Book Review  

Beginning Group Theory for Chemistry by Paul H Walton

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Beginning Group Theory for Chemistry, by Paul H Walton. Workbooks in Chemistry Series, Oxford University Press: Oxford, England. 144 pp, £16.95.

Group theory is an "all or nothing" topic within university chemistry courses: students either fail completely to get through its rather abstract nature or, alternatively, score very highly on problem-oriented exam questions. Any book that can transform the former into the latter will be welcomed by staff and students alike, and in general the Walton workbook will help do this. In content it is midway between two other established texts. It has the textual, explanatory component of Davidson's Group Theory for Chemists combined with the "self-learning through problems" method initially adopted by Vincent in his book Molecular Symmetry and Group Theory. In combining the two approaches, Walton has produced a book that contains sufficient explanatory material to act as a textbook to support a traditional lecture course, while giving students the chance to evaluate their understanding through regular self-assessment questions, supported by model answers which follow immediately. Additional problems at the end of each chapter that require the students to apply their acquired knowledge would have been useful, but this does not markedly detract from the format, as these can be found in other widely available texts.

After a brief Introduction, Chapter 2 deals with symmetry operations and point groups. Provided students have purchased model kits, this topic should not prove too difficult and the chapter follows traditional lines. Chapters 3 and 4 discuss groups and their representations, and form the interface between symmetry (and its application in vibrational spectroscopy) and bonding. This is usually the point at which students become lost, and the correct balance is important. It is also the most difficult task facing anyone teaching group theory and is inevitably the most contentious. The idea of "a group" is introduced using the analogy of soldiers on parade who can turn about, left, or right; this is an uncomplicated way of showing how operations can combine. The jump to replacing operations by integers requires the student to guess valid representations and, as this can be a little hit or miss, it may frustrate the unlucky guesser. Asking students to demonstrate that a certain collection of integers is a faithful representation of the operations would take students to the same point a little quicker.

Chapter 4 covers matrices, characters, and their use as representations and is a chapter that the author suggests (but does not recommend) nonmathematically minded students can omit. I can't agree with this. Matrices are the mathematical link between symmetry operations and groups and are not so difficult they cannot be explained with some simple examples. What is more difficult is for students who have ignored this chapter to take at face value a series of statements which define basis, reducible and irreducible representations, and characters at the outset of Chapter 5, which is entitled "The Heart of Group Theory." I can't help feeling that this latter crucial chapter is lost before it is begun without Chapter 4. Chapter 4 also introduces the idea of chi per unshifted atom and I find it surprising that, in a book that asks students to verify their understanding through regular self-assessments, this important concept is dismissed with the phrase "these equations can be easily verified by writing out the appropriate full matrix" but without further discussion. Perhaps it is in anticipation of many students omitting this chapter, but when the same concept reappears later in the book, its origin must be a puzzle to many.

Notwithstanding the above comments, Chapter 5 describes the use of the reduction formula in a clear manner which students will find particularly helpful. It is then used to derive irreducible representations for the hydrogen 1s orbitals in H₂O, which immediately makes a link with MO theory and places a rather abstract concept into a chemical context. Chapter 6 continues the applications theme and describes the use of group theory in the analysis of vibrational spectra. In so far as it reinforces the use of the reduction formula, this chapter is well presented. This is followed by the use of projection operators to describe the form of the various vibrational modes and here I am less sure how well this fits into a beginners' course. It is helpful to show how a pictorial view of stretching–bending modes can be derived, but in simple cases (e.g., the symmetric and antisymmetric stretching modes for H₂O) the A₁ and B₂ symmetries are easy to demonstrate by other means. I can't see much point in extending this to degenerate cases where the choice of generating coordinate is, as the author quite rightly says, beyond this text. This then leaves the student with a knowledge of the methodology but not the ability to apply it generally. I think this is a question of where one draws the line defining a beginners' course, and for those courses that emphasize the use of projection operators, what is incorporated in the workbook is of value.

The issue of relevance continues through the latter part of the chapter, where selection rules for IR and Raman spectroscopy are demonstrated using direct products. The same discussion applies here as to projection operations, but whatever one's view on this I can't understand why the basic rules are not more clearly spelled out: IR modes have the same symmetry as T_x, T_y, or T_z and Raman active modes the symmetry of one of the binary terms. In practice, this is what is actually used.

The workbook concludes with a more detailed look at the application of group theory to molecular orbital problems and much the same comments apply to this chapter as to that on vibrational spectroscopy. The generation of reducible representations and their combination in the correct manner to form MOs is clearly explained, but the shortcut that the symmetries of orbitals on the central atom can be read directly from the character table lies buried in Chapter 5. It seems a general weakness of the workbook that these key shortcuts are not highlighted more forcefully. Projection operators again figure heavily in this chapter, both for terminal atom SALC combinations and for molecules without a central atom. There is nothing wrong with this, other than its value in a beginners' book. Personally, I find that reasonably good pictorial representations can be arrived at simply by using the nodal patterns for linear arrays of orbitals, and increasing the number of nodes symmetrically as energy increases, cyclic arrays being derived from these by joining the ends of the appropriate linear array (there are very useful diagrams in, for example, Inorganic Chemistry by Purcell and Kotz). While such an approach is not as rigorous as the projection operator method, it is far easier for students to assimilate.

Overall, it is easier to be critical of a book than to praise it, and this review might sound more negative than intended. No book is read by students in isolation and without recourse to lecture material. As a book that students can work though to support lectures it has much to commend it and, as the author points out, group theory is one of those topics where practice does indeed make perfect. The book is logically arranged and contains a good balance of background, vibrational spectroscopy, and bonding. It is not unreasonably priced at about £16 and I shall certainly be asking our bookstore to stock it. I will also recommend it to my students as a book they should consider using as part of their learning strategy.