Elements and Coupling Coefficients

Although we have mainly talked about states it is clear that operators may just as well be classified by their behaviour under the various symmetry operations. Using the eigenvalue tables it is then straight forward to derive quantitative relations between matrix 

Consider an arbitrary matrix element (a c b). Since a matrix element is a number it must be unchanged by any symmetry operation. If both states a) and b) and the operator c have eigenvalues L, /.b, Ac respectively under some symmetry operator, then the matrix element can only be non zero if kb Xc = a. If the states or the operator are not eigenstates of the symmetry operator but instead are transformed into other states/ operators then quantitative relationships between the matrix elements are obtained. 

The same results may also be formulated in terms of coupling coefficients. From the eigenvalue tables we can immediately see how a product of two states transforms under the various symmetry operations. Let us consider the product of two E states in the point group D3. We have the four product states EiEi), jExE-j), lE Ej) and lE-jE-i). 

Under the C3 operation these states are again eigenstates with eigenvalues equal to the product of the individual eigenvalues, whereas operation with C2 connects jEjEj) with E iE i) and EiE j ) with lE—iEi). Forming the symmetric and antisymmetric products 

Acknowledgment. Kjeld Rasmussen, Kemisk Laboratorium A, Danmarks Tekniske H0jskole, is thanked for drawing my attention to the work of Altmann. 

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