• 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) collisionsOlivier 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:

ns

ns

p

p

)2(89.32

)3(462.30

2/1

2/3

6

6

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)

32 )()( ee rrrrV Anharmonic oscillator

...)21()21()21()( 22 vyvxvvG eeeee Energy levels: « term energies »

Non-rigid rotator (Herzberg 1950)

...)1()1()( 22 JDJJBJJF

Centrifugal distorsion constant (CDC)

234 BD

Rotational constant

22 2 ee RB

Coupled to each other…

Dunham expansion (2)

Dunham coefficients

2/1 RBv ...)21(

...)21(

vDD

vBB

eev

eev

...)1()1(...)21()21()21(

)()(2222

JJDJJBvyvxv

JFvGT

vveeeee

v

ml

mmllm JJvYT

,

)1()2/1(

...

1130

0220

0110

eee

eee

ee

YyY

DYxY

BYY

4144124 3200 eeeeeeee xBBBY

Note: zero-point energy correction

Determination of the Dunham coefficients

Minimization of the reduced standard error (dimensionless) by adjustment on measured term energies

2/12

1 )(

)()(1

N

i

obscalc

iu

iyiy

MN

N measured term energiesM Dunham coefficients to fit

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

47 Dunham coefficients to represent

16900 transitions, obtained by analysis of 348 fluorescence series excited with 21 wave lengths

r.m.s = 0.0011cm-1

)05.1(

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

;...)(;)(

21)1()(

32

)(

0

max

vv

vl

l

llm

mm

HvKDvK

vYJJvK

Centrifugal distorsion constants

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)

2

1

2/1)(2

2

1 R

R JvJ RVEdRv

Bohr-Sommerfeld quantification for a particle with mass in a potential V

Classical inner and outer turning points

2

12/1)(2

2 R

RJvJ RVE

dR

dE

dv

v

vvv GG

dvvRvR

0 2/1'

21

'

2

2)()(

inversion

0)( 0 vE

RKR-1

RKR approach (2)

2

1

2/1)(2

2

1 R

R JvJ RVEdRv

2

1 2/12 )(22

1 R

RJvJ

vRVER

dR

dE

dvB

v

vvv

v

GG

dvB

vRvR 0 2/1'

'

21

'22

)(

1

)(

1

0)1(/),( Jv JJJvEB

22)1()()( RJJRVRVJ

inversion

RKR-2

RKR potential curve

• Use Gv and Bv 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

v

vvv GG

dvvRvR

0 2/1'

21

'

2

2)()(

RKR-1

v

vvv

v

GG

dvB

vRvR 0 2/1'

'

21

'22

)(

1

)(

1

RKR-2

2/3

2/1

)(

)(

296

1)(

2

2

1 2

1 RVE

RVdRRVEdRv

JvJ

JR

R JvJ

Note: extension with 3rd order quantification: (C. Schwartz and R. J. Le Roy 1984 J. Chem. Phys. 81, 3996 )

Near-dissociation expansion (NDE)

Fit (a subset of) Gv and Bv with an expansion incorporating the

long-range behavior of the potential (Cn/Rn)

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

)2)2/(2())(()( mnnDmm vvnXvK

)2/(12

)()( n

nn

mm

C

nXnX

]2)2/(2[11

)2/(200

))(()()(

))(()()(

nn

D

nnD

vvnXvKvB

vvnXDvKvG

MLvvnXDvG nnD

NDE /))(()( )2/(20

M

j

jjM

L

i

iiL

zqQ

zpP

1

1

1

1

1

1

More elaborate form, for more flexibility

« outer Padé expression » ML QPML

)( vvz D

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).

New input for RKR analysis

Ex:

IPA: Inverted perturbation approach (1)R. J. Le Roy and J. van Kranendonk 1974 J. Chem. Phys. 61, 4750W. M. Kosman and J. Hinze 1975 J. Mol. Spectrosc. 56, 93C. R. Vidal and H. Scheingraber 1977 J. Mol. Spectrosc. 65, 46.

Expansion: i

ii RcRV )()(

i

vJivJivJvJvJvJvJ cRVEEE )0()0()0()0()0( )(

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).

Modified energies

Zero-order eigenfunctions Generally over-determinedLeast-square fit

IPA (2)

Choice of basis functions: )exp()()( 2mii xxPR

Legendre polynomials

Cut-off functio

n

RRRRRRRR

RRRRx

eoieoi

oie

22))((

))((

Functional relation, useful for strongly anharmonic potentials

Inner turning point

Outer turning point

Equlibrium distance 0

1

1

xRR

xRR

xRR

e

o

i

New determination of Gv, Bv

No unique solution

i

ijij XcEStandard error on ci,

through the covariance matrix

IPA: exampleC.R. Vidal, Comments At. Mol. Phys. 17, 173 (1986)

RKR

IPA

Energy differences

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

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

2)(1)( eM RReSMO eDRV

2))((1)( eGMO RRReGMO eDRV

2

2)(

1

1)(

MMO

MMO

e

eDRV

zz

eMMO

2))((1)( eEMO RRzeEMO eDRV

zz

ne

eMLJMLJe

R

RDRV )(1)(

in

i m

miG bRR

RRaRU

0

)(

Morse family simple

generalized

modified

extended

Modified Lennard-JonesBetter asymptotic behavior

General power expansion

DPF (2)

Pure long-range states in alkali dimers (e.g. double-well state in Cs2)

,...8,6,3

)(n

exchnn

LR VR

CRV

g0

(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: exampl

e

DPF:Example

:

3580 transitions resulting in 924 levels

Short distances

Large distances

Note: 1st estimate for the Ca scattering length