The advantages and disadvantages of a plane wave (PW) basis set were discussed in Chapter , Section . One further limitation of the PW approach is that the extended basis states do not provide a natural way of quantifying local atomic properties. In this Chapter we discuss a technique to overcome this deficiency using a projection of the PW states onto a localised basis set. A population analysis of these projected states can then be used to determine quantities such as atomic charges and bond populations. Both the formalisms of Mulliken  and Löwdin are used  in order to perform this analysis. These techniques are commonly applied in the analysis of calculations performed using localised basis sets and hence are widely used in Chemistry and Biochemistry. However, they have not been routinely applied to PW calculations. It is widely accepted that the absolute magnitude of the atomic charges have little physical meaning, as they display an extreme sensitivity to the atomic basis set with which they are calculated . In this chapter we will see that consideration of relative values of populations, in contrast to their absolute magnitudes, can yield useful information.
Section of this chapter describes the underlying theory of this technique. This is followed by an example of its application to some simple molecules in Section . Section describes the application of the methods to the analysis of a system of practical interest, namely the adsorption of a molecule onto a zeolite catalyst. These calculations were performed by Dr R. Shah . Section presents results for several simple bulk crystals. Previous work by Garcia and Cohen  considered the link between total valence charge density and measures of ionicity and electronegativity. We will discuss the use of Mulliken bond populations and valence charges in this context. Section examines the relationship between bond population and bond modulus and Section demonstrates the use of population analysis in addressing questions of spin populations. Finally, Section summarises the results presented in this chapter.