Representations for Cyclic and Related Groups

What about the Is AOs on the H s Can the representation table tell us in which MOs these will appear It turns out that it can, and that they go into both a -and e-type MOs, but we will defer showing how this can be told from the table until later in the chapter. 

Cyclic groups are the groups C2, C3, C4. C containing only the — 1 rotation operations and the identity operation E. We devote a separate section to these because there are some special problems connected with finding and labeling representations for these groups. (This section is off the mainstream of development of this chapter and may be skipped if desired.) 

Let us consider the operations associated with the n-fold proper axis oriented along the z axis and ask what will become of the function /=exp(n ) as it is rotated clockwise about this axis by Inin radians. (We entertain the idea that exp(i ) might be a convenient basis for a representation since such functions were found to be eigenfunctions for the particle-in-a-ring problem in Chapter 2.) Since the clockwise direction is opposite to the normal direction of the / coordinate, the effect of the rotation is to put /((f ) where f j) — 27tfn) used to be. To see how the function after rotation compares to that before rotation, we must compare exp(/ ) with exp[/( — 2nln)]. That is, the representation R/, such that exp(/ / ) = Rfexp i l ), is given by exp(/ )/exp[/( — 27t/ )], or 

The important point here is that /=exp(/ / ) is a basis for a one-dimensional representation for C since a rotation turns / into a constant times / and not into some other function. For the C3 group, then, we could write a partial representation table as shown in Table 13-14a. Now exp(47rz/3) is equal to exp(—2 7r//3), and exp(—Ttt//3) is the complex conjugate of exp(2 r / /3). If we let e = exp(2 7r i /3), we can write the table as shown in Table 13-14b. If / = exp(/(/ ) is a satisfactory basis, f = exp(—/ ) is also acceptable, since it is linearly independent of /. If one calculates the representations for = 1,2, 3. as before, one finds that the numbers R/ are the complex conjugates of Rf. Therefore, we can immediately expand our table as shown in Table 13-14c. Since the existence of exp(/(/ ) as a basis for a one-dimensional representation always implies that exp(—z ) exists as a basis, these sorts of one-dimensional representations always occur in pairs. It is conventional to combine these with braces and refer to them with the symbol E, which we claimed earlier is reserved for two-dimensional representations. Note, however, that a pair of one-dimensional representations is not the same as a two-dimensional representation, and we must broaden our definition of the symbol E to include both types of situation. 

Our representation table for the C3 group now looks almost the way one would find it in a standard tabulation. However, instead of listing exp(/ / ) and exp(—/(/ ) as bases. 

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