The net electrostatic force acting on a charged particle with index i {\displaystyle i} contained within a collection of particles is given as:
F ( r ) = ∑ j ≠ i F ( r ) r ^ ; F ( r ) = q i q j 4 π ε 0 r 2 {\displaystyle \mathbf {F} (\mathbf {r} )=\sum _{j\neq i}F(r)\mathbf {\hat {r}} \,\,\,;\,\,F(r)={\frac {q_{i}q_{j}}{4\pi \varepsilon _{0}r^{2}}}}
where r {\displaystyle \mathbf {r} } is the spatial coordinate, j {\displaystyle j} is a particle index, r {\displaystyle r} is the separation distance between particles i {\displaystyle i} and j {\displaystyle j} , r ^ {\displaystyle \mathbf {\hat {r}} } is the unit vector from particle j {\displaystyle j} to particle i {\displaystyle i} , F ( r ) {\displaystyle F(r)} is the force magnitude, and q i {\displaystyle q_{i}} and q j {\displaystyle q_{j}} are the charges of particles i {\displaystyle i} and j {\displaystyle j} , respectively. With the electrostatic force being proportional to r − 2 {\displaystyle r^{-2}} , individual particle-particle interactions are long-range in nature, presenting a challenging computational problem in the simulation of particulate systems. To determine the net forces acting on particles, the Ewald or Lekner summation methods are generally employed. One alternative and usually computationally faster technique based on the notion that interactions over large distances (e.g. > 1 nm) are insignificant to the net forces acting in certain systems is the method of spherical truncation. The equations for basic truncation are:
F C U T ( r ) = { q i q j 4 π ε 0 r 2 for r ≤ r c 0 for r > r c . {\displaystyle \displaystyle F_{CUT}(r)={\begin{cases}{\frac {q_{i}q_{j}}{4\pi \varepsilon _{0}r^{2}}}&{\text{for }}r\leq r_{c}\\0&{\text{for }}r>r_{c}.\end{cases}}}
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