A plane wave basis set must be used in conjunction with _periodic boundary conditions_. This requires that the external potential, and hence the ground state density, be periodic in space, i.e. \

$$
v_{ext}(\mathbf{r}+\mathbf{R})=v_{ext}(\mathbf{r}),
$$

and\
$$
\rho(\mathbf{r}+\mathbf{R})=\rho(\mathbf{r}),
$$
where $\mathbf{R}$ is any _real lattice vector_, defined by
$$
\mathbf{R}=l\mathbf{a}+m\mathbf{b}+n\mathbf{c}
$$
where $l$, $m$, and $n$ can each take any integer value, and $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$  are vectors defining 3 edges of the parallelepiped that forms the _unit cell_. In a similar way to the real lattice vectors, the _reciprocal lattice vectors_ are defined by
$$
\mathbf{G}=l\mathbf{a}^*+m\mathbf{b}^*+n\mathbf{c}^*
$$
  where the vectors $\mathbf{a}^*$, $\mathbf{b}^*$ and $\mathbf{c}^*$ are related to $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ by
$$
\mathbf{a}^*=2\pi\frac{\mathbf{b}\times\mathbf{c}}{\Omega}
$$
  $$
\mathbf{b}^*=2\pi\frac{\mathbf{c}\times\mathbf{a}}{\Omega}
$$
$$
\mathbf{c}^*=2\pi\frac{\mathbf{a}\times\mathbf{b}}{\Omega}
$$
and $\Omega$ is the volume of the cell, given by
$$
\Omega=\vert\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\vert.
$$