Ungerade / orbitals

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.

The g and u labels can be understood from the effect of the inversion operation i on the AOs d-orbitals are gerade and p-orbitals ungerade. 

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.

Note the absence of the g subscripts here. Although the d orbitals are still centro-symmetric, the tetrahedral environment lacks a centre of inversion. The d orbitals are therefore not classified with respect to a symmetry element which doesn t exist the absence of the g subscript does not imply the opposite - i.e. u (ungerade or odd). 

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. 

The / orbitals have not been considered previously except to note that they are ungerade (Chapter 2) and that they are split by an octahedral held into three levels, ant "2 (Chapter 11). A complete set of seven 4/orbitals is shown in Fig. 14.6. As with the d orbitals, there is no unique way of representing them, nor is there even a way which is optimum for all problems. Thus Fig. 14.6 presents two sets, a general set" and a cubic set." The latter is advantageous in considering the properties of the orbitals in cubic (i.e.. octahedral and tetrahedral) fields. 

Ultraviolet (UV) radiation, 244 Ungerade symmetry, 48. 160 Ungerade orbitals, 17-18, 160, 438... 

Other metal orbitals which are reasonably dose to, but more highly excited than the 3d and 4s levels arise from the 4p set. It may be shown that these are odd (or ungerade ) with respect to the operation of inversion—that is, they change sign. Moreover, that 4p orbital with m — 0 is invariant under a rotation a, whereas the nti = 1 orbitals are multiplied by factors . These orbitals are therefore called pau, pe respectively. 

Figure 26. Quantal deflection functions at higher energies for He (2 5) + He. Orbiting spikes result from trajectories that spiral into inner minimum of ungerade potential.

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. 

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