SYMMETRY OF CRYSTALS AND MOLECULES 2 2 2 − / − / − / ¯ The orthorhombic space group Fddd F d d d ; origin on 222, at 1 8, 1 8, 1 8 from 1. Symmetry of Crystals and Molecules MARK LADD Formerly Head of Chemical Physics, University of Surrey “A thing is symmetrical if there is something you can do to it so that after you have finished doing it, it looks the same as before.” Hermann Weyl (1983): Symmetry 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Mark Ladd 2014 The moral rights of the author have been asserted First Edition published in 2014 Impression: 1 All rights reserved. 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Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. Foreword The observation that stone-age people produced objects with a measure of symmetry is not as unexpected as often made out. The opposite would actu- ally be much more surprising. Totally asymmetric objects in the real world are extremely rare, if not altogether non-existent. Some objects are admit- tedly more symmetrical than others, and this rule applies without exception. Perfectly symmetrical objects are therefore as rare as the totally asymmetrical. The most perfectly fashioned sphere or the most exquisite crystal contains some blemish when examined in sufficient detail. The distinction drawn by chemists between chiral and achiral molecules is meaningless. I consider it all a matter of resolution; any molecule has a degree of chirality between zero and one. The powerful concept of symmetry applied with such good effect in science is an abstract idealization with precise definition only in the sense of logic. Like the absolute zero of temperature or the relativistic limit to material velocity it can be described mathematically, but remains physically unreachable. The mathematical tool that describes symmetry is known as group theory. In terms of Klein’s famous Erlangen program any geometry is characterized by a group of transformations under which its propositions remain valid. In this sense the Euclidean group is more restrictive than the projective group, and both of these are less general than the continuous topological group. Topology allows any amount of stretching or compression, without tearing; in this most general case the popular notion of symmetry disappears, but remains embedded in the group structure. To produce a monograph on a topic as wide as symmetry needs some cour- age. In doing so Mark Ladd wisely stipulated at the outset that the book is restricted to crystal and molecular structure, for which he is perhaps uniquely qualified. Molecular structure in this context strictly refers to the nuclear framework as revealed in crystallographic analyses. The appropriate sym- metry group in this case is the three-dimensional discrete Euclidean group, commonly referred to as space-group symmetry. As a seasoned crystallographer, albeit of forgotten vintage, I do not find much new in this presentation, but I marvel at the comprehensive detail and rigour. In the days of visually estimated x-ray intensities and Beevers–Lipson strips, crystal geometry and space-group symmetry were experienced as prac- tical necessities. It is not clear where and how the new generation, who twiddle a few knobs to produce a crystal structure, acquires this know-how. Working through the numerous examples could well provide the necessary hands-on experience. This is where I see this book becoming essential reading, if not prescribed for intensive study. Too many papers are being published around dubious symmetry arguments. vi Foreword Up to a point the analysis is based on Newton’s approach that relies only on conclusions which can be formulated mathematically—without the need of hypotheses. I notice in particular how the author wisely refrains from offering physical interpretations for observed aspects such as chirality, optical activity and other symmetry related phenomena, identified and manipulated purely as point group variables. Unfortunately, the temptation to reinterpret molecular symmetry in terms of hybrid orbitals, a concept popular in quantum chemistry, was clearly irresistible. It is common chemical practice to visualize Lewis type electron pair cova- lent interaction in terms of real orthogonal classical functions, called atomic orbitals. These functions, such as the so-called px, py and pz orbitals are often confused with quantum mechanical wave functions, which they resemble after a fashion. However, these functions cannot be described by quantum num- bers, conflict with the exclusion principle, do not represent surface harmonics, which are complex and hence cannot be used to describe electronic orbital angular momentum nor spin. Atomic orbitals are routinely used in linear com- bination, in the same way as Euclidean unit vectors, to visualize directed covalent interaction, inferred from crystallographic analyses, to derive point group symmetry of molecular species. These concepts are, admittedly, not eas- ily avoided, being so thoroughly entrenched in chemical group theory, but a word of warning is perhaps not misplaced. From the symmetry point of view, real px and py functions, which represent linear vibrations, can never be con- sidered equivalent to the rotation defined by a complex function. It is not the point group description of molecules that is at issue, but the way it is used to represent hybrid orbitals. Being of fundamental importance, it could serve as an encouragement for the reader to make an independent assessment, one way or the other. Professor Jan Boeyens University of Pretoria Preface The study of crystal and molecular symmetry is often considered difficult by those meeting it for the first time. I believe that there are two particular reasons for this problem. First, crystals and molecules, unlike the more familiar plane figures, are three-dimensional bodies and it can be difficult to take in simultan- eously all parts of such a body and see them in relation to the whole. In two dimensional figures, such as a rectangle or a regular hexagon, it can be done readily and we appreciate the symmetry without difficulty. The second reason, not unrelated to the first, lies in the unfamiliarity of three-dimensional rela- tionships, notwithstanding we encounter daily the three-dimensional spatial aspects of our space-time continuum. The purpose of this book is to present crystal and molecular symmetry in a straightforward and practical manner. It evolves from lecture courses given over many years, at undergraduate and postgraduate levels, to those pursuing crystallography, chemistry and materials science. It makes liberal use of ste- reoviews and computer aids to understanding the textual material. Most of the stereoscopic illustrations were drawn with the program PLUTO, by courtesy of Dr WDS Motherwell, University of Cambridge, and aids to stereoviewing are discussed in an appendix. Each chapter concludes with a set of problems and detailed tutorial solutions are provided. Computer programs relevant to the text and the problems have been devised, and are available via the publisher’s web site, <www.oup.co.uk/companion/ladd>. It is informative to study everyday objects in relation to symmetry: a glass and a beer jug; a chair and a table; a tiled floor and a brick wall; a Dalmatian and a Dobermann. In this way, symmetry can become clearer—it is not only about crystals and molecules. Two notations for symmetry are in use, the Hermann–Mauguin and Schönflies notations; each has a place in crystal and molecular symmetry, and each is discussed in detail. The study of the symmetry of crystals and molecules may be begun either with lattices or with a description of the external symmetry of crystals. The former has advantages in a short lecture course aimed at getting to grips with X-ray diffraction. However, a morphological approach has been chosen for this book because, in this way, the book should have a more general appeal and its pace a little slower. Time is needed in order to assimilate the concepts of symmetry and to consolidate them into a working knowledge of the sub- ject. It is hoped that this book will be helpful to all those meeting symmetry for the first time, whatever their specialization, and will prepare the reader for study of the current definitive texts on symmetry, the International tables for crystallography, Volumes A and A1. Pre-knowledge is confined mainly to mathematics and physical sciences, but no more than would be studied at A-level. A little more mathematics, along viii Preface lines indicated, is an advantage in certain contexts, and appropriate guidance is provided in appendices in order not to interrupt the flow of the text.