This chapter has described the calculation of local atomic properties from plane-wave electronic structure calculations. Section described the theory underlying these calculations and details of the implementation of this theory may be found in Appendix . Section gave some brief examples of the application of these techniques to simple molecules and Section demonstrated how they may be used in a system of practical interest. The application of population analysis to simple bulk crystals was described in Section and a relationship was found for overlap population and effective valence charge with covalency and ionicity respectively. The utility of overlap population as a measure of bond modulus was discussed in Section . The relationship between these quantities was found to be statistically significant. However, it was found that the overlap population can only be used as an approximate guide to bond modulus. Finally, Section showed that population analysis could be used to investigate the distribution of spin density in a spin-dependent calculation.
Section described how both Mulliken and Löwdin populations may be calculated from the overlap and density matrices. The calculation of these populations was demonstrated for simple molecules. However, in the remaining analyses presented in this chapter only the Mulliken populations have been calculated. Mulliken's definitions of these quantities were chosen as these are used more widely within the scientific community. Many other population analysis schemes have been proposed, such as those due to Roby  and Mayer [58, 59]. These analyses can also be performed using the overlap and density matrices calculated by the projection techniques described herein.
Provided a consistent choice of basis sets is made, population analysis offers an objective, reliable measure of differences in charge and spin distribution which can greatly aid in the detailed microscopic understanding of a system. Previously, analysis of plane wave calculations generally relied on subjective descriptions based on plots of charge density distributions. The overlap population offers a useful guide to bonding within a system. At the simplest level, it provides an quantitative criterion for the presence of a bond, in contrast to previous methods which relied on empirical measures such as interatomic distance or subjective examination of charge density distributions. However, the overlap population also provides additional information on the nature of the atomic interactions.
The work described in Section through was undertaken in collaboration with Dr R. Shah and C.J. Pickard and resulted in the publication of  and . The techniques developed have been used in the analysis of calculations of practical interest, examples of which may be found in ,  and the remainder of this thesis.