Translational normal coordinates

It is also quite straight forward to determine the character of rt,r. For a non-linear molecule there are three translational normal coordinates and three rotational normal coordinates and we write r = I r. The translational motion corresponds to a displacement of the molecule in some arbitrary direction and it can be depicted by a single vector showing the displacement of the centre of mass. Let this vector be e1+ye1+ S,... 

Since molecular translations do not change the dipole moment, all derivatives of die type dp/di, where r ate the translational normal coordinates, are zero. The dipole moment derivatives with respect to rotational coordinates are obtained according to the relation... 

Summarizing, in the crystal there are 36 Raman active internal modes (symmetry species Ug, hig, 2g> and 26 infrared active internal modes (biw b2w hsu) as well as 12 Raman active and 7 infrared active external vibrations (librations and translations). Vibrations of the type are inactive because there appears no dipole moment along the normal coordinates in these vibrations of the crystal. 

There are similar equations for the other operations of the point group. f Therefore, Ql forms a basis for the I 1 representation and is a vibrational normal coordinate, Q% and Qs together form a basis for the T representation and they are also vibrational normal coordinates, the rest correspond to X = 0 and are translational or rotational normal coordinates, Qt and Q6 belong to r, Q9 to r 1, Q7 to P 1 and and Q0 to r. ... 

We have taken the sums in (6.23) and (6.24) to include only 3N — 6 (rather than 3N) normal coordinates. The reasons for this are as follows. We know that there are only 3N — 6 (if the molecule is linear, read 5 for 6 in this and each subsequent expression) vibrational degrees of freedom. We didn t take the trouble to use the six relations between the s to eliminate six of them, but treated all 3N q s as if they were independent. It can be shown that the consequence of this is that six of the roots Xk will be zero these six vanishing roots correspond to translational and rotational degrees of freedom. We adopt the convention that the zero roots will be... 

Thus, Gajv-s. . Qsn 0 correspond to the three coordinates of translation, the tree coordinates of rotation, and the coordinate p. The requirement which specifies that the are the normal coordinates of vibration is that the potential energy expansion in Q is diagonal up to the quadratic terms [cf. Eq. (3.23)]... 

Figure 2.7-6 A Assignment of the Cartesian coordinate axes and the symmetry operations of a planar molecule of point group C2,.. B Character table, 1 symbol of the point group after Schoen-flies 2 international notation of the point group 3 symmetry species (irreducible representations) 4 symmetry operations 5 characters of the symmetry operations in the symmetry species +1 means symmetric, -1 antisymmetric 6 x, y, z assignment of the normal coordinates of the translations, direction of the change of the dipole moment by the infrared active vibrations, R, Ry, and R stand for rotations about the axes specified in the subscript 7 x, xy,. .. assign the Raman active species by the change of the components of the tensor of polarizability, aw, (Xxy,. ...

Six of these normal coordinates (five for a linear molecule) have a frequency eigenvalue identically equal to zero. These motions are translations and rotations of the molecule. Although the approach through Cartesian displacement coordinates is theoretically elegant, it is generally more practical to express the vibrational motions in terms of internal coordinates, such as bond stretches and distortions of bond angles. The method is discussed in detail in Chapter 4 of Wilson, Decius and Cross [57]. Since the distortions of the molecule can be described in terms of 3A — 6 of these internal coordinates there are no redundant dimensions to be removed when the analysis is complete. 

The function y/mx is the solution of the rotational problem. The vibrational part, is a function of the normal coordinates and is the vibrational wave function. Substituting Eq. (4.24) in Eq. (4.23) and ignoring the rotational and translational contributions, the Schrddinger equation for the vibrational wave function will be ... 

Another descriptor based on eigenvalues is the EVA descriptor [66,67] (where EVA stands for EigenVAlue). The EVA descriptor is an example of a 3D descriptor that does not require alignment of the 3D structures (i.e., it is invariant to translation or rotation of the structure). It is based on the infrared spectrum of a molecule, which is related to the 3D structure. The molecular vibrations are calculated using a normal coordinate analysis (NCA) of the energy-minimized structure (e.g., using MOPAC AMI), and the eigenvalues from the NCA are used to derive the EVA descriptor (see Ref. 67). 

Here the Q are the generalized normal coordinates and the to are the associated harmonic frequencies. They are obtained at each point on the path by diagonalizing the force constant matrix for which the reaction path direction as well as directions corresponding to rotations and translations have been projected out. The projected force constant matrix has seven zero eigenvalues corresponding to overall rotations, translations, and the reaction path direction. It also has 3N-7 nonzero eigenvalues corresponding to vibrations transverse to the path. 

Let a representation be written with the 3N rectangular coordinates of an A -atom molecule as its basis. If it is decomposed into its irreducible components, the basis for these irreducible representations must be the normal coordinates, and the number of appearances of the same irreducible representation must be equal to the number of normal vibrations belonging to the species represented by this irreducible representation. As stated previously, however, the 3A rectangular coordinates involve six (or five) coordinates, which correspond to the translational and rotational motions of the molecule as a whole. Therefore, the representations that have such coordinates as their basis must be subtracted from the result obtained above. Use of the character of the representation, rather than the representation itself, yields the same result. 

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