A fitting program for molecules with two equivalent methyl tops and C 2v point-group symmetry at equilibrium: Application to existing microwave, millimeter, and sub-millimeter wave measurements of acetone a Institute of Radio Astronomy of NASU, Kharkov, Ukraine. b NIST Guest Worker June – August 2009 & June – July c Sensor Science Division, NIST, Gaithersburg, MD , USA Vadim V. Ilyushin a,b, Jon. T. Hougen c

The PAM_C2v_2tops program makes use of an explicit two-dimensional potential function and carries out a global fit of rotational transitions in several torsional states simultaneously. A two step diagonalization procedure is used. Two groups are used in the program: Permutation-inversion group = G 36 with species A 1, A 2, A 3, A 4, E 1, E 2, E 3, E 4, and G having degeneracies 1, 2 and 4. Permutation group for 2 tops = G 9 is used to block- diagonalize Hamiltonian matrix in 4 blocks with (  A,  B )= (0,0), (0,1), (1,1), (1,2) (Groner) (0,0)  A 1, A 2, A 3, A 4 ; (0,1)  G, (1,1)  E 3, E 4 ; (1,2)  E 1,E 2

Structure of the computer program A general expression for the fitting Hamiltonian is written as H = (1/4)  knpqr1r2s1s2t1t2 B knpqr1r2s1s2t1t2  {J 2k J z n J x p J y q [p A r1 p B r2 cos(3s 1  A ) cos(3s 2  B )sin(3t 1  A )sin(3t 2  B ) + (-1) (n+q) p B r1 p A r2 cos(3s 1  B ) cos(3s 2  A )sin(3t 1  B )sin(3t 2  A )]                                                                                              + [(-1) (n+q) sin(3t 2  A )sin(3t 1  B )cos(3s 2  A )cos(3s 1  B )p A r2 p B r1 + sin(3t 2  B )sin(3t 1  A )cos(3s 2  B )cos(3s 1  A )p B r2 p A r1 ]J y q J x p J z n J 2k } The B’s are fitting parameters. Subscripts A and B denote the two methyl tops. Hamiltonian terms are requested by the user as input, via the integers k, n, p, q, r 1, r 2, s 1, s 2, t 1 and t 2 The program checks to see if all user requested terms are: (i) A 1 in PI group G 36 and (ii) invariant to time reversal.

Hamiltonian Reduction Nakagawa, Tsunekawa, Kojima JMS 126 (1987) 329 paper not yet done for two tops. BUT if we assume that their ordering scheme is applicable in current case then: 2 nd order – 8 terms are allowed 8 4 th order – 35 terms are allowed23 6 th order – 112 terms are allowed 9 8 th order – 293 terms are allowed as calculated from the difference between the total number of symmetry-allowed Hamiltonian terms of order n and the number of symmetry-allowed contact transformation terms of order n-1 Acetone fit

A check of the program code for different types of the higher-order terms was carried out by verifying that: Eigenvalues E and E’ of the Hamiltonians H and H +  H 2, satisfy E’ = E +  E 2 to machine round-off error Since in the program we encode ONE expression checking of a limited number of different types of terms validates correctness of all possible paths in Hamiltonian matrix setup routine

The acetone data set is from the recent literature [11] P. Groner, S. Albert, E. Herbst, F. C. De Lucia, F. J. Lovas, B. J. Drouin, J. C. Pearson, Ap. J. Supp. 142 (2002) ( = 0, J  60, K  30) [12] P. Groner, E. Herbst, F. C. De Lucia, B. J. Drouin, H. Mäder, J. Mol. Struct. 795 (2006) ( 12 = 1, J  38, K  16) [13] P. Groner, I. R. Medvedev, F. C. De Lucia, B. J. Drouin, J. Mol Spectrosc. 251 (2008) ( 17 = 1, J  31, K  8) We restrict the maximum value of J to 38 because such a fit clearly demonstrates the capabilities of the new program since it includes all 12 and 17 transitions that gave significant fitting problems in the literature.

By symmetry in G 9 By measurement uncertainty By torsional state ABAB # rms [kHz] Unc. [kHz] # rms [kHz] v#rms [MHz] J, K a max GS1002 (696 [11]) (0.156 [11]) 38, (671 [12]) (0.224 [12]) 38, (612 [13]) (0.485 [13]) 30, Overview of the data set and fit quality [25] I. Medvedev, M. Winnewisser, F. C. De Lucia, E. Herbst, E. Białkowska- Jaworska, L. Pszczółkowski, Z. Kisiel, J. Mol. Spectrosc. 228 (2004) 314–328.

UPPER STATELOWER STATE OBS-CALC SYM a JKAKA KCKC JKAKA KCKC ABbABb OBSERVED(UNC) c Ours[11] E (0.004) E (0.030) E (0.008) E2551E (0.008) E (0.200) E2550E (0.100) E31064E (0.050) E11064E (0.050) E1423E (0.050) E (0.200) G871G (0.020) E31082E (0.030) E31293E (0.050) E1844E (0.050) E1221E (0.050) A31293A (0.050) A31275A (0.050) E31275E (0.020) E21275E (0.020) Comparison of residuals for some ground-state lines of acetone which were assigned in [11] but excluded from both their separate-vibrational-state fit and our global fit

UPPER STATELOWER STATE OBS-CALC SYMJKAKA KCKC JKAKA KCKC ABAB OBSERVED(UNC)Ours[11] E230292E (0.200) G30292G (0.200) A330291A (0.200) A430292A (0.200) E330291E (0.200) E430292E (0.200) G30291G (0.200) E230291E (0.200) E130 1E (0.200) G30 1G (0.200) A130 0A (0.200) A230 1A (0.200) E330 0E (0.200) E430 1E (0.200) G30 0G (0.200) E130 0E (0.200) E131284E (0.200) G31284G (0.200) A131284A (0.200) A231283A (0.200) E331284E (0.200) Comparison of residuals for some ground-state lines of acetone which were assigned in [11] but excluded from both their separate-vibrational-state fit and our global fit

A plot against J(J+1) of reduced energies E red, i.e., energies calculated from E red = E – (1/2)(B+C)J(J+1), for torsion-rotation levels of acetone of species G in G J E red cm -1 v t =0 v t =1 v t =2 v t =3 v t =4 v t =5

Conclusions and Outlook A new program for fitting the spectra of molecules with two equivalent methyl rotors and C 2v symmetry at equilibrium has been developed, which allows carrying out a global fit of rotational transitions belonging to several torsional states simultaneously. Application of the new program to literature data on the ground and both fundamental torsional states of acetone shows that the program is capable of fitting modern spectral measurements to their experimental precision. A remeasurement campaign for acetone, to see if the problems remaining in our fit are caused by the model or by the measurements, is underway. The 49 – 149 GHz range is covered in Kharkov, about 6900 lines from new measurements are assigned. Future plans include application of the program to other molecules, perhaps starting with the important astrophysical molecule CH 3 -O-CH 3.

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