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The protons in the $\text{H}_2^{+}$ molecular ion are $0.106 \, \mathrm{nm}$ apart, and the binding energy of $\text{H}_2^{+}$ is $2.65\,\mathrm{eV} .$ What negative charge must be placed halfway between two protons this distance apart to give the same binding energy?

Question: In the above quote, does "the binding energy of $\mathrm{H}_2^{+}$" refer to the energy required to disassemble the whole system (two protons and one electron) or just the protons?

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Just the protons, basically - it's the energy required to take the system apart into a single proton and an isolated hydrogen atom.

Separating the proton and electron from that hydrogen atom would take a fair bit more energy - an additional 13.6 eV.

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