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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.

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  • 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


Inversion of spectroscopic data to extract molecular potential curves
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


Motivations
Motivations potential curves

  • 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)


Ex 1: potential curves

3580 transitions resulting in 924 levels


Ex 1: potential curves

3580 transitions resulting in 924 levels


Ex 1: potential curves

3580 transitions resulting in 924 levels


Ex 1: potential curves

3580 transitions resulting in 924 levels


Ex 2: potential curves


Ex 3: potential curves


Ex 3: potential curves


Ex 3: potential curves


Dunham expansion for energy levels
Dunham expansion for energy levels potential curves

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…


Dunham expansion 2
Dunham expansion (2) potential curves

Dunham coefficients

Note: zero-point energy correction


Determination of the dunham coefficients
Determination of the Dunham coefficients potential curves

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


47 potential curves Dunham coefficients

to represent

16900 transitions, obtained by analysis of 348 fluorescence series excited with

21 wave lengths

r.m.s = 0.0011cm-1


Dunham expansion summary
Dunham expansion: summary potential curves

  • 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



Rkr rydberg klein rees analysis 1
RKR: Rydberg-Klein-Rees analysis (1) potential curves

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


Rkr approach 2
RKR approach (2) potential curves

inversion

RKR-2


Rkr potential curve
RKR potential curve potential curves

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 )


Near dissociation expansion nde
Near-dissociation expansion (NDE) potential curves

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


Ex: potential curves


Ipa inverted perturbation approach 1
IPA: Inverted perturbation approach (1) potential curves

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


Ipa 2
IPA (2) potential curves

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


Ipa example
IPA: example potential curves

Energy differences

C.R. Vidal, Comments At. Mol. Phys. 17, 173 (1986)

RKR

IPA


Dpf direct potential fit 1
DPF: Direct potential fit (1) potential curves

  • 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


Dpf 2
DPF (2) potential curves

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


Dunham rkrnde ipa example
Dunham/RKRNDE/IPA: example potential curves


Dpf example
DPF: potential curvesexample


Dpf example1
DPF: potential curvesExample:

3580 transitions resulting in 924 levels

Short distances

Large distances

Note: 1st estimate for the Ca scattering length


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