Advances in computers have enabled the application of ab initio calculations based on density functional theory (DFT) to a wide range of systems . Many of these techniques, originating from band structure calculations, have used a plane wave (PW) basis set and ab initio pseudopotentials.
This approach has a number of advantages. The simplicity of the basis set enables straightforward implementation of codes and systematic convergence of results with respect to a single parameter, the cut-off energy for the plane wave basis set. Additionally, the use of a PW basis set enables efficient calculation of atomic forces allowing relaxation of atomic structure and dynamical simulation.
Although a PW basis set can be very large, the development of optimised pseudopotentials [2,3] has greatly reduced the number of plane waves needed for an accurate description of the electronic states. A major restriction of the range of systems to which PW methods may be applied is the need for periodic boundary conditions. However, an aperiodic system may be modeled using a supercell with careful consideration of electrostatics  and Brillouin zone sampling .
One remaining limitation of the PW technique is that the extended basis states do not provide a natural way of quantifying local atomic properties. In this paper we implement a technique to overcome this deficiency using a projection of the PW states onto a localised basis set. Population analysis of these projected states is then used to determine quantities such as atomic charges and bond populations. We use both the formalisms of Mulliken  and of Löwdin  in order to perform this analysis. The PW pseudopotential calculations were performed using the CASTEP code  with both local density and generalised gradient approximations for the exchange-correlation energy functional. Little difference was found between the results generated in each case.
The underlying theory of this technique is described in section 2. This is followed followed by an example of its application to some simple molecules in section 3. Section 4 describes the application of the methods to the analysis of a system of practical interest, namely the adsorption of a molecule onto a zeolite substrate. Finally, the results and their implications are discussed in section 5.