# How to find the scaling factors for minimum image convention?

Suppose that we have an arbitrary (not necessarily cubic) unit cell with cell parameters $$\mathbf{a}$$, $$\mathbf{b}$$, $$\mathbf{c}$$ (lattice vectors) and $$\alpha$$, $$\beta$$, $$\gamma$$ (lattice angles).

We have a particle at point $$\mathbf{r}_i$$ and we want to calculate its interactions with the other particles inside a cutoff radius of $$r$$. (E.g. interactions can be neglected outside this radius).

How can I find scaled versions of $$\mathbf{a}$$, $$\mathbf{b}$$, $$\mathbf{c}$$ such that the $$i$$-th particle interacts only with one periodic image of the (lets say) $$j$$-th particle?

According to this paper an approach is the following (see p. 6). We define the perpendicular widths of the cell as: $$W_a = \frac{|\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|}{|\mathbf{b} \times \mathbf{c}|}$$ $$W_b = \frac{|\mathbf{b} \cdot (\mathbf{c} \times \mathbf{a})|}{|\mathbf{c} \times \mathbf{a}|}$$ $$W_c = \frac{|\mathbf{c} \cdot (\mathbf{a} \times \mathbf{b})|}{|\mathbf{a} \times \mathbf{b}|}$$

($$W_a, W_b, W_c$$ are merely the shortest distance between opposing faces of the unit cell.)

and the condition to satisfy minimum image convention is:

$$r_c < \frac{1}{2} \min(W_a, W_b, W_c)$$

I am not able to follow the explanation for the validity proposed in Appendix (p. 7). Can someone provide a graphical illustration of the proof?

Special case - cubic unit cell

If the unit cell is cubic, then the above equations reduces to:

$$r_c < \frac{1}{2} L$$ where $$L$$ is the length of each side of the unit cell. Suppose that the distance between $$i$$ and $$j$$ particles is within the cutoff, that is $$r_{ij} \leq r_c$$. How do we prove that the distance between $$i$$ and a periodic image of $$j$$, call it $$j^*$$, is not within the cutoff?

I am adding the following visualization so it is easier to state my problem.

• What do you mean by "with cell parameters a, b, c". As phrased in the first paragraph they look like the length of the sides of the unit cell, while when defining Wa, Wb, Wc they look like lattice vectors, or at least vectors of some kind rather than scalar lengths. Which are they? Aug 2, 2023 at 15:47
• @IanBush Thanks for the correction. I have updated the OP. Aug 2, 2023 at 15:56