Pair potential

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

${\displaystyle E={\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=i+1}^{N}v\left(\left|{\vec {R}}_{i}-{\vec {R}}_{j}\right|\right)\ .}$

This expression uses the fact that interaction is symmetric between particles ${\displaystyle i}$ and ${\displaystyle j}$. It also avoids self-interaction by do not including the case when ${\displaystyle i==j}$.

A fundamental property of a pair potential is its range. It is expected that pair potentials go to zero for infinite distance as particles that are too far apart do not interact. In some cases the potential goes quickly to zero and the interaction for particles that are beyond a certain distance can be assumed to be zero, these are said to be short-range potentials. Other potentials, like the Coulomb or gravitational potential, are long range: they go slowly to zero and the contribution of particles at long distances still contributes to the total energy.

The total energy expression for pair potentials is quite simple to use for analytical and computational work. It has some limitations however, as the computational cost is proportional to the square of number of particles. This might be prohibitively expensive when the interaction between large groups of objects needs to be calculated.

For short-range potentials the sum can be restricted only to include particles that are close, reducing the cost to linearly proportional to the number of particles.

In some cases it is necessary to calculate the interaction between an infinite number of particles arranged in a periodic pattern.

Pair potentials are very common in physics and computational chemistry and biology; exceptions are very rare. An example of a potential energy function that is not a pair potential is the three-body Axilrod-Teller potential. Another example is the Stillinger-Weber potential for silicon, which includes the angle in a triangle of silicon atoms as an input parameter.[2][3]