2. - compact representation of large (unlimited) bodies of highly precise multi-isotopologue data - reliable extrapolation of V n (r) at large-r in accord with known dispersion coefficients (C n ) defining the potential at long-range - Parameters of V n (r) are adjusted such that the quantum mechanical eigenvalues reproduce the observed line positions Attributes of the DPF Approach for Diatomics - Hamiltonian H ni (r) for each state (n) and each isotopologue (i) contains a fully analytic potential V n (r) over entire radial domain (cf. IPA method) - fragmentary data, or data over small J­range (even for very large J) are easily and equally exploited - absence of non-physical behavior when extrapolated to small-r

3. - account of Born-Oppenheimer breakdown (BOB), if necessary - simultaneous fits for 2 (or more) states possible for traditional electronic band data - simultaneous fit of upper state terms and lower state potential for fluorescence data (or if perturbations present in one state) - extension possible to include fine structure effects (e.g. SO-coupling and Λ-type doubling - precise calculation of molecular constants (B v,D v,H v...) calculated a posteriori to high order - success depends critically on use of a simple, flexible and realistic analytic model

7. Despite success in a numerous challenging data analyses with high quality, extensive multi-isotopologue data sets, the MLJ-model lacks appeal – selection of the switching parameters δ and r 1/2 somewhat arbitrary in a trial-and error fashion Le Roy and Henderson (2007) – the Basic MLR Model The two innovations here are: 1. Use of u LR (r) with a set of C n coefficients in the pre-exponential factor 2. Invoking a polynomial expansion variable y p that can also serve as the switching function Note: z = y 1

9. The Basic MLR Model - Features and problems 1. Expansion in the long-range region So must choose p = n next – n 1, or at least p > n ℓ – n 1, use p = 5 or 6 such that the first implicit term is C 11 /r 11 or C 12 /r 12, respectively. ExplicitImplicit 2. Such large values of p mean that y p is a relatively “stiff” variable This leads to the need for very high-order polynomial representations and non-physical behavior on inner limb (use different N for inner and outer) 4. Rate of switching (analog of δ) is fixed by p 3. The pivot of the switching is fixed as r e – no analog of r 1/2 This can be too large (for large p) or too small (for small p)

11. Cs 2 (X)

13. The MLR3 Potential Model (Generalized Surkus variable) At long-range If m = p, and a = 1, collapses to extended MLR with 1/r (n 1 +p) leading implicit term If m > p, coefficient of leading implicit 1/r (n 1 +p) term is

14. LES/FTIR Spectroscopy of Cs 2 C. Amiot and O. Dulieu (2002) Cs 2 in a heat pipe at 600K Ti:Sa laser pumped 113 A state levels with J up to 287 Emission from 342 (v A,J) levels observed as P,R-doublets to v X = 0 -135 Doppler limited resolution (~0.01 cm -1 ) – no detection of hyperfine splittings at high v X FTIR spectra have measurement uncertainty of 0.001-0.003 cm -1 16900 measured transition wavenumbers v X = 135 is bound by ~28 cm -1 Fitted IPA potential in 8.5 – 10.8 Å range to determine C 6, C 8, C 10 (plus damping parameters) and D e, with fixed C 12 and exchange energies

15. In the present work, numerous MLR3 fits have been performed for various {pmq} models 1. p = 2: Fixed C 6 (C 8, C 10,,, = 0) 2. p = 4: Fixed C 6, C 8 (C 10, C 11,... = 0) 3. p = 5 or 6: Fixed C 6, C 8, C 10 (C 11, C 12... = 0) All fits are essentially of equal quality with drms values of 0.6063 ± 0.0003 (line position uncertainties were estimated conservatively) The remainder of this talk will discuss: 1. The selection of fixed values of C 6, C 8 and C 10 2. The selection of optimum models using “Le Roy long-range plots” as exemplified by {5mq} fits 3. A comparison of fitted V(r) with the potential of Amiot and Dulieu 4. The dissociation energy of Cs 2 (X)

17. Le Roy long-range plots for {5mq} MLR3 Fits For p = 5, with fixed C 6, C 8 and C 10, the MLR3 model at long-range expands as 1. m = p = 5 and a = 1 (Extended MLR ) Plots of LHS vs 1/r have intercept C 10 and variable limiting slopes depending on the set of fitted ϕ i -parameters 2. m > p Limiting slope has fixed value given by a, C 6, r e and ϕ ∞

22. Dissociation Energy of Cs 2 (X) From the selected MLR3 fits, we obtain the estimate D e = 3649.815 ± 0.02 cm -1 During the course of the present work, a precise estimate of D e with respect to the lowest hyperfine limit, Cs( 2 S 1/2,F=3) + Cs( 2 S 1/2,F=3), became available from an elegant two-photon transfer process (STIRAP) with ultracold Cs atoms J. G. Danzl et al. Science 321, 1062 (2008) D e = 3649.6847 ± 0.001 cm -1 How are these two values related?