## 4 Computational Details

### 4.1 Symmetry Properties of the Integrals

There are symmetries additional to the spatial symmetry that can be used to reduce the number of integrals to be computed and stored. For example, in the overlap matrix S(k), Eq. (6), the $χp(r)χqμ(r)$ integrals are invariant with respect to any integer shift of the origin for r, so that

$\begin{array}{l}\hfill {S}_{pq}\left(k\right)=\sum _{\mu =-\infty }^{\infty }exp\left(2\pi i\mu k\right)〈{\chi }_{p}\left(\mathbf{r}\right){\chi }_{q}\left(\mathbf{r}-\mu \stackrel{ˆ}{\mathbf{z}}\right)〉\\ =\sum _{\mu =-\infty }^{\infty }exp\left(2\pi i\mu k\right)〈{\chi }_{p}\left(\mathbf{r}+\mu \stackrel{ˆ}{\mathbf{z}}\right){\chi }_{q}\left(\mathbf{r}\right)〉.\hfill \end{array}$

By setting $μ→−μ$ and taking into account ...

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