Several in-plane vibrational coordinates of NiFJ

The two out-plane-modes (not illustrated in the figure, are also easy to characterize they are both non-degenerate In one a2u)-, the F atoms move together parallel to 2r, and the central Ni atom moves in the opposite direction so as to conserve the center of mass. In the other (62 ), one pair of rans-situated atoms moves up, the other pair moves down, and the central atom stays put. 

A comparative vibrational analysis of the CH- and NiF-stretching modes in ethylene and NiFj respectively illustrates the distinction between the characters of the irreps of commutative and non-commutative symmetry point groups. It also allows the introduction of two particularly useful group theoretical terms direct sum and projection operator. 

The four localized bond-stretching coordinates, illustrated in Fig. 4.11 for ethylene (D2h) and the nickel tetrafluoride dianion (D4/1), combine in both cases to the same four orthogonal linear combinations  

Intuitive considerations of this sort are often sufficient, but familiarity with a few simple rules for determining beforehand the irreps of the combinations of coordinates (or orbitals) in a given symmetry point group and constructing them formally will be useful in the more complicated cases discussed in the following chapters. Their implementation will be illustrated first with the CH-stretching modes of ethylene and then with the NiF-stretching modes of NiFJ. 

The X and y axes are rotated about z by 45 relative to those in Fig. 4.10, for consistency with the axis convention appropriate to ethylene. Big and B2g are interchanged as a result, as are Bi and B2u- Also the x and y components of Eu take different forms in the two axis coventions. 

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Figure 4.10. Several in-plane vibrational coordinates of NiFj"...

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