1. Predoc’ school, Les Houches,september 2004
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
with
n-n
e-n
e-e
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
reduced mass
for homonuclear molecules

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 nucleiZ q R Ri e- Y j
Kinetic momentum
of the nuclei 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 R 2 R 3 R 3 R 3 R 2 R 1 Z’’= Z Z’’’ q R 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 R 2 R 1

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

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 replacing Lx,y,z with jx,y,z=Lx,y,z+Sx,y,z
See further on…

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

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

15. Total angular momentum 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….Hv Hr Hc
Coriolis
interaction

19. No spatial representation for S
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 or adiabatic approximation:
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 energy Eb
Expressed in the adiabatic basis
< | >
Integration on electronic
coordinates
Set of differential coupled equations for Cba
Infinite sum
on a
J2 diagonal
BO approximation
non-adiabatic couplings

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

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

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

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
Useful
approximations
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…