Symmetry, axes translation

Once the state symmetries have been established it only remains to be shown that the direct product of the species of the ground state symmetry, the coordinate translational symmetries and the excited state symmetry belong to the totally symmetric species A. Let us take the n-wr transition in formaldehyde. The ground state total wave function has the symmetry Ax. The coordinate vectors x, y and z transform as By, Bf and Ax respectively (refer Character Table for C2k Section 2.9, Table 2.2), The excited state transforms as symmetry species A2. The direct products are ... 

The ratio of symmetry numbers s s° in equation 11.40 merely represents the relative probabilities of forming symmetrical and unsymmetrical molecules, and ni and nf are the masses of exchanging molecules (the translational contribution to the partition function ratio is at all T equal to the power ratio of the inverse molecular weight). Denoting as AX, the vibrational frequency shift from isotopically heavy to light molecules (i.e., AX, = X° — X ) and assuming AX, to be intrinsically positive, equation 11.40 can be transated into... 

In the crystal structures, neighboring doublehelices have the same rotational orientation and the same translation of half a fiber repeat as in the PARA 1 model. Only the Ax vector is slightly larger in the calculated interaction (1.077 nm) than in the observed ones 1.062 nm and 1.068 nm in the A type and B type, respectively. This may be due to the fact that in the crystal structures the helices depart slightly from perfect 6-fold symmetry. Also, no interpenetation of the van der Waals surfaces is allowed in the calculations, whereas some of them may occur in the cristallographic structure. It is quite interesting to note that the network of inter double-helices hydrogen bonds found in the calculated PARA 1 model reproduces those found in the crystalline structures. 

There are two totally symmetric ( ,) normal modes and one b2 normal mode. (The convention is to use lowercase letters for the symmetry species of the normal modes.) The symmetry species of the normal modes have been found without solving the vibrational secular equation. Moreover, since there is only one b2 normal mode, the form of this vibration must be determined from symmetry considerations together with the requirement that the vibration have no translational or rotational energy associated with it. Thus (Fig 6.1), any bent XYX molecule has a b2 normal mode with the X atoms vibrating along the X—Y bonds and the Y atom vibrating in the plane of the molecule and perpendicular to the symmetry axis. On the other hand, there are two ax symmetry coordinates and the two ax normal vibrations are linear combinations of the ax symmetry coordinates, where the coefficients are dependent on the nuclear masses and the force constants. Thus the angles between the displacement vectors of the X atoms and the X—Y bonds for the ax modes of a bent XYX molecule vary from molecule to molecule. 

The second term in Eqs. (9.75) and (9.76), die rotational atomic polarizability tensor reflects the contribution of molecular translation and rigid-body rotation to ax- The inclusion of the six external molecular coordinates in those equations - the diree translations Xy and X2, and the three rotations p, Py and P2, completes die set of molecular coordinates up to 3N. In diis vray polarizability dmivatives are transformed into quantities corresponding to a space-fixed Cartesian coordinate system. As already pointed out in section 4.1, the great advantage of such a step is that the imensity parameters defined in terms of a space-fixed coordinate system are independent on isotopic substitutions provided the symmetry of the molecule is preserved. This will be illustrated with an example in the succeeding section. By analogy with Eq. (9.77), die rotational polarizability tensor can be represented as... 

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