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 [43] and Löwdin are used
[44] 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 [45]. 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
[46]. Section presents results for several simple
bulk crystals. Previous work by Garcia and Cohen [47]
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.