• 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

Inversion of spectroscopic data to extract molecular potential curves

• Motivations • Apetizer: some examples • Rotating vibrator (or vibrating rotor!): Dunham expansion • RKR : semiclassical approach • NDE : towards the asymptotic limit • IPA : perturbative approach • DPF : brute force approach • Applications

Motivations

• Analysis of light/matter interaction • Gigantic amount of data: synthesis required • Yields informations on internal structure • Starting point: Born-Oppenheimer approximation • Other perturbations • Cold atoms: scattering length determination • Combined analysis with (less accurate) quantum chemistry calculations • Elaborate and efficient tools required • High resolution (on energies)

Ex 1: 3580 transitions resulting in 924 levels

Ex 1: 3580 transitions resulting in 924 levels

Ex 1: 3580 transitions resulting in 924 levels

Ex 1: 3580 transitions resulting in 924 levels

Ex 2:

 6

p

3 / 2  6

p

1 / 2  30 .

462 ( 3 )

ns

 32 .

89 ( 2 )

ns

Ex 3:

Ex 3:

Ex 3:

Dunham expansion for energy levels

« The energy levels of a rotating vibrator », J. L. Dunham, Phys. Rev. 41, 721 (1932) Anharmonic oscillator

V

  (

r



r e

) 2   (

r



r e

) 3 Energy levels: « term energies »

G

(

v

)  

e

(

v

 1 2 )  

e x e

(

v

 1 2 ) 2  

e y e

(

v

 1 2 ) 2  ...

Non-rigid rotator (Herzberg 1950)

F

(

J

) 

BJ

(

J

 1 ) 

DJ

2 (

J

 1 ) 2  ...

Rotational constant

B e

  2 2 

R e

2 Centrifugal distorsion constant (CDC)

D

 4

B

3  2 Coupled to each other…

B v

 1 /

R

2

Dunham expansion (2)

B v D v

 

B e D e

 

e

 

e

(

v

(

v

 1  1 2 ) 2 )   ...

...

T



G

(

v

) 

F v

(

J

)  

e

(

v

 1 2 )  

e x e

(

v

 1 2 ) 2  

e y e

(

v

 1 2 ) 2  ...

B v J

(

J

 1 ) 

D v J

2 (

J

 1 ) 2  ...

Y

10

Y

20

Y

30    

e



e



e x e y e T



l

, 

m Y lm

(

v

 1 / 2 )

l J m

(

J

 1 )

m

Dunham coefficients

Y

01

Y

02

Y

11 

B e

 

D e



e

...

Y

00 

B e

Note: zero-point energy correction 4  

e

12

B e

 

e



e

2 144

B e

3  

e x e

4

Determination of the Dunham coefficients

N M

measured term energies Dunham coefficients to fit

C. Amiot and O. Dulieu, 2002, J. Chem. Phys. 117, 5155

Minimization of the reduced standard error (dimensionless) by adjustment on measured term energies      

N

1 

M i N

  1   

y calc

(

i

) 

u

(

i

)

y obs

(

i

)    2     1 / 2

47 16900 obtained by analysis of 348 Dunham coefficients to represent transitions, fluorescence series excited with 21 wave lengths r.m.s = 0.0011cm

-1 (   1 .

05 )

Dunham expansion: summary

• Compact, accurate, empirical representation of a large number of energies • Not suitable for extrapolation at large distances • Not suitable for extrapolation at high J, for heavy molecules • High-order coefficients highly correlated , and not physically meaningful • No information on the molecular structure

Centrifugal distorsion constants

K m

(

v

)

K

2 (

v

)   

J

(

J



D v

;  1 ) 

m l

max

l

  ( 0

v

)

Y lm



v K

3 (

v

) 

H v

;...

 1 2 

l

RKR: Rydberg-Klein-Rees analysis (1)

R. Rydberg, Z. Phys. 73, 376 (1931); Z. Phys. 80, 514O (1933) Klein, Z. Phys. 76, 226 (1932); A. L. G. Rees, Proc. Phys. Soc. London 59, 998 (1947) Bohr-Sommerfeld quantification for a particle with mass  in a potential

V v

 1 2  2   

R

1

R

2

dR



E vJ



V J

(

R

)  1 / 2 Classical inner and outer turning points

dv dE

 2 2   

R

1

R

2 

E vJ dR



V J

(

R

)  1 / 2 inversion

R

1 (

v

) 

R

2 (

v

)  2 2  

v v

0 

G v



dv

'

G v

'  1 / 2

E

(

v

0 )  0

RKR-1

v

 1 2 

RKR approach (2)

2   

R

1

R

2

dR



E vJ



V J

(

R

)  1 / 2

V J

(

R

) 

V

(

R

)  

J

(

J

 1 ) 2 

R

2

B v

 

E

(

v

,

J

) /  

J

(

J

 1 ) 

J

 0

dv B v dE

  2  1 2  

R

2

R

1

R

2 

E vJ dR



V J

(

R

)  1 / 2 inversion 1

R

1 (

v

)  1

R

2 (

v

)  2 2  

v v

0 

G v B v

 '

dv

'

G v

'  1 / 2

RKR-2

RKR-1

R

1 (

v

) 

R

2 (

v

) 

RKR potential curve

2 2  

v v

0 

G v



dv

'

G v

'  1 / 2 1

R

1 (

v

)  1

R

2 (

v

)  2

RKR-2

2  

v v

0 

G v B v

 '

dv

'

G v

'  1 / 2 • Use

G v

and

B v

from experiment, Dunham expansion… • Extract a set of turning point for all energies • Specific codes (Le Roy’s group, U. Waterloo, Canada) • Limitations: smooth functions of v, poor extrapolation high v, or large distances Note: extension with 3rd order quantification: ( C. Schwartz and R. J. Le Roy 1984 J. Chem. Phys. 81, 3996

v

 1 2  2   

R

2

R

1

dR



E vJ



V J

(

R

)  1 / 2  96  1 2    ) 

E vJ dR V J

 ( 

V J

(

R

)

R

)  3 / 2

Near-dissociation expansion (NDE)

C. L. Beckel, R. B. Kwong, A. R. Hashemi-Attar, and R. J. Le Roy 1984 J. Chem. Phys. 81, 66

Fit (a subset of)

G v

and

B v

with an expansion incorporating the long-range behavior of the potential

(C n /R n )



K m

(

v

) 

X m

(

n

) 

X m

(

n

)(

v D

 

X m n C n

2 (  1

n

/(

n

)  2 ) 

v

) ( 2

n

/(

n

 2 )  2

m

)

G

 (

v

) 

B

 (

v

) 

K

0  (

v

)

K

1  (

v

)  

D



X

1 (

n X

0 (

n

)(

v D

)(

v D



v

) 2

n

/(

n

 2 ) 

v

) [ 2

n

/(

n

 2 )  2 ] R.J. Le Roy, R.B. Bernstein, J. Chem. Phys. 52, 3869 (1970) W.C. Stwalley, Chem. Phys. Lett. 6, 241 (1970); J. Chem. Phys. 58, 3867 (1973).

More elaborate form, for more flexibility

G NDE

(

v

) 

D



X

0 (

n

)(

v D



v

) 2

n

/(

n

 2 ) 

L

/

M

 

L M

 

P L Q M

New input for RKR analysis

P L

 1 

i L

  1

p i z

 

i

 1

Q M

 1 

j M

  1

q j z

 

j

 1

z

 (

v D



v

) 

Ex:

IPA: Inverted perturbation approach (1)

R. J. Le Roy and J. van Kranendonk 1974 J. Chem. Phys. 61, 4750 W. M. Kosman and J. Hinze 1975 J. Mol. Spectrosc. 56, 93 C. R. Vidal and H. Scheingraber 1977 J. Mol. Spectrosc. 65, 46.

Adjust an effective potential on experimental energies, no Dunham expansion Good initial approximation: RKR potential

V (0) (R)

.

Treat 

V(R)=V(R)-V (0) (R)

as a perturbation:

H=H (0)

+



V(R)

.

Expansion: 

V

(

R

)  

i c i



i

(

R

) Modified energies 

E vJ



E vJ

 (

E vJ

0 )   ( 0 )

vJ



V

(

R

)  ( 0 )

vJ

 

i c i

 ( 0 )

vJ



i

 ( 0 )

vJ

Zero-order eigenfunctions Generally over-determined Least-square fit

IPA (2)

Legendre polynomials Choice of basis functions: Cut-off functio n 

i

(

R

) 

P i

(

x

) exp( 

x

2

m

) Functional relation, useful for strongly anharmonic potentials Inner turning point Outer turning point Equlibrium distance Standard error on

c i

, through the covariance matrix

x

 (

R i



E j



R o

 

i c i X ij

( )(

R R

 

R e R e

)( )

R i

 

R

2

R i o R o

)  2

R e R R



R i R



R o R



R e

  

x x x

   1  0 1 New determination of No unique solution

G v , B v

IPA: example C.R. Vidal, Comments At. Mol. Phys. 17, 173 (1986) Energy differences RKR IPA

   

DPF: Direct potential fit (1)

Generalization of IPA approach Choose an analytical function to be fitted on experimental energies Need a good initial potential Package available: DSPotFit, from Le Roy’s group Morse family generalized modified

V MMO

( Y. Huang 2000, Chemical Physics Research Report 649, University of Waterloo.

simple

V SMO

(

R

) 

D e

 1 

e

 

M

(

R



R e

)  2

V GMO R

) 

D e

(

R

 1 )  

e

 1  

D e



e

  1

MMO

  ( 

MMO z

)

e z

 2    2

GMO

(

R

)(

R



R e V EMO

) 2  (

R

) 

D e

1  extended 

e

 

EMO

(

z

)(

R



R e

) 2 

V

Modified Lennard-Jones Better asymptotic behavior

MLJ

(

R

) 

D e

   1 

R R e n e

 

MLJ

(

z

)

z

   General power expansion

U G

(

R

) 

i n

  0

a i

 

R



R m R



bR m

 

i

DPF (2)

0 2 )

V LR

(

R

) 

n

 3 ,  6 , 8 ,...

C n R n



V exch

(See lecture on photoassociation) References: SMO: P. M. Morse 1929 Phys. Rev. 54, 57 GMO: J. A. Coxon and P. J. Hajigeorgiou 1991 J. Mol. Spectrosc. 150, 1 MMO: H. G. Hedderich, M. Dulick, and P. F. Bernath 1993, J. Chem. Phys. 99, 8363 EMO: E. G. Lee, J. Y. Seto, T. Hirao, P. F. Bernath, and R. J. Le Roy 1999 J. Mol. Spectrosc. 194, 197 MLJ: P. G. Hajigeorgiou and R. J. Le Roy 2000, J. Chem. Phys. 112, 3949 G: C. Samuelis, E. Tiesinga, T. Laue, M. Elbs, H. Knöckel, and E. Tiemann 2000, Phys. Rev. A, 63, 012710

Dunham/RKRNDE/IPA: example

DPF: example

DPF: Example: 3580 transitions resulting in 924 levels Short distances Large distances Note: 1 the Ca scattering length st estimate for