NEW MICROWAVE SPECTRUM AND GLOBAL FIT OF METHYL
NEW MICROWAVE SPECTRUM AND GLOBAL FIT OF METHYL ACETATE GROUND STATE I. KLEINER, M. TUDORIE Laboratoire Interuniversitaire des Systèmes Atmosphériques (LISA), Créteil, France J. HOUGEN National Institute for Standard and Technology (NIST), Gaithersburg, USA S. MELANDRI Dipartimento di Chimica "G. Ciamician", Bologna, Italy W. STAHL, L. SUTIKDJA Institut fur Physikalische Chemie, Raum Aachen, Germany
Columbus 2009: A NEW PROGRAM FOR NON-EQUIVALENT TWO-TOP INTERNAL ROTORS WITH A Cs FRAME n 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
Methyl Acetate: energy levels ± 1 ± 1 1 JKa. Kc ± 1 0 n 3 sets of internal rotation splittings : 0 ± 1 0 0 Without torsion Top 1 Top 2 Interaction Permutation-inversion group G 18 s 1 s 2 V 3 = 100 cm-1 1 = a few GHz -1 n (AA, AE). V 3 = 425 cm 2 = a few MHz n (AA, EE). Interaction between the 2 tops n (AA, EA). n a = 1. 64 D, b = 0. 06 D
The new code: BELGI-2 tops a new two-C 3 v-top program was written in 2009: 1. For low, medium or high barriers 2. With high accuracy (obs-calcs < 1 k. Hz) 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 suggested in 2009 ?
Theoretical Model: the global approach for one top RAM = Rho Axis Method (axis system) for a Cs (plane) frame : get rid of Jxpa HRAM = Hrot + Htor + Hint + Hc. d. Torsional operators and potential function V( ) Rotational Operators Constants 1 1 -cos 3 a p 2 a Japa 1 -cos 6 a p 4 a Ja p 3 a V 3/2 F r V 6/2 k 4 k 3 J 2 (B+C)/2* Fv Gv Lv Nv Mv k 3 J Ja 2 A-(B+C)/2* k 5 k 2 k 1 K 2 K 1 k 3 K Jb 2 - Jc 2 (B-C)/2* c 2 c 1 c 4 c 11 c 3 c 12 Ja. Jb+Jb. Ja Dab or Eab dab 6 ab ddab 1 Kirtman et al 1962 Lees and Baker, 1968 Herbst et al 1986 a = angle of torsion, r = couples internal rotation and global rotation, ratio of the moment of inertia of the top and the moment of inertia of the whole molecule Hougen, Kleiner, Godefroid JMS 1994
Two-step diagonalization for the two-top problem HRAM = Htor + Hrot + Hc. d + Hint 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: Hrot = AJa 2 + BRJb 2 +CRJc 2 + q 1 Jap 1 + q 2 Jap 2 + r 1 Jbp 1 + r 2 Jbp 2 Hc. d usual centrifugal distorsion terms Hint higher order torsional-rotational interactions terms : cos 3 1, cos 3 2 , p 1, p 2 and global rotational operators like Ja, Jb , Jc
Speeding up the code: suggested in 2009 n Arrange matrix blocks by K quantum number, If only K = 0, 1, 2 mixings are considered n. Unexpectedly: MKL library associated with the Intel Fortran compiler are faster than from Lapack subroutines written for banded matrices !
Speeding up the code: achievements in 2010 : two truncations n A) Matrix sizes for two-step diagonalization Step 1: n= (top 1) x (top 2) = (21) x (21) = 441 Step 2: n=(top 1) x (top 2) x (rotation) = 64 x(2 J+1) n x n = 2624 x 2624 for J = 20 (instead of n=18081) n B) Number of eigenvectors used in the derivative calculations : 18 lowest torsional states n C) Energy and derivatives stored only if included in the fit (Ilyushin, private communication) One iteration now takes 4’ 17” up to J=10 and 16’ up to J=20 Ohashi code: 20’ up to J=8
Previous studies on methyl acetate n -Williams, Owen, Sheridan J. Trans. Faraday Soc. 67, 922 (1970) : 13 -40 GHz n 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 in Sheridan et al (up to 10 MHz) Columbus 2009: 87 lines included (from Sheridan), fit with rms = 99 k. Hz
NEW MEASUREMENTS Wishes in Columbus 2009 : “We need more accurate and more extensive data to fully test the new program and the speed gain”. Sonia Melandri (Bologna): 512 lines newly measured with jetcooled molecular-beam Stark-modulated millimeterwave spectrometer 59. 8 - 77. 3 GHz, J 20, Ka 6 Only 2 lines from Sheridan left
NEW MEASUREMENTS L. Sutikdja, W. Stahl (Aachen): 309 lines newly measured with jet-cooled molecular-beam Fourier tranform microwave spectrometer n 8 - 40 GHz, J 8 , Ka 3
303← 212(A) n = 15976, 2413 MHz ~ 171 k. Hz 15976. 039 MHz 15976. 239 MHz 303← 212(E 1) n = 17322, 7869 MHz ~ 186 k. Hz 15976. 439 MHz 17322, 589 MHz 17322, 789 MHz 17322, 989 MHz Methyl acetate new measurements from Aachen
Overview of the data and quality of the fit FTMW lines (Aachen) c A E 1 E 2 E 3 E 4 Number rms 62 3. 2 62 4. 3 65 4. 5 58 4. 1 62 5. 3 Meas accuracy: 4 k. Hz Mm-wave lines (Bologna) Number 93 119 104 98 98 50 k. Hz rms 34 33 40 43 40
Results: 823 lines included rms = 18 k. Hz 29 floated parameters (cm-1) 4 Fixed parameters Rotational A B C DJK DK d. J d. K 0. 383408 (19) 0. 1403872 (30), 0. 1027736(23), 0. 000000023157(37), 0. 0000000390 (11), 0. 000001286 (87), 0. 000060(14), 0. 0000000663 (83), Top 1 V 31 (1/2)(1 -cos 3 1) Q 1 Jap 1 R 1 Jbp 1 B 1 Ja 2 p 12 Q 1 K Ja 3 p 1 F 1 p 12 V 31 K 97. 197(23), -0. 67456(13), -0. 122212(22), 0. 00001947 (85), -0. 00000917 (25), 5. 52465 (83), 0. 003614 (24),
Results (suite) 29 floated parameters Top 2 cm 4 Fixed parameters -1 Q 2 Jap 2 -0. 720355(60), R 2 Jbp 2 -0. 094276(25), B 2 Ja 2 p 12 0. 00000650(22), Q 2 K Ja 3 p 2 0. 0000210 (13), V 32 K -0. 007354(72), Top-Top Interaction * F 12 B 12 cm-1 V 32 = 417. 050 F 2 = 5. 52346 Fixed to N-methyl acetamide p 1 p 2 0. 65952(52), V 12 C = 0. 91427 p 1 p 2 Jx 2 -0. 000043525(41), V 12 S = - 2. 9885
New program gives good improvement: 1980 -2009 -2010 Upper Lower vt’ J’ Ka Kc vt” J” Ka Kc 0 1 1 1+ 0 0+ 0 5 3 2 - 0 5 2 3 + Obs. Freq. (MHz) 13484. 510(100) 13484. 540( 5) 31148. 034( 5) Obs-calc This work -0. 068 A 0. 006 A -0. 006 A SBB* (1980) -2. 622 ---- 0 5 -3 2 0 5 -2 3 34026. 900(100) 34026. 918( 5) -0. 052 E 1 -0. 005 4. 276 0 5 -3 2 0 5 -2 3 31149. 433( 5) -0. 003 E 2 ---- 0 5 -3 2 0 5 -2 3 0. 008 E 3 0. 003 -0. 094 E 4 -0. 009 4. 876 0 5 -3 2 33989. 850(100) 33989. 862( 5) 34039. 260(100) 34039. 300( 5) 0 3 +2 2 0 3 +1 3 23707. 700(100) 23707. 658( 5) *Sheridan, Bossert and Bauder, JMS 80 (1980) 0. 034 E 2 0. 008 5. 110 -7. 515
Conclusions 1. Since the new code for two inequivalent internal rotors seems to work well, it would be good to test vt = 1 of methyl acetate. 2. It may be a good time to begin a new measurement campaign to prepare a comprehensive astrophysical MW atlas of methyl acetate: would this species be detected ? ? ?
Aknowledgments n Vadim Ilyushyn n Nobukimi Ohashi n French ANR Program -08 -BLAN-0054
ENERGY LEVELS OF METHYL ACETATE
ENERGY LEVELS
Global approach for two tops : Ohashi’s model. n Htor = F 1 p 12 + F 2 p 22 + F 12 p 1 p 2 + (1/2) V 31 (1 -cos 3 1) + (1/2) V 32 (1 -cos 3 2) +V 12 c (1 -cos 3 1) ( 1 -cos 3 2) +V 12 s sin 3 1 sin 3 2 n Hrot = AJz 2 + BJx 2 + CJy 2 + cent. distorsion n Hint = r 1 Jxp 1 + r 2 Jx p 2 + q 1 Jzp 1 + q 2 Jzp 2 +B 1 p 12 Jx 2 + B 2 p 22 Jx 2 +B 12 p 1 p 2 Jx 2 + C 1 p 12 Jy 2 + C 2 p 22 Jy 2 + C 12 p 1 p 2 J y 2 +q p p (p +p ) J +q p p (p -p ) J +. . .
Expected time saving Matrix size for one-step diagonalization n = (top 1) x (top 2) x (rotation) = 21 x (2 J+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 (2 J+1) n x n = 4131 x 4131 for J = 25 Time n 3 (22491/4131)3 161 8 hours 3 min
Banded Matrix = Generalization of Tridiagonal Matrix : used in the 2 nd step Lapack Subroutines: DSBRDT and DSTEQR Bischof and Lang ACM 26, 602 (2000)
vt vt=0 vt=1 vt=2 vt=3 K -5 -4 -3 -2 -1 0 1 2 3 4 5 1 1 10 10 0 0 0 0 0 0 01 01 1 10 10 0 0 0 0 0 0 01 01 1 10 10 0 0 0 0 0 0 01 01 1 10 10 0 0 0 0 0 0 0 1 1 1 10 10 0 0 0 0 0 01 01 1 1 10 10 0 0 0 0 0 01 01 1 1 10 10 0 0 0 0 0 01 01 1 1 10 10 0 0 0 0 0 0 1 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 0 0 1 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 10 10 0 0 0 0 01 01 1 1 1 1 0 0 0 0 0 0 01 01 1 1 10 10 0 0 0 0 0 01 01 1 1 10 10 0 0 0 0 0 01 01 1 1 10 10 0 0 0 0 0 01 01 1 1 1 0 0 0 0 0 0 0 01 01 1 10 10 0 0 0 0 0 0 01 01 1 10 10 0 0 0 0 0 0 01 01 1 10 10 0 0 0 0 0 0 01 01 1 1 J=5 Courtesy of V. ILyushin
K K=-5 K=-4 K=-3 K=-2 K=-1 K=0 K=1 K=2 K=3 K=4 K=5 vt 0 1 2 3 0 1 2 3 0 1 2 3 Bandwidth=nvt ( Kmax+1)-1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 10 10 0 0 0 0 0 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 01 01 1 1 0 0 0 0 0 0 0 0 01 01 1 1 J=5 Courtesy of V. ILyushin
Checking the new code against Ohashi n Comparison of obs-calc values for NMA (N- methylacetamide molecule) calculated with no quadrupole terms with BELGI-2 -TOPS and with OHASHI’s code
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 Maeder up to one quad Often 1 MHz Obs-Calcs nucleus Ar-acetone, (CH 3)2 Si. F 2 ___________________________________ ERHAM Groner one or two Effective vt states fit internal rotors Fourier series for Torsional of sym. C 3 v or C 2 v Tunneling Splittings J up to 120. High Barrier acetone, di. MEether ___________________________________ SPFIT/ Pickett one or two internal Potential Function fit SPCAT rotors, sym or asym. propane ___________________________________ OHASHI Ohashi two C 3 v internal rotors Potential Function fit Hougen Cs or C 2 h Frame A and E species fit together 1 k. Hz accuracy, but very slow N-methylacetamide, biacetyl
Overview of Existing Two-Top Programs(suite) Name authors what it does? Method ___________________________________ JB 95 Plusquellic one internal rotor PAM but can be used for 2 tops in top-top interaction is small alanine dipeptide, peptide mimetics. . . graphical interface http: //physics. nist. gov/Divisions/Div 844/facilities/uvs/jb 95 userguide. htm
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