Cutoffs and the minimum image

The idea

N atoms have N(N−1)/2 pairs. For 100,000 atoms that is 5 × 10⁹ pair calculations every step: far too many. Luckily the Lennard-Jones attraction falls off as r⁻⁶. At 2.5σ it is only −0.016ε, about 1.6% of the well depth. So we introduce a cutoff r_c and simply ignore pairs beyond it.

Each atom then only interacts with neighbours inside a sphere of radius r_c. That is about ρ·(4/3)πr_c³ atoms, roughly 55 at 2.5σ in a liquid. The cost grows as r_c³: doubling the cutoff makes the run eight times slower.

How you cut matters:

• Truncate: V = 0 beyond r_c. The energy jumps when a pair crosses r_c, which spoils energy conservation (Lesson 1.6).
• Shift: subtract V(r_c) so the energy is continuous. The force still jumps at r_c.
• Force-shift: also subtract the force at r_c, so both energy and force go smoothly to zero. It changes the potential slightly everywhere.
• Switch: leave V untouched up to r_on, then smoothly bend it to zero by r_c. Common in force fields (CHARMM, GROMACS).

Pairs beyond r_c still add a small attraction. The tail correction adds it back analytically, assuming the liquid is uniform beyond r_c. It matters for energy and especially for pressure.

The algorithm

The force loop with a cutoff, using a shifted potential:

rc2 = rc * rc
V_rc = 4 * (rc**-12 - rc**-6)
for each pair i < j:
    d = min_image(pos[i] - pos[j])
    r2 = dot(d, d)
    if r2 >= rc2: continue              # the cutoff
    inv6 = 1 / r2**3
    U += 4 * (inv6**2 - inv6) - V_rc    # shifted energy
    f = 24 * (2 * inv6**2 - inv6) / r2 * d
    F[i] += f;  F[j] -= f

1. Compare r² with r_c² and avoid the square root, which is slow.
2. Keep r_c ≤ L/2 (Lesson 2.4).
3. Report energies with the tail correction, or state clearly which cutoff and scheme you used. Results are not comparable otherwise.
4. The loop above still checks all N² pairs. Lesson 2.6 uses neighbour lists so each atom only checks atoms that are nearby.