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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
 2 n '2  2 '2  2 '2
H 
i 
a 
b  V

2m i 1
2ma
2mb
electrons
nuclei
2
2
2



'2
 '2  '2
with  i 
'2
X i
Yi
Z i
and
n
2
n-n
e-e
e-n
n
2
2
Z ae
Zbe
Z a Zbe
e
40V  

 
rab
i 1 ria
i 1 rib
i  j rij
Relative distances
2
Separation of center-of-mass motion
• Origin=midpoint of the axis ≠center of mass
• Change of variables
Total mass:
M  ma  mb  i  m
 ma  ' mb  ' m
Rc 
Ra 
Rb 
M
M
M
 ' '
R  Ra  Rb
 ' 1 ' '
Ri  Ri  Ra  Rb
2


'
 Ri
n
i 1
ma
1 n
 
c   R   i
M
2 i 1
'
a
mb
1 n
 
c   R   i
M
2 i 1
m
 i'   i   c
M
'
b
Second Derivative Operator
1 n '2 1 '2 1 '2 1
1
i 
a 
b 
 c c   R R

m i 1
ma
mb
M

n
1 n 2 1 n
1
  i 
i j 
 R  i

m i 1
4 i , j 1

i 1
ma mb

ma  mb
reduced mass
ma mb
 
ma  mb
1  0
for homonuclear molecules
Hamiltonian in new coordinates
Radial relative motion
2 2
2  2 
O2

R  
R

2
2
2R R
R 2R 2
2 2 2 n 2
H 
R 
i  V

2
2m i 1
2

8
n
2
i j 
 R  i

2 
i , j 1
i 1
Electronic Hamiltonian
n
2 2

c
2M
Center-of-mass motion
Kinetic couplings  m/:
-Isotopic effect
-Origincenter of mass
Study of the internal Hamiltonian…
T in spherical coordinates: rotation of the nuclei
Z
2 2
2  2 
O2

R  
R

2
2
2R R
R 2R 2
q
R
Ri
eY
j
X
X R  R sin q cos j
YR  R sin q sin j
Z R  R cosq
 1 

1
 

O   
sin q
 2
2 
q sin q j 
 sin q q
2
2
Kinetic momentum
of the nuclei
  
O  R  R
i


 


 
 YR
    sin j

 ZR
 cot q cos j
i  Z R
YR 
i
q
j 


  

 
   cos j

OY   Z R
 XR
 cot q sin j
i  X R
Z R  i 
q
j 
OX 
OZ 


   
 X R
 
 YR
i
YR
X R  i j
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, /2}
X i   xi sin j  yi cos q cos j  zi sin q cos j
Yi  xi cos j  yi cos q sin j  zi sin q sin j
Molecular lab-fixed
Z i  yi sin q  zi cos q
xi   X i sin j  Yi cos j
Lab-fixed  molecular
yi   X i cosq cos j  Yi cosq sin j  Z i sin q
zi  X i sin q cos j  Yi sin q sin j  Z i cosq
R1
 X    cos j
  
 Y      sin j
 Z    0
  
R2
 X    cosq
  
 Y     0
 Z    sin q
  
R3
y=0 around Z’’’:
x=X’’’,y=Y’’’, z=Z’’’
Oy perp to OZz
Z’’’
Z’’= Z
sin j
0  X 
 
cos j 0  Y 
0
1  Z 
q
R
Y’’ =Y’’’
Y
j
X

R 3 R 2 R 1O
X ‘’
X’’’
 
i q

1

Oy 
i sin q j
Oz  0
Ox 
OR
0  sin q  X  
 
1
0  Y  
0 cosq  Z  
R3
y=/2 around Z’’’:
Ox perp to OZz
General case: y  0
and y   / 2
Z’’= Z
R1
 X    cos j
  
 Y      sin j
 Z    0
  
R2
 X    cosq
  
 Y     0
 Z    sin q
  
0  sin q  X  
 
1
0  Y  
0 cosq  Z  
sin y
R3
 x   cosy
  
 y     sin y
 z  0
  
Z’’’
y
sin j
0  X 
 
cos j 0  Y 
0
1  Z 
q
R
y
Y’’ =Y’’’
Y
j
X ‘’
X’’’ x
X

R 3R 2 R 1 O


cosy  


sin
y




q
sin
q

j




sin y  



cos
y


i 

q
sin
q

j


Ox 
Oy

i
Oz  0
cosy
0
0  X  
 
0  Y  
1  Z  
T in the molecular frame (1)
2 2
2  2 
2  1 

1
 



R  
R

sin q
 2
2
2 
2 
2
2R R
R 2R  sin q q
q sin q j 
With xi, yi, zi now depending on q and j.
  
 
 q  x , y , z
 

 q
 

 q
 

 q


 X ,Y , Z


 X ,Y , Z


 X ,Y , Z
 x  yi  zi  
  i



q yi q zi 
i 1  q xi
n
 
 
   zi
 yi


y

z
i 1 
i
i
i
 Lx
Total electronic angular

n
  
  
i
i



 
 cosq Lz  sin q Ly

 j  x , y , z  j  X ,Y , Z 
momentum in the
molecular frame
T in the molecular frame (2)
2 2
2  2 

R  
R
2
2
2 R R
R
vibration
2
2


  
  1  i
1 2
2




cot
q


L
cot
q

L

L
z
y
 2 x
2R 2  q 2
q  sin q j 



2 
i

i  1  i


 2 Lx
 2 Ly 
 Lz cot q  
2 
2 R 
 q
  sin q j 


2





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
n

 2 n 2
  2 n
2
H  
 i  V   
i j 
 R   i  He+H’e


2 
i 1
 2m i 1
  8 i , j 1

2  2 

R
2
2 R R
R
Hv
2
2



  
  1  i
1 2
2 

 cot q
 
 Lz cot q   2 Lx  Ly 
2
2

2 R  q
q  sin q j 




  H +H’
2 
i

i  1  i
  2 Lx
r
r

 2 Ly 
 Lz cot q  
2 
2 R 
 q
  sin q j 

2
Kinetic energy of the
nuclei in the
molecular frame
O2 : quite
complicated!
Total angular momentum in the molecular frame
  
J OL
2
J , J X , JY , J Z
Total angular momentum
Commute with H
(no external field)
 
i q

1

Oy 
i sin q j
Oz  0
Ox 
In the molecular frame
 
J x  O x  Lx 
i q
 1 
J y  Oy  Ly 
 cot q Lz
i sin q j
J z  Lz
  
 
 q  x , y , z
 

 q
 

 q
 

 q


 X ,Y , Z


 X ,Y , Z


 X ,Y , Z
 x  yi  zi  
  i



q yi q zi 
i 1  q xi
n
 
 
   zi
 yi

yi
zi 
i 1 
i
 Lx

n
  
  
i
i



 
 cosq Lz  sin q Ly

 j  x , y , z  j  X ,Y , Z 
Total angular momentum in the lab frame
In the lab frame


  cos j
 
J X    sin j
 cot q cos j
Lz
i
q
j  sin q


  sin j
 
J Y   cos j
 cot q sin j
Lz
i
q
j  sin q
JZ 
molecularlab
  sin j

 cos j
 0

 
i j
 cosq cos j sin q cos j 

 cosq sin j sin q sin j 
sin q
cosq 
Depends
only on Lz
J 2  J X2  J Y2  J Z2
 1 

1
2 
cot q

1
2


  
sin q
 2

2
i

L

L
z
z
2 
2
sin
q

q

q
sin
q

j
sin
q

j
sin
q


2
J 2  J x2  J y2  J z2   2 cot q

q
In the
molecular frame!!
Playing further on with angular momenta…
2
2



  1  i
1
O 2   2  2  cot q
 
 Lz cot q   2 L2x  L2y
 q
q  sin q j 
 



i

i  1  i
2

    2 Lx
 2 Ly 
 Lz cot q  
 q
  sin q j 








Playing further on with angular momenta…
 1 
2 
1



sin q
O   
2 
2
q sin q j 
 sin q q
2
2
1



 Lx  Lx 
  i cot q  2i
sin 2 q
q





 2i
 cos q Lz  sin q Ly cos q Lz  sin q Ly 
j


Compare with:
 1 

1
2 
cot q

1
2

J   
sin q
 2

2
i

L

L
z
z
2 
2
sin
q

q

q
sin
q

j
sin
q

j
sin
q


2
2
  i


2i
O  J  L  2i Lx

Ly 
 cosq Lz 
q sin q  j 

 
  i


i
2
L  J  Lz  iLx

Ly 
 cosq Lz 
Also via a
q sin q  j 

2
direct calculation:
2
2
 
  i


i
L  O   L2x  L2y  iLx

Ly 
 cosq Lz 
q sin q  j 

Yet another expression for H in the molecular
frame….
 
 

2
2
2
2
O  J  L  2 L  J  J  L  (2O  L )
   
 2
   
also : O  L  L  O
2
2
2
O  J  L  J  L  LJ  J L


H’e
He
n

 2 n 2
  2 n
2
H  
 i  V   
i j 
 R  i 


2 
i 1
 2m i 1
  8 i , j 1

 
2
2
2

 2 
J
L  2L  J

R


2
2
 

2 R R
R 2 R
2 R 2
Hv
Hr
Hc
Coriolis
interaction
also : 
L  (2O  L)
2R 2
Electronic spin
What about spin?

S
Notations:

I
Nuclear spin
   
J OLS
  
N OL
 
F J I
If S quantized in the molecular frame (i.e. strong coupling with
be replaced by
L), L should
j=L+S (with projection W) in all previous equations
But why…?
labmolecular
  sin j

  cosq cos j
 sin q cos j

cos j
 cosq sin j
sin q sin j
No spatial
0 

sin q 
cosq 
representation for
S
Rotation matrices:
mol  D 1lab  exp( iL z / 2) exp( iL yq /  ) exp( iL zj /  )lab
H mol  D 1 H lab D
S
mol
 exp( iS z / 2) exp( iS yq / ) exp( iS zj /  ) S
mol S
mol
lab
 exp( i ( Lz  S z ) / 2 ) exp( i ( Ly  S y )q /  )
exp( i ( Lz  S z )j /  )lab S
lab
Born-Oppenheimer approximation (1)
H=He+H’e+Hv+Hr+Hc.
m/>1800: approximate separation of electron/nuclei motion


H e (ri ; R)  U ( R) (ri ; R)
Potential curves:
R: separated atoms
R0: united atom
BO  y

( )


( R) ( R; ri )
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
( )
BO  y
( )


( R) ( R; ri )
 2
 ( )
 2 
1
2
( )


R



J


U
(
R
)


H

y
(
R
)

E
y
( R)






c



2
2
R
R
2R
 2R

Mean potential
All act on the electronic wave function
Validity of the BO approximation


 
 

 
1
 R; ri    C R  ri ; R
R 

Total wave function with energy E
Expressed in the adiabatic basis
 2 2

J2



H

H

E
e
c
    0

2
2
2R
 2 R

Set of differential coupled equations for
C
< | >
Integration on electronic
coordinates

 2 2

 J 2 

 U ( R)   H c   E  C ( R) 

2
2
2R
 2 R

Infinite sum
2

on 
 2

 2 '
 ' 





  H c  '  C ' ( R)

2

R

R R
 '  2 

BO approximation
J2 diagonal
non-adiabatic couplings
Non-adiabatic couplings (1)
 '
(1)

  
'
R
(1)
 
0
  ' 
(1)
proof
 V R  '
U  ' ( R)  U ( R)
• Ex: highly excited
potential curves in
Na2
Non-adiabatic couplings (2)
 2 
( 2)



 
R 2
 V R     V R  
   V R   


 (1     )



R  U   ( R)  U  ( R)    U   ( R)  U  ( R) U   ( R)  U   ( R) 
proof
Diagonal elements:
  '
(1)
   
2
 
 V R  
2
U  ( R)  U ( R)2
Non-adiabatic couplings (3)
2R  H c  
2
 
 2R  L  2 L  J  
2
2
  L2    2  L2z  
 2i  Lx  
 
  i


i
L  J  L2z  iLx

Ly 
 cosq Lz 
q sin q  j 


2i


 Ly  
q sin q
j
  2 cot q  Ly     Lz  
Lz , H e   0
 
Lz    
Diagonal elements
 H c  

1
2
2 2


L


2




2
2R
proof

L , H   0
2
e
« Improved » BO approximation
(also « adiabatic » approximation)

 2 2

 J 2  L2   2 2 2
 2
2

 U ( R) 

 E C ( R)  0

2
2
2
2 R
2
R
 2 R

Neglect all non-diagonal elements in the adiabatic basis |>
Unique by definition:
Diagonalizes He
Alternative: Diabatic basis
Neglect all (non-diagonal) couplings due to Hc

W (R)

 2 2

 J 2  L2   2 2 2

 U  ( R)  E  C  ( R) 

2
2
2 R
 2 R


2
 ( 2)
(1)  
    2 
 C  ' ( R)

M 
2   ' 
R 
if :
  M   (1)
R

Define a new basis which
cancels these couplings
~
   M  ( R)

~
~
   C  ( R) 
 2 2
~
~
~ ~


W

E
C


W

  
   
  C

2
  
 2 R

~
W    M W M 1


~
C ( R)   C  ( R) M  ( R)

proof
Couplings in the
potential matrix
Diabatic basis: facts
proof
• Not unique
• R-independent
• Definition at R=R0 (ex: R=)
« Nuclear » wave functions (1)
Adiabatic approximation:
Lz    
 2 2
 
 J 2 
 L2   2 2 2
2
 2


 U  ( R) 

 E C ( R )  0

2
2
2
2
2 R
2 R
2
R
 2 R

V(R)
1
  C 
R
Eigenfunctions of J2,
JZ, Lz (ou jz) ( Jz)
J
ou |JMW>

, H  J Z , H   0
J z , H c   0
J z , H BO   0
C.E.C.O
Wave functions: |JM>
2
proof
« Nuclear » wave functions (2)
C ( R,q , j )   ( R)R
J
M
(q , j )   ( R)e
iMj

J
M
(q )
 2 2

2
2


J
(
J

1
)



V
(
R
)

E


  ( R)  0
2
2
2R
 2 R



 1 
 M 2  2M cosq  2  J
J
sin q


(
q
)

J
(
J

1
)

M

M
 (q )


2
q
sin q
 sin q q

J 2R MJ  (q , j )   2 J ( J  1)R MJ  (q , j )
J Z R MJ  (q , j )  MR MJ  (q , j )
J zR MJ  (q , j )  R MJ  (q , j )
Rotational wave functions
Phase convention…
J X  iJ Y R MJ  (q , j )  J ( J  1)  M (M  1)1/ 2 R (JM 1)  (q , j )
R MJ (  0) (q , j )  YJM (q , j ); R (JM 0)  (q , j )  (1)  YJ (q , j )
(Condon&Shortley 1935, Messiah 1960)
…and normalization convention….!

2π
0

J
dj  sin qdqR  M  (q , j )R MJ  (q , j )   J J  M M
0
2J 1 J
R (q , j ) 
D M (q , j ,0)
4
DMJ  (q , j ,y )  e iMj d MJ  (q )e iy
J
M
d
J
M
(q ) 
4
J
M
 (q )
2J 1

JM

Up to now:
y/2….
Vibrational wave functions and energies (1)
No analytical solution
  2  2  2 J ( J  1)




V
(
R
)

E


  ( R)  0
2
2
2R
 2 R

Rigid rotator
 2 J ( J  1)
2 Re2
Useful approximations
Harmonic oscillator
1
V ( R)  De  k ( R  Re ) 2
2
EvJ  De  Be J ( J  1)  e (v  1 / 2)

 v ( R)  

2
1/ 4



2 v!
v
1 / 2
e
  2 ( R  Re ) 2 / 2
Equilibrium distance
2
Be 
; e 
2
2Re
k

H v  ( R  Re ) ;    e 
Vibrational wavefunctions and energies (2)
Deviation from the harmonic oscillator approximation:
Morse potential

V ( R)  De e 2 ( R  Re )  2e  ( R  Re )

1/ 2
 2D 
 e    e 
  
EvJ   De  Be J ( J  1)   e (v  1 2)   e xe (v  1 2) 2
xe 
Deviation from the rigid rotator approximation:
n

 2 J ( J  1)
1   Veff
~
Veff ( R)  V ( R) 
 Veff ( Re )  
2
n

2R
n  2 n!  R
 e
4 De

 Veff
~

( R  Re ) n ; 
 ~
 R
 R  Re


0
 R  R~e
EvJ  De  Bv J ( J  1)  De J 2 ( J  1) 2   e (v  1 2)   e xe (v  1 2) 2
Bv  Be   e (v  1 2)
4 Be3
De  2 2
 e
proof
Continuum states
Dissociation, fragmentation, collision…
 2 2

R   : 
 E   E ( R)  0
2
2
 2R R

Regular solution:
 E ( R)  CE sin( k E R  j E ) ; k E 
Influence of the potential
Normalization
2E
2
In wave numbers
 E ( R) 
2


sin( k E R  j E ) ;   E ( R)  E ( R)dR   (k E  k E )
0
In energy

2
 E ( R) 
sin( k E R  j E ) ;   E ( R)  E ( R)dR   ( E  E)
2
0
 k E
proof
Matrix elements of the rotational hamiltonian
2


1 2
i
  1 
 2   2
2 
 Lz cot q   2 Lx  Ly
 
 cot q
H r  H r  
2
2



q  sin q j 
2R q




i
i  1 

i
2 
1 2
2


 Lz cot q  
 2 Ly 
 2 Lx


L

L
2 
z
  sin q j 
 q
2R 

2




2
1  
 


L
J

L
J

L
J

L
J
x x
y y
2
2


Selection rule
  1(ou W  1)


Easy to evaluate in the BO basis:
L JM ou L S JMW
But in general, L and S are not
good quantum numbers…
…quantum chemistry is needed
Matrix elements of the vibrational hamiltonian
H vib
BO basis:
2  2 

R
2
2R R
R
H vib  v,v  Ev   vv  2
Vibrational
energy levels
2


 (r , R )
 

R

(  )
v
1 ( )
 v ( R)
R
 v( )
 2
2

 
R
2
R 2
 v( )  v( )
Interaction between
vibrational levels
Quantum chemistry is needed…