Transcript osu07.ppt

Applications of the Discrete Variable
Representation (DVR) for Modeling
Energy Levels of Alkali Dimer
Molecules
Tom Bergeman
SUNY Stony Brook
+ Many, many collaborators
Supported by NSF, ONR and ARO
A “New Paradigm” for
diatomic energy levels
Dunham/RKR → DVR + analytic potentials.
Yij’s → Short range + long range
potential functions
Vexch = q[I(n1.ℓ1,m1,n1’,ℓ1’,m1’|n2,ℓ2,m2,n2’,ℓ2’,m2’)]
The DVR method
•All mesh points used to obtain d2/dR2 – hence Ψ"
is as accurate as possible for a given mesh.
•Kinetic energy – full matrix over mesh points; potentials are
diagonal
•Hamiltonian matrix, n x n;
n = number of mesh points x number of channels.
•E(J,v) obtained by adding J(J+1)/R2 to potential –
hence no need to determine centrifugal distortion
parameters.
•Scaling function (Tiesinga et al., 1998) allows greater density
of points where potential is minimum.
•Basic reference: Colbert and Miller, JCP 96, 1982 (1992).
•TB thanks Paul Julienne and Barry Schneider for
introductions to DVR.
Applications of DVR
1. Analysis of new and previous data on the A
1Σ + and b 3Π states of Na (2007).
u
u
2
2. [As above, for K2 (2002).]
3. [As above, for RbCs (2003,+, incomplete).]
4. Analysis of photoassociation data on RbCs
(2004).
5. Photoassociation data from Rb2 (2006).
New Data on the Na2
A 1Σu+ State
Obtained by Peng Qi,
Jianmei Bai, Ergin Ahmed
and A. M. Lyyra, Temple
University, using subDoppler polarization
spectroscopy
Other co-authors:
S. Kotochigova, A. J.
Ross, C. Effantin, P.
Zalicki, J. Vigué. G.
Chawla, R. W. Field, T.-J.
Whang, W. C. Stwalley, H.
Knöckel, E. Tiemann, J.
Shang, L. Li, TB
Scheduled for publication in
J. Chem. Phys. July 07, ‘07
Summary of the data:
Potentials fitted to the data
Hamiltonian matrix:
Short range and long range potentials:
Spin-orbit
Functions
Relevant Potentials
Residuals from fit
Term
Values
Observed × and calculated •
term values less deperturbed
energy.
Dispersion terms, damping
(b) V3, V6, V8, and Vexch were fit to the data over the range R > 10.8 Å
(a) If these functions are extrapolated to smaller R, the “universal”
damping function, σn, such that ΔVn = σnVn, due to wavefunction
overlap must be considered. Here
σn = [1 – exp(-AnR/ρ-BnR2/ρ2)]n;
An = α0n-α1; Bn = β0exp(-β1n); αi, βi “universal” (from H2)
The exchange function, Vexch, for s + p atoms
(a) Top line: Vexch = Vdisp – Vsr
Top Line: Ratio a/c
(b) Middle line: (ab initio)(S. Kotochigova)
Bottom Line: Ratio b/c
(c)Bottom line: Bouty et al. (1995)
Conclusion: The standard theory gives lower
values for Vexch than empirical functions
Vexch
R(Angstroms)
The A 1Σ+ and b 3Π states of RbCs
A more dense spectrum, more severe
perturbations, less data available especially on
the b 3Π state. Results are of interest in efforts to
produce cold RbCs molecules (D. DeMille, Yale).
First report: TB, C. E. Fellows, R. F. Gutterres, C.
Amiot, PRA, 2003. (427 levels).
Additional data to a total of >1800 term values
more recently.
RbCs Data
Obtained by C. E.
Fellows
Solid lines: fitted rovibronic
structure
Variance vs. Te(b 3Π0)
RbCs Data
(cont’d)
Residuals from fit
Fraction of A 1Σ+ Character, J=0
Possible crossing with a b 3Π1 level
Photoassociation of cold Rb and
Cs atoms
Experimental data obtained by A. J. Kerman,
J. Sage, S. Sainis, and D. DeMille (Yale
University)
(Ultimately leading to the production of
cold RbCs molecules in the v=0 level of
the X 1Σ+ ground electronic state.)
Kerman et al., PRL 92, 033004; 153001 (2004); Sage et al.
PRL 94, 203001 (2005); TB et al., Eur. Phys. J. D 31, 179
(2004).
Photoassociation of RbCs*
Rb 5S1/2+ Cs 6P3/2
Rb 5S +
Cs 6P
Rb 5S1/2+Cs 6P1/2
Photoassociation
laser drives
free-bound
transition
RC
PA
Rb 5S + Cs 6S
molecules have ~same
translational temperature
as atoms
Condon points R C ~9-19Å
for detuning D from –9 to –80 cm-1
Excite at shorter range than homonuclear:
upper state potentials  r -6 rather than r -3
FC factors for free-bound PA transition
substantially smaller than for homonuclear
(at same detuning)
[Wang & Stwalley, JCP 108, 5767 (1998)]
Observed (at Yale) and Fitted B(v) Data
103 B(v) (cm-1)
Strong coupling between P1/2 and P3/2:
Weak coupling between P1/2 and P3/2:
Photoassociation of 85Rb Atoms into 0u+
States Near the 5S+5P Atomic Limits
Experimental data obtained by J. Qi, D.
Wang, Y. Huang, H. K. Pechkis, E. E.
Eyler, P. L. Gould, W. C. Stwalley, R.
A. Cline, J. D. Miller and D. J. Heinzen
TB et al., J. Phys. B 39, S813 (2006).
In this work, we adjust potentials and spin-orbit
functions to fit 0u+ band data below the Rb 5P1/2
and 5P3/2 limits
These potentials (before adjustments in the fit) were calculated by S. Lunell,
Uppsala, Sweden, and colleagues.
Photoassociation
data below the
85Rb 52S + 52P
1/2
Limit at 12579.00
cm-1(U. Conn.)
Trap loss spectra from a
dark SPOT (spontaneous
force optical trap).
Typical 0u+
bands
Asymptotic behavior with fine structure
Near the 52P dissociation limit, an asymptotic
expansion (Le Roy and Bernstein, 1970) applies:
vD – v = K C31/3[D-E(v)]1/6
However, here there is spin-orbit mixing:
V(1Σu+)
ΔΠΣ
H=
ΔΠΣ
V(3Π1u)-ΔΠΠ
So the asymptotic behavior becomes:
C (σ)
C (π)
V(1Σu+) E0 - 3 ; V(3Π1u) E0 - R33 ; C3(σ) = - 2C3(π)
R3
V(P1/2) →E0(P1/2) –
Thus
4C3(π)
3R3
V(P3/2) →E0(P3/2) - 5C3(π)
3R3
Fit of Rb2 0u+ levels below the 5S+5P1/2
limit to the Le Roy-Bernstein expression
v* = (v-vD) vs (energy)1/6 gives roughly a straight line,
but there are systematic deviations (in red).
A coupled channels approach is need to explain the residuals
The observed and fitted B(v) values exhibit the effects
of coupling between 0u+ states tending to the P1/2 and
P3/2 limits.
Maxima in the B(v) function indicate states with largest
“P3/2” character.
Conclusions
The DVR numerical method, with analytic
potential functions, is able to quantitatively
model diatomic energy levels, for single
channel, coupled channel and levels near
dissociation limits.
For the heavier alkali dimers, data is
presently inadequate for a detailed model
of the lowest excited states.
The A 1Σu+ and b 3Πu states of K2
M. R. Manaa, A. J.
Ross, F. Martin, P.
Crozet, A. M. Lyyra, L.
Li, C. Amiot, TB
(J. Chem. Phys.,2002)
Data less complete than
for Na2!
Recently, St. Falke, I.
Sherstov, E. Tiemann,
and C. Lisdat (J.
Chem. Phys. 2006)
have presented data
from very near the
dissociation limit.
K2 A and b state potentials and
typical term values
Photoassociation data
below the 52P3/2 limit at
12816.603 cm-1
(U. Texas; Cline, Miller
and Heinzen PRL 1994)
Trap loss spectra from a FORT
plus 1 MHz bandwidth laser.
All levels are broadened by
predissociation.
0u+ resonances
shaded.
Observed photoassociation spectra
RbCs and Cs2 rotational structure
(Ω = 0)
RbCs rotational + hyperfine structure
(Ω = 1,2)
•up to 70% depletion of Rb trap
•all observed lines can be saturated
•narrowest lines have G ~ 10 MHz
 dominated by radiative decay
All data indicate high rates
of molecule formation
Heavier alkali
dimers: NaRb