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Lest there be any misunderstanding, note that JD is plotting potential
energy for (charged) ions bound to each other, not neutral atoms bound to
each other. For the latter case, a well-known approximation is the Morse
potential energy (http://en.wikipedia.org/wiki/Morse_potential) for which a
harmonic oscillator well fits over a much wider range of distances than it
does for the two-ion situation.