Symmetry rotor group

Some decades ago, Swalen and Costain(1959), Myers and Wilson (1960) have extended the use of the symmetry point groups for studying the double internal rotation in acetone [1-2]. Dreizler generalized their considerations to two Csv rotor molecules with frames of a lower symmetry than C2U, and deduced their character tables [3]. 

In single rotor molecules such as toluene, phenol or F-phenols [31-32] where the NRG s coincide with some symmetry point groups, the deduction of the selection rules are straightformard. In double rotor molecules, this deduction is not so easy [22]. Let us consider again the acetone case to illustrate this application of the NRG s [58,59]. 

Table 2.8. Torsion angle transformations associated with the 24 frame symmetry operations of a CR4 molecule with a tetrahedral frame group F. For tetraphenylmethane, the complete set of isometric conformations is obtained by combining these 24 frame group transformations with a rotor group of order 16 to obtain a group of order 384...

If there arc two or more axes of greater than twofold symmetry, the molecule will be a spherical rotor (groups 3, 3a, 3 0, 6a, S). 

Figure 8.15. Correlation diagram between levels of a rigid rotor K = 0 (water dimer with Cs symmetry in the nontunneling limit), a rotor with internal rotation of the acceptor molecule around the C2 axis (permutation-inversion group G ), and group G16. The arrangement of levels is given in accordance with the hypothesis by Coudert et al. [1987], The arrows show the allowed dipole transitions observed in the (H20)2 spectrum. The pure rotational transitions E + (7 = 0) - E (J = 1) and E (7 = 1) <- E + (/ = 2) have frequencies 12 321 and 24 641 MHz, respectively. The frequencies of rotationtunneling transitions in the lower triplets AI (7 = 1) <- A,+ (7 = 2) and A," (7 = 3) <- A,+ (7 = 4) are equal to 4863 and 29 416 MHz. The transitions B2(7 = 0)<- B2(7 = l) and BJ(7 = 2) <- B2 (7 = 3) with frequencies 7355 and 17123 MHz occur in the higher multiplets.

As in the case of the simple switch, the existence of this operation is conditioned by the presence of symmetry planes in the rotors and the frame [33]. The character table of such a group may be written as shown in Table 4. 

The symmetry eigenvectors corresponding to the G36 acetone-like group are not so easily deduced by tryings and errors, from the character table, as in the case of the one-fold rotor molecules. The projector technique has to be used for [21,34] ... 

Fig. 8. The torsional potential and energy levels of a methyl-like rotor. The feasible group is isomorphic with C3. The three minima of the potential correspond to the three equilibrium orientations of the rotor in its molecular/crystal surrounding. The torsional levels come in triplets whose individual components transform according to the irreps A, Ea, and E, of C3. The E sublevels come in perfectly degenerate Kramers pairs those of A symmetry are shifted in energy from the Kramers sublevels by the tunnelling quanta the magnitudes of which rapidly grow with...

In asymmetric tops like methyl alcohol, CHjOH, and symmetric rotors like CH SiH, the methyl group can undergo internal rotation relative to the rest of the molecule, traditionally called the frame (LS59, OM07). Although various different tops are considered here, all have three-fold symmetry. In such cases, the potential l hindering the internal rotation can be written ... 

The symmetry number is easiest calculated by first treating the molecule as rigid and calculating the external symmetry number, cr. For CH4, cr = 12 for any other linear unbranched alkane, (7 = 6. For most branched alkanes, can be 1,2,3,4 or 6. Each internal CH3 rotor contributes a factor of 3 to each t-Bu rotor contributes a factor of 3" 3 for each CH3 group and 3 for overall symmetry of the three CH3 groups. 

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