The use of plane waves as a basis set may only be applied to a system exhibiting three dimensional translational symmetry. An aperiodic system would require a continuous plane wave basis set and hence an infinite number of plane wave basis functions.
This limitation may be overcome by embedding an aperiodic system such as a molecule in a periodic system of supercells (See Figure 2.2). Provided the supercell is sufficiently large that the interactions between the molecules are negligible, the energy per supercell will be identical to that of a single molecule. If the energy per supercell has converged sufficiently with respect to supercell size the interaction between molecules will be negligible.
Figure 2.2: A schematic
illustration of a supercell geometry for a molecule. The boundaries of
the supercell are depicted by dashed lines.