Symmetric Group Theoretic Approaches

The early workers, when treating two electron systems, usually made the observation that singlet states spin functions are antisymmetric while triplet spin functions are symmetric with respect to the interchange of particles, t. e., 

Consequently, for the total wave function to be properly antisymmetric, the spatial function to be multiplied by the spin functions must be symmetric or antisymmetric for singlet or triplet states, respectively. Satisfying these requirements may be made more explicit in the following way. 

We can use one of the spin eigenfunctions above, symbolizing it by j, and multiply it by an arbitrary spatial function, H, to obtain a function of both space and spin, 

We have given a short description of the two electron case. The important point is that there is a generalization of Eq. (12) to n electrons. It takes the general factored form 

The n-electron spin functions are sums of products of n a or functions that satisfy 

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Since the related Hamiltonian needs to remain invariant under all the symmetry operations of the molecular symmetry (point) group, the potential energy expansion, see equation (5), may contain only those terms which are totally symmetric under all symmetry operations. Consequently, a simple group theoretical approach, based principally on properties of the permutation groups can be devised, " which yields the number and symmetry classification of anharmonic force constants. The burgeoning number of force constants at higher orders can be appreciated from the entries given in Table 4. 

The group-theoretical approach is based on the fact that, if equation (5) contains products of internal (symmetry) coordinate displacements Sa(oi = /, y, k,...), then the product SaSfi - Sco and that obtained by any permutation of the indices are indistinguishable. Thus, the n-member products transform according to the permutation (symmetric) group Sn or in another, perhaps more appropriate, notation where... 

For the case of symmetric chromophores, what appears to be a fruitful theoretical approach is perhaps best illustrated by way of reference to a concrete example, and it will probably come as no surprise to the reader that the example we have chosen to use involves the carbonyl group of saturated ketones. From the point... 

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