1- Introduction, overview 2- Hamiltonian of a diatomic molecule

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:

10. Ex 3:

11. Ex 3:

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

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

14. Determination of the Dunham coefficients N measured term energies M Dunham coefficients to fit Minimization of the reduced standard error (dimensionless) by adjustment on measured term energies 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 m in a potential V Classical inner and outer turning points RKR-1 inversion

19. RKR approach (2) inversion RKR-2

20. RKR potential curve RKR-2 RKR-1 • Use Gv and Bv 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 Note: extension with 3rd order quantification: (C. Schwartz and R. J. Le Roy 1984 J. Chem. Phys. 81, 3996 )

21. Near-dissociation expansion (NDE) C. L. Beckel, R. B. Kwong, A. R. Hashemi-Attar, and R. J. Le Roy 1984 J. Chem. Phys. 81, 66 Fit (a subset of) Gv and Bv with an expansion incorporating the long-range behavior of the potential (Cn/Rn) 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). More elaborate form, for more flexibility « outer Padé expression » 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. Adjust an effective potential on experimental energies, no Dunham expansion Good initial approximation: RKR potential V(0)(R). Treat DV(R)=V(R)-V(0)(R)as aperturbation: H=H(0)+DV(R). Expansion: Modified energies Zero-order eigenfunctions Generally over-determined Least-square fit

24. IPA (2) Cut-off function Legendre polynomials Choice of basis functions: Functional relation, useful for strongly anharmonic potentials Inner turning point Outer turning point Equlibrium distance Standard error on ci, through the covariance matrix New determination ofGv, Bv No unique solution

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

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. simple Morse family generalized extended modified 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 Cs2) (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: 1st estimate for the Ca scattering length