3. Periodic cells, neighbors and electrostatics
Position in the course: Lesson 3 of 11. Complete the preceding derivation and use the explained exercises to check understanding.
1. A finite cell represents a repeated model
Periodic boundaries remove a free surface by repeating the simulation cell. They do not remove finite-size effects: a solute, defect or fluctuation interacts with its images, and the box excludes wavelengths longer than its dimensions. Assume a neutral three-dimensional bulk system unless the boundary convention is explicitly changed. A slab with vacuum is a different electrostatic problem, not simply a dilute bulk liquid.
Let cell vectors be columns of \(A\) and fractional coordinates be \(s=A^{-1}r\). Wrapping \(s\) into \([0,1)\) changes the stored image but not the physical particle identity. Keep unwrapped coordinates or image flags for transport; wrapped positions jump when a particle crosses a face.
2. Derive the minimum-image rule
For an orthorhombic cell, choose the image of a pair with shortest component separations:
For a short-range pair cutoff, require \(r_c<\min(L_x,L_y,L_z)/2\) for the simple single-image algorithm. Otherwise more than one image may fall inside the cutoff. In a highly skew cell, independently rounding fractional components does not always find the Euclidean nearest image. Use a validated triclinic algorithm rather than copying the orthorhombic formula.
3. Worked crossing and a neighbor-list bound
In a unit box, particles at \(x=0.95\) and \(0.05\) are separated by \(0.10\), not \(0.90\). If the first moves from \(0.95\) to wrapped \(0.02\), its true displacement can be \(+0.07\); subtracting wrapped coordinates would give \(-0.93\). This distinction is crucial for mean-square displacement.
A Verlet neighbor list stores pairs out to \(r_l=r_c+b\), where \(b\) is the skin. If every atom moves less than \(b/2\) from its list-building position, any pair distance changes by less than \(b\) by the triangle inequality. A pair originally outside \(r_l\) therefore cannot enter \(r_c\) before a rebuild. The condition uses displacement since the last build, not displacement in the most recent step.
4. Why Coulomb sums need a convention
The Ewald identity splits a long-range interaction into rapidly converging real and reciprocal parts:
The second contribution is smooth and can be treated in reciprocal space. The splitting parameter \(\alpha\) moves work between the sums; with both sums converged it does not change the physical answer. A self-interaction correction, treatment of the zero reciprocal vector and an external boundary convention are necessary. A net charged periodic cell requires an explicit convention such as a neutralizing background; its energy is not automatically an isolated-ion energy.
5. Separate numerical and finite-size convergence
At fixed box, refine reciprocal mesh, interpolation order and real-space precision. Then enlarge the physical cell at comparable numerical accuracy. Refining a mesh cannot remove image interactions. Match slab corrections and vacuum conventions to the geometry. For molecular liquids, inspect RDFs only at distances permitted by the box geometry and compare transport coefficients across sizes.
6. Exercises and solutions
A skin is \(0.4\) length units; one atom has moved \(0.25\) since the rebuild. Is the sufficient safety criterion satisfied? No: \(b/2=0.2\). This does not prove a missed pair already exists, but the guarantee has expired.
Does wrapping coordinates remove drift? No. It changes the representation. Removing center-of-mass momentum alters velocities and must be handled separately when defining a dynamical ensemble.
7. References
Original analytic teaching illustration under the stated assumptions; no simulation data.
8. Related theory and practice
Quantum mechanics · Monte Carlo · Quantum Monte Carlo · CP2K · Quantum-Espresso