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) Axial symmetry: 2  rotation Planar symmetry Central symmetry gerade ungerade spin

is not a good quantum number (precession around the axis) is a good quantum number if electrostatic interaction is dominant Symmetry properties of electronic functions (2) Ex: 2S+1: multiplicity  states: spin fixed in space, 2S+1 degenerate components  states: 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 Hund’s case a Herzberg 1950 L S J N   

Hund’s cases (2): vector precession model Hund’s case b Herzberg 1950 L S J N  K  not defined:  state -Spin weakly coupled

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

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

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

Hund’s case (3): interaction ordering (adapted from Lefebvre-Brion&Field) E.E. Nikitin & R.N. Zare, Mol. Phys. 82, 85 (1994) HeHe H SO HrHr (a) strongintermediateweak (b) strongweakintermediate (c) intermediatestrongweak (d) intermediateweakstrong (e) weakintermediatestrong

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

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

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

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

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

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

Selection rules for radiative transitions (1) Parallel transition f=if=i Perpendicular transition  f =  i ±1

Selection rules for radiative transitions (2) = 0 otherwise X If J f +J i +1 odd No Q line for  transition

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