Figure 8.

Violation of Newton’s third law for an illustrative 3-atom system approximated by the HCP. The three atoms have charges q₁, q₂, and q₃. Atoms 2 and 3 have equal and opposite charges, q₂ = −q₃ = e. Calculated forces between atoms i and j are shown as f_ij. For the exact all-atom calculation, the total force acting on the center-of-mass, ∑_i,j f_ij = 0 since f_ij = f_ji per Newton’s third law. For the HCP, let atoms 2 and 3 belong to the same group while atom 1 does not belong to the group. The approximate point charge for the group represented by a, is q_a = q₂ + q₃ = 0. For such a system the force acting on atom 1 due to the group is zero, while, in general, the net force on atoms 2 and 3, f₁₂ + f₁₃, is not zero. Therefore the HCP can result in a non-zero net force on a closed system, in violation of Newton’s third law.