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Renner-Teller Coupling in H2S+: Partitioning the Ro-vibronic and Spinorbit Coupling HamiltonianG. Duxbury1, Christian Jungen2 and Alex Alijah3

1Physics Department, University of Strathclyde, Glasgow, G4 0NG, UK

2LAC, Laboratoire Aime Cotton du CNRS, Universite de Paris -Sud, 91405 Orsay France

3GSMA, UMR CNRS 6089, Universite de Reims Champagne-Ardenne, B.P. 1039, 51687 Reims Cedex2, France

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Outline of presentation

• Resumé of the transformed Hamiltonian• Linear and Bent molecules• 1970 -1980, Key Dixon and Duxbury, and Jungen

and Merer papers• Adding stretching, the rise of the stretch-Bender.• Basic idea Christian Jungen• Bent molecule code, Alex Alijah, linear molecule

coding Horatiu Palivan

• Examples NH2 and H2S in next presentation.

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Basic approach to Renner-Teller Coupling• Molecules executing large amplitude motion• Electronic states are degenerate when linear• When nuclear frame is bent the degeneracy is broken• Two routes to formulation of the interaction Hamiltonian

• Basis functions in the linear limit, Jungen and Merer, ( J &M)

• Basis functions from the bent molecule limit, Barrow Dixon and Duxbury, (BDD)

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Partitioning the Renner-Teller Hamiltonian• Neglect Stretching: partition the bending-rotation into:• Bending plus a-axis rotation: linear triatomic limit• End over end rotation, b and c axes: degenerate in the linear

molecule limit• If a vibronic energy matrix is set up using a generalised one-

dimensional bending Hamiltonian, e.g. Hougen, Bunker and Johns J. Mol. Spectrosc. 34,136 (1970), the elements of the vibronic matrix are huge.

• Use a bending angle dependent coordinate transformation to separate the coupled equations into two one dimensional equations, i.e transform the Hamiltonian. BDD approach.

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Generate effective basis functions to create the final interaction matrix

• J & M, Mol. Phy. 40, 1 (1980), devised a generalised transformation matrix to couple the correct pairs of functions as the molecules bend from linear to strongly bent. In this approach the Hamiltonian matrix is transformed.

• Two transformations are used, S resembling the original Renner transformation, and T, a further correction to minimise the effects of the large energy splitting, in NH2 at the equilibrium geometry.

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The expectation value

Magnitude inferred from the extent to which the mixture of functions is unbalanced.

Below the barrier to linearity, proportional to K

(see BDD , Mol. Phys. 1974)

Above the barrier, dependant on the details of the resonant interactions (Jungen, Hallin and Merer, Mol. Phys. 1980)

Lz

+Λ and -Λ

Lz

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Separability of nuclear motion

• Triatomic molecules- Most coordinate systems used for large amplitude motion problems are based on a strict separation of stretching and bending. e.g. Jensen and Bunker MORBID (Morse Oscillator Rigid Bender)

• If the bonds stretch as the molecule bends, following the equilibrium path on the potential energy surface, then the description of the motion will involve an expansion in both bending and stretching functions.

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Separability of nuclear motion: stretch bender• JM and BDD used the semi-rigid bender, not the

rigid bender, to model the effect of bond length variation with large amplitude bending motion

• This minimised stretch-bend interaction.• Why not use the semi-rigid bender, not the rigid

bender, when including the effects of stretching explicitly? (Jungen 1989)

• This minimises the size of the interaction matrix, although leading to a lot of algebra on the way.

• This model is called the stretch-bender.

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Stretch-Bender co-ordinates

Symmetric stretch, Ss, Asymmetric stretch, Sa out of plane

rotation

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Write the a-axis rotation term in terms of an A "rotation constant”

H strb =− h2

2−gρρ f ρ( ) + gρρ ∂2

∂ρ2 + gss ∂2

∂Ss2 + gaa ∂2

∂Sa2

⎡

⎣⎢

⎤

⎦⎥

+ ′A J a2 +V ρ,Ss,Sa( )

with ′A=12gφφ and

Hrotb,c=′BJb2+′CJc2

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Switch to “dimensionless angular momentum operators

• define the functions

′ A ρ( ) = h

2

′ A ,

′ B ρ( ) = h

2

′ B and

′ C ρ( ) = h

2

′ C

which includes the factor of which originates from the normal angular momentum operators.

h2

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The form of the H” matrix

• elements H” (JM), zeroth order BDD. In the BDD model the re-ordering elements are identically zero.

• A test of the two methods for H2S+ gave the same numerical values for the vibronic interaction matrix

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Interaction super matrix

• The building blocks of the overall interaction matrix, or “ super matrix” are the H” matrices without or with the B and C axis rotation added.

• At this level the off diagonal interaction matrix elements within the super matrix may be block factorised.

• This enables a choice to be made of whether vibronic or ro-vibronic energies are to be calculated.

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Fig 1: Stretch bender

interaction matrix

•Factorisation•Vibrational resonances within a single half state, off diagonal in v2 and v1

•RT diagonal in v1

•H” matrices lie on the diagonal

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k , Σ( )

p = −

3

2

− 2 ,1

2( )

p = −

1

2

0 , −1

2( )

p =

1

2

0 ,1

2( )

p =

3

2

2 , −1

2( )

− 2 ,

1

2( ) 0 CR OR OR

0 , −

1

2( )

CR 0 SU 0

0 ,

1

2( ) OR SU 0 CR

2 , −

1

2( )

OR 0 CR 0

OR Overall Rotation terms SU Spin Uncoupling terms CR Cross terms ˆ

J −

ˆ S

+

+ˆ J

+

ˆ S

−

Figure 2

Block structure of the full interaction matrix for J = 3/2, S = 1/2 and K even. The diagonal elements of the matrices are shown in Figure 1 with the diagonal rotation terms added.

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CONCLUSIONSummary of the similarities of the two methods

• In both approaches the off-diagonal perturbation elements in the H” matrix arise from the use of a generalised transformation matrix which diagonalises only the potential part of the Hamiltonian

• This matrix does not commute with the nuclear kinetic energy operator.

• The off diagonal elements arise due to this non-commutation

• The block structure of the super-matrix is identical for both the JM and BDD methods

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References1 Barrow, T, Dixon, R.N and Duxbury, G, Mol. Phys. 27, 1217-1124 (1974)2 Duxbury, G and Dixon, R.N,Mol. Phys. 43, 255-274 (1974)3 Jungen, Ch and Merer, A.J. Mol. Phys. 40, 1-23 (1980), 4 Duxbury, G, Horani, M and Rostas, J., Proc. Roy. Soc.A 331,109-137, (1972) 5 Duxbury, G, Jungen, Ch and Rostas, J., Mol Phys. 48, 719-752 (1983)6 Duxbury, McDonald, Van Gogh, Alijah, Jungen and Palivan, J.Chem Phys 108, 2336, (1998)7 Alijah, A. and Duxbury, G. J. Mol. Spectrosc. 211, 1 (2002)8 Duxbury, G. and Reid, J.P. Mol. Phys. 105, 1603 (2007) 10.9 Hochlaf, M, Wietzel, K.-M. and Ng, C.Y., J. Chem. Phys. 120, 6944 (2004)10 Balzer, L. Karlsson, M. Lundquist, B. Wannberg, D.M.P. Holland and M.A. MacDonald, Chem. Phys. 195, 403-422 (1995)11 Han, S., Kang, T.Y. and Kim, S.K., J. Chem. Phys. 132, 124304 (2010)12 Webb, A.D., Dixon, R.N and Ashfold, M.N.R. J. Chem Phys. 127, 224307 (2007)13 Webb, A.D., Kawanaga, N., Dixon and Ashfold. J. Chem Phys.127, 224308 (2007)