Twice implied operation

In equations (l)-(7), vector notation is employed Vis the gradient operator, U is the unit tensor, two dots ( ) imply that the tensors are to be contracted twice, and the superscript T denotes the transpose of the tensor. The symbols appearing here are summarized in Table 1.1. 

Here, just as in equation (43) we have cut the number of terms in the sum in half, since each orbital appears twice in the sum shown in equation (36). In equation (44) we have included the shell index n explicitly in the label of the symmetry-preserving pair creation operator. In the individual one-electron creation operators of equation (44), this index is implied, as is the subshell index /. 

This also implies that the diagonal operator Pkk is idempotent, i.e., applying it twice gives exactly the same result as applying it once ... 

Improper axes can also be associated with several symmetry operations. We noted earlier that e, applied twice in succession, results in a simple 2tt/3 rotation about the S6 axis. In other words, we can write the set 5e, 5g, Sl, S, as Se, C3, S2, C. We stop at s because S = E due to the combination of C and an even number of reflections. 5g is equivalent to S2 because it contains three rotations by 2 r/6 and an odd number of reflections, and S2 means one rotation by n and one reflection. The operation S2, however, is easily shown to be equivalent to an inversion, and so we have, using a compressed notation, 25e, 2C3, i associated with the Se axis. Since we have already explicitly listed the elements C3 and i in our set of elements for staggered ethane (or any other system containing an Se axis) only the ISe operations are unique to the Se axis. The generalization of this case is that an S2n axis with odd n implies that elements C and i are also present. Of the 2 — 1 operations associated with S2 , — 1 are preempted by the C axis and 1 by the element i leaving — 1 operations to be attributed to the S2a axis. [If = 1, we have S2, C, and i as elements. But Ci = E, so we ignore it. There is only one operation here (21 — 1 = 1) and it is preempted by i. Therefore, S2 has no unique operations and it is not listed as a symmetry element.] For S2n with n even, i is not implied and there are n unique operations. 

---