Rotational normal coordinates

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. ... 

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,... 

For vei y small vibronic coupling, the quadratic terms in the power series expansion of the electronic Hamiltonian in normal coordinates (see Appendix E) may be considered to be negligible, and hence the potential energy surface has rotational symmetry but shows no separate minima at the bottom of the moat. In this case, the pair of vibronic levels Aj and A2 in < 3 become degenerate by accident, and the D3/, quantum numbers (vi,V2,/2) may be used to label the vibronic levels of the X3 molecule. When the coupling of the... 

Propagate by harmonic part of Hq for the time Arjl. This corresponds to the rotation of internal normal coordinates, P( and Q[, in the phase space by the corresponding vibrational frequency Ui... 

Table I Harmonic frequencies and cubic force constants (in the reduced normal coordinate representation) for the normal isotopomer (1) of Sij (top). Measured rotational constants and effective equilibrium values (in MHz) for the five isotopomers described in the text (bottom).

The relation between shear stress and shear strain can be established based on the relation between normal stress and normal strain. Equations (F.3) and (F.4). Actually, by rotating the coordinate system 45°, it becomes a problem of normal stress and normal strain. Using geometrical arguments, it can be shown that (see, for example, Timishenko and Goodier, 1970) ... 

By comparing their normal coordinate analysis results and their liquid experimental Raman spectra in Figure 12.6, Hamaguchi etal. [50,57,59,64,70] concluded that the two rotational isomers AA and GA must coexist in the IL state. According to the Raman spectra of all the liquids in Figure 12.6, both the key bands for the AA conformer (625 cm and 730 cm bands), and for the GA conformer (603 cm and 701 cm bands), respectively, appeared in the spectra. Therefore, the two isomers of rotational freedom around the C7-C8 and C8-C9 bonds, AA and GA, must coexist in these [C4CiIm]X liquids. 

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 the polar normal coordinate

linear molecule. Substitution of (6.89) into (6.88) and use of trigonometric identities gives... 

The extension of the trajectory calculations to a system with any number of atoms is straightforward except for the quantization of the vibrational and rotational states of the molecules. For a molecule with three different principal moments of inertia, there does not exist a simple analytical expression for the quantized rotational energy. This is only the case for molecules with some symmetry like a spherical top molecule, where all moments of inertia are identical, and a symmetric top, where two moments of inertia are identical and different from the third. For the vibrational modes, we may use a normal coordinate analysis to determine the normal modes (see Appendix E) and quantize those as for a one-dimensional oscillator. 

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