A NEW PROGRAM FOR NON-EQUIVALENT TWO-TOP INTERNAL ROTORS WITH A Cs FRAME

Isabelle KLEINERLaboratoire Interuniversitaire des Systèmes Atmosphériques

(LISA), Créteil, France

Jon HOUGENNational Institute for Standard and Technology (NIST),

Gaithersburg, USA

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

V3(1)=73 cm-1 V3(2)=79 cm-1 ;

Methyl Acetate : Williams et al, J. Trans. Faraday Soc 1970;Sheridan et al JMS 1980, KelleyAnd Blake, Ohio state 2006 : Astrophysical importance!

V3(1)=100 cm-1 V3(2)=425 cm-1

Methyl Acetate

JKaKc

3 sets of torsional splittings: (AA,EA). V3 = 100 cm-1

1 = a few GHz (AA,AE). V3 = 425 cm-1

2 = a few MHz (AA,EE). Interaction

between the 2 tops0 0

0 ±1

± 1 0

± 1 1

±1 ±1

1 2

Permutation-inversion group G18

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 1MHz Obs-Calcsnucleus Ar-acetone, (CH3)2SiF2

_____________________________________________________________________ERHAM Groner one or two Effective vt states fit

internal rotors Fourier series for Torsionalof sym. C3v or C2v Tunneling SplittingsJ up to 120. High Barrier

acetone, diMEether_____________________________________________________________________SPFIT/ Pickett one or two internal Potential Function fitSPCAT rotors, sym or asym. propane

_____________________________________________________________________OHASHI Ohashi two C3v internal rotors Potential Function fit Hougen Cs or C2h Frame A and E species fit together

1 kHz accuracy, but very slow N-methylacetamide, biacetyl

Overview of Existing Two-Top Programs(suite)

Name authors what it does? Method ______________________________________________________________________JB95 Plusquellic one internal rotor PAM

but can be usedfor 2 tops in top-topinteraction is small alanine dipeptide, peptide

mimetics ...

graphical interface

http://physics.nist.gov/Divisions/Div844/facilities/uvs/jb95userguide.htm

This work

Write a new two-C3v-top program :

1. For low, medium or high barriers2. 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

Theoretical Model: the global approach for one top

HRAM = Hrot + Htor + Hint + Hc.d.

RAM = Rho Axis Method (axis system) for a Cs (plane) frame : get rid of Jxp

Constants 1 1-cos3 p2 Jap 1-cos6 p4

Jap3

1 V3/2 F V6/2 k4 k3

J2 (B+C)/2* Fv Gv Lv Nv Mv k3J

Ja2 A-(B+C)/2* k5 k2 k1 K2 K1 k3K

Jb2 - Jc

2 (B-C)/2* c2 c1 c4 c11 c3 c12

JaJb+JbJa Dab or Eab dab ab ab dab6 ab ab

Torsional operators and potential function V()

Rot

atio

nal O

pera

tors

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 1962Lees and Baker, 1968 Herbst et al 1986

Two-step diagonalization

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 = AJa2 + BRJb

2 +CRJc2 + q1Jap1 + q2Jap2 + r1Jbp1 + r2Jbp2

Hc.d usual centrifugal distorsion termsHint higher order torsional-rotational interactions terms : cos3 cos32 , p1, p and global rotational operators like Ja, Jb , Jc

Htor = F1 p12 + F2 p2

2 + F12 p1p2 + (1/2) V31 (1-cos31) + (1/2) V32 (1-cos32) +V12c (1-cos31) ( 1-cos32) +V12s sin31sin32

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 n3 (22491/4131)3 161 8 hours 3 min

Expected time saving

Banded Matrix = Generalization of Tridiagonal Matrix : used in the 2nd step

Lapack Subroutines: DSBRDT and DSTEQRBischof and Lang ACM 26, 602 (2000)

J=5J=511111100000000000000001111110000000000000000111111000000000000000011111100000000000000001111111100000000000000111111110000000000000011111111000000000000001111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111110000000000000011111111000000000000001111111100000000000000111111110000000000000000111111000000000000000011111100000000000000001111110000000000000000111111111111000000000000000011111100000000000000001111110000000000000000111111000000000000000011111111000000000000001111111100000000000000111111110000000000000011111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111111000000000000111111111100000000000011111111110000000000001111111111000000000000001111111100000000000000111111110000000000000011111111000000000000001111111100000000000000001111110000000000000000111111000000000000000011111100000000000000001111111111110000000000000000111111000000000000000011111100000000000000001111110000000000000000111111110000000000000011111111000000000000001111111100000000000000111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111110000000000001111111111000000000000111111111100000000000011111111110000000000000011111111000000000000001111111100000000000000111111110000000000000011111111000000000000000011111100000000000000001111110000000000000000111111000000000000000011111111111100000000000000001111110000000000000000111111000000000000000011111100000000000000001111111100000000000000111111110000000000000011111111000000000000001111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111111100000000000011111111110000000000001111111111000000000000111111111100000000000000111111110000000000000011111111000000000000001111111100000000000000111111110000000000000000111111000000000000000011111100000000000000001111110000000000000000111111

vt=0

vt=1

vt=2

vt=3

-5-5-4-4-3-3-2-2-1-1001122334455

-5-5-4-4-3-3-2-2-1-1001122334455

-5-5-4-4-3-3-2-2-1-1001122334455

-5-5-4-4-3-3-2-2-1-1001122334455

Kvt

Courtesy ofV. ILyushin

11111111111111111111111100000000000000000000000000000000000000000000000000000000000000001111111111111111111111110000000000000000000000000000000000000000000000000000000000000000111111111111111111111111000000000000000000000000000000000000000000000000000000000000000011111111111111111111111100000000000000000000000000000000000000000000000000000000000000001111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111110000000000000000000000000000000000000000000000000000000011111111111111111111111111111111000000000000000000000000000000000000000000000000000000001111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000011111111111111111111111111111111111111110000000000000000000000000000000000000000000000001111111111111111111111111111111111111111000000000000000000000000000000000000000000000000111111111111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111110000000000000000000000000000000000000000000000000000000011111111111111111111111111111111000000000000000000000000000000000000000000000000000000001111111111111111111111111111111100000000000000000000000000000000000000000000000000000000111111111111111111111111111111110000000000000000000000000000000000000000000000000000000000000000111111111111111111111111000000000000000000000000000000000000000000000000000000000000000011111111111111111111111100000000000000000000000000000000000000000000000000000000000000001111111111111111111111110000000000000000000000000000000000000000000000000000000000000000111111111111111111111111

J=5J=50011223300112233001122330011223300112233001122330011223300112233001122330011223300112233

K=-5

K=-4

K=-3

K=-2

K=-1

K=0

K=1

K=2

K=3

K=4

K=5

K vtBandwidth=nvt(Kmax+1)-1

Courtesy ofV. ILyushin

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

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)

Results: 87 lines included (from Sheridan), rms = 99 kHz22 floated parameters (cm-1) 15 Fixed parameters (from NMA)

Rotational A 0.3969 (12)

B 0.14002(12), C 0.1025998(12), DJ 0.000000040(10), DJK -0.00000035(14), DK 0.000012(80),J 0.0000000080(55), 0.00000056(13),

Top 1 V31 (1/2)(1-cos31) 103.92(1.80),Q1 Jap1 -0.7166(58),R1 Jbp1 -0.12597(88),B1 Ja

2p12 0.0000419(45),

Q1K Ja3p1 0.000048(22),

F1 p12 5.761(89),

V31K 0.0022(21),

Results (suite)22 floated parameters 15 Fixed parameters (from NMA)

cm-1 cm-1

Top 2 Q2 Jap2 -0.7975(28), V32 = 425R2 Jbp2 -0.1010(14),B2 Ja

2p12 0.000056(21),

Q2K Ja3p2 0.00017(11),

V32K -0.0480(21),

Top-Top InteractionF12 p1p2 0.710(31), V12C = 0.91427 B12 0.000078(72), V12S = - 2.9885C12 -0.000122(11),

New program gives good improvement

Upper Lower Obs. Freq. (MHz) Obs-calc SBB*vt’ J’ Ka Kc vt” J” Ka Kc 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 -2.1460 5 2 3 + 0 5 1 4 - 16871.620(100) -0.038 A -6.6660 5 3 2 - 0 5 2 3 + 31148.030(100) 0.012 A -15.7040 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 5.1100 5 -3 2 0 5 -2 3 34026.900(100) -0.052 E1 4.2760 5 -3 2 0 5 -2 3 33989.850(100) 0.008 E3 4.8760 3 +2 2 0 3 +1 3 23707.700(100) 0.034 E2 -7.515

*Sheridan, Bossert and Bauder, JMS 80 (1980)

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.

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

Expected time saving

Arrange matrix blocks by K quantum numberIf 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 = 25n x n n x (n/10)

Expected time saving not researchedin literature yet （ごめんなさい）

PsPAM = Pseudo Principal Axis Method:Get rid of all JxJy , JyJz , and JzJx terms

Constants 1 1-cos3

p

2 Jap 1-cos6

p

4

Jap3

1 . V3/2 F V6/2 k4 k3

J2 . Bbar Fv Gv Lv Nv Mv k3J

Jz2 . A-Bbar k5 k2 k1 K2 K1 k3K

Jb2 - Jc

2 . (B-C)/2 c2 c1 c4 c11 c3 c12

JaJb+JbJa Dab dab ab ab dab6 ab ab

Torsional Operators = f(p p

Rot

atio

nal O

pera

tors

Kirtman et al. 1962; Lees and Baker 1968; Herbst et al. 1986

Operator = (rotation)x(torsion)

Global approach for two tops : Ohashi’s model.

Htor = F1 p12 + F2 p2

2 + F12 p1p2 + (1/2) V31 (1-cos31) + (1/2) V32 (1-cos32) +V12c (1-cos31) ( 1-cos32) +V12s sin31sin32

Hrot = AJz2 + BJx

2 + CJy2 + cent.distorsion

Hint = r1 Jxp1 + r2 Jx p2 + q1 Jzp1 + q2 Jzp2

+B1 p12Jx

2 + B2p22Jx

2 +B12 p1p2Jx2 + C1 p1

2Jy2 + C2 p2

2Jy2 + C12 p1p2Jy

2

+q12p p1p2 (p1+p2) Jz +q12m p1p2 (p1-p2) Jz + ...