How Great Is the Great Orthogonality Theorem? Carlos Contreras-Ortegal and Leonel Vera Universldad del Norte, Casllla 1280, Antofagasta, Chlle Eduardo Qulroz-Reyes Universldad Austral de Chlle, Casllla 567, Valdivla, Chile I t has been stated that all the properties of group representations and their characters can be derived from one basic theorem: the great orthogonality theorem (GOT2). Rules are derived from this theorem to work out the character table of a symmetry point group and are presented as the rules of the irreducible re~resentationsand their characters. All character tables must'adhere to these rules. It is a common helief that there is one and onlv one rharachere we ter tahle for each symmetry point group. present six tables for the symmetry point group D4h that hold perfectly well to the GOT rules (the reader can check this fact by using the rules as they are normally stated in classical textbooks in molecular symmetry3). If holding to the rules of the GOT is a proof of the correctness of a character table, we are then forced to say that the six tables are equally valid as character tables for the groupDu,. What has happened? Is there anything wrong with the GOT or, less drastically, with how the rules are normally set out in textbooks or in the number of them needed to get an unequivocal result? Or should something he kept in mind when using the rules that is normally forgotten or not sufficiently emphasized even in known classical textbooks in the field? In any case, there is the irrefutable fact that, no matter how the six tables have been constructed, they do exist. There they are! Our solution to the ahove paradox is as follows. The group D4h is formed when the Ci group operations are added (i and E ) to the D4 group operations. The D4h group elements are, besides the elements of bothgroups, those resultingfrom the products of symmetry operations between them. All the elements are generated from the multiplication of each of the elements of the groupD4 by a11 the elements of the group C;,that is,
ahove, the group Dab can be written as: D4h = D4 * Ci.The elements of mouD Dar, listed hv classes are shown a t the heading row of t i e tabies. ~ e c a k the e number of classes is 10, the number of irredurible representations is also 10. Fore better analysis, let us divideeach tahleas it is shown in table I). We ohserve that the classes on tor, of the upper left quadrant are the classes of group D4 (D4i9a subgro;~ of the group D d I t shows behavior that is independent from the rest of the group D4h; any operation between the group elements will give as a result an element in the group. Therefore, the matrices, or their characters, representing this set of classes of operations must be the same matrices, or their characters, representing the operations of group D4. Hence, this part of the character table can be considered independent from the rest of the table and their character taken from the character tahle of the group D4. The lower left quadrant must contain the traces of the matrices representing the operations of the group Dq for the same ahove reasons. As the irreducible representations cannot be repeated and five are needed, they must he, in this quadrant, the irreducible representations of group D4. Therefore, we repeat the character table of group Dq here. The operations, listed by classes, on top of the upper right quadrant result from the product between the inversion operation i and each of the operations of group D4. Consequently, the matrices representing these classes in each irreducible representation must be the product of the matrix representing i and each of the matrices representing the operations of D4 in the same representation. Which are the matrices that remesent the inversion or,eration i in each irreducible representation? First of all, let us remember that the matrices of an irreducible re~resentatiou operate over the function that is the base of the representa-
where R is any symmetry operation of group D4. From the
Author to whom correspondence should be addressed. 2Eyring. H.; Walter. J.; Kimball, G. E. Quantum Chemistry; Wiley: New York. 1944: o. 371. cotton, A. i.'Chemical Applications of Group Theory. 2nd ed.; Wiley: New York. 1971; p 78.
ow ever,
'
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Journal of Chemical Education
Character Tabbs*for the Symnwiry Polnt Group 4,
Mulllken symbols are plven in parentheses. Note that, according to mess symbalo, the ineducible repesentationr in all the tables are apparently the same. Also, all tables appear to obey me known rules derived from the Great Ormogonallfy mewem. Howevn, table D is the only one mat la correct.
tion. Let us name M(i) and M(R) the matrices representing the operation i and a given operation R of D, in a given irreducible representation I',. Let f, be a base function of this reoresentation. The action of M I R ) over f . will be to transform it in another function g,. ~he'posteri!or action of i will leave i t unchanged .. (e;) .., or will chanee its sien (-2.). . .... Let us assume that g, is symmetric to the inversion (i.e., it remains the same). Then deoendina on whether r,is a unidimensiona1 or a bidimensionil representation, the katrix for the operation i will he, respectively,
-
(their res~ectivecharacter beine 1and 2). Let us take now the irredLcible representation ?;;, which has its left half equal to the left half of rj. For tlus representation another function, fl, must exist which has to be antisymmetric with respect to mnversion. Otherwise. T;and r; would be eaual. and this cannot occur. Thus, the katrix tor i in Ti wiil be equal to the matrix for i in Tj,hut with opposite sign:
(its character will he -1 or -2, respectively). Hence, what the i operation does is to duplicate the representations of the moup - . Da: .. some of them symmetric with respect to inversion (the characters for i in allof them being pos%ive) and others antisymmetric with respect to i (the characters for i in all these representations being negative). Therefore, if we write the symmetric representations in the upper half of the character table. and the antisvmmetric onesin the lower half of it, the char&ersfor the i bperation in the severalirreducible reoresentations of the eroun. Dab. .... will he those shown in
able D.
Which are the characters for the remainine ooerations? First of all, the matrices for these operations are obtained by multiplvine the matrix for i in a ~ i v e nre~resentatioubv the matrcce; hilonging to the group^^ in tl;e same represents. tion and following the same order in which the classes are Volume 68 Number 3 March 1991
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r
listed in D4, that is: i E = i, i * (2CJ = (2S4),i * Cq = ah, i * (2C2) = (2a4), i * (2CJ = ( 2 ~ 4 . Second, we are interested m the characters of the product matrices. What the M(i) matrix does is to leave the M(R) matrix unrhanged whenits diagonal elementsareall+l, but chanee the siensof the M(R) matrix elements when itsdiaronal element; are all equ& to -1. I t is then easy to see that the trace Tr(M(i) r M(R)J = i Tr(M(R)), positive when Tr(M(i)) > 0 and negative if Tr(M(i)J < 0. Therefore, the upper right quadrant will contain the characters in the upper left quadrant, and the lower right quadrant will contain those in the lower left quadrant hut having opposite signs. Hence, the correct table will be the one having the following structure:
of the symmetry operations of the groups D, and C, or Ci, and of the groups C, and C, or Ci, respectively, where C, is the group containing the symmetry operations E and ah. They can be written as: D,h = D, r Ci and Cnh = Cn * Ci, when n is even and Dnh = D, * Cs and Cnh = Cn * Cs, when n is odd. Their character tables are built as follows:
Wesee that the onlvtahle satisfvina this relations hi^ is table D, and therefore t6is is the corr&tbne. Obviouslv the same above conclusion will be obtained for any group Eontaining a subgroup and the operation i. This occurs with groups Dnh and Cnh, where n is even. We note that the observation above is also valid for all groups containing a subgroup and the operation ah. In this case, the irreducible representations for the subgroups are also dupli~ a t e d(and enlarged in the characters of the new operations appearing), some symmetric and others antisymmetric with respect to the a h operation. This occurs with the groups D,h and Cnh, where n is odd. Everything so far discussed can he summarized in the followi& way: the symmetry operations of groups D,h and C,h (their group elements) can he generated as the products
where D. and C. are the character tables for those ~ o i n t groups. he mathematical foundations of the above i d e s can be found elsewhere4. The mathematical formulation confirms that the correct table is table D. However, it cannot tell us from a ~ u r e l vsvmmetric ~ o i nof t view, more conceptual and intuiiive, what is wrong with the other tables. The conclusion is that there is nothing wrong with the GOT or with the rules derived from it. However, agreement with these rules is not a proof by itself of thecorrectness of a character table, though in many cases it is. Symmetry rriteria should always be kept in mind for a continuous checking out of the results derived from the rules of the irreducible representations and their characters. A student working out a character table for a group containing a suhgroup, by only using the rules, could obtain a wrong result, as we did, and there would not he any good reason totell him or her that his or her result is incorrect by just claiming the GOT. Anyway, we stronalv recommend that they try to work out all the possible character tables of one of the analyzed groups just bv using the rules derived from the GOT. A challenge like that wo&d reveal to the student his degree of understanding of the subject.
Tinkhan, M. Group Theory and Quantum Mechanics; McGrawHill: New Yofk. 1964; P 44.
Acknowledgment The authors wish to thank A. Aizman for helpful discussions.
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31st Program in Applied Molecular Spectroscopy The 31st annual program in Applied Molecular Spectroscopy:Infrared to be offered by Arizona State University, July 22-26, 1991, has been designed to devote time to the use of mini- and microcomputers for data processing, search and
retrieval, and quantitative analysis in infrared spectroscopy, in addition to the more traditional topics, such as, sampling techniques, operating parameters, applications, and spectral interpretation. Lectures on FT-IR will be presented and hands-on use of FT-IR spectrometerswill he offered. The program includes basic theoretical considerations, hands-on instrumental training, and the interpretation of spectra. Four hours of lecture each morning will serve to present the theory, instrumentation, and applicationsof infrared spectroscopy. Some reference to Raman spectroscopy will also he included. Each student will spend every afternoon working in the laboratory under the direct guidance and supervision of experienced technical personnel. The instructional staff includes members of the Department of Chemistry at Arizona State University augmented by guest lecturers from industrial labortories. Enrollment in the course is limited and sufficient equipment is available to insure each student adequate time for personal operation of the instruments. The cost for the program is $700. For complete information, including descriptive brochure, please write Jacoh Fuchs, Director, Applied Molecular Spectroscopy, Department of Chemistry, Arizona State University, Tempe, Arizona 85287-1604.
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Journal of Chemical Education
