= Solution
Differentiate the <Lennard-Jones potential>. Its minimum satisfies
$$
-12\sigma^{12}r^{-13}+6\sigma^6r^{-7}=0,
\qquad\boxed{r_e=2^{1/6}\sigma,\quad V(r_e)=-\alpha.}
$$
For <dilithium>, the two wells consequently have
$$
\begin{array}{c|c|c}
(\sigma,\alpha)&r_e\text{ in bohr}&D_e\text{ in hartree}\\\hline
(3.4,0.06)&3.8164&0.060\\
(4.9,0.05)&5.5001&0.050
\end{array}
$$
\b[Under an additional same-orbital, Hund-type exchange comparison, assign $(3.4,0.06)$ to the triplet and $(4.9,0.05)$ to the singlet:] the lower-energy minimum is associated with the exchange-stabilized <triplet state>, as in part (a).
This assignment needs that extra orbital-parentage assumption. A scalar <Lennard-Jones potential> does not itself encode <spin>, and a common atomic dissociation limit does not specify a unique molecular electronic configuration or spatial symmetry. The two curves even cross at about $5.006$ bohr: neither lies below the other at every separation. Thus \b[the minima and well depths are determined unambiguously; the <spin> labels are a conditional qualitative inference], rather than a unique consequence of the four supplied parameters. Neither “a triplet must always have a longer bond” nor a universal ordering of unrelated excited states is justified.
\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2022/paper-1-lithium-wells.png]
{title=Original comparison of the two supplied model wells, showing their minima and the crossing without presupposing spin labels}
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