The new book Molecular Symmetry and Group Theory by R. L. Carter is comprehensive enough for graduate students but accessible enough for undergraduates. This readable text includes a qualitative development of group theory, with applications to bonding and vibrational spectroscopy as well as electronic spectroscopy of transition metal complexes. The introduction to group theory in the first four chapters neatly interweaves basic arguments needed to solve chemical problems in inorganic chemistry. Later chapters on vibrational and electron spectroscopy could be used independently or with connection to other texts.

The organization of the book makes it possible to adapt its use for two levels of students, either undergraduate or graduate (who have not seen group theory). For example, a qualitative procedure for forming SALCs (symmetry adapted linear combinations) and writing MO (molecular orbital) diagrams is outlined in Chapter 4. While this might provide the depth required of an undergraduate course, it also serves as a solid basis for the more detailed discussion on projection operators to generate SALCs in Chapter 5. More advanced topics such as double groups or F and G matrix treatment of molecular vibrations, however, are not included. Carter describes several pedagogically useful techniques within this text: for example, his worksheet method for reducing representations. This chartlike organization for reducing representations is not only useful as a teaching tool but would help minimize mistakes even for one practiced at reducing representations. In an analogous fashion, the projection operator method for generating SALCs is broken down into a sequence of steps, which are also mapped out in tabular form.

While the focus of the text is to present group theory without a full mathematical derivation, connections to the mathematical underpinnings are suggested. These bridges assume a fair amount of knowledge of linear algebra (e.g., there is only a brief treatment of matrix multiplication, block diagonalization, and similarity transforms), but they are not critical to understanding the book. One of the themes in Molecular Symmetry and Group Theory is group-subgroup relationships. Where other texts have left the utility of group-subgroup relationships to the reader, Carter illustrates the advantages in several contexts. He illustrates this principle with a number of examples of using rotational subgroups to write or reduce representations of high-order groups.

I found valuable explanations in Molecular Symmetry and Group Theory that are not commonly found in other textbooks. For example, Carter carefully explains how to reduce representations that involve imaginary characters or representations in infinite-order groups and illustrates several approaches to obtain SALCs for degenerate representations. Carter's book is far more approachable than Cotton's Chemical Applications of Group Theory and more easily adaptable to an undergraduate course. Carter carefully ties each discussion to relevant, concrete examples. He provides a fuller treatment of both group theory and its applications than George Davidson's Group Theory for Chemists, although Davidson nicely divides his development into short, concise, topical chapters. While equally readable, Carter's book follows a more standard approach than a book such as Symmetry and Structure, by S. F. A. Kettle (who develops point groups in parallel with writing SALCs and MO diagrams).

By contrast to almost all the other texts I have mentioned, I particularly appreciate the careful inclusion of data throughout Molecular Symmetry and Group Theory. For example, in writing molecular orbital diagrams, Carter provides the photoelectron spectra of example molecules and relates the peaks to the MO diagram. Likewise, the chapter on vibrational spectroscopy includes both infrared and Raman spectra of compounds discussed. Finally, in the chapter on electronic spectroscopy, UV-vis absorption spectra are illustrated for the d¹-d⁹ metals. Excellent problems accompany each chapter.

In terms of coverage and depth, Molecular Symmetry and Group Theory is well suited to an introductory group theory course. Although I would unbraid some of the arguments in the early chapters (particularly for an undergraduate course), on the whole I found the material very well presented. I enjoyed reading the book and am likely to adopt it in my own course.