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

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

Generalities on molecular symmetries • Determine the spectroscopy of the molecule • Guide the elaboration of dynamical models • Allow a complete classification of molecular states by: • Solving the Schrödinger equation • Looking at the separated atom limit (R) • Looking at the united atom limit (R0) • Adding electron one by one to build electronic configurations

Symmetry properties of electronic functions (1) spin Planar symmetry Axial symmetry: 2p rotation Central symmetry gerade ungerade

Symmetry properties of electronic functions (2) is not a good quantum number (precession around the axis) is a good quantum number if electrostatic interaction is dominant Ex: 2S+1: multiplicity Sstates: spin fixed in space, 2S+1 degenerate components Lstates: precession around the axis, multiplet structure, almost equidistant in energy

Symmetry properties of electronic functions (3) Otherwise:

Hund’s cases for a diatomic molecule (1) Rules for angular momenta couplings Determine the appropriate choice of basis functions This choice depends on the internuclear distance (recoupling) F. Hund, Z. Phys. 36, 657 (1926); 40, 742 (1927); 42, 93 (1927)

Hund’s cases (2): vector precession model Herzberg 1950 Hund’s case a J N W L S S L

Hund’s cases (2): vector precession model Herzberg 1950 Hund’s case b W not defined: • S state • Spin weakly coupled S K J N L L

Hund’s cases (2): vector precession model Herzberg 1950 Hund’s case c J N L W j L S

Hund’s cases (2): vector precession model Herzberg 1950 Hund’s case d L S K J N

Hund’s cases (2): vector precession model Herzberg 1950 Hund’s case e J N j L S

Hund’s case (3): interaction ordering E.E. Nikitin & R.N. Zare, Mol. Phys. 82, 85 (1994) (adapted from Lefebvre-Brion&Field)

Rotational energy for (a)-(e) cases (d), (e) cases: useful for Rydberg electrons (see Lefebvre-Brion&Field) Case (a) Case (b) Case (c)

Parity(ies) and phase convention(s) (1) Convention of ab-initio calculations On electron coordinates in the molecular frame: Convention of molecular spectroscopy « Condon&Shortley » lab mol One-electron orbital Many-electron wave function With s=1 for S- states, s=0 otherwise

Parity(ies) and phase convention(s) (2) Total parity: Parity of the total wavefunction: +/-

Parity(ies) and phase convention(s) (3) Total parity: Parity of the total wavefunction: +/- All states except Or –S+s+1/2

Radiative transitions (1) In the mol frame Absorption cross section: In the lab frame BO approximation

Radiative transitions (2) Absorption cross section: Dipole transition moment Hönl-London factor

Selection rules for radiative transitions (1) Parallel transition Lf=Li Perpendicular transition Lf=Li±1

Selection rules for radiative transitions (2) = 0 otherwise X IfJf+Ji+1odd NoQline forSStransition

Selection rules for radiative transitions (3) Franck-Condon factor Allowed Forbidden X Allowed Forbidden X X

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