and publishers do not warrant the information contained therein to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details, or other items may inadvertently be inaccurate.

Published jointly by VHCA, Verlag Helvetica Chimica Acta, Zürich (Switzerland) WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim (Federal Republic of Germany)

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Die Deutsche Bibliothek – CIP-Cataloguing-in-Publication-Data A catalogue record for this publication is available from Die Deutsche Bibliothek ISBN-10 3-906390-38-1 ISBN-13 978-3-906390-38-3

© Verlag Helvetica Chimica Acta, Postfach, CH–8042 Zürich, Switzerland, 2007 Printed on acid-free paper. All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printing: Konrad Triltsch, Print und Digitale Medien, D-97199 Ochsenfurt-Hohestadt Printed in Germany

Preface This book deals with the ubiquitous phenomenon of chirality and the distinction between image and mirror image. It is neither aimed at being a textbook nor a specialized scientific review. Rather, it is to be considered as an excursion through nature and the universe and an attempt at gaining an overview over the vast fields and expanses where chirality is encountered. The author being a physical chemist, the focus of the book is essentially molecular. The chapters of the book that deal with elementary particle physics, with astrophysics and cosmology, with biochemistry and biology, are consequently written by a nonspecialist who has also had to rely on secondary literature or on primary sources outside his specialty. If any errors have thereby crept in, they are exclusively the responsibility of the user of these sources. In general, the choice of scientific references is subjective, sometimes also based on pedagogical considerations and depending on immediate accessibility. An effort has been made to consider references that are as recent as possible. But where a given historic development has appeared particularly interesting and instructive, older references are also included. It is hoped that this text will be accessible to anyone who has had the equivalent of a first-year university instruction in physics and chemistry. Some elementary knowledge of crystallography will be helpful. On account of the variety of particular subjects, the focus differs from one chapter of the book to the next, as here indicated: Chapt. 1: Stereochemical and biochemical Chapt. 2 and 3: Physical Chapt. 4: Structural and physical; Sect. 4.1 and 4.2 contain some central ideas on symmetry Chapt. 5: Astrophysical Chapt. 6: Geophysical Chapt. 7: Structural, chemical, and physical Chapt. 8: Structural, chemical, and physical Chapt. 9: Structural and biochemical Chapt. 10: Astro- and geophysical; structural and biochemical This book needs not to be
linearly read from A to Z. Depending on inclination and taste, the reader may skip certain chapters, while concentrating on others. Mathematical equations may be overlooked. They are not essential to obtain a general view of the phenomenon and its significance. A glossary of scientific terms at the end of the text should be helpful for the nonspecialized

VI

Preface

reader. Although all aspects of chirality are related to each other in some way, there is no preset sequence to be followed. For excursions through nature, different trails may be chosen. The subject of natural optical activity and of chirality, discovered about two centuries ago, is presently as topical as ever. The writing of this book was a longheeded plan. It was stimulated by the possibility to participate in the research initiative
Chiral-2 on enantioselective chemistry of the Swiss National Science Foundation. I also much appreciate and herewith gratefully acknowledge the recent hospitality of the High Magnetic Field Laboratory in Grenoble, France, and the fruitful collaboration with Geert L. J. A. Rikken on the magnetochiral effect and related optical phenomena. I thank Andr
Dreiding and Karl-Heinz Ernst for drawing my attention to particular aspects of molecular chirality, as well as Bart van Tiggelen and Gnther Rasche for instructive conversations on symmetry laws in physics. Much inspiration was also given me by other colleagues and former students, most recently by Stanisław Woz´niak, Peter Kleindienst, and Nikolai Kalugin. I thank Barbara Huber who told me about the detective story of
The Documents in the Case by Dorothy L. Sayers, in which the mystery of a poisoning with muscarine was solved by a polarimetric measurement (Sect. 1.5). I am grateful to my wife Marie-Louise for her kind patience while I was writing this book, and for pointing out the Th
rse de Dillmont reference on lace and embroidery (Sect. 10.1). A book needs not only to be written. To present it to the reader, it must also be edited and printed. I am extremely grateful to M. Volkan Kisakrek for having outstandingly carried out this task. My thanks also go to Richard J. Smith and Thomas Kolitzus for their excellent assistance and counsel, and to Peter M. Wallimann for his careful technical support. Zurich, January 2007

Georges H. Wagnire

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V 1

1. 1.1. 1.2. 1.3. 1.4. 1.5.

Image and Mirror Image in Molecules . . . . . . . . . . . . . . . . . . . . . . . The Homochirality of Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Discovery of Natural Optical Activity . . . . . . . . . . . . . . . . . . . Chirality and the Birth of Stereochemistry . . . . . . . . . . . . . . . . . . . Absolute Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymmetric Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 6 8 12 16

2. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 2.9. 2.10.

The Violation of Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Through the Looking Glass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parity and the Laws of Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Time Reversal and the CPT Theorem . . . . . . . . . . . . . . . . . . . . . . . The Selectivity of the Weak Forces . . . . . . . . . . . . . . . . . . . . . . . . . . Parity Violation in a Nuclear Reaction . . . . . . . . . . . . . . . . . . . . . . . Parity and Selection Rules in Atoms . . . . . . . . . . . . . . . . . . . . . . . . The Violation of Parity in Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . The Violation of Parity in Molecules . . . . . . . . . . . . . . . . . . . . . . . . The Interconversion of Enantiomers . . . . . . . . . . . . . . . . . . . . . . . . From Meson Decay to Molecular Homochirality . . . . . . . . . . . . . .

23 23 25 28 31 33 35 38 41 44 46

3. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8.

Light, Magnetism, and Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electromagnetism and Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Interaction of Light with Molecules . . . . . . . . . . . . . . . . . . . Natural Optical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Optical Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Magnetochiral Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Magnetochiral Effect in an Atom . . . . . . . . . . . . . . . . . . . . . . . The Magnetochiral Effect in Molecular Fluids . . . . . . . . . . . . . . . . Magnetochiral Photochemistry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49 49 54 57 59 60 63 68 71

4. 4.1. 4.2. 4.3. 4.4. 4.5.

The PT Triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Triangle of P and T Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . The Inverse Magnetochiral Birefringence . . . . . . . . . . . . . . . . . . . . The Magnetochiral Effect in Electric Conduction . . . . . . . . . . . . . Magnetochirodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 78 80 82

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Contents

5. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7.

Journey into Outer Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Remarks on the History of Science . . . . . . . . . . . . . . . . . . . . . . . . . . The Evolution of the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Birth and Death of Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Is There a Temporal Evolution of the Basic Symmetries? . . . . . . . Galactic and Intergalactic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . Stellar and Galactic Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . The Fractal and Chiral Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85 85 88 91 92 94 95 97

6. 6.1. 6.2. 6.3. 6.4.

Return to Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Chaotic and Chiral Solar System . . . . . . . . . . . . . . . . . . . . . . . . Asymmetry on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chaos in the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geological and Mineralogical Chirality . . . . . . . . . . . . . . . . . . . . . .

101 101 104 106 110

7. 7.1. 7.2. 7.3. 7.4. 7.5.

Chirality at the Nano- and Micrometer Scale . . . . . . . . . . . . . . . . . On Chiral Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chirality in Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observing Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115 115 120 123 125 128

8. 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11.

Chiral Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Use of Models for the Description of Nature . . . . . . . . . . . . . . Dissecting a Cube and a Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Dihedral Angles to the Classification of Knots . . . . . . . . . . . From Ligand Partitions to Chirality Functions . . . . . . . . . . . . . . . . Sector Rules to Interpret CD and ORD Spectra . . . . . . . . . . . . . . . From Helices to Mçbius Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coupled and Asymmetric Classical Harmonic Oscillators . . . . . . An Early Quantum-Mechanical Model . . . . . . . . . . . . . . . . . . . . . . Are There Absolute Measures of Chirality? . . . . . . . . . . . . . . . . . . On Multipole Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetochiral Scattering of Light . . . . . . . . . . . . . . . . . . . . . . . . . . .

133 133 134 137 142 147 150 154 157 158 161 163

9. 9.1. 9.2. 9.3. 9.4. 9.5. 9.6.

Pathways to Homochirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chiral Intermolecular Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . Enantioselectivity at Phase Boundaries . . . . . . . . . . . . . . . . . . . . . . Enantioselective Adsorption on Chiral Surfaces . . . . . . . . . . . . . . . Chiral Chromatography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enantioselectivity by Achiral Chromatography . . . . . . . . . . . . . . . Enantiomeric Excess by Chiral Catalysis . . . . . . . . . . . . . . . . . . . . .

167 167 171 173 175 176 177

Contents

IX

9.7. 9.8. 9.9. 9.10. 9.11. 9.12.

Homochiral Polymerization: Oligopeptides . . . . . . . . . . . . . . . . . . Homochiral Polymerization: Polypeptides . . . . . . . . . . . . . . . . . . . . Biopolymer Generation on Mineral Surfaces . . . . . . . . . . . . . . . . . Selectivity in Biopolymer Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . Homochiral Organic Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Many Ways Lead to Rome? . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 182 184 186 190 192

10. 10.1 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9.

Prebiotic Evolution and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is Life? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Primary Origin of Absolute Enantiomeric Excess . . . . . . . . . Secondary Sources of Absolute Enantioselectivity . . . . . . . . . . . . . Prebiotic Evolution: From the Deep Sea to the Hadean Beach . . Enantiomeric Excess by Surface Geology and by Sunlight . . . . . . Prebiotic Evolution: Extraterrestrial Origins . . . . . . . . . . . . . . . . . Prebiotic Evolution: Looking for Research Strategies . . . . . . . . . . From Molecular Enantioselectivity to Biological Precision . . . . . . From Molecular Enantioselectivity to Macroscopic Biological Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

197 197 200 203 204 206 209 211 213 216

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

Introduction
Ah, not in knowledge is happiness, but in the acquisition of knowledge! Edgar Allan Poe (1809 – 1849)

That the world is asymmetric is a trivial observation. How can the world be anything but asymmetric? What would a totally symmetric universe be like? A homogeneous gas? A perfect crystal? Celestial bodies all of the same mass, regularly spaced, coherently moving on perfect circular orbits? The world is asymmetric, because anything else would be unthinkable, including our own human existence. This asymmetry being a fundamental property of the universe, there is, however, a basic question which, to this day, has not been satisfactorily answered. According to most laws of physics, any asymmetric object or any asymmetrically moving system could, in principle, exist with equal probability as the corresponding mirror object or mirror system. The laws of gravitation, of electromagnetism, and the strong interactions, governing the structure of atomic nuclei, predict that image and mirror image should have exactly the same energy, and consequently show the same probability of occurring. So then, why would we be living in the world as it is, and not in the mirror world? Or why could not the actual world and the mirror world coexist and be both simultaneously perceptible? Not before the second half of the 20th century was it discovered that the fourth kind of forces observable in the universe, the so-called weak interactions, indeed do distinguish between image and mirror image. But from the point of view of our everyday perception of reality, these weak forces are somewhat esoteric. They govern processes between elementary particles, such as b-decay or the emission of neutrinos, and they act on distances that are almost infinitesimally small. Is the macroscopic asymmetry of the world we live in really the – more or less direct or indirect – consequence of these weak interactions? If yes, how do these interactions, which are operative on such a small scale, then get amplified, so as to shape the whole world we live in, from submicroscopic domains to the dimensions of galaxies? The scientific endeavor of man aims at a better understanding of his own nature and of the universe he inhabits. One cannot study the one without considering the other. To obtain answers, one must ask questions. Aiming at new answers, one must ask questions deriving from previous answers. We here do not have the ambition to give any new answers, but merely to ask questions based on recently established scientific knowledge. On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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On Chirality and the Universal Asymmetry

What we here propose to do is an excursion through nature, in order to observe some of the many ways in which asymmetry manifests itself. We will then immediately realize that our understanding of asymmetry is based on a search for symmetry. This is the way our perception functions. Asymmetry can be understood only as a deviation, or a departure, from what is symmetric, ordered, and clear. We will start our excursion on Earth, in the chemistry and the physics laboratory. Then we shall venture out into deep space, and back in time. This should broaden our horizon. With an enlarged perspective, we will return to Earth. Our contemplation of the extraordinary richness and complexity of the world, which immediately surrounds us, and which we are part of, is directly connected to our wish and intention to understand it.

1. Image and Mirror Image in Molecules
C
est la dissymtrie qui cre le phnomne. Lorsque certains effets rvlent une certaine dissymtrie, cette dissymtrie doit se retrouver dans les causes qui lui ont donn naissance. (
It is the dissymmetry that creates the phenomenon. When certain effects reveal a certain dissymmetry, that dissymmetry must also be found in their causes.) Pierre Curie (1859 – 1906) (J. Phys. (Paris) 3me Srie 1894, 3, 393)

1.1. The Homochirality of Life One of our first experiences in life is the observation that our two hands – as well as our two feet – are similar but not identical. If we mirror the left hand, it appears to be identical with the right hand, and vice versa. If we take a right glove and turn it inside out, we obtain a left glove. The mirror image of a left shoe is a right shoe. Objects which are distinct from, that is, not superimposable onto their mirror image, such as left hand and right hand (Fig. 1.1), are said to be chiral. The word
chiral stems from the Greek word for hand, cei1. A chiral macroscopic object and its mirror image are said to be enantiomorphous. From everyday life, we know of other objects that are chiral, for instance, screws and propellers. A right-handed screw is the mirror image of a left-handed screw and vice versa. A right-handed and a left-handed propeller turning in the same sense would make a boat go in opposite directions. However, a righthanded propeller turning clockwise, say, will make a boat go in the same direction as a left-handed propeller turning counterclockwise. What we notice is that both a screw and a propeller combine a turning movement, or rotation, with a linear displacement, or translation, along the rotation axis. As we shall see in more detail later, the combination of these two kinds of movement is an essential aspect of dynamic chirality. But let us now return to the observation of our hands. Both hands have the same number of bones and muscles, and similar blood vessels. We now imagine that we surgically remove a single muscle cell from our left hand and another one from the corresponding muscle in our right hand. We compare these two cells under a microscope: are they also enantiomorphous? A biologist will immediately say that this can hardly be the case. In fact, it will probably be very hard to discern which cell came from which hand. If we now proceed further down, from the microscopic scale to the submicroscopic and finally to the molecular level, we discover that, in both hands, the conditions are exactly the On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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On Chirality and the Universal Asymmetry

Fig. 1.1. If the right hand draws the left hand, and the left hand draws the right hand, how then did it start? By M. C. Escher (Lithograph, 1948; taken from: D. R. Hofstadter, Gçdel, Escher, Bach, Basic Books, London, 1979; courtesy of M. C. Escher Foundation).

same. Both cells, from our left hand and from our right hand, contain B-DNA in the form of a right-handed double helix, all the proteins in these cells consist of l-amino acids, and most carbohydrates are derived from d-sugars.
l stands for laevum, meaning
left in Latin, and
d signifies dextrum,
right. The very complex molecules that make up living organisms, such as DNA, RNA, proteins, and sugars, are thus all chiral. RNA contains the carbohydrate moiety d-ribose and DNA its derivative d-2-deoxyribose. One of the most remarkable facts in biology is that the biomolecular chirality, be it in a virus, in a primitive bacterium, or in a human brain cell, is everywhere the same. A chiral molecule and its mirror image are called enantiomers. Life based on enantiomeric biomolecules, namely on left-handed B-DNA, d-amino acids, and lsugars, is nonexistent on Earth. This remarkable selectivity is called biological homochirality (Fig. 1.2). Suppose that in the universe there exists an as yet unknown planet, revolving around an as yet unidentified star, and on which atmospheric and

Reflections on Image and Mirror Image

5

Fig. 1.2. Left: Forms of biomolecules that occur in nature. Right-handed B-DNA, left-handed ZDNA, l-alanine as representative of l-amino acids. Right: Forms that do not occur in nature. Lefthanded B-DNA, right-handed Z-DNA; very seldom, d-amino acids (adapted from G. L. Zubay, Biochemistry, 4th edn., William C. Brown, Boston, 1998, p. 745).

climatic conditions resemble those on Earth. In the course of the history of this planet, life chemically resembling that on Earth has evolved. Will these living organisms be molecularly homochiral to those on Earth? In other words, is the biological homochirality the direct consequence of universal and thus fundamental laws, or is it due to chance – whatever that means – and thus mainly dependent on local conditions? This is indeed one of the first questions to be asked if ever extraterrestrial life is observed. From a chemical standpoint, a chiral molecule and its enantiomer should, under identical external conditions, have exactly the same energy. In a so-called thermodynamic equilibrium with their surroundings, both enantiomers would consequently have the same probability of existing. If, however, one also takes into account elementary particle interactions called the parity-violating weak forces, then one concludes that there must exist a very small energy difference favoring one chiral form with respect to the other. The question then is, do these weak interactions suffice to determine biological homochirality. In the following chapters, we shall refine our notion of chirality and examine which influences conceivably could be at the origin of the remarkable biological

6

On Chirality and the Universal Asymmetry

molecular selectivity. The development of definite homochirality in biopolymers of varying length and monomer sequence, and its maintenance in the course of complex reactions are a fundamental mystery in the origin of living organisms.

1.2. The Discovery of Natural Optical Activity In a chiral medium, the plane of polarization of linearly polarized light gets rotated. One calls this phenomenon natural optical rotation or natural optical activity. The angle of rotation is a specific molecular property of the medium and is furthermore proportional to the path length of light inside the probe. In the enantiomorphous – or enantiomeric – medium under the same conditions, the angle of rotation is exactly opposite. This very fundamental effect was discovered by two French scientists in the early 19th century [1]. In 1811, FranÅois Arago noticed optical rotation in slabs of a-quartz cut perpendicular and irradiated parallel to the optic axis. The observation that optical activity not only is a property of a particular crystal (see Fig. 1.3) but that it can also occur in liquids, for instance, in sugar solutions, was demonstrated four years later by Jean-Baptiste Biot. He thereby founded the method of saccharimetry and applied it also medically to the research on diabetes. The angle of rotation is, in general, measured with a so-called polarimeter. In principle, this is a simple apparatus consisting of a light source, a polarizer, the probe, a second adjustable polarizer called an analyzer, and a detector. If one measures optical rotation at different wavelengths of light, l , one observes what is called optical rotatory dispersion (abbreviated as ORD). The apparatus then is named a spectral polarimeter. While the optical activity of quartz is a consequence of the overall structure of the crystal, that of a sugar solution derives from the individual, randomly distributed molecules. The ORD spectrum

Fig. 1.3. Enantiomorphous left- and right-handed crystals of quartz (adapted from [2])

Reflections on Image and Mirror Image

7

exhibits the relatively largest, characteristic signals in the wavelength range in which the valence electrons respond most strongly to the incident radiation (ultraviolet/visible light). At wavelengths very short or very long compared to the molecular dimensions (short X-ray and below; far infrared and beyond), the spectrum gradually goes to zero. The wave theory of light, promoted by Augustin Jean Fresnel around 1824, led to the following conclusion: linearly polarized light may be considered as the superposition of left and right circularly polarized light of same frequency and amplitude. In a chiral medium, the index of refraction for left circularly polarized light nL is different from that for right circularly polarized light nR [3]. The rotation of the plane of linearly polarized light at a given wavelength l’ in the optically active medium may thus straightforwardly be shown to be proportional to the difference. For one enantiomer: nL(l’)
nR(l’) = + Dn(l’) „ 0, and for the other: nL(l’)
nR(l’) =
Dn(l’). For any molecule, there are characteristic wavelengths at which it absorbs light, more or less strongly. The strength of such an absorption is measured by the absorption coefficient e. The absorptions which are most sensitive to the molecular electronic structure, in general, lie in the ultraviolet-visible (UV/ VIS) wavelength range. Each absorption corresponds to a so-called electronic transition. This means that the molecular electrons are promoted, by light of the corresponding wavelength, from an energetically lower (ground) to a higher (excited) state (a!b). The energy difference thereby taken up from the radiation field, Eb
Ea, is related to the frequency of the exciting light n, or the wavelength l, by the famous Einstein–Planck relation: Eb
Ea = hn = hc/l where h stands for the very fundamental Plancks constant, and c for the light velocity. In chiral molecules, the nonvanishing difference of the refractive index for left and right circularly polarized light is connected to the fact that, inside absorption bands, the absorption coefficient for left and right circularly polarized light is also different: eL
eR = De „ 0. This phenomenon is called circular dichroism (abbreviated as CD). Every molecule shows a particular absorption spectrum, and every chiral molecule consequently exhibits a corresponding individual CD spectrum. For

8

On Chirality and the Universal Asymmetry

enantiomers, the CD spectrum at all wavelengths is equal in magnitude, but opposite in sign. Depending on the molecule, its chirality, and the strength of the absorption, the ratio De/e = (eL
eR)/e has an order of magnitude of ca. 10
2 at best, often only between 10
3 to 10
5. CD is thus a relatively small effect. The technology to measure it routinely with a circular dichrograph matured relatively late, in the 1960s. Commercial apparatus nowadays can measure De/e values down to ca. 10
6. Molecular optical activity has been widely studied and used as an analytic tool. A central question in the chemistry of chiral compounds is the determination of absolute configuration: given an ORD or CD spectrum, which one of the two possible enantiomers lies at its origin? For example, ORD has much contributed to the chemistry of cyclic carbonyl compounds, in particular of steroids [4], the analysis and synthesis of sex hormones, and the development of oral contraceptives. Here, enantioselectivity and the determination of absolute configuration play a decisive role for physiological reasons. The method of CD has the advantage over ORD that, on inspection, its signals can more directly be attributed to particular electronic transitions (see Sect. 1.4, 3.3, and 8.5). As already mentioned, in the course of the advancement of optical and electronic technology, CD has become the method of choice in many applications. It is, for instance, very useful in protein chemistry, as it allows to clearly distinguish between secondary polypeptide structures [5]: a-helix, b-pleated sheet, polyproline, random coil (see Fig. 1.4). The relation between the CD and the ORD spectrum has a fundamental basis: if one knows the CD spectrum of a molecule over the whole wavelength range, one may, in principle, directly deduce the ORD spectrum and vice versa. Mathematically speaking, CD and ORD are Kronig–Kramers transforms of each other. The same kind of relation also exists between the ordinary absorption spectrum and the dispersion spectrum of any molecular system.

1.3. Chirality and the Birth of Stereochemistry Chirality, as we here understand it, does not exist in two dimensions. In order to occur, chirality needs three spatial dimensions. The projection of a left hand onto a sheet of paper is identical with the projection of a right hand if you cut it out, lift it, and flip it over. Aragos and Biots discoveries in the early 19th century immediately stimulated much further work on the optical properties of matter. A breakthrough which proved particularly important for the future development of chemistry was achieved in 1848 by Louis Pasteur (1822 – 1895). He investigated solutions of sodium ammonium tartrate that were indifferent to polarized light, that is, not optically active. Letting the solutes crystallize, Pasteur discovered,

Reflections on Image and Mirror Image

9

Fig. 1.4. Circular dichroism (CD) spectrum of a polypeptide, like poly-l-alanine, in different, frequently occurring secondary structures. The secondary peptide (protein) structures, the a-helix and the b-pleated sheet, are both stabilized by hydrogen bonds between different amide residues (figure taken from: C. K. Mathews, K. E. van Holde, Biochemistry, Benjamin Cummings, Redwood City, 1990). The a-helix, discovered by Pauling and Corey in 1949, is right-handed, has a pitch of 0.54 nm, and contains 3.6 residues per turn.

however, that the crystals turned out to be hemihedral, indicating that they were chiral. Some crystals were hemihedral to the right, others hemihedral to the left, showing the presence of both enantiomorphous forms. Selectively redissolving the right-handed crystals, Pasteur found that the new solutions so obtained rotated the plane of linearly polarized light to one side, while the solutions made from the left-handed crystals rotated the plane of linearly polarized light to the other. The original, optically inactive solutions obviously were racemic, containing equal amounts of both enantiomers of the compound. Pasteur had, by these experiments, achieved the first resolution of a racemic mixture into its chiral components. Unfortunately, only very few racemic mixtures crystallize enantioselectively, and Pasteur was fortunate to have found this striking example. Very often, racemic solutions also form racemic crystals [6], and the resolution of the enantiomers has to be carried out by other, often less efficient procedures (see also Sect. 7.1 and 9.11).

10

On Chirality and the Universal Asymmetry

Pasteurs discovery really started the field of stereochemistry. If the two forms of the tartrate had exactly the same atomic composition but showed opposite rotations, then the difference had to be sought in the structure, namely, in the spatial arrangement of the atoms inside the molecule. It gradually became clear that molecules are not just strings of atoms, but that these atoms must have definite relative positions in space. In 1873, Johannes Wislicenus wrote:
If molecules can be structurally identical and yet possess dissimilar properties, this difference can be explained only on the ground that it is due to a different arrangement of the atoms in space [2]. In 1874, Jacobus Hendricus van t
Hoff in Holland and Joseph Achille Le Bel in France independently published a theory of the structure of saturated carbon compounds. According to it, the four valences of the carbon atom are directed to the corners of a tetrahedron, at the center of which the carbon atom is situated. If four different atoms, or groups of atoms, are attached at the four corners, then the molecule is asymmetric and capable of existing in two non-superimposable mirror-image forms (see Fig. 1.5, a and b). For some time, it was believed that the presence of an asymmetric carbon atom – or of several asymmetric carbon atoms connected in such a way that the molecule as a whole and its mirror image are different – was the prerequisite for the occurrence of optical activity. Lord Kelvin (William Thomson, 1824 – 1907), in his 1884 Baltimore Lectures [7], appears to have been the first to have formulated a general definition of structural chirality:
I call any geometrical figure, or group of points, chiral, and say that it has chirality, if its image in a plane mirror, ideally realized, cannot be brought to coincide with itself. From the point of view of crystallographic point group symmetry, one concludes that all three-dimensional bodies are chiral which lack any rotation– reflection (or improper rotation) axes Sn. This implies, in particular, that they contain no planes of reflection s, and no point of inversion i (see Sect. 4.1 for further details). All totally asymmetric bodies are, therefore, chiral, and in addition, all bodies that solely contain pure rotation symmetry axes Cn, such as a propeller, or a helix. The extension of three-dimensional structural thinking to inorganic chemistry was largely pioneered by Alfred Werner in Zurich in the time period from 1893 to ca. 1914. Noticing the existence of different complexes of transition metal ions of same composition but opposite optical rotation, Werner recognized that the enantiomerism could be rationalized on geometric grounds. Assuming essentially octahedral ligand arrangement around sixfold coordinated central metal ions, it could be concluded that, depending on the number and types of substituents, chiral forms were conceivable. Werner was then able to synthesize and to resolve some of these chiral forms and to prove his structural theory. A striking example is found, when the central metal ion contains

Reflections on Image and Mirror Image

11

Fig. 1.5. a), b) Different representations of the enantiomers of lactic acid (adapted from [2]). c) The [Cr(C2O4)3]3
complex (adapted from [6]). d) Hexahelicene (from: S. H. Pine, J. B. Hendrickson, D. J. Cram, G. S. Hammond, Organic Chemistry, Mc Graw Hill International, London, 1981).

three bidentate ligands, such as in the trisoxalatochromate(III) complex ([Cr(C2O4)3]3
). The octahedral ligand arrangement of crystallographic symmetry Oh is here distorted to the lower, chiral symmetry D3 (see Fig. 1.5, c). Oh is the symmetry of a perfect octahedron, D3 is the symmetry of a threebladed propeller.

12

On Chirality and the Universal Asymmetry

From what has been mentioned until now, it might appear that optical activity in a molecule is connected to the occurrence of one (or more) asymmetrically substituted atomic center(s). But this needs not be the case: a beautiful example for another type of chiral molecule is to be found in the helicenes, polycyclic cata-condensed aromatic hydrocarbons, which, due to steric hindrance, are forced to adopt a nonplanar structure. In a molecule such as hexahelicene [8], there are no asymmetric carbon atoms in the usual sense. Rather, the molecule as a whole is obliged to take on a helical conformation of C2-symmetry which can either be right-handed or left-handed (Fig. 1.5, d). The molecule thus exhibits intrinsic chirality, and the enantiomers show very strong CD signals in the near-UV part of the light spectrum. The element carbon plays an important role in nature. Until ca. 20 years ago, three forms of pure carbon were known to exist under normal conditions, diamond, graphite, and amorphous carbon. The discovery of the fullerenes, stable cage-like molecules of pure carbon [9], opened up vast new fields of investigation in structural chemistry. The molecule C60 is of icosahedral holohedral symmetry Ih and thus definitely achiral. However, higher fullerene homologues, such as C76, C78, and C84, exhibit chiral structures and, therefore, may occur in two enantiomeric forms. For instance, C84 is of symmetry D2. It may in principle transform from one enantiomer to the other via an achiral intermediate structure of higher symmetry D2d which, however, also lies so much higher in energy that the chiral forms are thermally stable (see Fig. 1.6, top). The molecule C84 is of course prepared, by contact-arc vaporization of carbon, as a racemic mixture. The resolution into enantiomers is difficult and has been achieved through a reaction with reagents that also are chiral [10]. The CD spectrum of the enantiomers should, as we have previously stated, at all wavelengths have exactly the same absolute magnitude but the opposite sign. Small deviations therefrom (Fig. 1.6, bottom) reflect the difficulty of obtaining the enantiomers in pure form.

1.4. Absolute Configuration What we call
left and what we call
right is a matter of convention. Once we have agreed upon what is what, it is easy to identify a
left glove or a
right shoe. But how is it with molecules? We have seen that enantiomers may be distinguished by their optical activity. It became customary to label chiral compounds by the sign of their optical rotation at a given wavelength, usually the sodium-D line at 589 nm. If (looking towards the light source) the plane of linearly polarized light is rotated clockwise by the molecular sample, it is said to be dextrorotatory and is designated as (+) or d. If the sample rotates the linearly polarized light counter-

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13

Fig. 1.6. Top: The enantiomers of C84, of symmetry D2. They are thermally stable, as the achiral intermediate form of symmetry D2d lies significantly higher in energy. Bottom: The CD spectra of the enantiomers. Ideally, the spectra should be exact mirror images with respect to the zero line. The minor inaccuracies reflect the difficulty in quantitatively separating the enantiomers (taken from [10]).

clockwise, it is called levorotatory and is designated by (
) or l. This label, of course, says nothing about the exact spatial arrangement of the atoms in the molecule, which is termed the absolute configuration. It may well occur that chemically unrelated compounds have similar rotations at the Na-D line, while others that differ by perhaps only one substituent around an asymmetric center may show different magnitudes and sign. In the course of the evolution of stereochemistry, it became evident that an unambiguous definition and designation of the absolute configuration was needed. A first stage in this development was reached within the chemistry of carbohydrates, amino acids and related compounds. An assumed and later experimentally confirmed absolute configuration attributed to the compound (+)-glyceraldehyde was designated by d. d-Glyceraldehyde was then chosen as the reference basis for the absolute configuration of the naturally occurring

14

On Chirality and the Universal Asymmetry

optically active sugars, irrespective of their optical rotations. In Fig. 1.5, a and b, we find (+)-l-lactic acid represented at left, and (
)-d-lactic acid at right. This exemplifies that d and l are not to be confounded with d and l as defined in the previous paragraph. If in the formulas for lactic acid, we replace the COOH group by CHO, and the CH3 group by CH2OH, then we obtain at left the structure of (
)-l-glyceraldehyde, at right that of the (+)-d-enantiomer. Starting from (+)-l-alanine (Fig. 1.2), we may likewise formally derive the other l-amino acids through replacement of the CH3 group bound to the central C(a)-atom by other substituents (side chains). The dynamic development of organic and inorganic chemistry in the 20th century soon led to the necessity of defining and designating the absolute configuration of molecules in a broader and more general manner: in 1966, Cahn, Ingold, and Prelog published their
Specification of molecular chirality [11]. To assign the absolute configuration to the three-dimensional structure of an asymmetrically substituted carbon atom, one defines a ranking of the substituents: the higher atomic number has precedence over the lower one; for structural details, see the above reference. Once the ranking has been established, we then look at the molecular structure (molecular model) in such a way that the lowest-ranking of the four substituents is at the greatest distance from the observer. A sequence of the three other substituents is established, beginning with the one of highest rank. If this sequence goes clockwise, the absolute configuration is (R), if it goes counterclockwise, the absolute configuration is (S). Thus, (+)-l-alanine is equivalent to (+)-(S)-alanine, and (
)-d-alanine to (
)(R)-alanine. If the chirality of a molecule is inherent, such as in hexahelicene, then a lefthanded helix is designated by (M) (Fig. 1.5, d; at left), a right-handed one by (P). The assignment of absolute configuration in the case of asymmetric octahedral binding with sixfold coordination, is somewhat more complicated and shall not be treated here in any detail. We merely state that the [Cr(C2O4)3]3
complex in Fig. 1.5, c, at right, has absolute configuration (R). The attentive reader will now think that we have hitherto eschewed the basic problem: given a bottle containing (R) and a bottle containing (S), how do we tell which is which? We must be able to determine the absolute configuration on the basis of some measurement that can be performed separately on both samples, and that can be unambiguously correlated with the absolute configuration. Many physical quantities, such as melting point, boiling point, conductivity, low-frequency dielectric constant, absorption coefficient for unpolarized light in the UV, VIS, or IR, etc., will be the same for (R) as for (S). The method that, of course, immediately suggests itself is optical activity, namely CD or ORD. As we just have seen, however, measuring the optical rotation merely at one wavelength, for instance at the Na-D line, provides

Reflections on Image and Mirror Image

15

insufficient information. Furthermore, from an entire CD or ORD spectrum, one cannot deduce the absolute configuration merely on inspection. It was first shown by Rosenfeld [12] that a direct connection could be established between the absolute configuration of a molecule and its optical activity through quantum-mechanical calculations. In particular, the circular dichroism De(a ! b) for the transition from a state a to a state b in a molecule of given structure is proportional to the so-called rotational strength ´(a ! b), which in principle is calculable. The rotational strength for a particular transition in a chosen compound of assumed structure thus predicts both the magnitude and the sign of the corresponding CD signal (in a fluid medium): it is, mathematically, equal to the dot product of the electric dipole transition moment for the transition a ! b with the magnetic dipole transition moment pffiffiffiffiffiffiffiffiffiffiffiffi for the transition b ! a, multiplied by
i ( 
ð
1Þ):

´(a ! b) 
i h aj m j b i · h bj m ja i. The vector quantities m and m represent the quantum-mechanical electric dipole and magnetic dipole operators, respectively. This indeed seems somewhat complicated! As we will discuss later on in more detail (see Sect. 3.2 and 3.3), the quantity m represents a time-even, parity-odd polar vector, while m stands for an (imaginary) time-odd, parity-even axial vector. The quantity (
i)m, thus, corresponds to a time-even, parity-even axial vector, and ´(a ! b) becomes a time-even, parity-odd pseudoscalar that changes its sign under mirror reflection. Such symmetry considerations will, later on, prove to be essential. Although the absolute configuration of a molecule may, in principle, be deduced from the CD (or ORD) spectrum in the way here briefly outlined, the task of computing the rotational strengths for the various transitions is by no means easy. There exist classes of substances where ingenious simplified quantum-mechanical models are successful, such as the sector rules for asymmetrically substituted carbonyl compounds [13] [14] (see Sect. 8.5), the coupled oscillator model for certain types of dimers and polymers [15] [16] (see Sect. 8.6), or helicity rules for inherently chiral compounds [17]. In other cases, even elaborate ab initio computations still lead to results fraught with great uncertainty. The chiroptic methods have played a decisive and historic role in the development of stereochemistry. More recent spectroscopic advances, such as vibrational CD [18] and Raman optical activity [19] extend the applicability of chiroptic measurements very significantly. In spite of these successes, chiroptic measurements, in general, are not the methods of immediate choice for the detailed elucidation of molecular structure.

16

On Chirality and the Universal Asymmetry

X-Ray analysis has the advantage that, from the numerous measured reflections, by Fourier transformations, the positions of atoms inside the unit cell of a crystal lattice may be deduced. However, if the light is scattered purely elastically, distances between all the atoms are obtainable, but not the absolute configuration which, as we have seen, depends also on the relative positions of atoms in space. This difficulty has to do with the
phase problem and the fact that one measures intensities and not directly structure factors [20]. Already in 1949, however, Bijvoet showed that it was possible to distinguish between enantiomorphous crystals when the wavelength of incident radiation coincided with, or was close to, the absorption wavelength of a given kind of atom in the lattice, leading to selective inelastic scattering, or anomalous dispersion [21]. In particular, Bijvoet and co-workers performed X-ray measurements on NaRb(+)-tartrate crystals using an emission line of Zr to excite the anomalous scattering of the Rb-atoms. The analysis then clearly confirmed that the absolute configuration of (+)-tartrate was the expected one, namely, in the more modern notation (2R,3R) (the numbers here designating the asymmetric carbon atoms in the tartrate). The refinement of the measuring procedures and the progress of computational methods have since made X-ray analysis a most powerful and widely applied method to determine molecular structure, including absolute configuration [20]. In summary, it may be stated that the problem of experimentally determining the absolute configuration of molecules has essentially been solved.

1.5. Asymmetric Synthesis Numerous simple chemical starting materials that either occur abundantly in nature or are synthesized industrially on a large scale, are achiral. Such are: water, carbon dioxide, ammonia, methane, ethane, ethene, ethine, methyl and ethyl alcohol, formaldehyde, etc. In an achiral environment, achiral reactants may never lead to a chiral product. If from achiral starting materials chiral molecules are formed, the product will always be obtained as a racemic, and, therefore, achiral, mixture of these molecules. A racemic mixture, consisting of enantiomers in the ratio 1 : 1, is achiral, because to every image present there also exists the mirror image. Likewise, a pair of gloves or a pair of shoes are achiral. To obtain a chiral product, several possibilities may be envisaged: 1.5.1. Enantioselective Separation of a Racemic Mixture into Isolated Enantiomers or Antipodes (Greek, anti = opposite, podes = feet). If one is lucky, as Pasteur was with the tartrate salt, and the racemic mixture crystallizes forming distinguishable chiral, enantiomorphous crystals (see also Sect. 7.1 and

Reflections on Image and Mirror Image

17

9.11), one may separate the crystals by inspection to obtain the molecular enantiomers (antipodes). Unfortunately, many racemic solutions under laboratory conditions form racemic crystals [6]. Furthermore, if chiral crystals indeed are formed, the procedures to select these crystals enantioselectively do not necessarily lend themselves well to separations on a larger scale. Another method of resolution relies on selective adsorption to an external chiral medium. A practical example is chiral chromatography. The chromatographic material (stationary phase) must be chemically inert, yet contain numerous, asymmetrically configured polar groups. Carbohydrate polymers such as cellulose or cyclodextrin, or macromolecules like chiral crown ethers, lend themselves well to this purpose (see Sect. 9.4). 1.5.2. Asymmetric Chemical Reactions. To obtain a chiral product by a chemical reaction, the chirality must by some means be introduced in the course of that reaction. This indeed occurs directly from the start, if at least one reactant is already chiral. Otherwise, there is the possibility of using a chiral catalyst. To be synthetically useful, such a chiral catalyst must be simple to prepare and efficient in its action. It is not an easy task to find the right architecture for the catalyst, and, in the preparation of the catalyst itself, the necessary asymmetry should be introduced by easily accessible chiral precursors. As the catalyst after the reaction may be reused, it does not have to be made in a very large quantity. However, if it works reliably and efficiently, it may render the production of specific chiral products on an industrial scale possible (see also Sect. 9.6). The centenary 2001 Nobel prize in chemistry was awarded to three pioneers in the design and application of chiral catalysts, underscoring the importance of the field. The catalysts used in this scientific work all have in common that they contain a transition metal center asymmetrically surrounded by carefully selected substituents. Such a chiral complex can then add to the achiral (prochiral) reactant, keeping it fixed, while the main reaction proceeds stereospecifically. Then, it releases the chiral product and reverts to its initial state. Thus, W. S. Knowles and R. Noyori, using rhodium [22] and ruthenium [23] complexes, respectively, developed methods for the stereospecific hydrogenation of double bonds and carbonyl moieties with enantiomeric excesses close to 100%. K. B. Sharpless discovered the feasibility of asymmetric epoxidation with a titanium catalyst. By carefully varying the architecture of the catalyst, it became applicable to a variety of reactions [24] [25]. It is noteworthy, that it made the total synthesis of the eight l-hexose sugars possible, the naturally occurring ones being all d, as we have seen. Many important biological reactions in vivo are catalyzed by biochemical catalysts, or enzymes. Such enzymes are mostly globular proteins with a highly specific amino acid sequence and a particular secondary and tertiary structure.

18

On Chirality and the Universal Asymmetry

They are of course chiral, and their function is summarized by the well-known scheme: S + E W X ES ! P + E The enzyme E reacts with the substrate S to form an enzyme–substrate complex, ES. The substrate, which may be chiral or achiral, fits into a particular part of the macromolecular enzyme like a key into a lock. The transformation of the substrate takes place stereospecifically within the enzyme, whereupon it is released as product P. As is taught in basic courses on chemical thermodynamics and kinetics, true catalysts may increase the speed of a reaction, but they do not change the position of the chemical equilibrium. This is of course also the case for chiral catalysis (see also Sect. 9.6). The product P usually is quickly consumed by following reactions, and an equilibrium situation between P and the reactant substrate S is thereby avoided. If the product started to accumulate in presence of the enzyme, the enzyme-catalyzed back-reaction would set in. The importance of chiral reactions in biology has been known for a long time. In the mystery novel
The Documents in the Case [26], the dramatic event is caused by the toxicity of muscarine. A crime is committed with a synthetic racemic mixture of both muscarine enantiomers. It turns out to be lethal and feigns a poisoning by the mushroom Amanita muscaria which, however, contains only the (+)-enantiomer. The tendency has existed in medicine to assume that, if one gave the human body medication which was racemic, the organism would pick out the
correct enantiomer, and discard or ignore the other. The danger of this opinion was dramatically demonstrated by the thalidomide case in the 1960s, as it was discovered, too late to avoid a catastrophe, that the (R)-enantiomer of the drug is indeed a soporific with anti-inflammatory activity, but that the (S)-enantiomer is teratogenic causing grave damage to human embryos. The situation is further complicated by the additional discovery that the liver contains an enzyme which converts the (R)- to the (S)-enantiomer [27]. The physiological sensitivity to chirality is nicely demonstrated by the olfactory sense. For instance: (+)-(R)-limonene smells like orange; the (
)-(S)enantiomer like lemon. (+)-(S)-Carvone smells like caraway; the (
)-(R)enantiomer like peppermint. The attention given to enantioselectivity in pharmacy has rapidly increased in recent years. For instance, the world production of pharmaceutical products as pure enantiomers of definite chirality has tripled within the five years between 1991 and 1996 [28]. Asymmetric synthesis is indeed at the core of modern chemistry.

Reflections on Image and Mirror Image

19

1.5.3. Enantioselective Photochemistry. The existence of circular dichroism (CD) in chiral media suggests the possibility of carrying out enantioselective photochemistry. Different types of transformations are conceivable with circularly polarized light: asymmetric photodestruction, partial photoresolution, and asymmetric photosynthesis. The first report on a successful enantioselective photochemical experiment is due to Kuhn and Braun. A racemic solution of a chiral organic ester was sealed into two separate cells. One cell was subjected to left circularly polarized, the other to right circularly polarized UV light of 280 nm wavelength. For circularly polarized light of given handedness, the rates of absorption of the enantiomers are unequal. The absorption at the wavelength here indicated leads to molecular dissociation into fragments which are achiral. When the raction had proceeded to ca. 50% under the influence of the radiation, the optical activity of the remaining ester was tested. The solution irradiated with left circularly polarized light showed optical rotation to the left, the other sample irradiated with right circularly polarized light, rotation to the right, in accord with expectations. The asymmetric photodestruction of the ester was thereby proven [29]. While, in asymmetric photodestruction, the optical activity finally disappears under continued irradiation, in partial photoresolution, starting from a racemic mixture, an equilibrium optical activity is attained. As an example, a racemic aqueous solution of the trisoxalatochromate(III) complex (see Fig. 1.5, c) was irradiated with right circularly polarized light of 546 nm, at which wavelength the complex absorbs strongly [30]. Due to the CD, one isomer absorbs more strongly than its enantiomer. The only photochemical reaction that a given isomer undergoes in the excited state is inversion to its enantiomer. The rate of other processes, such as thermal racemization and deactivation, are not enantioselective. Under irradiation, an equilibrium situation was consequently attained in which the concentration of one of the enantiomers is increased compared to that of the other. It may be shown that the maximum extent of photoresolution is governed by the dissymmetry factor (eL
eR)/e at the given wavelength. As we have seen (Sect. 1.2), this quantity is of the order of 0.1 – 1.0% at best (see also Sect. 3.8). Finally, we mention, as an important example, the photoinduced asymmetric synthesis of hexahelicene (Fig. 1.5, d). Use was made of circularly polarized light of 313 nm, coinciding with the regions of strong absorption, both of the achiral precursors and of the product. The method chosen for this synthesis, the photocyclization of achiral 1,2-diarylethenes to chiral dihydrohelicenes and simultaneous oxidation, proved applicable also to higher helicenes [31]. As expected, the optical yields so obtained were low, of the order of 0.2%. Compared to purely chemical, modern catalytic methods of asymmetric synthesis, the photochemical processes here outlined seem inefficient, and of

20

On Chirality and the Universal Asymmetry

practical use only in very particular cases. Yet, as we shall see, the influence of electromagnetic fields, in particular, of light, on chiral media is of fundamental interest, going beyond chemistry, and will become an important topic of later parts of this book. It is also to be assumed that photochemical transformations may have played a significant role in the development of the molecular homochirality of life.

REFERENCES [1] J. P. Mathieu,
Activit Optique Naturelle, Handbuch der Physik, 1957, Vol. 28, Spektroskopie II. [2] L. F. Fieser, M. Fieser, Organic Chemistry, D. C. Heath & Co., Boston, 1950, Chapt. 11. [3] L. Velluz, M. Legrand, M. Grosjean, Optical Circular Dichroism, Verlag Chemie, Weinheim, 1965. [4] C. Djerassi, Optical Rotatory Dispersion, Mc Graw-Hill, New York, 1960. [5] J. A. Schellman, P. Oriel,
Origin of the Cotton Effect of Helical Polypeptides, J. Chem. Phys. 1962, 37, 2114. [6] S. F. Mason, Molecular Optical Activity and the Chiral Discriminations, Cambridge University Press, Cambridge, 1982, Chapt. 9.4, p. 164. [7] Lord Kelvin (W. Thomson; 1884), The Baltimore Lectures (revised edn. 1904); see e.g., http:// chirality.ouvaton.org/research.htm. [8] M. S. Newman, D. Lednicer,
The Synthesis and Resolution of Hexahelicene, J. Am. Chem. Soc. 1956, 78, 4765. [9] H. W. Kroto, J. R. Heath, S. C. OBrien, R. F. Curl, R. E. Smalley,
C60 : Buckminsterfullerene, Nature 1985, 318, 162. [10] J. M. Hawkins, M. Nambu, A. Meyer,
Resolution and Configurational Stability of the Chiral Fullerenes C76, C78 and C84 : A Limit for the Activation Energy of the Stone-Walls Transformation, J. Am. Chem. Soc. 1994, 116, 7642. [11] R. S. Cahn, C. Ingold, V. Prelog,
Specification of Molecular Chirality, Angew. Chem., Int. Ed. 1966, 5, 385. [12] L. Rosenfeld,
Quantenmechanische Theorie der Natrlichen Optischen Aktivitt von Flssigkeiten und Gasen, Z. Phys. 1928, 52, 161. [13] W. Moffitt, R. B. Woodward, A. Moscowitz, W. Klyne, C. Djerassi,
Structure and the Optical Rotatory Dispersion of Saturated Ketones, J. Am. Chem. Soc. 1961, 83, 4013. [14] J. A. Schellman,
Symmetry Rules for Optical Rotation, J. Chem. Phys. 1966, 44, 55. [15] W. Moffitt,
Optical Rotatory Dispersion of Helical Polymers, J. Chem. Phys. 1956, 25, 467. [16] I. Tinoco Jr.,
Theoretical Aspects of Optical Activity. Part Two: Polymers, Adv. Chem. Phys. 1962, 4, 113. [17] W. Hug, G. Wagnire,
The Optical Activity of Chromophores of Symmetry C2, Tetrahedron 1972, 28, 1241. [18] L. A. Nafie, T. A. Keiderling, P. J. Stephens,
Vibrational Circular Dichroism, J. Am. Chem. Soc. 1976, 98, 2715. [19] L. D. Barron,
Raman Optical Activity: A New Probe of Stereochemistry and Magnetic Structure, Acc. Chem. Res. 1980, 13, 90. [20] J. Dunitz, X-Ray Analysis and the Structure of Organic Molecules, Cornell University Press, Ithaca, 1979. [21] J. M. Bijvoet, A. F. Peerdeman, A. J. van Bommel,
Determination of the Absolute Configuration of Optically Active Compounds by Means of X-Rays, Nature 1951, 168, 271. [22] W. S. Knowles,
Asymmetric Hydrogenation, Acc. Chem. Res. 1983, 16, 106. [23] R. Noyori, M. Yamakawa, S. Hashiguchi,
Metal-Ligand Bifunctional Catalysis: A Nonclassical Mechanism for Asymmetric Hydrogen Transfer between Alcohols and Carbonyl Compounds, J. Org. Chem. 2001, 66, 7931.

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[24] K. B. Sharpless,
The Discovery of the Asymmetric Epoxidation, Chem. Br. 1986, 22, 38. [25] H. C. Kolb, M. G. Finn, K. B. Sharpless,
Click Chemistry: Diverse Chemical Function from a Few Good Reactions, Angew. Chem., Int. Ed. 2001, 40, 2004. [26] D. L. Sayers, The Documents in the Case, Hodder & Stoughton, London, 1930. [27] T. D. Stephens,
Reinventing Thalidomide, Chem. Br. 2001, Nov., 38. [28]
Research Initiative on Chiral Chemistry of the Swiss National Science Foundation, Final Report, Switzerland, 1998. [29] W. Kuhn, E. Braun,
Photochemische Erzeugung Optisch Aktiver Stoffe
, Naturwissenschaften 1929, 17, 227. [30] K. L. Stevenson, J. F. Verdieck,
Partial Photoresolution. Preliminary Studies on some Oxalato Complexes of Chromium (III), J. Am. Chem. Soc. 1968, 90, 2974. [31] A. Moradpour, J. F. Nicoud, G. Balavoine, H. Kagan, G. Tsoucaris,
Photochemistry with Circularly Polarized Light. The Synthesis of Optically Active Hexahelicene, J. Am. Chem. Soc. 1971, 93, 2353.

2. The Violation of Parity
With every new language learned you gain a new soul Czech proverb, freely translated

2.1. Through the Looking Glass Charles Lutwidge Dodgson (1832 – 1898) was lecturer in mathematics at Christ Church, Oxford, from 1855 to 1881. Among his mathematical writings we find
An Elementary Treatise on Determinants,
Euclid, Book V, Proved Algebraically,
Euclid and his Modern Rivals, reflecting an outspoken interest in algebra and geometry. However, Dodgson gained fame less as a scientist, but rather as a writer of literary books under the pseudonym of Lewis Carroll. His stories, such as
Alice
s Adventures in Wonderland and
Through the Looking Glass, written for children of all ages and conveying a particular poetic charm, have become classics of world literature. In
Through the Looking Glass, Alice, looking into the mirror, ponders on how the world behind it might be (Fig. 2.1). Talking to her kitten, she wonders if
…Looking-glass milk…isn
t good to drink. Suddenly,
…the glass was beginning to melt away…In another moment Alice was through the glass, and had jumped lightly down into the Looking-glass room. [1].

Fig. 2.1. Alice crossing into the mirror world On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

24

On Chirality and the Universal Asymmetry

Fig. 2.2. When Alice holds the text to the mirror, she realizes that it is written in a language unknown to her [1]

The main observation Alice immediately makes is that the looking-glass world not only is the geometric mirror image of the real world, but that it appears as qualitatively different (Fig. 2.1). She enters a new world governed by particular laws and populated by bizarre individuals. When she tries to read the pages of a book, she realizes that she must hold it to a mirror first. But even so,
…it
s rather hard to understand! The poem is written in a special language (Fig. 2.2). There are instances where writers by intuition, or inspiration, raise questions which ultimately turn out to be of great scientific relevance, but which are only scientifically formulated, and possibly answered, much later. The beautiful story of Alice induces us to speculate: Suppose
enantiomorphous human beings existed, composed of d-amino acids, l-sugars and left-handed B-DNA, would their languages necessarily be different from those of the inhabitants of Earth? We know that language is not inherited, but learned. So it is very probably not directly related to the structure and absolute configuration of DNA and of proteins. On the other hand, it is to be assumed that most existing languages, be it English, Chinese, or Bantu, have some basic elements in their grammar in common [2], related to basic patterns of human thinking, which are connected to fundamental mechanisms in the brain and ultimately run on a molecular scale. Would these basic linguistic structures evolve differently in
enantiomorphous individuals?

Reflections on Image and Mirror Image

25

Leaving aside these interesting, but very speculative questions, to which we shall briefly return only at the end of this book (Sect. 10.9), we now consider more general aspects of basic physics: the idea that the Looking-glass world should be qualitatively different from the Real world appeared for a long time as mere fiction. Until 1956, it was generally believed that all physical laws were exactly the same for image as for mirror image; in other words, that parity was conserved. The theoretical prediction of the violation of parity and the experimental proof thereof one year later, was a surprise that even found its way into the daily press (see, for instance, The New York Times of January 16, 1957). For some eminent physicists, the news came as a rather shocking surprise. Wolfgang Pauli, a pioneer of the modern quantum theory of matter, discoverer of the Exclusion Principle, comments these developments critically. In a letter dated August 1957 to the famous psychologist C. G. Jung, Pauli remarks that
…So it is now certain that God is a weak left-hander…But His reasons we do not know…In such a possibility I never would have believed before January of this year [3].

2.2. Parity and the Laws of Physics The science of physics describes analytically the evolution of macroscopic and of microscopic objects in three-dimensional space and in time under the influence of forces. In the world of physics, one distinguishes between four kinds of forces: gravitation, electromagnetic forces, strong interactions, and weak interactions. For instance, gravitation governs the motion of the Moon around the Earth, and of the Earth around the Sun; electromagnetism manifests itself in an electric bulb, an electromotor, or any electronic device; the strong interactions hold the atomic nuclei together and are responsible for the release of nuclear energy; finally, the weak interactions are effective between elementary particles and become apparent in radioactive b-decay, or in meson decay. Formerly, it was assumed that it made no difference, if one described the laws governing these forces of nature by a right-handed or a left-handed spatial coordinate system (Fig. 2.3). In other words, one considered these laws all to be invariant under spatial reflection, or inversion, also called the parity operation P. While for gravitation, electromagnetism, and the strong interactions, this indeed appears to be the case, it was predicted by Lee and Yang in 1956 that, in weak interactions, parity might be violated [4]. The measurements by Wu et al. on the asymmetric spatial intensity distribution of the b-decay of magnetically polarized Cobalt-60 nuclei produced the convincing confirmation [5]. In the words of the previous section: there are physical processes where the
image and the
mirror image correspond to physically distinguishable situations. In the present case, one of the two situations occurs and is observable, the other does not, because it is forbidden by nature. But before we consider such

26

On Chirality and the Universal Asymmetry

Fig. 2.3. A right-handed cartesian coordinate system (i, j, k) is transformed by the parity operation (P) into a left-handed one (i’, j’, k’). Reflection in the j,k-plane also gives rise to a left-handed coordinate system (i’, j, k). It is related to (i’, j’, k’) by a rotation of 1808 around the i’-axis. Notice: the axis j points to the back of the image plane, the axis j’ to the front.

experiments in more detail, we wish to get better acquainted with what the parity operation means. The parity operation P consists in replacing the cartesian coordinates of a point in space, x, y, z, by
x,
y,
z . This corresponds to an inversion, or a reflection in the origin, of the coordinate system. We may also apply the operation to the coordinate system itself. If i, j, k are unit vectors in x-, y-, z-directions, respectively, we replace these unit vectors by i’ =
i, j’ =
j, k’ =
k. This leads from a right-handed to a left-handed coordinate system, as the reader may verify (Fig. 2.3). If r is the position vector of the point, we then have: r = x i + y j + z k =
x i’
y j’
z k’, and for the corresponding inverted point: r’ =
x i
y j
z k = x i’ + y j’ + z k’. We notice that in general, the parity operation P acting on r changes the sign of r: P r = r’ =
r.

Reflections on Image and Mirror Image

27

By definition, the position vector is a polar vector. It is odd with respect to parity. The linear velocity: v = vx i + vy j + vz k, or the linear momentum: mv = p = px i + py j + pz k

(m, the mass of the body),

are also polar vectors. We next consider the angular momentum l, defined as the vector product of r and p: l = r  p. By elementary algebra and vector geometry we find: l = x py (i  j) + x pz (i  k) +…, and, applying the parity operation: P l = l’ = x py (i’  j’) + x pz (i’  k’) +…, we notice the absence of a sign change. The parity operation acting on the angular momentum vector leaves it unchanged. More precisely, the angular momentum vector is an axial vector and is even with respect to parity: P l = l’ = l. The symmetry properties of these vectors are, of course, also valid for the corresponding quantum-mechanical vector operators and their expectation values. Furthermore, not only is the orbital angular momentum l a parity-even axial vector, but such is also the spin (of any particle) s. The function (or operator) describing the energy of any system in which gravitational, electromagnetic, or strong interactions are operative, is also even, or invariant, with respect to parity. But suppose a particular kind of energy depends on the dot product of spin s and linear momentum p, as is namely the case for the weak interactions: This dot product is a pseudoscalar and is odd with respect to parity. Indeed we have: P (s · p) = (P s) · (P p) = s’ · p’ =
s · p.

28

On Chirality and the Universal Asymmetry

Depending on if we describe it in a right-handed or a left-handed coordinate system, this type of interaction energy has a different sign. But, nature here makes a choice, and only one sign is admitted. The famous experiment by Wu and co-workers consisted in showing that the b-particles (electrons) emitted from the 60Co nuclei have their momentum p antiparallel to the spin of the parent nucleus I (evidently also an axial vector) and to their own electron spin s [6]. The asymmetry was detected in the laboratory by orienting the Co nuclei in a magnetic field. To obtain a sufficient degree of orientation for the measurement, the sample had to be cooled to a temperature of 5 K (kelvin). Wolfgang Paulis hesitation in accepting the violation of parity was apparently not shared by P. A. M. Dirac, the founder of relativistic quantum mechanics. Dirac is said to have expressed the view that the laws of physics should be invariant under transformations that may be considered as the sum of infinitesimal contributions, such as translations and rotations. Reflections and inversions, however, are discrete, finite operations. There is no a priori reason that the laws of nature should be invariant to these. Another kind of discrete transformation is charge conjugation, C, the replacement of all particles by their respective antiparticle. For instance, the replacement of a (negative) electron by a (positive) positron, or of a (positive) proton by a (negative) antiproton. Indeed, as we shall see in some detail in Sect. 2.4, the weak interactions are not invariant under charge conjugation, either.

2.3. Time Reversal and the CPT Theorem To reverse the time implies, in classical physics, to reverse all motions; in other words, to make the movie go backwards. Quantities which do not depend explicitly on time, such as the position vector(s) or a static potential energy, are invariant under time reversal. Velocities, momenta, and angular momenta, change their sign, of course. It is elementary to show this: let us consider an arbitrary component, px say, of the momentum vector p. Designating the operation of time reversal by T, which means replacing the time variable t by (
t), we may write: px = m dx/dt; T px = m dx/d(
t) =
m dx/dt =
px, similarly for the angular momentum: lx = y pz
z py ; T lx = y(
pz)
z(
py ) =
lx.

Reflections on Image and Mirror Image

29

Thus p and l are odd with respect to time reversal: T p =
p, and T l =
l. We also immediately conclude that quantities which are proportional to the square of the first derivative with respect to t, such as kinetic energy; or are proportional to the second derivative with respect to t, such as acceleration and force, are even with respect to time reversal. In quantum mechanics, the situation is basically the same, though rather different in detail. We use the same notation as in the classical description, but, instead of physical quantities, we here consider quantum-mechanical operators. For instance, the px component of the momentum operator p reads: px =
i
h @/@ x, where i is the imaginary unit (square root of
1),
h represents Plancks constant, h, divided by 2p, and the partial derivative is to be taken with respect to the space variable x. We notice the absence of any explicit time variable t. So we ask ourselves, how here the time is to be inverted. It is not entirely elementary to arrive at the conclusions stated below. For the derivation, we refer the reader to the specialized literature [7 – 10]. In quantum mechanics, the operation of time reversal T has to be translated into a corresponding time reversal operator which does not act directly on any time variable. T is now expressed as T = U K, where U is a unitary operator and K the operator of complex conjugation. The reader is here reminded that by
complex conjugation we mean
to change the sign if i in any complex or imaginary mathematical expression. Furthermore, the unitary operator U can, for our immediate purposes, be set equal to 1. Consequently: T px = + i
h @/@ x =
px and in general: T p=
p in analogy to the classical case. Similar considerations apply to the angular momentum operator l. An arbitrarily chosen component, lx say, is written: lx =
i
h (y @/@ z
z @/@ y) and thus:

30

On Chirality and the Universal Asymmetry

T lx =
lx. Then, in general, and, in analogy to the classical case: T l =
l. Physical quantities – or their corresponding quantum-mechanical operators – that are derived from r, p, and l by multiplication with physical constants transform under the operations of P and T like r, p, and l themselves. For instance: An (elementary) electric dipole moment: m = e r, P m =
m, T m = m (e, the elementary charge). An (elementary) magnetic dipole moment: m = (e/2m) l, P m = m, T m =
m (m, the mass of the particle carrying charge e). The wave vector (of a particle or photon): k = p/
h, P k =
k, T k =
k (the wave vector k is here not to be confounded with the unit spatial vector k). These transformation properties are all summarized in Table 2.1. We also take notice that the spin of any particle, collectively here denoted by s, transforms under T like an orbital angular momentum. The rigorous mathematical description of the spin properties is beyond the scope of this book. But, as in atoms and nuclei, orbital angular momenta and spin angular momenta may add to give constants of the motion, they quite evidently must behave the same way under both P and T. A fundamental connection between the operations P, C, and Twas shown to exist by Lders in 1954. An improved version of the
Proof of the TCP Theorem was published in 1957 [11]. It states that a wide class of quantized field theories which are invariant under the proper Lorentz group is also invariant with respect to the product of time reversal T, charge conjugation C, and parity P. In other words, all laws of physics, including the weak forces, should be invariant under the consecutive action of T, C, and P (the order being irrelevant). It cannot be excluded that in a Grand Unified Theory of the universe, which still remains to be discovered, the TCP theorem might be violated. But, up to the present time, there is no direct evidence for such a violation.

Reflections on Image and Mirror Image

31

Table 2.1. Transformation Properties (even (+) vs. odd (
)) of Some Basic Physical Quantities Under the Operations of Parity (P, space inversion), Charge Conjugation (C), and Time Reversal (T) Physical quantity

P

C

T

Position vector r Electric dipole moment m Orbital angular momentum l Spin angular momentum s Magnetic dipole moment m Momentum vector p Wave vector k Electric field vector E



+ + +




+
+ +
+ +


+ +




+

2.4. The Selectivity of the Weak Forces In this and the following section, we consider physical processes of different kinds in which parity violation manifests itself. Where the weak forces dominate, the selection rules due to parity violation are stringent. Our first example is the decay of a positively charged p+-meson into a positive muon m+ and a (neutral) neutrino nm : p+ ! m+ + nm, and of a negative p
-meson into a negative muon m
and an antineutrino
nm : p
! m
+
nm . Fig. 2.4 shows that the two equally allowed decay reactions, a and d, are related to each other by PC or CP, the consecutive operations of P and C, or vice versa. However, if we subject any one of these two reactions only to P, or only to C, we encounter processes which are not observed, that is, forbidden (Fig. 2.4, b and c) [12]. We also notice that, under P, the polar momentum vectors p of the product particles (in black) are inverted, whereas the axial spin vectors s (in red) remain unchanged. We now define a quantity called the helicity h. It is the scalar product of the polar vector p and of the concomitant axial vector s, divided by the arithmetic product of the absolute values of both vectors: h = s · p/(j s j · j p j). We conclude from Fig. 2.4 that the allowed processes either start from p+ with h =
1, or from p
with h = + 1; the other two possibilities being forbidden.

32

On Chirality and the Universal Asymmetry

Fig. 2.4. The selectivity of the weak interactions is apparent in p-meson decay. a) The observed process p+ ! m+ + nm ; b) the process resulting from a space inversion P, which is not observed; c) the process resulting from charge conjugation C, which is also not observed; and d) the observed nm , which is related to a by PC or CP. Black arrows represent the momentum process p
! m
+
vectors of the particles produced, red arrows the spin vectors.

Furthermore, in a given process, the two spin vectors are of equal magnitude, but opposite in sign, adding to zero. This illustrates the conservation of spin angular momentum, both parent particles, p+ and p
, having a spin of zero. For some time, it was believed that, in general, PC must be strictly conserved. But already in 1964, in the investigation of the decay modes of the Kmesons, the discovery was made that this is not the case [13] [14]. For instance, the long-lived, uncharged, and spinless K-meson KL shows the following two decay channels into either a p+- or p
-meson, an electron e
or a positron e+, and a neutrino n or antineutrino
nm : I) KL ! p
+ eþr + nl II) KL ! p+ + e
l +
nr The indices r/l (right/left) reflect the  helicity of the corresponding particles. The reader will notice that, in both channels, charge and spin are conserved, and that the channels are related to each other by the consecutive operations CP. Then, if the operations CP were strictly conserved, the decay rate for both channels would be exactly the same, and the difference of these rates would be zero. However, one measures a relative rate difference DR of:

Reflections on Image and Mirror Image

DR =

33

RðIÞ
RðIIÞ = 3.3  10
3 RðIÞ þ RðIIÞ

[14] [15]. This illustrates the fact that the observed violation of CP, or PC, symmetry in such processes precludes an all-or-nothing selectivity, but that the deviating influences are relatively small. It is apparently in contrast to the strict situation with respect to the violation of P only, or of C only, and the allowedness of PC shown in the previous example (Fig. 2.4). The violation of CP symmetry may, indeed, have very fundamental implications. If the prediction is true that CPT is universally conserved, then the violation of CP implies that the time reversal symmetry T is also violated. This then could mean that the physical evolution of the universe cannot be exactly the same if it goes forward or backward in time, thus suggesting the existence of a general arrow of time (see Sect. 5.4).

2.5. Parity Violation in a Nuclear Reaction We recall that only the weak interactions violate parity. Strong interactions which mainly govern the forces in atomic nuclei, and electromagnetic interactions that are involved in light absorption and emission, conserve parity. In physical processes in which these parity-conserving interactions dominate, it may indeed be difficult to unravel the weaker parity-violating effects. Nevertheless, the study of parity violation in nucleon–nucleon interactions has quickly become a vast field [16]. Out of it, we choose a single experiment which will turn out to be highly instructive for our later discussion. A particularly striking example of parity violation in a nuclear process has been reported by Adelberger et al. [17]: a beam of protons coming from an accelerator is transversely polarized, implying that the proton spins (with spin quantum number 1/2) are oriented perpendicular to the beam direction and are parallel to each other. In a gas cell, the protons collide with the Neon-22 isotope (containing 10 protons and 12 neutrons) to yield Fluorine-19 (9 protons and 10 neutrons) and an a-particle (Helium nucleus; 2 protons and 2 neutrons): 1 1

19 4 p(pol.) + 22 10 Ne ! 9 F*(pol.) + 2 He.

The F-nuclei, like the protons, possess a spin of 1/2. The transversely polarized protons are very efficient in transfering their polarization to the 199 F* nuclei. Furthermore, the F-nuclei are produced in an excited state (*) and emit electromagnetic g-radiation. Due to the parity-violating weak force in the nucleus and the polarization of the nuclear spins, one detects a fore-aft asymmetry in the intensity of the radiation, depending on if it is emitted parallel or antiparallel to

34

On Chirality and the Universal Asymmetry

Fig. 2.5. The spin-polarized protons hit the 22Ne target to produce 19F (and an a-particle; not shown). The spin polarization is transmitted to the 19F nuclei, thereby inducing a fore-aft asymmetry in the gemission, which is measured by the pair of detectors D1 and D2.

the direction of polarization (Fig. 2.5). In this situation, the polar vector is the wavevector k of the g-radiation, the axial vector the nuclear spin I of the Fnucleus. The helicity thus reads: k · I/(j k j · j I j), and is positive in one direction, negative in the other. The weak forces here manifest themselves by an – albeit small but significant – relative intensity difference DI registered by the detectors 1 and 2. It is of the order 1/10,000: DI =

jðI1
I2 Þj  18  10
5 (absolute value). ðI1 þ I2 Þ

The question which we now ask is if the parity-violating effect in the Fnucleus could also be detected in the absence of spin polarization. The parityviolating interaction inside it makes the nucleus in some sense chiral. Although the analogy may seem farfetched, we remember (Sect. 1.2) that a chiral molecule shows circular dichroism, namely, a different absorption coefficient for left and right circularly polarized light. For emission, it is similar, the intensity of left and right circularly emitted light is also different: IL
IR „ 0. In molecules, this is called circularly polarized luminescense. Does then the excited F-nucleus, in analogy, emit left and right circularly polarized g-radiation with different intensities? The answer is yes: parity violation results in circular polarization of g-emission from unpolarized nuclei or fore-aft asymmetry in gemission from polarized nuclei [12]. How these two manifestations of parity breaking in the emission of electromagnetic radiation are related with each other will be discussed in more detail in Sect. 3.6.

Reflections on Image and Mirror Image

35

2.6. Parity and Selection Rules in Atoms Historically, the first theory of atomic structure, the Bohr model, represented the electrons as revolving on particular orbits around the nucleus, like planets around the sun. The advent of wave mechanics in the 1920s modified this view fundamentally. As is today shown even in elementary textbooks, the atomic electrons are considered as standing waves of delocalized charge density. In the theory, this density is given by the absolute value squared of the electronic wave function. The electronic wave function itself is obtained as a solution of the time-independent Schrçdinger equation for the atom. In what immediately follows, we limit our discussion to the simplest atom, hydrogen atom, which consists of a proton as nucleus and of a single electron. The symmetry considerations which we will make later on are basically also applicable to many-electron atoms. One of the fundamental conclusions of the quantum theory is that an isolated atom at rest may exist only in particular states of discrete energy. The quantum states of the hydrogen atom essentially describe the internal dynamics of the electron in the electrostatic field of the nucleus. These states are labeled by a principal quantum number n, an orbital (or angular momentum) quantum number l, and a magnetic quantum number m. The quantum numbers may only take on integral values, and their ranges are: n = 1, 2, …, 1, l = 0, …, n
1, m =
l,
(l
1), …, + (l
1), + l

[(2l + 1) values].

In view of the fact that the electron not only carries mass and charge, but also a spin of 1/2, one has, in addition, to take into account the magnetic spin quantum number ms which can take on the values + 1/2 and
1/2. These four kinds of quantum numbers characterize the individual atomic electron. They are also applicable to the independent-particle model of many-electron atoms. For instance, the Pauli principle states then in a many-electron atom, no two electrons may have all four quantum numbers n, l, m, ms the same. Focusing on the hydrogen atom, it is worth noting that the energy levels En depend only on the principal quantum number n (in electron Volt; eV): En =
13.6

1 . n2

36

On Chirality and the Universal Asymmetry

Correspondingly, for the transition energies between states of different principal quantum number, n and n’, one may write:
 1 1 .
DE = En’
En =
13.6 n02 n2 We notice that if n = 1, and n’ = 1, we obtain DE = + 13.6 eV, which is the ionization energy of the hydrogen atom. The transition from an energetically lower state to a higher state may proceed via absorption of radiation; from a higher state to a lower state, via emission. Converting DE from energy units in eV to Joule, one finds the absorption (emission) wavelength l in meters from the relation: DE = hn =

hc (see also Sect. 1.2). l

For the settings: n = 1; n’ = 2, 3, …., 1, one obtains the wavelengths for the wellknown Lyman Series (in the ultraviolet region). For the setting n = 2; n’ = 3, 4, …, 1, the Balmer Series (in the visible) is obtained. The important point, which we now wish to mention, is that the absorption/ emission of radiation is not equally possible between arbitrary quantum states. The way in which the electromagnetic field acts on the atom to induce a transition depends decisively on the symmetry of the initial and final state. In this context, the orbital quantum number l becomes important, and the magnetic quantum number m also plays a role. In particular, one finds that all states with l even conserve their sign under the parity operation P, while those with l odd change it. It has become customary to designate states for which l = 0, as s; l = 1, as p; l = 2, as d; l = 3, as f; l = 4, as g; etc. Including also the principal quantum number n in the label, the sequence of lowest atomic states becomes: 1s; 2s, 2p; 3s, 3p, 3d; 4s, 4p, 4d, 4f; etc., as is to be found in any elementary textbook. Now concerning the symmetry under the parity operation, we find, for instance: P (ns) = (ns); P (np) =
(np); P (nd) = (nd); P (nf) =
(nf). For a radiative transition between two states to be allowed, the integrated product (the so-called transition matrix element) hinitial statej  (light interaction)  jfinal statei must be symmetric, or even, with respect to the parity operation P. Now, the strongest interaction possible between an atom (or molecule) and the light, is the so-called electric dipole interaction, labeled E1. From Table 2.1, we noticed that the electric dipole m (or the corresponding quantum-mechanical operator) changes sign under P. For the transition 1s !

Reflections on Image and Mirror Image

37

2s, we thus find P h1sj m j2si =
h1sj m j2si because even  odd  even equals odd; the transition is forbidden. For 1s ! 2p, on the other hand, i.e., P h1sj m j2pi = + h1sj m j2pi, the transition is electric dipole allowed. More generally, the selection rules for electric dipole transitions E1 in a hydrogen atom read Dl =  1, Dm = 0,  1, with Dn = n’
n being arbitrary. Where electric dipole transitions are forbidden, other types of radiationinduced transitions may be allowed, which depend on so-called higher-order interactions between the light field and the atom (see also Sect. 3.2). The next order is represented by magnetic dipole transitions M1 and by electric quadrupole transitions E2. These transitions, where allowed, are in general much weaker than the electric dipole transitions. The intensity of a transition is proportional to the absolute value of the transition matrix element squared. For an M1 transition and an E1 transition, the order of magnitude of the intensity ratio is: jhM1ij2 1 1 a2  = = 1.33  10
5, jhE1ij2 22 1372 4 where a = 1/137 is the so-called fine structure constant. The corresponding intensity ratio for E2 vs. E1 is wavelength-dependent. In the visible region around 500 nm it is of the order of 10
7. The magnetic dipole moment m (or the corresponding quantum-mechanical operator) is sign-invariant, or even, under P (Table 2.1). Consequently, the selection rules for M1 transitions are basically different from those for E1 transitions. We here state the selection rule Dl = 0. This condition is necessary, but not sufficient, however. Beside the orbital angular momentum, the spin angular momentum also comes into play. Due to the so-called spin–orbit coupling, a total angular momentum quantum number j has to be considered. In the hydrogen atom, j may take on the values l + s and l
s, where s = 1/2. The additional conditions then read [18]: Dj = 0,  1, Dmj = 0,  1. (The total magnetic angular momentum quantum number mj may take on the values
j,
j + 1, …, + j
1, + j). Without going into further details, we note that the electric quadrupole is also even with respect to parity, in analogy to the magnetic dipole. The selection rules for E2 transitions are: Dl = 0,  2, Dm = 0,  1,  2 (transitions between s states are forbidden).

38

On Chirality and the Universal Asymmetry

In summary: one-photon E1 transitions only occur between states of opposite parity (even ! odd, or odd ! even). This is called Laportes rule [18] [19]. In an atom, subjected to electromagnetic interactions, E1 transitions on one hand, M1 and E2 transitions on the other, are mutually exclusive: transitions that are E1-allowed are always M1- and E2-forbidden, and vice versa. This statement is also valid for many-electron atoms throughout the Periodic System. We now immediately guess that if an atom is influenced by parity-violating weak interactions, then these selection rules may get broken.

2.7. The Violation of Parity in Atoms Shortly after the experiments by Wu et al. confirmed the violation of parity in b-decay, Zel
dovich raised the question, if parity violation, or nonconservation, could also be observed in atomic electrons [20]. The predictions about the measurability of such effects were initially pessimistic, however. The problem was taken up seriously only in the later 1960s and early 70s. In this time period, the elementary-particle theory made spectacular progress. In particular, a unified theory of the electro-weak interactions was also developed [21]. Among the weak interactions, one distinguishes between charge-changing ones, mediated by charged W+ and W
bosons, and charge-preserving ones, mediated by neutral Z8 bosons. While, for instance, the p-meson decay, which we considered in Sect. 2.4, is governed essentially by the first type of interaction, the effects in atoms depend mainly on the so-called weak neutral currents. In the atomic processes, no charges are destroyed or created, and the parity violation is caused by the exchange of neutral Z8 bosons between atomic electrons and quarks in the nucleus. The development of a detailed theory of parity violation in atoms led to the conclusion that the effects should grow as the third power of the atomic number Z [22]. This Z3-law offered good hope of making successful experiments with heavy elements, such as with Cs, Tl, Bi, and Pb. From the experimental point of view, the systematic exploration of parity violation in atoms was decisively promoted through the application of the recently available tunable lasers. By exploiting the coherence, monochromaticity and intensity of the laser radiation at frequencies of ones choice, significant refinements in atomic spectroscopy proved possible that would permit the detection of weak parity-violating effects. The experiments, that have been performed in the decades since, essentially follow two avenues [23] [24]: 1) fluorescence experiments; 2) optical rotation experiments. In view of the general analogy with the molecular optical activity, we shall concentrate more on 2 than on 1. As already indicated, the effect of the parity-nonconserving, or parityviolating (PV), weak neutral currents in an atom, is to mix states of opposite

Reflections on Image and Mirror Image

39

parity. The quantum-mechanical expression for the energy of an electron under the PV influence has the form: pffiffiffi VPV = (GF a QW / 4 2) {s · p, d(r)}. The quantities inside the round brackets are constants: GF is the co-called Fermi weak coupling constant, a is the fine structure constant (see also Sect. 2.6), and QW is the weak nuclear charge, a quantity which is characteristic for a given atom. QW is the sum of the weak charges of all the constituents of the atomic nucleus, the up and down quarks [24]. For higher Z, QW is about equal to minus the neutron number. The curly bracket in the above formula is a so-called anticommutator. The expression s · p denotes the scalar product of the electron spin angular momentum operator s and the linear momentum operator of the electron p. From Sect. 2.3 and 2.4, we know that s is a parity-even, time-odd axial vector; p is a parity-odd, time-odd polar vector. Consequently, s · p is a parity-odd, timeeven pseudoscalar, and represents a chiral influence (see also Sect. 1.4). Finally, d(r) is a
delta function, which is supposed to be everywhere zero except at the site of the atomic nucleus. It expresses the fact that the PV weak nuclear currents only act at the nucleus itself, but not outside it. They can only influence the atomic electron(s) if the electronic charge density at the nucleus is nonvanishing. We now wish to discuss the
parity-mixing effect of VPV in some more detail. Let us consider a transition in an atom that goes from an initial state a to an end state b, both states being even with respect to parity. We furthermore assume that VPV
mixes into b a third state c that is odd: jbmixedi = j beveni + ib j coddi where ib is a (very small) imaginary coefficient. The quantity b is real, and i is the pffiffiffiffiffiffi ffi imaginary unit
1. We evidently find (see Sect. 2.6): haevenj m j beveni = 0. However: haevenj m j bmixedi = ib haeven j m j coddi „ 0; the parity-mixed transition obtains a small E1 component. We now distinguish: Case A. The transition aeven ! beven is also M1-forbidden: haeven j m j beveni = 0.

40

On Chirality and the Universal Asymmetry

The experimental search for the parity-violating effect consists in measuring the very weak E1 intensity in fluorescence. But it turns out to be difficult, requiring sophisticated experiments. Case B. The transition aeven ! beven is M1-allowed: haeven j m j beveni  hM1i „ 0; we similarly abbreviate: haeven j m j coddi  hE1i. It may then be shown that the absorption coefficient e of the transition aeven ! bmixed for left circularly polarized light is proportional to: eL  jhM1i + b hE1ij2 and for right circularly polarized light: eR  jhM1i
b hE1ij2 (the positive and negative signs have here only a relative meaning). Consequently: 2ðeL
eR Þ jhM1i þ bhE1ij2
jhM1i
bhE1ij2  . ðeL þ eR Þ jhM1ij2 To first order in b (assuming here all quantities real): De bhE1ihM1i bhE1i .   e hM1i hM1i2 The inequality eL „ eR implies that we have circular dichroism, and b hE1i · hM1i is the rotational strength of the transition (see Sect. 1.4). Concomitantly we have nL „ nR. This causes rotatory dispersion (outside resonances, normal; inside resonances, anomalous). We now remember that optical rotation is measured with a polarimeter (Fig. 2.6). In heavy elements, such as in Tl or Bi, the mixing parameter b may attain an order of 10
10 [23]. As the ratio hE1i/hM1i is roughly equal to 2/a = 2  137, where a is the fine-structure constant, one may expect De/e to be of the order of 10
7 at best, which is ca. 100 times smaller than the weakest signals measurable with a commercial CD apparatus. Rather, the measurement of the optical rotation, which is proportional to nL
nR, has been chosen as method. We mention here two relatively recent experiments and the values for the quantity b hE1i/hM1i (see above) that have been deduced therefrom: for the

Reflections on Image and Mirror Image

41

Fig. 2.6. The components of a polarimeter. Laser source, polarizer, probe compartment with chiral probe, analyzer; not shown is the detector. If the optical rotation measured in a) represents the actual experiment, then b) would correspond to the enantiomer (mirror image). If the probe in a is an atomic vapor, then the situation depicted by b is forbidden. The enantiomer of a chiral atom does not exist because of parity violation (or parity nonconservation) (taken from [23]).

876 nm (6p3 (J = 3/2 ! J’ = 3/2)) transition in bismuth [25]:
10.12  10
8 ; for the 1283 nm (6p (J = 1/2 ! J = 3/2)) transition in thallium [26]:
14.68  10
8. There is today no longer any doubt that parity-violating effects may be detected in atomic spectroscopy. These effects, though very small and difficult to measure, are of fundamental interest. They allow assessments of QW, the weak nuclear charge, that are complementary to those deduced from acceleratorbased elementary particle experiments. In the language of chemistry, one may say that atoms are weakly chiral, but that they occur only as one form of two conceivable enantiomers. The existence of the mirror-image atom is forbidden. On the grounds of the conditional, quasi-conservation of PC symmetry (Sect. 2.4), one can assume, however, that the mirror-image anti-atom (consisting of the corresponding antiparticles; carrying opposite charges) will be allowed. This question indeed invites experiments which, in the future, may become possible.

2.8. The Violation of Parity in Molecules The weak neutral currents that are responsible for the parity violation in atoms are, of course, also operative in molecules. The way in which these paritynonconserving effects manifest themselves in molecules is somewhat different from that in free atoms, however. The atoms which we have considered in the last section, are all paramagnetic, in other words, they have an open-shell ground state. Most molecules, in particular, also those of biological importance, have a closed-shell ground state and are diamagnetic. Also under these

42

On Chirality and the Universal Asymmetry

conditions, the weak nuclear currents are only operative in conjunction with the so-called spin–orbit coupling [27]. The coupling of the spins of the electrons with their orbital motion is necessary to render the molecule susceptible to the influence of the weak interactions. It may then be shown that the parityviolating effect causes a small energy shift, DEPV, in a molecule that is structurally chiral. Furthermore, it turns out that this shift is equal in magnitude but opposite in sign for the enantiomer. From this, we also conclude that the shift should be zero in any achiral molecule. In the previous section, we indicated that the parity-violating effect in an atom is proportional to GF a Z3. It may be shown that in an atom or molecule, spin–orbit effects grow roughly as a2 Z2. From the product of both, one may estimate the order of magnitude of the molecular-energy shift due to parity nonconservation: DEPV GF a3Z5h, where GF is the Fermi weak coupling constant, a the fine structure constant, Z the atomic number, and h a so-called asymmetry factor depending on the molecular connectivity and geometry. The Z5-dependence suggests that the effect will be strongly favored in molecules containing heavy atoms. An early estimate was that h may be of the order of 10
2. It was then concluded that for a molecule containing essentially carbon atoms: jDEPVj  10
18 a.u.  2.7  10
17 eV  2.6  10
15 kJ mol
1 = 2.6  10
12 J mol
1 (a.u.: atomic units; eV: electron Volts; J mol
1: Joule per mol). If one twists the hydrogen peroxide molecule, H2O2, around the O
O bond, the molecule becomes nonplanar and chiral, of symmetry C2. Early ab initio quantum-chemical calculations [28] predicted for a dihedral angle of 1208, for instance, and an absolute configuration (P) (see Sect. 1.4 concerning the absolute configuration of inherently chiral molecules; the symbol (P) here must of course not be confounded with the parity operation): DEPV  2.  10
20 a.u.  5.4  10
19 eV  5.2  10
17 kJ mol
1 = 5.2  10
14 J mol
1. The same authors similarly found for the difference between l- and d-alanine: 2  DEPV 
5  10
14 J mol
1. These values are indeed significantly smaller than some initial estimates. Since then, a great deal of effort has gone into the calculation of the molecular-energy shift due to parity violation. In general, these more recent results confirm the orders of magnitude of the early calculations, and follow, for

Reflections on Image and Mirror Image

43

molecules containing heavier atoms, approximately the Z 5-dependence. Such molecules become increasingly accessible to quite accurate quantum-chemical computations. As a next example, (P)-Cl2S2 (twisted around the Cl
Cl bond; and for a dihedral angle of 858) [29]: DEPV  + 0.8  10
12 cm
1  + 0.96  10
11 J mol
1. For the pyramidal molecule (R)-BiHFBr [30]: DEPV  + 7.1  10
15 a.u.  + 1.86  10
8 J mol
1. The theoretical prediction of the existence of a parity-violating energy shift in chiral molecules has immediately raised the question of its measurability. For instance, could it be possible to detect a small difference between enantiomers in the frequency of selected vibrations? Such experiments demand a very high accuracy and have until now been inconclusive [31]. An idea of central interest is, that DEPV may also influence the dynamics of the
image-to-mirror-image interconversion of enantiomers [32] [33]: (R) G H (S), or (P) G H (M). We shall at present briefly discuss this aspect.

2.9. The Interconversion of Enantiomers The conversion of one enantiomer (antipode) into the other is, in general, governed by a double minimum potential. In the absence of parity-violating interactions, this double minimum potential is symmetric. The two minima, corresponding to the ground-state energy of the antipodes, have equal values. The barrier between the two minima represents the potential energy that classically has to be overcome in order to go from one antipode to the other along the shortest reaction path. However, quantum mechanics teaches us that a given chiral molecule must not necessarily climb over the barrier to transform into the enantiomer, but that it can also tunnel through it. The broader and higher the barrier, the smaller is the tunneling probability. But the tunneling probability is, in principle, never zero if the dimensions of the barrier are finite. This implies that a given chiral form, in general, does not correspond to a true stationary state of the molecule, as there always remains a small probability of conversion to the enantiomer. The quantum-mechanical operator for the energy of an isolated (and nonparity-violated) molecule, the so-called molecular hamiltonian, only contains electromagnetic interactions and is even with respect to the parity operation P. This is also the case for the Schrçdinger equation derived from it. The solutions of the time-independent Schrçdinger equation, the molecular eigenfunctions,

44

On Chirality and the Universal Asymmetry

or stationary states, y may be either even or odd with respect to parity. The probability density jyj2 will always be even. We now immediately realize that the wave function describing a chiral molecule, yR say, cannot be an eigenfunction of the molecular Schrçdinger equation. It cannot describe a stationary state, for the physical reasons just stated, and, furthermore, as it is neither even nor odd with respect to P. The parity operation transforms yR into another function, namely that of the enantiomer yS. Thus: P yR = yS , P yS = yR. However, if we add or subtract these two equations, we find: P (yR + yS) = + (yR + yS), P (yR
yS) =
(yR
yS). On this basis we define the two functions y+ = N+ (yR + yS), y
= N
(yR
yS), P y+ = + y+, P y
=
y
. These functions y+ and y
(within the approximations of a two-level model) indeed prove to be solutions of the molecular Schrçdinger equation and describe stationary states (N+ and N
are so-called normalization constants and are of no importance to our argument). We denote the energy eigenvalues corresponding to y+ and y
, by E+ and E
, respectively. It may now be shown that, if we have a particular chiral molecule in the nonstationary state yR, say at time t = 0, then at time t’ = h/[2(E+
E
)] it will have transformed into the enantiomer yS . At time 2t’, the molecule will be back in the initial state yR. We consequently notice that, averaged over a sufficiently long time, the chiral molecule racemizes. However, depending on the structure of the molecule and the height of the barrier, the time t’ may vary over many orders of magnitude: in H2O2 with a very small barrier between the chiral minima [34]: DE = E+
E
 11 cm
1  130 J mol
1, t’  1.5  10
12 s. The molecule oscillates six hundred billion times per second. In an amino acid, such as alanine, where the transformation from one enantiomer to the other either requires the breaking of a bond or the passage through a sterically highly strained structure: DE = E+
E
 5.5  10
22 cm
1 

Reflections on Image and Mirror Image

45

Fig. 2.7. a) In order to be chiral, a molecule must possess at least four atoms and be nonplanar. The structure of type Y
X
X
Y is bent and twisted. The two enantiomeric energy minima have symmetry C2. By changing the dihedral angle around the X
X bond, one goes from one enantiomer to the other. b) Pyramidal four-atomic structure with a trivalent central atom A. To go from one enantiomer to the other, the atom A must move through the plane defined by the ligands B, C, and D. c) The double minimum potential governing the transition from one enantiomer to the other, and qualitatively applicable to our examples. The two minima, representing the ground-state energy of the enantiomers, are separated by a maximum corresponding to an intermediate achiral (planar) structure. Details about the conversion of one enantiomer to the other, are to be found in the text. In the absence of parity violation, the double minimum potential is symmetric.  DEPV raises one minimum and lowers the other (dashed lines).

6.6  10
21 J mol
1, t’  3.  1010 s  1000 years. The chiral molecule of given handedness appears essentially stable. In general, when the barrier is low, the splitting between the stationary state energies DE will tend to be comparatively high and t’ to be short; when the barrier is high, DE will be correspondingly small, and t’ will be large. Our central question now is: How does the violation of parity manifest itself in these situations? From what we have seen in the last section, the ground state energy of one enantiomer will now be shifted by the amount + DEPV, that of the other by
DEPV. The double-minimum potential will no longer be symmetric (Fig. 2.7). The dynamics of interconversion will evidently be influenced both by DE (for simplicity, not represented in the figure) as well as by DEPV. Eschewing

46

On Chirality and the Universal Asymmetry

mathematical details [33], we may qualitatively state: For DEPV  DE, the parity violation will have a negligible effect on the dynamics (case of H2O2). For DEPV  DE, the parity violation will quasi-statically stabilize one enantiomer with respect to the other (case of alanine). A series of possible experiments has been devised by which DEPV could be directly determined. These include measurements of quantum beats in optical activity [32], two-photon experiments with a high-lying intermediate state of well-defined parity, and kinetic experiments [35]. These experiments are all very difficult and require the capacity to measure beat periods of up to hours, namely of the order of h/(2 DEPV).

2.10. From Meson Decay to Molecular Homochirality In processes governed solely by the weak forces, parity violation manifests itself in a dominant way. For instance, in the decay of p-mesons (Sect. 2.4), parity-violating forces within the time scale of the order of only 10
8 s (the lifetime of the metastable parent particle) open selected reaction channels and keep others closed. In physical systems held together by electromagnetic forces, such as atoms and molecules, the parity-nonconserving interactions remain in the background and are difficult, if not hitherto impossible, to detect. It is all the more difficult to assess their possible chemical effects on any time scale, short or long. From that point of view, molecular homochirality in biology is a particularly remarkable and puzzling phenomenon. If, for instance, the molecule l-alanine is lower in energy than d-alanine by an amount of 10
14 J mol
1 (Sect. 2.8), then a mixture of 1 mol of each enantiomer in thermodynamic equilibrium at a temperature of 10 8C would contain an excess of the l-enantiomer of only ca. 106, or one million molecules, compared to an
astronomical total of 6  1023 molecules per mol. Such a small fraction is hardly a promising situation for attaining an almost 100% selection. If the parity-violating weak interactions are responsible for the biological homochirality, then this must be the result of an immense number of consecutive, relatively weakly selective chemical steps. And the cumulative effect of the previous steps must certainly not get lost, before the next step
takes over. If the age of the planet Earth is between 4 and 5 billion years, and the first geologically dated living organisms are 3 billion years old or more, then the period of prebiotic evolution has had to take place in about a billion years or less. This is an extremely long time compared to the life span of any living organism, but indeed not a long time compared to that of the biological evolution itself. We will return to these questions later on (Sect. 9.9 and 10.2 – 10.5).

Reflections on Image and Mirror Image

47

It has been pointed out [9] [15] [36] that CP enantiomers should be strictly degenerate (in spite of the fact that in the decay of certain mesons, CP is not strictly conserved). This implies that a chiral molecule of absolute configuration (R) should have exactly the same energy as the enantiomer of absolute configuration (S) made of antimatter (*) and vice versa: E(R) = E(S*), E(S) = E(R*). At present, such statements still seem rather academic. But if
chemical experiments with antimatter ever become feasible, they will indeed be of fundamental interest.

REFERENCES [1] Lewis Carroll, Through the Looking-Glass (illustrated by John Tenniel), Macmillan and Co., London, 1872. [2] D. Benedetto, E. Caglioti, V. Loreto,
Language Trees and Zipping, Phys. Rev. Lett. 2002, 88, 048702. [3] W. Pauli, C. G. Jung, Ein Briefwechsel: 1932 – 1958, Ed. C. A. Meier, Springer- Verlag, Berlin, 1992. [4] T. D. Lee, C. N. Yang,
Question of Parity Conservation in Weak Interactions, Phys. Rev. 1956, 104, 254. [5] C. S. Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, R. P. Hudson,
Experimental Test of Parity Conservation in Beta Decay, Phys. Rev. 1957, 105, 1413. [6] G. Musiol, J. Ranft, R. Reif, D. Seeliger, Kern- und Elementarteilchenphysik, VCH, Weinheim, 1988, Chapt. 9.5.7. [7] E. P. Wigner, Group Theory, Academic Press, New York, 1959, Chapt. 26. [8] A. Messiah, Mcanique Quantique, Dunod, Paris, 1964, Chapt. XV. [9] L. D. Barron, A. D. Buckingham,
Time Reversal and Molecular Properties, Acc. Chem. Res. 2001, 34, 781. [10] G. Rasche, Lectures on Quantum Mechanics, University of Zurich, 2000. [11] G. Lders,
Proof of the TCP Theorem, Ann. Phys. 1957, 2, 1; see also Dan. Mat. Fys. Medd. 1954, 28. [12] E. D. Commins, P. H. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge University Press, Cambridge, 1983, Sect. 1.5; Sect. 9.5. [13] J. H. Christenson, J. W. Cronin, V. L. Fitch, R. Turlay,
Evidence for the 2p Decay of the K2 Meson, Phys. Rev. Lett. 1964, 13, 138. [14] K. Gottfried, V. F. Weisskopf, Concepts of Particle Physics, Clarendon Press, Oxford, 1986, Vol. 1. [15] L. D. Barron,
CP Violation and Molecular Physics, Chem. Phys. Lett. 1994, 221, 311. [16] E. G. Adelberger, W. C. Haxton,
Parity Violation in Nucleon–Nucleon Interaction, Ann. Rev. Nucl. Part. Sci. 1985, 35, 501. [17] E. G. Adelberger, H. E. Swanson, M. D. Cooper, J. W. Tape, T. A. Trainor,
Parity Mixing in 19 F , Phys. Rev. Lett. 1975, 34, 402. [18] B. H. Bransden, C. J. Joachain, Physics of Atoms and Molecules, Longman, London, 1983, p. 178.

48

On Chirality and the Universal Asymmetry

[19] E. U. Condon, G. H. Shortley, The Theory of Atomic Spectra, Cambridge University Press, Cambridge, 1957. [20] Y. B. Zeldovich,
Parity Nonconservation in the First Order in the Weak-Interaction Constant in Electron Scattering and Other Effects, Sov. Phys. JETP 1959, 36(9), 682. [21] S. Weinberg,
A Model of Leptons, Phys. Rev. Lett. 1967, 19, 1264. [22] M. A. Bouchiat, C. C. Bouchiat,
Weak Neutral Currents in Atomic Physics, Phys. Lett. B 1974, 48, 111; T. P. Emmons, E. N. Fortson,
Parity Nonconservation in Atoms, in Progress in Atomic Spectroscopy: Part D, Eds H. J. Beyer, H. Kleinpoppen, Plenum Press, New York, 1987, p. 237. [23] D. N. Stacey,
Experiments on the Electro-Weak Interactions in Atoms, Physica Scripta 1992, T40, 15. [24] M. A. Bouchiat, C. Bouchiat,
Parity Violation in Atoms, Rep. Prog. Phys. 1997, 60, 1351. [25] M. J. D. Macpherson, K. P. Zetie, R. B. Warrington, D. N. Stacey, J. P. Hoare,
Precise Measurement of Parity Nonconserving Optical Rotation at 876 nm in Atomic Bismuth, Phys. Rev. Lett. 1991, 67, 2784. [26] P. A. Vetter, D. M. Meekhof, P. K. Majumder, S. K. Lamoreaux, E. N. Fortson,
Precise Test of Electroweak Theory from a New Measurement of Parity Nonconservation in Atomic Thallium, Phys. Rev. Lett. 1995, 74, 2658. [27] R. A. Hegstrom, D. W. Rein, P. G. H. Sandars,
Calculation of the Parity Nonconserving Energy Difference between Mirror-Image Molecules, J. Chem. Phys. 1980, 73, 2329; D. W. Rein, R. A. Hegstrom, P. G. H. Sandars, Parity Non-Conserving Energy Difference between Mirror Image Molecules, Phys. Lett. A 1979, 71, 499. [28] S. F. Mason, G. E. Tranter, The Parity-Violating Energy Difference between Enantiomeric Molecules, Chem. Phys. Lett. 1983, 94, 34; S. F. Mason,
Origins of Biomolecular Handedness, Nature 1984, 311, 19. [29] R. Berger, M. Gottselig, M. Quack, M. Willeke,
Parittsverletzung dominiert die Dynamik der Chiralitt in Dischwefeldichlorid, Angew. Chem. 2001, 113, 4342; Angew. Chem., Int. Ed. 2001, 40, 4195. [30] F. Faglioni, P. Lazzeretti,
Understanding Parity Violation in Molecular Systems, Phys. Rev. E: Stat. Phys., Plasma, Fluids, Relat. Interdiscip. Top. 2002, 65, 011904. [31] C. Chardonnet, T. Marrel, M. Ziskind, C. Daussy, A. Amy-Klein, C. J. Bord,
Spectroscopie de molcules chirales: recherche d
un effet de violation de la parit, J. Phys. IV 2000, 10, Pr8-45. [32] R. A. Harris, L. Stodolsky,
Quantum Beats in Optical Activity and Weak Interactions, Phys. Lett. B 1978, 78, 313. [33] M. Quack,
On the Measurement of the Parity Violating Energy Difference between Enantiomers, Chem. Phys. Lett. 1986, 132, 147. [34] M. Quack,
Molecules in Motion, Chimia 2001, 55, 753. [35] M. Quack,
Struktur und Dynamik Chiraler Molekle, Angew. Chem. 1989, 101, 588; Angew. Chem., Int. Ed. 1989, 28, 268. [36] M. Quack,
On the Measurement of CP-Violating Energy Differences in Matter–Antimatter Enantiomers, Chem. Phys. Lett. 1994, 231, 421.

3. Light, Magnetism, and Chirality
Quand les myst
res sont tr
s malins, ils se cachent dans la lumi
re (When mysteries are very clever, they hide in the light) Jean Giono (1895 – 1970)

3.1. Electromagnetism and Light As mentioned in Sect. 2.2, the electromagnetic forces do not violate parity. They conserve parity. This does not imply, however, that electromagnetic phenomena need to be achiral. We shall presently see that there are electromagnetic effects which appear to be chiral, but in fact are not. There are also electromagnetic effects which are truly chiral. But in those cases, the image and mirror image always correspond to situations of same energy, and thus are statistically equally probable to occur. One of the fundamental manifestations of electromagnetism is the Lorentz force F. When a (point) charge q moving at a velocity v enters a region with a magnetic field (magnetic induction) B assumed perpendicular to v, the trajectory of the charge gets deflected under the influence of a force F which is perpendicular to both v and B: F = q(v  B). In words: the force vector is equal to the charge times the cross product of the velocity and magnetic induction vectors. Suppose q is positive, v lies in the plane of the paper pointed away from the observer, and B is directed downward perpendicular to the plane. Then F points to the left. We now apply to F the parity operation P (space inversion). We notice: v is parity-odd, B is parityeven, F is consequently parity-odd. P F = P q(v  B) = q{(
v)  B} =
F. The deflection of the particle is now to the right. However, we also realize that both situations are completely equivalent, namely, they are related through a rotation of the experiment by 1808 around an axis parallel to B. We also examine the effect of the operation of time reversal T. T F = T q(v  B) = q{(
v)  (
B)} = F. On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

50

On Chirality and the Universal Asymmetry

We notice that the Lorentz force is even with respect to time reversal. But, as we shall also later see in more detail (Sect. 4.1), we do not have a chiral situation. In our next example, we consider a steady current I, of magnitude I, flowing through a thin conductor. The current generates a magnetic field. According to the Biot–Savart law, the magnetic field (magnetic induction) at a point P due to the current is given by: B(P) = (m0 I/4p) s dl  r/r3 where dl is an infinitesimal vector pointing along the conductor (wire) in the direction of the current, and r represents the distance vector between the point on the wire where dl is located, and the point P where B is measured. m0 denotes the magnetic field constant. The line integral is to be taken along the entire length of the wire. We now consider the situation, where the current flows through a long helical solenoid (Fig. 3.1), and we are interested in the direction of the magnetic field inside the solenoid, on the helix axis. Symmetry considerations lead us to the conclusion that the magnetic field is directed along the helix axis. As may be found in many physics textbooks, the field along the axis inside a quasi-infinite coil is: Bz = m0 I(N/l), N being the total number of helical turns, and l the length of the coil. What interests us here in particular, and seldom finds mention in textbooks, is that Bz has opposite sign, depending on if the coil is left-handed or righthanded [1]. We furthermore recognize that B is the cross product of two polar vectors, and, therefore, is an axial vector, making it even with respect to the parity operation. However, the component of B along a polar vector will transform as a pseudoscalar. Suppose k is a unit vector along the helix axis, we have (see also Sect. 2.2): P B = B, P (B · k) =
B · k. Indeed, we find (considering for simplicity the proportionality between the infinitesimal quantities): dB ~ dl  r, dBz ~ (dl  r) · k ~ df. where f is the rotation angle around the helix axis. f, or df, are positive or negative, depending on if the rotation is anticlockwise or clockwise. With the situation chosen in Fig. 3.1, we notice that Bz is parallel to the downward flow of current in the right-handed coil (at left in Fig. 3.1), and antiparallel in the lefthanded coil (at right in Fig. 3.1). Because of the chirality of the conductors, we have enantiomorphous situations, clearly represented by the opposite directions of the magnetic field. The helicity h may correspondingly be defined as:

Reflections on Image and Mirror Image

51

Fig. 3.1. Flow of current through two enantiomorphous helical coils. The conducting wire is assumed to be comparatively thin. The helix is a space curve on the surface of a regular cylinder. A movement along the helix combines a rotation with a translation. In our drawing, the translations in both coils are parallel, the rotations are opposite. The red arrows show the direction of the magnetic field inside the coils.

h = B · I/(j B j · j I j). It is positive in Fig. 3.1 at left, negative at right. Our next, and most important, example is the electromagnetic radiation. Although the wave–particle duality of light is well-established and common knowledge today, there has been a long controversy on the fundamental nature of light that lasted about two centuries. For instance, Isaac Newton (1642 – 1727) assumed that light consisted of particles, while later, in the 19th century, the wave nature of light became generally accepted. The wave description proved to give consistent explanations for such fundamental optical phenomena as interference, reflection, and refraction. The picture of the transversality of light emerged, namely, that the oscillations of freely propagating light waves are perpendicular to the direction of propagation. James Clerk Maxwell (1831 – 1879) deduced the equations named after him, generalizing and unifying all electromagnetic phenomena. Maxwells equations directly also lead to a wave equation for light, confirming its electromagnetic nature. It was shown experimentally that, independent of the wavelength, light waves in vacuum propagate with a constant velocity, called the speed of light. Today, we know that the classical electromagnetic wave description of light is widely applicable. The photon (particle) nature manifests itself primarily in the absorption/emission of light by matter and in so-called quantum optics. The success of quantum electrodynamics with the quantization of the radiation field shows us that the particle description is, indeed, more fundamental. For our

52

On Chirality and the Universal Asymmetry

Fig. 3.2. Representation of a linearly polarized light beam (plane wave) at constant time (from F. Kneubhl, Repetitorium der Physik, Teubner, Stuttgart, 1975)

purposes, oriented towards symmetry relations and selection rules, a (semi-)classical wave description of light is nevertheless useful and adequate. As is already taught in elementary physics courses, a light beam is characterized by an electric vector E, a magnetic vector B ( = m0 H), and a wave vector k pointed in the direction of propagation. A linearly polarized plane wave in vacuum, or in an isotropic medium, is represented in Fig. 3.2. In our example, the electric vector E always points in  x direction, the magnetic vector H always in  y direction, z denotes the direction of propagation. One of the most interesting properties of light is that it can be circularly polarized. Without going into too much mathematical detail, we here give the expressions for the field quantities in their dependence on the phase f. The unit vectors in x, y directions are denoted by i, j, respectively. In what follows, and in deviation from Sect. 2.2, however, we denote by k the wave, or propagation vector of the light. In Fig. 3.2, k happens to be also perpendicular to i and j. For a right (R) circularly polarized beam: ER = E0R (i cos f
j sin f), BR = B0R (i sin f+ j cos f). For a left (L) circularly polarized beam: EL = E0L (i cos f+ j sin f), BL = B0L (
i sin f+ j cos f). The phase f is a function of time t and of the distance of propagation z: f= wt
nkz.

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Fig. 3.3. If the time variable t is kept constant, the tip of the electric vector as a function of the distance z describes a right-handed helix for right circularly polarized light (at left), and a left-handed helix for left circularly polarized light (at right). The (black) wave vector is chosen to be directed downwards (in analogy to the current vector in Fig. 3.1). The (red) spin vector is antiparallel in the right circularly polarized wave, parallel in the left circularly polarized wave.

w = 2pn, k  j k j = 2p/l; w denotes the angular frequency, n the frequency, l the vacuum wavelength, and n the refractive index. In vacuum, n = 1. One notices that at a fixed time, the tip of the electric field vector (and correspondingly also of the magnetic vector) describes a helix. For right circularly polarized light, the helix is right-handed, for left circularly polarized light, left-handed (Fig. 3.3). From the above equations, the reader may verify the following: a) For a given circular polarization, E and H (or B) are always perpendicular to each other. b) The superposition of right and left circularly polarized light of same frequency and amplitude gives linearly polarized light. We realize that any movement along a helix combines a rotation with a translation. A rotation is connected to an angular momentum. Does then circularly polarized light
carry an angular momentum? The wave–particle duality of light allows us to always consider light also as an ensemble of photons. If circularly polarized light has angular momentum, do then the photons possess some kind of spin? This question has been considered by a number of investigators already in the early 1930s [2]. It found an answer in 1935 in a remarkable experiment [3]. The principles of this experiment are as follows: Left (or right) circularly polarized light falls onto a quartz plate of thickness l/2 (or an odd multiple thereof). It is thereby converted into circularly polarized light of opposite handedness, namely right (or left). The l/2 plate is suspended by a glass fiber. If, through the conversion of the polarization, there is also a change of the angular momentum of the light, the quartz plate should (because of the conservation of angular momentum) be subjected to a torque. The torque is determined by measuring the torsion angle around the suspending fiber about which the plate

54

On Chirality and the Universal Asymmetry

rotates. The measured torque was found to correspond to an angular momentum of 2 h
times the number of photons traversing the quartz plate per second (
h = h/2p; h, Plancks constant). This led to the conclusion that circularly polarized photons must have a spin s of  1
h, oriented parallel or antiparallel, respectively, to the direction of propagation. It also confirms that circularly polarized light is a chiral influence and that it may be characterized by a helicity: h = s · k/(j s j · j k j) where s here denotes the photon spin. With our definition, the helicity equals + 1 for left circularly polarized light,
1 for right circularly polarized light. It must be noted, furthermore, that a single photon cannot be in a zero spin state. Likewise, a photon has no rest mass, and it does not possess any magnetic moment [4]. This also has to do with the fact that a magnetic moment changes its sign under charge conjugation, but a photon has no charge, and is its own antiparticle. We also immediately notice that the parity operation, or a reflection in a plane perpendicular to the direction of propagation, changes right circularly polarized light into left circularly polarized light, and vice versa. This illustrates the fact that circularly polarized light is a truly chiral influence, capable of distinguishing between molecular enantiomers (see Sect. 1.2).

3.2. On the Interaction of Light with Molecules We must now refine our description. From the previous section, we recall that light may at the same time be described as spatially and temporally oscillating electromagnetic waves, and as a collection of photons of particle-like nature. Many of the effects caused by the interaction of light with matter may be satisfactorily understood on the basis of the classical wave-like picture. However, it is always necessary to describe the molecules themselves quantummechanically. According to the laws of quantum mechanics, an isolated molecule may only remain in discrete stationary energy states. When a light ray of angular frequency w strikes a material system, such as a molecule or a crystal, the subatomic particles of which it is composed (atomic nuclei, electrons) are influenced in their relative motion and begin to oscillate with that frequency. If we have resonance, that is, if the product
hw happens to coincide with the energy difference DEba between two stationary molecular states b and a, we may have absorption. If w lies outside such resonances, a frequency-dependent polarization will be induced in the molecule, and the material system becomes a secondary emitter of radiation, like an antenna. This phenomenon of quasisimultaneous reemission of incident radiation is called light scattering.

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The immediate assumption concerning such a scattering process is that the electric field of light induces in the molecule an oscillating electric dipole, termed electric dipole polarization. Although this indeed may be the dominant effect, such a picture is nonetheless an oversimplification. As we have seen, an electromagnetic wave has both an electric and a magnetic field vector. A molecular system under the influence of a sinusoidal electromagnetic wave is, in fact, subjected to a variety of interactions. One merely has to remember that moving charges give rise to magnetic fields, and oscillating magnetic fields may, in turn, induce movements of charge. To put some order into this situation, it proves meaningful to express the interaction between a molecule and the electromagnetic light field mathematically in terms of a multipole series. Each term in this series corresponds to a particular, well-defined kind of interaction. In principle, the series is infinite. But the larger the wavelength of light is compared to the molecular dimensions, the more is it justifiable (long-wavelength approximation) to consider only the few lowest terms. These are: E1 the electric dipole–electric field interaction M1 the magnetic dipole–magnetic field interaction E2 the electric quadrupole–electric field gradient interaction. Higher terms are generally negligible. In fact, most of atomic and molecular optics and spectroscopy deals only with the interactions of type E1. For our purposes, namely the study of molecular chirality, it is, however, essential to consider also M1 and E2 in conjunction with E1. In the language of quantum mechanics, the operator for the interaction energy between a molecule and the light field is written: Hinteraction =
m · E
m · B
Q : rE + … The symbols m, m, Q represent the electric dipole, magnetic dipole, and electric quadrupole operators of the molecular electrons, respectively. If an elastic light-scattering process occurs, then, by definition, no net energy is transferred from the radiation field to the molecule. Thus, if the molecule in question was in a given stationary state a before the interaction with the radiation field, it will again be in the state a after the scattering process has occurred. However, during the time of the scattering process, that is, while the interaction of the molecule with the photon of energy
hw lasts, it is not in a pure stationary state. The molecule is in a time-dependent nonstationary state which may be described as a rapidly evolving superposition of many stationary states. The duration of the scattering process may be estimated from the complementarity of frequency and time. If the angular frequency of the interacting radiation is w,

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On Chirality and the Universal Asymmetry

Fig. 3.4. Graphs for the radiation-induced molecular polarization in the optical dispersion/ absorption effects (named at right). The general form of the tensor products of the corresponding susceptibility are given: m stands for a parity-odd electric dipole interaction, m for a parity-even magnetic dipole interaction. i represents the imaginary unit. The overall symmetry with respect to parity P and time reversal T is also noted for each case.

then the time which such an interaction lasts is t  1/w. If, for instance, we have visible or near-UV light, then we find t  10
15 s. The interaction of light with a molecule – in our present case a scattering process
may be conveniently represented and classified in terms of graphs (Fig. 3.4). One may interpret these graphs as follows [5] [6]: The horizontal bar describes the temporal evolution of the material system (molecule) from the initial state a via one (or more) nonstationary virtual states k, (l, ….,) and back to the initial state a. An arrow pointing downward represents a particular interaction between the molecule and the electromagnetic field (
collision with a photon). Every interaction is characterized by a frequency w. The arrow pointing upward denotes the response of the system due to the induced polarization (
antenna;
quasi-simultaneous release of the photon). Ordinary, or Rayleigh, light scattering is caused by the electric dipole polarization of an atom or molecule induced by the electric dipole interaction (E1) with the radiation field (Fig. 3.4, a). The symmetry of the effect is formally

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represented by the product of two electric dipoles mm, a quantity which is even with respect to parity. Rayleigh scattering is an ubiquitous phenomenon. It occurs in all matter, and is comparatively dominant. In dense media, light scattering in the direction of the primary wave [7] causes refraction. In matter, the index of refraction n generally is different from 1, the value in vacuum. The dependence of the index of refraction on the frequency or wavelength of the radiation is called the dispersion. Inside absorption bands we have, subject to selection rules, absorption and anomalous dispersion. The absorption is measured in terms of the absorption coefficient e, as we remember.

3.3. Natural Optical Activity In Sect. 1.2, we mentioned the discovery of optical activity in the early 19th century and the importance of this phenomenon for the development of stereochemistry. The cause for natural optical activity resides in the fact that a chiral molecule reacts differently to enantiomorphous forms of electromagnetic radiation, namely left (L) and right (R) circularly polarized light. We recall from Sect. 3.1 that these chiral forms of light are characterized by the helicity h = s · k/(j s j · j k j) which, in the first case, has the value + 1, in the second
1. The helicity is a parity-odd, time-even pseudoscalar. Already in Sect. 1.4, we mentioned that the chiral response of a molecule undergoing the transition a ! b is measured by the rotational strength

´(a ! b) =
i h aj m jbi · h bj m jai ~ eL
eR. This product of an electric dipole transition moment m and of a magnetic dipole transition moment m times the imaginary unit i, is also a parity-odd, time-even pseudoscalar. The nonvanishing rotational strengths in chiral molecules arise through the property that, in addition to the pure electric dipole interactions leading to ordinary Rayleigh scattering and refraction, there concomitantly occur magnetic dipole (M1) and electric quadrupole (E2) interactions with the radiation field. These contributions, albeit relatively small, have the same absolute value but the opposite sign for enantiomers. For a given chiral molecule, they manifest themselves in a different light scattering power for left and right circularly polarized light. In refraction, they correspondingly lead to a difference in the refractive index: nL „ nR. And in absorption, to a difference in the absorption coefficient: eL „ eR.

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On Chirality and the Universal Asymmetry

As may straightforwardly be shown theoretically, and is confirmed experimentally, the electric quadrupole contribution (E2) vanishes to zero in fluid media (in which the molecules are randomly distributed). So natural optical activity, as one routinely measures it in the chemical laboratory in liquid (or gaseous) samples, is due to the superposition of the mechanisms E1 and M1 (denoted by w(M) in Fig. 3.4, b). Suppose that, in a given chiral molecular sample, at a particular wavelength, we measure: nL = n + dn, nR = n
dn; eL = e + de, eR = e
de, for the enantiomer we correspondingly find: nL = n
dn, nR = n + dn; eL = e
de, eR = e + de (Table 3.1). As we have already mentioned in Sect. 1.2, the measurement of Dn = nL
nR as a function of the wavelength of the incident light, is called Optical Rotatory Dispersion (ORD). The measurement of De = eL
eR is termed Circular Dichroism (CD). A linearly polarized light beam may always be considered as the superposition of a right and a left circularly polarized wave of same frequency and amplitude. In an optically active medium, one circularly polarized beam gets phase retarded, the other, opposite one, accelerated. By introducing in the light phase f (see the expressions for circularly polarized light in Sect. 3.1), the difference for the refractive index given above, the reader may, by simple algebra, verify that in the chiral medium the plane of polarization of linearly polarized light indeed gets rotated by an angle a = dn · kz = (p/l)(nL
nR)z. Similarly, if linearly polarized light enters a chiral absorbing medium, the light will emerge as elliptically polarized, due to the different absorption of the right and left circularly polarized components. Practically speaking, optical activity is characterized by measuring the optical rotation outside absorption bands, and the ellipticity, or the difference in the absorption coefficient, within. From a more general and fundamental point of view, ORD and CD contain the Table 3.1. Relative Changes of the Refractive Index (n) and the Absorption Coefficient (e) Due to Natural Optical Activity Light polarization

(R)-Enantiomer

(S)-Enantiomer

Left circular

n + dn e + de n
dn e
de

n
dn e
de n + dn e + de

Right circular

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same amount of information. From a mathematical standpoint, they are Kronig–Kramers transforms of each other. Such is similarly the case for ordinary dispersion and absorption. It may also be proven mathematically that both the ORD and CD spectrum of a chiral molecule integrate to zero over the whole wavelength range. The latter may formally be expressed as the sum of the rotational strengths over all excited states: Sb ´(a ! b) = 0. It is called the Kuhn–Condon sum rule.

3.4. Magnetic Optical Activity Magnetic optical activity was observed for the first time in 1845 by Michael Faraday (1791 – 1867). He found that the plane of polarization of linearly polarized light, traveling parallel to an applied static magnetic field in a sample of heavy (amorphous) glass, was rotated. The so-called Faraday effect proved to be fundamental in that a ray of light could be influenced by an independently applied magnetic field. Furthermore, it was soon realized that the effect occurs in all matter. Magnetic optical activity manifests itself as magnetic optical rotatory dispersion (MORD) and as magnetic circular dichroism (MCD). Although these circularly differential effects are measured in exactly the same way as natural ORD and CD, they have nothing to do with chirality. The circularly differential response is induced by the applied magnetic field and not by any intrinsic structural property of the medium. The amplitude of the signals of MORD and MCD is proportional to the strength of the applied magnetic field. As we have previously seen, when left circularly polarized light is reflected in a mirror perpendicular to the direction of propagation, it is transformed into right circularly polarized light, and vice versa. Whereas natural optical activity (ORD or CD) is cancelled upon specular reflection of the light in the sample, magnetic optical activity (MORD or MCD) is reinforced, or additive. The graph representing the Faraday effect is shown in Fig. 3.4, c. We notice that the interaction with the radiation field of frequency w is of the electric dipole, or E1 type, as in Rayleigh scattering and refraction. In addition, we have the magnetic dipole interaction with the static magnetic field of frequency 0 (denoted by 0(M)). The symmetry of the effect is represented by the product mmm, derived from the quantum-mechanical expression for the corresponding susceptibility. This product is even with respect to parity and odd with respect to time reversal, because of the coupling to the external magnetic field.

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On Chirality and the Universal Asymmetry

Table 3.2. Relative Changes of the Refractive Index and the Absorption Coefficient Due to Magnetic Optical Activity Light polarization

Parallel B0 field

Antiparallel B0 field

Left circular

n + dn e + de n
dn e
de

n
dn e
de n + dn e + de

Right circular

The effect occurs in all matter irrespective of its structural symmetry, as already mentioned. From Table 3.2, we see that the Faraday effect for left circularly polarized light parallel to the magnetic field has the same sign as that for right circularly polarized light antiparallel, and vice versa. This leads to the conclusion that, from the point of view of the spatial symmetry, the relative sign of magnetic optical activity signals depends on a quantity which is even with respect to parity, namely the scalar product of the applied magnetic field B0 and the photon spin s. We consequently may define a relative orientation coefficient o = B0 · s/(j B0 j · j s j) which is parity-even and may take on the values  1; it is not to be confounded with a helicity. It is an ordinary, parity-even scalar. The lineshape of the signals which one obtains for MORD and MCD depends on the occurrence, or not, of magnetic degeneracies in the medium. A diamagnetic medium of low symmetry exhibits only so-called B-terms. In cases where magnetic degeneracies do occur, one may find bisignate A-terms; if the molecular ground state is paramagnetic, also temperature-dependent C-terms [8]. These details are here mentioned more for the sake of completeness, and because we will encounter a somewhat analogous situation with the magnetochiral effect. B-Terms, which occur in any molecular system, are in general quite weak. If, for instance, one measures MCD with a commercial circular dichrograph, the magnetic field strength should be of the order of several tesla.

3.5. The Magnetochiral Effect As we now realize, magnetic optical activity may occur in all matter, irrespective of its symmetry. Therefore, in a chiral substance to which a magnetic field is applied, one encounters both natural and magnetic optical activity. The question then arises if both effects, though of entirely different origin, are strictly additive. In other words, does a magnetic field applied to a chiral molecule have an influence which is different from that on an achiral molecule?

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Do magneto-optical effects occur in chiral molecules that do not exist in achiral ones, or vice versa? In the course of working out the theory of magnetic optical activity, the influence of a magnetic field on a molecule was considered in a general way [9]. Higher order terms were obtained that only occur in noncentrosymmetric media. A
nonreciprocal magneto-optical effect linear in the magnetic field but occurring only in noncentrosymmetric media was later measured in crystals [10]. A
new linear magnetorefractive effect was predicted to occur also in liquids, and its measurement with circularly polarized light was suggested [11]. The prediction was then made that, in chiral molecules, a magnetic field collinear with the propagation direction of incident radiation induces a difference between the absorption coefficient of enantiomers [12] [13]. It was not only shown that the new effect survives isotropic averaging, but that, in a chiral fluid medium, it is independent of the polarization of the incident light. This means that the response in an isotropic solution (or gas) of molecules of given chirality is the same for left or right circularly polarized, linearly polarized, or unpolarized light. The name
magnetochiral was introduced by Barron and Vrbancich, who also worked out the relation between magnetochiral dichroism and magnetochiral birefringence [14] [15]. As stated, the magnetochiral effect is not circularly differential. In an isotropic chiral medium, the index of refraction and the absorption coefficient (of any transition) undergo a shift in value when a static magnetic field is applied collinearly with the direction of propagation of the incident radiation of arbitrary polarization. The sign of the shift changes if the relative direction of field and beam is inverted. For a given relative direction, the sign of the effect changes on going from one chiral medium to its enantiomer. This situation is summarized in Table 3.3. Looking up Fig. 3.4, d, we notice in the corresponding graph the magnetic dipole interaction with the static magnetic field, denoted by 0(M), as well as the magnetic dipole interaction with the radiation field, w(M); in analogy to natural optical activity, in contrast to magnetic optical activity. The product immm derived therefrom is odd with respect to parity and odd with respect to time reversal, due to the presence of the applied magnetic field. It is seen that the Table 3.3. Relative Changes of the Refractive Index and the Absorption Coefficient Due to the Magnetochiral Effect Magnetic-field configuration

(R)-Enantiomer

(S)-Enantiomer

Parallel

n + dn e + de n
dn e
de

n
dn e
de n + dn e + de

Antiparallel

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On Chirality and the Universal Asymmetry

effect occurs only in noncentrosymmetric media, implying that fluids (liquids, gases) must be composed of chiral molecules. It may also be shown that there is a contribution to the magnetochiral effect deriving from the electric quadrupole interaction with the radiation field w(Q). This is, however, not represented in Fig. 3.4. The magnetochiral effect in matter is proportional to the strength of the applied static magnetic field. We recall that magnetic optical activity is also linear in the magnetic field strength. Comparing c and d in Fig. 3.4, we can make a rough estimate of the order of magnitude of the magnetochiral effect. For a given magnetic field strength B0, the ratio between the difference n››
n›fl in the magnetochiral effect, and the difference nL
nR in magneto-optical activity should be of the order of jhmij/jhmij or equivalently hM1i/hE1i. As already mentioned in Sect. 2.7, this corresponds roughly to a/2, where a = 1/137 is the fine-structure constant. In conclusion, and for a given strength of the magnetic field, the magnetochiral signals in molecular media should be 10
3 to 10
2 times smaller than corresponding magneto-optical signals. As we shall immediately see, the measurement of the relatively small magnetochiral effects in molecules is not entirely trivial, be it in absorption or refraction. This is certainly one of the reasons why the phenomenon of magnetochirality has not been discovered earlier. In spite of this, and from the viewpoint of basic symmetry relations, the magnetochiral effect is of fundamental interest. As mentioned in the preceding sections, the chirality of circularly polarized light resides in the simultaneous influence of the axial, parity-even photon spin vector s and the polar, parity-odd propagation vector k. The corresponding combination of rotation and translation leads to the helicity. Unpolarized light may be viewed as the superposition of wave trains of left and right circularly polarized light of same frequency, but of random amplitude and phase. It may also be considered as the superposition of linearly polarized beams with the plane of polarization randomly (quasi-evenly) distributed around the propagation direction. Such light is unable to distinguish between molecular enantiomers. From the particle viewpoint, unpolarized light may be seen as an ensemble of photons in which the spins statistically cancel. In this sense, unpolarized light
carries no angular momentum, and is geometrically and dynamically only characterized by the propagation vector. Let us now, simultaneously with an unpolarized beam of light, apply a static magnetic field B0 collinear with the propagation vector. The missing axial influence of the photon spin is now replaced by B0. We now regain a chiral influence. Depending on if the vectors B0 and k are parallel or antiparallel in a medium of given chirality, the sign of the influence changes. In this sense, we may define a magnetochiral helicity: h = B0 · k/(j B0 j · j k j).

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We compare it with the helicity for circularly polarized light, relevant for natural optical activity: h = s · k/(j s j · j k j) and with what we have called the
orientation coefficient for magnetic optical activity: o = B0 · s/(j B0 j · j s j). We recognize three pairwise combinations of the vector quantities k, s, and B0. In this sense, the three optical effects that we have just discussed are closely related. Our study will be pursued from this point of view in the next sections, in particular, also in Sect. 4.2.

3.6. The Magnetochiral Effect in an Atom Because of its spherical symmetry, a free atom can only become chiral through the parity-violating weak forces in the nucleus, as we have seen in Sect. 2.7. This parity-nonconserving influence gives rise to atomic natural optical activity and, as we must assume in the presence of an external magnetic field, also to an atomic manifestation of the magnetochiral effect. We shall see in the following that, in an atom, both effects reveal themselves to be one and the same phenomenon. This conclusion is of fundamental interest and may, with due caution, also be applied to parity violation in g-ray-emitting nuclei. The following discussion will by necessity become rather technical. The reader who is less familiar with atomic physics and elementary quantum mechanics may thus skip the details and merely read the conclusions at the end of the section. For a hypothetical
thought experiment, we choose the hydrogen atom, because of the simplicity of its structure. In spite of this simplicity, the atom stands at the basis of atomic physics and provides the point of departure for understanding the Periodic Table of the elements. The relatively straightforward symmetry considerations that we here shall make, unquestionably may have a more general applicability. We consider the atom to be in the 2p electronic state. The principal quantum number n thus has the value 2, and the orbital quantum number l the value 1. The magnetic quantum number m may take on the values + 1, 0, and
1. We are reminded that the spin quantum number s of the electron equals 1/2, and that the magnetic spin quantum number ms may take on the values + 1/2 and
1/2 (see Sect. 2.6 and 2.7). If we only take into account the Coulomb interaction between the electron and the proton, the 2p state is sixfold degenerate. It consists of six quantum (sub-)states of same energy, of same quantum numbers n, l, but with different values for m and ms. However, this description is too crude for our

64

On Chirality and the Universal Asymmetry

present purposes. We must also take into consideration the atomic fine structure, that is, the additional coupling (interaction) between the orbital angular momentum l and the spin angular momentum s of the electron. The so-called spin–orbit coupling splits the sixfold degenerate 2p state into a fourfold degenerate 2P3/2 state and a doubly degenerate 2P1/2 state, separated by a relatively small energy of 0.365 cm
1 (ca. 4.5  10
5 eV) [16]. The 2P3/2 state has a total angular momentum quantum number j = 3/2 ( = 1 + 1/2 = l + s); the 2P1/2 state has j = 1/2 ( = 1
1/2 = l
s). To every j state belong (2 j + 1) magnetic substates with magnetic quantum numbers mj =
j,
j + 1, …, j
1, j. Obviously, the values of mj for the 2P3/2 state are:
3/2,
1/2, + 1/2, + 3/2; and for the 2 P1/2 state:
1/2, + 1/2. If we apply to the atom a magnetic field, then the degeneracy of the magnetic substates gets lifted, i.e., every quantum state obtains a different energy. If, furthermore, the magnetic field is not too strong, that is, if the corresponding energy shifts are not too large compared to the spin–orbit splitting, then one finds an energy level scheme as shown qualitatively in Fig. 3.5. Suppose that, through some external influence, the hydrogen atom is prepared in the 2P3/2 state(s), an emission of radiation might then be induced into the 2P1/2 state(s). These transitions would lie in the microwave region of the electromagnetic spectrum. They would be of relatively low intensity, because they are electric dipole forbidden and only magnetic dipole (M1) and electric quadrupole (E2) allowed. Nevertheless, they are of particular interest to us here, for reasons which will become apparent shortly. There are, in all, four possible transitions, which lead to emitted circularly polarized radiation. They obey the selection rules: Dmj = mj (lower level)
mj (upper level) =  1. As shown in Fig. 3.5, this corresponds to the transitions I, II, III, IV, with Dmj values
1, + 1,
1, + 1, respectively. The transitions I, IV, and II, III have pairwise the same intensity. We choose either one of these pairs, and now turn to Fig. 3.6. The direction of the circularly polarized emission is collinear with the applied magnetic field B0. By the transitions with Dmj = + 1, and Dmj =
1, a unit of angular momentum is transferred from the atom to the radiation field, which, in one case, is oriented antiparallel, in the other case, parallel to the magnetic field direction. In one case, the emitted photon has its spin s antiparallel to B0, in the other case parallel. The orientation coefficient o, defined in Sect. 3.4, correspondingly has respective values
1, and + 1. In the wave picture, the emitted light consists of a forward beam and a backward beam. In the first case, with Dmj = + 1, the forward beam is right (R) circularly polarized, the backward beam left (L) circularly polarized. In the case with Dmj =
1, it is

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65

Fig. 3.5. Energy splitting scheme of the hydrogen 2p electronic states under the influence of spin– orbit coupling and of an applied magnetic field. The magnetic dipole-/electric quadrupole-allowed transitions that give rise to circularly polarized emission are indicated.

the opposite (see Fig. 3.6,a). In either case, the selectivity between L and R is stringent, but the intensity of the forward and backward beam is always exactly the same: IL = IR  I. In summary, what we consider here, is the atomic manifestation of magnetic optical activity, called the Zeeman effect. Until now, we have taken into account the spin–orbit coupling and the influence of the applied static magnetic field, but we have ignored the parityviolating weak interaction. (Neither have we said anything about the hyperfine interaction between electron spin and nuclear spin, nor about relativistic effects; but this is of no direct importance to us in this context.) The parity-nonconserving interaction, on the other hand, is of course of central interest to us. In Sect. 2.7, we had already encountered the quantum-mechanical operator for the parity-violating weak interaction acting on an atomic electron. Written in adapted form: VPV = C {se · p, d(r)}

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On Chirality and the Universal Asymmetry

Fig. 3.6. Emission properties of an atom. a) In the presence of a magnetic field; b) in the presence of a magnetic field and of a parity-violating potential; c) only in the presence of a parity-violating potential. For details, see the text.

where se is the spin operator of the electron, p the momentum operator, d(r) the delta function of the position operator of the electron with respect to the nucleus, and the curly brackets indicate that we have to take the anticommutator. The factor C is the product of constants. Now, before noting how VPV acts on the individual atomic states, we consider some general quantummechanical relations: designating by H0 the hamiltonian (energy operator) for the atom, comprising Coulomb interaction and spin–orbit interaction, the following commutators are found to vanish: [H0, J2] = 0, [H0, Jz] = 0, [H0, P] = 0. It shows us that the operators of the total angular momentum J squared, of the component of angular momentum in direction of an applied magnetic field Jz, and the parity operator P are constants of the motion. This implies that the eigenfunctions of H0 are simultaneously also eigenfunctions of J2, of Jz, and of P.

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The corresponding quantum states are characterized by (beside n), the quantum numbers j and mj, and they are either even or odd with respect to P. It may similarly be shown that the parity-violating weak interaction operator VPV also commutes with the operators J2 and Jz, but quite evidently not with the parity operator: [VPV, J2] = 0, [VPV, Jz] = 0, [VPV, P] „ 0! One concludes therefrom that j and mj are still good quantum numbers. However, the perturbation VPV mixes states with the same j and mj values, but with different parity (different l values). In our case, in particular, the 2P1/2 state thereby obtains a small contribution from the energetically very close-lying 2S1/2 state (only separated by the Lamb shift [16]). This gives the M1- and E2-allowed transitions described above a weak electric dipole-allowed E1 component [17]. It may then be shown that this electric dipole-allowed influence leads to a fore-aft asymmetry in the intensity of the emitted light [18]. In particular, in the case Dmj = + 1, the intensity of the forward right circularly polarized beam decreases to a value I
DI, while that of the backward left circularly polarized beam increases correspondingly to I + DI. In the case Dmj =
1, the polarization of the forward and backward beam is interchanged, the fore-aft asymmetry is thereby reversed. This is evidently the atomic manifestation of magnetochiral dichroism (Fig. 3.6,b). The asymmetry resides in different orientations of B0 with respect to the wave vector k, corresponding to changes of the sign of the helicity h, as defined in the previous section. This also strikingly reminds us of the fore-aft asymmetry in the emission of g-rays by magnetically polarized nuclei (see Fig. 2.5). Finally, we consider the situation where the static magnetic field B0 is turned off and only VPV is active. The transitions Dmj =  1 both have now the same energy and appear at the same wavelength. The axis of quantization for the magnetic substates disappears with the fading magnetic field. The emission in any direction consists of a superposition of left and right circularly polarized light of different amplitude, that is, of elliptically polarized light. For the corresponding intensity, we find IL
IR = 2 DI. We thus expect to measure natural optical activity, revealed as circular dichroism in emission (Fig. 3.6,c). In the absence of parity violation and of the fore-aft asymmetry, an atomic sample would emit unpolarized light in any direction. By our exploratory experiment, we have encountered in one and the same simple hydrogen atom, magnetic optical activity, the magnetochiral effect, and natural optical activity. We notice that, in such a spin–orbit-coupled atomic system, the magnetochiral effect and natural optical activity are not only closely related, but basically the same phenomenon. The magnetochiral effect in atoms

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On Chirality and the Universal Asymmetry

is the manifestation of natural optical activity in pure, magnetically nondegenerate states. We wish to emphasize the formal analogy with atomic nuclei. Under the heading
Parity violation in nuclear forces in the book Weak Interactions of Leptons and Quarks by E. D. Commins and P. H. Bucksbaum [19] we find: Circular polarization and asymmetry of g-rays: Parity violation results in circular (in our atomic example, elliptical) polarization of g-emission from unpolarized nuclei or fore-aft asymmetry in g-emission from polarized nuclei. Although the hydrogen atom is theoretically well-suited to illustrate basic principles, the experimental investigation of atomic chirality becomes only promising in atoms of high Z number (see Sect. 2.7).

3.7. The Magnetochiral Effect in Molecular Fluids From our excursions into the physics of nuclei and of atoms, we now return to molecules which are chiral due to their particular geometric structure. Between the time when the magnetochiral effect in molecules was predicted, namely around 1980, and the dates of the first successful measurements, about 15 years elapsed. The experimental task is not easy, due to the fact that the magnetochiral signals at given external magnetic field strength are expected to be 10
3 to 10
2 times smaller than those for magnetic optical activity, possibly even smaller. The magnetochiral effect is overshadowed by the Faraday effect and, with the comparatively low magnetic fields accessible in the laboratory, also by natural optical activity. To measure magnetochiral dichroism and birefringence in a gas or fluid where molecules are randomly oriented, systematic advantage has to be taken of the already mentioned important property: the magnetochiral effect is independent of the state of polarization of the incident light. This may be well understood by the fact that the effect depends on the scalar product of the applied magnetic field B0 and the propagation vector k of the light. In an isotropic medium, k is perpendicular to both the electric field E(w) and the magnetic field B(w) of the radiation, and thus not directly affected by changes of polarization (Fig. 3.7). Magnetochiral dichroism will of course not only manifest itself in absorption but also in emission [15]. The first detection of the magnetochiral effect in a molecular probe was performed by measuring the emission of a solution of the highly chiral tris[3-(trifluoroacetyl)camphor]europium(3+) complex [20]. The sample was excited with unpolarized light of 350 nm, and the luminescence in the 590 – 620 nm wavelength region in the directions parallel and antiparallel to the applied alternating magnetic field was collected by optical fibers and compared. The relative intensity difference (fore-aft asymmetry) is given by:

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Fig. 3.7. Experimental setup for detecting the magnetochiral anisotropy in emission. The components of the apparatus are: light source L; sample S; monochromators M; phase-sensitive detector LA. The applied magnetic field B0 cos Wt has a maximal field strength of 0.9 Tand is modulated with the frequency W = 0.9 Hz (from [20]; copyright permission via CCC). Notice the analogy of the setup with the nuclear physics experiment schematically illustrated in Fig. 2.5.

DI/I =

I""
I"# . I"" þ I"#

It was found to be 5  10
3 for the 5D0 ! 7F1 transition, and 4  10
4 for the 5 D0 ! 7F2 transition in the investigated complex. This is surprisingly high and is due to the fact that these transitions correspond to so-called A-terms. The first measurement of magnetochiral birefringence followed soon after [21][22], using interferometry. The relative shift of the refractive index n is given by: Dn/n =

n""
n"# . n

This shift, expected in a diamagnetic molecular probe outside resonances in a magnetic field of 5 T, is estimated to be of the order of 10
8 to 10
6, which is indeed very small. Fig. 3.8 shows the experimental setup. An incident beam of linearly polarized light of 633 nm produced by a Helium-Neon laser is divided into two subbeams. The probe consists of two compartments, one containing the d-, the other the l-enantiomer of the substance to be investigated. The subbeams

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On Chirality and the Universal Asymmetry

Fig. 3.8. Interferometric magnetochiral Dn measurement. The subbeams 1 and 2 travel in opposite directions through both probe compartments, containing the d- and l-enantiomer, respectively. Correspondingly, the two beams travel parallel/antiparallel to the applied magnetic field (B  B0). Thereby, natural optical activity and Faraday rotation are compensated to zero, while the magnetochiral phase shift is enhanced. The experiment is repeated with opposite external magnetic field (B+, B
) (from [21]).

travel in opposite directions through both compartments in which there is a static axial magnetic field of 5 T. The experiment is conceived in such a way that natural and magnetic optical rotation are compensated to zero, while the magnetochiral signal is enhanced. One subbeam gets phase-retarded by the magnetochiral effect, the other one phase-accelerated. The two subbeams are brought to interference, and the magnitude and the sign of the relative phase shift are then measured. To enhance the weak signals, high concentrations of the enantiomeric material samples are required. Therefore, the first measurements were performed with the highly chiral compound 3-(trifluoroacetyl)camphor, which is liquid at room temperature and easily obtainable in both enantiomeric forms. The difference in refractive index so obtained was Dn = + 6  10
8 referred to the l-enantiomer in a static field configuration parallel to the direction of propagation of the incident light beams. Notice: For semantic and practical reasons, the light polarization is here, as in the previous sections, designated by L, R; the chirality of the molecular probe by d, l; see Sect. 1.4.

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3.8. Magnetochiral Photochemistry In Sect. 1.5 on asymmetric synthesis, we considered partial photoresolution with circularly polarized light as a possible means to obtain from a racemic mixture an excess of one enantiomer with respect to the other. It may be shown that, under constant irradiation of circularly polarized light of given handedness, L say, one can obtain a steady-state enrichment of one enantiomer with respect to the other due to the natural circular dichroism. One enantiomer, for instance d at a chosen wavelength, gets preferentially excited. From the excited state, d may either revert as d to the ground state, or convert to l. It may then be shown that this situation in time leads to an excess of l in the ground state. If the thermal racemization in the ground state is comparatively very slow compared to the photoracemization in the excited state, the excess is maximized. This excess may then be shown to be equal to the relative difference of the absorption coefficients: ðRÞ DcðLÞ clðLÞ
cðLÞ edðLÞ
elðLÞ eðLÞ edðLÞ
eðLÞ d d
ed l = 2 = 2 = = ðLÞ . ðLÞ ðLÞ ðLÞ ðLÞ ðRÞ c e cl þ cd ed þ el ed þ ed

cðLÞ l;d are the respective resulting ground state concentrations of the enantiomers, ðLÞ el;d the absorption coefficients, under left (L) circularly polarized light, e the average absorption coefficient of the racemic mixture. Under right (R) circularly polarized irradiation, the photoenrichment should of course have the same absolute value, but opposite sign: Dc(R) =
Dc(L). Using a racemic solution of the chiral trisoxalatochromate(III) complex (of symmetry D3), the partial photoresolution was experimentally confirmed [23] (see also Sect. 1.5). The existence of magnetochiral dichroism suggests a similar, albeit significantly smaller magnetochiral photoresolution under the influence of unpolarized light collinear with a static magnetic field. In analogy to the abovementioned situation, the maximum excess obtainable should be: Dcð""Þ cð""Þ
cð""Þ eð""Þ
eð""Þ eð""Þ
eð"#Þ eð""Þ
eð""Þ d d l d d d l = lð""Þ = 2 = 2 = ð""Þ ð""Þ ð""Þ ð""Þ ð"#Þ c e cl þ cd ed þ el ed þ ed

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On Chirality and the Universal Asymmetry

where the arrows ("") or ("#) denote parallel or antiparallel configuration of the applied magnetic field with respect to the direction of light propagation in the sample. Evidently we have: Dcð""Þ = Dcð##Þ =
Dcð"#Þ =
Dcð#"Þ . Repeating the experiment by Stevenson and Verdieck, using the same trisoxalatochromate(III) complex, Rikken and Raupach found, upon irradiation at 701 nm with circularly polarized light, the natural enantioselective partial photoresolution to be of absolute magnitude: jDcðRÞj jDcðLÞj = = 2.2  10
3. c c Using unpolarized light and an applied magnetic field of 7.5 T, the magnitude of the magnetochiral partial photoresolution was then determined [24]: ð""Þ Dc c

ð"#Þ Dc = = 1.7  10
6 T
1. c

This result confirms the order of magnitude expected from theory. The potential interest of enantioselective magnetochiral photochemistry scarcely lies in any foreseeable practical application. However, it may possibly be of some significance from an astrophysical or, rather, astrochemical point of view. It may suggest a mechanism in nature, until recently unknown, by which, starting from a primeval racemic mixture, the concentration of a given chiral species increases compared to that of its enantiomer [25] [26]. Previously, the following ways have mainly been envisaged for such nature-induced asymmetric changes in concentration: 1) A direct kinetic influence of the energy difference between enantiomers due to parity violation (see Sect. 2.8). 2) Chemical interaction of a racemic mixture with an external chiral material environment. 3) Spontaneous resolution of a racemic mixture through enantioselective crystallization. 4) Differential interaction of enantiomers with circularly polarized radiation of given handedness (see above). The first possibility will be reconsidered in Sect. 6.4, 10.2, and 10.5. The second lies at the origin of many known asymmetric syntheses (Sect. 1.5 and 9.6). The third possibility may lead to homochirality, but in itself does not give preference to any absolute handedness (Sect. 1.3, 7.1, and 9.11). The fourth, as we have seen above, is due to natural optical activity and may manifest itself as photoenrichment by photoisomerization, as photodestruction or as asymmetric photosynthesis (Sect. 1.5, 10.4 – 10.6).

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The particularity of the magnetochiral effect resides in its independence on light polarization in fluid media. The disadvantage lies in its relative smallness, except at very high field strengths. Based on the above-mentioned experiment by Rikken and Raupach, we conclude that in a magnetic field of 1 G = 10
4 T, of the order of the Earths magnetic field, the relative enrichment Dc/c reduces to about 10
10. It must also be emphasized that, in contrast to parity violation, which is assumed to be universal, the magnetochiral effect is also a
local influence. The induced enantiomeric excess depends on the relative direction of the magnetic field with respect to the direction of propagation of the incident light at the site of the molecules considered. On the other hand, with an energy difference between enantiomers due to parity nonconservation of the order of 10
14 J mol
1 (see Sect. 2.8), the thermal equilibrium between enantiomers there corresponds to a value of Dc/c of the order of only 10
18 to 10
17 at ambient temperature, which indeed is very small. The fundamental question, if the magnetochiral effect may have played a role in the prebiotic evolution of molecular homochirality, will be explored and discussed in an astrochemical and geochemical context later on (mainly in Sect. 10.5). As we then shall consider in more detail, it seems beyond doubt that the chiral selectivity needed for biological evolution must have developed through sequences of innumerable (photo-)chemical steps. To trace these steps back in time and to try to identify, stepwise, the dominant enantioselective influences, is indeed a rather formidable task.

REFERENCES [1] R. P. Feynman, Six Not-so-Easy Pieces, Penguin Books, London, 1997, Chapt. 2 (Symmetry in Physical Laws). [2] M. N. Saha, Y. Bhargava,
The Spin of the Photon, Nature 1931, 128, 870. [3] R. A. Beth,
Mechanical Detection and Measurement of the Angular Momentum of Light, Phys. Rev. 1936, 50, 115. [4] L. D. Barron,
Charge Conjugation Symmetry and the Nonexistence of the Photons Static Magnetic Field, Physica B 1993, 190, 307. [5] J. F. Ward,
Calculation of Nonlinear Optical Susceptibilities Using Diagrammatic Perturbation Theory, Rev. Mod. Phys. 1965, 37, 1. [6] G. Wagnire,
The Magnetochiral Effect and Related Optical Phenomena, Chem. Phys. 1999, 245, 165. [7] W. Kauzmann, Quantum Chemistry, Academic Press, New York, 1957. [8] A. D. Buckingham, P. J. Stephens,
Magnetic Optical Activity, Ann. Rev. Phys. Chem. 1966, 17, 399. [9] M. P. Groenewege,
A Theory of Magneto-optical Rotation in Diamagnetic Molecules of Low Symmetry, Mol. Phys. 1962, 5, 541. [10] V. A. Markelov, M. A. Novikov , A. A. Turkin,
Experimental Observation of a New Nonreciprocal Magneto-optical Effect, JETP Lett. 1977, 25, 378. [11] N. B. Baranova, B. Ya. Zeldovich,
Theory of a New Linear Magnetorefractive Effect in Liquids, Mol. Phys. 1979, 38, 1085.

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On Chirality and the Universal Asymmetry

[12] G. Wagnire, A. Meier,
The Influence of a Static Magnetic Field on the Absorption Coefficient of a Chiral Molecule, Chem. Phys. Lett. 1982, 93, 78. [13] S. Woz´niak, R. Zawodny,
Magneto-spatial Dispersional Variation of the Refractive Index in Molecular Fluids. I. Non-interacting Molecules, Acta Phys. Pol., A 1982, 61, 175. [14] L. D. Barron, J. Vrbancich,
Magneto-chiral Birefringence and Dichroism, Mol. Phys. 1984, 51, 715. [15] G. Wagnire,
Magnetochiral Dichroism in Emission. Photoselection and the Polarization of Transitions, Chem. Phys. Lett. 1984, 110, 546; G. Wagnire,
The Influence of a Static Magnetic Field on the Optical Properties of Chiral Molecules, Z. Naturforsch., A: Phys. Sci. 1984, 39, 254. [16] B. H. Bransden, C. J. Joachain, Physics of Atoms and Molecules, Longman, London, 1983, p. 202 and 232. [17] F. Curtis Michel,
Neutral Weak Interaction Currents, Phys. Rev B: Solid State 1965, 138, 408. [18] G. Wagnire,
Magnetochiral Dichroism as a Measure of Parity Nonconservation in an Atomic System, Z. Phys. D: At., Mol. Clusters 1988, 8, 229. [19] E. D. Commins, P. H. Bucksbaum, Weak Interactions of Leptons and Quarks, Cambridge University Press, Cambridge, 1983, Chapt. 9. [20] G. L. J. A. Rikken , E. Raupach,
Observation of Magneto-chiral Dichroism, Nature 1997, 390, 493. [21] P. Kleindienst, G. Wagnire,
Interferometric Detection of Magnetochiral Birefringence, Chem. Phys. Lett. 1998, 288, 89. [22] N. G. Kalugin, P. Kleindienst, G. Wagnire,
The Magnetochiral Birefringence in Diamagnetic Solutions and in Uniaxial Crystals, Chem. Phys. 1999, 248, 105. [23] K. L. Stevenson, J. F. Verdieck,
Partial Photoresolution. Preliminary Studies on Some Oxalato Complexes of Chromium(III), J. Am. Chem. Soc. 1968, 90, 2974; K. L. Stevenson, J. F. Verdieck,
Partial Photoresolution. II. Application to Some Chromium Complexes, Mol. Photochem. 1969, 1, 271. [24] G. L. J. A. Rikken, E. Raupach,
Enantioselective Magnetochiral Photochemistry, Nature 2000, 405, 932. [25] G. Wagnire, A. Meier,
Difference in the Absorption Coefficient of Enantiomers for Arbitrarily Polarized Light in a Magnetic Field: A Possible Source of Chirality in Molecular Evolution, Experientia 1983, 39, 1090. [26] L. D. Barron,
Chirality, Magnetism and Light, Nature 2000, 405, 895.

4. The PT Triangle
Ce n
est point le corps des lois que je cherche, mais leur me (
It is not the body of the laws that I seek, but their spirit) Charles de Montesquieu (1689 – 1755) (from LEsprit des Lois)

4.1. Definition of Chirality The study of crystal structure led to the notion of symmetry operation. Symmetry operations and their interrelations form the theoretical basis of the science of crystallography. Ren-Just Hay (1743 – 1826) is said to have accidentally dropped a large piece of calcite. It splintered into smaller pieces. He noticed that the smaller fragments had essentially the same form as the original piece (G. Cuvier, cited in [1]). Hay exclaimed:
Tout est trouv! (All is found!) Crystals, irrespective of their size, appear in definite symmetries, and to characterize these symmetries, the notion of symmetry operation, or symmetry element, is essential. Imagine a perfectly grown crystal fixed in space. One will find a certain number of spatial operations that bring the crystal into a new position indistinguishable from the original one. If an observer during the operation is blindfolded, he will not be able to know if the crystal has at all been moved or not. One essentially distinguishes between four kinds of elementary symmetry operations: 1) Rotations about one or more symmetry axes that lie inside the crystal. The existence of such a symmetry operation Cn implies that rotation by an angle of (3608/n) around the corresponding axis brings the crystal into a position that cannot be distinguished from the original one. 2) Reflections in one or more symmetry plane(s) s lying inside the crystal. If one moves any point of the crystal on one side of the plane, in a motion perpendicular to the plane, to the same distance on the other side, the crystal appears unchanged. 3) Reflection in a point in the crystal called a center of inversion i. In the presence of such a center of inversion, any line in the crystal going through i connects, at equal distance but in opposite direction from i, equivalent points of the crystal. Thus by such an inversion, the crystal as a whole remains unchanged. 4) Rotation–Reflections (also called improper rotations). Such a symmetry operation, denoted Sn, corresponds to a rotation by (3608/n) around a given On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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On Chirality and the Universal Asymmetry

axis inside the crystal followed by reflection in a plane perpendicular to that axis. The reader will notice that the operation S1 corresponds to reflection in a plane s, and that the operation S2 is identical with the inversion i. S3, applied an arbitrary number of times consecutively, is always equivalent to the sequence of two separate symmetry operations, namely C3 and s. But for n
4 and even, Sn represents independent symmetry operations. The higher the symmetry of a crystal, the more symmetry operations will it allow. The manifold of all symmetry operations of a crystal form the point group of the crystal. Every symmetry operation represents a symmetry element of that group. Among crystals in nature, a total of 32 different point groups occur. The symmetry criteria here summarized are, of course, not only applicable to crystals, but also to individual molecules, or to any three-dimensional body. To be chiral, a body may not contain any rotation–reflection axis Sn. In other words, it may have neither symmetry planes, nor any center of inversion, nor any higher rotation–reflection symmetry. But chirality does not necessarily imply total asymmetry. A chiral molecule or crystal may indeed exhibit true rotational symmetry. For instance, a regular finite helix exhibits a twofold symmetry axis C2. The symmetry axis goes through the midpoint of the helix and intersects the longitudinal axis of the helix at a right angle. As already mentioned (Sect. 1.3), an n-bladed propeller contains a Cn-axis collinear with the propeller axis, and n C20 -axes perpendicular to it. The corresponding point groups are designated as Dn. From a structural point of view, the criterion for the existence of chirality is easily stated. In particular, we notice that planes of reflection symmetry and inversion symmetry destroy chirality, whereas pure rotation symmetry is compatible with it. But from what we have seen in previous chapters of this book, a purely static definition of chirality is indeed insufficient. A dynamic definition, which, as a matter of fact, we have already several times encountered, is necessary. Barron, in a very thorough investigation on
Symmetry and Molecular Chirality proposes the following general definition [2]:
True chirality is exhibited by systems that exist in two distinct enantiomeric states that are interconverted by space inversion, but not by time reversal combined with any proper spatial rotation. The concept of chirality is thus connected to both the notions of space inversion, implying the parity operation P, and time reversal T. (By
proper spatial rotation, a rotation of the system as a whole is here meant; not to be confounded with rotational symmetry.) This allows us to speak of the chirality of any object in movement, irrespective of if this object exhibits structural chirality or not. As an example, Barron shows that a cone spinning around its axis is achiral, whereas a spinning cone translating along its axis is chiral. Such is also a

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translating spinning sphere, provided, of course, that the angular momentum vector of the rotation is not perpendicular to the momentum vector of the translation. From the definitions that we have already mentioned (Sect. 2.2 and 2.3), the reader should be able to verify the above statements for himself. The more general definition allows us to speak of the chirality of a neutrino or a muon (Sect. 2.4), of a metastable, parity-nonconserving fluorine nucleus (Sect. 2.5), or of a photon (Sect. 3.1). The existence of chirality may then be expressed by a nonvanishing helicity, a parity-odd, time-even pseudoscalar quantity. For elementary particles, an interesting aspect arises through the principle of relativity [2]: only particles that have no rest mass and travel with the speed of light, such as photons, have definite chirality or helicity. The helicity of particles with rest mass will depend on the relative speed of the observer. Depending on if the observer comes closer or moves away, the sign of the particles linear momentum will change, thereby also changing the relative sign of the helicity.

4.2. The Triangle of P and T Symmetry Any experiment that measures chirality distinguishes between enantiomers. The result of the measurement is odd with respect to the parity operation P applied to the object of the experiment, and even with respect to time reversal T. In Fig. 4.1, we denote this by P  T +. A chiral system, or interaction, arises in principle through the concomitant effect of linear movement in space, represented by linear momentum p, which is parity-odd and time-odd, P  T , and of rotatory movement or angular momentum l, which is parity-even and time-odd, P + T . In the case of natural optical activity, we consider the wave vector of light k, representing the transport of electromagnetic energy, and the spin of the photon s, corresponding to an inherent rotation of the photon. For the magnetochiral effect, the wave vector of light k is to be combined with the axial magnetic field B. As represented in Fig. 4.1, the concomitant influence of 1 and 2 results in 3. The question may now be raised, if triangular permutation of 1, 2, 3, may correspond to meaningful physical phenomena. For instance, the combined influence of 1 and 3 might result in 2. In words: the influence of unpolarized light in a chiral medium generates a magnetic field. Indeed, as we shall see in more detail, a mechanism exists by which this is realized. It leads to the effect of inverse magnetochiral birefringence, which we shall discuss in the following section. What does the combination of 2 and 3, resulting in 1, imply: a static magnetic field under a chiral influence may generate electromagnetic radiation? This is certainly not the kind of experiment that appears promising in the

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On Chirality and the Universal Asymmetry

Fig. 4.1. The PT Triangle (see text). The symbols represent the following quantities: p, the linear momentum; l, the angular momentum; I, the electric current; m, the magnetic moment; B, the magnetic field strength (induction); k, the wavevector of light; s the photon spin.

laboratory. But perhaps such a phenomenon might occur in an astrophysical situation (see Sect. 5.6). If, in our considerations, we replace the transport of electromagnetic energy by the transport of charge, in other words, by the current I, we will encounter some well-known phenomena. For instance, the influence of 1 and 3 resulting in 2, implying the generation of a magnetic field by current flowing through a chiral conductor, has been discussed in Sect. 3.1. On the other hand, the increase or decrease of a steady current in a chiral conductor by a static magnetic field parallel or antiparallel to the current, corresponding to 2 and 3 leading to 1, is a relatively novel effect and will be the topic of Sect. 4.4. We expect that what is allowed by symmetry should exist. But symmetry considerations alone do not indicate the detailed mechanisms by which processes occur, and they cannot predict if a given process will be big or small. A physical effect may in fact exist, but it may not be measurable in the laboratory, because, under laboratory conditions, it is too small to be detected. This, however, does not preclude the effect from occurring and being an influence in the universe where conditions are more extreme and the time is long.

4.3. The Inverse Magnetochiral Birefringence We now go back to Sect. 3.2 and recall what we noted concerning light scattering: when a light ray of angular frequency w strikes a material system, such as a molecule or a crystal, the charged particles of which it is composed

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(atomic nuclei, electrons) are influenced in their relative motion and start to oscillate with the frequency w. If the light frequency coincides with a resonance frequency of the system, the light, or part of it, may get absorbed. Otherwise, by this frequency-dependent induced polarization, the material system becomes a secondary emitter, like an antenna. This phenomenon of immediate reemission of radiation is called light scattering. In ordinary, or Rayleigh, light scattering, the incident radiation of frequency w will be reemitted at the same frequency w. When the intensity of the incident light is high, and in spite of the very short duration of such a scattering process, the molecule may behave as if it interacted quasi-simultaneously more than once with the radiation (photon) field, thereby producing scattered radiation, not only of the basic frequency w, but containing combinations thereof, caused by second harmonic generation: w + w = 2w; or third harmonic generation: w + w + w = 3w, etc. The frequency combinations may also be subtractive, for instance, w  w = 0, leading to optical rectification. These nonlinear optical effects [3] have not been detectable before the invention of the laser, in the 1960s. The magnitude of such effects depends significantly on the intensity and coherence of the incident radiation. Optical rectification corresponds to a static electric polarization of the medium induced by the incident light. It occurs only in noncentrosymmetric media. In all liquids, even in chiral ones, it is forbidden. Of more interest to us here are related difference-frequency effects that induce constant magnetic fields. If the incident radiation is circularly polarized, a static magnetization will be induced in matter of any symmetry. This so-called inverse Faraday effect [4] [5] has indeed been measured, also in liquids. However, our main attention belongs to a third difference-frequency phenomenon, the inverse magnetochiral birefringence: when a coherent beam of light of arbitrary polarization travels in a noncentrosymmetric fluid medium, composed of randomly oriented chiral particles, it is predicted to induce a constant magnetization parallel or antiparallel to the direction of propagation. The effect has opposite sign for enantiomers (Fig. 4.2). Under normal laboratory conditions, the induced magnetization should be very weak, yet detectable by modern means of measurement. With a radiation field intensity of 108 W cm2 in a diamagnetic fluid, a magnetic induction of the order of 1012 to 1011 T is estimated [6] [7]. The inverse magnetochiral birefringence is related to the inverse Faraday effect in the same way as
ordinary magnetochiral birefringence is related to the
ordinary Faraday effect. From the point of view of the interaction with the radiation, the inverse Faraday effect has pure electric dipole, E1, character. On the other hand, the inverse magnetochiral birefringence arises through a combined E1 and M1 (or E2) interaction of the medium with the light field (see Fig. 3.4). In a figurative sense, the magnetization, when induced by the inverse Faraday effect, may be thought of as arising through a transfer of photon spin

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Fig. 4.2. The inverse magnetochiral birefringence. When a coherent beam of unpolarized, or linearly polarized, light (incident light here drawn as blue wavy line) travels in a medium composed of randomly oriented chiral molecules, it is predicted to induce a constant magnetization parallel or antiparallel to the direction of propagation (red arrows). The absolute signs depend on the absolute configuration, (R) or (S), of the enantiomers.

angular momentum. In the case of the inverse magnetochiral birefringence, it is related to the magnetic dipole interaction with the radiation. The particular significance of the inverse magnetochiral birefringence resides in the fact that it is induced by arbitrarily polarized light. In principle, the effect can occur anywhere in the universe where there is light, and where parity is violated. Attempts to measure the effect are presently being undertaken with chiral molecular samples exposed to radiation of UV/VIS wavelengths. However, one may, for instance, imagine astrophysical situations where atomic nuclei, in which parity is violated, are exposed to g-radiation of extreme intensity and thereby get magnetically polarized. We shall return to this topic later on (Sect. 5.6).

4.4. The Magnetochiral Effect in Electric Conduction The electrical conductivity s is a measure of the mobility of charge carriers inside a conductor. The quantity relates the current density j (the current I per unit area; a parity-odd, time-odd vector) to the electric field E (a parity-odd, time-even vector): j = s E. As the vectors j and E have different properties under time reversal, and possess three spatial components each, s cannot be a simple scalar. s is, in general, a second-rank tensor, and its properties under time reversal may be deduced from the so-called Onsager reciprocity relations [8] [9]. In general, the conductivity in any material may be influenced by a magnetic field B. It may then be shown that for a two-terminal conductor, under time reversal, the following relation holds between diagonal tensor elements: sii(B) = sii( B). In this particular case, s may be considered as a scalar. The

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Fig. 4.3. The magnetochiral effect in conduction in enantiomers. B0, the externally applied field; BI, the magnetic self-field due to the current I. The total field B is the sum of B0 and of BI. In one case, the product B0 · I is positive, in the enantiomer it is negative, leading to a corresponding difference in the resistance R. To maintain a constant current in spite of this difference, the applied voltage must be changed accordingly (see also Fig. 4.4). To designate the enantiomers, various nomenclatures may be used: (S,R) or (M,P), or (l,d) or (l,d); the choice depends on the molecular structure, and is a matter of convenience; see Sect. 1.4.

equation implies that s must be an even function of B, for then ( B)2 = B2, etc. The same considerations hold for the resistivity 1 = s1, and the resistance R, which is the resistivity integrated over the cross-section of the conductor. By similar arguments as the foregoing, it can be shown that s in a chiral conductor must depend linearly on the parity-odd, time-even scalar product I · B. Consequently, the resistance R of the conductor may be written (higher terms assumed negligible) in the form: R = R0 (1 + b B2 + c(d,l) I · B), b and c being coefficients specific to the conducting material. In particular, c(d) = c(l), and in an achiral medium or in a racemic mixture, the third term vanishes. In a chiral two-terminal conductor in which I and B are collinear, the enantioselective influence of the magnetic field on the resistance R arises essentially through two mechanisms [10]: 1) The magnetic self-field mechanism (Fig. 4.3). A current in a chiral conductor of predominantly helical structure generates a magnetic self-field (see Fig. 3.1). This magnetic field BI is proportional to the current I, but with opposite relative sign for enantiomers. As the resistance R is proportional to B2 = (B0 + BI)2, where B0 is the externally applied magnetic field, one immediately deduces that an enantioselective contribution to the resistance proportional to B0 · I is induced.

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Fig. 4.4. Schematic representation of the measurement of the resistance anisotropy of a single-walled chiral nanotube under the influence of a static magnetic field (B0). The current is generated by an AC source, thereby periodically changing the sign of I · B0, and thus allowing for a phase-sensitive detection (picture adapted from [12]).

2) The second mechanism consists in an asymmetric scattering of the charge carriers inside the chiral conductor at the atomic/molecular level. This chiral scattering mechanism of the individual charge carriers may also be shown to contribute a term proportional to B0 · I. It is obvious that the relative importance of the two mechanisms depends heavily on the structure of the conductor. The magnetochiral anisotropy in conduction has been measured both in samples in which the chirality is essentially macroscopic, and in samples in which it is predominantly microscopic in origin. With helical bismuth wires, mainly the first mechanism could be verified [10] [11]. In samples consisting of chiral single-walled carbon nanotubes, the molecular mechanism is unquestionably predominant, although a contribution of the self-field mechanism at the molecular level cannot be ruled out entirely. The setup for the second experiment is schematically shown in Fig. 4.4 [12].

4.5. Magnetochirodynamics We realize that the existence of a magnetic moment, or a magnetic field, is closely connected to rotation of charge and to angular momentum. This fact is strikingly demonstrated by two gyromagnetic effects that have been discovered in the early 20th century [13] [14]. A) The Einstein–de Haas Effect. We consider a piece of iron carrying a spontaneous magnetization M. This macroscopic magnetization is the sum of the microscopic contributions of the individual electrons inside the metal, M = Si mi. The iron piece is at rest, but it is suspended in such a way that it can freely rotate around an axis parallel to M. An external magnetic field B is now applied antiparallel to M, thereby inducing the electrons to reorient their magnetic moments parallel to the external field. To compensate for the influ-

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ence of B, and to conserve the initial angular momentum inherent in M, the iron piece as a whole starts to rotate. The resulting induced mechanical angular momentum L is proportional to the applied magnetic field B: L = g B. We note that the parameter g is a parity-even, time-even scalar quantity. B) The Barnett Effect. It is the reverse of the Einstein–de Haas effect. Consider a cylinder of iron, or some other ferromagnetic material, suspended in such a way that it can freely rotate around its axis. We assume that the cylinder is initially at rest and demagnetized. Now the system is set in rapid motion around its axis. The result will be that the cylinder acquires a magnetization M, causing a magnetic field B. This resulting magnetization M is proportional to the imposed angular momentum L: M = g’ L. The parameter g’ is also a parity-even, time-even scalar quantity. We presently imagine performing similar experiments, but with a system that is chiral [15] [16]. In chiral systems, rotations are by symmetry coupled to translations and vice versa. We denote the acquired linear momentum of the system by p. In the case of the Einstein–de Haas effect, we may write: p = z B. And for the Barnett effect: M = z’ p. In words: in the first case, the resulting rotation is also accompanied by a translation. In the second case, an imposed translation contributes to the induced magnetic moment. We of course notice that the parameters z and z’ are parity-odd, time-even pseudoscalar quantities. Indeed, these quantities must then change sign for enantiomers (R, S): z(R) =  z(S), z’(R) =  z’(S). The Einstein–de Haas effect may have some chiral implications on a microscopic level. We assume a fluid containing suspended microscopic chiral ferromagnetic particles, or possibly a molecular solution of chiral paramagnetic molecules. An applied external magnetic field B will induce a precession of the unpaired molecular electrons around the field. As the particles/molecules are

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chiral, the induced angular momentum will be coupled to a translation. Conservation of angular and linear momentum will then cause the particles to both rotate and translate. The consequence should be an enantioselective anisotropic diffusion in the magnetic field, deriving from a positive or negative momentum increment DpB(R,S) along B, in addition to the random thermal momentum pT. The magnitude of the relative anisotropy will roughly be given by the quotient j DpB(R,S) j/j pT j. Similarly, it may be shown that chiral molecules subjected to an applied static magnetic field B and exposed to an isotropic radiation field, acquire a linear momentum along B. And conversely, a chiral molecule moving in an isotropic radiation field should acquire a magnetic moment along its velocity. The former effect derives from magnetochiral dichroism, the latter is related to the inverse magnetochiral birefringence [16].

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16]

J. J. Burckhardt, Die Symmetrie der Kristalle, Birkhuser Verlag, Basel, 1988. L. D. Barron,
Symmetry and Molecular Chirality, Chem. Soc. Rev. 1986, 15, 189. N. Bloembergen, Nonlinear Optics, W. A. Benjamin, Inc., New York, 1965. J. P. van der Ziel, P. S. Pershan, L. D. Malmstrom,
Optically-Induced Magnetization Resulting from the Inverse Faraday Effect, Phys. Rev. Lett. 1965, 15, 190. P. S. Pershan, J. P. van der Ziel, L. D. Malmstrom,
Theoretical Discussion of the Inverse Faraday Effect, Raman Scattering, and Related Phenomena, Phys. Rev. 1966, 143, 574. G. Wagnire,
Inverse Magnetochiral Birefringence, Phys. Rev. A: At., Mol., Opt. Phys. 1989, 40, 2437. S. Woz´niak, M. W. Evans, G. Wagnire,
Optically Induced Static Magnetization Near Optical Resonances in Molecular Systems. 2. Inverse Magnetochiral Birefringence, Mol. Phys. 1992, 75, 99. L. Onsager,
Reciprocal Relations in Irreversible Processes. I, Phys. Rev. 1931, 37, 405; L. Onsager,
Reciprocal Relations in Irreversible Processes. II, Phys. Rev. 1931, 38, 2265. L. D. Landau, E. M. Lifschitz, Statistical Physics, Part 1, 3rd edn. Pergamon Press, Oxford, 1980, paragraph 119; F. Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill Kogakusha Ltd; 1965; S. R. De Groot, Thermodynamics of Irreversible Processes, Interscience Publishers, New York, 1951. G. L. J. A. Rikken, J. Fçlling, P. Wyder,
Electrical Magnetochiral Anisotropy, Phys. Rev. Lett. 2001, 87, 236602. G. L. J. A. Rikken, E. Raupach, V. Krstic´, S. Roth,
Magnetochiral Anisotropy, Mol. Phys. 2002, 100, 1155. V. Krstic´, S. Roth, M. Burghard, K. Kern, G. L. J. A. Rikken,
Magneto-chiral Anisotropy in Charge Transport through Single-walled Carbon Nanotubes, J. Chem. Phys. 2002, 117, 11315. S. J. Barnett,
Magnetization by Rotation, Phys. Rev. 1915, 6, 239; C. J. Gorter, B. Kahn,
On the Theory of the Gyromagnetic Effects, Physica 1940, 7, 753. L. D. Landau, E. M. Lifschitz, Electrodynamics of Continuous Media, Pergamon Press, Oxford, 1984, § 35. V. Krstic´, G. Wagnire, G. L. J. A. Rikken,
Magneto-dynamics of Chiral Carbon Nanotubes, Chem. Phys. Lett. 2004, 390, 25. G. L. J. A. Rikken, B. A. van Tiggelen, V. Krstic´, G. Wagnire,
Light-Induced Dynamic Magnetochiral Anisotropy, Chem. Phys. Lett. 2005, 403, 298.

5. Journey into Outer Space
Il conna
t lunivers et ne se conna
t pas (
He knows the universe and does not know himself ) Jean de la Fontaine (1621 – 1695) (from Dmocrite et les Abdritains)

5.1. Remarks on the History of Science The outstanding qualities of the natural sciences are their rigor and consistency. What has been discovered and found to be correct is here to stay. Scientific knowledge is never exhaustive, it always requires improvement and refinement. But scientific progress can build on what is already known. Isaac Newton (1642 – 1727), the pioneer and architect of classical mechanics, said:
If I have seen farther than others, it is because I was standing on the shoulders of giants. How much more must this be true for scientists living today! The logic of Aristotle and the atomistic hypotheses of Democritus are yet considered as foundations of modern scientific thinking. Archimedes law of specific weight and hydrostatics is still valid today (Fig. 5.1). Almost two thousand years later, but still several centuries ago, the astronomical observations of Copernicus and Galilei, and Keplers laws of planetary motion laid the empirical basis for Newtons equations. Sending an artificial satellite around Mars in todays space exploration, essentially requires solving

Fig. 5.1. Archimedes already knew that, by combining a rotation with a chiral structure, or a chiral interaction, a translation is induced. This is probably one of the first applications of chirodynamics (picture and caption from Encyclopedia Britannica, Chicago, 1950). On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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Newtons equations of motion. These same equations are presently also applied in a completely different domain, namely, adapted to the molecular world, in biochemistry for the computation and prediction of protein folding. The myriads of electronic microcircuits that now pervade our daily life make use of knowledge that was mainly acquired in the 18th, 19th, and early 20th century. The laws of electric conduction were discovered in modestly equipped workshops and laboratories. The transistor was invented by studying the semiconducting properties of relatively large crystals of crude appearance. What, at first, may have seemed singular and clumsy, has rapidly grown into useful, multifarious, microscopic complexity. An achievement of historic significance was the unification of electromagnetism and optics in the middle of the 19th century. Maxwells equations not only serve as basis for understanding and applying electric and magnetic fields. The electromagnetic wave equation, directly derivable therefrom, enables us to quantitatively describe the propagation of light. It offers a practical means for the interpretation of numerous linear and nonlinear optical processes, many of which have only become observable thanks to the relatively recent invention of the laser. The early 20th century brought a scientific revolution, which, from a more general perspective, however, had all the attributes of continuity. The hypothesis of particle–wave duality led to wave or quantum mechanics. We must apply quantum mechanics to describe phenomena at the atomic and subatomic level. From the Bohr atom to the Schrçdinger atom: yet, the formalism of quantum mechanics, in many ways, directly grew out of the field of classical analytical mechanics. Furthermore, quantum mechanics has not made classical mechanics obsolete. Classical mechanics is valid and applicable at larger scales. Einsteins (1879 – 1955) theory of relativity: a unique combination of physical intuition and mathematical prowess. What at first seemed difficult to believe, and for many is still not easy to understand, has passed many tests of experimental verification. From the fine structure of the hydrogen atom to the gravitational redshift, the predictions have proven extraordinarily accurate. And as man ventures further and further out into space, new occasions for verification are encountered. Until now, the theory has not failed. The theoretical foundations of the modern physical sciences are, to a great extent, built on mathematical knowledge that was developed in previous centuries. Its inventors perhaps guessed, but did not know, to what extent this knowledge would become essential for the understanding of the real world. This is indeed a striking aspect of the consistency and continuity of human thinking. Mathematical knowledge that was gained a priori has also recently continued to be of central importance in further developments of theory. The evolution from elementary quantum mechanics to quantum electrodynamics and to field theory is a most striking example. Of course there is, perhaps increasingly, also

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the reverse, namely, a stimulation of research in pure mathematics through questions of physical origin. Until recently, one considered the physical sciences as essentially resting on two pillars, theory and experiment. In the course of the last decades, a third pillar has been added, namely computation. The computer, which initially was merely considered as a means to do fast arithmetic, has, by its accelerated development, rapidly grown into a role of its own. It has become an essential tool in scientific planning, in the steering of experiments, and in their interpretation and visualization. The computer has greatly contributed to the perfection of scientific observation, from the scales of the nanoworld and below, to those of deep space astronomy. The possibility to perform computations of high mathematical complexity on a reasonable timescale has not only advanced the physical sciences, but also mathematics itself. A quantitative increase in computational capacity has led to a qualitative step forward. If man ventures on to study outer space at bigger and bigger depth, he can move ahead with confidence. He must not fear that the rug on which he stands will be pulled from under his feet. There is every indication that the laws of nature, as deduced and formulated by man on earth, are valid throughout the universe. However, there is no certainty that these laws are in any way complete and exhaustive. Although we are primarily interested in the asymmetry on Earth and of its closer surroundings – in particular, in the molecular chirality of the biosphere – we shall now make a journey far into space and back in time. We realize that the asymmetry of the world closer to us unquestionably derives from origins far away and far back. We consider ourselves, therewith, as spectators in the domains where the study of the infinitely big, called astrophysics, meets the study of the infinitely small, namely elementary particle physics. For the short summary and brief discussion of the evolution of the universe that will now follow, we will have to use some terms of elementary particle physics. The beginning of the universe, as it is today generally accepted, must have been a rapid succession of highly complex interactions between these elementary particles. We assume that the reader is familiar with such notions as proton, neutron, and electron. Some other, truly elementary or composite, particles, we have already encountered previously (Sect. 2.4). Quarks, of which there exist various kinds, are strongly interacting elementary particles that are the constituents of protons, neutrons, and mesons. Protons and neutrons consist of three quarks. But the mass of a proton or neutron is not simply equal to the sum of the masses of the quarks of which it is composed. Most of the proton or neutron mass is contained in the binding energy, which is due to the strong interactions, carried by the so-called gluons. Mesons are composed of a quark and an antiquark. Free quarks have hitherto not been observed.

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Protons and neutrons belong to the wide class of particles called baryons. Baryons are characterized by a baryon number which for protons and neutrons has the value + 1. One of the fundamental conclusions of modern elementary particle theory is that to every particle there exists an antiparticle. An antibaryon has the same mass, spin, and absolute charge as its parent particle, but opposite sign of the charge and opposite baryon number. Consequently, the antiproton and antineutron have a baryon number of
1. When a particle and its antiparticle interact, they may annihilate generating electromagnetic radiation. Initially, theory started from the assumption that, in the early universe, the sum of the baryon numbers of all particles added to zero. This implies a complete equality of matter and antimatter. However, the universe accessible to our observation appears to consist entirely of matter. It, therefore, must be concluded that, in the early universe densely packed with baryons and antibaryons, there occurred a slight violation of this equality, leading after annihilation to the present surplus of matter. Leptons are a class of elementary particles that do not participate in the strong interactions, and include the electron, the muon, and the neutrino. Leptons have a baryon number of zero. The muon carries a negative charge like the electron but has a mass 207 times greater. The neutrino is an electrically neutral elementary particle with a spin and an extremely small mass. Neutrinos are only susceptible to weak and gravitational interactions.

5.2. The Evolution of the Universe The first third of the 20th century saw the birth of the theory of relativity and of quantum mechanics. It was also marked by discoveries in astronomy that would profoundly influence mans view of the evolution of the universe. In 1923, the american astronomer Edwin Hubble (1889 – 1953) and his co-workers became the first to establish that there were other galaxies in the universe beside our own. In 1929, Hubble made the observation that distant galaxies, independent of the direction of observation, are rapidly moving away from us and from each other. He formulated this as the law named after him: The apparent recession velocity of galaxies is proportional to their distance from the observer, no matter where the observer is. In other words, the universe must be expanding, the further away, the faster [1 – 3]. One of the basic notions of the theory of relativity is that light in space travels at a constant speed, namely at close to 300,000 km per second. Light emitted by a distant celestial body will thus take a certain finite time to reach an observer. The bigger the distance the light has to travel, the more the information carried by that light lies in the past. One may use the relation travel

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time  speed of light as a measure of distance. The most distant galaxies nowadays observable are about ten billion light years away. The light one receives today from them was correspondingly emitted about ten billion years ago. This consequently provides significant clues on the history of the universe. The experimental evidence drawn from astronomical observation shows that we are not living in a static universe which has been here forever and will last indefinitely. Instead, we are living in a dynamic universe that must have had a beginning and has a future, albeit not an exactly predictable one. If we let our imagination go backwards in time, we see the universe contracting. The early universe must correspondingly have been more confined. Already in the 1920s, mathematical models for an evolving, nonstationary universe were conceived in the theoretical frame of general relativity. These ideas later on proved significant for the interpretation of the history of the creation of the chemical elements. They suggested the model of a hot, dense early phase of the expanding universe in which the lighter elements, deuterium and helium, were formed first by fusion of protons and neutrons. Extrapolating even further back in time, an initial situation had to be assumed where the universe was completely localized in space–time and infinitely hot, resembling a singularity with an immense energy density. Starting from there, the cosmic expansion must have set in as a big explosion. This idea evolved into the so-called
Big Bang model of the universe, the main features of which were published around 1950 [4] [5]. Since then, intense research has been undertaken in that field of cosmology, and a vast amount of literature, both specialized and for a more general audience, has appeared on the subject [1] [3] [6] [7]. In an attempt to make some qualitative speculations on the conservation, or violation, of the basic symmetries C, P, T in the course of time, we try to briefly summarize what appears to be commonly accepted knowledge today. In the creation of the universe, there indeed must have been a time zero, but, from a physical point of view, the history begins at Planck time, 10
43 s later. What happened before is probably not accessible to scientific description, one reason being the quantum-mechanical complementarity of energy and time. In the beginning, all interactions, including gravitation, must have been unified. The following phases in the early history of the expanding universe are then considered [6] [8]: • Time around 10
35 s. The temperature of the universe is of the order of 1028 K (kelvin)! As the temperature decreases, the unified interaction between quarks and leptons separates into a strong interaction between quarks, leading to baryons and antibaryons, and an electroweak interaction between quarks and leptons. An initial, accelerated expansion of the universe takes

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• • • • •

•

•

•

•

On Chirality and the Universal Asymmetry

place, the so-called inflationary phase. Baryon–antibaryon asymmetry appears. Time around 10
15 s. The unified electroweak interaction between quarks and leptons separates into weak and electromagnetic interactions. Time around 10
6 s. The universe has
cooled to 1013 K. Still free quarks form hadrons (mesons). Time around 10
5 s. Mesons decay or annihilate with their antiparticles, converting energy into photons, electrons, and neutrinos. Time around 10
2 s. Neutrinos decouple from the other particles. Time between 10
1 s and 102 s. The temperature cools to 1010 K. Electron– positron annihilation continues, leaving an excess of relatively few electrons. Protons and neutrons start fusing to deuterium and helium nuclei. Time between 102 s and 103 s (the early minutes). The temperature descends to 109 K. Nucleogenesis is essentially complete. Thermonuclear fusion reactions lead to nuclei of further light elements, such as Li, Be, B. The cosmos consists of a plasma of nuclei, electrons, and photons. At ca. 1014 s (one million years). Other light elements have formed, and we witness the beginning of chemistry. Once the temperature descends below 104 K, molecules start to appear. At ca. 1016 s (one hundred million years). The temperature is now around only 170 K. Vast clouds of gas contain mainly hydrogen and helium. But one encounters also heavier elements and various molecules. Accretion sets in. Galaxy formation begins. At ca. 1018 s (ten billion years) up to the present. On distance scales exceeding 300 million light years, the universe still appears to the observer to be quite homogeneous. However, as we all know, todays universe contains innumerable local irregularities. These are thought to have developed from small differences in the density of the early universe, between one region and another. In the present-day universe, there are ca. 1011 (one hundred billion) galaxies. A galaxy, on an average, contains on the order of 1011 stars. Between these celestial bodies, there are clouds of both partly visible and of dark matter. The formation and extinction of stars is a dynamic process that still goes on today and lends credence to the assumption that we do not live in a static universe.

One of the remarkable predictions of the early
Big Bang model is the existence of cosmic electromagnetic background radiation [5]. Its history being finite in time, the universe should at present not have been able to cool down completely to zero kelvin temperature. By Plancks radiation law, any body or system at a temperature above zero emits electromagnetic radiation over a certain wavelength range. The lower the temperature, the longer the wavelength of the maximum of the emission. For an emitter at a temperature of a few

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kelvin, one should find the maximum in the far-infrared or microwave region. This cosmic microwave background radiation was indeed experimentally discovered at a wavelength of 7.3 cm with a radio telescope in 1964 [9], corresponding to a temperature of ca. 2.7 K. At first, it appeared that the spatial distribution of the intensity of this radiation was quite homogeneous. More recent and precise measurements indicate distinct inhomogeneities. This is at present a field of intense research. Will these inhomogeneities offer further clues on the origins of the universal asymmetry? On how the universe has
asymmetrically cooled from above 1030 K to below 3 K?

5.3. The Birth and Death of Stars Stars are formed from the condensation of clouds of gases and dust under the action of the mutual gravitational forces. These clouds may initially span distances of hundreds of light years. Gradually, under the influence of innumerable perturbations, they form cores of higher density in which the pull of gravity plays an increasing role. The pressure inside the core of such a protostar gradually becomes so high that the atomic nuclei (mainly of hydrogen) react with each other. The energy generated by these nuclear fusion reactions escapes partly as electromagnetic radiation, partly in the form of neutrinos. The temperature climbs to over 106 K. Inside the core, a pressure directed outwards builds up that for some time is in equilibrium with the gravitational pressure directed inwards. This quasi-stable situation is maintained as long as the star contains enough fuel to feed the nuclear reactions [2]. Low-mass stars, called
brown dwarfs, barely reach the threshold of nuclear fusion. They convert hydrogen to helium only very slowly, are comparatively hard to detect, and have a very long lifetime. Stars of intermediate mass, such as our Sun, still have a relatively low luminosity. They also
burn hydrogen to helium and live on the order of 10 billion years. Once the hydrogen is used up, a helium-
burning phase sets in. In the end, the star becomes a so-called red giant before ending as a white dwarf. High-mass stars, of more than 10 solar masses, are up to 104 times as luminous as the Sun. They live only millions of years. Once the hydrogen- and helium-
burning phases end, the high mass causes the core to compress sufficiently, and the temperature to rise to the point where even carbon and heavier elements begin nuclear fusion. This may go on until the core is made up of iron. Higher elements in the Periodic Table are not formed in great amounts, as they do not release energy upon nuclear fusion. Ultimately, the star collapses. This manifests itself by a cataclysmic burst called a supernova. The remains that are not propelled into outer space form a neutron star or a black hole.

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Fig. 5.2. a) The Andromeda Galaxy. It is the closest large galaxy outside the Milky Way, 2.4 million light years away (adapted from [2]). b) The nonexisting mirror image.

It must be emphasized that our aim here is not to discuss in detail the biography of stars of various sizes. Our question rather is how have the inhomogeneities of the early universe started, and by what kinds of mechanisms have they been amplified. Galaxies are asymmetric, chiral cosmic structures composed of billions of stars: why has a particular galaxy evolved as it is, and not into the enantiomorphous form? By energy requirements? Or just by
accident? As one may assume that for any chosen galaxy the exactly enantiomorphous galaxy does not exist (Fig. 5.2), the universe as a whole must be considered as very probably nonracemic and therefore as chiral.

5.4. Is There a Temporal Evolution of the Basic Symmetries? From experiments in particle physics, it appears that baryon number should be conserved: in general, the sum of the baryon numbers of all particles participating in a given elementary reaction is the same before and after that reaction. One would also expect that, in the early universe, the number of baryons and of antibaryons should have been equal, the total baryon number of the universe adding to zero, and remaining zero. Furthermore, the conservation of electric charge is considered one of the basic laws in physics. In addition, we have encountered the concept of charge conjugation C (Sect. 2.4): a process displays invariance with respect to C symmetry, if the rate of that process is identical to the rate of a similar process with all particles replaced by their antiparticles. The simultaneous existence of baryon-number conservation, charge conservation, and charge conjugation

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symmetry C, would imply that matter and antimatter in the universe are strictly equivalent, their relative amounts time invariant. In Sect. 2.4, we saw that the weak forces are not invariant with respect to C. If, then, instead of C symmetry, CP symmetry were strictly conserved, then matter and space-reflected antimatter would be equivalent. The observed surplus of dilute matter, or apparent nonexistence of antimatter, in the present-day universe suggests that baryon–antibaryon annihilation has taken place on a large scale, and that baryon number conservation must, in the course of time, have been violated. Furthermore, we know that, in the weak interactions, C symmetry is violated, and P symmetry is violated. CP Symmetry seems almost conserved, but with notable exceptions (Sect. 2.4). This means that, in fact, CP symmetry is indeed weakly violated. The most striking asymmetry in the universe is the imbalance of matter over antimatter. It, nonetheless, appears to be connected to the inequivalence between image and mirror image. In Sect. 2.3 we mentioned the CPT theorem: in the frame of the so-called standard model, all laws of physics should be invariant to the consecutive application, in arbitrary sequence, of charge conjugation C, the parity operation P, and time reversal T. However, to describe the very early phases of the universe where all interactions must have been unified, it is unquestionably necessary to go beyond the standard model and to devise a theory in which not only the strong, the weak, and the electromagnetic forces, but also gravitation are unified into a single general scheme. The question may be legitimately raised, if, in such a universal theory, CPT invariance still holds. The violation of CPT invariance would imply that there are no restrictions on the separate and independent violation of C, P, and T symmetry (Table 5.1). If CPT invariance holds, and CP invariance is only weakly violated, then this implies that the time reversal symmetry T cannot be strictly fulfilled. It then appears doubtful that T symmetry has ever been fulfilled. Rather, one would qualitatively suspect that T symmetry has been more strongly violated in early evolutionary stages of the universe than it is today. It is indeed difficult to Table 5.1. Qualitative and Somewhat Speculative Classification of the Basic Symmetries after the Big Bang. The symbols denote: sv, strongly violated; v, violated; wv, weakly violated; ac, almost conserved (with notable exceptions); c, conserved. Time [s]

CPT

CP

C

P

T

< 10
5 1 100 Todaya)

ac (?) c (?) c c

v v wv ac

v v v wv

v v wv wv

sv v wv ac

a

) Ca. 1017 s.

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believe that, during the rapid and fundamental transformations of the early universe, there was no arrow of time, and that going backwards might in any way have been possible or conceivable. In
A Brief History of Time, Hawking distinguishes between a psychological arrow of time, a thermodynamic arrow of time, and a cosmological arrow of time [1]. In early stages of the universe, the cosmological arrow of time must have been very pronounced, corresponding to what we would term a total absence of microscopic reversibility. In todays universe, the breakdown of time-reversal symmetry has, until now, only indirectly been detected in some, already briefly mentioned, meson decay reactions. Concerning the violation of parity P: if in the frame of the standard model, parity is violated within the weak interactions (Sect. 2.1 and 2.2), it is difficult to assume that parity should be conserved in a grand unified theory. In the early universe, there must consequently already have been a spatial bias in the evolution of microscopic inhomogeneities. This immediately leads to the question of how the chiral cosmic structures that exist in our universe today have evolved therefrom. By what mechanisms have the microscopic influences been amplified to manifest themselves so visibly and strikingly at a larger scale?

5.5. Galactic and Intergalactic Matter The total mass–energy density of the universe W is the sum of the density of matter WM, of the mass-equivalent energy of radiation WR, and of the vacuum WL [10]: W = WM + WR + WL (notice the mass–energy relation is given by the Einstein formula E = mc2). Assuming W to be normalized to unity, it has been concluded from different types of cosmological measurements, in particular from the observation of the dynamics of galaxies, that WM is only ca. 0.35. And as a further surprise, the density of baryonic matter WB (composed of protons and neutrons; including electrons as constituents of atoms and ions) makes only a small fraction of WM, namely WB  0.06. Baryonic matter comprises luminous baryonic matter WBL (mainly in stars) and dark baryonic matter WBD. In conclusion, the major part of the universe contains nonbaryonic dark matter of relative density WD : WM = WB + WD = WBL + WBD + WD. At least 85% of the matter of the universe is of as yet unknown form. Of what nonbaryonic dark matter consists, is still a matter of speculation.

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Elementary-particle theory provides some interesting suggestions. One of them is the postulated family of supersymmetric particles [11]. All known particles may be classified as either bosons with integer valued or zero spins, or as fermions with half-integer valued spins. The key idea then is that to every existing boson there should be a supersymmetric fermion, and to every existing fermion a supersymmetric boson. The list of elementary particles, consisting of the known particles and their supersymmetric partners, would thereby be doubled. It is predicted from theory that the lightest of these supersymmetric partners, the neutralino related to the photon, should be stable and have a mass of the order of 102 to 103 times that of the proton! Another candidate for nonbaryonic dark matter is the axion, a neutral, pseudoscalar boson. Its mass should only be on the order of 10
5 eV. From our point of view, the axion is particularly interesting, as its pseudoscalar nature gives it chiral symmetry properties (Sect. 3.3, 4.1, and 4.2). It has been proposed that axions could possibly be detected through their interaction with a strong magnetic field, by which they would be converted to photons. From our symmetry viewpoint, such a process would bear a similarity to magnetochiral emission: the parity-odd, time-even chiral element (particle) interacts with the parity-even, time-odd magnetic field to generate a parity-odd, time-odd photon.

5.6. Stellar and Galactic Magnetic Fields In Sect. 5.3, we mentioned that massive stars following a supernova explosion end their life as neutron stars – unless they eventually become black holes. Neutron stars, as the name implies, are composed mainly of bare neutrons packed at an extremely high density of ca. 1014 g cm
3 or equivalently 1014 tons per cubic meter [8]. If our sun became a neutron star – which it will not as its mass is too small – its radius would shrink to roughly only 14 km! The so-called pulsars, pulsating cosmic radio sources first observed in 1967, are interpreted as being neutron stars. When they are born in supernova explosions [12], pulsars are assumed to be rotating rapidly, on the order of 102 times per second, and to carry magnetic fields on the order of 1012 – 1013 gauss, or 108 – 109 T (tesla). The rapid rotation and the magnetic field combine to accelerate charged particles to nearly the speed of light and thereby to emit radiowaves of high intensity. Pulsars may also act as sources of high-energy photons in the g range. More recently, namely for the first time in 1979, some neutron stars have been detected showing emission characteristics distinctly different from those of pulsars. It has since been concluded that these
soft g-ray repeaters are probably neutron stars with exceptionally high magnetic fields, on the order of 1014 – 1015 gauss. They have been named magnetars. The

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particular emission properties of the magnetars are attributed to the fact that the observed radiation is not powered by the rotation of the star, but rather by a direct decay of the magnetic field; this magnetic field being strong enough to perturb the very structure of the vacuum [12] [13]. The exact mechanism of the radiative decay of the magnetic field of the magnetar is probably highly complex, and its elucidation is indeed the task of astrophysicists. The ideas that we will now express are no doubt superficial and speculative. We again go back to Sect. 4.2 and the PT triangle. Among the possibilities which we mentioned there, was the one of a static magnetic field being converted to electromagnetic radiation, via a chiral interaction. If such ideas have any applicability whatsoever here, they would imply that the radiative emission of a magnetar is directly dependent on the presence of parity-violating influences. The seeds of spatial asymmetry germinating in the early universe grew and were amplified by a variety of complex mechanisms. It does not lie within our possibilities to discuss these questions in any detail. However, it appears reasonable to assume that, in the course of time, electromagnetic interactions also played an important role. In spite of the fact that the universe as a whole is probably neutral, there have always existed plenty of free and moving charges and of electric and magnetic fields. And even though electromagnetic interactions conserve parity from an overall point of view, a particular interaction at a given place and at a certain time may well be chiral (Sect. 3.1 and 4.3). Such an interaction can then magnify a quasi-dormant asymmetric influence already present. Of general interest in this context are galactic magnetic fields. Large-scale magnetic fields are generated and maintained by helical turbulent motions of interstellar gas and by differential galactic rotation [14]. Seed fields for these turbulent motions may be outflows from supernovae and hot young stars. We may also mention further possibilities to induce magnetic seed fields, namely, those due to the interaction of electromagnetic radiation with microscopic systems in which parity is not conserved; for instance, intense g-radiation of arbitrary polarization interacting with atomic nuclei in which parity is violated, thereby inducing a magnetic polarization. This would be a consequence of the inverse magnetochiral effect (Sect. 4.3). Accordingly, wherever there are systems in which parity is broken and which interact with radiation, magnetic fields should in principle appear. Seed fields may be very small. Subsequently, there are various mechanisms by which magnetic fields may be amplified (see Fig. 5.3). These amplified fields, in turn, induce asymmetric movements of charged particles on a larger scale. And so the asymmetries grow.

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Fig. 5.3. Amplification of a galactic magnetic field via the mechanism of Zeldovichs rope dynamo. A loop of magnetic flux (a) is stretched to twice its radius (b), twisted (c), and doubled upon itself (d). After every cycle illustrated here, the magnetic flux of the loop is doubled. Repetition of the process may lead to exponential growth (picture and caption adapted from [14]).

5.7. The Fractal and Chiral Universe The universe is observed to contain disperse matter in the form of clusters of clusters. Between the clusters of lower order, the galaxies, and the clusters of higher order, the clusters of galaxies, there is a certain self-similarity. The universe is the largest-scale example of the fractal geometry of nature [15]. In his book, Mandelbrot likens the distribution of stars and galaxies in the sky with the distribution of raw diamonds in the Earths crust: the places where diamonds occur are sparsely and unevenly distributed. Similarly, within a given diamond-rich area, the concentration of the individual gems is sparse and irregular, both with respect to location and size. Imagine a sphere of variable radius R with origin anywhere in the universe, for instance at the Earths center. We denote the mass of the matter contained in this sphere as M(R). The matter density in the sphere is correspondingly M(R) divided by the spheres volume: M(R)/(4/3)pR3. If the matter within the sphere is homogeneous, then the density is constant and the mass M(R) grows as R3. If, as R increases, we enter regions of space which are
empty, then, within these regions, the mass as a function of R remains constant; M(R) there will be proportional to R0. Irrespective of where we place the origin of R, i.e., the center of the sphere, we may, as R goes to infinity, consider M(R) to be proportional to RD, where 0 < D < 3. This is a consequence of the inhomogeneity of the universe on smaller scales. For R ! 1, the exponent D, which may be considered as the fractal dimension, should asymptotically approach a certain definite value. Based on telescopic observations, an estimate has been made of D = 1.23. The chirality of the universe is not directly related to its fractality. But both the fractality and the chirality are due to the universes inhomogeneity on

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smaller scales. Already in 1926, Edwin Hubble classified galaxies according to their morphologies as ellipticals, irregulars, and spirals. Any dust cloud, any three-dimensional collection of randomly located particles, any inhomogeneous distribution of matter that lacks a rotation–reflection axis is inherently chiral. This is consequently also the case for any galaxy, irrespective of its morphology. A spiral galaxy consists of a circular disc of gas and stars in rotation about its center [10]. As all galaxies are receding, the further away the faster, they simultaneously perform a translation. The combination of the distinctly observable rotation with the translation conveys to spiral galaxies an additional dynamic chirality (Sect. 4.1). This leads to the important question, if the dynamic chirality, averaged over all galaxies of the universe, is significantly different from zero [16]. If this were the case, it would imply a resulting dynamic chirality for the entire universe. Spiral galaxies are designated as either S-like or Z-like, depending on the sense of their rotation. The Z-type galaxies are right-helical and the S-type galaxies are left-helical for any observer, as the recession velocity always points away from him. Kondepudi and Durand have performed a statistical investigation on 540 spiral galaxies which may be clearly classified as of either S- or Ztype. Depending on details of their structure, these galaxies may be further divided into different subclasses. In their study, the above-mentioned authors assign a (helicity) number of + 1 to the Z-type galaxies and of
1 to those of Stype. The sum of these numbers for N galaxies is denoted by DN. If in a sample there are equal numbers of Z and S galaxies, then DN = 0. If, furthermore, one assumes that the individual galaxies are uncorrelated in their motions, and that the numbers + 1 and
1 occur with equal probability, then for a sample of N galaxies the probability that DN has a particular value – meaning the statistical weight of a particular value of DN – is given by the binomial distribution. For N very large, the binomial distribution transforms into the normal pffiffiffiffi pffiffiffiffi distribution, and the statistically significant quantity is (DN= N ), where N is the standard deviation. Kondepudi pffiffiffiffi and Durand find for some subclasses of spiral galaxies values of (DN= N ) that significantly deviate from zero, but for the entire sample of N = 540, they obtain as result
0.17, corresponding to jDNj  4. This is, comparatively, very close to zero and to perfect equality. It leads to the conclusion that, on a larger cosmic scale, there should be approximate achirality. From this, we notice that pronounced asymmetry on a lower scale not necessarily entails the same degree of asymmetry on a higher scale. Nonetheless, the universe as a whole must undoubtedly be different from its spacereflected counterpart.

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REFERENCES [1] S. W. Hawking, A Brief History of Time, Bantam Books, New York, 1988. [2] R. Naeye, Through the Eyes of Hubble. The Birth, Life, and Violent Death of Stars, Institute of Physics Publishing, Bristol, 1998. [3] R. A. Alpher, R. Herman, Genesis of the Big Bang, Oxford University Press, Oxford, 2001. [4] R. A. Alpher, H. Bethe, G. Gamow,
The Origin of the Chemical Elements, Phys. Rev. 1948, 73, 803. [5] R. A. Alpher, J. W. Follin, R. C. Herman,
Physical Conditions in the Initial Stages of the Expanding Universe, Phys. Rev. 1953, 92, 1347. [6] R. A. Alpher, R. Herman,
Reflections on Early Work on
Big Bang Cosmology, Physics Today 1988, 41(8), 24. [7] S. Weinberg, The First Three Minutes, updated edn., Basic Books, New York, 1993. [8] H. R. Quinn,
The Asymmetry between Matter and Antimatter, Physics Today 2003, 56(2), 30; F. Wilczek,
The Cosmic Asymmetry between Matter and Antimatter, Sci. Am. 1980, 243(6), 60; G. Musiol, J. Ranft, R. Reif, D. Seeliger, Kern- und Elementarteilchenphysik, VCH Verlagsgesellschaft, Weinheim, 1988, Chapts. 13.3, 15.5, 15.9; I. S. Hughes, Elementary Particles, Cambridge University Press, Cambridge, 1991. [9] A. A. Penzias, R. W. Wilson,
A Measurement of Excess Antenna Temperature at 4080 Mc/s, Astrophys. J. 1965, 142, 419; J. Bernstein, Three Degrees above Zero, Charles Seribners Sons, New York, 1984. [10] D. J. Raine, E. G. Thomas, An Introduction to the Science of Cosmology, Institute of Physics Publishing, Bristol, 2001. [11] M. Fukugita,
The Dark Side, Nature 2003, 422, 489. [12] S. R. Kulkarni, C. Thompson,
A Star Powered by Magnetism, Nature 1998, 393, 215. [13] C. Kouveliotou, S. Dieters, T. Strohmayer, J. van Paradijs, G. J. Fishman, C. A. Meegan, K. Hurley, J. Kommers, I. Smith, D. Frail, T. Murakami,
An X-Ray Pulsar with a Superstrong Magnetic Field in the Soft g-Ray Repeater SGR 1806-20, Nature 1998, 393, 235. [14] A. Ruzmaikin, D. Sokoloff, A. Shukurov,
Magnetism of Spiral Galaxies, Nature 1988, 336, 341. [15] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman & Company, New York, 1977. [16] D. K. Kondepudi, D. J. Durand,
Chiral Asymmetry in Spiral Galaxies?, Chirality 2001, 13, 351.

6. Return to Earth
There lies a somnolent lake Under a noiseless sky, Where never the mornings break Nor the evenings die Trumbull Stickney (1874 – 1904) (In the Past)

6.1. The Chaotic and Chiral Solar System From metaphysical eternity to dynamical uncertainty: the sunrise and sunset, the daily succession of light and darkness, the rhythm of the tides, the near-monthly return of the moon, the yearly change of the seasons give us the impression of an irrevocable and eternal course of nature. Within the timescale of human history, nature changes relatively discreetly. On the cosmic scale of the evolution of the universe, the action is more dramatic. It is to be expected that, in a few billion years, the sun will have used up most of its nuclear fuel (mainly hydrogen) and that it will become a white dwarf (Sect. 5.3). This will, of course, also be the end of the solar system of planets, including the Earth, as we know it. But even before these cataclysmic events, the path of the planets, appearing to us as being prescribed forever, may have significantly changed. Recent very extensive computations, essentially based on Newtons laws of motion, point to an overall stability of the solar system of planets for the next billion years, but cannot exclude unexpected developments. Similarly, such computations are yet unable to go back in time sufficiently far to suggest more precisely how the Earth and its sister planets were created, and how they found their present orbits. It was for a long time assumed that the equations of motion for bodies under mutual gravitational attraction would make it possible to predict their trajectories to any degree of precision. The French mathematician Jules Henri Poincar
(1854 – 1912) later pointed out that, for systems of three bodies and more, reliable solutions could become very difficult to obtain, requiring the addition of very long series of mathematical terms that did not rapidly converge [1]. Today, one knows that, depending on the relative masses of the interacting bodies, initial conditions for positions and momenta that differ only very little may lead to vastly different orbits. The starting conditions would then have to be known to infinite precision to obtain reliable dynamical predictions. We encounter situations of inherent unpredictability and randomness in apparently completely deterministic systems. The mathematical phenomenon has been called
dynamical stochasticity or
deterministic chaos [2]. The systematic study On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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of deterministic chaos, which not only may manifest itself in the classical manybody problem but in numerous other kinetic phenomena (Sect. 6.3), has only within the last decades become an intense field of research of its own. In the first half of the 20th century, the attention of most physicists was rather exclusively directed towards atomic, solid state, and nuclear physics, and the development of quantum mechanics. In quantum mechanics, the theory that must be applied to the atomic and subatomic world, the question of limited predictability is of a different nature, at least partly. It is mainly connected to the particle–wave duality and the uncertainty principle, an aspect which does not immediately concern us here. The solar system consists of the Sun, the nine well-known planets, and numerous satellites or moons (numbers in brackets) [3]: Earth (1), Mars (2), Jupiter (60), Saturn (31), Uranus (22), Neptune (11), Pluto (1). If, in our survey, we take into account also the Asteroid Belt and the Rings of Saturn, etc., we find plenty of additional, smaller-sized residents of the solar system, each with its more or less predictable, in general periodic, orbit. All these bodies interact under the influence of gravitation. A solution of the laws of motion taking into account all these bigger and smaller interactions simultaneously is an impossible task. Rather, this mathematical undertaking has to be broken down into quasi-independent subproblems. And each one of these subproblems is in itself difficult enough to solve. This reveals that under a seeming overall symmetry hides a quite fundamental asymmetry. Due to their relatively high mass and the dominating influence of the gravitational field of the Sun, the planets have relatively stable orbits. These planetary orbits are in general elliptical, but with a relatively small eccentricity, and their inclination with respect to the ecliptic (the plane in which the center of gravity of the Earth and the Moon revolves around the Sun, taken as reference) is, with the exception of Pluto, between 0.8 and 7 degrees only. Pluto, the most distant planet, shows an inclination of 17 degrees. Furthermore, due to the relatively high excentricity, the orbit of Pluto at perihelion passage is closer to the Sun than that of Neptune. Computational analyses of the dynamics of Pluto predict minor fluctuations in the characteristics of the orbit which should repeat on a timescale of millions of years [1]. Such findings, though relatively insignificant for an immediate future, do not exclude the existence of further, chaotic irregularities which are evidently even more difficult to exactly foresee. As may be expected, a more complex picture is presented by the multifarious moons [3]: regular planetary satellites have relatively small, untilted circular orbits. They were probably formed out of disks of gas and dust that surrounded the big planets in their early stages. Nearly two thirds of the known moons are irregular satellites, however, orbiting far from their planets along tilted, elliptical paths. Irregular satellites have excentric orbits that can be highly inclined, with a prograde or retrograde sense of motion (meaning that the sense

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of rotation of the moon around its planet may be equal or opposite to that of the planet around the Sun). This suggests that these satellites may have been captured by their respective planets from independent orbits around the Sun, early in the history of the solar system. Computations show that irregular satellites are captured in a region of phase space where orbits are chaotic. Energy dissipation then switches these long-lived chaotic orbits into nearby regular zones from which escape is impossible [4]. We notice that coplanar orbits around one and the same central body, in general, define a plane of symmetry, and that the situation exhibited by such orbiting bodies at all times is achiral. Tilted and excentric orbits suggest the perturbing influence of many-body interactions and the possibility of a chaotic dynamics. Many-body systems, evolving on orbits tilted with respect to each other, may lead to situations that are structurally chiral, though, of course, timedependent. If a planet, in addition to its motion around the sun, exhibits a rotation of its own around a given axis, then additional dynamic chiral situations may arise (Sect. 4.1). But do such considerations of symmetry, asymmetry, and chirality have any deeper dynamical significance? An N-body dynamical system in three-dimensional space is described by 3N spatial coordinates q1, q2, q3, ….., q3N, and 3 N momenta p1, p2, p3, ….., p3N. These quantities are all functions of time t: qi(t), pi(t). To make the calculations tractable, one must, as already mentioned, focus on the body (planet, moon) of particular interest and attempt to reduce the task to an adapted two-, or even only one-body problem with as few variables as possible. In spite of this limitation, the best approximation possible is sought. The results of such calculations may be represented in phase space by two or three degrees of freedom for a given value of the energy [5]. For instance, one is then left with three variables, q1, q2, say, and p2. The results are visualized by so-called Poincar
maps: the points are displayed, where the computed trajectories q1 cut a given surface (q2, p2), for instance. From the two-dimensional patterns so obtained, one may distinguish between periodic, conditionally periodic and chaotic orbits (Fig. 6.1). It might, in general, prove instructive to analyze Poincar
maps with respect to space inversion or the parity operation, P. Under P, q1, q2, and p2 transform into
q1,
q2, and
p2. Comparing the results for q1, q2, and p2 with those for
q1,
q2, and
p2 in the form of difference maps could conceivably be a measure of the asymmetry or
chirality of the evolution of the system (Sect. 2.2). Similarly, applying the operation of time reversal T would leave q1 and q2 unchanged, but invert the sign of p2 (Sect. 2.3). Difference maps based on time inversion would then be a measure of the systems temporal evolution, or
arrow of time. Although these considerations are admittedly somewhat sketchy, they raise the question of generalizing the measure of the asymmetry of a dynamic system in space and time. We shall consider the question of the measure of structural chirality later on (Sect. 8.9 – 8.11).

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Fig. 6.1. Dynamic study of an irregular moon. Poincar
surface in the x,y-plane for two given values of the energy. Points on the surface are colored according to the sign of their angular momentum. The figure shows regions of chaotic (
shotgun pattern) and regular orbits (nested curves) (from [4]).

6.2. Asymmetry on Earth During their mission to circumnavigate the Moon, in December of 1968, the crew of Apollo 8 recorded the first pictures of Earth from distances of the order of 50,000 km or more. The beauty of these pictures is overwhelming. In contrast to the rather inhospitable aspect of the other planets, Earth appears as a jewel in the sky, with its blue oceans, irregularly structured continents, partly shrouded in varying patterns of white clouds. The Earth, of overall near-spherical symmetry, strikes us by the richness of the asymmetric details of its surface. Did the world that Alice entered, after passing through the Looking Glass (see Fig. 2.1), belong to an Earth that was the mirror image of the existing one (Fig. 6.2)? Under what conditions could such an enantiomorphous Earth have arisen? How would the solar system have had to evolve to lead to it? Are our existing earthly surroundings a direct consequence of the fundamental asymmetry of the early universe, or rather, are they to be considered merely as later
frozen accidents [6]? And suppose a macroscopically enantiomorphous Earth existed, would this affect the microscopic asymmetry on it, down to the level of the molecular chirality (Fig. 6.2)? Could this even have led to the emergence of opposite, enantiomeric biochemical homochirality? In previous chapters, we have considered the emergence of asymmetry on different scales of space and time: in the early universe, directly based on the laws of elementary particle physics (Sect. 5.2 – 5.4); at the scale of the later universe and at the level of the galaxies (Sect. 5.5 – 5.7); subsequently, in following the motion of individual celestial bodies of the solar system (Sect. 6.1). At present, we study the finer structure of a single and very particular celestial body, namely the one we inhabit. We would like to be able to follow continuously, through the different stages in space and time, the evolution and amplification of asymmetry in the universe:

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Fig. 6.2. a) The Planet Earth seen from a distance of ca. 50,000 km (adapted from Chapt. 5 in [2]). b) The nonexisting mirror image.

from the elementary and primeval to the cosmic; from the cosmic to the local macroscopic on Earth; from the macroscopic to the microscopic and molecular. Is there room for the fortuitous, for chance, or does the violation of parity at the elementary-particle level directly determine everything? Before returning, in our journey, to the molecular domain (Sect. 6.4 and following), we briefly recall some properties of Earth that, in particular, make it inhabitable for living organisms. The center of mass of the Earth and the Moon describes in the course of a year an elliptic path, with the center of the Sun at one focus. The eccentricity (difference of the greatest and least distances to the Sun divided by their sum) of the orbit is small, namely 0.017. The rotation of the Earth about its own polar axis is nearly uniform, the period being the sideral day of 23 h 56 min 4 s. The inclination of the Earths axis with respect to the normal to the ecliptic is 23827’. This inclination is at the origin of the seasons. The angle of incidence of the Suns radiation at a given time and point on the Earths surface depends both on the Earths rotation and on its position around the Sun. The angle of inclination is subject to small periodic fluctuations (precession, nutation, and variation of the latitude), due to the combined perturbing effect on Earth of the Moon and the Sun (three-body effects). A more immediately visible simultaneous influence of Sun and Moon are the ocean tides, the daily rise and fall of the water level of the sea. The body of the planet Earth consists of various layers of increasing density [7]: a Crust of thickness of 5 – 50 km (containing mainly the elements Si, Al, Ca, Mg, and O), an Upper Mantle reaching to a depth of ca. 400 km (rich in Si, Al, Ca, Mg, and small amounts of Fe) a Transition Zone followed by the Lower

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Mantle down to ca. 1000 km (mainly containing Si, Mg, and Fe). The Earths Core may be subdivided into an Outer Core (containing mainly Fe with smaller quantities of lighter elements) in a liquid state, and an Inner Core (consisting primarily of pure Fe plus some Ni) in a high-pressure solid phase. At a depth of 3000 km and beyond, the temperature is around 4000 K and above. The Earth exhibits a magnetic field that may be roughly interpreted as originating from a magnetic dipole placed at the Earths center. The poles of the field at the Earths surface are presently located at 788 N, 1048 W, and 658 S, 1398 E, respectively. On the Earths surface, the magnetic field varies between ca. 70 mT at the poles and 25 mT at the equator, corresponding to an average value of the order of 50 mT or 0.5 G. The primary component of this magnetic field, called the main field, is assumed to be due to the so-called dynamo process arising from electric currents flowing in the liquid Core. The Earths magnetic field is not static, but fluctuates in its strength over periods of centuries/milennia. The polarity of the field has reversed many times since the Earths formation, for the last time probably about a million years ago. In spite of these changes, it may be safely assumed that the Earths magnetic field is not independent of the Earths rotation. The axis of the Earths magnetic dipole field and the axis of the Earths angular momentum must manifestly be correlated. Has the Earths magnetic field played any role in prebiotic evolution? Possibly in conjunction with the incident sunlight, causing magnetochiral photochemistry, as suggested in Sect. 3.8? We will return to this question in Sect. 10.5.

6.3. Chaos in the Atmosphere The planet Earth has a relatively dense atmosphere (Greek: atmos = vapor, sfai1a = ball) composed of (in volume percent) 78% nitrogen, 21% oxygen, and minor fractions of argon, carbon dioxide, neon, and other gases. The pressure at sea level fluctuates around 1.013 bar (or hPa). According to the isothermal barometric formula, the pressure decreases exponentially with increasing altitude. From the same formula for a gas mixture, one may show that, at equilibrium, the relative abundance of heavier gases also decreases with height. But, as we know, the atmosphere is far from isothermal, and subject to a variety of perturbations. The atmosphere is vital for the existence of man, animals, and plants. Its content in oxygen is, compared to other planets, an anomaly. It is not known if the origin of the atmospheric oxygen mainly derives from plant photosynthesis or from the direct photodecomposition of water by UV light.

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To classify the height-dependent properties of the atmosphere, one subdivides it into different regions [8]: the troposphere (0 – 11 km), the stratosphere (up to 50 km), followed by the mesosphere (up to 85 km), the thermosphere and ionosphere (up to 300 km), and finally the exosphere (beyond 300 km). Of interest is the uneven decrease in average temperature with height: at sea level: 288 K (plus 15 8C); at 10 km: 223 K (
50 8C); at 20 km: 216 K (
57 8C); at 50 km: 270 K (only
3 8C!); at 80 km: 198 K (
75 8C). The unexpected intermediate maximum at 50 km occurs because ozone in this region absorbs solar UV radiation and heats the atmosphere. Ozone efficiently shields the Earths surface from this radiation, which may be harmful for living organisms. As is well known, the depletion of the ozone layer by perturbing influences, partly caused by man, is a matter of concern. Most of what we call
weather takes place in the troposphere and the lower stratosphere. The prediction of the weather is of central importance for much of human activity. In the course of time, man has come to realize that the weather is not just a local phenomenon, nor is it the expression of the whims of gods. Rather, it is the result of the worldwide interconnection of different influences on different terrestrial scales. The science of meteorology aims at systematizing the observation and interpretation of these effects. First, the General Circulation on a world scale: thermal drive and convection, the effect of the Earths rotation on atmospheric transport, the genesis of cyclones and anticyclones and of weather fronts. The Secondary Circulations: monsoon winds, tropical hurricanes. The Tertiary Circulations: convection produced by local heating, local showers, and thunderstorms. The basic problem of the weather forecaster is to extrapolate in some manner from an initial (recent) state of the atmosphere to some subsequent (future) state. How difficult this is, we realize daily, even in the modern era of weather satellites and worldwide networks of meteorological measurements. Nonetheless, the interplay of all these influences merits to be, and can be, better understood. Ideally, one would like to arrive at mathematical models capable of making long range forecasts. In view of all the elements that come into play, this is indeed an extremely demanding task. It turns out that even highly simplified model systems may reveal unexpected, sometimes chaotic, properties. One of these is the so-called Rayleigh–B
nard convection and the hydrodynamic equations used to describe it. The experiment consists of a fluid slab of finite thickness that is heated from below. A fixed temperature difference is maintained between the top cold surface and the bottom hot surface. Gravity acts downward, buoyancy, due to thermal expansion of the fluid, upward. The fluid motion (velocity field) is described by the Navier–Stokes equations. Furthermore, the conservation laws for mass and for heat must be fulfilled. Assuming only two-dimensional motion perpendicular to the two limiting

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Fig. 6.3. Rayleigh–Bnard Convection. a) Side-view of the convection currents between the lower (hot) plate and the upper (cold) plate (adapted from [10]). b) Top-view of some computer-simulated convection patterns (ascending and descending fluid regions) (from H.-W. Xi, J. D. Gunton, J. ViÇals,
Spiral-pattern Formation in Rayleigh–B
nard Convection, Phys. Rev. E 1993, 47, R2987).

surfaces (Fig. 6.3), the fluid flow can be characterized by two kinds of variables, namely, the stream function for the motion, and the deviation of the temperature profile from a linear decrease with height. To make even this simplified problem more easily tractable, the american meteorologist Edward Lorenz arrived at a system of coupled first-order differential equations for three dynamic variables [9], depending on three adjustable constant physical parameters. To his surprise, Lorenz noticed that even this set of simplified equations could lead to deterministic but nonperiodic, in present-day language, chaotic solutions. These time-dependent equations have, in the meantime, attracted so much attention, that it is worthwhile to briefly consider them. The variable x(t) is the amplitude of the convection motion, y(t) the temperature

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Fig. 6.4. Representation of the three-dimensional Lorenz attractor, rigorously proven to display chaotic motion. Notice the
chirality of the structure (from P. Bourke,
The Lorenz Attractor in 3D, http://astronomy.swin.edu.au/~pbourke/fractals/lorenz, 1997).

difference between ascending and descending currents, z(t) the distortion of the vertical temperature profile from linearity [10 – 12]. The equations read: dx/dt = a (y
x), dy/dt = x (b
z)
y, dz/dt = xy
cz. The constants a, b, and c characterize the flow properties of the system and are not directly relevant to our present discussion. The solutions of these coupled equations may be represented as trajectories in x, y, z space, and they describe a so-called 3D Lorenz Attractor (Fig. 6.4). It is immediately obvious that the illustration of the chaotic situation so obtained, is chiral. The above equations are invariant under the transformation x !
x, y !
y, z ! z, but they are not invariant under the parity operation P: x !
x, y !
y, z !
z. However, if this
chirality is more than just a curiosity, and if

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On Chirality and the Universal Asymmetry

it has any deeper physical meaning in our present context, may indeed be questioned. With this example, we notice how easily chaotic situations may occur in fluid systems. We will approach the hydrodynamic aspect from another point of view in Sect. 7.5, in connection with vorticity. What we wish to emphasize in conclusion here is that in the Earths atmosphere there always has been, and most probably always will be, plenty of room for unpredictable weather.

6.4. Geological and Mineralogical Chirality The surface of the Earth, the continents, mountains, hills, and landscapes are chiral. The world that immediately surrounds us is chiral. Numerous geomorphological structures are also fractal, from the coastline of Britain to the summits of the Himalayas. For instance, consider a mountain of volumedimension km3. You may find a single rock resembling this mountain, of volume-dimension m3, and even a grain of gravel of similar form, of volumedimension only mm3. The chirality in these cases is also connected to selfsimilarity at different scales. Choose a stone in your garden, or at the beach, or on a mountain excursion. This stone will almost certainly be chiral. Pick up at random a million more stones of comparable size. The more stones you consider, the bigger will the probability be that you find and include in your collection a stone nearly enantiomorphous to the first one. This suggests that a very large number of stones will tend to be less chiral than any single stone (unless the one considered happens to be nearly symmetrical (Sect. 4.1)). As the number of stones increases, the collection will approach a state that we might call
pseudoracemic. This does not mean
exactly racemic or achiral; only nearly so. It reminds us of the situation which we encountered in considering the universe as a whole (Sect. 5.7): on the largest cosmic scale, the universe appears nearly homogeneous, although it is made up of galaxies which, individually, are chiral. And as we recall, on the cosmic scale, we also encountered fractality. Stones are, in general, compact solid mixtures of minerals. The internal microscopic structure of stones is irregular. Minerals, on the other hand, are pure chemical compounds, characterized by a well-defined crystal structure. Therefore, to find the exact enantiomorph of a particular stone seems almost unfeasible. But to find the exact enantiomorph of a chiral mineral is, in principle, possible. About 12% of the volume of the Earths crust is composed of the mineral quartz. Quartz occurs in sands or, as compressed grains, in stones, and sometimes as large crystals. It is widely applied in technology because of its piezo-

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electric and particular optical properties. Well-grown crystal specimens are also used in jewelry and are considered as semiprecious stones. Quartz in the a form, stable under normal conditions, crystallizes in the symmetry group D3 and is, because of the absence of any rotation–reflection axes, chiral (Sect. 1.2, 1.3, and 4.1). Already in 1932, the question was investigated, if racemic mixtures of chiral organic molecules, interacting with either a d- or a l-quartz surface as catalyst, might chemically react to give an excess of one enantiomer in the product. Initial experiments led to contradictory results, however [13] [14]. A small imbalance of chirality was subsequently shown in the asymmetric adsorption of d- and l-alanine hydrochloride by quartz [15]. More recently, the d- and l-quartz-promoted, highly enantioselective synthesis of a chiral organic compound has been achieved [16]. The possibility that quartz might have been the source of chirality for the first optically active organic molecules on Earth has been envisaged and discussed [17] [18]. The question, if on Earth there might exist an excess of d- or l-quartz, is, therefore, of fundamental interest from several points of view. If such an excess really existed, it might give a clue as to how enantioselective biological homochirality gradually developed.

Fig. 6.5. Possible pathways of the amplification of asymmetry and of its induction at the molecular level. Essentially, there is the macroscopic transmission path (red arrows), and the direct microscopic induction (blue arrows).

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On Chirality and the Universal Asymmetry

Due to the energy difference between l- and d-quartz caused by the parityviolating weak interactions (Sect. 2.8), an excess of one enantiomorphous form in nature, namely of l-quartz, has been postulated [19] [20]. However, the results of the hitherto most exhaustive analyses of data on the distribution of left and right quartz in many locations of the surface of the Earth indicate that overall the enantiomorphous crystals are distributed in equal amounts [21]. It appears, therefore, unlikely that quartz-mediated reactions lie at the immediate origin of the enantioselective, prebiotic organic syntheses. This brings us back to the general and basic question of how the biochemical enantioselectivity has developed. Has it arisen directly from the parity-violating interactions within the individual reacting molecules (Sect. 2.8 and 2.9)? Or is it due to the overall macroscopic and microscopic asymmetry of the surroundings that has gradually evolved, in innumerable steps, from the symmetry-broken states of the early universe (Fig. 6.5)? We shall return to these questions and discuss them in more detail in Sect. 10.2 – 10.7.

REFERENCES [1] I. Peterson, Newtons Clock – Chaos in the Solar System, W. H. Freeman & Co., New York, 1993. [2] H. Bai-Lin, Chaos, World Scientific, Singapore, 1984. [3] D. P. Hamilton,
Jupiters Moonopoly, Nature 2003, 423, 235. [4] S. A. Astakhov, A. D. Burbanks, S. Wiggins, D. Farrelly,
Chaos-Assisted Capture of Irregular Moons, Nature 2003, 423, 264. [5] I. Prigogine, From Being to Becoming – Time and Complexity in Physical Sciences, Freeman, San Francisco, 1980. [6] M. Gell-Mann, The Quark and the Jaguar – Adventures in the Simple and the Complex, Abacus, London, 1994. [7] D. J. Doornbos,
Earth Interior, Encyclopedia of Science and Technology, McGraw Hill, New York, 1992. [8] R. G. Roble,
Atmospheric Structure, Encyclopedia of Applied Physics, Ed. G. L. Trigg, VCH, New York, 1991. [9] E. N. Lorenz,
Deterministic Nonperiodic Flow, J. Atmos. Sci. 1963, 20, 130. [10] A. J. Lichtenberg, M. A. Lieberman, Regular and Chaotic Dynamics, Springer Verlag, New York, 1992. [11] M. C. Cross, P. C. Hohenberg,
Pattern Formation Outside of Equilibrium, Rev. Mod. Phys. 1993, 65, 851. [12] H. D. Abarbanel, R. Brown, J. J. Sidorowich, L. S. Tsimring,
The Analysis of Observed Chaotic Data in Physical Systems, Rev. Mod. Phys. 1993, 65, 1331. [13] G.-M. Schwab, L. Rudolph,
Katalytische Spaltung von Racematen durch Rechts- und Linksquarz, Naturwissenschaften 1932, 21, 363. [14] A. Amariglio, H. Amariglio, X. Duval,
Essais de R
actions Dissym
triques sur Quartz Optiquement Actif(
Dissymmetric Reactions on Optically Active Quartz), Helv. Chim. Acta 1968, 51, 2110. [15] W. A. Bonner, P. R. Kavasmaneck,
Asymmetric Adsorption of d,l-Alanine Hydrochloride by Quartz, J. Org. Chem. 1976, 41, 2225. [16] K. Soai, S. Osanai, K. Kadowaki, S. Yonekubo, T. Shibata, I. Sato,
d- and l-Quartz-Promoted Highly Enantioselective Synthesis of a Chiral Organic Compound, J. Am. Chem. Soc. 1999, 121, 11235.

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[17] A. G. Cairns-Smith,
Chirality and the Common Ancestor Effect, Chem. Br. 1986, 22, 559. [18] W. A. Bonner,
The Quest for Chirality, AIP Conference Proc. 1996, 379, 17. [19] G. E. Tranter,
Parity-Violating Energy Differences of Chiral Minerals and the Origin of Biomolecular Homochirality, Nature 1985, 318, 172. [20] A. J. MacDermott,
The Weak Force and SETH: The Search for Extra-Terrestrial Homochirality, AIP Conference Proc. 1996, 379, 241. [21] E. Klabunovskii, W. Thiemann,
The Role of Quartz in the Origin of Optical Activity on Earth, Orig. Life Evol. Biosph. 2000, 30, 431.

7. Chirality at the Nano- and Micrometer Scale
L
invisible est dans le visible. Il nous aborde quand le visible nous parle, nous sollicite comme de lui-mÞme et sans raison (
The invisible is in the visible. It addresses us when the visible talks to us, challenges us as if by itself and without reason) Roger Munier (b. 1923)

7.1. On Chiral Crystals When the crystallization of a chemically uniform substance takes place under regular conditions and free from outside perturbations, the macroscopic shape of the developing crystals is determined solely by the particularities of the internal structure. Perfect crystals so grown display certain kinds of visible symmetry elements, such as (Sect. 4.1): rotation axes Cn ; rotation–reflection axes Sn. The totality of symmetry elements of a crystal represents the point group to which that crystal belongs. Depending on the composition, temperature, and pressure, any pure substance will crystallize in a particular, welldefined point group. In all, there occur in nature 32 different crystallographic point groups. Although we superficially tend to associate the notion of chirality with asymmetry, and that of a crystal with symmetry, there indeed also exist chiral crystals, namely, all those which as macroscopic symmetry elements only display rotations. The atoms/molecules of which a crystal is composed are arranged in space as a quasi-infinite lattice. The smallest portion of a lattice which represents the symmetry and structure of the entire lattice is called the unit cell. It is a parallelepiped containing one unit of the translationally repeating pattern of the lattice, and is thus the fundamental entity from which the entire lattice may be constructed by purely translational displacements. To a given point group there exist in general more than one compatible lattice. The point group alone is, therefore, insufficient to satisfactorily characterize the internal structure of a crystal. The way in which the atomic/ molecular elements of crystals are arranged in space is described by the socalled space groups, of which there are in all 230. To characterize quasi-infinite lattices, space groups not only contain the same symmetry operations as point groups, but, in addition, beside pure translations, also screw axes and glide planes. Consequently, there generally exists to a given point group more than one space group. On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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Of the 230 space groups, there are 65 that contain only pure rotation and screw axes (symmetry elements of the first kind), and are compatible with chiral crystal structures. Among these, 22 space groups are pairwise enantiomorphous, and are, therefore, truly chiral. The 43 other space groups, though compatible with lattices composed of chiral molecules of uniform handedness, transform into themselves by inversion through a point and are, as such, achiral. Using as notation the Schçnflies symbols, the chiral point groups are indicated below with (in brackets) the corresponding number of compatible space groups [1] [2]. Triclinic system: C1(1); monoclinic system: C2(3); orthorhombic system: D2(9); tetragonal system: C4(6), D4(10); trigonal system: C3(4), D3(7); hexagonal system: C6(6), D6(6); cubic system: T(5), O(8). Our main question now is: to what extent does chirality of the whole crystal imply chirality of the repetitive elements at the atomic/molecular scale? And to what extent does chirality of the elements imply chirality of the whole? The second question is easier to answer than the first. Suppose we have a molecular crystal composed exclusively of the (R)-enantiomer, say, of a given compound. It is straightforward to conclude that such an enantiomerically pure crystal must always be chiral. Any arrangement in space of the same enantiomer, even a random distribution, will always lack any rotation–reflection axis and will, therefore, be chiral (concerning hypothetical exceptions; see Sect. 8.2). But the reverse is not true. A chiral crystal may indeed form and exist which is composed of achiral atomic/molecular elements (Table 7.1). This may, at first sight, seem somewhat surprising, and will now be considered in more detail. As already indicated (Sect. 1.2 and 6.4), a-quartz crystallizes in the chiral point group D3 and occurs in two enantiomorphous space groups P3121 (right) and P3221 (left). The unit cell of quartz contains three Si-atoms and six O-atoms, corresponding to three SiO2 units. In the lattice, each Si-atom is at the center of a tetrahedron with four O-atoms at the corners, and each O-atom forms a bridge between two Si-atoms. If one considers a SiO4 tetrahedron, it will not be chiral by itself, neither will be a Si
O
Si bridge. The chirality is generated by the crystal lattice, namely, by the spatial arrangement of these units. One may say that the chirality exclusively resides in the space group and in the point group of the crystal as a whole. Indeed, if we consider only the Si-atoms, we notice that, in the lattice, they form a right-handed and left-handed helix, respectively, parallel Table 7.1. Restrictions in the Formation of Chiral/Achiral Crystals from Chiral/Achiral Atomic or Molecular Elements (for details, see, e.g., H. D. Flack,
Chiral and Achiral Crystal Structures, Helv. Chim. Acta 2003, 86, 905) Atomic or molecular element

Crystal

Chiral (pure (R) or (S)) Racemic ((R)/(S) 1 : 1) Achiral

Chiral (always) Achiral or chiral Achiral, chiral as notable exceptions

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Fig. 7.1. View down the trigonal c-axis of the crystal structure of tellurium. Notice the helices, here chosen to be right-handed (adapted from A. M. Schwartz,
Equivalence Principle Parity Test, 2002, http://www.mazepath.com/uncleal/eotvos.pdf).

to the threefold screw axis (c-axis). Similar helices are obtained, if along the same axis we follow the bridge O-atoms [3]. An even more striking example is that of a chiral atomic lattice. Metals and semimetals often form densely packed achiral lattices of high symmetry, such as the face-centered cubic (fcc) or hexagonal close-packed (hcp) lattices. An interesting exception is tellurium (Te) which, like quartz, crystallizes in the point group D3 and in the same enantiomorphous space groups. The unit cell of Te contains three atoms, and Fig. 7.1 shows their helical arrangement. What then causes this chiral reduction in symmetry? It may be shown that, in a hard-sphere fluid, volume exclusion and cylindrical confinement may lead to helical configurations, thus inducing chirality. In the absence of any other influences, both enantiomorphous configurations are of course equally probable. Similar effects have been theoretically shown to occur in a model of magnetorheological particles consisting of hard spheres with oriented (magnetic) dipole moments and short-ranged attractive (electrostatic) forces [4]. Such considerations, though instructive, do not help us much in trying to interpret the chiral structure of Te. Here, rather, electronic factors come into play: quantum-chemical calculations on the basis of the so-called density functional theory show that a hypothetical simple cubic structure of high symmetry (Oh) would be unstable with respect to the chiral distortions to D3 [5]. In other words, the chiral structures are lower in energy. The hypothetical cubic structure exhibits an orbitally degenerate ground state which is prone to socalled Peierls instability. This also explains the fact that Te is a semiconductor and not a conductor. Discounting the parity-violating weak interactions (Sect. 2.8), both enantiomorphous forms of Te evidently have the same energy and should occur with equal probability.

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As the helical structure shown (Fig. 7.1) suggests, and as is also interesting from a technological point of view, Tellurium lends itself to the synthesis of nanotubes [6] (see also Sect. 7.4). We continue this section by considering one more type of crystal where the building units are not chiral, but the crystal built from them is. The chlorate and bromate anions, ClO
3 and BrO
3 , are in solution trigonal-planar and thus achiral. Both ionic compounds NaClO3 and NaBrO3 crystallize in the chiral cubic point group T (containing exclusively threefold and twofold rotation axes) and in the nonenantiomorphous space group P213. The unit cell in both crystals contains four anionic units (with the corresponding cations), and the Clor Br-atoms occupy distorted octahedra [7] [8]. Aqueous solutions of NaClO3 or NaBrO3 are evidently achiral and thus non-optically active. If a supersaturated solution starts to crystallize, both enantiomorphous crystal forms are expected to occur in equal amounts. There have been attempts to induce an excess of one form with respect to the other by external factors, such as stirring of the solution. If a saturated solution of NaClO3 or NaBrO3 is agitated with levo- or dextrorotatory crystal powders of the same compound, the nucleation effect furnishes new crystals with a high degree of enantioselective purity [9]. An interesting experiment is the following [10]: a saturated solution of NaClO3 is made to flow over a NaBrO3 crystal of given handedness. It is then observed that the NaBrO3 crystal catalyzes the preferential formation of NaClO3 crystals of the same handedness or absolute configuration. In other words, chiral symmetry breaking has been induced through the flowing contact with a crystal of different composition but same type of lattice. The details of the nucleation kinetics, however, are probably very complex. There exists quite a number of mixed metal oxides of chiral crystal structure. These compounds are of interest in materials science [11]. After having focused our attention on chiral crystals built of achiral elementary units, we now consider crystals composed of molecules that are, themselves, chiral. In particular, we start from a racemic liquid mixture, containing equal amounts of the enantiomers d and l (or (R) and (S); our choice of designations for the absolute configuration is always a matter of convenience). Such a racemic mixture can solidify either as a racemic conglomerate, a racemic compound, or, rarely, as a racemic solid solution (Fig. 7.2). A conglomerate is a mechanical mixture of crystals of the two pure, thus homochiral, enantiomers. In a conglomerate, the crystal lattices are necessarily chiral. The enantiomorphous crystals appear as separate entities. This presents the situation of spontaneous resolution of a molecular racemate, as discovered by Pasteur (Sect. 1.3). The enantiomers may practically be separated by mechanical means. A racemic compound consists of crystals that are racemic, in the sense that the two molecular enantiomers are paired up in the unit cell of the crystal lattice. A racemic compound may, in principle, crystallize in any group,

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Fig. 7.2. Schematic representations of the enantiomorphous, homochiral crystals of a conglomerate (a) and of the same racemate as a solid racemic compound (b), with generalized liquid–solid phase diagram for the conglomerate (c) and the racemic compound (d) (adapted from J. Jacques, A. Collet, S. H. Wilen, Enantiomers, Racemates and Resolutions, John Wiley & Sons, New York, 1981).

even also a chiral one (Table 7.1). Finally, in a racemic solid solution, the two enantiomers coexist in variable ratios, in a comparatively unordered, amorphous solid state. In most cases, racemic mixtures of chiral molecules crystallize as racemic compounds. The thermodynamic equilibrium of the reaction l-crystal + d-crystal ! racemic compound usually lies to the right. Beside energetic influences that we shall consider in more detail later, the racemic compound is favored by entropic effects. The entropy of mixing per mol is seen to be R ln 2 (R, the gas constant) and the free energy of the reaction thus diminishes by
RT ln 2 (T, the absolute temperature), thereby shifting the equilibrium in favor of the racemic crystal. However, the formation of ordered, homochiral crystalline domains, and the prospect of relatively easy resolution of enantiomers by crystallization, makes the investigation of conglomerates a topic of high interest, to which we shall return (Sect. 9.11).

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7.2. Chirality in Liquid Crystals For most elements and for many smaller molecules, there is a sharp transition between the liquid and the solid state. However, for certain elongated, rod-like molecules, polymers, and amphiphilic compounds, one encounters a sequence of transitions on the way from the liquid to the solid, revealing the existence of intermediate phases. Such phases are called mesomorphic or liquidcrystalline. The structure of a liquid crystal evidently closely depends on the constitutional and steric properties of the molecules of which it is composed, and on the particular intermolecular interactions. A true liquid is isotropic in its mechanical and optical properties, the molecules being randomly oriented. A liquid crystal shows definite orientations of the constituent molecules and correspondingly an anisotropy of its physical properties. As we have seen in the last section, a real crystal in principle reveals perfect order in all three spatial dimensions. A liquid crystal possesses approximate order in only one or two dimensions. In the nematic phase (from the greek nhma = thread), the centers of gravity of the individual molecules are randomly distributed, but all molecules tend to be oriented parallel to some common axis or director n (Fig. 7.3). The direction of n is arbitrary in space, depending on minor external forces. The distribution is invariant with respect to the absolute sign of n. The nematic phase is consequently on the average centrosymmetric. The smectic phases (greek smhgma = soap) exhibit a layered structure with a well-defined interlayer spacing that can be measured by X-ray diffraction. Each layer behaves as a two-dimensional liquid, and there is inside a layer no longrange order of the molecular centers of gravity. One distinguishes mainly between the phases smectic A and smectic C. In the A phase, the molecular axes are on average perpendicular to the layers. In the C phase, the director is tilted by a certain angle. Depending on the molecular structure and dimensions, a variety of other liquid-crystalline phases also exist [12] [13], but for further information, the reader is referred to the specialized literature. What interests us, in particular, is how these phases are modified when the molecules are chiral. Perhaps the most striking phenomenon is the appearance of the cholesteric phase. The name
cholesteric derives from the fact that liquid crystals composed of chiral cholesterol esters tend to adopt this phase. These liquid crystals were among the first to be discovered and have been very thoroughly studied. The cholesteric phase may be considered as a helically distorted nematic phase (Fig. 7.3). Instead of pointing everywhere in the same, arbitrary direction, the director n precesses around a helical axis to which it is perpendicular. Because the physical situation for n and
n is equivalent, the spatial period of the helix is one half the pitch. Whereas the constituent molecules have

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Fig. 7.3. Left: Achiral liquid crystal phases (adapted from R. Macdonald, Liquid Crystals – Fascinating State of Matter or
Soft is Beautiful, http://moebius.physik.tu-berlin.de/lasergrp/lc/ lcs.html). Right: helical arrangement of molecules in a cholesteric phase. 2p/q0 is the pitch of the helix. The successive planes have been drawn for illustration purposes, but have here no direct physical meaning (adapted from [12]; Copyright permission, Oxford University Press).

lengths on the order of tens of nanometers or less, the spatial period of the helix is on the order of hundreds of nanometers or more; it is thus generally large compared to the molecular dimensions. The close relationship between the cholesteric phase and the nematic phase is well demonstrated in the observation that a nematic phase, composed of achiral molecules, may be transformed into a cholesteric phase by the addition of chiral molecules, even in relatively low concentration. This chiral induction is highly interesting from the point of view of the study of intermolecular interactions. The aim of such investigations is to correlate the molecular chirality of the guest, or solute, molecule with the sign and size of the pitch of the resulting host–guest, or solution, phase. It is important to note that under identical conditions, the guest enantiomer induces an enantiomorphous helix, of the same pitch but of opposite handedness. With racemic dopants, the solution phase quite evidently on the average remains nematic. Depending on the molecules

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involved, mol fractions of chiral dopants needed to induce a cholesteric phase may be as low as 0.0001 and even less [14]. This is a striking example of chiral symmetry breaking on a mesoscopic scale by relatively small perturbations. Cholesteric phases show interesting optical, electric, and magnetic properties. When light propagates along the helix axis, there is circular birefringence. Within a certain wavelength range, depending on the composition of the medium and the helical pitch, one circularly polarized light component (see also Sect. 3.1) may even get totally reflected, while the other of opposite handedness passes through unchanged [15]. This so-called Bragg scattering in cholesteric liquid crystals will be influenced by a magnetic field parallel to the helix axis, an effect essentially due to magnetochiral birefringence [16]. Many molecules forming nematic and/or cholesteric phases show very high dielectric and diamagnetic anisotropies. In some cases, the helix axis of a cholesteric phase will therefore tend to orient itself perpendicular to an applied external electric field. The effect of the field will then be to unwind the helix. This phenomenon has been put to use in chiroptical spectroscopy [17]: to measure the circular dichroism of linearly oriented samples of a chiral cholestenone derivative, a 3% solution of the sample was prepared in an achiral nematic liquid-crystal solvent. The cholesteric phase so induced was then returned to a nematic state by an applied AC electric field of the order of 106 V m
1. The anisotropic part of the molecular diamagnetic susceptibility gives rise to a coupling between an external magnetic field and a cholesteric structure. If, in zero magnetic field, the cholesteric liquid crystal has the characteristic helical molecular arrangement, in the presence of a magnetic field (of appropriate direction and strength) the structure may get distorted, and the pitch of the helix increases. If the field reaches and exceeds a certain critical limit (of the order tesla), there is a transition to the nematic phase [18] [19]. If, on one hand, the nematic phase may be made cholesteric by the addition of chiral solute molecules, the reverse is also observable [20]: for instance, achiral fluorescent molecules dissolved in an inherently cholesteric phase exhibit an induced chirality detectable by their circularly polarized luminescence. There, of course, also exist chiral smectic phases [12]. As we have just seen, if we try to obtain a nematic phase with a material that is chiral, there will generally occur a helical distortion to a cholesteric phase. Because of the confinement to layers, and the perpendicular orientation of the molecules within a layer, no such distortions can occur in a chiral smectic A phase. On the other hand, we have noticed that in a smectic C phase the molecules are tilted with respect to the normal to the plane of the layer (Fig. 7.3). In a chiral smectic C phase, the tilted directors will rotate, on going from one superposed layer to the next, around the common normal. In successive layers, the tilt directors will

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thereby describe a helix, and the common normal will become the helix axis. The reader is encouraged to make a drawing of such a structure and to notice the difference with respect to the cholesteric phase.

7.3. Chiral Surfaces A planar surface between two different homogeneous phases breaks the inversion symmetry of space but is in itself not chiral. To be chiral, an object must extend into the third spatial dimension. Therefore, the surface of a crystal may be chiral only because of the spatial arrangement of the atoms that make up that surface. If the bulk of a crystal is chiral, its surfaces should quite obviously also be chiral. If the bulk of a crystal is achiral and even highly symmetric, some of its surfaces may, astonishingly, also be chiral. This fact has been systematically explored only in the course of the last one or two decades. Novel technologies for the fabrication of surfaces under ultrahigh vacuum conditions, and new methods of microscopic observation, such as scanning tunneling microscopy (STM), transmission electron microscopy (TEM), or low-energy electron diffraction (LEED), have been instrumental in advancing this field of research. Of particular interest are metal surfaces in view of their electron exchange capacities and their use in heterogeneous chemical catalysis. Suitable metals, for instance, are nickel (Ni), copper (Cu), silver (Ag), gold (Au), rhodium (Rh), iridium (Ir), palladium (Pd), and platinum (Pt). It has been discovered that, even for crystals of very high bulk symmetry with a face-centered cubic (fcc) lattice, exposed surfaces may show a chiral structure. These surfaces tend to have high Miller indices (hkl). For instance, the fcc (643) surface of Pt may be viewed as made up of atomically flat fcc(111) terraces separated by steps one atom in height. The step edges are not straight lines. The crucial observation is that the spatial arrangement of these step edges constitutes a chiral structure (Fig. 7.4). Surfaces of this kind which are related by an odd number of spatial reflections within the crystal, are enantiomorphous. Thus (643), (643), (643), and (6 43) are equivalent; and they are enantiomorphous to (6 4 3), (643), (643), and (643) [21] [22]. Equivalent surfaces may be related by pure rotations. Enantiomorphous surfaces should react enantioselectively with a chiral adsorbent. As a reminder: the Miller indices (hkl) are directly related to the reciprocals of the lengths of the intercepts of a given crystal plane with the crystal axes. More precisely, a given triple (hkl) of Miller indices defines an infinite set of parallel planes at given angles with respect to the crystal axes. The higher the indices are, the smaller the distance between these planes, and the smaller this distance is, the more the local inhomogeneities due to the atomic structure of the crystal manifest themselves. In conclusion, the crystal with a highly symmetric bulk structure may nonetheless exhibit surfaces which are chiral.

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Fig. 7.4. Atomic ball models of the enantiomorphous fcc(643) and fcc(643) surfaces. The broken lines mark the step edges (adapted from [21]).

One half of these surfaces is enantiomorphous to the other half. From the point of view of the crystal as a whole, the surface structure is, therefore, racemic. We find here a striking example of local symmetry breaking, but of overall symmetry conservation. Inherently achiral surfaces, or surfaces with irregularities that are racemic, may be made chiral by adsorbing or binding molecules in such a way that the composite system lacks Sn-symmetry and is, therefore, chiral. Evidently, the adsorbed molecules must either themselves be chiral, or they must induce local chirality. In the context of molecules of biochemical interest, one may mention the example of (R,R)-tartaric acid on Cu(110) [23] or (S)-alanine on Cu(110) [24]. Attempts to bind (M)-[7]-helicene on Cu(111) [25] (see also Sect. 1.3) are interesting from the point of view of the pronounced helical structure of the adsorbate. However, the presence of a chiral adsorbate is not a prerequisite for inducing chirality on an achiral surface. The important condition is that the system (molecule + surface) be chiral. Consider, for example, a molecule which is planar and, therefore, achiral, but which may become chiral if an additional ligand is attached above (or below) the molecular plane. Such a molecule is termed prochiral. Bind a prochiral molecule onto a metal surface. The metal surface acts as the additional ligand, and the system as a whole becomes chiral. Chiral molecules adsorbed individually on metal surfaces often assemble into supramolecular ordered structures [26], mainly through direct lateral interactions, indirectly also via modification of the metal–metal bonds of the surface. An organisational chirality may thus be created which depends directly on the local chirality of the individual adsorbate. It may be verified that the enantiomorphous structure cannot develop from one and the same enantiomer. The corresponding mirror surface can obviously only be created via adsorption, or creation, of the opposite enantiomer (or

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antipode). The role of such supramolecular assemblies is of critical importance in inducing additional chirality, for instance, when the surface acts as a catalyst. When a racemic mixture of chiral molecules is bound to an achiral surface, supramolecular domains of enantiomorphous configuration may originate. One then observes spontaneous segregation into homochiral regions of opposite chirality. This surface resolution bears a resemblance to the enantioselective three-dimensional nucleation and the formation of crystalline conglomerates from a racemic solution, first observed by Pasteur (see Sect. 1.3, 7.1, and 9.2). These are important examples of how chirality at a higher level of organization depends on the presence, or the generation, of chirality at a more local, lower level. We have here focused our attention mainly on metal surfaces, and the interface between the solid and the liquid or gas phase. Chiral nucleation may also take place, and has been intensely studied, at the liquid–gas interface, in particular with Langmuir monolayers acting as templating agents. As already mentioned in Sect. 1.1, the generation of molecular homochirality is of fundamental interest. Surfaces and phase boundaries offer interesting possibilities for inducing enantioselective processes, eventually leading to homochiral structures. We will consider this general and important topic again in Sect. 9.2, 9.3, 9.7, 9.9, and 10.5, also in the context of prebiotic evolution.

7.4. Chiral Carbon Nanotubes Before the discovery of the fullerenes in the mid- and late 1980s, it was assumed that elemental carbon could exist only in two forms in the solid state: as graphite and as diamond. Soot was then believed to be just a mixture of planar graphite particles and of unsaturated hydrocarbons. The detection and isolation of new allotropic states of pure carbon in the form of hollow cage molecules became a major breakthrough both in basic chemistry and in applied materials science. As we have already mentioned, the first fullerene to be discovered, C60, shows icosahedral symmetry and is achiral. However, some higher fullerenes, such as C76 and C84, undergo a reduction in symmetry leading to a most stable ground-state conformation that is chiral (Sect. 1.3). They consequently exist as pairs of enantiomers. The gradual elongation of higher-order fullerenes may lead to molecules of tubular architecture, consisting of a single layer of the honeycomb graphite structure rolled up in form of a cylinder [27]. While the molecule C60 displays 12 five-membered carbon rings and only 20 six-membered rings, a carbon nanotube may show some five-membered rings at the ends, but the cylindrical walls have a regular graphene structure.

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Fig. 7.5. Definition of the chirality vector C for various values (n,m) of the chiral indices. For chiral angles q of 308 and whole multiples thereof, the nanotube is achiral. The reference vectors a1 and a2 are shown at the lower left. As an exercise, the reader may fill in the indices for additional sixmembered rings and calculate the chiral angles, simultaneously also determining the radius of the corresponding nanotube (adapted from [29]).

As one today knows, these cylindrical molecular carbon fibers may occur as single-walled (SWNT) or as multi-walled nanotubes (MWNT). In the first case the graphene sheet is rolled up once to form a seamless tube. The resulting structure is usually helical [28], but with notable exceptions. In the helical structure, all symmetry axes of the graphene sheet form an angle with respect to the cylinder axis, thereby defining a helical pitch. The structure of a SWNT is thus determined not only by its diameter, but also by the pitch, or chiral angle, of the helix. In MWNTs, there are essentially two architectures. 1) Several tubes of different diameter are concentrically nested inside each other. 2) One and the same graphene sheet is rolled up several times with uniform chirality to a scroll structure [29]. It is now well-established that the physical and chemical properties of carbon nanotubes depend critically on the helical parameters. For this reason, we wish to look at them more closely (Fig. 7.5). For a SWNT, a chirality (or circumference) vector C determines the circumference C, as well as the chirality (or helical pitch) angle q. The chirality vector is referred to two symmetry-adapted basis vectorspaffiffiffi1 and a2. These vectors enclose an angle of 608, and their absolute value a is 3d, where d is the carbon–carbon bond length (ca. 0.14 nm). Thus in a general way we may write: C = na1 + ma2, j a1 j = j a2 j  a,

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where n and m are integers. From simple vector geometry follows: j C j = C = a(n2 + m2 + nm)1/2. One must imagine the vector C in every case as being wrapped around the cylinder, perpendicularly to the cylinder axis. For the cases n „ 0, m = 0, the cylinder axis is parallel to one of the symmetry axes of the graphene sheet. For the cases n = m, likewise. Inbetween, there are variable degrees of skewness. The vector C may also be viewed as the sum of two perpendicular vectors, one along the general direction (n,0) ofplength (n + (m/2))a, the other at a right angle ffiffiffi to the first vector, and of length ( 3=2)ma. The tangent of the chiral angle q is then defined as the second quantity divided by the first, namely: pffiffiffi tan q = 3/(1 + (2n/m)). We then find: tan q ¼ 0, q ¼ 0 , pffiffiffi for ðn, nÞ, tan q ¼ 3=3, q ¼ 30 , pffiffiffi for ð0, mÞ, tan q ¼ 3, q ¼ 60 :

for ðn, 0Þ,

The reader will notice that, for all chirality angles of value = N · 308, where N is an integer, the skewness of the graphene sheet with respect to the cylinder axis vanishes, and so does the chirality. For N even (08, 608, etc.), one finds the socalled
zigzag structure, for N odd (308, 908, …), the
armchair structure. The indices (n,m) are not only of geometric significance. They are closely related to electronic [28] [30] and chemical [31] properties. For instance, it has been shown that if jn
mj is a multiple of three, then the nanotube is a onedimensional metal. Because n and m, in principle, can have any integral value, one third of the SWNTs are expected to be metallic, and the remaining two thirds to have an electronic energy bandgap making them semiconducting. Evidently, chiral nanotubes exhibit natural optical activity, in particular, the chiroptic properties that we have previously discussed (Sect. 1.2, 3.3, and 3.5), such as circular dichroism and magnetochiral dichroism. Furthermore, chiral nanotubes are expected to show conduction effects depending on the coupling of the electron wave vector along the tube principal axis and the orbital momentum around the tube circumference. In the photogalvanic effect, circularly polarized light incident along the helix axis induces a current along the axis. The effect may be considered as a transformation of the photon angular momentum into a translational motion of free charge carriers [32]. It is an electronic analog of a mechanical system which by rotatory motion induces

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linear translation, such as a screw or a propeller (see also Sect. 1.1, 4.1, and 5.1). The magnetochiral effect in electric conduction [33] (see Sect. 4.4) manifests itself as a change in conductivity along the nanotube, depending on a static magnetic field parallel or antiparallel to the helix axis. An interesting perspective is offered by the possibility that ultrathin carbon nanotubes may coalesce without suffering atomic defects, to form new nanotubes of different dimensions. Theoretical investigations by moleculardynamics simulations predict a sum rule for the chiral indices [31]: (n, m) + (n’, m’) = (n + n’, m + m’). The reader may easily ascertain that, in general, the chirality of the product (at right) may then be different from that of the reactants.

7.5. Observing Vortices By the word vortex, one designates a mass of streaming fluid, simultaneously undergoing a whirling, rotatory motion. We are daily observers of the formation and disappearance of such vortices: in water leaving a tub or sink by a narrow outlet at the bottom; when a drop of ink falls into a glass of water; behind the propellers of a boat; in air, when smoke rings form; when a swirl of wind blows up dust; behind the wings of an airplane (Fig. 7.6). Vortices are connected to the generation of turbulence, and turbulence is difficult to precisely describe. Turbulent motion may indeed be chaotic. Yet vortices all have something in common, namely, the simultaneous presence, or superposition, of circular and translational motion. From that dynamic point of view (Sect. 4.2 and 5.1), vortices possess a helicity and are correspondingly chiral. Vortices do not merely occur at a restricted macroscopic scale. They can be both macroscopic and microscopic. The spiral motion of galaxies may be viewed as the existence of vortices of cosmic dimensions. Vortices occur in the hot plasma of stars, as well as in the atmosphere of planets. On Earth, they manifest themselves as particular meteorological phenomena, such as hurricanes, tornadoes, and taifuns. In a molecular fluid, vortices may form down to nanometer dimensions, and, in superfluid liquid helium, to a still lower scale. There is evidence that vortices affect particle motion even inside the atomic nucleus [34]. Where there is viscous drag and friction, there is energy dissipation. In these conditions, vortex motion gradually dies out. Otherwise, it may persist, undergoing only transformations that preserve certain constants of the motion. Such is the case in superfluid media.

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Fig. 7.6. Numerical simulation of the trailing vortices of a transport aircraft. Notice the presence of enantiomorphous structures (taken from ONERA
s Science Pictures, http://www.onera.fr/ photos-en).

From the historic point of view, it is particularly interesting that already in the mid-19th century the laws of vortex motion were recognized to be inextricably linked to the mathematical theory of knots. In a 1858 paper, Hermann Helmholtz applied the new topological ideas to fluid mechanics [35] [36]. These ideas were taken up by other scientists, notably by Lord Kelvin, in relation to diverse fields of science. Lord Kelvin was so impressed by these notions that he postulated atoms, the fundamental constituents of nature, to be tiny vortex filaments embedded in an elastic-like fluid medium, called ether. Accordingly, the infinite variety of chemical compounds was made possible by the endless family of topological combinations of linked and knotted vortices. Another scientist to be influenced by these ideas was James Clerk Maxwell who formulated the electromagnetic equations named after him (Sect. 3.1), and who recognized the analogy between vortices and electromagnetic fields. Vortex filaments manifest themselves as linked and knotted structures, and, therefore, their mathematical description may exploit this analogy. One of the pioneers of knot theory is Peter Tait, who, in his scientific papers [37], laid some of the important foundations. Notably, he quantitatively defined the notion of
belinkedness or linking number which, following the custom, we shall designate by Lk. If a knotted string is laid down on a table, it will exhibit a certain number of crossings. One then assigns a direction to each curve and uses a chirality rule to designate a crossing by the number + 1 if it is left-handed,
1 if it is right-handed (for further details, see Sect. 8.3). The magnitude of the linking number is then defined as one half the sum of the numbers of all the crossings. The linking numbers for some simple examples are shown in Fig. 8.5 in Chapt. 8.

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For a nonviscous fluid, the Euler equations of hydrodynamics conserve certain physical quantities, such as kinetic energy, linear and angular momentum, and vortex strength. More recently, another conserved quantity has been found for vortices, called the helicity H [35] [38] [39]. The helicity of a vortex tube is defined as: H = s u · w dV where u is the flow velocity inside the vortex, and w=ru is called the vorticity. The integral is to be taken over the entire flow volume V. We easily recognize that from the point of view of basic symmetries, the helicity of a vortex is very similar to that of some of the electromagnetic phenomena described in Sect. 3.1 – 3.5 and 4.2 – 4.5. Indeed, u, as any velocity, is parity-odd, time-odd. And w, as any angular momentum, is parity-even, time-odd. So the product u · w is a parity-odd, time-even pseudoscalar, which characterizes a chiral effect. However, the mathematically inclined reader will notice that the helicity so defined is not normalized to unity, but is just a positive or negative quantity. A remarkable result of the more recent topological investigations of fluid mechanics is the direct connection between vortex helicity and
beknottedness: for n knotted and linked vortex tubes (i), each of constant strength (total vorticity) Fi, the helicity of the whole system can be expressed in terms of the linking numbers Lkij [35]: H =
ij LkijFiFj with i,j running from 1 to n. From the definition of the linking number, we may verify that Lkij = Lkji, and that Lkii is the linking number of the self-linked ith vortex tube. Each pair of tubes (i,j) is counted only once in the sum. This equation may indeed be applied to analyze flow structures. In magnetohydrodynamics, for instance in the observation of the solar atmosphere, the existence of cycle-invariant hemispheric helicity patterns has been established. A quantitative mathematical measure of the chiral properties of these structures is the magnetic helicity defined as Hm = s A · B dV where A represents the vector potential and

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B=rA is the magnetic field. Magnetic helicity quantifies how the magnetic field is sheared or twisted, and it is one of the few global quantities which is conserved [40]. Here as above, the vector A may be shown to be parity-odd, time-odd, and B is parity-even, time-odd, as we know. In the same field of magnetohydrodynamics, a current helicity is also defined: Hc = s j · B dV where j represents the current density: m0 j = r  B, m0 being the magnetic field constant. It usually turns out that Hm and Hc have the same sign. This also reminds us that we have already encountered a quantity similar to Hc in another context, namely in the magnetochiral effect in electric conduction (Sect. 3.1, 4.2, and 4.4).

REFERENCES [1] International Tables for Crystallography, 3rd edn., Ed. T. Hahn, Kluwer Academic, Dordrecht, 1992, Vol. A. [2] J. A. Salthouse, M. J. Ware, Point Group Character Tables and Related Data, Cambridge University Press, Cambridge, 1972. [3] K. L. Bartelmehs, Left/Right-Handed Quartz (SiO2), 1997 (see http://www.infotech.ns. utexas.edu/crystal/left_right.htm). [4] G. T. Pickett, M. Gross, H. Okuyama,
Spontaneous Chirality in Simple Systems, Phys. Rev. Lett. 2000, 85, 3652. [5] A. Decker, G. A. Landrum, R. Dronskowski,
Structural and Electronic Peierls Distortions in the Elements (A): The Crystal Structure of Tellurium, Z. Anorg. Allg. Chem. 2002, 628, 295. [6] B. Mayers, Y. Xia,
Formation of Tellurium Nanotubes Through Concentration Depletion at the Surface of Seeds, Adv. Mater. 2002, 14, 279. [7] S. C. Abrahams, J. L. Bernstein,
Remeasurement of Optically Active NaClO3 and NaBrO3, Acta Crystallogr., Sect. B 1977, 33, 3601. [8] M. E. Burke-Laing, K. N. Trueblood,
Sodium Chlorate: Precise Dimensions for the ClO3
Ion, Acta Crystallogr. Sect. B 1977, 33, 2698. [9] O. Vogl, M. Qin, J. Bartus, G. D. Jaycox ,
Chiral Nucleation, Monatsh. Chem. 1995, 126, 67. [10] T. Buhse, D. Durand, D. Kondepudi, J. Laudadio, S. Spilker,
Chiral Symmetry Breaking in Crystallization: The Role of Convection, Phys. Rev. Lett. 2000, 84, 4405. [11] P. S. Halasyamani, K. R. Poeppelmeier,
Noncentrosymmetric Oxides, Chem. Mater. 1998, 10, 2753. [12] P. G. de Gennes, J. Prost, The Physics of Liquid Crystals, 2nd edn., Clarendon Press, Oxford, 1993, Chapt. 1. [13] J. D. Litster, R. Shashidhar,
Liquid Crystals, Structure of, Encyclopedia of Applied Physics, Vol. 8, Ed. G. L. Trigg, VCH Publishers Inc., New York, 1994. [14] R. Memmer,
A Computer Simulation Study of Chiral Induction in Liquid Crystals, Science and Supercomputing at CINECA-Report, 2001, p. 670; D. J. Earl, M. R. Wilson,
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On Chirality and the Universal Asymmetry of Molecular Chirality and Helical Twisting Powers: A Theoretical Study, J. Chem. Phys. 2003, 119, 10280. S. Chandrasekhar, K. N. Srinivasa Rao,
Optical Rotatory Power of Liquid Crystals, Acta Crystallogr., Sect. A 1968, 24, 445. C. Koerdt, G. Dchs, G. L. J. A. Rikken,
Magnetochiral Anisotropy in Bragg Scattering, Phys. Rev. Lett. 2003, 91, 073902. H.-G. Kuball, B. Schultheis, M. Klasen, J. Frelek, A. Schçnhofer,
Circular Dichroism of Oriented Molecules, Tetrahedron: Asymmetry 1993, 4, 517. P. G. De Gennes,
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une Structure Cholestrique par un Champ Magntique, Solid State Commun. 1968, 6, 163. P. A. Pincus,
Magnetic Properties of Liquid Crystals, J. Appl. Phys. 1970, 41, 974. H. Stegemeyer, W. Stille, P. Pollmann,
Circular Fluorescence Polarization of Achiral Molecules in Cholesteric Liquid Crystals, Isr. J. Chem. 1979, 18, 312. D. S. Sholl, A. Asthagiri, T. D. Power,
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Electrochemical Studies of Enantioselectivity at Chiral Metal Surfaces, J. Phys. Chem., B 2001, 105, 3158. M. Ortega Lorenzo, S. Haq, T. Bertrams, P. Murray, R. Raval, C. J. Baddeley,
Creating Chiral Surfaces for Enantioselective Heterogeneous Catalysis: R,R-Tartaric Acid on Cu(110), J. Phys. Chem., B 1999, 103, 10661. R. Raval, C. J. Baddeley, S. Haq, S. Louafi, P. Murray, C. Muryn, M. Ortega Lorenzo, J. Williams,
Complexities and Dynamics of the Enantioselective Active Site in Heterogeneous Catalysis, Stud. Surf. Sci. Catal. 1999, 122, 11. K.-H. Ernst, Y. Kuster, R. Fasel, M. Muller, U. Ellerbeck,
Two-dimensional Separation of [7]Helicene Enantiomers on Cu(111), Chirality 2001, 13, 675. R. Raval,
Chiral Expressions at Metal Surfaces, Curr. Opin. Solid State Mater. Sci. 2003, 7, 67. M. S. Dresselhaus, G. Dresselhaus, R. Saito,
Carbon Fibers Based on C60 and their Symmetry, Phys. Rev. B: Condens. Matter 1992, 45, 6234. W. A. de Heer,
Carbon Nanotubes: Structure and Transport in Nanotubes, Nat. Mater. 2002, 1, 153. W. Ruland, A. K. Schaper, H. Hou, A. Greiner,
Multi-wall Carbon Nanotubes with Uniform Chirality: Evidence for Scroll Structures, Carbon 2003, 41, 423. R. A. Jishi, M. S. Dresselhaus, G. Dresselhaus,
Symmetry Properties of Chiral Carbon Nanotubes, Phys. Rev. B: Condens. Matter 1993, 47, 16671. T. Kawai, Y. Miyamoto, O. Sugino, Y. Koga,
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Chirality Effects in Carbon Nanotubes, Phys. Rev. B: Condens. Matter 2002, 66, 155404. G. L. J. A. Rikken, E. Raupach, V. Krstic´, S. Roth,
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Observations of Magnetic Helicity, 2003 (see http://www.solaire.obspm.fr/demoulin/03/review_H_cospar.pdf).

8. Chiral Models
… Ive learned that, in the description of nature, one has to tolerate approximations, and that even work with approximations can be interesting and can sometimes be beautiful P. A. M. Dirac (1902 – 1984) (in History of Twentieth Century Physics, Ed. C. Weiner, Academic Press, New York, 1977)

8.1. The Use of Models for the Description of Nature In the Concise Oxford Dictionary, the word approximate is described as:
very near; closely resembling; fairly correct. Correspondingly, approximations in science may be understood in different ways. We tend to describe nature by abstract notions that do not, as such, belong to the real world. Although we can define what a straight line is, we cannot draw it, even with the finest pencil and best ruler. The picture will never be strictly onedimensional, as it should be; the point of the pencil is actually not a point, but a surface; our drawn line will never be infinitely thin. Such considerations evidently apply to any geometrical object. Our descriptions of nature rest, in this sense, on idealizations. Although a perfectly grown quartz crystal has strict D3-symmetry, such a crystal does not exist. There will always be some imperfections in the crystal lattice, on the crystal surfaces, that have been caused by unpredictable influences of the surroundings but which, for our immediate purposes, may be irrelevant. Thus, we may consider the real crystal to be an approximation of the ideal crystal, or vice versa. A certain discrepancy between abstract, idealized model and reality does not necessarily diminish the usefulness of the model, on the contrary. Model building is essential for scientific thinking and understanding. The formulation of physical laws consists of mathematical relations of high precision. Yet the application of a given mathematical model may be restricted by certain conditions, for instance, limited to certain scales, beyond which its predictive power diminishes. The laws of classical mechanics are very precise when applied to the macroscopic world. If, however, we try to use them to describe the properties of atomic and subatomic particles, they fail due to the particle–wave duality. When the de Broglie wavelength of an object that we consider becomes of the same order of magnitude as the objects classical dimensions, another general model must be applied, namely quantum mechanics. Furthermore, precise equations may prove very difficult to solve exactly. This then makes it often unavoidable, for practical purposes, to seek approximate solutions. In the case of classical mechanics, the reader is reminded of the On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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difficulty of predicting planetary motion over very long periods of time (Sect. 6.1), even though the basic equations, from which the computations derive, are essentially exact. In quantum chemistry, for instance, the numerical solution of the Schrçdinger equation for many-electron systems is difficult and must be tackled by methods which, in general, are only asymptotically correct, that is, approximate. Considering thermodynamics: the ideal-gas law, by virtue of its simplicity, and in spite of the fact that an ideal gas does not exist, is essential to establish the foundations of the field. One uses the ideal-gas law not only as a starting point to describe real gases, but mainly also as a means to assess orders of magnitude of quantities, such as internal energy and heat capacity, that are basic for the understanding of the thermal properties of macroscopic matter. In the same spirit, the chiral models that we shall consider in the next sections are all rather abstract, possibly not immediately related to reality. The aim of this study is to refine our overall perceptions of chirality, in the hope that it might be of relevance to a better general understanding of the asymmetry in the real world. We will first consider chirality as a purely geometric property, then gradually move again closer to the physical description of chiral objects. We shall also ask the question, if a general measure of chirality is conceivable and may be meaningful.

8.2. Dissecting a Cube and a Sphere Consider a perfect cube. It shows the highest crystallographic point group symmetry, designated as Oh. In words: it is cubic holohedral. Now subdivide the cube into eight subcubes, each one representing an octant. Dissect the cube into two equal parts (halves) consisting of four subcubes each, which, by one operation, may be fitted together again to the original cube. This can essentially be done in four ways: a) the two halves of the original cube are achiral, of symmetry C3v . b) The two halves of the cube are achiral, of symmetry D4h. c) The two halves of the cube are chiral, of symmetry C2. As one then notices, the two parts are not only isometric but they are also homochiral. (We call here two figures
isometric, if they can be transformed into each other either by the identity operation or by reflection.) The halves may both either be of absolute configuration (P), or both of absolute configuration (M) (see Sect. 1.4). As may be easily ascertained with models, if the two parts were heterochiral, or enantiomorphous, they could not be fitted together to again form a cube. Expressed schematically (see also Fig. 8.1):

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Fig. 8.1. Dissection of a cube into halves consisting of four subcubes. The halves are of symmetry C3v, D4h, and C2(M or P). The reader may easily recognize that the two C2 forms here shown are enantiomorphous. However, to be fitted together to a cube, the two halves must be homochiral (adapted from [6]).

The expectation that both chiral parts should be enantiomorphous, rests on the – obviously wrong – assumption that if the whole is achiral, then so must be the sum of the two separate parts. A cube may in general be subdivided into n3 subcubes, where n is an integer. If every subcube is dissected in two chiral halves, as just described, then a cube may in principle give rise to 2n3 homochiral elements of symmetry C2. A cube may thus be dissected into an unlimited integral number of equal, homochiral parts. Similar, but not identical, considerations apply to a sphere. The sphere is the body of highest symmetry in three-dimensional space. It possesses an infinity of 1-fold rotation axes, and an infinity of planes of reflection. It must be noted, however, that, unlike a cube, a sphere cannot be subdivided compactly into an integral number of smaller, equal (sub)spheres. A sphere may, for instance, be dissected into two achiral hemispheres of symmetry C1v. The dissection follows a circle of symmetry D1h. A sphere may also be dissected into two isometric homochiral parts. This dissection, named La Coupe du Roi [1], follows the curves shown in Fig. 8.2. The points A, B, …., E are all located at the same distance from the center of a cartesian reference frame. They, therefore, all lie on the surface of a sphere. To dissect the sphere, we follow the surface curves connecting these six points, as shown. These
paths of dissection represent structures of symmetry D2, which, accordingly, occur in enantiomorphous

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Fig. 8.2. Top: Two chiral, enantiomorphous paths of dissection of a sphere, viewed in perspective. Bottom: Projection of these curves onto the plane containing the points A, C, D, F, viewed along the axis B – E. The reader may thus more easily verify the symmetry D2.

(enantiomeric) forms. The path at left leads to two homochiral halves of symmetry C2 and absolute configuration (M), the path at right to two halves of absolute configuration (P). Here again, we notice that an achiral body of high symmetry may be decomposed into homochiral parts (see also Fig. 8.3). Or in reverse, there exist homochiral forms that may be fitted together to a body of highest symmetry. This leads us to the following conjecture (to be proven by the reader): a sphere cannot be dissected into two isometric chiral parts that are enantiomorphous. It is not straightforward to find out if these purely geometric considerations have any deeper physical significance. The question has been discussed, if two homochiral molecules could be combined to give an achiral product [1], an achiral unit cell, or crystal lattice [2]. The chemical examples studied until now do not seem to allow any definite and generalized conclusions. To decompose a cube or a sphere into two chiral parts of given handedness, of course, presupposes the existence of a chiral influence at the outset. Otherwise there is nothing to give any preference to the decomposition into (M) + (M), rather than into (P) + (P). But once the initial step has taken place, we may imagine scenarios where the homochiral decomposition is perpetuated

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Fig. 8.3. Dissection (La Coupe du Roi) of an apple into two equal, homochiral halves

from one symmetric entity to the next. This may, in a very schematic and abstract way, suggest mechanisms of chiral amplification. The question remains, however, if such hypothetical mechanisms have any physical meaning.

8.3. From Dihedral Angles to the Classification of Knots Our point of view is now stereochemical: one atom defines a point, two atoms define a line. Three atoms may always be considered to lie in a plane, which thereby becomes a plane of symmetry. The smallest structurally chiral molecule must contain at least four atoms, as it ought to be nonplanar to lack any Sn-symmetry. One of the smallest potentially chiral molecules is hydrogen peroxide, HOOH. The HOO angle in the gas phase is 94.88 and in the crystalline state 101.98 [3]. Furthermore, the conformation of lowest energy corresponds to a twisted, and, therefore, nonplanar chiral structure. The degree of twisting is measured by the so-called dihedral angle, illustrated in Fig. 8.4. It is defined as the angle between the HOO and the OOH planes. In the gas phase, the dihedral angle of hydrogen peroxide is  111.58 and in the solid state  90.28. As previously mentioned (Sect. 2.9), the molecule oscillates so fast between the two energy minima that it is impossible to separate the enantiomers in the gas or liquid phase, and the crystal structure is racemic [4].

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Fig. 8.4. a) Definition of the sign of the dihedral angle. At left, we have a right-handed or (P)configuration, at right a left-handed or (M)-configuration. b) Designation of the dihedral angles in a peptide chain. The angle around the C(O)N bond of the amide groups is not shown. Because of the partial double-bond character, it is usually close to 1808 (trans), in some cases close to 08 (cis). The amide groups are generally quasi-planar.

By convention [5], the dihedral angle goes from 08 in the planar cisconformation, to 1808 in the planar trans-conformation, and the sign may be deduced from Fig. 8.4. Dihedral angles are very important quantities to characterize the secondary and tertiary structure of polypeptides and proteins. The potential energy threshold for internal rotation around the C(a)C(O) and N C(a) bonds (the C(a)-atoms carry the substituents) being relatively low, the polypeptide chain is quite flexible. It may then get stabilized, mainly by hydrogen-bond formation, in several typical secondary structures (a-helix, bpleated sheet, etc.; see Sect. 1.2). Knotted and linked structures may then also occur, and their complexity can make it difficult to systematically assess their chirality. Although the general criteria to recognize chirality formulated previously (Sect. 1.3 and 4.1) are always valid, the relatively flexible, intertwined structures call for an approach that is better adapted. Here, a topological point of view is welcome and may even be necessary. This aspect of molecular structure is rapidly becoming a field of intense investigation. A knot is a closed, non-self-intersecting curve in three-dimensional space [6]. A knot is topologically achiral if it can be continuously deformed to its mirror image (without interrupting or intersecting the curve). Otherwise it is topologically chiral [7] [8]. A link is an assembly of knots with mutual entanglement. Correspondingly, a knot is the special case of a link with only one component. A knot is oriented if we assign a direction to the curve. In general, we can do this in two ways. Quite obviously, if the number of components (individual knots) of a link is m, then there exist for this link 2m oriented structures or diagrams. A diagram representing a link or knot is a two-dimensional minimal projection showing with gaps undercrossings/overcrossings. For

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Fig. 8.5. a) Definition of the sign of the crossing number. Notice that, for a given absolute configuration, the sign is opposite to that of the corresponding dihedral angle (Fig. 8.4). This is the consequence of independent conventions and has no actual physical significance. b) The oriented Trefoil knot (311 ) with absolute configuration l. c) One of the two distinct oriented diagrams of the Hopf link (221 ) of absolute configuration d. d) An oriented diagram of the structurally chiral Whitehead link (521 ) of absolute configuration l. e) The Torus link (421 ). For each unoriented enantiomer there are two distinct oriented diagrams. This entails not only enantiomorphism, but also diastereoisomerism [9].

oriented knots and links, we characterize each crossing by a crossing number which can take on the values + 1 or  1, depending on criteria shown in Fig. 8.5. There is a certain analogy, though no coincidence, with the definition of the sign of a dihedral angle. One may also conceive of the crossing number as some kind of local helicity. In particular, we notice the following symmetry properties: spatial reflection or inversion of a crossing changes the sign of its crossing number. If a crossing belongs to one and the same uninterrupted space curve, that is, to one and the same component, the crossing number is invariant to a change of orientation. For a crossing between two different components, a change of orientation of one component with respect to the orientation of the other changes the sign of the crossing number. In Sect. 7.5, we encountered the notion of linking number in connection with the measure of vortex helicity. In general, the determination of the chirality of a link or knot of arbitrary complexity is far from trivial and, in some cases, may

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become ambiguous [9]. Some aspects of this question will here be briefly discussed. To this end, we shall here restate our definitions [10]. We consider the diagram of an oriented link between a component i and a component j. The linking number Lkij is one half the sum of the inter-component crossing numbers. Evidently Lkij = Lkji. The self-writhe si is the sum of the intracomponent crossing numbers of a given component. One may also write si = 2Lkii, and sj = 2Lkjj. The writhe wij is the sum of all crossing numbers in the linked and knotted components i and j. The following relation then holds: wij = si + sj + 2Lkij. Now consider the case of a link/knot consisting of many components. Going back to Sect. 7.5, we identify: X X X Lkij = 12 si + Lkij (i < j). ij

i

ij

The indices i and j are here to be viewed as variables, taking on the numbering of all components, but counting each pair only once. Knots are, in general, classified by the number of crossings, the number of components and a serial number. In Fig. 8.5, we show a few simple examples. In a one-component knot, such as the Trefoil knot 311 , a change of orientation has no effect on the sign of the writhe. The chirality is determined by the structural chirality of the unoriented structure (Table 8.1). In the so-called Hopf link 221 , the unoriented structure is achiral. It may easily be shown that the four oriented Table 8.1. Specifications of Some Simple Knots and Links. The topological chirality is due to
form, if the unoriented structure is chiral. If the unoriented structure is achiral, the chirality may only be due to
orientation. For further explanations, see the text.

Number of components Unoriented form Number of distinct oriented forms Linking Number d l Self-writhe d l Writhe d l Nature of topological chirality

Trefoil

Hopf

Whitehead

Torus

1 chiral 2H1

2 achiral 1H2

2 chiral 2H1

2 chiral 2H2

– –

+1 1

0 0

+ 2 ( 2)  2 (+ 2)

+3 3

0 0

+1 1

0 0

+3 3 form

+2 2 orientation

+1 1 form

+ 4 ( 4)  4 (+ 4) form and orientation

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Fig. 8.6. A synthetic organic molecule forming a Trefoil knot. The emergence of the knot is the striking result of a high degree of molecular self-organization (from [11]).

diagrams are pairwise equivalent with respect to rotations in three-dimensional space. The inequivalent diagrams give rise to topological enantiomorphism due to orientation. In the Whitehead link 521 , the linking number adds to zero. The structural chirality is represented by the self-writhe of the twisted central component. Finally, in what is called the Torus link 421 , there are two structural enantiomorphs of the unoriented link, and each of these gives rise to two independent oriented diagrams of opposite chirality. Consequently, we here encounter not only enantiomorphism, but also diastereoisomerism [9]. Within the last two decades, attempts to synthesize molecular knots have been successful. The instructive recent example of a trefoil is shown in Fig. 8.6 [11]. From the molecular point of view, the chirality of both unoriented and of oriented links proves significant. Returning to the example of polypeptide chains: these are, irrespective of the amino acid sequence, to be viewed as oriented. Indeed, it is topologically not equivalent if we proceed along the chain in the direction C(a) ! C(O) ! N ! C(a), or in the opposite direction C(a) ! N ! C(O) ! C(a). Two cyclo-polyglycene rings forming a Hopf link should correspondingly be chiral. This chain orientation is, for instance, also well-demonstrated by the stereochemical difference between the parallel and the antiparallel pleated sheet structure. The study of topological chirality, both in proteins [12] and in DNA structures [13], has been the object of intense research. The interested reader is encouraged to consult the specialized literature.

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8.4. From Ligand Partitions to Chirality Functions We have just discussed the chirality of knots and links, of intertwined continuous structures. Our point of view is here a completely different one. We presently consider a molecule as consisting of an achiral skeleton to which different achiral ligands are attached. The chirality of the molecule then arises through the asymmetric distribution of the ligands on the skeleton. An aim of this approach is to systematically classifiy the origin and magnitude of the chirality depending on a) the number of ligand sites on the skeleton, b) the point group symmetry of the skeleton (unsubstituted, or with all ligands equal), c) the kinds, numbers, and positions of ligands. This should offer the possibility to elaborate a chirality function to describe the relative influence of changes in the identity and arrangement of the ligands on physical quantities that measure chirality (such as ORD or CD; see Sect. 1.2 and 3.3). A chirality function depends on individual ligand parameters that, for a given chiral property, can, in principle, be either calculated ab initio or empirically calibrated. At the same time, the function reflects the geometric properties that determine the departure from achiral symmetry. This Chirality Algebra has been pioneered mainly by Ruch and co-workers in the late 1960s and 70s [14 – 17]. Although, in simple cases, the theory of chirality functions merely confirms stereochemical facts that can be deduced more pragmatically and easily otherwise, when the number of ligand sites increases, the more systematic approach unquestionably becomes necessary. Furthermore, the procedure leads to general classifications that other computational methods alone fail to point out so directly. It provides a group-theoretical view that may be useful to systematically assess and interpret the results of quantum-chemical computations. In our aim to acquaint ourselves with the principles underlying the Chirality Algebra, we are, however, presently obliged to limit ourselves to simple examples. We cannot possibly discuss here the general group-theoretical connections that are necessary for a more profound understanding of the method. Our point of departure is the group of permutations of the n ligand sites on the skeleton, Pn. As is known from elementary mathematics, the number of permutations of n distinguishable objects is n! = 1 · 2 · 3 · ……. · n. There are thus n! elements in the group. As examples, we now only consider skeletons with four ligand sites. It must be stressed that a skeleton must be three-dimensional, otherwise point-ligand substitution cannot possibly lead to chirality. Among our examples, we find the skeletons of point group symmetry Td (methane), D2d (allene); as also the symmetries C4v and C2v. We now first consider the permutations of four equal but distinguishable objects (ligands): 1,2,3,4; 1,2,4,3; ………..; 4,3,2,1. There are obviously 4! (= 24)

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Fig. 8.7. Young diagrams for the permutation group P4, representing the five irreducible representations. As P4 is isomorphic with the tetrahedral point group Td, the corresponding labels are also indicated.

such permutations. These 24 permutations may now be classified into five different classes: 1) All ligands remain where they are; no ligand may be permuted: (1)(2)(3)(4); it is designated as [1111], with one element in the class. 2) Two ligands are exchanged, two remain where they are: (12)(3)(4); it is designated as [211], with six elements in the class. 3) Two permutations take place pairwise: (12)(34); it is designated as [22], with three elements in the class. 4) Three ligands are exchanged, one remains unchanged: (123)(4); it is designated as [31], with eight elements in the class. 5) All four ligands are exchanged: (1234); it is designated as [4], with six elements in the class. From the above, the reader will of course verify that 1 + 6 + 3 + 8 + 6 = 24. (As an exercise, the reader may explicitly write out the 24 permutations and assign them to the different classes.) These five classes may be correlated with the different partitions of the number 4 and with so-called Young diagrams (Fig. 8.7). For what now follows, the reader is assumed to have an elementary knowledge of group theory. As the group P4 has 5 classes, it has 5 irreducible representations. And as the sum of the squares of the dimensions of the irreducible representations must be equal to the order of (number of elements in) the group, we obtain as the only possibility: 24 = 12 + 12 + 22 + 32 + 32. The irreducible representations of the group P4 consequently have the dimensions:

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1, 1, 2, 3, 3 It may furthermore be shown that the group P4 is isomorphic with the crystallographic point group Td of the tetrahedron. The corresponding irreducible representations of Td are labeled A1, A2, E, T1, T2 ; and the ones of P4 are designated [4], [1111], [22], [211], [31]. Scalar quantities measuring physical properties of achiral molecules, or properties of chiral molecules unrelated to their chirality, transform like the totally symmetric one-dimensional irreducible representation of the molecular point group. In the present case, this is A1 or [4], implying that the quantity under consideration is invariant with respect to any permutations of the ligands. On the other hand, the measure of any chiral property transforms like a pseudoscalar (as for instance the optical rotation or the rotational strength; see Sect. 3.3). This means that the value changes sign upon any rotation–reflection, entailing a space inversion. Such a quantity transforms in our case like the antisymmetric one-dimensional representation A2 or [1111]. Indeed, there must occur a change of sign upon the interchange of any two ligands. We conclude that for a tetrahedral skeleton, all four ligands must be different to sustain chirality. Now we assume that, from a skeleton of symmetry Td, we proceed to one of lower symmetry D2d, such as that of allene. Which ligand partitions will sustain chirality in this case? The descent in symmetry implies that the irreducible representations of the group Td transform into irreducible representations of D2d, or into sums thereof: Td

D2d

A1 [4]

! A1

A2 [1111]

! B1

E [22]

! A1 þ B1

T 1 [211]

! A2 þ E

T 2 [31]

! B2 þ E

By inspection of the character table of the group D2d, one easily finds that here the antisymmetric representation is not A2, but B1. From this and the above scheme, we conclude that chirality is sustained both by the ligand partition [1111] (all ligands different; as for the tetrahedral skeleton), and, additionally, by [22] (ligands pairwise the same), as illustrated in Fig. 8.8. This example should give a rough outline of how the Chirality Algebra is applied to determine chiral ligand partitions. Our next objective is to deduce the

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Fig. 8.8. Ligand partitions leading to chirality in the tetrahedral (methane) skeleton (a) and in the disphenoid (allene) skeleton (b). The corresponding, chirally active Young diagrams are also shown. The length of the highest row in these diagrams is called the order o of the diagram, a quantity we shall refer to in Sect. 8.5.

chirality functions. Here again, we state the general principles without going into mathematical details. The basic idea is that the value a of a pseudoscalar measurement on a molecule with n ligand sites may be expressed as a function of parameters li of the ligands at these sites: a = c(l1, l2,…….., ln). Permutations of the bound ligands corresponding to proper rotations of the molecule should leave the function invariant. Permutations corresponding to improper rotations, or rotation–reflections, should change the sign leaving the absolute value invariant. It seems reasonable to seek chirality functions in the form of polynomials. The exact procedure to deduce these polynomials cannot here be explained in detail. Again, we limit ourselves to stating a few examples which are not too difficult to interpret: for formal reasons we set li  si , and we find for the tetrahedral skeleton: c(l1, l2, l3, l4) = (s1  s2)(s1  s3)(s1  s4)(s2  s3)(s2  s4)(s3  s4).

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On Chirality and the Universal Asymmetry

We immediately notice that for any si = sj„i the chirality function becomes zero, demonstrating the disappearance of chirality when two ligands (substituents) are identical (see Fig. 8.8). For the disphenoid skeleton we must distinguish between two chiral partitions. For the partition [22], we write li  si and get: c’(l1, l2, l3, l4) = (s1  s3)(s2  s4). The function vanishes for s1 = s3 and for s2 = s4, but not necessarily for s1 = s2, or s1 = s4, etc. For the partition [1111], we write li  ti and find, in analogy to the tetrahedron: c’’(l1, l2, l3, l4) = (t1  t2)(t1  t3)(t1  t4)(t2  t3)(t2  t4)(t3  t4). A difference with respect to the tetrahedron is, that the functions c’ and c’’ alone are not qualitatively complete. In other words, to numerically classify and interpret the chirality of allene derivatives, a linear combination of both c’ and c’’ must be applied. Numerical examples on the optical rotation of such derivatives are to be found in [16]. A further basic question that the Chirality Algebra deals with is the following: Suppose you have a box full of shoes of various sizes, colors, and shapes. You are asked to separate all left shoes from all right shoes. The task is easily performed. Now suppose you have a crate full of potatoes, which of course are also chiral. The task of selecting left potatoes from right potatoes is obviously unmanageable, even meaningless. A similar situation is also encountered at the molecular level. The absolute configuration of different kinds of molecules may be correlated in a meaningful way only, if the molecules all are
shoe-like, and their skeletons thus belong to the category a. Within classes of such molecules, the determination of homochirality obviously makes sense. On the other hand,
potato-like molecules with skeletons belonging to the category b may not be similarly classified. These criteria require a grouptheoretical derivation, of which we again merely state the result: a molecular skeleton belongs to category a if, and only if, either the skeleton has only two sites for ligands, or the number n of sites is larger, but the symmetry of the skeleton contains mirror planes and each mirror plane contains n  2 ligand sites. Otherwise the skeleton belongs to category b. The reader is now invited to determine to which category the methane skeleton and the allene skeleton belong.

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8.5. Sector Rules to Interpret CD and ORD Spectra Circular dichroism (CD) and optical rotatory dispersion (ORD) are, in principle, the methods of choice to detect chirality. The weakness of these methods for structural analysis is that they do not provide atomic resolution. The difficulty in their application lies in the fact that the correlation of a given spectrum with a particular absolute configuration is unequivocal, but indirect. This contrasts unfavorably with X-ray analysis, from which atomic distances can be directly deduced and, in many instances, via anomalous scattering, also the absolute configuration. Notwithstanding, the chiroptic methods have played a fundamental role in the development of stereochemistry, as already indicated in Sect. 1.2 – 1.4. In this and in the following section, we briefly focus on some simple quantum-chemical models which, in the course of recent decades, have helped to elucidate the structure of different molecules of biological interest. The correlation of the CD/ORD spectrum of a molecule with its exact atomic geometry generally requires extensive quantum-chemical calculations. However, simple semiempirical models, though perhaps less accurate, may successfully reveal dominant structural features and render the deduction of absolute configurations possible. One of these models leads to the so-called Sector Rules. These rules are based on the following assumptions: 1) The electronic transitions of interest – usually including the longest-wavelength transition – are localized mainly in a particular part of the molecule. This part is called the chromophore (Greek c1wma, color; fe1ein, to carry). 2) The chromophore itself is considered to be achiral, carrying at least one symmetry element Sn. Depending on this symmetry, the space around the chromophore may be divided into different sectors. 3) The chirality of the molecule arises through the introduction of one or more substituent(s), making the molecule as a whole optically active. Depending on the sector in which a given substituent lies, its contribution to the longwavelength optical activity (longest-wavelength Cotton effect) will be positive or negative. 4) The contributions of more than one asymmetric substituent are assumed to be additive. 5) The absolute value and sign of the contribution of a given kind of substituent has to be empirically calibrated on a test molecule of known absolute configuration. The discovery of the Sector Rules resulted from the systematic study of asymmetrically substituted carbonyl chromophores [18] [19]. Keto-steroids play an important role in the chemistry of hormones [20]. The physiological activity of these compounds depends crucially on their absolute configuration,

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On Chirality and the Universal Asymmetry

Fig. 8.9. The Octant Rule applied to the carbonyl chromophore. Top: the coordinate system defining the octants. Bottom: projection of the positions of the carbon atoms of 3-methylcyclohexanone onto the xy plane; illustration of their relative positions in the four back octants. The measured value of the longest-wavelength n – p* Cotton effect induced by the substituent S is [A] = + 25 for a methyl group (adapted from [18]).

in particular, on the exact position of substituents with respect to the carbonyl moiety. The characteristic longest-wavelength transition of the carbonyl chromophore corresponds to the excitation of an electron from the nonbonding oxygen orbital n to an antibonding p* orbital of the double bond. The symmetry of the unsubstituted carbonyl choromophore is C2v. The nodal properties of the orbitals involved then lead to an octant (quadrant) rule (Fig. 8.9). This may be formulated as follows: the (relative) sign of the contribution which a given atom at point P(x,y,z) makes to the CD/ORD, or Cotton effect, of the n – p* transition, varies as the simple product x · y · z of its coordinates. The reader is thus encouraged to verify the signs in Fig. 8.10 from the data in Fig. 8.9. The reader will by now undoubtedly have recognized an analogy between the Sector Rules and the Chirality Algebra. The Chirality Algebra proves to be more general, but it is in particular instances less specific. However, the Sector Rules are implicitly contained in the more general theory. Consequently, it may be shown that not every skeleton gives rise to Sector Rules when ligands, or substituents, are asymmetrically attached to it. The Sector Rules are applicable only, when the optical activity may be induced by a single ligand, and when the influence of more than one ligand is, to a first approximation, additive. There are cases where the optical activity depends on the presence of at least two or more different ligands (substituents), and there the Sector Rules are not applicable (Fig. 8.11).

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Fig. 8.10. The back-octant projections of the carbon atoms in derivatives of trans-decalone. Compare with Fig. 8.9 to rationalize the relative signs of [A]. Notice that atoms localized in nodal planes of the coordinate system make contributions of zero to the Cotton effect. Equivalent atoms in octants of opposite sign make contributions which cancel (adapted from [18]).

Fig. 8.11. For achiral skeletons with four lattice sites, only those with chirality order o = n  1 = 3 lead to Sector Rules. This is indeed the case in the carbonyl chromophore. Pseudoscalar observations may in this case be considered as additive contributions from single ligands [16]. Compare with Fig. 8.8.

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On Chirality and the Universal Asymmetry

Table 8.2. Sector Rules for Some Simple Organic Chromophores Symmetry

Chromophore

Sector rule

CS C2h

lactone planar trans-diene trans-diazo compound carbonyl, nitroxide, planar s-cis-diene mono-olefin (C=C bond)

planara) planara)

C2v D2h

quadrantb) octant

a

)
Planar implies that space is divided into two sectors, above and below the plane of the chromophore. b) A quadrant rule gives the minimal number of different sectors: an octant rule is also allowed by symmetry.

This may be formulated as follows [16]: be n the number of ligand sites on the skeleton, and o the maximum number of equal ligands that may be attached to the skeleton without making the chirality vanish. Then, if: o = n  1, a Sector Rule applies.When o< n  1, a Sector Rule does not apply. Going back to Fig. 8.8, the reader will recognize that o is the order of the corresponding Young diagram. The Sector Rules may, on the basis of group theory, consequently be extended to chromophores of different symmetry [21]. The minimal number of sectors for the frequently occuring organic chromophores listed in Table 8.2 may therefrom be predicted. The stereochemically interested reader may want to consider in further detail the connection between absolute configuration (Sect. 1.4), permutation symmetry (Sect. 8.4), local and overall molecular symmetry, and chirality. This may be done by consulting the specialized literature, e.g., [22].

8.6. From Helices to Mçbius Strips For the interpretation of chiroptic spectra, the subdivision of a molecule into an achiral part – be it a
skeleton or a
chromophore – and into asymmetrically placed ligands or substituents, is not always meaningful. Suppose we consider a polymer built of many similar parts or monomers. The problem now consists in understanding the properties of the polymer as a whole on the basis of the attributes of the individual monomers. For instance, we wish to interpret the long-wavelength optical activity of a polypeptide on the basis of the electronic structure of the individual amide groups of which the peptide backbone is composed. Without going unduly into details, we state here the principle and application of the exciton, or quantum-mechanical coupled-oscillator, model: each amide chromophore exhibits in the wavelength region around 200 nm a socalled p – p* transition. The excitation of this transition in each monomer by UV

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Fig. 8.12. The 195-nm p – p* band of the polypeptide a-helix (see Fig. 1.4) may be interpreted as arising from the electrostatic coupling between the individual electric dipole p – p* transition moments, represented here by arrows, within each individual amide monomer (adapted from A. Wada,
Dielectric Properties of Polypeptides in Solution, in Poly-a-Amino Acids, Ed. G. D. Fasman, Marcel Dekker, Inc., New York, 1967).

radiation of the proper wavelength may be interpreted as the induction of a quasi-localized electric dipole transition moment (see also Sect. 1.4, 2.6, and 3.3). The individual transition moments in the monomers immediately couple electrostatically with each other (Fig. 8.12) to give resulting transition moments, which are characteristic for the structure of the polymer as a whole. From rotational strengths so deduced, the particular CD bands of the secondary structure of polypeptides and proteins (a-helix, b-pleated sheet, etc.; see Fig. 1.4) have been successfully interpreted [23] [24]. The exciton model rests on the assumption that the valence electrons in the polymer remain relatively localized within the individual monomers. From the point of view of intramolecular electron transport, this then corresponds to the properties of an insulator. One may, of course, also consider the other extreme, where electrons are free to move in the polymer, as in a conductor. An esthetically pleasing, simple quantum-mechanical model is that of free electrons on a helix. It is then found that a system of such randomly oriented right-handed helices has a positive rotational strength for the longest-wavelength absorption band [25], in qualitative accord with the predictions of the exciton model for the 195-nm p – p* band of the right-handed a-helix. This sign

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On Chirality and the Universal Asymmetry

Fig. 8.13. a) Building of a Mçbius strip from a paper band that has been twisted by 1808. The Mçbius strip is a one-sided surface and has maximum symmetry C2 (drawing from R. Courant, H. Robbins, Was ist Mathematik?, 3rd edn., Springer-Verlag, Berlin, 1973). b) A ladder-like molecule (I; with 3 rungs) is twisted by 1808 and bent to obtain a circular structure of Mçbius topology (II) (adapted from [28]).

prediction also agrees with the results of molecular orbital calculations on helicenes in the right-handed (P)-configuration [26]. Furthermore, the model has been successfully applied to interpret the sign of the long-wavelength rotatory dispersion of 1-cm copper helices in the microwave region [27]. From these stereochemical and spectroscopic considerations, we presently revert to a more topological point of view. Our interest focuses on the passage from an open helical structure to that of a Mçbius strip. We begin with a molecule of ladder-like structure (Fig. 8.13). The sides of the ladder, the strands AB and B’A’, are held together by a certain number of rungs, or bonding bridges. If we twist the ladder around the long-axis (parallel to the strands), then each one of the single strands will describe a helix, and the ladder as a whole will become a double helix, formally reminiscent of a short DNA chain. The ladder is now considered to be sufficiently flexible that it can also be bent, and that one end may be connected to the other. If previously the ladder has been twisted by 08or an even multiple of 1808 , the beginning of each strand will be bonded to its own endpoint. We then obtain a circular structure with two separate strands, always held together by the rungs. If, on the other hand, we twist the ladder by an odd multiple of 1808, the endpoint of one strand will be connected to the starting point of the other, A to A’ and B to B’, leading to a single strand of double length. The presence of the rungs will now prevent the newly formed single strand from forming a planar circle. Rather, the ladder will be forced to adopt the topology of a Mçbius strip [28]. The Mçbius strip is a one-sided surface obtained by cutting a band widthwise, giving it a half twist of 1808 and reattaching the two ends [6]. Before

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Fig. 8.14. Left: a HQckel and a Mçbius type system of pp centers. In the Mçbius system, the axis of the pp functions is gradually twisted around the ring to 1808. Right: two newly synthesized isomers. An achiral one of HDckel structure (above), and a chiral one with Mçbius topology (bottom) (from T. Kawase, M. Oda,
Mçbius Aromatic Hydrocarbons: Challenges for Theory and Synthesis, Angew. Chem., Int. Ed. 2004, 43, 4396).

specifically attracting the attention of chemists, the Mçbius strip had already fascinated both professional and hobby mathematicians. We are of course particularly attracted by it, because it is a basically chiral structure. A regular Mçbius strip, as does also a finite helix, shows a C2-symmetry. Not only ladder-type molecular structures may adopt a Mçbius topology, but also twisted p electron systems. Such p electrons generally occur in planar molecules with conjugated C=C bonds, as, for instance, in polyenes and in aromatic rings. It is well-known from HDckel molecular orbital theory that such molecules exhibit particular stability if they contain 4n + 2 (n, an integer) pp carbon centers, or the corresponding number of p valence electrons. Molecules with 4n centers, or 4n p electrons, are comparatively destabilized. This regularity is titled the
HDckel rule of aromaticity. As was then shown by Heilbronner, if such conjugated systems take on a Mçbius configuration, the rule is reversed, and 4n systems are stabilized [29]. The synthesis of such Mçbius molecules is far from trivial, however, as the required twist may lead to strain which raises the energy of the s valence elctrons. Fig. 8.14 shows a chiral Mçbius-type molecule with 36 pp centers, recently synthesized, in spite of these difficulties [30].

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8.7. Coupled and Asymmetric Classical Harmonic Oscillators Before the quantum-mechanical description of molecules became firmly established, an interpretation of the phenomenon of natural optical activity was attempted by classical methods. Possibly the simplest model conceivable is that of coupled oscillators: the electrons of a chiral molecule are collectively represented by two coupled classical harmonic oscillators, moving perpendicularly to each other and separated in the third spatial dimension by a finite distance d (Fig. 8.15). A particle of mass m1 and charge e1 has its equilibrium position at the origin of the coordinate system x, y, z. It is elasticially bound to this position by a restoring force and may only move in  x direction. Similarly, a second particle of mass m2 and charge e2 is bound elastically by a restoring force to a point d on the z-axis. It may only move along the  y direction [31]. The potential energy of the coupled system can then be written: V = (k1/2) x21 + (k2 /2) y22 + k12x1y2 , and the force acting on oscillators 1 and 2, respectively: F1 ¼ @V=@x1 ¼ k1 x1  k12 y2 ,

Fig. 8.15. Asymmetrically coupled classical oscillators. a) and c) Two opposite phases of the motion of one and the same oscillator. Notice that the chirality is maintained. b) and d) The same for the enantiomer.

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F2 ¼ @V=@y2 ¼ k2 y2  k12 x1 . The quantities k1, k2 are the corresponding harmonic oscillator force constants and k12 the coupling constant inducing the chirality. Taking, additionally, damping effects into account, proportional to the velocity of each individual oscillator and characterized by the constants of friction g1 and g2, as well as the interaction with the radiation field of frequency w, the equations of motion for the two oscillators then read: m1 ð@ 2 x1 =@t2 Þ ¼ F1  g1 ð@x1 =@tÞ þ Q1 expðiwtÞ, m2 ð@ 2 y2 =@t2 Þ ¼ F2  g2 ð@y2 =@tÞ þ Q2 expðiwtÞ. The quantities Q1 and Q2 represent not only the electric dipole interaction of the respective oscillator with the radiation field, but should also include spatial derivatives of the field with respect to the dimension z. This is necessary in order to take into account magnetic dipole and electric quadrupole contributions. Such interactions are essential to cause optical activity (see also Sect. 3.2 and 3.3). Eschewing further mathematical details [32], it can be concluded that the system exhibits Cotton effects which are proportional to the coupling constant k12. The reader may ascertain that a change in the sign of k12 entails a change of chirality. The asymmetric coupling adds to the motion of oscillator 1 in  x direction a small component in  y direction. Conversely, oscillator 2 obtains an increment of motion in  x direction. It is noteworthy that, over the years, the interest for this classical model has persisted, be it mainly for pedagogical reasons. As has recently been shown [33], the classical coupled oscillator model may similarly be applied to describe the Faraday effect and the magnetochiral effect (see Sect. 3.4 and 3.5), correctly predicting relative orders of magnitude, both for dispersion and absorption. In another recent investigation, the study of a single classical oscillator under the influence of a chiral potential is undertaken. The model consists in a three-dimensional harmonic oscillator constrained to move on a helicoid [34]. Thereby, the dynamic problem is mathematically reduced from three to two independent variables. The starting expression for the potential energy reads: V = (k/2) (x2 + y2) + (k’/2) z2, and the kinetic energy: T = (m/2) (v2x + v2y + v2z ),

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Fig. 8.16. The helicoid is the minimal surface having a helix as its boundary. It is the only ruled minimal surface other than the plane. The reader is invited to determine the absolute configuration ((P) or (M)) of the helicoid shown here, and to sketch the enantiomorphous structure (taken from [6]).

where vx, vy , and vz are the corresponding velocities. Use is then made of the equations for a helix of diameter r and pitch b: x = r cos (z/b), y = r sin (z/b), z = z. We notice that both x and y now depend on the coordinate z along the helix axis. If r, the radius of the helix, is kept constant, then the above equations describe a one-dimensional curve in three dimensions. If, however, r is considered as a continuous variable, the equations correspond to a helicoid, which is a twodimensional surface in three dimensions (see Fig. 8.16). Introducing these helicoidal constraints in the expressions for V and T, equations of motion in the variable pairs r, z and vr , vz are obtained. The problem in this way formally reduces to that of a two-dimensional harmonic oscillator, perturbed by a nonlinear term in r, vz, and depending parametrically on the helicoidal pitch b. Numerical analyses of these equations lead, within the domain of finite pitch, to chaotic solutions [34]. This then raises the question, if there exists any fundamental relation between chirality and classical chaos. In our brief considerations on the chaotic and chiral solar system (Sect. 6.1), we have already previously raised such questions, within another dynamical context, however. There quite obviously exist no simple answers to difficult and general questions. We note, for instance, that a system constrained to move in a plane, such as a double pendulum, may well behave chaotically, but cannot, according to our definitions, be chiral. On the other hand, the first example in this section

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[31] exhibits chirality but not chaos as, by an appropriate coordinate transformation, the two oscillators can be formally decoupled and thereby are shown to behave in a regular fashion. We close herewith our short excursion in the field of classical models and (re)turn to quantum mechanics.

8.8. An Early Quantum-Mechanical Model The success of quantum mechanics in interpreting atomic spectra and in predicting subatomic phenomena prompted also early attempts at a quantummechanical description of optical activity. In 1929, Rosenfeld deduced the expression for the rotational strength (Sect. 1.4, 2.7, 3.3), which allows the calculation of the isotropically averaged circular dichroism of any molecular transition between a state a and a state b, provided one knows the corresponding wave functions [35]. We must, however, bear in mind that quantum chemistry at that time was in its very early infancy. The historic paper on the hydrogen molecule [36] preceded this theoretical work on optical activity by only two years, and wave functions for chiral molecules were then not yet available. Therefore, the interest for simple, albeit somewhat artificial, quantum-chemical models, capable of imitating the behavior of electrons in a chiral molecule, was considerable. The computation of the wave functions for such a model would, of course, have to be relatively straightforward and easy. Starting from Kuhns classical work on two coupled oscillators (Sect. 8.7), the question was raised, if, in a quantum-mechanical frame, a potential could be devised, in which a single particle (electron) would behave chirally and manifest optical activity. Indeed, a model was found in a three-dimensional, asymmetric harmonic-oscillator potential, i.e., V0 perturbed by a noncentrosymmetric distortion Vnc [37]: V = V0 + Vnc = (1/2) (ka x2 + kby2 + kc z2) + Axyz, where ka „ kb „ kc „ ka, and A „ 0. Setting the first term V0 = constant, we find the equation for an ellipsoid with three unequal axes. This is in itself not yet a chiral structure, as it transforms like D2h. The additional, third-order term Axyz is also achiral, being invariant under S4, that is, a pure rotation by 908 inverts the sign, the following reflection in a plane perpendicular to the rotation axis restores it. This contradicts the definition of chirality. The sum of both V0 and Vnc, however, loses all planes of symmetry and rotation–reflection axes, and transforms like D2, a chiral point group containing only pure rotations (see also Sect. 8.2). The solution of the quantum-mechanical problem starts from the wellknown wave functions for the three-dimensional harmonic oscillator V0, and by treating the term Vnc as a small perturbation. Axyz may be visualized as a helical

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Fig. 8.17. a) Optically active model systems. Enantiomorphous, chirally distorted ellipsoids. b) The corresponding achiral, undistorted ellipsoid (plots from the Maple software; compare qualitatively with Fig. 1.6).

distortion of the ellipsoid (Fig. 8.17). Without going into mathematical details, one finds that the radiation-induced transitions which an electron in the achiral potential V0 can undergo are exclusively either electric dipole (E1)-, or magnetic dipole/electric quadrupole (M1/E2)-allowed. In the chirally perturbed potential V0 + Vnc, however, the transitions of type E1 indeed obtain a small M1/E2 contribution, and vice versa, leading for all allowed transitions to nonvanishing rotational strengths. The historic significance of this model was further brought to attention when, a few decades later, it served to describe the nonlinear optical polarization of chiral molecules in fluids [38]. In particular, it succeeded in clearly showing, why in chiral liquids sum frequency generation (w1 + w2 ; w1 „ w2) is allowed, while second harmonic generation (w + w = 2w) is forbidden.

8.9. Are There Absolute Measures of Chirality? How big is the chirality of a complex biomolecule? Of a potato? Of a swarm of bees? Of a galaxy? Is there an absolute measure of chirality? Many researchers in chemistry and physics have dealt with this quite fundamental problem and are still working on it. To these questions there are quite obviously no easy answers. The literature on this topic is correspondingly quite vast. Our aim here is not to give a detailed review of the problem but rather, based on a few examples, to indicate some general approaches. These will be the

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topics of this and the next two sections. We will mainly consider: a) geometric measures of asymmetry and chirality; b) mathematical analyses of characteristic physical quantities; and c) light-scattering properties of asymmetrically configurated clusters of point particles. Geometric Asymmetry Measures. A criterion that has intuitive appeal is that of minimal nonoverlap between enantiomers [39]. A three-dimensional chiral object cannot be made to completely overlap its mirror image by any continuous (rotational or translational) change of position. If a body is achiral, if image and mirror image are identical, they can, of course, be brought to full coverage. Then the maximum normalized overlap will be 1, and the minimal nonoverlap will be 0. However, if a molecule is chiral, it evidently cannot be brought to full coverage with its enantiomer. The maximum normalized overlap will be < 1, and the minimal nonoverlap > 0. The bigger the difference between enantiomers, the larger then the minimal nonoverlap should become. For instance, assume a chiral object described by a mass distribution (density) 1(x, y, z). We denote that of its enantiomer by 1* (x, y, z), and the total mass of one enantiomer by M. The minimum nonoverlap c will then by definition be given by the minimum of the following expression: Z 1 c ¼ min: j 1ðx; y; zÞ  1*ðx; y; zÞ j dxdydz; 2M V0

in which the integral goes over the total volume occupied by the two enantiomers. This may be illustrated by the two simple enantiomeric structures shown in Fig. 8.18: assuming for each enantiomer a unit total mass, we may write: 1 c = ðV  V*Þ. 2

Fig. 8.18. Enantiomers consisting each of four Hsubcubes (see also Fig. 8.1). Three subcubes may be brought to complete coverage, but not the fourth.

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The total volume of each enantiomer can be expressed as a sum of the volumes of the four corresponding subcubes: V = v1 + v2 + v3 + v4 ; V* = v1 + v2 + v3 + v 4 ; v 4 =  v4 ; jvnj = v. Three subcubes may be brought to full coverage, the fourth are differently oriented, and opposite signs may be assigned to their respective volumes. Thus we finally find: c = v. Another approach of particular interest is based on continuous symmetry measures [40]. This model is well-suited for objects representable by a given number of configuration points. The position vectors Pi of the points describing the chiral object are related to the position vectors P 0i which these points would have if they occupied the closest possible achiral configuration, of symmetry G. The continuous symmetry measure S refered to G is correspondingly defined as S(G) =

n 1 X jj Pi  P 0i jj2 n i¼1

where the quantity jj Pi  P 0i jj is the norm of the difference of these vectors, and the sum runs over all points of the structure. The problem thus mainly consists in finding the set of points P 0i which possesses the desired symmetry, such that the total distance from the original configuration Pi is minimal. Furthermore, the vectors Pi, respectively P 0i , are so normalized that 0 S 1. This procedure requires a series of geometric and mathematical steps which cannot be described here in detail [40]. The method may also be extended to the analysis of continuous contour lines and surfaces. A contour must be represented by a string of equally spaced points which may be chosen as closely as one wishes. Similarly, a surface is transformed into a dense polygon. This includes knots, Moebius strips, and catenanes. Large random objects, such as diffusion-limited aggregates, are also accessible to contour analysis, leading sometimes to rather unexpected results (Fig. 8.19). Illustrating a further approach: a quantitative comparison of the chirality of different tetrahedral shapes has been based on the so-called
Hausdorff chirality measure [41]. The question has been raised if topological chirality and geometric chirality always coincide [22] (see also Sect. 8.3). Recently, the problem has been examined for the case of the torus curves and knots. Taking the torsion as a criterion, it was then found that opposite geometric chiralities of knotted space curves may well correspond to one and the same topological chirality [42]. Mathematical Analyses of Chiral-Density Distributions. An object can be described by the multipole moments of its density (see also the next section).

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Fig. 8.19. a) Minimal point representation of a Mçbius strip. b) Its closest achiral structure. c) A chiral, diffusion-limited aggregate. d) The corresponding closest achiral figure (taken from [40]).

The chirality of the object may be characterized by pseudoscalar parameters that are constructed from products of these multipole moments. These pseudoscalar quantities change sign on going from one enantiomer to the other, and vanish for achiral systems. Their absolute magnitude indicates the degree of deviation of the density distribution from achiral symmetry [43]. However, there is a quasi-infinite hierarchy of such pseudoscalar quantities, and merely to consider the one of lowest order may not necessarily convey a sufficiently complete and informative picture.

8.10. On Multipole Expansions Any potential obeying an inverse-distance law may be expressed by a multipole expansion. For instance, the electrostatic potential of a charge distribution, 1(x,y,z)/d, may be described as the superposition of terms of definite symmetry (Fig. 8.20): monopole, dipole, quadrupole, octupole, etc. With increasing distance d from the charge distribution, the relative importance of the higher-order terms in the expansion should diminish, and the expansion should rapidly converge. Analogously, the magnetostatic potential of a system of permanent currents may be expanded in magnetic multipoles, and the gravitational potential of a system of masses, formally, into
gravitational multipoles.

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Fig. 8.20. Multipole expansion of a neutral electrostatic charge distribution. From left to right: dipole, quadrupole, octupole. Opposite charges are marked in different colors. Table 8.3. Symmetry Properties of Multipoles. For a general explanation, see text.

Monopole Dipole Quadrupole Octupole Hexadecapole

System of charges/ electrostatic potential

System of currents/ magnetostatic potential

Electromagnetic interaction (transition moment)

P+T+ PT+ P+T+ PT+ P+T+

– P+T P  T  a) P+T PT

– P  (E1) P + (M1/E2) P  (M2/E3) P + (M3/E4)

a

) The magnetic quadrupole moment includes also, in a toroidal topology, the anapole moment.

Table 8.3 indicates the symmetries with respect to parity (space inversion) P and time reversal T of the lower terms of multipole expansions. The important point is that, in chiral systems, quantities of different parity may coexist. An example of fundamental interest: in a neutron, a permanent electric dipole moment (hitherto unmeasured) and a magnetic dipole moment can only simultaneously occur if P symmetry as well as T symmetry are broken [44]. In Sect. 3.2, we had already mentioned the multipole expansion of the interaction energy of an electromagnetic radiation field with an atomic/ molecular system. Under the condition that the wavelength of the radiation is much greater than the dimensions of the molecule, the terms beyond M1 and E2 may usually be neglected. The pseudoscalar rotational strength (see Sect. 3.3.) is different from zero only for chiral systems, in which transitions are simultaneously E1- and M1/E2-allowed. The quantity is obviously a measure of chirality. However, in one and the same molecule, both the sign and the magnitude of the rotational strength may vary from one transition to the next. A circular dichroism (CD) spectrum contains, in general, both positive and negative signals, and, for certain wavelengths, it goes through zero. This, of course, very much restricts the applicability of the rotational strength as a general measure of chirality, in spite of its spectroscopic importance and usefulness. Another quantity that characterizes chirality is the so-called anapole moment, which is related to the magnetic quadrupole moment [45] [46]. The anapole moment can be pictured as originating from a toroidal electric current

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Fig. 8.21. The superposition of a circular current (magnetic dipole) and of a toroidal current (anapole) gives rise to a helico-toroidal current that can only exist if parity is broken (from A. Weis, Troisie`me Cycle Lectures, http://www.unifr.ch/physics/frap/; see in particular C. E. Wieman, quoted in Lecture 3, Part II).

which generates a circular magnetic field inside the torus. The circular magnetic field, in turn, corresponds to an axial (secondary) current. From that point of view, the anapole moment is parity-odd and time-odd. The simultaneous presence of a parity-even, time-odd magnetic dipole moment and of an anapole moment implies that parity is broken (Fig. 8.21). Recently, atomic spectra have revealed the existence of an anapole moment in the nucleus of 133Cs (cesium of atomic mass 133) due to parity violation [47] [48]. Magnetochiral effects have been detected in parity-nonconserving magneto-electric solids [49] and interpreted as being connected to the existence of a magnetic anapole moment. Finally, anapole moments have been predicted to occur in molecules of the fullerene type with a toroidal topology [45]. It is found that, in a chiral molecule of this general structure, the induction of a magnetic dipole moment along the principal axis of revolution also gives rise to a concomitant anapole moment.

8.11. Magnetochiral Scattering of Light We now return to the question of measuring the chirality of a random cluster of point particles. Beside the geometric procedure outlined in Sect. 8.9, there is an entirely different approach which, in principle, may be applied to any number of particles in any conceivable configuration. It is based on the computed multiple scattering of light from point-like scatterers, and is technically limited only by computer time. Evidently, if the configuration is chiral, the system as a whole should show natural optical activity. The scattering cross section calculated for left and right circularly polarized light should correspondingly be different. However, having to follow the evolution of the light polarization during a multiple scattering process is no easy mathematical task. This brings up the question, if the chirality may possibly be measured by an optical phenomenon which is independent of the state of polarization of the incident and scattered light. Indeed, as we know (Sect. 3.5), the magnetochiral

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Fig. 8.22. Distribution of the relative magnetochirality g of 1000 different realizations of ten scatterers randomly located in a sphere of constant volume. Notice that the distribution is symmetric with respect to the value zero (taken from [50]).

effect fulfills this condition. In a chiral medium, the magnetochiral scattering cross-section should be different for light incident parallel and antiparallel to an applied magnetic field, irrespective of how the light is polarized. This idea has been put to execution in a recent computational investigation [50]. In the model, the individual scatterers are assumed to possess a dielectric susceptibility making them capable of reemitting electric dipole radiation, and a diamagnetic susceptibility that may be interpreted as a Verdet constant. The system as a whole, provided it is chiral, thereby also becomes susceptible of exhibiting a magnetochiral effect. Indeed, the magnetochiral scattering crosssection of the system then reveals itself as a meaningful measure of the chirality. For instance, in a numerical application, an ensemble of 1000 different realizations of the positions of 10 scatterers, randomly located in a sphere of constant volume, shows for the magnetochiral effect a distribution of positive and negative values that peaks around zero, as one would intuitively expect (Fig. 8.22). Assuming that the ensemble average of many static systems corresponds to the time average of a single dynamic system, one realizes that a swarm of randomly moving noninteracting point particles, spatially confined as described above, should on the average be achiral. By assigning to the particles different electric and magnetic susceptibilities, the procedure should also lend itself to the study of the temporal evolution of the chirality of mixtures. It might provide a means of relating chirality to the measure of the entropy of mixing.

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9. Pathways to Homochirality
The world is chiral and clinal, enjoy symmetry wherever you find it Vladimir Prelog (1906 – 1998) (in J. D. Dunitz, E. Heilbronner, Reflections on Symmetry, Verlag Helvetica Chimica Acta, Basel)

9.1. Chiral Intermolecular Interactions Ideal gases are assumed to consist of noninteracting point-like particles. But real atoms or molecules interact with each other. Each real particle also occupies a volume of its own which is excluded for the others. The equation of state for a real gas, such as the van der Waals equation, takes these particle properties into account in terms of specific parameters. A real gas distinguishes itself from an ideal gas very markedly by the fact that it can liquefy. A real gas resembles an ideal gas only in the case of low pressure and of large average interparticle distances, where the interactions become insignificant. The question at the center of our attention concerns a gas of chiral molecules. Will, in general, a real gas composed of homochiral molecules behave differently from a racemic gas containing equal quantities of both enantiomers? What happens when the gas becomes more and more dense? Will, for instance, the boiling points at a given pressure be different, due to differences in the interactions between d and d (or l and l), as compared to d and l? In other words, to what extent are homochiral and heterochiral molecular interactions different? Are they of direct importance for the emergence of homochirality? Evidently, this question must be tackled primarily at the molecular level. We consider our chiral molecules to be neutral and, as an initial situation, assume the intermolecular distance to be so large (
1 nm = 109 m) and the temperature to be so high (ca. 300 K) that the molecules can quasi-freely rotate with respect to each other. At large distances, mainly electrostatic interactions come into play [1]. They are primarily of the kinds permanent dipole–permanent dipole, permanent dipole–induced dipole, and induced dipole–induced dipole. These interactions, when spatially Boltzmann-averaged over all mutual orientations, in general depend on the inverse sixth power of the intermolecular distance R, that is, they decrease as 1/R6. Between molecules that carry no permanent dipole moment, and also between atoms, there occur, out of the above enumeration, only induced dipole–induced dipole interactions. When two molecules or atoms approach each other, their electron clouds will distort, causing in each particle an induced On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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dipole moment. The concomitant interaction is universally attractive and gives rise to what one also calls dispersion forces [2]. Such encounters between atoms or molecules are, of course, highly dynamic, and, in some sense, resemble the interaction with electromagnetic radiation. To describe such dispersion effects satisfactorily, the quantum-mechanical nature of the particles involved must be taken into account. The dispersion energy between a molecule A and a molecule B may then be shown to depend on sums of products of electric-dipole transition moments (see Sect. 2.6 and 3.2), and on the excited-state energies of A and B. Will any of the interactions hitherto mentioned be chirally discriminating and be able to distinguish between dd (or ll) and ld? The answer is: definitely no. To explain the chiral discrimination between homochiral and heterochiral molecular pairs, the description of the intermolecular interaction has to be refined [3]. It has been shown that there indeed exist chirally selective electrostatic interactions, but they are of higher order. In the weakly-coupled limit, the electric quadrupole–electric octupole interactions are chirally discriminating (see Table 8.3, and the respective symmetries with respect to parity; one notices that the product is parity-odd, time-even). However, these interactions die off very fast, namely with the inverse 17th power of R (/ 1/R17) [4], so as to be of rather limited practical significance. On the other hand, the calculation of the dispersion forces may be complemented by including in the mathematical expression for the interaction energy also magnetic-dipole transition moments. One then obtains a contribution to the dispersion energy which depends on products of the rotational strengths of the molecules involved [5] [6] (see also Sect. 1.4, 2.7, and 3.3). If we compare the molecular pair dd with dl , the products of rotational strengths change sign; because ´l =  ´d, we find:

´d´d = ´l´l =  ´d´l. This kind of enantioselective interaction, albeit generally weak, depends on intermolecular distance as 1/R6 (for R in the range of a few nanometers). Over the years, a variety of different approaches have been followed to study enantioselective interactions at intermediate distances. The interaction energy between two chiral tetrahedral molecules, dd vs. dl (or, for instance, (R)(R) vs. (R)(S), according to Cahn, Ingold, and Prelog) has been studied in the frame of relatively simple quantum-chemical calculations. The differential energy, averaged over all relative orientations, DE = Edd  Edl, is then considered as a measure of the chiral discrimination. It may be interpreted as a sum of interactions between individual atoms on the two molecules. And it has then been shown that six-center forces between triplets of atoms, one triplet on

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Fig. 9.1. Enantiomers of tetrahedral molecules. The bond from the center atom to the substituent d formally defines a translation. The three substituents a, b, c formally define a rotation. The superposition of translation and rotation defines the chirality (see also Sect. 4.1 and 4.2). The chiral interaction between two such molecules cannot depend on less than three atoms on each molecule [7].

each molecule, are responsible for the enantioselective discrimination [7] (Fig. 9.1). The preference is found to be of the order of 104 to 102 times the total, rotationally averaged interaction energy at an intermolecular distance of 5  (= 0.5 nm), the heterochiral energy being in most cases lower. From the gas phase we now go into the liquid phase. The progress of the last decades in computation has very considerably improved the possibilities offered by molecular dynamics. Different models of molecular interaction can be numerically tested, also when molecules are in close contact. A quantity of interest, to understand the molecular arrangement in a fluid, is the radial distribution function. We pick out a molecule among many other ones in the fluid and ask how the surrounding molecules are distributed around it. Suppose we have a racemic mixture, and we pick out a molecule of absolute configuration d, will the other d- and the l-molecules be equally distributed around it? If the intermolecular interactions are chirally nondiscriminating, this should obviously be the case. However, if there is enantioselectivity, the radial distribution function gdd(R) should be different from gdl(R). Averaged over a large number of molecules, these are smooth functions of the distance, R, from the (arbitrarily chosen) central molecule. The difference between the homochiral and the heterochiral radial distribution function Dg(R) = gdd(R)  gdl(R) reflects the enantiomeric imbalance in the local fluid environment [8]. In such calculations, the individual molecules may enter as structures of bonded atoms, or they may be simulated by artificial electrostatic potentials exhibiting certain particular properties. In the above-cited reference, for instance, the individual molecules are represented by an ellipsoidal hard-core potential carrying in addition a given distribution of point charges, leading to corresponding multipole

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moments. (For enantiomers, these charge distributions are evidently mirror images of each other.) The influence of different charge distributions on Dg(R) is compared. It is found that particularly discriminating molecules have electric quadrupoles in a molecular plane (xy, say) perpendicular to electric dipoles (zdirected). (For ellipsoids made chiral, see also Sect. 8.8.) Using molecular-mechanics methods with Monte Carlo simulations, the chiral discrimination between the enantiomeric amino acids d- and l-alanine has been investigated in the liquid phase [9]. From calculations of the molecular partition function, enantioselective differences in the equation of state and in thermodynamic quantities were sought. It was found that these differences are very small, unless some internal degrees of freedom are frozen, and the molecules are represented by simplified models. In general, it was also found that the chiral preference, homochiral vs. heterochiral, can depend strongly on intermolecular distance. Similar calculations were later performed on the molecule 1,1,1-trifluoropropan-2-ol [10]. The conclusions are that the pure, homochiral fluid has a slightly higher density compared to the racemic mixture, indicating a more favorable packing interaction. On the other hand, the pure enantiomers have a lower enthalpy of vaporization, implying that the transition from the liquid to the gas phase requires less energy than in the racemate. Enantioselectivity in a liquid mixture of enantiomers implies a departure from ideality which, in principle, should manifest itself in macroscopic properties. The search for thermodynamic effects started already in the 19th century, and the literature on this topic is quite extensive. Comparative measurements of density, dielectric properties, surface tension, and viscosity have generally been inconclusive. Attempts at separating mixtures of enantiomers by distillation have not been successful. On the other hand, deviations from ideality have been found in measures of the optical rotation and in nuclear-magnetic-resonance (NMR) chemical shifts [11]: in mixtures of d- and l-antipodes of different percentage, the optical rotation should, in principle, vary linearly with the enantiomeric excess. But unambiguous deviations therefrom have been recorded. In an ideal d/l-mixture, the NMR chemical shifts should be independent of the relative composition. However, enantioselective effects have been observed there, as well. Enantiomeric compounds exhibiting such chiral discriminations have been observed to also show a measurable heat of mixing. In strongly chiral molecules, such as appropriately substituted biaryls and bior polyphenyls, the chiral discrimination may become so strong that cholesteric liquid crystalline phases will form (see Sect. 7.2). There the enantioselective interactions may lead to extended, partially anisotropic homochiral domains. Such processes may also be simulated numerically [12]. The intermolecular potential adopted for such calculations essentially represents two kinds of contributions:

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• a potential taking into account the molecular anisotropy in both the shape and the attractive forces • an additive chiral interaction proportional to: a) a pseudoscalar factor which changes sign on going from one chiral species to its enantiomer, b) another pseudoscalar factor which changes sign upon interchange of any two molecules of given spatial orientations in the medium.

9.2. Enantioselectivity at Phase Boundaries The boundary between two different, but homogeneous phases, such as gas– liquid, gas–solid, liquid–solid, or between two immiscible or partly miscible liquids, breaks the reflection symmetry in a plane and the point-inversion symmetry, but does not in itself imply the presence of chirality. However, the formation of chiral domains may be favored or enhanced for substances, even achiral ones, that aggregate at such a phase boundary. We find, for instance, a striking example of spontaneous chiral symmetry breaking by achiral molecules in a Langmuir–Blodgett film. Arachidic acid (CH3(CH2)18COOH) contains a long hydrophobic alkyl chain and a hydrophilic carboxy group. If a solution of the acid in the relatively hydrophobic solvent chloroform is spread onto the surface of an aqueous phase containing CaCl2, then the acid will tend to dissociate H+, and two arachidic anions will take Ca2+ as counter cation. The solute molecules will preferentially remain at the interface between the two solvents, forming a monomolecular film, with the hydrophobic tails immersed in the chloroform and the ionic moieties in the aqueous phase. By lateral compression of the molecular films, a so-called Langmuir–Blodgett (LB) layer is obtained. Such LB multilayers were deposited on mica substrates and examined by atomic-force microscopy. It was observed that the LB multilayers adopt a chiral, triclinic crystalline structure, exhibiting two distinct kinds of lattice domains that are enantiomorphous [13] (Fig. 9.2). It must be concluded that the symmetry breaking and induction of local chirality occurs from conflicting spatial requirements of the alkyl chains, on one hand, and of the carboxy groups and Ca2+ ions, on the other. One encounters the induction of chirality as a compromise. Particular constraints may make an achiral molecular arrangement energetically unfavorable. To achieve a more stable state of lower energy, the structure of the aggregate will distort, thereby making it chiral (see also Sect. 7.1). But, of course, as long as there is no overall bias favoring a given chiral structure, one obtains, on the average, enantiomorphous domains in equal percentage.

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Fig. 9.2. a) Schematic representation of the triclinic packing of arachidic acid calcium salt. The long hydrocarbon chains are assumed to lie as zig-zag structures perpendicular to the figure plane. There is a stacking fault after every four rows. The two enantiomorphous forms are shown, separated by a mirror plane (dotted line) (from [13]). b) Scanning-tunneling-microscopy picture of heptahelicene on a Cu(111) surface. Unit cells and the arrangement of molecules within them are outlined in red (adapted from [16] and [17]).

As a next example, we consider a chiral molecule, which is present as a racemic mixture and has proven to be very difficult to separate into enantiomers by usual means:

We here show a single antipode, and the reader is invited to draw the enantiomer. Compression of the oriented, LB-type racemic monolayer of this tetracyclic alcohol on the surface of an aqueous medium reveals the pressuredependent existence of several phases, one of which clearly separates into

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distinct, enantiomorphous chiral domains [14]. In other words, one finds a resolution of the racemate into separate enantiomers. This may be viewed as a pseudo-two-dimensional analogue of the spontaneous resolution of sodium ammonium tartrate by three-dimensional crystallization discovered by Pasteur (Sect. 1.3 and 7.1). The self-assembly of molecules on surfaces is governed both by lateral, direct interactions between the molecules, as also by influences due to the supporting substrate. This is, in particular, also the case when the substrate is a solid metal surface which is electronically highly polarizable. In analogy to our first example, it has been observed that achiral meso-tartaric acid on an achiral copper (110) surface may form two-dimensional enantiomorphous chiral structures, observable as mirror domains in low-energy electron diffraction [15]. In relation to our second example, the unsaturated hydrocarbon heptahelicene (structurally similar to hexahelicene, but with an additional fused benzene ring; see also Sect. 1.3 and 7.3) adsorbed on the achiral (111) surface of copper, reveals a certain mobility, even at lower temperatures. At close to full monolayer coverage, these molecules are forced into well-ordered arrangements [16] [17]. The deposition of pure (M)-heptahelicene in a closely packed arrangement results in a collective, chiral surface structure which is exactly enantiomorphous to the one similarly obtained with pure (P)-heptahelicene. It has also been observed that the adsorption of racemic heptahelicene tends to form enantiomorphous mirror domains in which the enantiomers are partially separated. This demonstrates that the molecular chirality, via combined intermolecular and molecule–substrate interaction, leads to extended surface chirality. As heptahelicene does neither bear any polar, nor any particularly polarizable substituents, it must be assumed that the collective chiral order is strongly influenced by short-range repulsive intermolecular forces. The adsorption of the same hydrocarbon on Ni(111) and Ni(100) does not reveal any collective chiral effects, due to tighter binding and lower mobility of the molecules on these surfaces.

9.3. Enantioselective Adsorption on Chiral Surfaces We have already previously considered chiral molecules selectively adsorbed on chiral solid surfaces, on quartz (Sect. 6.4) and on chiral surfaces of elementary metals (Sect. 7.3). We here return to this topic, concentrating on the mineral calcite (CaCO3), because of its particular properties and abundance. It is one of the most frequently occurring minerals, being part of some of the oldest known sedimentary formations. It is to be assumed that calcite-crystal surfaces have been widely present in prebiotic environments. Furthermore, calcite has a well-documented tendency to adsorb amino acids. Biomineralized calcite is

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strongly bonded to proteins in the shells of many invertebrates. These considerations underscore the general interest of the experiments and observations here summarized. Calcite in the rhombohedral space group R3c occurs commonly in a trigonal scalenohedral form featuring adjacent crystal faces with enantiomorphous surface structures. Surfaces of the same handedness are, for instance, (213), (132), (321), and of opposite handedness, (312), (123), (231). The chiral discrimination by these surfaces was demonstrated in the following measurements [18]: such crystals were immersed in a 0.05m solution of racemic aspartic acid, which strongly adsorbed onto the calcite. After equilibration, the adsorbed aspartic acid was removed from each individual crystal face, collected and the relative d/l abundances determined. Enantiomeric excesses of the order of (d  l)/(d + l)   0.5% were induced and measured. Of course, such a situation is yet far away from homochirality. Indeed, if enantioselective adsorption of this kind has played any role in the generation of the homochirality of biopolymers, one must assume that this only has been possible through a long series of consecutive steps, in which every step obviously was only partially selective. An interesting aspect from the biological point of view are modifications of the texture of calcite crystals by adsorbed proteins. It has been discovered that a wide variety of organisms, which form mineralized skeletons, are capable of controlling the textures of the constituent crystals [19]. Biologically formed calcite crystals are, for instance, found in the shells of Foraminifera. A striking example is that of the Spirillinidae, which form their entire shell, of a complex coiled microstructure, out of a single crystal of calcite. The c-axis of the single crystal is found to be perpendicular to the plane of the coil. Proteins play a regulatory role in the formation of calcite crystals also in higher organisms [20]: pancreatic juice is supersaturated with calcium carbonate. Calcite crystals, therefore, may form and obstruct pancreatic ducts. Human lithostathine, a protein synthesized by the pancreas, inhibits the growth of calcite crystals by modifying the crystal habit. The mechanism of this interaction has been investigated in detail by crystallographic methods and by moleculardynamics simulations. In particular, the role played by the N-terminal undecapeptide of lithostathine was investigated. It was found that such a protein inhibits calcium salt crystal growth due to its flexibility, enabling it to spread onto a crystal surface, and due to the strong coulombic interactions of the peptide backbone with that surface. Such interactions are selectively reinforced through the lateral chains of the peptide.

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9.4. Chiral Chromatography There are essentially two strategies for obtaining enantiomerically pure chiral compounds: a) asymmetric synthesis allowing easy isolation of the chiral product. b) Synthesis of a racemate and separation into antipodes. On both procedures, there exists a vast chemical literature, and it is neither possible for us, nor is it our aim, to attempt to review it in general. This would be an immense task. Rather, we limit our considerations to particular molecular and intermolecular phenomena that may have played a special role on the evolutionary road to biological homochirality in nature. We here briefly consider some aspects of chiral chromatography, because it is an important application in the chemical laboratory of enantioselective adsorption. It may, in principle, lead to the complete resolution of a racemic mixture into its constituent enantiomers. In comparison, the efficiency of the natural processes is certainly lower. In nature, as just indicated, we have no neatly packed chromatographic columns and no synthetic, taylor-made separating agents. Instead, in the course of prebiotic evolution, the innumerable adsorption phenomena must have taken place at the surface of minerals embedded in a quasi-infinite variety of grains of sand, of stones and rocks of different sizes. But the natural processes took place on a completely different time scale, thus leading, in particular cases, ultimately to a high degree of selectivity. In liquid chromatography in the laboratory, the racemic mixture to be separated is generally carried as a solute by an appropriate solvent. The solvent, or mobile phase, must be so chosen as to enhance the enantioselectivity of the chiral stationary phase. We here briefly consider some stationary phases which exhibit particular selectivity for molecules of biological interest [21]: 1) Proteins immobilized on, that is chemically bound to, silica gel: many small chiral molecules, such as amino acids and their derivatives, show stereoselective affinity to proteins, notably to serum albumin, selected glycoproteins, or a-chymotrypsin. The mobile phases are generally aqueous buffers, possibly with the addition of some organic modifiers. 2) Polysaccharide derivatives: compounds such as cellulose and amylose consist of d-glucose units linked by glucoside bonds. They form natural polymers with a highly ordered helical structure. Such carbohydrate polymers may be modified by selectively derivatizing the OH groups on each glucose, leading to taylor-made stationary phases. 3) Construction of chiral cavities into which the molecules from a solution may selectively fit: foremost in use are cyclodextrins, macrocyclic molecules containing a certain number of glycopyranose units: six for the a-form, seven for the b-form, eight for g-cyclodextrin.

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4) Stationary phases with p-donor/p-acceptor units: their function is based on the fact that aromatic rings with an electrophilic substituent (such as nitro) tend to form charge transfer complexes with aromatic rings bearing a nucleophilic substituent (such as amino). For instance, a stationary phase, to which chiral p-acceptor aromatic fragments are attached, will exhibit an enantioselective interaction with amino acid derivatives bearing an aromatic ring with electron-donating alkyl, ether, or amino substituents. Additional substituents, possibly leading to hydrogen-bond formation, may increase the specificity of the interaction. It is quite evident that the chromatographic separation of a racemic mixture into antipodes cannot be carried out with a stationary phase that is not chiral. What is surprising, however, is that with an achiral phase one may modify the relative composition of a nonracemic mixture containing unequal amounts of enantiomers. This is the topic of our next section.

9.5. Enantioselectivity by Achiral Chromatography We start with a mixture of unequal amounts of enantiomers. This may be considered as a racemic mixture, to which a certain quantity of one antipode has been added. An increase of the enantiomeric excess of one antipode with respect to the other entails a corresponding increase of the optical activity. Changes of the enantiomeric excess by achiral influences have been observed, initially and independently in (R)/(S)-mixtures of nicotine [22], of amino acid derivatives [23], and of cyclic unsaturated ketones [24]. It has since been shown that by achiral fractional chromatography, under favorable circumstances, the separation of the excess enantiomer from the residual racemic mixture should indeed be possible. The origin of this effect certainly resides in the fact that the enantioselective interaction between adsorbed molecules and solute molecules is different from that between solute molecules and other solute molecules. This differentiation should be influenced by the solvent which, like the stationary phase, is also achiral. As an example, we summarize a recent quantitative investigation [25]: the compound binaphthol ([1,1’-binaphthalene]-2,2-diol) is at the same time highly chiral and shows the tendency to dimerize in solution. It was found experimentally that initial mixtures of unequal amounts of (R) and (S) could indeed be chromatographically separated into a first fraction containing the excess enantiomer in practically pure form, followed by fractions that were nearly racemic. By optimizing the procedure, it revealed itself possible to asymptotically achieve a 100% separation of the antipode in excess, with a remaining racemic mixture. If one starts the experiment with a racemic

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solution, there is no separation, as expected, and all collected fractions are racemic. These results could be quantitatively interpreted on the basis of the following facts and assumptions: in a liquid mixture, containing both the (R)and (S)-enantiomers, there are three possible dimers: (RR), (SS), and (RS). It may be shown on statistical grounds that the equilibrium constant for the hetero-dimer must be on the order of twice that for the homo-dimers. All five species present in solution, (R), (S), (RR), (SS), (RS), can bind to the achiral stationary phase, but the bound monomers do no longer dimerize. On the basis of the data on the equilibria in solution, and of the competitive Langmuir isotherms for all five species, combined with the standard diffusion equations for a chromatographic column, the measured results could be satisfactorily reproduced by computations. What we here witness is a self-fractionation of the enantiomeric excess due to the differential association properties in the two separate achiral phases: liquid solvent and stationary solid.

9.6. Enantiomeric Excess by Chiral Catalysis As already mentioned in Sect. 1.5, if we start from achiral or racemic reactants, and wish to synthesize chiral products, we need an asymmetric catalyst or auxiliary. The enantiomeric excess of the product, Eprod = (R  S)/(R + S), will quite evidently attain a relative maximum, Eprod = Emax, if the auxiliary is enantiomerically pure. R and S here designate the mole fractions of the enantiomeric products. For practical reasons, it may often be difficult to obtain the auxiliary in enantiomerically pure form, the enantiomeric excess of the auxiliary correspondingly being lower than unity: Eaux < 1. One is then inclined to assume that there is a linear relationship between the enantiomeric excess of the auxiliary and of the product: Eprod = Emax · Eaux. In an ideal case, Eaux = 1, and Eprod = Emax, as shown above. Intense examination of a large number of asymmetric syntheses has, however, led to the conclusion that, in general, the above linear relationship does not hold, and that there may occur substantial deviations from linearity [26]. Such nonlinear effects in asymmetric synthesis can occur by a variety of mechanisms, and, in the following, we limit ourselves to considering one of the simplest as an illustrative example [27]. We consider here the case of a catalyst, or auxiliary, composed of a metal center M and of two chiral ligands LR and LS. The auxiliary may then exist in two homochiral enantiomeric forms LRMLR, LSMLS, and an achiral diastereoisomeric meso-form LRMLS. These three forms are assumed to be present in the reaction mixture in the steady-state concentrations designated by x, y, and

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z, respectively. The auxiliary reacts with an achiral reactant A, to form the three isomeric products PR, PS , and Pmeso : A + LRMLR ! PR (with velocity constant k) A + LSMLS ! PS (with velocity constant k) A + LRMLS ! Pmeso (with velocity constant k’) Notice that the reaction constant k’ for the achiral meso-form may be different from that for the chiral products, k. If the auxiliary is a true catalyst, as it should, it will of course be set free again, after the product formation. By elementary steady-state kinetics and a little algebra, one finds: Eprod 1þb ¼ Emax Eaux 1 þ gb where b = z/(x + y), the ratio of the concentration of the auxiliary in the mesoform to the sum of the concentrations of the chiral forms, and g = k’/k, the ratio of the velocity constant of formation of the meso-product to that of the chiral products. One immediately verifies that if z = 0, implying the absence of the meso-form, then the above relation is linear. The same holds if k’ = k. For b „ 0 and k’ @ k, one finds a relative lowering of the enantiomeric excess of the product; for k’ ! k, a relative increase (see also Fig. 9.3). The presence of the meso-form can, accordingly, both lead to a relative decrease or an increase of Eprod. It is, of course, evident that if we wish to obtain an enantiomerically pure product, we always need an auxiliary of highest possible homochiral purity and reactivity. A particular situation is encountered in asymmetric autocatalysis: in this case, the reaction is catalyzed by the product. We consider a bimolecular scheme by which two achiral reactants, A and B, add to form a chiral product P. In the case of ordinary, unselective catalysis (C), we have the situation: ðCÞ

C þ A þ B ƒ! P þ C In the case of enantioselective autocatalysis (PS, PR): ðPS Þ

PS þ A þ B ƒƒ! 2 PS ðPR Þ

PR þ A þ B ƒƒ! 2 PR

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Fig. 9.3. According to the mechanism by which the auxiliary works, the enantiomeric excess of the product (Eprod, here denoted as eeprod) will not depend linearly on that of the auxiliary (Eaux, here denoted as eeaux). Curve I: linear dependence of both quantities. Curve II: Asymmetric amplification. Curve III: Asymmetric depletion (from [26])

We immediately realize that here the amount of catalyst continuously increases. This leads to a corresponding acceleration of the reaction, at least as long as the concentration of reactants does not significantly diminish. As a consequence, an initial enantiomeric excess of PR over PS, or the reverse, can rapidly amplify [28]. Although, until now, only a limited number of reactions have been found to follow this enantioselective autocatalytic scheme, it certainly cannot be excluded that such kinetic processes may have played a role in prebiotic evolution. In Sect. 9.8, we will consider a particular example, where the fusion of two homochiral oligopeptides is autocatalytically promoted by the homochiral product. In the case of the corresponding heterochiral reaction, no such catalytic influence is found.

9.7. Homochiral Polymerization: Oligopeptides Amino acids polymerize to peptides by formation of amide links under loss of H2O: H2NCaHRCOOH + H2NCaHR’COOH ! H2NCaHRCONHCaHR’COOH + H2O The absolute configuration, l or d, of an amino acid depends on the relative spatial position of the side chain R, R’, at the respective C(a)-atom (see

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Fig. 1.2). Polymerization reactions of racemic mixtures of amino acids in isotropic solutions are expected to yield polymers comprising random sequences of l- and d-repeat units (n) in a binomial distribution: n = 1:

l

n = 2:

ll

n = 3:

lll

ld dl lld ldl dll

ldd dld ddl

d

1:1

dd

1:2 :1

ddd

1:3 :3 :1

As may be easily concluded, the probability of obtaining a polymer consisting of n units with a homochiral sequence of given handedness would then be 1/(2n). With growing n, this probability should indeed become negligible. The emergence of sizable concentrations of homochiral polypeptides must consequently be induced under particular, anisotropic conditions. From what we have seen so far, it appears promising to consider reactions at phase boundaries. Under the title
Chiral Amplification of Oligopeptides via Polymerization in Two-Dimensional Crystallites on Water [29], some interesting experiments at the air/water interface have been reported. Amino acids to which long hydrocarbon chains, e.g., C18H37, have been attached (see Fig. 9.4 and for further details the given references) tend to assemble at the air/water interface in an ordered fashion. The polar amino acids are hydrophilic and remain in the aqueous medium, while the hydrophobic alkyl chains arrange themselves above the surface. Racemic mixtures of molecules then form two-dimensional crystallites which, depending on details of the molecular structure, may appear in different modifications: 1) Conglomerates. Mixtures of enantiomorphous crystallites. Each crystallite consists of homochiral molecules of one or the opposite handedness. 2) Racemic crystallites. Each crystallite contains the molecular enantiomers in equal amounts. Under suitable conditions, polymerization reactions may be carried out. It has been observed that, within conglomerates, an enrichment with small polypeptides – also called oligopeptides – of homochiral sequence occurs. This is due to lattice-controlled polymerization within the bulk of the enantiomorphous domains. An enhanced formation of homochiral oligopeptides may also be formed in racemic crystallites, where the packing arrangement favors reactions between homochiral molecules along a translation axis. Nonracemic mixtures may resolve into a racemic mixture and into domains containing the enantiomer in excess. The homochiral enantiomers in excess,

Fig. 9.4. Amino acids esterified with octadecan-1-ol at the water/air interface. Formation of oligopeptides may proceed stepwise via a reaction between adjacent molecules, breaking the ester bond and releasing an alcohol molecule. a) Monolayer of the precursor. b) Formation of a dipeptide. c) The polymerization proceeds by
unzipping of the oligopeptide from the alcohol monolayer (from R. Eliash, I. Weissbuch, K. Kjaer, M. J. Weygand, T. R. Jensen, M. Lahav, L. Leiserowitz, Homochiral Peptides by
Unzipping Mechanism at the Air–Water Interface, Hasylab-Reports 2001 (http://www-hasylab.desy.de/science/annual_reports/2001_report/part1/contrib/43/5489.pdf).

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which crystallize in chiral crystallites, would evidently yield homochiral oligopeptides. There can also occur an amplification of chirality by nonlinear effects, that is, the enantiomeric excess of the product may be greater than that of the reactant (compare with the previous sections). This could be a further pathway to oligopeptides of given handedness. Polycondensation of the amino acid derivative tryptophan N-carboxyanhydride from a racemic mixture shows the enhanced formation of homochiral oligomers, both in solution and within artificial liposomes [30]. The liposomes tend to increase the length of the peptide sequences obtained. An observed relative over-representation of homochiral sequences also in bulk aqueous solution relative to the statistical expectations stated above, presents evidence for a homochiral molecular cooperative effect. Once a short …ll… (or …dd…) sequence is formed, the incorporation of a unit of same handedness is apparently favored. Of decisive importance is the question, if similar homochiral preferences exist in peptides composed not only of one, but of several different kinds of amino acids. What now immediately follows gives at least a partial answer to it.

9.8. Homochiral Polymerization: Polypeptides As is well-known, self-replicating polypeptides could have played a key role in the origin of biological homochirality on Earth. Here, we consider some experiments where oligomers composed of homochiral amino acid units fuse to give homochiral polymers of increased length. The reacting mixture is racemic in the sense that it contains enantiomorphous homochiral oligomers in equal amounts. The experiment is inspired by the structure and specific function of a particular segment of the yeast transcription factor, a protein that regulates the reading of the DNA of yeast cells [31] [32]. It is shown that an a-helically coiled homochiral peptide consisting of 32 amino acid residues and resembling in its sequence the so-called
leucine zipper domain of the yeast transcription factor GCN4 can act autocatalytically in templating and catalyzing its own synthesis. It accelerates the thioester-promoted amide-bond condensation of two helical fragments, consisting of 15 and 17 residues, respectively [33]. The 17-residue electrophilic oligomer is here designated by E, the 15-residue nucleophilic oligomer by N. The amino acid sequences (denoted by the usual abbreviations for the amino acid monomers, and including the activating end groups) are: E: N:

Ar-RVKQLEKKVSELLKKVA-COSBn CLEXEVARLKKLVGE-CONH2

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These fragments are synthesized in the laboratory as homochiral units, consisting entirely either of l-amino acids (El and Nl) or of d-amino acids (Ed and Nd). The experiments start from a
racemic mixture, consisting of equal amounts of El, Ed, and Nl, Nd. From this, the 32-residue product T can in principle be obtained in the form of four isomers, Tll, Tdl, Tld, and Tdd, according to the general scheme: El + Nl ! Tll El + Nd ! Tld Ed + Nl ! Tdl Ed + Nd ! Tdd As these experiments now reveal, there is autocatalysis of the homochiral reactions, but not of the heterochiral ones. Written in somewhat simplified form: El + Nl + Tll ! 2 Tll Ed + Nd + Tdd ! 2 Tdd Expressed in words, the homochiral products form preferentially, as they act as templates for their own synthesis. This is not the case for the heterochiral ones. To further test the selectivity and specificity of these reactions, experiments were carried out with
mutants. In the otherwise homochiral peptide fragment Nl, one among the fifteen l-amino acid residues was replaced by a d-amino acid. Such a modified peptide fragment is hereafter designated by NLd. It was found that equimolar quantities of El and NLd only give comparatively insignificant amounts of TLL d , the composite, 32-residue peptide carrying the single substitution. In a direct competition experiment, TLL d reveals itself not to be catalytically active in its own synthesis: ll Nl + NLd + El + Tll + TLL d ! 2T + ……

The dominant reaction is the formation of Tll, the unsubstituted, completely homochiral product. The reaction LL El + NLd + TLL d ! Td + ……

LL shows only background rates of TLL d formation. However, and remarkably, Td ll catalyzes the T formation:

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On Chirality and the Universal Asymmetry LL ll El + Nl + TLL d ! T + Td

This demonstrates a tendency of the error-carrying replicator TLL d to promote the formation of the perfectly homochiral product. One then notices that, even at the level of these relatively simple molecules, properties resembling those of living systems are encountered, namely: self-replication, propagation and maintenance of homochirality, resistance towards the accumulation of errors. The amino acid composition of the oligopeptides E and N was chosen to resemble that of specific sequences of a biologically active protein. The critical reader may remark that the conclusions just drawn on possible prebiotic mechanisms are in fact based on models inspired by later biological evolution. It may indeed be difficult to conceive of prebiotic molecular systems imitating biological evolution without previous knowledge of biological evolution. But the remarkable fact is that there at all exist some relatively small molecules that already show a behavior resembling biomolecular specificity.

9.9. Biopolymer Generation on Mineral Surfaces Mineral surfaces may not only selectively adsorb biomolecules (Sect. 9.3), but they may also promote their polymerization. As hydrolysis generally competes with polymerization, it has proven difficult to synthesize effectively biopolymers of greater length in bulk aqueous solution in the absence of specific catalysts. The formation of oligomers of the amino acid glutamate on the mineral illite, the polymerization of aspartic acid on the surface of hydroxylapatite (Ca5(PO4)3OH) have been quantitatively studied under laboratory conditions not too different from those that may have prevailed on prebiotic Earth [34]. Chain lengths of up to 55 units were obtained. There is substantial evidence that, in early prebiotic evolution, not only polyamino acids existed, but also polynucleotides in the form of RNA (Fig. 9.5; see also Sect. 9.10 and 10.1). The role of heterogeneous catalysis in prebiotic synthesis may be well exemplified by the montmorillonite clay-promoted synthesis of RNA oligomers [34 – 36]. Montmorillonite is an alumosilicate ([Al4 Si8O20(OH)4] · x H2O). Isomorphous substitution of Al3+ for Si4+, and of Mg2+ for Al3+, leads to the formation of ionic sheets. The reaction of activated monomers of nucleotides in aqueous solution containing Mg2+ in the presence of montmorillonite has yielded oligomers of 6 – 14 monomer units. The reaction takes place at the clay interlayers, through which there is control of the formation of the phosphodiester bonds linking the monomers. Both sequence selectivity and the tendency to generate an excess of homochiral products have been observed. Elongation of a primary decamer by stepwise further addition of monomers has led to chain lengths of ultimately ca. 40 monomer units.

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Fig. 9.5. a) An RNA oligomer composed of four nucleotides containing, respectively, the bases adenine (A), guanine (G), cytosine (C), and uracil (U) (adapted from, S. H. Pine, J. B. Hendrickson, D. J. Cram, G. S. Hammond, Organic Chemistry, 4th edn., McGraw-Hill, Auckland, 1980). b) Schematic representation of the primary strand (at right) and the complementary strand connected by H-bonds. In DNA, the base uracil is replaced by thymine (T), and the ribose sugars are replaced by deoxyribose units.

Activated monomers of biopolymers, present on the primitive Earth, would a priori be expected to have been racemic, containing equal quantities of d- and l-enantiomers. Analysis of the linear dimers of RNA formed under laboratory conditions on montmorillonite gave a ratio of 60 : 40 of d,d- and l,l-dimers, as compared to d,l- and l,d-dimers, showing some homochiral selectivity. However, control reactions in aqueous solution without montmorillonite resulted in a homochirality ratio of the dimers remarkably higher, namely of 94 : 6. On the other hand, by the surface-catalyzed reactions, absolute yields 6 – 19 times greater were obtained [35]. The unexpectedly high enantioselectivity observed in the uncatalyzed reactions may derive from homochiral stacking of the activated nucleotides in solution, leading preferentially to homochiral dimer formation.

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The importance of catalysts lies in the fact that they tend to direct a reaction along a few reaction pathways leading to a limited array of products. In the formation of oligonucleotides, in particular with more than two monomers, catalysts are, therefore, essential. In the absence of catalysis, the rate of hydrolysis of the activated monomers may become greater than the rate of polymerization. Heterogeneous catalysts like montmorillonite bind activated monomers in the interlayers between the clay sheets, converting them to a limited number of oligomeric products. Of course, montmorillonite is a much less efficient catalyst than the enzymes that promote the RNA synthesis in living organisms. There is evidence that in prebiotic chemistry, RNA itself acted not only as information carrier but also as catalyst. To fulfill this role, oligomers much longer than ten monomers are, however, necessary. Theoretical studies predict that 40 monomers or more are required for high-fidelity template-directed synthesis and catalysis [37]. Starting from oligomers produced by montmorillonite catalysis, longer homochiral RNA strands may indeed be formed by socalled
feeding reactions which, however, are relatively time-consuming. The synthesis of RNA essentially proceeds in two ways: 1) Production of primary strands by chain elongation, as discussed above. 2) Replication by template-directed formation of complementary strands (Fig. 9.5). Attempts to induce the replication of synthetic RNA oligonucleotides have been partly successful [38]. More on this aspect will follow in the next section.

9.10. Selectivity in Biopolymer Synthesis As we have seen in the preceding sections, the synthesis of biopolymers is a highly selective process. For the discussion of prebiotic evolution that will follow, it is perhaps worthwhile to briefly summarize at what levels of synthesis and by which criteria the molecular complexity and selectivity that we witness, develops and evolves. The Monomer Level. Assuming a prebiotic solution of free monomers, the first criterion to be considered is evidently the composition. Do we, from the beginning, have all 20 amino acids present in approximately equal amounts, or does a selection exist due to differing and variable ease of synthesis? Then, there is the fundamental criterion of the chirality. Must we assume initial mixtures that are racemic, containing both l- and d-amino acids in equal quantities, or is there already, from the start, an enantiomeric imbalance? The basic question of absolute enantioselectivity will continue to be at the center of our attention.

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It is interesting to note that the criteria of composition and chirality are probably not independent of each other, already at the monomer level. As has recently been pointed out [39], in autotrophic biological organisms, all synthetic pathways for amino acids, except that for glutamic acid, involve a transamination reaction with a glutamate donor. If these reactions conserve chirality, as may be assumed, then the homochirality of glutamate generates 18 amino acids of the same relative handedness (one amino acid, glycine, is achiral). Admittedly, this observation concerns biological, enzyme-catalyzed synthetic pathways. And as we have already mentioned (Sect. 9.8), it is questionable to speculate on early prebiotic reactions solely from the viewpoint of highly evolved mechanisms in living organisms. The danger of
back extrapolation may be, that one tends to attribute to early prebiotic processes an exaggerated degree of efficiency and selectivity. On the other hand, it is not only tempting, but certainly also justifiable, to consider prebiotic developments from the basis of biological evolution. Could not out of initially crude prebiotic, surfacecatalyzed amino acid syntheses, kinetic schemes have gradually emerged that, later on, acquired a central role in the metabolism of living organisms? The Oligomer Level. By
oligomer, we designate a strand of at least two monomers. The question is, from what number of monomers on, does an oligomer become a
polymer. From the point of view of enantiospecificity, this limit has perhaps been reached when the oligomer starts to act as an efficient autocatalytic template for its own synthesis. As we have seen in Sect. 9.8, Tll is a
polymer that catalyzes its own synthesis from the
oligomers El and Nl. While, in oligopeptides, we are primarily concerned with homochiral chain elongation, in oligonucleotides there occurs the additional functionality of complementary strand formation and replication. Probably mainly on philosophical grounds, one tends to consider existing natural biopolymers to represent optimal molecular architecture that cannot be surpassed in the laboratory. In this context, the synthesis of chemically modified RNA oligonucleotides and the investigation of their base-pairing properties are of particular interest. A nucleotide of RNA consists of a purine or pyrimidine base linked to a pentofuranose sugar, phosphorylated on one of the hydroxy groups of the sugar. In the polymer strand, the individual nucleotides are attached to each other by phosphodiester junctions (Fig. 9.5, a). Such a structure evidently is of great enantioselective specificity that is largely determined by the steric properties of the sugar. This should correspondingly also influence the base-pairing and replication properties. Assuming that the natural RNA structure originated through a combinatorial process with respect to the assembly of sugars and nucleic acid bases, the following investigations by Eschenmoser and co-workers are particularly instructive: The Watson–Crick base-pairing properties of oligonucleotides were

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compared, in which the natural pentofuranosyl ribose sugar moiety was alternatively synthetically replaced by hexopyranoses, pentopyranoses, and tetrofuranoses [40] [41]. The formation of base pairs by these derivatives with complementary oligonucleotide strands (Fig. 9.5, b) were quantitatively investigated in detail. It was, for instance, found that synthetic pentopyranosylribose RNA (p-RNA) was not only a much stronger pairing system than natural RNA, but also a more selective one with respect to pairing modes. In particular, the processes studied were highly chiroselective, opening pathways to largely homochiral p-RNA strands, starting from d- and l-ribose-derived diastereoisomeric tetramers. However, as also pointed out by the above-cited authors, the optimization of a variety of biochemical functions, not maximization of base-pairing strength alone, was certainly determinant for the natural selection of RNA. Furthermore, in the study of prebiotic molecular evolution, a central issue is the capability of nonenzymatic autocatalytic replication. High basepairing strength may facilitate the selective recognition of template sequences, but at the same time inhibit the production of new sequences. Another imaginative use of organic synthesis to study the transmission of enantioselectivity is the preparation of peptide nucleic acids. A peptide nucleic acid (PNA) is an achiral DNA (or RNA) mimic with an amide, pseudo-peptide backbone, structurally homomorphous with the deoxyribose-phosphate backbone. PNA consists of N-(2-aminoethyl)glycine units to which the same nucleobases are attached as in DNA. PNA oligomers form double helices with complementary DNA (or RNA) through Watson–Crick base pairing [42]:

It was then discovered that the (deoxy)ribose phosphate backbone of one of the strands is not an essential requirement for the formation of double-helical DNA-like structures. Two complementary PNA strands can hybridize to one another to form a helical duplex. To induce the chirality, it suffices to attach to the terminal backbone amide group a chiral l- or d-lysine carboxamide [43]. It was indeed found that the CD spectra (see Sect. 1.2 and 3.3) of the ll complexes are mirror images of those of the corresponding dd complexes. The amplitude of the CD spectra is comparable to that of the corresponding DNA–DNA double helices, confirming the assumption of prevalent doublehelical conformation. It was furthermore concluded that the association of two PNA oligomers begins with a fast base-pairing step resulting in a 1 : 1 racemic mixture of right- and left-handed double helices, followed by a relatively slow inversion of some of the helices to conform to the chirality of the terminal lysine residue. Similar enantioselection in a PNA–PNA duplex, in order to approach

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homochirality, may also be induced by other modifications, such as the replacement of the two N-terminus bases in only one of the strands of a double decamer by the corresponding d-deoxyribonucleotides [44]. These are striking examples of how local chirality in a single oligomer strand may, via Watson– Crick base pairing, get transmitted to a second, complementary strand. A next question would be, how far such local chiral influences may reach within a pair of longer polymers. The Polymer Level. Proteins are made up of 20 different amino acids. As an example, we consider a polypeptide containing 40 monomers, namely each amino acid once in the l- and in the d-form. (One amino acid, glycine, is indeed achiral, but, in the present context, we pretend that all monomers are chiral.) The question we pose is, how many distinguishable monomer combinations, or different sequences, of this polypeptide exist. It is obviously an elementary permutation problem, and the answer is 40!, which is equal to the astronomical number of 8.159…  1047. Now suppose we make the 40-mer homochiral, rendering the two amino acids of a given kind indistinguishable. The number of different distinguishable sequences will reduce to (40!)/(220), which gives 7.781…  1041 and is obviously still an astronomical number. What we have just considered here is admittedly a quite artificial situation, but the point we wish to make is that the condition of homochirality for the monomers does not drastically alter the permutational multiplicity of the polymer. As we already have concluded in discussing oligomers, the main influence of homochirality in biopolymers consists in imposing stringent steric selectivity. And only this steric selectivity can assure unambiguous sequence specificity. If proteins contained land d-amino acids in random fashion, it is hard to imagine how the specific structures and reactions of biological systems could ever occur and maintain themselves. Biochemical selectivity is essentially based on information recorded in terms of particular sequences of monomers, under the condition of strict enantioselectivity. In conclusion, the autocatalytic amplification of homochirality appears to be a sine qua non for biological organization. The selectivity of biopolymers depends to a large extent on H-bonds. HBonds stabilize both the secondary and tertiary structure of proteins. The Watson–Crick pairing of polynucleotides, and consequently the genetic code, is based on the selectivity of the two H-bonds between thymine and adenine in DNA (uracil and adenine in RNA), and on the three H-bonds between guanine and cytosine. In absence of the homochiral structure of the polymer, the longrange order induced by the H-bonded base pairs could not exist. Without homochirality, the number of base-pair mismatches would be expected to increase dramatically.

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9.11. Homochiral Organic Crystals Pasteurs discovery that a racemic mixture of sodium ammonium tartrate crystallizes as a conglomerate of enantiomorphous, homochiral crystals marked the beginning of stereochemistry (Sect. 1.3). The spontaneous resolution of a 1 : 1 mixture of molecular enantiomers by crystallization appears, in some sense, as a miraculous phenomenon. Why should some compounds crystallize from a racemate as an enantiomorphous conglomerate, while most others form racemic compounds? Indeed, racemic compounds, representing ca. 90% of all cases known today, are favored by entropy (Sect. 7.1). To obtain an enantiomorphous conglomerate, the particular intermolecular interactions in the crystal lattice must override this relative disadvantage. The exact calculation of molecular crystal lattice energies is a very demanding task of numeric quantum chemistry. However, during the last decades, for practical purposes, semiempirical procedures have been developed within the frame of classical mechanics. With present-day computer facilities, such molecular mechanics calculations can be quite easily handled for large systems, in particular, also where periodic boundary conditions apply. Even if the results of such calculations are not of the highest accuracy, they reveal themselves useful for systematic comparisons and in the prediction of trends. As an example, we consider the computational investigation of the crystals of some organic molecules of pharmacological interest [45]. In view of the necessity to prepare enantiomerically pure drugs of given chirality, the prospect of resolving enantiomers by crystallization is particularly interesting, as it may save much effort and investment in asymmetric syntheses. The point of departure in these calculations is the experimentally determined crystal structure both of the racemic compound and of a pure enantiomer. To refine the experimental data, the lattice energy is minimized with respect to the distances between the atoms belonging to different molecules in the unit cell. In those frequent cases where the racemic compound is formed, the (negative) lattice energy, E(rac), will tend to be lower than that of the pure enantiomer, E(enantio): DE(lattice) = E(rac)  E(enantio) < 0. But the computed data may give useful clues as to when the difference could become positive, possibly leading towards thermodynamic stability of an enantiomorphous conglomerate. As we had seen in Sect. 7.1, the free-energy term deriving from the mixing entropy favors the racemic compound. The lattice energy of the pure enantiomer must at least compensate for it to allow the conglomerate to approach thermodynamic stability.

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In these calculations, the lattice energy may be considered as the sum of essentially three kinds of contributions. These will be termed
van der Waals (vdW),
electrostatic (elstat) and
hydrogen bond (H): E(lattice) = EvdW + Eelstat + EH The first term takes into account nonbonded interactions between individual molecules of the kind described in Sect. 9.1; the second term represents additional electrostatic interactions in the lattice between free (ionic) or partial charges (due to high molecular polarity); and the third term stands for Hbonded interactions. This classification is, of course, a matter of appreciation and rests on technical considerations that cannot be discussed here in more detail. The interesting conclusions are [45] that, upon lattice formation, the van der Waals interactions play a key role in the chiral discrimination, with preference for the racemic compound. Where the lattice is formed of salts of the chiral organic compounds, the additional Coulomb interactions between the charges diminish in a relative sense the chiral discrimination, thereby favoring the formation of conglomerates. It has been found, for instance, that the amino acids alanine, leucine, and tryptophane crystallize as racemic compounds, whereas their benzenesulfonate salts form conglomerates [46]. As has also been predicted by calculations [47], some chiral amines can be transformed from racemic compounds into conglomerates when they are converted into salts with an achiral acid. Finally, we remember that sodium ammonium tartrate crystallizes as a conglomerate, while tartaric acid itself forms a racemic compound. A number of salts of chiral primary amines with achiral monocarboxylic acids crystallize as conglomerates [48]. It is found that the lattice structure of these crystals displays characteristic H-bonded columns around a twofold screw axis. The crystal structure is strongly influenced by inter-columnar van der Waals interactions. Depending on the presence, or absence, of these columns, and on the van der Waals interactions between them, either the racemic compound is favored, or the conglomerate structure is stabilized. In view of the possible importance of photochemical processes in prebiotic evolution, we also briefly consider the asymmetric induction in crystalline state photochemical reactions. An achiral, photoreactive carboxylic acid (or amine) and a chiral, optically pure amine (or acid) are joined to form a salt. The enantiomerically uniform substrate forces the photoreactive compound to crystallize in a chiral space group. The ensuing photochemical transformations within the crystal should be predominantly enantiospecific, leading to an enantiomeric excess of the chiral photoproduct [49].

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9.12. How Many Ways Lead to Rome? Before we turn to questions of prebiotic evolution, it seems appropriate to summarize some of our observations on the induction and amplification of chirality. It has been shown that, from a quantum-mechanical point of view, a molecular system under the influence of random perturbations may be stabilized in a (nonstationary) state of lower than maximum possible symmetry. In other words, a material system may become chiral under such conditions and by the influence of such external perturbations [50]. Elemental atoms or achiral molecules of relatively high symmetry may crystallize in chiral space groups. One finds that, for electronic reasons, there then exist two energy minima corresponding to enantiomorphous chiral structures. An achiral lattice of higher symmetry would in these cases be unstable (Sect. 7.1). But then, evidently, the existence of both enantiomorphous structures is equally probable, unless there is an additional, parity-breaking influence giving preference to a given chirality. The generation and propagation of enantiomorphous chiral structures appears to be relatively frequent. However, the systematic amplification and maintenance of an enantiomeric excess of definite handedness is due to particular influences. In the preceding sections, we have considered various further mechanisms to induce and amplify chirality. For instance, we have seen that chirality may arise as a local aspect, or manifestation, of an achiral entity: an achiral crystal can, due to the atomistic structure of its lattice, display surfaces that are chiral. But the achirality of the crystal as a whole is preserved by the fact that to every such surface there also exists another, enantiomorphous surface, related to the first one by a plane of symmetry or point of inversion inside the crystal (Sect. 7.3 and 9.3). Chiral surfaces may lead to enantiospecific adsorption processes, thereby enhancing and amplifying the local chirality. Racemic, and, therefore, achiral fluids, composed of equal quantities of enantiomers, can under given conditions spontaneously crystallize as conglomerates. A conglomerate crystal displays a high degree of chirality, the molecules in the unit cell of the crystal being homochiral. However, to every crystal of given handedness, and in the absence of any parity-breaking influences, the corresponding enantiomorphous crystal should form with equal probability and overall abundance (Sect. 7.1 and 9.11). Molecular chirality at a smaller scale is observed to induce chirality at a larger scale in particular achiral media. An achiral nematic liquid crystal can be made cholesteric by the addition of only a very small amount of chiral solute molecules (Sect. 7.2). On the other hand, chirality may be amplified by achiral surroundings and achiral influences. An achiral phase boundary or surface can favor the formation of homochiral aggregates and the synthesis of homochiral oligomers (Sect. 9.2 and 9.7). An amplification of enantiomeric excess may be

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induced by an achiral chromatographic phase (Sect. 9.5). And remarkably, there exists the possibility that the enantiomeric excess of the product of an asymmetric chemical reaction gets amplified with respect to that of the chiral catalyst by the additional influence of an achiral form of the catalyst (Sect. 9.6). A particularly important promotion of homochirality, from our point of view, occurs via polymerization of oligomers. Such reactions may display the striking effects of chiral autocatalysis. We have encountered the fusion of two oligopeptides to give a homochiral polymer in a reaction that is autocatalyzed by the product (Sect. 9.8). And we have seen that a homochiral oligonucleotide promotes its chirality by acting as template for the formation of the homochiral complementary strand (Sect. 9.10). We conclude that there are many ways by which chiral structures can originate, and chirality subsequently is amplified. As already stated, both enantiomorphous forms usually occur with practically equal probability. Where an enantiomeric excess of definite chirality develops, the preference for a given antipode over the other must be a consequence of the initial conditions of the process. This leads us to look for anterior parity-breaking mechanisms. Ultimately, it raises the question of how the universal violation of parity influences molecular processes (Sect. 2.2, 2.7 – 2.9, and 6.4). The highly selective preference of l- over d-amino acids, and of d- over l-sugars throughout the realm of biological chemistry, remains at the center of our attention (Sect. 1.1). How this stringent selectivity has developed, is still an open problem, which, at present, we are unable to solve. But after our recent excursion through the world of chirality, we are now better equipped to discuss the question.

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Asymmetric Autocatalysis and its Application to Chiral Discrimination, Chirality 2002, 14, 548, [29] I. Weissbuch, G. Bolbach, L. Leiserowitz, M. Lahav,
Chiral Amplification of Oligopeptides via Polymerization in Two-Dimensional Crystallites on Water, Orig. Life Evol. Biosph. 2004, 34, 79. [30] M. Blocher, T. Hitz, P. L. Luisi,
Stereoselectivity in the Oligomerization of Racemic Tryptophan-N-Carboxyanhydride (NCA-Trp) as Determined by Isotope Labeling and Mass Spectrometry, Helv. Chim. Acta 2001, 84, 842. [31] M. John, J.-P. Briand, M. Granger-Schnarr, M. Schnarr,
Two Pairs of Oppositely Charged Amino Acids from Jun and Fos Confer Heterodimerization to GCN4 Leucine Zipper, J. Biol. Chem. 1994, 269, 16247. [32] D. H. Lee, J. R. Granja, J. A. Martinez, K. Severin, M. R. Ghadiri,
A Self-Replicating Peptide, Nature 1996, 382, 525.

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[33] A. Saghatelian, Y. Yokobayashi, K. Soltani, M. R. Ghadiri,
A Chiroselective Peptide Replicator, Nature 2001, 409, 797. [34] J. P. Ferris, A. R. Hill, Jr., R. Liu, L. E. Orgel,
Synthesis of Long Prebiotic Oligomers on Mineral Surfaces, Nature 1996, 381, 59. [35] P. C. Joshi, S. Pitsch, J. P. Ferris,
Homochiral Selection in Montmorillonite-Catalyzed and Uncatalyzed Prebiotic Synthesis of RNA, Chem. Commun. 2000, 2497. [36] J. P. Ferris, P. C. Joshi, K.-J. Wang, S. Miyakawa, W. Huang,
Catalysis in Prebiotic Chemistry: Application to the Synthesis of RNA Oligomers, Adv. Space Res. 2004, 33, 100. [37] G. F. Joyce, L. E. Orgel,
Prospects for Understanding the Origin of the RNA World, Cold Spring Harbor Monograph Series 1999, 37 (RNA World, 2nd edn.), 49. [38] L. E. Orgel,
Molecular Replication, Nature 1992, 358, 203. [39] H. J. Morowitz,
The Conservation of Homochirality and the Prebiotic Synthesis of Amino Acids, 2001 (http://www.santafe.edu/sfi/publications/workingpapers/01-03-017.pdf). [40] A. Eschenmoser,
Chemical Etiology of Nucleic Acid Structure, Science 1999, 284, 2118. [41] M. Bolli, R. Micura, A. Eschenmoser,
Pyranosyl-RNA: Chiroselective Self-Assembly of Base Sequences by Ligative Oligomerization of Tetra Nucleotide-2’,3’- Cyclophosphates (with a Commentary Concerning the Origin of Biomolecular Homochirality), Chem. Biol. 1997, 4, 309. [42] M. Egholm, O. Buchardt, L. Christensen, C. Behrens, S. M. Freier, D. A. Driver, R. H. Berg, S. K. Kim, B. Nordn, P. E. Nielsen,
PNA Hybridizes to Complementary Oligonucleotides Obeying the Watson-Crick Hydrogen-Bonding Rules, Nature 1993, 365, 566. [43] P. Wittung, P. E. Nielsen, O. Buchardt, M. Egholm, B. Nordn,
DNA-Like Double Helix Formed by Peptide Nucleic Acid, Nature 1994, 368, 561. [44] I. A. Kozlov, L. E. Orgel, P. E. Nielsen,
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Relationship Between Physical Properties and Crystal Structures of Chiral Drugs, J. Pharm. Sci. 1997, 86, 1073. [47] H. Kimoto, K. Saigo, M. Hasegawa,
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Crystal Structures of the Salts of Chiral Primary Amines with Achiral Carboxylic Acids: Recognition of the CommonlyOccurring Supramolecular Assemblies of Hydrogen-Bond Networks and their Role in the Formation of Conglomerates, J. Am. Chem. Soc. 1996, 118, 3441. [49] J. R. Scheffer,
In the Footsteps of Pasteur: Asymmetric Induction in the Photochemistry of Crystalline Ammonium Carboxylate Salts, Can. J. Chem. 2001, 79, 349. [50] M. Simonius,
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10. Prebiotic Evolution and Beyond
Tout probl
me profane un myst
re;  son tour, le probl
me est profan par sa solution (
Every problem profanes a mystery; in turn, the problem is profaned by its solution) E. M. Cioran (1911 – 1995)

10.1. What Is Life? Under the title
What is Life? – The Physical Aspect of the Living Cell, Erwin Schrçdinger (1887 – 1961), one of the founders of quantum mechanics and father of the fundamental equation named after him, gave in 1943 a series of lectures at Trinity College in Dublin. Subsequently published as a book [1], this scientific essay concludes that the laws of biology are not alien to the laws of physics. Schrçdinger focuses mainly on a fundamental aspect of the living cell, the mechanism of replication. Already in the first chapter of the book he states:
… Let me anticipate… that the most essential part of a living cell – the chromosome fibre – may suitably be called an aperiodic crystal. In physics we have dealt hitherto only with periodic crystals …Yet, compared with the aperiodic crystal, they are rather plain and dull. The difference in structure is of the same kind as that between an ordinary wallpaper in which the same pattern is repeated again and again in regular periodicity and a masterpiece of embroidery, say a Raphael tapestry, which shows no dull repetition, but an elaborate, coherent, meaningful design traced by the great master. The vision of the chromosome as an aperiodic crystal was indeed quite strikingly confirmed ten years later in the structure of DNA [2]. And does not the comparison with the embroidery, in some sense, anticipate the chirality of the structure? Do not the multicolored, systematically knotted threads carrying the motif wind themselves in predetermined handedness around the underlying fabric [3]? To support and promote its capacity to replicate, the living cell must evidently also possess a metabolism. The cell has to obtain the free energy necessary to operate far away from thermodynamic equilibrium. The genetic apparatus has to be protected from external disturbances and from decay to be able to function with optimal efficiency. These diverse roles of protection and optimization are mainly played by proteins which both form structural elements, such as membranes and fibers, and perform chemical tasks in the role of enzymes. As is today well known, the prescription for the synthesis of this necessarily great variety of proteins is encoded in DNA. The reading of the encoded messages and the protein synthesis is then carried out by RNA. On Chiralty and the Universal Asymmetry. Georges H. Wagnière. © 2007 Verlag Helvetica Chimica Acta, Zürich ISBN: 978-3-906390-38-3

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Summarily stated, we find the following attribution of roles and division of labor within the living cell: DNA: RNA: Proteins:

Information carrier and replication. Information transfer and protein synthesis. Structure and metabolism.

Obviously, there is complete interdependence between these various actors. How then may the prebiotic molecular evolution have proceeded? Did these very complex molecules appear simultaneously? Or have they developed successively? And if so, has one kind of molecule taken precedence over the others? One easily recognizes that the link between replication and metabolism is mainly represented by RNA. To perform the complex tasks which this entails, RNA must be versatile and be able to occur in different forms. Indeed, the main types of RNA and their respective functions are [4]: mRNA (messenger): tRNA (transfer):

rRNA (ribosomal):

carries the genetic message copied from DNA to the ribosomes, where the proteins are synthesized; consists of shorter chains, which participate in the reading of the genetic code, correctly placing each amino acid in its sequence in the protein; provides both structural material and a catalytic center for peptide bond formation.

There are numerous additional forms of RNA performing a variety of supplementary cellular functions. Furthermore, and remarkably, the genomes of many viruses consist of RNA. Unlike DNA, which exists largely as double helices, RNA occurs as single chains, capable of folding into complex forms containing many bulges and loops. In 1982, it was discovered that a form of RNA existed with an active site capable of inducing its own splicing [5]. This catalytic form of RNA was named ribozyme. Previously, all biological catalysts known, the enzymes, were proteins. Based on the newly established properties of this RNA enzyme, a detailed model was then developed for the template-dependent synthesis, or partial self-replication, of RNA by an RNA polymerase, itself made of RNA [6]. Far from being exceptions, many ribozymes have since been discovered and their function investigated [7]. While in proteins the catalytic activity generally depends on the amino acid side chains, it must be concluded on stereochemical grounds that, in the base-paired secondary structures of ribozymes, these functions mainly involve the outside-directed ribose sugars and the phosphates of the backbone.

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Consequently, the remarkable properties of RNA have gradually lent support to the idea that a both information-encoding and catalytically active precursor of RNA may have played a key role in prebiotic evolution.

The RNA World

If one accepts the assumption that there was a prebiotic pool of randomsequence RNAs, including a replicase ribozyme of the order of 40 nucleotides and replicating with 90% fidelity, then one may conceive of ways in which RNAbased evolution might have started [8]. Based on the idea of a relatively primitive self-replicating polymer, statistical and kinetic models of evolutionary processes had already previously been conceived and explored mathematically [9] [10]. The results of these investigations are not only highly instructive, they lend support to the idea of the RNAWorld, and they delineate how further research in this direction may want to proceed. The model of the RNA World is essentially based on three assumptions: 1) At some time in prebiotic evolution, genetic continuity was ensured by the replication of RNA. 2) Watson–Crick base pairing was the key to replication. 3) Genetically encoded proteins were not involved as catalysts. The crucial question is, of course, how a prebiotic pool of (randomsequence) RNAs, including a replicase ribozyme could have originated. How and where was the necessary catalytic precursor to deliver the needed supply of activated b-d-nucleotides? Exploring in thoughts the primitive world, the perspectives are indeed vague. For instance, we are led to the possibilities considered in Sect. 9.9, and supported by recent experiments: biopolymer generation on mineral surfaces. But in order to get to the point where ribozymes may take over the catalytic action, the initially quite random processes must obviously have gone on for a very long time. Within the model of the RNA World, the amino acids and their polymerization to polypeptides should appear in a second stage of development [10]. It is then assumed that the catalytic properties of aggregates of the proto-ribozymes also act on amino acids, and the peptides so formed in turn surround and protect the ribozymes, thereby reducing external influences leading to destructive mutations. In this way, the development of proto-ribosomes may be envisaged. For a general outlook of how the further evolution may have led to information storage in DNA and to the genetic code, see the above-cited article by Kuhn [10].

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Metabolism First

There may be another point of view on prebiotic evolution, namely, that it started by metabolism rather than with precise replication [11] [12]. The basic ideas are essentially the following: Cells came first, enzymes second, genes much later. In more detail: 1) A cell is a confined volume of fluid containing small organic molecules such as amino acids. The cells interior surface contains active sites on which the monomers can adsorb (see, e.g., Sect. 9.2, 9.3, and 9.7). Monomers adsorbed onto neighboring sites will link together to form polymers. Among these polymers, some may turn out to have (auto)catalytic properties. Thus, a primitive form of metabolism will emerge. 2) Cells may divide, forming autonomous new cells. There is, of course, no exact replication, but merely reproduction. 3) In the course of time, independently formed polynucleotides (see Sect. 9.9) invaded the already existing cells and interfered with the metabolism. In this sense, the polynucleotides at first acted as parasites. Gradually, they were incorporated in the cells metabolic chemistry and evolved into the cells genetic apparatus. It lies neither within our present possibilities, nor is it our intention to try to give preference to either of the two models sketched here. We merely note that one model predicts
polynucleotides first – amino acids second, the other the reverse. To comparatively consider these hypotheses further in more detail lies beyond our range and aim. But in discussing the emergence of prebiotic enantioselectivity of definite handedness, we must evidently take into account both classes of molecules, (poly)nucleotides and (poly)amino acids.

10.2. The Primary Origin of Absolute Enantiomeric Excess As we have previously noted (Sect. 2.1 – 2.8 and 5.4), the fundamental cause for the inequivalence of image and mirror image in the universe, is the violation of parity induced by the weak forces. In molecules, this manifests itself by the weak neutral currents between electrons and nucleons, with the consequence that molecular enantiomers should have slightly different energies. The basic question for us here is, to what extent such small energy differences might directly influence the dynamic and chemical properties of chiral molecules. Will the kinetic behavior of one antipode measurably vary in comparison with the other? It is indeed conceivable that this asymmetry may lead to an enantiomeric excess of one antipode with respect to the other. Initially, the excess might be

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very small, but by autocatalytic kinetic amplification (see Sect. 9.6), one chiral species could ultimately dominate over the enantiomer [13]. In trying to understand the origin of biological homochirality (Sect. 1.1), namely the practically exclusive occurrence of l-amino acids and of d-(deoxy-)ribonucleotides, the direct influence of the weak forces is the primary cause to be considered. As already indicated in Sect. 2.8, quite extensive quantum-chemical studies have been undertaken on the energy difference due to parity violation between enantiomers of various chiral molecules, ranging from four to about a dozen atoms or more. A variety of quantum-chemical methods and computational procedures have been applied to this problem [14]. A series of calculations on a range of important biomolecules, carried out over a period of several years, reveal l-amino acids to be more stable than d-amino acids, and d-sugars to be more stable than l-sugars, in agreement with biochemical expectations [15] [16]. More recent investigations question the reliability of some of these conclusions, however [17]. When it comes to interpreting the significance of such predictions for the definite homochirality of the biopolymers, additional and far-reaching questions arise. If heterogeneous chiral catalysts, such as mineral surfaces, come into play (as suggested in Sect. 7.3, 9.3, 9.6, and 9.9), the effect of the weak forces on the catalyst will also directly influence the relative energy of the transition states, possibly in opposite sense, and correspondingly modify the expected kinetics of the polymerization. Such additional influences could, in particular, become significant with catalysts that contain heavier elements (see Sect. 2.8). To draw definite conclusions, quite extensive additional investigations seem, therefore, necessary. The energy difference due to parity violation (PVED) should, in principle, also influence the relative abundance of enantiomorphous crystals. An example which has attracted much attention is that of quartz (see Sect. 6.4). Presently existing quartz crystals in nature are the result of crystallization and recrystallization processes that have been going on for millions of years. It, therefore, would seem plausible that PVED effects could have influenced the relative natural abundance of the enentiomorphous forms. As already mentioned previously, the result of the most recent investigations tend to cast doubt on the statistical significance of measured small enantiomeric excesses [18 – 20] (see also Sect. 6.4). In Sect. 7.1, we had considered sodium chlorate and sodium bromate, salts that in solution are achiral, but which form chiral crystals. In the absence of any chiral bias, both enantiomorphous crystal forms should occur in equal amounts. The statistical evaluation of the experiments performed hitherto do not indicate any excess of one form over the other which could be attributed to PVED effects. By enantioselective seeding or stirring, however, symmetry breaking

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Fig. 10.1. Spontaneous resolution under racemizing conditions. Above: in the liquid phase, there is rapid equilibration between enantiomeric forms. Below: crystallization leads to conglomerates. If one assumes the crystallization to [ll] to be more rapid than to [dd], implying for the reaction constants kl+ > kd+, kl
< kd
, then [ll] may grow at the expense of [dd]. This scheme is a simplification. Homochiral nucleation is quite probably enantioselectively autocatalytic, making the kinetics more complex (M. Calvin, Chemical Evolution, Clarendon Press, Oxford, 1969; see also Sect. 9.6).

occurs during crystallization, and a large enantiomorphous excess may be generated. The crystallization from a racemic mixture may, under particular conditions, result in the spontaneous separation of only one of the enantiomers. In the liquid phase, at a higher temperature, rapid equilibration between enantiomers is here assumed to take place (Fig. 10.1). Crystallization should, furthermore, lead to conglomerates of which one enantiomorphous form is assumed to be favored with respect to the other. Complete crystallization to the preferred conglomerate may then gradually occur. The fixation and amplification of chirality thereby achieved should make such experiments suitable for the detection of PVED effects. In this respect, the experimental results hitherto obtained appear to be negative, however [19]. It is, of course, far from obvious if experiments that can be performed in the laboratory within a time of days, weeks, months, or even a few years, are at all capable of giving conclusive evidence on the long-range direct influence of the weak forces. One must bear in mind that, no matter what mechanism of selection between enantiomers ends up by being locally dominant, the weak forces are always operative, and it is indeed difficult to assess unequivocally their cumulative effect over the prebiotic time span of several hundred million years or more, and within complex spatio-temporal networks of billions of reaction steps. If, indeed, the direct influence of the weak forces is the decisive effect in determining the absolute handedness of the biomolecules, then the same handedness should be expected to also exist in eventual similar forms of life in other parts of the universe.

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10.3. Secondary Sources of Absolute Enantioselectivity We know that the manifestations of chirality in the universe, representing local or universal parity-breaking phenomena, are numerous and varied. They range from the structure and dynamics of galaxies, from the evolution of stars and of planets, down to the microstructure of the celestial bodies of differing sizes and compositions. They extend from cosmic dimensions down to those of subatomic entities. All the manifestations of chirality which originally are derivable from parity-violating forces may, in turn, become secondary sources of chirality, and so on, down a hierarchy that increasingly reflects particular, local conditions (see also Sect. 6.4). Returning to the question of prebiotic evolution and the origin of biochemical homochirality of definite handedness on Earth, we notice that there are several possible secondary sources of chirality. One of these sources may be radioactive decay or other origins of polarized elementary particles. As previously mentioned (Sect. 2.2), the electrons emitted by the b-decay of 60Co are polarized, in the sense that the electron spin is oriented antiparallel to the particle momentum. If such an electron impinges on matter, its kinetic energy will be partially transformed into light which will be (right) circularly polarized. According to the Vester–Ulbricht hypothesis, this so-called Bremsstrahlung (radiation due to deceleration) could then be absorbed by organic substrates, thereby inducing enantioselective synthetic or degradative reactions [21] [22]. Polarized elementary particles, carrying a spin that has a definite orientation with respect to the particles linear momentum, are chiral, and may, therefore, be characterized by a helicity (Sect. 2.4 and 4.1). In search of eventual enantioselective influences on molecules, several investigations have been performed with natural emitters of b-particles, such as with the isotopes 90Sr (strontium), 32 P (phosphorus), and 14C (carbon). Polarized particles generated artificially by accelerators, including electrons, protons, muons, and positrons, should, in principle, also exert chiroselective influences. These particles may interact with molecular matter by a variety of mechanisms, which cannot be reviewed here in any further detail. Yet many experiments performed in search of molecular enantiomeric excesses due to polarized elementary particles have, until now, led to negative or inconclusive results [19]. The next possible (secondary) source of chirality to be considered is electromagnetic radiation, of extraterrestrial origin. As has been discussed in Sect. 3.1 and 3.2, electromagnetic forces do not violate parity, but there are forms of electromagnetic radiation, such as circularly polarized light, that are chiral. Where circular or elliptic polarization comes into play, we may encounter circular dichroism and enantiomeric excesses produced by circular differential photochemistry (Sect. 1.5 and 3.3). In addition, we know that the concomitant influence of light of arbitrary polarization and of a magnetic field parallel or

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antiparallel to the direction of light propagation also represents a chiral influence which may lead to magnetochiral dichroism and magnetochiral photochemistry (Sect. 3.5 and 3.8). These aspects will be further treated in Sect. 10.5. Then, we must also consider chiral material influences. We have already encountered a variety of such mechanisms: chiral aggregation at phase boundaries, adsorption on chiral surfaces, chiral catalysis, induced chirality in liquid crystalline environments (Sect. 7.2, 9.2, 9.3, and 9.6). We have mentioned the effect of mechanical stirring and the chirality of vortices (Sect. 7.1 and 7.5). Such effects have in common that the induced chirality always reflects particular, local conditions, which originally derive from, but in general may not be directly related to, the universal parity-violating elementary forces (see Sect. 2.4, 2.8, 5.4, and 6.4).

10.4. Prebiotic Evolution: From the Deep Sea to the Hadean Beach The Solar System is ca. 4.6 billion years old. There is evidence that ca. 4.5 billion years ago (at minus 4.5 Giga years, or
4.5 Gyr), the proto-Earth was struck by another planet about the size of Mars. The impact spun and tilted the Earth, leading to the present day–night cycles and to the seasons. These properties were probably crucial for sustaining life in the earliest stages, and still are so to this day for making the Earth biologically habitable. By these initial cataclysmic events, enormous amounts of molten mantle were ejected into orbit, some of which coalesced to form the Moon. The oldest known rocks, the Acasta gneiss from northwest Canada, are dated about
4 Gyr. Until
3.8 Gyr, the Earth was subjected to very heavy meteorite bombardment, in particular, from Mars, caused by substantial material exchange between the nearby planets [23]. Geological evidence, in particular, what may be the oldest-known water-lain sediments, shows that life on Earth probably began before
3.8 Gyr. Filamentous microfossils are known in a
3.2 Gyr volcanogenic massive, deepwater sulfide deposit in Western Australia, implying that life must have existed in Archaean mid-ocean ridges. In rocks of the late-Archaean period, around
2.7 Gyr, the evidence for life becomes abundant, including molecular fossils of biological lipids. Oxygenic photosynthesis was then conceivably already occurring [23]. From such facts, it may be concluded that, in the very early microbial community, life was hot, ocean-bound, and chemotrophic. Accordingly, life would have begun in a hydrothermal setting, as proposed by the Oparin Ocean Scenario [11] [12] (see Sect. 10.1). The model assumes that the primitive oceans were shelter and repository for the great variety of chemicals produced in the

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primitive nonoxidizing atmosphere through the action of UV light and electric discharges, mainly on water vapor, carbon dioxide, and nitrogen. In this way, the primitive ocean became a
soup of energy-rich biochemicals [24]. The first organisms are considered to have been anaerobic, heterotrophic procaryotes, which later spawned photosynthetic and autotrophic sublines. The question is, of course, justified, if primary forms of life could plausibly have been extreme forms of life, or if, on the contrary, organisms living under extreme conditions have developed by secondary adaptation. The recent discovery of large submarine hydrothermal systems, resembling possible environments for prebiotic chemical evolution, has also incited laboratory experiments simulating such natural processes. For instance, it has been investigated, if amino acids could be synthesized under these conditions. Starting materials were simple, H-, C-, N-, O-containing compounds, such as CO2, CH4, H2, N2, NH3 in the presence of water and mineral buffers (pyrite (FeS2), pyrrhotite (FeS), magnetite (Fe3O4), fayalite (Fe2SiO4), and quartz). Experiments were carried out in the temperature range of 150 – 200 8C [25], and, as expected, the amino acids so obtained in relatively high yield were all racemic. By other experiments, modeling volcanic or hydrothermal settings in a CO and H2S atmosphere, and in the presence of an aqueous precipitate of the sulfides NiS and FeS as catalysts, it was demonstrated that at 100 8C amino acids could be activated to form di- and even tripeptides [26]. But no systematic trend towards homochirality could be detected there. At elevated temperatures, the tendency for amino acids to racemize increases. Racemates have a greater mixing entropy than homochiral systems, and, at a higher temperature, the mixing of enantiomers will correspondingly be favored. This suggests that deepsea hydrothermal conditions are indeed not optimal to promote homochirality. For such reasons, we now also consider hydrothermal surroundings that are less exclusively marine. Based on a series of laboratory experiments combined with computer simulations, it has recently been shown that between a dry phase containing photochemically generated nitrogen oxides NOx, and an aqueous phase with a steady input of a-amino acids, peptides could be continuously generated and elongated stepwise [27]. Such processes might have taken place on tidal beaches, requiring a buffered ocean, emerged land, and a nitrosating atmosphere. The experiments, which were carried out at ambient temperature (25 8C), tended to show that the products were essentially racemic, but that homochiral sequences (ll…, dd…) were kinetically favored over heterochiral ones (ld…, dl…). It then seems likely that the emerging, progressively homochiral peptides could quickly begin to act as catalysts for autoreplication. Quite obviously, however, if an absolute and nonlocal enantiomeric excess were to be induced, it would mainly have to derive from enantioselectivity in the steady input of the amino acids (see also Sect. 9.7 – 9.10, and 10.6).

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In conclusion, the question may be duly asked, if the hydrothermal conditions of the Deep Sea really fulfill the requirements for the emergence of biological complexity. It seems that high temperatures and possibly erratic hydrodynamic currents are not particularly conducive to generate homochirality and stable enantiomeric excesses, even on a local scale. If prebiotic evolution really started in the depths of the sea, it very soon also had to be continued at particular locations on the land, in direct contact with the atmosphere, and probably also with light. In What is Life, Schrçdinger, considering the role of metabolism, speaks of the necessity to feed on
negative entropy. Elementary thermodynamics teaches us that entropy reduction requires not heat, but work. We realize that work on a molecular scale may, for instance, be performed by electromagnetic radiation.

10.5. Enantiomeric Excess by Surface Geology and by Sunlight Accumulated recent evidence suggests that the more advanced stages of prebiotic evolution took place at the Earths surface, on beaches, at lagoons, in shallow lakes, at riversides. Aqueous solutions of a variety of simple organic molecules, among them amino acids, possibly also purine and pyrimidine bases [28], and carbohydrates [29], were washed onto sands and rocks where they initially became, in quite random fashion, partly adsorbed. The microcrystalline mineral surfaces imposed steric constraints on these, more or less tightly bound, molecules. As we have previously seen (Sect. 7.3 and 9.9), the adsorbing sites may, in general, have been chiral, and, therefore, enantioselective. But on an average, even taken within a rather restricted spatial domain, such interactions must have been pseudo-racemic (Sect. 6.4). From that point of view, it seems unlikely that conditions of surface geology alone could have generated stable enantiomeric excesses of given absolute handedness and of direct significance for any global development. Next, we, therefore, consider the direct influence of sunlight. The molecules, floating freely in diverse natural waters, or adsorbed on sands and rocks, have from the beginning been exposed to the light of the Sun, subjected to the dayand-night cycle and to seasonal changes. Consequently, it seems justified to envisage chiral photochemical effects induced by the incident sunlight. As we well remember (Sect. 1.5, 3.3, and 3.8), due to circular dichroism (CD), enantioselective synthesis and decomposition may be caused by circularly polarized light of particular handedness. A small percentage of the solar radiation reaching the Earths surface, mainly at twilight, is circularly polarized. But these polarizations will average to zero over the whole sky and over the diurnal cycle [30]. Alternative weak sources of circularly polarized light are either due to the Faraday effect in the atmosphere, induced by the Earths magnetic field [31], or

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due to particular phenomena on the Sun itself [32]. In summary, the resulting amount of circularly polarized light reaching the Earths surface does not appear to be really significant for enantioselective photochemistry. Furthermore, and as previously mentioned, the CD spectrum of any molecule, comprising both positive and negative signals, integrates to zero over the whole wavelength range, in accordance with the Kuhn–Condon sum rule. It may thus be concluded that the enantioselective photochemical influence on any chiral molecule of polychromatic
white light, containing all wavelengths in approximately equal amounts, should vanish or be close to zero [16]. Therefore, directly incident circularly polarized sunlight appears to be an unlikely source of prebiotic enantiomeric excess on Earth. As a further possibility for light-induced enantioselectivity, we consider the magnetochiral effect (Sect. 3.5 and 3.8) in the Earths magnetic field. The magnetochiral dichroism at a particular geographic location is proportional to the scalar product of the magnetic field vector and of the wave vector of the incident light. We remember that the effect is independent of the state of polarization of the electromagnetic radiation. As previously mentioned (Sect. 6.2), and depending on the location, the Earths magnetic field is only of the order of 0.5  10
4 T. It is not really constant in time, but fluctuates over periods of millennia. The polarity of the field has reversed many times since the Earths formation. But in spite of these changes, it may be assumed that the Earths magnetic dipole field axis on the average lies close to the Earths geographic rotation axis [33] (see also Sect. 6.2). Considering present-day magnetic field lines and the illumination of the Earth by the Sun (Fig. 10.2), we conclude that the magnetochiral dichroism vanishes in all geographic locations at sunset and sunrise, and that it does so similarly at all times near the equator. Under these conditions, the magnetic field is perpendicular to the light wave vector, and the scalar product of these two vectors is zero. As the reader may also immediately infer, the magnetochiral effect to the north of the magnetic equator should, at all seasons, be opposite to that to the south of it. Depending on location, an opposite enantiomeric excess might consequently have developed. On the other hand, as already at
4 Gyr the geological conditions at the Earths surface may have been quite different on the northern and southern hemisphere, the integrated influence of the magnetochiral effect could have been nonvanishing. The order of magnitude of the partial magnetochiral photoresolution was recently determined to be on the order of 10
6 T
1 [34] [35]. The degree of enrichment jDc/cj due to the terrestrial magnetochiral effect would consequently be ca. 10
11 to 10
10. In comparison, at thermal equilibrium, the same quantity due to the weak nuclear forces at ambient temperature would be on the order of only 10
17 to 10
14, depending on the kind of molecule (see Sect. 2.8 and 3.8). In conclusion, and irrespective of the relative magnitudes, the sign of an

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Fig. 10.2. a) Irradiation by the Sun (not to scale!) of the Earth on its orbit at the positions of June 21 (left) and December 21 (right). b) The Earths magnetic field (adapted from Meyers Neues Lexikon,
Erdmagnetismus, Bibliographisches Institut Mannheim, Meyers Lexikonverlag, 1979).

enantiomeric excess induced by the magnetochiral effect would be determined by local factors, while, for the weak forces, it must be universally the same. Prebiotic evolution on the surface of the Earth has taken place under very special conditions. The surface is protected by a dense atmosphere, yet subjected to the diurnal rhythm of light and darkness. The temperature scarcely ever reaches the boiling point of water, except at volcanic sites, and at many places it does not fall significantly below freezing. Photo-activation of molecular processes during the day has, over hundreds of millions of years, given way to partial equilibration at night. As a consequence, a rich variety of reactions, following a multitude of different kinetic paths, has been induced and sustained. An immense network of interdependent chemical processes has arisen, on microscopic scales at first, gradually amplifying to macroscopic complexity. There is a rich variety of meteorological conditions. In addition come further influences, such as those of the tides. Compared to other planetary bodies in the

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solar system, the Earth has a relatively large magnetic field. What role has this magnetic field played, beside in the magnetochiral effect just mentioned? In any case, if one considers the richness of dynamic natural phenomena at the Earths surface, the conditions on other celestial bodies of the solar system, be it on Mars or on the moons of Jupiter [36], seem very inhospitable, also in a historic perspective. On the other hand, it is difficult to discern at the Earths surface any single, dominant prebiotic mechanism apt to induce the enantiomeric excess necessary for the emergence of biological homochirality.

10.6. Prebiotic Evolution: Extraterrestrial Origins When, in everyday life, one speaks of meteoritic impacts, one tends to imagine singular, more or less destructive events. What modern research has established, and is yet not widely realized, is that matter from comets and asteroids has significantly contributed to shape the details of the surface of the Earth in early times. It has also been suggested that comets or carbonaceous asteroids have brought large amounts of organic matter to the primitive Earth, and thus possibly played an important role in prebiotic evolution. Of course, organic matter cannot survive the extremely high temperatures of 104 K or more, reached on impact of larger terrestrial bodies, which atomize the projectile and break all chemical bonds. On the other hand, meteors as small as 10
12 to 10
6 g are decelerated by the atmosphere more gently, and may deliver their organic matter intact. And there are exceptions, where much larger extraterrestrial objects which landed on Earth have been found to contain substantial amounts of organic matter. From the infall rate of meteoritic matter, the amount of such
soft-landed organic carbon compounds can be estimated to be at present roughly 300 tons per year. In the early days of prebiotic evolution, the influx rate of organic matter was much higher, possibly on the order of 105 tons per year [37 – 39]. The meteorite that struck Earth in 1969 near Murchison in Australia, called the
Murchison meteorite, is believed to be the remnant of a spent comet. In the fragments of this carbonaceous chondrite, a significant number of organic compounds of extraterrestrial origin have been identified, among them several a-amino acids. Analysis of their optical activity revealed enantiomeric excesses of up to 9% in favor of the l-form [30] [40] [41]. Similar findings have been made on the carbonaceous
Murray meteorite that landed in Kentucky in 1950. It has been estimated that on the order of 6  105 million tons of amino acids from such extraterrestrial sources arrived on Earth, during a period where the terrestrial conditions would have already allowed the persistence of the landed compounds [39]. If these amino acids all contained a significant excess of the l-

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form, then they might have initiated the evolution towards enantioselective lamino acid biochemistry on Earth. It is noteworthy that, beside amino acids, also sugar-related organic compounds have been detected in such meteoritic fragments. Analysis of water extracts indicate that extraterrestrial processes, including photolysis and formaldehyde chemistry, could account for the observed compounds [42]. If an excess of d-sugars over l-sugars were ascertained in those samples, this would be a further indication of the eventual prebiotic significance of such extraterrestrial seeding. These recent findings of course raise important questions: 1) Are the analyses carried out on meteoritic matter until now really sufficient to infer a global l-amino acid (and possible d-sugar) excess of extraterrestrial origin that might have uniformly influenced the course of prebiotic chemical evolution on Earth? 2) If the enantiomeric excess in organic matter of meteoritic origin really is a general phenomenon, then what were the extraterrestrial causes of this uniformity? We indeed assume that the first question may be answered in the affirmative, and suppose that a large amount of organic matter, such as amino acids and (or) sugars of definite enantiomeric excess, landed on Earth via meteoritic impact. As a next step, we must try to find out how these compounds were incorporated into the other, terrestrial prebiotic processes going on. For instance, did the synthesis of homochiral biopolymers directly start on the meteoritic material, or were the amino acids first washed out and carried elsewhere, possibly serving as starting materials for purely terrestrial reactions (as described in previous chapters), inducing an enantiomeric excess in otherwise racemic processes? On the other hand, we may not forget the existence of purely autochthonous geological sources of local enantiomeric excess, which might have started to compete with the chiral influences of extraterrestrial origin. The outcome of such competitions probably depended very much on local conditions. This leads to situations of great complexity, where clear-cut answers are hard to find. It makes unambiguous interpretations of the sources of enantioselective biochemical homochirality very difficult to arrive at. The exact cosmic origins of the myriads of smaller and larger extraterrestrial fragments which landed, and still land, on Earth are uncertain. How did the enantioselectivity in the organic compounds that were carried along arise? Ideas about eventual previously existing life on celestial bodies in not too distant solar, or extrasolar, regions are speculative at best, probably science fiction. What remains are: 1) weak forces, 2) circularly polarized light, and 3) the magnetochiral effect. First, the possible direct influence of the weak forces on

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molecular enantioselectivity has already been discussed previously, leading to uncertain conclusions (Sect. 10.2). Second, there are numerous sources of circularly polarized light in space [30], suggesting the possibility of circularly differential photochemistry based on natural optical activity (CD). But for reasons spelled out in the previous section, the irradiating light should not cover the whole spectrum; it would have to coincide in wavelength mainly with particular photochemically active molecular electronic transitions. Third, the magnetochiral effect might, in principle, also have played an enantioselective role in these extraterrestrial bodies and fragments. But in the presence of sizable intensities of circularly polarized radiation, the magnetic field has to be extremely high in order to make the magnetochiral effect competitive with natural optical activity. If an excess of d-sugars, notably of ribose, were to be found in meteoritic material, this would be of particular interest. As we remember, d-ribose is an important constituent of RNA (Sect. 9.9 and 9.10). This might then indicate that RNA was also among the earliest bioactive molecules to have occurred on Earth, and it would speak in favor of an RNAWorld, as opposed to an exclusive, initial Metabolism-First World. However, after what we have seen, it may safely be asked, if both
worlds did not start more or less simultaneously.

10.7. Prebiotic Evolution: Looking for Research Strategies The reader will by now have noticed that, in principle, there exist different possible prebiotic scenarios, and that none is entirely convincing. Unless the lamino acid/d-sugar homochirality can be directly traced back to the weak interactions, it may even prove impossible to unambiguously determine its origin. There is too much complexity involved, acting over too long a period of time. Nevertheless, all possible leads should be systematically followed. Thereby, new and unsuspected mechanisms may be discovered and the perception of the physico-chemical basis of life will be sharpened. If biological homochirality is due to influences of particular surroundings, and not to basic forces, then life of opposite handedness might in principle be possible. Life on Earth may have evolved as a competition between molecular entities of opposite handedness, the outcome having been due to influences that may be termed as
chance. On other celestial bodies – planets – with conditions comparable to those on Earth, life of opposite handedness consequently might have developed. Life has mainly blossomed at the surface of the Earth. Although the Deep Sea scenario and the Meteoritic Extraterrestrial Seeding scenario cannot simply be discarded, one rather obvious question must be raised: why should life not have originated where it is doubtlessly best sustained, namely, at or close to the

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Earths surface, directly exposed for hundreds of millions of years to the atmosphere and to the periodic effects of filtered sunlight? We have seen that there are many mechanisms and pathways leading to molecular homochirality (Sect. 9). The more there are such possibilities, the more difficult will it be to recognize the dominant ones, and the more elusive could it become to detect with certainty the origin of the absolute biochemical handedness. But even if an absolute and definite final answer remains unobtainable, such investigations are of foremost general scientific interest and, therefore, worth the undertaking. The research on prebiotic evolution will, in any case, continue to proceed along several, well-known lines: a) The search for extraterrestrial life and for extraterrestrial signs of prebiotic evolution. This implies: the analysis of meteorites that have landed on Earth; the launching of space probes with various missions of direct observation; the – both earth-bound and extraterrestrial – telescopic investigations over a broad spectrum of electromagnetic radiation. b) Physico-chemical experiments in the laboratory, where attempts will be made to improve the simulation of prebiotic conditions. In particular, efforts will focus on molecular self-organization and self-replication. One of the main shortcomings of such experiments will always be their limited duration and short time scale. c) Computer simulations of complex dynamic molecular processes over periods of time of increasing length. Most certainly, significant additional progress in computer hardware, software, and systematic handling of extensive information is yet to be expected. On the other hand, efforts to model large-scale prebiotic chemical evolution may meet certain fundamental difficulties, resembling, for instance, attempts to simulate the Earths weather over long time periods. We are now nearing the end of our excursion through Nature. Our aim has not been to solve any basic problems. Rather, we have tried to gain a general overview over certain facts, namely the varied manifestations of chirality in nature, and possibly to ask some related questions. Our excursion is, however, not yet concluded. Except briefly, in some of the early chapters, we have not contemplated biology itself. We have not yet attempted to consider how chirality manifests itself in biological organelles, in living cells, in macroscopic organisms. Of course, this is an almost unlimited undertaking. At best, we may expect to obtain a cursory impression, to get a short glimpse of this vast domain. From the prebiotic period, we, therefore, now make a big leap ahead in time into the world of biology. The homochirality of the biopolymers and of their

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constituents certainly manifests itself most remarkably in the architecture and function of biochemical reaction centers, organelles and cells. Among numerous possible examples, we here merely choose two, which play particularly important biochemical roles, namely, the photosynthetic reaction center in bacteria and plants, and the ribosomes, centers of protein synthesis in living cells. Finally, we shall also briefly consider, from the point of view of chirality, the morphology of some higher living beings.

10.8. From Molecular Enantioselectivity to Biological Precision Photosynthesis plays a central role in biology. It transforms light energy into chemical work. Through the capacity to perform chemical work, stored in substances with a high free-energy content, such as ATP and NADPH, living organisms attain a state of relative
negative entropy, of which Schrçdinger spoke [1]. By this
negative entropy, an organism may maintain a higher degree of order, preserving it from decay and enabling it to grow and replicate. After its isolation and the determination of its composition in the late 1970s, the crystallization of the photosynthetic reaction center of the bacterium Rhodopseudomonas viridis succeeded in 1982 [43]. The X-ray crystal-structure analysis of this protein–pigment complex was then carried out and revealed a most remarkable architecture [44]. The next step consisted in gradually elucidating the mechanism of energy transfer and transformation [45]. The bacterial reaction center (RC) is composed of a bacteriochlorophyll dimer (BChl)2, two monomeric bacteriochlorophylls (BChl), two bacteriopheophytins (BPh), and two quinone molecules QA and QB. These molecules are anchored in the L-, M-, and H-protein subunits of the transmembrane RC complex (Fig. 10.3). Directly, or via light-harvesting antenna pigments, the light excitation is first transmitted to the so-called
special pair (BChl)2, in which the two closely spaced bacteriochlorophyll molecules occupy a chiral conformation of symmetry C2. The light excitation leads in the special pair to a charge separation. A positive charge initially remains on the special pair, and a negative charge (electron) is transmitted via one of the molecular chains to the quinone molecules as here shown [46]: 3 ps

200 ps

100 ms

ðBChlÞ2 ! ðBChlÞ ƒ!ðBPhÞ ƒƒƒ! QA ƒƒƒ! QB Thereby, the ionic species are generated, Q
A, followed by Q
B . When fully reduced, the quinone QB is responsible for establishing the H+ gradient necessary for ATP synthesis. This reaction pathway and the time of the different steps here indicated (1 ps = 10
12 s, 1 ms = 10
6 s) were determined by a combi-

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Fig. 10.3. Arrangement of the cofactors (molecules responsible for the transformation of electronic excitation into charge separation) in the photosynthetic reaction center of Rhodobacter sphaeroides, as determined by X-ray diffraction. Rb. sphaeroides contains bacteriochlorophyll a and bacteriopheophytin a; Rps. viridis displays the structurally slightly different b-forms. The carrier-protein subunits M, L, and H, consisting of several hundred amino acids each, are not represented in detail. Arrows show electron-transfer steps (see text) (adapted from [46]).

nation of pulsed optical spectroscopy and magnetic resonance spectroscopy. It revealed that the specific binding of the porphinoid pigments to the carrier proteins plays an important role in determining the dynamics, irreversibility, and efficiency of the charge transfer. The point which we here wish to make is that such mechanisms would be unthinkable in surroundings that are not highly stereoselective and, in particular, enantioselective. The ribosomes are small cytoplasmic organelles (20 to 30 nm in diameter) that are the sites of protein synthesis within the living cells, both of prokaryotes (cells without nucleus; bacteria) and of eukaryotes (cells with nucleus; higher organisms). The ribosomes, of which there are many thousands within a cell, consist of complexes of large proteins and of a certain number of rRNA molecules. The ribosomes attach themselves to a given segment of messenger RNA (mRNA; see Sect. 9.9 and 10.1), gradually moving along the polynucleotide chain. The amino acids for the protein synthesis are carried to the ribosomal sites by transfer RNA (tRNA) molecules. Each tRNA contains at one end a nucleotide triplet, or anticodon, which binds to the complementary

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Fig. 10.4. a) Micrograph of ribosomes attached to mRNA in the process of synthesizing polypeptide chains (also visible). b) Schematic representation of the functioning of ribosomes moving along a given mRNA segment, starting at the initiator codon and ending at the stop codon. The tRNA molecules, one after another, bring in the amino acids to be incorporated into the growing peptide. After a ribosome reaches the stop codon, the last tRNA is released before the ribosome itself leaves the mRNA chain, splitting into two subunits [48].

sequence, or codon, on the mRNA. At its opposite end, the tRNA molecule carries the corresponding amino acid, in conformity with the genetic code. The ribosome moves gradually along the mRNA chain enabling each tRNA, one after another, to place its amino acid in such a way that a peptide bond with the previous peptidyl amino acid is formed. A peptide chain is thereby generated. Clusters of ribosomes may sit on a single mRNA chain, each reading a particular chain segment and synthesizing the corresponding polypeptide strand (Fig. 10.4) [47] [48]. The whole process depends on very delicate control mechanisms which evidently are extremely sensitive to steric molecular details. The genetic code correlates a DNA nucleotide sequence – or, rather, a mRNA sequence – with a protein amino acid sequence. In general, a mRNA nucleotide triplet, or codon, corresponds to a particular amino acid. As there exist four different RNA nucleotides (or bases), A, U, G, and C, there are in all 4  4  4 = 43 = 64 possible codons. On the other hand, proteins contain only 20 different kinds of amino acids. In the course of evolution, this has led to the situation that different codons may code for one and the same amino acid. For

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instance, the triplets GCA, GCC, GCG, GCU all code for alanine. However, only UGG codes for tryptophan. There are four triplets that do not code for any amino acid, but which denote the beginning and the end of a peptide sequence: the initiator codon AUG, and the terminator or stop codons, UAA, UAG, and UGA. Each tRNA molecule itself is built of ca. 75 nucleotides, consisting of four base-paired stems with three loops. The specific parts of the molecule are the anticodon loop, and the amino acid acceptor arm. As mentioned, and viewed very summarily, tRNA molecules take part in the following reactions: • Attachment of the correct amino acid to each tRNA molecule, in accord with its anticodon. • Incorporation of a tRNA with correct anticodon in the ribosome to meet (temporarily bind to) the corresponding codon of mRNA. • Deposition of the amino acid and simultaneous peptide bond formation. • The release of the corresponding tRNA after amino acid deposition. It does not take much imagination to realize the immense complexity of these processes. We indeed see that here the specific homochirality of the polynucleotides generates and sustains that of the polypeptides. Lack of stringent enantioselectivity would entirely prohibit such reactions from taking place.

10.9. From Molecular Enantioselectivity to Macroscopic Biological Chirality All biological organisms, all plants and animals including man, are chiral. In some organisms the morphological chirality is very striking, while in others one encounters elements of pseudo-symmetry, like approximate reflection planes that feign to destroy chirality. In some plants the leaves are helically disposed along the stem, or in some flowers the petals exhibit chiral patterns, while others, at a first glance, appear to be achiral. But a closer examination always reveals asymmetric details. The tremendous variety of macroscopic structures, in some ways, contrasts with the homochiral uniformity at the molecular level. This leads to the general observation that chirality at a lower level of biological structure does not necessarily imply that the same chirality must manifest itself at a higher level. Concomitantly, the reverse must also be true. The absence, or near-absence of chirality at a higher, macroscopic, level does not preclude that at a lower level chirality may be a dominant feature. Cell differentiation, although originating in the homochiral molecular clockwork precision of the single cell, implicitly contains a quasi-infinity of morphological possibilities.

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Fig. 10.5. The flower of Brassica oleracea Romanesco (Botanica, Random House, Australia, 1997). Notice the chirality of the main structure and of the substructures. This variety of cabbage has a taste between that of cauliflower and of broccoli. It is not necessarily all childrens favorite dish, but carefully cooked, well seasoned and peppered, may be quite edible.

Fig. 10.5 shows the flower of Brassica oleracea Romanesco, a variety of cabbage. The main conical spiral – here of left-handed chirality – is composed of subunits which themselves appear as conical spirals. This indeed resembles a fractal structure. But the reader will notice that the subunits often do not display the same chirality as the main spiral. It appears that the smaller the structural unit is, the less well is the macroscopic chirality definable. This, of course, clearly shows that the macroscopic part of a plant does not directly mirror the uniform microscopic chirality at the molecular level, and that, from such a point of view, cell differentiation follows rules of its own. Moving from the plant world to the animal kingdom, the observer of field and garden will also be aware of the chirality of the shells of snails. Usually, shells of snails are clearly either dextral (right-handed) or sinistral (left-handed). Most species are dextral, a clear minority are sinistral, and a few species are polymorphic [49]. There is no evidence that right-handedness is functionally superior or inferior to left-handedness [50]. Opposite direction of coiling may, however, lead to reproductive isolation. This can make the genetic trees complicated. Recent genetic investigations on the Japanese land snail Euhadra (Fig. 10.6) come to the conclusion that a single gene gives rise to the mirror-image form of

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Fig. 10.6. Enantiomorphous forms of the shell of the land snail Euhadra

the snails body plan. That is to say, the different mirror-image forms have evolved in favor of the genetically dominant handedness as a result of singlegene speciation [51]. Here again, as in the previous example, the macroscopic handedness is determined by a particular DNA sequence, and evidently not by the absolute chirality of the DNA molecules themselves. We may now return to Sect. 1.1, and emphasize what we actually knew from the start: left hand and right hand cannot possibly mirror molecular chirality, but they are the result of DNA coding, that is, nucleotide sequence. Unquestionably, however, the molecular homochirality is a prerequisite for the unambiguous polynucleotide codability. This brings us to the last question that we ask at the end of our excursion. Has the evolution of the genetic code in any way been directly influenced by the l-amino acid/d-sugar homochirality? In other words, if a d-amino acid/l-sugar life had started, would the genetic code have evolved differently, possibly then leading to different kinds of organisms? Of course, after what we have just seen, there is no immediate reason why the absolute molecular homochirality of proteins and nucleic acids should have had any influence on the actual structure of the genetic code. As we fully realize, the code connects nucleotide sequence with amino acid composition, and consequently does not directly depend on to the absolute configuration of the molecular elements involved. Nevertheless, and in final conclusion, the seemingly obvious might perhaps still hide the unexpected. Our journey through nature has reminded us that everything in the universe is interdependent, intertwined, and basically inseparable. Scientific research, however, is obliged to partition, to interpret similar phenomena separately, to isolate them from the rest. Only in this way, and by creating different welldefined disciplines, may science gain systematic insights. However, one should not forget that, to regain an overview, the distinct parts must be fitted back into

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an overall picture, and that the description of nature finally is an interdisciplinary undertaking. Thus we are reminded of the words of the famous French philosopher Henri Bergson (1859 – 1941) who in LEvolution Cratrice writes:
Lunivers dure. Plus nous approfondissons la nature du temps, plus nous comprendrons que dure signifie invention, cration de formes, laboration continue de labsolument nouveau. Les syst
mes dlimits par la science ne durent que parce quils sont indissolublement lis au reste de lunivers. (
The universe is lasting. The more we deepen our study of time, the more we realize that duration implies invention, creation of forms, continuous elaboration of the absolutely novel. The systems defined by science last only because they are inseparably linked to the rest of the universe.)

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Evolution and Distribution of Left-handed and Planispiral Coiling in Snails Nature 1975, 254, 419. [51] R. Ueshima, T. Asami,
Single-Gene Speciation by Left–Right Reversal, Nature 2003, 425, 679.

Glossary absolute configuration: Sense of the handedness, or chirality, of a molecule. Describes the arrangement in space of all the atoms in the molecule. See also stereochemistry. absolute temperature: Temperature measured on the kelvin scale. Temperature in Celsius plus 273.15 equals temperature in kelvin. At zero kelvin, an ideal gas under constant pressure should have volume zero. absorption of radiation: Uptake of the energy of a photon by an atom or a molecule. The atom or molecule thereby undergoes a transition. It gets promoted from a lower (ground) energy state to a higher (excited) energy state, the energy difference being equal to the energy of the absorbed photon. See also photon. absorption band: Characteristic wavelength range in which light is absorbed by a given material. Within this wavelength range, the absorption coefficient usually goes through a relative maximum. At the atomic/molecular scale, an absorption band is related to a particular transition, or a group of close-lying transitions. absorption coefficient: A measure of the amount of light (number of photons) of given wavelength that a unit concentration of matter (for instance, 1 mol per liter) can absorb per unit pathlength of light inside the medium, and per unit incident light intensity. accretion: Growing together under the influence of mutual attraction. For instance, formation of a galaxy under the influence of gravity. adsorption: Weak binding of an atom, molecule, or ion to a particular material, for instance a crystalline surface. This type of binding is usually due to so-called nonbonded interactions. ADP: Adenosine-5’-diphosphate. See also ATP. allotropic form: Modification; physical structure in which a chemical element occurs. For instance, the element carbon may occur as graphite or as diamond.

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amino acid: An organic acid that contains an amino moiety and may also show basic properties. Amino acids polymerize to chains called peptides. Proteins consist of large peptides. Amino acids thus are the monomer units of peptides and proteins. There occur about 20 different amino acids in nature. See also protein. anaerobic: Capable of living in the absence of air, in particular, of oxygen. anapole moment: Characterizes the electromagnetic interaction of a toroidal electric current that gives rise to a ring-like magnetic field. angular momentum: See momentum, angular. anisotropic: Showing physical properties that are direction-dependent. anomalous dispersion: Refraction of light inside an absorption band. See also refractive index. antipode: Molecule of opposite sense of chirality, or handedness. Mirror image. Also designated as enantiomer. aromatic: Chemically related to benzene. Benzene contains six carbon atoms in a cyclic hexagonal arrangement. Hence: aromatic ring. Aromatic compounds may contain one or several linked or fused aromatic rings. asymptotic: Approaching a definite value after an infinity of converging mathematical steps. atomic number: Number of protons in the atomic nucleus. Determines the number of atomic electrons and the chemical identity. Designated as Z. ATP: Adenosine-5’-triphosphate. Its interconversion to ADP is the major source of energy in biochemical processes. autocatalysis: Chemical reaction that is catalyzed by the reaction product. See catalyst. autotrophic: Capable of synthesizing its own food from inorganic substances only. In particular, deriving carbon for its metabolism from carbon dioxide. Using light or chemical energy: green plants, algae, and certain bacteria. See also chemotrophic, phototrophic.

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birefringent: Refracting light differently, depending on the direction of incidence and the light polarization. See also refraction of light; polarization of light. biological cell: Morphological and physiological unit of all living organisms. Enclosed by a cell membrane. Exhibits a metabolism. Capable of replication. buffered: Keeping a chemical equilibrium close to neutral, avoiding to become too acidic or too basic.

catalyst: Chemical element or compound that accelerates the course of a chemical reaction without being itself affected. At the end of the reaction, a true catalyst remains chemically unmodified, or returns to the initial state. cell (biological): See biological cell. center of mass: Midpoint of the mass of a body, or of a group of several bodies. Under external forces, the center of mass moves as if the whole mass of the system were concentrated there. centrosymmetric: Containing a center of inversion. Showing invariance with respect to reflection in that point. chaos, deterministic: Occurs in a system following the laws of classical mechanics, yet exhibiting an irregular, unpredictable behavior. In its dynamical evolution, very sensitive to the initial conditions. charge conjugation: Reversal of electric charge. Change of the sign of all charges in a reaction between subatomic (elementary) particles. Relation between matter and antimatter. chemotrophic: Gaining the energy for its metabolism from chemical reactions, generally in the absence of air; see anaerobic. chiral:
Handed. Property of being not superimposable onto its mirror image. chromatography: Separation of a mixture into its components by using differences in the adsorption properties of the components. The mixture usually is in form of a solution (mobile phase). It may also be a gas. The adsorption of the components takes place on solid material (stationary phase), usually in powdered form. See also adsorption.

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circular birefringence: Displaying a different refractive index for incident leftand right-circularly polarized light of the same wavelength. Circular birefringence is always accompanied by circular dichroism inside absorption bands. See also Cotton effect; optical activity; polarization of light. circular dichroism (CD): The difference of the absorption coefficient for incident left- and right-circularly polarized light of the same wavelength. It is measured with a circular dichrograph. See also Cotton effect; optical activity; polarization of light. circular differential: Responding differently to left- and right-circularly polarized light. classical mechanics: Mathematical description of the forces acting on bodies and of the ensuing motions. Classical mechanics is applicable to macroscopic and microscopic bodies; but only conditionally to nanometric objects, and not to subatomic particles. Classical mechanics is also called
Newtonian mechanics. complex conjugation: Change of the sign of the imaginary part of a complex number. A complex number consists of the sum of a real number and an imaginary number. convection: Transference of a mass of fluid against the force of gravity, for instance, under the influence of heating. Cotton effect: Concomitant occurence of anomalous optical rotatory dispersion (ORD) and of circular dichroism (CD) inside an absorption band. Coulomb interaction: Electric forces between point-like electric charges. crystal: Solid state of matter exhibiting long-range atomic or molecular order in three dimensions. A crystal as a whole shows definite (macroscopic) symmetry properties. crystallite, two-dimensional: Two-dimensional domain, or layer, in which longrange molecular order prevails.

deuterium: Isotope of hydrogen. The atomic nucleus of deuterium is composed of a proton and a neutron. Deuterium behaves chemically essentially like hydrogen.

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diamagnetic: Nonmagnetic. Repelled by a magnetic field. differential equation: An equation containing the derivatives of a function. The solution of the equation gives the function itself. dimer: Entity composed of two subunits, or monomers. dipole moment, electric: See electric dipole moment. dipole moment, magnetic: See magnetic dipole moment.

electric conduction: Transport of electric charge (electrons, ions) under the influence of an electric potential. The electric potential gives rise to an electric field that exerts a force on the charges, thereby inducing their migration. electric dipole moment: System composed of two opposite electric charges of equal absolute magnitude, fixed at a given distance from each other. The electric dipole moment as a whole is neutral, but it tends to align itself parallel to an external electric field. A dipole moment is represented by a polar vector. See also vector. electric field: Generated by electric charges. The electric field of one charge exerts a force on any other charge. Measured at a given point in space, the resulting electric field is described by a polar electric field vector, representing a value and a direction. See also vector. electric quadrupole moment: Arrangement of two positive and two negative charges, such that the whole is neutral, and that it does not possess any resulting electric dipole moment. Interacts with the gradient of an electric field. electromagnetic field: Interplay of an electric and a magnetic field. May manifest itself as electromagnetic waves, or light. electromagnetic radiation: Generalized notion for light radiation. See electromagnetic field. electron: Smallest freely detectable and stable negatively charged elementary particle. Carrier of the negative elementary charge. The mass of the electron is 1/1836 times that of the proton. In atoms, electrons constitute the outer atomic shells, balancing the positive charges of the protons in the nucleus. The electron

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has a spin of 1/2 , and belongs to the elementary particle family of leptons. See also proton; neutron. electron diffraction: Due to their wavelike properties, electrons may get similarly diffracted by atoms as light. The phenomenon is the basis of electron microscopy. The electron wavelength may be made very short, leading to a very high resolution, rendering objects visible down to nanometric dimensions. electron microscopy: See electron diffraction. electronic transition: Absorption or emission of light, affecting, in particular, the electrons in an atom or molecule. See also absorption of radiation. electrostatic potential: Potential energy per unit charge due to the influence of other, nonmoving charge(s). See also energy. emission of radiation: An atom or molecule in a higher energy state may revert to a lower energy state by emitting a photon of corresponding energy. There are two kinds of emission: spontaneous (or incoherent); stimulated (or coherent). Coherent emission is the basis of laser action. See also absorption of radiation. enantiomer, enantiomeric: Of opposite sense of chirality. (Term usually applied to single molecules.) enantiomorph, enantiomorphous: Of opposite chirality. (Term usually applied to crystals and to macroscopic objects.) energy: Occurs generally as potential energy, and as kinetic energy. Potential energy may be viewed as the capacity to perform work. Potential energy is measured as force times distance. Kinetic energy is related to the movement of a body. It is measured as one half the mass times the velocity squared. entropy: Quantity measuring the degree of disorder vs. order of a system. Relates the energy forms of heat and work. entropy of mixing: Entropy form that increases upon mixing of different chemical components. enzyme: Biochemical catalyst. Generally a protein. There exist also RNA molecules that act as enzymes. See also catalyst.

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fluorescence: Spontaneous emission of radiation. In molecules, fluorescence usually occurs as transition from the lowest excited electronic state to the ground state. force: Capacity to induce a motion of a body. Measured as mass times acceleration. Also defined as change of momentum per unit time. fractal: Self-similar. The parts of a fractal object exhibit a structure similar to the whole. The same holds for the parts of the parts; in principle, ad infinitum. Geometric fractal: shape that is recursively constructed and appears as selfsimilar at all scales of magnification. Random fractal: generated by stochastic, rather than deterministic processes; for example, fractal landscapes; mountains and rocks; coastlines. fusion, nuclear: The collision of the nuclei of the lighter elements, such as those of hydrogen and deuterium, leading to their fusion to heavier nuclei, is accompanied by the release of energy. The process evidently can only take place if the electrostatic repulsion between the positive nuclear charges is overcome by external forces, such as strong gravitational attraction inside stars. It is this energy release that makes stars shine, and which manifests itself in the explosion of the hydrogen bomb. The energy given off in such exothermal nuclear reactions is much greater than that for exothermal chemical reactions, because the binding energy of quarks inside nuclei is much greater than that for electrons inside molecules.

galaxy: A gravitationally bound assemblage of millions or billions of stars. Often of disklike shape and of spiral structure. gamma (g-) rays: Electromagnetic radiation of ultrashort wavelength, on the order of a thousandth of a nm and below. Gets emitted and absorbed by atomic nuclei. Ionizes atoms. Very penetrating. genetic code: The biological code that translates a DNA sequence into a protein (poly(amino acid)) sequence. globular proteins: Soluble proteins that often play a chemical role in living organisms. As opposed to fibrous proteins that serve as structural material. group: Mathematical notion, defined as a set of operands that fulfil certain conditions of multiplication. The set of all symmetry elements of a crystal form a group, the so-called point group of the crystal.

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harmonic oscillator: Mass point oscillating back and forth, periodically going through a point of equilibrium. The restoring force is proportional to the distance of the mass point to the point of equilibrium. helium: Second element in the Periodic Table of Elements. The atomic nucleus contains two protons and two neutrons, the negative shell two electrons. helix: A line coiling around a cylinder in a regular spiral fashion describes a helix. The pitch of the helix measures the steepness of the coil. heterogeneous: Spatially nonuniform. Subdivided by boundaries. heterotrophic: Capable of deriving energy for life processes only from the oxidation of organic compounds already present. See, for comparison, autotrophic. homogeneous: Spatially uniform, containing no boundaries. homochiral: Of same handedness or chirality. hydrodynamics: The field of (classical) physics that deals with the motion of fluids. hydrogen: First and lightest element in the Periodic Table. The atom consists of a single proton as nucleus and an electron. hydrolysis: Dissolution of a covalent chemical bond with the concomitant addition of a molecule of water. In particular: decomposition of a peptide into free amino acids. hydrophilic: Water-attracting. Binding water easily. hydrophobic: Water-repelling. Not prone to binding water. The contrary of hydrophilic.

ideal gas: Gas composed of noninteracting point-like particles that possess only mass and kinetic energy. index of refraction: See refractive index.

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infrared light (IR): Light in the wavelength range of micrometers. Heat radiation. ion: Electrically charged atom or molecule. isomer: Chemical compound belonging to a set of molecules having the same composition (same kinds and numbers of atoms; same stoichiometric formula), but of a different structure (different kinds of bonds; and/or different spatial arrangement of atoms; different structural formula). Isomers evidently have the same molecular mass. They differ in their chemical and spectroscopic properties. isomerization: Adoption of a new molecular structure, while maintaining the composition. Molecular rearrangement. isotope: Atom containing a nucleus with the same number of protons, but a different number of neutrons. For instance: deuterium is an isotope of hydrogen. See also neutron; proton. isotropic: Spatially uniform. Showing physical properties that are not directiondependent, such as a gas or a liquid. Homogeneous.

kinetic energy: Energy of motion. See energy. kelvin temperature: See absolute temperature.

Langmuir–Blodgett film: Thin film of (usually large) molecules spread onto the surface of a liquid subphase. The density of the film is maximized through lateral compression, eventually inducing two-dimensional order inside the film. light: Electromagnetic radiation, propagating at the constant speed of 300,000 km per second in vacuum. The relation holds: wavelength of light times frequency of light equals speed of propagation. light scattering: Scattering of light that falls onto a material object. One distinguishes between elastic and inelastic scattering. In elastic scattering, there is no lasting internal energy transfer between the radiation and the scatterer. Elastic scattering is also called Rayleigh scattering. The intensity of Rayleigh scattering is proportional to the fourth power of the light frequeny.

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magnetic dipole moment: May be considered as originating from the rotation of electric charge. The axial magnetic dipole vector is perpendicular to the plane of charge rotation. A magnetic dipole moment tends to align itself parallel to an external magnetic field; see also momentum, angular. magnetic field: Generated by moving electric charges, according to the Biot– Savart law. Time-varying magnetic fields may induce electric currents, according to the law of induction. The magnetic field at a given point in space is characterized by the value and direction of an axial vector; see also momentum, angular. magnetic induction: Magnetic flux density. In vacuum, the magnetic flux density and the magnetic field are parallel and proportional to each other. magnetostatic: Concerning the interaction between static magnetized bodies. meson: A family of elementary particles. Elementary particles that are subjected to the strong interactions are called hadrons. These are subdivided into baryons and mesons. There exist different kinds of mesons, for instance, pmesons (pions) and K-mesons (kaons). micrometer (mm): One millionth of a meter. Objects of this order of magnitude and larger, but not smaller, are distinguishable with conventional visible-light microscopes. mobile phase: Fluid phase carrying the mixture to be chromatographically separated. See chromatography. molecular dynamics: Computational method, combining both classical and quantum mechanics, to calculate the spatial structure and the motion of large molecules. Much used in modern biochemistry. momentum, linear: Quantity describing the linear motion of a body. Defined as mass of the body times its velocity. It is represented by a polar vector. momentum, angular: Quantity describing the rotational motion of a body. It is represented by an axial vector, and is defined as the vector product of the position vector and the linear momentum vector. monolayer: Layer of thickness of a single molecule. See also Langmuir– Blodgett film.

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monomer: Single repetitive unit of a polymer. Monte Carlo method: Statistical method sometimes used in moleculardynamics calculations.

NADPH: Reduced form of nicotinamide adenine dinucleotide phosphate. Plays an important role in biochemical processes. nanometer (nm): One billionth of a meter; one thousandth of a micrometer. Objects of this order of magnitude, such as large single atoms and single molecules, may be made indirectly visible by the various methods of electron microscopy and of scanning microscopy. nanotube: Tube-shaped structure of nanometric dimensions. Best known and most intensely investigated are carbon nanotubes. neutrino: Elementary particles belonging to the lepton family. Neutrinos have no charge, a very small mass, and a spin of 1/2 . They experience only weak and gravitational interactions. neutron: Neutral particle. Constituent of atomic nuclei (with the exception of hydrogen). The mass of the neutron is ca. 1.001 times that of the proton. The neutrons, therefore, contribute significantly to the atomic mass. The sum of the number of protons and the number of neutrons in an atom is termed the mass number. Neutrons and protons are both baryons. They both carry a spin of 1/2 . See also atomic number; proton. nucleic acid: General designation of the biopolymers RNA (ribonucleic acid) and DNA (deoxyribonucleic acid). nucleon: Particle constituting the atomic nucleus; proton, neutron. nucleoside: Biomolecule composed of a purine or a pyrimidine base to which a sugar (d-ribose, or d-2-deoxyribose) is attached. nucleotide: Purine and pyrimidine nucleotides are the monomeric precursors of RNA and DNA. Purine ribonucleotides occur also in other biochemical systems, such as ATP. Nucleotides are nucleosides phosphorylated on one or more of the hydroxy groups of the sugar (ribose or deoxyribose). See also ATP; nucleoside.

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oligomer: Composed of a few monomers. Small polymer. operator, quantum-mechanical: Mathematical operator (instruction for a mathematical operation) related to a physical quantity, such as momentum, angular momentum, energy. By acting with a given operator on the wave function of a physical system, the expectation value of the corresponding physical quantity may be deduced. Physical quantities that are described by vectors are, in quantum mechanics, represented by vector operators. See also vector; wave function. optical activity: Optical property of chiral substances, in particular of chiral fluids, consisting in the rotation of the plane of polarization of linearly polarized light. It is a direct consequence of circular birefringence. Optical activity inside absorption bands leads to anomalous optical rotatory dispersion and to circular dichroism. See also Cotton effect; polarization of light. optical rotation: Rotation of the plane of polarization of linearly polarized light. It is measured with an optical polarimeter. See optical activity. optical rotatory dispersion (ORD): Wavelength dependence of the optical rotation. It is measured with a spectral polarimeter. oscillator: See harmonic oscillator.

paramagnetic: Property of a substance carrying free (atomic or molecular) magnetic dipole moments capable of orienting themselves in an externally applied magnetic field. Paramagnetic substances are drawn into an external magnetic field, as opposed to diamagnetic substances which are repelled. parity operation: Mathematical operation consisting in changing the sign of all position variables. A right-handed Cartesian coordinate system is thereby transformed into a left-handed one, and vice versa. A polar vector has its direction reversed. partition function: Statistical quantity relating the energy states of microscopic (atomic or molecular) particles to the thermodynamic properties of a very high number (macroscopic sample) of such particles. peptide: Small protein. See polypeptide.

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permutation: Every arrangement of n distinguishable objects (or elements) in n distinguishable boxes (or positions) is a permutation. For n such elements and positions there are in all n! different permutations. phase, of a material medium: Part of a material system that has uniform physical properties. For instance: gas phase, liquid phase, crystalline phase of uniform lattice structure. phase space: Abstract representation of both the positions and the momenta of a given number of particles. For one particle in three-dimensional space, the phase space is six-dimensional. For N particles, it is 6N-dimensional. photochemistry: Field of chemistry concerning reactions occuring under the action of light. photon: Elementary light particle, carrying a spin of 1, but no charge and no rest mass. It may be considered as a quantum of electromagnetic energy. The photon energy is equal to the light frequency times Plancks constant. See also absorption of radiation. photosynthesis: Chemical synthesis under the action of light. In biology: conversion of light energy into chemical energy; in particular, the synthesis in green plants and in some bacteria of hexose sugars from carbon dioxide and water. phototrophic: Obtaining its metabolic energy via photosynthesis. See also autotrophic; chemotrophic. polarimeter: Apparatus to measure the optical activity, or optical rotation. polarization of light: Directions in space and ways in which the electric and magnetic field of a light ray oscillate. Hence: linearly polarized; circularly polarized. polarization induced in matter: Under the influence of an external electromagnetic field, such as a light wave, the positive and negative charges in a material medium (atom, molecule) are periodically shifted from their original equilibrium positions, thereby creating a resulting oscillating electric dipole moment, or (and) higher-order electric and magnetic moments. These induced electric and magnetic moments are the source of secondary (scattered) light.

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polarizable: Prone to being polarized by electric, or electromagnetic fields. See polarization induced in matter. potential energy: See energy. potential: See as example, electric potential. polyene: Organic compound displaying single and double bonds in alternate sequence. polymer: Large molecule consisting of many equal, or similar molecular repeating subunits, called monomers. See also oligomer. polynucleotide: Biopolymer, such as RNA or DNA, consisting of many nucleotide monomers. See nucleotide. polypeptide: Poly(amino acid); consisting of many amino acids linked through amide bonds. See peptide; protein. protein: Class of biopolymers playing a key role in the metabolism and structure of living organisms. See globular proteins; polypeptide. proton: Constituent of atomic nuclei. Nucleus of the element hydrogen. Carries a positive elementary charge, of same absolute magnitude as that of the electron, but of opposite sign. Displays a spin of 1/2 . See also electron; neutron. pseudoscalar: The scalar product of a polar vector and an axial vector. The quantity keeps its absolute value, but changes its sign under the parity operation.

quantum mechanics: Mathematically based physical theory to calculate the energy and time evolution of molecular, atomic, and subatomic objects. Assumes that material particles and electromagnetic radiation both possess wave-like and particle–like properties. One, therefore, speaks of the
particle– wave duality. See also uncertainty relation; wave function. racemic mixture: Mixture containing exactly equal amounts of both enantiomers of a chiral compound. A racemic fluid mixture as a whole is invariant under any reflection, or any rotation–reflection, and is, therefore, achiral. refraction of light: Result of light scattering in the direction of the incident light beam in a material medium. The resulting light beam travels at a different speed

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than in vacuum. Refraction may manifest itself as the bending of a light ray when it enters a medium where its speed is different. The bending angle depends also on the angle of incidence. refractive index: Measure of the speed of light in a material medium as compared to the speed of light in vacuum. The refractive index depends on the wavelength, or frequency, of the incident light, as well as on the chemical composition and structure of the medium. The wavelength dependence of the refractive index causes light dispersion. resolution, of a racemic mixture: Separation of enantiomers; separation of antipodes. rotational strength, of a transition: Quantum-mechanically computed measure of the circular dichroism due to a particular transition in a chiral molecule. Predicts the difference of the absorption coefficient for left- and right-circularly polarized light due to that transition. See also optical activity. The rotational strength for an isotropic chiral molecular medium, such as a chiral liquid solution, is a pseudoscalar quantity.

scalar: Mathematical quantity characterized by an absolute value and a sign. secondary structure, of proteins: The primary structure is the amino acid sequence. Nonbonded interactions and hydrogen bonds between the individual monomers induce chracteristic secondary structures of the peptide chains. In particular: the a-helix, the parallel and antiparallel b-pleated sheet, etc. selection rule: Conditions that must be met for a well-defined physical process to occur, for instance, a light absorption or emission process in an atom or molecule. self-similarity: See fractal. semiconductor: Solid crystalline material in which the electric conductivity is intermediate between that of a metal and that of an insulator. Semiconductors exhibit conduction properties that may strongly depend on external factors, such as temperature or externally applied voltage. Conduction may set in only after certain threshold values have been reached. Semiconductors are used in electronics as transistors, rectifiers, etc.

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spectroscopy: General term for the investigation of the interaction of radiation with matter. Subdivided into many subfields, depending on the kind of radiation and energy range involved. spin: Angular momentum carried by a particle. Characterized by a spin quantum number. The electron and proton both have a spin of 1/2 . In particles with nonzero rest mass, the spin leads to an intrinsic magnetic dipole moment. The photon has a spin of 1, but no rest mass; therefore, it has no magnetic dipole moment. stationary phase: Solid adsorbing phase in a chromatographic separation column. See chromatography. stereochemistry: Field of chemistry that focuses particularly on the spatial position of atoms inside molecules. Study of the configuration and conformation of molecules. supramolecular: Scale going beyond that of individual, smaller molecules. Nanometric to micrometric scale. symmetry breaking: Destroying a (possibly existing) symmetry. symmetry element: Operation in space carrying a body from an initial position to a new position that is indistinguishable from the initial one. See also group.

thermodynamics: The field of physics and chemistry that deals with the conversion of different forms of energy, for instance, chemical energy into heat and work. time reversal: Letting a motion go backwards. On the cosmic scale: reversing all motions in the universe. topology: Mathematical study of the properties of objects that are preserved through deformations, twistings, and stretchings; tearing, however, is not allowed. The theory of knots and links is a subfield of topology. torus: Donut-shaped surface. toroidal: Of similar form as, related to, a torus.

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transition: Change of the energy state of an atom or molecule under an external influence, such as the absorption or emission of light. See absorption of radiation. transition moment: Quantum-mechanically computed measure of light absorption (or emission) for a particular transition in an atom or molecule. The square of the transition moment is proportional to the absorption coefficient for that transition. translation: Unbounded, linear movement.

ultraviolet light (UV): Light of wavelength shorter than that of visible light. The wavelength range of visible light extends from 400 to 700 nm; that of UV light is below 400 nm. uncertainty relation: According to the laws of quantum mechanics, certain properties of an object (particle) cannot be measured simultaneously to arbitrarily high precision. This is, for instance, the case for the position and the momentum of a particle in one and the same direction in space. The more precisely one measures the position, the less exactly one knows the momentum, and vice versa. The uncertainty relation, or uncertainty principle, may be considered as a consequence of the particle–wave duality. See also quantum mechanics. An analogous situation of complementarity exists between time and energy: the shorter the time, or duration, of the state of an object, the more inaccurate is the determination of the energy of that state; the longer the time, the less uncertain the energy measurement.

vacuum: Empty space, containing no matter. vector: Mathematical quantity possessing both magnitude and direction. Physical quantities, such as position in space (with respect to a reference point), velocity, momentum, force, are described by vectors. Vectors may be represented by arrows of length proportional to the absolute value. See also momentum, linear; momentum, angular. In quantum mechanics, such quantities are represented by vector operators. See operator, quantum-mechanical.

wave function: Mathematical solution of the quantum-mechanical Schrçdinger equation. The absolute value of the wave function squared has the significance of a probability density. By acting onto the wave function with a quantum-

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mechanical operator, the expectation value of the corresponding physical quantity may be deduced via an integration. See operator, quantum-mechanical; and quantum mechanics. X-rays: Electromagnetic radiation with wavelength on the order of nanometers, or below. The scattering of X-rays by atoms in crystals may indirectly reveal the structure of the lattice, i.e., the position of the individual atoms in the lattice. This is the subject of X-ray analysis. X-ray analysis: Deduction of the atomic structure of a crystal lattice from the spatial distribution and pattern of scattering intensities, and via mathematical Fourier transformations. Determination of the absolute configuration requires also measuring anomalous scattering intensities.

Subject Index

A Absolute configuration (P)- and (M)- 14 (R)- and (S)- 14 Absolute measures of chirality 158 Absorption 7, 54 Achiral fractional chromatography 176 Alanine d- 5, 42 l- 5, 42, 124 Amino acids, l- 5, 44, 186 Amplification of asymmetry 111, 180 Anapole current 163 Anapole moment 162 Angular momentum 27, 53, 78 Anisotropic conditions 123, 171 Anomalous dispersion 57 Antimatter 47, 93 Anticodon 216 Antiparticle 92 Antipodes 16 Aperiodic crystal 197 Arrow of time, cosmological 94 Asteroid 103 Asymmetric adsorption 124, 171, 184 autocatalysis 17, 178 chemical reaction 17, 177 photodestruction 19 photosynthesis 19, 213 synthesis 16, 177 Asymmetry, amplification of 111, 192 Atoms, parity violation in 38 Autocatalysis, asymmetric 178, 183 Auxiliary, chiral 177 Axial vector 27, 31, 77 Axion 95

B Barnett effect 83 Baryon number 88, 92 Baryon–antibaryon asymmetry 90 Base pairing 185, 187, 188 Big Bang model of the universe 89 Bijvoet method 16 Biochemical enantioselectivity 4, 112 Biot–Savart law 50 Birefringence, magnetochiral 69 Bromate lattice 118

C Cahn–Ingold–Prelog rules 14 Cartesian coordinate system 26 Catalyst, chiral 177 Cell differentiation 217 Center of inversion 75 Chain elongation 187 Chance 211 Chaos classical 101, 104 deterministic 101, 104 in the atmosphere 108, 109 Chaotic dynamics 156 Charge conjugation (C) 28, 32, 92 Charge separation 213 Charge transfer 214 Chemotrophic 204 Chiral atomic lattice 117 auxiliary 177 carbon nanotubes 126 catalysis 17, 177 chromatography 17, 175 crystal 116, 190 discrimination 168

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domains 124, 173, 180 indices 128 induction 121 interaction 54, 78, 85 nanotube 82, 126 object 3, 10 point group 116 potential 155, 157 smectic phase 122 solid surface 123, 172, 173 structure 5, 85, 98, 116, 125, 135, 158, 217 surface layer 181 symmetry breaking 27, 39, 92, 111, 121, 168, 180 Chirality absolute measures of 159 dynamic 76 extended surface 124, 174 geological 110, intrinsic 14 morphological 216 of potatoes 146, 158 of shoes 3, 146 organizational 125 structural definition 10, 76 topological 138, 140 Chirality algebra 142 Chirality function 145 Chirality vector 126 Chirally distorted ellipsoid 158 Chlorate lattice 118 Cholesteric phase 121 Chromatography achiral fractional 176 chiral 17, 175 Chromophore, carbonyl 147 Chromosome 197 Circular dichroism (CD) 7, 58 Circular polarization 34, 52, 68 Circularly polarized light 34, 52, 68, 203, 206

Subject Index

Codon 215 Comet 102, 209 Complementary strand 185 Conductivity 80 Conglomerate 118, 180, 191 Coulomb interaction 191 Coupled-oscillator model classical 154 quantum-mechanical 150 CP Enantiomers 47 CP Symmetry 32, 93 Crossing number 129, 139 Crystal aperiodic 197 chiral 6, 9, 116, 190, 201 Crystallographic symmetry 75, 115 Current helicity 131

D d and l 12 d and l 13 Deoxyribonucleic acid see DNA Deoxyribose 185, 201 Dextrorotatory 12 Diastereoisomerism, topological 139, 141 Difference maps 103 Dihedral angle 137 Dipole moment, electric 30, 31, 162, 167 Dispersion, anomalous 57 Dispersion forces 168 Dissection of a cube 135 of a sphere 136 DNA (deoxyribonucleic acid) 5, 185, 197, 215 B- 5 Z- 5 Double helix 5, 152, 185, 188 Double-minimum potential 45 Dynamical stochasticity 101 Dynamics, chaotic 101, 108

Subject Index

E E1 (electric dipole allowed) 37, 55 E2 (electric quadrupole allowed) 37, 55 Earth asymmetry on 104 magnetic field of 106, 208 mirror image of 105 Einstein–de Haas effect 82 Electric dipole interaction 55, 167 Electric dipole moment 30 Electric dipole transition moment 15, 36, 57 Electric field 52, 55 Electric quadrupole 37, 55, 162 Electromagnetic background radiation 90 Electromagnetic forces 25, 49 Electronic transition 7, 36, 57, 147 Elementary particles, polarized 32, 33, 203 Elliptically polarized light 67 Emission 36, 69, 122 Enantiomer 4, 159 (R)- 14 (S)- 14 Enantiomeric 6, 159 Enantiomeric excess 46, 71, 174, 177, 206 Enantiomers, interconversion of 44 Enantiomorphism, topological 140 Enantiomorphous 3, 24, 158, 171 Enantiomorphous surfaces 123 Enantioselective influence 72 photochemistry 19, 72, 191 reaction 17, 177, 207 separation 16, 175, 176 Entropy negative 206 of mixing 119, 164, 190 Enzyme 18, 197, 198 Enzyme–substrate complex 18 Exciton model 150

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Extended surface chirality 173 Extraterrestrial life 24, 202, 212

F Faraday effect 59, 66 Flow of current 51, 81 Fore-aft asymmetry 34, 66, 69 Fractality 97, 110, 217 Fractional chromatography, achiral Free electrons on a helix 151 Fullerenes, chiral 12, 13

G Galaxy, spiral 98 Galaxy formation 90, 96 Gene 217 Genetic code 215 Geological chirality 110 Geometric asymmetry measures Glide plane 115 Globular protein 17, 197 Glyceraldehyde 13 Graphene sheet 126 Gravitation 25, 102

H Helicity 31, 34, 51, 54, 57, 62, 128 current 131 magnetic 130 Helicoid 156 Helicotoroidal current 163 a-Helix 9, 151 Hexose, l- 17 Homochiral organic crystal 119, 190 polymerization 182, 184, 187 polypeptides 182 regions 125 sequences 182, 205

176

159

244

Homochirality 3, 189, 193 Hopf link 140 Hubble
s law 88 Hydrogen bond 185, 189, 191 Hydrothermal systems 204

I Information carrier 186, 198 Inherent unpredictability 101, 108 Intergalactic matter 94 Intermolecular interactions, chiral 72, 168, 182, 186 Intermolecular potential, chiral 168, 170, 191 Inverse magnetochiral birefringence 79 Inverse magnetochiral effect 79, 96 Inversion i 26, 75 IR (infrared) 14, 15 Irradiation 203, 206 Irregular satellites 102 Isometric chiral parts 134, 136

K Kepler
s laws of planetary motion 85 Knots 129, 138 oriented 138 theory of 140 Kronig–Kramers transform 8, 59 Kuhn–Condon sum rule 59, 207

L l and d 12 l and d 13 La Coupe du Roi
135 Lactic acid 11 Langmuir–Blodgett layer 171 Left-handed 26 Left-handed helicity (M) 5, 14, 135, 137 Levorotatory 13

Subject Index

Lewis Carroll 23 Ligand partitions 142 Light circularly polarized 52, 64, 66 elliptically polarized 67 linearly polarized 52 Light scattering, Rayleigh 54, 56 Linearly polarized light 5, 52 Link 138 Linking number 129, 139 Liquid crystal 120, 170 Lord Kelvin 10, 129 Lorentz force 49 Lorenz attractor 109

M M1 (magnetic dipole allowed) 37, 55 Magnetar 95 Magnetic circular dichroism (MCD) 59 Magnetic dipole interaction 55 Magnetic dipole transition moment 15, 37, 57 Magnetic field galactic 96 of stars 95 of the Earth 106, 207 Magnetic helicity 130 Magnetic optical activity 59 Magnetochiral anisotropy in electric conduction 80 in emission 69 Magnetochiral birefringence 70 Magnetochiral effect 56, 60 in absorption 61, 68 in birefringence 70 in conduction 80 in emission 67, 69 inverse 79 light scattering 56, 163 photochemical 72, 207 Magnetochiral inverse effect 79, 96

Subject Index

Magnetochiral light scattering 163 Magnetochiral photoresolution 72, 207 Magnetochirodynamics 82 Meson K- 32 p+- 32 Mesoscopic scale 122 Messenger RNA (mRNA) see RNA Metabolism 197, 206 Metabolism First 200 Meteorite Murchison 209 Murray 209 Miller indices 123 Mineralogical chirality 110 Minimal projection of a knot/link 138 Mirror image 5, 26, 92, 105 Mçbius configuration 153 Mçbius strip 152 Molecular dynamics/mechanics 169, 190 Molecular knot 138, 141 Molecular PV energy shift 42, 45 Molecules, parity violation in 41, 200 Momentum vector 27, 31 Monte Carlo simulations 170 Moon 25 Moons 102 Morphological chirality 3, 216 mRNA (messenger RNA) see RNA Multipole expansions 55, 161 Multipole moments 162 Muscarine 18

N Nanotube, chiral 82, 118, 125 Nematic phase 120 Neutrino 31, 88, 90 Neutron star 91 Newton 51, 85 Newton
s equations of motion 85, 101 Nuclear reaction, parity violation in 33

245

O Ocean-bound 204 Octant rule 148 Olfactory sense 18 Oligomer 179, 181, 185, 187 Oligonucleotide 186 Oparin Ocean Scenario 204 Optical activity magnetic 56, 59, 66 natural 6, 40, 56, 57 Optical rotation, natural 6, 41 Optical rotatory dispersion (ORD) 6, 8, 58 Organic crystal, homochiral 119, 180, 190 Oriented knot 138

P Parity 25 Parity mixing 39 Parity nonconservation 38, 41 Parity operation (P) 26, 31 Parity violation 5, 25, 31, 33, 38, 41, 65, 93, 200 Pasteur 8, 16, 118, 190 PC Symmetry 32, 47, 93 Peptide bond formation 179, 181, 216 Peptide chain 138, 182, 215 Peptide nucleic acid 188 Permutation group 142 Phase boundary 171 Phase problem 16 Photogalvanic effect 127 Photon spin 54, 57, 60 Photoresolution, partial 19, 71, 207 Photosynthesis 205, 213 Photosynthetic reaction center 213 Planetary motion 85, 102 Planets 102 b-Pleated sheet 9, 141 Poincar
map 103

246

Point group 76, 115, 116 Polar vector 27, 31 Polarimeter 6, 41 Polarized elementary particles 28, 32, 33, 203 Polymerization, homochiral 180, 182, 187, 189, 215 Polynucleotides 184, 187 Polypeptide a-helix 9, 151 Polypeptides 9 homochiral 181, 182 self-replicating 182 Prebiotic evolution 199, 200 Propeller 3, 76 Protein synthesis, biological 215 Pseudoracemic 110 Pseudoscalar 27, 39, 57 PT Symmetry 78

Q Quadrant rule 150 Quartz 6, 111, 201

R Racemate resolution 16, 71, 125, 175, 176, 180, 190 Racemic 9, 16 compound 119 liquid mixture 9, 118 mixture 9, 125 solid solution 119 Radial distribution function 169 Raman optical activity 15 Rayleigh light scattering 56 Rayleigh–B
nard convection 107 Reflection s, i 26, 75 Refractive index 7, 53, 57, 69 Replication 184, 198 Ribosomal RNA (rRNA) see RNA Ribosome 214

Subject Index

Ribozyme 198 Right-handed 26 Right-handed helicity (P) 5, 14, 135 RNA (ribonucleic acid) 4 messenger (mRNA) 198 p- 188 ribosomal (rRNA) 198 transfer (tRNA) 198 RNA World 199 Rotation 3, 76, 105, 128 Rotation symmetry Cn 75, 76, 115 Rotational strength 15, 40, 57, 59 Rotation–reflection Sn 75, 115, 157 rRNA (ribosomal RNA) see RNA

S Satellites, irregular 102 Schrçdinger 197 Screw axis 116 Sector rules 15, 147, 150 Selection rules 37, 64 Self-organization 141, 212 Self-replication in oligonucleotides 187, 198, 199 in polypeptides 182 Self-similarity 97, 110 Self-writhe 140 Sequence specificity 189, 218 Smectic phase 121, 122 Solar system 102, 208 Space groups 115 Space-reflected universe 92 Spin 27 electron 28, 39, 63, 65, 66 elementary particle 31 nuclear 28, 34 photon 54, 77, 78 Spin–orbit coupling 42, 64 Spin-polarized 33 Spiral galaxy 98

Subject Index

Spontaneous resolution 9, 118, 125, 173, 180, 191 Star formation and disintegration 91, 95 Stereochemistry 8, 16, 148 Steric selectivity 189 Stochasticity, dynamical 101 Strong interactions 25, 33, 87 Sugars 6, 17, 185, 188, 210 Sum frequency generation 79, 158 Supersymmetric particles 95 Surface chirality, extended 124, 171, 174, 180, 184 Surfaces chiral 123, 173 enantiomorphous 123, 173 Symmetry, basic 93 Symmetry element/operation 75 Symmetry plane s 75

T Tartrate, sodium ammonium 8, 191 TCP Theorem 30 Tellurium lattice 117 Template-directed synthesis 186 Thalidomide 18 Three-body problem 101 Time reversal (T) 28, 33, 56, 76, 77, 80, 93, 103, 111, 162 Topological chirality 129, 138, 140 Torus link 139 Transfer RNA (tRNA) see RNA Transition matrix element 36 Transition moment electric dipole 37, 39, 55, 162 magnetic dipole 37, 39, 55, 162 Translation 3, 78, 115, 128 Trefoil knot 139, 141 Trisoxalatochromate(III) ion 11, 72

247

tRNA see RNA Turbulent motion 107, 128 Twisted p-electron system 153 Two-dimensional crystallites 180

U Unit cell 115 Universe chiral 92, 93, 98 expanding 88 fractal 97 space-reflected 92 Unpredictable orbits 102, 104 UV (ultraviolet) 14, 36

V van der Waals interactions 167, 191 Vibrational optical activity 15 Vortex 128, 130 Vorticity 130

W Water–air interface 181 Watson–Crick base pairing 185, 187, 188 Weak forces/interactions 5, 25, 31, 38, 39, 65, 200 Whitehead link 139, 140 Writhe 140

X X-Ray analysis 16

Z Zeeman effect 65, 66