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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, 61002 Kharkov, Ukraine. b NIST Guest Worker June – August 2009 & June – July 2013. c Sensor Science Division, NIST, Gaithersburg, MD 20899-8441, USA Vadim V. Ilyushin a,b, Jon. T. Hougen c

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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

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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.

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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

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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

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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) 145-151. ( = 0, J 60, K 30) [12] P. Groner, E. Herbst, F. C. De Lucia, B. J. Drouin, H. Mäder, J. Mol. Struct. 795 (2006) 173-178. ( 12 = 1, J 38, K 16) [13] P. Groner, I. R. Medvedev, F. C. De Lucia, B. J. Drouin, J. Mol Spectrosc. 251 (2008) 180-184. ( 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.

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By symmetry in G 9 By measurement uncertainty By torsional state ABAB # rms [kHz] Unc. [kHz] # rms [kHz] v#rms [MHz] J, K a max 0077397.54142.54GS1002 (696 [11]) 0.119 (0.156 [11]) 38, 21 0196590.58183.85 12 966 (671 [12]) 0.102 (0.224 [12]) 38, 16 11631104.01061.74 17 1030 (612 [13]) 0.068 (0.485 [13]) 30, 8 12629104.7204011.67 305720.05 506940.38 100147992.50 20037263.76 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.

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UPPER STATELOWER STATE OBS-CALC SYM a JKAKA KCKC JKAKA KCKC ABbABb OBSERVED(UNC) c Ours[11] E4321 3121110730.960(0.004)-0.380-0.400 E4752 7431117489.680(0.030)-0.032-0.208 E3862 8531120966.160(0.008)-0.035-0.458 E2551E15421225468.080(0.008)0.0050.216 E4972 9631126643.540(0.200)-0.014-0.773 E2550E15421228313.410(0.100)-0.0180.206 E31064E310551128632.430(0.050)-0.0060.250 E11064E210551228660.410(0.050)0.0340.272 E1423E23301231416.450(0.050)-0.227-0.303 E3871 8621132219.840(0.200)-0.101-0.367 G871G8620133044.710(0.020)0.2730.285 E31082E310731133826.590(0.030)-0.033-1.041 E31293E312841134777.620(0.050)-0.097-2.157 E1844E28351234790.490(0.050)-0.279-0.141 E1221E21101235219.940(0.050)-1.431-1.445 A31293A112840035274.810(0.050)0.0340.593 A31275A112660036179.750(0.050)-0.007-0.313 E31275E312661136254.390(0.020)0.0790.812 E21275E11266 36284.650(0.020)0.0430.756 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

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UPPER STATELOWER STATE OBS-CALC SYMJKAKA KCKC JKAKA KCKC ABAB OBSERVED(UNC)Ours[11] E230292E12928212599753.540(0.200)0.053-3.833 G30292G 28201599826.870(0.200)0.030-3.007 A330291A12928200599900.630(0.200)-0.235-2.462 A430292A22928100599900.630(0.200)-0.235-2.462 E330291E32928211600005.580(0.200)0.304-0.422 E430292E42928111600005.580(0.200)0.304-0.422 G30291G 28101600079.420(0.200)0.1930.312 E230291E12928112600247.210(0.200)-9.694-7.241 E130 1E229 112605568.400(0.200)0.051-8.609 G30 1G29 101605620.270(0.200)0.347-7.069 A130 0A329 100605672.010(0.200)0.092-6.101 A230 1A429 000605672.010(0.200)0.092-6.101 E330 0E329 111605832.310(0.200)-0.009-5.500 E430 1E429 011605832.310(0.200)-0.009-5.500 G30 0G29 001605884.500(0.200)-0.163-4.631 E130 0E229 012606097.000(0.200)0.425-2.128 E131284E23027312608325.630(0.200)-0.7010.075 G31284G3027301608400.590(0.200)-0.7170.762 A131284A33027300608476.340(0.200)-0.6931.804 A231283A43027400608476.340(0.200)-0.6931.804 E331284E33027311608541.170(0.200)-0.3174.962 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

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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 36. 05101520253035J 60 180 120 240 300 E red cm -1 v t =0 v t =1 v t =2 v t =3 v t =4 v t =5

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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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Thank you for your attention

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