Orbitals gerade/ungerade

Fig. 2.9 Angular wave Tunctions of s, p, d, and / orbitals illustrating gerade and ungerade symmeir> (a] > orbnaL yerade , (b) p orbital, ungerade, (c) pictorial representation of symmetry of p orbital (d) orbital, gerade (c) piaonul representation of symmetry of d orbital (f) d.i orbital, gerade (g)/,i orbital, ungerade.

Fig. 10.18 The inversion (gerade/ ungerade) character of a bonding and antibonding orbitals.

Fig. 2. Simplified molecular orbital diagram for a low spia octahedral complex, such as [Co(NH3 )g, where A = energy difference a, e, and t may be antisymmetric (subscript ungerade) or centrosymmetric (subscript, gerade) symmetry orbitals. See text.

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.

A core of two MOs was kept doubly occupied, namely, the lag,lbiu orbitals corresponding to carbon K shells (g and u denote gerade and ungerade, respectively). The remaining 12 electrons were left available for fractional occupation of the 24 MOs. 

Symmetry Notation.—A state is described in terms of the behavior of the electronic wave function under the symmetry operations of the point group to which the molecule belongs. The characters of the one-electron orbitals are determined by inspection of the character table the product of the characters of the singly occupied orbitals gives the character of the molecular wave function. A superscript is added on the left side of the principal symbol to show the multiplicity of the state. Where appropriate, the subscript letters g (gerade) and u (ungerade) are added to the symbol to show whether or not the molecular wave function is symmetric with respect to inversion through a center of symmetry. 

Another useful way to look at the faa is at their symmetry. Subscripts g (gerade) and u (ungerade) are labels specifically associated with the presence or absence of the inversion symmetry element in the given orbital. In (11) and Fig. 1, the three MO s fa, tfi2, ips of allyl, a w-system, may be designated as u, g, and u, respectively. 

Unless (v iG)spin = ( i E)spin. then the spin component is zero and the transition is spin-forbidden. Nevertheless, spin-forbidden transitions are observed as weak features (as in Fig. 2.18) typically with 10 -10 the intensity of fully allowed transitions. This is because of the interaction between the electron spin magnetic moment and the magnetic moment due to the orbital motion of the electron (spin-orbit coupling). The La-porte selection rule, furthermore, states that only transitions between wave functions with one having gerade and the other ungerade character are allowed (hence all d-d transitions are Laporte forbidden). This arises since the spatial component can be further broken down ... 

The dipole moment operator ( x) has associated with it ungerade character so the integral will be zero if v i[ and v g are both either gerade or ungerade. Again, Laporte-forbidden transitions do occur (with 10 -10 the intensity of fully allowed transitions) because of mixing of the orbitals in the excited state in noncentrosymmetric sites, and even in centrosym-metric sites as a result of vibrations of the metal atoms away from the center of symmetry (vibronic coupling). 

---