1. Torsion-Rotation Program for Six-Fold Barrier Molecules Toluene MW Fit Vadim V. Ilyushin 1, Zbigniew Kisiel 2, Lech Pszczolkowski 2, Heinrich Mäder 3, Jon T. Hougen 4 1 Institute of Radio Astronomy of NASU, Kharkov, Ukraine 2 Institute of Physics, Polish Acad. of Sci., Warsaw, Poland 3 Institute for Phys. Chem., Kiel University, Kiel, Germany 4 Optical Technology Division, NIST, Gaithersburg, MD

2. H1 H2 H3 H4 H6 H5 H7 H8 Cf Cg Ca Cb CeCc Cd z x Toluene PI Group Operations Toluene  C3 top C2 frame (23)* (123) (ab)(cd)x (45)(67) G12  C6v A1, A2, B1, B2, E1, E2 Methanol  C3 top C2 frame (23)* (123) none G6  C3v A1, A2, E

3. Transformation properties of the torsional and rotational variables under various operations of the PI group G12 for toluene torsional rotational E  , ,  (123)  + 2  /3 , ,  (ab)()()()     + , ,  (ab)()()()(123)   2  /6  + , ,  (23)*    ,   ,  + 

4. Symmetry species  in G12 and time reversal symmetry (+) or (  )  Torsional Momenta |JKaKc  A1cos6  (+) ee A2 sin6  (+)Jz (  ), p  (  )eo B1 cos3  (+)Jy (  ) oe B2 sin3  (+)Jx (  ) oo E1exp(  i  ) E2 exp(  2i  )

5. Character table for the subgroup G6 of G12 needed to understand basis set construction in the two-step diagonalization procedure basis functions: exp(6k+  )i  |JKM> E (ab)(123) (ab)(123) 2 (123) (123) 2 (ab)  K A 1 1 1 1 1 1 0 even B 1 -1 -1 1 1 -1 3 odd E1+ 1 -  -  *   * -1 +1 E1  1 -  * -   *  -1 -1 E2+ 2  *   *  1 +2 E2  2   *   * 1 -2  = exp(2  i/3)

6. Where are we in the fitting procedure? Answer: In the middle of it. 234 transitions for m = 0, 1, 2, +3, -3 21 parameters standard deviation = 5.9 kHz J  25 for m = 0, but J  8 for m = 1,2,+3,-3 Ka, Kc labels are problematic (next slides)

7. Current understanding of the toluene spectrum: J = 2 ← 1 region Observed (waveguide FTMW) Calculated = |m|≤3 in fit = |m|>3 assigned, not yet fitted

8. Specimen fit:

14. Conclusions: 1.Our 6 kHz fit is excellent, but: 2.Assignments need to be extended into higher J and K problem regions (see Zbigniew Kisiel at this meeting). 3. Measure J = 1  0 region (see Heinrich Mäder at this meeting). 4 Labeling algorithm in program needs to be able to deal with Ka,Kc (high-barrier) and  K (low barrier) labels.