Cartesian displacement vectors

We shall now illustrate this, using CO5" as an example. A number of further examples will be found in Section 10.7. The first step must be to determine the symmetry group to which the molecule belongs, as described in Chapter 3, especially Section 3.14. We find that CO5" belongs to the DVt group. Figure 10.3 shows the CO3 ion with the sets of Cartesian displacement vectors attached to.each atom. There are of course 3n - 12 in all, and the representation will therefore be of dimension 12. 

We now apply a threefold rotation to the set of Cartesian displacement vectors with the results pictured in Figure 10.5. Again we wish to construct the matrix expressing these results. This procedure is a trifle tedious but requires no more than the simplest trigonometry. For example, as Figure 10.6 shows, X[ can be expressed as - X2 - (V3/2)Y2, and this result has been... 

Figure 10.3 The set of 3n = 12 Cartesian displacement vectors used in determining the reducible representation spanning the irreducible representation of the normal modes of COj".

Figure 10.5 Diagrams showing the effect of a threefold rotation on the set of Cartesian displacement vectors.

Figure 10.8 Abbreviated matrix for the operation C2 on the Cartesian displacement vectors of COL-...

The details of the F-G matrix procedure are best explained by working through a simple example, such as the water molecule. This belongs to the point group C2l.. The nine Cartesian displacement vectors, three on each atom, give rise to the representation... 

The set of 12 Cartesian displacement vectors for the entire molecule generates the following reducible representation ... 

Such molecules belong to the point group Olt. They have 3(7) - 6 = 15 degrees of internal freedom. The 21 Cartesian displacement vectors generate the representation T given in the table below ... 

Let us take now a more complicated basis, and consider all the nuclear coordinates of HNNH shown in Figure 4-8a. These are the so-called Cartesian displacement vectors and will be discussed in Chapter 5 on molecular vibrations. Let us find the matrix representation of the crh operation (see Figure 4-8b). The horizontal mirror plane leaves all x and y coordinates unchanged while all z coordinates will go into their negative selves. In matrix notation this is expressed in the following way ... 

With the 12-dimensional reducible representation of the Cartesian displacement vectors of HNNH, the inspection method probably does not work. However, the reduction formula can be used. The reducible representation is ... 

The first step in the symmetry determination of the dynamic properties is the selection of the appropriate basis. Appropriate here means the correct representation of the changes in the properties examined. In the investigation of molecular vibrations (Chapter 5), either Cartesian displacement vectors or internal coordinate vectors are used. In the description of the molecular electronic structure (Chapter 6), the angular components of the atomic orbitals are frequently used... 

First, an appropriate basis set has to be found. Considering that a molecule has 37V degrees of motional freedom, a system of 37V so-called Cartesian displacement vectors is a convenient choice. A set of such vectors is shown in Figure 5-5 for the water molecule. A separate Cartesian coordinate system is attached to each atom of the molecule, with the atoms at the origin. The orientation of the axes is the same in each system. Any displacement of the atoms can be expressed by a vector, and in turn this vector can be expressed as the vector sum of the Cartesian displacement vectors. 

Next, the set of Cartesian displacement vectors is used as a basis for the representation of the point group. As discussed in Chapter 4, the vectors connected with atoms that change their position during an operation will not contribute to the character and thus they can be ignored. 

Continuing with the water molecule as an example, the basis of the Cartesian displacement vectors will consist of nine vectors... 

Figure 5-5. Cartesian displacement vectors as basis for representation of the water molecule.

Our first task is to generate the representation of the Cartesian displacement vectors of the four atoms of the molecule (see Figure 4-8a-c). As was shown in Chapter 4 (Section 4.7), the representation is... 

The set of Cartesian displacement vectors as basis for a representation is shown in Figure 5-9. The symmetry operations of the point group are also shown. The D h character table is given in Table 5-3. Recall (Chapter 4) that the matrix of rotation by an angle is... 

Cuony and Hug70) and Hug38) have discussed sum rules for vibrational ROA in the limited context of molecules that are chiral due to isotopic substitution. By expressing the normal coordinates in (8aafj/0Qp)o etc. in terms of atomic cartesian displacement vectors, they were able to show that the optical activity in the fundamentals sums to zero if the parent achiral group is other than and Cj. Their proof requires the assumption that the isotopic substitution does not affect the equilibrium electronic distribution, in which case the corresponding Rayleigh optical activity is zero. 

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