Ungerade, orbital symmetry

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

Section 4-3-3 g for gerade, orbitals symmetric to inversion, and u for ungerade, orbitals antisymmetric to inversion (those whose signs change on inversion). The g or m notation describes the symmetry of the orbitals without a judgment as to their relative eneigies. 

Within the space spanned by the minimal basis set, the exact wave functions will be linear combinations of these six determinants. The Hartree-Fock ground state has two electrons in a gerade orbital and is of g symmetry (plus times plus equals plus). The doubly excited determinant has two electrons in an ungerade orbital and hence is also of g symmetry (minus times minus equals plus). The singly excited determinants, on the other hand, have one electron in a gerade orbital and one electron in an ungerade orbital and, therefore, are of u symmetry (plus times minus equals minus). The exact groimd state wave function of minimal basis H2, Oo)> Hartree-Fock... 

Fig. 5.5 Magnetic orbitals of gemde (left) and ungerade (right) symmetry with important delocalization tails on the ligands...

Since both the operator and the/orbitals have u (ungerade, odd) symmetry, electric dipole transitions are forbidden by the parity rule (see section below on Judd-Ofelt theory and induced electric dipole transitions). The selection rales for these transitions are summarised in Table 1.13. 

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.

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). 

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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