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I have been learning about lennard jones potential as a model of interatomic bonds:

$$V(r) = 4ε\left[\left(\frac{σ}{r}\right)^q - \left(\frac{σ}{r}\right)^p\right]$$

But, I've been struggling to understand the physical significance of $V(r)$. I understand that it is a form of energy stored in the bond. What I don't understand is what is meant by its sign: what does it mean to have negative energy? I have also heard $V(r)$ described as the energy required to fuse the atoms together, and as the energy required to bring them an infinite distance apart, but I don't see how these definitions are reconcilable.

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By convention we set the energy when the atoms are very far away as zero. As we have to supply energy to separate the atoms of the molecule, the equilibrium must correspond to a negative value of energy.

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As you have already mentioned, $V(r)$ describes the energy within the system due to the interaction of those two atoms. Negative values of $V$ could be interpreted as a reduction of the overall energy in the system due to the atom-atom interaction.

In general, however, negative energies have no intrinsic meaning as the absolute value of energies is usually not interesting in physics; only energy differences really matter.

Now, let's say the potential has its minimum at position $r_0$, so the two atoms would prefer taking this distance while in a bond. If you move them infinitely far apart, their potential energy vanishes as $V(r) \rightarrow 0$ as $r \rightarrow 0$. Thus the energy it would take to do this would be $E = \left| V(r_0) - V(r = \infty) \right| = V(r_0)$

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