Ungerade molecular 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.

Some of the possible combinations of atomic orbitals are shown in Fig. 5.11. Those orbitals which are cylindrically symmetrical about the internuclear axis are called cr orbitals, analogous to an s orbital, the atomic orbital of highest symmetry. If the internuclear axis lies in a nodal plane, a n bond results. In S bonds (Chapter 16) the internuclear axis lies in two mutually perpendicular nodal planes. All antibonding orbitals (identified with an ) possess an additional nodal plane perpendicular to the internuclear axis and lying between the nuclei. In addition, the molecular orbitals may or may not have a center of symmetry. Of particular interest in this regard are orbitals, which are ungerade, and tt orbitals, which are gerade. 

More advanced applications of symmetry (not discussed here) involve the behaviour of molecular wavefunctions under symmetry operations. For example in a molecule with a centre of inversion (such as a homonuclear diatomic, see Topic C4). molecular orbitals are classified as u or g (from the German, ungerade and gerade) according to whether or not they change sign under inversion. In... 

Now consider the ways in which two electrons can occupy these molecular orbitals. To follow convention, call the spatially symmetric orbital Og (g = gerade = even) and the spatially antisymmetric orbital ou (u = ungerade = odd). The two electrons will obey the exclusion principle if they are in any of the wave functions shown in Table 6.1. 

Fig. 4.4 Upper part gerade and ungerade magnetic molecular orbitals for a M-L-M angle of 90°. The dxy orbitals on the metal centers have an equal overlap with the px and py orbitals on the ligand. Lower part In a system with a larger M-L-M angle, the overlap is larger for Px than for py...

FIGURE 15.11 Molecular orbitals of Oj. Simple diagrams like those in Figure 15.10 make it easy to determine which orbitals are gerade and which are ungerade. 

Here, the plus combinahon produces the ungerade anhbonding orbital. Perpendicular to the molecular axis (in either the y or z plane) are the following combinahons of p orbitals. 

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. 

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