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

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

2. Main steps: • Definition of the exact Hamiltonian • Definition of a complete set of basis functions • Matrix representation of finite dimension+perturbations • Comparison to observations to determine molecular parameters

3. Non-relativistic Hamiltonian for 2 nuclei and n electrons in the lab-fixed frame electrons nuclei with n-n e-e e-n and Relative distances

4. Separation of center-of-mass motion • Origin=midpoint of the axis ≠center of mass • Change of variables Total mass:

5. Second Derivative Operator for homonuclear molecules reduced mass

6. Hamiltonian in new coordinates Radial relative motion Electronic Hamiltonian • Kinetic couplings  m/m: • Isotopic effect • Origincenter of mass Study of the internal Hamiltonian… Center-of-mass motion

7. T in spherical coordinates: rotation of the nuclei Z q Kinetic momentum of the nuclei R Ri e- Y j X

8. Rotating or molecular frame • Specific role of the interatomic axis • Potential energy greatly simplified, independent of the molecule orientation • Euler transformation with a specific convention: { j, q, p/2} Molecular lab-fixed Lab-fixed molecular

9. R 1 Z’’’ Z’’= Z q R R 2 Y’’ =Y’’’ Y j y=0 around Z’’’: x=X’’’,y=Y’’’, z=Z’’’ Oy perp to OZz R 3 X ‘’ X X’’’ R 3 y=p/2 around Z’’’: Ox perp to OZz OR R 3 R2R1

10. General case: R 1 Z’’’ Z’’= Z y q y R R 2 Y’’ =Y’’’ Y j R 3 X ‘’ X x X’’’ R 3R2R1

11. T in the molecular frame (1) With xi, yi, zi now depending on q and j. Total electronic angular momentum in the molecular frame

12. T in the molecular frame (2) vibration rotation Electronic spin can be introduced by replacingLx,y,zwithjx,y,z=Lx,y,z+Sx,y,z See further on…

13. Hamiltonian in the molecular frame He+H’e Hv Hr+H’r O2 : quite complicated! Kinetic energy of the nuclei in the molecular frame

14. Total angular momentum in the molecular frame Total angular momentum Commute with H (no external field) In the molecular frame

15. Total angular momentum in the lab frame In the lab frame molecularlab Depends only on Lz In the molecular frame!!

16. Playing further on with angular momenta…

17. Playing further on with angular momenta… Compare with: Also via a direct calculation:

18. Yet another expression for H in the molecular frame…. H’e He Hv Hr Hc Coriolis interaction

19. What about spin? Electronic spin Notations: Nuclear spin If S quantized in the molecular frame (i.e. strong coupling with L), L should be replaced by j=L+S (with projection W) in all previous equations But why…? labmolecular No spatial representation for S Rotation matrices:

20. Born-Oppenheimer approximation (1) H=He+H’e+Hv+Hr+Hc. m/m>1800: approximate separation of electron/nuclei motion Potential curves: R: separated atoms R0: united atom BO or adiabatic approximation: factorization of the total wave function

21. Born-Oppenheimer approximation (2) H=He+H’e+Hv+Hr+Hc. BO oradiabaticapproximation: factorization of the total wave function Mean potential All act on the electronic wave function

22. Validity of the BO approximation Total wave function with energyEb Expressed in the adiabatic basis < | > Integration on electronic coordinates Set of differential coupled equations for Cba Infinite sum on a J2diagonal BO approximation non-adiabatic couplings

23. Non-adiabatic couplings (1) • Ex: highly excited potential curves in Na2 proof

24. Non-adiabatic couplings (2) Diagonal elements: proof

25. Non-adiabatic couplings (3) Diagonal elements proof

26. « Improved » BO approximation(also « adiabatic » approximation) Neglect all non-diagonal elements in the adiabatic basis |fa> Unique by definition: Diagonalizes He

27. Alternative: Diabatic basis Neglect all (non-diagonal) couplings due to Hc Define a new basis which cancels these couplings Couplings in the potential matrix proof

28. Diabatic basis: facts • Not unique • R-independent • Definition at R=R0 (ex: R=) proof

29. « Nuclear » wave functions (1) Adiabatic approximation: VL(R) Eigenfunctions of J2, JZ, Lz (ou jz) ( Jz) C.E.C.O proof Wave functions: |JML> ou |JMW>

30. « Nuclear » wave functions (2)

31. Rotational wave functions Phase convention… (Condon&Shortley 1935, Messiah 1960) …and normalization convention….! Up to now: y=p/2….

32. Vibrational wave functions and energies (1) No analytical solution approximations Useful Rigid rotator Harmonic oscillator Equilibrium distance

33. Vibrational wavefunctions and energies (2) Deviation from the harmonic oscillator approximation: Morse potential Deviation from the rigid rotator approximation: proof

34. Continuum states Dissociation, fragmentation, collision… Regular solution: Influence of the potential Normalization In wave numbers proof In energy

35. Matrix elements of the rotational hamiltonian Easy to evaluate in the BO basis: But in general, L and S are not good quantum numbers… …quantum chemistry is needed Selection rule

36. Matrix elements of the vibrational hamiltonian BO basis: Vibrational energy levels Interaction between vibrational levels Quantum chemistry is needed…