17 Basis Sets and Plane Waves

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17.1 Plane Wave Basis Sets

\[ \phi_i(\mathbf{r}) = \sum_{\mathbf{G}} c_{i,\mathbf{G}}\, e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}} \tag{17.1}\]

where \(\mathbf{G}\) are reciprocal lattice vectors and the sum is truncated at a kinetic energy cutoff \(E_{\text{cut}}\):

\[ \frac{|\mathbf{k} + \mathbf{G}|^2}{2} \leq E_{\text{cut}} \tag{17.2}\]

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title: "Basis Sets and Plane Waves"
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*Content coming soon.*

## Plane Wave Basis Sets {#sec-planewaves}

$$
\phi_i(\mathbf{r}) = \sum_{\mathbf{G}} c_{i,\mathbf{G}}\, e^{i(\mathbf{k}+\mathbf{G})\cdot\mathbf{r}}
$$ {#eq-planewave}

where $\mathbf{G}$ are reciprocal lattice vectors and the sum is truncated at
a kinetic energy cutoff $E_{\text{cut}}$:

$$
\frac{|\mathbf{k} + \mathbf{G}|^2}{2} \leq E_{\text{cut}}
$$ {#eq-ecut}