Gerade parity

We can now determine the symmetry contributions from the spin functions. The So function clearly transforms according to Tq. The rotational symmetry of the triplet spin functions follows from their properties as / = 1 angular momentum functions. In addition we need to consider the effect of inversion. But the product of two spin functions must always be of even, or gerade, parity. Thus the symmetrized spin basis... 

Fig. 22.3 Energy levels and associated optical transitions of (a) positive polaron. P (b) bipolaron, BP and (c) ir dimer, PD. The full and dashed arrows represent allowed and forbidden optical transitions, respectively. H, S, and L are HOMO, SUMO, and LUMO levels, respectively, and 14 and are odd (ungerade) and even (gerade) parity representations, respectively. 2o>o( ) and 2cj BP) are assigned note that wo( ) > o)n(BP).

Magnetic dipole (MD) transitions are allowed in both centrosymmetric and non-centrosymmetric point groups. The magnetic dipole operator has a gerade parity. Transitions allowed by the magnetic dipole mechanism are however sparse, because of the selection rule A7=0 or AJ = 1 (but not 0<-+0). The best-known MD transitions are Do Fi and D, Fo of Eu -. ... 

Figure B A qualitative molecular orbital diagram for ferrocene. The subscripts g and u refer to the parity of the orbitals g (German gerade, even) indicates that the orbital (or orbital combination) is symmetric with respect to inversion, whereas the subscript u (ungerade, odd) indicates that it is antisymmetric with respect to inversion. Only orbitals with the same parity can combine.

Table 1. Wavenumbers in kK ( = 1000 cnr1) of internal transitions in the partly filled shell of octahedral d3-sy stems and tetrahedral d -systems. In the latter cases, the parity is not defined, and the g (— "gerade") should be removed from the symmetry type symbols. The numerous references to the literature are only given if they cannot be found readily in recent compilations,1- ... 

The first selection rule, known as the Laporte rule, states that the only allowed transitions are those with a change of parity gerade to ungerade and ungerade... 

The various combinations are named according to the degree of degeneracy and to the g ( gerade , i.e. even) or u ( ungerade , i.e. odd) parity (see page 75) the aig and the two Cg combinations have an even number of nodes, whereas each of the three tiu combinations have just one node. 

The subscript g means gerade and the subscript u means ungerade. Gerade and ungerade designate the behaviour of the wavefxmction under the operation of inversion, and denote the parity (even or odd) of an orbital. 

When space-degeneracy effects play no role and the atomic states have definite parities (e.g. S and P states of alkaline and alkaline-earth elements), the superpositions allowed by the electronic inversion symmetry of X 2 are the symmetric (constructively interfering) and antisymmetric (destructive y interfering) eAgB>i gA B / corresponding to ungerade and gerade XS states, respectively. 

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