symmetry-adapted tensorial formalism to model rovibrational and rovibronic molecular spectra

Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006

Symmetry-Adapted Tensorial Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Formalism to Model Rovibrational and

Rovibronic Molecular SpectraRovibronic Molecular Spectra

Symmetry-Adapted Tensorial Symmetry-Adapted Tensorial Formalism to Model Rovibrational and Formalism to Model Rovibrational and

Rovibronic Molecular SpectraRovibronic Molecular Spectra

Vincent BOUDON

Laboratoire de Physique de l’Université de Bourgogne – CNRS UMR 5027

9 Av. A. Savary, BP 47870, F-21078 DIJON, FRANCE

E-mail : [email protected]

Web : http://www.u-bourgogne.fr/LPUB/tSM.html

mailto:[email protected]





ContentsContentsI. Introduction & general ideas

II. Symmetry adaptation

III. Rovibrational spectroscopy

Spherical tops: CH4, SF6, …

Quasi-spherical tops

Other symmetric and asymmetric tops

IV. Rovibronic spectroscopy

Jahn-Teller effect, (ro)vibronic couplings, …

Electronic operators

Application to some open-shell systems

V. Conclusion & perspectives


I. Introduction & general ideasI. Introduction & general ideas


Why tensorial formalism ?Why tensorial formalism ?

• Take molecular symmetry into account

Simplify the problem (block diagonalization, …)

Also consider approximate symmetries

• Systematic development of rovibrational/rovibronic interactions, for any polyad scheme

Effective Hamiltonian and transition moments construction

Global analyses


II. Symmetry adaptationII. Symmetry adaptation


Case of a symmetric topCase of a symmetric top

Quantum number

=irreducible representation

of a groupz z

O(3) ⊃ C∞v ⊃ C3v

Cv symmetrization: Wang basisC3v symmetrization: use of projection methods

z

€

J(J +1)

€

rJ J, K ,C K = k


Spherical tops: the O(3) G group chainSpherical tops: the O(3) G group chain

j,nCσ = ( j )GnCσm j,m

m∑

Tσ( j ,nC ) = ( j )GnCσ

m Tm( j )

m∑

G

Sphere

Lie group O(3)(or SU(2)CI)

Molecule

Point group G(or G

S)


What do all these indexes mean ?What do all these indexes mean ?Rank / O(3) symmetry (irrep)

z-axis projection / component

Symmetry / G irrep

Component

Multiplicity index

O(3) Tm( j )j, m

G j,nCσ Tσ( j ,nC )


Example 1: “Octahedral harmonic” of rank 4Example 1: “Octahedral harmonic” of rank 4

=5

24+ i

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

+5

24− i

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

+7

12

Y (4,A1 )(θ,φ) = (4 )GA14 Y4

(4)(θ,φ) + (4 )GA10 Y0

(4)(θ,φ) + (4 )GA1−4Y−4

(4)(θ,φ)


Example 2: Rank 3 harmonic, A2 symmetryExample 2: Rank 3 harmonic, A2 symmetry

> 0 < 0 Antisymmetric function

Y (3,A2 )(θ,φ) = (3)GA2

2 Y2(3)(θ,φ) + (3)GA2

−2Y−2(3)(θ,φ)

D(3) ↓O=A2 ⊕ F1 ⊕ F2


G matrix: Principle of the calculationG matrix: Principle of the calculation

H (4,A1 ) = (4 )GA1q

q=−4,0,4∑ Hq

(4 )

The idea consists in diagonalizing a typical octahedral (or tetrahedral) splitting term:

A( j)mm ' =(−1) j−m (4 )GA1

q

q∑ V

4 j jq −m m'

⎛⎝⎜

⎞⎠⎟

In the standard |j,m> basis this amounts to diagonalize the matrix:

[ j]−1/2FA1 nCσ ( j )(4 j ) nCσ

The eigenvectors lead to the G matrix; the eigenvalues are oriented Clebsch-Gordan coefficients:


Use of the G matrixUse of the G matrix

F( j1 j2 j)

p1 p2 p= ( j1 )Gp1

m1 ( j2 )Gp2m2 ( j )Gp

mVj1 j2 jm1 m2 m

⎛⎝⎜

⎞⎠⎟m1 ,m2 ,m

∑• Calculate symmetry-adapted coupling coefficients:

3jm (Wigner)3jp (p = nCσ)

TnCσ( j ) =Tσ

( j ,nC ) = ( j )GnCσm Tm

( j )

m∑

• Build symmetry-adapted tensorial operators:

Construction of Hamiltonian and transition-momentoperators

Coupling of operators, calculation of matrix elements


Matrix elements: the Wigner-Eckart theoremMatrix elements: the Wigner-Eckart theorem

Reduced matrix element« physical part »

Matrix element (p = nCσ)

Coupling coefficient« geometric part »

Group-dependantphase factor

ψ p '( j ') Tp0

(k ) ψ p '( j ') = Ξ j ' p ' F (k j j ')

p0 p pj ' T (k ) j

• In the O(3) G group chain:

• In the G group: ψ σ '(C ') Tγ

(Γ ) ψ σ '(C ') = ΞC 'σ ' F (Γ C C ')

γ σ σC ' T (Γ ) C

• Recoupling formulas: Using 6C, 9C, 12C coefficients, etc.


Quasi-spherical tops: ReorientationQuasi-spherical tops: Reorientation

j,nC,σ = ( j )GnCσm j,m

m∑

j,nC,C,σ = (C ) ′GCσσ j,nC,σ

σ∑~ ~

~ ~

O(3) ⊃ Td ⊃ C2v


III. Rovibrational spectroscopyIII. Rovibrational spectroscopy

S4

C3


Rotational & vibrational operatorsRotational & vibrational operators• Rotation, recursive construction of Moret-Bailly & Zhilinskií:

R1(1) =2J (1) =2rJ Elementary operator

R2(0) =(R1(1) ⊗ R1(1))(0) =−43

J 2 Scalar operator

RK (K ) =(RK−1(K−1) ⊗ R1(1))(K ) Recursive construction

RΩ(K ) = R2(0)( )Ω−K2 ×RK (K ) Maximum degree Ω, rank K

RσΩ(K ,nC) = (K )GnCσ

m RmΩ(K )

m∑ Symmetrized operator

• Vibration, construction of Champion:

Each normal mode ⇒ ai(C ), ai

+(C )

εV s{ } ′s{ }ΓvΓ ′v (Γ) = f ai

(C ), ai+(C ){ }( ), ε =time-reversal parity


Effective tensorial HamiltonianEffective tensorial Hamiltonian

H Pk{ }

= t s{ } s'{ }Ω K ,nΓ( )ΓvΓv'β RΩ K ,nΓ( ) ⊗ εV s{ } s'{ }

ΓvΓv' Γ( )⎡⎣ ⎤⎦all indexes∑ (A1g )

Systematic tensorial development

H Pn =P Pn HP Pn =H GS{ }

Pn + ...+ H Pk{ }Pn + ...+ H Pn{ }

Pn

Effective Hamiltonian and vibrational extrapolation

ΨrJ ,nC( ) ⊗Ψ v

Cv( )⎡⎣

⎤⎦

Γ( )

Coupled rovibrational basis

H =H P0 ≡GS{ }

+ H P1{ }+ ...+ H Pn{ }

+ ...

Polyad structure

P0

P1

P2

P3

Rotation Vibration


Transition momentsTransition moments

μΘ(Γ0 ) = 1;m Θ [Γ] μ {i} C (1g ,Γ ) ⊗M ({i},Γ )⎡

⎣⎤⎦

(Γ0 )

Γ∑

{i}∑

m∑

Dipole moment

αΘ1Θ2

(Γ0 ) = L;m Θ1Θ2 [Γ]α {i} C (Lg ,Γ ) ⊗P({i},Γ )⎡⎣

⎤⎦

(Γ0 )

Γ∑

{i}∑

m∑

L =0,2∑

Polarizability

Rovibrationaloperators

Direction cosines tensor

ParametersStone coefficients


Spherical topsSpherical topsS4

C3

ν1 ν2 ν3 ν4

A1 E F2 F2

Raman Raman IR/Ra. IR/Ra.

Stretch Bend Stretch Bend

O(3) ⊃Td

C3

C4

O(3) ⊃Oh

1 2 3 4 5 6

A1g Eg F1u F1u F2g F2u

Raman Raman IR IR Raman Hyper-Raman

Stretch Bend


The polyads of CH4The polyads of CH4

Global fit

n =2v1 + v2 + 2v3 + v4Polyad Pn:


Recent spherical top analysesRecent spherical top analyses

12CH4, 13CH4, 12CD4

Analysis of high polyads, intensities

GeH4, GeD4, GeF4

Fundamental bands (isotopic samples)

P4

ν3 stretching band

32SF6, 34SF6

First vibrational levels, hot bands

SeF6, WF6

Fundamental bands

Mo(CO)6

ν6 (CO stretch) band


The ν2+ν4 combination band of SF6The ν2+ν4 combination band of SF6

1.0

0.8

0.6

0.4

0.2

0.0

12641262126012581256Wavenumber / cm-1

Simulation

Experiment

Overview of the Q-branches region

F1u

F2u


Quasi-spherical tops: SO2F2 and SF5ClQuasi-spherical tops: SO2F2 and SF5Cl1600

1400

1200

1000

800

600

400

200

0

SO42− SO2F2

GS GS

ν2 (E)

ν4 (F2)

ν1 (A1)

ν3 (F2)

ν4 (a1)

ν5 (a2)

ν9 (b2)

ν3 (a1)ν7 (b1)

ν2 (a1)ν8 (b2)

ν1 (a1)

ν6 (b1)

O(3) ⊃Td ⊃C2v O(3) ⊃Oh ⊃C4v


Other moleculesOther molecules

O(3) ⊃D2h group chain

X2Y4 molecules

Example: Ethylene, C2H4

O(3) ⊃C∞v ⊃C3v group chain

XY3Z molecules

Examples: CH3D, CH3Cl, …


IV. Rovibronic spectroscopyIV. Rovibronic spectroscopy


The problem: Degenerate electronic statesThe problem: Degenerate electronic statesOpen-shell molecules have degenerate electronic states.

We only consider rovibronic transitions inside a single isolated degenerate electronic state.

Transition-metal hexafluorides (ReF6, IrF6, NpF6, …), hexacarbonyles (V(CO)6, …), radicals (CH3O, CH3S, …), etc have a degenerate electronic ground state.

In this case, the Born-Oppenheimer approximation is no more valid. There are complex rovibronic couplings (Jahn-Teller, …).

Molecules with an odd number of electrons have half-integer angular momenta: use of spinorial representations.


Modified Born-Oppenheimer approximationModified Born-Oppenheimer approximation

H (r,Q) = H(r)Electronic kinetic energy+ electronic interactions

{ + V(r,Q)Electron-nuclei and nuclei-nuclei

Coulomb interaction

1 2 3 + T(Q)Nuclear kinetic

energy

{

H(r) +V(r,Q0 )( )ψ n(r) =Enψ n(r), Ψ(r,Q) = χn(Q)ψ n(r)n∑

T(Q)δm,n +Umn(Q)( )χn(Q) =Eχm(Q)n∑

Umn(Q) = ψm* (r) H(r) +V(r,Q)( )∫ ψ n(r)dr

Degenerate electronic state Γ: sum restricted to the [Γ] degenerate states

• Inclusion of non-adiabatic interactions among the [Γ] multiplet• Non-adiabatic interactions with other states neglected


The Jahn-Teller « effect » The Jahn-Teller « effect »

U(Q) =U(Q0 ) +∂U(Q)∂Qi

⎛

⎝⎜⎞

⎠⎟Q0

Qi +12i=1

3N−6

∑ ∂2U(Q)∂Qi∂Qj

⎛

⎝⎜

⎞

⎠⎟

Q0

QiQji, j=1

3N−6

∑ +L

H =12

hν i Pi2 +Qi

2( )i=1

3N−6

∑ +L

+∂V(Q)∂Qi

⎛

⎝⎜⎞

⎠⎟Q0

Qii=1

3N−6

∑

Linear Jahn-Teller coupling1 24 44 34 4 4

+12

∂2V(Q)∂Qi∂Qj

⎛

⎝⎜

⎞

⎠⎟

Q0

QiQji, j=1

3N−6

∑

Quadratic Jahn-Teller coupling1 24 4 44 34 4 4 4

+L

After some rearrangements:

!Q0 is usually not an equilibrium configuration

Electronic energy = 0Electronic operators

Hermann Arthur JAHNHermann Arthur JAHN(1907 – 1979)(1907 – 1979)

Edward TELLEREdward TELLER(1908 – 2003)(1908 – 2003)


EE problem : linear JT levelsEE problem : linear JT levels

l =v,v−2,L ,−v

j = l ±1 / 2 ⇒ j blocks

H ( j) =hν ×

j +1 / 2 D(2 j +1)

D(2 j +1) j + 3 / 2 2D

2D j + 5 / 3 D(2 j + 3)

D(2 j + 3) j + 7 / 2 4D

4D O O

O O

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟

• Infinite matrices truncation

• HJT is non perturbative !!


If we include the molecular rotation …If we include the molecular rotation …Example of the G’g F1u problem (ReF6)Example of the G’g F1u problem (ReF6)

45,000 45,000for J = 28.5 only !45,000 45,000

for J = 28.5 only !

… the problem becomes intractable !


Constructing electronic operatorsConstructing electronic operators

EKe (Ke ,neΓe) = EKe−1(Ke−1) ⊗E(1)( )(Ke,neΓe) , E(1) =2J e

Linearly independant operators ⇒ Ke ≤2J e

Electronic state Γ Electronic angular momentum Je

•Γe = E’ : Je = 1/2, operators

• Γe = F : Je = 1, operators

•Γe = G’ : Je = 3/2, operators

E0(0,A1 ) =I , E1(1,F1 ) =2J e

E0(0,A1 ) =I , E1(1,F1 ) =2J e , E2(2,E ) ,

E2(2,F2 )

E0(0,A1 ) =I , E1(1,F1 ) =2J e , E2(2,E ) ,

E2(2,F2 ), E3(3, A2 ) , E3(3, F1 ) , E3(3, F2 )


Rovibronic operatorsRovibronic operators

H {Pk } = t{s} { s'}

Ω(K ,nΓ)(Ke,Γe )Γ1Γ2 (Γv )

all indexes∑ RΩ(K ,nΓ) ⊗ EKe(Ke,Γe ) ⊗ εV{ s}{ s'}

Γ1Γ2 (Γv )⎡⎣ ⎤⎦(Γ)⎡

⎣⎢⎤⎦⎥(A1 )

Rovibronic effective Hamiltonian

Ψr(J ,nCr ) ⊗ Ψ e

(Je ,Ce ) ⊗Ψ v(Cv )⎡⎣ ⎤⎦

(βevCev )⎡⎣

⎤⎦σ

(βC )

Coupled rovibronic basis

Rovibronic transition moments

Mθi{ } ,F1u( ) = RΩ K ,nΓ( ) ⊗ EKe Ke,Γe( ) ⊗V s{ } s'{ }

Γ1Γ2 Γv( )⎡⎣

⎤⎦

Γev( )⎡⎣⎢

⎤⎦⎥

F1u( )Dipole moment:

And similarly for the polarizability …

Rotation VibrationElectronic


The ν6 (C–O stretch) band of V(CO)6The ν6 (C–O stretch) band of V(CO)6Threefold degenerate electronic stateThreefold degenerate electronic state

1.0

0.8

0.6

0.4

0.2

0.0

2020201020001990198019701960Wavenumber / cm-1

Room temperature spectrum

Supersonic jet (13 K)P. Asselin et al.

SimulationM. Rey

J. Chem. Phys.114, 10773–10779(2001)


Half-integer states: spinorial representationsHalf-integer states: spinorial representations

)D Ri R j( ) =

)D Ri( )

)D Rj( )

• Vectorial (standard) representations:

(D Ri R j( ) = Ri ,Rj

©™ ¨Æ≠(D Ri( )

(D Rj( )

Ri , R j

©™

¨Æ≠=±1, projective factor

• Projective representations:

• Spinorial representations:

Projective representations that allow to symmetrizeSU(2) CI representations (spin states) into a subgroup G

“ Group ” GS


Example: The Oh “group”Example: The Oh “group”SS

OS E 4C3, 4C3−1 3C4

2 3C4 , 3C4−1 6C2

A1 1 1 1 1 1A2 1 1 1 −1 −1E 2 −1 2 0 0F1 3 0 −1 1 −1F2 3 0 −1 −1 1

′E1 2 1 0 2 0

′E2 2 1 0 − 2 0′G 4 −1 0 0 0

D1/2( ) ⊃ ′E1, D 3/2( ) ⊃ ′G , D 5 /2( ) ⊃ ′E2 ⊕ ′G , L

OhS =OS ⊗CI ⇒ ′E1τ , ′E2τ , ′Gτ ,L τ =g, u( )


Rhenium hexafluorideRhenium hexafluoride

(d)1

2F2g

2Eg G’g (b)

G’g (X)

E’2g (a)

Re6+ Voct Hso>> (0 cm-1)

(5015 cm-1)

(29430 cm-1)

0.80

0.75

0.70

0.65

0.60

65006000550050004500Wavenumber / cm-1

0.66

0.64

0.62

0.60

31000300002900028000Wavenumber / cm-1

• 129 electrons

• Strong spin orbit coupling

• Half-integer angular momenta


1.5

1.0

0.5

0.0

750740730720710700Wavenumber / cm-1

Experiment,(H. Hollenstein,M. Quack,ETH Zürich)

Simulation,(M. Rey,ETH Zürich)

E’1 E’2 G’ G’

ν3

ν2 + ν6

The The νν33 band of ReF band of ReF66The The νν33 band of ReF band of ReF66


C3v and its spinorial representations: C3vC3v and its spinorial representations: C3vSS

C3vS E C3,C3

−1 3σ v

A1 1 1 1A2 1 1 −1E 2 −1 0′A1 1 −1 i′A2 1 −1 −i′E 2 1 0

ψ ′A2( ) H A1( ) ψ ′A1( ) = ψ ′A1( ) H A1( ) ψ ′A2( )*

′A1, ′A2 are not physically discernable

′A2 = ′A1( )*

ψ C( ) ⇒ ψC*

( )

Complex irreps:


Cv and its spinorial representations: CvCv and its spinorial representations: CvSS

C∞vS E L 2C ϕ( ) L ∞σ v

Σ+ ≡0+ 1 L 1 L 1Σ+ ≡0− 1 L 1 L −1Π≡1 2 L 2cos ϕ( ) L 0Δ ≡ 2 2 L 2cos 2ϕ( ) L 0Φ ≡ 3 2 L 2cos 3ϕ( ) L 0

M M M M M M

1 / 2 2 L 2cos ϕ / 2( ) L 03 / 2 2 L 2cos 3ϕ / 2( ) L 05 / 2 2 L 2cos 5ϕ / 2( ) L 0

M M M M M M

Wang basis:

J,K ,± =12

J ,K ± J ,K( ) K = k( )

J ,0+ = J ,0 (even integer J ), J ,0− = J ,0 (odd integer J )


The ground electronic state of CH3OThe ground electronic state of CH3O

CH3O: 1a1( )22a1( )

23a1( )

24a1( )

21e( )4 5a1( )

22e( )3

2e( )3 ≡ 2e( )1 ⇒ %X 2E ground electronic term

S =1 / 2 ⇒ CS = ′E and CL =E, Ce ⊂CS ⊗CL = ′A1 ⊕ ′A2 ⊕ ′E

D 3/2( ) ⊃D 1/2( ) ⊕D 3/2( ) ⊃D Ce( ) = ′E ⊕ ′A1 ⊕ ′A2( )

⇒rJ =

rS+

rL, J e =3 / 2, Ke =1 / 2 , 3 / 2 , Ce = ′E , ′A1, ′A2

Basis set: ψ eσe

Ce( ) = 3 / 2,Ke,Ce,σ e

or 1 / 2, 1 / 2 , ′E( )⊗ 1, 1 ,E( ),Ce,σ e

or 1 / 2, 1 / 2( ) ⊗ 1, 1( ) ,Ke,Ce,σ e

Spin Orbit


Electronic operators for CH3OElectronic operators for CH3O

Spin: ES1 1( ) =2S 1( ),

Orbit: EL1 1( ) =2L 1( ), EL

2 2( ) = EL1 1( ) ⊗EL

1 1( )( )2( )

H GS{ }GS =t0 ES

1 1,0−( ) ⊗EL1 1,0−( )

( )0+ ,A1( )=4t0SzLz (spin-orbit coupling)

One order 0 non-trivial operator for the ground state:

′E ⊗E 4t0 ; 62 cm_1

1 / 2, 1 / 2( )⊗ 1, 1( ), 3 / 2 , ′A1 ⊕ ′A2

1 / 2, 1 / 2( )⊗ 1, 1( ), 1 / 2 , ′E


Vibronic operators for CH3OVibronic operators for CH3OCase of a doubly degenerate vibrationCase of a doubly degenerate vibration

H v=1{ }v=1 =t1 ES

0 0,0+( ) ⊗EL0 0,0+( )

( )0+( )⊗ +V1{ } 1{ }

1 1 0+( )⎛⎝

⎞⎠

0+ ,A1( ) → Harmonic oscillator

+t2 ES1 1,0−( ) ⊗EL

0 0,0+( )( )

0−( )⊗ −V1{ } 1{ }

1 1 0−( )⎛⎝

⎞⎠

0+ ,A1( ) → 2Szl (spin-vib., small)

+t3 ES0 0,0+( ) ⊗EL

1 1,0−( )( )

0−( )⊗ −V1{ } 1{ }

1 1 0−( )⎛⎝

⎞⎠

0+ ,A1( ) → 2Lzl (orb.-vib. ≡ JTL)

+t4 ES1 1,0−( ) ⊗EL

1 1,0−( )( )

0+( ) ⊗ +V1{ } 1{ }

1 1 0+( )⎛⎝

⎞⎠

0+ ,A1( ) → 4SzLz (spin-orb., v=1 correction)

+t5 ES0 0,0+( ) ⊗EL

2 2, 2( )( )

2( ) ⊗ +V1{ } 1{ }1 1 2( )⎛

⎝⎞⎠

0+ ,A1( ) → Vibronic ( ≡ JTQ)

+t6 ES1 1, 1( ) ⊗EL

1 1,0−( )( )

1( ) ⊗ +V1{ } 1{ }1 1 2( )⎛

⎝⎞⎠

3 ,A1( ) → Vibronic (spin-orb.-vib., small)

+ L

6 non-trivial operators up to order 2 for v = 1:

Presumably 3 main contributions : t1, t3 and t5


Vibronic levels for an E-mode of CH3OVibronic levels for an E-mode of CH3O

′Ce* ⊗ ′Cv = A1( )( ) ′Ce

* μ 0{ } 1{ }Γ( ) Ce ⊗ Cv = E( )( )C ≠ 0 ⇒

C ⊂ ′Ce* ⊗Γ

E ⊂ A1 ⊗Γ = Γ

⎧⎨⎩

Γ = E (perpendicular band) ⇒3 bands from C = ′A1 ⊕ ′A2

4 bands from C = ′E

⎧⎨⎩

μ 0{ } 1{ }E( ) μ 0{ } 1{ }

E( )

Cv =A1

v=0

l =0

Cv =Ev=1

l =1

Ke =3 / 2,Ce = ′A1 ⊕ ′A2

Ke =1 / 2,Ce = ′E

′A1 ⊕ ′A2

′A1 ⊕ ′A2

′E

′E

′E′E

Vibration Spin-orbit Spin-Vib. + Orb.-Vib. + … JT

Ke =3 / 2,Ce = ′A1 ⊕ ′A2

Ke =1 / 2,Ce = ′E

~ 62 cm-1


Rovibronic operators and basis for CH3ORovibronic operators and basis for CH3O

J, Kr( )⊗ 1 / 2, 1 / 2( )⊗ 1, 1( ),Ke( )⊗ v, l( ),Kev( ),K ,C,σ

Coupled rovibronic basis:

Spin Orbit

VibrationalElectronic

Rotational Vibronic

Rovibronic

RΩ Lr ,Λr( ) ⊗ ESLS LS ,ΛS( ) ⊗EL

LL LL ,ΛL( )( )

Λe( )⊗ εV ns{ } ms{ }K1K2 Λv( )

( )Λev( )

( )Λ,Γ( )

Hamiltonian: Γ =A1 Dipole moment: Γ =A1 or EPolarizability: Γ =A1 or A2 or E

Rovibrational operators


Comparison with the “usual” approachComparison with the “usual” approach

•E electronic state associated to the L = 1, KL = 1 effective quantum numbers through symmetry reasons only (KL does not need to be identified to Λ)

•Separate JT calculation replaced by tensorial operators built on powers of L and S

•Global spin-orbital-vibrational-rotational calculation

•All vibronic levels in a given polyad considered as a whole

•Method based on symmetry (construction of invariants in a group chain); the link to the “usual” physical (JT) problem is not straightforward


V. Conclusion & perspectivesV. Conclusion & perspectives


Future developmentsFuture developments

• Rovibronic transitions between different electronic states

General rovibronic model

• Analytic derivation of the effective Hamiltonian and transition moments from an ab initio potential energy surface

Analytic contact transformations

Cf. work of Vl. Tyuterev (Reims) on triatomics


Programs STDS& Co.

Programs STDS& Co.

Spherical TopData System

www.u-bourgogne.fr/LPUB/shTDS.html

• Molecular parameter database

• Calculation and analysis programs

• XTDS : Java interface


AcknowledgmentsAcknowledgments

• M. Rotger, A. El Hilali, M. Loëte, N. Zvereva-Loëte, Ch. Wenger, J.-P. Champion, F. Michelot (Dijon)

• M. Rey (Reims)

• D. Sadovskií, B. Zhilinskií (Dunkerque)

• M. Quack et al. (Zürich)

• …

symmetry-adapted tensorial formalism to model rovibrational and rovibronic molecular spectra

Documents

molecular symmetry

rovibronic transitions

c coefficients

problem block diagonalization

molecular rotation example

gg f1u problem ref6

band of vco6threefold

symmetry adaptationcase