Exact Diatomic Potentials from Direct Fitting: A New Approach by Joel Tellinghuisen and Marcus H. Mendenhall Vanderbilt University Nashville, TN 37235

The D 0 + u ( 3 P 2 ) State of I 2 * One of 6 lowest-lying ion-pair states. One of 6 lowest-lying ion-pair states. At large R, U = D e – C 1 /R –C 4 /R 4 + C 3 /R 3 … At large R, U = D e – C 1 /R –C 4 /R 4 + C 3 /R 3 … Fit to this plus a few higher-order terms in 1/R. Oops! Fit to this plus a few higher-order terms in 1/R. Oops! * J. Mol. Spectrosc. 217, 212 (2003).

Maybe the Generalized Morse Oscillator of Coxon and Hajigeorgiou … Maybe the Generalized Morse Oscillator of Coxon and Hajigeorgiou …  O.K., then surely the Modified Lennard-Jones oscillator of Hajigeorgiou and Le Roy …

With some tinkering, we can get a satisfactory fit with the polynomial in 1/R, with “reasonable” With some tinkering, we can get a satisfactory fit with the polynomial in 1/R, with “reasonable” behavior on the repulsive wall. But now agreement deteriorates on the attractive branch! One answer: Switching functions.

Alternative Approach Explored Here   Start with approximate potential having the correct behavior at small and large R.  Fit corrections to this zeroth-order potential using functions that are confined to the “business region,” so they cannot corrupt the potential in the wings.  What to use for these functions? Why not use eigenfunctions of the potential itself?  Not only are these properly confined on the R-axis, they are also orthonormal, ensuring success in fitting.  Can adjust range of coverage and number of functions by changing E max and “mass.”

Start with selected points. Illustratio n, same D state of I 2 Fit to some suitable function — here a Rittner- type potential.

Generate wavefunctions. Alter maximum E and mass to adjust coverage — number of functions and R range.

First try  modest success with 19 functions extending to E = cm –1 (22 adjustable parameters).

Residuals look O.K., but qualification needed.

Ln(R ) display suggests an alternative basis.

Much better results with only 15 functions!

Other possible scalings: 1/R or 1/R 2. Can also scale resulting basis functions. For example, 1/R 4 weighting emphasizes small-R region of repulsive branch. All of these can improve fit quality. They (and variable data weights) also neutralize the orthogonality of the basis set.

Other Bases Harmonic oscillator, without and with scaling (ln R, 1/R). Now location (R e ) also adjustable. Greater risk of damaging potential in wings … which can even occur with RKR-like basis. A

Other potentials tried I 2 (A) − known to 94% of dissociation. Treated like I 2 (D), with weighted fitting using computed RKR uncertainties. Comparable performance. I 2 (A) − known to 94% of dissociation. Treated like I 2 (D), with weighted fitting using computed RKR uncertainties. Comparable performance. Cs 2 (X) − known to 99.2% of dissociation. Coverage so close to dissociation may require new approach. Morse curve dissociating to higher D e, perhaps? Cs 2 (X) − known to 99.2% of dissociation. Coverage so close to dissociation may require new approach. Morse curve dissociating to higher D e, perhaps?

Summary Actual fitting is easy: Sometimes slow convergence but divergence only rarely. This includes cases having 25 basis functions and 30 adjustable parameters total. Equivalent fitting (linear!) to polynomials in 1/R required quad precision. Typical optimal representation: ~15 basis functions + ~10 other parameters. However, because of low correlation, decimal digit count comparable to other methods. Downside: Potential rendered in numerical rather than closed form. However, tools for “black box” implementation now widely available to spectroscopists.