Columbus 2005, 20/6/05 -- 24/6/05 ANALYSIS OF THE 3 / 7 / 9 BENDING TRIAD OF THE QUASI-SPHERICAL TOP MOLECULE SO 2 F 2 M. Rotger, V. Boudon, M. Lo ë te,

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ANALYSIS OF THE ANALYSIS OF THE 33/ / 77/ / 9 9 BENDING TRIAD OF THE BENDING TRIAD OF THE

QUASI-SPHERICAL TOP MOLECULE SOQUASI-SPHERICAL TOP MOLECULE SO22FF22

MM. . RotgerRotger, , VV. . BoudonBoudon,, M.M. LoLoëte,ëte, L.L. Margulès,Margulès,

J.J. Demaison,Demaison, F.F. Hegelund,Hegelund,I.I. MerkeMerke and and H.H. BürgerBürgerSOSO44

22

SOSO22FF22SFSF55ClCl

SFSF66

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Theoretical aspectsTheoretical aspects

Spectra analysis: the ground-stateSpectra analysis: the ground-state and the bending triad (510-580 cmand the bending triad (510-580 cm-1-1)) of SOof SO22FF22

PerspectivesPerspectives

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MAIN IDEASMAIN IDEAS

Consider a symmetric or asymmetric top as Consider a symmetric or asymmetric top as deriving from a spherical topderiving from a spherical top

Examples:Examples:SFSF55ClCl can be seen as deriving from thecan be seen as deriving from the SFSF6 6 molecule.molecule.

SOSO22FF22 can be seen as deriving from thecan be seen as deriving from the SOSO4422molecular ion.molecular ion.

Associate a group chain to these moleculesAssociate a group chain to these molecules

For exampleFor example, the following group chains, the following group chains

O(3)O(3) TTd d CC2v2v oror O(3)O(3) OOhh C C4v4v.

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GROUP CHAINGROUP CHAIN

Group of rotations in space : Group of rotations in space : O(3)O(3)

Spherical Spherical intermediate intermediate

groupgroup

Group of the moleculeGroup of the molecule

Direct orientationDirect orientation

(Watson)(Watson)

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TENSORIAL FORMALISM AND TENSORIAL FORMALISM AND EFFECTIVE HAMILTONIANEFFECTIVE HAMILTONIAN

ti : parameters,: parameters,

R andand V are the rotational and vibrational operators.are the rotational and vibrational operators.

Systematic development of all the operators at any orderSystematic development of all the operators at any orderand for any polyad, and for any polyad,

Vibrational extrapolation of the parametersVibrational extrapolation of the parameters,,

Suitable for programing.Suitable for programing.

H effective = tii∑ R Ci( ) ⊗V Ci( )

( )

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A SIMPLE CASE,A SIMPLE CASE,

THE GROUND STATE OF SOTHE GROUND STATE OF SO22FF22: v = 0: v = 0

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Watson’s HamiltonianWatson’s Hamiltonian::

HWatson =B200 J2 + B020 J Z

2 +T400 J 2( )2+T220 J

2 J Z2 +T040 J Z

4

+Φ600 J 2( )3+Φ420 J 2( )

2J Z2 +Φ240 J

2 J Z4 +Φ060 J Z

2 +12[B002 +T202 J

2 +T022 J Z2

+Φ402 J 2( )2+Φ222 J

2 J Z2 +Φ024 J Z

4 , J +2 + J −

2 ]+ +12[T004 +Φ204 J

2 +Φ024 J Z2 , J +

4 + J −4 ]+

+Φ006 (J +6 + J −

6 ).

THEORYTHEORY

Tensorial HamiltonianTensorial Hamiltonian::

HMoret-Bailly = tΩ(K ,nΓ,Γ)allindices∑ β RΩ(K ,nΓ,Γ)

RΩ(K ) =RΩ−K (0) ×RK (K ),

whereRΩ−K (0) =((R1(1) ×R1(1))(0))(Ω−K )/2

andRK (K ) =(RK−1(K−1) ×R1(1))(K ).

~~

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ROTATIONAL TENSORSROTATIONAL TENSORS

€

O(3) ⊃Td ⊃C2vGroups Chain:Groups Chain:

R σΩ(K ,nΓ) = (K )GnΓσ

M R MΩ(K )

M∑ RΩ(K ,nΓ,Γ) = (Γ)GΓ

'σ

σ∑ R σ

Ω(K ,nΓ)

G andand G’ are some are some orientation coefficientsorientation coefficients..

~

~

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MATRIX ELEMENTS OF THE MATRIX ELEMENTS OF THE TENSORIAL HAMILTONIANTENSORIAL HAMILTONIAN

Matrix elements of the pure rotational operator:Matrix elements of the pure rotational operator:

Ψ(J ',n 'Cr' ,Cr

' ) RΩ(K ,nr Γ r ,Γ ) Ψ (J ,nCr ,Cr ) = K 'Γ Cr Cr

'( )

Γ r Cr Cr'

( ) Knr Γ r nCr n 'Cr

'( )

K J J '( ) Ψ J ' RΩ(K ) Ψ J

Pure rotational basis:Pure rotational basis: Ψ(J ,nCr ,Cr )~

Rotational labels:Rotational labels: (J,Cr ,α)~~

~~ ~

~ ~ ~

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COMPARISON AT ORDER ZEROCOMPARISON AT ORDER ZERO

HWatson = B200 + 2B002( ) J X2 + B200 −2B002( ) J Y

2 + B020 + B200( ) J Z2 + ...

HMoret-Bailly = t1 −223t2

⎛

⎝⎜⎞

⎠⎟J x2 + t1 −2

23t2

⎛

⎝⎜⎞

⎠⎟J y2 + t1 + 4

23t2

⎛

⎝⎜⎞

⎠⎟J z2

+ 2 2t3 J xJ y + J yJ x( ) + ...

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JX =12J x + J y( ),

J Y =−12J x−J y( ),

J Z =J z,

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

oror

Jx =12J X −J Y( ),

J y =12J X + J Y( ),

J z =J Z.

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

PRINCIPAL AXES PRINCIPAL AXES

WatsonWatson(OXYZ)(OXYZ)

Moret-BaillyMoret-Bailly(Oxyz)(Oxyz)

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LINKS BETWEEN TENSORIAL AND LINKS BETWEEN TENSORIAL AND WATSON’S MODELSWATSON’S MODELS

J, Ka , KcKets :Kets :

tiParameters :Parameters : B002 , B020 , B200 ,

For all reduction types (For all reduction types (AA,, SS oror 66)) and till the and till the

66 degree. degree.

……

J, C,α~ ~

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DETAILS OF THE ANALYSIS (v = 0 )DETAILS OF THE ANALYSIS (v = 0 )

S-ReductionS-Reduction 6-Reduction6-Reduction†† tt99 = 0 = 0 opt. topt. t99

RMSRMS(kH(kHz)z)

102.85102.85 79.2779.27 72.8572.85 69.869.877

σσ(kHz)(kHz) 2626 2121 23.4723.47 23.223.2

991071 assigned lines (assigned lines (JJmaxmax = 99). = 99).

σσ:: standard deviation calculated from the median of absolute standard deviation calculated from the median of absolute residuals.residuals.

†† K. Sarka et al., Journal of Molecular Spectroscopy, 200, 55 – 64, (2000).K. Sarka et al., Journal of Molecular Spectroscopy, 200, 55 – 64, (2000).

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A MORE COMPLEX CASE,A MORE COMPLEX CASE,

THE (THE (33, , 77, , 99) BENDING TRIAD) BENDING TRIAD

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DETAILS OF THE ANALYSISDETAILS OF THE ANALYSIS

« Classical »« Classical »

Model Model †† Tensorial ModelTensorial Model

Nber of Nber of parametersparameters

79 (3 79 (3 interactions)interactions)

42 (10 42 (10 interactions)interactions)

Degree of Degree of developmentdevelopment 88 66

JJMAXMAX 8686 8686

Number of IR Number of IR assignmentsassignments 48064806 4732 + 1484732 + 148

Number of MW Number of MW assignmentsassignments 499499 384384

RMSRMSIRIR(mk)(mk) 0,5530,553 0,6080,608

RMSRMSMWMW(.10(.10-6-6 cm cm--

11))5,25,2 2,12,1

†† H. Bürger et al., Journal of Molecular Structure, 612, 133 – 141, (2002).H. Bürger et al., Journal of Molecular Structure, 612, 133 – 141, (2002).

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OVERVIEWOVERVIEW

IR spectrum: H. BIR spectrum: H. Bürger (Wuppertal)ürger (Wuppertal)

1.0

0.5

0.0

580570560550540530520510Wavenumber / cm-1

Simulation

Experiment

7(b1) 9(b2) 3(a1)

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THE THE 77 BANDBAND

1.0

0.8

0.6

524.4524.2524.0523.8523.6Wavenumber / cm-1

P(45) P(44) P(43)Simulation

Experiment

Simulation (Bürger et al.)

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THE THE QQ BRANCH OF THE BRANCH OF THE 77 BAND BAND1.0

0.8

0.6

0.4

0.2

0.0

539.1539.0538.9538.8538.7Wavenumber / cm-1

Simulation

Experiment

Simulation (Bürger et al.)

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PERSPECTIVESPERSPECTIVES

Analysis of the Analysis of the 55((aa22)/)/44((aa11) dyad of SO) dyad of SO22FF22

deriving from the deriving from the 22((EE) band of SO) band of SO4422

Global analysis of the first vibrational Global analysis of the first vibrational levels: the ground-state, the dyad and the triadlevels: the ground-state, the dyad and the triad

Ref.Ref. : : - - M. Rotger, V. Boudon et M. Loëte, J. Mol. Spectrosc., 216, 297–307, (2002).M. Rotger, V. Boudon et M. Loëte, J. Mol. Spectrosc., 216, 297–307, (2002). - M. Rotger, V. Boudon, M. Loëte, L. Margulès, J. Demaison, H. Mäder, - M. Rotger, V. Boudon, M. Loëte, L. Margulès, J. Demaison, H. Mäder, G. Winnewisser et H.S.P. Müller, J. Mol. Spectrosc., 222, 172–179, (2003).G. Winnewisser et H.S.P. Müller, J. Mol. Spectrosc., 222, 172–179, (2003). - Ch. Wenger, M. Rotger et V. Boudon, JQSRT, 93, 429-446, (2004)- Ch. Wenger, M. Rotger et V. Boudon, JQSRT, 93, 429-446, (2004) ..

Other quasi-spherical top molecules: IOFOther quasi-spherical top molecules: IOF55 ( (CC44vv),),

HH22SOSO44 (atmosphere of Venus, (atmosphere of Venus, CC22), …), …

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CC2v2vTDS DATABASETDS DATABASE

http://www.u-bourgogne.fr/LPUB/c2vTDS.html

CC2v2vTDS for XYTDS for XY22ZZ2 2 molecules.molecules.

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CC4v4vTDS DATABASE TDS DATABASE

http://www.u-bourgogne.fr/LPUB/c4vTDS.html

CC4v4vTDS for XYTDS for XY55Z molecules.Z molecules.

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MANY THANKS TO THESE PERSONS:MANY THANKS TO THESE PERSONS:

« Classical » analysis: I. Merke and F. Hegelund,« Classical » analysis: I. Merke and F. Hegelund,

IR spectra: H. Bürger,IR spectra: H. Bürger,

Millimeter and microwave spectra: L. Margulès and J. Demaison,Millimeter and microwave spectra: L. Margulès and J. Demaison, H. Mäder, H.S.P. Müller and G. Winnewisser,H. Mäder, H.S.P. Müller and G. Winnewisser,

V. Boudon, J.-P. Champion, M. Loëte, N. Zvereva-Loëte and V. Boudon, J.-P. Champion, M. Loëte, N. Zvereva-Loëte and Ch. Wenger.Ch. Wenger.