Symmetry operations vertical mirror plane

Any set with the four properties (a)-(d) forms a group therefore the set G is a group for which the group elements are point symmetry operators. This point group is called C3v or 3m, because the pyramid has these symmetry elements a three-fold principal axis and a vertical mirror plane. (If there is one vertical plane then there must be three, because of the three-fold symmetry axis.)... 

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 symmetry elements of the molecule are shown in the figure. The symmetry operators that belong to the nuclear framework are E, C3, C3 and the three reflection operators corresponding to vertical mirror planes passing through each of the three hydrogen nuclei, which we call da, Oh, and dc- The square of the C3 operator is included because we must include the inverse of all operators in the group and C3 is the inverse of C3. 

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 symmetry elements of the point group D i, can be identified by the labels for the axes and planes shown. For example, QfT) corresponds to a Q rotation around axis 4 (the X axis), while <7i is reflection through the vertical mirror plane that contains the z axis and axis 1 (the yz plane). Find the single symmetry operator that is equivalent to... 

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,. 

C2 axis and two vertical mirror planes, ctv and ay. These molecules do not have any other rotation axes or mirror planes and do not have an inversion centre so, according to the operations met so far, this set completely describes the symmetry of both molecules. The two molecules are said to belong to the same symmetry point group because they have exactly the same set of valid symmetry operations. 

In the example of H2O above we used the idea of a global axis system, X, Y, Z. This axis system is used to define the positions of the symmetry elements of the molecule and, once set, the global axis system is not moved by any operations that are carried out. This means that the symmetry elements should be considered immovable and symmetry operations only move the atoms in the molecule. This becomes especially important when molecules with more symmetry elements are considered. For example, ammonia (NH3) has a principal axis of order 3 and three vertical mirror planes, as shown in Figure 2.3. 

The whole set of operations produces an axial molecule in which there are four equivalent substituents at either end. The black and white points are at different distances from the axis, and so the radii of the circles drawn to show the relationship between images in Figure 3.6a are different, emphasizing that there are no symmetry-equivalent points at opposite ends of the object. Below the three-dimensional sketch is a plan view of the same structure. For the points at either end of the object, we could also think to introduce vertical mirror planes relating the points to their opposite images within the same terminal group. However, the offset in orientation between the two ends is such that these mirror planes would not be symmetry elements of the object as a whole. 

BF3 is a planar molecule with a C3 principal axis and three vertical mirror planes. Use the products of the operations accompanying these symmetry elements to show that the point group for the molecule contains a total of 12 operations and list them. 

There are no operations which swap atoms over in C ov symmetry, there is only the identity E, the infinite number of operations associated with the axis and the infinite number of vertical mirror planes that contain the molecular axis. Conventionally, we align the Z axis with the principal axis, along the molecular bond. The character table in Appendix 12 shows that Is, 2s and 2p AOs belong to the irreducible representation, while 2p t and 2py are degenerate, belonging to the n representation. 

A group that consists of point symmetry operations is called a point group. A unique symbol called a Schoenflies symbol is assigned to each point group. The Schoenflies symbol of the point group of the H2O molecule is C2 . This symbol indicates that the molecule has a C2 rotation axis and two vertical mirror planes. You can specify the symmetry properties of a molecule like H2O or SO2 quickly to a knowledgeable person... 

It is equivalent to describe the symmetry class of the tetrahedron as 3/2-m or 3/4. The skew line relating two axes means that they are not orthogonal. The symbol 3/2-m denotes a threefold axis, and a twofold axis which are not perpendicular and a symmetry plane which includes these axes. These three symmetry elements are indicated in Figure 2-50. The symmetry class 3/2-m is equivalent to a combination of a threefold axis and a fourfold mirror-rotation axis. In both cases the threefold axes connect one of the vertices of the tetrahedron with the midpoint of the opposite face. The fourfold mirror-rotation axes coincide with the twofold axes. The presence of the fourfold mirror-rotation axis is easily seen if the tetrahedron is rotated by a quarter of rotation about a twofold axis and is then reflected by a symmetry plane perpendicular to this axis. The symmetry operations chosen as basic will then generate the remaining symmetry elements. Thus, the two descriptions are equivalent. 

If reflection of all parts of a molecule through a plane produces an indistinguishable configuration, the plane is a plane of symmetry the S5Tnmetry operation is one of reflection and the S5anmetry element is the mirror plane (denoted by a). For BF3, the plane containing the molecular framework (the yellow plane shown in Figure 3.2) is a mirror plane. In this case, the plane lies perpendicular to the vertical principal axis and is denoted by the symbol cxh. 

The symmetry element is the plane itself, since all points in the plane remain unchanged by the operation of reflection. For the water molecule there are two planes of symmetry, as shown in Figure 1.11. These are distinguished by labelling the plane perpendicular to the molecule a and the plane of the molecule itself a. The C2 axis of water is the only axis, and so it is also the principal axis defining the vertical direction. This means that the mirror planes are standing vertically, and so a subscript is added to remind us of this, giving ay and CTv. ... 

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... 

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