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 V exch = q[I(n 1.ℓ 1,m 1,n 1 ’,ℓ 1 ’,m 1 ’|n 2,ℓ 2,m 2,n 2 ’,ℓ 2 ’,m 2 ’)]

The DVR method All mesh points used to obtain d 2 /dR 2 – 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)/R 2 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 1 Σ u + and b 3 Π u states of Na 2 (2007). 2.[As above, for K 2 (2002).] 3.[As above, for RbCs (2003,+, incomplete).] 4.Analysis of photoassociation data on RbCs (2004). 5.Photoassociation data from Rb 2 (2006).

New Data on the Na 2 A 1 Σ u + State Obtained by Peng Qi, Jianmei Bai, Ergin Ahmed and A. M. Lyyra, Temple University, using sub- Doppler 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 Summary of the data: Scheduled for publication in J. Chem. Phys. July 07, ‘07

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) V 3, V 6, V 8, and V exch 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, σ n, such that ΔV n = σ n V n, due to wavefunction overlap must be considered. Here σ n = [1 – exp(-A n R/ρ-B n R 2 /ρ 2 )] n ; A n = α 0 n -α1 ; B n = β 0 exp(-β 1 n); α i, β i “universal” (from H 2 )

The exchange function, V exch, for s + p atoms (a) Top line: V exch = V disp – V sr (b) Middle line: (ab initio)(S. Kotochigova) (c)Bottom line: Bouty et al. (1995) V exch Top Line: Ratio a/c Bottom Line: Ratio b/c R(Angstroms) Conclusion: The standard theory gives lower values for V exch than empirical functions

The A 1 Σ + and b 3 Π states of RbCs A more dense spectrum, more severe perturbations, less data available especially on the b 3 Π 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, (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 3 Π 0 )

RbCs Data (cont’d) Residuals from fit Fraction of A 1 Σ + Character, J=0 Possible crossing with a b 3 Π 1 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, ; (2004); Sage et al. PRL 94, (2005); TB et al., Eur. Phys. J. D 31, 179 (2004).

Photoassociation of RbCs* Rb 5S + Cs 6P Rb 5S + Cs 6S Photoassociation laser drives free-bound transition molecules have ~same translational temperature as atoms PA Condon points R C ~9-19 Å for detuning  from –9 to –80 cm -1 RCRC Rb 5S 1/2 +Cs 6P 1/2 Rb 5S 1/2 + Cs 6P 3/2 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 Strong coupling between P 1/2 and P 3/2 : Weak coupling between P 1/2 and P 3/2 : 10 3 B(v) (cm -1 )

Photoassociation of 85 Rb Atoms into 0 u + 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 0 u + band data below the Rb 5P 1/2 and 5P 3/2 limits These potentials (before adjustments in the fit) were calculated by S. Lunell, Uppsala, Sweden, and colleagues.

Photoassociation data below the 85 Rb 5 2 S P 1/2 Limit at cm -1 (U. Conn.) Typical 0 u + bands Trap loss spectra from a dark SPOT (spontaneous force optical trap).

Asymptotic behavior with fine structure Near the 5 2 P dissociation limit, an asymptotic expansion (Le Roy and Bernstein, 1970) applies: v D – v = K C 3 1/3 [D-E(v)] 1/6 However, here there is spin-orbit mixing: V( 1 Σ u + ) Δ ΠΣ Δ ΠΣ V( 3 Π 1u )-Δ ΠΠ So the asymptotic behavior becomes: V( 1 Σ u + ) E 0 - ; V( 3 Π 1u ) E 0 - ; C 3 (σ) = - 2C 3 (π) H = C3(σ)C3(σ) R3R3 C3(π)C3(π) R3R3 Thus V(P 1/2 ) →E 0 (P 1/2 ) – V(P 3/2 ) →E 0 (P 3/2 ) - 4C 3 (π) 3R 3 5C 3 (π) 3R 3

Fit of Rb 2 0 u + levels below the 5S+5P 1/2 limit to the Le Roy-Bernstein expression v* = (v-v D ) 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 0 u + states tending to the P 1/2 and P 3/2 limits. Maxima in the B(v) function indicate states with largest “P 3/2 ” 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 1 Σ u + and b 3 Π u states of K 2 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 Na 2 ! Recently, St. Falke, I. Sherstov, E. Tiemann, and C. Lisdat (J. Chem. Phys. 2006) have presented data from very near the dissociation limit.

K 2 A and b state potentials and typical term values

Photoassociation data below the 5 2 P 3/2 limit at cm-1 (U. Texas; Cline, Miller and Heinzen PRL 1994) 0 u + resonances shaded. All levels are broadened by predissociation. Trap loss spectra from a FORT plus 1 MHz bandwidth laser.

Observed photoassociation spectra RbCs and Cs 2 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  ~ 10 MHz  dominated by radiative decay All data indicate high rates of molecule formation

Heavier alkali dimers: NaRb