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system size for nuclear-centred Gaussian functions, i.e. plane wave basis sets become more favourable for large systems.

While plane wave basis sets have primarily been used for periodic systems, they can also be used for molecular species by using a supercell approach, where the molecule is placed in a sufficiently large unit cell such that it does not interact with its own image in the neighbouring cells.39 Placing a relatively small molecule in a large supercell to avoid self-interaction consequently requires many plane wave functions, and such cases are handled more efficiently by localized Gaussian functions. A three-dimensional periodic system, on the other hand, may be better described by a plane wave basis than with nuclear-centred basis functions.

Plane wave basis functions are ideal for describing delocalized slowly varying electron densities, such as the valence bands in a metal. The core electrons, however, are strongly localized around the nuclei, and the valence orbitals have a number of rapid oscillations in the core region to maintain orthogonality. Describing the core region adequately thus requires a large number of rapidly oscillating functions, i.e. a plane wave basis with very large kmax. The singularity of the nucleus-electron potential is furthermore almost impossible to describe in a plane wave basis, and this type of basis set is therefore used in connection with pseudopotentials (Section 5.9) for smearing the nuclear charge and modelling the effect of the core electrons. Note that a pseudopotential is also required for smearing the potential near the nucleus in hydrogen, even though hydrogen does not have core electrons.

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