+Next to their simplicity, plane waves have several advantages.
+The basis set is orthonormal by construction and, as mentioned above, it is simple to check for convergence.
+The biggest advantage, however, is the ability to perform exact calculations by a discrete sum over a numerical grid.
+This is due to the related construction of the grid and the PW basis.
+Ofcourse, exactness is restricted by the fact that the PW basis set is finite.
+The simple form of the PW representation of the KS equations
+\begin{equation}
+\sum_{\vec{G}'} \left[
+ \frac{\hbar^2}{2m}|\vec{k}+\vec{G}|^2 \delta_{\vec{GG}'}
+ + \tilde{V}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{H}}(\vec{G}-\vec{G}')
+ + \tilde{V}_{\text{xc}}(\vec{G}-\vec{G}')
+\right] c_{i,\vec{k}+\vec{G}} = \epsilon_i c_{i,\vec{k}+\vec{G}}
+\label{eq:basics:pwks}
+\end{equation}
+reveals further advantages.
+The various potentials are described in terms of their Fourier transforms.
+Equation \eqref{eq:basics:pwks} is solved by diagonalization of the Hamiltonian matrix $H_{\vec{k}+\vec{G},\vec{k}+\vec{G}'}$ given by the terms in the brackets.
+The gradient operator is diagonal in reciprocal space whereas the exchange-correlation potential has a diagonal representation in real space.
+This suggests to carry out different operations in real and reciprocal space, which requires frequent Fourier transformations.
+These, however, can be efficiently achieved by the fast Fourier transformation (FFT) algorithm.
+
+There are likewise disadvantages associated with the PW representation.
+By construction, PW calculations require a periodic system.
+This does not pose a severe problem since non-periodic systems can still be described by a suitable choice of the simulation cell.
+Describing a defect, for instance, requires the inclusion of enough bulk material in the simulation to prevent or reduce the interaction with its periodic, artificial images.
+As a consequence the number of atoms involved in the calculations are increased.
+To describe surfaces, sufficiently thick vacuum layers need to be included to avoid interaction of adjacent crystal slabs.
+Clearly, to appropriately approximate the wave functions and the respective charge density of a system composed of vacuum in addition to the solid in a PW basis, an increase of the cut-off energy is required.
+According to equation \eqref{eq:basics:pwks} the size of the Hamiltonian depends on the cut-off energy and, therefore, the computational effort is likewise increased.
+For the same reason, the description of tightly bound core electrons and the respective, highly localized charge density is hindered.
+However, a much more profound problem exists whenever wave functions for the core as well as the valence electrons need to be calculated within a PW basis set.
+Wave functions of the valence electrons exhibit rapid oscillations in the region occupied by the core electrons near the nuclei.
+The oscillations maintain the orthogonality between the wave functions of the core and valence electrons, which is compulsory due to the exclusion principle.
+Accurately approximating these oscillations demands for an extremely large PW basis set, which is too large for practical use.
+Fortunately, the impossibility to model the core in addition to the valence electrons is eliminated in the pseudopotential approach discussed in the next section.