Pair potential

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Some interactions, like Coulomb's law in electrodynamics or Newton's law of universal gravitation in mechanics naturally have this form for simple spherical objects. For other types of more complex interactions or objects it is useful and common to approximate the interaction by a pair potential, for example interatomic potentials in physics and computational chemistry that use approximations like the Lennard-Jones and Morse potentials.

Functional form

The total energy of a system of $N$ objects at positions ${\vec{R}}_{i}$ , that interact through pair potential $v$ is given by

E = \frac{1}{2} \sum_{i = 1}^{N} \sum_{j \neq i}^{N} v (| {\vec{R}}_{i} - {\vec{R}}_{j} |) .

Equivalently, this can be expressed as

E = \sum_{i = 1}^{N} \sum_{j = i + 1}^{N} v (| {\vec{R}}_{i} - {\vec{R}}_{j} |) .

This expression uses the fact that interaction is symmetric between particles $i$ and $j$ . It also avoids self-interaction by not including the case where $i = j$ .

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Functional form

Potential range

Computational cost

Infinitely periodic systems

Beyond pair potentials

Common pair potentials

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