Operator identity

Atoms have complete spherical synnnetry, and the angidar momentum states can be considered as different synnnetry classes of that spherical symmetry. The nuclear framework of a molecule has a much lower synnnetry. Synnnetry operations for the molecule are transfonnations such as rotations about an axis, reflection in a plane, or inversion tlnough a point at the centre of the molecule, which leave the molecule in an equivalent configuration. Every molecule has one such operation, the identity operation, which just leaves the molecule alone. Many molecules have one or more additional operations. The set of operations for a molecule fonn a mathematical group, and the methods of group theory provide a way to classify electronic and vibrational states according to whatever symmetry does exist. That classification leads to selection rules for transitions between those states. A complete discussion of the methods is beyond the scope of this chapter, but we will consider a few illustrative examples. Additional details will also be found in section A 1.4 on molecular symmetry. 

Now we do one of the standard quantum mechanical tricks, inserting the identity operator as a complete sum of states in the coordinate representation ... 

E, the identity operation which leaves the molecule unchanged. 

For the one-electron operator only the identity operator can give a non-zero contribution. For coordinate 1 this yields... 

IN being the identity operator in the subspace Then we have the expansion... 

If the group O contains the time reversal operator itself, we can choose Y0 = E, the identity operator, and Eq. (12-27) reduces to... 

By introducing the transition operator T T yk = Vk+i and the identity operator E Eyk = yk and obvious rearranging of the left-hand side of (82) as a product... 

The operator S is called the transition operator (from one layer to another). In addition to the canonical form (4), alternative forms of writing will appear in the sequel for two-layer schemes By = C y t if oi B = C j/ -h ry , where C = E — t A, E being the identity operator. In the case B = E, scheme (3) is called an explicit two-layer scheme ... 

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