Gerade, orbital symmetry

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... 

Since Omd has g (gerade, even) symmetry and the/orbitals u, magnetic dipole transitions are allowed in centrosymmetric and noncentrosymmettie point groups. However, the selection rules AJ=Q, 1 (but not O-m-0) are followed (Table 1.13), and so few magnetic dipole transitions, such as the Eu " Dq Fj transition, are known. 

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.

Fig. 2.9 Angular wave functions of s, p, d, and / orbitals illustrating gerade and ungerode symmetry (a] s orbital, gerade, (b) p orbital, ungeradai (c) pictorial representation of symmetry of p orbital (d) dx> orbital, gerade (c) pictorial representation of symmetry of d orbital (f) d.i orbital, gerade (g) /,i orbital, ungerode.

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. 

Crystal field, or d-d, transitions are defined as transitions from levels that are exclusively perturbed d orbitals to levels of the same type. In other words, the electron is originally localized at the central metal ion and remains so in the excited state. When the system has ( symmetry, Laporte s rule says that an electric-dipole allowed transition must be between a g state and an u state, i.e., u - g. Since all the crystal field electronic states are gerade ( g ), no electric-dipole allowed transitions are possible. In short, all d-d transitions are symmetry forbidden and hence have low intensities. The fact that the d-d transitions are observed at all is due to the interaction between the electronic motion and the molecular vibration. We will discuss this (vibronic) interaction later (Section 8.10). 

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