Of Cold Molecules

1
Of Cold Molecules
The Molecular Side
Olivier Dulieu Laboratoire Aimé Cotton, CNRS,
Campus dOrsay, Orsay, France olivier.dulieu_at_lac.u
-psud.fr
COMOLHPRN-CT-2002-00290
2
COMOL
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Molecules are really cool!

• The subtle flirtation of ultracold atoms
  ,Science, 280, 200 (1998) If high-energy
  accelarators make the rap music of physics-with
  their whirling particles and rapid-free smashups,
  then collisions between ultracold atoms are its
  Wagnerian opera
• Molecules are cool , J. DoyleB. Friedrich,
  Nature 401, 749 (1999) Under ordinary
  conditions, atoms and molecules of a gas zigzag
  in all directionsand are most likely to move
  at the speed of riffle bullets The emergence of
  methods for slowing and trapping gaseous species
  has lead to a renaissance in atomic physics,
  which is now progressing into molecular/chemical
  physics as well Just as atom cooling is opening
  up new avenues of research, it is likely that the
  same will happen with molecular cooling with
  repercussions for chemistry, and even, perhaps,
  in biology.
• Hot prospects for ultracold molecules , B.
  Goss Levi, Physics Today, Sept. 2000, p.46
• Quantum encounters of the cold kind , K.
  Burnett, P.D. Lett, E. Tiesinga, P.Julienne, C.J.
  Williams, Nature 416, 225 (2002) We have
  already seen our dreams of controlling
  interactions on the quantum level come true, and
  the exquisite nature of this control has proved
  remarkable. These achievements have come from
  experimental and theoretical developments that
  have been a joy to be involved in, and their
  impact on new physics, chemistry and quantum
  computation has only just begun.
• Really cool molecules , P. S. Julienne,
  Nature, 424, 24 (2003)

4
(Envisioned?) Applications of cold molecules

• Ultra-high resolution spectroscopy ?
• Test of fundamental theories ?
• Superchemistry ?
• Ultracold Photochemistry ?
• Quantum properties, BCS, superfluidity
• Ultimate control of reactive collisions
• Quantum information
• Biology

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Which properties do characterize a molecule?
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Excited state
Chemical bond
Ground state
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How to get them?
Ask your favorite molecular spectroscopist
because he/she seems to speak like an
experimentalist Or
Ask your favorite theoretician, who may happen to
be a quantum chemist because you have no other
solution!
In both cases, you should be prepared to
understand what he/she is going to talk about
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• 1- Introduction, overview
• 2- Molecular spectroscopy
• 3- Asymptotic approach for long-range properties
• 4- Hamiltonian of a diatomic molecule, Molecular
  symmetries Hunds cases Basics of quantum
  chemistry

9
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

10
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

12
Ex 1

• 3580 transitions resulting in 924 levels

13
Ex 1

• 3580 transitions resulting in 924 levels

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Ex 1

• 3580 transitions resulting in 924 levels

15
Ex 2
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Dunham expansion for energy levels

• The energy levels of a rotating vibrator , J.
  L. Dunham, Phys. Rev. 41, 721 (1932)

Anharmonic oscillator
Energy levels term energies
Non-rigid rotator (Herzberg 1950)
Rotational constant
Centrifugal distorsion constant (CDC)
Coupled to each other
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Dunham expansion (2)

• Dunham coefficients

Note zero-point energy correction
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Determination of the Dunham coefficients
N measured term energies M 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

19

• 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

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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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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)

Bohr-Sommerfeld quantification for a particle
with mass m in a potential V
Classical inner and outer turning points
RKR-1
inversion
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RKR approach (2)
inversion
RKR-2
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RKR potential curve
RKR-2
RKR-1

• Use Gv and Bv from experiment, Dunham expansion
• Extract a set of turning point for all energies
• Specific codes (Le Roys 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 )
25
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) Gv and Bv with an expansion
  incorporating the long-range behavior of the
  potential (Cn/Rn)

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
outer Padé expression
New input for RKR analysis
26
Ex
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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 DV(R)V(R)-V(0)(R) as a perturbation
HH(0)DV(R).
Expansion
Modified energies
Zero-order eigenfunctions
Generally over-determined Least-square fit
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IPA (2)
Cut-off function
Legendre polynomials

• Choice of basis functions

Functional relation, useful for strongly
anharmonic potentials
Inner turning point
Outer turning point
Equlibrium distance
Standard error on ci, through the covariance
matrix
New determination of Gv, Bv No unique solution
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IPA example

• C.R. Vidal, Comments At. Mol. Phys. 17, 173 (1986)

Energy differences
RKR
IPA
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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 Roys group

Y. Huang 2000, Chemical Physics Research Report
649, University of Waterloo.
simple
Morse family
generalized
extended
modified
Modified Lennard-Jones Better 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)

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 example
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DPFExample

• 3580 transitions resulting in 924 levels

Short distances
Large distances
Note 1st estimate for the Ca scattering length
35

• 1- Introduction, overview
• 2- Molecular spectroscopy
• 3- Asymptotic approach for long-range properties
• 4- Hamiltonian of a diatomic molecule, Molecular
  symmetries Hunds cases Basics of quantum
  chemistry

36
Very elaborated calculations

• Bussery, Aubert-Frécon, JCP 82, 3224 (1985)
• Marinescu, Sadeghpour, Dalgarno, PRA 49, 982
  (1994)
• Patil, Tang, JCP 106, 2298 (1997)
• Derevianko, Johnson, Safronova, Babb, PRL 82,
  3589 (1999)
• Derevianko, Babb, Dalgarno, PRA 63, 052704 (2001)
• Derevianko, Porsev, PRA 65, 053403 (2002)

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General formalism
Asymptotic expansion of the electrostatic
interaction between two ground state atoms
2l-pole dynamic polarizability w0 static pol.
Electric multipole operators
Sum over a complete set of atomic states
(N electrons)
Accurate determination of atomic wave functions
and energies needed
Spherical harmonics
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Most advanced and accurate developments by
Derevianko and coll

• Separation of atomic states in summations

Excitation of valence electrons
of core electrons
of core electrons tooccupied valence states

• Relativistic many-body perturbation theory

Relativistic frozen core Hamiltonian
Corrections due to core excitations core
polarization term
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Typical results

• Contribution of core polarization term to
  4 (Li), 38 (Cs)
• Contribution of to the total 0.01
  (Li), 0.8 (Cs)
• Estimated precision on 0.3 (Li), 3.7
  (Cs)
• C8 0.5 (Li), 4 (Cs)
• C10 10
• Contribution of to C8 0.2 (Li), 10 (Cs)

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• 1- Introduction, overview
• 2- Molecular spectroscopy
• 3- Asymptotic approach for long-range properties
• 4- Hamiltonian of a diatomic molecule, Molecular
  symmetries Hunds cases Basics of quantum
  chemistry

41
Main steps

• Definition of the exact Hamiltonian
• Definition of a complete set of basis functions
• Matrix representation of finite
  dimensionperturbations
• Comparison to observations to determine molecular
  parameters

42
Non-relativistic Hamiltonian for 2 nuclei and n
electrons in the lab-fixed frame

• General problem
• Leaads to a non separable equation due to
  interactions
• Born-Oppenheimer approximation nuclei/electrons
• Configuration Interaction electronic correlation

electrons
nuclei
with
n-n
e-e
e-n
and
Relative distances
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Separation of center-of-mass motion

• Originmidpoint of the axis ?center of mass
• Change of variables

Total mass
Reduced mass
for homonuclear molecules
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Hamiltonian in new coordinates
Radial relative motion
Hvib HrotHCoriolis
Electronic Hamiltonian
He

• Kinetic couplings ? m/m
• Isotopic effect
• Origin?center of mass

He
Study of the internal Hamiltonian
Center-of-mass motion
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Born-Oppenheimer approximation

• HHeHeHvibHrotHCoriolis.

m/mgt1800 approximate separation of
electron/nuclei motion
Potential curves
BO or adiabatic approximation factorization of
the total wave function
Mean potential
All act on the electronic wave function
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General form of the electronic solution
Slater determinant eigenfunction of He,
neglecting e-e interaction

• Interaction of configurations built on Slater
  determinants for the electronic correlation
• Two options ab initio or semi-empirical
  approaches
• Here treatment of small systems, ie small number
  of electrons
• Semi-empirical approach polarizable ionic core
  valence electrons
• Atomic calculation definition of the core
  potential, atomic orbital basis
• Molecular Hartree-Fock calculation, molecular
  orbitals
• Configuration Interaction
• CIPSI package from the quantum chemistry group in
  Toulouse (coll. P. Millié, F. Spiegelmann)

47
Atomic calculation effective potential for the
core representation

• Empirical Model Potential (Klapisch 1969), with
  parameters fitted on the atomic experimental
  spectrum
• Ab-initio pseudo-potential (Barthelat, Durand
  1975), with parameters fitted on the
  full-electron Hartree-Fock wave functions outside
  the core

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Core polarization representation

• BottcherDalgarno (1966), Foucrault, Millié,
  Daudey (1992)
• Electric field seen by a core embedded in a
  molecular system,due to valence electrons and
  other cores
• Induced dipole moment on the core l
• Core Polarization Potential
• cut-off functions, with l-dependent cut-off
  radii fitted on the atomic spectrum

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Self-consistent field approximation Molecular
Hartree-Fock calculation

• Partly includes e-e correlation as a mean field
  contribution, maintaining then an orbital product
  description of the wavefunction.
• Fock operatoreffective one-electron hamiltonian
• Starting from a large set of trial orbitals
  (built from gaussian atomic orbital centered at
  each core), iterative resolution of the
  Hartree-Fock equation for each electron (ie
  spectrum of the Fock operator), to get the HF
  potential, until convergence self-consistent
  field .
• Hartree-Fock ground state F0gt Slater
  determinant built from the Nv orbitals with
  lowest energies (occupied orbitals)

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Configuration interaction

• Determination of the correlation energy
• Diagonalization of He in a space of
  configurations obtained from F0gt by promoting an
  electron in an unoccupied orbital.
• Full CI configurations for N electrons,
  from K molecular orbitals complete active space
  (CAS)
• Tractable only for 2-3 electrons.
• Potential energy curves

From Vcpp
Empirical terms Pavolini et al, JPB 22, 1721
(1989)
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(No Transcript)
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Example GTOs for alkali atoms

• Updated static polarizabilities
• Careful adjustments of cut-off radii
• Systematic calculations upon request for all
  alkali pairs, through an automated procedure

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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

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Symmetry properties of electronic functions (1)
spin
Planar symmetry

• Axial symmetry 2p rotation

Central symmetry
gerade
ungerade
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Symmetry properties of electronic functions (2)

• is not a good quantum number (precession around
  the axis)
• is a good quantum number if electrostatic
  interaction is dominant

Ex
2S1 multiplicity
S states spin fixed in space, 2S1 degenerate
components L states precession around the axis,
multiplet structure, almost equidistant in energy
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Symmetry properties of electronic functions (3)
Otherwise
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References

• H. Lefebvre-BrionR.W. Field, The Spectra and
  Dynamics of Diatomic Molecules , Elsevier
  Academic Press, 2004
• G. Herzberg, The Spectra od Diatomic Molecules
  , Van Nostrand-Reinhold, Princeton, 1950,
  reprinted in 1989 by Krieger, Malabar.
• P.R. BunkerP. Jensen, Fundamentals of
  molecular symmetry , IOP series in Chemical
  Physics, 2005.
• Tutorials from G. Amat (Paris), A. Beswick
  (Orsay), C. Jungen (Orsay)
• Bibliographic Databases
• DiRef a bibliographic database for diatomic
  molecules, J. Mol. Spectrosc. 207, 287 (2001)
  http//diref.uwaterloo.ca
• Cold Molecules http//www.lac.u-psud.fr/coldmolec
  ules, in progress