## book reviews

**Group theoretical methods and applications to molecules and crystals.** By Shoon K. Kim. Cambridge: Cambridge University Press, 1999. Pp. xii + 492. Price £95.00. ISBN 0 521 64062 8

^{a}Institüt für Theoretische Physik Technische, Universität Wien, Wiedner Hauptstrasse 8-10/136, A-1040 Wien, Austria

According to the author's preface, this book is intended to explain the basic aspects of symmetry groups and their applications to problems in physics and chemistry by using an approach pioneered and developed by the author. The aim is to work out explicitly the structure of symmetry groups and their representations by eliminating the unduly abstract nature of standard group-theoretical methods.

The applied strategy, emphasized by the author, relies upon the fact that all space groups, unitary as well as anti-unitary ones, can be reconstructed from suitably chosen algebraic defining relations of point groups instead of introducing them, as he claims is done in almost all textbooks and monographs on solid-state physics, without any proof as `god-given'. The matrix representations of space groups are determined by projective representations of their associated point groups, whereas the representations of point groups are subduced from representations of the rotation group. Symmetry-adapted linear combinations of equivalent basis functions transforming according to unitary irreducible representations of point groups are deduced by applying the so-called *correspondence theorem*. This method is used not only to form molecular orbitals and symmetry coordinates of molecules but also to construct the energy-band eigenfunctions of Hamiltonians which are invariant with respect to space groups.

The *Preface* states that the book will be of great interest to graduate students and professionals in solid-state physics, chemistry, mathematics and geology, and especially to those who are interested in magnetic crystal structures.

The book is organized as follows: Chapter 1, *Linear transformations*, discusses the basic concepts of vector spaces and linear transformations and related topics. Chapter 2, *The theory of matrix transformations*, introduces general matrix transformations by putting the main emphasis on so-called involutional transformations. Chapter 3, *Elements of abstract group theory*, describes the basic concepts of groups, subgroups and mappings between groups. Chapter 4, *Unitary and orthogonal groups*, discusses the definition and properties of the unitary group *U*(*n*), of the orthogonal groups *O*(*n*, **C**), *O*(*n*, **R**) including the special case *O*(3, **R**). Chapter 5, *The point groups of finite order*, lists all possible finite subgroups of the rotation group including their basic properties. Chapter 6, *Theory of group representations*, treats the basic concepts of carrier spaces, like Hilbert spaces, and linear operators, as well as matrix representations of groups by including the properties of irreducible matrix representations. This concept is used to construct, amongst others, symmetry-adapted functions either by means of generating operators or by means of projection operators. Chapter 7, *Construction of symmetry-adapted linear combinations based on the correspondence theorem*, introduces an alternative method for the construction of symmetry-adapted states which is applied to compute hybrid atomic orbitals or, likewise, symmetry coordinates of molecular vibrations. Chapter 8, *Subduced and induced representations*, treats the construction of subduced and induced representations. Special emphasis is put on the situation where the of a given is an invariant one, since for all such cases a step-by-step procedure exists that allows the systematic construction of induced representations for the supergroups that are irreducible. Chapter 9, *Elements of continuous groups*, presents some basic aspects, like the parameterization of group elements, the invariant integration, Lie algebras, the connectedness and the multivalued representations of continuous groups. Chapter 10, *The representations of the rotation group*, discusses the structure of the special unitary group *SU*(2), since the latter is the universal covering group of the special rotation group *SO*(3, **R**), which explains why single- and double-valued representations of *SO*(3, **R**) exist. Possible representations are realized by introducing generalized spinors and angular momentum eigenfunctions and by using coupling coefficients. In Chapter 11, *Single- and double-valued representations of point groups*, the double-valued representations of point groups are represented by projective representations. As an alternative, ordinary (vector) representations of double point groups are discussed where, apart from the point groups of finite order, the most general point groups of the type *C*_{∞} and *D*_{∞} are considered. Chapter 12, *Projective representations*, introduces the basic concepts of projective representations of groups by demonstrating how covering groups and so-called representation groups are interrelated. Chapter 13, *The 230 space groups*, introduces the basic definition of space groups. Special emphasis is put on the definition of the 14 types and the 32 crystal classes. These properties are used to define 32 minimal generator sets in order to deduce the 230 space-group types. Chapter 14, *Representations of the space groups*, introduces reciprocal lattices and their to specify the general induction procedure for the construction of unitary irreducible matrix representations of space groups by favouring the use of projective representations of the underlying point groups. Chapter 15, *Applications of unirreps of space groups to energy bands and vibrational modes of crystals*, uses the free-electron model as the simplest model to construct symmetry-adapted wave functions that are composed of plane waves and transform according to unitary irreducible matrix representations of space groups. To show the different structure of symmetrized wave functions, one example deals with a symmorphic another with the symmetrization of the basis functions with respect to a non-symmorphic The construction of symmetry coordinates of vibration for a crystal with the diamond structure is discussed. Chapter 16, *Time reversal, anti-unitary point groups and their co-representations*, introduces the concept of time-reversal symmetry in classical and quantum mechanics. The definition of so-called anti-unitary point groups together with their co-representations is treated by putting special emphasis on the construction of unitary irreducible co-representations. Chapter 17, *Anti-unitary space groups and their co-representations*, introduces some specific extensions of space groups, here called anti-unitary space groups of the first kind and of the second kind (but presumably better known as Shubnikov or magnetic space groups) by extending the minimal generator sets for space groups correspondingly. The unitary irreducible co-representations of anti-unitary space groups are constructed *via* projective unitary irreducible co-representations of the underlying point groups. As possible applications, selection rules of Hamiltonians that are invariant with respect to anti-unitary space groups are discussed in general terms, but without reference to a single concrete example.

At first glance, on the basis of the concepts presented and the material discussed, the book would seem to qualify as an extremely ambitious project. The merits of the book are that it offers many details on the algebraic properties and matrix representations of

and magnetic space groups as well as applications to a number of physical problems such as the symmetry adaptation of functions needed for quantum-mechanical calculations.Unfortunately, however, a closer look reveals many drawbacks. The mathematical standards are on a rather modest level, presumably intended by the author to achieve better readability. Even so, some statements are misleading or even wrong as can be judged from the following examples, which are far from being complete. In Section 1.1, there is no proper distinction between vector spaces and point spaces since both are put on an equal footing throughout the whole text. In Section 6.1 (p. 102), one reads that a Hilbert space is called *compact* if the limit of any sequence of vectors in the space, if one exists, belongs to the space. In Subsection 6.1.2, linear operators are introduced without any hint that every operator possesses a domain of definition. On p. 104, one reads that the eigenfunctions of a single observable define a complete orthonormalized basis system of the Hilbert space. On p. 124, one finds that a group *G* = {*g _{i}*} is simultaneously regarded as group space whose elements are functions with

*G*as domain of definition and

**C**as range of variation. On p. 132, one discovers in Tables 6.4, 6.5 and 6.6 elementary bases of the form cos(

*n*θ) and sin(

*n*θ), although bases of this type were initially introduced as homogeneous polynomials of a certain degree. In Subsection 7.2.1, an equivalent

*S*

^{(n)}is introduced, which in the mathematical literature is known as the

*G*-orbit. On p. 153, one finds the curiosity that

*S*

^{(n)}is simultaneously regarded as

*n*-dimensional without any further assumptions. On p. 154, some normalized

*s*-orbitals (

*s*

_{1},

*s*

_{2},

*s*

_{3},

*s*

_{4}) are introduced (without explicit specification), which apparently are assumed to be located on the points of a

*G*-orbit,

*S*

^{(4)}, and thus are assumed to be mutually orthogonal, which, from the mathematical point of view, is wrong, since

*s*orbitals are Gaussians whose overlap integrals are non-zero. On p. 342, one finds the rather strange statement that the imposition of periodic boundary conditions, which are due to Born–von Karman, does not modify the physics although, for instance, the continuous has to reduce, owing to the boundary periodic conditions, to a finite set of special

**k**vectors. On p. 365, one finds a statement that for realistic periodic potentials no accidental degeneracy may occur for the energy eigenvalues, which is in absolute contrast to well known results in the mathematical literature dealing with the spectral properties of periodic Hamiltonians.

The readability of the book is rather cumbersome, which is not only because of the conventions adopted and the notations used but also because of the presence of a great number of misprints. Although the majority of the typographical errors are obvious, some of them are extremely misleading, especially for non-experts. As regards the notations and related terminology, several inconsistencies are unavoidable. For instance, it might be rather astonishing for every crystallographer to find that the book talks about 230 space groups and definitely not about 230 space-group types or, similarly, to regard it as not worth mentioning that the 230 space-group types are subdivided into 73 arithmetic classes. This and some other statements might make it hard, especially for a crystallographer, to get access to the methods employed in the case of space groups and their extensions to magnetic space groups.

For a textbook, it is unfortunate that there are also some statements that are at least misleading or even plain wrong. For instance, in Section 6.9, one finds that formula (6.9.2) clearly shows that the so-called multiplicity problem is disregarded since no comment is made in this respect. This type of problem appears when the carrier space of a representation of a group contains an irreducible representation more than once. In this context, a criticism must also be offered that for the alternative method being used to construct symmetrized states, the so-called *correspondence theorem*, the multiplicity problem is likewise entirely disregarded. There is no hint how to treat such problems. Another misleading or even wrong statement can be found on pp. 157 and 158, where Bloch and Wannier functions are discussed. The Bloch functions defined by (7.2.17a) are composed of Gaussian orbitals, which are localized at points. The normalization constants occurring in (7.2.17a) are assumed to take values that come not only from the periodic boundary conditions but also from the erroneous assumption that the localized orbitals are mutually orthogonal. This is wrong, since the overlap integrals of Gaussian orbitals are non-zero. By virtue of these wrong assumptions, the Wannier functions defined by (7.2.17b) are wrong too since they coincide with the original localized orbitals, which cannot be the case because the non-zero overlap integrals of the Gaussian orbitals are neglected. In this context, the classical paper by G. H. Wannier [*Phys. Rev.* (1937), **52**, 191] should be read in order to get a clear insight into the proper description of this subject. Likewise, it is misleading to use on p. 190 the phrase `… it is safe to assume that the set {*S˚ _{i}*ψ} is linearly independent', because these functions have to be linearly independent, as otherwise the induced representations would be of another form. On p. 376, one finds the statement that the simple application of the correspondence theorem does not yield the correct linear coefficient for the partner function. Moreover, on p. 388, one finds the extremely misleading comment that the degenerate bases are orthogonal if their overlap integrals are neglected, although at that instant the symmetry coordinates of the vibrations of a crystal are the subject of discussion.

The general strategy used throughout this book is to employ results that are very often stated without a complete proof or are proven only in later sections. Another aspect concerns the sometimes rather misleading labelling of symmetry-adapted functions, since bases of the type *z*, *x − y*, *x*^{2} − *y*^{2} are used as symmetry labels, even in cases such as Table 6.9 on p. 140, where the basis functions are composed of plane waves. Apart from this, as already pointed out, the discussion of the multiplicity problem is not properly carried out despite its crucial importance in almost all realistic applications. In addition, subgroups of space groups are treated in an extremely unbalanced manner, since only so-called *translationsgleiche* subgroups are discussed, whereas neither *klassengleiche* nor even more general subgroups are mentioned at all. Nevertheless, the author does not hesitate to regard his brief comments on *translationsgleiche* subgroups on p. 336 as valuable information on structural phase transitions. Similarly, the discussion of symmetries of magnetic structures is restricted to a few sentences on p. 429, where some brief comments are made on ferromagnetic, antiferromagnetic and ferrimagnetic structures. In this context, one cannot find a single remark that there exist more complicated magnetic structures whose symmetries cannot be described by Shubnikov space groups. Finally, the book suffers from a lack of concrete applications as regards the computation of selection rules. Thus, the examples considered are only treated in general terms and never tied to a concrete interaction, such as the Coulomb interaction, or the interaction with an electric field or with a magnetic field, or the phonon–electron interaction, to name but a few representative ones.

The publisher recommends the book to mathematicians, among others, which is rather questionable given its defects. Its recommendation to geologists is curious, as one cannnot find a single hint in the whole book as to which methods or presented applications are of particular interest to geologists. Finally, to recommend this textbook to physicists or chemists who are in particular interested in magnetic crystal structures is unjustified, as there is no single concrete example where the symmetries of ferromagnetic or anti-ferromagnetic structures are discussed in detail. Merely three sentences on p. 429 are devoted to this type of problem without any further hints on how to treat the symmetries of such magnetic structures.

The citation of standard references in the field of group theory with applications to physical problems is unfortunately treated by the author in an unsatisfactory manner. Standard references, such as Curtis, C. W. & Reiner, I. (1962). *Representation Theory of Finite Groups and Associative Algebras.* New York: Interscience; Jansen, L. & Boon, M. (1967). *Theory of Finite Groups. Applications in Physics.* Amsterdam: North-Holland; Hahn, Th. (1998). Editor, *International Tables for Crystallography*, Volume A: *Space-Group Symmetry*. Dordrecht: Kluwer; Butler, P. H. (1981). *Point Group Symmetry Applications: Methods and Tables.* New York: Plenum; Altmann, S. L. & Herzig, P. (1994). *Point Group Theory Tables.* Oxford University Press; Bradley, C. J. & Cracknell, A. P. (1972). *The Mathematical Theory of Symmetry in Solids, Representations Theory for Point Groups and Space Groups.* Oxford: Clarendon Press; Birman, J. L. (1974). *Theory of Crystal Space Groups and Infra-Red and Raman Lattice Processes of Insulating Crystals. Handbuch der Physik*, **XXV/2b**. Berlin: Springer; Opechowski, W. (1986). *Crystallographic and Metacrystallographic Groups.* Amsterdam: North-Holland, are, if given at all, cited in inappropriate locations. For instance, not to quote *International Tables for Crystallography* Volume A at the beginning of the discussion of space groups, in a book directed at crystallographers, is odd. Apart from this, some topics, like the symmetries of magnetic structures that can be described by Shubnikov space groups are, as already pointed out, not precisely enough nor sufficiently well discussed by the author. In this context, for instance, the book of Opechowski has to be consulted in order to get the desired insight into these types of problems.

**Note**: this review is an extended version of the review published in the print version of the journal.