vibronic coupling by the spin-orbit operator in molecules and … · 2009-10-05 · 17.09.2009...

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


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)


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


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)


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


Symmetry group of the spin-orbit operator

example: the equilateral triangle (D3h), single unpaired electron

symmetry operations:

for example:


The symmetry operations of the spin double group D3h'


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

^


unitary transformation:

The 2ExE Jahn-Teller Hamiltonian with spin-orbit coupling


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:


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


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


-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


Transition-Metal Trifluorides: Electronic Spectra

MnF3 5E'

without SO coupling

with SO coupling


Transition-Metal Trifluorides: Electronic Spectra

CoF3 5E'

without SO coupling

with SO coupling


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


Relativistic JT Effect in Tetrahedral Systems


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:


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


Relativistic and Electrostatic JT Coupling Parameters of P4

+, As4+, Sb4

+, Bi4+


P4+ As4

+


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)


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)


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


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


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


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)

vibronic coupling by the spin-orbit operator in molecules and … · 2009-10-05 · 17.09.2009...

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