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Book Review Group Theory in Chemistry and Spectroscopy. A Simple Guide to Advanced Usage. By B. S. Tsukerblat

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systems far from equilibrium”. But it is
much more than this, for Professor
Cramer easily straddles the gap between
the two cultures, in a virtuoso display of
cross-references and interconnections.
Taking his own research as a starting
point he aims at the general picture. deftly
crossing the boundaries between scientific
disciplines and artistic fields. Poems by
Holderlin and paintings by Kandinsky are
used to make a point about the delicate
transition between order and chaos. He is
right to use this approach, for it is at the
fuzzy borders where different disciplines
meet that innovation thrives. The hook is
also a very personal work (Professor
Cramer even confesses his reluctance to
part with it). and thus more universal in its
In Chum cine/ Order some fundamental
questions are asked-- what is life? how do
ideas arise in our brain? what does evolution mean?- but the author is careful to
separate the methods of science and philosophy. As he puts it. “scientific research
cannot yield moral values and cannot
teach us the meaning of life”. Only children ask “why?” and only the selfish start
their enquiries with “what for?”; the scientist’s business is to answer the “how?”
According to Goethe, what cannot be investigated is better left to contemplation.
But Professor Cramer does not eschew
ethical matters, stating very forcefully
that “gene therapy is a violation of the
personal rights of future human beings”.
He also does not shy away from denouncing the dangers of a capitalistic exploitation of publicly funded research with its
potential for the prostitution of science.
These matters are all the more important
because, as he persuasively argues in the
last chapter, intellectual evolution has
now overtaken biological evolution. We
do not develop any more organs; we invent domestic appliances instead. Information is passed from generation to generation not only through genes but also
through libraries, data banks, social traditions, etc.
It is only fitting that a book about
structure should itself be highly structured. Each chapter starts with a partly
fictitious dialogue and ends with a poem
(authors ranging from Yeats to Chemistry
Nobel Prize winner Roald Hoffmann).
The dialogues are constructed around an
assortment of scientists (Einstein, Pauli.
Darwin, the 18th century Gottingen
physicist Lichtenberg, etc.), writers
(Kleist, Goethe), philosophers (Socrates
and Wittgenstein) and various fictional
characters (Shakespeare’s Hamlet or Bottom and Carroll’s Alice, the latter because
scientific nonsense can be very meaning-
ful). Clicio.~
miel Order can thus be read on
several levels as a collection of dialogues
intended to show that difficult problems
are neither new nor difficult to discuss, as
an anthology of poetry (in the author’s
words “poetic language is often capable
of describing truths in a more valid. concise and enduring form”, which is not unlike mathematics) or as a masterly essay
on the complex structure of living systems.
Nonlinear systems with feedback coupling and a strong dependence on the initial conditions have the potential for
chaotic behavior. Inevitably Professor
Cramer invokes Wallace Stevens’s poem
Conr7ois.scwr of Chuos. which proclaims
that a violent order and a great disorder
are one and the same thing. One of the
book’s most impressive sections shows
that esthetic harmony can be found at the
irrational boundary separating order
from disorder. The evidence that the
famed golden section is somehow related
to this transition strikes us as a true coup
de theatre. The whole book, even the
more difficult sections, is a pleasure to
read, and is admirably illustrated with examples and illuminating parallels. Thus a
tree can be looked upon as an upsidedown bolt of lightning which propagates
very slowly, and the phenomenon of the
sudden boiling of superheated water is not
all that different from a stock market
crash (both are cusp catastrophes).
I was expecting the book to start --like
some modern-day genes is^ with chaos,
but Professor Cramer devotes the first
chapter to life, seen as a dynamic system
between order and decay. It is a process
governed by entropy, the “arrow of time”
(Eddington). This is the “leitmotiv” of the
book. Once taken apart, living systems die
and cannot be reassembled; in other
words, the whole of life is always greater
than the sum of its parts. As Far as life is
concerned. the Cartesian method should
thus be applied with caution. All living
matter is made from the same building
blocks: for instance, there are more than
10000 different proteins in the human organism, but they are made from only
twenty aminoacids! A sort of atomistic
model seems to prevail, and the various
ways of combining the structural units
make for a tremendous information capability. It is clear why life is too complex to
have been left to chance. The molecules of
life and cells have developed recognition
mechanisms of the lock and key type;
these can be rather selective and sophisticated. given the complex geometry of
molecular surfaces. N o wonder some of
the fundamental problems in biology are
topological (and fractal)!
Much of the book is involved with selforganization and evolution. Along with
heaviness (mass). self-organization is a
basic property of matter. Eigen’s theory
of hypercycles (a mathematical formulation of all possible evolution systems) is
Cramer insists that. like quantum mechanics, Eigen’s formulation is not an explanation but a description, but surely
with the loss of visualization (“anschaulichkeit”) that was brought about
by modern physics, mathematical description became the equivalent of meaning.
Then there is the question of complexity
of highly-organized systems which can be
quantified (a reincarnation of Boltzmann’s equation for the entropy). The
occurrence of bifurcation or fulguration
points in the evolutionary schemes confers a kind of history on the process, and
the systems become adaptable. even intelligent. In the old order of Newtonian systems, forever following their predictable
paths, aging was inconceivable. If equilibrium is abandoned and bifurcations are
possible, the future of the systems cannot
be predicted, and Leibnitz’s famous assertion “Natura non facit saltus” no longer
holds. Instead of following determinism,
the player has to learn the rules of the
game. Predictability has ceased to be a
criterion for scientific truth. But, as Professor Cramer explains in another context. growing old brings wisdom and freedom; ambitions are no longer necessary.
There is a moral to this book, too.
Jorgc C. G. Galado
Department of Chemistry
and Chemical Engineering
Instituto Superior Teknico
Lisbon (Portugal)
Group Theory in Chemistry and Spectroscopy. A Simple Guide to Advanced
Usage. By B. S. Tmkerbht. Academic Press, London, 1994, 430 pp..
The fundamentals of symmetry methods and their applications to chemistry
and spectroscopy are already well covered
by many excellent textbooks and monographs. In the book reviewed here B. S.
Tsukerblat aims to give an account of the
principles and applications of the theory
of finite groups, illustrated by detailed discussion of simple examples. and to enable
the reader to solve practical problems using these methods. The main fields of application covered are the symmetry aspects of the ligand field theory, including
spin-orbit interaction, and of molecular
orbital theoi-!. selection rules. vibrational
spectroscopy. electron spin resonance
spectroscop! (:in area that has hitherto
seldom been treated from this standpoint). and exchange interaction in coord i n;i t i on corn po u nds.
The choice of topics and the methods
~isedare well suited to serve the author's
aims. II'more cart' had been taken in writing the book i t might have successfully
lilled ii large gap that exists in the present
range of' student textbooks on chemistry
and spectroscopy. There is an unusually
high incidence of printing errors. mistakes. and incorrect statements. and the
fact that about IS'%, of the literature citations relcr t o publications in Russian must
lie ;I serious obstacle to readers of a book
i n English. I t should at least have been
possible to rel'er the reader t o traiislations.
or to English-l~iiiguage books covering
s i in i la I- pro ti nci .
The book begins with three introductory chapterx dealing with symmetry oper:ition\. point groups. and their representations. which ;iccordiiig to the preface
"might w e l l he assimilated at ;I pre-univerh i t y levcl". Here we already find numerous printing crrors. such ;IS those in
Table 2.3 listing the 32 crystal point
groups. aiid i n the acheme showing their
sub-groups (Table 2 . 5 ) , examples of poor
choice 01' words o r ambiguous phrases.
such a s "retlection within a plane" (p. 7 ) .
and i I I -chozc'ii exam ples ; t h us, t tie chair
lohexnne is used to illustrate a
body with S, symmetry (Fig. 2.12).
whereas there is no mention of it a s an
example of;^ body with the higher symmetry D %,,
. Many of the statements are simply I\ rong: x m e examples are: "CLvhas
three cI;isscs" (p. 33) this should be
four: oil page 29 it is stated that for odd
valiiec o f ~ the
i operation S,, is "not a new
symmetry oixration". despite the fact
Ihat the g r o u p C',,,, are not introduced until later: on page 43 we read "all planar
moIcciiIes pos~essingii C',, axis belong to
D,,,," the last symbol should be C,,,,. Errors s ~ i c h:I\ these arc confusing to the
novice. Equally :innoying are the occasional inconsistencies: for example. in
Equation (3.8X) a symmetry operation is
applied to the variables of a function
rather than. as previously defined in
Equation ( 3 57). to the function itself.
Also the setting up o f character tables is
not cltiitc x o simple as is implied in Section 3.6 ; the quadratic equations used
here ha\ e some redundant solutions.
Some iinportant topics of representation theor\i. such as the classification of
energy eigenfunctions. the direct product
of representations. Clebsch-Cordan coefficients. the Wigner - Eckart theorem.
projection operators. and the basis functions of irreducible representations are
treated in connection with the ligand field
theory (Chapters 4 -6). The solution to
the problem of classifying energy eigenfunctions in terms of irreducible representations is "proved" on page 99, despite
the fact that the concept of irreducibility
must be introduced as a postulate or via
the definition of "accidental degeneracy".
Even some statements that are given special prominence by being enclosed in a
"box" iire badly formulated. For example, in restricting a group to a subgroup, an irreducible representation does
not necessarily give ii reducible one
(p. 101). Clebsch-Gordan coefficients
are introduced by means of tables
(pp. 135. 232). but the discussion does not
include ii brief explanation of how they
are calculated. which would make the
tables understandable. These important
topics. the results of which are needed frequently in later parts of the book. deserved to be treated in a separate chapter.
The description of the ligand field theory
is not very satisfactory. On many aspects.
such a s the calculation of energy level
splittings. the relationships are not properly explained. and the reader must be
content with mere literature references.
Even some minor aspects are not discussed thoroughly enough. for example
the effect of configuration mixing on the
energy terms of ii d' ion (p. 148).
I n Chapter 7 on directed valences, the
orbitals given for a trigonal-hipyramidal
structure such a s that of PF, include not
only the correct hybrid orbitals of the
sp"d and spd3 types but also some incorrect ones, including sp3 (p. 186). I n Chapter 8 it is explained how molecular orbitals are classified according to group
theory; here the author exen manages to
make mistakes in calculating the n-MOs
of benzene (p. 207).
Selection rules cuii be expressed in the
form of conditions applied to the integrands of the corresponding transition
matrix elements. However. in Chapter 9
on optical transitions. it is disturbing to
find that the descriptions of "necessary"
and "sufficient" conditions ("only if' and
"if', respectively) are often used incorrectly (pp. 225. 238). Furthermore, through the
uncritical use of the formalism of representational theory. the Laporte rule forbidding pure d-d dipole transitions is
wrongly described as a "supplementary
restriction" in the case of \?stems without
an inversion center (p. 235). On the other
hand. the discussion of tu o-photon transitions in terms ofsymmetry i n Section 9.5
is very useful, especially ;I\ this is unlikely
to be found in other book\ on the subject.
Chapter 10 is devoted t o double
groups. Chapter 11 then builds on that
discussion by treating the >pin-orbit interaction. Chapters 12 and 13, dealing respectively with the symmetry aspects of
ESR spectroscopy and Mith exchange interaction i n coordination compounds. iire
the best ones in the book. after one has
corrected a few errors 111 the signs and
terms of some of the fornmulas. I n particular, the author constructs effective Hamiltonian operators for ii paramagnetic ion
in hgand fields of different symmetries,
and those for describing electric ticld effects in ESR spectroscop!. which is especially interesting. The use of group theory
for the classification of exchange multiplets in polynuclear complexes and of systems with ions in different valence states is
also explained. with many examples. The
main text ends with Chapter 14. dealing
with normal vibrations mid the JahnTeller effect. and with vibrational coupling in mixed-valence systems.
The appendix containx the matrices of
the irreducible representations for various
important point groups. the associated
basis functions in the form of symmetryadapted linear combinations o f harmonic
functions. the decomposition of product
representations into thcir components.
and symmetry-adapted Iiainiltonian operators for special ESR problems. It
would also have been iisef~ilhere to iiiclude answers to the exercix problems.
The work is not, as claimed in the preface, a handbook on the subject. Unfortunately it contains too many errors. far
more than have been mentioned here. and
too often the author fiiils to explain the
relationships correctly. I t cannot be recommended for beginners. except for gaining a general impression of the extent to
which group theory a n d representation
theory can be succesdully applied to
problems of chemistry anti spectroscopy.
On the other hand. the more advanced
reader who is prepared lo tolerate the errors in the book will find in it many interesting examples and problems that iire not
treated in other books on the subject. 1ltia.w
Ins t i t LI t I'iir Ph y si k a I i sche
und Thcoretische Chemie
der Freien Universitiit Berlin (Germany)
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