This weighted regression plot, with error bars too small to see, came from (ten replicates of) an experiment using the Diffusion Monte Carlo technique to solve the Schrödinger equation for the hydrogen-molecule ion (two protons, one electron). The model shown is not one that is found in the literature but is a good fit nonetheless (better than the familiar Morse curve).

This example requires a User-defined model in Regress+. The model, typed into a dialog (or cut-and-pasted from a textfile library), is parsed, saved and thereafter interpreted.

This example also illustrates one of the pitfalls of mathematical modeling. Looking at the formula, one wonders, "Why not the simpler, algebraically equivalent model shown below?"

y = 2.661 (1 − exp(−1.484 (x − 1.040)))^2.796 − 16.27

The reason is that, for x < 1.040, the latter requires raising a negative number to an arbitrary power, giving an imaginary result.

This dataset is one of the examples included in the Regress+ software package.