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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 : Vincent.Boudon@u-bourgogne.fr
Web : http://www.u-bourgogne.fr/LPUB/tSM.html
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
I. Introduction & general ideasI. Introduction & general ideas
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
II. Symmetry adaptationII. Symmetry adaptation
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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)
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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 )
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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)
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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:
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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.
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
III. Rovibrational spectroscopyIII. Rovibrational spectroscopy
S4
C3
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
The polyads of CH4The polyads of CH4
Global fit
n =2v1 + v2 + 2v3 + v4Polyad Pn:
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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, …
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
IV. Rovibronic spectroscopyIV. Rovibronic spectroscopy
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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.
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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)
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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 !!
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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 !
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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 )
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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)
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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( )
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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:
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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 )
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
V. Conclusion & perspectivesV. Conclusion & perspectives
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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
Mathematical Methods for Ab Initio Quantum Chemistry • Nice • October 20–21, 2006
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)
• …