Symmetry operations horizontal mirror plane

Consider the effects of the symmetry operations of the Cji, point group on the set of x, y, and z coordinates. [The set of p orbitals p, Py, Pz) behaves the same way, so this is a useful exercise.] The water molecule is an example of a molecule having C2, symmetry. It has a C2 axis through the oxygen and in the plane of the molecule, no perpendicular C2 axes, and no horizontal mirror plane, but it does have two vertical mirror planes. 

The improper rotation consists of a rotation and reflection. Even though the axis and horizontal plane need not be elements themselves, if they are present then the improper rotation will also be a symmetry element. For example, the planar molecule BF3 has a principal C3 axis and a horizontal mirror plane and so there is also an S, axis collinear with the C3. For planar molecules, the reflection in ai, does not alter any atom positions however, if we place a vertical arrow on one of the F atoms, then it will be reversed by the reflection. In later chapters, the addition of arrows like this will be used in the analysis of molecular vibrations and is referred to as a basis. A basis allows us to study the effect of symmetry operations not only on the atom positions but also their motion. In Figure 2.9 the idea is simpler we add the arrow to highlight operations which turn the molecular... 

The representations of other point groups label the MOs of all nonlinear molecules. For linear molecule wavefunctions, we considered only four symmetry operations rotation about the z axis, inversion, reflection in the xy horizontal plane, and reflection in the vertical planes. Remember that when these operators act on the wavefunction, they may change if/ but not if/. The same principle remains true when we move on to polyatomic molecules, now with other possible symmetry elements. The symmetry properties of the orbital are denoted by the representation used to label the orbital. The Uj MOs of the Cjy molecule F2O, for example, have electronic wavefunctions that are antisymmetric with respect to reflection in either of the two vertical mirror planes (Fig. 6.10). 

The highest order axis present is taken to be the principal axis and gives us the vertical direction. So, BF3 has three vertical mirror planes, each of which contains a B—Fbond an example of a vertical mirror plane in BF3 is shown in Figure 1.13a. The C3 operations wiU move the fluorine atoms between these planes, but each will always contain one fluorine atom and reflect the other two into one another. So, although there are three vertical planes, they are identical, requiring only the single label and there are three operations. The plane of the molecule for BF3 is also a plane of symmetry, as illustrated in Figure 1.13b. This contains all three of the B—F bonds, but not the principal axis. In fact, the plane is perpendicular to the C3 axis, i.e. the plane is horizontal and so is labelled cti,. 

An improper rotation S is actually a combination of two operations a rotation about a Cn axis and then a reflection through a plane which is horizontal with respect to the axis. This operation is defined as the two procedures together. The molecule has an S axis of symmetry if the combined rotation-reflection gives a result indistinguishable from the start point. After just the rotation the structure may be completely different from the start point neither the C axis nor the mirror plane need be symmetry elements themselves. 

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