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
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
Non-relativistic Hamiltonian for 2 nuclei and n electrons in the lab-fixed frame with n-n e-n e-e and Relative distances
Separation of center-of-mass motion Origin=midpoint of the axis ≠center of mass Change of variables Total mass:
Second Derivative Operator reduced mass for homonuclear molecules
Hamiltonian in new coordinates Radial relative motion Electronic Hamiltonian Kinetic couplings m/m: Isotopic effect Origincenter of mass Study of the internal Hamiltonian… Center-of-mass motion
T in spherical coordinates: rotation of the nuclei Z q R Ri e- Y j Kinetic momentum of the nuclei X
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
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
R 1 R 2 R 3 R 3 R 2 R 1 General case: Z’’= Z Z’’’ y q R y Y’’ =Y’’’ Y j R 3 X ‘’ X X’’’ x R 3 R 2 R 1
T in the molecular frame (1) With xi, yi, zi now depending on q and j. Total electronic angular momentum in the molecular frame
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…
Hamiltonian in the molecular frame He+H’e Hv Hr+H’r Kinetic energy of the nuclei in the molecular frame O2 : quite complicated!
Total angular momentum in the molecular frame Commute with H (no external field) In the molecular frame
Total angular momentum in the lab frame molecularlab Depends only on Lz In the molecular frame!!
Playing further on with angular momenta…
Playing further on with angular momenta… Compare with: Also via a direct calculation:
Yet another expression for H in the molecular frame…. Hv Hr Hc Coriolis interaction
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…? labmolecular No spatial representation for S Rotation matrices:
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 R0: united atom BO or adiabatic approximation: factorization of the total wave function
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
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
Non-adiabatic couplings (1) proof Ex: highly excited potential curves in Na2
Non-adiabatic couplings (2) proof Diagonal elements:
Non-adiabatic couplings (3) proof Diagonal elements
« Improved » BO approximation (also « adiabatic » approximation) Neglect all non-diagonal elements in the adiabatic basis |fa> Unique by definition: Diagonalizes He
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
Diabatic basis: facts Not unique R-independent Definition at R=R0 (ex: R=) proof
« 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>
« Nuclear » wave functions (2)
Rotational wave functions Phase convention… (Condon&Shortley 1935, Messiah 1960) …and normalization convention….! Up to now: y=p/2….
Vibrational wave functions and energies (1) No analytical solution Useful approximations Rigid rotator Harmonic oscillator Equilibrium distance
Vibrational wavefunctions and energies (2) Deviation from the harmonic oscillator approximation: Morse potential Deviation from the rigid rotator approximation: proof
Continuum states Dissociation, fragmentation, collision… Regular solution: Influence of the potential Normalization In wave numbers proof In energy
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
Matrix elements of the vibrational hamiltonian BO basis: Vibrational energy levels Interaction between vibrational levels Quantum chemistry is needed…