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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
2 S 1 
p

 vxz  ,   0    ,   0
 vxz  ,    vxz   ,    ,

ri  (ri , i ,  i )
~1  1 ; ~i   i  1
1 i~1
e  ri ,  i ,  i i 1 
2p


i

Lz ;
~
1 



 Lz   

2    , 

Central symmetry
iˆ(ri ,i , i )  (ri , p i , p  i )
  0   1   2   3   4 ...


 vxz  ,    ,
Axial symmetry:
2p rotation
 
 vxz (ri , i ,  i )  (ri , i , i )
Planar symmetry
...
iˆ  g    g
gerade
iˆ  u    u
ungerade
Symmetry properties of electronic functions (2)
2
L
2
S
is not a good quantum number (precession around the axis)
is a good quantum number if electrostatic interaction is dominant


2 S 1 
p 

Ex:
2
:  ,    , , S ,  ,     
g ,1 g ,3 u ,1  g ,1 u ,3  g ,3 u ,5 g ,...
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:
p :  , 
0g ,0g ,0u ,0u ,1g ,1u ,...
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


L

S
Hund’s cases (2): vector precession model
Hund’s case b
S
Herzberg 1950
 not defined:
 state
-Spin weakly coupled
K
J
N

L
Hund’s cases (2): vector precession model
Herzberg 1950
Hund’s case c
J
N

L

j
S
Hund’s cases (2): vector precession model
Hund’s case d
L
S
K
N
J
Herzberg 1950
Hund’s cases (2): vector precession model
Herzberg 1950
Hund’s case e
J
L
N
j
S
Hund’s case (3): interaction ordering
E.E. Nikitin & R.N. Zare, Mol. Phys. 82, 85 (1994)
H  H e  H r  H SO
(a)
(b)
(c)
(d)
(e)
He
HSO
Hr
strong
intermediate
weak
strong
weak
intermediate
intermediate
strong
weak
intermediate
weak
strong
weak
intermediate
strong
(adapted from Lefebvre-Brion&Field)
Rotational energy for (a)-(e) cases
     
O2
J  O  L  S ( N  S )
Hr 
2
2R
Case (a)

JMS
(a)
Erot
 Bv J ( J  1)   2  S ( S  1)   2
Case (b)
Case (c)
JM

(c)
Erot
 Bv J ( J  1)  2 2


(b )
Erot
 Bv N ( N  1)  2

(d), (e) cases: useful for Rydberg electrons (see Lefebvre-Brion&Field)

Parity(ies) and phase convention(s) (1)
On electron coordinates in
the molecular frame:
Convention of ab-initio calculations
 (ri , i ,  i )  (ri , i , i )
 vxz  ,    ,
xz
v
Convention of molecular spectroscopy
 vxz  ,    1  ,
« Condon&Shortley »
 vxz AM A   1A AM A ( )
lab
mol
A( M A  1) AX  iAY AM A   A( A  1)  M A ( M A  1)
AM A (  1) Ax  iAy AM A   A( A  1)   (  1)
One-electron orbital
Many-electron
wave function
 vxz  , s  1 2 ,    1 1 2   , s  1 2 ,
 vxz  S   1  S   s   S  
 vxz JM S   1J     S   s JM     S  
With s=1 for - states, s=0 otherwise
Parity(ies) and phase convention(s) (2)
Total parity:
Parity of the total wavefunction:
+/-
(1) J or (1) J 1 2 : e states
(1) J 1 or (1) J 1 2 : f states
  0,   0
 vxz J ,   0,   0 s , S ,   0   1J  S  s J ,   0,   0 s , S ,   0
1  3  5 

even
J
:
 0 ,  0 ,  0 ,...
2 S 1 s
 0 , : 

1  3  5 
odd
J
:

,

,

,...
0
0
0


2 S 1
 odd J :  ,  ,  ,... 
 , : 

even J :  ,  ,  ,...
 3  5 
0
0
0
1  3  5 
0
0
0
1
s
0
2 S 1
 0s , J , e :
2 S 1
 0s , J , f :
1
 0 ,3  0 ,5  0 ,...
1
 0 ,3  0 ,5  0 ,...
Parity(ies) and phase convention(s) (3)
Total parity:
Parity of the total wavefunction:
+/-
(1) J or (1) J 1 2 : e states
(1) J 1 or (1) J 1 2 : f states
All states except   0,   0



 vxz Js S  J    s S     1J  S  s Js S  J    s S  

s , J ,  Js S   1
2 S 1
2 S 1
 , J,
s

e
f
J S s

 Js S   1
S s
J    s S  

2
J    s S  

2
Or –S+s+1/2

Radiative transitions (1)
Absorption cross section:

 f  . i
In the lab frame
 
 . 

  e ri
p , q  0 , 1
n
i 1
 0   z ;  1   ( x  i y )
2
 (1)
In the mol frame


 .   X  X  Y Y   Z Z
 0   Z ;  1   ( X  iY )
2
p
  p  q D
 ( , ,0)
1 *
pq
BO approximation
v J M 
 v ( R) 2 J  1  J


D M  ( ,  ,0) ri ; R 
R
4p
2
Radiative transitions (2)
Absorption cross section:


 f i
p , q  0 , 1

 f i
q
(1) p   p  v f  q

(2 J i  1)( 2 J f  1) 
 Mf
Jf
v
i
J i  J f

p M i    f
1
Ji 

q  i 
1



( R)   d rk  f rk ; R i rk ; R  q   f  q i
3
Dipole transition moment
(2 J i  1)( 2 J f  1)
4p

2p
0
1
d  d (cos  ) DMff  f ( ,  ,0) D pq ( ,  ,0) D M i i ( ,  ,0)
J
1
Hönl-London factor
1
Ji
2
Selection rules for radiative transitions (1)
n
 n 

Lz , 0     ;e rk cos k    0
k 1
 i k 1  k

 Lz , 0    0   f   i   0 
f
i
f
 0 if  f   i
i
Parallel transition
f=i
 n 
e n
i k 
Lz ,  1     ;  rk sin  ke    1
2 k 1
 i k 1  k

 Lz ,  1       1    f  i    1 
f
i
f
i
 0 if  f   i  1
  0,1
f
i
Perpendicular
transition
f=i±1
Selection rules for radiative transitions (2)

2p
0
d e
iM f  ip iM i
e e
 2p if M i  p  M f  0
M  0,1
= 0 otherwise
 Jf

 M
f

J i  J f

p M i    f
1
Ji 
0
q  i 
1
J  0,1
Ji  0 
X Jf 0
 Jf

 0
1 Ji 
  0
0 0
If Jf+Ji+1 odd
No Q line for  transition
 1 P line

J  J f  J i  0 Q line
 1 R line

Selection rules for radiative transitions (3)
v 
f
 f i
q
v  
i
 f i
q
( Re )  v f  vi






 vxz  0   0   

X




Allowed
Forbidden
ug

iˆ q    q  
u X u, g X g
Allowed
Forbidden
FranckCondon factor