1. 1- Introduction, overview 2- Hamiltonian of a diatomic molecule 3- Molecular symmetries; Hund’s cases 4- Molecular spectroscopy 5- Photoassociation of cold atoms 6- Ultracold (elastic) collisions Olivier Dulieu Predoc’ school, Les Houches,september 2004

2. Inversion of spectroscopic data to extract molecular potential curves Motivations Apetizer: some examples Rotating vibrator (or vibrating rotor!): Dunham expansion RKR: semiclassical approach NDE: towards the asymptotic limit IPA: perturbative approach DPF: brute force approach Applications

3. Motivations Analysis of light/matter interaction Gigantic amount of data: synthesis required Yields informations on internal structure Starting point: Born-Oppenheimer approximation Other perturbations Cold atoms: scattering length determination Combined analysis with (less accurate) quantum chemistry calculations Elaborate and efficient tools required High resolution (on energies)

4. Ex 1: 3580 transitions resulting in 924 levels

5. Ex 1: 3580 transitions resulting in 924 levels

6. Ex 1: 3580 transitions resulting in 924 levels

7. Ex 1: 3580 transitions resulting in 924 levels

8. Ex 2:

9. Ex 3:

12. Dunham expansion for energy levels « The energy levels of a rotating vibrator », J. L. Dunham, Phys. Rev. 41, 721 (1932) Anharmonic oscillator Energy levels: « term energies » Non-rigid rotator (Herzberg 1950) Centrifugal distorsion constant (CDC) Rotational constant Coupled to each other…

13. Dunham expansion (2) Dunham coefficients Note: zero-point energy correction

14. Determination of the Dunham coefficients Minimization of the reduced standard error (dimensionless) by adjustment on measured term energies N measured term energies M Dunham coefficients to fit C. Amiot and O. Dulieu, 2002, J. Chem. Phys. 117, 5155

15. 47 Dunham coefficients to represent 16900 transitions, obtained by analysis of 348 fluorescence series excited with 21 wave lengths r.m.s = 0.0011cm -1

16. Dunham expansion: summary Compact, accurate, empirical representation of a large number of energies Not suitable for extrapolation at large distances Not suitable for extrapolation at high J, for heavy molecules High-order coefficients highly correlated, and not physically meaningful No information on the molecular structure

17. Centrifugal distorsion constants

18. RKR: Rydberg-Klein-Rees analysis (1) R. Rydberg, Z. Phys. 73, 376 (1931); Z. Phys. 80, 514O (1933) Klein, Z. Phys. 76, 226 (1932); A. L. G. Rees, Proc. Phys. Soc. London 59, 998 (1947) Bohr-Sommerfeld quantification for a particle with mass  in a potential V Classical inner and outer turning points inversion RKR-1

19. RKR approach (2) inversion RKR-2

20. RKR potential curve Use G v and B v from experiment, Dunham expansion… Extract a set of turning point for all energies Specific codes (Le Roy’s group, U. Waterloo, Canada) Limitations: smooth functions of v, poor extrapolation high v, or large distances RKR-1 RKR-2 Note: extension with 3rd order quantification: ( C. Schwartz and R. J. Le Roy 1984 J. Chem. Phys. 81, 3996 )

21. Near-dissociation expansion (NDE) Fit (a subset of) G v and B v with an expansion incorporating the long-range behavior of the potential (C n /R n ) C. L. Beckel, R. B. Kwong, A. R. Hashemi-Attar, and R. J. Le Roy 1984 J. Chem. Phys. 81, 66 More elaborate form, for more flexibility « outer Padé expression » R.J. Le Roy, R.B. Bernstein, J. Chem. Phys. 52, 3869 (1970) W.C. Stwalley, Chem. Phys. Lett. 6, 241 (1970); J. Chem. Phys. 58, 3867 (1973). New input for RKR analysis

22. Ex:

23. IPA: Inverted perturbation approach (1) R. J. Le Roy and J. van Kranendonk 1974 J. Chem. Phys. 61, 4750 W. M. Kosman and J. Hinze 1975 J. Mol. Spectrosc. 56, 93 C. R. Vidal and H. Scheingraber 1977 J. Mol. Spectrosc. 65, 46. Expansion: Adjust an effective potential on experimental energies, no Dunham expansion Good initial approximation: RKR potential V (0) (R). Treat  V(R)=V(R)-V (0) (R) as a perturbation: H=H (0) +  V(R). Modified energies Zero-order eigenfunctions Generally over-determined Least-square fit

24. IPA (2) Choice of basis functions: Legendre polynomials Cut-off functio n Functional relation, useful for strongly anharmonic potentials Inner turning point Outer turning point Equlibrium distance New determination of G v, B v No unique solution Standard error on c i, through the covariance matrix

25. IPA: example C.R. Vidal, Comments At. Mol. Phys. 17, 173 (1986) RKR IPA Energy differences

26. DPF: Direct potential fit (1) Generalization of IPA approach Choose an analytical function to be fitted on experimental energies Need a good initial potential Package available: DSPotFit, from Le Roy’s group Y. Huang 2000, Chemical Physics Research Report 649, University of Waterloo. Morse family simple generalized modified extended Modified Lennard-Jones Better asymptotic behavior General power expansion

27. DPF (2) Pure long-range states in alkali dimers (e.g. double-well state in Cs 2 ) (See lecture on photoassociation) References: SMO: P. M. Morse 1929 Phys. Rev. 54, 57 GMO: J. A. Coxon and P. J. Hajigeorgiou 1991 J. Mol. Spectrosc. 150, 1 MMO: H. G. Hedderich, M. Dulick, and P. F. Bernath 1993, J. Chem. Phys. 99, 8363 EMO: E. G. Lee, J. Y. Seto, T. Hirao, P. F. Bernath, and R. J. Le Roy 1999 J. Mol. Spectrosc. 194, 197 MLJ: P. G. Hajigeorgiou and R. J. Le Roy 2000, J. Chem. Phys. 112, 3949 G: C. Samuelis, E. Tiesinga, T. Laue, M. Elbs, H. Knöckel, and E. Tiemann 2000, Phys. Rev. A, 63, 012710

28. Dunham/RKRNDE/IPA: example

29. DPF: example

30. DPF: Example: 3580 transitions resulting in 924 levels Short distances Large distances Note: 1 st estimate for the Ca scattering length