G. gerade

Modes of flu symmetry are inactive, g = gerade modes are Raman active, u = ungerade modes are IR active... 

In the MuUiken notation, the subscripts u (ungerade = odd) and g (gerade = even) indicate whether an irreducible representation is symmetric (g) or anti-symmetric(M), in respect to the inversion operation (/). 

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. 

Before discussing other examples, we note here that, for a centrosymmet-ric molecule (one with an inversion center), rx, ry, and rz are u (from the German word ungerade, meaning odd) species, while binary products of x, y, and z have g (gerade, meaning even) symmetry. Thus infrared active modes will be Raman forbidden, and Raman active modes will be infrared forbidden. In other words, there are no coincident infrared and Raman bands for a centrosymmetric molecule. This relationship is known as the mle of mutual exclusion. 

Table 1. Wavenumbers in kK ( = 1000 cnr1) of internal transitions in the partly filled shell of octahedral d3-sy stems and tetrahedral d -systems. In the latter cases, the parity is not defined, and the g (— "gerade") should be removed from the symmetry type symbols. The numerous references to the literature are only given if they cannot be found readily in recent compilations,1- ... 

The various combinations are named according to the degree of degeneracy and to the g ( gerade , i.e. even) or u ( ungerade , i.e. odd) parity (see page 75) the aig and the two Cg combinations have an even number of nodes, whereas each of the three tiu combinations have just one node. 

In Molekiilen mit einem Inversionszentrum konnen die Orbitale iiber ihr PartYofeverhalten charakte-risiert werden. Betrachtet man ein s-Orbital, so hat die Orbitalfunktion am Punkt (x, y, z) das gleiche Vorzeichen wie am Punkt (-x, -y, -z), den man durch Inversion am Symmetriezentrum erhalt. Ein s-Orbital wird deshalb mit g (gerade) gekennzeichnet. 

With two electrons and four available spin orbitals, 2 determinantal collective states may be then built. However, Pauli s exclusion principle coupled to the notion of particle indiscernibiUty contributes to reduce this number to 6. Let us label Xs,sz) the collective states X = U (ungerade) or X = G (gerade) refers to the symmetry of the orbital part with respect to the interchange of A) and S) S and describe the total spin configuration. We shall denote V,[Pg.236]

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. 

Subscripts may be applied as follows. If there are C2 axes perpendicular to the principal axis, then a subscript 1 (2) means that the basis for the representation is symmetric (antisymmetric) for such a rotation. In the absence of such C2 axes, vertical reflection planes are used instead, if present. If there is an inversion center, subscripts g (gerade) and u (ungerade) refer to the basis for the representation being respectively symmetric or antisymmetric for inversion. 

Similarly, the super-indices ( ) and (") are added for the symmetry, respectively antisymmetry toward the plane (if applieable) and again as sub-indices are added the symbol g (gerade = even, in German) and m ... 

The A and B symbols attached to these representations are obtained as follows One-dimensional representations are designated by A if they are symmetric to rotation by In/n radians about the principal n-fold rotation axis (n = 2 for a 180° rotation in this case) and are designated by B if they are antisymmetric to this rotation. The subscripts 1 and 2 designate whether (in this case) they are symmetric or antisymmetric to reflection in a vertical plane. A two-dimensional representation is designated by E (not to be confused with the identity operation), and a three-dimensional representation is designated by T. Subscripts g and u are sometimes added to specify the symmetry with respect to inversion (g = gerade = even u = ungerade = odd).Arepresentation with all characters equal to 1, like the Ai representation in this case, is called the totally symmetric representation. 

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. 

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