Space-, Spin- and Overall Symmetry

Let us return to the isomerization of CBD in Section 8.4, and compare the symmetry properties of the orbital products that can be constructed from / 2 and (/5 3 in D2/1 and in D4/1. We recall from Section 3.2.2 that the irrep of a product of orbitals is the product of the irreps of the occupied orbitals, each taken once for every occupying electron. 

In T 2h t 2 is more stable than 3, so the orbital products are increasing energy  

The two totally symmetric orbital products can only be occupied by a pair of electrons with opposite spin, so - since all of the other electrons in the molecule are also paired - they represent closed shell singlet states. The configuration gives rise to two open shell states, Big and Big, that have the same energy at the orbital level of approximation but split when electron interaction is taken into account. 

The Pauli exclusion principle, which we have not yet had occasion to apply explicitly, must now be taken into account. It requires the overall wave function, expressed as the space wave function, - represented here by its dominant orbital product - multiplied by the spin function Ai, to be antisymmetric. Since the singlet and triplet spin functions are respectively antisymmetric and symmetric, the space functions associated with them have to be of opposite parity. The two totally symmetric closed shell space functions are obviously symmetric to interchange of the two electrons in a single orbital. 2 in Equation 9.8 stands for two equivalent open shell alternatives with the orbital product (j 2 f 3- 2 hg) l) 3( 2 )(2), in which the first electron is in 02 and the second in 03, 

We take note of the fact that the two closed shell singlets differ in energy and are not interconvertible both are totally symmetric, so the lower is labeled l Ag and the upper 2 Ag. Since (j)2 transforms like yz and j 3 like xz their product, like xyz (or xy) belongs to the irrep Big] so do the sum and difference, which transform like y(l)x(2) y(2)x(l). The open shell singlet and triplet thus have the same space symmetry and are labeled Big and Big respectively. 

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