Book Reviews Anwendung der Gruppentheorie in der Quantenmechanik. Authors : M. J. Petrac-

shen and E. D. Trifonow. Published by: Akademie-Verlag, Berlin, 1969. No. of pages: 240. (Translated from the Russian original npHMCHeHHC TeopHir rpynn B I C B ~ H T O B U ~ ~

DICXRHKKC M. 12. IIETPAIiIEHb E. 2. TPEIQOHOB Verlag Nauka, Maskau. Also avaiIable as: Applications of Group Theory in Quantum Mechanics. M.I.T. Press, Cambridge, Massachusetts, 1969 and Applications de la thiorie des groupes & la mCcanique quantique, Masson et Cie, Paris, 1970.)

Within the limited space of 240 pages the authors have succeeded in giving a very concise survey of all the main applications of group theory in quantum mechanics. About a third of the book is devoted to the theory of abstract groups and their representa- tions, including continuous groups. Point groups and space groups are described in con- nection with their use for classifying vibrations and electronic states of molecules and crystals. Four chapters are devoted to the three-dimensional rotation group and its applica- tion to atomic problems. As one could expect in a book from Leningrad, Fock’s explanation of the accidental degeneracy for the hydrogen atom and the corresponding treatment for the isotropic harmonic oscillator receive The symmetric groups and the symmetry properties of many electron wave functions are carefully treated. Two chapters deal with the use of group theory in perturbation theory and the construction of selection rules. In the two final chapters the Lorentz group and the symmetry properties of the Dirac equation are discussed.

Most chapters are provided with a few exercises. Hints for the solution of the more difficult ones are given at the end of the book.

One must admire the authors ability to restrict the topics to just those aspects of the subject which are really necessary for a consistent treatment. Despite the resulting con- centrated text the book is not at all difficult to read. I t can be recommended both to beginners and to students who already know some group theory,

Quantum Chemistry Group Uppsala University Uppsala, Sweden Received November 28, 1973.

due attention.

JEAN-LOUIS CALAIS

The Jerusalem Symposia on Quantum Chemistry and Biochemistry. Edited by E. D. Bergmann and B. Pullman. Published by: The Jerusalem Academy of Sciences and Humanities. Jerusalem 1970, 1971, and 1972.

The three Jerusalem International Symposia for the years 1969-1971 have dealt with the subjects of “Quantum Aspects of Heterocyclic Compounds in Chemistry and Bio- chemistry’’, “Aromaticity, Pseudo-aromaticity, and Anti-aromaticity”, and “The Purines, Theory and Experiment”. The scope of the presented papers comprises a rather pervasive microcosm of theoretically and experimentally oriented interpretive techniques with a primary emphasis on the middle ground. As such the 115 papers presented in these three volumes should provide both chemically oriented theorists and experimentalists with a reservoir of results and data which can be a valuable synopsis and supplement to their knowledge. As with most such conclaves it is not possible to maintain uniform standards and the judgement of the quality of individual discourses, which on the whole was stimu- lating, must be left to the discerning reader.

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A central theme of the 1969 Symposium on Quantum Chemistry and Biochemistry was the problem of intelligibility, simplicity, expense, precision, and accuracy in molecular calculations. Most of the participants, while recognizing the importance and theoretical preeminence of a6 initio calculations, sought to rationalize experimental results by means of parametrization methods which preserved the essrntial feacures of the more refined but lengthier and more expensive methods.

A question always in the background seemed to be whether it was prefcrable to utilize existing computer facilities for a relatively small number of precise non-empirical calcula- tions, which would survive the test of time, or to simplify these calculations by means of some self-consistent parametrization and extend the work to far larger categories of mole- cules. The content of the presented papers was overwhelmingly in favor of the latter approach. The adequacy of a given method provrs to depend largely on the properties being studied with a particularly large division occurring for ground state properties, electronic transitions, and chemical reactivity.

In the introductory paper to the heterocycle symposium A. Pullman presented a series of comparative calculations for adenine, quanine, cytosine, and uracil. Though the details often differed from one method to another, it was found that the essential features of such quantities as dipole moments, ionization energies, and charge densities were similar and compared favorably with experimental results and ab initio calculations.

I n this paper as in several others the gradation in refinement among the various approxi- mation methods has been analyscd. While the original Huckel method, which treated only a-electrons in a non-specifically interacting manner, served to explain many of the salient features of homocyclic aromatic molecules, its failure to treat u and a electron polarizations separately in heterocyclic compounds requires a drastic alteration in the method. Accord- ingly the “Extended Huckel Theory” (EHT) takes explicitly into account the polarizability of the u-core by the introduction of a suitably chosen set of empirical parameters which gives best agreement for a single calculation on a given system. This method is further improved by allowing the integrals to be dependent on the effective charges of the atoms. Since these depend on the final results of the calculation, a n iterative procedure is required; this com- prises the “Iterative Extended Huckel Theory” (IEHT).

An instructive exposition of the IEHT method and its application is provided by R. Rein et aI. wherein the details of the computational mcthod are thoroughly presented. The absence of explicit electron repulsion terms is underscored by the use of the Cusachs approximation

Hij = (a + b ISijl )Sij(Hii + Hij)

for individual atomic orbitals, di and dj, where H i s the usual empirical one electron Hamil- tonian, characteristic of the Huckel method.

Explicit accounting for the dependence of the charge distribution on the nct charges of the individual atoms is achieved by writing

where mio) and a: are fixed parameters and Qi is the net atomic charge of the atom of orbital i determined iteratively. The charges are calculated by dividing overlap charges between two orbitals in such a way as to use orbital geometries in a logical fashion rather than dividing the Sij contribution equally among the two centers. This avoids the errors implicit in atomic point charge approximations.

The authors conclude that IEHT methods are valuable for charge densities and orbital populations and therefore will be of value for analysing inductive and resonance effects in aromatic substitution. Needless to say any such method will be inadequate for describing

BOOK REVIEWS 1223

properties dependent on electron correlation and hence many of the features of spectra; however, a good relative ordering of excited states, often superior to ab znitio SCF methods, is claimed.

By contrast other semi-empirical methods, most notably the Pariser-Pople-Parr (PPP)

and “Complete NegIect of Differential Overlap” (CNDO) explicitly include eIectron repuI- sion terms appearing in the Hartree-Fock formalism, neglecting some and approximating others. I n the discussion of a paper employing the PPP method E. McWeeny and H. Jaffe make the following comments which seem to summarizc many of the divergences between the two approaches:

R. McWeeny: “I think an essential requirement of any semi-empirical theory is that it should be able to reproduce in good approximation the results of non-empirical calculations. In setting up a semi-empirical scheme, we have to neglect vast numbers of small quantities (e.g., 3- and 4-centre integrals); and unless we can be sure that this is justifiable within the theoretical framework we are using, we cannot have any confidence in the validity of the theory. When we are sure that the quantities we have retained are the ones that are physically important, we can consider the possibility of semi-empirical adjustment, knowing that the structure of the theory is reasonable.”

N. Jaffe: “McWeeny says we are making semi-empirical calculations trying to reproduce non-empirical calculations. This was Pople’s approach in developing CNDO, and there are a number of possibilities. I think the ones that have been discussed . . are more attempts to reproduce experimental data than to reproduce a priori calculation . . . as semi-empiricists we have great advantages over the true theoretician in that we . . . simply put everything into the parameters. We cannot find a single calculation that will do everything for us, and we have to use different parametrization and sometimes diiferent calculations for different purposes.”

Following up this line of thought, it would seem that the Jerusalem Symposia, con- cerned as they are with semi-empirical methods, have provided a valuable compendium of interpretive techniques which delineate the basic philosophy behind each of them and to a degree explores their strengths and weaknesses. A brief and by no means exhaustive synop- sis of the 1969 symposium includes in addition to the above mentioned:

1. R. McWeeny: “SCF Theory and Heterocyclic Systems” The problems of rs - T separation in conjugated systems are discussed and a system of

coupled SCF equations for the rs and T electrons is developed within the framework of the Roothaan equations. This allows separate adjustment of the cr and T electrons to the molecular environment without losing the advantages inherent in a formal separation. It is stated, for example, that the non-empirical results, which show markedly different rs and T electron electronegativities for certain atoms, are reproduced in a simpler and more intelligible manner.

2, R. Bonaccorsi et al.: “Manimal Basis set SCF Calculations f o r Three-Membered Ring Molecules” Although the nature of the basis set was unable to provide the authors with a definite

calculation, the qualitative nature of the orbitals gives an interesting background for a discussion of “strained” ring compounds, (CH,),, (CH,),NH, (CH,),O, (CH,),NH$, (CH,),CH,. Particularly noteworthy are the charge density maps, which show the outward displacement of the center of charge for.the C-C bonds and the potential energy curves, which indicate the tendencies for the influence of charge perturbations on these molecules.

3. J . Bertran et al.: “The Relation between the Structure and the Thermochemical or Photochemical Reactivity o f Heterocycles’ ’

Chemical reactivity is often interpreted in terms of static or dynamic indices, such as ground state charge density or partially delocalized orbitals under the infhence of the

1224 BOOK REVIEWS

attacking agent. Based on a simple molecular orbital approach it is shown that the modified r-energy at the 2 and 4 positions of the quinolinium ion is lowest for nucleophilic reagents as is borne out by experiment. This result unifies the delocalized and hyperconjugative ap- proaches to the problem, which has seemingly led to divergent interpretations.

4. 3. Pullman: “ ~ a u t o m e r ~ s ~ n in Purines” Both the PPP-CI and CNDO methods are employed in a n intensive study of this problem.

The question of which tautomers predominate in solution cannot be answered from crystal- lographic data. This investigation not only studies the properties of a variety of tautomers but also the interactions among tautomers which may lead to stabilization in both solid and liquid phases. The importance of dipole moment computations in distinguishing among the various isomers is stressed.

The Aromaticity Symposium of 1970 wrestles with the problem of rationalizing classical and intuitive concepts with more quantitative quantum mechanical ideas. Some of the criteria for aromaticity were expressed in terms of double-bond fixation, ring currents, low entropy of formation, delocalization, and chemical reactivity.

The most sensible conclusions regarding this aspect of the subject emphasised the need for flexibility and a desire to avoid narrowing the criteria to an exclusive consideration of either ground state or particular excited state properties.

The papers presented in this session include:

1 . J . J . C. Mulder and L. J . Oosterhoff: “Permutation Symmetry Control in Concerted Reactions” The LVoodward-Hoffmann rules are discussed from the standpoint of aromatic and

anti-aromatic transition states. I t is found that if 2n is the number of electrons in the transition state and v is the number of negative overlap integrals, a reaction is favoured for n -k v odd and unfavoured for n + v even.

2. E. Heilbronner : “Homoconjugation in Bridged Annulenes” A variety of interacting double bonded systems is investigated including spiroconjuga-

tion and aromatic G - n interaction in strained systems, such as paracyclophane. Starting from the simple concept of LCMO orbitals for interacting double bonds this idea is expanded by means of energy correlation diagrams and illustrated with experimental data.

3. M. Cais : “Aromaticity Problems in Metallocenes” The contrast between the chemical reactivity and delocalization concepts of aromaticity

is underscored in the properties of this class of compounds. Particularly noteworthy is the fact that the electronic spectra of 1,l’ disubstituted ferrocenes is almost identical with the monosubstituted derivatives, indicating an insulating effect by the metal atom. O n the other hand, not only is the reactivity of the one ring in a monosubstituted compound greatly enhanced but that of the unsubstituted ring is almost equally affected.

4. V. Boekelheide and R. H. Mztchell: “Novel Molecules Bearing Substituents within the Cavity of the n-Electron Clouds”

As its title implies, this paper should be of interest to both experimentalists and theorists in the series of compounds synthesized.

5. G. Berthier et al. : “All-Electron Calculations on Benzene Isomers” Benzene and its 3 planar isomers, fulvene, dimethylene-cyclobutene, trimethylene-

cyclopropane along with the non-planar Dewar benzene, benzvalene, and prismane were studied by a technique which first estimated the nuclear configurations by a n approximate method followed by a more refined SCF calculation once the molecular geometry had been assumed. Despite obvious anomalies in the attack on this difficult problem the results achieved a preliminary understanding and provided a stimulus to further experimental and theoretical work.

BOOK REVIEWS 1225

In conclusion it may be said that the collected papers of these symposia give many valuable insights towards the problems of interpretive chemistry, both from the experi- mental and theoretical viewpoint. There are many readable applications of the various approximations to the lengthy Hartree-Fock procedure ranging from neglect of all specific electron interactions in the EHT and IEHT methods to the negtect of certain type of interactions as in the PPP and CNDO methods.

Perhaps the most outstanding unanswered question (which may hopefully be the topic of a future Jerusalem Symposium) is related to the physical and mathematical significance of calculations which on the one hand follow the framework of an exact formalism, but on the other evaluate certain quantities by experimental parameters. As has been pointed out on occasion during the discussions, there is always the danger of counting some terms more than once and others not at all, the moment one looses track of all the terms which appear in the exact theory. Mere agreement with experiment and definitive ab initio calculations, while encouraging and often useful, is not sufficient to ensure the construction of a fully reliable theory. I t is to be hoped that a system may eventually evolve which combines the facility of the semi-empirical with the precision of the ab initio calculations in an intelligible and rigorous manner.

Quantum Chemistry Group Uppsala University Uppsala, Sweden

DENNIS CALDWELL