Gerade orbitals

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.

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. 

Now these 3d orbitals have certain interesting properties. In the first place, suppose we invert them in the origin, namdy, the atomic nudeus. That is, let us construct a new set of functions whose values at the point (x, y, z) are the same as those of the original set, evaluated at the point (—x, —y, —z). Then an examination of Figs. 1-4, or an inspection of equation (1) shows that the new orbitals are identical in every way with the old ones the orbitals are simply reproduced, without change of sign, under the inversion. For this reason, they are said to be even (or "gerade") and we use subscripts g in their specification. This is an example of what are known as transformation properties. It is easy to see that the 4s orbital is also "even" in this sense. 

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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