Water symmetry operations

The molecules shown in Fig. 1 are planar thus, the paper on which they are drawn is an element of symmetry and the reflection of all points through the plane yields an equivalent (congruent) structure. The process of carrying out the reflection is referred to as the symmetry operation a. However, as the atoms of these molecules are essentially point masses, the reflection operations are in each case simply the inversion of the coordinate perpendicular to the plane of symmetry. Following certain conventions, the reflection operation corresponds to z + z for BF3 and benzene, as it is the z axis that is chq ep perpendicular to die plane, while it is jc —> —x for water. It should be evident that the symmetry operation has an effect on the chosen coordinate systems, but not on the molecule itself. 

Although all molecules are in constant thermal motion, when all of their atoms are at their equilibrium positions, a specific geometrical structure can usually be assigned to a given molecule. In this sense these molecules are said to be rigid. The first step in the analysis of the structure of a molecule is the determination of the group of operations that characterizes its symmetry. Each symmetry operation (aside from the trivial one, E) is associated with an element of symmetry. Thus for example, certain molecules are said to be planar. Well known examples are water, boron trifluoride and benzene, whose structures can be drawn on paper in the forms shown in Fig. 1. 

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

The four operations which form the symmetry group for the water molecule are represented in Fig. 2. It can be easily verified that the multiplication table for these symmetry operations is that already developed (Table 2). Thus, the symmetry group of the water molecule is isomorphic with the four-group. 

The internal coordinates for the water molecule are chosen as changes in the structural parameters defined in Fig. 3. The effect of each symmetry operation of the symmetry group ( 2 on these internal coordinates is specified in Table 2. Clearly, the internal coordinate Ace is totally symmetric, as the characters xy(Aa) correspond to those given for the irreducible representation (IR) Ai. On die other hand, the characters x/(Ar), as shown, can not be identified with a specific IR. By inspection of Table 2, however, it is apparent that the direct sum Ai B2 corresponds to the correct symmetry of these coordinates. In more complicated cases the magic formula can always be employed to achieve the correct reduction of the representation in question. 

Figure 2.1. Rotation by 180° is a symmetry operation of the water molecule.

Table 2.1 summarizes the symmetry operations of the water molecule, and records the final position of each atom after each operation. 

Figure 2.3. C2 is a symmetry operation of both water and hydrogen peroxide, but hydrogen peroxide has no plane of mirror symmetry. The orientation of the symmetry axis in H2O2 is perpendicular to the 0-0 bond and bisects the angle between the H-0-0 and O-O-H planes.

Axes of rotation are among the most common of molecular symmetry operations. A onefold axis is a rotation by a full turn, equivalent to the identity. A twofold rotation axis, as in the example of the water molecule, is sometimes called a dyad. Cyclopropane has a threefold axis perpendicular to the plane containing the carbon atoms it also has three twofold axes. Can you visualize them ... 

Each of the symmetry operations we have defined geometrically can be represented by a matrix. The elements of the matrices depend on the choice of coordinate system. Consider a water molecule and a coordinate system so oriented that the three atoms lie in the x-z plane, with the z—axis passing through the oxygen atom and bisecting the H-O-H angle, as shown in Figure 5.1. 

Because the traces contain sufficient information to decompose Ftot into irreducible representations, it is necessary to compute only the diagonal elements of the matrices of the representation. If a particular atom changes position under a symmetry operation, its displacements can contribute no diagonal elements to the matrix therefore, for that symmetry operation, such an atom may be ignored. For example, the displacements of the hydrogen atoms in water do not contribute to the character of C2 in Ftot- The displacement of Hi means that the elements (1,1), (2,2) and (3,3) of the matrix are zero. 

Let us examine the behaviour of py orbital m water under the symmetry operation of the point group C2 (Figure 2.13a). Rotation around the z-axis changes sign of the wave function, hence under Ca, Py orbital is... 

Like water, it has the symmetry of point group C2e. When the MOs of formaldehyde molecule as given in the figure are subjected to the symmetry operations of this group, n-orbital is observed to transform as bx and n orbital as ba as shown below ... 

Consider the symmetry behavior of the normal modes of H20 and COz. For water, the symmetry operations are E, C2 b), ov(ab), and av(bc), where the a and b principal axes lie in the molecular plane the bond angle in water is relatively obtuse, so that the C2 axis is the axis of intermediate moment of inertia. Each of these operations converts the displacement vectors of p, in Fig. 6.1 to an indistinguishable configuration the same is true for v2. Normal modes that are symmetric with respect to all the molecular symmetry operations are called totally symmetric. For H20, the modes and v2 are totally symmetric, but v3 is not. Figure 6.3 shows that v3 is antisymmetric with respect to C2(b). We can tabulate the behavior of the H20 normal modes with respect to the molecular symmetry operations ... 

Fig. 3.21 Normal modes of vibration of [he water molecule (a) symmetrical stretching mode. A,i (b> bending mode. <4, (c) andsyiranetncal stretching mode. Bt, and their transformations under C symmetry operations.

The symmetry operations of the water molecule are collected in Table 6-3, together with the transformation properties of the sundry orbitals classified under C2V, the point group of water (see Figure 6-9). 

To deal with the molecular orbitals for water it is useful to examine the effect of certain symmetry operations on the molecule. We choose a set of rectangular Cartesian axes (Fig. 4) so that the x axis bisects the angle between the bonds and the z axis is perpendicular to the nuclear plane. 

The third internal coordinate which can be considered in the water molecule is the bond angle, H-O-H. Its change will be the bending mode. All symmetry operations leave this basis unchanged, so the representation is ... 

Figure 6-17. The C2v symmetry operations applied to the two hydrogen I s orbitals of water as basis functions.

A predominant motif in the hydrogen-bonding structure in a-cyclodex-trin-7.57H20 is a ribbon of fused four- and six-membered antidromic cycles, shown in Figs. 18.5 c and 18.7 c, d. This is generated by the symmetry operation of a screw axis in the b direction. It contains an infinite chain of water molecules,... 

Thus, the force constants of the bonds, the masses of the atoms, and the molecular geometry determine the frequencies and the relative motions of the atoms. Fig. 2.1-3 shows the three normal vibrations of the water molecule, the symmetric and the antisymmetric stretching vibration of the OH bonds, and Va, and the deformation vibration 6. The normal frequencies and normal coordinates, even of crystals and macromolecules, may be calculated as described in Sec. 5.2. In a symmetric molecule, the motion of symmetrically equivalent atoms is either symmetric or antisymmetric with respect to the symmetry operations (see Section 2.7). Since in the case of normal vibrations the center of gravity and the orientation of the molecular axes remain stationary, equivalent atoms move with the same amplitude. 

TrAnslations (and p orbitals) along the x and z axes in the water moleculB conform to difTerent symmetry patterns than the one just developed for the y axis. When the , Cj, ojixz), and cj yz) operations are applied, in that order, to a unit vector pointing in the +x direction, the labels + 1, — 1, + 1, and -- I are generated. A vector pointing in the +z direction will be unchanged by any of the symmetry operations and thus will be described by the set -H, +1, +1, -M. 

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