Improper Rotations S6 for Oh

• there are 6S4 operations in the Oh point group o each pair of C4 axis and σd mirror plane has an associated improper rotation, the S4 axis is coincident with the C4 axis. o phase changes are important for improper rotations and it is best to work with pAOs rather than sAO when drawing out these operations. There are two sets of symmetry related

orbitals; the equatorial and axial pπ orbitals. These subsets only transform amongst themselves under the S4, Figure 1

C4

σ d

Figure 1 Sets of symmetry related orbitals

1 o Consider a S4 operation which consists of a rotation of 90º around a C4 axis and then 1 reflection in the associated σh plane. An example of the S4 operation is given in Figure 2. 1 For simplicity the effect of the S4 operation has been shown for a single equatorial and axial

pπ orbital. Notice that if you considered only the equatorial pπ orbitals you might conclude 1 that S4 = C2′′ , which would be incorrect.

C4 C4 C4

σ d C4 σ d

S2 = C1 4 2 1 Figure 2 S4 operation

o working through the various improper rotation symmetry operations results in identifying that 2 4 S4 = C2 and S4 = E , Figure 3, thus for each C4 axis (of which there are three) there are two 1 3 −1 unique S4 operations ( S4 S4 = S4 ) and hence there are 6S4 operations in Oh.

C4 C4 C4

σ d 1 2 S4 S4

C C 4 4 3 S4

4 S4

Figure 3 All S4 operations

1

• there are 8S6 operations

o each C axis can also be thought of as having a 3 coincident C6 and σ h mirror plane perpendicular C3 to this axis. L o these are not symmetry elements of O because of L L h the staggered arrangement of the three ligands M σ h o in Figure 4, a simplified model is shown with has L L black dots on the sites that are occupied by ligands L 1 Figure 4 simplified model for octahedral o consider S6 , a rotation of 60º and then reflection

complex emphasising the C3 axis in the associated σ h plane perpendicular to the axis, Figure 5

o all of the S6 operations are shown in Figure 6, 1 5 −1 only two are unique S6 and S6 = S6 .

o thus for each C3 axis (of which there are four)

there are two associated S6 operations and hence

there are 8S6 operations in Oh

C6 σ h

1 S6

1 Figure 5 Components of the S6 operation

1 S2 3 S6 6 S6

4 S6 6 5 S6 S6

Figure 6 The S6 operations

2