Dennis P. Strommen Carthage College Kenosha. Wisconsin 53140
A Pedagogical Approach to the Direct Product
One of t h e most important concepts developed i n introductory group therapy for chemists is t h a t of the direct product representation-a representation of a point group generated by a set of basis functions which are themselves products of simpler basis functions. T h e representations ohtained in this manner a r e involved in many physically significant calculations, such as t h e determination of non-zero energy matrix elements, spectral transition probabilities, hybridization schemes for bonding orbitals, and correlation diagrams in ligandfield theory, t o mention only a few. A typical problem involves t h e determination of the characters of a direct product representation under a given point group symmetry followed by a determination of the irreducible representations spanned by the direct product. It is relatively simple t o perform t h e calculations required in t h e above examples since they are so highly formalized; however, i t has heen mv exnerience t h a t i t is auite difficult for the heainning s t u d e n t t o Anderstand t h e haiic concepts t h a t underlie these onerations. P a r t of t h e problem lies i n t h e abstract nature of group theory i t ~ e l fhut , a major reason is thecursory treatment eivcn t o t h e develwment of the direct product in most texts 7 1 4 ) . This is unfortunate since an understanding of the direct product can he used to enhance the students'grasp of the basis for a representation, which is, after all, t h e very thing t h a t makes croup theory s o useful in chemistry. T h e general formulati& o f t h e direct product, a s presented i n a number of standard texts, involves t h e use of some rather involved suhscript notation (1, 2, 6). Once this formulation is obtained, examples of direct product applications follow in quick succession, leaving a large gap t h a t the novice cannot easily bridge by himself. Therefore t h e remainder of this paper will he devoted t o t h e presentation of a few examples which will allow t h e beginning student t o develop a better understanding of just what constitutes a direct product. It will he useful to consider a rather involved example first; following this treatment the simpler applications will he more easily understood.
nalization requires the application of a similarity transformation (13).Again, a paint which is unclear in thestandard texts is that the similarity transformation matrix need not be amember of thegroup as it must be in the determination of classes. Indeed, in this case it cannot be. One matrix that will diagonalize the above matrices is shown below.
Application of (2) and its inverse to the set of matrices (1)results in the following transformed matrices:
rrpre*entatiun\ in ~ i r c l to r ~irr it aport t'rom !he uthrrs. ('ompdrism of the enrin4t.d nunlllrrs uith thwr in the ('r rhnrwtpr tahle given I , e l ~ ~ r r . c l r . ~ r l v s hthat ~ ~ u thw ~ a are ~ d r n ~ ~ r n l icharmer-ofthe nth~ XI representation. The other diagonal elements moving down the rows correspond to Bz,Ap, and A, symmetries, respectively. The direct oroduct reoresentation E X E under Ca.. svmrnetrv can now be seen i o man A, A. + R , +no. This is exacti; the same;esult as obtained , ~~.'+ ~. t h n u ~ hthe itandah treotrntnt iwd\ inK jusr rhe rhararrer, uf thr rrprcwntation. .\Ithough there iq murr wurk in\c,lved in the present method I feel that an example or two of this kind is warranted in a beginning course. ~
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Example 1. A Direct Product Involving More Than One Degenerate Representation E X E under C4, symmetry. One basis for the E representation is (x,y)while another is (xz,yz).
Therefore. the direct oroduct reoresentation mav. he .. senerated hv ( ~ - 2 , x \ 2 , ) r : . y ? ) . Notire that thcordrr uf the pndurt.< ia imlrmant, that ir, .r,z is folloutd In >x:, and thc
