1- Introduction, overview 2- Hamiltonian of a diatomic molecule
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Transcript of 1- Introduction, overview 2- Hamiltonian of a diatomic molecule
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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) collisionsOlivier Dulieu
Predoc’ school, Les Houches,september 2004
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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
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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)
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Ex 1:
3580 transitions resulting in 924 levels
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Ex 1:
3580 transitions resulting in 924 levels
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Ex 1:
3580 transitions resulting in 924 levels
![Page 7: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.fdocuments.in/reader035/viewer/2022062517/56812f54550346895d94e52a/html5/thumbnails/7.jpg)
Ex 1:
3580 transitions resulting in 924 levels
![Page 8: 1- Introduction, overview 2- Hamiltonian of a diatomic molecule](https://reader035.fdocuments.in/reader035/viewer/2022062517/56812f54550346895d94e52a/html5/thumbnails/8.jpg)
Ex 2:
ns
ns
p
p
)2(89.32
)3(462.30
2/1
2/3
6
6
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Ex 3:
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Ex 3:
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Ex 3:
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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…
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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
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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
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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(
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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
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;...)(;)(
21)1()(
32
)(
0
max
vv
vl
l
llm
mm
HvKDvK
vYJJvK
Centrifugal distorsion constants
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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
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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
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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 )
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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
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Ex:
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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
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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
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IPA: exampleC.R. Vidal, Comments At. Mol. Phys. 17, 173 (1986)
RKR
IPA
Energy differences
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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
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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
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Dunham/RKRNDE/IPA: example
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DPF: exampl
e
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DPF:Example
:
3580 transitions resulting in 924 levels
Short distances
Large distances
Note: 1st estimate for the Ca scattering length