1. A NEW PROGRAM FOR NON- EQUIVALENT TWO-TOP INTERNAL ROTORS WITH A C s FRAME Isabelle KLEINER Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA), Créteil, France Jon HOUGEN National Institute for Standard and Technology (NIST), Gaithersburg, USA

2. What kind of molecules ? N-methylacetamide: N. Ohashi, J. T. Hougen, R. D. Suenram, F. J. Lovas, Y. Kawashima, M. Fujitake, and J. Pyka, JMS 2004 V 3 (1)=73 cm -1 V 3 (2)=79 cm -1 ; Methyl Acetate : Williams et al, J. Trans. Faraday Soc 1970; Sheridan et al JMS 1980, Kelley And Blake, Ohio state 2006 : Astrophysical importance! V 3 (1)=100 cm-1 V 3 (2)=425 cm-1

3. Methyl Acetate JK a K c 3 sets of torsional splittings: (AA,EA). V 3 = 100 cm -1  1 = a few GHz (AA,AE). V 3 = 425 cm -1  2 = a few MHz (AA,EE). Interaction between the 2 tops 0 0 ±1 ± 1 0 ± 1 1 ±1  1  2 Permutation-inversion group G 18

4. Overview of Existing Two-Top Programs Name Authors What it does? Method http://info.ifpan.edu.pl/~kisiel/prospe.htm: programs for rotational spectroscopy (Z. Kisiel) _____________________________________________________________________ XIAM Hartwig up to 3 sym tops « IAM » Potential Function fit Maederup to one quad Often 1MHz Obs-Calcs nucleusAr-acetone, (CH 3 ) 2 SiF 2 _____________________________________________________________________ ERHAM Gronerone or two Effective v t states fit internal rotors Fourier series for Torsional of sym. C 3v or C 2v Tunneling Splittings J up to 120. High Barrier acetone, diMEether _____________________________________________________________________ SPFIT/ Pickettone or two internalPotential Function fit SPCATrotors, sym or asym.propane _____________________________________________________________________ OHASHI Ohashitwo C 3v internal rotorsPotential Function fit Hougen C s or C 2h Frame A and E species fit together 1 kHz accuracy, but very slow N-methylacetamide, biacetyl

5. Overview of Existing Two-Top Programs(suite) Nameauthors what it does? Method ______________________________________________________________________ JB95Plusquellic one internal rotorPAM but can be used for 2 tops in top-top interaction is smallalanine dipeptide, peptide mimetics... graphical interface http://physics.nist.gov/Divisions/Div844/facilities/uvs/jb95userguide.htm

6. This work Write a new two-C 3v -top program : 1. For low, medium or high barriers 2. With high accuracy (obs-calcs < 1 kHz) 3. With high computational speed Begin with Ohashi’s two-top program, but use: 1. Two-step diagonalization (Herbst, BELGI) 2. Banded matrix computational methods

7. Theoretical Model: the global approach for one top H RAM = H rot + H tor + H int + H c.d. RAM = Rho Axis Method (axis system) for a C s (plane) frame : get rid of J x p  Constants1 1-cos3  p2p2 JapJap 1-cos6  p4p4 Jap3Jap3 1V 3 /2F  V 6 /2k4k4 k3k3 J2J2 (B+C)/2*FvFv GvGv LvLv NvNv MvMv k 3J Ja2Ja2 A-(B+C)/2*k5k5 k2k2 k1k1 K2K2 K1K1 k 3K J b 2 - J c 2 (B-C)/2*c2c2 c1c1 c4c4 c 11 c3c3 c 12 JaJb+JbJaJaJb+JbJa D ab or E ab d ab  ab  ab d ab6  ab  ab Torsional operators and potential function V(  ) Rotational Operators Hougen, Kleiner, Godefroid JMS 1994  = angle of torsion,  = couples internal rotation and global rotation, ratio of the moment of inertia of the top and the moment of inertia of the whole molecule Kirtman et al 1962 Lees and Baker, 1968 Herbst et al 1986

8. Two-step diagonalization H RAM = H tor + H rot + H c.d + H int 1) Diagonalization of the torsional part of the Hamiltonian : Eigenvalues = torsional energies 2) A low set of torsional Eigenvectors x rotational wavefunctions are then used to set up the matrix of the rest of the Hamiltonian: H rot = AJ a 2 + B R J b 2 +C R J c 2 + q 1 J a p 1 + q 2 J a p 2 + r 1 J b p 1 + r 2 J b p 2 H c.d usual centrifugal distorsion terms H int higher order torsional-rotational interactions terms : cos3    cos3  2, p 1, p  and global rotational operators like J a, J b, J c

9. Matrix size for one-step diagonalization n = (top 1) x (top 2) x (rotation) = 21 x 21 x (2J+1) n x n = 22491 x 22491 for J = 25 Matrix sizes for two-step diagonalization Step 1: n= (top 1) x (top 2) = 441 Step 2: n=(top 1) x (top 2) x (rotation) = 9 x 9 x (2J+1) n x n = 4131 x 4131 for J = 25 Time  n 3 (22491/4131) 3  161 8 hours  3 min Expected time saving

10. Banded Matrix = Generalization of Tridiagonal Matrix : used in the 2nd step Lapack Subroutines: DSBRDT and DSTEQR Bischof and Lang ACM 26, 602 (2000)

11. J=5 11100000000111000000001110000000011100000000 11110000000111100000001111000000011110000000 11111000000111110000001111100000011111000000 01111100000011111000000111110000001111100000 00111110000001111100000011111000000111110000 00011111000000111110000001111100000011111000 00001111100000011111000000111110000001111100 00000111110000001111100000011111000000111110 00000011111000000111110000001111100000011111 00000001111000000011110000000111100000001111 00000000111000000001110000000011100000000111 11100000000111000000001110000000011100000000 11110000000111100000001111000000011110000000 11111000000111110000001111100000011111000000 01111100000011111000000111110000001111100000 00111110000001111100000011111000000111110000 00011111000000111110000001111100000011111000 00001111100000011111000000111110000001111100 00000111110000001111100000011111000000111110 00000011111000000111110000001111100000011111 00000001111000000011110000000111100000001111 00000000111000000001110000000011100000000111 11100000000111000000001110000000011100000000 11110000000111100000001111000000011110000000 11111000000111110000001111100000011111000000 01111100000011111000000111110000001111100000 00111110000001111100000011111000000111110000 00011111000000111110000001111100000011111000 00001111100000011111000000111110000001111100 00000111110000001111100000011111000000111110 00000011111000000111110000001111100000011111 00000001111000000011110000000111100000001111 00000000111000000001110000000011100000000111 11100000000111000000001110000000011100000000 11110000000111100000001111000000011110000000 11111000000111110000001111100000011111000000 01111100000011111000000111110000001111100000 00111110000001111100000011111000000111110000 00011111000000111110000001111100000011111000 00001111100000011111000000111110000001111100 00000111110000001111100000011111000000111110 00000011111000000111110000001111100000011111 00000001111000000011110000000111100000001111 00000000111000000001110000000011100000000111 v t =0 v t =1 v t =2 v t =3 -5-4-3-2012345-5-4-3-2012345-5-4-3-2012345-5-4-3-2012345 Kvtvt Courtesy of V. ILyushin

12. 11111111111100000000000000000000000000000000 11111111111111110000000000000000000000000000 11111111111111111111000000000000000000000000 00001111111111111111111100000000000000000000 00000000111111111111111111110000000000000000 00000000000011111111111111111111000000000000 00000000000000001111111111111111111100000000 00000000000000000000111111111111111111110000 00000000000000000000000011111111111111111111 00000000000000000000000000001111111111111111 00000000000000000000000000000000111111111111 J=501230123012301230123012301230123012301230123 K=-5 K=-4 K=-3 K=-2 K=-1 K=0 K=1 K=2 K=3 K=4 K=5 Kvtvt Bandwidth=n vt  (  K max +1)-1 Courtesy of V. ILyushin

13. Checking the new code against Ohashi Comparison of obs-calc values for NMA (N- methylacetamide molecule) calculated with no quadrupole terms with BELGI-2-TOPS and with OHASHI’s code

14. Previous studies on methyl acetate -Williams, Owen, Sheridan J. Trans. Faraday Soc. 67, 922 (1970) : 13-40 GHz Sheridan, Bossert, Bauder JMS 80, 1 (1980) : 8-40 GHz. MW-MW double resonance + theoretical treatment for internal rotors Data up to J = 5, large obs-calc values (up to 10 MHz)

15. Results: 87 lines included (from Sheridan), rms = 99 kHz 22 floated parameters (cm -1 ) 15 Fixed parameters (from NMA) Rotational A 0.3969 (12) B 0.14002(12), C 0.1025998(12), D J 0.000000040(10), D JK -0.00000035(14), D K 0.000012(80),  J 0.0000000080(55),   0.00000056(13), Top 1 V 31 (1/2)(1-cos3  1 ) 103.92(1.80), Q 1 J a p 1 -0.7166(58), R 1 J b p 1 -0.12597(88), B 1 J a 2 p 1 2 0.0000419(45), Q 1K J a 3 p 1 0.000048(22), F 1 p 1 2 5.761(89), V 31K 0.0022(21),

16. Results (suite) 22 floated parameters 15 Fixed parameters (from NMA) cm -1 cm -1 Top 2 Q 2 J a p 2 -0.7975(28),V 32 = 425 R 2 J b p 2 -0.1010(14), B 2 J a 2 p 1 2 0.000056(21), Q 2K J a 3 p 2 0.00017(11), V 32K -0.0480(21), Top-Top Interaction F 12 p 1 p 2 0.710(31),V 12C = 0.91427 B 12 0.000078(72), V 12S = - 2.9885 C 12 -0.000122(11),

17. New program gives good improvement Upper Lower Obs. Freq. (MHz) Obs-calc SBB* v t ’ J’ K a K c v t ” J” K a K c This work (1980) 0 1 1 1 + 0 0 0 0 + 13484.510(100) -0.068 A -2.622 0 3 1 3 + 0 2 0 2 + 25296.400(100) -0.129 A -2.146 0 5 2 3 + 0 5 1 4 - 16871.620(100) -0.038 A -6.666 0 5 3 2 - 0 5 2 3 + 31148.030(100) 0.012 A -15.704 0 4 2 2 + 0 4 1 3 - 17005.600(100) 0.034 A -8.367 0 5 -3 2 0 5 -2 3 34039.260(100) -0.094 E4 5.110 0 5 -3 2 0 5 -2 3 34026.900(100) -0.052 E1 4.276 0 5 -3 2 0 5 -2 3 33989.850(100) 0.008 E3 4.876 0 3 +2 2 0 3 +1 3 23707.700(100) 0.034 E2 -7.515 *Sheridan, Bossert and Bauder, JMS 80 (1980)

18. Conclusions 1.We need more accurate and more extensive data to fully test the new program and the speed gain. 2.It may be a good time to begin a new measurement campaign to prepare a comprehensive astrophysical MW atlas of methyl acetate.

19. Aknowledgments Vadim Ilyushyn Nobukimi Ohashi French ANR Program -08-BLAN-0054 : POST-DOC POSITION available in our lab (LISA, Créteil/paris, France) to work on that problem

20. Expected time saving Arrange matrix blocks by K quantum number If only  K = 0,  1,  2 mixings are considered -J …. 4 5 6 7 8 … +J Band width = 5 Saving = 5/(2J+1)  1/10 at J = 25 n x n  n x (n/10) Expected time saving not researched in literature yet （ごめんなさい）

21. PsPAM = Pseudo Principal Axis Method: Get rid of all J x J y, J y J z, and J z J x terms Constants 1 1-cos3   p  2 J a p  1-cos6   p  4 J a p  3 1.V 3 /2F  V 6 /2k4k4 k3k3 J 2.B bar FvFv GvGv LvLv NvNv MvMv k 3J J z 2. A-B bar k5k5 k2k2 k1k1 K2K2 K1K1 k 3K J b 2 - J c 2. (B-C)/2c2c2 c1c1 c4c4 c 11 c3c3 c 12 JaJb+JbJaJaJb+JbJa D ab d ab  ab  ab d ab6  ab  ab Torsional Operators = f(      p   p   Rotational Operators Kirtman et al. 1962; Lees and Baker 1968; Herbst et al. 1986 Operator = (rotation)x(torsion)

22. Global approach for two tops : Ohashi’s model. H tor = F 1 p 1 2 + F 2 p 2 2 + F 12 p 1 p 2 + (1/2) V 31 (1-cos3  1 ) + (1/2) V 32 (1-cos3  2 ) +V 12c (1-cos3  1 ) ( 1-cos3  2 ) +V 12s sin3  1 sin3  2 H rot = AJ z 2 + BJ x 2 + CJ y 2 + cent.distorsion H int = r 1 J x p 1 + r 2 J x p 2 + q 1 J z p 1 + q 2 J z p 2 +B 1 p 1 2 J x 2 + B 2 p 2 2 J x 2 +B 12 p 1 p 2 J x 2 + C 1 p 1 2 J y 2 + C 2 p 2 2 J y 2 + C 12 p 1 p 2 J y 2 +q 12p p 1 p 2 (p 1 +p 2 ) J z +q 12m p 1 p 2 (p 1 -p 2 ) J z +...