Vibronic Coupling by the Spin-Orbit Operator in Molecules and … · 2009-10-05 · 17.09.2009 Indo-German Workshop 2009 Vibronic Coupling by the Spin-Orbit Operator in Molecules
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17.09.2009 Indo-German Workshop 2009
Vibronic Coupling by the Spin-Orbit Operator
in Molecules and Clusters
Wolfgang DomckeTechnical University of Munich
Leonid V. PoluyanovRussian Academy of Sciences, Moscow
17.09.2009 Indo-German Workshop 2009
The Jahn-Teller TheoremThe Jahn-Teller TheoremThe Jahn-Teller TheoremThe Jahn-Teller Theorem
„A configuration of a polyatomic molecule for an electronic state having orbital degeneracy cannot be stable with respect to all
displacements of the nuclei unless in the original configuration the nuclei all lie on a straight line“
H. A. Jahn and E. Teller, Proc. Roy. Soc. A, 161, 220 (1937)
Stability of Polyatomic Molecules in Degenerate Electronic states. II. Spin Degeneracy
„It is not possible for the spin-orbit interaction to cause instability of an orbitally degenerate state“
H. A. Jahn, Proc. Roy. Soc. A 164, 117 (1938)
17.09.2009 Indo-German Workshop 2009
Vibronic Coupling
R. Renner 1934H. A. Jahn and E. Teller 1937W. Moffit and W. Thorson 1957H. C. Longuet-Higgins et al. 1958R. L. Fulton and M. Gouterman 1961
(i) Representation of the nonrelativistic electronic Hamiltonian in a(quasi)diabatic basis
(ii) Taylor expansion of the Hamiltonian in normal coordinates at a suitable reference geometry (up to second order)
(iii) Symmetry selection rules for the matrix elements
17.09.2009 Indo-German Workshop 2009
Spin-Orbit Couplingspin-orbit coupling is a relativistic effect
retardation andmagnetic interaction
Dirac - Coulomb - Breit Hamiltonian
NonrelativisticCoulomb int.
exact one-electron
kinematics
Reduction from four totwo components
Breit-Pauli operator:
W. Heisenberg, Z. Physik 39, 514 (1926)W. Pauli, Z. Physik 43, 601 (1927)G. Breit, Phys. Rev. 34, 553 (1929)
17.09.2009 Indo-German Workshop 2009
Spin-Orbit Vibronic Coupling
(i) Representation of the Breit-Pauli spin-orbit operator in a(nonrelativistic) quasidiabatic electronic basis
(ii) Taylor expansion of the spin-orbit operator in normalcoordinates at a suitable reference geometry (up to second order)
(iii) Symmetry selection rules for the matrix elements
17.09.2009 Indo-German Workshop 2009
Symmetry group of the spin-orbit operator
example: the equilateral triangle (D3h), single unpaired electron
symmetry operations:
for example:
17.09.2009 Indo-German Workshop 2009
The symmetry operations of the spin double group D3h'
17.09.2009 Indo-German Workshop 2009
Time-Reversal Symmetry
E. Wigner, Group Theory (Academic Press, NY, 1959)
ccT ...0110
0110ˆ
21⎟⎟⎠
⎞⎜⎜⎝
⎛ −⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
⎟⎟⎠
⎞⎜⎜⎝
⎛Ψ=⎟⎟
⎠
⎞⎜⎜⎝
⎛Ψ
10
01ˆ *T⎟⎟
⎠
⎞⎜⎜⎝
⎛Ψ−=⎟⎟
⎠
⎞⎜⎜⎝
⎛Ψ
01
10ˆ *T
even number of electrons:
odd number of electrons:
1ˆ 2 =T
ΨT̂ is orthogonal to : Kramers degeneracyΨ
1ˆ 2 −=T
^
17.09.2009 Indo-German Workshop 2009
unitary transformation:
The 2ExE Jahn-Teller Hamiltonian with spin-orbit coupling
17.09.2009 Indo-German Workshop 2009
Spin-Orbit Coupling in High-Spin E States of Trigonal Systems: the 3E×E, 4E×E, 5E×E Jahn-Teller Hamiltonians
MS = 1
MS = 0
MS = -1
3E state:
17.09.2009 Indo-German Workshop 2009
L. V. Poluyanov, W. Domcke, Chem. Phys. 352, 125 (2008)
3E×E JT Effect: Adiabatic Potentials
3E
G. Herzberg,Molecular Spectra and Molecular Structure
Vol III (Van Nostrand, 1966), p. 52
17.09.2009 Indo-German Workshop 2009
even spin multiplicity:
two-fold Kramers degeneracy,JT coupling is quenched by strong SO coupling
4E×E, 5E×E, ... JT Effect: Adiabatic Potentials
odd spin multiplicity:
There exists a pair of potential-energy functions which are strictly unaffected by SO coupling
5E
4E
17.09.2009 Indo-German Workshop 2009
-1.0 -0.5 0.0 0.5 1.0
-2000
0
2000
4000
6000
-1.0 -0.5 0.0 0.5 1.00
2000
4000
6000
4E' NiF3
5E" CoF3
-1.0 -0.5 0.0 0.5 1.0
-2000
0
2000
4000
6000
4E' CrF3
5E' MnF3
Pote
ntia
l Ene
rgy
(cm
-1)
ΔQ (Angstrom)
Transition-Metal Trifluorides: CASSCF Adiabatic Potentials
-0.5 0.0 0.5 1.0
-1000
0
1000
2000
30005E' CoF3
17.09.2009 Indo-German Workshop 2009
Transition-Metal Trifluorides: Electronic Spectra
MnF3 5E'
without SO coupling
with SO coupling
17.09.2009 Indo-German Workshop 2009
Transition-Metal Trifluorides: Electronic Spectra
CoF3 5E'
without SO coupling
with SO coupling
17.09.2009 Indo-German Workshop 2009
Relativistic Jahn-Teller Effect in Tetrahedral SystemsJahn-Teller selection rules for tetrahedral systems:
(H. A. Jahn and E. Teller, Proc. Roy. Soc. A 161, 220 (1937)
[E]2 = E + A (E modes are JT-active)
[T1]2 = [T2]2 = T2 + E + A (T2 and E modes are JT active)
Γ8 Γ6
with zeroth-order SO coupling:
Matrix elements of the SO operator are zero for a 2E state in tetrahedral systems
A 2T state splits into a four-fold degenerate (Γ8) and a two-fold degenerate (Γ6) manifold
17.09.2009 Indo-German Workshop 2009
Relativistic JT Effect in Tetrahedral Systems
17.09.2009 Indo-German Workshop 2009
Adiabatic Potential-Energy Surfaces of the Linear Relativistic E×T2and T2×(T2+E) JT Effects
three-dimensional „Mexican Hat“
four-dimensional „Mexican Hat“
ExT2:× ×
Γ8xE:
Γ8xT2:
17.09.2009 Indo-German Workshop 2009
X4+ tetrahedral radial cations of group V elements (X = P, As, Sb, Bi)
CASSCF,nonrelativistic
CASSCF,relativistic
2T 2T
2T 2T
2E 2E
2E 2E
17.09.2009 Indo-German Workshop 2009
Relativistic and Electrostatic JT Coupling Parameters of P4
+, As4+, Sb4
+, Bi4+
17.09.2009 Indo-German Workshop 2009
P4+ As4
+
17.09.2009 Indo-German Workshop 2009
isolated 2Π state (Λ=1)
2Π
W+
W-
The Renner-Teller Effect
( )( )yx
i
yx
iQQQ
i
±==
±=
±±
±
e 2
1
φρ
ψψψ diabatic electronic basis states
degenerate bending mode
Π−
Π
EececE
i
i
φ
φ
ρρ
22
22
ψ+ ψ−
ψ+
ψ−
HRT
RTN HTH +⎟⎠⎞
⎜⎝⎛ += 2
2
21 1ωρ
c : RT couplingconstant
symmetry properties:[ ]
number. quantum good a is
0,
lK
iij
jH
+Λ=∂∂
−∂∂
−=
=
φθhh
E. Renner, Z. Physik 92, 172 (1934)
17.09.2009 Indo-German Workshop 2009
Symmetry properties of
2Π Renner-Teller Hamiltonian with Spin-Orbit Coupling
SORTN HTH +⎟
⎠⎞
⎜⎝⎛ += 4
2
21 1ωρ
c ̂c 0110ˆ⎟⎟⎠
⎞⎜⎜⎝
⎛ −=
+∂∂
−=
T
iJ zz σθ
hh
2/3,22
222/1,
2/1,22
222/3,
00
00
Π−
Π
Π−
Π
EececE
EececE
i
i
i
i
φ
φ
φ
φ
ρρ
ρρ
φ
φ
φ
φ
ρρ
ρρ
i
i
i
i
eded
eded
−
−
−
−
d : linear spin-orbit vibronic coupling (SO-VC) constant
ζ = EΠ,3/2 - EΠ,1/2= SO splitting
ψ+ α ψ− α
ψ+ α
ψ− α
ψ+ β
ψ− β
ψ+ β ψ− βSORTH
SORTH
[ ]
[ ] 0ˆ,
0,
=
=
TH
JH
SORT
zSORT
L. V. Poluyanov and W. DomckeChem. Phys. 301, 111 (2004)
17.09.2009 Indo-German Workshop 2009
S.-G. He et al. J. Chem. Phys. 119, 10115 (2003)
“Sears Resonances“ in GeCH
complexity of the spectrum:
1. Renner-Teller effect2. strong spin-orbit coupling3. Fermi resonances
2Π ground state of GeCH
17.09.2009 Indo-German Workshop 2009
41.06
-325.0
-0.094-0.104
500.7500.0
CalculatedValue
---40.48d (cm-1)
-334.6-348.7ζ (cm-1)
-0.113-0.109ε
434.8435.6ω (cm-1)
Expt. FitValue
Fitted Value
Parameter
“Sears Resonances“ in GeCH
-34.5-4.81193.0κ 2Π1/2020
28.8-1.6889.8μ 2Π1/2020
12.5-1.4869.7μ 2Π3/2020
-13.9-1.1765.12Δ3/2010
-34.1-5.5755.6κ 2Σ1/2010
15.1-1.0444.5μ 2Σ1/2010
4.16.1440.02Δ5/2010
-12.61.8334.92Π1/2000
Without SO-VCWith SO-VC
Obs. energy -Calc. energy in cm-1Obs. energyin cm-1
Stateν1ν2ν3,
S. Mishra, V. Vallet, L. V. Poluyanov, W. Domcke, J. Chem. Phys. 123, 124104 (2005)
22
2cc
+=
ωωε
17.09.2009 Indo-German Workshop 2009
Conclusions
The „spin-orbit constant“ is not a constant!
Relativistic JT coupling is significant for systems containing heavy atoms.
Matrix elements of the Breit-Pauli operator can nowadays routinely becalculated with a number of ab initio packages.
There exists a wide world of novel vibronic-coupling phenomenabeyond the nonrelativistic approximation, for example:
(i) multiplicity-dependent Jahn-Teller effect in ME states of trigonalsystems
(ii) relativistic Jahn-Teller effect in 2E and 2T states of tetrahedral,cubic and octahedral systems.
(iii) linear relativistic vibronic coupling in spatially degenerate statesof linear molecules
17.09.2009 Indo-German Workshop 2009
TUM (PhD student, now University of Zürich)
Coworkers
TUM (PhD student)
Leonid V. Poluyanov
Sabyashachi Mishra
Institute of Chemical Physics,Academy of Sciences, Moscow
Padmabati Mondal
Daniel Opalka TUM (PhD student)
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