Coordinates symmetry

Modes of different symmetry never mix, even if they are close in energy. (This is a general rule which will have its analogous version for the transitions among electronic states as will be seen later in Chapters 6 and 7.) 

The above analysis of the types of normal modes brings us to the limit where simple symmetry considerations can take us. Nothing yet has been said about the pictorial manifestation of the various normal modes. Above we deduced, for example, that the B2 normal mode of the water molecule is a pure stretch. The question may also be asked, how does it look This question can be answered with the help of symmetry coordinates. 

The symmetry coordinates are symmetry-adapted linear combinations of the internal coordinates. They always transform as one or another irreducible representation of the molecular point group. 

Symmetry coordinates can be generated from the internal coordinates by the use of the projection operator introduced in Chapter 4. Both the symmetry coordinates and the normal modes of vibration belong to an irreducible representation of the point group of the molecule. A symmetry coordinate is always associated with one or another type of internal coordinate—that is pure stretch, pure bend, etc.—whereas a normal mode can be a mixture of different internal coordinate changes of the same symmetry. In some cases, as in H20, the symmetry coordinates are good representations of the normal vibrations. In other cases they are not. An example for the latter is Au2C16 where the pure symmetry coordinate vibrations would be close in energy, so the real normal vibrations are mixtures of the different vibrations of the same symmetry type [7], The relationship between the symmetry coordinates and the normal vibrations can be 

Return now to the symmetry coordinates of the water molecule. They can be generated using the projection operator. As has been mentioned before, here we are interested only in the symmetry aspects of the symmetry coordinates. Thus, the numerical factors are 

The second delta function on the RS of eq. (31) is zero unless K =K, when the exponential factors cancel and 

The determination of the eigenvalues wy(q)2 may be simplified by an orthogonal transformation to symmetry coordinates, which are linear combinations of the Cartesian displacements of the atoms which represent the actual displacements of the atoms in the unit cell. Simultaneously, the eigenvectors undergo the same orthogonal transformation (see Section 9.4, especially eqs. (9.4.4) and (9.4.6)). In matrix notation, 

The eigenvalues q)2 in eq. (2) are not necessarily all distinct, so the index j will now be replaced by the double index ak where a labels the distinct eigenvalues of D(q) and k= 1, 2,. .., 1(a) labels the linearly independent (LI) eigenvectors associated with the degenerate eigenvalue a. In this notation, 

Equation (2) shows that T(q, R) c(qnA.)) must be a linear combination of the LI eigenvectors of D(q) with eigenvalue tiv(q)2, 

Since there may be more than one IR TCT(q, R) of the same symmetry, an additional index (j,= l,2, c T may be needed to label the different eigenvalues of the same symmetry rr. 

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In the harmonic approximation, V does not involve the cross-term pj-Pe because pj- and are the symmetry coordinates. It is thus of the form... 

Symmetry coordinates Vibrational quantum numbers FiJ V, k (varies) Fy = d V/dS,dSj... 

FIGURE 12. Molecular internal coordinates (bond lengths and angles) to be combined in symmetry coordinates (see Table 2). 

TABLE 2. Analytical expressions for some symmetry coordinates" in dimethyl sulphoxide and dimethyl sulphone11... 

Symmetry-adapted combinations 6 Symmetry coordinates 13 Synthons 767, 788... 

Coordinates such as these, which have the symmetry properties of the point group are known as symmetry coordinates. As they transform in the same manner as the IRs when used as basis coordinates, they factor the secular determinant into block-diagonal form. Thus, while normal coordinates most be found to diagonalize the secular determinant, the factorization resulting horn the use of symmetry coordinates often provides considerable simplification of the vibrational problem. Furthermore, symmetry coordinates can be chosen a priori by a simple analysis of the molecular structure. 

In the example considered above, Arj - A/s is the only symmetry coordinate of species B2. Thus, it results in a factor of degree one in the completely reduced secular determinant It is therefore a normal coordinate. On the other hand, the two normal coordinates of species Ai are linear combinations of the symmetry coordinates Acr and Arj + Ar2. They can only be found by solution of the secular equations. 

It is usually convenient to normalize the symmetry coordinates. Hence, for the example considered here, the three symmetry coordinates take the form... 

It should be noted that when the symmetry coordinates have been normalized, V is orthogonal that is, V = U l. 

To see how use is made of symmetry coordinates as the bases of the vibrational problem, reconsider the kinetic and potential energies as given earlier, e.g. 

As both F and G are partitioned by the use of symmetry coordinates, the secular determinant is factored accordingly. The problem of calculating the vibrational frequencies is thus divided into two parts solution of a linear equation for the single frequency of species B2 and of a quadratic equation for the pair of frequencies of species Aj. 

It has been shown that the potential energy distribution provides an approximate method to evaluate the relative contribution of each symmetry coordinate to a given normal mode of vibration. From the definition of the symmetry coordinates, the relation... 

For symmetric tetraatomic molecules (ABBA), it is convenient to introduce the symmetry coordinates corresponding to the trans and cis bending by the vector relations... 

Pj and p2 represent the displacement vectors of the nuclei A and D (the corresponding polar coordinates are p1 cji, and p2, < )2, respectively) p, and pc are the displacement vectors and pT, r and pc, <[)f the corresponding polar coordinates of the terminal nuclei at the (collective) trans-bending and cis-bending vibrations, respectively. As a consequence of the use of these symmetry coordinates the nuclear kinetic energy operator for small-amplitude bending vibrations represents the kinetic energy of two uncoupled 2D harmonic oscillators ... 

The Hamiltonian operators can be written in terms of the symmetry coordinates... 

The two most useful sets are the bond displacements themselves, and the symmetry coordinates. The use of the latter leads naturally to a scheme in which the Hamiltonian for bent molecules is no longer diagonal in the total 0(4) quantum numbers (ti, x2), and thus one loses the simple form of the secular equation (Figure 4.11). The secular equation must be now diagonalized in the full space with dimensions that become rapidly larger. This scheme, developed by Leviatan and Kirson (1988), can be implemented only if the vibron numbers N are relatively small, N < 10. 

A consequence of the symmetry of the molecule is that states must transform according to representations of the appropriate symmetry group. In terms of coordinates, this implies that one must form internal symmetry coordinates. These are linear combinations of the internal coordinates. For example, denoting in Fig. 6.1 by sx, s2, s3,, v4, j5, s6 the stretching coordinates of the six C-H bonds, the internal symmetry coordinates are linear combinations... 

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